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+{"text":"\\section{Introduction}\n\nIn this paper, we study the following kinetic operator \n\\begin{equation}\\label{yo2}\nP=\\partial_t+v \\cdot \\partial_x  + a(t,x,v)(-\\tilde{\\Delta}_v)^{\\sigma}, \\ t \\in \\mathbb{R}, \\ x,v \\in \\mathbb{R}^{n},\n\\end{equation}\nwhere $0<\\sigma<1$ is a constant parameter, $x \\cdot y$ stands for the standard dot-product on $\\mathbb{R}^n$ and $a$ denotes a $C_b^{\\infty}(\\mathbb{R}^{2n+1})$ function satisfying \n\\begin{equation}\\label{eq0.5}\n\\exists a_0>0, \\forall (t,x,v) \\in \\mathbb{R}^{2n+1}, \\ a(t,x,v) \\geq a_0>0.\n\\end{equation}\nHere the notation $C_b^{\\infty}(\\mathbb{R}^{2n+1})$ stands for the space of $C^{\\infty}(\\mathbb{R}^{2n+1})$ functions whose derivatives of any order are bounded on $\\mathbb{R}^{2n+1}$ and \n$(-\\tilde{\\Delta}_v)^{\\sigma}$ is the Fourier multiplier with symbol\n\\begin{equation}\\label{eq0}\nF(\\eta)=|\\eta|^{2\\sigma}w(\\eta)+|\\eta|^2\\big(1-w(\\eta)\\big), \\ \\eta \\in \\mathbb{R}^n,\n\\end{equation}\nwith $|\\cdot|$ being the Euclidean norm, $w$ a  $C^{\\infty}(\\mathbb{R}^n)$ function satisfying $0 \\leq w \\leq 1$, $w(\\eta)=1$ if $|\\eta| \\geq 2$, $w(\\eta)=0$ if $|\\eta| \\leq 1$; and $D_t=(2\\pi i)^{-1}\\partial_t$, $D_x=(2\\pi i)^{-1}\\partial_x$, $D_v=(2\\pi i)^{-1}\\partial_v$.\n\n\nWhen $\\sigma=1$, this operator reduces to the so-called Vlasov-Fokker-Planck operator, whereas  when $0<\\sigma<1$, it stands for a simplified linear model of the spatially inhomogeneous Boltzmann equation without angular cutoff (see the end of this introduction together with section~\\ref{kkboltz} in appendix). This is our motivation for studying the regularizing properties of this linear model and establishing hypoelliptic estimates with optimal loss of derivatives with respect to the exponent $0<\\sigma<1$ of the fractional Laplacian $(-\\tilde{\\Delta}_v)^{\\sigma}$. This linear model has the familiar structure \n$$\\textrm{Transport part in the }(t,x) \\textrm{ variables}\\ + \\textrm{ Elliptic part in the }v \\textrm{ variable}$$ \nand it is easy to get the regularity in the $v$ variable. The non-commutation of the transport part (the skew-adjoint part) with the self-adjoint elliptic part $(-\\tilde{\\Delta}_v)^{\\sigma}$ will produce the regularizing effect in the $x$ variable.  \n\n\nRegarding this linear model, the existence and the $C^{\\infty}$ regularity for the solutions of the Cauchy problem to linear and semi-linear equations associated with the operator (\\ref{yo2}) were proved in \\cite{MoXu}. H.~Chen, W.-X.~Li and C.-J.~Xu have also recently studied its Gevrey hypoellipticity. More specifically, they established in~\\cite{chen} (Proposition~2.1) the following hypoelliptic estimate. Let $P$ be the operator defined in (\\ref{yo2}) and $K$ a compact subset of $\\mathbb{R}^{2n+1}$. For any $s \\geq 0$, there exists a positive constant $C_{K,s}>0$  such that for any $u \\in C_0^{\\infty}(K)$,\n\\begin{equation}\\label{yo4}  \n\\|u\\|_{s+\\delta} \\leq C_{K,s}\\big(\\|Pu\\|_{s}+\\|u\\|_{s}\\big),\n\\end{equation}\nwith $\\|\\cdot\\|_{s}$ standing for the $H^s(\\mathbb{R}^{2n+1})$ Sobolev norm and \n\\begin{equation}\\label{yo4.5}\n\\delta=\\textrm{max}\\Big(\\frac{\\sigma}{4},\\frac{\\sigma}{2}-\\frac{1}{6}\\Big)>0.\n\\end{equation}\nThe notation $C_0^{\\infty}(K)$ stands for the set of $C_0^{\\infty}(\\mathbb{R}^{2n+1})$ functions with support in $K$. This hypoelliptic estimate with loss of \n$$\\mbox{\\rm max}(2\\sigma,1)-\\delta>0,$$ \nderivatives is then a key instrumental ingredient for their investigation of the Gevrey hypoellipticity of the operator $P$. \nHowever, this hypoelliptic estimate (\\ref{yo4}) is not optimal. \nIn the present work, we are interested in establishing hypoelliptic estimates with optimal loss of derivatives with respect to the exponent $0<\\sigma<1$ of the fractional Laplacian $(-\\tilde{\\Delta}_v)^{\\sigma}$. More specifically, we shall show by using different microlocal techniques that the operator $P$ is hypoelliptic with a loss of \n$$\\frac{\\mbox{\\rm max}(4\\sigma^2,1)}{(2\\sigma+1)}>0,$$ \nderivatives, that is, that the hypoelliptic estimates (\\ref{yo4}) hold with the new positive gain \n\\begin{equation}\\label{yo6}\n\\delta=\\frac{2\\sigma}{2\\sigma+1}>0,\n\\end{equation}\nwhich improves for any $0<\\sigma<1$ the gain provided by (\\ref{yo4.5}),\n$$\\frac{2\\sigma}{2\\sigma+1}>\\textrm{max}\\Big(\\frac{\\sigma}{4},\\frac{\\sigma}{2}-\\frac{1}{6}\\Big).$$\nOur main result reads as follows.\n\n\n\\bigskip\n\n\n\\begin{theorem}\\label{TTH1}\nLet $P$ be the operator defined in \\emph{(\\ref{yo2})}, $K$ be a compact subset of $\\mathbb{R}^{2n+1}$ and $s \\in \\mathbb{R}$. Then, there exists a positive constant $C_{K,s}>0$ such that for all $u \\in C_0^{\\infty}(K)$,\n\\begin{equation}\\label{yo5}\n\\big\\|(1+|D_t|^{\\frac{2\\sigma}{2\\sigma+1}}+|D_x|^{\\frac{2\\sigma}{2\\sigma+1}}+|D_{v}|^{2\\sigma})u\\big\\|_{s} \\leq C_{K,s} \\big(\\|Pu\\|_{s}+\\|u\\|_{s}\\big),\n\\end{equation}\nwith  $\\|\\cdot\\|_{s}$ being the $H^s(\\mathbb{R}^{2n+1})$ Sobolev norm.\n\\end{theorem}\n\n\\bigskip\n\nThe hypoelliptic estimates (\\ref{yo5}) are optimal in term of the exponents of derivative terms appearing in their left-hand-side, namely, $2\\sigma\/(2\\sigma+1)$ for the regularity in the time and space variables and $2\\sigma$ for the regularity in the velocity variable. The exponent $2\\sigma$ for the regularity in the velocity variable has indeed the same growth as the diffusive part of the kinetic operator (\\ref{yo2}). Regarding the optimality of the exponent $2\\sigma\/(2\\sigma+1)$ for the regularity in the time and space variables, we first notice that Theorem~\\ref{TTH1} is a natural extension for the values of the parameter $0<\\sigma<1$ of the well-known optimal hypoelliptic estimates with loss of $4\/3$ derivatives known for the Vlasov-Fokker-Planck operator, case $\\sigma=1$, (see~\\cite{bouchut,chen1,perthame}),\n$$\\big\\|(1+|D_t|^{2\/3}+|D_x|^{2\/3}+|D_{v}|^{2})u\\big\\|_{s} \\leq C_{K,s} \\big(\\|Pu\\|_{s}+\\|u\\|_{s}\\big).$$\nSee also~\\cite{landau} for general microlocal methods for proving optimal hypoelliptic estimates with loss of $4\/3$ derivatives for certain classes of kinetic equations. \n\n\nWe deduce the optimality of the exponent $2\\sigma\/(2\\sigma+1)$ for the regularity in the time and space variables by using a simple scaling argument. Indeed, let us consider the specific case when the function $a$ appearing in the definition of the kinetic operator $P$ is constant and assume that the hypoelliptic estimates (\\ref{yo4}) hold for a positive gain $\\delta>0$. It follows that the estimate \n$$\\|(|D_t|^{\\delta}+|D_x|^{\\delta}+|D_{v}|^{\\delta})u\\|_{L^2} \\leq C\\big(\\|iD_tu+iv \\cdot D_xu  + |D_v|^{2\\sigma}u\\|_{L^2}+\\|u\\|_{L^2}\\big),$$\nholds for any $u \\in C_0^{\\infty}(\\mathbb{R}^{2n+1})$  with support in the closed unit Euclidean ball $B_1=\\overline{B(0,1)}$ in $\\mathbb{R}^{2n+1}$. The symplectic invariance of the Weyl quantization (Theorem~18.5.9 in~\\cite{hormander}) shows that for any $\\lambda \\geq 1$, \n$$T_{\\lambda}^{-1}\\big(iD_t+iv \\cdot D_x  + |D_v|^{2\\sigma}\\big)T_{\\lambda}=\\lambda^{\\frac{2\\sigma}{2\\sigma+1}}\\big(iD_t+iv \\cdot D_x  + |D_v|^{2\\sigma}\\big)$$\nand\n$$T_{\\lambda}^{-1}\\big(|D_t|^{\\delta}+|D_x|^{\\delta}+|D_{v}|^{\\delta}\\big)T_{\\lambda}=\\lambda^{\\frac{2\\sigma\\delta}{2\\sigma+1}}|D_t|^{\\delta}+\\lambda^{\\delta}|D_x|^{\\delta}+\\lambda^{\\frac{\\delta}{2\\sigma+1}}|D_{v}|^{\\delta},$$\nwhere $T_{\\lambda}$ stands for the following unitary transformation on $L^{2}(\\mathbb{R}^{2n+1})$, \n$$T_{\\lambda}u(t,x,v)=\\lambda^{\\frac{2\\sigma+n}{2(2\\sigma+1)}+\\frac{n}{2}}u(\\lambda^{\\frac{2\\sigma}{2\\sigma+1}}t,\\lambda x,\\lambda^{\\frac{1}{2\\sigma+1}}v).$$\nRecalling that $\\lambda \\geq 1$, we notice that the $C_0^{\\infty}(\\mathbb{R})$ function $T_{\\lambda}u$ is supported in $B_1$ if the $C_0^{\\infty}(\\mathbb{R})$ function $u$ is supported in $B_1$.\nBy applying the previous a priori estimate to functions $T_{\\lambda}u$, we obtain that \n$$\\|(|D_t|^{\\delta}+|D_x|^{\\delta}+|D_{v}|^{\\delta})T_{\\lambda}u\\|_{L^2} \\leq C\\big(\\|(iD_t+iv \\cdot D_x  + |D_v|^{2\\sigma})T_{\\lambda}u\\|_{L^2}+\\|T_{\\lambda}u\\|_{L^2}\\big).$$\nBy using that $T_{\\lambda}^{-1}$ is a unitary transformation on $L^{2}(\\mathbb{R}^{2n+1})$, we deduce that for any $\\lambda \\geq 1$, \n\\begin{multline*}\n\\|(\\lambda^{\\frac{2\\sigma\\delta}{2\\sigma+1}}|D_t|^{\\delta}+\\lambda^{\\delta}|D_x|^{\\delta}+\\lambda^{\\frac{\\delta}{2\\sigma+1}}|D_{v}|^{\\delta})u\\|_{L^2} \\\\\\ \\leq C\\big(\\lambda^{\\frac{2\\sigma}{2\\sigma+1}}\\|(iD_t+iv \\cdot D_x  + |D_v|^{2\\sigma})u\\|_{L^2}+\\|u\\|_{L^2}\\big).\n\\end{multline*}\nThis estimate may hold for any $\\lambda \\geq 1$ only if \n$$\\delta \\leq \\frac{2\\sigma}{2\\sigma+1}.$$\nThis scaling argument shows that the positive gain $2\\sigma\/(2\\sigma+1)>0$ in the hypoelliptic estimates (\\ref{yo4}) is the optimal possible one.\n\n\nAs a consequence of these optimal hypoelliptic estimates, we obtain the following result where we write\n$$f \\in H^s_{\\textrm{loc}, (t_0,x_0)}(\\mathbb{R}^{2n+1}_{t,x,v}),$$\nif there exists an open neighborhood $U$ of the point $(t_0,x_0)$ in $\\mathbb{R}^{n+1}$ such that\n$\\phi(t,x) f\\in H^s(\\mathbb{R}_{t,x,v}^{2n+1})$\nfor any $\\phi \\in C_0^\\infty(U)$. \n\n\n\\bigskip\n\n\\begin{Corollary}\\label{ev1}\nLet $P$ be the operator defined in \\emph{(\\ref{yo2})} and $N \\in \\mathbb{N}$. \nIf $u \\in H_{-N}(\\mathbb{R}_{t,x,v}^{2n+1})$ and $Pu \\in H^s_{\\emph{\\textrm{loc}}, (t_0,x_0)}(\\mathbb{R}^{2n+1}_{t,x,v})$ with $s \\ge 0$, then there exists \nan integer $k \\geq 1$ such that \n\\[\n\\frac{u}{\\langle  v \\rangle^k} \\in H^{s+\\frac{2\\sigma}{2\\sigma+1}}_{\\emph{\\textrm{loc}}, (t_0,x_0)}(\\mathbb{R}^{2n+1}_{t,x,v}),\n\\]\nwhere $\\langle v \\rangle=(1+|v|^2)^{1\/2}$.\nIn particular, if $u \\in H_{-N}(\\mathbb{R}^{2n+1})$ and $Pu \\in H^{\\infty}(\\mathbb{R}^{2n+1})$ then $u \\in C^{\\infty}(\\mathbb{R}^{2n+1})$.\n\\end{Corollary}\n\n\\bigskip\n\n\n\n\nCorollary~\\ref{ev1} allows to recover the $C^{\\infty}$ hypoellipticity proved in \\cite{MoXu} (Theorem~1.2) with now optimal loss of derivatives. Notice that the equation (\\ref{yo2}) is not a classical pseudodifferential equation. Indeed, the coefficient $v$ in (\\ref{yo2}) is unbounded and the fractional Laplacian $(-\\tilde{\\Delta}_v)^{\\sigma}$ is a classical pseudo-differential operator in the velocity variable $v$ but not in all the variables $t,x,v$. This accounts for parts of the difficulties encountered when studying this kinetic operator in particular when using cutoff functions in the velocity variable. This also accounts for the weight $\\langle  v \\rangle^{-k}$ appearing in the statement of Corollary~\\ref{ev1}. Notice that the proof of this result is constructive and that one may derive an explicit (possibly not sharp) bound on the integer $k \\geq 1$.\n\n\n\nThe proof of Theorem~\\ref{TTH1} is relying on some microlocal techniques developed by N.~Lerner for proving energy estimates while using the Wick quantization \\cite{cubo}. Let us mention that some of these techniques were already used in the work~\\cite{lms}. The strategy for proving Theorem~\\ref{TTH1} is the following. We want to consider this equation as an evolution equation along the characteristic curves of $\\partial_t+v \\cdot \\partial_x$; for that purpose, we straighten this vector field and get the normal form\n$$iD_t+a(t,D_{x_2},D_{x_1}+tD_{x_2})F(x_2-tx_1).$$\nThis normal form suggests to derive some a priori estimates for the one-dimensional first-order differential operator \n$$iD_t+a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1),$$\ndepending on the parameters $x_1,x_2,\\xi_1,\\xi_2 \\in \\mathbb{R}^n$. We then deduce from those a priori estimates some a priori estimates for the operator \n$$iD_t+\\big[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)\\big]^{\\textrm{Wick}},$$\ndefined by using the Wick quantization. As a last step, we need to control some remainder terms in order to come back to the standard quantization and derive a priori estimates for the original operator\n$$iD_t+a(t,D_{x_2},D_{x_1}+tD_{x_2})F(x_2-tx_1).$$\nFor the sake of completeness and to keep the paper essentially self-contained, the definition and all the main features of the Wick quantization are recalled in appendix (Section~\\ref{appendix}). Next section is devoted to the proof of a key hypoelliptic estimate (Proposition~\\ref{th1}) which is the core of the present work. This key estimate is then the main ingredient in Section~\\ref{section3} for proving Theorem~\\ref{TTH1}. Corollary~\\ref{ev1} is established in Section~\\ref{section4}. \n\n\n\n\n\nBefore ending this introduction, we give some references and comments about the hypoelliptic properties of the non-cutoff Boltzmann equation. Following the rigorous derivation of the lower bound for the non-cutoff Boltzmann collision operator in~\\cite{alexandre0}, there have been many works on the regularity for the solutions of the Boltzmann equation in both \nspatially homogeneous (see \\cite{alexandre1, al-saf-1, desv1,HMUY, MU,Mo2} and references therein) and\ninhomogeneous cases (see \\cite{amuxy3, amuxy4,  amuxy5-3}).\nIn all those works, it was highlighted that the Boltzmann collision operator behaves essentially as a fractional Laplacian $(-\\tilde{\\Delta}_v)^{\\sigma}$  \nunder the angular singularity assumption on the collision cross-section (see \\cite{alexandre0, amuxy3}). We refer the reader to Section~\\ref{kkboltz} in appendix for comprehensive explanations about the relevance of this model. In the spatially homogeneous case, this diffusive structure implies a $C^{\\infty}$ smoothing effect for the weak solutions to the Cauchy problem constructed in~\\cite{Villani} (See \\cite{amuxy7, HMUY}). Furthermore, as in the case of the heat equation, the spatially homogeneous Boltzmann equation without angular cutoff enjoys a\nsmoothing effect in the Gevrey class of order $\\sigma$ (see \\cite{ L-X, MU, Mo2}). Related to this Gevrey smoothing effect for the spatially homogeneous Boltzmann equation, an  \nultra-analytic smoothing effect was proved in~\\cite{Mo1} for both non-linear homogeneous Landau equations and inhomogeneous linear Landau equations.\nRegarding the study of the Boltzmann equation in Gevrey spaces, we also refer the reader to the seminal work \\cite{Ukai} which establishes the existence and uniqueness in Gevrey classes of a local solution to the Cauchy problem for the Boltzmann equation in both spatially homogeneous and inhomogeneous cases.\nConsidering now the spatially inhomogeneous Boltzmann equation without angular cutoff, \nthe $C^\\infty$ hypoellipticity was established in \\cite{amuxy3,amuxy4,amuxy5-3} by using the coercivity estimate for proving the regularity with respect to the velocity variable $v$ (see~\\cite{alexandre0,amuxy3}) and a version of the uncertainty principle for proving the regularity in the time and space variables $t,x$ via bootstrap arguments (see~\\cite{amuxy2}). However, it should be noted that those works do not provide any optimal hypoelliptic estimates for the spatially inhomogeneous Boltzmann equation without angular cutoff. As an attempt to understand further the smoothing effect induced by the Boltzmann collision operator and to relate exactly the structure of the angular singularity in the collision cross-section to this regularizing effect, the present work studies the hypoelliptic properties of the simplified linear model (\\ref{yo2}) and aims at giving insights on the hypoelliptic properties what may be expected for the general spatially inhomogeneous Boltzmann equation without angular cutoff. \n\n\n\n\n\n\n\n\n\\section{A key hypoelliptic estimate}\\label{section2}\n\nTheorem~\\ref{TTH1} will be derived from the following key hypoelliptic estimate: \n\n\n\n\\bigskip\n\n\n\\begin{proposition}\\label{th1}\nLet $P$ be the operator defined in~\\emph{(\\ref{yo2})} and $T>0$ be a positive constant. Then, there exists a positive constant $C_T>0$ such that for all $u \\in \\mathscr{S}(\\mathbb{R}_{t,x,v}^{2n+1})$ satisfying\n\\begin{equation}\\label{hypo0}\n\\emph{\\textrm{supp }} u(\\cdot,x,v) \\subset [-T,T], \\ (x,v) \\in \\mathbb{R}^{2n},\n\\end{equation}\nwe have\n\\begin{equation}\\label{hypo}\n\\big\\|(1+|D_x|^{\\frac{2\\sigma}{2\\sigma+1}}+|D_{v}|^{2\\sigma})u\\big\\|_{L^2(\\mathbb{R}^{2n+1})} \\leq C_T \\big(\\|Pu\\|_{L^2(\\mathbb{R}^{2n+1})}+\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}\\big).\n\\end{equation}\n\\end{proposition}\n\n\\bigskip\n\n\\noindent\n\nIn the following, we use standard notations for symbol classes, see~\\cite{hormander} (Chapter~18) or \\cite{birkhauser}.\nThe symbol class $S(m,\\Gamma)$ associated to the order function $m$ and metric\n$$\\Gamma=\\frac{dx^2}{\\varphi(x,\\xi)^2}+\\frac{d\\xi^2}{\\Phi(x,\\xi)^2},$$\nwith $\\varphi$, $\\Phi$ given positive functions,\nstands for the set of functions $a\\in C^{\\infty}(\\mathbb{R}_{x,\\xi}^{2n},\\mathbb{C})$ satisfying for all $\\alpha \\in \\mathbb{N}^{n}$ and $\\beta \\in \\mathbb{N}^n$, \n\\begin{equation}\\label{ro1}\n\\exists C_{\\alpha, \\beta}>0, \\forall (x,\\xi) \\in \\mathbb{R}^{2n}, \\  |\\partial_{x}^{\\alpha}\\partial_{\\xi}^{\\beta} a(x,\\xi)| \\leq C_{\\alpha, \\beta} m(x,\\xi) \\varphi(x,\\xi)^{-|\\alpha|}\\Phi(x,\\xi)^{-|\\beta|}.\n\\end{equation}\n\n\n\n\\subsection{Some symbol reductions} We begin by few symbol reductions in order to reduce the symbol of the operator to a convenient normal form. For convenience only, we shall use the Weyl quantization rather than the standard one.\nNotice that it will be sufficient to prove the hypoelliptic estimate (\\ref{hypo}) for the operator $p^w$ defined by the Weyl quantization of the symbol \n\\begin{equation}\\label{eq1}\np(t,x,v;\\tau,\\xi,\\eta)=2\\pi i\\tau+2\\pi iv\\cdot \\xi+a(t,x,v)F(2\\pi\\eta),\n\\end{equation}\nwith $\\tau$, $\\xi$, $\\eta$ respectively standing for the dual variables of the variables $t$, $x$, $v$ and $F$ being the function defined in (\\ref{eq0}),\nwhile using the following normalization for the Weyl quantization\n$$(a^wu)(x)=\\int_{\\mathbb{R}^{2n}}e^{2i\\pi(x-y)\\cdot \\xi}a\\big(\\frac{x+y}{2},\\xi\\big)u(y)dyd\\xi.$$\nIndeed, symbolic calculus shows that the operator\n$$R=P-p^w+\\frac{1}{2i}\\nabla_va(t,x,v) \\cdot (\\nabla F)(2\\pi D_v),$$\nis bounded on $L^2$. This is a direct consequence of the $L^2$ continuity theorem in the class $S_{00}^0$ (case $m=\\varphi=\\Phi=1$ in (\\ref{ro1})) after noticing that the Weyl symbol of the operator $R$ together with all its derivatives of any order are bounded on $\\mathbb{R}^{4n+2}$, since all the derivatives of order greater or equal to 2 of the symbol $F$ are bounded given the choice of the positive parameter $0 < \\sigma <1$. We refer the reader to formula (2.1.26) in \\cite{birkhauser} for explicit constants in the composition formula with the normalization of the Weyl quantization chosen here. It then remains to notice that for any $\\varepsilon>0$, there is a positive constant $C_{\\varepsilon}>0$ such that \n\\begin{multline*}\n\\big\\|\\nabla_va(t,x,v) \\cdot (\\nabla F)(2\\pi D_v)u\\big\\|_{L^2(\\mathbb{R}^{2n+1})} \\lesssim \\big\\|(\\nabla F)(2\\pi D_v)u\\big\\|_{L^2(\\mathbb{R}^{2n+1})}\\\\ \\lesssim \\|(1+|D_v|^{2\\sigma-1})u\\|_{L^2(\\mathbb{R}^{2n+1})}  \\leq \\varepsilon  \\|(1+|D_v|^{2\\sigma})u\\|_{L^2(\\mathbb{R}^{2n+1})} +C_{\\varepsilon}\\|u\\|_{L^2(\\mathbb{R}^{2n+1})},\n\\end{multline*}   \nsince the function $\\nabla_va$ is bounded on $\\mathbb{R}^{2n+1}$. When studying the operator $p^w$, it is convenient to work on the Fourier side in the variables $x,v$. This is equivalent to study the operator defined by the Weyl quantization of the symbol\n$$2\\pi i\\tau-2\\pi i x \\cdot \\eta +a(t,-\\xi,-\\eta)F(2\\pi v).$$\nAfter relabeling the variables and simplifying the notations, we are reduced to study the operator $P=p^w$ defined by the Weyl quantization of the symbol\n\\begin{equation}\\label{eq2}\np(t,x,v;\\tau,\\xi,\\eta)= i\\tau- iy \\cdot \\xi+a(t,\\xi,\\eta)F(x),\n\\end{equation}\nwhere $a$ stands for a $C_b^{\\infty}(\\mathbb{R}^{2n+1})$ function satisfying \n\\begin{equation}\\label{eq00.5}\n\\exists a_0>0, \\ \\forall (t,\\xi,\\eta) \\in \\mathbb{R}^{2n+1}, \\ a(t,\\xi,\\eta) \\geq a_0>0,\n\\end{equation}\nand $F$ is the function\n\\begin{equation}\\label{eq00}\nF(x)=|x|^{2\\sigma}w(x)+|x|^2\\big(1-w(x)\\big),\n\\end{equation}\nwith $w$ a new function having experienced a small homothetic transformation compared to the function appearing in (\\ref{eq0}).\nFor the sake of simplicity, we shall keep the definition given in (\\ref{eq0}) and assume that $w \\in C^{\\infty}(\\mathbb{R}^n)$, $0 \\leq w \\leq 1$, $w(\\eta)=1$ if $|\\eta| \\geq 2$, and $w(\\eta)=0$ if $|\\eta| \\leq 1$. \nUsing these notations, Proposition~\\ref{th1} is equivalent to the proof of the following a priori estimate. For any $T>0$, there exists a positive constant $C_T>0$ such that for all $u \\in \\mathscr{S}(\\mathbb{R}_{t,x,y}^{2n+1})$ satisfying\n\\begin{equation}\\label{eq3}\n\\textrm{supp } u(\\cdot,x,y) \\subset [-T,T], \\ (x,y) \\in \\mathbb{R}^{2n},\n\\end{equation}\nwe have\n\\begin{equation}\\label{hypo2}\n\\big\\|(1+|x|^{2\\sigma}+|y|^{\\frac{2\\sigma}{2\\sigma+1}})u\\big\\|_{L^2(\\mathbb{R}^{2n+1})} \\leq C_T \\big(\\|Pu\\|_{L^2(\\mathbb{R}^{2n+1})}+\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}\\big).\n\\end{equation}\nIn order to do so, we consider the new variables \n$$(x_1,x_2)=(y,x+ty),$$ \nwith $t \\in \\mathbb{R}$ fixed. We define $A(x,y)=(y,x+ty)$. Associated to this linear change of variables are the real linear symplectic transformation\n$$(x_1,x_2;\\xi_1,\\xi_2)=\\chi(x,y;\\xi,\\eta)=(A(x,y);(A^{-1})^T(\\xi,\\eta))=(y,x+ty;\\eta-t\\xi,\\xi),$$\nand the two unitary operators on $L^2(\\mathbb{R}^{2n})$\n$$(M_tu)(x_1,x_2)=u(x_2-tx_1,x_1); \\ (M_t^{-1}u)(x,y)=u(y,x+ty).$$  \nKeeping on considering the $t$-variable as a parameter and defining the symbol\n$$b_t(x,y;\\xi,\\eta)=- iy \\cdot \\xi+a(t,\\xi,\\eta)F(x),$$\nwe have\n$$(b_t \\circ \\chi^{-1})(x_1,x_2;\\xi_1,\\xi_2)=-ix_1 \\cdot \\xi_2+a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1),$$\nand the symplectic invariance of the Weyl quantization (Theorem~18.5.9 in~\\cite{hormander}) or a simple direct calculation\n$$M_t^{-1}a^wM_t=(a \\circ \\chi)^w,$$ \nshow that \n$$M_t b_t^w(x,y,D_x,D_y)M_t^{-1}=-ix_1 \\cdot D_{x_2}+\\big[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)\\big]^w.$$\nWe then consider the two unitary operators acting on $L^2(\\mathbb{R}_{t,x,y}^{2n+1})$,\n$$(Mu)(t,x_1,x_2)=u(t,x_2-tx_1,x_1), \\ (M^{-1}u)(t,x,y)=u(t,y,x+ty),$$ \nand notice that \n$$(MiD_t M^{-1}u)(t,x_1,x_2)=iD_t u(t,x_1,x_2)+ix_1 \\cdot D_{x_2}u(t,x_1,x_2).$$\nIt follows that \n$$iD_t+\\big[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)\\big]^w=MPM^{-1}.$$\nNotice also that \n$$1+|x_2-tx_1|^{2\\sigma}+|x_1|^{\\frac{2\\sigma}{2\\sigma+1}}=M(1+|x|^{2\\sigma}+|y|^{\\frac{2\\sigma}{2\\sigma+1}})M^{-1}.$$\nWe can therefore reduce the proof of Proposition~\\ref{th1} to the proof of the following a priori estimate. For any $T>0$, there exists a positive constant $C_T>0$ such that for all $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying\n\\begin{equation}\\label{eq4}\n\\textrm{supp } u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},\n\\end{equation}\nwe have\n\\begin{equation}\\label{eq5}\n\\big\\|(1+|x_2-tx_1|^{2\\sigma}+|x_1|^{\\frac{2\\sigma}{2\\sigma+1}})u\\big\\|_{L^2(\\mathbb{R}^{2n+1})} \\leq C_T \\big(\\|Pu\\|_{L^2(\\mathbb{R}^{2n+1})}+\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}\\big),\n\\end{equation}\nfor the operator \n\\begin{equation}\\label{eq6}\nP=iD_t+\\big[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)\\big]^w.\n\\end{equation}\n\n\\bigskip\n\n\n\\subsection{Energy estimates via the Wick quantization}\nIn order to establish the a priori estimate (\\ref{eq5}), we shall use some techniques developed in~\\cite{cubo} for proving energy estimates via the Wick quantization. We recall in appendix (see Section~\\ref{appendix}) the definition and all the main features of the Wick quantization which will be used here. \n\n\nA first step in adapting this approach is to study the first-order differential operator on $L^2(\\mathbb{R}_t)$,\n$$\\tilde{P}=iD_t+a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1),$$\nwhere the variables $x_1,x_2,\\xi_1,\\xi_2$ are considered as parameters, and prove some a priori estimates with respect to those parameters.  \nWe begin by noticing from (\\ref{eq00.5}) that for any $u \\in \\mathscr{S}(\\mathbb{R}_t)$, \n$$\\textrm{Re}(\\tilde{P}u,u)_{L^2(\\mathbb{R}_t)}=\\int_{\\mathbb{R}}a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)|u(t)|^2dt \n\\geq a_0 \\big\\|F(x_2-tx_1)^{1\/2}u\\big\\|_{L^2(\\mathbb{R}_t)}^2,$$\nsince the operator $iD_t$ is skew-adjoint.\nIt follows from the Cauchy-Schwarz inequality that\n\\begin{equation}\\label{eq7}\na_0 \\big\\|F(x_2-tx_1)^{1\/2}u\\big\\|_{L^2(\\mathbb{R}_t)}^2 \\leq \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}\\|u\\|_{L^2(\\mathbb{R}_t)}.\n\\end{equation}\nBy denoting $H={\\mathrm{1~\\hspace{-1.4ex}l}}_{\\mathbb{R}_+}$ the Heaviside function, we may write for any $T \\in \\mathbb{R}$,\n\\begin{multline}\\label{eq8}\n 2\\textrm{Re}(\\tilde{P}u,-H(t-T)u)_{L^2(\\mathbb{R}_t)} = 2\\textrm{Re}(D_tu,iH(t-T)u)_{L^2(\\mathbb{R}_t)} \\\\\n -2\\int_{\\mathbb{R}}H(t-T)a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)|u(t)|^2dt,\n\\end{multline}\nand obtain by a simple integration by parts that \n$$2\\textrm{Re}(D_tu,iH(t-T)u)_{L^2(\\mathbb{R}_t)}=([D_t,iH(t-T)]u,u)_{L^2(\\mathbb{R}_t)}=\\frac{1}{2\\pi}|u(T)|^2.$$\nRecalling that $a \\in L^{\\infty}(\\mathbb{R}^{2n+1})$, one may find a positive constant $C_0>0$ such that \n$$\\Big|2\\int_{\\mathbb{R}}H(t-T)a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)|u(t)|^2dt\\Big| \\leq C_0\\big\\|F(x_2-tx_1)^{1\/2}u\\big\\|_{L^2(\\mathbb{R}_t)}^2.$$\nIt follows from (\\ref{eq7}), (\\ref{eq8}) and the Cauchy-Schwarz inequality that there exists a constant $C_1>0$ such that for all $u \\in \\mathscr{S}(\\mathbb{R}_t)$ and $(x_1,x_2,\\xi_1,\\xi_2) \\in \\mathbb{R}^{4n}$,\n\\begin{equation}\\label{eq9}\n\\|u\\|_{L^{\\infty}(\\mathbb{R}_t)}^2 \\leq C_1\\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}\\|u\\|_{L^2(\\mathbb{R}_t)}.\n\\end{equation}\nFollowing~\\cite{cubo}, we split-up the $L^2(\\mathbb{R}_t)$-norm of $u$,\n\\begin{multline}\\label{eq10}\n |x_1|^{\\frac{2\\sigma}{2\\sigma+1}}\\|u\\|_{L^2(\\mathbb{R}_t)}^2\n=  \\int_{\\{t \\in \\mathbb{R} :\\ |x_1|^{\\frac{2\\sigma}{2\\sigma+1}} > |x_2-tx_1|^{2\\sigma}\\}}|x_1|^{\\frac{2\\sigma}{2\\sigma+1}}|u(t)|^2dt \\\\\n+\\int_{\\{t \\in \\mathbb{R} :\\ |x_1|^{\\frac{2\\sigma}{2\\sigma+1}} \\leq |x_2-tx_1|^{2\\sigma}\\}}|x_1|^{\\frac{2\\sigma}{2\\sigma+1}} |u(t)|^2dt ,\n\\end{multline}\nand estimate from above the first integral as\n\\begin{multline*}\n\\int_{\\{t \\in \\mathbb{R} :\\ |x_1|^{\\frac{2\\sigma}{2\\sigma+1}} > |x_2-tx_1|^{2\\sigma}\\}}|x_1|^{\\frac{2\\sigma}{2\\sigma+1}}|u(t)|^2dt \\\\\n\\leq |x_1|^{\\frac{2\\sigma}{2\\sigma+1}} m\\big(\\{t \\in \\mathbb{R} :\\ |x_1|^{\\frac{2\\sigma}{2\\sigma+1}} > |x_2-tx_1|^{2\\sigma}\\}\\big) \\|u\\|_{L^{\\infty}(\\mathbb{R}_t)}^2 \\leq 2\\|u\\|_{L^{\\infty}(\\mathbb{R}_t)}^2,\n\\end{multline*}\nwith $m$ standing for the Lebesgue measure on $\\mathbb{R}$.\nIt follows from (\\ref{eq9}) that this first integral can be estimated from above as\n\\begin{equation}\\label{eq10.5}\n\\int_{\\{t \\in \\mathbb{R} :\\ |x_1|^{\\frac{2\\sigma}{2\\sigma+1}} > |x_2-tx_1|^{2\\sigma}\\}}|x_1|^{\\frac{2\\sigma}{2\\sigma+1}}|u(t)|^2dt \\leq 2C_1\\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}\\|u\\|_{L^2(\\mathbb{R}_t)}.\n\\end{equation}\nWhile estimating from above the second integral in the right-hand-side of (\\ref{eq10}), we may first write that \n\\begin{multline*}\n\\int_{\\{t \\in \\mathbb{R} :\\ |x_1|^{\\frac{2\\sigma}{2\\sigma+1}} \\leq |x_2-tx_1|^{2\\sigma}\\}}|x_1|^{\\frac{2\\sigma}{2\\sigma+1}} |u(t)|^2dt\\\\ \\leq \\int_{\\{t \\in \\mathbb{R} :\\ |x_1|^{\\frac{2\\sigma}{2\\sigma+1}} \\leq |x_2-tx_1|^{2\\sigma}\\}}|x_2-tx_1|^{2\\sigma} |u(t)|^2dt\n\\leq \\int_{\\mathbb{R}}|x_2-tx_1|^{2\\sigma} |u(t)|^2dt,\n\\end{multline*}\nand then notice from (\\ref{eq00}) that there exists a positive constant $C_2>0$ such that for all $(t,x_1,x_2,\\xi_1,\\xi_2) \\in \\mathbb{R}^{4n+1}$,\n$$|x_2-tx_1|^{2\\sigma} \\leq F(x_2-tx_1)+C_2.$$\nIt follows from (\\ref{eq7}) that \n\\begin{multline}\\label{eq11}\n\\int_{\\{t \\in \\mathbb{R} :\\ |x_1|^{\\frac{2\\sigma}{2\\sigma+1}} \\leq |x_2-tx_1|^{2\\sigma}\\}}|x_1|^{\\frac{2\\sigma}{2\\sigma+1}} |u(t)|^2dt \\\\ \\leq \\|F(x_2-tx_1)^{1\/2}u\\|_{L^2(\\mathbb{R}_t)}^2+C_2\\|u\\|_{L^2(\\mathbb{R}_t)}^2 \\leq\na_0^{-1} \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}\\|u\\|_{L^2(\\mathbb{R}_t)}+C_2\\|u\\|_{L^2(\\mathbb{R}_t)}^2.\n\\end{multline}\nWe deduce from (\\ref{eq10}), (\\ref{eq10.5}) and (\\ref{eq11}) that there exist some positive constants $C_3>0$ and $C_4>0$ such that for all $u \\in \\mathscr{S}(\\mathbb{R}_t)$ and $(x_1,x_2,\\xi_1,\\xi_2) \\in \\mathbb{R}^{4n}$,\n$$ |x_1|^{\\frac{2\\sigma}{2\\sigma+1}}\\|u\\|_{L^2(\\mathbb{R}_t)}^2 \\leq C_3 \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}\\|u\\|_{L^2(\\mathbb{R}_t)}+C_3\\|u\\|_{L^2(\\mathbb{R}_t)}^2,$$\nthat is\n$$|x_1|^{\\frac{2\\sigma}{2\\sigma+1}}\\|u\\|_{L^2(\\mathbb{R}_t)} \\leq C_3 \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}+C_3\\|u\\|_{L^2(\\mathbb{R}_t)},$$\nwhich implies that \n\\begin{equation}\\label{eq12}\n \\|\\langle x_1\\rangle^{\\frac{2\\sigma}{2\\sigma+1}}u\\|_{L^2(\\mathbb{R}_t)}^2 \\leq C_4 \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}^2+C_4\\|u\\|_{L^2(\\mathbb{R}_t)}^2,\n\\end{equation}\nwith $\\langle x \\rangle=(1+|x|^2)^{1\/2}$.\nWe shall now prove that there exists a positive constant $C_5>0$ such that for all $u \\in \\mathscr{S}(\\mathbb{R}_t)$ and $(x_1,x_2,\\xi_1,\\xi_2) \\in \\mathbb{R}^{4n}$,\n\\begin{equation}\\label{eq13}\n\\|\\langle x_1\\rangle^{\\frac{2\\sigma}{2\\sigma+1}}u\\|_{L^2(\\mathbb{R}_t)}^2+ \\||x_2-tx_1|^{2\\sigma}u\\|_{L^2(\\mathbb{R}_t)}^2 \\leq C_5 \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}^2+C_5\\|u\\|_{L^2(\\mathbb{R}_t)}^2.\n\\end{equation}\nIn order to do so, let $\\varepsilon_0 \\in \\{\\pm1\\}$ and write the components of the variables $x_1,x_2$ as\n$$x_1=(x_{1,1},x_{1,2},...,x_{1,n}) \\textrm{ and } x_2=(x_{2,1},x_{2,2},...,x_{2,n}).$$ \nLet $j \\in \\{1,...,n\\}$. We shall first study the case when the real parameter \n\\begin{equation}\\label{oyo1}\nx_{1,j} \\neq 0.\n\\end{equation}\nWhile expanding the following $L^2(\\mathbb{R}_t)$ dot-product where $H$ still denotes the Heaviside function\n\\begin{align*}\n& \\ 2\\textrm{Re}\\big(\\tilde{P}u,|x_{2,j}-tx_{1,j}|^{2\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})u\\big)_{L^2(\\mathbb{R}_t)} \\\\\n=& \\ 2\\textrm{Re}\\big(D_tu,-i |x_{2,j}-tx_{1,j}|^{2\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})u\\big)_{L^2(\\mathbb{R}_t)} \\\\ \n+ & \\  2 \\int_{\\mathbb{R}}a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)|x_{2,j}-t x_{1,j}|^{2\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})|u(t)|^2dt,\n\\end{align*}\nwe notice that \n\\begin{multline*}\n 2\\textrm{Re}\\big(D_tu,-i |x_{2,j}-tx_{1,j}|^{2\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})u\\big)_{L^2(\\mathbb{R}_t)}\\\\\n= \\big([D_t,-i |x_{2,j}-tx_{1,j}|^{2\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})]u,u\\big)_{L^2(\\mathbb{R}_t)}. \n\\end{multline*}\nA direct computation gives that \n\\begin{align*}\n& \\ \\big([D_t,-i |x_{2,j}-tx_{1,j}|^{2\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})]u,u\\big)_{L^2(\\mathbb{R}_t)}\\\\\n=& \\ \\frac{\\varepsilon_0 \\sigma}{\\pi}\\int_{\\mathbb{R}} x_{1,j} |x_{2,j}-tx_{1,j}|^{2\\sigma-1}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})|u(t)|^2dt\\\\\n& \\ + \\varepsilon_0\\frac{2^{2\\sigma-1} }{\\pi} \\frac{x_{1,j}}{|x_{1,j}|} \\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}|u(T)|^2,\n\\end{align*}\nwith\n$$T=x_{2,j}x_{1,j}^{-1}-2\\varepsilon_0 x_{1,j}^{-1}\\langle x_1\\rangle^{\\frac{1}{2\\sigma+1}}$$\nand we deduce from (\\ref{eq9}) that \n\\begin{align*}\n& \\ \\big|\\big([D_t,-i |x_{2,j}-tx_{1,j}|^{2\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})]u,u\\big)_{L^2(\\mathbb{R}_t)}\\big|\\\\\n\\leq & \\ \\frac{2^{2\\sigma-1}}{\\pi}C_1 \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}\\|\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}u\\|_{L^2(\\mathbb{R}_t)}\\\\\n+& \\ \\frac{\\sigma}{\\pi}\\int_{\\mathbb{R}} |x_{1,j}| |x_{2,j}-tx_{1,j}|^{2\\sigma-1}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})|u(t)|^2dt.\n\\end{align*}\nSince from (\\ref{eq00.5}),  \n\\begin{multline*}\n2a_0\\|F(x_2-tx_1)^{1\/2}|x_{2,j}-t x_{1,j}|^{\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})u\\|_{L^2(\\mathbb{R}_t)}^2 \\\\ \\leq 2 \\int_{\\mathbb{R}}a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)|x_{2,j}-t x_{1,j}|^{2\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})|u(t)|^2dt,\n\\end{multline*}\nwe deduce from the Cauchy-Schwarz inequality; and all the previous identities and estimates obtained after (\\ref{oyo1}) that \n\\begin{align*}\n& \\ 2a_0\\|F(x_2-tx_1)^{1\/2}|x_{2,j}-t x_{1,j}|^{\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})u\\|_{L^2(\\mathbb{R}_t)}^2\\\\\n\\leq & \\ \\frac{2^{2\\sigma-1}}{\\pi}C_1 \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}\\|\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}u\\|_{L^2(\\mathbb{R}_t)}\\\\\n+ & \\ \\frac{\\sigma}{\\pi}\\int_{\\mathbb{R}} |x_{1,j}| |x_{2,j}-tx_{1,j}|^{2\\sigma-1}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})|u(t)|^2dt\\\\\n+& \\ 2\\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}\\||x_{2,j}-tx_{1,j}|^{2\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})u\\|_{L^2(\\mathbb{R}_t)}.\n\\end{align*}\nSince from (\\ref{eq00}), we have\n$$F(x_2-tx_1)=|x_2-t x_1|^{2\\sigma} \\geq |x_{2,j}-t x_{1,j}|^{2\\sigma},$$\non the support of the function $H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})$, we then deduce from the previous estimate that there exists a positive constant $C_6>0$ such that for all $u \\in \\mathscr{S}(\\mathbb{R}_t)$ and $(x_1,x_2,\\xi_1,\\xi_2) \\in \\mathbb{R}^{4n}$, $x_{1,j} \\neq 0$, \n\\begin{multline}\\label{eq14}\n C_6^{-1}\\||x_{2,j}-t x_{1,j}|^{2\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})u\\|_{L^2(\\mathbb{R}_t)}^2\n\\leq   \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}^2\\\\\n+ \\|\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}u\\|_{L^2(\\mathbb{R}_t)}^2\n+  \\int_{\\mathbb{R}} |x_{1,j}| |x_{2,j}-tx_{1,j}|^{2\\sigma-1}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})|u(t)|^2dt.\n\\end{multline}\nBy using that \n$$|x_{2,j}-tx_{1,j}|^{-1} \\leq \\frac{1}{2} \\langle x_1 \\rangle^{-\\frac{1}{2\\sigma+1}},$$\non the support of the function $H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})$, one can estimate from above the following integral as\n\\begin{align*}\n& \\ \\int_{\\mathbb{R}} |x_{1,j}| |x_{2,j}-tx_{1,j}|^{2\\sigma-1}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})|u(t)|^2dt \\\\ \n\\leq & \\ 2^{-1} \\int_{\\mathbb{R}} \\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}|x_{2,j}-tx_{1,j}|^{2\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})|u(t)|^2dt\\\\\n\\leq & \\ 2^{-1} \\|\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}u\\|_{L^2(\\mathbb{R}_t)}\\||x_{2,j}-tx_{1,j}|^{2\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})u\\|_{L^2(\\mathbb{R}_t)}.\n\\end{align*}\nAccording to (\\ref{eq14}), this implies  that there exists a positive constant $C_7>0$ such that for all $u \\in \\mathscr{S}(\\mathbb{R}_t)$ and $(x_1,x_2,\\xi_1,\\xi_2) \\in\\mathbb{R}^{4n}$, $x_{1,j} \\neq 0$, \n\\begin{multline}\\label{eq15}\n C_7^{-1}\\||x_{2,j}-t x_{1,j}|^{2\\sigma}H(\\varepsilon_0(x_{2,j}-tx_{1,j})-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})u\\|_{L^2(\\mathbb{R}_t)}^2\\\\\n\\leq   \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}^2\n+ \\|\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}u\\|_{L^2(\\mathbb{R}_t)}^2.\n\\end{multline}\nFinally, since\n\\begin{multline*}\n|x_{2,j}-t x_{1,j}|^{2\\sigma} \\leq |x_{2,j}-t x_{1,j}|^{2\\sigma}H(x_{2,j}-tx_{1,j}-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})\\\\\n+ |x_{2,j}-t x_{1,j}|^{2\\sigma}H(-x_{2,j}+tx_{1,j}-2\\langle x_1 \\rangle^{\\frac{1}{2\\sigma+1}})+2^{2\\sigma}\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}},\n\\end{multline*}\nwe deduce from (\\ref{eq15}) that there exists a positive constant $C_8>0$ such that for all $u \\in \\mathscr{S}(\\mathbb{R}_t)$ and $(x_1,x_2,\\xi_1,\\xi_2) \\in \\mathbb{R}^{4n}$, $x_{1,j} \\neq 0$, \n\\begin{equation}\\label{eq16}\n\\||x_{2,j}-t x_{1,j}|^{2\\sigma}u\\|_{L^2(\\mathbb{R}_t)}^2\\\\\n\\leq  C_8 \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}^2 + C_8\\|\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}u\\|_{L^2(\\mathbb{R}_t)}^2,\n\\end{equation}\nwhich together with (\\ref{eq12}) proves the a priori estimate\n\\begin{equation}\\label{eq13.11}\n\\|\\langle x_1\\rangle^{\\frac{2\\sigma}{2\\sigma+1}}u\\|_{L^2(\\mathbb{R}_t)}^2+ \\||x_{2,j}-tx_{1,j}|^{2\\sigma}u\\|_{L^2(\\mathbb{R}_t)}^2 \\leq C_9 \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}^2+C_9\\|u\\|_{L^2(\\mathbb{R}_t)}^2,\n\\end{equation}\nwith $C_9>0$ a positive constant; in the case when $x_{1,j} \\neq 0$. Assume now that $x_{1,j}=0$. In this case, a direct computation using (\\ref{eq00.5}) shows that \n\\begin{multline*}\n \\textrm{Re}(\\tilde{P}u,|x_{2,j}|^{2\\sigma}u)_{L^2(\\mathbb{R}_t)}=\\int_{\\mathbb{R}}a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-t x_1)|x_{2,j}|^{2\\sigma}|u(t)|^2dt \\\\\n\\geq a_0\\|F(x_2-t x_1)^{1\/2}|x_{2,j}|^{\\sigma}u\\|_{L^2(\\mathbb{R}_t)}^2,\n\\end{multline*}\nbecause $iD_t$ is a skew-adjoint operator.\nSince \n$$F(x_2-tx_1)=|x_2-t x_1|^{2\\sigma} \\geq |x_{2,j}|^{2\\sigma},$$ \nwhen $|x_{2,j}| \\geq 2$, we first deduce from the Cauchy-Schwarz inequality that for all $u \\in \\mathscr{S}(\\mathbb{R}_t)$ and $(x_1,x_2,\\xi_1,\\xi_2) \\in \\mathbb{R}^{4n}$, $x_{1,j}=0$, $|x_{2,j}| \\geq 2$,\n$$ a_0\\||x_{2,j}|^{2\\sigma}u\\|_{L^2(\\mathbb{R}_t)} \\leq \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)},$$\nwhich also implies that  \n$$ a_0^2\\||x_{2,j}|^{2\\sigma}u\\|_{L^2(\\mathbb{R}_t)}^2 \\leq \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}^2+2^{4\\sigma}a_0^2\\|u\\|_{L^2(\\mathbb{R}_t)}^2.$$\nNotice that this second estimate also holds when $|x_{2,j}| \\leq 2$. One can then deduce from another use of (\\ref{eq12}) that the estimate (\\ref{eq13.11}) also holds when $x_{1,j}=0$. This proves the following lemma.\n\n\\bigskip\n\n\\begin{lemma}\\label{prop0}\nConsider the first-order differential operator   \n$$\\tilde{P}=iD_t+a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1),$$\nwith parameters $(x_1,x_2,\\xi_1,\\xi_2) \\in \\mathbb{R}^{4n}$, and $a$ and $F$ standing for the functions defined in \\emph{(\\ref{eq00.5})} and \\emph{(\\ref{eq00})}. Then, there exists a positive constant $C>0$ such that for all $u \\in \\mathscr{S}(\\mathbb{R}_t)$ and $(x_1,x_2,\\xi_1,\\xi_2) \\in \\mathbb{R}^{4n}$,\n\\begin{equation}\\label{eq17}\n\\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})u\\|_{L^2(\\mathbb{R}_t)} \\leq C\\big(\\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}+\\|u\\|_{L^2(\\mathbb{R}_t)}\\big).\n\\end{equation}\n\\end{lemma}\n\n\n\\bigskip\n\n\\subsection{From a priori estimates for the one-dimensional operator with parameters to a priori estimates in Wick quantization}\nBy applying Lemma~\\ref{prop0}\n$$\\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})u\\|_{L^2(\\mathbb{R}_t)}^2 \\lesssim \\|\\tilde{P}u\\|_{L^2(\\mathbb{R}_t)}^2+\\|u\\|_{L^2(\\mathbb{R}_t)}^2,$$\nto a function $\\Phi(t,x_1,x_2,\\xi_1,\\xi_2) \\in \\mathscr{S}(\\mathbb{R}^{4n+1})$ and integrating this a priori estimate with respect to the variables $(x_1,x_2,\\xi_1,\\xi_2) \\in \\mathbb{R}^{4n}$, we obtain that we may find a positive constant $C_1>0$ such that for all $\\Phi \\in \\mathscr{S}(\\mathbb{R}^{4n+1})$,\n\\begin{equation}\\label{eq01}\n\\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})\\Phi\\|_{L^2(\\mathbb{R}^{4n+1})}^2 \\leq C_1\\big(\\|\\tilde{P}\\Phi\\|_{L^2(\\mathbb{R}^{4n+1})}^2+\\|\\Phi\\|_{L^2(\\mathbb{R}^{4n+1})}^2\\big).\n\\end{equation}\nLet $T>0$ be a positive constant. For any $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying\n$$\\textrm{supp } u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe shall consider its wave-packets transform in the variables $x_1,x_2$ with parameter $0<\\lambda \\leq 1$ defined as\n$$(W_{\\lambda}u)(t,Y)=\\big(u(t,\\cdot),\\varphi_{Y}^{\\lambda}\\big)_{L^2(\\mathbb{R}_{x_1,x_2}^{2n})} \\in \\mathscr{S}(\\mathbb{R}^{4n+1}), \\ Y=(y_1,y_2;\\eta_1,\\eta_2) \\in \\mathbb{R}^{4n},$$\nwith \n$$\\varphi_{Y}^{\\lambda}(x_1,x_2)=(2\\lambda)^{\\frac{n}{2}}e^{-\\pi \\lambda (|x_1-y_1|^2+|x_2-y_2|^2)}e^{2i \\pi((x_1-y_1)\\cdot \\eta_1+(x_2-y_2)\\cdot \\eta_2)}.$$\nWe apply the a priori estimate \n$$\\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})\\Phi\\|_{L^2(\\mathbb{R}^{4n+1})}^2 \\leq C_1\\big(\\|\\tilde{P}\\Phi\\|_{L^2(\\mathbb{R}^{4n+1})}^2+\\|\\Phi\\|_{L^2(\\mathbb{R}^{4n+1})}^2\\big),$$\nto the function $$\\Phi(t,X)=(W_{\\lambda}u)(t,X),$$ \nwith $X=(x_1,x_2;\\xi_1,\\xi_2) \\in \\mathbb{R}^{4n}$. By using the fact that the wave-packets transform is an isometric mapping from $L^2(\\mathbb{R}_{x_1,x_2}^{2n})$ to $L^2(\\mathbb{R}_{x_1,x_2,\\xi_1,\\xi_2}^{4n})$, see appendix (Section~\\ref{appendix}), we obtain that \n\\begin{multline}\\label{eq02}\n\\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2 \\\\ \\leq C_1\\big(\\|\\tilde{P}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2+\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}^2\\big).\n\\end{multline}\nWe recall from the appendix (Section~\\ref{appendix}) that the Wick quantization with parameter $0<\\lambda \\leq 1$ of a symbol $a$ is formally given by\n\\begin{equation}\\label{eq02.5}\na^{\\textrm{Wick}(\\lambda)}=W_{\\lambda}^*a^{\\mu}W_{\\lambda},\\ 1^{\\textrm{Wick}(\\lambda)}=W_{\\lambda}^*W_{\\lambda}=\\textrm{Id}_{L^2},\n\\end{equation}\nwhere $a^{\\mu}$ stands for the multiplication operator by the function $a$ on $L^2$, and that the operator \n$$\\pi_{\\lambda}=W_{\\lambda}W_{\\lambda}^*,$$ \nis an orthogonal projection on a closed proper subspace of $L^2$. It follows that \n\\begin{align*}\n& \\ \\|\\pi_{\\lambda}(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}\n+ \\langle x_2-tx_1 \\rangle^{2\\sigma})W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2 \\\\\n\\leq & \\  \\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2.\n\\end{align*}\nMoreover, since the wave-packets transform is an isometric mapping from $L^2(\\mathbb{R}_{x_1,x_2}^{2n})$ to $L^2(\\mathbb{R}_{x_1,x_2,\\xi_1,\\xi_2}^{4n})$, we may write that \n\\begin{align*}\n& \\ \\|\\pi_{\\lambda}(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}\\\\\n= & \\ \\|W_{\\lambda}W_{\\lambda}^*(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}\\\\\n= & \\ \\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})^{\\textrm{Wick}(\\lambda)}u\\|_{L^2(\\mathbb{R}^{2n+1})}.\n\\end{align*}\nIt follows from (\\ref{eq02}) that \n\\begin{align*}\n& \\ \\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})^{\\textrm{Wick}(\\lambda)}u\\|_{L^2(\\mathbb{R}^{2n+1})}^2 \\\\\n& \\ +\\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2\\\\\n\\leq & \\  2C_1\\big(\\|\\pi_{\\lambda}\\tilde{P}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2+\\|(1-\\pi_{\\lambda})\\tilde{P}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2+\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}^2\\big).\n\\end{align*}\nBy using similar arguments as previously, namely (\\ref{eq02.5}) and the fact that the wave-packets transform is an isometric mapping from $L^2(\\mathbb{R}_{x_1,x_2}^{2n})$ to $L^2(\\mathbb{R}_{x_1,x_2,\\xi_1,\\xi_2}^{4n})$, together with recalling that we only consider here the Wick quantization the variables $x_1,x_2$, and not in the $t$-variable, we obtain that \n\\begin{align*}\n& \\ \\|\\pi_{\\lambda}\\tilde{P}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}\\\\\n= & \\ \\|W_{\\lambda}W_{\\lambda}^*\\tilde{P}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}\\\\\n= & \\ \\|W_{\\lambda}W_{\\lambda}^*[iD_t+a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}\\\\\n= & \\ \\|W_{\\lambda}(iD_t+[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]^{\\textrm{Wick}(\\lambda)})u\\|_{L^2(\\mathbb{R}^{4n+1})}\\\\\n= & \\ \\|iD_tu+[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]^{\\textrm{Wick}(\\lambda)}u\\|_{L^2(\\mathbb{R}^{2n+1})}.\n\\end{align*}\nOn the other hand, notice that \n\\begin{multline*}\n(1-\\pi_{\\lambda})\\tilde{P}W_{\\lambda}=(1-\\pi_{\\lambda})[iD_t+a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]W_{\\lambda}\\\\\n=(1-\\pi_{\\lambda})a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)W_{\\lambda},\n\\end{multline*}\nbecause \n$$(1-\\pi_{\\lambda})iD_tW_{\\lambda}=(1-W_{\\lambda}W_{\\lambda}^*)W_{\\lambda}iD_t=W_{\\lambda}(1-W_{\\lambda}^*W_{\\lambda})iD_t=0,$$\nsince $D_t$ commutes with the wave-packets transform in the variables $x_1,x_2$, and that $W_{\\lambda}^*W_{\\lambda}=\\textrm{Id}_{L^2}$.\nWe may then write that \n\\begin{multline*}\n(1-\\pi_{\\lambda})a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)W_{\\lambda}=a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)W_{\\lambda}\\\\\n-[\\pi_{\\lambda},a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]W_{\\lambda}-a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)\\pi_{\\lambda}W_{\\lambda}.\n\\end{multline*}\nSince from (\\ref{eq02.5}), \n$$\\pi_{\\lambda}W_{\\lambda}=W_{\\lambda}W_{\\lambda}^*W_{\\lambda}=W_{\\lambda},$$\nit follows that \n$$(1-\\pi_{\\lambda})a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)W_{\\lambda}=-[\\pi_{\\lambda},a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]W_{\\lambda}.$$\nWe then deduce that for all $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying\n$$\\textrm{supp } u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n\\begin{multline*}\n (2C_1)^{-1}\\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})^{\\textrm{Wick}(\\lambda)}u\\|_{L^2(\\mathbb{R}^{2n+1})}^2 \\\\\n  + (2C_1)^{-1}\\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2\\\\\n\\leq  \\|iD_tu+[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]^{\\textrm{Wick}(\\lambda)}u\\|_{L^2(\\mathbb{R}^{2n+1})}^2\\\\\n + \\|[\\pi_{\\lambda},a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2+\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}^2.\n\\end{multline*}\nWe now need to study the commutator term\n\\begin{multline*}\n[\\pi_{\\lambda},a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]=[\\pi_{\\lambda},a(t,\\xi_2,\\xi_1+t\\xi_2)]F(x_2-tx_1)\\\\ +a(t,\\xi_2,\\xi_1+t\\xi_2)[\\pi_{\\lambda},F(x_2-tx_1)].\n\\end{multline*}\nIn order to do so, we recall from (\\ref{yo1}) in appendix that the kernel of the orthogonal projection $\\pi_{\\lambda}$ is given by \n\\begin{equation}\\label{EQ1}\ne^{-\\frac{\\pi}{2}\\Gamma_{\\lambda}(X-Y)}e^{i\\pi (x-y)\\cdot(\\xi+ \\eta)},\n\\end{equation} \nwith \n\\begin{equation}\\label{EQ2}\n\\Gamma_{\\lambda}(X)=\\lambda|x|^2+\\frac{|\\xi|^2}{\\lambda}, \\ X=(x,\\xi) \\in \\mathbb{R}^{4n}; \\ x=(x_1,x_2),\\ \\xi=(\\xi_1,\\xi_2) \\in \\mathbb{R}^{2n}.\n\\end{equation}\n\n\n\\bigskip\n\n\n\\begin{lemma}\\label{LEM1}\nLet $T>0$ be a positive constant. Then, there exists a positive constant $C>0$ such that for all $0<\\lambda \\leq 1$ and $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying \n$$\\emph{\\textrm{supp }} u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n\\begin{multline*}\n\\|[\\pi_{\\lambda},a(t,\\xi_2,\\xi_1+t\\xi_2)]F(x_2-tx_1)W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})} \\\\\n\\leq C\\lambda^{1\/2}\\|\\langle x_2-tx_1\\rangle^{2\\sigma}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}+C\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}.\n\\end{multline*}\n\\end{lemma}\n\n\\bigskip\n\n\\noindent\n\\textit{Proof of Lemma~\\ref{LEM1}}. Notice from (\\ref{EQ1}) that the kernel of the commutator \n$$[\\pi_{\\lambda},a(t,\\xi_2,\\xi_1+t\\xi_2)],$$ \nis given by\n$$K_{t,\\lambda}(X,Y)=e^{-\\frac{\\pi}{2}\\Gamma_{\\lambda}(X-Y)}e^{i\\pi (x-y)\\cdot(\\xi+ \\eta)}\\big(a(t,\\eta_2,\\eta_1+t\\eta_2)-a(t,\\xi_2,\\xi_1+t\\xi_2)\\big).$$\nRecalling that $a$ is a $C_b^{\\infty}(\\mathbb{R}^{2n+1})$ function and therefore a Lipschitz function, we may therefore find a positive constant $C_2>0$ such that for all $t \\in [-T,T]$, $0<\\lambda \\leq 1$ and $X,Y \\in \\mathbb{R}^{4n}$,\n\\begin{equation}\\label{E0}\n|K_{t,\\lambda}(X,Y)| \\leq C_2e^{-\\frac{\\pi}{2}\\Gamma_{\\lambda}(X-Y)}|\\eta-\\xi|.\n\\end{equation}\nNotice that \n\\begin{multline*}\n\\int_{\\mathbb{R}^{4n}}|K_{t,\\lambda}(X,Y)|dY \\leq C_2\\int_{\\mathbb{R}^{4n}}e^{-\\frac{\\pi}{2}\\Gamma_{\\lambda}(X-Y)}|\\eta-\\xi|dY=C_2\\int_{\\mathbb{R}^{4n}}e^{-\\frac{\\pi}{2}\\Gamma_{\\lambda}(Y)}|\\eta|dY\\\\\n=C_2\\lambda^{1\/2}\\int_{\\mathbb{R}^{4n}}e^{-\\frac{\\pi}{2}|Y|^2}|\\eta|dY=C_3\\lambda^{1\/2},\n\\end{multline*}\nwith $C_3>0$ a positive constant. \nBy symmetry of the estimate (\\ref{E0}), we also have \n$$\\int_{\\mathbb{R}^{4n}}|K_{t,\\lambda}(X,Y)|dX \\leq C_3\\lambda^{1\/2}.$$\nSchur test for integral operators together with (\\ref{eq00}) show that there exists a positive constant $C_4>0$ such that for all $0<\\lambda \\leq 1$ and $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying\n$$\\textrm{supp } u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n\\begin{multline*}\n\\|[\\pi_{\\lambda},a(t,\\xi_2,\\xi_1+t\\xi_2)]F(x_2-tx_1)W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})} \\leq C_3\\lambda^{1\/2}\\|F(x_2-tx_1)W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}\\\\\n\\leq C_4\\lambda^{1\/2}\\|\\langle x_2-tx_1\\rangle^{2\\sigma}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}+C_4\\|u\\|_{L^2(\\mathbb{R}^{2n+1})},\n\\end{multline*}\nsince we recall that for any $0<\\lambda \\leq 1$,\n$$\\|W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}=\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}. \\ \\Box$$\n\n\n\\bigskip\n\n\n\\begin{lemma}\\label{LEM2}\nLet $T>0$ be a positive constant. Then, there exists a positive constant $C>0$ such that for all $0<\\lambda \\leq 1$ and $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying \n$$\\emph{\\textrm{supp }} u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n\\begin{multline*}\n\\|a(t,\\xi_2,\\xi_1+t\\xi_2)[\\pi_{\\lambda},F(x_2-tx_1)]W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})} \\\\\n\\leq C \\lambda^{-\\frac{1+(2\\sigma-1)_+}{2}} \\|\\langle x_2-tx_1\\rangle^{(2\\sigma-1)_+}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})},\n\\end{multline*}\nwith $(2\\sigma-1)_+=\\emph{\\textrm{max}}(2\\sigma-1,0)$.\n\\end{lemma}\n\n\\bigskip\n\n\\noindent\n\\textit{Proof of Lemma~\\ref{LEM2}}. We first notice that \n$$\\|a(t,\\xi_2,\\xi_1+t\\xi_2)[\\pi_{\\lambda},F(x_2-tx_1)]W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})} \\lesssim \\|[\\pi_{\\lambda},F(x_2-tx_1)]W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})},$$\nsince $a$ is a $L^{\\infty}(\\mathbb{R}^{2n+1})$ function. Arguing as in previous lemma, we notice that the kernel of the commutator $[\\pi_{\\lambda},F(x_2-tx_1)]$ \nis given by\n$$K_{t,\\lambda}(X,Y)=e^{-\\frac{\\pi}{2}\\Gamma_{\\lambda}(X-Y)}e^{i\\pi (x-y)\\cdot(\\xi+ \\eta)}\\big(F(y_2-ty_1)-F(x_2-tx_1)\\big).$$\nRecall from (\\ref{eq00}) that there exists a positive constant $C_5>0$ such that for all $x \\in \\mathbb{R}^n$,\n\\begin{equation}\\label{eq271}\n|\\nabla F(x)| \\leq C_5 \\langle x \\rangle^{(2\\sigma-1)_+}\n\\end{equation}\nwith $(2\\sigma-1)_+=\\textrm{max}(2\\sigma -1,0)$. Writing \n\\begin{multline*}\nF(y_2-ty_1)-F(x_2-tx_1)\\\\ =\\big(y_2-x_2-t(y_1-x_1)\\big) \\cdot \\int_0^1\\nabla F\\big((1-\\theta)(x_2-y_2-t(x_1-y_1))+y_2-ty_1)\\big)d\\theta,\n\\end{multline*}\nwe have for all $(t,x_1,x_2,y_1,y_2) \\in [-T,T] \\times \\mathbb{R}^{4n}$,\n\\begin{align*}\n& \\ |F(y_2-ty_1)-F(x_2-tx_1)|\\\\\n \\lesssim & \\ |x-y| \\int_0^1\\langle (1-\\theta)(x_2-y_2-t(x_1-y_1))+y_2-ty_1)\\rangle^{(2\\sigma-1)_+} d\\theta\\\\\n\\lesssim & \\ |x-y| \\int_0^1  \\langle (1-\\theta)(x_2-y_2-t(x_1-y_1))\\rangle^{(2\\sigma-1)_+}  \\langle y_2-ty_1\\rangle^{(2\\sigma-1)_+} d\\theta\\\\\n\\lesssim & \\ |x-y| \\langle y_2-ty_1\\rangle^{(2\\sigma-1)_+} \\langle x_2-y_2-t(x_1-y_1)\\rangle^{(2\\sigma-1)_+}.\n\\end{align*}\nSetting\n$$\\tilde{K}_{t,\\lambda}(X,Y)=\\langle y_2-ty_1\\rangle^{-(2\\sigma-1)_+}K_{t,\\lambda}(X,Y),$$\nwe have \n\\begin{equation}\\label{E2}\n|\\tilde{K}_{t,\\lambda}(X,Y)| \\lesssim e^{-\\frac{\\pi}{2}\\Gamma_{\\lambda}(X-Y)} |x-y|\\langle x_2-y_2-t(x_1-y_1)\\rangle^{(2\\sigma-1)_+}.\n\\end{equation}\nWhile using a change of variables, we notice that for all $t \\in [-T,T]$ and $0<\\lambda \\leq 1$,\n\\begin{align*}\n\\int_{\\mathbb{R}^{4n}}|\\tilde{K}_{t,\\lambda}(X,Y)|dY \\lesssim & \\ \\int_{\\mathbb{R}^{4n}}e^{-\\frac{\\pi}{2}\\Gamma_{\\lambda}(X-Y)}|x-y|\\langle x_2-y_2-t(x_1-y_1)\\rangle^{(2\\sigma-1)_+}dY\\\\\n\\lesssim & \\ \\int_{\\mathbb{R}^{4n}}e^{-\\frac{\\pi}{2}\\Gamma_{\\lambda}(Y)}|y|\\langle y_2-ty_1\\rangle^{(2\\sigma-1)_+}dY \\\\\n\\lesssim & \\ \\int_{\\mathbb{R}^{4n}}e^{-\\frac{\\pi}{2}|Y|^2}|\\lambda^{-1\/2}y|\\langle \\lambda^{-1\/2}(y_2-ty_1)\\rangle^{(2\\sigma-1)_+}dY \\lesssim \\lambda^{-\\frac{1+(2\\sigma-1)_+}{2}},\n\\end{align*}\nsince we have $\\langle \\mu x \\rangle \\leq \\mu \\langle x \\rangle$, when $\\mu \\geq 1$. By symmetry of the estimate (\\ref{E2}), we also have \n$$\\int_{\\mathbb{R}^{4n}}|\\tilde{K}_{t,\\lambda}(X,Y)|dX \\lesssim \\lambda^{-\\frac{1+(2\\sigma-1)_+}{2}}.$$\nSchur test for integral operators then shows that for all $0<\\lambda \\leq 1$ and $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying \n$$\\textrm{supp } u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n$$\\|[\\pi_{\\lambda},F(x_2-tx_1)]W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})} \\lesssim \\lambda^{-\\frac{1+(2\\sigma-1)_+}{2}} \\|\\langle x_2-tx_1\\rangle^{(2\\sigma-1)_+}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})},$$\nwhich proves Lemma~\\ref{LEM2}.~$\\Box$\n\n\n\\bigskip\n\n\\noindent\nWe then deduce from Lemmas~\\ref{LEM1} and \\ref{LEM2} that there exists a positive constant $C_6>0$ such that for all $0<\\lambda \\leq 1$ and $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying\n$$\\textrm{supp } u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n\\begin{align*}\n& \\ C_6^{-1}\\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})^{\\textrm{Wick}(\\lambda)}u\\|_{L^2(\\mathbb{R}^{2n+1})}^2 \\\\\n+ & \\ C_6^{-1}\\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2\\\\\n\\leq & \\  \\|iD_tu+[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]^{\\textrm{Wick}(\\lambda)}u\\|_{L^2(\\mathbb{R}^{2n+1})}^2+\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}^2\\\\\n+& \\ \\lambda^{-1-(2\\sigma-1)_+} \\|\\langle x_2-tx_1\\rangle^{(2\\sigma-1)_+}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2+\\lambda\\|\\langle x_2-tx_1\\rangle^{2\\sigma}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2.\n\\end{align*}\nBy choosing a positive constant $0<\\lambda_0 \\leq 1$ such that \n$$\\lambda_0 \\leq \\frac{1}{2C_6},$$\nwe notice that one may estimate from above the following term as \n$$\\lambda\\|\\langle x_2-tx_1\\rangle^{2\\sigma}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2 \\leq (2C_6)^{-1}\\|(1+\\langle x_1 \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle x_2-tx_1 \\rangle^{2\\sigma})W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}^2,$$\nfor all $0<\\lambda \\leq \\lambda_0$. We then notice that for any $\\varepsilon>0$, there exists a positive constant $C_{\\varepsilon}>0$ such that for all $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying\n$$\\textrm{supp } u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n$$ \\|\\langle x_2-tx_1\\rangle^{(2\\sigma-1)_+}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})} \\leq \\varepsilon\\|\\langle x_2-tx_1\\rangle^{2\\sigma}W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}+C_{\\varepsilon}\\|W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}.$$ By recalling that \n$$\\|W_{\\lambda}u\\|_{L^2(\\mathbb{R}^{4n+1})}=\\|u\\|_{L^2(\\mathbb{R}^{2n+1})},$$ \nwe finally deduce from these estimates that for any $T>0$, there exists a positive constant $c_T>0$ independent of the parameter $0<\\lambda \\leq \\lambda_0$; and a second positive constant $C_T(\\lambda)>0$, which may depend on $\\lambda$ such that for all $0<\\lambda \\leq \\lambda_0$ and $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying \n$$\\textrm{supp } u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n\\begin{multline}\\label{eq18}\n\\|[\\langle x_1\\rangle^{\\frac{2\\sigma}{2\\sigma+1}}]^{\\textrm{Wick}(\\lambda)}u\\|_{L^2(\\mathbb{R}^{2n+1})} +\\|[\\langle x_2-tx_1\\rangle^{2\\sigma}]^{\\textrm{Wick}(\\lambda)}u\\|_{L^2(\\mathbb{R}^{2n+1})} \\\\ \\leq c_T\\|iD_tu+[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]^{\\textrm{Wick}(\\lambda)}u\\|_{L^2(\\mathbb{R}^{2n+1})}+C_T(\\lambda)\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}.\n\\end{multline}\n\n\\bigskip\n\n\n\\subsection{From a priori estimates in Wick quantization to a priori estimates in Weyl quantization}\nIn the previous section, we established an a priori estimate with symbols quantized in the Wick quantization with parameter $0<\\lambda \\leq \\lambda_0 \\leq 1$. We shall need the following lemma to estimate error terms incurred by coming back from Wick quantization with parameter $0<\\lambda \\leq \\lambda_0$ to the Weyl quantization for symbols appearing in the left-hand-side of (\\ref{eq18}).\n\n\\bigskip\n\n\n\\begin{lemma}\\label{lem1}\nLet $a \\in C^{\\infty}(\\mathbb{R}^{2n})$ be a symbol whose derivatives of order $ \\geq 2$ are bounded on $\\mathbb{R}^{2n}$. Then, there exists a positive constant $C>0$ depending only on the $L^{\\infty}$-norms of a finite number of derivatives of order greater or equal to 2 of the symbol $a$; such that for all $\\lambda > 0$ and $u \\in \\mathscr{S}(\\mathbb{R}^n)$,\n$$\\|a^{\\emph{Wick}(\\lambda)}u-a^wu\\|_{L^2(\\mathbb{R}^n)} \\leq C\\Big(\\lambda+\\frac{1}{\\lambda}\\Big)\\|u\\|_{L^2(\\mathbb{R}^n)}.$$\n\\end{lemma}\n\n\\bigskip\n\n\\noindent\n\\textit{Proof of Lemma~\\ref{lem1}.} We notice from (\\ref{lay1}) and (\\ref{lay2}) in appendix that one may write that\n$$a^{\\textrm{Wick}(\\lambda)}=a^w+R_{\\lambda}^w,$$\nwith \n$$R_{\\lambda}(X)=\\int_0^1\\int_{\\mathbb{R}^{2n}}(1-\\theta)a''(X+\\theta Y)Y^2e^{-2\\pi\\Gamma_{\\lambda}(Y)}2^ndYd\\theta, \\ X=(x,\\xi) \\in \\mathbb{R}^{2n},$$\nwhere \n$$\\Gamma_{\\lambda}(Y)=\\lambda |y|^2+\\frac{|\\eta|^2}{\\lambda}, \\ Y=(y,\\eta) \\in \\mathbb{R}^{2n}.$$\nIt follows that any derivative of the symbol $R_{\\lambda}$, \n$$\\partial_{X}^{\\alpha}R_{\\lambda}(X)=\\int_0^1\\int_{\\mathbb{R}^{2n}}(1-\\theta)\\partial_X^{\\alpha}a''(X+\\theta Y)Y^2e^{-2\\pi\\Gamma_{\\lambda}(Y)}2^ndYd\\theta,$$\ncan be estimated from above as\n$$\\|\\partial_{X}^{\\alpha}R_{\\lambda}\\|_{L^{\\infty}(\\mathbb{R}^{2n})} \\leq \\|\\partial_X^{\\alpha}a''\\|_{L^{\\infty}(\\mathbb{R}^{2n})}\\int_{\\mathbb{R}^{2n}}(|y|^2+|\\eta|^2)e^{-2\\pi\\lambda |y|^2}e^{-2\\pi\\frac{|\\eta|^2}{\\lambda}}2^ndyd\\eta.$$\nSince \n\\begin{multline*}\n\\int_{\\mathbb{R}^{2n}}(|y|^2+|\\eta|^2)e^{-2\\pi\\lambda |y|^2}e^{-2\\pi\\frac{|\\eta|^2}{\\lambda}}2^ndyd\\eta\\\\\n=\\int_{\\mathbb{R}^{2n}}\\Big(\\frac{|y|^2}{\\lambda}+\\lambda|\\eta|^2\\Big)e^{-2\\pi (|y|^2+|\\eta|^2)}2^ndyd\\eta \n=\\mathcal{O}\\Big(\\lambda+\\frac{1}{\\lambda}\\Big),\n\\end{multline*}\nLemma~\\ref{lem1} is then a direct consequence of the $L^2$ continuity theorem in the class $S_{00}^0$.~$\\Box$\n\n\\bigskip\n\n\n\\noindent\nNotice that we may apply Lemma~\\ref{lem1} to the two symbols seen as functions of the variables $(x_1,x_2,\\xi_1,\\xi_2) \\in \\mathbb{R}^{4n}$, \n$$\\langle x_1\\rangle^{\\frac{2\\sigma}{2\\sigma+1}} \\textrm{ and } \\langle x_2-tx_1\\rangle^{2\\sigma}\\chi_0(t),$$\nwith $\\chi_0 \\in C_0^{\\infty}(\\mathbb{R})$, $\\chi_0=1$ on $[-T,T]$, since $0<\\sigma<1$. We recall that the $t$-variable is seen as a parameter and that  \nwe only consider here the Wick and Weyl quantizations in the variables $x_1,x_2$. It follows from Lemma~\\ref{lem1} and (\\ref{eq18}) that for any fixed $T>0$, there exist a positive constant $c_T>0$ independent of the parameter $0<\\lambda \\leq \\lambda_0$, and a second positive constant $C_T(\\lambda)>0$, which may depend on $\\lambda$ such that for all $0<\\lambda \\leq \\lambda_0$ and $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying \n$$\\textrm{supp } u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n\\begin{multline}\\label{eq19}\n\\big\\|(1+|x_2-tx_1|^{2\\sigma}+|x_1|^{\\frac{2\\sigma}{2\\sigma+1}})u\\big\\|_{L^2(\\mathbb{R}^{2n+1})} \\\\ \\leq c_T\\|iD_tu+[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]^{\\textrm{Wick}(\\lambda)}u\\|_{L^2(\\mathbb{R}^{2n+1})}+C_T(\\lambda)\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}.\n\\end{multline}\nWe shall now establish a priori estimates for error terms incurred by coming back from Wick quantization with parameter $0< \\lambda \\leq \\lambda_0$ to the Weyl quantization for the symbol $$a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1).$$ \n\n\n\\bigskip\n\n\n\\begin{lemma}\\label{prop1}\nLet $T>0$ be a positive constant. Then, there exists a positive constant $C>0$ such that for all $0< \\lambda \\leq \\lambda_0$ and $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying \n$$\\emph{\\textrm{supp }} u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n\\begin{multline*}\nC^{-1}\\|[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]^{\\emph{\\textrm{Wick}}(\\lambda)}u-[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]^wu\\|_{L^2(\\mathbb{R}^{2n+1})} \\\\\n\\leq  \\lambda^{1-\\sigma} \\|\\langle x_2-tx_1\\rangle^{2\\sigma}u\\|_{L^2(\\mathbb{R}^{2n+1})}+ \\lambda^{-\\sigma} \\|\\langle x_2-tx_1\\rangle^{(2\\sigma-1)_+}u\\|_{L^2(\\mathbb{R}^{2n+1})}+ \\lambda^{-1}\\|u\\|_{L^2(\\mathbb{R}^{2n+1})},\n\\end{multline*}\nwith $(2\\sigma-1)_+=\\emph{\\textrm{max}}(2\\sigma-1,0)$.\n\\end{lemma}\n\n\\bigskip\n\n\\noindent\n\\textit{Proof of Lemma~\\ref{prop1}.} Let $\\chi_0$ be a $C_0^{\\infty}(\\mathbb{R})$ function satisfying $\\chi_0=1$ on $[-T,T]$. By using again (\\ref{lay1}) and (\\ref{lay2}) in appendix, one may write that\n\\begin{multline}\\label{eq20}\n\\chi_0(t)[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]^{\\textrm{Wick}(\\lambda)}\\\\ =\\chi_0(t)[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]^w +R_{t,\\lambda}^w,\n\\end{multline}\nwith \n$$R_{t,\\lambda}(X)=\\int_0^1\\int_{\\mathbb{R}^{4n}}(1-\\theta)\\chi_0(t)r_t''(X+\\theta Y)Y^2e^{-2\\pi\\Gamma_{\\lambda}(Y)}2^ndYd\\theta, $$\nwhere \n$$r_t(x_1,x_2;\\xi_1,\\xi_2)=a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1), \\  X=(x_1,x_2;\\xi_1,\\xi_2) \\in \\mathbb{R}^{4n},$$\nand\n$$\\Gamma_{\\lambda}(Y)=\\lambda (|y_1|^2+|y_2|^2)+\\frac{1}{\\lambda}(|\\eta_1|^2+|\\eta_2|^2), \\ Y=(y_1,y_2;\\eta_1,\\eta_2) \\in \\mathbb{R}^{4n}.$$\nDefine\n\\begin{equation}\\label{eq23}\n\\tilde{r}_{1,t}(X,Y,\\theta)=\\chi_0(t)a\\big(t,\\xi_2+\\theta \\eta_2,\\xi_1+t\\xi_2+\\theta(\\eta_1+t \\eta_2)\\big) (\\nabla_x^2B_t)(x+\\theta y).y^2,\n\\end{equation}\n\\begin{equation}\\label{eq22}\n\\tilde{r}_{2,t}(X,Y,\\theta)=\\chi_0(t)(\\nabla_{\\xi}A_t)(\\xi+\\theta \\eta).\\eta \\ (\\nabla_xB_t)(x+\\theta y).y,\n\\end{equation}\n\\begin{equation}\\label{eq21}\n\\tilde{r}_{3,t}(X,Y,\\theta)=\\chi_0(t)(\\nabla_{\\xi}^2A_t)(\\xi+\\theta \\eta).\\eta^2 \\ F(x_2-tx_1+\\theta(y_2-ty_1)),\n\\end{equation}\nwith\n\\begin{equation}\\label{eq5678}\nA_t(\\xi)=a(t,\\xi_2,\\xi_1+t\\xi_2), \\ \\xi=(\\xi_1,\\xi_2) \\textrm{ and } B_t(x)=F(x_2-tx_1), \\ x=(x_1,x_2).\n\\end{equation}\nWe also define\n\\begin{equation}\\label{eq24}\nR_{1,t,\\lambda}(X)=\\int_0^1\\int_{\\mathbb{R}^{4n}}(1-\\theta)\\tilde{r}_{1,t}(X,Y,\\theta)e^{-2\\pi\\Gamma_{\\lambda}(Y)}2^ndYd\\theta,\n\\end{equation}\n\\begin{equation}\\label{eq25}\nR_{2,t,\\lambda}(X)=\\int_0^1\\int_{\\mathbb{R}^{4n}}(1-\\theta)\\tilde{r}_{2,t}(X,Y,\\theta)e^{-2\\pi\\Gamma_{\\lambda}(Y)}2^ndYd\\theta,\n\\end{equation}\n\\begin{equation}\\label{eq26}\nR_{3,t,\\lambda}(X)=\\int_0^1\\int_{\\mathbb{R}^{4n}}(1-\\theta)\\tilde{r}_{3,t}(X,Y,\\theta)e^{-2\\pi\\Gamma_{\\lambda}(Y)}2^ndYd\\theta.\n\\end{equation}\nRecall from (\\ref{eq00}) that there exists a positive constant $C_1>0$ such that for all $x \\in \\mathbb{R}^n$,\n\\begin{equation}\\label{eq27}\n|F(x)| \\leq C_1 \\langle x \\rangle^{2\\sigma}, \\ \\ |F'(x)| \\leq C_1 \\langle x \\rangle^{(2\\sigma-1)_+}, \\ \\ \\forall k \\geq 2, \\ F^{(k)} \\in L^{\\infty}(\\mathbb{R}^n),\n\\end{equation}\nwith $(2\\sigma-1)_+=\\textrm{max}(2\\sigma-1,0)$, since $0<\\sigma <1$. \n\n\n\\bigskip\n\n\n\\begin{lemma}\\label{lem2}\nThere exists a positive constant $C>0$ such that \n$$\\forall \\ 0<\\lambda \\leq \\lambda_0, \\forall u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1}),\\ \\|R_{1,t,\\lambda}^wu\\|_{L^2(\\mathbb{R}^{2n+1})} \\leq C \\lambda^{-1}\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}.$$\n\\end{lemma}\n\n\\bigskip\n\n\\noindent\n\\textit{Proof of Lemma~\\ref{lem2}}. According to the  $L^2$ continuity theorem in the class $S_{00}^0$, it is sufficient to prove that for all $\\alpha \\in \\mathbb{N}^{4n}$, there exists a positive constant $C_{2,\\alpha}>0$ such that \n$$\\forall t \\in \\mathbb{R}, \\forall \\ 0< \\lambda \\leq \\lambda_0,\\ \\|\\partial_X^{\\alpha}R_{1,t,\\lambda}\\|_{L^{\\infty}(\\mathbb{R}_X^{4n})} \\leq C_{2,\\alpha}\\lambda^{-1}.$$\nRecalling that $a$ is a $C_b^{\\infty}(\\mathbb{R}^{2n+1})$ function, we deduce from (\\ref{eq23}), (\\ref{eq5678}), (\\ref{eq24}), (\\ref{eq27}) and a change of variables that for any $\\alpha \\in \\mathbb{N}^{4n}$, there exist some positive constants $C_{3,\\alpha},C_{4,\\alpha}>0$ such that\n\\begin{multline*}\n\\forall t \\in \\mathbb{R}, \\forall \\ 0<\\lambda \\leq \\lambda_0,\\ \\|\\partial_X^{\\alpha}R_{1,t,\\lambda}\\|_{L^{\\infty}(\\mathbb{R}_X^{4n})} \\\\ \\leq C_{3,\\alpha}\\int_{\\mathbb{R}^{4n}}\\big(|y_1|^2+|y_2|^2\\big)\ne^{-2\\pi(\\lambda (|y_1|^2+|y_2|^2)+\\lambda^{-1}(|\\eta_1|^2+|\\eta_2|^2))}dy_1dy_2d\\eta_1d\\eta_2=C_{4,\\alpha}\\lambda^{-1}.\n\\end{multline*}\n\n\n\\bigskip\n\n\n\\begin{lemma}\\label{lem3}\nLet $T>0$ be a positive constant. Then, there exists a positive constant $C>0$ such that for all $0<\\lambda \\leq \\lambda_0$ and $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying \n$$\\emph{\\textrm{supp }} u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n$$\\|R_{2,t,\\lambda}^wu\\|_{L^2(\\mathbb{R}^{2n+1})} \\leq C \\lambda^{-\\sigma} \\|\\langle x_2-tx_1\\rangle^{(2\\sigma-1)_+}u\\|_{L^2(\\mathbb{R}^{2n+1})},$$\nwith $(2\\sigma-1)_+=\\emph{\\textrm{max}}(2\\sigma-1,0) \\geq 0.$ \n\\end{lemma}\n\n\\bigskip\n\n\\noindent\n\\textit{Proof of Lemma~\\ref{lem3}}. We notice from (\\ref{eq5678}) and (\\ref{eq27}) that \n\\begin{multline*}\n\\big| (\\nabla_xB_t)(x+\\theta y)\\big| \\lesssim \\langle x_2-tx_1+\\theta(y_2-ty_1)\\rangle^{(2\\sigma -1)_+}\\\\ \\lesssim \\langle x_2-tx_1\\rangle^{(2\\sigma -1)_+}\n \\langle \\theta(y_2-ty_1)\\rangle^{(2\\sigma -1)_+}\n \\lesssim \\langle x_2-tx_1\\rangle^{(2\\sigma -1)_+} \\langle y_2-ty_1\\rangle^{(2\\sigma -1)_+},\n \\end{multline*}\nwhen $0 \\leq \\theta \\leq 1$ and $t \\in \\textrm{supp }\\chi_0$. \nRecalling that $a$ is a $C_b^{\\infty}(\\mathbb{R}^{2n+1})$ function, it follows from (\\ref{eq22}), (\\ref{eq5678}) and (\\ref{eq25}) that for all $0<\\lambda \\leq \\lambda_0$,\n\\begin{align*}\n|R_{2,t,\\lambda}(X)| \\lesssim & \\ \\chi_0(t) \\langle x_2-tx_1\\rangle^{(2\\sigma -1)_+}\\int_{\\mathbb{R}^{4n}}|\\eta||y|\\langle y_2-ty_1\\rangle^{(2\\sigma -1)_+}e^{-2\\pi\\Gamma_{\\lambda}(Y)}dY \\\\\n\\lesssim & \\  \\chi_0(t)\\langle x_2-tx_1\\rangle^{(2\\sigma -1)_+}\\int_{\\mathbb{R}^{4n}}|\\eta||y|\\langle \\lambda^{-\\frac{1}{2}}(y_2-ty_1)\\rangle^{(2\\sigma -1)_+}e^{-2\\pi|Y|^2}dY\\\\\n\\lesssim & \\  \\lambda^{-\\frac{(2\\sigma -1)_+}{2}}\\chi_0(t)\\langle x_2-tx_1\\rangle^{(2\\sigma -1)_+},\n\\end{align*}\nsince we have $\\langle \\mu x \\rangle \\leq \\mu \\langle x \\rangle$, when $\\mu \\geq 1$. \nAfter differentiating (\\ref{eq22}) with respect to the variable $X=(x,\\xi)$, and using the same kind of estimates, we obtain from (\\ref{eq27}) that for all $0<\\lambda \\leq \\lambda_0$,\n\\begin{align*}\n& \\ |\\partial_{x}^{\\alpha}\\partial_{\\xi}^{\\beta}R_{2,t,\\lambda}(X)| \\\\\n\\lesssim & \\ \\chi_0(t) \\langle x_2-tx_1\\rangle^{(2\\sigma -|\\alpha|-1)_+}\\int_{\\mathbb{R}^{4n}}|\\eta||y|\\langle y_2-ty_1\\rangle^{(2\\sigma -|\\alpha|-1)_+}e^{-2\\pi\\Gamma_{\\lambda}(Y)}dY \\\\\n\\lesssim & \\  \\chi_0(t)\\langle x_2-tx_1\\rangle^{(2\\sigma -|\\alpha|-1)_+}\\int_{\\mathbb{R}^{4n}}|\\eta||y|\\langle \\lambda^{-\\frac{1}{2}}(y_2-ty_1)\\rangle^{(2\\sigma -|\\alpha|-1)_+}e^{-2\\pi|Y|^2}dY\\\\\n\\lesssim & \\  \\lambda^{-\\frac{(2\\sigma -|\\alpha|-1)_+}{2}}\\chi_0(t)\\langle x_2-tx_1\\rangle^{(2\\sigma -|\\alpha|-1)_+},\n\\end{align*}\nwith \n$$(2\\sigma -|\\alpha|-1)_+=\\textrm{max}(2\\sigma -|\\alpha|-1,0).$$ \nIt follows from those estimates that the Weyl symbol\n$$\\tilde{a}_{\\lambda}(t,X)=\\lambda^{\\sigma}R_{2,t,\\lambda}(X) \\sharp[\\chi_0(t) \\langle x_2-tx_1\\rangle^{-(2\\sigma-1)_+}],$$ \nof the operator obtained by composition\n$$\\lambda^{\\sigma}R_{2,t,\\lambda}^w [\\chi_0(t) \\langle x_2-tx_1\\rangle^{-(2\\sigma-1)_+}]^w,$$\nbelongs to the class $S(1,dt^2+d\\tau^2+dX^2)$ uniformly with respect to the parameter $0<\\lambda \\leq \\lambda_0$. We may then deduce from the  $L^2$ continuity theorem in the class $S_{00}^0$ that for all $0<\\lambda \\leq \\lambda_0$ and $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying \n$$\\textrm{supp } u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n\\begin{align*}\n\\|R_{2,t,\\lambda}^wu\\|_{L^2(\\mathbb{R}^{2n+1})} = & \\ \\|\\lambda^{-\\sigma}\\tilde{a}_{\\lambda}^w\\langle x_2-tx_1\\rangle^{(2\\sigma-1)_+}u\\|_{L^2(\\mathbb{R}^{2n+1})} \\\\ \\lesssim & \\  \n\\lambda^{-\\sigma} \\|\\langle x_2-tx_1\\rangle^{(2\\sigma-1)_+}u\\|_{L^2(\\mathbb{R}^{2n+1})}.\n\\end{align*}\nThis ends the proof of Lemma~\\ref{lem3}.~$\\Box$\n\n\\bigskip\n\n\n\\begin{lemma}\\label{lem4}\nLet $T>0$ be a positive constant. Then, there exists a positive constant $C>0$ such that for all $0<\\lambda \\leq \\lambda_0$ and $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying \n$$\\emph{\\textrm{supp }} u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n$$\\|R_{3,t,\\lambda}^wu\\|_{L^2(\\mathbb{R}^{2n+1})} \\leq C \\lambda^{1-\\sigma} \\|\\langle x_2-tx_1\\rangle^{2\\sigma}u\\|_{L^2(\\mathbb{R}^{2n+1})}.$$ \n\\end{lemma}\n\n\\bigskip\n\n\\noindent\n\\textit{Proof of Lemma~\\ref{lem4}}. By using similar kind of estimates as in the previous lemma together with the fact that \n\\begin{multline*}\n|F(x_2-tx_1+\\theta(y_2-ty_1))| \\lesssim \\langle x_2-tx_1+\\theta(y_2-ty_1)\\rangle^{2\\sigma}\\\\ \\lesssim \\langle x_2-tx_1\\rangle^{2\\sigma}\n \\langle \\theta(y_2-ty_1)\\rangle^{2\\sigma}\n \\lesssim \\langle x_2-tx_1\\rangle^{2\\sigma} \\langle y_2-ty_1\\rangle^{2\\sigma},\n \\end{multline*}\nwhen $0 \\leq \\theta \\leq 1$;\nit follows from (\\ref{eq21}), (\\ref{eq5678}) and (\\ref{eq26}) that for all $0<\\lambda \\leq \\lambda_0$,\n\\begin{align*}\n|R_{3,t,\\lambda}(X)| \\lesssim & \\ \\chi_0(t) \\langle x_2-tx_1\\rangle^{2\\sigma}\\int_{\\mathbb{R}^{4n}}|\\eta|^2\\langle y_2-ty_1\\rangle^{2\\sigma}e^{-2\\pi\\Gamma_{\\lambda}(Y)}dY \\\\\n\\lesssim & \\  \\chi_0(t)\\langle x_2-tx_1\\rangle^{2\\sigma}\\int_{\\mathbb{R}^{4n}}\\lambda|\\eta|^2\\langle \\lambda^{-\\frac{1}{2}}(y_2-ty_1)\\rangle^{2\\sigma}e^{-2\\pi|Y|^2}dY\\\\\n\\lesssim & \\  \\lambda^{1-\\sigma}\\chi_0(t)\\langle x_2-tx_1\\rangle^{2\\sigma}.\n\\end{align*}\nAfter differentiating (\\ref{eq21}) and (\\ref{eq26}) with respect to the variable $X=(x,\\xi)$ and using similar types of estimates, we obtain from (\\ref{eq27}) that for all $0<\\lambda \\leq \\lambda_0$,\n\\begin{align*}\n& \\ |\\partial_{x}^{\\alpha}\\partial_{\\xi}^{\\beta}R_{3,t,\\lambda}(X)| \\\\\n\\lesssim & \\ \\chi_0(t) \\langle x_2-tx_1\\rangle^{(2\\sigma -|\\alpha|)_+}\\int_{\\mathbb{R}^{4n}}|\\eta|^2\\langle y_2-ty_1\\rangle^{(2\\sigma -|\\alpha|)_+}e^{-2\\pi\\Gamma_{\\lambda}(Y)}dY \\\\\n\\lesssim & \\  \\chi_0(t)\\langle x_2-tx_1\\rangle^{(2\\sigma -|\\alpha|)_+}\\int_{\\mathbb{R}^{4n}}\\lambda|\\eta|^2\\langle \\lambda^{-\\frac{1}{2}}(y_2-ty_1)\\rangle^{(2\\sigma -|\\alpha|)_+}e^{-2\\pi|Y|^2}dY\\\\\n\\lesssim & \\  \\lambda^{1-\\frac{(2\\sigma -|\\alpha|)_+}{2}}\\chi_0(t)\\langle x_2-tx_1\\rangle^{(2\\sigma -|\\alpha|)_+}.\n\\end{align*} \nIt follows from those estimates that the Weyl symbol\n$$\\tilde{a}_{\\lambda}(t,X)=\\lambda^{\\sigma-1}R_{3,t,\\lambda}(X) \\sharp[\\chi_0(t) \\langle x_2-tx_1\\rangle^{-2\\sigma}],$$ \nof the operator obtained by composition\n$$\\lambda^{\\sigma-1}R_{3,t,\\lambda}^w [\\chi_0(t) \\langle x_2-tx_1\\rangle^{-2\\sigma}]^w,$$\nbelongs to the class $S(1,dt^2+d\\tau^2+dX^2)$ uniformly with respect to the parameter $0<\\lambda \\leq \\lambda_0$. We may then deduce from the  $L^2$ continuity theorem in the class $S_{00}^0$ that for all $0<\\lambda \\leq \\lambda_0$ and $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying \n$$\\textrm{supp } u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n$$\\|R_{3,t,\\lambda}^wu\\|_{L^2(\\mathbb{R}^{2n+1})} =\\|\\lambda^{1-\\sigma}\\tilde{a}_{\\lambda}^w\\langle x_2-tx_1\\rangle^{2\\sigma}u\\|_{L^2(\\mathbb{R}^{2n+1})} \\lesssim \n\\lambda^{1-\\sigma} \\|\\langle x_2-tx_1\\rangle^{2\\sigma}u\\|_{L^2(\\mathbb{R}^{2n+1})}.$$\nThis ends the proof of Lemma~\\ref{lem4}.~$\\Box$\n\n\\bigskip\n\n\\noindent\nBy coming back to (\\ref{eq20}), Lemma~\\ref{prop1} is then a direct consequence of Lemmas~\\ref{lem2}, \\ref{lem3} and~\\ref{lem4}.~$\\Box$\n\n\\bigskip\n\nNotice that the power $1-\\sigma$ appearing in the right-hand-side of the estimate given by Lemma~\\ref{prop1} is positive by assumption, since $0<\\sigma<1$.\nWe deduce from Lemma~\\ref{prop1} and (\\ref{eq19}) that for any fixed $T>0$, one may choose a new positive parameter $0< \\lambda_0 \\leq 1$ indexing the Wick quantization sufficiently small so that the term \n$$\\lambda^{1-\\sigma} \\|\\langle x_2-tx_1\\rangle^{2\\sigma}u\\|_{L^2(\\mathbb{R}^{2n+1})},$$\nappearing in the right-hand-side of the estimate given by Lemma~\\ref{prop1}\ncan be controlled by one half times the left-hand-side terms appearing in the a priori estimate (\\ref{eq19}). For any fixed $T>0$,   \nthere therefore exists a positive constant $c_T>0$ such that for all $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying \n$$\\textrm{supp } u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n\\begin{multline*}\nc_T^{-1}\\big\\|(1+|x_2-tx_1|^{2\\sigma}+|x_1|^{\\frac{2\\sigma}{2\\sigma+1}})u\\big\\|_{L^2(\\mathbb{R}^{2n+1})}  \\leq \\|\\langle x_2-tx_1\\rangle^{(2\\sigma-1)_+}u\\|_{L^2(\\mathbb{R}^{2n+1})}\\\\\n+\\|iD_tu+[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]^{w}u\\|_{L^2(\\mathbb{R}^{2n+1})}+\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}.\n\\end{multline*}\nBy noticing that, for any $\\varepsilon>0$, there exists a positive constant $C_{\\varepsilon}>0$ such that for all $(t,x_1,x_2) \\in [-T,T] \\times \\mathbb{R}^{2n}$,\n$$\\langle x_2-tx_1\\rangle^{(2\\sigma-1)_+} \\leq \\varepsilon \\langle x_2-tx_1\\rangle^{2\\sigma}+C_{\\varepsilon},$$ \nwe finally obtain that for any fixed $T>0$, there exists a positive constant $c_T>0$ such that for all $u \\in \\mathscr{S}(\\mathbb{R}_{t,x_1,x_2}^{2n+1})$ satisfying \n$$\\textrm{supp } u(\\cdot,x_1,x_2) \\subset [-T,T], \\ (x_1,x_2) \\in \\mathbb{R}^{2n},$$\nwe have\n\\begin{multline*}\nc_T^{-1}\\big\\|(1+|x_2-tx_1|^{2\\sigma}+|x_1|^{\\frac{2\\sigma}{2\\sigma+1}})u\\big\\|_{L^2(\\mathbb{R}^{2n+1})}  \\\\ \n\\leq \\|iD_tu+[a(t,\\xi_2,\\xi_1+t\\xi_2)F(x_2-tx_1)]^wu\\|_{L^2(\\mathbb{R}^{2n+1})}+\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}.\n\\end{multline*}\nThis proves the a priori estimate (\\ref{eq5}) and ends the proof of Proposition~\\ref{th1}.~$\\Box$\n\n\n\n\\section{Proof of Theorem~\\ref{TTH1}}\\label{section3}\n\n\\noindent\nThe next step in the proof of Theorem~\\ref{TTH1} is given by establishing the following hypoelliptic estimate:\n\n\\bigskip\n\n\\begin{proposition}\\label{cor1}\nLet $P$ be the operator defined in \\emph{(\\ref{yo2})} and $K$ a compact subset of~$\\mathbb{R}^{2n+1}$. Then, there exists a positive constant $C_K>0$ such that for any $u \\in C_0^{\\infty}(K)$,\n$$\\big\\|(1+|D_t|^{\\frac{2\\sigma}{2\\sigma+1}}+|D_x|^{\\frac{2\\sigma}{2\\sigma+1}}+|D_{v}|^{2\\sigma})u\\big\\|_{L^2(\\mathbb{R}^{2n+1})} \\leq C_K \\big(\\|Pu\\|_{L^2(\\mathbb{R}^{2n+1})}+\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}\\big).$$\n\\end{proposition}\n\n\\bigskip\n\n\n\\noindent\n\\textit{Proof of Proposition~\\ref{cor1}}. \nLet $K$ be a compact subset of $\\mathbb{R}^{2n+1}$ and denote by $\\mathcal{F}_{t,x}$ the partial Fourier transform with respect to the $t,x$ variables. Then, for any $u \\in C_0^{\\infty}(K)$, we may write that\n$$\\||D_t|^{\\frac{2\\sigma}{2\\sigma+1}}u\\|_{L^2(\\mathbb{R}^{2n+1})} \\lesssim \\||\\tau+v\\cdot \\xi|^{\\frac{2\\sigma}{2\\sigma+1}}\\mathcal{F}_{t,x}u\\|_{L^2(\\mathbb{R}^{2n+1})}+\\||v\\cdot \\xi|^{\\frac{2\\sigma}{2\\sigma+1}}\\mathcal{F}_{t,x}u\\|_{L^2(\\mathbb{R}^{2n+1})}.$$\nNotice that there exists a positive constant $C_{K}>0$ such that for all $u \\in C_0^{\\infty}(K)$,\n$$\\||v\\cdot \\xi|^{\\frac{2\\sigma}{2\\sigma+1}}\\mathcal{F}_{t,x}u\\|_{L^2(\\mathbb{R}^{2n+1})} \\leq C_{K}  \\||D_x|^{\\frac{2\\sigma}{2\\sigma+1}}u\\|_{L^2(\\mathbb{R}^{2n+1})}$$\nand that \n\\begin{multline*}\n\\||\\tau+v\\cdot \\xi|^{\\frac{2\\sigma}{2\\sigma+1}}\\mathcal{F}_{t,x}u\\|_{L^2(\\mathbb{R}^{2n+1})} \\lesssim \\|(\\tau+v\\cdot \\xi)\\mathcal{F}_{t,x}u\\|_{L^2(\\mathbb{R}^{2n+1})}+\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}\\\\\n\\lesssim \\|Pu\\|_{L^2(\\mathbb{R}^{2n+1})}+\\|a(t,x,v)(-\\tilde{\\Delta}_v)^{\\sigma}u\\|_{L^2(\\mathbb{R}^{2n+1})}+\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}.\n\\end{multline*}\nRecalling that $a \\in L^{\\infty}(\\mathbb{R}^{2n+1})$, we obtain that for all $u \\in C_0^{\\infty}(K)$,\n\\begin{multline*}\n\\||D_t|^{\\frac{2\\sigma}{2\\sigma+1}}u\\|_{L^2(\\mathbb{R}^{2n+1})} \\lesssim  \\|Pu\\|_{L^2(\\mathbb{R}^{2n+1})}+ \\||D_x|^{\\frac{2\\sigma}{2\\sigma+1}}u\\|_{L^2(\\mathbb{R}^{2n+1})}\\\\ +\\|(-\\tilde{\\Delta}_v)^{\\sigma}u\\|_{L^2(\\mathbb{R}^{2n+1})}+\\|u\\|_{L^2(\\mathbb{R}^{2n+1})}.\n\\end{multline*}\nProposition~\\ref{cor1} is then a direct consequence of Proposition~\\ref{th1}.~$\\Box$\n\n\n\n\n\\bigskip\n\nBy using Proposition~\\ref{cor1}, one can now complete the proof of Theorem~\\ref{TTH1}. Let $K$ be a compact subset of $\\mathbb{R}^{2n+1}$, $s \\in \\mathbb{R}$ and $\\chi$ be a  $C_0^{\\infty}(\\mathbb{R}^{2n+1})$ function satisfying $\\chi=1$ on $K$. Setting\n$$\\Lambda=(1+|D_t|^2+|D_x|^2+|D_v|^2)^{\\frac{1}{2}},$$\nwe apply Proposition~\\ref{cor1} to the function $\\chi \\Lambda^su$ with $u \\in C_0^{\\infty}(K)$,\n$$\\big\\|(1+\\langle D_t\\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle D_x\\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle D_{v}\\rangle ^{2\\sigma})\\chi \\Lambda^su\\big\\|_{L^2} \\leq C_K \\big(\\|P\\chi \\Lambda^su\\|_{L^2}+\\|\\chi \\Lambda^su\\|_{L^2}\\big).$$\nNotice that \n$$\\chi \\Lambda^su= \\Lambda^s(\\chi u)+[\\chi,\\Lambda^s]u=\\Lambda^su+[\\chi,\\Lambda^s]u,$$\nsince $\\chi=1$ on $K$ and $u \\in C_0^{\\infty}(K)$. \nWe have \n$$\\|\\chi \\Lambda^su\\|_{L^2} \\leq \\|u\\|_{s}.$$\nSymbolic calculus shows that the Weyl symbol of the operator $[\\chi,\\Lambda^s]$ belongs to the class\n$S(\\lambda^{s-1},\\tilde{\\Gamma})$ with\n$$\\lambda=(1+|\\tau|^2+|\\xi|^2+|\\eta|^2)^{\\frac{1}{2}} \\textrm{ and } \\tilde{\\Gamma}=dt^2+dx^2+dy^2+\\lambda^{-2}(d\\tau^2+d\\xi^2+d\\eta^2).$$\nIt therefore follows that \n\\begin{multline*}\n\\big|\\big\\|(1+\\langle D_t \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle D_x\\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle D_{v}\\rangle^{2\\sigma})\\chi \\Lambda^su\\big\\|_{L^2} \n\\\\ -\\big\\|(1+\\langle D_t \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle D_x\\rangle^{\\frac{2\\sigma}{2\\sigma+1}}+\\langle D_{v}\\rangle^{2\\sigma})u\\big\\|_{s}\\big| \n\\lesssim  \\|\\langle D_{v}\\rangle ^{(2\\sigma-1)_+}u\\|_{s},\n\\end{multline*}\nwith $(2\\sigma-1)_+=\\textrm{max}(2\\sigma-1,0)$.\nOn the other hand, we have  \n\\begin{align*}\n\\|P\\chi \\Lambda^su\\|_{L^2} \\leq & \\ \\|P \\Lambda^su\\|_{L^2}+\\|P[\\chi,\\Lambda^s]u\\|_{L^2}\\\\ \\leq & \\ \\|Pu\\|_{s}+ \\|[P,\\Lambda^s]u\\|_{L^2} +\\|[P,[\\chi,\\Lambda^s]]u\\|_{L^2}+\\|[\\chi,\\Lambda^s]Pu\\|_{L^2}.\n\\end{align*}\nNotice that \n$$[P,\\Lambda^s]=[v,\\Lambda^s] \\cdot \\partial_x+[a(t,x,v),\\Lambda^s](-\\tilde{\\Delta}_v)^{\\sigma}.$$\nSymbolic calculus directly shows that \n$$ \\|[P,\\Lambda^s]u\\|_{L^2} \\lesssim \\|\\langle D_{v}\\rangle ^{(2\\sigma-1)_+}u\\|_{s}$$\nand \n$$\\|[\\chi,\\Lambda^s]Pu\\|_{L^2}=\\|[\\chi,\\Lambda^s]\\Lambda^{-s}\\Lambda^sPu\\|_{L^2} \\lesssim \\|Pu\\|_{s}.$$\nIt follows that \n\\begin{align*}\n& \\ \\big\\|(1+|D_t|^{\\frac{2\\sigma}{2\\sigma+1}}+|D_x|^{\\frac{2\\sigma}{2\\sigma+1}}+|D_{v}|^{2\\sigma})u\\big\\|_{s} \\\\ \\lesssim & \\\n \\|Pu\\|_{s}+\\|[P,[\\chi,\\Lambda^s]]u\\|_{L^2}\n+\\|\\langle D_{v}\\rangle ^{(2\\sigma-1)_+}u\\|_{s}.\n\\end{align*}\nIt remains to study the double commutator \n\\begin{multline*}\n[P,[\\chi,\\Lambda^s]]=[\\partial_t,[\\chi,\\Lambda^s]]+[v \\cdot \\partial_x,[\\chi,\\Lambda^s]]+[a(t,x,v)(-\\tilde{\\Delta}_v)^{\\sigma},[\\chi,\\Lambda^s]]\n=[\\partial_t,[\\chi,\\Lambda^s]]\\\\ +v[\\partial_x,[\\chi,\\Lambda^s]]+[v,[\\chi,\\Lambda^s]]\\partial_x+a(t,x,v)[(-\\tilde{\\Delta}_v)^{\\sigma},[\\chi,\\Lambda^s]]+[a(t,x,v),[\\chi,\\Lambda^s]](-\\tilde{\\Delta}_v)^{\\sigma}.\n\\end{multline*}\nBy using the fact that $a$ is a $C_b^{\\infty}(\\mathbb{R}^{2n+1})$ function and $0<\\sigma <1$, symbolic calculus directly shows that \n\\begin{multline*}\n\\|[\\partial_t,[\\chi,\\Lambda^s]]u\\|_{L^2}+\\|[v,[\\chi,\\Lambda^s]]\\partial_xu\\|_{L^2}+\\|a(t,x,v)[(-\\tilde{\\Delta}_v)^{\\sigma},[\\chi,\\Lambda^s]]u\\|_{L^2}\\\\ +\\|[a(t,x,v),[\\chi,\\Lambda^s]](-\\tilde{\\Delta}_v)^{\\sigma}u\\|_{L^2} \\lesssim \\|u\\|_{s},\n\\end{multline*}\nsince we recall that the Weyl symbol of the operator $[\\chi,\\Lambda^s]$ belongs to the class $S(\\lambda^{s-1},\\tilde{\\Gamma})$. It follows that \n\\begin{multline*}\n\\|[P,[\\chi,\\Lambda^s]]u\\|_{L^2} \\lesssim \\|[v,[\\partial_x,[\\chi,\\Lambda^s]]]u\\|_{L^2}+\\|[\\partial_x,[\\chi,\\Lambda^s]]]vu\\|_{L^2}+\\|u\\|_{s}\\\\\n\\lesssim \\|vu\\|_{s}+\\|u\\|_{s}\\lesssim \\|\\chi vu\\|_{s}+\\|u\\|_{s} \\lesssim \\|u\\|_{s},\n\\end{multline*}\nsince $\\chi=1$ on $K$ and $u \\in C_0^{\\infty}(K)$.\nBy using that for any $\\varepsilon>0$, there exists a positive constant $C_{\\varepsilon}>0$ such that\n$$\\|\\langle D_{v}\\rangle ^{(2\\sigma-1)_+}u\\|_{s} \\leq \\varepsilon \\||D_{v}|^{2\\sigma}u\\|_{s}+C_{\\varepsilon}\\|u\\|_{s},$$\nwe finally obtain that there exists a positive constant $C_K>0$ such that for any $u \\in C_0^{\\infty}(K)$,\n$$\\big\\|(1+|D_t|^{\\frac{2\\sigma}{2\\sigma+1}}+|D_x|^{\\frac{2\\sigma}{2\\sigma+1}}+|D_{v}|^{2\\sigma})u\\big\\|_{s} \\leq C_K \\big(\\|Pu\\|_{s}+\\|u\\|_{s}\\big).$$\nThis ends the proof of Theorem~\\ref{TTH1}.~$\\Box$\n\n\n\\section{Proof of Corollary~\\ref{ev1}}\\label{section4}\n\n\nThis section is devoted to the proof of Corollary~\\ref{ev1}. Let $P$ be the operator defined in (\\ref{yo2}) and $N_0 \\in \\mathbb{N}$. For simplicity, we shall assume that $(t_0,x_0) =(0,0)$ and consider a function $u \\in H^{-N_0}(\\mathbb{R}^{2n+1})$ satisfying \n\\begin{equation}\\label{sev20}\nPu \\in H^s_{\\textrm{loc}, (t_0,x_0)}(\\mathbb{R}^{2n+1}_{t,x,v}),\n\\end{equation}\nwith $s \\geq 0$. Let $\\varphi$ be a $C_0^\\infty(\\mathbb{R}_{t,x}^{n+1})$ function supported in the set \n$$\\{(t,x) \\in \\mathbb{R}^{n+1} : |(t,x)|<2\\}$$\nsatisfying $\\varphi=1$ in a neighborhood of $(t_0,x_0) =(0,0)$ and \n\\begin{equation}\\label{ts9}\n\\varphi(t,x) Pu \\in H^s(\\mathbb{R}_{t,x,v}^{2n+1}).\n\\end{equation}\nFor convenience only, we shall assume that \n\\begin{equation}\\label{ts8}\n \\varphi(t,x)=1,\n\\end{equation}\nfor all $|(t,x)| \\leq 3\/2$.\nWe consider $\\chi$ a $C^\\infty(\\mathbb{R}^{n+1}_{t,x})$ function satisfying $-N_0-2 \\le \\chi \\le  s$, \n$\\chi = s$ on  $\\{|(t,x)| \\leq  1\\}$, $\\chi = -N_0-2$ when $|(t,x)| \\ge 3\/2$, and define\n\\begin{equation}\\label{sev6}\nM_\\delta(t,x, \\zeta) =\\frac{\\langle \\zeta \\rangle^{\\chi(t,x)}}{\n(1+ \\delta \\langle \\zeta \\rangle)^{N_0+s+2}}, \\ \\zeta=(\\tau,\\xi,\\eta) \\in \\mathbb{R}^{2n+1},\n\\end{equation}\nwith $\\langle \\zeta \\rangle = (1+|\\zeta|^2)^{1\/2}$ and $0 <\\delta \\leq 1$. We recall that $\\tau$, $\\xi$ and $\\eta$ stand respectively for the dual variables of $t$, $x$ and $v$. It directly follows from this definition that for any $\\alpha \\in \\mathbb{N}^{n+1}$, $\\beta \\in \\mathbb{N}^{2n+1}$, there exist some positive constants $C_{\\alpha, \\beta} >0$ such that for all $0< \\delta \\leq 1$, $(t,x) \\in \\mathbb{R}^{n+1}$ and $\\zeta \\in \\mathbb{R}^{2n+1}$,\n\\begin{align}\\label{symbol-est}\n\\big| \\partial^\\alpha_{t,x} \\partial_{\\zeta}^\\beta\nM_\\delta(t,x, \\zeta) \\big|\\le C_{\\alpha, \\beta} \n\\big(\\log \\langle \\zeta \\rangle\\big)^{|\\alpha|} \\langle \\zeta \\rangle^{-|\\beta|}\nM_\\delta(t,x, \\zeta).\n\\end{align}\nNotice that \n$$M_{\\delta}(t,x,\\zeta) \\leq \\langle \\zeta \\rangle^s.$$\nLet $\\varepsilon>0$ be a positive constant and $k \\geq 0$ a nonnegative integer. The symbol $M_{\\delta}$ therefore belongs to the symbol class $S^{s+\\varepsilon}$ uniformly with respect to the parameter $0<\\delta \\leq 1$. We recall that the notation $S^m$, with $m \\in \\mathbb{R}$, stands for the symbol class\n$$S^{m}=S(\\langle \\zeta \\rangle^m,\\Gamma), \\quad  \\Gamma=dz^2+\\frac{d\\zeta^2}{\\langle \\zeta \\rangle^{2}},$$\nwith $z=(t,x,v) \\in \\mathbb{R}^{2n+1}$. The symbol $M_{\\delta}$ also belongs to the symbol class $\\tilde{S}^{s}$,\n\\begin{equation}\\label{tt1}\n\\tilde{S}^{s}=S(\\langle \\zeta \\rangle^s,\\tilde{\\Gamma}), \\quad  \\tilde{\\Gamma}=\\langle \\zeta \\rangle dz^2+\\frac{d\\zeta^2}{\\langle \\zeta \\rangle^{2}},\n\\end{equation}\nuniformly with respect to the parameter $0<\\delta \\leq 1$.\nFurthermore, notice that \n$$M_{\\delta}(t,x,\\zeta) \\leq \\delta^{-N_0-s-2} \\langle \\zeta \\rangle^{-N_0-2},$$\nwhen $0<\\delta \\leq 1$. It follows that for each fixed $0<\\delta \\leq 1$, the symbol $M_\\delta$ also belongs to the class\n$S^{-N_0-2+\\varepsilon}$. \nMore specifically, we deduce from (\\ref{sev6}) and the Leibniz formula that for any $\\alpha \\in \\mathbb{N}^{2n+1}$, $\\beta \\in \\mathbb{N}^{2n+1}$, we may write \n\\begin{equation}\\label{symbol-est1}\n\\partial^\\alpha_{z} \\partial_{\\zeta}^\\beta\\big[\\langle v \\rangle^{-k} M_\\delta(t,x, \\zeta)\\big]=a_{\\alpha,\\beta, \\delta}(z,\\zeta)\\langle v \\rangle^{-k}M_\\delta(t,x, \\zeta),\n\\end{equation}\nwith $a_{\\alpha,\\beta, \\delta}$ a symbol belonging to the class $S^{\\varepsilon-|\\beta|}$\nuniformly with respect to the parameter $0<\\delta \\leq 1$.  \nWe begin by establishing the following lemma:\n\n\n\n\n\\bigskip\n\n\\begin{lemma}\\label{llop1}\nLet $0<\\varepsilon<1$, $k \\geq 0$, $N \\in \\mathbb{N}$ and $A \\in S^m$, $m \\in \\mathbb{R}$. Then, there exists a symbol $B_{\\delta}$ belonging to the class $S^{m-(1-\\varepsilon)}$ uniformly with respect to the parameter $0<\\delta \\leq 1$ such that \n$$\\emph{\\textrm{Op}}(A\\langle v \\rangle^{-k}M_{\\delta})-A(z,D_z)\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)-\\emph{\\textrm{Op}}(B_{\\delta}\\langle v \\rangle^{-k}M_{\\delta}) \\in \\emph{\\textrm{Op}}(S^{-N}),$$\nuniformly with respect to the parameter $0<\\delta \\leq 1$.\n\n\\end{lemma}\n\n\\bigskip\n\n\\noindent\nHere $A(z,D_z)$ or $\\textrm{Op}(A)$ stands for the standard quantization of the symbol $A(z,\\zeta)$,\n$$A(z,D_z)u(z)=\\int_{\\mathbb{R}^{2n+1}}e^{2\\pi i(z-\\tilde{z}).\\zeta}A(z,\\zeta)u(\\tilde{z})d\\zeta d\\tilde{z}, \\ z=(t,x,v) \\in \\mathbb{R}^{2n+1}.$$\n\n\n\\bigskip\n\n\\noindent\n\\textit{Proof of Lemma~\\ref{llop1}}. Since $A \\in S^m$ and $M_{\\delta} \\in S^{s+\\varepsilon}$ uniformly with respect to the parameter $0<\\delta \\leq 1$, symbolic calculus shows that $\\langle v \\rangle^{-k}M_{\\delta} \\in S^{s+\\varepsilon}$ uniformly with respect to the parameter $0<\\delta \\leq 1$ and\n$$A(z,D_z)\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z) -\\sum_{0\\le|\\alpha|\\leq [m+s+\\varepsilon]+N}\\frac{1}{\\alpha !}\\textrm{Op}\\big((\\partial^\\alpha_\\zeta A) D_z^\\alpha\\big(\\langle v \\rangle^{-k}M_{\\delta}\\big)\\big) \\in \\textrm{Op}(S^{-N}),$$\nwith $[m+s+\\varepsilon]$ being the integer part of $m+s+\\varepsilon \\in \\mathbb{R}$. We may therefore write from (\\ref{symbol-est1}) that \n$$\\textrm{Op}(A\\langle v \\rangle^{-k}M_{\\delta})-A(z,D_z)\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)-\\textrm{Op}(B_{\\delta}\\langle v \\rangle^{-k}M_{\\delta}) \\in \\textrm{Op}(S^{-N}),$$\nwith \n$$B_{\\delta}(z,\\zeta)=-\\sum_{1\\le|\\alpha|\\leq [m+s+\\varepsilon]+N}\\frac{1}{\\alpha !}(\\partial^\\alpha_\\zeta A)(z,\\zeta)a_{\\alpha,0,\\delta}(z,\\zeta).$$\nSince $\\partial^\\alpha_\\zeta A \\in S^{m-|\\alpha|}$ and $a_{\\alpha,0,\\delta} \\in S^{\\varepsilon}$ uniformly with respect to $0<\\delta \\leq 1$, the symbol \n$B_{\\delta}$ belongs to the class $S^{m-(1-\\varepsilon)}$ uniformly with respect to $0<\\delta \\leq 1$. This proves Lemma~\\ref{llop1}.~$\\Box$\n\n\n\\bigskip\n\\noindent\n\n\n\\bigskip\n\n\\begin{lemma}\\label{llop2}\nLet $k \\geq 0$, $N \\in \\mathbb{N}$ and $A \\in S^m$, $m \\in \\mathbb{R}$. Then, there exists a symbol $A_{\\delta}$ belonging to the class $S^{m}$ uniformly with respect to the parameter $0<\\delta \\leq 1$ such that \n$$\\emph{\\textrm{Op}}(A\\langle v \\rangle^{-k}M_{\\delta})-A_{\\delta}(z,D_z)\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z) \\in \\emph{\\textrm{Op}}(S^{-N}),$$\nuniformly with respect to the parameter $0<\\delta \\leq 1$.\n\n\\end{lemma}\n\n\\bigskip\n\n\\noindent\n\\textit{Proof of Lemma~\\ref{llop2}}. Let $0<\\varepsilon<1$. Recalling that the symbol $\\langle v \\rangle^{-k}M_{\\delta}$  belongs to the class $S^{s+\\varepsilon}$ uniformly with respect to the parameter $0<\\delta \\leq 1$, Lemma~\\ref{llop2} follows from $k_0$ iterations of Lemma~\\ref{llop1} with $m+s+\\varepsilon-k_0(1-\\varepsilon)<-N$ on the successive remainder terms $\\textrm{Op}(B_{\\delta}\\langle v \\rangle^{-k}M_{\\delta})$.~$\\Box$\n\n\n\n\n\n\n\n\\bigskip\n\n\\noindent\nWe shall now use the following commutator argument: \n\n\n\n\n\\bigskip\n\n\\begin{lemma}\\label{lop1}\nLet $0<\\varepsilon<1$, $k \\geq 1$ and $u$ the function defined in \\emph{(\\ref{sev20})}. Then, there exists a positive constant $C_{k,\\varepsilon}>0$ such that for all $0<\\delta \\leq 1$,  \n$$\\|[\\partial_t + v\\cdot \\nabla_x \\, , \\langle v \\rangle^{-k} M_\\delta(t,x,D_z)] u\\|_{L^2} \\leq C_{k,\\varepsilon}  \\| \\langle v \\rangle^{-k+1} \nM_\\delta(t,x,D_z) u \\|_{\\varepsilon} +C_{k,\\varepsilon}\\|u\\|_{-N_0}.$$\n\\end{lemma}\n\n\\bigskip\n\n\n\\noindent\n\\textit{Proof of Lemma~\\ref{lop1}}. Let $k \\geq 1$ be a positive integer and $0<\\varepsilon<1$. We consider the commutator \n$$[\\partial_t + v\\cdot \\nabla_x \\, , \\langle v \\rangle^{-k} M_\\delta(t,x,D_z)].$$\nSymbolic calculus shows that the symbol of the commutator \n$$ [\\partial_t + v\\cdot \\nabla_x , \\langle v \\rangle^{-k}M_\\delta(t,x,D_z)]=\\langle v \\rangle^{-k} [\\partial_t + v\\cdot \\nabla_x , M_\\delta(t,x,D_z)],$$ \nin the standard quantization is \n$$\\langle v \\rangle^{-k} \\big((\\partial_t \\chi)(t,x) + v\\cdot (\\nabla_x\\chi)(t,x)\\big) M_\\delta(t,x, \\zeta)\\log \\langle \\zeta \\rangle-\\langle v \\rangle^{-k} \\xi \\cdot (\\nabla_{\\eta}M_{\\delta})(t,x,\\zeta).$$\nThis symbol may be written as \n\\begin{multline}\\label{sev21b}\n\\log \\langle \\zeta \\rangle\\Big(\\langle v \\rangle^{-1}(\\partial_t \\chi)(t,x) + \\frac{v}{\\langle v \\rangle}\\cdot (\\nabla_x\\chi)(t,x)\\Big) \\langle v \\rangle^{-(k-1)}M_\\delta(t,x, \\zeta)\\\\ - \\langle v \\rangle^{-1}\\xi \\cdot (\\nabla_{\\eta})\\big[\\langle v \\rangle^{-(k-1)}M_{\\delta}(t,x,\\zeta)\\big].\n\\end{multline} \nBy using that $\\log \\langle \\zeta \\rangle \\in S^{\\varepsilon}$, it follows from (\\ref{symbol-est1}) and (\\ref{sev21b}) that this symbol may also be written as\n \\begin{equation}\\label{sev21}\nA_{\\delta}(z,\\zeta)\\langle v \\rangle^{-(k-1)}M_{\\delta}(t,x,\\zeta),\n\\end{equation} \nwith $A_{\\delta}$ belonging to the class $S^{\\varepsilon}$ uniformly with respect to $0<\\delta \\leq 1$. Then, we deduce from Lemma~\\ref{llop2} that \n\\begin{equation}\\label{sev1}\n[\\partial_t + v\\cdot \\nabla_x ,\\langle v \\rangle^{-k} M_\\delta(t,x,D_z)]-\\tilde{A}_{\\delta}(z,D_z)\\langle v \\rangle^{-(k-1)}M_{\\delta}(t,x,D_z) \\in \\textrm{Op}(S^{-N_0}),\n\\end{equation}\nwith $\\tilde{A}_{\\delta}$ a new symbol belonging to the class $S^{\\varepsilon}$ uniformly with respect to $0<\\delta \\leq 1$.\nThis implies that \n\\begin{multline}\\label{sev2}\n\\|[\\partial_t + v\\cdot \\nabla_x , \\langle v \\rangle^{-k}M_\\delta(t,x,D_z)]u-\\tilde{A}_{\\delta}(z,D_z)\\langle v \\rangle^{-(k-1)}M_{\\delta}(t,x,D_z)u\\|_{L^2} \\\\ \\lesssim \\|u\\|_{-N_0}.\n\\end{multline}\nOn the other hand, we have\n\\begin{equation}\\label{rum2}\n\\|\\tilde{A}_{\\delta}(z,D_z)\\langle v \\rangle^{-(k-1)}M_{\\delta}(t,x,D_z)u\\|_{L^2} \\lesssim \\|\\langle v \\rangle^{-(k-1)}M_{\\delta}(t,x,D_z)u\\|_{\\varepsilon},\n\\end{equation}\nsince $\\tilde{A}_{\\delta} \\in S^{\\varepsilon}$ uniformly with respect to $0<\\delta \\leq 1$.\nIt follows from (\\ref{sev2}), (\\ref{rum2}) and the triangular inequality that the estimate \n$$\\|[\\partial_t + v\\cdot \\nabla_x \\, , \\langle v \\rangle^{-k} M_\\delta(t,x,D_z)] u\\|_{L^2} \\lesssim  \\| \\langle v \\rangle^{-k+1} \nM_\\delta(t,x,D_z) u \\|_{\\varepsilon} +\\|u\\|_{-N_0},$$\nholds uniformly with respect to the parameter $0<\\delta \\leq 1$. This proves Lemma~\\ref{lop1}.~$\\Box$\n\n\n\n\\bigskip\n\n\\noindent\nOne can now estimate from above the following second commutator: \n\n\n\n\n\\bigskip\n\n\\begin{lemma}\\label{lop2}\nLet $0<\\varepsilon<1$, $k \\geq 0$ and $u$ be the function defined in \\emph{(\\ref{sev20})}. Then, there exists a positive constant $C_{k,\\varepsilon}>0$ such that for all $0<\\delta \\leq 1$,  \n\\begin{multline*}\n\\|[a(t,x,v)(-\\tilde{\\Delta}_v)^{\\sigma}, \\langle v \\rangle^{-k} M_\\delta(t,x,D_z)] u\\|_{L^2}\\\\ \\leq C_{k,\\varepsilon} \n \\| \\langle D_v\\rangle^{(2\\sigma-1)_++\\varepsilon} \\langle v \\rangle^{-k} \nM_\\delta(t,x,D_z) u \\|_{L^2}+C_{k,\\varepsilon}\\|u\\|_{-N_0},\n\\end{multline*}\nwith $(2\\sigma-1)_+=\\mbox{\\rm max}(2\\sigma-1,0)$.\n\\end{lemma}\n\n\\bigskip\n\n\n\\noindent\n\\textit{Proof of Lemma~\\ref{lop2}}. Notice that the symbol of the operator $a(t,x,v)(-\\tilde{\\Delta}_v)^{\\sigma}$ in the standard quantization does not belong to the class $S^{2\\sigma}$ in $\\mathbb{R}_{z,\\zeta}^{4n+2}$. Handling this commutator term \n$$[a(t,x,v)(-\\tilde{\\Delta}_v)^{\\sigma}, \\langle v \\rangle^{-k} M_\\delta(t,x,D_z)]$$\nneeds therefore some caution. By using that \n$$[a(t,x,v),\\langle v \\rangle^{-k}]=0 \\textrm{ and } [(-\\tilde{\\Delta}_v)^{\\sigma},M_\\delta(t,x,D_z)]=0,$$ \nwe may write that \n\\begin{multline}\\label{ryu1}\n[a(t,x,v)(-\\tilde{\\Delta}_v)^{\\sigma}, \\langle v \\rangle^{-k} M_\\delta(t,x,D_z)]=a(t,x,v)[(-\\tilde{\\Delta}_v)^{\\sigma}, \\langle v \\rangle^{-k}]M_\\delta(t,x,D_z)\\\\\n+ \\langle v \\rangle^{-k} [a(t,x,v), M_\\delta(t,x,D_z)](-\\tilde{\\Delta}_v)^{\\sigma}.\n\\end{multline}\nLet $0<\\varepsilon<1$. Regarding the first term in the right-hand-side of (\\ref{ryu1}), we first notice that the symbol of the operator \n$[(-\\tilde{\\Delta}_v)^{\\sigma}, \\langle v \\rangle^{-k}]\\langle v \\rangle^{k} \\langle D_v \\rangle^{-(2\\sigma-1)_+-\\varepsilon}$ \nbelongs to the class\n$$S(1,dv^2+\\langle \\eta \\rangle^{-2}d\\eta^2).$$\nIndeed, the symbol of the operator $(-\\tilde{\\Delta}_v)^{\\sigma}$ belongs to $S(\\langle \\eta \\rangle^{2\\sigma},dv^2+\\langle \\eta \\rangle^{-2}d\\eta^2)$, the symbol of the operator \n$\\langle D_v \\rangle^{-(2\\sigma-1)_+-\\varepsilon}$ belongs to $S(\\langle \\eta \\rangle^{-(2\\sigma-1)_+-\\varepsilon},dv^2+\\langle \\eta \\rangle^{-2}d\\eta^2)$\n and\n$\\langle v \\rangle^{m} \\in S(\\langle v \\rangle^{m},dv^2+\\langle \\eta \\rangle^{-2}d\\eta^2)$ when $m \\in \\mathbb{R}$. Symbolic calculus shows that the symbol of the commutator \n$[(-\\tilde{\\Delta}_v)^{\\sigma}, \\langle v \\rangle^{-k}]$ belongs to $S(\\langle \\eta \\rangle^{2\\sigma-1}\\langle v \\rangle^{-k},dv^2+\\langle \\eta \\rangle^{-2}d\\eta^2)$ and therefore that the symbol of the operator $[(-\\tilde{\\Delta}_v)^{\\sigma}, \\langle v \\rangle^{-k}]\\langle v \\rangle^{k}\\langle D_v \\rangle^{-(2\\sigma-1)_+-\\varepsilon}$ \nbelongs to the class\n$$S(\\langle \\eta \\rangle^{2\\sigma-1-(2\\sigma-1)_+-\\varepsilon},dv^2+\\langle \\eta \\rangle^{-2}d\\eta^2) \\subset S(1,dv^2+\\langle \\eta \\rangle^{-2}d\\eta^2).$$\nRecalling that by assumption $a \\in C_b^{\\infty}(\\mathbb{R}_{t,x,v}^{2n+1})$, we obtain that \n\\begin{align}\\label{ryu2}\n& \\ \\|a(t,x,v)[(-\\tilde{\\Delta}_v)^{\\sigma}, \\langle v \\rangle^{-k}]M_\\delta(t,x,D_z)u\\|_{L^2} \\\\ \\notag\n\\lesssim & \\ \\|[(-\\tilde{\\Delta}_v)^{\\sigma}, \\langle v \\rangle^{-k}] \\langle v \\rangle^{k}\\langle D_v \\rangle^{-(2\\sigma-1)_+-\\varepsilon}\\langle D_v \\rangle^{(2\\sigma-1)_++\\varepsilon} \\langle v \\rangle^{-k}M_\\delta(t,x,D_z)u\\|_{L^2} \\\\ \\notag\n\\lesssim & \\  \\|\\langle D_v\\rangle^{(2\\sigma-1)_++\\varepsilon}  \\langle v \\rangle^{-k}M_\\delta(t,x,D_z)u\\|_{L^2}.\n\\end{align}\nRecalling that $M_{\\delta} \\in S^{s+\\varepsilon}$ uniformly with respect to $0<\\delta \\leq 1$ and that by assumption $a \\in S^0$, symbolic calculus shows that the symbol of the operator \n$$\\langle v \\rangle^{-k}[a(t,x,v), M_\\delta(t,x,D_z)],$$ \nwrites as\n\\begin{align}\\label{ryu3}\n& \\ -\\langle v \\rangle^{-k} \\sum_{1\\le|\\alpha|\\leq [s+\\varepsilon]+N_0+2}\\frac{1}{\\alpha !}(\\partial^\\alpha_\\zeta M_{\\delta}) D_z^\\alpha a\\\\ \n= & \\ -\\sum_{1\\le|\\alpha|\\leq [s+\\varepsilon]+N_0+2}\\frac{1}{\\alpha !}(D_z^\\alpha a)\\partial^\\alpha_\\zeta\\big[\\langle v \\rangle^{-k} M_{\\delta}\\big],\\notag\n\\end{align}\nup to a term in $S^{-N_0-2}$. This latter term in the class $S^{-N_0-2}$ gives rise to a term bounded by \n$$\\|(-\\tilde{\\Delta}_v)^{\\sigma}u\\|_{-N_0-2} \\lesssim \\|u\\|_{-N_0},$$\nbecause $0<\\sigma <1$,\nwhile estimating from above the term \n$$\\|\\langle v \\rangle^{-k} [a(t,x,v), M_\\delta(t,x,D_z)](-\\tilde{\\Delta}_v)^{\\sigma}u\\|_{L^2}.$$\nBy coming back to (\\ref{ryu3}), we deduce from (\\ref{symbol-est1}) that this symbol reduces to a term $A_{\\delta}(z,\\zeta)\\langle v \\rangle^{-k} M_{\\delta}$ with \n$A_{\\delta}$ belonging to $S^{\\varepsilon-1}$ uniformly with respect to $0<\\delta \\leq 1$. By using Lemma~\\ref{llop2}, we notice that \n$$\\textrm{Op}(A_{\\delta}\\langle v \\rangle^{-k}M_{\\delta})-\\tilde{A}_{\\delta}(z,D_z)\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z) \\in \\textrm{Op}(S^{-N_0-2}),$$\nwith $\\tilde{A}_{\\delta}$ a new symbol belonging to the class $S^{\\varepsilon-1}$ uniformly with respect to $0<\\delta \\leq 1$.\nIt follows that \n\\begin{multline}\\label{ryu4}\n\\|\\textrm{Op}(A_{\\delta}\\langle v \\rangle^{-k}M_{\\delta})(-\\tilde{\\Delta}_v)^{\\sigma}u\\|_{L^2} \\lesssim \\|(-\\tilde{\\Delta}_v)^{\\sigma}u\\|_{-N_0-2}\\\\ + \\|\\tilde{A}_{\\delta}(z,D_z)\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)(-\\tilde{\\Delta}_v)^{\\sigma}u\\|_{L^2}. \n\\end{multline}\nSince $[M_{\\delta}(t,x,D_z),(-\\tilde{\\Delta}_v)^{\\sigma}]=0$ and $\\tilde{A}_{\\delta} \\in S^{\\varepsilon-1}$ uniformly with respect to $0<\\delta \\leq 1$, we deduce from (\\ref{ryu4}) that\n\\begin{multline}\\label{ryu5}\n\\|\\textrm{Op}(A_{\\delta}\\langle v \\rangle^{-k}M_{\\delta})(-\\tilde{\\Delta}_v)^{\\sigma}u\\|_{L^2} \\lesssim \\|u\\|_{-N_0} + \\|\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)(-\\tilde{\\Delta}_v)^{\\sigma}u\\|_{\\varepsilon-1}\\\\\n\\lesssim \\|u\\|_{-N_0} + \\|\\langle v \\rangle^{-k}(-\\tilde{\\Delta}_v)^{\\sigma}\\langle v \\rangle^{k}\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{\\varepsilon-1}. \n\\end{multline}\nNotice that \n\\begin{multline}\\label{ryu6}\n\\|\\langle v \\rangle^{-k}(-\\tilde{\\Delta}_v)^{\\sigma}\\langle v \\rangle^{k}\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{\\varepsilon-1} \n\\leq   \\|(-\\tilde{\\Delta}_v)^{\\sigma}\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{\\varepsilon-1}\\\\\n+\\|[\\langle v \\rangle^{-k},(-\\tilde{\\Delta}_v)^{\\sigma}]\\langle v \\rangle^{k}\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{\\varepsilon-1}.\n\\end{multline}\nWe may write\n\\begin{align}\\label{ryu111}\n& \\ \\|(-\\tilde{\\Delta}_v)^{\\sigma}\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{\\varepsilon-1}  \\\\ \n\\lesssim & \\ \\|\\langle D_{z}\\rangle^{-(1-\\varepsilon)}(-\\tilde{\\Delta}_v)^{\\sigma}\\langle D_v\\rangle^{-2\\sigma+1-\\varepsilon}\\langle D_v\\rangle^{2\\sigma-1+\\varepsilon}\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{L^2} \\notag \\\\\n\\lesssim & \\ \\|\\langle D_v\\rangle^{2\\sigma-1+\\varepsilon}\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{L^2}\\lesssim  \\|\\langle D_v\\rangle^{(2\\sigma-1)_++\\varepsilon}\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{L^2}, \\notag\n\\end{align}\nwith $\\langle D_z \\rangle=(1+D_t^2+|D_x|^2+|D_v|^2)^{1\/2}$, since the operator \n$$\\langle D_{z}\\rangle^{-(1-\\varepsilon)}(-\\tilde{\\Delta}_v)^{\\sigma}\\langle D_v\\rangle^{-2\\sigma+1-\\varepsilon},$$ \nis bounded on $L^2$.\nNotice also that\n\\begin{align}\\label{ryu112}\n& \\  \\|[\\langle v \\rangle^{-k},(-\\tilde{\\Delta}_v)^{\\sigma}]\\langle v \\rangle^{k}\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{\\varepsilon-1} \\\\\n\\lesssim & \\ \\notag  \\|\\langle D_{z}\\rangle^{-(1-\\varepsilon)}[\\langle v \\rangle^{-k},(-\\tilde{\\Delta}_v)^{\\sigma}]\\langle v \\rangle^{k}\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{L^2}\\\\\n\\lesssim & \\ \\notag  \\|\\langle D_{z}\\rangle^{-(1-\\varepsilon)}[\\langle v \\rangle^{-k},(-\\tilde{\\Delta}_v)^{\\sigma}]\\langle v \\rangle^{k}\\langle D_v\\rangle^{-2\\sigma+1-\\varepsilon}\\langle D_v\\rangle^{2\\sigma-1+\\varepsilon}\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{L^2} \\\\\n\\lesssim & \\ \\|\\langle D_v\\rangle^{2\\sigma-1+\\varepsilon}\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{L^2}\\lesssim  \\|\\langle D_v\\rangle^{(2\\sigma-1)_++\\varepsilon}\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{L^2}. \\notag\n\\end{align}\nIndeed, we just need to check that the operator \n\\begin{multline*}\n\\langle D_{z}\\rangle^{-(1-\\varepsilon)}[\\langle v \\rangle^{-k},(-\\tilde{\\Delta}_v)^{\\sigma}]\\langle v \\rangle^{k}\\langle D_v\\rangle^{-2\\sigma+1-\\varepsilon}\\\\\n=\\langle D_{z}\\rangle^{-(1-\\varepsilon)}\\langle D_v\\rangle^{1-\\varepsilon} \\langle D_v\\rangle^{-1+\\varepsilon} [\\langle v \\rangle^{-k},(-\\tilde{\\Delta}_v)^{\\sigma}]\\langle v \\rangle^{k}\\langle D_v\\rangle^{-2\\sigma+1-\\varepsilon},\n\\end{multline*}\nis bounded on $L^2$. This is actually the case since the operator \n$$\\langle D_{z}\\rangle^{-(1-\\varepsilon)}\\langle D_v\\rangle^{1-\\varepsilon},$$\nis obviously bounded on $L^2$ since $0<1-\\varepsilon<1$, and that symbolic calculus easily shows that the symbol of the operator \n$$\\langle D_v\\rangle^{-1+\\varepsilon} [\\langle v \\rangle^{-k},(-\\tilde{\\Delta}_v)^{\\sigma}]\\langle v \\rangle^{k}\\langle D_v\\rangle^{-2\\sigma+1-\\varepsilon},$$\nbelongs to the class $S(1,dv^2+\\langle \\eta \\rangle^{-2}d\\eta^2)$. It follows from (\\ref{ryu5}), (\\ref{ryu6}), (\\ref{ryu111}) and (\\ref{ryu112}) that \n\\begin{align}\\label{ryu7}\n& \\ \\|\\langle v \\rangle^{-k} [a(t,x,v), M_\\delta(t,x,D_z)](-\\tilde{\\Delta}_v)^{\\sigma}u\\|_{L^2} \\\\ \\lesssim & \\ \n\\|\\langle D_v\\rangle^{(2\\sigma-1)_++\\varepsilon} \\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)u\\|_{L^2}+ \\|u\\|_{-N_0}. \\notag\n\\end{align}\nTogether with (\\ref{ryu1}) and (\\ref{ryu2}), we finally obtain the estimate\n\\begin{align*}\n&\\|[a(t,x,v)(-\\tilde{\\Delta}_v)^{\\sigma}, \\langle v \\rangle^{-k} M_\\delta(t,x,D_z)] u\\|_{L^2} \\\\\n&\\lesssim  \\| \\langle D_v\\rangle^{(2\\sigma-1)_++\\varepsilon} \\langle v \\rangle^{-k} \nM_\\delta(t,x,D_z) u \\|_{L^2} +\\|u\\|_{-N_0},\n\\end{align*}\nwhich proves Lemma~\\ref{lop2}.~$\\Box$\n\n\\bigskip\n\n\\noindent\nSumming up the two previous lemmas provides the following estimate:\n\n\n\\bigskip\n\n\\begin{lemma}\\label{lop3}\nLet $0<\\varepsilon<1$, $k \\geq 1$, $P$ be the operator defined in \\emph{(\\ref{yo2})} and $u$ the function defined in \\emph{(\\ref{sev20})}. Then, there exists a positive constant $C_{k,\\varepsilon}>0$ such that for all $0<\\delta \\leq 1$,  \n\\begin{multline*}\n\\|[P, \\langle v \\rangle^{-k} M_\\delta(t,x,D_z)] u\\|_{L^2} \\leq C_{k,\\varepsilon}  \\|\\langle D_v\\rangle^{(2\\sigma-1)_++\\varepsilon}\\langle v \\rangle^{-k} \nM_\\delta(t,x,D_z) u \\|_{L^2} \\\\ +C_{k,\\varepsilon}  \\| \\langle v \\rangle^{-k+1} \nM_\\delta(t,x,D_z) u \\|_{\\varepsilon} +C_{k,\\varepsilon}\\|u\\|_{-N_0},\n\\end{multline*}\nwith $(2\\sigma-1)_+=\\mbox{\\rm max}(2\\sigma-1,0)$.\n\\end{lemma}\n\n\\bigskip\n\n\\noindent\nResuming our proof of Corollary~\\ref{ev1}, we may first deduce from Proposition~\\ref{th1} that for all $w \\in \\mathscr{S}(\\mathbb{R}_{t,x,v}^{2n+1})$ satisfying\n\\begin{equation}\\label{rie1}\n\\textrm{supp } w(\\cdot,x,v) \\subset [-3,3], \\ (x,v) \\in \\mathbb{R}^{2n},\n\\end{equation}\nwe have\n\\begin{equation}\\label{rie2}\n\\|\\langle D_x \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}w\\|_{L^2}+\\|\\langle D_v \\rangle^{2\\sigma}w\\|_{L^2}  \\lesssim \\|Pw\\|_{L^2}+\\|w\\|_{L^2}.\n\\end{equation}\nThese estimates are maximal hypoelliptic estimates and we therefore get an upper bound for the transport term\n\\begin{multline}\\label{rie3}\n\\|(\\partial_t+ v\\cdot \\nabla_x)w\\|_{L^2} \\lesssim \\|P w \\|_{L^2}+\\|a(t,x,v)(-\\tilde{\\Delta}_v^\\sigma)w\\|_{L^2}\\\\ \\lesssim \\|P w \\|_{L^2}+\\|\\langle D_v\\rangle^{2\\sigma}w\\|_{L^2}\n\\lesssim \\|Pw\\|_{L^2}+\\|w\\|_{L^2},\n\\end{multline}\nsince $a \\in C_b^{\\infty}(\\mathbb{R}^{2n+1})$. Notice from (\\ref{rie2}) and (\\ref{rie3}) that for those functions $w$, \n\\begin{align}\\label{rie4}\n& \\ \\|\\langle v \\rangle^{-1}\\Lambda_{t,x}^{-\\frac{1}{2\\sigma +1}}D_t w\\|_{L^2}\\\\\n\\lesssim & \\ \\|\\langle v \\rangle^{-1}\\Lambda_{t,x}^{-\\frac{1}{2\\sigma +1}}(\\partial_t+ v\\cdot \\nabla_x) w\\|_{L^2}+ \\|\\langle v \\rangle^{-1}\\Lambda_{t,x}^{-\\frac{1}{2\\sigma +1}} v\\cdot \\nabla_x w\\|_{L^2} \\notag \\\\ \\notag\n\\lesssim  & \\ \\|(\\partial_t+ v\\cdot \\nabla_x) w \\|_{L^2} + \\|\\langle D_x\\rangle^{\\frac{2\\sigma}{2\\sigma+1}} w\\|_{L^2} \\lesssim  \n \\|Pw\\|_{L^2}+\\|w\\|_{L^2},\n\\end{align}\nwith $\\Lambda_{t,x}=(1+D_t^2+|D_x|^2)^{1\/2}$. It follows from (\\ref{rie2}) and (\\ref{rie4}) that \n\\begin{multline*}\n\\|\\Lambda_{t,x}^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-1}w\\|_{L^2}\n\\lesssim  \\|\\langle v \\rangle^{-1}\\Lambda_{t,x}^{-\\frac{1}{2\\sigma +1}}D_tw\\|_{L^2}+\\|\\langle v \\rangle^{-1}\\Lambda_{t,x}^{-\\frac{1}{2\\sigma +1}}D_xw\\|_{L^2}\\\\ \\lesssim\n \\|Pw\\|_{L^2}+\\|w\\|_{L^2}+\\|\\langle D_x \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}w\\|_{L^2} \\lesssim \\|Pw\\|_{L^2}+\\|w\\|_{L^2}.\n\\end{multline*}\nAnother use of (\\ref{rie2}) shows that \n\\begin{equation}\\label{rie5}\n\\|\\Lambda_{t,x}^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-1}w\\|_{L^2}+\\|\\langle D_v \\rangle^{2\\sigma}w\\|_{L^2}  \\lesssim \\|Pw\\|_{L^2}+\\|w\\|_{L^2}.\n\\end{equation}\nNotice furthermore that \n\\begin{multline*}\n\\|\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-1}w\\|_{L^2}  \\lesssim \\|\\Lambda_{t,x}^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-1}w\\|_{L^2}+\n\\|\\langle D_v \\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-1}w\\|_{L^2}  \n\\lesssim \\|\\Lambda_{t,x}^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-1}w\\|_{L^2} \\\\ +\n\\|[\\langle D_v \\rangle^{\\frac{2\\sigma}{2\\sigma +1}},\\langle v \\rangle^{-1}]w\\|_{L^2}+\\|\\langle v \\rangle^{-1}\\langle D_v \\rangle^{\\frac{2\\sigma}{2\\sigma +1}}w\\|_{L^2}\n\\lesssim \\|Pw\\|_{L^2}+\\|w\\|_{L^2},\n\\end{multline*}\nbecause \n$$\\|\\langle v \\rangle^{-1}\\langle D_v \\rangle^{\\frac{2\\sigma}{2\\sigma +1}}w\\|_{L^2} \\lesssim \\|\\langle D_v \\rangle^{\\frac{2\\sigma}{2\\sigma +1}}w\\|_{L^2} \\lesssim \\|\\langle D_v \\rangle^{2\\sigma}w\\|_{L^2}  \\lesssim \\|Pw\\|_{L^2}+\\|w\\|_{L^2}$$\nand that the operator $[\\langle D_v \\rangle^{\\frac{2\\sigma}{2\\sigma +1}},\\langle v \\rangle^{-1}]$ is bounded on $L^2$ since symbolic calculus shows that its symbol belongs to the class $S^0$.\nWe therefore obtain from (\\ref{rie5}) the estimate\n\\begin{equation}\\label{rie6}\n\\|\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-1}w\\|_{L^2}+\\|\\langle D_v \\rangle^{2\\sigma}w\\|_{L^2}  \\lesssim  \\|Pw\\|_{L^2}+\\|w\\|_{L^2},\n\\end{equation}\nwhich may be extended to any function $w \\in H^{2-\\varepsilon}(\\mathbb{R}^{2n+1})$ satisfying \\eqref{rie1} and \n$$v \\cdot \\nabla_x w \\in L^2(\\mathbb{R}^{2n+1}),$$ \nwith $\\varepsilon>0$ such that $\\mbox{\\rm max}(2\\sigma,1) \\leq 2-\\varepsilon$. This is possible since $0<\\sigma <1$.\nTake now a $C_0^{\\infty}(\\mathbb{R})$ function $\\psi$ such that \n$$\\textrm{supp }\\psi \\subset [-3,3] \\textrm{ and } \\psi(t) = 1 \\textrm{ if }|t| < 2.$$ \nRecalling that the function $u$ defined in (\\ref{sev20}) belongs to $H^{-N_0}(\\mathbb{R}^{2n+1})$ and that for any $\\varepsilon >0$ such that $\\mbox{\\rm max}(2\\sigma,1) \\leq 2-\\varepsilon$, the symbol $M_{\\delta}$ belongs to $S^{-N_0-2+\\varepsilon}$ for each fixed $0<\\delta \\leq 1$, we notice that\n$$M_{\\delta}(t,x,D_z)u \\in H^{2-\\varepsilon}(\\mathbb{R}^{2n+1}),$$\nfor any $0<\\delta \\leq 1$.\nIt follows that the estimate (\\ref{rie6}) may be applied to function\n\\begin{equation}\\label{ts2}\n\\psi(t)\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u,\n\\end{equation}\nwith an integer $k \\geq 1$. We obtain that \n\\begin{multline}\\label{rie7}\n\\|\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\psi(t)\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{L^2}+\\|\\langle D_v \\rangle^{2\\sigma}\\psi(t)\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2}  \\\\ \n\\lesssim  \\|P\\psi(t)\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2}+\\|\\psi(t)\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2}.\n\\end{multline}\nNotice that the choice of function $\\chi$ in (\\ref{sev6}) implies that the symbol \n$$(1-\\psi(t))\\langle v \\rangle^{-l}M_{\\delta}(t,x,\\zeta),$$ \nwith $l \\geq 0$, belongs to the class $S^{-N_0-2}$ uniformly with respect to the parameter $0<\\delta \\leq 1$. Recalling that $0<\\sigma<1$, it follows that \n\\begin{align}\\label{rie8}\n\\|\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}(1-\\psi(t))\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{L^2} \\\\ \\lesssim \\|(1-\\psi(t))\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{\\frac{2\\sigma}{2\\sigma +1}} \\lesssim \\|u\\|_{-N_0} \\notag\n\\end{align}\nand\n\\begin{align}\\label{rie9}\n\\|\\langle D_v \\rangle^{2\\sigma}(1-\\psi(t))\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2} \\\\ \\lesssim \\|(1-\\psi(t))\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{2\\sigma} \\lesssim \\|u\\|_{-N_0}, \\notag\n\\end{align}\nuniformly with respect to $0<\\delta \\leq 1$. Moreover, since the commutator \n$$[P,\\psi(t)]=\\psi'(t),$$ \nis bounded on $L^2$ and that \n\\begin{align*}\n& \\ \\|P\\psi(t)\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2} \\\\\n\\leq & \\ \\|\\psi(t)P\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2} + \\|[P,\\psi(t)]\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2} \\\\\n\\lesssim & \\ \\|P\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2}+ \\|\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2},\n\\end{align*}\nwe deduce from (\\ref{rie7}), (\\ref{rie8}), (\\ref{rie9}) and the triangular inequality that\n\\begin{multline}\\label{rie10}\n\\|\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{L^2}+\\|\\langle D_v \\rangle^{2\\sigma}\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2}  \\\\ \n\\lesssim  \\|P\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2}+\\|\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2}+ \\|u\\|_{-N_0},\n\\end{multline}\nuniformly with respect to $0<\\delta \\leq 1$. Let $0<\\varepsilon_0<1$ such that \n\\begin{equation}\\label{ts1} \n0 < \\varepsilon_0 < \\frac{2\\sigma}{2\\sigma+1} \\textrm{ and } (2\\sigma-1)_++\\varepsilon_0 <2\\sigma.\n\\end{equation}\nWe may use Lemma~\\ref{lop3} to obtain the estimate\n\\begin{align}\n\\label{rie11} & \\ \\|P\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2} \\\\ \\notag\n \\leq  & \\ \\|\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) Pu\\|_{L^2} +  \\|[P,\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z)]u\\|_{L^2}\\\\ \\notag\n \\lesssim  & \\   \\|\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) Pu\\|_{L^2}+ \\|\\langle D_v\\rangle^{(2\\sigma-1)_++\\varepsilon_0}\\langle v \\rangle^{-k} \nM_\\delta(t,x,D_z) u \\|_{L^2}  \\\\ \\notag\n& \\ \\quad + \\| \\langle v \\rangle^{-k+1} \nM_\\delta(t,x,D_z) u \\|_{\\varepsilon_0} +\\|u\\|_{-N_0},\n\\end{align}\nwhich holds uniformly with respect to $0<\\delta \\leq 1$. It follows from (\\ref{rie10}) and (\\ref{rie11}) that\n\\begin{align}\\label{rie12}\n& \\ \\|\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{L^2}+\\|\\langle D_v \\rangle^{2\\sigma}\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2}  \\\\ \n\\lesssim & \\ \\notag \\|\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) Pu\\|_{L^2}+ \\|\\langle D_v\\rangle^{(2\\sigma-1)_++\\varepsilon_0}\\langle v \\rangle^{-k} \nM_\\delta(t,x,D_z) u \\|_{L^2} \\\\\n& \\ \\quad \\notag + \\|\\langle D_z \\rangle^{\\varepsilon_0} \\langle v \\rangle^{-k+1} \nM_\\delta(t,x,D_z) u \\|_{L^2} \n+ \\|u\\|_{-N_0},\n\\end{align}\nuniformly with respect to $0<\\delta \\leq 1$. Since for any $\\varepsilon>0$, there exists $C_{\\varepsilon}>0$ such that \n\\begin{multline*}\n\\|\\langle D_v\\rangle^{(2\\sigma-1)_++\\varepsilon_0}\\langle v \\rangle^{-k} M_\\delta(t,x,D_z) u \\|_{L^2}\\\\ \\leq \\varepsilon \\|\\langle D_v \\rangle^{2\\sigma}\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2}\n+C_\\varepsilon \\|\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2},\n\\end{multline*}\nbecause $(2\\sigma-1)_++\\varepsilon_0 <2\\sigma$, and\n$$ \\|\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) u\\|_{L^2} \\lesssim \\|\\langle v \\rangle^{-k+1}  M_\\delta(t,x,D_z) u\\|_{L^2} \\lesssim \\|\\langle D_z \\rangle^{\\varepsilon_0}\\langle v \\rangle^{-k+1}  M_\\delta(t,x,D_z) u\\|_{L^2},$$\nbecause $\\varepsilon_0>0$, we deduce from (\\ref{rie12}) that \n\\begin{multline}\\label{rie13}\n \\|\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{L^2}  \\\\\n\\lesssim  \\|\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) Pu\\|_{L^2} + \\|\\langle D_z \\rangle^{\\varepsilon_0} \\langle v \\rangle^{-k+1} \nM_\\delta(t,x,D_z) u \\|_{L^2} \n+ \\|u\\|_{-N_0},\n\\end{multline}\nuniformly with respect to $0<\\delta \\leq 1$. We shall need the following instrumental lemma:\n\n\\bigskip\n\n\n\\begin{lemma}\\label{lop4} \nLet $s_1<s_2$, $N \\geq 0$ and an integer $l \\geq 2$. Then, there exists an integer $m \\geq 2$ such that\n$$\\forall \\varepsilon >0, \\exists C_{\\varepsilon}>0,  \\forall w \\in \\mathscr{S}(\\mathbb{R}^{2n+1}), \\ \\|\\langle  v \\rangle^{l} w\\|_{s_1}  \\leq \\varepsilon \\| w \\|_{s_2} + C_{\\varepsilon} \\|\\langle v \\rangle^m w\\|_{-N}.$$\n\\end{lemma}\n\n\\bigskip\n\n\\noindent\n\\textit{Proof of Lemma~\\ref{lop4}}. Let $m \\in \\mathbb{R}$. Since the symbol of the operator $\\langle D_z \\rangle^{m}$ belongs to the class $S(\\langle \\zeta \\rangle^{m},dz^2+\\langle \\zeta \\rangle^{-2}d\\zeta^2)$ and \n$\\langle v \\rangle^m \\in S(\\langle v \\rangle^{m},dz^2+\\langle \\zeta \\rangle^{-2}d\\zeta^2),$\nsymbolic calculus shows that the symbol of the operator \n$$\\langle D_{z}\\rangle^{-s_2} \\langle  v \\rangle^{l}\\langle D_{z}\\rangle^{2s_1 }\\langle  v \\rangle^{-l},$$ \nbelongs to the class $S(\\langle \\zeta \\rangle^{2s_1-s_2},dz^2+\\langle \\zeta \\rangle^{-2}d\\zeta^2)$.\nIt follows from the Cauchy-Schwarz inequality that for any $\\varepsilon>0$, there exists $C_{\\varepsilon}>0$ such that \n\\begin{align*}\n \\|\\langle  v \\rangle^{l} w\\|_{s_1}^2 = & \\ \\|\\langle D_{z}\\rangle^{s_1} \\langle  v \\rangle^{l} w \\|_{L^2}^2 =  (\\langle D_{z}\\rangle^{s_2} w,\\langle D_{z}\\rangle^{-s_2} \\langle  v \\rangle^{l}\\langle D_{z}\\rangle^{2s_1 }\\langle  v \\rangle^{l} w)_{L^2}\\\\\n= & \\  (\\langle D_{z}\\rangle^{s_2} w,\\langle D_{z}\\rangle^{-s_2} \\langle  v \\rangle^{l}\\langle D_{z}\\rangle^{2s_1 }\\langle  v \\rangle^{-l}\\langle  v \\rangle^{2l} w)_{L^2}\\\\\n\\leq & \\  \\|\\langle D_{z}\\rangle^{s_2} w\\|_{L^2}\\|\\langle D_{z}\\rangle^{-s_2} \\langle  v \\rangle^{l}\\langle D_{z}\\rangle^{2s_1 }\\langle  v \\rangle^{-l}\\langle  v \\rangle^{2l} w\\|_{L^2}\\\\\n\\lesssim & \\  \\|w\\|_{s_2}\\|\\langle  v \\rangle^{2l} w\\|_{s_1-(s_2-s_1)}\\\\\n\\lesssim & \\  \\varepsilon \\|w\\|_{s_2}^2+C_{\\varepsilon}\\|\\langle  v \\rangle^{2l} w\\|_{s_1-(s_2-s_1)}^2.\n\\end{align*}\nNotice that by assumption $s_1-(s_2-s_1)<s_1$. Lemma~\\ref{lop4} then follows by $k_0$ iterations of this procedure with $s_1-k_0(s_2-s_1)<-N$.~$\\Box$\n\n\n\\bigskip\n\n\n\\noindent\nRecalling (\\ref{ts1}), we may apply Lemma \\ref{lop4} with $l=2$, $s_1=\\varepsilon_0$, $s_2=2\\sigma\/(2\\sigma+1)$ and $N=N_0+s+1$. There therefore exists an integer $m \\geq 2$ such that for all $\\varepsilon >0$,\n\\begin{equation}\\label{ts4}\n \\exists C_{\\varepsilon}>0,  \\forall w \\in \\mathscr{S}(\\mathbb{R}^{2n+1}), \\ \\|\\langle  v \\rangle^{2} w\\|_{\\varepsilon_0}  \\leq \\varepsilon \\| w \\|_{\\frac{2\\sigma}{2\\sigma+1}} + C_{\\varepsilon} \\|\\langle v \\rangle^m w\\|_{-N_0-s-1}.\n\\end{equation}\nWe then choose the integer $k=m-1 \\geq 1$ in (\\ref{ts2}). Recalling that the symbol $M_{\\delta}$ belongs to the class $S^{s+1}$ uniformly with respect to $0<\\delta \\leq 1$, we obtain that \n$$ \\|\\langle v \\rangle^m w\\|_{-N_0-s-1}=\\|M_\\delta(t,x,D_z) u\\|_{-N_0-s-1} \\lesssim \\|u\\|_{-N_0},$$\n uniformly with respect to $0<\\delta \\leq 1$ with $w=\\langle v \\rangle^{-(k+1)} M_\\delta(t,x,D_z) u.$ \nThe estimate (\\ref{ts4}) extends by density. It follows that \n\\begin{multline}\\label{ts5}\n\\|\\langle D_z \\rangle^{\\varepsilon_0} \\langle v \\rangle^{-k+1} M_\\delta(t,x,D_z) u \\|_{L^2}=\\|\\langle v \\rangle^{2}w \\|_{\\varepsilon_0}  \\leq \\varepsilon \\| w \\|_{\\frac{2\\sigma}{2\\sigma+1}}\\\\ + C_{\\varepsilon} \\|\\langle v \\rangle^m w\\|_{-N_0-s-1}\n\\lesssim  \\varepsilon \\| \\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-(k+1)} M_\\delta(t,x,D_z) u \\|_{L^2} + C_{\\varepsilon}  \\|u\\|_{-N_0}.\n\\end{multline} \nWe deduce from (\\ref{rie13}) and (\\ref{ts5}) that \n\\begin{equation}\\label{rie14}\n \\|\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{L^2}  \\\\\n\\lesssim  \\|\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) Pu\\|_{L^2} + \\|u\\|_{-N_0},\n\\end{equation}\nuniformly with respect to $0<\\delta \\leq 1$.  Notice from (\\ref{ts8}) and (\\ref{sev6}) that the symbol of the operator  \n$$\\langle v \\rangle^{-k}M_{\\delta}(t,x,D_z)(1-\\varphi(t,x))\\langle v \\rangle,$$ \nwith $k \\geq 1$, belongs to the class $S^{-N_0-2}$ uniformly with respect to $0<\\delta \\leq 1$. Recalling from (\\ref{tt1}) that $M_{\\delta} \\in \\tilde{S}^s$ uniformly with respect to $0<\\delta \\leq 1$, we obtain that \n\\begin{align}\\label{ts10}\n& \\ \\|\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z) Pu\\|_{L^2} \\\\ \\notag\n\\leq & \\  \\|\\langle v \\rangle^{-k}  M_\\delta(t,x,D_z)\\varphi Pu\\|_{L^2}+\\|\\langle v \\rangle^{-k} M_\\delta(t,x,D_z)(1-\\varphi)Pu\\|_{L^2} \n\\\\ \\notag\n\\leq & \\  \\|M_\\delta(t,x,D_z)\\varphi Pu\\|_{L^2}+\\|\\langle v \\rangle^{-k} M_\\delta(t,x,D_z)(1-\\varphi)\\langle v \\rangle\\langle v \\rangle^{-1}Pu\\|_{L^2}\n\\\\ \\notag\n\\lesssim & \\ \\|\\varphi Pu\\|_{s} +\\|\\langle v \\rangle^{-1}Pu\\|_{-N_0-2}.\n\\end{align}\nSince obviously $\\|\\langle v \\rangle^{-1}Pu\\|_{-N_0-2} \\lesssim \\|u\\|_{-N_0}$, we deduce from (\\ref{rie14}) and (\\ref{ts10}) that for all $0<\\delta \\leq 1$,\n\\begin{equation}\\label{tt2}\n \\|\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{L^2}  \n\\lesssim   \\|\\varphi Pu\\|_{s} + \\|u\\|_{-N_0} <+\\infty,\n\\end{equation}\nsince by assumption $u \\in H^{-N_0}(\\mathbb{R}^{2n+1})$ and $\\varphi Pu \\in H^{s}(\\mathbb{R}^{2n+1})$. Let $\\phi$ be $C_0^\\infty(\\mathbb{R}_{t,x}^{n+1})$ function such that $\\textrm{supp }\\phi \\subset \\{|(t,x)|< 1\\}$. Since the commutator $[\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}},\\phi(t,x)]$ is obviously bounded on $L^2$, we deduce from (\\ref{tt2}) that for all $0<\\delta \\leq 1$, \n\\begin{align}\\label{tt3}\n& \\   \\|\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\phi(t,x)\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{L^2} \\\\ \\notag\n \\leq & \\  \\|[\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}},\\phi(t,x)]\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{L^2} \\\\ \\notag\n & \\ \\quad \\quad  +\\|\\phi(t,x)\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{L^2}\\\\ \\notag\n \\lesssim & \\  \\|\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{L^2}\n +\\|\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{L^2} \\\\\\notag\n \\lesssim & \\ \\|\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\langle v \\rangle^{-(k+1)}  M_\\delta(t,x,D_z) u\\|_{L^2} \\lesssim   \\|\\varphi Pu\\|_{s} + \\|u\\|_{-N_0} <+\\infty.\n\\end{align}\nNotice from (\\ref{sev6}) that \n$$\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\frac{\\phi(t,x)}{\\langle v \\rangle^{k+1}} M_\\delta(t,x,D_z) u=\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\frac{\\phi(t,x)}{\\langle v \\rangle^{k+1}}\\frac{\\langle D_z \\rangle^{s} }{\n(1+ \\delta \\langle D_z \\rangle)^{N_0+s+2}}  u$$\nand that \n$$\\langle D_{z}\\rangle^{\\frac{2\\sigma}{2\\sigma +1}}\\frac{\\phi(t,x)}{\\langle v \\rangle^{k+1}} M_\\delta(t,x,D_z) u \\longrightarrow\n\\langle D_z \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}\\frac{\\phi(t,x)}{\\langle v \\rangle^{k+1}} \\langle D_z \\rangle^{s}u,$$\nin $\\mathscr{S}'(\\mathbb{R}^{2n+1})$ when $\\delta \\to 0$. Because of the weak compactness of the unit ball in $L^2$, it follows from (\\ref{tt3}) that  \n\\begin{equation}\\label{aay10}\n\\langle D_z \\rangle^{\\frac{2\\sigma}{2\\sigma+1}}\\frac{\\phi(t,x)}{\\langle v \\rangle^{k+1}} \\langle D_z \\rangle^{s}u \\in L^2(\\mathbb{R}^{2n+1}),\n\\end{equation}\nfor any $C_0^\\infty(\\mathbb{R}_{t,x}^{n+1})$ function $\\phi$ such that $\\textrm{supp }\\phi \\subset U_0=\\{|(t,x)|< 1\\}$.\n\n\nTo complete the proof of Corollary~\\ref{ev1}, we notice that the following two assertions \n\\begin{equation}\\label{aay1}\n\\forall \\phi \\in C_0^\\infty(U_0), \\  \\frac{\\phi(t,x)}{\\langle v \\rangle^k}  u \\in H^{r+m}(\\mathbb{R}^{2n+1}_{t,x,v})\n\\end{equation}\nand\n\\begin{equation}\\label{aay2}\n\\forall \\phi \\in C_0^\\infty(U_0), \\ \\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\langle D_z \\rangle^{m}  u \\in H^r(\\mathbb{R}_{t,x,v}^{2n+1})\n\\end{equation}\nare equivalent if $u \\in H_{-N_0}(\\mathbb{R}^{2n+1}_{t,x,v})$ and $m, r \\in \\mathbb{R}$. Assume first that (\\ref{aay1}) holds.  Let $\\phi$ be a $C_0^{\\infty}(U_0)$ function and $\\tilde{\\phi}$ another $C_0^{\\infty}(U_0)$ function such that $\\tilde{\\phi}=1$ on a neighborhood of $\\textrm{supp }\\phi$. Symbolic calculus allows to write\n\\begin{align}\\label{aay3}\n & \\ \\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\langle D_z \\rangle^{m}-\\langle D_z \\rangle^{m} \\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\\\ \\notag\n= & \\ -\\sum_{0 < |\\alpha| < m +N_0+[r]+1} \\frac{1}{\\alpha!}  \\textrm{Op}\\Big(D_z^{\\alpha}\\left(\\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\right) \\partial_{\\zeta}^{\\alpha} \\big(\\langle \\zeta \\rangle^{m}\\big)\\Big) +R \\\\ \\notag\n= & \\ -\\sum_{0 < |\\alpha| < m +N_0+[r]+1} \\frac{1}{\\alpha!}  \\textrm{Op}\\Big(D_z^{\\alpha}\\left(\\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\right) \\partial_{\\zeta}^{\\alpha} \\big(\\langle \\zeta \\rangle^{m}\\big)\\Big)\\tilde{\\phi} +\\tilde{R}\n\\end{align}\nwith $R, \\tilde{R} \\in \\textrm{Op}(S^{-N_0-[r]-1})$ and therefore \n\\begin{equation}\\label{aay4}\n\\|\\tilde{R} u\\|_{r} \\lesssim \\|u\\|_{-N_0-[r]-1+r} \\lesssim \\|u\\|_{-N_0} <+\\infty. \n\\end{equation}\nIn order to prove that (\\ref{aay2}) holds, it is sufficient to check that \n\\begin{multline*}\n\\Big\\| \\textrm{Op}\\Big(D_z^{\\alpha}\\left(\\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\right) \\partial_{\\zeta}^{\\alpha} \\big(\\langle \\zeta \\rangle^{m}\\big)\\Big)\\tilde{\\phi}u\\Big\\|_r \\\\\n=\\Big\\| \\textrm{Op}\\Big(D_z^{\\alpha}\\left(\\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\right) \\partial_{\\zeta}^{\\alpha} \\big(\\langle \\zeta \\rangle^{m}\\big)\\Big)\\langle v \\rangle^{k}\\langle D_z \\rangle^{-m}\\langle D_z \\rangle^{m}\\frac{\\tilde{\\phi}(t,x)}{\\langle v \\rangle^k}u\\Big\\|_r\n<+\\infty,\n\\end{multline*}\nwhen $0 < |\\alpha| < m +N_0+[r]+1$. This is actually the case since symbolic calculus shows that the symbol of the operator \n$$\\textrm{Op}\\Big(D_z^{\\alpha}\\left(\\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\right) \\partial_{\\zeta}^{\\alpha} \\big(\\langle \\zeta \\rangle^{m}\\big)\\Big)\\langle v \\rangle^{k}\\langle D_z \\rangle^{-m},$$\nbelongs to $S^0$ when $0 < |\\alpha| < m +N_0+[r]+1$ and that by (\\ref{aay1}),\n$$\\langle D_z \\rangle^{m}\\frac{\\tilde{\\phi}(t,x)}{\\langle v \\rangle^k}u \\in H^r(\\mathbb{R}_{t,x,v}^{2n+1}).$$\nIt follows that (\\ref{aay1}) implies (\\ref{aay2}).\nConversely, we may write for any function $\\phi$ in $C_0^{\\infty}(U_0)$, \n\\begin{align*}\n & \\ \\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\langle D_z \\rangle^{m}-\\langle D_z \\rangle^{m} \\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\\\ \\notag\n= & \\ -\\sum_{0 < |\\alpha| < m +N_0+[r]+1} \\frac{1}{\\alpha!}  \\textrm{Op}\\Big(D_z^{\\alpha}\\left(\\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\right) \\partial_{\\zeta}^{\\alpha} \\big(\\langle \\zeta \\rangle^{m}\\big)\\Big)\\langle D_z \\rangle^{-m}\\langle v \\rangle^{k} \\langle v \\rangle^{-k} \\langle D_z \\rangle^{m}+R \\\\ \\notag\n= & \\ -\\sum_{0 < |\\alpha| < m +N_0+[r]+1} \\frac{1}{\\alpha!}  \\textrm{Op}\\Big(D_z^{\\alpha}\\left(\\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\right) \\partial_{\\zeta}^{\\alpha} \\big(\\langle \\zeta \\rangle^{m}\\big)\\Big)\\langle D_z \\rangle^{-m}\\langle v \\rangle^{k}\\frac{\\tilde{\\phi}(t,x)}{\\langle v \\rangle^{k}} \\langle D_z \\rangle^{m} +\\tilde{R}\n\\end{align*}\nwith $R, \\tilde{R} \\in \\textrm{Op}(S^{-N_0-[r]-1})$ if  $\\tilde{\\phi}$ is a $C_0^{\\infty}(U_0)$ function satisfying $\\tilde{\\phi}=1$ on a neighborhood of $\\textrm{supp }\\phi$. This implies that \n$$\\|\\tilde{R} u\\|_{r} \\lesssim \\|u\\|_{-N_0-[r]-1+r} \\lesssim \\|u\\|_{-N_0} <+\\infty. $$\nIn order to prove that (\\ref{aay1}) holds, it is sufficient to check that \n$$\\Big\\|\\textrm{Op}\\Big(D_z^{\\alpha}\\left(\\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\right) \\partial_{\\zeta}^{\\alpha} \\big(\\langle \\zeta \\rangle^{m}\\big)\\Big)\\langle D_z \\rangle^{-m}\\langle v \\rangle^{k}\\frac{\\tilde{\\phi}(t,x)}{\\langle v \\rangle^{k}} \\langle D_z \\rangle^{m}u\\Big\\|_r\n<+\\infty,$$\nwhen $0 < |\\alpha| < m +N_0+[r]+1$. This is actually the case since symbolic calculus shows that the symbol of the operator \n$$\\textrm{Op}\\Big(D_z^{\\alpha}\\left(\\frac{ \\phi(t,x) }{\\langle v \\rangle^k}\\right) \\partial_{\\zeta}^{\\alpha} \\big(\\langle \\zeta \\rangle^{m}\\big)\\Big)\\langle D_z \\rangle^{-m}\\langle v \\rangle^{k},$$\nbelongs to $S^0$ when $0 < |\\alpha| < m +N_0+[r]+1$ and that by (\\ref{aay2}),\n$$\\frac{\\tilde{\\phi}(t,x)}{\\langle v \\rangle^k}\\langle D_z \\rangle^{m}u \\in H^r(\\mathbb{R}_{t,x,v}^{2n+1}).$$\nIt follows that (\\ref{aay2}) implies (\\ref{aay1}).\n\n\nBy finally coming back to (\\ref{aay10}), we deduce that the function defined in (\\ref{sev20}) satisfies \n$$\\frac{\\phi(t,x)}{\\langle v \\rangle^{k+1}} u \\in H^{s+\\frac{2\\sigma}{2\\sigma+1}}(\\mathbb{R}^{2n+1}),$$\nfor any $C_0^\\infty(\\mathbb{R}_{t,x}^{n+1})$ function $\\phi$ such that $\\textrm{supp }\\phi \\subset U_0=\\{|(t,x)|< 1\\}$, that is \n$$\\frac{u}{\\langle  v \\rangle^{k+1}} \\in H^{s+\\frac{2\\sigma}{2\\sigma+1}}_{\\textrm{loc}, (t_0,x_0)}(\\mathbb{R}^{2n+1}_{t,x,v}),$$\nwith $(t_0,x_0)=(0,0)$. This ends the proof of Corollary~\\ref{ev1}.~$\\Box$\n\n\n\n\n\n\\section{Appendix}\n\n\\subsection{The Boltzmann equation}\\label{kkboltz}\nThe Boltzmann equation \\cite{17,19} describes the behavior of a dilute gas when the only interactions taken into account are binary collisions. It reads as the evolution equation \n\\begin{equation}\\label{xiaoe1}\n\\begin{cases}\n\\partial_tf+v\\cdot\\nabla_{x}f=Q(f,f)\\\\\nf|_{t=0}=f_0,\n\\end{cases}\n\\end{equation}\nfor the density distribution of the particles $f=f(t,x,v) \\geq 0$  at time $t$, having position $x \\in \\mathbb{R}^d$ and velocity $v \\in \\mathbb{R}^d$.\nThe term appearing in the right-hand-side of this equation $Q(f,f)$ is the so-called quadratic Boltzmann collision operator associated to the Boltzmann bilinear operator \n\\begin{equation}\\label{xiaoeq1}\nQ(g, f)=\\int_{\\mathbb{R}^d}\\int_{S^{d-1}}B(v-v_{*},\\sigma) \\big(g'_* f'-g_*f\\big)d\\sigma dv_*,\n\\end{equation}\nacting on $L^1(\\mathbb{R}^d) \\times L^1(\\mathbb{R}^d)$, $d \\geq 2$, \nwhere $f'_*=f(t,x,v'_*)$, $f'=f(t,x,v')$, $f_*=f(t,x,v_*)$, $f=f(t,x,v)$, \n$$v'=\\frac{v+v_*}{2}+\\frac{|v-v_*|}{2}\\sigma,\\quad   v_*'=\\frac{v+v_*}{2}-\\frac{|v-v_*|}{2}\\sigma,$$\nfor $\\sigma$ belonging to the unit sphere $S^{d-1}$. Those relations between pre and post collisional velocities follow from the conservations of momentum and kinetic energy\n$$\\quad v+v_{\\ast}=v'+v_{\\ast}', \\quad |v|^2+|v_{\\ast}|^2=|v'|^2+|v_{\\ast}'|^2,$$\nin the binary collisions.\nThe Boltzmann operator has the fundamental properties of conserving mass, momentum and energy\n$$\\int_{\\mathbb{R}^d}Q(f,f)\\phi(v)dv=0, \\quad \\phi(v)=1,v,|v|^2,$$\nand to satisfy to the so-called Boltzmann's H theorem\n$$-\\frac{d}{dt} \\int_{\\mathbb{R}^d}f \\log f dv=-\\int_{\\mathbb{R}^d}Q(f,f)\\log f dv \\geq 0.$$\nThe functional $-\\int f \\log f$ is the entropy of the solution and the Boltzmann's H theorem implies that at some point $x \\in \\mathbb{R}^d$, any equilibrium distribution function, i.e., any function maximizing the entropy, has the form of a locally Maxwellian distribution\n$$M(\\rho,u,T)(v)=\\frac{\\rho}{(2\\pi T)^{d\/2}}e^{-\\frac{|u-v|^2}{2T}},$$\nwhere $\\rho, u, T$ are respectively the density, mean velocity and temperature of the gas at the point $x$, defined by\n$$\\rho=\\int_{\\mathbb{R}^d}f(v)dv, \\quad u=\\frac{1}{\\rho}\\int_{\\mathbb{R}^d}vf(v)dv, \\quad T=\\frac{1}{N\\rho}\\int_{\\mathbb{R}^d}|u-v|^2f(v)dv.$$ \nFor further details on the physical background and derivation of the Boltzmann equation, we refer the reader to the extensive expositions \\cite{17,19,villani2}. \n\n\nFor monatomic gas, the non-negative cross section $B(z,\\sigma)$ defining the collision operator (\\ref{xiaoeq1}) depends only on $|z|$ and the scalar product \n$\\frac{z}{|z|}\\cdot \\sigma.$\nFurthermore, the cross section is assumed to be supported in the set where $\\frac{z}{|z|} \\cdot \\sigma \\geq 0$. This condition is not a restriction when dealing with the quadratic Boltzmann operator $Q(f,f)$. As noticed in~\\cite{alexandre0}, one may indeed reduce to this case after a symmetrization of the cross section since the term $f'f_*'$ appearing in the Boltzmann operator is invariant under the mapping $\\sigma \\rightarrow -\\sigma$.  \n\n\nMore specifically, we consider cross sections of  the type\n\\begin{equation}\\label{xiaoeq1.01}\nB(v-v_*,\\sigma)=\\Phi(|v-v_*|)b\\Big(\\frac{v-v_*}{|v-v_*|} \\cdot \\sigma\\Big), \\quad \\cos \\theta=\\frac{v-v_*}{|v-v_*|} \\cdot \\sigma, \\quad 0 \\leq \\theta \\leq \\frac{\\pi}{2},\n\\end{equation}\nwith a kinetic factor\n\\begin{equation}\\label{xiaosa0}\n\\Phi(|v-v_*|)=|v-v_*|^{\\gamma}, \\quad  \\gamma \\in ]-d,+\\infty[,\n\\end{equation}\nand a factor related to the collision angle with a singularity \n\\begin{equation}\\label{xiaosa1}\n(\\sin \\theta)^{d-2} b(\\cos \\theta) \\sim K \\theta^{-1-2s},\n\\end{equation} \nwhen $\\theta \\to 0$, $\\theta>0$, for some constants $K>0$ and $0 < s <1$. Notice that this singularity is not integrable\n$$\\int_0^{\\frac{\\pi}{2}}(\\sin \\theta)^{d-2}b(\\cos \\theta)d\\theta=+\\infty.$$\nThis non-integrability property plays a major r\\^ole regarding the qualitative behavior of the solutions of the Boltzmann equation. Indeed, as first observed by L.~Desvillettes for the Kac equation~\\cite{D95}, grazing collisions that account for the non-integrability of the angular factor near $\\theta \\sim 0^+$, \ndo induce smoothing effects for the solutions of the non-cutoff Kac equation, or more generally for the solutions of the non-cutoff Boltzmann equation; whereas those solutions are at most as regular as the initial data, see e.g. \\cite{36}, when the collision cross section is assumed to be integrable, or after removing the singularity by using a cutoff function (Grad's angular cutoff assumption).\n\n\n\n\nBeing concerned with a close to equilibrium framework, the setting of the problem can be formulated as follows. First of all, without loss of generality, we consider the fluctuation around\n$$\\mu(v)=(2\\pi)^{-\\frac{d}{2}}e^{-\\frac{|v|^2}{2}},$$\nthe unique normalized equilibrium with mass 1, momentum 0 and temperature 1, by setting \n$$f=\\mu+\\sqrt{\\mu}g.$$ \nSince $Q(\\mu,\\mu)=0$ by the conservation of the kinetic energy\n$$|v|^2+|v_{\\ast}|^2=|v'|^2+|v_{\\ast}'|^2,$$\nthe Boltzmann collision operator can be split into three terms\n$$Q(\\mu+\\sqrt{\\mu}g,\\mu+\\sqrt{\\mu}g)=Q(\\mu,\\sqrt{\\mu}g)+Q(\\sqrt{\\mu}g,\\mu)+Q(\\sqrt{\\mu}g,\\sqrt{\\mu}g),$$\nwhose linearized part is \n$$Q(\\mu,\\sqrt{\\mu}g)+Q(\\sqrt{\\mu}g,\\mu).$$\nDefine\n$$\\mathcal{L}g=\\mathcal{L}_1g+\\mathcal{L}_2g,$$\nwith \n$$\\mathcal{L}_1g=-\\mu^{-1\/2}Q(\\mu,\\mu^{1\/2}g), \\quad \\mathcal{L}_2g=-\\mu^{-1\/2}Q(\\mu^{1\/2}g,\\mu),$$\nthe original Boltzmann equation is reduced to the Cauchy problem for the fluctuation\n$$\\begin{cases}\n\\partial_tg+v\\cdot\\nabla_{x}g+\\mathcal{L}g=\\mu^{-1\/2}Q(\\sqrt{\\mu}g,\\sqrt{\\mu}g)\\\\\ng|_{t=0}=g_0.\n\\end{cases}$$\nThis linearized Boltzmann operator $\\mathcal{L}$ is known (see e.g. \\cite{17}) to be an unbounded symmetric operator on $L^2$ (acting in the velocity variable) such that its Dirichlet form satisfies to\n$$\\mathbf{D}g=(\\mathcal{L}g,g)_{L^2} \\geq 0,$$\nand that \n$$\\mathbf{D}g=0 \\Leftrightarrow g=\\mathbf{P}g,$$ \nwhere\n$$\\mathbf{P}g=(a+b \\cdot v+c|v|^2)\\mu^{1\/2},$$ \nwith $a,c \\in \\mathbb{R}$, $b \\in \\mathbb{R}^d$, being the orthogonal projection onto the space of the so-called collisional invariants\n$$\\textrm{Span}\\big\\{\\mu^{1\/2},v_1 \\mu^{1\/2},...,v_d\\mu^{1\/2},|v|^2\\mu^{1\/2}\\big\\}.$$  \nFurthermore, recent works in particular those by P.L.~Lions \\cite{lions}, R.~Alexandre, L.~Desvillettes, C.~Villani, B.~Wennberg \\cite{alexandre0}, C.~Mouhot~\\cite{44}, C.~Mouhot, R.M.~Strain \\cite{strain},\nR. Alexandre, Y. Morimoto, C.-J. Xu, S. Ukai, T. Yang \\cite{amuxy5-1}\n or P.T.~Gressman, R.M. Strain \\cite{gress1,gress2} have highlighted that the non-cutoff Boltzmann operator enjoys remarkable coercive properties. Indeed, the results proved in~\\cite{44,strain} show that the linearized Boltzmann operator enjoys the following coercive estimate \n\\begin{equation}\\label{xiaosa2}\n(\\mathcal{L}g,g)_{L^2(\\mathbb{R}^d)} \\gtrsim \\|\\langle v \\rangle^{s+\\frac{\\gamma}{2}}(g-\\mathbf{P}g)\\|_{L^2(\\mathbb{R}^d)}^2+\\|g-\\mathbf{P}g\\|_{H^s_{\\textrm{loc}}(\\mathbb{R}^d)}^2,\n\\end{equation}\nwith $\\langle v \\rangle=(1+|v|^2)^{1\/2}$. This coercive estimate was improved in \\cite{amuxy5-1} by showing that the latter term $\\|g-\\mathbf{P}g\\|_{H^s_{\\textrm{loc}}(\\mathbb{R}^d)}^2$ in (\\ref{xiaosa2}) may be substituted by the global improvement \n$$\\|\\langle v \\rangle^{\\frac{\\gamma}{2}}(g-\\mathbf{P}g)\\|_{H^s(\\mathbb{R}^d)}^2.$$\nSee~\\cite{amuxy5-1,gress1,gress2} for optimal coercive estimates in appropriate weighted anisotropic Sobolev spaces. Regarding those coercive estimates,\nthe studies \\cite{alexandre0,lions} have decisively made clear that the smoothing\neffect of the non-cutoff Boltzmann equation is produced by the term\n$$\\int_{\\mathbb{R}^{2d}}\\int_{S^{d-1}}B(v-v_{*},\\sigma)g'_*(f- f')^2d\\sigma dvdv_*,$$\nthanks to the celebrated cancellation lemma, see~\\cite{alexandre0} (Lemma~1 and Proposition~2). The discovery of these special features of the non-cutoff Boltzmann operator have led those authors to conjecture that this collision operator behaves and induces smoothing effects as a fractional Laplacian $-(-\\Delta)^s$. See \\cite{alexandre0}, p.~331. This conjecture accounts for the choice of the operator (\\ref{yo2}) for the linearized non-cutoff Boltzmann equation. Notice that this operator is only a simplified model which does not account for the actual anisotropic features of the linearized non-cutoff Boltzmann equation and that current investigations are led to determine the exact microlocal structure of this operator. \nSee the first works in this direction by R.~Alexandre~\\cite{a1,a3,alexandre1}. \n\n\n\n\n\n\n\\subsection{Wick calculus}\\label{appendix}\n\n\n\nThe purpose of this appendix is to recall few facts and basic properties of the Wick quantization with parameter that are used in this paper. We refer the reader to the works of N.~Lerner \\cite{lerner}, \\cite{cubo} and \\cite{birkhauser} for thorough and extensive presentations of this quantization and some of its applications. \n\n\nThe main property of the Wick quantization is its property of positivity, i.e., that non-negative Hamiltonians define non-negative operators\n$$a \\geq 0 \\Rightarrow a^{\\textrm{Wick}(\\lambda)} \\geq 0.$$\nWe recall that this is not the case for the Weyl quantization nor the standard\nquantization and refer to \\cite{lerner} for an example of non-negative Hamiltonian defining an operator which is not non-negative. Before defining properly the Wick quantization, we first need to recall the definition of the wave-packets transform with parameter $\\lambda>0$ of a function $u \\in \\mathscr{S}(\\mathbb{R}^n)$, \n$$W_{\\lambda}u(y,\\eta)=(u,\\varphi_{y,\\eta}^{\\lambda})_{L^2(\\mathbb{R}^n)}, \\ (y,\\eta) \\in \\mathbb{R}^{2n}.$$\nwhere \n$$\\varphi_{y,\\eta}^{\\lambda}(x)=(2\\lambda)^{n\/4}e^{- \\pi \\lambda |x-y|^2}e^{2i \\pi (x-y) \\cdot \\eta}, \\ x \\in \\mathbb{R}^n,$$\nwith $|\\cdot|$ standing for the Euclidean norm and $x \\cdot y$ the standard dot product on $\\mathbb{R}^n$. With this definition, one can check (Lemma 1.3 in \\cite{cubo}) that \nthe mapping $u \\mapsto W_{\\lambda}u$ is continuous from $\\mathscr{S}(\\mathbb{R}^n)$ to $\\mathscr{S}(\\mathbb{R}^{2n})$, isometric from $L^{2}(\\mathbb{R}^n)$ to $L^2(\\mathbb{R}^{2n})$ and that we have the\nreconstruction formula\n\\begin{equation}\\label{lay0.1}\n\\forall u \\in \\mathscr{S}(\\mathbb{R}^n), \\forall x \\in \\mathbb{R}^n, \\ u(x)=\\int_{\\mathbb{R}^{2n}}{W_{\\lambda}u(y,\\eta)\\varphi_{y,\\eta}^{\\lambda}(x)dyd\\eta}.\n\\end{equation}\nBy denoting by $\\Sigma_Y^{\\lambda}$ the operator defined by the Weyl quantization of the symbol \n$$p_Y(X)=2^n e^{-2\\pi\\Gamma_{\\lambda}(X-Y)}, \\ X=(x,\\xi) \\in \\mathbb{R}^{2n}, \\ Y=(y,\\eta) \\in \\mathbb{R}^{2n},$$\nwith \n$$\\Gamma_{\\lambda}(X)=\\lambda |x|^2+\\frac{|\\xi|^2}{\\lambda}, \\ X=(x,\\xi) \\in \\mathbb{R}^{2n},$$\nwhich is a rank-one orthogonal projection,\n$$\\big{(}\\Sigma_Y^{\\lambda} u\\big{)}(x)=W_{\\lambda}u(Y)\\varphi_Y^{\\lambda}(x)=(u,\\varphi_Y^{\\lambda})_{L^2(\\mathbb{R}^n)}\\varphi_Y^{\\lambda}(x),$$\nwe define the Wick quantization with parameter $\\lambda>0$ of any $L^{\\infty}(\\mathbb{R}^{2n})$  symbol $a$ as\n\\begin{equation}\\label{lay0.2}\na^{\\textrm{Wick}(\\lambda)}=\\int_{\\mathbb{R}^{2n}}{a(Y)\\Sigma_Y^{\\lambda} dY}.\n\\end{equation}\nMore generally, one can extend this definition when the symbol $a$ belongs to $\\mathscr{S}'(\\mathbb{R}^{2n})$ by defining the operator $a^{\\textrm{Wick}(\\lambda)}$ for any $u$ and $v$ in $\\mathscr{S}(\\mathbb{R}^{n})$ by\n$$\\langle a^{\\textrm{Wick}(\\lambda)}u,\\overline{v}\\rangle_{\\mathscr{S}'(\\mathbb{R}^{n}), \\mathscr{S}(\\mathbb{R}^{n})}=\\langle a(Y),(\\Sigma_Y^{\\lambda}u,v)_{L^2(\\mathbb{R}^n)}\\rangle_{\\mathscr{S}'(\\mathbb{R}^{2n}),  \\mathscr{S}(\\mathbb{R}^{2n})},$$\nwhere $\\langle \\textrm{\\textperiodcentered},\\textrm{\\textperiodcentered}\\rangle_{\\mathscr{S}'(\\mathbb{R}^n),\\mathscr{S}(\\mathbb{R}^n)}$ denotes the duality bracket between the\nspaces $\\mathscr{S}'(\\mathbb{R}^n)$ and $\\mathscr{S}(\\mathbb{R}^n)$. \nNotice from Proposition~3.1 in~\\cite{cubo} that \n\\begin{equation}\\label{lay10}\na^{\\textrm{Wick}(\\lambda)}=W_{\\lambda}^*a^{\\mu}W_{\\lambda}, \\ 1^{\\textrm{Wick}(\\lambda)}=\\textrm{Id}_{L^2(\\mathbb{R}^n)},\n\\end{equation}\nwhere $a^{\\mu}$ stands for the multiplication operator by $a$ on $L^2(\\mathbb{R}^{2n})$.\nThe Wick quantization with parameter $\\lambda>0$ is a positive quantization\n\\begin{equation}\\label{lay0.5}\na \\geq 0 \\Rightarrow a^{\\textrm{Wick}(\\lambda)} \\geq 0. \n\\end{equation}\nIn particular, real Hamiltonians get quantized in this quantization by formally self-adjoint operators and one has (Proposition 3.1 in \\cite{cubo}) that $L^{\\infty}(\\mathbb{R}^{2n})$ symbols define bounded operators on $L^2(\\mathbb{R}^n)$ such that \n\\begin{equation}\\label{lay0}\n\\|a^{\\textrm{Wick}(\\lambda)}\\|_{\\mathcal{L}(L^2(\\mathbb{R}^n))} \\leq \\|a\\|_{L^{\\infty}(\\mathbb{R}^{2n})}.\n\\end{equation}\nAccording to Proposition~3.2 in~\\cite{cubo} (see (51) in~\\cite{cubo}), the Wick and Weyl quantizations of a symbol $a$ are linked by the following identities\n\\begin{equation}\\label{lay1bis}\na^{\\textrm{Wick}(\\lambda)}=\\tilde{a}^w,\n\\end{equation}\nwith\n\\begin{equation}\\label{lay2bis}\n\\tilde{a}(X)=\\int_{\\mathbb{R}^{2n}}{a(X+Y)e^{-2\\pi \\Gamma_{\\lambda}(Y)}2^ndY}, \\ X \\in \\mathbb{R}^{2n},\n\\end{equation}\nand\n\\begin{equation}\\label{lay1}\na^{\\textrm{Wick}(\\lambda)}=a^w+r_{\\lambda}(a)^w,\n\\end{equation}\nwhere $r_{\\lambda}(a)$ stands for the symbol\n\\begin{equation}\\label{lay2}\nr_{\\lambda}(a)(X)=\\int_0^1\\int_{\\mathbb{R}^{2n}}{(1-\\theta)a''(X+\\theta Y)Y^2e^{-2\\pi \\Gamma_{\\lambda}(Y)}2^ndYd\\theta}, \\ X \\in \\mathbb{R}^{2n}.\n\\end{equation}\nWe recall that we use here the following normalization for the Weyl quantization\n\\begin{equation}\\label{lay3}\n(a^wu)(x)=\\int_{\\mathbb{R}^{2n}}{e^{2i\\pi(x-y)\\cdot\\xi}a\\big(\\frac{x+y}{2},\\xi\\big)u(y)dyd\\xi}.\n\\end{equation}\nLet us finally mention that the operator $\\pi_{\\lambda}=W_{\\lambda}W_{\\lambda}^*$ is an orthogonal projection on a closed proper subspace of $L^2(\\mathbb{R}^{2n})$ whose kernel is given by\n\\begin{equation}\\label{yo1}\nK_{\\lambda}(X,Y)=e^{-\\frac{\\pi}{2}\\Gamma_{\\lambda}(X-Y)}e^{i\\pi (x-y)\\cdot(\\xi+ \\eta)}, \\ X=(x,\\xi) \\in \\mathbb{R}^{2n}, \\ Y=(y,\\eta) \\in \\mathbb{R}^{2n}. \n\\end{equation}\nIndeed, for any $u \\in \\mathscr{S}(\\mathbb{R}_x^n)$ and $v \\in \\mathscr{S}(\\mathbb{R}_Y^{2n})$, we may write\n\\begin{multline*}\n(W_{\\lambda}u,v)_{L^2(\\mathbb{R}_Y^{2n})}=\\int_{\\mathbb{R}_Y^{2n}}\\Big(\\int_{\\mathbb{R}_x^n}u(x)\\overline{\\varphi_{Y}^{\\lambda}}(x)dx\\Big)\\overline{v}(Y)dY\\\\ =\n\\int_{\\mathbb{R}_x^{n}}u(x)\\Big(\\overline{\\int_{\\mathbb{R}_Y^{2n}}\\varphi_{Y}^{\\lambda}(x)v(Y)dY}\\Big)dx\n=(u,W_{\\lambda}^*v)_{L^2(\\mathbb{R}_x^n)}.\n\\end{multline*}\nIt follows that \n$$(W_{\\lambda}^*v)(x)=\\int_{\\mathbb{R}_Y^{2n}}\\varphi_{Y}^{\\lambda}(x)v(Y)dY.$$\nWriting \n\\begin{multline*}\n(\\pi_{\\lambda}v)(X)=(W_{\\lambda}W_{\\lambda}^*v)(X)=\\int_{\\mathbb{R}_t^n}\\Big(\\int_{\\mathbb{R}_Y^{2n}}\\varphi_{Y}^{\\lambda}(t)v(Y)dY\\Big)\\overline{\\varphi_{X}^{\\lambda}}(t)dt\\\\\n=\\int_{\\mathbb{R}_Y^{2n}}\\Big(\\int_{\\mathbb{R}_t^{n}}\\varphi_{Y}^{\\lambda}(t)\\overline{\\varphi_{X}^{\\lambda}}(t)dt\\Big)v(Y)dY=\\int_{\\mathbb{R}_Y^{2n}}K_{\\lambda}(X,Y)v(Y)dY,\n\\end{multline*}\nwith \n$$K_{\\lambda}(X,Y)=\\int_{\\mathbb{R}_t^{n}}\\varphi_{Y}^{\\lambda}(t)\\overline{\\varphi_{X}^{\\lambda}}(t)dt.$$\nBy using the identity\n$$2|t-y|^2+2|t-x|^2=|x-y|^2+|x+y-2t|^2,$$\nan explicit computation gives that \n\\begin{align*}\nK_{\\lambda}(X,Y)=& \\ (2\\lambda)^{\\frac{n}{2}}\\int_{\\mathbb{R}_t^{n}}e^{-\\pi \\lambda |t-y|^2}e^{-\\pi \\lambda |t-x|^2}e^{2i\\pi(t-y)\\cdot \\eta}e^{-2i\\pi(t-x)\\cdot \\xi}dt\\\\\n= & \\ (2\\lambda)^{\\frac{n}{2}}e^{-\\frac{\\pi}{2}\\lambda |x-y|^2}e^{2i\\pi (x\\cdot \\xi-y\\cdot \\eta)}\\int_{\\mathbb{R}_t^{n}}e^{-\\frac{\\pi}{2} \\lambda |x+y-2t|^2}e^{2i\\pi t \\cdot (\\eta-\\xi)}dt.\n\\end{align*}\nSince \n\\begin{multline*}\n\\int_{\\mathbb{R}_t^{n}}e^{-\\frac{\\pi}{2} \\lambda |x+y-2t|^2}e^{2i\\pi t \\cdot (\\eta-\\xi)}dt=\\int_{\\mathbb{R}_t^{n}}e^{-2\\pi \\lambda |t-\\frac{x+y}{2}|^2}e^{2i\\pi t \\cdot (\\eta-\\xi)}dt\n\\\\ =e^{i\\pi (x+y) \\cdot (\\eta-\\xi)}\\int_{\\mathbb{R}_t^{n}}e^{-2\\pi \\lambda |t|^2}e^{2i\\pi t \\cdot (\\eta-\\xi)}dt=(2\\lambda)^{-\\frac{n}{2}}e^{i\\pi (x+y) \\cdot (\\eta-\\xi)}e^{-\\frac{\\pi}{2\\lambda} |\\eta-\\xi|^2},\n\\end{multline*}\nit follows that \n\\begin{align*}\nK_{\\lambda}(X,Y)= & \\ e^{-\\frac{\\pi}{2}\\lambda |x-y|^2}e^{-\\frac{\\pi}{2\\lambda} |\\eta-\\xi|^2}e^{2i\\pi (x\\cdot \\xi-y\\cdot \\eta)}e^{i\\pi (x+y) \\cdot (\\eta-\\xi)}\\\\\n= & \\ e^{-\\frac{\\pi}{2}\\Gamma_{\\lambda}(X-Y)}e^{i\\pi (x-y)\\cdot(\\xi+ \\eta)}.\n\\end{align*}\n\n\n\n\\bigskip\n\\bigskip\n\n\\noindent\n\\textbf{Acknowledgements.} \nThe research of the second author was\nsupported by  Grant-in-Aid for Scientific Research No.22540187,\nJapan Society of the Promotion of Science. \nThe third author is very grateful to the Japan Society for the Promotion of Science and JSPS London for supporting his stay at Kyoto University during the summer 2010.  He would like to thank Kyoto University and JSPS very warmly for their very kind hospitality and the exceptional working surroundings in which this work was done. Finally, all the three authors are very grateful to the referees for enriching comments and relevant suggestions which have helped to improve significantly the presentation of this article.\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\nEmbryonic development is driven by spatial patterns of gene expression\nthat determine the fate of each cell in the embryo. While gene\nexpression is often highly erratic, embryonic development is usually\nexceedingly precise. In particular, gene expression boundaries are\nrobust not only against intra-embryonic fluctuations such as noise in\ngene expression and protein diffusion, but also against\nembryo-to-embryo variations in the morphogen gradients, which provide\npositional information to the differentiating cells. How development\nis robust against intra- and inter-embryonic variations is not\nunderstood. A common motif in the gene regulation networks that\ncontrol embryonic development is mutual repression between pairs of\ngenes.\nTo assess the role of mutual repression in the robust formation\nof gene expression patterns, we have performed large-scale stochastic\nsimulations of a minimal model of two mutually repressing gap genes \nin {\\it Drosophila},  {\\it hunchback} (\\textit{hb}\\xspace) and {\\it knirps} (\\textit{kni}\\xspace).\nOur model includes not only mutual repression between \\textit{hb}\\xspace and \\textit{kni}\\xspace,\nbut also the stochastic and cooperative activation of \\textit{hb}\\xspace by the anterior\nmorphogen  Bicoid (Bcd\\xspace) and of \\textit{kni}\\xspace by the posterior morphogen Caudal\n(Cad\\xspace), as well as the diffusion of Hb\\xspace and Kni\\xspace between neighboring\nnuclei.\nOur analysis reveals that mutual repression can markedly increase the \nsteepness and precision of the gap gene expression boundaries. \nIn contrast to other mechanisms such as spatial averaging and cooperative \ngene activation, mutual repression thus allows for gene-expression \nboundaries that are both steep and precise. Moreover,\nmutual repression dramatically enhances their robustness against\nembryo-to-embryo variations in the morphogen levels. Finally, our\nsimulations reveal that diffusion of the gap proteins plays a critical\nrole not only in reducing the width of the gap gene expression\nboundaries via the mechanism of spatial averaging, but also in\nrepairing patterning errors that could arise because of the\nbistability induced by mutual repression.\n\n\n\\pagebreak\n\n\\section*{Introduction}\n\\pdfbookmark[1]{Introduction}{BookmarkIntro}\nThe development of multicellular organisms requires spatially\ncontrolled cell differentiation. The positional information for the\ndifferentiating cells is typically provided by spatial concentration gradients of\nmorphogen proteins.  In the\nclassical picture of morphogen-directed patterning, cells translate\nthe morphogen concentration into spatial gene-expression domains via\nsimple threshold-dependent readouts\n\\cite{Wolpert1969,Wolpert1994,Driever1988a,Driever1988b}.  Yet, while\nembryonic development is exceedingly precise, this mechanism is not\nvery robust against intra- and inter-embryonic variations\n\\cite{Houchmandzadeh2002,Gregor2007,Gregor2007b}: the spatial patterns\nof the target genes do not scale with the size of the embryo and the\nboundaries of the expression domains are susceptible to fluctuations\nin the morphogen levels and to the noise in gene\nexpression. Intriguingly, the target genes of morphogens often\nmutually repress each other, as in the gap-gene system of the fruit\nfly {\\em\n  Drosophila} \\cite{Jackle1986,Clyde2003,\n  Jaeger2004,Surkova2008,Manu2009PlosCompBiol,Manu2009PlosBiol,Vakulenko2009}.\nTo elucidate the role of mutual repression in the robust formation of\ngene expression patterns, we have performed extensive\nspatially-resolved stochastic simulations of the gap-gene system of\n{\\em Drosophila melanogaster}. Our results show that mutual repression between\ntarget genes can markedly enhance both the steepness and the precision\nof gene-expression boundaries. Furthermore, it makes them robust against\nembryo-to-embryo variations in the morphogen gradients.\n\nThe fruit fly \\textit{Drosophila melanogaster} is arguably the\nparadigm of morphogenesis. During the first 90 minutes after\nfertilization it is a syncytium, consisting of a cytoplasm that\ncontains rapidly diving nuclei, which are not yet encapsulated by\ncellular membranes.  Around cell cycle 10 the nuclei migrate towards\nthe cortex of the embryo and settle there to read out the\nconcentration gradient of the morphogen protein Bicoid (Bcd\\xspace), which\nforms from the anterior pole after fertilization \\cite{Driever1988a}.\nOne of the target genes of Bcd\\xspace is the gap gene {\\it hunchback} (\\textit{hb}\\xspace),\nwhich is expressed in the anterior half of the embryo. In spite of\nnoise in gene expression, the midembryo boundary of the \\textit{hb}\\xspace expression\ndomain is astonishingly sharp. By cell cycle 11, the \\textit{hb}\\xspace mRNA\nboundary varies by about one nuclear spacing only \\cite{Porcher2010,\nHe2011, Perry2011}, while by cell cycle 13 a similarly sharp\noundary is observed for the protein level \\cite{Houchmandzadeh2002,Gregor2007,He2008}.\nThis precision is higher than the best achievable precision for a \ntime-averaging based readout mechanism of the Bcd gradient \\cite{Gregor2007}.\nInterestingly, the\nstudy of Gregor {\\it et al.} revealed that the Hb concentrations in\nneighboring nuclei exhibit correlations and the authors suggested that\nthis implies a form of spatial averaging that enhances the precision\nof the posterior Hb boundary \\cite{Gregor2007}. Two recent simulation\nstudies suggest that the mechanism of spatial averaging is based on\nthe diffusion of Hb itself \\cite{Erdmann2009,Okabe-Oho2009}; as shown\nanalytically in \\cite{Erdmann2009}, Hb diffusion between neighboring\nnuclei reduces the super-Poissonian part of the noise in its\nconcentration. In essence, diffusion reduces noise by washing out\nbursts in gene expression. However, the mechanism of spatial averaging\ncomes at a cost: it tends to lessen the steepness of the expression\nboundaries.\n\nBcd induces the expression of not only \\textit{hb}\\xspace, but a number of gap genes,\nand pairs of gap genes tend to repress each other\nmutually. Interestingly, repression between directly neighboring gap\ngenes is weak, whereas repression between non-adjacent genes is strong\n\\cite{Kraut1991}.  \\textit{hb}\\xspace forms a strongly repressive pair with {\\it knirps}\n(\\textit{kni}\\xspace) which is expressed further towards the posterior pole; \nboth genes play a prominent role in the later positioning of\ndownstream pair-rule gene stripes \\cite{Clyde2003}.\nIt has\nbeen argued that mutual repression can enhance robustness to\nembryo-to-embryo variations in morphogen levels\n\\cite{Manu2009PlosCompBiol,Manu2009PlosBiol,Vakulenko2009} and sharpen\na morphogen-induced transition between the two mutually \nrepressing genes in a non-stochastic background \\cite{Saka2007,Ishihara2008}.\nHowever, mutual repression can also lead to bistability\n\\cite{Cherry2000,Kepler2001,Warren2004,Warren2005,Papatsenko2011}. While bistablity may buffer\nagainst inter-embryo variations and rapid intra-embryo fluctuations in\nmorphogen levels, it may also cause stochastic switching between\ndistinct gene expression patterns, which would be highly detrimental.\nTherefore, the precise role of mutual repression in the robust\nformation of gene-expression patterns remains to be elucidated.\n\nWhile the role of antagonistic interactions in the formation of\ngene-expression patterns has been studied using mean-field models\n\\cite{Manu2009PlosCompBiol,Papatsenko2011,Ishihara2005,Zinzen2006,Zinzen2007},\nto address the question whether mutual repression enhances the\nrobustness of these patterns against noise arising from the inherent\nstochasticity of biochemical reactions a stochastic model is\nessential. We have therefore performed large-scale stochastic\nsimulations of a minimal model of mutual repression between \\textit{hb}\\xspace and\n\\textit{kni}\\xspace. Our model includes the stochastic and cooperative activation\nof \\textit{hb}\\xspace by Bcd\\xspace and of \\textit{kni}\\xspace by the posterior morphogen Caudal (Cad\\xspace)\n\\cite{Rivera-Pomar1995,Schulz1995}. Moreover, Hb\\xspace and Kni\\xspace can diffuse\nbetween neighboring nuclei and repress each other's expression,\ngenerating two separate spatial domains interacting at midembryo (see\nFig.\\xspace \\ref{Fig1}). We analyze the stability of these domains by\nsystematically varying the diffusion constants of the Hb\\xspace and Kni\\xspace\nproteins, the strength of mutual repression and the Bcd\\xspace and Cad\\xspace\nactivator levels. To quantify the importance of mutual\nrepression, we compare the results to those of a system containing\nonly a single gap gene, which is regulated by its morphogen only;\nthis is the ``system without mutual repression''. While our model is\nsimplified---it neglects, {\\it e.g.}, the interactions of \\textit{hb}\\xspace and \\textit{kni}\\xspace\nwith {\\it kr\\\"{u}ppel} (\\textit{kr}\\xspace) and {\\it giant} (\\textit{gt}\\xspace)\n\\cite{Jaeger2011}---it does allow us to elucidate the mechanism by\nwhich mutual repression can enhance the robust formation of gene\nexpression patterns.\n\nOne of the key findings of our analysis is that mutual repression\nenhances the robustness of the gene expression domains against\nintra-embryonic fluctuations arising from the intrinsic\nstochasticity of biochemical reactions. Specifically, mutual\nrepression increases the precision of gene-expression\nboundaries: it reduces the variation $\\Delta x$ in their\npositions due to these fluctuations.  At the same time, mutual\nrepression also enhances the steepness of the expression boundaries.\nTo understand the interplay between steepness, precision and\nintra-embryonic fluctuations (biochemical noise), it is instructive\nto recall that the width $\\Delta x$ of a boundary of the expression\ndomain of a gene $g$ is, to first order, given by\n\\begin{equation}\n\\label{EqDeltaX}\n\\Delta x =\\frac{\\sigma_G (x_t)}{|\\Avg{G(x_t)}^\\prime|},\n\\end{equation}\nwhere $\\sigma_G(x_t)$ is the standard deviation of the copy number $G$\nof protein G\\xspace and $|\\Avg{G(x_t)}^\\prime|$ is the magnitude of the\ngradient of $G$ at the boundary position $x_t$\n\\cite{Gregor2007,Tostevin2007,Erdmann2009}. Steepness thus refers to\nthe slope of the average concentration profile,\n$|\\Avg{G(x_t)}^\\prime|$, while precision refers to $\\Delta x$,\nwhich is the standard deviation in the position at which $G$\ncrosses a specified threshold value, here taken to be the\nhalf-maximal average expression level of $G$. \n\nThe simulations reveal, perhaps surprisingly, that mutual repression\nhardly affects the noise $\\sigma_G(x_t)$ at the expression boundaries\nof \\textit{hb}\\xspace and \\textit{kni}\\xspace. Moreover, mutual repression can strongly enhance the\nsteepness $|\\Avg{G(x_t)}^\\prime|$ of these boundaries: the steepness\nof the boundaries in a system with mutual repression can, depending on\nthe diffusion constant, be twice as large as that in the system\nwithout mutual repression. Together with Eq. \\ref{EqDeltaX}, these\nobservations predict that mutual repression can significantly enhance\nthe precision of the boundaries, i.e. decrease $\\Delta x$, which\nis indeed precisely what the simulations reveal. Interestingly, there\nexists an optimal diffusion constant that minimizes the boundary width\n$\\Delta x$, as has been observed for a system without mutual\nrepression \\cite{Erdmann2009}. While the minimal $\\Delta x$ of the\nsystem with mutual repression is only marginally lower than that of\nthe system without it, this optimum is reached at a lower value of the\ndiffusion constant, where the steepness of the boundaries is much\nhigher. We find that these observations are robust,\ni.e. independent of the precise parameters of the model, such as\nmaximum expression level, size of the bursts of gene expression, and\nthe cooperativity of gene activation.\n\nOur results also show that mutual repression can strongly buffer\nagainst embryo-to-embryo variations in the morphogen levels by\nsuppressing boundary shifts via a mechanism that is akin to that of\n\\cite{Howard2005, Morishita2009}.  A more detailed analysis reveals\nthat when the regions where Bcd\\xspace and Cad\\xspace activate \\textit{hb}\\xspace and \\textit{kni}\\xspace\nrespectively overlap, bistability can arise in the overlap zone. Yet,\nthe mean waiting time for switching is longer than the lifetime of the\nmorphogen gradients, which means that the \\textit{hb}\\xspace and \\textit{kni}\\xspace expression\npatterns are stable on the relevant developmental time scales. This\nalso means, however, that when errors are formed during development,\nthese cannot be repaired. Here, our simulations reveal another\nimportant role for diffusion: without diffusion a spotty phenotype\nemerges in which the nuclei in the overlap zone randomly express either\nHb\\xspace or Kni\\xspace; diffusion can anneal these patterning defects, leading\nto well-defined expression domains of Hb\\xspace and Kni\\xspace. \nFinally, we also study a scenario where \\textit{hb}\\xspace and \\textit{kni}\\xspace are\nactivated by Bcd\\xspace only. While this scheme is not robust against\nembryo-to-embryo variations in the morphogen levels, mutual\nrepression does enhance boundary precision and steepness also\nin this scenario.\n\n\\section*{Results}\n\\pdfbookmark[1]{Results}{BookmarkResults}\n\\subsection*{Model}\n\\pdfbookmark[2]{Model}{BookmarkModel}\nWe consider the embryo in the syncytial blastoderm stage at late cell cycle\n14, ca. $2~\\unit{h}$ after fertilization. In this stage the majority of the nuclei\nforms a cortical layer and \\textit{hb}\\xspace and \\textit{kni}\\xspace expression\ncan be detected \\cite{Surkova2008}. Our model is an extension of the\none presented in \\cite{Erdmann2009}. It is based on a cylindrical\narray of diffusively coupled reaction volumes which represent the\nnuclei, with periodic boundary conditions in the angular ($\\phi$)\nand reflecting boundaries in the axial ($x$) direction.  The\ndimensions of the cortical array are $N_x=N_\\phi=64$, with equal\nspacing of the nuclei $\\ell=8.5 \\unit{\\mu m}$ in both directions. For a given embryo length\n$L$, this implies a cylinder radius\n$R=\\frac{L}{2\\pi}\\simeq\\frac{L}{6}$, which is close to the\nexperimentally observed ratio.\nThe resulting number of $N=4096$ nuclei roughly corresponds to the\nexpected number of cortical nuclei at cell cycle 14 if non-dividing\npolyploid yolk nuclei are taken into account \\cite{Foe1983} (see\nText S1\\xspace for details); we also emphasize, however, that none of\nthe results presented below depend on the precise number of nuclei.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=3.27in]{Figure1.pdf}\n\\end{center}\n\\caption{ {\\bf The model.}  \\subfig{A} Cartoon of our model.  Bcd\\xspace\n  activates \\textit{hb}\\xspace, while its antagonist \\textit{kni}\\xspace is activated by Cad\\xspace. The\n  gap genes \\textit{hb}\\xspace and \\textit{kni}\\xspace repress each other mutually.  In each nuclear\n  compartment we simulate the genetic promoters of both \\textit{hb}\\xspace and\n  \\textit{kni}\\xspace. Activation is cooperative: In the default setting, 5 morphogen proteins have to bind to\n  the promoter to initiate gene expression. Hb\\xspace and Kni\\xspace both form\n  homodimers, which can bind to the other gene's promoter to totally\n  block expression, irrespective of the number of bound morphogen\n  proteins.  Both dimers and monomers travel between neighboring\n  nuclear compartments via diffusion.  \\subfig{B} Protein copy number\n  profiles along the AP axis in a typical simulation in steady state,\n  with parameter values as in Table S2 in Text S1\\xspace.  Plotted are\n  the morphogen gradients Bcd\\xspace ($\\Avg{B}$, solid green line) and Cad\\xspace \n  ($\\Avg{C}$, solid red line) and the resulting Hb\\xspace ($H$) and Kni\\xspace ($K$) total copy \n  number profiles for different times. The dashed green and red lines show the\n  Hb\\xspace ($\\Avg{H}$) and Kni\\xspace ($\\Avg{K}$) profiles averaged over time and the \n  circumference of the (cylindrical) system.\n  }\n\\label{Fig1}\n\\end{figure}\n\nIn each nuclear volume we simulate the activation of the gap genes \\textit{hb}\\xspace\nand \\textit{kni}\\xspace by the morphogens Bcd\\xspace and Cad\\xspace, respectively, and mutual\nrepression between \\textit{hb}\\xspace and \\textit{kni}\\xspace (see Fig.\\xspace \\ref{Fig1}). In what\nfollows, we will refer to Hb\\xspace and Kni\\xspace as repressors and to Bcd\\xspace and\nCad\\xspace as activators.  Our model of gene regulation bears\nsimilarities to those of\n\\cite{Bolouri2003,Janssens2006,Zinzen2006,Zinzen2007,Papatsenko2011},\nin the sense that it is based on a statistical mechanical model of\ngene regulation by transcription factors, allowing the computation\nof promoter-site occupancies. However, the models of\n\\cite{Bolouri2003,Janssens2006,Zinzen2006,Zinzen2007,Papatsenko2011}\nare mean-field models, which cannot capture the effect of\nintra-embryonic fluctuations due to biochemical noise arising from\nthe inherent stochasticity of biochemical reactions. This requires a\nstochastic model; moreover, it necessitates a model in which the\ntransitions between the promoter states are taken into account\nexplicitly, since these transitions form a major source of noise in\ngene expression, as we will show. To limit the number of\ncombinatorial promoter states, we have therefore studied a minimal model\nthat only includes Bcd\\xspace, Cad\\xspace, Hb\\xspace and Kni\\xspace.\nFollowing \\cite{Erdmann2009}, we assume that Bcd\\xspace and Cad\\xspace bind\nstochastically and cooperatively to $n_{max}$ sites on their target\npromoters.  To obtain a lower bound on the precision of the \\textit{hb}\\xspace and\n\\textit{kni}\\xspace expression domains, we assume that the activating morphogens Bcd\\xspace\nand Cad\\xspace bind to their promoters with a diffusion-limited rate $k_{\\rm on}^{\\rm A}=4\\pi\\alpha D_A\/V$,\nwhere $\\alpha$ is the dimension of a binding site, $D_A$ is the \ndiffusion constant of the morphogen, and $V$ is the nuclear volume \n(see ``Materials \\& Methods'' for parameter values). Since\nthe morphogen-promoter association rate is assumed to be diffusion\nlimited, cooperativity of \\textit{hb}\\xspace and \\textit{kni}\\xspace activation is tuned via the\ndissociation rate $k_{\\rm off,n}^{\\rm A}=a\/b^n$, which decreases with\nincreasing number $n$ of promoter-bound morphogen molecules. The\nbaseline parameters are set such that the half-maximal activation\nlevel of \\textit{hb}\\xspace and \\textit{kni}\\xspace is at midembryo, and the effective Hill\ncoefficient for gene activation is around 5 \\cite{Erdmann2009}; while\nwe will vary the Hill coefficient, this is our baseline\nparameter. Again to obtain a lower bound on the precision of the\ngap-gene expression boundaries, transcription and translation is\nconcatenated in a single step. Mutual repression between \\textit{hb}\\xspace and \\textit{kni}\\xspace\noccurs via binding of Hb\\xspace to the \\textit{kni}\\xspace promoter, which blocks the\nexpression of \\textit{kni}\\xspace irrespective of the number of bound Cad\\xspace molecules,\nand vice versa.  To assess the importance of bistability, Hb\\xspace and Kni\\xspace\ncan homodimerize and bind to their target promoters only in their\ndimeric form, which is a prerequisite for bistability in the\nmean-field limit \\cite{Cherry2000}.  Both the monomers and dimers\ndiffuse between neighboring nuclei and are also degraded; the\neffective degradation rate $\\mu_{\\rm eff}$ is such that the gap-gene\nexpression domains can form sufficiently rapidly on the time scale of\nembryonic development ($\\approx 10-20~\\unit{min}$ \\cite{Foe1983}).\nIn the absence of mutual repression, our model\nbehaves very similarly to that of \\cite{Erdmann2009}, even though our\nmodel contains both monomers and dimers instead of only monomers.\n\n\nMotivated by experiment \\cite{Driever1988a,Houchmandzadeh2002,Gregor2007b},\nand in accordance with the diffusion-degradation model,\nwe adopt an exponential shape for the stationary Bcd\\xspace profile; we thus \ndo not model the establishment of the gradient \\cite{Sample2010}.\nTo elucidate the role of mutual repression,\nit will prove useful to take our model to be\nsymmetric: the Cad\\xspace profile is the mirror image of the Bcd\\xspace\nprofile, and \\textit{hb}\\xspace and \\textit{kni}\\xspace repress each other equally\nstrongly. Diffusion of Bcd\\xspace and Cad\\xspace between nuclei induce\nfluctuations in their copy numbers on the time scale\n$\\tau_d=\\ell^2\/(4D_A)\\simeq 6~\\unit{s}$. Because $\\tau_d$ is much\nsmaller than the time scale for promoter binding, $1\/k_{\\rm\n  on}^{\\rm A}\\simeq 360~\\unit{s}$, fluctuations in the copy number\nof Bcd\\xspace and Cad\\xspace are effectively averaged out by slow binding of Bcd\\xspace\nand Cad\\xspace to their respective promoters, \\textit{hb}\\xspace and \\textit{kni}\\xspace\n\\cite{Erdmann2009}.  To elucidate the importance of the threshold\npositions for \\textit{hb}\\xspace and \\textit{kni}\\xspace activation, we will scale the morphogen\ngradients by a global dosage factor $A$; this procedure will also\nallow us to study the robustness of the system against\nembryo-to-embryo variations in the morphogen levels.\n\nWe simulate the model using the Stochastic Simulation Algorithm (SSA)\nof Gillespie \\cite{Gillespie1976, Gillespie1977}.  Diffusion is\nimplemented into the scheme via the next-subvolume method used in\nMesoRD \\cite{Elf2004, Hattne2005}.\nA recent version of our code is available at GitHub and can be\naccessed via \\url{http:\/\/ggg.amolf.nl} .\n\n\\subsection*{Characteristics of gap-gene expression boundaries}\n\\pdfbookmark[2]{Characteristics of gap-gene expression boundaries}{BookmarkBoundaries}\nThree key characteristics of gene expression boundaries are 1) the\nnoise in the protein concentration at the boundary; 2) the steepness\nof the boundary; 3) the width of the boundary. While these quantities\nmay make intuitive sense, their definitions are not unambiguous.  Equally important, different definitions will\nreveal different properties of the system.\n\n\\mysubsubsection{Decomposing the noise} Let's consider the variance in the copy number $G$ of protein G\\xspace\n  at position $x$ along the anterior-posterior (AP) axis.  We define\n  its mean copy number, averaged over all embryos, circumferential positions\n  $\\phi$ and all times, at the anterior-posterior position $x$ as\n\\begin{eqnarray}\n\\ETPAvg{G}(x)&\\equiv&\\frac{1}{N_e}\\frac{1}{T}\\frac{1}{N_\\phi} \\sum_{e=0}^{N_e-1}\\sum_{t=0}^{T-1}\\sum_{\\phi=0}^{N_\\phi-1}G_e(\\phi,x,t),\n\\end{eqnarray}\nwhere $G_e(x,\\phi,t)$ is the copy number of protein G\\xspace in embryo $e$ at position $x$\n and angle $\\phi$ in the\ncircumferential direction (perpendicular to the AP-axis) at time\n$t$. Here, we introduce the convention that the overline denotes\nan average in time, while the ensemble brackets with\na subscript $\\phi$ \ndenote an average along the $\\phi$ direction and that with a\nsubscript $e$ an average over all embryos. The variance in the copy number\n$G\\equiv G_e(x,\\phi,t)$ is then given by\n\\begin{eqnarray}\n\\sigma^2_G(x)&=&\\ETPAvg{(G-\\ETPAvg{G})^2}\\\\\n&=&\\Avg{\\TPAvg{G^2}}_e-\\Avg{\\overline{\\Avg{G}_\\phi^2}}_e+\\Avg{\\overline{\\Avg{G}_\\phi^2}}_e-\\Avg{\\TPAvg{G}^2}_e+\\Avg{\\TPAvg{G}^2}_e-\\ETPAvg{G}^2\\\\\n&=&\\overbrace{\\Avg{\\overline{\\sigma^2_G}}_e(x)+\\Avg{\\sigma^2_{\\Avg{G}_\\phi}}_e(x)}^{\\text{mean\n    intra-embryonic\n    noise}}\\hspace{2.76cm}+\\overbrace{\\sigma^2_{\\Avg{\\overline{G}}_\\phi}(x)}^{\\text{inter-embryonic variations}}\n\\label{EqNoiseDecom}\n\\end{eqnarray}\nThe total variance in the copy number can thus be decomposed into\nintra-embryonic fluctuations averaged over all embryos and\ninter-embryonic variations. The former can, furthermore, be decomposed into\n$\\Avg{\\overline{\\sigma^2_G}}_e(x)$, which is the time-averaged mean of\nthe variance in $G$ along the circumferential direction,\n$\\overline{\\sigma^2_G}(x)$, averaged over all embryos, and\n$\\Avg{\\sigma^2_{\\Avg{G}_\\phi}}_e(x)$, which is the variance in time\nover the mean of $G$ along the circumferential direction,\n$\\sigma^2_{\\Avg{G}_\\phi}(x)$, again averaged over all embryos. These\nintra-embryonic terms capture different types of dynamics. If the\nexpression boundary is rough\nbut its average position does not fluctuate in time,\nthen $\\overline{\\sigma^2_G}(x)$ will be large yet\n$\\sigma^2_{\\Avg{G}_\\phi}(x)$ will be small. Conversely, when the\nboundary is smooth but its average position does fluctuate in time, then\n$\\overline{\\sigma^2_G}(x)$ will be small yet\n$\\sigma^2_{\\Avg{G}_\\phi}(x)$ will be large. Naturally, a combination\nof the two is also possible. The third term,\n$\\sigma^2_{\\Avg{\\overline{G}}_\\phi}(x)$, captures the embryo-to-embryo\nvariations in the average over time and $\\phi$ of the protein-copy\nnumber.  Similarly, we can decompose the fluctuations in the boundary\nposition $x_t$ as\n\\begin{eqnarray}\n\\Delta x &=&\\sigma_{x_t}\\\\\n&=&\\sqrt{\\Avg{\\overline{\\sigma^2_{x_t}}}_e+\\Avg{\\sigma^2_{\\Avg{x_t}_\\phi}}_e+\\sigma^2_{\\Avg{\\overline{x_t}}_\\phi}}\n\\label{EqXtDecom}\n\\end{eqnarray}\nThe two different contributions to the intra-embryonic variance, $\\Avg{\\overline{\\sigma^2_{x_t}}}_e+\\Avg{\\sigma^2_{\\Avg{x_t}_\\phi}}_e$, are illustrated\n in Fig.\\xspace \\ref{Fig2}.\nHere and in the next section, we will study the robustness of the\nsystem against intra-embryonic fluctuations, while in the section\n``Robustness to inter-embryonic variations: Mutual repression can\nbuffer against correlated morphogen level variations'' we will study\nthe robustness against inter-embryonic variations in the morphogen levels.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\vspace{0.33in}\n\\includegraphics[width=3.27in]{Figure2.pdf}\n\\end{center}\n\\caption{ {\\bf Two different contributions to the intra-embryonic\n      variance in the boundary position.}  The total variance of the\n    gap gene expression boundary position $x_t$ due to intra-embryonic\n    fluctuations, $\\sigma^2_{x_t, intra}$, can be decomposed into two\n    contributions: $\\sigma^2_{\\Avg{x_t}_\\phi}$, the variance in time\n    of the circumferential mean of $x_t$, and\n    $\\overline{\\sigma^2_{x_t}}$, the time-average of the variance of\n    $x_t$ along the circumference of the embryo.  The sketch illustrates\n    two extremal cases: If the boundary is very smooth along the\n    circumference at any moment in time, concerted movements of the boundary\n    will dominate the total variance, i.e. $\\sigma^2_{x_t, intra}\n    \\simeq \\sigma^2_{\\Avg{x_t}_\\phi}$ (left side).  If, in contrast,\n    the boundary is rough but its mean position does not fluctuate\n    much in time, then $\\sigma^2_{x_t, intra} \\simeq\n    \\overline{\\sigma^2_{x_t}}$ (right side).  Naturally, a combination\n    of the two types of fluctuations is possible.  }\n\\label{Fig2}\n\\end{figure}\n\n{\\mysubsubsection{Intra-embryonic fluctuations} Fig.\\xspace S2 in Text S1\\xspace\n  shows the decomposition of the noise in the Hb\\xspace copy number\n  $H$ and the threshold position $x_t$ of the Hb\\xspace boundary, as a\n  function of the diffusion constant. We show the intra-embryonic\n  fluctuations for one given embryo (with the baseline parameter set);\n  how $\\Delta x$ (the boundary variance originating from\n  intra-embryonic fluctuations) changes with embryo-to-embryo\n  variations in the morphogen levels is addressed in section ``Overlap\n  of morphogen activation domains does not corrupt robustness to\n  intrinsic fluctuations''. Fig.\\xspace S2 shows that by far the dominant\n  contribution to the intra-embryonic noise in the copy number and\n  threshold position is the time average of the variance in these\n  observables along the circumferential direction; the variance in\n  time of the $\\phi$-average of these quantities is indeed very\n  small. The picture that emerges is that the expression boundary is\n  rough, even when the diffusion constant $D$ is large,\n  i.e. $D=1~\\unit{\\mu m^2\/s}$. An analysis of the spatial correlation function\n  at midembryo $\\TPAvg{\\delta H(0)\\delta H(\\phi)}(x_t)$, where $\\delta\n  H(\\phi)=H(x_t,\\phi,t)-\\TPAvg{H}$, revealed that the correlation\n  length $\\xi_\\phi$ is on the order of a few nuclei, which corresponds\n  to the diffusion length $\\lambda=\\sqrt{D\/\\mu_{eff}}$ a protein can\n  diffuse with diffusion constant $D$ before it is degraded with a\n  rate $\\mu_{eff}$; the correlation length is thus small compared to\n  the circumference. One possible source of coherent\n  fluctuations in the mean copy number $\\Avg{X}_\\phi$ and boundary\n  position $\\Avg{x_t}_\\phi$ are temporal variations of the morphogen\n  profiles. However, in our model, these profiles are static---we\n  argued that the morphogen fluctuations are fast on the timescale\n  of gene expression, and are thus effectively integrated out. The\n  small correlation length $\\xi_\\phi$ then indeed means that the\n  varations in the mean over $\\phi$, $\\Avg{\\dots}_\\phi$, will be\n  small.   This leads to an interesting implication for\n  experiments, which we discuss in the Discussion section.\n\n\n  \\mysubsubsection{The boundary steepness} Now that we have\n    characterized the fluctuations in the copy number and the boundary\n    position, the next question is how fluctuations in the copy number\n    affect the steepness of the boundary.  In particular, a\n    gene-expression boundary can be shallow either because at each\n    moment in time the interface is shallow, or because at each moment\n    in time the interface is sharp yet the interface fluctuates in\n    time, leading to a smooth profile.  The question is thus how much\n    the gradient of the mean concentration profile,\n    $\\TPAvg{G}^\\prime$, and the mean of the gradient,\n    $\\TPAvg{G^\\prime}$, differ (here the prime denotes the spatial\n    derivative). Fig.\\xspace S3 in Text S1\\xspace shows both quantities as a function of the\n    diffusion constant. It is seen that while the average of the\n    gradient is larger than the gradient of the average (as it\n    should), the difference is around a factor of 2. We thus conclude that\n    the steepness of the expression boundary at each moment in time\n    does not differ very much from the steepness of the average\n    concentration profile.\n\nIn the rest of the manuscript, we will predominantly focus on the\nproperties of individual embryos, and average quantities are\ntypically averages over time and the circumference. For brevity, therefore, \n$\\Avg{\\dots}=\\TPAvg{\\dots}$, unless stated otherwise.\n\\subsection*{Robustness to intra-embryonic fluctuations: Mutual repression allows for steeper profiles without raising the noise level at the boundary}\n\\pdfbookmark[2]{Robustness to intra-embryonic fluctuations: Mutual repression allows for steeper profiles without raising the noise level at the boundary}{BookmarkRobustnessIntra}\n\n\\mysubsubsection{Mutual repression shifts boundaries apart}\nFig.\\xspace \\ref{Fig3}A shows the average Hb\\xspace and Kni\\xspace steady-state\nprofiles along the anterior-posterior (AP) axis as a function of their\ndiffusion constant $D$ for a system with mutual repression.  The inset\nshows the morphogen-activation profiles, which are the spatial profiles of\nthe probability that the \\textit{hb}\\xspace and \\textit{kni}\\xspace promoters have 5 copies of their\nrespective morphogens bound. Without mutual repression, thus when Hb\\xspace\nand Kni\\xspace cannot bind to their respective target promoters, these profiles\ndescribe the probability that \\textit{hb}\\xspace and \\textit{kni}\\xspace are activated by their\nrespective morphogens.  Indeed, without mutual repression and without\nHb\\xspace and Kni\\xspace diffusion, the Hb\\xspace and Kni\\xspace concentration profiles would\nbe proportional to their respective morphogen-activation profiles\n\\cite{Erdmann2009}, which means that they would precisely intersect at\nmidembryo.  In contrast, Fig.\\xspace \\ref{Fig3}A shows that the Hb\\xspace and Kni\\xspace\nconcentration profiles are shifted apart in the system with mutual\nrepression.  There is already a finite separation for $D=0$, which\nincreases further as $D$ is increased.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=6in]{Figure3.pdf}\n\\end{center}\n\\caption{ {\\bf The effect of mutual repression on the average protein\n    concentrations and their standard deviations.}  \\subfig{A} Time-\n  and circumference-averaged Hb\\xspace ($\\Avg{H}$, solid lines) and Kni\\xspace\n  ($\\Avg{K}$, dashed lines) total protein copy number profiles along\n  the AP axis for various diffusion constants $D$ in a system with\n  mutual repression.  The inset shows for both the \\textit{hb}\\xspace and the \\textit{kni}\\xspace\n  promoter the probability that the promoter binds 5 morphogen\n  proteins irrespective of whether the antagonistic gap protein is\n  bound to it (meaning that the promoter is activated by the\n  morphogen, even though it may be repressed by the antogonistic gap\n  protein); these ``morphogen-activation'' profiles are identical for\n  all $D$ values.  \\subfig{B} Profiles of the probability\n  $\\Avg{H^0_5}$ that the \\textit{hb}\\xspace promoter is induced, meaning that it has\n  5 copies of Bcd\\xspace bound to it and no Kni\\xspace dimer (solid lines), and the\n  probability $\\Avg{H^1_5}$ that \\textit{hb}\\xspace is activated by Bcd\\xspace yet\n  repressed by Kni\\xspace, in which case \\textit{hb}\\xspace is indeed not expressed (dashed\n  lines).  \\subfig{C} AP profiles of the time- and\n  circumference-averaged standard deviation of the total gap protein\n  copy number for Hb\\xspace ($\\sigma_H$, solid lines) and Kni\\xspace ($\\sigma_K$,\n  dashed lines).  \\subfig{D} Normalized standard deviation\n  $\\sigma_{H}(x)\/\\Avg{H}_{max}$ versus the normalized mean\n  $\\Avg{H}(x)\/\\Avg{H}_{max}$; $\\Avg{H}(x)$ is the averaged total\n  Hb\\xspace copy number at $x$ and $\\Avg{H}_{max}$ is the maximum of this\n  average over all $x$.  The grey dashed line represents the\n  Poissonian limit (PL) given by\n  $\\sqrt{(1+f_D)\\Avg{H}(x)}\/\\Avg{H}_{max}$, where $f_D$ is the\n  fraction of proteins in dimers.  }\n\\label{Fig3}\n\\end{figure}\n\nIn Fig.\\xspace \\ref{Fig3}B we show the profile of the probability\n$\\Avg{H^0_5}$ that the \\textit{hb}\\xspace promoter is induced, meaning that it has 5\ncopies of Bcd\\xspace bound to it and no Kni\\xspace, and the profile of the\nlikelihood $\\Avg{H^1_5}$ that \\textit{hb}\\xspace is activated by Bcd\\xspace, yet repressed\nby Kni\\xspace, in which case \\textit{hb}\\xspace is not expressed. It is seen that\nrepression by \\textit{kni}\\xspace almost fully inhibits \\textit{hb}\\xspace expression beyond the\nhalf-activation point, where \\textit{hb}\\xspace would be expressed without \\textit{kni}\\xspace\nrepression (see inset Panel A).  Indeed, mutual repression effectively\ncuts off protein production beyond midembryo. The production\nprobability therefore changes more abruptly along the AP axis, leading\nto a higher steepness of the protein profiles near midembryo.  For\n$D>0$, repressor influx over the midplane increases, and as a result\nthe regions of expression inhibiton are enlarged and the concentration\nprofiles shift apart further.\n\n\\mysubsubsection{Noise reduction via spatial averaging}\nFig.\\xspace \\ref{Fig3}C shows the standard deviation of the protein copy\nnumber along the AP axis for both Hb\\xspace ($\\sigma_H$) and Kni\\xspace\n($\\sigma_K$).  It is seen that the noise increases close to the\nhalf-activation point where promoter-state fluctuations are strongest\n\\cite{Tkacik2008,VanZon2006,So2011}.  This is also observed in Fig.\\xspace\n\\ref{Fig3}D, which shows the normalized standard deviation $\\sigma_H \/\n\\Avg{H}_{max}$ versus the normalized mean $\\Avg{H}\/\\Avg{H}_{max}$\nof the average Hb\\xspace copy number; here, $\\Avg{H}_{max}$ is the\nmaximum average concentration of Hb\\xspace. The noise maximum close to mid\nembryo diminishes with increasing $D$, approaching the\nPoissonian limit.  Note that the Poissonian limit here is given by\n$\\sigma_P=\\sqrt{(1+f_D)\\Avg{H}}$, where $f_D=2\\Avg{H_D}\/\\Avg{H}$ is\nthe fraction of dimerized Hb\\xspace proteins with respect to the total Hb\\xspace\ncopy number (see Text S1\\xspace for details).  Clearly, the\nspatial averaging mechanism described in\n\\cite{Erdmann2009,Okabe-Oho2009} reduces the noise also in our system,\nwhich differs from those in \\cite{Erdmann2009,Okabe-Oho2009}\nby the presence of both gap gene monomers and dimers instead\nof monomers only.\n\n\\mysubsubsection{Mutual repression reduces the boundary width by\n  increasing the steepness}\nFig.\\xspace \\ref{Fig4} quantifies the impact of spatial averaging and\nmutual repression on the Hb\\xspace boundary width $\\Delta x$, comparing it\nto that of the system without mutual repression. To first order, the\nboundary precision $\\Delta x$ is related to the standard\ndeviation in the protein copy number at the boundary, $\\sigma_H(x_t)$,\nand the steepness of the boundary, $|\\Avg{H(x_t)}^\\prime|$, via\nEq. \\ref{EqDeltaX} \\cite{Tostevin2007,Gregor2007,Erdmann2009}. The\nnoise $\\sigma_H(x_t)$ decreases with increasing $D$ due to spatial\naveraging in an almost identical manner for the systems with and\nwithout mutual repression (Fig.\\xspace \\ref{Fig4}, top panel); indeed,\nperhaps surprisingly, mutual repression has little effect on the noise\nat the boundary. Increasing $D$ also lessens the steepness of the\nprotein profiles, thus reducing the slope $|\\Avg{H(x_t)}^\\prime|$\n(Fig.\\xspace \\ref{Fig4}, middle panel).  While without mutual repression this\nreduction is monotonic, in the case with mutual repression the\nsteepness first rises because increasing $D$ increases the influx of\nthe antagonistic repressor into the regions where the gap genes are\nactivated by their respective morphogens, which, for low values of\n$D$, {\\em steepens} the effective gene-activation profile\n$\\Avg{H^1_5}(x)$ by most strongly reducing gene expression near\nmidembryo; after the steepness has reached its maximum at\n$D=0.032~\\unit{\\mu m^2\/s}$, it drops for higher diffusion constants, because\nthe diffusion of the gap-gene proteins now flattens their\nconcentration profiles.  Most importantly, with mutual repression\n$|\\Avg{H(x_t)}^\\prime|$ reaches significantly higher values for all\n$D\\leq 1.0~\\unit{\\mu m^2\/s}$.  At $D=0.032~\\unit{\\mu m^2\/s}$ the profile is roughly twice\nas steep as in the case without repression. Interestingly, for\n$D\\lesssim 0.1~\\unit{\\mu m^2\/s}$, our simulation results for the steepness of\nthe profiles as normalized by their maximal values agree with those\nmeasured experimentally by Surkova {\\em et al.} in cell cycle 14\n\\cite{Surkova2008}: In both simulation and experiment, the\nconcentration drops from 90\\% to 10\\% of the maximal values over\n5-10\\% of the embryo length.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=3.27in]{Figure4.pdf}\n\\end{center}\n\\caption{ {\\bf The effect of mutual repression on the precision and\n    steepness of the Hb\\xspace boundary.}  The figure shows the\n  time- and circumference-average of the standard deviation of the\n  total Hb\\xspace copy number at the boundary $\\sigma_H(x_t)$ (upper panel),\n  the slope of the total Hb\\xspace copy number profile at the boundary\n  $|\\Avg{H}'(x_t)|$ (middle panel) and the Hb\\xspace boundary width $\\Delta\n  x$ (lower panel) as a function of the diffusion constant $D$ of the\n  gap proteins.  Red solid lines show the case without (NR) and green\n  solid lines the case with mutual repression (R); the red and green\n  dashed lines show the limiting values without diffusion of the gap\n  proteins. The grey dashed lines in the boundary width plot are the\n  values based on the approximation $\\Delta\n  x=\\sigma_H(x_t)\/|\\Avg{H(x_t)}'|$.  Note that for $D<3.2~\\unit{\\mu m^2\/s}$,\n  mutual repression enhances the steepness of the boundary, which in\n  turn enhances the precision of the boundary.\n  The black dotted line marks the $D$-value\n  where the boundary is both steep and precise due to mutual repression.}\n\\label{Fig4}\n\\end{figure}\n\nBoth with and without Hb\\xspace-Kni\\xspace mutual repression the trade-off between\nnoise and steepness reduction leads to an optimal diffusion constant\n$D_{min}$ that maximizes boundary precision, i.e. minimizes $\\Delta x$\n(Fig.\\xspace \\ref{Fig4}, lower panel).  Mutual repression enhances the\nprecision for $D\\leq 1.0~\\unit{\\mu m^2\/s}$ because in this regime decreasing $D$\nincreases the steepness markedly while it has only little effect on\nthe noise as compared to the system without mutual repression.\nConversely, $\\Delta x$ is increased by mutual repression for $D\\geq\n10~\\unit{\\mu m^2\/s}$ because it reduces the steepness.  The minimum in the case\nwith repression is marginally lower than that without\n($D_{min,R}\/D_{min,NR}\\simeq0.86$), but located at a lower $D$-value\n($1.0~\\unit{\\mu m^2\/s}$ vs. $3.2~\\unit{\\mu m^2\/s}$).  Most importantly, at\n$D=0.32~\\unit{\\mu m^2\/s}$, the system with mutual repression produces a profile\nthat is twice as steep as that of the system without it at\n$D_{min,NR}=3.2~\\unit{\\mu m^2\/s}$, whereas the precision $\\Delta x$ is\nessentially the same in both cases.  Clearly, mutual repression can\nstrongly enhance the steepness of gene-expression boundaries without\ncompromising their precision.\n\n\\mysubsubsection{Influence of Hill coefficient}\nA key parameter controlling the precision of the gap-gene\nexpression boundaries, is the degree of cooperativity by which the\ngap genes are activated by their respective morphogens---this\ndetermines the profile steepness of the average gap-gene\npromoter activity. To investigate this, we have lowered the\neffective Hill coefficient from its baseline value of 5 by reducing\nthe number $n_{\\rm max}$ of morphogen molecules that are required to\nbind the promoter to activate gene expression. To isolate the effect\nof varying the {\\em mean} gene-activation profiles $\\Avg{H_{n_{\\rm\nmax}}^0}(x)$ and $\\Avg{K_{n_{\\rm max}}^0}(x)$, we varied, upon\nvarying $n_{\\rm max}$, the association and dissociation rates such\nthat 1) the average gene activation probabilities near midembryo,\n$\\Avg{H_{n_{\\rm max}}^0}(L\/2)$ and $\\Avg{K_{n_{\\rm max}}^0}(L\/2)$, are\nunchanged and 2) the waiting-time distribution for the gene\non-to-off transition is unchanged (since the average activation\nprobability is fixed, the mean off-to-on rate is also unchanged, although\nthe waiting-time distribution is not; see also Fig.\\xspace S5 in Text S1\\xspace).\nWe observe that  mutual repression\nmarkedly enhances the steepness of the gap-gene expression\nboundaries, also with a\nlower Hill coefficient for gene activation (Fig.\\xspace S6 in Text S1\\xspace). However, lowering the Hill coefficient\nreduces the steepness of the gene-activation profiles, causing the\ntwo antagonistic gene-activation profiles to overlap more. As a\nresult, in each of the two gap-gene expression domains, more of the\nantagonist is present, which tends to increase the noise in gene\nexpression by occassionally shutting off gene production. This, as\nexplained in more detail later, is\nparticularly detrimental when the diffusion constant is low. Indeed,\nwhen the effective Hill\ncoefficient of gene activation is 3 or lower, mutual repression {\\em\nincreases} $\\Delta x$ when the diffusion constant is low,\ni.e. below approximately $0.1~\\unit{\\mu m^2\/s}$. Nonetheless, the {\\em minimal} $\\Delta\nx$ is still lower with mutual repression, and, consequently, also\nwith a lower Hill coefficient for gene activation, mutual repression\ncan enhance both the steepness and the precision of gene-expression\nboundaries.\n\n\\mysubsubsection{Influence of the repression strength}\nAs a standard we assume very tight binding of the Hb\\xspace and Kni\\xspace dimers,\n``the repressors'', to their respective promoters. To test how this\nassumption affects our results we performed simulations in which we\nsystematically varied the repressor-promoter dissociation rate $k^{\\rm\n  R}_{off}$ in the range $[5.27\\cdot 10^{-4}\\unit{\/s},5.27\\cdot\n10^{2}\\unit{\/s}]$, keeping the diffusion constant at $D=1.0~\\unit{\\mu m^2\/s}$\n(the value that minimizes the boundary width at $k^{\\rm\n  R}_{off}=5.27\\cdot 10^{-3}\\unit{\/s}$) and all other parameters the\nsame as before.  Fig.\\xspace \\ref{Fig5} shows the noise, steepness and\nboundary precision as a function of the repressor-promoter\ndissociation rate. For high dissociation rates, these quantities equal\nthose in the system without mutual repression (dashed lines). Yet, as\nthe dissociation rate is decreased, the steepness rises markedly at\n$k^{\\rm R}_{off}=1\/{\\rm s}$. In contrast, the noise $\\sigma_H(x_t)$\nfirst decreases with decreasing $k^{\\rm R}_{off}$, passing through\na minimum at $k^{\\rm R}_{off}=0.1\/{\\rm s}$ before rising to a level that\nis higher than that in a system without mutual repression. This\nminimum arises because on the one hand increasing the affinity of\nthe repressor (the antagonist) makes the operator-state fluctuations\nof the activator (the morphogen) less important---increasing\nrepressor binding drives the concentration profiles of Hb\\xspace and Kni\\xspace\naway from midembryo, where the promoter-state fluctuations of the\nactivators are strongest; on the other hand, when the repressor\nbinds too strongly, then slow repressor unbinding leads to\nlong-lived promoter states where gene expression is shut off,\nincreasing noise in gene expression; this phenomenon is similar to\nwhat has been observed in Refs. \\cite{VanZon2006} and\n\\cite{Morelli2008}, where slower binding of the gene regulatory\nproteins to the promoter increases noise in gene expression and\ndecreases the stability of a toggle switch, respectively. The interplay between the\nnoise and the steepness yields a marked reduction of the boundary\nwidth $\\Delta x$; indeed, even in the limit of very tight repressor binding,\nmutual repression significantly enhances the precision of the\nboundary.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=3.27in]{Figure5.pdf}\n\\end{center}\n\\caption{ {\\bf The effect of varying repression strength on the precision\n    and steepness of the Hb\\xspace boundary.}  Shown are the time- and\n  circumference average of the standard deviation of the total Hb\\xspace copy\n  number at the boundary $\\sigma_H(x_t)$ (upper panel), the steepness\n  of the boundary $|\\Avg{H}'(x_t)|$ (middle panel) and the Hb\\xspace\n  boundary width $\\Delta x$ (lower panel) as a function of\n  $k_{\\rm off}^{\\rm R}$, the promoter-dissociation rate of Hb\\xspace and Kni\\xspace.  The solid green line are values obtained from the\n  boundary position distribution, the dashed grey line the ones\n  calculated from the approximation $\\Delta\n  x=\\sigma_H(x_t)\/|\\Avg{H(x_t)}'|$.  Straight dashed lines mark\n  the limits for the case without mutual repression ($k_{on}^{\\rm R}=k_{off}^{\\rm R}=0$).}\n\\label{Fig5}\n\\end{figure}\n\n\n\\mysubsubsection{Influence of expression level}\nSince the precise gap protein expression level is not known, we also varied the\nmaximal protein copy number $N$ by varying the maximal expression\nrate $\\beta$ (see Text S1\\xspace). Fig.\\xspace S9 in Text S1\\xspace shows the output\nnoise and slope at the boundary position, and the boundary precision $\\Delta x$, \nas a function of the diffusion constant for three different\nexpression levels. It is seen that for low diffusion constant, the\nprecision is independent of $N$, while for higher diffusion constant it scales roughly with\n$1\/\\sqrt{N}$. This can be understood by noting that the steepness of\nthe gene-expression boundary scales to a good approximation\nwith $N$ independently of $D$, while the noise $\\sigma$ scales with\n$N$ when the diffusion constant is small, but with $\\sqrt{N}$ when\nthe diffusion constant is large (see also Eq. \\ref{EqDeltaX}). The\nscaling of the noise with $N$ is due to the fact that for low $D$\nthe noise in the copy number is dominated by the noise coming from\nthe promoter-state fluctuations, which scales linearly with $N$,\nwhile for high $D$, diffusion washes out the expression bursts resulting from\nthe promoter-state flucutations, leaving only the noise coming from\nthe Poissonian fluctuations arising from transcription and\ntranslation, which scales with the square root of $N$\n\\cite{Erdmann2009}. In Text S1\\xspace we also study the importance of\nbursts arising in the transcription-translation step (see Fig.\\xspace S8 in Text S1\\xspace); however, we\nfind that for a typical burst size, these bursts do not dramatically\naffect boundary precision.\n\n\\FloatBarrier\n\n\\subsection*{Robustness to inter-embryonic variations: Mutual repression can buffer against correlated morphogen level variations}\n\\pdfbookmark[2]{Robustness to inter-embryonic variations: Mutual repression can buffer against correlated morphogen level variations}{BookmarkRobustnessInter}\nAlthough the Bcd\\xspace copy number at midembryo has been determined\nexperimentally \\cite{Gregor2007}, the measured value is not\nnecessarily the half-activation threshold of \\textit{hb}\\xspace.  Indeed, in vivo the\nHb\\xspace profile is shaped by other forces, like mutual repression. In the\n\\textit{kni}\\xspace-\\textit{kr}\\xspace double mutant, the Hb boundary at midembryo shifts\nposteriorly \\cite{Manu2009PlosBiol}. Moreover, gap gene domain\nformation has been observed at strongly reduced Bcd\\xspace levels,\nsuggesting that Bcd\\xspace might be present in excess\n\\cite{Ochoa-Espinosa2009}.  Also from a theoretical point of view it\nis not obvious that a precisely centered morphogen-activation\nthreshold is optimal, in terms of robustness against both\nintra-embryonic fluctuations and inter-embryonic variations. Here, we\nstudy the effect of changing the threshold position where \\textit{hb}\\xspace and \\textit{kni}\\xspace\nare half-maximally activated by their respective morphogens, Bcd\\xspace and\nCad\\xspace.  While the threshold positions could be varied by changing the\nthreshold morphogen concentrations for half-maximal gap-gene\nactivation (for example by changing the morphogen-promoter\ndissociation rates), we will vary these positions by changing the\namplitude of the morphogen profiles by a factor $A$.  This procedure\nnot only preserves the promoter-activation dynamics at the\nboundaries---a key determinant for the noise at the boundaries---but\nalso allows us to study the importance of mutual repression in\nensuring robustness against embryo-to-embryo variations. Indeed, we\nwill examine not only how changing the threshold position affects the\nprecision of the gap-gene expression boundaries, $\\Delta x(A)$,\nbut also how the average boundary positions vary with morphogen\ndosage, $x_t(A)$, and how the latter gives rise to embryo-to-embryo\nvariations in the boundary position $\\Delta x_t(\\Delta A)$ due to\nembryo-to-embryo variations in the morphogen dosage $\\Delta A$.\n\n\\mysubsubsection{Double-activation induces bistability}\nWe first consider the scenario in which the amplitudes of both\nmorphogens are scaled by the same factor $A$. When $A=1$, the\nposition at which \\textit{hb}\\xspace and \\textit{kni}\\xspace are half-maximally activated by their\nrespective morphogens coincide at midembryo, meaning that the\ndomains in which \\textit{hb}\\xspace and \\textit{kni}\\xspace are activated beyond half-maximum are\nadjoining, but do not overlap---this is the scenario discussed in\nthe previous sections. When $A>1$, the position at which \\textit{hb}\\xspace is\nhalf-maximally activated by its morphogen is shifted posteriorly,\nwhile that of \\textit{kni}\\xspace is shifted anteriorly, creating an overlap between\nthe two regions where \\textit{hb}\\xspace and \\textit{kni}\\xspace are activated.\nIn this ``double-activated region'' both \\textit{hb}\\xspace and \\textit{kni}\\xspace are\nactivated by their respective morphogens, yet they also mutually\nrepress each other. This may lead to bistability.  To probe whether\nthis is the case, we performed a bifurcation analysis of the\nmean-field chemical-rate equations of isolated nuclei, implying that\n$D=0$ (see Fig.\\xspace S1 in Text S1\\xspace). In addition, we performed stochastic\nsimulations of isolated nuclei with different morphogen levels\ncorresponding to different positions along the AP axis. All other\nparameter values were the same as in the full-scale simulation.  We\nrecorded long trajectories of the order parameter $\\Delta N \\equiv\nH-K$, the difference between the total Hb\\xspace and total Kni\\xspace copy numbers,\nin the stationary state.  From each trajectory we computed the\ndistribution $P(\\Delta N)$ of the probability that the system is in a\nstate with copy number difference $\\Delta N$.  This defines a ``free\nenergy'' $G(\\Delta N)\\equiv -\\ln P(\\Delta N)$, with minima of\n$G(\\Delta N)$ corresponding to maximally probable values of $\\Delta\nN$ \\cite{Warren2004,Warren2005}. For a bistable system, $G(\\Delta\nN)$ resembles a double-well potential with minima located at a\npositive value of $\\Delta N=\\Delta N_{\\rm H}$ and a negative value of\n$\\Delta N=\\Delta N_{\\rm K}$, respectively. At midembryo the morphogen\nlevels of Bcd\\xspace and Cad\\xspace are the same and hence the biochemical network\nin the nuclei in the midplane is symmetric, which means that, if this\nnetwork is bistable, $G(\\Delta N)$ resembles a symmetric double-well\npotential with $\\Delta N_H=-\\Delta N_K$ and $\\Delta G \\equiv G(\\Delta\nN_H) - G(\\Delta N_K)=0$. Away from the middle, the morphogen levels\ndiffer, and one state will become more stable than the other; if the\nother state is, however, still metastable, then $G(\\Delta N)$ will\nresemble an asymmetric double-well potential, with $\\Delta G$ being\nnegative if the \\textit{hb}\\xspace-dominant state is more stable than the\n\\textit{kni}\\xspace-dominant state, and vice versa. The emergence of such a\n``spatial switch'' along the AP axis is also captured by our\nmean-field, bifurcation analysis (see Text S1\\xspace) and was recently\nalso shown in the mean-field analysis of Papatsenko and Levine for the same pair of\nmutually repressing genes\\cite{Papatsenko2011}.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=3.27in]{Figure6.pdf}\n\\end{center}\n\\caption{ {\\bf Emergence of bistability in double-activated regions.}\n  The ``free energy'' difference $\\Delta G\\equiv G(\\Delta N_H) -\n  G(\\Delta N_K)$ as a function of $x$, the distance of the nucleus\n  from the anterior pole, for different amplitudes of the morphogen\n  gradients $A$; here, $G(\\Delta N)\\equiv -\\ln(P(\\Delta N))$, where\n  $P(\\Delta N)$ is the stationary distribution of the order parameter\n  $\\Delta N=H-K$; $\\Delta N_H\\simeq -\\Delta N_K \\approx 800$ correspond to\n  the minima of $G(\\Delta N)$.  Negative values of $\\Delta G$\n  represent a strong bias towards the high-Hb\\xspace state, while positive\n  values correspond to high-Kni\\xspace states.  The insets shows $G(\\Delta\n  N)$ as a function of $\\Delta N$ at the positions indicated by the\n  numbers in their corners (values in [\\unit{\\% EL}]; colors correspond to\n  main plot). The data is obtained from simulations of single nuclei\n  with morphogen levels corresponding to the ones at position $x$ in\n  the full system; this is equivalent to the full system without\n  diffusion between neighboring nuclei. Note the bistable behavior in\n  a wide region of the embryo for higher $A$ values.  }\n\\label{Fig6}\n\\end{figure}\n\nFig.\\xspace \\ref{Fig6} shows $\\Delta G$ as a function of the position along\nthe AP axis, for different amplitudes $A$ of the morphogen gradients. The\ninset shows the energy profiles $G(\\Delta N)$ for different positions\nalong the AP axis.  For $A=1$, $G(\\Delta N)$ always exhibits one\nminimum only, irrespective of the position along the AP axis; at midembryo,\nthis minimum is located at $\\Delta N=0$, while moving towards\nthe anterior (posterior) the energy minimum rapidly shifts to $\\Delta\nN \\approx +800 (-800)$, reflecting that in the anterior (posterior) half of\nthe embryo \\textit{hb}\\xspace (\\textit{kni}\\xspace) is essentially fully expressed. For $A=2$,\n$G(\\Delta N)$ develops into a double-well potential at midembryo,\nwith two pronounced minima at $\\Delta N\\approx 800$ and $\\Delta\nN\\approx-800$, respectively. These two minima correspond to a state in\nwhich \\textit{hb}\\xspace is highly expressed ($\\Avg{H}\\approx 800$) and \\textit{kni}\\xspace is\nstrongly repressed ($\\Avg{K}\\approx 0$) and another state in which\n\\textit{kni}\\xspace is highly expressed and \\textit{hb}\\xspace strongly repressed, respectively. The\nfact that the two energy mimima are equal indicates that both of these\nstates are equally likely. Moving away from midembryo, however, one\ngap-gene expression state rapidly becomes more stable than the other,\nand bistability is lost, yielding a potential with one minimum located\nat $\\Delta N\\approx 800$ in the anterior half and a potential with one minimum located\nat $\\Delta N\\approx -800$ in the posterior half of the embryo.\nInterestingly, for $A=4$\nand $A=8$ a wide region of bistability  develops\naround midembryo. In this region, $\\Delta G \\approx 0$, meaning that \nthe high-\\textit{hb}\\xspace---low-\\textit{kni}\\xspace state and the low-\\textit{hb}\\xspace---high-\\textit{kni}\\xspace state are\nequally stable. These two states are equally likely because in this\nregion both the \\textit{hb}\\xspace and \\textit{kni}\\xspace promoters are fully activated by their\nrespective morphogens. It can also be seen that the width of this\nbistable region increases with the amplitude of the morphogen\ngradients, as expected.\n\\pagebreak\n\n\\mysubsubsection{Slow switching ensures a low noise level while diffusion avoids error locking}\nThe bistability observed for $A>1$ and $D=0$ raises an important\nquestion, namely whether the nuclei can switch between the two\ngap-gene expression states on the time scale of embryonic\ndevelopment. This question is particularly pertinent for the higher\nmorphogen amplitudes, where these two states are equally likely ($\\Delta\nG\\approx 0$) over a wide region of the embryo (Fig.\\xspace \\ref{Fig6}): random\nswitching between the two distinct gap-gene expression states in this\nwide region would then lead to dramatic fluctuations in the positions\nof the \\textit{hb}\\xspace and \\textit{kni}\\xspace expression boundaries, which clearly would be\ndetrimental for development. We therefore computed \\cite{Warren2005} from the recorded\nswitching trajectories the average waiting time for switching,\n$\\tau_{s}$, at midembryo  ($\\Delta\nG\\simeq 0$) for different values of $A$; for $A\\geq 2$, we find\n$\\tau_{s} \\simeq\n6~\\unit{h}$ (see Table S1 in Text S1\\xspace). During cell cycle 14, approximately 2-3 hours\nafter fertilization, the Bcd\\xspace gradient disappears \\cite{Drocco2011}, suggesting that the\nspontaneous switching rate is indeed low on the relevant time scale of\ndevelopment.\n\nWith diffusion of Hb\\xspace and Kni\\xspace between neighboring nuclei ($D>0$), the\ntime scale for switching will be even longer. Diffusion couples\nneighboring nuclei, creating larger spatial domains with the same\ngap-gene expression state. This reduces the probability that a nucleus\nin the overlap region flips to the other gap-gene expression\nstate. The latter can be understood from the extensive studies on the\nswitching behavior of the ``general toggle switch''\n\\cite{Warren2004,Warren2005,Allen2005,Lipshtat2006,Loinger2007,Morelli2008}, which is highly\nsimilar to the system studied here---indeed, the toggle switch\nconsists of two genes that mutually repress each other. These studies\nhave revealed that the ensemble of transition states, which separate\nthe two stable states, is dominated by configurations where both\nantagonistic proteins are present in low copy numbers. Clearly, the\nprobability that in a given nucleus not only the minority gap protein,\nbut also the majority gap protein reaches a low copy number, is\nreduced by the diffusive influx of that majority species from the\nneighboring nuclei, which are in the same gap-gene expression\nstate. In essence, diffusion increases the effective system size, with\nits spatial dimension given by $\\lambda=\\sqrt{D\/\\mu_{\\rm eff}}$; in\nfact, since the stability of the toggle switch depends exponentially\non the system size \\cite{Warren2004,Warren2005}, we expect the stability $\\tau_{s}$\nto scale with the diffusion constant as $\\tau_{s}\\sim e^D$. We thus\nconclude that random switching between the two gap-gene expression\nstates, the high-\\textit{hb}\\xspace---low-\\textit{kni}\\xspace and low-\\textit{hb}\\xspace---high-\\textit{kni}\\xspace states, is not\nlikely to occur on the time scale of early development.\n\nThe observation that the switching rate is low raises another\nimportant question: if errors are formed during development, can they\nbe corrected? We observe in the simulations with $D=0$ that when we\nallow the gap-gene expression patterns to develop starting from\ninitial conditions in which the Hb\\xspace and Kni\\xspace copy numbers are both\nzero, in the overlap (bistable) region a spotty gap-gene expression\npattern emerges, consisting of nuclei that are either in the\nhigh-\\textit{hb}\\xspace---low-\\textit{kni}\\xspace state or in the low-\\textit{hb}\\xspace---high-\\textit{kni}\\xspace state. When\nthe diffusion constant of Hb\\xspace and Kni\\xspace is zero, then these defects are\nessentially frozen in, precisely because of the low switching\nrate. Interestingly, however, we find in the simulations that a finite\ndiffusion constant {\\em can} anneal these defects. This may seem to\ncontradict the statement made above that diffusion lowers the\nswitching rate. The resolution of this paradox is that while diffusion\nlowers the switching rate for nuclei that are surrounded by nuclei\nthat are in the same gap-gene expression state, it enhances the\nswitching rate for nuclei that are surrounded by nuclei with a\ndifferent gap-gene expression state; this is indeed akin to spins in\nan Ising system below the critical point. The mechanism for the\nformation of the gap-gene expression patterns, then, depends on the\ndiffusion constant. When $D$ is small yet finite, $0<D<0.1 \\unit{\\mu m^2\/s}$, in\nthe overlap region first small domains are formed consisting of nuclei\nthat are in the same gap-gene expression state; these domains then\ncoarsen analogously to Ostwald ripening of small crystallites in a\nliquid below the freezing temperature; ultimately, they combine with\nthe \\textit{hb}\\xspace or \\textit{kni}\\xspace expression domains that have formed in the meantime\noutside the overlap region, where \\textit{hb}\\xspace and \\textit{kni}\\xspace are activated by their\nrespective morphogens yet do not repress each other (see Videos S1 and S2).\nFor $D\\gtrsim 0.1 \\unit{\\mu m^2\/s}$, no ``crystallites'' are formed in\nthe overlap region (both the Hb\\xspace and Kni\\xspace copy numbers are low yet\nfinite and \\textit{hb}\\xspace and \\textit{kni}\\xspace simultatenously repress each other); instead,\nthe \\textit{hb}\\xspace and \\textit{kni}\\xspace domains formed near the poles slowly invade the\noverlap region (see Videos S3 and S4). Interestingly, even while in the\nabsence of Hb\\xspace and Kni\\xspace diffusion $\\Delta G\\approx 0$ in the overlap region, the\ninterface between the \\textit{hb}\\xspace and \\textit{kni}\\xspace expression domains does slowly\ndiffuse towards midembryo when $D>0$ and $A\\leq 4$, \ndue to the diffusive influx of Hb\\xspace and Kni\\xspace from the regions outside\nthe overlap region. When $A=8$, the \\textit{hb}\\xspace and \\textit{kni}\\xspace expression boundaries\nare not pinned to the middle of the embryo, and their positions\nexhibit slow and large fluctuations, presumably because the energetic driving\nforce is small, and the diffusive influx of Hb\\xspace and Kni\\xspace from the\nregions near the poles is negligible.  We will investigate this effect in\nmore detail in a forthcoming publication. \n\n\n\\mysubsubsection{Mutual repression inhibits boundary shifts}\nFig.\\xspace \\ref{Fig7}A shows the average gap-gene expression profiles for\n$A\\in\\lbrace 1,2,4\\rbrace$ and $D=1.0~\\unit{\\mu m^2\/s}$, which minimizes the\nboundary width $\\Delta x$ when $A=1$ (see Fig.\\xspace \\ref{Fig4}). While\nthe morphogen-activation thresholds shift beyond midembryo as $A$ is\nincreased beyond unity, leading to an overlap of the domains where the\ngap genes are activated by their respective morphogens (see inset),\nthe gap-gene expression boundaries overlap only marginally. This is\nquantified in panel B, which shows the Hb\\xspace boundary position $x_t$ as\na function of $A$ and as a function of $\\Delta x_A\\equiv\nx_{A,Kni}-x_{A,Hb}$, which is defined as the separation between the\npositions $x_{A,Kni}$ and $x_{A,Hb}$ where Kni\\xspace and Hb\\xspace are\nhalf-maximally activated by their respective morphogens; for $A=1$,\nwith adjoining morphogen activation regions, $\\Delta x_A=0$ and for\n$A>1$, with overlapping activation regions, $\\Delta x_A$ is\nnegative. Without mutual repression (red data), the Hb\\xspace boundary\nposition $x_t$ tracks the shift of the \\textit{hb}\\xspace activation threshold, as\nexpected.  In contrast, with mutual repression (green data) the\nboundary does not move beyond the position for $A=1$ as $A$ is\nincreased. The same robustness was also observed for other values\nof the Hill coefficient of gap-gene activation (see Fig.\\xspace S7 in Text S1\\xspace).\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=6in]{Figure7.pdf}\n\\end{center}\n\\caption{ {\\bf Mutual repression buffers against correlated variations\n    in the activator levels.}  \\subfig{A} Time- and\n  circumference-averaged Hb\\xspace ($\\Avg{H}$, solid lines) and Kni\\xspace\n  ($\\Avg{K}$, dashed lines) total copy-number profiles along the AP\n  axis for various morphogen dosage factors $A$.  Inset: the\n  corresponding average occupancy of the promoter states with five\n  bound morphogen molecules as a function of $x$.  \\subfig{B} The average \n  Hb\\xspace boundary position $x_t$ as a function of $\\Delta x_A$, the distance\n  between the Hb\\xspace and Kni\\xspace boundaries without mutual repression, for\n  the system with mutual repression (green,) and without it (red);\n  $\\Delta x_A$ is varied by changing the morphogen dosage factor\n  $A$. Note that mutual repression makes the gap-gene expression\n  boundaries essentially insensitive to correlated changes in\n  morphogen levels when $A>1$.  \\subfig{C} AP profiles of the average\n  standard deviation of the total Hb\\xspace ($\\sigma_H$, solid lines) and\n  Kni\\xspace ($\\sigma_K$, dashed lines) copy numbers.  Inset:\n  $\\sigma_{H}(x)\/\\Avg{H}_{max}$ as a function of\n  $\\Avg{H}(x)\/\\Avg{H}_{max}$, where $\\Avg{H}(x)$ is the average Hb\\xspace\n  copy number at $x$ and $\\Avg{H}_{max}$ its maximum over $x$.  The\n  grey dashed line represents the Poissonian limit.\\subfig{D} The Hb\\xspace\n  boundary width $\\Delta x$ as a function of $\\Delta x_A$ with (green)\n  and without (red) mutual repression. For $A=4$, it was impossible to\n  obtain a reliable error bar on $\\Delta x$, because of the weak\n  pinning force on the \\textit{hb}\\xspace and \\textit{kni}\\xspace expression boundaries. }\n\\label{Fig7}\n\\end{figure}\n\n\\mysubsubsection{Mutual repression enhances robustness to\n  embryo-to-embryo variations}\nThe fact that mutual repression can pin expression boundaries,\ndramatically enhances the robustness against embryo-to-embryo\nvariations in the morphogen levels. We did not sample inter-embryo\nvariations in $A$ explicitly, but made an estimate using $\\Delta x_t =\n(dx_t\/dA) \\Delta A$, where $d x_t\/dA$ was taken from Fig.\\xspace \\ref{Fig7}B.\nA correlated symmetric variation $\\delta_A\\equiv\\Delta A\/A=0.1$ of\nboth morphogen levels then would lead to $\\Delta\nx_t(\\delta_A)\\simeq0.82~\\unit{\\% EL}$ at $A=1$ and $\\Delta\nx_t(\\delta_A)\\simeq0.25~\\unit{\\% EL}$ at $A=2$.  Without mutual repression\n$\\Delta x_{t,NR}(\\delta_A)\\simeq2.2~\\unit{\\% EL}$. This analysis thus\nsuggests that mutual repression reduces boundary variations due to\nfluctuations in the morphogen levels by almost a factor of 10 if the\nhalf-activation threshold is slightly posterior to midembryo\n(e.g. $A=2$). If, on average, $A=1$, then mutual repression\nstill reduces $\\Delta x_t$ by inhibiting posterior shifts in those\nembryos in which $A>1$.  These results are consistent with those of\n\\cite{Howard2005,Vakulenko2009}.\n\n\\mysubsubsection{Overlap of morphogen activation domains does not\n  corrupt robustness to intrinsic fluctuations}\nWhile mutual repression proves beneficial in buffering against\nembryo-to-embryo variations in morphogen levels, the question arises\nwhether overlapping morphogen-activation domains does not impair\nrobustness to intrinsic fluctuations arising from noisy gene\nexpression and diffusion of gap gene proteins. We found that this\ndepends on the Hill coefficient of gap-gene\nactivation, which depends on the number $n_{\\rm max}$ of\nmorphogen binding sites on the promoter. Fig.\\xspace \\ref{Fig7}C shows, for $n_{\\rm max}=5$, that even though\nmutual repression increases the noise in gap-gene expression away\nfrom the boundaries, it has little effect on the noise at the boundaries\nwhen $A\\leq 2$. For $A> 2$, the noise does increase significantly;\nin fact, it was impossible to obtain reliable error bars, because of\nthe weak pinning force of the \\textit{hb}\\xspace-\\textit{kni}\\xspace interface. Moreover,\noverlapping morphogen activation domains decrease the steepness of\nthe expression boundaries (panel A), and this increases the boundary\nwidth $\\Delta x$ (panel D). Indeed, when $n_{\\rm max}=5$,\nmutual repression can enhance the precision of gene-expression\nboundaries, but only if the activation domains are adjoining\n($A=1$), or have a marginal overlap ($1<A<2$). For lower values of\n$n_{\\rm max}$, however, this enhancement of precision extends over a much\nbroader range of $A$ values; in fact, when $n_{\\rm max}<3$, mutual repression enhances\nprecision even up to $A=4$ (see Fig.\\xspace S7 in Text S1\\xspace).\n\n\\subsection*{Boundaries shift upon uncorrelated variations in morphogen levels, yet intrinsic noise remains unaltered}\n\\pdfbookmark[2]{Boundaries shift upon uncorrelated variations in morphogen levels, yet intrinsic noise remains unaltered}{BookmarkUncorrelated}\nSince correlated upregulation of both morphogen levels\nis a special case, we also studied the effect of uncorrelated\nactivator scaling.  To this end, only the Bcd\\xspace level was multiplied by\na global factor $A\\in\\lbrace0.5,1,2,3,4\\rbrace$, while\nother parameters were left unchanged.  Again we investigated the Hb\\xspace boundary\nposition $x_t$, its variance $\\Delta x_t(\\Delta A)$ due to extrinsic\n(embryo-to-embryo) variations in $A$ and the variance due to\nintrinsic (intra-embryo) fluctuations $\\Delta x(A)$.  Results for\n$D=1.0\\unit{\\mu m^2\/s}$ are summarized in Fig.\\xspace \\ref{Fig8}.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=6in]{Figure8.pdf}\n\\end{center}\n\\caption{ {\\bf Robustness of the gap-gene expression boundaries to\n    variations in the \\textit{bcd}\\xspace gene dosage.}  \\subfig{A} Time- and circumference-averaged Hb\\xspace\n  ($\\Avg{H}$, solid lines) and Kni\\xspace ($\\Avg{K}$, dashed lines)\n  total copy-nymber profiles along the AP axis for various \\textit{bcd}\\xspace gene dosage\n  factors $A_{Bcd}=A\\in\\lbrace 0.5, 1, 2, 3, 4\\rbrace$ and\n  $D=1.0~\\unit{\\mu m^2\/s}$.  Inset: the average occupancy of the promoter states\n  with five bound morphogen molecules as a function of $x$.\n  \\subfig{B} Comparison of the boundary position\n  $x_t$ as a function of $A$ for $D=1.0~\\unit{\\mu m^2\/s}$ to values measured by\n  Houchmanzadeh et al. \\cite{Houchmandzadeh2002} (black line).  The\n  red line shows the simulation results for the system without mutual\n  repression. Note the good agreement between the experimental data\n  and the simulation data of the system with mutual repression.   \\subfig{C} Profiles of the average standard deviation of the total Hb\\xspace\n  ($\\sigma_H$, solid lines) and Kni\\xspace ($\\sigma_K$, dashed lines) copy\n  number.  Inset: $\\sigma_{H}(x)\/\\Avg{H}_{max}$ as a function of\n  $\\Avg{H}(x)\/\\Avg{H}_{max}$.  The grey dashed line represents the\n  Poissonian limit.\n  \\subfig{D} The Hb\\xspace boundary width $\\Delta x$ as a function of $A$ and\n  $\\Delta x_A$, the separation between the Hb\\xspace and Kni\\xspace boundaries in\n  a system without mutual repression, for the system with (green)\n  and without (red) mutual repression. $\\Delta x_A$ is varied by\n  multiplying the Bcd\\xspace level by $A_{Bcd}$.\n  }\n\\label{Fig8}\n\\end{figure}\n\n\\mysubsubsection{The Hb\\xspace boundary shifts less with mutual repression}\nFig.\\xspace \\ref{Fig8}A shows that the \\textit{hb}\\xspace expression boundary shifts posteriorly\nwith increasing $A$, in contrast to the case of correlated\nactivator scaling. The Kni\\xspace profile retracts in concert with the\nadvance of the Hb\\xspace domain.  In Fig.\\xspace \\ref{Fig8}B we compare the Hb\\xspace\nboundary $x_t(A)$ to the data of Houchmandzadeh et\nal. \\cite{Houchmandzadeh2002}, assuming a 100\\% efficiency of the\nadditional \\textit{bcd}\\xspace gene copies. It is seen that the agreement between\nsimulation and experiment is very good: while $x_t(A)$ of the simulations\nhas a marginal offset as compared to the experimental data, the slope\nof $x_t(A)$ is essentially the same. Moreover, the slope is much lower\nthan that obtained without mutual repression, showing that mutual\nrepression can indeed buffer against uncorrelated variations in\nmorphogen levels. These results parallel those of \n\\cite{Howard2005}.\n\n\\mysubsubsection{Robustness to inter-embryo fluctuations} To estimate\nthe boundary variance due to inter-embryo variations in morphogen\nlevels, we fitted a generic logarithmic function $x_{t,fit}(A):=a \\log\n(A) + b$ to the simulation data, giving $a\\lesssim 15~\\unit{\\% EL}$ for all\nvalues of $D$ studied.  Hence $\\Delta x_t(\\Delta A)\\lesssim\n15~\\unit{\\% EL}~\\Delta A\/A$.  A $10\\%$ variability in $A$ around $A=1$ thus\nwould result in $\\Delta x_t(\\Delta A)\\lesssim 1.5~\\unit{\\% EL}$, which is\nhalf as much as predicted by the model in \\cite{Howard2005} for that\ncase.  Nevertheless, it is yet too large to correspond to the\nexperimental observations of Manu et al. that variations in the Bcd\\xspace\ngradient of $\\Delta A\/A\\approx 20\\%$ correspond to variations in the\nHb\\xspace boundary position of $\\Delta x_t(\\Delta A)\\lesssim 1.1~\\unit{\\% EL}$\n\\cite{Manu2009PlosBiol}. \nOur results therefore support their conjecture that\nhigher levels of Bcd\\xspace are correlated with upregulation of Kni\\xspace and Cad\\xspace.\n\n\\mysubsubsection{Robustness to intra-embryo fluctuations} The output\nnoise at the Hb\\xspace boundary remains largely unaffected (Fig.\\xspace \\ref{Fig8}C\nand inset) by Bcd\\xspace upregulation, whereas the slope is reduced by\napproximately $10\\%$ per doubling of $A$ (data not shown).  As a\nresult, the boundary width $\\Delta x$ stays close to $1~\\unit{\\% EL}$ for all\nconsidered $A$ (green data; Fig.\\xspace \\ref{Fig8}D), remaining lower than that obtained\nwithout mutual repression (red data; Fig.\\xspace \\ref{Fig8}D).\n\n\n\\subsection*{Mutual repression with one morphogen gradient}\n\\pdfbookmark[2]{Mutual repression with one morphogen gradient}{BookmarkOneGradient}\nIn the mutual repression motif discussed above, the two antagonistic\ngenes were activated by independent morphogens, one emanating from the\nanterior and the other from the posterior pole. An alternative mutual\nrepression motif is one in which the two genes are activated by the\nsame morphogen, {\\it e.g.} \\textit{hb}\\xspace and \\textit{kni}\\xspace both being activated by Bcd\\xspace\n\\cite{Saka2007,Cotterell2010}. \n\nWe simulated a system in which \\textit{hb}\\xspace and \\textit{kni}\\xspace mutually repress each\nother, yet both are activated by Bcd\\xspace, with \\textit{kni}\\xspace having a lower Bcd\\xspace\nactivation threshold than \\textit{hb}\\xspace. This generates a Hb\\xspace and Kni\\xspace domain,\nwith the latter being located towards the posterior of the former (see\nFig.\\xspace S4 in Text S1\\xspace). We systematically varied the mutual repression strength and\nthe diffusion constant, to elucidate how mutual repression and spatial\naveraging sculpt stable expression patterns in this motif. Our\nanalysis reveals that since \\textit{hb}\\xspace and \\textit{kni}\\xspace are both activated by the\nsame morphogen gradient, \\textit{hb}\\xspace should repress \\textit{kni}\\xspace more strongly than\nvice versa: with equal mutual repression strengths either a spotty\ngap-gene expression pattern emerges in the anterior half, namely when\nthe Hb\\xspace and Kni\\xspace diffusion constant are low ($D<0.1 \\unit{\\mu m^2\/s}$), or Kni\\xspace\ndominates or even squeezes out Hb\\xspace, namely when their diffusion\nconstant is large.  Nonetheless, for unequal mutual repression\nstrengths and sufficiently high $D$, the repression of \\textit{hb}\\xspace by \\textit{kni}\\xspace does enhance the precision\nand the steepness of the Hb\\xspace boundary, although the effect is smaller\nthan in the two-gradient motif (Fig.\\xspace S4 in Text S1\\xspace). Clearly, while the\none-morphogen-gradient motif cannot provide the robustness against\nembryo-to-embryo variations in morphogen levels that the\ntwo-morphogen-gradient motif can provide, mutual repression can\nenhance boundary precision also in this motif.\n\n\\pagebreak\n\\section*{Discussion}\n\\pdfbookmark[1]{Discussion}{BookmarkDiscussion}\nUsing large-scale stochastic simulations, we have examined the role of\nmutual repression in shaping spatial patterns of gene expression, with\na specific focus on the \\textit{hb}\\xspace-\\textit{kni}\\xspace system. \nOur principal findings are that mutual\nrepression enhances the robustness both against intra-embryonic\nfluctuations due to noise in gap-gene expression and\nembryo-to-embryo variations in morphogen levels.\n\nTo investigate the importance of mutual repression in shaping\ngene-expression patterns, we have systematically varied a large\nnumber of parameters: the strength of mutual repression, the\ndiffusion constant of the gap proteins, the maximum expression\nlevel, the Hill coefficient of gap-gene activation, and the\namplitude of the morphogen gradients. To elucidate how varying\nthese parameters changes the precision of the gap-gene boundaries,\nwe examined how they affect both the steepness of the\ngene-expression boundaries and the expression noise at these boundaries (see\nEq. \\ref{EqDeltaX}). The\neffect on the steepness is, to a good approximation, independent\nof the noise, and would therefore be more accessible\nexperimentally. We find that the steepness increases with\ndecreasing diffusion constant, but increases with increasing strength of mutual repression,\nmaximum expression level, and Hill coefficient of gap-gene\nactivation. Moreover, mutual repression shifts the expression\nboundaries apart and makes the system more robust to\nembryo-to-embryo variations in the morphogen levels. In contrast,\nthe noise at the expression boundaries decreases with increasing\ndiffusion constant,  decreasing expression level,    and decreasing Hill\ncoefficient, while the dependence on the strength of mutual\nrepression is non-monotonic, albeit not very large.\nThe interplay between noise and steepness means that the precision of the\ngap-gene expression boundaries  increases (i.e., $\\Delta\nx$ decreases) with increasing expression level.\nThe dependence of $\\Delta x$ on the diffusion constant and the strength of mutual\nrepression, on the other hand, is non-monotonic: there is an optimal diffusion\nconstant and repression strength that maximizes precision.\nThe effect of the Hill coefficient is conditional on the strength of mutual\nrepression: without mutual repression, the precision\nslightly decreases with increasing Hill coefficient, \nwhile with mutual repression the precision increases with\nincreasing Hill coefficient.\n\n\nWhile mutual repression has only a weak effect on the noise in the\nexpression levels at the gene-expression boundaries, it does markedly\nsteepen the boundaries, especially when the diffusion constant is low.\nIndeed, mutual repression can enhance the precision of gene expression\nboundaries by steepening them.  Nonetheless, even with mutual\nrepression spatial averaging \\cite{Okabe-Oho2009,Erdmann2009} appears\nto be a prerequisite for achieving precise expression boundaries:\nwithout diffusion of the gap proteins, the width of the \\textit{hb}\\xspace expression\nboundary is larger than that observed experimentally\n\\cite{Gregor2007}. Hence, while previous mean-field analysis found\ndiffusion not be important for setting up gene-expression patterns\n\\cite{Papatsenko2011,Manu2009PlosCompBiol}, our analysis underscores\nthe importance of diffusion in reducing copy-number\nfluctuations. In addition, diffusion can anneal patterning defects\nthat might arise from the bistability induced by mutual\nrepression. Diffusion is, indeed, a potent mechanism for\nreducing the effect of fluctuations, such that mean-field analyses\ncan accurately describe mean expression profiles.\n\nInterestingly, the minimum boundary width at the optimal diffusion\nconstant in a system with mutual repression is not much lower than\nthat in one without mutual repression. Yet, in the latter case the\nboundary width is already approximately one nuclear spacing, and there\ndoes not seem to be any need for reducing it further. However, with mutual\nrepression, the same boundary width can be obtained at a lower\ndiffusion constant, where the steepness of the boundaries is much\nhigher, approximately twice as high as that without mutual\nrepression. Our results thus predict that mutual repression allows for\ngap-gene expression boundaries that are both precise and steep. In\nfact, the width and steepness of the boundaries as prediced by our\nmodel are in accordance with those measured experimentally\n\\cite{Surkova2008}.\n\nOur observation that mutual repression increases the steepness\nof gene-expression boundaries without significantly raising the\nnoise, makes the mechanism distinct from other mechanisms for\nsteepening gene expression boundaries, such as lowering diffusion\nconstants \\cite{Erdmann2009} or increasing the cooperativity of\ngene activation (see Fig.\\xspace S6 in Text S1\\xspace). These mechanisms typically involve a\ntrade off between steepness and noise: lowering the diffusion\nconstant or increasing the Hill coefficient of gene activation\nsteepens the profiles but also raises the noise in protein\nlevels at the expression boundary. In fact, increasing the Hill\ncoefficient (without mutual repression) {\\em decreases} the\nprecision of gene-expression boundaries. This is because increasing the\nHill coefficient increases the width of the distribution of times\nduring which the promoter is off, leading to larger promoter-state\nfluctuations and thereby to larger noise in gene expression (see Fig.\\xspace S5 in Text S1\\xspace).\n\nAnother important role of mutual repression as suggested by our\nsimulations is to buffer against inter-embryonic variations in the\nmorphogen levels. Houchmandzadeh {\\it et al.} observed that in\n\\textit{bcd}\\xspace overdosage experiments the Hb\\xspace boundary does not shift as far\nposteriorly as predicted by the French flag model\n\\cite{Houchmandzadeh2002}. One possible explanation that has been put\nforward is that Bcd\\xspace is inactivated in the posterior half of the\nembryo via a co-repressor diffusing from the posterior pole\n\\cite{Howard2005}. More recently, it has been proposed that gap gene\ncross regulation underlies the resilience of the gap-gene expression\ndomains towards variations in the \\textit{bcd}\\xspace gene dosage\n\\cite{Manu2009PlosBiol,Manu2009PlosCompBiol}. Our analysis supports the latter\nhypothesis. In particular, our results show that when the regions in\nwhich \\textit{hb}\\xspace and \\textit{kni}\\xspace are acitvated by their respective morphogens\noverlap, the boundary positions are essentially insensitive to\ncorrelated variations in both morphogen levels, and very robust\nagainst variations of the Bcd\\xspace level only, with the latter being in\nquantitative agreement with what has been observed experimentally\n\\cite{Houchmandzadeh2002}. Moreover, when this overlap is about 0-20\\%\nof the embryo length, mutual repression confers robustness not only\nagainst inter-embryonic variations in morphogen levels, but also\nintra-embryonic fluctuations such as those due to noise in gene\nexpression.\n\nManu {\\it et al.} found that in the \\textit{kr}\\xspace;\\textit{kni}\\xspace double mutant,\nwhich lacks the mutual repression between \\textit{hb}\\xspace and \\textit{kni}\\xspace\/\\textit{kr}\\xspace, the Hb\\xspace\nmidembryo boundary is about twice as wide as that in the wild-type\nembryo \\cite{Manu2009PlosBiol}. This could be due to a reduced\nrobustness against embryo-to-embryo variations in morphogen levels,\nbut it could also be a consequence of a diminished robustness against\nintra-embryonic fluctuations. The analysis of Manu {\\it et al.}\nsuggests the former \\cite{Manu2009PlosBiol,Manu2009PlosCompBiol}, and\nalso our results are consistent with this hypothesis. However, our\nresults also support the latter scenario: for $D\\approx 0.3\\unit{\\mu m^2\/s}$,\nthe Hb\\xspace boundary width in the system without mutual repression is\nabout twice as large as that in the system with mutual repression (see\nFig.\\xspace \\ref{Fig4}C). Clearly, new experiments are needed to establish\nthe importance of intra-embryonic fluctuations versus inter-embryonic\nvariations in gene expression boundaries.\n\nTo probe the relative magnitudes of intra- vs inter-embryonic\nvariations, one ideally would like to measure an ensemble of embryos\nas a function of time; one could then measure the different\ncontributions to the noise in the quantity of interest following\nEq. \\ref{EqNoiseDecom}.  This, however, is not always possible;\nstaining, e.g., typically impedes performing measurements as a\nfunction of time.  The question then becomes: if one measures\ndifferent embryos at a given moment in time, are embryo-to-embryo\nvariations in the mean boundary position or protein copy number\n(thus averaged over the circumference) due to intra-embryonic\nfluctuations in time or due to systematic embryo-to-embryo\nvariations in e.g. the morphogen levels? Experiments performed on different embryos but at one\ntime point cannot answer this question. Our analysis, however,\nsuggests that the intra-embryonic fluctuations in the mean copy\nnumber or boundary position (i.e. averaged over $\\phi$) over time are very\nsmall, and that hence embryo-to-embryo variations in the mean\nquantity of interest are really due to systematic embryo-to-embryo\nvariations; these variations then correspond to \n$\\sigma^2_{\\Avg{\\overline{G}}_\\phi}$ or\n$\\sigma^2_{\\Avg{\\overline{x_t}}_\\phi}$ in Eq. \\ref{EqNoiseDecom}\nor Eq. \\ref{EqXtDecom}, respectively.  The\nintra-embryonic fluctuations, $\\Avg{\\overline{\\sigma^2_G}}_e(x)$\nor $\\Avg{\\overline{\\sigma^2_{x_t}}}_e(x)$, can then be measured\nby measuring the quantity of interest, $G$ or $x_t$, as a\nfunction of $\\phi$, and averaging the resulting variance over\nall embryos. We expect that these observations, in particular\nthe critical one that intra-embryonic fluctuations in the mean\nquantity of interest are small, also hold for\nnon-stationary systems, although this warrants further investigation.\n\nOur model does not include self-activation of the gap\ngenes. Auto-activation has been reported for \\textit{hb}\\xspace, \\textit{kr}\\xspace\nand \\textit{gt}\\xspace, but there seems to be no evidence in case of \\textit{kni}\\xspace\n\\cite{Treisman1989,Jaeger2011}.  The self-enhancement of gap genes has\nthe potential to steepen and sharpen expression domains even more by\namplifying local patterns \\cite{Lopes2008,Holloway2011}. Our\nresults suggest, however, that auto-activation is not necessary to reach the\nboundary steepness and precision as observed experimentally. \n\n Our results provide a new\nperspective on the Waddington picture of development\n\\cite{Waddington1942, Waddington1959}. Waddington argued that\ndevelopment is ``canalized'', by which he meant that cells\ndifferentiate into a well-defined state, despite variations and\nfluctuations in the underlying biochemical processes. It has been\nargued that canalization is a consequence of multistability\n\\cite{Manu2009PlosBiol,Manu2009PlosCompBiol,Papatsenko2011},\nwhich is the idea that cells are driven towards attractors, or basins of attraction in state\nspace. To determine whether a given system is multistable, it is\ncommon practice to perform a stability analysis at the level of\nsingle cells or nuclei. Our results show that this approach should be\nused with care: diffusion of proteins between cells or nuclei within\nthe organism can qualitatively change the energy landscape;\nspecifically, a cell that is truly bistable without diffusion might be\nmonostable with diffusion. Indeed, our results highlight that a\nstability analysis may have to be performed not at the single cell\nlevel, but rather at the tissue level, taking the diffusion of\nproteins between cells into account.\n\n\nFinally, while our results have shown that mutual repression can\nstabilize expression patterns of genes that are activated by morphogen\ngradients, one may wonder whether it is meaningful to ask the converse\nquestion: do morphogen gradients enhance the stability of expression\ndomains of genes that mutually repress each other? This question\npresupposes that stable gene expression patterns can be generated\nwithout morphogen gradients.\nAlthough it was shown that confined (though aberrant) gap gene patterns\nform in the absence of Bcd\\xspace \\cite{Huelskamp1989,Huelskamp1990,Struhl1992} and that\nHb\\xspace can partly substitute missing Bcd\\xspace in anterior embryo patterning\\cite{Simpson-Brose1994},\nit is not at all obvious how precise domain positioning could succeed in such a scenario.\nIn particular, one might expect that with mutual repression\nonly, thus without morphogen gradients, there is no force that pins\nthe expression boundaries. Our results for the large overlapping\nmorphogen-activation domains, with $A=8$, illustrate this problem: in\nthe overlap region, both \\textit{hb}\\xspace and \\textit{kni}\\xspace are essentially fully activated\nby their respective morphogens, as a result of which the morphogen\ngradients cannot determine the positions of the gap-gene boundaries\nwithin this region; indeed, mutual repression has to pin the\nexpression boundaries of \\textit{hb}\\xspace and \\textit{kni}\\xspace. Yet, our results show that in\nthis case the positions of the \\textit{hb}\\xspace and \\textit{kni}\\xspace expression boundaries\nexhibit large and slow fluctuations, suggesting that mutual repression\nalone cannot pin expression boundaries. Interestingly, however, with\n$A=4$, the region in which both genes are activated is still quite\nlarge, about 50\\% of the embryo, and yet even though the underlying\nenergy landscape is flat in this region, the interfaces do\nconsistently move towards the middle of the embryo, due to diffusive\ninflux of Hb\\xspace and Kni\\xspace from the polar regions. It is tempting to\nspeculate that mutual repression and diffusion can maintain stable\nexpression patterns, while morphogen gradients are needed to set up\nthe patterns, {\\it e.g.} by breaking the symmetry between the possible\npatterns that can be formed with mutual repression only.\n\n\\pagebreak\n\\section*{Materials and Methods}\n\\pdfbookmark[1]{Materials and Methods}{BookmarkMatMeth}\nIn the following we describe details of our parameter choice\nand sampling technique.\nTo unravel the mechanisms by which mutual repression shapes\ngene-expression patterns, it is useful to take the Cad\\xspace-Kni\\xspace-system\nto be a symmetric copy of the Bcd\\xspace-Hb\\xspace-system. Cad\\xspace thus\ninherits its parameters from Bcd\\xspace and Kni\\xspace from Hb\\xspace, if not otherwise\nstated.  Table S2 in Text S1\\xspace\ngives an overview of our standard parameter values.  Data from\nexperiments was used whenever possible.  When it was unavailable we\nmade reasonable estimates.\n\n\\mysubsubsection{Binding rates are diffusion limited}\nWe assume all promoter binding rates to be diffusion limited and\ncalculate them via $k_{on}^{\\rm X} =4\\pi\\alpha D_X\/V$.  Here\n$\\alpha=10~\\unit{nm}$ is the typical size of a binding site, $D_X$ is\nthe intranuclear diffusion constant of species $X$ and $V=143.8~\\mu{\\rm m}^3$\nis the nuclear volume. The precise values of $D_X$ for the different species\nin our system are not known.  Gregor et al. have shown experimentally that the\nnuclear concentration of Bcd\\xspace is in permanent and rapid dynamic\nequilibrium with the cytoplasm \\cite{Gregor2007b}, suggesting that\nnuclear and cytoplasmic diffusion constants can be taken for equal.\nThey have found $D_{Bcd}\\simeq 0.32~\\unit{\\mu m^2\/s}$ by FRAP measurements.\nThis value has been subject to controversy because it is too low to\nestablish the gradient before nuclear cycle 10 ($\\simeq\n90~\\unit{min}$) by diffusion and degradation only,  prompting\nalternative gradient formation models \\cite{Bergmann2007, Bialek2008,\n  Barkai2009, Spirov2009, Hecht2009, Porcher2010b}.  A more recent\nstudy revisited the problem experimentally via FCS, yielding\nsignificantly higher values for $D_{Bcd}$ up to $10~\\unit{\\mu m^2\/s}$ with a\nlower limit of $1~\\unit{\\mu m^2\/s}$ \\cite{Abu-Arish2010}.  We therefore have\nchosen a 10x higher value of $D_{Bcd}=D_{Cad}\\equiv D_A=3.2~\\unit{\\mu m^2\/s}$ as\ncompared to the earlier choice in \\cite{Erdmann2009}.  For simplicity,\nthis value is taken for all binding reactions occuring in our model,\nexcept for the dimerization reaction rate $k_{on}^{\\rm D}$, which is taken\nto be higher by a factor of 2 to account for the fact that both reaction\npartners diffuse freely. \n\nTo model cooperative activation of \\textit{hb}\\xspace and \\textit{kni}\\xspace by their respective\nmorphogens, the morphogen-promoter dissociation rate is given by\n$k_{off,n}^{\\rm A}=a\/b^n \/ {\\rm s}$, where $n$ is the number of\nmorphogen molecules that are bound to the promoter; for our standard\ncooperativity $n_{max}=5$ the values of\n$a=410$ and $b=6$ have been chosen such that the threshold\nconcentration for promoter activation (in the absence of repression)\nequals the observed average number of morphogen molecules at midembryo (when\n$A=1$, see below). $n_{max}$ is varied in some simulations; we describe in\nText S1\\xspace how $a$ and $b$ are chosen in these cases.\nThe promoter unbinding rate of \\textit{hb}\\xspace and \\textit{kni}\\xspace (the\nrepressor-promoter unbinding rate) $k_{off}^{\\rm R}$ is a parameter\nthat we vary systematically. To study the potential role of\nbistability we decided to set $k_{off}^{\\rm R}$ to a value which\nensures bistable behavior when both \\textit{hb}\\xspace and \\textit{kni}\\xspace are fully activated\nby their respective morphogens (meaning that all five binding\nmorphogen-binding sites on the promoter are occupied).  This requires\ntight repression, yielding dissociation constants $\\sim\n10^{-2}~\\unit{nM}$ (but see also below).  The dimer dissociation rate is set to be\n$k_{off}^{\\rm D}=k_{on}^{\\rm D}\/V$, which is motivated by the choice\nfor the toggle switch models studied in \\cite{Warren2004, Warren2005} and\n\\cite{Morelli2008}, and asserts that at any moment in time the\nmajority of the gap proteins is dimerized.  This is a precondition for\nbistability in the mean-field limit \\cite{Cherry2000, Warren2004, Warren2005}.\n\nThe parameters of the exponential morphogen gradients are chosen such\nthat the number of morphogen molecules at midembryo and the decay\nlength of the gradient are close to the experimentally observed values\nfor Bcd\\xspace, 690 and $\\lambda=119.5~\\unit{\\mu m}$, respectively\n\\cite{Gregor2007}.\n\n\\mysubsubsection{Production and degradation dynamics}\nThe copy numbers of both monomers and dimers and the effective gap\ngene degradation rate $\\mu_{eff}$ depend in a nontrivial manner on\nproduction, degradation and dimerization rates.  However, for constant\nproduction rate $\\beta$, without diffusion and neglecting promoter\ndynamics, an analytical estimate for the monomer and dimer copy\nnumbers can be obtained from steady state solutions of the rate\nequations (see Text S1\\xspace).  Based on this we have made a\nchoice for $\\beta$ and the monomeric ($\\mu_M$) and dimeric ($\\mu_D$)\ndecay rates that leads to reasonable copy numbers and $\\mu_{eff}$ (see\nTable S2 in Text S1\\xspace).  The latter is defined as the mean of\n$\\mu_M$ and $\\mu_D$ weighted by the species fractions.  $\\mu_M$ and\n$\\mu_D$ are set such that $\\mu_{eff}\\simeq\n4.34\\cdot10^{-3}~\\unit{1\/s}$, which corresponds to an effective\nprotein lifetime of $\\sim 4~\\unit{min}$.  This is close to values used\nearlier \\cite{Howard2005,Erdmann2009} and allows for the rapid\nestablishment of the protein profiles observed in experiments.  The\ndimers have a substantially lower degradation rate than monomers,\nwhich enhances bistability \\cite{Buchler2005}.  The lower decay rate\nof the dimers may be attributed to a stabilizing effect of\noligomerization (cooperative stability) \\cite{Buchler2005}.\n\n\\mysubsubsection{Free parameters}\nOne of the key parameters that we vary systematically is the\ninternuclear gap gene diffusion constant $D$, which defines a nuclear\nexchange rate $k_{ex}=4D\/\\ell^2$ ($\\ell$ = internuclear distance).  To\nstudy the effect of embryo-to-embryo variations in the morphogen\nlevels, the latter are scaled globally by a dosage factor $A$.  We\nconsidered two scenarios: scaling both gradients by the same $A$\n(``correlated variations'') or scaling the Bcd\\xspace gradient only\n(``uncorrelated variations'').  To test how strongly the assumption of\nstrong repressor-promoter binding affects our results, we also varied\nthe repressor-promoter dissociation rate $k_{off}^{\\rm R}$. Moreover,\nto study the dependence of our results on the gap-gene copy numbers,\nwe also increased the protein production rate $\\beta$. These\nsimulations are much more computationally demanding; therefore we limited\nourselves to simulations with $\\beta=2\\beta_0$ and $\\beta=4\\beta_0$ where\n$\\beta_0$ is our baseline value. Finally we\nalso studied a system where both gap genes are activated by the same\ngradient (Bcd\\xspace), varying both the diffusion constant $D$ and the Kni\\xspace\nrepressor off-rate $k_{off}^{\\rm R,Kni\\xspace}$, while keeping $k_{off}^{\\rm\n  R,Hb\\xspace}$ at the standard value.\n\n\\mysubsubsection{Algorithmic details}\nAll simulations are split into a relaxation and a measurement run.\nDuring the relaxation run we propagate the system towards the steady\nstate without data collection.  To reach steady state, as a standard\nwe run $1\\cdot10^9-3\\cdot10^9$ Gillespie steps (ca. $2\\cdot10^5-7\\cdot 10^5$ updates per\nnucleus).  The measurement run is performed with twice the number of\nsteps ($2\\cdot10^9-6\\cdot10^9$). The simulations are started from exponential\nmorphogen gradients and step profiles of the gap proteins; however,\nwe verified that the final result was independent of the precise\ninitial condition, and that the system reached\nsteady state after the equilibration run. The results for\n$A=4$ (Fig.\\xspace \\ref{Fig7}) form, however, an exception: here it was impossible to obtain a\nreliable error bar, because of the weak pinning force on the \\textit{hb}\\xspace and\n\\textit{kni}\\xspace expression boundaries.\n\nIn steady state, we record for each row of nuclei and with a\nmeasurement interval of $\\tau_{m}=100~\\unit{s}$ the Hb\\xspace boundary\nposition $x_t$, i.e. the position where $H$ drops to half of the average\nsteady-state value measured at its plateau close to the anterior pole,\nwhich in our simulations is equal to the maximum average total Hb\\xspace level $\\Avg{H}_{max}$.\nFrom the corresponding histogram we obtain the boundary width $\\Delta x$ by\ncomputing the standard deviation.  Additionally, after runtime we calculate \nan approximation for $\\Delta x$ from the standard deviation\nof $H$ divided by the slope of the averaged $H$ profile, both\nquantities taken at $x_t$, see Eq. \\ref{EqDeltaX} \\cite{Gregor2007,\n  Tostevin2007, Erdmann2009}.  Further details of boundary measurement\nare described in Text S1\\xspace.\n\nError bars for a given quantity are estimated from the standard\ndeviation among $N_B=10$ block averages (block length $6\\cdot10^8$)\ndivided by $\\sqrt{N_B-1}$, following the procedure described in\n\\cite{Frenkel+Smit}.  We verified that estimates with smaller and\nlarger block sizes yield similar estimates for a representative set of\nsimulations.\n\n\\section*{Acknowledgments}\n\\pdfbookmark[1]{Acknowledgements}{BookmarkAcknowlegdements}\nWe thank N. Becker for fruitful discussions and a critical reading \nof the manuscript.\n\n\\pagebreak\n\\pdfbookmark[1]{References}{BookmarkReferences}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn breast examinations, such as mammography, detected actionable tumors are further examined through invasive histology.\nObjective interpretation of these modalities is fraught with high inter-observer variability and limited reproducibility~\\cite{nature}. In this context, a reference based assessment, such as presenting prior cases with similar disease manifestations (termed Content Based Image Retrieval (CBIR)) could be used to circumvent discrepancies in cancer grading. With growing sizes of clinical databases, such a CBIR system ought to be both scalable and accurate. Towards this, hashing approaches for CBIR are being actively investigated for representing images as compact binary codes that can be used for fast and accurate retrieval~\\cite{iupui2,KSH,deepresidual}.  \n\nMalignant carcinomas are often co-located with\nbenign looking manifestations and suspect normal tissues. In such cases, describing the whole image with a single label is often inadequate for objective machine learning and alternatively requires expert annotations delineating the exact location of the \nregion of interest. This argument extends to screening modalities like mammograms, where multiple anatomical views are acquired. In such scenarios, the status of the tumor is best represented to a CBIR system by constituting a bag of all associated images, thus veritably becoming multiple instance (MI) in nature. This is illustrated in Fig.~\\ref{fig:exampleBag}. With this as our premise we present, for the first time, a novel deep learning based MI hashing method, termed as Robust Multiple Instance Hashing (RMIH). \n\n\\begin{figure}[t]\n\\includegraphics[width=\\textwidth]{MIBagExampleNew}\n\\caption{Schematic representation of bag representation. Here, an example of a large region of interest ($\\sim$ 6K $\\times$ 4K) labeled as \\textit{malignant} is shown wherein a few patches overlapping with the actual tumor represent the underlying malignant concept while proximal patches are potentially benign or less discriminative connective \/ lipidic tissues. It must be noted that these are not individually identified and only a bag-level weak annotation is available for learning.}\n\\label{fig:exampleBag}\n\\end{figure}\n\n\n Seminal works on shallow learning-based hashing include Iterative Quantization (ITQ)~\\cite{ITQ}, Kernel Sensitive Hashing (KSH)~\\cite{KSH} \\textit{etc.} that propose a two-stage framework involving extraction of hand-crafted features followed by binarization. Yang \\textit{et al.} extend these methods to MI learning scenarios with two variants: Instance Level MI Hashing (IMIH) and Bag Level MI Hashing (BMIH)~\\cite{bmih}. However, these approaches are not end-to-end and are susceptible to semantic gap between features and associated concepts. Alternatively, deep hashing methods such as simultaneous feature learning and hashing (SFLH)~\\cite{lai2015}, deep hashing networks (DHN)~\\cite{zhuloss} and deep residual hashing (DRH)~\\cite{deepresidual} to name a few, propose the learning of representations and hash codes in an end-to-end fashion, in effect bridging this semantic gap. It must be noted that all the above deep hashing works targeted single instance (SI) hashing scenarios and an extension to MI hashing was not investigated.\n \n\\begin{figure}[t]\n\\includegraphics[width=\\textwidth]{breasttexMIPOOLpdf}\n\\caption{Overview of RMIH for end-to-end generation of bag-level hash codes. Breast anatomy image is attributed to Cancer Research UK\/Wikimedia Commons.}\n\\label{fig:overview}\n\\end{figure} \n \n\n\nEarlier works on MI deep learning in computer vision include work by Wu \\textit{et al.}~\\cite{wu2015deep}, where the concept of an MI pooling (MIPool) layer is introduced to aggregate representations for multi-label classification. Yan \\textit{et al.} leveraged MI deep learning for efficient body part recognition~\\cite{bodypart}. Unlike MI classification that potentially substitutes the decision of the clinician, retrieval aims at presenting them with richer contextual information similar to the case at hand to facilitate decision-making.  \nRMIH effectively bridges the two concepts for CBIR systems by combining the representation learning strength of deep MI learning with the potential for scalability arising from hashing. Within CBIR for breast cancer, notable prior art includes work on mammogram image retrieval by Jiang \\textit{et al.}~\\cite{mammogram1} and large-scale histology retrieval by Zhang \\textit{et al.}~\\cite{iupui2}. Both these works pose CBIR as an SI retrieval problem.\nContrasting with~\\cite{mammogram1} and~\\cite{iupui2}, within RMIH we create a bag of images\nto represent a particular pathological case and generate a bag-level hash code, as shown in Fig.~\\ref{fig:overview}. Our contributions in this paper include: \\textbf{1)} introduction of a robust supervised retrieval loss for learning in presence of weak labels and potential outliers; \\textbf{2)} propose training with an auxiliary SI arm with gradual loss trade-off for improved trainability; and \\textbf{3)} incorporation of the MIPool layer to aggregate representations across variable number of instances within a bag, generating bag-level discriminative hash codes. \n\n\n\n\n\n  \n\\section{Methodology}\n\\begin{wrapfigure}[20]{r}{0.39\\textwidth}\n\\vspace{-17pt}\n\\centering\n\\includegraphics[width=0.38\\textwidth]{resnet50}\n\\caption{RMIH Architecture with ResNet-50~\\cite{residualHe} as the \\textit{Deep CNN} model.}\n\\label{fig:architecture}\n\\end{wrapfigure}\nLets consider database $\\mathcal{B} = \\{B_1, \\dots, B_{N_{B}}\\}$ with $N_{B}$ bags. Each bag, $B_{i}$, with varying number ($n_{i}$) of instances ($I_{i}$) is denoted as $B_i = \\{I_1,\\dots,I_{n_{i}}\\}$. We aim at learning $\\mathcal{H}$ that maps each bag to a $K$-d Hamming space $\\mathcal{H}:\\mathcal{B} \\rightarrow\\{-1,1\\}^K$, such that \\textit{bags with similar instances and labels are mapped to similar codes}. For supervised learning of $\\mathcal{H}$, we define a bag-level pairwise similarity matrix $\\mathcal{S}^{\\text{MI}}=\\{s_{ij}\\}^{N_{B}}_{ij=1}$, such that $s_{ij}=1$ if the bags are similar and zero otherwise. \nIn applications, such as this one, where retrieval ground truth is unavailable\nwe can use classification labels as a surrogate for generating $\\mathcal{S}^{\\text{MI}}$. \n\n\n\n\n\n\\noindent\n\\textbf{Architecture}: As shown in Fig.~\\ref{fig:architecture}, the proposed RMIH framework consists of a deep CNN terminating in a fully connected layer (FCL). Its outputs $\\left \\{ z_{ij} \\right \\}_{j=1}^{n_{i}}$ are fed into the MIPool layer to generate the aggregated representation $\\hat{z}_{i}$ that is pooled ($\\text{max}_{\\forall j}\\left \\{ z_{ij} \\right \\}_{j=1}^{n_{i}}$, mean$(\\cdot)$, \\textit{etc.}\n) across instances within the bag. \n$\\hat{z}_{i}$ is an embedding in the space of the bags and is the input of a fully connected MI hashing layer. The output of this layer is squashed to $[-1, 1]$ by passing it through a tanh$\\{\\cdot\\}$ function to generate $h_i^{\\text{MI}}$, which is quantized to produce bag-level hash codes as $\\mathbf{b}_{i}^{\\text{MI}}=\\ \\text{sgn}\\ (\\mathbf{h}_{i}^{\\text{MI}})$. The deep CNN mentioned earlier could\nbe a pretrained network, such as VGGF \\cite{vgg}, GoogleNet~\\cite{goingdeeper}, ResNet50 (R50)~\\cite{residualHe} or an application specific network. \n\nDuring training of RMIH, we introduce an auxiliary SI hashing (aux-SI) arm, as shown in Fig.~\\ref{fig:architecture}. It taps off at the FCL layer and feeds directly into a fully connected SI hashing layer with tanh$\\{\\cdot\\}$ activation to generate instance level non-quantized hash codes, denoted as $\\{h_{ij}^{\\text{SI}}\\}_{j=1}^{n_{i}}$. While training RMIH using backpropagation, the MIPool layer significantly sparsifies the gradients (analogous to using very high dropout while training CNNs), thus limiting the trainability of the preceding layers. The SI hashing arm helps to potentially mitigate this by producing auxiliary instance level gradients. \n\n\\noindent\n\\textbf{Model Learning and Robust Optimization}: \nTo learn similarity preserving hash codes, we propose a robust version of supervised retrieval loss based on neighborhood component analysis employed by~\\cite{torralba}. The motivation to introduce robustness within the loss function is two-fold: (1) robustness induces immunity to potentially noisy\nlabels due to high inter-observer variability and limited reproducibility for the applications at hand~\\cite{nature}; (2) it can effectively counter ambiguous label assignment while training with the aux-SI hashing arm. \n\nGiven $\\mathcal{S}^{\\text{MI}}$, the robust supervised retrieval loss $J_{\\mathcal{S}}^{\\text{MI}}$ is defined as: \n\n\\begin{equation}\n{J_{\\mathcal{S}}^{\\text{MI}} = 1 - \\frac{1}{N_{B}^{2}} \\sum_{i,j = 1}^{N_{B}} s_{ij}p_{ij}}\n\\end{equation}\n\\noindent\nwhere  $p_{ij}$ is the probability that two bags (indexed as $i$ and $j$) are neighbors. Given hash codes $\\mathbf{h_i} = \\left \\{ h_{i}^{k} \\right \\}_{k=1}^{K}$ and $\\mathbf{h_j}$, we define a bit-wise residual operation $r_{ij}$ as $r_{ij}^k = (h_i^k - h_j^k)$. We estimate $p_{ij}$ as:\n\\begin{equation}\np_{ij} = \\frac{e^{-\\mathcal{L}_{\\text{Huber}}(\\mathbf{h_i},\\mathbf{h_j})}}{\\sum_{i\\neq l}^{N_{B}}e^{-\\mathcal{L}_{\\text{Huber}}(\\mathbf{h_i},\\mathbf{h_l})}},\\text{  where } \\mathcal{L}_{\\text{Huber}}(\\mathbf{h_i},\\mathbf{h_j}) = \\sum_{\\forall k}\\rho_k(r_{ij}^{k})\n\\end{equation}\n\\noindent\nThe Huber norm's robustness operation $\\rho_k$ is defined as: \n\\begin{equation}\n\\rho_k(r_{ij}^k) = \\begin{cases}\n\\dfrac{1}{2}(r_{ij}^k)^2, &\\text{if\\ $\\mid r_{ij}^k \\mid \\leqslant c_k$}\\\\\nc_k \\mid r_{ij}^k \\mid -  \\dfrac{1}{2} c_k^2, &\\text{if $\\mid r_{ij}^k \\mid > c_k$}\n\\end{cases}\n\\label{eq:huber}\n\\end{equation}\n\\noindent\nIn Eq.~\\eqref{eq:huber}, the tuning factor $c_k$ is estimated inherently from the data and is set to $c_k=1.345 \\times \\sigma_k$. The factor of $1.345$ is chosen to provide approximately 95\\% asymptotic efficiency and $\\sigma_k$ is a robust measure of bit-wise variance of $r_{ij}^k$. Specifically, $\\sigma_k $ is estimated as $1.485$ times the median absolute deviation of $r_{ij}^k$ as empirically suggested in~\\cite{robusthuber}. This robust formulation provides immunity to outliers during training by clipping their gradients. For training with the aux-SI hashing arm, we employ a similar robust retrieval loss $J_{\\mathcal{S}}^{\\text{SI}}$ defined over single instances with bag-labels assigned to member instances.\n\nTo minimize loss of retrieval quality due to quantization, we use a differentiable quantization loss $J_{Q} = \\sum_{i=1}^M(\\text{log}\\ \\text{cosh}(\\mid\\mathbf{h_i}\\mid - \\ \\mathbf{1}))$ proposed in~\\cite{zhuloss}. This loss also counters the effect of using continuous relaxation in definition of $p_{ij}$ over using Hamming distance. \nAs a standard practice in deep learning, we also add an additional weight decay regularization term $R_{W}$, which is the Frobenius norm of the weights and biases, to regularize the cost function and avoid over-fitting. \n\n\n\n\\begin{wrapfigure}[15]{r}[0pt]{0pt}\n\\resizebox{0.4\\textwidth}{!}{\n\\begin{tikzpicture}\n\\begin{axis}[\n\txlabel= \\Large{t (Epochs)},ylabel= {\\Large{$\\lambda_{\\text{MI\/SI}}^t$}},ylabel shift = -1 pt,legend style={at={(0.46,0.5)},anchor=west,draw=none}]\n\\addplot[smooth,blue,ultra thick] coordinates 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Trade-off.}}\\label{wrap-weights}\n\\end{wrapfigure}\n\n\\noindent\nThe following composite loss is used to train RMIH: \n\\begin{equation}\nJ = \\lambda_{\\text{MI}}^{t} J_{\\mathcal{S}}^{\\text{MI}}+ \\lambda_{\\text{SI}}^{t} J_{\\mathcal{S}}^{\\text{SI}}+ \\lambda_q J_{Q} + \\lambda_w R_{W}\n\\end{equation}\nwhere $ \\lambda_{\\text{MI}}^{t}$, $ \\lambda_{\\text{SI}}^{t}$, $\\lambda_q$ and $\\lambda_w$ are hyper-parameters that control the contribution of each of the loss terms. Specifically, $ \\lambda_{\\text{MI}}^{t}$ and $ \\lambda_{\\text{SI}}^{t}$ control the trade-off between the MI and SI hashing losses. The SI arm plays a significant role only in the early stages of training and can be traded off eventually to avoid sub-optimal MI hashing. For this we introduce a weight trade-off formulation that gradually down-regulates  $ \\lambda_{\\text{SI}}^{t}$, while simultaneously up-regulating $ \\lambda_{\\text{MI}}^{t}$. Here, we use $\\lambda_{\\text{SI}}^{t} = 1 - 0.5\\left ( 1 - \\nicefrac{t}{t_{\\text{max}}} \\right )^{2}$ and $\\lambda_{\\text{MI}}^{t} = 1 - \\lambda_{\\text{SI}}^{t}$, where $t$ is the current epoch and $t_{\\text{max}}$ is the maximum number of epochs (see Fig.~\\ref{wrap-weights}). We train RMIH with mini-batch stochastic gradient descent (SGD) with momentum. Due to potential outliers that can occur at the beginning of training, we scale $c_{k}$ up by a factor of 7 for $t = 1$ to allow a stable state to be reached. \nSpecifically, the gradient of $J_{\\mathcal{S}}^{(\\cdot)}$ w.r.t. to $\\mathbf{h}_{i}$ is derived as: \n\\begin{align}\n\\frac{\\partial J_{\\mathcal{S}}^{(\\cdot)}}{\\partial \\mathbf{h}_{i}} = &\\left ( \\sum_{l:s_{li} > 0} p_{li}\\mathbf{\\mathcal{L'_H}}(\\mathbf{h}_l,\\mathbf{h}_i) - \\sum_{l \\neq i}\\left ( \\sum_{q:s_{lq} > 0} p_{lq} \\right ) p_{li}\\mathbf{\\mathcal{L'_H}}(\\mathbf{h}_l,\\mathbf{h}_i) \\right ) \\nonumber \\\\ \n- &\\left ( \\sum_{j:s_{ij} > 0} p_{ij}\\mathbf{\\mathcal{L'_H}}(\\mathbf{h}_i,\\mathbf{h}_j) - \\sum_{j:s_{ij} > 0} p_{ij}\\left ( \\sum_{z \\neq i} p_{iz}\\mathbf{\\mathcal{L'_H}}(\\mathbf{h}_i,\\mathbf{h}_z) \\right ) \\right ) \n\\label{eq:derJSbig}\n\\end{align}\n\\noindent\nwhere ${\\mathcal{L_H'}}(\\mathbf{h_i},\\mathbf{h_j}) = \\{ {\\rho_k'}(r_{ij}^k)\\}_{k=1}^{k}$. The derivative of the huber term ${\\rho_k}'(r_{ij}^k)$ can be computed as:\n\\begin{equation}\n{\\rho_k'}(r_{ij}^k) = \\begin{cases}\nr_{ij}^k, &\\text{if\\ $\\mid r_{ij}^k \\mid \\leqslant c_k$}\\\\\nc_k\\ \\text{sgn}(r_{ij}^k), &\\text{if $\\mid r_{ij}^k \\mid > c_k$}\n\\end{cases}\n\\end{equation}\n\nRegarding the quantization loss function, the derivative can be computed by $\\frac{\\partial J_{Q}}{\\partial \\mathbf{h}_{i}} = \\text{tanh}\\left ( \\left | \\mathbf{h}_{i} \\right | - \\mathbf{1} \\right )\\text{sgn}\\left ( \\mathbf{h}_{i} \\right )$. Having computed these gradients, we use backpropagation to compute the derivatives of the preceding layers.\n\n\n\\begin{figure}[t]\n\\centering\n\\begin{minipage}{\\textwidth}\n  \\begin{minipage}[t]{0.6\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{RandomImages}\n  \\end{minipage}\n  \\hfill\n  \\begin{minipage}[t]{0.36\\textwidth}\n\\vspace{-3.4cm}\n\\captionof{figure}{Select images from the \\textit{IPUHL} and \\textit{DDSM} datasets to showcase the degree of anatomical variability within and across the classes.}\n\\label{fig:randomimages}\n    \\end{minipage}\n  \\end{minipage}\n  \\vspace{-0.7cm}\n\\end{figure} \n\n\n\n\n\\section{Experiments}\n\n\\textbf{Databases:} Clinical applicability of RMIH has been validated on two large scale datasets, namely, Digital Database for Screening Mammography (DDSM)~\\cite{mammogram1,mammogram2} and a retrospectively acquired histology dataset from the Indiana University Health Pathology Lab (\\textit{IUPHL})~\\cite{iupui1,iupui2}. The \\textit{DDSM} dataset comprises of 11,617 expert selected regions of interest (ROI) curated from 1861 patients. Multiple ROIs associated with a single breast from anatomical views constitute a bag (size: 1-12; median: 2), which has been annotated as normal, benign or malignant by expert radiologists.\nA bag labeled \\textit{malignant} could potentially contain multiple suspect normal and benign masses, which have not been individually identified. The \\textit{IUPHL} dataset is a collection of 653 ROIs from histology slides from 40 patients (20 with precancerous ductal hyperplasia (UDH) and rest with ductal carcinoma \\textit{in situ} (DCIS)) with ROI level annotations done by expert histopathologists. Due to high variability in sizes of these ROIs (upto 9K $\\times$ 8K pixels), we extract multiple patches (of size $1024\\times1024$) and populate a ROI-level bag (size: 1-15; median: 8). \nAs cellular and nuclei level characteristics are important to distinguishing DCIS from UDH, it is not recommended to rescale these images to standard input sizes used by CNNs (typically, 244 $\\times$ 224 in~\\cite{vgg, residualHe, goingdeeper}). Fig.~\\ref{fig:randomimages} illustrates select images from the two datasets to showcase anatomical variability within and across the constituent classes.\nFrom both the datasets, we use patient-level\nnon-overlapping splits to constitute the training (80\\%) and testing (20\\%) sets. \n\n\\textbf{Model Settings and Validations}: To validate proposed contributions, namely robustness within NCA loss and trade-off from the aux-SI arm, we perform ablative testing with combinations of their baseline variants by fine-tuning multiple network architectures. Additionally, we compare RMIH against four state-of-the art methods: ITQ~\\cite{ITQ}, KSH~\\cite{KSH}, SFLH~\\cite{lai2015} and DHN~\\cite{zhuloss}. For a fair comparison, we use R50 for both SFLH and DHN, since as discussed later it performs the best. Since SFLH and DHN were originally proposed for SI hashing, we introduce additional MI variants by hashing through the MIPool layer.\nFor ITQ and KSH, we further create two comparative settings: \\textbf{1)} Using IMIH~\\cite{bmih} that learns instance-level hash codes followed by bag-level distance computation and \\textbf{2)} Utilizing BMIH~\\cite{bmih} using bag-level kernalized representations followed by binarization.\nFor IMIH and SI variants of SFLH, DHN and RMIH, given two bags  $B_p$ and $B_q$ with SI hash codes, say  $\\mathcal{H}(B_q) = \\{h_{q1},\\dots,h_{qM}\\}$ and $\\mathcal{H}(B_p) = \\{h_{p1},\\dots,h_{pN}\\}$, the bag-level distance is computed as: \n\\begin{equation}\nd(B_p,B_q) = \\frac{1}{M}\\sum_{i=1}^{M}(\\min_{\\forall j }\\ \\text{Hamming}(h_{pi},h_{qj})).\n\\label{eq:bagDist}\n\\end{equation}\n\n\n\n\n\nAll images were resized to $224 \\times 224$ and training data were augmented\nto create equally balanced classes. $ \\lambda_{\\text{MI}}^{t}$ and $ \\lambda_{\\text{SI}}^{t}$ were set assuming $t_{\\text{max}}$ as 150 epoch; $\\lambda_q$ and $\\lambda_w$ were set at 0.05 and 0.001 respectively. The momentum term within SGD was set to 0.9 and batch size to 128 for \\textit{DDSM} and 32 for \\textit{IUPHL}. For efficient learning, we use an exponentially decaying learning rate initialized at 0.01. The RMIH framework was implemented in MatConvNet~\\cite{matconvnet}. We use standard retrieval quality metrics: nearest neighbor classification accuracy (nnCA) and precision-recall (PR) curves to perform the aforementioned comparisons. The results (nnCA)\nfrom ablative testing and comparative methods are tabulated in Table~\\ref{wrap-tab:ablative} and Table~\\ref{table:CM} respectively. Within Table~\\ref{table:CM}, methods were evaluated at two different code sizes (16 bits and 32 bits). We also present the PR curves of select bag-level methods (32 bits) in Fig. \\ref{fig:PRCurves}.\n\n\n\n\n\n\n\n\n\\section{Results and Discussion} \\label{results}\n\n\\begin{figure}[t]\n\\resizebox{\\textwidth}{!}{\\begin{tabular}{|c|c|C|C|c|c|c|c|c|c|} \\hline\n\\multicolumn{2}{|c|}{\\multirow{2}{*}{\\textbf{Method}}} & \\multicolumn{2}{c|}{\\thead{\\textbf{Variants}}} & \\multicolumn{3}{c|}{\\thead{\\textit{\\textbf{DDSM}}}} & \\multicolumn{3}{c|}{\\thead{\\textit{\\textbf{IUPHL}}}}\\\\ \\cline{3-10}\n\\multicolumn{2}{|c|}{\\multirow{2}{*}{}} & \\thead{R} &\\thead{T} &\\thead{\\ VGGF$\\ $} &\\thead{\\ R50$\\ $} &\\thead{\\ GN$\\ $ }&\\thead{\\ VGGF$\\ $} &\\thead{\\ R50$\\ $} &\\thead{\\ GN$\\ $} \\\\ \\hline\n\\multirow{5}{*}{\\makecell{{\\small \\textbf{Ablative}}\\\\ {\\small \\textbf{Testing}}}} & A & $\\circ$ & $\\circ$ & 68.65 & 72.76  & 71.70 & 83.85  & 85.42  &  82.29  \\\\ \\cline{2-10}\n& B & $\\circ$ & $\\bullet$ & 75.38 & 77.34  & 72.92  & 85.94  & 90.10  & 88.02  \\\\ \\cline{2-10}\n& C & $\\bullet$ & $\\circ$  & 70.65 & 76.63 & 70.02 & 83.33  & 85.94  &  86.46  \\\\ \\cline{2-10}\n& D & $\\circ$ & \\tiny{$\\blacksquare$} & 66.65 & 69.67 & 68.26 & 83.33  & 88.54  &  84.90  \\\\ \\cline{2-10}\n& E & $\\bullet$ & \\tiny{$\\blacksquare$} & 67.05 & 76.59 & 72.84 & 84.38  & 89.58  &  85.42  \\\\\\hline\n\\multicolumn{2}{|c|}{\\textbf{RMIH-mean}} & $\\bullet$ & $\\bullet$ & 78.67 & 82.31 & 76.83  &  87.50 & 89.58  & \\textbf{89.06} \\\\ \\hline\n\\multicolumn{2}{|c|}{\\textbf{RMIH-max}}& $\\bullet$ & $\\bullet$ & \\textbf{81.21} & \\textbf{85.68}  & \\textbf{78.67}  & \\textbf{91.67} & \\textbf{95.83} & 88.02 \\\\ \\hline\n\\multicolumn{2}{|c|}{\\textbf{RMIH}($\\lambda_q=0$)} & $\\bullet$ & $\\bullet$ & 75.34 & 79.88 & 73.06 & 87.50 & 89.58  &  88.51 \\\\ \\hline\n\\multicolumn{2}{|c|}{\\textbf{RMIH NB}}  &  $\\bullet$ & $\\bullet$ & 83.25 & 88.02 & 79.06  & 94.79  & 96.35 & 92.71 \\\\ \\hline\n\\multirow{4}{*}{\\textbf{Legend}}  \n& \\multicolumn{3}{r|}{\\textbf{R(Robustness)}} & \\multicolumn{6}{l|}{ $\\circ=L_2,\\  \\bullet=L_{\\text{Huber}}$}\\\\ \n& \\multicolumn{3}{r|}{\\makecell[r]{\\textbf{T(Trade-off)}}} & \\multicolumn{6}{l|}{\\makecell[l]{$\\circ=\\text{\\small{Equal weights,}}\\ \\bullet=\\text{\\small{Decaying SIL weights,}}$ \\\\ \\scriptsize{$\\blacksquare$} \\small{$=\\text{No SIL branch}$}}}\\\\ \n& \\multicolumn{3}{r|}{\\textbf{Networks}} & \\multicolumn{6}{l|}{R50: ResNet50,\\ GN: GoogleNet} \\\\  \\hline\n\\end{tabular}}\n\\caption{Performance of ablative testing at code size of 16 bits. We report the nearest neighbor classification accuracy (nnCA) estimated over unseen test data. Letters A-E are introduced for easier comparisons, discussed in Section \\ref{results}.}\n\\label{wrap-tab:ablative}\n \\vspace{-0.5cm}\n\\end{figure}\n\n\n\\textbf{Effect of aux-SI Loss}: To justify using the aux-SI loss, we introduce a variant of RMIH without it (E in Table~\\ref{wrap-tab:ablative}), which\nleads to a significant decline of 3\\% to 14\\% \nin contrast to RMIH. This could be potentially attributed to the prevention of the gradient sparsification caused by the MIPool layer. From Table~\\ref{wrap-tab:ablative}, we observe a 3\\%-10\\% increase in performance, comparing cases with gradual decaying trade-off (B)\nagainst baseline setting ($\\lambda^{t}_{\\text{MI}} = \\lambda^{t}_{\\text{SI}} = 0.5$, A,C).\n\n\n\\textbf{Effect of Robustness}: For robust-NCA, we compared against the original NCA formulation proposed in~\\cite{torralba} (A,B,D in Table~\\ref{wrap-tab:ablative}). Robustness helps handle potentially noisy MI labels, inconsistencies within a bag (like non-informative patches) and the ambiguity in assigning SI labels. Comparing the effect of robustness \nfor baselines sans the SI hashing arm (D \\textit{vs.} E) \nwe observe marginally positive improvement across the architectures and datasets, with a substantial 7\\% in ResNet50 for \\textit{DDSM}. Robustness contributes more with the addition of the aux-SI hash arm (proposed \\textit{vs.} E)\nwith improved performance in the range of 4\\%-5\\% across all settings. This observation further validates our prior argument. \n\n\\textbf{Effect of Quantization}: To assess the effect of quantization, we define two baselines: (1) setting $\\lambda_{q} = 0$ and (2) using non-quantized hash codes for retrieval (RMIH - NB). The latter potentially acts as an upper bound for performance evaluation. From Table~\\ref{wrap-tab:ablative}, we observe a consistent increase in performance by margins of 3\\%-5\\% if RMIH is learnt with an explicit quantization loss to limit the associated error. It must also be noted that comparing with RMIH - NB, there is only a marginal fall in performance (2\\%-4\\%), which is desired. \n\n\nComparing max \\textit{vs.} mean MI Pool variants, we observe that max achieves marginally better performance, since it is more selective than mean, which is particularly important in cases of detecting malignancy. \n\n\\begin{figure}[t]\n\\includegraphics[width=\\textwidth]{RetrievalLarge_compressed}\n\\caption{Retrieval results for RMIH at code size 16 bits. +1 indicates retrieval from class consistent with query and -1 indicates otherwise.}\n\\label{fig:retrievedimages}\n\\vspace{-8pt}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\begin{minipage}{\\textwidth}\n  \\begin{minipage}[t]{0.7\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{PR_Curves-cropped}\n  \\end{minipage}\n  \\hfill\n  \\begin{minipage}[t]{0.26\\textwidth}\n\\vspace{-2.7cm}\n\\captionof{figure}{PR curves for \\textit{DDSM} and \\textit{IUPHL} datasets at code size of 32.}\n\\label{fig:PRCurves}\n    \\end{minipage}\n  \\end{minipage}\n  \\vspace{-0.7cm}\n\\end{figure} \n\n\n\n\nAs a whole, the two-pronged proposed approach, including robustness and trade-off, along with quantization loss delivers the highest performance, proving that RMIH is able to learn effectively, despite the ambiguity induced by the SI hashing arm. Fig.~\\ref{fig:retrievedimages} demonstrates the retrieval performance of RMIH on the target databases. For \\textit{IUPHL}, the retrieved images are semantically similar to the query as consistent anatomical signatures\nare evident in the retrieved neighbors. For \\textit{DDSM}, in the cancer and normal cases the retrieved neighbors are consistent, however it is hard to distinguish between benign and malignant. The retrieval  time for a single query for RMIH was observed at 31.62 ms (for \\textit{IUPHL}) and 17.48 ms (for \\textit{DDSM}) showing potential for fast and scalable search.  \n\n\n\n\n\n\\begin{wraptable}[18]{L}{6.0cm}\n\\vspace{-0.4cm}\n    \\centering\n   \\resizebox{5.9cm}{!}{\\begin{tabular}{x{0.4cm}|x{2.2cm}|x{1cm}|x{0.5cm}|x{1cm}|x{1cm}|x{1.2cm}|x{1cm}|}\n\\multicolumn{2}{c|}{\\multirow{2}{*}{\\textbf{Method}}} & \\multirow{2}{*}{\\thead{\\textbf{A\/F}}} & \\multirow{2}{*}{\\thead{\\textbf{L}}} & \\multicolumn{2}{c|}{\\thead{\\textit{\\textbf{DDSM}}}} & \\multicolumn{2}{c|}{\\thead{\\textit{\\textbf{IUPHL}}}} \\tabularnewline \\cline{5-8}  \n\\multicolumn{2}{c|}{} & & & 16-bit &  32-bit &  16-bit &  32-bit \\tabularnewline \\cline{1-8}\n\\multirow{8}{*}{\\rotatebox[origin=c]{90}{\\textbf{Shallow}}} & \\multirow{4}{*}{ITQ \\cite{ITQ}} & R50 & $\\circ$ & 66.35 & 67.71 & 78.58 &  80.28  \\tabularnewline \\cline{3-8}\n& & R50 & $\\bullet$ & 64.56 & 71.98 & 89.58 & 79.69 \\tabularnewline \\cline{3-8} \n& & G & $\\circ$ & 65.22 & 66.55 & 51.79  & 51.42    \\tabularnewline  \\cline{3-8}\n&  & G & $\\bullet$ & 59.73 & 61.03 &  57.29 &  58.85  \\tabularnewline \\cline{2-8}\n& \\multirow{4}{*}{KSH \\cite{KSH}} &  R50 & $\\circ$  & 61.88 & 64.81 & 87.74 & 86.51 \\tabularnewline \\cline{3-8}\n& &  R50 & $\\bullet$ & 59.81 & 72.17 & 70.83 & 80.21 \\tabularnewline \\cline{3-8}\n& & G & $\\circ$  & 60.50 & 61.91  & 57.36  & 57.83 \\tabularnewline \\cline{3-8}\n& & G & $\\bullet$ & 55.34 & 55.67 & 60.94 &  58.85 \\tabularnewline \\hline\n\\multirow{6}{*}{\\rotatebox[origin=c]{90}{\\textbf{Deep}}} & \\multirow{2}{*}{SFLH \\cite{lai2015}} & R50  & $\\circ$  & 73.54 & 77.46 & 83.33 & 85.94 \\tabularnewline \\cline{3-8}\n&  &  R50M & \\tiny{$\\blacksquare$} & 71.98 & 75.93 & 85.42 & 88.54  \\tabularnewline \\cline{2-8}\n& \\multirow{2}{*}{DHN \\cite{zhuloss}} &  R50 & $\\circ$ & 65.64 & 74.79 & 82.29 & 86.46  \\tabularnewline \\cline{3-8}\n&  &  R50M & \\tiny{$\\blacksquare$} & 72.88 & 80.43 & 88.02 & 90.62  \\tabularnewline  \\cline{2-8}\n& RMIH-SIL &  R50 & $\\circ$ & 76.02 & 78.37 & 87.92 & 88.58 \\tabularnewline  \\cline{2-8}\n& \\textbf{RMIH} & R50M  & \\tiny{$\\blacksquare$} & \\textbf{85.68} & \\textbf{89.47} & \\textbf{95.83} & \\textbf{93.23}  \\tabularnewline \\hline\n\\multirow{3}{*}{\\rotatebox[origin=c]{90}{\\textbf{Legend}}} & \\makecell*[r]{\\textbf{A\/F:}} & \\multicolumn{6}{l|}{\\makecell*[l]{\\small{\\textbf{A}: Architecture,\\ \\textbf{F}: Features} \\tabularnewline \\scriptsize{R50: ResNet50, R50M: ResNet50$+$MIPool, G: GIST}}} \\tabularnewline \\cline{2-8}\n& \\makecell*[r]{\\textbf{L:}} & \\multicolumn{6}{l|}{\\makecell*[l]{$\\circ=\\text{IMIH}$,\\ $\\bullet=\\text{BMIH}$,\\ \\footnotesize{$\\ \\blacksquare$\\ }=\\small{\\ End-to-end}}} \\tabularnewline \\hline\n\\end{tabular}}\n\\caption{Results of comparison with state-of-the art hashing methods. }\\label{table:CM}\n\\end{wraptable}\n\n\\noindent\n\\textbf{Comparative Methods}\nIn the contrastive experiments against ITQ and KSH, hand-crafted GIST \\cite{gist} features underperformed significantly, while the improvement with the R50 features ranged from 5\\%-30\\%. However, RMIH still performed 10\\%-25\\% better, proving that even if deep learnt features severely boost the performance, the shallow methods cannot fully breach the gap to the deep ones.\nComparing the SI with the MI variations of DHN, SFLH and RMIH, it is observed that the performance improved in the range of 3\\%-11\\%, suggesting that end-to-end learning of MI hash codes is preferred over two-stage hashing \\textit{i.e.} hashing at SI level and comparing at bag level with Eq.~\\eqref{eq:bagDist}. However, RMIH fares comparably better than both the SI and MI versions of SFLH and DHN, owing to the robustness of the proposed retrieval loss function to potentially noisy labels and inconsistent instances within bags (also evident from PR curves in Fig. \\ref{fig:PRCurves}). In all fairness, the concepts of training with aux-SI hashing arm with gradual trade-off and robustness could be potentially adapted to SFLH and DHN to improve their MI hashing performance.  \nAs also seen from the associated PR curves in Fig. \\ref{fig:PRCurves}, the performance gap between shallow and deep hashing methods remains significant despite using R50 features. Comparative results strongly support our premise that end-to-end learning of MI hash codes is preferred over conventional two-stage approaches. \n\n\n\n\n\n\n\\section{Conclusion}\nIn this paper, for the first time, we proposed an end-to-end deep robust hashing framework, termed RMIH, for retrieval under a multiple instance setting. We incorporate the MIPool layer to aggregate representations across instances to generate a bag-level discriminative hash code. We introduce the notion of robustness into our supervised retrieval loss and improve the trainability of RMIH by utilizing an aux-SI hashing arm regulated by a trade-off. Extensive validations and ablative testing on two public breast cancer datasets\ndemonstrate the superiority of RMIH and its potential for future extension to other MI applications. \n\\vspace{-0.15cm}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Introduction}\n\\medskip\n\n\nThe mean curvature flow (abbreviated as MCF) of a submanifold $M \\subset \\mathbb{R}^{N}$\nover an interval $I$ is a map\n$ f: I \\times M \\longrightarrow \\mathbb{R}^{N} $\nsuch that for all $t \\in I$ and $x \\in M$, $\\frac{\\partial}{\\partial t} f(t, x)$ is equal\nto the mean curvature vector  of $M(t)= f(t, M)$ at the point\n$x(t) = f(t, x)$.\nMean curvature flows of convex hypersurfaces have been extensively studied in the literature\n(cf. \\cite{GH}, \\cite{Hu}).\nAn exposition of the work in this area was given in the book \\cite{Zhu}.\nComparatively, the behavior of mean curvature flows of submanifolds with higher codimension\nis less understood (cf. \\cite{W}). This is partly due to the lack of understanding of collapsing\nand the formation of singularities of the flow equations in the higher codimensional case.\n\n\nA submanifold $M$ of a Riemannian manifold is {\\it isoparametric} if\nits normal bundle is flat and principal curvatures along any parallel normal\nvector field are constant. The codimension of $M$ is called the {\\it rank} of $M$.\n An isoparametric submanifold $M$ in $\\mathbb{R} ^N$ is {\\it full} if it is not contained in any proper\n hyperplane, and\nis {\\it irreducible\\\/} if it is not a product of two isoparametric submanifolds.\nWe refer to \\cite{Terng} for the basic properties and structure\ntheories for isoparametric submanifolds.\nPrincipal orbits of isotropy representations of symmetric spaces are isoparametric,  they are the only compact homogeneous isoparametric submanifolds in Euclidean space\n(cf. \\cite{PT}), and are called {\\it generalized flag manifolds}.\nThere are also infinite families of non-homogeneous isoparametric submanifolds which arise from\nrepresentations of Clifford algebras (cf. \\cite{FKM}).\nAll these non-homogeneous examples have rank 2. A theorem of Thorbergsson \\cite{Th} asserts\nthat compact full irreducible isoparametric submanifolds with rank bigger than 2 are always\nhomogeneous.\n\nA complete isoparametric submanifold of $\\mathbb{R}^{N}$ can be\ndecomposed as the product of a compact, irreducible, isoparametric\nsubmanifold and a subspace of $\\mathbb{R}^{N}$. Since mean\ncurvature flows with affine subspaces of $\\mathbb{R}^{N}$ as\ninitial data is trivial and the mean curvature flow starting from\na product submanifold stays as product, we will only consider\ncompact, full, irreducible isoparametric submanifolds.\n\nLet $M$ be an isoparametric submanifold of $\\mathbb{R} ^N$, and $\\xi$ a parallel normal vector field\non $M$. Then\n$M_\\xi=\\{x+ \\xi(x)\\ \\vert\\  x\\in M\\}$\nis again a smooth submanifold (may have higher codimension), and the map $M\\to M_\\xi$ is either a\ndiffeomorphism or a fibration with a generalized flag manifold as fiber.\nThe family of these parallel sets forms a singular foliation of the ambient Euclidean space $\\mathbb{R} ^N$.\nTop dimensional leaves are all isoparametric in $\\mathbb{R}^{N}$, and they are called {\\it parallel\nisoparametric submanifolds}.\nLower dimensional leaves are no longer isoparametric, and they are called\n{\\it focal submanifolds\\\/}.\n\n\nWe show that if $f:M\\times [0, T)\\to \\mathbb{R} ^N$ is a solution of the MCF in $\\mathbb{R} ^N$ with\n$f(\\cdot, 0)$ isoparametric then $f(\\cdot, t)$ is isoparametric for all $t\\in [0,T)$, i.e.,\nthe MCF preserves isoparametric condition.  This reduces the MCF to a system of ordinary\ndifferential equations. There is a Weyl group $W$ associated to each isoparametric submanifold $M$ that acts on the normal plane $p+ \\nu_p M$. The ODE given by the mean curvature flow with initial data an isoparametric submanifold is given by a vector field $H$ smoothly defined on the interior of the Weyl chamber $C$ of $W$ but blows up at the boundary of $C$.  However, we can use generators of $W$-invariant polynomials to change coordinate so that the vector field $H$ becomes a polynomial vector field and its flows can be solved explicitly.\n\nEvery compact isoparametric submanifold is contained in a sphere. This sphere is\nalso foliated by parallel isoparametric submanifolds and focal submanifolds.\nEach isoparametric foliation contains a unique isoparametric submanifold which is a minimal\nsubmanifold of this sphere. The mean curvature flow in $\\mathbb{R} ^N$ with initial data a minimal submanifold\nin $S^{N-1}$ behaves like the mean curvature flow of  a sphere, i.e. it just shrinks homothetically\nalong the radial direction and collapses to a point in finite time\n(cf. Lemma \\ref{miniso}). If $M$ is an isoparametric submanifold in $\\mathbb{R} ^N$\nwhich is not minimal in the sphere, then its mean curvature flow will converge to a focal\nsubmanifold $F$ of positive dimension (cf. Corollary \\ref{cor:miniso}).\nIn fact, $M$ is a fibration over $F$ with each fiber a homogeneous isoparametric submanifold of a\nlower dimensional Euclidean space.\nEach fiber of this fibration collapses\nto a point under the mean curvature flow in a finite time.\n\nWe summarize some of the main results of this paper in the following Theorem (cf. Theorem \\ref{thm:finiteconv}, Theorem \\ref{thm:focalconv}, and Proposition \\ref{bm}):\n\n\\begin{thm}\\label{bo}\nThe mean curvature flow in $\\mathbb{R} ^N$ with initial data a compact isoparametric submanifold\n\\begin{enumerate}\n\\item converges to a focal submanifold in finite time $T$,\n\\item if the fibration from the initial isoparametric submanifold to the limiting focal\nsubmanifold is a sphere fibration (this is the generic case), then the mean curvature flow\n$M(t)$ has type I singularity, i.e., there is a constant $c_0$ such that\n$||{\\rm II\\\/}(t)||_\\infty^2(T-t)\\leq c_0$ for all $t\\in [0,T)$, where $||{\\rm II\\\/}(t)||_\\infty$ is\nthe sup norm of the second fundamental form of $M(t)$,\n\\item every focal submanifold is the limit of\nthe mean curvature flow with some parallel isoparametric submanifold as initial data,\n\\item  if $M_1$ and $M_2$ are distinct parallel full isoparametric submanifolds in\n$\\mathbb{R} ^N$ that lie in the same sphere. Then the mean curvature flows in $\\mathbb{R} ^N$ with initial data\n$M_1$ and $M_2$ collapse to two distinct focal submanifolds.\n\\end{enumerate}\n\\end{thm}\n\n The mean curvature flow in $S^{N-1}$ with initial data an isoparametric  submanifold behaves very similarly to\nthe Euclidean mean curvature flow. In particular we have the following theorem:\n\n\\begin{thm} \\label{thm:MCFisopS}\nLet $M$ be an isoparametric submanifold of $S^{N-1}$.  Then the mean curvature flow in\n$S^{N-1}$ with $M$ as initial data\n\\begin{enumerate}\n\\item  is constant if $M$ is minimal in $S^{N-1}$, or\n\\item converges to a focal submanifold of positive dimension in finite\ntime if $M$ is not minimal.\n\\end{enumerate}\n\\end{thm}\n\nAn isometric action of $G$ on a Riemannian manifold $N$ is {\\it polar\\\/} if there exists a closed embedded submanifold $\\Sigma$ of $N$ that meets all $G$-orbits and meets orthogonally. Such $\\Sigma$ is called a {\\it section\\\/} of the polar action.  Principal orbits of a polar action in $\\mathbb{R} ^n$ and $S^n$ are isoparametric (cf. \\cite{PT}).  We prove that if the $G$-action on $N$ is polar then the mean curvature flow preserves $G$-orbits and the flow becomes an ordinary differential equation on the section $\\Sigma$.  We expect that methods developed in this paper can be applied to study mean curvature flows for orbits of polar actions with flat sections in symmetric spaces.\n\nThis paper is organized as follows:  We give a brief review of properties of isoparametric submanifolds that are needed in section \\ref{sec:prelim}, present proofs of results stated in Theorem \\ref{bo} in section \\ref{sec:Basic}, construct explicit solutions of the MCF in $\\mathbb{R} ^N$ with initial data an isoparametric submanifold in section \\ref{sec:solution}.  Since focal submanifolds are smooth manifolds, we can consider their mean curvature\nflow. Most properties of the mean curvature flows for isoparametric submanifolds\nalso hold for focal submanifolds. This will be briefly discussed in section \\ref{sec:focal}.\nWe describe MCF in spheres with initial data an isoparametric submanifold in spheres in section \\ref{sec:sphere}, and in the last section we discuss MCF in  a Riemannian manifold $N$ with initial data a principal orbit of a polar action on $N$.\n\n\nThe authors like to thank Mu-Tao Wang, Yng-Ing Lee, and Mao-Pei Tsui for discussions on types of singularities of the mean curvature flows.\n\n\\bigskip\n\\section{Preliminaries}\n\\label{sec:prelim}\n\nGeometric and topological properties of isoparametric submanifolds can be found in\n\\cite{Terng}. In this section we\nbriefly review the properties which will be used in this paper.\nLet $M \\subset \\mathbb{R}^{N}$ be a full compact\nisoparametric submanifold of rank $k$.\n\n\\subsection{Curvature spheres and curvature normals}\n\n\\hspace{100pt}\\\\\nThe tangent bundle of $M$ can be decomposed into\northogonal sums of {\\it curvature distributions} $\\{ E_{i} \\mid i=1, \\cdots, g\\}$\nfor some integer $g>0$. At each point of $M$, $E_{i}$ is a common eigenspace\nof the shape operators of $M$ at that point.\n There are parallel normal vector fields ${\\bf n}_{i}$\nsuch that the shape operator $A_{\\xi}$ has the property\n\\[ A_{\\xi} |_{E_{i}} = \\langle \\xi, {\\bf n}_{i}\\rangle  {\\rm Id\\\/}_{E_i} \\]\nfor all normal vector $\\xi$.\nEach vector field ${\\bf n}_{i}$ is called the {\\it curvature normal} of $E_{i}$. The\nrank of $E_{i}$ is called the {\\it multiplicity} of ${\\bf n}_{i}$,\nwhich will be  denoted by $m_{i}$.\nEach $E_{i}$ is an integrable distribution whose leaves are $m_{i}$-dimensional\nround spheres with radius $1\/\\|{\\bf n}_{i}\\|$. Such spheres are called\n{\\it curvature spheres}.\n\n\n\n\\subsection{Weyl group associated to $M$}\\label{ae}\n\n\\hspace{100pt}\\\\\nFor each $i \\in \\{1, \\cdots, g\\}$, let $\\sigma_{i}(x)$ be the antipodal point in the\n$i$-th curvature sphere passing through $x$. Then $\\sigma_{i}$ is an involution on $M$.\nThe group $W$ generated by $\\sigma_{1}, \\cdots, \\sigma_{g}$ is  a crystallographic {\\it Coxeter group}.\nIt is known that $M$ is irreducible if and only if $W$ is irreducible. For each $x \\in M$,\n$W$ also acts as a reflection group\non the affine normal space $x+\\nu_{x}M$\ngenerated by reflections along hyperplanes\n\\[ L_{i} := \\{ x+\\xi \\mid \\xi \\in \\nu_{x}M, \\,\\,\\, 1- \\langle \\xi, {\\bf n}_{i}(x)\\rangle = 0 \\} \\]\nfor $i=1, \\cdots, g$.\n\nThe intersection $\\bigcap_{i=1}^{g} L_{i}$ consists of a single constant point\nwhich is denoted by  $a$. Then $M$ is contained in a sphere which is centered at $a$.\nWithout loss of generality, we always {\\bf assume} that $a=0$, i.e.,  $M$ is contained in a sphere\ncentered at the origin of $\\mathbb{R}^{N}$.  This condition is equivalent to\n\\begin{equation} \\label{eqn:centerO}\n \\langle -x, \\, {\\bf n}_{i}(x)\\rangle \\,\\, =  \\,\\, 1\n\\end{equation}\nfor all $x \\in M$ and $i=1, \\cdots, g$ (cf. \\cite[Corrolary 1.17]{Terng}).\n\n\n\\subsection{Parallel submanifold}\n\n\\hspace{100pt} \\\\\n For any parallel normal vector field $\\xi$ on $M$, define\n\\[ M_{\\xi}:= \\{ x+\\xi(x) \\mid x \\in M \\}. \\]\nIf\n\\begin{equation} \\label{eqn:noncollcond}\n 1- \\langle\\xi(x), {\\bf n}_{i}(x)\\rangle \\neq 0\n\\end{equation}\n for $i=1, \\cdots, g$, then\n$M_{\\xi}$ is again an isoparametric submanifold with the same dimension as $M$.\n$M_{\\xi}$ is called the {\\it parallel isoparametric submanifold} of $M$ defined by $\\xi$.\nThe curvature normals of $M_{\\xi}$ at the point $x+ \\xi(x)$ are given\nby\n\\[ \\frac{{\\bf n}_{i}(x) }{1- \\langle\\xi(x), {\\bf n}_{i}(x)\\rangle} \\]\nwith same multiplicities $m_{i}$ for $i=1, \\cdots, g$.\nThe mean curvature vector of $M_{\\xi}$ at $x+ \\xi(x)$ is given\nby\n\\begin{equation} \\label{eqn:MCV}\n H(x+\\xi(x)) = \\sum_{i=1}^{g} \\frac{m_{i}{\\bf n}_{i}(x) }{1- \\langle\\xi(x), {\\bf n}_{i}(x)\\rangle}.\n\\end{equation}\n\nWhen condition \\eqref{eqn:noncollcond} fails,  $M_{\\xi}$ is still\na smooth submanifold of $\\mathbb{R}^{N}$, but it is no longer\nisoparametric. This submanifold is called\na focal submanifold of $M$. The dimension of $M_{\\xi}$ is strictly\nsmaller than that of $M$. The map\n\\[ \\begin{array}{crcl}\n\\pi:& M & \\longrightarrow & M_{\\xi} \\\\\n    & x & \\mapsto & x+\\xi(x)\n\\end{array} \\]\nis a fibration over $M_{\\xi}$ with each fiber an isoparametric submanifold in the normal space\nof $M_{\\xi}$ at $\\pi(x)$.\n In fact, fix $x_0\\in M$, let $C$ denote the Weyl chamber of $W$ on $x_0+ \\nu_{x_0}M$\n containing $x_0$, i.e.,\n $$C=\\{x_0+\\xi\\ \\vert\\  \\xi\\in \\nu_{x_0}M, \\, \\langle \\xi, n_i\\rangle <1\\}.$$\nIf $y_0=x_0+\\xi(x_0)$ lies in the boundary of $C$ and $y_0\\not=0$, then the fiber of the\nfibration $M\\to M_\\xi$ is a generalized flag manifold with Weyl group $W_{y_0}$,\nthe isotropy subgroup of $W$ at $y_0$.\n\n For any parallel normal vector field $\\xi$ on $M$, the intersection of $M_{\\xi}$ with\n$x + \\nu_{x}M$ is an orbit of $W$. In particular,  if $M_{\\xi}$\nis a parallel isoparametric submanifold, then it intersects each open Weyl chamber of\n$W$ exactly once. Moreover, $M_{\\xi}$ is a focal submanifold if and only if\n$ x+\\xi(x)$ is contained in $\\bigcup_{i=1}^{g} L_{i}$.\n\n\n\\subsection{Isoparametric map and $W$-invariant polynomials}\n\n\\hspace{100pt} \\\\\nGiven a $W$-invariant polynomial $f$ on $V=x_0+\\nu_{x_0}M$, there is a unique extension\n$\\Psi(f)$ on $\\mathbb{R} ^N$ such that $\\Psi(f)$ is constant along any parallel submanifold $M_\\xi$\nand $\\Psi(f)\\ \\vert\\ _{V}= f$.  Moreover, $\\Psi(f)$ is also a polynomial.\n\nLet $\\tilde \\triangle$ and $\\triangle$ denote the Laplacian in $\\mathbb{R} ^N$ and $V$ respectively.\nThen by Lemma 3.2 of \\cite{Terng},\n\\begin{equation}\\label{eqn:mcfpoly}\nF(x)=\\tilde \\triangle \\Psi (f)(x)- \\Psi(\\triangle f)(x)= \\sum_{i=1}^g \\frac{m_i \\langle \\nabla f(x), {\\bf n}_i\\rangle}{\\langle x, {\\bf n}_i\\rangle}.\n\\end{equation}\nis a polynomial on $\\mathbb{R} ^N$ and is constant along parallel submanifolds of $M$.\nMoreover,  if $f$ is a homogeneous $W$-invariant polynomial of degree $m$ on $V$,\nthen  $F$ is a homogeneous polynomial of degree $m-2$ on $\\mathbb{R} ^N$.\n\n\\bigskip\n\\section{Mean curvature flows for general isoparametric submanifolds}\n\\label{sec:Basic}\n\nLet $M$ be an isoparametric submanifold of $\\mathbb{R} ^N$, fix $x_0\\in M$, and $W$ the Coxeter group associated to $M$.\nWe prove that the MCF stays isoparametric and the MCF equation becomes a flow equation of a vector field $H$ defined in the interior of the Weyl chamber of $W$ containing $x_0$ in $x_0+ \\nu_{x_0}M$ and the vector field $H$ tends to infinity at the boundary of the Weyl chamber.  We prove that solutions of the ODE $x'= H(x)$ only exists for finite time. To see the finer structure of the behavior of the blow-up of MCF,  we use $W$-invariant polynomials to construct a new coordinate system for the Weyl chamber so that the corresponding vector field $H$ becomes a polynomial vector field. We then analyse the behavior of flows of this polynomial vector field to obtain informations on the collapsing of the MCF.\n\nFix $x_{0} \\in M$.  Since $\\nu M$\nis globally flat, we can identify a vector $v\\in \\nu_{x_0}M$ and the unique parallel normal field $\\hat v$\nalong $M$ defined by $\\hat v(x_0)=v$.\nLet ${\\bf n}_{i}$ be curvature normals of $M$ with multiplicity $m_{i}$ for $i=1, \\ldots, g$.\nWe may view ${\\bf n}_{i}$ either as a global parallel normal vector field along $M$ or\nan element in $\\nu_{x_0}M$. The precise meaning should be clear from the context.\n\nLet $\\xi(t) \\in \\nu_{x_0}M$ be a one parameter family of normal vectors satisfying the flow equation\n\\begin{equation} \\label{eqn:mcf}\n \\dot{\\xi}(t) = \\sum_{i=1}^{g} \\frac{m_{i}{\\bf n}_i}{1- \\langle\\xi(t), {\\bf n}_{i}\\rangle}, \\qquad\n    \\xi(0) = 0.\n\\end{equation}\nIt follows from  \\eqref{eqn:MCV} that $\\xi$ is a solution of \\eqref{eqn:mcf} if and only if\n the one parameter family of parallel submanifolds $M(t) := M_{\\xi(t)}$ satisfy the\n{\\it mean curvature flow} equation  with $M(0) = M$.\nIn other words, the MCF preserves the isoparametric condition:\n\n\\begin{prop}\nIf $f:M\\times [0,T)\\to \\mathbb{R} ^N$ satisfies the mean curvature flow in $\\mathbb{R} ^N$ and $f(\\cdot, 0)$ is isoparametric,\nthen $f(\\cdot, t)$ is isoparametric for all $t\\in [0,T)$.\n\\end{prop}\n\nEquation \\eqref{eqn:mcf} does not make sense if $\\langle\\xi(t), {\\bf n}_{i}\\rangle = 1$ for some $i$.\nWe will only study the flow equation under the condition\n\\[ \\langle\\xi(t), {\\bf n}_{i}\\rangle \\,\\, <  \\,\\, 1 \\]\nfor all $i=1, \\ldots, g$. In other words, we require that $x_{0} + \\xi(t)$ stays in the same\nWeyl chamber as $x_{0}$ for all $t$. Under this condition, all $M(t)$ are still isoparametric.\n\nNote that \\eqref{eqn:mcf} is a system of non-linear ODE given by a vector field  defined on the\nWeyl chamber $C$ containing $x_0$ and the vector field blows up along the boundary of $C$.\nThe study of MCF with isoparametric submanifolds as initial data reduces to the study of\nsolutions of this ODE system.\n\n\\begin{thm} \\label{thm:radialMCV}\nLet $M\\subset S^{N-1}(r_0)$ be an $n$-dimensional isoparametric submanifold in $\\mathbb{R} ^N$, and $x_0\\in M$.\nIf $\\xi(t)$ satisfies the mean curvature flow equation \\eqref{eqn:mcf}, then $x(t)=x_0+\\xi(t)$\nsatisfies\n\\begin{equation}\\label{aa}\nx'(t)= -\\sum_{i=1}^g \\frac{m_i {\\bf n}_i}{\\langle x(t), {\\bf n}_i\\rangle},\n\\end{equation}\nwith $x(0)=x_0$.\nLet $H(t)$ be the mean curvature vector of $M(t)=M_{\\xi(t)}$ at the point $x(t)$. Then\n\\begin{enumerate}\n\\item[(a)] $\\langle x(t), \\, H(t)\\rangle = -  n$,\n\\item[(b)] $|| x(t) ||^{2} = || x(0)||^{2} - 2nt$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nBy equation \\eqref{eqn:centerO},\n$$ \\langle x(t), \\, {\\bf n}_{i}\\rangle  =\\langle x(0), {\\bf n}_{i}\\rangle  + \\langle \\xi(t), {\\bf n}_{i}\\rangle \\nonumber = -1+ \\langle \\xi(t), {\\bf n}_{i}\\rangle$$\nfor all $i=1, \\cdots, g$.\nSince\n\\begin{equation} \\label{eqn:MCVcenterO}\nH(t) = - \\sum_{i=1}^{g} \\frac{m_{i}{\\bf n}_i}{\\langle x(t), {\\bf n}_{i}\\rangle},\n\\end{equation}\nwe have $\\langle x(t), \\, H(t)\\rangle = -  \\sum_{i=1}^{g} m_{i} = -n$.\nThis proves part (a). Part (b) follows from integrating the following formula\n\\[ \\frac{d}{dt} \\|x(t)\\|^{2}\n\\,\\,  = \\,\\, 2 \\langle x(t), \\, x^{\\prime}(t)\\rangle\n\\,\\,=\\,\\, 2 \\langle x(t), \\, H(t)\\rangle\n\\,\\,= \\,\\, -  2n.\\]\n\\end{proof}\n\nHence we have\n\\begin{cor} \\label{cor:finiteInt}\nThe  mean curvature flow in $\\mathbb{R} ^N$ with initial data an isoparametric submanifold in $S^{N-1}(r_0)$ exists only for finite time with maximal interval $[0, T)$, where\n$0<T \\leq T_{0}=\\frac{r_0^2}{ 2n}$.\n\\end{cor}\n\nThe following Theorem is the key in proving\n\\begin{enumerate}\n\\item the limits of two flows of \\eqref{aa} have two different limit on the boundary $\\partial C$ of the Weyl chamber $C$,\n\\item every point of $\\partial C$ is a limit of some flow of \\eqref{aa}.\n\\end{enumerate}\n\n\n\\begin{thm}\\label{ba}\nLet $M$ be a compact isoparametric submanifold in $\\mathbb{R} ^N$, $W$ the Weyl group associated to $M$, $x_0\\in M$ a fixed point, and $V=x_0+\\nu_{x_0}M$.\nLet $P_1, \\ldots, P_k$ be a set of generators of the ring $\\mathbb{R} [V]^W$ of $W$-invariant polynomials\non $V$ such that $P_i$ are homogeneous polynomials of degree $s_i$ with\n$P_1(x)=||x||^2$ and $s_1\\leq s_2\\cdots \\leq s_k$, and $C$ the Weyl chamber of $W$\ncontaining $x_0$ in $V$.  Let $P:\\bar C\\to \\mathbb{R} ^k$ be the map defined by\n$P(x)= (P_1(x), \\ldots, P_k(x))$.  Then\n $P$ is a homeomorphism from $\\bar C$ to a closed subset $B=P(\\bar C)$. Moreover, there\n is a polynomial map\n $$\\eta=(\\eta_1, \\ldots, \\eta_k):\\mathbb{R} ^k\\to \\mathbb{R} ^k$$\n with $\\eta_1=-2n$ and $\\eta_j$ is a polynomial in $\\eta_1, \\ldots, \\eta_{j-1}$ such that\n if $x:[0,T)\\to C$ is a solution of \\eqref{aa}, then $y(t)= P(x(t))$ is a solution of\n$$y'(t)= \\eta(y(t))= (\\eta_1(y(t)), \\ldots, \\eta_k(y(t))).$$\n\\end{thm}\n\n\\begin{proof}\nSince each orbit of $W$ intersect $\\overline{C}$\nexactly once and the algebra of invariant polynomials separate $W$ orbits,\nthe map\n\\[ \\begin{array}{crcl}\n    P: & \\overline{C}\\cap D(1) & \\longrightarrow & \\mathbb{R}^{k} \\\\\n        & x & \\longmapsto &  (P_{1}(x), \\cdots, P_{k}(x))\n        \\end{array}   \\]\nis an injective continuous  map.\nSince $P$ is injective and proper,  $B=P(\\overline{C})$ is a closed subset of $\\mathbb{R}^{k}$ and\n $P$ is a homeomorphism from $\\overline{C}$ to $B$.\n\nThe Coxeter group $W$ acts on $V$. If $f$ is a $W$-invariant\nhomogeneous polynomial of degree $j$ on $V$,\nthen by equation \\eqref{eqn:mcfpoly},\n$$ F(x) := \\sum_{i=1}^{g} m_{i}\\,\\, \\frac{\\langle \\nabla f(x), \\, {\\bf n}_{i}\\rangle}{\\langle x, \\, {\\bf n}_{i}\\rangle }$$\nis a $W$-invariant homogenous polynomial of degree $j-2$.\nLet $x(t)$ be a solution of \\eqref{aa}, and $f(t)=f(x(t))$.\nBy equation \\eqref{eqn:MCVcenterO},\n\\[ f^{\\prime}(t) = \\langle \\nabla f(x(t)), x^{\\prime}(t)\\rangle\n    = \\langle \\nabla f(x(t)), H(t)\\rangle = - F(x(t)). \\]\nHence $f^{\\prime}(t)$ is the value of a $W$-invariant polynomial\nof degree $k-2$ evaluated at $x(t)$.\nIn particular, $\\frac{d^{j}}{dt^{j}} f(t) = 0$ if $j>k\/2$.\nTherefore $f(t)$ is a polynomial in $t$.\n\nLet $y(t)= (y_1(t), \\ldots, y_k(t))= P(x(t))$, and\n $$F_i(x)= \\sum_{i=1}^g m_{i} \\, \\frac{\\langle \\nabla P_i(x), {\\bf n}_i\\rangle}{\\langle x, {\\bf n}_i\\rangle}.$$\nBy equation \\eqref{eqn:mcfpoly}, $F_i$ is a $W$-invariant homogeneous polynomial on\n$V$ of degree $s_i-2$. Since $\\mathbb{R} [V]^W=\\mathbb{R} [P_1, \\ldots, P_k]$,\n    $$F_i= - \\eta_i(P_1, \\ldots, P_{i-1})$$\nfor some polynomial $\\eta_i$.  But we have shown above that $y_{i}'(t)= - F_i(x(t))$, so\n$$y_i'(t)= - F_{i}(x(t)) = \\eta_i(y_1(t), \\ldots, y_{i-1}(t)).$$\n\nThis shows that $y(t)$ is an integral curve of the polynomial vector field $\\eta$ on $\\mathbb{R} ^k$.\nSince $y_1(t)= ||x(0)||^2 -2nt$, solution $y$ can be solved explicitly by integrations.\n \\end{proof}\n\nThe MCF equation \\eqref{aa} is given by the vector field\n $$H(x)= -\\sum_{i=1}^g \\frac{m_i {\\bf n}_i}{\\langle x, {\\bf n}_i\\rangle},$$ which is smoothly defined\n on the Weyl chamber $C$ of $x_0+\\nu_{x_0}M$ and blows up at the boundary $\\partial C$.\n If we use generators of $W$-invariant polynomials on $x_0+\\nu_{x_0}M$ to change coordinates\n to $P$ as in Theorem \\ref{ba}, then the vector field $H$ becomes the polynomial vector field $\\eta$ on $P(C)$.\n Moreover, the flow of $\\eta$  can be solved explicitly and globally. Then apply $P^{-1}$\n to flows of $\\eta$ to get flows of \\eqref{aa}.\n\n\\begin{thm} \\label{thm:finiteconv}\nFor any compact isoparametric submanifold $M$ in $\\mathbb{R} ^N$,  the mean curvature flow always converges\nto a focal submanifold at a finite time.  Moreover, if $M_{1}$ and $M_{2}$ are parallel full\nisoparametric submanifolds which\n are contained in the same sphere, then mean curvature flows with initial data $M_1$ and $M_2$\n never intersect and they\n converge to two distinct focal submanifolds.\n\\end{thm}\n\n\\begin{proof}\n We use the same notation as Theorem \\ref{ba}.\nLet $x:[0, T)\\to C$ be the maximal interval for a solution of the\nmean curvature flow equation \\eqref{aa}. Note that\n$P_i(t)=P_{i}(x(t))$ are well defined since $P_{i}(t)$ are\npolynomials in $t$. Therefore the mean curvature flow of $x_{0}\n\\in M$ must converge to $P^{-1}(P_{1}(T), \\cdots, P_{k}(T))$ which\nlies on the boundary of $\\overline{C}$. The mean curvature flow of\n$M$ then converges to the focal submanifold passing through this\npoint.\n\nWe may assume that $x_i(0)$ lies in the unit sphere.  Let $T_i$ denote the maximum time for the solution $x_i(t)$. If $T_1\\not= T_2$, then $||x_i(t)||^2= 1- 2n t$, so $\\lim_{t\\to T_1^-}||x_1(t)||^2\\not=\\lim_{t\\to T_2^-} ||x_2(t)||^2$.  If $T_1=T_2= T$, then since $x_i(t)$ are solutions of \\eqref{aa} and $\\langle x_i(t), n_j\\rangle <0$, we have\n\\begin{equation}\\label{bp}\n\\frac{1}{2} \\frac{d}{dt}\\, \\big| ||x_1(t)-x_2(t)||^2= \\sum_{i=1}^g m_i\\, \\frac{\\langle x_1(t)-x_2(t), {\\bf n}_i\\rangle^2}{\\langle x_1(t), n_i\\rangle \\langle x_2(t), {\\bf n}_i\\rangle} \\geq 0.\n\\end{equation}\nThis implies that $||x_1(t)-x_2(t)||^2$ increases in $t\\in [0, T)$, hence\n$$\\lim_{t\\to T^-} x_1(t)\\not= \\lim_{t\\to T^-} x_2(t).$$\n\\end{proof}\n\n\n\\begin{thm}\\label{thm:focalconv}\nEvery focal submanifold is a limit of the mean curvature flow of certain isoparametric submanifold.\n\\end{thm}\n\nWe need a couple Lemmas.\nFirst a simple Lemma on scaling and the proof is obvious:\n\n\\begin{lem}\\label{ac}\nIf $f:M\\times [0,T)\\to \\mathbb{R} ^N$ is a solution to the mean curvature flow with $f(x,0)= f_0(x)$,\nthen given any $r\\not=0$, $\\tilde f(x,t)= rf(x, r^{-2}t)$ is a solution with $\\tilde f(x,0)= rf_0(x)$.\n\\end{lem}\n\n\\begin{lem} \\label{miniso}\nLet $f:M^{n} \\longrightarrow S^{N-1}(r_{0})$ be an immersed minimal submanifold of\na sphere with radius $r_{0}$.\nFor any $x \\in M$, the solution to the mean curvature flow equation  in $\\mathbb{R}^{N}$\nwith initial data $M$ is given by\n$$F(x, t) = \\sqrt{1 - (2nt\/r_{0}^{2}) } \\, f(x).$$\nIn particularly, the mean curvature flow of $M$  shrinks to a point homothetically\n in finite time\n$T_{0} \\,\\, = \\,\\, r_{0}^{2}\/(2n)$.\n\\end{lem}\n\n\\begin{proof}\nFor minimal submanifolds of the sphere $S^{N-1}(r)$ with radius $r$,\nthe mean curvature vector at a point $x$ is $- n x\/r^{2}$.\nLet $F(x, t)= r(t) f(x)$ for $x \\in M$ with $r(t) \\geq 0$. Then the mean curvature\nvector field of $F(\\cdot, t)$ at point $x$ is given by\n$- \\frac{n}{ r_{0}^{2} r(t)} \\, f(x)$.\nSo $F(x, t)$ satisfies the mean curvature flow equation for $f$ if and only if\n\\[ r^{\\prime}(t) = - n\/( r_{0}^{2} r(t)) \\hspace{20pt}\n    {\\rm and} \\hspace{20pt} r(0)=1.\\]\nIt follows that $ r(t) = \\sqrt{1 - (2nt\/r_{0}^{2})}$.\n\\end{proof}\n\n\\medskip\n\\noindent {\\bf Proof of Theorem \\ref{thm:focalconv}}\n\nIn each isoparametric family, there exists a unique isoparametric submanifold $M\\subset S^{N-1}(1)$,\nwhich is minimal in $S^{N-1}(1)$. Let $x_0\\in M$. The mean curvature flow for minimal\n submanifold in spheres can be solved explicitly as in Lemma \\ref{miniso}, i.e.,\n $x(t)= \\sqrt{1-2nt}\\, x_0$ is a solution of \\eqref{aa} and $x(t)\\in C$ for all\n $t\\in [0,\\frac{1}{2n})$.\n\n Recall that integral curves of $H(x)= -\\sum_{i=1}^g \\frac{m_i{\\bf n}_i}{\\langle x, {\\bf n}_i\\rangle}$ map to\n integral curves of the polynomial vector field $\\eta$ under the homeomorphism $P$ defined in\n Theorem \\ref{ba}.  Since the integral curve starting from $x_0$ lies in $C$, the flow of $\\eta$\n starting at $P(x_0)$ lies in $P(C)$.  But $-\\eta$ is a polynomial vector field and the one-parameter\n subgroup $\\phi_t$ generated by $-\\eta$ is a globally defined polynomial map.  So there exists $\\delta>0$\n and an open subset $\\mathcal U$ of $P(\\bar C)$ such that $\\mathcal U$ contains the origin and\n $\\phi_t(z)\\in P(C)$ for $t\\in (0,\\delta)$ and\n $z\\in {\\mathcal U}$.  This shows that the flow of $-\\eta$ starting at the boundary of $\\mathcal U$\n points inward in $P(C)$.   Hence any boundary point of $P^{-1}({\\mathcal U})$ is a limit of some\n MCF with initial data in $C$. It follows from Lemma \\ref{ac} that any focal submanifold can be a\n limit of some MCF with some initial isoparametric submanifold.\n $\\Box$\n\n\nAs consequence of Theorem  \\ref{thm:finiteconv} and Lemma \\ref{miniso}, we have\n\n\\begin{cor} \\label{cor:miniso}\nLet $M$ be an isoparametric submanifold.\nThe mean curvature flow of $M$ converges to a point if and only\nif it is minimal in the sphere containing it.\n\\end{cor}\n\nBelow we describe the rate of collapsing of the MCF for isoparametric submanifolds.  Recall that a MCF, $M_t$, collapses at time $T<\\infty$ is said to have {\\it type I singularity\\\/} (cf. \\cite{W}) if there is a constant $c_0$ such that\n$$||{\\rm II\\\/}(t)||_\\infty^2(T-t)\\leq c_0$$\nfor all $t\\in [0,T)$, where $||{\\rm II\\\/}(t)||_\\infty$ is the sup norm of the second fundamental form for $M_t$.\n\n\\begin{prop}\\label{bm}\nLet  $x(t)$ be a solution of the MCF \\eqref{aa}, and $T$ is the maxmial time.  Then\n\\begin{enumerate}\n\\item $x(T):=\\lim_{t\\to T^-} x(t)$ exists and belong to the boundary $\\partial C$ of the Weyl chamber $C$,\n\\item $\\lim_{t\\to T^-} \\frac{||x(t)-x(T)||^2}{T-t} = 2m$, where $m=\\dim(M_{x(0)})-\\dim(M_{x(T)})$,\n\\item if $x(T)$ lies in a highest dimensional stratum of $\\partial C$, then the MCF has type I singularity.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nWe have proved (1) in Theorem \\ref{thm:finiteconv}. Statement (2) follows from the L'Hopital law:\n\\begin{align*}\n& \\lim_{t\\to T^-} \\frac{||x(t)-x(T)||^2}{T-t}\n=\\lim_{t\\to T^{-}} \\frac{2 \\langle x(t)-x(T), \\,  x'(t)\\rangle}{-1} \\\\\n&= 2\\lim_{t\\to T^{-1}} \\sum_{i=1}^{g} \\langle x(t)-x(T), \\, \\frac{m_i {\\bf n}_i}{\\langle x(t), \\, {\\bf n}_i\\rangle}\\rangle\\\\\n&=2 \\lim_{t\\to T^{-1}} \\sum_{i\\not\\in I} m_i \\frac{\\langle x(t)-x(T), \\, {\\bf n}_i\\rangle}{\\langle x(t), \\, {\\bf n}_i\\rangle}\n + 2\\sum_{i\\in I} m_i\\frac{\\langle x(t), \\, {\\bf n}_i\\rangle }{\\langle x(t), \\, {\\bf n}_i\\rangle} = 2 \\sum_{i\\in I} m_i,\n\\end{align*}\nwhich is the dimension of the fiber of $M_{x(0)}\\to M_{x(T)}$.\nHere $I=\\{1\\leq i\\leq g\\ \\vert\\  \\langle x(T), \\, {\\bf n}_i\\rangle=0\\}$.\n\nWe now prove statement (3).\nIf $x(T)$ lies in a highest dimensional stratum of $\\partial C$, then there exists a  unique $i$ such that $x(T)$ lies\nin the hyperplane defined by ${\\bf n}_i$, i.e.,  $\\langle x(T), {\\bf n}_i\\rangle=0$.  We may assume $i=1$.\nNote that the norm square of the second fundamental form of $M_{x(t)}$ satisfies\n\\begin{align*}\n& ||{\\rm II\\\/}(x(t))||^2(T-t) \\leq \\sum_{i=1}^g \\frac{m_i ||{\\bf n}_i||^2}{\\langle x(t), {\\bf n}_i\\rangle^2}(T-t) \\\\\n&= \\frac{m_1||{\\bf n}_1||^2(T-t)}{\\langle x(t), {\\bf n}_1\\rangle^2}+\\sum_{i=2}^g \\frac{m_i||{\\bf n}_i||^2}{\\langle x(t), {\\bf n}_i\\rangle^2}(T-t).\n\\end{align*}\nAs $t\\to T^-$, the second term tends to zero because $\\langle x(T), {\\bf n}_i\\rangle\\not=0$ for\nall $i\\geq 2$, and by the l'Hopital law the first term has the same limit as\n$$\\frac{-m_1||{\\bf n}_1||^2}{-2\\langle x(t), {\\bf n}_1\\rangle \\sum_{i=1}^g m_i \\frac{\\langle{\\bf n}_i, {\\bf n}_1\\rangle}{\\langle x(t), {\\bf n}_i\\rangle}}.$$\nBut the denominator tends to $-2m_1||{\\bf n}_1||^2$, so the limit is $1\/2$.\n\\end{proof}\n\nWe remark that there is an open dense subset ${\\mathcal{O}}$ of the Weyl chamber $C$ such that the solution $x(t)$ of \\eqref{aa} with $x(0)\\in {\\mathcal{O}}$ converges to a point in a highest dimensional stratum of $\\partial C$.\n\n\n\\bigskip\n\\section{Solutions to the mean curvature flow equation}\n\\label{sec:solution}\n\nIn this section, we use Theorem \\ref{ba} to construct explicit solutions of the MCF \\eqref{aa} by\nselecting a set of generators $P_1, \\ldots, P_k$ for the $W$-invariant polynomials and calculating\nflows of the polynomial vector field $\\eta$.\n\nWe use the root system of the Coxeter group given in \\cite{GB}.\nLet $M$ be a compact, irreducible isoparametric submanifold in $\\mathbb{R} ^N$, $W$ its Weyl group, and\n${\\bf n}_i$ its curvature normals.\nLet $\\Pi$ denote a set of simple roots of $W$, and $\\Delta_{+}$ the set of positive roots\ndefined by $\\Pi$.   Then $\\{\\mathbb{R} {\\bf n}_i\\ \\vert\\  1\\leq i\\leq g\\}$ is equal to $\\{\\mathbb{R} \\alpha\\ \\vert\\  \\alpha\\in \\Delta_+\\}$.  So\nthe Weyl chamber $C$ containing $x_0$ is precisely given by\n\\[ C = \\{ x \\in V \\mid\n    -\\langle x, \\alpha\\rangle > 0 \\,\\,\\, {\\rm for \\,\\,\\, all} \\,\\,\\, \\alpha \\in \\Pi \\}. \\]\nThe closure of $C$ is\n\\[ \\overline{C} = \\{ x \\in V \\mid\n    -\\langle x, \\alpha\\rangle \\geq 0 \\,\\,\\, {\\rm for \\,\\,\\, all} \\,\\,\\, \\alpha \\in \\Pi \\}, \\]\nand the MCF \\eqref{aa} becomes\n\\begin{equation}\\label{ab}\nx'(t)=-\\sum_{\\alpha\\in \\Delta^+} \\frac{m_\\alpha }{\\langle x(t),\\alpha\\rangle}\\,\\, \\alpha\n\\end{equation}\nwhere $m_{\\alpha}$ is the multiplicity of the curvature normal which is parallel to $\\alpha$.\nSince \\eqref{ab} is invariant under re-scaling of each $\\alpha$, we may\nnormalize roots of the Coxeter group to be of {\\it unit length}.\n\nIf $M = M_{1} \\times M_{2}$ with $M_{i}$ an isoparametric submanifold of $\\mathbb{R}^{N_{i}}$\nfor $i=1, 2$, then the Weyl group of $M$ is the product of the Weyl groups of $M_1$ and $M_2$\nand the mean curvature flow of $M$ is the product\nof the mean curvature flows of $M_{1}$ and $M_{2}$.\n So without loss of generality,\nwe may assume that $M$ is an irreducible isoparametric submanifold.\nIn the rest of this section, we work out explicit solutions for mean curvature flow equations\nfor compact isoparametric submanifolds whose Coxeter group are $A_{k}$, $B_{k}$,\n$D_{k}$ and $G_{2}$.\n\n\\begin{eg} {\\bf The $A_{k}$ case}\\par\n\n\\smallskip\n\n\nSuppose that $k \\geq 2$.\nLet $\\{ e_{1}, \\cdots e_{k+1} \\}$ be the standard orthonormal basis of $\\mathbb{R}^{k+1}$\nand $(x_{1}, \\cdots , x_{k+1})$ the corresponding coordinate.\nThe set\n$\\frac{1}{\\sqrt{2}} \\left(e_{i} - e_{i+1} \\right)$ with $1 \\leq i \\leq k$ is a simple root\nsystem of $A_k$, and\nthe set of positive roots is\n$\\frac{1}{\\sqrt{2}} \\left(e_{i} - e_{j} \\right)$ with $1 \\leq i<j \\leq k+1$.\nLet\n\\[ V:= \\{(x_{1}, \\cdots, x_{k+1}) \\in \\mathbb{R}^{k+1} \\mid x_{1}+\\cdots+x_{k+1} = 0 \\}. \\]\nThe normal space of an isoparametric submanifold of type $A_{k}$ can be identified with $V$.\nThe Coxeter group acts on\n$V$ and is generated by all permutations of $\\{e_{1}, \\cdots, e_{k+1}\\}$ .\nThe open positive Weyl chamber $C$ containing $x_0$ is\n\\[ C=\\{ (x_{1}, \\cdots, x_{k+1}) \\in V \\mid x_{1} < x_{2} < \\cdots < x_{k+1}  \\}.\\]\n Since the multiplicities of\ncurvature spheres are invariant under the action of the Coxeter group, there is only one\npossible multiplicity which we denote by $m$.  So \\eqref{ab} is\n\\begin{equation} \\label{ak}\nx'= - \\sum_{i<j} m\\, \\frac{e_i-e_j}{x_i-x_j},\n\\end{equation}\nwhich implies that\n\\begin{eqnarray} \\label{eqn:MCFA}\n\\frac{1}{m} \\frac{d}{dt} x_{i}\n&=&  \\sum_{j \\neq i} \\frac{1}{x_{j} - x_{i}}, \\quad 1\\leq i\\leq k+1.\n\\end{eqnarray}\nIf $x(0) \\in V$, then\n$x(t) \\in V$ for all $t$. This follows from the simple fact that\n$ \\frac{d}{dt} (x_{1} + \\cdots x_{k+1}) = 0$.\n\nLet $\\sigma_{r}$ be the $r$-th elementary symmetric polynomial in $x_{1}, \\cdots, x_{k+1}$:\n$$\\sigma_r=\\sum_{1\\leq i_1<\\cdots < i_r\\leq k+1} x_{i_1}\\cdots x_{i_r},$$\nand $\\sigma_{0}=1$.\nLet $x(t)$ be a solution of \\eqref{ak}, and\n$$y_r(t)= \\sigma_r(x(t)).$$\nWe claim that\n\\begin{equation}\\label{ak1}\n\\begin{cases} y_2'= n,\\\\\ny_r'= \\frac{1}{2}\\, m(k-r+3)(k-r+2) y_{r-2}.\\end{cases}\n\\end{equation}\n\n\nTo see this, we compute\n\\begin{eqnarray*}\n\\frac{r!}{m} \\frac{d}{dt} y_{r}\n&=& \\frac{1}{m} \\frac{d}{dt} \\sum_{i_{1} \\neq \\cdots \\neq i_{r}} x_{i_{1}} \\cdots x_{i_{r}} \\\\\n&=& \\sum_{i_{1} \\neq \\cdots \\neq i_{r}} \\sum_{q=1}^{r}\n        x_{i_{1}} \\cdots \\,\\, \\widehat{x_{i_{q}}} \\,\\, \\cdots x_{i_{r}}\n        \\sum_{j \\neq i_{q}} \\frac{1}{x_{j}-x_{i_{q}}}\n\\end{eqnarray*}\nWrite\n\\[ \\sum_{j \\neq i_{q}} \\frac{1}{x_{j}-x_{i_{q}}}\n    = \\sum_{j \\neq i_{1}, \\cdots, i_{r}} \\frac{1}{x_{j}-x_{i_{q}}}\n        + \\sum_{1 \\leq p \\leq r, \\, p \\neq q} \\frac{1}{x_{i_{p}}-x_{i_{q}}}. \\]\nFor fixed indices $i_{1}, \\cdots, \\widehat{i_{q}}, \\cdots, i_{r}$,\n\\[ \\sum \\frac{1}{x_{j}-x_{i_{q}}} = 0 \\]\n if the summation is running over all\npossible values for $i_{q}$ and $j$ such that $i_{q} \\neq j$ and\nboth of them are not equal to $i_{1}, \\cdots, \\widehat{i_{q}}, \\cdots, i_{r}$.\nTherefore\n\\begin{eqnarray}\\label{eqn:dersrA}\n\\frac{r!}{m} \\frac{d}{dt} y_{r}\n&=& \\sum_{i_{1} \\neq \\cdots \\neq i_{r}}\n        x_{i_{1}} \\cdots  x_{i_{r}}\n        \\sum_{1 \\leq p, q \\leq r, \\, p \\neq q} \\frac{1}{x_{i_{q}}(x_{i_{p}}-x_{i_{q}})}.\n\\end{eqnarray}\nWrite\n\\[ \\sum_{1 \\leq p, q \\leq r, \\, p \\neq q} \\frac{1}{x_{i_{q}}(x_{i_{p}}-x_{i_{q}})}\n    = \\sum_{p>q} \\frac{1}{x_{i_{q}}(x_{i_{p}}-x_{i_{q}})}\n        + \\sum_{p<q} \\frac{1}{x_{i_{q}}(x_{i_{p}}-x_{i_{q}})}. \\]\nSwitch the indices $p$ and $q$ in the second term and adding the first term, we have\n\\begin{eqnarray*}\n\\sum_{1 \\leq p, q \\leq r, \\, p \\neq q} \\frac{1}{x_{i_{q}}(x_{i_{p}}-x_{i_{q}})}\n&=& \\sum_{p>q} \\frac{1}{x_{i_{p}}-x_{i_{q}}} \\left(\\frac{1}{x_{i_{q}}} - \\frac{1}{x_{i_{p}}} \\right) \\\\\n&=&  \\sum_{p>q} \\frac{1}{x_{i_{p}} x_{i_{q}}}.\n\\end{eqnarray*}\nHence by equation \\eqref{eqn:dersrA}\n\\begin{eqnarray*}\n \\frac{r!}{m} \\frac{d}{dt} y_{r}\n&=&  \\sum_{i_{1} \\neq \\cdots \\neq i_{r}} \\,\\, \\, \\sum_{1 \\leq p, q \\leq r, \\, p > q}\n        x_{i_{1}} \\cdots \\,\\, \\widehat{x_{i_{q}}} \\,\\, \\cdots \\,\\,\n            \\widehat{x_{i_{p}}} \\,\\,\\cdots x_{i_{r}}  \\\\\n&=& \\frac{1}{2} r(r-1) (k-r+3)(k-r+2) (r-2)! y_{r-2}\n\\end{eqnarray*}\nThis proves the claim.\n\nThe explicit formula for $y_{r}(t)$ can be obtained from \\eqref{ak1}\nrecursively, and  it is  a polynomial\nin $t$ and initial conditions $x_{1}(0), \\cdots, x_{k+1}(0)$.\nFor each $t$, we can obtain $x_{1}(t), \\cdots, x_{k+1}(t)$ as the $k+1$ solutions\nof the following polynomial equation in $z$:\n\\begin{equation} \\label{eqn:rootssymAk}\n \\sum_{r=0}^{k+1} \\,\\, (-1)^{k+1-r} \\,\\, y_{k+1-r}(t) \\, z^{r} = 0,\n\\end{equation}\nwith the property\n\\[ x_{1}(t) < x_{2}(t) < \\cdots < x_{k+1}(t). \\]\n\\end{eg}\n\n\\begin{eg} \\label{sec:Bk} {\\bf The $B_{k}$ case}\\par\n\n\\smallskip\nSuppose that $k \\geq 2$.\nLet $\\{ e_{1}, \\cdots e_{k} \\}$ be the standard orthonormal basis of $\\mathbb{R}^{k}$\nand $(x_{1}, \\cdots , x_{k})$ the corresponding coordinate.\nWe identify $\\mathbb{R}^{k}$ with a normal space of an isoparametric submanifold of type $B_{k}$.\nThe set  $e_{k}$ and\n$\\frac{1}{\\sqrt{2}} \\left(e_{i} - e_{i+1} \\right)$ with $1 \\leq i \\leq k-1$ is a simple root system of $B_k$, and the set of positive roots are $e_{i}$ with $1 \\leq i \\leq k$ and\n$\\frac{1}{\\sqrt{2}} \\left(e_{i} \\pm e_{j} \\right)$ with $1 \\leq i<j \\leq k$.\nThe Coxeter group is generated by all permutations and sign changes of $\\{e_{1}, \\cdots, e_{k}\\}$.\nSince the multiplicities of curvature spheres are invariant under the action of the Coxeter group,\nthere are only two possible multiplicities.\nLet $m_{1}$ be the multiplicities of curvature spheres corresponding to\n$\\frac{1}{\\sqrt{2}} \\left(e_{i} \\pm e_{j} \\right)$ , and $m_{2}$\nthe multiplicities of curvature spheres corresponding to $e_{i}$.  So the MCF \\eqref{ab} is\n$$-x'= m_1\\left(\\sum_{i<j} \\frac{e_i+e_j}{x_i+x_j} + \\frac{e_i-e_j}{x_i-x_j}\\right)\n+ m_2\\sum_{i} \\frac{e_i}{x_i}.$$\nSo\n$$-x_i'= \\frac{m_2}{x_i}+ m_1\\left(\\frac{1}{x_i-x_j}+ \\frac{1}{x_i+x_j}\\right).$$\nSet $y_i= x_i^2$. Then we have\n\\begin{equation} \\label{eqn:MCFB}\n \\frac{1}{2} \\frac{d}{dt} y_{i} =  - m_{2}\n    - m_{1} \\sum_{j \\neq i} \\frac{2 y_{i}}{y_{i} - y_{j}}.\n\\end{equation}\nLet $s_0=1$,  $s_{i}$ the $i$-th elementary symmetric polynomial\nof $y_1, \\ldots, y_k$, and\n$$\\zeta_r(t)= s_r(x(t)).$$\nWe claim that\n\\begin{equation}\\label{bk1}\n\\zeta_j'= -2(k-j+1)(m_2+m_1(k-j)) \\zeta_{j-1},\\quad 1\\leq j\\leq k.\n\\end{equation}\nNote that when $j=1$ the right hand side is $-2k(m_2+m_1(k-1))$, which is equal to $-2n$.\nTo prove this claim, we compute as follows:\nFirst\n\\begin{align*}\n&\\frac{1}{2} \\frac{d}{dt} \\left( y_{i_{1}} \\cdots y_{i_{j}} \\right)\n= \\sum_{l=1}^{j} \\frac{y_{i_{1}} \\cdots y_{i_{j}}}{y_{i_{l}}}\n    \\left( - m_{2} - 2 m_{1} \\sum_{p \\neq i_{l}} \\frac{y_{i_{l}}}{y_{i_{l}} - y_{p}} \\right) \\\\\n&\\,\\, =  y_{i_{1}} \\cdots y_{i_{j}}\n    \\left( - m_{2} \\sum_{l=1}^{j} \\frac{1}{y_{i_{l}}}\n    - 2 m_{1} \\sum_{l=1}^{j} \\sum_{p \\neq i_{l}} \\frac{1}{y_{i_{l}}-y_{p}}\n    \\right).\n\\end{align*}\nSince\n\\begin{eqnarray*}\n\\sum_{l=1}^{j} \\sum_{p \\neq i_{l}} \\frac{1}{y_{i_{l}}-y_{p}}\n&=& \\sum_{l=1}^{j} \\sum_{1 \\leq q \\leq j, \\,\\, q \\neq l} \\frac{1}{y_{i_{l}}-y_{i_{q}}}\n    + \\sum_{l=1}^{j} \\sum_{p \\neq i_{1}, \\cdots, i_{j}} \\frac{1}{y_{i_{l}}-y_{p}},\n\\end{eqnarray*}\nand the first term on the right hand side is 0, we have\n\\begin{align}\n&\\frac{d}{dt} s_{j}\n= \\frac{1}{j!} \\frac{d}{dt} \\left(\\sum_{i_{1} \\neq \\cdots \\neq i_{j}} y_{i_{1}} \\cdots y_{i_{j}}\n\\right)  \\nonumber \\\\\n&\\,\\,= - \\frac{2 m_{2}}{j!}  \\sum_{l=1}^{j} \\sum_{i_{1} \\neq \\cdots \\neq i_{j}}\n    \\frac{y_{i_{1}} \\cdots y_{i_{j}}}{y_{i_{l}}}\n    - \\frac{4 m_{1}}{j!}\\sum_{l=1}^{j}  \\sum_{i_{1} \\neq \\cdots \\neq i_{j} \\neq p}\n    \\frac{y_{i_{1}} \\cdots y_{i_{j}}}{y_{i_{l}}-y_{p}}. \\label{eqn:dersigma}\n\\end{align}\nThe first term on the right hand side of this equation is\n$ -2 m_{2} (k-j+1) \\zeta_{j-1}$.\n To compute the second term, we notice that\n\\[\\sum_{i_{1} \\neq \\cdots \\neq i_{j} \\neq p}\n    \\frac{y_{i_{1}} \\cdots y_{i_{j}}}{y_{i_{l}}-y_{p}}\n    \\]\ncan be written as\n\\[\n \\sum_{ i_{1} \\neq \\cdots \\neq i_{j}, i_{l} > p }\n       \\frac{( y_{i_{1}} \\cdots \\widehat{y_{i_{l}}} \\cdots y_{i_{j}})y_{i_{l}}}{y_{i_{l}}-y_{p}}\n    + \\sum_{ i_{1} \\neq \\cdots \\neq i_{j},  i_{l} < p}\n    \\frac{ (y_{i_{1}} \\cdots \\widehat{ y_{i_{l}}} \\cdots y_{i_{j}})y_{i_{l}}}{y_{i_{l}}-y_{p}}.\n\\]\nSwitching the indices $i_{l}$ and $p$ in the second term and adding to the first term, we have\n\\begin{eqnarray*}\n\\sum_{i_{1} \\neq \\cdots \\neq i_{j} \\neq p}\n    \\frac{y_{i_{1}} \\cdots y_{i_{j}}}{y_{i_{l}}-y_{p}}\n&=& \\sum_{\\begin{array}{c} i_{1} \\neq \\cdots \\neq i_{j} \\neq p \\\\ i_{l} > p \\end{array}}\n    y_{i_{1}} \\cdots \\widehat{y_{i_{l}}} \\cdots y_{i_{j}}\n\\end{eqnarray*}\nTherefore the second term on the right hand side of equation \\eqref{eqn:dersigma} is\n\\[ -2 m_{1}(k-j+1)(k-j) s_{j-1}.\\]\nThis proves the claim.\n\nNote that explicit formula for $\\zeta_{i}(t)$ can be obtained from \\eqref{bk1} recursively, and\nit is always a\ndegree $i$ polynomial\nin $t$ and initial conditions $y_{1}(0), \\cdots, y_{k}(0)$.\n Let $y_{1}(t), \\cdots y_{k}(t)$\nbe the $k$ roots of\n\\begin{equation} \\label{eqn:rootssymBk}\n \\sum_{r=0}^{k} \\,\\, (-1)^{k-r} \\,\\, s_{k-r}(t) \\, z^{r} = 0,\n\\end{equation}\nwith the property $y_{1}(t) > y_{2}(t) > \\cdots > y_{k}(t) > 0$.\nThen $x_{i}(t) = - \\sqrt{y_{i}(t)}$ for $i=1, \\cdots k$.\n\\end{eg}\n\n\\begin{eg} \\label{sec:Dk} {The $D_{k}$ case}\\par\n\n\\smallskip\nSuppose that $k \\geq 4$.\nLet $\\{ e_{1}, \\cdots e_{k} \\}$ be the standard orthonormal basis of $\\mathbb{R}^{k}$\nand $(x_{1}, \\cdots , x_{k})$ the corresponding coordinate.\nWe identify $\\mathbb{R}^{k}$ with a normal space of an isoparametric submanifold of type $D_{k}$.\nThe set of simple roots are $\\frac{1}{\\sqrt{2}} \\left(e_{k-1} + e_{k} \\right)$ and\n$\\frac{1}{\\sqrt{2}} \\left(e_{i} - e_{i+1} \\right)$ with $1 \\leq i \\leq k-1$, and the set\nof positive roots is\n$\\{\\frac{1}{\\sqrt{2}} \\left(e_{i} \\pm e_{j} \\right)\\ \\vert\\  1 \\leq i<j \\leq k\\}$.\nThe Coxeter group is generated by all permutations and even number of sign changes of\n$\\{e_{1}, \\cdots, e_{k}\\}$.\nThe open positive Weyl chamber $C$ is\n\\[ C := \\{ x \\in \\mathbb{R}^{k} \\mid x_{1} < x_{2} < \\cdots < x_{k} \\,\\,\\,\n    {\\rm and} \\,\\,\\,  x_{k-1}+x_{k} < 0 \\}. \\]\nAll multiplicity for curvature spheres are equal and will be denoted by $m$.\nThe mean curvature flow equation \\eqref{ab} becomes\n\\begin{equation}\\label{dk1}\n-x'= m\\, \\sum_{i<j} \\frac{e_i-e_j}{x_i-x_j} + \\frac{e_i+e_j}{x_i+x_j}.\n\\end{equation}\n\\[ \\frac{1}{m} \\frac{d}{dt} x_{i}\n    = \\sum_{q \\neq i} \\frac{2 x_{i}}{x_{q}^{2}-x_{i}^{2}}. \\]\nMultiply both sides by $\\frac{x_{i}}{2}$ to obtain\n\\begin{equation} \\label{eqn:MCFD}\n \\frac{1}{4m} \\frac{d}{dt} y_{i} =  \\sum_{q \\neq i} \\frac{ y_{i}}{y_{q} - y_{i}}\n\\end{equation}\nwhere\n$y_{i} :=  x_{i}^{2}$ for all $i=1, \\cdots, k$.\n\nLet $s_{i}$ be the $i$-th elementary symmetric polynomial\nof $y_{1}, \\cdots, y_{k}$ and we set $s_{0}=1$.\nThen $s_{1}, \\cdots, s_{k-1}$ and $\\sqrt{s_{k}}$\ngenerate the algebra of polynomials invariant under the action of the Coxeter\ngroup.\nNote that equation \\eqref{eqn:MCFD} is a special case of the equation\n\\eqref{eqn:MCFB} with $m_{2}=0$ and $m_{1} = m$.\nSet  $\\zeta_j(t)= s_j(x(t))$.\nThe proof of \\eqref{bk1} also works here, and we obtain\n$$\\zeta_{r}' = -2 m (k-r+1)(k-r)\\zeta_{r-1}, \\qquad 1\\leq r\\leq k.$$\n\\end{eg}\n\n\\medskip\n\\begin{eg} {\\bf The rank $2$ cases}\\par\n\n\\smallskip\nWe use a different set of generators for the ring of $W$-invariant polynomials to compute explicit\nsolutions of the MCF \\eqref{aa} for the rank $2$ cases.  The Weyl group is the dihedral group with\n$2g$ elements. We identify the normal space of a rank $2$ isoparametric submanifold with $\\mathbb{R} ^2=\\mathbb{C}$,\nand use $e^{\\frac{ik\\pi}{g}}$ with $0\\leq k< g$ as positive roots.  Then\n$$P_1(x)= x_1^2+x_2^2, \\quad P_2(x)= {\\rm Re}((x_1+ix_2)^g)$$\nform a set of generator of the ring of invariant polynomials\n($g=3$ for $A_2$, $g=4$ for $B_2$, and $g=6$ for $G_2$). It is known (cf. [PTb]) that\n\\begin{enumerate}\n\\item if  $g=3, 6$, then all multiplicities are equal and are either $1$ or $2$; and if $g=4$, then\nthere are two positive integers $m_1\\leq m_2$ such that the multiplicity corresponding\nto $\\mathbb{R}  {\\bf n}_j= \\mathbb{R}  e^{\\frac{j \\pi i}{4}}$ is $m_1$ for $j$ even and is $m_2$ for $j$ odd.\n\\item Let $F_i= \\sum_{i=1}^g \\frac{m_i\\langle \\nabla P_i, {\\bf n}_i\\rangle}{\\langle x, {\\bf n}_i\\rangle}$ for $i=1, 2$ be as in Theorem \\ref{ba}. Then\n$$F_1(x)=2n,\\quad F_2(x)=\\begin{cases} 0 &\\, {\\rm if\\,\\, } g=3 \\,\\, {\\rm or \\,\\, 6 \\\/},\n\\\\ 8(m_2-m_1)(x_1^2+x_2^2), & \\,{\\rm if\\,\\,} g=4,  \\end{cases}$$\n\\end{enumerate}\nIf $x(t)$ is a solution of the MCF, then it follows from Theorem \\ref{ba} that $y(t) =(P_1(x(t)), P_2(x(t))$ satisfies\n\\begin{equation}\\label{ah}\n\\begin{cases} y_1'= -2n,\\\\\ny_2'=\\begin{cases} 0, & {\\rm if\\,} g=3, 6,\\\\\n-8(m_2-m_1) y_1, & {\\rm if\\, } g=4.\n\\end{cases}\\eca\n\\end{equation}\nBy Lemma \\ref{ac}, it suffices to consider initial data $x_0=e^{i\\theta_0}$ for the MCF.\nThe corresponding solution for \\eqref{ah} with initial data $y(0)=(1, \\cos g \\theta_0)$ is\n\\begin{align*}\n& y_1(t)= 1-2nt, \\quad y_2(t)= \\cos(g\\theta_0), \\quad {\\rm if\\,\\,} g= 3, 6,\\\\\n& y_1(t)=1-2nt, \\quad y_2(t)= \\cos 4\\theta_0 - 8(m_2-m_1)(t-nt^2), \\quad {\\rm if\\,\\,} g=4.\n\\end{align*}\n  Set $x(t)= r(t)e^{i\\theta(t)}$. Since $y_1(t)= r^2(t)$ and $y_2(t)= r^g(t) \\cos g\\theta(t)$,\n  solution of the MCF for rank $2$ case with initial data $e^{i\\theta_0}$  is\n$r(t)= (1-2nt)^{\\frac{1}{2}}$ and\n$$ \\theta(t)=\\begin{cases} \\frac{1}{g}\\,\\cos^{-1}\\left(\\frac{\\cos g\\theta_0}{(1-2nt)^{\\frac{g}{2}}}\\right), & g= 3, 6,\\\\\n\\frac{1}{4}\\, \\cos^{-1} \\left(\\frac{\\cos 4\\theta_0 - 8(m_2-m_1) (t-nt^2)}{(1-2nt)^2}\\right), & g=4.\n\\end{cases}$$\n\nWe claim that the isoparametric submanifold through $x_0= e^{i\\theta_g}$ is minimal in sphere, where\n\\begin{equation}\\label{bb}\n\\theta_g= \\begin{cases} \\frac{\\pi}{2g}, &  g=3, 6,\\\\ \\frac{1}{4}\\cos^{-1}(\\frac{m_2-m_1}{m_2+m_1}), & g=4.\\end{cases}\n\\end{equation}\nTo see this,   by Lemma \\ref{miniso}, the the polar angle of the MCF flow starting at a minimal\nsubmanifold in sphere must be some constant $\\theta_0$.  So:\n\\begin{enumerate}\n\\item[(i)] For $g=3, 6$, we have $y_2(t)= r(t)^g \\cos g\\theta_0= \\cos g\\theta_0$.\nThis implies that $\\cos g\\theta_0= 0$, hence $\\theta_0= \\frac{\\pi}{2g}$.\n\\item[(ii)] For $g=4$,  $y_2(t)= y_1^2(t) \\cos 4\\theta_0$ implies that\n$$y_2'= 2y_1y_1'\\cos 4\\theta_0= -4n y_1\\cos 4\\theta_0= -8(m_2-m_1)y_1.$$\nHence $\\cos 4\\theta_0=\\frac{2(m_2-m_1)}{n}=\\frac{m_2-m_1}{m_1+m_2}$.\nThis proves the claim.\n\\end{enumerate}\n\\end{eg}\n\nNext we compute the maximal time $T$ for the MCF for the above examples.\nThe flow blows up  when $\\cos g\\theta(t)= \\pm 1$.\nIf $g=3$ or $6$, then\n$$y_2(t)= r(t)^g \\cos g\\theta(t) = (1-2nt)^{g\/2} \\cos g\\theta(t)= \\cos g\\theta_0.$$\nHence  $T= \\frac{1}{2\\pi} (1-|\\cos g\\theta_0)|^{2\/g})$.\nFor $g=4$, we have\n$$y_2(t)= r(t)^4 \\cos 4\\theta(t) = (1-2nt)^2 \\cos 4\\theta(t) = \\cos 4\\theta_0 - 8(m_2-m_1)(t-nt^2).$$\nThen $\\cos g\\theta(T)=1$ if $\\theta_0\\in (0, \\theta_4)$, and $\\cos g\\theta(T)=-1$ if $\\theta_0\\in (\\theta_4, \\frac{\\pi}{4})$.\nThis solves $T$ and we get:\n\n\\begin{prop} Let $\\theta_g$ be the constant defined by \\eqref{bb}.  Then\n\\begin{enumerate}\n\\item The MCF through $e^{i\\theta_g}$  homothetically converges to $0$ at time $T=\\frac{1}{2n}$.\n\\item For $g= 3, 6$, the maximal time for the MCF with initial data $e^{i\\theta_0}$ is\n$T= \\frac{1}{2n}(1-|\\cos g\\theta_0|^{\\frac{2}{g}})$. For $g=4$, the maximal time is\n\\[ T =\\left\\{ \\begin{array}{ll}\n    \\frac{1}{2n} \\left(1-\\sqrt{\\frac{m_{1}+m_{2}}{2 m_{1}}\n                (\\cos 4\\theta_0- \\frac{m_2-m_1}{m_2+m_1})}\\right),\n        & {\\rm if} \\,\\,\\, \\theta_{0} \\in (0, \\theta_{4}), \\\\ \\\\\n    \\frac{1}{2n} \\left(1-\\sqrt{\\frac{m_{1}+m_{2}}{2 m_{2}}\n                (-\\cos 4\\theta_0+ \\frac{m_2-m_1}{m_2+m_1})}\\right),\n        & {\\rm if} \\,\\,\\, \\theta_{0} \\in (\\theta_{4}, \\frac{\\pi}{4}).\n        \\end{array} \\right. \\]\n\\item\n$$\\lim_{t\\to T^-} \\theta(t)= \\begin{cases} 0, &{\\rm if \\,\\, } \\theta_0\\in (0, \\theta_g),\\\\\n        \\frac{\\pi}{g}, & {\\rm if\\,\\,} \\theta_0\\in (\\theta_g, \\frac{\\pi}{g}).\\end{cases}$$\n\\end{enumerate}\n\\end{prop}\n\n\\bigskip\n\n\\section{Mean curvature flows of focal submanifolds}\n\\label{sec:focal}\n\nWe consider the mean curvature flows\nof focal submanifolds of isoparametric submanifolds. They behave very similarly to the\nmean curvature flows of isoparametric submanifolds. With slight modification,\nmost results in section \\ref{sec:Basic}\nalso hold for mean curvature flows of focal submanifolds.\n\n\nLet $M_{0}$ be an isoparametric submanifold and $p \\in M_{0}$ a fixed point.\nAs before, we assume that $M_{0}$ is contained in a sphere centered at the origin.\nLet $C \\subset p+ \\nu_{p}M_{0}$ be the Weyl chamber containing $p$, and\n$\\Delta^{+}$ the set of positive roots of the Weyl group $W$ associated to $M_0$.\n Then\n\\begin{enumerate}\n\\item $\\bar  C$ is a stratified space,\n\\item the isotropy subgroup of any two points in the same stratum $\\sigma$ are the same,\nand will be denoted by $W_\\sigma$,\n\\item for $x\\in \\partial C$, let\n$$\\triangle^+(x)=\\{\\alpha\\in \\triangle^+\\ \\vert\\  \\langle x, \\alpha\\rangle >0\\},$$ then $\\triangle^+(x_1)=\\triangle^+(x_2)$ if and only if\n$x_1, x_2$ lie in the same stratum $\\sigma$, and will be denoted by $\\triangle^+(\\sigma)$,\n\\item $\\sigma$ is the Weyl chamber of $W_x$ for $x\\in \\sigma$ and $\\sigma$ is an\nopen simplicial cone in the following  linear subspace\n$$V(\\sigma)=\\{x\\in p+\\nu_pM_0\\ \\vert\\  \\langle x, \\alpha\\rangle\n=0, \\,\\, {\\rm for\\, all\\,} \\, \\alpha\\in \\triangle^+\\setminus \\triangle^+(\\sigma)\\},$$\n\\end{enumerate}\n\nLet $\\sigma$ be a stratum in $\\partial C$, $x_0\\in \\sigma$, and $M$ the focal submanifold of\n$M_0$ through $x_0$.\nBy \\cite[Theorem 4.1]{Terng},\nthe mean curvature vector field of $M$ at $x_0$ is given by\n\\begin{equation} \\label{eqn:focalMCV}\nH(x_0) = - \\sum_{\\alpha \\in \\Delta^{+}(\\sigma)} \\frac{m_{\\alpha}}{\\langle x_0, \\alpha\\rangle} \\,\\, \\alpha\n\\end{equation}\nwhere $m_{\\alpha}$ are multiplicities of curvature spheres of $M_{0}$.\nMoreover\n\\begin{equation} \\label{eqn:focalxH}\n \\langle x_0, \\alpha\\rangle \\,\\, =\\,\\, \\langle H(x_0), \\alpha\\rangle \\,\\,= \\,\\,0\n\\end{equation}\nfor all $\\alpha \\in \\Delta^{+} \\setminus \\Delta^{+}(\\sigma)$.\nThe mean curvature flow equation of $M$ is the following ODE on $\\sigma$:\n\\begin{equation}  \\label{eqn:focalMCF}\n\\frac{dx}{dt}  = - \\sum_{\\alpha \\in \\Delta^{+}(\\sigma)}\n\\frac{m_{\\alpha}}{\\langle x, \\alpha\\rangle} \\,\\, \\alpha.\n\\end{equation}\n\nThe analogue of Theorem \\ref{thm:radialMCV} also holds for this case. In particular,\nif $x(t)$ satisfies the flow equation \\eqref{eqn:focalMCF} then\n\\begin{equation}\n \\|x(t)\\|^{2} = \\|x(0)\\|^{2} - 2nt\n\\end{equation}\nwhere $n = \\sum_{\\alpha \\in \\Delta^{+}(\\sigma)} m_{\\alpha}$ is the dimension of $M$.\nTherefore we have\n\\begin{thm}\nThe maximal interval for the solution of the mean curvature flow equation for any focal submanifold\nis finite.\n\\end{thm}\n\nSuppose $x(t)$ and $y(t)$ satisfy equation \\eqref{eqn:focalMCF} on $\\sigma$ and $x(0)\\neq y(0)$. Use the same computation for \\eqref{bp} to get\n\\begin{align}\\label{bd}\n\\frac{1}{2} \\frac{d}{dt} \\|x(t) - y(t) \\|^{2}\n&= \\langle x(t) - y(t),  \\,\\, x^{\\prime}(t) - y^{\\prime}(t)\\rangle   \\nonumber \\\\\n&= \\sum_{\\alpha\\in \\triangle_+(\\sigma)}\n    m_{\\alpha}\\frac{\\langle x(t)-y(t), \\alpha \\rangle^2}{\\langle x(t), \\alpha \\rangle \\langle y(t), \\alpha \\rangle}\\, > 0.\n    \\end{align}\nThen proofs given in section \\ref{sec:Basic} works, so we have\n\n\\begin{thm} \\label{bc}\nLet $M^n\\subset S^{n+k-1}$ be a compact isoparametric submanifold in $\\mathbb{R} ^{n+k}$, $W$ its Weyl group,\n$C$ the Weyl chamber in $x_0+\\nu(M)_{x_0}$ containing $x_0\\in M$, and $M_y$ the submanifold parallel\nto $M$ through $y$.  If $\\sigma \\subset \\overline{C}$ is a stratum, then\n\\begin{enumerate}\n\\item there is a unique $y_\\sigma\\in \\sigma$ such that the focal submanifold $M_{y_\\sigma}$ is minimal\nin $S^{n+k-1}$, and the MCF in $\\mathbb{R} ^{n+k}$ with initial data $M_{y_{\\sigma}}$ homothetically shrinks\nto a point,\n\\item if $y_0\\in \\sigma \\bigcap S^{n+k-1}-\\{y_\\sigma\\}$, then the MCF in $\\mathbb{R} ^{n+k}$ with $M_{y_0}$\nas initial data blows up in finite time $T < \\frac{1}{2n}$, $x(t)\\in \\sigma$ for all $t\\in [0, T)$,\nand\n$\\lim_{t\\to T^-} x(t)\\,\\in \\partial \\sigma$, in particular, the limit is a focal submanifold with\nlower dimension,\n\\item if $y_1, y_2\\in \\sigma \\bigcap S^{n+k-1}$ are distinct, then\nthe MCF in $\\mathbb{R} ^{n+k}$ with initial data $M_{y_1}$ and $M_{y_2}$  converge to\ndistinct focal submanifolds of lower dimensions.\n\\end{enumerate}\n\\end{thm}\n\n\\bigskip\n\n\\section{Mean curvature flows for isoparametric submanifolds in spheres}\n\\label{sec:sphere}\n\nIf $M^{n}\\subset S^{n+k-1}$ is an isoparametric submanifold in   $\\mathbb{R} ^{n+k}$, then\n$M$ is also isoparametric\nin $\\mathbb{R}^{n+k}$. So basic structure theory for isoparametric submanifolds in Euclidean\nspaces also applies to $M$.\nFor $x \\in M$, let $H(x)$ and $H_{E}(x)$ be the mean curvature vector fields of $M$ at $x$\nas a submanifold of $S^{n+k-1}$ and $\\mathbb{R}^{n+k}$ respectively.\nThen $H(x)$ is the orthogonal projection of $H_{E}(x)$ to $T_{x}S^{n+k-1}$. More precisely\n\\[ H(x) = H_{E}(x) + n x \\]\nfor all $x \\in M$. In particular, $H$ is again a parallel normal vector field along $M$.\nThe mean curvature flow of $M$ as a submanifold of $S^{n+k-1}$ behaves similarly\nto its flow as a submanifold of $\\mathbb{R}^{n+k}$. With slight modifications, most results\nfor mean curvature flows for isoparametric submanifolds in the Euclidean spaces also hold\nfor isoparametric submanifolds in spheres. We only need to explain how to deal with\nthe arguments in the Euclidean case which can not be applied directly to the spherical case.\n\nFix $x_{0} \\in M$ and let $V = x_0+ \\nu_{x_{0}}M$ be\nthe normal space of $M$ as a submanifold of $\\mathbb{R}^{N}$ at the point $x_{0}$, $W$ its\nCoxeter group, and $C\\subset V$ the Weyl chamber containing $x_0$.\nThe mean curvature flow of $M$ in $S^{n+k-1}$ is uniquely determined by the flow of $x_{0}$ in\n$S := C \\bigcap S^{k-1}$:\n\\begin{equation} \\label{be}\nx'(t)= -\\sum_{\\alpha\\in \\triangle_+} \\frac{m_\\alpha \\alpha}{\\langle x(t),\\alpha\\rangle} \\, + n x(t).\n\\end{equation}\nThe set $S$ is a geodesic $(k-1)$-simplex on $S^{k-1}$.\nLet $x(t) \\in S$ be a solution to equation \\eqref{be} with initial condition $x_{0}$.\nThen\n\\[ y(t) = \\sqrt{1-2nt} \\,\\,\\,\\, x\\left(- \\frac{1}{2n} \\log(1-2nt)\\right) \\]\nsatisfies the Euclidean mean curvature flow equation \\eqref{ab} with initial condition\n$y(0) = x_{0}$.\nLet $[0, T_{x})$ and $[0, T_{y})$ be the maximal intervals for the domains of $x(t)$ and $y(t)$\nrespectively. Then\n\\[ T_{x} = - \\frac{1}{2n} \\log(1-2n T_{y}) \\]\nand\n\\[ \\lim_{t \\rightarrow T_{y}^{-}} \\,\\, y(t)\n    \\,\\,=\\,\\, \\sqrt{1-2n T_{y}}\\,\\,  \\lim_{t \\rightarrow T_{x}^{-}} \\,\\, x(t). \\]\nNote that by Theorems \\ref{thm:radialMCV} and Corollary \\ref{cor:miniso},\n$T_{y} \\leq \\frac{1}{2n}$ and the equality holds if and only if\nthe isoparametric submanifold $M_{0}$ passing $x_{0}$ is minimal in the sphere $S^{n+k-1}$.\nSo by Theorem \\ref{thm:finiteconv}, if $M_{0}$ is not minimal in the sphere, then\n$x(t)$ converges to a focal submanifold at a finite time $T_{x}$.\nThis proves Theorem \\ref{thm:MCFisopS}.\n\n\nIf $x_{1}(t) \\in S$ and $x_{2}(t) \\in S$ satisfy the spherical mean curvature flow equation\n\\eqref{be}, then\n\\begin{eqnarray*}\n\\frac{1}{2} \\frac{d}{dt} \\|x_{1}(t) - x_{2}(t)\\|^{2}\n&=& \\langle x_{1}-x_{2}, \\left(H_{E}(x_{1}) + n x_{1} \\right)\n- \\left(H_{E}(x_{2}) + n x_{2} \\right)\\rangle \\\\\n&=& n \\|x_{1}-x_{2}\\|^{2} + \\langle x_{1}-x_{2}, H_{E}(x_{1})  - H_{E}(x_{2})\\rangle.\n\\end{eqnarray*}\nBy \\eqref{bp}, $\\langle x_{1}-x_{2}, H_{E}(x_{1}) - H_{E}(x_{2})\\rangle \\,\\,\\geq \\,\\, 0$.\nTherefore\n\\begin{equation} \\label{eqn:departS}\n \\frac{d}{dt} \\|x_{1}(t) - x_{2}(t)\\|^{2} \\,\\,\\geq \\,\\, 2n \\|x_{1}(t)-x_{2}(t)\\|^{2}.\n\\end{equation}\nWe use \\eqref{eqn:departS} to give an estimate of the maximal interval $[0, T)$\nfor the spherical mean curvature flow $x(t)$.\nLet $p_{0}$ be the unique point in $S$ such that the isoparametric submanifold\npassing $p_{0}$ is minimal in the sphere $S^{n+k-1}$.\nSet $x_{1}(t) = x(t)$ and $x_{2}(t) = p_{0}$ in equation \\eqref{eqn:departS}. Since $x_2(t)$ exists for all $t>0$, we obtain\n\\[ \\|x(t) - p_{0} \\| \\geq e^{nt} \\|x(0) - p_{0} \\| \\]\n for all $t$ as long as $x(t)\\in S$.\nLet $D$ be the diameter of $S$, then $D \\leq 2$ and\n\\[ T \\leq \\frac{1}{n} \\log \\frac{D}{\\|x(0) - p_{0} \\|}.\\]\n\nNow we discuss the behavior of invariant polynomials under the spherical mean curvature flow.\nLet $x(t) \\in S$ be the mean curvature flow of $x_{0}$.\nFor any function $f$ on $V$, let $f(t) = f(x(t))$. Then\n\\[ f^{\\prime}(t) \\,\\,=\\,\\, \\langle \\nabla f(x(t)), H(x(t))\\rangle \\,\\,=\\,\\,\n    \\langle \\nabla f(x(t)), H_{E}(x(t))\\rangle + n \\langle \\nabla f(x(t)), x(t)\\rangle. \\]\nIf $f$ is a homogenous polynomial of degree $k$ which is invariant under the action\nof the Coxeter group $W$, then as in the proof of Theorem \\ref{thm:finiteconv},\n\\begin{equation} \\label{eqn:invpolES}\n f^{\\prime}(t) \\,\\,=\\,\\, - F(x(t)) + nk f(t)\n\\end{equation}\nwhere $F$ is defined by equation \\eqref{eqn:mcfpoly}\nand it is an invariant polynomial of degree $k-2$.\nIf we have computed $F(t) := F(x(t))$, then we can solve $f(t)$ from equation \\eqref{eqn:invpolES}\nand obtain\n\\begin{equation} \\label{eqn:intinvpolS}\n f(t) = - e^{knt} \\int e^{-knt} F(t) \\, dt.\n\\end{equation}\nNote that there is no homogeneous invariant polynomial of degree 1. By induction on the degree,\nwe obtain the following\n\\begin{thm}\nIf $x(t)$ satisfies the spherical mean curvature flow equation \\eqref{be} and $f$ is a\n$W$-invariant polynomial, then $f(t) = f(x(t))= c_1 e^{knt} +h(t)$ for some constant $c_1$  and polynomial $h$.\n\\end{thm}\n\nIn particular $f(t)$ is well defined for all $t \\in \\mathbb{R}$.\nIn section \\ref{sec:solution}, we have given  explicit formulas for\n$F_i$ for invariant homogeneous polynomials $P_i$ for isoparametric submanifolds.\nWe can use these formula and \\eqref{eqn:intinvpolS} to construct  explicit\nsolutions to the spherical mean curvature flow equation for isoparametric submanifolds in spheres.\n\n\\begin{eg} {\\bf Phase portrait for rank $2$ cases}\\par\n\nLet $M^n\\subset S^{n+1}\\subset \\mathbb{R} ^{n+2}$ be an isoparametric hypersurface with $g$ distinct principal curvatures.  Then the Weyl group associated to $M$ as a rank $2$ isoparametric submanifold in $\\mathbb{R} ^{n+2}$ is the dihedral group of $2g$ elements. Let $C$ denote the Weyl chamber containing $x_0\\in M$, and $D$ the intersection of $C$ and the normal circle at $x_0$ in $S^{n+1}$.  Let $p_1, p_2$ denote the end points of $D$.  The  arc $D=\\widehat{p_1p_2}$ has length $\\pi\/g$. For $y\\in \\bar C$, let $M_y$ denote the submanifold through $y$ that is parallel to $M$ (a leaf of the isoparametric foliation).  There exists a unique $p_0\\in D$ such that $M_{p_0}$ is minimal in $S^{n+1}$.\n\\begin{enumerate}\n\\item The spherical MCF \\eqref{be} has three orbits: a stationary point $p_0$, the orbit $\\widehat{p_0p_1}$ with one end tends to $p_0$ and the other end tends to $p_1$, and the orbit $\\widehat{p_0p_2}$ with one end tends to $p_0$ and the other end tends to $p_2$.\n\\item The MCF \\eqref{aa} in $\\mathbb{R} ^{n+2}$ starting at $M_y$ degenerates homothetically to one point (the origin) if $y=p_0$, to $M_{rp_2}$ for some $0<r<1$ if $y\\in \\widehat{p_0p_2}$, and to $M_{rp_1}$ for some $1<r<1$ if $y\\in \\widehat{p_1p_0}$.\n\\end{enumerate}\n\\end{eg}\n\n\\begin{eg} {\\bf Phase portrait for the $A_3$ cases}\\par\n\n\\smallskip\nLet $M^n\\subset S^{n+2}$ be an isoparametric submanifold with Weyl group $A_3$ and uniform\nmultiplicity $m$, and $x_0\\in M$. Let $C$ denote the Weyl chamber containing $x_0$, and $D$\nthe intersection of $C$ and the normal sphere at $x_0$.  Then $D$ is a geodesic triangle\nwith  vertices $p_1, p_2, p_3$ and interior angles $\\frac{\\pi}{3}, \\frac{\\pi}{3}, \\frac{\\pi}{2}$.\nThe phase spaces of spherical MCF \\eqref{be} and Euclidean MCF \\eqref{aa} are $D$ and $C$ respectively.   We describe\nthe phase portraits:\n\n(1) There exists a unique $p_0$ in the interior of $D$ such that $p_0$ is the fixed point\nof the ODE \\eqref{be}.  This implies that the spherical MCF starting at the parallel\nsubmanifold $M_{p_0}$ is stationary and $M_{p_0}$ is minimal in $S^{n+k-1}$.\n\n(2) For each $1\\leq i\\leq 3$, there is a unique flow $\\ell_i$ that\nat one end approaches  $p_0$ and at the other end approaches\n$p_i$. This implies that for $y\\in \\ell_i$, the spherical MCF\nstarting at $M_y$ collapses in finite time to the focal\nsubmanifold $M_{p_i}$ by collapsing fibers of the fibration\n$M_y\\to M_{p_i}$. The fibers are isoparametric submanifolds with\nWeyl group $A_2$ for $i=1, 2$ and $A_1\\times A_1$ for $i=3$.  In\nparticular, when $m=2$, $M_y$ is diffeomorphic to the manifold of\nflags in $\\mathbb{C}^4$ and collapsing is along complex flag manifolds of\n$\\mathbb{C}^3$ in $M_y$ for $i=1, 2$ and $S^2\\times S^2$ in $M_y$ for\n$i=3$.\n\n(3) For distinct $i, j, k$, let $D_k$ denote the triangle  with vertices $p_i, p_j, p_0$ and\nedges $\\ell_i, \\ell_j$, and geodesic segment $\\widehat{p_ip_j}$ in the sphere. The flow for \\eqref{be} starting\nat a point in the interior $D_k^0$ of $D_k$ exists for finite time and converges to a point on\nthe interior of $\\widehat{p_ip_j}$.  This implies that for $y\\in D_k^0$, the spherical MCF\nstarting at $M_y$ converges in finite time to a focal submanifold $M_q$ with\n$q\\in\\widehat{p_ip_j}\\setminus \\{p_i, p_j\\}$ by collapsing one family of curvature spheres.\n\n(4) The flow of \\eqref{aa} on $C$ starting at $p_0$ is the straight line joining the\norigin to $p_0$, the flow starting from a point in $D_k^0$ converges to a point on the\nwall containing $p_i, p_j$ ($i, j, k$ distinct), and the flow starting at a point on $\\ell_i$\nconverges to a point on the line segment $\\overline{Op_i}$.   This implies that the Euclidean\nMCF with initial data $M_y$\n\\begin{enumerate}\n\\item[(i)] shrinks homothetically to the origin if $y=p_0$,\n\\item[(ii)] converges to a focal submanifold $M_q$ for some $q$ lies in the open cone spanned\nby $\\widehat{p_ip_j}$ and the collapsing is along a curvature $m$-sphere if $y\\in D_k^0$,\n\\item[(iii)] converges to a focal submanifold $M_q$ with $q\\in \\overline{Op_i}$ for $y\\in \\ell_i$,\nmoreover, the collapsing is along fibers of the fibration $M_y\\to M_q$ and the fibers are\nisoparametric submanifolds with Weyl group $A_2, A_2$, and $A_1\\times A_1$ respectively for\n$i=1, 2, 3$.\n\\end{enumerate}\n\\end{eg}\n\n\\bigskip\n\\section{Mean curvature flow for polar action orbits}\n\\label{sec:polar}\n\nLet $G$ act on a Riemannian manifold $N$ isometrically, and $G\\cdot p$ be a principal orbit\nthrough $p$.  If $v\\in \\nu(G\\cdot p)_p$, then $\\hat v(g\\cdot p)= dg_p(v)$ is a globally defined\nnormal vector field on $G\\cdot p$ and is called a {\\it $G$-equivariant normal field}.\nAn isometric action of a compact Lie group $G$ on a Riemannian manifold $N$ is called\n{\\it polar\\\/} if there is a totally geodesic submanifold $\\Sigma$ that meets all $G$-orbits\nand meets orthogonally.  Such $\\Sigma$ is called a {\\it section\\\/}.  We list some properties\nof polar actions (cf. \\cite{PT}):\n\\begin{enumerate}\n\\item The action $G$ is polar if and only if every $G$-equivariant normal field is parallel\nwith respect to the induced normal connection on $G\\cdot p$ as a submanifold of $N$.\n\\item Let $N(\\Sigma)=\\{g\\in G\\ \\vert\\  g\\cdot \\Sigma=\\Sigma\\}$ and\n$Z(\\Sigma)=\\{ g\\in G\\ \\vert\\  g\\cdot x= x \\,\\, \\forall\\,\\, x\\in \\Sigma\\}$ denote the normalizer and\ncentralizer of $\\Sigma$ respectively. Then the quotient group $W(\\Sigma)=N(\\Sigma)\/Z(\\Sigma)$ is a finite\ngroup acting on $\\Sigma$, and is called {\\it the generalized Weyl group associated to the\npolar action\\\/}.\n\\item The orbit space $\\Sigma\/W$ is isomorphic to $N\/G$ and the ring of smooth $G$-invariant\nfunctions on $N$ is isomorphic to the ring of $W$-invariant functions on $\\Sigma$ under the\nrestriction map.\n\\item If $p_0\\in \\Sigma$ is a singular point, i.e., $G\\cdot p_0$ is a singular orbit, then\nthe slice representation of $G_{p_0}$ on the normal space of the orbit at $p_0$ is a polar\nrepresentation.\n\\end{enumerate}\n\n\\begin{thm}\nSuppose the isometric action $G$ on $N$ is polar, and $\\Sigma$ is a section.  Then\n\\begin{enumerate}\n \\item[(i)] if $x\\in \\Sigma$, then the mean curvature vector $\\xi(x)$ of $G\\cdot x$ at $x$\n is tangent to $\\Sigma$ at $x$,\n \\item[(ii)] if $x'(t)= \\xi(x(t))$ with $x(t)$ regular (i.e., $G\\cdot x(t)$ is a principal orbit),\n then $G\\cdot x(t)$ satisfies the MCF in $N$, in other words, the MCF in $N$ with a principal\n $G$-orbit as initial data flows among principal $G$-orbits.\n \\end{enumerate}\n\\end{thm}\n\nFor general polar action, $W$ need not be a Coxeter group and the orbit space of the\n$W$-action on the section can be complicated. In fact,\ngiven any finite group $W$ and any compact $W$-manifold, there exist a\nRiemannian manifold $N$, a compact group $G$, and an isometric  polar $G$-action on $N$  such that the induced action on the section is the given $W$-action (cf. \\cite{PT}).\n Hence the behavior of the MCF for general polar\nactions is not as clear as in the sphere and Euclidean case.\n\nA polar action on a symmetric space is {\\it hyperpolar\\\/} if the sections are flat. In this case\nthe fundamental domain of the $W$-action on a section is a geodesic simplex.  A submanifold in a symmetric space is called {\\it equifocal\\\/} if its normal bundle is flat, exponential of each normal space is a flat, and the focal radii along a parallel normal field are constant.  It was proved in \\cite {TT} that principal orbits of a hyperpolar action on symmetric space are equifocal, and  parallel foliation of an equifocal submanifold is an orbit-like foliation. Moreover, generators of the ring of smooth functions that are constant along parallel leaves were constructed in \\cite{HLO}.    Hence we believe that methods developed in this paper can be used to solve the MCF starting with an equifocal submanifold  in symmetric spaces.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nThe neutrino oscillation physics enters the precision era: notably, the NOVA~\\cite{NOvA:2021nfi} and T2K~\\cite{patrick_dunne_2020_3959558} long-baseline experiments are featuring measurements of the neutrino mixing angle $\\theta_{23}$ and of the largest mass splitting in the atmospheric sector at few percent precision. Sensitivity studies combining future T2K and NOVA data together with measurements at reactor experiments, show the potential of more than 3$\\sigma$ significance for possible charge-parity (CP) violation hints and mass ordering determination~\\cite{Cabrera:2020own, PhysRevD.103.112010}. The next-generation experiments DUNE~\\cite{DUNE:2020lwj} and HyperKamiokande~\\cite{Abe:2018uyc} are aiming to establish the mass ordering and possibly discover charge-parity violation with 5$\\sigma$ significance, as well as to measure the value of the CP-parameterizing phase ($\\delta_{CP}$) with a precision better than 15 degrees. Such results will be enabled by unprecedented statistics of produced and detected neutrinos, requiring an exceptionally robust and precise control of systematic uncertainties.\n\nThe largest and most complex systematic uncertainty in present neutrino long-baseline experiments stems from the modeling of neutrino-nucleus interactions. The so called ``near detectors'', placed near the neutrino source before any standard neutrino oscillation can occur, are designed to characterize the neutrino flux and to measure the neutrino-nucleus cross section in order to tune the interaction models and minimize the corresponding uncertainties.\nIn order to cope with the increasing needs in precision, a new generation of near detectors is being developed based on the concept of a precise and exclusive reconstruction of all the final-state particles produced in neutrino-nucleus interactions. This concept is at the core of the design of the upgrade of T2K near detector ND280~\\cite{T2K:2019bbb}. A new active target~\\cite{Blondel:2017orl} will allow 3-dimensional reconstruction capabilities for low momentum tracks. Such a detector also enable the reconstruction of neutron kinetic energy by measuring time-of-flight between the neutrino interaction vertex and the first neutron secondary interaction~\\cite{Munteanu:2019llq}.\nThe sensitivity of this type of detector for the most relevant systematic uncertainties in neutrino-nucleus interactions below 1~GeV has been recently documented in Ref.~\\cite{Dolan:2021hbw}. Notably,  the exclusive reconstruction of the hadronic part of the final state of neutrino-nucleus interactions permits a more precise reconstruction of the neutrino energy on an event-by-event basis, yet it poses new challenges on the modeling of such interactions. Neutrino-nucleus interaction models have historically been developed and tuned to describe inclusive processes, thus they characterize the cross section as a function of the outgoing lepton kinematics. A new effort is ongoing to expand their predictivity to the kinematics of the hadrons for an exclusive description of the final state~\\cite{Dolan:2019bxf,Gonzalez-Jimenez:2021ohu}. \n\nMost of the available neutrino-interaction models and Monte Carlo simulations describe the initial nuclear state and the fundamental interaction separately from the re-interactions of the final-state hadrons with the nucleus. \nTypically, final-state interactions (FSI) of resulting hadrons are simulated with a cascade mechanism. It is worth mentioning that many nuclear models, originally developed to describe electron-nucleus scattering, include the effect of an outgoing hadrons distortion in a given nuclear potential while solving the lepton-nucleus interaction ~\\cite{Benhar:1994hw,Benhar:2005dj,Ankowski:2007uy, Boffi:1993gs, Meucci:2003uy, Meucci:2012yq, Kim:2003mp, Dickhoff:2004xx, Kelly:2005is, Udias:1993xy, Udias:2001tc, Martinez:2005xe, Gonzalez-Jimenez:2012snp, Pandey:2014tza, Nikolakopoulos:2019qcr, Gonzalez-Jimenez:2019qhq, Gonzalez-Jimenez:2019ejf, Nikolakopoulos:2022qkq}. Extensive feasibility studies of implementing such models into generators are nowadays performed in the community, e.g., in Ref.~\\cite{Nikolakopoulos:2022qkq}. While such an improvement is certainly needed to cope with the precision of the next generation of long-baseline experiments, we focus on FSI modeling with currently available tools given the needs of running experiments, notably the analysis of the data from the upgraded ND280. \nThis paper presents the study of FSI with two different nuclear models implemented in the IntraNuclear Cascade Liege (INCL) code~\\cite{Boudard:2012wc, Mancusi:2014fba, Rodriguez-Sanchez:2017odk, Hirtz2018} and in NuWro~\\cite{Golan:2012wx}. Fundamental neutrino-nucleus interactions without pion production are studied, where pions in the final state can only be produced through nucleon FSI. This work can be expanded in the future to antineutrino interactions (focusing on neutrons FSI) and to (anti)neutrino interactions with pion production (focusing on pion FSI). The study presented here and its possible further developments aim at evaluating in a detailed way the possible uncertainties inherent to the FSI simulation in neutrino-nucleus scattering. This study is also a step forward into a more precise and complete implementation of FSI effects in Monte Carlo simulations of neutrino-nucleus interactions.\n\nAn important novelty in this paper is a discussion of the production of nuclear clusters (like $\\alpha$ particles and deuterons) which is modeled in the cascade mechanism of INCL, for the first time, in neutrino-nucleus interactions. Such predictions may significantly impact  the analysis and interpretation of the experimental data, notably for the identification of low momentum particles and the measurement of energy deposits around the neutrino vertex (also known as vertex activity).\n\nThe NuWro and INCL nuclear models are described in Section~\\ref{sec:nucmod}. The procedure to simulate fundamental neutrino-nucleus interaction events and, subsequently, simulate the intra-nuclear cascade with different models in INCL or NuWro, is described in Section~\\ref{sec:sim}. Section~\\ref{sec:analysis} introduces the studied variables and the analysis strategy. Section~\\ref{sec:res} presents the results from the simulations of the different nuclear models and their comparison. We study different variables to characterize and quantify the impact of FSI on the kinematics of the leading proton, with particular focus on Single Transverse Variables (STV) introduced in Ref.~\\cite{Lu:2015tcr} and on the visible energy deposited around the vertex. Section~\\ref{sec:data} reports a reasoned comparison of the studied simulations with available measurements of STV.\nFinally the main conclusions are reported in Section~\\ref{sec:conclusions}.\n\n\\section{Nuclear models}\n\\label{sec:nucmod}\n\n\\subsection{NuWro}\nSince 2005, the theoretical group of the University of Wroc\u0142aw has developed NuWro as a comprehensive Monte Carlo lepton-nucleus event generator~\\cite{NuWroREPO}, optimizing it for use in accelerator-based neutrino oscillation experiments, i.e., the few-GeV energy region.\nDepending on the energy transferred from the interacting leptonic probe to the hadronic system, NuWro provides quasielastic~\\cite{Juszczak:2005wk}, hyperon production~\\cite{Thorpe:2020tym}, single-pion production and more inelastic channels (DIS)~\\cite{Juszczak:2005zs} \nfor scattering off free nucleons. After including complex nuclear targets, additional channels such as two-body processes~\\cite{Bonus:2020yrd}, coherent pion production~\\cite{Berger:2008xs}, and neutrino scattering off atomic electrons~\\cite{Zhuridov:2020hqu} are included.\nThe framework utilizes various nuclear models to provide predictions for the dynamics of target nucleons (e.g., global or local Fermi gas, spectral functions~\\cite{Benhar:1994hw,Ankowski:2007uy}, or a momentum-dependent nuclear potential~\\cite{Juszczak:2005wk}). Finally, FSI are simulated by an intranuclear cascade which propagates the outgoing nucleons~\\cite{Niewczas:2019fro} and produced pions~\\cite{Golan:2012wx} through the residual nucleus. \nIn the context of this work, the technical aspects of modeling quasielastic neutrino-nucleus scattering and the subsequent FSI are emphasized. More details on the used NuWro version (19.02.2) could be found in Refs.~\\cite{Niewczas:2019fro,Niewczas:2020fev}.\n\nIn the quasielastic interaction channel, nuclear modeling enters utilizing the {\\it plane-wave impulse approximation} that factorizes the one nucleon knock-out processes into the interaction on a single off-shell nucleon convoluted with a particular {\\it hole spectral function}, i.e., the probability of leaving the residual system with specific excitation energy and recoil.\nThe approach relies on the calculation by {\\it O. Benhar et al.}~\\cite{Benhar:1994hw} that takes into account the electron scattering input to the single-particle wave functions and adding the correlated part evaluated within the local-density approximation.\nAdditionally, in this model, the prescription by {\\it A. Ankowski et al.}~\\cite{Ankowski:2014yfa} is applied to go beyond the factorized picture and account for the effects of distorting the final nucleon wave function by an optical potential.\nAlternatively, the target nucleons can be treated as constituing the ideal Fermi gas, parametrized through nuclear density or its average value and referred to as {\\it local} (LFG) or {\\it relativistic Fermi gas} (RFG), respectively.\nFinally, the primary interaction vertex is constrained by the conserved vector current (CVC) and partially conserved axial current (PCAC) hypotheses.\nThe vector form factors are provided by the BBBA05 parametrization~\\cite{Bradford:2006yz}, while the axial form factor has a dipole shape with $g_A=1.267$ and the axial mass parameter $M_A = 1.03 \\ \\mathrm{GeV\/c^2}$, according to the discussion in Ref.~\\cite{Megias:2019qdv}.\n\nModeling final-state interactions is a challenging many-body problem that bears a tension between numerical efficiency and accuracy of nuclear calculations.\nNuWro solution is based on seminal papers by {\\it N. Metropolis et al.}~\\cite{Metropolis:1958wvo,Metropolis:1958sb}, which describe an algorithm of the space-like cascade model, and applied up-to-date physics ingredients.\nIn this approach, mean free paths are attributed to the particles propagated in straight lines with steps of $\\Delta x$ through a continuous medium.\nSuch Monte Carlo sampling uses the standard non-interaction probability formula\n\\begin{equation}\nP(\\Delta x) = \\exp(-\\Delta x\/\\lambda),    \n\\end{equation}\nwhere $\\lambda = (\\rho \\sigma)^{-1}$ is the mean free path calculated locally, expressed in nuclear density $\\rho$ and an effective interaction cross section $\\sigma$.\nThe maximal step of $\\Delta x = 0.2 \\ \\mathrm{fm}$ is sufficient to grasp the structure of commonly used density profiles.\nBy default, the nucleons constituting the nuclear medium originate from the LFG model, and therefore, meet its Pauli blocking rules (applied on an event-by-event basis).\nThe cascade terminates when all the moving hadrons leave the nucleus or do not have enough kinetic energy and are stuck in nuclear potential (with the separation energy of $7 \\ \\mathrm{MeV}$).\nThe remnant nucleus is in an excited state, and its deexcitation is not modeled.\n\nIn Ref.~\\cite{Niewczas:2019fro}, the nucleon part of the NuWro cascade has been exhaustively tested, aiming to reproduce the nuclear transparency in exclusive $(e,e'p)$ scattering experiments.\nThe essence of this model lies in nucleon-nucleon cross sections, which replicate the PDG dataset~\\cite{ParticleDataGroup:2016lqr}, the fraction of single-pion production adjusted to follow the fits of Ref.~\\cite{Bystricky}, and the center-of-momentum frame angular distributions of Ref.~\\cite{Cugnon:1996kh}.\nAdditionally, the cross sections are modified with in-medium corrections~\\cite{Pandharipande:1992zz,Klakow:1993dj} and two-nucleon correlation effects~\\cite{Niewczas:2019fro}.\nFinally, the pion-nucleon interaction dynamics is taken from the model of {\\it L.L. Salcedo et al.}~\\cite{Salcedo:1987md}.\nThis aspect, together with the formation zone effect for the inelastic scattering channels, has been presented and compared to data in Ref.~\\cite{Golan:2012wx}.\n\n\\subsection{INCL}\n\nINCL is a nuclear model dedicated to the simulation of the reactions induced by baryons (nucleons, $\\Lambda$, $\\Sigma$), mesons (pions and Kaons) or light nuclei (A$\\leqslant$18) on a target nucleus. It shows a remarkable agreement with an exhaustive list of experimental data of Refs.~\\cite{2015, IAEA}. As an example, the comparison to cross section data of proton interactions with fixed $^{12}$C target is shown in Fig.~\\ref{fig:p_benchmark}. The INCL cascade is usually coupled to the deexcitation code ABLA~\\cite{ABLA} but can be combined also to the SMM~\\cite{Botvina:1994vj, Bondorf:1995ua} or GEMINI++~\\cite{Mancusi:2010tg, Charity:1988zz} deexcitation codes. INCL is a very flexible tool that has been implemented in GEANT4~\\cite{Allison:2016lfl} and GENIE~\\cite{PhysRevD.104.053006}.\nIn this study the stand-alone version of INCL is studied and the subsequent deexcitation model is disabled.\n\n\\begin{figure}[ht]\n\\centering \n\\includegraphics[width=0.98\\linewidth]{plots\/proton_benchmark.pdf}\n\\caption{\\label{fig:p_benchmark} Cross section of proton-$^{12}$C interactions as a function of the proton momentum: comparison of the INCL and NuWro models to available experimental data~\\cite{Auce, Ingemarsson:1999sra, Trzaska:1991ww, Slaus:1975zz, Menet1, Menet2, Pollock, Wilkins, Meyer, GOODING1959241, BURGE1959511, Cassels, RENBERG197281, Dicello, Kirkby, MAKINO1964145, TAYLOR1961642, GOLOSKIE1962474, Jaros}.  Results of the NuWro 19.02.2  simulation were taken from Ref.~\\cite{PhysRevD.104.053006}.}\n\\end{figure}\n\nThe standard version for the projectile employs the impact parameter formalism taking into account the Coulomb distortion. \nA `working sphere', where all events take place,  is defined with radius $R_{max}$:\n\n\\begin{equation}\n    R_{max} = \\left\\{\n    \\begin{array}{lll}\n        R_0 + 8a & \\mbox{for } A\\: \\textgreater\\: 19\\\\\n        5.5 + 0.3\\left( A - 6 \\right)\/12 & \\mbox{for } 6\\: \\leq A\\: \\leq\\: 19\\\\\n        R_0 + 4.5 & \\mbox{for } A \\geq 2\n    \\end{array}\n\\right.\n\\end{equation}\nwhere $R_0$ and $a$ are radius and diffuseness of the target nucleus density respectively. For carbon, $R_{max}$~=~5.7~fm.\n If the trajectory of the projectile, defined by a randomly chosen asymptotic impact parameter and modified by the Coulomb field of the target, intersects the working sphere, the projectile enters the nucleus and can interact with the nucleons or not (a \"transparent\" event).\n\nThe simulation of neutrino interactions in INCL is not yet available. In this study, the neutrino-nucleon interaction inside the nucleus is enforced by using the neutrino vertex from the NuWro code. Then, the produced particles from the neutrino-nucleon interaction may undergo final-state interactions and initiate a cascade.  The exact matching procedure to interface NuWro and INCL will be described in Sec.~\\ref{sec:sim}. The overall normalization is taken from the total cross section calculated by NuWro. \n\nThe INCL nuclear model is essentially classical, with some additional ingredients to mimic quantum effects. Each nucleon in the nucleus has its position and momentum and moves freely in a square potential well. The radius of the potential well depends on the kinetic energy of the nucleon. Nucleon momenta are distributed uniformly in a sphere whose radius is defined by the maximal Fermi momentum. Position and momentum are correlated in INCL. In such a {\\it classical} picture, the particle is allowed to move in a sphere with the maximum radius fixed by its momentum, and so its position is sampled in this sphere. This picture has been refined to take into account the quantum properties of the wave functions. Based on a HFB (Hartree-Fock-Bogoliubov) formalism, the correlation is still valid but less strict, i.e. the nucleon has a non-zero probability for going beyond the maximum radius. Further details can be found in Ref.~\\cite{Rodriguez-Sanchez:2017odk}.\n\nThe cascade reaction can be described as an avalanche of independent binary collisions. Different types of events inside the cascade could occur: collisions, decays and, at the surface, reflections or transmissions. The particles travel along straight lines and the different possible fates are calculated:\n\\begin{itemize}\n\\setlength\\itemsep{-0.4em}\n    \\item two particles reach the minimal distance to interact,\n    \\item a particle hits the border of the potential well then it deflects back or leaves the nucleus,\n    \\item a particle decays (e.g. a $\\Delta$ resonance or $\\omega$ meson).\n\\end{itemize}\nThe fate with the shortest time is chosen.\n\nNucleons that do not participate in the cascade are defined as \\textit{spectators}. In the beginning, all nucleons are spectators except for the projectile. A spectator could become a participant while interacting with the projectile or another participant.\nTo prevent spontaneous nucleon emission (nuclear boiling), the spectator nucleons of the target cannot interact between themselves. The projectile feels the nuclear potential of the target. A participant nucleon can become a spectator again if its energy decreases below a threshold (details are in~Ref.~\\cite{Boudard:2012wc}). While such an option has been included to  improve the agreement with some sets of data~\\cite{Boudard:2012wc}, the modeling of very low energy nucleons is still an open issue for cascade mechanisms. In the scope of this paper, this feature is disabled. \nParticipants can be absorbed at the end of the cascade when the stopping time is reached and the nucleus is thermalized through equipartition of energy. In this case, the participant's energy is left in the nucleus, putting it in an excited state.\n\nThe cascade lasts until one of these conditions is satisfied:\n\\begin{itemize}\n\\setlength\\itemsep{-0.4em}\n    \\item the stopping time determined by the model is reached,\n    \\item there are no more participants,\n    \\item the mass number of the target is less than 4,\n    \\item the projectile leaves without an interaction (transparent event).\n\\end{itemize}\nConcerning a participant nucleon at the surface, it can be reflected or emitted. It leaves the nucleus if its energy is higher than the Fermi energy plus the value of its separation energy that is taken from mass tables based on experimental data~\\cite{Tuli}. However, a notable feature of INCL is that an outgoing nucleon (subject to FSI) with some probability could clusterize with other nucleons and leave the nucleus as a nuclear cluster (e.g., $\\alpha$ particle).  The emission of nuclear clusters was extensively compared with representative experimental data. One can find more information about cluster emission in INCL in Ref.~\\cite{Mancusi:2014fba}.\n\nIn addition to quantum effects embedded in the elementary cross sections used and the one previously quoted with the refined position-momentum correlation related to the quantum behavior of the wave function, Pauli-blocking is considered and checked. \nIf the collision is blocked, the particle propagates until it reaches the next allowed interaction in the cascade and the products of that reaction are further propagated.\nThere are two main Pauli blocking models implemented in INCL: the strict one, forbidding the interaction if the projectile momentum is below the Fermi momentum, and the statistical model that considers only nearby nucleons in the phase-space volume and acts upon the calculated occupation probability.\nThe default option applied in INCL is a compromise between the strict Pauli blocking for the first collision and the statistical one for the subsequent ones. Motivation and details of this procedure can be found in Ref.~\\cite{Boudard:2012wc, Henrotte}. In this paper, the first interaction is the neutrino interaction with the nucleus taken from NuWro. So for the first interaction, the Pauli blocking from NuWro is used, then for subsequent proton interactions statistical Pauli blocking is applied.\n\n\\section{Simulation procedure}\n\n\\label{sec:sim}\nThe study focuses on the comparison between NuWro and INCL of proton-induced FSI cascades in Charged-Current Quasi-Elastic (CCQE) neutrino interactions on Carbon using the T2K neutrino energy flux at the near detector from Ref.~\\cite{T2K:2015sqm}. The initial state and the neutrino-nucleus interaction is simulated with NuWro with the Spectral Function (SF) approach. About 350000 CCQE events have been simulated with NuWro. In each event, the leading proton exiting the interaction of a neutrino on a neutron is then injected in the INCL nuclear model and FSI is simulated with INCL. The results are compared to the FSI cascade simulation done in NuWro. \n\nIn the INCL simulation, the neutron with the closest momentum-vector to the initial neutron simulated by NuWro for the neutrino interaction is selected.  This procedure is independent of the exact values of neutron momenta in NuWro and INCL and is applied event-by-event for the whole sample. The chosen INCL neutron is then substituted with the outgoing particles (muon and proton) preserving their kinematic properties calculated from the neutrino-neutron interaction by NuWro. The Pauli blocking in the neutrino interaction is simulated by NuWro, while the Pauli Blocking along the proton FSI cascade is simulated differently by NuWro and INCL, as explained in Sec.~\\ref{sec:nucmod}.\n\n\n\\begin{figure}[ht]\n\\centering \n\\includegraphics[width=0.98\\linewidth]{plots\/neutron_momentum_all.pdf}\n\\caption{\\label{fig:n_mom} Momentum distribution of neutrons in the nucleus for NuWro SF and RFG and INCL nuclear models (shape comparison).}\n\\end{figure}\n\nAlthough driven by the same experimental charge density distribution, NuWro and INCL differ in their approaches to model the nucleus that can result in small inconsistencies within our algorithm. The momentum distribution of nucleons in SF and INCL is quite different, as can be seen in Fig.~\\ref{fig:n_mom}: the high-momentum tail due to nucleon-nucleon correlations is missing in INCL (work is ongoing to include them, and we defer such results for a future paper).  Another discrepancy stems from the lack of correlation between position and momentum in the spectral function approach, while having the Hartree-Foch-Bogoliubov formalism implemented in INCL, as presented in Fig.~\\ref{fig:mom_pos1}.\nThus, the INCL position distribution sampled starting from the simulated NuWro interactions is slightly different than the intrinsic nucleon distribution in INCL, as shown in Fig.~\\ref{fig:n_mom2}.\nIn general, neutrons with high momentum in the SF tail tend to be attributed to the peripheral region of the nucleus, according to the INCL nuclear model shown in Fig.~\\ref{fig:mom_pos1}, while in the NuWro SF model, there is no intrinsic position-momentum correlation for the target neutrons.\nStill, as shown in Fig.~\\ref{fig:n_mom2}, the chosen INCL neutron tends to be at smaller radius with respect to the general neutron distribution in INCL: this is due to the fact that the momentum-vector (and not the momentum magnitude) is used to find the INCL neutron which best matches the NuWro neutron.\n\n\\begin{figure}[ht]\n\\centering\n\\begin{minipage}[h]{0.98\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/pos_mom_other.pdf} \\\\}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.98\\linewidth}\n\\center{\\hspace{-0.2cm}\\includegraphics[width=0.95\\linewidth]{plots\/mom_pos_neutron_NuWro_LFG.pdf} \\\\}\n\\end{minipage}\n\\caption{\\label{fig:mom_pos1} Radial coordinate and momentum distribution inside the nucleus for INCL (top), and NuWro LFG (bottom) nuclear models} (z axis in arbitrary units).\n\\end{figure}\n\nIt is worth mentioning that NuWro uses the local Fermi gas for nuclear model in the cascade itself. This model does include a momentum-position correlation, which is presented at the bottom of Fig.~\\ref{fig:mom_pos1}. One can see that there exist a contrary dependence relative to the INCL solution, which is an important point in comparing these two cascades regardless of the primary vertex simulation.\n\nIn order to check that the results on FSI characterization are robust against the approximations of the simulation procedure above and, more in general, against assumptions on the initial state nuclear model, similar studies are also performed using Relativistic Fermi Gas and using a simulation representing the INCL nuclear model in the initial state. This is discussed in Appendix~\\ref{app:pn}.\n\nIn the factorized models considered here, the FSI interaction cannot change the fundamental neutrino-nucleus interaction cross section. Thus for all studies presented here the NuWro cross section from SF (or from RFG for the results in Appendix~\\ref{app:pn}) is considered.\n\nThe procedure to match NuWro and INCL models described above has been made possible by the high-level of modularity of Monte Carlo  implementations. It also opens the road to possible further improvement of FSI modeling into such simulation programs.\n\n\\begin{figure}[ht!]\n\\centering \n\\includegraphics[width=0.98\\linewidth]{plots\/position.pdf}\n\\caption{\\label{fig:n_mom2} Position distribution of neutrons in INCL and NuWro and of the INCL neutron chosen to match the SF neutron simulated in NuWro in our matching algorithm.} \n\n\\end{figure}\n\n\n\n\\section{Analysis strategy}\n\\label{sec:analysis}\nIn this section, the analysis strategy to compare and characterize the FSI effects in INCL and NuWro models is described.\n\nVarious channels are possible, depending on the particles leaving the nucleus: their probability is quantified in the two models. The FSI effects on the leading proton are then characterized by comparing its kinematics before and after FSI. Additionally, the Single Transverse Variables (STV), as introduced in Ref.~\\cite{Lu:2015tcr}, are studied and compared between the two models. Resolution effects or thresholds are not applied in the results of Sec.~\\ref{subsec:leadprot} and some of the studied variables are not observable experimentally. Comparison to data is deferred to Section~\\ref{sec:data}, where direct comparison to STV measurements from T2K~\\cite{T2K:2018rnz} and MINERvA~\\cite{MINERvA:2018hba} are shown. In this case, acceptance cuts are applied to match the phase space covered by the two experiments.\n\n\n\\begin{figure}\n    \\centering\n\\includegraphics[width=0.98\\linewidth]{plots\/stv.pdf}\n    \\caption{STV representation in the plane perpendicular to the neutrino direction.}\n    \\label{fig:stv_fig}\n\\end{figure}\n\nThe study focuses on the following STV:\n\\begin{equation}\n\\begin{gathered}\n   \\delta \\alpha_{T} = arccos\\frac{-\\vec{p^{\\mu}}_{T} \\cdot \\delta \\vec{p}_{T}}{p^{\\mu}_{T} \\cdot \\delta p_{T}} \\\\\n   |\\delta \\vec{p_{T}}| = |\\vec{p^{p}}_{T} + \\vec{p^{\\mu}}_{T}| \\\\\n \n\\end{gathered}\n\\end{equation}\nwhere $\\vec{p^{p}}_{T}$ is the component of the proton momentum projected into the plane transverse to the neutrino direction (transverse component) and $\\vec{p^{\\mu}}_{T}$ is the transverse component of the muon momentum.\nThe illustration of the STV definition is presented in Fig.~\\ref{fig:stv_fig}. \nThe variable $\\delta \\vec{p_{T}}$ could be considered as the ``missing transverse momentum'' and, in absence of FSI in quasi-elastic events, it represents the Fermi motion of the initial nucleon.\nThe variable $\\delta \\alpha_{T}$ is especially sensitive to the FSI of the leading proton. In transparent events (where the proton leaves the nucleus without FSI), the $\\delta \\alpha_{T}$ distribution depends only on the stochastic Fermi motion of the initial neutron and therefore is expected to be uniform. FSI tends to decrease the proton transverse momentum, inducing larger values of missing transverse momentum and $\\delta \\alpha_{T}$. \n\nThe production of clusters is an important novelty brought by INCL: no other cascade model used within neutrino studies is able to simulate cluster production. Not only the kinematics of the simulated clusters but also their identification probability in scintillating detectors is studied.\nTo simulate clusters' interactions inside a detector, a simple Geant4 model~\\cite{AGOSTINELLI2003250, Geant4, Allison:2016lfl}\\footnote{For electromagnetic processes the standard physics list was used, for hadron processes we used  G4HadronPhysicsINCLXX. Additionally, G4DecayPhysics and G4IonINCLXXPhysics were used.} is built. A uniform, fully active block of hydrocarbon (with 1.06 g\/cm$^3$ density corresponding to polystyrene) with dimensions big enough to contain a whole track (1900$\\times$1900$\\times$3000~cm$^3$ size) is simulated.  \nThe clusters are generated with momentum distribution as simulated by INCL FSI production. \n\nA simplified analysis is performed to gauge the observability of clusters: an identification algorithm is developed based on the path traversed by clusters and the energy deposited in the detector, as shown in the simple schematic of Fig.~\\ref{fig:pid_scheme}.  \n\n The main idea of this algorithm is to try to identify the type of the particle based on its ionization curve. \nThe measured deposited energy is used as an estimation of the particle momentum along its trajectory. The measured local deposits of energy ($dE\/dx$) along the track are compared to the expected ones for different clusters \\{$\\alpha$, D, T, $^3$He, p\\} at the estimated momenta. Finally the algorithm selects the cluster hypothesis that best matches the observed $dE\/dx$ along the track.\n\\begin{center}\n\\begin{figure}[ht!]\n\\includegraphics[width=0.98\\linewidth]{plots\/pid.pdf}\n\\caption{Scheme of the particles simulation and identification algorithm in a segmented scintillating detector. The shaded line indicates the cluster track traversing multiple scintillating cubes. The cluster has initial kinetic energy $E^0$ and deposits energy $E^{dep}_i$ in each cube. The last cube, where the cluster stops, is not considered in the identification algorithm.}\n\\label{fig:pid_scheme}\n\\end{figure}\n\\end{center}\n\nActual identification capabilities in an experiment depend highly on the detector geometry, granularity, and scintillator material. Here we consider a detector granularity of 1~cm$^3$ cubes, corresponding, for instance, to the geometry of the ND280 upgrade scintillating target superFGD~\\cite{T2K:2019bbb}. We do not take into account any reflecting material or border effects.\n\nTracks are simulated as straight lines parallel to one of the cube faces and starting from the cube boundary. The last part of the track, which is shorter than 1~cm, is not used in the analysis since the exact length of the last step is unknown, given the simulated granularity. Thus we effectively remove the Bragg peak from this simplified identification algorithm. In more sophisticated analyses, the Bragg peak study could bring further information to help particle identification. Moreover, at the end of the track, a fraction of events undergoes inelastic interactions with the creation of secondary particles. We leave the investigation of such secondary interactions and their detector signature for future work: we do not include the energy of secondary particles in the analysis since we remove the last cube. \n\nWe consider only the energy deposited by the primary particle by ionization and we apply Birks quenching. Depending on the material, an additional overall suppression factor should be considered in energy to scintillation light conversion but is omitted in the present study.\n\nVisible energy in a given simulation step is therefore calculated as\n\\begin{equation}\n    \\mathrm{E}^{vis}_{step} = \\frac{\\mathrm{E}^{dep}_{step}}{1 + k_B \\cdot \\frac{\\mathrm{E}^{dep}_{step}}{\\mathrm{L}_{step}}}.\n\\end{equation}\nwhere $\\mathrm{E}^{dep}_{step}$ is the energy deposited in the step, k$_B$ is the Birks coefficient, and $\\mathrm{L}_{step}$ is the step length.\nBirks coefficient for protons is taken from Ref.~\\cite{T2KND280FGD:2012umz}, where $k_B$ = 0.0208~(cm\/MeV); according to Ref.~\\cite{GSO}, the coefficients for protons and deuterons are assumed to be the same. The same Birks coefficient is also assumed for tritium. For $\\alpha$ particles the results from Ref.~\\cite{Sarazin:2012ppa} are used, where $k_B$ = 0.0085~(cm\/MeV).  \n\nThe total visible energy is evaluated as the sum of the visible energy in each cube is\n\\begin{equation}\nE^{vis} = \\sum_{i=1}^{n} E^{vis}_i.\n\\end{equation}\nThis visible energy is a proxy for the total kinetic energy of the particle. However, it is not a perfect estimator since the energy lost in the last cube, where inelastic events may happen, is discarded from the analysis. As an analogous estimator of the remaining kinetic energy of the cluster at each step along the track, the total `residual' visible energy is estimated as\n\\begin{equation}\n\\label{eq:evis}\nE^{vis, res}_i = E^{vis} - \\sum_{m=1}^{i} E^{vis}_m.\n\\end{equation}\nThe distribution of the visible energy deposited in each step $E^{vis}_i$, as a function of the left visible energy $E^{vis, res}_i$, is used to build the expectations for each of the five particle hypotheses  \\{$\\alpha$, D, T, $^3$He, p\\}. For each particle $k$, the mean of the simulated visible energy at a given step is estimated ($E^{exp,k}_i$) for the corresponding value of remaining visible energy ($E^{vis, res}_i$). The width of such distribution is also evaluated ($\\sigma^k_i$).\nFinally for each step of the simulated cluster track, the visible energy is compared to the expected one for each particle hypothesis: a $\\chi^2$ is built for each particle hypothesis\n\\begin{equation}\n\\chi_k^2= \\sum_{i=1}^{n} \\frac{(E^{vis}_i - E^{exp,k}_i)^2}{(\\sigma^{k}_i)^2}\n\\end{equation}\nand the hypothesis with lowest $\\chi^2$ is chosen to identify the particle.\n\\begin{comment}\nActual identification capabilities in a real experiment will be highly dependent on the detector geometry, granularity and scintillator material. A detector granularity of 1~cm$^3$ cubes is considered, corresponding, for instance, to the geometry of the ND280 upgrade scintillating target superFGD~\\cite{T2K:2019bbb}. No reflecting material or border effects are considered.\nTracks are simulated as straight lines parallel to one of the cube face and starting from the cube boundary.\nThe last part of the track, which is shorter than 1~cm, is not used in the analysis, since the exact length of the last step is not known, given the simulated granularity. Thus we effectively remove the Bragg peak from this simplified identification algorithm. In more sophisticated analyses, the study of the Bragg peak could bring further information to help particle identification. Moreover, in the last part of the track a fraction of events undergoes inelastic interactions with creation of additional particles. We leave the investigation of such secondary interactions and their detector signature for a future work: we do not include the energy of secondary particles in the analysis, since we remove the last cube. Only the energy deposited by the primary particle by ionization is considered and Birks quenching is applied. An additional overall suppression factor, depending on the scintillator material, should be considered in the conversion from energy to scintillation light but is not included in the present study. Visible energy in a given simulation step is therefore calculated as\n\\begin{equation}\n    \\mathrm{E}^{vis} = \\frac{\\mathrm{E}^{dep}}{1 + k_B \\cdot \\frac{\\mathrm{E}^{dep}}{\\mathrm{L}_{step}}}.\n\\end{equation}\nwhere $\\mathrm{E}^{dep}$ is the energy deposited in the step, k$_B$ is the Birks coefficient and $\\mathrm{L}_{step}$ is the step length.\nBirks coefficient for protons is taken from Ref.~\\cite{T2KND280FGD:2012umz}, where $k_B$ = 0.0208~(cm\/MeV); according to Ref.~\\cite{GSO}, the coefficients for protons and deuterons are assumed to be the same. The same Birks coefficient is also assumed for tritium. For $\\alpha$ particles the results from Ref.~\\cite{Sarazin:2012ppa} are used, where $k_B$ = 0.0085~(cm\/MeV).  \n\nThe total visible energy is evaluated as the sum of the visible energy in each cube\n\\begin{equation}\nE^{vis} = \\sum_{i=1}^{n} E^{vis}_i ,\n\\end{equation}\nThis visible energy is a proxy to the total kinetic energy of the particle, but it is not a perfect estimator since the energy lost in the last cube, where inelastic events may happen, is not included. As an analogous estimator of the remaining kinetic energy of the cluster at each step along the track, the total `residual' visible energy is estimated as\n\\begin{equation}\n\\label{eq:evis}\nE^{vis, res}_i = E^{vis} - \\sum_{m=1}^{i} E^{vis}_m.\n\\end{equation}\nThe distribution of the visible energy deposited in each step $E^{vis}_i$, as a function of the left visible energy $E^{vis, left}_i$, is used to build the expectations for each of the 5 particle hypotheses  \\{$\\alpha$, D, T, $^3$He, p\\}. For each particle $k$, the mean of the simulated visible energy at a given step is estimated ($E^{exp,k}_i$) for the corresponding value of remaining visible energy ($E^{vis, left}_i$). The width of such distribution is also evaluated ($\\sigma^k_i$).\nFinally for each step of the simulated cluster track, the visible energy is compared to the expected one for each particle hypothesis: a $\\chi^2$ is built for each particle hypothesis\n\\begin{equation}\n\\chi_k^2= \\sum_{i=1}^{n} \\frac{(E^{vis}_i - E^{exp,k}_i)^2}{(\\sigma^{k}_i)^2}\n\\end{equation}\nand the hypothesis with lowest $\\chi^2$ is chosen to identify the particle.\n\\end{comment}\n\n\\section{Results}\n\\label{sec:res}\n\nIn Tab.~\\ref{tab:channelsSF} the fraction of different final states produced by INCL and NuWro FSI simulation from CCQE interactions are reported. \n In the NuWro SF simulation, due to short-range correlations, the neutrino vertex contains two outgoing protons in 15\\% of events.\nIn the INCL vertex, only the leading proton, defined as the proton with higher kinetic energy, is retained and triggers the cascade.\n We have also tested that complete removal of events with 2-protons  at the neutrino vertex does not impact the conclusions on the characterization of the FSI cascade of the leading proton in NuWro and INCL.  We consider only the leading proton and its secondary hadrons for the channel characterization in Tab.~\\ref{tab:channelsSF}. If the proton has the same energy before and after FSI, the event is characterized as ``no FSI'' (a transparent event). \n \n \\begin{table}[htbp]\n\\centering\n\\caption{\\label{tab:channelsSF}  Fractions of the different FSI channels, i.e. fraction of events with different final state particles after the FSI cascade, in CCQE events with T2K neutrino energy flux. Fractions of events with and without protons in the final state are quoted separately, while in the text the percentages with respect to the total number of events are quoted. }\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n&\\multirow{2}{*}{\\textbf{Channel}} & \\multirow{2}{*}{\\textbf{NuWro SF}} & \\textbf{INCL}\\\\\n&&&\\textbf{+NuWro SF}\\\\\\hline\n& no protons & 1.37\\% &19.47\\% \\\\\n& protons & 98.63\\% &80.53\\%  \\\\\n\\hline\n\\hline\n\\multicolumn{1}{|c|}{\\multirow{4}{*}{\\rotatebox{90}{\\textbf{no proton}}}}& absorption  & 4.45\\% & 39.49\\% \\\\ \n\\multicolumn{1}{|c|}{}&neutron + $\\pi$ production  & 3.40\\%  &0.60\\% \\\\  \n\\multicolumn{1}{|c|}{}&$\\pi$ production& 0.21\\% & 0\\% \\\\\n\n\\multicolumn{1}{|c|}{}&neutron knock-out& 91.4\\% &29.58\\% \\\\\n\\multicolumn{1}{|c|}{}&cluster knock-out & 0\\%  & 30.33\\% \\\\\n\\hline\n\\hline\n\\multicolumn{1}{|c|}{\\multirow{4}{*}{\\rotatebox{90}{\\textbf{proton}}}}& 1 proton, no FSI   & 70.38\\% & 68.49\\% \\\\ \n\\multicolumn{1}{|c|}{}&1 proton only with FSI  & 2.45\\%  &19.21\\% \\\\  \n\\multicolumn{1}{|c|}{}&1p + other nucleons or clusters& 26.21\\% &11.68\\% \\\\\n\\multicolumn{1}{|c|}{}&proton(s)+ $\\pi$ production  & 0.96\\%  &0.62\\% \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\nINCL features an evident enhancement of events with absorption of the proton and only a muon in the final state:\n7.7\\% of the total events in INCL against less than 0.1\\% in NuWro. \nIndeed, in the NuWro cascade, the vast majority of particles produced, and not Pauli blocked, leave the nucleus. The INCL nuclear model features larger probability of reabsorbing particles in the nucleus during FSI. This tendency is also confirmed by the smaller fraction of events with more than 1 proton in the final state (multinucleon production by FSI results into 25.6\\%  of total events in NuWro and 9\\% in INCL).\n\n\n We have also compared both INCL and NuWro cascades against existing transparency data. Here we define transparency as the percentage of events without FSI interaction on the leading proton exiting from the interaction vertex. Fig.~\\ref{fig:transp} shows that both models are in good agreement with data while having a divergence in their prediction. Especially for the low momentum protons, INCL features smaller transparency. The difference between the two models is dominated by the impact of the short-range correlations on the FSI definition, as also shown in Refs.~\\cite{Niewczas:2019fro} and~\\cite{PhysRevD.104.053006}. In the region of interest for the present study ($0.2-1$~GeV of proton momentum), the predictions vary the most, while there are very sparse data and no clear preference for one model over the other. \n\nThe larger FSI strength in INCL suggests a larger dissipation of energy across the nucleus through interactions that tend to be more `soft'. \nOn the other hand, the nuclear model of INCL includes the probability to form  clusters during the attempt of the nucleon to leave the nucleus, as explained in Sec.~\\ref{sec:sim}.\n Indeed, the events with no proton in the final state are in the large majority due to charge exchange in NuWro (91\\% of neutron production) while in INCL the probability of cluster production and neutron production in events without protons in the final state is similar (around 30\\% each).\n\n\n\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.98\\linewidth]{plots\/transparency_new.pdf}\n\\caption{\\label{fig:transp} Nuclear transparency of $^{12}$C (percentage of events without FSI) as a function of proton momentum modelled by NuWro and INCL.}\n\\end{figure}\n\n\n\\subsection{Leading proton kinematics}\n\\label{subsec:leadprot}\nThe leading proton momentum before and after FSI is shown in Fig.~\\ref{fig:mom_bef}. Events with the production of other particles, e.g., clusters, but no protons in the final state, are included only in the distribution of proton momentum before FSI. \n The proton momentum before FSI has by construction the same distribution for NuWro and INCL, as explained in Sec.~\\ref{sec:sim}. In Fig.~\\ref{fig:mom_bef} the relative fraction of the different final states as a function of proton momentum before and after FSI is also shown. A larger diversity of channels is visible in the INCL case. As previously discussed, the smaller transparency of INCL induces a smaller number of events with at least one proton in the final state after FSI. In particular, all the events with full proton absorption are concentrated at low  proton momentum before FSI, while the events with cluster production populate the distribution just below the main peak of initial proton momentum (around 350~MeV\/c). \nAgain, due to the smaller nuclear transparency of INCL, the events where the proton leaves the nucleus are much less often accompanied by additional nucleons. Thus in NuWro, the protons at very low momenta after FSI are almost always accompanied by other nucleons, while in INCL, they are mostly events with only one proton.\n\n\\begin{figure*}[h!]\n\\centering\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/proton_mom_bef_stacked_INCL.pdf} \\\\}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/proton_mom_bef_stacked_NuWro.pdf} \\\\}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/proton_mom_aft_stacked_INCL.pdf} \\\\}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/proton_mom_aft_stacked_NuWro.pdf} \\\\}\n\\end{minipage}\n\\caption{\\label{fig:mom_bef} Top: proton momentum before FSI for INCL (left), and NuWro (right) cascade models in CCQE events with T2K neutrino energy flux.  Sub-processes correspond to the fate of the proton after FSI. The shape of proton momentum before FSI is by definition  identical for INCL and NuWro cascades. Bottom: leading proton momentum after FSI for INCL (left), and NuWro SF (right) nuclear models. The fractions of different FSI sub-processes as listed in Tab.~\\ref{tab:channelsSF} is also shown. The 0 proton channel in NuWro includes muon only and pion and neutron production. There is no cluster production in NuWro.}\n\\end{figure*}\n\n\\begin{figure*}[t!]\n\\centering\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/p_bef_aft.pdf} \\\\}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/cos_theta_bef_aft.pdf} \\\\}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/dp_superimposed.pdf} \\\\}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/dcosT_superimposed.pdf} \\\\}\n\\end{minipage}\n\\caption{\\label{fig:pdcosT_befaft} Distribution of leading proton's momentum in NuWro SF and NuWro+INCL before and after FSI (top left, shape comparison). Distribution of difference in proton's momentum before and after FSI ($\\Delta P=P_{after} - P_{before}$) in NuWro SF and NuWro+INCL (bottom left, shape comparison). Distribution of leading proton $\\cos(\\Theta$) in NuWro SF and NuWro+INCL before and after FSI (top right, shape comparison). Distribution of difference in proton's $\\cos(\\Theta$) before and after FSI ($\\Delta \\cos\\Theta =\\cos\\Theta_{after} - \\cos\\Theta_{before}$) in NuWro SF and NuWro+INCL (bottom left, shape comparison). The ``no FSI'' channel is excluded. CCQE events with T2K neutrino energy flux are simulated.}\n\\end{figure*}\n\nMore direct FSI quantification on the leading proton is done by comparing the kinematics of the proton before and after FSI, as shown in Fig.~\\ref{fig:pdcosT_befaft}.\nAs expected, FSI decelerate the leading proton and induce smearing compared to the pre-FSI angular distribution that is peaked in the forward direction (but this effect is less evident in INCL). \nInterestingly, INCL tends, event-by-event, to decelerate  protons more but still the distribution of momenta of all the protons leaving the nucleus after FSI tends to be larger in INCL, since the low momenta protons are absorbed in the nucleus.\n\nIn order to disentangle FSI effects from the physics of the initial nuclear state, STV are used. As explained in Sec.~\\ref{sec:analysis}, the identified experimental observable most sensitive to FSI is $\\delta\\alpha_T$, while $\\delta p_T$ is primarily sensitive to the initial nuclear state. This is confirmed by the results in Fig.~\\ref{fig:dpt}, where such distributions are  compared between NuWro and INCL. The shape of $\\delta p_T$ is very similar between the two models since the initial nuclear momentum is taken from the same model.\nThe most visible feature is the suppression of the large $\\delta\\alpha_T$ values in INCL, corresponding to a suppression of low momentum protons. This conclusion is robust, independently of the nuclear model used for the fundamental interaction: we observe a similar behavior using RFG, as shown in Appendix~\\ref{app:pn} in Fig.~\\ref{fig:rfg}.\nPart of the low momentum protons lost in INCL are emitted as clusters. Such clusters will not have the same detector signature of protons, as discussed in the following, and they do not fill the suppression at very large values of $\\delta\\alpha_T$, as can be seen in Fig.~\\ref{fig:dpt}: indeed, there is still a sizable fraction of events, at low proton momenta, with full proton absorption and only a muon in the final state.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.98\\linewidth]{plots\/dpt_superimposed.pdf}\n\\vfill\n\\includegraphics[width=0.98\\linewidth]{plots\/dalphat_superimposed.pdf}\n\\caption{\\label{fig:dpt} Top: $\\delta$p$_T$ distribution in NuWro SF and NuWro+INCL (shape comparison). Bottom:  $\\delta\\alpha_T$ distribution in NuWro SF and NuWro+INCL (shape comparison). The effect on  $\\delta\\alpha_T$ shape of including also the leading nuclear cluster in INCL events is shown. CCQE events with T2K neutrino energy flux are simulated.}\n\\end{figure}\n\n\n\\begin{comment}\n\\begin{figure}[ht]\n\\centering\n\\begin{minipage}[h]{0.98\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/dalphat_stacked_NuWro.pdf} \\\\}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.98\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/dalphat_stacked_INCL.pdf} \\\\}\n\\end{minipage}\n\\caption{\\label{fig:dat} $\\delta\\alpha_T$  for NuWro SF (top) and NuWro+INCL (bottom) divided by the sub-processes listed in Tab.~\\ref{tab:channelsSF} in CCQE events with T2K neutrino energy flux.}\n\\end{figure}\n\\end{comment}\n\n\\subsection{Nuclear cluster production}\n\\label{subsec:nucl_clus}\nIn events with proton absorption {\\and multiple nucleons production} in INCL, different particles can leave the nucleus, as shown in Fig.~\\ref{fig:part0p}.  As one can see, INCL features a notable production of deuterons, $\\alpha$-particles, $^3$He isotopes and tritons. In this paper, we focus on studying production of these nuclear clusters.\nThe production of clusters in nucleon scattering on nuclei is a well established evidence in nuclear physics, see for instance Ref.~\\cite{neutron_clus}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.98\\linewidth]{plots\/0p_part_namesINCL17_0INCL.pdf}\n\\caption{\\label{fig:part0p} Particles leaving the nucleus in events without proton in the final state in INCL.}\n\\end{figure}\n\nThe production of clusters by FSI in neutrino interactions is a novelty of this study. While the presence of clusters can change the interpretation of various measurements in data (proton multiplicity, vertex activity, single transverse variables, etc.) the signature of such clusters is very detector-dependent, notably regarding granularity and threshold of measurable energy deposits. We provide here the fundamental information to characterize the observability and signature of such clusters in different detectors. In particular, the path length of cluster tracks, their mis-identification probabilities, evaluated with a simplified identification algorithm, and the energy released around the neutrino vertex are quantified.\n\nFigure~\\ref{fig:e_step} shows the length of the tracks produced by different particles as a function of their visible energy, as defined in Eq.~\\ref{eq:evis}. The sparse population of events at lower track lengths correspond to inelastic processes, which are included in the simulation but not in the characterization of visible energy, as explained in Sec.~\\ref{sec:analysis}. In figure~\\ref{fig:dEdX_birks} we show the momentum distribution and the deposited  ionization energy  with Birks correction as a function of momentum. \nThese distributions are the result of the simulation and the input to the algorithm described in Sec.~\\ref{sec:analysis}.\nTable~\\ref{tab:PIDtr} shows the fraction of events where clusters travel more than 1 or 3~cm in the detector. While $\\alpha$ and $^3$He do not travel enough and will mostly contribute to vertex activity (energy deposited around the vertex), deuteron and tritium leave a visible track in a sizable fraction of events and, if searched for, can be quite cleanly identified. \n\nThe primary source of misidentification are secondary interactions through inelastic processes, which typically happen at the end of the cluster track. In such inelastic interactions the cluster looses a large amount of energy and breaks the nucleus. The observed energy released by ionization along the cluster track, before the end of the track, does not include the energy released by secondary interactions and it is therefore  smaller than what is expected for a cluster of that energy. As a consequence, particles could be misidentified with less ionizing ones. Therefore, tritium could be misidentified ($\\sim$10\\% of events) as proton or deuteron with similar probability, deuteron could be misidentified ($\\sim$20\\% of events) as protons, while protons cannot be misidentified as nuclear clusters. $^3$He and $\\alpha$ rarely travel more than 1 cm and when they do, they can be easily identified due to their very large energy deposit. The characterization of the inelastic interactions, the particles produced and their observability in the detector are left for future studies.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.98\\linewidth]{plots\/step_e_all-1.png}\n\\caption{\\label{fig:e_step} Total track length of nuclear clusters as a function of visible energy of the particles. The $^3$He distribution overlaps with the $\\alpha$ distribution.}\n\\end{figure}\n\n\\begin{comment}\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.98\\linewidth]{plots\/dEdX_full_mom.png}\n\\caption{\\label{fig:dEdX_full} Momentum distribution of clusters produced by FSI in INCL (solid line) and energy loss by ionization as a function of their momentum (scatter plot).}\n\\end{figure}\n\\end{comment}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.98\\linewidth]{plots\/dEdX_birks_mom.png}\n\\caption{\\label{fig:dEdX_birks} Momentum distribution of clusters produced by FSI in INCL (solid line) and visible energy loss by ionization (with Birks correction) as a function of their momentum (scatter plot).}\n\\end{figure}\n\n\n\\begin{table}[htbp]\n\\centering\n\\caption{\\label{tab:PIDtr} Percentage of clusters travelling more than 1 (3) cm} \n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n & \\textbf{$\\alpha$} & \\textbf{$^3$He} & \\textbf{T} & \\textbf{D} & \\textbf{proton} \\\\\n\\hline\nTravels more than 1 cm, \\% & 0.3 & 1.3 & 60 & 72 & 87\\\\\nTravels more than 3 cm, \\% & 0   & 0   & 34 & 51 & 74\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nThe particles that are below tracking threshold still contribute to the total energy deposited around the vertex. To compute the contribution of clusters to the vertex activity, the visible energy deposited by ionization in a sphere of 1 (3)~cm radius around the vertex is calculated, as shown in Fig.~\\ref{fig:VA} and Fig.~\\ref{fig:VA_all}. The distributions have two components depending if the particles leave the sphere or if they release all their energy inside the sphere.\nThe long tail of INCL comes from the energy deposits due to clusters. There are 8~(11)\\% of the events with more than 15~(20)~MeV energy deposited in the 1~(3)~cm sphere. The tail of the NuWro distribution comes from multinucleon neutrino interactions and FSI inelastic processes with production of multiple protons. There are 10~(10)\\% of the events with more than 15~(20)~MeV energy deposited in the 1~(3)~cm sphere.\nThe energy deposited by inelastic interactions is not included, since its observability depends on the particles produced by such interactions. Inelastic interactions happen within 1 (3)~cm from the primary vertex in about 3\\% (8\\%) of tritium and deuteron events, 1\\% (3\\%) of proton events, while they happen in a negligible fraction of events with $\\alpha$ and $^3$He.\n\n\\begin{figure}[t!]\n\\centering\n\\center{\\includegraphics[width=0.98\\linewidth]{plots\/VertexActivity3cm_mu_logINCL17_0INCL.pdf} \\\\}\n\\caption{\\label{fig:VA} Visible energy deposited by ionization by different particles in a sphere of 3~cm radius around the vertex in events simulated with INCL.}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.98\\linewidth]{plots\/VA_all_log.pdf}\n\\caption{\\label{fig:VA_all} Total visible energy deposited by ionization by all the particles in a sphere of 1 (3)~cm radius around the vertex. Distributions from events simulated with INCL and NuWro are compared with the same overall normalization.}\n\\end{figure}\n\n\\section{Comparison to data}\n\\label{sec:data}\nThe prediction of NuWro and INCL are compared to STV measurements ($\\delta \\alpha_T$ and $\\delta p_T$) of T2K~\\cite{T2K:2018rnz} and Miner$\\nu$a~\\cite{MINERvA:2018hba} in Figs.~\\ref{fig:dataSF_T2K} and~\\ref{fig:dataSF}.\nThe contribution from non-QE interactions is simulated with NuWro and added on top of INCL predictions. Simulated events are selected to cope with the acceptance of the detectors. Such acceptance cuts, notably requiring large enough proton momentum to be reconstructed, tend to suppress the difference between the models. We look forward to measurements with lower tracking threshold, as expected for instance with ND280 upgrade~\\cite{T2K:2019bbb} and the detectors of the SBN program~\\cite{Antonello:2015lea}.\nThe different shape in $\\delta\\alpha_T$ is washed out, still the lower overall normalization of INCL, due to proton absorption, is visible and tends to slightly improve the comparison to data. The impact of nuclear clusters in the distributions is shown assuming perfect identification and same acceptance as protons. With present momentum threshold, nuclear clusters tend to distribute at high $\\delta\\alpha_T$, similarly to protons affected by FSI. Given the relatively high momentum threshold for proton tracking, the different proton momentum distributions after FSI in the two cascade models has a subleading effect.\n\n\\begin{figure*}[ht]\n\\centering\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/dat_NuWro_SF.pdf} \\\\}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/dpt_NuWro_SF.pdf} \\\\}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/dat_INCL_SF_cl.pdf} \\\\}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/dpt_INCL_SF.pdf} \\\\}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/dat_SF_shape.pdf} \\\\}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/dpt_SF_shape.pdf} \\\\}\n\\end{minipage}\n\\vfill\n\\center{\\includegraphics[width=1\\linewidth]{plots\/ratio_SF.pdf} \\\\}\n\n\\caption{\\label{fig:dataSF_T2K} Top to bottom:  NuWro SF comparison to T2K data;  INCL + NuWro SF comparison to T2K data; comparison of NuWro, INCL + NuWro SF and data; ratio of NuWro SF and INCL + NuWro SF models of QE channel. Left: $\\delta\\alpha_T$, right: $\\delta$p$_T$}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n\\centering\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/MINERvA\/dat_NuWro_SF.pdf} \\\\}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/MINERvA\/dpt_NuWro_SF.pdf} \\\\}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/MINERvA\/dat_INCL_SF.pdf} \\\\}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/MINERvA\/dpt_INCL_SF.pdf} \\\\}\n\\end{minipage}\n\\vfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/MINERvA\/dat_SF_shape.pdf} \\\\}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[h]{0.49\\linewidth}\n\\center{\\includegraphics[width=1\\linewidth]{plots\/MINERvA\/dpt_SF_shape.pdf} \\\\}\n\\end{minipage}\n\\vfill\n\\center{\\includegraphics[width=1\\linewidth]{plots\/MINERvA\/ratio_SF.pdf} \\\\}\n\\caption{\\label{fig:dataSF} Top to bottom: NuWro SF comparison to Miner$\\nu$a data;  INCL + NuWro SF comparison to Miner$\\nu$a data; comparison of NuWro, INCL + NuWro SF and data; ratio of NuWro SF and INCL + NuWro SF models of QE channel. Left: $\\delta\\alpha_T$, right: $\\delta$p$_T$}\n\\end{figure*}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have compared the simulation of the final-state interactions between the NuWro and INCL cascade models in CCQE events. We have characterized the produced particles and the change of kinematics of the leading proton induced by FSI. We have quantified the impact on observables like STV and vertex activity. We have also compared the results of the simulations with the existing experimental data provided by T2K and MINER$\\nu$A collaborations.\n\nAn essential novelty of this study is the simulation of cluster production by INCL in FSI of neutrino interactions. Such clusters can be detected as highly ionizing tracks or, if below the tracking threshold, can contribute to the energy deposited around the vertex (vertex activity). Nuclear clusters behave differently than protons from many points of view. Since nuclear clusters release more energy by ionization, the detector tracking threshold tends to require the clusters to have larger momentum than protons in order to be reconstructed. When identified as tracks, the particle momentum could be measured by its curvature, and the corresponding kinetic energy depends on the actual particle mass. Generally, the kinetic energy of a cluster leaving the nucleus is different from the kinetic energy of the initial proton producing the cluster. \nMoreover, for a complete recollection of all the energy of the particles leaving the nucleus, the correct reconstruction of their secondary interactions is needed. While we leave the detailed characterization of secondary interactions for future work, we have considered the fraction of energy lost in secondary interactions and their probability, which depends on the nature of the particles leaving the nucleus. Notably, emitted nuclear clusters have different probability of secondary interactions with respect to protons.\nFor all these reasons, the accurate modeling and identification of particles in the final state, including nuclear clusters, are crucial to estimate correctly all the energy leaving the nucleus and the FSI corrections to it.\nThus, the modeling of cluster production by FSI is relevant for correct interpretation of data to constrain nuclear models and it may affect the calorimetric method of total neutrino energy reconstruction and how it is shared between the leptonic and hadronic part of the final state.  The precise neutrino energy reconstruction is a vital step for the new generation of oscillation experiments. The calorimetric method of the neutrino energy reconstruction, which takes into account all emitted particles after the neutrino interaction, allows for better neutrino energy resolution but also demands better modeling of the hadronic compound of the neutrino-nucleus interactions. Understanding the production and kinematical properties of nuclear clusters is crucial for improving the existing neutrino-nucleus interaction models and controlling the corresponding systematic uncertainties.\n\nRegarding the effects of the final-state interactions on the leading proton in neutrino-nucleus interactions, the nuclear model of INCL features much smaller transparency than the NuWro cascade. Transparency is a measure that allows estimating the FSI strength in a given model. Therefore, such results originate in a characteristic combination of the nucleon-nucleon cross sections,  Pauli blocking, and the specifics of the used nuclear model. In particular, INCL FSI simulation features a significant fraction of events without a proton in the final state, especially while propagating low momentum protons from the primary vertex. The INCL model also tends to re-absorb other particles produced during the FSI cascade: events with low momentum protons are mostly accompanied in NuWro with additional nucleons but not in INCL.\n\nThe accurate modeling of the rate of events without protons in the final state directly affects the correct interpretation of experimental data in long-baseline neutrino oscillation experiments. Regarding specific observables, the depletion of low momentum protons causes suppression of large values of $\\delta \\alpha_T$. Unfortunately, the proton tracking threshold in the detector induces a similar effect:  as we show in the comparison to T2K and Minerva measurements, the available data have very limited sensitivity to the difference between the two FSI models due to the proton momentum threshold of present detectors. These results demonstrate the importance of measuring low momentum protons to characterize their FSI. Currently running and upcoming experiments are planning to perform such measurements by providing low tracking thresholds and precise calorimetry~\\cite{steven_gardiner_2022_6717687}.\n\nSummarizing, the study presented here is a progress along the road of a complete and precise estimation of uncertainties due to the modeling of final-state interactions effects of neutrino-scattering simulation. It is also a useful example of the modularity of Monte Carlo generators and a step forward toward improving the simulation of FSI in neutrino-nucleus interactions.\n\n\n\\section*{Acknowledgements}\nWe have conducted this work within the T2K Near Detector upgrade project and gratefully acknowledge all productive meetings with our colleagues in this context.\nWe appreciate the fruitful discussions at the INCL meetings, particularly those with S.~Leray, D.~Mancusi, D.~Zharenov, and J.~L.~Rodr\u00edguez-S\u00e1nchez.\nThis work was supported by P2IO LabEx (ANR-10-LABX-0038 \u2013 Project ``BSMNu'') in the framework ``Investissements d'Avenir'' (ANR-11-IDEX-0003-01), managed by the Agence Nationale de la Recherche (ANR), France.\nWe acknowledge the support of CNRS\/IN2P3 and CEA. J.T.~Sobczyk and K.~Niewczas were partially supported by the Polish Ministry of Science and Higher Education grant No. DIR\/WK\/2017\/05, and under the Polish National Science Centre (NCN) grants No. 2021\/41\/B\/ST2\/02778 and No. 2020\/37\/N\/ST2\/01751, respectively.\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}