{"text":"\\section{Introduction}\n\n\\subsection{The Models}\n\nThe Fokker-Planck equations arise in many areas of sciences, including\nprobability, statistical physics, plasma physics, gas and stellar dynamics.\nThe term \\textquotedblleft Fokker-Planck\\textquotedblright\\ is widely used to\nrepresent various diffusion processes (Brownian motion).\n\nIn this paper, we study the kinetic Fokker-Planck equation with potential in $\\mathbb{R}^{3}$. It reads\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l\n\\displaystyle\\pa_{t}F+v\\cdot\\nabla_{x}F=\\nabla_{v}\\cdot\\left[ \\nabla\n_{v}F+(\\nabla_{v}\\Phi)F\\right] \\,,\\quad x,v\\in\\mathbb{R}^{3},\\ t>0,\\\\\n\\\\\n\\displaystyle F(0,x,v)=F_{0}(x,v)\\,,\n\\end{array}\n\\right. \\label{in.1.a\n\\end{equation}\nwhere the potential $\\Phi(v)$ is of the form\n\\[\n\\Phi=\\frac{1}{\\ga}\\left\\langle v\\right\\rangle ^{\\ga}+\\Phi_{0}\\,,\\ \\ga>0,\n\\]\nfor some constant $\\Phi_{0}.$ We define\n\\[\n\\mathcal{M}(v)=e^{-\\Phi(v)}\\,,\n\\]\nwith $\\Phi_{0}\\in\\mathbb{R}\\ $such that $\\mathcal{M}$ is a probability\nmeasure. It is easy to see that $\\mathcal{M}$ is a steady state to the\nFokker-Planck equation (\\ref{in.1.a}). Thus it is natural to study the\nfluctuation of the Fokker-Planck equation (\\ref{in.1.a}) around $\\mathcal{M\n(v)$, with the standard perturbation $f(t,x,v)$ to $\\mathcal{M}$ as\n\\[\nF=\\mathcal{M}+\\mathcal{M}^{1\/2}f\\,.\n\\]\nThe Fokker-Planck equation for $f(t,x,v)=\\mathbb{G}^{t}f_{0}$ now takes the\nform\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l\n\\displaystyle\\pa_{t}f+v\\cdot\\nabla_{x}f=\\Delta_{v}f-\\frac{1}{4}|v|^{2\n\\left\\langle v\\right\\rangle ^{2\\ga-4}f+\\left( \\frac{3}{2}\\left\\langle\nv\\right\\rangle ^{\\ga-2}+\\frac{\\ga-2}{2}|v|^{2}\\left\\langle v\\right\\rangle\n^{\\ga-4}\\right) f=Lf\\,,\\\\\n\\\\\nf(0,x,v)=f_{0}(x,v)\\,,\\ \\ \\ \\left( x,v\\right) \\in\\mathbb{R}^{3\n\\times\\mathbb{R}^{3}\\,.\n\\end{array}\n\\right. \\label{in.1.c\n\\end{equation}\nHere $\\mathbb{G}^{t}$ is the solution operator of the Fokker-Planck equation\n(\\ref{in.1.c}). It is obvious that $L$ is a non-positive self-adjoint operator\non $L_{v}^{2}.$ More precisely, its Dirichlet form is given by\n\\[\n\\left\\langle Lf,f\\right\\rangle _{v}=-\\int_{\\mathbb{R}^{3}}\\left\\vert\n\\nabla_{v}f+\\frac{\\nabla\\Phi}{2}f\\right\\vert ^{2}dv=-\\int_{\\mathbb{R}^{3\n}\\left\\vert \\nabla_{v}\\left( \\frac{f}{\\sqrt{\\mathcal{M}}}\\right) \\right\\vert\n^{2}\\mathcal{M}dv.\n\\]\nTherefore, the null space of $L$ is given by\n\\[\nKer(L)=\\hbox{span}\\left\\{ E_{D}\\right\\} \\,,\n\\]\nwhere $E_{D}=\\sqrt{\\mathcal{M}}$. Based on this property, we can introduce the\nmacro-micro decomposition as follows: the macro projection $\\mathrm{P}_{0}$ is\nthe orthogonal projection with respect to the $L_{v}^{2}$ inner product onto\n$\\mathrm{Ker}(L)$, and the micro projection $\\mathrm{P}_{1}\\equiv\n\\mathrm{Id}-\\mathrm{P}_{0}$.\n\n\\subsection{Main theorem}\n\nBefore the presentation of the main theorem, let us define some notation in\nthis paper. We denote $\\left\\langle v\\right\\rangle ^{s}=(1+|v|^{2})^{s\/2}$,\n$s\\in{\\mathbb{R}}$. For the microscopic variable $v$, we denote\n\\[\n|f|_{L_{v}^{2}}=\\Big(\\int_{{\\mathbb{R}}^{3}}|f|^{2}dv\\Big)^{1\/2},\n\\]\nand the weighted norms $|\\cdot|_{L_{v}^{2}(m)}$ and $|\\cdot|_{L_{\\theta}^{2}}$\ncan be defined by\n\\[\n|f|_{L^{2}(m)}=\\Big(\\int_{\\mathbb{R}^{3}}|f|^{2}mdv\\Big)^{1\/2}\\,,\\quad\n|f|_{L_{\\theta}^{2}}=\\Big(\\int_{\\mathbb{R}^{3}}\\left\\langle v\\right\\rangle\n^{2\\theta}|f|^{2}dv\\Big)^{1\/2},\n\\]\nrespectively, where $m=m\\left( t,x,v\\right) $ is a weight function. The\n$L_{v}^{2}$ inner product in ${\\mathbb{R}}^{3}$ will be denoted by\n$\\left\\langle \\cdot,\\cdot\\right\\rangle _{v}$,\n\\[\n\\left\\langle f,g\\right\\rangle _{v}=\\int_{{\\mathbb{R}}^{3}}f(v)\\overline\n{g(v)}dv.\n\\]\nFor the space variable $x$, we have the similar notation. In fact, $L_{x}^{2}$\nis the classical Hilbert space with norm\n\\[\n|f|_{L_{x}^{2}}=\\Big(\\int_{\\mathbb{R}^{3}}|f|^{2}dx\\Big)^{1\/2}\\,.\n\\]\nWe denote the supremum norm as\n\\[\n|f|_{L_{x}^{\\infty}}=\\sup_{x\\in{\\mathbb{R}^{3}}}|f(x)|\\,.\n\\]\nThe standard inner product in $\\mathbb{R}^{3}$ will be denoted by\n$(\\cdot,\\cdot)$. For the Fokker-Planck equation, the natural space in the $v$\nvariable is equipped with the norm $|\\cdot|_{L_{\\sigma}^{2}}$, which is\ndefined as\n\\[\n|f|_{L_{\\sigma}^{2}}^{2}=|\\big^{\\ga-1}f|_{L_{v}^{2}}^{2}+|\\nabla\n_{v}f|_{L_{v}^{2}}^{2}\\,,\n\\]\nand the corresponding weighted norms are defined as\n\\[\n|f|_{L_{\\sigma}^{2}(m)}^{2}=|\\big^{\\ga-1}f|_{L_{v}^{2}(m)}^{2\n+|\\nabla_{v}f|_{L_{v}^{2}(m)}^{2}\\,,\\quad|f|_{L_{\\sigma,\\theta}^{2}\n^{2}=|\\big^{\\ga-1}f|_{L_{\\theta}^{2}}^{2}+|\\nabla_{v}f|_{L_{\\theta\n^{2}}^{2}\\,.\n\\]\nMoreover, we define\n\\[\n\\Vert f\\Vert_{L^{2}}^{2}=\\int_{{\\mathbb{R}}^{3}}|f|_{L_{v}^{2}}^{2\ndx\\,,\\quad\\Vert f\\Vert_{L_{\\sigma}^{2}}^{2}=\\int_{{\\mathbb{R}}^{3\n}|f|_{L_{\\sigma}^{2}}^{2}dx\\,,\n\\]\nand\n\\[\n\\Vert f\\Vert_{L_{x}^{\\infty}L_{v}^{2}}=\\sup_{x\\in{\\mathbb{R}^{3}}\n|f|_{L_{v}^{2}}\\,,\\quad\\Vert f\\Vert_{L_{x}^{1}L_{v}^{2}}=\\int_{{\\mathbb{R\n^{3}}}|f|_{L_{v}^{2}}dx\\,.\n\\]\nFinally, we define the high order Sobolev norm in $x$ variable: let\n$k\\in{\\mathbb{N}}$ and let $\\alpha$ be any multi-index,\n\\[\n\\left\\Vert f\\right\\Vert _{H_{x}^{k}L_{v}^{2}}:=\\sum_{|\\alpha|\\leq k}\\left\\Vert\n\\partial_{x}^{\\alpha}f\\right\\Vert _{L^{2}}\\,.\n\\]\nThe weighted spaces in the $(x,v)$-variable can be defined in a similar way.\n\nFor multi-indices $\\alpha$, $\\beta_{j}(j=1,\\dots,s)\\in\\mathbb{N}_{0}^{3}$ with\n$\\alpha=\\sum\\limits_{j=1}^{s}\\beta_{j}$, we denote the multinomial\ncoefficients by\n\\[\n\\binom{\\alpha}{\\beta_{1}\\,\\beta_{2}\\,\\dots\\,\\beta_{s}}=\\prod_{k=1}^{3\n\\frac{\\alpha_{k}!}{\\prod_{j=1}^{s}(\\beta_{j})_{k}!}\\,.\n\\]\n\n\nThe domain decomposition plays an essential role in our analysis, hence we\ndefine a cut-off function $\\chi:{\\mathbb{R}}\\rightarrow{\\mathbb{R}}$, which is\na smooth non-increasing function, $\\chi(s)=1$ for $s\\leq1$, $\\chi(s)=0$ for\n$s\\geq2$ and $0\\leq\\chi\\leq1$. Moreover, we define $\\chi_{R}(s)=\\chi(s\/R)$.\n\nFor simplicity of notation, hereafter, we abbreviate \\textquotedblleft{\\ $\\leq\nC$} \\textquotedblright\\ to \\textquotedblleft{\\ $\\lesssim$ }\\textquotedblright,\nwhere $C$ is a positive constant depending only upon fixed numbers.\n\nHere is the precise description of our main results (combining theorem \\ref{time-like region for gamma larger or equal than 1}, theorem \\ref{time-like region for gamma less than 1}, theorem \\ref{space-like region for gamma greater or equal to 3\/2} and theorem \\ref{space-like region for 00$, there exists a positive constant $C$ such that\nthe solution $f$ satisfies\n\n\\begin{enumerate}\n\\item For $\\left\\langle x\\right\\rangle \\leq2Mt$,\n\\[\n\\left\\vert f(t,x)\\right\\vert _{L_{v}^{2}}\\lesssim\\left[ (1+t^{-9\/4\n)e^{-Ct}+(1+t)^{-3\/2}\\Big(1+\\frac{|x|^{2}}{1+t}\\Big)^{-N} \\right] \\Vert\nf_{0}\\Vert_{L_{x}^{\\infty}L_{v}^{2}}\\,.\n\\]\n\n\n\\item For $\\left\\langle x\\right\\rangle \\geq2Mt$,\n\\[\n\\left\\vert f(t,x)\\right\\vert _{L_{v}^{2}}(1+t^{-9\/4})e^{-C(\\left\\langle\nx\\right\\rangle +t)^{\\frac{\\gamma}{3-\\gamma}}}\\Vert f_{0}\\Vert_{L^{2\n(e^{4\\alpha\\left\\langle v\\right\\rangle ^{\\gamma}})}\\,.\n\\]\n\n\\end{enumerate}\n\n\\item As $0<\\gamma<1$, for any sufficiently small $\\alpha>0$, there exists a\npositive constant $C$ such that the solution $f$ satisfies\n\n\\begin{enumerate}\n\\item For $\\left\\langle x\\right\\rangle \\leq2Mt$,\n\\[\n\\left\\vert f(t,x)\\right\\vert _{L_{v}^{2}}\\lesssim\\left[ (1+t^{-9\/4\n)e^{-Ct^{\\frac{\\gamma}{2-\\gamma}}}+(1+t)^{-3\/2}\\right] \\left\\Vert\nf_{0}\\right\\Vert _{L^{2}(e^{4\\alpha\\left\\langle v\\right\\rangle ^{\\gamma}\n)}\\,.\n\\]\n\n\n\\item For $\\left\\langle x\\right\\rangle \\geq2Mt$,\n\\[\n\\left\\vert f(t,x)\\right\\vert _{L_{v}^{2}}\\lesssim(1+t^{-9\/4\n)e^{-C(\\left\\langle x\\right\\rangle +t)^{\\frac{\\gamma}{3-\\gamma}}}\\Vert\nf_{0}\\Vert_{L^{2}(e^{4\\alpha\\left\\langle v\\right\\rangle ^{\\gamma}})}\\,.\n\\]\n\n\\end{enumerate}\n\\end{enumerate}\n\\end{theorem}\n\n\\subsection{Review of previous works and significant points of the paper}\n\nThe study of the Fokker-Planck equation can be traced back to 1930's. When the\npotential $\\Phi=0$, the equation (\\ref{in.1.a}) is known as the\nKolmogorov-Fokker-Planck equation.\nIn 1934 Kolmogorov \\cite{[Kolmogorov]} derived the Green function for the\nwhole space problem. The explicit formula surprisingly showed that the\nsolution becomes smooth in the $t,x,v$ variables when $t>0$ immediately.\n\nLater the regularization effect has been investigated further and been\nrecovered by some more general and robust methods. For example, it is known\nthat the Fokker-Planck operator $-v\\cdot\\nabla_{x}+\\Delta_{v}$ is a\nhypoelliptic operator. So one can apply H\\\"{o}rmander's commutator\n\\cite{[Hormander]} to the linear Fokker-Planck operator to obtain that\ndiffusion in $v$ together with the transport term $v\\cdot\\nabla_{x}$ has a\nregularizing effect on solutions not only in $v$ but also in $t$ and $x$. It\ncan also be obtained through the functional method, see \\cite{[Herau],Villani\n. On the other hand, the Fokker-Planck operator is also known as a\nhypocoercive operator, which concerns the rate of convergence to equilibrium.\nIndeed, the trend to equilibria with a certain rate has been investigated in\nmany papers (cf. \\cite{[Desvillettes],[Duan],[Herau],[Herau-Nier], [MisMou],\nmouNeu}) for the close to Maxwellian regime in the whole space or in the\nperiodic box.\n\nLet us point out the recent important results constructed by Mouhot and\nMischler \\cite{[MisMou]}. They developed an abstract method for deriving decay\nestimates of the semigroup associated to non-symmetric operators in Banach\nspaces. Applying this method to the kinetic Fokker-Planck equation in the\ntorus with potential in the close to equilibrium setting, they obtained\nspectral gap estimates for the associated semigroup in various norms,\nincluding Lebesgue norms, negative Sobolev norms, and the\nMonge-Kantorovich-Wasserstein distance $W_{1}$.\n\nIn this paper, we study the Fokker-Planck equation with potential\nin the close to equilibrium setting. In the literature, this kind of problem\nbasically focuses on the rate of convergence to equilibrium (see the reference\nlisted above). Instead, in this paper we supply an explicit description of the\nsolution in the sense of pointwise estimate. It turns out the structure of the\nsolution sensitively depends on the potential function. Let us illustrate the\nnovelties of the paper:\n\n\\begin{itemize}\n\\item We obtain the global picture of the solution, which consists of three\nparts: the time-like region (large time behavior), the space-like region\n(spatially asymptotic behavior) and the small time region (the evolution of\ninitial singularity).\n\n\\begin{enumerate}\n\\item In the time-like region, we have distinctly different descriptions\naccording to potential functions. For $\\gamma\\geq1$, thanks to the spectrum\nanalysis,\nwe have a pointwise fluid structure, which is more precise than previous\nresults. The leading term of the wave propagation has been recognized. More\nspecifically, for $\\gamma\\geq3\/2$ the leading term is a diffusion wave with\nheat kernel type, while for $1\\leq\\gamma<3\/2$ the diffusion wave is of\nalgebraic type. By contrast, the spectral information is missing for\n$0<\\gamma<1$ due to the weak damping for large velocity, which leads to the\nunavailability of pointwise structure. Nevertheless, we can apply Kawashima's\nargument \\cite{[Kawashima],[Strain]} to get a uniform time decay rate.\n\n\n\n\\item Concerning the space-like region, we have exponential decay for\n$\\gamma\\geq3\/2$ and sub-exponential decay for $0<\\gamma<3\/2$. The results are\nconsistent with the wave behaviors inside the time-like region for different\n$\\gamma$'s respectively. To our knowledge, this is the first result for the\nasymptotic behavior of the Fokker-Planck equation with potential.\n\n\\item Owing to the regularization effect, the initial singularity is\neliminated instantaneously.\n\n\n\\end{enumerate}\n\n\\medskip\n\n\\item The regularization estimate is a key ingredient of this paper (see Lemma\n\\ref{regularization} and Lemma \\ref{second-der}), which enables us to obtain\nthe pointwise estimate without regularity assumptions on the initial\ncondition.\nIn the literature, the regularization estimates for the kinetic Fokker-Planck\nequation and Landau equation have been proved for various purposes, see for\ninstance \\cite{[Herau]}, \\cite{[MisMou]}, \\cite{Villani} (Appendix A.21.2) for\nthe Fokker-Planck case and \\cite{[CTK]} for the Landau case. {The\nabove-mentioned regularization estimates are sufficient for studying the time\ndecay of the solution. However, to gain understanding of the spatially\nasymptotic behavior, one needs to analyze the solution in some appropriate\nweighted spaces. Taking this into account, we construct the regularization\nestimates in suitable weighted spaces. The calculation of the estimates is\ninteresting and more sophisticated than before. Moreover, this type of\nregularization estimate is itself new.} \\medskip\n\n\\item {The pointwise estimate of the solution in the space-like region is\nconstructed by the weighted energy estimate. The time-dependent weight\nfunctions are chosen according to different confinement potentials. For\n$\\gamma\\geq3\/2$, from estimate in the time-like region, the solution decays\nexponentially along the wave cone, i.e., $|x|=Mt$, suggesting the exponential\ndecay at the spatial infinity. It turns out that a simple weight function is\nsatisfactory (see Proposition \\ref{weig_1}). However, when $0<\\gamma<3\/2$, we\nnotice that in \\eqref{in.1.c} the exponent of damping coefficient($\\sim\n\\left\\langle v\\right\\rangle ^{2(\\gamma-1)}$) is less than 1. From the scaling\nof transport equation, we cannot expect exponential decay in the spatial\nvariable. In fact, motivating by the transport equation with weak damping, we\ndevise appropriate weight functions, introduce a refined space-velocity domain\ndecomposition and eventually show the sub-exponential decay for $0<\\gamma<3\/2$\n(Proposition \\ref{weig_2}).} \\medskip\n\n\\item We believe that our idea in this paper can have potential applications\nin other important kinetic equations, such as the Landau equation or Boltzmann\nequation without angular cutoff. In fact, these projects are in progress.\n\\end{itemize}\n\n{To the best of our knowledge, the first pointwise result of the kinetic type\nequation is the Boltzmann equation for hard sphere\n\\cite{[LiuYu],[LiuYu2],[LiuYu1]}; the authors have established important\nresults regarding the pointwise behavior of the Green function and completed\nthe nonlinear problem. Later, the result was generalized to the Boltzmann\nequation with cutoff hard potentials \\cite{[LeeLiuYu]}.\nVery recently, the authors of the current paper extend the pointwise result to\nmore general potentials, the range $-2<\\gamma<1$, and obtain an explicit\nrelation between the decay rate and velocity weight assumption\n\\cite{[LinWangWu]}. Let us point out some similarities and differences between\nthe Fokker-Planck equation with potential and the Boltzmann equation\nwith hard sphere or cutoff hard potentials.}\n\n\\begin{itemize}\n\\item The solutions of both equations in large time are dominated by the fluid\npart. For the Fokker-Planck with $\\gamma\\geq1$ and for the Boltzmann with hard\nsphere or hard potentials with cutoff, the fluid parts are characterized by\ndiffusion waves. To extract them, both need the long wave-short wave\ndecomposition. However the wave structures of them are quite different. For\nthe Boltzmann equation, there are diffusion waves propagating with different\nspeeds: one with the background speed of the global Maxwellian while the other\nwith the superposed speed of the background speed and the sound speed. In\ncomparison, there is only one diffusion wave for the Fokker-Planck equation.\nThe fluid behavior can be seen formally from the Chapman-Enskog expansion,\nwhich indicates that the macroscopic part (the fluid part) of the solution\nsatisfies the viscous system of conservation laws\\textbf{.} For the Boltzmann\nequation there are conservation laws of mass, momentum and energy, while the\nFokker-Planck equation only preserves the mass, explaining the difference of\ntheir wave structures.\n\\medskip\n\n\\item Since the leading term of the solution in large time is the fluid part\nand it essentially has finite propagation speed, the solution in the\nspace-like region,\\ compared to the leading part, should be much smaller. In\nfact it is shown that the asymptotic behaviors exponentially or\nsub-exponentially decay. This is similar to the solution of the Boltzmann\nequation outside the finite Mach number region. \\medskip\n\n\\item The regularization mechanism of the Fokker-Planck equation is distinct\nfrom that of the Boltzmann equation. For the Boltzmann equation, the initial\nsingularity will be preserved (although decays in time very fast), one has to\nsingle them out. Since the singular waves satisfy a damped transport equation,\nthere is an explicit solution formula, from which the pointwise structure can\nbe deduced. Then the regularity of the resulting remainder part comes from the\ncompact part of the collision operator (see the Mixture Lemma in\n\\cite{[LeeLiuYu]}, \\cite{[LiuYu]} and \\cite{[LiuYu2]}). By contrast, for the\nFokker-Planck equation the regularity comes from the combined effect of\nellipticity in the velocity variable $v$ and the transport term (see Lemma\n\\ref{regularization} and Lemma \\ref{second-der}). The initial singularities\nhave been identified. However, there is no explicit formula for singular\nwaves. Instead, they are accurately estimated by suitable weighted energy estimates.\n\\end{itemize}\n\n\\subsection{Method of proof and plan of the paper}\n\nThe main idea of this paper is to combine the long wave-short wave\ndecomposition, the wave-remainder decomposition, the weighted energy estimate\nand the regularization estimate together to analyze the solution. The long\nwave-short wave decomposition, based on the Fourier transform, gives the fluid\nstructure of the solution. The wave-remainder decomposition is used for\nextracting the initial singularity. The weighted energy estimate is used for\nthe pointwise estimate of solution inside the space-like region, where the\nregularization estimate is also used. We explain the idea in more detail as below.\n\nIn the time-like region (inside the region $|x|\\leq Mt$ for some $M$), the\nsolution is dominated by the fluid part, which is contained in the long wave\npart. In order to obtain its estimate, we devise different methods for\n$\\gamma\\geq1$ and $0<\\gamma<1$ respectively. For $\\gamma\\geq1$, taking\nadvantage of the spectrum information of the Fokker-Planck operator\n\\cite{[LuoYu]} (in fact, the paper \\cite{[LuoYu]} only studies the case\n$\\gamma=2$ and we can extend it to the case $\\gamma\\geq1$), the complex\nanalytic or Fourier multiplier techniques can be applied to obtain pointwise\nstructure of the fluid part. However, for $0<\\gamma<1$, the spectrum\ninformation is missing due to the weak damping for large velocity. Instead, we\nuse Kawashima's argument \\cite{[Kawashima]} to get the optimal decay only in\ntime. It is shown that the $L^{2}$ norm of the short wave exponentially decays\nin time for $\\gamma\\geq1$ essentially due to the spectrum gap, while it decays\nonly algebraically for $0<\\gamma<1$ if imposing certain velocity weight on\ninitial data.\n\nWe use the wave-remainder decomposition to extract the possible initial\nsingularity in the short wave. This decomposition is based on a Picard-type\niteration. The first several terms in the iteration contain the most singular\npart of the solution, and they are the so-called wave part.\nBy functional methods, we prove the iteration equation has a regularization\neffect, which enables us to show the remainder becomes more regular. Noticing\nthe singularity will disappear after initial time, the regularization estimate\ntogether with $L^{2}$ decay of the short wave yields the $L^{\\infty}$ decay of\nthe short wave. Combing this with the long wave, we finish the pointwise\nstructure inside the wave cone.\n\nTo get the global structure of the solution, we need the estimate outside the\nwave cone, i.e., inside the space-like region. The weighted energy estimates\nplay a decisive role here. The weight functions are carefully chosen for\ndifferent $\\gamma$'s. It is noted that the sufficient understanding of the\nstructure of the wave part obtained previously, is essential in the estimate.\nMoreover, the regularization effect makes it possible to do the higher order\nweighted energy estimate. Then the desired pointwise estimate follows from the\nSobolev inequality.\n\nThe rest of this paper is organized as follows: We first prepare some\nimportant properties in Section \\ref{pre} for the long wave-short wave\ndecomposition, the wave-remainder decomposition and regularization estimates.\nThen we study the large time behavior in Section \\ref{large-time}. Finally, we\nstudy the initial layer and the asymptotic behavior in Section \\ref{layer}.\n\n\\section{Preliminary}\n\n\\label{pre}\n\n\\subsection{The operator $L$}\n\nFirst, we introduce a new norm $\\left\\vert \\cdot\\right\\vert _{L_{\\widetilde\n{\\sigma},\\theta}^{2}}$:\n\\[\n\\left\\vert g\\right\\vert _{L_{\\widetilde{\\sigma},\\theta}^{2}}^{2\n:=\\int\\left\\langle v\\right\\rangle ^{2\\theta}\\left\\vert \\nabla g\\right\\vert\n^{2}dv+\\int\\left\\langle v\\right\\rangle ^{2\\theta}\\frac{\\left\\vert v\\right\\vert\n^{2}\\left\\langle v\\right\\rangle ^{2\\gamma-4}}{2}\\left\\vert g\\right\\vert\n^{2}dv,\\ \\ \\ \\theta\\in\\mathbb{R},\\ \\gamma>0,\n\\]\nwhich is equivalent to the natural norm $\\left\\vert \\cdot\\right\\vert\n_{L_{\\sigma,\\theta}}$. Through this equivalent norm, we can derive the\ncoercivity of the operator $L$ for all $\\gamma>0,$ as below. The proof is\nanalogous to the Landau case \\cite{[Guo]}.\n\n\\begin{lemma}\n[Coercivity]\\label{co} Let $\\theta\\in\\mathbb{R}$, $\\gamma>0$. For any $m>1,$\nthere is $00$ such tha\n\\begin{equation}\n\\left\\langle -Lg,g\\right\\rangle _{v}\\geq\\nu_{0}\\left\\vert \\mathrm{P\n_{1}g\\right\\vert _{L_{\\sigma}^{2}}^{2}. \\label{coercivity\n\\end{equation}\n\n\\end{lemma}\n\nNow, let us decompose the collision operator $L=-\\Lambda+K$, where\n\\[\n\\Lambda=-L+\\varpi\\chi_{R}\\left( |v|\\right) \\,,\\quad K=\\varpi\\chi_{R}\\left(\n|v|\\right) \\,,\n\\]\nhere $\\varpi>0$ and $R>0$ are as large as desired.\n\nRegarding the behavior of solutions to equation (\\ref{in.1.c}) in the\nspace-like region, the following weight functions $\\mu(x,v)$ will be taken\ninto account:\n\\[\n\\mu(x,v)=1\\quad\\hbox{or}\\quad\\exp\\left( \\left\\langle x\\right\\rangle\n\/D\\right) \\quad\\hbox{if}\\quad\\ga\\geq3\/2\\,,\n\\]\nfor $D$ large, and\n\\[\n\\mu(x,v)=1\\quad\\hbox{or}\\quad\\exp\\left( \\alpha c(x,v)\\right) \\quad\n\\hbox{if}\\quad0<\\ga<3\/2\\,,\n\\]\nwhere\n\\begin{align*}\nc(x,v) & =5\\Big(\\de\\left\\langle x\\right\\rangle \\Big)^{\\frac{\\ga}{3-\\ga\n}\\left( 1-\\chi\\left( \\de\\left\\langle x\\right\\rangle \\left\\langle\nv\\right\\rangle ^{\\ga-3}\\right) \\right) \\\\\n& \\quad+\\bigg[\\left( 1-\\chi\\left( \\de\\left\\langle x\\right\\rangle\n\\left\\langle v\\right\\rangle ^{\\ga-3}\\right) \\right) \\de\\left\\langle\nx\\right\\rangle \\left\\langle v\\right\\rangle ^{2\\ga-3}+3\\left\\langle\nv\\right\\rangle ^{\\ga}\\bigg]\\chi\\left( \\de\\left\\langle x\\right\\rangle\n\\left\\langle v\\right\\rangle ^{\\ga-3}\\right) \\,,\n\\end{align*}\nthe positive constants $\\delta$ and $\\alpha$ being determined later.\n\n\\begin{lemma}\n\\label{prop1} Assuming that $\\gamma>0$, we have the following properties of\nthe operators $\\Lambda$ and $K$.\n\n\\noindent\\textrm{(i)} There exists $c>0$ such that\n\\[\n\\int\\left( \\Lambda g\\right) g\\mu dxdv\\geq c\\left\\Vert g\\right\\Vert\n_{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2}.\n\\]\n\\noindent\\textrm{(ii)}\n\\[\n\\int\\left( Kg\\right) g\\mu dxdv\\leq\\varpi\\left\\Vert g\\right\\Vert\n_{L^{2}\\left( \\mu\\right) }^{2}.\n\\]\n\n\\end{lemma}\n\n\\begin{proof}\nWe only prove part (i) when $\\mu\\left( x,v\\right) =e^{\\alpha c\\left(\nx,v\\right) },$ since the other cases of part (i) and part (ii) are trivial.\nNotice that there is a constant $c_{1}>0$ such tha\n\\[\n\\left\\Vert \\nabla_{v}g\\right\\Vert _{L^{2}\\left( \\mu\\right) }^{2}+\\int\\left[\n\\frac{\\left\\vert v\\right\\vert ^{2}\\left\\langle v\\right\\rangle ^{2\\gamma-4}\n{4}-\\left( \\frac{3}{2}\\left\\langle v\\right\\rangle ^{\\gamma-2}+\\frac{\\left(\n\\gamma-2\\right) }{2}\\left\\vert v\\right\\vert ^{2}\\left\\langle v\\right\\rangle\n^{\\gamma-4}\\right) +\\varpi\\chi_{R}\\right] g^{2}\\mu dxdv\\geq c_{1}\\left\\Vert\ng\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2\n\\]\nwhenever $\\varpi,$ $R>0$ are sufficiently large. On the other hand, it follows\nfrom\n\\begin{equation}\n\\left\\vert \\nabla_{v}c\\left( x,v\\right) \\right\\vert \\leq C\\left(\n\\gamma\\right) \\left\\langle v\\right\\rangle ^{\\gamma-1}\\left( 1+\\left\\vert\n\\chi^{\\prime}\\left( \\left[ \\delta\\left\\langle x\\right\\rangle \\right]\n\\left\\langle v\\right\\rangle ^{\\gamma-3}\\right) \\right\\vert \\right) ,\n\\label{v-derivtive-c\n\\end{equation}\nthat\n\\[\n\\left\\vert \\int\\nabla_{v}g\\cdot\\nabla_{v}\\left( \\mu\\right) gdxdv\\right\\vert\n\\leq\\alpha C\\left( \\gamma\\right) \\sup\\left( 1+\\left\\vert \\chi^{\\prime\n}\\right\\vert \\right) \\int\\left\\langle v\\right\\rangle ^{\\gamma-1}\\left\\vert\ng\\right\\vert \\left\\vert \\nabla_{v}g\\right\\vert \\mu dxdv\\leq\\frac{\\alpha Q\n{2}\\left\\Vert g\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2},\n\\]\nwhere $Q=\\left[ C\\left( \\gamma\\right) \\sup\\left( 1+\\left\\vert \\chi\n^{\\prime}\\right\\vert \\right) \\right] .$ Therefore, we choose $\\alpha>0$\nsufficiently small with $\\alpha Q0$ such that if\n$|\\eta|>\\de$,\n\\begin{equation}\n\\hbox{\\rm Spec}(\\eta)\\subset\\{z\\in\\mathbb{C}:\\mathrm{Re}(z)<-\\tau\\}\\,.\n\\label{pre.ab.e\n\\end{equation}\n\\newline\\noindent\\textrm{(ii)} If $|\\eta|<\\de$,\n\\begin{equation}\n\\hbox{\\rm Spec}(\\eta)\\cap\\{z\\in\\mathbb{C}:\\mathrm{Re}(z)>-\\tau\\}=\\{\\la(\\eta\n)\\}\\,, \\label{pre.ab.f\n\\end{equation}\nwhere $\\lambda(\\eta)$ is the eigenvalue of $L_{\\eta}$ which is real and smooth\nin $\\eta$ only through $|\\eta|^{2}$, i.e., $\\lambda(\\eta)=\\mathscr{A}(|\\eta\n|^{2})$ for some real smooth function $\\mathscr{A}$; the eigenfunction\n$e_{D}(v,\\eta)$ is smooth in $\\eta$ as well. In addition, they are analytic in\n$\\eta$ if $\\ga\\geq3\/2$. Their asymptotic expansions are given as below:\n\\begin{equation\n\\begin{array}\n[c]{l\n\\label{pre.ab.h}\\displaystyle\\la(\\eta)=-a_{\\ga}|\\eta|^{2}+O(|\\eta|^{4})\\,,\\\\\n\\\\\n\\displaystyle e_{D}(\\eta)=E_{D}+iE_{D,1}|\\eta|+O(|\\eta|^{2})\\,,\n\\end{array}\n\\end{equation}\nwith $a_{\\ga}>0$, $E_{D,1}=L^{-1}(v\\cdot\\om E_{D})$, $\\om=\\eta\/|\\eta|$. Here\n$\\{e_{D}(\\eta)\\}$ can be normalized by\n\\[\n\\big_{v}=1\\,.\n\\]\n\\newline\\noindent\\textrm{(iii)} Moreover, the semigroup $e^{(-i\\eta\\cdot\nv+L)t}$ can be decomposed as\n\\begin{equation}\n\\displaystyle e^{(-i\\eta\\cdot v+L)t}f=e^{(-i\\eta\\cdot v+L)t}\\Pi_{\\eta\n^{D\\perp}f+\\mathbf{1}_{\\{|\\eta|<\\de\\}}e^{\\la(\\eta)t}\\big_{v}e_{D}(\\eta)\\,, \\label{pre.ab.g\n\\end{equation}\nwhere $\\mathbf{1}_{D}$ is the characteristic function of the domain $D$, and\nthere exist $a(\\tau)>0$ and $\\overline{a}>0$ such that $|e^{(-i\\eta\\cdot\nv+L)t}\\Pi_{\\eta}^{D\\perp}|_{L_{v}^{2}}\\lesssim e^{-a(\\tau)t}$ and\n$|e^{\\la(\\eta)t}|\\leq e^{-\\overline{a}|\\eta|^{2}t}$. \\medskip\n\\end{lemma}\n\n\\begin{proof}\nLet $L=-\\Lambda+K$ wit\n\\[\n\\Lambda_{\\eta}f=\\left( -\\Lambda-i\\eta\\cdot v\\right) f,\\ \\ \\ \\ \\ L_{\\eta\n}=\\left( L-i\\eta\\cdot v\\right) f,\n\\]\nHere $f\\in D\\left( \\Lambda_{\\eta}\\right) =\\left\\{ f\\in L_{v}^{2\n;\\Lambda_{\\eta}f\\in L_{v}^{2}\\right\\} $ and $D\\left( \\Lambda_{\\eta}\\right)\n=D\\left( L_{\\eta}\\right) .$ Since $K$ is a bounded operator in $L_{v}^{2},$\n$L_{\\eta}\\ $is regarded as a bounded perturbation of $\\Lambda_{\\eta}.$ We\nshall verify that such a decomposition satisfies the four hypotheses\n\\textbf{H1-H4} stated in \\cite{[YangYu]}. Under the assumptions \\textbf{H1-H4,\n}using semigroup theory and linear operator perturbation theory,\\textbf{\n}Theorem 1.1 in\\textbf{ \\cite{[YangYu]} }asserts that\\textbf{ }the spectrum of\n$L_{\\eta}$\\textbf{ }has the similar structure of the Boltzmann\\textbf{\n}equation with cutoff hard potential. Since the null space of the linear\nFokker-Planck operator is one-dimensional, for $\\left\\vert \\eta\\right\\vert $\nsmall enough, we only obtain one smooth eigenvalue of $L_{\\eta}$ while there\nare five smooth eigenvalues for the Boltzmann equation with cutoff hard\npotentials. As to the verification of \\textbf{H1-H4, }the proof is a slight\nmodification of the paper \\cite{[LuoYu]} and hence we omit the details. The\nhypothesis \\textbf{H1} is worthy of being mentioned, for $\\varpi$ sufficiently\nlarge, there exists a constant $c>0$ such that\n\\[\n\\left\\langle \\Lambda f,f\\right\\rangle _{v}\\geq c\\left\\vert f\\right\\vert\n_{L_{\\sigma}^{2}}^{2}\\geq c\\left\\vert f\\right\\vert _{L_{v}^{2}}^{2},\n\\]\nfor all $\\gamma\\geq1,$ the last inequality holds since $\\left\\vert\nf\\right\\vert _{L_{\\sigma}^{2}}$ is stronger than $\\left\\vert f\\right\\vert\n_{L_{v}^{2}}\\ $as $\\gamma\\geq1.\\ $This is why we miss the spectrum structure\nfor the case $0<\\gamma<1.$\n\n\n\nTo prove (ii), we need to explore the symmetric properties of $\\lambda(\\eta)$\nand $e_{D}(\\eta)$. Here we follow the framework of section 7.3 in\n\\cite{[LiuYu1]}. First we notice there is a natural three dimensional\northogonal group $O(3)$-action on $L_{v}^{2}$: Let $a\\in O(3)$, $f\\in\nL_{v}^{2}$,\n\\[\n(a\\circ f)(v)\\equiv f(a^{-1}v).\n\\]\nThen it is easy to check the $O(3)$-action commutes with operators $L$,\n$\\mathrm{P}_{0}$ and $\\mathrm{P}_{1}$. Consider the eigenvalue problem\n\\begin{equation}\nL_{\\eta}e_{D}(\\eta)=(-iv\\cdot\\eta+L)e_{D}(\\eta)=\\lambda(\\eta)e_{D}(\\eta).\n\\label{eigenprob\n\\end{equation}\nApply $a\\in O(3)$ to \\eqref{eigenprob}, by commutative properties and the fact\nthat $a$ preserves the vector inner product in $\\mathbb{R}^{3}$,\n\\[\n(-iv\\cdot(a\\eta)+L)(a\\circ e_{D}(\\eta))=\\lambda(\\eta)(a\\circ e_{D})(\\eta).\n\\]\nThen $\\lambda(a\\eta)=\\lambda(\\eta)$, $e_{D}(a\\eta)=a\\circ e(\\eta)$, which\nimplies that $\\lambda(\\eta)$ is dependent only upon $|\\eta|$. Now let $a\\in\nO(3)$ be an orthogonal transformation that sends $\\frac{\\eta}{|\\eta|}$ to\n$(1,0,0)^{T}$. Thus the original eigenvalue problem \\eqref{eigenprob} is\nreduced to\n\\begin{equation}\n(-iv_{1}|\\eta|+L)e(|\\eta|)=\\lambda(|\\eta|)e(|\\eta|), \\label{eigenprob1\n\\end{equation}\nwith $\\lambda(\\eta)=\\lambda(|\\eta|)$, $e_{D}(\\eta)=a^{-1}\\circ e_{D}(|\\eta\n|)$.\nApply the Macro-Micro decomposition to \\eqref{eigenprob1} to yield\n\\begin{subequations}\n\\label{eq:L6.1\n\\begin{align}\n& -i|\\eta|\\mathrm{P}_{0}v_{1}\\big(\\mathrm{P}_{0}e+\\mathrm{P}_{1\ne\\big)=\\lambda\\mathrm{P}_{0}e,\\label{eq:L6.1a}\\\\\n& -i|\\eta|\\mathrm{P}_{1}v_{1}\\mathrm{P}_{0}e-i|\\eta|\\mathrm{P}_{1\nv_{1}\\mathrm{P}_{1}e+L\\mathrm{P}_{1}e=\\lambda\\mathrm{P}_{1}e. \\label{eq:L6.1b\n\\end{align}\nSet $\\lambda(|\\eta|)=i|\\eta|\\zeta(|\\eta|).$ We can solve $\\mathrm{P}_{1}e$ in\nterms of $\\mathrm{P}_{0}e$ from \\eqref{eq:L6.1b},\n\\end{subequations}\n\\begin{equation}\n\\mathrm{P}_{1}e=i|\\eta|\\big[L-i|\\eta|\\mathrm{P}_{1}v_{1}-i|\\eta|\\zeta\n(|\\eta|)\\big]^{-1}\\mathrm{P}_{1}v_{1}\\mathrm{P}_{0}e, \\label{eq:L6.2\n\\end{equation}\nthen substitute this back to \\eqref{eq:L6.1a} to get\n\\begin{equation}\n\\Big(\\mathrm{P}_{0}v_{1}+i|\\eta|\\mathrm{P}_{0}\\big[L-i|\\eta|\\mathrm{P\n_{1}v_{1}-i|\\eta|\\zeta(|\\eta|)\\big]^{-1}\\mathrm{P}_{1}v_{1}\\Big)\\mathrm{P\n_{0}e=-\\zeta\\mathrm{P}_{0}e. \\label{eigenprob_fte\n\\end{equation}\nWe notice that this is actually a finite dimensional eigenvalue problem. The\nsolvability of it and the asymptotic expansions of eigenvalue and\neigenfunction for $|\\eta|\\ll1$ are essentially due to the implicit function\ntheorem. The procedure is basically the same as the case $\\gamma=2,$ we refer\nthe readers to Theorem 3.2 in \\cite{[LuoYu]} for details. We obtain\n$\\lambda(|\\eta|)$ and $\\mathrm{P}_{0}e(|\\eta|)=\\beta(|\\eta|)E_{D}$ with\n$\\lambda$ and $\\beta$ being smooth functions. Furthermore, $\\lambda(|\\eta|)$\nand $\\beta(|\\eta|)$ are not merely smooth but analytic for $\\gamma\\geq3\/2$. To\nprove this, it suffices to check that the perturbation $ivf$ is $L$-bounded,\ni.e.,\n\\[\n|vf|_{L_{v}^{2}}^{2}\\leq C_{1}|Lf|_{L_{v}^{2}}^{2}+C_{2}|f|_{L_{v}^{2}\n^{2}\\,.\n\\]\nThen the Kato-Rellich theorem guarantees the operator $B(z)=-iv_{1}z+L$ is in\nthe analytic family of Type (A), see \\cite{[Kato]}, which in turn implies the\neigenvalue and eigenfunction associated with \\eqref{eigenprob1} are analytic\nin $|\\eta|$, cf. \\cite{[DeLe]}. Now, let us calculate $\\big<\\Lambda f,\\Lambda\nf\\big>_{v}$ first. For simplicity of notation, let\n\\[\n\\psi(v)=\\frac{1}{4}|v|^{2}\\left\\langle v\\right\\rangle ^{2\\ga-4}-\\left(\n\\frac{3}{2}\\left\\langle v\\right\\rangle ^{\\ga-2}+\\frac{\\ga-2}{2}|v|^{2\n\\left\\langle v\\right\\rangle ^{\\ga-4}\\right) +\\varpi\\chi_{R}(|v|)\\,,\n\\]\nthen\n\\begin{align*}\n\\big<\\Lambda f,\\Lambda f\\big>_{v} & =|\\Delta_{v}f|_{L_{v}^{2}}^{2\n+|\\psi(v)f|_{L_{v}^{2}}^{2}\\\\\n& \\quad+2\\big<\\psi(v),\\big(\\nabla_{v}f,\\nabla_{v}f\\big)\\big>_{v\n+2\\big_{v}\\,.\n\\end{align*}\nBy the Cauchy inequality, we have\n\\[\n\\big|\\big_{v}\\big|\\leq\n\\big<\\psi(v),\\big(\\nabla_{v}f,\\nabla_{v}f\\big)\\big>_{v}+\\frac{1}{4\n\\Big<\\frac{f^{2}}{\\psi(v)},\\big(\\nabla_{v}\\psi(v),\\nabla_{v}\\psi\n(v)\\big)\\Big>_{v}\\,.\n\\]\nLet us compare $\\big(\\nabla_{v}\\psi(v),\\nabla_{v}\\psi(v)\\big)$ and $\\psi\n^{3}(v)$. For $|v|$ large, we have\n\\[\n\\big(\\nabla_{v}\\psi(v),\\nabla_{v}\\psi(v)\\big)\\approx|v|^{4\\ga-6\n\\]\nand\n\\[\n\\psi^{3}(v)\\approx|v|^{6\\ga-6}\\,.\n\\]\nFor $|v|$ small, one can choose $\\varpi$ large enough such that\n\\[\n\\big(\\nabla_{v}\\psi(v),\\nabla_{v}\\psi(v)\\big)\\ll\\psi^{3}(v)\\,.\n\\]\nThis means\n\\[\n\\big<\\Lambda f,\\Lambda f\\big>_{v}\\geq|\\Delta_{v}f|_{L_{v}^{2}}^{2}+\\frac{1\n{2}|\\psi(v)f|_{L_{v}^{2}}^{2}\\gtrsim|\\left\\langle v\\right\\rangle\n^{2\\ga-2}f|_{L_{v}^{2}}^{2}\\,.\n\\]\nHence if $\\ga\\geq3\/2$,\n\\begin{align*}\n|vf|_{L_{v}^{2}}^{2}\\leq C\\big<\\Lambda f,\\Lambda f\\big>_{v} &\n=C\\big_{v}\\\\\n& \\leq C_{1}|Lf|_{L_{v}^{2}}^{2}+C_{2}|f|_{L_{v}^{2}}^{2}\\,.\n\\end{align*}\nHowever, we cannot simply deduce smoothness (analyticity) in $\\eta$ from\nsmoothness (analyticity) in $|\\eta|$. Our goal is to show $\\lambda(z)$ and\n$\\beta(z)$ are in fact even functions in $z$. If so, due to a classical\ntheorem of Whitney \\cite{Whitney}, we have\n\\[\n\\lambda(|\\eta|)=\\mathscr{A}(|\\eta|^{2}),\\qquad\\beta(|\\eta|)=\\mathscr{B}(|\\eta\n|^{2}),\n\\]\nfor some smooth or analytic functions $\\mathscr{A}$ and $\\mathscr{B}$ provided\n$\\lambda(|\\eta|)$ and $\\beta(|\\eta|)$ are smooth or analytic respectively. To\nshow they are even, let us define a map $\\mathcal{R}:(v_{1},v_{2\n,v_{3})\\mapsto(-v_{1},v_{2},v_{3})$, then obviously $\\mathcal{R}\\in O(3)$. We\napply $\\mathcal{R}$ to \\eqref{eigenprob1},\n\\[\n(-iv_{1}(-|\\eta|)+L)(\\mathcal{R}\\circ e(|\\eta|))=\\lambda(|\\eta|)(\\mathcal{R\n\\circ e(|\\eta|)),\n\\]\nwhich is an eigenvalue problem with $|\\eta|\\rightarrow-|\\eta|$. This follows\nthat the eigenpair $\\{\\lambda(|\\eta|),\\mathcal{R}\\circ e(|\\eta|)\\}$ coincides\nwith $\\{(\\lambda(-|\\eta|),e(-|\\eta|))\\}$. Hence\n\\begin{equation}\n\\lambda(|\\eta|)=\\lambda(-|\\eta|),\\qquad\\mathcal{R}\\circ e(|\\eta|)=e(-|\\eta|).\n\\label{eq:L6.3\n\\end{equation}\nIn addition, use $\\mathcal{R}\\circ\\mathrm{P}_{0}e(|\\eta|)=\\mathrm{P\n_{0}\\mathcal{R}\\circ e(|\\eta|)=\\mathrm{P}_{0}e(-|\\eta|)$ and $\\mathcal{R}\\circ\nE_{D}=E_{D}$ to find $\\beta(|\\eta|)=\\beta(-|\\eta|)$, namely $\\beta$ is also an\neven function. We can show\n\\begin{equation}\n\\overline{\\lambda(|\\eta|)}=\\lambda(-|\\eta|),\\qquad\\overline{e(|\\eta\n|)}=e(-|\\eta|), \\label{eq:L6.4\n\\end{equation}\nby taking the complex conjugate of \\eqref{eigenprob1}. This together with\n\\eqref{eq:L6.3} shows $\\lambda(|\\eta|)$ and $\\beta(|\\eta|)$ are real\nfunctions. By \\eqref{eq:L6.2}, we can construct $e(|\\eta|)$ from\n$\\mathrm{P}_{0}e(|\\eta|)$,\n\\[\ne(|\\eta|)=\\mathrm{P}_{0}e(|\\eta|)+\\mathrm{P}_{1}e(|\\eta|)=\\Big(1+i|\\eta\n|\\big[L-i|\\eta|\\mathrm{P}_{1}v_{1}-\\lambda(|\\eta|)\\big]^{-1}\\mathrm{P\n_{1}v_{1}\\Big)\\beta(|\\eta|)E_{D}.\n\\]\nThe eigenfunction $e_{D}(\\eta)$ to the original eigenvalue-problem\n\\eqref{eigenprob} can be recovered by applying $a^{-1}$, namely\n\\[\ne_{D}(\\eta)=a^{-1}\\circ e(|\\eta|)=\\Big(1+\\big[L-\\mathrm{P}_{1}i\\eta\\cdot\nv-\\mathscr{A}(|\\eta|^{2})\\big]^{-1}\\mathrm{P}_{1}i\\eta\\cdot\nv\\Big)\\mathscr{B}(|\\eta|^{2})E_{D}.\n\\]\nTherefore the proof is complete.\n\\end{proof}\n\n\\subsection{The semigroup operator $e^{t\\mathcal{L}}$}\n\nNow, let $h$ be the solution of the equation\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l\n\\pa_{t}h=\\mathcal{L}h,\\quad\\hbox{where}\\quad\\mathcal{L}h=-v\\cdot\\nabla\n_{x}h-\\Lambda h\\,,\\\\[4mm\nh(0,x,v)=h_{0}(x,v)\\,.\n\\end{array}\n\\right.\n\\end{equation}\nIn this subsection, we will study some properties of the semigroup operator\n$e^{t\\mathcal{L}}$.\n\n\\begin{lemma}\n\\label{x-energy}For any $k\\in{\\mathbb{N}}\\cup\\{0\\},$\n\n\\noindent\\textrm{(i)} If $\\gamma\\geq1$, there exists $C>0$ such that\n\\begin{equation}\n\\left\\Vert e^{t\\mathcal{L}}h_{0}\\right\\Vert _{H_{x}^{k}L_{v}^{2}(\\mu)}\\lesssim\ne^{-Ct}\\left\\Vert h_{0}\\right\\Vert _{H_{x}^{k}L_{v}^{2}(\\mu)}\\,.\n\\label{energy1\n\\end{equation}\n\\noindent\\textrm{(ii)} If $0<\\gamma<1$, we have\n\\begin{equation}\n\\left\\Vert e^{t\\mathcal{L}}h_{0}\\right\\Vert _{H_{x}^{k}L_{v}^{2}(\\mu)\n\\lesssim\\left\\Vert h_{0}\\right\\Vert _{H_{x}^{k}L_{v}^{2}(\\mu)}\\,.\n\\label{energy2\n\\end{equation}\n\n\\end{lemma}\n\n\\begin{proof}\nIt suffices to show that there exists $c_{0}>0$ such that for any multi-index\n$\\beta,\n\\begin{equation}\n\\frac{d}{dt}\\left\\Vert \\partial_{x}^{\\beta}h\\right\\Vert _{L^{2}\\left(\n\\mu\\right) }^{2}\\leq-c_{0}\\left\\Vert \\partial_{x}^{\\beta}h\\right\\Vert\n_{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2}. \\label{Energy-evolu\n\\end{equation}\nIn view of Lemma \\ref{prop1}, we have\n\\begin{align*}\n\\frac{1}{2}\\frac{d}{dt}\\left\\Vert \\partial_{x}^{\\beta}h\\right\\Vert\n_{L^{2}\\left( \\mu\\right) }^{2} & =\\int-v\\cdot\\nabla_{x}\\left(\n\\partial_{x}^{\\beta}h\\right) \\partial_{x}^{\\beta}h\\mu dxdv-\\int\\Lambda\\left(\n\\partial_{x}^{\\beta}h\\right) \\partial_{x}^{\\beta}h\\mu dxdv\\\\\n& \\leq\\int\\frac{1}{2}\\left( \\partial_{x}^{\\beta}h\\right) ^{2}v\\cdot\n\\nabla_{x}\\mu dxdv-c_{0}\\left\\Vert \\partial_{x}^{\\beta}h\\right\\Vert\n_{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2}.\n\\end{align*}\n\n\nIf $\\mu\\left( x,v\\right) \\equiv1,$ $\\left( \\ref{Energy-evolu}\\right) $ is\nobvious since $\\int\\frac{1}{2}\\left( \\partial_{x}^{\\beta}h\\right) ^{2\nv\\cdot\\nabla_{x}\\mu dxdv=0.$\n\nIf $\\mu\\left( x,v\\right) =\\exp\\left( \\left\\langle x\\right\\rangle \/D\\right)\n$\\ and $\\gamma\\geq3\/2$, we choose $D$ sufficiently large such that\n$1\/D<\\min\\{c_{0},1\\}\\ $and thus obtain\n\\begin{align*}\n\\left\\vert \\int\\frac{1}{2}\\left( \\partial_{x}^{\\beta}h\\right) ^{2\nv\\cdot\\nabla_{x}\\mu dxdv\\right\\vert & =\\left\\vert \\int\\frac{1}{2}\\left(\n\\partial_{x}^{\\beta}h\\right) ^{2}v\\cdot\\frac{x}{D\\left\\langle x\\right\\rangle\n}\\mu dxdv\\right\\vert \\\\\n& \\leq\\frac{1}{2D}\\int\\left\\langle v\\right\\rangle ^{2\\gamma-2}\\left(\n\\partial_{x}^{\\beta}h\\right) ^{2}\\mu dxdv\\leq\\frac{c_{0}}{2}\\left\\Vert\n\\partial_{x}^{\\beta}h\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2}.\n\\end{align*}\n\n\nIf $\\mu\\left( x,v\\right) =e^{\\alpha c\\left( x,v\\right) }$ and $0<$\n$\\gamma<3\/2,$ since\n\\[\n\\left\\vert \\nabla_{x}c\\left( x,v\\right) \\right\\vert \\leq\\delta C\\left\\langle\nv\\right\\rangle ^{2\\gamma-3},\n\\]\nfor some constant $C>0$ depending only upon $\\gamma,$ we have\n\\[\n\\left\\vert \\int\\frac{1}{2}\\left( \\partial_{x}^{\\beta}h\\right) ^{2\nv\\cdot\\nabla_{x}\\mu dxdv\\right\\vert \\leq\\frac{\\delta C\\alpha}{2\n\\int\\left\\langle v\\right\\rangle ^{2\\gamma-2}\\left( \\partial_{x}^{\\beta\n}h\\right) ^{2}\\mu dxdv\\leq\\frac{c_{0}}{2}\\left\\Vert \\partial_{x}^{\\beta\n}h\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2\n\\]\nby choosing $\\alpha,\\delta$ $>0\\ $small enough with $0<$ $\\delta C\\alpha\n<\\min\\{c_{0},1\\}.$\n\nGrouping the above discussions, we obtain $\\left( \\ref{Energy-evolu}\\right)\n$ and thus deduce that for $\\gamma>0,$ $k\\in{\\mathbb{N\\cup\\{}}0{\\mathbb{\\}}\n$,\n\\[\n\\left\\Vert e^{t\\mathcal{L}}h_{0}\\right\\Vert _{H_{x}^{k}L_{v}^{2}(\\mu)\n\\leq\\left\\Vert h_{0}\\right\\Vert _{H_{x}^{k}L_{v}^{2}(\\mu)}.\n\\]\nMoreover, if $\\gamma\\geq1,$ then $\\left( \\ref{Energy-evolu}\\right) $\nbecomes\n\\[\n\\frac{1}{2}\\frac{d}{dt}\\left\\Vert \\partial_{x}^{\\beta}h\\right\\Vert\n_{L^{2}\\left( \\mu\\right) }^{2}\\leq-\\frac{c_{0}}{2}\\left\\Vert \\partial\n_{x}^{\\beta}h\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2}\\leq\n-\\frac{c_{0}}{2}\\left\\Vert \\partial_{x}^{\\beta}h\\right\\Vert _{L^{2}\\left(\n\\mu\\right) }^{2},\n\\]\nwhich leads to the exponential time decay of all $x$-derivatives of the\nsolution $e^{t\\mathcal{L}}h_{0}$ in the weighted $L^{2}$ norm.\n\\end{proof}\n\nThe following is the regularization estimate of the semigroup operator\n$e^{t\\mathcal{L}}$ in small time.\n\n\\begin{lemma}\n[Regularization estimate]\\label{regularization} For $\\gamma>0$ and $00$ and $\\varpi>0$ sufficiently large such tha\n\\begin{equation}\n\\ \\left\\{\n\\begin{array}\n[c]{l\n\\frac{1}{4}\\left\\vert \\nabla_{v}\\Phi\\right\\vert ^{2}-\\frac{1}{2}\\triangle\n_{v}\\Phi\\leq\\frac{1}{3}\\left\\langle v\\right\\rangle ^{2\\gamma-2\n,\\ \\ \\ \\left\\vert \\nabla_{v}\\left( \\frac{1}{4}\\left\\vert \\nabla_{v\n\\Phi\\right\\vert ^{2}-\\frac{1}{2}\\triangle_{v}\\Phi\\right) \\right\\vert\n\\lesssim\\left\\langle v\\right\\rangle ^{2\\gamma-3}\\ \\ \\text{for }\\left\\vert\nv\\right\\vert >R,\\vspace{3mm}\\\\\n\\left\\vert \\frac{1}{4}\\left\\vert \\nabla_{v}\\Phi\\right\\vert ^{2}-\\frac{1\n{2}\\triangle_{v}\\Phi\\right\\vert ,\\ \\left\\vert \\nabla_{v}\\left( \\frac{1\n{4}\\left\\vert \\nabla_{v}\\Phi\\right\\vert ^{2}-\\frac{1}{2}\\triangle_{v\n\\Phi\\right) \\right\\vert <\\frac{\\varpi}{2}\\ \\ \\text{for }\\left\\vert\nv\\right\\vert \\leq R.\n\\end{array}\n\\right. \\label{*\n\\end{equation}\nNow, define the energy functional\n\\[\n\\mathcal{F}\\left( t,h_{t}\\right) :=A\\left\\Vert h\\right\\Vert _{L^{2}\\left(\n\\mu\\right) }^{2}+at\\left\\Vert \\nabla_{v}h\\right\\Vert _{L^{2}\\left(\n\\mu\\right) }^{2}+2ct^{2}\\left\\langle \\nabla_{x}h,\\nabla_{v}h\\right\\rangle\n_{L^{2}\\left( \\mu\\right) }+bt^{3}\\left\\Vert \\nabla_{x}h\\right\\Vert\n_{L^{2}\\left( \\mu\\right) }^{2},\n\\]\nwith $a,$ $b,$ $c>0$ and $c<\\sqrt{ab}$ (positive definite) and $A>0$\nsufficiently large. We shall show that $d\\mathcal{F}\/dt\\leq0,\\ t\\in\\left(\n0,1\\right) ,$ via choosing suitable positive constants $A,$ $a,$ $b$ and $c.$\n\nIn $\\left( \\ref{Energy-evolu}\\right) ,$ it has been shown that\n\\begin{equation}\n\\frac{d}{dt}\\left\\Vert h\\right\\Vert _{L^{2}(\\mu)}^{2}\\leq-c_{0}\\left\\Vert\nh\\right\\Vert _{L_{\\sigma}^{2}(\\mu)}^{2},\n\\end{equation\n\\begin{equation}\n\\frac{d}{dt}\\left\\Vert \\partial_{x_{i}}h\\right\\Vert _{L^{2}(\\mu)}^{2\n\\leq-c_{0}\\left\\Vert \\partial_{x_{i}}h\\right\\Vert _{L_{\\sigma}^{2}(\\mu)}^{2}.\n\\end{equation}\nNext, we show that\n\\begin{equation}\n\\frac{1}{2}\\frac{d}{dt}\\left\\Vert \\partial_{v_{i}}h\\right\\Vert _{L^{2}\\left(\n\\mu\\right) }^{2}\\leq-\\int\\partial_{x_{i}}h\\partial_{v_{i}}h\\mu dxdv-\\frac\n{c_{0}}{2}\\left\\Vert \\partial_{v_{i}}h\\right\\Vert _{L_{\\sigma}^{2}\\left(\n\\mu\\right) }^{2}+C_{\\varepsilon}\\left\\Vert h\\right\\Vert _{L_{\\sigma\n^{2}\\left( \\mu\\right) }^{2}+\\varepsilon\\left\\Vert \\partial_{v_{i\n}h\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2},\n\\end{equation}\nwhere $\\varepsilon>0$ is arbitrarily small and $C_{\\varepsilon}=O\\left(\n1\/\\varepsilon\\right) .$ Comput\n\\begin{align*}\n\\frac{1}{2}\\frac{d}{dt}\\left\\Vert \\partial_{v_{i}}h\\right\\Vert _{L^{2}\\left(\n\\mu\\right) }^{2} & =-\\int\\partial_{x_{i}}h\\partial_{v_{i}}h\\mu\ndxdv+\\int\\left[ \\frac{v}{2}\\left( \\partial_{v_{i}}h\\right) ^{2}\\right]\n\\cdot\\nabla_{x}\\mu dxdv\\\\\n& \\quad-\\int\\left( \\Lambda\\partial_{v_{i}}h\\right) \\partial_{v_{i}}h\\mu\ndxdv-\\int\\left( \\left[ \\partial_{v_{i}},\\Lambda\\right] h\\right)\n\\partial_{v_{i}}h\\mu dxdv,\n\\end{align*}\nwher\n\\[\n\\left[ \\partial_{v_{i}},\\Lambda\\right] h=\\left[ \\partial_{v_{i}}\\left(\n\\frac{1}{4}\\left\\vert \\nabla_{v}\\Phi\\right\\vert ^{2}-\\frac{1}{2}\\triangle\n_{v}\\Phi\\right) \\right] h+\\varpi\\partial_{v_{i}}\\left( \\chi_{R}\\right) h.\n\\]\nIn the course of the proof of Lemma \\ref{x-energy}, one can see that\n\\begin{equation}\n\\left\\vert \\nabla_{x}\\mu\\right\\vert \\leq\\min\\{c_{0},1\\}\\cdot\\left\\langle\nv\\right\\rangle ^{2\\gamma-3}\\mu\\ \\ \\ \\text{and}\\ \\ \\ \\ \\ \\ \\left\\vert\n\\nabla_{x}\\mu\\right\\vert \\leq\\min\\{c_{0},1\\}\\mu, \\label{x-derivative-mu\n\\end{equation}\nso\n\\[\n\\left\\vert \\int\\left[ \\frac{v}{2}\\left( \\partial_{v_{i}}h\\right)\n^{2}\\right] \\cdot\\nabla_{x}\\mu dxdv\\right\\vert \\leq\\frac{c_{0}}{2\n\\int\\left\\langle v\\right\\rangle ^{2\\gamma-2}\\left( \\partial_{v_{i}}h\\right)\n^{2}\\mu dxdv\\leq\\frac{c_{0}}{2}\\left\\Vert \\partial_{v_{i}}h\\right\\Vert\n_{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2}.\n\\]\nFurthermore, by $\\left( \\ref{*}\\right) $ we obtain\n\\begin{align*}\n\\left\\vert \\int\\left( \\left[ \\partial_{v_{i}},\\Lambda\\right] h\\right)\n\\partial_{v_{i}}h\\mu dxdv\\right\\vert & \\leq C\\int\\left\\langle v\\right\\rangle\n^{2\\gamma-2}\\left\\vert h\\partial_{v_{i}}h\\right\\vert \\mu dxdv+\\frac{\\varpi\n{2}\\int\\chi_{R}\\left\\vert h\\partial_{v_{i}}h\\right\\vert \\mu dxdv+\\frac{\\varpi\n}{R}\\int\\left\\vert \\chi_{R}^{\\prime}\\right\\vert \\left\\vert h\\partial_{v_{i\n}h\\right\\vert \\mu dxdv\\\\\n& \\leq C^{\\prime}\\int\\left\\langle v\\right\\rangle ^{2\\gamma-2}\\left\\vert\nh\\partial_{v_{i}}h\\right\\vert \\mu dxdv\\leq C_{\\varepsilon}\\left\\Vert\nh\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2}+\\varepsilon\\left\\Vert\n\\partial_{v_{i}}h\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2},\n\\end{align*}\nwhere $\\varepsilon>0$ is arbitrary small and $C_{\\varepsilon}=O\\left(\n1\/\\varepsilon\\right) .$ It turns out that\n\\[\n\\frac{1}{2}\\frac{d}{dt}\\left\\Vert \\partial_{v_{i}}h\\right\\Vert _{L^{2}(\\mu\n)}^{2}\\leq-\\int\\partial_{x_{i}}h\\partial_{v_{i}}h\\mu dxdv-\\frac{c_{0}\n{2}\\left\\Vert \\partial_{v_{i}}h\\right\\Vert _{L_{\\sigma}^{2}(\\mu)\n^{2}+C_{\\varepsilon}\\left\\Vert h\\right\\Vert _{L_{\\sigma}^{2}\\left(\n\\mu\\right) }^{2}+\\varepsilon\\left\\Vert \\partial_{v_{i}}h\\right\\Vert\n_{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2}.\n\\]\n\n\nFinally, direct computation gives\n\\begin{align*}\n& \\quad\\frac{d}{dt}\\int\\partial_{x_{i}}h\\partial_{v_{i}}h\\mu dxdv\\\\\n& =-\\int\\left\\vert \\partial_{x_{i}}h\\right\\vert ^{2}\\mu dxdv-2\\int\\nabla\n_{v}\\left( \\partial_{x_{i}}h\\right) \\cdot\\nabla_{v}\\left( \\partial_{v_{i\n}h\\right) \\mu dxdv-2\\int\\left( \\frac{\\left\\vert \\nabla_{v}\\Phi\\right\\vert\n^{2}}{4}-\\frac{\\triangle_{v}\\Phi}{2}+\\varpi\\chi_{R}\\right) \\partial_{x_{i\n}h\\partial_{v_{i}}h\\mu dxdv\\\\\n& \\quad+\\int\\left( v\\cdot\\nabla_{x}\\mu\\right) \\partial_{v_{i}\nh\\partial_{x_{i}}hdxdv+\\frac{1}{2}\\int\\partial_{v_{i}}\\left( \\frac{\\left\\vert\n\\nabla_{v}\\Phi\\right\\vert ^{2}}{4}-\\frac{\\triangle_{v}\\Phi}{2}+\\varpi\\chi\n_{R}\\right) h^{2}\\partial_{x_{i}}\\mu dxdv\\\\\n& \\quad-\\int\\nabla_{v}\\left( \\partial_{x_{i}}h\\right) \\cdot\\nabla\n_{v}\\left( \\mu\\right) \\partial_{v_{i}}hdxdv-\\int\\nabla_{v}\\left(\n\\partial_{v_{i}}h\\right) \\cdot\\nabla_{v}\\left( \\mu\\right) \\partial_{x_{i\n}hdxdv.\n\\end{align*}\nFrom $\\left( \\ref{v-derivtive-c}\\right) $, it follows tha\n\\begin{align*}\n& \\quad\\left\\vert \\int\\nabla_{v}\\left( \\partial_{x_{i}}h\\right) \\cdot\n\\nabla_{v}\\left( \\mu\\right) \\partial_{v_{i}}hdxdv+\\int\\nabla_{v}\\left(\n\\partial_{v_{i}}h\\right) \\cdot\\nabla_{v}\\left( \\mu\\right) \\partial_{x_{i\n}hdxdv\\right\\vert \\\\\n& \\leq\\alpha C\\left( \\gamma\\right) \\sup\\left( 1+\\left\\vert \\chi^{\\prime\n}\\right\\vert \\right) \\int\\left\\langle v\\right\\rangle ^{\\gamma-1}\\left(\n\\left\\vert \\nabla_{v}\\left( \\partial_{x_{i}}h\\right) \\right\\vert \\left\\vert\n\\partial_{v_{i}}h\\right\\vert +\\left\\vert \\nabla_{v}\\left( \\partial_{v_{i\n}h\\right) \\right\\vert \\left\\vert \\partial_{x_{i}}h\\right\\vert \\right) \\mu\ndxdv\\\\\n& \\leq\\int\\left( \\left\\langle v\\right\\rangle ^{\\gamma-1}\\left\\vert\n\\nabla_{v}\\left( \\partial_{x_{i}}h\\right) \\right\\vert \\left\\vert\n\\partial_{v_{i}}h\\right\\vert +\\left\\langle v\\right\\rangle ^{\\gamma\n-1}\\left\\vert \\nabla_{v}\\left( \\partial_{v_{i}}h\\right) \\right\\vert\n\\left\\vert \\partial_{x_{i}}h\\right\\vert \\right) \\mu dxdv,\n\\end{align*}\nfrom $\\alpha C\\left( \\gamma\\right) \\sup\\left( 1+\\left\\vert \\chi^{\\prime\n}\\right\\vert \\right) =\\alpha Q<1.$ Note that this inequality is valid in the\ncases $\\mu\\left( x,v\\right) =1$ and $\\mu\\left( x,v\\right) =\\exp\\left(\n\\left\\langle x\\right\\rangle \/D\\right) $ as well, since $\\nabla_{v}\\mu=0$ in\nboth cases. Therefore\n\\begin{align*}\n& \\quad\\frac{d}{dt}\\int\\partial_{x_{i}}h\\partial_{v_{i}}h\\mu dxdv\\\\\n& \\leq-\\int\\left\\vert \\partial_{x_{i}}h\\right\\vert ^{2}\\mu dxdv-2\\int\n\\nabla_{v}\\left( \\partial_{x_{i}}h\\right) \\cdot\\nabla_{v}\\left(\n\\partial_{v_{i}}h\\right) \\mu dxdv-2\\int\\left( \\frac{\\left\\vert \\nabla\n_{v}\\Phi\\right\\vert ^{2}}{4}-\\frac{\\triangle_{v}\\Phi}{2}+\\varpi\\chi\n_{R}\\right) \\partial_{x_{i}}h\\partial_{v_{i}}h\\mu dxdv\\\\\n& \\quad+\\int\\left( v\\cdot\\nabla_{x}\\mu\\right) \\partial_{v_{i}\nh\\partial_{x_{i}}hdxdv+\\frac{1}{2}\\int\\partial_{v_{i}}\\left( \\frac{\\left\\vert\n\\nabla_{v}\\Phi\\right\\vert ^{2}}{4}-\\frac{\\triangle_{v}\\Phi}{2}+\\varpi\\chi\n_{R}\\right) h^{2}\\partial_{x_{i}}\\mu dxdv\\\\\n& \\quad+\\int\\left( \\left\\langle v\\right\\rangle ^{\\gamma-1}\\left\\vert\n\\nabla_{v}\\left( \\partial_{x_{i}}h\\right) \\right\\vert \\left\\vert\n\\partial_{v_{i}}h\\right\\vert +\\left\\langle v\\right\\rangle ^{\\gamma\n-1}\\left\\vert \\nabla_{v}\\left( \\partial_{v_{i}}h\\right) \\right\\vert\n\\left\\vert \\partial_{x_{i}}h\\right\\vert \\right) \\mu dxdv.\n\\end{align*}\n\n\nCollecting terms give\n\\begin{align*}\n& \\quad\\frac{d}{dt}\\mathcal{F}\\left( t,h_{t}\\right) \\\\\n& \\leq-c_{0}A\\left\\Vert h\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right)\n}^{2}+a\\left\\Vert \\nabla_{v}h\\right\\Vert _{L^{2}\\left( \\mu\\right) \n^{2}+4ct\\left\\langle \\nabla_{x}h,\\nabla_{v}h\\right\\rangle _{L^{2}\\left(\n\\mu\\right) }+3bt^{2}\\left\\Vert \\nabla_{x}h\\right\\Vert _{L^{2}\\left(\n\\mu\\right) }^{2}\\\\\n& \\quad+2at\\left[ -\\sum_{i=1}^{3}\\int\\partial_{x_{i}}h\\partial_{v_{i}}h\\mu\ndxdv-\\frac{c_{0}}{2}\\left\\Vert \\nabla_{v}h\\right\\Vert _{L_{\\sigma}^{2}\\left(\n\\mu\\right) }^{2}+3C_{\\varepsilon}\\left\\Vert h\\right\\Vert _{L_{\\sigma\n^{2}\\left( \\mu\\right) }^{2}+\\varepsilon\\left\\Vert \\nabla_{v}h\\right\\Vert\n_{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2}\\right] \\\\\n& \\quad+2ct^{2}\\left[ -\\int\\left\\vert \\nabla_{x}h\\right\\vert ^{2}\\mu\ndxdv-2\\sum_{i=1}^{3}\\int\\nabla_{v}\\left( \\partial_{x_{i}}h\\right)\n\\cdot\\nabla_{v}\\left( \\partial_{v_{i}}h\\right) \\mu dxdv\\right. \\\\\n& \\quad-2\\sum_{i=1}^{3}\\int\\left( \\frac{\\left\\vert \\nabla_{v}\\Phi\\right\\vert\n^{2}}{4}-\\frac{\\triangle_{v}\\Phi}{2}+\\varpi\\chi_{R}\\right) \\partial_{x_{i\n}h\\partial_{v_{i}}h\\mu dxdv-\\sum_{i=1}^{3}\\int\\left( v\\cdot\\nabla_{x\n\\mu\\right) \\partial_{v_{i}}h\\partial_{x_{i}}hdxdv\\\\\n& \\quad+\\frac{1}{2}\\sum_{i=1}^{3}\\int\\partial_{v_{i}}\\left( \\frac{\\left\\vert\n\\nabla_{v}\\Phi\\right\\vert ^{2}}{4}-\\frac{\\triangle_{v}\\Phi}{2}+\\varpi\\chi\n_{R}\\right) h^{2}\\partial_{x_{i}}\\mu dxdv\\\\\n& \\quad\\left. +\\sum_{i=1}^{3}\\int\\left( \\left\\langle v\\right\\rangle\n^{\\gamma-1}\\left\\vert \\nabla_{v}\\left( \\partial_{x_{i}}h\\right) \\right\\vert\n\\left\\vert \\partial_{v_{i}}h\\right\\vert +\\left\\langle v\\right\\rangle\n^{\\gamma-1}\\left\\vert \\nabla_{v}\\left( \\partial_{v_{i}}h\\right) \\right\\vert\n\\left\\vert \\partial_{x_{i}}h\\right\\vert \\right) \\mu dxdv\\right] -bc_{0\nt^{3}\\left\\Vert \\nabla_{x}h\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right)\n}^{2}.\n\\end{align*}\nBy $\\left( \\ref{*}\\right) $ and $\\left( \\ref{x-derivative-mu}\\right) $,\n\\begin{align*}\n& \\quad\\left\\vert -2\\sum_{i=1}^{3}\\int\\left( \\frac{\\left\\vert \\nabla_{v\n\\Phi\\right\\vert ^{2}}{4}-\\frac{\\triangle_{v}\\Phi}{2}+\\varpi\\chi_{R}\\right)\n\\partial_{x_{i}}h\\partial_{v_{i}}h\\mu dxdv+\\sum_{i=1}^{3}\\int\\left(\nv\\cdot\\nabla_{x}\\mu\\right) \\partial_{v_{i}}h\\partial_{x_{i}}hdxdv\\right\\vert\n\\\\\n& \\leq\\sum_{i=1}^{3}2\\int\\left\\langle v\\right\\rangle ^{2\\gamma-2}\\left\\vert\n\\partial_{x_{i}}h\\partial_{v_{i}}h\\right\\vert \\mu dxdv+3\\varpi\\int\\chi\n_{R}\\left\\vert \\nabla_{x}h\\cdot\\nabla_{v}h\\right\\vert \\mu dxdv\\\\\n& \\leq\\sum_{i=1}^{3}\\left( \\frac{bc_{0}}{8c}t\\left\\Vert \\left\\langle\nv\\right\\rangle ^{\\gamma-1}\\left( \\partial_{x_{i}}h\\right) \\right\\Vert\n_{L^{2}\\left( \\mu\\right) }^{2}+\\frac{8c}{bc_{0}t}\\left\\Vert \\left\\langle\nv\\right\\rangle ^{\\gamma-1}\\left( \\partial_{v_{i}}h\\right) \\right\\Vert\n_{L^{2}\\left( \\mu\\right) }^{2}\\right) +3\\varpi\\int\\chi_{R}\\left\\vert\n\\nabla_{x}h\\cdot\\nabla_{v}h\\right\\vert \\mu dxdv\\\\\n& =\\frac{bc_{0}}{8c}t\\left\\Vert \\nabla_{x}h\\right\\Vert _{L_{\\sigma\n^{2}\\left( \\mu\\right) }^{2}+\\frac{8c}{bc_{0}t}\\left\\Vert \\nabla\n_{v}h\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2}+3\\varpi\\int\\chi\n_{R}\\left\\vert \\nabla_{x}h\\cdot\\nabla_{v}h\\right\\vert \\mu dxdv.\n\\end{align*}\nBy the Cauchy-Schwartz inequality,\n\\begin{align*}\n\\left\\vert 2\\sum_{i=1}^{3}\\int\\nabla_{v}\\left( \\partial_{x_{i}}h\\right)\n\\cdot\\nabla_{v}\\left( \\partial_{v_{i}}h\\right) \\mu dxdv\\right\\vert &\n\\leq\\sum_{i=1}^{3}\\left( \\frac{bc_{0}}{8c}t\\left\\Vert \\nabla_{v}\\left(\n\\partial_{x_{i}}h\\right) \\right\\Vert _{L^{2}\\left( \\mu\\right) }^{2\n+\\frac{8c}{bc_{0}t}\\left\\Vert \\nabla_{v}\\left( \\partial_{v_{i}}h\\right)\n\\right\\Vert _{L^{2}\\left( \\mu\\right) }^{2}\\right) \\\\\n& \\leq\\frac{bc_{0}}{8c}t\\left\\Vert \\nabla_{x}h\\right\\Vert _{L_{\\sigma\n^{2}\\left( \\mu\\right) }^{2}+\\frac{8c}{bc_{0}t}\\left\\Vert \\nabla\n_{v}h\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2},\n\\end{align*\n\\begin{align*}\n& \\quad\\left\\vert \\left( 4ct-2at\\right) \\int\\nabla_{x}h\\cdot\\nabla_{v}h\\mu\ndxdv\\right\\vert +\\int6c\\varpi t^{2}\\chi_{R}\\left\\vert \\nabla_{x}h\\cdot\n\\nabla_{v}h\\right\\vert \\mu dxdv\\\\\n& \\leq\\left( 4c+2a+6c\\varpi t\\right) t\\left[ \\varepsilon t\\left\\Vert\n\\nabla_{x}h\\right\\Vert _{L^{2}\\left( \\mu\\right) }^{2}+\\frac{C_{\\varepsilon\n}{t}\\left\\Vert \\nabla_{v}h\\right\\Vert _{L^{2}\\left( \\mu\\right) }^{2}\\right]\n,\\ \\ \\ C_{\\varepsilon}=O\\left( \\frac{1}{\\varepsilon}\\right) ,\n\\end{align*}\nan\n\\begin{align*}\n& \\quad\\left\\vert \\sum_{i=1}^{3}\\int\\left( \\left\\langle v\\right\\rangle\n^{\\gamma-1}\\left\\vert \\nabla_{v}\\left( \\partial_{x_{i}}h\\right) \\right\\vert\n\\left\\vert \\partial_{v_{i}}h\\right\\vert +\\left\\langle v\\right\\rangle\n^{\\gamma-1}\\left\\vert \\nabla_{v}\\left( \\partial_{v_{i}}h\\right) \\right\\vert\n\\left\\vert \\partial_{x_{i}}h\\right\\vert \\right) \\mu dxdv\\right\\vert \\\\\n& \\leq\\sum_{i=1}^{3}\\left( \\frac{2c}{bc_{0}t}\\int\\left\\langle v\\right\\rangle\n^{2\\gamma-2}\\left\\vert \\partial_{v_{i}}h\\right\\vert ^{2}\\mu dxdv+\\frac{bc_{0\n}{8c}t\\int\\left\\vert \\nabla_{v}\\left( \\partial_{x_{i}}h\\right) \\right\\vert\n^{2}\\mu dxdv\\right) \\\\\n& \\quad\\ +\\sum_{i=1}^{3}\\left( \\frac{bc_{0}}{8c}t\\int\\left\\langle\nv\\right\\rangle ^{2\\gamma-2}\\left\\vert \\partial_{x_{i}}h\\right\\vert ^{2}\\mu\ndxdv+\\frac{2c}{bc_{0}t}\\int\\left\\vert \\nabla_{v}\\left( \\partial_{v_{i\n}h\\right) \\right\\vert ^{2}\\mu dxdv\\right) \\\\\n& \\leq\\frac{2c}{bc_{0}t}\\left\\Vert \\nabla_{v}h\\right\\Vert _{L_{\\sigma\n^{2}\\left( \\mu\\right) }^{2}+\\frac{bc_{0}}{8c}t\\left\\Vert \\nabla\n_{x}h\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2}.\n\\end{align*}\nIn view of $\\left( \\ref{*}\\right) $ and $\\left( \\ref{x-derivative-mu\n\\right) $,\n\\begin{align*}\n& \\quad\\left\\vert \\frac{1}{2}\\sum_{i=1}^{3}\\int\\partial_{v_{i}}\\left(\n\\frac{\\left\\vert \\nabla_{v}\\Phi\\right\\vert ^{2}}{4}-\\frac{\\triangle_{v}\\Phi\n}{2}+\\varpi\\chi_{R}\\right) h^{2}\\partial_{x_{i}}\\mu dxdv\\right\\vert \\\\\n& \\leq\\frac{C}{2}\\int\\left\\langle v\\right\\rangle ^{2\\gamma-3}h^{2}\\mu\ndxdv+\\frac{\\varpi}{4}\\int\\left\\langle v\\right\\rangle ^{2\\gamma-3}\\chi_{R\nh^{2}\\mu dxdv+\\frac{\\varpi}{2R}\\int\\left\\langle v\\right\\rangle ^{2\\gamma\n-3}\\left\\vert \\chi_{R}^{\\prime}\\right\\vert h^{2}\\mu dxdv\\\\\n& \\leq M^{\\prime}\\int\\left\\langle v\\right\\rangle ^{2\\gamma-2}h^{2}\\mu\ndxdv\\leq M^{\\prime}\\left\\Vert h\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right)\n}^{2},\n\\end{align*}\nwhere $M^{\\prime}>0$ is dependent only upon $R$ and $\\varpi.$ Gathering the\nabove estimates, we therefore deduc\n\\begin{align*}\n\\frac{d}{dt}\\mathcal{F}\\left( t,h_{t}\\right) & \\leq\\left\\Vert h\\right\\Vert\n_{L_{\\sigma}^{2}\\left( \\mu\\right) }^{2}\\left[ -c_{0}A+a+6atC_{\\varepsilon\n}+\\left( 4c+2a+6c\\varpi t\\right) C_{\\varepsilon}+2cM^{\\prime}t^{2}\\right] \\\\\n& \\quad+\\left\\Vert \\nabla_{x}h\\right\\Vert _{L^{2}\\left( \\mu\\right) \n^{2}\\left( -2c+3b+\\left( 4c+2a+6c\\varpi t\\right) \\varepsilon\\right)\nt^{2}\\\\\n& \\quad+\\left\\Vert \\nabla_{x}h\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right)\n}^{2}\\left( -\\frac{bc_{0}}{4}\\right) t^{3}\\\\\n& \\quad+\\left\\Vert \\nabla_{v}h\\right\\Vert _{L_{\\sigma}^{2}\\left( \\mu\\right)\n}^{2}\\left( -ac_{0}+2a\\varepsilon+\\frac{36c^{2}}{bc_{0}}\\right) t.\n\\end{align*}\nSet $a=\\varepsilon,$ $4b=c=\\varepsilon^{3\/2}.\\ $After choosing $A>0$\nsufficiently large and $\\varepsilon>0$ sufficiently small, we obtai\n\\[\n\\frac{d}{dt}\\mathcal{F}\\left( t,h_{t}\\right) \\leq0,\\ \\ \\ t\\in\\left(\n0,1\\right) ,\n\\]\nwhich implies that\n\\[\n\\mathcal{F}\\left( t,h_{t}\\right) \\leq\\mathcal{F}\\left( 0,h_{0}\\right)\n=A\\left\\Vert h_{0}\\right\\Vert _{L^{2}}^{2},\\ \\ \\ t\\in\\left[ 0,1\\right] .\n\\]\nThis completes the proof of the lemma.\n\\end{proof}\n\nBefore the end of this section, we introduce the wave-remainder decomposition,\nwhich is the key decomposition in our paper. The strategy is to design a\nPicard-type iteration, treating $Kf$ as a source term. The zero order\napproximation of the Fokker-Planck equation (\\ref{in.1.c}) is\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l\n\\pa_{t}h^{(0)}+v\\cdot\\nabla_{x}h^{(0)}+\\Lambda h^{(0)}=0\\,,\\\\[4mm\nh^{(0)}(0,x,v)=f_{0}(x,v)\\,.\n\\end{array}\n\\right. \\label{bot.3.b\n\\end{equation}\nThus the difference $f-h^{(0)}$ satisfies\n\\[\n\\left\\{\n\\begin{array}\n[c]{l\n\\pa_{t}(f-h^{(0)})+v\\cdot\\nabla_{x}(f-h^{(0)})+\\Lambda(f-h^{(0)\n)=K(f-h^{(0)})+Kh^{(0)}\\,,\\\\[4mm\n(f-h^{(0)})(0,x,v)=0\\,.\n\\end{array}\n\\right.\n\\]\nTherefore the first order approximation $h^{(1)}$ can be defined by\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l\n\\pa_{t}h^{(1)}+v\\cdot\\nabla_{x}h^{(1)}+\\Lambda h^{(1)}=Kh^{(0)}\\,,\\\\[4mm\nh^{(1)}(0,x,v)=0\\,.\n\\end{array}\n\\right. \\label{bot.3.c\n\\end{equation}\nIn general, we can define the $j^{\\mathrm{th}}$ order approximation $h^{(j)}$,\n$j\\geq1$, as\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l\n\\pa_{t}h^{(j)}+v\\cdot\\nabla_{x}h^{(j)}+\\Lambda h^{(j)}=Kh^{(j-1)}\\,,\\\\[4mm\nh^{(j)}(0,x,v)=0\\,.\n\\end{array}\n\\right. \\label{bot.3.d\n\\end{equation}\n\n\nThe wave part and the remainder part can be defined as follows:\n\\[\nW^{(3)}=\\sum_{j=0}^{3}h^{(j)}\\,,\\quad\\mathcal{R}^{(3)}=f-W^{(3)}\\,,\n\\]\n$\\mathcal{R}^{(3)}$ solving the equation:\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l\n\\pa_{t}\\mathcal{R}^{(3)}+v\\cdot\\nabla_{x}\\mathcal{R}^{(3)}=L\\mathcal{R\n^{(3)}+Kh^{(3)}\\,,\\\\[4mm\n\\mathcal{R}^{(3)}(0,x,v)=0\\,.\n\\end{array}\n\\right.\n\\end{equation}\nThe next two lemmas are the fundamental properties of $h^{(j)},0\\leq j\\leq3$.\n\n\\begin{lemma}\n[$L^{2}$ estimate of $h^{(j)}$, $0\\leq j\\leq3$]\\label{initial-sing} For all\n$0\\leq j\\leq3$ and $t>0$,\n\n\\noindent\\textrm{(i)} If $\\gamma\\geq1$, there exists $C>0$ such that\n\\[\n\\Vert h^{(j)}\\Vert_{L^{2}(\\mu)}\\lesssim t^{j}e^{-Ct}\\Vert f_{0}\\Vert\n_{L^{2}(\\mu)}\\,,\n\\]\n\\noindent\\textrm{(ii)\\ }If $0<\\gamma<1$, we have\n\\[\n\\Vert h^{(j)}\\Vert_{L^{2}(\\mu)}\\lesssim t^{j}\\Vert f_{0}\\Vert_{L^{2}(\\mu)}\\,.\n\\]\n\n\\end{lemma}\n\nThis lemma is immediate from Lemma \\ref{x-energy} and hence we omit the details.\n\n\\begin{lemma}\n[$x$-derivative estimate of $h^{(j)}$, $0\\leq j\\leq3$]\\label{second-der} Let\n$\\gamma>0$ and $k=1,2$. Then\n\n\\noindent\\textrm{(i)} For $01$, we have that if $\\gamma\\geq1$,\n\\[\n\\Vert\\nabla_{x}^{k}h^{(j)}\\Vert_{L^{2}(\\mu)}\\lesssim t^{j}e^{-Ct}\\Vert\nf_{0}\\Vert_{L^{2}(\\mu)}\\,,\n\\]\nand if $0<\\gamma<1$,\n\\[\n\\Vert\\nabla_{x}^{k}h^{(j)}\\Vert_{L^{2}(\\mu)}\\lesssim t^{j}\\Vert f_{0\n\\Vert_{L^{2}(\\mu)}\\,.\n\\]\n\n\\end{lemma}\n\n\\begin{proof}\nWe divide our proof into several steps: \\newline Step 1: First $x$-derivative\nof $h^{(j)}$, $0\\leq j\\leq3$ in small time. We want to show that for\n$01$,\n\\[\n\\Vert\\nabla_{x}h^{(j)}\\Vert_{L^{2}(\\mu)}\\lesssim t^{j}e^{-Ct}\\Vert f_{0\n\\Vert_{L^{2}(\\mu)}\\,,\\ \\ 0\\leq j\\leq3.\n\\]\nIn view of Lemma \\ref{x-energy}, we have\n\\begin{equation}\n\\left\\Vert \\nabla_{x}h^{(0)}\\right\\Vert _{L^{2}(\\mu)}(t)\\leq e^{-C(t-1)\n\\left\\Vert \\nabla_{x}h^{(0)}\\right\\Vert _{L^{2}(\\mu)}(1)\\lesssim e^{-Ct}\\Vert\nf_{0}\\Vert_{L^{2}(\\mu)}\\,,\\ \\ t>1. \\label{h0\n\\end{equation}\nFor $h^{(1)}$, we have\n\\[\nh^{(1)}(t,x,v)=e^{(t-1)\\mathcal{L}}h^{(1)}(1,x,v)+\\int_{1}^{t\ne^{(t-s)\\mathcal{L}}\\left[ Kh^{(0)}(s,x,v)\\right] ds\\,,\\ \\ t>1.\n\\]\nUsing Lemma \\ref{x-energy} and (\\ref{h0}) gives\n\\begin{align*}\n\\left\\Vert \\nabla_{x}h^{(1)}\\right\\Vert _{L^{2}(\\mu)}(t) & \\leq\ne^{-C(t-1)}\\left\\Vert \\nabla_{x}h^{(1)}\\right\\Vert _{L^{2}(\\mu)}\\left(\n1\\right) +\\int_{1}^{t}e^{-C(t-s)}\\left\\Vert \\nabla_{x}h^{(0)}\\right\\Vert\n_{L^{2}(\\mu)}(s)ds\\\\\n& \\lesssim te^{-Ct}\\Vert f_{0}\\Vert_{L^{2}(\\mu)}\\,,\\ \\ \\ \\ \\ t>1,\n\\end{align*}\nand similarly for $\\nabla_{x}h^{(2)}$ and $\\nabla_{x}h^{(3)}.$\\newline Step 4:\nSecond $x$-derivatives of $h^{(j)}$, $0\\leq j\\leq3$ in large time for\n$\\gamma\\geq1$. We demonstrate that for $t>1$,\n\\[\n\\Vert\\nabla_{x}^{2}h^{(j)}\\Vert_{L^{2}(\\mu)}\\lesssim t^{j}e^{-Ct}\\Vert\nf_{0}\\Vert_{L^{2}(\\mu)}\\,,\n\\]\nwhose proof is similar to Step 3.\n\\end{proof}\n\n\n\n\\section{In the time-like region}\n\n\\label{large-time}\n\nIn this section, we will see the large time behavior of solutions to equation\n(\\ref{in.1.c}). In the sequel, we separate our discussion for the case\n$\\gamma\\geq1$ and the case $0<\\gamma<1$.\n\n\\subsection{The case $\\gamma\\geq1$}\n\nAccording to Lemma \\ref{second-der}, together with the Sobolev inequality\n\\cite[Theorem 5.8]{[Adams]} \n\\[\n\\left\\Vert f\\right\\Vert _{L_{x}^{\\infty}L_{v}^{2}}\\lesssim\\left\\Vert\n\\nabla_{x}^{2}f\\right\\Vert _{L^{2}}^{3\/4}\\left\\Vert f\\right\\Vert _{L^{2\n}^{1\/4},\n\\]\nwe immediate obtain the behavior of the wave part as follows.\n\n\\begin{proposition}\n\\label{pointwise-wave part >=1}Assume that $\\gamma\\geq1$. Then for $0\\leq\nj\\leq3$ and $t>0$, there exists $C>0$ such that\n\\[\n|h^{(j)}|_{L_{v}^{2}}\\lesssim e^{-Ct}t^{j-\\frac{9}{4}}\\Vert f_{0}\\Vert_{L^{2\n}\\,.\n\\]\n\n\\end{proposition}\n\nBased on the wave-remainder decomposition, it remains to study the large time\nbehavior of the remainder part. By the Fourier transform with respect to the\n$x$ variable, the solution of the Fokker-Planck equation (\\ref{in.1.c}) can be\nrepresented as\n\\begin{equation}\n\\displaystyle f(t,x,v)=\\int_{{\\mathbb{R}}^{3}}e^{i\\eta\\cdot x+(-iv\\cdot\n\\eta+L)t}\\widehat{f}_{0}(\\eta,v)d\\eta\\,.\n\\end{equation}\nWe can decompose the solution $f$ into the long wave part $f_{L}$ and the\nshort wave part $f_{S}$ given respectively by\n\\begin{equation\n\\begin{array}\n[c]{l\n\\label{bot.2.e}\\displaystyle f_{L}=\\int_{|\\eta|<\\de}e^{i\\eta\\cdot\nx+(-iv\\cdot\\eta+L)t}\\widehat{f}_{0}(\\eta,v)d\\eta\\,,\\\\\n\\\\\n\\displaystyle f_{S}=\\int_{|\\eta|>\\de}e^{i\\eta\\cdot x+(-iv\\cdot\\eta\n+L)t}\\widehat{f}_{0}(\\eta,v)d\\eta\\,.\n\\end{array}\n\\end{equation}\nThe following short wave analysis relies on spectral analysis (Lemma\n\\ref{pr12}).\n\n\\begin{proposition}\n[Short wave $f_{S}$]Assume that $\\gamma\\geq1$\\ and $f_{0}\\in L^{2}$. Then\n\\begin{equation}\n\\Vert f_{S}\\Vert_{L^{2}}\\lesssim e^{-a(\\tau)t}\\Vert f_{0}\\Vert_{L^{2}}\\,.\n\\label{bot.2.f\n\\end{equation}\n\n\\end{proposition}\n\nIn order to study the long wave part $f_{L}$ for $\\gamma\\geq1$, we need to\nfurther decompose the long wave part as the fluid part and the nonfluid part,\ni.e., $f_{L}=f_{L;0}+f_{L;\\perp}$, where\n\\begin{equation\n\\begin{array}\n[c]{l\n\\label{bot.2.g}\\displaystyle f_{L;0}=\\int_{|\\eta|<\\de}e^{\\lambda(\\eta\n)t}e^{i\\eta\\cdot x}\\big_{v}e_{D}(\\eta)d\\eta\\,,\\\\\n\\\\\n\\displaystyle f_{L;\\perp}=\\int_{|\\eta|<\\de}e^{i\\eta\\cdot x}e^{(-iv\\cdot\n\\eta+L)t}\\Pi_{\\eta}^{D\\perp}\\hat{f}_{0}d\\eta\\,.\n\\end{array}\n\\end{equation}\n\n\nUsing Lemma \\ref{pr12}, we obtain the exponential decay of the nonfluid long\nwave part.\n\n\\begin{proposition}\n[Non fluid long wave $f_{L;\\perp}$]Assume that $\\gamma\\geq1$ and $f_{0}\\in\nL^{2}$. Then for $s>0$,\n\\begin{equation}\n\\Vert f_{L;\\perp}\\Vert_{H_{x}^{s}L_{v}^{2}}\\lesssim e^{-a(\\tau)t}\\Vert\nf_{0}\\Vert_{L^{2}}\\,. \\label{bot.2.h\n\\end{equation}\n\n\\end{proposition}\n\nFor the fluid part, we have the following structure:\n\n\\begin{proposition}\n[Fluid long wave $f_{L;0}$]\\label{fluid} For $\\ga\\geq3\/2$ and any given $M>1,$\nthere exists $C>0$ such that for $|x|\\leq Mt$,\n\\begin{equation}\n\\left\\vert f_{L;0}(t,x,v)\\right\\vert _{L_{v}^{2}}\\leq C\\left[ (1+t)^{-3\/2\ne^{-\\frac{|x|^{2}}{C(t+1)}}+e^{-t\/C}\\right] \\Vert f_{0}\\Vert_{L_{x}^{1\nL_{v}^{2}}\\,.\n\\end{equation}\nOn the other hand, for $1\\leq\\ga<3\/2$ and any given positive integer $N$,\nthere exists a positive constant $C$ depending on $N$ such that\n\\begin{equation}\n\\left\\vert f_{L;0}(t,x,v)\\right\\vert _{L_{v}^{2}}\\leq C\\left[ (1+t)^{-3\/2\n\\Big(1+\\frac{|x|^{2}}{1+t}\\Big)^{-N}+e^{-t\/C}\\right] \\Vert f_{0}\\Vert\n_{L_{x}^{1}L_{v}^{2}}\\,.\n\\end{equation}\n\n\\end{proposition}\n\n\\begin{proof}\nBefore the proof of this proposition, we need the following two lemmas:\n\n\\begin{lemma}\n[{Lemma 7.11, \\cite{[LiuYu1]}}]\\label{heatkernel_ana} Suppose that\n$g(t,\\eta,v)$ is analytic in $\\eta$ for $|\\eta|<\\delta\\ll1$ and satisfies\n\\[\n|g(t,\\eta,v)|_{L_{v}^{2}}\\lesssim e^{-A|\\eta|^{2}t+O(|\\eta|^{4})t}\\,,\n\\]\nfor some constant $A>0$. Then in the region of $|x|<(\\mathfrak{M}+1)t$, where\n$\\mathfrak{M}$ is any given positive constant, there exists a constant $C>0$\nsuch that the following inequality holds:\n\\[\n\\displaystyle\\left\\vert \\int_{|\\eta|<\\delta}e^{ix\\cdot\\eta}g(t,\\eta\n,v)d\\eta\\right\\vert _{L_{v}^{2}}\\leq C\\left[ (1+t)^{-\\frac{3}{2}\ne^{-\\frac{|x|^{2}}{Ct}}+e^{-t\/C}\\right] .\n\\]\n\n\\end{lemma}\n\n\\begin{lemma}\n[{Lemma 2.2, \\cite{[LiuWang]}}]\\label{heatkernel_smo} Let $x,\\eta\n,v\\in\\mathbb{R}^{3}$. Suppose $g(t,\\eta,v)$ is smooth and has compact support\nin the variable $\\eta$, and there exists a constant $b>0$ such that\n$g(t,\\eta,v)$ satisfies\n\\[\n\\left\\vert D_{\\eta}^{\\beta}(g(t,\\eta,v))\\right\\vert _{L_{v}^{2}}\\leq C_{\\beta\n}(1+t)^{|\\beta|\/2}e^{-b|\\eta|^{2}t},\n\\]\nfor any multi-indexes $\\beta$ with $|\\beta|\\leq2N$. Then there exists positive\nconstants $C_{N}$ such that\n\\[\n\\left\\vert \\int_{{\\mathbb{R}}^{3}}e^{ix\\cdot\\eta}g(t,\\eta,v)d\\eta\\right\\vert\n_{L_{v}^{2}}\\leq C_{N}\\left[ (1+t)^{-3\/2}B_{N}(|x|,t)\\right] ,\n\\]\nwhere $N$ is any fixed integer and\n\\[\nB_{N}(|x|,t)=\\left( 1+\\frac{|x|^{2}}{1+t}\\right) ^{-N}.\n\\]\n\n\\end{lemma}\n\nWe now return to the proof of this proposition. Notice that\n\\[\nf_{L;0}(t,x,v)=\\int_{|\\eta|<\\de}e^{i\\eta\\cdot x}e^{\\lambda(\\eta)t\n\\big_{v}e_{D}(\\eta)d\\eta\\,.\n\\]\nLet\n\\[\ng(t,\\eta,v)=e^{\\lambda(\\eta)t}\\big_{v}e_{D\n(\\eta)\\cdot\\mathbf{1}_{\\left\\{ \\left\\vert \\eta\\right\\vert <\\delta\\right\\}\n},\n\\]\nwhere $\\mathbf{1}_{D}$ is the characteristic function of the domain $D$. When\n$\\gamma\\geq3\/2$, the eigenvalue $\\lambda(\\eta)$ and eigenvector $e_{D}(\\eta)$\nare analytic in $\\eta$. Owing to the asymptotic expansion of $\\lambda(\\eta)$\nin \\eqref{pre.ab.h},\\ we have\n\\[\n|g(t,\\eta,v)|_{L_{v}^{2}}\\leq e^{-a_{\\ga}|\\eta|^{2}t+O(|\\eta|^{4})t}\\left\\Vert\nf_{0}\\right\\Vert _{L_{x}^{1}L_{v}^{2}}.\n\\]\nFrom Lemma \\ref{heatkernel_ana} it follows\n\\[\n\\left\\vert f_{L;0}(t,x,v)\\right\\vert _{L_{v}^{2}}\\lesssim\\left[\n(1+t)^{-3\/2}e^{-\\frac{|x|^{2}}{C(t+1)}}+e^{-t\/C}\\right] \\left\\Vert\nf_{0}\\right\\Vert _{L_{x}^{1}L_{v}^{2}}\\,.\n\\]\nAs for $1\\leq\\gamma<3\/2$, the eigenvalue and eigenvector are only smooth in\n$\\eta$. In this case, one can see that\n\\[\n\\left\\vert D_{\\eta}^{\\beta}g(t,\\eta,v)\\right\\vert _{L_{v}^{2}}\\lesssim\\left(\n1+t^{\\left\\vert \\beta\\right\\vert \/2}\\right) \\left( 1+\\left\\vert\n\\eta\\right\\vert ^{2}t\\right) ^{\\left\\vert \\beta\\right\\vert \/2}e^{-a_{\\gamma\n}\\left\\vert \\eta\\right\\vert ^{2}t\/2}\\left\\Vert f_{0}\\right\\Vert _{L_{x\n^{1}L_{v}^{2}},\n\\]\nsince $f_{0}$ has compact support in the $x$ variable. Note that the\npolynomial growth $\\left( 1+\\left\\vert \\eta\\right\\vert ^{2}t\\right)\n^{\\left\\vert \\beta\\right\\vert \/2}$ can be absorbed by the exponential decay,\nhence we can conclude tha\n\\[\n\\left\\vert f_{L;0}(t,x,v)\\right\\vert _{L_{v}^{2}}\\lesssim\\left[\n(1+t)^{-3\/2}\\Big(1+\\frac{|x|^{2}}{1+t}\\Big)^{-N}\\right] \\Vert f_{0\n\\Vert_{L_{x}^{1}L_{v}^{2}}\\,,\n\\]\nin accordance with Lemma \\ref{heatkernel_smo}.\n\\end{proof}\n\nWe define the fluid part as $f_{F}=f_{L,0}$ and the nonfluid part as $f_{\\ast\n}=f-f_{F}=f_{L;\\perp}+f_{S}$. By the fluid-nonfluid decomposition and the\nwave-remainder decomposition, we have\n\\[\nf=f_{F}+f_{\\ast}=W^{(3)}+\\mathcal{R}^{(3)}\\,.\n\\]\nWe can define the tail part as $f_{R}=\\mathcal{R}^{(3)}-f_{F}$ and so $f$ can\nbe written as $f=W^{(3)}+f_{F}+f_{R}.$\n\nIt follows from Proposition \\ref{pointwise-wave part >=1} and \\ref{fluid} that\nthe estimates of wave part $W^{(3)}$ and the fluid part $f_{F}$ inside the\ntime-like region is completed. Hence, it remains to study the tail part\n$f_{R}$. From Lemma \\ref{second-der} and the fact that {$\\mathbb{G}^{t},K$ are\nbounded operators on $L^{2},$ }$\\mathcal{R}^{(3)}$ has the following estimate\n\\begin{equation}\n\\Vert\\mathcal{R}^{(3)}(t)\\Vert_{H_{x}^{2}L_{v}^{2}}\\leq\\int_{0}^{t}\\left\\Vert\nh^{(3)}(s)\\right\\Vert _{H_{x}^{2}L_{v}^{2}}ds\\lesssim\\Vert f_{0}\\Vert_{L^{2\n}\\,. \\label{R3\n\\end{equation}\nIn view of (\\ref{bot.2.f}), (\\ref{bot.2.h}), (\\ref{R3}) and Lemma\n\\ref{initial-sing}, there exists $C>0$ such that\n\\[\n\\Vert f_{R}\\Vert_{L^{2}}=\\Vert f_{\\ast}-W^{(3)}\\Vert_{L^{2}}\\lesssim\ne^{-Ct}\\Vert f_{0}\\Vert_{L^{2}}\\,,\n\\]\nan\n\\[\n\\Vert f_{R}\\Vert_{H_{x}^{2}L_{v}^{2}}=\\Vert\\mathcal{R}^{(3)}-f_{F}\\Vert\n_{H_{x}^{2}L_{v}^{2}}\\lesssim\\Vert f_{0}\\Vert_{L^{2}}\\,.\n\\]\nThe Sobolev inequality \\cite[Theorem 5.8]{[Adams]} implies\n\\begin{equation}\n\\left\\vert f_{R}\\right\\vert _{L_{v}^{2}}\\leq\\Vert f_{R}\\Vert_{L_{x}^{\\infty\n}L_{v}^{2}}\\lesssim\\Vert f_{R}\\Vert_{H_{x}^{2}L_{v}^{2}}^{3\/4}\\Vert f_{R\n\\Vert_{L^{2}}^{1\/4}\\lesssim e^{-Ct}\\Vert f_{0}\\Vert_{L^{2}}\\,, \\label{fR\n\\end{equation}\nfor some constant $C>0$. In conclusion, we have that for the time-like region,\nif $\\gamma\\geq3\/2$, there exists a constant $C>0$ such that\n\\begin{equation}\n\\left\\vert \\mathcal{R}^{(3)}\\right\\vert _{L_{v}^{2}}\\lesssim\\left[\n(1+t)^{-3\/2}e^{-C\\frac{|x|^{2}}{t+1}}+e^{-Ct}\\right] \\Vert f_{0}\\Vert\n_{L_{x}^{\\infty}L_{v}^{2}}\\,; \\label{R-larger than 3\/2\n\\end{equation}\nand if $1\\leq\\gamma<3\/2$, any given $N>0$, there exists a constant $C>0$ such\nthat\n\\begin{equation}\n\\left\\vert \\mathcal{R}^{(3)}\\right\\vert _{L_{v}^{2}}\\lesssim\\left[\n(1+t)^{-3\/2}\\Big(1+\\frac{|x|^{2}}{1+t}\\Big)^{-N}+e^{-Ct}\\right] \\Vert\nf_{0}\\Vert_{L_{x}^{\\infty}L_{v}^{2}}\\,. \\label{R-less than 3\/2\n\\end{equation}\n\n\nCombining Proposition \\ref{pointwise-wave part >=1}, (\\ref{fR}),\n(\\ref{R-larger than 3\/2}) and (\\ref{R-less than 3\/2}), we obtain the pointwise\nestimate for the solution in the time-like region.\n\n\\begin{theorem}\n[Time-like region for $\\gamma\\geq1$\n\\label{time-like region for gamma larger or equal than 1}Let $\\gamma\\geq1$ and\nlet $f$ be the solution to equation (\\ref{in.1.c}). Assume that the initial\ncondition $f_{0}$ has compact support in the $x$ variable and is bounded in\n$L_{v}^{2}$. Then for any given $M>1$ and $|x|\\leq Mt$,\n\n\\noindent\\textrm{(i)} As $\\gamma\\geq3\/2$, there exists a positive constant $C$\nsuch that\n\\begin{equation}\n\\left\\vert f\\right\\vert _{L_{v}^{2}}\\lesssim\\left[ (1+t)^{-3\/2\ne^{-C\\frac{|x|^{2}}{t+1}}+(1+t^{-9\/4})e^{-Ct}\\right] \\Vert f_{0}\\Vert\n_{L_{x}^{\\infty}L_{v}^{2}}\\,;\n\\end{equation}\n\\newline\\textrm{(ii) }As $1\\leq\\gamma<3\/2$, any given $N>0$, there exists a\nconstant $C>0$ such that\n\\begin{equation}\n\\left\\vert f\\right\\vert _{L_{v}^{2}}\\lesssim\\left[ (1+t)^{-3\/2\n\\Big(1+\\frac{|x|^{2}}{1+t}\\Big)^{-N}+(1+t^{-9\/4})e^{-Ct}\\right] \\Vert\nf_{0}\\Vert_{L_{x}^{\\infty}L_{v}^{2}}\\,.\n\\end{equation}\n\n\\end{theorem}\n\n\\subsection{The case $0<\\gamma<1$}\n\nFirst, we introduce the $L^{2}$ estimate and the pointwise estimate of the\nwave part.\n\n\\begin{proposition}\n\\label{pointwise-wave part <1}Assume that $0<\\gamma<1$. Then for $0\\leq\nj\\leq3,$ and $t>0$, there exists $c_{\\gamma}>0$ such that\n\\begin{equation}\n\\left\\Vert h^{(j)}\\right\\Vert _{L^{2}}\\lesssim t^{j}e^{-c_{\\gamma\n\\alpha^{\\frac{2\\left( 1-\\gamma\\right) }{2-\\gamma}}t^{\\frac{\\gamma}{2-\\gamma\n}}}\\Vert f_{0}\\Vert_{L^{2}\\left( e^{\\left( j+1\\right) \\alpha\\left\\langle\nv\\right\\rangle ^{\\gamma}}\\right) }\\,, \\label{L2-esti-gamma<1\n\\end{equation}\nan\n\\begin{equation}\n\\left\\vert h^{(j)}\\left( t,x,v\\right) \\right\\vert _{L_{v}^{2}}\\lesssim\nt^{j-\\frac{9}{4}}e^{-\\frac{1}{4}c_{\\gamma}\\alpha^{\\frac{2\\left(\n1-\\gamma\\right) }{2-\\gamma}}t^{\\frac{\\gamma}{2-\\gamma}}}\\Vert f_{0\n\\Vert_{L^{2}\\left( e^{\\left( j+1\\right) \\alpha\\left\\langle v\\right\\rangle\n^{\\gamma}}\\right) }.\\, \\label{Wave-ptw-gamma<1\n\\end{equation}\n\n\\end{proposition}\n\n\\begin{proof}\nWe first consider the $L^{2}$ estimate for $h^{\\left( 0\\right) }.\\ $It is\neasy to see that (\\ref{Energy-evolu}) is still valid if setting $\\mu\\left(\nt,x,v\\right) =e^{\\alpha\\left\\langle v\\right\\rangle ^{\\gamma}}$, namely,\n\\begin{equation}\n\\frac{d}{dt}\\left\\Vert h^{\\left( 0\\right) }\\right\\Vert _{L^{2}\n^{2}+C\\left\\Vert h^{\\left( 0\\right) }\\right\\Vert _{L_{\\sigma}^{2}}^{2}\\leq0,\n\\label{3\n\\end{equation}\nand\n\\begin{equation}\n\\frac{d}{dt}\\left\\Vert e^{\\alpha\\left\\langle v\\right\\rangle ^{\\gamma\n}h^{\\left( 0\\right) }\\right\\Vert _{L^{2}}^{2}+C\\left\\Vert e^{\\alpha\n\\left\\langle v\\right\\rangle ^{\\gamma}}h^{\\left( 0\\right) }\\right\\Vert\n_{L_{\\sigma}^{2}}^{2}\\leq0. \\label{4\n\\end{equation}\nHence, $\\left\\Vert h^{\\left( 0\\right) }\\right\\Vert _{L^{2}}\\leq\\Vert\nf_{0}\\Vert_{L^{2}}\\ $for $t\\geq0$ and it suffices to show that for $t\\geq1,$\n\\[\n\\left\\Vert h^{(0)}\\right\\Vert _{L^{2}}\\lesssim e^{-c_{\\gamma}\\alpha\n^{\\frac{2\\left( 1-\\gamma\\right) }{2-\\gamma}}t^{\\frac{\\gamma}{2-\\gamma}\n}\\Vert f_{0}\\Vert_{L^{2}\\left( e^{\\alpha\\left\\langle v\\right\\rangle ^{\\gamma\n}}\\right) }.\n\\]\nAs in the work of Caflisch \\cite{[Caflisch]}, we consider a time-dependent low\nvelocity part\n\\[\nE=\\{\\left\\langle v\\right\\rangle \\leq\\beta t^{p^{\\prime}}\\},\n\\]\nand its complementary high velocity part $E^{c}=\\{\\left\\langle v\\right\\rangle\n>\\beta t^{p^{\\prime}}\\},$ where $p^{\\prime}>0$ and $\\beta>0$ will be\ndetermined later. Following the argument as in Section 5 of\n\\cite{[Strain-Guo]}, together with (\\ref{3}) and (\\ref{4}), we obtai\n\\[\n\\left\\Vert h^{\\left( 0\\right) }\\right\\Vert _{L^{2}}\\lesssim e^{-c_{\\gamma\n}\\alpha^{\\frac{2\\left( 1-\\gamma\\right) }{2-\\gamma}}t^{\\frac{\\gamma\n{2-\\gamma}}}\\left\\Vert f_{0}\\right\\Vert _{L^{2}(e^{\\alpha\\left\\langle\nv\\right\\rangle ^{\\gamma}})}\\,,\n\\]\nfor some constant $c_{\\gamma}>0,$ after choosing $p^{\\prime}=\\frac{1\n{2-\\gamma}$ in the Fokker -Planck case and $\\beta>0$ sufficiently large. This\ncompletes the $L^{2}$ estimate for $h^{\\left( 0\\right) }$.\n\nThrough the Duhumel Principle, we immediately obtain (\\ref{L2-esti-gamma<1})\nfor $1\\leq j\\leq3.$ Furthermore, the Sobolev inequality, together with Lemma\n\\ref{second-der} and (\\ref{L2-esti-gamma<1}), implies the desired pointwise\nestimate for the wave part $h^{\\left( j\\right) },\\ 0\\leq j\\leq3$.\n\\end{proof}\n\nNext, we are concerned with the pointwise behavior of the remainder part in\nthe time-like region. In virtue of the lack of the spectral analysis for\n$0<\\gamma<1$, we will instead use the method of the weighted $L^{2}$ estimate\nin the Fourier transformed variable and the interpolation argument to deal\nwith the time decay of the solution $f$ to equation $\\left( \\ref{in.1.c\n\\right) $ in this case. The main idea is to construct the desired weighted\ntime-frequency Lyapunov functional to capture the total energy dissipation\nrate. In the course of the proof we have to take great care to estimate the\nmicroscopic and macroscopic parts for $\\left\\vert \\eta\\right\\vert \\leq1$ and\n$\\left\\vert \\eta\\right\\vert >1$ respectively. Consider $\\left( \\ref{in.1.c\n\\right) ,$ taking the Fourier transform with respect to the $x$ variable\nleads to\n\\begin{equation}\n\\partial_{t}\\widehat{f}+iv\\cdot\\eta\\widehat{f}=L\\widehat{f}. \\label{FT\n\\end{equation}\nWe first calculate the $L^{2}$ estimate.\n\n\\begin{proposition}\n[$L^{2}$ estimate]Let $f$ be the solution to equation $\\left( \\ref{in.1.c\n\\right) $ . Then there exists a time-frequency functional $\\mathcal{E}\\left(\nt,\\eta\\right) $ such that\n\\begin{equation}\n\\mathcal{E}\\left( t,\\eta\\right) \\approx\\left\\vert \\widehat{f}\\left(\nt,\\eta,v\\right) \\right\\vert _{L_{v}^{2}}^{2},\n\\end{equation}\nwhere for any $t>0$ and $\\eta\\in\\mathbb{R}^{3}$, we have\n\\begin{equation}\n\\partial_{t}\\mathcal{E}\\left( t,\\eta\\right) +\\sigma\\widehat{\\rho}\\left(\n\\eta\\right) \\left\\vert \\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert\n_{L_{\\gamma-1}^{2}}^{2}\\leq0. \\label{E-N\n\\end{equation}\nHere we use the notation $\\widehat{\\rho}\\left( \\eta\\right) :=\\min\n\\{1,\\left\\vert \\eta\\right\\vert ^{2}\\}.$\n\\end{proposition}\n\n\\begin{proof}\nWe multiply equation $\\left( \\ref{FT}\\right) $ by $\\overline{\\widehat\n{f}\\left( t,\\eta,v\\right) }$ and integrate over $v$ to obtai\n\\[\n\\frac{1}{2}\\frac{d}{dt}\\left\\vert \\widehat{f}\\left( t,\\eta,v\\right)\n\\right\\vert _{L_{v}^{2}}^{2}-\\operatorname{Re}\\left\\langle L\\widehat\n{f},\\widehat{f}\\right\\rangle =0\n\\]\nFrom the coercivity in Lemma \\ref{co}, it follows tha\n\\begin{equation}\n\\frac{1}{2}\\frac{d}{dt}\\left\\vert \\widehat{f}\\left( t,\\eta,v\\right)\n\\right\\vert _{L_{v}^{2}}^{2}+\\nu_{0}\\left\\vert \\mathrm{P}_{1}\\widehat\n{f}\\right\\vert _{L_{\\sigma}^{2}}^{2}\\leq0. \\label{Fourier-energy -wo-weigh\n\\end{equation}\nNow, we need the estimate of $\\mathrm{P}_{0}\\widehat{f}$. In the sequel, we\nwill apply Strain's argument to estimate the macroscopic dissipation, in the\nspirit of Kawashima's work on dissipation of the hyperbolic-parabolic system.\nLet $a=\\left\\langle \\mathcal{M}^{1\/2},f\\right\\rangle _{v}$ and $b=\\left(\nb_{1},b_{2},b_{3}\\right) \\ $with $b_{i}=\\left\\langle v_{i}\\mathcal{M\n^{1\/2},f\\right\\rangle _{v}=\\left\\langle v_{i}\\mathcal{M}^{1\/2},\\mathrm{P\n_{1}f\\right\\rangle _{v}.$ Then $P_{0}f=a\\mathcal{M}^{1\/2}$ and from $\\left(\n\\ref{in.1.c}\\right) ,$ $a$ and $b$ satisfy the fuild-type syste\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l\n\\partial_{t}a+\\nabla_{x}\\cdot b=0\\vspace{3mm}\\\\\n\\partial_{t}b+\\alpha\\nabla_{x}a+\\nabla_{x}\\cdot\\Gamma\\left( \\mathrm{P\n_{1}f\\right) =-\\int\\left( \\mathcal{M}^{1\/2}\\nabla_{v}\\Phi\\right)\n\\mathrm{P}_{1}fdv,\n\\end{array}\n\\right. \\label{Evolu-ab\n\\end{equation}\nwher\n\\[\n\\alpha=\\frac{1}{3}\\int\\left\\vert v\\right\\vert ^{2}\\mathcal{M}dv>0,\n\\]\nand $\\Gamma=\\left( \\Gamma_{ij}\\right) _{3\\times3}$ is the moment function\ndefined by\n\\[\n\\Gamma_{ij}\\left( g\\right) =\\left\\langle \\left( v_{i}v_{j}-1\\right)\n\\mathcal{M}^{1\/2},g\\right\\rangle _{v},\\ \\ \\ \\ 1\\leq i,\\ j\\leq3.\n\\]\nNote by the definition of $P_{0}$ that $\\Gamma\\left( \\mathrm{P}_{1}f\\right)\n=\\int\\left( v\\otimes v\\right) \\mathcal{M}^{1\/2}\\mathrm{P}_{1}fdv.$ Taking\nthe Fourier transform with respect to $x$ of $\\left( \\ref{Evolu-ab}\\right)\n,$ we hav\n\\begin{align*}\n\\left\\vert \\eta\\right\\vert ^{2}\\left\\vert \\widehat{a}\\right\\vert ^{2} &\n=\\left( i\\eta\\widehat{a},i\\eta\\widehat{a}\\right) =\\frac{1}{\\alpha}\\left(\ni\\eta\\widehat{a},-\\partial_{t}\\widehat{b}-i\\Gamma\\left( \\mathrm{P\n_{1}\\widehat{f}\\right) \\eta-\\int\\left( \\mathcal{M}^{1\/2}\\nabla_{v\n\\Phi\\right) \\mathrm{P}_{1}\\widehat{f}dv\\right) \\\\\n& =\\frac{1}{\\alpha}\\left[ -\\left( i\\eta\\widehat{a},\\widehat{b}\\right)\n_{t}+\\left| \\eta\\cdot\\widehat{b}\\right| ^{2}-\\left( i\\eta\\widehat\n{a},i\\Gamma\\left( \\mathrm{P}_{1}\\widehat{f}\\right) \\eta\\right) -\\left(\ni\\eta\\widehat{a},\\int\\left( \\mathcal{M}^{1\/2}\\nabla_{v}\\Phi\\right)\n\\mathrm{P}_{1}\\widehat{f}dv\\right) \\right] .\n\\end{align*}\n\\newline Invoking on the rapid decay of $\\mathcal{M}^{1\/2}$ and using the\nCauchy-Schwartz inequality, we have\n\\[\n\\left\\vert \\int\\left( \\mathcal{M}^{1\/2}\\nabla_{v}\\Phi\\right) \\mathrm{P\n_{1}\\widehat{f}dv\\right\\vert ^{2}\\leq\\left\\vert \\mathcal{M}^{1\/2}v\\left\\langle\nv\\right\\rangle ^{-1}\\right\\vert _{L_{v}^{2}}^{2}\\left\\vert \\left\\langle\nv\\right\\rangle ^{\\gamma-1}\\mathrm{P}_{1}\\widehat{f}\\right\\vert _{L_{v}^{2\n}^{2}\\leq3\\alpha\\left\\vert \\mathrm{P}_{1}\\widehat{f}\\right\\vert _{L_{\\gamma\n-1}^{2}}^{2},\n\\]\nand\n\\[\n\\left\\vert \\left( i\\eta\\widehat{a},i\\Gamma\\left( \\mathrm{P}_{1}\\widehat\n{f}\\right) \\eta\\right) \\right\\vert \\leq\\epsilon\\left\\vert \\eta\\right\\vert\n^{2}\\left\\vert \\widehat{a}\\right\\vert ^{2}+C_{\\epsilon}\\left\\vert\n\\eta\\right\\vert ^{2}\\left\\vert \\mathrm{P}_{1}\\widehat{f}\\right\\vert\n_{L_{\\gamma-1}^{2}}^{2},\n\\]\nfor any small $\\epsilon>0.$ Therefore, we can conclud\n\\begin{equation}\n\\partial_{t}\\operatorname{Re}\\frac{\\left( i\\eta\\widehat{a},\\widehat\n{b}\\right) }{1+\\left\\vert \\eta\\right\\vert ^{2}}+\\frac{\\sigma\\left\\vert\n\\eta\\right\\vert ^{2}}{1+\\left\\vert \\eta\\right\\vert ^{2}}\\left\\vert \\widehat\n{a}\\right\\vert ^{2}\\leq C\\left\\vert \\mathrm{P}_{1}\\widehat{f}\\right\\vert\n_{L_{\\gamma-1}^{2}}^{2}, \\label{Fourier energy est.\n\\end{equation}\nfor some $\\sigma>0.$ Now, we define\n\\begin{equation}\n\\mathcal{E}\\left( t,\\eta\\right) =\\left\\vert \\widehat{f}\\left(\nt,\\eta,v\\right) \\right\\vert _{L_{v}^{2}}^{2}+\\kappa_{3}\\operatorname{Re\n\\frac{\\left( i\\eta\\widehat{a},\\widehat{b}\\right) }{1+\\left\\vert\n\\eta\\right\\vert ^{2}}, \\label{Lyapu 1\n\\end{equation}\nfor a constant $\\kappa_{3}>0$ to be determined later. One can fix $\\kappa\n_{3}>0$ small enough such that $\\mathcal{E}\\left( t,\\eta\\right)\n\\approx\\left\\vert \\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert _{L_{v}^{2\n}^{2}.$ Furthermore, according to Lemma \\ref{co} and $\\left(\n\\ref{Fourier energy est.}\\right) ,$ we choose $\\kappa_{3}>0$ sufficiently\nsmall such tha\n\\begin{equation}\n\\partial_{t}\\mathcal{E}\\left( t,\\eta\\right) +\\sigma\\left\\vert \\mathrm{P\n_{1}\\widehat{f}\\right\\vert _{L_{\\gamma-1}^{2}}^{2}+\\frac{2\\sigma\\left\\vert\n\\eta\\right\\vert ^{2}}{1+\\left\\vert \\eta\\right\\vert ^{2}}\\left\\vert \\widehat\n{a}\\right\\vert ^{2}\\leq0, \\label{E\n\\end{equation}\nfor some $\\sigma>0.$ In conclusion, we now have\n\\[\n\\partial_{t}\\mathcal{E}\\left( t,\\eta\\right) +\\sigma\\widehat{\\rho}\\left(\n\\eta\\right) \\left\\vert \\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert\n_{L_{\\gamma-1}^{2}}^{2}\\leq0.\n\\]\nHere we use the notation $\\widehat{\\rho}\\left( \\eta\\right) :=\\min\n\\{1,\\left\\vert \\eta\\right\\vert ^{2}\\}$.\n\\end{proof}\n\nSince $\\gamma-1<0,$ it is insufficient to gain the time decay of the total\nenergy of the solution $f.$ Therefore, in order to capture the total energy\ndissipation rate, we need to make further energy estimates on the microscopic\npart $\\mathrm{P}_{1}f$ and the macroscopic part $\\mathrm{P}_{0}f$.\n\n\\begin{proposition}\nLet $f$ be the solution to equation $\\left( \\ref{in.1.c}\\right) $ . Then\nthere exists a weighted time-frequency functional $\\widetilde{\\mathcal{E\n}\\left( t,\\eta\\right) $ such that\n\\begin{equation}\n\\widetilde{\\mathcal{E}}\\left( t,\\eta\\right) \\approx\\left\\vert e^{\\frac\n{\\alpha}{2}\\left\\langle v\\right\\rangle ^{\\gamma}}\\widehat{f}\\left(\nt,\\eta,v\\right) \\right\\vert _{L_{v}^{2}}^{2},\n\\end{equation}\nwhere $0<\\alpha\\gamma<1\/20\\ $and for any $t>0$ and $\\eta\\in\\mathbb{R}^{3}$ we\nhave\n\\begin{equation}\n\\partial_{t}\\widetilde{\\mathcal{E}}\\left( t,\\eta\\right) +\\sigma\\widehat\n{\\rho}\\left( \\eta\\right) \\left\\vert e^{\\frac{\\alpha}{2}\\left\\langle\nv\\right\\rangle ^{\\gamma}}\\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert\n_{L_{\\gamma-1}^{2}}^{2}\\leq0. \\label{EL\n\\end{equation}\nHere we use the notation $\\widehat{\\rho}\\left( \\eta\\right) :=\\min\\{1,$\n$\\left\\vert \\eta\\right\\vert ^{2}\\}.$\n\\end{proposition}\n\n\\begin{proof}\nFirstly, we shall prove the following Lyapunov inequality with a velocity\nweight $e^{\\alpha\\left\\langle v\\right\\rangle ^{\\gamma}},\\ 0<\\alpha\\gamma\n<1\/20$\n\\begin{equation}\n\\frac{d}{dt}\\left\\vert e^{\\frac{\\alpha}{2}\\left\\langle v\\right\\rangle\n^{\\gamma}}\\mathrm{P}_{1}\\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert\n_{L_{v}^{2}}^{2}+\\sigma\\left\\vert e^{\\frac{\\alpha}{2}\\left\\langle\nv\\right\\rangle ^{\\gamma}}\\mathrm{P}_{1}\\widehat{f}\\left( t,\\eta,v\\right)\n\\right\\vert _{L_{\\gamma-1}^{2}}^{2}\\leq C_{\\sigma}\\left\\vert \\eta\\right\\vert\n^{2}\\left\\vert \\widehat{f}\\right\\vert _{L_{\\gamma-1}^{2}}^{2}+C_{\\gamma\n}\\left\\vert \\mathrm{P}_{1}\\widehat{f}\\right\\vert _{L_{v}^{2}\\left(\nB_{2R}\\right) }^{2}, \\label{Micro-weighted ineq\n\\end{equation}\nwhere the constants $C_{\\gamma}>0$ and $R>0$ are dependent only upon $\\gamma.$\nWe split the solution $f$ into two parts: $f=P_{0}f+\\mathrm{P}_{1}f,$ and then\napply $\\mathrm{P}_{1}$ to equation $\\left( \\ref{FT}\\right) $\n\\[\n\\partial_{t}\\mathrm{P}_{1}\\widehat{f}+iv\\cdot\\eta\\mathrm{P}_{1}\\widehat\n{f}-L\\mathrm{P}_{1}\\widehat{f}=-\\mathrm{P}_{1}\\left( iv\\cdot\\eta\nP_{0}\\widehat{f}\\right) +P_{0}\\left( iv\\cdot\\eta\\mathrm{P}_{1}\\widehat\n{f}\\right) .\n\\]\nMultiply the above equation by $e^{\\alpha\\left\\langle v\\right\\rangle ^{\\gamma\n}}\\overline{\\mathrm{P}_{1}\\widehat{f}}$ and integrate with respect to $v$ to\nobtai\n\\[\n\\frac{1}{2}\\frac{d}{dt}\\left\\vert e^{\\frac{\\alpha}{2}\\left\\langle\nv\\right\\rangle ^{\\gamma}}\\mathrm{P}_{1}\\widehat{f}\\left( t,\\eta,v\\right)\n\\right\\vert _{L_{v}^{2}}^{2}-\\operatorname{Re}\\left\\langle e^{\\alpha\n\\left\\langle v\\right\\rangle ^{\\gamma}}L\\mathrm{P}_{1}\\widehat{f\n,\\mathrm{P}_{1}\\widehat{f}\\right\\rangle _{v}=\\Gamma,\n\\]\nwhere\n\\[\n\\Gamma=-\\operatorname{Re}\\left\\langle \\mathrm{P}_{1}\\left( iv\\cdot\\eta\nP_{0}\\widehat{f}\\right) ,e^{\\alpha\\left\\langle v\\right\\rangle ^{\\gamma\n}\\mathrm{P}_{1}\\widehat{f}\\right\\rangle +\\operatorname{Re}\\left\\langle\nP_{0}\\left( iv\\cdot\\eta\\mathrm{P}_{1}\\widehat{f}\\right) ,e^{\\alpha\n\\left\\langle v\\right\\rangle ^{\\gamma}}\\mathrm{P}_{1}\\widehat{f}\\right\\rangle\n.\n\\]\nOwing to the rapid decay of $\\mathcal{M}^{1\/2},$ we obtai\n\\[\n\\left\\vert \\Gamma\\right\\vert \\leq\\epsilon\\left\\vert e^{\\frac{\\alpha\n{2}\\left\\langle v\\right\\rangle ^{\\gamma}}\\mathrm{P}_{1}\\widehat{f}\\left(\nt,\\eta,v\\right) \\right\\vert _{L_{\\gamma-1}^{2}}^{2}+C_{\\epsilon}\\left\\vert\n\\eta\\right\\vert ^{2}\\left( \\left\\vert \\mathrm{P}_{1}\\widehat{f}\\left(\nt,\\eta,v\\right) \\right\\vert _{L_{\\gamma-1}^{2}}^{2}+\\left\\vert P_{0\n\\widehat{f}\\right\\vert _{L_{v}^{2}}^{2}\\right) ,\n\\]\nwhich holds for any small $\\epsilon>0.$ On the other hand, we rewrite\n$L=-\\Lambda+K,$ $K=\\varpi\\chi_{R}\\left( \\left\\vert v\\right\\vert \\right) ,$\nwhere $R>0$ and $\\varpi>0\\ $are chosen sufficiently large such tha\n\\[\n\\frac{\\left\\vert v\\right\\vert ^{2}\\left\\langle v\\right\\rangle ^{2\\gamma-4}\n{4}-\\frac{3}{2}\\left\\langle v\\right\\rangle ^{\\gamma-2}-\\frac{\\left(\n\\gamma-2\\right) }{2}\\left\\vert v\\right\\vert ^{2}\\left\\langle v\\right\\rangle\n^{\\gamma-4}+\\varpi\\chi_{R}\\left( \\left\\vert v\\right\\vert \\right) \\geq\n\\frac{1}{5}\\left\\langle v\\right\\rangle ^{2\\gamma-2}.\n\\]\nHence, we hav\n\\begin{align*}\n& \\quad-\\operatorname{Re}\\left\\langle e^{\\alpha\\left\\langle v\\right\\rangle\n^{\\gamma}}L\\mathrm{P}_{1}\\widehat{f},\\mathrm{P}_{1}\\widehat{f}\\right\\rangle\n_{v}\\\\\n& =\\operatorname{Re}\\int e^{\\alpha\\left\\langle v\\right\\rangle ^{\\gamma\n}\\left[ \\left( \\Lambda-K\\right) \\mathrm{P}_{1}\\widehat{f}\\right]\n\\mathrm{P}_{1}\\overline{\\widehat{f}}dv\\\\\n& \\geq\\int e^{\\alpha\\left\\langle v\\right\\rangle ^{\\gamma}}\\left\\vert\n\\nabla_{v}\\mathrm{P}_{1}\\widehat{f}\\right\\vert ^{2}+\\operatorname{Re\n\\int\\alpha\\gamma\\left\\langle v\\right\\rangle ^{\\gamma-2}e^{\\alpha\\left\\langle\nv\\right\\rangle ^{\\gamma}}\\left( v\\cdot\\nabla_{v}\\mathrm{P}_{1}\\widehat\n{f}\\right) \\mathrm{P}_{1}\\overline{\\widehat{f}}dv\\\\\n& \\quad+\\frac{1}{5}\\int\\left\\langle v\\right\\rangle ^{2\\gamma-2\ne^{\\alpha\\left\\langle v\\right\\rangle ^{\\gamma}}\\left\\vert \\mathrm{P\n_{1}\\widehat{f}\\right\\vert ^{2}dv-C^{\\prime}\\left\\vert \\mathrm{P}_{1\n\\widehat{f}\\right\\vert _{L_{v}^{2}\\left( B_{2R}\\right) }^{2},\n\\end{align*}\nwhere $C^{\\prime}=C^{\\prime}\\left( \\alpha,\\gamma,R\\right) .$ Note that\n$\\alpha\\gamma<1\/20,$ the Cauchy-Schwartz inequality implies\n\\begin{align*}\n& \\quad\\left\\vert \\operatorname{Re}\\int\\alpha\\gamma\\left\\langle\nv\\right\\rangle ^{\\gamma-2}e^{\\alpha\\left\\langle v\\right\\rangle ^{\\gamma\n}\\left( v\\cdot\\nabla_{v}\\mathrm{P}_{1}\\widehat{f}\\right) \\mathrm{P\n_{1}\\overline{\\widehat{f}}dv\\right\\vert \\\\\n& \\leq\\int\\alpha\\gamma\\left\\langle v\\right\\rangle ^{\\gamma-1}e^{\\alpha\n\\left\\langle v\\right\\rangle ^{\\gamma}}\\left\\vert \\nabla_{v}\\mathrm{P\n_{1}\\widehat{f}\\right\\vert \\left\\vert \\mathrm{P}_{1}\\widehat{f}\\right\\vert\ndv\\\\\n& \\leq\\int e^{\\alpha\\left\\langle v\\right\\rangle ^{\\gamma}}\\left\\vert\n\\nabla_{v}\\mathrm{P}_{1}\\widehat{f}\\right\\vert ^{2}dv+\\frac{1}{80\n\\int\\left\\langle v\\right\\rangle ^{2\\gamma-2}e^{\\alpha\\left\\langle\nv\\right\\rangle ^{\\gamma}}\\left\\vert \\mathrm{P}_{1}\\widehat{f}\\right\\vert\n^{2}dv,\n\\end{align*}\nso we deduce\n\\[\n-\\operatorname{Re}\\left\\langle \\left\\langle v\\right\\rangle ^{2\\ell\nL\\mathrm{P}_{1}\\widehat{f},\\mathrm{P}_{1}\\widehat{f}\\right\\rangle _{v\n\\geq\\frac{1}{6}\\int\\left\\langle v\\right\\rangle ^{2\\gamma-2}e^{\\alpha\n\\left\\langle v\\right\\rangle ^{\\gamma}}\\left\\vert \\mathrm{P}_{1}\\widehat\n{f}\\right\\vert ^{2}dv-\\widetilde{C}\\left( R,\\gamma,\\alpha\\right) \\left\\vert\n\\mathrm{P}_{1}\\widehat{f}\\right\\vert _{L_{v}^{2}\\left( B_{2R}\\right) }^{2},\n\\]\nwhere $\\widetilde{C}\\left( R,\\ell,\\alpha\\right) >0.\\ $Consequently,\n\\[\n\\frac{d}{dt}\\left\\vert e^{\\frac{\\alpha}{2}\\left\\langle v\\right\\rangle\n^{\\gamma}}\\mathrm{P}_{1}\\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert\n_{L_{v}^{2}}^{2}+\\sigma\\left\\vert e^{\\frac{\\alpha}{2}\\left\\langle\nv\\right\\rangle ^{\\gamma}}\\mathrm{P}_{1}\\widehat{f}\\left( t,\\eta,v\\right)\n\\right\\vert _{L_{\\gamma-1}^{2}}^{2}\\leq C_{\\sigma}\\left\\vert \\eta\\right\\vert\n^{2}\\left\\vert \\widehat{f}\\right\\vert _{L_{\\gamma-1}^{2}}^{2}+C_{\\gamma\n}\\left\\vert \\mathrm{P}_{1}\\widehat{f}\\right\\vert _{L_{v}^{2}\\left(\nB_{2R}\\right) }^{2},\n\\]\nfor some constant $\\sigma>0.$ \\bigskip In addition, if we multiply $\\left(\n\\ref{FT}\\right) $ with $e^{\\alpha\\left\\langle v\\right\\rangle ^{\\gamma\n}\\overline{\\widehat{f}\\left( t,\\eta,v\\right) },$ integrate in $v$ and use\nthe same procedure as above, we also obtai\n\\begin{equation}\n\\frac{1}{2}\\frac{d}{dt}\\left\\vert e^{\\frac{\\alpha}{2}\\left\\langle\nv\\right\\rangle ^{\\gamma}}\\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert\n_{L_{v}^{2}}^{2}+\\sigma\\left\\vert e^{\\frac{\\alpha}{2}\\left\\langle\nv\\right\\rangle ^{\\gamma}}\\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert\n_{L_{\\gamma-1}^{2}}^{2}\\leq C_{\\gamma}\\left\\vert \\widehat{f}\\right\\vert\n_{L^{2}\\left( B_{2R}\\right) }^{2}. \\label{weighted ineq\n\\end{equation}\n{\\ }\n\nTo do the weighted estimate, we introduce a new energy as follows :\n\\[\n\\widetilde{\\mathcal{E}}\\left( t,\\eta\\right) :=\\widetilde{\\mathcal{E}\n^{0}\\left( t,\\eta\\right) +\\widetilde{\\mathcal{E}}^{1}\\left( t,\\eta\\right)\n,\n\\]\nwit\n\\[\n\\widetilde{\\mathcal{E}}^{0}\\left( t,\\eta\\right) =1_{\\left\\vert\n\\eta\\right\\vert \\leq1}\\left( \\mathcal{E}\\left( t,\\eta\\right) +\\kappa\n_{4}\\left\\vert e^{\\frac{\\alpha}{2}\\left\\langle v\\right\\rangle ^{\\gamma\n}\\mathrm{P}_{1}\\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert _{L_{v}^{2\n}^{2}\\right) ,\n\\\n\\[\n\\widetilde{\\mathcal{E}}^{1}\\left( t,\\eta\\right) =1_{\\left\\vert\n\\eta\\right\\vert >1}\\left( \\mathcal{E}\\left( t,\\eta\\right) +\\kappa\n_{5}\\left\\vert e^{\\frac{\\alpha}{2}\\left\\langle v\\right\\rangle ^{\\gamma\n}\\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert _{L_{v}^{2}}^{2}\\right) ,\n\\]\nwhere $\\mathcal{E}\\left( t,\\eta\\right) $ is defined as in $\\left(\n\\ref{Lyapu 1}\\right) $ and the constants $\\kappa_{4},$ $\\kappa_{5}>0$ will be\nchosen small enough. Notice further that $\\left\\vert \\widehat{a}\\right\\vert\n^{2}=\\left\\vert P_{0}\\widehat{f}\\right\\vert _{L_{v}^{2}}^{2}\\gtrsim\\left\\vert\ne^{\\frac{\\alpha}{2}\\left\\langle v\\right\\rangle ^{\\gamma}}P_{0}\\widehat\n{f}\\right\\vert _{L_{v}^{2}}^{2}$ for all $0<\\alpha\\gamma<1\/20,$ and so\n$\\widetilde{\\mathcal{E}}\\left( t,\\eta\\right) \\approx\\left\\vert\ne^{\\frac{\\alpha}{2}\\left\\langle v\\right\\rangle ^{\\gamma}}\\widehat\n{f}\\right\\vert _{L_{v}^{2}}^{2}.$\n\nFor $\\widetilde{\\mathcal{E}}^{1}\\left( t,\\eta\\right) ,$ we combine $\\left(\n\\ref{E}\\right) $ and $\\left( \\ref{weighted ineq}\\right) $ for $\\left\\vert\n\\eta\\right\\vert >1$ to obtain\n\\begin{equation}\n\\partial_{t}\\widetilde{\\mathcal{E}}^{1}\\left( t,\\eta\\right) +\\sigma\n\\left\\vert e^{\\frac{\\alpha}{2}\\left\\langle v\\right\\rangle ^{\\gamma}\n\\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert _{L_{\\gamma-1}^{2}\n^{2}1_{\\left\\vert \\eta\\right\\vert >1}\\leq0, \\label{E0\n\\end{equation}\nfor $\\kappa_{5}>0$ small enough, since $\\left\\vert \\eta\\right\\vert\n^{2}\/\\left( 1+\\left\\vert \\eta\\right\\vert ^{2}\\right) \\geq\\frac{1}{2}.$\n\nFor $\\widetilde{\\mathcal{E}}^{0}\\left( t,\\eta\\right) ,$ since $\\left\\vert\n\\eta\\right\\vert ^{2}\/\\left( 1+\\left\\vert \\eta\\right\\vert ^{2}\\right)\n\\geq\\frac{\\left\\vert \\eta\\right\\vert ^{2}}{2}$ for $\\left\\vert \\eta\\right\\vert\n\\leq1$ and $\\left\\vert \\widehat{a}\\right\\vert ^{2}\\gtrsim\\left\\vert\ne^{\\frac{\\alpha}{2}\\left\\langle v\\right\\rangle ^{\\gamma}}P_{0}\\widehat\n{f}\\right\\vert _{L_{v}^{2}}^{2}$ for all $\\alpha\\gamma<1\/20,$ combining\n$\\left( \\ref{E}\\right) $ and $\\left( \\ref{Micro-weighted ineq}\\right) $\nfor $\\left\\vert \\eta\\right\\vert \\leq1$ gives\n\\begin{equation}\n\\partial_{t}\\widetilde{\\mathcal{E}}^{0}\\left( t,\\eta\\right) +\\sigma\n\\left\\vert \\eta\\right\\vert ^{2}\\left\\vert e^{\\frac{\\alpha}{2}\\left\\langle\nv\\right\\rangle ^{\\gamma}}\\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert\n_{L_{\\gamma-1}^{2}}^{2}1_{\\left\\vert \\eta\\right\\vert \\leq1}\\leq0, \\label{E1\n\\end{equation}\nfor $\\kappa_{4}>0$ small enough. This completes the proof.\n\\end{proof}\n\nNow, it is enough to prove the estimate in the time-like region. We apply the\nH\\\"{o}lder inequality to obtain that for $j\\geq1,${\n\\begin{align*}\n\\mathcal{E}\\left( t,\\eta\\right) & \\lesssim\\left\\vert \\widehat{f}\\left(\nt,\\eta,v\\right) \\right\\vert _{L_{v}^{2}}^{2}=\\int\\left( \\left\\vert\n\\widehat{f}(t,\\eta,v)\\right\\vert ^{2}e^{\\alpha\\left\\langle v\\right\\rangle\n^{\\gamma}}\\right) ^{\\frac{1}{j+1}}\\left( \\left\\vert \\widehat{f\n(t,\\eta,v)\\right\\vert ^{2}e^{-\\frac{\\alpha}{j}\\left\\langle v\\right\\rangle\n^{\\gamma}}\\right) ^{\\frac{j}{j+1}}dv\\\\\n& \\leq\\left( \\int e^{-\\frac{\\alpha}{j}\\left\\langle v\\right\\rangle ^{\\gamma\n}\\left\\vert \\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert ^{2}dv\\right)\n^{j\/\\left( j+1\\right) }\\left( \\int e^{\\alpha\\left\\langle v\\right\\rangle\n^{\\gamma}}\\left\\vert \\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert\n^{2}dv\\right) ^{1\/\\left( j+1\\right) }\\\\\n& \\lesssim\\left\\vert \\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert\n_{L_{\\gamma-1}^{2}}^{2j\/\\left( j+1\\right) }\\widetilde{\\mathcal{E\n}^{1\/\\left( j+1\\right) }\\left( t,\\eta\\right) \\,.\n\\end{align*}\n} Thus we conclude that\n\\[\n\\mathcal{E}^{\\left( j+1\\right) \/j}\\left( t,\\eta\\right) \\lesssim\\left\\vert\n\\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert _{L_{\\gamma-1}^{2}\n^{2}\\widetilde{\\mathcal{E}}^{1\/j}\\left( t,\\eta\\right) \\lesssim\\left\\vert\n\\widehat{f}\\left( t,\\eta,v\\right) \\right\\vert _{L_{\\gamma-1}^{2}\n^{2}\\widetilde{\\mathcal{E}}^{1\/j}\\left( 0,\\eta\\right) .\n\\]\nNow we can rewrite $\\left( \\ref{E}\\right) ,$ for any $\\eta\\in\\mathbb{R\n^{3},$ as\n\\[\n\\partial_{t}\\mathcal{E}\\left( t,\\eta\\right) +\\sigma\\widehat{\\rho}\\left(\n\\eta\\right) \\mathcal{E}^{\\left( j+1\\right) \/j}\\left( t,\\eta\\right)\n\\widetilde{\\mathcal{E}}^{-1\/j}\\left( 0,\\eta\\right) \\leq0.\n\\]\nIntegrating this over time, we obtai\n\\[\nj\\mathcal{E}^{-1\/j}\\left( 0,\\eta\\right) -j\\mathcal{E}^{-1\/j}\\left(\nt,\\eta\\right) \\lesssim-t\\widehat{\\rho}\\left( \\eta\\right) \\widetilde\n{\\mathcal{E}}^{-1\/j}\\left( 0,\\eta\\right) .\n\\]\nAs a consequence, for any $j\\geq1,$ uniformly in $\\eta\\in\\mathbb{R}^{3},$ we\nget\n\\[\n\\mathcal{E}\\left( t,\\eta\\right) \\lesssim\\widetilde{\\mathcal{E}}\\left(\n0,\\eta\\right) \\left( \\frac{t\\widehat{\\rho}\\left( \\eta\\right) \n{j}+1\\right) ^{-j}.\n\\]\n\n\nRecall that the long wave part $f_{L}$ and the short wave part $f_{S}$ of the\nsolution $f$ are given respectively by\n\\[\nf_{L}=\\int_{\\left\\vert \\eta\\right\\vert \\leq1}e^{i\\eta\\cdot x+\\left(\n-iv\\cdot\\eta+L\\right) t}\\widehat{f_{0}}\\left( \\eta,v\\right) d\\eta,\n\\\n\\[\nf_{S}=\\int_{\\left\\vert \\eta\\right\\vert >1}e^{i\\eta\\cdot x+\\left( -iv\\cdot\n\\eta+L\\right) t}\\widehat{f_{0}}\\left( \\eta,v\\right) d\\eta,\n\\]\nWhen $\\left\\vert \\eta\\right\\vert \\leq1$ and $k\\in{\\mathbb{N\\cup\\{}}0\\}$,\nsince\n\\[\n\\int_{\\left\\vert \\eta\\right\\vert \\leq1}|\\eta|^{2k}\\left( \\frac{t\\left\\vert\n\\eta\\right\\vert ^{2}}{j}+1\\right) ^{-j}d\\eta\\lesssim(1+t)^{-\\frac{3}{2\n-k}\\text{ if }j>\\frac{3}{2}+k,\n\\]\nwe obtain\n\\begin{align}\n\\int_{\\left\\vert \\eta\\right\\vert \\leq1}|\\eta|^{2k}\\mathcal{E}\\left(\nt,\\eta\\right) d\\eta & \\lesssim\\int_{\\left\\vert \\eta\\right\\vert \\leq1\n|\\eta|^{2k}\\left( \\frac{t\\left\\vert \\eta\\right\\vert ^{2}}{j}+1\\right)\n^{-j}\\widetilde{\\mathcal{E}}\\left( 0,\\eta\\right) d\\eta\\label{Elong}\\\\\n& \\lesssim(1+t)^{-\\frac{3}{2}-k}\\left\\Vert e^{\\frac{\\alpha}{2}\\left\\langle\nv\\right\\rangle ^{\\gamma}}f_{0}\\right\\Vert _{L_{x}^{1}L_{v}^{2}}^{2},\\nonumber\n\\end{align}\nwhich implies that\n\\[\n\\left\\Vert \\nabla_{x}^{k}f_{L}\\right\\Vert _{L^{2}}\\lesssim(1+t)^{-\\frac{3\n{4}-\\frac{k}{2}}\\left\\Vert e^{\\frac{\\alpha}{2}\\left\\langle v\\right\\rangle\n^{\\gamma}}f_{0}\\right\\Vert _{L_{x}^{1}L_{v}^{2}}.\n\\]\nBy the Sobolev inequality, we get\n\\[\n\\left\\Vert f_{L}\\right\\Vert _{L_{x}^{\\infty}L_{v}^{2}}\\lesssim\\left\\Vert\n\\nabla_{x}^{2}f_{L}\\right\\Vert _{L^{2}}^{3\/4}\\left\\Vert f_{L}\\right\\Vert\n_{L^{2}}^{1\/4}\\lesssim(1+t)^{-\\frac{3}{2}}\\left\\Vert f_{0}\\right\\Vert\n_{L_{x}^{1}L_{v}^{2}\\left( e^{\\alpha\\left\\langle v\\right\\rangle ^{\\gamma\n}\\right) }.\n\\]\n\n\n{When $\\left\\vert \\eta\\right\\vert >1,$ we note that the equations \\ref{E-N}\nand \\ref{EL} for $f_{S}$ are similar to \\ref{3} and \\ref{4} for $h^{(0)}$.\nThen following the similar procedure of the proof,} it implies\n\\[\n\\left\\Vert f_{S}\\right\\Vert _{L^{2}}\\lesssim e^{-c_{\\gamma}\\alpha\n^{\\frac{2\\left( 1-\\gamma\\right) }{2-\\gamma}}t^{\\frac{\\gamma}{2-\\gamma}\n}\\left\\Vert f_{0}\\right\\Vert _{L^{2}(e^{\\alpha\\left\\langle v\\right\\rangle\n^{\\gamma}})}\\,,\\ \\ t\\geq0,\n\\]\nfor some constant $c_{\\gamma}>0.$\n\nTo sum up, we have the following proposition:\n\n\\begin{proposition}\n\\label{LS-estimate2}Let $0<\\gamma<1\\ $and let $f$ be the solution of equation\n$\\left( \\ref{in.1.c}\\right) .$ For any $\\alpha>0$ small with $\\alpha\n\\gamma<1\/20$, we have \\newline\\noindent\\textrm{$(i)$ (Long wave $f_{L}$)}\n\\begin{equation}\n\\left\\Vert f_{L}\\right\\Vert _{L_{x}^{\\infty}L_{v}^{2}}\\lesssim(1+t)^{-\\frac\n{3}{2}}\\left\\Vert f_{0}\\right\\Vert _{L_{x}^{1}L_{v}^{2}\\left( e^{\\alpha\n\\left\\langle v\\right\\rangle ^{\\gamma}}\\right) }. \\label{f_S-long\n\\end{equation}\n\n\n\\noindent\\textrm{$(ii)$ (Short wave $f_{S}$)} There exists $c_{\\gamma}>0$ such\nthat\n\\begin{equation}\n\\left\\Vert f_{S}\\right\\Vert _{L^{2}}\\lesssim e^{-c_{\\gamma}\\alpha\n^{\\frac{2\\left( 1-\\gamma\\right) }{2-\\gamma}}t^{\\frac{\\gamma}{2-\\gamma}\n}\\left\\Vert f_{0}\\right\\Vert _{L^{2}(e^{\\alpha\\left\\langle v\\right\\rangle\n^{\\gamma}})}\\,. \\label{f_S-soft\n\\end{equation}\n\n\\end{proposition}\n\nBased on the long wave-short wave decomposition and wave -remainder\ndecomposition, i.e.,\n\\[\nf=f_{L}+f_{S}=W^{\\left( 3\\right) }+\\mathcal{R}^{(3)},\n\\]\nwe now define the tail part as $f_{R}=\\mathcal{R}^{(3)}-f_{L}=f_{S}-W^{\\left(\n3\\right) },$ which leads to that $f$ can be written as $f=W^{\\left(\n3\\right) }+f_{L}+f_{R}.$ From Lemma \\ref{second-der},\n\\begin{equation}\n\\left\\Vert \\mathcal{R}^{(3)}(t)\\right\\Vert _{H_{x}^{2}L_{v}^{2}}\\lesssim\n\\int_{0}^{t}\\left\\Vert h^{(3)}(s)\\right\\Vert _{H_{x}^{2}L_{v}^{2}\nds\\lesssim\\left( 1+t^{4}\\right) \\left\\Vert f_{0}\\right\\Vert _{L^{2}},\n\\label{Prop19-1\n\\end{equation}\nand so\n\\[\n\\left\\Vert f_{R}\\right\\Vert _{H_{x}^{2}L_{v}^{2}}=\\left\\Vert \\mathcal{R\n^{(3)}-f_{L}\\right\\Vert _{H_{x}^{2}L_{v}^{2}}\\lesssim\\left( 1+t^{4}\\right)\n\\left\\Vert f_{0}\\right\\Vert _{L^{2}},\\ \\ t>0.\n\\]\nIn view of Proposition \\ref{pointwise-wave part <1} and Proposition\n\\ref{LS-estimate2},\n\\[\n\\left\\Vert f_{R}\\right\\Vert _{L^{2}}=\\left\\Vert f_{S}-W^{\\left( 3\\right)\n}\\right\\Vert _{L^{2}}\\lesssim e^{-\\frac{c_{\\gamma}}{2}\\alpha^{\\frac{2\\left(\n1-\\gamma\\right) }{2-\\gamma}}t^{\\frac{\\gamma}{2-\\gamma}}}\\left\\Vert\nf_{0}\\right\\Vert _{L^{2}(e^{4\\alpha\\left\\langle v\\right\\rangle ^{\\gamma}\n)},\\ \\ t>0.\\\n\\]\nThe Sobolev inequality implies\n\\begin{equation}\n\\left\\vert f_{R}\\right\\vert _{L_{v}^{2}}\\leq\\left\\Vert f_{R}\\right\\Vert\n_{L_{v}^{2}L_{x}^{\\infty}}\\lesssim\\left\\Vert f_{R}\\right\\Vert _{H_{x}^{2\nL_{v}^{2}}^{3\/4}\\left\\Vert f_{R}\\right\\Vert _{L^{2}}^{1\/4}\\lesssim\ne^{-\\frac{c_{\\gamma}}{16}\\alpha^{\\frac{2\\left( 1-\\gamma\\right) }{2-\\gamma\n}t^{\\frac{\\gamma}{2-\\gamma}}}\\left\\Vert f_{0}\\right\\Vert _{L^{2\n(e^{4\\alpha\\left\\langle v\\right\\rangle ^{\\gamma}})},\\ \\ t>0.\n\\label{fR-gamma<1\n\\end{equation}\n\n\nCombining (\\ref{Wave-ptw-gamma<1}), (\\ref{f_S-long}) and (\\ref{fR-gamma<1}),\nwe obtain the pointwise estimate for the solution in the time-like region.\n\n\\begin{theorem}\n[Time-like region for $0<\\gamma<1$\n\\label{time-like region for gamma less than 1} Let $0<\\gamma<1$ and let $f$ be\nthe solution to equation (\\ref{in.1.c}). Assume that the initial condition\n$f_{0}$ has compact support in the $x$ variable and is bounded in $L_{v\n^{2}(e^{4\\alpha\\left\\langle v\\right\\rangle ^{\\gamma}})$. Then for $\\alpha>0$\nis small enough, there exists a positive constant $c_{\\gamma}$ such that\n\\begin{equation}\n\\left\\vert f\\right\\vert _{L_{v}^{2}}\\lesssim\\left[ (1+t)^{-3\/2\n+(1+t^{-9\/4})e^{-c_{\\gamma}\\alpha^{\\frac{2\\left( 1-\\gamma\\right) }{2-\\gamma\n}}t^{\\frac{\\gamma}{2-\\gamma}}}\\right] \\Vert f_{0}\\Vert_{L_{x}^{\\infty\nL_{v}^{2}(e^{4\\alpha\\left\\langle v\\right\\rangle ^{\\gamma}})}\\,.\n\\end{equation}\n\n\\end{theorem}\n\n\\section{In the space-like region}\n\n\\label{layer} We have finished the estimate of solution inside the time-like\nregion. To have the global picture of the space-time structure of solution, we\nstill need to investigate the solution in the space-like region. To this end,\nwe shall estimate the wave part $W^{(3)}$ and the remainder part\n$\\mathcal{R}^{(3)}$ separately. Here, the weighted energy estimate plays a\ndecisive role.\n\n\n\n\\subsection{The case $\\gamma\\geq3\/2$: Exponential decay}\n\n\\begin{proposition}\n\\label{weig_1} Consider the weight functions\n\\[\nw(x,t)=e^{\\frac{\\left\\langle x\\right\\rangle -Mt}{2D}}\\,,\\quad\\mu\n(x)=e^{\\frac{\\left\\langle x\\right\\rangle }{D}}\\,,\n\\]\nwhere $D$ and $M$ are chosen sufficiently large. Then for $0\\leq j\\leq3$, we\nhav\n\\begin{equation}\n\\Vert wh^{(j)}\\Vert_{H_{x}^{2}L_{v}^{2}}\\lesssim t^{-3+j}\\Vert f_{0\n\\Vert_{L^{2}(\\mu)},\\ \\ \\ \\ \\ 01, \\label{energy 2\n\\end{equation}\nand\n\\begin{equation}\n\\Vert w\\mathcal{R}^{(3)}\\Vert_{H_{x}^{2}L_{v}^{2}}\\lesssim\\Vert f_{0\n\\Vert_{L^{2}(\\mu)},\\ \\ \\ \\ \\ \\ t>0. \\label{energy 3\n\\end{equation}\n\n\\end{proposition}\n\n\\begin{proof}\nIn view of that $w(x,t)$ is non-increasing in $t$, it is not hard to verify\nthat\n\\[\n\\left\\Vert wg(t)\\right\\Vert _{H_{x}^{2}L_{v}^{2}}\\lesssim\\left\\Vert\ng(t)\\right\\Vert _{H_{x}^{2}L_{v}^{2}(\\mu)}\\,.\n\\]\nThen the weighted energy inequalities (\\ref{energy 1}) and (\\ref{energy 2})\nfollow from Lemma \\ref{second-der} directly.\n\nIt remains to show the weighted energy estimates for the remainder part\n$\\mathcal{R}^{(3)}$, $t>0$. We shall demonstrate that\n\\[\n\\Vert w\\mathcal{R}^{(3)}\\Vert_{H_{x}^{2}L_{v}^{2}}\\lesssim\\left( 1+t\\right)\n\\Vert f_{0}\\Vert_{L^{2}(\\mu)},\\ \\ \\ \\ \\ \\ t>0.\n\\]\nTo see this, let $u=w\\mathcal{R}^{(3)}$and then $\\partial_{x}^{\\beta}u,$ where\n$\\beta$ is a multi-index, solves the equation\n\\begin{align*}\n\\partial_{t}\\left( \\partial_{x}^{\\beta}u\\right) & =-v\\cdot\\nabla\n_{x}\\left( \\partial_{x}^{\\beta}u\\right) -\\frac{1}{2D}\\left( M-\\frac{x\\cdot\nv}{\\left\\langle x\\right\\rangle }\\right) \\partial_{x}^{\\beta}u+L\\partial\n_{x}^{\\beta}u+K\\partial_{x}^{\\beta}\\left( wh^{(3)}\\right) \\,\\\\\n& \\quad+\\frac{1}{2D}\\sum_{\\substack{\\beta_{1}+\\beta_{2}=\\beta\\\\\\left\\vert\n\\beta_{1}\\right\\vert \\geq1}}\\binom{\\beta}{\\beta_{1}\\,\\beta_{2}}\\partial\n_{x}^{\\beta_{1}}\\left( \\frac{x}{\\left\\langle x\\right\\rangle }\\right) \\cdot\nv\\partial_{x}^{\\beta_{2}}u.\n\\end{align*}\nThe energy estimate gives\n\\begin{align*}\n\\frac{1}{2}\\partial_{t}\\left\\Vert \\partial_{x}^{\\beta}u\\right\\Vert _{L^{2\n}^{2} & =-\\frac{1}{2D}\\int\\left( M-\\frac{x\\cdot v}{\\left\\langle\nx\\right\\rangle }\\right) \\left\\vert \\partial_{x}^{\\beta}u\\right\\vert\n^{2}dxdv+\\int\\left( L\\partial_{x}^{\\beta}u\\right) \\partial_{x}^{\\beta\n}udxdv\\\\\n& \\quad+\\frac{1}{2D}\\int\\sum_{\\substack{\\beta_{1}+\\beta_{2}=\\beta\\\\\\left\\vert\n\\beta_{1}\\right\\vert \\geq1}}\\binom{\\beta}{\\beta_{1}\\,\\beta_{2}}\\partial\n_{x}^{\\beta_{1}}\\left( \\frac{x}{\\left\\langle x\\right\\rangle }\\right) \\cdot\nv\\partial_{x}^{\\beta_{2}}u\\partial_{x}^{\\beta}udxdv\\\\\n& \\quad+\\int\\partial_{x}^{\\beta}uK\\partial_{x}^{\\beta}\\left( wh^{(3)\n\\right) dxdv.\n\\end{align*}\nNote that $2\\gamma-2\\geq1$ if $\\gamma\\geq3\/2,$ and recall that $\\Lambda=-L+K,$\nhenc\n\\begin{align*}\n\\left\\vert \\int\\frac{x\\cdot v}{\\left\\langle x\\right\\rangle }\\left\\vert\n\\partial_{x}^{\\beta}u\\right\\vert ^{2}dxdv\\right\\vert & \\leq\\int\\left\\langle\nv\\right\\rangle ^{2\\gamma-2}\\left\\vert \\partial_{x}^{\\beta}u\\right\\vert\n^{2}dxdv\\lesssim\\int\\left( \\Lambda\\partial_{x}^{\\beta}u\\right) \\partial\n_{x}^{\\beta}udxdv\\\\\n& \\lesssim-\\int\\left( L\\partial_{x}^{\\beta}u\\right) \\partial_{x}^{\\beta\n}udxdv+\\int\\left\\vert \\partial_{x}^{\\beta}u\\right\\vert ^{2}dxdv,\n\\end{align*}\nand\n\\begin{align*}\n& \\quad\\left\\vert \\int\\sum_{\\substack{\\beta_{1}+\\beta_{2}=\\beta\\\\\\left\\vert\n\\beta_{1}\\right\\vert \\geq1}}\\binom{\\beta}{\\beta_{1}\\,\\beta_{2}}\\partial\n_{x}^{\\beta_{1}}\\left( \\frac{x}{\\left\\langle x\\right\\rangle }\\right) \\cdot\nv\\partial_{x}^{\\beta_{2}}u\\partial_{x}^{\\beta}udxdv\\right\\vert \\\\\n& \\lesssim\\int\\sum_{\\substack{\\beta_{1}+\\beta_{2}=\\beta\\\\\\left\\vert \\beta\n_{1}\\right\\vert \\geq1}}\\left\\langle v\\right\\rangle ^{2\\gamma-2}\\left\\vert\n\\partial_{x}^{\\beta_{2}}u\\partial_{x}^{\\beta}u\\right\\vert dxdv\\\\\n& \\lesssim\\int\\sum_{\\substack{\\beta_{1}+\\beta_{2}=\\beta\\\\\\left\\vert \\beta\n_{1}\\right\\vert \\geq1}}\\left( \\left( -L\\partial_{x}^{\\beta_{2}}u\\right)\n\\partial_{x}^{\\beta_{2}}u+\\left( -L\\partial_{x}^{\\beta}u\\right) \\partial\n_{x}^{\\beta}u\\right) dxdv+\\int\\sum_{\\substack{\\beta_{1}+\\beta_{2\n=\\beta\\\\\\left\\vert \\beta_{1}\\right\\vert \\geq1}}\\left( \\left\\vert \\partial\n_{x}^{\\beta_{2}}u\\right\\vert ^{2}+\\left\\vert \\partial_{x}^{\\beta}u\\right\\vert\n^{2}\\right) dxdv.\n\\end{align*}\nAlso,\n\\[\n\\left\\vert \\int\\partial_{x}^{\\beta}uK\\partial_{x}^{\\beta}\\left(\nwh^{(3)}\\right) dxdv\\right\\vert \\lesssim\\Vert\\partial_{x}^{\\beta\nu\\Vert_{_{L^{2}}}\\Vert\\partial_{x}^{\\beta}\\left( wh^{(3)}\\right)\n\\Vert_{_{L^{2}}}.\n\\]\nAfter choosing $D$ and $M$ large enough, we have\n\\[\n\\frac{d}{dt}\\Vert u\\Vert_{H_{x}^{2}L_{v}^{2}}\\lesssim\\Vert wh^{(3)\n\\Vert_{_{H_{x}^{2}L_{v}^{2}}}\\,.\n\\]\nHence, it follows from (\\ref{energy 1}) and (\\ref{energy 2}) that\n\\[\n\\Vert u\\Vert_{H_{x}^{2}L_{v}^{2}}\\left( t\\right) \\lesssim\\int_{0}^{t}\\Vert\nwh^{(3)}\\Vert_{_{H_{x}^{2}L_{v}^{2}}}\\,\\left( s\\right) ds\\lesssim\\Vert\nf_{0}\\Vert_{L^{2}(\\mu)}\\,.\n\\]\n\n\\end{proof}\n\nNote that $w(x,t)\\geq e^{\\frac{\\left\\langle x\\right\\rangle +2Mt}{8D}}$ if\n$\\left\\langle x\\right\\rangle \\geq2Mt$, hence for $\\gamma\\geq3\/2$, the Sobolev\ninequality implie\n\\begin{align*}\ne^{\\frac{\\left\\langle x\\right\\rangle +2Mt}{8D}}|f|_{L_{v}^{2}} & \\leq\n\\sum_{j=0}^{3}\\left\\vert wh^{\\left( j\\right) }\\right\\vert _{L_{v}^{2\n}+\\left\\vert w\\mathcal{R}^{(3)}\\right\\vert _{L_{v}^{2}}\\\\\n& \\lesssim\\sum_{j=0}^{3}\\left\\Vert wh^{\\left( j\\right) }\\right\\Vert\n_{H_{x}^{2}L_{v}^{2}}^{3\/4}\\left\\Vert wh^{\\left( j\\right) }\\right\\Vert\n_{L^{2}}^{1\/4}+\\left\\Vert w\\mathcal{R}^{\\left( 3\\right) }\\right\\Vert\n_{H_{x}^{2}L_{v}^{2}}^{3\/4}\\left\\Vert w\\mathcal{R}^{\\left( 3\\right)\n}\\right\\Vert _{L^{2}}^{1\/4}\\\\\n& \\lesssim\\left( t^{-9\/4}+1\\right) \\Vert f_{0}\\Vert_{L^{2}(\\mu)}\\\\\n& \\lesssim\\left( t^{-9\/4}+1\\right) \\Vert f_{0}\\Vert_{L^{2}}\\,.\n\\end{align*}\nThe last inequality is due to the compact support assumption of the initial data.\n\n\\begin{theorem}\n[Space-like region for $\\gamma\\geq3\/2$\n\\label{space-like region for gamma greater or equal to 3\/2} Let $\\gamma\n\\geq3\/2$ and let $f$ be the solution to equation (\\ref{in.1.c}). Assume that\nthe initial condition $f_{0}$ has compact support in the $x$ variable and is\nbounded in $L_{v}^{2}$. Then there exists a large positive constant $M$ such\nthat if $\\left\\langle x\\right\\rangle >2Mt$, we have\n\\[\n\\left\\vert f\\right\\vert _{L_{v}^{2}}\\lesssim(1+t^{-9\/4})e^{-C\\left(\n\\left\\langle x\\right\\rangle +t\\right) }\\Vert f_{0}\\Vert_{L^{2}}\\,,\n\\]\nhere $C=C(M)$ is a positive constant.\n\\end{theorem}\n\n\\subsection{The case $0<\\gamma<3\/2$: Subexponential decay}\n\nIf $0<\\gamma<3\/2$, we consider the weight functions\n\\[\nw(t,x,v)=e^{\\frac{\\alpha\\rho(t,x,v)}{2}}\\,,\\quad\\mu(x,v)=e^{\\alpha c(x,v)}\\,,\n\\]\nwhere\n\\begin{align*}\n\\rho(t,x,v) & =5\\left( \\delta(\\left\\langle x\\right\\rangle -Mt)\\right)\n^{\\frac{\\gamma}{3-\\gamma}}\\left( 1-\\chi\\left( \\delta\\left( \\left\\langle\nx\\right\\rangle -Mt\\right) \\left\\langle v\\right\\rangle ^{\\gamma-3}\\right)\n\\right) \\\\\n& \\quad+\\left[ \\left( 1-\\chi\\left( \\delta\\left( \\left\\langle\nx\\right\\rangle -Mt\\right) \\left\\langle v\\right\\rangle ^{\\gamma-3}\\right)\n\\right) \\delta(\\left\\langle x\\right\\rangle -Mt)\\left\\langle v\\right\\rangle\n^{2\\gamma-3}+3\\left\\langle v\\right\\rangle ^{\\gamma}\\right] \\chi\\left(\n\\delta\\left( \\left\\langle x\\right\\rangle -Mt\\right) \\left\\langle\nv\\right\\rangle ^{\\gamma-3}\\right) \\,,\n\\end{align*}\nand\n\\begin{align*}\nc(x,v) & =5\\left( \\delta\\left\\langle x\\right\\rangle \\right) ^{\\frac\n{\\gamma}{3-\\gamma}}\\left( 1-\\chi\\left( \\delta\\left\\langle x\\right\\rangle\n\\left\\langle v\\right\\rangle ^{\\gamma-3}\\right) \\right) \\\\\n& \\quad+\\left[ \\left( 1-\\chi\\left( \\delta\\left\\langle x\\right\\rangle\n\\left\\langle v\\right\\rangle ^{\\gamma-3}\\right) \\right) \\delta\\left\\langle\nx\\right\\rangle \\left\\langle v\\right\\rangle ^{2\\gamma-3}+3\\left\\langle\nv\\right\\rangle ^{\\gamma}\\right] \\chi\\left( \\delta\\left\\langle x\\right\\rangle\n\\left\\langle v\\right\\rangle ^{\\gamma-3}\\right) \\,.\n\\end{align*}\nHere $M$ is a large positive constant, $\\delta,$ $\\alpha$ are small positive\nconstants; all of them will be chosen later. We introduce the following\nspace-velocity decomposition:\n\\[\nH_{+}=\\{(x,v):[\\delta(\\left\\langle x\\right\\rangle -Mt)]\\geq2\\left\\langle\nv\\right\\rangle ^{3-\\gamma}\\}\\,,\n\\\n\\[\nH_{0}=\\{(x,v):\\left\\langle v\\right\\rangle ^{3-\\gamma}<[\\delta(\\left\\langle\nx\\right\\rangle -Mt)]<2\\left\\langle v\\right\\rangle ^{3-\\gamma}\\}\\,,\n\\]\nand\n\\[\nH_{-}=\\{(x,v):[\\delta(\\left\\langle x\\right\\rangle -Mt)]\\leq\\left\\langle\nv\\right\\rangle ^{3-\\gamma}\\}\\,.\n\\]\n\n\n\\begin{proposition}\n\\label{weig_2}Consider the weight functions\n\\[\nw(t,x,v)=e^{\\frac{\\alpha\\rho(t,x,v)}{2}}\\,\\ \\text{\\ \\ and}\\ \\ \\ \\ \\mu\n(x,v)=e^{\\alpha c(x,v)}\\,,\n\\]\nwhere $\\alpha>0$ is sufficiently small with $\\alpha\\gamma<1\/20.$ Then\n\n\\noindent\\textrm{(i) }For $0\\leq j\\leq3,$\\textrm{ \n\\[\n\\Vert wh^{(j)}\\Vert_{H_{x}^{2}L_{v}^{2}}\\lesssim t^{-3+j}\\Vert f_{0\n\\Vert_{L^{2}(\\mu)},\\ \\ \\ \\ \\ 01.\n\\]\n\n\n\\noindent\\textrm{(ii)} \\textrm{ }For $1\\leq\\gamma<3\/2$,\n\\[\n\\Vert w\\mathcal{R}^{(3)}\\Vert_{H_{x}^{2}L_{v}^{2}}\\lesssim t(1+t)\\Vert\nf_{0}\\Vert_{L^{2}(\\mu)}\\,,\\,\\ \\ \\ t>0,\n\\]\nand for $0<\\gamma<1$,\n\\[\n\\Vert w\\mathcal{R}^{(3)}\\Vert_{H_{x}^{2}L_{v}^{2}}\\lesssim t(1+t^{4})\\Vert\nf_{0}\\Vert_{L^{2}(\\mu)}\\,,\\,\\ \\ \\ t>0.\n\\]\n\n\\end{proposition}\n\n\\begin{proof}\nIt is similar to Proposition \\ref{weig_1} that the weighted energy estimate of\nthe wave parts $h^{(j)}$ is a consequence of Lemma \\ref{second-der} in virtue\nof $\\rho(t,x,v)$ being non-increasing in $t$ and $\\rho(0,x,v)=c(x,v)$.\n\nWe shall focus on the weighted energy estimate for the remainder part\n$\\mathcal{R}^{(3)}$, $t>0$. We want to show that for $1\\leq\\gamma<3\/2$,\n\\[\n\\Vert w\\mathcal{R}^{(3)}\\Vert_{H_{x}^{2}L_{v}^{2}}\\lesssim t(1+t)\\Vert\nf_{0}\\Vert_{L^{2}(\\mu)}\\,,\\,\\ \\ \\ t>0,\n\\]\nand for $0<\\gamma<1$,\n\\[\n\\Vert w\\mathcal{R}^{(3)}\\Vert_{H_{x}^{2}L_{v}^{2}}\\lesssim t(1+t^{4}))\\Vert\nf_{0}\\Vert_{L^{2}(\\mu)}\\,,\\,\\ \\ \\ t>0.\n\\]\n\n\nLet $u=w\\mathcal{R}^{(3)}=e^{\\frac{\\alpha\\rho}{2}}\\mathcal{R}^{(3)}$, and then\n$\\partial_{x}^{\\beta}u,$ where $\\beta$ is a multi-index, solves the equatio\n\\begin{align*}\n\\partial_{t}\\left( \\partial_{x}^{\\beta}u\\right) & =-v\\cdot\\nabla\n_{x}\\left( \\partial_{x}^{\\beta}u\\right) +\\frac{\\alpha}{2}(\\partial_{t\n\\rho+v\\cdot\\nabla_{x}\\rho)\\partial_{x}^{\\beta}u+e^{\\frac{\\alpha\\rho}{2\n}L\\left( e^{-\\frac{\\alpha\\rho}{2}}\\partial_{x}^{\\beta}u\\right) \\\\\n& \\quad+\\frac{\\alpha}{2}\\sum_{\\substack{\\beta_{1}+\\beta_{2}=\\beta\\\\\\left\\vert\n\\beta_{1}\\right\\vert \\geq1}}\\binom{\\beta}{\\beta_{1}\\,\\beta_{2}}(\\partial\n_{t}\\partial_{x}^{\\beta_{1}}\\rho+v\\cdot\\nabla_{x}\\partial_{x}^{\\beta_{1}\n\\rho)\\partial_{x}^{\\beta_{2}}u\\\\\n& \\quad+\\sum_{\\substack{\\beta_{1}+\\beta_{2}+\\beta_{3}=\\beta\\\\\\left\\vert\n\\beta_{3}\\right\\vert <\\left\\vert \\beta\\right\\vert }}\\binom{\\beta}{\\beta\n_{1}\\,\\beta_{2}\\,\\beta_{3}}(\\partial_{x}^{\\beta_{1}}e^{\\frac{\\alpha\\rho}{2\n})L\\left( (\\partial_{x}^{\\beta_{2}}e^{-\\frac{\\alpha\\rho}{2}})\\partial\n_{x}^{\\beta_{3}}u\\right) \\\\\n& \\quad+K\\partial_{x}^{\\beta}\\left( wh^{(3)}\\right) .\n\\end{align*}\n\n\nThe energy estimate gives\n\\[\n\\begin{aligned} & \\quad \\frac{1}{2}\\frac{d}{dt}\\left\\Vert \\partial_{x}^{\\beta}u\\right\\Vert _{L^{2}}^{2}\\\\ & =\\int_{{\\mathbb{R}}^{3}}\\left\\langle e^{\\frac{\\alpha \\rho}{2}}L\\left( e^{-\\frac{\\alpha\\rho}{2}}\\partial_{x}^{\\beta}u\\right) ,\\partial_{x}^{\\beta}u\\right\\rangle _{v}dx+\\frac{\\alpha}{2}\\int_{{\\mathbb{R}}^{3}}\\left\\langle (\\partial_{t}\\rho+v\\cdot\\nabla_{x}\\rho)\\partial_{x}^{\\beta}u,\\partial_{x}^{\\beta }u\\right\\rangle _{v}dx\\\\ & \\quad+\\frac{\\alpha}{2}\\sum_{\\substack{\\beta_{1}+\\beta_{2} =\\beta \\\\ \\left\\vert \\beta_{1}\\right\\vert \\geq1} }\\binom{\\beta}{\\beta_{1} \\,\\beta_{2}}\\int_{{\\mathbb{R}}^{3}}\\left\\langle (\\partial_{t}\\partial_{x}^{\\beta_{1}}\\rho+v\\cdot \\nabla_{x}\\partial_{x}^{\\beta_{1}}\\rho)\\partial_{x}^{\\beta_{2}}u,\\partial _{x}^{\\beta}u\\right\\rangle _{v}dx\\\\ & \\quad +\\sum_{\\substack{\\beta_{1}+\\beta_{2}+\\beta_{3} =\\beta \\\\ \\left\\vert \\beta_{3}\\right\\vert <|\\beta|} }\\binom{\\beta}{\\beta_{1} \\,\\beta_{2}\\, \\beta_{3}}\\int_{{\\mathbb{R}}^{3}}\\left\\langle (\\partial _{x}^{\\beta_{1}}e^{\\frac{\\alpha\\rho}{2}})L\\left( (\\partial_{x}^{\\beta_{2}}e^{-\\frac{\\alpha\\rho}{2}})\\partial_{x}^{\\beta_{3}}u\\right),\\partial _{x}^{\\beta}u\\right\\rangle _{v}dx\\\\ & \\quad + \\int_{{\\mathbb{R}}^{3}}\\left\\langle K\\partial_{x}^{\\beta}\\left( wh^{(3)}\\right) ,\\partial_{x}^{\\beta}u\\right\\rangle _{v}dx\\\\ &\\quad := (I_1)+(I_2)+(I_3)+(I_4)+(I_5)\\,. \\end{aligned}\n\\]\nWe shall estimate $(I_{i})\\,,\\ i=1,\\ldots,5,$ term by term. \\medskip\n\nFor $(I_{1})$, it is easy to see that\n\\[\n\\left\\langle g,e^{\\frac{\\alpha\\rho}{2}}L\\left( e^{-\\frac{\\alpha\\rho}{2\n}g\\right) \\right\\rangle _{v}=\\left\\langle g,e^{-\\frac{\\alpha\\rho}{2}}L\\left(\ne^{\\frac{\\alpha\\rho}{2}}g\\right) \\right\\rangle _{v}=\\left\\langle\ng,Lg\\right\\rangle _{v}+\\frac{\\alpha^{2}}{4}\\left\\langle g^{2},\\left\\vert\n\\nabla_{v}\\rho\\right\\vert ^{2}\\right\\rangle _{v}\\,.\n\\]\nIn addition, direct calculation gives\n\\begin{align*}\n\\nabla_{v}\\rho & =\\left[ (\\gamma-3)(1-2\\chi)\\delta(\\left\\langle\nx\\right\\rangle -Mt)\\left\\langle v\\right\\rangle ^{2\\gamma-3}+3(\\gamma\n-3)\\left\\langle v\\right\\rangle ^{\\gamma}-5(\\gamma-3)\\left( \\delta\n(\\left\\langle x\\right\\rangle -Mt)\\right) ^{\\frac{\\gamma}{3-\\gamma}}\\right] \\\\\n& \\quad\\times\\left[ \\delta(\\left\\langle x\\right\\rangle -Mt)\\left\\langle\nv\\right\\rangle ^{\\gamma-4}\\right] \\frac{v}{\\left\\langle v\\right\\rangle \n\\chi^{\\prime}\\\\\n& \\quad+\\left[ (2\\gamma-3)\\delta(\\left\\langle x\\right\\rangle\n-Mt)\\left\\langle v\\right\\rangle ^{2\\gamma-4}\\right] \\frac{v}{\\left\\langle\nv\\right\\rangle }(1-\\chi)\\chi+3\\gamma\\left\\langle v\\right\\rangle ^{\\gamma\n-1}\\frac{v}{\\left\\langle v\\right\\rangle }\\chi\\,.\n\\end{align*}\nThis implies\n\\[\n|\\nabla_{v}\\rho|\\lesssim\\left\\langle v\\right\\rangle ^{\\gamma-1}\\quad\n\\text{on}\\quad H_{0}\\,\\cup H_{-}\\,,\n\\]\nand\n\\[\n\\nabla_{v}\\rho=0\\quad\\text{on}\\quad H_{+}\\,.\n\\]\nTherefore,\n\\begin{equation}\n\\label{formula-I1}\\begin{aligned} (I_1) &=\\int_{{\\mathbb{R}}^{3}}\\left\\langle e^{\\frac{\\alpha\\rho}{2}}L\\left( e^{-\\frac{\\alpha\\rho}{2}}\\partial_{x}^{\\beta}u\\right) ,\\partial_{x}^{\\beta }u\\right\\rangle _{v}dx\\\\ &\\leq-\\left( \\nu_{0}-\\frac{\\alpha^{2}C}{4}\\right) \\int_{{\\mathbb{R}}^{3}}\\left\\vert \\mathrm{P}_{1}\\partial_{x}^{\\beta }u\\right\\vert _{L_{\\sigma}^{2}}^{2}dx+\\frac{\\alpha^{2}C}{4}\\int_{H_{0}\\cup H_{-}}\\left\\vert \\mathrm{P}_{0}\\partial_{x}^{\\beta}u\\right\\vert ^{2}dxdv, \\end{aligned}\n\\end{equation}\nfor some constant $\\nu_{0}>0.$ \\medskip\n\nFor $(I_{2})$ and $(I_{3})$, we need the estimates of derivatives of\n$\\rho(t,x,v)$. Direct computation give\n\\begin{align*}\n\\partial_{t}\\rho & =-\\delta M\\left\\langle v\\right\\rangle ^{2\\gamma-3}\\left(\n\\frac{5\\gamma}{3-\\gamma}\\left[ \\delta(\\left\\langle x\\right\\rangle\n-Mt)\\left\\langle v\\right\\rangle ^{\\gamma-3}\\right] ^{\\frac{2\\gamma\n-3}{3-\\gamma}}\\left( 1-\\chi\\right) +\\chi(1-\\chi)\\right) \\\\\n& \\quad+\\delta M\\left( 5\\left[ \\delta(\\left\\langle x\\right\\rangle\n-Mt)\\left\\langle v\\right\\rangle ^{\\gamma-3}\\right] ^{\\frac{\\gamma}{3-\\gamma\n}-(1-2\\chi)\\left[ \\delta(\\left\\langle x\\right\\rangle -Mt)\\left\\langle\nv\\right\\rangle ^{\\gamma-3}\\right] -3\\right) \\left\\langle v\\right\\rangle\n^{2\\gamma-3}\\chi^{\\prime}\\,\\leq0,\n\\end{align*}\n(the constants $5$ and $3$ are chosen artificially such that the quantity in\nthe latter bracket is nonnegative on $H_{0}$) and,\n\\begin{align*}\n\\nabla_{x}\\rho & =\\delta\\left( \\nabla_{x}\\left\\langle x\\right\\rangle\n\\right) \\left\\langle v\\right\\rangle ^{2\\gamma-3}\\left( \\frac{5\\gamma\n}{3-\\gamma}\\left[ \\delta(\\left\\langle x\\right\\rangle -Mt)\\left\\langle\nv\\right\\rangle ^{\\gamma-3}\\right] ^{\\frac{2\\gamma-3}{3-\\gamma}}\\left(\n1-\\chi\\right) +\\chi(1-\\chi)\\right) \\\\\n& \\quad-\\delta\\left( \\nabla_{x}\\left\\langle x\\right\\rangle \\right) \\left(\n5\\left[ \\delta(\\left\\langle x\\right\\rangle -Mt)\\left\\langle v\\right\\rangle\n^{\\gamma-3}\\right] ^{\\frac{\\gamma}{3-\\gamma}}-(1-2\\chi)\\left[ \\delta\n(\\left\\langle x\\right\\rangle -Mt)\\left\\langle v\\right\\rangle ^{\\gamma\n-3}\\right] -3\\right) \\left\\langle v\\right\\rangle ^{2\\gamma-3}\\chi^{\\prime\n}\\,,\n\\end{align*}\nso\n\\[\n\\partial_{t}\\rho=v\\cdot\\nabla_{x}\\rho=0\\quad\\text{on}\\ H_{-}\\,\\text{,\n\\\n\\[\n|\\partial_{t}\\rho|\\lesssim\\delta M\\left\\langle v\\right\\rangle ^{2\\gamma\n-3},\\ \\ \\ \\ \\ |v\\cdot\\nabla_{x}\\rho|\\lesssim\\delta\\left\\langle v\\right\\rangle\n^{2\\gamma-2}\\ \\quad\\text{on\\ }H_{0}\\text{\\thinspace,}\\ \\ \\\n\\\n\\[\n\\partial_{t}\\rho=-\\frac{5\\delta M\\gamma}{3-\\gamma}\\left[ \\delta(\\left\\langle\nx\\right\\rangle -Mt)\\right] ^{\\frac{2\\gamma-3}{3-\\gamma}},\\quad v\\cdot\n\\nabla_{x}\\rho=\\frac{5\\delta\\gamma}{3-\\gamma}\\frac{v\\cdot x}{\\left\\langle\nx\\right\\rangle }\\left[ \\delta(\\left\\langle x\\right\\rangle -Mt)\\right]\n^{\\frac{2\\gamma-3}{3-\\gamma}}\\ \\ \\ \\text{on\\ }H_{+}\\text{\\thinspace.\n\\]\nFurthermore, we can also obtain that for $\\left\\vert \\beta_{1}\\right\\vert\n\\geq1,$\n\\[\n\\partial_{t}\\partial_{x}^{\\beta_{1}}\\rho=\\nabla_{x}\\partial_{x}^{\\beta_{1\n}\\rho=0\\ \\ \\ \\ \\text{on\\ }H_{-}\\text{,\n\\\n\\[\n\\left\\vert \\partial_{t}\\partial_{x}^{\\beta_{1}}\\rho\\right\\vert \\lesssim\n\\delta^{2}M\\left\\langle v\\right\\rangle ^{\\gamma+\\left( \\left\\vert \\beta\n_{1}\\right\\vert +1\\right) \\left( \\gamma-3\\right) },\\ \\ \\ \\ \\left\\vert\n\\nabla_{x}\\partial_{x}^{\\beta_{1}}\\rho\\right\\vert \\lesssim\\delta\n^{2}\\left\\langle v\\right\\rangle ^{\\gamma+\\left( \\left\\vert \\beta\n_{1}\\right\\vert +1\\right) \\left( \\gamma-3\\right) }\\ \\ \\ \\ \\text{on\\ \nH_{0}\\cup H_{+}\\text{.\n\\]\nFrom these, there exist constants $C>0$ and $C^{\\prime}>0$ such that\n\\begin{equation}\n\\label{formula-I2-1}\\begin{aligned} \\alpha\\left\\vert \\int_{{\\mathbb{R}}^{3}}\\left\\langle v\\cdot\\nabla_{x}\\rho\\partial_{x}^{\\beta}u,\\partial_{x}^{\\beta}u\\right\\rangle _{v}dx\\right\\vert & \\leq \\alpha\\delta C\\left( \\int_{\\mathbb{R}^{3}}|\\left\\langle v\\right\\rangle ^{\\gamma-1}\\mathrm{P}_{1}\\partial_{x}^{\\beta}u|_{L_{v}^{2}}^{2}dx+\\int_{H_{0}}|\\mathrm{P}_{0}\\partial_{x}^{\\beta}u|^{2}dxdv\\right. \\\\ & \\qquad\\left.+\\int_{H_{+}}\\left[ \\delta(\\left\\langle x\\right\\rangle -Mt)\\right] ^{\\frac{2\\gamma-3}{3-\\gamma}}|\\mathrm{P}_{0}\\partial_{x}^{\\beta}u|^{2}dxdv\\right)\\, , \\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\label{formula-I2-2}\\begin{aligned} \\alpha\\int_{\\mathbb{R}^{3}}\\left\\langle (\\partial_{t}\\rho)\\partial _{x}^{\\beta}u,\\partial_{x}^{\\beta}u\\right\\rangle _{v}dx & \\leq -\\alpha\\delta MC^{\\prime}\\int_{H_{+}}\\left[ \\delta(\\left\\langle x\\right\\rangle -Mt)\\right] ^{\\frac{2\\gamma-3}{3-\\gamma}}|\\mathrm{P}_{0}\\partial_{x}^{\\beta}u|^{2}dxdv\\\\ & \\quad + \\alpha\\delta MC\\left( \\int_{\\mathbb{R}^{3}}|\\left\\langle v\\right\\rangle ^{\\gamma-1}\\mathrm{P}_{1}\\partial_{x}^{\\beta}u|_{L_{v}^{2}}^{2}dx+\\int_{H_{0}}|\\mathrm{P}_{0}\\partial_{x}^{\\beta}u|^{2}dxdv\\right)\\, , \\end{aligned}\n\\end{equation}\nand for $\\left\\vert \\beta_{1}\\right\\vert \\geq1,\n\\begin{equation}\n\\label{formula-I3}\\begin{aligned} &\\quad \\frac{\\alpha}{2}\\left\\vert \\int_{\\mathbb{R}^{3}}\\left\\langle (\\partial _{t}\\partial_{x}^{\\beta_{1}}\\rho+v\\cdot\\nabla_{x}\\partial_{x}^{\\beta_{1}}\\rho)\\partial_{x}^{\\beta_{2}}u,\\partial_{x}^{\\beta}u\\right\\rangle _{v}dx\\right\\vert \\\\ & \\leq\\frac{\\alpha\\delta^{2}M}{2}C\\left[ \\int_{\\mathbb{R}^{3}} \\left( \\left|\\left\\langle v\\right\\rangle ^{\\gamma-1}\\mathrm{P}_{1}\\partial_{x}^{\\beta_{2}}u\\right|_{L_{v}^{2}}^{2}+|\\left\\langle v\\right\\rangle ^{\\gamma-1}\\mathrm{P}_{1}\\partial _{x}^{\\beta}u|_{L_{v}^{2}}^{2}\\right) dx \\right. \\\\ &\\qquad +\\int_{H_{0}}\\left( \\left\\vert \\mathrm{P}_{0}\\partial_{x}^{\\beta_{2}}u\\right\\vert ^{2}+\\left\\vert \\mathrm{P}_{0}\\partial_{x}^{\\beta}u\\right\\vert ^{2}\\right) dxdv\\\\ & \\qquad\\left. +\\int_{H_{+}}\\left[ \\delta(\\left\\langle x\\right\\rangle -Mt)\\right] ^{\\frac{2\\gamma-3}{3-\\gamma}}\\left( \\left\\vert \\mathrm{P}_{0}\\partial _{x}^{\\beta_{2}}u\\right\\vert ^{2}+\\left\\vert \\mathrm{P}_{0}\\partial_{x}^{\\beta}u\\right\\vert ^{2}\\right) dxdv \\right] \\,. \\end{aligned}\n\\end{equation}\n\n\nAs for $(I_{4})$, $|\\beta_{1}|+|\\beta_{2}|\\geq1$, we have\n\\begin{equation}\n\\label{formula-I4}\\begin{aligned} &\\quad \\left| \\int_{{\\mathbb{R}}^{3}}\\left\\langle (\\partial _{x}^{\\beta_{1}}e^{\\frac{\\alpha\\rho}{2}})L\\left( (\\partial_{x}^{\\beta_{2}}e^{-\\frac{\\alpha\\rho}{2}})\\partial_{x}^{\\beta_{3}}u\\right),\\partial _{x}^{\\beta}u\\right\\rangle _{v}dx \\right|\\\\ & \\leq\\frac{\\alpha\\delta C}{2}\\left[ \\int_{\\mathbb{R}^{3}}\\left( \\left\\vert \\mathrm{P}_{1}\\partial_{x}^{\\beta_{3}}u\\right\\vert _{L_{\\sigma}^{2}}^{2}+\\left\\vert \\mathrm{P}_{1}\\partial_{x}^{\\beta}u\\right\\vert _{L_{\\sigma }^{2}}^{2}\\right) dx+\\int_{H_{0}}\\left\\vert \\mathrm{P}_{0}\\partial_{x}^{\\beta_{3}}u\\right\\vert ^{2}+\\left\\vert \\mathrm{P}_{0}\\partial_{x}^{\\beta }u\\right\\vert ^{2}dxdv\\right. \\\\ & \\qquad\\left. +\\int_{H_{+}}\\left[ \\delta(\\left\\langle x\\right\\rangle -Mt)\\right] ^{\\frac{2\\gamma-3}{3-\\gamma}}\\left( \\left\\vert \\mathrm{P}_{0}\\partial_{x}^{\\beta_{3}}u\\right\\vert ^{2}+\\left\\vert \\mathrm{P}_{0}\\partial_{x}^{\\beta}u\\right\\vert ^{2}\\right) dxdv\\right] . \\end{aligned}\n\\end{equation}\n\n\nLastly,\n\\begin{equation}\n(I_{5})\\leq\\left\\vert \\int_{{\\mathbb{R}}^{3}}\\left\\langle K\\partial_{x\n^{\\beta}\\left( wh^{(3)}\\right) ,\\partial_{x}^{\\beta}u\\right\\rangle\n_{v}dx\\right\\vert \\lesssim\\left\\Vert \\partial_{x}^{\\beta}u\\right\\Vert _{L^{2\n}\\left\\Vert \\partial_{x}^{\\beta}\\left( wh^{(3)}\\right) \\right\\Vert _{L^{2}}.\n\\label{formula-I5\n\\end{equation}\n\n\nGathering the terms (\\ref{formula-I1})--(\\ref{formula-I5}), we fin\n\\begin{align*}\n\\frac{d}{dt}\\Vert u\\Vert_{H_{x}^{2}L_{v}^{2}}^{2} & \\lesssim\\Vert\nu\\Vert_{H_{x}^{2}L_{v}^{2}}\\Vert wh^{(3)}\\Vert_{H_{x}^{2}L_{v}^{2}\n\\,+\\int_{H_{0}\\cup H_{-}}\\left( \\left\\vert \\mathrm{P}_{0}\\nabla_{x\n^{2}u\\right\\vert ^{2}+\\left\\vert \\mathrm{P}_{0}\\nabla_{x}u\\right\\vert\n^{2}+\\left\\vert \\mathrm{P}_{0}u\\right\\vert ^{2}\\right) dxdv\\\\\n& \\lesssim\\Vert u\\Vert_{H_{x}^{2}L_{v}^{2}}\\Vert wh^{(3)}\\Vert_{H_{x\n^{2}L_{v}^{2}}+\\Vert u\\Vert_{H_{x}^{2}L_{v}^{2}}\\Vert\\mathcal{R}^{(3)\n\\Vert_{H_{x}^{2}L_{v}^{2}}\\\\\n& \\lesssim\\Vert u\\Vert_{H_{x}^{2}L_{v}^{2}}\\left( \\Vert h^{(3)}\\Vert\n_{H_{x}^{2}L_{v}^{2}\\left( \\mu\\right) }+\\Vert\\mathcal{R}^{(3)}\\Vert\n_{H_{x}^{2}L_{v}^{2}}\\right) ,\n\\end{align*}\nafter choosing $\\delta$, $\\alpha>0$ small and $M$ large enough with\n$\\alpha\\gamma<1\/20$. Hence, it follows from Lemmas \\ref{initial-sing} and\n\\ref{second-der} that for $1\\leq\\gamma<3\/2$,\n\\[\n\\Vert w\\mathcal{R}^{(3)}\\Vert_{H_{x}^{2}L_{v}^{2}}=\\Vert u\\Vert_{H_{x\n^{2}L_{v}^{2}}\\lesssim t(1+t)\\left\\Vert f_{0}\\right\\Vert _{L^{2}\\left(\n\\mu\\right) },\n\\]\nand for $0<\\gamma<1,\n\\[\n\\Vert w\\mathcal{R}^{(3)}\\Vert_{H_{x}^{2}L_{v}^{2}}=\\Vert u\\Vert_{H_{x\n^{2}L_{v}^{2}}\\lesssim t(1+t^{4})\\left\\Vert f_{0}\\right\\Vert _{L^{2}\\left(\n\\mu\\right) }.\n\\]\nThis completes the proof of the proposition.\n\\end{proof}\n\nObserve that for $\\left\\langle x\\right\\rangle >2Mt$,\n\\[\n\\rho(t,x,v)\\gtrsim\\left( \\delta(\\left\\langle x\\right\\rangle -Mt)\\right)\n^{\\frac{\\gamma}{3-\\gamma}}.\n\\]\nand\n\\[\n\\left\\langle x\\right\\rangle -Mt>\\frac{\\left\\langle x\\right\\rangle }{3\n+\\frac{Mt}{3}\\,.\n\\]\nThe Sobolev inequality immediately gets\n\n\\begin{theorem}\n[Space-like region for $0<\\gamma<3\/2$\n\\label{space-like region for 00$ small\nenough. Then there exists a positive constant $C=C(M)$ such that for\n$\\left\\langle x\\right\\rangle \\geq2Mt$,\n\\[\n\\left\\vert f\\right\\vert _{L_{v}^{2}}\\lesssim(1+t^{-9\/4})e^{-C(\\left\\langle\nx\\right\\rangle +t)^{\\frac{\\gamma}{3-\\gamma}}}\\Vert f_{0}\\Vert_{L^{2\n(e^{4\\alpha\\left\\langle v\\right\\rangle ^{\\gamma}})}\\,.\n\\]\n\n\\end{theorem}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Basic concepts}\n\\textit{Min-entanglement entropy---}\nVarious measures of the entanglement entropy exist depending on the partition of the system (here, we consider only bi-partite entanglement entropies) and on the information it carries.\\cite{vedral1998entanglement, vedral1997quantifying, horodecki2001entanglement, eisert1999comparison}. A generally accepted and widely used entanglement measure for bi-partite systems is the quantum Renyi entropy\\cite{eisert2008area, horodecki2009quantum}\n\\begin{equation}\\label{eq:renyi_entropy}\nS_\\alpha = \\frac{1}{1-\\alpha}\\log \\left(\\Tr \\hat{\\rho}^\\alpha\\right),\n\\end{equation} \nwhere $\\hat{\\rho}$ is the density matrix of the bi-partite system and the trace is performed over the subsystem. This entanglement measure parametrises the whole family of measures depending on the order parameter $\\alpha\\geq 0$ and $\\alpha\\neq 1$. \n\nThe limit $\\alpha \\rightarrow \\infty$ defines a very informative and useful entanglement measure which is called \\textit{min-entropy}.\\cite{konig2009operational, smith2011quantifying, espinoza2013min} It defines a lower bound on entanglement entropy that is equal to the maximal eigenvalue of the reduced density matrix. This kind of entanglement entropy has a deeper physical meaning, namely, it represents the informational content that is most likely to come out during the random experiment.\\cite{konig2009operational} There is also an interpretation of the min entanglement entropy as a single-copy distillation entropy.\\cite{horodecki2001entanglement}\n\nScaling of the entanglement entropy with the subsystem size can characterize the macroscopical fundamental properties of the system such as criticality, topological properties, the measure of quantum chaos in the system, etc. The most typical scaling is a so-called \\textit{volume law}, which is inherent to the thermal states, excited states and quantum chaotical systems. A much less common case is the \\textit{area law} scaling which is typical of ground states. Usually, it is found in the literature that the entanglement entropy satisfies an area law if $S(\\rho_I) = \\mathcal{O}(s(I))$, where $s(I)$ is the length of the boundary of the subsystem $I$. \\cite{eisert2008area}\n\nOne can expect that the addition of weak interactions to the system will not qualitatively influence the leading order of the scaling law. It can, however, introduce a pre-factor and\/or give rise to the sub-leading terms with a different scaling. The current work aims to check this conjecture. \n\n\\textit{Flow equation---}\nThe main focus of this paper is the application of the flow equation holography method\\cite{kehrein2017flow} to the calculation of the min entanglement entropy of a weakly interacting system. The idea behing this method resembles the so-called flow equation method\\cite{wegner1994flow,glazek1993renormalization}, which is an analytical tool for the Hamiltonian diagonalization. This technique allows us to find the approximate set of eigenmodes in a perturbative way. \n\nThe flow equation method is based on the energy scale separation concept, where we iteratively reduce the ultraviolet (UV)-cutoff $\\Lambda_{\\text{RG}}$ in \\textit{energy differences}, or frequencies.\\cite{kehrein2007flow, wegner1994flow, glazek1993renormalization} These energies correspond to the off-diagonal matrix elements of $\\hat{H}$. In contrast to the conventional scaling methods, here all the energy scales are preserved (see Fig. \\ref{fig:rg_vs_flow}). \n\n\\begin{figure}[htb!]\n\\centering\n \\includegraphics[width=0.4\\textwidth]{rg_vs_flow}\n \\caption{Schematic representation of the conventional scaling methods (left) and of the flow equation approach (right). In the former, one integrates out high energy degrees of freedom and arrives at an effective Hamiltonian that describes the low-energy sector. In the latter, one gets rid of the off-diagonal elements that correspond to high energy \\textit{differences} with the help of a unitary transformation.}\n \\label{fig:rg_vs_flow}\n\\end{figure}\n\nThe task then boils down to finding the generator that will diagonalize the Hamiltonian during the flow, \\textit{i.e.,} the generator that will transform the Hamiltonian in a way to satisfy \n\\begin{equation}\\label{eq:flow_generator_condition}\\frac{d}{dB} \\Tr \\left( \\hat{H}_{\\text{int}}^2\\right)\\leq 0.\n\\end{equation}\n\nThis can be solved by the Jacobi method giving the following generator\n\\begin{equation}\\label{eq:flow_generator}\n\\hat{\\eta}(B) = \\left[\\hat{H}_0(B), \\hat{H}_{\\text{int}}(B) \\right],\n\\end{equation}\nwhere $\\hat{H}_0(B)$ denotes the diagonal part of the Hamiltonian during the flow, and $ \\hat{H}_{\\text{int}}(B) $ the interaction part that we want to eliminate. Ideally, in the limit of infinitely long flow, we get diagonal Hamiltonian. In practice, however, there can be some difficulties to reach the final diagonal matrix. The reason is that the flow becomes much harder to calculate when the states are almost degenerate.\\cite{moeckel2008interaction}\nTypically, the expression can diverge starting from some moment in the flow, $B^*$.\nIn this case, we say that the Hamiltonian of such a system can be diagonalized only to \\textit{some} extent. The parameter $B^*$ also depends on the energy of the quasi-particles, \\textit{e.g.}, for some systems the diagonalization can be exact only on the Fermi surface, while away from it one gets degeneracies which indicate non-perturbative effects\nThis will be discussed in more detail in the subsection called ``Disentangling flow\" of section \\ref{sec:min_ent}.\n\n\\textit{Flow equation holography---}\nWhen it comes to an explicit calculation of the entanglement entropy, the situation is very similar to the Hamiltonian diagonalization: very little can be done analytically even for non-interacting systems.\\cite{eisert2008area, peschel2012special}\nHowever, the scientific community keeps coming up with new clever ways to bypass complications and bottlenecks, and extract the information needed.\\cite{peschel2012special}\nRecently, a new powerful method called \\textit{flow equation holography} was introduced \\cite{kehrein2017flow} that allows for a non-perturbative, analytical calculation of the min entanglement\nentropy of the general many-body system.\nThis method provides a systematic procedure which connects the entanglement properties of eigenstates of a generic quantum many-body Hamiltonian to a disentangling flow in an emergent RG-like dimension. The method is conceptually similar to the flow equation approach and uses the same RG-like flow, here, however, in a very different manner.\n\nTo be more precise, we divide our Hilbert space $\\mathcal{H}$ into two sub-spaces $\\mathcal{A}$ and $\\mathcal{B}$ which are, generally, entangled between each other (see Fig. \\ref{fig:system}). Correspondingly, the Hamiltonian of the whole system is \n\\begin{equation}\\label{eq:Hamiltonian_partition}\\hat{H} = \\hat{H}_{\\mathcal{A}}\\bigotimes\\mathds{1}_{\\mathcal{B}}+ \\mathds{1}_{\\mathcal{A}}\\bigotimes\\hat{H}_{\\mathcal{B}} + g\\hat{H}_{\\text{ent}}.\n\\end{equation} \n\n The first two terms act in a non-trivial way only on their corresponding subsystems, while the part $\\hat{H}_{\\text{ent}}$ is responsible for the entanglement between two subsystems.\n The coefficient $g\\ll1$ is called \\textit{weak link parameter} that couples two subsystems.\n\\begin{figure}[h!]\n\\centering\n \\includegraphics[width=0.25\\textwidth]{system}\n \\caption{Schematic representation of a bi-partite system with a weak link, that is represented by a dotted line.}\n \\label{fig:system}\n\\end{figure}\n\nNow, in the very same way as we got rid of the non-diagonal part of the Hamiltonian in the flow equation approach, we use the unitary flow to eliminate the entangling part of the Hamiltonian $\\hat{H}_{\\text{ent}}$ here. In the end, when the flow parameter is sent to infinity $B \\rightarrow \\infty$, our Hilbert space is a direct sum of two sub-spaces that correspond to two subsystems $\\mathcal{H} = \\mathcal{H}_{\\mathcal{A}\\mathcal{B}} = \\mathcal{H}_{\\mathcal{A}}\\bigoplus\\mathcal{H}_{\\mathcal{A}} $ and the subsystems are completely decoupled. By building the analogy between the flow equation and flow holography methods, one can easily deduce the flow generator in this case \n\\begin{equation}\\label{eq:entangling_flow_generator}\n\\hat{\\eta}(B) = \\left[ \\hat{H}(B), g\\hat{H}_{\\text{ent}}(B) \\right].\n\\end{equation}\n\nThe structure of the generator implies that the flow is iterative in its nature and each step is a separate unitary transformation that eliminates the corresponding off-diagonal element labelled by some energy difference.\nWhich means that the transformation acts on sites that are further and further apart from the boundary between regions $\\mathcal{A}$ and $\\mathcal{B}$ as the flow parameter increases. A schematic representation of the process can be seen in Fig. \\ref{fig:factorization}, where, however, the flow is reversed.\n\\begin{figure}[h!]\n\\centering\n \\includegraphics[width=0.5\\textwidth]{factorisation}\n \\caption{Schematic evolution of the state during the \\textit{entangling} flow starting from the product state. The scheme tracks the largest Schmidt coefficient of the state. The figure is taken from \\cite{kehrein2017flow}.}\n \\label{fig:factorization}\n\\end{figure}\n\nBy using this crucial property, one can derive the analytical expression of the min-entropy \\cite{kehrein2017flow}\n\\begin{equation}\\label{eq:main}\nS_{\\text{min}} = -4\\int_{\\frac{1}{\\Lambda^2}}^{\\infty} BdB \\bra{\\psi}\\hat{\\eta}^2(B)\\ket{\\psi} + \\mathcal{O}(g^3).\n\\end{equation}\nHere, $\\Lambda$ corresponds to the ultra-violet cut-off, which reflects the fact that we consider the system on a lattice with a finite lattice constant $a$ and $\\ket{\\psi}$ is a ground state of the decoupled system. This is needed in order to get rid of the divergences that arise at $B=0$. \n\n\\section{Min entanglement entropy calculation}\n\\label{sec:min_ent}\n\\textit{Model---}\nLet us consider the bi-partite system ($\\mathcal{A}$ and $\\mathcal{B}$, see Fig. \\ref{fig:system}) that is filled with locally interacting fermions. These two sub-regions are weakly coupled by spin-symmetric hopping terms. For the sake of simplicity, the interaction is turned on only in the region $\\mathcal{B}$. This does not influence the final result qualitatively and affects only the numerical pre-factor. The Hamiltonian reads\n\\begin{equation*}\\label{eq:hubbard}\n\\hat{H}_\\mathcal{A} = -\\sum_i \\sum_{\\sigma} \\hat{a}^{\\dagger}_{\\sigma,i} \\hat{a}_{\\sigma,i+1} + U\\sum_i \\hat{n}^{\\uparrow}_i \\hat{n}^{\\downarrow}_i + \\text{h.c.}\n\\end{equation*}\n\\begin{equation*}\n\\hat{H}_\\mathcal{B} = -\\sum_i \\sum_{\\sigma} \\hat{b}^{\\dagger}_{\\sigma,i} \\hat{b}_{\\sigma,i+1} + \\text{h.c.}\n\\end{equation*}\n\\begin{equation}\n\\hat{H}_{\\text{ent}} = -g \\sum_{\\sigma} \\hat{a}^{\\dagger}_{\\sigma,0} \\hat{b}_{\\sigma,1} + \\text{h.c.}\n\\end{equation}\n\nFor what concerns the non-interacting part of the Hamiltonian, this would lead to the same result as for free fermions (see appendix \\ref{appendix:free}). The presence of the interacting term in the Hamiltonian $\\mathcal{A}$ creates a different situation where the entanglement of the subsystem is defined by the competition between the kinetic part, that would lead to growing entanglement entropy, and the interacting one where the entanglement is zero because the state is an insulating one.\n\n\\textit{Treatement of the interactions using the flow equation technique---}\nBefore we start the consideration of the disentangling flow, we first need to diagonalize the interacting Hamiltonian of the subsystem $\\mathcal{A}$. In this way, we will obtain the eigenmodes which are needed for the considerations that will follow, such as the calculation of the expectation value of the flow generator for the disentangling flow. The detailed calculations for this section can be found in appendix \\ref{appendix:diag}, here, we only sketch the main steps to give a general idea.\n\nSince we assumed the subsystem $\\mathcal{B}$ to be non-interacting, the corresponding fermionic degrees of freedom can be described using an eigenbasis composed of plane waves.\nThe dispersion relation is then the usual $\\epsilon_k = -2\\cos k$, where $k$ is the wave vector. The part of the Hamiltonian that acts on the subsystem $\\mathcal{B}$ can be then written as\n\\begin{equation}\n\\hat{H}_\\mathcal{B} = -\\sum_{\\sigma}\\sum_k \\varepsilon_k\\hat{b}^{\\dagger}_{\\sigma,k}\\hat{b}_{\\sigma,k} .\n\\end{equation}\n\n\nIf we naively apply the same transformation to the Fermi-Hubbard Hamiltonian of subsystem $\\mathcal{A}$, the presence of local interactions in the real space would make our Hamiltonian non-diagonal in the energy basis\n\\begin{equation}\n\\hat{H}_\\mathcal{A} = -\\sum_{\\sigma}\\sum_k \\varepsilon_k\\hat{a}^{\\dagger}_{\\sigma,k}\\hat{a}_{\\sigma,k}+ U\\sum_{\\alpha,\\beta,\\gamma,\\delta}\\hat{a}^{\\uparrow\\dagger}_{\\alpha}\\hat{a}^{\\uparrow}_{\\beta}\\hat{a}^{\\downarrow\\dagger}_{\\gamma}\\hat{a}^{\\downarrow}_{\\delta} \\delta^{\\alpha+\\gamma}_{\\beta + \\delta}.\n\\end{equation}\n\nTherefore, one has to diagonalize $\\hat{H}_\\mathcal{A}$ with the flow equation method \\cite{wegner1994flow}.\n Fermi gas in the thermodynamic limit represents a many-particle problem with an infinite number of degrees of freedom. This implies that many different energy scales contribute to the Hamiltonian. It is obvious that the interaction generates occupation in many different excited eigenstates of the interacting Hamiltonian. Therefore we implement the diagonalizing transformation such that a controlled treatment of different energy scales is possible \\cite{moeckel2008interaction}.\n This is exactly what the flow equation method can offer us.\n\nBefore to look into the technical details, one can develop an intuitive picture of the process: during the flow one-particle creation and annihilation operators transform and acquire multi-particle coating (see Fig. \\ref{fig:quasi}). Also, different spins mix between each other such that the particle with some spin $\\sigma$ becomes a multi-particle excitation of different spins. Such excitation is called \\textit{quasi-particle} and is denoted by new creation and annihilation operators ($\\hat{c}, \\hat{c}^{\\dagger}$ in the subsystem A). In the basis of these operators, our Hamiltonian has a diagonal form. However, in the same fashion as for the conventional RG-flow, not only the operational structure of the Hamiltonian changes, but also the parameters in the Hamiltonian become ``flowing\", see Eq. \\ref{eq:diagonal_H}. \n\n\\vspace{0.3cm}\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=0.5\\textwidth]{quasi}\n \\caption{The schematic representation of the multi-particle structure that the quasi-particle acquire during the diagonalization flow.}\n \\label{fig:quasi}\n\\end{figure}\n\nOne can write down a physical ansatz for the quasparticles\\cite{moeckel2008interaction}, where the functional dependence is deduced from the flow differential equations (see appendix \\ref{appendix:diag}) and get the expression for the diagonalized Hamiltonian\n\\begin{widetext}\n\\begin{align}\n\\nonumber \\hat{H} = -\\sum_{\\sigma}\\sum_k \\varepsilon_k\\hat{a}^{\\uparrow\\dagger}_{k,\\sigma}\\hat{a}^{\\uparrow}_{k,\\sigma} -\\sum_{\\sigma}\\sum_k \\varepsilon_k\\hat{b}^{\\uparrow\\dagger}_{k,\\sigma}\\hat{b}^{\\uparrow}_{k,\\sigma} &-\\sum_{\\sigma}\\sum_{l,m}\\dfrac{2}{N+1}\\sin m \\sin l \\left( h_l(\\infty)g_{l,m}(B)\\hat{a}^{\\dagger}_{l,\\sigma} \\hat{b}_{m,\\sigma} + \\right.\\\\\n&\\left.+\\sum_{p' q' p} M^l_{p'q' p}(\\infty)g_{p'q' p,l,m}(B) :\\hat{a}^{\\dagger}_{p',\\sigma}\\hat{a}^{\\dagger}_{q',-\\sigma}\\hat{a}_{p,-\\sigma}:\\hat{b}_{m,\\sigma} + h.c \\right)\n\\end{align}\nwith\n\\begin{align}\n M^l_{p'q' p}(B) &= U\\rho\\int^{\\infty}_{-\\infty} dE\\dfrac{ \\delta(E - \\varepsilon_p' - \\varepsilon_q' + \\varepsilon_p)}{(E-\\varepsilon_l)} \\left( 1 - e^{-B(E-\\varepsilon_l)^2} \\right)\\\\\n h_l(B) &= 1 - U^2\\rho^3\\int^{\\infty}_{-\\infty} dE\\dfrac{(E-\\varepsilon_F)^2}{(E-\\varepsilon_l)^2} \\left( 1 - e^{-B(E-\\varepsilon_l)^2} \\right).\n\\end{align}\n\\end{widetext}\n\nThe explicit calculation is done in appendix \\ref{appendix:diag}. It is important to emphasise that the flow function $h_l(B)$ is connected to the quasi-particle residue via $h_{k_F} = \\sqrt{Z(B)}$ and depicts the coherent overlap of the physical fermion with the related interaction-free momentum mode (quasi-particle) of the current representation.\n\n\\textit{Disentangling flow---}\nIn the previous section, we considered the flow that diagonalized the Fermi Hubbard Hamiltonian. The corresponding flow parameter $B$ was sent up to infinity in order to reach perfect diagonalization. However, as was mentioned in the introduction, for some systems such procedure may diverge which means that the corresponding Hamiltonian can be diagonalized only up to a certain extent. In order to control all the divergencies, we will keep the final value of the previous diagonalising flow parameter at some fixed (big enough in order to assume that the Hamiltonian is diagonal) value $B^*$ and proceed to the next (disentangling) flow with the corresponding parameter $B$. Then, in the end, we will see whether we can send $B^*$ having all the expressions finite. And if we can not: what are the limitations for it.\n\nIn the end, we want to see how the interactions influence the entanglement entropy, therefore, we are keeping track of all the difference with the same calculations for free fermions.\n\nKeeping in mind the commutation relations for the fermions one can work out the generator of the flow. This will, however, only give the operational form of it. In order to find the functional dependence for the $g_{l,m}(B), g_{p'q' p,l,m}(B)$ on should solve the differential equation\\cite{kehrein2017flow}\n\\begin{equation}\\label{eq:diff_eq_flow}\nd\\hat{H}\/dB = -\\left[\\hat{H}(B), \\hat{\\eta}(B)\\right]\n\\end{equation} \nobtaining\n\\begin{align}\\label{eq:solutions_g}\n g_{l,m}(B) &= g e^{-B(\\varepsilon_l- \\varepsilon_m)^2}\\\\\n g_{p'q' p,l,m}(B) &= g e^{-B(\\varepsilon_{p'} +\\varepsilon_{q'}- \\varepsilon_p-\\varepsilon_m)^2}.\n\\end{align}\nPlugging the solutions \\ref{eq:solutions_g} in the Eq. \\ref{eq:flow_generator} we obtain the full form of the flow generator\n\n\\begin{widetext}\n\\begin{align}\\label{eq:flow_fin}\n\\nonumber& \\hat{\\eta}(B) = \\sum_{\\sigma}\\sum_{m,l}\\dfrac{2g}{N+1}\\sin m \\sin l \\left[h_le^{-B(\\varepsilon_l- \\varepsilon_m)^2} \\left( \\hat{a}^{\\dagger}_{l,\\sigma}\\hat{b}_{m,\\sigma} - \\hat{a}_{l,\\sigma}\\hat{b}^{\\dagger}_{m,\\sigma}\\right)(\\varepsilon_l - \\varepsilon_m) + \\right.\\\\\n+ \\sum_{p' q' p} &\\left. M^l_{p'q' p}e^{-B(\\varepsilon_{p'} +\\varepsilon_{q'}- \\varepsilon_p-\\varepsilon_m)^2}\\left( :\\hat{a}^{\\dagger}_{p',\\sigma}\\hat{a}^{\\dagger}_{q',-\\sigma}\\hat{a}_{p,-\\sigma}\\hat{b}_{m,\\sigma}:-:\\hat{a}_{p',\\sigma}\\hat{a}^{\\dagger}_{p,-\\sigma}\\hat{a}_{q',-\\sigma} \\hat{b}^{\\dagger}_{m,\\sigma}: \\right)(\\varepsilon_{p'} +\\varepsilon_{q'}- \\varepsilon_p-\\varepsilon_m) \\right].\n\\end{align}\n\\end{widetext}\n\nFlow generator that we obtained enters the expression for the entanglement entropy in the form of the following expectation value: $\\bra{\\psi_0}\\hat{\\eta}^2\\ket{\\psi_0}$. Term $\\sim h\\times h$ will give the same contribution as in the case for the free fermions but multiplied by the quasi-particle residue. Cross-terms (the ones $\\sim h\\times M$) will not contribute to the expectation value at all as there are the unequal amount of creation and annihilation operators in them and therefore the expectation value vanishes.\n\nThe only new and ``non-trivial\" contribution to $\\langle\\hat{\\eta}^2\\rangle$ can arise from the $ M\\times M$ term, which is $\\sim U^2$, which one can work out with the help of the Wick theorem. By considering this term, we will calculate the correction $\\delta S = S_{\\text{interacting}} - ZS_{\\text{free}}.$\nThe scaling of this additional term with the subsystem size is not known and is of a fundamental interest. By inserting the corresponding part of the $\\hat{\\eta}^2$ in Eq. \\ref{eq:main} and performing the integration (see appendix \\ref{appendix:int}) we obtain\n\n\\begin{widetext}\n\\begin{align}\\label{eq:entropy_correction}\n \\delta S &= \\dfrac{6g^2U^2 \\rho}{\\pi^2\\sqrt{2\\pi}} \\left(\\dfrac{\\Lambda}{l} - \\Lambda\\right)\\int^{\\infty}_{-\\infty} d \\tilde{\\varepsilon_l} \\int_{0}^{\\infty} d\\tilde{E} \\dfrac{\\tilde{E}^2}{(\\tilde{E}+\\tilde{\\varepsilon_l})^2}\\text{erfc}(\\tilde{E}).\n\\end{align}\n\\end{widetext}\n\n\\textit{Final remarks---}\nIf one naively looks at the Eq.~\\ref{eq:entropy_correction}, they will immediately see that the expression has a pole of the second order at $-\\tilde{E} = \\tilde{\\varepsilon}_l$. The origin of this is hidden in the first diagonalization flow procedure. As was mentioned before, for some energies of the quasi-particle we can not diagonalize the Hamiltonian precisely, but only make it ``nearly\" diagonal. Therefore, in order to avoid divergences one has to try the same procedure, but cutting the diagonalization flow of $\\hat{H}_A$ at some value $B^*$. Presumably, this threshold $B^*$ can depend on the energy $\\varepsilon$. The further we go from the Fermi energy, the smaller is the lifetime of the quasi-particle $\\tau \\sim (\\varepsilon-\\varepsilon_F)^{-2}$. That is why those quasi-particles with a very small lifetime just do not manage to penetrate far enough into the other subsystem in order to entangle because they decay before doing it.\n\nIn order to perform the same calculation keeping all the divergences under control, we act iteratively: diagonalize the Hamiltonian up to some extent, then entangle only up to some energies from the Fermi level, then proceed to better diagonalization, then entangle further and so on. Indeed, if one re-considers all the calculations keeping $B^*$ finite, one would see that additional pre-factor that arises cancels the pole when $\\tilde{E} \\rightarrow -\\tilde{\\varepsilon}_l$. The regularised result has the same scaling and we leave the technical details of the analysis in appendix \\ref{appendix:int}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\n\\section{ introduction and main results}\nLet $A_n$ be an $n\\times n$ matrix with real or complex entries. The linear statistics of\neigenvalues $\\lambda_1,\\lambda_2,\\ldots, \\lambda_n$ of $A_n$ is a\nfunction of the form\n\\begin{equation} \\label{eqn:1}\n\\frac{1}{n}\\sum_{k=1}^{n}f(\\lambda_k)\n\\end{equation}\nwhere $f$ is some fixed function. The function $f$ is known as the test function. One of the interesting \nobject to study in random matrix theory is the fluctuation of linear\nstatistics of eigenvalues of random matrices. The study of fluctuation of linear statistics of eigenvalues was initiated by Arharov \\cite{arharov} in 1971 for sample covariance matrices. In 1975 Girko \\cite{girko} studied the central limit theorem (CLT) of the traces of the Wigner and sample covariance matrices using martingale techniques. In\n1982, Jonsson \\cite{jonsson} proved the \nCLT of linear eigenvalue statistics for Wishart matrices using method of moments. After that the fluctuations of\neigenvalues for various random matrices have been extensively studied by various people. \nFor new results on fluctuations of linear eigenvalue statistics of Wigner and sample covariance matrices, see \\cite{johansson1998}, \\cite{soshnikov1998tracecentral}, \\cite{bai2004clt}, \\cite{lytova2009central}, \\cite{shcherbina2011central}. For band and sparse random matrices, see \\cite{anderson2006clt}, \\cite{jana2014}, \\cite{li2013central}, \\cite{shcherbina2015} and for Toeplitz and band Toeplitz matrices, see \\cite{chatterjee2009fluctuations} and \\cite{liu2012}.\n\nIn a recent article \\cite{adhikari_saha2017}, the CLT for linear eigenvalue statistics has been established in total variation norm for circulant, symmetric circulant and reverse circulant matrices with Gaussian entries. In a subsequent article \\cite{adhikari_saha2018}, the authors extended their results for independent entries which are smooth functions of Gaussian variables. Here we consider the fluctuation problem for symmetric circulant matrices with general entries which are independent and satisfy some moment condition.\n \n\n\nA sequence is said to be an {\\it input sequence} if the matrices are constructed from the given sequence. We consider the input sequence of the form $\\{x_i: i\\geq 0\\}$\nand the symmetric circulant matrix is defined as \n$$\nSC_n=\\left(\\begin{array}{cccccc}\nx_0 & x_1 & x_2 & \\cdots & x_{2} & x_1 \\\\\nx_1 & x_0 & x_1 & \\cdots & x_{3} & x_{2}\\\\\nx_{2} & x_1 & x_0 & \\cdots & x_{4} & x_{3}\\\\\n\\vdots & \\vdots & {\\vdots} & \\ddots & {\\vdots} & \\vdots \\\\\nx_1 & x_2 & x_3 & \\cdots & x_1 & x_0\n\\end{array}\\right).\n$$\nFor $j=1,2,\\ldots, n-1$, its $(j+1)$-th row is obtained by giving its $j$-th row a right circular shift by one positions and the (i,\\;j)-th element of the matrix is $x_{\\frac{n}{2}-|\\frac{n}{2}-|i-j||}$. Also note that the symmetric circulant matrix is a Toeplitz matrix with the restriction that $x_{n-j}=x_j$.\n\n \nNow we consider linear eigenvalue statistics as defined in \\eqref{eqn:1} for $SC_n$ with test function $f(x)=x^{p}$, $p\\geq 2$. Therefore\n$$ \\sum_{k=1}^{n}f(\\lambda_k)= \\sum_{k=1}^{n}(\\lambda_k)^{p}= \\mbox{\\rm Tr}(SC_n)^{p},$$\nwhere $\\lambda_1,\\lambda_2,\\ldots,\\lambda_n$ are the eigenvalues of $SC_n$. We scale and centre $\\mbox{\\rm Tr}(SC_n)^{p}$ to study its fluctuation, and define \n\\begin{equation}\\label{eqn:SCw_p}\nw_p := \\frac{1}{\\sqrt{n}} \\bigl\\{ \\mbox{\\rm Tr}(SC_n)^{p} - \\mbox{E}[\\mbox{\\rm Tr}(SC_n)^{p}]\\bigr\\}. \n\\end{equation}\nFor a given real polynomial $$Q(x)=\\sum_{k=1}^da_kx^{k}$$ \nwith degree $d$ where $d\\geq 2$, we define \n\\begin{equation}\\label{eqn:RCw_Q}\nw_Q := \\frac{1}{\\sqrt{n}} \\bigl\\{ \\mbox{\\rm Tr}(Q(SC_n)) - \\mbox{E}[\\mbox{\\rm Tr}(Q(SC_n))]\\bigr\\}. \n\\end{equation}\n \nNote that $w_Q$ and $w_p$ depends on $n$. But we suppress $n$ to keep the notation simple. In our first result, we calculate the covariance between $w_p$ and $w_q$ as $n \\rightarrow \\infty$. \n\\begin{theorem}\\label{thm:symcircovar}\n Suppose $SC_n$ is the symmetric circulant matrix with independent input sequence $\\{\\frac{X_i}{\\sqrt n}\\}_{i\\geq 0}$ such that\n \\begin{equation}\\label{eqn:condition}\n\\mbox{E}(X_i)=0, \\mbox{E}(X_i^2)=1, \\mbox{E}(X_i^4)=\\mbox{E}(X_1^4)\\ \\mbox{and}\\ \\sup_{i\\geq 1}\\mbox{E}(|X_i|^k)=\\alpha_k<\\infty \\ \\mbox{for}\\ k\\geq 3.\n\\end{equation} \n Then for $p,q \\geq2$,\n \\begin{align} \\label{eqn:sigma_p,q}\n\t\\sigma_{p,q}&:=\\lim_{n\\to\\infty} \\mbox{\\rm Cov}\\big(w_p,w_q\\big) \\nonumber \\\\ \n\t&= \\left\\{\\begin{array}{ll} \t \n\t\t \t\\displaystyle\\frac{a_1}{2^{\\frac{p+q-4}{2}} } (\\mbox{E} X^4_1- 1) + \\sum_{r=2}^{ \\min\\{ \\frac{p}{2},\\frac{q}{2} \\} } \\frac{a_r}{2^{\\frac{p+q-4r}{2}} } \\sum_{s=0}^{2r}\\binom{2r}{s}^2 s!(2r-s)! \\ h_{2r}(s) & \\text{if}\\ p, q \\mbox{ both are even,}\\\\\\\\\n\t\t \t\\displaystyle\\sum_{r=0}^{ \\min\\{ \\frac{p-1}{2},\\frac{q-1}{2} \\} } \\frac{b_r}{2^{\\frac{p+q-4r-2}{2}} } \\sum_{s=0}^{2r+1}\\binom{2r+1}{s}^2 s!(2r+1-s)! \\ h_{2r+1}(s) \\\\\t\t \t\n\t\t\\displaystyle+ pq \\binom{p-1}{(p-1)\/2}\\binom{q-1}{(q-1)\/2} \\left((p-1)\/2\\right)! \\left((q-1)\/2\\right)!\\frac{1}{2^{(\\frac{p+q}{2}-1)}} \t & \\text{if}\\ p, q \\mbox{ both are odd,}\\\\\\\\\n\t\t\t0 & \\text{otherwise}, \t \t \n\t\t \t \\end{array}\\right.\t\t\n\t\t \t \n\t\t \t \n\\end{align}\nwhere $a_r$ and $b_r$ are appropriate constants, will be given in proof, and $h_d(s)$ is given as\n$$h_d(s)=\n \\frac{1}{(d-1)!}\\sum_{i=-\\lceil\\frac{d-s}{2}\\rceil}^{\\lfloor\\frac{s}{2}\\rfloor}\\sum_{j=0}^{2i+d-s}(-1)^q\\binom{d}{j}\\left(\\frac{2i+d-s-j}{2}\\right)^{d-1}.$$\n \\end{theorem}\nIf $p=q$ then we denote $\\sigma_{p,q}$ by $\\sigma^{2}_{p}$.\nIn our second result, we see the fluctuation of linear eigenvalue statistics of symmetric circulant matrices with polynomial test functions.\n\\begin{theorem}\\label{thm:symcirpoly} \nSuppose input entry of $SC_n$ is independent sequence $\\{\\frac{X_i}{\\sqrt n}\\}_{i\\geq 1}$ which satisfy \\ref{eqn:condition}.\nThen, as $n\\to \\infty$,\n\\begin{align*}\nw_Q \\stackrel{d}{\\longrightarrow} N(0,\\sigma_{Q}^2).\n\\end{align*}\nIn particular, for $Q(x)= x^{p}$\n\\begin{align*}\n w_p \\stackrel{d}{\\longrightarrow} N(0,\\sigma_{p}^2),\n\\end{align*}\nwhere \n$$\\sigma_{Q}^2= \\sum_{\\ell=1}^d \\sum_{k=1}^d a_{\\ell} a_k \\sigma_{\\ell,k}, \\ \\ \\sigma^2_{p}= \\sigma_{p,p}$$\nand $\\sigma_{p,q}$ is as given in (\\ref{eqn:sigma_p,q}). \n\\end{theorem}\n\n\t\n\\begin{remark}\nIn the above theorems we have considered the fluctuation of $w_p$ for $p\\geq2$. For $p=0$, \n\\begin{align*}\nw_0 = \\frac{1}{\\sqrt{n}} \\bigl\\{ \\mbox{\\rm Tr}(I) - \\mbox{E}[\\mbox{\\rm Tr}(I)]\\bigr\\} = \\frac{1}{\\sqrt{n}}[n-n] = 0\n\\end{align*}\nand hence it has no fluctuation.\nFor $p=1$,\n\\begin{align*}\nw_1 = \\frac{1}{\\sqrt{n}} \\bigl\\{ \\mbox{\\rm Tr}(SC_n) - \\mbox{E}[\\mbox{\\rm Tr}(SC_n)]\\bigr\\} = \\frac{1}{\\sqrt{n}} \\big[n \\frac{X_0}{\\sqrt n} - \\mbox{E}(n\\frac{X_0}{\\sqrt n})\\big] = X_0,\n\\end{align*}\nas $\\mbox{E}(X_0)=0$. So $w_1$ is distributed as $X_0$ and its distribution does not depend on $n$. So we ignore these two cases, for $p=0$ and $p=1$. \n\\end{remark}\t\n\t\n\nIn Section \\ref{sec:cov} we prove Theorem \\ref{thm:symcircovar}. We derived trace formula and state some results which will be used to prove Theorem \\ref{thm:symcircovar}. In Section \\ref{sec:poly} we prove Theorem \\ref{thm:symcirpoly}. We use method of moments and Wick's formula to prove Theorem \\ref{thm:symcirpoly}. \n \n\n\n\n\\section{Proof of Theorem \\ref{thm:symcircovar}}\\label{sec:cov}\n\nWe first define some notation which will be used in the proof of Theorem \\ref{thm:symcirpoly}.\n\\begin{align} \\label{def:A_p_SC}\nA_{p}&=\\{(j_1,\\ldots,j_{p})\\; : \\; \\sum_{i=1}^{p}\\epsilon_i j_i=0\\; \\mbox{(mod n)}, \\epsilon_i\\in\\{+1,-1\\}, 1\\le j_1,\\ldots,j_{p}\\le \\frac{n}{2}\\}, \\\\\n\\tilde{A}_{k}&=\\{(j_1,\\ldots,j_{k})\\; : \\; \\sum_{i=1}^{k}\\epsilon_i j_i=0\\; \\mbox{(mod } \\frac{n}{2}) \\mbox{ and } \\sum_{i=1}^{k}\\epsilon_i j_i \\neq 0\\; \\mbox{(mod }n), \\epsilon_i\\in\\{+1,-1\\}, 1\\le j_1,\\ldots,j_{k}\\le \\frac{n}{2}\\}, \\nonumber\n\\\\A_p^{(k)}&=\\{(j_1,\\ldots,j_p)\\in A_p\\; : \\; j_1+\\cdots+j_k - j_{k+1}-\\cdots -j_p=0 \\;\\mbox{ (mod $n$)}\\}, \\nonumber\n\\\\A_p'^{(k)}&=\\{(j_1,\\ldots,j_p)\\in A_p\\; : \\; j_1+\\cdots+j_k - j_{k+1}-\\cdots -j_p=0 \\;\\mbox{ (mod $n$)}, j_1\\neq \\cdots \\neq j_k \\}, \\nonumber\n\\\\A_{p,s}^{(k)}&=\\{(j_1,\\ldots,j_p)\\in A_p\\; : \\; j_1+\\cdots+j_k - j_{k+1}-\\cdots -j_p=sn\\}. \\nonumber\n\\end{align}\nIn set $A_p$ and $\\tilde{A}_p$, we collect $(j_1,\\ldots,j_{p})$ according to their multiplicity. \n\n Now we derive a convenient formula of trace for symmetric circulant matrices. First suppose $n$ is odd positive integers. We write $ n\/2$ instead of $\\lfloor n\/2\\rfloor $, as asymptotic is same as $n\\to\\infty$. Then\n\\begin{align*}\n\\mbox{\\rm Tr}(SC_n^p)&=\\sum_{\\ell=0}^{n-1}\\lambda_\\ell^p\n=\\sum_{\\ell=0}^{n-1}\\left(X_0+2\\sum_{j=1}^{n\/2}X_j\\cos(\\omega_\\ell j)\\right)^p\n\\\\&=\\sum_{k=0}^{p}\\binom{p}{k}X_0^{p-k}\\sum_{\\ell=0}^{n-1} \\left(\\sum_{j=1}^{n\/2}X_j(e^{i\\omega_\\ell j}+e^{-i\\omega_\\ell j})\\right)^{k},\n\\end{align*}\nwhere $\\omega_\\ell=\\frac{2\\pi \\ell}{n}$. Since $\\sum_{\\ell=0}^{n-1}e^{i\\omega_\\ell j}=0$ for $j\\in \\mathbb Z\\backslash \\{0\\}$, we have \n\\begin{align}\\label{trace formula_SC_odd}\n\\mbox{\\rm Tr}(SC_n^p)=n\\sum_{k=0}^{p}\\binom{p}{\\ell}X_0^{p-k}\\sum_{A_{k}} X_{j_1}X_{j_2}\\ldots X_{j_{k}},\n\\end{align}\nwhere $A_{k}$ for $k=1,\\ldots,p$ is given by\n\\begin{align*}\nA_{k}:=\\left\\{(j_1,\\ldots,j_{k})\\; : \\; \\sum_{i=1}^{k}\\epsilon_i j_i=0\\; \\mbox{(mod n)}, \\epsilon_i\\in\\{+1,-1\\}, 1\\le j_1,\\ldots,j_{k}\\le \\frac{n}{2}\\right\\}\n\\end{align*} \nand $A_0$ is an empty set with the understanding that the contribution from the sum corresponding to $A_0$ is 1. Note, in $A_{k}$, $(j_1,\\ldots,j_{k})$ are collected according to their multiplicity. \n\n Now suppose $n$ is even positive integers. We write $ n\/2$ instead of $ n\/2-1$, as asymptotic is same as $n\\to\\infty$. Then\n \\begin{align*}\n\\mbox{\\rm Tr}(SC_n^p)=\\sum_{\\ell=0}^{n-1}\\lambda_\\ell^p\n&=\\sum_{\\ell=0}^{n-1} \\Big\\{X_0 + (-1)^\\ell X_{\\frac{n}{2}}+2\\sum_{j=1}^{n\/2}X_j\\cos(\\omega_\\ell j) \\Big\\}^p\\\\\n&=\\sum_{\\ell=0}^{n-1} \\sum_{k=0}^{p}\\binom{p}{k} {(X_0 + (-1)^\\ell X_{\\frac{n}{2}})}^{p-k} \\Big\\{ \\sum_{j=1}^{n\/2}X_j(e^{i\\omega_\\ell j}+e^{-i\\omega_\\ell j}) \\Big\\}^{k} \\\\\n&=\\sum_{k=0}^{p}\\binom{p}{k} \\Big\\{ {(X_0 + X_{\\frac{n}{2}})}^{p-k} \\sum_{\\ell=0, even}^{n-1} \\Big[ \\sum_{j=1}^{n\/2}X_j (e^{i\\omega_\\ell j}+e^{-i\\omega_\\ell j}) \\Big] ^{k} \\\\\n& + {(X_0 - X_{\\frac{n}{2}} )}^{p-k} \\sum_{\\ell=0, odd}^{n-1} \n\\Big[ \\sum_{j=1}^{n\/2}X_j (e^{i\\omega_\\ell j}+e^{-i\\omega_\\ell j}) \\Big] ^{k} \\Big\\},\n\\end{align*}\nwhere $\\omega_\\ell=\\frac{2\\pi \\ell}{n}$. Since we know\n \\begin{align*} \n\t\\sum_{\\ell=0, even}^{n-1} e^{i\\omega_\\ell j}\n&= \\left\\{\\begin{array}{ccc} \t \n\t\t \\frac{n}{2} & \\text{if}& j =0\\; \\mbox{(mod } \\frac{n}{2}) \\\\\\\\\n\t\t \t0 & \\text{if}& j \\neq0\\; \\mbox{(mod } \\frac{n}{2}),\t \n\t\t \t \\end{array}\\right.\t \t \n\\end{align*}\nand \n\\begin{align*} \n\t\\sum_{\\ell=0, odd}^{n-1} e^{i\\omega_\\ell j}\n&= \\left\\{\\begin{array}{ccc} \t \n\t\t \\frac{n}{2} & \\text{if}& j =0\\; \\mbox{(mod } \\frac{n}{2}) \\mbox{ and } j = 0\\; \\mbox{(mod n)} \\\\\\\\\n\t\t -\\frac{n}{2} & \\text{if}& j =0\\; \\mbox{(mod } \\frac{n}{2}) \\mbox{ and } j \\neq0\\; \\mbox{(mod n)} \\\\\\\\\n\t\t \t0 & \\text{if}& j \\neq0\\; \\mbox{(mod } \\frac{n}{2}) ,\t \n\t\t \t \\end{array}\\right.\t \t \n\\end{align*} \nTherefore from the above last two observations, $\\mbox{\\rm Tr}(SC_n^p)$ will be\n\\begin{align} \\label{trace formula_SC_even}\n \\mbox{\\rm Tr}(SC_n^p) &=\\frac{n}{2} \\sum_{k=0}^{p}\\binom{p}{k} \\Big[ \\Big\\{ {(X_0 + X_{\\frac{n}{2}})}^{p-k} + {(X_0 - X_{\\frac{n}{2}})}^{p-k} \\Big\\} \\sum_{A_k} X_{j_1}X_{j_2}\\ldots X_{j_{k}} \\nonumber\\\\\n& \\qquad + \\Big\\{ {(X_0 + X_{\\frac{n}{2}})}^{p-k} - {(X_0 - X_{\\frac{n}{2}})}^{p-k} \\Big\\} \\sum_{ \\tilde{A}_k} X_{j_1}X_{j_2}\\ldots X_{j_{k}} \\Big] \\nonumber \\\\\n& = \\frac{n}{2} \\sum_{k=0}^{p}\\binom{p}{k} \\Big[ Y_k \\sum_{A_k} X_{J_{k}} + \\tilde{Y}_k \\sum_{ \\tilde{A}_k} X_{J_{k}} \\Big] , \\mbox{ say}\n\\end{align}\nwhere for each $k=0, 1, 2,\\ldots, p$, $A_{k}$ is same as $A_{k}$ of $n$ odd case \nand $\\tilde{A}_{k}$ for $k=1,\\ldots,p$ is given by\n\\begin{align*}\n\\tilde{A}_{k}:=\\left\\{(j_1,\\ldots,j_{k})\\; : \\; \\sum_{i=1}^{k}\\epsilon_i j_i=0\\; \\mbox{(mod } \\frac{n}{2}) \\mbox{ and } \\sum_{i=1}^{k}\\epsilon_i j_i \\neq 0\\; \\mbox{(mod }n), \\epsilon_i\\in\\{+1,-1\\}, 1\\le j_1,\\ldots,j_{k}\\le \\frac{n}{2}\\right\\}.\n\\end{align*} \nHere note that $\\tilde{A}_0$ is an empty set with the understanding that the contribution from the sum corresponding to $\\tilde{A}_0$ is $1$ and in $\\tilde{A}_k$, $(j_1,\\ldots,j_{k})$ are collected according to their multiplicity. Also\n\\begin{align} \\label{eqn:Y_k}\nY_k & = {(X_0 + X_{\\frac{n}{2}})}^{p-k} + {(X_0 - X_{\\frac{n}{2}})}^{p-k}, \\tilde{Y}_k = {(X_0 + X_{\\frac{n}{2}})}^{p-k} - {(X_0 - X_{\\frac{n}{2}})}^{p-k} \\\\\nX_{J_{k}} & = X_{j_1}X_{j_2}\\ldots X_{j_{k}}. \\nonumber \n\\end{align}\nFrom the definition of $A_k$ and $\\tilde{A}_k$, observe that $|A_k|= O(n^{k-1})$, because the entries of $A_k$ has one constraint, whereas $|\\tilde{A}_k|= O(n^{k-2})$, because the entries of $\\tilde{A}_k$ has two constraints. Therefore\n\\begin{equation} \\label{eqn:A,tildeA}\n|\\tilde{A}_k| < |A_k|.\n\\end{equation}\n\nThe following result will be used in the proof of Theorem \\ref{thm:symcircovar}. \n\n\\begin{result} \\label{result:def_h}\n\tSuppose $|A_{p,s}|$ denotes the cardinality of $A_{p,s}$. Then \n\t$$\n\th_p(k):=\n\\lim_{n\\to \\infty}\\frac{|A_p^{(k)}|}{n^{p-1}}= \\frac{1}{(p-1)!}\\sum_{s=-\\lceil\\frac{p-k}{2}\\rceil}^{\\lfloor\\frac{k}{2}\\rfloor}\\sum_{q=0}^{2s+p-k}(-1)^q\\binom{p}{q}\\left(\\frac{2s+p-k-q}{2}\\right)^{p-1}, \n$$\nwhere $\\lceil x\\rceil$ denotes the smallest integer not less than $x$.\n\n\t\n\\end{result}\nFor the proof of Result \\ref{result:def_h}, we refer to \\cite[Lemma 14]{adhikari_saha2017}. \nNow for a given vector $(j_1, j_2, \\ldots, j_p)$, we define a term called {\\it opposite sign pair matched} elements of the vector. \n\\begin{definition}\\label{def:odd-even}\n\tSuppose $(j_1, j_2, \\ldots, j_p) \\in A_p$.\n\n\t We say $j_k, j_\\ell$ is {\\it opposite sign pair matched}, if $\\epsilon_k$ and $\\epsilon_\\ell$ corresponding to $j_k$ and $j_\\ell$, respectively, are of opposite sign and $j_k= j_\\ell$, where $\\epsilon_k$ and $\\epsilon_\\ell$ are corresponds to (\\ref{def:A_p_SC}).\n\t For example; In $(2,3,5,2)$, entry $2$ is {\\it opposite sign pair matched}, if $\\epsilon_1=1$ and $\\epsilon_4=-1$ or $\\epsilon_1=-1$ and $\\epsilon_4=1$ whereas if $\\epsilon_1$ = $\\epsilon_4= 1$ or $\\epsilon_1$ = $\\epsilon_4= -1$, then $2$ is not {\\it opposite sign pair matched}. Similarly, we can also define {\\it opposite sign pair matched} elements of $\\tilde{A}_p$. We shall call, vector $(j_1, j_2, \\ldots, j_p)$ is {\\it opposite sign pair matched}, if all the entries of $(j_1, j_2, \\ldots, j_p)$ are {\\it opposite sign pair matched}.\n\\end{definition}\n Observe that, if $(j_1, j_2, \\ldots, j_p)\\in A_{p}$, that is, $\\sum_{i=1}^{p}\\epsilon_i j_i=0 \\mbox{ (mod $n$) }$ and each entry of $\\{j_1, j_2, \\ldots, j_p\\}$ has multiplicity greater than or equal to two. Then the maximum number of free variable in $(j_1, j_2, \\ldots, j_p)$ will be $\\frac{p}{2}$ only when $p$ is even and $(j_1, j_2, \\ldots, j_p)$ is {\\it opposite sign pair matched}. We shall use this observation in proof of Theorem \\ref{thm:symcircovar}, for maximum contribution.\n\nNow assuming the above Result, we proceed to prove Theorem \\ref{thm:symcircovar}. \nWe shall use trace formula of $SC_n$ to prove \\ref{thm:symcircovar}. Since for odd and even value of $n$, we have different trace formula, therefore we shall prove \\ref{thm:symcircovar} in two steps. In Step 1, we calculate limit of $\\mbox{\\rm Cov}\\big(w_p,w_q\\big)$ as $n\\to\\infty$ with odd $n$ and in Step 2, we calculate limit of $\\mbox{\\rm Cov}\\big(w_p,w_q\\big)$ as $n\\to\\infty$ with even $n$. We shall show that for both the cases, even and odd value of $n$, limit of $\\mbox{\\rm Cov}\\big(w_p,w_q\\big)$ is same.\n\\begin{proof}[Proof of Theorem \\ref{thm:symcircovar}]\n\tSince $\\mbox{E}(w_p)=\\mbox{E}(w_q)=0$, therefore we get\n\t\\begin{align*}\n\t\\mbox{\\rm Cov}\\big(w_p,w_q\\big)&=\\mbox{E}[w_p w_q]\n\t= \\frac{1}{n} \\Big\\{ \\mbox{E}[\\mbox{\\rm Tr}(SC_n)^{p}\\mbox{\\rm Tr}(SC_n)^{q} ]- \\mbox{E}[\\mbox{\\rm Tr}(SC_n)^{p}]\\mbox{E}[\\mbox{\\rm Tr}(SC_n)^{q}] \\Big\\}. \n\t\\end{align*}\n\tFirst we suppose $\\mbox{\\rm Cov}\\big(w_p,w_q\\big)$ for odd value of $n$. \\\\\\\n\t\n\t\\noindent \\textbf{Step 1.} Suppose $n$ is odd, then by the trace formula (\\ref{trace formula_SC_odd}), we get\n\t\\begin{align*}\n\t\\mbox{E}[\\mbox{\\rm Tr}(SC_n)^{p}]&= \\mbox{E}\\Big[n\\sum_{k=0}^{p}\\binom{p}{k}X_0^{p-k}\\sum_{A_{k}}\\frac{X_{i_1}}{ \\sqrt{n}} \\cdots \\frac{X_{i_{k}}}{ \\sqrt{n}} \\Big]\n\t= \\frac{1}{ n^ {\\frac{p}{2}-1}} \\mbox{E}\\Big[ \\sum_{k=0}^{p}\\binom{p}{k}X_0^{p-k}\\sum_{A_{k}} X_{i_1} \\cdots X_{i_{k}}\\Big].\t\n\t\\end{align*}\n\tTherefore\n\t\\begin{align} \\label{eqn:T_1+T_2_SC}\n\t\\mbox{\\rm Cov}\\big(w_p,w_q\\big)&=\\mbox{E}[w_p w_q] \\nonumber\\\\\n\t&= \\frac{1}{n^{\\frac{p+q}{2}-1}} \\Big[ \\mbox{E}\\Big\\{ \\Big( \\sum_{k=0}^{p}\\binom{p}{k} X_0^{p-k}\\sum_{A_{k}} X_{i_1} \\cdots X_{i_{k}} \\Big) \\Big( \\sum_{\\ell=0}^{q}\\binom{q}{\\ell}X_0^{q-\\ell}\\sum_{A_{\\ell}} X_{j_1} \\cdots X_{j_{\\ell}} \\Big) \\Big\\}\\nonumber \\\\\n\t &\\qquad - \\mbox{E} \\Big( \\sum_{k=0}^{p}\\binom{p}{k}X_0^{p-k}\\sum_{A_{k}} X_{i_1} \\cdots X_{i_{k}} \\Big) \\mbox{E} \\Big( \\sum_{\\ell=0}^{q}\\binom{q}{\\ell}X_0^{q-\\ell}\\sum_{A_{\\ell}} X_{j_1} \\cdots X_{j_{\\ell}} \\Big) \\Big] \\nonumber\\\\\n\t &=\\frac{1}{n^{\\frac{p+q}{2}-1}} \\sum_{k, \\ell =0}^{p, q}\\binom{p}{k} \\binom{q}{\\ell} \\sum_{A_{k}, A_{\\ell}} \\Big\\{\\mbox{E}[ X_0^{p+q-k-\\ell}] \\mbox{E}[X_{i_1} \\cdots X_{i_{k}} X_{j_1} X_{j_2}\\cdots X_{j_{\\ell}} ]\\\\\n\t &\\qquad -\\mbox{E}[X_0^{p-k}]\\mbox{E}[ X_{i_1} \\cdots X_{i_{k}} ] \\mbox{E}[X_0^{q-\\ell}]\\mbox{E}[ X_{j_1} \\cdots X_{j_{\\ell}} ] \\Big\\}.\t \\nonumber \n\t\\end{align}\n\tDepending on the values of $k$ and $\\ell$, the following two cases arise. \\\\\\\n\t\n\t\\noindent \\textbf{Case I.} \\textbf{Either $k=p, \\ell \\leq q$ or $\\ell=q, k \\leq p$ :} Since in this case, we always get $\\mbox{E}[ X_0^{p+q-k-\\ell}]= \\mbox{E}[X_0^{p-k}] \\mbox{E}[X_0^{q-\\ell}]$. Therefore, if $\\{i_1,i_2,\\ldots,i_{k}\\}\\cap \\{j_1,j_2,\\ldots,j_{\\ell}\\}=\\emptyset$ then from independence of $X_i$'s, we get\n\t$$\\mbox{E}[ X_0^{p+q-k-\\ell}] \\mbox{E}[X_{i_1} \\cdots X_{i_{k}} X_{j_1} X_{j_2}\\cdots X_{j_{\\ell}} ] -\\mbox{E}[X_0^{p-k}]\\mbox{E}[ X_{i_1} \\cdots X_{i_{k}} ] \\mbox{E}[X_0^{q-\\ell}]\\mbox{E}[ X_{j_1} \\cdots X_{j_{\\ell}} ] =0.$$\n\n\tHence in this case, we can get non-zero contribution from (\\ref{eqn:T_1+T_2_SC}) only when there is at least one cross-matching among $\\{i_1,\\ldots,i_{k}\\}$ and $\\{j_1,\\ldots,j_{\\ell}\\}$, i.e., $\\{i_1,i_2,\\ldots,i_{k}\\}\\cap \\{j_1,j_2,\\ldots,j_{\\ell}\\}\\neq\\emptyset$ for some $k =0, 1, \\ldots, p$ and $\\ell=0, 1, \\ldots, q$. \n\tSo from the above observation, (\\ref{eqn:T_1+T_2_SC}) can be written as\n\t\\begin{align} \\label{eqn:T_k,l_SC}\n\t\\lim_{n\\to\\infty} \\mbox{\\rm Cov}\\big(w_p,w_q\\big) &=\\lim_{n\\to\\infty} \\frac{1}{n^{\\frac{p+q}{2}-1}} \\sum_{k, \\ell =0}^{p, q}\\binom{p}{k} \\binom{q}{\\ell} \\sum_{m=1}^{ \\min \\{k,\\ell\\} } \\sum_{I_m} \\Big\\{\\mbox{E}[ X_0^{p+q-k-\\ell}] \\mbox{E}[X_{i_1} \\cdots X_{i_{k}} X_{j_1} \\cdots X_{j_{\\ell}} ]\\nonumber \\\\\n\t &\\qquad -\\mbox{E}[X_0^{p-k}]\\mbox{E}[ X_{i_1} \\cdots X_{i_{k}} ] \\mbox{E}[X_0^{q-\\ell}]\\mbox{E}[ X_{j_1} \\cdots X_{j_{\\ell}} ] \\Big\\}\t \\nonumber \\\\\t\n\t& =\\lim_{n\\to\\infty} \\frac{1}{n^{\\frac{p+q}{2}-1}} \\sum_{k, \\ell =0}^{p, q}\\binom{p}{k} \\binom{q}{\\ell} \\sum_{m=1}^{ \\min \\{k,\\ell\\} } T^m_{k,\\ell}, \\mbox{ say},\n\\end{align}\t\t \n\twhere for each $m=1,2, \\ldots, \\min\\{p,q\\}$, $I_m$ is defined as\n\t\\begin{equation}\\label{def:I_k}\n\tI_{m}:=\\{((i_1,\\ldots,i_{k}),(j_1,\\ldots,j_{\\ell}))\\in A_{k}\\times A_{\\ell}\\; : \\; |\\{i_1,\\ldots,i_{k}\\}\\cap \\{j_1,\\ldots,j_{\\ell}\\}|=m\\}.\n\t\\end{equation}\n\tNow we calculate the contribution due to the typical term $T^m_{k,\\ell}$ of (\\ref{eqn:T_k,l_SC}) for some fixed value of $k= 1, 2, \\ldots, p$, $\\ell= 1, 2, \\ldots, q$ and $m =1,2, \\ldots, \\min\\{k,\\ell\\}$. Since from (\\ref{eqn:condition}), we have \n\t\\begin{equation*}\n\t\\mbox{E}[X_i]= 0, \\ \\mbox{E}(X^2_i)=1 \\mbox{ and } \\sup_{i \\geq 1}\\mbox{E}(|X_i|^{k})= \\alpha_k < \\infty \\mbox{ for } k \\geq 3.\n\t\\end{equation*} \n\tTherefore there exist $\\gamma >0$, which depends only on $k$ and $\\ell$, such that \n\t\\begin{align} \\label{eqn:gamma_1}\n\t|T^m_{k,\\ell}| & = \\sum_{I_m} |\\mbox{E}[ X_0^{p+q-k-\\ell} X_{i_1} \\cdots X_{i_{k}} X_{j_1} \\cdots X_{j_{\\ell}} ] -\\mbox{E}[X_0^{p-k} X_{i_1} \\cdots X_{i_{k}} ] \\mbox{E}[X_0^{q-\\ell} X_{j_1} \\cdots X_{j_{\\ell}} ] | \\nonumber \\\\ \n\t& \\leq \\gamma |B_{k,\\ell}|,\n\t\\end{align}\n\twhere $B_{k,\\ell} \\subseteq A_k \\times A_\\ell$ with conditions that $\\{ i_1, i_2,\\ldots,i_k \\} \\cap \\{ j_1, j_2,\\ldots,j_\\ell \\} \\neq \\emptyset$ and each element of set $\\{ i_1, i_2,\\ldots,i_{k}\\} \\cup \\{ j_1, j_2, \\ldots, j_\\ell\\} $ has multiplicity greater than or equal to two. So, to solve (\\ref{eqn:gamma_1}), it is enough to calculate the cardinality of $B_{k,\\ell}$. Suppose $((i_1, i_2,\\ldots,i_{k}), (j_1,j_2,\\ldots,j_{\\ell}))\\in B_{k,\\ell}$ with $|\\{i_1,\\ldots,i_{k}\\}\\cap \\{j_1,\\ldots,j_{\\ell}\\}|=m$, for some $m=1,2,\\ldots, \\min\\{k,\\ell\\}$, \n\n\t where $|\\{\\cdot\\}|$ denotes cardinality of the set $\\{\\cdot\\}$. Therefore typical element of $B_{k,\\ell}$ will look like $$((d_1, d_2,\\ldots, d_m, i_{m+1}, \\ldots, i_{k}), (d_1,d_2,\\ldots, d_m, j_{m+1}, \\ldots, j_{\\ell})).$$\n\tObserve that, we shall get maximum number of free entries in $B_{k,\\ell}$, if following conditions hold\n\t\n\t\\begin{enumerate}\n\t\t\\item [(i)] each elements of $ \\{d_1, d_2, \\ldots, d_{m}\\}$ are distinct,\n\t\t\\item [(ii)] if $k-m$ is even. Then \n\t\t$(i_{m+1}, \\ldots, i_{k})$ is {\\it opposite sign pair matched}\nwith $ \\{d_1, d_2, \\ldots, d_{m}\\} \\cap \\{i_{m+1},\\ldots,i_{k}\\}=\\emptyset$. Similar condition also hold when $\\ell-m$ is even,\n\t\t\\item [(iii)] if $k-m$ is odd. Then\n$\\{i_{m+1},\\ldots,i_{k}\\} \\setminus \\{i^*\\}$ is {\\it opposite sign pair matched} and\n $ \\{d_1, d_2, \\ldots, d_{m-2}\\} \\cap \\{i_{m+1},\\ldots,i_{k}\\} \\setminus \\{i^*\\}=\\emptyset$, where $i^{*}$ is {\\it opposite sign pair matched} with $d_s$ for some $s=1,2, \\ldots, m$.\n\n Similar condition also hold when $\\ell-m$ is odd.\n\t \\end{enumerate}\nUnder the above assumption, the cardinality of $B_{k,\\ell}$ will be\n\t\n\t \\begin{align}\\label{card_B_k_l}\n\t |B_{k,\\ell}| & = \\left\\{\\begin{array}{ll} \t \n\t\t \t O(n^{m-1+\\frac{k-m}{2} +\\frac{\\ell -m}{2}}) & \\text{if}\\ (k-m) \\mbox{ and } (\\ell-m) \\mbox{ both are even,}\\\\\\\\\n\t\t \t O(n^{m-3+\\frac{k-m+1}{2} +\\frac{\\ell -m+1}{2}}) & \\text{if}\\ (k-m) \\mbox{ and } (\\ell-m) \\mbox{ both are odd,}\\\\\\\\\n\t\t\tO(n^{m-2+\\frac{k-m+\\ell -m+1}{2}})& \\text{otherwise}, \t \t \n\t\t \t \\end{array}\\right.\t \\nonumber \\\\\n\t & = \\left\\{\\begin{array}{ll} \t \n\t\t \t O(n^{\\frac{k+\\ell}{2}-1}) & \\text{if}\\ (k-m) \\mbox{ and } (\\ell-m) \\mbox{ both are even,}\\\\\\\\\n\t\t\to(n^{\\frac{k+\\ell}{2}-1}) & \\text{otherwise}. \t \t \n\t\t \t \\end{array}\\right.\t\t \n\t \\end{align}\nNow from (\\ref{eqn:gamma_1}) and (\\ref{card_B_k_l}), we get\n\t\\begin{align} \\label{card_T_k,l}\n\t|T^m_{k,\\ell}| & = \\left\\{\\begin{array}{ll} \t \n\t\t \t O(n^{\\frac{k+\\ell}{2}-1}) & \\text{if}\\ k, \\ell \\mbox{ and } m \\mbox{ all are even or } k, \\ell \\mbox{ and } m \\mbox{ all are odd},\\\\\\\\\n\t\t\to(n^{\\frac{k+\\ell}{2}-1}) & \\text{otherwise}. \t \t \n\t\t \t \\end{array}\\right.\t\n\t\\end{align}\n\tOn using (\\ref{eqn:T_k,l_SC}) and (\\ref{card_T_k,l}), we get that $T^m_{k,\\ell}$ has non-zero contribution in (\\ref{eqn:T_k,l_SC}) only when $k=p$ and $\\ell=q$. In fact $T^m_{k,\\ell}$ has non-zero contribution only when either $p,q, m$ all are even or $p,q, m$ all are odd. So, if we use (\\ref{card_T_k,l}) in (\\ref{eqn:T_k,l_SC}), we get \n\t\\begin{align} \\label{eqn:lim_cov_SC}\n\t&\\lim_{n\\to\\infty} \\mbox{\\rm Cov}\\big(w_p,w_q\\big) \\nonumber \\\\\n\t&= \\displaystyle\\lim_{n\\to\\infty} \\frac{1}{n^{\\frac{p+q}{2}-1}} \\sum_{m=1}^{ \\min \\{p,q\\} } \\sum_{I_m} \\Big\\{\\mbox{E}[X_{i_1} \\cdots X_{i_{p}} X_{j_1} \\cdots X_{j_{q}} ] -\\mbox{E}[ X_{i_1} \\cdots X_{i_{p}} ] \\mbox{E}[ X_{j_1} \\cdots X_{j_{q}} ] \\Big\\}\t\\nonumber \\\\\n\t&= \\left\\{\\begin{array}{ll} \t \n\t \\displaystyle\\lim_{n\\to\\infty} \\frac{1}{n^{\\frac{p+q}{2}-1}} \\sum_{r=1}^{ \\min\\{ \\frac{p}{2},\\frac{q}{2} \\} } \\sum_{I_{2r}} \\big( \\mbox{E}[X_{i_1} \\cdots X_{i_{p}} X_{j_1} \\cdots X_{j_{q}} ]\\\\\n\t\\quad -\\mbox{E}[ X_{i_1} \\cdots X_{i_{p}} ] \\mbox{E}[ X_{j_1} \\cdots X_{j_{q}} ] \\big) & \\text{if}\\ p, q \\mbox{ both are even,}\\\\\\\\\n\t\t \t \\displaystyle\\lim_{n\\to\\infty}\\frac{1}{n^{\\frac{p+q}{2}-1}} \\sum_{r=0}^{ \\min\\{ \\frac{p-1}{2},\\frac{q-1}{2} \\} } \\sum_{I_{2r+1}} \\big( \\mbox{E}[X_{i_1} \\cdots X_{i_{p}} X_{j_1} \\cdots X_{j_{q}} ]\\\\\n\t\\qquad -\\mbox{E}[ X_{i_1} \\cdots X_{i_{p}} ] \\mbox{E}[ X_{j_1} \\cdots X_{j_{q}} ] \\big) & \\text{if}\\ p, q \\mbox{ both are odd,}\\\\\\\\\n\t\t\t0 & \\text{otherwise}. \t \t \n\t\t \t \\end{array}\\right.\t\t\t \t \n\t\\end{align}\nNow we calculate right hand side of (\\ref{eqn:lim_cov_SC}). Depending on values of $p,q$, following two subcases arise.\\\\\\\\\n\t\\noindent \\textbf{subcase (i)} \\textbf{$p,q$ both are even:} \n\tFirst recall, the typical term of $I_{2r}$ is \n\t$$((d_1, d_2,\\ldots, d_{2r}, i_{2r+1}, \\ldots, i_{p}), (d_1,d_2,\\ldots, d_{2r}, j_{2r+1}, \\ldots, j_{q})).$$\n\tFor such an element of $I_{2r}$, the number of free entries in $I_{2r}$ will be maximum, if following conditions hold\n\t\\begin{enumerate}\n\n\t\t\\item [(i)] $ \\{d_1, d_2, \\ldots, d_{2r}\\} \\cap \\{i_{2r+1},\\ldots,i_{p}\\}\\cap \\{j_{2r+1},\\ldots,j_{q}\\}=\\emptyset$, \n\t\t\\item [(ii)] $(i_{2r+1}, \\ldots, i_{p})$ and $(j_{2r+1}, \\ldots, j_{q})$ are {\\it opposite sign pair matched}.\n\t\\end{enumerate}\n\tDue to the above consideration, the constraints, $\\sum_{t=1}^{2r}\\epsilon_t d_t +\\sum_{t=2r+1}^{p}\\epsilon_t i_t =0\\; \\mbox{(mod n})$ and $\\sum_{t=1}^{2r}\\epsilon_t d_t + \\sum_{t=2r+1}^{q}\\epsilon_t j_t =0\\; \\mbox{(mod n})$ will change into one constraint \n\\begin{equation}\\label{eqn:constraint_d_even}\n\\sum_{t=1}^{2r}\\epsilon_t d_t =0\\; \\mbox{(mod n}).\n\\end{equation}\t\nNow first we consider $r\\geq 2$, later we shall deal $r=1$ case.\n\tNote that for $r=2,3, \\ldots, \\min\\{p,q\\}$, if we assume each entries of $ \\{d_1, d_2, \\ldots, d_{2r}\\}$ are distinct, then cardinality of $I_{2r}$ will be of the order $O(n^{2r-1 +\\frac{p-2r}{2} +\\frac{q-2r}{2} })= O(n^{\\frac{p+q}{2} -1}),$ where $(-1)$ is arising due to (\\ref{eqn:constraint_d_even}). In any other situation,\n\n\tcardinality of $I_{2r}$ will be $o(n^{p+q-1}).$ Also note that, as each entries of $ \\{d_1, d_2, \\ldots, d_{2r}\\}$ are distinct, therefore\n\t$$ \\mbox{E}[ X_{i_1} \\cdots X_{i_{p}} ] \\mbox{E}[ X_{j_1} \\cdots X_{j_{q}} ] =0. $$\n\tHence for each fixed $r\\geq 2$, first part of (\\ref{eqn:lim_cov_SC}) ($p,q$ both even) will be \n\t\\begin{align} \\label{eqn:I_2k2_SC}\n\t& \\lim_{n\\to\\infty} \\frac{1}{n^{\\frac{p+q}{2}-1}} \\sum_{I_{2r}} \\mbox{E}[X_{i_1} \\cdots X_{i_{p}} X_{j_1} \\cdots X_{j_{q}} ] \\nonumber \\\\\n\t& = \\lim_{n\\to\\infty} \\frac{1}{n^{\\frac{p+q}{2}-1}} a_r (n\/2)^{\\frac{p-2r}{2}+\\frac{q-2r}{2}} \\sum_{A_{2r}, A_{2r}} \\mbox{E}[X_{i_1} \\cdots X_{i_{2r}} X_{j_1} \\cdots X_{j_{2r}} ] \\nonumber \\\\\n\t& = \\frac{a_r}{2^{\\frac{p+q-4r}{2}} } \\lim_{n\\to\\infty} \\frac{1}{n^{2r-1}} \\sum_{A_{2r}, A_{2r}} \\mbox{E}[X_{i_1} \\cdots X_{i_{2r}} X_{j_1} \\cdots X_{j_{2r}} ],\n\t\\end{align}\n\twhere $ a_r=\t\\binom{p}{p-2r}\\binom{p-2r}{\\frac{p-2r}{2}} (\\frac{p-2r}{2})!\\binom{q}{q-2r}\\binom{q-2r}{\\frac{q-2r}{2}} (\\frac{q-2r}{2})!$.\n $a_r$ factor is arising for pair-matching of $(p-2r)$ many variables in $(i_1,i_2,\\ldots,i_{p})$ and $(j_1,j_2,\\ldots, j_{q})$ both with opposite sign. In $(i_1,i_2,\\ldots,i_{p})$, we can choose $(p-2r)$ variables in $\\binom{p}{p-2r}$ many ways. Out of $(p-2r)$ variables, $(\\frac{p-2r}{2})$ many variables can be chosen with positive sign in $\\binom{p-2r}{\\frac{p-2r}{2}}$ many ways. After free choice of $(\\frac{p-2r}{2})$ variables with positive sign, rest of the $(\\frac{p-2r}{2})$ variables with negative sign can be chosen in $(\\frac{p-2r}{2})!$ ways. Therefore for pair matching of $(p-2r)$ many variables in $(i_1,i_2,\\ldots,i_{p})$ with opposite sign, we get $(\\binom{p}{p-2r}\\binom{p-2r}{\\frac{p-2r}{2}} (\\frac{p-2r}{2})!)$ factor. Similarly from $(j_1,j_2,\\ldots,j_{q})$, we get $(\\binom{q}{q-2r}\\binom{q-2r}{\\frac{q-2r}{2}} (\\frac{q-2r}{2})!)$ factor. \nNow from \\eqref{eqn:I_2k2_SC}, we get\n\n\\begin{align*}\n\\lim_{n\\to\\infty} \\frac{1}{n^{\\frac{p+q}{2}-1}} \\sum_{I_{2r}} \\mbox{E}[X_{i_1} \\cdots X_{i_{p}} X_{j_1} \\cdots X_{j_{q}} ]\n& = \\frac{a_r}{2^{\\frac{p+q-4r}{2}} } \\lim_{n\\to\\infty} \\frac{1}{n^{2r-1}} \\sum_{A_{2r}, A_{2r}} \\mbox{E}[X_{i_1} \\cdots X_{i_{2r}} X_{j_1} \\cdots X_{j_{2r}} ] \\\\\n&=\\frac{a_r}{2^{\\frac{p+q-4r}{2}} } \\lim_{n\\to \\infty}\\frac{1}{n^{2r-1}}\\sum_{s=0}^{2r}\\binom{2r}{s}^2 s!(2r-s)! |A_{2r}'^{(s)}|\\\\\n&= \\frac{a_r}{2^{\\frac{p+q-4r}{2}} } \\sum_{s=0}^{2r}\\binom{2r}{s}^2 s!(2r-s)! \\lim_{n\\to \\infty}\\frac{|A_{2r}^{(s)}|}{n^{2r-1}}. \n\\end{align*}\nThe factor $\\binom{2r}{s}^2$ appeared because in $\\binom{2r}{s}$ ways we can choose $s$ many $+1$ from $\\{\\epsilon_1,\\ldots,\\epsilon_{2r}\\}$ in one $A_{2r}'$. The factor $(s!(2r-s)!)$ appeared because for each choice of $(i_1,\\ldots,i_{2r})$ we have $(s!(2r-s)!)$ many choice for $(j_1,\\ldots,j_{2r})$. \nNow using Result \\ref{result:def_h} in right side of the last above equality, we get\n\\begin{align}\\label{eqn:p,q_even1}\n\\lim_{n\\to\\infty} \\frac{1}{n^{\\frac{p+q}{2}-1}} \\sum_{I_{2r}} \\mbox{E}[X_{i_1} \\cdots X_{i_{p}} X_{j_1} \\cdots X_{j_{q}} ]\n&= \\frac{a_r}{2^{\\frac{p+q-4r}{2}} } \\sum_{s=0}^{2r}\\binom{2r}{s}^2 s!(2r-s)! \\ h_{2r}(s),\n\\end{align}\t\nwhere $h_{2r}(s)$ is defined in Result \\ref{result:def_h}.\n\nNow we calculate first part of (\\ref{eqn:lim_cov_SC}) ($p,q$ both even) for $r=1$.\n\tNote that, if $r=1$ in this Case, then from (\\ref{eqn:constraint_d_even}), we get $d_1=d_2$, and hence \n\t\\begin{align}\\label{eqn:p,q_even2}\n\\lim_{n\\to\\infty} \\frac{1}{n^{\\frac{p+q}{2}-1}} \\sum_{I_{2}} \\mbox{E}[X_{i_1} \\cdots X_{i_{p}} X_{j_1} \\cdots X_{j_{q}} ]- \\mbox{E}[ X_{i_1} \\cdots X_{i_{p}} ] \\mbox{E}[ X_{j_1} \\cdots X_{j_{q}} ]\n&= \\frac{a_1}{2^{\\frac{p+q-4}{2}} } (\\mbox{E} X^4_1- (\\mbox{E} X^2_1)^2) \\nonumber \\\\\n&= \\frac{a_1}{2^{\\frac{p+q-4}{2}} } (\\mbox{E} X^4_1- 1),\n\\end{align}\nwhere $a_1=(\\binom{p}{p-2}\\binom{p-2}{\\frac{p-2}{2}} (\\frac{p-2}{2})!)(\\binom{q}{q-2}\\binom{q-2}{\\frac{q-2}{2}} (\\frac{q-2}{2})!)$. Therefore from (\\ref{eqn:lim_cov_SC}), (\\ref{eqn:p,q_even1}) and (\\ref{eqn:p,q_even2}), we get\n\\begin{align} \\label{eqn:p,q_even}\n \\lim_{n\\to\\infty} &\\frac{1}{n^{\\frac{p+q}{2}-1}} \\sum_{r=1}^{ \\min\\{ \\frac{p}{2},\\frac{q}{2} \\} } \\sum_{I_{2r}} \\big( \\mbox{E}[X_{i_1} \\cdots X_{i_{p}} X_{j_1} \\cdots X_{j_{q}} ] -\\mbox{E}[ X_{i_1} \\cdots X_{i_{p}} ] \\mbox{E}[ X_{j_1} \\cdots X_{j_{q}} ] \\big) \\\\\n &= \\frac{a_1}{2^{\\frac{p+q-4}{2}} } (\\mbox{E} X^4_1- 1) + \\sum_{r=2}^{ \\min\\{ \\frac{p}{2},\\frac{q}{2} \\} } \\frac{a_r}{2^{\\frac{p+q-4r}{2}} } \\sum_{s=0}^{2r}\\binom{2r}{s}^2 s!(2r-s)! \\ h_{2r}(s).\\nonumber\n\\end{align}\n\t\n\t\\noindent \\textbf{subcase II.} \\textbf{$p,q$ both are odd:} In this case we calculate right hand side of (\\ref{eqn:lim_cov_SC}) for odd value of $p, q$ and $m$. If $m=2r+1$ for $r=0,1, \\ldots, \\min\\{\\frac{p-1}{2}, \\frac{q-1}{2}\\}$, then the typical term of $I_{2r+1}$ looks like\n\t$$((d_1, d_2,\\ldots, d_{2r+1}, i_{2r+2}, \\ldots, i_{p}), (d_1,d_2,\\ldots, d_{2r+1}, j_{2r+2}, \\ldots, j_{q}))$$\n\tand for such an elements of $I_{2r+1}$, the number of free entries in $I_{2r+1}$ will be maximum, if following conditions hold\n\t\\begin{enumerate}\n\t\\item [(i)] each entries of $ \\{d_1, d_2, \\ldots, d_{2r+1}\\}$ are distinct,\n\t\t\\item [(ii)] $ \\{d_1, d_2, \\ldots, d_{2r+1}\\} \\cap \\{i_{2r+2},\\ldots,i_{p}\\}\\cap \\{j_{2r+2},\\ldots,j_{q}\\}=\\emptyset$,\n\t\t\\item [(iii)] $(i_{2r+2}, \\ldots, i_{p})$ and $(j_{2r+2}, \\ldots, j_{q})$ are {\\it opposite sign pair matched},\n\t\\end{enumerate}\n and the contribution will be of the order $O(n^{2r+1-1 +\\frac{p-2r-1}{2} +\\frac{q-2r-1}{2} })= O(n^{\\frac{p+q}{2} -1})$, where $(-1)$ is arising due to the constraint, $\\sum_{t=1}^{2r+1}\\epsilon_t d_t =0\\; \\mbox{(mod n}).$ In any other situation, the cardinality of $I_{2r+1}$ will be $o(n^{p+q-1}).$ Since each entries of $ \\{d_1, d_2, \\ldots, d_{2r+1}\\}$ are distinct, as (ii) holds. Therefore\n\t$$ \\mbox{E}[ X_{i_1} \\cdots X_{i_{p}} ] \\mbox{E}[ X_{j_1} \\cdots X_{j_{q}} ] =0. $$\n\tNow by the similar calculations as we have done in Case I, second part of (\\ref{eqn:lim_cov_SC}) ($p,q$ both odd) will be \n\t\\begin{align} \\label{eqn:p,q_odd}\n &\\lim_{n\\to\\infty} \\frac{1}{n^{\\frac{p+q}{2}-1}} \\sum_{r=0}^{ \\min\\{ \\frac{p-1}{2},\\frac{q-1}{2} \\} } \\sum_{I_{2r+1}} \\big( \\mbox{E}[X_{i_1} \\cdots X_{i_{p}} X_{j_1} \\cdots X_{j_{q}} ] -\\mbox{E}[ X_{i_1} \\cdots X_{i_{p}} ] \\mbox{E}[ X_{j_1} \\cdots X_{j_{q}} ] \\big) \\nonumber\\\\\n\t& = \\lim_{n\\to\\infty} \\frac{1}{n^{\\frac{p+q}{2}-1}} \\sum_{r=0}^{ \\min\\{ \\frac{p-1}{2},\\frac{q-1}{2} \\} } \\sum_{I_{2r+1}} \\mbox{E}[X_{i_1} \\cdots X_{i_{p}} X_{j_1} \\cdots X_{j_{q}} ] \\nonumber\\\\\n\t& = \\sum_{r=0}^{ \\min\\{ \\frac{p-1}{2},\\frac{q-1}{2} \\} } \\frac{b_r}{2^{\\frac{p+q-4r-2}{2}} } \\sum_{s=0}^{2r+1}\\binom{2r+1}{s}^2 s!(2r+1-s)! \\ h_{2r+1}(s) ,\n\t\\end{align}\n\t where $h_{2r+1}(s)$ is defined in Result \\ref{result:def_h} and $b_r=\t\\binom{p}{p-2r-1}\\binom{p-2r-1}{\\frac{p-2r-1}{2}} (\\frac{p-2r-1}{2})!\\binom{q}{q-2r-1}\\binom{q-2r-1}{\\frac{q-2r-1}{2}} (\\frac{q-2r-1}{2})!.$\n\t \nNow, after combining both the sub-cases I and II, using (\\ref{eqn:p,q_even}) and (\\ref{eqn:p,q_odd}) in (\\ref{eqn:lim_cov_SC}, we get \n\\begin{align} \\label{eqn:cov_k=p_odd}\n\t&\\lim_{n\\to\\infty} \\mbox{\\rm Cov}\\big(w_p,w_q\\big) \\nonumber \\\\ \n\t&= \\left\\{\\begin{array}{ll} \t \n\t\t \t\\displaystyle\\frac{a_1}{2^{\\frac{p+q-4}{2}} } (\\mbox{E} X^4_1- 1) + \\sum_{r=2}^{ \\min\\{ \\frac{p}{2},\\frac{q}{2} \\} } \\frac{a_r}{2^{\\frac{p+q-4r}{2}} } \\sum_{s=0}^{2r}\\binom{2r}{s}^2 s!(2r-s)! \\ h_{2r}(s) & \\text{if}\\ p, \\mbox{} q \\mbox{ both are even},\\\\\\\\\n\t\t \t\\displaystyle\\sum_{r=0}^{ \\min\\{ \\frac{p-1}{2},\\frac{q-1}{2} \\} } \\frac{b_r}{2^{\\frac{p+q-4r-2}{2}} } \\sum_{s=0}^{2r+1}\\binom{2r+1}{s}^2 s!(2r+1-s)! \\ h_{2r+1}(s) & \\text{if}\\ p, \\mbox{} q \\mbox{ both are odd},\\\\\\\\\n\t\t\t0 & \\text{otherwise}. \t \t \n\t\t \t \\end{array}\\right.\t\t\t \t \n\t\\end{align}\n\t\\noindent \\textbf{Case II.} \\textbf{$ko(n^{\\frac{p_1+p_2}{2}-1})$.\n\\end{remark}\n\n\t\n\nThe following lemma is an easy consequence of Lemma \\ref{lem:cluster}.\n\\begin{lemma}\\label{lem:maincluster}\n\tSuppose $\\{J_1, J_2, \\ldots, J_\\ell \\} $ form a cluster where $J_i\\in A_{p_i}$ with $p_i\\geq 2$ for $1\\leq i\\leq \\ell$ and $\\{X_i\\}_{i \\geq 1}$ is independent which satisfies (\\ref{eqn:condition}). Then for $\\ell \\geq 3,$\n\t\\begin{equation}\\label{equation:maincluster}\n\t\\frac{1}{ n^{\\frac{p_1+p_2+ \\cdots + p_\\ell - \\ell}{2}}} \\sum_{A_{p_1}, \\ldots, A_{p_\\ell}} \\mbox{E}\\Big[\\prod_{k=1}^{\\ell}\\Big(X_{J_k} - \\mbox{E}(X_{J_k})\\Big)\\Big] = o(1),\n\t\\end{equation}\n\twhere \n\t$$J_k = (j^{k}_{1}, j^{k}_{2}, \\ldots, j^{k}_{p_k} ) \\ \\mbox{and} \\ X_{J_k} = X_{j^{k}_{1}} X_{j^{k}_{2}} \\cdots X_{j^{k}_{p_k}}.$$\t\n\\end{lemma}\n\\begin{proof} First observe that $\\mbox{E}\\Big[\\prod_{k=1}^{\\ell}\\Big(X_{J_k} - \\mbox{E}(X_{J_k})\\Big)\\Big]$ will be non-zero only if each $X_i$ appears at least twice in the collection $\\{ X_{j^{k}_{1}}, X_{j^{k}_{2}}, \\ldots ,X_{j^{k}_{2p_k}} ; 1\\leq k\\leq \\ell\\}$, because $\\mbox{E}(X_i)=0$ for each $i$. Therefore\n\t\\begin{equation}\\label{eqn:equality_reduction}\n\t\\sum_{A_{p_1}, \\ldots, A_{p_\\ell}} \\hspace{-3pt}\\mbox{E}\\Big[\\prod_{k=1}^{\\ell}\\Big(X_{J_k} - \\mbox{E}(X_{J_k})\\Big)\\Big]=\\sum_{(J_1,\\ldots,J_\\ell)\\in B_{P_\\ell}} \\hspace{-3pt} \\mbox{E}\\Big[\\prod_{k=1}^{\\ell}\\Big(X_{J_k} - \\mbox{E}(X_{J_k})\\Big)\\Big],\n\t\\end{equation} \n\twhere $B_{P_\\ell}$ as in Definition \\ref{def:B_{P_l}}. Since from (\\ref{eqn:condition}), we have \n\t\\begin{equation*}\\label{eqn:higher moment finite}\n\t\\mbox{E}(X^2_i)=1 \\mbox{ and } \\sup_{i \\geq 1}\\mbox{E}(|X_i|^{k})= \\alpha_k < \\infty \\mbox{ for } k \\geq 3.\n\t\\end{equation*} \n\t Therefore for $p_1,p_2,\\ldots,p_\\ell \\geq 2$, there exists $\\beta_\\ell>0$, which depends only on $p_1,p_2,\\ldots,p_\\ell$, such that \n\t\\begin{equation}\\label{eqn:modulus finite}\n\t\\Big|\\mbox{E}\\big[\\prod_{k=1}^{\\ell}\\big(X_{J_k} - \\mbox{E}(X_{J_k})\\big)\\big]\\Big|\\leq \\beta_\\ell\n\t\\end{equation} \n\tfor all $(J_1, J_2, \\ldots, J_\\ell)\\in A_{p_1}\\times A_{p_2}\\times \\cdots \\times A_{p_\\ell}$.\n\t\n\tNow using \\eqref{eqn:equality_reduction} and \\eqref{eqn:modulus finite}, we have\n\t\\begin{align*}\n\t\\sum_{A_{p_1}, \\ldots, A_{p_\\ell}} \\Big|\\mbox{E}\\big[\\prod_{k=1}^{\\ell}\\big(X_{J_k} - \\mbox{E}(X_{J_k})\\big)\\big]\\Big|\n\t& \\leq \\sum_{(J_1,J_2,\\ldots,J_\\ell)\\in B_{P_\\ell}} \\beta_{\\ell} \n\t\\ = |B_{p_\\ell}| \\ \\beta_\\ell.\n\t\\end{align*}\n\tBy using Lemma \\ref{lem:cluster} in above expression, we get\n\t\\begin{align*}\n\t\\sum_{A_{p_1}, \\ldots, A_{p_\\ell}} \\Big|\\mbox{E}\\big[\\prod_{k=1}^{\\ell}\\big(X_{J_k} - \\mbox{E}(X_{J_k})\\big)\\big]\\Big| \\leq o \\big(n^{\\frac{p_1+p_2 + \\cdots + p_\\ell -\\ell}{2} }\\big),\n\t\\end{align*}\n\tand hence \n\t\\begin{equation*}\n\t\\frac{1}{ n^{\\frac{p_1+p_2+ \\cdots + p_\\ell - \\ell}{2}}} \\sum_{A_{p_1}, \\ldots, A_{p_\\ell}} \\mbox{E}\\Big[\\prod_{k=1}^{\\ell}\\Big(X_{J_k} - \\mbox{E}(X_{J_k})\\Big)\\Big] = o(1).\n\t\\end{equation*}\n\n\t\n\n\tThis completes the proof of lemma. \n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:cluster,decompose}\n\tSuppose $J_i\\in A_{d_i}$ with $d_i\\geq 2$ for $1\\leq i\\leq \\ell$ and $\\{X_i\\}_{i \\geq 1}$ is independent which satisfies (\\ref{eqn:condition}). Then \n\t\\begin{align*}\n\t\\sum_{A_{d_1}, \\ldots, A_{d_\\ell}} \\mbox{E}\\Big[\\prod_{k=1}^{\\ell}\\Big(X_{J_k} - \\mbox{E}(X_{J_k})\\Big)\\Big]\n\t &= \\left\\{\\begin{array}{ll} \n\t\t \tO( n^{\\frac{d_1+d_2+ \\cdots + d_\\ell - \\ell}{2}}) & \\text{if} \\ \\{ J_1, J_2, \\ldots, J_\\ell\\} \\mbox{ decomposes } \\\\ \n\t\t \t& \\mbox{ into clusters of length } 2 \\\\\\\\\n\t\t \t o( n^{\\frac{d_1+d_2+ \\cdots + d_\\ell - \\ell}{2}}) & \\text{otherwise}, \t \n\t\t \t \\end{array}\\right.\n\t\\end{align*}\n\twhere \n\n\t$$J_k = (j^{k}_{1}, j^{k}_{2}, \\ldots, j^{k}_{d_k} ) \\ \\mbox{and} \\ X_{J_k} = X_{j^{k}_{1}} X_{j^{k}_{2}} \\cdots X_{j^{k}_{d_k}}.$$\t\n\\end{lemma}\n\\begin{proof} First observe that for a fixed $J_1,J_2,\\ldots,J_\\ell$, if there exists a $k\\in\\{1,2,\\ldots,\\ell\\}$ such that $J_k$ is not connected with any $J_i$ for $i\\neq k$, then \n\t$$\\mbox{E}\\Big[\\prod_{k=1}^{\\ell}\\Big(X_{J_k} - \\mbox{E}(X_{J_k})\\Big)\\Big]=0,$$\n\tdue to the independence of $\\{X_i\\}_{i\\geq 1}$.\n\t\n\tTherefore for non-zero contribution, $J_1,J_2,\\ldots,J_\\ell$ must form clusters with each cluster length greater than or equal to two, that is, each cluster should contain at least two vectors. Suppose $G_1,G_2,\\ldots,G_s$ are the clusters formed by vectors $J_1,J_2,\\ldots,J_\\ell$ and $|G_i|\\geq 2$ for all $1\\leq i \\leq s$ where $|G_i|$ denotes the length of the cluster $G_i$. Observe that $\\sum_{i=1}^s |G_i|=\\ell$. \n\t\n\tIf there exists a cluster $G_j$ among $G_1,G_2,\\ldots,G_s$ such that $|G_j|\\geq 3$, then from Theorem \\ref{thm:symcircovar} and Lemma \\ref{lem:maincluster}, we have \n\t\\begin{align*}\n\t \\sum_{A_{d_1}, \\ldots, A_{d_\\ell}} \\mbox{E}\\Big[\\prod_{k=1}^{\\ell}\\Big(X_{J_k} - \\mbox{E}(X_{J_k})\\Big)\\Big]=o(n^{\\frac{d_1 + d_2 + \\cdots +d_\\ell -\\ell}{2}}).\n\t\\end{align*} \n\tTherefore, if $\\ell$ is odd then there will be a cluster of odd length and hence \n\t\\begin{align*}\n\t \\sum_{A_{d_1}, \\ldots, A_{d_\\ell}} \\mbox{E}\\Big[\\prod_{k=1}^{\\ell}\\Big(X_{J_k} - \\mbox{E}(X_{J_k})\\Big)\\Big]=o(n^{\\frac{d_1 + d_2 + \\cdots +d_\\ell -\\ell}{2}}).\n\t\\end{align*} \n\t\n\tSimilarly, if $\\ell$ is even then from Theorem \\ref{thm:symcircovar} and Lemma \\ref{lem:maincluster}, the contribution is $O(n^{\\frac{d_1 + d_2 + \\cdots +d_\\ell -\\ell}{2}})$ only when $\\{ J_1, J_2, \\ldots, J_\\ell\\}$ decomposes into clusters of length 2. \n\t\nThis completes the proof of lemma. \n\\end{proof}\n\n\\begin{remark} \\label{rem:main_cluster,even}\n \tSuppose $J_i\\in F_{d_i}$ with $d_i\\geq 2$ for $1\\leq i\\leq \\ell$ and $\\{X_i\\}_{i \\geq 1}$ is independent which satisfies (\\ref{eqn:condition}), where $F_{d_i}$ is $A_{d_i}$ or $\\tilde{A}_{d_i}$. Then from (\\ref{eqn:A,tildeA}) and Lemma \\ref{lem:cluster,decompose}, we get\n\t\\begin{align}\n\t\\sum_{F_{d_1}, \\ldots, F_{d_\\ell}} \\mbox{E}\\Big[\\prod_{k=1}^{\\ell}\\Big(X_{J_k} - \\mbox{E}(X_{J_k})\\Big)\\Big]\n\t &= \\left\\{\\begin{array}{ll} \n\t\t \tO( n^{\\frac{d_1+d_2+ \\cdots + d_\\ell - \\ell}{2}}) & \\text{if} \\ \\{ J_1, J_2, \\ldots, J_\\ell\\} \\mbox{ decomposes into clusters } \\\\ \n\t\t \t& \\mbox{ of length }2 \\mbox{ and } F_{d_i} =A_{d_i} \\forall \\ i=1, 2, \\ldots, \\ell \\\\\\\\\n\t\t \t o( n^{\\frac{d_1+d_2+ \\cdots + d_\\ell - \\ell}{2}}) & \\text{otherwise.}\t \t \n\t\t \t \\end{array}\\right.\n\t\\end{align}\n\\end{remark}\nWe shall use the above lemmata, Remarks and Theorem \\ref{thm:symcircovar} to prove Theorem \\ref{thm:symcirpoly}. \n\n\\begin{proof}[Proof of Theorem \\ref{thm:symcirpoly}] We use method of moments and Wick's formula to prove Theorem \\ref{thm:symcirpoly}. Recall that from the method of moments, to prove $w_Q \\stackrel{d}{\\longrightarrow} N(0,\\sigma_{Q}^2)$, it is sufficient to show that\n\t\\begin{align*} \n\t\\lim_{n\\to\\infty} \\mbox{E}[ (w_Q)^\\ell] = \\mbox{E}[ (N(0, \\sigma^2_{Q}))^\\ell ] \\ \\ \\forall \\ \\ell=1,2, \\ldots.\n\t\\end{align*}\n\t\n\tNow to prove above equation, it is enough to show that, for $p_1, p_2, \\ldots , p_\\ell \\geq 2$,\n\t\\begin{align} \\label{eqn:moment w_Q}\n\t\\lim_{n\\to\\infty}\\mbox{E}[w_{p_1}w_{p_2} \\cdots w_{p_\\ell}]=\\mbox{E}[N_{p_1}N_{p_2} \\cdots N_{p_\\ell}],\n\t\\end{align}\n\twhere $\\{N_{p}\\}_{p \\geq 1}$ is a centered Gaussian family with covariance $\\sigma_{p,q}$, that is, $\\mbox{E}[N_{p},N_{q}]= \\sigma_{p,p}$, where $\\sigma_{p,q}$ as in (\\ref{eqn:sigma_p,q}). Since for odd and even value of $n$, we have different trace formula, therefore we show (\\ref{eqn:moment w_Q}) is true for odd and even value of $n$. \n\t \n\tFirst suppose $n$ is odd. Since from trace formula (\\ref{trace formula_SC_odd}), we have \n\t\\begin{align*}\n\tw_{p_k} & = \\frac{1}{\\sqrt{n}} \\Big(\\mbox{\\rm Tr}(SC_n)^{p_k} - \\mbox{E}[\\mbox{\\rm Tr}(SC_n)^{p_k}]\\Big)\\\\\n\n\t&= \\frac{1}{n^{\\frac{p_k -1}{2}}} \\sum_{d_k=0}^{p_k}\\binom{p_k}{d_k}\\sum_{A_{d_k}} (X_0^{p_k-d_k} X_{J_{d_k}} - \\mbox{E} [X_0^{p_k-d_k} X_{J_{d_k}}] ).\n\t\\end{align*}\n\tTherefore \n\t\\begin{align}\\label{eqn:expectation_thm2} \n\t& \\lim_{n\\rightarrow \\infty} \\mbox{E}[w_{p_1}w_{p_2} \\cdots w_{p_\\ell}] \\nonumber\\\\\n\t&=\\lim_{n\\rightarrow \\infty} \\frac{1}{n^{\\frac{p_1 + p_2 + \\cdots +p_\\ell -\\ell}{2}}} \\sum_{d_1=0}^{p_1} \\cdots \\sum_{d_\\ell=0}^{p_\\ell} \\binom{p_1}{d_1} \\cdots \\binom{p_\\ell}{d_\\ell} \\sum_{A_{d_1}, \\ldots, A_{d_\\ell}} \\mbox{E} \\Big[ \\prod_{k=1}^{\\ell} \\Big(X_0^{p_k-d_k} X_{J_{d_k}} - \\mbox{E} [X_0^{p_k-d_k} X_{J_{d_k}}\\Big) \\Big] \\nonumber \\\\ \n\t& = \\lim_{n\\rightarrow \\infty} \\frac{1}{n^{\\frac{p_1 + p_2 + \\cdots +p_\\ell -\\ell}{2}}} \\sum_{A_{p_1}, \\ldots, A_{p_\\ell}} \\mbox{E}\\Big[\\prod_{k=1}^{\\ell}\\Big(X_{J_k} - \\mbox{E}(X_{J_k})\\Big)\\Big],\n\t\\end{align}\n\twhere the last equality comes due to Lemma \\ref{lem:cluster,decompose}. Because\n\t$$\\sum_{A_{d_1}, \\ldots, A_{d_\\ell}} \\mbox{E} \\Big[ \\prod_{k=1}^{\\ell} \\Big(X_0^{p_k-d_k} X_{J_{d_k}} - \\mbox{E} [X_0^{p_k-d_k} X_{J_{d_k}}\\Big) \\Big] \\leq O(n^{\\frac{d_1 + d_2 + \\cdots +d_\\ell -\\ell}{2}}).$$\n\t\n Now combining Lemma \\ref{lem:cluster,decompose} for $d_i=p_i$ and \\eqref{eqn:expectation_thm2}, we get\n\t\\begin{align*} \\label{eq:multisplit}\n\t& \\quad \\lim_{n\\rightarrow \\infty} \\mbox{E}[w_{p_1} w_{p_2}\\cdots w_{p_\\ell}]\\\\\n\t& \\quad =\\lim_{n\\to\\infty} \\frac{1}{n^{\\frac{p_1 + p_2 + \\cdots +p_\\ell -\\ell}{2}}} \\sum_{A_{p_1}, \\ldots, A_{p_\\ell}} \\mbox{E}\\Big[\\prod_{k=1}^{\\ell}\\Big(X_{J_k} - \\mbox{E}(X_{J_k})\\Big)\\Big]\\\\\n\t& \\quad =\\lim_{n\\to\\infty} \\frac{1}{n^{\\frac{p_1 + p_2 + \\cdots +p_\\ell -\\ell}{2}}} \\sum_{\\pi \\in \\mathcal P_2(\\ell)} \\prod_{i=1}^{\\frac{\\ell}{2}} \\sum_{A_{p_{y(i)}},\\ A_{p_{z(i)}}} \\mbox{E}\\big[ (X_{J_{y(i)}} - \\mbox{E} X_{J_{y(i)}}) (X_{J_{z(i)}} - \\mbox{E} X_{J_{z(i)}})\\big],\n\t\\end{align*}\n\twhere $\\pi = \\big\\{ \\{y(1), z(1) \\}, \\ldots , \\{y(\\frac{\\ell}{2}), z(\\frac{\\ell}{2}) \\} \\big\\}\\in \\mathcal P_2(\\ell)$ and $\\mathcal P_2(\\ell)$ is the set of all pair partition of $ \\{1, 2, \\ldots, \\ell\\} $. Using Theorem \\ref{thm:symcircovar}, from the last equation, we get\n\t\\begin{equation}\\label{eqn:product of expectation}\n\t\\lim_{n\\rightarrow \\infty} \\mbox{E}[w_{p_1}w_{p_2} \\cdots w_{p_\\ell}]\n\t=\\sum_{\\pi \\in P_2(\\ell)} \\prod_{i=1}^{\\frac{\\ell}{2}} \\lim_{n\\rightarrow \\infty} \\mbox{E}[w_{p_{y(i)}} w_{p_{z(i)}}].\n\t\\end{equation}\n\n Since from Theorem \\ref{thm:symcircovar}, we have \n\t$$\\lim_{n\\to\\infty}\\mbox{E}(w_p w_q) = \\sigma_{p,q} (= \\mbox{E}(N_p N_q)).$$\n\tTherefore using Wick's formula, from \\eqref{eqn:product of expectation} we get \n\t\\begin{align*}\n\t\\lim_{ \\substack{{n\\to\\infty} \\\\ {n \\mbox{ odd} } }} \\mbox{E}[w_{p_1}w_{p_2} \\cdots w_{p_\\ell}]\n\t&=\\sum_{\\pi \\in P_2(\\ell)} \\prod_{i=1}^{\\frac{\\ell}{2}} \\lim_{n\\rightarrow \\infty} \\mbox{E}[w_{p_{y(i)}} w_{p_{z(i)}}] \\\\\n\t& =\\sum_{\\pi \\in \\mathcal P_2(\\ell)} \\prod_{i=1}^{\\frac{\\ell}{2}} \\mbox{E}[N_{p_{y(i)}} N_{p_{z(i)}} ] \\\\\n\t&=\\mbox{E}[ N_{p_1}N_{p_2} \\cdots N_{p_\\ell} ].\n\t\\end{align*} \n\t\n\tNow suppose $n$ is even. Then by using trace formula (\\ref{trace formula_SC_even}), we get\n \\begin{align*}\n\tw_{p_k} & = \\frac{1}{n^{\\frac{p_k -1}{2}}} \\sum_{d_k=0}^{p_k}\\binom{p_k}{d_k} \\Big[ Y_k \\sum_{A_{d_k}} X_{J_{d_k}} + \\tilde{Y}_{d_k} \\sum_{ \\tilde{A}_{d_k}} X_{J_{d_k}} - \\mbox{E} [Y_{d_k} \\sum_{A_{d_k}} X_{J_{d_k}} + \\tilde{Y}_{d_k} \\sum_{ \\tilde{A}_{d_k}} X_{J_{d_k}} ] \\Big] \\\\\n\t & = \\frac{1}{n^{\\frac{p_k -1}{2}}} \\sum_{d_k=0}^{p_k}\\binom{p_k}{d_k} \\Big[ \\sum_{A_{d_k}} Y_{d_k} X_{J_{d_k}} - \\mbox{E}[Y_{d_k} X_{J_{d_k}} ] + \\sum_{ \\tilde{A}_{d_k}} \\tilde{Y}_{d_k} X_{J_{d_k}} - \\mbox{E} [\\tilde{Y}_{d_k} X_{J_{d_k}}] \\Big],\\\\\n\t\\end{align*}\n and therefore \n\t\\begin{align}\\label{eqn:expectation_thm2,even} \n\t& \\lim_{n\\rightarrow \\infty} \\mbox{E}[w_{p_1}w_{p_2} \\cdots w_{p_\\ell}] \\nonumber\\\\\n\t&= \\lim_{n\\rightarrow \\infty} \\frac{1}{n^{\\frac{p_1 + p_2 + \\cdots +p_\\ell -\\ell}{2}}} \\sum_{d_1=0}^{p_1} \\cdots \\sum_{d_\\ell=0}^{p_\\ell} \\binom{p_1}{d_1} \\cdots \\binom{p_\\ell}{d_\\ell} \\mbox{E}\\Big[ \\prod_{k=1}^{\\ell} \\Big( \\sum_{A_{d_k}} Y_{d_k} X_{J_{d_k}} - \\mbox{E}[Y_{d_k} X_{J_{d_k}} ] \\nonumber \\\\\n\t& \\qquad + \\sum_{ \\tilde{A}_{d_k}} \\tilde{Y}_{d_k} X_{J_{d_k}} - \\mbox{E} [\\tilde{Y}_{d_k} X_{J_{d_k}}] \\Big) \\Big]\\nonumber \\\\\n\t& = \\lim_{n\\rightarrow \\infty} \\frac{1}{n^{\\frac{p_1 + p_2 + \\cdots +p_\\ell -\\ell}{2}}} \\sum_{A_{p_1}, \\ldots, A_{p_\\ell}} \\mbox{E}\\Big[\\prod_{k=1}^{\\ell}\\Big(X_{J_k} - \\mbox{E}(X_{J_k})\\Big)\\Big],\n\t\\end{align}\n\twhere the last equality comes due to Lemma \\ref{lem:cluster,decompose} and Remark \\ref{rem:main_cluster,even}. Since (\\ref{eqn:expectation_thm2,even}) is same as (\\ref{eqn:expectation_thm2}), therefore by the the similar calculation as we have done for $n$ odd case, we get\n\t$$\\lim_{ \\substack{{n\\to\\infty} \\\\ {n \\mbox{ even} } }} \\mbox{E}[w_{p_1}w_{p_2} \\cdots w_{p_\\ell}]\n\t=\\mbox{E}[ N_{p_1}N_{p_2} \\cdots N_{p_\\ell} ].$$\n\n\tThis completes the proof of Theorem \\ref{thm:symcirpoly} after combining odd and even cases of $n$.\n\\end{proof}\n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\nWith its unprecedented values of luminosity and center-of-mass energy, the Large Hadron Collider (LHC) has all the features of a top factory: $t\\bar{t}$ events are produced abundantly allowing to study the properties of top quarks with high precision. \nThe cross section of the inclusive process $pp \\to t\\bar{t}+X$ is an important benchmark of the Standard Model (SM) with a wealth of phenomenological applications. Precision tests of perturbative QCD \\cite{Czakon:2013goa}, constraints on large-$x$ parton distribution functions (PDF) \\cite{Czakon:2016olj} and accurate determinations of SM parameters related to the top quark are just selected examples which underline the importance of this channel.\nAlso, it should be noticed that a significant fraction of the inclusive $t\\bar{t}$ sample is accompanied by additional SM particles, either electroweak bosons, leptons or highly energetic jets. \nLet us focus our attention on the associated production of top-quark pairs with one hard jet (hereafter denoted $t\\bar{t}j$). Besides representing a QCD background for Higgs boson searches in the Vector Boson Fusion and $t\\bar{t}H$ channels, this process plays also a role in searches of physics beyond the SM (for example signals from decay chains of SUSY particles).\nThe $t\\bar{t}j$ production process is also important for the precision measurement of the top quark mass at the LHC \\cite{Alioli:2013mxa, Fuster:2017rev}.\nWe would like to stress that a precise determination of $m_t$ is crucial not only because it affects predictions of cross sections that are indispensable to study Higgs boson properties or new signals from BSM physics, but also because the stability of the electroweak vacuum depends crucially on the actual value of this parameter \\cite{Degrassi:2012ry,Alekhin:2012py}. Last but not least, the top quark mass is an ingredient for global electroweak fits which are important consistency checks of the SM \\cite{Baak:2014ora}.\nIt is not surprising that $t\\bar{t}j$ received considerable attention in the last years \\cite{Dittmaier:2007wz, Dittmaier:2008uj, Melnikov:2010iu, Melnikov:2011qx, Kardos:2011qa, Alioli:2011as, Czakon:2015cla, Bevilacqua:2015qha, Bevilacqua:2016jfk}.\nIn this contribution we discuss the impact of off-shell effects in $t\\bar{t}j$ production and show selected results from our recent work \\cite{Bevilacqua:2017ipv} where we explore this channel in relation to the extraction of $m_t$ at the LHC Run II, taking the viewpoint of a full calculation and comparing with different levels of on-shell approximations. We analyse in particular two observables, $\\rho_s$ and $M_{be^+}$, which have been widely investigated both theoretically \\cite{Alioli:2013mxa,Fuster:2017rev,Denner:2012yc,AlcarazMaestre:2012vp,Heinrich:2013qaa,Heinrich:2017bqp} and experimentally, see \\textit{e.g.} \\cite{Aad:2015waa,CMS:2016khu,Aaboud:2016igd,Sirunyan:2017idq}, with the goal of assessing their sensitivity to the top quark mass.\n\n\n\n\\section{NLO analysis of $pp \\to e^+\\nu_e\\mu^-\\bar{\\nu}_\\mu b \\bar{b}j$}\nWe present here selected results for the LHC Run II, specifically NLO QCD predictions at the perturbative order $\\mathcal{O}(\\alpha^4\\alpha_s^4)$ for the center-of-mass energy $\\sqrt{s}=13$ TeV. \nDetails about the SM parameters, the jet algorithm and the kinematical cuts used for the calculation are described in Ref. \\cite{Bevilacqua:2017ipv}.\nWe employ the CT14 \\cite{Dulat:2015mca}, NNPDF3 \\cite{Ball:2014uwa} and MMHT2014 \\cite{Harland-Lang:2014zoa} parton distribution functions, following the PDF4LHC recommendations for the LHC Run II \\cite{Butterworth:2015oua}.\nFor the renormalization and factorization scales we consider three possibilities. The first one is the \\textit{fixed} scale $\\mu_R = \\mu_F = \\mu_0 = m_t$ while the remaining two are \\textit{dynamical} scales: $\\mu_R = \\mu_F = \\mu_0 = E_T\/2$ and $\\mu_R = \\mu_F = \\mu_0 = H_T\/2$,\n\\begin{eqnarray}\nE_T &=&\\sqrt{m^2_{t}+p_T^2(t)}+\\sqrt{m^2_{t}+p_T^2(\\,\\bar{t}\\,)}\\,,\\\\ \nH_T&=&p_T(e^+) + p_T(\\mu^-) + p_T(j_{b_1}) + p_T(j_{b_2}) + p_T(j_1) +\np_T^{miss}\\,,\n\\end{eqnarray}\nwhere $p_T(t), \\, p_T(\\bar{t})$ denote the transverse momenta of the top quarks reconstructed from their decay products. Theoretical uncertainties stemming from the scale dependence of the cross section are estimated by simultaneously varying $\\mu_R$ and $\\mu_F$ by a factor 2 around their central value $\\mu_0$. \n\nThe goal of our analysis is to perform a systematic comparison of three distinct approaches to the calculation of $t\\bar{t}j$ production in the di-lepton channel, based on different levels of approximation. The first approach, dubbed \\textit{Full}, consists of a complete $\\mathcal{O}(\\alpha^4\\alpha_s^4)$ calculation of the process $pp \\to e^+\\nu_e\\mu^-\\bar{\\nu}_\\mu b \\bar{b}j$ where all possible contributions, \\textit{i.e.} double-top , single-top, and non-resonant diagrams, are taken into account \\cite{Bevilacqua:2015qha, Bevilacqua:2016jfk}. This calculation has been performed with the help of the program \\texttt{HELAC-NLO} \\cite{Bevilacqua:2011xh}, which comprises \\texttt{HELAC-1LOOP} \\cite{vanHameren:2009dr} and \\texttt{HELAC-DIPOLES} \\cite{Czakon:2009ss, Bevilacqua:2013iha}. The second approach, dubbed \\textit{$\\mbox{NWA}_{Prod}$}, considers on-shell top quarks and $W$ bosons and restricts the computation to the decay chain $pp \\to t\\bar{t}j \\to e^+\\nu_e\\mu^-\\bar{\\nu}_\\mu b \\bar{b}j$ as described in Ref. \\cite{Melnikov:2010iu}. This means considering on-shell $t\\bar{t}j$ production at NLO while modeling spin-correlated top quark decays at LO. The third and last approach, dubbed \\textit{$\\mbox{NWA}$}, is a more sophisticated and complete version of narrow-width approximation which includes QCD corrections and jet radiation into top quark decays as well. This requires to take consistently into account the additional decay chain $pp \\to t\\bar{t} \\to e^+\\nu_e\\mu^-\\bar{\\nu}_\\mu b \\bar{b}j$, as described in Ref. \\cite{Melnikov:2011qx}.\nLet us stress that this analysis is carried out at fixed order, namely effects of parton shower and hadronization are not taken into account at this stage.\n\nThe performance of different prescriptions for the renormalization and factorization scales has been extensively studied in the context of the \\textit{Full} calculation. Indeed the genuine nature of $pp \\to e^+\\nu_e\\mu^-\\bar{\\nu}_\\mu b \\bar{b}j$ as a multi-scale process suggests that a judicious choice of dynamical scales could help to capture effects from higher orders and minimize shape distortions induced by radiative corrections, thus improving the perturbative stability of our predictions. \nA comparative analysis of predictions based on different scale choices has been performed in \\cite{Bevilacqua:2016jfk} considering a wide spectrum of observables. It has been shown that the fixed scale $\\mu_0= m_t$ does not always ensure a stable shape when going from LO to NLO, and significant distortions have been observed particularly in the case of $p_T$ and invariant mass distributions. Also, in the fixed-scale setup the NLO error bands do not generally fit well within the LO ones as one would expect from a well-behaved perturbative expansion. Using dynamical scales, instead, the QCD corrections are positive and vary from rather small to moderate in the whole considered range.\nWe believe that the dynamical scale $\\mu_0 = H_T\/2$ performs reasonably well in accounting for the multi-scale nature of the process, at least for the kinematical setup considered in our analysis. We promote $\\mu_0=H_T\/2$ as the reference scale for our benchmark predictions based on the most accurate calculation, \\textit{i.e.} the \\textit{Full} approach. \n\nThe overall impact of the off-shell effects related to top quark and $W$ boson decays, as comes from comparing the total NLO cross section in the \\textit{Full} and \\textit{NWA} approaches, is at the level of 2\\%. This is fully consistent with the size of NWA effects, \\textit{i.e.} $\\mathcal{O}(\\Gamma_t\/m_t)$. It is well known, on the other hand, that this kind of effects can be dramatically enhanced in specific regions of the phase space and might play a much more relevant role at the differential level.\nTo assess their size on a more exclusive ground, we focus on two observables which have been widely investigated in the context of precision measurements of $m_t$ at the LHC. The first one, denoted $\\mathcal{R}(m_t^{pole},\\rho_s)$, is the normalized differential cross section as a function of the inverse invariant mass of the $t\\bar{t}j$ system, $M_{t\\bar{t}j}$ \\cite{Alioli:2013mxa}:\n\\begin{equation}\n{\\cal R}(m_t^{pole},\\rho_s) \\equiv \\frac{1}{\\sigma_{t\\bar{t}j}} \n\\frac{d\\sigma_{t\\bar{t}j}}{d\\rho_s} \\,, \n~~~{\\rm with} ~~~~\n\\rho_s = \\frac{2m_0}{M_{t\\bar{t}j}}\\,,\n\\end{equation}\nwhere $m_0 = 170$ GeV is a scale parameter of the order of the top quark mass. \nThe second observable is the normalized differential cross section as a function of $M_{be^+}$:\n\\begin{equation}\n{\\cal R}(m_t^{pole},M_{be^+}) \\equiv \\frac{1}{\\sigma_{t\\bar{t}j}} \n\\frac{d\\sigma_{t\\bar{t}j}}{dM_{be^+}} \\,, \n~~~{\\rm with} ~~~~\nM_{be^+} = \\min \\left\\{ M_{b_1 e^+} , M_{b_2 e^+} \\right\\} \\,,\n\\end{equation}\nwhere $b_1$ and $b_2$ denote the two $b$-jets in the final state. \nIn Figure \\ref{Fig:Rho_Mbl_Full_vs_NWA} we compare the NLO predictions for $\\mathcal{R}(m_t^{pole},\\rho_s)$ and $\\mathcal{R}(m_t^{pole},M_{be^+})$ obtained with the three different approaches. Also shown is the relative size of NLO QCD corrections and off-shell contributions on the shape of the two distributions in the full kinematical range. In the case of $\\mathcal{R}(m_t^{pole},\\rho_s)$, one can observe that deviations of \\textit{NWA} from the \\textit{Full} result are below 15\\% in the most sensitive region. On the other hand, substantial differences of the order of 50\\%\\---100\\% are visible for \\textit{$\\mbox{NWA}_{Prod}$} in the same region. Given that $\\rho_s \\approx 1$ corresponds to the threshold of $t\\bar{t}$ production, which is by its own nature most sensitive to the value of $m_t$, these differences should have a considerable impact on the extraction of $m_t$ when the $\\mathcal{R}(m_t^{pole},\\rho_s)$ distribution is used as template for fits.\nIn the case of the normalized $M_{be^+}$ distribution, a remarkably different behavior can be noticed in the regions below and above the critical value defined by $M_{be^+} = \\sqrt{m_{t}^2-m_{W}^2} \\approx 153$ GeV. It should be noticed that, in the \\textit{$\\mbox{NWA}_{Prod}$} case, this value corresponds to a kinematical endpoint for the observable at hand. When QCD radiation is included in the modeling of top quark decays, or alternatively when off-shell contributions are taken into account, the kinematical endpoint is smeared. This is the region where off-shell effects have a pretty large impact of the order of 50\\% (see Figure \\ref{Fig:Rho_Mbl_Full_vs_NWA}). On the contrary, they have an almost negligible size in the range below the kinematical endpoint.\n\\begin{figure}[h!tb]\n\\centerline{%\n\\includegraphics[width=0.885\\textwidth]{Rho.pdf}\n\\put(-210,103){\\tiny CT14 PDF}\n\\put(-248,33){\\scriptsize $\\mu = m_t$}\n}\n\\centerline{%\n\\hspace{-0.5cm} \\includegraphics[width=0.95\\textwidth]{Mbl.pdf}\n\\put(-218,106){\\tiny CT14 PDF}\n\\put(-248,35){\\scriptsize $\\mu = m_t$}\n}\n\\caption{Normalized distributions of the observables $\\rho_s$ (upper plots) and $M_{be^+}$ (lower plots) at NLO QCD. Also shown is the relative size of the QCD corrections (upper-right panels) and of the off-shell effects (lower-right panels).}\n\\label{Fig:Rho_Mbl_Full_vs_NWA}\n\\end{figure}\n\n\n\n\\section{Top quark mass extraction with template methods}\nThe sensitivity of the shape of differential cross sections to the top quark mass can be exploited to extract the latter parameter from fits to data: this is the basic concept of the \\textit{template method}. Figure \\ref{Fig:rho_Mbl_massdep} gives an idea of the expected variation in shape of $\\mathcal{R}(m_t^{pole},\\rho_s)$ and $\\mathcal{R}(m_t^{pole},M_{be^+})$ for five different values of the input mass used in the calculation, ranging in steps of 2.5 GeV from $m_t=168.2$ GeV up to $m_t=178.2$ GeV.\nIn the case of $\\mathcal{R}(m_t^{pole},\\rho_s)$, as given by the most accurate predictions with $\\mu_R=\\mu_F=H_T\/2$, a significant mass dependence can be observed in the ranges $0.25 < \\rho_s < 0.45$ and $\\rho_s > 0.6$.\nIn the case of $\\mathcal{R}(m_t^{pole},M_{be^+})$ one of the most sensitive regions is the one centered around the kinematical endpoint, $140 \\mbox{ GeV} < M_{be^+} <160 \\mbox{ GeV}$.\n\nTo quantify the impact of the off-shell effects in this context, in the first step we generate pseudo-data for a given value of collider luminosity. These are generated according to our most accurate prediction, \\textit{i.e.} the \\textit{Full} calculation with $\\mu_R=\\mu_F=H_T\/2$. Let us call this prediction \"theory input\" for brevity. In the second step, the pseudo-data are fitted with a template distribution, namely a prediction from either one of the three approaches that we have considered: \\textit{Full}, \\textit{NWA} or \\textit{$\\mbox{NWA}_{Prod}$}. The position of the minimum of the $\\chi^2$ distribution is used to extract the numerical value of the top quark mass. To account for statistical fluctuations, the whole procedure is iterated $1000$ times. In the end a distribution of extracted masses is obtained, whose average value and spread at 68\\% C.L. define the final result of the fit in the form $m_t^{out} \\pm \\delta m_t^{out}$.\nWe refer to Ref. \\cite{Bevilacqua:2017ipv} for a more detailed description of the statistical procedure.\n\nIn Table \\ref{Tab:fit_Rho_Mbl} we report a comparative analysis of the results of the fit obtained for the three different approaches to the calculation, as well as for different scale choices. Results refer to the two observables described in the paper for two reference values of luminosity, $2.5 \\mbox{ fb}^{-1}$ and $25 \\mbox{ fb}^{-1}$, which correspond approximately to 5400 and 54000 events respectively (we have included a multiplicity factor of 4 which accounts for all combinations of charged leptons of the first two generations).\nTogether with the extracted mass and its uncertainty, we monitor also the quality of the fit and the mass shift with respect to the input value $m_t^{in}$ used for the pseudo-data. \nA few comments are in order. The first thing one can notice, looking at the $\\rho_s$ observable in the low-luminosity case, is an overall agreement of the pseudo-data, below 1.2$\\sigma$, with any of the three approaches considered. Normally the quality of the fit would start to be questionable when an agreement worse than $2\\sigma$ is found. Despite the good agreement, different mass shifts of the order of 1 GeV, 2 GeV and 3.8 GeV are observed for the \\textit{Full}, \\textit{NWA} and \\textit{$\\mbox{NWA}_{Prod}$} cases respectively. This should be compared with the statistical uncertainty $\\delta m_t^{out}$ found at low luminosity, which is of the order of 1 GeV. When the high-luminosity setup is considered, $\\mathcal{L}=25 \\mbox{ fb}^{-1}$, the quality of fits based on NWA results gets visibly worse. Also the template based on the \\textit{Full} calculation with $\\mu_0=m_t$ does not adequately describe the pseudo-data. We note that the mass shift does not change significantly with respect to our previous findings, while the statistical uncertainty $\\delta m_t^{out}$ is dramatically smaller as expected. Thus, at $\\mathcal{L}=25 \\mbox{ fb}^{-1}$ the pseudo-data start to become sensitive to off-shell effects as well as to the scale choice. Looking now at the $M_{be^+}$ results, one can note a reduced statistical uncertainty $\\delta m_t^{out}$ in comparison with $\\rho_s$ for a given value of luminosity. Also lower mass shifts are observed with respect to $\\rho_s$, of the order 0.2 GeV, 0.7 GeV and 0.6 GeV for \\textit{Full}, \\textit{NWA} and \\textit{$\\mbox{NWA}_{Prod}$} respectively. As observed in the case of $\\rho_s$, at $\\mathcal{L} = 25 \\mbox{ fb}^{-1}$ the pseudo-data start to resolve off-shell and scale effects, thus templates based on the narrow-width approximation as well as the \\textit{Full} case with $\\mu_0 = m_t$ are clearly disfavored.\nThe impact of systematic uncertainties stemming from scale and PDF variations has also been estimated taking the \\textit{Full} case as a benchmark. Concerning the $\\rho_s$ observable, scale uncertainties are of the order of 2 GeV for the $\\mu_0=m_t$ setup, while they are reduced to 0.6 \\-- 1.2 GeV when dynamical scales are used. In the case of $M_{be^+}$ they are smaller, of the order of 1 GeV for the fixed scale and only 0.05 GeV for dynamical scales. On the other hand, PDF uncertainties are at the level of 0.4 \\-- 0.7 GeV for $\\rho_s$ and 0.02 \\-- 0.03 GeV for $M_{be^+}$ independently on the scale choice, therefore they are well below the dominant scale systematics.\n\n\\begin{figure}[h!tb]\n\\centerline{%\n\\includegraphics[width=0.55\\textwidth]{rho_OFFSHELL_NLO_HT_CT14_40bins.pdf}\n\\put(-149,126){\\tiny \\textit{Full} NLO}\n\\put(-148,117){\\tiny $\\mu = H_T\/2$}\n\\put(-56,88){\\tiny CT14 PDF}\n\\includegraphics[width=0.55\\textwidth]{Mbl_OFFSHELL_NLO_HT_CT14_40bins.pdf}\n\\put(-149,126){\\tiny \\textit{Full} NLO}\n\\put(-148,117){\\tiny $\\mu = H_T\/2$}\n\\put(-56,88){\\tiny CT14 PDF}\n}\\caption{Normalized distribution of the observables $\\rho_s$ (left plot) and $M_{be^+}$ (right plot) for five different values of the input $m_t$ used for the calculation.}\n\\label{Fig:rho_Mbl_massdep}\n\\end{figure}\n\n\\begin{table}[t!]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\hline\\hline\n&&&&\\\\\n{Theory, NLO QCD}& $m^{out}_t \\pm \\delta m^{out}_t$\n& Average & Probability &\n$m_t^{in} -m_t^{out}$ \\\\ {CT14 PDF} & [GeV] &\n$\\chi^2\/{\\rm d.o.f.}$\n& {\\it p-value} & [GeV] \\\\\n&&&&\\\\\n\\hline\\hline\n&&&&\\\\\n$\\rho_s$ &$\\mathcal{L} = 2.5 \\mbox{ fb}^{-1}$&&&\\\\ \\vspace{-0.3cm}\n&&&&\\\\\n\\hline\\hline\n{\\it Full}, $\\mu_0=H_T\/2$& 173.05 $\\pm$ 1.31 & 0.99 \n&0.42 (0.8$\\sigma$)& $+0.15$\\\\\n{\\it Full}, $\\mu_0=E_T\/2$ & 172.19 $\\pm$ 1.34 & 1.05\n & 0.39 (0.9$\\sigma$) &$+1.01$\\\\\n{\\it Full}, $\\mu_0=m_t$ & 173.86 $\\pm$ 1.39 & 1.42 \n& 0.21 (1.2$\\sigma$) & $-0.66$\\\\\n\\hline\\hline \n{\\it NWA}, $\\mu_0=m_t$ & 175.22 $\\pm$ 1.15 &\n1.38 & 0.23 (1.2$\\sigma$)& $-2.02$\\\\\n{\\it NWA}${}_{Prod.}$, $\\mu_0=m_t$ & 169.39 $\\pm$ 1.46 &\n1.12 & 0.35 (0.9$\\sigma$) & $+3.81$\\\\\n\\hline\\hline\n&&&&\\\\\n$\\rho_s$ &$\\mathcal{L} = 25 \\mbox{ fb}^{-1}$&&&\\\\ \\vspace{-0.3cm}\n&&&&\\\\\n\\hline\\hline\n{\\it Full}, $\\mu_0=H_T\/2$& 173.06 $\\pm$ 0.44 & 0.97 \n&0.44 (0.8$\\sigma$)& $+0.14$\\\\\n{\\it Full}, $\\mu_0=E_T\/2$ & 172.36 $\\pm$ 0.44 & 1.38\n & 0.23 (1.2$\\sigma$) &$+0.84$\\\\\n{\\it Full}, $\\mu_0=m_t$ & 173.84 $\\pm$ 0.42 & 5.12 \n& 1 $\\cdot 10^{-4}$ (3.9$\\sigma$) & $-0.64$\\\\\n\\hline\\hline \n{\\it NWA}, $\\mu_0=m_t$ & 175.23 $\\pm$ 0.37 &\n5.28 & 7 $\\cdot 10^{-5}$ (4.0$\\sigma$)& $-2.03$\\\\\n{\\it NWA}${}_{Prod.}$, $\\mu_0=m_t$ & 169.43 $\\pm$ 0.50 &\n2.61 & 0.02 (2.3$\\sigma$) & $+3.77$\\\\\n\\hline\\hline \n&&&&\\\\\n$M_{be^+}$ & $\\mathcal{L} = 2.5 \\mbox{ fb}^{-1}$ &&&\\\\ \\vspace{-0.3cm}\n&&&&\\\\\n\\hline\\hline\n{\\it Full}, $\\mu_0=H_T\/2$&173.09 $\\pm$ 0.48 & 1.05 \n& 0.38 (0.9$\\sigma$) & $+$0.11\\\\\n{\\it Full}, $\\mu_0=E_T\/2$& 173.01 $\\pm$ 0.50 & 1.06\n& 0.37 (0.9$\\sigma$) & $+$0.19\\\\\n{\\it Full}, $\\mu_0=m_t$& 173.07 $\\pm$ 0.49 & 1.22 \n& 0.18 (1.3$\\sigma$) &$+$0.13\\\\\n\\hline\\hline\n{\\it NWA}, $\\mu_0=m_t$&173.90 $\\pm$ 0.50& 1.11\n& 0.30 (1.0$\\sigma$) &$-$0.70 \\\\\n{\\it NWA}$_{\\rm Prod.}$, $\\mu_0=m_t$& 172.56 $\\pm$ 0.54& 1.64\n& 0.01 (2.6$\\sigma$) &$+$0.64 \\\\\n\\hline\\hline\n&&&&\\\\\n$M_{be^+}$ & $\\mathcal{L} = 25 \\mbox{ fb}^{-1}$ &&&\\\\ \\vspace{-0.3cm}\n&&&&\\\\\n\\hline\\hline\n{\\it Full}, $\\mu_0=H_T\/2$& 173.18 $\\pm$ 0.15 &\n1.02 & 0.42 (0.8$\\sigma$) & $+$0.02 \\\\\n{\\it Full}, $\\mu_0=E_T\/2$& 173.23 $\\pm$ 0.15 & 1.03\n& 0.41 (0.8$\\sigma$)&$-$0.03\\\\\n{\\it Full}, $\\mu_0=m_t$& 173.22 $\\pm$ 0.16 & 1.78\n& 0.005 (2.8$\\sigma$) &$-$0.02\\\\\n\\hline\\hline\n{\\it NWA}, $\\mu_0=m_t$& 173.98 $\\pm$ 0.16 & 2.56 \n& 5 $\\cdot 10^{-6}$ (4.6$\\sigma$) & $-$0.78\\\\\n{\\it NWA}$_{\\rm Prod.}$, $\\mu_0=m_t$& 172.62 $\\pm$ 0.17 & 8.23\n& 0 ($\\gg 5\\sigma$) & $+$0.58\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\it Top quark mass fits obtained using the normalized $\\rho_s$ and $M_{be^+}$ distributions as templates. Results are shown for the two reference luminosities of 2.5 $fb^{-1}$ and 25 $fb^{-1}$. From left to right: mean value of the top quark mass ($m_t^{out}$) obtained from 1000 pseudo-data sets together with its 68$\\,\\%$ C.L. statistical error ($\\delta m_t^{out}$); average minimum $\\chi^2\/d.o.f$; $p$-value with the corresponding number of standard deviations; top quark mass shift ($m_t^{in} -m_t^{out}$). For the $\\rho_s$ distribution, the histogram binning of Ref.\\cite{Aad:2015waa} is considered.}\n\\label{Tab:fit_Rho_Mbl}\n\\end{table}\n\n\n\\section{Conclusions}\nIn this paper we have studied the normalized distributions of two observables of interest in the study of $t\\bar{t}j$ production with leptonic decays at the LHC, namely $\\rho_s$ and $M_{be^+}$. Through a systematic comparison of the full fixed-order calculation with results based on NWA, we have found that the off-shell effects play an important role in kinematical regions which are relevant for the extraction of $m_t$. Overall, the $M_{be^+}$ observable shows the best performance in terms of the statistical uncertainty on the extracted top quark mass as well as the systematics related to scale and PDF variations. Our fixed-order analysis indicates that off-shell effects have an impact on the fits particularly in the high-luminosity case considered, $\\mathcal{L}=25 \\mbox{ fb}^{-1}$. Templates based on the narrow-width approximation induce visible mass shifts which are relevant also at lower luminosities. \nThe results presented in this paper are part of a wider phenomenological study aimed at exploring the relevance of the off-shell effects on a more extensive set of kinematical observables. Although not explicitly shown here, we have found other observables, such as $M_{t\\bar{t}}$ and $H_T$, which exhibit competitive performances for the purpose of extracting $m_t$ from $t\\bar{t}j$ \\cite{Bevilacqua:2017ipv}.\n\n\\medskip \\medskip\nThe research of G.~B. was supported by grant K 125105 of the National Research, Development and Innovation Office in Hungary. The work of M.~W. and H.~B.~H. was supported in part by the German Research Foundation (DFG) under Grant no. WO 1900\/2 \\-- \\textit{Top Quarks under the LHCs Magnifying Glass: From Process Modeling to Parameter Extraction}.\nThe work of H.~B.~H. was also supported by a Rutherford Grant ST\/M004104\/1.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction and statement of the main results}\n\nAccording to \\cite{relatives}, two K\\\"ahler manifolds are called {\\em relatives} when they share a common K\\\"ahler submanifold, i.e. if a complex submanifold of one of them with the induced metric is biholomorphically isometric to a complex submanifold of the other one with the induced metric. \nIn his seminal paper \\cite{calabi}, Calabi determined a criterion which characterizes K\\\"ahler manifolds admitting a K\\\"ahler immersion into finite or infinite dimensional complex space forms. The main tool he introduced is the {\\em diastasis function} associated to a real analytic K\\\"ahler manifold, namely a particular K\\\"ahler potential characterized by being invariant under pull--back through a holomorphic map. \nThanks to this property, the diastasis plays a key role in studying when two K\\\"ahler manifolds are relatives.\nIn \\cite{umehara} Umehara proved that two finite dimensional complex space forms with holomorphic sectional curvatures of different signs can not be relatives. Although, as firstly pointed out by Bochner in \\cite{bochner}, when the ambient space is allowed to be infinite dimensional, the situation is different: any K\\\"ahler submanifold of the infinite dimensional flat space $\\ell^2(\\mathds{C})$ admits a K\\\"ahler immersion into the infinite dimensional complex projective space.\nUmehara's work has been generalized in the recent paper by X. Cheng and A. J. Di Scala \\cite{fubinirelatives}, where the authors state necessary and sufficient conditions for \ncomplex space forms of finite dimension and different curvatures to not be relative to each others.\nIn \\cite{relatives} A. J. Di Scala and A. Loi prove that a Hermitian symmetric space of noncompact type endowed with its Bergman metric is not relative to a projective K\\\"ahler manifold, i.e. a K\\\"ahler manifold which admit a local holomorphic and isometric (from now on {\\em K\\\"ahler}) immersion into the {\\em finite} dimensional complex projective space (see also \\cite{huang} for the case of Hermitian symmetric spaces of noncompact type and Euclidean spaces), and their result has been generalized in \\cite{mossa} to homogeneous bounded domains of $\\mathds{C}^n$.\nThroughout the paper, we say that a K\\\"ahler manifold is {\\em projectively induced} when it admits a K\\\"ahler immersion into $\\mathds{C}{\\rm P}^{N\\leq\\infty}$. When we also specify that it is {\\em infinite projectively induced}, we mean that the K\\\"ahler immersion is full into $\\mathds{C}{\\rm P}^{\\infty}$.\n\nIn this paper we are interested in studying when a K\\\"ahler manifold $(M,g)$ is {\\em strongly not relative} to any projective K\\\"ahler manifold, that is, when $(M,c\\, g)$ is not relative to any projective K\\\"ahler manifold for any value of the constant $c>0$ multiplying the metric.\n\n\n Our first result can be viewed as a generalization of the results in \\cite{relatives, mossa} and can be stated as follows:\n\\begin{theor}\\label{hbd}\nLet $(M,g)$ be a K\\\"ahler manifold such that $(M,\\beta g)$ is infinite projectively induced for any $\\beta>\\beta_0\\geq 0$. If $(M,g)$ and $\\mathds{C}{\\rm P}^n$ are not relatives for any $n<\\infty$, then $(M,g)$ is strongly not relative to any projective K\\\"ahler manifold.\n\\end{theor}\nObserve that in general there are not reasons for a K\\\"ahler manifold which is not relative to another K\\\"ahler manifold to remain so when its metric is rescaled. For example, consider that the complex projective space $(\\mathds{C}{\\rm P}^2,c\\,g_{FS})$ where $g_{FS}$ is the Fubini--Study metric, for $c=\\frac23$ is not relative to $(\\mathds{C}{\\rm P}^2,g_{FS})$, while for positive integer values of $c$ it is (see \\cite{fubinirelatives} for a proof).\n\nIn order to state our second result, consider a $d$-dimensional K\\\"ahler manifold $(M,g)$ which admits global coordinates $\\{z_1,\\dots, z_d\\}$ and denote by $M_{j}$ the $1$-dimensional submanifold of $M$ defined by:\n$$\nM_j=\\{ z\\in M|\\, z_1=\\dots=z_{j-1}=z_{j+1}=\\dots=z_d=0\\}.\n$$\nWhen exists, a K\\\"ahler immersion $f\\!:M\\rightarrow \\mathds{C}{\\rm P}^\\infty$ is said to be {\\em transversally full} when for any $j=1,\\dots, d$, the immersion restricted to $M_{j}$ is full into $\\mathds{C}{\\rm P}^\\infty$.\n\n\\begin{theor}\\label{trfull}\nLet $(M, g)$ be a K\\\"ahler manifold infinite projectively induced through a transversally full map. If for any $\\alpha\\geq \\alpha_0>0$, $(M,\\alpha\\, g)$ is infinite projectively induced then $(M,g)$ is strongly not relative to any projective K\\\"ahler manifold.\n\\end{theor}\n\nAs a consequence of Theorem \\ref{hbd} and Theorem \\ref{trfull} we get that the $1$-parameter families of Bergman--Hartogs and Fock--Bargmann--Hartogs domains, which we describe in Section \\ref{bbh}, are strongly not relative to any projective K\\\"ahler manifold (see corollaries \\ref{chrel} and \\ref{fbhrel} below).\\\\\n\n\n\nThe paper consts of three more sections. In the first one we briefly recall the definition of diastasis function and its properties we need and in the second one we prove Theorem \\ref{hbd} and Theorem \\ref{trfull}. Finally, in the third and last section we apply our results to Bergman--Hartogs and Fock--Bargmann--Hartogs domains.\n\n\n\nThe author is very grateful to Prof. Andrea Loi for all the interesting discussions and comments that helped her to improve the contents and the exposition.\n\\section{Calabi's diastasis function}\nConsider a real analytic K\\\"ahler manifold $(M,g)$ and let $\\varphi\\!: U\\rightarrow \\mathds{R}$ be a K\\\"ahler potential for $g$ defined on a coordinate neighborhood $U$ around a point $p\\in M$. Consider the analytic extension $\\tilde\\varphi\\!: W\\rightarrow \\mathds{R}$, $\\tilde\\varphi(z,\\bar z)=\\varphi(z)$, of $\\varphi$ on a neighborhood $W$ of the diagonal in $U\\times \\bar U$. The {\\em diastasis function} ${\\rm D}(z,w)$ is defined by the formula:\n\\begin{equation}\\label{diast}\n{\\rm D}(z,w):=\\tilde\\varphi(z,\\bar z)+\\tilde\\varphi(w,\\bar w)-\\tilde\\varphi(z,\\bar w)-\\tilde\\varphi(w,\\bar z).\n\\end{equation}\nObserve that since:\n$$\n\\frac{\\partial^2}{\\partial z\\partial \\bar z}{\\rm D}(z,w)=\\frac{\\partial^2}{\\partial z\\partial \\bar z}\\tilde \\varphi\\left(z,\\bar z\\right)=\\frac{\\partial^2}{\\partial z\\partial \\bar z}\\varphi\\left(z\\right),\n$$\nonce one of its two entries is fixed, the diastasis is a K\\\"ahler potential for $g$. We denote by ${\\rm D}_0(z)$ the diastasis centered at the origin. The following theorem due to Calabi \\cite{calabi}, expresses the diastasis' property which is fundamental for our purpose.\n\\begin{theor}[E. Calabi]\\label{induceddiast}\n Let $(M, g)$ and $(S,G)$ be K\\\"ahler manifolds and assume $G$ to be real analytic. Denote by $\\omega$ and $\\Omega$ the K\\\"ahler forms associated to $g$ and $G$ respectively. If there exists a holomorphic map $f\\!:(M,g)\\rightarrow (S,G)$ such that $f^*\\Omega=\\omega$, then the metric $g$ is real analytic. Further, denoted by ${\\rm D}^M_p\\!:U\\rightarrow \\mathds R$ and ${\\rm D}^S_{f(p)}\\!:V\\rightarrow\\mathds R$ the diastasis functions of $(M,g)$ and $(S,G)$ around $p$ and $f(p)$ respectively, we have ${\\rm D}_{f(p)}^S\\circ f={\\rm D}^M_p$ on $f^{-1}(V)\\cap U$.\n\\end{theor}\n\n\nConsider the complex projective space $\\mathds{C}{\\rm P}^N_b$ of complex dimension $N\\leq \\infty$, with the Fubini-Study metric $g_{b}$ of holomorphic bisectional curvature $4b$ for $b>0$. When $b=1$ we denote by $g_{FS}$ and $\\omega_{FS}$ the Fubini-Study metric and the Fubini-Study form respectively. Let $[Z_0,\\dots,Z_N]$ be homogeneous coordinates,\n$p=[1,0,\\dots,0]$ and $U_0=\\{Z_0\\neq 0\\}$. Define affine coordinates $z_1,\\dots, z_N$ on $U_0$ by $z_j=Z_j\/(\\sqrt{b}Z_0)$. The diastasis on $U_0$ centered at the origin reads:\n\\begin{equation}\\label{diastcp}\n{\\rm D}^b_0(z)=\\frac{1}{b}\\log\\left(1+b\\sum_{j=1}^N|z_j|^2\\right).\n\\end{equation}\n\nDue to Th. \\ref{induceddiast} and the expression of $\\mathds{C}{\\rm P}^N_b$'s diastasis \\eqref{diastcp}, if $f\\!:S\\rightarrow \\mathds{C}{\\rm P}^N_b$ is a holomorphic map, $f(z)=[f_0(z),f_1(z),\\dots, f_N(z)]$, then the induced diastasis ${\\rm D}^S_0$ in a neighborhood of a point $p\\in S$ is given by:\n$$\n{\\rm D}^S_0(z)=\\frac{1}{b}\\log\\left(1+b\\sum_{j=1}^N|f_j(z)|^2\\right).\n$$\nFurther, if the K\\\"ahler map $f$ is assumed to be {\\em full}, i.e. the image $f(S)$ is not contained into any lower dimensional totally geodesic submanifold of $\\mathds{C}{\\rm P}_b^N$, then $f$ is univocally determined up to rigid motion of $\\mathds{C}{\\rm P}^N_b$ \\cite[pp. 18]{calabi}:\n\\begin{theor}[Calabi's Rigidity]\\label{local rigidityb}\nIf a neighborhood $V$ of a point $p$ admits a full K\\\"ahler immersion into $(\\mathds{C}{\\rm P}^N_b,g_b)$, then $N$ is univocally determined by the metric and the immersion is unique up to rigid motions of $(\\mathds{C}{\\rm P}^N_b,g_b)$.\n\\end{theor} \nObserve that by Th. \\ref{local rigidityb} above, a K\\\"ahler manifold which is infinite projectively induced does not admit a K\\\"ahler immersion into any finite dimensional complex projective space.\n\n\n\\section{Proof of Theorems \\ref{hbd} and \\ref{trfull}}\n\\begin{proof}[Proof of Theorem \\ref{hbd}]\nObserve first that due to Th. \\ref{induceddiast} it is enough to prove that $(M, c\\,g)$ is not relative to $\\mathds{C}{\\rm P}^n$ for any finite $n$ and any $c>0$. For any $c>0$, we can choose a positive integer $\\alpha$ such that $c\\alpha>\\beta_0$. Denote by $\\omega$ the K\\\"ahler form on $M$ associated to $g$. Let $F\\!:M\\rightarrow \\mathds{C}{\\rm P}^\\infty$ be a full K\\\"ahler map such that $F^*\\omega_{FS}=c\\alpha\\, \\omega$. Then $\\tilde F=F\/\\sqrt{\\alpha}$ is a K\\\"ahler map of $(M,c\\, g)$ into $ \\mathds{C}{\\rm P}^\\infty_\\alpha$.\n Let $S$ be a common K\\\"ahler submanifold of $(M,c\\,g)$ and $\\mathds{C}{\\rm P}^n$. Then by Th. \\ref{induceddiast} for any $p\\in S$ there exist a neighborhood $U$ and two holomorhic maps $f\\!:U\\rightarrow M$ and $h\\!:U\\rightarrow \\mathds{C}{\\rm P}^n$, such that $f^*(c\\omega)|_U=( \\tilde F\\circ f)^*\\omega_{FS}|_U=h^*\\omega_{FS}|_U$.\n \nThus, by \\eqref{diastcp} one has:\n$$\n\\log\\left(1+\\sum_{j=1}^n|h_j|^2\\right)=\\frac1\\alpha\\log\\left(1+\\sum_{j=1}^\\infty|(F\\circ f)_j)|^2\\right).\n$$\ni.e.:\n\\begin{equation}\\label{contradiction}\n\\alpha\\log\\left(1+\\sum_{j=1}^n|h_j|^2\\right)=\\log\\left(1+\\sum_{j=1}^\\infty|(F\\circ f)_j)|^2\\right).\n\\end{equation}\nSince $F\\circ f$ is full and $\\alpha$ is a positive integer, this last equality and Calabi rigidity Theorem \\ref{local rigidityb} imply $n=\\infty$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{trfull}]\nDue to Th. \\ref{hbd} and Th. \\ref{induceddiast} we need only to prove that a if a K\\\"ahler manifold is infinite projectively induced through a transversally full immersion then it is not relative to $\\mathds{C}{\\rm P}^n$ for any $n$. \nAssume that $S$ is a $1$-dimensional K\\\"ahler submanifold of both $\\mathds{C}{\\rm P}^n$ and $(M, g)$.\nThen around each point $p\\in S$ there exist an open neighborhood $U$ and two holomorphic maps $\\psi\\!:U\\rightarrow \\mathds{C}{\\rm P}^n$ and $\\varphi\\!:U\\rightarrow M$, $\\varphi(\\xi)=(\\varphi_1(\\xi),\\dots,\\varphi_d(\\xi))$ where $\\xi$ are coordinates on $U$, such that $\\psi^*\\omega_{FS}|_U=\\varphi^*(c\\omega)|_U$. Without loss of generality we can assume $\\frac{\\partial\\varphi_1(\\xi)}{\\partial \\xi}(0)\\neq 0$.\n Let $f\\!:M\\rightarrow \\mathds{C}{\\rm P}^\\infty$ be a K\\\"ahler map from $(M, g)$ into $\\mathds{C}{\\rm P}^\\infty$. Since by assumption $f$ is transversally full, $f=[f_0,\\dots, f_j,\\dots]$ contains for any $m=1,2,3,\\dots$, a subsequence $\\left\\{f_{j_1},\\dots, f_{j_m}\\right\\}$ of functions which restricted to $M_1$ are linearly independent.\nThe map $f\\circ\\varphi\\!:U\\rightarrow \\mathds{C}{\\rm P}^\\infty$ is full, in fact $f|_{M_1}\\circ \\varphi$ is full since $\\varphi_1(\\xi)$ is not constant and for any $m=1,2,3,\\dots$, $\\left\\{f_{j_1}(\\varphi_1(\\xi)),\\dots, f_{j_m}(\\varphi_1(\\xi))\\right\\}$ is a subsequence of $\\{f|_{M_1}\\circ \\varphi\\}$ of linearly independent functions. Conclusion follows by Calabi's rigidity Theorem \\ref{local rigidityb}.\n\\end{proof}\n\n\n\n\n\\section{Applications}\\label{bbh}\n\nLet $(\\Omega, \\beta g_B)$, $\\beta >0$, denote a bounded domain of $\\mathds{C}^d$ endowed with a positive multiple of its Bergman metric $g_B$. Recall that $g_B$ is the K\\\"ahler metric on $\\Omega$ whose associated K\\\"ahler form $\\omega_B$ is given by $\\omega_B=\\frac{i}{2}\\partial\\bar\\partial\\log \\K(z,z)$, where $\\K(z,z)$ is the reproducing kernel for the Hilbert space:\n$$\\mathcal{H} =\\left\\{\\varphi\\in{\\rm hol}(\\Omega),\\ \\int_\\Omega |\\varphi|^{2}\\ \\frac{\\omega_0^d}{d!}<\\infty\\right\\},$$\nwhere $\\omega_0=\\frac{i}{2} \\sum_{j=1}^d dz_j\\wedge d\\bar z_j$ is the standard K\\\"ahler form of $\\mathds{C}^d$. It follows by \\eqref{diast} that the diastasis function for $g_B$ is given by:\n\\begin{equation}\\label{diastomega}\n{\\rm D}_0^\\Omega(z)=\\log\\frac{\\K(z,z)\\K(0,0)}{|\\K(z,0)|^2}.\n\\end{equation}\nObserve that the Bergman metric $g_B$ admits a natural K\\\"ahler immersion into the infinite dimensional complex projective space (cfr. \\cite{kodomain}). More precisely, if $\\K(z,z)=\\sum_{j=0}^\\infty |\\varphi_j(z)|^2$, the map:\n\\begin{equation}\\label{bergmanimm}\n\\varphi\\!:\\Omega\\rightarrow\\mathds{C}{\\rm P}^\\infty,\\quad \\varphi=(\\varphi_0,\\dots, \\varphi_j,\\dots),\n\\end{equation}\nis a K\\\"ahler immersion of $(\\Omega, g_B)$ into $\\mathds{C}{\\rm P}^\\infty$, for $\\varphi^*g_{FS}=g_B$, as it follows by:\n$$\n\\omega_B=\\frac i2\\partial\\bar\\partial\\log(\\K(z,z))=\\frac i2\\partial\\bar\\partial\\log\\left(\\sum_{j=0}^\\infty |\\varphi_j(z)|^2\\right)=\\varphi^*\\omega_{FS}.\n$$\nFurther, such immersion is full since $\\{\\varphi_j\\}$ is a basis for the Hilbert space $\\mathcal H$ and a bounded domain does not admit a K\\\"ahler immersion into a finite dimensional complex projective space even when the metric is rescaled. Although the existence of a K\\\"ahler immersion of $(\\Omega, \\beta g_B)$ into $\\mathds{C}{\\rm P}^\\infty$ is strictly related to the constant $\\beta$ which multiplies the metric (see \\cite{articwall} for the case when $\\Omega$ is symmetric). In \\cite{ishi} it is proven that the only homogeneous bounded domain which is projectively induced for all positive values of the constant multiplying the metric is a product of complex hyperbolic spaces.\nAlthough, the property of being projectively induced for a large enough constant is not so unusual and the following holds \\cite{loimossaber}:\n\\begin{theor}[A. Loi, R. Mossa]\\label{loimossaimm}\nLet $(\\Omega,g)$ be a homogeneous bounded domain. Then, there exists $\\alpha_0>0$ such that $(\\Omega,\\alpha g)$ is projectively induced for any $\\alpha\\geq\\alpha_0>0$. \n\\end{theor}\nNotice that it is an open question if the same statement holds dropping the homogeneous assumption. \n\nRegarding the property of being relative to some projective K\\\"ahler manifold, we recall the following result due to A. J. Di Scala and A. Loi in \\cite{relatives}, which plays a key role in the proof of Corollary \\ref{chrel}.\n\\begin{theor}[A. J. Di Scala, A. Loi ]\\label{loidiscalahbd}\nA bounded domain of $\\mathds{C}^n$ endowed with its Bergman metric and a projective K\\\"ahler manifold are not relatives.\n\\end{theor}\nObserve that due to theorems \\ref{loimossaimm} and \\ref{loidiscalahbd}, Theorem \\ref{hbd} implies that a bounded domain of $\\mathds{C}^n$ endowed with its Bergman metric and a projective K\\\"ahler manifold are {\\em strongly} not relatives. Althought, this result has been proven in a more general context by R. Mossa in \\cite{mossa}, where he shows that a homogeneous bounded domain and a projective K\\\"ahler manifold are not relatives.\\\\\n\n\nLet us now describe the family of Bergman--Hartogs domains. \nFor all positive real numbers $\\mu$ a {\\em Bergman-Hartogs domain} is defined by:\n\\begin{equation}\\label{defm}\nM_{\\Omega}(\\mu)=\\left\\{(z,w)\\in \\Omega\\times\\mathds{C},\\ |w|^2<\\tilde \\K(z, z)^{-\\mu}\\right\\},\n\\end{equation}\nwhere $\\tilde \\K(z, z)=\\frac{\\K(z,z)\\K(0,0)}{|\\K(z,0)|^2}$ with $\\K$ the Bergman kernel of $\\Omega$.\nConsider on $M_{\\Omega}(\\mu)$ the metric $g(\\mu)$ whose associated K\\\"ahler form $\\omega(\\mu)$ can be described by the (globally defined) K\\\"ahler potential centered at the origin\n\\begin{equation}\\label{diastM}\n\\Phi(z,w)=-\\log(\\tilde\\K(z, z)^{-\\mu}-|w|^2).\n\\end{equation}\nThe domain $\\Omega$ is called the {\\em base} of the Bergman--Hartogs domain \n$M_{\\Omega}(\\mu)$ (one also says that \n$M_{\\Omega}(\\mu)$ is based on $\\Omega$). Observe that these domains include and are a natural generalization of Cartan--Hartogs domains which have been studied under several points of view (see e.g. \\cite{fengtubalanced,berezinCH} and references therein). To the author knowledge, Bergman-Hartogs domains has been already considered in \\cite{hao,hao2,hao3}.\n\nIn \\cite{articwall} the author of the present paper jointly with A. Loi proved that when the base domain is symmetric $(M_{\\Omega}(\\mu),c\\,g(\\mu))$ admits a K\\\"ahler immersion into the infinite dimensional complex projective space if and only if $(\\Omega, (c+m)\\mu g_B)$ does for every integer $m\\geq0$. As pointed out in \\cite{hao}, a totally similar proof holds also when the base is a homogeneous bounded domain. This fact together with Theorem \\ref{loimossaimm} proves that a Bergman--Hartogs domain $(M_{\\Omega}(\\mu),c\\,g(\\mu))$ is projectively induced for all large enough values of the constant $c$ multiplying the metric. Further, the immersion can be written explicitely as follows (cfr. \\cite[Lemma 8]{balancedch}):\n\\begin{lemma}\\label{chimm}\nLet $\\alpha$ be a positive real number such that the Bergman--Hartogs domain $(M_{\\Omega}(\\mu),\\alpha\\, g(\\mu))$ is projectively induced. Then, the K\\\"ahler map $f$ from $(M_{\\Omega}(\\mu),\\alpha\\,g(\\mu))$ into $\\mathds{C}{\\rm P}^\\infty$, up to unitary transformation of $\\mathds{C}{\\rm P}^\\infty$, is given by:\n\\begin{equation}\\label{immf}\nf=\\left[ 1, s, h_{\\mu\\, \\alpha},\\dots,\\sqrt{\\frac{(m+ \\alpha-1)!}{(\\alpha-1)!m!}}h_{\\mu(\\alpha +m)}w^m,\\dots\\right],\n\\end{equation}\nwhere $s=(s_1,\\dots, s_m,\\dots)$ with\n$$s_m=\\sqrt{\\frac{(m+ \\alpha-1)!}{(\\alpha-1)!m!}}w^m,$$\nand $h_k=(h_k^1,\\dots,h_k^j,\\dots)$ denotes the sequence of holomorphic maps on $\\Omega$ such that the immersion $\\tilde h_k=(1,h_k^1,\\dots, h_k^j,\\dots)$, $\\tilde h_k\\!:\\Omega\\rightarrow\\mathds{C}{\\rm P}^\\infty$, satisfies $\\tilde h_k^*\\omega_{FS}=k \\omega_B$, i.e. \n\\begin{equation\n1+\\sum_{j=1}^{\\infty}|h_k^j|^2=\\tilde \\K^{-k}.\\nonumber\n\\end{equation} \n\\end{lemma}\n\\begin{proof}\nThe proof follows essentially that of \\cite[Lemma 8]{balancedch} once considered that $\\Phi(z,w)=-\\log(\\tilde \\K(z, z)^{-\\mu}-|w|^2)$ is the diastasis function for $(M_\\Omega(\\mu), g(\\mu))$ as follows readily applying \\eqref{diast}.\n\\end{proof}\n\nObserve that such map is full, as can be easily seen for example by considering that for any $m=1,2,3,\\dots,$ the subsequence $\\{s_1,\\dots, s_m\\}$ is composed by linearly independent functions.\n\nAs a consequence of theorems \\ref{hbd}, \\ref{trfull}, \\ref{loidiscalahbd} and Lemma \\ref{chimm}, we get the following:\n\\begin{cor}\\label{chrel}\nFor any $\\mu>0$, a Bergman--Hartogs domain $(M_\\Omega(\\mu), g(\\mu))$ is strongly not relative to any projective manifold. \n\\end{cor}\n\\begin{proof}\nObserve first that due to Th. \\ref{induceddiast} it is enough to prove that $(M_\\Omega(\\mu),\\alpha g(\\mu))$ is not relative to $\\mathds{C}{\\rm P}^n$ for any finite $n$. Further, by Th. \\ref{hbd} and Th. \\ref{loidiscalahbd}, a common submanifold $S$ of both $(M_\\Omega(\\mu),\\alpha g(\\mu))$ and $\\mathds{C}{\\rm P}^n$ is not contained into $(\\Omega,\\alpha g(\\mu)|_\\Omega)$, since $\\alpha g(\\mu)|_\\Omega=\\frac{\\alpha\\mu}\\gamma g_B$ is a multiple of the Bergman metric on $\\Omega$. Thus, due to arguments totally similar to those in the proof of Th. \\ref{trfull}, it is enough to check that the K\\\"ahler immersion $f\\!:M_\\Omega(\\mu)\\rightarrow \\mathds{C}{\\rm P}^\\infty$ is transversally full with respect to the $w$ coordinate. Conclusion follows then by \\eqref{immf}. \n\\end{proof}\n\n\nFinally, we describe what we need about the $1$-parameter family of Fock--Bargmann--Hartogs domains, referring the reader to \\cite{fbh} and reference therein for details and further results. For any value of $\\mu>0$, a Fock--Bargmann--Hartogs domain $D_{n,m}(\\mu)$ is a strongly pseudoconvex, nonhomogeneous unbounded domains in $\\mathds{C}^{n+m}$ with smooth real-analytic boundary, given by:\n$$\nD_{n,m}(\\mu):=\\{(z,w)\\in \\mathds{C}^{n+m}: ||w||^2< e^{-\\mu||z||^2}\\}.\n$$\nOne can define a K\\\"ahler metric $\\omega(\\mu;\\nu)$, $\\nu>-1$ on $D_{n,m}(\\mu)$ through the globally defined K\\\"ahler potential:\n$$\n\\Phi(z,w):=\\nu\\mu||z||^2-\\log(e^{-\\mu||z||^2}-||w||^2).\n$$\nIn \\cite{fbh}, E. Bi, Z. Feng and Z. Tu prove that when $n=1$ and $\\nu=-\\frac{1}{m+1}$, the metric $\\omega(\\mu;\\nu)$ is infinite projectively induced whenever it is rescaled by a big enough constant. More precisely they prove the following:\n\\begin{theor}[E. Bi, Z. Feng, Z. Tu]\\label{fbhth}\nThe metric $\\alpha g(\\mu;\\nu)$ on the Fock--Bargmann--Hartogs domain $D_{n,m}(\\mu)$ is balanced if and only if $\\alpha>m+n$, $n=1$, $\\nu=-\\frac1{m+1}$.\n\\end{theor}\nRecall that a balanced K\\\"ahler metric is a particular projectively induced metric such that the immersion map is defined by a orthonormal basis of a weighted Hilbert space (see e.g. \\cite{balancedch}). \n\nIn order to apply Th. \\ref{trfull} to Fock--Bargmann--Hartogs domains we need the following lemma:\n\\begin{lemma}\\label{focktr}\nFor any $\\mu>0$ and any $\\alpha>m+1$, a Fock--Bargmann--Hartogs domain $\\left(D_{1,m}(\\mu),\\alpha\\omega(\\mu;-\\frac1{m+1})\\right)$ admits a transversally full K\\\"ahler immersion into $\\mathds{C}{\\rm P}^\\infty$.\n\\end{lemma}\n\\begin{proof}\nA K\\\"ahler immersion exists due to Th. \\ref{fbhth}. In order to see that it is transversally full, observe that when $w_1=\\dots= w_m=0$, $\\alpha\\omega(\\mu;-\\frac1{m+1})|_{M_1}$ is a multiple of the flat metric, and when only one $w_j$ is different from zero $\\alpha\\omega(\\mu;-\\frac1{m+1})|_{M_j}$ is a multiple of the hyperbolic metric.\n\\end{proof}\n\\begin{cor}\\label{fbhrel}\nFor any $\\mu>0$, a Fock--Bargmann--Hartogs domain $\\left(D_{1,m}(\\mu),\\omega(\\mu;-\\frac1{m+1})\\right)$ is strongly not relative to any projective manifold. \n\\end{cor}\n\\begin{proof}\nIf follows directly from Th. \\ref{trfull} and Lemma \\ref{focktr}.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}