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+{"text":"\\section{Introduction}\n\nIt is well-known that the space of ordinary differential operators of\nthe form $\\pa^n + u_1 \\pa^{n-2} + \\ldots + u_{n-1}$ has a remarkable\nPoisson structure, often called the (second) Adler-Gelfand-Dickey\nbracket \\cite{A,GD}. Drinfeld-Sokolov reduction \\cite{DS} gives a\nnatural realization of this Poisson structure via the hamiltonian\nreduction of the dual space to the affine Kac-Moody algebra\n$\\sun$. Drinfeld and Sokolov \\cite{DS} have applied an analogous\nreduction procedure to the dual space of the affinization $\\G$ of an\narbitrary semisimple Lie algebra $\\g$. The Poisson algebra ${\\cal\nW}(\\g)$ of functionals on the corresponding reduced space is called\nthe classical ${\\cal W}$--algebra. Thus, one can associate a classical\n${\\cal W}$--algebra to an arbitrary semisimple Lie algebra $\\g$. In\nparticular, the classical ${\\cal W}$--algebra associated to $\\sw_2$ is\nnothing but the classical Virasoro algebra, i.e. the Poisson algebra\nof functionals on the dual space to the Virasoro algebra (see, e.g.,\n\\cite{FR}).\n\nIt is interesting that ${\\cal W}(\\g)$ admits another description as\nthe center of the universal enveloping algebra of an affine\nalgebra. More precisely, let $Z(\\widehat{{\\frak g}})_{-h^{\\vee }}$ be\nthe center of a completion of the universal enveloping algebra\n$U(\\G)_{-\\k}$ at the critical level $k=-h^{\\vee }$ (minus the dual\nCoxeter number). This center has a canonical Poisson structure. It was\nconjectured by V.~Drinfeld and proved by B.~Feigin and E.~Frenkel\n\\cite{FF,EF} that as Poisson algebra $Z(\\G)_{-\\k}$ is isomorphic to the\nclassical ${\\cal W}$--algebra ${\\cal W}(^L\\!\\g)$ associated with the\nLanglands dual Lie algebra $\\gL$ of $\\g$.\n\nIn \\cite{FR} two of the authors used this second realization of\n$\\W$--algebras to obtain their $q$--deformations. For instance the\n$q$--deformation $\\W_q(\\sw_n)$ of ${\\cal W}(\\sw_n)$ was defined as the\ncenter $Z_q(\\sun)$ of a completion of the quantized universal\nenveloping algebra $U_q(\\sun)_{-\\k}$. The Poisson structure on\n$Z_q(\\sun)$ was explicitly described in \\cite{FR} using results of\n\\cite{ext}. It was shown that the underlying Poisson manifold of\n$Z_q(\\sun)={\\cal W}_q(\\sw_n)$ is the space of $q$--difference\noperators of the form $D_q^n + t_1 D_q^{n-1} + \\ldots + t_{n-1} D_q +\n1$. Furthermore, in \\cite{FR} a $q$--deformation of the Miura\ntransformation, i.e. a homomorphism from $\\W_q(\\sw_n)$ to a\nHeisenberg-Poisson algebra was defined. The construction \\cite{FR} of\n${\\cal W}_q(\\sw_n)$ was followed by further developments: it was\nquantized \\cite{SKAO,ell,AKOS} and the quantum algebra was used in the\nstudy of lattice models \\cite{LP,Miwa}; Yangian analogue of ${\\cal\nW}_q(\\sw_2)$ was considered in \\cite{Hou}; $q$--deformations of the\ngeneralized KdV hierarchies were introduced \\cite{Fr}.\n\nIn this paper we first formulate the results of \\cite{FR} in terms of\nfirst order $q$--difference operators and $q$--gauge action. This\nnaturally leads us to a generalization of the Drinfeld-Sokolov scheme\nto the setting of $q$--difference operators. The initial Poisson\nmanifold is the loop group $LSL_n$ of $SL_n$, or more generally, the\nloop group of a simply-connected simple Lie group $G$. Much of the\nneeded Poisson formalism has already been developed by one of the\nauthors in \\cite{RIMS,dual}. Results of these works allow us to\ndefine a Poisson structure on the loop group, with respect to which the\n$q$--gauge action is Poisson. We then have to perform a reduction of\nthis Poisson manifold with respect to the $q$--gauge action of the\nloop group $LN$ of the unipotent subgroup $N$ of $G$.\n\nAt this point we encounter a new kind of anomaly in the Poisson\nbracket relations, unfamiliar from the linear, i.e. undeformed,\nsituation. To describe it in physical terms, recall that the reduction\nprocedure consists of two steps: (1) imposing the constraints and (2)\npassing to the quotient by the gauge group. An important point in the\nordinary Drinfeld-Sokolov reduction is that these constraints are of\nfirst class, according to Dirac, i.e., their Poisson bracket vanishes\non the constraint surface. In the $q$--difference case we have to\nchoose carefully the classical $r$--matrix defining the initial\nPoisson structure on the loop group so as to make all constraints\nfirst class. If we use the standard $r$--matrix, some of the\nconstraints are of second class, and so we have to modify it.\n\nIn this paper we do that in the case of $SL_2$. We show that there is\nessentially a unique classical $r$--matrix compatible with the\n$q$--difference Drinfeld-Sokolov scheme. To the best of our knowledge,\nthis classical $r$--matrix is new; it yields an elliptic deformation\nof the Lie bialgebra structure on the loop algebra of $\\sw_2$\nassociated with the Drinfeld ``new'' realization of quantized affine\nalgebras \\cite{nr}, \\cite{kh-t}. The result of the Drinfeld-Sokolov\nreduction is the $q$--deformation of the classical Virasoro algebra\ndefined in \\cite{FR}.\n\nWe also construct a finite difference version of the Drinfeld-Sokolov\nreduction in the case of $SL_2$. This construction gives us a discrete\nversion of the (classical) Virasoro algebra. We explain in detail the\nconnection between our discrete Virasoro algebra and the lattice\nVirasoro algebra of Faddeev--Takhtajan--Volkov \\cite{ft,v1,v2,fv}. We\nhope that our results will help to clarify further the meaning of the\ndiscrete Virasoro algebra and its relation to various integrable\nmodels.\n\nThe construction presented here can be generalized to the case of an\narbitrary simply-connected simple Lie group. This is done in the\nsecond part of the paper \\cite{SS} written by A.~Sevostyanov and one\nof us.\n\nThe paper is arranged as follows. In Sect.~2 we recall the relevant\nfacts of \\cite{DS} and \\cite{FR}. In Sect.~3 we interpret the results\nof \\cite{FR} from the point of view of $q$--gauge transformations.\nSect.~4 reviews some background material on Poisson structures on Lie\ngroups following \\cite{RIMS,dual}. In Sect.~5 we apply the results of\nSect.~4 to the $q$--deformation of the Drinfeld-Sokolov reduction in\nthe case of $SL_2$. In Sect.~6 we discuss the finite difference\nanalogue of this reduction and compare its results with the\nFaddeev-Takhtajan-Volkov algebra.\n\n\\subsubsection*{Acknowledgements.} E.~Frenkel thanks\nP.~Schapira for his hospitality at Universit\\'{e} Paris VI in June of\n1996, when this collaboration began. Some of the results of this paper\nhave been reported in E.~Frenkel's lecture course on Soliton Theory\ngiven at Harvard University in the Spring of 1996.\n\nThe research of E.~Frenkel was supported by grants from the Packard\nand Sloan Foundations, and by the NSF grants DMS 9501414 and DMS\n9304580. The research of N.~Reshetikhin was supported by the NSF grant\nDMS 9296120.\n\n\\section{Preliminaries}\n\n\\subsection{The differential Drinfeld-Sokolov reduction in the case of\n$\\sw_n$}    \\label{diff1}\n\nLet ${\\cal M}_n$ be the manifold of differential operators of the form\n\\beq\n\\label{L} L = \\pa^n + u_1(s) \\pa^{n-2} + \\ldots +\nu_{n-2}(s) \\pa + u_{n-1}(s),\n\\end{equation}\nwhere $u_i(s) \\in \\C((s))$.\n\nAdler \\cite{A} and Gelfand-Dickey \\cite{GD} have defined a remarkable\ntwo-parameter family of Poisson structures on ${\\cal M}_n$, with\nrespect to which the corresponding KdV hierarchy is hamiltonian. In\nthis paper we will only consider one of them, the so-called second\nbracket. There is a simple realization of this structure in terms of\nthe Drinfeld-Sokolov reduction \\cite{DS}, Sect.~6.5. Let us briefly\nrecall this realization.\n\nConsider the affine Kac-Moody algebra $\\sun$ associated to $\\sw_n$;\nthis is the central extension $$0 \\arr \\C K \\arr \\sun \\arr \\lsln \\arr\n0,$$ see \\cite{Kac}. Let $M_n$ be the hyperplane in the dual space to\n$\\sun$, which consists of linear functionals taking value $1$ on\n$K$. Using the differential $dt$ and the bilinear form $\\on{tr} AB$ on\n$\\sw_n$, we identify $M_n$ with the manifold of first order\ndifferential operators\n$$\\pa_s + A(s), \\quad \\quad A(s) \\in \\lsln.$$ The coadjoint action of\nthe Lie group $\\wh{SL}_n$ on $\\sun^*$ factors through the loop group\n$LSL_n$ and preserves the hyperplane $M_n$. The corresponding action\nof $g(s) \\in LSL_n$ on $M_n$ is given by \\beq \\label{coadjoint} g(s)\n\\cdot (\\pa_s + A(s)) = g(s)(\\pa_s + A(s))g(s)^{-1},\n\\end{equation} or $$A(s) \\mapsto g(s) A(s) g(s)^{-1} - \\pa_s g(s) \\cdot\ng(s)^{-1}.$$\n\nConsider now the submanifold $M^J_n$ of $M_n$ wich consists of\noperators $\\pa_s + A(s)$, where $A(s)$ is a traceless matrix of the form\n\\beq    \\label{upper}\n\\begin{pmatrix}\n* & * & * & \\hdots & * & * \\\\\n-1 & * & * & \\hdots & * & * \\\\\n0 & -1 & * & \\hdots & * & * \\\\\n\\hdotsfor{6} \\\\\n0 & 0 & 0 & \\hdots & * & * \\\\\n0 & 0 & 0 & \\hdots & -1 & *\n\\end{pmatrix}\n\\end{equation}\n\nTo each element ${\\cal L}$ of $M^J_n$ one can naturally attach an\n$n$th order scalar differential operator as follows. Consider the\nequation ${\\cal L} \\cdot \\Psi = 0$, where $$\\Psi = \\begin{pmatrix}\n\\Psi_n \\\\ \\Psi_{n-1} \\\\ \\hdots \\\\ \\Psi_1 \\end{pmatrix}.$$\nDue to the special form \\eqref{upper} of ${\\cal L}$, this equation is\nequivalent to an $n$th order differential equation $L \\cdot \\Psi_1 =\n0$, where $L$ is of the form \\eqref{L}. Thus, we obtain a map $\\pi:\nM^J_n \\arr \\M$ sending ${\\cal L}$ to $L$.\n\nLet $N$ be the subgroup of $SL_n$ consisting of the upper triangular\nmatrices, and $LN$ be its loop group. If $g \\in LN$ and $\\Psi$ is a\nsolution of ${\\cal L} \\cdot \\Psi = 0$, then $\\Psi'= g \\Psi$ is a\nsolution of ${\\cal L}' \\cdot \\Psi' = 0$, where ${\\cal L}' = g {\\cal L}\ng^{-1}$. But $\\Psi_1$ does not change under the action of\n$LN$. Therefore $\\pi({\\cal L}') = \\pi({\\cal L})$, and we see that\n$\\pi$ factors through the quotient of $M^J_n$ by the action of\n$LN$. The following proposition describes this quotient.\n\n\\begin{prop}[\\cite{DS}, Proposition 3.1]    \\label{free}\nThe action of $LN$ on $M^J_n$ is free, and each orbit contains a\nunique operator of the form\n\\beq    \\label{special}\n\\pa_s + \\begin{pmatrix}\n0 & u_1 & u_2 & \\hdots & u_{n-2} & u_{n-1} \\\\\n-1 & 0 & 0 & \\hdots & 0 & 0 \\\\\n0 & -1 & 0 & \\hdots & 0 & 0 \\\\\n\\hdotsfor{6} \\\\\n0 & 0 & 0 & \\hdots & 0 & 0 \\\\\n0 & 0 & 0 & \\hdots & -1 & 0\n\\end{pmatrix}.\n\\end{equation}\n\\end{prop}\n\nBut for ${\\cal L}$ of the form \\eqref{special}, $\\pi({\\cal L})$ is\nequal to the operator $L$ given by formula \\eqref{L}. Thus, we have\nidentified the map $\\pi$ with the quotient of $M^J_n$ by $LN$ and\nidentified ${\\cal M}_n$ with $M^J_n\/LN$.\n\nThe quotient $M^J_n\/LN$ can actually be interpreted as the result of\nhamiltonian reduction. The manifold $M_n$ has a canonical Poisson\nstructure, which is the restriction of the Lie-Poisson structure on\n$\\sun^*$ (such a structure exists on the dual space to any Lie\nalgebra). The coadjoint action of $LN$ on $M_n$ is hamiltonian with\nrespect to this structure. The corresponding moment map $\\mu: M_n \\arr\nL\\n_- \\simeq L\\n^*$ sends $\\pa_s + A(s)$ to the lower-triangular part\nof $A(s)$. Consider the one-point orbit of $LN$,\n$$J = \\begin{pmatrix} 0 & 0 & 0 & \\hdots & 0 & 0 \\\\ -1 & 0 & 0 &\n\\hdots & 0 & 0 \\\\ 0 & -1 & 0 & \\hdots & 0 & 0 \\\\ \\hdotsfor{6} \\\\ 0 & 0\n& 0 & \\hdots & 0 & 0 \\\\ 0 & 0 & 0 & \\hdots & -1 & 0\n\\end{pmatrix}.$$ Then $M^J_n = \\mu^{-1}(J)$. Hence $M_n$ is the\nresult of hamiltonian reduction of $M_n$ by $LN$ with respect to the\none-point orbit $J$.\n\nThe Lie-Poisson structure on $M_n$ gives rise to a canonical Poisson\nstructure on ${\\cal M}_n$, which coincides with the second\nAdler-Gelfand-Dickey bracket, see \\cite{DS}, Sect.~6.5. The Poisson\nalgebra of local functionals on ${\\cal M}_n$ is called the classical\n$\\W$--algebra associated to $\\sw_n$, and is denoted by $\\W(\\sw_n)$.\n\n\\begin{rem}    \\label{kostant}\nFor $\\al \\in \\C$, let $M_{\\al,n}$ be the hyperplane in the dual space\nto $\\sun$, which consists of linear functionals on $\\sun$ taking value\n$\\al$ on $K$. In the same way as above (for $\\al=1$) we identify\n$M_{\\al,n}$ with the space of first order differential operators\n$$\\al \\pa_s + A(s), \\quad \\quad A(s) \\in \\lsln.$$ The coadjoint action\nis given by the formula $$A(s) \\mapsto g(s) A(s) g(s)^{-1} - \\al \\pa_s\ng(s) \\cdot g(s)^{-1}.$$ The straightforward generalization of\n\\propref{free} is true for any $\\al \\in \\C$. In particular, for\n$\\al=0$ we obtain a description of the orbits in $M_n^J$ under the\nadjoint action of $LN$. This result is due to B.~Kostant \\cite{Ko}.\n\nDrinfeld and Sokolov \\cite{DS} gave a generalization of \\propref{free}\nwhen $\\sw_n$ is replaced by an arbitrary semisimple Lie algebra\n$\\g$. The special case of their result, corresponding to $\\al=0$,\nis also due to Kostant \\cite{Ko}.\\qed\n\\end{rem}\n\nThe Drinfeld-Sokolov reduction can be summarized by the following\ndiagram.\n\n\\setlength{\\unitlength}{1mm}\n\n\\begin{center}\n\\begin{picture}(125,40)(-35,-33)     \n\\put(0,-1){\\vector(0,-1){26}}\n\\put(62,-1){\\vector(0,-1){26}}\n\\put(9,3){\\vector(1,0){45}}\n\\put(9,-31){\\vector(1,0){45}}\n\\put(8.9,3.9){\\oval(1,1)[l]\n\\put(8.9,-30.1){\\oval(1,1)[l]\n\\put(-2,2){$M_n^J$}\n\\put(-23,-32){$\\M=M_n^J\/LN$}\n\\put(59,2){$M_n$}\n\\put(57,-32){$M_n\/LN$}\n\n\\end{picture}\n\\end{center}\n\n\nThere are three essential properties of the Lie-Poisson structure on\n$M_n$ that make the reduction work:\n\n\\begin{itemize}\n\n\\item[(i)] The coadjoint action of $LSL_n$ on $M_n$ is hamiltonian\nwith respect to this structure;\n\n\\item[(ii)] the subgroup $LN$ of $LSL_n$ is admissible in the sense that\nthe space $S$ of $LN$--invariant functionals on $M_n$ is a Poisson\nsubalgebra of the space of all functionals on $M_n$;\n\n\\item[(iii)] Denote by $\\mu_{ij}$ the function on $M_n$, whose\nvalue at $\\pa + A \\in M_n$ equals the $(i,j)$ entry of $A$. The ideal\nin $S$ generated by $\\mu_{ij} + \\delta_{i-1,j}, i>j$, is a Poisson\nideal.\n\n\\end{itemize}\n\nWe will generalize this picture to the $q$--difference case.\n\n\\subsection{The Miura transformation}\n\nLet $\\F$ be the manifold of differential operators of the\nform\n\\begin{equation}    \\label{cartan}\n\\pa_s +\n\\begin{pmatrix}\nv_1 & 0 & 0 & \\hdots & 0 & 0 \\\\\n-1 & v_2 & 0 & \\hdots & 0 & 0 \\\\\n0 & -1 & v_3 & \\hdots & 0 & 0 \\\\\n\\hdotsfor{6} \\\\\n0 & 0 & 0 & \\hdots & v_{n-1} & 0 \\\\\n0 & 0 & 0 & \\hdots & -1 & v_n\n\\end{pmatrix},\n\\end{equation}\nwhere $\\sum_{i=1}^n v_i = 0$.\n\nWe have a map $\\muu: \\F \\arr \\M$, which is the composition of the\nembedding $\\F \\arr M^J_n$ and the projection $\\pi: M^J_n \\arr \\M$.\n\nUsing the definition of $\\pi$ above, $\\muu$ can be described explicitly\nas follows: the image of the operator \\eqref{cartan} under $\\muu$ is\nthe $n$th order differential operator\n$$\\pa_s^n + u_1(s) \\pa_s^{n-2} + \\ldots + u_{n-1}(s) = (\\pa_s +\nv_1(s)) \\ldots (\\pa_s + v_n(s)).$$\n\nThe map $\\muu$ is called the Miura transformation.\n\nWe want to describe the Poisson structure on $\\F$ with respect to\nwhich the Miura transformation is Poisson. To this end, let us\nconsider the restriction of the gauge action (\\ref{coadjoint}) to the\nopposite triangular subgroup $LN_{-};$ let $\\ol{\\mu} :M_n\\rightarrow\nL {\\frak n}_+ \\simeq L{\\frak n}_{-}^{*}$ be the corresponding moment\nmap. The manifold $\\F$ is the intersection of two level surfaces,\n$\\F = \\mu^{-1}(J)\\cap \\ol{\\mu}^{-1}(0).$ It is easy to see that it\ngives a local cross-section for both actions (in other words, the\norbits of $LN$ and $LN_{-}$ are transversal to $F_n$). Hence $F_n$\nsimultaneously provides a local model for the reduced spaces\n$M_n=\\mu^{-1}(J)\/LN$ and $\\ol{\\mu}^{-1}(0)\/LN_{-}$. The Poisson\nbracket on $\\F$ that we need to define is the so-called Dirac\nbracket (see, e.g., \\cite{Flato}), where we may regard the matrix\ncoefficients of $\\ol{\\mu}$ as subsidiary conditions, which fix the\nlocal gauge. The computation of the Dirac bracket for the diagonal\nmatrix coefficients $v_i$ is very simple, since their Poisson brackets\nwith the matrix coefficients of $\\ol{\\mu}$ all vanish on $\\F$. The\nonly correction arises due to the constraint $\\sum_{i=1}^N v_i = 0$.\n\nDenote by $v_{i,n}$ the linear functional on $\\F$, whose value on the\noperator \\eqref{cartan} is the $n$th Fourier coefficient of\n$v_i(s)$. We obtain the following formula for the Dirac bracket on\n$\\F$:\n\\begin{align*}\n\\{ v_{i,n},v_{i,m} \\} &= \\frac{N-1}{N} n \\delta_{n,-m}, \\\\ \\{\nv_{i,n},v_{j,m} \\} &= - \\frac{1}{N} n \\delta_{n,-m}, \\quad i<j.\n\\end{align*}\n\nSince $\\F$ and $\\M$ both are models of the same reduced space, we\nimmediately obtain:\n\n\\begin{prop}[\\cite{DS}, Proposition 3.26]\nWith respect to this Poisson structure the map $\\muu: \\F \\arr \\M$ is\nPoisson.\n\\end{prop}\n\n\\subsection{The $q$--deformations of $\\W(\\sw_n)$ and Miura transformation}\n\\label{qdef}\n\nIn this section we summarize relevant results of \\cite{FR}.\n\nConsider the space ${\\cal M}_{n,q}$ of $q$--difference operators of\nthe form \\beq \\label{qL} L = D^n + t_1(s) D^{n-1} + \\ldots +\nt_{n-1}(s) D + 1,\n\\end{equation}\nwhere $t_i(s) \\in \\C((s))$ for each $i=1,\\ldots,n$, and $[D \\cdot\nf](s) = f(sq)$.\n\nDenote by $t_{i,m}$ the functional on $\\Mq$, whose value at $L$ is the\n$m$th Fourier coefficient of $t_i(s)$. Let $\\RN$ be the completion\nof the ring of polynomials in $t_{i,m}, i=1,\\ldots,N-1; m \\in \\Z$,\nwhich consists of finite linear combinations of expressions of the\nform \\beq\n\\label{qloc} \\sum_{m_1+\\ldots +m_k=M} c(m_1,\\ldots,m_k) \\cdot\nt_{i_1,m_1} \\ldots t_{i_k,m_k},\n\\end{equation}\nwhere $c(m_1,\\ldots,m_k) \\in \\C$. Given an operator of the form\n\\eqref{qL}, we can substitute the coefficients $t_{i,m}$ into an\nexpression like \\eqref{qloc} and get a number. Therefore elements of\n$\\RN$ define functionals on the space $\\Mq$.\n\nIn order to define the Poisson structure on $\\Mq$, it suffices to\nspecify the Poisson brackets between the generators $t_{i,m}$. Let\n$T_i(z)$ be the generating series of the functionals $t_{i,m}$:\n$$T_i(z) = \\sum_{m\\in\\Z} t_{i,m} z^{-m}.$$ We define the Poisson\nbrackets between $t_{i,m}$'s by the formulas \\cite{FR}\n\\begin{align} \\notag\n\\{ T_i(z),T_j(w) \\} = & \\sum_{m\\in\\Z} \\left( \\frac{w}{z}\n\\right)^m \\frac{(1-q^{im})(1-q^{m(N-j)})}{1-q^{mN}} T_i(z) T_j(w) \\\\\n\\label{p2} &+ \\sum_{r=1}^{\\on{min}(i,N-j)} \\delta \\left( \\frac{wq^r}{z}\n\\right) T_{i-r}(w) T_{j+r}(z) \\\\ \\notag &- \\sum_{r=1}^{\\on{min}(i,N-j)}\n\\delta\\left( \\frac{w}{zq^{j-i+r}} \\right) T_{i-r}(z) T_{j+r}(w), \\quad\ni\\leq j.\n\\end{align}\nIn these formulas $\\delta(x) = \\sum_{m\\in\\Z} x^m$, and we use the\nconvention that $t_0(z) \\equiv 1$.\n\n\\begin{rem}\nThe parameter $q$ in formula \\eqref{p2} corresponds to $q^{-2}$ in the\nnotation of \\cite{FR}.\\qed\n\\end{rem} \\medskip\n\nNow consider the space $\\Fq$ of $n$--tuples of $q$--difference operators\n\\beq    \\label{ntuple}\n(D+\\la_1(s),\\ldots,D+\\la_n(s)).\n\\end{equation}\nDenote by $\\la_{i,m}$ the functional on $\\Mq$, whose value is the\n$m$th Fourier coefficient of $\\la_i(s)$. We will denote by\n$\\La_i(z)$ the generating series of the functionals $\\la_{i,m}$:\n$$\\La_i(z) = \\sum_{m\\in\\Z} \\la_{i,m} z^{-m}.$$\n\nWe define a Poisson structure on $\\Fq$ by the formulas \\cite{FR}:\n\\begin{align}    \\label{pbg1}\n\\{ \\La_i(z),\\La_i(w) \\} & = \\sum_{m\\in\\Z} \\left( \\frac{w}{z} \\right)^m\n\\frac{(1-q^m)(1-q^{m(N-1)})}{1-q^{mN}} \\La_i(z) \\La_i(w), \\\\ \\label{pbg2}\n\\{ \\La_i(z),\\La_j(w) \\} & = - \\sum_{m\\in\\Z} \\left(\n\\frac{wq^{N-1}}{z} \\right)^m \\frac{(1-q^m)^2}{1-q^{mN}} \\La_i(z)\n\\La_j(w), \\quad i<j.\n\\end{align}\n\nNow we define the $q$--deformation of the Miura transformation as the\nmap $\\muu_q: \\Fq \\arr \\Mq$, which sends the $n$--tuple \\eqref{ntuple}\nto \\beq\n\\label{Li} L = (D+\\la_1(s))(D+\\la_2(sq^{-1})) \\ldots\n(D+\\la_N(sq^{-N+1})),\n\\end{equation}\ni.e.\n\\beq    \\label{formulai}\nt_i(s) = \\sum_{j_1 < \\ldots < j_i} \\la_{j_1}(s) \\la_{j_2}(sq^{-1})\n\\ldots \\la_{j_i}(sq^{-i+1}).\n\\end{equation}\n\n\\begin{prop}[\\cite{FR}]\nThe map $\\muu_q$ is Poisson.\n\\end{prop}\n\n\\subsection{$q$--deformation of the Virasoro algebra}\nHere we specialize the formulas of the previous subsection to the\ncase of $\\sw_2$ (we will omit the index $1$ in these formulas). We\nhave the following Poisson bracket on $T(z)$:\n\\beq    \\label{vira}\n\\{ T(z),T(w) \\} = \\sum_{m\\in\\Z} \\left( \\frac{w}{z}\n\\right)^m \\frac{1-q^m}{1+q^m} T(z) T(w) + \\delta \\left( \\frac{wq}{z}\n\\right) - \\delta \\left( \\frac{w}{zq} \\right).\n\\end{equation}\n\nThe $q$--deformed Miura transformation reads:\n\\beq    \\label{virmiura}\n\\La(z) \\mapsto T(z) = \\La(z) + \\La(zq)^{-1}.\n\\end{equation}\n\nThe Poisson bracket on $\\La(z)$:\n\\beq    \\label{virpois}\n\\{ \\La(z),\\La(w) \\} = \\sum_{m\\in\\Z} \\left( \\frac{w}{z}\n\\right)^m \\frac{1-q^m}{1+q^m} \\La(z) \\La(w).\n\\end{equation}\n\n\\section{Connection with $q$--gauge transformations}    \\label{qgauge}\n\nIn this section we present the results of \\cite{FR} in the form\nof $q$--difference Drinfeld-Sokolov reduction.\n\n\\subsection{Presentation via first order $q$--difference operators}\n\nBy analogy with the differential case, it is natural to consider the\nmanifold $M_{n,q}$ of first order difference operators $D + A(s)$,\nwhere $A(s)$ is an element of the loop group $LSL_n$ of $SL_n$. The\ngroup $LSL_n$ acts on this manifold by the $q$--gauge transformations\n\\beq \\label{qadjoint} g(s) \\cdot (D + A(s)) = g(sq)(D +\nA(s))g(s)^{-1},\n\\end{equation}\ni.e. $g(s) \\cdot A(s) = g(sq) A(s) g(s)^{-1}$.\n\nNow we consider the submanifold $M^J_{n,q} \\subset M_{n,q}$ which\nconsists of operators $D+A(s)$, where $A(s)$ is of the form\n\\eqref{upper}. It is preserved under the $q$--gauge action of the\ngroup $LN$.\n\nIn the same way as in the differential case, we define a map $\\pi_q:\nM^J_{n,q} \\arr \\Mq$, which sends each element of $M^J_{n,q}$ to an\n$n$th order $q$--difference operator $L$ of the form \\eqref{qL}. It is\nclear that the map $\\pi_q$ factors through the quotient of $M^J_{n,q}$\nby $LN$. Now we state the $q$--difference analogue of \\propref{free}.\n\n\\begin{lem}    \\label{qfree}\nThe action of $LN$ on $M^J_{n,q}$ is free and each orbit contains a\nunique operator of the form \\beq \\label{qcan} D +\n\\begin{pmatrix} t_1 & t_2 & t_3 & \\hdots & t_{n-1} & 1 \\\\ -1 & 0 & 0 &\n\\hdots & 0 & 0 \\\\ 0 & -1 & 0 & \\hdots & 0 & 0 \\\\ \\hdotsfor{6} \\\\ 0 & 0\n& 0 & \\hdots & 0 & 0 \\\\ 0 & 0 & 0 & \\hdots & -1 & 0\n\\end{pmatrix}.\n\\end{equation}\n\\end{lem}\n\n\\noindent {\\em Proof} is an exercise in elementary matrix algebra. For\n$\\al=1,\\ldots,n$, denote by $M^\\al_{n,q}$ the subset of matrices from\n$M^J_{n,q}$ satisfying the property that all entries in their rows $i=\n\\al+1,\\ldots,n$ are zero except for the $(i,i-1)$ entry equal to\n$-1$. We will prove that given $A(s) \\in M^\\al_{n,q}, \\al>1$, there\nexists $g(s) \\in LN$, such that $g(sq) A(s) g(s)^{-1} \\in\nM^{\\al-1}_{n,q}$. Since the condition is vacuous for $\\al=n$, i.e.\n$M^n_{n,q} = M^J_{n,q}$, this will imply that each $LN$--orbit in\n$M^J_{n,q}$ contains an element of the form \\eqref{qcan}.\n\nTo prove the statement for a given $\\al$, we will recursively\neliminate all entries of the $\\al$th row of $A(s)$ (except the\n$(\\al,\\al-1)$ entry), from right to left using elementary unipotent\nmatrices. Denote by $E_{i,j}(x)$ the upper unipotent matrix whose only\nnon-zero entry above the diagonal is the $(i,j)$ entry equal to\n$x$. At the first step, we eliminate the $(\\al,n)$ entry $A_{\\al,n}$\nof $A(s)$ by applying the $q$--gauge transformation \\eqref{qadjoint}\nwith $g(s) = E_{\\al-1,n}(-A_{\\al,n}(s))$. Then we obtain a new matrix\n$A'(s)$, which still belongs to $M^\\al_{n,q}$, but whose $(\\al,n)$\nentry is equal to $0$. Next, we apply the $q$--gauge transformation by\n$E_{\\al-1,n-1}(-A'_{\\al,n-1}(s))$ to eliminate the $(\\al,n-1)$ entry\nof $A'(s)$, etc. It is clear that at each step we do not spoil the\nentries that have already been set to $0$. The product of the\nelementary unipotent matrices constructed at each step gives us an\nelement $g(s) \\in LN$ with the desired property that $g(sq) A(s)\ng(s)^{-1} \\in M^{\\al-1}_{n,q}$.\n\nTo complete the proof, it suffices to remark that if $A(s)$ and\n$A'(s)$ are of the form \\eqref{qcan}, and $g(sq) A(s) g(s)^{-1} =\nA'(s)$ for some $g(s) \\in LN$, then $A(s)=A'(s)$ and $g(s)=1$.\\qed\n\nFor ${\\cal L}$ of the form \\eqref{special}, $p({\\cal L})$ equals the\noperator $L$ given by formula \\eqref{L}. Thus, we have identified the\nmap $\\pi_q$ with the quotient of $M^J_{n,q}$ by $LN$ and $\\Mq$ with\n$M^J_{n,q}\/LN$.\n\n\\begin{rem}\nIn the same way as above we can prove the following more general\nstatement. Let $R$ be a ring with an automorphism $\\tau$. It rives\nrise to an automorphism of $SL_n(R)$ denoted by the same\ncharacter. Define $M^J_{\\tau,n}$ as the set of elements of $SL_n(R)$\nof the form \\eqref{upper}. Let the group $N(R)$ act on\n$M^J_{\\tau,n}(R)$ by the formula $g \\cdot A = (\\tau \\cdot g) A\ng^{-1}$. Then this action of $N(R)$ is free, and the quotient is\nisomorphic to the set ${\\cal M}^J_{\\tau,n}(R)$ of elements of\n$SL_n(R)$ of the form \\eqref{qcan} (i.e. each orbit contains a unique\nelement of the form \\eqref{qcan}). Note that the proof is not sensible\nto whether $\\tau=\\on{Id}$ or not.\n\nWhen $\\tau=\\on{Id}$, this result is well-known. It gives the classical\nnormal form of a linear operator. Moreover, in that case R.~Steinberg\nhas proved that the subset ${\\cal M}^J_{\\on{Id},n}(K)$ of $SL_n(K)$,\nwhere $K$ is an algebraically closed field, is a cross-section of the\ncollection of regular conjugacy classes in $SL_n(K)$ \\cite{St},\nTheorem 1.4. Steinberg defined an analogous cross-section for any\nsimply-connected semisimple algebraic group \\cite{St}. His results can\nbe viewed as group analogues of Kostant's results on semisimple Lie\nalgebras \\cite{Ko} (cf. \\remref{kostant}). Steinberg's cross-section\nis used in the definition of the discrete Drinfeld-Sokolov reduction\nin the general semisimple case (see \\cite{SS}).\\footnote{We are\nindebted to B.~Kostant for drawing our attention to \\cite{St}}\\qed\n\\end{rem}\n\n\\subsection{Deformed Miura transformation via $q$--gauge action}\n\nLet us attach to each element of $\\Fq$ the $q$--difference operator\n\\beq    \\label{Lambda}\n\\La = D + \\begin{pmatrix}\n\\la_1(s) & 0 & \\hdots & 0 & 0 \\\\\n-1 & \\la_2(sq^{-1}) & \\hdots & 0 & 0 \\\\\n\\hdotsfor{5} \\\\\n0 & 0 & \\hdots & \\la_{n-1}(sq^{-n+2}) & 0 \\\\\n0 & 0 & \\hdots & -1 & \\la_n(sq^{-n+1})\n\\end{pmatrix},\n\\end{equation}\nwhere $\\prod_{i=1}^n \\la_i(sq^{-i+1}) = 1$.\n\nLet $\\wt{\\muu}_q: \\Fq \\arr \\Mq$ be the composition of the embedding\n$\\Fq \\arr M^J_{n,q}$ and $\\pi_q: M^J_{n,q} \\arr M^J_{n,q}\/LN \\simeq\n\\Mq$. Using the definition of $\\pi_q$ above, one easily finds that for\n$\\La$ given by \\eqref{Lambda}, $\\wt{\\muu}_q(\\La)$ is the operator\n\\eqref{qcan}, where $t_i(s)$ is given by formula \\eqref{formulai}.\n\nTherefore we obtain\n\n\\begin{lem}\nThe map $\\wt{\\muu}_q$ coincides with the $q$--deformed Miura\ntransformation $\\muu_q$.\n\\end{lem}\n\n\\begin{rem}    \\label{qchar}\nLet $G$ be a simply-connected semisimple algebraic group over $\\C$.\nLet $V_i$ be the $i$th fundamental representation of $G$ (in the case\n$G=SL_n$, $V_i = \\Lambda^i \\C^n$), and $\\chi_i: G \\arr \\C$ be the\ncorresponding character, $\\chi_i(g) = \\on{Tr}(g,V_i)$. Define a map\n$p: G \\arr \\C^N$ by the formula $p(g) =\n(\\chi_1(g),\\ldots,\\chi_n(g))$. By construction, $p$ is constant on\nconjugacy classes. In the case $G=SL_n$ the map $p$ has a\ncross-section $r: \\C^n \\arr SL_n(\\C)$:\n$$(a_1,\\ldots,a_n) \\mapsto \\begin{pmatrix} a_1 & a_2 & a_3 & \\hdots &\na_{n-1} & 1 \\\\ -1 & 0 & 0 & \\hdots & 0 & 0 \\\\ 0 & -1 & 0 & \\hdots & 0\n& 0 \\\\ \\hdotsfor{6} \\\\ 0 & 0 & 0 & \\hdots & 0 & 0 \\\\ 0 & 0 & 0 &\n\\hdots & -1 & 0\n\\end{pmatrix}.$$ The composition $r \\circ p$, restricted to\n$M^J_{n,1}$ coincides with the map $\\pi_1$. Moreover, $\\wt{\\muu}_1$\ncan be interpreted as the restriction of $p$ to the subset of $SL_n$\nconsisting of matrices of the form \\eqref{Lambda}. Hence $\\wt{\\muu}_1$\nsends $(\\la_1,\\ldots,\\la_n)$ to the elementary symmetric polynomials\n$$t_i = \\sum_{j_1 < \\ldots < j_i} \\la_{j_1} \\la_{j_2} \\ldots\n\\la_{j_i},$$ which are the characters of the fundamental\nrepresentations of $SL_n$. As we mentioned above, Steinberg has defined\nan analogue of the cross-section $r$ for an arbitrary simply-connected\nsemisimple algebraic group \\cite{St}.\n\nFormula \\eqref{formulai} means that in terms of $\\la_j(z)$ the\ngenerators $t_i(z)$ of $W_q(\\sw_n)$ can be thought of as\n$q$--deformations of the characters of fundamental representations of\n$SL_n$. It is interesting that the same interpretation is also\nsuggested by the definintion of $\\W_q(\\sw_n)$ as the center of a\ncompletion of the quantized universal enveloping algebra\n$U_q(\\sun)_{-\\k}$ \\cite{FR}. Namely, $t_i(z)$ is then defined as the\n($q$--deformed) trace of the so-called $L$--operator acting on\n$\\Lambda^i \\C^n$ considered as a representation of $U_q(\\sun)$, see\n\\cite{ext,FR} (note also that $t_i(z)$ is closely connected with a\ntransfer-matrix of the corresponding integrable spin model).\\qed\n\\end{rem}\n\nThus, we have now represented $\\Mq$ as the quotient of the submanifold\n$M^J_{n,q}$ of the manifold $M_{n,q}$ of first order $q$--difference\noperators by the action of the group $LN$ (acting by $q$--gauge\ntransformations). We have also interpreted the $q$--deformed Miura\ntransformation in these terms. In the next sections we discuss the\nPoisson structure on $M_{n,q}$, which gives rise to the Poisson\nstructure on $\\Mq$ given by explicit formula \\eqref{p2}.\n\n\\section{Poisson structures}    \\label{P}\n\n\\subsection{Overview}    \\label{over}\nIn view of the previous section, the following diagram is the\n$q$--difference analogue of the diagram presented at the end of\n\\secref{diff1}.\n\n\n\\begin{center}\n\\begin{picture}(125,40)(-30,-33)     \n\\put(0,-1){\\vector(0,-1){26}}\n\\put(62,-1){\\vector(0,-1){26}}\n\\put(9,3){\\vector(1,0){45}}\n\\put(9,-31){\\vector(1,0){45}}\n\\put(8.9,3.9){\\oval(1,1)[l]\n\\put(8.9,-30.1){\\oval(1,1)[l]\n\\put(-5,2){$M_{n,q}^J$}\n\\put(-26,-32){$\\Mq=M_{n,q}^J\/LN$}\n\\put(57,2){$M_{n,q}=(LG,\\,\\eta_*^q)$}\n\\put(57,-32){$M_{n,q}\/LN$}\n\n\\end{picture}\n\\end{center}\n\n\nAs in the differential case, in order to define a $q$--deformation of\nthe Drinfeld-Sokolov reduction we need to find a Poisson structure\n$\\eta_*^q$ on $M_{n,q}$ and a Poisson-Lie structure $\\eta$ on $LSL_n$\nsatisfying the following properties.\n\n\\begin{itemize}\n\n\\item[(i)] the action $LSL_n \\times M_{n,q} \\arr M_{n,q}$ by\n$q$--gauge transformations is Poisson;\n\n\\item[(ii)] the subgroup $LN$ of $LSL_n$ is admissible in the sense\nthat the algebra $S_q$ of $LN$--invariant functionals on $M_{n,q}$\nis a Poisson subalgebra of the algebra of all functionals on $M_{n,q}$;\n\n\\item[(iii)] Denote by $\\mu_{ij}$ the function on $M_{n,q}$, whose\nvalue at $D + A \\in M_{n,q}$ equals the $(i,j)$ entry of $A$. The\nideal in $S_q$ generated by $\\mu_{ij} + \\delta_{i-1,j}, i>j$, is a\nPoisson ideal.\n\n\\end{itemize}\n\nGeometrically, the last condition means that $\\Mq$ is a Poisson\nsubmanifold of the quotient $M_{n,q}\/LN$.\n\nFor the sake of completeness, we recall the notions mentioned\nabove. Let $M$ be a Poisson manifold, and $H$ be a Lie group, which is\nitself a Poisson manifold. An action of $H$ on $M$ is called Poisson\nif $H \\times M\\rightarrow M$ is a Poisson map (here we equip $H\\times\nM$ with the product Poisson structure). In particular, if the\nmultiplication map $H \\times H \\arr H$ is Poisson, then $H$ is called\na Poisson-Lie group.\n\nIn this section we describe the general formalism concerning problems\n(i)--(iii) above. Then in the next section we specialize to $M_{2,q}$\nand give an explicit solution of these problems.\n\n\\subsection{Lie bialgebras}    \\label{bialg}\nLet $\\g$ be a Lie algebra. Recall \\cite{Dr} that $\\g$ is called a Lie\nbialgebra, if $\\g^*$ also has a Lie algebra structure, such that the dual\nmap $\\delta: \\g \\arr \\Lambda^2 \\g$ is a one-cocycle. We will consider {\\em\nfactorizable} Lie bialgebras ($\\g,\\delta$) satisfying the following\nconditions:\n\n\\begin{itemize}\n\n\\item[(1)] There exists a linear map $r_+: \\g^* \\arr \\g$, such that both\n$r_+$ and $r_-= -r_+^*$ are Lie algebra homomorphisms.\n\n\\item[(2)] The endomorphism $t = r_+ -r_-$ is $\\g$-equivariant and\ninduces a linear isomorphism $\\g^*\\arr\\g$.\n\n\\end{itemize}\n\nInstead of the linear operator $r_+ \\in \\on{Hom}(\\g^*,\\g)$ one often\nconsiders the corresponding element $r$ of $\\g^{\\ot 2}$ (or a\ncompletion of $\\g^{\\ot 2}$ if $\\g$ is infinite-dimensional). The\nelement $r$ (or its image in the tensor square of a particular\nrepresentation of $\\g$) is called classical $r$--matrix. It satisfies\nthe classical Yang-Baxter equation:\n\\begin{equation}    \\label{yb}\n[r_{12},r_{13}] + [r_{12},r_{23}] + [r_{13},r_{23}] = 0.\n\\end{equation}\nIn terms or $r$, $\\delta(x)=[r,x], \\forall x \\in \\g$ (here $[a \\otimes\nb,x] = [a,x] \\otimes b + a \\otimes [b,x]$). The maps $r_\\pm: \\g^* \\arr\n\\g$ are given by the formulas: $r_+(y) = (y \\otimes \\on{id})(r),\nr_-(y) = - (\\on{id} \\otimes y)(r)$.\n\nProperty (2) above means that $r+\\sigma(r)$, where $\\sigma(a \\otimes\nb) = b \\otimes a$ is a non-degenerate $\\g$--invariant symmetric\nbilinear form on $\\g^*$.\n\nSet $\\g_\\pm = \\on{Im}(r_\\pm)$. Property (1) above implies that\n$\\g_\\pm \\subset \\g$ is a Lie subalgebra. The following statement is\nessentially contained in \\cite{BD} (cf. also \\cite{rmatr}).\n\n\\begin{lem}\nLet $(\\g, \\g^*)$ be a factorizable Lie bialgebra. Then\n\n(1) The subspace $\\n_\\pm = r_\\pm(\\on{Ker} r_\\mp)$ is a Lie ideal in\n$\\g_\\pm$.\n\n(2) The map $\\theta: \\g_+\/\\n_+\\arr\\g_-\/\\n_-$ which sends the\nresidue class of $r_+(X), X\\in\\g^*$, modulo $\\n_+$ to that of $r_-(X)$\nmodulo $\\n_-$ is a well-defined isomorphism of Lie algebras.\n\n(3) Let $\\D=\\g\\oplus\\g$ be the direct sum of two copies of\n$\\g$. The map $$i: \\g^* \\arr \\D, \\quad \\quad X \\mapsto\n(r_+(X),r_-(X))$$ is a Lie algebra embedding; its image $\\g^* \\subset\n\\D$ is $$\\g^* = \\{(X_+,X_-) \\in \\g_+ \\oplus \\g_- \\subset \\D | \\ol{X}_- =\n\\theta(\\ol{X}_+) \\},$$ where $\\ol{Y}_\\pm = Y \\on{mod} \\n_\\pm$.\n\\end{lem}\n\n\\begin{rem} The connection between our notation and that of\n\\cite{RIMS} is as follows: the operator $r \\in \\on{End}\\g$ of\n\\cite{RIMS} coincides with the composition of $r_+ + r_-$ up to the\nisomorphism $t = r_+ - r_-: \\g^* \\arr \\g$; the bilinear form used in\n\\cite{RIMS} is induced by $t$.\\qed\n\\end{rem}\n\n\\subsection{Poisson-Lie groups and gauge transformations}\n\\label{adj}\nLet ($G,\\eta$) (resp., \\newline ($G^*,\\eta^*$)) be a Poisson-Lie group\nwith factorizable tangent Lie bialgebra ($\\g,\\delta$) (resp.,\n($\\g^*,\\delta^*$)). Let $G_\\pm$ and $N_\\pm$ be the Lie subgroups of\n$G$ corresponding to the Lie subalgebras $\\g_\\pm$ and $\\n_\\pm$. We\ndenote by the same symbol $\\theta$ the isomorphism $G_+\/N_+ \\arr\nG_-\/N_-$ induced by $\\theta: \\g_+\/\\n_+ \\arr \\g_-\/\\n_-$. Then the group\n$G^*$ is isomorphic to $$\\{ (g_+,g_-) \\in G_+ \\times G_- |\n\\theta(\\ol{g}_+) = \\ol{g}_- \\},$$ and we have a map $i: G^* \\arr G$\ngiven by $i((g_+,g_-)) = g_+ (g_-)^{-1}$.\n\nExplicitly, Poisson bracket on ($G,\\eta$) can be written as follows:\n\\beq \\label{skl} \\{ \\vf,\\psi \\} = \\langle r,\\nb \\vf \\wedge \\nb \\psi -\n\\nb' \\vf \\wedge \\nb' \\psi \\rangle,\n\\end{equation}\nwhere for $x \\in G$, $\\nb \\vf(x), \\nb' \\vf(x) \\in \\g^*$ are defined by\nthe formulas:\n\\begin{align}    \\label{nabla}\n\\langle \\nb \\vf(x),\\xi \\rangle &= \\frac{d}{dt} \\vf\\left( e^{t\\xi} x\n\\right)|_{t=0},\\\\ \\langle \\nb' \\vf(x),\\xi \\rangle &= \\frac{d}{dt}\n\\vf\\left( x e^{t\\xi} \\right)|_{t=0},\n\\end{align}\nfor all $\\xi \\in \\g$. Analogous formula can be written for the\nPoisson bracket on ($G^*,\\eta^*$). In formula \\eqref{skl} we use the\nstandard notation $a \\wedge b = (a \\ot b - b \\ot a)\/2$.\n\nBy definition, the action of $G$ on itself by left translations is a\nPoisson group action. There is another Poisson structure $\\eta_*$ on\n$G$ which is covariant with respect to the adjoint action of $G$ on\nitself and such that the map $i: (G^*,\\eta^*) \\arr (G,\\eta_*)$ is\nPoisson. It is given by the formula \\beq \\label{another} \\{ \\vf,\\psi\n\\} = \\langle r,\\nb \\vf \\wedge \\nb \\psi + \\nb' \\vf \\wedge \\nb' \\psi\n\\rangle - \\langle r, \\nb' \\vf \\ot \\nb \\psi - \\nb' \\psi \\ot \\nb \\vf\n\\rangle.\n\\end{equation}\n\n\\begin{prop}    \\label{embed}\n(1) The map $i: G^* \\arr G$ is a Poisson map between the Poisson\nmanifolds ($G^*,\\eta^*$) and ($G,\\eta_*$);\n\n(2) The Poisson structure $\\eta_*$ on $G$ is covariant with respect to\nthe adjoint action, i.e. the map\n$$(G,\\eta) \\times (G,\\eta_*) \\arr (G,\\eta_*): (g,h) \\mapsto g h\ng^{-1}$$ is a Poisson map.\n\\end{prop}\n\nThese results are proved in \\cite{RIMS}, \\S~3 (see also \\cite{dual},\n\\S~2), using the notion of the Heisenberg double of $G$. Formula\n\\eqref{another} can also be obtained directly from the explicit\nformulas for the Poisson structure $\\eta^*$ and for the embedding $i$.\n\nMore generally, let $\\tau$ be an automorphism of $G$, such that the\ncorresponding automorphism of $\\g$ satisfies $(\\tau \\ot \\tau)(r) =\nr$. Define a twisted Poisson structure $\\eta_*^\\tau$ on $G$ by the\nformula\n\\begin{align}\n\\label{mainb1} \\{ \\vf,\\psi \\} &= \\langle r,\\nb \\vf \\wedge \\nb \\psi + \\nb'\n\\vf \\wedge \\nb' \\psi \\rangle \\\\ \\notag &- \\langle (\\tau \\otimes\n\\on{id})(r), \\nb' \\vf \\ot \\nb \\psi - \\nb' \\psi \\ot \\nb \\vf \\rangle,\n\\end{align}\nand the twisted adjoint action of $G$ on itself by the formula $g\n\\cdot h = \\tau(g) h g^{-1}$.\n\n\\begin{thm}    \\label{act1}\n{\\em The Poisson structure $\\eta_*^\\tau$ on $G$ is covariant with\nrespect to the twisted adjoint action, i.e. the map\n$$(G,\\eta) \\times (G,\\eta_*^\\tau) \\arr (G,\\eta_*^\\tau): (g,h) \\mapsto\n\\tau(g) h g^{-1}$$ is a Poisson map.}\n\\end{thm}\n\nThis result was proved in \\cite{RIMS}, \\S~3 (see also \\cite{dual},\n\\S~2), using the notion of the twisted Heisenberg double of $G$. We\nwill use \\thmref{act1} in two cases. In the first, $G$ is the loop\ngroup of a finite-dimensional simple Lie group $\\ol{G}$, and $\\tau$ is\nthe automorphism $g(s) \\arr g(sq), q \\in \\C^\\times$. In the second, $G\n= \\ol{G}^{\\Z\/N\\Z}$, and $\\tau$ is the automorphism $(\\tau(g))_i \\arr\ng_{i+1}$. In the first case twisted conjugations coincide with\n$q$--gauge transformations, and in the second case they coincide with\nlattice gauge transformations.\n\n\\subsection{Admissibility and constraints}    \\label{admcon}\n\nLet $M$ be a Poisson manifold, $G$ a Poisson Lie group and $G \\times M\n\\arr M$ be a Poisson action. A subgroup $H\\subset G$ is called\nadmissible if the space $C^\\infty(M)^H$ of $H$--invariant functions on\n$M$ is a Poisson subalgebra in the space $C^\\infty(M)$ of all\nfunctions on $M$.\n\n\\begin{prop}[\\cite{RIMS},Theorem 6]    \\label{admiss}\nLet $\\left( {\\frak g},{\\frak g}^{*}\\right) $ be the tangent\nLie bialgebra of $G.$ A connected Lie subgroup $H\\subset G$ with Lie algebra \n${\\frak h}\\subset {\\frak g}$ is admissible if ${\\frak h}^{\\perp }\\subset\n{\\frak g}^{*}$ is a Lie subalgebra.\n\\end{prop}\n\nIn particular, $G$ itself is admissible. Note that $H\\subset G$ is a\nPoisson subgroup if and only if ${\\frak h}^{\\perp }\\subset {\\frak\ng}^{*}$ is an ideal; in that case the tangent Lie bialgebra of $H$ is\n$\\left( {\\frak h},{\\frak g}^{*}\/{\\frak h}^{\\bot }\\right)$.\n\nLet $H\\subset G$ be an admissible subgroup, and $I$ be a Poisson ideal\nin $C^\\infty(M)^H$, i.e. $I$ is an ideal in the ring $C^\\infty(M)^H$,\nand $\\{ f,g \\} \\in C^\\infty(M)^H$ for all $f \\in I, g \\in\nC^\\infty(M)^H$. Then $C^\\infty(M)^H\/I$ is a Poisson algebra.\n\nMore geometrically, the Poisson structure on $C^\\infty(M)^H\/I$ can be\ndescribed as follows. Assume that the quotient $M\/H$ exists as a smooth\nmanifold. Then there exists a Poisson structure on $M\/H$ such that the\ncanonical projection $\\pi: M\\rightarrow M\/H$ is a Poisson map. Hamiltonian\nvector fields $\\xi_\\varphi ,\\varphi \\in \\pi ^{*}C^\\infty(M\/H),$ generate an\nintegrable distribution ${\\frak H}_\\pi$ in $TM$. The following result is\nstraightforward.\n\n\\begin{lem}\n\\label{poisson}\nLet $V\\subset M$ be a submanifold preserved by $H$. Then $V\/H$ is a\nPoisson submanifold of $M\/H$ if and only if $V$ is an integral\nmanifold of ${\\frak H}_\\pi.$\n\\end{lem}\n\nThe integrality condition means precisely that the ideal $I$ of all\n$H$--invariant functions on $M$ vanishing on $V$ is a Poisson ideal in\n$C^\\infty(M)^H$, and $C^\\infty(V\/H) = C^\\infty(V)^H$ $=\nC^\\infty(M)^H\/I$. If this property holds, we will say that the Poisson\nstructure on $M\/H$ can be restricted to $V\/H$. \n\n\\begin{center}\n\\begin{picture}(125,25)(-30,-18)     \n\\put(0,0){\\vector(0,-1){12}}\n\\put(50,0){\\vector(0,-1){12}}\n\\put(9,3){\\vector(1,0){35}}\n\\put(9,-16){\\vector(1,0){35}}\n\\put(8.9,3.9){\\oval(1,1)[l]\n\\put(8.9,-15.1){\\oval(1,1)[l]\n\\put(-1,2){$V$}\n\\put(-3,-17){$V\/H$}\n\\put(48,2){$M$}\n\\put(47,-17){$M\/H$}\n\n\\end{picture}\n\\end{center}\n\nThe Poisson structure on $V\/H$ can be described as follows. Let\n$N_V\\subset T^{*}M\\mid_V$ be the conormal bundle of $V$. Clearly,\n$T^{*}V\\simeq T^{*}M\\mid _V\/N_V.$ Let $\\varphi ,\\psi \\in C(V)^H$ and\n$\\overline{d\\varphi},\\overline{d\\psi}\\in T^{*}M\\mid _V$ be any\nrepresentatives of $d\\varphi,d\\psi \\in T^{*}V.$ Let $P_M\\in\n\\bigwedge^2T\\,M$ be the Poisson tensor on $M$.\n\n\\begin{lem}\n\\label{Reduce}\nWe have \n\\begin{equation}\n\\left\\{ \\varphi ,\\psi \\right\\} = \\left\\langle\nP_M,\\overline{d\\varphi }\\ot \\overline{d\\psi}\\right\\rangle ;\n\\label{Pbr}\n\\end{equation}\nin particular, the right hand side does not depend on the choice of\n$\\overline{d\\varphi},\\overline{d\\psi}.$\n\\end{lem}\n\n\\begin{rem}\nIn the case of Hamiltonian action (i.e. when the Poisson structure on\n$H$ is trivial), one can construct submanifolds $V$ satisfying the\ncondition of \\lemref{poisson} using the moment map. Although a similar\nnotion of the nonabelian moment map in the context of Poisson group\ntheory is also available \\cite{Lu}, it is less convenient. The reason\nis that the nonabelian moment map is ``less functorial'' than the\nordinary moment map. Namely, if $G\\times M\\rightarrow M$ is a\nHamiltonian action with moment map $\\mu_G: M\\rightarrow {\\frak\ng}^{*},$ its restriction to a subgroup $H\\subset G$ is also\nHamiltonian with moment $\\mu_H=p \\circ \\mu_G$ (here $p: {\\frak\ng}^{*}\\rightarrow {\\frak h}^{*}$ is the canonical projection). If $G$\nis a Poisson-Lie group, $G^{*}$ its dual, $G\\times M\\rightarrow M$ a\nPoisson group action with moment $\\mu_G: M\\rightarrow G^{*},$ and\n$H\\subset G$ a Poisson subgroup, the action of $H$ still admits a\nmoment map. But if $H\\subset G$ is only admissible, then the\nrestricted action does not usually have a moment map. This is\nprecisely the case which is encountered in the study of the\n$q$--deformed Drinfeld-Sokolov reduction.\\qed\n\\end{rem}\n\n\\section{The $q$--deformed Drinfeld-Sokolov reduction in the case of\n$SL_2$}\n\nIn this section we apply the general results of the previous section\nto formulate a $q$--analogue of the Drinfeld-Sokolov reduction when\n$G=SL_2$.\n\n\\subsection{Choice of $r$--matrix} Let $\\g = L\\sw_2$. We would like\nto define a factorizable Lie bialgebra structure on $\\g$ in such a way\nthat the resulting Poisson-Lie structure $\\eta$ on $LSL_2$ and the\nPoisson structure $\\eta_*^q$ on $M_{2,q}$ satisfy the conditions\n(ii)--(iii) of \\secref{over}.\n\nLet $\\{ E,H,F \\}$ be the standard basis in $\\sw_2$ and $\\{ E_n,H_n,F_n\n\\}$ be the corresponding (topological) basis of $L\\sw_2 = \\sw_2 \\ot\n\\C((s))$ (here for each $A \\in \\sw_2$ we set $A_n = A \\ot s^n \\in\nL\\sw_2$). Let $\\tau$ be the automorphism of $L\\sw_2$ defined by the\nformula $\\tau(A(s)) = A(sq)$ (we assume that $q$ is generic). We have:\n$\\tau \\cdot A_n = q^n A_n$. To be able to use \\thmref{act1}, the\n$r$--matrix $r \\in L\\sw_2^{\\ot 2}$ defining the Lie bialgebra\nstructure on $L\\sw_2$ has to satisfy the condition $(\\tau \\ot \\tau)(r)\n= r$. Hence the invariant bilinear form on $L\\sw_2$ defined by the\nsymmetric part of $r$ should also be $\\tau$--invariant.\n\nThe Lie algebra $L\\sw_2$ has a unique (up to a non-zero constant\nmultiple) invariant non-degenerate bilinear form, which is invariant\nunder $\\tau$. It is defined by the formulas\n$$(E_n,F_m) = \\delta_{n,-m}, \\quad \\quad (H_n,H_m) = 2\n\\delta_{n,-m},$$ with all other pairings between the basis elements\nare $0$. This fixes the symmetric part of the element $r$. Another\ncondition on $r$ is that the subgroup $LN$ is admissible. According to\n\\propref{admiss}, this means that $L\\n_+^\\perp$ should be a Lie\nsubalgebra of $L\\sw_2^*$.\n\nA natural example of $r$ satisfying these two conditions is given by\nthe formula: \\beq \\label{new} r_0 = \\sum_{n \\in \\Z} E_n \\ot F_{-n} +\n\\frac{1}{4} H_0 \\ot H_0 + \\pol \\sum_{n>0} H_n \\ot H_{-n}.\n\\end{equation}\nIt is easy to verify that this element defines a factorizable Lie\nbialgebra structure on $\\g$. We remark that this Lie bialgebra\nstructure gives rise to Drinfeld's ``new'' realization of the\nquantized enveloping algebra associated to $L\\sw_2$\n\\cite{nr,kh-t,ked}. As we will see in the next subsection, $r_0$ can\nnot be used for the $q$--deformed Drinfeld-Sokolov reduction. However,\nthe following crucial fact will enable us to perform the\nreduction. Let $L{\\frak h}$ be the loop algebra of the Cartan\nsubalgebra ${\\frak h}$ of $\\sw_2$.\n\n\\begin{lem}    \\label{defi}\nFor any $\\rho \\in \\wedge^2 L{\\frak h}$, $r_0 + \\rho$ defines a\nfactorizable Lie bialgebra structure on $L\\sw_2$, such that\n$L\\n_+^\\perp$ is a Lie subalgebra of $L\\sw_2^*$.\n\\end{lem}\n\nThe fact that $r_0 +\\rho$ still satisfies the classical Yang-Baxter\nequation is a general property of factorizable $r$--matrices\ndiscovered in \\cite{BD}. \\lemref{defi} allows us to consider the\nclass of elements $r$ given by the formula\n\\beq \\label{class1} r = \\sum_{n \\in\n\\Z} E_n \\ot F_{-n} + \\frac{1}{2} \\sum_{m,n \\in \\Z} \\phi_{n,m} \\cdot\nH_n \\ot H_m,\n\\end{equation}\nwhere $\\phi_{n,m} + \\phi_{m,n} = \\delta_{n,-m}$.  The condition $(\\tau\n\\ot \\tau)(r) = r$ imposes the restriction $\\phi_{n,m} = \\phi_n\n\\delta_{n,-m}$, so that \\eqref{class1} takes the form \\beq\n\\label{class} r = \\sum_{n \\in \\Z} E_n \\ot F_{-n} + \\frac{1}{2} \\sum_{n\n\\in \\Z} \\phi_n \\cdot H_n \\ot H_{-n},\n\\end{equation}\nwhere $\\phi_n + \\phi_{-n} = 1$.\n\n\\subsection{The reduction}\nRecall that $M_{2,q} = LSL_2 = SL_2((s))$ consists of $2 \\times 2$\nmatrices\n\\beq    \\label{two}\nM(s) = \\begin{pmatrix} a(s) & b(s) \\\\ c(s) & d(s) \\end{pmatrix}, \\quad\n\\quad ad - bc = 1.\n\\end{equation}\nWe want to impose the constraint $c(s) = -1$, i.e. consider the\nsubmanifold $M_{2,q}^J$ and take its quotient by the (free) action of\nthe group $$LN = \\left\\{ \\begin{pmatrix} 1 & x(s) \\\\ 0 & 1\n\\end{pmatrix} \\right\\}.$$ Let $\\eta$ be the Poisson-Lie structure on\n$LSL_2$ induced by $r$ given by formula \\eqref{class}.\n\nLet $\\eta_*^q$ be the Poisson structure on $M_{2,q}$ defined by\nformula \\eqref{mainb1}, corresponding to the automorphism $\\tau: g(s)\n\\arr g(sq)$. The following is an immediate corollary of \\thmref{act1},\n\\propref{admiss} and \\lemref{defi}.\n\n\\begin{prop}\n(1) The $q$--gauge action of ($LSL_2,\\eta$) on ($M_{2,q},\\eta^q_*$)\ngiven by formula $g(s) \\cdot M(s) = g(sq) M(s) g(s)^{-1}$ is Poisson;\n\n(2) The subgroup $LN \\subset LSL_2$ is admissible.\n\\end{prop}\n\nThus, we have satisfied properties (i) and (ii) of \\secref{over}. Now\nwe have to choose the remaining free parameters $\\phi_n$ so as to\nsatisfy property (iii).\n\nThe Fourier coefficients of the matrix elements of the matrix $M(s)$\ngiven by \\eqref{two} define functions on $M_{2,q}$. We will use the\nnotation $a_m$ for the $m$th Fourier coefficient of $a(s)$. Let\n$R_{2,q}$ be the completion of the ring of polynomials in $a_m, b_m,\nc_m, d_m, m \\in \\Z$, defined in the same way as the ring $\\RN$ of\nSect.~2.3. Let $S_{2,q} \\subset R_{2,q}$ be the subalgebra of\n$LN$--invariant functions. Denote by $I$ be the ideal of $S_{2,q}$\ngenerated by $\\{ c_n + \\delta_{n,0}, n\\in\\Z \\}$ (the defining ideal of\n$M_{2,q}^J$).\n\nProperty (iii) means that $I$ is a Poisson ideal of $s_{2,q}$, which\nis equivalent to the condition that $\\{ c_n,c_m \\} \\in I$, i.e. that\nif $\\{ c_n,c_m \\}$ vanishes on $M_{2,q}^J$. This condition means that\nthe Poisson bracket of the constraint functions vanishes on the\nconstraint surface, i.e. the constraints are of first class according\nto Dirac.\n\nLet us compute the Poisson bracket between $c_n$'s. First, we list the\nleft and right gradients for the functions $a_n,b_n,c_n,d_n$ (for this\ncomputation we only need the gradients of $c_n$'s, but we will soon\nneed other gradients as well). It will be convenient for us to\nidentify $L\\sw_2$ with its dual using the bilinear form introduced in\nthe previous section. Note that with respect to this bilinear form the\ndual basis elements to $E_n, H_n$, and $F_n$ are $F_{-n}, H_{-n}\/2$,\nand $E_{-n}$, respectively.\n\nExplicit computation gives (for shorthand, we write $a$ for $a(s)$,\netc.):\n$$\\nb a_m = s^{-m} \\begin{pmatrix} \\pol a & 0 \\\\ c & - \\pol a\n\\end{pmatrix}, \\quad \\quad \\nb b_m = s^{-m} \\begin{pmatrix} \\pol b & 0\n\\\\ d & - \\pol b \\end{pmatrix},$$\n$$\\nb c_m = s^{-m} \\begin{pmatrix} - \\pol c & a \\\\ 0 & \\pol c\n\\end{pmatrix}, \\quad \\quad \\nb d_m = s^{-m} \\begin{pmatrix} - \\pol d & b\n\\\\ 0 & \\pol b \\end{pmatrix},$$\n$$\\nb' a_m = s^{-m} \\begin{pmatrix} \\pol a & b \\\\ 0 & - \\pol a\n\\end{pmatrix}, \\quad \\quad \\nb' b_m = s^{-m} \\begin{pmatrix} - \\pol b\n& 0 \\\\ a & \\pol b \\end{pmatrix},$$\n$$\\nb' c_m = s^{-m} \\begin{pmatrix} \\pol c & d \\\\ 0 & - \\pol c\n\\end{pmatrix}, \\quad \\quad \\nb' d_m = s^{-m} \\begin{pmatrix} - \\pol d & 0\n\\\\ c & - \\pol d \\end{pmatrix}.$$\n\nNow we can compute the Poisson bracket between $c_n$'s using formula\n\\eqref{mainb1}: \\beq \\label{cbracket} \\{ c_m,c_k \\} = \\pol \\sum_{n \\in\n\\Z} \\left( \\phi_n - \\phi_{-n} + \\phi_n q^n - \\phi_{-n} q^{-n} \\right)\nc_{-n+m} c_{n+k}.\n\\end{equation}\n\nRestricting to $M_{2,q}^J$, i.e. setting $c_n = -\\delta_{n,0}$, we\nobtain:\n$$\\{ c_m,c_k \\}|_{M_{2,q}^J} = \\pol \\sum_{n \\in \\Z} \\left( \\phi_m -\n\\phi_{-m} + \\phi_m q^m - \\phi_{-m} q^{-m} \\right) \\delta_{m,-k}.$$ This\ngives us the following equation on $\\phi_m$'s: $$\\phi_m - \\phi_{-m} +\n\\phi_m q^m - \\phi_{-m} q^{-m} = 0.$$ Together with the previous condition\n$\\phi_m + \\phi_{-m} = 1$, this determines $\\phi_m$'s uniquely:\n\n\\begin{thm}    \\label{third}\n{\\em The Poisson structure $\\eta_*^q$ satisfies property (iii) of the\n$q$--deformed Drin\\-feld-Sokolov reduction if and only if} $$\\phi_n =\n\\frac{1}{1+q^n}.$$\n\\end{thm}\n\nConsider the $r$--matrix \\eqref{class1} with $\\phi_n = (1+q^n)^{-1}$.\nFor this $r$--matrix, the Lie algebras defined in section\n\\secref{bialg} are as follows: $\\g_\\pm = L{\\frak b}_\\mp, {\\frak n}_\\pm\n= L{\\frak n}_\\mp$, where ${\\frak n}_+ = \\C E, {\\frak n}_- = \\C F,\n{\\frak b}_\\pm = {\\frak h} \\oplus {\\frak n}_\\pm$. We have:\n$\\g_\\pm\/{\\frak n}_\\pm \\simeq L{\\frak h}$. The transformation $\\theta$\non $L{\\frak h}$ induced by this $r$--matrix is equal to $-\\tau$.\n\nExplicitly, on the tensor product of the two $2$--dimensional\nrepresentations of $\\sw_2((t))$, the $r$--matrix looks as follows:\n\\begin{equation}    \\label{explicit}\n\\begin{pmatrix} \\vff & 0 & 0 & 0 \\\\ 0 & - \\vff & \\delta \\left(\n\\frac{t}{s} \\right) & 0 \\\\ 0 & 0 & - \\vff & 0 \\\\ 0 & 0 & 0 & \\vff\n\\end{pmatrix},\n\\end{equation}\nwhere $$\\phi(x) = \\pol \\sum_{n \\in \\Z} \\frac{1}{1+q^n} x^n.$$ Note\nthat $2 \\pi \\phi(xq^{1\/2})$ coincides with the power series expansion\nof the Jacobi elliptic function $dn$ (delta of amplitude).\n\nNow we have satisfied all the necessary properties on the Poisson\nstructures and hence can perform the $q$--Drinfeld-Sokolov reduction\nof \\secref{qgauge} at the level of Poisson algebras. In the next\nsubsection we check that it indeed gives us the Poisson bracket\n\\eqref{vira} on the reduced space ${\\cal M}_{2,q} = M_{2,q}^J\/LN$.\n\n\\subsection{Explicit computation of the Poisson brackets}\n\nIntroduce the generating series $$A(z) = \\sum_{n \\in \\Z} a_n z^{-n},$$\nand the same for other matrix elements of $M(s)$ given by formula\n\\eqref{two}. We fix the element $r$ by setting $\\phi_n = (1+q^n)^{-1}$\nin formula \\eqref{class} in accordance with \\thmref{third}. Denote\n\\beq\n\\label{fi} \\vf(z) = \\sum_{n \\in \\Z} (\\phi_n - \\phi_{-n}) z^n = \\sum_{n \\in\n\\Z} \\frac{1-q^n}{1+q^n} z^n.\n\\end{equation}\nUsing the formulas for the gradients of the matrix elements given in\nthe previous section and formula \\eqref{mainb1} for the Poisson\nbracket, we obtain the following explicit formulas for the Poisson\nbrackets.\n\n\\begin{align*}\n\\{ A(z),A(w) \\} &= \\vf \\left( \\frac{w}{z} \\right) A(z) A(w), \\\\\n\\{ A(z),B(w) \\} &= - \\delta \\left( \\frac{w}{z} \\right) A(z) B(w), \\\\\n\\{ A(z),C(w) \\} &= \\delta \\left( \\frac{w}{z} \\right) A(z) C(w), \\\\\n\\{ A(z),D(w) \\} &= - \\vf \\left( \\frac{w}{z} \\right) A(z) D(w),\\\\\n\\{ B(z),B(w) \\} &= 0, \\\\\n\\{ B(z),C(w) \\} &= \\delta \\left( \\frac{w}{z} \\right) A(z) D(w) -\n\\delta \\left( \\frac{wq}{z} \\right) A(z) A(w), \\\\\n\\{ B(z),D(w) \\} &= - \\delta \\left( \\frac{wq}{z} \\right) A(z) B(w), \\\\\n\\{ C(z),C(w) \\} &= 0, \\\\\n\\{ C(z),D(w) \\} &= \\delta \\left( \\frac{w}{zq} \\right) A(z) C(w),\\\\\n\\{ D(z),D(w) \\} &= \\vf \\left( \\frac{w}{z} \\right) D(z) D(w) - \\delta\n\\left( \\frac{wq}{z} \\right) C(z) B(w) + \\delta \\left( \\frac{w}{zq}\n\\right) B(z) C(w).\n\\end{align*}\n\n\\begin{rem}\nThe relations above can be presented in matrix form as follows. Let\n$$L(z) = \\begin{pmatrix} A(z) & B(z) \\\\ C(z) & D(z) \\end{pmatrix},$$\nand consider the operators $L_1 = L \\ot \\on{id}, L_2 = \\on{id} \\ot L$\nacting on $\\C^2 \\ot \\C^2$. The $r$--matrix \\eqref{explicit} also acts\non $\\C^2 \\ot \\C^2$. Formula \\eqref{mainb1} can be written as follows:\n\\begin{align*}\n\\{ L_1(z),L_2(w) \\} &= \\pol r^- \\left( \\frac{w}{z} \\right) L_1(z)\nL_2(w) + \\pol L_1(z) L_2(w) r^- \\left( \\frac{w}{z} \\right) \\\\ &-\nL_1(z) r \\left( \\frac{wq}{z} \\right) L_2(w) + L_2(w) \\sigma(r) \\left(\n\\frac{zq}{w} \\right) L_1(z),\n\\end{align*}\nwhere $$r^- \\left( \\frac{w}{z} \\right) = r \\left( \\frac{w}{z} \\right)\n- \\sigma(r) \\left( \\frac{z}{w} \\right) = \\begin{pmatrix} \\pol \\vpp & 0\n& 0 & 0 \\\\ 0 & - \\pol \\vpp & \\delta \\left( \\frac{w}{z} \\right) & 0 \\\\\n0 & - \\delta \\left( \\frac{w}{z} \\right) & - \\pol \\vpp & 0 \\\\ 0 & 0 & 0\n& \\pol \\vpp\n\\end{pmatrix}.$$\\qed\n\\end{rem}\n\n\\subsection{Reduced Poisson structure}\nWe know that ${\\cal M}_{2,q} = M_{2,q}^J\/LN$ is isomorphic to\n$$\\left\\{\n\\begin{pmatrix} t(s) & 1 \\\\ -1 & 0 \\end{pmatrix} \\right\\}$$ (see\nSect.~3). The ring ${\\cal R}_{2,q}$ of functionals on ${\\cal M}_{2,q}$\nis generated by the Fourier coefficients of $t(s)$. In order to\ncompute the reduced Poisson bracket between them, we have to extend\nthem to $LN$--invariant functions on the whole $M_{2,q}$. Set \\beq\n\\wt{t}(s) = a(s) c(sq) + d(sq) c(s).\n\\end{equation}\nIt is easy to check that the Fourier coefficients $\\wt{t}_m$ of\n$\\wt{t}(s)$ are $LN$--invariant, and their restrictions to $M_{2,q}^J$\ncoincide with the corresponding Fourier coefficients of $t(s)$.\n\nLet us compute the Poisson bracket between $\\wt{t}_m$'s. Set\n$$\\wt{T}(z) = \\sum_{m \\in \\Z} \\wt{t}_m z^{-m}.$$ Using the explicit\nformulas above, we find\n\\begin{align} \\notag\n\\{ \\wt{T}(z),\\wt{T}(w) \\} &= \\vf \\left( \\frac{w}{z} \\right) \\wt{T}(z)\n\\wt{T}(w) \\\\ \\label{tildet} &+ \\delta \\left( \\frac{wq}{z} \\right)\n\\Delta(z) c(w) c(wq^2) - \\delta \\left( \\frac{w}{zq} \\right) \\Delta(w)\nc(z) c(zq^2),\n\\end{align}\nwhere $\\Delta(z) = A(z)D(z) - B(z)C(z) = 1$. Hence, restricting to\n$M_{2,q}^J$ (i.e. setting $c(z)=1$ in formula \\eqref{tildet}), we\nobtain:\n$$\\{ T(z),T(w) \\} = \\vf \\left( \\frac{w}{z} \\right) T(z) T(w) + \\delta\n\\left( \\frac{wq}{z} \\right) - \\delta \\left( \\frac{w}{zq} \\right).$$\nThis indeed coincides with formula \\eqref{vira}.\n\n\\begin{rem}\nConsider the subring $\\wt{S}_{2,q}$ of the ring $R_{2,q}$, generated by\n$c_m, \\wt{t}_m, m \\in \\Z$. The ring $\\wt{S}_{2,q}$ consists of\n$LN$--invariant functionals on $M_{2,q}$ and hence it can serve as a\nsubstitute for the ring of functions on $M_{2,q}\/LN$. Let us compute\nthe Poisson brackets in $\\wt{S}_{2,q}$. The Poisson brackets of\n$\\wt{t}_m$'s are given by formula \\eqref{tildet}, and by construction\n$\\{ c_m,c_k \\} = 0$. It is also easy to find that $\\{ c_m,\\wt{t}_k \\}\n= 0$. Hence $\\wt{S}_{2,q}$ is a Poisson subalgebra of $R_{2,q}$. Thus, the\n$q$--deformed Drinfeld-Sokolov reduction can be interpreted as\nfollows. The initial Poisson algebra is $R_{2,q}$. We consider its\nPoisson subalgebra $\\wt{S}_{2,q}$ generated by $c_m$'s and\n$\\wt{t}_m$'s. The ideal $I$ of $\\wt{S}_{2,q}$ generated by $\\{ c_m +\n\\delta_{m,0}, m \\in \\Z \\}$ is a Poisson ideal. The quotient\n$\\wt{S}_{2,q}\/I$ is isomorphic to the $q$--Virasoro algebra ${\\cal\nR}_{2,q}$ defined in Sect.~3.\\qed\n\\end{rem}\n\n\\subsection{$q$--deformation of Miura transformation}\n\nAs was explained in Sect.~3.2, the $q$--Miura transformation of\n\\cite{FR} is the map between two (local) cross-sections of the\nprojection $\\pi_q: M_{n,q}^J \\arr M_{n,q}^J\/LN$. In the case of\n$LSL_2$, the first cross-section $$\\left\\{ \\begin{pmatrix} \\la(s) & 0\n\\\\ -1 & \\la(s)^{-1} \\end{pmatrix} \\right \\}$$ is defined by the\nsubsidiary constraint $b(s)=0$, and the second $$\\left\\{\n\\begin{pmatrix} t(s) & 1 \\\\ -1 & 0 \\end{pmatrix} \\right\n\\}$$ is defined by the subsidiary constraint $d(s)=0$. The map between\nthem is given by the formula $$\\muu_q: \\la(s) \\mapsto t(s) = \\la(s) +\n\\la(sq)^{-1}.$$ Now we want to recover formula \\eqref{virpois} for the\nPoisson brackets between the Fourier coefficients $\\la_n$ of $\\la(s)$,\nwhich makes the map $\\muu_q$ Poisson.\n\nWe have already computed the Poisson bracket on the second (canonical)\ncross-section from the point of view of Poisson reduction. Now we need\nto compute the Poisson bracket between the functions $a_n$'s on the\nfirst cross-section, with respect to which the map $\\muu_q$ is\nPoisson. This computation is essentially similar to the one outlined\nin Sect.~3.2. The Poisson structure on the local cross-section is\ngiven by the Dirac bracket, which is determined by the choice of the\nsubsidiary conditions, which fix the cross-section.\n\nThe Dirac bracket has the following property (see\n\\cite{Flato}). Suppose we are given constraints $\\xi_n, n \\in I$, and\nsubsidiary conditions $\\eta_n, n \\in I$, on a Poisson manifold $M$,\nsuch that $\\{ \\xi_k,\\xi_l \\} = \\{ \\eta_k,\\eta_l \\} = 0, \\forall k,l\n\\in I$. Let $f, g$ be two functions on $M$, such that $\\{ f,\\xi_k \\}$\nand $\\{ g,\\xi_k \\}$ vanish on the common level surface of all $\\xi_k,\n\\eta_k$. Then the Dirac bracket of $f$ and $g$ coincides with their\nordinary Poisson bracket.\n\nIn our case, the constraint functions are $c_m+\\delta_{m,0}, m \\in\n\\Z$, and the subsidiary conditions $b_m, m \\in \\Z$, which fix the\nlocal model of the reduced space. We have: $\\{ b_m,b_k \\} = 0$, $\\{\nc_m,c_k \\} = 0$, and $\\{ a_m,b_k \\} = 0$, if we set $b_m=0, \\forall m\n\\in \\Z$. Therefore we are in the situation described above, and the\nDirac bracket between $a_m$ and $a_k$ coincides with their ordinary\nbracket. In terms of the generating function $A(z) = \\sum_{m \\in \\Z}\na_m z^{-m}$ it is given by the formula $$\\{ A(z),A(w) \\} = \\vf \\left(\n\\frac{w}{z} \\right) A(z) A(w),$$ which coincides with formula\n\\eqref{virpois}. Thus, we have proved the Poisson property of the\n$q$--deformation of the Miura transformation from the point of view of\nthe deformed Drinfeld-Sokolov reduction.\n\n\\section{Lattice Virasoro algebra}\n\nIn this section we consider the lattice counterpart of the\nDrinfeld-Sokolov reduction. Our group is thus $\\GG = (SL_2)^{\\Z\/N\\Z}$,\nwhere $N$ is an integer, and $\\tau$ is the automorphism of $G$, which\nmaps $(g_i)$ to $(g_{i+1})$. Poisson structures on $\\GG$ which are\ncovariant with respect to lattice gauge transformations $x_n \\mapsto\ng_{n+1} x_n g_n^{-1}$ have been studied already in \\cite{RIMS}\n(cf. also \\cite{AFS}). In order to make the reduction over the\nnilpotent subgroup $\\NN \\subset \\GG$ feasible, we have to be careful\nin our choice of the $r$--matrix.\n\n\\subsection{Discrete Drinfeld-Sokolov reduction}\nBy analogy with the continuous case, we choose the element $r$ defining the\nLie bialgebra structure on $\\g = \\sw_2^{\\oplus \\Z\/N\\Z}$ as follows: $$r =\n\\sum_{n \\in \\Z\/N\\Z} E_n \\ot F_n + \\frac{1}{4} \\sum_{m,n \\in \\Z\/N\\Z}\n\\phi_{n,m} H_n \\ot H_m,$$ where $\\phi_{n,m} + \\phi_{m,n} = 2\n\\delta_{m,n}$. It is easy to see that $r$ defines a factorizable Lie\nbialgebra structure on $\\g$. For \\thmref{act1} to be applicable, $r$ has to\nsatisfy the condition $(\\tau \\ot \\tau)(r) = r$, which implies that\n$\\phi_{n,m} = \\phi_{n-m}$.\n\nAn element of $\\GG$ is an $N$--tuple $(g_i)$ of elements of $SL_2$:\n$$g_k = \\begin{pmatrix} a_k & b_k \\\\ c_k & d_k \\end{pmatrix}.$$ We\nconsider $a_k,b_k,c_k,d_k, k \\in \\Z\/N\\Z$, as the generators of the\nring of functions on $\\GG$.\n\nThe discrete analogue of the Drinfeld-Sokolov reduction consists of\ntaking the quotient $\\MM=\\GG^J\/\\NN$, where $\\GG^J = (G^J)^{\\ZN}$,\n$$G^J = \\left\\{ \\begin{pmatrix} a & b \\\\ -1 & d \\end{pmatrix}\n\\right\\},$$ and $\\NN = N^{\\ZN}$, acting on $G^J$ by the formula \\beq\n\\label{dgauge} (h_i) \\cdot (g_i) = (h_{i+1} g_i h_i^{-1}).\n\\end{equation}\nIt is easy to see that $$\\MM \\simeq \\left \\{ \\begin{pmatrix} t_i &\n1 \\\\ -1 & 0 \\end{pmatrix}_{i \\in \\ZN} \\right\\}.$$\n\nThe element $r$ with $\\phi_{n,m} = \\phi_{n-m}, \\phi_k + \\phi_{-k} =\n2\\delta_{k,0}$, defines a Lie bialgebra structure on $\\g$ and Poisson\nstructures $\\eta, \\eta_*^\\tau$ on $\\GG$. According to \\thmref{act1},\nthe action of ($\\GG,\\eta$) on ($\\GG,\\eta_*^\\tau$) given by formula\n\\eqref{dgauge} is Poisson.\n\nAs in the continuous case, for the Poisson structure $\\eta_*^\\tau$ to\nbe compatible with the discrete Drinfeld-Sokolov reduction, we must\nhave:\n\\beq    \\label{vanish}\n\\{ c_n,c_m \\}|_{\\GG^J} = 0.\n\\end{equation}\nExplicit calculation analogous to the one made in the previous\nsubsection shows that \\eqref{vanish} holds if and only if $$\\phi_{n-1}\n+ 2\\phi_n + \\phi_{n+1} = 2 \\delta_{n,0} + 2 \\delta_{n+1,0}.$$ The\ninitial condition $\\phi_0 = 1$ and periodicity condition give us a\nunique solution: for odd $N$, $\\phi_k = (-1)^k$; for even $N$, $\\phi_k\n= (-1)^k \\left( 1 - \\dfrac{2k}{N} \\right)$. In what follows we\nrestrict ourselves to the case of odd $N$ (note that in this case the\nlinear operator $\\on{id}+\\tau$ is invertible).\n\nContinuing as in the previous subsection, we define $$\\wt{t}_n = a_n\nc_{n+1} + d_{n+1} c_n, \\quad \\quad n \\in \\ZN.$$ These are\n$\\NN$--invariant functions on $\\GG$. We find in the same way as in the\ncontinuous case:\n\\beq    \\label{wtt}\n\\{ \\wt{t}_n,\\wt{t}_m \\} = \\vf_{n-m} \\wt{t}_n \\wt{t}_m + \\delta_{n,m+1}\nc_m c_{m+2} - \\delta_{n+1,m} c_n c_{n+2},\n\\end{equation}\n$$\\{ \\wt{t}_n,c_m \\} = 0, \\quad \\quad \\{ c_n,c_m \\} = 0,$$ where\n$$\\vf_k = \\pol (\\phi_k - \\phi_{-k}) = \\left\\{ \\begin{array}{cc} 0, &\nk=0, \\\\ (-1)^k, & k\\neq 0.\n\\end{array} \\right.$$\nThe discrete Virasoro algebra $\\C[t_i]_{i \\in \\ZN}$ is the quotient of\nthe Poisson algebra $\\C[\\wt{t}_i,c_i]_{i \\in \\ZN}$ by its Poisson\nideal generated by $c_i+1, i \\in \\ZN$. From formula \\eqref{wtt} we\nobtain the following Poisson bracket between the generators $t_i$:\n\\beq\n\\label{dvir} \\{ t_n,t_m \\} = \\vf_{n-m} t_n t_m + \\delta_{n,m+1} -\n\\delta_{n+1,m}.\n\\end{equation}\n\nThe discrete Miura transformation is the map from the local\ncross-section $$\\left\\{ \\begin{pmatrix} \\la_n & 0 \\\\ -1 & \\la_n^{-1}\n\\end{pmatrix} \\right \\}$$ to $\\MM$,\n\\beq    \\label{dmiura}\n\\la_n \\mapsto t_n = \\la_n + \\la_{n+1}^{-1}.\n\\end{equation}\nIt defines a Poisson map $\\C[\\la_i^\\pm]_{i \\in \\ZN} \\arr \\C[t_i]_{i\n\\in \\ZN}$, where the Poisson structure on the latter is given by the\nformula \\beq \\label{dheis} \\{ \\la_n,\\la_m \\} = \\vf_{n-m} \\la_n \\la_m.\n\\end{equation}\n\n\\begin{rem}\nThe Poisson algebra $\\C[t_i]_{i \\in \\ZN}$ can be considered as a\nregularized version of the $q$--deformed Virasoro algebra when\n$q=\\ep$, where $\\ep$ is a primitive $N$th root of unity. Indeed, we\ncan then consider $t(\\ep^i), i \\in \\ZN$, as generators and truncate in\nall power series appearing in the relations, summations over $\\Z$ to\nsummations over $\\ZN$ divided by $N$. This means that we replace\n$\\phi(\\ep^n)$ given by formula \\eqref{vira} by\n$$\\wt{\\phi}(\\ep^n) = \\frac{1}{N} \\sum_{i \\in \\ZN}\n\\frac{1-\\ep^i}{1+\\ep^i} \\ep^{ni},$$ and $\\delta(\\ep^n)$ by\n$\\delta_{n,0}$. The formula for the Poisson bracket then becomes: $$\\{\nt(\\ep^n),t(\\ep^m) \\} = \\wt{\\phi}(\\ep^{m-n}) t(\\ep^n) t(\\ep^m) +\n\\delta_{n,m+1} - \\delta_{n+1,m}.$$ If we set $t(\\ep^i)=t_i$, we\nrecover the Poisson bracket \\eqref{dvir}, since it is easy to check\nthat $\\wt{\\phi}(\\ep^{m-n}) = \\vf_{n-m}$.\n\nOne can apply the same procedure to the $q$--deformed $\\W$--algebras\nassociated to $\\sw_N$ and obtain lattice Poisson algebras. It would be\ninteresting to see whether they are related to the lattice\n$\\W$--algebras studied in the literature, e.g., in \\cite{Be,Bo}. In\nthe case of $\\sw_2$, this connection is described in the next\nsubsection.\\qed\n\\end{rem}\n\n\\subsection{Connection with Faddeev-Takhtajan-Volkov algebra}\nThe Poisson structures \\eqref{dvir} and \\eqref{dheis} are nonlocal,\ni.e. the Poisson brackets between distant neighbors on the lattice\nare nonzero. However, one can define closely connected Poisson\nalgebras possessing local Poisson brackets; these Poisson algebras can\nactually be identified with those studied by L.~Faddeev,\nL.~Takhtajan, and A.~Volkov.\n\nLet us first recall some results of \\cite{FR} concerning the\ncontinuous case. As was explained in \\cite{FR}, one can associate a\ngenerating series of elements of the $q$--Virasoro algebra to an\narbitrary finite-dimensional representation of $\\sw_2$. The series\n$T(z)$ considered in this paper corresponds to the two-dimensional\nrepresentation. Let $T^{(2)}(z)$ be the series corresponding to the\nthree-dimensional irreducible representation of $sw_2$. We have the\nfollowing identity \\cite{FR}\n$$T(z) T(zq) = T^{(2)}(z) + 1,$$ which can be taken as the definition of\n$T^{(2)}(z)$. From formula \\eqref{virmiura} we obtain:\n\\begin{align*}\nT^{(2)}(z) &= \\La(z) \\La(zq) + \\La(z) \\La(zq^2)^{-1} + \\La(zq)^{-1}\n\\La(zq^2)^{-1} \\\\ &= A(z) + A(z) A(zq)^{-1} + A(zq)^{-1},\n\\end{align*}\nwhere\n\\beq    \\label{az}\nA(z) = \\La(z) \\La(zq)\n\\end{equation}\n(the series $A(z)$ was introduced in Sect.~7 of \\cite{FR}). From\nformula \\eqref{virpois} we find:\n$$\\{ A(z),A(w) \\} = \\left( \\delta \\left( \\frac{w}{zq} \\right) - \\delta\n\\left( \\frac{wq}{z} \\right) \\right) A(z) A(w).$$ It is also easy to find\n\\begin{align*}    \\label{three}\n\\{ T^{(2)}(z),T^{(2)}(w) \\} &= \\left( \\delta \\left( \\frac{w}{zq}\n\\right) - \\delta \\left( \\frac{wq}{z} \\right) \\right) \\left( T^{(2)}(z)\nT^{(2)}(w) - 1 \\right) \\\\ &+ \\delta \\left( \\frac{wq^2}{z} \\right) T(w)\nT(wq^3) - \\delta \\left( \\frac{w}{zq^2} \\right) T(z) T(zq^3).\n\\end{align*}\n\nWe can use the same idea in the lattice case. Let $\\nu_n = \\la_n\n\\la_{n+1}$; this is the analogue of $A(z)$. We have: \\beq \\label{nun}\n\\{ \\nu_n,\\nu_m \\} = (\\delta_{n+1,m} - \\delta_{n,m+1}) \\nu_n \\nu_m,\n\\end{equation}\nand hence $\\C[\\nu_i^\\pm]$ is a Poisson subalgebra of $\\C[\\la_n^\\pm]$\nwith local Poisson brackets. We can also define $t^{(2)}_n = t_n\nt_{n+1} - 1$. The Poisson bracket of $t^{(2)}_n$'s is local:\n\\begin{align}    \\label{t2}\n\\{ t^{(2)}_n,t^{(2)}_m \\} &= \\left( \\delta_{n+1,m} - \\delta_{n,m+1}\n\\right) \\left( t^{(2)}_n t^{(2)}_m - 1 \\right) \\\\ \\notag &+\n\\delta_{n,m+2} t_m t_{m+3} - \\delta_{n+2,m} t_n t_{n+3}.\n\\end{align}\nUnfortunately, it does not close on $t^{(2)}_n$'s, so that\n$\\C[t^{(2)}_i]$ is not a Poisson subalgebra of $\\C[t_i]$. But let us\ndefine formally\n\\beq    \\label{sn}\ns_n = \\frac{1}{1+t^{(2)}_n} = t_n^{-1} t_{n+1}^{-1} =\n\\frac{1}{(1+\\nu_n)(1+\\nu_{n+1}^{-1})}.\n\\end{equation}\nThen from formulas \\eqref{sn} and \\eqref{nun} we find:\n\\begin{align}    \\label{fad}\n\\{ s_n,s_m \\} = & s_n s_m \\big( (\\delta_{n+1,m} - \\delta_{n,m+1})(1 -\ns_n - s_m) - \\\\ \\notag & - s_{n+1} \\delta_{n+2,m} + s_{m+1}\n\\delta_{n,m+2} \\big).\n\\end{align}\nThus, the Poisson bracket closes among $s_n$'s and defines a Poisson\nstructure on $\\C[s_i]_{i\\in\\ZN}$.\n\nThe Poisson algebra $\\C[s_i]_{i\\in\\ZN}$ with Poisson bracket\n\\eqref{fad} was first introduced by Faddeev and Takhtajan in \\cite{ft}\n(see formula (54)). We see that it is connected with our version of\nthe discrete Virasoro algebra, $\\C[t_i]$, by a change of variables\n\\eqref{sn}. The Poisson algebra $\\C[\\nu_i^\\pm]$ and the Poisson map\n$\\C[\\nu_i^\\pm] \\arr \\C[s_n]$ given by formula \\eqref{sn} were\nintroduced by Volkov in \\cite{v1} (see formulas (2) and (23))\nfollowing \\cite{ft}; see also related papers \\cite{v2,fv}. This map is\nconnected with our version \\eqref{dmiura} of the discrete Miura\ntransformation by a change of variables.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nPre-trained Language Models (PLMs) have been successfully adapted to a wide range of Natural Language Processing (NLP) tasks using \\emph{prompt-based} learning~\\cite{Radford:2018,GPT-2,GPT3,petroni-etal-2019-language} such as sentiment classification~\\cite{gao-etal-2021-making}, natural language inference (NLI)~\\cite{schick-schutze-2021-exploiting,schick-schutze-2022-true}, relation extraction~\\cite{shin-etal-2020-autoprompt}, cross-lingual inference~\\cite{qi-etal-2022-enhancing}.\nHowever, manually writing prompts that generalize well is very challenging for several reasons such as (a) it might not always be possible to recruit domain-expert human annotators, \n(b) human annotators might not be able to cover all corner cases by writing prompts, and\n(c) there can be disagreements between human annotators regarding the coverage of a particular prompt.\nTo address these challenges, automatic learning of discrete prompts has been proposed such as AdvTrigger~\\cite{wallace-etal-2019-universal}, AutoPrompt~\\cite[\\textbf{AP};][]{shin-etal-2020-autoprompt}, WARP~\\cite{hambardzumyan-etal-2021-warp}, and RLPrompt~\\cite{DBLP:journals\/corr\/abs-2205-12548}.\n\n\n\\begin{table*}[t]\n    \\centering\n    \\small\n    \\begin{tabular}{lllc}\n        \\toprule\n         \\textbf{Relation} &  \\textbf{Method} & \\textbf{Prompt} & \\textbf{P@1} \\\\ \\midrule\n         \\textsf{native-language-of} (P103)   & Manual            & \\texttt{The native language of [X] is [Y]} & 74.54\\\\\n                                            & AP BERT   & \\texttt{[X]PA communerug speaks proper [Y]} & \\textbf{84.87}\\\\\n                                            & AP RoBERTa & \\texttt{[X]neau optionally fluent!?\\\" traditional [Y]} & 81.61\\\\ \\midrule\n         \\textsf{profession-of} (P106) & Manual & \\texttt{[X] is a [Y] by profession} & 0.73 \\\\\n                                            & AP BERT & \\texttt{[X] supporters studied politicians musician turned [Y]} & 15.83 \\\\\n                                            & AP RoBERTa & \\texttt{[X] (), astronomers businessman\u00b7former [Y]} & \\textbf{19.24} \\\\ \\midrule\n         \\textsf{music-played-by} (P136) & Manual & \\texttt{[X] plays [Y] music} & 0.7\\\\\n                                         & AP BERT & \\texttt{[X] freaking genre orchestra fiction acid [Y]} & \\textbf{59.95} \\\\\n                                          & AP RoBERTa & \\texttt{[X] blends postwar hostage drama sax [Y]} & 52.97 \\\\\n         \\bottomrule\n    \\end{tabular}\n    \\caption{Examples of prompts learnt by AP for the fact retrieval task for BERT and RoBERTa PLMs and the human-written manual prompts. T-REx relation ids are shown with brackets for each relation type. Precision@1 (P@1) scores are shown when each prompt is used in fact retrieval.}\n    \\label{tbl:autoprompt-examples}\n\\end{table*}\n\nAlthough discrete prompt learning methods have achieved good performance in numerous downstream tasks by automatically learnt prompts, such automatic prompts seem to be significantly different from the manually-written ones.\nFor example, \\autoref{tbl:autoprompt-examples} shows manually-written and AP-learnt prompts for fact retrieval~\\cite{petroni-etal-2019-language}.\nWe see that the AP-learnt prompts for BERT~\\cite{devlin-etal-2019-bert} and RoBERTa~\\cite{RoBERTa} outperform the manual prompts in precision\\@1 (P@1) scores.\nHowever, the AP-learnt prompts contain various counter-intuitive language constructs such as punctuation (e.g. `(', `?', `!', `)'), spelling errors (e.g. \\emph{commuenrug}) etc., which seem unrelated to the target relation.\nSimilar cases can be observed for AP-learnt prompts for other tasks as well (see Appendix in \\newcite{shin-etal-2020-autoprompt}).\nIt is unrealistic that a human annotator would be able to write such prompts even if they were able to see the same training instances as used by automatic methods.\n\nConsidering the fact that discrete prompt learning methods are trained in a few-shot setting where they use only a small number of training instances, the seemingly counter-intuitive nature of the discrete prompts learnt by automatic methods raises concerns about their robustness.\nFor example, \\emph{How will the performance of a target task change if we add small random perturbations to the prompts learnt by AP?} and \\emph{Whether the prompts learnt by AP generalize to out-of-domain data?}.\nTo study these issues, in this paper we evaluate the robustness of discrete prompts learnt by automatic prompt learning methods and compare that with manually-written prompts and direct fine-tuning of PLMs.\n\nAn evaluation of the robustness of discrete prompts is important for two main reasons.\nFirst, given that discrete prompt learning methods are learning those prompts from a small set of training instances, it is important that they cover the core patterns that generalize to the target task and not simply capture some random artefacts in the training samples.\nSecond, unlike embedding-based continuous prompts~\\cite{li-liang-2021-prefix,lester-etal-2021-power}, discrete prompts~\\cite{wallace-etal-2019-universal,shin-etal-2020-autoprompt,DBLP:journals\/corr\/abs-2205-12548} are represented in natural language and supposed to be interpretable.\nHowever, if a discrete prompt learning method is less robust, a seemingly harmless perturbation such as removing a punctuation character can significantly alter the performance of a downstream task.\n\nIn contrast to the numerous work that has used prompts for fine-tuning PLMs, to the best of our knowledge, the robustness of discrete prompts to random or adversarial perturbations has not been systematically studied.\nTo address this gap, we use AP as a concrete example of a widely-used method and evaluate its robustness under different types of carefully designed perturbations.\nHowever, we note that our perturbation techniques are not limited to AP and can be used for any discrete prompt learning method.\nWe compare the performance of AP-learnt prompts against fine-tuning using Manually-written Prompts (MP), and Head-based Fine-Tuning (HFT), where we fine-tune both the classifier head and the PLM parameters. \n\nFrom our evaluation, we find several interesting facts about the robustness of discrete prompts as summarized below.\n\\begin{itemize}\n    \\item Overall, when the number of training instances is increased, MP outperforms both AP and HFT on CB~\\cite{de23commitmentbank} and MNLI~\\cite{DBLP:conf\/naacl\/WilliamsNB18}, two independent benchmark datasets for NLI (\\autoref{sec:exp:datasize}).\n    In particular, the performance of AP on MNLI is much worse than that on CB. This is in contrast to the superior performance of AP on SICK-E~\\cite{DBLP:conf\/lrec\/MarelliMBBBZ14}, another NLI dataset, as reported by \\newcite{shin-etal-2020-autoprompt}.\n    \n    \\item Moreover, we see a performance drop when we use discrete prompts learnt from CB for MNLI and vice-versa (\\autoref{sec:exp-cross-data}).\n    These results indicate that the performance of discrete prompts learnt by AP is highly dataset-dependent and such discrete prompts do not generalize well across datasets.\n    \n    \\item Compared to MP, AP-learnt discrete prompts turn out to be highly sensitive to the ordering of prompt tokens (\\autoref{sec:exp:reorder}).\n    \n    \\item Random deletion of prompt tokens decreases performance in both AP and MP (\\autoref{sec:exp:deletion}).\n    \n    \\item  We create an adversarial NLI dataset from randomly-sampled test instances from MNLI and CB, and manually modify the hypothesis sentences with keeping the corresponding premise sentences unchanged, such that (a) the target label would not change, and (b) would reverse an entailment label to a contradiction (or vice-versa).\n    Both AP and MP remain relatively robust against the perturbations that do not change the target label, but the performance of MP drops significantly in the label-changing setting (\\autoref{sec:exp:input-perturbation}).\n    This shows that AP is relatively more robust against adversarial perturbations than MP, which explains AP's superior performance in various tasks. \n\\end{itemize}\n\n\n\\section{Related Work}\n\\paragraph{Prompting Methods:}\nPrompting or \\emph{in-context-learning} has received wide attention as an efficient method to extract knowledge from PLMs ~\\cite{GPT3,petroni-etal-2019-language,cui-etal-2021-template}.\nHowever, to manually write prompts one must possess task-specific domain knowledge.\nAs an alternative, methods that can automatically learn prompts from training data have been proposed.\nTwo distinct types of prompts have been learnt in prior work:\ndiscrete prompts (learns lexical sequences), and continuous prompts (learns embeddings).\nContinuous prompts~\\cite{li-liang-2021-prefix,lester-etal-2021-power} are parameter efficient because they learn generalizable task-specific embeddings, with performance comparable to PLM fine-tuning.\nHowever, continuous prompts cannot be learnt when a PLM is publicly unavailable and the only access to it is via an API~\\cite{GPT3}.\nMoreover, compared to discrete prompts, continuous prompts are difficult to interpret.\nLearning discrete prompts~\\cite{wallace-etal-2019-universal,shin-etal-2020-autoprompt,DBLP:journals\/corr\/abs-2205-12548} does not suffer from these limitations of continuous prompts and can be used with diverse NLP tasks.\nEspecially, fine-tuning massive PLMs has become computationally costly, which has made discrete prompt learning an attractive alternative.\n\n\n\\paragraph{Analysis of Prompting Methods:}\nPrior work has analyzed prompts from various viewpoints. \n\\citet{DBLP:conf\/naacl\/ScaoR21} studied the effect of training dataset size on fixed-prompt PLM fine-tuning and head-based fine-tuning and found that prompting is often worth 100s of instances on average across classification tasks.\n\\citet{kavumba-etal-2022-prompt} showed that the performance of prompt-based models varies significantly depending on the surface cues in the sentence.\n\\citet{lu-etal-2022-fantastically} found that ordering of task input significantly affects the performance.\n\\citet{utama-etal-2021-avoiding} focused on the reliance on lexical overlap in sentence pair classification and showed that prompt-based models fail to make predictions dependent on the lexical overlap.\nTo the best of our knowledge, the robustness of discrete prompts under different types of perturbations has not been studied in prior work, which is the main focus of this paper.\n\n\n\\section{Experiments}\n\\label{sec:experiments}\nLet us first describe experimental settings common to all experiments.\n\n\\paragraph{Prompting and Fine-Tuning Methods: }\nWe compared the following methods. \n\n\\begin{itemize}\n    \\item \\textbf{AutoPrompt}~\\cite[\\textbf{AP};][]{shin-etal-2020-autoprompt} is a representative method of discrete prompt learning.\n    The learning strategy is based on fill-in-the-blank task~\\cite{devlin-etal-2019-bert}.\n    First, a manually created prompt template (e.g., \\texttt{[X] <MASK> <T> ... <T> [Y]}) is given, and a prompt token (called a trigger token) is learnt by replacing \\texttt{<T>}, which is a special token representing a trigger token.\n    In the search for trigger tokens, the probability of \\texttt{<MASK>} is converted into class probability by using label tokens (e.g., \\{`\\emph{nobody}', `\\emph{nor}'\\} for contradiction~\\cite{shin-etal-2020-autoprompt}), and trigger tokens are searched by gradient-guided search~\\cite{wallace-etal-2019-universal} to find a candidate set consisting of trigger tokens from a vocabulary of the language model. \n    As a template for NLI, we used the one given by \\citet{shin-etal-2020-autoprompt}, and the prompt tokens were learnt from the training dataset. \n    In our experiments, we used the official implementation.\\footnote{\\url{https:\/\/github.com\/ucinlp\/autoprompt}}\n    \n    \\item \\textbf{Manually-written Prompts}~\\cite[\\textbf{MP};][]{schick-schutze-2021-exploiting} is a method for fine-tuning the entire masked language model with training data using manually-written prompts as the input and predicting the \\texttt{<MASK>} tokens for the labels (e.g., `\\emph{yes}' for entailment).\n    We used the template \\texttt{\\{hypothesis\\}? | <MASK>, \\{premise\\}} and  verbalizer (`\\emph{yes}' for entailment, `\\emph{no}' for contradiction, `\\emph{maybe}' for neutral) following prior work \\cite{schick-schutze-2021-exploiting,DBLP:conf\/naacl\/ScaoR21}.\n    \\citet{schick-schutze-2021-exploiting} proposed an ensemble-based method with multiple rounds of fine-tuning using different templates.\n    However, because a single template is used in AP, for a fair comparison in our experiments, we fine-tuned a PLM using one MP template.\n\n    \\item \\textbf{Head-based Fine-Tuning}~\\cite[\\textbf{HFT};][]{devlin-etal-2019-bert} fine-tunes the PLM with a classifier head.\n    We report the head-based results trained by \\newcite{DBLP:conf\/naacl\/ScaoR21}. \n    They trained HFT with a low learning rate ($10^{-5}$) and always with a large number of steps (at least 250), following the recommendations in prior work~\\cite{DBLP:conf\/iclr\/MosbachAK21,DBLP:conf\/iclr\/0007WKWA21}.\n    Note that HFT is not a prompt-based method, so it was excluded from some experiments on the robustness of discrete prompts.\n\\end{itemize}\n\n\\paragraph{Datasets:}\nWe used NLI as an evaluation task to compare the robustness of discrete prompting methods.\nThe NLI task has been used in multiple previous studies to evaluate and\/or propose novel prompt learning methods because it is a fundamental task related to many NLP applications~\\cite{shin-etal-2020-autoprompt,DBLP:conf\/naacl\/ScaoR21,DBLP:conf\/naacl\/WebsonP22}.\nIt is important to use the same NLI task and datasets in our experiments to facilitate fair comparisons and reach reproducible conclusions. \nWe used the two datasets: CommitmentBank \\cite[\\textbf{CB};][]{de23commitmentbank}\\footnote{\\url{https:\/\/super.gluebenchmark.com\/tasks}} (a corpus of short texts), and Multi-Genre Natural Language Inference Corpus \\cite[\\textbf{MNLI};][]{DBLP:conf\/naacl\/WilliamsNB18}\\footnote{\\url{https:\/\/cims.nyu.edu\/~sbowman\/multinli\/}} (a crowdsourced collection of sentence pairs for NLI).\nEach sentence pair is labelled with \\emph{entailment}, \\emph{neutral}, or \\emph{contradiction}.\n\n\\paragraph{PLM:}\nIn our experiments, we used the same pre-trained language model to evaluate AP, MP, and HFT equally.\nSpecifically, we used RoBERTa-large (355M parameters) \\footnote{\\url{https:\/\/huggingface.co\/roberta-large}}~\\cite{RoBERTa}, which has been used in much prior work in prompt learning~\\cite{shin-etal-2020-autoprompt,DBLP:conf\/naacl\/ScaoR21}.\nThe PLM was trained on five datasets, including BookCorpus\\footnote{\\url{https:\/\/yknzhu.wixsite.com\/mbweb}}, English Wikipedia\\footnote{\\url{https:\/\/en.wikipedia.org\/wiki\/English_Wikipedia}}, CC-News\\footnote{\\url{https:\/\/commoncrawl.org\/2016\/10\/news-dataset-available\/}}, OpenWebText\\footnote{\\url{https:\/\/github.com\/jcpeterson\/openwebtext}}, and Stories\\footnote{\\url{https:\/\/arxiv.org\/abs\/1806.02847}}.\nThe texts were tokenised using a byte-level Byte-Pair Encoding \\cite[BPE;][]{DBLP:conf\/acl\/SennrichHB16a} vocabulary of size 50,000.\n\n\\paragraph{Evaluating the Robustness of Prompts: }\nWe used \\emph{rate of degradation} (\\textbf{RoD})~\\cite{Meyers:2020} to evaluate robustness, which is defined as the decrease in accuracy of the target task due to the perturbations added to the prompt.\nIf the RoD of a model is small after the inclusion of a perturbation, the model is considered to be robust against that perturbation.\nSpecifically, we first calculate the respective accuracies $\\textrm{acc}_x$ and $\\textrm{acc}_{x^\\ast}$ on the same evaluation set for both prompt $x$ and its perturbated version $x^{\\ast}$.\nUsing the average accuracies $\\textrm{avg-acc}_x$ and $\\textrm{avg-acc}_{x^\\ast}$ over $M$ prompts ${x_1, ..., x_M}$, we calculate the RoD as $(\\textrm{avg-acc}_x - \\textrm{avg-acc}_{x^\\ast}) \/ \\textrm{avg-acc}_x = 1 - \\textrm{avg-acc}_{x^\\ast} \/ \\textrm{avg-acc}_x$.\n\n\n\\subsection{Effect of the Training Dataset Size}\n\\label{sec:exp:datasize} \nBefore moving on to robustness experiments, we first investigate the number of training instances on which AP and MP perform best, and used the best-performing AP and MP to evaluate their robustness in the subsequent experiments.\n\n\\paragraph{Experimental Settings:} \nWe gradually increased the size of the training dataset following the experimental setup of~\\citet{DBLP:conf\/naacl\/ScaoR21}. \nSpecifically, we experimented with randomly sampled subsets of the training dataset having varying numbers of instances in $\\{10, 15, 20, 30, 50, 70, 100, 150, 200\\}$.\nBecause the performance of few-shot learning methods often varies due to the high feature variance in the training data, we randomly sampled four subsets per each dataset size and used them independently for training the models\\footnote{NVIDIA RTX A5000 was mainly used.} (i.e. trigger tokens and label tokens for AP, or fine-tuned language model for MP and HFT) for each subset and report the average accuracy on the validation data for the four models ($M = 4$).\nWe used the matched (example from the same source as the training set) validation set for MNLI.\nFor CB, we held out 50 training instances for development as in \\citet{DBLP:conf\/naacl\/ScaoR21} and evaluated the original validation set as test data.\n\nWe searched for the optimal values for the following hyperparameters: the number of trigger tokens in \\{3, 5, 10\\}, the number of label tokens in \\{3, 5, 10\\}, and the number of tokens in a candidate set in \\{10, 50\\}. \nWe evaluated the test accuracy using the hyperparameters that had the highest accuracy on the validation data for each dataset size.\nIn the training of MP, we used AdamW optimizer~\\cite{DBLP:conf\/iclr\/LoshchilovH19} with an initial learning rate of $10^{-5}$ and a learning step of 1,000 following \\citet{DBLP:conf\/iclr\/MosbachAK21}.\n\n\n\\begin{table*}[t]\n    \\centering\n    \\small\n    \\begin{tabular}{ccccccc}\n        \\toprule\n        \\textbf{Method} & \\textbf{\\#Train} & \\textbf{Template} & \\textbf{\\#Prompt tokens} & \\textbf{\\#Label tokens per class} & \\multicolumn{2}{c}{\\textbf{Avg. accuracy}} \\\\\n        & & & & & \\textbf{CB} & \\textbf{MNLI} \\\\\n        \\midrule\n        AP & 200 & \\texttt{\\red{p} <MASK> \\blue{<T> ... <T>} \\red{h}} & 10 & 3 & 68.3 & 37.7 \\\\ \n        MP & 200 & \\texttt{\\red{h}\\blue{? |} <MASK>\\blue{,} \\red{p}} & 3 & 1 & \\uline{95.1} & \\uline{65.5} \\\\ \n        HFT & - & \\texttt{<CLS> \\red{p} <SEP> \\red{h}} & 0 & - & - & - \\\\ \n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n    The average accuracy of the experiment with four training subsets of 200 instances.\n    \\red{Red} represents the task inputs, \\red{\\texttt{h}} represents the hypothesis, \\red{\\texttt{p}} represents the premise, \\blue{blue} represents the prompt tokens, and \\blue{\\texttt{<T>}} represents a trigger token.\n    Unreported values were marked with `-'. \n    }\n    \\label{tab:pre-train-best}\n\\end{table*}\n\n\n\\paragraph{Main Results:}\n\\autoref{fig:pre-train} shows the performance\\footnote{HFT results were obtained from \\citet{DBLP:conf\/naacl\/ScaoR21}, F1-macro for CB and accuracy for MNLI.} against the training dataset size.\nWe see that in both CB and MNLI \\textbf{MP is always superior to AP}.\nFor example, with a dataset of size 200, AP and MP achieved the best accuracy in CB, MP's accuracy was 92.7\\%, while that of AP was lower at 54.2\\%.\n\n\\begin{figure}[t]\n  \\centering\n    \\includegraphics[clip, width=8cm]{datapoints.pdf}\n    \\caption{\n    Performance of AutoPrompt (AP), Manually-written Prompt (MP), and Head-based Fine-Tuning (HFT) on the scale of dataset size for CB and MNLI.\n    Means and their 95\\% confidence intervals are plotted.\n    The accuracy of HFT for dataset size for CB was not plotted because the accuracy was not reported.\n    }\n    \\label{fig:pre-train}\n\\end{figure}\n\nOur results also suggest that \\textbf{the performance of discrete prompts learnt by AP is highly dataset dependent.}\n\\citet{shin-etal-2020-autoprompt} reported results for AP and HFT on SICK-E~\\cite{DBLP:conf\/lrec\/MarelliMBBBZ14}, which is another NLI dataset.\nThey concluded that AP was always superior to HFT up to training dataset sizes of 1,000 for the same RoBERTa-large PLM that we use.\nHowever, our experiments show the opposite trend (i.e. HFT is superior to AP).\nThis suggests that even if AP is superior to HFT on a given dataset, it is not guaranteed to be superior in a different dataset for the same task.\nThis may be due to the differences in the domain and annotation guidelines for each dataset.\nFor example, the accuracy of MNLI was quite low on AP, which contrasts with that of CB.\nThis result suggests that the discrepancies in domains and annotation guidelines make it difficult for AP to perform consistently.\n\n\\paragraph{Best Prompts:}\n\\autoref{tab:pre-train-best} shows the average accuracy of models trained on 200 instances that performed well in both CB and MNLI.\nNote that there are four training subsets for each dataset size, resulting in corresponding four trained AP prompts and four PLMs fine-tuned by MP. \\footnote{We show the four best prompts learnt by AP in \\autoref{sec:supplementary}.}\nIn the robustness evaluations in \\autoref{sec:exp:reorder} through \\autoref{sec:exp:input-perturbation}, we used these learnt APs and MPs.\nIn this paper, (a) trigger tokens learnt by AP, and (b) manually-written prompts excluding the task inputs and  mask tokens are collectively referred to as the \\emph{prompt tokens}.\n\n\n\\subsection{Token Reordering}\n\\label{sec:exp:reorder}\nAs seen from \\autoref{tbl:autoprompt-examples}, compared to MPs where the ordering of tokens in a prompt is manually determined, discrete prompts learnt by AP appear to have no obvious ordering among their tokens.\nTo empirically investigate the importance of the token order in a discrete prompt, we conduct an experiment where we randomly shuffle the prompt tokens and measure the effect on the downstream task performance.\n\n\\paragraph{Experimental Procedure:}\nGiven a discrete prompt, we first randomly reordered its prompt tokens (e.g. shaded in blue in \\autoref{tab:pre-train-best}).\nNext, we used the reordered prompt with the PLM to make entailment predictions for the test instances in the CB and MNLI datasets.\nFinally, the entailment prediction accuracy (Acc) obtained with the reordered prompts was computed.\nWe repeated this evaluation 10 times for each prompt and report the averaged values and the corresponding RoD values.\n\n\\paragraph{Main Results:}\nFrom \\autoref{tab:reorder_result} we see that the accuracy drops for both AP and MP when the prompt tokens are randomly reordered.\nIn particular, the accuracy of AP drops significantly compared to that of MP.\nFor example, the accuracy of AP on CB drops by ca. 14\\% due to token reordering, while that for MP drops only by ca. 2\\%.\nIntuitively, one would expect that changing the order of prompt tokens in MP would result in a significant drop in accuracy because the meaning of the prompts would change.\nHowever, we see that this is not the case. \nThis result shows that \\textbf{the discrete prompts learnt by AP strongly rely on the token order}.\n\n\\paragraph{Additional Analysis:}\nTo further analyze the relationship between the level of perturbation introduced by reordering prompt tokens in AP and its effect on the performance, we computed the token-level edit distance \\citep[Levenshtein distance;][]{levenshtein1966binary} between each prompt and its token-shuffled version as shown in \\autoref{fig:scatter_reodering}.\nFor all four AP prompts, we see that the accuracy drops when the perturbation noise (i.e. measured by edit distance) increases.\nThis reconfirms the lack of robustness in discrete prompts learnt by AP to the random shuffling of prompt tokens.\n\n\\begin{table}[t]\n    \\centering\n    \\small\n    \\begin{tabular}{clllccc}\n        \\toprule\n        \\textbf{Method} & \\textbf{Metrics} & \\textbf{CB} & \\textbf{MNLI} \\\\\n        \\midrule\n        \\multirow{2}{*}{AP} & Acc & \\textbf{54.2} & \\textbf{34.3} \\\\ \n                            & RoD & \\textbf{0.21} & \\textbf{0.10} \\\\ \n        \\midrule\n        \\multirow{2}{*}{MP} & Acc & \\uline{92.7} & \\uline{59.3} \\\\\n                            & RoD & \\uline{0.03} & \\uline{0.09} \\\\ \n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n    Performance of reordered prompts.\n    Acc denotes accuracy; RoD denotes the RoD from before the reordering (\\autoref{tab:pre-train-best}).\n    The largest drops in accuracy are \\textbf{bolded} and the smallest drops are \\uline{underlined} for each method and dataset.\n    AP relies more strongly on word order than MP.\n    }\n    \\label{tab:reorder_result}\n\\end{table}\n\n\n\\subsection{Token Deletion} \n\\label{sec:exp:deletion}\nAs seen from \\autoref{tbl:autoprompt-examples}, the discrete prompts learnt by AP perform better than MP.\nHowever, it is often difficult to determine the importance of prompt tokens to the target task due to their lack of interpretability (e.g. prompt token `\\emph{neau}' in \\autoref{tbl:autoprompt-examples}).\nTo understand the significance of individual prompt tokens to the overall discrete prompt, we conducted an experiment where we systematically deleted one or more prompt tokens at various positions from a given discrete prompt and measure the drop (if any) in the performance of the NLI task.\n\n\\begin{figure}[t]\n  \\centering\n    \\includegraphics[clip, width=7.5cm]{reordering_edit_distance.png}\n    \\caption{Edit distance and accuracy of the reordered trigger tokens. We evaluated them on the validation data of CB. \n    The prompts numbered 0 through 3 each represent the four prompts learnt by AP (shown in \\autoref{tab:best-ap-prompts-cb}).\n    Note that a point with an edit distance of zero indicates accuracy with the original trigger token.\n    }\n    \\label{fig:scatter_reodering}\n    \\vspace{-3mm}\n\\end{figure}\n\n\\paragraph{Experimental Procedure:}\nWe evaluated two settings of prompt deletion: \\emph{single} and \\emph{multiple} token deletion.\nIn the single token deletion setting, we deleted one token at different positions in a given prompt.\nFor AP, we repeated this with each of the four discrete prompts (shown in \\autoref{tab:pre-train-best}) and report the average accuracy.\nIn the multiple token deletion setting, we delete $n \\in \\{1, 3, 5, 7\\}$ prompt tokens following three strategies:\n\\emph{Random-deletion} deletes $n$ prompt tokens randomly,\n\\emph{Front-deletion} deletes $n$ consecutive prompt tokens from the beginning of the prompt, and \\emph{Back-deletion} deletes $n$ tokens counted backwards from the end of the prompt.\nIn random-deletion, we ran 100 trials and report the average accuracy.\nAs in the previous experiments, we used four prompts for AP and report the averaged results.\n\n\\begin{table*}[t!]\n    \\setlength{\\tabcolsep}{1.9mm} \n    \\centering\n    \\small\n    \\begin{tabular}{cccccccccccccc}\n    \\toprule\n    \\multirow{2}{*}{\\textbf{Task}} & \\multirow{2}{*}{\\textbf{Method}} & \\multirow{2}{*}{\\textbf{Metrics}}  & \\multicolumn{10}{c}{\\textbf{Position of the deleted prompt token}} & \\multirow{2}{*}{\\textbf{Orig.}} \\\\\n    & & & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n    \\midrule\n    \\multirow{4}{*}{CB} & \n        \\multirow{2}{*}{AP} & Acc & 62.1 & \\textbf{61.6} & 63.4 & 59.4 & \\uline{65.6} & \\uline{65.6} & 62.1 & 63.8 & 62.1 & 62.9 & 68.3\\\\\n        & & RoD & 0.09 & \\textbf{0.10} & 0.07 & 0.13 & \\uline{0.04} & \\uline{0.04} & 0.09 & 0.07 & 0.09 & 0.08 & - \\\\\n        \\cmidrule(lr){2-14}\n        & \\multirow{2}{*}{MP} & Acc & 93.8 & \\textbf{93.3} & \\uline{96.0} & - & - & - & - & - & - & - & 95.1 \\\\\n        & & RoD & 0.01 & \\textbf{0.02} & \\uline{-0.01} & - & - & - & - & - & - & - & - \\\\\n        \\midrule\n    \\multirow{4}{*}{MNLI} & \n        \\multirow{2}{*}{AP} & Acc & \\uline{37.9} & 37.8 & \\textbf{36.6} & 37.5 & 37.5 & 37.2 & 37.5 & 37.4 & 37.5 & 37.1 & 37.7 \\\\\n        & & RoD & \\uline{-0.01} & 0.00 & \\textbf{0.03} & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & 0.02 & - \\\\\n        \\cmidrule(lr){2-14}\n        & \\multirow{2}{*}{MP} & Acc & 64.5 & \\uline{65.4} & \\textbf{55.4} & - & - & - & - & - & - & - & 65.5 \\\\\n        & & RoD & 0.02 & \\uline{0.00} & \\textbf{0.15} & - & - & - & - & - & - & - & - \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{\n    Average accuracy was obtained after deleting a single token at different positions of a given prompt.\n    The largest drops in accuracy over the deletion positions are \\textbf{bolded} and the smallest drops are \\uline{underlined} for each method and dataset.\n    Column `Orig.' shows the performance of the original prompt.\n    }\n    \\label{tab:deletion_one}\n\\end{table*}\n\n\\paragraph{Results:}\nFrom \\autoref{tab:deletion_one} we see that the \\textbf{accuracy of both AP and MP drops even when a single token is deleted} at specific positions.\nHowever, the observed trends differ in CB and MNLI.\nFor example, AP resulted in higher RoD values in CB compared to MNLI.\nThis shows that the robustness of AP under single token deletion heavily depends on the dataset.\n\\autoref{tab:deletion_multi} shows the results for the multiple token deletion setting.\nWe see that \\textbf{the performance of both AP and MP degrades when more tokens are deleted.}\nInterestingly, the accuracy drop in CB is very small for MP even when all prompt tokens are deleted (i.e., only the task inputs and \\texttt{<MASK>} were used as the input).\nThis suggests that the performance on CB is less reliant on the prompt tokens in MP.\n\n\n\\subsection{Cross-Dataset Evaluation}\n\\label{sec:exp-cross-data}\nGiven that discrete prompt learning methods such as AP learn prompts from a small set of training instances, it is important that the learnt prompts encode generalizable task-specific features and not random artefacts in the training sample used.\nTo study the transferability of the learnt discrete prompts from one dataset to another, we conduct a cross-dataset evaluation as described next.\n\n\\begin{table}[t!]\n    \\setlength{\\tabcolsep}{0.9mm} %\n    \\centering\n    \\small\n    \\begin{tabular}{ccccccccccccc}\n        \\toprule\n        \\multirow{2}{*}{\\textbf{Strategy}} & \\multirow{2}{*}{\\textbf{Method}} & \\multirow{2}{*}{\\textbf{Metrics}} & \\multicolumn{4}{c}{\\textbf{\\#Deleted Tokens}} & \\multirow{2}{*}{\\textbf{Orig.}} \\\\\n        & & & 1 & 3 & 5 & 7 \\\\\n        \\midrule\n        \\midrule\n        \\multicolumn{8}{c}{\\textbf{CB}} \\\\\n        \\midrule\n       \n        \\multirow{4}{*}{Random}\n            & \\multirow{2}{*}{AP}\n            & Acc & \\uline{56.7} & 56.0 & 55.4 & \\textbf{54.8} & 68.3 \\\\\n        & & RoD & \\uline{0.17} & 0.18 & 0.19 & \\textbf{0.20} & - \\\\\n            \\cmidrule(lr){2-8}\n            & \\multirow{2}{*}{MP}\n            & Acc & \\textbf{93.3} & \\uline{94.6} & - & - & 95.1\\\\\n        & & RoD & \\textbf{0.02} & \\uline{0.01} & - & - & - \\\\\n        \\midrule\n        \\multirow{4}{*}{Front}\n            & \\multirow{2}{*}{AP}\n            & Acc & \\uline{62.1} & \\textbf{49.1} & 57.6 & 57.6 & 68.3 \\\\\n        & & RoD & \\uline{0.09} & \\textbf{0.28} & 0.16 & 0.16 & - \\\\\n            \\cmidrule(lr){2-8}\n            & \\multirow{2}{*}{MP}\n            & Acc & \\textbf{93.8} & \\uline{94.6} & - & - & 95.1\\\\\n        & & RoD & \\textbf{0.01} & \\uline{0.01} & - & - & - \\\\\n        \\midrule\n        \\multirow{4}{*}{Back}\n            & \\multirow{2}{*}{AP}\n            & Acc & \\uline{62.9} & 57.6 & 55.8 & \\textbf{51.3} & 68.3 \\\\\n        & & RoD & \\uline{0.08} & 0.16 & 0.18 & \\textbf{0.25} & - \\\\\n            \\cmidrule(lr){2-8}\n            & \\multirow{2}{*}{MP}\n            & Acc & \\uline{96.0} & \\textbf{94.6} & - & - & 95.1 \\\\\n        & & RoD & \\uline{-0.01} & \\textbf{0.01} & - & - & - \\\\\n        \\midrule\n        \\midrule\n       \n        \\multicolumn{8}{c}{\\textbf{MNLI}} \\\\\n        \\midrule\n            \\multirow{4}{*}{Random}\n            & \\multirow{2}{*}{AP}\n            & Acc & \\textbf{35.8} & \\textbf{35.8} & 36.0 & \\uline{36.2} & 37.7 \\\\\n        & & RoD & \\textbf{0.05} & \\textbf{0.05} & 0.05 & \\uline{0.04} & - \\\\\n            \\cmidrule(lr){2-8}\n            & \\multirow{2}{*}{MP}\n            & Acc & \\uline{65.4} & \\textbf{52.6} & - & - & 65.5 \\\\\n        & & RoD & \\uline{0.0} & \\textbf{0.20} & - & - & - \\\\\n        \\midrule\n        \\multirow{4}{*}{Front}\n            & \\multirow{2}{*}{AP}\n            & Acc & \\uline{37.9} & 36.5 & 36.2 & \\textbf{36.0} & 37.7 \\\\\n        & & RoD & \\uline{-0.01} & 0.03 & 0.04 & \\textbf{0.05} & - \\\\\n            \\cmidrule(lr){2-8}\n            & \\multirow{2}{*}{MP}\n            & Acc & \\uline{64.5} & \\textbf{52.6} & - & - & 65.5 \\\\\n        & & RoD & \\uline{0.02} & \\textbf{0.20} & - & - & - \\\\\n        \\midrule\n        \\multirow{4}{*}{Back}\n            & \\multirow{2}{*}{AP}\n            & Acc & \\uline{37.1} & 36.7 & 35.7 & \\textbf{36.5} & 37.7 \\\\\n        & & RoD & \\uline{0.02} & 0.03 & 0.05 & \\textbf{0.03} & - \\\\\n            \\cmidrule(lr){2-8}\n            & \\multirow{2}{*}{MP}\n            & Acc & \\uline{55.4} & \\textbf{52.6} & - & - & 65.5 \\\\\n        & & RoD & \\uline{0.15} & \\textbf{0.20}  & - & - & - \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n    Average accuracy was obtained after deleting multiple tokens from a given prompt.\n    The largest drops in accuracy over the deleted tokens  are \\textbf{bolded} and the smallest drops are \\uline{underlined} for each strategy and method.\n    }\n    \\label{tab:deletion_multi}\n\\end{table}\n\n\\paragraph{Experimental Procedure:}\nWe used one NLI dataset (e.g. CB) to learn the prompts and then use them to make entailment predictions in another NLI dataset (e.g. MNLI).\nWe then measured the drop in accuracy using RoD for this cross-dataset transferability task with respect to the accuracy of test data from the same dataset.\n\n\\paragraph{Results:}\nAs seen from \\autoref{tab:cross-dataset-eval}, \\textbf{AP-based prompts do not generalize well across datasets.}\nFor both AP and MP, RoD is larger in the transfer from CB to MNLI than in the opposite direction.\nThis implies that MNLI is a better dataset for fine-tuning a PLM for NLI using discrete prompts.\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{lccccc}\n\\toprule\n\\multirow{2}{*}{\\textbf{Method}}& \\multicolumn{2}{c}{\\textbf{Test Dataset}} & \\multirow{2}{*}{\\textbf{RoD}}\\\\\n& \\textbf{CB} & \\textbf{MNLI} & \\\\\n\\midrule\nAP trained on CB   & 68.3 & 36.1  & \\uline{0.47} \\\\ \nAP trained on MNLI & 42.9 & 37.7  & \\uline{0.12} \\\\ \n\\midrule\nMP trained on CB   & 95.1 & 43.4 & \\textbf{0.54} \\\\ \nMP trained on MNLI & 43.8 & 65.5 & \\textbf{0.33} \\\\ \n\\bottomrule\n\\end{tabular}\n\\caption{\nAccuracy and RoD for the cross-dataset evaluation where a method (AP\/MP) is trained on one NLI dataset (CB\/MNLI) and the learnt prompts are used to make entailment predictions in a different NLI dataset.\n}\n\\label{tab:cross-dataset-eval}\n\\end{table}\n\n\n\\subsection{Adversarial Perturbations}\n\\label{sec:exp:input-perturbation}\nIntroducing carefully designed adversarial perturbations to the test instances such as modifications to sentences that might or might not alter the original target labels have been used as a technique for probing the robustness of models~\\cite{DBLP:journals\/corr\/GoodfellowSS14}. \nPrevious studies~\\cite{DBLP:journals\/corr\/SamantaM17,DBLP:conf\/aaai\/JinJZS20} have shown that pre-trained models can be easily fooled to make incorrect predictions with seemingly innocuous perturbations to the test instances. Therefore, we evaluate discrete prompt-based NLI models for their robustness against adversarially perturbated test instances.\n\n\\paragraph{Evaluation Dataset:}\nFor this purpose, we asked two annotators to manually edit hypothesis sentences in NLI test data considering two types of perturbations:\n(1) perturbations that do not change reference labels, and (2) perturbations that change reference labels.\nAn example is shown in  \\autoref{tab:perturbation-data}. \n\nFor the first type of perturbation, we edited a hypothesis sentence such that its relationship with the corresponding premise remains unchanged.\nFor the second type, we edited a hypothesis sentence such that its relationship (e.g., from \\emph{entailment} to \\emph{contradiction}) will be reversed.\nThe premise and hypothesis pairs were sampled from CB (validation set) and MNLI (test set).\nBecause there are ca. 10,000 test instances in MNLI and it is costly to manually edit sentences, we used 100 randomly-chosen sentence pairs covering MNLI and CB.\n\n\\begin{table}[t]\n    \\centering\n    \\small\n    \\begin{tabular}{lp{7em}p{5em}}\n        \\toprule\n        \\textbf{} & \\textbf{Hypothesis} & \\textbf{Label} \\\\ \n        \\midrule\n        Original & The Wither's only had daughters. & contradiction \\\\\n        \\midrule\n        Perturbation & \\\\\n        \\quad w\/o label changes & The Wither's did not have sons. & contradiction \\\\\n        \\quad w\/ label changes & The Wither's had a boy. & entailment \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n    Examples of our evaluation set consisting of task inputs with perturbations. \n    The premise sentence is `\\emph{The Wither's eldest boy, one of the four of the town militia, saluted in the old style with his stick sword.}'\n    } \n    \\label{tab:perturbation-data}\n\\end{table}\n\n\\iffalse\n\\begin{table}[t]\n    \\setlength{\\tabcolsep}{1mm} %\n    \\centering\n    \\begin{tabular}{lccccc}\n        \\toprule\n        & \\multicolumn{2}{c}{\\textbf{\\#Perturbation}} \\\\\n        & \\textbf{w\/o label changes} & \\textbf{w\/ label changes} \\\\ \n        \\midrule\n        CB & 50 & 49 \\\\ \n        MNLI & 48 & 47 \\\\ \n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n    The dataset sizes (measured by the number of instances) for the adversarial dataset.\n    }\n    \\label{tab:perturbation-dataset-stat}\n\\end{table}\n\\fi\n\n\\paragraph{Experimental Procedure:}\nWe computed the RoD of average accuracies obtained with original and adversarial test instances.\nSpecifically, we used the AP prompts in \\autoref{tab:pre-train-best} under three settings:\n(a) original (without perturbations), \n(b) perturbations without label changes, \nand (c) perturbations with label changes.\nThen, we calculate RoD from (a) to (b) and (a) to (c) as shown in \\autoref{tab:task-input-perturbation}.\n\n\\begin{table}[t!]\n\\small\n\\setlength{\\tabcolsep}{0.5mm} %\n\\centering\n\\begin{tabular}{ccccc}\n\\toprule\n\\textbf{Perturbation} & \\textbf{Method} & \\textbf{Metrics} & \\textbf{CB} & \\textbf{MNLI} \\\\\n\\midrule\n\\multirow{4}{*}{Original}\n    & \\multirow{2}{*}{AP} \n      & Acc & 54.5 & 40.5 \\\\\n    & & RoD & - & - \\\\\n    \\cmidrule(l){2-5}\n    & \\multirow{2}{*}{MP} \n      & Acc & 95.5 & 71.0 \\\\\n    & & RoD & - & - \\\\\n\\midrule\n\\multirow{4}{*}{\\begin{tabular}{c}Perturbation \\\\w\/o label changes\\end{tabular}}\n    & \\multirow{2}{*}{AP} \n      & Acc & 55.5 & 43.2 \\\\\n    & & RoD & \\uline{-0.02} & \\uline{-0.07} \\\\\n    \\cmidrule(l){2-5}\n    & \\multirow{2}{*}{MP} \n      & Acc & 93.0 & 66.7 \\\\\n    & & RoD & \\textbf{0.03} & \\textbf{0.06} \\\\\n\\midrule\n\\multirow{4}{*}{\\begin{tabular}{c}Perturbation \\\\w\/ label changes\\end{tabular}}\n    & \\multirow{2}{*}{AP} \n      & Acc & 42.3 & 39.4 \\\\\n    & & RoD & \\uline{0.22} & \\uline{0.03} \\\\\n    \\cmidrule(l){2-5}\n    & \\multirow{2}{*}{MP} \n      & Acc & 41.8 & 61.2 \\\\\n    & & RoD & \\textbf{0.56} & \\textbf{0.14} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{\nAccuracy and RoD in prompts for task inputs that include perturbations. The RoD here is the rate of degradation in the average accuracy from the original without perturbations to perturbations without label changes or perturbations with label changes.\nThe largest drops in accuracy are \\textbf{bolded} and the smallest drops are \\uline{underlined} for each perturbation and method.\n}\n\\label{tab:task-input-perturbation}\n\\end{table}\n\n\n\\paragraph{Results:}\nOverall, we see that the RoD of AP is consistently smaller than that of MP in both CB and MNLI under both types of perturbations.\nHowever, it is also clear that the accuracy obtained with AP is much smaller than that with MP.\nFor the perturbations without label changes, both AP and MP show small RoD values, compared to those with label changes.\\footnote{w\/o label change modifications slightly increase the average length of a hypothesis and AP seems to better exploit this extra information for inference resulting in a slight improvement in accuracy (negative RoD).}\nThis shows that both AP and MP are relatively robust against modifications to the hypotheses that do not significantly alter the meaning.\nHowever, when stronger perturbations are introduced that would result in label changes, the accuracy of both AP and MP drops significantly. \\footnote{MP is less robust compared to AP, likely as a result of overfitting to strongly perturbed training data during fine-tuning the PLM.}\nThis is a concern because it shows that \\textbf{neither AP nor MP is sufficiently robust to correctly predict the target labels when the hypothesis sentences in test data are adversarially modified.}\n\n\n\\section{Conclusion}\nWe investigated the robustness of discrete prompts under different perturbations.\nWe found that although discrete prompts remain relatively robust against token deletion, it is highly sensitive to other types of perturbations such as token shuffling.\nFor adversarial perturbations to the input, discrete prompts were robust to weak perturbations without label changes, but AP was more robust than MP for perturbations with label changes.\nMoreover, they generalize poorly across different datasets annotated for NLI.\nWe hope our analysis will inspire future work to develop methods that learn both accurate as well as robust discrete prompts.\n\n\n\\section{Limitations}\nPossible limitations of this work are:\n\\begin{itemize}\n\\item We chose popular discrete prompt methods of AP and MP and did not investigate other methods in this work. Our analysis procedure can still be applied to other discrete prompts such as AvgTrigger~\\cite{wallace-etal-2019-universal}.\n\\item We chose RoBERTa-large following previous studies of HFT~\\cite{DBLP:conf\/naacl\/ScaoR21} and AP~\\cite{shin-etal-2020-autoprompt} for reproducible and identical comparisons with them. Other PLMs would lead to different results, but they can also be investigated in the same way as in this work.\n\\item This work focuses on NLI because it is a fundamental natural language understanding task and still difficult even with PLMs~\\cite{GPT3}. Other complex downstream tasks are worth investigating for a deeper understanding of prompt-based approaches in future work.\n\\item The results and conclusions are from the English datasets and would differ in other languages. However, our methodologies do not depend on English and can be applied to other languages as important future studies. \n\\item Since there was a performance gap between MP\/HFT and AP, the accuracies by the perturbations could be affected. However, this work does not aim to find the best prompt learning method but to analyze the robustness of discrete prompts for a deeper understanding of them.\n\\end{itemize}\n\n\n\\section{Ethical Considerations}\nOur adversarial dataset came from existing datasets of CB and MNLI.\nWe visually checked the instances in the data development and found no instances with ethical concerns.\n\nOne should also be aware of social biases (e.g. gender stereotypes) in PLM. \nRoBERTa, the PLM we used in our experiments, is known to have gender biases~\\cite{DBLP:journals\/corr\/abs-2105-05541}. \nSince we used it as-is in order to follow the experimental conditions of previous studies using RoBERTa, our current results are possibly influenced by such biases.\nHowever, the consideration of the prompt robustness of this work would not pose or magnify such ethical concerns.\n\n\n\\section{Acknowledgments}\nThis research was supported by the JSPS KAKENHI (Grants-in-Aid for Scientific Research) JP22H03654.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section*{Supplementary Material}\\label{supplementary_material}\n\n\n\\section{Introduction}\n\\label{introduction}\n\n\nSeveral studies about Hybrid Unmanned Aerial Underwater Vehicles (HUAUVs) have been published recently~\\cite{drews2014hybrid, neto2015attitude, da2018comparative, maia2017design, lu2019multimodal, mercado2019aerial, horn20, aoki2021}. These types of vehicles enable an interesting range of new applications due to their capability to operate both in the air and underwater. These include inspection and mapping of partly submerged areas in industrial facilities, search and rescue and others. Most of the literature in the field is still focused on vehicle design, with few published works on the theme of autonomous navigation~\\cite{bedin2021deep}. The ability to navigate in both environments and successfully transit from one to another imposes additional challenges that must be addressed.\n\nLately, approaches based on Deep-RL have been enhanced to address navigation-related tasks for a range of mobile vehicles, including ground mobile robots~\\cite{ota2020efficient}, aerial robots~\\cite{tong2021uav,grando2022double} and underwater robots~\\cite{carlucho2018}. Based on actor-critic methods and multi-layer network structures, these approaches have achieved interesting results in mapless navigation, obstacle avoidance, even including media transitioning for HUAUVs~\\cite{bedin2021deep, de2022depth}. However, the challenges faced by this kind of vehicle make these existing approaches still too limited, with poor generalization through different scenarios.\n\n\nIn this work, we present two new double-critic Deep-RL approaches in the context of HUAUVs to perform navigation-related tasks in a continuous state-space environment:  (1)~a deterministic approach based on Twin Delayed Deep Deterministic Policy Gradient (TD3)~\\cite{fujimoto2018addressing}; and\n (2)~a stochastic approach based on Soft Actor-Critic (SAC)~\\cite{haarnoja2018soft}. We show we are capable of training agents that are consistently better than state-of-the-art in generalizing through different simulated scenarios, with improved stability in mapless navigation, obstacle avoidance and medium transitions. Our evaluation tasks included both air-to-water and water-to-air transitions. We compared our methods with other single critic approaches and with an adapted version of a traditional Behavior-Based Algorithm (BBA)~\\cite{marino2016minimalistic} used in aerial vehicles. Fig.~\\ref{fig:simenv} shows a snapshot of our simulation environment.\n \n\\begin{figure}[tbp!]\n    \\vspace{-2mm}\n    \\centering\n    \\includegraphics[width=\\linewidth]{img\/sonar_v5.png}\n    \\caption{Our HUAUV underwater in the first scenario (left) and its respective sonar readings (right).}\n    \\label{fig:simenv}\n    \\vspace{-4mm}\n\\end{figure}\n\n\nThis work provides the following main contributions:\n\n\\begin{itemize}\n\n\\item We show that our agents present a robust capacity for generalization through different environments, achieving a good performance in a complex and completely unknown environment. The robot also performs the medium transition, being capable of arriving at the desired target and avoiding collisions.\n\n\\item We show that a Long Short Term Memory (LSTM) architecture can achieve better overall performance and capacity for generalization than the state-of-the-art Multi-Layer Perceptron (MLP) architectures\n\n\\end{itemize}\n\nThis work has the following structure: the related works are discussed in the following section (Sec. \\ref{related_works}). Following it, we present our methodology in Sec. \\ref{methodology}. The results are presented in Sec. \\ref{results} and discussed in Sec. \\ref{conclusion}.\n\n\\section{Related Work}\n\\label{related_works}\n\n\nFor more traditional types of vehicles, several works have been published demonstrating how efficiently Deep-RL can solve the mapless navigation problem~\\cite{tobin2017domain}. For a ground robot, Tai~\\emph{et al.}~\\cite{tai2017virtual} demonstrated a mapless motion planner based on the DDPG algorithm employing a 10-dimensional range finder combined with the relative distance to the target as inputs and continuous steering signals as outputs. Recently, Deep-RL methods have also been successfully used by Ota~\\emph{et al.}~\\cite{ota2020efficient}, de Jesus~\\emph{et al.}~\\cite{jesus2019deep,jesus2021soft} and others, to accomplish mapless navigation-related tasks for terrestrial mobile robots. Singh and Thongam~\\cite{singh2018mobile} demonstrated efficient near-optimal navigation for a ground robot in dynamic environments employing an MLP to perform speed control while choosing collision-free path segments.\n\nFor UAVs, Kelchtermans and Tuytelaars \\cite{kelchtermans2017hard} demonstrated how memory could help Deep Neural Networks (DNN) for navigation in a simulated room crossing task. Tong~\\emph{et al.}~\\cite{tong2021uav} showed better than state-of-the-art convergence and effectiveness in adopting a DRL-based method combined with a LSTM to navigate a UAV in highly dynamic environments, with numerous obstacles moving fast.\n\nWhen it comes to problems involving specifically mapless navigation for UAVs, few works examine the effectiveness of Deep-RL. Grando~\\emph{et al.}~\\cite{grando2020visual} explored a Deep-RL architecture, however, navigation was constrained to a 2D space. Rodriguez \\emph{et al.}~\\cite{rodriguez2018deep} employed a DDPG-based strategy to solve the problem of landing UAVs on a moving platform. Similar to our work, they employed RotorS framework~\\cite{furrer2016rotors} combined with the Gazebo simulator. Sampedro~\\emph{et al.}~\\cite{sampedro2019fully} proposed a DDPG-based strategy for search and rescue missions in indoor environments, utilizing real and simulated visual data. Kang~\\emph{et al.}~\\cite{kang2019generalization} also used visual information, although he focused on the subject of collision avoidance. In a go-to-target task, Barros~\\emph{et al.}~\\cite{2020arXiv201002293M} applied a SAC-based method for the low-level control of a UAV. Double critic-based Deep-RL approaches similar to the one proposed here have also been shown to yield good results\\cite{grando2022double}.\n\nThe HUAUV literature is still mostly concerned with vehicle design and modeling \\cite{drews2014hybrid, neto2015attitude, da2018comparative, maia2017design, lu2019multimodal, mercado2019aerial, horn20}. Two works have recently tackled the navigation problem with the medium transition of HUAUVs~\\cite{pinheiro2021trajectory}, \\cite{bedin2021deep}. Pinheiro~\\emph{et al.} \\cite{pinheiro2021trajectory} focused on smoothing the medium transition problem in a simulated model on MATLAB. Grando~\\emph{et al.}~\\cite{bedin2021deep} developed Deep-RL actor-critic approaches and a MLP architecture. These two works were developed using generic distance sensing information for aerial and underwater navigation. In contrast, our work relies on more realistic sensing data, with the simulated LIDAR and sonar being both based on real-world devices.\n\nThe HUAUV presented in this paper is based on Drews-Jr~\\emph{et al.} \\cite{drews2014hybrid} model, which Neto~\\emph{et al.}~\\cite{neto2015attitude} has largely expanded. Our work differs from the previously discussed works by only using the vehicle's relative localization data and not its explicit localization data. We also present Deep-RL approaches based on double critic techniques instead of single critic, with RNN structures instead of MLP, traditionally used for mapless navigation of mobile robots. We compare our approaches with state-of-the-art Deep-RL approaches and with a behavior-based algorithm \\cite{marino2016minimalistic} adapted for hybrid vehicles to show that our new methodology improves the overall capability to generalize through distinct environments.\n\n\\section{Methodology}\n\\label{methodology}\n\nIn this section, we describe our simulation environment, our hybrid vehicle, and the proposed Deep-RL, detailing the network structure for both deterministic and stochastic agents. We also introduce the task that the vehicle must accomplish autonomously and the respective reward function.\n            \n\\subsection{Deterministic Deep RL\n\nDeveloping on the DQN~\\cite{mnih2013playing}, Deep Deterministic Policy Gradient (DDPG)~\\cite{lillicrap2015continuous} employs an actor-network where the output is a vector of real values representing the chosen action, and a second neural network to learn the target function, providing stability and making it ideal for mobile robots~\\cite{jesus2019deep}. While it provides good results, DDPG still has its problems, like overestimating the Q-values, which leads to policy breaking. TD3~\\cite{fujimoto2018addressing} uses DDPG as its backbone, adding some improvements, such as clipped double-Q~learning with two neural networks as targets for the Bellman error loss functions, delayed policy updates, and Gaussian noise on the target action, raising its performance.    \n\nOur deterministic approach is based on the TD3 technique. The pseudocode can be seen in Algorithm~\\ref{alg:docrl_d}.\n\n\\begin{algorithm}[!htb]\n    \\algsetup{linenosize=\\tiny}\n    \\scriptsize\n    \\caption{Deep Reinforcement Learning Deterministic}\n    \\label{alg:docrl_d}\n    \\begin{algorithmic}[1]\n        \\STATE Initialize params of critic networks $\\theta_{1}$, $\\theta_{2}$ , and actor network $\\phi$\n        \\STATE Initialize params of target networks $\\phi^{\\prime}\\leftarrow\\phi$, $\\theta_{1}^{\\prime}\\leftarrow\\theta_{1}$, $\\theta_{2}^{\\prime}\\leftarrow\\theta_{2}$\n        \\STATE Initialize replay buffer $\\beta$\n        \\FOR{$ep = 1$ to $max\\_eps$}\n            \\STATE reset environment state\n            \\FOR{$t = 0$ to $max\\_steps$}\n                \\IF {$t < start\\_steps$}\n                    \\STATE $a_{t} \\leftarrow $ env.action\\_space.sample() \n                %\n                \\ELSE\n                    \\STATE $a_{t}\\leftarrow\\mu_{\\phi}(s_t)+\\epsilon,\\ \\epsilon\\sim \\mathcal{N}(0,OU)$\n                %\n                \\ENDIF\n                \n                \\STATE $s_{t+1}$, $r_{t}$, $d_{t}$, \\_ $\\leftarrow$ env.step($a_{t}$)\n                \n                \\STATE store the new transition $(s_{t}, a_{t}, r_{t}, s_{t+1}, d_{t})$ into $\\beta$\n                \n                \\IF{$t > start\\_steps$}\n                    \\STATE Sample mini-batch $B$ of $N$ transitions $(s_{t}, a_{t}, r_{t}, s_{t+1}, d_{t})$ from $\\beta$\n                    \n                    \\STATE $a'\\leftarrow\\mu_{\\phi^{\\prime}}(s^{\\prime})+\\epsilon,\\ \\epsilon\\sim clip(\\mathcal{N}(0,\\tilde{\\sigma}), -c,\\ c)$ \n                    \n                    \\STATE Computes target: \\\\ $Q_{t} \\leftarrow r+\\gamma*\\min_{i=1,2}Q_{\\theta_i}(s', a')$\n                    \n                   \n                    \n                    \\STATE Update double critics with one step gradient descent:\\\\\n                   $\\nabla_{\\theta_i} \\frac{1}{N} \\sum_{i \\in B}(Q_t - Q_{\\theta_{i}(s_{t},a_{t})})^2$  \\qquad for i=1,2\n                    \n                    \\IF {t \\% $policy\\_freq(t)$ == 0}\n                        \\STATE Update policy with one step gradient descent:\\\\             \n                        $\\nabla_{\\phi}\\frac{1}{N} \\sum_i[\\nabla_{a_{t}}Q_{\\theta_{1}}(s_{t},a_{t})\\vert _{a_{t}=\\mu(\\phi)}\\nabla_{\\phi}\\mu_{\\phi}(s_{t})]$\n                        \n                        Soft update for the target networks: \\\\\n                        \\STATE $\\phi^{\\prime}\\leftarrow\\tau\\phi+(1-\\tau)\\phi^{\\prime}$\n                        \\STATE $\\theta_{i}^{\\prime}\\leftarrow\\tau\\theta_{i}+(1-\\tau)\\theta_{i}^{\\prime}$ \\qquad for i=1,2\n                        \n                        \n                    \\ENDIF\n                \\ENDIF\n            \\ENDFOR\n        \\ENDFOR\n  \\end{algorithmic}\n\\end{algorithm}\n\nWe train for $max\\_steps$ steps in $max\\_eps$ episodes. Our approach starts by exploring random actions for the initial $start\\_steps$ steps. We use an LSTM as the actor-network $\\phi$ and $\\phi^{\\prime}$ as its target. The double critics are also LSTM networks, denoted by $\\theta_{1}$ and $\\theta_{2}$, with $\\theta_{1}^{\\prime}$ and $\\theta_{2}^{\\prime}$ as their targets. The learning of both networks happens simultaneously, addressing approximation error, reducing the bias, and finding the highest Q-values. The actor target chooses the action $a^{\\prime}$ based on the state $s^{\\prime}$, and we add Ornstein-Uhlenbeck noise to it. The double critic targets take the tuple ($s^{\\prime}$, $a^{\\prime}$) and return two Q-values as outputs, from which only the minimum of the two is considered. The loss is calculated with the Mean Squared Error of the approximate value from the target networks and the value from the critic networks. We use Adaptive Moment Estimation (Adam) to minimize the loss.\n\nWe update the policy network less frequently than the value network, taking into account a $policy\\_freq$ factor that increases over time by the following rule:\n\n\\begin{equation*} policy\\_freq(t)=\\left\\lfloor\\left(0.5- \\frac{t}{max\\_steps \\times 3}\\right)^{-1}\\right\\rfloor\\end{equation*}\n\n\\subsection{Stochastic Deep RL\n\nWe also introduce a bias-stochastic actor-critic algorithm based on SAC~\\cite{haarnoja2018soft}, that combines off-policy updates with a stochastic actor-critic method to learn continuous action space policies. It uses neural networks as approximation functions to learn a policy and two Q-values functions similarly to TD3. However, SAC utilizes the current stochastic policy to act without noise, providing better stability and performance, maximizing the reward and the policy's entropy, encouraging the agent to explore new states and improving training speed. We use the soft Bellman equation with neural networks as a function approximation to maximize entropy. The pseudocode can be seen in Algorithm \\ref{alg:docrl_s}.\n\n\\begin{algorithm}[!htb]\n    \\algsetup{linenosize=\\tiny}\n    \\scriptsize\n    \\caption{Deep Reinforcement Learning Stochastic}\n    \\label{alg:docrl_s}\n    \\begin{algorithmic}[1]\n        \\STATE Initialize params of critic networks $\\theta_{1}$, $\\theta_{2}$ , and actor network $\\phi$\n        \\STATE Initialize params of target networks $\\phi^{\\prime}\\leftarrow\\phi$, $\\theta_{1}^{\\prime}\\leftarrow\\theta_{1}$, $\\theta_{2}^{\\prime}\\leftarrow\\theta_{2}$\n        \\STATE Initialize replay buffer $\\beta$\n        \\FOR{$ep = 1$ to $max\\_eps$}\n            \\STATE reset environment state\n            \\FOR{$t = 0$ to $max\\_steps$}\n                \\IF {$t < start\\_steps$}\n                    \\STATE $a_{t} \\leftarrow $ env.action\\_space.sample() \n                %\n                \\ELSE\n                    \\STATE $a_t\\leftarrow \\text{sample from } \\pi_{\\phi}(\\cdot|s_t)$\n                %\n                \\ENDIF\n                \n                \\STATE $s_{t+1}$, $r_{t}$, $d_{t}$, \\_ $\\leftarrow$ env.step($a_{t}$)\n                \n                \\STATE store the new transition $(s_{t}, a_{t}, r_{t}, s_{t+1}, d_{t})$ into $\\beta$\n                \n                \\IF{$t > start\\_steps$}\n                    \\STATE Sample mini-batch $B$ of $N$ transitions $(s_{t}, a_{t}, r_{t}, s_{t+1}, d_{t})$ from $\\beta$\n\n                    \\STATE $\\tilde{a}_{t} \\leftarrow \\text{sample from } \\pi_{\\phi}(\\cdot|s_t)$\n                    \n                    \\STATE $double = ([min_{i=1,2}( Q_{\\theta'_{i}}({s_{t}},{\\tilde{a}_{t}}))-\\alpha \\log \\tilde{a}_{t})])$\n                    \n                    \\STATE $Q_t=r({s_{t}},{a_{t}})+\\gamma(1-d_{t})*double$ \n                    \n                    \\STATE Update double critics with one step gradient descent:\\\\\n                    $\\nabla_{\\theta_i} \\frac{1}{N} \\sum_{s_t \\in B}(Q_t - Q_{\\theta_{i}}(s_{t},a_{t}))^2 \\text{ for } i=1,2$\n                    \n                    \\IF {t \\% $policy\\_freq(t)$ == 0}\n                        \n                        \\STATE Update policy with one step gradient descent:\\\\\n                        $ \\nabla_{\\phi} \\frac{1}{N} \\sum_{s_t \\in B} ([min_{i=1,2}( Q_{\\theta_{i}}({s_{t}},{\\tilde{a}_{t}}))-\\alpha \\log \\tilde{a}_{t}])$\n                        \n                        \\STATE Soft update for the target networks: \\\\\n                        \\STATE $\\phi^{\\prime}\\leftarrow\\tau\\phi+(1-\\tau)\\phi^{\\prime}$\n                        \\STATE $\\theta_{i}^{\\prime}\\leftarrow\\tau\\theta_{i}+(1-\\tau)\\theta_{i}^{\\prime}$ \\qquad for i=1,2\n\n                    \\ENDIF\n                \\ENDIF\n            \\ENDFOR\n        \\ENDFOR\n  \\end{algorithmic}\n\\end{algorithm}\n\nLike before, here we train for ($max\\_steps$) steps in ($max\\_eps$) episodes as well, exploring random actions for the first ($start\\_steps$) steps. An LSTM structure was used for the policy network $\\phi$. After sampling a batch $B$ from the memory $\\beta$, we compute the targets for the Q-functions $Q_t({r_{t}},{s_{t+1}},{d_{t}})$, and update the Q-functions. Also, here we update the policy less frequently than the value network, using the same $policy\\_freq$ factor we used in our deterministic approach. \n\n\\subsection{Simulated Environments\n\nOur experiments were conducted on the Gazebo simulator together with ROS, using the RotorS framework \\cite{furrer2016rotors} to allow the simulation of aerial vehicles with different command levels, such as angular rates, attitude, location control and the simulation of wind with an Ornstein-Uhlenbeck noise. The underwater simulation is enabled by the UUV simulator \\cite{manhaes2016uuv}, which allows the simulation of hydrostatic and hydrodynamic effects, as well as thrusters, sensors, and external perturbations. With this framework, we define the vehicle's underwater model with parameters such as the volume,  additional mass, center of buoyancy, etc., as well as the characteristics of the underwater environment itself.\n\nWe developed two environments that simulate a walled water tank, with dimensions of 10$\\times$10$\\times$6 meters and a one-meter water column. The first environment has four cylindrical columns representing subsea drilling risers. The second environment simulates complex structures, like those found in sea platforms, and contains several elements, such as walls, half walls and pipes (Figure ~\\ref{fig:env2} ).\n\n\\begin{figure}[tbp!]\n    \\vspace{-2mm}\n    \\centering\n    \\includegraphics[width=\\linewidth]{img\/env_3_v2.png}\n    \\caption{Our HUAUV performing in the second scenario.}\n    \\label{fig:env2}\n    \\vspace{-4mm}\n\\end{figure}\n\n\\subsection{HUAUV Description}\n\n\nOur vehicle was based on the model presented by Drews-Jr \\emph{et al.} \\cite{drews2014hybrid}, Neto \\emph{et al.} \\cite{neto2015attitude} and \\emph{et al.} \\cite{horn2019study}. We described it using its actual mechanical settings, including inertia, motor coefficients, mass, rotor velocity, and others. A ROS package containing the vehicle's description plus the Deep-RL agents can be found in the \\nameref{supplementary_material}.\n\nThe vehicle sensing was optimized to mimic real-world LIDAR and Sonar. The described LIDAR is based on the UST 10LX model. It provides a 10 meters distance sensing with $270$\\degree~of range and $0.25$\\degree~of resolution, simulated using the plugin ray of Gazebo. Our simulated FLS sonar was based on the sonar simulation plugin developed by Cerqueira \\emph{et al.}~\\cite{cerqueira2017novel}. We described a FLS sonar with 20 meters of range, with a bin count of 1000 and a beam count of 256. The width and height angles of the beam were $90$\\degree~and $15$\\degree~, respectively. We obtained these values from the relative localization data using Rotors' geometric controller. In the real world, localization information can be obtained from a combination of standard localization sensing of hybrid vehicles like Global Positioning System (GPS) and Ultra Short Baseline (USBL).\n\n\\subsection{Network Structure and Rewarding System} \\label{secapproach}\n\n\n\n\nThe structure of both our approaches has a total of 26 dimensions for the state, 20 samples for the distance sensors, the three previous actions and three values related to the target goal, which are the vehicle's relative position to the target and relative angles to the target in the x-y plane and the z-distance plane. When in the air, 20 samples come from the LIDAR. We get these samples equally spaced by $13.5\\degree$ in the $270\\degree$ LIDAR. When underwater, the distance information comes from the Sonar. We also get 20 beams equally spaced among the total of 256, and we take the highest bin in each beam. This conversion based on the range gives us the distance towards the obstacle or the tank's wall \\cite{Santos18,Santos19}. The actions are scaled between $0$ and $0.25$ $m\/s$ for the linear velocity, from $-0.25$ $m\/s$ to $0.25$ $m\/s$ for the altitude velocity and from $-0.25$ to $0.25$ $rad$ for the $\\Delta$ yaw.\n\n\\subsubsection{Reward Function}\n\nWe proposed a binary rewarding function that yields a positive reward in case of success or a negative reward in case of failure or in case the episode ($ep$) ends at the 500 steps limit:\n\n\\vspace{-5mm}\n\\begin{equation}\nr(s_t, a_t)= \n\\begin{cases}\n    r_{arrive}           & \\text{if } d_t < c_d\\\\\n    r_{collide}          & \\text{if } min_x < c_o\\ ||\\ ep = 500\\\\\n   \n   \n   \n\\end{cases}\n\\end{equation}\n\nThe reward $r_{arrive}$ was set to 100, while the negative reward $r_{collide}$ was set to -10. Both $c_d$ and $c_o$ distances were set to $0.5$ meters.\n\\section{Experimental Results}\n\\label{results}\n\nIn this section, the results obtained during our evaluation are shown. During the training phase, we created a randomly generated goal towards which the agent should navigate. The agents trained for a maximum of 500 steps or until they collided with an obstacle or with the tank's border. In case of reaching the goal before the limit of episodes, a new random goal was generated, allowing the total amount of reward to eventually exceed 100. A learning rate of $10^{-3}$ was used, with a minibatch of 256 samples and the Adam optimizer for all approaches, including the compared methods. We limited the number of episodes trained to 1500 episodes. The limits for the episode number ($max\\_steps$) were used based on the stagnation of the maximum average reward received.\n\nFor each scenario and model, an extensive amount of statistics were collected. The task addressed is goal-oriented navigation considering medium transition, where the robot must navigate from a starting point to an endpoint. This task was addressed in two ways in our tests: (1) starting in the air, performing the medium transition and navigating to a target underwater; and the other way around, (2) starting underwater, performing the medium transition and navigating to a target in the air. We collected the statistics for each of our proposed models (Det. and Sto.) and compared them with the performance of the state-of-the-art deterministic (Det.) and stochastic (Sto.) Deep-RL methods for HUAUVs, \nas well as a behavior-based algorithm \\cite{marino2016minimalistic} .\nThese tasks were performed for 100 trials each and we recorded the total of successful trials, the average time for both underwater ($t\\_water$) and aerial ($t\\_air$) navigation and their standard deviations. \n\nThe models were all trained in the first environment and evaluated in both first (same as trained) and second (never seen) environments. We aim to outline one of the main contributions of this work, \\textit{i.e.} the robust capacity to generalize of our method across environments, in this case performing in a second, unknown and more complex environment. We set the initial position for the Air-Water (A-W) trials to (0.0, 0.0, 2.5) in the Gazebo Cartesian coordinates for the two scenarios. The target position used was (3.6, -2.4, -1.0). In both environments, the target was defined in a path with obstacles on the way. Table \\ref{table:mean_std} shows the results obtained for each environment for 100 navigation trials.\n\n\n\n\\begin{table}[bp!]\n\\vspace{-5mm}\n\\centering\n\\setlength{\\tabcolsep}{0.8pt}\n\\caption{Mean and standard deviation metrics over 100 navigation trials for all approaches in all scenarios.}\n\\label{table:mean_std}\n\\begin{tabular}{c c c c c} \n\\toprule\nEnv & Test & $t_{air}$ (s) & $t_{water}$ (s) & Success \\\\\n\\midrule\n1 & A-W Det. & $76.28$ $\\pm$ $63.20$ & $12.51$ $\\pm$ $20.71$ & 94 \\\\\n1 & A-W Sto. & $21.79$ $\\pm$ $4.57$ & $25.58$ $\\pm$ $5.70$ & $100$ \\\\\n1 & A-W Sto. Grando \\emph{et al.} \\cite{bedin2021deep} & $42.46$ $\\pm$ $62.94$ & $13.13$ $\\pm$ $15.15$ & $42$ \\\\\n1 &\\textbf{ A-W Det. Grando \\emph{et al.} \\cite{bedin2021deep}} & $\\textbf{13.84}$ $\\pm$ $\\textbf{2.11}$ & $\\textbf{5.44}$ $\\pm$ $\\textbf{1.73}$ & $\\textbf{100}$ \\\\\n1 & A-W BBA & $32.42$ $\\pm$ $1.79$ & $21.27$ $\\pm$ $0.18$ & $100$ \\\\\n1 & W-A Det. & $24.66$ $\\pm$ $10.06$ & $5.0$ $\\pm$ $0.71$ & $83$ \\\\\n1 & W-A Sto. & $79.73$ $\\pm$ $27.91$ & $5.41$ $\\pm$ $0.34$ & $100$ \\\\\n\n2 & \\textbf{A-W Det.} & $61.94$ $\\pm$ $45.29$ & $\\textbf{8.44}$ $\\pm$ $\\textbf{9.09}$ & $\\textbf{73}$ \\\\\n2 & \\textbf{A-W Sto.} & $\\textbf{14.89}$ $\\pm$ $\\textbf{1.120}$ & $18.48$ $\\pm$ $6.24$ & $\\textbf{94}$ \\\\\n2 & A-W Sto. Grando \\emph{et al.} \\cite{bedin2021deep} & - & - & $0$ \\\\\n2 & A-W Det. Grando \\emph{et al.} \\cite{bedin2021deep} & - & - & $0$ \\\\\n2 & A-W BBA & $39.69$ $\\pm$ $21.92$ & $11.32$ $\\pm$ $7.46$ & $28$ \\\\\n2 & \\textbf{W-A Det.} & $\\textbf{8.54}$ $\\pm$ $\\textbf{4.44}$ & $\\textbf{4.27}$ $\\pm$ $\\textbf{0.47}$ & $\\textbf{8}$ \\\\\n2 & \\textbf{W-A Sto.} & $\\textbf{15.43}$ $\\pm$ $\\textbf{13.39}$ & $\\textbf{6.60}$ $\\pm$ $\\textbf{1.75}$ & $\\textbf{10}$ \\\\\n2 & W-A Sto. Grando \\emph{et al.} \\cite{bedin2021deep} & - & - & $0$ \\\\\n2 & W-A Det. Grando \\emph{et al.} \\cite{bedin2021deep} & - & - & $0$ \\\\\n2 & W-A BBA & $34.3$ $\\pm$ $22.93$ & $6.13$ $\\pm$ $17.22$ & $8$ \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nWe also performed a complementary comparison in the second scenario. We used the models trained in the second environment to collect statistics. For a better analysis, we also performed a comparison between models in this second environment. First, we collected the data for Deterministic and Stochastic models trained only in the first environment for 1500 episodes (Env1), as shown before. Then, we trained these models for 500 more episodes in the second environment (Both). Lastly, we compared them with Deterministic and Stochastic trained only in the second environment for 1500 episodes. Table \\ref{table:Comparistion_lstm} shows the obtained results.\n\n\n\\begin{table}[tp!]\n\\centering\n\\setlength{\\tabcolsep}{2.5pt}\n\\caption{Mean and standard deviation metrics over 100 navigation trials tested in the second simulated environment, for both deterministic and stochastic models trained only in the first environment (Env1), in both first and second environments (Both), and only in the second environment (Env2).}\n\\label{table:Comparistion_lstm}\n\\begin{tabular}{c c c c c} \n\\toprule\nModel & $t_{air}$ (s) & $t_{water}$ (s) & Success \\\\\n\\midrule\n\\textbf{A-W Det. (Env1)} & $61.94$ $\\pm$ $45.29$ & $\\textbf{8.44}$ $\\pm$ $\\textbf{9.09}$ & $\\textbf{73}$ \\\\\n\\textbf{A-W Sto. (Env1)} & $\\textbf{14.89}$ $\\pm$ $\\textbf{1.120}$ & $18.48$ $\\pm$ $6.24$ & $\\textbf{94}$ \\\\\n\\textbf{A-W Det. (Both)} & $\\textbf{14.14}$ $\\pm$ $\\textbf{3.77}$ & $\\textbf{8.69}$ $\\pm$ $\\textbf{3.17}$ & $\\textbf{99}$ \\\\\n\\textbf{A-W Sto. (Both)} & $16.82$ $\\pm$ $2.12$ & $14.92$ $\\pm$ $3.60$ & $\\textbf{100}$ \\\\\nA-W Det. (Env2) & $23.17$ $\\pm$ $31.12$ & $32.53$ $\\pm$ $60.28$ & $21$ \\\\\nA-W Sto. (Env2) & $19.98$ $\\pm$ $15.99$ & $49.61$ $\\pm$ $27.86$ & $87$ \\\\\n\nW-A Det. (Env1) & $8.54$ $\\pm$ $4.44$ & $4.27$ $\\pm$ $0.47$ & $8$ \\\\\nW-A Sto. (Env1) & $15.43$ $\\pm$ $13.39$ & $6.60$ $\\pm$ $1.75$ & $10$ \\\\\n\\textbf{W-A Det. (Both)} & $\\textbf{25.09}$ $\\pm$ $\\textbf{38.86}$ & $\\textbf{4.62}$ $\\pm$ $\\textbf{0.51}$ & $\\textbf{34}$ \\\\\n\\textbf{W-A Sto. (Both)} & $33.41$ $\\pm$ $11.82$ & $11.40$ $\\pm$ $2.38$ & $\\textbf{83}$ \\\\\nW-A Det. (Env2) & - & - & $0$ \\\\\nW-A Sto. (Env2) & $3.73$ $\\pm$ $2.97$ & $30.47$ $\\pm$ $9.47$ & $1$ \\\\\n\\bottomrule\n\\end{tabular}\n\\vspace{-4mm}\n\\end{table}\n\n\n\\section{Discussion}\n\\label{discussion}\n\n\nThe evaluation shows an overall increase in performance in navigation through both environments. It is possible to see that our approaches achieve a consistent performance of 100 successful air-to-water navigation trials with also a consistent navigation time ($14.55 \\pm 0.87$ and $11.19 \\pm 2.86$). In this same scenario, the stochastic performed a little worse in air-to-water navigation but outperformed the deterministic approach in water-to-air navigation. \nIn the second scenario, we can see more clearly that a double-critic-based approach with an RNN structure also has a better ability to learn and generalize the environment, including the obstacles and the medium transition. While the state-of-the-art approaches with a MLP structure were not capable of performing the task, our approaches presented once again a consistent performance, especially in air-to-water navigation. Our approaches showed an excellent ability to learn the tasks and the environmental difficulties, not only the scenario itself. That was further addressed in our additional evaluation with agents trained in the first environment only, both first and second environments and the second environment only. Overall, we can conclude that double critic approaches with recurrent neural networks present a consistent ability to learn through scenarios and environments and to generalize between them. Also, our approaches outperformed the BBA algorithm \nin the rate of successful trials and average time in almost all situations.\n\nIt is important to mention that these approaches are extensively evaluated in a realistic simulation, including control issues and disturbances such as wind. Thus, the results indicate that our approach may achieve real-world application if the correct data from the sensing and the relative localization are correctly ensured. Finally, it is also possible to analyze that these new RNN-based approaches provided a more consistent average course of action throughout the environments.\n\n\n\n\n\\section{Conclusions}\n\\label{conclusion}\n\nThe evaluation shows an overall increase in performance in navigation through both environments. It is possible to see that our approaches achieve a consistent performance of 100 successful air-to-water navigation trials with also a consistent navigation time ($14.55 \\pm 0.87$ and $11.19 \\pm 2.86$). In this same scenario, the stochastic performed a little worse in air-to-water navigation but outperformed the deterministic approach in water-to-air navigation. \nIn the second scenario, we can see more clearly that a double-critic-based approach with an RNN structure also has a better ability to learn and generalize the environment, including the obstacles and the medium transition. While the state-of-the-art approaches with a MLP structure were not capable of performing the task, our approaches presented once again a consistent performance, especially in air-to-water navigation. Our approaches showed an excellent ability to learn the tasks and the environmental difficulties, not only the scenario itself. That was further addressed in our additional evaluation with agents trained in the first environment only, both first and second environments and the second environment only. Overall, we can conclude that double critic approaches with recurrent neural networks present a consistent ability to learn through scenarios and environments and to generalize between them. Also, our approaches outperformed the BBA algorithm \nin the rate of successful trials and average time in almost all situations.\n\nIt is important to mention that these approaches are extensively evaluated in a realistic simulation, including control issues and disturbances such as wind. Thus, the results indicate that our approach may achieve real-world application if the correct data from the sensing and the relative localization are correctly ensured. Finally, it is also possible to analyze that these new RNN-based approaches provided a more consistent average course of action throughout the environments.The evaluation shows an overall increase in performance in navigation through both environments. It is possible to see that our approaches achieve a consistent performance of 100 successful air-to-water navigation trials with also a consistent navigation time ($14.55 \\pm 0.87$ and $11.19 \\pm 2.86$). In this same scenario, the stochastic performed a little worse in air-to-water navigation but outperformed the deterministic approach in water-to-air navigation. \nIn the second scenario, we can see more clearly that a double-critic-based approach with an RNN structure also has a better ability to learn and generalize the environment, including the obstacles and the medium transition. While the state-of-the-art approaches with a MLP structure were not capable of performing the task, our approaches presented once again a consistent performance, especially in air-to-water navigation. Our approaches showed an excellent ability to learn the tasks and the environmental difficulties, not only the scenario itself. That was further addressed in our additional evaluation with agents trained in the first environment only, both first and second environments and the second environment only. Overall, we can conclude that double critic approaches with recurrent neural networks present a consistent ability to learn through scenarios and environments and to generalize between them. Also, our approaches outperformed the BBA algorithm \nin the rate of successful trials and average time in almost all situations.\n\nIt is important to mention that these approaches are extensively evaluated in a realistic simulation, including control issues and disturbances such as wind. Thus, the results indicate that our approach may achieve real-world application if the correct data from the sensing and the relative localization are correctly ensured. Finally, it is also possible to analyze that these new RNN-based approaches provided a more consistent average course of action throughout the environments.\n\n\n\nBy using physically realistic simulation in several water-tank-based scenarios, we showed that our approaches achieved an overall better capability to perform autonomous navigation, obstacle avoidance and medium transition than other approaches. Disturbances such as wind were successfully assimilated and good generalization through different scenarios was achieved. With our simple and realistic sensing approach that took into account only the range information, we presented overall better performance than the state-of-the-art and classical behavior-like algorithm. Future studies with our real HUAUV are on the way. \n\n\\section*{Acknowledgment}\n\n\nThe authors would like to thank the VersusAI team. This work was partly supported by the CAPES, CNPq and PRH-ANP.\n\n\\vspace{-2mm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAngular momentum (AM) is a fundamental parameter in the evolution of galaxies. A dark matter (DM) halo spinning up in the early universe is subject to the same tidal torques as the baryons at its centre, so the total AM of both components is linked, and the specific AM, $j = J\/M$ of the baryons is well matched to $j$ of the DM, (e.g \\cite[Catalan \\& Theuns 1996]{Catalan+1996}). \n$j$ is connected to photometric morphology via the stellar mass -- specific AM -- morphology plane, first shown by \\cite[Fall 1983]{Fall83}, such that galaxies with higher $M_*$ have higher $j$, modulo morphology, with the relation for earlier type galaxies offset to lower $j$. This has since also been shown by \\cite[Romanowsky \\& Fall (2012)]{RF12}, \\cite[Obreschkow \\& Glazebrook (2014)]{OG14}, \\cite[Cortese et al. (2016)]{Cortese+2016}, \\cite[Posti et al. (2018)]{Posti+2018}, and \\cite[Sweet et al. (2018)]{Sweet+2018}.\n\nAlthough the total $j$ for baryons and DM is linked, further physical processes affect the distribution of $j$ for baryons. \\cite[van den Bosch et al. (2001)]{vdb+01} studied the probability density function of $j$ normalised to the mean of the galaxy, \\pdf, and found that dwarf galaxies had a deficit of high-$j$ and of low-$j$ material with respect to the prediction for a DM halo. They attributed this to tidal stripping of the outer, rapidly-rotating material and feedback ejecting the inner, dispersion-dominated material respectively. \\cite[Sharma \\& Steinmetz (2005)]{SS05} then predicted the \\pdf\\\/ for baryonic components (see Fig.\\,\\ref{ss05}), where bulges, which are dominated by random motions, exhibit a peak at $j=0$, and disks have a \\pdf\\\/ of the form $x$exp$(-kx)$ due to their well-ordered rotation. Also see our updated predictions using the NIHAO simulations (Wang+ in prep)).\n\nThe \\pdf\\\/ encodes more physical information than photometry alone, so in this work I am investigating the utility of \\pdf\\\/ as a kinematic tracer of morphology and kinematic decomposition tool.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.4\\linewidth]{SS05.png} \n \\caption{Predictions from \\cite[Sharma \\& Steinmetz 2005]{SS05} for baryonic galaxy components. The \\pdf\\\/ for bulge peaks at $j=0$, while disk components have an exponential profile.}\n   \\label{ss05}\n\\end{center}\n\\end{figure}\n\n\\section{\\pdf\\\/}\n\nI have constructed \\pdf\\\/ for a subset of the Calar Alto Legacy Integral Field Area survey (CALIFA, \\cite[Sanchez et al. 2012]{Sanchez+2012}), using observations of 25 galaxies where the stellar kinematics reach to three times the effective radius. I calculate $j_i = r_i \\times v_i$ in every spaxel $i$, where the velocity $v_i = sqrt( v_{i,circ}^2 + v_{i,disp}^2)$ includes the circular velocity $v_{i,circ}$ and dispersion $v_{i,disp}$ added in quadrature. The map of $j$ is then weighted by stellar surface density, as a proxy for mass, and the histogram plotted as the \\pdf\\\/. \n\nA spanning set of local examples is shown in Fig.\\,\\ref{examples}. The colour represents radial distance, with lighter colours indicating material nearer the centre. NGC 6063 is a late-type spiral galaxy with low bulge-to-total light ratio B\/T = 0.04. Its \\pdf\\\/ is broad and symmetric, and peaks near 1, reminiscent of the predictions by  \\cite[Sharma \\& Steinmetz (2005)]{SS05} for pure disks. NGC 2592 is an early type galaxy with large B\/T = 0.54; its \\pdf\\\/ is strongly-skewed and peaks near $j=0$ like the spheroidal components in \\cite[Sharma \\& Steinmetz (2005)]{SS05}. Intermediate between these two extremes, the \\pdf\\\/ for NGC 7653 (B\/T = 0.33) has characteristics of both disk and bulge. Unfortunately the spatial resolution, which dictates the number of bins, is not sufficient to resolve separate components in the \\pdf\\\/.\n\nI also show a clumpy disk galaxy at $z\\sim 1.5$, using a combination of OSIRIS adaptive optics integral field spectroscopy to mitigate the effects of beam smearing in the centre of the galaxy, with deeper KMOS seeing-limited data to trace the velocity field out to higher multiples of the effective radius, using a method described in Sweet et al. (in prep.). The shape is intermediate between the pure disk and bulge-dominated local examples, likely due to the typical high-$z$ morphology of dispersion-dominated clumps embedded in a strongly-rotating disk.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.495\\linewidth]{6063.png} \n \\includegraphics[width=0.495\\linewidth]{2592.png} \n \\includegraphics[width=0.495\\linewidth]{7653.png} \n \\includegraphics[width=0.495\\linewidth]{COSMOS_127977.png} \n \\caption{Example \\pdf\\\/ for local galaxies in the CALIFA sample and one disk at $z\\sim 1.5$. Top left: late-type spiral NGC 6063 with low B\/T ratio has a broad, symmetric \\pdf\\\/ which peaks near 1. Top right: early type NGC 2592 with high B\/T ratio has a strongly-skewed \\pdf\\\/ which peaks nearer 0. Bottom left: NGC 7653 with moderate B\/T ratio has a \\pdf\\\/ which is intermediate between the two extremes. Bottom right: $z\\sim 1.5$ clumpy disk galaxy COSMOS 127977 also has an intermediate \\pdf\\\/.}\n   \\label{examples}\n\\end{center}\n\\end{figure}\n\nThere is an apparent trend whereby \\pdf that are more skewed and peak nearer $j=0$ correspond to galaxies that are earlier in type and have bigger bulges. This is quantified in the correlation between shape of \\pdf\\\/ (traced by skewness or kurtosis) and morphology (traced by Hubble type and B\/T ratio). For instance, Fig.\\,\\ref{t-skew} illustrates the relation between skewness and Hubble type. Much of the scatter in this correlation arises from the difficulties inherent in photometric classification of Hubble type and quantifying bulge-to-total ratio. Arguably, \\pdf\\\/ as a kinematic quantity encodes more physical information than photometry alone, so may be a more robust tracer of morphology.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.45\\linewidth]{t-skew.png} \n \\caption{Correlation between morphology and shape of \\pdf\\\/. Galaxies with earlier Hubble type are more strongly negatively skewed.}\n   \\label{t-skew}\n\\end{center}\n\\end{figure}\n\nI also see that the bulge is linked with \\pdf\\\/.  Fig.\\,\\ref{bulgediskkurt} demonstrates that the shape of \\pdf\\\/ is moderately correlated with the surface brightness of the bulge, such that galaxies with bigger bulges have more strongly-tailed \\pdf. On the other hand, the \\pdf\\\/ shape is not at all correlated with the central surface brightness of the disk. This indicates that the size of the bulge is related to the distribution of $j$ within a galaxy, but disks of all sizes have similar \\pdf\\\/.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.8\\linewidth]{bulgediskkurt.png} \n \\caption{Correlation between galaxy components and shape of \\pdf\\\/. Galaxies with bigger bulges have more strongly-tailed \\pdf\\\/, but the shape of \\pdf\\\/ is the same for disks of all sizes.}\n   \\label{bulgediskkurt}\n\\end{center}\n\\end{figure}\n\n\n\\section{Conclusion: Utility of PDF(j)}\nThe \\pdf\\\/ traces kinematic morphology of a galaxy. It encodes more physical information than photometry alone, and is a product of the evolutionary history of the galaxy. In future, as spatial resolution increases, I predict that \\pdf\\\/ will be useful to separate out kinematic components: thin disk from thick disk and bulge, clumps from bulges, and pseudobulges from classical bulges.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper, we study the quantitative long time dynamics for the spherically symmetric dispersive spacetimes satisfying the Einstein-scalar field equations. More precisely, these are spherically symmetric solutions $(\\calM,\\bfg,\\phi)$ to the Einstein-scalar field system, where $\\bfg$ is a Lorentzian metric and $\\phi$ is a real valued function on a $3+1$ dimensional manifold $\\calM$, such that $(\\calM,\\bfg)$ is future causally geodesically complete and $\\phi$ scatters locally in the scale-invariant bounded-variation (BV) norm. For these spacetimes, we establish a Price-law type decay for the scalar field $\\phi$, the Christoffel symbols associated to $\\bfg$ and all of their derivatives. To obtain the decay results, we do not need to assume any smallness of the initial data. Moreover, we show that the decay rates in this paper are sharp.\n\nThe spherically symmetric Einstein-scalar field system, being one of the simplest model of self-gravitating matter in this symmetry class, has been studied extensively both numerically and mathematically. In a seminal series of papers by Christodoulou \\cite{Christodoulou:1987ta}, \\cite{Christodoulou:1991}, \\cite{Christodoulou:1993bt}, \\cite{Christodoulou:1994}, \\cite{Christodoulou:1999}, he achieved a complete understanding of the singularity structure of spherically symmetric spacetime solutions to this system. The culmination of the results shows that generic\\footnote{in the BV class, i.e., the initial data for $\\partial_v(r\\phi)$ has bounded variation. More precisely, Christodoulou showed that the non-generic set of initial data has co-dimension at least two in the BV topology.} spherically symmetric initial data with one asymptotically flat end give rise to a spacetime whose global geometry is either dispersive (with a Penrose diagram represented by Figure 1) or contains a black hole region $\\mathcal B\\mathcal H$ which terminates in a spacelike curvature singularity $\\mathcal S$ (with a Penrose diagram represented by Figure 2). In particular, in either of these generic scenarios, the spacetime possesses a complete null infinity $\\mathcal I^+$ and thus obeys the weak cosmic censorship conjecture. Moreover, in either case the maximal Cauchy development of the data is inextendible with a $C^2$ Lorentzian metric and therefore also verifies the strong cosmic censorship conjecture. We refer the readers to \\cite{Kommemi} for a comprehensive discussion on general singularity structures for spherically symmetric spacetimes.\n\n\\begin{figure}[htbp] \\label{fig.disp}\n\\begin{center}\n \n\\includegraphics{fig_I.pdf}\n \n\\caption{The dispersive case}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n \n\\includegraphics{fig_II.pdf}\n \n\\caption{The black hole case}\n\\end{center}\n\\end{figure}\n\nThe remarkable resolution of the cosmic censorship conjectures however gives very little information on the long time dynamics for these spacetimes except for the small data\\footnote{i.e., when the initial data is close to that of Minkowski space.} case  \\cite{Christodoulou:1993bt}. In particular, not much is known about the asymptotic decay of the scalar field as measured by a far-away observer at null infinity. In the dispersive case, Christodoulou showed that the Bondi mass decays to zero along null infinity without an explicit decay rate. In the black hole case, he showed that the Bondi mass approaches the mass of the black hole, from which one can infer the non-quantitative decay for the scalar field along null infinity \\cite{Christodoulou:1993bt}.\n \nThe long time dynamics in the case where the spacetime settles to a black hole was subsequently studied in the seminal work\\footnote{In fact, they considered the more general Einstein-Maxwell-scalar field equations.} of Dafermos-Rodnianski \\cite{DR} . They proved a quantitative decay rate for the scalar field (and its derivatives) in the spacetime including along null infinity $\\mathcal I^+$ and the event horizon $\\mathcal H^+$. The proof is based on the local conservation of energy, which is subcritical, together with techniques exploiting the conformal geometry of the spacetime and the celebrated red-shift effect along the event horizon. The result in particular justified, in a nonlinear setting, the heuristics of Price \\cite{Price}. It turns out that the quantitative decay rates, when combined with the results of \\cite{D}, also have interesting consequences for the strong cosmic censorship conjecture in the context of the spherically symmetric Einstein-Maxwell-scalar field system.\n\nIn this paper, we study the other generic scenerio, i.e., spherically symmetric dispersive spacetime solutions to the Einstein-scalar field system. Unlike in the black hole case, the monotonic Hawking mass is \\emph{supercritical} and provides no control over the dynamics of the solution. We thus do not expect to be able to obtain quantitative decay rates for large solutions without imposing extra assumptions. Instead, we assume \\emph{a priori} the non-quantitative decay of a \\emph{critical} quantity - the BV norm\\footnote{Solutions of bounded variation have been first studied by Christodoulou \\cite{Christodoulou:1993bt} and plays an important role in the proof of the cosmic censorship conjectures \\cite{Christodoulou:1999}.} - but only locally in a region where the area of the orbit of the symmetry group $SO(3)$ remains uniformly bounded. Under this assumption of local BV scattering, we show that the scalar field and all its derivatives decay with a quantitative rate, reminescent of the Price law decay rates in the black hole case. (We refer the readers to the statement of the main theorems in Section \\ref{sec.main.thm} for the precise rates that we obtain.) We prove, in particular, a quantitative decay rate for the scalar field along null infinity.\n\nOur results apply in particular to the class of solutions arising from initial data with small BV norm. Christodoulou \\cite{Christodoulou:1993bt} showed that these spacetimes are future causally geodesically complete. One can easily deduce from \\cite{Christodoulou:1993bt} that in fact these spacetimes satisfy the BV scattering assumption and therefore the solutions obey the quantitative decay estimates of our main theorem (see Theorem \\ref{thm:smallData}). On the other hand, our results do not require any smallness assumptions on the initial data. We conjecture that indeed our class of spacetimes contains those arising from large data:\n\\begin{conjecture}\\label{large.sol.conj}\nThere exists initial data of arbitrarily large BV norm whose maximal global development scatters locally in the BV norm.\n\\end{conjecture}\n\nIn addition to the upper bounds that we obtain in our main theorem, we also construct examples where we prove lower bounds for the solutions with the same rates as the upper bounds. In particular, there exists a class of initial data with compactly supported scalar field whose future development saturates the decay estimates in the main theorem. This shows that the decay rates are sharp. We note that the decay rate is also consistent with the numerical study of Biz\\'on-Chmaj-Rostworowski \\cite{BCR}.\n\nAs a corollary of the main result on decay, we show the following dichotomy: either the quantitative decay rates are satisfied or the solution blows up at infinity. The latter are solutions such that some scale-invariant spacetime norms become infinite (see precise definition in Definition \\ref{def.blow.up.infty}).\n\nThe decay result in this paper easily implies that the locally BV scattering solutions that we consider are stable against small, regular, \\emph{spherically symmetric} perturbations. More ambitiously, one may conjecture that\n\\begin{conjecture}\\label{stab.conj}\nSpherically symmetric locally BV scattering dispersive solutions to the Einstein-scalar field equations are stable against \\emph{non-spherically symmetric} perturbations.\n\\end{conjecture}\n\nConjecture \\ref{stab.conj}, if true, will generalize the monumental theorem on the nonlinear stability of Minkowski spacetime of Christodoulou-Klainerman \\cite{CK} (see also a simpler proof in \\cite{LR}). For nonlinear wave equations satisfying the null condition, it is known \\cite{Alinhac}, \\cite{Yang} that \\emph{large} solutions decaying sufficiently fast are globally stable against small perturbations. On the other hand, our main theorem shows a quantitative decay rate for spherically symmetric locally BV scattering dispersive spacetimes. Conjecture \\ref{stab.conj} can therefore be viewed as an attempt to generalize the results in \\cite{Alinhac}, \\cite{Yang} to the Einstein-scalar field system. We will address both Conjectures \\ref{large.sol.conj} and \\ref{stab.conj} in future works.\n\\\\\n\\\\\n\\noindent{\\bf Acknowledgements.} The authors would like to thank Mihalis Dafermos and Igor Rodnianski for valuable discussions. We also thank Jonathan Kommemi for providing the Penrose diagrams. \n\nJ. Luk is supported by the NSF Postdoctoral Fellowship DMS-1204493. S.-J. Oh is a Miller Research Fellow, and thanks the Miller Institute at UC Berkeley for the support.\n\n\\subsection{Outline of the paper}\nWe outline the remainder of the paper. In Section \\ref{sec.setup}, we discuss the set-up of the problem and in particular define the class of solutions considered in the main theorem, i.e., the locally BV scattering solutions. In Section \\ref{sec.main.thm}, we state the main theorems in the paper (Theorems \\ref{main.thm.1} and \\ref{main.thm.2}), their consequences and additional theorems on the optimality of the decay rates. In the same section, we outline the main ideas of the proof. In Sections \\ref{sec.anal.prop} and \\ref{sec.geom}, we explain the main analytic features of the equations and the geometry of the class of spacetimes that we consider.\n \nSections \\ref{sec.decay1} and \\ref{sec.decay2} consist of the main content of this paper. In Section \\ref{sec.decay1}, we prove the decay estimates for $\\phi$, $\\partial_v(r\\phi)$ and $\\partial_u(r\\phi)$, i.e., the first main theorem (Theorem \\ref{main.thm.1}). In Section \\ref{sec.decay2}, using in particular the results in Section \\ref{sec.decay1}, we derive the decay bounds for the second derivatives for $r\\phi$ and the metric components, i.e., the second main theorem (Theorem \\ref{main.thm.2}). \n\nIn the remaining sections of the paper, we turn to other theorems stated in Section \\ref{sec.main.thm}. In Section \\ref{sec.dichotomy}, we give a proof of the dichotomy alluded to above, i.e., either the quantitative decay holds or the spacetime blows up at infinity. In Section \\ref{sec:smallData}, we sketch a proof of a refinement of the conclusions of the main theorems in the small data case. Finally, in Section \\ref{sec.opt}, we prove optimality of the decay rates asserted by the main theorems.\n\n\\section{Set-up}\\label{sec.setup}\nIn this section, we define the set-up, formulate the equations in a double null coordinate system and explain the characteristic initial value problem. This will allow us to state the main theorem in the next section.\n\n\\subsection{Spherically Symmetric Einstein-Scalar-Field System (SSESF)} \\label{subsec:derivation}\nWe begin with a brief discussion on the derivation of the Spherically Symmetric Einstein-Scalar-Field System \\eqref{eq:SSESF} from the $(3+1)$-dimensional Einstein-scalar-field system. \n\nSolutions to the Einstein-scalar field equations can be represented by a triplet $(\\calM, \\bfg_{\\mu \\nu},\\phi)$, where $(\\calM, \\bfg_{\\mu \\nu})$ is a $(3+1)$-dimensional Lorentzian manifold and $\\phi$ a real-valued function on $\\calM$. The spacetime metric $\\bfg_{\\mu \\nu}$ and the scalar field $\\phi$ satisfy the Einstein-scalar-field system:\n\\begin{equation} \\label{eq:ES}\n\\left\\{\n\\begin{aligned}\n\t\\bfR_{\\mu \\nu} - \\frac{1}{2} \\bfg_{\\mu \\nu} R =& 2 \\bfT_{\\mu \\nu}, \\\\\n\t\\nb^{\\mu} \\partial_{\\mu} \\phi =& 0.\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere $\\bfR_{\\mu \\nu}$ is the Ricci curvature of $\\bfg_{\\mu \\nu}$, $R$ is the scalar curvature, and $\\nb_{\\mu}$ is the covariant derivative given by the Levi-Civita connection on $(\\calM, \\bfg)$. The energy-momentum tensor $\\bfT_{\\mu \\nu}$ is given by the scalar field $\\phi$, i.e.\n\\begin{equation}\\label{eq:T} \n\t\\bfT_{\\mu\\nu} = \\partial_{\\mu} \\phi \\partial_{\\nu} \\phi - \\frac{1}{2} \\bfg_{\\mu \\nu} \\partial^{\\lambda} \\phi \\partial_{\\lambda} \\phi.\n\\end{equation}\n\nAssume that the solution $(\\calM, \\bfg_{\\mu \\nu}, \\phi)$ is spherically symmetric, i.e., the group $\\mathrm{SO}(3)$ of three dimensional rotations acts smoothly and isometrically on $(\\calM, \\bfg)$, where each orbit is either a point or is isometric to $\\mathbb S^{2}$ with a round metric. The scalar field $\\phi$ is required to be constant on each of the orbits. These assumptions are propagated by \\eqref{eq:ES}; hence, if $(\\calM, \\bfg_{\\mu \\nu}, \\phi)$ is a Cauchy development, then it suffices to assume spherical symmetry only on the initial data.\n\nThe quotient $\\mathcal M \/ \\mathrm{SO}(3)$ gives rise to a (1+1)-dimensional Lorentzian manifold with boundary, which we denote by $(\\calQ, g_{ab})$. The boundary $\\Gamma$ consists of fixed points of the group action. We define the \\emph{area radius function} $r$ on $\\calQ$ to be\n\\begin{equation*}\n\t r := \\sqrt{\\frac{\\mbox{Area of symmetry sphere}}{4 \\pi}}.\n\\end{equation*}\nand $r=0$ at $\\Gamma$. Note that each component of $\\Gamma$ is a timelike geodesic.\n\nWe assume that $\\Gamma$ is non-empty and connected, and moreover that there exists a \\emph{global double null coordinates} $(u,v)$, i.e. a coordinate system $(u,v)$ covering $\\calQ$ in which the metric takes the form\n\\begin{equation} \\label{eq:defn4Met}\n\tg_{ab} \\mathrm{d} x^{a} \\cdot \\mathrm{d} x^{b} = - \\Omega^{2} \\mathrm{d} u \\cdot \\mathrm{d} v\n\\end{equation}\nfor some $\\Omega > 0$. We remark that both assumptions are easily justified if $(\\mathcal M, {\\bf g})$ is a Cauchy development of a spacelike hypersurface homeomorphic to $\\mathbb R^{3}$. \n\n\nThe metric ${\\bf g}_{\\mu \\nu}$ of $\\mathcal M$ is characterized by $\\Omega$ and $r$ and takes the form\n\\begin{equation}\n\t{\\bf g}_{\\mu \\nu} \\mathrm{d} x^{\\mu} \\cdot \\mathrm{d} x^{\\nu} = - \\Omega^{2} \\mathrm{d} u \\cdot \\mathrm{d} v + r^{2} \\mathrm{d} s^{2}_{\\mathbb S^{2}}\\label{metric}\n\\end{equation}\nwhere $\\mathrm{d} s^{2}_{\\mathbb S^{2}}$ is the standard line element on the unit sphere $\\mathbb S^{2}$. Therefore, we may reformulate the \\emph{spherically symmetric Einstein-scalar-field system} \\eqref{eq:SSESF} in terms of the triplet $(\\phi, r, \\Omega)$ as \n\\begin{equation} \\label{eq:SSESF} \\tag{SSESF}\n\\left\\{\n\\begin{aligned}\n\tr \\partial_{u} \\partial_{v} r =& - \\partial_{u} r \\partial_{v} r - \\frac{1}{4} \\Omega^{2}, \\\\\n\tr^{2} \\partial_{u} \\partial_{v} \\log \\Omega =& \\, \\partial_{u} r\\partial_{v} r + \\frac{1}{4} \\Omega^{2} -  r^{2} \\partial_{u} \\phi \\partial_{v} \\phi, \\\\\n\tr \\partial_{u} \\partial_{v} \\phi =& - \\partial_{u} r \\partial_{v} \\phi - \\partial_{v} r \\partial_{u} \\phi, \\\\\n\t2 \\Omega^{-1} \\partial_{u} r\\,  \\partial_{u} \\Omega =& \\, \\partial^{2}_{u} r + r (\\partial_{u} \\phi)^{2}, \\\\\n\t2 \\Omega^{-1} \\partial_{v} r\\,  \\partial_{v} \\Omega =& \\, \\partial^{2}_{v} r + r (\\partial_{v} \\phi)^{2},\n\\end{aligned}\n\\right.\n\\end{equation}\nwith the boundary condition $r=0$ along $\\Gamma$. \n\n\\subsection{Basic assumptions, notations and conventions}\nIn this subsection, we introduce the basic assumptions on the base manifold $\\calQ$, as well as some notations and conventions that will be used in the rest of the paper.\n\n\\subsubsection*{Definition of $\\calQ$ and $\\calM$}\nDenote by $\\mathbb R^{1+1}$ the (1+1)-dimensional Minkowski space, with the standard double null coordinates $(u,v)$. Let $\\calQ$ be a (1+1)-dimensional Lorentzian manifold which is conformally embedded into $\\mathbb R^{1+1}$ with $\\mathrm{d} s^{2}_{\\calQ} = - \\Omega^{2} \\mathrm{d} u \\cdot \\mathrm{d} v$. Given a non-negative function $r$ on $\\calQ$, we define the set $\\Gamma := \\set{(u,v) \\in \\calQ : r(u,v) = 0}$, called the \\emph{axis of symmetry}. We also define $(\\calM, \\bfg_{\\mu \\nu})$ to be the (1+3)-dimensional Lorentzian manifold with $\\calM = \\calQ \\times \\mathbb S^{2}$ and $\\bfg_{\\mu \\nu}$ given by \\eqref{eq:defn4Met}; this is to be thought of as the full spacetime before the symmetry reduction. (We refer to \\S \\ref{subsec:derivation} for the full interpretation.)\n\n\\subsubsection*{Assumptions on the conformal geometry of $\\calQ$}\nWe assume that $\\Gamma \\subset \\calQ$ is a connected set, which is the image of a future-directed timelike curve emanating from the point $(1,1)$.\nWe also assume that $C_{1} \\subset \\calQ$, where\n\\begin{equation*}\nC_{1} = \\set{(u,v) \\in \\mathbb R^{1+1} : u=1, \\, 1 \\leq v < \\infty}.\n\\end{equation*}\n \nFurthermore, $\\calQ$ is assumed to be the domain of dependence of $\\Gamma$ and $C_{1}$ to the future, in the sense that every causal curve in $\\calQ$\nhas its past endpoint on either $\\Gamma$ or $C_{1}$. \n\n\\subsubsection*{Notations for the conformal geometry of $\\calQ$}\nDenote by $C_{u}$ (resp. $\\underline{C}_{v}$) the constant $u$ (resp. $v$) curve in $\\calQ$. Note that these are null curves in $\\calQ$.\n\nGiven $(u_{0}, v_{0}) \\in \\calQ$, we define the \\emph{domain of dependence} of the line segment $C_{u_{0}} \\cap \\set{v \\leq v_{0}}$, denoted by $\\mathcal D(u_{0}, v_{0})$, to be the set of all points $p \\in \\calQ$ such that all past-directed causal curves passing $p$ intersects $\\Gamma \\cup (C_{u_{0}} \\cap \\set{v \\leq v_{0}})$, plus the line segment $(C_{u_{0}} \\cap \\set{v \\leq v_{0}})$ itself. \n\nAlso, we define the \\emph{future null infinity} $\\mathcal I^{+}$ to be the set of ideal points $(u, +\\infty)$ such that $\\sup_{C_{u}} r = \\infty$. \n\n\\subsubsection*{Integration over null curves}\nWhenever we integrate over a subset of $C_{u}$ (resp. $\\underline{C}_{v}$), we will use the standard line element $\\mathrm{d} v$ ($\\mathrm{d} u$) for integrals over, i.e.,  \n\\begin{align*}\n\t\\int_{\\underline{C}_v\\cap\\{u_1\\leq u\\leq u_2\\}} f = \\int_{u_1}^{u_2} f(u',v) \\, \\mathrm{d} u'  \\\\ \n\t\\int_{C_u\\cap\\{v_1\\leq v\\leq v_2\\}} f = \\int_{v_1}^{v_2} f(u,v') \\, \\mathrm{d} v',\n\\end{align*} \nrespectively.\n\n\\subsubsection*{Functions of bounded variation}\nUnless otherwise specified, functions of bounded variation (BV) considered in this paper will be assumed to be right-continuous. By convention,\n\\begin{equation*}\n\t\\partial_{v} f \\, \\mathrm{d} v  \\hbox{ or } \\partial_{v} f\n\\end{equation*}\nwill refer to the distributional derivative of $f$, which is a finite signed measure, and\n\\begin{equation*}\n\t\\abs{\\partial_{v} f} \\, \\mathrm{d} v  \\hbox{ or } \\abs{\\partial_{v} f}\n\\end{equation*}\nwill denote the total variation measure. Unless otherwise specified, these measures will be evaluated on intervals of the form $(v_{1}, v_{2}]$. Thus, according to our conventions,\n\\begin{align*}\n\t\\int_{v_{1}}^{v_{2}} \\partial_{v} f (v) \\, \\mathrm{d} v =& f(v_{2}) - f(v_{1}), \\\\\n\t\\int_{v_{1}}^{v_{2}} \\abs{\\partial_{v} f (v)} \\, \\mathrm{d} v = & \\mathrm{T.V.}_{(v_{1}, v_{2}]} [f]. \n\\end{align*}\n\n\\subsubsection*{New variables}\nWe introduce the following notation for the directional derivatives of $r$:\n\\begin{equation*}\n\t\\lmb := \\frac{\\partial r}{\\partial v}, \\quad \\nu := \\frac{\\partial r}{\\partial u},\n\\end{equation*}\n\nThe \\emph{Hawking mass} $m(u,v)$ is defined by the relation\n\\begin{equation}\\label{mdef}\n1 - \\frac{2m}{r} = \\partial^{a} r \\partial_{a} r = - 4 \\Omega^{-2} \\partial_{u} r \\partial_{v} r.\n\\end{equation}\n\nFor a solution to \\eqref{eq:SSESF}, the quantity $m$ possesses useful monotonicity properties (see Lemma \\ref{lem:mntn4m}), which will be one of the key ingredients of our analysis. We define the \\emph{mass ratio} to be\n\\begin{equation*}\n\t\\mu := \\frac{2m}{r}.\n\\end{equation*}\n\nWe also define the \\emph{Bondi mass} on $C_{u}$ by $M(u) := \\lim_{v \\to \\infty} m(u, v)$, provided the limit exists. The Bondi mass $M_{i} := M(1) = \\lim_{v \\to \\infty} m(1, v)$ on the initial curve $C_{1}$ is called the \\emph{initial Bondi mass}.\n\n\\subsection{Refomulation in terms of the Hawking mass}\nThe Hawking mass as defined in \\eqref{mdef} turns out to obey useful monotonicity (See \\S \\ref{subsec:monotonicity}). We therefore reformulate \\eqref{eq:SSESF} in terms of $m$ and eliminate $\\Omega$. Notice that by \\eqref{metric} and \\eqref{mdef}, the metric is determined by $r$ and $m$.\n\nWe say that \\emph{$(\\phi, r, m)$ on $\\calQ$ is a solution to }(SSESF) if the following equations hold:\n\\begin{equation} \\label{eq:SSESF:dr}\n\\left\\{\n\\begin{aligned}\n\\partial_{u} \\lmb = & \\frac{\\mu}{(1-\\mu) r} \\lmb \\nu, \\\\\n\\partial_{v} \\nu = & \\frac{\\mu}{(1-\\mu) r} \\lmb \\nu,\n\\end{aligned}\n\\right.\n\\end{equation}\n\\begin{equation} \\label{eq:SSESF:dm}\n\\left\\{\n\\begin{aligned}\n2 \\nu \\partial_{u} m = &  (1-\\mu) r^{2} (\\partial_{u} \\phi)^{2}, \\\\\n2 \\lmb \\partial_{v} m = &  (1-\\mu) r^{2} (\\partial_{v} \\phi)^{2},\n\\end{aligned}\n\\right.\n\\end{equation}\n\\begin{equation} \\label{eq:SSESF:dphi}\n\\partial_{u} \\partial_{v} (r \\phi) = \\frac{\\mu \\lmb \\nu}{(1-\\mu)r} \\phi,\n\\end{equation}\nand moreover, the following boundary conditions hold:\n\\begin{equation*}\n\tr = 0 \\hbox{ and } m = 0 \\hbox{ along } \\Gamma.\n\\end{equation*}\n\nWe remark that using \\eqref{eq:SSESF:dr}, the wave equation \\eqref{eq:SSESF:dphi} for $\\phi$ may be rewritten in either of the following two equivalent forms:\n\\begin{align} \n\\partial_{u} (\\partial_{v} (r \\phi)) = (\\partial_{u} \\lmb) \\phi, \\label{eq:SSESF:dphi'} \\tag{\\ref{eq:SSESF:dphi}$'$} \\\\\n\\partial_{v} (\\partial_{u} (r \\phi)) = (\\partial_{v} \\nu) \\phi. \\label{eq:SSESF:dphi''} \\tag{\\ref{eq:SSESF:dphi}$''$}\n\\end{align}\n\n\\subsection{Choice of coordinates} \\label{subsec:coordSys}\nNote that $\\calQ$ is ruled by the family of null curves $C_{u}$. Since a null curve $C_{u}$ with $u \\neq 1$ cannot intersect $C_{1}$, its past endpoint must be on $\\Gamma$. Therefore, our assumptions so far impose the following conditions on the double null coordinates $(u,v)$ on $\\calQ$: $u$ is constant on each future-directed null curve emanating from $\\Gamma$ and $v$ is constant on each conjugate null curve. However, these conditions are insufficient to give a unique choice of a coordinate system, as the system \\eqref{eq:SSESF} and assumptions so far are invariant under the change of coordinates\n\\begin{equation*}\n\tu \\mapsto U(u), \\quad v \\mapsto V(v), \\quad U(1) = V(1) = 1,\n\\end{equation*}\nfor any strictly increasing functions $U$ and $V$. To remove this ambiguity, we fix the choice of the coordinate system, once and for all, as follows.\n\nWe first fix $v$ on $C_{1}$ relating it with the function $r$. Specifically, we will require that $v = 2r + 1$ on $C_1$, which in particular implies that \n\\begin{equation} \\label{eq:id4dvr}\n\\lmb(1, v) = \\frac{1}{2}.\n\\end{equation}\n\nNext, in order to fix $u$, we prescribe $u$ such that $\\Gamma = \\set{(u,v) : u = v}$. To do so, for every outgoing null curve $\\underline{C}$ in $\\calQ$, follow the incoming null curve to the past starting from $\\underline{C}\\cap \\Gamma$ until the point $p_*$ where it intersects the initial curve $C_1$. We then define the $u$-coordinate value for $\\underline{C}$ to be the $v$-coordinate value for $p_*$. \n\nUnder such coordinate choice, $\\mathcal D(u_{0}, v_{0})$ may be expressed as\n\\begin{equation*}\n\t\\mathcal D(u_{0}, v_{0}) = \\set{(u,v) \\in \\calQ : u \\in [u_{0}, v_{0}], v \\in [u, v_{0}]}.\n\\end{equation*}\n\nMoreover, if $r$ and $\\phi$ are sufficiently regular functions on $\\calQ$, then our coordinate choice leads to $\\lim_{v \\to u+}(\\lmb + \\nu) (u,v) = \\lim_{u \\to v-}(\\lmb + \\nu) (u,v) = 0$ and $\\lim_{v \\to u+} (\\partial_{v} + \\partial_{u}) (r \\phi)(u,v) = \\lim_{u \\to v-} (\\partial_{v} + \\partial_{u}) (r \\phi)(u,v) = 0$. These conditions will be incorporated into precise formulations of solutions to \\eqref{eq:SSESF} with limited regularity in the following subsection.\n\n\\subsection{Characteristic initial value problem}\nIn this paper, we study the characteristic initial value problem for \\eqref{eq:SSESF} with data prescribed on $C_{1}$, under quite general assumptions on the regularity. In this subsection, we give precise formulations of initial data and solutions to \\eqref{eq:SSESF} to be considered in this paper.\n\nWe begin with a discussion on the constraint imposed by \\eqref{eq:SSESF} (more precisely, \\eqref{eq:SSESF:dr}--\\eqref{eq:SSESF:dphi}) on initial data for $(\\phi, r, m)$. In fact, the constraint is very simple, thanks to the fact that they are prescribed on a characteristic (i.e., null) curve $C_{1}$. Once we prescribe $\\phi$ on $C_{1}$, the coordinate condition \\eqref{eq:id4dvr} dictates the initial values of $r$, and the initial values of $m$ are then determined by the constraint \\eqref{eq:SSESF:dm} along the $v$ direction, as well as the boundary condition $m(1, 1) = 0$.  In other words, initial data for $(\\phi, r, m)$ possess only one degree of freedom, namely the prescription of a single real-valued function $\\phi(1, v)$, or equivalently, $\\partial_{v} (r \\phi)(1, v)$.\n\nFollowing Christodoulou \\cite{Christodoulou:1993bt}, we say that an initial data set for $(\\phi, r, m)$ is of \\emph{bounded variation} (BV) if $\\partial_{v}(r \\phi)(1, \\cdot)$ is a (right-continuous) BV function on $[1, \\infty)$ with finite total variation on $(1, \\infty)$. We also define the notion of \\emph{solution of bounded variation} to \\eqref{eq:SSESF} as follows.\n\n\\begin{definition}[Bounded variation solutions to \\eqref{eq:SSESF}] \\label{def:BVsolution}\nA solution $(\\phi, r, m)$ to \\eqref{eq:SSESF} is called a \\emph{solution of bounded variation} on $\\calQ$ if on every compact domain of dependence $\\mathcal D(u_{0}, v_{0})$, the following conditions hold:\n\\begin{enumerate}\n\\item $\\sup_{\\mathcal D(u_{0}, v_{0})} (-\\nu) < \\infty$ and $\\sup_{\\mathcal D(u_{0}, v_{0})} \\lmb^{-1} < \\infty$.\n\\item $\\lmb$ is BV on each $C_{u} \\cap \\mathcal D(u_{0}, v_{0})$ uniformly in $u$, and $\\nu$ is BV on each $\\underline{C}_{v} \\cap \\mathcal D(u_{0}, v_{0})$ uniformly in $v$.\n\\item For each $a$ with $(a, a) \\in \\Gamma$,\n\\begin{equation*}\n\t\\lim_{\\epsilon \\to 0+} (\\nu + \\lmb)(a, a+\\epsilon) = 0.\n\\end{equation*}\n\\item $\\phi$ is an absolutely continuous function on each $C_{u} \\cap \\mathcal D(u_{0}, v_{0})$ with total variation bounded uniformly in $u$, and also an absolutely continuous function on each $\\underline{C}_{v} \\cap \\mathcal D(u_{0}, v_{0})$ with total variation bounded uniformly in $v$.\n\\item For each $a$ with $(a, a) \\in \\Gamma$,\n\\begin{align*}\n\\lim_{\\epsilon \\to 0} \\sup_{0 < \\delta \\leq \\epsilon} \\mathrm{T.V.}_{\\set{a-\\delta} \\times (a-\\delta, a)} [\\phi] =0, \n& \\qquad \\lim_{\\epsilon \\to 0} \\sup_{0 < \\delta \\leq \\epsilon} \\mathrm{T.V.}_{(a-\\epsilon, a-\\delta) \\times \\set{a-\\delta}} [\\phi] =0,  \\\\\n\\lim_{\\epsilon \\to 0} \\sup_{0 < \\delta \\leq \\epsilon} \\mathrm{T.V.}_{(a, a+\\delta) \\times \\set{a+\\delta}} [\\phi] =0, \n& \\qquad \\lim_{\\epsilon \\to 0} \\sup_{0 < \\delta \\leq \\epsilon} \\mathrm{T.V.}_{\\set{a+\\delta} \\times (a+\\delta, a+\\epsilon)} [\\phi] =0.\n\\end{align*}\n\\item $\\partial_{v}(r \\phi)$ is BV on each $C_{u} \\cap \\mathcal D(u_{0}, v_{0})$ uniformly in $u$, and $\\partial_{u}(r \\phi)$ is BV on each $\\underline{C}_{v} \\cap \\mathcal D(u_{0}, v_{0})$ uniformly in $v$.\n\\item For each $a$ with $(a, a) \\in \\Gamma$,\n\\begin{equation*}\n\t\\lim_{\\epsilon \\to 0+} \\big( \\partial_{v}(r \\phi) + \\partial_{u}(r \\phi) \\big) (a, a+\\epsilon) = 0.\n\\end{equation*}\n\\end{enumerate}\n\\end{definition}\n\nWe also consider more regular data and solutions, as follows. We say that an initial data set for $(\\phi, r, m)$ is $C^{1}$ if $\\partial_{v}(r \\phi)(1, \\cdot)$ is $C^{1}$ on $[1, \\infty)$ with $\\sup_{C_{1}} \\abs{\\partial_{v}^{2}(r \\phi)} < \\infty$. In the following definition, we define the corresponding notion of a \\emph{$C^{1}$ solution} to \\eqref{eq:SSESF}.\n\n\\begin{definition}[$C^{1}$ solutions to \\eqref{eq:SSESF}] \\label{def:C1solution}\nA solution $(\\phi, r, m)$ to \\eqref{eq:SSESF} is called a \\emph{$C^{1}$ solution} on $\\calQ$ if the following conditions hold on every compact domain of dependence $\\mathcal D(u_{0}, v_{0})$:\n\\begin{enumerate}\n\\item $\\sup_{\\mathcal D(u_{0}, v_{0})} (-\\nu) < \\infty$ and $\\sup_{\\mathcal D(u_{0}, v_{0})} \\lmb^{-1} < \\infty$.\n\\item $\\lmb$, $\\nu$ are $C^{1}$ on $\\mathcal D(u_{0}, v_{0})$.\n\\item For each $a$ with $(a, a) \\in \\Gamma$,\n\\begin{equation*}\n\t\\lim_{\\epsilon \\to 0+} (\\nu + \\lmb)(a, a+\\epsilon) = \\lim_{\\epsilon \\to 0+} (\\nu + \\lmb)(a-\\epsilon, a) = 0.\n\\end{equation*}\n\\item $\\partial_{v}(r \\phi)$ and $\\partial_{u} (r \\phi)$ are $C^{1}$ on $\\mathcal D(u_{0}, v_{0})$.\n\\item For each $a$ with $(a, a) \\in \\Gamma$,\n\\begin{equation*}\n\t\\lim_{\\epsilon \\to 0+} \\big( \\partial_{v}(r \\phi) + \\partial_{u}(r \\phi) \\big) (a, a+\\epsilon) \n\t= \\lim_{\\epsilon \\to 0+} \\big( \\partial_{v} (r \\phi) + \\partial_{v}(r \\phi) \\big) (a - \\epsilon, a)\n\t= 0.\n\\end{equation*}\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark} \\label{rem:wp}\nBy \\cite[Theorem 6.3]{Christodoulou:1993bt}, a BV initial data set leads to a unique BV solution to \\eqref{eq:SSESF} on $\\set{(u,v) : 1 \\leq u \\leq 1+\\delta, v \\geq u}$ for some $\\delta > 0$.\nIf the initial data set is furthermore $C^{1}$, then it is not difficult to see that the corresponding solution is also $C^{1}$ (persistence of regularity). In fact, this statement follows from the arguments in Section \\ref{sec.decay2} of this paper; see, in particular, the proof of Lemma \\ref{lem:decay2:key4nullStr}.\n\\end{remark}\n\n\n\\subsection{Local scattering in BV and asymptotic flatness}\nWe are now ready to formulate the precise notion of \\emph{locally BV scattering solutions} to \\eqref{eq:SSESF}, which is the class of solutions that we consider in this paper. In particular, for this class of solutions, we make a priori assumptions on its global geometry. \n\n\\begin{definition}[Local scattering in BV] \\label{def:locBVScat}\nWe say that a BV solution $(\\phi, r, m)$ to \\eqref{eq:SSESF} is \\emph{locally scattering in the bounded variation norm (BV)}, or a \\emph{locally BV scattering solution}, if the following conditions hold:\n\\begin{enumerate}\n\\item \\emph{Future completeness of radial null geodesics}: Every incoming null geodesic in $\\calQ$ has its future endpoint on $\\Gamma$, and every outgoing null geodesic in $\\calQ$ is infinite towards the future  in the affine parameter. \nMoreover, there exists a global system of null coordinates $(u,v)$ and $\\calQ$ is given by\n\\begin{equation} \\label{eq:globalCoords}\n\t\\calQ = \\set{(u,v) : u \\in [1, \\infty), v \\in [u, \\infty)}.\n\\end{equation}\n\n\\item \\emph{Vanishing final Bondi mass}: The final Bondi mass vanishes, i.e.,\n\\begin{equation} \\label{eq:zeroMf}\nM_{f} := \\lim_{u \\to \\infty} M(u) = 0.\n\\end{equation}\n\n\\item \\emph{Scattering in BV in a compact $r$-region}: There exists $R > 0$ such that for $\\PD_{\\mathrm{cpt}}$ defined to be the region $\\set{(u,v) \\in \\calQ : r(u,v) \\leq R}$, we have \n\\begin{equation} \\label{eq:locBVScat}\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v}^{2} (r \\phi)} \\to 0, \\quad\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} \\log \\lmb} \\to 0\n\\end{equation}\nas $u \\to \\infty$.\n\\end{enumerate}\n\\end{definition}\n\nSeveral remarks concerning Definition~\\ref{def:locBVScat} are in order.\n\\begin{remark} \nIn fact, the condition \\eqref{eq:globalCoords} is a consequence of future completeness of radial null geodesics and the preceding assumptions. To see this, first recall our assumption that $C_{1} = \\set{(u,v) : u = 1, v \\in [1, \\infty)}$. Hence from our choice of the coordinate $u$ and future completeness of incoming radial null geodesics, it follows that the range of $u$ must be $[1, \\infty)$. Furthermore, for each $u \\in [1, \\infty)$, the range of $v$ on $C_{u}$ is $[u, \\infty)$ by future completeness of outgoing radial null geodesics and Definition~\\ref{def:BVsolution}. More precisely, future completeness of $C_{u}$ implies that it can be continued past $\\set{u} \\times [u, v_{0}]$ as long as $\\int_{u}^{v_{0}} \\Omega^{2} \\, \\mathrm{d} v < \\infty$, and Definition~\\ref{def:BVsolution} implies\\footnote{We refer to the proof of Proposition~\\ref{prop:geomLocBVScat} below for details of estimating $\\frac{-\\nu}{1-\\mu}$ in terms of assumptions on $\\phi$, $\\partial_{v}(r \\phi)$ and $\\lmb$.} that $\\Omega^{2} = - \\frac{4 \\lambda \\nu}{1-\\mu}$ indeed remains bounded on $\\set{u} \\times [u, v_{0}]$ for every finite $v_{0}$.\n\\end{remark}\n\n\\begin{remark}  \\label{rem:FCGC}\nFor more regular (e.g., $C^{1}$) asymptotically flat solutions, the conditions $(1)$ and $(2)$ in Definition \\ref{def:locBVScat} may be replaced by a single equivalent condition, namely requiring the full spacetime $(\\calM, \\bfg)$ to be \\emph{future casually geodesically complete} as a (1+3)-dimensional Lorentzian manifold. In particular, (2) follows from the deep work \\cite{Christodoulou:1987ta} of Christodoulou, in which it was proved that if $M_{f} > 0$ then the space-time necessarily contains a black-hole and thus is not future causally geodesically complete.\n\\end{remark}\n\n\\begin{remark}\\label{rmk.unif.int}\nAs we will see in the proof, there exists a universal $\\tilde{\\epsilon}_0$ such that (3) in Definition \\ref{def:locBVScat} can be replaced by the weaker requirement that there exists $R>0$ and $U>0$ such that\n\\begin{equation*}\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v}^{2} (r \\phi)} \\leq \\tilde{\\epsilon}_0, \\quad\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} \\log \\lmb} \\leq \\tilde{\\epsilon}_0\n\\end{equation*}\nfor $u\\geq U$. To simplify the exposition, we will omit the proof of this improvement.\n\\end{remark}\n\n\\begin{remark} \nFor a sufficiently regular, asymptotically flat solution to \\eqref{eq:SSESF}, Condition $(1)$ in Definition \\ref{def:locBVScat} is equivalent to requiring that the conformal compactification of $\\calQ$ is depicted by a Penrose diagram as in Figure \\ref{fig.disp} (in the introduction). For more discussion on Penrose diagrams, we refer the reader to \\cite[Appendix C]{DR} and \\cite{Kommemi}. In fact, this equivalence follows easily from the classification of all possible Penrose diagrams for the system \\eqref{eq:SSESF} given in the latter reference.\n\\end{remark}\n\n\nWe also define the precise notion of \\emph{asymptotic flatness} for initial data with BV or $C^{1}$ regularity. As we shall see soon in the main theorems, the rate of decay for the initial data, measured in $r$, is directly related to the rate of decay of the corresponding solution, in both $u$ and $r$.\n\n\\begin{definition}[Asymptotic flatness of order $\\omega'$ in BV or $C^{1}$] \\label{def:AF}\nFor $\\omega' >1$, we make the following definition.\n\n\\begin{enumerate}\n\\item We say that an initial data set is \\emph{asymptotically flat of order $\\omega'$ in BV} if $\\partial_{v} (r \\phi)(1, \\cdot) \\in \\mathrm{BV}[1, \\infty)$ and there exists $\\mathcal I_{1} > 0$ such that \n\\begin{equation}\n\t\\sup_{C_{1}} (1+r)^{\\omega'} \\abs{\\partial_{v}(r \\phi)} \\leq \\mathcal I_{1} < \\infty.\n\\end{equation}\n\n\\item We say that an initial data set is \\emph{asymptotically flat of order $\\omega'$ in $C^{1}$} if $\\partial_{v} (r \\phi)(1, \\cdot) \\in C^{1}[1, \\infty)$ and there exist $\\mathcal I_{2} > 0$ such that \n\\begin{equation}\n\t\\sup_{C_{1}} (1+r)^{\\omega'} \\abs{\\partial_{v}(r \\phi)} + \\sup_{C_{1}} (1+r)^{\\omega'+1} \\abs{\\partial_{v}^{2}(r \\phi)} \\leq \\mathcal I_{2} < \\infty.\n\\end{equation}\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark} \nThe initial Bondi mass $M_{i} := \\lim_{v \\to \\infty} m(1, v)$ can be easily bounded by $\\leq C \\mathcal I_{1}^{2}$; see Lemma \\ref{lem:bnd4Mi}. \n\\end{remark}\n\n\\begin{remark} \\label{rem:PhiIsZero}\nObserve that both conditions imply that $(r \\phi)(1,v)$ tends to a finite limit as $v \\to \\infty$; in particular, $\\lim_{v \\to \\infty} \\phi(1, v) = 0.$\nThis serves to fix the gauge freedom $(\\phi, r, m) \\mapsto (\\phi + c, r, m)$ for solutions to \\eqref{eq:SSESF}.\n\\end{remark}\n\n\\section{Main results}\\label{sec.main.thm}\n\\subsection{Main theorems}\nWith the definitions of locally BV scattering solutions and asymptotically flat initial data, we now have the necessary means to state the main theorems of this paper. Roughly speaking, these theorems say that locally BV scattering solutions with asymptotically flat initial data exhibits quantitative decay rates, which can be read off from the rate $\\omega'$ in Definition \\ref{def:AF}. The first theorem is for initial data and solutions in BV.\n\n\\begin{theorem}[Main Theorem in BV] \\label{main.thm.1}\nLet $(\\phi, r, m)$ be a locally BV scattering solution to \\eqref{eq:SSESF} with asymptotically flat initial data of order $\\omega'$ in BV. Then for $\\omega := \\min \\set{\\omega', 3}$, there exists a constant $A_{1} > 0$ such that \n\\begin{align} \n\t\\abs{\\phi} \\leq & A_{1} \\min \\set{u^{-\\omega}, r^{-1} u^{-(\\omega-1)}}, \\label{eq:decay1:1} \\\\\n\t\\abs{\\partial_{v}(r \\phi)} \\leq & A_{1} \\min \\set{u^{-\\omega}, r^{-\\omega}}, \\label{eq:decay1:2} \\\\\n\t\\abs{\\partial_{u} (r \\phi)} \\leq & A_{1} u^{-\\omega}. \\label{eq:decay1:3}\n\\end{align}\n\\end{theorem}\n\nThe second theorem is for initial data and solutions in $C^{1}$. \n\n\\begin{theorem}[Main Theorem in $C^{1}$] \\label{main.thm.2}\nLet $(\\phi, r, m)$ be a locally BV scattering solution to \\eqref{eq:SSESF} with asymptotically flat initial data of order $\\omega'$ in $C^{1}$. Then, in addition to the bounds \\eqref{eq:decay1:1}-\\eqref{eq:decay1:3}, there exists a constant $A_{2} > 0$ such that \n\\begin{align} \n\t\\abs{\\partial_{v}^{2} (r \\phi)} \\leq & A_{2} \\min \\set{u^{-(\\omega+1)}, r^{-(\\omega+1)}}, \\label{eq:decay2:1} \\\\\n\t\\abs{\\partial_{u}^{2} (r \\phi)} \\leq & A_{2} u^{-(\\omega+1)}, \\label{eq:decay2:2} \\\\\n\t\\abs{\\partial_{v} \\lmb} \\leq & A_{2} \\min \\set{u^{-3}, r^{-3}}, \\label{eq:decay2:3} \\\\\n\t\\abs{\\partial_{u} \\nu} \\leq & A_{2} u^{-3}. \\label{eq:decay2:4}\n\\end{align}\nfor $\\omega := \\min \\set{\\omega', 3}$.\n\\end{theorem}\n\nSome remarks regarding the main theorems are in order.\n\n\\begin{remark}\nNotice that in Theorem \\ref{main.thm.2}, the decay rates for $\\partial_v\\lambda$ and $\\partial_u\\nu$ are independent of the order $\\omega'$ of asymptotic flatness of the initial data. This is because the scalar field terms enter the equations for $\\partial_u\\partial_v\\log\\lambda$ and $\\partial_v\\partial_u\\log\\nu$ quadratically (see equations \\eqref{eq:eq4dvdvr:normal} and \\eqref{eq:eq4dudur:normal}) and thus as long as $\\omega'>1$, their contributions to the decay rates of $\\partial_v\\lambda$ and $\\partial_u\\nu$ are lower order compared to the term involving the Hawking mass.\n\\end{remark}\n\n\\begin{remark}\nBy Remark \\ref{rem:wp}, a $C^{1}$ initial data set gives rise to a $C^{1}$ solution. Hence Remark \\ref{rem:FCGC} applies, and the conditions (1)--(2) of Definition \\ref{def:locBVScat} may be replaced by a single equivalent condition of \\emph{future causal geodesic completeness} of $(\\calM, {\\bf g})$ in the case of Theorem \\ref{main.thm.2}. \n\\end{remark}\n\n\\begin{remark}\nIn general, the constants $A_1$ and $A_2$ depend not only on the size of the initial data (e.g., $\\mathcal I_{1}$, $\\mathcal I_{2}$), but rather on the full profile of the solution. Nevertheless, for the special case of small initial total variation of $\\partial_{v}(r \\phi)$, $A_{1}$ and $A_{2}$ \\emph{do} depend only on the size of the initial data; see \\S \\ref{sec:mainThm:smallData}.\n\\end{remark}\n\n\\begin{remark}\nIf the initial data also verify higher derivative estimates, then the techniques in proving Theorems \\ref{main.thm.1} and \\ref{main.thm.2} also allow us to derive decay bounds for higher order derivatives. The proof of the higher derivative decay estimates is in fact easier than the proofs of the first and second derivative decay bounds since we have already obtained sufficiently strong control of the scalar field and the geometry of the spacetime. We will omit the details.\n\\end{remark}\n\n\\begin{remark}\nThe decay rates that we obtain in these variables imply as immediate corollaries decay rates for $\\partial_v \\phi$, $\\partial_u \\phi$, etc. See Corollaries \\ref{cor:decay1} and \\ref{cor:decay2}.\n\\end{remark}\n\n\\begin{remark} \\label{rem:coord}\nThe decay rates in the main theorems are measured with respect to the double null coordinates $(u,v)$ normalized at the initial curve and the axis $\\Gamma$ as in \\S \\ref{subsec:coordSys}. To measure the decay rate along null infinity, one can alternatively normalize the $u$ coordinate\\footnote{In particular, this normalization is used in \\cite{DR} for the black hole case. By changing the null coordinates, we can thus more easily compare the decay rates in our setting and that in \\cite{DR}.} by requiring $\\partial_{u} r=-\\frac 12$ at future null infinity. As we will show in \\S \\ref{sec.coord}, for the class of spacetimes considered in this paper, the decay rates with respect to this new system of null coordinates are the same up to a constant multiplicative factor.\n\\end{remark}\n\n\\begin{remark}\nIn view of Remark \\ref{rmk.unif.int}, the assumption of local BV scattering can be replaced by the \\emph{boundedness} of the \\emph{subcritical} quantities\n$$\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v}^{2} (r \\phi)}^p \\leq C, \\quad\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} \\log \\lmb}^p \\leq C,\\quad\\mbox{for }p>1.$$\nThis is because for fixed $\\tilde{\\epsilon}_0$, one can choose $R$ to be sufficiently small (depending on $C$) and apply H\\\"older's inequality to ensure\tthat\n$$\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v}^{2} (r \\phi)} \\leq \\tilde{\\epsilon}_0, \\quad\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} \\log \\lmb} \\leq \\tilde{\\epsilon}_0.$$\n\\end{remark}\n\n\\begin{remark}\nWe also notice that the proof of our main theorem can be carried out in an identical manner for locally BV scattering solutions arising from asymptotically flat \\emph{Cauchy data}. More precisely, we can consider initial data given on a Cauchy hypersurface\n$$v=f(u),\\quad\\mbox{with }C^{-1}\\leq -f'(u)\\leq C$$\nand satisfying the constraint equation together with the following bounds on the initial data:\n$$(1+r)^{\\omega'}(|\\phi|+|\\partial_v(r\\phi)|+|\\partial_u(r\\phi)|+|\\lambda-\\frac 12|+|\\nu+\\frac 12|)\\leq \\tilde{\\mathcal I}_1,$$\nand\n$$(1+r)^{\\omega'+1}(|\\partial_v^2(r\\phi)|+|\\partial_u^2(r\\phi)|+|\\partial_v \\log\\lambda|+|\\partial_u\\log\\nu|)\\leq \\tilde{\\mathcal I}_2.$$ \nThen, if we assume in addition that the spacetime is locally BV scattering to the future, the conclusions of Theorems \\ref{main.thm.1} and \\ref{main.thm.2} hold.\n\\end{remark}\n\n\\begin{remark}\nOur main theorems can also be viewed as results on upgrading qualitative decay to quantitative decay estimates. Such problems have been widely studied in the \\emph{linear} setting (without the assumption on spherical symmetry) on nontrapping asymptotically flat Lorentzian manifolds \\cite{DR2}, \\cite{Ta}, \\cite{MTT}, as well as for the obstacle problem on Minkowski space \\cite{Morawetz}, \\cite{Strauss}. In the \\emph{nonlinear} setting, we mention the work of Christodoulou-Tahvildar--Zadeh \\cite{CTZ}, who studied the energy critical 2-dimensional spherically symmetric wave map system and proved asymptotic decay for the solution and its derivatives.\n\\end{remark}\n\n\\subsection{BV scattering and the blow-up at infinity scenerio}\n\nThe condition of local BV scattering in the main theorems follows if one rules out, a priori, a blow-up at infinity scenario. More precisely, we say that a solution blows up at infinity if some scale-invariant spacetime norms are infinite as follows:\n\n\\begin{definition}\\label{def.blow.up.infty}\nLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF} such that the condition $(1)$ of Definition \\ref{def:locBVScat} (future completeness of radial null geodesics) holds. We say that the solution \\emph{blows up at infinity} if at least one of the following holds:\n\\begin{enumerate}\n\\item $\\displaystyle{\\sup \\lambda_{\\Gamma}^{-1} = \\infty}$, where $\\displaystyle{\\lambda_{\\Gamma}(u) := \\lim_{v \\to u+} \\lmb(u,v)}$.\n\\item $\\displaystyle{\\int_1^{\\infty}\\int_u^{\\infty}|\\partial_v\\lambda\\partial_u\\phi-\\partial_u\\lambda\\partial_v\\phi| \\mathrm{d} v \\mathrm{d} u =\\infty}$,\n\\item $\\displaystyle{\\int_1^{\\infty}\\int_u^{\\infty}|\\partial_u\\phi\\partial_v(\\nu^{-1}\\partial_u(r\\phi))-\\partial_v\\phi\\partial_u(\\nu^{-1}\\partial_u(r\\phi))| \\mathrm{d} v \\mathrm{d} u =\\infty}$.\n\\end{enumerate}\n\\end{definition} \n\n\\begin{remark}\nWe do not prove in the paper the existence or non-existence of solutions that blow up at infinity. We remark that this is analogous to the blow-up at infinity scenarios which have recently been constructed in some simpler \\emph{semilinear}, \\emph{critical} wave equations \\cite{DK}.\n\\end{remark}\n\nIt follows from our main theorem that if a solution does not blow up at infinity, it obeys quantitative decay estimates. More precisely, we have\n\\begin{theorem}[Dichotomy between blow-up at infinity and BV scattering] \\label{thm.dichotomy}\nLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF} such that the condition $(1)$ of Definition \\ref{def:locBVScat} (future completeness of radial null geodesics) holds. Assume furthermore that the initial data for $(\\phi, r, m)$ obey the condition\\footnote{By Remark \\ref{rem:PhiIsZero}, note that this is the only condition on $\\lim_{v \\to \\infty} \\phi(1, v)$ which is consistent with asymptotic flatness.} $\\lim_{v \\to \\infty} \\phi(1,v) = 0$ and\n\\begin{equation} \\label{dichotomy.hyp}\n\\int_{C_{1}} \\abs{\\partial_{v}^{2} (r \\phi)} \\, \\mathrm{d} v + \\sup_{C_{1}} \\abs{\\partial_{v}(r \\phi)} < \\infty.\n\\end{equation}\n\nThen either\n\\begin{enumerate}\n\\item the solution blows up at infinity; or\n\\item the solution is globally BV scattering, in the sense that the conditions $(2)$ and $(3)$ of Definition \\ref{def:locBVScat} hold with $R =\\infty$.\n\\end{enumerate}\n\\end{theorem}\n\nThis theorem is established in Section \\ref{sec.dichotomy}. It then follows from our main theorems (Theorems \\ref{main.thm.1} and \\ref{main.thm.2}) that if a BV solution does not blow up at infinity and possesses asymptotically flat initial data, then it obeys quantitative decay estimates.\n\n\\subsection{Refinement in the small data in BV case} \\label{sec:mainThm:smallData}\nBy a theorem of Christodoulou \\cite{Christodoulou:1993bt}, the maximal development of data with small BV norms does not blow up at infinity. The previous theorem applies, and thus the corresponding solution is globally BV scattering, in the sense described in Theorem \\ref{thm.dichotomy}. Moreover, a closer inspection of the proof of the main theorems reveals that the following stronger conclusion holds in this case.\n\n\\begin{theorem} [Sharp decay for data with small BV norm] \\label{thm:smallData}\nThere exists a universal $\\epsilon_{0} > 0$ such that for $0 < \\epsilon \\leq \\epsilon_{0}$, the following statements hold.\n\n\\begin{enumerate}\n\\item If the initial data set is asymptotically flat of order $\\omega'$ in BV and\n\\begin{equation*}\n\t\\int_{C_{1}} \\abs{\\partial_{v}^{2} (r \\phi)} < \\epsilon,\n\\end{equation*}\nthen the maximal development $(\\phi, r, m)$ is globally BV scattering, in the sense that Definition \\ref{def:locBVScat} holds with arbitrarily large $R > 0$. Moreover, it satisfies the estimates \\eqref{eq:decay1:1}--\\eqref{eq:decay1:3} with $A_{1} \\leq C_{\\mathcal I_{1}} (\\mathcal I_{1} + \\epsilon)$.\n\n\\noindent Here (and similarly in (2)), we use the convention that $C_{\\mathcal I_{1}}$ depends on $\\mathcal I_1$ in a non-decreasing fashion.\\footnote{In particular, for $\\mathcal I_1$ sufficiently small, we have the estimate $A_1\\leq C(\\mathcal I_1+\\epsilon)$ for some absolute constant $C$.}\n\n\\item If, in addition, the initial data set is asymptotically flat of order $\\omega'$ in $C^{1}$, then the maximal development also satisfies \\eqref{eq:decay2:1}--\\eqref{eq:decay2:4} with $A_{2} \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon)$.\n\\end{enumerate}\n\\end{theorem}\n\nThe point of this theorem is that we only need to know that the initial total variation to be small in order to conclude pointwise decay rates; in particular, $\\mathcal I_{1}$, $\\mathcal I_{2}$ can be arbitrarily large. In this sense, Theorem \\ref{thm:smallData} generalizes both the small BV global well-posedness theorem \\cite[Theorem 6.2]{Christodoulou:1993bt} and the earlier small data scattering theorem \\cite{Christodoulou:1986ue} for data that are small in a weighted $C^1$ norm. A proof of this theorem will be sketched in Section \\ref{sec:smallData}.\n\n\\subsection{Optimality of the decay rates}\n\nOur main theorems show upper bounds for the decay rates of the scalar field $\\phi$ and its derivatives both towards null infinity (i.e., in $r$) and along null infinity (i.e., in $u$). For $\\omega' = \\omega<3$, if the decay rate of the initial data towards null infinity satisfies also a lower bound, then we can show that both the $r$ and $u$ decay rates in Theorem \\ref{main.thm.1} are saturated. More precisely,\n\n\\begin{theorem} [Sharpness of $t^{-\\omega}$ tail for $1 < \\omega < 3$] \\label{thm.opt.1}\nLet $1 < \\omega < 3$. Suppose, in addition to the assumptions of Theorem \\ref{main.thm.1}, that there exists $V\\geq 1$ such that the initial data set satisfies the lower bound\n\\begin{equation*}\n\tr^{\\omega} \\partial_{v}(r \\phi) (1, v) \\geq L > 0,\n\\end{equation*}\nfor $v\\geq V$.\n\nThen there exists a constant $L_{\\omega} >0$ such that\n\\begin{align*}\n\t\\partial_{v} (r \\phi)(u, v) \\geq & L_{\\omega} \\min\\set{r^{-\\omega}, u^{-\\omega}}, \\\\\n\t- \\partial_{u} (r \\phi)(u, v) \\geq & L_{\\omega} u^{-\\omega},\n\\end{align*}\nfor $u$ sufficiently large.\n\\end{theorem}\n\\begin{remark}\nOne can also infer the sharpness of the decay of $\\phi$ from that of its derivatives. We will omit the details.\n\\end{remark}\n\n\nThis theorem will be proved in \\S \\ref{subsec.opt.1}. In fact, the proof of this theorem is similar to the proof of the upper bounds in the first main theorem (Theorem \\ref{main.thm.1}). We will show that after restricting to $u$ sufficiently large, the initial lower bound propagates and the nonlinear terms only give lower order contributions. Notice also that the analogous statement is false for $\\omega'\\geq 3$, since the nonlinear terms may dominate the contribution of the initial data.\n\nFor $\\omega' \\geq 3$, we can show that the decay rates in Theorem \\ref{main.thm.1} are sharp in the following sense:\n\\begin{theorem} [Sharpness of $t^{-3}$ tail] \\label{thm.opt.2}\nFor arbitrarily small $\\epsilon > 0$, there exists a locally BV scattering solution $(\\phi, r, m)$ to \\eqref{eq:SSESF} which satisfies the following properties:\n\\begin{enumerate}\n\\item $\\partial_{v} (r \\phi)(1, v)$ is smooth, compactly supported in the $v$-variable and has total variation less than $\\epsilon$, i.e.,\n\\begin{equation*}\n\t\\int_{C_{1}} \\abs{\\partial_{v}^{2} (r \\phi)} < \\epsilon.\n\\end{equation*}\n\\item There exists a constant $L_{3} > 0$ such that\n\\begin{align*}\n\t\\partial_{v} (r \\phi) (u, v) \\geq & L_{3} \\min \\set{r^{-3}, u^{-3}}, \\\\\n\t- \\partial_{u} (r \\phi) (u, v) \\geq & L_{3} u^{-3},\n\\end{align*}\nfor $u$ sufficiently large.\n\\end{enumerate}\n\\end{theorem}\n\n\nTo prove Theorem \\ref{thm.opt.2}, we will first establish a sufficient condition for the desired lower bounds in terms of (non-vanishing of) a single real number $\\mathfrak{L}$, which is computed from information at the null infinity. This result (Lemma \\ref{lem:LB}) is proved using the decay rates proved in the main theorems, and we believe it might be of independent interest. In \\S \\ref{subsec.opt.2}, we will complete the proof of Theorem \\ref{thm.opt.2} by constructing an initial data set for which $\\mathfrak{L}$ can be bounded away from zero. This can be achieved by showing that the solution is close to that of a corresponding linear problem and controlling the error terms after taking $\\epsilon > 0$ to be sufficiently small and using Theorem \\ref{thm:smallData}.\n\n\\subsection{Strategy of the proof of the main theorems}\n\nRoughly speaking, the proof of decay of $\\phi$ and its derivatives can be split into three steps. In the first two steps, we control the incoming part\\footnote{We call these variables `incoming' because they obey a transport equation in the $\\partial_u$ direction.} of the derivatives of the scalar field and metric components, i.e., $\\partial_v(r\\phi)$, $\\partial_v^2(r\\phi)$ and $\\partial_v\\lmb$. To this end, we split the spacetime into the exterior region $\\PD_{\\mathrm{ext}}:=\\{(u,v)\\in \\mathcal Q: v\\geq 3u\\}$ and the interior region $\\PD_{\\mathrm{int}}:=\\{(u,v)\\in \\mathcal Q: v\\leq 3u\\}$. In the first step, we control the incoming part of the solution in the exterior region. In this region, we have $r \\gtrsim v, u$, thus the negative $r$ weights in the equations give the required decay of $\\phi$ and its derivatives. We then prove bounds in the interior region in the second step. Here, we exploit certain (non-quantitative) smallness in the spacetimes quantities as $u \\to \\infty$ given by the assumption of local BV scattering to propagate the decay estimates from the exterior region to the interior region all the way up to the axis. Finally, in the third step, we control the outgoing part of the solution, i.e., $\\partial_u(r\\phi)$, $\\partial_u^2(r\\phi)$ and $\\partial_u\\nu$, by showing that the decay bounds that we have proved along the axis can be propagated in the outgoing direction.\n\nWe remind the readers that the above sketch is only a heuristic argument and is not true if taken literally. In particular, in order to carry out this procedue, we need to first show that the local BV scattering assumption provides some control over the spacetime geometry. As we will show below, the estimates are derived in slightly different fashions for the first and the second derivatives of $r\\phi$. We note in particular that carrying out this general scheme relies heavily on the analytic structure of the Einstein-scalar field equations, including the montonicity properties as well as the null structure of the (renormalized) equations.\n\n\\subsubsection{Estimates for first derivatives of $r\\phi$}\n\nTo obtain decay bounds for the first derivatives of $r\\phi$, we will rely on the wave equation\n$$\\partial_u \\partial_v(r\\phi)=\\frac{2 m \\lambda \\nu}{(1-\\mu)r^2}\\phi.$$\nNotice that when we solve for the incoming radiation $\\partial_v(r\\phi)$ using this as a transport equation in $u$, the right hand side does not depend explicitly on the outgoing radiation $\\partial_u(r\\phi)$. Instead, the right hand side consists of terms that are either lower order (in terms of derivatives) or satisfy a certain monotonicity property.\n\nIn particular, this equations shows that as long as $\\phi$ can be controlled, we can estimate $\\partial_v(r\\phi)$ by integrating along the incoming $u$ direction. On the other hand, we can also control $\\phi$ once a bound on $\\partial_v(r\\phi)$ is known by integrating along the outgoing $v$ direction.\n\nTo achieve the desired decay rates for $\\phi$, $\\partial_v(r\\phi)$ and $\\partial_u(r\\phi)$, we follow the three steps outlined above:\n\n\\begin{enumerate}\n\\item [(1)] Bounds\\footnote{The estimates in this region are similar to the corresponding bounds for the black hole case in \\cite{DR}. There, it was observed that the quantity $\\partial_v(r \\phi)$, which Dafermos-Rodnianski called an almost Riemann invariant, verifies an equation such that the right hand side has useful weights in $r$ and give the desired decay rates.} for $\\partial_v(r\\phi)$ and $\\phi$ in $v\\geq 3u$: In the exterior region, we have $r\\gtrsim u,v$, it is therefore sufficient to prove the decay in $r$. First, we  prove that $\\sup_{C_u}(1+r)\\phi$ is bounded. This is achieved in a compact region by continuity of the solution\\footnote{In particular, since we are simply using compactness, the constants in Theorem \\ref{main.thm.1} depend not only on the size of the initial data.} and in the region of large $r$ by integrating $\\partial_v(r\\phi)$ in the outgoing direction from the compact region. Since $\\partial_v(r\\phi)$ can in turn be controlled by $\\phi$, we get the desired bound. To improve over this bound we define\n\\begin{equation*}\n\\mathcal B_{1}(U) := \\sup_{u \\in [1, U]} \\sup_{C_{u}} \\Big( u^{\\omega} \\abs{\\phi} + r u^{\\omega-1} \\abs{\\phi} \\Big)\n\\end{equation*}\nand show via the wave equation that\n$$r^\\omega|\\partial_v(r\\phi)|\\leq C(u_1)+\\epsilon(u_1)\\mathcal{B}_1(U),$$\nwhere $\\epsilon\\to 0$ as $u_1\\to \\infty$. This gives the optimal decay rate for $\\partial_v(r\\phi)$ in the exterior region up to an arbitrarily small loss, which can be estimated once $\\mathcal{B}_1(U)$ can be controlled.\n\n\\item [(2)] Bounds for $\\partial_v(r\\phi)$ and $\\phi$ in $v\\leq 3u$: For the decay of the first derivatives, the interior region $\\{v\\leq 3u\\}$ is further divided into the intermediate region $\\{r\\geq R\\}$ and the compact region $\\{r\\leq R\\}$. In these two regions, the $r$ weight in the equation is not sufficient to give the sharp decay rate. Instead, we start from the decay rate $\\partial_v(r\\phi)$ obtained in the first step in the exterior region and propagate this decay estimate inwards. To achieve this, we need to show that $\\int \\frac{2 m \\lambda \\nu}{(1-\\mu)r^2}$ is small when $u$ is sufficiently large. \n\n\\item [(2a)] $r\\geq R$ and $v\\leq 3u$: In the intermediate region where we still have a lower bound on $r$, the required smallness is given by the \\emph{qualitative} information that the Hawking mass approaches $0$. Thus, from some large time onwards, $\\int \\frac{2 m \\lambda \\nu}{(1-\\mu)r^2}$ becomes sufficiently small and we can integrate the wave equation directly to obtain the desired decay bounds.\n\n\\item [(2b)] $r\\leq R$ and $v\\leq 3u$: In this region, we use the local BV scattering assumption to show that $\\int_{\\{r\\leq R\\}} \\frac{2 m \\lambda \\nu}{(1-\\mu)r^2}\\to 0$ as $u\\to\\infty$. This smallness allows us to propagate the decay estimates from the curve $r=R$ to the region $r< R$. At this point, we can also recover the control for $\\mathcal{B}_1(U)$ and close the estimates in step 1. This allows us to derive all the optimal decay rates for $\\phi$ and $\\partial_v(r\\phi)$\n\n\\item [(3)] Bounds for $\\partial_u(r\\phi)$: To achieve the bounds for $\\partial_u(r\\phi)$, first note that along the axis we have $\\partial_u(r\\phi)=-\\partial_v(r\\phi)$. Thus, by the previous derived control for $\\partial_v(r\\phi)$, we also have the decay of $\\partial_u(r\\phi)$ along the axis. We then consider the wave equation as a transport equation in the outgoing direction for $\\partial_u(r\\phi)$ to obtain the sharp decay for $\\partial_u(r\\phi)$ in the whole spacetime. \n\n\\end{enumerate}\n\n\\subsubsection{Estimates for second derivatives of $r\\phi$}\n\nAs for the first derivatives, we control the second derivatives by first integrating the equation in the exterior region up to a curve $v=3u$. We then propagate the decay bounds from the exterior region to the interior region using the estimates already derived for the first derivative of $\\phi$, as well as the local BV scattering assumption. However, at this level of derivatives, some new difficulties arise as we now describe.\n\\\\\n\\fparagraph{Renormalization and the null structure}\nThe assumption of local BV scattering implies that\n\\begin{eqnarray}\n\\int_{C_u\\cap\\{r\\leq R\\}} (|\\partial_v\\phi|+|\\partial_v^2(r\\phi)|)\\to 0 \\label{BV.small.1}\n\\end{eqnarray}\nas $u\\to \\infty$. When combined with Christodoulou's BV theory, this also implies that as $v\\to \\infty$, we have\n\\begin{eqnarray}\n\\int_{\\underline{C}_v\\cap\\{r\\leq R\\}} (|\\partial_u\\phi|+|\\partial_u^2(r\\phi)|) \\to 0.  \\label{BV.small.2}\n\\end{eqnarray}\nNotice that on $C_u$ (resp. $\\underline{C}_v$), we only control the integral of $\\partial_v^2(r\\phi)$ and $\\partial_v\\phi$ (resp. $\\partial_u^2(r\\phi)$ and $\\partial_u\\phi$).\n\nSuppose when integrating along the incoming direction to control $\\partial_v^2(r\\phi)$ and $\\partial_v\\lmb$, we need to estimate terms of the form\n$$\\int_{\\underline{C}_v\\cap\\{r\\leq R\\}} |\\partial_u\\phi \\partial_v\\phi|.$$\nWe can apply the BV theory to show that for $v$ sufficiently large,\n$$\\int_{\\underline{C}_v\\cap\\{r\\leq R\\}} |\\partial_u\\phi| \\leq \\epsilon.$$\nOn the other hand, one can show that\n$$\\sup_{\\underline{C}_v\\cap\\{r\\leq R\\}} |\\partial_v\\phi|\\leq C \\sup_{J^-(\\underline{C}_v\\cap\\PD_{\\mathrm{cpt}})} |\\partial_v^2(r\\phi)|$$\nwhich can be controlled by the quantity that we are estimating.\n\nHowever, in equation \\eqref{eq:eq4dvdvrphi:normal} for $\\partial_v^2(r\\phi)$ derived by differentiating \\eqref{eq:SSESF:dphi}, there are terms of the form\n$$\\partial_v\\phi \\partial_v\\phi$$\nsuch that neither of the factors can be controlled a priori in $L^1$ by the local BV scattering assumption. In other words, the equation does not obey any null condition.\n\nTo deal with this problem, we follow \\cite{Christodoulou:1993bt} and introduce the renormalized variables\n$ \\partial_{v}^{2} (r \\phi) - (\\partial_{v} \\lmb) \\phi, $\n\t$ \\partial_{u}^{2} (r \\phi) - (\\partial_{u} \\nu) \\phi, $\n\t$ \\partial_{v} \\log \\lmb - \\frac{\\lmb}{(1-\\mu)} \\frac{\\mu}{r} + \\partial_{v} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} ( r \\phi) \\Big), $\n\t$ \\partial_{u} \\log (-\\nu) - \\frac{\\nu}{(1-\\mu)} \\frac{\\mu}{r} + \\partial_{u} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} (r \\phi) \\Big)$\nwhich have the property that the nonlinear terms arising in the equations for these variables in fact have a null structure. In particular, we can apply the above heuristic procedure to obtain decay estimates in the compact region $r\\leq R$.\n\\\\\n\\fparagraph{Non-renormalized variables and decay towards null infinity}\n\nWhile the renormalization allows us to apply the BV theory in the interior region, it does not give the optimal $r$ decay rates in the exterior region. For example, the renormalized quantity\n$$\\partial_{v} \\log \\lmb - \\frac{\\mu}{(1-\\mu)} \\frac{\\lmb}{r} + \\partial_{v} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi)  - \\nu^{-1} \\partial_{u} (r \\phi) \\Big)$$\ndecays only as $r^{-2}$ towards null infinity due to the contribution of $\\frac{\\mu}{(1-\\mu)} \\frac{\\lmb}{r}$, which is weaker than the desired $r^{-3}$ decay for $\\partial_v\\log \\lmb$. Therefore, in order to obtain the optimal estimates everywhere in the spacetime, we need to use the variables $\\partial_v^2(r\\phi)$, $\\partial_u^2(r\\phi)$, $\\partial_v\\lmb$ and $\\partial_u\\nu$ together with their renormalized versions.\n\\\\\n\\fparagraph{Coupling of the incoming and outgoing parts}\n\nFinally, an additional challenge is that unlike the estimates for the first derivatives of the scalar field, the bounds for the incoming part of the solution $\\partial_v^2(r\\phi)$ and $\\partial_v\\lmb$ are coupled to that for the outgoing part $\\partial_u^2(r\\phi)$ and $\\partial_u\\nu$. Likewise, to control $\\partial_u^2(r\\phi)$, we need estimates for $\\partial_v^2(r\\phi)$ and $\\partial_v\\lmb$. For example, in the equation for $\\partial_{v} \\log \\lmb - \\frac{\\mu}{(1-\\mu)} \\frac{\\lmb}{r} + \\partial_{v} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi)  - \\nu^{-1} \\partial_{u} (r \\phi) \\Big)$, there is a term involving $\\partial_u^2(r\\phi)$ on the right hand side. In particular, in order to obtain the desired decay for $\\partial_v \\lmb$, we need to at the same time prove the decay for $\\partial_u^2(r\\phi)$.\n\\\\\n\\fparagraph{Strategy for obtaining the decay estimates}\nWith the above difficulties in mind, we can now give a very rough sketch of the strategy of the proof.\n\n\\begin{enumerate}\n\\item [(1)] Bounds for $\\partial_v^2(r\\phi)$ and $\\partial_v\\lmb$ for large $r$: As in the case for the first derivatives, we first prove the optimal $r$ decay for $\\partial_v^2(r\\phi)$ and $\\partial_v\\lmb$ in the exterior region. To this end, we integrate the equations satisfied by the \\emph{non-renormalized} variables. We note that the error terms can all be bounded using the local BV scattering assumption and the decay estimates already proved for the first derivatives.\n\n\\item [(2)] Bounds for all second derivatives: Steps 2 and 3 for the decay bounds for the first derivatives are now coupled. Define \\begin{align*}\n\t\\mathcal B_{2}(U) := \\sup_{u \\in [1, U]} \\sup_{C_{u}} \\Big( & u^{\\omega} \\abs{\\partial_{v}^{2} (r \\phi)} +u^{\\omega} \\abs{\\partial_{u}^{2} (r \\phi)} \n\t\t\t\t\t+ u^{\\omega} \\abs{\\partial_{v} \\lmb} + u^{\\omega} \\abs{\\partial_{u} \\nu} \\Big).\n\\end{align*}\nWe then show that $\\mathcal B_{2}(U)$ can control the error terms arising from integrating the \\emph{renormalized} equations in the sense that we can obtain an inequality of the form\n$$\\abs{\\mbox{weighted renormalized variables}}\\leq C(u_2)+\\epsilon(u_2)\\mathcal B_{2}(U),$$\nwhere $\\epsilon(u_2)\\to 0$ as $u_2\\to \\infty$. We then prove that the renormalized variables in fact control all the weighted second derivatives in $\\mathcal B_2$. After choosing $u_2$ to be sufficiently large, we show that $\\mathcal B_{2}(U)$ is bounded independent of $U$ and thus all the second derivatives have $u^{-\\omega}$ decay.\n\\item [(3)] Optimal bounds in terms of $u$ decay: While we have obtained $u^{-\\omega}$ decay for the second derivatives, the decay rates are not the sharp rates claimed in the main theorem. To finally obtained the desired bounds, we integrate the equations of the \\emph{non-renormalized} variables and use the preliminary estimates obtained in (1) and (2) above. Here, we make use of the fact that the estimates obtained in step (2) above are sufficiently strong (both in terms of regularity and decay) to control the error terms in the non-renormalized equations.\n\\end{enumerate}\n\n\\section{Analytic properties of \\eqref{eq:SSESF}}\\label{sec.anal.prop}\nIn this section, we discuss the analytic properties of \\eqref{eq:SSESF}. These include scaling, monotonicity and the null structure of the system. All these features will play crucial roles in the analysis.\n\\subsection{Scaling}\nFor $a>0$, \\eqref{eq:SSESF} is invariant under the scaling of the coordinate system\n$$ u \\mapsto au,\\quad v\\mapsto av$$\ntogether with the scaling of the functions\n$$r \\mapsto ar,\\quad m\\mapsto am,\\quad \\Omega\\mapsto \\Omega,\\quad\\phi\\mapsto\\phi.$$\nThis in particular implies that the BV norms\n\\begin{equation*}\n\\int_u^{\\infty} |\\partial_v^2(r\\phi)(u,v')| \\mathrm{d} v'\n\\hbox{ and }\n\\int_u^{\\infty} |\\partial_v \\lambda(u,v')| \\mathrm{d} v'\n\\end{equation*}\nare scale invariant. Thus the a priori assumptions \\eqref{eq:locBVScat} are taken with respect to localized versions of scale invariant norms.\n\n\\subsection{Monotonicity properties} \\label{subsec:monotonicity}\nWe first begin with basic monotonicity properties of $r$.\n\\begin{lemma}[Monotonicity of $r$] \\label{lem:mntn4r}\nLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF}. Then we have \n\\begin{equation*}\n\t\\nu < 0 \\hbox{ in } \\calQ,\n\\end{equation*}\nand\n\\begin{equation*}\n\t\\left\\{\n\t\\begin{aligned}\n\t\\lmb > 0 & \\hbox{ when } 1-\\mu > 0, \\\\\n\t\\lmb = 0 & \\hbox{ when } 1-\\mu = 0, \\\\\n\t\\lmb < 0 & \\hbox{ when } 1-\\mu < 0.\n\t\\end{aligned}\t\n\t\\right.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof} \nThis was proved in \\cite[Propositions 1.1 and 1.2]{Christodoulou:1993bt}; we reproduce the proof for the reader's convenience. Note the equation\n\\begin{equation*}\n\t\\partial_{u} \\partial_{v} (r^{2}) = - \\frac{1}{2} \\Omega^{2}.\n \\end{equation*}\nwhich easily follows from \\eqref{eq:SSESF}. As $\\partial_{u} r^{2} = 2 r \\partial_{u} r= 0$ on $\\Gamma$ and $r > 0$ on $\\calQ$, we easily see that $\\nu < 0$. Then from the definition of $1-\\mu$, the second conclusion also follows. \\qedhere\n\\end{proof}\n\nAccording to the sign of $\\lmb$, a general Penrose diagram $\\calQ$ is divided into three subregions as follows:\n\\begin{align*}\n\t\\calT := \\set{(u,v) \\in \\calQ : \\lmb < 0}, \\quad \\calA := \\set{(u,v) \\in \\calQ : \\lmb = 0}, \\quad \\calR := \\set{(u,v) \\in \\calQ : \\lmb > 0}.\n\\end{align*}\n\nThese are called the \\emph{trapped region}, \\emph{apparent horizon}, and \\emph{regular region}, respectively. The next lemma, which we borrow from \\cite{Christodoulou:1993bt}, shows that the solutions to \\eqref{eq:SSESF} considered in this paper consist only of the regular region $\\calR$. Therefore, extensive discussion of $\\calT$ and $\\calA$ will be suppressed.\n\n\\begin{lemma}[{\\cite[Proposition 1.4]{Christodoulou:1993bt}}] \\label{lem:regR}\nLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF}. Then the causal past of $\\Gamma$ in $\\calQ$ is contained in $\\calR$.\nIn particular, $\\calQ = \\calR$ if $(\\phi, r, m)$ satisfies the condition $(1)$ in Definition \\ref{def:locBVScat} (future completeness of radial null geodesics).\n\\end{lemma}\n\n\nNext, we turn to monotonicity properties of the Hawking mass $m$, which will play an important role in our paper. The following lemma is an obvious consequence of \\eqref{eq:SSESF:dm}.\n\n\\begin{lemma}[Monotonicity of $m$] \\label{lem:mntn4m}\nFor a BV solution $(\\phi, r, m)$ to \\eqref{eq:SSESF}, we have\n\\begin{equation*}\n\t\\partial_{v} m \\geq 0, \\quad  \\partial_{u} m \\leq 0 \\hbox{ in } \\calR.\n\\end{equation*}\n\\end{lemma}\n\nBy the monotonicity $\\partial_{v} m \\geq 0$, the limit $M(u):=\\lim_{v \\to \\infty} m(u,v)$ exists (possibly $+\\infty$ at this point) for each $u$. This is called the \\emph{Bondi mass} at retarded time $u$. The following statement is an easy corollary of the preceding lemma.\n\n\\begin{corollary} [Monotonicity of the Bondi mass] \\label{cor:mntn4Bondi}\nLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF}, and suppose that $C_{u} \\subset \\calR$ for $u \\in [u_{1}, u_{2}]$. Then the Bondi mass $M(u)$ is a non-increasing function on $[u_{1}, u_{2}]$.\n\\end{corollary}\n\nThe following lemma shows that $M_{i} < \\infty$ for initial data sets considered in this paper. \n\\begin{lemma} \\label{lem:bnd4Mi}\nSuppose that $\\partial_{v}(r \\phi)(1, \\cdot)$ is asymptotically flat or order $\\omega' > 1$ in the sense of Definition \\ref{def:AF}. Then we have\n\\begin{equation} \\label{eq:bnd4Mi}\n\tM_{i} := \\lim_{v \\to \\infty} m(1, v) \\leq C \\mathcal I_{1}^{2}.\n\\end{equation}\n\\end{lemma}\n\nThis is an easy consequence of \\eqref{eq:SSESF:dm} and Lemma \\ref{lem:mntn4r}; we omit its proof.\nBy the preceding corollary, it follows that $M(u) < \\infty$ for each $u$.\n\nWe conclude this subsection with additional monotonicity properties of solutions to \\eqref{eq:SSESF}, useful for controlling the geometry of locally BV scattering solutions to \\eqref{eq:SSESF}.\n\n\\begin{lemma} \\label{lem:mntn4kpp}\n\tLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF}. For $(u,v) \\in \\calR$, we have\n\t\\begin{equation*}\n\t\t\\frac{\\lmb}{1-\\mu}(u,v) \\leq \\frac{\\lmb}{1-\\mu}(1, v),\n\t\\end{equation*}\n\t\n\t$$\\partial_u\\lmb =\\partial_v\\nu \\leq 0.$$\n\t\\end{lemma}\n\n\\begin{proof} \nThe lemma follows from the formula\n\\begin{equation*}\n\t\\partial_{u} \\log \\abs{\\frac{\\lmb}{1-\\mu}} = - (- \\nu)^{-1} r (\\partial_{u} \\phi)^{2}\n\\end{equation*} \nand \\eqref{eq:SSESF:dr}. \\qedhere\n\\end{proof}\n\n\\subsection{Null structure of the evolution equations} \\label{subsec:nullStr}\nIn this subsection, we follow \\cite{Christodoulou:1993bt} and demonstrate that the evolution equations verify a form of null structure. In particular, the null structure occurs in the equations for the second derivatives of the scalar field and the metric. However, it is not apparent if we simply take the derivatives of the equations \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dphi}. Instead, we rewrite the equations in renormalized variables for which the null structure is manifest. We will perform this renormalization separately for the wave equations for $\\phi$ and for the equations for $\\lambda$ and $\\nu$.\n\n\\vspace{.1in}\n{\\it - The wave equation for $\\phi$.}\nTaking $\\partial_{v}$ of the equation \\eqref{eq:SSESF:dphi}, we obtain\n\\begin{equation*}\n\t\\partial_{u} (\\partial_{v}^{2} (r \\phi)) = \\partial_{v} (\\partial_{u} \\lmb \\, \\phi) = \\partial_{u} \\lmb \\, \\partial_{v} \\phi + (\\partial_{v} \\partial_{u} \\lmb) \\phi,\n\\end{equation*}\nor equivalently, after substituting in the first equation in \\eqref{eq:SSESF:dr},\n\\begin{equation} \\label{eq:eq4dvdvrphi:normal}\n\\partial_{u} (\\partial_{v}^{2} (r \\phi)) = \n\\frac{2m \\lmb \\nu}{(1-\\mu) r^{2}} \\, \\partial_{v} \\phi + \\frac{ \\nu}{(1-\\mu) }  (\\partial_{v} \\phi)^{2} \\phi \n + \\frac{2m \\nu}{(1-\\mu) r^{2}} (\\partial_{v} \\lmb) \\phi - \\frac{4m}{(1-\\mu) r^{3}} \\lmb^{2} \\nu \\phi.\n\\end{equation}\n\nSome terms on the right hand side, such as $(1-\\mu)^{-1} \\nu (\\partial_{v} \\phi)^{2} \\phi$, do not exhibit null structure and are dangerous near $\\Gamma$. To tackle this, we rewrite\n\\begin{equation*}\n\t(\\partial_{v} \\partial_{u} \\lmb) \\phi = \\partial_{u} [(\\partial_{v} \\lmb) \\phi ] - \\partial_{v} \\lmb \\, \\partial_{u} \\phi.\n\\end{equation*}\n\nThus, from the first equation, we derive\n\\begin{equation} \\label{eq:eq4dvdvrphi}\n\t\\partial_{u} [\\partial_{v}^{2} (r \\phi) - (\\partial_{v} \\lmb) \\phi] = \\partial_{u} \\lmb \\, \\partial_{v} \\phi - \\partial_{v} \\lmb \\, \\partial_{u} \\phi.\n\\end{equation}\nBy switching $u$ and $v$, we obtain the following analogous equations in the conjugate direction.\n\\begin{equation} \\label{eq:eq4dudurphi:normal}\n\\partial_{v} (\\partial_{u}^{2} (r \\phi)) = \n\\frac{2m \\lmb \\nu}{(1-\\mu) r^{2}} \\, \\partial_{u} \\phi + \\frac{\\lmb }{(1-\\mu) }  (\\partial_{u} \\phi)^{2} \\phi \n + \\frac{2m \\lmb}{(1-\\mu) r^{2}} (\\partial_{u} \\nu) \\phi - \\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu^{2} \\phi.\n\\end{equation}\n\n\\begin{equation} \\label{eq:eq4dudurphi}\n\t\\partial_{v} [\\partial_{u}^{2} (r \\phi) - (\\partial_{u} \\nu) \\phi] = \\partial_{v} \\nu \\, \\partial_{u} \\phi - \\partial_{u} \\nu \\, \\partial_{v} \\phi.\n\\end{equation}\n\n\n\n\\vspace{.1in}\n{\\it - The equations for $\\lmb$ and $\\nu$.}\nFrom \\eqref{eq:SSESF:dr}, we have\n\\begin{equation*}\n\t\\partial_{u} \\log \\lmb = \\frac{\\mu}{(1-\\mu) r} \\nu, \\quad \\partial_{v} \\log (-\\nu) = \\frac{\\mu}{(1-\\mu) r} \\lmb.\n\\end{equation*}\n\nTake $\\partial_{v}$, $\\partial_{u}$ of the first and second equations respectively. Using \\eqref{eq:SSESF:dr}, it is not difficult to verify that\n\\begin{align}\n\\partial_{u} \\partial_{v} \\log \\lmb\n=& \\frac{1}{(1-\\mu) }  \\lmb^{-1} \\nu (\\partial_{v} \\phi)^{2} - \\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu, \\label{eq:eq4dvdvr:normal} \\\\\n\\partial_{v} \\partial_{u} \\log (-\\nu)\n=& \\frac{1}{(1-\\mu) }  \\nu^{-1} \\lmb (\\partial_{u} \\phi)^{2} - \\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu. \\label{eq:eq4dudur:normal}\n\\end{align}\n\nTo reveal the null structure, we must carry out the renormalization as we have done for \\eqref{eq:eq4dvdvrphi}, \\eqref{eq:eq4dudurphi}. Following Christodoulou \\cite{Christodoulou:1993bt}, it is easy to check that the above two equations are equivalent to\n\\begin{equation} \\label{eq:eq4dvdvr}\n\\begin{aligned}\n& \\partial_{u} \\Big[ \\partial_{v} \\log \\lmb - \\frac{\\mu}{(1-\\mu)} \\frac{\\lmb}{r} + \\partial_{v} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi)  - \\nu^{-1} \\partial_{u} (r \\phi) \\Big) \\Big] \\\\\n& \\qquad = \\partial_{u} \\phi \\, \\partial_{v}\\Big( \\nu^{-1} \\partial_{u} (r \\phi) \\Big)- \\partial_{v} \\phi \\, \\partial_{u} \\Big( \\nu^{-1} \\partial_{u} (r \\phi) \\Big),\n\\end{aligned}\n\\end{equation}\nand the conjugate equation\n\\begin{equation} \\label{eq:eq4dudur}\n\\begin{aligned}\n& \\partial_{v} \\Big[ \\partial_{u} \\log (-\\nu) - \\frac{\\mu}{(1-\\mu)} \\frac{\\nu}{r} + \\partial_{u} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} (r \\phi) \\Big) \\Big]  \\\\\n&\\qquad = - \\partial_{u} \\phi \\, \\partial_{v}\\Big( \\lmb^{-1} \\partial_{v} (r \\phi) \\Big) + \\partial_{v} \\phi \\, \\partial_{u} \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) \\Big).\n\\end{aligned}\n\\end{equation}\n\n\n\\section{Basic estimates for locally BV scattering solutions} \\label{sec.geom}\nIn this section, we gather some basic estimates concerning locally BV scattering solutions. These estimates will apply, in particular, to solutions satisfying the hypotheses of Theorem \\ref{main.thm.1}.\n\n\\subsection{Integration lemmas for $\\phi$} \\label{subsec:est4phi}\nWe first derive some basic inequalities for $\\phi$, $\\lmb^{-1} \\partial_v(r\\phi)$ and $\\partial_{v} \\phi$. We remark that these are functional inequalities which hold under very general assumptions, and in particular does not rely on the locally BV scattering assumption.\n\n\\begin{lemma} \\label{lem:est4phi}\nLet $\\phi(u, \\cdot)$ and $r(u, \\cdot)$ be Lipschitz functions on $[u,v]$ with $\\lmb>0$ and $r(u, u) = 0$. \nThen the following inequality holds.\n\\begin{equation} \\label{eq:intEst4phi:1}\n\\abs{\\phi(u,v)} \\leq  \\sup_{v' \\in [u, v]} \\Big\\vert \\frac{\\partial_{v}(r \\phi)}{\\lmb}(u, v') \\Big\\vert.\n\\end{equation}\n\nMore generally, for $u \\leq v_{1} \\leq v_{2}$, we have\n\\begin{equation} \\label{eq:intEst4phi:2}\n\\abs{r \\phi(u,v_{1}) - r \\phi(u, v_{2})} \\leq  \\Big( r(u, v_{2}) - r(u, v_{1}) \\Big) \\sup_{v' \\in [v_{1}, v_{2}]} \\Big\\vert \\frac{\\partial_{v}(r \\phi)}{\\lmb}(u, v') \\Big\\vert.\n\\end{equation}\n\n\n\\end{lemma}\n\n\\begin{proof} \nWe shall prove \\eqref{eq:intEst4phi:2}, since \\eqref{eq:intEst4phi:1} then follows as a special case. Integrating $\\partial_{v} (r \\phi)(u, v')$ over $v' \\in [v_{1}, v_{2}]$, we get\n\\begin{align*}\n\t\\abs{r \\phi(u,v_{1}) - r \\phi (u, v_{2})} \n\t\\leq & \\int_{v_{1}}^{v_{2}} \\abs{\\partial_{v} (r \\phi)(u, v')} \\, \\mathrm{d} v' \\\\\n\t\\leq & \\sup_{v' \\in [v_{1}, v_{2}]} \\Big\\vert \\frac{\\partial_{v}(r \\phi)}{\\lmb} (u, v') \\Big\\vert \\, \\times \\int_{v_{1}}^{v_{2}} \\lmb(u, v') \\, \\mathrm{d} v' \\\\\n\t=& \\Big( r(u, v_{2}) - r(u, v_{1}) \\Big) \\sup_{v' \\in [v_{1}, v_{2}]} \\Big\\vert \\frac{\\partial_{v}(r \\phi)}{\\lmb} (u, v') \\Big\\vert . \\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{lemma} \\label{lem:est4dvphi}\nLet $\\phi(u, \\cdot)$ and $r(u, \\cdot)$ be functions on $[u, v]$ such that $\\partial_v\\phi$ is integrable, $r$ is Lipschitz with $\\lambda>0$ and $r(u, u) = 0$. \nSuppose furthermore that $\\lmb^{-1} \\partial_{v} (r \\phi)(u, \\cdot)$ is BV on $[u, v]$. Then the following statements hold.\n\\begin{enumerate}\n\\item We have\n\\begin{equation} \\label{eq:est4dvphi:2}\n\\int_{u}^{v} \\abs{\\partial_{v} \\phi(u,v')} \\, \\mathrm{d} v' \\leq \\int_{u}^{v}\\abs{\\partial_{v}(\\lmb^{-1} \\partial_{v} ( r \\phi))(u,v')} \\, \\mathrm{d} v'.\n\\end{equation}\n\n\\item Suppose, in addition, that $\\lmb^{-1} \\partial_{v}(r \\phi)(u, \\cdot)$ is Lipschitz on $[u,v]$. Then we have\n\\begin{equation} \\label{eq:est4dvphi:1}\n\\abs{\\partial_{v} \\phi(u,v)} \\leq \\frac{1}{2} \\frac{\\sup_{v' \\in [u,v]} \\lmb(u, v')}{\\inf_{v' \\in [u,v]} \\lmb(u, v')}  \\sup_{v' \\in [u, v]} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} ( r \\phi))(u, v')}.\n\\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} \nWe proceed formally to compute\n\\begin{align*}\n\t\\partial_{v} \\phi(u,v) \n\t=& \\frac{\\lmb}{r} \\Big( \\lmb^{-1} \\partial_{v} ( r \\phi) - \\phi\\Big) (u,v) \\\\\n\t=& \\frac{\\lmb}{r^{2}} (u, v) \\int_{u}^{v} \\Big( \\int_{v'}^{v} \\partial_{v} (\\lmb^{-1} \\partial_{v} ( r \\phi)) (u, v'') \\, \\mathrm{d} v'' \\Big) \\lmb(u, v')\\, \\mathrm{d} v' \\\\\n\t=& \\frac{\\lmb}{r^{2}} (u,v) \\int_{u}^{v} r(u, v'') \\partial_{v} (\\lmb^{-1} \\partial_{v} ( r \\phi)) (u, v'') \\, \\mathrm{d} v''.\n\\end{align*}\n\nThe above computation is justified thanks to the hypotheses, where we interpret \n\\begin{equation*}\n\t\\partial_{v} (\\lmb^{-1} \\partial_{v} ( r \\phi)) (u, v'') \\, \\mathrm{d} v''\n\\end{equation*}\nto be the the weak derivative of $\\lmb^{-1} \\partial_{v} ( r \\phi)$, which is a finite signed measure. For a fixed $(u, v)$, observe that\n\\begin{equation*}\n\t\\sup_{v'' \\in [u, v]} r(u, v'') \\int_{v''}^{v} \\frac{\\lmb(u,v')}{r^{2}(u,v')} \\, \\mathrm{d} v' \\leq 1.\n\\end{equation*}\n\nThis proves \\eqref{eq:est4dvphi:2}. For \\eqref{eq:est4dvphi:1}, note that the function $\\lmb^{-1} \\partial_{v} (r \\phi)$ is absolutely continuous on $[u,v]$, so $\\partial_{v}(\\lmb^{-1} \\partial_{v} ( r \\phi)(u, \\cdot))$ exists almost everywhere on $[u, v]$; moreover, it belongs to $L^{\\infty}$ by the Lipschitz assumption. Noting that\n\\begin{equation*}\n\t\\sup_{v' \\in [u, v]} \\frac{\\lmb(u,v')}{r^{2}(u,v')} \\int_{u}^{v'} r(u, v'') \\, \\mathrm{d} v'' \\leq \\frac{1}{2} \\frac{\\sup_{v' \\in [u,v]} \\lmb(u, v')}{\\inf_{v' \\in [u,v]} \\lmb(u, v')}\n\\end{equation*}\nwe obtain \\eqref{eq:est4dvphi:1}.\n\\end{proof}\n\n\\subsection{Geometry of locally BV scattering solutions}\nThe goal of this subsection is to prove the following proposition.\n\\begin{proposition} \\label{prop:geomLocBVScat}\nLet $(\\phi, r, m)$ be a locally BV scattering solution to \\eqref{eq:SSESF} as in Definition \\ref{def:locBVScat}. Assume furthermore that on the initial slice $C_{1}$, we have $\\lmb(1, \\cdot) = \\frac{1}{2}$ and\n\\begin{equation*}\n\t\\sup_{C_{1}} \\abs{\\partial_{v}(r \\phi)} + M_{i} < \\infty.\n\\end{equation*}\n\t\nThen there exist finite constants $K, \\Lambda > 0$ such that the following bounds hold for all $(u, v) \\in \\calQ$:\n\\begin{gather}\n\t\\Lambda^{-1} \\leq \\lmb(u,v) \\leq \\frac{1}{2} \\label{eq:bnd4dvr} \\\\\n\t\\Lambda^{-1} \\leq - \\nu(u,v) \\leq K \\label{eq:bnd4dur} \\\\\n\t1 \\leq (1-\\mu(u,v))^{-1} \\leq K \\Lambda. \\label{eq:bnd4mu}\\\\\n\t0 < \\frac{- \\nu}{1-\\mu(u,v)} \\leq K. \\label{eq:bnd4conjKpp}\n\\end{gather}\n\nMoreover, there exists a finite constant $\\Psi > 0$ such that for all $(u,v) \\in \\calQ$, we have\n\\begin{gather}\n\t\\abs{\\partial_{v}(r \\phi)(u,v)} \\leq \\Psi, \\label{eq:bnd4dvrphi} \\\\\n\t\\abs{\\phi(u,v)} \\leq \\Lambda \\Psi. \\label{eq:bnd4phi}\n\\end{gather}\n\\end{proposition}\n\n{\\bf Once we have this proposition, we will denote by $\\Lambda$, $K$ and $\\Psi$ the best constants such that \\eqref{eq:bnd4dvr}-\\eqref{eq:bnd4phi} hold.}\n\nBy Lemma \\ref{lem:regR}, we already know that $\\lmb > 0$, $- \\nu > 0$ and $(1-\\mu)^{-1} < \\infty$. The first three bounds, namely \\eqref{eq:bnd4dvr}--\\eqref{eq:bnd4mu}, ensure that these bounds concerning the geometry of the spacetime does not degenerate anywhere, in particular along the axis $\\Gamma$. They will be very useful in the analysis in the later section of the paper. \n\nThe proof of Proposition \\ref{prop:geomLocBVScat} will consist of several steps. We begin with elementary bounds for $\\lmb$ and $\\nu$.\n\n\\begin{lemma} \\label{lem:basicEst4dr}\nLet $(\\phi, r, m)$ be a BV solution to \\eqref{eq:SSESF} with $\\calQ = \\calR$. Then for every $(u,v) \\in \\calQ$, we have\n\\begin{align}\n\t\\lmb(u,v) \\leq& \\, \\lmb(1, v), \\label{eq:basicEst4dr:1} \\\\\n\t\\lmb^{-1}(u,v) \\leq& \\, \\lim_{u' \\to v-} \\lmb^{-1}(u',v), \\label{eq:basicEst4dr:2} \\\\\n\t\\nu(u,v) \\leq& - \\lim_{v' \\to u+} \\lmb(u, v'). \\label{eq:basicEst4dr:3}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n By \\eqref{eq:SSESF:dr}, we have\n\\begin{equation*}\n\\begin{aligned}\n\t\\lmb(u,v) =& \\lmb(1, v) \\exp \\Big( \\int_{1}^{u} \\Big( \\frac{2m}{(1-\\mu)r^{2}} \\nu \\Big) (u', v) \\, \\mathrm{d} u' \\Big), \\\\\n\t\\lmb^{-1}(u,v) =& \\lim_{u' \\to v-} \\lmb(u', v)^{-1} \\exp \\Big( \\int_{u}^{v} \\Big( \\frac{2m}{(1-\\mu)r^{2}} \\nu \\Big) (u', v) \\, \\mathrm{d} u' \\Big), \\\\\n\t\\nu(u,v) =& \\lim_{v' \\to u+} \\nu(u, v') \\exp \\Big( \\int_{u}^{v} \\Big( \\frac{2m}{(1-\\mu)r^{2}} \\lmb \\Big) (u, v') \\, \\mathrm{d} v' \\Big).\n\\end{aligned}\n\\end{equation*}\n\nSince $-\\nu, (1-\\mu) > 0$ everywhere, \\eqref{eq:basicEst4dr:1} and \\eqref{eq:basicEst4dr:2} follow. Moreover, since \n\\begin{equation*}\n\\lim_{v' \\to u+} \\nu(u,v') = - \\lim_{v' \\to u+}\\lmb(u,v'),\n\\end{equation*}\nand $\\lmb > 0$ on $\\calQ$, \\eqref{eq:basicEst4dr:3} follows as well. \\qedhere\n\\end{proof}\n\nBy Lemma \\ref{lem:regR}, $\\calQ = \\calR$ holds for a solution \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. As an immediate corollary, we have the following easy upper bound for $\\lmb$.\n\\begin{corollary}  \\label{cor:est4dr}\nLet $(\\phi, r, m)$ be a solution to \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. Then by the coordinate condition $\\lmb(1, v) = \\frac{1}{2}$ and \\eqref{eq:basicEst4dr:1}, we have\n\\begin{equation*}\n\t\\sup_{\\calQ} \\lmb \\leq \\frac{1}{2} \\, .\n\\end{equation*}\n\\end{corollary}\n\nNext, we proceed to prove the lower bounds of \\eqref{eq:bnd4dvr} and \\eqref{eq:bnd4dur}. We begin with a technical lemma concerning a large-$r$ region, which will also be useful in our proof of \\eqref{eq:bnd4dvrphi} and \\eqref{eq:bnd4phi}.\n\n\\begin{lemma} \\label{lem:babySmllPtnl:1}\nLet $(\\phi, r, m)$ be a solution to \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. Then for arbitrarily small $\\epsilon > 0$, there exists $r_{0} > 1$ such that\n\\begin{align} \n\t\\sup_{(u,v) \\in \\set{r \\geq r_{0}}} \\int_{1}^{u} \\abs{\\frac{\\mu}{(1-\\mu)} \\frac{\\nu}{r} (u', v)} \\, \\mathrm{d} u' <&  \\epsilon \\, . \\label{eq:babySmllPtnl:1}\n\\end{align}\n\\end{lemma}\n\\begin{proof} \nFor $(u,v) \\in \\set{r \\geq r_{0}}$, we begin by simply estimating as follows:\n\\begin{equation*}\n\t\\abs{\\frac{\\mu}{(1-\\mu)} \\frac{\\nu}{r}} \\leq \\frac{2 M_{i}}{(1-\\frac{2M_{i}}{r_{0}})} \\frac{(- \\nu)}{r^{2}}\n\\end{equation*}\n\nThe above inequality holds as long as\\footnote{Indeed, it suffices to choose $r_0>2M_i$ here. The condition $r_0> R$ will be used in the proof of Lemma \\ref{lem:babySmllPtnl:2}.} we choose $r_{0} > \\max \\set{2 M_{i}, R}$. Note that if $(u, v) \\in \\set{r \\geq r_{0}}$, then the null curve $\\set{(u', v) : u' \\in [1, u]}$ from the initial slice $C_{1}$ to $(u,v)$ lies entirely in $\\set{r \\geq r_{0}}$. Integrating along this curve, we obtain for $(u, v) \\in \\set{r \\geq r_{0}}$\n\\begin{equation*}\n\t\\int_{1}^{u} \\abs{\\frac{\\mu}{(1-\\mu)} \\frac{\\nu}{r}(u',v)} \\, \\mathrm{d} u' < \\frac{2 M_{i}}{(1-\\frac{2M_{i}}{r_{0}})} \\frac{1}{r_{0}}\n\\end{equation*}\n\nTaking $r_{0}$ sufficiently large, \\eqref{eq:babySmllPtnl:1} follows. \\qedhere\n\n\\end{proof}\n\n\nNext, we prove an analogous result in a large $u$ region. Key to its proof will be the identity \\eqref{eq:babySmllPtnl:pf:0} below, which will also be used to relate \\eqref{eq:babySmllPtnl:1} and \\eqref{eq:babySmllPtnl:2} to the desired lower bounds of $\\lmb$ and $-\\nu$.\n\n\\begin{lemma} \\label{lem:babySmllPtnl:2}\nLet $(\\phi, r, m)$ be a solution to \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. Then for arbitrarily small $\\epsilon > 0$, there exists $U > 1$ such that\n\\begin{align} \n\t\\sup_{v \\geq U} \\int_{U}^{v} \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r} (u', v)} \\, \\mathrm{d} u' <&  \\epsilon \\, . \\label{eq:babySmllPtnl:2}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof} \nLet $\\epsilon > 0$ be an arbitrary positive number. Using \\eqref{eq:SSESF:dr} and the fact that $1-\\mu > 0, -\\nu > 0$ on $\\calQ$, we have for any $1 \\leq u_{1} \\leq u_{2} < v$,\n\\begin{equation} \\label{eq:babySmllPtnl:pf:0}\n\t\\int_{u_{1}}^{u_{2}} \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r} (u', v)} \\, \\mathrm{d} u' = \\log \\lmb(u_{1}, v) - \\log \\lmb(u_{2}, v).\n\\end{equation}\n\nIn order to prove \\eqref{eq:babySmllPtnl:2}, it therefore suffices to exhibit $U > 1$ such that \n\\begin{equation} \\label{eq:babySmllPtnl:pf:1}\n\t\\sup_{(u, v), (u', v') \\in \\set{u \\geq U}} \\abs{\\log \\lmb(u, v) - \\log \\lmb(u', v')} < \\epsilon.\n\\end{equation} \n\nIn order to proceed, we divide $\\calQ$ into three regions: $\\PD_{\\mathrm{cpt}} := \\set{r \\leq R}$, $\\calQ_{[R, r_{0}]} := \\set{R \\leq r \\leq r_{0}}$ and $\\calQ_{[r_{0}, \\infty)} := \\set{r \\geq r_{0}}$, where $r_{0} > \\max\\{2M_i,R\\}$ is chosen via Lemma \\ref{lem:babySmllPtnl:1} so that\n\\begin{equation*} \n\t\\sup_{(u,v) \\in \\calQ_{[r_{0}, \\infty)}}\\int_{1}^{u} \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r}(u',v)} \\, \\mathrm{d} u' < \\frac{\\epsilon}{8}.\n\\end{equation*}\n\nUsing \\eqref{eq:babySmllPtnl:pf:0} and the fact that $\\log \\lmb(1, v) = \\frac{1}{2}$, the preceding inequality is equivalent to\n\\begin{equation} \\label{eq:babySmllPtnl:pf:2}\n\t\\sup_{(u,v) \\in \\calQ_{[r_{0}, \\infty)} } \\abs{\\log \\lmb(u,v) - \\frac{1}{2}} < \\frac{\\epsilon}{8}.\n\\end{equation}\n\nNext, we turn to the region $\\calQ_{[R, r_{0}]}$; here we exploit the vanishing of the final Bondi mass. Indeed, taking $U_{1}$ large enough so that $2 M(U_{1}) < R$, we may estimate\n\\begin{equation*}\n\t\\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r}} \\leq \\frac{2 M(U_{1})}{(1-\\frac{2 M(U_{1})}{R}) R^{2}} (-\\nu)\\quad\\mbox{for }u\\geq U_1.\n\\end{equation*}\n\nConsider now the time-like curve given by $\\gamma_{0} := \\set{(u',v') : r(u', v') = r_{0}}$. On $\\gamma_{0} \\cap \\set{(u,v) : u \\geq U_{1}}$, note that \\eqref{eq:babySmllPtnl:pf:2} holds. Integrating the preceding inequality along incoming null curves emanating from $\\gamma_{0} \\cap \\set{(u,v) : u \\geq U_{1}}$, we obtain for $(u, v) \\in \\calQ_{[R, r_{0}]} \\cap \\set{(u,v) : u \\geq U_{2}}$\n\\begin{equation*} \n\t\\abs{\\log \\lmb(u,v) - \\frac{1}{2}} < \\frac{\\epsilon}{8} + \\frac{2 M(U_{1}) (r_{0} - R)}{(1- \\frac{2 M(U_{1})}{R}) R^{2}} .\n\\end{equation*}\nwhere $U_{2} = U_{2}(U_{1}, r_{0})$ is the future endpoint of the incoming null curve in $\\calQ_{[R, r_{0}]}$ from the past endpoint of $\\gamma_{0} \\cap \\set{(u,v) : u \\geq U_{1}}$; more precisely, $U_{2} = \\sup \\set{u : r(u, V_{1}) \\geq R}$, where $V_{1}$ is the defined by $r(U_{1}, V_{1}) = r_{0}$. Choosing $U_{1}$ sufficiently large, we then obtain\n\\begin{equation} \\label{eq:babySmllPtnl:pf:3}\n\t\\sup_{(u, v) \\in \\calQ_{[R, r_{0}]} \\cap \\set{u \\geq U_{2}}} \\abs{\\log \\lmb(u,v) - \\frac{1}{2}} < \\frac{\\epsilon}{4}.\n\\end{equation}\n\nFinally, in $\\PD_{\\mathrm{cpt}}$, we use the local BV scattering condition \\eqref{eq:locBVScat} to choose $U \\geq U_{2}$ large enough so that we have\n\\begin{equation} \\label{eq:babySmllPtnl:pf:4}\n\t\\sup_{(u, v), (u, v') \\in \\PD_{\\mathrm{cpt}} \\cap \\set{u \\geq U}}\\abs{\\log \\lmb (u, v) - \\log \\lmb(u, v')} < \\frac{\\epsilon}{4}.\n\\end{equation}\n\nTo compare $\\log \\lmb(u, v)$ and $\\log \\lmb(u', v')$ with $u \\neq u'$, we use \\eqref{eq:babySmllPtnl:pf:3}, \\eqref{eq:babySmllPtnl:pf:4} and the triangle inequality. Thus, the desired conclusion \\eqref{eq:babySmllPtnl:pf:1} follows. \\qedhere\n\\end{proof}\n\nAs a corollary of the preceding lemmas and \\eqref{eq:babySmllPtnl:pf:0} (or, more directly, \\eqref{eq:babySmllPtnl:pf:1} and \\eqref{eq:babySmllPtnl:pf:2}), we immediately see that $\\lmb$ and $-\\nu$ is uniformly bounded away from zero.\n\\begin{corollary} \\label{cor:lowerBnd4dvr}\nLet $(\\phi, r, m)$ be a solution to \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. Then there exists $0 < \\Lambda < \\infty$ such that for all $(u, v) \\in \\calQ$, we have\n\\begin{equation*}\n\t\\Lambda^{-1} \\leq \\lmb(u, v), \\quad\n\t\\Lambda^{-1} \\leq - \\nu(u,v).\n\\end{equation*}\n\\end{corollary}\n\nTogether with Corollary \\ref{cor:est4dr}, this concludes the proof of \\eqref{eq:bnd4dvr}. Next, using Lemmas \\ref{lem:est4phi}, \\ref{lem:babySmllPtnl:1}, \\ref{lem:babySmllPtnl:2} and the wave equation \\eqref{eq:SSESF:dphi} for $\\phi$, we prove \\eqref{eq:bnd4dvrphi}, \\eqref{eq:bnd4phi} in the following lemma.\n\n\\begin{lemma}  \\label{lem:bnd4dvrphiphi}\nLet $(\\phi, r, m)$ be a solution to \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. Then there exists a constant $0 < \\Psi < \\infty$ such that\n\\begin{equation} \\label{eq:bnd4dvrphiphi}\n\t\\sup_{\\calQ} \\abs{\\partial_{v}(r \\phi)} \\leq \\Psi, \\quad \n\t\\sup_{\\calQ} \\abs{\\phi} \\leq \\Lambda \\Psi,\n\\end{equation}\nwhere $\\Lambda$ is the best constant such that Corollary \\ref{cor:lowerBnd4dvr} holds.\n\\end{lemma}\n\\begin{proof} \nNote that the second inequality of \\eqref{eq:bnd4dvrphiphi} is an immediate consequence of the first inequality, Lemma \\ref{lem:est4phi} and Corollary \\ref{cor:lowerBnd4dvr}. The proof of the first inequality will proceed in two steps: First, we shall show that $\\partial_{v}(r \\phi)$ is uniformly bounded on the large $r$ region, essentially via Lemma \\ref{lem:babySmllPtnl:1}. By compactness, it immediately follows that $\\partial_{v}(r \\phi)$ is uniformly bounded on the finite $u$ region. Then in the second step, we shall show that $\\partial_{v}(r \\phi)$ is uniformly bounded on a large $u$ region as well using Lemma \\ref{lem:babySmllPtnl:2}.\n\nBy Lemma \\ref{lem:babySmllPtnl:1}, choose $r_{0} > 0$ so that \n\\begin{equation} \\label{eq:bdd4dvrphiphi:pf:1}\n\t\\sup_{(u,v) \\in \\set{r \\geq r_{0}}} \\int_{1}^{u} \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r} (u', v)} \\, \\mathrm{d} u' < \\frac{1}{10 \\Lambda}.\n\\end{equation}\n\nWe also borrow the notation $\\calQ_{[r_{0}, \\infty)} := \\set{(u,v) : r(u,v) \\geq r_{0}}$ from the proof of Lemma \\ref{lem:babySmllPtnl:2}. Given $U \\geq 1$, define $\\Psi_{[r_{0}, \\infty)}(U)$ to be\n\\begin{equation*}\n\t\\Psi_{[r_{0}, \\infty)}(U) := \\sup_{(u, v) \\in \\calQ_{[r_{0}, \\infty)} \\cap \\set{1 \\leq u \\leq U}} \\abs{\\partial_{v} (r \\phi)(u, v)}.  \n\\end{equation*}\n\nLet $(u,v) \\in \\calQ_{[r_{0}, \\infty)}$. Using \\eqref{eq:SSESF:dphi}, we then write\n\\begin{align*}\n\t\\partial_{u} \\partial_{v} ( r \\phi)\n\t= & \\frac{\\mu}{1-\\mu} \\frac{\\nu}{r} \\Big( \\frac{\\lmb}{r} (r \\phi - r_{0} \\phi_{r_{0}}) + \\frac{\\lmb}{r} r_{0} \\phi_{r_{0}} \\Big).\n\\end{align*} \n\nHere, $\\phi_{r_{0}}(u,v) := \\phi(u, v^{\\star}_{0}(u))$, where $v^{\\star}_{0}(u)$ is the unique $v$-value for which $r(u, v^{\\star}_{0}(u)) = r_{0}$. Note that the outgoing null curve from $(u, v^{\\star}_{0}(u))$ to $(u,v) \\in \\calQ_{[r_{0}, \\infty)}$ lies entirely in $\\calQ_{[r_{0}, \\infty)}$. Thus, by Lemma \\ref{lem:est4phi} and \\eqref{eq:bnd4dvr}, we see that for $(u, v) \\in \\calQ_{[r_{0}, \\infty)}$ with $1 \\leq u \\leq U$, \n\\begin{align*}\n\t\\abs{\\partial_{u} \\partial_{v} ( r \\phi)} \n\t\\leq & \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r}} \\Big( \\frac{(r - r_{0})}{2 r} \\Lambda \\Psi_{[r_{0}, \\infty)}(U) + \\frac{r_{0}}{2 r} \\abs{\\phi_{r_{0}}} \\Big) \\\\\n\t\\leq & \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r}} \\Big( \\Lambda \\Psi_{[r_{0}, \\infty)}(U) + \\abs{\\phi_{r_{0}}} \\Big).\n\\end{align*}\n\nIntegrating this equation over the incoming null curve from $(1, v)$ to $(u, v)$ (which lies in $\\calQ_{[r_{0}, \\infty)} \\cap \\set{1 \\leq u \\leq U}$) and using Lemma \\ref{lem:babySmllPtnl:1}, we then obtain\n\\begin{align*}\n\t\\Psi_{[r_{0}, \\infty)}(U)\t\\leq \\sup_{C_{1} \\cap \\calQ_{[r_{0}, \\infty)}} \\abs{\\partial_{v} (r \\phi)} + \\frac{1}{10} \\Psi_{[r_{0}, \\infty)}(U) + \\frac{1}{10 \\Lambda} \\sup_{\\gamma_{0} \\cap \\set{1 \\leq u \\leq U}} \\abs{\\phi}\n\\end{align*}\nwhere $\\gamma_{0}$ is the time-like curve $\\set{(u,v) : r(u,v) = r_{0}}$. Note that the first term on the right-hand side is finite by the assumptions on the initial data, whereas the last term is finite for every $1 \\leq U < \\infty$ by compactness of $\\gamma_{0} \\cap \\set{(u,v) : 1 \\leq u \\leq U}$ and continuity of $\\phi$. Then, by a simple continuity argument, it follows that $\\Psi_{[r_{0}, \\infty)}(U) < \\infty$ for every $1 \\leq U < \\infty$. Moreover, by compactness of $\\set{(u, v) : r(u,v) \\leq r_{0}, \\, 1 \\leq u \\leq U}$, as well as the uniform BV assumption on $\\partial_{v}(r \\phi)$, we also have\n\\begin{equation*}\n\t\\Psi_{[0, \\infty)}(U) := \\sup_{(u,v) \\in \\set{1 \\leq u \\leq U}} \\abs{\\partial_{v}(r \\phi)(u,v)} < \\infty.\n\\end{equation*}\n\nWe now proceed to deal with the large-$u$ region, namely $\\set{(u,v) : u \\geq U}$. Using Lemma \\ref{lem:babySmllPtnl:2}, we choose $U_{0} \\geq 1$ sufficiently large so that\n\\begin{equation}\n\t\\sup_{v \\geq U_{0}} \\int_{U_{0}}^{v} \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r} (u', v)} \\, \\mathrm{d} u' < \\frac{1}{10 \\Lambda}.\n\\end{equation}\n\nProceeding as before via Lemma \\ref{lem:est4phi}, we estimate for $(u,v) \\in \\set{(u,v) : u \\geq U_{0}}$ \n\\begin{align*}\n\t\\abs{\\partial_{u} \\partial_{v} (r \\phi)(u,v)} \\leq \\abs{\\frac{\\mu}{1-\\mu} \\frac{\\nu}{r}} \\, \\Lambda \\sup_{v' \\in [u, v]} \\abs{\\partial_{v}(r \\phi)(u, v')} .\n\\end{align*}\n\nIntegrating along incoming null curves from $C_{U_{0}}$, we see that\n\\begin{equation*}\n\t\\Psi_{[0, \\infty)}(U) \\leq \\Psi_{[0, \\infty)}(U_{0}) + \\frac{1}{10} \\Psi_{[0, \\infty)}(U)\n\\end{equation*}\nfor any $U \\geq U_{0}$. Absorbing the second term on the right-hand side into the left-hand side and taking $U \\to \\infty$, we obtain \\eqref{eq:bnd4dvrphiphi} with $\\Psi \\leq \\frac{10}{9} \\Psi_{[0, \\infty)}(U_{0}) < \\infty$. \\qedhere\n\\end{proof}\n\nWe are finally ready to conclude the proof of Proposition \\ref{prop:geomLocBVScat}, by proving \\eqref{eq:bnd4conjKpp}. Indeed, the upper bounds in \\eqref{eq:bnd4dur} and \\eqref{eq:bnd4mu} would then follow immediately. Moreover, the lower bound in \\eqref{eq:bnd4mu} is trivial, as $\\mu = \\frac{2m}{r} \\geq 0$.\n\n\\begin{lemma} \\label{lem:est4dur}\nLet $(\\phi, r, m)$ be a solution to \\eqref{eq:SSESF} satisfying the hypotheses of Proposition \\ref{prop:geomLocBVScat}. Then there exists a finite constant $K > 0$ such that for all $(u,v) \\in \\calQ$,\n\\begin{equation} \\label{eq:est4dur:key}\t\n \\frac{- \\nu}{1-\\mu} (u,v) \\leq K.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof} \nTo prove \\eqref{eq:est4dur:key}, we shall rely on the equation\n\\begin{equation} \\label{eq:mntn4durOver1-mu}\n\t\\partial_{v} \\log \\Big( \\frac{-\\nu}{1-\\mu} \\Big) = \\lmb^{-1} r (\\partial_{v} \\phi)^{2},\n\\end{equation}\nwhich may be easily derived from \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dm}.\n\nFor $(u, v) \\in \\calQ$, we begin by integrating \\eqref{eq:mntn4durOver1-mu} on the outgoing null curve from $(u,u) \\in \\Gamma$ to $(u,v)$, which gives\n\\begin{equation*}\n\t\\Big( \\frac{-\\nu}{1-\\mu} \\Big)(u,v) \\leq \\Big( \\lim_{v' \\to u+} \\Big( \\frac{-\\nu}{1-\\mu} \\Big) (u, v') \\Big)  \\exp \\Big( \\int_{u}^{v} \\lmb^{-1} r (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v' \\Big).\n\\end{equation*}\n\nWe claim that $\\lim_{v' \\to u+} (- \\nu) (u, v') = \\lim_{v' \\to u+} \\lmb (u, v')\\leq \\frac 12$ and $\\lim_{v' \\to u+} \\mu(u,v') = 0$. The first assertion is obvious. To prove the second one, we first use \\eqref{eq:SSESF:dm} to write\n\\begin{equation*}\nm(u,v) \\leq \\tfrac{1}{2} \\Big( \\sup_{v' \\in [u, v]} \\abs{r^{2} \\partial_{v} \\phi}(u, v') \\Big) \\int_{u}^{v} \\abs{\\partial_{v} \\phi(u, v')} \\, \\mathrm{d} v'.\n\\end{equation*}\nNow observe that $\\sup_{v' \\in [u, v]} \\abs{r^{2} \\partial_{v} \\phi}(u, v') \\leq C r(u,v) \\sup_{v' \\in [u,v]} \\abs{\\partial_{v}(r \\phi)}$, and the remaining integral goes to $0$ as $v \\to u+$, since $\\phi$ is assumed to be absolutely continuous on $C_{u}$ near the axis by Definition \\ref{def:BVsolution}. \n\nBy the above claim, we have\n\\begin{equation*}\n\t\\Big( \\frac{-\\nu}{1-\\mu} \\Big)(u,v) \\leq \\frac{1}{2} \\exp \\Big( \\int_{u}^{v} \\lmb^{-1} r (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v' \\Big).\n\\end{equation*}\n\nThe lemma would therefore follow if we could prove\n\\begin{equation*}\n\\sup_{(u,v) \\in \\calQ} \\int_{u}^{v} \\lmb^{-1} r (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v' < \\infty.\n\\end{equation*}\n\nTo achieve this, we shall divide the integral into two parts, one in $\\PD_{\\mathrm{cpt}}$ and the other in its complement $\\PD_{\\mathrm{cpt}}^c$. Indeed, defining $v^{\\star}(u)$ to be the unique $v$ value such that $r(u, v^{\\star}(u)) = R$, we divide the integral into $\\int_{u}^{v^{\\star}(u)}$ and $\\int_{v^{\\star}(u)}^{v}$. If $v < v^{\\star}(u)$, the latter integral will be taken to be zero.\n\nFor the first integral, let us begin by pulling out $\\lmb^{-1} r \\partial_{v} \\phi$ from the integral. Using the identity $\\lmb^{-1} r \\partial_{v} \\phi = \\lmb^{-1} \\partial_{v} (r \\phi) - \\phi$ we have\n\\begin{align*}\n&\\int_{u}^{v^{\\star}(u)} \\lmb^{-1} r (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v'\\\\\n& \\quad \\leq \\sup_{v' \\in [u, v^{\\star}(u)]} \\Big( \\lmb^{-1} \\abs{\\partial_{v}(r\\phi)}(u, v') + \\abs{\\phi}(u, v') \\Big) \\int_{u}^{v^{\\star}(u)} \\abs{\\partial_{v} \\phi(u,v')} \\, \\mathrm{d} v'.\n\\end{align*}\n\nThen by Lemmas \\ref{lem:est4dvphi}, \\ref{lem:bnd4dvrphiphi} and the local BV scattering assumption, the right-hand side is uniformly bounded in $u$ from above, as desired. For the second integral, note that, by Lemma \\ref{lem:mntn4kpp} and Corollary \\ref{cor:lowerBnd4dvr}, we have\n\\begin{equation*}\n(1-\\mu)^{-1}(u,v) \\leq \\Lambda \\frac{\\lmb}{1-\\mu}(u,v) \\leq \\frac{\\Lambda}{2} \\sup_{C_{1}} (1-\\mu)^{-1}. \n\\end{equation*}\n\nNotice that the quantity $\\sup_{C_{1}}(1-\\mu)^{-1}$ for the initial data is finite, since $1-\\mu > 0$ everywhere and $1-\\mu(1, v) \\to 1$ as $v \\to \\infty$.\nMoreover, for $v \\geq v^{\\star}(u)$, we have $r(u, v) \\geq R$. Therefore, in view of \\eqref{eq:SSESF:dm}, we may estimate\n\\begin{align*}\n\t\\int_{v^{\\star}(u)}^{v} \\lmb^{-1} r (\\partial_{v} \\phi)^{2} \\, \\mathrm{d} v'\n\t\\leq & \\frac{\\Lambda}{R} \\sup_{C_{1}} (1-\\mu)^{-1} \\int_{v^{\\star}(u)}^{v} \\frac{1}{2} \\lmb^{-1} (1-\\mu) r^{2} (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v' \\\\\n\t\\leq & \\frac{\\Lambda}{R} \\sup_{C_{1}} (1-\\mu)^{-1} (m(u, v) - m(u, v^{\\star}(u))) \\\\\n\t\\leq & C_{\\Lambda, R, M_{i}, \\sup_{C_{1}} (1-\\mu)^{-1}} < \\infty,\n\\end{align*}\nfrom which the lemma follows. \\qedhere\n\\end{proof}\n\nWe conclude this subsection with a pair of identities which are useful for estimating $\\int\\abs{\\partial_{u} \\lmb} \\, \\mathrm{d} u$ and $\\int \\abs{\\partial_{v} \\nu} \\, \\mathrm{d} v$ in terms of information on $\\phi$.\n\\begin{lemma} \\label{lem:auxEqs}\nFrom \\eqref{eq:SSESF}, the following identities hold:\n\\begin{align}\n\t\\int_{u}^{v} \\frac{\\mu}{1-\\mu} \\frac{\\lmb}{r} (u,v') \\, \\mathrm{d} v' = & \\log(1-\\mu)(u,v) + \\int_{u}^{v} \\lmb^{-1} r (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v', \\label{eq:auxEqs:1} \\\\\n\t\\int_{u}^{v} \\frac{\\mu}{1-\\mu} \\frac{(-\\nu)}{r} (u',v) \\, \\mathrm{d} u' = & \\log(1-\\mu)(u,v) + \\int_{u}^{v} (-\\nu)^{-1} r (\\partial_{u} \\phi)^{2} (u', v) \\, \\mathrm{d} u'. \\label{eq:auxEqs:2}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof} \n\tWe shall prove \\eqref{eq:auxEqs:1}, leaving the similar proof of \\eqref{eq:auxEqs:2} to the reader. \n\tFrom the proof of Lemma \\ref{lem:basicEst4dr}, we have\n\t\\begin{equation*}\n\t\t\\int_{u}^{v} \\frac{\\mu}{1-\\mu} \\frac{\\lmb}{r}  (u,v') = \\log \\frac{\\nu(u, v)}{\\lim_{v' \\to u+} \\nu(u,v')}.\n\t\\end{equation*}\n\t\n\tComparing with the integral of \\eqref{eq:mntn4durOver1-mu}, along with the fact that $\\lim_{v' \\to u+} (1-\\mu)(u, v') = 1$, we arrive at \\eqref{eq:auxEqs:1}. \\qedhere\n\\end{proof}\n\n\\subsection{Normalization of the coordinate system}\\label{sec.coord}\n\nIn \\S \\ref{subsec:coordSys}, the coordinates are normalized such that $\\lmb$ is constant on the initial hypersurface $\\{u=1\\}$. Alternatively, one can introduce a new coordinate system $(u_{\\infty},v_{\\infty})$ which is normalized at future null infinity by requiring that $\\nu_{\\infty}\\to-\\frac 12$ along each outgoing null curve towards null infinity and require, as before, that $\\Gamma=\\{(u,v):u=v\\}$. We will show that the coordinate functions $u$ and $u_{\\infty}$ are comparable and thus the main theorem on the decay rates can also be stated in this alternatively normalized coordinate system.\n\nWe can compute explicitly the coordinate change, which is given by\n\\begin{equation*}\n\\frac{du_{\\infty}}{du}(u)=-2\\lim_{v\\to\\infty}\\nu(u,v),\\quad u_\\infty(1)=1\n\\hbox{ and }\nv_{\\infty}(v)=u_{\\infty}(v).\n\\end{equation*}\n\nNotice that the limit $\\displaystyle\\lim_{v\\to\\infty}\\nu(u,v)$ is well-defined due to the monotonicity of $\\nu$.\n\n\\begin{equation*}\n\tu_{\\infty}(u) = - 2\\int_{1}^{u} \\Big(\\lim_{v\\to\\infty}\\nu(u',v)\\Big) \\, \\mathrm{d} u' + 1,\n\\end{equation*}\n\nBy Proposition \\ref{prop:geomLocBVScat}, the following estimate holds:\n\\begin{equation*}\n\t2(\\Lambda)^{-1} (u-1) \\leq u_{\\infty}-1 \\leq 2 K (u-1).\n\\end{equation*}\n\n\\subsection{Consequence of local BV scattering}\nIn this subsection, we give some estimates for $\\partial_u^2(r\\phi)$, $\\partial_u\\phi$ and $\\partial_u \\nu$ that follow from from the local BV scattering assumption. To this end, we will need the analysis for solutions to \\eqref{eq:SSESF} with small bounded variation norm by Christodoulou in \\cite{Christodoulou:1993bt}. In particular, Christodoulou proved\n\\begin{theorem}[{Christodoulou \\cite[Theorem 6.2]{Christodoulou:1993bt}}]\\label{Chr.BV.Thm}\nThere exists universal constants $\\epsilon_0$ and $C_0$ such that for $\\epsilon<\\epsilon_0$, if $\\lmb(1, \\cdot) = \\frac{1}{2}$ and $\\partial_{v} (r \\phi)(1, \\cdot)$ is of bounded variation with\n\\begin{equation} \\label{Chr.BV.Thm.hyp}\n\\int_{C_1} |\\partial_v^2(r\\phi)| <\\epsilon,\n\\end{equation}\nthen its maximal development $(\\phi, r, m)$ satisfies condition $(1)$ in Definition \\ref{def:locBVScat} (future completeness of radial null geodesics) and obeys\n\\begin{gather}\n\t\\frac{1}{3} \\leq \\lmb \\leq \\frac{1}{2}, \\quad \n\t\\frac{1}{3} \\leq - \\nu \\leq \\frac{2}{3}, \\quad\n\t\\frac{2}{3} \\leq (1-\\mu) \\leq 1, \\label{Chr.BV.Thm.geom} \\\\\n\t\\sup_{u \\geq 1} \\int_{C_{u}} \\Big( \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))} + \\abs{\\partial_{v} \\phi} + \\abs{\\partial_{v} \\log \\lmb} \\Big) < C_{0} \\epsilon, \\label{Chr.BV.Thm.dv} \\\\\n\t\\sup_{v \\geq 1} \\int_{\\underline{C}_{v}} \\Big( \\abs{\\partial_{u} (\\nu^{-1} \\partial_{u} (r \\phi))} + \\abs{\\partial_{u} \\phi} + \\abs{\\partial_{u} \\log \\nu} \\Big) < C_{0} \\epsilon. \\label{Chr.BV.Thm.du}\n\\end{gather}\n\\end{theorem}\n\n\\begin{remark} \nWe remark that in \\cite[Theorem 6.2]{Christodoulou:1993bt}, it is implicitly assumed that $\\phi(1, 1) = 0$; see \\cite[Section 4]{Christodoulou:1993bt}. \nNote, however, that the bounds in the above theorem are stated in such a way that they are invariant under the transform $(\\phi, r, m) \\mapsto (\\phi + c, r, m)$, under which \\eqref{eq:SSESF} is also invariant. Any solution may be then transformed to satisfy $\\phi(1, 1) = 0$. As a consequence, we do not need to check $\\phi(1,1) = 0$ in order to apply the theorem.\n\\end{remark}\nUsing Theorem \\ref{Chr.BV.Thm}, we prove the following bound for locally BV scattering solution to \\eqref{eq:SSESF}.\n\n\\begin{theorem} \\label{thm:decayInCpt}\nLet $(\\phi, r, m)$ be a locally BV scattering solution to \\eqref{eq:SSESF}. For every $\\epsilon > 0$, there exists $u_{0} > 1$ such that the following estimate holds.\n\\begin{align*}\n\t\\sup_{v \\in [u_{0}, \\infty)} \\Big( \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{0}}\\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{u}^{2} (r \\phi)} \n\t+ \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{0}} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{u} \\phi} \n\t+ \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{0}} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{u} \\log \\nu} \\Big) < \\epsilon.\n\\end{align*}\n\nMoreover, we also have\n\\begin{equation} \\label{eq:bnd4durphi}\n\t\\sup_{\\calQ} \\abs{\\partial_{u} (r \\phi)} \\leq C_{K, \\Lambda} \\Psi.\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nWe first show that for a locally BV scattering solution to \\eqref{eq:SSESF},\n\\begin{equation*}\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))} \\to 0 \\hbox{ as } u \\to \\infty,\n\\end{equation*}\nExpanding this expression, we have\n\\begin{eqnarray*}\n\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))}\n\\leq \\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\lmb^{-1} (\\abs{ \\partial_{v}^2 (r \\phi)}+\\abs{ (\\partial_{v} \\log \\lambda) \\partial_v(r \\phi)})\n\\end{eqnarray*}\nBy \\eqref{eq:bnd4dvr} and \\eqref{eq:bnd4dvrphi}, we have\n$$\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))}\\leq \nC_{\\Lambda, \\Psi}\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{ \\partial_{v}^2 (r \\phi)}+\\abs{ \\partial_{v} \\log \\lambda },$$\nwhich by \\eqref{eq:locBVScat} in Definition \\ref{def:locBVScat} (Scattering in BV in a compact $r$-region) tends to 0 as $u\\to \\infty$.\nNotice that the quantity $\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))}$ which we have controlled is invariant under any rescaling of the coordinate $v$, and also under the transform $(\\phi, r, m) \\to (\\phi + c, r, m)$.\n\nWe now proceed to the proof of the theorem. Let $v_0$ be sufficiently large and $u^{\\star}(v_0)$ be the unique $r(u^{\\star}(v_0),v_0)=R$. By the finite speed of propagation of the equations, the solution on $\\underline{C}_{v_0}\\cap\\PD_{\\mathrm{cpt}}$ depends only on the data on $C_{u^{\\star}(v_0)} \\cap\\PD_{\\mathrm{cpt}}$.\n\nIn order to apply Theorem \\ref{Chr.BV.Thm}, we change coordinates $(u,v)\\mapsto (U(u),V(v))$ in the region bounded by $C_{u^{\\star}(v_0)}$ and $\\underline{C}_{v_0}$ to a new double null coordinate $(U,V)$ such that for $U^{\\star}=U(u^{\\star}(v_0))$, we have $\\lambda(U^{\\star},V)=\\frac 12$. To this end, define $V(v)$ by\n$$\\frac{dV}{dv}=2\\lmb(u^{\\star}(v_0),v),\\quad V(v_0)=v_0.$$\nNotice that this is acceptable since $\\lmb>0$. In order for the condition $U = V$ to hold on $\\Gamma$, we require\n$U(u)=V(u).$\nThen with respect to the coordinate $V$\n$$\\partial_V r(U^{\\star},V)=\\frac 12.$$\nBy \\eqref{eq:bnd4dvr}, we have\n$$\\Lambda^{-1}\\leq \\frac{dV}{dv}, \\frac{dU}{du}\\leq \\frac 12.$$\nMoreover,\n$$\\int_{u^{\\star}(v_0)}^{v_0} |\\frac{d^2V}{dv^2}(v')|dv'\\leq 2\\int_{u^{\\star}(v_0)}^{v_0} |\\partial_v\\lambda(u^{\\star}(v_0),v')| \\mathrm{d} v',$$\nwhich tends to $0$ as $v_0\\to \\infty$ by the assumption of local BV scattering. For $v_0$ sufficiently large, in the $(U,V)$ coordinate system, $\\int_{C_{u^{\\star}(v_0)} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{V} ((\\partial_{V} r)^{-1} \\partial_{V} (r \\phi))} dV$ is small and $\\partial_V r=\\frac 12$. The data satisfy the assumptions of Theorem \\ref{Chr.BV.Thm} and therefore\\footnote{More precisely, we apply Theorem \\ref{Chr.BV.Thm} to the truncated initial data $$ \\partial_{V} (r \\widetilde{\\phi})(U^{\\star}, V) = \\left\\{ \\begin{array}{cc} \\partial_{V}(r \\phi)(U^{\\star}, V) & \\hbox{ for } V < v_{0} \\\\ \\partial_{V}(r \\phi)(U^{\\star}, v_{0}) & \\hbox{ for } V \\geq v_{0} \\end{array} \\right.$$}\n$$\\int_{\\underline{C}_{v_0}\\cap\\PD_{\\mathrm{cpt}}}(\\abs{\\partial_{U} ((\\partial_{U} r)^{-1} \\partial_{U} (r \\phi))} + \\abs{\\partial_{U} \\phi} + \\abs{\\partial_{U} \\log \\partial_{U} r} ) dU \\to 0$$\nas $v_{0} \\to \\infty$. Returning to the original coordinate system $(u,v)$, the first statement easily follows.\n\nFinally, for the $L^\\infty$ estimate for $\\partial_{u}(r\\phi)$, notice that $\\abs{\\partial_{u}(r\\phi)} \\leq \\Psi$ at the axis by \\eqref{eq:bnd4dvrphi} and $(7)$ of Definition \\ref{def:BVsolution} (BV solutions to \\eqref{eq:SSESF}). Using \\eqref{eq:SSESF:dphi''}, \\eqref{eq:SSESF:dr} (in particular, the fact that $\\partial_{v} \\nu \\leq 0$), \\eqref{eq:bnd4phi} and \\eqref{eq:bnd4dur}, we have\n\\begin{equation*}\n\\abs{\\partial_{u}(r\\phi)(u,v)} \\leq \\Psi+ \\Lambda \\Psi \\int_{u}^{v} (-\\partial_{v} \\nu) \\, \\mathrm{d} v' \\leq C_{K, \\Lambda} \\Psi. \\qedhere\n\\end{equation*}\n\\end{proof}\n\n\n\\section{Decay of $\\phi$ and its first derivatives}\\label{sec.decay1}\nIn this section, we prove the first main theorem (Theorem \\ref{main.thm.1}). Throughout this section, we assume that $(\\phi, r, m)$ is a locally BV scattering solution to \\eqref{eq:SSESF} with asymptotically flat initial data of order $\\omega'$ in BV, as in Definitions \\ref{def:locBVScat} and \\ref{def:AF}, respectively. Let $\\omega = \\min \\set{\\omega', 3}$.\n\n\\subsection{Preparatory lemmas}\nThe following lemma will play a key role in the proof of both Theorems \\ref{main.thm.1} and \\ref{main.thm.2}. It is a consequence of the scattering assumption \\eqref{eq:locBVScat} and vanishing of the final Bondi mass.\n\n\\begin{lemma} \\label{lem:smallPtnl}\nLet $\\epsilon > 0$ be an arbitrary positive number. For $u_{1} > 1$ sufficiently large, we have\n\\begin{align} \n\t\\sup_{v \\in [u_{1}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{1}}} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}}}   < \\epsilon, \\label{eq:smallPtnl:u} \\\\ \n\t\\sup_{u \\in [u_{1}, \\infty)} \\int_{C_{u}} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}}} < \\epsilon. \\label{eq:smallPtnl:v}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof} \nThe first statement \\eqref{eq:smallPtnl:u} was proved in Lemma \\ref{lem:babySmllPtnl:2}; thus it only remains to prove \\eqref{eq:smallPtnl:v}.\n\nDivide $\\calQ$ into $\\PD_{\\mathrm{cpt}} =\\calQ\\cap\\set{r \\leq R}$ and $\\PD_{\\mathrm{cpt}}^{c} := \\calQ \\setminus \\PD_{\\mathrm{cpt}}$. First, note that by \\eqref{eq:locBVScat} we have\n\\begin{align*}\n\t\\sup_{u \\in [u_{1}, \\infty)} \\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}}} < \\epsilon\/2\n\\end{align*}\nfor $u_{1}$ sufficiently large. \nNext, we consider $\\PD_{\\mathrm{cpt}}^{c}$. Define $v^{\\star}(u) := \\sup \\set{v \\in [u, \\infty) : r(u,v) \\geq R}$; note that $r(u, v^{\\star}(u)) = R$ by continuity. We now compute\n\\begin{align*}\n\t\\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}^{c}} \\abs{\\frac{2 m \\lmb}{(1-\\mu) r^{2}}}  \n\t= & \\int_{v^{\\star}(u)}^{\\infty} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}}(u,v')} \\, \\mathrm{d} v' \\\\\n\t\\leq & 2 K \\Lambda M(u_{1}) \\int_{v^{\\star}(u)}^{\\infty} \\frac{\\lmb}{r^{2}}(u, v') \\, \\mathrm{d} v' \\\\\n\t\\leq & 2 R^{-1} K \\Lambda M(u_{1}).\n\\end{align*}\nuniformly in $u \\geq u_{1}$. As $\\lim_{u_{1} \\to \\infty} M(u_{1}) = 0$ by \\eqref{eq:zeroMf} (vanishing final Bondi mass), the last line can be made arbitrarily small by taking $u_{1}$ sufficiently large. This proves \\eqref{eq:smallPtnl:v}. \\qedhere\n\\end{proof}\n\nThe following lemma allows us to estimate $\\phi$ in terms of $\\abs{\\partial_{v} (r \\phi)}$.\n\\begin{lemma} \\label{lem:intEst4phi}\nThe following estimates hold.\n\\begin{align*} \n\t\\abs{\\phi}(u,v) \\leq & \\Lambda \\sup_{C_{u} } \\abs{\\partial_{v} (r \\phi)} , \\\\\n\tr u^{\\omega-1} \\abs{\\phi}(u,v) \\leq & C \\Lambda \\Big( \\sup_{C_{u}} u^{\\omega} \\abs{\\partial_{v} (r \\phi)} + \\sup_{C_{u}} r^{\\omega} \\abs{\\partial_{v} (r \\phi)} \\Big). \n\\end{align*}\n\\end{lemma}\n\n\\begin{proof} \nThe first estimate follows from Lemma \\ref{lem:est4phi} and Proposition \\ref{prop:geomLocBVScat}. The second estimate is a consequence of the first when $r(u,v) \\leq u$, so it suffices to assume $r(u,v) \\geq u$. Introducing a parameter $v_{1} \\in [u, v]$, we estimate\n\\begin{align*}\n\tr u^{\\omega-1} \\abs{\\phi}(u,v) \n\t\\leq & u^{\\omega-1} \\int_{u}^{v} \\abs{\\partial_{v} (r \\phi)(u, v')} \\, \\mathrm{d} v' \\\\\n\t\\leq & \\Lambda u^{\\omega-1}(\\sup_{C_u}|\\partial_v(r\\phi)|)\\int_u^{v_1}\\lmb(u,v')\\mathrm{d} v'+\\Lambda u^{\\omega-1}(\\sup_{C_u}r^{\\omega}|\\partial_v(r\\phi)|)\\int_{v_1}^{v}\\frac{\\lmb}{r^{\\omega}}(u,v')\\mathrm{d} v'\\\\\n\t\\leq & \\Lambda (r(u, v_{1})\/u) \\sup_{C_{u}} u^{\\omega} \\abs{\\partial_{v} (r \\phi)} + \\frac{\\Lambda}{\\omega-1} (u^{\\omega-1}\/r(u, v_{1})^{\\omega-1}) \\sup_{C_{u}} r^{\\omega} \\abs{\\partial_{v} (r \\phi)}.\n\\end{align*}\n\nChoosing $v_{1}$ so that $r(u, v_{1}) = u$ (which is possible since $r(u,v) \\geq u$), the desired estimate follows.\n\\end{proof}\n\n\\subsection{Preliminary $r$-decay for $\\phi$} \\label{subsec:decay1:rDecay}\nIn this subsection, we derive bounds for $\\phi$ which are sharp in terms of $r$-weights. As a consequence, they give sharp decay rates towards null infinity.\n\n\\begin{lemma} \\label{lem:decay1:cptu:0}\n\tThere exists a constant $0 < H_{1} < \\infty$ such that the following estimate holds.\n\t\\begin{equation} \\label{eq:decay1:cptu:0}\n\t\t\\sup_{\\calQ} (1+r) \\abs{\\phi}   \\leq H_{1}.\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{proof} \nLet $r_{1} > 0$ be a large number to be chosen below. Different arguments will be used in $\\set{r \\geq r_{1}}$ and $\\set{r \\leq r_{1}}$. For each $u \\geq 1$ let $v^{\\star}_{1}(u)$ be the unique $v$-value for which $r(u, v_{1}^{\\star}(u)) = r_{1}$. By the fundamental theorem of calculus, we have\n\\begin{equation} \\label{eq:decay1:cptu:0:pf:1}\n\tr \\phi = r_{1} \\phi(u, v^{\\star}_{1} (u)) + \\int_{v^{\\star}_{1}(u)}^{v} \\partial_{v} (r \\phi) (u, v') \\, \\mathrm{d} v'.\n\\end{equation} \n\nIntegrate \\eqref{eq:SSESF:dphi} along the incoming direction from $(1,v)$ to $(u,v)$. By Corollary \\ref{cor:mntn4Bondi} and Proposition \\ref{prop:geomLocBVScat}, we have\n\\begin{align*}\n\t\\abs{\\partial_{v} (r \\phi)(u,v)}\n\t\\leq& \\abs{\\partial_{v} (r \\phi)(1, v)} + \\abs{\\int_{1}^{u} \\frac{2m \\lmb \\nu}{(1-\\mu) r^{3}} (r\\phi) (u', v) \\, \\mathrm{d} u'} \\\\\n\t\\leq &\\abs{\\partial_{v} (r \\phi)(1, v)} + \\frac{K \\Lambda M_{i}}{2} \\frac{1}{r^{2}(u,v)}   \\sup_{u' \\in [1, u]} \\abs{r \\phi(u', v)}.\n\\end{align*}\n\nSubstituting the preceding bound into \\eqref{eq:decay1:cptu:0:pf:1}, we obtain\n\\begin{equation} \\label{eq:decay1:cptu:0:pf:2}\n\\begin{aligned}\n\t\\sup_{C_{u} \\cap \\set{r \\geq r_{1}}} \\abs{r \\phi} \n\t\\leq & \\abs{r_{1} \\phi(u, v^{\\star}_{1} (u))} + \\int_{v_{1}^{\\star}(u)}^{v} \\abs{\\partial_{v} (r \\phi)(1, v')} \\, \\mathrm{d} v' \\\\\n\t& + \\frac{K \\Lambda^{2} M_{i}}{2 r_{1}}  \\sup_{u' \\in [1, u]} \\sup_{C_{u'} \\cap \\set{r \\geq r_{1}}} \\abs{r \\phi}.\n\\end{aligned}\n\\end{equation}\n\nThe first term on the right-hand side is bounded by $r_{1} \\Lambda \\Psi$ by \\eqref{eq:bnd4phi}, whereas the second term depends only on the initial data and can be estimated in terms of $\\mathcal I_{1}$ as follows:\n\\begin{equation*}\n\t\\int_{v_{1}^{\\star}(u)}^{v} \\abs{\\partial_{v} (r \\phi)(1, v')} \\, \\mathrm{d} v' \\leq \\Lambda \\mathcal I_{1} \\int_{1}^{\\infty}  (1+r(1, v'))^{-\\omega'} \\lmb(1,v') \\, \\mathrm{d} v' \\leq \\frac{\\Lambda}{\\omega'-1} \\mathcal I_{1}.\n\\end{equation*}\n\nMoreover, choosing $r_{1}$ to be large enough so that \n\\begin{equation*}\n\\frac{K \\Lambda^{2}  M_{i}}{2 r_{1}} \\leq \\frac{1}{2},\n\\end{equation*}\nthe last term of \\eqref{eq:decay1:cptu:0:pf:2} can be absorbed in to the left-hand side and we conclude\n\\begin{equation*}\n\t\\sup_{\\set{r \\geq r_{1}}} \\abs{r \\phi} \\leq 2 r_{1} \\Lambda \\Psi + \\frac{2}{\\omega'-1}\\Lambda \\mathcal I_{1}.\n\\end{equation*}\n\nOn the other hand, in $\\set{r \\leq r_{1}}$ we have\n\\begin{equation*}\n\t\\sup_{\\set{r \\leq r_{1}}} \\abs{r \\phi} \\leq r_{1} \\Lambda \\Psi\n\\end{equation*}\nby \\eqref{eq:bnd4phi}. Combining the bounds in $\\{r\\geq r_1\\}$ and $\\{r\\leq r_1\\}$, the lemma follows. \\qedhere\n\\end{proof}\n\n\\begin{remark} \nThe preceding argument shows that Lemma \\ref{lem:decay1:cptu:0} holds with\\footnote{Notice that while the constant $C_{\\mathcal I_{1}, K, \\Lambda}$ depends on $\\mathcal I_1$, the preceding argument moreover allows us to choose $C_{\\mathcal I_{1}, K, \\Lambda}$ to be non-decreasing in $\\mathcal I_1$. In particular, \\emph{for $\\mathcal I_1$ sufficiently small}, we have $H_{1} \\leq C_{K, \\Lambda} \\, (\\mathcal I_{1} + \\Psi)$. It is for this reason that we prefer to write the expression $C_{\\mathcal I_{1}, K, \\Lambda} \\, (\\mathcal I_{1} + \\Psi)$ instead of the more general $C_{\\mathcal I_{1}, K, \\Lambda, \\Psi}$.}\n\\begin{equation} \\label{eq:decay1:H1}\n\tH_{1} \\leq C_{\\mathcal I_{1}, K, \\Lambda} \\, (\\mathcal I_{1} + \\Psi).\n\\end{equation}\n\\end{remark}\n\n\n\\subsection{Propagation of $u$-decay for $\\partial_{u} (r \\phi)$}\nHere, we show that $u$-decay estimates proved for $\\partial_{v} (r \\phi)$ and $\\phi$ may be `transferred' to $\\partial_{u} (r \\phi)$; this reduces the proof of Theorem \\ref{main.thm.1} to showing only \\eqref{eq:decay1:1} and \\eqref{eq:decay1:2}. To this end, we integrate $\\partial_{v} \\partial_{u} (r \\phi)$ from the axis $\\Gamma$, along which $\\partial_{u} (r \\phi) = - \\partial_{v} (r \\phi)$. \n\n\\begin{lemma} \\label{lem:decay1:uDecay4durphi}\nSuppose that there exists a finite positive constant $A$ such that \n\\begin{equation*}\n\t\\sup_{\\calQ} \\abs{\\phi} \\leq A u^{-\\omega}, \\qquad \n\t\\sup_{\\calQ} \\abs{\\partial_{v} (r \\phi)} \\leq A u^{-\\omega}.\n\\end{equation*}\n\nThen the following estimate holds.\n\\begin{equation*}\n\t\\sup_{\\calQ} \\abs{\\partial_{u} (r \\phi)} \\leq (1+K)A u^{-\\omega}.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nFix $u \\geq 1$ and $v \\geq u$. Integrate \\eqref{eq:SSESF:dphi''} along the outgoing direction from $(u, u)$ to $(u, v)$ and take the absolute value. Using $(7)$ of Definition \\ref{def:BVsolution} (BV solutions to \\eqref{eq:SSESF}), \\eqref{eq:SSESF:dr} (in particular, $\\partial_{v} \\nu\\leq 0$), \\eqref{eq:bnd4dur} and the hypotheses, we have\n\\begin{align*}\n\t\\abs{\\partial_{u}(r\\phi)(u,v)} \n\t\\leq & \\lim_{v' \\to u+} \\abs{\\partial_{v} (r \\phi)(u, v')} + \\sup_{u \\leq v' \\leq v} \\abs{\\phi(u,v')} \\int_{u}^{v} (-\\partial_{v} \\nu) \\, \\mathrm{d} v' \\\\\n\t\\leq & A u^{-\\omega} + K A u^{-\\omega}. \\qedhere\n\\end{align*}\n\\end{proof}\n\n\\subsection{Full decay for $\\phi$ and $\\partial_{v} (r \\phi)$}\\label{sec.full.decay.1}\nIn this subsection, we finish the proof of Theorem \\ref{main.thm.1}. By Lemma \\ref{lem:decay1:uDecay4durphi}, it suffices to establish the full decay of $\\phi$ and $\\partial_{v} ( r \\phi)$, i.e., \\eqref{eq:decay1:1} and \\eqref{eq:decay1:2}. For the convenience of the reader, we recall these estimates below:\n\\begin{align*}\n\t\t\\abs{\\phi} \\leq & A \\min \\set{u^{-\\omega}, r^{-1} u^{-(\\omega-1)}}, \\tag{\\ref{eq:decay1:1}} \\\\\n\t\t\\abs{\\partial_{v} (r \\phi)} \\leq & A \\min \\set{u^{-\\omega}, r^{-\\omega}}. \\tag{\\ref{eq:decay1:2}}\n\\end{align*}\n\nFor $U > 1$, let\n\\begin{equation*}\n\\mathcal B_{1}(U) := \\sup_{u \\in [1, U]} \\sup_{C_{u}} \\Big( u^{\\omega} \\abs{\\phi} + r u^{\\omega-1} \\abs{\\phi}  \\Big).\n\\end{equation*}\n\nNotice that this is finite for every fixed $U$ by Lemma \\ref{lem:decay1:cptu:0}. To establish the decay estimate \\eqref{eq:decay1:1}, it suffices to prove that $\\mathcal B_{1}(U)$ is bounded by a finite constant which is \\emph{independent of $U$}. We will show that this implies also \\eqref{eq:decay1:2}. Divide $\\calQ$ into $\\PD_{\\mathrm{ext}} \\cup \\PD_{\\mathrm{int}}$, defined by\n\\begin{equation*}\n\t\\PD_{\\mathrm{ext}} := \\set{(u,v) \\in \\calQ : v \\geq 3u}, \\quad \\PD_{\\mathrm{int}} := \\set{(u,v) \\in \\calQ : v \\leq 3u}.\n\\end{equation*}\n\nWe first establish a bound for $\\partial_{v} (r \\phi)$ with the sharp $r$-weight, which thus gives the sharp decay rate in $\\PD_{\\mathrm{ext}}$. \n\\begin{lemma} \\label{lem:decay1:extr}\nLet $u_{1} > 1$. Then for $u_{1}\\leq u\\leq U$, the following estimate holds.\n\\begin{equation} \\label{eq:decay1:extr}\n\t\\sup_{C_{u}} r^{\\omega} \\abs{\\partial_{v} (r \\phi)} \\leq \\mathcal I_{1} +   C_{K, M_{i}} \\, u_{1} H_{1} + C u_{1}^{-1} K M_{i} \\, \\mathcal B_{1}(U).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof} \nWe separate the proof into cases $\\omega \\geq 2 $ and $1<\\omega\\leq 2$.\n\n\\noindent{\\bf Case 1: $\\omega\\geq 2$}\n\nFirst, notice that\n$$|\\phi|\\leq \\mathcal B_1(U) (r^{-1}u^{-(\\omega-1)})^{\\omega-2} (u^{-\\omega})^{1-(\\omega-2)}\\leq \\mathcal B_1(U) r^{-(\\omega-2)}u^{-2}.$$\nApplying Lemma \\ref{lem:decay1:cptu:0}, we also have\n$$|\\phi|\\leq (1+r)^{-1}H_1.$$\nBy Corollary \\ref{cor:mntn4Bondi} and Proposition \\ref{prop:geomLocBVScat}, we have the following pointwise bounds:\n\\begin{equation*}\n\\sup_{u'\\in [1,u_1]} |\\frac{m\\lmb\\nu}{1-\\mu}|\\leq \\frac{K M_i}{2}\\, , \\quad\n\\sup_{u'\\in [u_{1},\\infty)} |\\frac{m\\lmb\\nu}{1-\\mu}|\\leq \\frac{K M(u_{1})}{2}\\, .\n\\end{equation*}\nTherefore, integrating \\eqref{eq:SSESF:dphi} along the incoming direction from $(1, v)$ to $(u, v)$, we have\n\\begin{align*}\n\t&\\abs{\\partial_{v} (r \\phi)(u,v)}\\\\\n\t& \\quad \\leq \\abs{\\partial_{v} (r \\phi)(1, v)} + \\abs{\\int_{1}^{u} \\frac{2m \\lmb \\nu\\phi}{(1-\\mu) r^{2}}  (u', v) \\, \\mathrm{d} u'} \\\\\n\t& \\quad \\leq \\abs{\\partial_{v} (r \\phi)(1, v)} + \\frac{K M_{i}}{r^2(u,v)(1+r(u,v))} H_{1} \\int_{1}^{u_{1}}  \\, \\mathrm{d} u' + \\frac{K M(u_1)}{r^{\\omega}(u,v)} \\mathcal B_{1}(U) \\int_{u_{1}}^{u} (u')^{-2} \\, \\mathrm{d} u' \\\\\n\t& \\quad \\leq \\abs{\\partial_{v} (r \\phi)(1, v)} + \\frac{u_{1} K M_{i}}{r^2(u,v)(1+r(u,v))} H_{1} + \\frac{K M(u_{1})}{u_{1} r^{\\omega}(u,v)} \\mathcal B_{1}(U).\n\\end{align*}\n\nMultiplying both sides by $r^{\\omega}(u,v)$ and using the fact that $r(u,v) \\leq r(1, v)$, we conclude\n\\begin{align*}\n\tr^{\\omega} \\abs{\\partial_{v} (r \\phi)}(u,v) \n\t\\leq & r^{\\omega}\\abs{\\partial_{v} (r \\phi)}(1,v) + u_{1} \\frac{r^{\\omega-2}}{(1+r)} K M_{i} \\, H_{1}+ u_{1}^{-1} K M(u_{1}) \\, \\mathcal B_{1}(U) \\\\\n\t\\leq & \\mathcal I_{1}+ C_{u_{1}, K, M_{i}}  H_{1} + u_{1}^{-1} K M_{i} \\, \\mathcal B_{1}(U).\n\\end{align*}\n\n\\noindent{\\bf Case 2: $1<\\omega\\leq 2$}\n\nWe will use the following bounds for $\\phi$. First, \n$$|\\phi|\\leq \\mathcal B_1(U) (r^{-1}u^{-(\\omega-1)})^{\\omega-1} (u^{-\\omega})^{(2-\\omega)}\\leq \\mathcal B_1(U) r^{-(\\omega-1)}u^{-1}.$$\n\nAlso, Lemma \\ref{lem:decay1:cptu:0} implies\n$$|\\phi|\\leq (1+r)^{-1}H_1.$$\n\nAs in Case 1 we integrate \\eqref{eq:SSESF:dphi} along the incoming direction from $(1, v)$ to $(u, v)$:\n\\begin{align*}\n\t&\\abs{\\partial_{v} (r \\phi)(u,v)}\\\\\n\t& \\quad \\leq \\abs{\\partial_{v} (r \\phi)(1, v)} + \\abs{\\int_{1}^{u} \\frac{2m \\lmb \\nu\\phi}{(1-\\mu) r^{2}}  (u', v) \\, \\mathrm{d} u'} \\\\\n\t& \\quad \\leq \\abs{\\partial_{v} (r \\phi)(1, v)} + \\frac{K \\Lambda M_{i} H_{1}}{(1+r)} \\int_{1}^{u_{1}} \\frac{-\\nu}{r^2}  \\, \\mathrm{d} u' \n\t\t\t\t\t\t\t\t+ \\frac{K \\Lambda M(u_1)}{u_1} \\mathcal B_{1}(U) \\int_{u_{1}}^{u}  \\frac{-\\nu}{r^{\\omega+1}} \\, \\mathrm{d} u' \\\\\n\t& \\quad \\leq \\abs{\\partial_{v} (r \\phi)(1, v)} + \\frac{\\omega K \\Lambda M_{i}}{r(u,v)(1+r(u,v))} H_{1} + \\frac{\\omega K \\Lambda M(u_{1})}{u_{1} r^{\\omega}(u,v)} \\mathcal B_{1}(U).\n\\end{align*}\n\nMultiply both sides by $r^{\\omega}$ to arrive at the conclusion as in Case 1. In this case, note that the second term is a bit better than what is claimed, as there is no dependence on $u_{1} \\geq 1$. \\qedhere\n\\end{proof}\n\n\\begin{remark} \nNote that the proof of this lemma limits $\\omega$ to be $\\leq 3$. More precisely, this limitation comes from the contribution of the right-hand side of \\eqref{eq:SSESF:dphi}\n\\end{remark}\n\n\nWe are now ready to prove bounds \\eqref{eq:decay1:1} and \\eqref{eq:decay1:2}. The idea is  to `propagate' the exterior decay estimate \\eqref{eq:decay1:extr} into $\\PD_{\\mathrm{int}}$ to obtain decay in $u$, using the smallness coming from Lemma \\ref{lem:smallPtnl} in the region where $u$ is sufficiently large. On the other hand, the preliminary $r$-decay estimates proved in \\S \\ref{subsec:decay1:rDecay} will give the desired $r$-decay rates in rest of the space-time.\n\n\\begin{proof}[Proof of \\eqref{eq:decay1:1} and \\eqref{eq:decay1:2}] \nLet $1 \\leq u_{1} \\leq U$. For $(u, v) \\in \\calQ$ with $u \\in [3 u_{1}, U]$, integrate \\eqref{eq:SSESF:dphi} along the incoming direction from $(u\/3, v)$ to $(u, v)$. Then\n\\begin{equation} \\label{eq:decay1:intr:pf:1}\n\\begin{aligned}\n\t\\abs{\\partial_{v} (r \\phi) (u,v)} \\leq \n\t& \\abs{\\partial_{v} (r \\phi) (u\/3,v)} \\\\\n\t& + \\frac{1}{2} (\\sup_{u' \\in [u\/3, u]} \\sup_{C_{u'}} \\abs{\\phi}) \\int_{u\/3}^{u} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u'.\n\\end{aligned}\n\\end{equation} \n\nMultiply both sides by $u^{\\omega}$ and estimate each term on the right-hand side.  For the first term, the key observation is the following: For $v \\geq u$, the point $(u\/3, v)$ lies in $\\PD_{\\mathrm{ext}}$, where \\eqref{eq:decay1:extr} is effective. Indeed, note that\n\\begin{equation*}\n\t(2\/3\\Lambda)  u \\leq \\Lambda^{-1} ( v- (u\/3) ) \\leq r(u\/3, v).\n\\end{equation*}\n\nThus, by \\eqref{eq:decay1:extr},\n\\begin{align*}\n\tu^{\\omega} \\abs{\\partial_{v} (r \\phi) (u\/3,v)} \n\t\\leq & (3 \\Lambda \/2)^{\\omega} \\Big( r^{\\omega}(u\/3, v) \\abs{\\partial_{v} (r \\phi) (u\/3,v)} \\Big) \\\\\n\t\\leq & (3 \\Lambda\/2)^{\\omega} \\Big( \\mathcal I_{1} +   C_{u_{1}, K, M_{i}}  H_{1} + C u_{1}^{-1} K M_{i} \\, \\mathcal B_{1}(U) \\Big) \\\\\n\t\\leq & C_{u_{1}, K, \\Lambda, M_{i}} (\\mathcal I_{1} + H_{1}) + C_{K, \\Lambda} M_{i} u_{1}^{-1} \\, \\mathcal B_{1}(U).\n\\end{align*}\n\nFor the second term on the right-hand side of \\eqref{eq:decay1:intr:pf:1}, we have\n\\begin{align*}\n\\frac{u^{\\omega}}{2} (\\sup_{u' \\in [u\/3, u]} \\sup_{C_{u'}} \\abs{\\phi}) \\int_{u\/3}^{u} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u'\n\\leq & \\frac{3^{\\omega}}{2} \\Big( \\int_{u\/3}^{u} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u' \\Big) \\mathcal B_{1}(U).\n\\end{align*}\n\nCombining these estimates, we deduce\n\\begin{equation} \\label{eq:decay1:intr:pf:2}\n\\begin{aligned}\n\t\\sup_{C_{u}} u^{\\omega} \\abs{\\partial_{v}(r \\phi)(u, v)} \\leq & C_{u_{1}, K, \\Lambda, M_{i}} (\\mathcal I_{1} + H_{1}) \\\\ \n\t&+ \\Big( C_{K, \\Lambda} M_{i} u_{1}^{-1} + C \\int_{u\/3}^{u}  \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u' \\Big) \\, \\mathcal B_{1}(U).\n\\end{aligned}\n\\end{equation}\n\nRecalling the bounds of $\\phi$ in terms of $\\partial_v(r\\phi)$ in Lemmas \\ref{lem:intEst4phi}, we have\n\\begin{align*}\n\t\\mathcal B_{1}(U) \n\t\\leq & (1+2\\Lambda) \\sup_{u \\in [1, U]} \\sup_{C_{u}} \\Big( u^{\\omega} \\abs{\\partial_{v}(r \\phi)} + r^{\\omega} \\abs{\\partial_{v} (r \\phi)} \\Big).\n\\end{align*}\nThe right-hand side can be controlled by \\eqref{eq:decay1:intr:pf:2} and \\eqref{eq:decay1:extr}, from which we conclude\n\\begin{equation} \\label{eq:decay1:intr:pf:key}\n\t\\mathcal B_{1}(U) \\leq C_{u_{1}, K, \\Lambda, M_{i}} (\\mathcal I_{1} + H_{1}) + \\Big( C_{K, \\Lambda} M_{i} u_{1}^{-1} + C \\int_{u\/3}^{u}  \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u'\\Big) \\mathcal B_{1}(U).\n\\end{equation}\n\nAs a consequence of Lemma \\ref{lem:smallPtnl}, the entire coefficient in front of $\\mathcal B_{1}(U)$ can made to be smaller than (say) $1\/2$ by taking $u_{1}$ sufficiently large. Since $\\mathcal B_{1}(U) < \\infty$, we can then absorb this term into the left-hand side. Observing that this bound is independent of $U > 1$, we have thus obtained \\eqref{eq:decay1:1}.\n\nTo prove \\eqref{eq:decay1:2}, simply apply \\eqref{eq:decay1:intr:pf:2} and \\eqref{eq:decay1:extr}, which shows that\n\\begin{align*}\n&\\sup_{u \\in [1, U]} \\sup_{C_{u}} \\Big( u^{\\omega} \\abs{\\partial_{v}(r \\phi)} + r^{\\omega} \\abs{\\partial_{v} (r \\phi)} \\Big)\\\\\n& \\quad \\leq  C_{u_{1}, K, \\Lambda, M_{i}} (\\mathcal I_{1} + H_{1}) + \\Big( C_{K, \\Lambda} M_{i} u_{1}^{-1} + C \\int_{u\/3}^{u}  \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u'\\Big) \\mathcal B_{1}(U).\n\\end{align*}\nThis boundedness of $\\mathcal B_1(U)$ that we just proved thus implies \\eqref{eq:decay1:2}.\n\\qedhere\n\\end{proof}\n\n\\begin{remark} \nAccording to the proof that we have just given, the constant $A_{1} > 0$ depends on our choice of $u_{1} > 1$, which in turn depends on how fast the coefficient in front of $\\mathcal B_{1}(U)$ in \\eqref{eq:decay1:intr:pf:key} vanishes as $u_{1} \\to \\infty$. This explains why $A_{1} > 0$ does not depend only on the size of the initial data, as remarked in Section \\ref{sec.main.thm}. Controlling the size of $u_{1} > 1$ under an additional small data assumption will be key to proving Statement (1) of Theorem \\ref{thm:smallData} in Section \\ref{sec:smallData}.\n\\end{remark}\n\n\\subsection{Additional decay estimates}\nWe end this section with the following decay estimates for $\\partial_{v} \\phi$, $\\partial_{u} \\phi$ and $m$.\n\\begin{corollary} \\label{cor:decay1}\nLet $(\\phi, r, m)$ be a locally BV scattering solution to \\eqref{eq:SSESF} with asymptotically flat initial data of order $\\omega'$ in BV, and define $\\omega = \\min \\set{\\omega', 3}$.\nLet $A_{1}$ be the constant in Theorem \\ref{main.thm.1}. Then the following decay estimates hold.\n\\begin{align} \n\t\\abs{\\partial_{v} \\phi} \\leq & C A_{1} \\min \\set{r^{-1} u^{-\\omega}, r^{-2} u^{-(\\omega-1)}}, \\label{eq:decay1:4} \\\\\n\t\\abs{\\partial_{u} \\phi} \\leq & C_{K} A_{1} \\, r^{-1} u^{-\\omega}, \\label{eq:decay1:5} \\\\\n\tm \\leq & C_{\\Lambda} A_{1}^{2} \\min \\set{r u^{-2\\omega}, u^{-(2\\omega-1)}}.  \\label{eq:decay1:6}\n\\end{align}\n\\end{corollary}\n\\begin{proof} \n\tLet $u \\geq 1$ and $v \\geq u$. Since\n\t\\begin{equation*}\n\t\tr \\partial_{v} \\phi = \\partial_{v} ( r \\phi) - \\lmb \\phi, \\qquad\n\t\tr \\partial_{u} \\phi = \\partial_{u} ( r \\phi) - \\nu \\phi,\n\t\\end{equation*}\n\tthe estimates \\eqref{eq:decay1:4}, \\eqref{eq:decay1:5} follow from \\eqref{eq:decay1:1}--\\eqref{eq:decay1:3} and the fact that $\\sup_{\\calQ} \\abs{\\lmb} \\leq 1\/2$, $\\sup_{\\calQ} \\abs{\\nu} \\leq K$. \n\t\n\tOn the other hand, by \\eqref{eq:SSESF:dm}, we have \n\t\\begin{equation} \\label{eq:decay1:6:pf:1}\n\t\tm(u,v) = \\frac{1}{2} \\int_{u}^{v} \\lmb^{-1} (1-\\mu) r^{2} (\\partial_{v} \\phi)^{2} (u, v')\\, \\mathrm{d} v'.\n\t\\end{equation}\n\t\n\tUsing $\\abs{\\partial_{v} \\phi (u,v) } \\leq C A_{1} r^{-1} u^{-\\omega}$ (which has just been established), we obtain\n\t\\begin{equation*}\n\t\tm(u,v) \\leq C_{\\Lambda} A_{1}^{2} \\, r u^{-2\\omega},\n\t\\end{equation*}\n\twhich proves a `half' of \\eqref{eq:decay1:6}. \n\tTo prove the other `half', let us introduce a parameter $r_{1} > 0$ (to be determined later) and define $v_{1}^{\\star}(u)$ to be the unique $v$-value such that $r(u, v^{\\star}_{1}(u)) = r_{1}$. For $v \\geq v^{\\star}_{1}(u)$, divide the $v'$-integral in \\eqref{eq:decay1:6:pf:1} into $\\int_{u}^{v^{\\star}_{1}(u)} + \\int_{v^{\\star}_{1}(u)}^{v}$ and use $\\abs{\\partial_{v} \\phi (u,v) } \\leq C A_{1} \\, r^{-1} u^{-\\omega}$ for the former and $\\abs{\\partial_{v} \\phi (u,v) } \\leq C A_{1} \\, r^{-2} u^{-(\\omega-1)}$ for the latter. As $m(u,v)$ is non-decreasing in $v$, we then arrive at the estimate\n\t\\begin{equation*}\n\t\t\\sup_{C_{u}} m \\leq C_{\\Lambda} A_{1}^{2} \\, r_{1} u^{-2\\omega} + C_{\\Lambda} A_{1}^{2} \\, r_{1}^{-1} u^{-2(\\omega-1)}.\n\t\\end{equation*}\n\t\nChoosing $r_{1} = u$, we obtain \\eqref{eq:decay1:6}. \\qedhere\n\\end{proof}\n\n\\section{Decay of second derivatives}\\label{sec.decay2}\nIn this section, we establish our second main theorem (Theorem \\ref{main.thm.2}). Throughout the section, we assume that $(\\phi, r, m)$ is a locally BV scattering solution to \\eqref{eq:SSESF} with asymptotically flat initial data of order $\\omega'$ in $C^{1}$, as in Definitions \\ref{def:locBVScat} and \\ref{def:AF}. As discussed in Remark \\ref{rem:wp}, $(\\phi, r, m)$ is then a $C^{1}$ solution to \\eqref{eq:SSESF}. As before, let $\\omega = \\min\\set{\\omega', 3}$. \n\n\\subsection{Preparatory lemmas}\nThe following lemma, along with Lemma \\ref{lem:smallPtnl}, provides the crucial smallness for our proof of Theorem \\ref{main.thm.2}.\n\\begin{lemma} \\label{lem:smallDphi}\nFor every $\\epsilon > 0$, there exists $u_{2} > 1$ such that\n\\begin{align}\n\t\\sup_{v \\in [u_{2}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}}} \\abs{\\partial_{u} \\phi}  < \\epsilon, \\label{eq:smallDphi:u} \\\\ \n\t\\sup_{u \\in [u_{2}, \\infty)} \\int_{C_{u}} \\abs{\\partial_{v} \\phi} < \\epsilon. \\label{eq:smallDphi:v}\n\\end{align}\n\\end{lemma}\n\\begin{proof} \nWe will only prove \\eqref{eq:smallDphi:u}, leaving the similar proof of \\eqref{eq:smallDphi:v} to the reader. As in the proof of Lemma \\ref{lem:smallPtnl}, we divide $\\calQ$ into $\\PD_{\\mathrm{cpt}}:=\\calQ\\cap\\{r\\leq R\\}$ and $\\PD_{\\mathrm{cpt}}^{c} := \\calQ \\setminus \\PD_{\\mathrm{cpt}}$, and argue separately. First, by Theorem \\ref{thm:decayInCpt}, we have\n\\begin{equation*}\n\t\\sup_{v \\in [u_{2}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{u} \\phi}  < \\epsilon\/2, \n\\end{equation*}\nfor $u_{2}$ sufficiently large. Next, to derive \\eqref{eq:smallDphi:u} in $\\PD_{\\mathrm{cpt}}^{c}$, we define $u^{\\star}(v) := \\sup \\set{u \\in [u_{2}, v] : r(u,v) \\geq R}$, where we use the convention $u^{\\star}(v) = u_{2}$ when the set is empty. Then using Proposition \\ref{prop:geomLocBVScat} and Schwarz, we compute\n\\begin{align*}\n\t\\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}} \\cap \\PD_{\\mathrm{cpt}}^{c}} \\abs{\\partial_{u} \\phi} \n\t= & \\int_{u_{2}}^{u^{\\star}(v)} \\abs{\\partial_{u} \\phi(u',v)} \\, \\mathrm{d} u' \\\\\n\t\\leq & \\sqrt{\\frac{2 K \\Lambda}{R}} \\sqrt{\\int_{u_{2}}^{u^{\\star}(v)} \\frac{1}{2}(-\\nu)^{-1} (1-\\mu) r^{2} (\\partial_{u} \\phi)^{2} (u', v) \\, \\mathrm{d} u'} \\\\\n\t\\leq & \\sqrt{\\frac{2 K \\Lambda}{R} m(u_{2}, v)} \\leq \\sqrt{\\frac{2 K \\Lambda}{R} M(u_{2})}.\n\\end{align*}\n\nBy \\eqref{eq:zeroMf} (Vanishing final Bondi mass), $\\lim_{u_{2} \\to \\infty} M(u_{2}) = 0$. \\eqref{eq:smallDphi:u} thus follows. \\qedhere\n\\end{proof}\n\nThe next lemma allows us to estimate the first derivative of $\\phi$ at $(u,v)$ in terms of information on $C_{u} \\cap \\set{(u, v'): u \\leq v' \\leq v}$.\n\\begin{lemma} \\label{lem:dphi}\nFor every $(u,v) \\in \\calQ$, the following inequalities hold.\n\\begin{align*}\n\t& \\abs{\\partial_{v} \\phi(u,v)} \\leq \t\t\t\\frac{\\Lambda^{2}}{4} \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v}^{2}(r\\phi)(u, v')} \\\\\n\t& \\phantom{\\abs{\\partial_{v} \\phi(u,v)} \\leq} \t+ \\frac{\\Lambda^{3}}{4} \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v} (r \\phi)(u, v')} \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v} \\lmb (u, v')}, \\\\\n\t& \\abs{\\partial_{u} \\phi(u,v)} \\leq \\Lambda  \\sup_{u \\leq v' \\leq v} (- \\nu)(u, v') \\abs{\\partial_{v} \\phi(u,v')}.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof} \nThe first is an easy consequence of \\eqref{eq:est4dvphi:1} in \\S \\ref{subsec:est4phi}. To prove the second inequality, we start from the equation\n\\begin{equation*}\n\t\\partial_{v} (r \\partial_{u} \\phi) = - \\nu \\partial_{v} \\phi,\n\\end{equation*}\nwhich follows from \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dphi}. Therefore, we have\n\\begin{align*}\n\t\\abs{\\partial_{u} \\phi (u,v)} \n\t\\leq & \\frac{1}{r(u,v)} \\int_{u}^{v} (-\\nu) \\abs{\\partial_{v} \\phi} (u, v') \\, \\mathrm{d} v',\n\\end{align*}\nfrom which the second inequality easily follows. \\qedhere\n\\end{proof}\n\nIn the next lemma, we show that improved estimates for $m$ near $\\Gamma$ hold if we assume an $L^{\\infty}$ control of ${\\partial_{v} \\phi}$.\n\\begin{lemma} \\label{lem:muOverR}\nFor every $(u,v) \\in \\calQ$, the following inequalities hold:\n\t\\begin{align} \n\t\\frac{\\mu}{r}(u,v) \\leq & \\Lambda^{2} \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v} (r \\phi)(u, v')} \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v} \\phi (u, v')}, \\label{eq:muOverR:1} \\\\\n\t\\frac{\\mu}{r^{2}}(u,v) \\leq & \\frac{\\Lambda^{2}}{3} \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v} \\phi(u, v')}^{2}. \\label{eq:muOverR:2}\n\t\\end{align}\n\\end{lemma}\n\\begin{proof} \n\tRecall $\\mu = 2m\/r$. By \\eqref{eq:SSESF:dm}, we have\n\t\\begin{equation*}\n\t\t2 m(u,v) = \\int_{u}^{v} (1-\\mu) \\lmb^{-1} r^{2} (\\partial_{v} \\phi)^{2} (u, v') \\, \\mathrm{d} v'.\n\t\\end{equation*}\n\t\n\tPulling everything except $r^{2} \\lmb$ outside the integral and using $\\int_{u}^{v} r^{2} \\lmb(u, v') \\, \\mathrm{d} v' = (1\/3) r^{3}(u,v)$, we obtain \\eqref{eq:muOverR:2}. \n\tOn the other hand, using $\\lmb^{-1} r \\partial_{v} \\phi = \\lmb^{-1} \\partial_{v} (r \\phi) - \\phi$ and $\\int_{u}^{v} r \\lmb(u, v') \\, \\mathrm{d} v' = (1\/2) r^{2}(u,v)$, we easily deduce\n\\begin{equation*}\n\t\\frac{\\mu}{r}(u,v) \\leq \\frac{1}{2} \\sup_{u \\leq v' \\leq v} \\Big( \\Lambda^{2} \\abs{\\partial_{v} (r \\phi)(u, v')} + \\Lambda \\abs{\\phi(u, v')} \\Big) \\abs{\\partial_{v} \\phi(u, v')}.\n\\end{equation*}\n\t\n\tFrom the fact that $\\abs{\\phi(u,v)} \\leq \\Lambda \\sup_{u \\leq v' \\leq v} \\abs{\\partial_{v} (r \\phi)(u, v')}$, \\eqref{eq:muOverR:1} easily follows.\\qedhere\n\\end{proof}\n\n\\subsection{Preliminary $r$-decay for $\\partial_{v}^{2} (r \\phi)$ and $\\partial_{v} \\lmb$}\nIn this subsection, we establish decay estimates for $\\partial_{v}^{2} (r \\phi)$ and $\\partial_{v} \\lmb$ which are sharp in terms of $r$-weights in the region $\\PD_{\\mathrm{ext}}$. We remind the reader the decomposition $\\calQ = \\PD_{\\mathrm{ext}} \\cup \\PD_{\\mathrm{int}}$, where\n\\begin{equation*}\n\t\\PD_{\\mathrm{ext}} = \\set{(u,v) \\in \\calQ : v \\geq 3u}, \\quad \\PD_{\\mathrm{int}} = \\set{(u,v) \\in \\calQ : v \\leq 3u}.\n\\end{equation*}\n\nIn particular, note that $r \\geq 2 \\Lambda^{-1} u > 0$ in $\\PD_{\\mathrm{ext}}$.\n\n\\begin{lemma} \\label{lem:decay2:rDecay}\nThe following estimates hold.\n\\begin{align}\n\t \\sup_{\\PD_{\\mathrm{ext}}} r^{3} \\abs{\\partial_{v} \\lmb} \\leq & C_{K, \\Lambda} A_{1}^{2}, \t \\label{eq:decay2:rDecay:1} \\\\\n\t\\sup_{\\PD_{\\mathrm{ext}}} r^{\\omega+1} \\abs{\\partial_{v}^{2} (r \\phi)} \\leq & C \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3}. \\label{eq:decay2:rDecay:2}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof} \nWe begin by proving \\eqref{eq:decay2:rDecay:1}. Recall \\eqref{eq:eq4dvdvr:normal}:\n\\begin{equation*} \\tag{\\ref{eq:eq4dvdvr:normal}}\n\\partial_{u} \\partial_{v} \\log \\lmb\n= \\frac{1}{(1-\\mu)}  \\lmb^{-1} \\nu (\\partial_{v} \\phi)^{2} - \\frac{4 m}{(1-\\mu) r^{3}} \\lmb \\nu.\n\\end{equation*}\n\nNote that $\\partial_{v} \\log \\lmb= 0$ on $C_{1}$ by our choice of coordinates. Therefore, integrating the preceding equation along the incoming direction from $(1,v)$ to $(u,v)$, we have\n\\begin{equation*}\n\t\\abs{\\partial_{v} \\log \\lmb(u, v)} \\leq \\int_{1}^{u} \\abs{\\frac{1}{(1-\\mu)}  \\lmb^{-1} \\nu (\\partial_{v} \\phi)^{2} (u', v)} \\, \\mathrm{d} u' + \\int_{1}^{u} \\abs{\\frac{4 m}{(1-\\mu) r^{3}} \\lmb \\nu (u', v)} \\, \\mathrm{d} u'.\n\\end{equation*}\n\nThen \\eqref{eq:decay2:rDecay:1} follows using Proposition \\ref{prop:geomLocBVScat}, \\eqref{eq:decay1:4} and \\eqref{eq:decay1:6}. We remark that the power of $r$ is dictated by the second integral.\n\nThe proof of \\eqref{eq:decay2:rDecay:2} is very similar. We start by recalling \\eqref{eq:eq4dvdvrphi:normal}:\n\\begin{equation*} \\tag{\\ref{eq:eq4dvdvrphi:normal}}\n\\partial_{u} (\\partial_{v}^{2} (r \\phi)) = \n\\frac{2m \\lmb \\nu}{(1-\\mu) r^{2}} \\, \\partial_{v} \\phi + \\frac{ \\nu}{(1-\\mu) }  (\\partial_{v} \\phi)^{2} \\phi \n + \\frac{2m \\nu}{(1-\\mu) r^{2}} (\\partial_{v} \\lmb) \\phi - \\frac{4m}{(1-\\mu) r^{3}} \\lmb^{2} \\nu \\phi.\n\\end{equation*}\n\nFor $u \\geq 1$, we have $r(u,v) \\leq r(1,v)$; moreover, by hypothesis, we have the estimate for the initial data term \n$$(1+r(1,v))^{\\omega'+1} \\abs{\\partial_{v}^{2}(r \\phi)(1,v)} \\leq \\mathcal I_{2} \\, .$$\nTherefore, by the fundamental theorem of calculus, it suffices to bound\n\\begin{align*}\n& \\int_{1}^{u} \\abs{\\frac{2m \\lmb \\nu}{(1-\\mu) r^{2}} \\, \\partial_{v} \\phi (u', v)} \\, \\mathrm{d} u' + \\int_{1}^{u} \\abs{\\frac{ \\nu}{(1-\\mu) }  (\\partial_{v} \\phi)^{2} \\phi (u', v)} \\, \\mathrm{d} u' \\\\\n& \\quad + \\int_{1}^{u} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (\\partial_{v} \\lmb) \\phi (u', v) } \\, \\mathrm{d} u'  + \\int_{1}^{u} \\abs{\\frac{4m}{(1-\\mu) r^{3}} \\lmb^{2} \\nu \\phi (u', v)} \\, \\mathrm{d} u'\n\\end{align*}\nby $C_{K, \\Lambda, M_{i}} A_{1}^{3} r^{-(\\omega+1)}$. This is an easy consequence of Proposition \\ref{prop:geomLocBVScat}, \\eqref{eq:decay1:1}, \\eqref{eq:decay1:4}, \\eqref{eq:decay1:6} and also \\eqref{eq:decay2:rDecay:1} that has just been established. Note that the last term is what limits $\\omega \\leq 3$. \\qedhere\n\\end{proof}\n\n\\subsection{Propagation of $u$-decay for $\\partial_{u}^{2} (r \\phi)$ and $\\partial_{u} \\nu$}\nHere, we show that certain $u$-decay for $\\partial_{u}^{2} (r \\phi)$ and $\\partial_{u} \\nu$ proved in $\\PD_{\\mathrm{int}}$ can be propagated to $\\calQ$. The technique employed is very similar to that in the previous subsection.\n\\begin{lemma} \\label{lem:decay2:uDecayInExtr}\nFor $U \\geq 1$, suppose that there exists a finite positive constant $A, k_{1}, k_{2}$ such that\n\\begin{equation*}\n\t0 \\leq k_{1} \\leq 2\\omega+1, \\quad\n\t0 \\leq k_{2} \\leq 3\\omega+1, \n\\end{equation*}\nand for $u \\in [1, U]$, we have\n\\begin{equation*}\n\t\\sup_{C_{u} \\cap \\PD_{\\mathrm{int}}} u^{k_{1}} \\abs{\\partial_{u} \\nu} \\leq A, \\quad\n\t\\sup_{C_{u} \\cap \\PD_{\\mathrm{int}}} u^{k_{2}} \\abs{\\partial_{u}^{2}(r\\phi)} \\leq A.\n\\end{equation*}\n\nThen for $u \\in [1, U]$, the following estimates hold.\n\\begin{align}\n\t\\sup_{C_{u}} u^{k_{1}} \\abs{\\partial_{u} \\nu} \\leq & C_{K, \\Lambda} A + C_{K, \\Lambda} A_{1}^{2}, \\label{eq:decay2:uDecayInExtr:1} \\\\\n\t\\sup_{C_{u}} u^{k_{2}} \\abs{\\partial_{u}^{2}(r\\phi)} \\leq & A + C_{K, \\Lambda} A_{1}^{3} + C_{K, \\Lambda} A_{1}^{3} \\, \\sup_{C_{u}} u \\abs{\\partial_{u} \\nu} . \\label{eq:decay2:uDecayInExtr:2}\n\\end{align}\n\nFurthermore, the following alternative to \\eqref{eq:decay2:uDecayInExtr:2} also holds.\n\\begin{equation} \\label{eq:decay2:uDecayInExtr:3}\n\\sup_{C_{u}} u^{k_{2}} \\abs{\\partial_{u}^{2}(r\\phi)} \n\t\\leq A + C_{K, \\Lambda} A_{1}^{3} + \n\tC_{K, \\Lambda} \\Psi \\int_{3u}^{\\infty} \\abs{\\frac{2 m \\lmb}{(1-\\mu) r^{2}}}(u, v') \\, \\mathrm{d} v' \\cdot \\sup_{C_{u}} u^{k_{2}} \\abs{\\partial_{u} \\nu}. \n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nLet us begin with \\eqref{eq:decay2:uDecayInExtr:1}. Recall \\eqref{eq:eq4dudur:normal}\n\\begin{equation*} \\tag{\\ref{eq:eq4dudur:normal}}\n\\partial_{v} \\partial_{u} \\log \\nu\n= \\frac{1}{(1-\\mu)}  \\lmb \\nu^{-1} (\\partial_{u} \\phi)^{2} - \\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu.\n\\end{equation*}\n\nGiven $(u,v) \\in \\PD_{\\mathrm{ext}}$ (with $u \\in [1,U]$), let us integrate this equation along the outgoing direction from $(u, 3u)$ to $(u,v)$, take the absolute value and multiply by $u^{k_{1}}$. Using the hypothesis\n\\begin{equation*}\n\t\\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [1,U]}} u^{k_{1}} \\abs{\\partial_{u} \\nu} \\leq A,\n\\end{equation*}\n\\eqref{eq:decay2:uDecayInExtr:1} is reduced to showing\n\\begin{align} \n\tu^{k_{1}} \\int_{3u}^{\\infty} \\abs{\\frac{1}{(1-\\mu)} \\lmb \\nu^{-1} (\\partial_{u} \\phi)^{2} (u, v)} \\, \\mathrm{d} v \\leq & C_{K, \\Lambda} A_{1}^{2}, \\label{eq:decay2:uDecayInExtr:pf:1} \\\\ \n\tu^{k_{1}} \\int_{3u}^{\\infty} \\abs{\\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu (u, v)} \\, \\mathrm{d} v \\leq & C_{K, \\Lambda} A_{1}^{2}, \\label{eq:decay2:uDecayInExtr:pf:2}\n\\end{align}\nfor $u \\in [1, U]$.\n\nUsing Proposition \\ref{prop:geomLocBVScat} and \\eqref{eq:decay1:5}, the left-hand side of \\eqref{eq:decay2:uDecayInExtr:pf:1} is bounded by\n\\begin{equation*}\n\tC_{K, \\Lambda} A_{1}^{2} \\, u^{k_{1} - 2\\omega} \\int_{3u}^{\\infty} \\frac{1}{r^{2}} \\lmb \\, \\mathrm{d} v \n\t\\leq C_{K, \\Lambda} A_{1}^{2} \\, u^{k_{1}- 2\\omega} r^{-1}(u, 3u).\n\\end{equation*}\n\nAs $u \\geq 1$ and $r(u, 3u) \\geq 2 \\Lambda^{-1} u$, \\eqref{eq:decay2:uDecayInExtr:pf:1} follows. Similarly, by \\eqref{eq:bnd4mu} and \\eqref{eq:decay1:6}, the left-hand side of \\eqref{eq:decay2:uDecayInExtr:pf:2} is also bounded by  $C_{K, \\Lambda} A_{1}^{2} \\, u^{k_{1}- 2\\omega} r^{-1}(u, 3u)$, from which \\eqref{eq:decay2:uDecayInExtr:pf:2} immediately follows.\n\nNext, we turn to \\eqref{eq:decay2:uDecayInExtr:2} and \\eqref{eq:decay2:uDecayInExtr:3}; as they are proved similarly as before, we will only outline the main points. Recall \\eqref{eq:eq4dudurphi:normal}:\n\\begin{equation*} \\tag{\\ref{eq:eq4dudurphi:normal}}\n\\partial_{v} (\\partial_{u}^{2} (r \\phi)) = \n\\frac{2m \\lmb \\nu}{(1-\\mu) r^{2}} \\, \\partial_{u} \\phi + \\frac{\\lmb }{(1-\\mu) }  (\\partial_{u} \\phi)^{2} \\phi \n + \\frac{2m \\lmb}{(1-\\mu) r^{2}} (\\partial_{u} \\nu) \\phi - \\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu^{2} \\phi.\n\\end{equation*}\n\nFix $(u,v) \\in \\PD_{\\mathrm{ext}}$ with $u \\in [1,U]$. We then integrate the preceding equation along the outgoing direction from $(u, 3u)$ to $(u, v)$, take the absolute value and multiply by $u^{k_{2}}$. In order to prove \\eqref{eq:decay2:uDecayInExtr:2}, in view of the hypothesis\n\\begin{equation*}\n\t\\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u, v) \\in \\calQ : u \\in [1, U]}} u^{k_{2}} \\abs{\\partial_{u}^{2} (r \\phi)} \\leq A,\n\\end{equation*}\nit suffices to establish the following estimates for $u \\in [1, U]$:\n\\begin{align*}\n\tu^{k_{2}} \\int_{3u}^{\\infty} \\abs{\\frac{2m \\lmb \\nu}{(1-\\mu) r^{2}} \\, \\partial_{u} \\phi (u,v)}\\, \\mathrm{d} v \n\t\\leq & C_{K, \\Lambda} A_{1}^{3}, \\\\\n\tu^{k_{2}} \\int_{3u}^{\\infty} \\abs{\\frac{\\lmb }{(1-\\mu) }  (\\partial_{u} \\phi)^{2} \\phi  (u,v)}\\, \\mathrm{d} v \n\t\\leq & C_{K, \\Lambda} A_{1}^{3}, \\\\\n\tu^{k_{2}} \\int_{3u}^{\\infty} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}} (\\partial_{u} \\nu) \\phi  (u,v)}\\, \\mathrm{d} v \n\t\\leq & C_{K, \\Lambda} A_{1}^{3} \\, \\sup_{\\calQ} u \\abs{\\partial_{u} \\nu}, \\\\\n\tu^{k_{2}} \\int_{3u}^{\\infty} \\abs{\\frac{4m}{(1-\\mu) r^{3}} \\lmb \\nu^{2} \\phi (u,v)} \\, \\mathrm{d} v\n\t\\leq & C_{K, \\Lambda} A_{1}^{3}.\n\\end{align*}\n\nThe proof of these estimates are similar to that of \\eqref{eq:decay2:uDecayInExtr:pf:1}, \\eqref{eq:decay2:uDecayInExtr:pf:2}; we omit the details. To prove \\eqref{eq:decay2:uDecayInExtr:3}, we replace the third estimate by\n\\begin{equation*}\n\tu^{k_{2}} \\int_{3u}^{\\infty} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}} (\\partial_{u} \\nu) \\phi  (u,v)}\\, \\mathrm{d} v \n\t\\leq C_{K, \\Lambda} \\Psi \\int_{3u}^{\\infty} \\abs{\\frac{2 m \\lmb}{(1-\\mu) r^{2}} }(u, v') \\, \\mathrm{d} v' \\cdot \\sup_{C_{u}} u^{k_{2}} \\abs{\\partial_{u} \\nu},\n\\end{equation*}\nwhich is an easy consequence of Proposition \\ref{prop:geomLocBVScat}.\n\\qedhere\n\\end{proof}\n\n\n\\subsection{Full decay for $\\partial_{v}^{2} (r \\phi)$, $\\partial_{u}^{2} (r \\phi)$, $\\partial_{v} \\lmb$ and $\\partial_{u} \\nu$} \\label{sec.full.decay.2}\nWith all the preparations so far, we are finally ready to prove Theorem \\ref{main.thm.2}. Our proof consists of two steps.\nThe first step is use the local BV scattering assumption to prove a preliminary decay rate of $u^{-\\omega}$ for $\\partial_{v}^{2} (r \\phi)$, $\\partial_{u}^{2} (r \\phi)$, $\\partial_{v} \\lmb$ and $\\partial_{u} \\nu$. In this step, it is crucial to pass to the \\emph{renormalized variables} and exploit the null structure of \\eqref{eq:SSESF}, in order to utilize the a priori bounds in the local BV scattering assumption. The second step to upgrade these decay rates to those that are claimed in Theorem \\ref{main.thm.2}. Thanks to the preliminary $u^{-\\omega}$ decay from the first step, the null structure is not necessary at this point.\n\nWe now begin with the first step. The null structure of \\eqref{eq:SSESF} as demonstrated in \\S \\ref{subsec:nullStr} is used in an essential way.\n\n\\begin{proposition} \\label{prop:decay2:nullStr}\nThere exists a finite constant $A_{2}' > 0$ such that the following estimates hold.\n\\begin{align*}\n\t& \\abs{\\partial_{v}^{2} (r \\phi)} \\leq A_{2}' u^{-\\omega}, \\quad\n\t\\abs{\\partial_{u}^{2} (r \\phi)} \\leq A_{2}' u^{-\\omega},  \\\\\n\t& \\abs{\\partial_{v} \\lmb} \\leq A_{2}' u^{-\\omega},  \\qquad\n\t\\abs{\\partial_{u} \\nu} \\leq A_{2}' u^{-\\omega}.\n\\end{align*}\n\\end{proposition}\n\n\\begin{proof} \nFor $U > 1$, we define \n\\begin{equation} \\label{eq:decay2:def4B2}\n\t\\mathcal B_{2}(U) := \\sup_{u \\in [1, U]} \\sup_{C_{u}} \\Big(  u^{\\omega} \\abs{\\partial_{v}^{2} (r \\phi)} +u^{\\omega} \\abs{\\partial_{u}^{2} (r \\phi)} \n\t\t\t\t\t+ u^{\\omega} \\abs{\\partial_{v} \\lmb} + u^{\\omega} \\abs{\\partial_{u} \\nu} \\Big).\n\\end{equation}\n \n\n\nNotice that the above is finite for every fixed $U$ due to Lemmas \\ref{lem:decay2:rDecay} and \\ref{lem:decay2:uDecayInExtr}. As indicated earlier, we need to use the null structure of the system \\eqref{eq:SSESF} as in \\S \\ref{subsec:nullStr}. For convenience, we define the shorthands\n\\begin{align*}\n\tF_{1} := & \\partial_{v}^{2} (r \\phi) - (\\partial_{v} \\lmb) \\phi, \\\\\n\tG_{1} := & \\partial_{u}^{2} (r \\phi) - (\\partial_{u} \\nu) \\phi, \n\\end{align*}\nand\n\\begin{align*}\n\tF_{2} := & \\partial_{v} \\log \\lmb - \\frac{\\lmb}{(1-\\mu)} \\frac{\\mu}{r} + \\partial_{v} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} ( r \\phi) \\Big), \\\\\n\tG_{2} := & \\partial_{u} \\log (-\\nu) - \\frac{\\nu}{(1-\\mu)} \\frac{\\mu}{r} + \\partial_{u} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} (r \\phi) \\Big).\n\\end{align*}\n\nThen \\eqref{eq:eq4dvdvrphi}, \\eqref{eq:eq4dudurphi}, \\eqref{eq:eq4dvdvr} and \\eqref{eq:eq4dudur} may be rewritten in the following fashion.\n\\begin{align} \n& \\partial_{u} F_{1} = \\partial_{u} \\lmb \\, \\partial_{v} \\phi - \\partial_{v} \\lmb \\, \\partial_{u} \\phi, \\label{eq:decay2:nullStr:pf:1} \\\\\n& \\partial_{u} F_{2} = \\partial_{u} \\phi \\, \\partial_{v}\\Big( \\nu^{-1} \\partial_{u} (r \\phi) \\Big)- \\partial_{v} \\phi \\, \\partial_{u} \\Big( \\nu^{-1} \\partial_{u} (r \\phi) \\Big),  \\label{eq:decay2:nullStr:pf:2}  \\\\\n& \\partial_{v} G_{1} = \\partial_{v} \\nu \\, \\partial_{u} \\phi - \\partial_{u} \\nu \\, \\partial_{v} \\phi,\\label{eq:decay2:nullStr:pf:3}  \\\\\n& \\partial_{v} G_{2} = - \\partial_{u} \\phi \\, \\partial_{v} \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) \\Big) + \\partial_{v} \\phi \\, \\partial_{u} \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) \\Big). \\label{eq:decay2:nullStr:pf:4}\n\\end{align}\n\nThe following lemma is the key technical component of the proof.\n\n\\begin{lemma} \\label{lem:decay2:key4nullStr}\nThere exists a finite positive constant $C = C_{A_{1}, \\mathcal I_{2}, K, \\Lambda}$ and positive function $\\epsilon(u)$ satisfying\n\\begin{equation*}\n\t\\epsilon(u) \\to 0 \\hbox{ as } u \\to \\infty\n\\end{equation*}\nsuch that the following inequalities holds for $1 \\leq u_{2} \\leq U$:\n\\begin{align}\n\t\\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [3 u_{2}, U]}} \\Big( u^{\\omega} \\abs{F_{1}} + u^{\\omega} \\abs{G_{1}} \\Big)\n\t \\leq & C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + \\epsilon(u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:1} \\\\\n\t \\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [3 u_{2}, U]}} \\Big( u^{\\omega} \\abs{F_{2}} + u^{\\omega} \\abs{G_{2}} \\Big)\n\t  \\leq & C_{K, \\Lambda} A_{1}^{2} + \\epsilon(u_{2}) \\mathcal B_{2}(U). \\label{eq:decay2:key4nullStr:2} \n\\end{align}\n\\end{lemma}\n\nWe defer the proof of this lemma until later. Instead, we first finish the proof of Proposition \\ref{prop:decay2:nullStr}, assuming Lemma \\ref{lem:decay2:key4nullStr}.\n\n\\noindent\\emph{Proof of Proposition \\ref{prop:decay2:nullStr}.}\nFirst, we claim that \\eqref{eq:decay2:key4nullStr:1} and \\eqref{eq:decay2:key4nullStr:2} imply\n\\begin{equation} \\label{eq:decay2:nullStr:pf:5}\n\t\\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [3 u_{2}, U]}} u^{\\omega} \\Big( \\abs{\\partial_{v}^{2} (r \\phi)} + \\abs{\\partial_{u}^{2} (r \\phi)} + \\abs{\\partial_{v} \\lmb} + \\abs{\\partial_{u} \\nu} \\Big)\n\t\\leq H_{2} + (\\epsilon + \\epsilon')(u_{2}) \\mathcal B_{2}(U).\n\\end{equation}\nfor some constant $0 < H_{2} < \\infty$ and some positive function $\\epsilon'(u_{2})$ which tends to zero as $u_{2} \\to \\infty$.\n\nThe point is that $F_{1}, F_{2}, G_{1}, G_{2}$ controls $\\partial_{v}^{2} (r \\phi)$, $\\partial_{u}^{2} (r \\phi)$, $\\partial_{v} \\lmb$, $\\partial_{u} \\nu$, respectively, up to higher order terms, which may be absorbed into the second term on the right-hand side. Indeed, consider $u \\in [3 u_{2}, U]$. For $F_{1}$ and $G_{1}$, we estimate\n\\begin{align*}\n\t& u^{\\omega} \\abs{\\partial_{v}^{2} (r \\phi)(u,v)} = u^{\\omega} \\abs{F_{1} +  (\\partial_{v} \\lmb) \\phi}(u,v) \\leq u^{\\omega} \\abs{F_{1}(u,v)} + \\sup_{C_{u}} \\abs{\\phi} \\cdot \\mathcal B_{2}(U), \\\\\n\t& u^{\\omega} \\abs{\\partial_{u}^{2} (r \\phi)(u,v)} = u^{\\omega} \\abs{G_{1} + (\\partial_{u} \\nu) \\phi}(u,v) \\leq u^{\\omega} \\abs{G_{1}(u,v)} + \\sup_{C_{u}} \\abs{\\phi} \\cdot \\mathcal B_{2}(U),\n\\end{align*}\nwhich are acceptable, as $\\sup_{C_{u}} \\abs{\\phi} \\to 0$ as $u \\geq 3 u_{2} \\to \\infty$ by Theorem \\ref{main.thm.1}. \nFor $F_{2}$, we use Proposition \\ref{prop:geomLocBVScat} to estimate\n\\begin{align*}\n\tu^{\\omega} \\abs{\\partial_{v} \\lmb} = & u^{\\omega} \\lmb \\Big\\vert F_{2} + \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} + \\partial_{v} \\phi ( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} (r \\phi) ) \\Big\\vert \\\\\n\t\\leq &\\frac{1}{2} u^{\\omega} \\abs{F_{2}} + \\frac{K \\Lambda}{4} u^{\\omega} \\abs{\\frac{\\mu}{r}} + \\frac{\\Lambda}{2} u^{\\omega} \\abs{\\partial_{v} \\phi} \\Big( \\abs{\\partial_{v} (r \\phi)} + \\abs{\\partial_{u} (r \\phi)} \\Big).\n\\end{align*}\n\nApplying \\eqref{eq:muOverR:1} (from Lemma \\ref{lem:muOverR}) to the second term on the last line, and then using Lemma \\ref{lem:dphi} to control $u^{\\omega} \\abs{\\partial_{v} \\phi}$, we arrive at \n\\begin{equation*}\nu^{\\omega} \\abs{\\partial_{v} \\lmb(u,v)} \\leq \\frac{1}{2} u^{\\omega} \\abs{F_{2}(u,v)} + C_{K, \\Lambda} \\, \\Psi \\sup_{C_{u}} \\Big( \\abs{\\partial_{v} (r \\phi)} + \\abs{\\partial_{u} (r \\phi)} \\Big) \\cdot \\mathcal B_{2}(U),\n\\end{equation*}\nwhich is acceptable in view of Theorem \\ref{main.thm.1}. Proceeding similarly, but also using the second inequality of Lemma \\ref{lem:dphi} to control $\\abs{\\partial_{u} \\phi}$, we obtain\n\\begin{equation*}\nu^{\\omega} \\abs{\\partial_{u} \\nu(u,v)} \\leq K u^{\\omega} \\abs{G_{2}(u,v)} + C_{K, \\Lambda} \\, \\Psi \\sup_{C_{u}} \\Big( \\abs{\\partial_{v} (r \\phi)} + \\abs{\\partial_{u} (r \\phi)} \\Big) \\cdot \\mathcal B_{2}(U).\n\\end{equation*}\n\nCombining these estimates with \\eqref{eq:decay2:key4nullStr:1} and \\eqref{eq:decay2:key4nullStr:2}, we conclude \\eqref{eq:decay2:nullStr:pf:5} with\n\\begin{align} \n\tH_{2} =& C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + C_{K, \\Lambda} A_{1}^{2}, \n\t\\label{eq:decay2:H2} \\\\\n\t\\epsilon'(u_{2}) =& C \\sup_{u \\geq 3u_{2}} \\abs{\\phi} + C_{K, \\Lambda} \\Psi \\sup_{u \\geq 3u_{2}} \\Big( \\abs{\\partial_{v}(r \\phi)} + \\abs{\\partial_{u}(r \\phi)} \\Big).\n\t\\label{eq:decay2:eps'}\n\\end{align}\n\nNext, note that the (non-decreasing) function\n\\begin{equation} \\label{eq:decay2:def4H'2}\n\tH'_{2}(u_{2}) := \\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [1, 3u_{2}]}} u^{\\omega} \\Big( \\abs{\\partial_{v}^{2} (r \\phi)} + \\abs{\\partial_{u}^{2} (r \\phi)} + \\abs{\\partial_{v} \\lmb} + \\abs{\\partial_{u} \\nu} \\Big) \\geq 0\n\\end{equation}\nis always \\emph{finite} for any fixed $u_{2} \\geq 1$, as the region $\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [1, 3u_{2}]}$ is compact and each of these terms is a continuous function, since $(\\phi, r, m)$ is a $C^{1}$ solution (see Definition \\ref{def:C1solution}). Combining with \\eqref{eq:decay2:nullStr:pf:5}, we obtain\n\\begin{equation*} \n\t\\sup_{\\PD_{\\mathrm{int}} \\cap \\set{(u,v) \\in \\calQ : u \\in [1, U]}} u^{\\omega} \\Big( \\abs{\\partial_{v}^{2} (r \\phi)} + \\abs{\\partial_{u}^{2} (r \\phi)} + \\abs{\\partial_{v} \\lmb} + \\abs{\\partial_{u} \\nu} \\Big)\n\t\\leq H_{2} + H'_{2}(u_{2}) + (\\epsilon + \\epsilon')(u_{2}) \\mathcal B_{2}(U),\n\\end{equation*}\nfor every $u_2\\in [1,U]$.\n\nNow apply \\eqref{eq:decay2:uDecayInExtr:1}, \\eqref{eq:decay2:uDecayInExtr:3} in Lemma \\ref{lem:decay2:uDecayInExtr} to $\\partial_{u}^{2}(r \\phi)$, $\\partial_{u} \\nu$. Apply also Lemma \\ref{lem:decay2:rDecay} (along with the fact that $r(u,v) \\geq 2 \\Lambda^{-1} u$ in $\\PD_{\\mathrm{ext}}$ and $\\omega \\leq 3$) to $\\partial_{v}^{2} (r \\phi)$, $\\partial_{v} \\lmb$ in $\\PD_{\\mathrm{ext}}$. Then we see that there exist a non-negative and non-decreasing function $H''_{2}(u_2)$ and a positive function $\\epsilon''(u_{2})$ such that\n\\begin{equation*} \n\t\\mathcal B_{2}(U) \\leq H_{2}''(u_2) +  \\epsilon''(u_{2}) \\mathcal B_{2}(U),\n\\end{equation*}\nand $\\epsilon''(u_{2}) \\to 0$ as $u_{2} \\to \\infty$. Taking $u_{2}$ sufficiently large, the second term on the right-hand side can be absorbed into the left-hand side; then we conclude that $\\mathcal B_{2}(U) \\leq C_{A_{1}, K, \\Lambda} H''_{2}(u_{2})$. As this bound is independent of $U$, Proposition \\ref{prop:decay2:nullStr} then follows. \\qedhere\n\\end{proof}\n\n\\begin{remark} \nUsing \\eqref{eq:decay2:uDecayInExtr:1}, \\eqref{eq:decay2:uDecayInExtr:3} in Lemma \\ref{lem:decay2:uDecayInExtr} and \\eqref{eq:decay2:rDecay:1}, \\eqref{eq:decay2:rDecay:2} in Lemma \\ref{lem:decay2:rDecay}, the functions $H_{2}''(u_{2})$ and $\\epsilon''(u_{2})$ can be explicitly bounded from the above as follows:\n\\begin{align}\n\tH_{2}''(u_2) \\leq & C_{K, \\Lambda} \\Big( 1 + \\Psi \\int_{3}^{\\infty} \\abs{\\frac{2 m \\lmb}{(1-\\mu) r^{2}}}(u, v') \\, \\mathrm{d} v' \\Big) \\cdot (H_{2} + H'(u_{2}) + A_{1}^{2} + A_{1}^{3}) \t\n\t\\label{eq:decay2:H2''} \\\\\n\t\t\t& + C \\mathcal I_{2} + C_{K, \\Lambda} A_{1}^{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} \n\t\\notag \\\\\n\t\\epsilon''(u_{2}) \\leq & C_{K, \\Lambda} \\Big( 1 + \\Psi \\int_{3}^{\\infty} \\abs{\\frac{2 m \\lmb}{(1-\\mu) r^{2}}}(u, v') \\, \\mathrm{d} v' \\Big) \\cdot (\\epsilon + \\epsilon')(u_{2}).\n\t\\label{eq:decay2:eps''}\n\\end{align}\n\nThese bounds will be useful in our proof of Theorem \\ref{thm:smallData} in Section \\ref{sec:smallData}.\n\\end{remark}\n\nAt this point, in order to complete the proof of Proposition \\ref{prop:decay2:nullStr}, we are only left to prove Lemma \\ref{lem:decay2:key4nullStr}.\n\\begin{proof}[Proof of Lemma \\ref{lem:decay2:key4nullStr}]\nLet $(u, v) \\in \\PD_{\\mathrm{int}}$ (i.e., $v \\in [u, 3u]$) with $u \\in [3 u_{2}, U]$. In this proof, we will use the notation $\\epsilon(u_{2})$ to refer to a positive quantity which may be made arbitrarily small by choosing $u_{2}$ large enough, which may vary from line to line.\n\nWe first estimate $F_1$ and $F_2$. Integrating \\eqref{eq:decay2:nullStr:pf:1} and \\eqref{eq:decay2:nullStr:pf:2} along the incoming direction from $(u\/3, v)$ to $(u,v)$, we obtain\n\\begin{align*}\n\t\\abs{F_{1}(u,v)} \\leq & \\abs{F_{1}(u\/3, v)} + \\int_{u\/3}^{u} \\abs{\\partial_{u} \\lmb \\, \\partial_{v} \\phi(u', v)} + \\abs{\\partial_{v} \\lmb \\, \\partial_{u} \\phi (u', v)} \\, \\mathrm{d} u',  \\\\\n\t\\abs{F_{2}(u,v)} \\leq & \\abs{F_{2}(u\/3, v)} + \\int_{u\/3}^{u} \\abs{\\partial_{u} \\phi \\, \\partial_{v}( \\nu^{-1} \\partial_{u} (r \\phi) )(u', v)} + \\abs{\\partial_{v} \\phi \\, \\partial_{u} ( \\nu^{-1} \\partial_{u} (r \\phi) )(u', v)}  \\, \\mathrm{d} u'.\n\\end{align*}\n\nMultiply both sides of these inequalities by $u^{\\omega}$. For $v \\in [u, 3u]$, note that $(u\/3, v) \\in \\PD_{\\mathrm{ext}}$ and $u \\leq (3 \\Lambda\/2) r(u\/3, v)$. \nTherefore, using Theorem \\ref{main.thm.1} for $\\phi$, $\\partial_v(r\\phi)$, Corollary \\ref{cor:decay1} for $\\partial_{v} \\phi$, Lemma \\ref{lem:muOverR} for $\\mu\/r$ and Lemma \\ref{lem:decay2:rDecay} for $\\partial_{v}^{2}(r\\phi)$, $\\partial_{v} \\lmb$, we have\n\\begin{align*}\n\tu^{\\omega} \\abs{F_{1} (u\/3, v)} \n\t\t\\leq & C_{\\Lambda} r^{\\omega} \\Big( \\abs{\\partial_{v}^{2} (r \\phi)} + \\abs{(\\partial_{v} \\lmb) \\phi} \\Big) (u\/3, v)  \\\\\n\t\t\\leq & C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3}, \\\\\n\tu^{\\omega} \\abs{F_{2} (u\/3, v)} \n\t\t\\leq & C_{\\Lambda} r^{\\omega} \\Big( \\abs{\\lmb^{-1} \\partial_{v} \\lmb} + \\frac{\\mu}{(1-\\mu)} \\frac{\\lmb}{r} + \\abs{\\partial_{v} \\phi ( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} ( r \\phi) )} \\Big) (u\/3, v)\\\\\n\t\t\\leq & C_{K, \\Lambda} A_{1}^{2}.\n\\end{align*}\n\nTherefore, we only need to deal with the $u'$-integrals. For $u \\in [3 u_{2}, U]$, we claim that\n\\begin{align}\n\tu^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{u} \\lmb(u', v)} \\abs{\\partial_{v} \\phi(u',v)} \\, \\mathrm{d} u' \\leq & \\epsilon(u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:pf:1} \\\\\n\tu^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{v} \\lmb(u', v)} \\abs{\\partial_{u} \\phi(u',v)}\\, \\mathrm{d} u' \\leq & \\epsilon(u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:pf:2} \\\\\n\tu^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{u} \\phi(u', v)} \\abs{\\partial_{v} (\\nu^{-1} \\partial_{u} (r \\phi))(u', v)} \\, \\mathrm{d} u' \\leq & \\epsilon (u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:pf:3} \\\\\n\tu^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{v} \\phi(u', v)} \\abs{\\partial_{u} (\\nu^{-1} \\partial_{u} (r \\phi))(u', v)} \\, \\mathrm{d} u' \\leq &  \\epsilon (u_{2}) \\mathcal B_{2}(U). \\label{eq:decay2:key4nullStr:pf:4}\n\\end{align}\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:1}}\nWe proceed similarly as in the proof of Theorem \\ref{main.thm.1}. By \\eqref{eq:SSESF:dr}, \\eqref{eq:bnd4dvrphi} and Lemma \\ref{lem:dphi}, we estimate\n\\begin{align*}\n& u^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{u} \\lmb(u', v)} \\abs{\\partial_{v} \\phi(u',v)} \\, \\mathrm{d} u' \\\\\n& \\quad \\leq C_{\\Lambda} \\Big( \\int_{u_{2}}^{v} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}}(u', v)} \\, \\mathrm{d} u' \\Big) \\sup_{u' \\in [u\/3, u]} \\sup_{C_{u'}} (u')^{\\omega} (\\abs{\\partial^{2}_{v}(r \\phi)} + \\Psi \\abs{\\partial_{v} \\lmb} ) \\\\\n& \\quad \\leq C_{\\Lambda, \\Psi} \\Big( \\int_{u_{2}}^{v} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}}(u', v)} \\, \\mathrm{d} u' \\Big) \\mathcal B_{2}(U).\n\\end{align*}\n\nThus \\eqref{eq:decay2:key4nullStr:pf:1} follows by Lemma \\ref{lem:smallPtnl}.\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:2}}\nWe have\n\\begin{align*}\nu^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{v} \\lmb(u', v)} \\abs{\\partial_{u} \\phi(u',v)}\\, \\mathrm{d} u'\n\\leq & C \\Big( \\int_{u_{2}}^{v} \\abs{\\partial_{u} \\phi(u', v)} \\, \\mathrm{d} u' \\Big) \\sup_{u' \\in [u\/3, u]} \\sup_{C_{u'}} (u')^{\\omega} \\abs{\\partial_{v} \\lmb} \\\\\n\\leq & C \\Big( \\int_{u_{2}}^{v} \\abs{\\partial_{u} \\phi(u', v)} \\, \\mathrm{d} u' \\Big) \\mathcal B_{2}(U).\n\\end{align*}\n\nThus \\eqref{eq:decay2:key4nullStr:pf:2} follows by Lemma \\ref{lem:smallDphi}.\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:3}}\nWe start with the identity\n\\begin{equation*}\n\t\\partial_{v} (\\nu^{-1} \\partial_{u} (r\\phi)) = - \\frac{2m}{(1-\\mu)r^{2}} \\lmb ( \\nu^{-1} \\partial_{u}(r \\phi) - \\phi).\n\\end{equation*}\nwhich is readily verifiable using \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dphi}. By \\eqref{eq:bnd4phi} and \\eqref{eq:bnd4durphi}, we estimate\n\\begin{align*}\n& u^{\\omega} \\int_{u\/3}^{u} \\abs{\\partial_{u} \\phi(u', v)} \\abs{\\partial_{v} (\\nu^{-1} \\partial_{u} (r \\phi))(u', v)} \\, \\mathrm{d} u' \\\\\n& \\quad \\leq C_{K, \\Lambda} \\Psi \\Big( \\int_{u_{2}}^{v} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u' \\Big) \\,  \\sup_{u' \\in [u\/3, u]} \\sup_{C_{u'}} (u')^{\\omega}\\abs{\\partial_{u} \\phi}.\n\\end{align*}\n\nThe $u'$-integral vanishes as $u_{2} \\to \\infty$ by Lemma \\ref{lem:smallPtnl}. On the other hand, by Lemma \\ref{lem:dphi} and Proposition \\ref{prop:geomLocBVScat}, we have\n\\begin{equation} \\label{eq:decay2:key4nullStr:pf:3:1}\n\\sup_{C_{u'}} (u')^{\\omega}\\abs{\\partial_{u} \\phi}\n\\leq C_{K, \\Lambda} \\sup_{C_{u'}} (u')^{\\omega}\\abs{\\partial_{v} \\phi} \n\\leq C_{K, \\Lambda, \\Psi} \\mathcal B_{2}(U),\n\\end{equation}\nfor any $u' \\in [1, U]$.\nTherefore, \\eqref{eq:decay2:key4nullStr:pf:3} follows.\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:4}}\nHere we divide the integral into two, one in $\\PD_{\\mathrm{cpt}}$ and the other outside. Recall the notation $u^{\\star}(v) = \\sup \\set{u \\in [1, v] : r (u,v) \\geq R}$. Below, we will consider the case $u^{\\star}(v) \\in [u\/3, u]$, i.e., when the line segment $\\set{(u', v) \\in \\calQ : u' \\in [u\/3, u]}$ crosses $\\set{r = R}$; the other case is easier, and can be handled with a minor modification.\n\nWe first deal with the integral over the portion in $\\PD_{\\mathrm{cpt}}$. We claim that\n\\begin{equation*}\n\tu^{\\omega} \\int_{u^{\\star}(v)}^{u} \\abs{\\partial_{v} \\phi(u', v)} \\abs{\\partial_{u} (\\nu^{-1} \\partial_{u} (r \\phi))(u', v)} \\, \\mathrm{d} u' \\leq  \\epsilon (u_{2}) \\mathcal B_{2}(U). \n\\end{equation*}\n\nThis is an easy consequence of the bound for $|\\partial_v\\phi|$ in Lemma \\ref{lem:dphi}, the fact that $u$, $u'$ are comparable over the domain of integration and\n\\begin{equation*}\n\t\\sup_{v \\in [u_{2}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{u} (\\nu^{-1} \\partial_{u} (r \\phi))}   \\to 0 \\hbox{ as } u_{2} \\to \\infty\n\\end{equation*}\nwhich follows from \\eqref{eq:decay1:3}, \\eqref{eq:bnd4dur} and Theorem \\ref{thm:decayInCpt}.\n\nWe now consider the remaining contribution to the integral. We begin as follows.\n\\begin{align*}\n\t& u^{\\omega} \\int_{u\/3}^{u^{\\star}(v)} \\abs{\\partial_{v} \\phi(u', v)} \\abs{\\partial_{u} (\\nu^{-1} \\partial_{u} (r \\phi))(u', v)} \\, \\mathrm{d} u'  \\\\\n\t& \\quad \\leq C_{K, \\Lambda} \\Big( \\int_{u\/3}^{u^{\\star}(v)} \\abs{\\partial_{v} \\phi(u', v)} \\, \\mathrm{d} u' \\Big) \\sup_{u' \\in [u\/3, u^{\\star}(v)]} \\sup_{C_{u'}} (u')^{\\omega} (\\abs{\\partial_{u}^{2}(r \\phi)} + \\Psi \\abs{\\partial_{u} \\nu}) \\\\\n\t& \\quad \\leq C_{K, \\Lambda, \\Psi} \\Big( \\int_{u\/3}^{u^{\\star}(v)} \\abs{\\partial_{v} \\phi(u', v)} \\, \\mathrm{d} u' \\Big) \\mathcal B_{2}(U).\n\\end{align*}\n\nFor $u' \\in [u\/3, u^{\\star}(v)]$, we have $r (u', v) \\geq R$. Thus, by \\eqref{eq:decay1:4}, we have\n\\begin{equation*}\n\t\\int_{u\/3}^{u^{\\star}(v)} \\abs{\\partial_{v} \\phi(u', v)} \\, \\mathrm{d} u' \\leq \\frac{C_{K} A_{1}}{R} \\int_{u_{2}}^{\\infty} (u')^{-\\omega} \\, \\mathrm{d} u'  \\leq \\frac{C_{K} A_{1}}{R} u_{2}^{-(\\omega-1)},\n\\end{equation*}\nwhich vanishes as $u_{2} \\to \\infty$. Therefore, in the case under consideration, \\eqref{eq:decay2:key4nullStr:pf:4} follows.\n\n\\vspace{.5em}\n\nWe have therefore obtained the desired bounds for $F_1$ and $F_2$. Next, estimate $G_1$ and $G_2$. Let us integrate \\eqref{eq:decay2:nullStr:pf:3} and \\eqref{eq:decay2:nullStr:pf:4} along the outgoing direction from $(u, u)$ on the axis to $(u,v)$. Then we obtain\n\\begin{align*}\n\t\\abs{G_{1}(u,v)} \\leq & \\lim_{v' \\to u+} \\abs{G_{1}(u, v')} + \\int_{u}^{v} \\abs{\\partial_{v} \\nu \\, \\partial_{u} \\phi (u, v')} + \\abs{\\partial_{u} \\nu \\, \\partial_{v} \\phi (u, v')} \\, \\mathrm{d} v',  \\\\\n\t\\abs{G_{2}(u,v)} \\leq & \\lim_{v' \\to u+} \\abs{G_{2}(u, v')} + \\int_{u}^{v} \\abs{\\partial_{v} \\phi \\, \\partial_{u} ( \\lmb^{-1} \\partial_{v} (r \\phi) ) (u, v')} + \\abs{\\partial_{u} \\phi \\, \\partial_{v} ( \\lmb^{-1} \\partial_{v} (r \\phi) ) (u, v')} \\, \\mathrm{d} v'.\n\\end{align*}\n\nNote that\n\\begin{equation*}\n\t\\lim_{v \\to u+} \\frac{\\mu}{r} (u, v) = 0, \\quad\n\t\\lim_{v \\to u+} \\Big( \\lmb^{-1} \\partial_{v} (r \\phi)(u,v) - \\nu^{-1} \\partial_{u} ( r \\phi) (u,v) \\Big) = 0,\n\\end{equation*}\nsince $(\\phi, r, m)$ is a $C^{1}$ solution. It follows that $\\lim_{v \\to u+} \\partial_{u} \\partial_{v} (r \\phi)(u, v) = 0$ and $\\lim_{v \\to u+} \\partial_{u} \\partial_{v} r (u,v) = 0$. Moreover, we also have\n\\begin{align*}\n\t& \\lim_{v \\to u+} \\partial_{v}^{2} (r \\phi) (u,v) = - \\lim_{v \\to u+} \\partial_{u}^{2} (r \\phi) (u,v), \\quad\n\t\\lim_{v \\to u+} \\partial_{v} \\lmb (u, v)= - \\lim_{v \\to u+} \\partial_{u} \\nu (u,v).\n\\end{align*}\n\nAs a consequence, \n\\begin{equation*}\n\t\\lim_{v' \\to u+} G_{1}(u, v') = - \\lim_{v' \\to u+} F_{1}(u, v'), \\quad\n\t\\lim_{v' \\to u+} G_{2}(u, v') = \\lim_{v' \\to u+} F_{2}(u, v').\n\\end{equation*}\n\nTherefore, by the previous estimates for $F_{1}, F_{2}$, we have\n\\begin{align*}\n\t& u^{\\omega} \\lim_{v' \\to u+} \\abs{G_{1}(u, v')} \\leq C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + \\epsilon(u_{2}) \\mathcal B_{2}(U), \\\\\n\t& u^{\\omega} \\lim_{v' \\to u+} \\abs{G_{2}(u, v')} \\leq C_{K, \\Lambda} A_{1}^{2} + \\epsilon(u_{2}) \\mathcal B_{2}(U),\n\\end{align*}\nwhich are acceptable. Recalling that we are considering $(u,v) \\in \\PD_{\\mathrm{int}}$, hence $v \\in [u, 3u]$, we are now left to establish the following estimates: \n\\begin{align}\n\tu^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{v} \\nu(u, v')} \\abs{\\partial_{u} \\phi(u,v')} \\, \\mathrm{d} v' \\leq & \\epsilon(u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:pf:5} \\\\\n\tu^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{u} \\nu(u, v')} \\abs{\\partial_{v} \\phi(u,v')}\\, \\mathrm{d} v' \\leq & \\epsilon(u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:pf:6} \\\\\n\tu^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{v} \\phi(u, v')} \\abs{\\partial_{u} (\\lmb^{-1} \\partial_{v} (r \\phi))(u, v')} \\, \\mathrm{d} v' \\leq & \\epsilon (u_{2}) \\mathcal B_{2}(U), \\label{eq:decay2:key4nullStr:pf:7} \\\\\n\tu^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{u} \\phi(u, v')} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))(u, v')} \\, \\mathrm{d} v' \\leq &  \\epsilon (u_{2}) \\mathcal B_{2}(U). \\label{eq:decay2:key4nullStr:pf:8}\n\\end{align}\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:5}}\nSubstituting $\\partial_{v} \\nu$ by \\eqref{eq:SSESF:dr} and using \\eqref{eq:decay2:key4nullStr:pf:3:1}, we have\n\\begin{align*}\nu^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{v} \\nu(u, v')} \\abs{\\partial_{u} \\phi(u,v')} \\, \\mathrm{d} v'\n\\leq & K \\Big( \\int_{u}^{\\infty} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}}(u, v')} \\, \\mathrm{d} v' \\Big) \\sup_{v' \\in [u, 3u]} u^{\\omega} \\abs{\\partial_{u} \\phi(u,v')}  \\\\\n\\leq & C_{K, \\Lambda, \\Psi} \\Big( \\sup_{u \\geq 3u_{2}} \\int_{u}^{\\infty} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}}(u, v')} \\, \\mathrm{d} v' \\Big) \\mathcal B_{2}(U).\n\\end{align*}\n\nThus \\eqref{eq:decay2:key4nullStr:pf:5} follows by Lemma \\ref{lem:smallPtnl}.\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:6}}\nWe have\n\\begin{align*}\nu^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{u} \\nu(u, v')} \\abs{\\partial_{v} \\phi(u,v')}\\, \\mathrm{d} v'\n\\leq & \\int_{u}^{\\infty} \\abs{\\partial_{v} \\phi(u, v')} \\, \\mathrm{d} v'  \\sup_{v' \\in [u, 3u]} u^{\\omega} \\abs{\\partial_{u} \\nu(u, v')}  \\\\\n\\leq & \\Big( \\sup_{u \\geq 3u_{2}} \\int_{u}^{\\infty} \\abs{\\partial_{v} \\phi(u, v')} \\, \\mathrm{d} v'  \\Big) \\mathcal B_{2}(U).\n\\end{align*}\n\nThus \\eqref{eq:decay2:key4nullStr:pf:6} follows by Lemma \\ref{lem:smallDphi}.\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:7}}\nBy \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dphi}, we have the identity\n\\begin{equation*}\n\t\\partial_{u} (\\lmb^{-1} \\partial_{v} (r\\phi)) = - \\frac{2m}{(1-\\mu)r^{2}} \\nu ( \\lmb^{-1} \\partial_{v}(r \\phi) - \\phi).\n\\end{equation*}\n\nThen by Proposition \\ref{prop:geomLocBVScat}, we have\n\\begin{align*}\n&u^{\\omega} \\int_{u}^{3u} \\abs{\\partial_{v} \\phi(u, v')} \\abs{\\partial_{u} (\\lmb^{-1} \\partial_{v} (r \\phi))(u, v')} \\, \\mathrm{d} v' \\\\\n& \\quad \\leq C_{K, \\Lambda} \\Psi \\Big( \\int_{u}^{\\infty} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}} (u, v')} \\, \\mathrm{d} v' \\Big)  \\sup_{v' \\in [u, 3u]} u^{\\omega} \\abs{\\partial_{v} \\phi(u,v')}.\n\\end{align*}\n\nNow \\eqref{eq:decay2:key4nullStr:pf:7} follows by Lemmas \\ref{lem:smallPtnl} and \\ref{lem:dphi} and \\eqref{eq:bnd4dvrphi}.\n\n\\pfstep{Proof of \\eqref{eq:decay2:key4nullStr:pf:8}}\nAs in the proof of \\eqref{eq:decay2:key4nullStr:pf:4}, we will divide the integral into two pieces. More precisely, let us define $v^{\\star}(u)$ to be the unique $v$-value such that $r(u, v^{\\star}(u)) = R$. Assuming $v^{\\star}(u) \\in [u, 3u]$, the integral $\\int_{u}^{3u}$ will be divided into $\\int_{u}^{v^{\\star}(u)}$ and $\\int_{v^{\\star}(u)}^{3u}$. The remaining case $v^{\\star}(u) > 3u$ can be dealt with by adapting the argument for the first integral.\n\nFor the first integral, we claim that\n\\begin{equation*}\n\tu^{\\omega} \\int_{u}^{v^{\\star}(u)} \\abs{\\partial_{u} \\phi(u, v')} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))(u, v')} \\, \\mathrm{d} v' \\leq \\epsilon(u_{2}) \\mathcal B_{2}(U).\n\\end{equation*}\n\nFrom the locally BV scattering assumption \\eqref{eq:locBVScat}, we have\n\\begin{equation*}\n\t\\sup_{u \\in [3 u_{2}, \\infty)} \\int_{C_{u} \\cap \\PD_{\\mathrm{cpt}}} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))}  \\to 0 \\hbox{ as } u_{2} \\to \\infty.\n\\end{equation*}\n\nCombined with \\eqref{eq:decay2:key4nullStr:pf:3:1}, the claim follows.\n\nNext, we turn to the second integral. By \\eqref{eq:bnd4dvr} and \\eqref{eq:bnd4dvrphi}, we estimate\n\\begin{align*}\n& u^{\\omega} \\int_{v^{\\star}(u)}^{3u} \\abs{\\partial_{u} \\phi(u, v')} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))(u, v')} \\, \\mathrm{d} v'  \\\\\n& \\quad \\leq \\sup_{v' \\in [v^{\\star}(u), 3u]} u^{\\omega} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))(u, v')} \\int_{v^{\\star}(u)}^{3u} \\abs{\\partial_{u} \\phi(u, v')} \\, \\mathrm{d} v' \\\\\n& \\quad \\leq C_{\\Lambda, \\Psi} \\mathcal B_{2}(U) \\int_{v^{\\star}(u)}^{3u} \\abs{\\partial_{u} \\phi(u, v')} \\, \\mathrm{d} v'.\n\\end{align*}\n\nFor $v' \\in [v^{\\star}(u), 3u]$, we have $r (u, v) \\geq R$. Thus, by \\eqref{eq:decay1:5}, we have\n\\begin{equation*}\n\t\\int_{v^{\\star}(u)}^{3u} \\abs{\\partial_{u} \\phi(u, v')} \\, \\mathrm{d} v' \\leq \\frac{C_{K} A_{1}}{R} \\int_{u}^{3u} u^{-\\omega} \\, \\mathrm{d} v'  \\leq \\frac{C_{K} A_{1}}{R} u_{2}^{-(\\omega-1)},\n\\end{equation*}\nwhich vanishes as $u_{2} \\to \\infty$, and therefore finishes the proof of \\eqref{eq:decay2:key4nullStr:pf:8}. We remark that the fact that we are in $\\PD_{\\mathrm{int}}$ is used crucially here, as otherwise the integral would not be convergent. \n\\end{proof}\n\n\\begin{remark} \nIn the case where we have global BV scattering (i.e., conditions $(2)$ and $(3)$ of Definition \\ref{def:locBVScat} are satisfied with $R=\\infty$), we can take $R = \\infty$ in the preceding argument to obtain the following explicit upper bound on $\\epsilon(u_{2})$:\n\\begin{equation} \\label{eq:decay2:eps}\n\\begin{aligned}\n\t\\epsilon(u_{2}) \n\t\\leq & C_{K, \\Lambda, \\Psi} \\sup_{v \\in [u_{2}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}}} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}}}\n\t\t+C \\sup_{v \\in [u_{2}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}}} \\abs{\\partial_{u} \\phi} \\\\\n\t\t& + C_{K, \\Lambda, \\Psi} \\sup_{v \\in [u_{2}, \\infty)} \\int_{\\underline{C}_{v} \\cap \\set{u \\geq u_{2}}} \\abs{\\partial_{u} (\\nu^{-1} \\partial_{u} (r \\phi))} \\\\\n\t\t& +C_{K, \\Lambda, \\Psi}  \\sup_{u \\geq 3u_{2}} \\int_{C_{u}} \\abs{\\frac{2m \\lmb}{(1-\\mu) r^{2}}}\n\t\t+C  \\sup_{u \\geq 3u_{2}} \\int_{C_{u}} \\abs{\\partial_{v} \\phi}  \\\\\n\t\t& +C_{K, \\Lambda, \\Psi} \\sup_{u \\in [3 u_{2}, \\infty)} \\int_{C_{u}} \\abs{\\partial_{v} (\\lmb^{-1} \\partial_{v} (r \\phi))}.\n\\end{aligned}\n\\end{equation}\n\nThis will be useful in our proof of the sharp decay rate in the case of small BV norm (Theorem \\ref{thm:smallData}) in Section \\ref{sec:smallData}.\n\\end{remark}\n\nIn the second step of our proof of Theorem \\ref{main.thm.2}, we use the preliminary $u^{-\\omega}$ decay proved in Proposition \\ref{prop:decay2:nullStr} to obtain the optimal the $u$-decay. Key to this step is the following proposition, which claims optimal $u$-decay in $\\PD_{\\mathrm{int}}$.\n\n\\begin{proposition} \\label{prop:decay2:final}\nThere exists a constant $0 < A_{2}'' < \\infty$ such that the following estimates hold.\n\\begin{align} \n\\sup_{\\PD_{\\mathrm{int}}} u^{\\omega+1} \\abs{\\partial_{v}^{2} (r \\phi)} \\leq & A_{2}'',\n\t\\label{eq:decay2:final:1} \\\\\n\\sup_{\\PD_{\\mathrm{int}}} u^{\\omega+1} \\abs{\\partial_{u}^{2} (r \\phi)} \\leq & A_{2}'', \n\t\\label{eq:decay2:final:2} \\\\\n\\sup_{\\PD_{\\mathrm{int}}} u^{3} \\abs{\\partial_{v} \\lmb} \\leq & A_{2}'', \n\t\\label{eq:decay2:final:3} \\\\\n\\sup_{\\PD_{\\mathrm{int}}} u^{3} \\abs{\\partial_{u} \\nu} \\leq & A_{2}''. \n\t\\label{eq:decay2:final:4}\n\\end{align}\n\\end{proposition}\n\nOnce we establish Proposition \\ref{prop:decay2:final}, the desired decay for $\\partial_{v}^{2}(r \\phi)$ and $\\partial_{v} \\lmb$ follow from Lemma \\ref{lem:decay2:rDecay} and the fact that $r \\geq 2 \\Lambda^{-1} u$ in $\\PD_{\\mathrm{ext}}$. Furthermore, the desired decay for $\\partial_{u}^{2}(r \\phi)$ and $\\partial_{u} \\nu$ follow from Lemma \\ref{lem:decay2:uDecayInExtr}.\n\n\\begin{proof} \nThanks to the fact that we have pointwise bounds for sufficient number of derivatives (albeit with sub-optimal decay) near $\\Gamma$ at this point, it suffices to work with the `non-renormalized' equations \\eqref{eq:eq4dvdvrphi:normal}, \\eqref{eq:eq4dudurphi:normal}, \\eqref{eq:eq4dvdvr:normal} and \\eqref{eq:eq4dudur:normal}. In particular, we need not utilize the null structure of \\eqref{eq:SSESF}. \n\nLet $(u,v) \\in \\PD_{\\mathrm{int}}$ (i.e., $v \\in [u, 3u]$) with $u \\geq 3$. We begin with \\eqref{eq:decay2:final:1}. Integrating $\\partial_{u} \\partial_{v}^{2}(r\\phi)$ in the $u$-direction from $u\/3$ to $u$, multiplying by $u^{\\omega+1}$ and using $r(u\/3, v) \\geq (2\/3) \\Lambda^{-1} u$, we obtain\n\\begin{equation} \\label{eq:decay2:final:1:pf}\n\tu^{\\omega+1}\\abs{\\partial_{v}^{2}(r \\phi)}(u,v)\n\t\\leq C_{\\Lambda} r^{\\omega+1} \\abs{\\partial_{v}^{2}(r \\phi)}(u\/3, v) + u^{\\omega+1} \\int_{u\/3}^{u} \\abs{\\partial_{u} \\partial_{v}^{2}(r \\phi)}(u', v) \\, \\mathrm{d} u'.\n\\end{equation}\n\nSince $(u\/3, v) \\in \\PD_{\\mathrm{ext}}$, the first term on the right-hand side is bounded by $\\leq C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3}$, thanks to Lemma \\ref{lem:decay2:rDecay}. To estimate the $u'$-integral, we substitute $\\partial_{u} \\partial_{v}^{2} (r \\phi)$ by \\eqref{eq:eq4dvdvrphi:normal}. Then applying Proposition \\ref{prop:geomLocBVScat}, Lemma \\ref{lem:dphi}, Lemma \\ref{lem:muOverR}, Theorem \\ref{main.thm.1} and Proposition \\ref{prop:decay2:nullStr}, we obtain\n\\begin{equation*}\n\t\\abs{\\partial_{u} \\partial_{v}^{2}(r \\phi)}(u', v)\n\t\\leq C_{A_{1}, K, \\Lambda} (u')^{-3\\omega} A_{1} (A_{2}')^{2}.\n\\end{equation*}\n\nThus we have\n\\begin{equation} \\label{eq:decay2:final:1:explicit}\n\tu^{\\omega+1}\\abs{\\partial_{v}^{2}(r \\phi)}(u,v)\n\t\\leq C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + C_{A_{1}, K, \\Lambda} A_{1} (A_{2}')^{2},\n\\end{equation}\nwhere we have used the fact that $\\omega > 1$, and thus $3 \\omega - 1 > \\omega + 1$ to throw away the $u$-weight in the last term. This proves \\eqref{eq:decay2:final:1}.\n\nNext, we prove \\eqref{eq:decay2:final:2}. Integrating $\\partial_{v} \\partial_{u}^{2} (r \\phi)$ in the $v$-direction from $u+$ to $v$ and multiplying by $u^{\\omega+1}$, we have\n\\begin{equation} \\label{eq:decay2:final:2:pf}\n\tu^{\\omega+1} \\abs{\\partial_{u}^{2}(r \\phi)}(u,v)\n\t\\leq \\lim_{v' \\to u+} u^{\\omega+1} \\abs{\\partial_{u}^{2} (r \\phi)}(u, v') + u^{\\omega+1} \\int_{u}^{3u} \\abs{\\partial_{v} \\partial_{u}^{2} (r \\phi)}(u, v') \\, \\mathrm{d} v'.\n\\end{equation}\n\nRecall that $\\lim_{v' \\to u+} \\partial_{u}^{2} (r \\phi)(u, v') = \\lim_{v' \\to u+} \\partial_{v}^{2} (r \\phi)(u, v')$, as $(\\phi, r, m)$ is a $C^{1}$ solution. Thus the first term on the right-hand side can be estimated via \\eqref{eq:decay2:final:1:explicit}. Substitute $\\partial_{v} \\partial_{u}^{2} (r \\phi)$ by \\eqref{eq:eq4dudurphi:normal} and apply, as before, Proposition \\ref{prop:geomLocBVScat}, Lemma \\ref{lem:dphi}, Lemma \\ref{lem:muOverR}, Theorem \\ref{main.thm.1} and Proposition \\ref{prop:decay2:nullStr}. Then we have\n\\begin{equation*}\n\t\\abs{\\partial_{v} \\partial_{u}^{2} (r \\phi)}(u, v')\n\t\\leq C_{A_{1}, K, \\Lambda} u^{-3\\omega} A_{1} (A_{2}')^{2}.\n\\end{equation*}\n\nIt now follows that \n\\begin{equation} \\label{eq:decay2:final:2:explicit}\n\tu^{\\omega+1}\\abs{\\partial_{u}^{2}(r \\phi)}(u,v)\n\t\\leq C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + C_{A_{1}, K, \\Lambda} A_{1} (A_{2}')^{2},\n\\end{equation}\nwhich proves \\eqref{eq:decay2:final:2}.\n\nAt this point, combining Lemma \\ref{lem:dphi}, Theorem \\ref{main.thm.1}, Lemma \\ref{lem:decay2:rDecay} and \\eqref{eq:decay2:final:1:explicit}, note that we have the following improved $u$-decay for $\\partial_{v} \\phi$:\n\\begin{equation} \\label{eq:decay2:final:impDvphi} \n\\begin{aligned}\n\t\\sup_{\\calQ} u^{\\omega+1} \\abs{\\partial_{v}\\phi} \n\t\\leq C_{\\Lambda} \\sup_{\\calQ} (u^{\\omega+1} \\abs{\\partial_{v}^{2}(r \\phi)} + u A_{1} \\abs{\\partial_{v} \\lmb})\n\t\\leq B\n\\end{aligned}\n\\end{equation}\nwhere \n\\begin{equation} \\label{eq:decay2:final:aux}\nB := C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + C_{A_{1}, K, \\Lambda} A_{1} (A_{2}')^{2} + C_{\\Lambda} A_{1} A_{2}'.\n\\end{equation}\n\nWe now turn to \\eqref{eq:decay2:final:3}. Integrating $\\partial_{u} \\partial_{v} \\log \\lmb$ in the $u$-direction from $u\/3$ to $u$, multiplying by $u^{3}$ and using $r(u\/3, v) \\geq (2\/3) \\Lambda^{-1} u$, we obtain\n\\begin{equation} \\label{eq:decay2:final:3:pf}\n\tu^{3} \\abs{\\partial_{v} \\log \\lmb }(u, v) \\leq \n\tC r^{3} \\abs{\\partial_{v} \\log \\lmb}(u\/3, v) + u^{3} \\int_{u\/3}^{u} \\abs{\\partial_{u} \\partial_{v} \\log \\lmb}(u', v) \\, \\mathrm{d} u'\n\\end{equation}\n\nSince $(u\/3, v) \\in \\PD_{\\mathrm{ext}}$, the first term on the right-hand side is estimated $\\leq C_{K, \\Lambda} A_{1}^{2}$ by Lemma \\ref{lem:decay2:rDecay} and the fact that $\\lmb^{-1} \\leq \\Lambda$. Next, substituting $\\partial_{u} \\log \\lmb$ by \\eqref{eq:eq4dvdvr:normal}, applying Proposition \\ref{prop:geomLocBVScat}, Lemma \\ref{lem:muOverR}, Lemma \\ref{lem:dphi} and using the improved bound \\eqref{eq:decay2:final:impDvphi}, we have \n\\begin{equation*}\n\t\\abs{\\partial_{u} \\partial_{v} \\log \\lmb}(u',v) \\leq C_{K, \\Lambda} B^{2} (u')^{-2(\\omega+1)}.\n\\end{equation*}\n\nTherefore\n\\begin{equation} \\label{eq:decay2:final:3:explicit}\n\tu^{3} \\abs{\\partial_{v} \\lmb}(u,v) \\leq C_{K, \\Lambda} A_{1}^{2} + C_{K, \\Lambda} B^{2},\n\\end{equation}\nwhere we used $2(\\omega + 1) - 1 > 3$ to throw away the $u$-weight in the last term. This proves \\eqref{eq:decay2:final:3}.\n\nFinally, we prove \\eqref{eq:decay2:final:4}. Integrating $\\partial_{v} \\partial_{u} \\log \\nu$ in the $v$-direction from $u+$ to $v$ and multiplying by $u^{3}$, we have\n\\begin{equation} \\label{eq:decay2:final:4:pf}\n\tu^{3} \\abs{\\partial_{u} \\log \\nu}(u,v) \\leq \\lim_{v' \\to u+} u^{3} \\abs{\\partial_{u} \\log \\nu}(u, v') + u^{3} \\int_{u}^{3u} \\abs{\\partial_{v} \\partial_{u} \\log \\nu}(u, v') \\, \\mathrm{d} v'.\n\\end{equation}\n\nSince $\\lim_{v' \\to u+} \\partial_{u} \\nu(u, v') = - \\lim_{v' \\to u+} \\partial_{v} \\lmb(u, v')$, the first term is bounded by \\eqref{eq:decay2:final:3:explicit}. Furthermore, substituting $\\partial_{v} \\partial_{u} \\log \\nu$ by \\eqref{eq:eq4dudur:normal} and applying Proposition \\ref{prop:geomLocBVScat}, Lemma \\ref{lem:muOverR}, Lemma \\ref{lem:dphi} and using the improved bound \\eqref{eq:decay2:final:impDvphi}, we have \n\\begin{equation*}\n\t\\abs{\\partial_{v} \\partial_{u} \\log \\nu}(u, v') \\leq C_{K, \\Lambda} B^{2} u^{-2(\\omega+1)}.\n\\end{equation*}\n\nAs before, it follows that\n\\begin{equation} \\label{eq:decay2:final:4:explicit}\n\tu^{3} \\abs{\\partial_{u} \\nu}(u,v) \\leq C_{K, \\Lambda} A_{1}^{2} + C_{K, \\Lambda} B^{2},\n\\end{equation}\nwhich proves \\eqref{eq:decay2:final:4}.\n\\end{proof}\n\n\\begin{remark} \nCombining \\eqref{eq:decay2:final:1:explicit}, \\eqref{eq:decay2:final:2:explicit}, \\eqref{eq:decay2:final:3:explicit} and \\eqref{eq:decay2:final:4:explicit}, we see that Proposition \\ref{prop:decay2:final} holds with\n\\begin{equation} \\label{eq:decay2:A2''}\n\tA_{2}'' \\leq C_{\\Lambda} \\mathcal I_{2} + C_{K, \\Lambda, M_{i}} A_{1}^{3} + C_{A_{1}, K, \\Lambda} A_{1} (A_{2}')^{2}\n\t\t+ C_{K, \\Lambda} A_{1}^{2} + C_{K, \\Lambda} B^{2}\n\\end{equation}\nwhere $B$ is as in \\eqref{eq:decay2:final:aux}.\n\\end{remark}\n\n\\begin{remark} \nAccording to the argument of this subsection, note that the size of $A_{2}'$ in Proposition \\ref{prop:decay2:nullStr} depends on the choice of $u_{2}$ through the term $H_{2}'(u_{2})$, where the size of $u_{2}$ depends on the rate of convergence of $\\epsilon''(u_{2}) \\to 0$ as $u_{2} \\to \\infty$. This explains why $A_{2}$ does not depend only on the size of the initial data, as remarked in Section \\ref{sec.main.thm}. On the other hand, as stated in Statement (2) of Theorem \\ref{thm:smallData}, we shall show that in the case of small BV initial data, $A_2$ depends only on the size of the initial data. To achieve this, we show in Section \\ref{sec:smallData} that we may take $u_{2} = 1$ under this small data assumption. \n\\end{remark}\n\n\n\\subsection{Additional decay estimates}\nAs in the previous section, we conclude this section by providing additional decay rates concerning second derivatives of $\\phi$, $r$ and improved decay for $m$ near $\\Gamma$.\n\n\\begin{corollary} \\label{cor:decay2}\nLet $(\\phi, r, m)$ be a locally BV scattering solution to \\eqref{eq:SSESF} with asymptotically flat $C^{1}$ initial data of order $\\omega'$.\nLet $A_{1}$ and $A_{2}$ be the constants in Theorems \\ref{main.thm.1} and \\ref{main.thm.2}, respectively. Then \nthe following bounds hold.\n\\begin{align} \n\t\\abs{\\partial_{v} \\phi} \\leq & C_{\\Lambda} (A_{1} + A_{2} + A_{1} A_{2}) \\min \\set{u^{-(\\omega+1)}, r^{-2} u^{-(\\omega-1)}} \\label{eq:decay2:5} \\\\\n\t\\abs{\\partial_{u} \\phi} \\leq & C_{K, \\Lambda} (A_{1} + A_{2} + A_{1} A_{2}) \\min \\set{u^{-(\\omega+1)}, r^{-1} u^{-\\omega}} \\label{eq:decay2:6} \\\\\n\t\\abs{\\partial_{v}^{2} \\phi} \\leq & C_{\\Lambda} (A_{1} + A_{2} + A_{1} A_{2}) \\min \\set{r^{-1} u^{-(\\omega+1)}, r^{-3} u^{-(\\omega-1)}}, \\label{eq:decay2:7}\\\\\n\t\\abs{\\partial_{u} \\partial_{v} \\phi} \\leq & C_{K, \\Lambda} (A_{1} + A_{2} + A_{1} A_{2}) \\min \\set{r^{-1} u^{-(\\omega+1)}, r^{-2} u^{-\\omega}}, \\label{eq:decay2:8}\\\\\n\t\\abs{\\partial_{u}^{2} \\phi} \\leq & C_{K, \\Lambda} (A_{1} + A_{2} + A_{1} A_{2}) \\, r^{-1} u^{-(\\omega+1)}, \\label{eq:decay2:9}\\\\\n\t\\abs{\\partial_{u} \\partial_{v} r} \\leq & C_{K, \\Lambda} (A_{1} + A_{2} + A_{1} A_{2})^{2} \\min \\set{r u^{-(2\\omega+2)}, r^{-2} u^{-(2\\omega-1)}}, \\label{eq:decay2:10} \\\\\n\tm \\leq & C_{K, \\Lambda} (A_{1} + A_{2} + A_{1} A_{2})^{2} \\min \\set{r^{3} u^{-(2\\omega+2)}, u^{-(2\\omega-1)}}. \\label{eq:decay2:11}\n\\end{align}\n\\end{corollary}\n\nThis corollary follows immediately from the estimates derived in Theorem \\ref{main.thm.2}. We sketch the proof below.\n\n\n\\begin{proof} \nFirst, note that \\eqref{eq:decay2:5} and \\eqref{eq:decay2:6} follows from Corollary \\ref{cor:decay1}, Theorem \\ref{main.thm.2} and Lemma \\ref{lem:dphi}. \nNext, \\eqref{eq:decay2:7} and \\eqref{eq:decay2:9} are easy consequences of the preceding estimates, Theorems \\ref{main.thm.1}, \\ref{main.thm.2} and the identities\n\\begin{equation*}\n\tr \\partial_{v}^{2} \\phi = \\partial_{v}^{2} ( r\\phi) - (\\partial_{v} \\lmb) \\phi - 2 \\lmb \\partial_{v} \\phi, \\quad\n\tr \\partial_{u}^{2} \\phi = \\partial_{u}^{2} ( r\\phi) - (\\partial_{u} \\nu) \\phi - 2 \\nu \\partial_{u} \\phi.\n\\end{equation*}\n\nOn the other hand, for \\eqref{eq:decay2:8}, we use the identity\n\\begin{equation*}\n\tr \\partial_{u} \\partial_{v} \\phi = - \\lmb \\partial_{u} \\phi - \\nu \\partial_{v} \\phi,\n\\end{equation*}\nwhich may be verified from \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dphi}.\n\nNext, \\eqref{eq:decay2:11} follows from Corollary \\ref{cor:decay1}, Lemma \\ref{lem:muOverR} and \\eqref{eq:decay2:5}. Finally, using Corollary \\ref{cor:est4dr}, Lemma \\ref{lem:est4dur}, \\eqref{eq:decay2:11} and the equation \\eqref{eq:SSESF:dr}, we conclude \\eqref{eq:decay2:10}. \\qedhere\n\\end{proof}\n\n\\section{Decay and blow up at infinity}\\label{sec.dichotomy}\n\nIn this section, we prove Theorem \\ref{thm.dichotomy}, i.e., unless the solution blows up at infinity, a `future causally geodesically complete' solution scatters in BV.\n\nTake a BV solution to \\eqref{eq:SSESF} satisfying the hypotheses of Theorem \\ref{thm.dichotomy}, which does not blow up at infinity. Note, in particular, that $\\calQ = \\calR$ by $(1)$ of Definition \\ref{def:locBVScat} and Lemma \\ref{lem:regR}. In order to prove Theorem \\ref{thm.dichotomy}, our goal is to show that such a spacetime is in fact BV scattering, i.e., (1), (2) and (3) in Definition \\ref{def:locBVScat} hold and moreover (3) holds with $R=\\infty$.\n\nThe main step will be to show that there exists a constant $C_{\\Lambda}$ such that for every $\\epsilon > 0$, there exists $U$ such that for every $u\\geq U$, we have\n\\begin{equation}\\label{dichotomy.goal}\n\\int_{C_u}|\\partial_v^2(r\\phi)| + \\int_{C_u}|\\partial_v \\lambda| \\leq C_{\\Lambda} \\epsilon.\n\\end{equation}\nThis will be achieved in a sequence of Lemmas and Propositions below.\n\nBefore we proceed, we first prove a preliminary bound on $\\lmb$:\n\\begin{proposition}\nThere exists $0<\\Lambda<\\infty$ such that\n\\begin{equation} \\label{dic.bnd4dvr}\n\t\\Lambda^{-1} \\leq \\lmb(u,v) \\leq \\frac{1}{2}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nBy $(1)$ in Definition \\ref{def.blow.up.infty}, there exists $0 < \\Lambda < \\infty$ such that $\\sup \\lmb_{\\Gamma}^{-1} \\leq \\Lambda$. As $\\lim_{u \\to v-} \\lmb_{\\Gamma}(u) = \\lim_{u \\to v-} \\lmb(u', v)$ (see \\cite[Section 7]{Christodoulou:1993bt}), it follows from Lemma \\ref{lem:basicEst4dr} that for every $(u,v) \\in \\calQ$, we have the estimate \\eqref{dic.bnd4dvr}.\n\\end{proof}\n\nWe now proceed to show \\eqref{dichotomy.goal}. The first step is to show that for $u$ sufficiently large, the integrals along $C_u$ of $|F_1|$ and $|F_2|$ are small. Here, we recall the notation in the proof of Proposition \\ref{prop:decay2:nullStr}, i.e.,\n\\begin{align*}\nF_{1} := & \\partial_{v}^{2} (r \\phi) - (\\partial_{v} \\lmb) \\phi, \\\\\n\tF_{2} := & \\partial_{v} \\log \\lmb - \\frac{\\lmb}{(1-\\mu)} \\frac{\\mu}{r} + \\partial_{v} \\phi \\Big( \\lmb^{-1} \\partial_{v} (r \\phi) - \\nu^{-1} \\partial_{u} ( r \\phi) \\Big).\n\\end{align*}\nOnce we obtain the desired bounds for $F_1$ and $F_2$, we then derive \\eqref{dichotomy.goal} from these bounds. This will be the most technical part (see discussions in Remark \\ref{technical.rmk.dichotomy}).\n\nFirst, we bound the integrals of $F_1$ and $F_2$ in the following proposition:\n\\begin{proposition}\\label{8.1.1}\nFor every $\\epsilon>0$, there exists $V$ sufficiently large such that the following bound holds for $u\\geq V$:\n\\begin{equation} \\label{dichotomy.beginning}\n\\int_{C_u} (|F_1|+|F_2|)(u,v)  \\leq 3\\epsilon.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nBy $(2)$ and $(3)$ in Definition \\ref{def.blow.up.infty}, we have\n$$\\int_1^{\\infty}\\int_u^{\\infty}|\\partial_v\\lambda\\partial_u\\phi-  \\partial_u\\lambda\\partial_v\\phi  | \\mathrm{d} v \\mathrm{d} u <\\infty,$$\nand\n$$\\int_1^{\\infty}\\int_u^{\\infty}|\\partial_u\\phi\\partial_v(\\nu^{-1}\\partial_u(r\\phi))-\\partial_v\\phi\\partial_u(\\nu^{-1}\\partial_u(r\\phi))| \\mathrm{d} v \\mathrm{d} u <\\infty.$$\n\nThus, by choosing $V$ sufficiently large, we have\n\\begin{equation}\n\\int_V^{\\infty}\\int_u^{\\infty}|\\partial_v\\lambda\\partial_u\\phi-\\partial_v\\lambda\\partial_v\\phi| \\mathrm{d} v \\mathrm{d} u <\\epsilon,\\label{non.blowup.1}\n\\end{equation}\nand\n\\begin{equation}\n\\int_V^{\\infty}\\int_u^{\\infty}|\\partial_u\\phi\\partial_v(\\nu^{-1}\\partial_u(r\\phi))-\\partial_v\\phi\\partial_u(\\nu^{-1}\\partial_u(r\\phi))| \\mathrm{d} v \\mathrm{d} u <\\epsilon.\\label{non.blowup.2}\n\\end{equation}\n\n\n\nFrom the initial conditions, we easily see that $F_{1}(1, \\cdot)$, $F_{2}(1, \\cdot)$ obey $\\int_{C_{1}} \\abs{F_{1}} + \\abs{F_{2}} < \\infty$. \nThus, by choosing $V$ larger if necessary, we have\n\\begin{equation}\n\\int_V^{\\infty} (|F_1|+|F_2|)(1,v) \\mathrm{d} v\\leq \\epsilon.\\label{data.BV.bd}\n\\end{equation}\nNotice that by equations \\eqref{eq:decay2:nullStr:pf:1} and \\eqref{eq:decay2:nullStr:pf:2}, the estimates \\eqref{non.blowup.1} and \\eqref{non.blowup.2} control $\\iint |\\partial_u F_1| \\mathrm{d} u \\mathrm{d} v$ and $\\iint |\\partial_u F_2| \\mathrm{d} u \\mathrm{d} v$. Thus, we have\n$$\\int_{\\max\\set{u, V}}^{\\infty} (|F_1|+|F_2|)(u,v) \\mathrm{d} v\\leq 3\\epsilon.$$\nfor every $u\\geq 1$. In particular, for $u\\geq V$, we have\n\\begin{equation*}\n\\int_{C_u} (|F_1|+|F_2|)(u,v)  \\leq 3\\epsilon,\n\\end{equation*}\nas desired.\n\\end{proof}\n\nThe inequality \\eqref{dichotomy.beginning} is the starting point for our proof of \\eqref{dichotomy.goal}. More precisely, our basic strategy is to use a continuous induction on $v$, beginning from the axis, to remove the quadratic and higher terms from \\eqref{dichotomy.beginning} and infer \\eqref{dichotomy.goal}. \n\n\\begin{remark} \\label{technical.rmk.dichotomy}\nBefore beginning the proof in earnest, we would like to point out two technical nuisances that we confront: \nFirst, in order to estimate the scalar field $\\phi$ itself from $F_{1}$ and $F_{2}$, we need to integrate essentially from null infinity\\footnote{More precisely, $\\phi$ is determined from $\\partial_{v} (r \\phi)$, which in turn can be determined from $\\int \\abs{\\partial_{v}^{2}(r \\phi)}$ by integrating from $v = \\infty$. Another conceptual reason why information near $v = \\infty$ is relevant for estimating $\\phi$ is that the initial condition $\\lim_{v \\to \\infty} \\phi(1, v) = 0$ implies that $\\lim_{v \\to \\infty} \\phi(u, v)= 0$ for every $u \\geq 1$. See the discussion before \\eqref{dic.0.2}.}, which is opposite to the direction of our method of continuity. Second, as $\\partial_{v}(r \\phi)$ is only assumed to be BV, the left-hand side of \\eqref{dichotomy.goal} is not continuous in $v$ in general. To overcome the first, we make use of the invariance of \\eqref{eq:SSESF} and $F_{1}$, $F_{2}$ under the change $\\phi \\mapsto \\phi + c$. To take care of the second, we carefully keep track of the evolution of discontinuities of $\\partial_{v}(r \\phi)$.\n\\end{remark}\n\nNotice that in order to obtain \\eqref{dichotomy.goal} from \\eqref{dichotomy.beginning}, we only need to integrate on a \\emph{fixed} hypersurface $C_u$. We now fix $u_0\\geq V$ and define a new function $\\overline{\\phi}_{u_{0}}$ by\n\\begin{equation} \\label{dic.0.0}\n\t\\overline{\\phi}_{u_{0}}(u,v) := \\phi(u,v) - \\lim_{v' \\to u_{0} +} \\phi(u_{0}, v').\n\\end{equation}\n\nAs remarked before, note that \\eqref{eq:SSESF} is invariant under the change $(\\phi, r, m) \\mapsto (\\overline{\\phi}_{u_{0}}, r, m)$, i.e., $(\\overline{\\phi}_{u_{0}}, r, m)$ is still a solution to \\eqref{eq:SSESF}. Moreover, it is easy to check that $F_{1}$ and $F_{2}$ are also invariant under this change, i.e.,\n\\begin{equation} \\label{eq:inv4F12}\n\\begin{aligned}\n\tF_{1} =& \\partial_{v}^{2} (r \\overline{\\phi}_{u_{0}}) - (\\partial_{v} \\lmb) \\overline{\\phi}_{u_{0}} \\\\\n\tF_{2} =& \\partial_{v} \\log \\lmb - \\frac{\\lmb}{(1-\\mu)} \\frac{\\mu}{r} +  \\partial_{v} \\overline{\\phi}_{u_{0}} \\Big( \\lmb^{-1} \\partial_{v} (r \\overline{\\phi}_{u_{0}}) - \\nu^{-1} \\partial_{u} ( r \\overline{\\phi}_{u_{0}}) \\Big).\n\\end{aligned}\n\\end{equation}\n\nThe new scalar field has been chosen so that $\\overline{\\phi}_{u_{0}}(u_{0}, \\cdot)$ and $\\partial_{v}(r \\overline{\\phi}_{u_{0}})(u_{0}, \\cdot)$ vanish at the axis, i.e.,\n\\begin{equation} \\label{dic.0.1}\n\\lim_{v \\to u_{0}+}\\overline{\\phi}_{u_{0}}(u_{0},v) = \\lim_{v \\to u_{0}+}\\partial_{v}(r \\overline{\\phi}_{u_{0}}) (u_{0},v) = \\lim_{v \\to u_{0}+} \\partial_{u}(r \\overline{\\phi}_{u_{0}})(u_{0},v) = 0.\n\\end{equation}\n\nWe claim that the original scalar field $\\phi(u, v)$ obeys the condition \n\\begin{equation}\\label{claim.vanishing}\n\\lim_{v \\to \\infty} \\phi(u_{0}, v) = 0\n\\end{equation} \nfor every $u_{0} \\geq 1$.\nTherefore, by the definition given in \\eqref{dic.0.0}, we see that $\\phi$ and $\\overline{\\phi}_{u_0}$ are also related by\n\\begin{equation} \\label{dic.0.2}\n\t\\phi(u,v) = \\overline{\\phi}_{u_{0}}(u,v) - \\lim_{v' \\to \\infty} \\overline{\\phi}_{u_{0}}(u, v').\n\\end{equation}\n\n\nTo establish the claim \\eqref{claim.vanishing}, we proceed as in the proof of Lemma~\\ref{lem:decay1:cptu:0}, but work with $\\phi$ rather than $r \\phi$. \nFix $u_{0} > 1$ and let $r_{1} > 0$ be a large number to be determined. For each $u \\geq 1$, let $v^{\\star}_{1}(u)$ be the unique $v$-value such that $r(u, v^{\\star}_{1}(u)) = r_{1}$. Consider $(u, v) \\in \\set{1 \\leq u \\leq u_{0}} \\cap \\set{r \\geq r_{1}}$. Using the uniform bound of $m$ and $\\frac{\\lmb}{1-\\mu}$ in terms of the data at $u= 1$ (which holds thanks to monotonicity), we may integrate \\eqref{eq:SSESF:dphi} along the incoming direction to estimate\n\\begin{equation*}\n\t\\abs{\\partial_{v} (r \\phi) (u, v) - \\partial_{v} (r \\phi)(1, v)} \n\t\\leq \\frac{C_{0}}{r(u, v)} \\sup_{u' \\in [1, u]} \\abs{\\phi(u', v)},\n\\end{equation*}\nwhere $C_{0}$ depends only on the data at $u=1$. Integrating both sides in the outgoing direction from $v_{1}^{\\star}(u)$ to $v$ (using Lemma~\\ref{dic.bnd4dvr} for the right-hand side) and dividing by $r = r(u, v)$, we obtain\n\\begin{equation} \\label{claim.vanishing.key}\n\\begin{aligned}\n\t\\abs{\\phi(u, v)} \\leq & \\frac{r_{1}}{r} \\abs{\\phi(u, v^{\\star}_{1}(u))} + \\frac{r(1, v^{\\star}_{1}(u))}{r} \\abs{\\phi(1, v^{\\star}_{1}(u))} \\\\\n\t& + \\frac{r(1, v)}{r} \\abs{\\phi(1, v)}\t\t+ \\frac{C_{0} \\Lambda}{r} \\log \\Big( \\frac{r}{r_{1}}\\Big) \\sup_{1 \\leq u' \\leq u, \\, v \\geq v^{\\ast}_{1}(u)} \\abs{\\phi}.\n\\end{aligned}\n\\end{equation}\nNow the idea is to use \\eqref{claim.vanishing.key} to first show that $\\phi$ is bounded on the region $\\set{1 \\leq u \\leq u_{0}}$, and then use \\eqref{claim.vanishing.key} again with the additional boundedness of $\\phi$ to conclude that \\eqref{claim.vanishing} holds.\nTo begin with, observe that $\\phi$ is bounded on each set compact subset of $\\calQ$, since it is a BV solution in the sense of Definition~\\ref{def:BVsolution}. Combined with the hypothesis that $\\phi(1, v) \\to 0$ as $v \\to \\infty$, we see that the first three terms are bounded by a constant that depends on $r_{1}$. On the other hand, by taking $r_{1}$ sufficiently large, the coefficient $(C_{0} \\Lambda \/ r) \\log ( r \/ r_{1} )$ of the last term can be made arbitrarily small for $r \\geq r_{1}$. This smallness allows us to absorb the last term to the left-hand side, and conclude the desired boundedness of $\\phi$ on the region $\\set{1 \\leq u \\leq u_{0}}$.\nThen plugging in $u = u_{0}$ and the uniform bound for $\\phi$ into \\eqref{claim.vanishing.key}, the claim \\eqref{claim.vanishing} follows from the hypothesis $\\lim_{v \\to \\infty} \\phi(1, v)  = 0$.\n\n\n\n\n\n\nLet\n\\begin{eqnarray*}\nI_1(u, v)&:=&\\int_{u}^v |\\partial_v^2(r \\overline\\phi_{u_{0}})| (u,v')dv'  ,\\quad I_2(u, v):=\\int_{u}^v |\\partial_v\\lambda | (u,v')dv' .\n\\end{eqnarray*}\n\nIn the following two lemmas, we will show that\n\\begin{align} \nI_1(u_0, v) \\leq & 3\\epsilon+ C_{\\Lambda} I_{1}(u_0,v) I_2(u_0,v) , \\label{main.ineq.dichotomy.1} \\\\\nI_2(u_0, v)\\leq & 3\\epsilon+ C_{\\Lambda} I_{1}(u_0,v)^{2} (1+I_{1}(u_0,v))^{2} (1+I_{2}(u_0,v))^{2} e^{C_{\\Lambda} I_{1}(u_0,v)^{2} (1+I_{2}(u_0,v))} \\label{main.ineq.dichotomy.2}\n\\end{align}\nfor every $V \\leq u_0 \\leq v$, with $C_{\\Lambda}$ independent of $u_0$ and $v$.\n\n\\begin{lemma} \\label{lem.dic.1}\nThere exists a constant $C_{\\Lambda} > 0$ such that for every $V \\leq u_0 \\leq v$,\n\\begin{equation*}\nI_1(u_0,v) \\leq 3\\epsilon+ C_{\\Lambda} I_{1}(u_0,v) I_2(u_0,v).\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nIn this proof, we fix $u_0 \\geq V$ and use the abbreviations\n\\begin{equation} \\label{eq:abbrev-overline-phi}\n\\overline{\\phi} := \\overline{\\phi}_{u_{0}} , \\quad\n\\partial_{v}(r \\overline{\\phi}) := \\partial_{v}(r \\overline{\\phi}_{u_{0}}) \n\\ \\hbox{ and } \\ \n\\partial_{v}^{2}(r \\overline{\\phi}) := \\partial_{v}^{2}(r \\overline{\\phi}_{u_{0}}).\n\\end{equation}\nBy Lemma \\ref{lem:est4phi}, we have\n\\begin{equation} \\label{phi.est}\n\t\\abs{\\overline{\\phi}(u_0,v)} \n\t\\leq \\frac{1}{r} \\int_{u_0}^{v} \\partial_{v}(r \\overline{\\phi}) (u_0, v') \\, \\mathrm{d} v' \n\t\\leq \\Lambda \\sup_{u_0 \\leq v' \\leq v} \\abs{\\partial_{v}(r \\overline{\\phi})(u_0,v')}.\n\\end{equation}\n\nBy the fundamental theorem of calculus and \\eqref{dic.0.1}, note that\n\\begin{equation} \\label{dic.1.0}\n\\sup_{u_0 \\leq v' \\leq v} |\\partial_{v}(r \\overline{\\phi})(u_0,v')|\\leq I_1(u_0,v).\n\\end{equation}\n\nThus, recalling the definition of $F_1$ in \\eqref{eq:inv4F12}, we have\n\\begin{equation*}\n\\begin{split}\nI_1(u_0,v) \\leq &\\int_{u_0}^v |F_1(u_0,v')|dv'+\\int_{u_0}^v|\\partial_v\\lmb||\\overline{\\phi}|(u_0,v')dv' \\\\\n\\leq &\\int_{u_0}^v |F_1(u_0,v')|dv'+ \\Lambda\\,  I_1(u_0,v) I_2(u_0,v) \n\t\t\\leq  3\\epsilon+C_{\\Lambda} I_{1}(u_0,v) I_2(u_0,v). \\qedhere\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\nWe now move on to estimate $I_2(u_0,v)$.\n\\begin{lemma} \\label{lem.dic.2}\nThere exists a constant $C_{\\Lambda} > 0$ such that for every $V \\leq u_0 \\leq v$,\n\\begin{equation*} \nI_2(u_0,v)\\leq 3\\epsilon+ C_{\\Lambda} I_{1}(u_0,v)^{2} (1+I_{1}(u_0,v))^{2} (1+I_{2}(u_0,v))^{2} e^{C_{\\Lambda} I_{1}(u_0,v)^{2} (1+I_{2}(u_0,v))}. \n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nAgain, we fix $u_0 \\geq V$ and use the abbreviation \\eqref{eq:abbrev-overline-phi}, as well as\n\\begin{equation} \\label{eq:abbrev-du-overline-phi}\n\t\\partial_{u} (r \\overline{\\phi}) (u,v) := \\partial_{u} (r \\overline{\\phi}_{u_{0}})(u,v) .\n\\end{equation}\nRecalling the equation for $F_2$ in \\eqref{eq:inv4F12}, in order to control $I_2(u_0,v)$ from $F_2$, we need to estimate $\\int_{u_0}^{v} (\\frac{\\lambda}{1-\\mu} \\frac{\\mu}{r})(u_0,v') \\mathrm{d} v'$ and $\\int_{u_0}^v \\partial_{v} \\overline{\\phi} ( \\lmb^{-1} \\partial_{v} (r \\overline{\\phi}) - \\nu^{-1} \\partial_{u} ( r \\overline{\\phi})) (u_0,v') \\mathrm{d} v'.$\nBy Lemma \\ref{lem:auxEqs},\n$$\\int_{u_0}^{v} (\\frac{\\lambda}{1-\\mu} \\frac{\\mu}{r})(u_0,v') \\mathrm{d} v' = \\log (1-\\mu(u_0,v))+\\int_{u_0}^v \\frac{r(\\partial_v \\overline{\\phi})^2}{\\lambda}(u_0,v') \\mathrm{d} v'.$$\nSince $\\calQ =\\calR$, the integrand on the left-hand side is non-negative. Notice furthermore that since $\\mu \\geq 0$, $\\log (1-\\mu(u_0,v)) <0$. Thus, \n\\begin{eqnarray*}\n\\int_{u_0}^{v} (\\frac{\\lambda}{1-\\mu} \\frac{\\mu}{r})(u_0,v') \\mathrm{d} v'\n&\\leq & |\\int_{u_0}^v \\frac{r(\\partial_v \\overline{\\phi})^2}{\\lambda}({u_0},v') \\mathrm{d} v'|\\\\\n&\\leq & \\int_{u_0}^v |\\partial_v \\overline{\\phi}({u_0},v')||(\\frac{\\partial_v(r\\overline{\\phi})}{\\lambda}- \\overline{\\phi})({u_0},v')| \\mathrm{d} v'\\\\\n&\\leq & 2 \\Lambda I_{1}({u_0},v)  \\int_{u_0}^v |\\partial_v \\overline{\\phi}({u_0},v')| \\, \\mathrm{d} v' ,\n\\end{eqnarray*}\nwhere we have used \\eqref{phi.est} and \\eqref{dic.1.0} on the last line. Using Lemma \\ref{lem:est4dvphi}, we estimate the integral on the last line by\n\\begin{equation} \\label{dic.2.0}\n\\int_{u_0}^v |\\partial_v \\overline{\\phi}({u_0},v')|dv' \\leq\n\\int_{u_0}^v|\\partial_v(\\lambda^{-1}\\partial_v(r \\overline{\\phi}))({u_0},v')| \\mathrm{d} v' ,\n\\end{equation}\nand the right hand side can in turn be estimated using \\eqref{dic.1.0} by\n\\begin{eqnarray}\n\\int_{u_0}^v|\\partial_v(\\lambda^{-1}\\partial_v(r \\overline{\\phi}))({u_0},v')| \\mathrm{d} v' \n&\\leq &\\int_{u_0}^v \\lambda^{-1}|\\partial_v^2(r \\overline{\\phi})({u_0},v')| \\mathrm{d} v'+\\int_{u_0}^v \\lambda^{-2}|\\partial_v\\lambda\\partial_v(r \\overline{\\phi})({u_0},v')| \\mathrm{d} v'\\notag\\\\\n&\\leq &\\Lambda I_1({u_0},v)+ \\Lambda^2 I_{1}({u_0},v) I_2({u_0},v). \\notag\n\\end{eqnarray}\n\nTherefore, we have\n\\begin{equation}\\label{dic.2.1}\n\\int_{u_0}^{v} (\\frac{\\lambda}{1-\\mu} \\frac{\\mu}{r})({u_0},v') \\mathrm{d} v'\\leq  C_{\\Lambda} I_1({u_0},v)^{2} (1+ I_{2}({u_0},v)).\n\\end{equation}\n\n\n\nWe now move on to bound $\\int_{u_0}^v \\partial_{v} \\overline{\\phi} \\, \\lmb^{-1} \\partial_{v} (r \\overline{\\phi}) ({u_0}, v') \\mathrm{d} v'$. Using \\eqref{dic.1.0} and \\eqref{dic.2.0}, we easily estimate\n\\begin{align}\n\\int_{u_0}^v \\abs{\\partial_{v} \\overline{\\phi} \\, \\lmb^{-1} \\partial_{v} ( r \\overline{\\phi}) ({u_0},v')} \\mathrm{d} v' \\notag\n\\leq & \\Lambda \\int_{{u_0}}^{v} \\abs{\\partial_{v} \\overline{\\phi}}({u_0}, v') \\, \\mathrm{d} v' \\sup_{{u_0} \\leq v' \\leq v} \\abs{\\partial_{v} ( r \\overline{\\phi}) ({u_0},v')} \\\\\n\\leq & C_{\\Lambda} I_{1}({u_0},v)^{2} (1+I_{2}({u_0},v)). \\label{dic.2.2}\n\\end{align}\n\nFinally, we are only left to bound $- \\int_{u_0}^v \\partial_{v} \\overline{\\phi} \\, \\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) ({u_0},v') \\mathrm{d} v'$. As before, we begin by estimating\n\\begin{align}\n\\int_{u_0}^v \\abs{\\partial_{v} \\overline{\\phi} \\, \\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) ({u_0},v')} \\mathrm{d} v'\n\\leq & \\int_{{u_0}}^{v} \\abs{\\partial_{v} \\overline{\\phi}}({u_0}, v') \\, \\mathrm{d} v' \\sup_{{u_0} \\leq v' \\leq v} \\abs{\\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) ({u_0},v')} \\notag \\\\\n\\leq & C_{\\Lambda} I_{1}({u_0},v) (1+I_{2}({u_0},v)) \\sup_{{u_0} \\leq v' \\leq v} \\abs{\\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) ({u_0},v')}. \\label{dic.2.3}\n\\end{align}\n\nIn this case, we do not wish to pull out $\\nu$ as we have not assumed any bound on it. Instead, we consider $\\nu^{-1} \\partial_{u} ( r \\overline{\\phi})$ as a whole and note that\n\\begin{equation}\\label{inv.wave.eqn}\n\\partial_{v}(\\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) ) \n= \t- \\Big( \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} \\Big) \\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) \n\t+ \\Big( \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} \\Big) \\overline{\\phi}.\n\\end{equation}\nThe equation \\eqref{inv.wave.eqn} holds since by \\eqref{eq:SSESF:dr} and \\eqref{eq:SSESF:dphi}, we have \n\\begin{equation*}\n\\partial_{v}(\\nu^{-1} \\partial_{u} ( r {\\phi}) ) \n= \t- \\Big( \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} \\Big) \\nu^{-1} \\partial_{u} ( r {\\phi}) \n\t+ \\Big( \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} \\Big) {\\phi}\n\\end{equation*}\nand moreover both the left hand side and the right hand side of the equation are invariant under the transformation $\\phi\\to \\phi+c$.\n\nTherefore, by the variation of constants formula and \\eqref{dic.0.1}, we have\n\\begin{equation*}\n\t\\nu^{-1} \\partial_{u} (r \\overline{\\phi})(u_0, v)\n\t= e^{-J({u_0},v)} \\int_{{u_0}}^{v} e^{J({u_0},v')}  \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} \\,  \\overline{\\phi} ({u_0}, v') \\, \\mathrm{d} v',\n\\end{equation*}\nwhere\n\\begin{equation*}\n\tJ({u_0},v) := \\int_{{u_0}}^{v} \\frac{\\lmb}{1-\\mu} \\frac{\\mu}{r} ({u_0}, v') \\, \\mathrm{d} v'.\n\\end{equation*}\n\nBy \\eqref{dic.1.0} and \\eqref{dic.2.1}, we have\n\\begin{equation*}\n\t\\sup_{{u_0} \\leq v' \\leq v}\\abs{\\nu^{-1} \\partial_{u} (r \\overline{\\phi}) ({u_0},v')} \n\t\\leq C_{\\Lambda} I_{1}({u_0},v)^{3} (1+I_{2}({u_0},v)) e^{C_{\\Lambda} I_{1}({u_0},v)^{2} (1+I_{2}({u_0},v))}.\n\\end{equation*}\n\nThen by \\eqref{dic.2.3}, we conclude that \n\\begin{equation} \\label{dic.2.4}\n\\int_{u_0}^v \\abs{\\partial_{v} \\overline{\\phi} \\, \\nu^{-1} \\partial_{u} ( r \\overline{\\phi}) ({u_0},v')} \\mathrm{d} v'\n\t\\leq C_{\\Lambda} I_{1}({u_0},v)^{4} (1+I_{2}({u_0},v))^{2} e^{C_{\\Lambda} I_{1}({u_0},v)^{2} (1+I_{2}({u_0},v))}.\n\\end{equation}\n\nCombining \\eqref{dic.2.1}, \\eqref{dic.2.2} and \\eqref{dic.2.4}, we conclude that \\eqref{main.ineq.dichotomy.2} holds. \\qedhere\n\\end{proof}\n\nNext, we apply \\eqref{main.ineq.dichotomy.1} and \\eqref{main.ineq.dichotomy.2} to show that \n\\begin{proposition} \\label{dic.main.prop}\nFor ${u_0}$ sufficiently large and $v\\geq {u_0}$, we have\n$$I_1({u_0},v)+I_2({u_0},v)\\leq C_{\\Lambda} \\epsilon.$$\n\\end{proposition}\n\\begin{remark}\nIf it is the case that $\\int_{u_0}^v(|\\partial_v^2(r\\overline{\\phi})| + |\\partial_v \\lambda|)({u_0},v')dv'$ is continuous in $v$ for each fixed ${u_0}$, then the desired conclusion follows from \\eqref{main.ineq.dichotomy.1} and \\eqref{main.ineq.dichotomy.2} via a simple continuity argument in $v$. In particular, the conclusion follows in the case where the initial data of $\\partial_v(r \\phi)$ are in $W^{1,1}$ or $C^{1}$. The only remaining difficulty is therefore to control the size of the delta function singularities in $\\partial_v^2(r\\overline{\\phi})$ in the general case where we only have a BV solution.\n\\end{remark}\n\\begin{proof}\nWe begin by studying the propagation of discontinuities for a BV solution to \\eqref{eq:SSESF}. In the general case where $\\partial_v(r \\phi)(1, \\cdot)$ is only in BV and contains jump discontinuities (at which $\\partial_{v}(r \\phi)(1, \\cdot)$ is assumed to be right-continuous), notice that the jump discontinuities for a BV function are discrete, i.e., they occur only at a (possibly infinite) sequence of points $V<v_1<v_2< v_3 < \\cdots$. On the other hand, note that by the initial condition $r = 2v$ on $C_{1}$, we have $\\lmb(1,v) = \\frac{1}{2}$; in particular, $\\lmb$ is continuous initially.\n\nThanks to the initial condition, it follows that $\\lmb$ does not possess any discontinuities outside $\\Gamma$. Indeed, from the definition of a BV solution, $m$ and $r$ are continuous. Then, by \\eqref{eq:SSESF:dr}, we see that that $\\nu$ is Lipschitz in the $v$ direction outside of $\\Gamma$, with bounded Lipschitz constant on each compact interval of $u$. Looking back at \\eqref{eq:SSESF:dr} and recalling that $\\lmb(1,v) = \\frac{1}{2}$, we then see that $\\lmb$ does not possess any discontinuities outside $\\Gamma$ as desired. Since $\\lmb$ is a priori in BV, it follows that $\\int_{u_0}^{v} \\abs{\\partial_{v} \\lmb({u_0},v')} \\, \\mathrm{d} v'$ is continuous in $v \\in ({u_0}, \\infty)$ with $\\int_{{u_0}}^{v} \\abs{\\partial_{v} \\lmb({u_0},v')} \\, \\mathrm{d} v' \\to 0$ as $v \\to {u_0}+$. \n\nBy the above regularity statements and equation \\eqref{eq:SSESF:dphi}, as well as the fact that $\\phi$ is continuous outside $\\Gamma$ by the definition of a BV solution, it now follows that the jump discontinuities of $\\partial_{v}(r \\phi)$ are propagated along constant $v_{i}$ curves. Therefore, for ${u_0} \\geq V$, we see that $\\partial_v (r \\phi)({u_0},v)$ is a right-continuous BV function on $({u_0}, \\infty)$ with jump discontinuities at ${u_0} < v_1 < v_2 < v_3 < \\cdots$ with the same sizes as $\\partial_{v} (r \\phi)(1,v)$. \nFrom \\eqref{dic.0.0}, notice that, using the abbreviations in \\eqref{eq:abbrev-overline-phi} and \\eqref{eq:abbrev-du-overline-phi},\n\\begin{equation*}\n\t\\int_{{u_0}}^{v} \\abs{\\partial_{v}^{2} (r \\overline{\\phi})({u_0},v')} \\, \\mathrm{d} v' = \\int_{{u_0}}^{v} \\abs{(\\partial_{v}^{2} (r \\phi) - c \\partial_{v} \\lmb)({u_0},v')} \\, \\mathrm{d} v',\n\\end{equation*}\nfor the constant $c = \\lim_{v \\to {u_0}+} \\phi({u_0},v)$, which is independent of $v$. Thanks to the continuity property of $\\lmb({u_0}, \\cdot)$, we see that the integral of $\\abs{\\partial_{v}^{2}(r \\overline{\\phi})({u_0}, \\cdot)}$ has the same jump discontinuities as $\\abs{\\partial_{v}^{2} (r \\phi)({u_0}, \\cdot)}$. In particular, by \\eqref{data.BV.bd}, each jump of $I_{1}({u_0},v)$ is at most of size $\\epsilon$.\n\nFix $C_{\\Lambda} > 1$ to be larger than the maximum of the constants from \\eqref{main.ineq.dichotomy.1} and \\eqref{main.ineq.dichotomy.2}. First, a standard continuity argument using \\eqref{main.ineq.dichotomy.1} and \\eqref{main.ineq.dichotomy.2} implies that if \n$$\\lim_{v\\to v_i+}\\int_{u_0}^v |\\partial_v^2(r\\overline{\\phi}) ({u_0},v')| \\mathrm{d} v'\\leq 5 C_{\\Lambda}\\epsilon\n\\hbox{ and }\n\\lim_{v\\to v_i+}\\int_{u_0}^v |\\partial_v \\lambda({u_0},v')| \\mathrm{d} v'\\leq 5\\epsilon,$$\n(with the convention $v_{0} := {u_0}$) then\n$$\\int_{u_0}^v |\\partial_v^2(r\\overline{\\phi}) ({u_0},v')| \\mathrm{d} v'\\leq 4 C_{\\Lambda}\\epsilon\n\\hbox{ and }\n\\int_{u_0}^v |\\partial_v \\lambda({u_0},v')| \\mathrm{d} v'\\leq 4 \\epsilon,$$\nfor $v_i< v< v_{i+1}$.\n\nAssume, for the sake of contradiction, that the conclusion of the proposition is not satisfied. Recall that the integral of $\\abs{\\partial_v\\lambda}$ is continuous. Thus, we have that for some $v_i$ with $i > 0$\n$$\\lim_{v\\to v_i-}\\int_{u_0}^v |\\partial_v^2(r\\overline{\\phi}) ({u_0},v')| \\mathrm{d} v'\\leq 4 C_{\\Lambda}\\epsilon,$$\nholds, but at the same time\n$$\\lim_{v\\to v_i+}\\int_{u_0}^v |\\partial_v^2(r\\overline{\\phi}) ({u_0},v')| \\mathrm{d} v' > 5 C_{\\Lambda}\\epsilon.$$\n\nHowever, we have seen that the size of the jump in $I_1({u_0},v)$ is bounded by $\\epsilon$, which is smaller than $C_{\\Lambda} \\epsilon$ if $C_{\\Lambda}>1$. This leads to a contradiction and thus the conclusion of the proposition holds.\n\\end{proof}\n\nWe are now ready to conclude the proof of Theorem \\ref{thm.dichotomy}.\n\\begin{proof} [Conclusion of Proof of Theorem \\ref{thm.dichotomy}]\nWe first establish \\eqref{dichotomy.goal}. In what follows, we use the abbreviations in \\eqref{eq:abbrev-overline-phi} and \\eqref{eq:abbrev-du-overline-phi}, such as $\\overline{\\phi} = \\overline{\\phi}_{u_{0}}$ etc. The idea is to transform back to $(\\overline{\\phi}, r, m) \\mapsto (\\phi, r, m)$ using \\eqref{dic.0.2}. Note that $\\abs{\\partial_{v} \\lmb}$ remains the same under this change, so it suffices to estimate $\\abs{\\partial_{v}^{2} (r \\phi)}$. By \\eqref{dic.2.0} and Proposition \\ref{dic.main.prop}, for sufficiently large ${u_0}$, the limit $\\overline{\\phi}({u_0}, \\infty) := \\lim_{v \\to \\infty} \\overline{\\phi}({u_0},v)$ exists and satisfies\n\\begin{equation*}\n\t\\abs{\\overline{\\phi}({u_0}, \\infty)} \\leq C_{\\Lambda} \\epsilon,\n\\end{equation*}\nwhere we note that $C_{\\Lambda}$ is independent of ${u_0}$.\n\nBy \\eqref{dic.0.2}, we have $\\phi(u,v) = \\overline{\\phi}(u,v) - \\overline{\\phi}(u, \\infty)$ for all $u$. Thus, using Proposition \\ref{dic.main.prop}, we estimate\n\\begin{align*}\n\t\\int_{{u_0}}^{\\infty} \\abs{\\partial_{v}^{2} ( r\\phi)({u_0},v)} \\, \\mathrm{d} v\n\t= & \\int_{{u_0}}^{\\infty} \\abs{\\partial_{v}^{2} ( r \\overline{\\phi}({u_0},v) - r \\overline{\\phi}({u_0}, \\infty))} \\, \\mathrm{d} v \\\\\n\t\\leq & \\int_{{u_0}}^{\\infty} \\abs{\\partial_{v}^{2} (r \\overline{\\phi})({u_0},v)} \\, \\mathrm{d} v \n\t\t+ \\abs{\\overline{\\phi}({u_0},\\infty)} \\int_{{u_0}}^{\\infty} \\abs{\\partial_{v} \\lmb ({u_0},v)} \\, \\mathrm{d} v \\\\\n\t\\leq & C_{\\Lambda} (\\epsilon + \\epsilon^{2}).\n\\end{align*}\nSince $u_0\\geq V$ is arbitrary, this proves \\eqref{dichotomy.goal}.\n\nFinally, we prove that the conditions $(2)$ and $(3)$ of Definition \\ref{def:locBVScat} hold.\nIndeed, since $\\partial_{v} \\log \\lmb = \\lmb^{-1} \\partial_{v} \\lmb$, $(3)$ in Definition \\ref{def:locBVScat} follows from \\eqref{dichotomy.goal} and \\eqref{dic.bnd4dvr}. In fact, $(3)$ in Definition \\ref{def:locBVScat} holds with arbitrarily large $R > 0$. Next, by \\eqref{eq:SSESF:dm}, non-negativity of $1-\\mu$ and $\\mu$ (by Lemma \\ref{lem:mntn4r}) and the fact that $m$ is invariant under $\\phi \\mapsto \\overline{\\phi}$,\n\\begin{equation*}\n\tm(u_0, v) \n\t\\leq \\frac{1}{2} \\sup_{u_0 \\leq v' \\leq v} \\abs{(\\lmb^{-1} \\partial_{v}(r \\overline{\\phi}) - \\overline{\\phi})(u_0, v')} \\int_{u_0}^{v} \\abs{\\partial_{v} \\overline{\\phi}(u_0, v')} \\, \\mathrm{d} v'\n\\end{equation*}\nwhere the right-hand side is $\\leq C_{\\epsilon, \\Lambda} \\epsilon$ (with $C_{\\epsilon, \\Lambda}$ non-decreasing in $\\epsilon$) by the estimates proved so far. Therefore, $(2)$ of Definition \\ref{def:locBVScat} follows.\nThis concludes the proof of Theorem \\ref{thm.dichotomy}. \\qedhere\n\\end{proof}\n\n\\section{Refinement in the small data case} \\label{sec:smallData}\nIn this section, we sketch a proof of Theorem \\ref{thm:smallData}. The idea is to revisit the proofs of the main theorems (Theorems \\ref{main.thm.1} and \\ref{main.thm.2}), and notice that all the required smallness can be obtained by taking initial total variation of $\\partial_{v}(r \\phi)$ small. Key to this idea is the following lemma.\n\n\\begin{lemma} \\label{lem:smallData}\nThere exist universal constants $\\epsilon_{0}$ and $C_{0}$ such that for $\\epsilon < \\epsilon_{0}$, the following holds. Suppose that $\\lmb(1, \\cdot) = \\frac{1}{2}$ and $\\partial_{v}(r \\phi)(1, \\cdot)$ is of bounded variation with\n\\begin{equation} \\label{eq:smallData:hyp}\n\t\\int_{C_{1}} \\abs{\\partial_{v}^{2} (r \\phi)} < \\epsilon.\n\\end{equation}\n\nSuppose furthermore that $\\lim_{v \\to \\infty} \\phi(1, v) = 0$. Then the maximal development  $(\\phi, r, m)$ satisfies condition $(1)$ of Definition \\ref{def:locBVScat} (future completeness of radial null geodesics) and obeys\n\\begin{gather}\n\t\\sup_{v \\in [1, \\infty)} \\int_{\\underline{C}_{v}} \\abs{\\frac{\\mu }{(1-\\mu)} \\frac{\\nu}{r}} \n\t+ \\sup_{u \\in [1, \\infty)} \\int_{C_{u}} \\abs{\\frac{\\mu }{(1-\\mu)} \\frac{\\lmb}{r}} \\leq C_{0} \\epsilon^{2},\t\\label{eq:smallData:smallPtnl} \\\\\n\t\\sup_{v \\in [1, \\infty)} \\int_{\\underline{C}_{v}} \\abs{\\partial_{u} \\phi}\n\t+ \\sup_{u \\in [1, \\infty)} \\int_{C_{u}} \\abs{\\partial_{v} \\phi} \\leq C_{0} \\epsilon,\t\t\t\t \\label{eq:smallData:smallDphi} \\\\\n\t\\sup_{v \\in [1, \\infty)} \\int_{\\underline{C}_{v}} (\\abs{\\partial_{u}^{2} (r \\phi)} + \\partial_{u} \\log \\nu)\n\t+ \\sup_{u \\in [1, \\infty)} \\int_{C_{u}} (\\abs{\\partial_{v}^{2} (r \\phi)} + \\partial_{v} \\log \\lmb) \\leq C_{0} \\epsilon.\t\t\t\t \\label{eq:smallData:smallTV} \n\\end{gather}\n\nMoreover, the bounds in Proposition \\ref{prop:geomLocBVScat} hold with\n\\begin{equation}  \\label{eq:smallData:geom}\n\tK + \\Lambda \\leq C_{0}, \\quad\n\t\\Psi \\leq C_{0} \\epsilon.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof} \nThis lemma is an easy consequence of Theorem \\ref{Chr.BV.Thm} and Lemma \\ref{lem:auxEqs} once we show \n\\begin{equation*}\n\\sup_{\\calQ} \\abs{\\partial_{v} (r \\phi)} \\leq C_{0} \\epsilon,\n\\end{equation*}\nusing the additional condition $\\lim_{v \\to \\infty} \\phi(1, v) = 0$. By Lemma \\ref{lem:est4dvphi}, note that $\\int_{C_{1}} \\abs{\\partial_{v} \\phi} \\leq C \\epsilon$; therefore, integrating from $v = \\infty$, we have $\\lim_{v \\to 1+} \\abs{\\phi(1, v)} \\leq C \\epsilon$. Then using \\eqref{eq:smallData:hyp} to integrate from $v = 1$, where we note that $\\lim_{v \\to 1+}\\phi(1, v) = \\lim_{v \\to 1+} \\partial_{v}(r \\phi)(1, v)$, we obtain\n\\begin{equation*}\n\t\\sup_{C_{1}} \\abs{\\partial_{v} (r \\phi)} \\leq C \\epsilon.\n\\end{equation*}\n\nUsing \\eqref{eq:SSESF:dphi'}, $\\partial_{u} \\lmb \\leq 0$, Lemma \\ref{lem:est4phi} (to control $\\abs{\\phi}$ from $\\abs{\\partial_{v}(r \\phi)}$) and $\\frac{1}{3} \\leq \\lmb \\leq \\frac{1}{2}$ (by Theorem \\ref{Chr.BV.Thm}), it follows that\n\\begin{align*}\n\t\\sup_{\\mathcal D(1, v)}\\abs{\\partial_{v}(r \\phi)} \n\t\\leq & \\sup_{1 \\leq v' \\leq v} \\abs{\\partial_{v} (r \\phi)(1, v')} + \\sup_{(u,v) \\in \\mathcal D(1,v)} \\sup_{1 \\leq u' \\leq u} \\abs{\\phi(u', v)} \\int_{1}^{u} (- \\partial_{u} \\lmb)(u', v) \\, \\mathrm{d} u' \\\\\n\t\\leq & C \\epsilon + \\frac{1}{2} \\sup_{\\mathcal D(1,v)} \\abs{\\partial_{v}(r \\phi)},\n\\end{align*} \nwhich proves $\\sup_{\\calQ} \\abs{\\partial_{v} (r \\phi)} \\leq C_{0} \\epsilon$, as desired. \\qedhere\n\\end{proof}\n\nEquipped with Lemma \\ref{lem:smallData}, we now proceed to outline the proof of Theorem \\ref{thm:smallData}.\n\\begin{proof} [Proof of $(1)$ in Theorem \\ref{thm:smallData}]\nThat $(\\phi, r, m)$ is globally BV scattering follows from Theorem \\ref{thm.dichotomy} and the fact that initial data with small total variation cannot lead to a development which blows up at infinity; the latter fact follows from the results proved by Christodoulou \\cite[Section 4, Theorem 6.2]{Christodoulou:1993bt}. \n\nIt remains to prove that \\eqref{eq:decay1:1}--\\eqref{eq:decay1:3} hold with $A_{1} \\leq C_{\\mathcal I_{1}} (\\mathcal I_{1} + \\epsilon)$ if $\\epsilon > 0$ is sufficiently small. By \\eqref{eq:decay1:H1}, it follows Lemma \\ref{lem:decay1:cptu:0} holds with $H_{1} \\leq C_{\\mathcal I_{1}} (\\mathcal I_{1} + \\epsilon)$, and \\eqref{eq:decay1:extr} in Lemma \\ref{lem:decay1:extr} becomes\ufffd\n\\begin{equation*} \\tag{\\ref{eq:decay1:extr}$'$}\n\t\\sup_{C_{u}} r^{\\omega} \\abs{\\partial_{v} (r \\phi)} \\leq C_{\\mathcal I_{1}} u_{1} (\\mathcal I_{1} + \\epsilon) + C M_{i} u_{1}^{-1} \\mathcal B_{1}(U),\n\\end{equation*}\n\nNote that $M_{i} \\leq C \\mathcal I_{1}^{2}$. Then repeating the arguments in \\S \\ref{sec.full.decay.1}, we see that \\eqref{eq:decay1:intr:pf:key} becomes\n\\begin{equation*} \\tag{\\ref{eq:decay1:intr:pf:key}$'$}\n\t\\mathcal B_{1}(U) \\leq C_{\\mathcal I_{1}} u_{1} (\\mathcal I_{1} + \\epsilon) + C (\\mathcal I_{1}^{2} u_{1}^{-1} + \\epsilon^{2}) \\mathcal B_{1}(U). \n\\end{equation*}\n\nIt is important to note that the constant $C$ in the last term does not depend on $\\mathcal I_{1}$. Take $u_{1} = 1000 C (1+\\mathcal I_{15})^{2}$. Then for $\\epsilon > 0$ sufficiently small (independent of $\\mathcal I_{1}$), we derive\n\\begin{equation*}\n\t\\mathcal B_{1}(U) \\leq C_{\\mathcal I_{1}} (\\mathcal I_{1} + \\epsilon)\n\\end{equation*}\n\nIt then follows that \\eqref{eq:decay1:1} and \\eqref{eq:decay1:2} hold with $A_{1} \\leq C_{\\mathcal I_{1}} (\\mathcal I_{1} + \\epsilon)$. Applying Lemma \\ref{lem:decay1:uDecay4durphi}, we conclude that \\eqref{eq:decay1:3} holds with $A_{1} \\leq C_{\\mathcal I_{1}} (\\mathcal I_{1} + \\epsilon)$ as well. \\qedhere\n\\end{proof}\n\n\\begin{proof} [Proof of $(2)$ in Theorem \\ref{thm:smallData}]\nWe need to prove that \\eqref{eq:decay2:1}--\\eqref{eq:decay2:4} hold with $A_{2} \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon)$. The key is to show that Proposition \\ref{prop:decay2:nullStr} holds with\n\\begin{equation} \\label{eq:smallData:pf2:key}\n\tA_{2}' \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon)\n\\end{equation}\n\nIndeed, by the explicit bounds on the constants (in particular, \\eqref{eq:decay2:rDecay:1}, \\eqref{eq:decay2:rDecay:2}, \\eqref{eq:decay2:uDecayInExtr:1}, \\eqref{eq:decay2:uDecayInExtr:2}, \\eqref{eq:decay2:final:aux} and \\eqref{eq:decay2:A2''}), the desired conclusion easily follows once \\eqref{eq:smallData:pf2:key} is established.\n\nNote that $\\mathcal I_{1} \\leq \\mathcal I_{2}$ by definition, and thus $A_{1} \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon)$ by the preceding proof. We furthermore claim that the following statements hold:\n\\begin{itemize}\n\\item Lemma \\ref{lem:decay2:key4nullStr} holds with\n\\begin{equation} \\label{eq:smallData:pf2:1}\n\t \\epsilon(u_{2}) \\leq C \\epsilon,\n\\end{equation}\nfor every $u_{2} \\geq 1$.\n\n\\item We have\n\\begin{equation} \\label{eq:smallData:pf2:2}\n\t H_{2}'(1) \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon),\n\\end{equation}\nwhere we remind the reader that\n\\begin{equation*}\nH_{2}'(1) = \\sup_{\\set{(u,v) : u \\in [1, 3] \\, v \\in [u, 3u]}} u^{\\omega} \\Big( \\abs{\\partial_{v}^{2}(r \\phi)} + \\abs{\\partial_{u}^{2}(r \\phi)} + \\abs{\\partial_{v} \\lmb} + \\abs{\\partial_{u} \\nu}\\Big)\n\\end{equation*}\naccording to \\eqref{eq:decay2:def4H'2}.\n\\end{itemize}\n\nThe first claim follows easily from Lemma \\ref{lem:smallData} and \\eqref{eq:decay2:eps}. For the second claim, since $1 \\leq u \\leq 3$, it suffices to prove\\footnote{Recall that $\\mathcal D(1,9) = \\set{(u,v) : u \\in [1, 3] \\, v \\in [u, 3u]}$ is the domain of dependence of $C_{1} \\cap \\set{1 \\leq v \\leq 9}$.}\n\\begin{equation*}\n\t \\sup_{\\mathcal D(1,9)} \\Big( \\abs{\\partial_{v}^{2}(r \\phi)} + \\abs{\\partial_{u}^{2}(r \\phi)} + \\abs{\\partial_{v} \\lmb} + \\abs{\\partial_{u} \\nu}\\Big)\\leq C (\\mathcal I_{2} + \\epsilon),\n\\end{equation*}\nwhich follows from a persistence of regularity argument, similar to our proof of Lemma \\ref{lem:decay2:key4nullStr}. \n\nTo conclude the proof, recall that we had\n\\begin{equation*}\n\t\\mathcal B_{2}(U) \\leq H_{2}''(u_2) + \\epsilon''(u_{2}) \\mathcal B_{2}(U)\n\\end{equation*}\nwhere $\\mathcal B_{2}(U)$ was defined in \\eqref{eq:decay2:def4B2}, and $H_{2}''(u_2)$, $\\epsilon''(u_{2})$ obey the bounds in \\eqref{eq:decay2:H2''} and \\eqref{eq:decay2:eps''} respectively. Thanks to \\eqref{eq:smallData:pf2:1}, it follows that we may take $u_{2} = 1$ and $\\epsilon''(1) \\leq C\\epsilon$, where $C$ does not depend on $\\mathcal I_{2}$. Next, since $u_{2} = 1$, we see that $H_{2}''(1) \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon)$ by \\eqref{eq:smallData:pf2:2}. Therefore, for $\\epsilon > 0$ sufficiently small (independent of $\\mathcal I_{2}$), we conclude that\n\\begin{equation*}\n\t\\mathcal B_{2}(U) \\leq C_{\\mathcal I_{2}} (\\mathcal I_{2} + \\epsilon),\n\\end{equation*}\nwhich proves that Proposition \\ref{prop:decay2:nullStr} holds with \\eqref{eq:smallData:pf2:key}, as desired.  \\qedhere\n\\end{proof}\n\n\n\\section{Optimality of the decay rates} \\label{sec.opt}\nIn this section, we show the optimality of the decay rates obtained above, i.e., we prove Theorems \\ref{thm.opt.1} and \\ref{thm.opt.2}. \n\n\\subsection{Optimality of the decay rates, in the case $1 < \\omega' < 3$}  \\label{subsec.opt.1}\nIn this subsection, we prove Theorem \\ref{thm.opt.1}. More precisely, we will demonstrate that the proof of the upper bounds for $\\phi$ and its derivatives can in fact be sharpened to give also lower bounds for $\\partial_v(r\\phi)$ and $\\partial_u(r\\phi)$ if the initial data satisfy appropriate lower bounds for $\\omega<3$.\n\n\\begin{proof}[Proof of Theorem \\ref{thm.opt.1}]\n\nWe first prove the lower bound for $\\partial_v(r\\phi)$. We split the spacetime into the exterior region $\\PD_{\\mathrm{ext}}$ and interior region $\\PD_{\\mathrm{int}}$ as before. Notice that in the exterior region, $u\\lesssim r$ and it suffices to prove a lower bound for $r^\\omega\\partial_v(r\\phi)$. Similarly, in the interior region, $r\\lesssim u$ and it suffices to prove a lower bound for $u^\\omega\\partial_v(r\\phi)$.\n\nRevisiting the proof of Lemma \\ref{lem:decay1:extr}, we note that instead of controlling $\\partial_v(r\\phi)$ by the initial data and error terms, we can bound the difference between $\\partial_v(r\\phi)(u,v)$ and the corresponding initial value of $\\partial_v(r\\phi)(1,v)$. More precisely, from the proof of Lemma \\ref{lem:decay1:extr}, we have\n\n\\begin{align*}\n\t\\abs{\\partial_{v} (r \\phi)(u,v)-\\partial_{v} (r \\phi)(1, v)}\n\t\\leq  \\frac{u_{1} K M_{i}}{r^2(u,v)(1+r(u,v))} H_{1} + \\frac{K M(u_{1})}{u_{1} r^{\\omega}(u,v)} \\mathcal B_{1}(U)\n\\end{align*}\nin the case $2<\\omega<3$ and\n\\begin{align*}\n\t\\abs{\\partial_{v} (r \\phi)(u,v)-\\partial_{v} (r \\phi)(1, v)}\n\t\t\\leq \\frac{\\omega K M_{i}}{r(u,v)(1+r(u,v))} H_{1} + \\frac{\\omega K M(u_{1})}{u_{1} r^{\\omega}(u,v)} \\mathcal B_{1}(U).\n\\end{align*}\nin the case $1<\\omega\\leq 2$. By the decay results proved in \\S \\ref{sec.full.decay.1}, we have\n$$\\sup_u(H_{1}+ \\mathcal B_{1}(u))\\leq A$$\nfor some constant $A$. Therefore, by choosing $u_1$ sufficiently large, we have in the region $3u\\leq v$,\n$$r^{\\omega}\\abs{\\partial_{v} (r \\phi)(u,v)-\\partial_{v} (r \\phi)(1, v)}\\leq \\frac{L}{4},$$\nas long as $u\\geq u_1$.\nWe now apply the assumption on the lower bound for the initial data $r^{\\omega} \\partial_{v} (r \\phi)(1, v)\\geq L$ for $v\\geq V$. Choosing $u$ larger if necessary, we can assume that $u\\geq V$. Then, we derive that in $3u\\leq v$,\n$$r^{\\omega}\\partial_{v} (r \\phi)(u,v)\\geq \\frac{L}{2}.$$\n\nWe now move to the interior region where $3u \\geq v$. To this end, we improve the bounds in \\eqref{eq:decay1:intr:pf:1}. First, notice that the lower bound in the exterior region implies that there exists $L'$ such that \n\\begin{eqnarray}\nu^{\\omega}\\partial_{v} (r \\phi)(u,v)\\geq L'\\label{lower.bd.L1}\n\\end{eqnarray}\nfor $3u \\leq v$. Then, integrating \\eqref{eq:SSESF:dphi} along the incoming direction from $(u\/3, v)$ to $(u, v)$, we get\n\\begin{equation*}\n\\begin{aligned}\n\t\\abs{\\partial_{v} (r \\phi) (u,v)-\\partial_{v} (r \\phi) (u\/3,v)} \n\t\\leq  \\frac{1}{2} (\\sup_{u' \\in [u\/3, u]} \\sup_{C_{u'}} \\abs{\\phi}) \\int_{u\/3}^{u} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u'.\n\\end{aligned}\n\\end{equation*} \nBy Theorem \\ref{main.thm.1}, we have \n$$\\sup_{C_u}|\\phi|\\leq A_{1} u^{-\\omega}$$\nfor some $A_{1} > 0$. Lemma \\ref{lem:smallPtnl} implies that \n$$\\int_{u\/3}^{u} \\abs{\\frac{2m \\nu}{(1-\\mu) r^{2}} (u', v)} \\, \\mathrm{d} u'\\to 0$$\nas $u\\to \\infty$. Thus the right hand side can be bounded by $\\frac{L' u^{\\omega}}{2}$ after choosing $u$ to be sufficiently large.\nCombining this with the lower bound \\eqref{lower.bd.L1}, we have\n$$u^{\\omega}\\partial_{v} (r \\phi) (u,v) \\geq \\frac{L'}{2}$$\nfor $3u \\leq v$ and $u$ sufficiently large.\n\nWe now proceed to obtain the lower bound for $\\partial_u(r\\phi)$ by revisiting the proof of Lemma \\ref{lem:decay1:uDecay4durphi}. Integrating \\eqref{eq:SSESF:dphi} along the outgoing direction from $(u, u)$ to $(u, v)$, we have\n\\begin{align}\\label{est.durphi.diff}\n\t\\abs{\\partial_{u} (r \\phi)(u,v) - \\lim_{v' \\to u+} \\partial_{u} (r \\phi)(u, v')}\n\t\\leq & \\int_{C_{u}} \\abs{\\frac{\\mu\\lmb\\nu}{(1-\\mu)r}\\phi}.\n\\end{align}\nAs before, we Theorem \\ref{main.thm.1}, i.e.,\n$$\\sup_{C_u}|\\phi|\\leq A_{1} u^{-\\omega}$$\nfor some $A_{1} > 0$. By Lemma \\ref{lem:smallPtnl} and the upper bound \\eqref{eq:bnd4dur} for $|\\nu|$, we have\n$$\\int_{C_{u}} \\abs{\\frac{\\mu\\lmb\\nu}{(1-\\mu)r}} \\to 0, \\quad\\mbox{as }u\\to\\infty.$$\nTherefore, we can choose $u$ sufficiently large such that\n$$u^{\\omega} \\int_{C_{u}} \\abs{\\frac{\\mu\\lmb\\nu}{(1-\\mu)r}\\phi}\\leq \\frac{L'}{4}.$$\nReturning to \\eqref{est.durphi.diff} and recalling that for $u$ large,\n$$-\\lim_{v' \\to u+} \\partial_{u} (r \\phi)(u, v')=\\lim_{v' \\to u+} \\partial_{v} (r \\phi)(u, v')\\geq \\frac{L'}{2}u^{-\\omega},$$\nwe have\n$$-\\partial_{u} (r \\phi)(u,v) \\geq \\frac{L'}{4}u^{-\\omega}$$\nfor $u$ sufficiently large, as desired. \\qedhere\n\n\\end{proof}\n\n\\subsection{Key lower bound lemma}\nThe goal of the remainder of this section is to prove Theorem \\ref{thm.opt.2}. In this subsection we establish the following result, which provides a sufficient condition for the desired lower bounds on the decay of $\\phi$ in terms of a number (called $\\mathfrak{L}$) computed on $\\mathcal I^{+}$. This will be an important ingredient for our proof of Theorem \\ref{thm.opt.2} in the next subsection.\n\n\\begin{lemma}[Key lower bound lemma] \\label{lem:LB}\nLet $(\\phi, r, m)$ be a $C^{1}$ solution to \\eqref{eq:SSESF} which is locally BV scattering and asymptotically flat initial data of order $\\omega = 3$ in $C^{1}$. \nSuppose furthermore that \n\\begin{equation*}\n\t\\mathfrak{L} := \\lim_{v \\to \\infty} r^{3} \\partial_{v} (r \\phi)(1, v) + \\int_{1}^{\\infty} (M \\nu_{\\infty} \\Phi)(u)  \\, \\mathrm{d} u \\neq 0,\n\\end{equation*}\nwhere $M(u) := \\lim_{v \\to \\infty} m(u, v)$, $\\nu_{\\infty}(u) := \\lim_{v \\to \\infty} \\nu(u,v)$ and $\\Phi(u) := \\lim_{v \\to \\infty} r \\phi(u,v)$. \nThen there exist constants $U, L_{3} > 0$ such that the following lower bounds for the decay of $\\partial_{v}(r \\phi)$, $\\partial_{u} (r \\phi)$ hold on $\\set{(u, v)  : u \\geq U}$.\n\\begin{align} \n\t\\abs{\\partial_{v}(r \\phi)(u, v)} \\geq & L_{3} \\min \\set{r(u,v)^{-3}, u^{-3}}, \\label{eq:LB:1} \\\\\n\t\\abs{\\partial_{u}(r \\phi)(u, v)} \\geq & L_{3} u^{-3}. \\label{eq:LB:2}\n\\end{align}\n\\end{lemma}\n\n\\begin{remark} \nNote that $\\partial_{v}(r \\phi)$ and $\\partial_{u}(r \\phi)$ have definite signs by \\eqref{eq:LB:1}, \\eqref{eq:LB:2}. In fact, the proof below shows that the signs of $\\partial_{v} (r \\phi)$ and $-\\partial_{u} (r \\phi)$ agree with that of $\\mathfrak{L}$.\n\\end{remark}\n\n\n\\begin{proof} \nWithout loss of generality, assume that $\\mathfrak{L} > 0$. For $0 < \\eta \\leq 1$, define the \\emph{$\\eta$-exterior region} by\n\\begin{equation*}\n\t\\PD_{\\mathrm{ext}}^{\\eta} := \\set{(u,v) \\in \\calQ : u \\leq \\eta v}.\n\\end{equation*}\n\n\\pfstep{Step 1} In the first step, we make precise the relation between $r$ and $v$ in $\\PD_{\\mathrm{ext}}^{\\eta}$ for small $\\eta$. We claim that $r \\sim v\/2$ in this region; more precisely,\n\\begin{equation} \\label{eq:LB:pf:1}\n\\abs{\\frac{r(u,v)}{v} - \\frac{1}{2}} \\leq \\eta C_{A_{1}, A_{2}, K, \\Lambda}.\n\\end{equation}\n\nIntegrating by parts, we have\n\\begin{align*}\nr(u,v) \n= \\int_{u}^{v} \\lmb(u, v') \\, \\mathrm{d} v' \n= - \\int_{u}^{v} \\partial_{v} \\lmb(u, v') v' \\, \\mathrm{d} v' + v \\lmb(u, v) - u \\lmb(u, u).\n\\end{align*}\n\nTo make the leading term $v \\lmb(u,v)$ and small number $\\frac{u}{v}$ explicit, we rewrite the last expression as follows:\n\\begin{equation*}\n\tr(u,v) = v \\Big[ \\lmb(u,v)  - \\frac{u}{v} \\Big( \\lmb(u,u) + \\int_{u}^{v} \\partial_{v} \\lmb(u, v') \\frac{v'}{u} \\, \\mathrm{d} v' \\Big)  \\Big].\n\\end{equation*}\n\nRecall that $\\lmb$ is uniformly bounded from the above and below on $\\calQ$, i.e., $\\Lambda^{-1} \\leq \\lmb \\leq 1\/2$. Moreover, by the decay estimates for $\\partial_{v} \\lmb$ proved in Theorem \\ref{main.thm.2}, we have\n\\begin{equation*}\n\\sup_{(u,v) \\in \\calQ} \\int_{u}^{v} \\abs{\\partial_{v} \\lmb(u, v')} \\frac{v'}{u} \\, \\mathrm{d} v' \\leq C_{A_{2}}.\n\\end{equation*}\n\nAs a consequence,\n\\begin{equation*}\n\\abs{\\frac{r(u,v)}{v} - \\lmb(u,v)} \\leq \\eta C_{A_{2}, \\Lambda}.\n\\end{equation*}\n\nThus \\eqref{eq:LB:pf:1} will follow once we establish\n\\begin{equation} \\label{eq:LB:pf:1:1}\n\t\\abs{\\lmb(u,v) - \\frac{1}{2}} \\leq \\eta^{2} C_{A_{1}, A_{2}, K, \\Lambda}.\n\\end{equation}\n\nThis inequality is proved by integrating the decay estimate \\eqref{eq:decay2:10} for $\\partial_{u} \\lmb = \\partial_{u} \\partial_{v} r$ along the incoming direction, starting from the normalization $\\lmb(1, v) = 1\/2$. \nHere, we use the easy geometric fact that if $(u,v)$ lies in $\\PD_{\\mathrm{ext}}^{\\eta}$, then so does the incoming null curve from $(1,v)$ to $(u,v)$. \n\n\\pfstep{Step 2} We claim that for $U_{1} \\geq 1$ sufficiently large and $0 < \\eta \\leq 1$ suitably small, we have\n\\begin{equation} \\label{eq:LB:pf:2}\n\t\\partial_{v} (r \\phi)(u,v) \\geq \\frac{\\mathfrak{L}}{2} \\Big( \\frac{v}{2} \\Big)^{-3}\n\\end{equation}\nfor $(u, v) \\in \\PD_{\\mathrm{ext}}^{\\eta} \\cap \\set{u \\geq U_{1}}$.\n\nWe begin with\n\\begin{equation} \\label{eq:LB:pf:2:1}\n\t\\Big( \\frac{v}{2} \\Big)^{3} \\partial_{v}(r \\phi)(u,v) = \\Big( \\frac{v}{2} \\Big)^{3} \\partial_{v}(r \\phi)(1, v) + \\Big( \\frac{v}{2} \\Big)^{3} \\int_{1}^{u} \\frac{2m \\lmb \\nu}{(1-\\mu) r^{3}} r \\phi(u', v) \\, \\mathrm{d} u',\n\\end{equation}\nobtained by integrating the $\\partial_{u} \\partial_{v} (r \\phi)$ equation and multiplying by $(v\/2)^{3}$. To prove \\eqref{eq:LB:pf:2}, it suffices to show that the right-hand side of \\eqref{eq:LB:pf:2:1} is bounded from below by $\\mathfrak{L}\/2$ for $(u, v) \\in \\PD_{\\mathrm{ext}}^{\\eta} \\cap \\set{u \\geq U_{1}}$ with sufficiently large $U_{1} \\geq 1$ and small $0 < \\eta \\leq 1$.\n\nNote that $r = \\frac{v-1}{2}$ on $C_{1}$, and $v \\geq \\eta^{-1}$ if $(u,v) \\in \\PD_{\\mathrm{ext}}^{\\eta}$. Thus for $(u,v) \\in \\PD_{\\mathrm{ext}}^{\\eta}$ and $0 < \\eta \\leq 1$ sufficiently small, we have\n\\begin{equation*}\n\\abs{\\Big( \\frac{v}{2} \\Big)^{3} \\partial_{v} (r \\phi)(1, v) - \\lim_{v \\to \\infty} r^{3} \\partial_{v} (r \\phi)(1, v)} < \\frac{\\mathfrak{L}}{8}.\n\\end{equation*}\n\nIn order to proceed, it is useful to keep in mind the following technical point: For $U_{1} \\geq 1$, by the decay estimates \\eqref{eq:decay1:1} and \\eqref{eq:decay1:6}, we have\n\\begin{equation} \\label{eq:LB:pf:2:2}\n\t\\sup_{v \\geq U_{1}} \\int_{U_{1}}^{v} \\abs{\\frac{2m \\lmb \\nu}{1-\\mu} r \\phi (u', v)} \\, \\mathrm{d} u' \\leq U_{1}^{-6} C_{A_{1}, \\Lambda}.\n\\end{equation}\n\nIn what follows, let $(u, v) \\in \\PD_{\\mathrm{ext}}^{\\eta} \\cap \\set{u \\geq U_{1}}$. Using \\eqref{eq:LB:pf:1}, \\eqref{eq:LB:pf:2:2} and the fact that the null segment from $(1, v)$ to $(u, v)$ lies in $\\PD_{\\mathrm{ext}}^{\\eta}$, we get\n\\begin{equation*}\n\\Big\\vert \\Big( \\frac{v}{2} \\Big)^{3} \\int_{1}^{u} \\frac{2m \\lmb \\nu}{(1-\\mu) r^{3}} r \\phi(u', v) \\, \\mathrm{d} u' - \\int_{1}^{u} \\frac{2m \\lmb \\nu}{1-\\mu} r \\phi(u', v) \\, \\mathrm{d} u'\\Big\\vert\n\\leq \\eta C_{A_{1}, A_{2}, K, \\Lambda}.\n\\end{equation*}\n\nTaking $U_{1} \\geq 1$ large enough and using \\eqref{eq:LB:pf:2:2}, we may arrange \n\\begin{equation*}\n\t\\sup_{v \\geq U_{1}} \\int_{U_{1}}^{v} \\abs{\\frac{2m \\lmb \\nu}{1-\\mu} r \\phi (u', v)} \\, \\mathrm{d} u' + \\int_{U_{1}}^{\\infty} \\abs{M \\nu_{\\infty} \\Phi (u')} \\, \\mathrm{d} u' < \\frac{\\mathfrak{L}}{8}.\n\\end{equation*} \n\nOn the other hand, note that $2m \\lmb \\nu (1-\\mu)^{-1} r \\phi (u, v) \\to M \\nu_{\\infty} \\Phi(u)$ for each $u \\geq 1$ as $v \\to \\infty$. Therefore, by the dominated convergence theorem, for $0 < \\eta \\leq 1$ sufficiently small (so that $v$ is large), we have\n\\begin{align*}\n\t\\abs{\\int_{1}^{U_{1}} \\frac{2m \\lmb \\nu}{1-\\mu} r \\phi (u', v) \\, \\mathrm{d} u' - \\int_{1}^{U_{1}} M \\nu_{\\infty} \\Phi(u')  \\, \\mathrm{d} u'} < \\frac{\\mathfrak{L}}{8}.\n\\end{align*}\n\nPutting these together and taking $0 < \\eta \\leq 1$ sufficiently small, we conclude \\eqref{eq:LB:pf:2}.\n\n\n\\pfstep{Step 3} Next, we claim that there exists $U_{2} = U_{2}(U_{1}, A_{2}, \\Lambda, K, \\eta) \\geq 1$ such that $U_{2} \\geq U_{1}$ and for $(u,v) \\in (\\calQ \\setminus \\PD_{\\mathrm{ext}}^{\\eta}) \\cap \\set{u \\geq U_{2}}$, we have\n\\begin{align} \n\t\\partial_{v}(r \\phi)(u, v) \\geq 2 \\eta^{3} \\mathfrak{L} \\, u^{-3}.\t\t\\label{eq:LB:pf:3}\n\\end{align}\n\nCombined with \\eqref{eq:LB:pf:2} (keeping in mind that $r \\sim v\/2$ in $\\PD_{\\mathrm{ext}}^{\\eta}$ by \\eqref{eq:LB:pf:1}), this would establish \\eqref{eq:LB:1}.\n\nTake $U_{2} \\geq \\eta^{-1} U_{1}$, and consider $(u, v) \\in (\\calQ \\setminus \\PD_{\\mathrm{ext}}^{\\eta}) \\cap \\set{u \\geq U_{2}}$. Integrating \\eqref{eq:SSESF:dphi}, we have\n\\begin{equation*}\n\t\\partial_{v} (r \\phi)(u,v) = \\partial_{v} (r \\phi)(\\eta u, v) + \\int_{\\eta u}^{u} \\frac{2m \\lmb \\nu}{r^{2}} \\phi(u', v) \\, \\mathrm{d} u'.\n\\end{equation*}\n \nNote that $(\\eta u, v) \\in \\PD_{\\mathrm{ext}}^{\\eta} \\cap \\set{u \\geq U_{1}}$ since $v \\geq u$ and $\\eta u \\geq \\eta U_{2} \\geq  U_{1}$. Therefore, by \\eqref{eq:LB:pf:2} and the fact that $\\eta^{-1} u > v$ (as $(u,v) \\in \\calQ \\setminus \\PD_{\\mathrm{ext}}^{\\eta}$), the first term on the right-hand side obeys the lower bound\n\\begin{equation*}\n\t\\partial_{v} (r \\phi)(\\eta u, v) \\geq \\Big( \\frac{\\mathfrak{L}}{2} \\Big) \\Big( \\frac{v}{2} \\Big)^{-3} > 4 \\eta^{3} \\mathfrak{L} \\, u^{-3}.\n\\end{equation*}\n\nOn the other hand, using \\eqref{eq:decay1:1} and \\eqref{eq:decay2:11}, we have\n\\begin{equation*}\n\t\\abs{\\int_{\\eta u}^{u} \\frac{2m \\lmb \\nu}{r^{2}} \\phi(u', v) \\, \\mathrm{d} u'}\n\t\\leq C_{A_{1}, A_{2}, \\Lambda, K} \\int_{\\eta u}^{u} \\frac{1}{(u')^{10}} \\, \\mathrm{d} u' \n\t\\leq C_{A_{1}, A_{2}, \\Lambda, K, \\eta} \\, u^{-9}.\n\\end{equation*}\n\nTaking $U_{2}$ large enough, we conclude that \\eqref{eq:LB:pf:3} holds.\n\n\\pfstep{Step 4} Finally, we claim that there exists $U = U(U_{2}, A_{2}, \\Lambda, K, \\eta) \\geq 1$ such that $U \\geq U_{2} \\geq U_{1}$ and for $(u,v) \\in \\set{u \\geq U}$, we have\n\\begin{align} \n\t- \\partial_{u}(r \\phi)(u, v) \\geq \\eta^{3} \\mathfrak{L} \\, u^{-3}.\t\t\\label{eq:LB:pf:4}\n\\end{align}\n\nThis would prove \\eqref{eq:LB:2}, thereby completing the proof of Lemma \\ref{lem:LB}.\n\nOur argument will be very similar to the previous step. Take $U \\geq U_{2}$ and consider $(u,v) \\in \\set{u \\geq U}$. Integrating \\eqref{eq:SSESF:dphi} along the outgoing direction, we have\n\\begin{equation*}\n\t- \\partial_{u}(r \\phi)(u,v) = - \\partial_{u}(r \\phi)(u, u) - \\int_{u}^{v} \\frac{2 m \\lmb \\nu}{r^{2}} \\phi(u, v') \\, \\mathrm{d} v'.\n\\end{equation*}\n\nRecall that $\\lim_{v \\to u+} \\partial_{u} (r \\phi)(u, v) = - \\lim_{v \\to u+} \\partial_{v}(r \\phi)(u,v)$. By \\eqref{eq:LB:pf:3} and the fact that $u \\geq U \\geq U_{2}$, we see that the first term on the right-hand side obeys the lower bound\n\\begin{equation*}\n\t- \\partial_{u} (r \\phi)(u,u) \\geq 2 \\eta^{3} \\mathfrak{L} \\, u^{-3}.\n\\end{equation*}\n\nOn the other hand, using \\eqref{eq:decay1:1} and \\eqref{eq:decay2:11}, we have\n\\begin{equation*}\n\t\\abs{\\int_{u}^{v} \\frac{2 m \\lmb \\nu}{r^{2}} \\phi(u, v') \\, \\mathrm{d} v'}\n\t\\leq C_{A_{1}, A_{2}, K} \\int_{u}^{v} \\min \\set{u^{-10}, r^{-2} u^{-8}} \\, \\lmb \\, \\mathrm{d} v' \n\t\\leq C_{A_{1}, A_{2}, K} \\, u^{-9}.\n\\end{equation*}\n\nTaking $U$ sufficiently large, we conclude that \\eqref{eq:LB:pf:4} holds. \\qedhere\n\\end{proof}\n\n\\subsection{Optimality of the decay rates, in the case $\\omega' \\geq 3$}  \\label{subsec.opt.2}\nIn this subsection, we prove Theorem \\ref{thm.opt.2} by studying the solution to \\eqref{eq:SSESF} arising from the initial value\n\\begin{equation*}\n\t\\partial_{v}(r \\phi)(1, v) = \\epsilon \\widetilde{\\chi}\\Big( \\frac{v - v_{0}}{N} \\Big),\n\\end{equation*}\nwhere $\\widetilde{\\chi} : (-\\infty, \\infty) \\to [0, \\infty)$ is a smooth function such that\n\\begin{equation*}\n\\mathrm{supp} \\, \\widetilde{\\chi} \\subset (-1\/2, 1\/2), \\quad\n\\int_{\\mathbb R} \\widetilde{\\chi} = 1.\n\\end{equation*}\n\nWe also require that $v_{0} \\geq 2$ and $N \\leq v_{0}$. With such data, the initial total variation is of size $\\leq C\\epsilon$, i.e.,\n\\begin{equation*}\n\t\\int_{1}^{\\infty} \\abs{\\partial_{v}^{2} (r \\phi) (1, v)} \\, \\mathrm{d} v \\leq \\epsilon \\int_{-\\infty}^{\\infty} \\abs{ \\widetilde{\\chi}\\,' \\Big( \\frac{v - v_{0}}{N} \\Big) } \\, \\frac{\\mathrm{d} v}{N} \\leq C \\epsilon.\n\\end{equation*}\n\nWe also see that $\\mathcal I_{1} \\leq C \\epsilon v_{0}^{3}$ and $\\mathcal I_{2} \\leq C \\epsilon v_{0}^{4} \/ N$ with $\\omega' = 3$, as\n\\begin{equation*}\n\t\\sup_{v \\in [1, \\infty)} (1+r)^{3} \\abs{\\partial_{v} (r \\phi)}(1,v) \\leq C \\epsilon v_{0}^{3}, \\quad\n\t\\sup_{v \\in [1, \\infty)} (1+r)^{4} \\abs{\\partial_{v}^{2} (r \\phi)}(1,v) \\leq C \\epsilon \\frac{v_{0}^{4}}{N}.\n\\end{equation*}\n\nWe are now ready to give a proof of Theorem \\ref{thm.opt.2}. The idea is to compute $\\mathfrak{L}$ to the leading order (which turns out to be $- c\\epsilon^{3}$ for some $c > 0$), and then control the lower order terms by taking $\\epsilon > 0$ sufficiently small and applying Theorem \\ref{thm:smallData}.\n\n\\begin{proof} [Proof of Theorem \\ref{thm.opt.2}]\nFor this proof, we fix $v_{0} = 4$ and $N = 1$. We use the shorthand\n\\begin{equation*}\n\t\\chi(v) := \\widetilde{\\chi}(v-4).\n\\end{equation*}\n\nBy the preceding discussion on the size of initial data, we see that Theorem \\ref{thm:smallData} applies when $\\epsilon > 0$ is sufficiently small. Therefore, there exists a constant $C > 0$ independent of $\\epsilon > 0$ such that Theorems \\ref{main.thm.1}, \\ref{main.thm.2} and Proposition \\ref{prop:geomLocBVScat} hold with\n\\begin{equation} \\label{eq:opt2:eps}\n\tA_{1}, A_{2} \\leq C \\epsilon, \\quad K, \\Lambda \\leq C.\n\\end{equation}\n\nWe begin by showing\n\\begin{equation} \\label{eq:opt2:dvrphi}\n\t\\partial_{v}(r \\phi)(u,v) = \\epsilon \\chi(v) + \\mathrm{Err}_{1}(u,v),\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:opt2:Err1}\n\t\\abs{\\mathrm{Err}_{1}(u,v)} \\leq C \\epsilon^{3} \\min \\set{u^{-3}, r(u,v)^{-3}}.\n\\end{equation}\n\nThe argument is similar to the proof of Theorem \\ref{thm.opt.1}, but this time we rely on Theorem \\ref{thm:smallData} to make the dependence of $\\mathrm{Err}_{1}$ on $\\epsilon$ explicit. Indeed, by \\eqref{eq:SSESF:dphi}, we have\n\\begin{equation*}\n\t\\abs{\\mathrm{Err}_{1}(u,v)}\n\t\\leq \\int_{1}^{u} \\abs{\\frac{\\mu \\lmb \\nu}{(1-\\mu)r}  \\phi} (u', v) \\, \\mathrm{d} u'.\n\\end{equation*}\n\nThen estimating the right-hand side using Theorem \\ref{main.thm.1}, Proposition \\ref{prop:geomLocBVScat} and Corollary \\ref{cor:decay2}, and using \\eqref{eq:opt2:eps} make the $\\epsilon$-dependence explicit, \\eqref{eq:opt2:Err1} follows.\n\nIntegrating \\eqref{eq:opt2:dvrphi}, we also have\n\\begin{align*} \n\tr\\phi(u,v) \t= & \\int_{u}^{v} \\partial_{v}(r \\phi)(u, v') \\, \\mathrm{d} v' \\\\\n\t\t\t= & \\epsilon \\int_{u}^{v} \\chi(v') \\, \\mathrm{d} v' + \\int_{u}^{v} \\mathrm{Err}_{1}(u, v') \\, \\mathrm{d} v' \\\\\n\t\t\t= & \\epsilon X(u,v) + \\mathrm{Err}_{2}(u,v)\n\\end{align*}\nwhere $X(u, v) := \\int_{u}^{v} \\chi(v') \\, \\mathrm{d} v'$ and $\\mathrm{Err}_{2}(u,v) := \\int_{u}^{v} \\mathrm{Err}_{1}(u, v') \\, \\mathrm{d} v'$. Integrating \\eqref{eq:opt2:Err1}, and using the bound $C^{-1} \\leq \\lmb \\leq 1\/2$, we easily obtain\n\\begin{equation} \\label{eq:opt2:Err2}\n\t\\abs{\\mathrm{Err}_{2}(u,v)} \\leq C \\epsilon^{3} \\min \\set{r u^{-3}, u^{-2}}.\t\n\\end{equation}\n\nIn particular, taking $v \\to \\infty$, we see that\n\\begin{equation} \\label{eq:opt2:Phi}\n\t\\abs{\\Phi(u) - \\epsilon X(u, \\infty)} \\leq C \\epsilon^{3} u^{-2}.\n\\end{equation}\n\nWe now proceed to estimate $M(u)$. We begin with the easy observation\n\\begin{equation} \\label{eq:opt2:UBforM}\n\tM(u) \\leq C \\epsilon^{2} u^{-5},\n\\end{equation}\nwhich follows from Corollary \\ref{cor:decay2} and \\eqref{eq:opt2:eps}. On the other hand, recalling the definition of $M(u)$ from \\eqref{eq:SSESF:dm} and using the elementary inequality $(a+b)^{2} \\geq \\frac{1}{2} a^{2} - b^{2}$,\n\\begin{align*}\n\tM(u)\t= & \\frac{1}{2} \\int_{u}^{\\infty} \\frac{1-\\mu}{\\lmb} [ \\partial_{v}(r\\phi) - \\frac{\\lmb}{r} (r \\phi) ]^{2} (u,v) \\, \\mathrm{d} v  \\\\\n\t\t\\geq & \\frac{\\epsilon^{2}}{4} \\int_{u}^{\\infty} \\frac{1-\\mu}{\\lmb}(u,v) [\\chi (v) - \\frac{\\lmb}{r} X (u,v) ]^{2} \\, \\mathrm{d} v \n\t\t\t- \\frac{1}{2}\\int_{u}^{\\infty} \\frac{1-\\mu}{\\lmb} [\\mathrm{Err}_{1} - \\frac{\\lmb}{r} \\mathrm{Err}_{2}]^{2} (u,v) \\, \\mathrm{d} v .\n\\end{align*}\n\nBy \\eqref{eq:opt2:eps}, \\eqref{eq:opt2:Err1} and \\eqref{eq:opt2:Err2}, we have\n\\begin{equation*}\n\\abs{\\frac{1}{2} \\int_{u}^{\\infty} \\frac{1-\\mu}{\\lmb} [\\mathrm{Err}_{1} - \\frac{\\lmb}{r} \\mathrm{Err}_{2}]^{2} (u,v) \\, \\mathrm{d} v} \\leq C \\epsilon^{6}.\n\\end{equation*}\n\nFurthermore, note that $(1-\\mu) \\geq (K \\Lambda)^{-1} \\geq C^{-1} > 0$ by Proposition \\ref{prop:geomLocBVScat} and \\eqref{eq:opt2:eps}. Also, for $(u,v) \\in [1,2] \\times [8, \\infty)$, note that $\\chi(v) = 0$ and $X(u,v) = 1$. Therefore, for $1 \\leq u \\leq 2$, there exists $c > 0$ (independent of $\\epsilon > 0$) such that\n\\begin{align*}\n\\frac{1}{4}\\int_{u}^{\\infty} \\frac{1-\\mu}{\\lmb}(u,v) [\\chi - \\frac{\\lmb}{r} X]^{2} (u,v) \\, \\mathrm{d} v\n\\geq & (4 C)^{-1} \\int_{u}^{\\infty}  [\\chi - \\frac{\\lmb}{r} X]^{2} (u,v) \\, \\lmb^{-1} (u,v)  \\mathrm{d} v \\\\\n\\geq & (4 C)^{-1} \\int_{8}^{\\infty} \\frac{\\lmb}{r^{2}} (u,v) \\, \\mathrm{d} v \\\\\n\\geq & c\\,.\n\\end{align*}\n\nTherefore, we conclude that\n\\begin{equation} \\label{eq:opt2:LBforM}\n\tM(u) \\geq c \\epsilon^{2} - C \\epsilon^{6} \\qquad \\hbox{ for } 1 \\leq u \\leq 2.\n\\end{equation}\n\nWe are now ready to compute $\\mathfrak{L}$ and complete the proof. We begin by observing that \n\\begin{equation*}\n\\lim_{v \\to \\infty} r^{3} \\abs{\\partial_{v}(r \\phi)(1,v)} = 0\n\\end{equation*}\nby our choice of data. Therefore,\n\\begin{align*}\n\t- \\mathfrak{L}\n\t\t = &\\int_{1}^{\\infty} M (-\\nu_{\\infty}) \\Phi(u) \\, \\mathrm{d} u \\\\\n\t\t= & \\epsilon \\int_{1}^{\\infty} M (u) (-\\nu_{\\infty}) (u) X(u,\\infty) \\, \\mathrm{d} u \n\t\t\t+ \\int_{1}^{\\infty} M(u) (-\\nu_{\\infty})(u) \\mathrm{Err}_{2}(u, \\infty) \\, \\mathrm{d} u. \n\\end{align*}\n\nBy Proposition \\ref{prop:geomLocBVScat}, \\eqref{eq:opt2:eps}, \\eqref{eq:opt2:Err2} and \\eqref{eq:opt2:UBforM}, we have\n\\begin{equation*}\n\\abs{\\int_{1}^{\\infty} M(u) (-\\nu_{\\infty})(u) \\mathrm{Err}_{2}(u, \\infty) \\, \\mathrm{d} u } \\leq C \\epsilon^{5}.\n\\end{equation*}\n\nOn the other hand, by Proposition \\ref{prop:geomLocBVScat}, \\eqref{eq:opt2:eps} and \\eqref{eq:opt2:LBforM}, we have (taking $c > 0$ smaller if necessary)\n\\begin{align*}\n\t\t\\epsilon \\int_{1}^{\\infty} M(u) (-\\nu_{\\infty})(u) X(u, \\infty) \\, \\mathrm{d} u \n\t\t\\geq & \\epsilon \\int_{1}^{2} M(u) (-\\nu_{\\infty})(u) X(u, \\infty) \\, \\mathrm{d} u \\\\\n\t\t\\geq & \\Lambda^{-1} \\epsilon \\int_{1}^{2} M(u) \\, \\mathrm{d} u\n\t\t\\geq c \\epsilon^{3} - C \\epsilon^{7}.\n\\end{align*}\n\nTherefore, taking $\\epsilon > 0$ sufficiently small, we see that $- \\mathfrak{L} > \\frac{c}{2} \\epsilon^{3} > 0$. \\qedhere\n\\end{proof}\n\n\\bibliographystyle{amsplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}