\documentclass[reqno]{amsart} \usepackage{amssymb} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 10, pp. 1--46.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/10\hfil Asymptotic representation of solutions] {Asymptotic representation of solutions to the Dirichlet problem for elliptic systems with discontinuous coefficients near the boundary} \author[V. Kozlov\hfil EJDE-2006/10\hfilneg] {Vladimir Kozlov} \address{Vladimir Kozlov \hfill\break Department of Mathematics, University of Link\"oping, SE-581 83 Link\"oping, Sweden} \email{vlkoz@mai.liu.se} \date{} \thanks{Submitted April 24, 2005. Published January 24, 2006.} \thanks{Supported by the Swedish Research Council (VR) (Link\"oping)} \subjclass[2000]{35B40, 35B65, 35J15, 35D10} \keywords{Asymptotic behaviour of solutions; elliptic systems; \hfill\break\indent Dirichlet problem; measurable coefficients} \begin{abstract} We consider variational solutions to the Dirichlet problem for elliptic systems of arbitrary order. It is assumed that the coefficients of the principal part of the system have small, in an integral sense, local oscillations near a boundary point and other coefficients may have singularities at this point. We obtain an asymptotic representation for these solutions and derive sharp estimates for them which explicitly contain information on the coefficients. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newcommand{\Wcirc}{{\mathaccent"7017 W}} \section{Introduction} Let $B_+(\delta )=\mathbb{R}^n_+\cap B(\delta )$, where $\mathbb{R}^n_+=\{ x=(x',x_n): x_n>0\}$ and $B(\delta )$ is the ball with the center at the origin and with the radius $\delta >0$. We consider solutions to the Dirichlet problem \begin{gather}\label{4.2zas1} \mathcal{L}(x,D_x)u=0 \quad \mbox{on }B_+(\delta ),\\ \label{4.2zb1} \partial _{x_n}^ku\big |_{x_n=0}=0\quad\mbox{for } k=0,1,\dots ,m-1,\;|x'|<\delta \end{gather} for the differential operator $$\label{4.2zbw} \mathcal{L}(x,D_x)u=L(D_x)u-N(x,D _x)u\, ,$$ where $D_x=-i\partial_x$ and $L(D _x)$ is a strongly elliptic differential operator with constant $d\times d$-matrix coefficients. The operator $$\label{NN5ad} N(x,D _x)u=\sum _{|\alpha |,|\beta |\leq m }D _x^\alpha\big ( N_{\alpha \beta}(x)D_x^\beta u)$$ will be treated as a perturbation operator and we shall characterize it by the function $$\label{Ocen1Intr} \kappa (x)=\sum _{|\alpha|, |\beta |\leq m} x_n^{2m-|\alpha +\beta |}|N_{\alpha \beta }(x)|.$$ The function $\kappa$ is assumed to be bounded and the function \label{88h} r\to\int_{r/e<|y|0} \kappa^2(y )|y|^{-n}dy+c\omega_0^2\, , $$where c_1 and c_2 are two constants independent of \delta. For more explicit formulation of relation (\ref{Intr223}) as well as for the corresponding relations for derivatives of u we refer to Theorems \ref{TTT12} and \ref{TTT11}. A direct consequence of (\ref{Intr223}) is the asymptotic formula u(x)\sim {\bf c}x_n^m for solution u to (\ref{4.2zas1}), (\ref{4.2zb1}) proved in Corollary \ref{kTTT1}, under the assumption that$$ \int_{B_+(\delta )}\kappa (x)|x|^{-n}dx <\infty\, . $$We note that actually this result is proved without smallness assumption on the function \kappa. Another application is the following. It is well known that any variational solution belongs to the Sobolev space (W^{m,p} )^d with sufficiently large p, depending on \mathop{\rm ess\,sup}\kappa (see \cite{ADN} and \cite{GT} for second order equations). We prove the following estimate for solutions to problem (\ref{4.2zas1}), (\ref{4.2zb1}) from (W^{m,2}(B_+(\delta))^d: $$\label{Intr11} |\nabla_ku(x)|\leq C J(u,\delta ) |x|^{m-k}\exp\Big (\int_{|x|<|y|<\delta,y_n>0} (\lambda_+(y)+c\kappa^2(y )|y|^{-n})dy\Big )$$ for |x|<\delta /2 and k=0,1,\dots ,m-1. Here \nabla _ku is the vector \{ \partial _x^\alpha u\} _{|\alpha |=k}, \lambda_+(y) is the maximal eigenvalue of the matrix \Re \mathcal{Q}(y). The constants C and c in (\ref {Intr11}) depend only on the operator L(D _x) and n, and$$ J(u,\delta )=\delta^{-n/2}\Big (\int_{|y|<\delta}|\nabla_mu|^2dy\Big )^{1/2}\, . Estimate (\ref{Intr11}) follows directly from Remark 2, Corollary \ref{TTT1} with p>n and the Sobolev imbedded theorem. The case of a scalar operator \mathcal{L}(x,D_x) was treated in \cite{KM2} under the smallness assumption of the function \kappa. But even in the scalar case this work improves asymptotic formulae from \cite{KM2} in the following way. In the exponent in (\ref{Intr11}) and in similar formulae in Corollaries \ref{Le12a}--\ref{TTT1} we have a remainder term of the form \kappa^2(y), whereas in \cite{KM2} instead of this term we have (\mathop{\rm ess\,sup}_{|y|/e<|\xi|<|y|}\kappa (\xi))^2. The importance of this improvement will be demonstrated in the forthcoming paper on estimates of solutions to Dirichlet boundary value problem for elliptic systems in convex domains. Let us describe the idea of the proof. We use the same reduction to the first order evolution system as in \cite{KM2}--\cite{KM6} and transfer the study of behavior of solutions near the boundary point to the study of behavior of solutions to the evolution system at infinity. The next step is a reduction of the infinite dimensional system to a finite system of ordinary differential equations perturbed by nonlocal integro-differential operator, which was used in \cite{KM2}-\cite{KM6}. The new feature here is a more refine study of this finite dimensional system, which is performed in Section \ref{Sect88f}--\ref{Subs113}. \section{Preliminaries and formulation of main results}\label{Section1} \subsection{Assumptions and some functional spaces}\label{State1a} In parallel to (\ref{4.2zas1}), (\ref{4.2zb1}) we shall consider the Dirichlet problem $$\label{4.2zas} \mathcal{L}(x,D_x)u=f(x) \quad \mbox{in {\mathbb{R}}_+^n,}$$ $$\label{4.2zb} \partial _{x_n}^ku\big |_{x_n=0}=0\quad \mbox{for k=0,1,\dots ,m-1} \quad \mbox{on {\mathbb{R}}^{n-1}\setminus \mathcal{O}.}$$ We suppose that $$\label{RTR1} \mathcal{L}(x,D_x)u= \sum _{|\alpha |,|\beta |\leq m }D _x^\alpha\big ( \mathcal{L}_{\alpha \beta}(x)D_x^\beta u)\, ,$$ where the coefficients \mathcal{L}_{\alpha \beta } are measurable complex valued d\times d-matrix functions on \mathbb{R}_+^n. We write the operator L(D_x) in (\ref{4.2zbw}) as $$\label{LL1s} L(D _x)=\sum _{|\alpha |=|\beta |= m }L_{\alpha\beta }D_x^{\alpha +\beta}$$ and assume that the matrix \Re L(\xi ) is positively definite for all \xi\in {\mathbb{R}}^n\setminus \mathcal{ O}. According to (\ref{4.2zbw}) and (\ref{LL1s}) $$\label{NN5a} N_{\alpha\beta}= \begin{cases} L_{\alpha\beta}-\mathcal{L}_{\alpha\beta} & \mbox{if } |\alpha|=|\beta|=m\\ -\mathcal{L}_{\alpha\beta} &\mbox{if }|\alpha|+|\beta|<2m. \end{cases}$$ We consider solutions u from the space (\Wcirc^{m,p}_{\rm loc} (\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}))^d, where \Wcirc^{m,p}_{\rm loc} (\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}), 10, where K_{\rho ,r}=\{ x\in {\mathbb{R}}_+^n\, :\; \rho <|x|1. We shall suppose that b_0\geq 1 and that \label{PEPEa} p_1\leq \left\{\begin{array}{ll} np/(n-p) & \mbox{if 2p0.} \begin{itemize} \item[(H3)] (\emph{Smallness of N}) We shall require that $$\label{4.8aq8hh} b_0\kappa_{\frac{p_1p}{p_1-p}}(r)\leq\omega_0,$$ where p_1 and p are the same as in (H2) and \omega_0 is a small constant depending on m, n, p, \gamma and on the unperturbed operator L. \end{itemize} \begin{remark}\label{Rem5tax} \rm We note that in the case p_1>p, (H3) follows from boundedness of \kappa and smallness of \kappa_1, because of {\rm (\ref{TTrr2ad})}. From {\rm (\ref{PEPEa})} it follows that p_1\leq 2p and hence p_1\leq p_1p/(p_1-p). This together with {\rm (\ref{4.8aq8hh})} implies, in particular, that $$\label{PEPEb} b_0\kappa_1(r)+b_0\kappa_{p_1}(r)\leq c\omega_0$$ with c depending on n. Assumption {\rm (\ref{PEPEa})} implies also that p_1\leq pn/(n-p) if p0. Let also Z\in (\Wcirc^{m,p}_{\rm loc} (\overline{\mathbb{R}_+^n}\setminus \mathcal{O}))^d be a solution of \mathcal{L}(x,D_x)Z=0 on \mathbb{R}_+^n\setminus \mathcal{O} subject to $$\label{Kyt1a} \mathfrak{M}^{m}_p(Z;K_{r/e,r})= o\Big ( r^{m-n}\exp\Big (-\mathcal{ C}\int _r^\delta\Omega (\rho )\frac{d\rho }{\rho }\Big )\Big )$$ as r\to 0 and $$\label{Kyt1a2} \mathfrak{M}^{m}_p(Z;K_{r/e,r})= o\Big ( r^{m+1}\exp\Big (-\mathcal{ C}\int _\delta^r\Omega (\rho )\frac{d\rho }{\rho }\Big )\Big )$$ as r\to \infty. Then Z\in (\Wcirc^{m,p_1}_{\rm loc} (\overline{\mathbb{R}_+^n}\setminus \mathcal{O}))^d and $$\label{Kyt2am} (r\partial _r )^k Z(x) = J_Z\exp\Big (\int _r^\delta\Upsilon (\rho )\frac{d\rho }{\rho }\Big ) \big (x _n^mm^k{\bf q}(r) +r^mv_k(x)\big )$$ for k=0,1,\dots ,m. Here r=|x| and $$\label{2005c} \Upsilon (r)=\big ({\bf R}(r){\bf q}(r),{\bf q}(r)\big )+\Upsilon_1(r)\, ,$$ where {\bf R}(r) is given by {\rm (\ref{U2117})}, the vector function {\bf q}  and the scalar function \Upsilon_1 are measurable and satisfy |{\bf q}(r)|=1 and $$\label{2005d} \int_{r/e}^r|\partial_\rho {\bf q}(\rho)|d\rho\leq c(\kappa_1(r)+\chi(r)),\quad \int_{r/e}^r|\Upsilon_1(\rho)|\frac{d\rho}{\rho}\leq c\chi(r)$$ for all r >0 with \label{OOPPa} \begin{aligned} \chi(r ) &=b_0\kappa_{p_1'} (r)\Big (r^{-n}\int _0^re^{\mathcal{ C}\int _\rho ^r\Omega (s)\frac{ds }{s}} \kappa_{p_1} (\rho )\rho ^{n-1}d\rho \\ &\quad +r\int _r^e e^{\mathcal{C}\int _r ^\rho\Omega (s)\frac{ds }{s}}\kappa_{p_1} (\rho )\rho ^{-2}d\rho \Big ), \end{aligned} where p_1'=p_1/(p_1-1). The constant J_Z in {\rm (\ref{Kyt2am})} admits the estimates $$\label{2005e} c_1\mathfrak{M}_2^0(Z;K_{\delta /e,\delta })\leq |J_Z|\delta^m\leq c_2\mathfrak{M}_2^0(Z;K_{\delta /e,\delta })\, .$$ The functions v_k belong to L^{p_1}_{\rm loc} ((0,\infty );(\Wcirc^{m-k,p_1}(S_+^{n-1}))^d) and satisfy \label{Ququ3za} \begin{aligned} &\Big (\int _{r /e}^r \big (\|v_k(\rho ,\cdot )\|_{W^{m-k,p_1}(S_+^{n-1})}^{p_1}+ \|\rho \partial _\rho v_k(\rho ,\cdot )\|_{W^{m-k-1,p_1}(S_+^{n-1})}^{p_1}\big ) \frac{d\rho }{\rho }\Big )^{1/p_1}\\ &\leq cb_0\Big (r^{-n}\int _0^re^{\mathcal{C}\int _\rho ^r\Omega (s)\frac{ds }{s}} \kappa_{p_1} (\rho )\rho ^{n-1}d\rho +r\int _r^e e^{\mathcal{C}\int _r ^\rho\Omega (s)\frac{ds }{s}}\kappa_{p_1} (\rho )\rho ^{-2}d\rho \Big ) \end{aligned} where k=0,\dots ,m-1 and \Wcirc^{m-k,p_1}(S_+^{n-1}) is the completion of C_0^\infty (S_+^{n-1}) in the norm of the Sobolev space W^{m-k,p_1}(S_+^{n-1}). In the case k=m estimate {\rm (\ref{Ququ3za})} holds without the second norm in the left-hand side. The dimension of the space of such solutions Z is equal to d. \end{theorem} We note that by (\ref{PEPEb}) the left-hand side of (\ref{Ququ3za}) is small. Let us formulate a local version of the above theorem. \begin{theorem}\label{TTT12} Assume that (H1)--(H3) are fulfilled. Let u\in (\Wcirc^{m,p}_{\rm loc} (\overline{\mathbb{R}_+^n}\setminus \mathcal{ O}))^d be a solution of \mathcal{L}(x,D_x)u=0 on B_{\delta }^+, \delta >0, subject to $$\label{Kyt1aaa} \mathfrak{M}^{m}_p(u;K_{r/e,r})= o\Big ( r^{m-n}\exp\Big (-\mathcal{ C}\int _r^\delta\Omega (\rho )\frac{d\rho }{\rho }\Big )\Big )$$ as r\to 0. Then $$\label{Zet1} u=Z+w\, ,$$ where Z is a special solution from Theorem \ref{TTT11}, which admits the asymptotic representation {\rm (\ref{Kyt2am})} with $$\label{I4.7q2} |J_Z|\leq cb_0\delta^{-m}\mathfrak{M}^m_{p_1}(u;K_{\delta/4,\delta})$$ and $$\label{I4.7q1} \mathfrak{M}^m_{p_1}(w;K_{r/e,r})\leq c b_0 \Big (\frac{r}{\delta }\Big )^{m+1}e^{\mathcal{C}\int_r^\delta\Omega (s)\frac{ds}{s}}\mathfrak{ M}^m_{p_1}(u;K_{\delta/4,\delta})$$ for r<\delta. \end{theorem} The proofs of these theorems are presented in Sections \ref{Section2}--\ref{Section3}. \subsection{Corollaries of the main results}\label{Corofmain} In this section we present several corollaries of Theorems \ref{TTT11} and \ref{TTT12} concerning the case when (\ref{4.8aq8hhh}) is satisfied with sufficiently small \omega_0. \begin{corollary}\label{Le12a} Let p\geq 2. There exists \omega_0>0 depending on n, p and L such that if {\rm (\ref{4.8aq8hhh})} is satisfied then the following assertion is valid. If Z\in (\Wcirc^{m,2}_{\rm loc} (\overline{\mathbb{R}_+^n}\setminus \mathcal{O}))^d is a solution of \mathcal{L}(x,D_x)Z=0 on \mathbb{R}_+^n\setminus \mathcal{O} subject to {\rm (\ref{Kyt1a})} and {\rm (\ref{Kyt1a2})} with p replaced by 2, then Z\in (\Wcirc^{m,p}_{\rm loc} (\overline{\mathbb{R}_+^n}\setminus \mathcal{O}))^d and for every \delta >0 representation {\rm (\ref{Kyt2am})} holds for k=0,1,\dots ,m, where \Upsilon is given by {\rm (\ref{2005c})} with the same {\bf R} and {\bf q}. Moreover, estimates {\rm (\ref{2005d})} are fulfilled with \label{OOPPah} \begin{aligned} \chi(r ) &=\kappa_{2} (r)\Big (r^{-n}\int _0^re^{\mathcal{C}\int _\rho ^r\Omega (s)\frac{ds }{s}} \kappa_{2} (\rho )\rho ^{n-1}d\rho \\ &\quad +r\int _r^e e^{\mathcal{C}\int _r ^\rho\Omega (s)\frac{ds }{s}}\kappa_{2} (\rho )\rho ^{-2}d\rho \Big ). \end{aligned} The coefficient J_Z satisfies {\rm (\ref{2005e})} and the remainder term v_k is subject to {\rm (\ref{Ququ3za})} with p_1 replaced by p. The dimension of the space of such solutions Z is equal to d. \end{corollary} \begin{corollary}\label{TTT1h} Let p\geq 2 and \delta >0. There exists \omega_0>0 depending on n, p and L such that if {\rm (\ref{4.8aq8hhh})} is satisfied for r<\delta  then the following assertion is valid. Let u\in (\Wcirc^{m,2}_{\rm loc} (\overline{\mathbb{R}_+^n}\setminus \mathcal{O}))^d be a solution of \mathcal{L}(x,D_x)u=0 on B_{\delta }^+ subject to {\rm (\ref{Kyt1aaa})} with p replaced by 2. Then u\in (\Wcirc^{m,p}_{\rm loc} (\overline{\mathbb{R}_+^n}\setminus \mathcal{ O}))^d and satisfies {\rm (\ref{Zet1})}, where Z is a special solution from {\rm Corollary \ref{Le12a}}, which admits the asymptotic representation {\rm (\ref{Kyt2am})} with \begin{gather}\label{I4.7q2d} |J_Z|\leq c\delta^{-m}\mathfrak{M}^m_2(u;K_{\delta/16,\delta}),\\ \label{I4.7q1b} \mathfrak{M}^m_{p}(w;K_{r/e,r})\leq c \Big (\frac{r}{\delta }\Big )^{m+1}e^{\mathcal{C}\int_r^\delta\Omega (s)\frac{ds}{s}}\mathfrak{ M}^m_2(u;K_{\delta/16,\delta}) \end{gather} for r<\delta/2. \end{corollary} The next two corollaries give a rougher but more explicit description of solutions to \mathcal{L}u=0. We denote by \Upsilon _-(\rho ) and \Upsilon _+(\rho ) the minimal and maximal eigenvalue of the matrix {\bf R}(\rho ). \begin{corollary}\label{Le12} Let {\rm (\ref{4.8aq8hhh})} be fulfilled with sufficiently small constant \omega_0 depending on n, p and L. Let also Z\in (\Wcirc^{m,2}_{\rm loc} (\overline{\mathbb{R}_+^n}\setminus \mathcal{O}))^d be a solution of \mathcal{L}(x,D_x)Z=0 on \mathbb{R}_+^n\setminus \mathcal{O} subject to {\rm (\ref{Kyt1a})} and {\rm (\ref{Kyt1a2})} with p replaced by 2. Then Z\in (\Wcirc^{m,p}_{\rm loc} (\overline{\mathbb{R}_+^n}\setminus \mathcal{O}))^d and for every \delta >0 \label{Ququ3z1} \begin{aligned} &C_1J(Z)\Big (\frac{r}{\delta}\Big )^m\exp\Big (\int_r^\delta(\Upsilon_-(\rho)-c\nu(\rho ))\frac{d\rho}{\rho}\Big)\\ &\leq \mathfrak{M}^m_p(Z;K_{r/e,r})\\ &\leq C_2J(Z)\Big (\frac{r}{\delta}\Big )^m\exp\Big (\int_r^\delta (\Upsilon_+(\rho)+c\nu (\rho ))\frac{d\rho}{\rho}\Big ) \end{aligned} for r<\delta . Here $$\label{U22} \nu (\rho )=\int_{S_+^{n-1}}\kappa^2 (\xi ) d\theta\, ,\quad \rho =|\xi|,\quad \theta =\xi /|\xi|,$$ and  J(Z)=\mathfrak{M}^0_2(Z;K_{\delta/e,\delta}). The dimension of the space of such solutions Z is equal to d. \end{corollary} \begin{corollary}\label{TTT1} Let {\rm (\ref{4.8aq8hhh})} be fulfilled with sufficiently small constant \omega_0 depending on n, p and L. Let u\in (\Wcirc^{m,2}_{\rm loc} (\overline{\mathbb{R}_+^n}\setminus \mathcal{O}))^d be a solution of \mathcal{L}(x,D_x)u=0 on B_{\delta }^+, \delta >0, subject to {\rm (\ref{Kyt1aaa})} with p replaced by 2. Then u\in (\Wcirc^{m,p}_{\rm loc} (\overline{\mathbb{R}_+^n}\setminus \mathcal{ O}))^d and $$\label{Ququ3z} \mathfrak{M}^m_p(u;K_{r/e,r})\leq CJ_m(u)\Big (\frac{r}{\delta }\Big )^m\exp\Big (\int_r^\delta (\Upsilon_+(\rho)+c\nu (\rho ))\frac{d\rho}{\rho}\Big )$$ for r<\delta /2. Here $$\label{Zet2} J_m(u)\leq c\mathfrak{M}^m_2(u;K_{\delta/16,\delta})\, .$$ \end{corollary} The following consequence of Corollary \ref{TTT1h} treats the case when u has the same asymptotics as in the constant coefficient case. \begin{corollary}\label{kTTT1} Let (H1) be valid and let $$\label{Using1} \int_{B_+(\delta )}\kappa (x)|x|^{-n}dx <\infty\, .$$ Then there exists p_1>2, depending on L, m, n and \gamma such that if u\in (\Wcirc^{m,2} (\mathbb{R}_+^n))^d be a solution of \mathcal{L}(x,D_x)u=0 on B_{2\delta }^+, \delta >0, then u\in (\Wcirc^{m,p_1} (B_{\delta }^+))^d and u(x)={\bf c}x_n^m+v(x)\, , where {\bf c} is a constant vector and v satisfies the relation $$\label{Using167} \mathfrak{M}^m_{p_1}(v;K_{r/e,r})=o(r^m)$$ as r\to 0. \end{corollary} \medskip The proofs of these corollaries can be found in Section \ref{Section3}. \subsection{Solvability results for the Dirichlet problem in \mathbb{R}_+^n}\label{PertProb} The next statement for d=1 and a=e is proved in \cite[Proposition 1]{KM2}. The proof for arbitrary d and a>1 is the same since the arguments there do not use the facts d=1 and a=e. \begin{proposition}\label{T4.2z} {\rm (i)} Let f\in (W^{-m,q}_{\rm loc} (\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}))^d, q\in (1,\infty ), be subject to $$\label{4.6q} \int _0^1 \rho ^m \mathfrak{M}^{-m}_q(f;K_{\rho /a,\rho} )\frac{d\rho }{\rho } +\int _1^\infty \rho ^{m-1}\mathfrak{M}^{-m}_q(f;K_{\rho /a,\rho} )\frac{d\rho }{\rho } <\infty \; ,$$ where a>1. Then the system $$\label{DD2} L(D_x)u=f\quad \mbox{in \mathbb{R}_+^n}$$ has a solution u\in (\Wcirc^{m,q}_{{\rm loc} }(\overline{{\mathbb{R}}_+^n}\setminus \mathcal{ O}))^d satisfying \label{4.7q} \begin{aligned} \mathfrak{M}^m_q(u;K_{r/a,r})&\leq c\Big (\int _0^r r^m\rho ^m\mathfrak{M}^{-m}_q( f;K_{\rho /a,\rho} )\frac{d\rho }{\rho }\\ &\quad +\int _r^\infty r^{m+1}\rho ^{m-1}\mathfrak{M}^{-m}_q( f;K_{\rho /a,\rho} )\frac{d\rho }{\rho }\Big ). \end{aligned} Estimate {\rm (\ref{4.7q})} implies $$\label{4.7aq} \mathfrak{M}^m_q(u;K_{r/a,r})=\begin{cases} o(r^m ) & \mbox{if } r\to 0\\ o(r^{m+1}) & \mbox{if } r\to\infty . \end{cases}$$ Solution u\in (\Wcirc^{m,q}_{{\rm loc} }(\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}))^d of equation {\rm (\ref{DD2})} subject to {\rm (\ref{4.7aq})} is unique. {\rm (ii)} Let f\in (W^{-m,q}_{\rm loc} (\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}))^d be subject to $$\label{4.8q} \int _0^1 \rho ^{m+n} \mathfrak{M}^{-m}_q( f;K_{\rho /a,\rho} )\frac{d\rho }{\rho }+\int _1^\infty \rho ^m\mathfrak{M}^{-m}_q( f:K_{\rho /a,\rho} )\frac{d\rho }{\rho } <\infty \; .$$ Then system {\rm (\ref{DD2})} has a solution u\in (\Wcirc^{m,q}_{{\rm loc} }(\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}))^d satisfying \label{4.9q} \begin{aligned} \mathfrak{M}^m_q(u;K_{r/a,r})&\leq c\Big (\int _0^r r^{m-n} \rho ^{m+n} \mathfrak{M}^{-m}_q( f;K_{\rho /a,\rho} )\frac{d\rho }{\rho } \\ &\quad +\int _r^\infty r^m \rho ^m\mathfrak{M}^{-m}_q( f;K_{\rho /a,\rho} )\frac{d\rho }{\rho }\Big ). \end{aligned} Estimate {\rm (\ref{4.9q})} implies $$\label{4.9aq} \mathfrak{M}^m_q(u;K_{r/a,r})=\begin{cases} o(r^{m-n}) & \mbox{if } r\to 0\\ o(r^{m}) & \mbox{if }r\to\infty . \end{cases}$$ The solution u\in (\Wcirc^{m,q}_{{\rm loc} }(\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}))^d of equation {\rm (\ref{DD2})} subject to {\rm (\ref{4.9aq})} is unique. \end{proposition} The next proposition contains a solvability result for problem (\ref{4.2zas}), (\ref{4.2zb}). \begin{proposition}\label{T4.2zz} Let (H1)--(H3) be fulfilled and let f\in (W^{-m,p_1}_{\rm loc} (\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}))^d be subject to \label{4.8qz} \begin{aligned} &\int _0^1 \rho ^{m+n}e^{\mathcal{C}\int_\rho^1\Omega (y)\frac{dy}{y}} \mathfrak{M}^{-m}_{p_1}( f;K_{\rho /e,\rho} )\frac{d\rho }{\rho }\\ &+\int _1^\infty \rho ^me^{\mathcal{C}\int_1^\rho\Omega (y)\frac{dy}{y}}\mathfrak{M}^{-m}_{p_1}( f:K_{\rho /e,\rho} )\frac{d\rho }{\rho } <\infty \; . \end{aligned} Then there exists a solution u\in (\Wcirc^{m,p_1}_{{\rm loc} }(\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}))^d of $$\label{VAK2} \mathcal{L}(x,D_x)u=f\quad \mbox{in } \mathbb{R}_+^n$$ satisfying \label{4.9qz} \begin{aligned} \mathfrak{M}^m_{p_1}(u;K_{r/e,r})&\leq cb_0\Big (\int _0^r r^{m-n} \rho ^{m+n}e^{\mathcal{C}\int_\rho^r\Omega (y)\frac{dy}{y}} \mathfrak{M}^{-m}_{p_1}( f;K_{\rho /e,\rho} )\frac{d\rho }{\rho }\\ &\quad +\int _r^\infty r^m \rho ^me^{\mathcal{C}\int_r^\rho\Omega (y)\frac{dy}{y}}\mathfrak{M}^{-m}_{p_1}( f;K_{\rho /e,\rho} )\frac{d\rho }{\rho }\Big ). \end{aligned} Estimate {\rm (\ref{4.9qz})} implies $$\label{4.9aqz} \mathfrak{M}^m_p(u;K_{r/e,r})=\begin{cases} o(r^{m-n}e^{-\mathcal{C}\int_r^1\Omega (s)\frac{ds}{s}}) & \mbox{if } r\to 0 \\ o(r^{m}e^{-\mathcal{C}\int_1^r\Omega (s)\frac{ds}{s}}) & \mbox{if } r\to\infty . \end{cases}$$ The solution u\in (\Wcirc^{m,p}_{{\rm loc} }(\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}))^d of problem {\rm (\ref{VAK2})} subject to {\rm (\ref{4.9aqz})} is unique. \end{proposition} \begin{proof} (1) \emph{Solvability in (\Wcirc^{m,2}(\mathbb{R}_+^n))^d.} Using Lax-Milgram Theorem together with (H1) we obtain unique solvability of problem (\ref{VAK2}) in the space (\Wcirc^{m,2}(\mathbb{R}_+^n))^d for every f\in (W^{-m,2}(\mathbb{R}_+^n))^d. (2) \emph{Solvability in (\Wcirc^{m,p_1}_{{\rm loc}}(\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}))^d.} Since f\in (W^{-m,p_1}_{\rm loc} (\overline{{\mathbb{R}}_+^n} \setminus \mathcal{O}))^d can be approximated by functions from (W^{-m,2}(\mathbb{R}_+^n))^d with compact supports in the norm defined by the left-hand side in (\ref{4.8qz}), for establishing the existence result together with estimate (\ref{4.9qz}) it suffices to prove (\ref{4.9qz}) for solutions from (1). We start with estimating the norm \mathfrak{M}^{-m}_{p}( Nu;K_{r /a,r} ). By H\"older and Hardy inequalities, we have \begin{align*} &\Big |\int_{K_{r /a,r}}\sum _{|\alpha |,|\beta |\leq m }\big ( N_{\alpha \beta}(x)D_x^\beta u,D _x^\alpha v\big )dx\Big | \\ &\leq C\Big (\int_{K_{r /a,r}} \kappa (x)^sdx\Big)^{1/s} \Big (\int_{K_{r /a,r}}|\nabla_mu|^{p_1}dx\Big )^{1/p_1}\Big (\int_{K_{r /a,r}}|\nabla_mv|^{p'}dx\Big )^{1/p'}, \end{align*} where p'=p/(p-1) and s=p_1p/(p_1-p). This leads to $$\label{4.7qq} r^{2m}\mathfrak{M}^{-m}_{p}( Nu;K_{r /a,r} )\leq C\kappa_{s,a}(r)\mathfrak{M}^{m}_{p_1}( u;K_{r /a,r} )$$ with \kappa_{s,a}=\|\kappa \|_{L^s(K_{r/a,r})}. We write equation (\ref{VAK2}) in the form Lu=f_1 with f_1=f+Nu. One can check that the function f_1 satisfies (\ref{4.8q}). Applying Proposition \ref{T4.2z}(ii) with q=p, we obtain that \begin{align*} \mathfrak{M}^m_p(u;K_{r/a,r}) &\leq c\Big (\int _0^r r^{m-n}\rho ^{m+n}\mathfrak{M}^{-m}_p( f+Nu;K_{\rho /a,\rho} )\frac{d\rho }{\rho }\\ &\quad+\int _r^\infty r^{m}\rho ^{m}\mathfrak{M}^{-m}_p( f+Nu;K_{\rho /a,\rho} )\frac{d\rho }{\rho }\Big ). \end{align*} Now, using this estimate with a close to 1 together with (\ref{4.7qq}) and (\ref{4.7aq7hh}) we arrive at \begin{align*} &\mathfrak{M}^m_{p_1}(u;K_{r/e,r})\\ &\leq cb_0\Big (\int_0^r \Big (\frac{r}{\rho}\Big ) ^{m-n}\big (\rho^{2m}\mathfrak{M}^{-m}_{p_1}( f;K_{\rho /e,\rho} ) +\kappa_s(\rho)\mathfrak{M}^m_{p_1}(u;K_{\rho/e,\rho})\big ) \frac{d\rho }{\rho }\\ &\quad +\int _r^\infty \Big (\frac{r}{\rho}\Big )^{m}\big (\rho^{2m}\mathfrak{M}^{-m}_{p_1}( f;K_{\rho /e,\rho} )+\kappa_s(\rho)\mathfrak{M}^m_{p_1}(u;K_{\rho/e,\rho})\big )\frac{d\rho }{\rho }\Big ). \end{align*} Iterating this estimate we obtain $$\label{TTkk1} \mathfrak{M}^m_{p_1}(u;K_{r/e,r}) \leq cb_0\int_0^\infty g_s(r,\rho )\rho^{2m}\mathfrak{M}^{-m}_{p_1}( f;K_{\rho /e,\rho} )\frac{d\rho }{\rho },$$ where \begin{align*} g(e^{-t},e^{-\tau})&=\mu (t-\tau )+\sum_{k=1}^\infty (cb_0)^k\int_{\mathbb{R}^k}\mu(t-\tau_1)\kappa_s(e^{-\tau_1}) \mu(\tau_1-\tau_2)\\ &\quad\dots \kappa_s(e^{-\tau_k})\mu(\tau_k-\tau)d\tau_1\dots d\tau_k. \end{align*} Here \mu(t)=e^{-mt}/n if t\geq 0 and \mu(t)=e^{(n-m)t}/n if t< 0. Since (m-n-\partial_t)(\partial_t+m)\mu(t)=\delta(t), it can be checked that the function \mu_s(t,\tau)=g(e^{-t},e^{-\tau}) satisfies \big ((m-n-\partial_t)(\partial_t+m)-cb_0\kappa_s(e^{-t})\big )\mu_s(t,\tau)=\delta (t-\tau). Using \cite[Proposition 6.3.1]{KM1}, we obtain \begin{gather*} \mu_s(t,\tau)\leq C\exp\Big (-m(t-\tau)+cb_0\int_\tau^t\kappa_s(y)dy\Big )\quad \mbox{if } t\geq \tau\,, \\ \mu_s(t,\tau)\leq C\exp\Big ((n-m)(t-\tau)+cb_0\int_t^\tau\kappa_s(y)dy\Big )\quad \mbox{if } t< \tau\,. \end{gather*} These estimates together with (\ref{TTkk1}) give (\ref{4.9qz}) with the norm \mathfrak{M}^m_p(u;K_{r/e,r}) in the left-hand side. The last norm can be replaced by \mathfrak{M}^m_{p_1}(u;K_{r/e,r}) by using (\ref{4.7aq7hh}). (3) \emph{Uniqueness}. Let \mathcal{L}u=0, u\in (\Wcirc^{m,p}_{{\rm loc} }(\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}))^d and let u be subject to (\ref{4.9aqz}). By (\ref{4.7aq7hh}) one can replace p by p_1 here. Let R be a large positive number and let \eta_R(r) be smooth function equals 1 for R^{-1}\leq r\leq R and 0 for r\leq (Re)^{-1} and for r\geq Re. We can suppose that |\partial_r^k\eta_R(r)|\leq c_kr^{-k} with c_k independent of R. Since \eta_Ru\in (\Wcirc^{m,2}(\mathbb{R}_+^n))^d, we can apply uniqueness result from (1) and obtain from (2) that \begin{align*} \mathfrak{M}^m_{p_1}(\eta_Ru;K_{r/e,r}) &\leq cb_0\Big (\int _0^r r^{m-n} \rho ^{m+n}e^{\mathcal{C}\int_\rho^r\Omega (y)\frac{dy}{y}} \mathfrak{M}^{-m}_{p_1}( \mathcal{L}(\eta_Ru);K_{\rho /e,\rho} ) \frac{d\rho }{\rho }\\ &\quad+\int _r^\infty r^m \rho ^me^{\mathcal{C}\int_r^\rho\Omega (y)\frac{dy}{y}}\mathfrak{M}^{-m}_{p_1}( \mathcal{L}(\eta_Ru);K_{\rho /e,\rho} )\frac{d\rho }{\rho }\Big ), \end{align*} which implies \begin{align*} \mathfrak{M}^m_{p_1}(u;K_{r/e,r}) &\leq cb_0\Big ( r^{m-n} R ^{m-n}e^{\mathcal{C}\int_{1/R}^r\Omega (y)\frac{dy}{y}} \mathfrak{M}^{m}_{p_1}( u);K_{1/(Re),1/R} ) \\ &\quad +r^m R ^{-m}e^{\mathcal{C}\int_r^R\Omega (y)\frac{dy}{y}} \mathfrak{M}^{m}_{p_1}( u;K_{R ,Re} )\Big ) \end{align*} for eR^{-1}\leq r\leq R. By (\ref{4.9aqz}) the right-hand side tends to 0 as R\to 0. Therefore u=0. \end{proof} \section{First order system associated with (\protect\ref{4.2zas}), (\protect\ref{4.2zb})}\label{Section2} \subsection{Reduction of problem (\protect\ref{4.2zas}), (\protect\ref{4.2zb}) to the Dirichlet problem in a cylinder}\label{SX6} We shall use the variables $$\label{5.1} t=-\log |x|\quad \mbox{and}\quad \theta =x/|x|.$$ The mapping x\to (\theta, t) transforms {\mathbb{R}}_+^n onto the cylinder \Pi = S^{n-1}_+\times {\mathbb{R}}. The images of \Wcirc^{m,p}_{\rm loc} (\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}) and W^{-m,p}_{\rm loc} (\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}) under mapping (\ref{5.1}) we denote by \Wcirc^{m,p}_{{\rm loc} }(\Pi ) and W^{-m,p}_{{\rm loc} }(\Pi ). These spaces can be defined independently as follows. The space \Wcirc^{m,p}_{{\rm loc} }(\Pi ) consists of functions whose derivatives up to order m belong to L^p(D) for every compact subset D of \overline{\Pi } and whose derivatives up to order m-1 vanish on \partial\Pi . The seminorm \mathfrak{M}^m_p(u;K_{e^{-a-t},e^{-t}}) in \Wcirc^{m,p}_{\rm loc} (\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}) is equivalent to the seminorm \|u\|_{W^{m,p}(\Pi _{t,t+a})}\; ,\;\; t\in {\mathbb{R}}\; , $$where$$ \Pi _{t,t+a}=\{ (\theta ,\tau )\in\Pi: \tau\in (t,t+a)\}. If a=1 than we shall use also the notation \Pi _{t} for \Pi _{t,t+1}. The space W^{-m,p}_{{\rm loc} }(\Pi ) consists of the distributions f on \Pi  such that the seminorm $$\label{5.4} \|f\|_{W^{-m,p}(\Pi _t)}=\sup \Big |\int _{\Pi _t} f\, \overline{v}d\tau d\theta \Big |$$ is finite for every t\in {\mathbb{R}}. The supremum in (\ref{5.4}) is taken over all v\in \Wcirc^{m,p'}_{{\rm loc} }(\Pi ), p'=p/(p-1), supported in \overline{\Pi _t} and subject to \|v\|_{W^{m,p'}(\Pi _t)}\leq 1. The seminorm (\ref{5.4}) is equivalent to \mathfrak{M}_p^{-m}(f;K_{e^{-1-t},e^{-t}}). In the variables (\theta ,t) the operator L takes the form $$\label{5.2} L(D _x)=e^{2mt}{\bf A}(\theta ,D _\theta ,D_t)\; ,$$ where {\bf A} is an elliptic partial differential operator of order 2m on \Pi  with smooth matrix coefficients. We introduce the operator \mathbb{N} by $$\label{NnNn} L(D_x)-\mathcal{L}(x,D_x)=e^{2mt}\mathbb{N}(\theta ,t,D_\theta ,D_t).$$ Now problem (\ref{4.2zas}), (\ref{4.2zb}) can be written as $$\label{5.3} \begin{gathered} {\bf A}(\theta ,D_\theta ,D _t)u=\mathbb{N}(\theta ,t,D _\theta ,D_t)u+ e^{-2mt}f \quad \mbox{on \Pi }\\ u\in (\Wcirc^{p,m}_{\rm loc} (\Pi ))^N\, , \end{gathered}$$ where f\in (W^{-m,p}_{{\rm loc} }(\Pi ))^d. By (\ref{KUp1a}), the operator \mathbb{N} satisfies \begin{equation*} \|\mathbb{N}\|_{\Wcirc^{m,p}(\Pi_t)\to W^{-m,p}(\Pi_t)}\leq c\,\kappa_\infty (e^{-t})\leq c\gamma \end{equation*} with the same \gamma as in (H1). We put $$\label{Kruto2a} \omega (t)=\Omega (e^{-t})\, ,$$ where \Omega is given by (\ref{VAK1}). Clearly, \omega (t)\leq\omega_0, where \omega_0 is the same as in (H3). By the change of variables (\ref{5.1}) we can formulate Proposition \ref{T4.2zz} as follows \begin{proposition}\label{T5.2z} Let (H1)--(H3) be fulfilled and let f\in (W^{-m,p_1}_{{\rm loc} }(\Pi ))^d be subject to \label{5.8q} \begin{aligned} &\int _0^{\infty } e^{-(m+n)\tau +\mathcal{C}\int _0^\tau \omega (s)ds}\|f\|_{W^{-m,p_1}(\Pi _\tau )}d\tau \\ &+\int _{-\infty }^0 e^{-m\tau +\mathcal{C}\int _\tau ^0\omega (s)ds} \|f\|_{W^{-m,p_1}(\Pi _\tau )}d\tau <\infty \; . \end{aligned} Then problem {\rm (\ref{5.3})} has a solution u\in (\Wcirc^{m,p_1}_{{\rm loc} }(\Pi ))^d satisfying the estimate \label{5.9q} \begin{aligned} \|u\|_{W^{m,p_1}(\Pi _t)} &\leq c\Big (\int _t^\infty e^{(n-m)t-(m+n)\tau +\mathcal{C}\int _t^\tau \omega (s)ds} \|f\|_{W^{-m,p_1}(\Pi _\tau )}d\tau \\ &\quad +\int _{-\infty }^t e^{-m(t+\tau )+\mathcal{C}\int _\tau ^t\omega (s)ds}\|f\|_{W^{-m,p_1}(\Pi _\tau )}d\tau \Big ). \end{aligned} Estimate {\rm (\ref{5.9q})} implies $$\label{5.9aq} \|u\|_{W^{m,p}(\Pi _t)}=\begin{cases} o(e^{(n-m)t-\mathcal{C}\int _0^t \omega (s)ds}) & \mbox{if t\to +\infty }\\ o(e^{-mt-\mathcal{C}\int _t^0 \omega (s)ds}) & \mbox{if t\to -\infty.} \end{cases}$$ The solution u\in (\Wcirc^{m,p}_{{\rm loc} }(\Pi ))^d of problem {\rm (\ref{5.3})} subject to {\rm (\ref{5.9aq})} is unique. \end{proposition} Let W^{-m,p}(S_+^{n-1}) denote the dual of \Wcirc^{m,q}(S_+^{n-1}), q=p/(p-1), with respect to the inner product in L^2(S_+^{n-1}). We introduce the operator pencil $$\label{4.2a} \mathcal{A}(\lambda ): (\Wcirc ^{m,p}(S_+^{n-1}))^d\to (W^{-m,p}(S_+^{n-1}))^d$$ by $$\label{4.2ab} \mathcal{A}(\lambda )U(\theta )=r^{i\lambda +2m}L(D _x)r^{-i\lambda }U(\theta ) ={\bf A}(\theta ,D_\theta ,\lambda )U(\theta ).$$ The following properties of \mathcal{A} and its adjoint are standard and their proofs can be found, for example in \cite[Section 10.3]{KM4}. The operator (\ref{4.2a}) is Fredholm for all \lambda\in \mathbb{C} and its spectrum consists of eigenvalues with finite geometric multiplicities. These eigenvalues are $$\label{Spec1a} i(m+k)\quad \mbox{and}\quad i(m-n-k)\quad \mbox{for k=0,1,\dots },$$ and there are no generalized eigenvectors. The eigenvectors corresponding to the eigenvalue im are  c|x|^{-m}x_n^m=c\theta _n^m, where c\in \mathbb{C}^d. We introduce the operator pencil \mathcal{A}^*(\lambda ) defined on \Wcirc ^{m,p}(S_+^{n-1}) by the formula \mathcal{A}^*(\lambda )U(\theta )=r^{i\lambda +2m}L^*(D _x)r^{-i\lambda }U(\theta )\, . $$This pencil has the same eigenvalues as the pencil \mathcal{A}(\lambda ). Eigenvector corresponding to the eigenvalue i(m-n) are linear combinations of |x|^{n-m}E_j(x)=E_j(\theta), where E_j are defined in Section \ref{SectAsym}. Moreover, the following biorthogonality condition holds: $$\label{4.2c} \int _{{\mathbb{R}}_+^n}(L\big (D_x)(e_k\zeta x_n^m),E_j(x))dx= m!\delta_j^k\; ,$$ where \zeta  is a smooth function equal to 1 in a neighborhood of the origin and zero for large |x|. This relation can be checked by integration by parts. Using the definitions of the above pencils and Green's formula for L and \overline{L} one can show that $$\label{Adj1} (\mathcal{A}(\lambda ))^*=\mathcal{A}^*(\overline{\lambda }+(2m-n)i)\, ,$$ where * in the left-hand side denotes passage to the adjoint operator in (L^2(S_+^{n-1}))^d. This implies, in particular, $$\label{Adj1a} (\mathcal{A}(im))^*E_j(\theta )=0\quad \mbox{for j=1,\dots ,d.}$$ \subsection{Reduction of problem (\protect\ref{5.3}) to a first order system in t}\label{Subs22} To reduce problem (\ref{5.3}) to a first order system, first we represent the right-hand side f\in W^{-m,p}_{\rm loc} (\Pi ) as $$\label{Rep1} f=e^{2mt}\sum _{j=0}^mD _t^{m-j}f_j\;,$$ where f_j\in L^p_{\rm loc} ({\mathbb{R}};W^{-j,p}(S_+^{n-1})). This representation can be chosen to satisfy$$ c_1\mathfrak{M}^{-m}_p(f;K_{e^{-1-t},e^{-t}})\leq e^{2mt}\sum _{j=0}^m\|f_j\|_{W^{-j,p}(\Pi _t)} \leq c_2\mathfrak{ M}^{-m}_p(f;K_{e^{-2-t},e^{1-t}})\, , $$where c_1 and c_2 are constants depending only on n, m and p (see \cite[Lemma 1]{KM2}). Next, we represent the operators r^{|\alpha |}D_x^\alpha and r^{2m}D_x^\alpha (r^{-2m+|\alpha |}\;\cdot\; ) as polynomials with respect to -rD_r we obtain \begin{gather*} r^{|\alpha |}D _x^\alpha u=\sum _{l=0}^{|\alpha |}Q_{\alpha l}(\theta ,D_\theta )(-rD_r)^lu\,, \\ r^{2m}D _x^\alpha (r^{-2m+|\alpha |}u) =\sum _{l=0}^{|\alpha |}P_{\alpha l}(\theta ,D_\theta )(-rD_r)^lu\, , \end{gather*} where Q_{\alpha l}(\theta ,D_\theta ) and P_{\alpha l}(\theta ,D_\theta ) are differential operators of order |\alpha |-l with smooth coefficients. Furthermore, integrating by parts in$$ \int _{\mathbb{R}_+^n}D _x^\alpha (r^{-2m+|\alpha |}u)r^{2m-n}\overline{v}dx, $$we obtain $$\label{KLPN} \sum_{l=0}^{|\alpha|} Q_{\alpha l}(-rD _r+i(2m-n))^l =\sum _{l=0}^{|\alpha |}P^*_{\alpha l}(-rD_r)^l\, ,$$ where P^*_{\alpha l} is the differential operator on S^{n-1} adjoint to P_{\alpha l}. Now we write {\bf A} in the form$$ {\bf A}(\theta ,D_\theta ,D_t)=\sum _{j=0}^mD _t^{m-j}\mathcal{A}_j(D _t)\, , $$where$$ \mathcal{A}_j(D _t)=\sum _{k=0}^mA_{jk}D _t^{m-k} $$with$$ A_{jk}=\sum_{|\alpha |=|\beta |=m}P_{\alpha ,m-j}(\theta ,D_\theta ) L_{\alpha\beta} Q_{\beta ,m-k}(\theta ,D_\theta )\, . $$It is clear that $$\label{AAAA1} A_{jk}:(\Wcirc^{k,p}(S_+^{n-1}))^d\to (W^{-j,p}(S_+^{n-1}))^d\;$$ are differential operators of order \leq j+k on S_+^{n-1} with smooth matrix coefficients. We also write $$\label{OperN1} \mathbb{N}(\theta ,t,D_\theta ,D_t)u=\sum _{j=0}^mD _t^{m-j} \big (\mathcal{N}_j(t,D_t)u\big )\, ,$$ where $$\label{OperN2} \mathcal{N}_j(t,D_t)=\sum _{k=0}^m\mathcal{N}_{jk}(t)D _t^{m-k}\,$$ with $$\label{OperN22} \mathcal{N}_{jk}=\sum_{m-j\leq|\alpha |\leq m}\sum_{m-k\leq |\beta |\leq m}P_{\alpha ,m-j}e^{-(2m-|\alpha|-|\beta|)t}N_{\alpha\beta} Q_{\beta ,m-k}\, ,$$ where N_{\alpha\beta} is defined by (\ref{NN5a}). By (\ref{OperN22}) the operators$$ \mathcal{N}_{jk}(t)\, :\,(\Wcirc^{k,p}(S_+^{n-1}))^d\to (W^{-j,p}(S_+^{n-1}))^d\; are continuous. By (\ref{OperN22}) and (\ref{KLPN}), for almost all r>0 \label{UHO1} \begin{aligned} &\int _{S_+^{n-1}}\sum_{j=0}^m\Big (\mathcal{N}_{j}(D_t)u,D_t^{m-j} (e^{(2m-n)t}v)\Big )d\theta \\ &=\int _{S_+^{n-1}}\sum_{j,k\leq m}\sum_{m-j\leq|\alpha |\leq m}\sum_{m-k\leq |\beta |\leq m} \Big (N_{\alpha\beta} Q_{\beta ,m-k}D_t^{m-k}u, \\ &\quad e^{-(2m-|\alpha|-|\beta|)t}P^*_{\alpha ,m-j}D_t^{m-j}(e^{(2m-n)t}v)\Big )d\theta \\ &=r^n\int_{S_+^{n-1}}\sum_{|\alpha |,|\beta |\leq m}\Big (N_{\alpha\beta}(x)D_x^\beta u, D_x^\alpha v\Big )d\theta\, , \end{aligned} where u and v are in  (\Wcirc^{m,p}_{\rm loc} (\overline{\mathbb{R}_+^n\setminus \mathcal{O}}) )^d. Using the operators \mathcal{A}_j(D_t) and \mathcal{N}_j(t,D _t), and (\ref{Rep1}) we write problem (\ref{5.3}) in the form $$\label{6.1} \sum _{j=0}^mD_t^{m-j}\mathcal{A}_j(D_t)u(t) =\sum _{j=0}^mD_t^{m-j}\big (\mathcal{N}_j(t,D _t)u+f_j(t)\big )\quad \mbox{on {\mathbb{R}},}$$ where we consider u and f_j as functions on \mathbb{R} taking values in function spaces (\Wcirc^{m,p}( S_+^{n-1}))^d and (W^{-j,p}(S_+^{n-1}))^d respectively. By (\ref{Ocen1}) and (\ref{OperN22}) $$\label{Ocen2} \|\mathcal{N}_{jk}(t)\|_{(\Wcirc^{k,p}(S_+^{n-1}))^d\to (W^{-j,p}(S_+^{n-1}))^d}\leq \; c\,\kappa_\infty(e^{-t})\; .$$ Therefore, \mathcal{N}_j acts from (\Wcirc^{m,p}_{\rm loc} (\Pi ))^d to (L^p_{\rm loc} ({\mathbb{R}};W^{-j,p}(S_+^{n-1})))^d. The local estimate (H2) can be reformulated now as follows \begin{itemize} \item[(H2a)] Let p and p_1 be the same as in (H2) and u\in (\Wcirc^{m,p}_{\rm loc} (\Pi ))^d satisfies (\ref{6.1}) with f_j\in (L^{p_1}_{\rm loc} ({\mathbb{R}};W^{-j,p_1}(S_+^{n-1})))^d, then u\in (\Wcirc^{m,p_1}_{\rm loc} (\Pi ))^d and \label{VAK3} \begin{aligned} &\|u\|_{W^{m,p_1} (\Pi _{t,t+a})} \\ &\leq cb_0\big(\sum_{j=0}^m\|f_j\|_{L^{p_1} (t-a,t+2a;W^{-j,p_1}(S_+^{n-1}))}+\|u\|_{W^{m,p} (\Pi _{t-a,t+2a})}\big ), \end{aligned} where c may depend on a>0. \end{itemize} Let \mathcal{U}=\mathop{\rm col} (\mathcal{U}_1,\dots ,\mathcal{U}_{2m}), where \begin{gather}\label{X6.2} \mathcal{U}_k=D _t^{k-1}u,\quad k=1,\dots ,m, \\ \label{X6.3} \mathcal{U}_{m+1}=\mathcal{A}_0(D _t)u -\mathcal{N}_0(t,D _t)u-f_0\,, \\ \label{X6.4} \mathcal{U}_{m+j}=D_t\mathcal{U}_{m+j-1}+\mathcal{A}_{j-1}(D_t)u -\mathcal{ N}_{j-1}(t,D_t)u-f_{j-1} \end{gather} for j=2,\dots ,m. With this notation (\ref{6.1}) takes the form $$\label{X6.5} D_t\mathcal{U}_{2m }+\mathcal{A}_m(D_t)u-\mathcal{N}_m(t,D _t)u-f_m=0.$$ Using (\ref{X6.2}) we write (\ref{X6.3}) as $$\label{Yab} (A_{00}-\mathcal{N}_{00}(t))D_t^mu =\mathcal{U}_{m+1}-\sum _{k=0}^{m-1}(A_{0,m-k}-\mathcal{N}_{0,m-k}(t))\mathcal{U}_{k+1}+f_0\; .$$ Since Q_{\alpha ,|\alpha |}=P_{\alpha ,|\alpha |}=\theta^\alpha , we have A_{00}-\mathcal{N}_{00}=L(\theta )- \sum _{|\alpha |=|\beta |=m}N_{\alpha \beta }(e^{-t}\theta )\theta ^{\alpha +\beta }, $$and by (H1) the matrix A_{00}-\mathcal{N}_{00} is invertible, and the norm of the inverse matrix is bounded by a constant times \gamma^{-1} . Thus equation (\ref{Yab}) is uniquely solvable with respect to D_t^mu, and $$\label{Yac} D_t^mu=\mathcal{S}(t)\hat{\mathcal{U}}\; ,$$ where $$\label{Yacz} \mathcal{S}(t)\hat{\mathcal{U}}= (A_{00}-\mathcal{N}_{00}(t))^{-1} \Big (\mathcal{U}_{m+1}-\sum _{k=0}^{m-1}(A_{0,m-k}-\mathcal{ N}_{0,m-k}(t))\mathcal{U}_{k+1}+f_0\Big )$$ and \hat{\mathcal{U}}=(\mathcal{U}_1,\dots ,\mathcal{U}_m,\mathcal{ U}_{m+1}). If we introduce the following two operators \begin{gather}\label{Yacz1a} \mathcal{S}_0\hat{\mathcal{U}}= A_{00}^{-1} \Big (\mathcal{U}_{m+1}-\sum _{k=0}^{m-1}A_{0,m-k}\mathcal{U}_{k+1}\Big )\,\\ \label{Yacz2a} \mathcal{S}'(t)\hat{\mathcal{U}}=\mathcal{S}(t)\hat{\mathcal{U}}- (A_{00}-\mathcal{N}_{00}(t))^{-1} f_0\,, \end{gather} then $$\label{Yacz1b} \mathcal{S}'(t)\hat{\mathcal{U}}-\mathcal{S}_0(t)\hat{\mathcal{ U}}=A_{00}^{-1}\Big (\sum _{k=0}^{m-1}\mathcal{N}_{0,m-k}(t)\mathcal{ U}_{k+1}+\mathcal{N}_{00}(t)\mathcal{S}'(t)\hat{\mathcal{U}}\Big ).$$ One verifies directly that $$\label{2005a} \|\mathcal{S}'(t)\hat{\mathcal{U}}\|_{L^q(S_+^{n-1})}\leq C\sum_{j=0}^m\|\mathcal{U}_{j+1}\|_{W^{m-j,q}(S_+^{n-1})}$$ and$$ \|\mathcal{S}_0(t)\hat{\mathcal{U}}\|_{L^q(S_+^{n-1})}\leq C\sum_{j=0}^m\|\mathcal{U}_{j+1}\|_{W^{m-j,q}(S_+^{n-1})} for q\in (1,\infty). From (\ref{X6.2}) it follows that $$\label{Y1} D_t\mathcal{U}_k=\mathcal{U}_{k+1}\quad \mbox{for k=1,\dots ,m-1.}$$ By (\ref{Yac}) we have $$\label{Y1z} D_t\mathcal{U}_m=\mathcal{S}(t)\hat{\mathcal{U}}.$$ Using (\ref{Yac}), we write (\ref{X6.4}) as \label{Y2} \begin{aligned} D_t\mathcal{U}_{m+j} &=\mathcal{U}_{m+j+1}-\sum _{k=0}^{m-1}(A_{j,m-k}-\mathcal{N}_{j,m-k}(t)) \mathcal{U}_{k+1} \\ &\quad-(A_{j0}-\mathcal{N}_{j0}(t))\mathcal{S}(t)\hat{\mathcal{U}}+f_j \end{aligned} for j=1,\dots ,m-1 and (\ref{X6.5}) takes the form $$\label{Y3} D_t\mathcal{U}_{2m}+\sum _{k=0}^{m-1}(A_{m,m-k}-\mathcal{N}_{m,m-k}(t)) \mathcal{U}_{k+1} +(A_{m0}-\mathcal{N}_{m0}(t))\mathcal{S}(t)\hat{\mathcal{U}}-f_m=0.$$ Relations (\ref{Y1})--(\ref{Y3}) can be written as the first order evolution system $$\label{X6.10} (\mathcal{I}D _t+\mathfrak{A})\mathcal{U}(t)-\mathfrak{N}(t) \widehat{\mathcal{U}}(t)=\mathcal{F}(t)\quad \mbox{on {\mathbb{R}},}$$ where $$\label{FFF1a} \mathcal{F}(t)= \mathop{\rm col} (0,\dots ,0, \mathcal{F}_{m}(t),\mathcal{ F}_{m+1}(t),\dots ,\mathcal{F}_{2m}(t))$$ with $$\label{FFF1b} \mathcal{F}_{m}(t)=(A_{00}-\mathcal{N}_{00}(t))^{-1}f_0(t),$$ $$\label{FFF1c} \mathcal{F}_{m+j}(t)=f_j(t)-(A_{j0}-\mathcal{N}_{j0}(t))(A_{00}-\mathcal{ N}_{00}(t))^{-1}f_0(t),\quad j=1,\dots ,m\; .$$ The operator \mathfrak{N} is given by $$\label{101} \mathfrak{N}(t)\widehat{\mathcal{U}}=\mathop{\rm col} (0,\dots ,0, \mathfrak{ N}_m(t)\mathcal{U},\mathfrak{N}_{m+1}(t)\mathcal{U}, \dots ,\mathfrak{ N}_{2m}(t)\mathcal{U}),$$ where $$\label{102} \mathfrak{N}_m(t)\widehat{\mathcal{U}}=A_{00}^{-1} \Big (\sum _{k=0}^{m-1}\mathcal{N}_{0,m-k}(t)\mathcal{U}_{k+1}+ \mathcal{ N}_{00}(t)\mathcal{S}'(t)\hat{\mathcal{U}}\Big )$$ and \label{103} \begin{aligned} \mathfrak{N}_{m+j}(t)\widehat{\mathcal{U}} &=\sum _{k=0}^{m-1}\mathcal{N}_{j,m-k}(t)\mathcal{U}_{k+1} +\mathcal{N}_{j0}(t)\mathcal{S}'(t)\hat{\mathcal{U}} \\ &\quad -A_{j0}A_{00}^{-1}\Big (\sum _{k=0}^{m-1}\mathcal{ N}_{0,m-k}(t)\mathcal{U}_{k+1} +\mathcal{N}_{00}(t)\mathcal{ S}'(t)\hat{\mathcal{U}}\Big ) \end{aligned} for j=1,\dots ,m. In (\ref{X6.10}), by \mathcal{I}, we denote the identity operator. We also use the operator matrix $$\label{Ququ2} \mathfrak{A}=-\mathcal{J}+\mathfrak{L}\;$$ with \mathcal{J}=\{ \mathcal{J}_{jk}\} _{j,k=1}^{2m}  given by \let\c=\cdots \let\v=\vdots \let\d=\ddots \let\*=\cdot \bordermatrix{&&&&&(m+1)&\cr &0&I&0&\c&\c&\c&0\cr &0&0&I&\c&\v&\c&0\cr &\c&\c&\c&\d&\v&\c&\c\cr (m)&0&\c&\c&\c&A_{00}^{-1}&\c&0\cr &\c&\c&\c&\c&\c&\d&\c\cr &0&0&0&\c&\c&\c&I\cr &0&0&0&\c&\c&\c&0\cr} $$and with \mathfrak{L}=\{ \mathfrak{L}_{jk}\} _{j,k=1}^{2m}  equal to$$ \let\c=\cdots \let\v=\vdots \let\d=\ddots \let\*=\cdot \bordermatrix{&&&&&&\cr &0&\c&0&0&\c&0\cr &\v&\d&\v&\c&\v&\v\cr &0&\c&0&0&\c&0\cr &A_{00}^{-1}A_{0,m}&\c&A_{00}^{-1}A_{0,1}&0&\c&0\cr \smallskip &A_{1,m}-A_{1,0}A_{00}^{-1}A_{0,m}&\c&A_{1,1}-A_{1,0}A_{00}^{-1}A_{0,1}&A_{10}A_{00}^{-1}&\c&0\cr &\v&\c&\v&\c&\d&\v\cr &A_{m,m}-A_{m,0}A_{00}^{-1}A_{0,m}&\c&A_{m,1}-A_{m,0}A_{00}^{-1}A_{0,1}&A_{m0}A_{00}^{-1}&\c&0\cr} $$We put \begin{gather}\label{TTN1} \mathcal{T}=B_m\times\cdots\times B_1\times B_0 \times (B_{-m})^{m-1}, \\ \label{TTN2} \mathcal{R}=B_{m-1}\times\cdots\times B_1\times B_0 \times (B_{-m})^m\; , \end{gather} where $$\label{2001} \begin{gathered} B_j=(\Wcirc^{j,p}(S_+^{n-1}))^d\quad \mbox{for j=1,\dots ,m,}\quad B_0=(L^p(S_+^{n-1}))^d\,, \\ B_{-j}=(W^{-j,p}(S_+^{n-1}))^d\quad \mbox{for j=1,\dots ,m.} \end{gathered}$$ By (\ref{AAAA1}) the operator \mathfrak{A}: \mathcal{T}\to \mathcal{ R} is continuous. Sometimes we shall write B_j^p, \mathcal{T}_p and \mathcal{R}_p in order to mark the dependence of these spaces on p. \subsection{Spectral properties of the pencil \lambda \mathcal{I}+\mathfrak{A}}\label{S7} We introduce the operator matrix \mathcal{E}(\lambda )=\{ \mathcal{ E}_{pq}(\lambda )\} _{p,q=1}^{2m } as $$\label{Inv2} \let\c=\cdots \let\v=\vdots \let\d=\ddots \let\*=\cdot \bordermatrix{&&&&(m)&&\cr &e_1(\lambda )&e_2(\lambda )&\c&e_m(\lambda )&\c&e_{2m-1}(\lambda )&e_{2m}(\lambda )\cr &-I&0&\c&0&\c&0&0\cr &0&-I&\c&0&\c&0&0\cr &\v&\v&\c&\v&\c&\v&\v\cr (m+1)&0&0&\c&- A_{00}&\c&0&0\cr &\v&\v&\c&\v&\c&\v&\v\cr &0&0&\c&0&\c&-I&0\cr}$$ where \begin{gather*} e_{2m-j}(\lambda )=\lambda^j,\quad j=0,\dots ,m-1,\\ e_{m}=\sum_{j=0}^m\lambda^{m-j}A_{j0}, \\ e_{m-k}(\lambda )=\sum_{s=0}^k\sum_{j=0}^m\lambda^{k+m-s-j}A_{js},\quad k=1,\dots,m-1. \end{gather*} Let also$$ J(\lambda )=\left ( \begin{array}{ccccc} I&0&\dots&0&0\\ -\lambda &I&\dots&0&0\\ \vdots&\vdots& &\vdots&\vdots\\ 0&0&\dots&I&0\\ 0&0&\dots&-\lambda&I \end{array} \right ),\; \mathcal{M} =\left ( \begin{array}{ccccc} 0&0&\dots&0&0\\ A_{10} A_{00}^{-1}&0&\dots&0&0\\ A_{20} A_{00}^{-1}&0&\dots&0&0\\ \vdots&\vdots&&\vdots&\vdots\\ A_{m-1,0}A_{00}^{-1}&0&\dots&0&0 \end{array} \right ) $$be two m\times m-matrices. One can check directly that \mathcal{ E}^{-1}(\lambda ) is given by$$ \let\c=\cdots \let\v=\vdots \let\d=\ddots \let\*=\cdot \bordermatrix{&&&&(m+1)&&\cr &0&-I&\c&0&\c&0&0\cr &0&0&\c&0&\c&0&0\cr &\v&\v&\c&\v&\c&\v&\v\cr (m)&0&0&\c&- A^{-1}_{00}&\c&0&0\cr &\v&\v&\c&\v&\c&\v&\v\cr &0&0&\c&0&\c&0&-I\cr &I&e_1(\lambda )&\c&e_m(\lambda )&\c&e_{2m-2}(\lambda )&e_{2m-1}(\lambda )\cr} and that $$\label{Inv1} J(\lambda )^{-1}=\left ( \begin{array}{cccccc} I&0&0&\dots&0&0\\ \lambda &I&0&\dots&\vdots &\vdots\\ \lambda ^2&\lambda &I&\dots &\vdots &\vdots \\ \vdots&\vdots& \vdots&\dots &I&0\\ \lambda ^{m-1}&\lambda ^{m-2}&\lambda ^{m-3}&\dots&\lambda&I \end{array} \right )\; .$$ The following assertion is proved in \cite{KM2} \begin{proposition}\label{L6.2} For all \lambda\in \mathbb{C} $$\label{6.15} \mathcal{E}(\lambda )(\lambda \mathcal{I}+\mathfrak{A}) =\mathop{\rm diag} (\mathcal{A}(\lambda ),I,\dots ,I) \left ( \begin{array}{cc} J(\lambda )&0\\ -\mathcal{B}(\lambda )& J(\lambda )-\mathcal{M} \end{array}\right )$$ where the m\times m-matrix \mathcal{B}(\lambda ) is defined by \begin{align*} \mathcal{B}(\lambda )&=\left ( \begin{array}{ccccc} A_{0m}&\dots &A_{02}& A_{01}\\ A_{1,m}&\dots &A_{12}&A_{11}\\ \vdots&\dots &\vdots &\vdots \\ A_{m-1,m}& \dots &A_{m-1,2}& A_{m-1,1} \end{array} \right )\\ &\quad -\left ( \begin{array}{ccccc} 0&\dots &0&-\lambda A_{00}\\ A_{10}A_{00}^{-1}A_{0,m}& \dots &A_{10}A_{00}^{-1}A_{02}&A_{10}A_{00}^{-1}A_{01}\\ \vdots &\dots &\vdots &\vdots \\ A_{m-1,0}A_{00}^{-1}A_{0,m}&\dots & A_{m-1,0}A_{00}^{-1}A_{02}&A_{m-1,0}A_{00}^{-1}A_{01} \end{array} \right ) \end{align*} Moreover, $$\label{6.16} \left ( \begin{array}{cc} J(\lambda )&0\\ -\mathcal{B}(\lambda )&J(\lambda )-\mathcal{M} \end{array}\right )^{-1}= \left ( \begin{array}{cc} J^{-1}(\lambda )&0\\ Q(\lambda )&J^{-1}(\lambda )(I+\mathcal{M}) \end{array}\right ) ,$$ where the elements of the matrix Q(\lambda )=\{ Q_{jk}(\lambda )\}_{j,k=1}^m are given by $$\label{6.18} Q_{jk}(\lambda )=\sum _{l=0}^{j-1}\sum _{q=k-1}^{m}\lambda ^{q+l+1-k}A_{j-l-1,m -q}.$$ \end{proposition} The relation (\ref{6.15}) allows one to establish the following correspondence between \mathcal{A}(\lambda ) and the linear pencil \lambda \mathcal{I}+\mathfrak{A}. \begin{proposition}\label{P6.1} {\rm (see \cite{KM2})} {\rm (i)} The operator $$\label{6.19} \lambda \mathcal{I}+\mathfrak{A}:\mathcal{T}\to \mathcal{R}$$ is Fredholm for all \lambda\in \mathbb{C}. {\rm (ii)} The spectra of the operator \mathfrak{A} and the pencil \mathcal{A}(\lambda ) coincide and consist of eigenvalues of the same multiplicity. \end{proposition} We put \phi _j(\theta )=\theta _n^m\, e_j \; \;\;\mbox{and}\quad \psi _j (\theta )=(m!)^{-1}E_j(\theta )\quad j=1,\dots ,d, $$where e_j and E_j are defined in the beginning of Section \ref{SectAsym}. By (\ref{4.2c}) and (\ref{4.2ab}) $$\label{Ort1} \int _{\Pi }(\mathcal{A}(D _t)(\eta (t)e^{-mt}\phi _k(\theta ))\, , e^{mt}\psi _j(\theta ))d\theta dt =\delta _k^j\; ,$$ where \eta  is a smooth function equal to 1 for large positive t and 0 for large negative t. The equality (\ref{Ort1}) can be written as $$\label{Ort1a} \int _{S_+^{n-1}}(\mathcal{A}'(im)\phi _k(\theta )\, ,\psi _j(\theta ))d\theta =i\delta _k^j\; .$$ We introduce the vector functions $$\label{Phi1a} \Phi _j=\mathop{\rm col} (\Phi_{jl})_{l=1}^{2m} =\left (\begin{array}{cc} J^{-1}(im)&0\\ Q(im)&J^{-1}(im)(I+\mathcal{M}) \end{array}\right ) \mathop{\rm col} (\phi _j,0,\dots ,0).$$ Owing to (\ref{6.15}) and (\ref{6.16}) we obtain $$\label{Eigen1} (im\mathcal{I}+\mathfrak{A})\Phi _j=0\; .$$ Using (\ref{Inv1}) and the definitions of the matrices \mathcal{M} and \mathcal{B} we get $$\label{7.2} \Phi _{jl}=(im)^{l-1}\phi _j,\quad l=1,\dots ,m,$$ $$\label{7.3} \Phi _{j,m+l}=\sum _{p=0}^{l-1}\sum _{q=0}^mA_{l-p-1,m-q}(im)^{p+q}\phi_j$$ for l=1,\dots ,m. We introduce the vector \Psi _j=\mathop{\rm col} (\Psi _{jl})_{l=1}^{2m} , by $$\label{7.4} \Psi _j=\mathcal{E}^*(-im)\mathop{\rm col} (\psi _j,0,\dots ,0)\;$$ where \mathcal{E}^*(\lambda ) is the adjoint of \mathcal{ E}(\overline{\lambda }). Since \psi _j is the eigenfunction of the pencil (\mathcal{A}(\lambda ))^* corresponding to the eigenvalue \lambda =im (see (\ref{Adj1a})), it follows from (\ref{6.15}) that $$\label{Eigen2} (-im\mathcal{I}+\mathfrak{A}^*)\Psi_j =0.$$ By (\ref{Inv2})$$ \Psi _{jl}=\sum _{p=0}^{m-l}\sum _{q=0}^mA_{qp}^*(-im)^{2m -l-q-p}\psi_j for l=1,\dots ,m-1, \begin{gather}\label{7.4d} \Psi _{jm}=\sum _{q=0}^mA_{q0}^*(-im)^{m-q}\psi_j\,, \\ \label{7.4e} \Psi _{j,m+l}=(-im)^{m-l}\psi_j \end{gather} for l=1,\dots ,m. Clearly, \Phi _j\in \mathcal{T},\quad \Psi _j\in \mathcal{R}^*, where \begin{align*} \mathcal{R}^*&=(W^{1-m,q}(S_+^{n-1}))^d\times\cdots\times (W^{-1,q}(S_+^{n-1}))^d\\ &\quad \times (L^q(S_+^{n-1}) )^d\times ((\Wcirc^{m,q}(S_+^{n-1}))^d)^m\; . \end{align*} \begin{proposition} The biorthogonality condition $$\label{7.5a} \langle\Phi_k ,\Psi _j\rangle =i\delta_k^j$$ is valid, where \langle\Phi_k ,\Psi _j\rangle =\sum _{s=1}^{2m}\int_{S_+^{n-1}}(\Phi_{ks} ,\Psi _{js})d\theta \, . $$\end{proposition} \begin{proof} We put$$ \Phi_{k\lambda}=\left (\begin{array}{cc} J^{-1}(\lambda )&0\\ Q(\lambda )&J^{-1}(\lambda )(I+\mathcal{M}) \end{array}\right ) \mathop{\rm col} (\phi _k,0,\dots ,0) $$and \Psi_{j\lambda}=\mathcal{E}^*(\overline{\lambda })\mathop{\rm col} (\psi_j,0,\dots ,0). Then by (\ref{6.15}) and (\ref{6.16})$$ \langle (\lambda \mathcal{I}+\mathfrak{A})\Phi_{k\lambda} ,\Psi _{j\lambda}\rangle =\int_{S^{n-1}_+}( \mathcal{A}(\lambda )\phi_k,\psi_j)d\theta\, . $$Differentiating this equality with respect to \lambda, setting \lambda =im and using (\ref{Eigen1}), (\ref{Eigen2}) together with (\ref{Ort1a}) we obtain (\ref{7.5a}). \end{proof} We introduce the spectral projector \mathcal{P} corresponding to the eigenvalue \lambda =im: $$\label{7.6} \mathcal{P}\mathcal{F}=-i\sum _{q=1}^d\langle \mathcal{F},\Psi _q\rangle \Phi _q\; .$$ This operator maps \mathcal{R} into \mathcal{T}. Using (\ref{Eigen1}) we obtain $$\label{TDHa} \mathfrak{A}\mathcal{P}=-im\mathcal{P}\, .$$ \subsection{Equivalence of equation (\protect\ref{6.1}) and system (\protect\ref{X6.10})}\label{S8} Here we collect definitions of some spaces which are used in the sequel. Let \mathcal{T} and \mathcal{R} be the spaces defined by (\ref{TTN1}) and (\ref{TTN2}). We introduce the space \mathbb{ T}(a,b) of vector functions \mathcal{U}=\mathop{\rm col} (\mathcal{U}_j)^{2m} _{j=1} defined on (a,b), taking values in \mathcal{T} and supplied with the norm$$ \| \mathcal{U}\| _{\mathbb{T}(a,b)}= \Big (\int ^b_a\big (\| \mathcal{U}(\tau )\| ^p_\mathcal{ T}+ \| D_\tau \mathcal{U}(\tau )\| ^p_\mathcal{R}\big )d\tau \Big )^{1/p}. $$This definition is equivalent to $$\label{15.4.2} \mathbb{T}(a,b)=\big \lbrace \mathcal{U}:\mathcal{U}\in L_p(a,b;\mathcal{T}), D_t\mathcal{U}\in L_p(a,b;\mathcal{R})\big \rbrace.$$ Here p is the same number as in the definition of the spaces B_k. By \mathbb{T}'(a,b) we denote the space of vector-functions$$ \mathcal{U}'= (\mathcal{U}_1,\dots ,\mathcal{U}_{m})\, , $$with values in B_{m}\times\cdots \times B_1 which is endowed with the norm$$ \| \mathcal{U}'\| _{\mathbb{T}'(a,b)}= \Big (\sum _{j=1}^{m}\int ^{b}_a\big (\| \mathcal{U}_{j}(\tau )\| ^p_{B_{m-j+1}} +\| D_\tau \mathcal{U}_{j}(\tau )\| ^p_{B_{m-j}}\big )d\tau \Big )^{1/p}\, , $$where B_j is defined by (\ref{2001}). Also let \hat{\mathbb{T}}(a,b) be the product \mathbb{T}'(a,b)\times L_p(a,b;B_0), which consists of the vector functions$$ \hat{\mathcal{U}}=(\mathcal{U}',\, \mathcal{U}_{m+1})\, $$with the norm$$ \|\hat{\mathcal{U}}\|_{\hat{\mathbb{T}}(a,b)}=(\| \mathcal{ U}'\| _{\mathbb{T}'(a,b)}^p+\|\mathcal{ U}_{m+1}\|_{L_p(a,b;B_0)}^p)^{1/p}. $$Furthermore, we use the spaces \mathbb{T}'_{\rm loc} (\mathbb{R}) and \hat{\mathbb{T}}_{{\rm loc} }(\mathbb{R}) endowed with the seminorms \| \mathcal{U}'\| _{\mathbb{T}'(t,t+1)} and \| \hat{\mathcal{U}}\| _{\hat{\mathbb{T}}(t,t+1)}, t\in\mathbb{R}. If u\in \Wcirc_{p,{\rm loc}}^{m}(\Pi) then by setting \mathcal{ U}_j=D_t^{j-1}u we see that$$ \mathcal{U}'\in\mathbb{T}'_{\rm loc} (\mathbb{R})\quad \mbox{and}\quad \hat{\mathcal{U}}\in \hat{\mathbb{T}}_{\rm loc} (\mathbb{R}) and $$\label{Norm1} 2^{-1/p}\|\hat{\mathcal{U}}\|_{\hat{\mathbb{T}}(t,t+1)}\leq \|u\|_{W_{p}^{m}(t,t+1;\{ B_k\}_{k=0}^{m})}\leq \|\mathcal{ U}'\|_{\mathbb{T'}(t,t+1)}\, .$$ Let W^{m,p}_0(\Pi_{a,b}) be the closure of the space of smooth functions u defined on \Pi_{a,b} and equal zero in a neighborhood of \partial\Pi\cap\overline{\Pi}_{a,b}. By {\bf S}(a,b) we denote the space of all vector functions \mathcal{U}(t) represented in the form $$\label{15.4.3a} \mathcal{U}(t)=\mathop{\rm col}\;\big (u(t),\dots ,D_t^{m-1}u(t), u_{m+1}(t),\dots ,u_{2m} (t)\big )$$ where u\in W^{m,p}_0(\Pi_{a,b}), \begin{gather}\label{15.4.3a1} u_{m+1}\in L_p(a,b;B_0),\;\; D_t u_{m+1}\in L_p(a,b;B_{-m}) \\ \label{15.4.3a2} u_{m+j}, D_t u_{m+j}\in L_p(a,b;B_{-m})\, ,\quad j=2,\dots ,m. \end{gather} We equip the space {\bf S}(a,b) with the norm \begin{align*} \|\mathcal{U}\|_{{\bf S}(a,b)} &=\Big (\|u\|^p_{W^{m,p}(\Pi_{a,b})} +\|u_{m+1}\|^p_{L_p(a,b;B_0)} +\|D_tu_{m+1}\|^p_{L_p(a,b;B_{-m})}\\ &\quad +\sum_{j=2}^m(\|u_{m+j}\|^p_{L_p(a,b;B_{-m})} +\|D_tu_{m+j}\|^p_{L_p(a,b;B_{-m})})\Big)^{1/p} \end{align*} Clearly, {\bf S}(a,b)\subset \mathbb{T}(a,b) and $$\label{Space1} \|\mathcal{U}\|_{\mathbb{T}(a,b)}\leq c\|\mathcal{U}\|_{{\bf S}(a,b)}$$ for all \mathcal{U}\in {\bf S}(a,b). Furthermore, if m=1, then {\bf S}(a,b)= \mathbb{T}(a,b) and the norms are equivalent. We use the notation \mathbb{T}_{{\rm loc} }({\mathbb{R}}) for the space of vector functions defined on \mathbb{R} whose restrictions to an arbitrary finite interval (a,b) belong to \mathbb{T}(a,b). In the same way, the space {\bf S}_{{\rm loc} }({\mathbb{R}}) is defined. \begin{lemma}\label{LOce1} {\rm (i)} If u\in (\Wcirc^{m,p}_{\rm loc} (\overline{\Pi}))^d is a solution of {\rm (\ref{6.1})}, then the vector function \mathcal{U} given by {\rm (\ref{X6.2})}--{\rm (\ref{X6.4})} belongs to {\bf S}_{\rm loc} (\mathbb{R}) and satisfies {\rm (\ref{X6.10})}. {\rm (ii)} If \mathcal{U}\in\mathbb{T}_{\rm loc} (\mathbb{R}) is a solution of {\rm (\ref{X6.10})}, then \mathcal{U}\in {\bf S}_{\rm loc} (\mathbb{R}) and the vector function u=\mathcal{U}_1 belongs to (\Wcirc^{m,p}_{\rm loc} (\overline{\Pi}))^d satisfies {\rm (\ref{6.1})}. \end{lemma} For the proof of the above lemma see \cite{KM2}. Sometimes, to emphasize the dependence of the spaces \mathbb{T} and {\bf S} on p we shall write \mathbb{T}_p and {\bf S}_p respectively. Let us show that for solutions of (\ref{X6.10}) the following local estimate is valid \begin{itemize} \item[(H2b)] Let p and p_1 be the same as in (H2) and (H2a) and let \mathcal{U}\in \mathbb{T}_{p,{\rm loc} }(\mathbb{R}) be a solution of (\ref{X6.10}) with \mathcal{F}_{m+j}\in (L^{p_1}_{\rm loc} ({\mathbb{R}};W^{-j,p_1}(S_+^{n-1})))^d, then \mathcal{U}\in\mathbb{T}_{p_1,{\rm loc} }(\mathbb{R}) and \|\hat{\mathcal{U}}\|_{\hat{ \mathbb{T}}_{p_1}(t,t+a)}\leq cb_0\big (\sum_{j=0}^m\|\mathcal{F}_{m+j}\|_{L_{p_1} (t-a,t+2a;W^{-j,p_1}(S_+^{n-1}))}+\|\hat{\mathcal{U}}\|_{\hat{ \mathbb{T}}_{p}(t-a,t+2a)}\big ). $$\end{itemize} Let \mathcal{U}\in \mathbb{T}_{p,{\rm loc} }(\mathbb{R}) be a solution of (\ref{X6.10}) with \mathcal{F}_{m+j}\in (L_{p_1,{\rm loc}} ({\mathbb{R}};W^{-j,p_1}(S_+^{n-1})))^d. Then by Lemma \ref{LOce1}(ii) the function u=\mathcal{U}_1 is a solution of {\rm (\ref{6.1})}. Clearly, the functions f_j in (\ref{FFF1b}) and (\ref{FFF1c}) belong to (L_{p_1,{\rm loc}} ({\mathbb{R}};W^{-j,p_1}(S_+^{n-1})))^d. By (H2a) u\in (\Wcirc^{m,p_1}_{\rm loc} (\overline{\mathbb{R}_+^n}))^d and (\ref{VAK3}) holds. This together with (\ref{FFF1b}) and (\ref{FFF1c}) gives (H2b). \section{Description of solutions to the homogeneous system \protect\eqref{X6.10}}\label{SSect5} Our goal here is to describe all solutions \mathcal{U}\in \mathbb{T}_{p,{\rm loc}} (\mathbb{R} ) to equation $$\label{X6.10h} (\mathcal{I}D _t+\mathfrak{A})\mathcal{U}(t)-\mathfrak{N}(t) \widehat{\mathcal{U}}(t)=\mathcal{O}\quad \mbox{on {\mathbb{R}},}$$ subject to $$\label{183.1a} \|\mathcal{U}\|_{\mathbb{T}_p(t,t+1)}=\begin{cases} o\big (e^{(n-m)t-c_0\int _0^t\omega (s)ds}\big ) &\mbox{as t\to +\infty }\\ o\big (e^{-(m+1)t-c_0\int _t^0\omega (s)ds}\big ) &\mbox{as t\to -\infty ,} \end{cases}$$ where \omega is given by (\ref{Kruto2a}) and c_0 is a sufficiently large constant. The main theorem is contained in Section \ref{Subs113}. \subsection{Spaces \mathbb{X} and \mathbb{Y}}\label{SubSectSp} Here we add some new function spaces to spaces \mathbb{T}, \mathbb{T}', \hat{\mathbb{T}} and {\bf S}. By \mathbb{X}(a,b) we denote the space of all vector functions $$\label{15.4.3} \mathcal{U}(t)=(\mathcal{I} -\mathcal{P})\mathcal{V}(t)$$ with \mathcal{V}\in {\bf S}(a,b). We define the space \mathbb{X}_{{\rm loc} }({\mathbb{R}}) of all vector functions on {\mathbb{R}} which are represented in the form (\ref{15.4.3}) with a certain \mathcal{V}\in {\bf S} _{{\rm loc} }({\mathbb{R}}). Clearly, \mathbb{X}_{{\rm loc} }({\mathbb{R}})\subset \mathbb{T}_{{\rm loc} }({\mathbb{R}}) and we shall use seminorms \|\cdot \|_{\mathbb{T}(t,t+1)} in \mathbb{X}_{{\rm loc} }({\mathbb{R}}). If m=1 then \mathbb{X}(a,b) is a closed subspace in \mathbb{T}(a,b) consisting of functions {\bf v}\in\mathbb{T}(a,b) satisfying (\mathcal{ I}-\mathcal{P}){\bf v}(t)={\bf v}(t) almost for all t\in (a,b). For the case m\geq 2 we prove the following \begin{lemma}\label{Lem35} Let m\geq 2. Then \noindent{\rm (i)} if (\mathcal{I}-\mathcal{P})\mathcal{U}=0 with \mathcal{U}\in {\bf S}(a,b) then $$\label{15.4.3aa} \mathcal{U}=e^{-mt}\sum_{j=1}^dc_j\Phi_j$$ with some constants c_j; \noindent{\rm (ii)} if {\bf v}\in \mathbb{X}(a,b) then there exists \mathcal{ U}\in {\bf S}(a,b) such that {\bf v} =(\mathcal{I}-\mathcal{P})\mathcal{ U} and $$\label{15.4.3ab} \|\mathcal{U}\|_{\mathbb{T}(a,b)}\leq c\|{\bf v}\|_{\mathbb{T}(a,b)}$$ with constant c depending only on b-a, n, m, p and \mathcal{P}. \end{lemma} \begin{proof} (i) The equality \mathcal{U}=\mathcal{P}\mathcal{U} implies$$ \mathcal{U}(t)=\sum_{k=1}^dh_k(t)\Phi_k. $$Since \mathcal{U}_2(t)=D_t\mathcal{U}_1(t) and \Phi_{k1}=\phi_k, \Phi_{k2}=im\phi_k we have that$$ \sum_{k=1}^dD_th_k(t)\phi_k=im\sum_{k=1}^dh_k(t)\phi_k. $$Using linear independence of the functions \phi_k we obtain that D_th_k(t)=imh_k(t) or h_k(t)=c_ke^{-mt}. (ii) We introduce the factor space \mathbb{T}_0=\mathbb{T}(a,b)/K, where K is the subspace of elements of the form (\ref{15.4.3aa}). The norm is defined by$$ \|\mathcal{U}\|_{\mathbb{T}_0}=\min_{\mathcal{V}\in K}\|\mathcal{U}+\mathcal{ V}\|_{\mathbb{T}_0}. $$Clearly the minimum is attained for a certain \mathcal{V}. Suppose that the assertion (ii) is not valid. Then there exist functions \mathcal{U}_j such that \|\mathcal{U}_j\|_{\mathbb{ T}(a,b)}=\|\mathcal{U}_j\|_{\mathbb{T}_0}=1 and \|(\mathcal{I}-\mathcal{ P})\mathcal{U}_j\|_{\mathbb{T}(a,b)}\to 0 as j\to \infty. We write$$ \mathcal{P}\mathcal{U}_j(t)=\sum_{k=1}^dh_k^{(j)}(t)\Phi_k. $$Using \begin{gather*} \|\mathcal{U}_{j1}-\sum_{k=1}^dh_k^{(j)}(t)\phi_k\|_{L_p(a,b;B_m)}\to 0,\\ \|\mathcal{U}_{j2}-im\sum_{k=1}^dh_k^{(j)}(t)\phi_k\|_{L_p(a,b;B_{m-1})} \to 0, \\ \|D_t\mathcal{U}_{j1}-\sum_{k=1}^dD_th_k^{(j)}(t)\phi_k\|_{L_p(a,b;B_{m-1})} \to 0 \end{gather*} together with D_t\mathcal{U}_{j1}=\mathcal{U}_{j2}, we obtain that$$ \|D_th_k^{(j)}-imh_k^{(j)}\|_{L^p(a,b)}\to 0\quad \mbox{as $j\to\infty$.} $$Putting f_k^{(j)}=D_th_k^{(j)}-imh_k^{(j)}, we obtain$$ h_k^{(j)}(t)=c_k^{(j)}e^{-mt}+F_k^{(j)}(t)\quad \mbox{with}\quad F_k^{(j)}(t)=\int_a^te^{-m(t-\tau)}f_k^{(j)}(\tau)d\tau, $$where c_k^{(j)} are constants. Clearly, F_k^{(j)}\to 0 in W^{1,p}(a,b). If we introduce$$ \mathcal{U}_j'=\mathcal{U}_j-e^{-mt}\sum_{k=1}^dc_k^{(j)}\Phi_k\, , $$then \|\mathcal{U}_j'\|_{\mathbb{T}(a,b)}\geq 1 and \|\mathcal{P}\mathcal{ U}_j'\|_{\mathbb{T}(a,b)}\to 0 as j\to 0. Since (\mathcal{I}-\mathcal{ P})\mathcal{U}_j=(\mathcal{I}-\mathcal{P})\mathcal{U}_j' we have also \|(\mathcal{I}-\mathcal{P})\mathcal{U}_j\|_{\mathbb{T}(a,b)}\to 0. This implies that \|\mathcal{U}_j'\|_{\mathbb{T}(a,b)}\to 0 as j\to 0. This contradiction proves (ii). \end{proof} \begin{corollary}\label{KrK3s} The space \mathbb{X}_{\rm loc} (\mathbb{R}) is closed in \mathbb{T}_{\rm loc} (\mathbb{R}). \end{corollary} \begin{proof} For m=1 this is obvious. Let m\geq 2 and let {\bf v}_j\in\mathbb{X}_{\rm loc} (\mathbb{R}) and {\bf v}_j\to {\bf v} in \mathbb{T}_{\rm loc} (\mathbb{R}). We put \delta_k=(k,k+3/2). Then {\bf v}_j\to {\bf v} in \mathbb{T}_{\rm loc} (\delta_k) for each k\in \mathbb{Z}. By Lemma \ref{Lem35}(ii) there exists \mathcal{ U}_j^{(k)}\in \mathbb{T}_{\rm loc} (\delta_k) such that (\mathcal{I}-\mathcal{ P})\mathcal{U}_j^{(k)}={\bf v}_j on \delta_k and estimate (\ref{15.4.3ab}) holds for the interval \delta_k. Therefore the sequence \{ \mathcal{U}_j^{(k)}\} has the limit \mathcal{U}^{(k)} in \mathbb{T}_{\rm loc} (\delta_k) and (\mathcal{I}-\mathcal{P})\mathcal{ U}^{(k)}={\bf v}. By Lemma \ref{Lem35}(i)$$ \mathcal{U}_{k+1}-\mathcal{U}_k=e^{-mt}\sum_{j=1}^dc_j^{(k)}\Phi_j $$with some constants c_j^{(k)}. This implies that there exists \mathcal{U}\in \mathbb{T}_{\rm loc} (\mathbb{R}) such that (\mathcal{I}-\mathcal{ P})\mathcal{U}={\bf v} on \mathbb{R} and \mathcal{U}-\mathcal{U}_j has the same form as the right-hand side of (\ref{15.4.3aa}) on each \delta_k. Therefore, {\bf v}\in\mathbb{X}_{\rm loc} (\mathbb{R}). \end{proof} We shall also use the space \mathbb{Y}_{{\rm loc} }({\mathbb{R}}) of vector functions $$\label{15.4.6} \mathcal{F}(t)=\mathop{\rm col}\big (0,\dots ,0,\mathcal{F}_{m}(t),\mathcal{ F}_{m+1}(t),\dots ,\mathcal{F}_{2m}\big )$$ with some \mathcal{F}_{m+j}\in L_{p,{\rm loc} }({\mathbb{R}};B_{-j}), j=0,\dots ,m. We equip this space with the seminorms$$ \|\mathcal{F}\|_{\mathbb{Y}(t,t+1)}=\Big (\sum_{j=0}^m\|\mathcal{ F}_{m+j}\|_{L_p(t,t+1;B_{-j})}^p\Big )^{1/p}\, . $$We put \begin{gather}\label{9.7.4a} \widehat{\mathcal{T}}=B_m\times B_{m-1}\times\cdots\times B_1\times B_0 \,,\\ \label{Estim11a} \varkappa_s(t)=\kappa_s(e^{-t}). \end{gather} We shall use also the notation \mathbb{X}_p, \mathbb{Y}_p, B_k^p and \mathcal{T}_p parallel to \mathbb{X}, \mathbb{Y}, B_k and \mathcal{T} in order to indicate their dependence on p. Let us prove the following estimate \begin{lemma}\label{LEm1} Let q\geq p, \delta >0 and let \widehat{\mathcal{U}}\in L_{q}(t,t+\delta;\hat{\mathcal{T}}_q). Then $$\label{Estim11b} \|\mathfrak{N}\widehat{\mathcal{U}}\|_{\mathbb{Y}_{p}(t,t+\delta)}\leq c\varkappa_{s,\delta}(t)\|\widehat{\mathcal{U}}\|_{L_q(t,t+\delta;\hat{\mathcal{T}}_q)},$$ where $$\label{Estim11bg} \varkappa_{s,\delta}(t)=\Big (\int_{K_{e^{-\delta-t},e^{-t}}}\kappa^s(x)|x|^{-n}dx\Big )^{1/s}$$ and s=qp/(q-p). \end{lemma} \begin{proof} Using definitions (\ref{102}) and (\ref{Yacz2a}) of the operators \mathfrak{N}_m and \mathcal{S}', we have $$\label{Estim11c} \|\mathfrak{N}_m\widehat{\mathcal{U}}\|_{L_p(t,t+\delta;B_0^p)}\leq c\Big (\sum_{k=0}^{m-1}\|\mathcal{N}_{0,m-k}\mathcal{ U}_{k+1}\|_{L_p(t,t+\delta;B_0^p)}+\|\mathcal{N}_{00}\mathcal{ S}'\widehat{\mathcal{U}}\|_{L_p(t,t+\delta;B_0^p)}\Big ).$$ By (\ref{2005a}) and H\"older's inequality, we get$$ \|\mathcal{N}_{00}\mathcal{S}'\widehat{\mathcal{U}}\|_{L_p(t,t+\delta;B_0^p)}\leq c\varkappa_{s,\delta}(t)\|\widehat{\mathcal{U}} \|_{L_q(t,t+\delta;\hat{\mathcal{T}}_q)}. $$By (\ref{OperN22}) the sum in the right-hand side in (\ref{Estim11c}) is estimated by$$ c\sum_{k=0}^{m-1}\sum_{|\alpha|=m}\sum_{k\leq |\beta|\leq m}e^{(|\beta|-m)t}\|N_{\alpha\beta}Q_{\beta k}\mathcal{ U}_{k+1}\|_{L_p(t,t+\delta;B_0^p)}. $$Using Hardy's and H\"older's inequalities we estimate this sum by the right-hand side in (\ref{Estim11b}). Thus the norm of \mathfrak{ N}_m\widehat{\mathcal{U}} is estimated. The corresponding norms of \mathfrak{N}_{m+j}\widehat{\mathcal{U}}, j=1,\dots ,m, are estimated analogously. \end{proof} \subsection{Spectral splitting of system (\protect\ref{X6.10})} Let $$\label{9.4} {\bf u}(t)=\mathcal{P}\mathcal{U}(t),\quad {\bf v}(t)=(\mathcal{I}-\mathcal{ P})\mathcal{U}(t).$$ Then $$\label{9.4g} \mathcal{U}(t)={\bf u}(t)+{\bf v}(t)\, .$$ Also, let \hat{{\bf u}}=\mathop{\rm col} ({\bf u}_1,\dots , {\bf u}_{m+1}) and \hat{{\bf v}}=\mathop{\rm col} ({\bf v}_1,\dots , {\bf v}_{m+1}). Applying \mathcal{P} to equation (\ref{X6.10}) and using (\ref{TDHa}) we arrive at $$\label{9.2z} (D_t-im){\bf u}-\mathcal{P}\mathfrak{N}(t)(\hat{{\bf u}}+\hat{{\bf v}}) =\mathcal{P}\mathcal{F}\quad \mbox{on {\mathbb{R}}.}$$ Applying \mathcal{I}-\mathcal{P} to (\ref{X6.10}) we obtain $$\label{9.3z} (\mathcal{I}D_t+\mathfrak{A}){\bf v}-(\mathcal{I}-\mathcal{P})\mathfrak{ N}(t)\hat{{\bf v}} =(\mathcal{I}-\mathcal{P})(\mathcal{F}+\mathfrak{ N}(t)\hat{{\bf u}})\quad \mbox{on {\mathbb{R}}.}$$ Thus we have split system (\ref{X6.10}) into the finite-dimensional system (\ref{9.2z}) and the infinite-dimensional system (\ref{9.3z}). Clearly, \mathcal{U}\in \mathbb{T}_{p,{\rm loc}} (\mathbb{R}) implies that {\bf u} and D_t{\bf u} belong to  L_{p,{\rm loc} }(\mathbb{R};\mathcal{T}_q) for all q\geq p. The next proposition shows the equivalence of (\ref{X6.10}) and the split system (\ref{9.2z}), (\ref{9.3z}). \begin{proposition}\label{T9.1} {\rm (i)} Let \mathcal{U}\in\mathbb{T}_{\rm loc} (\mathbb{R}) be a solution of {\rm (\ref{X6.10})}. Then \mathcal{U}\in {\bf S}_{\rm loc} (\mathbb{R}) and the pair {\bf u}, {\bf v} given by {\rm (\ref{9.4})} satisfy systems {\rm (\ref{9.2z})}, {\rm (\ref{9.3z})}. {\rm (ii)} Let {\bf u} and {\bf v} belong to \mathbb{T}_{\rm loc} (\mathbb{R}), satisfy {\rm (\ref{9.2z})}, {\rm (\ref{9.3z})} and be subject to {\bf u}(t)=\mathcal{P}{\bf u}(t) and {\bf v}(t)=(\mathcal{I}-\mathcal{P}){\bf v}(t) on \mathbb{R}. Then the function {\rm (\ref{9.4g})} satisfies system {\rm (\ref{X6.10})}. \end{proposition} The proof of the above proposition is obvious. This proposition, combined with Lemma \ref{LOce1}, ensures the equivalence of equation (\ref{6.1}) and the split system (\ref{9.2z}), (\ref{9.3z}). \subsection{The infinite-dimensional part of the split system}\label{Sect88f} We start with the case \mathfrak{N}=0, i.e. we consider the system $$\label{8.4} (\mathcal{I}D _t+\mathfrak{A}){\bf v}=(\mathcal{I}-\mathcal{P})F\quad \mbox{on {\mathbb{R}}.}$$ We put $$\label{Mu19} \mu (t )=\begin{cases} e^{-(m+1)t} & \mbox{for t\geq 0 }\\ e^{(n-m)t} & \mbox{for t<0.} \end{cases}$$ The following result is proved in [KM2, Lemma 8]. \begin{lemma} \label{T8.1a} {\rm (i) (Existence)} Let F\in\mathbb{Y}_{q,{\rm loc}} (\mathbb{R}), q\in (1,\infty ) and \delta >0. Suppose that $$\label{8.1aa} \int _{\mathbb{R}} \mu (-\tau )\|F\|_{\mathbb{Y}_q(\tau ,\tau +\delta)}d\tau <\infty\; .$$ Then {\rm (\ref{8.4})} has a solution {\bf v}\in \mathbb{ X}_{q,{\rm loc} }({\mathbb{R}}) satisfying $$\label{8.5} \|{\bf v}\|_{\mathbb{T}_q(t,t+\delta)}\leq c\, \int _{\mathbb{R}} \mu (t-\tau )\|F\|_{\mathbb{Y}_q(\tau ,\tau +\delta)}d\tau ,$$ where c is a constant independent of F. {\rm (ii)} {\rm (} \emph{Uniqueness}{\rm )} Let {\bf v}\in \mathbb{ T}_{q,{\rm loc} }({\mathbb{R}}) satisfy {\rm (\ref{8.4})} with F=0 and \mathcal{P}{\bf v}(t)=0 for almost every t\in\mathbb{R}. Also let $$\label{8.6a} \|{\bf v}\|_{\mathbb{T}_q(t,t+\delta)} =\begin{cases} o(e^{(n-m)t} ) & \mbox{if t\to +\infty }\\ o(e^{-(m+1)t}) & \mbox{if t\to -\infty } \end{cases}$$ be valid. Then {\bf v}=0. \end{lemma} Now, we study the system $$\label{QQQ4} (\mathcal{I}D _t +\mathfrak{A}){\bf v}-(\mathcal{I}-\mathcal{P})\mathfrak{ N}(t)\hat{\bf v} =(\mathcal{I}- \mathcal{P})F\quad \mbox{on {\mathbb{R}}.}$$ We introduce the function $$\label{Mu1} \mu_\omega (t,\tau )=\begin{cases} \exp\big (-(m+1)(t-\tau )+c_0\int_\tau^t\omega (s)ds\big ) & \mbox{for t\geq \tau }\\ \exp\big ((n-m)(t-\tau )+c_0\int_t^\tau\omega (s)ds\big ) & \mbox{for t<\tau ,} \end{cases}$$ where c_0 is a sufficiently large positive constant depending on n, m, p, \gamma and L. \begin{proposition}\label{T9.1a} Let assumptions (H1)--(H3) be fulfilled and let p and p_1 be the same as in (H2). Then the following assertions are valid: {\rm (i)} Let F belong to \mathbb{Y}_{p_1,{\rm loc}} ({\mathbb{R}}) and let $$\label{QQQ3} \int _{\mathbb{R}}\mu_\omega (0,\tau )\|F\|_{\mathbb{Y}_{p_1}(\tau ,\tau +1)}d\tau <\infty\, .$$ Then system {\rm (\ref{QQQ4})} has a solution {\bf v}\in \mathbb{ X}_{p_1,{\rm loc}} ({\mathbb{R}}) satisfying \begin{gather}\label{QQQ5} \|{\bf v}\|_{\mathbb{T}_{p}(t,t+1)}\leq c\,\int _{\mathbb{R}} \mu_\omega (t,\tau )\|F\|_{\mathbb{Y}_{p_1}(\tau ,\tau +1)}d\tau\,, \\ \label{QQQ5d} \|\hat{{\bf v}}\|_{\hat{\mathbb{T}}_{p_1}(t,t+1)}\leq cb_0\,\int _{\mathbb{R}} \mu_\omega (t,\tau )\|F\|_{\mathbb{Y}_{p_1}(\tau ,\tau +1)}d\tau\, . \end{gather} {\rm (ii)} The solution {\bf v}\in \mathbb{X}_{p,{\rm loc} }({\mathbb{R}}) to {\rm (\ref{QQQ4})} subject to $$\label{QQQ2} \|{\bf v}\|_{\mathbb{T}_p(t,t+1)}=\begin{cases} o\Big (e^{(n-m)t-c_0\int _0^t\omega (\tau )d\tau }\Big ) & \mbox{as t\to +\infty }\\ o\Big (e^{-(m+1)t-c_0\int _t^0\omega (\tau )d\tau }\Big )& \mbox{as t\to -\infty } \end{cases}$$ is unique. (We note that {\rm (\ref{QQQ3})} together with {\rm (\ref{QQQ5})} imply {\rm (\ref{QQQ2})}.) \end{proposition} \begin{proof} (1). \emph{Solvability in \mathbb{X}_{2}(\mathbb{R}).} We introduce the space \mathbb{T}_2(\mathbb{R}), which consists of vector functions \mathcal{U}\in\mathbb{T}_{2,{\rm loc}}(\mathbb{R}) with finite norm$$ \|\mathcal{U}\|_{\mathbb{T}_2(\mathbb{R})}=\Big (\int_{\mathbb{R}}e^{(2m-n)t}\|\mathcal{U}\|_{\mathbb{ T}_2(t,t+1))}^2dt\Big )^{1/2}. $$The space {\bf S}_2(\mathbb{R}) contains vector functions (\ref{15.4.3a}) with finite norm$$ \|\mathcal{U}\|_{{\bf S}_2(\mathbb{R})}=\Big (\int_{\mathbb{R}}e^{(2m-n)t}\|\mathcal{U}\|_{{\bf S}_2(t,t+1))}^2dt\Big )^{1/2}. $$The space \mathbb{X}_{2}(\mathbb{R}) consists of {\bf v} represented as (\mathcal{I}-\mathcal{P})\mathcal{U} with \mathcal{U}\in {\bf S}_2(\mathbb{R}). Let also \mathbb{Y}_2(\mathbb{R}) consists of vector functions \mathcal{F} from \mathbb{Y}_{2,{\rm loc}}(\mathbb{R}) with finite norm$$ \|\mathcal{F}\|_{\mathbb{Y}_2(\mathbb{R})}=\Big (\int_{\mathbb{R}}e^{(2m-n)t}\|\mathcal{F}\|_{\mathbb{ Y}_2(t,t+1))}^2dt\Big )^{1/2}. $$Consider first problem (\ref{X6.10}) with \mathcal{F}\in \mathbb{Y}_2(\mathbb{R}). Using the solvability result for (\ref{VAK2}) from (1) in the proof of Proposition \ref{T4.2zz} and connection of problems (\ref{VAK2}) and (\ref{X6.10}) established in Section \ref{Section2} we obtain that for every \mathcal{F}\in \mathbb{Y}_2(\mathbb{R}) there exists the unique \mathcal{U}\in \mathbb{T}_{2}(\mathbb{R}) solving (\ref{X6.10}) and it satisfies$$ \|\mathcal{U}\|_{\mathbb{T}_{2}(\mathbb{R})}\leq c \|\mathcal{F}\|_{\mathbb{ Y}_2(\mathbb{R})}. $$Therefore the function {\bf v}=(\mathcal{I}-\mathcal{P})\mathcal{U} belongs to \mathbb{X}_{2}(\mathbb{R}) solves (\ref{QQQ4}) with F=\mathcal{F}. (2). \emph{Local estimate for {\bf v}}. Let {\bf v}=(\mathcal{ I}-\mathcal{P})\mathcal{U} with \mathcal{U}\in {\bf S}_{p,{\rm loc}} (\mathbb{R}) satisfy (\ref{QQQ4}) with F\in \mathbb{Y}_{p_1,{\rm loc}}(\mathbb{R}) and let m\geq 2. According to Lemma \ref{Lem35}(ii) we can suppose that \mathcal{U} is subject to (\ref{15.4.3ab}) with a=t-\delta and b=t+2\delta, where t and \delta are fixed. We write equation (\ref{QQQ4}) as $$\label{QQQ234} (\mathcal{I}D _t +\mathfrak{A})\mathcal{U}-\mathfrak{N}(t)\hat{\bf v} =F+\mathcal{ P}G,$$ where$$ G=(\mathcal{I}D_t +\mathfrak{A})\mathcal{U}-\mathfrak{N}(t)\hat{\bf v} -F. $$System (\ref{QQQ234}) consists of 2m equations. Since \mathcal{ U}\in {\bf S}_{p,{\rm loc}} (\mathbb{R}) and the first components of \mathfrak{N}(t)\hat{\bf v} and F are zero, we have (\mathcal{ P}G)_1=0. But$$ (\mathcal{P}G)(t)=\sum_{j=1}^dh_j(t)\Phi_j $$and the vector functions (\Phi_j)_1=\phi_j are linear independent, which implies h_j=0 and hence \mathcal{P}G=0. Thus system (\ref{QQQ234}) becomes $$\label{QQQ234a} (\mathcal{I}D_t +\mathfrak{A})\mathcal{U}-\mathfrak{N}(t)\hat{\mathcal{U}} =F-\mathfrak{N}(t)\widehat{\mathcal{P}\mathcal{U}}.$$ Since \mathcal{U}\in {\bf S}_{p,{\rm loc}} (\mathbb{R})\subset \mathbb{ T}_{p,{\rm loc} }(\mathbb{R}) it follows that \mathcal{P}\mathcal{U} and \partial_t\mathcal{P}\mathcal{U} belong to L_{p,{\rm loc}}(\mathbb{R};\mathcal{T}_q) for all q\geq p and$$ \|\mathcal{P}\mathcal{U}\|_{L_p(t,t+\delta;\mathcal{T}_q)}+\|\partial_\tau \mathcal{P}\mathcal{U}\|_{L_p(t,t+\delta;\mathcal{T}_q)}\leq c\|\mathcal{ U}\|_{\mathbb{T}_p(t,t+\delta)}. $$This implies $$\label{QQQ234am} \|\mathcal{P}\mathcal{U}\|_{L_q(t,t+\delta;\mathcal{T}_q)}\leq c\|\mathcal{ U}\|_{\mathbb{T}_p(t,t+\delta)}.$$ Now applying (H2b) to (\ref{QQQ234a}) and using the last inequality together with Lemma \ref{LEm1}, we obtain that \mathcal{ U}\in \mathbb{T}_{p_1,{\rm loc} }(\mathbb{R}) and$$ \|\hat{\mathcal{U}}\|_{\hat{ \mathbb{T}}_{p_1}(t,t+\delta)}\leq cb_0\big (\sum_{j=0}^m\|F_{m+j}\|_{L_{p_1} (t-\delta,t+2\delta;W^{-j,p_1}(S_+^{n-1}))} +\|\hat{\mathcal{U}}\|_{\hat{ \mathbb{T}}_{p}(t-\delta,t+2\delta)}\big ). Now using (\ref{15.4.3ab}) we arrive at $$\label{QQQ534a} \|\hat{\bf v}\|_{\hat{ \mathbb{T}}_{p_1}(t,t+\delta)}\leq cb_0\big (\|F\|_{\mathbb{Y}_{p_1} (t-\delta,t+2\delta)}+\|{\bf v}\|_{ \mathbb{T}_{p}(t-\delta,t+2\delta)}\big ).$$ When m=1, a direct application of (H2b) to the system (\mathcal{I}D_t +\mathfrak{A}){\bf v}-\mathfrak{N}(t)\hat{\bf v} =F-\mathcal{ P}\mathfrak{N}(t)\hat{\bf v} gives \begin{align*} \|\hat{\bf v}\|_{\hat{ \mathbb{T}}_{p_1}(t,t+\delta)} &\leq cb_0\big(\sum_{j=0}^1\|(F-\mathcal{P}\mathfrak{N}(t)\hat{\bf v})_{1+j}\|_{L_{p_1} (t-\delta,t+2\delta;W^{-j,p_1}(S_+^{n-1}))}\\ &\quad +\|{\bf v}\|_{ \mathbb{T}_{p}(t-\delta,t+2\delta)}\big ). \end{align*} This together with (\ref{QQQ234am}) implies (\ref{QQQ534a}) for m=1. The local estimate (\ref{QQQ534a}) together with (\ref{Estim11b}), with q=p_1, gives $$\label{QQQ534av} \|\mathfrak{N}\hat{\bf v}\|_{\mathbb{Y}_{p}(t,t+\delta)}\leq c\big (\omega_0\| F\|_{\mathbb{Y}_{p_1} (t-\delta,t+2\delta)}+b_0\varkappa_{s,\delta}(t)\|{\bf v}\|_{\mathbb{ T}_{p}(t-\delta,t+2\delta)}\big ),$$ where s=p_1p/(p_1-p) and \varkappa_{s,\delta} is given by (\ref{Estim11bg}). (3). \emph{Existence of solution.} One can verify that every vector function F from \mathbb{Y}_{p_1,{\rm loc}} ({\mathbb{R}}) subject to (\ref{QQQ3}) can be approximated by vector functions from \mathbb{Y}_2(\mathbb{R}) with compact support in the norm defined by the left-hand side in (\ref{QQQ3}). Therefore it suffices to prove estimate (\ref{QQQ5}) and (\ref{QQQ5d}) for solutions from (1). We write system (\ref{QQQ4}) in the form (\mathcal{I}D _t +\mathfrak{A}){\bf v}=(\mathcal{I}- \mathcal{P})(F+\mathfrak{ N}(t)\hat{\bf v}). $$Using {\bf v}\in\mathbb{X}_{2}(\mathbb{R}) one can check that the right-hand side satisfies (\ref{8.1aa}). Applying to this equation Lemma \ref{T8.1a} with q=p and using (\ref{QQQ534av}), we arrive at$$ \|{\bf v}\|_{\mathbb{T}_p(t,t+\delta)}\leq c\, \int _{\mathbb{R}} \mu (t-\tau )\big (\| F\|_{\mathbb{Y}_{p_1} (t-\delta,t+2\delta)}+b_0\varkappa_{s,\delta}(t)\|{\bf v}\|_{\mathbb{ T}_{p}(t-\delta,t+2\delta)}\big )d\tau . $$Taking here \delta sufficiently small we derive the estimate$$ \|{\bf v}\|_{\mathbb{T}_p(t,t+1)}\leq c\, \int _{\mathbb{R}} \mu (t-\tau )\big (\| F\|_{\mathbb{Y}_{p_1} (t,t+1)}+b_0\varkappa_s(t)\|{\bf v}\|_{\mathbb{T}_{p}(t,t+1)}\big )d\tau . Iterating this inequality we obtain $$\label{8.5kk} \|{\bf v}\|_{\mathbb{T}_p(t,t+1)}\leq c\, \int _{\mathbb{R}} g_\omega (t,\tau )\| F\|_{\mathbb{Y}_{p_1} (t,t+1)}d\tau ,$$ where \begin{align*} & g_\omega (t,\tau )\\ & =\mu (t-\tau )+\sum_{k=1}^\infty (cb_0)^k\int_{\mathbb{R}^k}\mu(t-\tau_1)\varkappa_s(\tau_1) \mu(\tau_1-\tau_2)\dots\varkappa_s(\tau_k)\mu(\tau_k-\tau)d\tau_1\dots d\tau_k. \end{align*} Since (m-n-\partial_t)(\partial_t+m+1)\mu(t)=(n+1)\delta(t), we can check that \big ((m-n-\partial_t)(\partial_t+m+1)-(n+1)cb_0\kappa_s(e^{-t})\big )g_\omega (t,\tau)=(n+1) \delta (t-\tau). $$Using \cite[Proposition 6.3.1]{KM1}, we obtain$$ g_\omega(t,\tau)\leq C\mu_\omega (t,\tau). $$This together with (\ref{8.5kk}) leads to (\ref{QQQ5}). Estimate (\ref{QQQ5}) together with the local estimate (\ref{QQQ534a}) gives (\ref{QQQ5d}). (4) \emph{Uniqueness}. First we observe that we can start in (3) from a solution {\bf v}\in \mathbb{X}_{p,{\rm loc}}(\mathbb{R}) subject to a certain growth restrictions at \pm\infty, for example {\bf v} has a compact support with respect to t, and reasoning as above we will arrive at estimates (\ref{QQQ5}) and (\ref{QQQ5d}) for such {\bf v}. This leads to uniqueness in the class of functions with compact support. The uniqueness in the class of functions subject to (\ref{QQQ2}) is proved in the same way as in Proposition \ref{T4.2zz}. \end{proof} \subsection{The finite dimensional system}\label{Sfds} By Proposition \ref{T9.1a} we can introduce the operator \mathfrak{M}:F\to \hat{{\bf v}} which is defined on F\in\mathbb{Y}_{p_1,{\rm loc}} (\mathbb{R}) subject to (\ref{QQQ3}) and \mathfrak{ M}(F)=\hat{{\bf v}} where {\bf v} is the solution of (\ref{QQQ4}) from Proposition \ref{T9.1a}. By (\ref{QQQ5}) we have $$\label{QQQ5h} \|\mathfrak{M}(F)\|_{\hat{\mathbb{T}}_{p_1}(t,t+1)}\leq cb_0\,\int _{\mathbb{R}} \mu_\omega (t,\tau )\|F\|_{\mathbb{Y}_{p_1}(\tau ,\tau +1)}d\tau \; .$$ Using the operator \mathfrak{M}, we write (\ref{9.2z}) in the form$$ (D_t-im){\bf u}-\mathcal{P}\mathfrak{N}(\hat{{\bf u}}+\mathfrak{M}(\mathfrak{ N}\hat{{\bf u}}))(t) =\mathcal{P} (\mathcal{F}+\mathfrak{N}\mathfrak{M}(\mathcal{ F}))(t)\quad \mbox{on ${\mathbb{R}}$}. We rewrite this system as $$\label{9.2zab} (D_t-im){\bf u}-\mathcal{P}\mathfrak{N}^{(0)}\hat{{\bf u}}-\mathcal{P}\mathcal{ K}(\hat{{\bf u}}) =\mathcal{P}(\mathcal{F}+\mathfrak{N}\mathfrak{M}(\mathcal{ F}))(t)\quad \mbox{on {\mathbb{R}}},$$ where \begin{gather}\label{KKh1} \mathcal{K}[\hat{{\bf u}}](t)=(\mathfrak{N}(t)-\mathfrak{ N}^{(0)}(t))\hat{{\bf u}}+\mathfrak{N}(t)\mathfrak{M}(\mathfrak{N}\hat{{\bf u}})\,,\\ \label{101v} \mathfrak{N}^{(0)}(t)\hat{\mathcal{U}}=\mathop{\rm col} (0,\dots ,0, \mathfrak{ N}^{(0)}_m(t)\hat{\mathcal{U}},\mathfrak{N}^{(0)}_{m+1}(t)\hat{\mathcal{ U}}, \dots ,\mathfrak{N}^{(0)}_{2m}(t)\hat{\mathcal{U}}) \end{gather} with $$\label{102s} \mathfrak{N}^{(0)}_m(t)\hat{\mathcal{U}}=A_{00}^{-1} \Big (\sum _{k=0}^{m-1}\mathcal{N}_{0,m-k}(t)\mathcal{U}_{k+1}+ \mathcal{ N}_{00}(t)\mathcal{S}_0(t)\hat{\mathcal{U}}\Big )$$ and \label{103s} \begin{aligned} \mathfrak{N}^{(0)}_{m+j}(t)\hat{\mathcal{U}} &=\sum _{k=0}^{m-1}\mathcal{ N}_{j,m-k}(t)\mathcal{U}_{k+1} +\mathcal{N}_{j0}(t)\mathcal{S}_0(t)\hat{\mathcal{U}} \\ &\quad -A_{j0}A_{00}^{-1}\Big (\sum _{k=0}^{m-1}\mathcal{ N}_{0,m-k}(t)\mathcal{U}_{k+1} +\mathcal{N}_{00}(t) \mathcal{S}_0(t)\hat{\mathcal{U}}\Big ) \end{aligned} for j=1,\dots ,m. Here \mathcal{S}_0 is given by (\ref{Yacz1a}). The vector function {\bf u} can be represented as $$\label{PLMb1a} {\bf u}(t)=e^{-mt}\sum_{j=1}^dh_j(t)\Phi_j\, ,$$ where \Phi_j is given by (\ref{7.2}) and (\ref{7.3}). Inserting (\ref{PLMb1a}) into (\ref{9.2zab}), multiplying then (\ref{PLMb1}) by vectors (\ref{7.4}) and using (\ref{7.5a}) and (\ref{7.6}), we obtain the system for the vector function {\bf h}=\mathop{\rm col} (h_1,\dots , h_d) $$\label{PLMb1b} \partial_t{\bf h}(t)-\mathcal{R}(t){\bf h}(t)-(\mathcal{M}{\bf h})(t)={\bf g}\, ,$$ where (\mathcal{R}(t){\bf h}(t))_k=\sum_{j=1}^N\mathcal{R}_{kj}(t)h_j(t) $$with$$ \mathcal{R}_{kj}(t)=\langle \mathfrak{N}^{(0)}(t)\Phi_j,\Psi_k\rangle \quad\mbox{and}\quad (\mathcal{M}{\bf h})(t)=((\mathcal{M}{\bf h})_1(t),\dots ,\mathcal{ M}{\bf h})_d(t)), $$where $$\label{JLK1} (\mathcal{M}{\bf h})_k(t)=e^{mt}\langle\mathcal{K}(e^{-m\tau }\sum_{j=1}^dh_j\hat{\Phi}_j)(t),\Psi _k\rangle\, .$$ The right-hand side {\bf g}(t)=(g_1(t),\dots ,g_d(t)) is defined by $$\label{JLK19} g_k(t)=e^{mt}\langle \mathcal{F}(t)+\mathfrak{N}\mathfrak{M}(\mathcal{ F})(t),\Psi _k\rangle \, .$$ Using (\ref{7.2}), (\ref{7.3}) and (\ref{Yacz1a}) we obtain that \mathcal{S}_0\Phi_j=(im)^m\varphi_j. Therefore, \begin{gather*} \mathfrak{N}^{(0)}_m(t)\Phi_j=A_{00}^{-1}\mathcal{N}_0(t,im)\varphi_j\,, \\ \mathfrak{N}^{(0)}_{m+q}(t)\Phi_j=\mathcal{ N}_q(t,im)\varphi_j-A_{q0}A_{00}^{-1}\mathcal{N}_0(t,im)\varphi_j \end{gather*} for q=1,\dots ,m, where we have used notation (\ref{OperN2}) and formulae (\ref{102s}), (\ref{103s}). This together with (\ref{7.4d}) and (\ref{7.4e}) gives $$\label{PLMb1c} \mathcal{R}_{kj}(t)=\sum_{q=0}^m\int_{S_+^{n-1}}(\mathcal{ N}_q(t,im)\varphi_j,(-im)^{m-q}\psi_k)d\theta\, .$$ Furthermore, {\bf u}\in L_{{q,\rm loc} }(\mathbb{R};\mathcal{T}_q) if and only if {\bf h}\in (L^q_{{\rm loc}}(\mathbb{R}))^d and $$\label{HHp} C_1 \|{\bf h}\|_{L^q(t,t+1)}\leq e^{mt}\|{\bf u}\|_{L_q(t ,t +1;\mathcal{T}_q)}\leq C_2 \|{\bf h}\|_{L^q(t,t+1)}$$ with constants independent of t. To derive estimates for \mathcal{M} we need some formulae. Using definition (\ref{Yacz2a}) of the operator \mathcal{S}' together with (\ref{7.2}) and (\ref{7.3}) we obtain $$\label{for1} \mathcal{S}'(t)\hat{\Phi}_j=(im)^m\varphi_j+(A_{00}-\mathcal{ N}_{00})^{-1}\mathcal{N}_0(t,im)\varphi_j\, .$$ Therefore,$$ (\mathfrak{N}_m(t)-\mathfrak{N}_m^{(0)}(t))\hat{\Phi}_j=A_{00}^{-1}\mathcal{ N}_{00}(t)(A_{00}-\mathcal{N}_{00})^{-1}\mathcal{N}_0(t,im)\varphi_j and \begin{align*} (\mathfrak{N}_{m+q}(t)-\mathfrak{N}_{m+q}^{(0)}(t))\hat{\Phi}_j &=\mathcal{N}_{q0}(t)(A_{00}-\mathcal{N}_{00})^{-1} \mathcal{N}_0(t,im)\varphi_j\\ &\quad -A_{q0}A_{00}^{-1}\mathcal{N}_{00}(A_{00}-\mathcal{N}_{00})^{-1}\mathcal{ N}_0(t,im)\varphi_j \end{align*} for q=1,\dots ,m. These relations together with (\ref{7.4d}) and (\ref{7.4e}) give \label{POpa3} \begin{aligned} &\langle (\mathfrak{N}(t)-\mathfrak{ N}^{(0)}(t))\hat{\Phi}_j,\Psi_k\rangle \\ &=\int_{S_+^{n-1}}((A_{00}-\mathcal{N}_{00})^{-1}\mathcal{ N}_0(t,im)\varphi_j,\sum_{q=0}^m\mathcal{ N}_{q0}^*(t)(-im)^{m-q}\psi_k)d\theta\, . \end{aligned} We also need the formulae \begin{gather}\label{for3} \mathfrak{N}_m\Phi_j=(A_{00}-\mathcal{N}_{00})^{-1}\mathcal{ N}_0(t,im)\varphi_j\,,\\ \label{for4} \mathfrak{N}_{m+q}\Phi_j=\mathcal{N}_{q}(t,im)\varphi_j-(A_{q0}-\mathcal{ N}_{q0})(A_{00}-\mathcal{N}_{00})^{-1}\mathcal{N}_0(t,im)\varphi_j \end{gather} for q=1,\dots ,m, which can be checked directly by using the definitions of the operator \mathfrak{N}, the vector function \Phi_j and (\ref{for1}). Using again definitions of the operator \mathfrak{N} and the vector functions \Psi_k one verifies that \label{Kap1} \begin{aligned} \langle \mathfrak{N}(t)\hat{\mathcal{U}}, \Psi_k\rangle &=\sum_{s=0}^m\sum_{q=0}^{m-1}\int_{S_+^{n-1}}(\mathcal{ N}_{s,m-q}\mathcal{ U}_{q+1},(-im)^{m-s}\psi_k)d\theta \\ &\quad+\sum_{s=0}^m\int_{S_+^{n-1}}(\mathcal{N}_{s0}\mathcal{S}'\hat{\mathcal{ U}},(-im)^{m-s}\psi_k)d\theta . \end{aligned} Now we estimate the norm of the operator \mathcal{M}. We use the notation $$\label{Kappa19} \tilde{\mu}_\omega (t,\tau )=\mu_\omega (t,\tau )e^{m(t-\tau)}\, .$$ Using (\ref{QQQ5h}) and the definition of \mathcal{M} one can derive the following estimate $$\label{Kappa193} \|\mathcal{M}({\bf h})\|_{L^{p_1}(t,t+1)}\leq c\int_{\mathbb{R}}\tilde{\mu}_\omega (t,\tau )\|{\bf h}\|_{L^{p_1}(\tau ,\tau +1)}d\tau\, ,$$ which is valid for {\bf h} subject to \int_{\mathbb{R}}\tilde{\mu}_\omega (0,\tau )\|{\bf h}\|_{L^{p_1}(\tau ,\tau +1)}d\tau <\infty\, . $$In what follows we shall need also another estimate for \mathcal{M}. \begin{lemma}\label{Lem4r} For all {\bf h}\in(L^\infty _{\rm loc} (\mathbb{R}))^d subject to $$\label{KKhf} \int_{\mathbb{R}}\tilde{\mu}_\omega (0,\tau )\varkappa_{p_1}(\tau )\|{\bf h}\|_{L^\infty(\tau ,\tau +1)}d\tau <\infty$$ the following estimate holds: $$\label{KKh} \|\mathcal{M}({\bf h})\|_{L^1(t,t+1)}\leq cb_0\varkappa _{p_1'}(t)\int_{\mathbb{R}}\tilde{\mu}_\omega (t,\tau )\varkappa_{p_1}(\tau )\|{\bf h}\|_{L^\infty(\tau ,\tau +1)}d\tau\, ,$$ where p_1'=p_1/(p_1-1) and \varkappa_s is defined by {\rm (\ref{Estim11a})} and {\rm (\ref{TTrr2a})}. \end{lemma} \begin{proof} We start with proving the estimates \begin{gather}\label{Kap1a} \|\mathfrak{N}_{m+k}(t)\hat{\Phi}_j\|_{W^{-k,s}(S_+^{n-1})}\leq c\overline{\varkappa }_s(t)\,, \\ \label{Kap1b} \Big |\langle \mathfrak{N}(t )\hat{\mathcal{U}},\Psi_k\rangle\Big |\leq c\overline{\varkappa }_{s'}(t )\sum_{j=1}^{m+1}\|\mathcal{U}_j(t )\|_{W^{m-j+1^,s}(S_+^{n-1})}\, , \end{gather} where s\in (1,\infty ), 1/s'=1-1/s, $$\label{SSS1} \overline{\varkappa }_s(\tau )=\sum_{|\alpha|,\beta|\leq m}\Big (\int_{S_+^{n-1}}((e^{-t}\theta_n)^{2m-|\alpha|-|\beta|}|N_{\alpha\beta}(e^{-\tau}\theta )|)^sd\theta\Big )^{1/s}$$ and the constant c depends on n, s and coefficients L_{\alpha\beta}. By (\ref{for3}) and (\ref{for4})$$ \|\mathfrak{N}_{m+k}(t)\hat{\Phi}_j\|_{W^{-k,s}(S_+^{n-1})}\leq c\big (\|\mathcal{ N}_0(t ,im)\varphi _j\|_{L^s(S_+^{n-1})}+\|\mathcal{N}_k(t ,im)\varphi _j\|_{W^{-k,s}(S_+^{n-1})}\big )\, . Using (\ref{OperN2}), (\ref{OperN22}) and that \varphi_j(\theta )=\theta_n^me_j, we can estimate the right-hand side by \begin{align*} &c\sum_{|\alpha|\leq m}\sum_{|\beta|\leq m}e^{-(2m-|\alpha|-|\beta|)t}\|N_{\alpha\beta}(e^{-t}\theta) \theta_n^{m-|\beta|}\|_{W^{|\alpha|-m,s}(S_+^{n-1}))}\\ &\leq c\sum_{|\alpha|\leq m}\sum_{|\beta|\leq m}e^{-(2m-|\alpha|-|\beta|)t}\|N_{\alpha\beta}(e^{-t}\theta) \theta_n^{2m-|\beta|-|\alpha|}\|_{L^s(S_+^{n-1})}\,, \end{align*} which is estimated by c\overline{\varkappa}_s. Inequality (\ref{Kap1a}) is proved. Using (\ref{Kap1}) and (\ref{Yacz1b}) together with (\ref{OperN22}) and observing that \psi_k is equal to \phi_k times a smooth function, we arrive at \begin{align*} \Big |\langle \mathfrak{N}(t )\hat{\mathcal{U}},\Psi_k\rangle\Big |&\leq c\sum _{q=0}^m\sum_{|\alpha|\leq m}\sum_{q\leq|\beta|\leq m}e^{-(2m-|\alpha|-|\beta|)t}\\ &\quad\times \|N_{\alpha\beta}^*\theta_n^{2m -|\alpha|-|\beta|}\|_{L^{s'}(S_+^{n-1})}\|\theta_n^{|\beta |-m}Q_{\beta,q}\mathcal{U}_{q+1}\|_{L^s(S_+^{n-1})}\, . \end{align*} Using the Hardy inequality for estimating the last factor we arrive at (\ref{Kap1b}). We represent \mathcal{M} as the sum \mathcal{M}_1+\mathcal{M}_2, where \begin{gather*} \mathcal{M}_1({\bf h})=\sum_{j=1}^d(\langle\mathfrak{N}(t)-\mathfrak{ N}^{(0)}(t))\hat{\Phi}_j,\Psi_k\rangle h_j(t) \,,\\ \mathcal{M}_2({\bf h})_k=\langle \mathfrak{N}(t)\mathfrak{M}(\mathfrak{ N}\sum_{j=1}^dh_j(t)\hat{\Phi}_j),\Psi_k\rangle \, . \end{gather*} From (\ref{POpa3}) and (\ref{OperN2}), (\ref{OperN22}) it follows that \begin{align*} |\mathcal{M}_1({\bf h})| &\leq c\sum_{|\alpha |=m}\sum_{m-k\leq |\beta|}e^{-(2m-|\alpha|-|\beta|)t}\|N_{\alpha\beta}\theta_n^{2m-|\beta|-k}\|_{L^{p_1'}(S_+^{n-1})}\\ &\quad \times\sum_{|\beta |=m}\sum_{m-j\leq |\alpha|}e^{-(2m-|\alpha|-|\beta|)t}\|N^*_{\alpha\beta}\theta_n^{2m-|\alpha|-j}\|_{L^{p_1}(S_+^{n-1})}|{\bf h}(t)|\, , \end{align*} which implies $$\label{1009} |\mathcal{M}_1({\bf h})(t)|\leq c\overline{\varkappa}_{p_1'}(t)\overline{\varkappa}_{p_1}(t)|{\bf h}(t)|\, .$$ Furthermore, by (\ref{Kap1b}) |\mathcal{M}_2({\bf h})(t)|\leq C\overline{\varkappa }_{p_1'}(t )\sum_{j=1}^{m+1}\sum_{k=1}^d\|\mathfrak{M}(\mathfrak{ N}h_k(t)\hat{\Phi}_k)_j\|_{W^{m-j+1^,p_1}(S_+^{n-1})}\, . $$Now, using (\ref{QQQ5h}) together with (\ref{Kap1a}) with s=p_1 and (\ref{HHp}), we obtain$$ \|\mathcal{M}_2({\bf h})\|_{L^1(t,t+1)}\leq Cb_0\|\overline{\varkappa} _{p_1'}\|_{L^{p_1'}(t,t+1)}\int_{\mathbb{R}}\tilde{\mu}_\omega (t,\tau )\varkappa_{p_1}(\tau )\|{\bf h}\|_{L^\infty (\tau ,\tau +1)}d\tau\, . $$From (\ref{1009}) it follows the same estimate for \|\mathcal{M}_1({\bf h})\|_{L^1(t,t+1)}. These two estimates give (\ref{KKh}). The proof is complete. \end{proof} \subsection{Homogeneous equation \protect\eqref{9.2zab}} Here we shall study the homogeneous equation (\ref{PLMb1b}), i.e. $$\label{PLMb1} \partial_t{\bf h}(t)-\mathcal{R}(t){\bf h}(t)-(\mathcal{M}{\bf h})(t)=0\quad \mbox{on {\mathbb{R}}}.$$ We start with a uniqueness result. \begin{lemma}\label{LS9w} If {\bf h}\in (W^{1,1}_{\rm loc} (\mathbb{R}))^d is a solution of {\rm (\ref{PLMb1})} subject to $$\label{18.3.1s} |{\bf h}(t)|=\begin{cases} o\Big (e^{nt-c_1\int _0^t\omega (s)ds}\Big ) &\mbox{as t\to +\infty }\\ o\Big (e^{-t-c_1\int _t^0\omega (s)ds}\Big ) &\mbox{as t\to -\infty ,} \end{cases}$$ with sufficient large c_1 and {\bf h}(t_0)=0 for some t_0 then {\bf h}(t)=0 for all t\in\mathbb{R}. \end{lemma} \begin{proof} Without loss of generality we can assume that t_0=0. By (\ref{18.3.1s}) the function {\bf h} satisfies (\ref{KKhf}). Integrating (\ref{PLMb1}) from 0 to t and using (\ref{KKh}) together with the inequality$$ \int_a^b|f(t)|dt\leq\int_{a-1}^b\int_t^{t+1}|f(\tau)|d\tau dt, we arrive at \begin{align*} &|{\bf h}(t)|\\ &\leq c\int_{-1}^t\varkappa_1(\tau)\|{\bf h}\|_{L^\infty (\tau,\tau+1)}d\tau +cb_0\int_{-1}^t\varkappa_{p_1'}(\tau)\int_{\mathbb{R}} \tilde{\mu}_\omega (\tau,s )\varkappa_{p_1}(s)\|{\bf h}\|_{L^\infty(s ,s+1)}ds \end{align*} if t\leq 0 and \begin{align*} &|{\bf h}(t)|\\ &\leq c\int_{t-1}^0\varkappa_1(\tau)\|{\bf h}\|_{L^\infty (\tau,\tau+1)}d\tau +cb_0\int_{t-1}^0\varkappa _{p_1'}(\tau)\int_{\mathbb{R}}\tilde{\mu}_\omega (\tau,s )\varkappa_{p_1}(s)\|{\bf h}\|_{L^\infty(s ,s+1)}ds \end{align*} if t<0. Now repeating the proof of \cite[Lemma 12]{KM2} we obtain {\bf h}=0. \end{proof} \begin{lemma}\label{LS9q} For each {\bf a}\in\mathbb{C}^d\setminus O equation {\rm (\ref{PLMb1})} has a solution {\bf h}\in (W^{1,p}_{\rm loc} (\mathbb{R}))^d which has the form $$\label{888f} {\bf h}(t)=|{\bf a}|\exp{\int_0^t\Lambda (\tau )d\tau }\;\Theta (t)\, ,$$ where |\Theta (t)|=1 for all t\in\mathbb{R}, \Theta (0)={\bf a}/|{\bf a}|, and $$\label{TNT1} \Lambda (t)=\Re\big (\mathcal{R}(t)\Theta (t),\Theta (t)\big )+\Lambda_1(t)$$ where \Lambda _1 satisfies the estimate $$\label{Ukr1} \|\Lambda_1\|_{L^1(t,t+1)}\leq cb_0\varkappa_{p_1'} (t)\int_{\mathbb{R}}\tilde{\mu}_\omega (t,\tau )\varkappa_{p_1} (\tau )d\tau\, ,$$ where p_1' and p_1 are the same as in {\rm Lemma \ref{Lem4r}}. Furthermore, $$\label{302} \| \Theta '\|_{L^1(t,t+1)}\leq C\wp (t)\, ,$$ where \Theta '(t)=d\Theta (t)/dt and $$\label{Ukr19} \wp (t)= \varkappa_1 (t)+b_0\varkappa_{p_1'} (t)\int_{\mathbb{R}}\tilde{\mu}_\omega (t,\tau )\varkappa_{p_1} (\tau )d\tau\, .$$ \end{lemma} \begin{proof} It suffices to prove the assertion for vectors {\bf a} with |{\bf a}|=1. Let us first prove the existence of a solution {\bf h} subject to the estimate $$\label{4.9.4a} |{\bf h}(t)|\leq C\exp{c_1\Big |\int_0^t\wp (\tau )d\tau \Big |}$$ with some positive constants c_1 and C. In order to construct a solution we shall use the following iterative procedure: {\bf h}_0(t)={\bf a}, and {\bf h}_{k+1}={\bf a}+\int_0^t(\mathcal{R}(\tau ){\bf h}_k(\tau )+(\mathcal{M}{\bf h}_k)(\tau ))d\tau $$for k=0,1,\dots  We introduce the Banach space B_{\wp }  which consists of measurable vector functions {\bf h}=(h_1,\dots ,h_d) on \mathbb{R} with the norm$$ \|{\bf h}\|_{B_{\wp}}=\sup_{t\in\mathbb{R}}\Big ( \exp{\Big (-c_1\Big |\int_0^t\wp (\tau )d\tau \Big |\Big )}\|{\bf h}\|_{L^\infty (t,t+1)}\Big )\, , $$where the constant c_1 will be chosen later. Let us show that the sequence \{ {\bf h}_k\}_{k=0}^\infty is convergent in B_\wp if c_1 is sufficiently large. Using (\ref{PLMb1c}) we obtain |\mathcal{R}(t)|\leq c\overline{\varkappa}_1(t), where \overline{\varkappa}_s is defined by (\ref{SSS1}). By the last estimate and by (\ref{KKh}) $$\label{BB1a} \|{\bf h}_{k+1}-{\bf h}_k\|_{L^\infty (t,t+1)}\leq c_2\int_{-1}^t\wp (\tau )\|{\bf h}_{k+1}-{\bf h}_k\|_{L^\infty (\tau ,\tau +1)}d\tau$$ for t\geq 0 and$$ \|{\bf h}_{k+1}-{\bf h}_k\|_{L^\infty (t,t+1)}\leq c_2\int_{t-1}^0\wp (\tau )\|{\bf h}_{k+1}-{\bf h}_k\|_{L^\infty (\tau ,\tau +1)}d\tau $$for t\leq 0. Let t\geq 0. Then (\ref{BB1a}) implies $$\label{BB1b} \|{\bf h}_{k+1}-{\bf h}_{k}\|_{B_\wp}\leq \frac{c_2}{c_1}\|{\bf h}_k-{\bf h}_{k-1}\|_{B_{\wp}}\, .$$ Analogously, for t\leq 0 we have $$\label{BB1c} \|{\bf h}_{k+1}-{\bf h}_{k}\|_{B_\wp}\leq \frac{c_2}{c_1}\sup_{t\leq 0 }e^{c_1\int_{t-1}^t\wp (\tau )d\tau }\|{\bf h}_k-{\bf h}_{k-1}\|_{B_{\wp}}\, .$$ By (\ref{PEPEb}) and b_0\geq 1, the function \wp does not exceed \omega_0 times a constant. Therefore, we can choose c_1 sufficiently large and \omega_0 sufficiently small so that the constants in the right-head sides in (\ref{BB1b}) and (\ref{BB1c}) are less than 1. This guarantees the convergence of \{ {\bf h}_k\}_{k=0}^\infty to {\bf h}\in B_{\wp}. Clearly this vector function satisfies the equation$$ {\bf h}={\bf a}+\int_0^t(\mathcal{R}(\tau ){\bf h}(\tau )+(\mathcal{ M}{\bf h})(\tau ))d\tau\, , $$which is equivalent to (\ref{PLMb1}). We define q(t)=|{\bf h}(t)| and \Theta (t)={\bf h}(t)/q(t). We note that by Lemma \ref{LS9w} the function q is positive for all t. Multiplying equation (\ref{PLMb1}) by \Theta (t) and taking the real part we obtain $$\label{4.9.4b} \frac{dq}{dt}(t)-a(t)q(t)-(\mathcal{M}_sq)(t)=0\, ,$$ where $$\label{KKh1aaa} a(t)=\Re (\mathcal{R}(t)\Theta (t),\Theta (t))\quad \mbox{and}\quad (\mathcal{M}_sq)(t)=\Re (\mathcal{M} (\Theta q)(t),\Theta (t))\, .$$ >From (\ref{KKh}) it follows that $$\label{KKh1aa} \|\mathcal{M}_sq\|_{L^1(t,t+1)}\leq cb_0\varkappa_{p_1'} (t)\int_{\mathbb{R}}\tilde{\mu}_\omega (t,\tau )\varkappa_{p_1} (\tau )\|q\|_{L^\infty (t ,t +1)}d\tau\, .$$ Let us show that $$\label{4.9.4c} q(t)=\exp{\int_0^t\Lambda (\tau )d\tau}\, ,$$ where \Lambda is a measurable function satisfying estimate (\ref{Ukr1}). We shall consider (\ref{4.9.4b}) as an equation with respect to q only, supposing \Theta be fixed. Making substitution q(t)=\exp\big (\int_0^ta(\tau )d\tau\big )\,z(t) we arrive at the equation $$\label{4.9.4bs} \frac{dz}{dt}(t)-(\mathcal{M}_2z)(t)=0\, ,$$ where (\mathcal{M}_2z)(t)=(\mathcal{M}_s)_{\tau\to t}(\exp\big (\int_t^\tau a(\tau )d\tau\big )z(\tau )). One can check directly that the operator \mathcal{M}_2 also satisfies the estimate (\ref{KKh1aa}), possibly with another constant c_0 in definition (\ref{Mu1}) of the function \mu. Equation (\ref{4.9.4bs}) has the form (173) in \cite{KM2}, but the operator \mathcal{M}_2 is estimated in different norms. Therefore the representation z(t)=\exp\big (\int_0^t\Lambda_1(\tau )d\tau\big ) with \Lambda_1 subject to (\ref{Ukr1}) follows actually from \cite[Lemma 13]{KM2} if one makes there evident changes caused by the only available L^1-estimate for \mathcal{M}_2. It remains to prove (\ref{302}). Expressing {\bf h}' from (\ref{PLMb1}) and using (\ref{888f}) and (\ref{KKh}) for estimating the second and the third terms in the left-hand side of (\ref{PLMb1}) we arrive at$$ \|{\bf h} '\|_{L^1(t,t+1)}\leq |a|C\wp (t)\exp{\int_0^t\Lambda (\tau )d\tau }\, , $$From representation (\ref{888f}) and this estimate we derive (\ref{302}). The proof is complete. \end{proof} \subsection{Solutions to the homogeneous system \protect\eqref{X6.10}}\label{Subs113} Using (\ref{JLK1}), (\ref{KKh1aaa}) and (\ref{KKh1}) we can represent the operator \mathcal{M}_s in (\ref{4.9.4b}) as $$\label{OPSa} \mathcal{M}_s(q)(t)=\Re\langle (\mathfrak{N}(t)-\mathfrak{ N}^{(0)}(t))\sum_{j=1}^d\Theta_j\hat{\Phi}_j,\sum_{k=1}^d\Theta_k\Psi_k\rangle q +e^{mt}\Re\langle \mathfrak{N}(t)\hat{\bf v},\sum_{k=1}^d\Theta_k\Psi_k\rangle\, ,$$ where the vector function {\bf v} satisfies (\ref{9.3z}) with \mathcal{F}=0 and \hat{\bf u}=e^{-mt}\sum_{j=1}^d\Theta_j\hat{\Phi}_jq. Let {\bf a}\in\mathbb{C} and |{\bf a}|=1. We denote by {\bf h} the unique solution of (\ref{PLMb1}) having the form (\ref{888f}). Then q(t):=|h(t)|=\exp{\int_0^t\Lambda (\tau)d\tau} and this function satisfies (\ref{4.9.4b}). We represent the vector function {\bf v} as $$\label{441} {\bf v}=\exp{\Big (-mt+\int_0^t\Lambda (\tau )d\tau\Big ) }\; {\bf V}(t)\, .$$ Inserting these q and \bf v into (\ref{PLMb1}) and using (\ref{OPSa}) we arrive at $$\label{PLNa} \Lambda (t)-a(t)-b(t,\Lambda )=0\, ,$$ where a is given by (\ref{KKh1aaa}) and $$\label{POpa2} b(t,\Lambda )=\Re\langle (\mathfrak{N}(t)-\mathfrak{ N}^{(0)}(t))\sum_{j=1}^d\Theta_j\hat{\Phi}_j,\sum_{k=1}^d\Theta_k\hat{\Psi}_k\rangle +\Re\langle \mathfrak{N}(t)\hat{\bf V},\sum_{k=1}^d\Theta_k\hat{\Psi}_k\rangle$$ with {\bf V} satisfying $$\label{9.3zz} (\mathcal{I}D_t +\mathfrak{A}+im-i\Lambda ){\bf V}-(\mathcal{I}-\mathcal{P})\mathfrak{ N}\hat{{\bf V}} =\sum_{j=1}^d(\mathcal{I}-\mathcal{P})\Theta_j\mathfrak{N} \widehat{\Phi_j}\quad \mbox{on {\mathbb{R}}.}$$ Using estimate (\ref{QQQ5}) for the function {\bf v} and observing that {\bf v} satisfies (\ref{QQQ4}) with$$ F=\exp\Big (-mt+\int_0^t\Lambda (\tau )d\tau\Big )\sum_{j=1}^d\Theta_j\mathfrak{N} \widehat{\Phi_j}, $$we arrive at the following estimate for {\bf V}:$$ \|\hat{{\bf V}}\|_{\hat{\mathbb{T}}_{p_1}(t,t+1)}+\|{\bf V}\|_{\mathbb{ T}_p(t,t+1)}\leq cb_0\sum_{j=1}^d\int_{\mathbb{R}}\tilde{\mu}_\omega (t,\tau )\|\mathfrak{N}\hat{\Phi}_j\|_{\mathbb{Y}_{p_1}(\tau ,\tau +1)}d\tau\, . This together with (\ref{Kap1a}) gives the estimate $$\label{for5} \|\hat{{\bf V}}\|_{\hat{\mathbb{T}}_{p_1}(t,t+1)}+\|{\bf V}\|_{\mathbb{ T}_p(t,t+1)}\leq cb_0\int_{\mathbb{R}}\mu_\omega (t,\tau )\varkappa_{p_1}(\tau ) d\tau\, ,$$ where \tilde{\mu}_\omega is given by (\ref{Kappa19}) and (\ref{Mu1}) possibly with a larger constant c_0. Here we used the inequalities |\Lambda (\tau)|\leq c\wp(\tau)\leq c\omega(\tau). Now we are in position to formulate and prove an assertion about solutions to (\ref{X6.10h}) subject to (\ref{183.1a}). \begin{lemma}\label{Teor12} Let \mathcal{U}\in \mathbb{T}_{p,{\rm loc} }(\mathbb{R} ) be a solution to system {\rm (\ref{X6.10h})} subject to {\rm (\ref{183.1a})}. Then $$\label{183.1b} \mathcal{U}(t)=c\exp\Big (-mt+\int_0^t\Lambda (\tau )d\tau\Big )\,\Big (\sum _{j=1}^d\Theta_j\Phi_j+{\bf V}\Big )\, ,$$ where c is a constant, the function  \Lambda (t) admits representation {\rm (\ref{TNT1})}, where \Lambda_1 satisfies {\rm (\ref{Ukr1})}, the vector function \Theta subject to |\Theta (t)|=1 and {\rm (\ref{302})}, and the function {\bf V}\in \mathbb{T}_{p,{\rm loc} }(\mathbb{R} ) satisfies equation {\rm (\ref{9.3zz})} and estimate {\rm (\ref{for5})}. \end{lemma} \begin{proof} By (\ref{9.4g}) and (\ref{PLMb1a}) we obtain $$\label{183.1c} \mathcal{U}=e^{-mt}\sum_{j=1}^dh_j(t)\Phi_j+{\bf v}\, .$$ Using (\ref{888f}) together with (\ref{441}) we arrive at the representation (\ref{183.1b}) with {\bf V} solving equation (\ref{9.3zz}) and satisfying estimate (\ref{for5}). The proof is complete. \end{proof} Let us denote by \Lambda_+(t) and \Lambda_-(t) the largest and the least eigenvalue of the matrix \Re \mathcal{R}. We finish this section by the following two-side estimates for \Lambda. \begin{lemma}\label{Lem9a} {\rm (i)} The function \Lambda satisfies the estimates $$\label{kra1} \Lambda_-(t )-c_1\wp_0(t) \leq\Lambda (t)\leq \Lambda_+(t )+c_2\wp_0(t)\, ,$$ where $$\label{wp2} \wp_0(t)=b_0\varkappa_{p_1'} (t)\int_{\mathbb{R}}\tilde{\mu}_\omega (t,\tau )\varkappa_{p_1} (\tau )d\tau\, .$$ {\rm (ii)} Furthermore, if {\rm (\ref{4.8aq8hhh})} is fulfilled with a sufficiently small \omega_0 depending on m, n, p, \gamma and L then $$\label{Ner3a} \int_a^b\Lambda (\tau )d\tau \leq\int_a^b\Lambda_+(\tau )d\tau +c\int_a^b\varkappa_2^2(\tau )d\tau +c\omega_0^2$$ and $$\label{Ner3b} \int_a^b\Lambda (\tau )d\tau \geq\int_a^b\Lambda_-(\tau )d\tau -c\int_a^b\varkappa_2^2(\tau )d\tau -c\omega_0^2$$ for a2\delta /3. We can choose \eta_\delta such that |d^k\eta_\delta (r)/dr^k|\leq C_k\delta^{-k}. Let \zeta_\delta be another smooth function, which is equal to 1 in a neighborhood of [\delta /2,2\delta /3] and is zero outside the interval (\delta /3,3\delta /4). We can suppose that |d^k\zeta_\delta (r)/dr^k|\leq C_k\delta^{-k}. We set u_\delta =\eta_\delta u. Then u_\delta  satisfies \label{RTR1f} \begin{aligned} \mathcal{L}(x,D_x)u_\delta &= \sum _{|\alpha |,|\beta |\leq m }D _x^\alpha\big ( \mathcal{L}_{\alpha \beta}(x)(D_x^\beta \eta_\delta \zeta_\delta u-\eta_\delta D_x^\beta\zeta_\delta u)\big ) \\ &\quad +\sum _{|\alpha |,|\beta |\leq m }(D _x^\alpha \eta_\delta-\eta_\delta D _x^\alpha )\big ( \mathcal{L}_{\alpha \beta}(x)D_x^\beta\zeta_\delta u\big ). \end{aligned} If we denote the functional corresponding to the left-hand side in (\ref{RTR1f}) by f then it is supported by \delta /2\leq |x|\leq 2\delta /3 and, by Sobolev imbedding theorem and by (\ref{PEPEa}), \mathfrak{M}^{-m}_{p_2}(f;K_{r/e,r})\leq c\delta^{-2m}\mathfrak{ M}^m_p(\zeta_\delta u;K_{r/e,r}), $$where p_1=\min ( 2n/(n-2),p) if n>2 and p_1=p if n=2. We can verify that all requirements of Proposition {T4.2zzZ} are fulfilled and therefore there exists a solution v\in (\Wcirc^{m,p_1}_{{\rm loc} }(\overline{{\mathbb{R}}_+^n}\setminus \mathcal{O}))^d of problem (\ref{4.2zas}), (\ref{4.2zb}) satisfying estimate (\ref{4.9qz2}), which takes, in our case, the form $$\label{4.7q1} \mathfrak{M}^m_{p_1}(v;K_{r/e,r})\leq c b_0 \Big (\frac{r}{\delta }\Big )^{m+1}e^{\mathcal{C}\int_r^\delta\Omega (s)\frac{ds}{s}}\mathfrak{ M}^m_{p_1}(u;K_{\delta/4,\delta})$$ for r\leq\delta and $$\label{4.7q2} \mathfrak{M}^m_{p_1}(v;K_{r/e,r})\leq c b_0\Big (\frac{r}{\delta }\Big )^me^{\mathcal{C}\int_\delta^r\Omega (s)\frac{ds}{s}}\mathfrak{ M}^m_{p_1}(u;K_{\delta/4,\delta})$$ for r>\delta. The function Z=u_\delta -v satisfies all conditions of Theorem \ref{TTT11} and hence, admits representation (\ref{Kyt2am}). Thus we arrive at representation (\ref{Zet1}) with w=v+(1-\eta_\delta)u and estimate (\ref{I4.7q1}) follows from (\ref{4.7q1}). In order to prove (\ref{I4.7q2}) we observe that$$ \mathfrak{M}^0_2(Z;K_{\delta/e,\delta})\leq \mathfrak{M}^0_2(u_\delta ;K_{\delta/e,\delta})+\mathfrak{M}^0_2(v;K_{\delta/e,\delta}) $$and using (\ref{4.7q1}) we obtain$$ \mathfrak{M}^0_2(Z;K_{\delta/e,\delta})\leq cb_0\mathfrak{ M}^m_p(u;K_{\delta/4,\delta}). Using this estimate together with the right-hand inequality in (\ref{2005e}), we arrive at (\ref{I4.7q2}). This completes the proof of Theorem \ref{TTT12}. \end{proof} \subsection{Proof of Corollaries \protect\ref{Le12a} and \protect\ref{TTT1h}} \begin{proof}[Proof of Corollaries \ref{Le12a}] As it was noted in Remark \ref{Rem5ta} under assumption (\ref{4.8aq8hhh}) all conditions {\bf H1}--{\bf H3} are satisfied with p_1=p as well as with p_1=p=2 and b_0=1 provided \omega_0 is sufficiently small. Inclusion Z\in (\Wcirc^{m,2}_{\rm loc} (\overline{\mathbb{R}_+^n}\setminus \mathcal{O}))^d together with (\ref{4.7aq7hh8}) implies Z\in (\Wcirc^{m,p}_{\rm loc} (\overline{\mathbb{R}_+^n}\setminus \mathcal{O}))^d as well as (\ref{Kyt1a}) and (\ref{Kyt1a2}). Thus we can apply Theorem \ref{TTT11}. Choosing in (\ref{OOPPa}) p_1=p=2 we arrive at (\ref{2005d}) with \chi given by (\ref{OOPPah}). The required estimate for v_k follows from (\ref{Ququ3za}) if we take there p_1=p. \end{proof} Corollaries \ref{TTT1h} is proved similarly. The only new element here is that first we obtained (\ref{I4.7q2d}) and (\ref{I4.7q1b}) with \mathfrak{M}^m_{p_1}(u;K_{\delta/8,\delta/2}) instead of \mathfrak{M}^m_2(u;K_{\delta/16,\delta}) but using the local estimate for u we arrive at the required estimates. \subsection{Proof of Corollary \protect\ref{Le12}} To prove this assertion we use Corollary \ref{Le12a}. In our case p_1=p, b_0=1 and \kappa_p(r)\leq c\omega_0. Therefore from the asymptotic representation (\ref{Kyt2am}) we derive the estimates \begin{align*} c_1|J_Z| r^m\exp\Big (\int_r^1 \Upsilon (\rho)\frac{d\rho}{\rho}\Big ) &\leq \mathfrak{M}^m_p(Z;K_{r/e,r})\\ &\leq c_2|J_Z|r^m\exp\Big (\int_r^1 \Upsilon (\rho)\frac{d\rho}{\rho}\Big )\, . \end{align*} Using (\ref{2005e}) for estimating the constant J_Z we obtain \label{Stockh1} \begin{aligned} &C_1J(Z)\Big (\frac{r}{\delta}\Big )^m\exp\Big (\int_r^\delta \Upsilon (\rho)\frac{d\rho}{\rho}\Big )\\ &\leq \mathfrak{M}^m_p(Z;K_{r/e,r}) \\ &\leq C_2J(Z)\Big (\frac{r}{\delta}\Big )^m\exp\Big (\int_r^\delta \Upsilon (\rho )\frac{d\rho}{\rho}\Big ). \end{aligned} Since \Upsilon (\rho )=\Lambda\big (\log\rho^{-1}\big) and \Upsilon_\pm (\rho )=\Lambda_\pm\big(\log\rho^{-1}\big ), estimates (\ref{Ququ3z1}) follows from (\ref{Ner3a}), (\ref{Ner3b}) and (\ref{Stockh1}). \subsection{Proof of Corollary \protect\ref{TTT1}}\label{Subs115} Here we apply Corollary \ref{TTT1h} for proving this assertion. Since \frac{r}{\delta}\exp\Big (\mathcal{C}\int_r^\delta\Omega (s)\frac{ds}{s}\Big )\leq c\exp\Big (\int_r^\delta (\Upsilon_+(\rho)+c\nu (\rho ))\frac{d\rho}{\rho}\Big ), estimate (\ref{I4.7q1b}) for the remainder w in (\ref{Zet1}) implies $$\label{BNH1a} \mathfrak{M}^m_{p}(w;K_{r/e,r})\leq c \, \mathfrak{ M}_2^m(u;K_{\delta/16,\delta})\Big (\frac{r}{\delta }\Big )^m\exp\Big (\int_r^\delta (\Upsilon_+(\rho)+c\nu (\rho ))\frac{d\rho}{\rho}\Big ).$$ Using the right-hand inequality in (\ref{Ququ3z1}) and this estimate, we obtain \label{BNH1} \begin{aligned} \mathfrak{M}_p^m(u;K_{r/e,r}) &\leq c\big (\mathfrak{M}_2^m(u;K_{\delta/16,\delta})+\mathfrak{ M}_2^0(Z;K_{\delta/e,\delta})\big )\Big (\frac{r}{\delta }\Big )^m \\ &\quad \times\exp\Big (\int_r^\delta (\Upsilon_+(\rho)+c\nu (\rho ))\frac{d\rho}{\rho}\Big ). \end{aligned} Since \mathfrak{M}_2^0(Z;K_{\delta/e,\delta})\leq \mathfrak{ M}_2^0(u;K_{\delta/e,\delta})+\mathfrak{ M}_2^0(w;K_{\delta/e,\delta}), $$we apply (\ref{BNH1a}) to estimate the last term and obtain$$ \mathfrak{M}_2^0(w;K_{\delta/e,\delta})\leq c\mathfrak{ M}_2^m(u;K_{\delta/16,\delta}). $$This along with (\ref{BNH1}) proves (\ref{Ququ3z}). \subsection{Proof of Corollary\protect\ref{kTTT1}} (1) \emph{Existence of p_1 and validity of (H2)}. Let us show first that there exists p_1>2, depending on m, n, \gamma and L such that the following local estimate is valid: if u\in\Wcirc^{m,2}_{\rm loc} (K) solves problem {\rm (\ref{4.2zas})}, {\rm (\ref{4.2zb})} with f\in W^{-m,p_1}_{\rm loc} (K), then u\in\Wcirc^{m,p_1}_{\rm loc} (K) and $$\label{4.7aq7hhv} \mathfrak{M}_{p_1}^m(u;K_{r/e,r})\leq b_0 \big (r^{2m}\mathfrak{M}_{p_1}^{-m}(f;K_{r/e^2,er})+\mathfrak{M}_{2}^m(u;K_{r/e^2,er})\big ),$$ where b_0 is a constant depending on m, n, \gamma and L. We note that this is (H2) condition with p=2. Indeed, consider the operator $$\label{4.7aq7hhw} \mathcal{L}\, :\, (\Wcirc^{m,p} ({\mathbb{R}}_+^n))^d\to (\Wcirc^{-m,p} ({\mathbb{R}}_+^n))^d\quad \mbox{with p\in [q',q],}$$ where q>2 and q'=q/(q-1). The norm of this operator is bounded by a constant depending on the constants \gamma, m, n and q. By (\ref{RTR19}) this operator has inverse for p=2 with the norm which depends on the same constants. Using Shneiberg result \cite{Sh} (see also \cite{KM4} and references there), we conclude that there exist constants p_1>2 and C depending on \gamma, m, n and q, such that the operator (\ref{4.7aq7hhw}) is invertible for p\in [p_1',p_1] and the norms of inverse operators are bounded by C. Let \eta=\eta(\tau) be a smooth function on (0,\infty) such that \eta(\tau)=1 for \tau\in [e^{-1},1] and \eta(\tau)=0 outside [e^{-2},e]. Let also \eta_r(\tau)=\eta(\tau/r). Then \mathcal{L}(\eta_ru)=\eta_rf+(\mathcal{L}\eta_r -\eta_r \mathcal{L}) u. Using that the operator (\ref{4.7aq7hhw}) is isomorphism for p_1\in [p_2',p_2] we obtain$$ \|\eta_r u\|_{\Wcirc^{m,p_1}(\mathbb{R}^n_+)}\leq c (\|\eta_rf\|_{W^{-m,p_1}(\mathbb{R}^n_+)}+\|(\mathcal{L}\eta_r -\eta_r \mathcal{L}) u\|_{W^{-m,p_1}(\mathbb{R}^n_+)}). $$This estimate together with Sobolev's imbedding theorem implies (\ref{4.7aq7hhv}). (2) \emph{Validity of (H3)}. From (\ref{Using1}) it follows that \kappa_1(r)\to 0 as r\to 0. This together with boundedness of \kappa implies (H3) because of (\ref{TTrr2ad}). (3) \emph{Application of Theorem \ref{TTT12}}. Since (H1)--(H3) are valid we can apply to solution u Theorem \ref{TTT12} and we obtain the asymptotic representation (\ref{Zet1}) with Z satisfying (\ref{Kyt2am}) and w subject to (\ref{I4.7q1}). The first inequality in (\ref{2005d}) implies that the vector function {\bf q}(r) has a limit as r\to 0 because of (\ref{Using1}), which we denote by {\bf q}_0. Let us show that the integral$$ \int _r^\delta\Upsilon (\rho )\frac{d\rho }{\rho } has a limit as r\to 0. Using the definition (\ref{U2117}) of {\bf R}(\rho), we check that $$\label{using3} |({\bf R}(\rho ){\bf q}(\rho),{\bf q}(\rho))|\leq c\int_{S_+^{n-1}}\kappa (y)d\theta$$ where \rho =|y| and \theta =y/|y|. From (\ref{NTG4}) and the definition (\ref{OOPPah}) of the function \chi, it follows \label{using2} \begin{aligned} \int_0^\delta|\chi (\rho )|\frac{d\rho}{\rho} &\leq c\Big (\int_0^\delta\kappa_{p_1'}^{p_1'}(\rho )\frac{d\rho}{\rho}\Big )^{1/p_1'}\Big (\int_0^\delta\kappa_{p_1}^{p_1}(\rho )\frac{d\rho}{\rho}\Big )^{1/p_1}+c\omega_0^2 \\ &\leq C\int_0^\delta\kappa_1(\rho )\frac{d\rho}{\rho} +c\omega_0^2\, . \end{aligned} Therefore, \int_0^\delta |\Upsilon (\rho )|\frac{d\rho}{\rho}<\infty $$because of (\ref{Using1}), and consequently, $$\label{using4} \int_r^\delta\Upsilon (\rho )\frac{d\rho}{\rho}=C-\int_0^r\Upsilon (\rho )\frac{d\rho}{\rho}\, ,$$ where the last integral is absolutely convergent and, hence, is o(1) as r\to 0. This leads to $$\label{Slut1} \exp\Big (\int_r^\delta\Upsilon (\rho )\frac{d\rho}{\rho}\Big )=C_1+o(1)\quad \mbox{as r\to 0}.$$ We put v=u-{\bf c}x_n^m, where {\bf c}=J_ZC_1{\bf q}_0 and J_Z is the constant in (\ref{Kyt2am}). Due to (\ref{I4.7q1}) in order to show that v satisfies (\ref{Using167}) it suffices to show that Z-{\bf c}x_n^m satisfies (\ref{Using167}). By (\ref{Kyt2am}) we have$$ (r\partial_r)^k(Z-{\bf c}x_n^m)=J_Zm^kx_n^m\Big (\exp\Big (\int _r^\delta\Upsilon (\rho )\frac{d\rho }{\rho }\Big ) {\bf q}(r)-C_1{\bf q}_0\Big ) +J_Zr^mv_k(x)\, .  By (\ref{Ququ3za}), ${\bf q}(r)\to {\bf q}_0$ as $r\to 0$ and by (\ref{Slut1}) this implies (\ref{Using167}). Now the result follows from Corollary \ref{TTT12} and from the asymptotic representation (\ref{Kyt2am}) for the first term in the right-hand side in (\ref{Zet1}). \begin{thebibliography}{00} \bibitem{ADN} S. Agmon, A. Douglis and L. Nirinberg, \emph{Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I}, Comm. Pure Appl. Math. {\bf 12}(1959), 623-727. \bibitem{GT} D. Gilbarg and N. Trudinger, \emph{Elliptic Partial Differential Equations of Second Order}, (2nd ed.), Springer, 1983. \bibitem{KM1} Kozlov, V., Maz'ya, V.: \emph{Differential Equations with Operator Coefficients} (with Applications to Boundary Value Problems for Partial Differential Equations), Monographs in Mathematics, Springer-Verlag, 1999. \bibitem{KM2} Kozlov, V., Maz'ya, V., Asymptotic formula for solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients near the boundary, Ann. Scuola Norm. Sup. Pisa, {\bf 2}(2003), 3, 551-600. \bibitem{KM3} Kozlov, V., Maz'ya, V., Asymptotics of a singular solution to the Dirichlet problem for elliptic equations with discontinuous coefficients near the boundary, Function Spaces, Differential Operators and Nonlinear Analysis, 2003, 75-115. \bibitem{KM4} Kozlov, V., Maz'ya, V., Asymptotic formula for solutions to elliptic equations near the Lipschitz boundary, Annali di Matematica Pura ed Applicata, to appear. \bibitem{KM5} Kozlov, V., Maz'ya, V., Sharp conditions for classical asymptotic behaviour near a point for solutions to quasilinear elliptic systems, Asymptotic Analysis, to appear. \bibitem{KM6} Kozlov, V., Maz'ya, V., An asymptotic theory of higher-order operator differential equations with nonsmooth nonlinearities, Journal of Functional Analysis, {\bf 217} (2004), 448-488. \bibitem{KM7} Kozlov, V., Maz'ya, V. and Rossmann, J.: \emph{ Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations}, Mathematical Surveys and Monographs {\bf 85} (2001), Amer. Math. Soc. \bibitem{KM8} Krugljak, N. and Milman, M., A distance between orbits that controls commutator estimates and invertibility of operators, Advances in Mathematics, {\bf 182}(2004), 78-123. \bibitem{Sh} Shneiberg, I.Ya., On the solvability of linear equations in interpolation families of Banach spaces, Soviet Math. Dokl. {\bf 14}(1973), 1328-1331. \end{thebibliography} \end{document}