\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 101, pp. 1--21.\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-2006/101\hfil BMO estimates near the boundary]
{BMO estimates near the boundary for solutions of elliptic systems}
\author[A. El Baraka\hfil EJDE-2006/101\hfilneg]
{Azzeddine El Baraka}

\address{Azzeddine El Baraka \newline
University Sidi Mohamed Ben Abdellah \\
FST Fez, BP 2202, Route Immouzer \\
30000 Fez, Morocco}
\email{aelbaraka@yahoo.com}

\date{}
\thanks{Submitted March 2, 2006. Published August 31, 2006.}
\subjclass[2000]{35J45, 35J55}
\keywords{Elliptic systems; BMO-Triebel-Lizorkin spaces; Campanato spaces}

\begin{abstract}
 In this paper we show that the scale of  Sobolev-Campanato spaces
 $\mathcal{L}^{p,\lambda,s}$ contain the general
 BMO-Triebel-Lizorkin spaces $F_{\infty,p}^{s}$ as special cases,
 so that the conjecture by Triebel regarding estimates for
 solutions of scalar regular elliptic boundary value problems in
 $F_{\infty,p}^{s}$ spaces (solved in the case $p=2$ in a previous
 work) is completely solved now.

 Also we prove that the method used for the scalar case works
 for systems, and we give a priori estimates near the boundary for
 solutions of regular elliptic systems in the general spaces
 $\mathcal{L}^{p,\lambda,s}$ containing BMO, $F_{\infty,p}^{s}$,
 and Morrey-Campanato spaces $\mathcal{L}^{2,\lambda}$
 as special cases. This result extends the work by the author
 in the scalar case.
 \end{abstract}

\dedicatory{Dedicated to  all the civilian victims of the catalogue of
horror in the Middle East}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

The aim of this paper is to give the regularity for solutions of
regular elliptic systems in the John and Nirenberg space BMO and
more generaly in Morrey-Campanato spaces $\mathcal{L}^{2,\lambda}$
and their local versions $bmo$ and $l^{2,\lambda}$. So, this paper
is the continuation of \cite{az} where we got the regularity for
solutions of a scalar regular elliptic boundary value probem in
$\mathcal{L}^{p,\lambda,s}(\Omega)$ spaces
containing
$\dot {F}_{\infty,p}^{s}$, BMO, $\mathcal{L}^{2,\lambda}$, and
their local versions as special cases.

Firstly, we mention the well known work for variational systems of
Campanato \cite{c3} who obtained some results concerning local and
global (under Dirichlet boundary conditions) regularity for
solutions $u\in H_{0}^{1} (\Omega,\mathbb{R}^{N})$ of second order
linear strongly elliptic systems of the form
\[
\sum_{i,j=1}^{n}
\int_{\Omega}\langle A_{ij}(x).D_{j}u|D_{i}\phi\rangle dx=
\sum_{i=1}^{n}
\int_{\Omega}\langle f_{i}(x)|D_{i}\phi\rangle dx
\]
for any $\phi\in C_{0}^{\infty}(\Omega,\mathbb{R}^{N})$. He showed that if
$f\in BMO(\Omega,\mathbb{R}^{nN})$ then $Du\in BMO(\Omega,\mathbb{R}^{nN})$
provided the cofficients $A_{ij}$ are H\"{o}lder continuous in $\overline
{\Omega}$ and $\partial\Omega$ is H\"{o}lder differentiable, and he got the
a-priori estimate
\[
\|Du\|_{BMO}\leq C\|f\|_{BMO}
\]

An inspection of Campanato's proof gave a refinement of this result for
(nonregular) elliptic systems with coefficients just belonging to the class of
small multipliers of $BMO(\Omega)$, cf. \cite{a}.

In this paper we deal with non-variational and inhomogeneous systems. For
instance, let us take the classical regular second order elliptic system
\begin{equation}
\begin{gathered}
Au=f\quad \text{in }\Omega\\
u|_{\Omega}=\varphi\quad \text{on }\partial\Omega\,,
\end{gathered} \label{ex}
\end{equation}
where

\begin{itemize}
\item $\Omega$ is a regular bounded open set of $\mathbb{R}^{n}$.

\item $A=\sum_{|\alpha|\leq2}a_{\alpha}(x)D_{x}^{\alpha},\;a_{\alpha
}(x)$ is the $N\times N$ matrix $(a_{\alpha}^{ij}(x))_{i,j=1\dots N}$
with smooth coefficients on $\overline{\Omega}$, here
$\alpha=(\alpha_{1},\dots ,\alpha_{n})$ is a multi-index,
$|\alpha|=\alpha_{1}+\dots +\alpha_{n}$ is the length of $\alpha$,
and
$D_{x}^{\alpha}=D_{x_{1}}^{\alpha_{1}}\dots D_{x_{n}}^{\alpha_{n}}$
is the derivation, with
$D_{x_{j}}^{\alpha_{j}}=\frac{1}{i^{\alpha_{j}}
}\frac{\partial^{\alpha_{j}}}{\partial x_{j}^{\alpha_{j}}}$.

\item $u,f,\varphi$ are vector-valued functions in $\mathbb{R}^{N}$.
\end{itemize}

We prove in this paper that under the proper ellipticity of $A$, if
$f\in BMO(\Omega,\mathbb{R}^{N})$ and $\varphi\in BMO^{3/2}(\partial\Omega
,\mathbb{R}^{N})$ then the solution $u$ of the system (\ref{ex}) belongs to
$BMO^{2}(\Omega,\mathbb{R}^{N})$, that is $u,Du$ \textit{and} $D^{2}u\in BMO$,
and we give the estimate
\[
\|u\|_{BMO}+\|Du\|_{BMO}+\|
D^{2} u\|_{BMO}\leq C\{\|f\|_{BMO}+\|
\varphi\|_{BMO^{3/2}}\}
\]
In addition, this work generalizes the result of the above example to the
elliptic systems in the sense of Douglis and Nirenberg \cite{dn} and to the
general spaces $\mathcal{L}^{p,\lambda,s}$ defined in \cite{e4} and \cite{e6}.

In \cite{cd} we showed that the Sobolev-Campanato spaces
$\mathcal{L}^{p,\lambda,s}$ contain $BMO,\mathcal{L}^{2,\lambda}$
and their local versions $bmo,l^{2,\lambda}$ as special cases. In
this paper we give a generalization of some results of \cite{cd}
by showing that $\mathcal{L} ^{p,\lambda,s}$ spaces contain the
general BMO-Triebel-Lizorkin spaces $F_{\infty,p}^{s}$ as special
cases cf. Theorem \ref{TL}. We want to attract attention in this
paper that, with the result of Theorem \ref{TL} in mind and the a
priori estimates of \cite{az} relative to the scalar case,
Triebel's conjecture \cite[section 4.3.4]{tr} previously solved in
the case $p=2$ in \cite{az} is completely solved now in the
general spaces $F_{\infty,p}^{s}$.

We show that the method used for the scalar case in \cite{az} can
be adapted for elliptic systems. The plan of the paper is as
follows: in the first section we give the main definitions and
results (Theorems \ref{141} and \ref{13}), section 2 contains an
index theorem for a system of ordinary differential equations
needed in the proof. In section 3 we identify the space
$\mathcal{L}^{p,\lambda,s}$ for $\lambda=n$ and we give a partial
result on the topological dual of
$\mathaccent"7017{F}_{\infty,p}^{s}$ (the closure of Schwartz class
$\mathcal{S}(\mathbb{R}^{n})$ in BMO-Triebel-Lizorkin spaces
$F_{\infty,p}^{s})$, and next we recall some results proved in
previous papers concerning intermediate derivatives, compactness,
interpolation and traces. Finally, section 4 deals with the proof
of the main result: we follow one Peetre's method used in the
scalar case \cite{az}. This method was described in the case of
Sobolev spaces $H^{s}$ for a class of degenerate elliptic systems
in \cite{bc}, and consists in doing a partial Fourier transform
with respect to the tangential direction on the system of
equations, and reducing the problem to an isomorphism theorem for
a system of ordinary differential operators. Thereby we estimate
the ``almost tangential derivatives'' of the solution in some
vector-valued $L^{p}-$spaces, built on $\mathcal{L}
^{p,\lambda,s}(\mathbb{R}^{n-1})$\ in the sense of Bochner's
integrals (Proposition \ref{prop}). Next, we make use of an
interpolation lemma to estimate the normal derivatives of the
solution.

From these BMO estimates we can get again the classical $L^{p}$
estimates \cite{adn} via an interpolation theorem due to
Stampacchia \cite{st}.

To make the paper self contained, we recall the definitions of the spaces
$\mathcal{L}^{p,\lambda,s}$. For this we need a Littlewood-Paley partition of
unity: Denote $x=(t,x')\in\mathbb{R}\times
\mathbb{R}^{n-1}$ and $\xi=(\tau,\xi')$ its dual variable.

Let $\varphi\in C_{0}^{\infty}(\mathbb{R}^{n})$, $\varphi\geq0$
and $\varphi$ equal to $1$ for $|\xi|  \leq1$, $0$ for $|
\xi|  \geq2$. Putting $\theta(\xi)=\varphi(\xi)-\varphi(2\xi)$, we have
$\mathop{\rm supp}\theta\subset\{\frac{1}{2}\leq|\xi|  \leq2\}$. For
$j\in\mathbb{Z}$ we set
\[
\dot {\Delta}_{j}u=\theta(2^{-j}D_{x})u,\,\dot {\Delta}
_{j}'u=\theta(2^{-j}D_{x'})u
\]
which means that $\dot {\Delta}_{j}$ and
$\dot {\Delta} _{j}'$ are the convolution operators
with symbols $\theta(2^{-j}\xi)$ and
$\theta(0,2^{-j}\xi')$; and denoting $\psi_{j}(\xi)=\sum
_{k\leq j}\theta(2^{-k}\xi)(=\varphi(2^{-j}\xi)$  for
$\xi\neq0)$ we set
\[
\dot {S}_{j}u=\psi_{j}(D_{x})
u,\,\dot {S} _{j}'u=\psi_{j}(
D_{x'})u
\]
with the same meaning as above.
If $j\geq1$ we set also
\begin{gather*}
[c]{l}
\Delta_{j}u=\dot {\Delta}_{j}u,\,\Delta_{j}'u=\dot
{\Delta}_{j}'u,\\
S_{j}u=\varphi(2^{-j}D_{x})u,\,S_{j}'u=\varphi(2^{-j}D_{x'})u,
\\
S_{0}u=\Delta_{0}u=\varphi(D_{x})u,\,S_{0}'u=\Delta_{0}^{\prime
}u=\varphi(D_{x'})u
\end{gather*}

\begin{remark} \label{ca} \rm
For $u\in\mathcal{S}'(\mathbb{R}^{n})$ we have
\[
u=\sum_{k\geq0}\Delta_{k}u=\Delta_{0}u+\sum_{k\geq1}\Delta_{k}u=S_{j}
u+\sum_{k\geq j+1}\Delta_{k}u\quad \text{for }j\in\mathbb{N}.
\]
If $0\notin \mathop{\rm supp}\mathcal{F}u$, then
\[
u=\sum_{k\in\mathbb{Z}}\dot {\Delta}_{k}u=\dot {S}_{j}
u+\sum_{k\geq
j+1}\dot {\Delta}_{k}u\quad \text{for }j\in\mathbb{Z}.
\]
If we remove the condition $0\notin \mathop{\rm supp}\mathcal{F}u$,
the above formula
remains valid modulo polynomials, cf. \cite[Lemma 2.6]{e6}.
\end{remark}

\begin{definition}\label{def1} \rm
Let $s\in\mathbb{R}$, $\lambda\geq0$ and $1\leq p<+\infty$. The
space $\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n})$ denotes the set of all
tempered distributions $u\in\mathcal{S}'(\mathbb{R}^{n})$ such that
\begin{equation}
\|u\|_{\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n})}
=\big\{
\sup_{J,B}\frac{1}{|B|^{\lambda/n}} \sum_{k\geq J^{+}}2^{kps}\|
\Delta_{k}u\|_{L^{p}(B)}^{p}\big\}  ^{1/p}<+\infty\label{def11}
\end{equation}
where $J^{+}=\max(J,0),|B|  $ is the measure of
$B$ and the supremum is taken over all $J\in\mathbb{Z}$ and all balls $B$ of
$\mathbb{R}^{n}$ of radius $2^{-J}$.
\end{definition}

The space $\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n})$ equipped with the norm
(\ref{def11}) is a Banach space.
If $\Omega$ is either $\mathbb{R}_{+}^{n}$ or a bounded
$C^{\infty}$-domain in $\mathbb{R}^{n}$,
$\mathcal{L}^{p,\lambda,s}(\Omega)$ denotes the space of all
restrictions to $\Omega$ of elements of $\mathcal{L}^{p,\lambda,s}
(\mathbb{R}^{n})$.

To give the homogeneous counterpart of the spaces
$\mathcal{L}^{p,\lambda ,s}(\mathbb{R}^{n})$, we recall the notation
 of \cite[chapter 5]{tr}. Let
\[
Z(\mathbb{R}^{n})=\{\varphi\in\mathcal{S}(\mathbb{R}^{n});(
D^{\alpha}\mathcal{F}\varphi)(0)=0 \text{ for every multi-index }\alpha\}
\]
$Z(\mathbb{R}^{n})$ is considered as a subspace of
$\mathcal{S(}\mathbb{R} ^{n}\mathcal{)}$ with the same topology,
and $Z'(\mathbb{R}^{n})$ is the topological dual of
$Z(\mathbb{R}^{n})$. We may identify $Z^{\prime }(\mathbb{R}^{n})$
and $\mathcal{S}'(\mathbb{R}
^{n}\mathcal{)}/\mathcal{P}$, where $\mathcal{P}$ is the set of
all polynomials of $\mathbb{R}^{n}$ with complex coefficients.
$Z^{\prime }(\mathbb{R}^{n})$ is interpreted as
$\mathcal{S}'( \mathbb{R}^{n}\mathcal{)}$ modulo
polynomials.

\begin{definition} \label{def2} \rm
Let $s\in\mathbb{R}$, $\lambda\geq0$ and $1\leq p<+\infty$. The
dotted space $\dot {\mathcal{L}}^{p,\lambda,s}(\mathbb{R}^{n})$ denotes
the set of all $u\in Z'(\mathbb{R}^{n})$ such that
\begin{equation}
\|u\|
_{\dot {\mathcal{L}}^{p,\lambda,s}(\mathbb{R}^{n} )}=\big\{
\sup_{J,B}\frac{1}{|B|  ^{\frac{\lambda}
{n}}}\sum_{k\geq J}2^{kps}\|
\dot {\Delta}_{k}u\|
_{L^{p}(B)}^{p}\big\}  ^{1/p}<+\infty\label{def21}
\end{equation}
where the supremum is taken over all $J\in\mathbb{Z}$ and all balls $B$ of
$\mathbb{R}^{n}$ of radius $2^{-J}$.
\end{definition}

If $P$ is a polynomial of $\mathcal{P}$ and $u\in\mathcal{S}'(\mathbb{R}^{n})$,
it follows immediately that
\[
\|u+P\|_{\dot {\mathcal{L}}^{p,\lambda,s}(\mathbb{R}
^{n})}=\|u\|_{\dot {\mathcal{L}}^{p,\lambda,s}
(\mathbb{R}^{n})}
\]
This shows that the norm (\ref{def21}) is well defined. Further, the space
$\dot {\mathcal{L}}^{p,\lambda,s}(\mathbb{R}^{n})$ equipped with this
norm is a Banach space.

\begin{remark} \label{rmk4} \rm
The supremum in expressions (\ref{def11}) and (\ref{def21}) can be taken over
all $J\in\mathbb{Z}$ and all cubes of $\mathbb{R}^{n}$ of sidelength $2^{-J}$.
\end{remark}

The reader can find the properties of these spaces in
\cite{e4,e6,cd}. We recall that the space
$\mathcal{L}^{p,\lambda,s}$ coincides with Campanato space
$l^{2,\lambda}=\mathcal{L}_{2}^{\frac{\lambda-n}{2}}$ when
$s=0,p=2$ and $0\leq\lambda<n+2$, itself equals $bmo$ for
$\lambda=n$, cf. \cite{cd}; and we will see in Theorem \ref{TL}
that in the case $\lambda=n$, the space
$\dot {\mathcal{L}}^{p,n,s}$ coincides with the homogeneous
BMO-Triebel-Lizorkin space $\dot {F}_{\infty,p}^{s}(1\leq
p<+\infty)$, itself equals BMO when $s=0$ and $p=2$.

We return to our main goal which is the estimates for solutions of
regular elliptic systems in $C^{\infty}-$bounded open sets
$\Omega$\ in $\mathbb{R} ^{n}$. This problem can be reduced, via a
partition of unity, to a priori estimates for solutions of regular
elliptic systems in the upper-half space
$\mathbb{R}_{+}^{n}=\{  x=(t,x');\;t>0\}$. We denote
 $\xi=(\tau,\xi')$ the dual variable of $x=(t,x')
\in\mathbb{R}^{n}=\mathbb{R}\times\mathbb{R}^{n-1}$.

Consider $L$ the following differential system in the sense of Douglis and
Nirenberg \cite{dn}:
\begin{equation}
L=L(t,x';D_{x'},D_{t})=(L_{ij}(x;D_{x}))_{i,j=1,\dots ,N}
\label{sys}
\end{equation}
where
\begin{equation}
L_{ij}(x;D_{x})u(x)=L_{ij}u(x)=\sum_{|\alpha|  \leq s_{i}+t_{j}
}a_{\alpha}^{ij}(x)D_{x}^{\alpha}u(x)\label{sys1}
\end{equation}
here $\alpha=(\alpha_{1},\dots ,\alpha_{n})$ is a multi-index,
$|\alpha |=\alpha_{1}+\dots +\alpha_{n}$ is the length of $\alpha$,
and $D_{x}^{\alpha}=D_{x_{1}}^{\alpha_{1}}\dots D_{x_{n}}^{\alpha_{n}}$
is the derivation, with
$D_{x_{j}}^{\alpha_{j}}=\frac{1}{i^{\alpha_{j}}}\frac{\partial^{\alpha_{j}}
}{\partial x_{j}^{\alpha_{j}}}$. The coefficients
$a_{\alpha}^{ij}$ are assumed to be in
$C^{\infty}(\overline{\mathbb{R}_{+}^{n}})$ and $s_{i},t_{j}$ are
integers which we can suppose satisfying $s_{i}\leq0$ and
$t_{j}\geq0$. In the definition of $L_{ij}$, it is to be
understood that if $s_{i}+t_{j}<0$ then $L_{ij}=0$. Let
$L_{ij}^{0}(x;D_{x})$ represent the principal part of
$L_{ij}(x,D_{x})$, which is the sum of the terms in
$L_{ij}(x;D_{x})$ which are exactly of the order $s_{i}+t_{j}$.

We suppose that $L$ is elliptic in the following classical sense.
\begin{itemize}
\item[(E1)] For any $(\tau,\xi')\in\mathbb{R}^{n}\setminus\{0\}$,
\[
\det(L_{ij}^{0}(0;\xi',\tau))_{i,j=1,\dots ,N}\neq0
\]
and the number $m_{+}(\xi')$ of the roots with positive imaginary
parts of the polynomial
\[
P(\tau)=\det(L_{ij}^{0}(0;\xi',\tau))_{i,j=1,\dots ,N}
\]
in the complex variable $\tau$, is constant and equals $m_{+}$.
\end{itemize}
Now we will define the traces. For each
$u\in\mathcal{L}^{p,\lambda,s+t_{j} }(\mathbb{R}_{+}^{n})$ we
consider $t_{j}$ traces of $u$ defined for
$x'\in\mathbb{R}^{n-1}$ by:
\[
\gamma_{l}u(x')=D_{t}^{l}u(0,x')\text{ for }l=0,\dots ,t_{j}-1.
\]

Set $\gamma=(\gamma_{0},\dots ,\gamma_{t_{j}-1})$. We showed in
\cite[Theorem 3.1]{az} that the operator $\gamma$ is continuous
from $\mathcal{L} ^{p,\lambda,s+t_{j}}(\mathbb{R}_{+}^{n})$ to
$\prod_{l=0}^{t_{j}
-1}\mathcal{L}^{p,\lambda,s+t_{j}-l-1/p}(\mathbb{R}^{n-1})$.

If $m_{+}=0$ there is no boundary conditions for our problem.
If $m_{+}>0$, for each $i=1,\dots ,m_{+}$, let $\sigma_{i}$ be an
integer  $\sigma_{i}\leq -1$. Set
\[
(B_{ij}(x',D_{x'})\cdot\gamma)u=\sum_{l=0}^{t_{j}
-1}B_{ijl}(x',D_{x'})\gamma_{l}u,
\]
for $i=1,\dots ,m_{+}$, $j=1,\dots ,N$;
where $B_{ijl}(x',D_{x'})$ is a differential operator of
degree less than or equal $\sigma_{i}+t_{j}-l$, with smooth coefficients
bounded with their derivatives. If $\sigma_{i}+t_{j}-l<0$ we put $B_{ijl}=0$.

Denote $B\gamma$ the matrix $B(x',D_{x'})\gamma=(B_{ij}
(x',D_{x'})\cdot\gamma)_{i=1,\dots ,m_{+}; j=1,\dots ,N}$.
This operator is continuous from $\prod_{j=1}
^{N}\mathcal{L}^{p,\lambda,s+t_{j}}(\mathbb{R}_{+}^{n})$ into
$\prod _{i=1}^{m_{+}}\mathcal{L}^{p,\lambda,s-\sigma_{i}-1/p}(\mathbb{R}
^{n-1})$. Let $B_{ij}^{0}\cdot\gamma$ represent the principal part
of $B_{ij}\cdot\gamma$, which is the sum of the terms in
$B_{ij}(x^{\prime },D_{x'})\cdot\gamma$ which are exactly
of the order $\sigma_{i} +t_{j}-l$ in
$B_{ijl}(x',D_{x'})$. We denote $B^{0}
\gamma=B^{0}(x',D_{x'})\gamma=(B_{ij}^{0}\cdot\gamma
)_{\substack{i=1,\dots ,m_{+}\\j=1,\dots ,N}}$.

We assume that
\begin{itemize}
\item[(E2)]  For any $\xi'\in\mathbb{R}^{n-1},|\xi'|=1$, the problem
\begin{gather*}
L^{0}(t,0;\xi',D_{t})v(t)=0\\
B^{0}(0,\xi')\gamma v=0
\end{gather*}
has only the trivial solution $v=0$ in
$\prod_{j=1}^{N}W^{t_{j} ,p}(\mathbb{R}_{+})$.
\end{itemize}

The first part of our main result is the following theorem.

\begin{theorem} \label{141}
Let $s$ and $\lambda$ be two nonnegative real numbers
and $1\leq p<+\infty$. Under hypotheses (E1) and (E2),
for any compact set $K$ of
$\overline{\mathbb{R}_{+}^{n}}$, there is a constant $C_{K}>0$,
such that for any
$u\in\prod_{j=1}^{N}\mathcal{L}^{p,\lambda,s+t_{j}
}(\mathbb{R}_{+}^{n})$ with $\mathop{\rm supp}u\subset K$, we get
\begin{align*}
 \|u\|_{\prod_{j=1}^{N}\mathcal{L}^{p,\lambda,s+t_{j}
}(\mathbb{R}_{+}^{n})}
& \leq  C\big\{  \|Lu\|_{\prod_{i=1}^{N}\mathcal{L}^{p,\lambda,
s-s_{i}}(\mathbb{R}_{+}^{n})}\\
&\quad +\|B\gamma u\|_{\prod_{i=1}^{m_{+}}
\mathcal{L}^{p,\lambda ,s-\sigma_{i}-\frac{1}{p}}(\mathbb{R}^{n-1})}
+ \|u\|_{\prod_{j=1}^{N}\mathcal{L}
^{p,\lambda,s+t_{j}-1}(\mathbb{R}_{+}^{n})}\big\}
\end{align*}
This statement remains true if we replace $\mathcal{L}$ by the dotted space
$\dot {\mathcal{L}}$.
\end{theorem}

If $\lambda=n$ we get the estimates in the spaces of
BMO-Triebel-Lizorkin $F_{\infty,p}^{s}$ and
$\dot {F}_{\infty,p}^{s}$. If $0\leq\lambda<n+2$ and $p=2$
we get the estimates in Morrey-Campanato spaces
$\mathcal{\ L}^{2,\lambda}$.

Now, let $\Omega$ be a $C^{\infty}-$bounded open set of $\mathbb{R}^{n}$,
$\Gamma$ its boundary. Let $L$ be the differential system defined by
(\ref{sys}) and (\ref{sys1}), where the coefficients $a_{\alpha}^{ij}$ are
belonging to $C^{\infty}(\overline{\Omega})$.

We assume the following hypotheses:
\begin{itemize}
\item[(H1)] For any $x\in\Gamma$ and any
$\xi\in\mathbb{R}^{n}\setminus\{0\}$,
$\det(L_{ij}^{0}(x;\xi))_{i,j=1,\dots ,N}\neq0$
and for any $\xi_{x}\in\mathbb{R}^{n}\setminus\{0\}$ tangent to
$\Gamma$ at $x$, the number $m_{+}(x,\xi)$ of the roots with positive
imaginary parts of the polynomial
\[
P(\tau)=\det(L_{ij}^{0}(x;\xi_{x}+\tau\nu_{x}))_{i,j=1,\dots ,N}
\]
in the complex variable $\tau$, is constant and equals $m_{+}$.
Here $\nu_{x}$ is the inward unit normal vector to the boundary $\Gamma$
at $x$.
\end{itemize}
To give the complementing condition we define the traces. For each
$u\in\mathcal{L}^{p,\lambda,s+t_{j}}(\Omega)$ we consider $t_{j}$ traces of
$u$ defined by
\[
\gamma_{l}u= \frac{\partial^{l}u}{\partial\nu^{l}}\big| _{\Gamma}
\quad \text{for }l=0,\dots ,t_{j}-1.
\]
The operator $\gamma=(\gamma_{0},\dots ,\gamma_{t_{j}-1})$ is a continuous
operator from $\mathcal{L}^{p,\lambda,s+t_{j}}(\Omega)$ to
$\prod_{l=0}^{t_{j}-1}\mathcal{L}^{p,\lambda,s+t_{j}-l-1/p}(\Gamma)$.
Let $\sigma_{i}$ be an integer $\leq-1$. We define the following differential
operators on $\Gamma$,
\[
(B_{ij}(x,D)\cdot\gamma)u=\sum_{l=0}^{t_{j}-1}B_{ijl}(x,D)\gamma_{l}u
\]
where $B_{ijl}(x,D)$ is a differential operator on $\Gamma$ with smooth
coefficients on $\Gamma$, of degree $\leq\sigma_{i}+t_{j}-l$.
If $\sigma_{i}+t_{j}-l<0$ we put $B_{ijl}=0$.

Denote $B\gamma=B(x,D)\gamma=(B_{ij}\cdot\gamma)_{i=1,\dots ,m_{+};
j=1,\dots ,N}$. This operator is continuous from
$\prod_{j=1}^{N}\mathcal{L}^{p,\lambda,s+t_{j}}(\Omega)$ into
$\prod_{i=1}^{m_{+}}\mathcal{L}^{p,\lambda,s-\sigma_{i}-1/p}(\Gamma)$. Let
$B_{ij}^{0}\cdot \gamma$ represent the principal part of
$B_{ij}\cdot\gamma$. We denote
$B^{0}\gamma=B^{0}(x;D_{x})\gamma=(B_{ij}^{0}\cdot\gamma
)_{i=1,\dots ,m_{+};j=1,\dots ,N}$.

Finally we suppose:
\begin{itemize}
\item[(H2)] For any $x\in\Gamma$ and any
$\xi_{x}\in\mathbb{R}^{n}\setminus\{0\},|\xi_{x}|=1$, tangent
to $\Gamma$ at $x$, the problem,
\begin{gather*}
L^{0}(x;\xi_{x}+\nu_{x}D_{t})v(t)=0\\
B^{0}(x;\xi_{x})\gamma v=0
\end{gather*}
has only the trivial solution $v=0$ in
$\prod_{j=1}^{N}W^{t_{j} ,p}(\mathbb{R}_{+})$.
\end{itemize}

\begin{theorem} \label{13}
Let $s$ and $\lambda$ be two nonnegative real numbers and
$1\leq p<+\infty$. Under hypotheses (H1) and (H2), there exists a
constant $C>0$ such that for any
$u\in\prod_{j=1}^{N}\mathcal{L}^{p,\lambda,s+t_{j}}(\Omega)$, we get
\begin{align*}
&\|u\|_{\prod_{j=1}^{N}\mathcal{L}^{p,\lambda,s+t_{j} }(\Omega)}\\
& \leq  C\big\{  \|Lu\|_{\prod_{i=1}^{N}\mathcal{L}^{p,
\lambda,s-s_{i}}(\Omega)}+\|B\gamma u\|_{\prod_{i=1}^{m_{+}}
\mathcal{L}^{p,\lambda,s-\sigma_{i}-\frac{1}{p}}(\Gamma)}
+ \|u\|_{\prod_{j=1}^{N}\mathcal{L}^{p,\lambda,s+t_{j}-1}(\Omega)}\big\}
\end{align*}
\end{theorem}

This theorem extends the scalar case work \cite{az}.
From this theorem and the interpolation theorem of Stampacchia, cf. \cite{st}
or \cite[Theorem 4.6]{g}, we get in the same manner as in \cite{az}, the
classical $L^{p}$ estimates for solutions of regular elliptic systems
\cite{adn}.

\section{A system of ordinary differential equations}

Let $L=(L_{ij})_{i,j=1,\dots ,N}$ be a system of ordinary differential
operators with constant coefficients defined on $\mathbb{R}_{+}$ by:
\[
L_{ij}u=L_{ij}(D_{t})u=\sum_{k=0}^{s_{i}+t_{j}}a_{k}^{ij}D_{t}^{k}u,
\quad \text{for }i,j=1,\dots ,N.
\]
where $a_{k}^{ij}\in\mathbb{C}$ and $u$ is a function of the variable
$t\in\mathbb{R}_{+}$.

In this section we are interested in the system $Lu(t)=f(t)$, where
\[
u=\begin{pmatrix} u_{1}\\ \vdots \\ u_{N} \end{pmatrix}\quad
\text{and}\quad
f=\begin{pmatrix} f_{1}\\ \vdots \\ f_{N} \end{pmatrix}
\]
are vectors of functions defined on $\mathbb{R}_{+}$. The operator
$L$ is bounded from the space $\prod_{j=1}^{N}W^{t_{j},p}(\mathbb{R}_{+})$
to $\prod_{i=1}^{N}W^{-s_{i},p}(\mathbb{R}_{+})$.
Let $L_{ij}^{0}$ be the principal part of $L_{ij}$. We assume that
the polynomial
\[
P(\tau)=\det(L_{ij}^{0}(\tau))_{i,j}
\]
is not vanishing on the real line $\mathbb{R}$. Let $m_{+}$ be the number of
the roots of $P(\tau)$ satisfying $\mathop{\rm Im}\tau>0$.

\begin{theorem} \label{ODS}
Under the above  assumption the operator
\[
L:\prod_{j=1}^{N}W^{t_{j},p}(\mathbb{R}_{+})\longrightarrow
\prod_{i=1}^{N}W^{-s_{i},p}(\mathbb{R}_{+})
\]
is a Fredholm operator and its index is equal to $m_{+}$.
\end{theorem}

The proof of this theorem is classical see for example \cite{adn, bc}.
We study $L$ on a neighborhood of $0$ and next on a neighborhood
of $+\infty$.

\section{Some preliminary results\label{sec3}}

In \cite{cd} we established  the connection between
$\mathcal{L}^{p,\lambda,s}$ spaces, BMO, Campanato spaces
$\mathcal{L}^{p,\lambda}$ and their local versions. Also, we
showed directly that for $p=2$ and $\lambda=n$ the space
$\mathcal{L}^{2,n,s}$ coincides with the space $F_{\infty,2}^{s}$
itself equals $I^{s}(bmo)$, where $I^{s}$ is the Riesz potential
operator. The following theorem shows that the general
BMO-Triebel-Lizorkin spaces $F_{\infty,p}^{s}$ are a particular
case of $\mathcal{L}^{p,\lambda,s}$ spaces. The definition of the
spaces $F_{\infty,p}^{s}$ and their homogeneous version
$\dot {F}_{\infty,p}^{s}$ are respectively given in
\cite[section 2.3.4]{tr} and \cite[section 5.1.4]{tr}.

\begin{theorem} \label{TL}
Let $s\in\mathbb{R}$ and $1\leq p<+\infty$.
The space
$F_{\infty,p}^{s}(\mathbb{R}^{n})$
[respectively $\dot{F}_{\infty,p}^{s}(\mathbb{R}^{n})]$ coincides
algebraically and topologically with the space
$\mathcal{L}^{p,n,s}(\mathbb{R}^{n})$
[respectively $\dot {\mathcal{L}}^{p,n,s}(\mathbb{R}^{n})]$.
\end{theorem}

In particular, for $p=2$ we have a result in \cite{cd}.

\begin{proof}
The proof is a consequence of some results of \cite{fj}. Firstly, we remark
that \cite[(5.1) and (5.2)]{fj} gives the homogeneous part of the theorem.
To show the inhomogeneous counterpart, we mention the following equivalent
norm for $F_{\infty,p}^{s}$ space, cf. \cite[(12.8)]{fj}
\begin{equation}
\|f\|_{F_{\infty,p}^{s}}\approx\big\{
 \sup_{J\geq0,B(2^{-J})} \frac{1}{|B|  }{\int_{B}}
\sum_{k\geq J}2^{kps}|\Delta_{k}f|  ^{p}dx\big\}  ^{1/p}
\label{fj1}
\end{equation}
the supremum is taken over all nonnegative integers $J$\ and over all cubes
$B$ of $\mathbb{R}^{n}$ of sidelength $2^{-J}$. This equivalence yields the
continuous embedding $\mathcal{L}^{p,n,s}(\mathbb{R}^{n})\hookrightarrow
F_{\infty,p}^{s}(\mathbb{R}^{n})$.

Conversely, let $f\in F_{\infty,p}^{s}(\mathbb{R}^{n})$. Let $B$ a cube of
$\mathbb{R}^{n}$ of sidelength $2^{-J}$, with $J$ a negative integer. We
divide the cube $B$ into $2^{-nJ}$ nonoverlapping cubes $Q_{i}$ of sidelength
equal to $1$. Thus
\begin{align*}
\frac{1}{|B|}{\int_{B}}\sum_{k\geq0=J^{+}}2^{kps}|\Delta_{k}f|  ^{p}dx
&  \leq 2^{nJ}\sum_{i}\frac{1}{|Q_{i}|  }{\int_{Q_{i}}}
\sum_{k\geq0}2^{kps}|\Delta_{k}f|  ^{p}dx\\
&  \leq\sup_{Q}\frac{1}{|Q|  }{\int_{Q}}
\sum_{k\geq0}2^{kps}|\Delta_{k}f|  ^{p}dx
\end{align*}
the supremum is taken over all cubes $Q$ of sidelength equal to
$1$. The last term is obviously bounded from above by the right
hand side of (\ref{fj1}). Hence
$f\in\mathcal{L}^{p,n,s}(\mathbb{R}^{n})$ and we have the
continuous embedding
$F_{\infty,p}^{s}(\mathbb{R}^{n})\hookrightarrow\mathcal{L}
^{p,n,s}(\mathbb{R}^{n})$.
The proof of Theorem \ref{TL} is complete.
\end{proof}

Let us denote by $\mathaccent"7017{F}_{\infty,p}^{s}$ (respectively
$cmo$) the closure of Schwartz class $\mathcal{S}(\mathbb{R}^{n})$
in BMO -Triebel-Lizorkin spaces $F_{\infty,p}^{s}$ (respectively
$bmo=F_{\infty ,2}^{0}$). The following result shed some light on
the topological dual of
$\mathaccent"7017{F}_{\infty,p}^{s}$.

\begin{corollary}
Let $s\in\mathbb{R},1\leq p<+\infty$, and $1<p'\leq+\infty$ with
$\frac{1}{p}+\frac{1}{p'}=1$. We have
\[
F_{1,1}^{-s}(\mathbb{R}^{n})\hookrightarrow(\mathaccent"7017{F}_{\infty,p}
^{s}(\mathbb{R}^{n}))'\hookrightarrow F_{p',p' }^{-s-\frac{n}{p}}
(\mathbb{R}^{n})
\]
in particular
\[
F_{1,1}^{0}(\mathbb{R}^{n})\hookrightarrow(cmo)^{\prime
}\hookrightarrow F_{2,2}^{-\frac{n}{2}}(\mathbb{R}^{n})\,.
\]
We have a similar result for the homogeneous spaces.
\end{corollary}

The proof of this corollary is a simple consequence of \cite[(3.3)]{e6} and
Theorem \ref{TL}.

Now we recall some lemmas needed in the proof of Theorem \ref{141}. The
following lemma is proved in \cite{e6}.

\begin{lemma}\label{sup}
Let $1\leq p<+\infty$, and $A<0$. If $(a_{j\nu})_{j,\nu}$ is a
sequence of positive real numbers satisfying $(a_{j\nu})_{j}\in l^{p}$ for any
$\nu\geq 1$, then there is a constant $C>0$ such that
\[
{\sum_{j\geq1}}{\sum_{\nu\geq1}}
2^{\nu A}a_{j\nu})^{p}\leq C\sup_{\nu\geq1}
{\sum_{j\geq1}}a_{j\nu}^{p}
\]
holds.
\end{lemma}

We introduce the following spaces needed in the proof.

\begin{definition} \label{def11a}\rm
We denote $W^{t_{j},p}(\mathbb{R})$ the classical Sobolev space of
all $u\in L^{p}(\mathbb{R})$ satisfying $D_{t}^{k}u\in L^{p}(
\mathbb{R})\;$for $1\leq k\leq t_{j}$.
By $W^{t_{j},p}(\mathbb{R};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))$
we denote the functions $u$ in
$L^{p}(\mathbb{R};\mathcal{L}^{p,\lambda,s+t_{j}}
(\mathbb{R}^{n-1}))$ satisfying
$D_{t}^{k}u\in L^{p}(\mathbb{R};\mathcal{L}^{p,\lambda,s+t_{j}-k}
(\mathbb{R}^{n-1}))$ for $k=1,\dots ,t_{j}$.
\end{definition}

\begin{remark} \label{rmk13} \rm
The most convenient norm of
$L^{p}(\mathbb{R};\mathcal{L}^{p,\lambda ,s}(\mathbb{R}^{n-1}))$ for
this purpose is
\[
\|u\|_{L^{p}(\mathbb{R};\mathcal{L}^{p,\lambda
,s}(\mathbb{R}^{n-1}))}=\{\sup_{J,B}\frac{1}{|B|
^{\frac{\lambda}{n-1}}}\sum_{k\geq J^{+}}2^{ksp}\|\Delta_{k}^{^{\prime
}}u\|_{L^{p}(\mathbb{R}\times B)}^{p}\}^{1/p}
\]
where the supremum is taken over all $J\in\mathbb{Z}$ and all balls $B$ of
$\mathbb{R}^{n-1}$ with radius $2^{-J}$.
\end{remark}

Here are some results proved in \cite{az} regarding intermediate derivatives,
compactness and interpolation.

\begin{lemma}\label{242}
There exists $C>0$ such that for any $\varepsilon>0$ and any
$u\in \mathcal{L}^{p,\lambda,s+t_{j}}(\mathbb{R}^{n})$
[respectively $W^{t_{j},p}(\mathbb{R};\mathcal{L}^{p,\lambda,s}(\mathbb{R}
^{n-1}))$]   we get for $k=0,\dots ,t_{j}-1$,
\[
\|D_{t}^{k}u\|_{\mathcal{L}^{p,\lambda,s+t_{j}-j}
(\mathbb{R}^{n})}\leq C\{\varepsilon\|D_{t}^{t_{j}}u\|
_{\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n})}+\varepsilon^{-\frac{k}{t_{j}-k}
}\|u\|
_{\mathcal{L}^{p,\lambda,s+t_{j}}(\mathbb{R}^{n})}\}
\]
[respectively
\begin{align*}
&\|D_{t}^{k}u\|_{L^{p}(\mathbb{R};\mathcal{L}^{p,\lambda,s+t_{j}-j}
(\mathbb{R}^{n-1}))}\\
& \leq  C\big\{  \varepsilon\|D_{t}^{t_{j}}u\|_{L^{p}(
\mathbb{R};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))}
+\varepsilon^{-\frac{k}{t_{j}-k}}\|u\|_{L^{p}(\mathbb{R};
\mathcal{L}^{p,\lambda,s+t_{j}}(\mathbb{R}^{n-1}))}\big\}.]
\end{align*}
We have the same result when we replace $\mathcal{L}$ by
$\dot {\mathcal{L}}$.
\end{lemma}

\begin{lemma} \label{243}
Let $s_{1}\leq s_{2}<s_{3}$ be three real numbers, $\lambda\geq0$ and
$1\leq p<+\infty$. There exists a constant $C>0$ such that for any
$\varepsilon >0$, and $u\in\mathcal{L}^{p,\lambda,s_{3}}(\mathbb{R}^{n})$
[respectively $L^{p}(\mathbb{R};\mathcal{L}^{p,\lambda,s_{3}}(\mathbb{R}
^{n-1}))$] we get
\[
 \|u\|_{\mathcal{L}^{p,\lambda,s_{2}}(\mathbb{R}^{n})}\leq C\big\{
\varepsilon\|u\|_{\mathcal{L}^{p,\lambda,s_{3}
}(\mathbb{R}^{n})}+\varepsilon^{-\frac{s_{2}-s_{1}}{s_{3}-s_{2}}}\|
u\|_{\mathcal{L}^{p,\lambda,s_{1}}(\mathbb{R}^{n})}\big\}
\]
[respectively
\begin{align*}
&\|u\|_{L^{p}(\mathbb{R};\mathcal{L} ^{p,\lambda,s_{2}}(\mathbb{R}^{n-1}))}\\
&\leq C\big\{  \varepsilon \|u\|_{L^{p}(\mathbb{R};\mathcal{L}^{p,\lambda,s_{3}
}(\mathbb{R}^{n-1}))}
+\varepsilon^{-\frac{s_{2}-s_{1}}{s_{3}-s_{2}}}\|
u\|_{L^{p}(
\mathbb{R};\mathcal{L}^{p,\lambda,s_{1}}(\mathbb{R} ^{n-1}))
}\big\}.  ]
\end{align*}
We have the same result if we replace $\mathcal{L}$ by
$\dot {\mathcal{L}}$.
\end{lemma}

\begin{lemma} \label{2310}
Let $\lambda\geq0$ and $1\leq p<+\infty$. Let $m$ be
an integer $\geq1$, and $s$ be a real $<m$. If $u\in L^{p}(
\mathbb{R} ;\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))$
is such that
$D_{t}^{m}u\in L^{p}(\mathbb{R};\mathcal{L}^{p,\lambda,s-m}(\mathbb{R}^{n-1}))$
then $u\in\mathcal{L}^{p,\lambda ,s}(\mathbb{R}^{n})$, and there is a
constant $C>0$ independent of $u$ such that
\[
\|u\|_{\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n})}
\leq C\big\{ \|D_{t}^{m}u\|_{L^{p}(\mathbb{R};\mathcal{L}
^{p,\lambda,s-m}(\mathbb{R}^{n-1}))
}+\|u\|_{L^{p}(\mathbb{R};\mathcal{L}^{p,\lambda,s}
(\mathbb{R}^{n-1}))}\big\}\,.
\]
This statement remains true if we replace $\mathcal{L}$ by
$\dot {\mathcal{L}}$ provided $-m<s<m$.
\end{lemma}

\begin{lemma} \label{244}
There exists $C_{0}>0$ such that for any
$\varphi\in\mathcal{S}(\mathbb{R}^{n})$, there exists $C_{1}>0$
satisfying for any
$u\in\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n})$
[respectively $L^{p}(\mathbb{R};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))$]
\[
\|\varphi u\|_{\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n})}\leq C_{0}\|
\varphi\|_{L^{\infty}(\mathbb{R}^{n})
}\|u\|_{\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n})}
+C_{1}\|u\|_{\mathcal{L}^{p,\lambda,s-1}(\mathbb{R}^{n})}
\]
[respectively
\begin{align*}
&\|\varphi u\|_{L^{p}(\mathbb{R};\mathcal{L}^{p,\lambda,s}
(\mathbb{R}^{n-1}))}\\
&\leq C_{0}\|\varphi\|_{L^{\infty}(
\mathbb{R}^{n})}\|u\|_{L^{p}(\mathbb{R};\mathcal{L}^{p,\lambda
,s}(\mathbb{R}^{n-1}))}+
C_{1}\|u\|_{L^{p}(\mathbb{R};\mathcal{L}^{p,\lambda
,s-1}(\mathbb{R}^{n-1}))}. ]
\end{align*}
This statement remains true if we replace $\mathcal{L}$ by
$\dot{\mathcal{L}}$.
\end{lemma}

The characterization of the traces for elements of the spaces involved in this
paper is given in the following theorem proved in \cite{az}.

\begin{theorem} \label{251}
Let $t_{j}$ be an integer
$\geq1,l\in\{0,1,\dots ,t_{j}-1\},s\in\mathbb{R} ,\lambda\geq0$ and
$1\leq p<+\infty$. For $u\in W_{loc}^{t_{j},p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))$,
the series
$\sum_{k\geq0}D_{t}^{l}\Delta_{k}'u(0,.)$ converges in
$\mathcal{S}'(\mathbb{R}^{n-1})$ and define an element $\gamma_{l}u$
belonging to the space
$\mathcal{L}^{p,\lambda,s+t_{j}-l-\frac{1}{p}}(\mathbb{R}^{n-1})$. Further,
the map $u\longmapsto\gamma_{l}u$ is a continuous operator from
$W_{loc}^{t_{j} ,p}(\mathbb{R}_{+};
\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))$
 to $\mathcal{L}^{p,\lambda,s+t_{j}-l-\frac{1}{p}}(\mathbb{R}^{n-1})$
and there exists an extension operator $R_{l}$ from the space
$\mathcal{L}^{p,\lambda ,s+t_{j}-l-\frac{1}{p}}(\mathbb{R}^{n-1})$
to the space
$W^{t_{j},p}(\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))$
such that
\[
\gamma_{l}\circ R_{l}=id_{\mathcal{L}^{p,\lambda,s+t_{j}-l-\frac{1}{p}
}(\mathbb{R}^{n-1})}.
\]
In particular, if $s\geq0$, then the operator $\gamma_{l}$ maps
$\mathcal{L} ^{p,\lambda,s+t_{j}}(\mathbb{R}_{+}^{n})$ into the space
$\mathcal{L}^{p,\lambda ,s+t_{j}-l-\frac{1}{p}}(\mathbb{R}^{n-1})$.
We have the same results if we replace $\mathcal{L}$ by
$\dot {\mathcal{L}}$.
\end{theorem}

\begin{remark} \label{rmk18} \rm
Let $s>0,\lambda=n$ and $l=0$.
In the case $p=2$, Strichartz \cite{str2} showed that the trace
$\gamma_{0}$ of functions in
$I^{s}(BMO)(=\dot {\mathcal{L}}^{2,n,s}(\mathbb{R}^{n})$ cf.
\cite{cd}) must be in the homogeneous H\"{o}lder space $C^{s}
(\mathbb{R}^{n})$. In addition, he proved that $\gamma_{0}$ is
surjective by showing that the extension operator
$R_{0}f(x)=\mathcal{F}^{-1}(e^{-t^{2} |\xi|
^{2}}\mathcal{F}f),x=(t,x')$, maps $C^{s}
(\mathbb{R}^{n})$ into $I^{s}(BMO)$. In the case $1\leq
p<+\infty$, Frazier and Jawerth \cite[Theorem 11.2]{fj}
generalized the last result by showing that the space of traces of
functions in $\dot {F}_{\infty,p}
^{s}(\mathbb{R}^{n})(=\dot {\mathcal{L}}^{p,n,s}(\mathbb{R}^{n})$
cf. Theorem \ref{TL} above) is independent of $p$ and coincides
with $C^{s}(\mathbb{R}^{n})$ as well.
\end{remark}

\section{Proof of Theorem \ref{141}}

The first step in the proof is the following statement.

\begin{proposition} \label{prop}
Let $s\in\mathbb{R},\lambda\geq0$ and $1\leq p<+\infty$.
Under hypotheses (E1) and (E2), for any compact
set $K$ of $\overline{\mathbb{R}_{+}^{n}}$, there is a constant
$C_{K}>0$ such that for any
$u\in\prod_{j=1}^{N}W^{t_{j},p}(\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s}
(\mathbb{R}^{n-1}))$ with $\mathop{\rm supp}u\subset K$, we get
\begin{align*}
\|u\|_{\prod_{j=1}^{N}W^{t_{j},p}(\mathbb{R} _{+};\mathcal{L}^{p,\lambda,s}
(\mathbb{R}^{n-1}))}
&\leq C_{K}\Big\{  \|Lu\|_{\prod_{i=1}^{N}W^{-s_{i},p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))}\\
&\quad + \|B\gamma u\|_{\prod_{i=1}^{m_{+}}
\mathcal{L}^{p,\lambda,s-\sigma_{i}-\frac{1}{p}}(\mathbb{R}^{n-1})}\\
&\quad +\|u\|_{\prod_{j=1}^{N}L^{p}(\mathbb{R}_{+};\mathcal{L}
^{p,\lambda,s+t_{j}-1}(\mathbb{R}^{n-1}))}\Big\}
\end{align*}
We have the same result if we replace $\mathcal{L}$ by
$\dot{\mathcal{L}}$.
\end{proposition}

\begin{proof}
It is classical that with the aid of Lemmas \ref{242}, \ref{243} and
\ref{244}, and with the freezing technique of the coefficients of $L$, we can
restrict ourselves to proving the last proposition for the following
homogeneous system of operators with constant coefficients
\[
L^{0}=L^{0}(D_{x'},D_{t})=(L_{ij}^{0}(0;D_{x'},D_{t}
))_{i,j=1\dots N}\;and\;B^{0}\gamma=B^{0}(0,D_{x'})\gamma
\]
where
\[
L_{ij}^{0}(0;D_{x'},D_{t})=\sum_{k+|
\alpha'| =s_{i}+t_{j}}a_{\alpha}^{ij}(
0)D_{x'}^{\alpha' }D_{t}^{k}.
\]
With the aid of Theorem \ref{ODS}, we prove as in
\cite{lm,bc,az} that under hypotheses (E1) and (E2),
 for every $\xi'\in\mathbb{R}^{n-1}\setminus\{0\}$, the operator
$(L^{0}(\xi',D_{t}),B^{0}(0,\xi ')\gamma)$ is invertible from
$\prod_{j=1}^{N} W^{t_{j},p}(\mathbb{R}_{+})$
to $\prod_{i=1} ^{N}W^{-s_{i},p}(\mathbb{R}_{+})\times\mathbb{C}^{m_{+}}$
and if $K_{\xi'}$ denotes its inverse, then the mapping
$\xi^{\prime }\longmapsto K_{\xi'}$ is $C^{\infty}$ from
$\mathbb{R}^{n-1} \setminus\{0\}$ to $\mathcal{L}(
\prod_{i=1}^{N}W^{-s_{i} ,p}(\mathbb{R}_{+})
\times\mathbb{C}^{m_{+}};\prod _{j=1}^{N}W^{t_{j},p}(
\mathbb{R}_{+}))$ and for any multi-index
$\alpha'$, there exists $C_{\alpha'}>0$ such that
for any $\xi'$, $\frac{1}{2}\leq|\xi'|\leq2$, and
any $(f,g)\in\prod_{i=1}^{N}W^{-s_{i},p}(\mathbb{R}_{+})
\times\mathbb{C}^{m_{+}}$,
\begin{equation}
\|D_{\xi'}^{\alpha'}K_{\xi'}(f,g)
\|_{\prod_{j=1}^{N}W^{t_{j},p}(\mathbb{R}_{+})
}\leq C_{\alpha'}\|(f,g)\|_{\prod
_{i=1}^{N}W^{-s_{i},p}(\mathbb{R}_{+})\times
\mathbb{C}^{m_{+}}}. \label{322}
\end{equation}

First of all, we will prove that for any integer $M\geq1$ sufficiently large,
there exists a constant $C>0$ such that for any ball $B$ of $\mathbb{R}^{n-1}$
of radius $2^{-J},J\in\mathbb{Z}$, centered at $x_{0}'\in
\mathbb{R}^{n-1}$,
\begin{equation}
\begin{aligned}
&\|u\|_{\prod_{j=1}^{N}L^{p}(B;W^{t_{j},p}(\mathbb{R}_{+}))} \\
&\leq C\Big\{\|L^{0}u\|_{\prod_{i=1}^{N}L^{p}(2B;W^{-s_{i},p}(
\mathbb{R}_{+}))}
 +\|B^{0}\gamma u\|_{\prod_{i=1}^{m_{+}}L^{p}(2B)} \\
&\quad + |B|^{1/p}{\sum_{\nu\geq-J+1}}
2^{-2\nu M}|F_{\nu}|^{1-\frac{1}{p}}\Big(\|
L^{0}u\|_{\prod_{i=1}^{N}L^{p}(F_{\nu};W^{-s_{i},p}(\mathbb{R} _{+}))}\\
&\quad +\|B^{0}\gamma u\|_{\prod_{i=1}^{m_{+} }L^{p}(F_{\nu})}\Big)\Big\}
\end{aligned}
\label{323}
\end{equation}
holds for any $u\in\prod_{j=1}^{N}\mathcal{S}(\mathbb{R} ^{n-1};W^{t_{j},p}
(\mathbb{R}_{+}))$ whose tangential spectrum (i.e. the
support of the tangential Fourier transform of $u$) belongs to the annulus
$\frac{1}{2}\leq|\xi'|\leq2$, here
$F_{\nu}=\{x'\in\mathbb{R}^{n-1};\;2^{\nu}\leq|x'-x_{0}'
|\leq2.2^{\nu}\}$.

For this, we apply the operator $(L^{0}(\xi'
,D_{t}),B(0,\xi')\gamma)$ to the relation
\[
\mathcal{F}u(.,\xi')=\int e^{-iy'\cdot \xi'}u(.,y')dy'
\]
to obtain the system
\begin{gather*}
L^{0}(\xi',D_{t}) \mathcal{F}u(.,\xi^{\prime })
= \mathcal{F}L^{0}u(.,\xi') =  \int
e^{-iy'\cdot\xi'}(L^{0}u)(.,y^{\prime })dy'\\
B^{0}(0,\xi')\gamma\mathcal{F}u(.,\xi')
= \mathcal{F}B^{0}\gamma u(\xi')
=  \int e^{-iy'\cdot\xi'}(B^{0}\gamma u)(y')dy'
\end{gather*}
Then we apply $K_{\xi'}$ to this system,
\[
\mathcal{F}u(.,\xi')=\int e^{-iy'\cdot
\xi'}K_{\xi'}(L^{0}u(.,y'),B^{0}\gamma u(y'))dy'
\]
Since $u(.,x')=\int e^{ix'\cdot\xi' }\Phi(\xi')
\mathcal{F}u(.,\xi')\frac{d\xi'}{(2\pi)^{n-1}}$,
$\Phi\in C_{0}^{\infty }(\mathbb{R}^{n-1})$ is equal to $1$
for $\frac{1}{2}\leq |\xi'|\leq2$ and its support belongs
to an annulus, we integrate by
parts with respect to $\xi'$, then
\begin{align*}
&u(.,x')\\
&=\iint\frac{e^{i(x'-y')\cdot
\xi'}}{(1+|x'-y'|^{2})^{M}}
(I-\Delta_{\xi'})^{M}\{\Phi(\xi')K_{\xi'}
(L^{0}u(.,y'),B^{0}\gamma u(y'))\}\frac{dy'
d\xi'}{(2\pi)^{n-1}}.
\end{align*}
Inequality (\ref{322}) yields
\begin{align*}
&\|u(.,x')\|_{\prod_{j=1}^{N}W^{t_{j},p}(\mathbb{R}_{+})}\\
&\leq C\int_{\mathbb{R}^{n-1}}\frac{1}{(1+|x'-y'
|^{2})^{M}}\|(L^{0}u(.,y'),B^{0}\gamma u(y')
)\|_{\prod _{i=1}^{N}W^{-s_{i},p}(
\mathbb{R}_{+})\times
\mathbb{C}^{m_{+}}}dy'\,.
\end{align*}
We integrate with respect to $x'\in B$,
\begin{align*}
&\|u\|_{L^{p}(B;\prod_{j=1}^{N}W^{t_{j},p}(\mathbb{R}_{+}))}\\
&\leq C\Big\{{\int_{x'\in B}}
\Big({\int_{y'\in\mathbb{R}^{n-1}}}\frac{1}{(1+|x'-y'|^{2})^{M}}\\
&\quad \times \|(L^{0}u(.,y'),B^{0}\gamma u(y'))\|
_{\prod_{i=1}^{N}W^{-s_{i},p}(\mathbb{R} _{+})
\times\mathbb{C}^{m_{+}}}dy'\Big)^{p}dx'\Big\}^{1/p}
\end{align*}
We decompose $\mathbb{R}^{n-1}=(2B)\cup\bigcup_{\nu
\geq-J+1}F_{\nu}$, where
$2B=\{  y';\,|y'-x_{0}'|\leq2.2^{-J}\}  $ and
$F_{\nu}=\{  y';\,2^{\nu}\leq|y'-x_{0}'|\leq2.2^{\nu}$, $\nu\geq-J+1$.
Thus
\begin{align*}
&\|u\|_{L^{p}(B;\prod_{j=1}^{N}W^{t_{j},p}(\mathbb{R}_{+}))}\\
&\leq C\Big\{  \int_{x'\in B}\Big(\int_{2B}\frac{1}{(1
 +|x'-y'|^{2})^{M}}\chi_{2B}(y')\\
&\quad \times \|(L^{0}u(.,y'),B^{0}\gamma u(y'))\|_{\prod_{i=1}^{N}
W^{-s_{i},p}(\mathbb{R}_{+})\times\mathbb{C}^{m_{+}}}dy'\Big)^{p}dx'
\Big\} ^{1/p} \\
&\quad +C\Big\{  \int_{x'\in B}\Big(\sum_{\nu\geq-J+1}\int_{F_{\nu
}}\frac{1}{(1+|x'-y'|^{2})^{M}}\\
&\quad\times \|(L^{0}u(.,y'),B^{0}\gamma u(y'))\|_{\prod_{i=1}^{N}
W^{-s_{i},p}(\mathbb{R}_{+})\times\mathbb{C}^{m_{+}}}dy'\Big)
^{p}dx'\Big\}  ^{1/p}
\end{align*}
The first term of the right hand side of the above inequality is
an $L^{p}-$ norm of a convolution product of a function of
$L^{1}(\mathbb{R} ^{n-1})$ (for $M$ large) and a
function of $L^{p}(\mathbb{R}^{n-1})$; on the other
hand, for the second term we remark that for $x'\in
B\;$and $y'\in F_{\nu},\nu\geq-J+1$, we have
$|x'-y'|\sim|x_{0}'-y'|\sim2^{\nu}$. Hence
\begin{align*}
&\|u\|_{L^{p}(B;\prod_{j=1}^{N}W^{t_{j},p}(\mathbb{R}_{+}))}\\
&\leq C\{\|L^{0}u\|_{L^{p}(2B;\prod_{i=1}^{N}W^{-s_{i},p}(
\mathbb{R}_{+}))}+\|B^{0}\gamma u\|
_{_{\prod_{i=1}^{m_{+}}L^{p}(2B)}}\}\\
&\quad +C|B|^{1/p}
{\sum_{\nu\geq-J+1}}2^{-2\nu M}{\int_{y'\in F_{\nu}}}
\|(L^{0}u(.,y'),B^{0}\gamma u(y'))\|_{\prod_{i=1}^{N}W^{-s_{i}
,p}(\mathbb{R}_{+})\times\mathbb{C}^{m_{+}}}dy'\\
&\leq C\{\|L^{0}u\|
_{L^{p}(2B;\prod_{i=1}^{N}W^{-s_{i} ,p}(
\mathbb{R}_{+}))}+\|B^{0}\gamma u\|
_{_{\prod_{i=1}^{m_{+}}L^{p}(2B)}}\}\\
&\quad +C|B|^{1/p}
 {\sum_{\nu\geq-J+1}}2^{-2\nu M}|F_{\nu}|^{1-\frac{1}{p}}\\
&\quad\times \Big({\int_{y'\in F_{\nu}}}\|(L^{0}u(.,y'),B^{0}\gamma
 u(y'))\|_{\prod_{i=1}^{N}W^{-s_{i} ,p}(\mathbb{R}_{+})
 \times\mathbb{C}^{m_{+}}}^{p}dy'\Big)^{1/p}.
\end{align*}
So that inequality (\ref{323}) is proved.
Let $u=\begin{pmatrix} u^{1}\\ \vdots \\ u^{N}\end{pmatrix}
\in\prod_{j=1}^{N}W^{t_{j},p}(\mathbb{R}_{+} ;\mathcal{L}^{p,\lambda,s}(
\mathbb{R}^{n-1}))$ with
$\mathop{\rm supp} u\subset K$, $K$ is
a compact set of $\overline{\mathbb{R}_{+}^{n}}$. For
$k\in\mathbb{N},$ we set
$u_{k}(x)=\Delta_{k}'u(2^{-k}x)$. If $k\geq1$, then
$u_{k}\in \prod_{j=1}^{N}\mathcal{S}(\mathbb{R}^{n-1};
W^{t_{j},p}(\mathbb{R}_{+}))$ and its tangential spectrum
(i.e. the support of its tangential Fourier transform)
belongs to the annulus $\{\frac{1}{2}\leq|\xi'|\leq2\}$. We have
\begin{equation}
(L_{ij}^{0}u^{j})_{k}=2^{k(s_{i}+t_{j})}L_{ij}^{0}u_{k} ^{j}\quad
\mbox{and}\quad (B_{ij}^{0}\gamma u^{j})_{k}=2^{k(\sigma
_{i}+t_{j})}B_{ij}^{0}\gamma u_{k}^{j}\,. \label{324}
\end{equation}
We apply inequality (\ref{323}) for each $u_{k},k\geq1$,
\begin{equation}
\begin{aligned}
&\|u_{k}\|_{\prod_{j=1}^{N}L^{p}(B;W^{t_{j} ,p}(\mathbb{R}_{+}))}\\
&\leq C\Big\{  \|L^{0} u_{k}\|_{\prod_{i=1}^{N}L^{p}(2B;W^{-s_{i},p}(
\mathbb{R}_{+}))}+\|B^{0}\gamma u_{k}\|
_{\prod_{i=1}^{m_{+}}L^{p}(2B)}\\
&\quad +  |B|^{1/p}{\sum_{\nu\geq-J+1}}
2^{-2\nu M}|F_{\nu}|^{1-\frac{1}{p}}\Big(\|L^{0}u_{k}\|
_{\prod_{i=1}^{N}L^{p}(F_{\nu};W^{-s_{i},p}(\mathbb{R} _{+}))}\\
&\quad +\|B^{0}\gamma u_{k}\|_{\prod_{i=1} ^{m_{+}}L^{p}(F_{\nu})}\Big)
\Big\}
\end{aligned}\label{325}
\end{equation}
The operator $\Delta_{k}'$ commutes with the derivation and then with
the constant coefficients operator $L^{0}$, so with the aid of
(\ref{324}) we have
\begin{align*}
\|u_{k}\|_{\prod_{j=1}^{N}L^{p}(B;W^{t_{j},p}(\mathbb{R}_{+}))}
&  ={\sum_{j=1}^{N}}\|u_{k}^{j}\|_{L^{p}(B;W^{t_{j},p}
(\mathbb{R}_{+}))}\\
&  = {\sum_{j=1}^{N}} {\sum_{r=0}^{t_{j}}}
\|D_{t}^{r}u_{k}^{j}\|_{L^{p}(B;L^{p}(\mathbb{R}_{+}))}\\
&  = {\sum_{j=1}^{N}} {\sum_{r=0}^{t_{j}}}
2^{kn/p}2^{-kr}\|\Delta_{k}'D_{t}^{r}u^{j}\|
_{L^{p}(\mathbb{R}_{+}\times2^{-k}B)}
\end{align*}
and
\begin{align*}
&\|L^{0}u_{k}\|_{\prod_{i=1}^{N}L^{p}(2B;W^{-s_{i},p}(\mathbb{R}_{+}))}\\
&={\sum_{i=1}^{N}}\|(L^{0}u_{k})^{i}\|_{L^{p}(2B;W^{-s_{i},p}(
\mathbb{R}_{+}))}\\
&={\sum_{i=1}^{N}} \|{\sum_{j=1}^{N}}
2^{-k(s_{i}+t_{j})}(L_{ij}^{0}u^{j})_{k}\|_{L^{p}(
2B;W^{-s_{i},p}(\mathbb{R}_{+}))}\\
&={\sum_{i=1}^{N}}{\sum_{r=0}^{-s_{i}}} 2^{kn/p}\|
{\sum_{j=1}^{N}}
2^{-k(s_{i}+t_{j}+r)}\Delta_{k}'D_{t}^{r}(L_{ij}^{0}u^{j})\|
_{L^{p}(\mathbb{R}_{+}\times2^{-k+1}B)}
\end{align*}
and finally
\begin{align*}
& \|B^{0}\gamma u_{k}\|_{\prod_{i=1}^{m_{+}}L^{p}(2B)}\\
& ={\sum_{i=1}^{m_{+}}}\|(B^{0}\gamma u_{k})^{i}\|_{L^{p}(2B)}\\
&={\sum_{i=1}^{m_{+}}}\|{\sum_{j=1}^{N}}
2^{-k(\sigma_{i}+t_{j})}(B_{ij}^{0}\gamma u^{j})_{k}\|_{L^{p}(2B)}\\
&={\sum_{i=1}^{m_{+}}}2^{k(n-1)/p}\|  {\sum_{j=1}^{N}}
2^{-k(\sigma_{i}+t_{j})}\Delta_{k}'B_{ij}^{0}\gamma u^{j}\|
_{L^{p}(2^{-k+1}B)}
\end{align*}
Substituting the above equalities in (\ref{325}) gives
\begin{align*}
&{\sum_{j=1}^{N}}{\sum_{r=0}^{t_{j}}}
2^{-kr}\|\Delta_{k}'D_{t}^{r}u^{j}\|_{L^{p}(
\mathbb{R}_{+}\times2^{-k}B)}\\
& \leq C\Big\{{\sum_{i=1}^{N}}
{\sum_{r=0}^{-s_{i}}}\| {\sum_{j=1}^{N}}
2^{-k(s_{i}+t_{j}+r)}\Delta_{k}'D_{t}^{r}L_{ij}^{0}u^{j}\|
_{L^{p}(\mathbb{R}_{+}\times2^{-k+1}B)}\\
&\quad + {\sum_{i=1}^{m_{+}}}
 \|{\sum_{j=1}^{N}}2^{-k(\sigma_{i}+t_{j}+1)}\Delta_{k}'B_{ij}^{0}\gamma u^{j}\|
 _{L^{p}(2^{-k+1}B)}\\
&\quad +|B|^{1/p}{\sum_{\nu\geq-J+1}}
 2^{-2\nu M}|F_{\nu}|^{1-\frac{1}{p}}
 \Big[{\sum_{i=1}^{N}} {\sum_{r=0}^{-s_{i}}}
 \| {\sum_{j=1}^{N}} 2^{-k(s_{i}+t_{j}+r)}\Delta_{k}'D_{t}^{r}L_{ij}^{0}u^{j}\|
 _{L^{p}(\mathbb{R}_{+}\times2^{-k}F_{\nu})}\\
&\quad + {\sum_{i=1}^{m_{+}}}
\|{\sum_{j=1}^{N}} 2^{-k(\sigma_{i}+t_{j}+1)}\Delta_{k}'B_{ij}^{0}
\gamma u^{j}\|_{L^{p}(2^{-k}F_{\nu})}]\Big\}
\end{align*}
Now we replace $u^{j}$ by $2^{k(s+t_{j})}u^{j}$ to get
\begin{align*}
&{\sum_{j=1}^{N}} {\sum_{r=0}^{t_{j}}}
2^{k(s+t_{j}-r)}\|\Delta_{k}'D_{t}^{r}u^{j}\|
_{L^{p}(\mathbb{R}_{+}\times2^{-k}B)}\\
&\leq C\Big\{{\sum_{i=1}^{N}} {\sum_{r=0}^{-s_{i}}}
2^{k(s-s_{i}-r)}\|{\sum_{j=1}^{N}}
\Delta_{k}'D_{t}^{r}L_{ij}^{0}u^{j}\|_{L^{p}(
\mathbb{R}_{+}\times2^{-k+1}B)}\\
&\quad + {\sum_{i=1}^{m_{+}}} 2^{k(s-\sigma_{i}-1/p)}\|
{\sum_{j=1}^{N}}
\Delta_{k}'B_{ij}^{0}\gamma u^{j}\|_{L^{p}(2^{-k+1}B)}\\
&\quad +|B|^{1/p}
{\sum_{\nu\geq-J+1}} 2^{-2\nu M}|F_{\nu}|^{1-\frac{1}{p}}
\Big[{\sum_{i=1}^{N}}{\sum_{r=0}^{-s_{i}}}
2^{k(s-s_{i}-r)}\| {\sum_{j=1}^{N}}
\Delta_{k}'D_{t}^{r}L_{ij}^{0}u^{j}\|_{L^{p}(
\mathbb{R}_{+}\times2^{-k}F_{\nu})}\\
&\quad + {\sum_{i=1}^{m_{+}}}
2^{k(s-\sigma_{i}-1/p)}\|{\sum_{j=1}^{N}}
\Delta_{k}'B_{ij}^{0}\gamma u^{j}\|_{L^{p}(
2^{-k} F_{\nu})}\Big]\Big\}
\end{align*}
Hence
\begin{align*}
&{\sum_{j=1}^{N}}{\sum_{r=0}^{t_{j}}}
2^{k(s+t_{j}-r)}\|\Delta_{k}'D_{t}^{r}u^{j}\|
_{L^{p}(\mathbb{R}_{+}\times2^{-k}B)}\\
&\leq C\Big\{{\sum_{i=1}^{N}}{\sum_{r=0}^{-s_{i}}}
2^{k(s-s_{i}-r)}\|\Delta_{k}'D_{t}^{r}(L^{0}u)^{i}\|
_{L^{p}(\mathbb{R}_{+}\times2^{-k+1}B)}\\
&\quad + {\sum_{i=1}^{m_{+}}}
2^{k(s-\sigma_{i}-1/p)}\|\Delta_{k}'(B^{0}\gamma u)^{i}
\|_{L^{p}(2^{-k+1}B)}\\
&\quad +|B|^{1/p} {\sum_{\nu\geq-J+1}}
2^{-2\nu M}|F_{\nu}|^{1-\frac{1}{p}}
\Big[{\sum_{i=1}^{N}}{\sum_{r=0}^{-s_{i}}}
2^{k(s-s_{i}-r)}\|\Delta_{k}'D_{t}^{r}(L^{0}u)^{i}\|
_{L^{p}(\mathbb{R}_{+}\times2^{-k}F_{\nu})}\\
&\quad + {\sum_{i=1}^{m_{+}}}
2^{k(s-\sigma_{i}-1/p)}\|\Delta_{k}'(B^{0}\gamma u)^{i}
\|_{L^{p}(2^{-k}F_{\nu})}\Big]\Big\}
\end{align*}
Set $K=J+k\in\mathbb{Z}$ and $\mu=\nu-k\in\mathbb{Z}$, then the
ball $2^{-k}B$ becomes the ball $B_{K}$ of $\mathbb{R}^{n-1}$ of
radius $2^{-K}$, the annulus $2^{-k}F_{\nu}$ becomes the annulus
$F_{\mu}$ of $\mathbb{R}^{n-1}$, and we
deduce
\begin{align*}
&{\sum_{j=1}^{N}} {\sum_{r=0}^{t_{j}}}
2^{k(s+t_{j}-r)p}\|\Delta_{k}'D_{t}^{r}u^{j}\|
_{L^{p}(\mathbb{R}_{+}\times B_{K})}^{p}\\
&\leq C\Big\{{\sum_{i=1}^{N}}{\sum_{r=0}^{-s_{i}}}
2^{k(s-s_{i}-r)p}\|\Delta_{k}'D_{t}^{r}(L^{0}u)^{i}\|
_{L^{p}(\mathbb{R}_{+}\times2B_{K})}^{p}\\
&+ {\sum_{i=1}^{m_{+}}}
2^{k(s-\sigma_{i}-1/p)p}\|\Delta_{k}'(B^{0}\gamma
u)^{i}\|_{L^{p}(2B_{K})}^{p}\\
&\quad +|2^{k}B_{K}|\Big({\sum_{\mu\geq-K+1}}
2^{(\mu+K)(-2M+(n-1)(1-\frac{1}{p}))}2^{\mu\frac{\lambda}{p}}\\
&\quad\times \Big[{\sum_{i=1}^{N}}
{\sum_{r=0}^{-s_{i}}}
\frac{2^{k(s-s_{i}-r)}}{|F_{\mu}|
^{\frac{\lambda}{(n-1)p}} }\|
\Delta_{k}'D_{t}^{r}(L^{0}u)^{i}\|_{L^{p}(
\mathbb{R}_{+}\times F_{\mu})}\\
&\quad +{\sum_{i=1}^{m_{+}}}
\frac{2^{k(s-\sigma_{i}-1/p)}}{|F_{\mu}|  ^{\frac{\lambda
}{(n-1)p}}}\|\Delta_{k}'(B^{0}\gamma u)^{i}\|
_{L^{p}(F_{\mu})}]\Big)^{p}\Big\}
\end{align*}
A simple calculation yields
\begin{align*}
&{\sum_{j=1}^{N}}{\sum_{r=0}^{t_{j}}}
2^{k(s+t_{j}-r)p}\|\Delta_{k}'D_{t}^{r}u^{j}\|
_{L^{p}(\mathbb{R}_{+}\times B_{K})}^{p}\\
&\leq C\Big\{{\sum_{i=1}^{N}}{\sum_{r=0}^{-s_{i}}}
2^{k(s-s_{i}-r)p}\|\Delta_{k}'D_{t}^{r}(L^{0}u)^{i}\|
_{L^{p}(\mathbb{R}_{+}\times2B_{K})}^{p}\\
&\quad + {\sum_{i=1}^{m_{+}}}
2^{k(s-\sigma_{i}-1/p)p}\|\Delta_{k}'(B^{0}\gamma
u)^{i}\|_{L^{p}(2B_{K})}^{p}\\
&\quad + 2^{(k-K)(-2N+n-1)p}2^{-K\lambda}(
{\sum_{\mu\geq1}}
2^{\mu(-2M+(n-1)(1-\frac{1}{p})+\frac{\lambda}{p})}\\
&\quad \Big[{\sum_{i=1}^{N}} {\sum_{r=0}^{-s_{i}}}
\frac{2^{k(s-s_{i}-r)}}{|F_{\mu-K}|
^{\frac{\lambda}{(n-1)p}} }\|
\Delta_{k}'D_{t}^{r}(L^{0}u)^{i}\|_{L^{p}(
\mathbb{R}_{+}\times F_{\mu-K})}\\
&\quad + {\sum_{i=1}^{m_{+}}}
\frac{2^{k(s-\sigma_{i}-1/p)}}{|F_{\mu-K}|  ^{\frac{\lambda
}{(n-1)p}}}\|\Delta_{k}'(B^{0}\gamma u)^{i}\|
_{L^{p}(F_{\mu-K})}])^{p}\Big\}
\end{align*}
Set $A_{M}=-2M+(n-1)(1-\frac{1}{p})+\frac{\lambda}{p}$.
Multiply by $1/|B_{K}|  ^{\frac{\lambda}{n-1}}$ and sum
over $j$, $k\geq \max(K^{+},1)$,
\begin{align*}
&\frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{k\geq\max(K^{+},1)}}{\sum_{j=1}^{N}}
{\sum_{r=0}^{t_{j}}}
2^{k(s+t_{j}-r)p}\|\Delta_{k}'D_{t}^{r}u^{j}\|
_{L^{p}(\mathbb{R}_{+}\times B_{K})}^{p}\\
&\leq C\Big\{  \frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{k\geq K^{+}}}{\sum_{i=1}^{N}}
{\sum_{r=0}^{-s_{i}}}
2^{k(s-s_{i}-r)p}\|\Delta_{k}'D_{t}^{r}(L^{0}u)^{i}\|
_{L^{p}(\mathbb{R}_{+}\times2B_{K})}^{p}\\
&\quad + \frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{k\geq K^{+}}}{\sum_{i=1}^{m_{+}}}
2^{k(s-\sigma_{i}-1/p)p}\|\Delta_{k}'(B^{0}\gamma
u)^{i}\|_{L^{p}(2B_{K})}^{p}\\
&\quad +
{\sum_{k\geq K^{+}}}\Big({\sum_{\mu\geq1}}
2^{\mu A_{M}}\Big[{\sum_{i=1}^{N}}
{\sum_{r=0}^{-s_{i}}}
\frac{2^{k(s-s_{i}-r)}}{|F_{\mu-K}|
^{\frac{\lambda}{(n-1)p}} }\|
\Delta_{k}'D_{t}^{r}(L^{0}u)^{i}\|_{L^{p}(
\mathbb{R}_{+}\times F_{\mu-K})}\\
&\quad +{\sum_{i=1}^{m_{+}}}
\frac{2^{k(s-\sigma_{i}-1/p)}}{|F_{\mu-K}|  ^{\frac{\lambda
}{(n-1)p}}}\|\Delta_{k}'(B^{0}\gamma u)^{i}\|
_{L^{p}(F_{\mu-K})}\Big]\Big)^{p}\Big\}
\end{align*}
Now we use Lemma \ref{sup} for the last sum
$k\geq K^{+}$,
\begin{align*}
&\frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{k\geq\max(K^{+},1)}}{\sum_{j=1}^{N}}
{\sum_{r=0}^{t_{j}}} 2^{k(s+t_{j}-r)p}\|\Delta_{k}'D_{t}^{r}u^{j}\|
_{L^{p}(\mathbb{R}_{+}\times B_{K})}^{p}\\
&\leq C\Big\{  \frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{k\geq K^{+}}}{\sum_{i=1}^{N}}
{\sum_{r=0}^{-s_{i}}}
2^{k(s-s_{i}-r)p}\|\Delta_{k}'D_{t}^{r}(L^{0}u)^{i}\|
_{L^{p}(\mathbb{R}_{+}\times2B_{K})}^{p}\\
&\quad + \frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{k\geq K^{+}}}{\sum_{i=1}^{m_{+}}}
2^{k(s-\sigma_{i}-1/p)p}\|\Delta_{k}'(B^{0}\gamma
u)^{i}\|_{L^{p}(2B_{K})}^{p}\\
&\quad +\sup_{\mu\geq1}{\sum_{k\geq K^{+}}}
\Big[{\sum_{i=1}^{N}}{\sum_{r=0}^{-s_{i}}}
\frac{2^{k(s-s_{i}-r)p}}{|F_{\mu-K}|
^{\frac{\lambda}{n-1}} }\|
\Delta_{k}'D_{t}^{r}(L^{0}u)^{i}\|_{L^{p}(
\mathbb{R}_{+}\times F_{\mu-K})}^{p}\\
&\quad +{\sum_{i=1}^{m_{+}}}
\frac{2^{k(s-\sigma_{i}-1/p)p}}{|F_{\mu-K}|  ^{\frac{\lambda
}{n-1}}}\|\Delta_{k}'(B^{0}\gamma u)^{i}\|
_{L^{p}(F_{\mu-K})}^{p}\Big]\Big\}
\end{align*}

On the left hand side of the above inequality we add the terms associated to
$k=0$, and since $F_{\mu-K}\subset B_{K-\mu-1}$ we deduce
\begin{align*}
&\frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{k\geq K^{+}}}{\sum_{j=1}^{N}}
{\sum_{r=0}^{t_{j}}}
2^{k(s+t_{j}-r)p}\|\Delta_{k}'D_{t}^{r}u^{j}\|
_{L^{p}(\mathbb{R}_{+}\times B_{K})}^{p}\\
&\leq C\Big\{\frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{k\geq K^{+}}}{\sum_{i=1}^{N}}
{\sum_{r=0}^{-s_{i}}}
2^{k(s-s_{i}-r)p}\|\Delta_{k}'D_{t}^{r}(L^{0}u)^{i}\|
_{L^{p}(\mathbb{R}_{+}\times2B_{K})}^{p}\\
&\quad + \frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{k\geq K^{+}}}{\sum_{i=1}^{m_{+}}}
2^{k(s-\sigma_{i}-1/p)p}\|\Delta_{k}'(B^{0}\gamma
u)^{i}\|_{L^{p}(2B_{K})}^{p}\\
&\quad +\sup_{\mu\geq1}{\sum_{k\geq(K-\mu-1)^{+}}}{\sum_{i=1}^{N}}
\Big[{\sum_{r=0}^{-s_{i}}}
\frac{2^{k(s-s_{i}-r)p}}{|F_{\mu-K}|
^{\frac{\lambda}{n-1}} }\|
\Delta_{k}'D_{t}^{r}(L^{0}u)^{i}\|_{L^{p}(
\mathbb{R}_{+}\times B_{K-\mu-1})}^{p}\\
&\quad +{\sum_{i=1}^{m_{+}}}
\frac{2^{k(s-\sigma_{i}-1/p)p}}{|F_{\mu-K}|  ^{\frac{\lambda
}{n-1}}}\|\Delta_{k}'(B^{0}\gamma u)^{i}\|
_{L^{p}(B_{K-\mu-1})}^{p}\Big]\Big\}+R_{0}^{K},
\end{align*}
where
\[
R_{0}^{K}=\frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{j=1}^{N}}{\sum_{r=0}^{t_{j}}}
\|\Delta_{0}'D_{t}^{r}u^{j}\|_{L^{p}(
\mathbb{R}_{+}\times B_{K})}^{p}\,.
\]
Taking the supremum over $K$ and $B_{K}$ yields
\begin{align*}
&{\sum_{j=1}^{N}} {\sum_{r=0}^{t_{j}}}
\|D_{t}^{r}u^{j}\|_{L^{p}(\mathbb{R}_{+};\mathcal{L}
^{p,\lambda,s+t_{j}-r}(\mathbb{R}^{n-1}))}^{p}\\
&\leq C\Big\{{\sum_{i=1}^{N}}{\sum_{r=0}^{-s_{i}}}
\|D_{t}^{r}(L^{0}u)^{i}\|_{L^{p}(\mathbb{R}
_{+};\mathcal{L}^{p,\lambda,s-s_{i}-r}(\mathbb{R}^{n-1}))}^{p}\\
&\quad +{\sum_{i=1}^{m_{+}}}
\|(B^{0}\gamma u)^{i}\|_{\mathcal{L}^{p,\lambda,s-\sigma
_{i}-1/p}(\mathbb{R}^{n-1})}^{p}\Big\}  +R_{0},
\end{align*}
where
\[
R_{0}=\sup_{K,B_{K}}\frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{j=1}^{N}}{\sum_{r=0}^{t_{j}}}
\|\Delta_{0}'D_{t}^{r}u^{j}\|_{L^{p}(
\mathbb{R}_{+}\times B_{K})}^{p}
\]
Finally
\begin{equation} \label{326}
\begin{aligned}
\|u\|_{\prod_{j=1}^{N}W^{t_{j},p}(\mathbb{R} _{+};
\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1})) }^{p}
& \leq C\Big\{\|L^{0}u\|_{\prod_{i=1}^{N}W^{-s_{i},p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))} ^{p}\\
&\quad + \|B^{0}\gamma u\|_{\prod_{i=1}^{m_{+}
}\mathcal{L}^{p,\lambda,s-\sigma_{i}-\frac{1}{p}}(\mathbb{R}^{n-1})}
^{p}\Big\}  +R_{0}
\end{aligned}
\end{equation}
To estimate from above the remainder term $R_{0}$, we write
\begin{equation}
R_{0}^{K}=\frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{j=1}^{N}}{\sum_{r=0}^{t_{j}-1}}
\|\Delta_{0}'D_{t}^{r}u^{j}\|_{L^{p}(
\mathbb{R}_{+}\times B_{K})}^{p}+\frac{1}{|B_{K}|
^{\frac{\lambda}{n-1}}}{\sum_{j=1}^{N}}
\|\Delta_{0}'D_{t}^{t_{j}}u^{j}\|_{L^{p}(
\mathbb{R}_{+}\times B_{K})}^{p} \label{3211}
\end{equation}
For the first term in $R_{0}^{K}$, we use Lemma \ref{242} to get a
constant $C>0$ such that for any $\varepsilon>0$,
\begin{equation}
\begin{aligned}
&\frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{j=1}^{N}}{\sum_{r=0}^{t_{j}-1}}
\|\Delta_{0}'D_{t}^{r}u^{j}\|_{L^{p}(
\mathbb{R}_{+}\times B_{K})}^{p}\\
&\leq {\sum_{j=1}^{N}} {\sum_{r=0}^{t_{j}-1}}
\|D_{t}^{r}u^{j}\|_{L^{p}(\mathbb{R}_{+};\mathcal{L}
^{p,\lambda,s+t_{j}-r-1}(\mathbb{R}^{n-1}))}^{p}\\
&\leq C {\sum_{j=1}^{N}}
\Big\{  \varepsilon^{p}\|D_{t}^{t_{j}}u^{j}\|_{L^{p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))}^{p}+
{\sum_{r=0}^{t_{j}-1}}
\varepsilon^{-\frac{rp}{t_{j}-r}}\|u^{j}\|_{L^{p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s+t_{j}-1}(\mathbb{R}^{n-1}))
}^{p}\Big\} \\
&\leq C {\sum_{j=1}^{N}} \Big\{  \varepsilon^{p}\|u^{j}\|
_{W^{t_{j},p}(\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))
} ^{p}+C_{\varepsilon}'\|u^{j}\|_{L^{p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s+t_{j}-1}(\mathbb{R}^{n-1}))
}^{p}\Big\}.
\end{aligned} \label{327}
\end{equation}
To estimate the second term of $R_{0}^{K}$ we  return to the
equation $L^{0}u=f$. For each $i=1,\dots ,N$,
${\sum_{j=1}^{N}}L_{ij}^{0}u^{j}=f^{i}$, so
${\sum_{j=1}^{N}}{\sum_{r+|\alpha'|  =s_{i}+t_{j}}}
a_{r,\alpha'}^{ij}(0)D_{x'}^{\alpha'}D_{t}^{r}u^{j}=f^{i}$,
here $f^{i}=(L^{0}u)^{i}$. Thus
\[
{\sum_{j=1}^{N}}
a_{s_{i}+t_{j},\alpha'}^{ij}(0)D_{t}^{s_{i}+t_{j}}
u^{j}=f^{i}-{\sum_{j=1}^{N}}
{\sum_{\substack{r+|\alpha'|
=s_{i}+t_{j}\\0\leq r\leq s_{i}+t_{j}-1}}}
a_{r,\alpha'}^{ij}(0)D_{x'}^{\alpha'
}D_{t}^{r}u^{j}
\]
Applying $D_{t}^{-s_{i}}$ to the both sides gives
\begin{equation}
{\sum_{j=1}^{N}}
a_{s_{i}+t_{j},\alpha'}^{ij}(0)D_{t}^{t_{j}}u^{j}
=D_{t}^{-s_{i}}f^{i}-{\sum_{j=1}^{N}}
{\sum_{\substack{r'+|\alpha'  =t_{j}\\-s_{i}\leq r'\leq t_{j}-1}}}
a_{r'+s_{i},\alpha'}^{ij}(0)D_{x'
}^{\alpha'}D_{t}^{r'}u^{j} \label{328}
\end{equation}

The ellipticity condition gives that the constant matrix
$A=(a_{s_{i} +t_{j},\alpha'}^{ij}(0))_{i,j}$ is invertible.
Let us denote $D_{t}^{T}u$ the vector
$(D_{t}^{t_{j}}u^{j})_{j}$, $D_{t}^{-S}f$ the vector
$(D_{t}^{-s_{i}}f^{i})_{i}$ and $V$ the vector $(v^{i})_{i}$ where
$v^{i}={\sum_{j=1}^{N}}
{\sum_{\substack{r'+|\alpha^{\prime
}|  =t_{j}\\-s_{i}\leq r'\leq t_{j}-1}}}
a_{r'+s_{i},\alpha'}^{ij}(0)D_{x'
}^{\alpha'}D_{t}^{r'}u^{j}$. From (\ref{328}) we obtain
\begin{equation}
D_{t}^{T}u=A^{-1}D_{t}^{-S}f-A^{-1}V \label{3214}
\end{equation}
and then
\[
\Delta_{0}'D_{t}^{T}u=A^{-1}\Delta_{0}'D_{t}^{-S}
f-A^{-1}\Delta_{0}'V
\]
For the second term of $R_{0}^{K}$, we write
\begin{equation}
\begin{aligned}
&\frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}
{\sum_{j=1}^{N}}\|\Delta_{0}'D_{t}^{t_{j}}u^{j}\|_{L^{p}(
\mathbb{R}_{+}\times B_{K})}^{p}\\
&=\frac{1}{|B_{K}|^{\frac{\lambda}{n-1}}}\|\Delta_{0}'D_{t}^{T}u\|
_{\prod_{j=1}^{N}L^{p}(\mathbb{R}_{+}\times B_{K})}^{p}\\
&\leq C\{\frac{1}{|B_{K}|  ^{\frac{\lambda}{n-1}}}\|\Delta
_{0}'D_{t}^{-S}f\|_{\prod_{i=1}^{N}L^{p}(
\mathbb{R}_{+}\times B_{K})}^{p}+\frac{1}{|B_{K}|
^{\frac{\lambda}{n-1}}}\|\Delta_{0}'V\|_{\prod
_{j=1}^{N}L^{p}(\mathbb{R}_{+}\times B_{K})}^{p}\}\\
&\leq C\{\|D_{t}^{-S}f\|_{\prod_{i=1}^{N}L^{p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))}
^{p}+\|V\|_{\prod_{j=1}^{N}L^{p}(\mathbb{R}
_{+};\mathcal{L}^{p,\lambda,s-1}(\mathbb{R}^{n-1}))}^{p}\}\\
&\leq C\{\|f\|_{\prod_{i=1}^{N}W^{-s_{i},p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))
} ^{p}+\|V\|_{\prod_{j=1}^{N}L^{p}(\mathbb{R}_{+};
\mathcal{L}^{p,\lambda,s-1}(\mathbb{R}^{n-1}))}^{p}\}
\end{aligned} \label{329}
\end{equation}
Now
\begin{align*}
\|V\|_{\prod_{j=1}^{N}L^{p}(\mathbb{R}
_{+};\mathcal{L}^{p,\lambda,s-1}(\mathbb{R}^{n-1}))}^{p}
&  \leq C {\sum_{j=1}^{N}}{\sum_{\substack{r'+|\alpha'|
=t_{j}\\0\leq r'\leq t_{j}-1}}}
\|D_{x'}^{\alpha'}D_{t}^{r'}u^{j}\|
_{L^{p}(\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s-1}(\mathbb{R}
^{n-1}))}^{p}\\
&  \leq C {\sum_{j=1}^{N}}{\sum_{0\leq r'\leq t_{j}-1}}
\|D_{t}^{r'}u^{j}\|_{L^{p}(\mathbb{R}
_{+};\mathcal{L}^{p,\lambda,s+t_{j}-r'-1}(\mathbb{R}^{n-1}))}^{p}
\end{align*}
In the same way as for (\ref{327}) there is a constant $C>0$
such that for any $\varepsilon>0$,
\begin{equation}
\begin{aligned}
&\|V\|_{\prod_{j=1}^{N}L^{p}(\mathbb{R}_{+};
\mathcal{L}^{p,\lambda,s-1}(\mathbb{R}^{n-1}))}^{p}\\
&\leq C {\sum_{j=1}^{N}}
\Big\{  \varepsilon^{p}\|D_{t}^{t_{j}}u^{j}\|_{L^{p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))}^{p}+
{\sum_{r'=0}^{t_{j}-1}}
\varepsilon^{-\frac{r'p}{t_{j}-r}}\|u^{j}\|
_{L^{p}(\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s+t_{j}-1}(\mathbb{R}
^{n-1}))}^{p}\Big\} \\
&\leq C {\sum_{j=1}^{N}}
\Big\{  \varepsilon^{p}\|u^{j}\|_{W^{t_{j},p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))
} ^{p}+C_{\varepsilon}'\|u^{j}\|_{L^{p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s+t_{j}-1}(\mathbb{R}^{n-1}))
}^{p}\Big\}\,.
\end{aligned} \label{3210}
\end{equation}
Finally (\ref{3211})--(\ref{3210}) give
\begin{align*}
R_{0} &  \leq C\{\|f\|_{\prod_{i=1}^{N}W^{-s_{i} ,p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))
}^{p}+\varepsilon^{p}\|u\|_{\prod_{j=1}^{N}W^{t_{j} ,p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))}^{p}\\
&\quad + C_{\varepsilon}'\|u\|_{\prod_{j=1}^{N}L^{p}(
\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s+t_{j}-1}(\mathbb{R}^{n-1}))
}^{p}\}
\end{align*}
Substituting the above inequality in (\ref{326}) and choosing $\varepsilon>0$
arbitrarily small we get Proposition \ref{prop} for the system
$(L^{0},B^{0}\gamma)$.
\end{proof}

To complete the proof of Theorem \ref{141}, we have to estimate the normal
derivatives of the solution.

\begin{lemma} \label{normal}
Let $s$ and $\lambda$ be two real numbers $\geq0$ and $1\leq p<+\infty$.
For any compact set $K$ of $\overline{\mathbb{R}_{+}^{n}}$, there
exists a constant $C_{K}>0$ such that for any
$\,u\in\prod_{j=1}^{N}
\mathcal{L}^{p,\lambda,s+t_{j}}(\mathbb{R}_{+}^{n})$ with
$\mathop{\rm supp}u\subset K$ we
get
\begin{equation}
\|u\|_{\prod_{j=1}^{N}\mathcal{L}^{p,\lambda,s+t_{j}
}(\mathbb{R}_{+}^{n})}
\leq  C_{K}\{\|Lu\|_{\prod_{i=1}^{N}\mathcal{L}^{p,\lambda,s-s_{i}}
(\mathbb{R}_{+}^{n})}+\|u\|_{\prod_{j=1}^{N}W^{t_{j},p}(\mathbb{R}
_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))}\}
\label{331}
\end{equation}
\end{lemma}

\begin{proof}
As in the proof of Proposition \ref{prop} we can restrict ourselves to the
operator $L^{0}$.
Firstly let us take $0\leq s<1$. We have
\[
\|u^{j}\|_{\mathcal{L}^{p,\lambda,s+t_{j}}(\mathbb{R}_{+}^{n}
)}=\|D_{t}^{t_{j}}u^{j}\|_{\mathcal{L}^{p,\lambda,s}
(\mathbb{R}_{+}^{n})}+{\sum_{\substack{k+|\alpha'|
\leq t_{j}\\0\leq k\leq t_{j}-1}}}
\|D_{x'}^{\alpha'}D_{t}^{k}u^{j}\|
_{\mathcal{L}^{p,\lambda,s}(\mathbb{R}_{+}^{n})}
\]
The interpolation lemma \ref{2310} gives for
$k+|\alpha'|\leq t_{j}$, $0\leq k\leq t_{j}-1$,
\begin{equation}
\begin{aligned}
&\| D_{x'}^{\alpha'}D_{t}^{k}u^{j}\|_{\mathcal{L}^{p,
\lambda,s}(\mathbb{R}_{+}^{n})}\\
&\leq C\Big\{  \|D_{x'}^{\alpha'}D_{t}^{k}u^{j}\|
_{L^{p}(\mathbb{R}_{+;}\mathcal{L}^{p,\lambda,s}(\mathbb{R}
^{n-1}))}+\|D_{x'}^{\alpha'}D_{t}^{k+1} u^{j}\|
_{L^{p}(\mathbb{R}_{+;}\mathcal{L}^{p,\lambda
,s-1}(\mathbb{R}^{n-1}))}\Big\} \\
& \leq C\Big\{  \|D_{t}^{k}u^{j}\|_{L^{p}(\mathbb{R}
_{+;}\mathcal{L}^{p,\lambda,s+t_{j}-k}(\mathbb{R}^{n-1}))
}+\|D_{t}^{k+1}u^{j}\|_{L^{p}(\mathbb{R}_{+;}\mathcal{L}
^{p,\lambda,s+t_{j}-k-1}(\mathbb{R}^{n-1}))}\Big\} \\
& \leq C\|u^{j}\|_{W^{t_{j},p}(\mathbb{R}_{+;}\mathcal{L}
^{p,\lambda,s}(\mathbb{R}^{n-1}))}
\end{aligned} \label{332}
\end{equation}
To estimate $\|D_{t}^{t_{j}}u^{j}\|_{\mathcal{L}^{p,\lambda
,s}(\mathbb{R}_{+}^{n})}$, we return again to the equation $L^{0}u=f$,
(\ref{3214}), to get
\[
D_{t}^{T}u=A^{-1}D_{t}^{-S}f-A^{-1}V
\]
and then with the aid of (\ref{332}),
\begin{equation}
\begin{aligned}
&\|D_{t}^{T}u\|_{\prod_{j=1}^{N}\mathcal{L}^{p,\lambda
,s}(\mathbb{R}_{+}^{n})}\\
&={\sum_{j=1}^{N}}
\|D_{t}^{t_{j}}u^{j}\|_{\mathcal{L}^{p,\lambda,s}
(\mathbb{R}_{+}^{n})}\\
&\leq C\Big \{{\sum_{i=1}^{N}}
\|D_{t}^{-s_{i}}(L^{0}u)^{i}\|_{\mathcal{L}^{p,\lambda
,s}(\mathbb{R}_{+}^{n})}+
{\sum_{\substack{k+|\alpha'|
=t_{j}\\0\leq k\leq t_{j}-1}}}
\|D_{x'}^{\alpha'}D_{t}^{k}u^{j}\|
_{\mathcal{L}^{p,\lambda,s}(\mathbb{R}_{+}^{n})}\Big\}\\
& \leq C\Big\{ {\sum_{i=1}^{N}}
\|(L^{0}u)^{i}\|_{\mathcal{L}^{p,\lambda,s-s_{i}}
(\mathbb{R}_{+}^{n})}+{\sum_{j=1}^{N}}
\|u^{j}\|_{W^{t_{j},p}(
\mathbb{R}_{+;}\mathcal{L} ^{p,\lambda,s}(\mathbb{R}^{n-1}))
}\Big\}
\end{aligned} \label{3300}
\end{equation}

The lemma is proved for $0\leq s<1$. For the general case
$s\geq0$, we write $s=q+r$ with $q\in\mathbb{N}$ and $0\leq r<1$,
and we do an induction on $q.$This is true for the case $q=0$.
Assuming that the estimation (\ref{331}) is true for any $q$, we
show that it holds for $q+1$. Let $K$ be a compact set of
$\overline{\mathbb{R}_{+}^{n}}$ and $u\in\prod_{j=1}^{N}
\mathcal{L}^{p,\lambda,s+1+t_{j}}(\mathbb{R}_{+}^{n})$ with
$\mathop{\rm supp}u\subset K$.

We remark that $u^{j}\in\mathcal{L}^{p,\lambda,s+1+t_{j}}(\mathbb{R}_{+}^{n})$
if and only if $u^{j}\in\mathcal{L}^{p,\lambda,s+t_{j}}(\mathbb{R}_{+}^{n})$
and $D_{x_{k}}u^{j}\in\mathcal{L}^{p,\lambda,s+t_{j}}(\mathbb{R}_{+}^{n})$ for
any $1\leq k\leq n-1$, and $D_{t}^{t_{j}}u^{j}\in\mathcal{L}^{p,\lambda
,s+1}(\mathbb{R}_{+}^{n})$. Moreover we have
\begin{equation}
\begin{aligned}
&\|u\|_{\prod_{j=1}^{N}\mathcal{L}^{p,\lambda,s+1+t_{j}
}(\mathbb{R}_{+}^{n})} \\
&\leq C\Big\{  \|u\|_{\prod_{j=1}^{N}\mathcal{L}^{p,\lambda,
s+t_{j}}(\mathbb{R}_{+}^{n})}\\
&\quad + \sum_{k=1}^{n-1}\|D_{x_{k}}u\|
_{\prod_{j=1}^{N}\mathcal{L}^{p,\lambda,s+t_{j}
}(\mathbb{R}_{+}^{n})}+\|D_{t}^{T}u\|_{\prod_{j=1}
^{N}\mathcal{L}^{p,\lambda,s+1}(\mathbb{R}_{+}^{n})}\Big\}
\end{aligned} \label{fond1}
\end{equation}
where $D_{t}^{T}u$ is defined in (\ref{3214}).
By the induction hypothesis,
\begin{equation}
\|u\|_{\prod_{j=1}^{N}\mathcal{L}^{p,\lambda,s+t_{j}
}(\mathbb{R}_{+}^{n})}
 \leq  C_{K}\{\|L^{0}u\|_{\prod_{i=1}^{N}\mathcal{L}^{p,\lambda,s-s_{i}}
(\mathbb{R}_{+}^{n})}+\|u\|_{\prod_{j=1}^{N}W^{t_{j},p}(\mathbb{R}
_{+};\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n-1}))}\}
\label{334}
\end{equation}
We substitute in (\ref{334}) $u$ by $D_{x_{k}}u$, $1\leq k\leq n-1$, and the
coefficients of $L^{0}$ are constant, so
\begin{equation}
\begin{aligned}
&\|D_{x_{k}}u\|_{\prod_{j=1}^{N}\mathcal{L}^{p,\lambda,s+t_{j}}
(\mathbb{R}_{+}^{n})}\\
&\leq C_{K}\Big\{\|L^{0}u\|_{\prod_{i=1}^{N}\mathcal{L}^{p,\lambda,s-s_{i}
+1}(\mathbb{R}_{+}^{n})}+\|u\|_{\prod_{j=1}^{N}
W^{t_{j},p}(\mathbb{R}_{+};\mathcal{L}^{p,\lambda,s+1}(\mathbb{R}
^{n-1}))}\Big\}
\end{aligned} \label{335}
\end{equation}
In the same way as for (\ref{3300}), with $s+1$ instead of $s$,
 we return to the equation $L^{0}u=f$ to get
\begin{equation}
\begin{aligned}
&\|D_{t}^{T}u\|_{\prod_{j=1}^{N}\mathcal{L}^{p,\lambda,s+1}
(\mathbb{R}_{+}^{n})}\\
&\leq C\Big\{{\sum_{i=1}^{N}}\|(L^{0}u)^{i}\|_{\mathcal{L}
^{p,\lambda,s-s_{i}+1}(\mathbb{R}_{+}^{n})}+{\sum_{j=1}^{N}}
\|u^{j}\|_{W^{t_{j},p}(\mathbb{R}_{+;}\mathcal{L}
^{p,\lambda,s+1}(\mathbb{R}^{n-1}))}\}
\end{aligned}\label{3360}
\end{equation}
Finally we substitute inequalities (\ref{334}), (\ref{335}) and (\ref{3360})
in (\ref{fond1}) to get (\ref{331}) for $s+1=q+1+r$.
\end{proof}

Note that Theorem \ref{141} is a consequence of
Proposition \ref{prop} and Lemma \ref{normal}.


\subsection*{Acknowledgements}
the author would like to thank the anonymous referee for his/her
 valuable comments regarding elliptic systems.

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\end{document}