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Elliptic systems, anisotropic Roumieu classes
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\documentclass{article}
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%TCIDATA{Version=5.50.0.2890}
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%TCIDATA{BibliographyScheme=Manual}
%TCIDATA{Created=Monday, October 04, 2010 10:28:53}
%TCIDATA{LastRevised=Monday, October 04, 2010 10:34:18}
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%TCIDATA{CSTFile=40 LaTeX article.cst}
\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\begin{document}
\title{Systems of differential operators in anisotropic Roumieu classes}
\author{R. CHAILI \\
%EndAName
U.S.T.M.B. ORAN,\ ALGERIE}
\date{}
\maketitle
\begin{abstract}
The aim of this work is to show that he ellipticity is a
sufficient condition for the elliptic iterate property to hold for
systems of linear differential operators in the anisotropic
Roumieu classes
\end{abstract}
\section{Introduction}
In this work we show that the ellipticity is a sufficient condition for the
elliptic iterate property to hold for systems of linear differential
operators in the anisotropic classes $\left\{ M_{p}\right\} $ of functions
(also called Roumieu classes). The obtained result is an extension of the
results of Nelson \cite{N}, Kotake-Narasimhan \cite{KN}, Bolley-Camus \cite%
{BC} and Zanghirati \cite{Z}. Also we give here an answer to a problem asked
by Lions-Magenes \cite{LM}.
Let $\Omega $ be an open subset of $\mathbb{R}^{n},$ $P_{j}\left( x,D\right)
,\,j=1,..,N,$ henceforth denoted $\left( P_{j}\right) _{j=1}^{N},$ $\,$%
linear differential operators of order $m$,
\begin{equation*}
P_{j}\left( x,D\right) =\sum\limits_{\left\vert \alpha \right\vert \leq
m}a_{j\alpha }\left( x\right) D^{\alpha }\ .
\end{equation*}%
We define the principal symbol of the operator $P_{j}\left( x,D\right) $ by
\begin{equation*}
P_{jm}\left( x,\xi \right) =\sum\limits_{\left\vert \alpha \right\vert
=m}a_{j\alpha }\left( x\right) \xi ^{\alpha }\ .
\end{equation*}
Let $\left( M_{p}\right) $ be a sequence of positive real numbers satisfying
the following conditions
\begin{eqnarray}
&&M_{p}^{2}\leq M_{p+1}M_{p-1},\ \forall p\in \mathbb{N}^{\ast };
\label{1.1} \\
&&\exists A>0,\exists H>0,\exists c>0:cC_{p}^{j}M_{p-j}M_{j}\leq M_{p}\leq
AH^{p}M_{p-j}M_{j},\ \forall p\in \mathbb{N},j\leq p; \label{1.2} \\
&&\forall m\geq 2,\exists d>0,\forall p\in \mathbb{N},\ 0\leq q\leq m:\left(
M_{pm}\right) ^{m-q}\left( M_{pm+m}\right) ^{q}\leq d\left( M_{pm+q}\right)
^{m}. \label{1.3}
\end{eqnarray}
\begin{definition}
We call vector of class $\left\{ M_{p}\right\} $ of the system $\left(
P_{j}\right) _{j=1}^{N}$\ every function $u\in C^{\infty }\left( \Omega
\right) $\ such that%
\begin{equation}
\forall K\ compact\ of\ \Omega ,\ \exists C>0,\ \forall l\in \mathbb{N}%
:\left\Vert P_{i_{1}}...P_{i_{l}}u\right\Vert _{L^{2}\left( K\right) }\leq
C^{lm+1}M_{lm}\ , \label{1.4}
\end{equation}%
where $1\leq i_{j}\leq N,\ 1\leq j\leq l.$
The space of these functions is denoted $R_{M}\left( \Omega ,\left(
P_{j}\right) _{j=1}^{N}\right) .$
\end{definition}
\begin{definition}
Let $M^{1},..,M^{n}$\textit{\ be sequences satisfying }$\left( \ref{1.1}%
\right) -\left( \ref{1.3}\right) .$ An anisotropic Roumieu space in $\Omega $
denoted $R_{M^{1},...,M^{n}}\left( \Omega \right) ,$\ is the space of
functions $u\in C^{\infty }\left( \Omega \right) $\ such that
\begin{equation}
\forall K\ compact\ de\ \Omega ,\;\exists C>0:\left\Vert D^{\alpha
}u\right\Vert _{L^{2}\left( K\right) }\leq C^{\left\vert \alpha \right\vert
+1}M_{\alpha _{1}}^{1}...M_{\alpha _{n}}^{n}\ . \label{1.5}
\end{equation}
\end{definition}
\begin{definition}
the system $\left( P_{j}\right) _{j=1}^{N}$ is said elliptic in $\Omega $ if
for every $x_{0}\in \Omega ,$ we have
\[
\sum_{j=1}^{N}\left\vert P_{jm}\left( x_{0},\xi \right) \right\vert \neq 0,\
\forall \xi \in \mathbb{R}^{n}\backslash \left\{ 0\right\} \ .
\]
\end{definition}
The elliptic iterate property for the system $\left( P_{j}\right) _{j=1}^{N}$
in the anisotropic Roumieu classes means the inclusion
\begin{equation*}
R_{M}\left( \Omega ,\left( P_{j}\right) _{j=1}^{N}\right) \subset
R_{M^{1},...,M^{n}}\left( \Omega \right) ,
\end{equation*}%
more precisely, we will prove the following theorem.
\begin{theorem}
Let $M^{1},..,M^{n},$ $M$\ be sequences satisfying $\left( \ref{1.1}\right)
-\left( \ref{1.3}\right) $ and
\begin{equation}
\frac{\left( i_{1}+...+i_{n}\right) !}{i_{1}!...i_{n}!}%
M_{i_{1}}^{1}...M_{i_{n}}^{n}\leq M_{i_{1}+...+i_{n}}\leq
B^{i_{1}+...+i_{n}}M_{i_{1}}^{1}...M_{i_{n}}^{n},\ \label{1.6}
\end{equation}%
for some $B>0$ and for all $i_{j}\in \mathbb{N},\ j=1,...,n.$\newline
Let $\left( P_{j}\right) _{j=1}^{N}$ be a system of differential operators
with coefficients in $R_{M^{1},...,M^{n}}\left( \Omega \right) ,$ then
\[
\left( P_{j}\right) _{j=1}^{N}\ \text{\textit{is elliptic in} }\Omega
\Rightarrow R_{M}\left( \Omega ,\left( P_{j}\right) _{j=1}^{N}\right)
\subset R_{M^{1},...,M^{n}}\left( \Omega \right) \ .
\]
\end{theorem}
\section{Preliminary lemmas}
Let $x_{0}\in \Omega ,$ and let $00,\
\forall u\in C^{\infty }\left( \Omega \right) ,$%
\begin{equation}
\left\vert u\right\vert _{pm+q,\rho }\leq \varepsilon \left\vert
u\right\vert _{pm+m,\rho }+C_{m}\varepsilon ^{-\frac{q}{m-q}}\left\vert
u\right\vert _{pm,\rho },\ \forall p,q\in \mathbb{Z}_{+},\ q0$ such that
$\forall u\in C^{\infty }\left( \omega \right) ,\forall \rho >0,\forall
\delta \in ]0,1[$ with $\overline{B}_{\rho +\delta }\subset \omega ,\forall
p\in \mathbb{N},$%
\begin{equation}
\left\vert u\right\vert _{\left( p+1\right) m,\rho }\leq C\left(
\sum_{j=1}^{N}\left\vert P_{j}u\right\vert _{pm,\rho +\delta }+\delta
^{-m}\left\vert u\right\vert _{pm,\rho +\delta
}+\sum_{j=1}^{N}\sum_{\left\vert \alpha \right\vert =pm}\frac{\left(
pm\right) !}{\alpha !}\left\vert \left[ P_{j},D^{\alpha }\right]
u\right\vert _{0,\rho +\delta }\right) \ . \label{2.2}
\end{equation}
\end{lemma}
We give a convenient form to this lemma for systems with coefficients in $%
R_{M^{1},...,M^{n}}\left( \Omega \right) .$
\begin{lemma}
\label{lem2.3}Let $\left( P_{j}\right) _{j=1}^{N}$\ be an elliptic system
with coefficients in $R_{M^{1},...,M^{n}}\left( \Omega \right) ,$\ $\omega $
as in lemma \ref{lem2.2}, $\varepsilon \in ]0,1[,$ so there exists $C_{1}>0$
and $C_{2}\left( \varepsilon \right) >0$ such that
$\forall u\in C^{\infty }\left( \omega \right) ,\forall \rho >0,\forall
\delta \in ]0,1[$ with $\overline{B}_{\rho +\delta }\subset \omega ,\forall
p\in \mathbb{Z}_{+}^{\ast },$%
\begin{eqnarray}
\left\vert u\right\vert _{\left( p+1\right) m,\rho } &\leq &C_{1}\left(
\sum_{j=1}^{N}\left\vert P_{j}u\right\vert _{pm,\rho +\delta }+\left(
\varepsilon \delta \right) ^{-m}\left\vert u\right\vert _{pm,\rho +\delta
}+\varepsilon \left\vert u\right\vert _{pm+m,\rho +\delta }\right) \nonumber
\\
&&\ +\sum_{h=0}^{p}\frac{M_{pm+m}}{M_{hm}}C_{2}\left( \varepsilon \right)
^{\left( p+1-h\right) m}\left\vert u\right\vert _{hm,\rho +\delta }\ ,
\label{2.3}
\end{eqnarray}%
and
\begin{equation}
\left\vert u\right\vert _{m,\rho }\leq C_{1}\left( \sum_{j=1}^{N}\left\vert
P_{j}u\right\vert _{0,\rho +\delta }+\left( \varepsilon \delta \right)
^{-m}\left\vert u\right\vert _{0,\rho +\delta }+\varepsilon \left\vert
u\right\vert _{m,\rho +\delta }\right) \label{2.3b}
\end{equation}
\end{lemma}
\begin{proof}
The estimate $\left( \ref{2.3b}\right) $ follows from $\left( \ref{2.2}%
\right) $ for $p=0.$ For $\left( \ref{2.3}\right) $ we have to estimate the
third term of the right-hand side of $\left( \ref{2.2}\right) .$ By Leibniz
formula , we have
\begin{equation}
\left\vert \left[ a_{j\beta },D^{\alpha }\right] u\right\vert _{0,\rho
+\delta }\leq \sum\limits_{\gamma <\alpha }\left( _{\gamma }^{\alpha
}\right) \left\vert D^{\gamma }uD^{\alpha -\gamma }a_{j\beta }\right\vert
_{0,\rho +\delta }\ . \label{2.4}
\end{equation}%
As the coefficients $a_{j\beta }$ are in $R_{M^{1},...,M^{n}}\left( \Omega
\right) ,$ then
\begin{equation*}
\exists C_{1}>0,\forall \left\vert \beta \right\vert \leq m,\ \forall
j=1,..,N,\ \forall \gamma \in \mathbb{Z}_{+}^{n}:\sup_{B_{R}}\left\vert
D^{\gamma }a_{j\beta }\right\vert \leq C_{1}^{\left\vert \gamma \right\vert
+1}\prod_{i=1}^{n}M_{\gamma _{i}}^{i}\ .
\end{equation*}%
Substituting in $\left( \ref{2.4}\right) $, and taking account of $\left( %
\ref{1.6}\right) $ and $\left( \ref{1.2}\right) $ we obtain
\begin{eqnarray*}
\frac{\left\vert \alpha \right\vert !}{\alpha !}\left\vert \left[ a_{j\beta
},D^{\alpha }\right] u\right\vert _{0,\rho +\delta } &\leq
&\sum\limits_{\gamma <\alpha }\frac{\left\vert \alpha \right\vert !}{\left(
\alpha -\gamma \right) !\gamma !}C_{1}^{\left\vert \alpha -\gamma
\right\vert +1}\prod_{i=1}^{n}M_{\alpha _{i}-\gamma _{i}}^{i}\left\vert
D^{\gamma }u\right\vert _{0,\rho +\delta } \\
&\leq &\sum\limits_{\gamma <\alpha }\frac{\left\vert \alpha \right\vert !}{%
\left\vert \alpha -\gamma \right\vert !\gamma !}C_{1}^{\left\vert \alpha
-\gamma \right\vert +1}M_{\left\vert \alpha -\gamma \right\vert }\left\vert
D^{\gamma }u\right\vert _{0,\rho +\delta } \\
&\leq &\sum\limits_{\gamma <\alpha }C_{1}^{\left\vert \alpha -\gamma
\right\vert +1}\frac{1}{c}\frac{\left\vert \gamma \right\vert !}{\gamma !}%
\frac{M_{\left\vert \alpha \right\vert }}{M_{\left\vert \gamma \right\vert }}%
\left\vert D^{\gamma }u\right\vert _{0,\rho +\delta }\ ,
\end{eqnarray*}%
hence for every $p\in \mathbb{Z}_{+}^{\ast },$ we have
\begin{equation}
\sum\limits_{\left\vert \alpha \right\vert =pm}\frac{\left( pm\right) !}{%
\alpha !}\left\vert \left[ a_{j\beta },D^{\alpha }\right] u\right\vert
_{0,\rho +\delta }\leq \sum_{h=0}^{pm-1}C_{2}^{pm-h+1}\frac{M_{pm}}{M_{h}}%
\left\vert u\right\vert _{h,\rho +\delta }\ . \label{2.5}
\end{equation}%
Now
\begin{equation*}
\lbrack P_{j},D^{\alpha }]u=\sum\limits_{\left\vert \beta \right\vert \leq
m}\left( a_{j\beta }\left( x\right) D^{\beta +\alpha }u-D^{\alpha }\left(
a_{j\beta }\left( x\right) D^{\beta }u\right) \right)
=\sum\limits_{\left\vert \beta \right\vert \leq m}[a_{j\beta },D^{\alpha
}]D^{\beta }u\ ,
\end{equation*}%
so from $\left( \ref{2.5}\right) ,$ we get
\begin{eqnarray}
\sum\limits_{\left\vert \alpha \right\vert =pm}\frac{\left( pm\right) !}{%
\alpha !}\left\vert [P_{j},D^{\alpha }]u\right\vert _{0,\rho +\delta } &\leq
&\sum\limits_{\left\vert \beta \right\vert \leq m}\sum\limits_{\left\vert
\alpha \right\vert =pm}\frac{\left( pm\right) !}{\alpha !}\left\vert
[a_{j\beta },D^{\alpha }]D^{\beta }u\right\vert _{0,\rho +\delta } \notag \\
\ &\leq &\sum\limits_{k=0}^{m}\sum\limits_{\left\vert \beta \right\vert
=k}\sum_{h=0}^{pm-1}C_{2}^{pm-h+1}\frac{M_{pm}}{M_{h}}\left\vert D^{\beta
}u\right\vert _{h,\rho +\delta } \notag \\
&\leq &\sum\limits_{k=0}^{m}\sum_{h=k}^{pm+k-1}C_{2}^{pm+k-h+1}\frac{M_{pm}}{%
M_{h-k}}\left\vert u\right\vert _{h,\rho +\delta }\ . \label{2.6}
\end{eqnarray}%
In the other hand, it easy to check that the condition $\left( \ref{1.1}%
\right) $ gives\
\begin{equation}
\forall h\leq p,\ \forall i\in \mathbb{Z}_{+}:\frac{M_{p}}{M_{h}}\leq \frac{%
M_{p+i}}{M_{h+i}}\ . \label{2.7}
\end{equation}%
In particular
\begin{equation*}
\frac{M_{pm}}{M_{h-k}}\leq \frac{M_{pm+m}}{M_{h-k+m}}\leq \frac{M_{pm+m}}{%
M_{h}}\ ,
\end{equation*}%
which gives with $\left( \ref{2.6}\right) \ $%
\begin{eqnarray}
\sum\limits_{\left\vert \alpha \right\vert =pm}\frac{\left( pm\right) !}{%
\alpha !}\left\vert [P_{j},D^{\alpha }]u\right\vert _{0,\rho +\delta } &\leq
&\left( m+1\right) \sum_{h=0}^{pm+m-1}C_{2}^{pm+m-h+1}\frac{M_{pm+m}}{M_{h}}%
\left\vert u\right\vert _{h,\rho +\delta } \notag \\
&\leq &\sum_{l=0}^{p}\sum_{q=0}^{m-1}C_{3}^{pm+m-lm-q+1}\frac{M_{pm+m}}{%
M_{lm+q}}\left\vert u\right\vert _{lm+q,\rho +\delta }\ . \label{2.8}
\end{eqnarray}%
From lemma \ref{lem2.1}, we have for every $\varepsilon ^{\prime }\in \left[
0,1\right] ,$%
\begin{equation*}
\left\vert u\right\vert _{lm+q,\rho +\delta }\leq \varepsilon ^{\prime
}\left\vert u\right\vert _{lm+m,\rho +\delta }+C_{m}\varepsilon ^{\prime -%
\frac{q}{m-q}}\left\vert u\right\vert _{lm,\rho +\delta }\ .
\end{equation*}%
Set $\varepsilon ^{\prime }=\varepsilon \dfrac{M_{lm+q}}{M_{lm+m}}%
C_{3}^{-m+q},$ so from $\left( \ref{1.3}\right) ,$ we obtain
\begin{eqnarray*}
C_{3}^{m-q}\frac{M_{pm+m}}{M_{lm+q}}\left\vert u\right\vert _{lm+q,\rho
+\delta } &\leq &\varepsilon \frac{M_{pm+m}}{M_{lm+m}}\left\vert
u\right\vert _{lm+m,\rho +\delta } \\
&&+C_{m}\varepsilon ^{-\frac{q}{m-q}}C_{3}^{m}\frac{M_{pm+m}}{M_{lm+q}}%
\left( \frac{M_{lm+m}}{M_{lm+q}}\right) ^{\frac{q}{m-q}}\left\vert
u\right\vert _{lm,\rho +\delta } \\
&\leq &\varepsilon \frac{M_{pm+m}}{M_{lm+m}}\left\vert u\right\vert
_{lm+m,\rho +\delta }+C_{m}\varepsilon ^{-m}C_{3}^{m}d^{\prime }\frac{%
M_{pm+m}}{M_{lm}}\left\vert u\right\vert _{lm,\rho +\delta }\ .
\end{eqnarray*}%
With $\left( \ref{2.8}\right) $ we obtain
\begin{eqnarray*}
\sum\limits_{\left\vert \alpha \right\vert =pm}\frac{\left( pm\right) !}{%
\alpha !}\left\vert [P_{j},D^{\alpha }]u\right\vert _{0,\rho +\delta } &\leq
&m\varepsilon \sum_{l=0}^{p}C_{3}^{pm-lm+1}\frac{M_{pm+m}}{M_{lm+m}}%
\left\vert u\right\vert _{lm+m,\rho +\delta } \\
&&+mC_{m}\varepsilon ^{-m}d^{\prime }\sum_{l=0}^{p}C_{3}^{pm+m-lm+1}\frac{%
M_{pm+m}}{M_{lm}}\left\vert u\right\vert _{lm,\rho +\delta } \\
&\leq &m\varepsilon C_{3}\left\vert u\right\vert _{pm+m,\rho +\delta } \\
&&+m\left( \varepsilon +C_{m}\varepsilon ^{-m}d^{\prime }\right)
\sum_{l=0}^{p}C_{3}^{pm+m-lm+1}\frac{M_{pm+m}}{M_{lm}}\left\vert
u\right\vert _{lm,\rho +\delta }\ .
\end{eqnarray*}%
The last estimate and $\left( \ref{2.2}\right) $ give the requested estimate
$\left( \ref{2.3}\right) .$
\end{proof}
\section{The theorem}
For every$\ \lambda >0,$ define
\begin{equation*}
\sigma _{p}\left( u,\lambda \right) =\frac{1}{\lambda ^{p}M_{pm}}\underset{%
R/2\leq \rho 0\ ($depending only of $\left( P_{j}\right) _{j=1}^{N}$
and $R),\ \forall \lambda \geq \lambda _{0},\ \forall u\in C^{\infty }\left(
\omega \right) ,\ \forall p\in \mathbb{Z}_{+},$%
\begin{equation}
\sigma _{p+1}\left( u,\lambda \right) \leq \frac{M_{pm}}{M_{pm+m}}%
\sum\limits_{j=1}^{N}\sigma _{p}\left( P_{j}u,\lambda \right)
+\sum\limits_{h=0}^{p}\sigma _{h}\left( u,\lambda \right) \ . \label{3.1}
\end{equation}
\end{lemma}
\begin{proof}
First we prove this lemma for $p\geq 1.$ For every $\rho \in \lbrack R/2,R[$
we take $\delta =\frac{R-\rho }{p+1},$ then we have $\rho +\delta \in
\lbrack R/2,R[$ and $\left( \frac{R-\rho }{R-\rho -\delta }\right)
^{pm}0$ and for all $i_{j}\in \mathbb{N},\ j=1,...,n,$ and let $%
\left( P_{j}\right) _{j=1}^{N}$ be an elliptic system with coefficients in $%
R_{M^{1},...,M^{n}}\left( \Omega \right) ,$ so
\[
\left( P_{j}\right) _{j=1}^{N}\ \text{\textit{is elliptic in } }\Omega
\Rightarrow R_{M}\left( \Omega ,\left( P_{j}\right) _{j=1}^{N}\right)
\subset R_{M^{1},...,M^{n}}\left( \Omega \right) \ .
\]
\end{theorem}
\begin{proof}
It suffies to show the\ estimate $\left( \ref{1.5}\right) $ in the
neighborhood of every point of $\Omega .$ Let $x_{0}\in \Omega ,$ so there
exists a compact neighborhood $\omega $ of $x_{0}$ for which the above
lemmas hold. Suppose that $u\in R_{M}\left( \Omega ,\left( P_{j}\right)
_{j=1}^{N}\right) ,$ then there exists $C>0$ such that
\begin{equation}
\left\Vert P_{i_{0}}\cdot \cdot P_{i_{h}}u\right\Vert _{L^{2}\left( \omega
\right) }\leq C^{h+1}M_{hm}\ . \label{3.6}
\end{equation}%
Let $R\in ]0,1[$ such that $B_{R}\subset \omega ,$ so from $\left( \ref{3.6}%
\right) ,$ we have
\begin{equation*}
\sigma _{0}\left( P_{i_{0}}\cdot \cdot P_{i_{h}}u,\lambda _{0}\right) \leq
C^{h+1}M_{hm}\ ,
\end{equation*}%
consequently from $\left( \ref{3.4}\right) ,$ we obtain
\begin{eqnarray*}
\sigma _{p+1}\left( u,\lambda _{0}\right) &\leq
&2^{p+1}C+\sum\limits_{h=1}^{p+1}2^{p+1-h}\left( _{~h}^{p+1}\right)
N^{h}C^{h+1} \\
&\leq &C\left( 2+NC\right) ^{p+1}\ ,
\end{eqnarray*}%
which gives
\begin{eqnarray}
\left\vert u\right\vert _{\left( p+1\right) m,R/2} &\leq &\lambda
_{0}^{p+1}M_{pm+m}\left( 2+NC\right) ^{p+1}\left( \frac{2}{R}\right) ^{pm}
\notag \\
\ &\leq &C_{1}^{\left( p+1\right) m}M_{pm+m}\ . \label{3.7}
\end{eqnarray}
Now, let $\alpha \in \mathbb{Z}_{+}^{n},$ so there exists positive integers $%
p,q$ such that $\left\vert \alpha \right\vert =pm+q,$ $q\leq m-1.$ Set $%
\varepsilon =C_{1}^{q-m}\frac{M_{pm+q}}{M_{pm+m}},$ from $\left( \ref{1.3}%
\right) $ and $\left( \ref{2.1}\right) ,$ we have
\begin{eqnarray*}
\left\vert u\right\vert _{pm+q,R/2} &\leq &C_{1}^{q-m}\frac{M_{pm+q}}{%
M_{pm+m}}\left\vert u\right\vert _{pm+m,R/2}+C_{m}C_{1}^{q}\left( \frac{%
M_{pm+m}}{M_{pm+q}}\right) ^{\frac{q}{m-q}}\left\vert u\right\vert _{pm,R/2}
\\
&\leq &C_{1}^{q-m}\frac{M_{pm+q}}{M_{pm+m}}\left\vert u\right\vert
_{pm+m,R/2}+C_{1}^{q}C_{m}d^{\prime }\frac{M_{pm+q}}{M_{pm}}\left\vert
u\right\vert _{pm,R/2}\ .
\end{eqnarray*}%
Taking account of $\left( \ref{3.5}\right) ,$ we obtain from $\left( \ref%
{3.7}\right) $ and the last estimate
\begin{eqnarray*}
\left\Vert D^{\alpha }u\right\Vert _{L^{2}\left( B_{R/2}\right) } &\leq
&\left\vert u\right\vert _{pm+q,R/2} \\
&\leq &C_{1}^{pm+q}M_{pm+q}+C_{1}^{pm+q}C_{m}d^{\prime }M_{pm+q} \\
&\leq &C_{2}^{\left\vert \alpha \right\vert +1}M_{\left\vert \alpha
\right\vert } \\
&\leq &C_{2}^{\left\vert \alpha \right\vert +1}B^{\left\vert \alpha
\right\vert }M_{\alpha _{1}}^{1}...M_{\alpha _{n}}^{n},
\end{eqnarray*}%
which shows that $u\in R_{M^{1},...,M^{n}}\left( B_{R/2}\right) .$
\end{proof}
As consequences of Theorem \ref{th3.1} we obtain a result of Roumieu
-Hypoellipticity of elliptic systems and the principal result of \cite{Z} in
the scalar case where $\left( M_{p}\right) $ is the Gevrey sequence $\left(
p!\right) ^{s}$.
\begin{corollary}
Under the assumptions of Theorem \ref{th3.1}, the following assertions are
equivalent.
$\left( i\right) $ $u\in \mathcal{D}^{\prime }\left( \Omega \right) $ and $%
P_{j}u\in R_{M^{1},...,M^{n}}\left( \Omega \right) ,\ \forall j=1,...,N.$
$\left( ii\right) \ u\in R_{M^{1},...,M^{n}}\left( \Omega \right) .$
\end{corollary}
\begin{corollary}
Let $\Omega $ be an open subset of $\mathbb{R}^{n}$ and $s\geq 1,$ and let $%
P $ be a linear differential operator with cofficients in $G^{s,q}\left(
\Omega \right) ,$ then
\[
P\ \text{is }q-\text{quasi-elliptic in }\Omega \Longrightarrow G^{s}\left(
\Omega ,P\right) \subset G^{s,q}\left( \Omega \right)
\]
\end{corollary}
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\end{document}
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