Content-Type: multipart/mixed; boundary="-------------1010051230384" This is a multi-part message in MIME format. ---------------1010051230384 Content-Type: text/plain; name="10-165.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="10-165.comments" classification: 35H30, 35H10 e-mail: chaili_rachid@yahoo.fr ---------------1010051230384 Content-Type: text/plain; name="10-165.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="10-165.keywords" Elliptic systems, anisotropic Roumieu classes ---------------1010051230384 Content-Type: application/x-tex; name="Preprint.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Preprint.tex" \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amsfonts} \usepackage{amsmath} \setcounter{MaxMatrixCols}{10} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.50.0.2890} %TCIDATA{} %TCIDATA{BibliographyScheme=Manual} %TCIDATA{Created=Monday, October 04, 2010 10:28:53} %TCIDATA{LastRevised=Monday, October 04, 2010 10:34:18} %TCIDATA{} %TCIDATA{} %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} \begin{thebibliography}{9} \bibitem{BC} Bolley P.,Camus J., Powers and Gevrey regularity for a system of differential operators, Czechoslovak Math. J., Praha, 29 (104), (1979), 649-661. \bibitem{BCh} Bouzar C., Cha\"{\i}li R., R\'{e}gularit\'{e} des vecteurs de Beurling de syst\`{e}mes elliptiques, Maghreb Math. Rev., Vol. 9, no. 1 \& 2, (2000), 43-53. \bibitem{J} John O., Sulla regolarit\`{a} delle soluzioni delle equazioni lineari ellittiche nelle classi di Beurling, Boll. U.M.I., 4, 2, (1969), 183-195. \bibitem{KN} Kotake T., Narasimhan M.S., Regularity theorems for fractional powers of a linear elliptic operator, Bull. Soc. Math. France, 90, (1962), 449-471. \bibitem{LM} Lions J. L., Magenes E., Probl\`{e}mes aux limites non homogenes et appliquations, Vol 3, Dunod, Paris, 1970. \bibitem{N} Nelson E., Analytic vectors, Ann. of Math., Vol. 70, no. 3, (1959), 572-615. \bibitem{Z} Zanghirati L., Iterati di operatori quasi-ellittici e classi di Gevrey, Bollettino U.M.I., Vol. 5, no. 18-B, (1981), 411-428. \end{thebibliography} \end{document} ---------------1010051230384--