\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 156, pp. 1--3.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/156\hfil Lateral estimates] {Lateral estimates for iterated elliptic operators and analyticity} \author[S. Tarama\hfil EJDE-2009/156\hfilneg] {Shigeo Tarama} \address{Shigeo Tarama \newline Laboratory of Applied Mathematics, Faculty of Engineering, Osaka City University, Osaka 558-8585, Japan} \email{starama@mech.eng.osaka-cu.ac.jp} \thanks{Submitted November 3, 2009. Published December 1, 2009.} \subjclass[2000]{35L30, 16D10} \keywords{Elliptic operators; analyticity} \begin{abstract} Analyticity of functions satisfying the lateral estimates for iterated elliptic operators is shown. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \section{Introduction} Bernstein \cite{BR} showed that a function $f(x)$ satisfying the inequalities $\frac{d^k}{dx^k}f(x)\le 0 \quad \text{on (a,b) for any integer k\ge 0}$ is real analytic on $(a,b)$. According to \cite {BO}, to obtain the analyticity, it is sufficient to have the above inequalities only for an increasing sequence $k_j$ satisfying $k_{j+1} \le Ak_j$ with some $A>0$. Lelong \cite {L} showed as an extension to a multidimensional case, that the inequalities for the iterated Laplacian $\Delta^k$: for any $k=0,1,2,\dots$, $\Delta^ku(x)\le 0 \quad \text{on a domain D in \mathbb{R}^n }$ imply the analyticity of $u(x)$ on $D$. Novickii \cite{N} showed the above assertion is still valid if the Laplacian $\Delta$ is replaced by a second order strongly elliptic operator $L$ with real-valued and real analytic coefficients, as a corollary of his representation theorem for L-superharmonic functions. On the other hand, Kotake and Narasimhan \cite{KN} showed that the analyticity of $u(x)$ on $D$ follows from the estimates: For any $k=0,1,2,\dots$ $$\label{BCM} \|P^{k}u\|_{L^2(D)}\le C_0C^{mk}(mk)!^{mk},$$ for an ellipitc operator of order $m$ with real analytic coefficients. Bolley, Camus and Metivier \cite{BCM} (see also \cite{BM}) showed the above assertion is still valid if we have the estimates \eqref{BCM} for an increasing sequence of natural numbers $k_j$ satisfying $k_{j+1}\le A k_j$ with some $A>0$. We note that they showed in \cite{BCM} that the conclusion holds even if $P$ is a principal type and hypoelliptic operator with real analytic coefficients. In this short note, we show that in the case where $P$ is an elliptic operator with real-valued and real analytic coefficients, the above assertion is still valid if the estimates \eqref{BCM} are replaced by lateral estimates. \begin{theorem} \label{thm1} Let $D$ be an open set in $\mathbb{R}^n$. Let $P$ be an elliptic operator of order $m$ with real valued and real analytic coefficients. Assume that the inequalities $$\label{let} P^{k_j}u(x)\le C_0C^{mk_j}(mk_j)!^{mk_j} \quad \text{on D}$$ hold for an increasing sequence of natural numbers $k_j$ satisfying $k_{j+1}\le A k_j$ with some $A>0$. Then the function $u(x)$ is real analytic on $D$. \end{theorem} \section{Proof of Theorem} \begin{proof} Indeed the theorem follows from simple integration by parts and Bolley-Camus-Metivier's theorem mentioned above. Since the argument is local, we may consider the case where $D$ is an open ball with center at the origin, and it is sufficient to show that $u(x)$ is real analytic near the origin. Then we assume that $D=B(r)$ where $B(r)=\{ x\in \mathbb{R}^n\ |\ |x|0$. First of all, we remark that $u(x)$ is $C^{\infty}$ even if the inequalities \eqref{let} are satisfied in distribution sense. Indeed since \eqref{let} implies that $P^{k_j}u$ is a measure and $P^{k_j}$ is a $mk_j$-th order elliptic operator, we see that $u(x)$ belongs to the Sobolev space $H^{mk_j-(n+1)/2}_{loc}(D)$. We use cut-off functions $\chi_k(x)$. Let $\chi_k(x)$ ($k=1,2,3,\dots$) be non-negative smooth functions satisfying the following conditions: \begin{itemize} \item[(P-1)] $1\ge \chi_k(x)\ge 0$, $\chi_k(x)=1$ for $|x|\le r/2$ and $\chi_k(x)=0$ for $|x|\le 2r/3$ \item[(P-2)] For any $\alpha$ with $|\alpha|\le k$, we have $$\label{test} \big|\frac{d^{\alpha}}{dx^{\alpha}}\chi_k(x)\big| \le C_0C_1^{|\alpha|}k^{|\alpha|}\quad \text{on D.}$$ where the constants $C_0,C_1$ are independent of $k$ and $\alpha$. (See \cite{H}) \end{itemize} Then, noting that $P^{k_j}u(x)- C_0C_1^{mk_j}(mk_j)!^{mk_j}\le0$ and (P-1), we have \label{est} \begin{aligned} &\int_D \chi_{mk_j}(x)\Bigl(P^{k_j}u(x)- C_0C_1^{mk_j}(mk_j)!^{mk_j} \Bigr)\,dx\\ &\le \int_{|x|\le r/2} \Bigl(P^{k_j}u(x)- C_0C_1^{mk_j}(mk_j)!^{mk_j}\Bigr) \,dx\le 0. \end{aligned} Through the integration by parts, we see that the left hand side is equal to $\int_D \Bigl((^tP)^{k_j}\chi_{mk_j}(x)\Bigr)u(x)\,dx - CC_0C_1^{mk_j}(mk_j)!^{mk_j}$ where $^tP$ is the transposed operator of $P$. Since the coefficients of $P$ are real analytic, it follows from \eqref{test} that $\big|(^tP)^{k_j}\chi_{mk_j}(x)\big|\le K_0K_1^{mk_j}(mk_j)^{mk_j},$ with some constants $K_0, K_1$, see for example \cite[Lemma 8.6.3]{H}. Then we see that the absolute value of the left hand side of \eqref{est} is not greater than $K_0K_1^{mk_j}(mk_j)^{mk_j}|D|(\|u(x)\|_{L^{\infty}(D)}+1).$ Here we replace the constants $K_0,K_1$ by larger constants, if necessary. While $P^{k_j}u(x)- C_0C_1^{mk_j}(mk_j)!^{mk_j}\le0$ implies \begin{align*} &\int_{|x|\le r/2}|P^{k_j}u(x)|\,dx\\ &\le (-1)\int_{|x|\le r/2}\Bigl( P^{k_j}u(x)- C_0C_1^{mk_j}(mk_j)!^{mk_j} \Bigr)\,dx +C_rC_0C_1^{mk_j}(mk_j)!^{mk_j}, \end{align*} where the first term of the right hand side is not greater than $K_0K_1^{mk_j}(mk_j)^{mk_j}|D|(\|u(x)\|_{L^{\infty}(D)}+1).$ Hence we have $\int_{|x|\le r/2}|P^{k_j}u(x)|\,dx \le K_0K_1^{mk_j}(mk_j)^{mk_j}|D|(\|u(x)\|_{L^{\infty}(D)}+1).$ with some positive constants $K_0, K_1$. From the above $L^1$-estimates, we see that $u(x)$ is real analytic on a neighborhood of the origin thanks to Bolley-Camus-Metivier's theorem \cite{BCM}. Indeed, according to \cite[Theorem 1.2]{BM}, we see that \cite[Proposition 3.3]{BCM} is still valid using $L^1$ estimates for $P^nu$. Then we have the desired conclusion. The proof is complete. \end{proof} \begin{thebibliography}{0} \bibitem{BR} S. N. 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Lelong; \emph{Sur les fonctions ind\'efiniment d\'erivables de plusieurs variables dont les laplaciens successifs ont des signes altern\'es}, Duke Math. J., 14(1947), 143--149 \bibitem{N} M. V. Novickii\; \emph{Representation of completely L-superharmonic functions}, Izv. Akad. Nauk SSSR, Ser. Mat. Tom 39(1975), 1279--1296 \end{thebibliography} \end{document}