\documentclass[reqno]{amsart}
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\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 $k=0,1,2,\dots$
\begin{equation}\label{BCM}
\|P^{k}u\|_{L^2(D)}\le C_0C^{mk}(mk)!^{mk},
\end{equation}
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
\begin{equation}\label{let}
P^{k_j}u(x)\le C_0C^{mk_j}(mk_j)!^{mk_j} \quad \text{on $D$}
\end{equation}
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
$B(r)=\{ x\in \mathbb{R}^n\ |\ |x|<r\}$ with $r>0$. 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
 \begin{equation}\label{test}
\big|\frac{d^{\alpha}}{dx^{\alpha}}\chi_k(x)\big|
\le C_0C_1^{|\alpha|}k^{|\alpha|}\quad \text{on $D$.}
\end{equation}
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
\begin{equation}\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}
\end{equation}
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}


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\end{thebibliography}
\end{document}