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%%% START HERE %%%

\extraline{First author supported in part by a grant 
           from the National Science Foundation}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}
\newunnumbered{remark}{Remark}
\newunnumbered{remarks}{Remarks}
\newunnumbered{notation}{Notation}
\newunnumbered{definition}{Definition}

\classno{35B60}
\begin{document}

\title[Dirichlet's Problem When The Data Is An Entire Function]
      {Dirichlet's Problem When The Data\\ Is An Entire Function}

\author{Dmitry Khavinson \and\ Harold S. Shapiro}

\maketitle

\section{Introduction}
\label{sec-Introduction}

This paper may be regarded as a sequel (and correction) to \cite{Incorrect
version}, and we use similar notations. Thus $x=(x_1,\ldots,x_n)$ and
$y=(y_1,\dots,y_n)$ denote points of~${\Bbb R}^n$ and $z=x+iy$ a point
of~${\Bbb C}^n$. We use standard multi-index notations; thus for
$\alpha=(\alpha_1,\dots,\alpha_n)$ with $\alpha_j$ non-negative integers,
$z^\alpha=z_1^{\alpha_1}\!\dots z_n^{\alpha_n}$,
$|\alpha|=\alpha_1+\dots+\alpha_n$, $\alpha!=\alpha_1!\dots\alpha_n!$ and
$|z|=(|z_1|^2+\dots|z_n|^2)^{1/2}$.

$D$ denotes $(D_1,\dots,D_n)$ with $D_j=\partial/\partial z_j$, and
$\partial$~denotes $(\partial_1,\ldots,\partial_n)$ with
$\partial_j=\partial/\partial x_j$ (or $\partial/\partial y_j$,
etc., as the case may be).

${\scr P}_{m,n}$ denotes the set of polynomials on $n$~letters with complex
coefficients, of degree at most~$m$, and ${\scr H}_{m,n}$ the set of
homogeneous polynomials of degree~$m$ in ${\scr P}_{m,n}$ augmented by~$0$
(so that ${\scr H}_{m,n}$ is a vector space over $\Bbb C$). The number of
variables~($n$) will usually be suppressed in the notation, and we shall then
write ${\scr P}_m{\scr H}_m$.

For $f\in{\scr P}_m$, $f^*$~is the polynomial obtained from~$f$ by
conjugating its coefficients.

$E_n$ denotes the set of entire functions on~${\Bbb C}^n$, and $X_n$ the entire
functions of exponential type. $E_n$~and~$X_n$ may be considered as topological
vector spaces, so as to be duals of one another, in a standard way (compare
\cite{Incorrect version}). $F_n$~is the Hilbert space of entire functions~$f$
on~${\Bbb C}^n$, $f=\sum c_\alpha z^\alpha$ normed by
\begin{equation}
\|f\|^2 = \sum \alpha! \, |c_\alpha|^2.
\label{eq:Fischer norm}
\end{equation}

{\em Whenever\/ $\|\,{\cdot}\,\|$ appears in this paper, it designates this
norm.} We denote by~$\langle\,\, , \,\rangle$ the corresponding inner product
in~$F_n$. Finally, for $z$~and~$w$ in~${\Bbb C}^n$, $z\cdot w$~denotes
$\sum_1^n z_j w_j$.

The main objective of this paper is to prove Theorem~\ref{theorem-Ellipsoid}
below. The special case $a_1=\dots=a_n=1$ (or, rather, a formulation equivalent
to this) is stated as Theorem~2 in \cite{Incorrect version}. Unfortunately, the
proof offered there is incorrect (the error, on p.~522, lies in applying
Lemma~1 to the series of polynomials $\sum h_m$: here $h_m$~is in~${\scr
P}_m$, but not homogeneous, so Lemma~1 is not applicable). Even more
unfortunately, the generalization of Theorem~2 of \cite{Incorrect version}
presented as the Corollary on p.~525 is also based on an invalid deduction. The
error here is the assertion that the analog of Theorem~3 for the space~$X_n$
rather than $E_n$ can be proved by a similar argument---it cannot. Thus far we
do not know whether this Corollary is true as stated, or not.

The (we hope) correct proof of Theorem~\ref{theorem-Ellipsoid} below (Theorem~2
of \cite{Incorrect version}) is based on elementary potential theory, not using
techniques of \cite{Incorrect version} based on the Fischer
norm~(\ref{eq:Fischer norm}). Using the latter technique, we have not succeeded
in proving this theorem in full generality, but only within classes of entire
functions of limited growth (Theorem~\ref{theorem-Ellipsoid} below). But, in
return, we obtain an analogous result not only for the Laplace operator, but
for a fairly large class of differential operators. Thus, the Corollary on
p.~525 of \cite{Incorrect version} is shown to be true for each homogeneous
polynomial~$P$ of a certain `amenable' class (see Section~\ref{sec-Ellipsoid}
below), provided the given~$f$ is restricted to an appropriate subclass of the
entire functions. This is done in Theorem~\ref{theorem-Ellipsoid} of the
present paper.

\section{Dirichlet's problem for the ellipsoid}
\label{sec-Ellipsoid}

\begin{theorem}
\label{theorem-Ellipsoid}
Let\/ $\Omega=\{x\in{\Bbb R}^n : \sum_{j=1}^n a_j^{-1}x_j^2 < 1\}$, where
$a_j>0$. If $f$~is entire on\/ ${\Bbb C}^n$, the solution of the Dirichlet
problem
\begin{equation} 
\vcenter{\openup\jot\ialign
         {\strut\hfil$\displaystyle#$&$\displaystyle{}#$\hfil&\quad#\hfil\cr
\Delta u &= 0 &in\/ $\Omega$,\cr
       u &= f &on $\partial\Omega$,\cr}}
\label{eq:Dirichlet}
\end{equation}
extends to a harmonic function on\/~${\Bbb R}^n$. {\rm (}Hence it extends to
an entire function on\/~${\Bbb C}^n$ satisfying\/ $\sum_1^n D_j^2 u = 0$, and
equal to~$f$ on the variety\/ $\{z\in{\Bbb C}^n : \sum_1^n a_j^{-2}z_j^2 =
1\}$.{\rm $\,$)}
\end{theorem}

\begin{proof}
We can write the Taylor expansion of~$f$ as $f=\sum_0^\infty f_m$, where
$f_m\in{\scr H}_m$. The Dirichlet problem analogous to~(\ref{eq:Dirichlet})
with $f_m$ in place of~$f$ has a unique solution $u_m\in{\scr P}_m$. (This
is well known, and is recalled for the reader's convenience in
Section~\ref{sec-Conclusion} below.) To complete the proof, we shall show that
$\sum_0^\infty u_m$ converges uniformly on compact subsets of ${\Bbb R}^n$.

Let $\Gamma$ denote $\partial\Omega$. Let
\begin{equation}
u_m = u_{m,0}+u_{m,1}+\dots+u_{m,m}
\label{eq:u definition}
\end{equation}
denote the decomposition of $u_m$ into homogeneous polynomials; thus $u_{m,j}$
is in~${\scr H}_j$ and harmonic.

We shall now prove that, for every $R>0$, there is a constant $A(R)$ such that
\begin{equation}
\sum_{m=0}^\infty \sum_{k=0}^m |u_{m,k}(x)| \le A(R),
\quad \mbox{for $|x| \le R$},
\label{eq:A(R) definition}
\end{equation}
which implies the desired convergence of~$\sum_0^\infty u_m$.
\end{proof}

\begin{lemma}
\label{lemma-max limit}
Let $F_m=\max\{|f_m(x)| : x\in\Gamma\}$. Then $F_m^{1/m}\to 0$.
\end{lemma}

\begin{proof}
The proof uses only that $\Gamma$~is a compact subset of~${\Bbb C}^n$,
contained in, say, the ball $B$:~$\{|z|\le\rho\}$.

We have for $t\in{\Bbb C}$,
\[ 
f(tz) = \sum_0^\infty t^m f_m(z). 
\]
Fixing $z\in B$, $f_m(z)$~are the Taylor coefficients of the entire function
$t\mapsto f(tz)$ on~$\Bbb C$. By the Cauchy--Hadamard estimate,
\[ 
|f_m(z)| \le \frac{\max \{|f(tz)| : t \le T\}}{T^m} 
\]
holds for all $T>0$. Hence,
\[ 
\max_{x \in B} |f_m(z)|\le\frac{\max\{|f(\zeta)|:|\zeta|\le\rho T\}}{T^m} 
\]
Taking $m$th~roots and letting $m\to\infty$ gives
\[ 
\limsup_{m\to\infty}\bigl(\max_{x\in B} |f_m(z)|\bigr)^{1/m} \le T^{-1} 
\]
for arbitrary~$T$, implying the assertion.
\end{proof}

\begin{remark}
The referee has remarked that it would be of interest to obtain a sharp form of
Lemma~\ref{lemma-max limit}, and has kindly supplied a proof that the
exponent~$n/2$ in~(\ref{eq:A(R) definition}) can be improved to $(n-2)/2$.
\end{remark}

\begin{corollary*}
Let $v$, $v_k$ and $\Sigma$ be as in Lemma\/~{\rm \ref{lemma-max limit}}, and let
$D$~be a bounded open set in\/~${\Bbb R}^n$ containing the ball\/
$\{|x|\le\rho\}$. Then, for $x\in\Sigma$,
\begin{equation}
|v_k(x)| \le {\scr C}_n k^{n/2} \rho^{-k} \cdot
\max_{x\in\partial D} |v(x)|, \quad k \ge 1.
\label{eq:bound on v}
\end{equation}
Also, $|v_0(x)| = |v_0(0)| \le \max\{|v(x)| : x\in\partial D\}$.
\end{corollary*}

\begin{proof}
The statement concerning~$v_0$ is obvious, so suppose $k\ge1$, and without
loss of generality, assume $\max\{|v(x)| : x\in\partial D\}$ is~$1$.

Then $|v(x)| \le 1$ for $|x|=\rho$, by the maximum principle, so $|v(\rho x)|
\le 1$ for $x\in\Sigma$. By the lemma, we have for $x\in\Sigma$,
\[ 
|v_k(\rho x)| \le {\scr C}_n k^{n/2}, 
\]
which gives (\ref{eq:bound on v}), since $v_k \in {\scr H}_k$.
\end{proof}

\begin{proof}[of Theorem, completed] 
We have, for $x\in\Gamma$,
\[ 
|f_m(x)| \le \varepsilon_m^m, 
\]
where $\varepsilon_m$ is a sequence which tends to~$0$. Hence, for
$x\in\Gamma$, $|u_m(x)|\le\varepsilon_m^m$. By the Corollary, the $u_{m,k}$
in~(\ref{eq:u definition}) satisfy
\[ 
|u_{m,k}(x)| \le {\scr C}_n k^{n/2} \rho^{-k} \varepsilon_m^m |x|^k,
   \quad k \ge 1, 
\]
and $|u_{m,0}| \le {\scr C}_n \varepsilon_m^m$, for all $x \in {\Bbb R}^n$,
where $\rho=\min_i a_i$. In particular, for $|x| \le R$, 
\[ 
|u_{m,k}(x)| \le {\scr C}'_n \cdot A^k\varepsilon_m^m R^k 
\] 
holds for every choice of
$A>\rho^{-1}$. Thus (\ref{eq:A(R) definition})~follows since $\sum_{m=0}^\infty
\varepsilon_m^m \sum_{k=0}^m (AR)^k$ is clearly convergent (for assuming, as we
may, $AR>1$, the inner sum is $\le C(AR)^{m+1}$, etc.). This completes the
proof of Theorem~\ref{theorem-Ellipsoid}.
\end{proof}

\begin{remarks}
Variants of the theorem can easily be obtained from the above estimates, for
example, {\em if $f$~is of exponential type, so is~$u$}; indeed, $f$~is of
exponential type if and only~if $\max\{|f_m(z)| : z\in K\}$ does not exceed
$(A/m)^m$ (where $A=A(K)$~is some constant), holds for some (hence every)
compact~$K$ having $0$~as an interior point. The proof now follows in the
same way as before.
\end{remarks}

\begin{definition}
Let $\Lambda$~denote the class of positive sequences $\{\lambda_m\}_0^\infty$
with $\lambda_m \searrow 0$. To each sequence $\lambda=\{\lambda_m\}$
in~$\Lambda$ we define $B_\lambda$ to be the set of all entire functions
$f = \sum_0^\infty f_m$ on~${\Bbb C}^n$ (where, as usual, $f_m \in {\scr
H}_m$) such that
\begin{equation}
\|f_m\| = o(\lambda_m^m)m^{m/2}, \quad m\to\infty,
\end{equation}
and
\begin{equation}
\|f\|_\lambda = \sup \lambda_m^{-m}m^{-m/2}\|f_m\|.
\label{eq:lambda definition}
\end{equation}
It is easy to check that $B_\lambda$ is a Banach space with the norm
$\|f\|_\lambda$. Moreover, it is separable, indeed $\sum_{m=0}^k f_m$
converges to~$f$ as $k\to\infty$, for all $f \in B_\lambda$.
\end{definition}

\begin{lemma}
If $g \in {\scr H}_m$, then
\[ 
\sum_1^n \|D_j g\|^2 = m\|g\|^2. 
\]
\end{lemma}

\begin{proof}
$mg = \sum_1^n z_j D_j g$ by Euler's formula, so
\[ 
m\|g\|^2 = \sum_1^n \langle z_jD_jg,g \rangle = \sum_1^n \|D_jg\|^2. 
\]
\end{proof}

\begin{notation}
Throughout this section, $Q$ denotes $\sum_1^n z_j^2$.
\end{notation}

\begin{remark}
In terms of the Dirichlet problem, Theorem~\ref{theorem-Ellipsoid} says the
following. {\em For~$f\in B_\lambda$, where $\lambda$~satisfies (\ref{eq:lambda
definition})} (and hence, see the following remark, {\em for every entire~$f$
of order~$<4$}), {\em the problem~(\ref{eq:lambda definition}), where\/
$\Omega$~is the unit sphere, has a solution~$u$ that is (the restriction
to\/~$\Omega$ of) an entire function in~$B_\lambda$.} It would not be hard to
modify the proof to obtain an analogous result for ellipsoids rather than
spheres (and $B_\lambda$ replaced by some related class of entire functions),
by modifying Fischer's inner product so that $\sum_1^n D_j^2$ and
multiplication by $\sum_1^n a_j^{-2}z_j^2$ become adjoint operators. However,
we have been unable to obtain Theorem~\ref{theorem-Ellipsoid} (even for
spheres) by such methods. On the other hand (and this is the point of the
following section) these methods allow a generalization from $\sum_1^n D_j^2$
to a large class of differential operators~$P(D)$ ($P$~being a homogeneous
polynomial).
\end{remark}

\begin{remark}
To give some feeling for what (\ref{eq:lambda definition})~means, let us show:
{\em every entire function of order less than four is in~$B_\lambda$, for some
$\lambda\in\Lambda$ satisfying\/} (\ref{eq:lambda definition}). Indeed, suppose
$f$~is entire and
\begin{equation}
|f(z)| \le Ae^{|z|^\rho}, \quad z\in{\Bbb C}^n,
\end{equation}
where $A$ and $\rho$ are positive constants and $\rho<4$.
\end{remark}

\section{Concluding remarks}
\label{sec-Conclusion}

\subsection{}

The basic question underlying this paper is that of finding global continuation
of the solution to Dirichlet's problem when such continuation is known both for
the equation of~$\partial\Omega$ and for the `data function'~$f$. Even when
extreme regularity is assumed, for example, $\partial\Omega$~algebraic and
$f$~entire, few results are known (even in two dimensions) about the maximal
domain to which the solution extends harmonically, let alone the nature of the
singularities that may arise. This is in contrast to the situation for Cauchy's
problem, where, for example, complete results are known in two dimensions,
based on the Schwarz function (compare \cite{Ref9,Ref12}). Moreover, G.~Johnson
\cite{Ref8} has obtained complete results for the Cauchy problem when the
initial data is an entire function restricted to a quadric surface (this for a
class of differential operators including the Laplacian). So far, there are no
results of this precision available for the Dirichlet problem.

We have already spoken of the question of whether ellipsoids are characterized
by Theorem~\ref{theorem-Ellipsoid}. In this connection, recall that when
$\Omega$~is an ellipsoid, the solution of Dirichlet's problem with data
in~${\scr P}_m$ also lies in~${\scr P}_m$ (this is a classical result
from the study of ellipsoidal harmonics, and we used it in proving
Theorem~\ref{theorem-Ellipsoid}). This has a kind of converse, which one
readily sees as follows.

\subsection{}

Concerning the material in Sections
\ref{sec-Introduction}~and~\ref{sec-Ellipsoid}, some questions remain.
Especially, it seems of interest to know when the set of solutions
of~$P^*(D)(P-1)f=0$, $f\in B_\lambda$, not merely is finite-dimensional (for
which we gave sufficient conditions, in terms of $P$~and~$\lambda$) but
consists of $0$~alone. Perhaps the uniqueness assumption in
Theorem~\ref{theorem-Ellipsoid} could be omitted---we know of no
counterexample.

\begin{acknowledgements}
This work was done while the first author was visiting Stockholm in the spring
of~1991. The first author is indebted to the Royal Institute of~Technology for
support and for providing a congenial research environment.
\end{acknowledgements}

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%
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\end{thebibliography}

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\affiliationone{Department of Mathematics\\ 
University of Arkansas\\ Fayetteville, AR 72701\\ USA}                                             
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\affiliationtwo{Mathematiska Institutionen\\ 
Kungl.\ Tekniska H\"ogskolan\\ S-100 44 Stockholm\\ Sweden} 

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