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\newcommand{\bld}[1] {{\mathbf {#1}}} \fi \ifnfsstwo \newcommand{\mitbf}[1] {\hbox{\mathversion{bold}${#1}$}} \newcommand{\rmn}[1] {{\mathrm {#1}}} \newcommand{\itl}[1] {{\mathit {#1}}} \newcommand{\bld}[1] {{\mathbf {#1}}} \fi %%% 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} \begin{thebibliography}{99} \bibitem{Ref1} {\bibname V. Bargmann}, `On a Hilbert space of analytic functions and an associated integral transform', {\it Comm.\ Pure Appl.\ Math.}\ (1961) 187--214. % \bibitem{Ref2} {\bibname F. A. Berezin}, `Covariant and contravariant symbols of operators', {\it Math.\ USSR-Izv.}\ 6 (1972) 1117--1151. % \bibitem{Ref3} {\bibname C. A. Berger \and L. A. Coburn}, `Toeplitz operators and quantum mechanics', {\it J. Funct.\ Anal.}\ 68 (1986) 273--299. % \bibitem{Ref4} {\bibname C. A. Berger \and L. A. Coburn}, `Toeplitz operators on the Segal--Bargmann space', {\it Trans.\ Amer.\ Math.\ Soc.}\ 301 (1987) 813--829. % \bibitem{Ref5} {\bibname V. Guillemin}, `Toeplitz operators in $n$~dimensions', {\it Integral Equations Operator Theory\/} 7 (1984) 154--205. % \bibitem{Ref6} {\bibname J. Janas}, `Toeplitz and Hankel operators on Bargmann spaces', {\it Glasgow Math.\ J.} 30 (1988) 315--323. % \bibitem{Ref7} {\bibname J. Janas}, `Unbounded Toeplitz operators in the Bargmann--Segal space', {\it Studia Math.}, to appear. % \bibitem{Ref8} {\bibname G. Johnson}, `The Cauchy problem in ${\Bbb C}^n$ for second-order PDE with data on a quadric surface', in preparation. % \bibitem{Ref9} {\bibname D. Khavinson \and H. S. Shapiro}, {\it The Schwarz potential in\/ ${\Bbb R}^n$ and Cauchy's problem for the Laplace equation}, Research Report TRITA-MAT-1989-36 (Royal Institute of Technology, 1989). % \bibitem{Ref10} {\bibname D. J. Newman \and H. S. Shapiro}, `A Hilbert space of entire functions related to the operational calculus', mimeographed, Ann Arbor, 1964. % \bibitem{Ref11} {\bibname D. J. Newman \and H. S. Shapiro}, {\it Fischer spaces of entire functions}, Proc.\ Sympos.\ Pure Math II (Amer.\ Math.\ Soc., Providence, RI, 1968) 360--369. % \bibitem{Ref12} {\bibname H. S. Shapiro}, {\it The Schwarz function and its generalization to higher dimensions\/} (Wiley, 1991). % \bibitem{Incorrect version} {\bibname H. S. Shapiro}, `An algebraic theorem of E.~Fischer, and the holomorphic Goursat problem', {\it Bull.\ London Math.\ Soc.}\ 21 (1989) 513--535. % \bibitem{Ref14} {\bibname H. S. Shapiro}, `Analytic continuation of the solution to Dirichlet's problem', in preparation. % \bibitem{Ref15} {\bibname B. Yu.\ Sternin \and V. E. Shatalov}, `Continuation of solutions of elliptic equations and localization of singularities', preprint, 1991. \end{thebibliography} %% Note that there is no paragraph spacing between two affiliations %% on one line \affiliationone{Department of Mathematics\\ University of Arkansas\\ Fayetteville, AR 72701\\ USA} % \affiliationtwo{Mathematiska Institutionen\\ Kungl.\ Tekniska H\"ogskolan\\ S-100 44 Stockholm\\ Sweden} \end{document}