\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{A polyharmonic analogue of a Lelong theorem } { Mohamed Boutaleb } \begin{document} \setcounter{page}{77} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 2002-Fez conference on Partial Differential Equations,\newline Electronic Journal of Differential Equations, Conference 09, 2002, pp 77--92. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A polyharmonic analogue of a Lelong theorem and polyhedric harmonicity cells % \thanks{ {\em Mathematics Subject Classifications:} 31A30, 31B30, 35J30. \hfil\break\indent {\em Key words:} Harmonicity cells, polyharmonic functions, extremal points, \hfil\break\indent Lelong transformation. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published December 28, 2002.} } \date{} \author{Mohamed Boutaleb} \maketitle \begin{abstract} We prove a polyharmonic analogue of a Lelong theorem using the topological method presented by Siciak for harmonic functions. Then we establish the harmonicity cells of a union, intersection, and limit of domains of $\mathbb{R}^n$. We also determine explicitly all the extremal points and support hyperplanes of polyhedric harmonicity cells in $\mathbb{C}^2$. \end{abstract} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} Throughout this paper, $D$ denotes a domain (a connected open) in $\mathbb{R}^n$ with $n\geq 2$, where $D$ and $\partial D$ are not empty. Since 1936, $p$-polyharmonic functions in $D$ have been used in elasticity calculus \cite{n1}. These functions are $C^\infty$-solutions of the partial differential equation \begin{equation*} \Delta ^pf(x)=\sum_{|\alpha|=p} \frac{p!}{\alpha !} \frac{\partial ^{2| \alpha | }f(x)}{\partial x_1^{2\alpha_1}\dots \partial x_n^{2\alpha _n}}=0,\quad p\in N^{*},\quad x\in D. \end{equation*} To study the singularities of these functions in $D$, Aronzajn \cite{a1,a2} considered the connected component $\mathcal{H}(D)$, containing $D$, of the open set $\mathbb{C}^n\setminus \cup_{t\in \partial D}\Gamma (t)$, where $\Gamma(t)=\{w\in \mathbb{C}^n: \sum_{j=1}^n (w_j-t_j)^2=0\}$. $\mathcal{H}(D)$ is called the harmonicity cell of $D$. Lelong \cite{l2,l3} proved that $\mathcal{H}(D)$ coincides with the set of points $w\in \mathbb{C}^n$ such that there exists a path $\gamma$ satisfying: $\gamma (0)=w$, $\gamma (1)\in D$ and $T[\gamma (\tau )]\subset D$ for every $\tau$ in $[0,1]$, where $T$ is the Lelong transformation, mapping points $w=x+iy\in \mathbb{C}^n$ to $(n-2)$-spheres $\mathbb{S}^{n-2}(x,\| y\| )$ of the hyperplane of $\mathbb{R}^n$ defined by: $\langle t-x,y\rangle =0$. This work can be divided into three sections: the first one treats a result on polyharmonic functions, the second some general properties on $\mathcal{H}(D)$, and the last one deals with a geometrical description of polyhedric harmonicity cells in $\mathbb{C}^2$. Pierre Lelong \cite{l1} proved in addition that for every bounded domain $D$ of $\mathbb{R}^n$, there exists a harmonic function $f$ in $D$ such that its domain of holomorphy $( X_f,\Phi )$ over $\mathbb{C}^n$satisfies $\Phi ( X_f) =\mathcal{H}( D)$, see also \cite{a4}. A concise proof of this result is given in Siciak's paper \cite{s1} in the case of the Euclidean ball $B_n^r=\{x\in \mathbb{R}^n;\| x\| <1\}$. In \cite{b1}, we established that the former method can be applied to arbitrary domains. Also, V.Avanissian noted in \cite{a4} that the equality: $( X_f,\Phi ) =( \mathcal{H}( D) ,Id)$ holds in the following cases: $D$ is starshaped with respect to some point $x_0$ of $D$, or $D$ is a C-domain ( that is $D$ contains the convex hull of any $(n-2)$ dimensional-sphere included in $D)$, or $D\subset \mathbb{R}^n$ with $n$ even and $n\geq 4$. The object of Section $2$ is to use a topological argument \cite{s1} to prove an analogous result for polyharmonic functions in $D$. As a consequence of this generalization we shall get \begin{quote} For every integer $1\leq p\leq [\frac n2]$ and suitable domain $D$ (say $D$ is a C-domain, or in particular a convex domain), the harmonicity cell $\mathcal{H}(D)$ is nothing else but the greatest (in the inclusion sense) domain of $\mathbb{C}^n$ whose trace on $\mathbb{R}^n$ is $D$ and to which all p-polyharmonic functions in $D$ extends holomorphically. \end{quote} In Section 3, we establish the harmonicity cell of an intersection, a union, and a limit of domains of $\mathbb{R}^n$, $n\geq 2$. We give next in Section 4 some results about plane domains, prove the existence of polyhedric harmonicity cells in $\mathbb{C}^2$, and we calculate all extremal points of the harmonicity cell of a regular polygon. For an arbitrary convex polygon $P_n$, with $n$ edges, we show that $\mathcal{H}(P_n)$ has exactly $2n$ faces in $\mathbb{R}^4$ completely determined by means of the $n$ support lines of $P_n$. It is well known by \cite{j1} that if we are given a complex analytic homeomorphism $f:D_1\to D_2$, where $D_1$, $D_2$ are domains of $\mathbb{R}^2$, $D_1,D_2$ not equal to $\mathbb{R}^2$ and $\mathbb{R}^2\simeq \mathbb{C}$, then $\mathcal{H}(D_1)$ and $\mathcal{H}(D_2)$ are analytically homeomorphic in $\mathbb{C}^2$. The holomorphic map $Jf:\mathcal{H}(D_1)\to \mathcal{H}(D_{\mathbf{2}})$ defined by $w\mapsto w'$ with: \begin{equation*} w_1'=\frac{f(w_1+iw_2)+\overline{f(\overline{w}_1+i\overline{w_2})}} 2,\quad w_2' =\frac{f(w_1+iw_2)-\overline{f(\overline{w}_1+i\overline{w_2})}}{2i} \end{equation*} realizes this homeomorphism. In proposition 4.4, we show the continuity, according to the compact uniform topology, of the above Jarnicki extension $f\mapsto Jf$ and estimate $\| (Jf)(w)\| ,w\in \mathcal{H}(D)$ by means of $\sup_{z\in D}| f(z)|$. As applications, we find the harmonicity cells of half strips and arbitrary convex plane polygonal domains (owing to an explicit calculation of their support function). \section{A polyharmonic analogue of Lelong theorem} Recall that any polyharmonic function $u$ in $D$, being in particular analytic in $D$, has a holomorphic continuation $\widetilde{u}$ in a corresponding domain $D^u$ of $\mathbb{C}^n$ whose trace with $\mathbb{R}^n$ is $D$. Therefore, given any integer $p$ ($00$ in $D$. Note that $\widetilde{h_\xi ^p}(z)$ is holomorphic in $\mathcal{H}(D)$ and infinite in any neighborhood of $\xi$. By a similar calculus, we find for every $x\in D$, \begin{gather*} \Delta ^p[\widetilde{h_\xi ^p}| D(x)]=0 \\ \Delta ^{p-1}[\widetilde{h_\xi ^p}| D(x)]\neq 0\,. \end{gather*} \paragraph{Existence of $h$:} In the following we shall make use of the lemma. \begin{lemma} \label{lm2.2} Let $\mathcal{O}[\mathcal{H}(D)]$ denote the Fr\'{e}chet space of all holomorphic functions on $\mathcal{H}(D)$, if it is endowed with the topology $(\tau )$ of uniform convergence on compact subsets of $\mathcal{H}(D)$. Then for all integer $p=1,2,\dots$, the set \begin{equation*} \mathcal{O}^p[\mathcal{H}(D)]=\{F\in \mathcal{O}[\mathcal{H}(D)];\; F| D\in {H}^p(D)\} \end{equation*} is a close subspace of $\mathcal{O}[\mathcal{H}(D)]$, and therefore it is itself a Fr\'{e}chet space. \end{lemma} \paragraph{Proof} Let us consider $F_1,F_2,\dots$. a sequence in $\mathcal{O}^p[\mathcal{H}(D)]\subset \mathcal{O}[\mathcal{H}(D)]$ converging to a function $F$ , uniformly on every compact $K'$ of $\mathcal{H}(D)$. It is well known by a theorem of Weierstrass that $F$ is also holomorphic in $\mathcal{H}(D)$, it remains thus to verify that $\Delta ^p(F| D)=0$, $p=1,2,\dots$. By \cite{c1}, page 161, for all multi-index $\beta =(\beta _1,\dots ,\beta _n)\in \mathbb{N}^n$: $D^\beta F_j\to D^\beta F$, uniformly on every compact $K'$ of $\mathcal{H}(D)$; in particular we also have $(D^\beta F_j)| D\to (D^\beta F)| D$ uniformly on any compact $K\subset D$ since we may treat all $K'\cap \mathbb{R}^n\neq \emptyset$ as compact subsets of the real subspace in the complex $(z_1,\dots ,z_n)-$space. Now, note that \begin{align*} (D^\beta F_j)| D&=(D_z^\beta F_j)| D\\ &=(\frac{\partial ^{| \beta | }F_j}{\partial z_1^{\beta _1}\dots \partial z_n^{\beta _n}})| D=\frac{\partial ^{| \beta | }}{\partial x_1^{\beta _1}\dots \partial x_n^{\beta_n}}(F_j| D)=D_x^\beta (F_j| D), \end{align*} where $z_j=x_j+iy_j$ , $j=1,\dots ,n$. Then for $q=1,2,\dots ,p-1$, the sequence \begin{align*} (\Delta _z^qF_j)| D&=[(\sum_{j=1}^n \frac{\partial ^2}{\partial z_j^2})^qF_j]| D =(\sum_{| \alpha | =q} \frac{q!}{\alpha !}D_z^{2\alpha }F_j)| D \\ &=\sum_{| \alpha | =q} \frac{q!}{\alpha !}D_x^{2\alpha }(F_j| D) =\Delta _x^q(F_j| D), \end{align*} being a finite sum of derivatives $(D^\beta F_j)| D$, we have $\Delta_x^q(F_j| D)\to \Delta _x^q(F| D)$, uniformly on every compact $K$ of $D$. Putting $F_j| D=f_j$ and $F| D=f$, we have also for every $x\in D$: $\lim_{j\to \infty } [\Delta ^qf_j(x)]=\Delta ^qf(x)$, $q=1,2,\dots ,p-1$. Since each $f_j$ is supposed $p$-polyharmonic in $D$ for $1\leq p\leq [ \frac n2]$, we have $f_j\in \mathbf{C}_{\mathbb{R}}^{2p}(D)$ and $f_j$ satisfies the appropriate mean value property, see \cite{a4}: $$\label{1} \lambda (f_j,x,R)=f_j(x)+\sum_{q=1}^{p-1} a_qR^{2q}\Delta ^qf_j(x)\quad$$ for all x $\in D$, and $R>0$ so small that $B_n^r(x,R)=\{y\in \mathbb{R}^n;\| y-x\| 0$, $D\subset B_n^r(0,R)=\{x\in \mathbb{R}^n;\| x\| 0$ is small enough for $w_0$ to belong at $B_n^c$ and for $T(w_0)$ to contain a certain $\xi _0\in \mathbb{R}^n$ with $\| \xi _0\|$ $\geq 1$. Taking $[n+2$ $\sqrt{n-1}]^{-1/2}<\rho <1/n$ and writing $w_0=x_0+i$ $y_0$ we see that a $\xi _0$ satisfying \begin{equation*} [ \langle \xi _0-x_0,y_0\rangle =0,\quad \| \xi _0-x_0\| =\| y_0\|, \quad\text{and}\quad \| \xi _0\| \geq 1] ; \end{equation*} that is, \begin{equation*} \rho \xi _1=0,\quad \xi _1^2+(\xi _2-\rho )^2+\dots (\xi _n-\rho )^2=\rho ^2\quad \text{and}\quad \xi _1^2+\dots +\xi _n^2\geq 1 \end{equation*} is given by: $\xi _0=\rho [1+(n-1)^{\frac{-1}2}](0,1,\dots ,1)$. \begin{remark} \label{rmk3.5} \rm Due to propositions 3.1 and 3.2 above, the definition of a harmonicity cell may be naturally extended to arbitrary open sets of $\mathbb{R}^n$ for $n\geq 1$ as follows $\mathcal{H}(\emptyset )=\emptyset$, $\mathcal{H}(\mathbb{R}^n)=\mathbb{C}^n$, $\mathcal{H}(]a,b[)=\mathbb{C}$ for $]a,b[\subset \mathbb{R}$, and $\mathcal{H}(O)=\cup_{i\in I}\mathcal{H}(O_i)$, where $O$ is an open set of $\mathbb{R}^n$, $(O_i)_{i\in I}$ the family of the connected components of $O$. \end{remark} \begin{remark} \label{rmk3.6} \rm Some properties are not always preserved by $D\mapsto \mathcal{H}(D)$; this is especially the case if: \begin{itemize} \item[(i)] $D$ is simply connected in $\mathbb{R}^n$ with $n\geq 3$. Indeed, the two domains $D=\mathbb{R}^n-\{0\}$ and $\mathcal{H}(D)=\mathbb{C}^n-\{z\in \mathbb{C}^n;z_1^2+\dots +z_n^2=0\}$, having $0$ and $\mathbb{Z}$ respectively as fundamental groups, they offer then an example of a not simply connected harmonicity cell corresponding to a real simply connected domain; for $\pi _1[\mathcal{H}(D)]=\mathbb{Z}$, see \cite{b2}. \item[(ii)] $D$ is strictly convex in $\mathbb{R}^n$with $n\geq 2$. An example is given by the harmonicity cell of the unit ball $B_n^r$ of $\mathbb{R}% ^n$. If $\mathcal{E}( \overline{V})$ denotes the set of all extremal points of a convex $V$ we have $\mathcal{E}( \overline{B_n^r}% ) =\partial B_n^r$ since these two sets coincide with the unit Euclidean sphere $S^{n-1}$ of $\mathbb{R}^n$. Nevertheless, by \cite{h2}: $\mathcal{E}( \overline{\mathcal{H}( B_n^r) }) =\partial ^{\vee }[ \mathcal{H}( B_n^r) ] =\{w=xe^{i\theta }\in \mathbb{C} ^n;x\in S^{n-1},\theta \in \mathbb{R\}}$, where $\partial ^{\vee }U$ denotes the \^Silov boundary of $U\subset \mathbb{C}^n$; thus: $\mathcal{E}( \overline{\mathcal{H}( B_n^r) }) \stackrel{\neq }{\subset }\partial [ \mathcal{H}( B_n^r) ]$. \item[(iii)] $D$ is partially - circled in $\mathbb{C}^n\simeq \mathbb{R}^{2n}$, $n\geq 2$, that is (for instance): $z\in D\Rightarrow (z_1,\dots ,z_{n-1},e^{i\theta}z_n)\in D$, for all $\theta \in \mathbb{R}$. Indeed if $D=B_n^c=\{z\in \mathbb{C}^n;\| z\| <1\}$, $\mathcal{H}( B_n^c)$ is not partially - circled in $\mathbb{C}^{2n}$ with respect to $w_{2n}$ since $w_0=\sqrt{1+2n}(1,\dots ,1)\in \mathbb{C}^{2n}$ satisfies $L(w_0)=\sqrt{2n/(1+2n)}<1$, but $L[(2n+1)^{\frac{-1}2},\dots ,(2n+1)^{\frac{-1}2},i(2n+1)^{\frac{-1} 2}]=[2n+2\sqrt{2n-2}]^{\frac 12}(2n+1)^{\frac{-1}2}>1$. On the other hand, $B_n^c$ is even circled (at the origin).\end{itemize} \end{remark}\section{Harmonicity cells of polygonal plane domains}The case $n=2$ is rather special since the Lelong map $T$ is given by: $T(z)=\{z_1+iz_2$, $\overline{z_1}+i\overline{z_2}\}$, where $z\in \mathbb{C}^2$ and $\mathbb{R}^2\simeq \mathbb{C}$. So, in \cite{b1}, we have determined explicitly the harmonicity cells of some plane domains and shed light on the close connection between the set $\mathcal{E}(\overline{D})$, of all the extremal points of a convex domain $D$ of $\mathbb{R}^2$, and the set $\mathcal{E}(\overline{\mathcal{H}(D)})$, see also \cite{a4}. We will give now some properties and constructions which are proper to the complex plane. More precisions on the Jarnicki extension given in Section 1 will also be established.\begin{proposition} \label{prop4.1} The operator $\mathcal{H}:\mathfrak{D}^2\to \mathfrak{C}_s^2$ satisfies \begin{itemize} \item[a)] If $D$ is circled at $z_0\in \mathbb{C}$, balanced at $z_0\in D$, or simply connected, then so is $\mathcal{H}(D)$ respectively.\item[b)] If $P_n^a$ is an arbitrary convex polygon with $n$ edges, then the harmonicity cell $\mathcal{H}(P_n^a)$ is of polyhedric form in $\mathbb{C}^2$ with $2n$ faces and $n^2$ vertices. Furthermore, identifying $\mathbb{C}^2$ with $\mathbb{R}^4$ by writing $y$ $=(x_3,x_4)$ and $x+iy=(x_1,x_2,x_3,x_4)$, each support line of $P_n^a$ defined, for a certain $j=1,\dots ,n$, by $a_jx_1+b_jx_2-\alpha _j=0$, $(a_j,b_j,\alpha _j\in \mathbb{R})$, generates two support hyperplanes of $\mathcal{H}(P_n^a)$ of respective equations: \begin{equation*} a_jx_1+b_jx_2+b_jx_3-a_jx_4-\alpha _j=0\quad \text{and} \quad a_jx_1+b_jx_2-b_jx_3+a_jx_4-\alpha _j=0. \end{equation*} \item[c)] Let $P_n^r$ denote the regular polygon which vertices are $\omega _k=e^{2ik\pi /n}$, $k=0,\dots ,n-1$. Then \begin{align*} \mathcal{H}(P_n^r)=&\Big\{w=x+iy\in \mathbb{C}^2: x_1\cos (2k+1)\frac \pi n+x_2\sin (2k+1)\frac \pi n\\ &+\sqrt{\| y\| ^2-[y_1\cos (2k+1)\frac \pi n+y_2\sin (2k+1)\frac \pi n]^2}<\cos \frac \pi n,\\ &k=0,\dots ,n-1\Big\}. \end{align*}\item[d)] The $n^2$ vertices of $\overline{\mathcal{H}(P_n^r)}$ are given by $\omega _{km}=x_{km}+iy_{km}$ and $\overline{\omega _{km}}=x_{km}-i y_{km}$, $(0\leq k\leq m\leq n-1)$, where \begin{gather*} x_{km}=\frac 12(\cos \frac{2k\pi }n+\cos \frac{2m\pi }n,\sin \frac{2k\pi } n+\sin \frac{2m\pi }n),\\ y_{kk}=0,\; k=0,\dots ,n-1, \\ y_{km}=\frac{\sin \pi (m-k)\text{ }/n}{\sqrt{2}[1-\cos 2\pi (m-k)\text{ }/n ]^{1/2}}\\ \times(\sin \frac{2\pi m}n-\sin \frac{2\pi k}n,\cos \frac{2\pi k} n-\cos \frac{2\pi m}n). \end{gather*} \end{itemize} \end{proposition} \paragraph{Proof} a) For $\theta \in \mathbb{R}$, $z_0=a+ib\in \mathbb{C}$, and $w=(w_1,w_2)\in \mathcal{H}(D)$, we see that $z_0+e^{i\theta }w$ remains in $\mathcal{H}(D)$. Since $T(z_0+e^{i\theta }w)=\{a+e^{i\theta }w_1+i(b+e^{i\theta }w_2)$, $a+e^{-i\theta }\overline{w_1}+i(b+e^{-i\theta }\overline{w_2} )\}=\{z_0+e^{i\theta }(w_1+iw_2),z_0+e^{-i\theta }(\overline{w_1} +i\overline{w_2})\}$, and as $D$ is circled with respect to $z_0$, we have $T(z_0+e^{i\theta }w)\subset D$. If the above circled domain $D$ is supposed starshaped at $z_0$ too, then $\mathcal{H}(D)$ is also starshaped at $z_0$ (by 3.1.d) that is, $\mathcal{H}(D)$ is balanced at $z_{0}$. Let $D\in \mathcal{D}^2$ be a simply connected domain and $f$ a holomorphic one-one map sending $D$ onto $B=\{z\in \mathbb{C};| z| <1\}$. By Jarnicki Theorem , $f$ extends to a holomorphic homeomorphism $Jf:\mathcal{H}(D)\to \mathcal{H}(B)$. Now, by \cite{a4}, $\mathcal{H}(B)$ is the unit disk of $(\mathbb{C}^2,L)$, where $L$ is the Lie norm; this means that $\mathcal{H}(B)$ is convex and in particular simply connected. Since $Jf$ is a homeomorphism, $\mathcal{H}(D)$ is also simply connected. \noindent b) Suppose that $P_n^a$ is defined by: $$P_n^a=\{x=x_1+ix_2\in \mathbb{R}^2;\langle x,V^j\rangle <\alpha _j ,j=1,\dots ,n\},$$ with given vectors $V^j=(a_j,b_j)\in \mathbb{R}^2$ and scalars $\alpha _j\in \mathbb{R}$. By 3.1.d, one has $w=x+iy\in \mathcal{H}(P_n^a)\Longleftrightarrow x+T(iy)\subset P_n^a\Longleftrightarrow x+\xi \in P_n^a,\forall \xi \in T(iy)\Longleftrightarrow \langle x,V^j\rangle +\max_{\xi \in T(iy)}\langle \xi ,V^j\rangle <\alpha _j$, $j=1,\dots ,n$. Since $T(iy)=\{(-y_2,y_1),(y_2,-y_1)\}$, we have \begin{equation*} \mathcal{H}(P_n^a)=\{w=x+iy\in \mathbb{C}^2;\langle w,U^j\rangle <\alpha _j\text{ and } \langle w,W^j\rangle <\alpha _j,\; j=1,\dots ,n\}, \end{equation*} where $w=(x_1,x_2,x_3,x_4)$, $y=(x_3,x_4)$, $U^j=(a_j,b_j,-b_j,a_j)$, and $W^j=(a_j,b_j,b_j,-a_j)$, while $\langle,\rangle$ denotes the usual scalar product in $\mathbb{R}^4$. From the expression above, we deduce that the harmonicity cell of an arbitrary convex polygon (not necessarily bounded) with $n$ edges is a polyhedron of $\mathbb{C}^2\simeq \mathbb{R}^4$ having $2n$ faces and by \cite{b1}, $n^2$ vertices. \noindent c) For the regular polygon $P_n^r$, we have also another expression of its harmonicity cell. Indeed, if $\mathbb{C}\simeq \mathbb{R}^2$, we put $\omega _n=\omega _0,\omega _k=(\cos \frac{2k\pi }n,\sin \frac{2k\pi }n)$, and $V^k=\omega _{k+1}-\omega _k=(a_k,b_k)$, $k=0,\dots ,n-1$. By (b) we have \begin{equation*} \mathcal{H}(P_n^r)=\big\{x\in \mathbb{R}^2;\langle x,V^k\rangle +\max_{\xi \in T(iy)}\langle \xi ,V^k\rangle <\cos \frac \pi n,k=0,\dots ,n-1\big\}. \end{equation*} By the method of Lagrange multipliers \cite{a4}, we find $\max_{\xi \in T(iy)}\langle \xi ,V^k\rangle =[\| y\|^2 -\langle y,V^k\rangle^2]^{1/2}$; the announced expression of $\mathcal{H}(P_n^r)$ follows. \noindent d) Applying the following two lemmas proved in \cite{b1}, (see also \cite{a4}) we obtain all the extremal points of $\overline{\mathcal{H}(P_n^r)}$ by means of those of $\overline{P_n^r}$ \hfill$\square$ \begin{lemma} \label{lm4.2} If $D$ is a non empty convex domain of $\mathbb{R}^n$, $n\geq 2$, $\partial D\neq\emptyset$, then $\mathcal{E}(\overline{D)}\subset \mathcal{E}(\overline{\mathcal{H}(D)})$. \end{lemma} \begin{lemma} \label{lm4.3} Let $D$ be a non empty convex domain, $\partial D\neq \emptyset$, in $\mathbb{C}\simeq \mathbb{R}^2$. \\ a) Every point $w\in \mathcal{E}(\overline{\mathcal{H}(D)})$ satisfies $T(w)\subset \mathcal{E}(\overline{D)}$. \\ b)Conversely, given arbitrary points $a$ and $b$ of $\mathcal{E}(\overline{D)}$, there exists $w\in \mathcal{E}(\overline{\mathcal{H}(D)})$ such that $T(w)=\{a,b\}$. \end{lemma} Let $U,V$ be two domains of $\mathbb{C}^n$, $n\geq 1$. we denote $\hom (U,V)$ the set of all holomorphic homeomorphisms $F:U\to V$, and $\hom _r(\mathcal{H}(D),\mathcal{H}(D'))$ the set of all $F\in \hom (\mathcal{H}(D),\mathcal{H}(D'))$ of which the restriction $F|_D$ belongs to $\hom (D,D')$, where $D,D'\in \mathcal{D}^2$ and $\mathbb{C}\simeq \mathbb{R}^2$. \begin{proposition} \label{prop4.4} Let $D,D'\subset \mathbb{C}$ be two non empty domains with $D\neq \mathbb{C}$, $D'\neq \mathbb{C}$. The Jarnicki extension $J$ is an injective continuous mapping from $\hom (D,D')$ onto $\hom _r(\mathcal{H}(D),\mathcal{H}(D'))$ according to the compact uniform topology ($\tau$). Furthermore, $\hom _r(\mathcal{H}(D),\mathcal{H}(D'))\simeq \hom (D,D')$ (topologically homeomorphic); and for a holomorphic homeomorphism $f:D\to D'$ we have the estimate \begin{equation*} \| Jf(w)\| \leq \sup_{z\in D}| f(z)| ,\quad \text{ for every }w\in \mathcal{H}(D). \end{equation*} \end{proposition} \paragraph{Proof} If $f$ and $f'$ are such that $Jf=Jf'$ on $\mathcal{H}(D)$ then by \cite{j1}, $f=(Jf)| D=(Jf')| D=f'$ on $D$. Let $(f_n)_{n\geq 1}$ be a convergent sequence in $(\hom (D,D'),\tau )$. By 3.2.b,to test $(J$ $f_n)_{n\geq 1}$ for compact uniform convergence in the harmonicity cell of $D$ it is not really necessary to check uniform convergence on every compact set $K$ in $\mathcal{H}(D)$ - checking it on the closed harmonicity cells $\overline{\mathcal{H}(D_0)}$ where $D_0$ is an arbitrary relatively compact domain in $D$ is enough. Now if $w_0\in \mathcal{H}(D_0)$ with $w_0=(w_1^0,w_2^0)$: $$\| Jf_n(w_0)-Jf(w_0)\|^2=A_n^2(w)+B_n^2(w),$$ where $f=\lim_{n\to \infty }f_n$, and \begin{gather*} A_n=\frac 12| [f_n(w_1^0+iw_2^0)-f(w_1^0+iw_2^0)] +[\overline{f_n(\overline{w_1^0}+i\overline{w_2^0})} -\overline{f(\overline{w_1^0}+i\overline{w_2^0})}]|,\\ B_{n=}\frac 12| [f_n(w_1^0+iw_2^0)-f(w_1^0+iw_2^0)] -[\overline{f_n(\overline{w_1^0}+i\overline{w_2^0})} -\overline{f(\overline{w_1^0}+i\overline{w_2^0})}]|. \end{gather*} Both $A_n$ and $B_n$ are bounded above by $\frac 12\sup_{w\in \mathcal{H}(D_0)}|f_n(w_1+iw_2)-f(w_1+iw_2)| +\frac 12\sup_{w\in \mathcal{H}(D_0)}| f_n(\overline{w_1} +i\overline{w_2})-f(\overline{w_1}+i\overline{w_2})$. By 3.1.h: $w\in \mathcal{H}(D_0)$ if and only if $w_1+iw_2\in D_0$ and $\overline{w_1}+i\overline{w_2}\in D_0$. Thus: \begin{gather*} A_n\leq \sup_{z\in D_0}| f_n(z)-f(z)|, \quad B_n\leq \sup_{z\in D_0}| f_n(z)-f(z)|, \\ \sup_{w\in \overline{\mathcal{H}(D_0)}} \|Jf_n(w)-Jf(w)\| \leq \sqrt{2}\sup_{z\in \overline{D_0}} | f_n(z)-f(z)|. \end{gather*} Since $\lim_{n\to \infty }\sup_{z\in\overline{D_0}}| f_n(z)-f(z)| =0$, we have $J f_n\to Jf$, according to ($\tau$). The mapping $J:\hom (D,D')\to \hom _r(\mathcal{H}(D),\mathcal{H}(D'))$ is continuous and injective. To see that this mapping is onto, take $F\in \hom _r(\mathcal{H}(D),\mathcal{H}(D'))$ and observe that (by \cite{j1}) $J(F|D)$ and $F$ are both holomorphic homeomorphisms from $\mathcal{H}(D)$ onto $\mathcal{H}(D')$ having the same restriction on $D:(J(F|D))|D=F|D$. So by the uniqueness principle of analytic extension in $\mathbb{C}^n:J(F|D)=F$. Conversely, putting: $R=J^{-1}$and making use of 3.1.c, e and 3.2.b, we have for every $D_0\subset D$ with $\overline{D_o}$ compact: $\sup_{\overline{\mathcal{H}(D_0)}}\| F_n-F\| \geq \sup_{\overline{D_0}}|RF_n-RF|$, which implies that $R$ is also continuous. Finally,we have \begin{align*} \| Jf(w)\|^2=&\frac 14| f(w_1+iw_2)+\overline{ f(\overline{w_1}+i\overline{w_2})}| ^2+\frac 14| f(w_1+iw_2)-\overline{ f(\overline{w_1}+i\overline{w_2})}| ^2\\ =&\frac 12[| f(w_1+iw_2)| ^2+| f(\overline{w_1}+i \overline{w_2})| ^2]\\ \leq& \frac 12\big[(\sup_{\overline{D}}| f| )^2 +(\sup_{\overline{D}}| f| )^2\big] =(\sup_{\overline{D}} | f| )^2. \end{align*} \quad \hfill$\square$ \begin{remark} \label{rmk4.5} \rm The notion of harmonicity cells has a functorial aspects; indeed let $\mathfrak{D}^2$ still denote the category of all domains $D$ of $\mathbb{R}^2\simeq \mathbb{C}$, $D\neq \emptyset$, $\partial D\neq \emptyset$ with arrows in $\hom (D_1,D_2)$, and $\mathfrak{C}_s^2$ the category of all domains $U$ of $\mathbb{C}^2$ which are symmetric with respect to $\mathbb{R}^2$, with arrows $F$ in $\hom (U_1,U_2)$. Then, by the uniqueness theorem of holomorphic continuation in $\mathbb{C}^n$, to the composition: $D_1\stackrel{f}{\to }D_2\overset{g}{\to }D_3$ corresponds $\mathcal{H}(D_1)\overset{Jf}{\to } \mathcal{H}(D_2)\overset{Jg}{\to }\mathcal{H}(D_3)$ such that: $J(g\circ f)=( Jg) \circ ( Jf)$; next $f=Id$ in Jarnicki Theorem (Section 1) gives: \\ $J$ $Id_D=Id_{\mathcal{H}(D)}$. This means that the operator: $D\in \mathfrak{D}^2\mapsto \mathcal{H}(D)\in \mathfrak{C} _s^2$ and $f\in \hom (D_1,D_2)\mapsto \mathcal{H}(f)=Jf\in \hom [\mathcal{H}(D_1),\mathcal{H}(D_2)]$ may be considered as a covariant functor between the said categories. The representability of this functor and its classifying object will be discussed in a further paper. \end{remark} \paragraph{Example} If $V$ is an arbitrary half strip of $\mathbb{R}^2$, there exists an usual transformation $f$, mapping $V$ onto $V'=\{x\in \mathbb{R}^2 : x_1>a, k_10$, $k_1,k_2\in \mathbb{R}$. Now by \cite{a4,c1}, we have for all convex domains $U$ of $\mathbb{R}^n$ ($n\geq 2$): $\mathcal{H}(U)=\big\{w=x+iy\in \mathbb{C}^n;\max_{t\in T(iy)} [ \max_{\xi \in S^{n-1}}( \langle x+t,\xi\rangle-\sup_{u\in U} \langle \xi ,u\rangle) ] <0\}.$ This formula gives $\mathcal{H}(U)$ by means of the support function of $U:\delta _U( \xi ) =\sup_{u\in U}\langle \xi ,u\rangle$. Making use of the fact that the function $u\mapsto \xi _1u_1+\xi _2u_2$, being harmonic in $V'$, attains its supremum at some point of $\partial V'$. We find by simple calculations that $\delta _{V'}( \xi ) =\begin{cases} +\infty & \text{if }\xi _1> 0 \\ a\xi _1+k_2\xi _2 & \text{if }\xi _1\leq 0 \text{ and } \xi _2\geq 0 \\ a\xi _1+k_1\xi _2 & \text{if }\xi _1\leq 0 \text{ and } \xi _2\leq 0 \end{cases}$ where $\xi \in \Gamma$, the unit circle of $\mathbb{C}$. Next, to search the supremum on $\Gamma$ of the function $g(\xi_1,\xi _2)=\langle x+t,\xi \rangle -\delta _{V'}( \xi )$, we restrict the study to $\{\xi \in \Gamma :\xi _1\leq 0\}$. Since $g(\xi _1,\xi _2)=g(\xi _1,\pm \sqrt{1-\xi _1^2})$, with $\xi _1\in [-1,0]$, we put $g_1(\xi _1)=g(\xi _1,\sqrt{1-\xi _1^2})=\alpha _1\xi _1+\alpha _2\sqrt{1-\xi _1^2}\quad \text{and}\quad g_2(\xi _1)=\alpha _1\xi _1-\beta \sqrt{1-\xi _1^2},$ where $\alpha _1=x_1+t_1-a$, $\alpha _2=x_2-t_2-k_2$, $\beta =x_2-t_2-k_1$. One obtains that $g_1'(\xi _1)=0$ if $\xi _1=\pm \alpha _1/\sqrt{\alpha _1^2+\alpha _2^2}$ (when $\alpha _1\neq 0$ or $\alpha _2\neq 0$). In addition, the study of variations of $g_1(\xi_1)$, in $-1\leq \xi _1\leq 0$, in each of the three cases: $\alpha _1\leq 0$, ($\alpha _1\geq 0$ and $\alpha _2\leq 0)$, and ($\alpha _1\geq 0$ and $\alpha_2\geq 0)$ leads to $\max_{-1\leq \xi _1\leq 0}g_1( \xi_1) =\max (-\alpha _1,\alpha _2)$. Obviously, this equality holds even if $\alpha _1=\alpha _2=0$. A similar calculus for $g_2( \xi _1)$ gives $\max_{-1\leq \xi _1\leq 0}g_2( \xi _1) =\max(-\beta ,-\alpha _2)$. Putting $\gamma =\max ($ $-\alpha _1,\alpha _2)$, $\delta =-\min(\beta ,\alpha _2)$, and as $T(iy)=\{(-y_2,y_1),(y_2,-y_1)\}$, we obtain the equivalence $\max (\gamma ,\delta )<0\Leftrightarrow \left\{\begin{array}{c} a-x_1+y_2<0,x_2+y_1-k_2<0,k_1-x_2-y_1<0, \\ a-x_1-y_2<0,x_2-y_1-k_2<0,k_1-x_2+y_1<0. \end{array} \right.$ At last, writing $\min (u,v)=\frac 12(u+v-| u-v| )$ , and by the Jarnicki extension $f\mapsto Jf=\widetilde{f}$ (see section 1), we deduce $\mathcal{H}(V)=(\widetilde{f})^{-1}[\mathcal{H}(V')]$ , where \[ \mathcal{H}(V')=\{w=x+iy\in \mathbb{C}^2;| y_1| <\frac{k_2-k_1}2-| x_2-\frac{k_1+k_2}2| ,| y_2| 0$. Next, applying successively the translation$\tau _{-\beta }$, the homothety$h_{\frac 1R}$and a suitable rotation$\rho _\theta $, we obtain$P_{n}^r=\rho_\theta h_{1/R}\tau _{-\beta }P_{n,r}'$which is studied in Proposition \ref{prop4.1}.c. Note that the same process applies to arbitrary regular polyhedrons in$\mathbb{R}^n$,$n\geq 3$. \begin{thebibliography}{00} \frenchspacing \bibitem{a1} N. Aronszajn: Sur les d\'{e}compositions des fonctions analytiques uniformes et sur leurs applications, Acta. math. 65 (1935) 1-156. \bibitem{a2} N. Aronszajn, M. C. Thomas, J. L.Leonard: Polyharmonic functions, Clarendon. Press. Oxford(1983). \bibitem{a3} V. Avanissian: Sur les fonctions harmoniques d'ordre quelconque et leur prolongement analytique dans$\mathbb{C}^n$. S\'{e}minaire P. Lelong-H.Skoda, Lecture Notes in Math,$n^0919$, Springer-Verlag, Berlin (1981) 192-281. \bibitem{a4} V. Avanissian: Cellule d'harmonicit\'{e} et prolongement analytique complexe, Travaux en cours, Hermann, Paris (1985). \bibitem{b1} M. Boutaleb: Sur la cellule d'harmonicit\'{e} de la boule unit\'{e} de$\mathbb{R}^n$- Doctorat de 3$^0\$cycle, U.L.P. Strasbourg, France (1983). \bibitem{b2} E. Brieskorn: Beispiele zur Differentialtopologie von Singularit\"{a}ten, Inv.Math.2 (1966) 1-14. \bibitem{c1} R.Coquereaux, A.Jadczyk: Conformal Theories, Curved phase spaces, Relativistic wavelets and the Geometry of complex domains, Centre de physique th\'{e}orique, Section 2, Case 907. Luminy, 13288. Marseille, France (1990). \bibitem{h1} M. Herv\'{e}: Les fonctions analytiques, Presses Universitaires de France (1982), Paris. \bibitem{h2} L. K. Hua: Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Transl of Math. Monographs 6, Amer. Math. Soc. Provid. R. I.(1963). \bibitem{j1} M. Jarnicki: Analytic Continuation of harmonic functions, Zesz. Nauk. U J, Pr. Mat 17, (1975) 93-104. \bibitem{l1}P. Lelong: Sur la d\'{e}finition des fonctions harmoniques d'ordre infini, C. R. Acad. Sci. Paris 223 (1946) 372-374. \bibitem{l2} P. Lelong: Prolongement analytique et singularit\'{e}s complexes des fonctions harmoniques, Bull. Soc. Math. Belg. 7 (1954-55) 10-23.\bibitem{l3} P. Lelong: Sur les singularit\'{e}s complexes d'une fonction harmonique, C. R. Acad. Sci. Paris 232 (1951) 1895-1897. \bibitem{n1} M. Nicolesco: Les fonctions polyharmoniques, Hermann Paris(1936). \bibitem{r1} W. Rudin: Functional Analysis, Mc Graw Hill (1973). \bibitem{s1} J. Siciak, Holomorphic continuation of harmonic functions. Ann. Pol. Math. XXIX (1974) 67-73.1. \end{thebibliography} \noindent \textsc{Mohamed Boutaleb} \\ D\'epartement de Math\'ematiques et Informatique\\ Facult\'e des Sciences Dhar-Mahraz \\ B. P. 1796 Atlas, F\`es, Maroc\\ e-mail: mboutalebmoh@yahoo.fr \end{document}