\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 $p$ ($0<p<+\infty )$ and any domain $D$
of $\mathbb{R}^n$, one can associate a domain $\mathcal{NH}(D)$ of
$\mathbb{C}^n$ (depending on $D$ only) such that the whole class
${H}^p(D)$, of all $p$-polyharmonic functions in $D$, extends
holomorphically to $\mathcal{NH}(D)$. This last complex domain, called the
kernel of $\mathcal{H}(D) $, coincides with the set of all
$z\in \mathcal{H}( D) $ satisfying $C_h[T(z)]\subset D$, where $C_h[T(z)]$
denotes the convex hull of $T(z)$. For more details see \cite{a4}.

Making use of a topological argument appearing in \cite{s1}, we will show now the
following theorem.

\begin{theorem} \label{thm2.1}
Let $D$ be a bounded domain of $\mathbb{R}^n$, $n\geq 2$, $D\neq \emptyset$,
$\partial D\neq \emptyset$, and $\mathcal{H}( D)$ its harmonicity cell.
Then for all integer $1\leq p\leq [ \frac n2] $, $([ \frac n2] $
is the integer part of $\frac n2)$ and all domains
$\widetilde{\Omega }\supset \mathcal{H}( D) $ the problem for the
$2p$-order linear partial differential operator $\Delta ^p$
\begin{gather*}
\Delta ^pu=0 \quad \text{ in }D \\
\overline{D_1}\widetilde{u}=\dots =\overline{D_n}\widetilde{u}=0 \quad
\text{in }\mathcal{H}( D)
\end{gather*}
has a solution $h\in {H}^p(D)$ which cannot be
holomorphically continued in $\widetilde{\Omega }$.
Here   $\Delta =\Delta _x=\partial _{x_1x_1}+\dots +\partial _{x_nx_n}$
is the usual Laplacian of $\mathbb{R}^n,\overline{D_j}=\frac \partial {\partial
\overline{z_j}}$ , $j=1,2,\dots ,n$.
\end{theorem}

\paragraph{Proof} Let $\xi \in \partial \mathcal{H}( D) $
and $1\leq p\leq [ \frac n2] $. Firstly, we will construct
explicitly a $p$-polyharmonic function $h_\xi $ in $D$ whose holomorphic
continuation $\widetilde{h_\xi }$ in $\mathfrak{N}\mathcal{H}(D)$ extends to the
whole of $\mathcal{H}( D) $; however $\widetilde{h_\xi }$ cannot be
holomorphically continued in a neighborhood of $\xi$.
Next, we will deduce by a topological reasoning the
existence of a $p$-polyharmonic function $h$ in $D$ such that
$\mathcal{H}(D) $ is the domain of holomorphy of $\widetilde{h}$.

\paragraph{Construction of $h_\xi$.}
1) $D\subset \mathbb{C\simeq R}^2$: by \cite{l2}, the boundary point $\xi $
belongs to some isotropic cone of vertex a $t\in \partial D$ , i.e.
$\xi \in \Gamma (t)$, or
$t\in T(\xi )=\{\xi _1+i\xi _2,\overline{\xi _1}+i\overline{\xi _2}\}$.\\
a) If $t=\xi _1+i\xi _2$ , we consider the function
\begin{equation*}
\widetilde{h_\xi }(z)=\mathop{\rm Ln}\big\{[(\xi _1+i\xi _2)-(z_1+iz_2)]
[\overline{(\xi _1+i\xi _2)-(\overline{z_1}+i\overline{z_2})}]\big\},
\end{equation*}
where the branch is chosen such that $\widetilde{h_\xi }$ is real in
$D$. Note that $\widetilde{h_\xi }$ is holomorphic in
$\mathcal{H}(D)$, its restriction $\widetilde{(h_\xi }|D)(x)
=2\mathop{\rm Ln}\| x-t\|$,
where $x=x_1+ix_{2}$ is harmonic in $D$, and
$\lim_{z\to  \xi }| \widetilde{h_\xi }(z)| =\infty $.
Hence $\widetilde{h_\xi }$ cannot be holomorphically continued beyond $\xi $.
\\
b) If $t=\overline{\xi _1}+i\overline{\xi _2}$ , the function
\begin{equation*}
\widetilde{h_\xi }(z)=\mathop{\rm Ln}
\big\{[(\overline{\xi _1}+i\overline{\xi _2})
-(z_1+iz_2)][\overline{(\overline{\xi _1}+i\overline{\xi _2})
-(\overline{z_1}+i\overline{z_2})}]\big\},
\end{equation*}
satisfies the same requirements of (a).
\noindent
2) $D\subset $ $\mathbb{R}^n$, $n\geq 3$:
\\
a) Suppose $n$ even $\geq 4$. There exists by \cite{l2} a point $t\in \partial D$
such that $\sum_{j=1}^n (\xi _j-t_j)^2=0$. Consider then
$\widetilde{h_\xi ^p}:\mathcal{H}( D) \to  \mathbb{C}$,
$z=(z_1,\dots ,z_n)\mapsto \widetilde{h_\xi ^p}(z)$ with
\begin{equation*}
\widetilde{h_\xi ^p}(z)=
\begin{cases}
\frac 1{[(z_1-t_1)^2+\dots (z_n-t_n)^2]^{\frac n2-p}} &  \text{when } 1\leq
p\leq \frac n2-1  \\
\mathop{\rm Ln}\sum_{j=1}^n(z_j-t_j)^2 &
\text{when }  p=\frac n2 \end{cases}
\end{equation*}
(The branch is chosen in the complex logarithm such that
($\widetilde{h_\xi}| D$) is real in $D$). Since
$\mathcal{H}(D$ is the connected component containing $D$ of
$\mathbb{C}^n\backslash \cup_{t\in \partial D}
\{z\in \mathbb{C}^n \sum_{j=1}^n (z_j-t_j)^2=0\}$, we see that
$\widetilde{h_\xi ^p}$ is defined and holomorphic in $\mathcal{H}(D)$
and that $\lim_{z\to  \xi }| \widetilde{h_\xi ^p}(z)| =\infty $.
It remains thus to prove that the restriction $\widetilde{h_\xi ^p}| D$
is actually $p$-polyharmonic in $D$.

\noindent 2.a.i: \quad $1\leq p\leq \frac n2-1$.
Since for every $x\in D:(\widetilde{h_\xi ^p}| D)( x)
=1/(r^{n-2p})$ depends only on $r=\| x-t\|$ the proof can be
carried out directly. Indeed, it is simplest to introduce polar coordinates
with $t$ as origin and to use
\begin{equation*}
\Delta ^p=\big( \frac{\partial ^2}{\partial r^2}+\frac{n-1}r\frac \partial
{\partial r}+\frac 1{r^2}B\big) ^p,
\end{equation*}
where $B$ is the Beltrami operator containing only derivatives
with respect to the angles variables. Now by induction on $q=1,2,\dots$.
we find after some calculus that for an arbitrary (complex) $\alpha$,
\begin{equation*}
\Delta ^q( r^\alpha ) =\alpha ( \alpha +n-2) (
\alpha -2) ( \alpha +n-4) \dots (\alpha -2q+2)( \alpha
+n-2q) r^{\alpha -2q}.
\end{equation*}
Observe that if $\alpha =2p-n$ we obtain
\begin{equation*}
\Delta ^q( r^{2p-n}) =( 2p-n) ( 2p-2)
\dots (2p-n-2q+2)( 2p-2q) r^{2p-n-2q},
\end{equation*}
which gives respectively for $q=p$ and $q=p-1$:
\begin{align*}
\Delta ^p(r^{2p-n})=&0, \\
\Delta ^{p-1}( r^{2p-n}) =&(2p-n) ( 2p-2) ( 2p-n-2) ( 2p-4)
\dots ( 4-n) 2r^{2-n}.
\end{align*}
Note that $\Delta ^{p-1}( r^{2p-n}) \neq 0$ if $n$ is even and greater than
or equal to $6$; in addition, the case $n=4$ involves $p=1$, and so the
last equality holds since $\Delta (\frac 1{r^2})=0$ ,
$\Delta ^0(\frac 1{r^2})=1$.
\\
2.a.ii:\quad  $p=\frac n2$: \quad Since for every $x\in D$, the restriction
$\widetilde{h_\xi ^p}| D:x\mapsto 2\mathop{\rm Ln}r$, where $r=\| x-t\| $,
is a radial function, we use the same process to verify that $\mathop{\rm Ln}$ $r$ is a
$\frac n2$-polyharmonic function in $\mathbb{R}^n-\{0\}$ for all $n=2p\geq 4$.
 As $\Delta(\mathop{\rm Ln}r)=( 2p-2) r^{-2}$ , and
$\Delta ^q(\mathop{\rm Ln}r) =( 2q-2) \Delta ^{q-1}(r^{-2}) $, we can make
use of the corresponding formula of 2.a.(i)
with $\alpha =-2$ to have
\begin{equation*}
\Delta ^q( \mathop{\rm Ln}r) =(-1)^{q-1}2^q(q-1)[(q-1)!](n-4)(n-6)\dots
(n-2q)\frac 1{r^{2q}}.
\end{equation*}
The last equality holds actually for all $n\geq 2$ and $q\geq 1$ since by
the case (1) above this result is true for $n=2$. Observe that if
$n=2p\geq 4 $ one obtains
\begin{equation*}
\Delta ^q( \mathop{\rm Ln}r)
=(-1)^{q-1}2^q(q-1)[(q-1)!](2p-4)\dots (2p-2q)\frac 1{r^{2q}}\quad
\text{in}\quad \mathbb{R}^{2p}-\{0\}.
\end{equation*}
Thus $\Delta ^q( \mathop{\rm Ln}r) \neq 0$ for $q=1,2,\dots ,p-1$,
and $\Delta ^q( \mathop{\rm Ln}r) =0$ if $q=p$.
\\
b) Suppose $n$ is odd, $n\geq 3$. We consider again
\begin{equation*}
\widetilde{h_\xi ^p}(z)=\frac 1{[[\xi _1-z_1)^2+\dots
+[\xi _n-z_n)^2]^{\frac n2-p}}
\end{equation*}
with $1\leq p\leq [ \frac n2]-1$, where the chosen branch is
such that $\widetilde{h_\xi ^p}| D$, $(x)>0$ 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}:
\begin{equation} \label{1}
\lambda (f_j,x,R)=f_j(x)+\sum_{q=1}^{p-1} a_qR^{2q}\Delta ^qf_j(x)\quad
\end{equation}
for all x $\in D$, and $R>0$ so small that
$B_n^r(x,R)=\{y\in \mathbb{R}^n;\| y-x\| <R\}\subset D$, where
$\lambda (f_j,x,R)$ denotes the integral mean values over the surface
$\partial B_n^r(x,R)$:
$$
\lambda (f_j,x,R)=\frac{\Gamma (\frac n2)}{2\sqrt{\pi ^n}}
\int_{\| a\| =1}f_j(x+Ra)d\sigma (a)\,,
$$
with $d\sigma (a)$ an element of surface differential on the sphere
$S^{n-1}(O,1)$ and $a_q=\Gamma (\frac n2)/(2^{2q}q!\Gamma (q+\frac n2))$.
As $f_j$ converges to $f$ uniformly on the compact set $S^{n-1}(x,R)$,
the limit process applied to \eqref{1} yields of course \eqref{1} for $f$:
\begin{equation*}
\lambda (f,x,R)=f(x)+\sum_{q=1}^{p-1} a_qR^{2q}\Delta^qf(x);
\end{equation*}
that is, $f=\lim_{j\to  \infty } f_j$ has the mean value
property \eqref{1} in $D$. Thus $f$ is $p$-polyharmonic in $D$.

To prove the existence of the aforesaid function $h$, let
$z^{(1)},\dots ,z^{(j)}$. be a denumerable dense subset of the compact set
$\partial \mathcal{H}(D)$. For every $(j,k)\in \mathbb{N}^{*^2}$,
let $B_n^c(z^{(j)},\frac 1k)=\{w\in \mathbb{C}^n;\|w-z^{(j)}\| <\frac 1k\}$
denote the hermitian ball of $\mathbb{C}^n$
centered at $z^{(j)}$ and of radius$\frac 1k$, and put
\begin{equation*}
\mathcal{H}_{j,k}(D)=\mathcal{H}(D)\cup B_n^c(z^{(j)},\frac 1k).
\end{equation*}
Due to the density of $\{z^{(j)}\}_{j\in N^{*}}$ in
$\partial \mathcal{H}(D)$, it is enough to prove the existence of a function belonging
to $\mathcal{O}^p[\mathcal{H}(D)]$ which cannot be holomorphically
continued beyond $\mathcal{H}(D)$. This amounts to show the existence of a
functions belonging to $\mathcal{O}^p[\mathcal{H}(D)]$ which cannot be
holomorphically continued to any domain $\mathcal{H}_{j,k}(D)$.
Thanks to the construction step, for all $j,k\in \mathbb{N}^{*}$
and every $p=1,2,\dots ,[ \frac n2] $, we have:
$$\mathcal{O}^p[\mathcal{H}(D)]\setminus \mathbf{R}_{j,k}
\{\mathcal{O}^p[\mathcal{H}_{j,k}(D)]\}\neq \emptyset ,
$$
where $\mathbf{R}_{j,k}$ denotes the restriction mapping from
$\mathcal{O}^p[\mathcal{H}_{j,k}(D)]$ to $\mathcal{O}^p[\mathcal{H}(D)]$,
and $\mathcal{O}^p$ $[\mathcal{H}_{j,k}(D)]$ the space of all holomorphic
functions in $\mathcal{H}_{j,k}(D)$ whose trace on $\mathbb{R}^n$ is
$p$-polyharmonic in $D$. The spaces $\mathcal{O}^p[\mathcal{H}(D)]$
and $\mathcal{O}^p[\mathcal{H}_{j,k}(D)]$ being Fr\'{e}chet spaces by the
Lemma above, and the linear and continuous mapping $\mathbf{R}_{j,k}$ being not
onto, we deduce owing to a Banach Theorem \cite{n1} that the range of
$\mathbf{R}_{j,k}$ is a subset of the first category of
$\mathcal{O}^p[\mathcal{H}(D)]$; that is,
\begin{equation*}
\mathbf{R}_{j,k}\{\mathcal{O}^p[\mathcal{H}_{j,k}(D)]\}
=\cup_{m=1}^\infty X_{j,k}^m,
\end{equation*}
where $X_{j,k}^m$, $m=1,2,\dots $ are subsets of
$\mathcal{O}^p[\mathcal{H}(D)]$ satisfying
$( \overline{X_{j,k}^m}) ^0=\emptyset$, with respect
to the topology $(\tau )$. Observe that
\begin{equation*}
\cup_{j,k=1}^\infty \mathbf{R}_{j,k}\{\mathcal{O}^p[\mathcal{H}_{j,k}(D)]\}
=\cup_{j,k=1}^\infty (\cup_{m=1}^\infty X_{j,k}^m)
=\cup_{j,k,m=1}^\infty X_{j,k}^m
\end{equation*}
is also of the first category in $\mathcal{O}^p[\mathcal{H}(D)]$.
Since $\mathcal{O}^p[\mathcal{H}(D)]$ is in particular a Baire space, we
have, of course,
\begin{equation*}
\mathcal{O}^p[\mathcal{H}(D)]\setminus \cup_{j,k,m=1}^\infty X_{j,k}^m
=\mathcal{O}^p[\mathcal{H}(D)]\setminus \cup_{j,k=1}^\infty
\mathbf{R}_{j,k}\{\mathcal{O}^p[\mathcal{H}_{j,k}(D)]\}\neq \emptyset ,
\end{equation*}
so we can pick up an element $h$ of $\mathcal{O}^p[\mathcal{H}(D)]$
which cannot be continued holomorphically through $\partial \mathcal{H}(D)$.
\hfill$\square$

\begin{corollary} \label{coro2.3}
Let $D$ be a $C$-domain of $\mathbb{R}^n(n\geq 2)$, or $D$ be a convex
domain of $\mathbb{R}^n$. Then for every integer $p$, $(1\leq p\leq [ \frac
n2]$, the integer part of $\frac n2)$, the harmonicity cell of $D$
satisfies
\begin{equation}
\mathcal{H}(D)=[ \cap_{u\in {H}^p(D)} D^u] ^0 \label{2}
\end{equation}
where ${H}^p(D)=\{u\in \mathbf{C}^\infty (D);$ $\Delta
^pu=0$ in $D\}$ and $D^u$ is the complex domain of $\mathbb{C}^n$ to which a
polyharmonic function $u$ extends holomorphically.
\end{corollary}

\paragraph{Proof} By \cite{a4} Lemma 1.1.2, each p-polyharmonic function
$u $ in $D$, $p\in \mathbb{N}^{*}$, is the restriction of a holomorphic
function $\widetilde{u}$ in $D^u\subset \mathbb{C}^n$ such that
$D^u\cap \mathbb{R}^n=D$. The former property is actually a consequence of
the analyticity of $u$. In
addition, the $p$-polyharmonicity of $u$ implies more precisely that the
kernel of $\mathcal{H}(D)$ is included in $D^u($see \cite{a4} Theorem 5.2.6). If
$u $ wanders through the whole class ${H}^p(D)$, we obtain:
$\mathcal{NH}(D)\subset [ \cap _{u\in {H}^p(D)} D^u] ^0$
(note here that the kernel of a harmonicity cell is a connected open of
$\mathbb{C}^n$).

Inversely, due to Theorem \ref{thm2.1} above, we can associate to every $1\leq
p_0\leq [ \frac n2] $ a function $f_{p_0}\in {H}^{p_0}(D)$
satisfying $D^{f_{p_0}}=\mathcal{H}(D)$. So, if
$1\leq p\leq [ \frac n2] $ we get obviously the inclusion:
$[ \cap_{u\in {H}^p(D)} D^u] ^0\subset D^{f_{p_0}}$.
Hence, one deduces
$\mathcal{NH}(D)\subset [ \cap_{u\in {H}^p(D)}D^u] ^0\subset
\mathcal{H}(D)$. Now the assumption on $D$ guarantees
that $\mathcal{NH}(D)=\mathcal{H}(D)$; the desired equality follows.
\hfill$\square$ \smallskip

Observe that it is an unexpected result that the right-hand side of Equality
\eqref{2} does not depend on the choice of $p$. This allows us thus
to give the following result.

\begin{corollary} \label{coro2.4}
For every bounded domain $D$ of $\mathbb{R}^n$, $n\geq 2$, with $D\neq
\emptyset $, and $\partial D\neq \emptyset $, we have
\begin{equation*}
\mathcal{H}(D)=\cap_{1\leq p\leq [ \frac n2] }[
\cap_{u\in {H}^p(D)}D^u] ^0.
\end{equation*}
\end{corollary}

\begin{remark} \label{rmk2.5} \rm
Putting $p=1$ in Corollary \ref{coro2.3}, we find again an Avanissian s' result
(cf. \cite{a4} p.67): Let $\mathbf{A}(D)$ $(\mathbf{H}a(D))$ be the class
of all real analytic (harmonic) functions on $D\subset \mathbb{R}^n$.
For $f\in\mathbf{A}(D)$, we denote $\widetilde{f}:D^f\to  \mathbb{C}$ the
holomorphic extension of $f$ to the maximal domain $D^f$ of $\mathbb{C}^n$
(in the inclusion meaning). Then the sets:
$A=\cap_{f\in \mathbf{A}(D)} D^f$ and
$B=\cap_{f\in \mathbf{Ha}(D)} D^f$ satisfy
$\stackrel{\circ}{A}=\emptyset$,
$\stackrel{\circ}{B}=\mathcal{H}(D)$.
\end{remark}

\section{Some properties of harmonicity cells}

In \cite{a4}, Avanissian established the following general results about the
operation $D\mapsto \mathcal{H}(D)$; see also \cite{l3}.

\begin{proposition} \label{prop3.1}
The harmonicity cells of domains of $\mathbb{R}^n,n\geq 2$, satisfy
\begin{itemize}
\item[a)] If $D_1\cap D_2=\emptyset $, $\mathcal{H}(D_1)\cap \mathcal{H}
(D_2)=\emptyset $; if $D_1\subset D_2$, $\mathcal{H}(D_1)\subset \mathcal{H}
(D_2)$.

\item[b)] $\mathcal{H}(\cup_{\nu \in J} D_\nu )
=\cup_{\nu\in J}\mathcal{H}(D_\nu )$ for every exhaustive increasing family
of domains $D_\nu $.

\item[c)] $\mathcal{H}(D)\cap \mathbb{R}^n=D$; $\mathcal{H}(D)$ is symmetric
with respect to $\mathbb{R}^n$; and if $D$ is convex then so is
$\mathcal{H}(D)$.

\item[d)] If $D$ is starshaped at $a_0$, then $\mathcal{H}(D)$ is
starshaped at $a_0$ and $\mathcal{H}(D)=\{z\in \mathbb{C}^n$;
$T(z)\subset D\}$.

\item[e)] $\partial D\subset \partial \mathcal{H}(D)$ ; if $z\in
\overline{\mathcal{H}(D)}$ , $T(z)\subset \overline{D}$ ; and if $z\in
\partial \mathcal{H}(D)$ , $T(z)\cap \partial D\neq \emptyset $.

\item[f)]
$\delta [\mathcal{H}(D)]\leq 2[\frac n{2n+2}]^{\frac 12}\delta (D)$,
where $\delta (D)$ denotes the diameter of D.

\item[g)] $\mathcal{H}(D)$ may be explicitly obtained  when $D$ is a ball,
a cube, or a difference of two balls.

\item[h)] If $n=2$ and $\mathbb{R}^2\simeq \mathbb{C}$,
$\mathcal{H}(D)=\{z\in \mathbb{C}^2;$ $z_1+iz_2\in D$,
$\overline{z_1}+i\overline{z_2}\in D\}$.
\end{itemize}
\end{proposition}

In the following, we establish supplementary results. Let $\mathfrak{D}^n$
denote henceforth the family of all domains $D$ of $\mathbb{R}^n$, $D\neq
\emptyset $, $\partial D\neq \emptyset $, and $\mathfrak{C}_s^n$ the family of
all domains of $\mathbb{C}^n$ which are symmetric with respect to
$\mathbb{R}^n=\{x+iy\in \mathbb{C}^n$; $y=0\}$.

\begin{proposition} \label{prop3.2}
The mapping $\mathcal{H}$ : $D\in \mathfrak{D}^n\mapsto \mathcal{H}(D)\in \mathfrak{C%
}_s^n$ satisfies:
\begin{itemize}
\item[a)] $\mathcal{H}$ is injective ; $\mathcal{H}(D)$ is bounded if and
only if $D$ is bounded.

\item[b)] For every compact set $K\subset \mathcal{H}(D)$, there exists a
domain $D_1\in \mathfrak{D}^n$ such that $D_1$ is relatively compact in $D$ and
$K\subset \mathcal{H}(D_1)$.

\item[c)] If $D_1,D_2\in \mathfrak{D}^2$ are such that $D_1\cap D_2$ is
connected then
$\mathcal{H}(D_1\cap D_2)=\mathcal{H}(D_1)\cap \mathcal{H}(D_2)$.
If $(D_j)_{j\in J}$ is a family of starshaped domains in
$\mathbb{R}^n(n\geq 2)$ such that $\cap_{j\in J}D_j$ is a
starshaped domain, or if $(D_j)_{j\in J}$ is a family of convex domains
in $\mathbb{R}^n$ such that $\cap_{j\in J} D_j$ is open, then
$\mathcal{H}(\cap_{j\in J} D_j)=\cap_{j\in J} \mathcal{H}(D_j)$.

\item[d)]If $D_1,D_2\in \mathfrak{D}^2$ with $D_1\cap D_2\neq \emptyset $
then $\mathcal{H}(D_1\cup D_2)\supset \mathcal{H}(D_1)\cup \mathcal{H}(D_2)$,
and the equality holds if and only if $D_1\subset D_2$ or $D_2\subset D_1$.
More generally, if $(D_j)_{j\in J}\subset \mathfrak{D}^n(n\geq 2)$ is such that
$D_i\cap D_j\neq \emptyset $ for all $i,j\in J$ then
$\mathcal{H}(\cup_{j\in J} D_j)\supset \cup_{j\in J} \mathcal{H}(D_j)$.
The equality holds if $\cup_{j\in J} D_j=D_{j_0}$ for a certain $j_0\in J$.
\end{itemize}
\end{proposition}

\paragraph{Proof} a)
By Proposition \ref{prop3.1}, $\mathcal{H}$ is well defined on $\mathfrak{D}^n$ with
values in $\mathfrak{C}_s^n$; and if two harmonicity cells $\mathcal{H}(D)$
and $\mathcal{H}(D')$ coincide in $\mathbb{C}^n$, then their traces
on $\mathbb{R}^n$ coincide also, that is $D=D'$. Besides, suppose that
for some $R>0$, $D\subset B_n^r(0,R)=\{x\in \mathbb{R}^n;\| x\| <R\}$.
Then $\mathcal{H}(D)\subset \mathcal{H}[B_n^r(0,R)]=LB(0,R)
=\{z\in \mathbb{C}^n;L(z)<R\}$ (the Lie ball of $\mathbb{C}^n$),
see \cite{a3,b1,h2}, where
$$
L(z)=\Big[\| z\|^2+\sqrt{\| z\| ^4-| {\textstyle \sum_{j=1}^n} z_j^2| ^2}\;\Big]^{1/2}.
$$
Since $\| z\|\leq L(z)$, we have
$LB(0,R)\subset B_n^c(0,R)=\{z\in \mathbb{C}^n;\| z\| <R\}$ and
 $\mathcal{H}(D)$ is bounded in $\mathbb{C}^n$. The converse is
obvious since $D\subset \mathcal{H}(D)$.

\noindent b) Let us consider an increasing exhaustive family $(D_\nu )_{\nu
\in J}$ ($J$ is a fixed indices set) of bounded domains
$D_\nu \in \mathfrak{D}^n $ such that $D=\cup_{\nu \in J} D_\nu $.
Due to 3.1.b, the family $(\mathcal{H}(D_\nu ))_{\nu \in J}$ of
harmonicity cells of $(D_\nu)_{\nu \in J}$ is also increasing, exhaustive
and satisfies $\mathcal{H}(D)=\cup_{\nu \in J}\mathcal{H}(D_\nu )$.
We then have: $K\subset\cup_{\nu \in J}\mathcal{H}(D_\nu )$. Since $K$ is a
compact set, we can extract from this open covering of $K$, a finite
sub-covering of $K$:
 $K\subset \cup_{k=1}^n \mathcal{H}(D_{\nu _k})$. Afterwards we
have by 3.1.a: $\cup_{k=1}^n \mathcal{H}(D_{\nu _k})\subset
\mathcal{H}(\cup_{k=1}^n D_{\nu _k})$ and
$K\subset \mathcal{H}(\cup_{k=1}^n D_{\nu _k})$. Seeing that $D'$ is a
relatively compact domain in $D$ and taking
$D'=\cup_{k=1}^n D_{\nu _k}$, we obtain the desired result.

\noindent c) The inclusion $\mathcal{H}(D_i\cap D_j)\subset \mathcal{H}
(D_i)\cap \mathcal{H}(D_j)$ is obvious from $D_i\cap D_j\subset D_i$,
$D_i\cap D_j\subset D_j$. If $w\in \mathcal{H}(D_i)\cap \mathcal{H}(D_j)$,
$T(w)\subset D_i\cap D_j$, that is $w\in \mathcal{H}(D_i\cap D_j)$. By
similar arguments we obtain the general case.

\noindent d) $D_1\cap D_2\neq \emptyset$ guarantees that $D_1\cup D_2\in
\mathfrak{D}^n$. Since $D_i\subset D_1\cup D_2$, $i=1,2$,
$\mathcal{H}(D_1)\cup \mathcal{H}(D_2)\subset \mathcal{H}(D_1\cup D_2)$.
Suppose now that $D_1$ is neither included in $D_2$, nor $D_2 $ in $D_1$.
If $a$ and $b$ are arbitrarily chosen in $D_1\setminus D_2$
and $D_2\setminus D_1$ respectively, and if $\mathbb{R}^2\simeq \mathbb{C}$,
then the point $w=(\frac{a+\overline{b}}2,\frac{a-\overline{b}}{2i})$
of $\mathbb{C}^2$ satisfies $T(w)=\{a,b\}\subset D_1\cup D_2$.
Now, the last hypothesis on $D_1 $ and $D_2$ involves that
$w\notin \mathcal{H}(D_1)\cup \mathcal{H}(D_2) $.
 Besides, as $D_i\cap D_j\neq \emptyset $ we have
 $\cup_{j\in J} D_j\in \mathfrak{D}^n$ and thus this union does possess
a harmonicity cell in $\mathbb{C}^n$. The given inclusion is evident
since $D_i\subset \cup_{j\in J} D_j$. Suppose in addition that
$D_{j_0}=\cup_{j\in J}D_j$. From
$\mathcal{H}(D_{j_0})\subset \cup_{j\in J}\mathcal{H}(D_j)$ and
$\mathcal{H}(\cup_{j\in J}D_j)\supset \cup_{j\in J}\mathcal{H}(D_j)$,
we deduce the equality
$\mathcal{H}(\cup_{j\in J}D_j)=\cup_{j\in J}\mathcal{H}(D_j)$.
\hfill$\square$

\begin{corollary} \label{coro3.3}
If $(D_j)_{j\geq 1}$ is a monotonous sequence in $\mathfrak{D}^n$,  so is
$(\mathcal{H}(D_j))_{j\geq 1}$ in $\mathfrak{C}_s^n$; and writing
$D=\lim_{j\to  \infty }D_j$, we have
$\lim_{n\to\infty } \mathcal{H}(D_n)=\mathcal{H}(D)$ under the assumptions
that: $\cup_{j\geq 1}D_j\neq \mathbb{R}^n$ if the sequence $(D_j)_{j\geq 1}$
is increasing, and that $\cap_{j\in J}D_j\in \mathfrak{D}^n$ in the decreasing
case.
\end{corollary}

\paragraph{Proof}
If the sequence is increasing then
$\liminf_{n\to  \infty }D_n=\cup_{n\geq 1}(\cap_{k\geq n}D_k)
=\cup_{n\geq 1}D_n$,
$\limsup_{n\to \infty} D_n=\cap_{n\geq 1}(\cup_{k\geq n}D_k)=
\cap_{n\geq 1}(\cup_{k\geq n}D_k)=\cup_{k\geq 1}D_k$, so \break
$\lim_{n\to  \infty }D_{n}=\cup_{n\geq 1}D_n$.
Since $\cup_{n\geq 1}D_n\neq \emptyset $ and
$\cup_{n\geq 1}D_n\neq \mathbb{R}^n$,  we deduce that
$\lim_{n\geq 1}D_n$ is an element of $\mathfrak{D}^n$.

Next, $(\mathcal{H}(D_n))_{n\geq 1}$ being also increasing,
$\lim_{n\to  \infty }\mathcal{H}(D_n)=\cup_{n\geq 1}\mathcal{H}(D_n)$.
Now by 3.2.d:
$\cup_{n\geq 1}\mathcal{H}(D_n)\subset \mathcal{H}(\cup_{n\geq 1}D_n)$.
Moreover, if $w_0\in \mathcal{H}(\cup_{n\geq 1}D_n)$ one has
$T(w_0)\subset \cup_{n\geq 1}D_n$; then by 3.2.b and the fact that
$T(z)$ is a compact set for every $z\in \mathbb{C}^n$, there exists
$n_0\geq 0$ such that $T(w_0)\subset D_{n_0}$ i.e.
$w_0\in \mathcal{H}(D_{n_0})\subset \cup_{n\geq 1}\mathcal{H}(D_n)$.
Thus $\mathcal{H}(\lim_{n\to \infty}D_n)\subset
\lim_{n\to\infty }\mathcal{H}(D_n)$, which involves the aforesaid equality.
In case of a decreasing sequence $(D_n)_{n\geq 1}$ one has
 $\liminf_{n\to  \infty }D_n=\cup_{n\geq 1}(\cap_{k\geq n}D_k)
 =\cup_{n\geq 1}(\cap_{k\geq 1}D_k)=\cap_{k\geq 1}D_k$, and
 $\limsup_{n\to  \infty }D_n=\cap_{n\geq 1}(\cup_{k\geq n}D_k)=
\cap_{n\geq 1}D_n$. So $\lim_{n\to \infty }D_n=\cap_{n\geq 1}D_n$,
which is in $\mathfrak{D}^n$ by hypothesis. Now, $(\mathcal{H}(D_n))_{n\geq 1}$
being decreasing one also has:
$\lim_{n\to  \infty }\mathcal{H}(D_n)=\mathcal{H}(\lim_{n\to \infty}D_n)$.
\hfill$\square$

\begin{corollary} \label{coro3.4}
The mapping $D\mapsto \mathcal{H}(D)$ is not a surjective operator:
The unit hermitian ball $B_n^c=\{z\in \mathbb{C}^n;\| z\| <1\}$\ does
not represent a harmonicity cell in $\mathbb{C}^n$.
\end{corollary}

\paragraph{Proof.}
Since $B_n^c\cap \mathbb{R}^n=B_n^r=\{x\in \mathbb{R}^n;\| x\| <1\}$ is
a convex domain of $\mathbb{R}^n$ (according to the induced topology),
we have to find a point $w_0\in B_n^c$ for which the Lelong sphere
$T(w_0)$ is not contained into $B_n^r$. For, put $w_0=\rho$ ($i,1,\dots ,1)\in
\mathbb{C}^n$ where $\rho >0$ 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
$V'=\{x\in \mathbb{R}^2 : x_1>a, k_1<x_2<k_2\}$, for some $a>0$,
$k_1,k_2\in \mathbb{R}$. 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
\[
\mathcal{H}(V')=\{w=x+iy\in \mathbb{C}^2;| y_1| 
<\frac{k_2-k_1}2-| x_2-\frac{k_1+k_2}2| ,| y_2| <x_1-a\}.
\]

\paragraph{Example}
The harmonicity cell of an arbitrary convex polygon $P_n'$ may be
explicited by means of the $n$ vertices $\omega _0',\dots , \omega _{n-1}'$.
 For, put $\alpha =\frac{\omega _0'+\omega_2'}2$ and consider the
translation $\tau _{-\alpha }:z\mapsto z-\alpha $. The domain
$P_n=\tau _{-\alpha }(P_n')$ is also a convex polygon, with
$O\in P_n$ and $n$ vertices $\omega _0,\dots ,\omega _{n-1}$,
given by $\omega _k=\omega _k'-\alpha $.
Making use of (d) and (h) in Proposition \ref{prop3.1}, we find after
calculus and simplifications:
\begin{align*}
\mathcal{H}(P_n)=\Big\{&w=x+iy\in \mathbb{C}^2: \mathop{\rm sgn}
(\mathop{\rm Im}\overline{\omega _k}
\omega _{k+1})\mathop{\rm Im}\overline{x}(\omega _{k+1}-\omega _k)\\
&+\sqrt{| y| ^2| \omega _{k+1}-\omega _k| ^2-\mathop{\rm Im}{}^2
\overline{y}(\omega _{k+1}-\omega _k)}\\
&< | \mathop{\rm Im}\overline{\omega _k}\omega_{k+1}| ,\; k=0,1,\dots ,n-1\Big\}
\end{align*}
with $\mathbb{R}^2\simeq \mathbb{C}$, $\mathop{\rm Im}z$ is the imaginary
part of $z$, and $\mathop{\rm sgn}\alpha $ is the sign of $\alpha$.
Note that $P_n'=\tau _\alpha P_n$ means that
$[ w'\in \mathcal{H}(P_n')] $ if and only if
$[ w'-\alpha \in \mathcal{H}(P_n)] $. If now $P_{n,r}'$ is some regular
polygon, it is enough to consider its circumscribed circle
$\mathcal{C}( \beta,R) $, centered at $\beta \in $ $\mathbb{R}^2$, with radius
$R>0$. 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}