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# A Generalized Poisson Transform of an $L^{p}$-Function over the Shilov Boundary of the $n$-Dimensional Lie Ball

Published:2010-08-18
Printed: Dec 2010
• Fouzia El Wassouli,
Department of Mathematics, Faculty of Sciences, University Ibn Tofail, Kenitra, Morocco.
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## Abstract

Let $\mathcal{D}$ be the $n$-dimensional Lie ball and let $\mathbf{B}(S)$ be the space of hyperfunctions on the Shilov boundary $S$ of $\mathcal{D}$. The aim of this paper is to give a necessary and sufficient condition on the generalized Poisson transform $P_{l,\lambda}f$ of an element $f$ in the space $\mathbf{B}(S)$ for $f$ to be in $L^{p}(S)$, $1 > p > \infty.$ Namely, if $F$ is the Poisson transform of some $f\in \mathbf{B}(S)$ (i.e., $F=P_{l,\lambda}f$), then for any $l\in \mathbb{Z}$ and $\lambda\in \mathbb{C}$ such that $\mathcal{R}e[i\lambda] > \frac{n}{2}-1$, we show that $f\in L^{p}(S)$ if and only if $f$ satisfies the growth condition $$\|F\|_{\lambda,p}=\sup_{0\leq r < 1}(1-r^{2})^{\mathcal{R}e[i\lambda]-\frac{n}{2}+l}\Big[\int_{S}|F(ru)|^{p}du \Big]^{\frac{1}{p}} < +\infty.$$
 Keywords: Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundle
 MSC Classifications: 32A45 - Hyperfunctions [See also 46F15] 30E20 - Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx] 33C67 - Hypergeometric functions associated with root systems 33C60 - Hypergeometric integrals and functions defined by them ($E$, $G$, $H$ and $I$ functions) 33C55 - Spherical harmonics 32A25 - Integral representations; canonical kernels (Szegoo, Bergman, etc.) 33C75 - Elliptic integrals as hypergeometric functions 11K70 - Harmonic analysis and almost periodicity