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


Published:20100818
Printed: Dec 2010
Fouzia El Wassouli,
Department of Mathematics, Faculty of Sciences, University Ibn Tofail, Kenitra, Morocco.
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
$$
\F\_{\lambda,p}=\sup_{0\leq r
< 1}(1r^{2})^{\mathcal{R}e[i\lambda]\frac{n}{2}+l}\Big[\int_{S}F(ru)^{p}du
\Big]^{\frac{1}{p}} < +\infty.
$$
Keywords: 
Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundle
Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundle

MSC Classifications: 
32A45, 30E20, 33C67, 33C60, 33C55, 32A25, 33C75, 11K70 show english descriptions
Hyperfunctions [See also 46F15] Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx] Hypergeometric functions associated with root systems Hypergeometric integrals and functions defined by them ($E$, $G$, $H$ and $I$ functions) Spherical harmonics Integral representations; canonical kernels (Szegoo, Bergman, etc.) Elliptic integrals as hypergeometric functions Harmonic analysis and almost periodicity
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
