Journal of Lie Theory 12 (2002), No. 1, 001--014
Copyright Heldermann Verlag 2002
Characterization of the Lp-Range of the Poisson Transform in Hyperbolic Spaces B(Fn)
A. Boussejra
Dept. of Mathematics, Faculty of Sciences, University Ibn Tofail, Kénitra, Morocco
H. Sami
Dept. of Mathematics, Faculty of Sciences, University Hassan II, Casablanca, Morocco
[Abstract-pdf]
\newcommand{\sC}{{\mathbb C}} \newcommand{\sF}{{\mathbb F}} \newcommand{\sR}{{\mathbb R}} \newcommand{\sH}{{\mathbb H}} The aim of this paper is to give, in a unified manner, the characterization of the $L^p$-range ($p\geq 2$) of the Poisson transform $P_{\lambda}$ for the hyperbolic spaces $B({\sF}^n)$ over ${\sF}=\sR, \, \sC$ or the quaternions $\sH$. Namely, if $\Delta $ is the Laplace-Beltrami operator of $B({\sF}^n)$ and $sF$ a $\sC$-valued function on $B({\sF}^n)$ satisfying $\Delta F=-(\lambda ^2+\sigma ^2)F; \lambda \in \sR ^{*}$ then we establish: i) F is the Poisson transform of some $f\in L^2(\partial B({\sF}^n))$ (ie $P_{\lambda}f=F$) if and only if it satisfies the growth condition: $$ \sup _{t >0}\frac{1}{t}\int_{B(0,t)} 'F(x)'^2d \mu (x)<+\infty,$$ where $B(0,t)$ is the ball of radius $t$ centered at $0$ and $d\mu $ the invariant measure on $B({\sF}^n)$. ii) F is the Poisson transform of some $f\in L^p(\partial B({\sF}^n))$, $p\geq 2$; if and only if it satisfies the following Hardy-type growth condition: $$ \sup _{0\leq r <1} (1-r^2)^{-\frac{\sigma }{2}}\left ( \int_{\partial B({\sF}^n)} 'F(r\theta)'^p d\theta ) \right ) ^{\frac{1}{p}} <+\infty .$$
Keywords: Hyperbolic spaces, Poisson transform, Calderon Zygumund estimates, Jacobi functions.
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