Characterization of the ${\bf L}^p$-Range of the Poisson Transform in Hyperbolic Spaces ${\bf B}(\F^n)$

A. Boussejra and H. Sami

Abstract: 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 $B({\F}^n)$ over ${\F}=\R, \, \C$ or the quaternions $\HH$. Namely, if $\Delta $ is the Laplace-Beltrami operator of $B({\F}^n)$ and $F$ a $\C$-valued function on $B({\F}^n)$ satisfying $\Delta F=-(\lambda ^2+\sigma ^2)F; \lambda \in \R ^{*}$ then we establish: i) F is the Poisson transform of some $f\in L^2(\partial B({\F}^n))$ (ie $P_{\lambda}f=F$) if and only if it satisfies the growth condition: $$ \sup _{t >0}{1\over t}\int_{B(0,t)}{\left|{F(x)}\right|}^2d\mu (x)<+\infty,$$where $B(0,t)$ is the ball of radius $t$ centered at $0$ and $d\mu $ the invariant measure on $B({\F}^n)$. ii) F is the Poisson transform of some $f\in L^p(\partial B({\F}^n))$, $p\geq 2$; if and only if it satisfies the following Hardy-type growth condition: $$ \sup _{0\leq r <1} (1-r^2)^{-{\sigma\over2}}\left ( \int_{\partial B({\F}^n)}{\left|{F(r\theta )}\right|}^p d\theta ) \right ) ^{1\over p} <+\infty .$$

Keywords: hyperbolic spaces, Poisson transform, Calderon Zygmund estimates, Jacobi functions

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