Journal of Lie Theory

\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: $$ \sup _{0\leq r 1} (1-r^2)^{-\frac{\sigma }{2}}\left ( \int_{\partial B({\sF}^n)} 'F(r\theta)'^p d\theta ) \right ) ^{\frac{1}{p}} +\infty .$$