On Poisson transforms of differential forms on real hyperbolic spaces

Abstract

This paper is concerned with the Poisson transform of differential forms on the hyperbolic space $H^n(\mathbb{R})$. Consider an integer $p$ such that $1⩽p⩽n$ and let $q$ be either $p-1$ or $p$. For $1<r<\infty$, we prove that the Poisson transform is a topological isomorphism from the space of $L^r$-differential $q$-forms on the boundary $\partial H^n(\mathbb{R})$ onto a Hardy-type subspace of $p$-eigenforms of the Hodge-de Rham Laplacian on $H^n(\mathbb{R})$.