On Poisson transforms of differential forms on real hyperbolic spaces

TL;DR

This paper proves that the Poisson transform creates a topological isomorphism between boundary differential forms and eigenforms of the Laplacian on hyperbolic spaces, enriching understanding of harmonic analysis on these spaces.

Contribution

It establishes the topological isomorphism of the Poisson transform for differential forms on real hyperbolic spaces, extending classical results to a broader setting.

Findings

Poisson transform is a topological isomorphism for certain differential forms

Identifies Hardy-type subspaces of eigenforms on hyperbolic space

Connects boundary forms with eigenforms of the Laplacian

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 $1\leqslant p\leqslant n$ 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 $H^n(\mathbb R)$.