A characterization of the $L^2$-range of the generalized spectral projections related to the Hodge-de Rham Laplacian

TL;DR

This paper characterizes the $L^2$-range of spectral projections related to the Hodge-de Rham Laplacian on hyperbolic space, confirming a conjecture of Strichartz for differential forms.

Contribution

It provides a complete description of the $L^2$-range of spectral projections and Poisson transform on differential forms over hyperbolic space, solving a conjecture by Strichartz.

Findings

Characterization of the $L^2$-range of spectral projections on hyperbolic space.

Positive resolution of Strichartz's conjecture for differential forms.

Insights into the structure of the Poisson transform on the boundary of hyperbolic space.

Abstract

Let $H^n(\mathbb R)$ be the real hyperbolic space. In this paper, we present a characterization of the $L^2$-range of the generalized spectral projections on the bundle of differential forms over $H^n(\mathbb R)$. As an underlying result we show a characterization of the $L^2$-range of the Poisson transform on the bundle of differential forms on the boundary $\partial H^n(\mathbb R)$. This gives a positive answer to a conjecture of Strichartz on differential forms.