Riesz Transform on Locally Symmetric Spaces and Riemannian Manifolds with a Spectral Gap

Abstract

In this paper we study the Riesz transform on complete and connected Riemannian manifolds $M$ with a certain spectral gap in the $L^2$ spectrum of the Laplacian. We show that on such manifolds the Riesz transform is $L^p$ bounded for all $p \in (1,\infty)$. This generalizes a result by Mandouvalos and Marias and extends a result by Auscher, Coulhon, Duong, and Hofmann to the case where zero is an isolated point of the $L^2$ spectrum of the Laplacian.