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

Lizhen Ji; Peer Kunstmann; Andreas Weber

doi:10.1016/j.bulsci.2009.09.003

paper

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

arXiv:1005.2975 · doi:10.1016/j.bulsci.2009.09.003

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.

8 pp