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

TL;DR

This paper proves that on certain Riemannian manifolds with a spectral gap, the Riesz transform is bounded on all L^p spaces, extending previous results to cases with an isolated zero in the spectrum.

Contribution

It generalizes existing boundedness results of the Riesz transform to manifolds with a spectral gap in the Laplacian spectrum, including isolated zero points.

Findings

Riesz transform is L^p bounded for p in (1,∞) on manifolds with spectral gap

Extension of previous results to cases with zero as an isolated spectral point

Generalization of boundedness results to broader class of Riemannian manifolds

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.