A perturbation result for the Riesz transform

TL;DR

This paper establishes a perturbation theorem for the boundedness of the Riesz transform on Riemannian manifolds, showing that under certain geometric conditions, boundedness properties are preserved when manifolds are close outside a compact set.

Contribution

It provides a new perturbation result linking the boundedness of the Riesz transform on a manifold to that on a nearby manifold with specific geometric properties.

Findings

Boundedness of Riesz transform on M_0 implies boundedness on M under conditions

Results apply when M is p-hyperbolic or has a single end

Extends understanding of Riesz transform stability under geometric perturbations

Abstract

We show a perturbation result for the boundedness of the Riesz transform : if $M$ and $M_0$ are complete Riemannian manifolds satisfying a Sobolev inequality of dimension $n$, which are isometric outside a compact set, and if the Riesz transform on $M_0$ is bounded on $L^q$, then for all $\frac{n}{n-2}, the Riesz transform on $M$ is bounded on $L^p$ provided that $M$ is p-hyperbolic OR $M$ has only one end.