Riesz meets Sobolev

TL;DR

This paper establishes an equivalence between the boundedness of the Riesz transform in L^p spaces and Sobolev inequalities on certain Riemannian manifolds, linking heat kernel estimates to geometric analysis.

Contribution

It provides a new characterization of Riesz transform boundedness via Sobolev inequalities and relates heat kernel gradient estimates to polynomial growth manifolds.

Findings

Riesz transform boundedness for p>2 is equivalent to Sobolev inequalities.

Characterization of heat kernel gradient estimates on polynomial growth manifolds.

Connections between heat kernel estimates and geometric properties of manifolds.

Abstract

We show that the $L^p$ boundedness, $p>2$, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize in such terms the heat kernel gradient upper estimate on manifolds with polynomial growth.