Boundedness of Riesz transforms on $\RCD(K, \infty)$ spaces

TL;DR

This paper proves the boundedness of Riesz transforms on a broad class of metric measure spaces with Ricci curvature bounds, extending previous results and improving the understanding of Sobolev-Lipschitz approximations.

Contribution

It establishes the $L^p$-boundedness of Riesz transforms on $ ext{RCD}(K, ext{infty})$ spaces for all $p$, without dimension restrictions, using analytic Bellman function techniques.

Findings

Proved $L^p$-boundedness of Riesz transforms on $ ext{RCD}(K, ext{infty})$ spaces.

Extended Sobolev-Lipschitz approximation results to a larger class of spaces.

Applied analytic methods with explicit Bellman functions to achieve results.

Abstract

For $1<p<\infty$, we prove the $L^p$-boundedness of the Riesz transform operators on metric measure spaces with Riemannian Ricci curvature bounded from below, without any restriction on their dimension. This large class of spaces include e.g. that of Hilbert spaces endowed with a log-concave probability measure. As a consequence, we extend the range of the validity of the Lusin-type approximation of Sobolev by Lipschitz functions, previously obtained by L. Ambrosio, E. Bru\`e and the third author in the quadratic case, i.e. $p=2$. The proofs are analytic and rely on computations on an explicit Bellman function.