A multiplicative inequality of Riesz transform type on general Riemannian manifolds

TL;DR

This paper establishes a dimension-free multiplicative inequality involving Riesz transforms on complete Riemannian manifolds, extending to abstract sub-Markov semigroup generators, with implications for analysis on manifolds.

Contribution

It proves a new multiplicative inequality for Riesz transforms on general Riemannian manifolds and in abstract semigroup settings, generalizing previous results.

Findings

Dimension-free inequality for Riesz transforms on manifolds

Extension to generators of sub-Markov semigroups

Applicable for all p in (1, 2] and any epsilon > 0

Abstract

Given any complete Riemannian manifold $M$, we prove that for every $p \in (1, 2]$ and every $\epsilon > 0$, $$ \| \nabla f \|_p^2 \le C_\epsilon \| \Delta^{\frac{1}{2} + \epsilon} f \|_{p}\| \Delta^{\frac{1}{2} - \epsilon} f \|_{p}.$$The estimate is dimension free. This inequality is even proved in the abstract setting of generators of sub-Markov semigroups.