Riesz transforms through reverse H\"older and Poincar\'e inequalities

TL;DR

This paper investigates the boundedness of Riesz transforms on doubling metric measure spaces with gradient operators, showing new results under weaker assumptions like reverse H"older inequalities and elliptic perturbations.

Contribution

It extends previous work by weakening the assumptions needed for Riesz transform boundedness, replacing Poincaré inequalities with reverse H"older inequalities and analyzing elliptic perturbations.

Findings

Riesz transforms are bounded on $L^p$ for $p$ near 2 under reverse H"older conditions.

Gradient estimates for semigroups imply Riesz transform boundedness in a range of $p$.

Results improve prior work by relaxing Poincaré inequality assumptions.

Abstract

We study the boundedness of Riesz transforms in $L^p$ for $p>2$ on a doubling metric measure space endowed with a gradient operator and an injective, $\omega$-accretive operator $L$ satisfying Davies-Gaffney estimates. If $L$ is non-negative self-adjoint, we show that under a reverse H\"older inequality, the Riesz transform is always bounded on $L^p$ for $p$ in some interval $[2,2+\varepsilon)$, and that $L^p$ gradient estimates for the semigroup imply boundedness of the Riesz transform in $L^q$ for $q \in [2,p)$. This improves results of \cite{ACDH} and \cite{AC}, where the stronger assumption of a Poincar\'e inequality and the assumption $e^{-tL}(1)=1$ were made. The Poincar\'e inequality assumption is also weakened in the setting of a sectorial operator $L$. In the last section, we study elliptic perturbations of Riesz transforms.