Riesz transform for $1 \leq p \le 2$ without Gaussian heat kernel bound

TL;DR

This paper investigates the boundedness of the Riesz transform on Riemannian manifolds and graphs under volume doubling and sub-Gaussian heat kernel bounds, revealing new conditions for $L^p$ boundedness without Gaussian estimates.

Contribution

It establishes $L^p$ boundedness of the Riesz transform for $1 < p < 2$ without requiring Gaussian heat kernel bounds, and analyzes the reverse inequality on specific manifolds and graphs.

Findings

Riesz transform is $L^p$-bounded for $1 < p < 2$ without Gaussian heat kernel bounds.

Reverse inequality fails on Vicsek manifolds and graphs for $1 < p < 2$.

The $p$-range behavior differs significantly from Euclidean spaces.

Abstract

We study the $L^p$ boundedness of Riesz transform as well as the reverse inequality on Riemannian manifolds and graphs under the volume doubling property and a sub-Gaussian heat kernel upper bound. We prove that the Riesz transform is then bounded on $L^p$ for $1 \textless{} p \textless{} 2$, which shows that Gaussian estimates of the heat kernel are not a necessary condition for this.In the particular case of Vicsek manifolds and graphs, we show that the reverse inequality does not hold for $1 \textless{} p \textless{} 2$. This yields a full picture of the ranges of $p\in (1,+\infty)$ for which respectively the Riesz transform is $L^p$ -bounded and the reverse inequality holds on $L^p$ on such manifolds and graphs. This picture is strikingly different from the Euclidean one.