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arXiv · 1510.08275

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

Li Chen·Thierry Coulhon·Joseph Feneuil·Emmanuel Russ
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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.

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BibTeXRIS

Li Chen, Thierry Coulhon, Joseph Feneuil, Emmanuel Russ. 2015-10-28. Riesz transform for $1 \leq p \le 2$ without Gaussian heat kernel bound. https://arxiv.org/abs/1510.08275

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