Riesz Transform for 1 ≤ p≤ 2 Without Gaussian Heat Kernel Bound

Li Chen; Thierry Coulhon; Joseph Feneuil; Emmanuel Russ

doi:10.1007/s12220-016-9728-5

Riesz Transform for 1 ≤ p≤ 2 Without Gaussian Heat Kernel Bound

Li Chen, Thierry Coulhon, Joseph Feneuil, Emmanuel Russ

Mathematics

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

We study the L^p boundedness of the 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 < p< 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 < p< 2. This yields a full picture of the ranges of p∈ (1 , + ∞) 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.

Original language	English (US)
Pages (from-to)	1489-1514
Number of pages	26
Journal	Journal of Geometric Analysis
Volume	27
Issue number	2
DOIs	https://doi.org/10.1007/s12220-016-9728-5
State	Published - Apr 1 2017

Bibliographical note

Funding Information:
The Li Chen has been supported in part by ICMAT Severo Ochoa project SEV-2011-0087 and she acknowledges that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Agreement No. 615112 HAPDEGMT. Li Chen, Joseph Feneuil and Emmanuel Russ are supported by the French ANR project HAB (no. ANR-12-BS01-0013).

Publisher Copyright:
© 2016, Mathematica Josephina, Inc.

Keywords

Graphs
Heat kernel
Riemannian manifolds
Riesz transforms
Sub-Gaussian estimates

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abstract = "We study the Lp boundedness of the 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 Lp for 1 < p< 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 < p< 2. This yields a full picture of the ranges of p∈ (1 , + ∞) for which respectively the Riesz transform is Lp-bounded and the reverse inequality holds on Lp on such manifolds and graphs. This picture is strikingly different from the Euclidean one.",

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N2 - We study the Lp boundedness of the 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 Lp for 1 < p< 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 < p< 2. This yields a full picture of the ranges of p∈ (1 , + ∞) for which respectively the Riesz transform is Lp-bounded and the reverse inequality holds on Lp on such manifolds and graphs. This picture is strikingly different from the Euclidean one.

AB - We study the Lp boundedness of the 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 Lp for 1 < p< 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 < p< 2. This yields a full picture of the ranges of p∈ (1 , + ∞) for which respectively the Riesz transform is Lp-bounded and the reverse inequality holds on Lp on such manifolds and graphs. This picture is strikingly different from the Euclidean one.

KW - Graphs

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KW - Sub-Gaussian estimates

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Riesz Transform for 1 ≤ p≤ 2 Without Gaussian Heat Kernel Bound

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