Riesz meets Sobolev

Thierry Coulhon; Adam Sikora

doi:10.4064/cm118-2-20

Riesz meets Sobolev

Thierry Coulhon, Adam Sikora

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

Abstract

We show that the Lp boundedness, p>2, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize in such terms the heat kernel gradient upper estimate on manifolds with polynomial growth.

Original language	English
Pages (from-to)	685-704
Number of pages	20
Journal	Colloquium Mathematicum
Volume	118
Issue number	2
DOIs	https://doi.org/10.4064/cm118-2-20
Publication status	Published - 2010

Keywords

heat kernels
Riesz transform
Sobolev types inequalities

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abstract = "We show that the Lp boundedness, p>2, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize in such terms the heat kernel gradient upper estimate on manifolds with polynomial growth.",

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author = "Thierry Coulhon and Adam Sikora",

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AB - We show that the Lp boundedness, p>2, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize in such terms the heat kernel gradient upper estimate on manifolds with polynomial growth.

KW - heat kernels

KW - Riesz transform

KW - Sobolev types inequalities

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DO - 10.4064/cm118-2-20

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JO - Colloquium Mathematicum

JF - Colloquium Mathematicum

IS - 2

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Riesz meets Sobolev

Abstract

Keywords

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