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

Li Chen, Thierry Coulhon, Joseph Feneuil, Emmanuel Russ

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)1489-1514
Number of pages26
JournalJournal of Geometric Analysis
Volume27
Issue number2
DOIs
StatePublished - Apr 1 2017

Keywords

  • Graphs
  • Heat kernel
  • Riemannian manifolds
  • Riesz transforms
  • Sub-Gaussian estimates

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