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

22 Scopus citations


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
Issue number2
StatePublished - Apr 1 2017

Bibliographical note

Publisher Copyright:
© 2016, Mathematica Josephina, Inc.

Copyright 2017 Elsevier B.V., All rights reserved.


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

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