Riesz meets Sobolev

Thierry Coulhon, Adam Sikora

    Research output: Contribution to journalArticlepeer-review


    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 languageEnglish
    Pages (from-to)685-704
    Number of pages20
    JournalColloquium Mathematicum
    Issue number2
    Publication statusPublished - 2010


    • heat kernels
    • Riesz transform
    • Sobolev types inequalities


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