Riesz meets Sobolev

Volume 118 / 2010

Thierry Coulhon, Adam Sikora Colloquium Mathematicum 118 (2010), 685-704 MSC: 58J35, 42B20, 46E35. DOI: 10.4064/cm118-2-20


We show that the $L^p$ 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.


  • Thierry CoulhonDépartement de Mathématiques
    Université de Cergy-Pontoise
    Site de Saint-Martin
    2, rue Adolphe Chauvin
    F-95302 Cergy-Pontoise Cedex, France
  • Adam SikoraDepartment of Mathematics
    Macquarie University
    Sydney, NSW 2109, Australia

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