Riesz meets Sobolev
Volume 118 / 2010
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.