Riesz transform on manifolds and heat kernel regularity

One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L-P bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L-P estimate in the same interval of p's. (c) 2004 Elsevier SAS.