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. 
Pascal Auscher (LM-Orsay), Thierry Coulhon (AGM), Xuan Thinh Duong,   Steve Hofmann