Riesz transform on manifolds and heat kernel regularity

@inproceedings{Auscher2004RieszTO,
  title={Riesz transform on manifolds and heat kernel regularity},
  author={Pascal Auscher and Thierry Coulhon and Xuan Thinh Duong and Steve Hofmann},
  year={2004}
}
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. 

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