Riesz transforms for

It has been asked (see R. Strichartz, Analysis of the Laplacian, J. Funct. Anal. 52 (1983), 48-79) whether one could extend to a reasonable class of non-compact Riemannian manifolds the boundedness of the Riesz transforms that holds in . Several partial answers have been given since. In the present paper, we give positive results for under very weak assumptions, namely the doubling volume property and an optimal on-diagonal heat kernel estimate. In particular, we do not make any hypothesis on the space derivatives of the heat kernel. We also prove that the result cannot hold for under the same assumptions. Finally, we prove a similar result for the Riesz transforms on arbitrary domains of .