### Abstract

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 L^{p} boundedness of the Riesz transforms that holds in ℝ^{n}. Several partial answers have been given since. In the present paper, we give positive results for 1 ≤ p ≤ 2 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 p > 2 under the same assumptions. Finally, we prove a similar result for the Riesz transforms on arbitrary domains of ℝ^{n}.

Original language | English |
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Pages (from-to) | 1151-1169 |

Number of pages | 19 |

Journal | Transactions of the American Mathematical Society |

Volume | 351 |

Issue number | 3 |

Publication status | Published - 1999 |

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## Cite this

*Transactions of the American Mathematical Society*,

*351*(3), 1151-1169.