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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Riesz transforms for $1\leq p\le 2$


Authors: Thierry Coulhon and Xuan Thinh Duong
Journal: Trans. Amer. Math. Soc. 351 (1999), 1151-1169
MSC (1991): Primary 42B20, 58G11
MathSciNet review: 1458299
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Abstract: It has been asked (see R. Strichartz, Analysis of the Laplacian$\dotsc$, 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 ${\mathbb R}^n$. Several partial answers have been given since. In the present paper, we give positive results for $1\leq p\leq 2$ 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 ${\mathbb R}^n$.


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Additional Information

Thierry Coulhon
Affiliation: Départment de Mathématiques, Université de Cergy-Pontoise, 95302 Cergy Pontoise, France
Email: coulhon@u-cergy.fr

Xuan Thinh Duong
Affiliation: Department of Mathematics, Macquarie University, North Ryde NSW 2113, Australia
Email: duong@macadam.mpce.mq.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02090-5
PII: S 0002-9947(99)02090-5
Received by editor(s): October 1, 1996
Received by editor(s) in revised form: March 20, 1997
Article copyright: © Copyright 1999 American Mathematical Society