Riesz transforms for $1\le p\le 2$
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- by Thierry Coulhon and Xuan Thinh Duong PDF
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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 $\mathbb {R}^n$.References
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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
- MR Author ID: 271083
- Email: duong@macadam.mpce.mq.edu.au
- Received by editor(s): October 1, 1996
- Received by editor(s) in revised form: March 20, 1997
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 1151-1169
- MSC (1991): Primary 42B20, 58G11
- DOI: https://doi.org/10.1090/S0002-9947-99-02090-5
- MathSciNet review: 1458299