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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Riesz transforms for $1\le p\le 2$
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by Thierry Coulhon and Xuan Thinh Duong PDF
Trans. Amer. Math. Soc. 351 (1999), 1151-1169 Request permission

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
  • 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