Riesz transforms for 1 ≤ p ≤ 2

Thierry Coulhon*, Xuan Thinh Duong

*Corresponding author for this work

    Research output: Contribution to journalArticle

    145 Citations (Scopus)

    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 Lp boundedness of the Riesz transforms that holds in ℝn. 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 ℝn.

    Original languageEnglish
    Pages (from-to)1151-1169
    Number of pages19
    JournalTransactions of the American Mathematical Society
    Volume351
    Issue number3
    Publication statusPublished - 1999

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