Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2

Thierry Coulhon*, Xuan Think Duong, Xiang Dong Li

*Corresponding author for this work

    Research output: Contribution to journalArticle

    21 Citations (Scopus)

    Abstract

    We study the weak type (1,1) and the Lp-boundedness, 1 < p ≤ 2, of the so-called vertical (i.e. involving space derivatives) Littlewood-Paley-Stein functions G and H, respectively associated with the Poisson semigroup and the heat semigroup on a complete Riemannian manifold M. Without any assumption on M, we observe that G and H are bounded in Lp, 1 < p ≤ 2. We also consider modified Littlewood-Paley-Stein functions that take into account the positivity of the bottom of the spectrum. Assuming that M satisfies the doubling volume property and an optimal on-diagonal heat kernel estimate, we prove that G and H (as well as the corresponding horizontal functions, i.e. involving time derivatives) are of weak type (1, 1). Finally, we apply our methods to divergence form operators on arbitrary domains of ℝn.

    Original languageEnglish
    Pages (from-to)37-57
    Number of pages21
    JournalStudia Mathematica
    Volume154
    Issue number1
    Publication statusPublished - 2003

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