Littlewood–Paley–Stein functions on complete Riemannian manifolds for $1\leq p\leq 2$

Volume 154 / 2003

Thierry Coulhon, Xuan Thinh Duong, Xiang Dong Li Studia Mathematica 154 (2003), 37-57 MSC: 42B25, 58J35. DOI: 10.4064/sm154-1-4

Abstract

We study the weak type $(1, 1)$ and the $L^p$-boundedness, $1< p\le 2$, of the so-called vertical (i.e. involving space derivatives) Littlewood–Paley–Stein functions ${\cal G}$ and ${\cal H}$ respectively associated with the Poisson semigroup and the heat semigroup on a complete Riemannian manifold $M$. Without any assumption on $M$, we observe that ${\cal G}$ and ${\cal H}$ are bounded in $L^p$, $1< p\leq 2$. 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 ${\cal G}$ and ${\cal H}$ (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 ${\mathbb R}^n$.

Authors

  • Thierry CoulhonUniversité de Cergy-Pontoise
    95302 Pontoise, France
    e-mail
  • Xuan Thinh DuongMacquarie University
    North Ryde, NSW 2113
    Australia
    e-mail
  • Xiang Dong LiUniversity of Oxford
    Oxford OX1 3LB, United Kingdom
    e-mail

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