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

Thierry Coulhon¹ ; Xuan Thinh Duong² ; Xiang Dong Li³

¹ Université de Cergy-Pontoise 95302 Pontoise, France
² Macquarie University North Ryde, NSW 2113 Australia
³ University of Oxford Oxford OX1 3LB, United Kingdom

Studia Mathematica, Tome 154 (2003) no. 1, pp. 37-57

Résumé

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 ${\mathbb R}^n$.

Détail

DOI : 10.4064/sm154-1-4

Keywords: study weak type p boundedness so called vertical involving space derivatives littlewood paley stein functions cal cal respectively associated poisson semigroup heat semigroup complete riemannian manifold without assumption observe cal cal bounded leq consider modified littlewood paley stein functions account positivity bottom spectrum assuming satisfies doubling volume property optimal on diagonal heat kernel estimate prove cal cal corresponding horizontal functions involving time derivatives weak type finally apply methods divergence form operators arbitrary domains mathbb