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

• Autorzy: Thierry Coulhon , Xuan Thinh Duong , Xiang Dong Li
• Języki publikacji: EN

Abstrakty

EN

We study the weak type (1,1) and the $L^{p}$-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 $L^{p}$, 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.

Kategorie tematyczne

• 42B25: Maximal functions, Littlewood-Paley theory
• 58J35: Heat and other parabolic equation methods

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2003
• Tom: 154
• Numer: 1
• Strony: 37-57

Daty

wydano

2003

Twórcy

autor

Thierry Coulhon

• Université de Cergy-Pontoise, 95302 Pontoise, France

autor

Xuan Thinh Duong

• Macquarie University, North Ryde, NSW 2113, Australia

autor

Xiang Dong Li

• University of Oxford, Oxford OX1 3LB, United Kingdom

Identyfikatory

DOI

10.4064/sm154-1-4