We study the weak type (1,1) and the -boundedness, 1 < p ≤ 2, of the so-called vertical (i.e. involving space derivatives) Littlewood-Paley-Stein functions 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 and ℋ are bounded in , 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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm154-1-4,
author = {Thierry Coulhon and Xuan Thinh Duong and Xiang Dong Li},
title = {Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 $\leq$ p $\leq$ 2},
journal = {Studia Mathematica},
volume = {157},
year = {2003},
pages = {37-57},
zbl = {1035.42014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm154-1-4}
}
Thierry Coulhon; Xuan Thinh Duong; Xiang Dong Li. Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2. Studia Mathematica, Tome 157 (2003) pp. 37-57. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm154-1-4/