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

Thierry Coulhon, Xuan Thinh Duong, and Xiang Dong Li. "Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2." Studia Mathematica 154.1 (2003): 37-57. <http://eudml.org/doc/284405>.

@article{ThierryCoulhon2003,
abstract = {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 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 $L^\{p\}$, 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 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.},
author = {Thierry Coulhon, Xuan Thinh Duong, Xiang Dong Li},
journal = {Studia Mathematica},
keywords = {heat kernel; Poisson kernel; Littlewood-Paley-Stein functionals; weak type (1,1)},
language = {eng},
number = {1},
pages = {37-57},
title = {Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2},
url = {http://eudml.org/doc/284405},
volume = {154},
year = {2003},
}

TY - JOUR
AU - Thierry Coulhon
AU - Xuan Thinh Duong
AU - Xiang Dong Li
TI - Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2
JO - Studia Mathematica
PY - 2003
VL - 154
IS - 1
SP - 37
EP - 57
AB - 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 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 $L^{p}$, 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 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.
LA - eng
KW - heat kernel; Poisson kernel; Littlewood-Paley-Stein functionals; weak type (1,1)
UR - http://eudml.org/doc/284405
ER -