Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 141379, 25 pages
doi:10.1155/2008/141379
Research Article
Boundedness of Parametrized Littlewood-Paley Operators with Nondoubling Measures
1School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China
2School of Information, Renmin University of China, Beijing 100872, China
Received 2 April 2008; Accepted 30 July 2008
Academic Editor: Siegfried Carl
Copyright © 2008 Haibo Lin and Yan Meng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let μ be a nonnegative Radon measure on ℝd which only
satisfies the following growth condition that there exists a
positive constant C such that μ(B(x,r))≤Crn for all
x∈ℝd, r>0 and some fixed n∈(0,d]. In this paper, the
authors prove that for suitable indexes ρ and λ, the
parametrized gλ∗ function ℳλ∗,ρ is bounded
on Lp(μ) for p∈[2,∞) with the assumption that the kernel
of the operator ℳλ∗,ρ
satisfies some Hörmander-type condition, and is bounded from
L1(μ) into weak L1(μ) with the assumption that the kernel
satisfies certain slightly stronger Hörmander-type condition. As a
corollary, ℳλ∗,ρ with the kernel satisfying the above stronger
Hörmander-type condition is bounded on Lp(μ) for p∈(1,2).
Moreover, the authors prove that for suitable indexes ρ and
λ,ℳλ∗,ρ is bounded from L∞(μ) into RBLO(μ)
(the space of regular bounded lower oscillation functions) if the
kernel satisfies the Hörmander-type condition, and from the Hardy
space H1(μ) into L1(μ) if the kernel satisfies the above
stronger Hörmander-type condition. The corresponding properties
for the parametrized area integral ℳSρ are also established in this paper.