Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 141379, 25 pages
Research Article

Boundedness of Parametrized Littlewood-Paley Operators with Nondoubling Measures

Haibo Lin1 and Yan Meng2

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.


Let μ be a nonnegative Radon measure on d which only satisfies the following growth condition that there exists a positive constant C such that μ(B(x,r))Crn for all xd,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 λ,ρ is bounded on Lp(μ) for p[2,) with the assumption that the kernel of the operator λ,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, λ,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,λ,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 Sρ are also established in this paper.