Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 175230, 28 pages
Research Article

Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures

Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Ministry of Education, Beijing 100875, China

Received 18 April 2009; Accepted 18 August 2009

Academic Editor: Shusen Ding

Copyright © 2009 Qingying Xue and Juyang Zhang. 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 positive Radon measure on d which may be nondoubling. The only condition that μ satisfies is μ(B(x,r))C0rn for all xd, r>0, and some fixed constant C0. In this paper, we introduce the operator gλ,μ related to such a measure and assume it is bounded on L2(μ). We then establish its boundedness, respectively, from the Lebesgue space L1(μ) to the weak Lebesgue space L1,(μ), from the Hardy space H1(μ) to L1(μ) and from the Lesesgue space L(μ) to the space RBLO(μ). As a corollary, we obtain the boundedness of gλ,μ in the Lebesgue space Lp(μ) with p(1,).