Open Access

Boundedness of Parametrized Littlewood-Paley Operators with Nondoubling Measures

Journal of Inequalities and Applications20082008:141379

DOI: 10.1155/2008/141379

Received: 2 April 2008

Accepted: 30 July 2008

Published: 31 July 2008

Abstract

Let be a nonnegative Radon measure on which only satisfies the following growth condition that there exists a positive constant such that for all and some fixed . In this paper, the authors prove that for suitable indexes and , the parametrized function is bounded on for with the assumption that the kernel of the operator satisfies some Hörmander-type condition, and is bounded from into weak with the assumption that the kernel satisfies certain slightly stronger Hörmander-type condition. As a corollary, with the kernel satisfying the above stronger Hörmander-type condition is bounded on for . Moreover, the authors prove that for suitable indexes and is bounded from into (the space of regular bounded lower oscillation functions) if the kernel satisfies the Hörmander-type condition, and from the Hardy space into if the kernel satisfies the above stronger Hörmander-type condition. The corresponding properties for the parametrized area integral are also established in this paper.

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Authors’ Affiliations

(1)
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education
(2)
School of Information, Renmin University of China

Copyright

© H. Lin and Y. Meng. 2008

This article is published under license to BioMed Central Ltd. 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.