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  • Research Article
  • Open Access

Boundedness of Littlewood-Paley Operators Associated with Gauss Measures

Journal of Inequalities and Applications20102010:643948

https://doi.org/10.1155/2010/643948

  • Received: 16 December 2009
  • Accepted: 17 March 2010
  • Published:

Abstract

Modeled on the Gauss measure, the authors introduce the locally doubling measure metric space , which means that the set is endowed with a metric and a locally doubling regular Borel measure satisfying doubling and reverse doubling conditions on admissible balls defined via the metric and certain admissible function . The authors then construct an approximation of the identity on , which further induces a Calderón reproducing formula in for . Using this Calderón reproducing formula and a locally variant of the vector-valued singular integral theory, the authors characterize the space for in terms of the Littlewood-Paley -function which is defined via the constructed approximation of the identity. Moreover, the authors also establish the Fefferman-Stein vector-valued maximal inequality for the local Hardy-Littlewood maximal function on . All results in this paper can apply to various settings including the Gauss measure metric spaces with certain admissible functions related to the Ornstein-Uhlenbeck operator, and Euclidean spaces and nilpotent Lie groups of polynomial growth with certain admissible functions related to Schrödinger operators.

Keywords

  • Full Article
  • Gauss Measure
  • Publisher Note
  • Operator Associate

Publisher note

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Authors’ Affiliations

(1)
School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University, Beijing, 100875, China

Copyright

© L. Liu and D. Yang. 2010

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.

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