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  • Research Article
  • Open Access

Boundedness of Littlewood-Paley Operators Associated with Gauss Measures

Journal of Inequalities and Applications20102010:643948

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


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.


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

Publisher note

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Authors’ Affiliations

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