Boundedness of Littlewood-Paley Operators Associated with Gauss Measures

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.

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