Boundedness of Littlewood-Paley Operators Associated with Gauss Measures

Modeled on the Gauss measure, the authors introduce the locally doubling measure metric space X, d, μ ρ, which means that the set X is endowed with a metric d and a locally doubling regular Borel measure μ satisfying doubling and reverse doubling conditions on admissible balls defined via the metric d and certain admissible function ρ. The authors then construct an approximation of the identity on X, d, μ ρ, which further induces a Calderón reproducing formula in L X for p ∈ 1,∞ . Using this Calderón reproducing formula and a locally variant of the vector-valued singular integral theory, the authors characterize the space L X for p ∈ 1,∞ in terms of the Littlewood-Paley g-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 X, d, μ ρ. 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.