Copyright © 2010 Liguang Liu and Dachun Yang. 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.

Abstract

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 Lp(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 Lp(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.