Localized Morrey-Campanato Spaces on Metric Measure Spaces and Applications to Schrödinger Operators

Let X be a space of homogeneous type in the sense of Coifman and Weiss and D a collection of balls in X . The authors introduce the localized atomic Hardy space H q D (X ) with p ∈ (0, 1] and q ∈ [1,∞] ∩ (p,∞], the localized Morrey-Campanato space E p D (X ) and the localized Morrey-Campanato-BLO space Ẽ p D (X ) with α ∈ R and p ∈ (0,∞) and establish their basic properties including H q D (X ) = H D (X ) and several equivalent characterizations for E p D (X ) and Ẽ p D (X ). Especially, the authors prove that when α > 0 and p ∈ [1, ∞), then Ẽ p D (X ) = E p D (X ) = LipD(α; X ), and when p ∈ (0, 1], then the dual space of H D (X ) is E 1 D (X ). Let ρ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces E p D (X ) and Ẽ p D (X ), respectively, by Eα, p ρ (X ) and Ẽα, p ρ (X ), when D is determined by ρ. The authors then obtain the boundedness from Eα, p ρ (X ) to Ẽα, p ρ (X ) of the radial and the Poisson semigroup maximal functions and the LittlewoodPaley g-function which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, to the Schrödinger operator or the degenerate Schrödinger operator on R, or the subLaplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.