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Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrüdinger operators

Published online by Cambridge University Press:  11 January 2016

Dachun Yang ,
Dongyong Yang  and
Yuan Zhou
Dachun Yang
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s, Republic of Chinadcyang@bnu.edu.cn
Dongyong Yang
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s, Republic of Chinadyyang@mail.bnu.edu.cn
Yuan Zhou
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s, Republic of Chinayuanzhou@mail.bnu.edu.cn
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Abstract

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Let be a space of homogeneous type in the sense of Coifman and Weiss, and let be a collection of balls in . The authors introduce the localized atomic Hardy space the localized Morrey-Campanato space and the localized Morrey-Campanato-BLO (bounded lower oscillation) space with α ∊ ℝ and p ∊ (0, ∞) , and they establish their basic properties, including and several equivalent characterizations for In particular, the authors prove that when α > 0 and p ∊ [1, ∞), then and when p ∈(0,1], then the dual space of is Let ρ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces and , respectively, by and when is determined by ρ. The authors then obtain the boundedness from of the radial and the Poisson semigroup maximal functions and the Littlewood-Paley 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, the Schrödinger operator or the degenerate Schrödinger operator on ℝd, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 198 , June 2010 , pp. 77 - 119
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2010

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