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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Localized Hardy spaces $ H^1$ related to admissible functions on RD-spaces and applications to Schrödinger operators


Authors: Dachun Yang and Yuan Zhou
Journal: Trans. Amer. Math. Soc. 363 (2011), 1197-1239
MSC (2010): Primary 42B30; Secondary 42B20, 42B25, 42B35, 42B37
Published electronically: October 22, 2010
MathSciNet review: 2737263
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Abstract: Let $ {\mathcal X}$ be an RD-space, which means that $ {\mathcal X}$ is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $ {\mathcal X}$. In this paper, the authors first introduce the notion of admissible functions $ \rho$ and then develop a theory of localized Hardy spaces $ H^1_\rho ({\mathcal X})$ associated with $ \rho$, which includes several maximal function characterizations of $ H^1_\rho ({\mathcal X})$, the relations between $ H^1_\rho ({\mathcal X})$ and the classical Hardy space $ H^1({\mathcal X})$ via constructing a kernel function related to $ \rho$, the atomic decomposition characterization of $ H^1_\rho ({\mathcal X})$, and the boundedness of certain localized singular integrals on $ H^1_\rho({\mathcal X})$ via a finite atomic decomposition characterization of some dense subspace of $ H^1_\rho ({\mathcal X})$. This theory has a wide range of applications. Even when this theory is applied, respectively, to the Schrödinger operator or the degenerate Schrödinger operator on $ \mathbb{R}^n$, or to the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups, some new results are also obtained. The Schrödinger operators considered here are associated with nonnegative potentials satisfying the reverse Hölder inequality.


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Additional Information

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 China
Email: dcyang@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 China
Email: yuanzhou@mail.bnu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05201-8
PII: S 0002-9947(2010)05201-8
Keywords: Space of homogeneous type, Heisenberg group, connected and simply connected Lie group, admissible function, Hardy space, maximal function, atom, Schrödinger operator, reverse Hölder inequality, Riesz transform.
Received by editor(s): August 26, 2008
Published electronically: October 22, 2010
Additional Notes: The first author was supported by the National Natural Science Foundation (Grant No. 10871025) of China.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.