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Transactions of the American Mathematical Society

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Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type


Authors: The Anh Bui, Xuan Thinh Duong and Fu Ken Ly
Journal: Trans. Amer. Math. Soc. 370 (2018), 7229-7292
MSC (2010): Primary 42B30, 42B35, 47B38
DOI: https://doi.org/10.1090/tran/7289
Published electronically: July 5, 2018
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Abstract: Let $ X$ be a space of homogeneous type and let $ \mathfrak{L}$ be a nonnegative self-adjoint operator on $ L^2(X)$ enjoying Gaussian estimates. The main aim of this paper is twofold. Firstly, we prove (local) nontangential and radial maximal function characterizations for the local Hardy spaces associated to $ \mathfrak{L}$. This gives the maximal function characterization for local Hardy spaces in the sense of Coifman and Weiss provided that $ \mathfrak{L}$ satisfies certain extra conditions. Secondly we introduce local Hardy spaces associated with a critical function $ \rho $ which are motivated by the theory of Hardy spaces related to Schrödinger operators and of which include the local Hardy spaces of Coifman and Weiss as a special case. We then prove that these local Hardy spaces can be characterized by (local) nontangential and radial maximal functions related to $ \mathfrak{L}$ and $ \rho $, and by global maximal functions associated to `perturbations' of $ \mathfrak{L}$. We apply our theory to obtain a number of new results on maximal characterizations for the local Hardy type spaces in various settings ranging from Schrödinger operators on manifolds to Schrödinger operators on connected and simply connected nilpotent Lie groups.


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Additional Information

The Anh Bui
Affiliation: Department of Mathematics, Macquarie University, NSW 2109, Australia
Email: the.bui@mq.edu.au, bt_anh80@yahoo.com

Xuan Thinh Duong
Affiliation: Department of Mathematics, Macquarie University, NSW 2109, Australia
Email: xuan.duong@mq.edu.au

Fu Ken Ly
Affiliation: The School of Mathematics and Statistics, Faculty of Science and the Mathematics Learning Center, Education Portfolio, University of Sydney, NSW 2006, Australia
Email: ken.ly@sydney.edu.au

DOI: https://doi.org/10.1090/tran/7289
Keywords: Hardy space, atomic decomposition, the nontangential maximal function, the radial maximal function, critical function, Schr\"odinger operator.
Received by editor(s): November 3, 2016
Received by editor(s) in revised form: March 17, 2017
Published electronically: July 5, 2018
Additional Notes: The first and second authors were supported by the research grant ARC DP140100649 from the Australian Research Council.
Article copyright: © Copyright 2018 American Mathematical Society

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