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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Hardy spaces associated to the discrete Laplacians on graphs and boundedness of singular integrals


Authors: The Anh Bui and Xuan Thinh Duong
Journal: Trans. Amer. Math. Soc. 366 (2014), 3451-3485
MSC (2010): Primary 42B20, 42B25, 60J10
DOI: https://doi.org/10.1090/S0002-9947-2014-05915-1
Published electronically: February 17, 2014
MathSciNet review: 3192603
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Abstract: Let $\Gamma$ be a graph with a weight $\sigma$. Let $d$ and $\mu$ be the distance and the measure associated with $\sigma$ such that $(\Gamma , d, \mu )$ is a doubling space. Let $p$ be the natural reversible Markov kernel associated with $\sigma$ and $\mu$ and $P$ be the associated operator defined by $Pf(x) = \sum _{y} p(x, y)f(y)$. Denote by $L=I-P$ the discrete Laplacian on $\Gamma$. In this paper we develop the theory of Hardy spaces associated to the discrete Laplacian $H^p_L$ for $0<p\leq 1$. We obtain square function characterization and atomic decompositions for functions in the Hardy spaces $H^p_L$, then establish the dual spaces of the Hardy spaces $H^p_L, 0<p\leq 1$. Without the assumption of Poincaré inequality, we show the boundedness of certain singular integrals on $\Gamma$ such as square functions, spectral multipliers and Riesz transforms on the Hardy spaces $H^p_L$, $0<p\leq 1$.


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Additional Information

The Anh Bui
Affiliation: Department of Mathematics, Macquarie University, NSW 2109, Australia – and – Department of Mathematics, University of Pedagogy, Ho Chi Minh City, Vietnam
MR Author ID: 799948
Email: the.bui@mq.ed.au, bt_anh80@yahoo.com

Xuan Thinh Duong
Affiliation: Department of Mathematics, Macquarie University, NSW 2109, Australia
MR Author ID: 271083
Email: xuan.duong@mq.edu.au

Keywords: Graphs, discrete Laplacian, Hardy spaces, spectral multipliers, square functions, Riesz transforms
Received by editor(s): April 24, 2012
Received by editor(s) in revised form: June 22, 2012
Published electronically: February 17, 2014
Additional Notes: The first author was supported by a Macquarie University scholarship
The second author was supported by an ARC Discovery grant
Article copyright: © Copyright 2014 American Mathematical Society