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Transactions of the American Mathematical Society
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
Published electronically: February 17, 2014
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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
Email: the.bui@mq.ed.au, bt{\textunderscore}anh80@yahoo.com

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

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-05915-1
PII: S 0002-9947(2014)05915-1
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