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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Hardy spaces associated to the discrete Laplacians on graphs and boundedness of singular integrals
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by The Anh Bui and Xuan Thinh Duong PDF
Trans. Amer. Math. Soc. 366 (2014), 3451-3485 Request permission

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
  • 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
  • © Copyright 2014 American Mathematical Society
  • 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
  • MathSciNet review: 3192603