## 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 $H^p_L$, $0<p\leq 1$.## References

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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