Potential Analysis

, Volume 41, Issue 3, pp 817–848 | Cite as

Weighted Hardy Spaces Associated to Discrete Laplacians on Graphs and Applications



Let Γ be a infinite graph with a weight μ and let d and m be the distance and the measure associated with μ such that (Γ,d,m) is a space of homogeneous type. Let p(·,·) be the natural reversible Markov kernel on (Γ,d,m) and its associated operator be defined by \(Pf(x) = \sum _{y} p(x, y)f(y)\). Then the discrete Laplacian on L 2(Γ) is defined by L=IP. In this paper we investigate the theory of weighted Hardy spaces \({H^{p}_{L}}(\Gamma , w)\) associated to the discrete Laplacian L for 0<p≤1 and \(w\in A_{\infty }\). Like the classical results, we prove that the weighted Hardy spaces \({H^{p}_{L}}(\Gamma , w)\) can be characterized in terms of discrete area operators and atomic decompositions as well. As applications, we study the boundedness of singular integrals on (Γ,d,m) such as square functions, spectral multipliers and Riesz transforms on these weighted Hardy spaces \({H^{p}_{L}}(\Gamma ,w)\).


Graphs Discrete Laplacian Hardy spaces Spectral multipliers Square functions Riesz transforms 

Mathematics Subject Classifications (2010)

60J10 42B20 42B25 


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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsMacquarie UniversityNSWAustralia
  2. 2.Department of MathematicsUniversity of PedagogyHo Chi Minh CityVietnam

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