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

The Anh Bui*, Xuan Thinh Duong

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    9 Citations (Scopus)

    Abstract

    Let Γ be a graph with a weight σ. Let d and μ be the distance and the measure associated with σ such that (Γ, d, μ) is a doubling space. Let p be the natural reversible Markov kernel associated with σ and μ and P be the associated operator defined by Pf(x) =∑yp(x, y)f(y). Denote by L = I −P the discrete Laplacian on Γ. In this paper we develop the theory of Hardy spaces associated to the discrete Laplacian HL pfor 0 < p ≤ 1. We obtain square function characterization and atomic decompositions for functions in the Hardy spaces HL p, then establish the dual spaces of the Hardy spaces HL p, 0 < p ≤ 1. Without the assumption of Poincaré inequality, we show the boundedness of certain singular integrals on Γ such as square functions, spectral multipliers and Riesz transforms on the Hardy spaces HL p, 0 < p ≤ 1.

    Original languageEnglish
    Pages (from-to)3451-3485
    Number of pages35
    JournalTransactions of the American Mathematical Society
    Volume366
    Issue number7
    DOIs
    Publication statusPublished - 2014

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