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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.
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