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

Let Γ Γ be a graph with a weight σ σ . Let d d and μ μ be the distance and the measure associated with σ σ such that ( Γ , d , μ ) (Γ , d, μ ) is a doubling space. Let p p be the natural reversible Markov kernel associated with σ σ and μ μ and P P be the associated operator defined by P f ( x ) = ∑ y p ( x , y ) f ( y ) Pf(x) = ∑ y p(x, y)f(y) . Denote by L = I − P L=I-P the discrete Laplacian on Γ Γ . In this paper we develop the theory of Hardy spaces associated to the discrete Laplacian H</mml