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=I?P. 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 (win 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)) .