HARDY SPACES ASSOCIATED TO THE DISCRETE LAPLACIANS ON GRAPHS AND BOUNDEDNESS OF SINGULAR INTEGRALS

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