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

摘要

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) = Σ_y p(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 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 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 H_L~p, 0 < p ≤ 1.