Weighted Norm Inequalities on Graphs

Abstract

Let (Γ,μ) be an infinite graph endowed with a reversible Markov kernel p and let P be the corresponding operator. We also consider the associated discrete gradient ∇. We assume that μ is doubling, a uniform lower bound for p(x,y) when p(x,y)>0, and gaussian upper estimates for the iterates of p. Under these conditions (and in some cases assuming further some Poincare inequality) we study the comparability of (I−P)1/2 f and ∇f in Lebesgue spaces with Muckenhoupt weights. Also, we establish weighted norm inequalities for a Littlewood–Paley–Stein square function, its formal adjoint, and commutators of the Riesz transform with bounded mean oscillation functions.