Weighted Norm Inequalities for Spectral Multipliers on Graphs

Let Γ be a graph endowed with a reversible Markov kernel p, whose associated operator P is defined by (Pf(x) = {sum }_{y} p(x, y)f(y)). We assume that the kernels p_n(x, y) associated to Pⁿ satisfy Gaussian upper bounds but do not assume they satisfy the Hölder continuity property and the temporal regularity. Denote by L = I ? P the discrete Laplacian on Γ. This article shows the weighted weak type (1, 1) estimates and the weighted L^p norm inequalities for the spectral multipliers of L. We also obtain the weighted L^p norm inequalities for the commutators of the spectral multipliers of L with BMO functions which are new even for the unweighted case.