Littlewood–Paley and Spectral Multipliers on Weighted L p Spaces

Abstract

Let L be a non-negative self-adjoint operator acting on L 2(X), where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e −tL whose kernel p t (x,y) has a Gaussian upper bound but there is no assumption on the regularity in variables x and y. In this article we study weighted L p -norm inequalities for spectral multipliers of L. We show that a weighted Hormander-type spectral multiplier theorem follows from weighted L p -norm inequalities for the Lusin and Littlewood–Paley functions, Gaussian heat kernel bounds, and appropriate L 2 estimates of the kernels of the spectral multipliers.