Plancherel-type estimates and sharp spectral multipliers

Abstract

We study general spectral multiplier theorems for self-adjoint positive definite operators on L 2( X, μ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmander-type spectral multiplier theorems follow from the appropriate estimates of the L 2 norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding heat kernel. The sharp Hörmander-type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the Riesz means summability, which we also study here. We discuss several examples, which include sharp spectral multiplier theorems for a class of scattering operators on R 3 and new spectral multiplier theorems for the Laguerre and Hermite expansions.