Spectral multipliers for self-adjoint operators

Abstract

In this article, we give a survey of spectral multipliers and present (without proof) sharp Hörmander-type multiplier theorems for a self adjoint operator $A$ under the assumption that $A$ has Gaussian heat kernel bounds and satisfies appropriate estimates of the $L^2$ norm of the kernels of spectral multipliers. Our theorems imply several important, previously known results on spectral multipliers and give new results for sharp estimates for the critical exponent for the Riesz means summability.