CMA Research Report

MRR01-002

Plancherel Type Estimates and Sharp Spectral Multipliers

Abstract:  We study general spectral multiplier theorems for self-adjoint positive definite operators on L²(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 L² 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 R³ and new spectral multiplier theorems for Laguerre and Hermite expansions.

AMS Classification:  42B15, 35P99
Date:  3 September 2000

Download PDF   ( 428k )


Return to MRR01 contents