Elsevier

Journal of Functional Analysis

Volume 196, Issue 2, 20 December 2002, Pages 443-485
Journal of Functional Analysis

Plancherel-type estimates and sharp spectral multipliers

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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 the Laguerre and Hermite expansions.

MSC

primary 42B15
secondary 35P99

Keywords

Spectral multipliers

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