Spectral multipliers of self-adjoint operators on Besov and Triebel--Lizorkin spaces associated to operators

TL;DR

This paper establishes a spectral multiplier theorem for self-adjoint operators on Besov and Triebel--Lizorkin spaces, extending boundedness results beyond classical L^p and Hardy spaces to these more refined function spaces.

Contribution

It provides the first proof of spectral multiplier boundedness on Besov and Triebel--Lizorkin spaces associated to operators satisfying Gaussian heat kernel estimates.

Findings

Proves a Hörmander type spectral multiplier theorem for these spaces.

Recovers boundedness on L^p and Hardy spaces.

First to establish spectral multiplier boundedness on Besov and Triebel--Lizorkin spaces.

Abstract

Let $X$ be a space of homogeneous type and let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ which satisfies a Gaussian estimate on its heat kernel. In this paper we prove a H\"omander type spectral multiplier theorem for $L$ on the Besov and Triebel--Lizorkin spaces associated to $L$. Our work not only recovers the boundedness of the spectral multipliers on $L^p$ spaces and Hardy spaces associated to $L$, but also is the first one which proves the boundedness of a general spectral theorem on Besov and Triebel--Lizorkin spaces.