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Let $X$ be a space of homogeneous type and let $L$ be a nonnegativeself-adjoint operator on $L^2(X)$ which satisfies a Gaussian estimate on itsheat kernel. In this paper we prove a H\"omander type spectral multipliertheorem 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 whichproves the boundedness of a general spectral theorem on Besov andTriebel--Lizorkin spaces.

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