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Let   $X$   be a space of homogeneous type and let   $L$   be a nonnegative self-adjoint operator on   $L^2(X)$   that satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Hörmander-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 1st one that proves the boundedness of a general spectral multiplier theorem on Besov and Triebel–Lizorkin spaces.

Spectral Multipliers of Self-Adjoint Operators on Besov and Triebel–Lizorkin Spaces Associated to Operators