Spectral multipliers of self-adjoint operators on besov and triebel-lizorkin spaces associated to operators

Let X be a space of homogeneous type and let L be a nonnegative self-adjoint operator on L²(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 Lp 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.