Besov and Triebel–Lizorkin spaces associated with non-negative self-adjoint operators

Let ( X , ρ ) be a locally compact metric space endowed with a doubling measure μ, and L be a non-negative self-adjoint operator on L 2 ( X , d μ ) . Assume that the semigroup P t = e − t L generated by L consists of integral operators with (heat) kernel p t ( x , y ) enjoying Gaussian upper bound but having no information on the regularity in the variables x and y. In this paper, we introduce Besov and Triebel–Lizorkin spaces associated with L , and present an atomic decomposition of these function spaces.