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

Let (X,ρ)(X,ρ) be a locally compact metric space endowed with a doubling measure μ  , and LL be a non-negative self-adjoint operator on L2(X,dμ)L2(X,dμ). Assume that the semigroup Pt=e−tLPt=e−tL generated by LL consists of integral operators with (heat) kernel pt(x,y)pt(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 LL, and present an atomic decomposition of these function spaces.