Heat kernel and Lipschitz-Besov spaces

On a metric measure space (M, ρ, μ) we consider a family of the Lipschitz-Besov spaces Λ s,q that is defined only using the metric ρ and measure μ, and a family of Besov spaces B s p,q that is defined using an auxiliary self-adjoint operator L and the associated heat semigroup. Under certain assumptions about the heat kernel of L, we prove the identity of the two families of the function spaces.