Continuous characterizations of inhomogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators

Let $(M,\rho ,\mu )$ be a metric measure space satisfying the doubling, reverse doubling and non-collapsing conditions, and $\mathcal{L}$ be a self-adjoint operator on $L^2(M,d\mu )$ whose heat kernel $p_t(x,y)$ satisfy the small-time Gaussian upper bound, H"{o}lder continuity and Markov property. In this paper, we give characterizations of inhomogeneous “classical” and “non-classical” Besov and Triebel-Lizorkin spaces associated to $\mathcal{L}$ in terms of continuous Littlewood-Paley and Lusin area functions defined by the heat semigroup, for complete range of indices. This extends related classical results for Besov and Triebel-Lizorkin spaces on ${\mathbb{R}}^n$ to more general setting, and extends corresponding results in [Trans. Amer. Math Soc. 367 (2015), 121-189] to complete range of indices.