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

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