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

Abstract

Let $$(M, \rho , \mu )$$ be a metric measure space satisfying the doubling, reverse doubling and noncollapsing conditions, and let $$\mathscr {L}$$ be a nonnegative self-adjoint operator acting on $$L^2 (M, d\mu )$$ whose heat kernel satisfies the small-time Gaussian upper bound, Hölder continuity and Markov property. In this paper, we establish new characterizations of the “classical” and “nonclassical” Besov and Triebel-Lizorkin spaces associated to $$\mathscr {L}$$ introduced by Kerkyacharian and Petrushev. More precisely, we obtain characterizations of these spaces in terms of continuous Littlewood-Paley and Lusin functions associated to the heat semigroup generated by $$\mathscr {L}$$ , for complete range of indices. This extends related known results in the classical Euclidean setting to our general setting, and extends corresponding results in (Trans Am Math Soc, 367:121–189, 2015) to complete range of indices.