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Classical and non classical Besov and Triebel-Lizorkin spaces with completerange of indices are developed in the general setting of Dirichlet space with adoubling measure and local scale-invariant Poincar\'e inequality. This leads toHeat kernel with small time Gaussian bounds and H\"older continuity, which playa central role in this article. Frames with band limited elements ofsub-exponential space localization are developed, and frame and heat kernelcharacterizations of Besov and Triebel-Lizorkin spaces are established. Thistheory, in particular, allows to develop Besov and Triebel-Lizorkin spaces andtheir frame and heat kernel characterization in the context of Lie groups,Riemannian manifold, and other settings.

Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces