  Classical and non classical Besov and Triebel-Lizorkin spaces with complete
range of indices are developed in the general setting of Dirichlet space with a
doubling measure and local scale-invariant Poincar\'e inequality. This leads to
Heat kernel with small time Gaussian bounds and H\"older continuity, which play
a central role in this article. Frames with band limited elements of
sub-exponential space localization are developed, and frame and heat kernel
characterizations of Besov and Triebel-Lizorkin spaces are established. This
theory, in particular, allows to develop Besov and Triebel-Lizorkin spaces and
their frame and heat kernel characterization in the context of Lie groups,
Riemannian manifold, and other settings.
