HEAT KERNEL BASED DECOMPOSITION OF SPACES OF DISTRIBUTIONS IN THE FRAMEWORK OF DIRICHLET SPACES

Classical and nonclassical 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 Poincare inequality. This leads to a heat kernel with small time Gaussian bounds and Holder 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 the development of Besov and Triebel-Lizorkin spaces and their frame and heat kernel characterization in the context of Lie groups, Riemannian manifolds, and other settings.