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

Kerkyacharian, Gerard; Petrushev, Pencho

doi:10.1090/S0002-9947-2014-05993-X

Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces
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by Gerard Kerkyacharian and Pencho Petrushev PDF

Trans. Amer. Math. Soc. 367 (2015), 121-189 Request permission

Abstract:

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 Poincaré inequality. This leads to a heat kernel with small time Gaussian bounds and Hölder 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.

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