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


Authors: Gerard Kerkyacharian and Pencho Petrushev
Journal: Trans. Amer. Math. Soc. 367 (2015), 121-189
MSC (2010): Primary 58J35, 46E35; Secondary 42C15, 43A85
DOI: https://doi.org/10.1090/S0002-9947-2014-05993-X
Published electronically: June 18, 2014
MathSciNet review: 3271256
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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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Additional Information

Gerard Kerkyacharian
Affiliation: Laboratoire de Probabilités et Modèles Aléatoires, CNRS-UMR 7599, Université Paris Diderot, Batiment Sophie Germain, Avenue de France, Paris 75013, France
Email: kerk@math.univ-paris-diderot.fr

Pencho Petrushev
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: pencho@math.sc.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-05993-X
Keywords: Heat kernel, functional calculus, frames, Besov spaces, Triebel-Lizorkin spaces
Received by editor(s): February 28, 2012
Received by editor(s) in revised form: October 9, 2012
Published electronically: June 18, 2014
Additional Notes: The second author was supported by NSF Grant DMS-1211528.
Article copyright: © Copyright 2014 American Mathematical Society