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Besov and Triebel–Lizorkin spaces on spaces of homogeneous typewith applications to boundedness of Calderón–Zygmund operators

Volume 565 / 2021

Fan Wang, Yongsheng Han, Ziyi He, Dachun Yang Dissertationes Mathematicae 565 (2021), 1-113 MSC: Primary 46E35; Secondary 42B25, 42B20, 42B35, 30L99. DOI: 10.4064/dm821-4-2021 Published online: 12 August 2021

Abstract

In this article, the authors introduce Besov and Triebel–Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss, prove that these (inhomogeneous) Besov and Triebel–Lizorkin spaces are independent of the choices of both (inhomogeneous) approximations of the identity with exponential decay and underlying spaces of distributions, and give some basic properties of these spaces. As applications, the authors show that some known function spaces coincide with certain special cases of Besov and Triebel–Lizorkin spaces and, moreover, obtain the boundedness of Calderón–Zygmund operators on these Besov and Triebel–Lizorkin spaces. All these results strongly depend on the geometrical properties, reflected via dyadic cubes, of the relevant space of homogeneous type. Compared with the known theory of these spaces on metric measure spaces, a major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the measure under study of the underlying space.

Authors

  • Fan WangLaboratory of Mathematics and Complex Systems (Ministry of Education)
    School of Mathematical Sciences
    Beijing Normal University
    Beijing 100875, People’s Republic of China
    e-mail
  • Yongsheng HanDepartment of Mathematics
    Auburn University
    Auburn, AL 36849-5310, U.S.A.
    e-mail
  • Ziyi HeLaboratory of Mathematics and Complex Systems (Ministry of Education)
    School of Mathematical Sciences
    Beijing Normal University
    Beijing 100875, People’s Republic of China
    e-mail
  • Dachun YangLaboratory of Mathematics and Complex Systems (Ministry of Education)
    School of Mathematical Sciences
    Beijing Normal University
    Beijing 100875, People’s Republic of China
    e-mail

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