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Besov and Triebel-Lizorkin spaces on spaces of homogeneous type with applications to boundedness of Calderón-Zygmund operators. (English) Zbl 1489.46044

Let \((\mathcal{X},d,\mu)\) be a space of homogeneous type in the sense of Coifman and Weiss, where \(d\) is a quasi-metric and \(\mu\) is a doubling measure. This paper is devoted to developing a theory of Besov and Triebel-Lizorkin spaces on \((\mathcal{X},d,\mu)\) for full admissible ranges \(s\in(0,1)\) and \(p,q\in(0,\infty]\) in this setting. Via using approximations of the identity with exponential decay, the authors introduce the homogeneous Besov space \({\dot B}_{p,q}^s(\mathcal{X})\), the homogeneous Triebel-Lizorkin space \({\dot F}_{p,q}^s(\mathcal{X})\), as well as their inhomogeneous counterparts \(B_{p,q}^s(\mathcal{X})\) and \(F_{p,q}^s(\mathcal{X})\). All these spaces are proved to be independent of the choices of both approximations of the identity with exponential decay and the related spaces of distributions. Some basic embedding properties of these spaces are also given. Besides, it is proved that some known function spaces, including Lebesgue spaces, Hölder spaces, and the space of functions of bounded mean oscillation, coincide with certain special cases of these Besov and Triebel-Lizorkin spaces. As an application, the boundedness of certain Calderón-Zygmund operators on these Besov and Triebel-Lizorkin spaces is established. All these results get rid of the dependence on the reverse doubling assumption of the considered measure of the underlying space.
Reviewer: Wen Yuan (Beijing)

MSC:

46E36 Sobolev (and similar kinds of) spaces of functions on metric spaces; analysis on metric spaces
42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B35 Function spaces arising in harmonic analysis
30L99 Analysis on metric spaces
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