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A difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces. (English) Zbl 1171.42013

The authors discuss relations between various function spaces defined on RD-spaces. An RD-space is a space of homogeneous type in the sense of Coifman and Weiss (metric space with regular Borel measure satisfying doubling property) with additional property that a reverse doubling property holds. Spaces of Lipschitz type, both of homogeneous and inhomogeneous type, are introduce on RD-spaces. They are defined via local means of differences. Relation of the Lipschitz type spaces to homogeneous (respectively inhomogeneous) Besov and Triebel-Lizorkin spaces are discuss. As a result a characterization of the Besov and Triebel-Lizorkin spaces on RD-spaces via differences is obtained. Finally some relations between the Lipschitz type spaces and Hajłasz-Sobolev spaces or Korevaar-Schoen type Sobolev spaces are proved.

MSC:

42B35 Function spaces arising in harmonic analysis
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
43A99 Abstract harmonic analysis
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