Wavelet characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type and its applications. (English) Zbl 1477.46041
Authors’ abstract: In this article, the authors establish the wavelet characterization of Besov and Triebel-Lizorkin spaces on a given space \((X, d,\mu)\) of homogeneous type in the sense of Coifman and Weiss. Moreover, the authors introduce almost diagonal operators on Besov and Triebel-Lizorkin sequence spaces on \(X\), and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors obtain the molecular characterization of Besov and Triebel-Lizorkin spaces. Applying this molecular characterization, the authors further establish the Littlewood-Paley characterizations of Triebel-Lizorkin spaces on \(X\). The main novelty of this article is that all these results get rid of their dependence on the reverse doubling property of μ and also the triangle inequality of \(d\), by fully using the geometrical property of \(X\) expressed via its equipped quasi-metric \(d\), dyadic reference points, dyadic cubes, and wavelets.
Reviewer: Jie Xiao (St. John’s)
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46E36 | Sobolev (and similar kinds of) spaces of functions on metric spaces; analysis on metric spaces |
| 46E39 | Sobolev (and similar kinds of) spaces of functions of discrete variables |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 30L99 | Analysis on metric spaces |
Keywords:
space of homogeneous type; Besov space; Triebel-Lizorkin space; almost diagonal operator; wavelet; molecule; Littlewood-Paley functionReferences:
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