Abstract
We give characterizations for homogeneous and inhomogeneous Besov-Lizorkin-Triebel spaces (H. Triebel 1983, 1992, and 2006) in terms of continuous local means for the full range of parameters. In particular, we prove characterizations in terms of Lusin functions (tent spaces) and spaces involving the Peetre maximal function to apply the classical coorbit space theory according to Feichtinger and Gröchenig (H. G Feichtinger and K. Gröchenig 1988, 1989, and 1991). This results in atomic decompositions and wavelet bases for homogeneous spaces. In particular we give sufficient conditions for suitable wavelets in terms of moment, decay and smoothness conditions.
Highlights
This paper deals with Besov-Lizorkin-Triebel spaces Bps,q Rd and Fps,q Rd on the Euclidean space Rd and their interpretation as coorbits
Equivalent quasi- normings of this kind were first given by Triebel in 2
Once we have interpreted classical homogeneous Besov-Lizorkin-Triebel spaces as certain coorbits, we are able to benefit from the achievements of the abstract theory in 11– 15
Summary
This paper deals with Besov-Lizorkin-Triebel spaces Bps,q Rd and Fps,q Rd on the Euclidean space Rd and their interpretation as coorbits. We use the established characterizations for the homogeneous spaces in order to embed them in the abstract framework of coorbit space theory originally due to Feichtinger and Grochenig 11–15 in the 1980s This connection was already observed by them in 11, 14, 15. They worked with Triebel’s equivalent continuous normings from 2 and the results on tent spaces, which were introduced more or less at the same time by Coifman et al 10 to interpret Lizorkin-Triebel spaces as coorbits. Once we have interpreted classical homogeneous Besov-Lizorkin-Triebel spaces as certain coorbits, we are able to benefit from the achievements of the abstract theory in 11– 15. The underlying decay result of the continuous wavelet transform and some basic facts about orthonormal wavelet bases are shifted to the appendix
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