From Frazier-Jawerth characterizations of Besov spaces to Wavelets and Decomposition spaces

摘要

This article describes how the ideas promoted by the fundamental paperspublished by M. Frazier and B. Jawerth in the eighties have influencedsubsequent developments related to the theory of atomic decompositions andBanach frames for function spaces such as the modulation spaces andBesov-Triebel-Lizorkin spaces. Both of these classes of spaces arise as special cases of two different,general constructions of function spaces: coorbit spaces and decompositionspaces. Coorbit spaces are defined by imposing certain decay conditions on theso-called voice transform of the function/distribution under consideration. Asa concrete example, one might think of the wavelet transform, leading to thetheory of Besov-Triebel-Lizorkin spaces. Decomposition spaces, on the other hand, are defined using certaindecompositions in the Fourier domain. For Besov-Triebel-Lizorkin spaces, oneuses a dyadic decomposition, while a uniform decomposition yields modulationspaces. Only recently, the second author has established a fruitful connectionbetween modern variants of wavelet theory with respect to general dilationgroups (which can be treated in the context of coorbit theory) and a particularfamily of decomposition spaces. In this way, optimal inclusion results andinvariance properties for a variety of smoothness spaces can be established. Wewill present an outline of these connections and comment on the basic resultsarising in this context.