Embedding theorems for decomposition spaces with applications to wavelet coorbit spaces

• Einbettungssätze für Dekompositions-Räume mit Anwendungen auf Wavelet-Coorbit-Räume

Voigtlaender, Felix; Führ, Hartmut (Thesis advisor); Feichtinger, Hans G. (Thesis advisor); Rauhut, Holger (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2016)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2015

Abstract

The main topic of this thesis is the development of criteria for the (non)-existence of embeddings between decomposition spaces.A decomposition space is defined in terms of- a covering $\mathcal{Q}=(Q_{i})_{i\in I}$ of (a subset) of the frequency space $\mathbb{R}^{d}$,- an integrability exponent $p$ and- a certain discrete sequence space $Y$ on the index set $I$.The decomposition space norm of a distribution $f$ is then computed by decomposing the frequency content of $f$ according to the covering $\mathcal{Q}$, using a suitable partition of unity. Each of the localized pieces is measured in the Lebesgue space $L^{p}$ and the contributions of the individual pieces are aggregated using the discrete sequence space norm $\Vert \cdot\Vert_{Y}$. Given two decomposition spaces, it is of interest to know whether there is an embedding between these two spaces. Since both decomposition spaces are defined only in terms of the respective coverings, weights and discrete sequence spaces, it should be possible to decide the existence of the embedding only based on these quantities. Our findings will show that this is not only possible, but that the resulting criteria only involve discrete combinatorial considerations. In particular, no knowledge of Fourier analysis is needed for the application of these criteria. Finally, our results completely characterize the existence of the desired embedding under mild assumptions on the two coverings and sequence spaces. We apply our findings to a large number of concrete examples. Among others, we consider embeddings between- $\alpha$ -modulation spaces,- homogeneous and inhomogeneous Besov spaces and- shearlet-type coorbit spaces.In all cases, the known results for embeddings between these spaces turn out to be special cases of our criteria; in some cases, our new approach even yields stronger results than those previously known.For the discussion of shearlet-type coorbit spaces, we employ the second main result of this thesis which shows that the Fourier transform induces a natural isomorphism between a large class of wavelet coorbit spaces and certain decomposition spaces. This further emphasizes the scope of our embedding results for decomposition spaces.