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@PHDTHESIS{Voigtlaender:564979,
      author       = {Voigtlaender, Felix},
      othercontributors = {Führ, Hartmut and Feichtinger, Hans G. and Rauhut, Holger},
      title        = {{E}mbedding theorems for decomposition spaces with
                      applications to wavelet coorbit spaces},
      school       = {Aachen, Techn. Hochsch.},
      type         = {Dissertation},
      address      = {Aachen},
      publisher    = {Publikationsserver der RWTH Aachen University},
      reportid     = {RWTH-2015-07506},
      year         = {2016},
      note         = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
                      University 2016; 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.},
      cin          = {114320 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)114320_20140620$ / $I:(DE-82)110000_20140620$},
      typ          = {PUB:(DE-HGF)11},
      urn          = {urn:nbn:de:hbz:82-rwth-2015-075066},
      url          = {https://publications.rwth-aachen.de/record/564979},
}