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Embeddings of Decomposition Spaces into Sobolev and BV Spaces


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Voigtlaender, Felix:
Embeddings of Decomposition Spaces into Sobolev and BV Spaces.
2016. - 64 S.


Open Access
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In the present paper, we investigate whether an embedding of a decomposition space D(Q,Lp,Y) into a given Sobolev space Wk,q(Rd) exists. As special cases, this includes embeddings into Sobolev spaces of (homogeneous and inhomogeneous) Besov spaces, (α)-modulation spaces, shearlet smoothness spaces and also of a large class of wavelet coorbit spaces, in particular of shearlet-type coorbit spaces.
Precisely, we will show that under extremely mild assumptions on the covering Q=(Qi)i∈I, we have D(Q,Lp,Y)↪Wk,q(Rd) as soon as p≤q and Y↪ℓq▽u(k,p,q)(I) hold. Here, q▽=min{q,q′} and the weight u(k,p,q) can be easily computed, only based on the covering Q and on the parameters k,p,q.
Conversely, a necessary condition for existence of the embedding is that p≤q and Y∩ℓ0(I)↪ℓqu(k,p,q)(I) hold, where ℓ0(I) denotes the space of finitely supported sequences on I.
All in all, for the range q∈(0,2]∪{∞}, we obtain a complete characterization of existence of the embedding in terms of readily verifiable criteria. We can also completely characterize existence of an embedding of a decomposition space into a BV space.

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Publikationsform:Preprint, Working paper, Diskussionspapier
Sprache des Eintrags:Englisch
Institutionen der Universität:Mathematisch-Geographische Fakultät > Mathematik > Lehrstuhl für Reliable Machine Learning
DOI / URN / ID:arXiv:1601.02201
Open Access: Freie Zugänglichkeit des Volltexts?:Ja
Titel an der KU entstanden:Nein
Eingestellt am: 30. Mär 2022 14:19
Letzte Änderung: 31. Mär 2022 13:04
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