Embeddings of Decomposition Spaces into Sobolev and BV Spaces

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