Embeddings of Decomposition Spaces into Sobolev and BV Spaces

Abstract

In the present paper, we investigate whether an embedding of a decomposition space $\mathcal{D}\left(\mathcal{Q},L^p,Y\right)$ into a given Sobolev space $W^{k,q}({\mathbb{R}}^d)$ exists. As special cases, this includes embeddings into Sobolev spaces of (homogeneous and inhomogeneous) Besov spaces, ($\alpha$)-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 $\mathcal{Q}={\left(Q_i\right)}_{i\in I}$, we have $\mathcal{D}\left(\mathcal{Q},L^p,Y\right)\hookrightarrow W^{k,q}({\mathbb{R}}^d)$ as soon as $p\le q$ and $Y\hookrightarrow {\ell }_{u^{\left(k,p,q\right)}}^{q^{▽}}\left(I\right)$ hold. Here, $q^{▽}=\min \left\{q,q^{\prime }\right\}$ and the weight $u^{\left(k,p,q\right)}$ can be easily computed, only based on the covering $\mathcal{Q}$ and on the parameters $k,p,q$. Conversely, a necessary condition for existence of the embedding is that $p\le q$ and $Y\cap {\ell }_0\left(I\right)\hookrightarrow {\ell }_{u^{\left(k,p,q\right)}}^q\left(I\right)$ hold, where ${\ell }_0\left(I\right)$ denotes the space of finitely supported sequences on $I$. All in all, for the range $q\in (0,2]\cup \{\infty \}$, 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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