Embeddings of decomposition spaces

Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two decomposition spaces, is there an embedding between the two? A decomposition space $\mathcal{D}(\mathcal{Q}, L^p, Y)$ can be described using : a covering $\mathcal{Q}=(Q_{i})_{i\in I}$ of the frequency domain, an exponent $p$ and a sequence space $Y\subset\mathbb{C}^{I}$. Given these, the decomp. space norm of a distribution $g$ is $\| g\| _{\mathcal{D}(\mathcal{Q}, L^p, Y)}=\left\| \left(\left\| \mathcal{F}^{-1}\left(\varphi_{i}\widehat{g}\right)\right\| _{L^{p}}\right)_{i\in I}\right\| _{Y}$, where $(\varphi_{i})_{i\in I}$ is a suitable partition of unity for $\mathcal{Q}$. We establish readily verifiable criteria which ensure an embedding $\mathcal{D}(\mathcal{Q}, L^{p_1}, Y)\hookrightarrow\mathcal{D}(\mathcal{P}, L^{p_2}, Z)$, mostly concentrating on the case, $Y=\ell_{w}^{q_{1}}(I)$ and $Z=\ell_{v}^{q_{2}}(J)$. The relevant sufficient conditions are $p_{1}\leq p_{2}$, and finiteness of a norm of the form $$\left\| \left(\left\| (\alpha_{i}\,\beta_j \cdot v_{j}/w_{i})_{i\in I_{j}}\right\| _{\ell^{t}}\right)_{j\in J}\right\| _{\ell^{s}}<\infty,$$ where the $$I_{j}=\{ i\in I : Q_{i}\cap P_{j}\neq\emptyset\} \qquad\text{ for }j\in J$$ are defined in terms of the two coverings $\mathcal{Q}=(Q_{i})_{i\in I}$ and $\mathcal{P}=(P_{j})_{j\in J}$. We also show that these criteria are sharp: For almost arbitrary coverings and certain ranges of $p_{1},p_{2}$, our criteria yield a complete characterization. The same holds for arbitrary values of $p_{1},p_{2}$ under more strict assumptions on the coverings. We illustrate the resulting theory by applications to $\alpha$-modulation and Besov spaces. All known embedding results for these spaces are special cases of our approach; often, we improve considerably upon the state of the art.