Embeddings of decomposition spaces

Abstract: 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.