分解空间的嵌入
Embeddings of decomposition spaces
摘要：
调和分析中的许多光滑空间是分解空间。在本文中，我们问：给定两个分解空间，两者之间是否存在嵌入？分解空间$\mathcal{D}(\mathcal{Q}, L^p, Y)$可以使用以下来描述：频域的覆盖$\mathcal{Q}=(Q_{i})_{i\in I}$、指数$p$和序列空间$Y\subset\mathbb{C}^{I}$。根据这些腐烂程度分布$g$空间范数是$\| g\| _{\mathcal{D}(\mathcal{Q}, L^p, Y)}=\left\| \left(\left\| \mathcal{F}^{-1}\left(φ_{i}\widehat{g}\right)\right\| _{L^{p}}\right)_{i\in I}\right\| _{Y}$，其中$(φ_{i})_{i\in I}$是$\mathcal{Q}$的适当单位划分。我们建立易于验证的标准，确保嵌入$\mathcal{D}(\mathcal{Q}, L^{p_1}, Y)\hookrightarrow\mathcal{D}(\mathcal{P}, L^{p_2}, Z)$，主要集中在案件，$Y=\ell_{w}^{q_{1}}(I)$和$Z=\ell_{v}^{q_{2}}(J)$。相关的充分条件是$p_{1}\leq p_{2}$，和形式为\[ \left\| \left(\left\| (α_{i}\,β_j \cdot v_{j}/w_{i})_{i\in I_{j}}\right\| _{\ell^{t}}\right)_{j\in J}\right\| _{\ell^{s}}<\infty, \]的范数的有限性，其中\[ I_{j}=\{ i\in I : Q_{i}\cap P_{j}\neq\emptyset\} \qquad\text{ for }j\in J \]根据两个覆盖$\mathcal{Q}=(Q_{i})_{i\in I}$和$\mathcal{P}=(P_{j})_{j\in J}$定义。我们还表明，这些标准是尖锐的：几乎任意覆盖和一定范围的$p_{1},p_{2}$，我们的标准产生一个完整的表征。在覆盖上更严格的假设下，这同样适用于$p_{1},p_{2}$的任意值。我们说明了由此产生的理论的应用$α$调制和Besov空间。这些空间的所有已知嵌入结果都是我们方法的特殊情况;通常，我们在现有技术的基础上进行了相当大的改进。
英文摘要：
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 $\| g\| _{\mathcal{D}(\mathcal{Q}, L^p, Y)}=\left\| \left(\left\| \mathcal{F}^{-1}\left(φ_{i}\widehat{g}\right)\right\| _{L^{p}}\right)_{i\in I}\right\| _{Y}$, where $(φ_{i})_{i\in I}$ 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 \[ \left\| \left(\left\| (α_{i}\,β_j \cdot v_{j}/w_{i})_{i\in I_{j}}\right\| _{\ell^{t}}\right)_{j\in J}\right\| _{\ell^{s}}<\infty, \] 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 $α$-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.
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