arXiv 1601.022012016-01-10math.FA

分解空间在Sobolev和BV空间中的嵌入

Embeddings of Decomposition Spaces into Sobolev and BV Spaces

Felix Voigtlaender

摘要：

本文研究了分解空间$\mathcal{D}\left(\mathcal{Q},L^{p},Y\right)$在Sobolev空间$W^{k,q}(\mathbb{R}^{d})$中的嵌入是否存在.作为特殊情况，这包括嵌入到索伯列夫空间（齐次和非齐次）Besov空间，（$α$）调制空间，剪切光滑空间，也是一大类小波coorbit空间，特别是剪切型coorbit空间。确切地说，我们将证明，在对覆盖$\mathcal{Q}=\left(Q_{i}\right)_{i\in I}$的极其温和的假设下，只要$p\leq q$和$Y\hookrightarrow\ell_{u^{\left(k,p,q\right)}}^{q^{\triangledown}}\left(I\right)$成立，我们就有$\mathcal{D}\left(\mathcal{Q},L^{p},Y\right)\hookrightarrow W^{k,q}(\mathbb{R}^{d})$。这里，仅基于覆盖$\mathcal{Q}$和参数$k,p,q$，就可以容易地计算$q^{\triangledown}=\min\left\{ q,q'\right\} $和权重$u^{\left(k,p,q\right)}$。相反，嵌入存在的必要条件是$p\leq q$和$Y\cap\ell_{0}\left(I\right)\hookrightarrow\ell_{u^{\left(k,p,q\right)}}^{q}\left(I\right)$成立，其中$\ell_{0}\left(I\right)$表示$I$上的支持序列的空间。总而言之，对于范围$q \in (0,2]\cup\{\infty\}$，我们得到了一个完整的特征的嵌入的存在性方面的易于验证的标准。我们还可以完全刻画分解空间嵌入BV空间的存在性。

英文摘要：

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, ($α$)-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 $\mathcal{Q}=\left(Q_{i}\right)_{i\in I}$, we have $\mathcal{D}\left(\mathcal{Q},L^{p},Y\right)\hookrightarrow W^{k,q}(\mathbb{R}^{d})$ as soon as $p\leq q$ and $Y\hookrightarrow\ell_{u^{\left(k,p,q\right)}}^{q^{\triangledown}}\left(I\right)$ hold. Here, $q^{\triangledown}=\min\left\{ q,q'\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 $p\leq q$ 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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