论文标题
一般矢量值序列空间中的核嵌入，并应用于在准结合域上的功能空间的Sobolev嵌入
Nuclear embeddings in general vector-valued sequence spaces with an application to Sobolev embeddings of function spaces on quasi-bounded domains
论文作者
论文摘要
我们研究了在准结合的域$ω\ subset {\ mathbb r}^d $上定义的BESOV和Triebel-Lizorkin类型的空间的核嵌入。在有限域上定义的此类功能空间的对应物已长期考虑，并且直到最近才获得完整的答案。已经详细研究了在准结合域上定义的功能空间的紧凑型嵌入，也已经研究了其熵和$ s $ numbers。现在，我们在这种情况下证明了第一个完全完全的核性结果。 Grothendieck在1955年已经提出了核性的概念。我们的第二个主要贡献是著名的TOND结果（1969年）的概括，该结果表征了核对角线操作员在$ \ ell_r $ type的序列空间之间作用，$ 1 \ leq r \ leq r \ leq \ infty $。现在，我们可以将其扩展到$ \ ell_q（β_j\ ell_p^{m_j}）$的常规矢量值序列空间，并使用$ 1 \ leq p，q \ leq \ leq \ infty $，$ m_j \ In {特别是，我们证明了嵌入$id_β的标准：\ ell_ {q_1}（β_j\ ell_ {p_1}^{m_j}）\ hookrightArrow \ elwrightArrow \ ell_ {q_2}（\ ell_ ell _ el _ ell_ {p_2}}}^{m_j {m_j {m_j}）$。
We study nuclear embeddings for spaces of Besov and Triebel-Lizorkin type defined on quasi-bounded domains $Ω\subset {\mathbb R}^d$. The counterpart for such function spaces defined on bounded domains has been considered for a long time and the complete answer was obtained only recently. Compact embeddings for function spaces defined on quasi-bounded domains have been studied in detail already, also concerning their entropy and $s$-numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has been introduce by Grothendieck in 1955 already. Our second main contribution is the generalisation of the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of $\ell_r$ type, $1\leq r\leq\infty$. We can now extend this to the setting of general vector-valued sequence spaces of type $\ell_q(β_j \ell_p^{M_j})$ with $1\leq p,q\leq\infty$, $M_j\in {\mathbb N}_0$ and weight sequences with $β_j>0$. In particular, we prove a criterion for the embedding $id_β: \ell_{q_1}(β_j \ell_{p_1}^{M_j}) \hookrightarrow \ell_{q_2}(\ell_{p_2}^{M_j})$ to be nuclear.
