Nuclear embeddings in general vector-valued sequence spaces with an application to Sobolev embeddings of function spaces on quasi-bounded domains
| Autor: | Hans-Gerd Leopold, Dorothee D. Haroske, Leszek Skrzypczak |
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| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Numerical Analysis Sequence Mathematics::Functional Analysis Control and Optimization Algebra and Number Theory Function space Applied Mathematics General Mathematics Diagonal Context (language use) Type (model theory) Functional Analysis (math.FA) Combinatorics Sobolev space Mathematics - Functional Analysis Bounded function FOS: Mathematics Embedding 46E35 47B10 Mathematics |
| DOI: | 10.48550/arxiv.2009.00474 |
| Popis: | We study nuclear embeddings for spaces of Besov and Triebel-Lizorkin type defined on quasi-bounded domains $\Omega\subset {\mathbb R}^d$. 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 $\ell_q(\beta_j \ell_p^{M_j})$ with $1\leq p,q\leq\infty$, $M_j\in {\mathbb N}_0$ and weight sequences with $\beta_j>0$. In particular, we prove a criterion for the embedding $id_\beta : \ell_{q_1}(\beta_j \ell_{p_1}^{M_j}) \hookrightarrow \ell_{q_2}(\ell_{p_2}^{M_j})$ to be nuclear. Comment: arXiv admin note: text overlap with arXiv:2002.03136 (it is the revised = second version of arXiv:2009.00474) |
| Databáze: | OpenAIRE |
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