Nuclear embeddings in general vector-valued sequence spaces with an application to Sobolev embeddings of function spaces on quasi-bounded domains

TL;DR

This paper establishes the first comprehensive nuclearity results for embeddings of Besov and Triebel-Lizorkin spaces on quasi-bounded domains, extending classical operator characterizations to vector-valued sequence spaces.

Contribution

It provides the first complete nuclearity characterization for function space embeddings on quasi-bounded domains and generalizes Tong's theorem to vector-valued sequence spaces.

Findings

Proves nuclearity of embeddings for function spaces on quasi-bounded domains.

Extends Tong's theorem to general vector-valued sequence spaces.

Provides criteria for nuclearity of specific sequence space embeddings.

Abstract

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.