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

We study nuclear embeddings for spaces of Besov and Triebel-Lizorkin type defined on quasi-bounded domains Omega subset of 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 already been studied in detail, also concerning their entropy and s-numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has already been introduced by Grothendieck in 1955. Our second main contribution is the generalisation of the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of l(r) type, 1 <= r <= infinity. We can now extend this to the setting of general vector-valued sequence spaces of type l(q) (beta(j)l(p)M(j)) with 1 <= p, q <= infinity, M-j is an element of N-0 and a weight sequence with beta(j) > 0. In particular, we prove a criterion for the embedding id(beta) : lq(1)(beta(j)l(p1)M(j)) hooked right arrow l(q2) (l(p2)M(j)) to be nuclear. (C) 2021 The Author(s). Published by Elsevier Inc.

Automated summary

This article studies nuclear embeddings for function spaces of Besov and Triebel-Lizorkin type defined on quasi-bounded domains, and provides the first complete result on nuclearity in this context. Furthermore, the famous Tong result is generalized to general vector-valued sequence spaces.