Nuclear embeddings of Morrey sequence spaces and smoothness Morrey spaces

TL;DR

This paper establishes the first comprehensive nuclearity results for Morrey-type spaces, extending known results from Besov and Triebel-Lizorkin spaces using wavelet techniques and classical operator theory.

Contribution

It provides the first complete nuclearity characterization for Morrey sequence and smoothness spaces, filling a gap in the understanding of their embedding properties.

Findings

Proves nuclearity of Morrey spaces using wavelet decompositions.

Extends classical results to Morrey spaces, previously known only for Besov and Triebel-Lizorkin spaces.

Utilizes Tong's theorem to characterize nuclear operators in this context.

Abstract

We study nuclear embeddings for spaces of Morrey type, both in its sequence space version and as smoothness spaces of functions defined on a bounded domain $\Omega \subset {\mathbb R}^d$. This covers, in particular, the meanwhile well-known and completely answered situation for spaces of Besov and Triebel-Lizorkin type defined on bounded domains which has been considered for a long time. The complete result was obtained only recently. Compact embeddings for function spaces of Morrey type have already been studied in detail, also concerning their entropy and approximation 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. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of $\ell_r$ type, $1 \leq r \leq\infty$.