Nuclear and compact embeddings in function spaces of generalised smoothness

TL;DR

This paper investigates nuclear and compact embeddings of generalized smoothness function spaces on bounded Lipschitz domains, unifying various results and extending techniques using wavelet decompositions and nuclear operator theory.

Contribution

It introduces a new, unified approach to nuclear and compact embeddings for generalized smoothness spaces, extending existing results and applying wavelet techniques and nuclear operator theory.

Findings

Established nuclear embeddings for spaces of generalized smoothness.

Provided new criteria for compact embeddings in these function spaces.

Extended nuclear operator results to vector-valued settings.

Abstract

We study nuclear embeddings for function spaces of generalised smoothness defined on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^d$. This covers, in particular, the well-known situation for spaces of Besov and Triebel-Lizorkin spaces defined on bounded domains as well as some first results for function spaces of logarithmic smoothness. In addition, we provide some new, more general approach to compact embeddings for such function spaces, which also unifies earlier results in different settings, including also the study of their entropy numbers. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) about nuclear diagonal operators acting in $\ell_r$ spaces, which we could recently extend to the vector-valued setting needed here.