Compact embeddings of generalised Morrey smoothness spaces on bounded domains

TL;DR

This paper investigates the embeddings of generalized Morrey smoothness spaces on bounded domains, providing conditions for their continuity and compactness, and extends previous results using wavelet characterizations.

Contribution

It offers new sufficient and necessary conditions for embeddings in generalized Morrey spaces, improving upon classical results with wavelet-based methods.

Findings

Established conditions for continuity of embeddings.

Identified criteria for compactness of embeddings.

Generalized and improved classical Morrey space results.

Abstract

We study embeddings within different scales of generalised smoothness Morrey spaces defined on bounded smooth domains, i.e., in $\mathcal{N}^s_{\varphi,p,q}(\Omega)$, $\mathcal{E}^s_{\varphi,p,q}(\Omega)$, $B^{s,\varphi}_{p,q}(\Omega)$ and $F^{s,\varphi}_{p,q}(\Omega)$ spaces. We prove sufficient conditions for continuity and compactness of the embeddings. In some cases the conditions are also necessary. We generalise and even improve some earlier results known for the classical smoothness Morrey spaces. Our approach is based on wavelet characterisation of the function spaces.