Compact embeddings of generalised Morrey smoothness spaces on bounded domains

Abstract

We study embeddings within different scales of generalised smoothness Morrey spaces defined on bounded smooth domains, i.e., in ${\mathcal{N}}_{\phi ,p,q}^s(\Omega )$, ${\mathcal{E}}_{\phi ,p,q}^s(\Omega )$, $B_{p,q}^{s,\phi }(\Omega )$ and $F_{p,q}^{s,\phi }(\Omega )$ 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.