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Compact embeddings of generalised Morrey smoothness spaces on bounded domains

Dorothee D. Haroske, Susana D. Moura, Leszek Skrzypczak

Abstract

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

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 , , and 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.
Paper Structure (5 sections, 32 theorems, 188 equations)

This paper contains 5 sections, 32 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Embeddings of sequence spaces
  4. Embeddings of generalised Besov-Morrey spaces on domains
  5. Embeddings of further generalised Morrey smoothness spaces on domains

Key Result

Corollary 2.7

Let $s_i\in {\mathbb R}$, $0<q_i\leq\infty$, $i=1,2$.

Theorems & Definitions (87)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.5
  • Remark 2.6
  • Corollary 2.7: GHS-21
  • Remark 2.8
  • Remark 2.9
  • Definition 3.1
  • Remark 3.2
  • ...and 77 more