Compact embeddings in Besov-type and Triebel-Lizorkin-type Spaces on bounded domains

Abstract

We study embeddings of Besov-type and Triebel-Lizorkin-type spaces, $i{d}_{\tau }:B_{p_1,q_1}^{s_1,{\tau }_1}(\Omega )\hookrightarrow B_{p_2,q_2}^{s_2,{\tau }_2}(\Omega )$ and $i{d}_{\tau }:F_{p_1,q_1}^{s_1,{\tau }_1}(\Omega )\hookrightarrow F_{p_2,q_2}^{s_2,{\tau }_2}(\Omega )$, where $\Omega \subset {\mathbb{R}}^d$ is a bounded domain, and obtain necessary and sufficient conditions for the compactness of $i{d}_{\tau }$. Moreover, we characterise its entropy and approximation numbers. Surprisingly, these results are completely obtained via embeddings and the application of the corresponding results for classical Besov and Triebel-Lizorkin spaces as well as for Besov-Morrey and Triebel-Lizorkin-Morrey spaces.