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, $id_\tau : {B}{p_1,q_1}^{s_1,\tau_1}(\Omega) \hookrightarrow {B}{p_2,q_2}^{s_2,\tau_2}(\Omega)$ and $id_\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 $id_\tau$. 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.