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

We study embeddings of Besov-type and Triebel-Lizorkin-type spaces, $id_τ: {B}_{p_1,q_1}^{s_1,τ_1}(Ω) \hookrightarrow {B}_{p_2,q_2}^{s_2,τ_2}(Ω)$ and $id_τ: {F}_{p_1,q_1}^{s_1,τ_1}(Ω) \hookrightarrow {F}_{p_2,q_2}^{s_2,τ_2}(Ω)$, where $Ω\subset {\mathbb R}^d$ is a bounded domain, and obtain necessary and sufficient conditions for the compactness of $id_τ$. 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.

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