We study embeddings of Besov-type and Triebel–Lizorkin-type spaces,\({\text {id}}_\tau {:}\,{B}_{p_1,q_1}^{s_1,\tau _1}(\varOmega )\,\hookrightarrow \,{B}_{p_2,q_2}^{s_2,\tau _2}(\varOmega )\) and \({\text {id}}_\tau {:}\,{F}_{p_1,q_1}^{s_1,\tau _1}(\varOmega ) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}(\varOmega ) \), where \(\varOmega \subset {{\mathbb R}^d}\) is a bounded domain, and obtain necessary and sufficient conditions for the compactness of \({\text {id}}_\tau \). Moreover, we characterize 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.

我们研究Besov型和Triebel–Lizorkin型空间的嵌入，\（{\ text {id}} _ \ tau {：} \，{B} _ {p_1，q_1} ^ {s_1，\ tau _1}（ \ varOmega）\，\ hookrightarrow \，{B} _ {p_2，q_2} ^ {s_2，\ tau _2}（\ varOmega）\）和\（{\ text {id}} _ \ tau {：} \， {F} _ {p_1，q_1} ^ {s_1，\ tau _1}（\ varOmega）\ hookrightarrow {F} _ {p_2，q_2} ^ {s_2，\ tau _2}（\ varOmega）\），其中\（ \ varOmega \ subset {{\ mathbb R} ^ d} \）是一个有界域，并为\（{\ text {id}} _ \ tau \）的紧凑性获取必要和充分的条件。此外，我们表征了它的熵和近似数。出乎意料的是，这些结果是通过嵌入和将相应结果应用于经典Besov和Triebel–Lizorkin空间以及Besov–Morrey和Triebel–Lizorkin-Morrey空间而完全获得的。