Nuclear and Compact Embeddings in Function Spaces of Generalised Smoothness

We study nuclear embeddings for function spaces of generalised smoothness defined on a bounded Lipschitz domain Ω ⊂ ℝ^d. This covers, in particular, the well-known situation for spaces of Besov and Triebel–Lizorkin spaces defined on bounded domains as well as some first results for function spaces of logarithmic smoothness. In addition, we provide some new, more general approach to compact embeddings for such function spaces, which also unifies earlier results in different settings, including also the study of their entropy numbers. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) about nuclear diagonal operators acting in ∓_r spaces, which we could recently extend to the vector-valued setting needed here.