Continuity envelopes of spaces of generalised smoothness, entropy and approximation numbers (Q596810)

A positive function \(\Psi\) on \((0,1]\) is called slowly varying if \[ \lim_{t \to 0} \frac{\Psi (st)}{\Psi (t)} =1 \quad \text{for} \quad s \in (0,1]. \] The paper under review deals with the function spaces \(B^{(s, \Psi)}_{pq} (\mathbb{R}^n)\) and \(F^{(s, \Psi)}_{pq} (\mathbb{R}^n)\) in generalisation of the well-known scales \(B^s_{pq} (\mathbb{R}^n)\) and \(F^s_{pq} (\mathbb{R}^n)\) where \(s \in \mathbb{R}\), \(0< p,q \leq \infty\) (\(p< \infty\) in the \(F\)-case), where the slowly varying function \(\Psi\) might be considered as a perturbation of the smoothness \(s\). The authors give new characterisations of the \(B\)-spaces with \(s > n(1/p - 1)_+\) in terms of approximations and moduli of continuity. If \(n/p <s< n/p +1\), then sharp assertions for the embedding \[ B^{(s, \Psi)}_{pq} (\mathbb{R}^n) \hookrightarrow C(\mathbb{R}^n) \] in terms of the continuity envelopes are given. Finally, restricted to the unit ball \(U\), compact embeddings between these spaces \(B^{(s,\Psi)}_{pq} (U)\) are considered, expressed in terms of approximation numbers and entropy numbers.