Sharp embeddings of Besov spaces with logarithmic smoothness
The paper deals with generalized Besov spaces \(B^{\sigma, \alpha}_{p,r} ({\mathbb R}^n)\), where \(1\leq p,r \leq \infty\) refer to the integrability indices, \(\sigma >0\) stands for the usual (main) smoothness and \(\alpha\) indicates a logarithmic perturbation. The main aim is to study embeddings of type \[ B^{\sigma, \alpha}_{p,r} ({\mathbb R}^n) \hookrightarrow L_{q,s,\alpha} (\Omega), \] where \(\Omega\) is a bounded domain in \({\mathbb R}^n\) and \(L_{q,s,\alpha} (\Omega)\) stands for generalized Lorentz--Zygmund spaces. Necessary and sufficient conditions for these sharp embeddings are given in two relevant cases: (i) the subcritical case \(0 < \sigma < n/p\) (Theorem 2.1) and (ii) the critical case \(\sigma = n/p\) (Theorem 2.3). These results generalise corresponding earlier results by D.~D. Haroske, D.~E. Edmunds and the reviewer.
scientific article; zbMATH DE number 2187070