Gurka, Petr; Opic, Bohumír Sharp embeddings of Besov spaces with logarithmic smoothness. (English) Zbl 1083.46018 Rev. Mat. Complut. 18, No. 1, 81-110 (2005). 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. Reviewer: Hans Triebel (Jena) Cited in 1 ReviewCited in 10 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Keywords:generalized Besov spaces; Lorentz–Zygmund spaces; sharp embeddings; growth envelopes Citations:Zbl 0935.46031 PDFBibTeX XMLCite \textit{P. Gurka} and \textit{B. Opic}, Rev. Mat. Complut. 18, No. 1, 81--110 (2005; Zbl 1083.46018) Full Text: DOI EuDML