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Sharp embeddings of Besov spaces with logarithmic smoothness. (English) Zbl 1083.46018
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.

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.)
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