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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.

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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