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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:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Zbl 0935.46031
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