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Sharp embeddings of Besov spaces with logarithmic smoothness in sub-critical cases. (English) Zbl 1389.46031
Let \(B^{s,b}_{p,q} (Q)\) with \(Q= [0,1]^n\) be the logarithmic refinement of the classical Besov spaces \(B^s_{p,q} (\mathbb R^n)\) restricted to \(Q\). Let \(L^{(p,b,q} (Q)\) be a similar refinement of the \(L^p\)-spaces (called small Lebesgue spaces) and let \(L_{p,q} (\log L)_a (Q)\) be the Lorentz-Zygmund spaces. The paper deals with embeddings of \(B^{s,b}_{p,q} (Q)\) in the limiting situation \(s= n \max \big( \frac{1}{p} -1, 0 \big)\) into \(L^{(p,b,q} (Q)\) and \(L_{p,q} (\log L)_a (Q)\).

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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