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On embeddings of logarithmic Bessel potential spaces. (English) Zbl 0934.46036
The fractional Sobolev spaces \(H^s_p(\mathbb{R}^n)= J_{-s}L_p(\mathbb{R}^n)\) can be obtained from the Lebesgue spaces \(L_p(\mathbb{R}^n)\), \(1< p<\infty\), by application of the Bessel-potential operator \(J_{-s}\). The authors study in detail corresponding spaces where \(L_p\) is replaced by \(L_{p,q}\prod_j\log^{\alpha_j}_k(L)\), where \(L_{p,q}\) are the Lorentz spaces and \(\log_k(L)\) are iterated log-refinements with the Zygmund spaces \(L_p(\log L)^\alpha(\mathbb{R}^n)\) as an example. The indicated basic spaces also considered for their own sake (embeddings, duality, etc.). As for \(H^s L_{p,q}\prod_j \log^{\alpha_j}_k(L)\) embeddings are treated. Special attention is paid to limiting situations. Then spaces of type \(\exp\dots\exp(L)\), generalizing Trudinger spaces, come in. Also limiting embeddings in critical Hölder spaces are considered.
Reviewer: H.Triebel (Jena)

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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