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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:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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