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Compact and continuous embeddings of logarithmic Bessel potential spaces. (English) Zbl 1083.46017

The paper deals with continuous and compact embeddings of the refinement \(H^\sigma L_{p,q;\alpha_1, \dots, \alpha_m} (\Omega)\) of the Bessel potential spaces \(H^\sigma_p (\Omega)\) into generalised Lorentz–Zygmund spaces \(L_{r,q;\alpha_1, \dots, \alpha_m} (\Omega)\) and generalised Hölder spaces \(C^{0, \lambda (\cdot)} (\bar{\Omega})\). Here, \(1<p< \infty\) and \(\Omega\) stands for \(\mathbb R^n\) or a bounded domain in \(\mathbb R^n\). Furthermore, \(L_{p,q}\) are the usual Lorentz–Zygmund spaces, while the \(\alpha_1, \dots, \alpha_n\) refer to logarithmic refinements and \(\lambda\) indicates a generalised modulus of continuity.

MSC:

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