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Global limiting embeddings of logarithmic Bessel potential spaces. (English) Zbl 0934.46034

The paper is a continuation of a respectable series of papers written jointly by the authors and D. E. Edmunds. The main aim of the paper is to get global versions of local embeddings of Bessel potential spaces into Orlicz spaces. Namely, the authors modify the Young function which generates the target Orlicz space in order that the embedding holds on entire \(\mathbb R^n\) rather than on a bounded domain. The idea of modifying Young function near the origin in order that the domain in question can be of infinite measure had been used earlier, for example in the paper of D. E. Edmunds and W. D. Evans [Proc. R. Soc. Lond., Ser. A 342, 373-400 (1975; Zbl 0298.35028)] or in the book “Sobolev spaces” by R. A. Adams [Academic Press, New York (1975; Zbl 0314.46030)]. The paper under review uses different techniques, namely a general version of Hardy’s inequality.
Reviewer: L.Pick (Praha)

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.)
26D15 Inequalities for sums, series and integrals
46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables
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