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Sharpness of embeddings in logarithmic Bessel-potential spaces. (English) Zbl 0860.46024

Summary: This paper is a continuation of [Stud. Math. 115, No. 2, 151-181 (1995; Zbl 0829.47024)], where embeddings of certain logarithmic Bessel-potential spaces (modelled upon generalized Lorentz-Zygmund spaces) in appropriate Orlicz spaces (with Young functions of single and double exponential type) were derived. The aim of this paper is to show that these embedding results are sharp in the sense of [J. A. Hempel, G. R. Morris and N. S. Trudinger, Bull. Aust. Math. Soc. 3, 369-373 (1970; Zbl 0205.12801)].

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.)
47B38 Linear operators on function spaces (general)
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References:

[1] Edmunds, Studia Math. 115 pp 151– (1995)
[2] Bennett, Interpolation of Operators 129 (1988) · doi:10.1016/S0079-8169(08)60845-4
[3] Bennett, Dissertationes Math. 175 pp 1– (1980)
[4] DOI: 10.2307/1971445 · Zbl 0672.31008 · doi:10.2307/1971445
[5] DOI: 10.1007/978-1-4612-1015-3 · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3
[6] Trudinger, J. Math. Mech. 17 pp 473– (1967)
[7] Edmunds, Houston J. Math. 21 pp 119– (1995)
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[11] DOI: 10.1017/S0004972700046074 · Zbl 0205.12801 · doi:10.1017/S0004972700046074
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