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Zeitschrift für Analysis und ihre Anwendungen


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Volume 30, Issue 1, 2011, pp. 1–27
DOI: 10.4171/ZAA/1421

Published online: 2011-01-03

Compact Embeddings of Bessel-Potential-Type Spaces into Generalized Hölder Spaces Involving k-Modulus of Smoothness

Amiran Gogatishvili[1], Júlio S. Neves[2] and Bohumír Opic[3]

(1) Czech Academy of Sciences, Prague, Czech Republic
(2) Universidade de Coimbra, Portugal
(3) Czech Academy of Sciences, Prague, Czech Republic

We present conditions which are necessary and sufficient for compact embeddings of Bessel potential spaces HσX(ℝn), modelled upon a rearrangement-invariant Banach function spaces X(ℝn), into generalized Hölder spaces involving k-modulus of smoothness. To this end, we derive a characterization of compact subsets of generalized Hölder spaces. We apply our results to the case when X(ℝn) is a Lorentz–Karamata space Lp,q;b(ℝn). Applications cover both superlimiting and limiting cases.

Keywords: Bessel potentials, (fractional) Sobolev-type spaces, rearrangement-invariant Banach function spaces, generalized Hölder-type spaces, embedding theorems, slowly varying functions, Lorentz–Karamata spaces

Gogatishvili Amiran, Neves Júlio, Opic Bohumír: Compact Embeddings of Bessel-Potential-Type Spaces into Generalized Hölder Spaces Involving k-Modulus of Smoothness. Z. Anal. Anwend. 30 (2011), 1-27. doi: 10.4171/ZAA/1421