Modulus of continuity of the Mazur map between unit balls of Orlicz spaces and approximation by Hölder mappings. (English) Zbl 1084.41012

Let \(G\) be one of the sets \([0,1]\), \(\mathbb{N}\) or \((0,\infty)\), with the associated measure. Further, let \(L_M(G)\) and \(L_N(G)\) be the Orlicz spaces built on \(G\) by means of the Orlicz functions \(M: \mathbb{R}\to\mathbb{R}\) and \(N: \mathbb{R}\to\mathbb{R}\), respectively. The closed unit balls of these spaces are denoted by \(B_M(G)\) and \(B_N(G)\), respectively. By generalizing the S. Mazur map [Studia 1, 83–85 (1929; JFM 55.0242.01)] the author defines a mapping \(\phi_{MN}: L_M(G)\to L_N(G)\) by \[ \phi_{MN}(x)= N^{-1}\circ M(|x|)\text{sign}(x). \] Under the assumption that \(N^{-1}\circ M\) satisfies certain regularity conditions involving given constants \(\alpha\) and \(\beta\), where \(0< \alpha\leq\beta<\infty\), it is proved that \(\phi_{MN}\) is \(\alpha\wedge 1\)-Hölder on \(B_M(G)\) and that \(\phi^{-1}_{MN}= \phi_{NM}\) is \((1/\beta)\wedge 1\)-Hölder on \(B_N(G)\). By using the above-defined mapping \(\phi_{MN}\) the author extends a result due to I. G. Tsar’kov [Math. Notes 54, No. 3, 957–967 (1993); translation from Mat. Zametki 54, No. 3, 123–140 (1993; Zbl 0821.46037)], concerning the uniform approximation of uniformly continuous mappings by \(\alpha\)-Hölder mappings with \(\alpha\in (0,1]\) as large as possible, to the setting of Orlicz spaces. At the end of the paper the obtained approximation result is related to the Boyd indices and to the problem of isomorphic extension of Hölder mappings.


41A30 Approximation by other special function classes
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)