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Gâteaux derivatives and their applications to approximation in Lorentz spaces \(\Gamma_{p,w}\). (English) Zbl 1184.46031

The aim of the present paper is to study the strict convexity and the Gâteaux differentiability of the norm in the Lorentz space \(\Gamma_{p,w},\, 1\leq p<\infty\), of measurable functions \(f:[0,\alpha)\to\overline{\mathbb R}\) such that \(\|f\|_{\Gamma_{p,w}}:=\left(\int_0^\alpha f^{**p}w\right)^{1/p}<\infty\).Here, \(f^{**}(t)=t^{-1}\int_0^tf^*(s)\,ds,\, t>0,\) is the maximal function of the decreasing rearrangement \(f^*\) of \(f\) and \(w\) is a measurable non-negative weight function on \([0,\alpha)\) [see A.Kamińska and L.Maligranda, Isr.J.Math.140, 285–318 (2004; Zbl 1068.46019)]. The classical Lorentz spaces \(\Lambda_{p,w}\) are defined similarly, but with \(f^*\) instead of \(f^{**}\). Since \(f^*\leq f^{**},\) the space \(\Gamma_{p,w}\) is naturally embedded in \(\Lambda_{p,w}\).
The authors study in Sections 3 and 4 the strict convexity of the space \(\Gamma_{p,w}\) and the left- and right-hand Gâteaux derivatives of the norm, with applications to the characterizations of support functionals and smooth points in these spaces. In Sections 6 and 7, these results are applied to obtain characterizations of the best approximation elements from convex subsets of \(\Gamma_{p,w}\) in terms of the Gâteaux derivatives and of extreme points of the unit ball of the dual space \(\Gamma^*_{p,w}\). The characterizations are based on some properties of measure preserving transformations, given in Section 2.
Similar results in the case of Orlicz-Lorentz spaces \(\Lambda_{\phi,w}\) were obtained by F.E.Levis and H.H.Cuenya [Acta Math.Univ.Comen., New Ser.73, No.1, 31–41 (2004; Zbl 1097.46507) and Math.Nachr.280, No.11, 1282–1296 (2007; Zbl 1139.58005)]. Note that, for \(w(t)=t^p,\, \Lambda_{\phi,w}=\Lambda_{p,w}\).
The study done by the authors puts in evidence the differences between these two cases.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B20 Geometry and structure of normed linear spaces
46B70 Interpolation between normed linear spaces
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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