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On the \(K\)-functional of interpolation between \(L^ p\) and Orlicz spaces. (English) Zbl 0714.46019
The authors obtain an explicit form of the interpolation \(K\)-functional between \(L^ p\)-spaces \((1\leq p<\infty)\) and Orlicz spaces. The basic tool is an optimization method for a maximum problem (naturally related to the interpolating norm) and the differential relation derived from it.
In the particular case of the \(L^ p\)-\(L^ q\) interpolation, the \(K\)-functional studied in this paper is equivalent to the usual \(K\)-functional, explicitly described by P. Nilsson and J. Peetre, J. Approx. Theory 48, 322–327 (1986; Zbl 0617.46077)].
Reviewer: M. Putinar

MSC:
46B70 Interpolation between normed linear spaces
46M35 Abstract interpolation of topological vector spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:
[1] Bergh, I; Lofstrom, J, Interpolation spaces. an introduction, (1976), Springer-Verlag New York/Berlin · Zbl 0344.46071
[2] Holmstedt, T; Peetre, J, On certain functionals arising in theory of interpolation spaces, J. funct. anal., 4, 88-94, (1969) · Zbl 0175.42601
[3] Lindenstrauss, J; Tzafriri, L, Classical Banach spaces II, (1979), Springer-Verlag New York/Berlin · Zbl 0403.46022
[4] Nilson, P; Peetre, J, On the K-functional between L1 and L2 and some other K-functionals, J. approx. theory, 48, 322-327, (1986)
[5] Peetre, J, A theory of interpolation of normed spaces, Lecture notes brasilia notas mat., 39, 1-86, (1968) · Zbl 0162.44502
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