On the \(K\)-functional of interpolation between \(L^ p\) and Orlicz spaces. (English) Zbl 0714.46019

Bastero, Jesús; Raynaud, Yves

J. Approximation Theory 60, No. 1, 11-23 (1990).

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.)

Keywords:

interpolation K-functional between \(L^ p\)-spaces; Orlicz spaces; optimization method for a maximum problem

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