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Continuity properties of the superposition operator

Part of: Functions of several variables Classical measure theory Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

Jürgen Appell  and
Pjotr P. Zabrejko
Jürgen Appell
Affiliation:
Institut für Mathematik Universität WürzburgAm Hubland D-8700 Würzburg, West Germany
Pjotr P. Zabrejko
Affiliation:
Mech.-Mat. Fakultet Belgosuniversitet Minsk Pl. Lenina 1 SU-220080 Minsk, USSR
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Abstract

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Various continuity conditions (in norm, in measure, weakly etc.) for the nonlinear superposition operator F x(s) = f(s, x(s)) between spaces of measurable functions are established in terms of the generating function f = f(s, u). In particular, a simple proof is given for the fact that, if F is continuous in measure, then f may be replaced by a function f which generates the same superposition operator F and satisfies the Carathéodory conditions. Moreover, it is shown that integral functional associated with the function f are proved.

MSC classification

Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26B40: Representation and superposition of functions 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 2 , October 1989 , pp. 186 - 210
Copyright
Copyright © Australian Mathematical Society 1989

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