Summary
Let Ф denote a finite-valued Orlicz function vanishing only at zero and consider the Orlicz spaceL Ф over a non-atomic, σ-finite and infinite measure space with the norm ∥ · ∥ being either the Luxemburg norm or (in the case where\(\mathop {\lim }\limits_{u \to 0} \Phi (u)/(u) = 0\) and\(\mathop {\lim }\limits_{u \to + \infty } \Phi (u)/(u) = + \infty \) the Orlicz norm. Ifp ∈ [1, + ∞) and
for all disjoint setsA andB of positive and finite measure, thenL Ф is the Lebesgue spaceL p.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aczél, J.,Lectures on functional equations and their applications. [Mathematics in Science and Engineering, Vol. 19]. Academic Press, New York—London, 1966.
Aliprantis, Ch. andBurkinshaw, O.,Positive operators. Academic Press, Orlando—San Diego—New York—London—Toronto—Montreal—Sydney—Tokyo, 1985.
Grzaślewicz, R.,On isometric domains of positive operators on Orlicz spaces. Rend. Circ. Mat. Palermo (2)2 (1982), 131–134.
Krasnosel'skii, M. A. andRutickii, Ya. B.,Convex functions and Orlicz spaces. P. Noordhoff, Groningen, 1961.
Kuczma, M.,An introduction to the theory of functional equations and inequalities. Cauchy's equation and Jensen's inequality. [Prace Naukowe Uniwersytetu Ślaskiego w Katowicach Nr 489]. Państwowe Wydawnictwo Naukowe & Uniwersytet Ślaski, Warszawa—Kraków—Katowice, 1985.
Lindenstrauss, J. andTzafriri, L.,Classical Banach spaces II. Function spaces. [Ergeb. Math. Grenzgeb. 97]. Springer-Verlag, Berlin—Heidelberg—New York, 1979.
Musielak, J.,Orlicz spaces and modular spaces. [Lecture Notes in Math., No. 1034]. Springer-Verlag, Berlin—Heidelberg—New York—Tokyo, 1983.
Rao, M. M. andRen, Z. D.,Theory of Orlicz spaces. Marcel Dekker, New York—Basel—Hong Kong, 1991.
Zaanen, A. C.,Integration. North-Holland Publishing Co., Amsterdam, 1967.