Inclusion relations among Orlicz spaces

Abstract

This paper contains two results; the first extends to a wide class of Orlicz spaces the statement, due to Krasnosel'skii and Rutickii [1, p. 60], that L1 is the union of the Orlicz spaces which it contains properly; the second shows that for a wide class of spaces this is not true, i.e. there exists a set of Orlicz spaces no one of which is the union of the Orlicz spaces it contains properly. H-Iere the Orlicz spaces are defined on [0, 1 ] which is given Lebesgue measure ,u. 1. We give in this section several definitions together with some elementary results about Orlicz spaces and convex functions. Let e be the set of convex symmetric functions 4: (oo, co) > [0, oo) suqh that 4(0) = 0, lim8.0 'f(s)/s = 0 and lim,0 4?(s) = co. If 4) and Q are two elements of e, we say 4 < Q if there exist constants c and s0 such that 4P(s) <? (cs) for all s ? so. We say 4D-9 if F _Q i and _ <P; we say 4 <Q if ? <Q but Q :$. If 41'--42 and 1 <?Q1 ((1 <Q1) then 42 _ 02 (42 < 02). If 4Ee, then there exists a nondecreasing function q: [0, co) -4[, oo) such that 4)(0) = 0, lim, .O +(s) = oX and