ABSTRACT
IIPIILoo = II[PlIILoo = 1IIPIIILoo = IIIP leIlLoo' where e(t,s) == 1. This proves the assertion for the space Loo• In case of the space Ll the proof follows from Theorem 11.4 and equality (12.32). •
12.3. Orlicz spaces with mixed norm We give now some parallel results for Orlicz spaces which we already introduced in Subsection 4.1. Recall that, given a Young function M : lR - [0,00), the Orlicz space LM = LM(n) is defined by one of the (equivalent) norms (4.3) or (4.4). To state the theorems of this subsection, some further notions and results on Orlicz spaces are in order. For simplicity, we only consider Orlicz spaces over bounded domains n in this subsection. Given two Young functions M and N, we write M :5 N if there exist k > 0 and Uo ~ 0 such that
for every k > O. Of course, in case M(u) = lulP and N(u) = luI" (1 < p, q < 00) we have M :5 N if and only if p ~ q, and M -< N if
and only if p < q. In general, one can show that M ~ N is equivalent to the fact that LN is continuously imbedded in LM, and M -< N is equivalent to the fact that LN is absolutely continuously imbedded in LM (i.e., the unit ball of LN is an absolutely bounded subset of LM). Moreover, the inclusions Leo ~ LM ~ L1 are always true. Let U = LM(T) and V = LN(S) be two Orlicz spaces. We are interested in the Orlicz spaces with mixed norm [U -+ V] and [U +- V] defined by the formulas (12.1) and (12.2), respectively. These spaces are perfect ideal spaces. They are regular if and only if the Young functions M and N satisfy a a2-condition. IT M2, N2 ~ M1 , Nt the inclusions
are obvious. Moreover, the inclusions
follow from the Jensen integral inequality
and the definition of the norm in LM. In fact, for x E LM and k > 0 sufficiently large we have
Consequently, x E [L t -+ LM), and hence the left inclusion is proved. The right inclusion is proved analogously.
