The aim of this note is to study the IP-complemented subspaces in ORLICZ function spaces LV(0, co) over the infinite interval (0, co). Several results previously given by the authors in ([H-R,]) for the ORLICZ spaces 1, and Lq [O, 11 are extended to the class of ORLICZ function spaces Lv(O, 00).
The structure of the ORLICZ function spaces L"(0, co) has been studied by N. J. NIELSEN ([N]) and also by W. JOHNSON, B. MAUREY, G. SCHECHTMANN and L. TZAFRIRI in ([J-M-S-TI). Several interesting results on the 1 P-embedding into the ORLICZ spaces l+ ' and LV[O, 11, given by J. LINDENSTRAUSS and L. TZAFRIRI ([L-TI], [L-T,]), were extended to the case of spaces Lq(0, co) by NIELSEN in [N], where (also somewhat) unexpected peculiar properties of the spaces LV(O, co) are showed, which are specific for this class of function spaces over the infinite interval (0, a). For example, in Theorems 1.5 and 1.6 of [Nl, it is proved that the set of p's for which the unit vector basis of l P is equivalent to a sequence of functions with disjoint supports in the ORLICZ space L"'(0, co), is either the interval [a?, p,] (in the case of the associated indices verify &? < a,), or the union of the intervals [a,, p,] and [a;, p; ] (in the other cases).
Here, we show that the behavior of the ORLICZ function spaces L+'(O, co) in connection with the problem of 1 P-complemented embedding is quite different and very singular, like in the case of the spaces 1 +' and Lq [0, 11 (cf. [L-T,] , [H-P,] and[H-R,]). We notice that the set of p's for which 1 is isomorphic to a complemented subspace of a reflexive ORLICZ function space Lp(O, co) is always a subset of [a,, B,]U [a;, p;]U {2}, and that this set can be very arbitrary. Thus, Theorem 7 states that given a closed set F of positive numbers there exists an ORLICZ function space L+'(O, co) such that the set of values p for which an 1 P-space is complementably embedded into Lq(O, co) is just the set FU(2). The proof of this result uses an adapted method of the given in ([H-R,] Theorems 2 and 7).
Let us start with some definitions and notations. Given a positive measure space (s2,p) and an ORLICZ function cp (i.e. a continuous convex non-decreasing function defined for x > 0 so that cp(0) = 0 and q(1) = l), the ORLICZ function space Lq(p) is defined as the set of equivalence classes of p-measurable scalar functions on (a, p) such