INCLUSIONS OF HARDY ORLICZ SPACES

Let ϕ be a continuous positive increasing function defined on [0,∞) such that ϕ(x

(0) 0. The Hardy-Orlicz space generated by is denoted by H().In this paper, we prove that for # 4, if H() H() as sets, then H() H() as topological vector spaces.Some other results are given.
KEY WORDS AND PHRASES.Modulus function, Orlicz spaces.
Let T be the unit circle, and m be the Lebesgue measure on T. A complex valued measurable function f defined on T is called -integrable if fiIf(t) Idm(t) < .The space of all -integrable functions on T will be denoted by L().This space was first introduced by Orlicz, [8].Subsequent papers were written to study different aspects of L().Examples of these papers are Cater, [4], Gramsch, [5] and Pallashke [9].
In [6] and [7], Lesniewicz introduced the so called Hardy-Orlicz spaces H() for a given such function 9.The space H() was defined to be the space of all functions f e L() such that f is the radial limit of some function g analytic in the open unit disc and belongs to the Nevalinna class N. The relation between different H()spaces was studied by Deeb, Khalil and Marzug [3].In this paper, we show that the inclusion map between two H()-spaces is always continuous.Some other results are given.It should be remarked that in the work of Lesniewicz, [6], [7] and many other authors, is assumed to be a Q-convex function.In this paper it is not assumed so.

PRELIMINARIES AND NOTATIONS.
A function ' The functions (x) x p, 0 p and (x) Further, if bl and b2 are modulus functions, then l 2 and bl '}2 is a modulus function if is.
are modulus functions.Further, @-# Let T {z" Izl i}, A {z" Izl < i}.The space of analytic functions on & is denoted by H(4).Let H+(A) {f e H(4)" li f @ rX+ (re I exists a.e.@}We will consider H+(4) as a space of functions on T. For a given modulus function we define" If(e )Ido }. ,If(e ie) g(e i0) Id defines a metric on H(O), under which H(b) becomes a topological vector space.If one assumes that Olul is subharmonic for u e I-I(4), then tt.) turns out to be complete [21.For f e H(,), we write llfll T If(e )ld@" If ,(x) x p, 0 p i, f then H() H p and for (x) where N is the Nevalinna class.
In [2], it was shown that H H() for all modulus functions The authors in [3] were not able to show that the inclusion map H H() is continuous.
In this section we prove that H H() is continuous.Some other related questions are discussed.
THEOREM 2.1.Let and , be two modulus functions such that lira (  Let fn 0 in H().Thus the sequence (fn) is bounded in the metric of H(,) and consequently bounded in H().If possible let there exist a subsequence (f n k fnk fnk 0 (fnk) has a subsequence which such that II I I a > 0. Since I] I1 converges pointwise to the zero function.With no loss of generality, we can assume that f 0 a.e.Another application of the proof of Theorem 2.1, yields (x) n k (a) + b-Ixl for all x e [0,).Hence The sequence of functions gnk ,(a) / b 01fnk converges a.e. to ,(a) and f (t)dt T gnk Consequently, by the generalized Lebesgue convergence theorem, [10], we have lin 41fnk(t; Idt limnk 41 fnk (t) Idt 0. This is a contradiction.Thus, the point w 0 is the only limit point of the bounded sequence (11 fnll,)" Consequently, [11]  H() as sets.Theorem 2.2 implies that I: H() H() is an isomorphism.This ends the proof.
A linear map A" H() H() is called metrically bounded if for all f e H() and some X 0. Clearly every metrically bounded map is contanuous.The converse need not be true.However, for the inclusion map, we have the following: THEOREM 2.4.Let be any modulus function.Then there exists X 0 such that for all f e H I, PROOF.It is know, [2] (and easy to show) that H H() for all modulus func- tions .If f e H and If film m, then using the argument in Theorem 2.  for some X 0.
in(1 x) are examples of modulus functions.

2 .
Let lira --O.Then the inclusion map I" H() tt() is X-Oo continuous.PROOF.From the proof of Theorem 2.1, there exists a,b 0 such that IIfll (a) b ]lfl] for all f H().
TItEOREM 2.5.Let be a given modulus function such that H H(). If metric and topological bounded sets coincide in H(,), then Ilfll xllfll for al f e It(,), the sequence lie converges to