A N N A L E S SO C IE T A T IS M A TH EM A TIC A E PO LO N A E Series I : C O M M EN TA TIO N ES M A TH EM A TIC A E X I X (1977) R O C Z N IK I P O L SK IE G O TO W A RZY STW A M A TEM A TYCZNEGO
Séria I : P R A C E M A TEM A TY CZ N E X I X (1977)
A n d r z e j K o zek (W roclaw )
Orlicz spaces of functions with values in Banach spaces
0. Introduction and preliminaries. An idea to give a definition of a norm in spaces of measurable functions by means of convex functions originates in works of Orlicz [20], [21]. The spaces of functions called Orlicz spaces, are Banach ones and they have been an object of numerous investigations. The literature on this subject until 1957 can be found in [11]. Moreover, since the theory Orlicz spaces is a source of large number of miscellaneous functional spaces, it affords possibilities for extensions of a rang of applications of various theories (cf. [5], [ 6 ], [10], [11], [13], [23].
Then again, new extensions of the notion of Orlicz space have been made ([5], [11], [16], [ 2 2 ], [23], [29], [30], [31]). A recent developement in this direction is characterized by an use of a “normal convex integrand”
on B n instead of classical А -functions (cf. [5], [ 6 ], [22], [29], [30]).
The present paper deals with the theory of Orlicz spaces of functions with values in Banach spaces. In [3], [4] Dinculeanu investigated a similar problem. However, he restricted himself to the case of composition of classical A-f unctions on a real line and the norm whereas the notions of A ', A " and А -functions introduced in this paper includes a larger class of convex functions.
A treatment of the theory of Orlicz spaces presented here differs from those given in [11] or in [ 6 ], [7]. We start from the theory of mod
ular spaces (cf. [18], [19]) and next we define a modular 1Ф on a space of measurable functions. Thus, we consider the Orlicz spaces as a large subclass of modular spaces. Let us note that the approach was used by Nakano in [18] for Lp(t) spaces and by Musielak and Orlicz in [16] for numerous examples of then generalized modular spaces.
Let f{x ,t ) be a normal convex integrand (cf. Appendix) and let
T
be a convex' functional on the space of all measurable functions from a set T into a Banach space X . Often it is convenient to restrict the do
main of I f to such a non-trivial subspace of measurable functions that
in appropriate topologies I f becomes continuous, lower semicontinuous,
inf-compact or subdifferentiable. It is interesting that the theory of Orlicz spaces offers subspaces and topologies which are convenient for these purposes (cf. Sections 5, 6 and [5]-[7], [10]).
It seems that for a clarity of the exposition it is worth to notice that all theorems of this paper are valid if Ф is an AT-f unction (cf. Section 2) and if the following conditions are satisfied with Ï7* = X :
C o n d ition A: there exists an increasing sequence (T J of measurable
CO
sets, /л(Т{) < oo, i — 1 , 2, 3, ..., ( J T{ — T , and a sequence {Ï7J of neighbourhoods of zero in a Banach space X such that F includes all i= l
functions of the form X ' x A {t), where X a ( ‘ ) is an indicator function of A while A is a measurable subset of T{ for some i and x e ;
C o n d ition В : Ф(х, t) < / n(C for [| æ || < 7i, n — 1 , 2 , ..., where f n(') are measurable and /^-summable functions on every Ti7 i = 1 , 2 , . . .
(Let us note that if Condition В is fulfilled with respect to an in
creasing sequence {Tt) of sets (U = T), then there exists a family F such that Condition A with f7* = X is fulfilled. However, in general, the considered set F need not to satisfy Condition A even though Condition В is fulfilled, e.g., F may be a one element set. Moreover, Condition В implies that for a.e. t the function Ф( - , t) is finite on X, thus also contin
uous on X and hence condition f ) in the definition of A-function is fulfilled.
If X = B n and и{ = X, then Condition A implies Condition B.)
In the following we assume that measurable functions taking their val
ues in a Banach space X are strongly measurable. By a Pettis theorem if X is separable, then the strong measurability is equivalent to the weak one..
In Section 1 we recall known results and notions of the theory of modular spaces (cf. [19]). In Section 2 we define Or liez spaces from the modular spaces point of view and we prove that if Ф is an A^'-function, then the Orlicz spaces are Banach ones. Section 3 contains a definition of spaces Е ф and their basic properties. In Section 4 we investigate Orlicz spaces i f v, where W is the conjugate function of Ф. This section includes a theorem on a representation of linear continuous functionals on Е ф.
In Section 5 the subdifferentials д1ф,д1^, дФ(&(#), t) and dW[y(t), t) are investigated. In Section 6 we discuss properties of convex functionals on Orlicz spaces. Finally in Appendix we give definitions of the normal convex integrand, decomposable spaces and we prove Bockafellar’s theorem on conjugates of convex integral functionals under somewhat
weakened assumptions.
As it was pointed out, in the presented theory of Orlicz spaces we
use results of the theory of modular spaces developed by ISTakano. Moreover,
the ideas of proofs in some theorems of this paper are similar to those
given in [11], [15]. However, in the proofs we have had to use some more
recent results connected with theorems on selectors [ 2 ], [ 1 2 ], [28] and
Orlicz spaces o f functions 261
a general theory of convex functionals on locally convex spaces [14].
Symbols x, y (may be with indices) will be used for vectors from Banach spaces X and Y respectively. Functions taking their values in X and Y will be denoted x{ • ) and y( • ), may be with indices. However, to simplify the notation we will omit sometimes the brackets when it does not lead to a misunderstanding.
1. Modular spaces. The theory of modular spaces has been developed in many papers and we mention from among them the works of Xakano [18], [19] and Musielak, Orlicz [16], [17].
In this section we recall some notions and theorems proved in [19]
which we need in the next sections.
A function m on a linear space 8 is called a modular on 8, if for any x e S we have
1) 0 < m (x ) < + oo, 2 ) m( —x) = m(x), 3) inf m(£x) = 0 ,
! > 0
4) m(£x) < m(yx) for 0 < £ < y, 5) there is % > 0 such that
+ + m ix ® *)) for a11 a>ae/S.
If m is convex, then condition 5 is satisfied with # = 2 . The con
vexity of m will be always assumed in the sequel.
Let 8 be the linear space of all linear functionals on 8 and let 8 be the linear space of all bounded linear functionals on 8 for m, i.e., functionals -satisfying the condition
q>(x) < у (m(x) + l ) for some y > 0 and for all xe8.
A modular m is said to be normal if
m(x) — sup m(£x) for all x eS .
0 < £ < 1
A conjugate m of the modular m is defined on 8 as m((p) = sup {(p{x) — m(x)) for <pe8.
x e S
The conjugate is a normal, convex modular on 8. Two seminorms are defined on 8:
Жг{х) = sup <p{x) and
X 2{x) = inf(l/£: m ( £ » ) < l, ! > 0 ).
In the theory of Orlicz spaces these seminorms becomes norms and they are called the Orlicz and the Luxemburg norm, respectively. The seminorms N x and N 2 have the following properties
1 ) N x(x) = inf (l + m(£a?))] for xeS,
£>o \ Ç ]
2 ) N 2{x) < 1 if and only if m(x) < 1 , 3) if m{x) < 1, then m{x) < N 2(x), 4) if m(x) > 1, then m(x) > N 2(x).
In a similar way norms N x and N 2 are defined:
Nx((p) = sup(ç>(Æ): m{x) < 1) for <p<=8, N2(<p) = inf(l/£: m(£(p) < 1 , £ > 0) for <pe8.
Moreover, Nx{q>) < oo if and only if <peS. Since m is pure on S, i.e., the implication
m(£ç>) = 0 for all £ > 0 => <p = 0 holds, then N 2 is a norm on 8.
The Orlicz and Luxemburg norms are equivalent, i.e., N 2( x ) ^ N x( x ) ^ 2N2( x ) for XeS and
N 2((p) ^ Nx(<p) ^ %N2{(p) for (peS.
A convergence for a modular m (m) defined by xk^-x if m(£(a?fc —£»))->0 for all £ > 0 ,
if ™(H<Pk~(P))~>'Q for £ > 0 )
^ . jÿ.
is equivalent to the convergence xk -> x, i = 1 , 2 (<pk -A (p, i — 1 , 2 ), respectively.
The following relationships between the Orlicz and Luxemburg norms hold :
N x(x) = sup[<p{x): N 2(<p) < 1, (peS) for a?e$, A 2 ( æ ) = sup( 99 (£u): N x(<p) ^ 1 , for a?e$, A j C ç ?) = sup{<p(x): N 2( x ) < 1 , a?e$) for N 2{< p ) = sup(ç>(ü?): N x{œ )^ l, Xe8j for <pe8.
A modular m is reflexive, i.e.,
m(x) = sup (a?) — т(<р)} for all xeS
<peS
holds if and only if m is convex and normal.
Orlics spaces o f functions 263
Obviously, the following Holder inequalities hold:
|Ç>(®)I < Nz{(p), \<p{x)\ < N iW N M for <peS, XeS.
2. Orlicz spaces. We assume henceforth that (T, 51, [л] is a measure space, where 5fis a cr-algebra of subsets of T, fi is a positive c-finite com
plete measure on and that X is a Banach space.
D e f in it io n 2.1.1. A fu n ctio n 0 (x ,t ): X x f - > [ 0, oo] is said to be an A "-function if
a) Ф is 88 x 51 measurable, where 88 denotes the (Т-algebra of Borel subsets of X ,
b) Ф is lower semicontinuous on X for almost every tcT, c) Ф is convex for almost every teT,
d) Ф{ 0 , t) = 0 and Ф (х, t) = Ф (—х, t) for almost every teT.
D e f in it io n 2 . 1 . 2 . An A"-function Ф is called an N'-function if for almost every t
e) Ф(х, t) > a(t) > 0 for \\x\\ ^ X0{t), a{t), Ao(J)e(0, + oo) holds.
D e f in it io n 2.1.3. An A'-function Ф is called an N-function if X is a separable, reflexive Banach space and if for almost every teT
f) Ф(х, t) < £(t) < oo for ||a?|| < Q(t), £(t), g(t)e(0, + o o ) holds, i.e., if Ф(-,t) is continuous at zero.
In the following the symbol Ф will be used only for functions which are A''-functions at least.
Let us denote the set of 5f-measurable functions from T into X by J t x . We assume henceforth that A is a given non-empty subset of J t x such that
(2.1) I 0 (x{')) = j0 ( x ( t ) , t)p(dt) < oo
T
for all x ( ’ )eF and we denote the smallest linear space spanned on F by Lin A. Let J5? 0 (A) stand for the set of all x { ') e J t x for which there exists a sequence a?n(-)€LinA such that
( 2 . 2 ) lim 1ф(§(хп — x)) = 0 for every £ > 0
n->oo
holds, where 1Ф is given by ( 2 . 1 ).
Since
1Ф{£{аяп — ая)) = 1ф($\а\{хп — х)) and
1ф(Ихп + У п - х - у ) ) < \1ф(2£{хп- х )) + \1ф(2£{уп-у )]
for xn, уn, x, ye Л x holds, it is obvious that 3 0{F) is a linear space.
If F is the set of all functions x ( - ) e jf x such that (2.1) holds, then we use the symbol 3?ф instead of 3 0{F) and we call the space 3 ф the Orlicz space.
Let us note that from the proof of Proposition 2.2 given below it follows that in this case 3 ф = Lin I 7. This links our definition of Orlicz space with the original one given by W. Orlicz in [ 20 ].
P r o p o sit io n 2.2. 1Ф is a normal convex modular on 3 0{F).
P roof. It is evident that 1Ф satisfies conditions 1, 2 , 4 and 5 of the definition of modular (cf. Section 1) and condition 3 for x( • ^ L in F . Let xn{ • ^ L in F and let 1Ф [£{xn~ f o r every £ > 0 . We take £n€(0, -|) such that 1Ф(2£nxn) < oo and N 0 such that for n ^ N 0
1ф{хп — х) < OO.
Then we have for n > F 0
1Ф{ М < Н * ( 2 ^„(а?„-®)) + ^ # ( 2 | па?я)
< £ » **(« ,» - « ) d-i-MSI»®»») < °°-
Thus, in view of the convexity and continuity of 1Ф( £x) on ( — £N , £N ) condition 3 holds for every x( ') e 3 0{F).
Now, we prove that 1Ф is a normal modular. We have 1Ф(я) < jlu n 0 ( £ nx(t), t)fA(dt) < lim 1ф(£яа?)
T
for every sequence The first inequality follows from the lower semicontinuity of Ф and the second one from the Fat ou lemma. On the other hand 1ф{х) > 1ф{£х) for I < 1. Hence, 1ф(х) = sup 1ф{£х), i.e.,
1Ф is normal. 0<f<1
P r o p o sit io n 2.3. I f F is a linear space, then 1ф{х) < oo for every;
x { ') e 3 0{F).
P roof. Let a>n( * ) €LinF , and 1ф(£{хп- х ))->0 for all £ > 0. Then 1ф(х) < %1ф(2{хп- х )) + 11ф{2хп) < oo
for sufficiently large n and the proof is concluded.
In the following we will consider the spaces 3 0{F ) as modular ones with the modular 1Ф and with Orlicz and Luxemburg norms Жг and N 2 respectively (cf. Section 1).
T h e o r e m 2.4. I f Ф is an N'-function, then N{ (i = 1 , 2 ) are norms on 3 0{F) and 3 0{F) is a Banach space.
P roof. (i = 1 , 2 ) are norms. Since N {, i = 1 , 2 , are equivalent
seminorms it is enough to prove that if Жг(х) = 0 , then x(t) = 0 a.e.
Orlicz spaces o f functions 265
We have
0 = N ^x) = inf ^ ( 1 + 1 ф(£ж)) l >0 ç
and we infer that 1ф(£ж) = 0 for every £ > 0. However, if there exists e > 0 such that \\x(t)\\ > e for teA , /л(A) > 0 , then from Definition 2 . 1.2 follows the existence of a number JSf0 such that 0 (N oæ(t)) > 1 on a set В <= A, ju(B) > 0 and hence I 0 (Nox{ *)) > 0. Thus x{t) = 0 a.e.
J? 0(F) is a Banach space. It is enough to prove the completness of
£ 0{F) with the convergence for modular 1Ф in the case /л(Т) < oo.
In fact, because // is u-finite then there exists a positive and yit-sum- mable function, say f(t). Thus we can put Фг = Ф/f and dpx = fdfx. Фх is an W'-function and the modular I 0 given by
1фх(я{т))= / ^ i (»(*),
T
is equal to I 0{x(')) for every x{ * ). Therefore we can consider I 0i and /ux instead of I 0 and p which lead to the same spaces ££Ф{В) and £?01{F).
First, we will prove that if I 0 [£(xn — æm))-^-0 for every £ > 0 , then the sequence {xn( • )} satisfies the Cauchy condition for the convergence in measure. Suppose that {xn( • )} does not converge in measure. Then there exist e0 > 0 and rj0 > 0 such that for every M there exist nM > M and mM^ M such that n(t: \K M{t) - xmM{t)\\ > e0) > f)0.
Let
A M = {t: > £ob
We have
; M^- m ) > Vo
for every M. Let us take £0 sufficiently large. Then there exists a set В such that fi(B) > ц(Т) — r}0/2 and Ф{х, t) > 1 for || æ || > £0-£0. Hence /*{Ам п В ) > rj0j2 for every M and we have
I <p(£o(XnM-®mM)) > Vol% for every M.
Thus {#«(•)} satisfies the Cauchy condition for the convergence in measure and we obtain the existence of x(-)e J i x such that xn{ • ) in measure. Hence there exists a subsequence хщ { •) such that xnk{t)~>x(t) a.e. In view of the lower semicontinuity of Ф we have for all sufficiently large n
1Ф{Ихп-® )) < j lim t) p{dt)
А—нзо
< lim I 0 (£(xn- X nk)) < 6.
А—изо
Hence {#n(-)} is convergent to # (•) for modular I 0 and x (')e£?0(F).
By Properties 2 and 3 of norm N2 (cf. Section 1) every Cauchy sequence for the convergence in this norm is a Cauchy sequence for the convergence for modular 1Ф. This yields that £?0{F) is a Banach space.
Let us note that we have used the completeness of X only in the proof of the completeness of £?0{F).
3. The spaces Е ф.
D e f in it io n 3.1. Suppose Condition A (cf. Section 0) is fulfilled with и{ = X. Let F 0 be the set of all functions of the form x%A{t), where A c T{ and xeX . Then the space J£0{F O) is said to be Е ф.
It is evident, that it F F 0, then £? 0{F) => Е ф. In this section we will always assume that Е ф is well defined, i.e., that Condition A is fulfilled with Ui = X .
Let us observe that in the definition of the space Е ф a sequence of sets T{ is used. However, the dependence of Е ф on {T J is ostensible and we will prove it now (cf. also Corollary 3.3.1 given below).
Suppose that {T{} is an another sequence of sets which can be used in Definition 3.1 and that F 0 is the corresponding set of simple functions.
Moreover, let Е ф = J? 0(F O). If x ( ’ )eÉ 0, then there exists a sequence {xn{-)} such that 1ф(£(х — xn))->0 for every real £ and
where Д п is a measurable subset of some Tj.
By the property of E 0 given in Proposition 3.3.a) below and by dom
inated convergence theorem there exists Tjn an element of the sequence {Tj} such that
Хп(*) = J^Xi,nXDi>n(t), i—1
Moreover, let xn(-) be given by
= 2 ]Ч п Х о {'ПУ)
where Di n — Di nr\Tjn. Then xn( -)eLin ^ 0 and we have
I 0( £ ( x - x j) = 1ф( £ ( х - х п+х,'n Xn) ) ^ l l 0(2 £ {x -x nj) + l l 0[2£{xn- x n))-
In view of our assumption the first component of the right-hand side
of the inequality converges to zero whereas the second one is bounded
Orlicz spaces o f functions 267
by l j 2 n for ||| < n. Tims lim 1 ф( £ { х — x n)) = 0 for every real |, i.e.,
n-*- oo
x { • ) е Е ф and hence Е ф Ё ф. In view of the symmetry we obtain Е ф = Ё ф.
P roposition 3.2. I f C o n d itio n В i s s a t is f ie d , then every m e a su ra b le , bounded a n d v a n is h in g o u tsid e o f a f i n it e n u m b er o f T i fu n c tio n i s a n e l
em ent o f Е ф.
P roof. Suppose cc{-) has the desirable properties. The measurability of x {- ) implies the existence of a sequence {#„(•)} of step functions which are bounded, vanishing outside of a finite number of T { and convergent a.e.
to &(•). Since the functions x n { - ) are elements of Е ф, we hava, in view of Condition B, that
x ( t ) , <)) ^ fjy {t) f ° r U T N
and from the Lebesgue theorem we obtain the modular convergence of {«»(*)} to <»(*)- The number N depends here on œ( • ) and |. Thus х { ‘ )еЕ ф.
P roposition 3.3.a) I f х { - ) с Е ф, then 1 ф { х { - ) ) < oo.
b) I f C o n d itio n В i s fu lfille d a n d 1ф(!& (•))< oo f o r every I > 0,
then x ( ' ) с Е ф.
P roof. Let х ( - ) с Е ф. If x ( ‘ ) € F 0, then the implication follows from the definition of F 0, if a?(-)eLin.F0, then we obtain the desirable result from the convexity of 1Ф. Finally, taking L in F 0 instead of F 0, we obtain from Proposition 2.3 that 1 ф[ х ( - ) ) < oo for every х { ' ) € Е ф.
Now, let x { ’ ) e J t x and 1ф(£ г(-))< oo for every | > 0. If # (•) is bounded and vanishing outside of a finite number of T { , then х ( - ) е Е ф
in view of Proposition 3.2. If it is not the case, we put
(3.1) xn(t) x(t)
0
if ||#(C!I < n and tcTn, otherwise.
Obviously oon( ’ ) are measurable, bounded and ocn(t) is convergent to x(t) a.e. Moreover, Ф ( |(x(t) — xn(t), t)) < <f>[£x(t), t) and if
(3.2) An = {t: xn{t) = x(t)},
OO
then y ( T \ ( J A{) = 0.
i = l
Hence, and from the dominated convergence theorem we have lim 1ф^[х{-) — #„(•))) = Hm J 0(£x(t), t)y{dt) = 0
n
—>00П —MX) /
An for every | > 0 . Thus x (-)e E 0 .
From the proposition we obtain immediately
C o r o lla r y 3.3.1. I f Condition В is fulfilled, then Е ф is the largest subspace of dom l0. Moreover, if d o m l0 is a linear space, then f£ф — Е ф,
— P race M atem atyczne 19 z. 2
where
dom/ф = {x{-)e J£x : 1ф[х{‘ ) ) < oo}.
P roposition 3.4. I f х {-)е Е ф, then %A )— >-0, while p(A)-^-0.
P roof. Suppose there exists a sequence of sets {An} such that /u,(An)-+ 0 and N z(®(‘ )XAn(')) > « > 0 . Then JV "2 | i > 1 and from Section 1 we obtain that ТФ|— a>( • )Х а п( * )j > 1» i*e*»
J Ф x{t), /n(dt) > 1 .
But this is in contradiction with the continuity of the integral with respect to the measure p. Since the norms N x and N 2 are equivalent the proof is concluded.
D e f in it io n 3.5. A sequence хп{-)еУ ф{Е) is said to be convergent in the mean to х {')е £ ?ф{Е) if 1ф(хп — х) • n— > 0 .
P roposition 3.6. I f Condition В is fulfilled, then d o m l^ is contained in the closure of LinF 0 with respect to the convergence in the mean.
P roof. The proposition is obvious for х ( т)€Еф which are bounded and vanish outside of a finite number of T{ (cf. Proposition 3.2). If it» is not the case we repeat the proof of Proposition З.З.Ь for 1 = 1.
R em ark . If the assumptions of Proposition 3.6 are satisfied and if Ф satisfies the following condition Az
Ф(2х, t) < КФ(х, t) + h(t),
where h ( • ) is a non-negative and //-summable function, then the modular convergence and the convergence in the mean are equivalent.
If the spaces Е ф and I£ф are given, then we put
d [ x (- ),E 0) = inf (Ai (x — w): w eE0) for х {')е & ф and
П (Е Ф, r) = {х (- )€ ^ ф: d(x, Е ф) < r ] . P r o p o sit io n 3.7. We have
П (Е Ф, 1) c d om l0.
Moreover, if Condition В is fulfilled, then
do m /ф c cl Щ Е Ф, 1 ),
where the closure is for the modular convergence.
Orlics spaces o f functions 26 9
P ro o f. Let х0{-)е Е ф, x{-)eSC0 and let N x{x — x0) < 1 —a for some
1 lx — x о \ »
a > 0. Then — æ0( • )еЕ ф and _2УГХ I---1 < [l. By the Polar Theorem
a \ 1 — a f b
proved in [19] we have
Thus (1 —a) 1(x — x0) is an element of dom/ф and by the convexity of dom j^ we obtain that œ( • )€dom l0. Indeed, œ{ • ) = a — a?0( • ) + 1
+ (1 —a)((l — «)_1(a?( - ) — a?0( * )))* °
Now, let a?(• JtrdomJ#. In view of Proposition 3.6 for every e > 0 there exists хее Е ф such that I 0{x — xs) < s. Thus N ^ x — xe) < 1 + e and we have
d (E 0,x ) = inf [Nx(x — w ) : w c E 0) < Ef^x — xe) < l + «-
Since the inequalities for every s > 0 hold we obtain that d[E0, x(-)j
< 1 and the proposition is proved.
P ro po si t i o n 3 . 8 .
I f Condition В is fulfilled and if Ф is an EC-function,
then
lim ЕГг(x — xn) — d (E 0, x) for x*.f£ ф,
n-+oo
where xn( ‘ ) are given by (3.1).
P roof. In view of Proposition 3.2, xn( •) is an element of Е ф. Thus l i m l ^ æ - xn) ^ d (E 0, x).
Let us take e > 0 and let a > 0 satisfy
(d{E0, x) + 2e )- 1 < a < (d(E0, x )J!-e)~1.
Then we obtain
d (E 0, ax) = inf (N ^ax — to): WeE0)
= » ) : » е й ф) < d ^ * ]+ s < 1 .
Hence ax€dom l0 and by Proposition 3.6 for sufficiently large n the inequality
1ф(ах—ахп) < as
bolds. Thus N x{ax — axn) < 1 + as. Moreover, we have N x(x — x ) < --- b £ < d(E 0, x) + 3e.
a
We may take e small as we like and hence И т Х г(х — xn) < d (E 01 x).
Thus the proof is concluded.
4. The spaces We denote henceforth by J l Y the set of strongly 31-measurable functions from T into Y, where Y is the dual space of the Banach space X. Moreover, let <[x, y> stand for the value of the func
tional y e Y at the point xeX . Obviously, for x ( - ) e ^ x and y {')€ .Jt T the function (co{t), y{t)} is 31-measurable.
The set of functions y ( - ) € J t r such that
J y ( t ) } \ y ( d t ) < 00 f ° r every x (- )€ ^ 0(F)
t
is called
We recall that I f 0 {F) is a decomposable space (cf. Appendix) if every bounded function x{ • ) which vanishes outside of some T{ is an element of £f0 {F). Since for any such x( • ) there is g > 0 such that I 0(gx( • )) < oo, the function £ # (•) has a decomposability property. Therefore, to apply Theorem A5 from Appendix in the case L = SÛ0{F) and f(x, t) = Ф(х, t) it is enough to assume that £P0 {F) is decomposable.
P r o p o sit io n 4.1. I f Ф is an N'-function and K>0{F) is decomposable, then for every y(-)€£f%(F)
sup j j <«(<), y(t)yy{dt)} < oo
holds, where the supremum is taken over the set of all х { ') е ^ ф(Е) such that 1ф(х{ • )) < 1 .
P roof. Let us suppose that y (’ )€^%(F), xn( • )€£P0(F), 1ф{хп) < 1 and
/<®„(<),y(<)>/.(<«)> 2 ”.
We can assume that xn ( • ) are bounded functions with supports con
tained in suitable sets Tt and that (xn{t), y(t)) > 0 a.e. Since 10{<сп) < 1 we have X z(xn) < 1, where N 2 is the Luxemburg norm. Let
П 1 9nV) = ] ? J k ækW ‘
k=l
The sequence {gn} satisfies the Cauchy condition with respect to the norm jVa, thus, in view of Theorem2.4, gn is convergent to g { ’ )eS^0{F) and we have for some subsequence {%}
И/С
g(t) = lim ^ 2 ~nxn{t)
k-+<x> n = 1 a.e.
Orlics spaces o f functions 271
Moreover,
j<9n(t), > n
and for the sequence of fonctions (gn{t), y (<)) is an increasing one, we have (<g(t), y(t)>v(dt) = f \im ( g n{t), y{t)}p(dt)
n— > oo
= lim j< g n(t),y(t)}iu(dt) = oo.
?г->оо
Hence y{ •) is not an element of &%{F) and the proposition is proved.
Let ns note that the proposition allows ns to consider the elements of # 1 {F ) as the elements of the space of bounded linear functionals on
& Ф{В) (cf. Section 1). The pairing between £?Ф{В) and J£%{F) is given by a formula
Ç>(®(-)) = J <æ(t),y{t)yp{dt) for x{')e£e0{F),y(-)<i£e%(F).
However, it may happen that different functions from S£*0{F) define the same functional <p on <£0{F). In the next proposition we give suffi
cient conditions to avoid such cases.
P r o p o sit io n 4.2. Let X b e a separable Banach space and lety(-) €&%(F).
Suppose Condition A is fulfilled, then
(AI) J (x(t), y(t)yju(dt) = 0 for every х {-)е& ф{В) if and only if y(t) = 0 a.e.
P ro of. It is enough to prove the sufficiency. Let {a^} be a dense subset of X . If (4.1) holds for х^%А, œj from the dense, countable subset of X and for every 4 c f . , then (Xj, y{tf) = 0 outside of a set Bj of
OO
measure zero. Hence (fOj,y(t)y = 0 for j — 1 , 2 , ... outside of [ J Bt .
3 = 1
By a continuity we obtain (x, r/(t)> = 0 for t 4 \ J B j and for an arbitrary x iX , thus y(t) = 0 a.e. on Tt. Because ( J ^ i = T, we have proved the proposition.
Prom Propositions 4.1 and 4.2 we infer that if £P0 {F) is decom
posable, then there is a one to one correspondence between Sf%{F) and a subset of bounded functionals on JS%(P).
In the following part of this section we will always assume that X is a separable, reflexive Banach space.
Let W(y,t) be the conjugate of Ф{х, t) from T x Y into [0, + oo], i.e., W{y,t) = sup { { х ,у } - Ф { х ,Щ .
XeX
Proposition 2 in [28] implies that if Ф{х, t) is an ^''-function, then
'F{y,tj is .^''-function as well. The space defined by
& 'P — I У { ' ) e y : I p { ^ y ( ‘ )) — t)f*{dt) < 00
for some positive number | | is an Orlicz space. It follows from the definition of Orlicz space and from the first part of the proof of Proposition 2.2. The space i f v is said to be
conjugate to the Orlicz space i% .
P roposition 4.3. For every non-empty set F we have
P roof. Let y { - ) e ^ w. Then
£v<a(t), y(t) > < 0(№ {t), t) + W(riy(t), t) for every a>(')e£f0(F). For suitable £ and rj we have
/ <£®($), ПУ(1)>У(М) < !* ( № { • ) ) + I v ( w { ’ )) < 00 • Hence, y(')e<£*0{F).
P roposition 4.4. Suppose Ф is an A '-function. I f ££0{F) is a de
composable space, then
& % ( F ) = & V.
P roof. £?0{F) is a modular space (cf. Proposition 2 . 2 ). By Propo
sitions 4.1 and 4.2 we can consider £?*0{F) as a subset of the set of bounded functionals on i% (F ). By Bockafellar’s Theorem (cf. Appendix) I v is the conjugate of 1Ф. Because the conjugate is a modular on the set of bounded functionals, then for every y {’ )e<F*0{F) there exists £ > 0 such that I y(£y(*)) < 00 . Thus i f %(F) <= i f y.
The spaces & 0{F) and i f v satisfy assumption A 1 about spaces L and M in Rockafellar’s Theorem and we can state the following corollary:
P roposition 4.5. I f £P0(F) ( i f y) is decomposable, then I p{y(- )) = s u p ( J (œ (t),y(t)>y(d t)-I0 (æ{-))-,x( • )ci f 0 (F))
for all y (-)eify , ( * ф И *)) = s u p ( J * (x {t),y (t))p {d t)-Iv {y (-))’, y (-)eify )
for all x (-)€<?0(F)) provided X (T) is separable.
In the following part of the section we will deal with A-f unctions only. We have mentioned, that Ф is an A"-function if and only if V is an A''-function. A similar equivalence is true for A-f unctions as well.
P roposition 4.6. Ф is an Ж-function if and only if F is an Ж-function.
P roof. It is enough to prove that Ф satisfies conditions e) and f) of
the definition of A-function if and only if F does.
0 rliez spaces o f functions 273
Suppose Ф is an Y-f unction and let y(t) = 3 C(t)lg{t), where £(•) and £>(•) are from condition f) for Ф. If condition e) is not fulfilled by У7, then for the y(t) there exists a set A, /u(A) > 0, such that for teA
inf {¥ (y , f); ||y|| > y(t)} = 0 .
Take y(t) and next x{t) satisfying the following conditions for teA:
1) 112/WII = Y if) and W(y{t), #) < C(t), 2) ||д?(«)|| = g(t) and <x(t),y(t)) = e(t)y(t).
Now, we obtain by the Young inequality that for teA 3£{t) = Q{t)y{i) = <x{t),y(t)} < W(y{t), t) + 0(x{t),t)
< 2 C(t).
This contradicts the assumption that e) is not fulfilled by 4*.
Now, we will prove that 4* satisfies Condition f). We have T {y ,t) — sup[<a?, у} — Ф{х, t) ; xeX ]
< sup[<®, у } - Ф { х , t); ||a?|| < A0(*)] -h
+ sup[<a>, у ) - Ф ( х , #); INI > A0(«)] < A 0 (f) \\y\\ + + sup[cA0(C (x, y ) - 0 ( c ù o{t)x, f); c > l,||® || = 1 ]
< *o(t) 112/11 + sup[cA0(*) ?/>-сФ ’(Я 0 (#)а?, /); c > 1 , \\x\\ = 1 ].
( Put Q(t) = \a{t)]X0{t), where a(-) and A0(-) are from condition e) for Ф. Then we have
sup[A 0 (f) <a>, y } ‘, \\x\\ = 1 , ||y|| < g{t)] < A 0 (#)g(f) = and
inf [Ф(Я 0 (/)я?, /); INI = l] = a(t).
Therefore if \\y\\ < Q(t), then
sup [сЯ0(С (x, y } - c 0 (a ( t )x , t)j c > 1 , INI = l] < 0 and hence
sup[*P(r/,C; IN K Q(t)l < я0(#) * e(<) = \a {t),
i.e., Ф fulfils condition f) with £(/) = |a(f) and q{t) = fa(f)/A 0 (f). In view of a symmetry between X, Y and Ф, W the proof of the proposition is concluded.
T h e o r e m 4.7. Assume that Ф is an N -function and that J g Ф{Т) is a decomposable space. Then v is isometrically isomorphic to a closed subspace of the space of all bounded linear functionals on f£Ф{Т).
P roof. By Propositions 4.1-4.4 we identify fg y with a subspace of
the space of bounded linear functionals on <g0{F). Since W, the conjugate
of Ф, is an Y-function, then Theorem 2.4 implies that ^ 5 , is a Banach
space. Thus fg ^ is a closed set with modular convergence and with I v as
a modular. Now, Bockafellar’s Theoiem implies that is the conjugate of 1Ф on rj,. Thus, the modular convergence with the conjugate modular and with I v is the same on S£ у and the completeness of ST w implies that it is a closed set in the space of hounded functionals on ST0{E).
T h e o r e m 4.8. Let Ф be an N -function and let Conditions A and В be fulfilled. Then every continuous linear functional on Е ф is of the form
cp(cc(-)) = j(x {t),y (t)}y (d t), T
where y{ •
P roof. Let <p be a continuous linear functional on Е ф. The functions of the form x-%A( ') i A ci Tn, are elements of Е ф and we put
Ф ш Х а ( ‘ )) =
It is evident that f(x , A) is a linear function on X for every fixed A Tn and an additive function on measurable subsets of Tn for every fixed xeX . Since <p is continuous on Е ф, we have
I fix , A) I < N 2{ x -X a )N x {9) < n i {®'X a )N i {< p )-
Then, by Proposition 3 A f(x , •) is countably additive and absolutely continuous with respect to the measure у on Tn. Thus the Badon-Nikodym theorem implies the existence of a function k'(x,t) which is measurable in t for every x and such that
(4.2) f(x , A) = J h'(x, t)y(dt) for A c Tn and xeX .
A
Let X d be a countable, dense subset of X and let X d be the linear space over the field of rational numbers spanned on X d. Since X d is a countable set, k '(x, t) is additive and rationally homogenous on X d, i.e., linear on X d for all teTn\ N n, where N n is a set of measure zero.
Let us denote
[ Tc'(x, t) for x €X d and t€Tn\ X n, k(x, t) =
( 0 for XeXd and teNn.
Hence, for every teTn, Tc(x,t) is a linear functional on X d. We will prove that k (x,t) is a continuous linear functional on X d. Let
h(t) = sup(||a?ir 1 fc(ir, t): (CeXd) for teTn.
Then, h(t) is an ^(-measurable function. Let X d — {хг, œ2, • • •}.
Clearly, h{ •) is a non-negative function. If h( •) is summable over Tn, then Jc(x, t) is a continuous functional on X d for a.e. teTn. Suppose that on the contrary
J h(t)y{dt) = -f- oo.
Orlicz spaces o f functions 275
Let
Jik(t) = m ax{ll^ir 1 ^(a?*, t), г < &}.
The sequence {Jik( •)} is increasing and convergent to h( •). Thus for every M > 0 we can choose a number N(M) such that
‘ f h N (M )(t) f * { d t ) > A 1 .
We now define sets A{
А\ = { t e T n: llaqr1^ . , t) = m ax(0, hN(M)(<))}, i < N(M) and sets A{
A, = A [ , At = A'{\ U A3., i = 1 , . . . , ДГ(Ж).
Let ;=i
®o(<) = * = 1 , . . . , N ( i ¥ ) .
The function a ?0 (• ) is an element of Е ф because is bounded and van
ishes outside of Tn. Moreover, we have
M(N)
9 > K ( 0 ) = f \W\~1Jc(V i,t)fi(dt)^ j hN(M){t)/x{dt) > M.
i= l a ( т п
On the other hand, for every function <»(•) such that ||я?(<)|| < n and suppa?(-) c Tn the following inequalities hold
|ç>(®( *))| < т{ср) + 1ф{х( •)) < Щ<р)+ J f n(t)/*(dt) < +oo, тп
where m is a modular on Е ф conjugate to 1Ф and /„ (•) is the function from Condition B. This yields a contradiction.
Thus h(t) is finite for a.e. t*T n and k(a>,t) is continuous on X d for a.e. t€Tn, i.e.,
(4.3) k(œ,t) = (œ ,y{t)} for x eX d and t€Tn\ M n,
where /л{Мп) = 0 and y(t) eXd. In view’ of the continuity of y(t) on X d, y{t) can be univocally extended on X , thus we have y(t)eY. Moreover, for k{cc,t) is measurable for every xeX , then у ( • ) is weakly measurable and hence, a Pettis theorem yields that y ( ' ) c J f Y‘
Now, it is evident that since U T „ = r , we can define the function k{æ,t) for all tcT and that y(t) can be defined for a.e. teT in such a way that (4.3) holds for all xeX .
The vector space spanned on the set of all functions of the form
x
' X
a{ ')? where x e X and A <= Tnfor some n is a dense subset of Е ф. By (4-2) and (4.3) we have
(4.4) ç>(®(-)) = f <0(t),y(t)>/*(dt)
T
on the dense subset of Е ф. We will prove, that (4.4) for every x( ■ )еЕ ф holds.
Let (4.4) hold for хк( ') е Е ф and let
lim W 2 (a>(•) - % ( •) ) = 0 ,
к-юо
where х ( ') е Е ф. Suppose the convergence of œk(t) to œ(t) is a.e. Moreover, we can assume that JV2(% (•))< W2(^( • ))• Thus, by Fatou lemma, we have
|J <0(0» I < / |<0(O» y{t)>\t*(dt)
= flim l< 0 fc(O ,y ( O > l^ W
< l i m j \(xk{t), y(t)>\p{dt)
= lim f< ek(t)a>k(t), y(t))/z(dt)
= Пm<p{eksok) < lim Nx((p) Ж 2(ekcck)
< Ё х((р)]5Г2(х(-)) < oo, where
+ 1 if < 0 *(О»У(О>^ 0 , sk(t) —
—1 otherwise.
Thus by the continuity of <p it follows that (4.4) holds for all æ( • )еЕ ф.
Moreover, Proposition 4.4 implies that y{')e££4,.
On the other hand, Propositions 4.1 and 4.4 imply that every el
ement of i f y corresponds to a continuous functional on Е ф. Proposition 4.2 asserts that the correspondence is one to one. An isometry is obvious and follows from Propositions 4.5 and 4.6.
It was mentioned that an equality
1<р — ( 1 Ф)
for # (•)« ££v follows from Bockafellar’s Theorem. However, if we take Е ф instead of modular space 8 given in Section 1, we obtain that S = if^ . Thus, by Keflexivity Theorem (cf. [19]), we have
m = I ф = m = (I ,P)
without assuming that i f y is decomposable (cf. Proposition 4.5).
5. Subdifferentials. First, we recall some known results and notions of the theory of convex functions. It will be assumed that values of con
sidered convex functions are in ( — oo, +oo] and if /( •) is convex on a vector space B X1 then
dom/ = [х€Вг: f(æ) < + oo).
Orlicz spaces o f functions 277
Let vector spaces B x and B 2 form a dual pair with a pairing A vector y eB2 is said to be a subgradient of / at a point £c 0 edom/ if
f(x)>f(o o 0) + < x - x 0, y>
holds for every x eB x. The set of subgradients of / at the point a? 0 edom/
is called subdifferential of f at x0 and it is denoted by df(x0). /( • ) is said to be subdifferentiable at x0 if df(xQ) is a non-empty set. In the sequel if B x will stand for a Banach space, then B 2 will denote the dual space of B x and the natural pairing <•, •> between B x and B 2 will be used.
We will use the following known properties of convex functionals (cf. [14]):
P r o p o sit io n 5.1. a) I f f is a convex functional on a topological vector space B x and if int dom f is a non-empty set, then f is continuous on int dom f if and only if at least one же int dom fh as a neighbourhood on which f is bounded above,
b) if g is the conjugate of f, i.e., if
g(y) = sup (<«?, y }-f(æ )) for every yeB 2,
X tB x
then g is convex and lower semicontinuous on B 2 in a(B 2, B x) topology, c) if f and 9 are conjugate to each other, then
x €d g {y )o y € d f{x )o f{x )+ g {y ) = <x, y>.
d) df(x) is a convex, closed (may be empty) subset of B 2,
e) tf f is finite and continuous in an admissible topology at a point x e B x, then df(x) is a non-empty set and the subdifferential is compact in the weak topology a(B 2, B x),
f) / ( •) attains its infimum at a point x if and only if 0 edf(x).
Moreover, Proposition 2.2 and a) and e) above imply that
g) if Ф is an N'-function, then 1Ф defined on & 0 (F) is continuous and subdifferentiable (i.e., д !ф # 0 ) at every point a?0eint dom 1Ф.
In the further part of this section we will consider properties of subdifferentials of modulars 1Ф, and subdifferentials of normal convex integrands 0 (x ,t) and W(y,t).
P r o p o sit io n 5.2. Let Ф be an N'-function and let ^ 0(F) be decom
posable. I f Æo^edomltfi, where
1ф( х ( ') ) = f Ф ( х (1 ),1 )^ ) for X(-)eSe0(F), if and 2 / 0 (г)едФ(ж 0 (г), г), then Уо(-)ед1ф(х0(')).
P roof. Because Ф is an A'-function and Condition A is fulfilled,
& <
p( F ) is a Banach space (cf. Proposition2.2) and JS?V т а У be considered
as a subset of the dual space of £?0(F) (cf. Propositions 4.1 and 4.2). Since
y o( t ) € d 0 { œ o(t), t ), we have by the definition of a subgradient that
0 { x ( t ) , t ) ^ 0 ( æ o( t ) , t) + ( x ( t ) - x 0( t ) , y 0( t ) }
for every x ( • ) е £ ? ф(1 '). Hence we have
1 ФИ ^ 1 Ф{ х 0) + f < x ( t ) - x 0( t ) , y 0( t ) ) p ( d t )
and the right-hand side of the inequality is finite and well defined. Thus, y0( • ) ed iф (xQ ( • )), where д 1 ф (a ?0 ( • )) is a subset of the dual space of St? Ф{В).
P roposition 5.3. S u p p o s e X i s a s e p a r a b le , re fle x iv e B a n a c h sp a c e . I f x ( - )tint dom 1 Ф a n d
0 { x ( t ) + # , t) < 7c(2) < со a lm o st everyw here
f o r ||a?|| < r ( t ) (0 < r ( t ) < + o o ), then every ^ .-m e a su ra b le fu n c tio n y ( ‘ ) su c h th at y { t ) € d 0 [ x ( t ) , t ) i s a n elem ent o f dom
P roof. Because Ф ( x { t ) -\-x , t) is bounded above on an open set in X for a.e. t c T , then 0 ( x ( t ) + x , t) is continuous and subdifferentiable for a.e. t e T at the point x ( t ) . Hence a set P ( t ) given by
P ( t ) = d 0 ( œ { t ) , t )
is non-empty for a.e. t e T . A support function of the set P ( t ) is defined as follows
<*р(оИ*)) = suP(<®(0»y>: ytHif))- By formula 10.15 in [14] we have
ôm ( œW) = Hm + * ) - Ф ( ® ( * ) > t)\
Л -*0 Л '
< i ( 0 ( ( l + Ao)®(*), t ) - 0 ( x { t ) , <)).
to
The function on the right-hand side of the inequality is a summable one, while À0 is sufficiently small. Moreover, if y ( • ) is a measurable function and y ( t ) e P ( t ) for a.e. t e T , then
\ ( x ( t ) , y ( t)y \ ^ ô p y)[x(t)) -j- <5p(<)( — &(^))
^ ~ { 0 ( ( 1 + A 1 )o?( 0 , <) + Ф ((1 — ^i)a?(<)» t)-2 0 (x (t), <)) and the function on the right-hand side of the inequality is summable for an appropriate Ях. How, by the equality
' F( y( i) , t) = У ( Ф - Ф (a?(t), «)
we obtain that i/(-Jedomly,.
Let us note that the nature of Propositions 5.2 and 5.3 is essentially
the same although assumptions differ one from another.
Orlicz spaces o f functions 279
P r o p o sit io n 5.4. Let f n(-) be a summable function on Tn and sup Ф{х, t) < /„(«) for n = 1, 2 , ..., rn > 0 . IMI
I f æo( • )€m tàom l0 (clom 1Ф <=■ J ? 0(F)), then for a.e. teTthe set 0Ф [x0{t),t) is non-empty, convex and compact in o(Y , X) topology.
P roof. Since x0{ • )eintdomZ0 , then there exists A > 1 such that Xx0{ - ieintdonil,^. Thus
sup 0 (xo{t) + X-l {X - l)x , t) 0{Xxo{t),t)-\-X~1(X ~ l)fn{t) < oo.
\\x\\<rn Л
Hence, sup (ф (a? 0 (t) + œ, t) : ||л?|| < rn) is dominated by a function which is summable on Tn1 n — 1 ,2 ,..., thus it is finite almost everywhere on Tn and Ф ( •, < ) is bounded above on a neighbourhood of the point cc0(t). This implies a continuity of Ф(*, t) at x0{t) for a.e. teT. Now, by 5.1.e) we obtain that d0{xo{t), t) is non-empty for a.e. teT, convex and a{ ¥ f X) compact.
P r o p o sit io n 5.5. Let X be a separable, reflexive Banach space. I f and d0[x{t),t) Ф 0 for a.e. teT , then there exists a countable set { 3 /i(')}? 2 /г(‘ )€ ^ f > such that cl{y{(t)} = d0{x(t), t), where the closure is in an admissible topology of the pair (Y , X) and Y is the dual of X.
P roof. We prove that the set-valued mapping t-+d0(x(t), t'j is measurable, i.e., that {(y, t): yed0(x{t), t)} is x9I-measurable. This is the case, because the following equality holds
{(t, y): yed0(x{t),t)\ = {(y, t): 0{x(t), t) + W(y, t ) - ( x { t ) , y> = 0 ).
This set, called the graph of the mapping t-^-дФ (x{t), t) is measurable for 0{x{t), t), W(y, t) and (x{t), y> are x 31-measurable. Thus the propo
sition follows from Theorem lg in [28].
P r o p o sit io n 5.6. Let X be a separable, reflexive Banach space and let Conditions A and В be fulfilled. I f xo(')e ^ >0(F), N t(x0) < 1 and if y0( ') € su°h that yo(t)çd0{xo(t), t) for a.e. teT, then I T{yf) < 1 , i.e., y 0 €domly.
P roof. Let xn(-) and An be given by (3.1) and (3.2), respectively.
By Proposition 3.2 xn( •) is an element of £?0{F). If we put
I Уо^ if UAn’
I 0 otherwise,
then yn( - ) e J f Y and yn{t)ed0[xn{t), t) for a.e. teT. Let 4(2) = ^n( 2 )(l + e( 2 )),
where
e{t) = sgn( x jt ) , y jt ) ) .
By the definition of a subdifferential we obtain
# K ( * ) , *)+!<<»»(<)> y»oo>i-
Since Conditions A and В are fulfilled, we have 1ф{х'п( ‘ )) < oo and 1Ф(хп{ • )) < °°- Thus
У п ( Ф \ p № ) < oo
and we have
xF(yn(t) i t) < +
= !<«»(<)} y»(*)>l-
Thus Iv(yn( - ) ) < oo, l.e., Уfi ( * ) ^ dom f ^. Suppose that f ^(у f) ^ 1 >
Because W is lower semicontinuous on Y (ef. Proposition 6.1.b) for a.e.
teT we have by the Patou lemma
1 < J 4'(y0(t),t)/a(dt) ^ lim J W(yn{t), t)p(dt).
Hence for n sufficiently large Iy(yn) > 1. However,
t
Iv iV n X f<æ n(t),yn(t)>/*(dt)
< N^{yn)N x{xn) < N x(<cn) I v(yn).
Thus JSfx(xn) > 1. Because JYx(xn) < N x(x0) we obtain that X x{x0) ^ 1 and this contradicts our assumption.
Let us note that Proposition 6.6 implies that д !ф the subdifferential mapping from intdom 1Ф (dom7 0 <= £?Ф{В)) into the dual space of J5%(.F) has the following property
д1ф(оо0) n Ф 0
for x0(-) belonging to the open unit ball of space JS?0(F) with norm Жх.
Now, we extend this result.
P r o p o sit io n 5.7. Let X be a separable, reflexive Banach space and let Conditions A and В be fulfilled. Suppose that Ф is an N ’-function. I f a?(-)eintdom 1Ф (d o m ^ c jSf ф), then every measurable function y(-) such that
y{t)edO[x{t),t) a.e.
is an element of S£xp. Moreover, д !ф[х( ^ Ф 0, where 3?^ is identified with the corresponding set of linear functionals on У? ф.
Proof. We can assume that p(T) < oo. Let x( • )eintdomT(P and let y{-)e J i Y 2iïià.y{t)€d0{æ(t), t) for a.e. tcT (cf. Propositions 5.3 and 5.4).
By Proposition 3.8 we have that for sufficiently large n the inequality
■ tfiHo-;te0(*)) < i
S 0 = T \ { t : teTn and ||aj(i)||<n}.
holds, where
Orties spaces o f functions 281
The function œ{ ’ )X t \ s 0( ’ ) is an clement of Е ф (cf. Proposition 3.2), thus I 0 (2cc( -) xt \ s 0( ' )) < ° ° • Moreover, then there exists at most a finite number of atoms (г = 1, ..., p) and at most a finite number of disjoint, measurable sets Sj (j = 1, ..., r) such that the following conditions are satisfied :
Û = T,
i=о <=i
1 < J Ф(2 x(t), t)fi(dt) for i = 1, 2, ..., p , Zi
J Ф[2х(Ь), t)fi(dt) < 1 for i = 1, 2, ..., r.
Si
Then it follows that (2x( • ) xs{( ' )) < 1 and that N 1 (x( • ) ^ f( • )) < 1 (cf. Section 1).
Let ns put
Xi(t) = x(t)XjSi(t), and
Viit) ^yW XsiW for i = 1 , 2 , ..., r.
Since 0едФ(0, t), we have
у4(г)едФ(<в{{г), t) for i = 1, 2, ..., r.
Moreover, Proposition 5.5 implies y{( •)««£? у Obviously we have 'и(-)Хг4{ ' ) * Я ф and y ( Z i ) e ¥ ,
thus
!<«(<)» y(t)xzt(t )>I < 00•
Because
Iv(y( ')Хг{(-)) = f <0(t), y{t)Xzi ^ )'> ^ {d t)-I0{x{-)xz .{ • )) and the right-hand side is finite, we obtain
■ МзК • )Xz4(')) < °°-
Hence we obtain I 4>[y{')) < oo (cf. Proposition 5.6).
Ju st now we have proved that there exists y { ') e J t 7 such that y (*)€»2V and y[t)*№ (® {t),t) for a.e. teT. Hence the following equality is valid
М ^ ( * ) ) + - * ф И ’ )) = / <a>(t),y(t)>p{dt), T
i.e., y ( ’ ) is an element of д1ф[х (‘ )) (cf. Proposition 5.1.c). The proof is concluded.
Let us note that Proposition 5.7 implies that
M ® (*)) =su p(J<iB («),y(#)>^(d/)-Iy(y(«)): y(-)«^V )
while x{ • )<rintdomI,p. The result does not follow neither from Bockafellar’s Theorem nor from the Beflexiviiy Theorem of Nakano (cf. the end of Section 4).
Because the assumptions about spaces 3?ф and i f w are different, we obtain no information about d l v from the results stated above. Now, we will give some properties of the subdifferential dl,p.
P r o p o sit io n 5.8. Let Ф be an N -function and let 2 /(-)€ if^ . I f y (-)eintdomIxj, and if
'F(y{t) + y,t) < lc x (t) < OO a.e.
for \\y\\ < r^t), where r^t) is finite and positive for every teT, then dW{y(t), tj is a non-empty, convex and weakly compact subset of X. Moreover, there exists a countable sbt of measurable functions (жг(-)} such that дФ [у^),{)
= с1{д?г(£)} and every measurable function x(t) tdW{y(t), t) is an element of dom 1Ф.
P ro o f. By Proposition S.l.e the subdifferential dW[y(t), t) is a non
empty, convex and weakly compact set in X for W{y,t) is bounded above on an open set in Y containing the point y(t). Then, in the way used in the proof of Proposition 5.5 we obtain the existence of a count
able set {&*(•)} of measurable functions such that dW[y{t), t) = cf{&*(£)}.
To prove the last statement of the proposition we use the analogous argumentation to that given in the proof of Proposition 5.3.
6. Remarks on convex integral functionals on Orlicz spaces. It is impor
tant, that convex integral functionals on S£0(F ), if ф, Е ф, £?*0(F), i f v have nice properties such as continuity, lower semicontinuity, inf-compactness or an existence of subdifferentials. The properties are usefull in various fields [5]-[10] and it is one of the reasons that we have extended the notion of Orlicz space. Propositions given below are simple consequences of general theorems and theorems obtained in the preceding sections.
P r o p o sit io n 6.1.a) Suppose Ф is an N'-function and f(x , t) is a nor
mal convex integrand. I f
f(x , t) < 0 ( x - x o(t), t)+ z{t),
where x0( • )eI£0{F) and z{ •) is a non-negative and summable function, then I f{') given by
J / И ') ) = ff(% (t),t)y{dt), х { ‘ )€£? ф {Е),
t
