On some equivalent conditions in Musielak-Orlicz spaces

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIV (1984)

He n r y k Hu d z ik

(Poznan)

On some equivalent conditions in Musielak-Orlicz spaces

Abstract. We shall generalize Theorem 4 [9] to Musielak-Orlicz spaces and we shall give some sufficient conditions for reflexivity of Musielak-Orlicz spaces.

O. Introduction. N is the set of all natural numbers; (T, I , pi) is a space of non-negative, non-atomic, complete, and cr-finite measure; Z0 denotes the class of all sets A e X with pi(A)

R

(

со, oo), Re+ — [0, oo], R

= [0, oo), (X , || -||) is a real Banach space, X Ф [0}. Instead of the statement

“/(-almost every” we write shortly “ц-a.e.”.

We denote by S(X) — S(T, I , X) the linear space obtained from the set of all strongly /(-measurable functions / : T -* X by identifying the functions which are equal pi-almost everywhere.

A function M : T x X -> Re+ is called an Ж -function if : (a) M is I x ^-measurable {:$ is the Borel rr-algebra in X),

(b) for pi-a.e. t e T the function M ( t , ) : X - * R e+ is lower semicon- tinuous, convex, even, M(r, — x) = M(f, x) for every x e X , and M(t, 0) = 0,

(c) for every /(-measurable function (к T -> (0, x ) there exists a /(-measur- able function 2 : T ->(0, x ) such that for pi- a.e. t e T , if x e X and ||x|| ^ a (r), then M(t, x) ^ f(t),

(d) M{t,-) is a continuous function at zero for pi-a.e. teT.

For simplicity, but without loss of generality, we shall assume in proofs that the exceptional null sets appearing in various conditions imposed on M are empty.

Let M be an Л -function. Then the functional / = /* : S( X) - +Re+

defined by

/ ( / ) = J

T

is a convex pseudomodular [8] on S (AT). The pseudomodular I determines the Musielak-Orlicz space

ZiV = EM (T, I , X) = \ f e S { X) : I(rf) < 00 for some r > 0}.

Using the pseudomodular /, we define the Luxemburg norm N: EM- * R +, by

JV(/) = inf [r> 0: / ( / » < 1}.

We say that an Ж -function M satisfies the condition A2 if there exist a set T^eZo, a constant К > 0 and a /х-measurable, non-negative function h with J h(t)dfi < o o such that

T

s M(f, 2x) ^ KM(t, x) + h(t) for every r e T \ 7 i and x e X .

The following condition for Ж -functions will be used :

(B) There exists an increasing sequence [7]},* j of sets from Z with p(Tj) 00

< 0 0 for i = 1 , 2 , . . . such that p ( T \ (J / ) = 0 and there is a sequence

i = 1

{/«}*= i °f //-measurable non-negative functions such that J f„(t)dp < o o for every /', w = 1, 2, ... and M(r, x) ^ /„(f) for ||x|| < я, и = 1, 2, ... and for //- T a.e. t e Г.

0.1. D

. Let M be an Ж -function satisfying condition (B). Let L denote the set of all simple functions (that is, of functions which assume only a finite number of values on sets belonging to Z) f : T -» X with supports contained in a set T„ from condition (B). We define the subspace Е^м of the space EM as the closure of L in EM with respect to A/-norrn topology in EM.

Note that E%i is well defined, that is, the dependence of E^M on {/} and {/„} from condition (B) is not essential (see [6], p. 266).

For every Ж -function M: T x X -> Re+ we define the complementary function M : T x X * -» Re+ (X* denotes the dual space of X) by the formula

M(r, x*) = sup [x*(x) — M(t, x): x e X ] for every teT, x*eX*.

1. RESULTS

1.1. T

. Let M be an Ж -function. Consider the following assertions:

(a) M satisfies the condition A 2 ; (b) l ( f /N (/)) = 1 for 0 ^ / e EM ;

(c) There do not exist norm-one functions f and g with disjoint supports such that N ( f +g) — 1 ;

(d) No closed subspace of EM is isometric to /x ;

(e) No closed subspace of EM is isomorphic to /x ;

(f) No closed subspace of EM is isomorphic to c0.

Then (a) =>(b) =>(c) and (f) =>(e) =>(d). I f X is separable and M(t, •) is continuous function on X for p-a.e. t e T {1), then (b)=>(a), (c)=>(a), and (d)=>(a). Moreover, if X — R and M satisfies condition (B), then (a)=>(f).

Thus, under all assumptions from this theorem all assertions (a)-(f) are pairwise equivalent.

P roof. The implications (f) =>(e) =>(d) are trivial. We now show that (a) implies (b). It is obvious that 1(f) ^ 1 if and only if N ( f ) ^ 1. Since N ( f / N ( f ) ) = l for О Ф f e L M, so I ( f / N ( f ) ) ^ 1. Let us assume that We have / ( 2 / /N ( /|) < oo by assumption (a). Choose a sequence {k„} such that 1/2 ^ kn < 1 and k„ 1 1 Then, we have

I ( f / K N ( f ) ) ^ I ( 2 f / N( f ) ) < oo and f(t)/knN ( f ) U ( t ) / N ( f ) for /i-а.е. t e T. Thus, by the Lebesgue bounded-convergence theorem, we have

ton H f / K N ( f ) ) = H f / N ( f ) ) < l . n~+ 00

Hence, we have l { f / k nQN( f ) ) < 1 for a number knQe ![l/2i 1) and thus N ( f / N ( f ) ) < k„Q < 1, a contradiction.

In order to prove that (b) implies (a) let us assume that (a) is not satisfied. If additionally dom M(t, •) = X for p-a.Q. t eT, then (see the proof of Theorem 1.6 [4]) there exists a function / eS(X) such that 1(f ) < 1 and N( f ) = 1, i.e. condition (b) does not holds.

Now let us assume that (a) is not satisfied and that the condition dom M(t, •) = X for д-а.е. t e T

is not satisfied. Let us write

В — I te T : dom M(t, •) Ф X] = {t e T: M(t, x) = oo for some xeX }.

Let \xk)f=l be a dense subset of X. We have, by continuity of M(t, •),

00 y 00 00

B = U {t eT: M(t, xk) = оo} = (J П { ^ Т : M(t, xk) ^ n) e l .

k = 1 fc= 1 n = 1

We define on the set В a multifunction H by H(t) = { x e X: M(t, x) = oo}.

Obviously, H(t) = 0 and H(t) are closed sets for every t e B. Moreover, by condition (a) from the definition of Ж -function, we have

Gr H — {(r, x ) e T x X : xeH(t )} = {(t, x): M(t, x) = oo}

= f) {(r, x): M(t, x) ^ n ] e l x

(M The continuity of M (t,-) we understood in the extended sense, i.e. limits may be infinite.

Thus, by Theorem 5.2 [3], there exists a strongly //-measurable function (selector) h: В -* X such that h(t)eH(t), i.e. M(t, h(t)) = x for every te B.

Now, we define a function k: B - + ( 0, o o ) :

k{t) — inf {/ > 0 : M (?, kh{t)) = x }.

Here is applied condition (d) from definition of the . 1 -function M. First, we shall prove that the function к is //-measurable. For, it suffices to prove that for every a > 0 the set

Qa = {teB: a ^ k(t)}

belongs to I . Let us write

An = {teB: M(r, (a — 1 /n)h(t)) < x } , Amn = {teB: M(t, (a-l/n)h(t)) ^ mj

for n, m — 1, 2, ... We have Amn c An for every m , n e N and (J Amn = An.

me.\

The sets Amn and An are //-measurable. Now, we shall prove that Qa — 0 An.

n

Let t e Q a and n e N . Then a —l/n < k(t) and hence M(t, (a—l/n)h(t)) < x , i.e. t e A„. Thus Qa <= p A„. Conversely, let t<£Qa, i.e. a > k(t). There exists

H c: , \

n0e l \ such that a — \/n0 > k(t). Hence M\t, {a— l/n0)h(t)) ~ x , i.e. t$A„0 and so t$ П An. Thus f) c and further Qa — f] A„ eI . This means

n n n

that the function к is //-measurable. Let us write g(t) = k(t)h(t). This function is strongly //-measurable and possesses the following property:

for every в > 0 and t e B , M{t, (1— c)g{t)) < x and M(f, (1 +e)^(r)) = x . Let C be an arbitrary measurable subset of В of positive and finite measure and let u(t) = Хс(г)д(0- Let {Ck)f=1 be a sequence of pairwise disjoint subsets of C of positive measure. Let 0 < gk < 1 for к = 1, 2, ... and gk 1 1.

We define the sets

Q =

\ t e C k: M(t, gkg{t))

к, neN.

We have Ck cz Cl for every k, m, ne N, m ^ n. Moreover, ( J Cl = Ck and

n

0 < lim //( Q = //(Q) < x for every ke N. Hence, it follows that for every rt-^OO

k e N there exists n(k)eN such that //(C£(k>) > 0. Further, for every k e N there exists Dk c= Cl(k\ Dke l , such that

g(Dk)> 0 and J M(t, tjkg{t))dg ^ 2~k~l.

We define

ж,

fk(t) = 4kXDk(t)g(t) for к = 1, 2, ... and f (t) = ^ f k(t).

k= 1

We have 1(f) < 1/2 and for every r, 0 < r < 1, 7 (//r) = oc. Thus N( f ) = 1.

This finishes the proof of implication (b)=>(a).

Now, we shall prove that (c) implies (a). For, let us assume that (a) is not satisfied. Then we may find (see the proof of Theorem 1.6 [4] in the case dom M(t, •) = X for g-a.e. t e T and the proof of the implication (b)=>(a) in the opposite case) functions / and g with disjoint supports and such that

1( f ) < 1/2, 1(g) < 1/2 and N( f ) = N(g) = 1.

Thus we have I ( f +g) < 1. Hence it follows that N( f +g) ^ 1. Since also N( f +g) ^ max (N(f ), N (g)) = 1, we have N ( f +g) = 1 a contradiction.

(b) =>(c). Let/ and g be two functions from S(X) such that N( f ) = N(g)

= 1 and let the supports of / and g be disjoint. By (b), we have / (/) = 1(g) = 1, and hence / ( / + #) = H f ) + I(g) = 2 and thus, by (b), it must be

N ( f + g ) > l .

(d)=>(a). For, let us assume that (a) is not satisfied. We shall consider two cases.

1° dom M(r, •) = X for /л-а.е. teT. Then EM contains an isomorphic copy of /x (see [4], Corollary 1.8).

2° Opposite to 1°. Then there exists (see the proof of implication (b)=>(a)) a sequence of functions from S(X) with pairwise disjoint supports such that I(fk) ^ 2 k and I( f j r) = oo for every к e N and 0 < r < 1.

We define on Г the operator P by

(Px)(t)= X ckf k(t) for x = {ck} e /x .

k = 1

We have

i ( p x m \ j = x /(c„/t/iMu « x / ( л к £ 2~k = i .

k = 1 k = l fc=l

Moreover, for every 0 < r < 1 there exists k0e N such that ckJ r M L > 1, and hence

I(Px/r\\x\D ^ I(ckQf kJ r IW IJ = 00 .

Thus, we have Л/^Рх/ЦхЦ О = 1, i.e. N(Px) = ||x ||, . This means that the operator P is an isometry of /°° onto P/00 c EM, i.e. condition (d) does not hold.

(a) =>(f). Assume that (a) holds and (f) is not satisfied, i.e. c0 is isomorphic to a closed subspace of EM. Let f„ eE E correspond to the unit vector e„ in c0, where the ;th component of e„ is 1 if j — n, and is 0 otherwise. Now, we shall find a separable sub-<r-field I ' of I such that each f n is measurable with respect to I'. For, let denote the set of all rational numbers and let

A (k, n)= { t eT: fn(t) < wk] ; к, n = 1, 2, ...

Let sé denote the smalest field containing the set {A(k, n)}kt„=1. Obviously, the field sé is countable. Next, we denote by I ' the closure of Ж with respect to the pseudometric

(A, В) = p(A — В), where A — В = (A \B) v { B \A ). It may be proved that I ' is a separable er-field. The separability of I ' is obvious. Let us observe that all functions/„, n = 1, 2, ... are measurable with respect to I '. This follows from the equality

{ t e T: f n(t) <r} = U {reT: f n{t) < wk},

wk ^r

for every n e N, r e R and hence that {t eT: f„(t) < w*J e l ' for every k, neN.

Since (T, I', fi) is separable under the pseudometric

(A, В) = p{A — В) and M satisfies the condition A2, so (see [2], Theorem 3.2) EM(T, I', R) is separable. Moreover, by condition (B) for M, EM(T, Г', R) is the dual space of Efo(T, Г, R) (see [6]).

It is known (see [1], Theorem 3) that every dual space X of a Banach space X containing an isomorphic copy of c0, contains also an isomorphic copy of /°°. Thus, if EM(T, I , R) contains an isomorphic copy of c0, then the space /°° is separable, a contradiction. This completes the proof of the implication (a) => (0-

R E F L E X I V I T Y O F EM

There holds the following (for the method of the proof see [5], Lemma 3.2)

T

1.2. If M: T x X Re+ is an Ж -function and both complement­

ary functions M and M : T x X * -* Re+ satisfy the condition A2 and (B), then the space EM is reflexive.

P

1.3. I f X is finite dimensional and M : T x X — ► P+ is continuous on X for p-a.e. t e T Ж -function satisfying the condition A2, then M satisfies condition (B).

Pr oof. Since dim (X) < oo, so the closed ball В = { x e X : ||x|| < 1} is a compact set in X. Let {T„'}*=i a sequence of subsets of T such that

GO

7^' > 7^+i, p(T„) < oo for every n e N and (J T„ = T. Let us define the sets n — 1

Bn = { t e T: sup [M (r, x): ||x|| < 1] ^ и}, n = 1, 2, . . .

Since В is a compact set and M( t , - ) is continuous for /x-a.e. teT, so 00 sup [M(f, x): ||x|| < 1] < oo for ^-a.e. teT. Hence it follows that p{ (J B„)

n— 1

= p(T). Further, we define the sets Tn = Гл' n f o r n e N. Since B„ c= B„+i for every n e N, so Tn c= Tn+1 for every n e N. Denoting

f {t ) = sup [M (t, x): I j x| j ^ 1]

and applying the condition A2, we have

sup [M (t, x): j|x|| ^ 2] = sup \_M(t, 2x): ||x|| ^ 1]

^ К sup [M (r, x): ||x|| < l] + /t(f)

= K f(t) + h(t).

Now, we may prove by induction that

k - 1

sup [M(r, X): 11 x| I ^ 2k] ^ K kf( t) + ( £ K l)h(t), k e N . i= о

It is obvious that \ f (t) dp < oo for every ne IS. Denoting

T _{1 n}

Ш = Kkf( t) + ( £ K ')h(t) 1=0

we have M (t, x) ^ f k(t) for ||x|| ^ 2k and so M (t, x) < f k(t) for ||x|| ^ k.

Moreover, we have f f k(t)dfj.< o o for every k, n e N . Thus, the proof is finished. тп

From the last Proposition it follows by Theorem 1.2 the following T

1.4. I f M: T x X -> R+ is a continuous .V -function such that its complementary function M acts from T x X* into R + and also is continuous, if dim (X) < o c and both Ж -functions M and M satisfy the condition A 2, then the space EM is reflexive.(2)

P roof. It is known (see Theorem 4.8 [ 6 ]) that the space (££,)* is isometrically isomorphic to E^{T, I , X) with Orlicz norm (for definition see e.g. [ 6 ] and [7]). Moreover, for every .Ж-function M satisfying the condition A 2 there holds the equality = EM. For further details of the proof see Lemma 3.2 [5].

References

[1] C. B e s s a g a , A. P e lc z y n s k i, On bases and unconditional convergence o f series in Banach spaces, Studia Math. 17 (1958), 151-164.

[2] L. D r e w n o w s k i, A. K a m in s k a , Orlicz spaces o f vector functions generated by a fam ily o f measures, Comment. Math. 22 (1981), 175-186.

[3] C. J. H im m e lb e r g , Measurable relations, Fund. Math. 87 (1975), 53-72.

[4] H. H u d z ik , Strict convexity o f Musielak-Orlicz spaces with Luxemburg's norm, Bull. Acad.

Polon. Sci. 29 (1981), 235-247.

[5] —, The problems o f separability, duality, reflexivity and o f comparison for generalized Orlicz- Sobolev space Wj^(Q), Comment. Math. 21 (1979), 315-324.

(2) The assumption that C-functions M and M satisfy the condition A2 is also necessary in order for the space EM be reflexive (see [7]).

[6] A. K o z e k , Orlicz spaces o f functions with values in Banach spaces, ibidem 19 (1976), 279- 308.

[7 ] —, Convex integral functionals on Orlicz spaces, ibidem 21 (1980), 109-135.

[8] J. M u s ie la k , W. O r lic z , On modular spaces, Studia Math. 18 (1959k 49-65.

[9] B. Tu rett, Rotundity o f Orlicz spaces, Proceedings o f the Koninklijke Nederlandse Akademie van Watenschappen, Amsterdam, Series A, 79 (5), December 13 (1976), 462-469.

I N S T I T U T E O F M A T H E M A T I C S , A . M I C K I E W I C Z U N I V E R S I T Y , P O Z N A N