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ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COM MENT ATIONES MATHEMATICAE XXVIII (1988) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXVIII (1988)

Wo j c ie c h

M.

Ko z l o w s k i

(Krakôw)

Notes on modular function spaces. II

Abstract. This paper is a continuation o f the article Notes on modular function spaces. I (the same issue). Section 6 deals with a d 2-condition defined for modular function spaces. The case of non-monotone convex function modulars is considered in Section 7. Section 8 is a brief exposition of some special cases and examples; it contains many references to the mathematical literature. Section 9 deals with countably modulared function spaces and Section 10 collects some results about interpolation in modular function spaces.

6. Subspace Lce and the condition A2

6.1.

D e f i n i t i o n.

By we shall mean a class of all functions f e L e such that g ( f •) is order continuous. The smallest linear subspace of L e which contains L° will be denoted by U6, i.e., / eUQ iff there exists a A > 0 such that

The following remark is an immediate consequence of Definitions 4.1 and 6.1.

6 .2 . Pr o p o s i t i o n.

Ee

cr c=

UQ and is a linear space if and only if E6

= L° — Tc

Modifying slightly the proof of the Vitali Theorem 4.3 we obtain the following lemma.

6 .3 . Le m m a.

Let f neL® and f n

-► 0

g-a.e.; then the following conditions are equivalent:

(i) Q(fn) - 0 ,

(ii) g{fn, •) are order equicontinuous.

6 .4 . De f i n i t i o n.

We say that g satisfies the A2-condition if and only if for each sequence (/„) c Lc6 the following implication holds: p(/„, •) are order equicontinuous implies g{2f„, •) are order equicontinuous.

It is easy to check that L°e is a convex and balanced subset of L e. If we assume additionally that L® is absorbing in Lc, then clearly Lc6 = Le. In fact, m many special cases these spaces are identical.

1980 Mathematics Subject Classification: 46 E 30, 46 E 40.

Key words: function spaces, modular spaces, Orlicz spaces.

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The next theorem is the main result of this section.

6 .5 . Th e o r e m.

I f L® is absorbing, then the following conditions are equiv­

alent :

(a) g satisfies the A2-condition, (b) L°

q

is a linear subspace of L e, (c) E

q

= L°e = Le,

(d) the modular convergence is equivalent to the F-norm convergence in Le.

P roof, (a) =>(b) Let/ eL° and Ek e l , Ek \ 0 . Since/ eL° then g ( f Ek) ->0 and, by the d 2“condition g(2fi Ek) ->0, which implies that 2/ eL° and consequently is linear.

(b) => (c) Evident.

(c) =>(a) Suppose to the contrary that there exists a sequence of func­

tions (/„) c Le such that g{f„, •) are order equicontinuous while g{2f„, ) are not. It follows from Theorem 5.6 in [6] that g(2f„, •) are not uniformly exhausting. Thus, passing if necessary to a subsequence we may assume that there exists a sequence of disjoint sets Dk e l and a constant r\ > 0 such that ô(2fk, Dk) > rj for every ke!S. Observe that g( fk, Dk) ->0 because g ( f k, Dk)

^ sup@(/„, Dk) — 0. The latter convergence is a consequence of the fact that n

sup g(f„, ■) is order continuous, hence, exhausting as well. Let (f k.) be a П

00

subsequence of (f k) such that £ g( fk., Dk) ^ 1, therefore, defining sm

m

i= 1

oo

= Z f k i h i ^ L c = Ee and note that g { f - s m)

i= 1 i= 1

oo

< X Qifki* Dk{) The function f therefore, is a member of Le = Ee.

i = m l 1

Indeed, let e > 0 be given arbitrarily; for 0 < ek < j , ek ->0, 2k = 2sk we have Q (ел f ) ^ Q ( k ( / “ sm)) + Q (K sm) < в ( / - sm) + Q (2k sm).

We may take m0 such that g { f — smQ) < e/2 and k 0 such that g{Àk sm)

< е / 2 for all к ^ k0. Thus, g(ekf ) < e for к ^ k0, i.e., / e L e = Ee. Since Le

— Ee is linear, it follows from / e E e that 2/ is also a member of EQ. It was observed above, however, that g(2f, Dk) = g{2fk, Dk) > r\. Therefore, g{2f, •) is a cr-subadditive submeasure which is not exhausting, i.e., g{2f, •) must not be order continuous. The latter means that the function 2 / does not belong to Ee.

This contradiction completes this part of the proof.

(a)=>(d) It suffices to prove that g{f„) ->0 implies g(2f„) ->0 for f ne L e

(see [23], p. 18). Assume, therefore, that f ne L e and g(f„) ->0. There exists a

subsequence (gn) of (/„) such that gn ->0 ^-a.e. (cf. Propositions 3.2 and 3.3).

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Modular function spaces. II 103

By Lemma 6.3, we conclude that g(gn, •) are order equicontinuous; therefore, g(2g„, •) are also equicontinuous in virtue of the d 2-condition. Using Lemma 6.3 again we get g{2gn) -►0. Hence, g{2fn) -*0.

(d)=>(a) L et/„ eL e and let g(f„, •) be order equicontinuous. Assume to the contrary that there exists a sequence of sets Eke Z such that Ek \ 0 and sup@(2/„, Ek) does not tend to zero. Choose an e > 0 and a subsequence

П

(gk) of (/„) such that g(2gk, Ek) = g(2gk l Ek) > s. On the other hand, Q(Gk, Ek) ^ sup < g{fn, Ek) -+0. By (d) then g(2gk, Ek) = g(2gk l Ek) -+0. Con-

П tradiction.

7. The case of non-monotone convex function modulars

In Definition 2.1 we assumed that the function modular was monotone with respect to the norm | • | in the Banach space S (property (P2)). We may raise a question whether the theory developed through the precedings sections can be applied to the case of non-monotone functionals which have some similar properties to those considered in Sections 2, 3 and 4. This problem is of great importance to us since many “function modulars” like those introduced by Turett in [34] or by Kozek in [14], [15] are non­

monotone. The more accurate question can be formulated as follows: is it possible (under some reasonable assumptions) to equip the Banach space S with the norm Ц-Ц^ equivalent to the previous one such that g will be non­

decreasing with respect to the norm Ц-Ц^?

In this section we shall demonstrate that for the convex case the answer is affirmative.

7.1a.

Definition.

A functional g: M ( X , S) x l -+[0,

oo]

is called a non­

monotone convex function modular iff

(a^ for every f

eM ( X ,

S), g(f, •): X -*[(),

oo]

is a tr-submeasure, (a2) g(-, A): M ( X , S) ->[0,

oo]

is a convex modular,

(a3) (P4), (P5) and (P6) from Definition 2.1 are satisfied, (a4) c\ = Ee cz Le с: M ( X , S).

7.1b.

Definition.

A non-monotone convex function modular g is said to be accurate if and only if there exists a partition (X{) of X (Хге/^, X t are mutually disjoint) such that the modular q>: S -*[0,

oo]

defined by (p{r)

= supg( rlx , X n) satisfies the following conditions:

n

(b j (p is continuous at zero in S, (b2) (p(r) < oc for every roS,

(b3) for each rl5 r2 eS such that <p(ri) ^ <p(r2) there holds <p(ar1)

^ cp(otr2) for all a > 0,

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(b4) for

/ e

M( X, S), E

e

X there holds g(f, E) = sup \g(g, E

):

gE

<?,

(p(g(x))

^ <?(/(*))

'for every

x e

E}.

Observe that on account of convexity of q> it follows from (b2) that for every r e S the function R э i—► q> (rX) is continuous. Let g be an accurate convex function modular, then we can equip S with a new norm || • defined by the formula

IMLp = inf {a > 0: <p(r/a) < 1], reS .

Since q> is continuous at zero, it follows that the modular space Sv introduced by (p coincides with S. Note that by (b3), the fact <p(rj) ^ (p{r2) implies ^ ||r2||v . For a set A

e

& put ft(A) = {r l A:

t e

S}.

7.2. Lemma.

For every

A e ?A

the set ft (A) is closed in L e.

Proof. Let us note that it suffices to prove that S \ f t ( A ) is open since S is dense in the closed set Ee. Let us fix a function f

e

S\ f t ( A) ; since / i s simple, it follows that its range is a finite set \r{, ..., rk) a S. Let ô > 0 be such that \\r1—ri\\(p> 2 0 for i = 2, k. Then for given reS there holds either Ц г - г ^ > 3 or ||r-r,||„ > <5 for i = 2, ..., k. Thus, ||r -/(x )||„ > ô or Ik-r/ILp > S for i = 2 , . . . , k . Thus, ||r —/(x )||v > Ô for all x e E or lk -/W I I „ > ^ for all

x e F ,

where E = f ~ 1{\r1 ]), F = f ~ 1(\r1, rk)).

Let us suppose, for instance, that the first possibility arises. Hence, inf<a > 0: cp r - f ( x )

for all

x e E .

Therefore, for all

x eE

there holds

/ r - / W V .

‘/ “ Г 1-

Let fe<p_1([l}). Then

j v (rlA( x ) - f ( x ) \ ^ \ v [rlA( x ) - f { x ) \

<x e X :

--- -

---j >

1> = x e l : <pl

--- ---

I

> < p { t )

By (ai), (a2) and (b4) we obtain

e

>

в E J > e ( t h , E ) >

o.

Denoting

k E = g ( t l E,

E

)

we observe that two possibilities arise: if

k E ^ 1,

then l/j —

f W y > 3 ;

if

k E < 1,

then for all

x e E

e(rj ÿ ) >e(-f L) >1 forallxe£-

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Modular function spaces. II 105

The last inequality implies that \\rlA—f\ \ e > S - k E.

Finally, \\rlA—f\\e > min(<5, S - k E, S - k F) > 0, where kF = g( tlF, F). Con­

sequently, S \ C 6(A) is an open subset of Le.

7.3.

Le m m a.

(S,

Ц-Ц^)

is complete.

Proof. Let (rn) be a Cauchy sequence in (S , Ц-Ц,,). Let A = X k for a certain k e N . Denoting /„ = rn 1A e T, (Â) we observe that we have for all

y . > 0

Q(<*(fn-fm)) = <p(*(rn- r m))-> 0 as n, m ^ x .

Le is complete, therefore, there exists a function / e L e such that \\ f„- fm\\e -►0. By Lemma 7.2, however, f (A) is closed in Le, then / = r l A for a certain r eS . Finally, (p(a (rn — r)) = д (a (/„ — /) ) -> 0.

7.4.

Th e o r e m.

The norm

Ц-Ц^

is equivalent to |-|.

Proof. Let us consider the identity map e\ (S , | -|) ->(S, || -Ц^). We shall prove that e is continuous. Indeed, let |r„| -*■ 0, then for each a > 0, |ar„| ->■ 0.

Hence, llrJI^-^0 and e is continuous. Since e is a linear isomorphism and both spaces (S, |-|) and (S, ||- | y are complete then they are isomorphic, by the open mapping theorem. This completes the proof of the theorem.

As a consequence of properties (b2) and (b3) we obtain the next result.

7.5.

Proposition.

There holds

||rx

^

||r2||^,

if and only if tp{r

J

< (p(r2).

P roof. Since (р{гх) ^ (p(r2) implies Н г^ ^ ||r2||v, it suffices to prove that cpfa) = <p(r2) if \ \ г Х = ||r2||„. Let = ||r2||v = a. By the continuity of the function Д э 2 i—► (Ar) for every r e S , we conclude that (p(rja)

= <p{r2/a) = 1 and by (b3) we obtain (р{гг) — q>{r2).

The following theorem is the main result of this section. It is an immediate consequence of Theorem 7.4, Proposition 7.5 and property (b4).

7.6.

Th e o r e m.

Let

q

he an accurate non-monotone convex function modu­

lar. Then

q

is monotone with respect to the equivalent norm ||-||v, i.e., (1) g{f, E) = sup \д(д, E): g e â , \\g(x)||„ ^ / (x)||* for all x e E ) , which implies also

(2) if f , g e M ( X , S) and ||gf(x)||v < ||/(x )||v for all x e E , then Q ( g , E ) ^ g { f , E ) .

8. Special cases

8.1. Musielak-Orlicz spaces may be regarded as modular function spaces in the sense of Definition 2.1. Following Musielak [23], we shall recall some basic concepts of the theory of Musielak-Orlicz spaces.

Let (X, I , p) be a measure space. Assume p to be a non-negative a-

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finite measure and denote by the <5-ring of all sets of finite measure. Let us consider a function qy. X x R + —> R + satisfying the following conditions:

(i) for every x e X , (p(x, •) is non-decreasing, continuous function such that cp(x, 0)

=

0, cp(x, u) > 0 for и > 0, (p(x, u) —>

oc

as и ->

oo;

(ii) (?(-, u) is a Г-measurable, locally integrable function for all и ^ 0.

It is easily seen that

(üi) Q (f, E) = .!> ( * ,\f{x)\)dft

E

is a function modular.

The modular space introduced by

q

is called the Musielak-Orlicz space I f .

Our Theorem 3.6 corresponds to Theorem 7.7 in [23], Theorem 4.3 (the Vitali theorem) corresponds to Lemma 9.2 in [23], our Theorem 4.6 (the characterization of Ep by means of simple functions) and Theorem 5.3 (the separability theorem) are similar to Theorems 7.6 and 7.10 in [23].

Let us remark that in the theory of Musielak-Orlicz spaces the subspace Ev (Eg in our notation) is called the space of finite elements, i.e., f e Ev iff

e № < oo

for all л > 0. Furthermore,

L% ( L g

in our notation) is absorbing;

For the modular given by formula (iii) the following d 2-condition is con­

sidered (see [23], p. 52):

(p(x, 2

m) ^

Ktp(x, u) + h(x

)

for all и

^

0 and almost every x e X , where h is a non-negative, integrable function in X and К is a positive constant.

Modifying the proof of Theorem 8.4 from [23], one can observe that this condition is equivalent to A 2 from our Definition 6.4.

Let us note that our compactness Theorem 4.7 is analogous to Theorem 9.3 from [23] giving similar characterization of compact subsets of Ee.

8.2. L. Drewnowski and A. Kaminska introduced in [7] a modular / ф defined on the space of measurable vector-valued functions by the formula

1Ф( Л = sup (/(* ), x)dp,

цеМ x

where Ф: S x I - » [ 0 ,

oo]

is an -function (see [7], p. 178) and M is a family of countably additive non-negative measures on X.

If we assume additionally that Ф is non-decreasing, i.e., Ф{ги x) < Ф{f2, x) if Irj| ^ |r2|, then / ф may be regarded as a function modular in the sense of Section 1 of our paper.

Our Theorem 3.6 and Proposition 3.2 correspond to Theorem 2.2 in [7], Theorem 4.7 is similar to Theorem 1.2 from [12], our Theorem 5.3 to Theorem 3.1 from [7].

Observe that under some supplementary assumptions on the family M

(7)

Modular function spaces. 11 107

of measures the assumptions of monotonicity of Ф may be omitted and the methods of Section 7 may be applied.

The reader is referred to [7] for interesting list of examples. We want only to stress that Musielak and Waszak in [26] (see also [7], p. 120) investigated similar generalization of Orlicz spaces. We shall deal with such spaces in Section 9.

8.3. As it was stated, Fenchel-Orlicz spaces introduced by Turett in [34] may be regarded as accurate non-monotone convex function modular spaces in the sense of Definition 7.1. The reader is referred to [34] for the definition of Fenchel-Orlicz spaces; we would only note that in Turett’s main theorem (Theorem 2.20) there are included special versions of our Proposition 3.2 and Theorem 3.6. Some of ideas from [34] were employed by us in proving results of Section 7.

8.4. Dobrakov considered in [4], [5] linear operator valued measure m, i.e., m: Y) is countably additive in the strong operator topology.

n

For a simple function / = £ r, 1E. (r,eS, £ fe # ) the integral was defined

i —

1

n

there as follows: \ f d m = £ m(£ n £ ()r,-, where E belongs to I . Then the

E i

= 1

domain of integration was extended to the space Jt(m) of all f e M ( X , S) such that there exists a sequence of simple functions (/„) converging m-a.e. to f, for which the integrals ( [ /„ dm) are uniformly countably additive on I . In

(‘•>

Part II of his paper Dobrakov defined “the Lj-norm” on M ( X , 5) by m(g, E) = sup { | | | / d m ||: /

e

A, | / (

x

)| < |g(x)| for each x e E \ . Theorem 1 in

E

[5] states that m is a function modular in the sense of our paper. Our Proposition 3.3 corresponds to Lemma 4 from [5], the space Тх(т) in the sense of Dobrakov plays an identical role to EQ in the theory presented here, our Theorem 3.6 corresponds to Dobrakov’s Theorem 9, the Vitali and Lebesgue convergence theorems correspond to Theorems 16 and 17 and our Theorems 4.6 and 5.3 to Dobrakov’s Theorems 8 and 20, respectively.

Theorem 5.4 at last corresponds to Theorem 19 from [5].

8.5. Integration with respect to non-linear operator valued measures was considered by many authors in connection with representation of ortho­

gonally additive operators (see e.g. [9] and [1]). We use here terminology and notation taken from recent papers [16] and [32].

Let f be a Banach space. By N ( S , F ) we denote the space of all

mappings U: S -* F such that U0 = 0 and U are uniformly continuous on

bounded subsets of S. A set function pt\ fP^>N( S,F) is said to be an

operator valued measure if /л has the following properties:

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(Mi) /л{0) = 0,

(ц2) И is countably additive in the point-wise sense, (p3) for each Е

е

âP and a > 0 svô(na, E ) -> 0 as <5 0,

(p4) for each r e 5 the submeasure majorant for /i denoted by

ft

is order continuous on I .

Recall two definitions:

svô{pia, E) = sup ||| X In (£,) r,- - n (£,) r(-]|| : U £, c £ , Et e

/ = l < = i

W, |r;| ^ a, |rf — r;| ^ S, 1 ^ i < n, n e N J

n(E)r = sup !||/1(Л)г||: A c E, A e . J/\ .

Integration of simple functions and extension of integration to the class can be done similarly as in the linear case. Defining then

g(g, E) = sup !||j'/d/i||: f e t , \f{x)\ ^ \g{x)\ for each x e E )

E

and assuming additionally

(p5) if E e l , {fdf.i = Q for all f e 6 such that |/ ( ,\ ) |^ a in E, then

E

E is /i-null,

we obtain a function modular g. Properties (Px), (P2) and (P3) follow immediately from the definition of g while (p3) implies (P4), (p5) implies (P5) and property (P6) is a consequence of (p4).

The above introduced integral may be regarded as a non-linear operator J/(n)E>f \ f dfieF. From this point of view we note two interesting

x

properties of the space Ee and of the class L°.

8.5a. T

heorem

.

L°6 cz , Л { ц ) .

Proof. We claim that

\ \ \ f d/x||

^

g ( f ,

E) for every /

e ^ { j x )

and E e l . i:

Since/ E.//(n), it follows that there exists a sequence (s„) of simple functions such that s„ (x )-► / (x) and |s„(x)| ^ |/(x )| for all x e X and \snd n ~ + \ f d n .

E E

For given e > 0 we can choose a number n e N such that

||J' f d f i - f $n dfx\I < £-

E E

Hence,

\ \ { f dn\\ ^ £ + ||j’s„dfx\\ ^ e + g(f, E),

к к

which implies that ||f / dg|| ^ g( f, E) because e was chosen arbitrarily.

к

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Modular function spaces. II 109

Assume now that / e L °

ci

M ( X , S); let (s„) be such as it was mentioned above. Let EkGl , Ek \ 0 . For every n, к

g IS we have then

Il f sndn|| ^ g{sH, Ek) ^ g(f, Ek), 4

therefore, sup || ^ g(f, Ek) ->0 because /

gL°.

Thus, (|' sndp) are order

» к ('■)

equicontinuous and consequently /

e

. //(/d (cf. [32], Theorem 2.5).

8.5b.

Th e o r e m.

The integral is a continuous operator acting from Ee into F, i.e., if /„, /

e

E

q

and \fn- f \ Q ->0, then, for every E e l , [ f nd p ^ { f d p .

к к

Proof. Let E

g

Z, /„, /

g

E

q

and \fn—f \ e -*0. Then, by Propositions 3.2 and 3.3, there exists a subsequence (gn) of (/„) such that g„~* f

q

-a.e. By the Egoroff theorem, there exists a sequence (Hk) such that HkGf?, H k s' E and gn f on every Hk. For given s > 0 let us choose a natural number к such that g(2f E \ H k) < s/S and an A^

g

N such that there holds g(2(gn—f ) ) < e/8 for n ^ N t . Thus,

g(g„, E \ H t) « e (2 (» „ -/), E \ H t ) + g(2f, E \ H k) <s/4.

Let us consider the following inequalities:

\\\gnd p - { f d p \\ ^ \ \ \ gnd p - \ f dp\\ + \\ j gndp|| + || j f dp||

E E H k Hk E \ Hk E \ H k

« || f gnd n - f fdg\ \ + Q(g„ E \ H k) + e (f, E \ H k)

Hfc «к

< || f 9nd p — j / dp\\ + s/4 + e/& ^ || ( gnd p - J' fdp\\ + e/2

Hk Hk нк Hk

for n ^ ISk.

Let ( GJ be a sequence of sets from & such that Gm ? Hk and f l Gm are bounded for all m. Similarly as it was done above we can take an m and JV2

^ Ni such that g(2f, Hk \ G m) < e/4 for n ^ IS2. Since gni l f on Gm and all are bounded on Gm, it follows by (p3) that

|| f g „ d p - j fdp\\ < £ for n ^ iV3 ^ iV2 ^

dm Gm

Hence,

Il .f gnd p - j f dp\\ < £ + || J Ôf„^|| + || j f d p \ \ ^ s + s/2

H k H k H k \ Gm H kSGm

and finally

||\ g nd p - I f dp\\ < 2s.

к к

Since (/„) was an arbitrary sequence converging to / in Ee, it follows that

the integral is a continuous operator.

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Since many classical operators like Urysohn, Hammerstein or Nemytskii ones may be regarded as integrals with respect to suitable operator measures then Theorem 8.5b gives an answer to the question how to construct an F-space in which the given operator is continuous.

9. Countably modulared function spaces

Let (g„) be a sequence of function pseudomodulars, i.e., g„: M ( X , S ) x F ^ [ 0 , oo]

which satisfy conditions (Pt), (P2), -(Рз), (P4) and (P6). Assume that f ( x ) = 0 for all x

e

E, if Q„(f, E) = 0 for all n = 1 , 2 , . . . Following [26], [27], [23]

let us define

Q ( f ,

£) = I 2 - n= 1

Qn( f , E )

1 + <?.(/, £)*

в о i f ,

E) = supe„(/, F);

n

the convention

oo/(l + oo)

= 1 has been used here. Then Le and Leo are called the countably modulared space and the uniformly countably modulared space, respectively.

It is well known ([23], Theorem 15.2) that both

q

and

q

0 are modulars in oc

M ( X , S), L e = 0 L e. and Leo c L g. This embedding is continuous both with

i = 1

respect to modular convergence and F-norm convergence. We are, however, interested in a question, whether

q

and g0 are the function modulars in the sense of Definition 2.1.

9.1.

Th e o r e m.

The introduced above modular g has properties (PjHPe)- Proof. It follows immediately from properties of gn and of the real function [0, o o ) 3 t ^ t ( l + t ) ~ \ that g satisfies conditions (Pj), (P2), (P3) and (P5). In order to prove (P4) and (P6), we shall check first that

(9.1a) Indeed,

■À 1 + ( ё Л ( £ )

ga(E) = sup \g(g, E): g

e

S, \g(x)| ^ a for all

x e

E\

Q„(g,E) + Qn(g, E)

= sup<X 2 niT :' W/ cr for a11

X ^ E

(„=! 1 + gn(g,E)

<

£ 2 ~ n sup 17-7—7 : g e t ,

\ g ( x )

| for all

x e E

„=1

(1 + gn{g,E)

< у 2 - и ( Ш Е )

1Н Ш Е ) '

(11)

Modular function spaces. II 111

The last inequality is caused by the fact that the function 11—>f(l -K)-1 is non- decreasing. Since gn is a function pseudomodular, for every n separately (@n)a (E) —* 0 as a - > 0 +. Clearly,

1 2 - (вп)ЛЕ)

1 + (f?n)a(E) 0 as at -*■ 0 + ;

hence, by (9.1a) we obtain that ga(E) —> 0 as a 0 +. This completes the proof of (P4); (P6) may be proved similarly.

The next theorem deals with uniformly countably modulared spaces.

9.2. T

heorem

.

q

0 is a function modular if and only if the following conditions are satisfied, simultaneously:

(a) sup(Qk)a(En) -+ 0 for all a > 0, Ene # , En \ 0 , к

(b) sup(Qk)a(E) -о 0 as ot->0+, Ee0>.

к

It is clear that £0 satisfies conditions (РД (P2), (P3) and (P5) while (P4) and (P6) are assumed in (a) and (b).

An answer to the question under which conditions both spaces Le and Leo are equal may be formulated as follows:

9.3. T

heorem

. L e = L QQ if and only if condition (a) from Theorem 9.2 holds and also

(b') sup Qk (A„ f , E) —* 0 for Xn -+ 0 +, f e L e, E& .9.

к

We omit the easy proof and observe that (b') implies (b) therefore, the following theorem holds:

9.4. T

heorem

. I f L e = L eo, then

q0

is a function modular.

Let us note that there are examples of {gk)fL 1 such that = SUP Qk is a к

function modular while L Q

q

Ф L e.

9.5. E

xample

. Let m denote the Lebesgue measure in X = [0, 1) and let

<Pi(x) = |x|; for k > 2 put

i 0, x e [ 0, 1],

(pk(x) = л 2(m~ 1)2, x e [ m —l , m ) , 2 ^ m ^ k, ( 2kl, x ^ k

00

and let gk{f, E) = f (pk( f (x))dm. Put f = Y, nh ■ > where /„ с X are mutual-

E n= 1

ly disjoint and m(/„) = 1/ 2".

CO

We claim that / e L e = 0 Евк. Given a sequence ym ->0+, let us choose

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a subsequence (Я7) such that 2 ,^ 1 //. Then we have Qk (A//) = Л,- £ n

n= 1

00

m (/„) ^ cÀj -» 0, where c = £ n/2" < oo and for к ^ 2 n= 1

00

00 00

ft (Aj/) « ft (//У) ^ 2* I m(/„) = 2* • X 2 -” - 0 as

j

- со.

n = j n = j

We used here the fact that for n < j and i e / „ there holds f (x )/ j = n/j < 1 and, consequently, cpk( f (x)/j) = 0. On the other hand, for n ^ kj, к ^ 2, x e/„

there holds / (x)// = и// ^ k. Thus, (pk( f (x)//) = 2k2 and

00 00

supgk(f/j) > sup2fc2 X m(/„) = sup2fe2 £ 2'"

fc n= к j к n = k j

— sup2k2-2 -2 _fc7 = sup2fc2_fcj + 1 =

+ o o .

к к

Hence, / does not belong to Leo.

10. Interpolation of modular spaces

In this section we shall give an outline of an application of the Krbec interpolation method [18] to modular function spaces. Books [2] and [33]

deal with general interpolation theory.

For the sake of simplicity we shall restrict our consideration to the case S = R and convex function modulars g0, gl . Let a: (0,

oo)

->(0,

oo)

be a

00

measurable function such that | m in(l, t)o(t)dt <

oo.

For given g0 and g± we

b

construct a cr-interpolated convex modular by the formula 00

(10.1) f t ( / ) = J ^ (t,/)< 7 (t)* ,

b

where £F{t,f) = inf !e0(/o) + tei (/i): / = f o + f u fi e L e.}.

GO

We shall prove that ga{f, E) = j' < f { t , f l E)o{t)dt satisfies conditions

b

( Р ^ Р Д i.e., it is a function modular.

Let us start with three preliminary results.

10.2. P

roposition

. For every E e l we have Qa(0, E) = 0.

P roof. Observe that for given E e l , 01Ee L ei and 01Ee L QQ, therefore, 0 ^ SF{t,

01 E)

^

Q0 (0 1 E) +

tQl (

01

E) = 0o(O>

E ) +

tQx (0, E).

Hence, j^ ( t , 01E) = 0 and consequently, E) = 0.

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Modular function spaces. II 113 1 0 .3 .

P

roposition

. I f

| / ( x ) | = |g ( x ) |

for all x e X , then t , f

) =

fë(t, g

)

for all t > 0. In particular, there holds SF(t,f) = fëit, |/|).

Proof. Define E t = JxeX: f ( x ) = g(x)}, E 2 = \ x e X : g(x)^=0 and f{x)/g{x) — —1}; then £j u £ 2 = X, E l r \ E 2 = 0 , JEi and E 2 belong to X.

Let / = / 0+ /i, f e L e. and put gt = f l El - f I El (i = 0, 1).

Thus, gf eL e. and go + 0 i = 9- Observe that gfh) — gAj) for measurable functions h and j such that \h\ = \j\, therefore, go(0o) + ^ i (9i) = Qo(fo) + + ^ i( /i) - Hence, ^o(/o) + ^ i ( / i ) ^ ôf) and taking infimum over all partitions / = / 0+ /i, f ^ E e., we have t , f ) > f £ {t, g). The inverse in­

equality may be obtained similarly. Finally, ë ë ( t , f ) = f£(t,g).

1 0 .4 .

P

roposition

. Let for each x e X , 0 ^ f ( x ) < g(x); then

Q A f , X ) ^ Q 0(g,X).

Proof. It can easily be proved that to every partition h = ho + hx of a non-negative function h e L eo + Lei there corresponds a partition h = h0 + hl such that h(e Le., h{ ^ 0 and {?,•(/*,) <

q

AK). Hence, it suffices to take an arbitrary partition g = g0+ 9 i , 0,-eLei and g, ^ 0. Then, we have the fol­

lowing partition of the function/: / = f 0 + f , f 0 = min(g0, / ) , / i = f ~ f 0- Let us note that 0 < / < gt for i = 0, 1 and, therefore, / eL e . We have

■ ¥ { t , f ) ^ Q o ( f o ) + t Q i ( / i ) ^ Q 0 { 9 o ) + t Q i Ы ;

hence, ë ë ( t , f ) ^ f f {t , g) for all t > 0 and finally, e A f ) < ôAg)-

1 0 .5 .

T

heorem

. For every f е ё the set function ga{f,

- ) : X -*

[0, oo] is a о-subadditive submeasure.

P roof. It follows immediately from the definition of ga and from Proposition 10.4 that g A f , 0 ) = 0 and that g A f , A) = g A f 1

a

) ^ Q A f h )

~ Qaif-, B) for A, B e X such that A с B.

We claim that ga(g) is <T-subadditive on X for every fixed function g е ё . 00

Let us denote A = \J A n, where A„eX are mutually disjoint and f = g l A.

n= 1

For every n e IS we may choose an arbitrary partition of f l A i f l A - i f 1

a

)

o

+ + ( f l A) i such that ( f I A) t e L e. and ( f l A)i ^ 0 (i = 0, 1). Let us define the

00

function / by / = £ ( f l A)t. Let us note that f e L e.. This follows from the

n= 1

Lebesgue dominated convergence theorem for gt and from the fact that sk(x)

к

= £ ( f l A ) i ( x) ^ f i ( x) and 0 ^s„(x) while the function f e ë c z L e. for i

n= 1

= 0, 1. Since clearly / = / 0+ /i, then by <r-subadditivity of g0 and gk we obtain

8 — Roczniki PTM — Prace Matematyczne XXVIII

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^ « ,/ )«£ > o(/ o) + fÉ h (/ ,K I М (Я л „ )о ) + 'М (/Ч>1Й -

П= 1

Thus, Z ¥ ( t , f l AJ and

n= 1

0<r(0> Л) = вЛЛ

= j ^ { t j ) a { t ) d t ^ ]'( Z X ' ( t , f l AJ)a(t)dt

О О n~ 1

GO OO 00

= I ( f ^ ( f , f l A ) a ( t ) d t = X < ? „ ( / , Л „ )

n = l 0 n = l

00

= Z ÔÀ9, Л ).

и= 1

00

Finally we have: A) ^ Z which is the desired result.

n= 1

r 10.6. L

emma

. There exists a constant 0 < c < oo such that for every a > 0 and E

g

& there holds

(ôX( E) ^ с т а х (&)*(£).

i = 0,1

P roof. Fix a > 0 and Ee0>. Since

ol1e g e

it follows that (ga)a{E) 00

= Qa(odE) = f <f{t, (x.lE)o{t)dt. It can be easily seen that

b

& { t t a 1E) ^ t(ei)AE)

(Qo)AE)

for 0 ^ < 1 , for t > 1, m in(l, r)(6i)«(£)

m in(l, г)(ёо)«(£) < m in(l, t) max (@,)a(.E).

i= 0,1 Thus,

( Ш Е )

00 00

fA?{t, a l E)o(t)dt < max

(Qi)a ( E) -

f m in(l, t)o{t)dt.

Ь 0,1 b

00

Since J m in(l, t)(r(t)dt = c < oo then finally о

Ш А Е ) < с т а х{ д{)АЕ).

i = 0,1

The lemma is completely proved.

From Propositions 10.3, 10.4 and Theorem 10.5 it follows immediately

that

q

„ satisfies properties (PJ, (P2) and (P3). From Lemma 10.6 it follows

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Modular function spaces. II 115

that Qa satisfies also (P4), (P5) and (P6) since g{ (i = 0, 1) have those properties. Thus we proved the following theorem.

10.7.

Th e o r e m.

I f g0, are function modulars, then (Leo, L Qi)a = L Qa, where ga is given by (10.1); ga is a function modular in the sense of Definition

2.1.

10.8.

Th e o r e m.

I f

q0 , д г

are orthogonally additive, then ga given by (10.1) is orthogonally additive as well.

P roof. It suffices to prove that ga( f l A) + ga( f I B) < вЛЛ for A, B

e

I such that А г л В — 0 and 4 u B = X. Let / = / 0+ /i, f , e L e.. Then f l A

= / 0 1

a

+ /

i

^ and f l B — fo l B+ /i 1

b

- Let us fix a t > 0. Compute

■^(C/^/i)+ ^ ( L / 1

b

) ^ Qoifo I f f + tQ! {fi l A) + go(fo 1

r

) + ^

i

( /

i

= Q o ( f o ) + t Q i ( f i ) -

Thus, ^ , { t , f l A) - \ - f f ' { t , f I f ) ^ f f /{t,f), which gives the desired inequality:

e A f U + e , ( f i i ) * e . i f ) -

References

[1] J. B a tt, Nonlinear integral operators on C{S, E), Studia Math. 48 (1973), 145-177.

[2] J. B erg h and J. L o fstr o m , Interpolation Spaces, Springer-Verlag, A Series of Compre­

hensive Studies in Mathematics 223 (1976).

[3] Z. B ir n b a u m and W. O r lic z , (Jber die Verallgemeineruny des Begriffes der Zueinander Konjugierten Potenzen, Studia Math. 3 (1931), 1-67.

[4] I. D o b r a k o v , On integration in Banach spaces I, Czech. Math. J. 20 (1970), 511-536.

[5] —, On integration in Banach spaces II, ibidem 20 (1970), 680-695.

[6] L. D rew n o W sk i, Topological rings o f sets, continuous set functions, integration I, II, Bull.

Acad. Sci. Ser. Sci. Math. Astronom. Phys. 20 (1972), 269-286.

[7] — and A. К a m in ska, Orlicz spaces o f vector functions generated by a family o f measures, Comment. Math. 22 (1981), 175-186.

[8] N. D u n fo r d and J. S c h w a r tz , Linear Operators, Part I, Interscience, New York 1958.

[9] N. F r ie d m a n and A. E. T o n g , On additive operators, Canad. J. Math. 23 (1971), 468-480.

[10] P. R. H a lm o s, Measure Theory, D. Van Nostrand, New York 1956.

[11] H. H u d z ik , Musielak-Orlicz Spaces Isomorphic to Strictly Convex Spaces, Bull. Acad.

Polon. Sci. Ser. Sci. Math. Astronom. Phys. 29 (1981), 465-470.

[12] A. K a m in sk a , On some compactness criteria for Orlicz subspace Еф(й), Comment. Math.

22 (1982), 245-255.

[13] —, Strict Convexity o f Sequence Orlicz-Musielak Spaces with Orlicz Norm, J. Funct. Anal.

50 (1983), 285-305.

[14] A. K o z e k , Orlicz spaces o f functions with values in Banach spaces, Comment. Math. 19 (1976), 259-288.

[15] —, Convex integral functionals on Orlicz spaces, ibidem 21 (1979), 259- 288.

[16] W. M. K o z l o w s k i and T. S z c z y p ir is k i, Some remarks on the non-linear operator measures and integration. Coll. Math. Soc. Janos Bolyai 35 (1984). 751-756.

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[17] M. A. K r a s n o s e l s k i î and Ya. B. R u tic k iî, Convex functions and Orlicz spaces (in Russian), Moscow 1958.

[18] M. K rb ec, Modular Interpolation Spaces I, Zeitschrift für Analysis und ihre Anwendungen 1 (1982), 25-40.

[19] J. L in d e n s t r a u s s and L. T z a fr ir i, Classical Banach Spaces I, Sequence Spaces, Sprin­

ger-Verlag, A Series of Modern Surveys in Mathematics 92 (1977).

[20] —, —, Classical Banach Spaces II, Function Spaces, Springer-Verlag, A Series of Modern Surveys in Mathematics 97 (1979).

[21] W. A. J. L u x e m b u r g , Banach function spaces, Thesis, Delft 1955.

[22] — and A. C. Z a a n e n , Riesz spaces, Amsterdam 1971.

[23] J. M u s ie la k , Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics 1034, Springer-Verlag, 1983.

[24] — and W. O r lic z , On modular spaces, Studia Math. 18 (1959), 49-65.

[25] — and W. O r lic z , Some remarks on modular spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math.

Astronom. Phys. 7 (1959), 661-668.

[26] — and A. W a sz a k , Countably modulared spaces connected with equisplittable families o f measures, Comment. Math. 13 (1970), 267-274.

[27] - , —, Some new countably modulared spaces, ibidem 15 (1971), 203-215.

[28] H. N a k a n o , Modulared semi-ordered linear spaces, Tokyo 1950.

[29] W. O r lic z , (Iber eine gewisse klasse von Rdume vom Typus B, Bull. Acad. Polon. Sci. Ser.

A (1932), 207-220.

[30] - , Über Rdume LM, Bull. Polon. Sci. Ser. A (1936), 93-107.

[31] R. L. R o s e n b e r g , Orlicz spaces based on families o f measures, Studia Math. 35 (1970), 16- 49.

[32] T. S z c z y p in s k i, Non-linear operator valued measures and integration, to appear.

[33] H. T r ie b e l, Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin 1978.

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DEPARTMENT O F MATHEMATICS AND COMPUTER SCIENCES JAGIELLONIAN UNIVERSITY, KRAKOW

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