Continuity of the identity embedding of some Orlicz spaces (I)Abstract. Let

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIV (1984) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXIV (1984)

Ma r e k W i s l a ( P o z n a n )

Continuity of the identity embedding of some Orlicz spaces (I)

Abstract. Let E (M) be an Orlicz space of functions with values in a real linear-topological space X . We obtain necessary and sufficient conditions for the inclusion E ( M ) <= E (M) to hold.

The above problem has been studied by Matuszewska [5], Ishii [1], Kozek [4], and others with some different assumptions than we use here. Finally, we consider here the continuity of the mapping i : I f (M) -> E (M) such that for every f e E ( M ) .

1. Introduction. Let (T, Z, /*) be a measure space, where T is an abstract set, Г is a (т-algebra of subsets of the set Т,ц is a positive, or-finite and complete measure on Z.

(X, t) will denote a real linear-topological space. В will always be a base of neighbourhoods of 0 , where © is the origin of X . Moreover, we shall denote by A the smallest tr-algebra of subsets of X containing t.

Let M be the set of all functions / : X such that f ~ 1(U )e Z for every

CO

U e A . It is obvious that every function of the type ]Г xk Хлк belongs to the set

к = 1

M, where {xk: k e N } c= X , {Ak}keN is a family of measurable and disjoint subsets of T and Xa denotes the indicator function of the set A. We shall describe by M an arbitrary linear subset of the set M.

Definition1.1. A function Ф : 1 х Т - > [0, + oo] is said to be a Ф-function if there exists a set T0e Z , fi(T\T0) = 0 such that

(a) Ф is A x Z measurable ; (b) Ф (0, t) — 0 for every t e T0 ;

(c) Ф(х, t) = Ф( — x, t) for every x e X , t e T 0 ;

(d) Ф(их + ьу, t) ^ Ф(х, г) + Ф(у, t) for every и, v ^ 0, u + r = 1, x , y e X and t e T 0 ;

(e) Ф(\ t) is lower semi-continuous on X for every t e T 0, i.e. for every a < Ф(х, t) there is a set V eB , x e V, such that a < Ф(у, t) for all y e V.

À function Ф : X x T-> [0, -F oo] is called a convex Ф-function if it is а Ф- function satisfying the condition

(1.1) Ф{ux + vy, t) ^ иФ(х, г) + иФ(у, t)

for every u ,v ^ 0, и + » = 1, x , y e X and t e T0.

R e m a rk s , (i) A. Kozek has investigated Orlicz spaces of functions with values in a Banach space X , generated by an N"-function Ф: X x T

- [ О , + oo] (see [3], [4]). Let X be a Banach space. Then every convex Ф- function Ф is an iV"-function.

(ii) If X is the space of real numbers, then every continuous (i.e. for every e > 0 and x e X such that Ф(х, t) < oo there is a set VeB, x e V , such that IФ(х, t) — Ф(у, r)| < £ for all y e V ) Ф-function Ф with finite values which satisfies the condition

(1.2) V (Ф(х, t) = O o x — 0 )

t e T 0

is a well-known (^-function with a parameter which has been considered in many papers, for example in [1] and [7].

We shall denote by 1Ф the functional from M into [0, + oo] defined by 1ф( Л = t)dn. Let us note that 1Ф\М: M

->[

+

/ф|м(/) = /ф(/),

is a pseudomodular on M (see [7], [8]).

г

By the Orlicz space I f (M) we mean the set of all functions f e M such that U a f ) < +oo for some a > 0.

The function | • |ф defined by

(1.3) | / | ф = inf{w > 0: / ф(м- 1 / ) ^ u}

is an F-pseudonorm on lf(M ). If Ф is a convex Ф-function, then we can define a pseudonorm on I?(M) in the following way

(1.4) ||/ ||ф = inf{w > 0: 1}.

R e m a rk . Suppose that X = R, Ф satisfies (1.2) and M = M. Then we can consider this Ф-function as a family of modulars depending of parameter:

(p(t, x) = Ф(х, t), the functional / ф as a modular £s and the space I f (M) as a modular space X Qs (see [2]).

For other properties of the Orlicz space I f (M) (in the case where Ф is а Ф- function with finite values) see [9].

In the sequel we shall use the following notations (Ф, Ф are Ф-functions) : (1.5) d o m Ф(*, t) = ( x e l : Ф(х, t) < +oo}, where t e T ;

(1.6) dom 1Ф = {f e M : 1ф( /) < +oo};

(1.7) Р И)С(х) = (f e T: u4*(cx, t) > Ф(х, t) and Ф(х, t) < +oo}, where u , c > 0 and x e X ;

(1.8) a„,c(0 = sup Ф{cx, t)xP ,x){t), where u ,c > 0 , te T ;

x e X

(1.9) ' p u{t) = au>1(r) for every и > 0 and re T;

(1.10) yu{t) = ctUtU(t) and Pu(x) = Pu>u{x)

for every и > 0, t e T and x e X .

Identity embedding of some Or liez spaces 345

2. The inclusion I?(M) a Ü (M). First, we shall prove some lemmas.

Le m m a 2.1. I f X is a separable space, Ф, W are Ф-functions and Ф(-, t) is continuous on X for almost every t e T , then the function aUtC is measurable and (2.1) ctu>c(t) = sup Ф(cyk,t) ip ( } (f)

k e N

for almost every t e T , where the set {yk : k e N } is dense in the space X . P ro o f. The measurability of the function auc follows from (2.1). To prove the lemma it suffices to show that

(2.2) aUtC (f) ^ sup V (cyk, t) Xpu с(Ук) (t)

k e N

for almost every t e T . First, we prove that if t e P uc(x) for some x e X , then (2.3) V 3 yke x + V and t e P UtC{yk).

V e B k e N

Let t e P uc(x). Then 4*(cx,t) > u ~ x Ф{х, t) + a for some a > 0. The lower semicontinuity of W(-, t) implies that there is a set U e B such that T (z ,f) > и * 1 Ф(х, t) + a for every z e c x + l f . Moreover, there is a set We В such that Ф(г, t) < Ф{х, t) + au for every z e x + W by the continuity of Ф{-, t).

Thus Ф(г, t) < + oo for every z e x + (U n V n W ) . Since the space X is separable yfce x + (c_1 U n V n W) for some k e N . Hence Ф{ук, t) < +oo and

T (cyk, t) > u ~ 1 Ф(х, t) + a > u ~ l Ф(ук, t).

Thus condition (2.3) holds.

It is obvious that otu>c(t) = 0 implies (2.2).

Let us suppose that att(C(t) = + oo. Then there is a sequence {x„} such that t e P uc(x„) and ^ ( c x ^ t ) > n for each n e N . By the lower semicontinuity of T(-, t) we obtain the existence of a collection of sets {Vn} such that T (z, t) > n for every z e c x n + V„. Now, (2.3) implies that for every n e N we can find ykn such that ykne x n + c ~ l V„ and t e P u>c(ykt). Hence T(cykn, t) > n and t e P u>c(ykJ. So, inequality (2.2) holds.

If 0 < otu>c (t) < + oo, then there is a sequence {x n} such that t e Pu c (x„) and T (cxn, t) > otu>c{t) — n ~ x for each n e N . Since T(-, t) is lower semicontinuous, for every n e N there is a set V„eВ such that W{z, t) > otu<c(t) — n-1 for every z e c x n+ Vn. Therefore, by (2.3), we can find a sequence {ykn} such that ykne x n + - f c 1 Vn and t e P U'C(ykJ for each n e M . Thus

au.c (0 < ^ (Cyk„, t) Xpu c(yk ) (t) + 1 /n

and in this case inequality (2.2) holds. The proof is finished.

Since the measure p is cr-finite, there is an increasing sequence {7^: n e N }

of sets of finite measures such that (J Tn = T. If {yk : к e N] is a dense subset of П = 1

the space X , then putting

(2.4) Pu,c,n(x ) — {t E Tn: Ч*(сх, t) > u_1 Ф{х, t) and Ф(х, t) ^ n}

for и, c > 0, n e N , x e X and

(2 5) аи.с>и( 0 = sup ^ ( c y k f t)xpUte^ k)(t) l^k^n

for и, c > 0, n e N , t e T , we have p(Pu>c<n(x)) < 00 and

(2.6) аМ(С>1 (0 ^ <xu,Ct2 (*)< ••• and lim aUiC>„(0 = a„>c(t) И-*-аО

for every x e X , u , c > 0 and for these t e T for which Ф (•, t) is continuous on X . Now, we prove a useful lemma.

Lemma 2.2. Let X be a separable space. I f Ф, W are Ф-functions, then for every, и, c > 0, n e N there is a simple function g such that

(2.7) g e d o m /ф,

(2.8) V(cg{t),t) = <xu>C'„{t) for every te T , (2.9) 0(éf(t), t) < u*P(cg(t), t) for every te T .

P ro o f. Let us put

C1 = {te T : aU(C;„(t) = T{cyu t)xpuc<niyi){t)}, Ck = { t e T \ (J Tp: ccuc<„(t) = W(cyk, t)Xpuc n(yk){t)}

p=

i

for к = 2, 3 ,..., n and let

(2 1 °) g = Z ykXct ip UiC,„(yk)[t).

к= 1 Then the function g is measurable and

Л

№ (9(t),t)dp ^ £ f Ф{ук. t)dp ^ n2p{Tn) < + со.

T k = 1 Р иуС' П(Ук)

Thus g e d o m /ф.

It is easy to verify that (2.8) holds.

Let us suppose that g(t) Ф 0 . Then there is exactly one k e N such that t e C k n P u>c n{yk). Thus g(t) = yk and t e P UtC>n(g{t)). Now, (2.4) implies (2.9). If g{t) = 0 , then inequality (2.9) is obvious.

Kozek in [4] proved the following lemma (Lemma 1.7.3):

Lemma 2.3. Let {am: m e N} be a sequence of positive numbers and

\hm: m e N } a sequence of finite non-negative functions such that §hm(t)dp

T

Identity embedding of some Orlicz spaces 347

^ 2mam, m = 1 ,2 ,... I f p is non-atomic, then there exists a subsequence {mk} of integers and a collection {Gk: k e N } of disjoint subsets of T such that

J hmk{t)dg = amk for к = 1^, 2^{, . . .}

Gk

Now, we will establish relations among the following conditions:

(2.11) there are a set S of measure 0 and a number и > 0 and an integrable non-negative function h such that T {x, t) ^ u ~ 1 Ф{х, t) + h(t) for every x e X and every t e T \ S ,

(2.12) dom / ф c= d o m /y ,

(2.13) there is a number и > 0 such that j f i u(t)dp < + o o .

T

Th e o r e m 2.4. (a ) Condition (2.11) implies condition (2.12).

(b) I f X is a separable space, Ф, W are Ф-functions and Ф(-, t) is continuous on X for almost every te T , then condition (2.13) implies condition (2.11).

(c) If, moreover, p is a non-atomic measure, then condition (2.12) implies condition (2.13).

R e m a rk . If all assumptions of Theorem 2.4 are fulfilled, then conditions (2.11), (2.12), (2.13) are equivalent.

P ro o f. It is easy to verify that the first and the second part of the theorem hold, and the proof is left to the reader.

Let us suppose that dom / ф c= dom I y and §fiu(t)dp = +cc for every

T

и > 0. Then, in virtue of (2.6), we can find a sequence {nm} such that

(2.14) > 2”

Lemma 2.2 implies the existence of functions gm such that conditions (2.7), (2.8) and (2.9) hold (with и = 2~m, с = 1 and n = nm). The functions gm belong to dom I 0, so they also belong to dom 7r . Thus Iy(g m) < + oo and the values of the functions hm given by

(2.15) hm(t) = T (g m(t),t),

where t e T , are finite for almost every t e T . Therefore, by (2.8) and (2.14), f hm(t)dp ^ 2m for each m e N . Since the measure p is non-atomic, Lemma 2.3

t

implies that there are a sequence {mk} of integers and a collection {Gk} of disjoint subsets of T such that

J hmk(t)dp = 1 for к = 1, 2 ,...

Gk

(

2

16

00

Let us put f k = gmkXck and /== £ /*. Then, by (2.9), (2.15) and (2.16),

k = 1

00 00 00

M / ) = E 1 < E 2~ mk J 'P (9mk ( t ) , t ) d n = £ 2 ~ TOfc < + 0 0 .

k ~ 1 Gfc k = 1 /с — 1

'Hence / e d o m /ф. On the other hand, (2.15) and (2.16) imply that

00 00

(2.17) Ы / ) = E J ^ (9mk(th t)d ii= E 1 = + oo.

k = 1 Gk k = l

Thus, we nave obtained a function which belongs to the set dom / ф and does not belong to the set dom Iy. This contradiction ends the proof.

We shall denote by Рф the set of all functions from dom / ф which are of the

00

type E хкХлк> where {A k : k e N} is a collection of disjoint subsets of T and

k = 1

{xfc: k e N } is a sequence of elements of X .

Th e o r e m 2.5. (a) Let Ф, ¥ be Ф-functions and let the following condition hold:

(2.18) there are a set S of measure 0 and a number и > 0 and an integrable non- negative function h such that ¥ ( u x ,t) < u -1 Ф(х, t)-{-h(t) for every x e X and every te T \ S .

Then

(2.19) Z?(M) c i f (M) (i.e., Zf(M) is contained as a set in Ü (M)).

(b) I f X is a separable space, Ф, ¥ are Ф-functions and Ф(-, t) is continuous on X for almost every te T , then the condition

■ (2.20) there is a number и > 0 such that §yu(t)dp < +oo implies condition (2.18).

(c) I f moreover, p is a non-atomic measure and Рф a M, then condition (2.19) implies condition (2.20).

P ro o f. The first and the second parts of the theorem are obvious.

Let us suppose that if(M ) c= f f (M) and

J

T

Then, by (2.6), there exists a sequence {nm} of integers such that f a , _ m (t)dp ^ 2m for m = 1, 2 ,... Thus, by Lemma 2.2, for each m e N

J 2 , 2 , n m

we can find a step function gm such that conditions (2.7), (2.8) and (2.9) hold (with с = и = 2~m and n = nm).

Let us put A m = { t e T : ¥ ( 2 ~ mgm(t), t) = + oo},

(2.21) M O

¥ ( 2 ~mgm(t), t)

« 0

2

r)+M 0

if t e A m,

if t e A m and p (A m) = 0, if t e A m and p (A m) > 0,

Identity embedding of some Orlicz spaces 349

where Sm is a measurable non-negative function with finite values such that j ô m(t)dp ^ 2m. Then (2.9) implies that

T

(2.22) <P(«m( M « 2— АяМ

for every m e N and a.e. t e T , and, by (2.8) we obtain

(2.23) > hm(t)

for every me AT and a.e. t e T . Moreover, from (2.8), (2.21) we get §hm{t)dp

T

^ 2m for m e N . Now, we apply Lemma 2.3. There are a sequence {mk} of integers and a collection {Gk} of disjoint subsets of T such that j hmk(t)dpt = 1

f o r * = 1 ,2 ,... G*

00

Let us put f k = gmki Gk and / = Z /*; then by (2.22) we get

k = 1

00 00 00

U i f ) = Z 7фШ = Z I <P(9mk(t),t)dp ^ X 2_Wfc I hmk(t)dg < +00,

fc= 1 k = l G k k = 1 Gk

so / е Р ф, thus /е Т ф(М). On the other hand, / r (c/) = +oo for every c > 0.

Indeed, let k0 be an integer such that 2 mii ^ c for every к ^ k0. Then, by (2.23), we obtain

00 00

Ы c f) > Z $ 'F(2~mk9mk( t ) , t ) d n ^ Z J’ K k(t)dn = +oo.

k = k 0 Gk k = k 0 Gk

Hence, we have obtained a function which belongs to the set I?{M) and does not belong to the set lT(M). The proof is finished.

R e m a rk s , (i) Theorem 2.4 implies Theorem 1.7 in [4].

(ii) Let us denote by M x the set of all strongly measurable functions / : T -> X , where X is a Banach space, and let F be a subset of the set M x n dom / ф.

If M F is defined by

M F = { f e M x : 3 V I 0 { a ( f - f n)|) - 0},

{T„ }® = 1 <=linF*> 0

then M F is a linear space. Moreover, it is easy to verify that MF = L*(Mf ) = <£ф (F),

where iF 0 (F) is the space defined by Kozek in [3]. If F = M x n dom / ф, then I?{MX) — ^ Ф(Т) (in this case A. Kozek has denoted the space ^ 0 {F) by ф and has called it the Orlicz space; see also Proposition 1.2 in [3]). Thus, Theorem 1.8 in [4] can be obtained from our Theorem 2.5.

(iii) W. Matuszewska in [5] has considered the so-called ^-functions without parameter, i.e., functions q>: [0, + o o )-» [0 , +oo) which are non-

12 — Prace Matematyczne 24.2

decreasing, vanishing only for и —0 and such that q>(u) -* + oo for и -* + oo.

Thus, Theorem 2.5 in [5] is a special case of our Theorem 2.5. Moreover, Theorem 2.4 implies Theorem 3 in [5] (for (pv = q>, v = 1, 2 ,...).

3. The mapping /: i f (M) -»• f f (M), / ( / ) = / . We shall say that a sequence { fk} is norm convergent (n.c.) to 0 in the space lf(M ), iff ke M for each к e N and I<p(afk) -* 0 for к -> + oo, for every a > 0. In the sequel the mapping i : lf ( M ) -> Ü (M), i ( f ) = / , is called a continuous mapping if every sequence {fk} which is n.c. to 0 in the space 1Ф(М) is n.c. to 0 in the space ZT(M).

Let us note that if Ф is a Ф-function (a convex Ф-function), then a sequence {fk} is n.c. to 0 in I f (M) if and only if it converges to 0 in the topological space (lf(M ), N *) ((Z?(M), IHU), respectively) (cf. (1.3) and (1.4)).

We shall denote by 8Ф the set of all simple functions which belong to the set do m /ф.

Theorem 3.1. (a) I f Ф, W are Ф-functions and

(3.1) there is a set S of measure 0 such that for every s > 0 there are a number и > 0 and an integrable non-negative function hu such that J MO dp < в and 4/ ( u x , t ) ^ : u 1 Ф(х, t) + hu(t) for every x e X and

T

every t e T \S then

(3.2) the mapping i : lf(M ) -* lT(M), i ( / ) = / , is well defined {i.e., lf ( M ) c= i f (M)) and it is continuous.

(b) I f X is a separable space, Ф, W are Ф-functions and Ф(-, t) is continuous on X for almost every te T , then the condition

(3.3) V 3 J yu( t ) d p < e

e > 0 u > 0 T

implies condition (3.1).

(c) I f the above assumptions are fulfilled and, moreover, if p is a non-atomic measure and Бф a M , then condition (3.2) implies condition (3.3).

P ro o f. We shall prove the third part of Theorem 3.1 only.

Assume, for a contrary, that there is a number 1 > ^e > 0 such that J y„(t)dp > e for every и > 0. Then

^ 3 J'^2- m,2~m,

m e N n m e N т ’

( 0 dp ^ £.

nm

Lemma 2.2 implies that for every m e N we can find a simple function gm such that conditions (2.7), (2.8) and (2.9) hold (with n = nm and и = c = 2~m). In an analogous fashion, as in the proof of Theorem 2.5, there is a collection of functions {hmk} and a family {Gk} of disjoint sets such that the conditions

Identity embedding of some Orlicz spaces 351

(3.4) Ф(вщ (t), t) 2 ' "‘ A,», (0 for every k e N , and a.e. te T , (3.5) K k (0 (0). t) for every k e N , and a.e. te T,

(3.6) {t)dpt = £ for * = 1 ,2 ,...,

hold. Let us put

(3.7) /к = 2 “■ mu

< 4 XGk’ к = 1 , 2 ,. ..

Then (2.7) implies t h a t ^ e ^ . Thus f k e i f (M) for each ,k e N and the sequence {/k} is n.c. to Oin the space Z?(Af). Indeed, if a > 0 and k0 is an integer such that a ^ 2W* for every к ^ k0, then

l * m ^ 1ф(2”‘Л) = 1ф(дщ Хок) « e 2 '" ‘ for к ^ k 0. Thus, I 0 (afk) -> 0 for к -> -f oo.

On the other hand, assumption (3.2) implies that / k e IT(M) for к = 1, 2 ,...

Since £ < 1, , ,

1ч*(£ 1 fk) ^ fr(/k) ^ j hmk(t)dfi = £ Gk

for each k e N . So, we have obtained a sequence of functions which is n.c. to 0 in Z?(M) and is not n.c. to 0 in I f (M). This ends the proof.

In the sequel we shall denote by X 0 the family of all countable subsets of X . Moreover, to simplify the notation we describe by Condition I the following conditions :

Condition I. (a) I is a separable space;

(b) Ф and Ф are Ф-functions;

(c) Ф(-, t) is continuous on X for almost every t e T;

(d) Ц is a non-atomic measure;

(e) с M ;

(f) Zf(M) is contained as a set in Zf(M).

We shall say that a sequence {fk} is modular convergent (m.c.) to О in the space iff ke M for each k e N and there is a number a > 0 such that

^ф(лЛ) 0 for к -+ + oo.

R e m a rk s , (i) It is obvious that if {/k} is m.c. to & in the space lf ( M ) and condition (3.1) holds, then {/k} is m.c. to 0 in the space

(ii) Let us assume that Condition I is fulfilled. If every sequence which is n.c.

to 0 in L0 (M) is m.c. to 0 in i f (M), then condition (3.3) holds.

Let us put f k = 2 mkl1 gmkXGk-> к = 1 ,2 ,..., instead of (3.7) in the proof of Theorem 3.1. If a > 0 and k0 e N is such that a < 2”*'2 Гог к ^ k0, then, by (3.4) and (3.6),

1ф(“А ) « /* (2 mtl2f k) = 1ф(дщ х ч ) «: c 2 mt

for к ^ k0. Thus {/*} is n.c. to 0 in lf{M ). On the other hand, for every a > 0 we can find an integer /cj such that 2 mfc/2 ^ a for к ^ k x. Then, by (3.5) and (3.6),

M a /к) ^ M 2 mk'2 '1 k> ^ 8

for every к ^ k x. Hence, the sequence {^} is not m.c. to 0 in Ü (M).

Now, we deduce from Theorem 3.1 three propositions which are convenient for applications.

Proposition3.2. I f Condition I is fulfilled and the function W satisfies the condition

(3.8) 3 g(T\G) = 0&

G e l

V V ( 3 V t) ^ K t => V 3 V Y (их, t) < e),

t e G Z e X Q K t > 0 x e Z e > 0 ut > 0 x e Z

then the mapping i: 1?(M) -* i f (M) is continuous.

P ro o f. Theorem 2.5 implies that J yuo(t)dp < + oo for some u0 > 0. Thus,

T

the set {t e T : yUQ(t) = + oo} is of measure zero. To prove the proposition it is enough to show that lim yu{t) = 0 for every t e H , where H = {seG : 0 < y„0(s)

. u-> 0

< + 00 } .

Let t e H . Then, by (1.10) and (2.1),

0 < yUQ(t) = sup *P(u0y„, t)Xp (yn)(t) = sup W(u0yn, t) ^ K t

neN 0 ne J t, UQ

for some K t > 0, where J tu = { n e N : t e P u(yn)}. Let rj be a positive number.

Then condition (3.8) implies the existence of a number afe ( 0 ,1) such that 4*(at u0y„,t) < Y) for every n e J t<UQ. Taking Mi = at u0 we have iq < u0, so J t,u c Jt.u0 ^ог

every

yu(t) = sup P{uyn, t) < Г].

n e J tyU

for every и ^ ux. Thus limyu(r) = 0 for t e H . м-0

By the Lebesgue dominated convergence theorem J yu{t)dp < & for some

T

и ^ m1} thus condition (3.3) holds. By Theorem 3.1 the mapping i : If(M ) f f (M) is continuous.

Corollary3.2.1. I f Condition I holds and W is a convex Ф-function {and, moreover, if Ф is a convex Ф-functiori), then the mapping i from the space (Z?(M), I • I ф) ((L0 (M), || • ||ф) respectively) into the space ( I f (M),\\-\\4,) is continuous.

We shall denote henceforth by pL the Lebesgue measure on R, and by I L the о -algebra of all Lebesgue measurable subsets of R. Moreover, we shall denote by M L the set of all Lebesgue measurable functions from R into R.

Identity embedding o f some Orlicz spaces 353

Ex a m p l e 3.1. Let us put X = R, (T, I , fi) = (R , I L, p L), M = M L and let Ф(х, t) = |x| p(t), ¥ { x , t) = (|x|/(l + |x|))p(f), where p is an integrable and positive function. Then Condition I holds and ¥ does not satisfy (3.8).

Indeed, taking Z = IV we have ¥ { n , t ) ^ 1 for every n e N and te T , but W{un, t) = — — p{t) > ip(f)un

1 -fun for every te T , и > 0 and n > \/u.

On the other hand

¥{ux, t) = U^X\ p(t) ^ u _ 1 |x|p(t) = u ~ 1 Ф(х, t) 1+Ujxl

for \x\ ^ u — u ~ l and x = 0. Hence ¥ (u x, t) ^ u ~ l Ф{х, t)for и ^ l , x e X and te T . Thus the set Pu(x) is empty for и ^ 1 and x e X , and f yu(t)dp = 0 for

T

и ^ 1. By Theorem 3.1 the mapping i : I ? (M) -» Ü (M) is continuous. Thus the continuity vof the mapping i does not imply condition (3.8).

Co r o l l a r y 3.2.2. I f Condition I is fulfilled, X is a locally bounded space and the following conditions

(3.9) ¥(-, t) is continuous at 0 for almost every te T ,

(3.10) lim ¥ ( x , t ) = +oo for almost every t e T {more precisely:

I jc| — 00

supT (x„, t) ^ К implies sup|x„| < a for some a > 0)

n e N ne N

hold, then the mapping i : ТФ(М )-► lf(M ) is continuous.

P ro o f. Let us suppose that lim ¥ { x , t) = +oo and Z e X 0. If there is

I jc| -»oo

К > 0 such that ¥ { x , t) ^ К for x e Z , then (3.10) implies the existence of a number a > 0 such that |x| < a for every x e Z . Let £ be a positive number. The continuity of ¥{-, t) at 0 implies that we can find a number Ô > 0 such that

< £ for every |x| < Ô. Moreover, there is и > 0 such that |юс| < Ô for every x e Z (by the local boundedness of the space X).

Therefore ¥{ux, t) < e for every x e Z , so ¥ satisfies condition (3.8). Thus the mapping i is continuous.

R e m a rk . The continuity of the mapping i does not imply (3.10). Example 3.1 shows that there are Ф-functions Ф, ¥ such that the mapping i is continuous but

lim ¥ { x , t)= lim (|x|/(l + |x|))p(f) = p(t) < + oo

Ijc| -► 00 | x | -♦ 00

for almost every t e T .

Pr o p o s it io n 3.3. Assume that Condition I holds. Let Ф be a Ф-function such that

(3.11) the family o f sets Vt = {Vt (a) : a e(0, + oo)} is a base of neighbourhoods of 0 for almost every t e T , where Vt {a) = { x e X : Ф{х, t) < a}.

I f moreover, W is a Ф-function satisfying condition (3.9), then the mapping i : lf{ M ) —► ZT(M) is continuous.

P ro o f. We shall show that limy„(t) = 0 for almost every te H , where H = {seT: 0 < yMo(s) < + oo} and $yUQ(t)dp < -(-oo.

T

If J tu = {ne N : t e Pu (y„)}, where и > 0 and {yn: n e N} is a dense subset of X , then {y„: n e J ttUQ} e X 0 and sup W{u0y„, t) = yUQ (t) < К for some К > 0.

neJt , u0

By (3.9) and (3.11), for every £ > 0 there is a number a > 0 such that W(x, t)

< e/2 for every x e V t (a).

Suppose that yn$Vt (ci) and u ^ v = m in {a/K, l , u 0}. Then У&Уп, t) ^ П»оУп, t) ^ К ^ и ' 1 a ^ и ' 1 Ф(уп, t),

so the set Pu{yn) is empty for every и < v and yneVt(a). Thus, putting J tua

= {n e N : t e P u(yn) and y„eVt (a)j, where t e T and u , a > 0, we obtain yu(t) = sup T{uyn, t) ^ sup *F(y„, t) < sup W(x, t) < & < £

n€^t , u, a

for every u < v. This ends the proof.

Corollary 3.3.1. I f Condition I holds and, moreover, X = R, W satisfies (3.9) and Ф fulfils (1.2), then the mapping i : lf{ M ) —> IT(M) is continuous.

P ro o f. If U is a neighbourhood of 0, then { — a, a) c U for some a > 0.

Then (1.2) implies that the set Ц(Ф(а, t)) is non-empty for almost every te T . Moreover, Ф(х, t) < Ф(у, t) implies that |x| < |y| for t e T0 (by Definition 1.1).

Therefore Ц(Ф(а, t)) c= ( — a, a) c; U. Thus the family Vt is a base of neighbourhoods of 0 for almost every te T .

R e m a rk . The above corollary shows that Theorem 2.21 in [6] (when the measure p is non-atomic) is a special case of our Theorem 3.1.

Ex a m p l e 3.2. There are Ф-functions Ф, W and the space X such that the mapping i is continuous and the function Ф does not satisfy condition (3.11).

It suffices to take X — R 2, (T, I , p) = (R , I L, pL), M = M L and Ф((х, y), t)

= |xj p(t), T({x, y), t) = --- p(t), where p is an integrable and positive|X|

l + \x\

function. Then (see Example 3.1) yu(t) — 0 for every t e T and u < 1, thus the mapping i is continuous. But the family Vt = {R x ( — a, a) : a e(0, + oo)} is not a base of neighbourhoods of 0.

Pr o po sitio n 3.4. I f Condition I holds and Ф, 4* satisfy the following condition

Identity embedding o f some Orlicz spaces 355

(3.12) there is a measurable set G such that p(T\G) — 0 and V V ( 3 V Ф(х, t) ^ K t => V 3 V Ф(щ х , t) < e),

t eG Z e X g K t > 0 x e Z e > 0 ut > 0 x e Z

then the mapping i : f ? ( M ) -* Cf (M) is continuous.

P ro o f. Let t e { s e G : 0 < y„0(s) < + oo}, where u0 is a number such that j y UQ(t)dp < + o o . Then

T

У«о(0 = sup Ч^(и0у я, 1 ) ^ К

neJt,UQ

for some К > 0, where Juu = {n e N : t e P u(y„)}. Thus Ф(Ут t) ^ u0 P (u0y„, t) ^ u 0 К

for every n e J tyUQ. Let s > 0 be given. By (3.12), there is ut > 0 such that Ф(щуп, t) < e/2 for every n e J tyUQ. Therefore

yu(t) = sup *P(uy„, t) < sup T{ut y„, t) ^ e/2 < e

n e J t,u ne J t , u0

for every u ^ ut . Hence, by the Lebesgue dominated covergence theorem, J 7u(t)dp s for some и ^ Uq, so the proof is finished.

T

Corollary 3.4.1. I f Condition I holds, X is a locally bounded space, *P fulfils condition (3.9) and Ф satisfies the following condition:

(3.13) lim Ф(х, t) = + oo for almost every t e T ( c ( . (3.10)), then the mapping i: Z?(Af)->lT(M) is continuous.

The proof is easy and is left to the reader (cf. Corollary 3.2.2).

Example 3.3. The continuity of the mapping i: lf(M )-> lT (M ) implies neither (3.12) nor (3.13). Let us put X = R, (T, I , p) - (R , XL, p L), M = M L and Ф(х, t) = V (x, t) = 1*1

1+1*1p(t), where p is an integrable and non-negative function. It is obvious that the mapping i is continuous and Ф does not satisfy (3.13). Moreover, 0 ( n , t ) ^ p ( t ) for each n e N but W(un, t) ^ \p{t) for

n > w-1 . Thus the functions Ф, 4* do not satisfy condition (3.13).

Example 3.4. There are Ф-functions Ф and W, spaces X , (T ,X ,p ) and M such that I?{M) a f f (M) but the mapping i : lf ( M ) -> I ? (M), i{f) — f, is not continuous.

Indeed, let us put X = R 2, (T, X, p) = (R, XL, pL), M = M L and Ф({х, y), f)

= \x\p(t), *F{(x,y), t) function. Then

Ы 1 + Ы

p(t), where p is an integrable and non-negative

f Yu(t)dp = J sup 4'(u(x,y ),t)x p u{(x,y))(t)dp < | p(t)dp < + c o .

T T (x,y)eX T

Thus I?(M ) c= Ü (M) by Theorem 2.5.

On the other hand,

P„((*.>’)) = { ' eT ; 1*1 P(0 < l U+ u \y \P(t)}' Thus, putting x — и/ 2 and у — 2/и, we obtain

u2 (2/и) 2

---LLJ— =4u > \u.

l + u(2/n) 3 2 Hence Pu((u/2, 2/u)) = T. Therefore

.f yu(t)dfi > j *Р(п(м/2, 2/n ), t)xPu«ui2,2iu»(t)dLi

t T

= f 1 U^ n , ч P ( f) dP = i f p(t)dfi .

Г 1 +U (2/n) T

The arbitrariness of и implies that j y u(t)dfi does not converge to 0. By

г

Theorem 3.1 the mapping i : i f (M) -► Ü (M) is not continuous.

References

[1] J. Is h ii, On equivalence o f modular spaces, Proc. Japan Acad. 35 (1959), 551-556.

[2 ] T. M. J ç d r y k a , J. M u s ie la k , On a modular equation I, Functiones et Approximatif) 3 (1976),

101

111

[3 ] A. K o z e k , Orlicz spaces offunctions with values in Banach spaces, Comment. Math. 19 (1977), 259-288.

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

[5 ] W. M a tu s z e w s k a , On generalized Orlicz spaces, Bull. Acad. Polon. Sci. Sér. sci., math. astr. et phys. 8 (1960), 349-353.

[6 ] —, and W. O r lic z , A note on the theory o f s-normed spaces o f (p-integrahle functions, Studia Math. 21 (1961), 107-115.

[7 ] J. M u s ie la k and W. O r lic z , On modular spaces, ibidem 18 (1959), 49-65.

[8 ] H. N a k a n o , Generalized modular spaces, ibidem 31 (1968), 439-449.

[9 ] M. W is la , Separability and local boundedness o f Orlicz spaces o f functions with values in separable linear-topological spaces, in print.

INSTITUTE OF MATHEMATICS A. MICKIEWICZ UNIVERSITY POZNA14