Boundedness of the identity embedding of some Musielak-Orlicz spaces

R O CZNIK I P OLSKIEGO T O W A R Z Y ST W A M A T E M A T Y C ZN E G O Séria I: PRACE M A T E M A T Y C ZN E XXVII (1988)

Ma r ek Wisla (Poznan)

Boundedness of the identity

embedding of some Musielak-Orlicz spaces

Abstract. The aim o f this paper is to extend a theorem proved by P. Turpin in [4] which concerns the boundedness o f the identity embedding o f Musielak-Orlicz spaces.

1. Introduction. P. Turpin [4 ] has studied in details the identity embed­

ding of Musielak-Orlicz spaces o f functions with real values. Theorem 4 presented in that paper which gives conditions concerning the boundedness of the embedding i: I f i f , i ( f ) = f (cf. Definition 1.3 below) we extend to the case o f more generally Musielak-Orlicz spaces consisting of functions with values in linear-topological spaces.

After the preliminary comprehensions and auxiliary results o f this sec­

tion, in the second one we give fundamental theorems of this paper. In these theorems there are considered functions auc satisfying the condition

V V 3

г > О к > О с > 0 T

In the third section we study conditions which must be assumed in order that the above condition could be replaced by a weaker one:

V 3 j aM >c

о .

u > 0 c > 0 T

The last condition links the considerations included in this paper with the above-mentioned theorem of P. Turpin.

Let (T, I , ц) be a measure space, where T is an abstract set, I is a a- algebra of subsets of T, ц is a non-negative, complete, atomless and (Т-finite measure on I . (X , t) will denote a real linear-topological space. By & x we shall denote the сг-algebra of Borel subsets o f X.

Let 9Л(Т, X ) be the set of all measurable functions / : T -> X , i.e., functions satisfying f ~ l ( U ) e Z for every U e @ x . W e shall denote by M (T , X ) (or shortly M ) an arbitrary linear subset of the set Ш(Т, X ). In particular, S ( T, X ) will denote the set o f all simple functions, i.e., functions of

n

the type X xkxAk, where xke X , ц{Ак) <: T oo for к — 1, 2, ..., и.

k= 1

11 — Com m entationes M ath. 27.2

1.1. Definition. A function Ф : X x T ->[0, + o o ] is said to be а Ф- function if there is a set T0 of measure 0 such that:

(a) Ф is 0&x х Г -measurable;

(b) Ф(0, t) = 0 and Ф(х, t) = Ф( — х, t) for every x e X and te T\T0;

(c) Ф (’, t) is lower semicontinuous on X and continuous at 0 for every t e T \ T 0;

(d) Ф(их + vy, t) ^ Ф(x, t) + Ф(y , t) for every u, v ^ 0, u + v = l, x, y&X and t e T \ T 0.

If Ф(-, t) is continuous on (x : Ф (х, t) < + oo< (resp. convex on X ) for almost every (a.e.) t e T, we shall shortly write “ Ф is continuous” (resp.

convex). The functions which are Ф-functions at least we shall denote by Greek letters Ф, Ф, ...

The functional /ф: УЯ(Т, X) ->[0, + o o ] is defined by /ф(/ ) = (Ф (/ (г ), t)dfi. Let us note that /ф\M: M ->[0, + oo 1 , Ы м ( Л = 1ф(Л is a T

pseudomodular on M — M (T , X) (see [1], [2]).

1.2. Definition. By the Musielak-Orlicz space ЬФ(М (Т , A")) (or shortly ЬФ(М )) we mean the set o f all functions / eM (T , X) such that /ф(а/) < -f-oo

for some a > 0. \

In the space ЬФ(М ) we shall consider the F-seminorm defined by

|/|ф = inf {и > 0: 1ф(//и) < и}.

In the sequel the following notations will be used:

(1) Вф(г) = { /е Ь ф(М ): |/|ф < r}, for г > 0, (2) d om /ф = '/ е Щ Т , X ): 1Ф(/) < + о о ],

(3) Ри,с(х) — {f е Т : и¥(сх, t) > Ф(х, t) and Ф (х, t) < + о о } for и, с > 0, х е Х , (4) аи>с(0 = sup ¥ (сх , t)xpuc(x)(t) for и, с > 0, х е * .

х е Х

1.3. Definition. Let У, Z be linear-topological spaces. A linear operator G: Y - > Z is bounded if there is a neighbourhood U of 0 in Y such that the set G (U ) is bounded in Z (the set U may be not bounded in У).

The following assertions will be quoted in the sequel:

1.4. [5 ] I f X is a separable space and Ф is continuous, then the functions aUtC are measurable and

au,c(0 = sup ¥ (cxk, t) ^{Xp u} c ( x k )(0 for a.e. t e T,

k e N

where the set {x 1? x 2, ...} is dense in the space X.

1.5. [6 ] Assume that X is a separable space and Ф is continuous. If

meN T

for some sequences (um) and (cm) o f positive numbers, then there are sequences (mk) of numbers and (gk) o f simple functions such that

1 t e T : gk(t) Ф 0} n { t e T : g f t) Ф 0} = 0 for к Ф l,

1ф(вк)^итк and Iv{Cmkgk) > 1 f o r * = 1 ,2 ,...

1.6. [6 ] If the assumptions o f 1.5 are satisfied and V \ct.um,cm{ t )d g > £

meN T

for some e > 0 and some sequences (um), (cm) of positive numbers, then there is a sequence (gm) o f simple functions such that ^ e d o m /ф, 1ф(дт) ^ ume

and for m eN.

1.7. [4 ] The set A c L®(M ) is additively bounded, i.e., for every s > 0 there is n e N such that A c B0(s) + ^. + B0(s) (n terms B0(s), we shall denote this set by + " B 0 (s)) if and only if

3 sup §<P(af(t), t)dg < + oo.

e > 0 feA T

2. The following theorem is a generalization of Theorem 4 (b) in [4 ]:

2.1. Theorem, (a) I f

(B) there is a set T0 of measure 0 such that for all numbers e > 0 and К > 0 there are a number c > 0 and a measurable function h: T —> [0, + o o ] such that §h(t)dfj. < e and

T

Ч*(сх, t) ^ К Ф (х, t) + h(t) for all x e X and t e T \ T 0, then the embedding i : L 0(M ) i f (M ), i (/ ) = /, is bounded.

(b) I f X is a separable space, Ф is continuous on X , S ( T , X ) n n d o m /ф c M ( T , X ) and the embedding i : Ь Ф{М) -* 1 У (М ) is bounded, then condition (B) holds.

P r o o f, (a) W e shall show that the unit ball Вф(1) = { f e ЬФ(М ): |/|ф

< 1} is bounded in the space l f ( M ) , i.e., V 3 аВф(1) czByid).

d > 0 a > 0

Let d be an arbitrary positive number. In virtue o f (B) there are a set T0 of

measure 0, a number c > 0 and a function h: T -> [0, -l-oo] such that J h (t) dp d and

T

(5) W(cx, t) ^ jd4>(x, t) + h(t) for all x e X and t<£T0.

Now, let a = j c d and f е В ф{ 1). Then, by (5),

^ = ^ ^ § d.

Thus lafly^Tjd <d, so af s B ^ d ).

(b) In virtue o f 1.4 the functions au>c given by (4) are measurable.

Consider

(6)

V V 3

с > О и > 0 c > 0 T

Suppose that (6) does not hold. Then there are numbers e0 > 0 and и > 0 such that §0Luc(t)dp ^ £0 for every c > 0. Since the embedding i is bounded,

T

the ball ВФ(Ь) is bounded in the space L'P(M ) for some b > 0. Now, if e is a sufficiently small number, e.g. 0 < e < m in {e0, b/2u}, then j a uc( f )dp > s for

r

every c > 0 and the ball B0(2us) is bounded in the space L*P(M ). Hence аЁф(ие) c B«p(£) for some a > 0, where Вф{г) = { /е Ь ф(М ): |/|ф ^ г}, г > 0.

In virtue o f 1.6 (putting um = u, cm= au for each m = 1 ,2 ,...) there is a simple function g such that 1ф(д) ^ us and ly{aug) > e. Write / = £ -ug. Then Л к (/ М ) = K (g ) < uz, i.e., f e B 0(u£).

On the other hand Iyiaf/e) = Itf/(aug) > £, so а[фВч,(е) and we get a contradiction. Since (6)=>(B) (with K = u~x and h(t) = aM>c(f)) the proof is finished.

2.2. R em a rk . Conditions (B) and (6) are equivalent by the assumptions o f Theorem 2.1 (b).

2.3. Corollary, (a) I f

(LB ) there is a set T0 of measure 0 such that for arbitrary numbers £ > 0 and К > 0 we can find a number c > 0 and a measurable function h:

T -> [0 , + oo] such that \ h {t)d p < e and T

Ф(сх, t) ^ К Ф (х, t) + h(t) for every x e X and t e T \ T 0, then the Musielak-Orlicz space (Т Ф(М ), |-|ф) is locally bounded.

(b) I f X is a separable space, Ф is continuous, S(T, X) n d o m / 0 a M ( T , X) and the space (Ь Ф( М), |*|ф) is locally bounded, then the function Ф satisfies condition (LB).

2.4. Corollary. I f

( + ) there is a function x: [0, + o o )-+ [0 , + o o ) such that x(0) = 0, lim x(c) = 0 and Ф(сх, t) ^ х (с)Ф (х , t) for all c ^ 0, x e X and

c - * 0

a.e. t e T

(in particular: if Ф is a p-convex function on X), then the Musielak-Orlicz space (ЬФ(М ), |-|ф) is locally bounded.

2.5. R em a rk . If the function Ф satisfies condition ( + ), then either supФ{х, t) = Ч- c o or Ф (*, t) = 0 for a.e. te T .

x e X

P r o o f. Assume that Ф satisfies ( + ) for all гф T0 (p(T0) = 0). Moreover, let t<£T0 be fixed and suppose Ф (-, t) ф 0, i.e., Ф(у, t) > 0 for some ye X. By continuity o f the function x at 0 we obtain

V 3 V

n e N c „ > 0 x e X H

Let (dn) be a sequence o f positive numbers such that 1 ^ cndn^ cn+1 d„+1 for each n e N . Therefore

SUp Ф (x, t) ^ SUp Ф (dn ‘ y, t) ^ sup (n^Ф(cnd„yy t)) ^ 8ир(п-Ф(у, t)) = + 0 0-

x e X neN neN neN

2.6. Ex a m p l e. Denote by C(R, R) the space o f all continuous functions x: R -> R (R being the space o f real numbers) with topology yielded by the family o f sets o f the form

V{au ..., am; e) = {x e C (R , R):

V

U i « m

where al , . . . , a me R and e > 0. As the space o f parameters T we shall consider the space R with o--algebra I which consists o f all Lebesgue measurable subsets o f R and with the Lebesgue measure p on I . Define

S: C (R, R) x R -> [0, + o o ), (x, t) —> |x (r)(.

Then E is a convex Ф-function with finite values and, in virtue of Corollary 2.4, the Musielak-Orlicz space C(R, /?))), |-|s) is locally bounded.

2.7. P. Turpin in [4 ] has proved that in the case where X = R, Ф is a Musielak-Orlicz function (cf. [4 ] Definition 1, p. 71) M (T , R) is the space of all measurable functions from T into R, the embedding i : Ь Ф(М ) -» L'P(M ) is bounded if and only if

0 ) v 3 /с „ е С { М ) ,

м > 0 c > 0

where f CtU(t) = su p {x ^ 0: uTicx, t) > Ф(х, t)} (if the set on the right-hand side of the equality is empty, then we define X ,«(f) = 0)-

On the other hand, defining Ф( + oo, t) = lim W(x, t), we have

x -»+ 00

(8) att>c (0 = sup W(cx, t) ip w(t) = V(<cfCytt (f), t)

x > 0

for a.e. t e l Hence condition (B) takes the form (9)

V V

E > 0 u > 0 с > 0 Г

Therefore, are conditions (7) and (9) equivalent?

The implication (9) => (7) is obvious. Conversely, let и > 0. By (7), f V'lsfc.uiO, t)dfi < + o o for some c, s > 0. Since f v>u(t) ^ f c,u(t) for every T

0 < v ^ c and t e T,

•P(vfVjU{t), f) < t) for 0 < v < min !c, 5Î and a.e. f e 7 . Moreover, by continuity of the function t) at 0,

lim 4?(vf,tU{t), t) ^ lim 'f'(vfcu(t), t) = 0 for a.e. f gT,

v ->0 ’ v —*0

so, by the Lebesgue dominated convergence theorem, lim j V(vfv u(t), t)dfi

= 0. Thus, condition (9) holds. ° T

Let us note that in this case the assumption concerning the continuity of the function Ф on R may be omitted, since it is only used in the proof of the fact that the functions auc are measurable. The measurability o f these functions follows immediately from (8).

3. In this section we shall study a little more general problem than the above discussed, namely: what conditions must be fulfilled in order that conditions (B) and (B*), where

(B*) there is a set T0 o f measure 0 such that for every К > 0 there are a number c > 0 and a measurable function h: T -> [0, + o o ] such that Jh(t)dn < + oo and

T

W{cx, t) < К Ф (х, t) + h(t) for every x gX and t e T \ T 0, to be equivalent to each other?

T o simplify the notations, in the sequel we shall assume that X is a separable space and Ф is a continuous Ф-function.

3.1. Th e o r e m. The equivalence

(B )< ^ [(B *) and

V

и > 0 c ->0

holds.

P r o o f. The implication (B )=>(B *) is obvious. Let и > 0 be an arbitrary fixed number. Write

A n = 0 U e T : aU'1/m(t) > l/n}, A = {J A„.

m— 1 n= 1

If pi{A) = 0 then

V V 3 a„>1/m(f) ^ l/n,

t$A ne N m eN

so, lim au c (t) = 0 for a.e. te T . c- 0

Now, assume ц(А) > 0. Then ц (А п) > 0 for some n e N : and hence /m(t)dn > f 0Lu>1/m(t)dfi ^ (1 /n)(i{An)

T A „

for each m = 1 ,2 ,... Therefore, (B) is not true for 0 < s < (l/n)g(An) and we get a contradiction.

T o prove the converse implication we shall show at first that condition (B*) is equivalent to the following one:

(10) V 3 < + o o .

u > 0 c > 0 T

Suppose that (10) does not holds, i.e.,

3 V $<xuA/m{ t ) d n = + o o .

u > 0 m e N T

Then, by 1.5 (for sequences um = u, cm = l/m, m — 1, 2, ...), we can find a sequence of simple functions (gk) such that

( П )

\teT: gk(t) Ф 0} n { t e T : gt(t) Ф 0} = 0 for к Ф l,

/ФЫ < m and I y l — for к = ¹, 2, ...

\ГПь J

In virtue o f (B*) (for К = 1/2и) there are a set T0 of measure 0, a number c > 0 and a function h: T -> [0, + o o ] such that H = §h(t)dn < +oo and

^(cx, t) ^ ^ - Ф (х , t) + h{t) for every x e X and t e T \ T 0.

2 и

Let m e N be an arbitrary number. Then there is k0e N such that l/mk ^ c for к ^ k0. Write

k g + n — 1

/ « = £ 9k, n = 1 ,2 ,...

k = kg

Then

fu(L)+n = f ° I 1*Ш+на\п+н.

2u 2и k=k 2

On the other hand, by (11)

k Q + n - 1 / j \

I М “ 0 * р и’

fc=fc0 ymfc 1

so H ^ \n for all n e i V , i.e., §h(t)dp = + oo.

T

The obtained contradiction ends the proof o f the implication (B *) =>(10).

Since the converse one is obvious, the equivalence (B*)<=>(10) holds. Now, it suffices to use the Lebesgue dominated convergence theorem and Remark 2.2 to obtain the thesis.

3.2. Lemma. The following conditions are equivalent: (i) the identity embedding i: Ь Ф(М ) -* L f (M ) is bounded, (ii) every ball Вф(г) is bounded in the space L f { M ).

P ro o f. It suffices to prove the implication (i)=>(ii) only. Let s > 0 be such a number that the ball B0(s) is bounded in the space L'f,(M ) and let e, r > 0. By 1.7, the ball Вф(г) is additively bounded in L 0(M ), i.e., Вф(г) c

+ "B0(s) for some n e N . Moreover, we can find a > 0 such that a 'B 0(s) a B v {e/n), so

а 'В ф(г) c + naB0{s) <= + пВ ч,{г/п) c By(e).

Thus Вф(г) is a bounded set in L'f,(M ).

3.3. Corollary. The Musielak-Orlicz space ЬФ(М ) is locally bounded if and only if the base {Вф{г): r > 0} of the topology of the space ЬФ(М ) consists o f bounded sets.

O f course, the above lemma and corollary remain true for arbitrary space X and Ф-functions Ф and T.

In the next theorem the following condition will be used:

(12) for every sequence (x„) o f elements o f the space X there is a set T0 o f measure 0 such that the implication

sup Ф (x„, t) < 4- oo => lim sup Tivx,,, t) — 0 for every t gT\T0

n e N v - * 0 n e N

holds.

3.4. Theorem. (B )o [(B *) and (12)].

P r o o f. Assume that conditions (B*) and (12) are satisfied and let и > 0.

Then JaUyC(t)dfi < + oo for some c > 0. Write (cf. (3), (4))

T

A u,c = { t e T : 0 < a MiC(f) < + go}, Л,«,с= {k e N : t e P UtC{xk)},

where the set {xk: k e N ] is dense in the space X. Let T0 be the set taken from condition (12) for the sequence (xk). Then

a,i,c(t) = SUP Ф (cxk, t) ^ K t < + oo

ke J t,u,c

for t e A U'C\T0 and some number K t > 0 . Hence

sup Ф(хк, t) < sup u*P(cxk, t) ^ u -K t < +oo

keJt,u,c k e Jttu,c

for every t e A uc\T0. By (12)

lim au>t,(r) = lim sup *P(i;xk, t) ^ lim sup P (v x k, t) = 0

v -*0 v -*0 ke Jt u v v ->0 keJt u c

for t e A UtC\T0.

If t e T \ A UtC then aUtC(t) is equal either 0 or + o o . In the first case ccUtV(t)

= 0 for every 0 < v < c. Moreover, the set \t e T: aUjC(t) = +oo} is o f measure 0, so lim au>i;(f) = 0 for a.e. ts T . Hence, in virtue of Theorem 3.1,

v ~>0

condition (B) holds.

Now, we shall show that (B)=>(12). Suppose that condition (12) is not satisfied, i.e., there is a sequence (xk) o f elements o f X such that the set

G = { t e T : supФ(xk, t) < + oo and lim sup’P^Xfc, t) > 0]

k e N c-*0 k e N

is o f positive measure. Write

G„ = \teTn: s u p 0 (x k, t) ^ n and lim sup Ф(схк, t) > 1/n]

k e N c - » 0 k e N

for n = 1, 2, ..., where (T„) is a nondecreasing sequence o f subsets of T of

oo oo

finite measure and such that ц (Т \ U Tn) = 0. Since n(G\ \J Gn) — 0, there

n= 1 n = 1

is a number n such that 0 < n(G„) < + o o. Now, write / k = xk/G for к

= 1 ,2 ,... Then

1ф{/к) ^ nP (G n) for each k e N .

inf sup’FfcXfc, t) > 1/n for a.e. teG„,

c> 0 k e N

(13) Moreover,

because x, t) < 4/(c2x, t) for every ct ^ c 2, x e X and a.e. r eT. Hence

V 3 Ixf/i f km\^ ~ B{G„),

m eN kme N \ m ) n

SO

r =

inf I r ( - f k m) > 0.

m eN \ m m)

Let F = {f km: m e N }. By (13) F cz Вф{г), where 1 if nn(G„) ^ 1,

\n-n{Gn) otherwise.

By Lemma 3.2, the ball Вф(г) is bounded in the space Ь^(М), so V 3 a - F с a-B0(r) с: В^(е).

e > 0 a > 0

On the other hand, taking a number m0 such that

— and ее

(о,

iV(—

m0 \ meN \m "

we obtain

This contradiction ends the proof.

Write

(14) for every sequence (x„) there is a set T0 o f measure 0 such that

sup ^(Xn, t) < + oo => lim sup ^(vx,,, t) = 0

neN v ~*0 neN

for every t e T \ T 0.

35. Le m m a. [(B *) and (14)1 =>(B).

P r o o f. In virtue o f Theorem 3.1 it suffices to show that lim aUit;(r) = 0 for a.e. te T and every и > 0. Let и > 0 be fixed. By (B*), j au,c{t)dn < 4- oo

T

for some c > 0. Let us put A = {te T : 0 < au c(r) < + oo} and Jt.u.c = Ac e /V: t e P uc(xk)}, where (xfc) is a dense subset o f X. Then

0 < a«,c(f) = sup 4{схк, t)xpuc{xk){t)

k e N

= sup *Р(схк, t) < + oo for t e A .

k e J t,u,c

N ow let £ > 0 and te A \ T 0, where the set T0 is taken from (14) for the sequence (xfc). By (14) we obtain the existence of a number af e(0, 1) such that

¥{at cxk, t) < £ for k e j tu<c. Therefore

<xu,v(0 = sup ¥ (vxk, t) < £

ke Jt,u,v

for all v ^ a t c because Jtuv c J liM(1 Thus lim aH>l) (f) = 0 for a.e. teT .

’ ’ f r-0

3.6. Corollary. I f the function ¥ satisfies condition ( + ) introduced in Corollary 2.4 {for Ф = ¥), then conditions (B) and (B*) are equivalent.

3.7. Th e o r e m. Assume that the topology in the space X is generated by a p-homogeneous norm Ц-||.

(a) I f lim inf ¥ (x, t) > 0 for a.e. t e T, then

IU II - + ac

(B )= » [(B * ) and V (sup<£(x„, t) = + oo for a.e. t e T)].

I U n l l " e / V

(b) [(B *) and V (sup ¥ { x n, t) = + oo for a.e. tG T )]= > (B ).

IU n ll - « » n t \

P r o o f, (a) Suppose that (B) is satisfied and there are a sequence (x„) and a set A o f positive measure such that ||x„|| -> + o o and sup<P(x„, t) < + oo for

neN

t e A . Then

sup ¥ (cx„, t) ^ lim inf ¥ (x, t) > 0

neN IU II - * + oc

for every c > 0 and t e A . Hence condition (12) does not hold and we get a contradiction.

(b) We shall show that ¥ satisfies condition (14). Let (x„) be an arbitrary sequence o f elements of the space X.

(b l) Suppose that the sequence (||x„||) is bounded, i.e., sup||x„|| ^ a for

neN

some a > 0. Now, let T0 be the set taken from Definition 1.1 and assume that e > 0, t<£T0. By the continuity o f the function ¥{-, t) at 0 there is a number Ô > 0 such that ¥ {x , t) < e for all ||x|| < <5. Since ||ux„|| < Ô for 0 < v

<{ô/a)1/p, lim sup¥(vx„, t) = 0, so ¥ satisfies (14).

v -*0 ne N

(b2) The sequence (||x„||) is not bounded. Then there is a subsequence (||x H) such that ||x„J| ->• + o o as к -> T o o . Write

A = { t e T : sup ¥ { x t) < T o o }.

k e N

By assumption, the set A is o f measure 0. Let t$ A . Then + oo = sup ¥ { x n , t) < sup ¥ ( x n, t).

k e N neN

Thus the condition on the left-hand side o f implication (14) is false, so 4*

satisfies condition (14).

3.8. R e m a rk . I f X = {R, |*|), then conditions:

(15)

V

\x„\ -*ao neN

and

(16) lim ^ (x , t) = +oo for a.e. t e T

|x | - » + 00

are equivalent.

3.9. C^orollary. I f the topology of the space X is generated by a p- homogeneous norm ||*||, lim inf T (x , f) > 0 and sup Ф(х, t) < + oo for all

I N ! - * + у x e X

te G , where G e l is a set of positive measure, then the identity embedding i : ЬФ(М ) —> ЬФ(М ) is not bounded. In particular, if T = Ф, then the Musielak- Orlicz space (Ь Ф( М ), |*|ф) is not locally bounded.

Write:

(L B *) there is a set T0 o f measure 0 such that for every numbers e > 0 and К > 0 we can find a number c > 0 and an integrable function h: T - > [ 0, + o o ] such that

Ф(сх, t) ^ К Ф (х, t) + h(t) for every x e X and t e T \ T 0.

From the above theorems we obtain the following corollary:

3.10. Corollary, (a) (LB) о [(L B *) and V lim a*c(f) = 0 for a.e. te T ],

u > 0 c~* 0

where the function a *c is defined by (4) with W = Ф.

(b) (LB)<=>[(LB*j and (13*)], where by (13*) we denote condition (13) with 4* = Ф.

(c) I f the topology of the space X is generated by a p-homogeneous norm

||*|| and lim inf Ф(х, t) > 0 for a.e. te T , then

| | x | | - » + 00

(LB) о [(L B *) and

V

||jc„|| -♦00 neN

3.11. Assume X = R and let the functions Ф, 4* do not depend on the parameter. Then condition (B) (which is equivalent to (B*)) takes the form

V 3 3 V

0 < K <1 c > 0 a > 0 x ^ a

if p (T ) < + o o ; and if p (T ) = + o o :

V 3 V

0 < K < 1 c > 0 x ^ O

If, moreover, 0 < Ф(х) < + o o for x > 0, then the above conditions can be written in the foliowing way ([3 ], [4 ]):

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

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

[3 ] P. T u r p in , Opérateurs linéaires entre espaces <T Orlicz non localement convexes, ibidem 46 (1973), 153-165.

[4 ] —, Conditions de bornitude et espaces de fonctions mesurables, ibidem 56 (1976), 69-91.

[5 ] M. W is la , Continuity o f the identity embedding o f some Orlicz spaces. I, Comment. Math.

24 (1984), 171-184.

[6 ] —, Continuity o f the identity embedding o f some Orlicz spaces. I I , Bull. Acad. Polon. Sci.

31 (1983), 143-150.

INSTYTUT M A TEM AT YK I

U N IW ERSYTET im. A. M IC K IEW IC ZA, P O Z N A N INSTITUTE OF M A THEM ATICS

A. M IC K IE W IC Z UNIVERSITY, P O Z N A N

Л > 0 t Ф ( 0

where t -+ + oo if /i(X) < + o o and 0,

+ 00, if ц{Т) = + oo.

References