On inclusions between Orlicz spaces

ROCZN1KI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVI (1986)

Ma r ia n No w a k

(Poznan)

On inclusions between Orlicz spaces

Abstract. Let <p, ф be ^-functions and (E, I , ц) be a measure space with a positive measure. Let L*4’, L0,p -denote the Orlicz space and the space of finite elements, respectively. The main aim of the paper is to show that if there hold the equalities: L0* = (J L*v, L*v — f) L0^,

<peT &eS

where for every <pe T (феЕ) the proper inclusion L*41 ç L°* holds, then the set T (resp. S) must be uncountable.

In [2] and [7] there are two equalities proved:

( + ) L°* = (J

L*9,

L** = C\L°*,

4> * Ф

where the union (intersection) is taken over all ф-functions q> (resp. ф) such that for all c > 0: ф{си)/(р(и)-^> 0 as u-+ oo in the case where E is a compact subset of Rm with the usual Lebesgue measure or ф(си)/(р(и)-► 0 as u-> oo and и-* 0 in the case where E = Rm.

Equality ( + + ) is used in studying some linear topologies in Orlicz spaces (see [8], the proof of Theorem 5.1, [10], p. 45). In [1] the equality L1 = (J W is proved, where I f denotes the

<P

Orlicz class, E is a compact subset of Rm and the union is taken over all iV-functions q>.

From Theorem 3.1 of this paper it follows that in equalities ( + ) and ( + + ) the proper inclusions L*q> ç L0^ hold and hence, by Theorem 4.3 (Theorem 5.3), the above union (intersection) is taken over uncountable set of (^-functions </> (resp. ф).

PRELIMINARIES

0.1. ^-functions

0.1.1. It is said that a function <p\ [0, oo)-» [0, oo) is a (p-function if it is continuous, non-decreasing and such that q>(0) = 0, q>(u)> 0 for u > 0 and (/>(u)-> oo for u — > oo ([4]).

0.1.2. Let ф and (p be ^-functions. We shall denote ф -kq>

{ф ■< (p, ф -k <f>) if there exists a constant c > 0 such that lim inf (р(си)/ф(и) > 0 as u-* oo and u~* 0 (resp. as w—► oo, as n —► 0). In other words, the relation ф A (p (ф -< (p, ф -к (p) holds if there exist constants c, d > 0 such that ф(си) ^ dq>(u) for и ^ 0 (resp. for и ^ u0 ^ 0, for и ^ u0, u0 > 0) ([4], p. 123).

0.1.3. Let ф and q> be (^-functions. We shall denote ф <1 cp

(Ф < q>, ф < (p) if for an arbitrary c > 0: lim sup ф(си)/(р(и) < oo as u-+ oo

and и -+

(resp. as и -*

, as и -+

In other words, the relation ф

(ф

<p, ф

holds if for an arbitrary c

0 there exists a constant d

0 such that ф(си) ^ d<p(u) for и ^ 0 (resp. for и ^ u0 ^ 0, for w < u0,

0 > 0) ([8], p. 72).

0.2. Orlicz spaces

0.2.1. Let

be «^-function and let

be an arbitrary measure space with a cr-additive and positive measure p on a-algebra I . For a real valued, /i-measurable function x defined on

we write

Qv (x) = J <p(\x{t)\)dp.

E

oo

It is clear that if p is a purely atomic measure, i.e., E = (J e^, ea m being m= 1

different atoms, then ^-measurable functions x in E are constant in each of the sets eam. Then we have

M x) = Z m= 1 where x (t)= c m for fe e ".

By L*(p we denote the linear space of those functions

for which Q^iXx)

<

for some Я > 0, by L0<p the linear space of those functions

for which

qv

(Лх)

for all Л > 0 and by Lv the set of those

for which

The linear spaces L*<p and L0<p are called the Orlicz space and the space of finite elements, respectively. The set L? is called the Orlicz class. Moreover, it

is known that L0<p is the maximal linear subspace of L** which is contained in V ([6]).

Let (E, I , p) be a finite (infinite) measure space. Then by B(E) (B0(E)) we will denote the set of all functions defined and /{-measurable on E, which are bounded (are bounded and vanish outside of some set of finite measure).

0.2.2. The inclusion L*v <= L** holds if and only if

(a) ф -kcp (ф <p) if (E, I , p) is an infinite (finite) atomless measure space ([4], 2.4),

00

(b) ф -к. (p if E = (J ea m with 0 < inf p(ea m) < sup p{ea m) <

([3]).

m=

1

m

0.23. A sequence (x„) in L*v is called (p-modular convergent to x e L **, in symbols х„Д x, if there exists a constant X > 0 such that ^ (A(x„ — x)) — > 0 ([6])-

0.2.4. In L*v, an F-norm can be defined as follows: ||x||v

= inf {e > 0: Q^ix/e) ^ ej. Moreover, ||x„||v -*■ 0 implies x„ 0 for a sequence

(x„) in L*v ([6]).

R E S U L T S

1. In this section we shall give necessary and sufficient conditions for the inclusion L*v c= L0^.

1.1.

The inclusion L** a L0* holds and simultaneously x„-^0 implies ||x„|j^ -*• 0 for every sequence (x„) in L*<p if

(a) ф <1 (p {ф < (p) if (£, Г, p) is a measure space (a finite measure space), 00

(b) ф < (p if E = (J eam, where ea m are different atoms with inf p(e^) > 0.

m — 1 m

P roof, (a) Since ф < (p (ф < (p), for every natural number m there exists a constant dm > 1 such that ф{и) < dm (pm(u) for и ^ 0 (resp. for и > um ^ 0), where (pm{u) = (p(u/m). Hence by ([4], 2.23), we have immediately

GO

L*9 = (J l f m с Lf and since L°* is the maximal linear subspace of £**

m = 1

which is contained in Lf, we get L*< p c: L0*.

From ([8], Lemma 3.1) and ([5], 3.32), we have that x„-^0 implies

\\хп\\ф -* 0 for every sequence (x„) in £*ф.

(b) First, we shall prove that the inclusion L*v с L°* holds. Obviously, it sufficies to show that L** c i f . Indeed, let x e L * <p. We have x(t) = cm for tee°m and denote c = inf p{ea m) > 0. Hence there exists a number A0 > 0 such that

00 X

(1) с X <p(A0 \cm\)< X

> M c m\)p{ea m) = ^(A 0x )< oo.

m - 1 m = 1

Since ф <3 (p there exists a constant d > 1 and a number u0 > 0 such that

(2) ф(и) ^ d(p(À0u) for и < u0.

From (1) it follows that lim <p(A0|cJ) = 0 and hence there exists a natural

m->oo

number N such that (p(À0\cm\) < (p{À0u0) for N. Hence |c j <

0 for w ^ N. Therefore from (2) and (1) we obtain

вф(х)= X Ф (\cm\) p(ea J = X Ф(\Ст\)р(еа т)+ X Ф(\Ст\)РЮ

m = 1 m = 1 m= N

N - 1 oo

< X + d X <p(lo\cm\)p{ea m) <

.

m = 1 m = N

Thus x e L f.

Now, we shall show that 0 implies ||хп||^-> 0 for every sequence (x„) in L*v. Indeed, let x„-^ 0. We have x„(t) = c” for te e a m. Then there exists a constant A0 > 0 such that

= f <PttoK\)p(ea J - ^ 0 .

m = 1

(

3

We shall show that j|xj|^ -> 0. In fact, let г be an arbitrary positive number.

Since ф < (p, there exist a constant d > 1 and a number u0 > 0 such that

(4) ф(и/£) ^ d(p(À0u) for u ^ u 0.

Denote E„ = { te E : |x„(OI < «о} and x'n = xn-xEn, x'„' = -X

-

„- We have Qv^ox^) ^ (p{X0u0) ii ( E - E n) and hence by (3) there exists a natural number IVj such that p{E — E„) ^ (Я0*„)/<?(Я0uo) < c for N x.

Therefore p{E — En) = 0 for n ^ IV j and hence

(5) x" = 0 for n ^ N i .

On the other hand, from (3) it follows that there exists a natural number N 2 such that

CO

(6) = X И Л )к«1)М О <еА * for n ^ N 2.

m= 1

Then from (5), (4) and (6) for n ^ max(N lt N 2) we get

* 00

e*(xje) = Q^ix'Je) = £ ф (\c^\/e)p (C n En) m= 1

00 g

£ <pC*0ld ) ju ( O ^ d*- = £,

m = 1 d

i-e-, 1Ы* ^ 8-

1.2.

Suppose the inclusion L <=. L0* holds. Then

(a) ф <\ (p (ф < (p) if (E , Z, p) is an infinite (finite) atomless measure space,

CO

(b) ф < q> if E = (J ea m with 0 < inf/ 4 0 ^ sup / 4 0 < oo.

m = 1 m m

P ro o f, (a) Since L** <= L0^, we have L** = (J l f m c i f , where (pm(u)

m= 1

= (p{ujm). Hence by ([4], 2.23) for every natural number m there exists a constant dm> 0 such that ф(и) ^ dm(pm(u) for и ^ 0 (u ^ um ^ 0) and this means that ф <i <p (resp. ф < </>).

(b) Suppose L*v c L0^ and let us assume that ф < q> does not hold. Let

<5 be a number such that 0 < à < 1. Then there exists a constant c0 > 1 and a sequence (un) of positive numbers such that

(1) (p(un) ^ 3/(2nd) and ф(с0un) > 2"<p(u„), where d = sup / 4 0 -

m

Then there exists a sequence {EJ of pairwise disjoint sets in E such that

(2)

< / 4 0 ^

2 > (

„) <

Ы

Define

un for te E n, n = 1 ,2 ,...,

CO

0 for t e E — IJ E„.

Then

00

M * ) = Z Z < p W /(2> M = i >

and hence

L*v.

On the other hand, from (2), (3) and (1) we have

00 00

M c‘o *)= Z *Mco un)p(E n) ^ £ ф (co un)((1/2"(p{u„)) — d)

GO

> Z (1-<5) = 00>

and hence хфЬ0ф. Thus we are led to a contradiction.

1.3.

Suppose xn-^

implies

for every sequence

in B0(E) (B(E) if p(E) < oo). Then there hold conditions (a) and (b) from Theorem 1.2.

P ro o f, (a) It follows from ([11], Theorem 2.4).

(b) Suppose x„-^0 implies ||xn||^,->0 for every sequence (xn) in B0(E) and let us assume that the relation ф < tp does not hold. Then conditions (1) and (2) of the proof of the previous theorem are fulfilled. We shall show that there exists a sequence (xn) in B0(E) such that хпЛ 0 and HxJI^-y> 0.

Then xne B 0(E). We have e„(x„) < g (м„)/(2п <p (n„)) = 1/2" and hence x„-^0.

On the other hand, we have

<P*(c0 *„) = Ф(c0 un)p (£„) ^ ф(с0ип) - ( \ - 2"dtp(M„))/(2" <p(u„))

> l- < 5 > 0 and hence ||х„|Ц -f 0.

2. Now, we shall give a necessary condition for the inclusion Ь0ф c L*<p.

Namely, the following holds:

2.1.

There exists х е Ь 0ф such that хфЬ*4* if

(a) lim inf ф (cu)/q> (u) = 0 for all c > 0 as и -> оо or u~* 0 (и -> oo ) if {E, I , p) is an infinite {resp. finite) atomless measure space,

Define

for te E n,

for t$E„.

(b) lim inf ф(си)/ср(и) = 0 for all o O if E = (J ea m with

и -*0 m —

1

0 < inf ц{е^) < sup ц(е„) < oo.

m m

(c) lim \l/(cu)/(p(u) = 0 (lim ф(си)/(р(и) = 0) for all c > 0 if E = U ea m

u-*0 и

— *oo

1

with inf /i(e“) > 0 (inf ju(e“) = 0).

m m

P ro o f, (a) Under the assumption for every pair of natural numbers (i,j) there exists a number ui} > 0 such that

ф(21+]ии) < (p(uu/i)/i2

(resp. ф (Т+]ии) < (p(uu/i)/i2 and ф{2*'ч ии) ^ n 2/2).

Now, let Еи (i , j = 1, 2 ,...) be pairwise disjoint /z-measurable sets in E such that

И(Еи) = l/(/22 W +4 ) )

(resp. Ц (Eij) = Ц (E)/(i2 2J ф (2i+j u0))).

for teE ij, i , j = 1 , 2 ,..., for t e E - U Ü Еи .

«= l j=l

First, we shall show that x e L i.e., ^(A\x(t)\)dpi < oo for all A > 0.

E

Indeed, let Я > 0 be given and choose a natural number N such that Я < 2N.

Then we have Define

*(0

oo oo

^(A \x (t)\)d n = £ ( Z Ф№ij)v(Eij))

E i = 1 j = 1

Then

^ X (X ф(Аии)ц{Еи)) + £ (X Ф(2Nuij)и(Ец))-

i = 1 j = 1 i = N j = I

00 00 00 00

I d Ф(2"щ,)1л(Е„))*; £ d ^ (2 'tg /( ;22 w ' +' “«)))

00 00 00

^ Z ( Z VO’2^) = Z V*2 < °o

i

=

i i = iv

oo 00 00

( I d ^(2"иц)лх{£у)) ^ д(£) I 1/i2 < oo).

i = JV j = 1 i = N

Moreover, for every natural number i'0 with 1 ^ i 0 ^ N — l there exists a natural number j 0 such that Я < 2l° +Jo. Then we have

oo J 0 1 oo

Z ^(ÂulV)At(£/o/) < Z Z

J= 1 J= 1 •/'=

ф{2‘0+1ои,*

M M

J0~ 1

s: X 1ИЛ«,0, ) М М +

j - i Z ^(2 ,0+' 0«,ы)/(.а2 ^ (2 '0+^ 1Ы))

« I + P £ 1 / 2 ' < с о .

j=l *0 7=J0

oo -/'о- 1 u / £ \ oo

(I ♦ f c . / l l i l i v X I ⁺ S 1 /2 '< CO).

7=i

- 1 *

Hence

/V— 1 00

Z ( Z •И Я и .о у Ы ^ Н oo.

i = l J = 1

Now, we shall show that хфЬ*4*, i.e., § (p(2\x(t)\)dfi = oo for all Я > 0.

E

Indeed, let Я be an arbitrary positive number and let M be a natural number such that Я > 1/M. We have

{<р(Я|х(Г)|)4и= X ( Z Ф(Яму)^(£у))

7 = 1 i = l

00 00

^ Z ( Z Ww.j/O^ (£«■;))

7 = 1 i — M

00 00

^ Z ( Z i2 Ф (2‘+j “y)/(*2 2j Ф (2,+J «y))) 7= 1 i= A#

oo x x

= 1 ( 1 1/2') = Z 1 = °°>

7* = 1 i = M i = M

x *

(resp. |

(Я|

(0 | ) ^ ^ Z M£) = oo).

£ i=M

(b) Let Ô be a number such that 0 < <5 < 1. Since for all c > 0:

lim inf ф(си)/ф{и) = 0, for every pair of natural numbers ( i,/) there exists

и

-*0

a number ui} > 0 such that

Ф (2i+j ии) ^ S/(di2 2j) and ф (2 +j uu) < (p(uu/i)/i2,

where d = sup ц(еа т).

Then there exist pairwise disjoint sets

E such that

i2 2) ф (2‘+j ии) d ^ р(Ей) <

122 ф (21+)ии) {i,j = 1, 2, ...).

Define

*(0 = 0

for te E u, i , j = 1, 2, .., 00 00 for te JE - u

Eu.

i = l 7 = 1

Similarly as in the proof of (a) we show that and хфЬ*<р.

(c) Since for all c > 0: lim ф (cu)/(p (u) = 0 (lim ф(си)/<р{и) — 0), for

и-*0 u->oo

every pair of natural numbers (i, j) there exists a number vu > 0 such that ф(2'+]и) < <p(u/i)/i2 for и ^ i’и (resp. for и ^ vu).

Since inf p(e„) > 0 (resp. inf ju(e^) = 0) there exist pairwise disjoint sets

m m

Eu in E such that

n(Eij) ^ l/( r 2Jф(2‘+jvu)) (resp. p(Eu) < 1 /(/2 2J‘ф( 2 +jvu)))

for every pair (i, ;). Hence for every t>fj there exists a positive number му ^ i?y (resp. Uij^Vij) such that p (£y) — l/(i2 2j ф (2i+j ии)) and simultaneously, ф{2'+]ии) ^ (p{uu/i)/i2.

Define

for teEfj, i , j — 1, 2, ...,

for Û Ü %

i = l J = 1

Similarly as in the proof of (a) we show that x e L 0* and хфЬ**.

From Theorem 2.1, Definition 0.1.2 and Theorem 0.2.2 we get

2.2.

I f a measure space (E , I , p) is of the type (a) (b), (c), where (a) (E , I , p) is an inifnite atomless measure space,

(b) (E , I , p) is a finite atomless measure space, 00

(c) E = (J ea m with 0 < inf p(e“) < sup p(e„) < oo,

m = 1 m m

then the following conditions are equivalent : (i) L** cz L**,

(ii) L°* cz L**,

(iii) (р-кф ((р-кф, <р-кф) if (E, Z, p) is of the type (a) (resp. (b), (c)).

R em ark . The above theorem follows from Theorem 10 of [9] in the

case where E is the set of all natural numbers.

3. We shall present here a criterion for the proper inclusion L** L0ф.

3.1.

The proper inclusion L**

Ь0ф holds if and only if (a) ф < (p and lim inf ф (cu)/<p(u) = 0 for all c > 0 as и -> oo or и -*0 if (.E, Г, p) is an infinite atomless measure space,

(b) ф < <p and lim inf ф(си)/(р(и) — 0 for all c > 0 if (E, I , p) is a finite

u -►ao

atomless measure space,

00

(c) ф < q> and lim inf ф(си)/(р(и) — 0 for all c > 0 if E = У ea m with

и ->0 m = 1

0 < inf jU (0 ^ sup d(em) < 00 •

m m

P ro o f. Necessity follows from Theorem 1.2, Theorem 2.2 and Theorem 0.2.2. Sufficiency follows from Theorem 1.1 and Theorem 2.1.

4. Now, using the proof of Theorem 2.1, we prove a generalization of Theorem 2.24 of [4].

4.1.

Let ф and (pm be (p-functions for m — 1 ,2 ,... The inclusion 00

Т0ф cz (J i f (Pm holds if and only if there exist a natural number m0 and

m - 1

constants c, d > 0 such that the inequality

( + ) <Pm0 (u) ^ d ф (cm)

holds

(a) for и ^ 0 if (E, I , p) is an infinite atomless measure space, (b) for и ^ u0 ^ 0 if (E, I , p) is a finite atomless measure space,

00

(c) for и ^ U

,

0 > 0 if E = U eH m with 0 < inf p(ea m) ^ sup p{ea J < со.

m = 1 m m

P ro o f. Sufficiency being obvious, we prove the necessity. Suppose 00

Ь0ф cz (J l?* ”1 and assume that ( + ) does not hold. It means by Definition

m = 1

0.1.2 that for j

1, 2, ...: lim inf i

(cu)/(Pj(u) = 0 for all c > 0 as u-> oo or и -> 0 in case (a) (resp. as и —► oo in case (b), as и -> 0 in case (c)).

Then for every pair of natural numbers (i,j) there exists a number uu > 0 such that

<М2НЧ ) < (pj(Uij/i)/i2 in case (a),

»// (2‘+J Uij) < <Pj(ipj/iyi2 and ф(2,+]ии) ^ n2/2 in case (b), ф(2i +j My) < (pj(Uij/i)/i2 and ф(2i+jму) ^ S/(di2 2j),

where d = sup p{e^) and <5 is a number such that 0 < <5 < 1 in case (c).

for

i , j = 1, 2, for fe JE - û Ü Еф

i = i j= i

where sets are as in the proof of Theorem 2.1.

Similarly, as in the proof of Theorem 2.1 we show that х е Ь 0ф.

Now, we shall prove that х ф Ь for к = 1, 2, ... Indeed, let к be a fixed natural number. Let Я > 0 be given and choose a natural number N such that Я > 1/iV. Then in case (a) we have

j (pk (Я \x(t)\)dp = £ ( Z <Pk (Лии) n(Eij))

E

^ Z (Z Р*(м«У0/*(£«/•))2* f <p*(w.-*/0/*(£,-*)

j= i ;=/v j= v

£ (2

Ф

ф (2i+k uik)) ~

1 = ЛГ 1 = ЛГ

Hence хф Ь*^. Similarly we show that x£L*n in cases (b) and (c). Thus condition ( + ) holds.

00

4.2.

Suppose the equality Ь0ф — (J L*<Pm holds. Then there

m = 1

exists a natural number m0 such that Ьоф — L m° and ф satisfies the A2- condition

(a) for all и if (E. I , p) is an infinite atomless measure space, (b) for large и if (E, I , p) is a finite atomless measure space,

CO

(c) for small и if E = (J ea m with 0 < inf p{e™) ^ sup p(ea m,) < oo.

m = 1 m m

P ro o f. By Theorem 4.1, there exists a natural number m0 such that

<Pm0 ^ Ф (<Pm0 <Pm0 -к Ф) in case (a) (resp. (b), (c)). Hence from Theorem 0.2.2 we have Ь0ф с: Ь*ф cz L <Pm°. On the other hand, since L*<Pmо с: Ь0ф we have Ь0ф = L*Vm° and by Theorem 1.2, it follows ф <i (pMQ (resp. ф < q>mQ, ф < (Pm0). Hence ф <i ф {ф < ф, ф < ф) and this means that ф satisfies the d 2-condition for all и (resp. for large u, for small u).

From the above Theorem we obtain:

4.3.

Let (E, I , p) satisfy the assumptions as in previous theorems o f this section. Suppose the equality Ь0ф = (J L*v holds, where T is

<peT

In all cases define

x(t) =

an arbitrary set of (p-functions and there holds the proper inclusion L*4* Ç L°*

for every (peT.

Then the set T is uncountable.

4.4. T

. Suppose the equality Ь0ф = (J L*v holds and suppose ф

<peT

does not satisfy the A2-condition of types (a), (b), (c) of Theorem 4.2. Then the set T is uncountable.

5. Finally, we prove a generalization of Theorem 2.21 of [4].

5.1. T

. Let <p and фт be (p-functions for m = 1, 2, ... The inclusion L*4*

f] L 4>m holds if and only if there exists a natural number m0 and

m = 1

constants c, d > 0 such that the inequality

( + ) (p{u )^d sup(фх{си), ..., фто(си)) holds for и satisfying (a), (b), and (c) from Theorem 4.1.

P ro o f. Sufficiency. Denote фт(и) = supfi/^(м), ..., фт(и)). Then ( + )' means that (p к фто (<P ^ ^ $m0) in case (a) (resp. (b), (c)). Hence from Theorem 0.2.2 we have Ь°Фт cz L*v. Since фто{и) ^ ф1(и) + ... + фто(и), we have f| L°*m cz

and hence f)

cz

m = 1 oo m = 1

Necessity. Suppose L*<p

f) Ь Фт and assume that ( + ) does not hold.

m= 1

Then there exists a sequence (um) of positive numbers such that:

(p{um) > 2m\J/m{rn2um) in case (a),

(p(um) > 2тфт(т2ит), мт | oo and ^ 1(n1) > 1 in case (b), (p (

J > 2m фт (m2 um), um j 0 and фт (m2 um) ^ 0/{2m d), where d = sup p{e^) and Ô is a number such that 0 < <5 < 1 in case (c).

m

Then there exists a sequence (Em) of pairwise disjoint ju-measurable sets in £ such that:

n(Em) = 1/(2 тфт(т2ит)) p(Em) = p (£)/(2m фт (m2 um)) 1

2m фт(т2 um) In all cases define

— d ^ p(Em) ^ 1 2тфт(т2и,

in case (a), in case (b), in case (c).

x(t) = rnum 0

for te E m, m — 1, 2, ..., 00

for t e E — (J Em.

m= 1 We shall show that x e fj Е°Фт and x$ L * <p.

m= 1

(> - Prace Matematyczne 26.2

Indeed, let n be an arbitrary natural number and let X > 0. Choose a natural number m0 such that m0 ^ max(n, X). Since the sequence фт(и) is non-decreasing for и ^ 0, in the case (a) we have

00

Q^n (Лх) = Z (^mwj/(2m фт (m2 n J)

m = 1

™0~1 сю

< Z ФЛтоит)/(2тфт(т2ит))+ £ фт{т2ит)/(2тфт(т2ит))< со.

m = 1 m = mg

oo Since ф„(и) ^ ^„(«) (n = 1, 2, ...) we have xe f] •

и = 1 00

Similarly, we prove that x e

L0^" in cases (b) and (c). On the other

n = 1

hand, for an arbitrary X > 0 choose a natural number mj such that X > 1 /m l . Then in case (a) we get

oo 00

Z (Xmum)/(2m фт(т2 um)) ^ £ (p(um)/(2m фт(m2 u j)

m = 1 m= mj

oo

> Z 1 = 00 •

m =

Similarly, in case (b) and (c) we have

(Я

) = oo for all X > 0. Thus хфЬ**.

Thus we get a contradiction and hence ( + ) holds.

5.2.

Suppose the equality L*v = f] L Vm holds. Then there

m = 1 m 0

exists a natural number m0 such that L*<p = f] Ь Фт and q> satisfies the A2-

m — 1

condition of types (a), (b), (c) of Theorem 4.2.

00

P ro o f. Let L*v

L Vm. By Theorem 5.1, there exists a natural

m = 1

number m0 such that (p ^ фто {(p < фто, (p фто) in (a) (resp. (b), (c)), where Фт0(и) — sup(^i (n), ..., фт0(и)) for и ^ 0. Hence from Theorem 0.2.2 we have L 0 c L*<p. It is seen that L 0 = f] Ь Фт and therefore f] Ь Фт

m = 1 m= 1

— L*v. Since L*v cz L ^m°, from Theorem 1.2 we get фто<кр (фто< <p, Фт0 < Ф) in case (a) (resp. (b), (c)). Hence q> <\ q> (q> < (p, (p < tp) and this means that q> satisfies the A2-condition for all и (resp. for large n, for small w).

The following theorems follow directly from the above theorem.

5.3.

Let (E , I , fi) satisfy the assumptions as in this section and

let T be a set of (p-functions ф such that: ф1, ..., фте Т implies L*v L°^w.

Suppose the equality L*v = 0 L0^ holds. Then the set T is uncountable.

феТ

5.4.

Suppose the equality L*,p = f) L0* holds and (p does not феТ

satisfy the A2-condition of types (a), (b) (c) of Theorem 4.2. Then the set T is uncountable.

References

[1] M. A. K r a s n o s e l’s k ii and Ya. B. R u tic k ii, Convex functions and Orlicz spaces, Groningen 1961.

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

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[6] —, On generalized Orlicz spaces, Bull. Acad. Polon. Sci. 8 (1960), 349-353.

[7] M. N o w a k , On two equalities for Orlicz spaces, Funct. Approx. 10 (1980), 69-81.

[8] —, On two linear topologies on Orlicz spaces L*q>, Comment. Math. 23 (1983), 71-84.

[9] I. V. Sr a g in , Uslovia vlozeni klassov posledovatelnostei i sledstvia iz nih, Mat. Zam. 20 (1976), 681-692 (in Russian).

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131 (1976).

[11] M. W isla , Continuity o f the Identity Embedding of Some Orlicz Spaces II, Bull. Acad.

Polon. Sci. 31 (1983), 143-150.

INSTITUTE OF MATHEMATICS, A. MICKIEWICZ UNIVERSITY, Poznan INSTYTUT MATEMATYKI, UNIWERSYTET im. A. MICKIEWICZA, Poznan