Abstract. Let q> and ф be ^-functions or Orlicz functions. We shall say that <p increases essentially more rapidly than ф for all и (for large u) in symbols ф <p (ф <k <p) if for c > 0

Series I: COMMENTATIONES MATHEMATICAE XXV (1985) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXV (1985)

M. N o w a k (Poznan)

Inductive limit of a sequence of balanced topological spaces in Orlicz spaces (ц)

Abstract. Let q> and ф be ^-functions or Orlicz functions. We shall say that <p increases essentially more rapidly than ф for all и (for large u) in symbols ф <p (ф <k <p) if for c > 0

Ф(ы) л . .. Ф{<*) л Ф(с и) h m --- = 0 and hm — -— = 0 I hm --- - = 0

«->0 <p(u) ii ~* oo ф{и) \и-хю (p(u)

Let <p be (^-function and let (£, I , ц) be a measure space with a positive measure. In [7] we have considered some linear topology on (ц), denoted by which has a base of neighbour­

hoods of 0 consisting of all sets of the form: K^(r) n Ц?(ц), where r > 0 and ф is such that Ф <q>.

Let be the topology on L*EV (ц) of the strict inductive limit of the sequence of balanced topological spaces ((Kv(2"), {2B)): n ^ 0), where 0 is the topology of convergence in measure, i.e., .T j is the finest of afi Hnear topologies 2Г on LE 'P(/fi which satisfy the condition:

Я~\К9(2») = ^ о\Кф(2п) ^ОГ П ^ 0.

The main aim of present paper is to prove that the topology is identical with the topology .T t on L fU ji).

Introduction. Let ç be an Orlicz function and let (E, I , p) be a measure space with a positive measure. In [8 ] Ph. Turpin has considered the topology on ,L* ev (p), which we denote by of the strict inductive limit of the sequence of balanced topological spaces ((X^(2"), ^ "o li^ "))1 n ^ 0 ) , where 0 is the topology of convergence in measure. Ph. Turpin proved that in the case when cp is non-bounded and satisfies the d j-^ n d itio n and (E, I , /л) is a measure space with a positive, finite, atomless measure the topology ZTi is identical with some linear topology on L Ÿ (p) which has a base of neighbourhoods of 0 consisting of all sets of the form: К ф{г )п Ц ? {p), where r > 0 and ф is an Orlicz function such that lim ф(си)/(р{и) = 0 for every c > 0 and satisfies the d 2-condition

H-+CO

(Chapter I, Theorem 1.1.2).

0. Preliminaries

Throughout this paper we assume that (E, Z, fi) is a measure space with

a positive measure. We will denote the Orlicz spaces and the spaces of finite

elements respectively by L** and L°*.

296 M . N o w a k

0.1. Or liez spaces.

0.1.1. It is said that a function cp: [0, oo)-+ [0, oo) is a cp-function if it is continuous, non-decreasing and such that <p (0) = 0, <p (u) > 0 for и > 0 and <p (и) -*■ 00 for U -> X .

It is said that a function <p: [0, oo) -> [0, oo] is an Orlicz function if it is continuous in 0, non-decreasing and such that <p(0) = 0, and non-identical equal 0 ([8], p. 24).

0.1.2. ^T

. Let ip and ф be ip-functions. \ (a) I f ф < ip, then L *9 a Е >ф.

(b) I f ф h ip and //(E) < oo, then L*9 cz l?*.

The proof of this theorem may be found in ([6], the proof of Theorem 1) and in [4], assuming respectively in case (a) that E — Rm and pi is the Lebesgue measure and in case (b) that E is a bounded and closed set in Rm. In the case when (E, I , pi) is a measure space with a positive measure the proofs of (a) and (b) are the same and will be omitted here.

0.1.3. T h e o r e m . In L*9 an F-norm can b e defined as follow s:

N L = inf U > 0: C?„(x/A) ^ A}.

The space L*9 is complete with respect to the E-norm || ||v ([5]).

We shall denote by the topology on l ? 9 generated by the E-norm || - ||v.

0.1.4. T h e o r e m . L et ip be a ip-function. Then L*9 = П Ь0ф

Ф

where the intersection is taken over all (p-functions ф such that ф <£ <p.

The proof of the above equality is the same as in [6], where was assumed that

£ = IC and pi is the Lebesgue measure and will be omitted here. In the case when ip is an Orlicz function which is non-bounded and satisfies the z ^ ^ n d itio n and (E, I , pi) is a measure space with a positive, finite, atomless measure then the equality l * 9 = П where the intersection is taken over all Orlicz functions ф

i Ф . . .

such that ф ^ ip and which satisfy the A 2-condition, was obtained by Turpin ([8], Chapter I, Theorem 1.2.2). In the case when ip is a /p-function and E is a bounded and closed set in R“ the equality L*” = П L°*\ where denotes the set of all /р-functions ф such that ф -k (p, was obtained by Lesniewicz in [4].

0.2. Topology of convergence in measure.

0.2.1. Let S0 be the linear space consisting of all real valued functions

defined and //-measurable on £ which are almost everywhere finite valued and

bounded outside a set of finite measure. Then an F-norm || ||0 can be defined in S0 as follows:

IWIo = inf Je > 0 : p { t e E : [x(t)| > e} < e} ([8], p. 30).

We shall denote by the topology on S 0 generated by the F-norm || ||0. It is seen that a sequence (xn) in S0 is convergent to x e S 0 in ZT0 if and only if the sequence (x„) is convergent to x in measure (in symbols x n^>x).

Moreover, for every (^-function tp we have L *</> <= S0 and is strictly finer than .T 0 restricted to L*v([8], p. 30).

0.3. Linear topology on L**.

03.1. Let q> be (^-function. Denote by ¥ <q> the set of all (^-functions ф such that ф (p. Since L*’’ c= L0^ for every ф е ¥ <<р we have the linear projective system:

j+: L** с (Ь0ф, 'Т'ф), where ф е ¥ <<p

and denotes for every ф е ¥ <<р the usual topology ЗГф on L** restricted to L04>.

We shall denote by 3~<<p the linear topology of the above projective system.

The above definition was introduced in ([7], p. 74) by the assumptions that E

= RT and /г — the usual Lebesgue measure.

Now, we recall some theorems from [7] which we need in the paper.

03.2. T heorem . The topology has a base o f neighbourhoods o f 0 consisting o f all sets o f the fo r m :

К ф(г) n L**, where ф е ¥ <<р, r > 0 ([7], p. 74).

0 3 3 . T h e o r e m . The topology is coarser than the usual topology on L** ([7], p. 78).

0.3.4. T heorem . The space (L*tp, , T <<P) is complete ([7], p. 82).

The above theorems are proved in [7] in the case when (F , I , p) is a finite dimensional Euclidean space with the Lebesgue measure but they remain true when (F , I , p) is an arbitrary measure space with a positive measure. The proofs of these theorems are the same when (F , I , p) is a measure space with a positive measure.

0.4. Inductive limit of a sequence of balanced topological spaces. In this section we recall some notions and theorems proved in [8] which we need in the paper.

0.4.1. D e f in it io n . Let F be a linear space. Let ((F„, n > 0) be a sequence of topological spaces such that F„ are balanced subsets of F. Let the following conditions will be satisfied:

(0 F = U F*-

n > 0

298 M. Nowak

(ii) F n + F n a F n+1 and the function F n x F n3(x , y)\~* x + y e F n+l is con­

tinuous (hence the imbedding F n <=> F n+l is continuous).

(iii) the function D x F ns{À, x ) -> À x e F„, where D = { k e R : |A| < 1} is continuous.

Let be the finest of all linear topologies on F which induce on F n the topology coarser than &~n for every n ^ 0. Then the linear topological space (F , j) is called the inductive limit of the sequence of balanced topological spaces ((F n, ,T n): n ^ 0).

If, moreover, the following condition is satisfied:

(iv) for every n ^ 0 J r „+ г induces 3~n on F n, then (F , У т) is called the strict inductive limit of the sequence of balanced topological spaces ((F„, n

> 0)([8], Chapter I, 1.1.1).

0.4.2. T heorem . L et (F , j) he the inductive limit o f a sequence o f balanced topological spaces ((F „, n ^ 0). Then ,T l has a base o f neighbourhoods o f 0 consisting o f all sets o f the fo r m :

1 4 = û Œ _n П — 0 n= о к),

where V„ f o r every n ^ 0 is a neighbourhood o f 0 in (F„, F n) ([8], Chapter I, Theorem 1.1.6).

0.43. T heorem . L et (F , t) be the strict inductive limit o f a sequence o f balanced topological spaces ((F n, ,9~n): n ^ 0). Then fo r every n ^ 0 the topology

induces on F n the topology 3 ~n ([8], Chapter I, Theorem 1.1.8).

0.4.4. R e m a rk . From Theorem 0.4.3 it follows that in the case of the strict inductive limit the topology F j is the finest of all linear topologies ЗГ on F which satisfy the condition: F\Fn = 3Tn.

0.43. D efinition . Let (F , .T j) be the strict inductive limit of a sequence of balanced topological spaces ((F„, ^~„): n ^ 0). Then for every n ^ 0 the sets ((x, y ) e F „ x F „ : x —y e V), where F is a neighbourhood of 0 in (F„+1, dt~n+1) constitute a fundamental system of the uniform structure on F„. It is easy to check that this uniform structure coincides with the uniform structure on F„ induced by the canonical uniform structure on (F , 3Tj) ([8], Chapter I, Theorem 1.1.10).

0.4.6. T heorem . L et (F , &~j) be the strict inductive limit o f a sequence o f balanced topological spaces ((F„, ZTn)\ n ^ 0). Then, if F„ is com plete fo r the uniform structure described above, then the space (F , 3Tj) is com plete ([8], Chapter I, Theorem 1.1.10). 1

1. The strict inductive limit of the sequence ((X v(2"), (2B)): и ^ 0) in

Orlicz spaces L** *

1.1. D efinition . Let q> be a (^-function and let be the topology of

convergence in measure on S 0. Write K <p(2n) = { x e L * 41: ||x||v ^ 2n). Then the

sequence ((K v (2n), (2И)): n ^ 0) of balanced topological spaces satisfies conditions (i)-(iv) of Definition 0.4.1.

Hence, by Remark 0.4.4. one can define on L** the topology as the finest of all linear topologies iT on L** which satisfy the condition: n —

for every n ^ 0.

Moreover, from Theorem 0.4.6 it follows that the space (L**, 3Ti) is complete.

The main aim of this paper is to prove that У <<p = 2Tj. For this purpose we shall prove a few theorems and lemmas.

1.2. L emma . L et ф and (p be (p-functions such that ф -k(p. L et St, a {x e L*<p:

IMU ^ a f or som e a > 0}. Then a set & has uniformly absolutly continuous norms

|| ||^, i.e., fo r every e > 0 there exists Ô > 0 such that fo r every x e J there holds fo r p ( A ) < 6 .

( xa = X'X a > where X a 15 the characteristic function o f A.)

P r o o f. Indeed, let e > 0. Since ф < (p there exists a constant К > 0 such that

(1) — — — > 2a/e for u > K .

ф(аи/е) Substituting in (1), и = \x(t)\/a we get

(2) ф(\х(1)\/е) < e(p(\x(t)\/a)/2a. for \x{t)\/a > K . Let S = e/2ф (аК /е) and p(A ) < 3. Write

A l = { t e A : |x(t)|/a > K } and A 2 = {t e A : \x{t)\ja < K } . Thus, from (2) for x e M and p(A ) < 5 we get

$Ф( |x(OI/e)d/i= J ф(\x(t)\/e)dp+ $ ф { \ х { ^ ^ р

A A\ A 2

^ J (p (\x(t)\/a)dц + ф {aK /£) ц (А2) ^ e, i.e., | | xj* ^ e.

R e m a rk . The above lemma is proved in ([3], Lemma 13.2) in the case where ф and (p are N -functions and E is closed, bounded set in Я"1.

1 3 . T heorem . On the Orlicz spaces L *4* the topology 2Г0 o f convergence in measure is strictly coarser than

P r o o f. By Definition 0.3.1 the topology F <(p is the supremum of the family of topologies for ip e W * * . Since ^ ,| L** £ ^o\L*v for ф е Т <<р (0.2.1) we get

1.4. T heorem . L et Z c: L*v be a metric bounded set, i.e., sup{||x||v: x e Z ]

30 0 Mf. Nowak

< оо. Then on a set Z the topology is identical with the topology i-e ->

= ^ o lz .

P ro o f. By Theorem 1.3 it suffices to show that 3T<9\Z <= &~0\z , where sup x e Z } < со'. The sets

V (x0, rj) = { x e S 0 \ ll*-*o llo ^ *?}

= 1 x e S 0: p ({t: \ (x -x 0)(t)\ > r j})^ r j}, rj > 0, constitute a base of neighbourhoods of a point x 0 e S 0 in the topology on the space S0 . We shall show that for every x 0 e Z , Ф е ^Т<< р and r > 0 there exists rj о > 0 such that

V (x0, tjo) n Z а К ф (x0, r) n L*«\

Indeed, let x 0 e Z be fixed. Let ф be a (^-function such that ф < (p and let r be a positive number. Write

E*°(x) = { t e E : | x (r)-x 0 (t)| > q} for x e Z, G*°(x) = [ t e E : | x (f)-x 0 (f)| ^ *l) for x e Z , where rj > 0. Then

( 1 ) { X — JC oH O = (X - X o ) £ * o , „ ( t ) + ( x - Xo)e *o(x) ( f )

and

(2) qv J ^ L, where L = 2sup{\\x\\v : x e Z ] . Since ф -4 (p, there exists u0 > 0 such that

(3) ф(2и/г) ^ ^ ( p ( u / L ) f o r « < M 0 -

Next, in virtue of Lemma 1.2 there exists <50 > 0 such that for every

Z

there holds:

(4) l l ( * - x 0) J * < j r for p(A ) < (50 .

Let rjQ = m in(u0, S0) and

V ( x 0 , rj0) n Z . Then p(E*£(x)) ^ tj0 ^ <50, and hence from (4) we get

(5) l l ( x - * o W A < i r - *чок '

On the other hand, since rj0 ^ u 0, from (3) and (2), we obtain /2 <* - * о > с;о м \ ф П \ ( х - х 0т

вф

,*0(x)

'ю !) ^dp

r f\ (x -X 0){t)\\J r

^ Y l J 9 I --- ?--- ) d p < — L = 2 L 2’

from which it follows that

(6) II( ^x - * ^o ) ^g 3C 0 (,)||* < 1 r.

no Thus, from (1), (5) and (6) we get

ll(*-*o)ll* < II( x - X o )£* o ( j J*44I( x - x 0) g * o (jc)||* ^ r/3 + r/2 < r.

no no

1.5. C orollary . L et (p and ф be cp-functions such that ф <£ cp and let Z a I f * be such that sup{||x||v: x e Z } < oo. Then

^ o lz = ^ l z -

P r o o f. By definition of topology and from (0.2.1) we have or \ Oijjxp — r- <T \ <~ <T<(p ^

Hence, from Theorem 1.4, we have 0 'o\z = $~ф\г -

R e m a rk . The above corollary is proved in [8] (see Definition 0.3.8.1, Theorem 0.3.8.2 and Theorem 0.3.8.5(a)).

2. The equality of г and on L*4>

2.1. L emma . T he set 0 o f simple functions is dense in I f * in the sense o f (p-modular convergence.

P r o o f. Let 0 be the set of simple functions, i.e.,

m

0 = { x e S 0: x(t) = £ H t n H j = 0

i= 1

for i Ф j , p{H i) < oo, m eiV}.

Then 0 a L**. First, assume that x g L*’’ and x(t) ^ 0 almost everywhere.

Then there exists a sequence of simple functions (x„) such that 0 < x „ { t ) ] x ( t ) for t e E as n-> oo. Since x e L * 9, so ^ (A o*o) < 00 for some A0 > 0 and then (p({X0/2)\xn(t) — x (f)| )j0 almost everywhere.

On the other hand

(p((20/2)\x„{t)-x(t)\) ^ <p((Ao/2)|xi(t)-x(0l) ^ <p(A0 |x(t)|) and

j < 5 p((A0/ 2 )|x1(f)-x(OI)d/* ^ j (p{À0 \x{t)\)dfj. =

{ 2 0

) < oo.

E E

Thus, by the Lebesgue bounded convergence theorem ([2], p. 110) we get Q<P ((A0/2) (x„ — x)) -» 0, i.e., x „ -^ x .

At last, since x = x + — x _ , where x + and x_ are positive and negative portions of x, we apply the above for x + and x _ , respectively.

8 — Prace Matematyczne 25.2

30 2 M . N o w a k

Now, we shall prove that the set ^ is dense in and (L*v, 3T<V).

For the purpose we shall prove the following theorem.

2.2. T heorem . L et (xk) be a sequence in 1 * q>. Then

xk 0 implies xk 0.

P r o o f. Let xk 0. Then there exists A0 > 0 such that for every tj > 0 there exists a natural number К „ such that

**)<>/ for k ^ K n.

We shall show that xk -*■ 0. From Theorem 0.4.2 it follows that the topology S j has a base of neigbourhoods of 0 consisting of all sets of the form:

u ( £ K ,( 2 " ) n K ( 0 ) , N=О „=o

where £ „ > 0 , £„|0, V{s) = {xe-S 0: ||x|j0 ^ e} = { x e S 0: p ( { t e E : И 01 > e })= ^ e }.

Hence, it suffices to show that for every sequence (£„), where e„ > 0, e„ j 0 there exists a natural number К such that

¥ Ü ( S K v (2n)n V (e„ )) for к > К .

N — О п = О

F or this purpose we first prove that there exists a natural number N 0 such that for every sequence ( e j there exists a natural number K 0 such that

xk e Ky (2N) n V (sNo) for к ^ K 0.

Indeed, let N 0 be a natural number such that 2/A0 < 2N°. Let (e„) be an arbitrary sequence such that en > 0. Take such that

(1) *lo = mkif^AoeArJsjvo, 1).

Let K 0 be a natural number such that

(2) QvttoXk) < rio for к ^ K 0.

We shall show that

xke V(% 0) for к ^ K 0.

Indeed, let E x(e) = { t ç E : |x(t)| > s j for б > 0. Then

(p(À0 e )p (E x(e ))= J <p(A0 e)d/x^ J <p(A0 \x{t)\)dp ^ ^ (A 0 x).

E x (s) Ex (e)

Hence from (2) and (1) we have for k ^ K 0 :

A4 { E Xk ( e iV0 ) ) ^ Q<p ( * 0 Х к У Ф ( ^ 0 eN 0 ) < W Ф (Л 0 BNç ) <

0 •

Thus we obtain ||xk||0 < e Nq, i.e., xk e V{ e Nq) for K 0. Now, we shall show that

x . e K ^ 0) for k ^ K 0.

We have that, if 0«ДЯо х) ^ 1, then ||х||ф < 2N°. In fact, if ^ (Я 0х) ^ 1, then P o x IU ^ 1- Let N x be a natural number such that 1 ^ N 1 X0 < 2 (for 0 < Я0

< 1). Then we obtain

IMI, l|AMo*ll, « JV, ||A0 x||, « ЛГ, < 2/A0 « 2n°.

Hence, since ^ (Я 0хк) < rj0 < 1 for k ^ K 0, we have J , e K , ( 2*°) for k > K 0 . Thus, for к > К = K 0 there holds

J i e X , ( 2 " ) n n g c û (I ^ ( 2”) n F ( e J ) .

N= 0 „ = 0

2 3 . T heorem . T he set # o f simple functions is dense in (L*v, .Tj) and (L*<p, . T 4(p).

P r o o f. From Lemma 2.1 and Theorem 2.2 it follows that the set 0* is dense in (L*^, &~j). On the other hand, by definition of the topology and from Theorem 1.4 we have 3~<4>c z 3 'l . Hence the set & is also dense in (L**, .Г * * ).

Finally, we shall prove the main theorem of this paper.

2.4. T heorem . L et q> be a (p-function. Then on Orlicz spaces L *4> the topology &~4<p is identical with the topology ZTj, i.e.,

= *Tj.

P r o o f. The inclusion c ^ follows from Theorem 1.4, and by definition of the topology F l . We must show that F l c= . We know

that the set & of simple functions is dense in (L*v, (Theorem 2.3) and the spaces (L*v, ^ j ) and ( l? v, ST4<f>) are complete (Theorem 0.3.3). Therefore it suffices to show the inclusion

Π*T<V\9 ([1], Corollary of Lemma 4, p. 34).

From Theorem 0.4.2 we know that the topology ZTj has a base of neighbour­

hoods of 0 consisting of all sets of the form:

u ( £ K y (2") n V (e j),

N = 0 „ = 0

where £„ > 0 and V( e „) = {x eL *^ : p ( { t e E : |x(r)| > e„}) ^ £„}, n ^ 0. Hence,

it suffices to show that for every sequence (e„), where e„ > 0 there exists

304 M . N o w a k

a (^-function i J/ q e W ^ and a number r0 > 0 such that

00 N

К ФО( Г О) П 0 > С 2 U ( I K , ( 2")nF(£„)).

N= 0 n— 0

Without loss of generality we can take sequences (e„) such that

£„+ 1 < £и, £n j, 0, £ q 1, (eo) <'- 1 *

For a sequence (£„) we construct a subsequence with the following properties:

(3) Let k 2 > k 1 be such that:

(a) <p(£kl/2) > Ф (ek2X

(P) if <p(0 = l/fik2, then W (У) % < £3/2.

(4) For k n (n ^ 2) we take k„+1 > kn such that:

(a) i<p(ekJn) > <p(£kfl+1),

(p) if <p(t) = 1 jekn, then -■■■ <p{t/n) > l/ek|f_ 1, (У) % < $ £ n+1 .

It is clear that from (4(a)) it follows that ekJ 2 > £ kfi+1 for л > 1. Let us denote

N (^ô) = SU P ^ ^{nl h} ^ ^ W for ^ ^ ’

Now we shall construct the ^-function such that ф0 4 (p and (1) k 0 = 0.

(2) Let k x be the smallest natural number such that <p{£k^) < l/£t l .

N {t) = sup {n: ekn ^ t} for t 4: ek l, and

for t 4 £k2 and

for q>(t) ^ l/£*2.

Let t0 > Bkl be the smallest number such that (p(t0) = l/efcl. Let us denote A’„ = { t > 0 : ek n + l < t ^ EkJ , 1 ,

A = {t > 0: £kl < t < t0},

К = { t > 0 : i/£kn ^ <p{t) < l/ek„+1}, 1,

B'n = (s > 0: <p(ekn+1) < s ^ <p(ekff)} for n ^ 1, В = {5 > 0: <p(ekl) < s < <j0(fo)},

Б;' = {s > 0: ф(1/екя) ^ <p(s) < <p(l/ek|I+1)} for n ^ 1.

First, let us define the functions p: [0, oo)-> [0, oo), q : [0, oo) follows:

Г0 P (0 =

2 / (n -l) 1 . n - 1

for t — 0,

for te AJ,, n ^ 3,

for t g A 2 u A* u А и A'/, for fe A", n ^ 2,

[0, oo) as

<?($) = 0 2Д и -1) 1 n/2

for s = 0, for seB J,, n ^ 3,

for s e B ^ u B i u B u B'/, for s e B " , n ^ 2 .

Next, we define the function [0 ,o o )-+ [0 , oo) as follows:

, |X >) for n e [0 , t0], tf"(*<) for we [ t0, oo),

u f

where [0, t0] -> [0, u0] , *'(«) = J p{t)dt and u0 = f p(t)dt ^ p(t0) t 0 = *<>.

IÂo> oo)-> [u0, oo), (w) = « *(«), a(w) = t0 + j p (t)d t.

“0 In turn, let us put

l<T(tO for u e [0 , <p(w0)], V |<j;"(tO for v e[(p (u 0), oo),

<p(“0>

where [0, <p(w0)] -► [0, y0] , <j;'(i>) = f q (s)d s, v0 = j <?(s)ds,

ô 6

ç": [<р (w0), <»)-► [>0, oo), £"(*>) = ^q *(t;), ^q(v ) = <р(м0)+ j <?(s)ds.

306 M . N o w a k

<р(и о)

Since V q = J q(s)d s ^ q((p(u0))(p(u0) ^ 1 • (p(u0) = (p(u0) and u0 ^ t 0 we о

have

(1) v0 ^ (p (u 0) ^ ( p ( t 0).

At last, we define the (^-functions ф0 by the equality:

, , (* 1 0 .) = £ '(ç>(x'(«))) for U 6[0, t0], 4'o W \ Ш и ) = ^"(ч>(х"(и))) for K 6[(„, OO).

Now, we shall show that if/0 q>.

ф0 (си)

First, we shall show that lim — —— = 0 for an arbitrary c > 0, i.e., for

и -* О <P\U)

arbitrary c > 0, e > 0 there exists u'c<e > 0 such that

(2) Фо(си) ^ e(p(u) for all u ^ u 'ce.

In fact, we shall first show that for arbitrary c > 0 there exists u!c > 0 such that x'(cu) ^ u f°r all и ^u'c. Namely, if c ^ (n —1)/2 (n ^ 3), then for м ^ n'

= 2ekJ(n — 1) we have

cu

X'(cu) = J p(t)dt < p{cu)cu ^ p(ekn) i ( n - l ) u = u.

о Hence

(3) ф0 (си) = ф'о (eu) = Ç ((p (x' ( cm ))) ^ £' ((p ( m )) for и < u'c.

Now, let 2/(n— 1) < 6 ^ 1 (n ^ 3). Then, for и ^ n' = &kn we have:

(4) { '( * ( » ) )

< p ( u ) 2

= J q(s)ds ^ q((p(u))(p(u) < q(<p(£kr))(p(u) = — ~(p(u) ^ e(p(u).

0 n L

At last, from (3) and (4) for и ^ u'c<E = min (u'c, m ') we have Фо(си) ^ £<p(u).

Now, we shall show that lim = 0 for an arbitrary c > 0, i.e., for

«-00 <p(u)

arbitrary c > 0, e > 0 there exists м"е > 0 such that ф0 (си) < в(р(и) for all

^ u”e.

In fact, if c ^ (n —1)/2 (n ^ 2), then let m " > 0 be such that (p(u”/2) = l/s kn- Then, since

“o

j p(t)dt ^ p ( uq )-U q ^ m 0 ^ t0,

we have, for и ^ m "

If M

а(м) = J p(t)dt + t0 > j p{t)dt ^ p (u "/2)ju = ( n - l ) - ? u ^ cm ,

“ о о

and, since /'(u) = a -1 ( m ), we obtain

/'(см) ^ и for и ^ u".

Therefore, for all м > и" we have

(5) Mcu ) = ^ (c u ) = ( " Ы Г М ))'< £ > (« ))■

Now, let 4/n < £ < 1 and m " > 0 be such that <р(м") = n/2ekn. Since

” 0

j q(s)ds < q(v0)v 0 < 4 (ы0)) • (p (м0) ^ <р(м0),

о

so for

^

" there holds

e<p(u) E<p{u)

ц (e ( m )) = j q (s)d s + (p (u0) ^ j q ( s ) d s ^ q (etp (u)/2) • e<p (u)/2

V Q 0

1 ч )2 < р (и )1 п = < p ( u ).

Hence we have

(6) Ç” (ô(uj) < s(p(u) for all m ^ Mg'.

At last, from (5) and (6), for м ^ м"£ = шах(м'', м"), we obtain ф0 (си) < eq>(u).

Thus, we proved that ф0 <4 (p.

Now, we shall show that

Ф М ^ <p(t/N{t)) for t ^ e*2, where

iV(f) = sup {и: £kn ^ t}.

Write N (t) = n0. Then £*„o+1 < t ^ e ^ , i.e., t e A'„0, n0 ^ 2. We have

<p(x'(t ))

(7) Ф М = ф’ М = f'(<?(*'«)) = f «(*)* М)/2)-<р(/М)/2.

0

Since 7 £k x, > £t x * hq +1 *« q +27 so we have

r

/ ( 0 = J p(s)<2s ^ p{t/2 )t/2 ^ r/n0 for n0 ^ 2 .

(8) о

308 M . N o w a k

Hence, according to (*)4(a), we obtain

<p(ï(t)) > (p(t/n0) ^ </>(£k„0+1/«о) ^ <Р(*кПо+1/по + 1) > 2<р(екяо + 2), so

(9) q ( v ( ï( t ) ) / 2 ) > 2/((n0 + l ) - l ) = 2/n0.

Finally, from (7), (8) and (9) it follows that

<AoM > — <p{t/no).

n0 Now we shall show that

M ) > N ( l M t) ) + i y (t/N(1/y(,|)>

for t such that (p{t) ^ l/ek2, where N(l/<p(f)) = su p {n: &kn^ l/(p{t)}. Write N (l/(p(t)) = n0 . Then l/£fcno ^ q>(t) < 1 l 4 nQ+v i-e-, teA ''0, n0 ^ 2. We have

■4

(1 0 ) iAo(0 = *Ao(0 = £ "(< ?(*"W)), where / '( 0 = a - 1 (0, Г ( £) =

By condition (*)4(p) we have (p(t/n0) ^ 2(n0 + l)/ek x > V£kl = ф(го)> an<3 hence t/n0 > t0 .

Therefore we obtain:

t/п 0

a {t/n0) = j p(s)ds + t0 ^ p (t/n0) t/n0 + t0 ^ p ( t ) (t/n0) + 10

" 0

5$ (H0 -l)-t/ H 0 + t0 < f.

Hence x"(t) ^ t/n0, and from (10) we get

(11) ф0 (t) ^ Г(<Р(0*о))-

We shall show that

? ’ (<p{t/n0)) > <p{t/n0)/(n0 + l ) . According to ((*)4(p)) and (1) we obtain:

<P(t/n 0 )/{n 0 + l) > (p(t/n0)/2(n0 + l ) > l/efc|io_ 1 > l/ekl = <p(t0) ^ v0.

Moreover,

</>(f/n0 )/(«0+ !)

j <?(s)ds ^^(<jp(t/w0)/2(n0 + l))<jp(t/?io)/2(no + l)

0

^ <gr(l/ek„0_ A)<p(f/n0)/2(n0 -F 1) ^ <p{t/n0)/2(n0 + l) ^ l/ek(i(>_ x

> l/ekl = (p(t0) ^ <p(u0).

Therefore, from above two inequalities we get:

< P ( t / n 0 W n Q + 1 ) < p ( r / n 0 ) / ( n 0 + 1 )

>7((0*o)/(«o + l) )= J q(s)d s + <p(u0) ^ 2 f q(s)ds

^ 2 q(q> (t/n0)/(n0 + 1)) (p (t/n0)/(n0 + 1)

< 2q(<p(t))-<p(t/n0)/(n0 + l ) < (p(t/n0) and hence

(12) £"(<p{t/n0)) ^ (p(t/n0)/{n0 + 1).

Finally, from (11) and (12) it follows that

<Ao(0 ^ <p(t/n0)/(n0 + l).

Now, let ( e j be an arbitrary sequence such that

£„+ 1 <£„, £„ j 0, £0 <1> е0 (р{е0) ^ 1 . Let r0 = min(l/2, d0 s0 (p(c0)), where

(13)

We shall prove that

— = sup \<p(t)/ÿ0(t): t e A \ u A u A'{).

do

К ф(г0) п Р c U (Z 2")nF(£„)).

N = 0 n = 0

In fact, let * = Z иХн^К-Ф0 (ro)» I — a finite set. Write

ieJ

К = { i e l : \ti\eA\ u А и Л'/}, L = { i e / : |t£| ^ £*2},

J = { i e / : <p(|ti|) ^ l/fifc2} - Let

*' = Z x" = Z ЬХнг x>" = Z

ieA' ieL i e J

Since х е К Фо(г0) and r0 < 1 we have

(14) дФо{х) = Z = c ^ r 0 .

Write

Сг = Ф оШ )1*(Н д.

First, we shall show that

(15) x' e Кф ( 1) о V( eq ) .

310 M . N o w a k

Denote by B9 (r) = [ x e L * v: ^ ( x ) ^ r j. Then Х Д г) = rB^(r). Let r x

= E0 (p(eo) ^ 1. Then Д Д ^ ) <= V( e 0). Indeed, let E x ( e 0) = { t e E : |x(t)| > e0}

for x e B ^ ir j). Then

(p{e0) n ( E x {s0) ) = J <p{£0) d n < J <p(|x(OI)<*A*<G„(x)>

£x(e0) £x<E0) and hence

A* (Ex (fi0)) ^

(*)/<? ^{( e o )} <*o<P Ы !< Р ^{( e 0 )} = ^{« o >} i*e., * e F ^{( e 0 ) .} Thus £ „ 0 4 ) c= F (e0).

Since q ^0 { x ') = £ Ci ^ c ^ r 0 ^ d 0 E0 (p {E 0 ) = d0 ru so using (13) we have

ieK

Qv (*') = j ( X к Хщ (s)) dfx= Y, <P (l*il) H{Щ

E ieK ieK

^ -J- Z ^{Фо (Kl) = J -} Q<,0(x') < •

“ 0 ieK “ 0

Hence c F (e0).

Thus, we obtain x' g 5^(1) n F (£0) = (1) л F (£0).

Next, we shall show that

* i

(16) x " e £ K „ ( i 2 " ) n F ( i e J

n — 1

for some natural number N 1 ^ 1.

Denote by щ — sup{n: Ekn ^ |ff|) for i e L . Then 2 and e* < \k\ <

But we already proved

Фо (W) > — <P Ш /щ ) for i e L, Щ

from which it follows

(17) =

П: C;

Фо(\к\) <P(k№

for i e L .

Now, let sls . . . , smi denote the different natural numbers in the set (nf: i e L } and let be sx < s 2 < ... < smi. Write L, = { i e L : щ = s,}, where 1 ^ l ^ m1.

Then we have

Write

x" = Z ^кХщ = Z (Z ^{k l u ) -}

ie L 1= 1 ieL [

Уi = Z кХщ

ie L f

for 1 < / ^ mx.

Then eksj+1 < |f{| < £ksj for i e L t. Since

v(Eyi(eks)) = jy,(01 > e*,,}) = n(0) = 0 < ekj/,

from which it follows that

But ek < £S|/2 for Sj ^ 2 ((*)4(y)), hence

S| I

(18) y i s V ^ j l ) .

On the other hand from (17) we have

hence

M-ViAi) = Z < (Z Ф/ ^ si>

ieL\ ie L i

(19) y , e K v (,s,)<=Kv ( i ^ ) .

From (18) and (19) we obtain

y i e ^ t ë l V v t ë ü , , ) - Therefore for N t = smi we obtain

W1 ml

x" = Z ^ихщ = z (Z ^{UX h ^} Z ^{v (i} £s,)

ieL 1=1 teLj 1=1

*1

с X Z , ( i 2 ”) n V ( i s J . Finally, we shall show that

n

2

(20) х"' = 1 и х и ,е I K „(ï2")n K (b „)

i e J n= 1

for some natural number iV2 ^ 2.

Write n ^ s u p jn : £kfI ^ l/<y>(|tf|)} for i e J . Then n , 2 and l/£kn

< <?(W) < 1/ \ + Г

But we already proved that lAo(lhl) ^ ^(lhlM)/(ni + 1) for i e J , from which it follows that

( 21 ) < (”i + l) c i

^ < р Ы К ) for i e J .

Now, let j b • • •,jm2 denote the different natural numbers in the set {щ: i e J ) and let be j’i < ... < j m2. Write J t = { i e J : щ = у'г}, where 1 ^ ^ m2. Then we have

m2

X " ' = X ^г.Хн,- = Z (Z ^*.Хя,)-

is/ f = 1 ie/j

312 M . N o w a k

Write yt = £ и х щ for 1 < / < m2. Then 1/e^ ^ <p(|tj|) < 1/%г+1 for i e J t.

Therefore from (21), ((*)4((3)) and ((*)4(y)) we have

+ ^ .____ < ri^ p .

l i ( f i i ) ^ ^ ^ С.* ^

<^(|г.-1//|) ( р Ш / j i ) J l

from which it follows that

(22) у 1 6 V (i ( Z cf) e j c= K (i £,,).

i e J i

On the other hand, since £ с,- < c < r0 < 1/2, from (21) and (14) we obtain iei

Qcpiyi/ji) = Z 9>(l?il///))u(ffi)

ie/j

< Z < (Z c«)0'i+1) < 0'/+1)/2 ^<ji,

i e J t V W i w J l i e J {

hence

(23) yi e K v (j,)<=K v ( i ? ‘).

From (22) and (23) we obtain

y . e K J i ? ' ) n V (ie jt) for 1 « / s: m2.

Therefore for N 2 = j„ 2 we obtain m2

= Z = Z (Z

/е/ J = 1 je / j

m2 ^2

e l £ K „ ( i 2 " ) n K ( H ) .

i = l n= 1

Finally, from (15), (16) and (20) for N 0 = max ( N l , N 2) we have j = i ' + j " + T e K , ( l ) n f ( t 0)+ £ K ,(2”)nV(<s„) N°

n= 1

и ( Z ^<p(2") n V (£„)).

N=0 „=0 This proves that

oo JV _

K*0(ro)n^c= U ( Z ^ ( 2 fl) n F ( £J).

N=0 „=0

Thus, we proved that c=

References [1] R. C riste scu , Topological vector spaces, Leyden.

[2] P. H alm os, Measure theory, Springer-Verlag, New York 1974.

[3] M. A. K ra s n o s e l’skiT and Ya. B. R utickT, Convex functions and Orlicz spaces, Groningen 1961.

[4] R. L esn iew icz, On two equalities fo r Orlicz spaces, Bull. Acad. Polon. Sci. 27 (1979), 557- 560.

[5] W. M atu szew ska, On generalized Orlicz spaces, ibidem 8 (1960), 349-353.

[6] M. N ow ak, On two equalities fo r Orlicz spaces, Funct. et Approx. 10 (1980), UAM, 69-81.

[7] —, On two linear topobgips on Orlicz spaces 1?<р, I, Comment. Math. 23 (1981), 71-84.

[8] Ph. T u rp in , Convexités dans les espaces vectoriels topologiques généraux, Dissert. Math.

131 (1976).

INSTITUTE OF MATEMATICS, A. MICKIEWICZ UNIVERSITY, POZNAN

INSTYTUT MATEMATYKI UNIWERSYTETU im. A. MICKIEWICZA, POZNAN