On the translation in the Orlicz-Musielak sequence spaces

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIV (1984)

A. K

(Poznan)

On the translation in the Orlicz-Musielak sequence spaces

Abstract. Some notion of the translation and the modulus of smoothness in sequence spaces was given by J. Musielak in [5]. In this paper we present a new concept of these notions.

In the main theorems necessary and sufficient conditions for convergence to zero of the modulus of smoothness in Or liez-Musielak sequence spaces are given.

First we introduce the following notations. Let /V be the set of integers, R the set of real numbers, X the real linear space. Let ip = ((pj): X x N

— [0, + x ) be a sequence of Orlicz functions, i.e. q>j are convex, <рД0) = 0,

<Pj ( x

) = (pj( —

) for all j e N , x e X . On the family of all sequences x = (xj),

where Xj6X , is defined a functional / ф(х) = £ ^ (х ,), which is a pseudo- modular [ 6 ]. This functional generates the modular space j= i = {x = (х,): /^(Ax)

< oo for some A > 0} called Orlicz-Musielak space. Define also h(p, the linear subset of l(p, as follows: hv = (x = (х,): /^(Ax) < oo for every A > 0}.

The functional ||x|| = inf |e > 0: /^(x/e) ^ 1} is the Luxemburg norm in lv ([4], [ 6 ]). The family of unit vectors e, = (0, ..., 0,1, 0, ...) makes the

i Schauder basis of h9 [4].

Now, we will define two versions of translation of an arbitrary sequence x = (x 7). Let k, ie N, к ^ i; then the following operators are translations

Taking i, к fixed, the above operators ai k, are linear. Using the operators ai,k or ri>k one can define two kinds of moduli of smoothness со,, со* as follows :

o c

(T/,* x)j = 0

if j ^ i, if i < j < к if j > k.

co,-(x) = sup ||X ( 7 ,- k x||, со1(х) = sup ||х-т,.дх||.

In the following, we will answer the question when cofx) -*0 and со'(х) ->>0 in the Orlicz-Musielak spaces. If <pj = (pk for all i, k e N , i.e. q> is a usual Orlicz function, then the above conditions are fulfilled for all x e h 9. In the general case it does not need to be so, as we will see in remarks and examples 3. The exact considerations we will carry out for one of the operators, for example ai k. Define a new function q>i,k = (q>ÿ*) for к ^ i, as follows

[ ъ if j < /,

* = < 0 if i < j < k, к Vi-k + j if j > k.

Hence for all x = (х;) we have

= W (* )- The following functional

U (x) = sup W (* ) k^i

is a convex pseudomodular. The modular space defined by this pseudo- modular will be denoted by l{i). Let |H |t- (Ц-Щ) denote the Luxemburg norm in l(i) (/„i(k). It is easily seen that

I Wli = sup \\x\\itk = sup|[< 7 i>kx||.

k ^ i k ^ i

In the next theorem there will be given necessary and sufficient conditions in order that cu,(x)-»0 for all x e h <p. A similar result was obtained in the Orlicz-Musielak function spaces, where the translation of a function was defined in the usual way [ 2 ].

1. T

. The following conditions are equivalent :

( 1 ) there exist i0 e N, l, Ô > 0 , a family of non-negative sequences (cjfi such

00

that sup £ c) < oo and

k > i0 i

q>Tk (lx) < (pj(x) + tf for all j e N , к ^ i 0 if (pj(x) ^ <5,

( 2 ) there are i0e N , p > 0 such that INI.-o < p IMI

for all sequences x = (xfi, i.e. the spaces l^ and l(i0) are isomorphic, (3) for all x — (Xj) e h^ the condition

cofix) = sup ||x — ffi>kX|| -* 0

k ^ i

as i -> oo is satisfied.

P ro o f. (1)=>(2). It can be assumed that x e lr Consider the element x/||x||. Then / ф(ах/||х||) ^ Ô, where a = min (1, S). So, using condition (1) we obtain

(pj°,k (ltxxj/l\x\\) < (Pj(aXj/\\x\\) + tf 00

for all k , j G N , k ^ i 0. Denoting d = sup £ ck we get Iio(hx/\\x\\) k>i0 j=i

< / v(ax/||x||) + d . Let d > 1, then /, 0 (a/x/d||x||) < / v(ax/||x||) + l. Since

/ od x \ 2d

/„(ax/ЦхЦК 1, so IiQi — J ^ 1. Hence i|x||io ^ — ||x||. Putting p = 2d/h, ( 2 ) is obtained.

(2)=>(3). Let x = {x^ehp. Then for ie N

oo oo

s u p ||x — crI%k x|| ^ ||_k>« j-.= i + 1

I

+

= k+ I

X 00

B u t X X j &i - k + j &i,k ( Xj- e j)

and

j = k + 1 j = i + 1

OO X

s u p ||<7i)k( _{к > i j = i +}X ₁ X :J e M «

sup Ikio.* (

z

for i ^ i0. Hence and by (2) we have

00 00

S lip

II £ *;ei-t+j||<p|| Z xi ei\\-

k^i i = I + i ; = i +1

mi( x ) ^ ( l+ p ) || £ XjejW^O, as i -*■ oo, since x e h (p.

7=« + 1

(3) => (1). First we will show the following condition

( 1 . 1 ) there are i0, h g N, l, Ô > 0 and a family of non-negative sequences

X

(cj) such that sup £ c* < oo an(*

k>i O j = n

qjP'k {lx) ^ <jO y(x) + c}

for all k, j e N, к ^ i0 if (pj{x) ^ <5.

Let

C)’k = sup ! <pÿ* (( l/i4‘) x) - (Pj (x)}

^-( jc ) =$ 1 / 2 * for к ^ i, /с, i e N .

It can be shown that condition (1.1) is equivalent to the following one:

there are i, m e N such that sup ]T c),k < oo. Indeed, assume condition (1.1) k^i j=m

is fulfilled. Let i e N and i ^ max(z0, n ) be such that (l/i4‘) ^ l and (l/2‘) ^ <5.

Then

( 1.2) sup : (pj

<Piix)^ 1/2*'

;.о-л ((1 //4 ')х )-^ (х )} < sup {(pj°’k (lx)-(Pj{x)) < c kj

<Pj(x) < Ô

for each j, k e N , where i0. By definition of (pt,k we have

(1.3) (pf (x) ^ (pj0'£~i + i° (*)

for every x e X , all i ^ i0, s ^ i and j ^ i + 1 .

Putting к — s — i + i0 in (1.2) and using (1.3) we get c‘j s = sup ! (p)’s (( l/z'4‘) x)-(pj(x)] < Cj l + l°

(pj(x)

^ 1/2*

00 oo

for j ^ / + 1, s ^ i. Hence sup c},s ^ sup £ c) < oo. Putting m = z'-f 1,

s > i j = i + l s > - i 0 j = „

the desired implication is obtained. It is evident that с)’к = sup {(р)'к {(l/2i)x )-(p j (i2ix)}.

<py(i 2 ‘x)< 1 / 2 * Let us denote

d‘j k = sup {(p),k ((l/ 2 ')x): (pj(24x) < 1 / 2 ' and <plj k (2~l x) — (pj(2l ix) ^ 0 }.

Then с'-к ^ é-k for all i,j, k e N , where к ^ z. Suppose condition (1.1) is not satisfied. Then, by the above consideration

00

sup £

o o

к - Ï i j = m

for all /, m e N . One can choose a family of pairwise disjoint subsets of N in such a manner that Ni = (1, 2, ..., nj}, where zzj ^ 2 , N 2 = {пл + 1, ...

..., n2}, ...» JVf = {wf-_i + l, nt}, ... and

(1.4) sup £ 4* > z,

k > i j e N i

(1.5) sup £ dj,k ^ z

k ^ i j e N ^ i n p

for all z ^ 2, putting sup £ d)k = 0. By (1.4), we can find kt ^ z for z ^ 2, к > i j s0

such that Y j d) ' > z. Next, using the definition of ctj l, we find elements Xj

jeNi

possesing the following properties

(1.6) sup £ ¥}k (2 *'

> i k^i jeNi

for all i ^ 2 and

(1.7)

(2‘

^ l/2‘,

1 (2 1

—

(2‘ ?x,) ^ 0 for every i e N , j e Af,.

Let Я be an arbitrary positive number. There is an integer Ï ^ 2 such r — 1

that Я ^ i for i ^ i . Then, putting x — (xj) and ^ ^ ç ? j ( axj ) and

i — 1

using (1.5), (1.7) we obtain

/ <?(Я х )^ М + £ £ ^(iXj)

i = i" j e N , -

00

^ M + £ [( 1 / 2 ') X Vi (2' **/) + 0 *»f)]

i = i' je^,\{"il

< A f + X [( 1/20 sup X 4 ’* + ( 1 / 20 ]

; = > ' k > i jeNp{np

00

« м + X ((«■/ 2 l) + ( l/ 2 ‘) ) < œ . i = Г

Hence xeh y.

Now, let l > 0 and it e N be arbitrary. There exists i2 ^ max (2, it) such that 2~* < / for every i ^ i2. Then

Itl (lx) Si sup У V <pi'-k (2~‘

> sup I <pj"k <2~‘

i = i 2 j e N , - f c > 1 1 j e J V «

for every i ^ i 2. Analogously to (1.3), we have

<Pj (x) ^ (Pj (x)

for each x e X , i ^ ix, к ^ i and j ^ i + 1. Since it was assumed nx ^ 2, so Nt a { /+ 1 , i-f 2, ...}. Hence

i 1Ж / л — i — \ v i f Ac■“ i j + i y » — ï — \

<Pj (2 *;) > (pj (2 Xj) for all je A f, where i ^ i2 ^ ij. Then, by (1.6) it is

sup X ' 1 +‘( 2 ~‘х;)

*^î‘l jeJV,-

= sup X (pij's (2 ~ i Xj) > i x Z i jeNi

- Prace Matematyczne 24.1

for each i ^ i2. So, we have shown /,■ (lx) = oo for every ix eN , / > 0, which means that ||x||(- = oo for all ie N . Hence, it is seen that sup ||x — ai k x|| = oo

к & i

for every ie N , which contradicts assumption (3).

To finish the proof, it is enough to show that (1.1) implies (1). If n ^ i0, then the implication is evident. So, lef n > i0. Then, by (1.1), we have

<p?i + ”- i0(bc)^<pj(x) + c}

for all k , j e N , k ^ i 0, if (pj(x) ^ ô . Denoting s = k + n — i0 and putting e] — Cj for j ^ n + 1 and = 0 for j ^ n, we get

(p"’s ( l x )

^

00 <X>

( for each j e N , s ^ n if

< Ô. Also sup Y

= sup Y ck < oo.Thus

s ? n j = 1 0 j = n + 1

condition ( 1 ) is shown with i0 = n, what ends the proof of the theorem.

Now, we will give an analogous theorem for the operator xik . To this end let us introduce the following notations. Let Ф1,к = (Ф1 / к), where

фа = | % for ./'«<.

1 I n + j - i for j > i.

Then / (т,- kx) = / k

and

= sup

(x) is a new convex pseudo- modular. Let Ц'Ц‘ be the Luxemburg norm generated by f and L(i) the modular space defined by this pseudomodular. We have also

1 ИГ = sup ||x|| -tk = sup Цт.-дхЦ.

k2i k^i

2. T

. The following conditions are equivalent

( 1 ) there are i0e N , l, Ô > 0 , a family of non-negative sequences (ej)j such 00

that sup Y A < 00 an(l

ф)°’к (lx) < (Pj(x) + ek for all k , j e N if k ^ i0 and q>j(x) ^ ô,

( 2 ) there are i0 e N, d > 0 such that

\\x\(°^d\\x\\

for all x —(xj); which means that the spaces lv and L(i0) are isomorphic, (3) for every x — (xj) e hv it is satisfied

of (x) = sup ||x — zifkxl| — > 0 as i oo.

k>i

The proof of this theorem is analogous to that of Theorem 1, so we will omit it.

3. Remarks and Examples.

1. Let (Y, || U) be a Banach space with a subsymmetric Schauder basis (a,), [4]. Then there exists a norm || ||0, equivalent to || -||, defined as follows:

X CO

Il Z Yi«i||o= sup sup || Y fliY.-aJ i = 1 0,-=±l {n,} i= 1

for all у = (yf)e Y. We have со1 (у) -> 0 for all y e Y. Indeed, first let us note X

t i<ky = Y y jaJ ’ where (пк)} is some subsequence of N. Then

j = i 1

CO1 (y) = sup ||y -T i>fcy|| ^

k^i _j=i+l I ^{y j aj\\ +}

+ sup|| Z >jak-i+jlN II Z Y/*/11 + 11 Z Y/«J|o-0

k ^ i j = i + 1 j — i+ 1 j = i + l

as i -> oo, since ||-||0 is equivalent to ||-||.

2. Let (pj) be a non-decreasing sequence tending to infinity and M(x): R -> [0, + oo) a usual Orlicz function. Let (pj(x) = M(pjX) for je N . Then condition (1) of Theorem 1 is satisfied evidently, since

(Pi-k+j(x) = M(xp,_fe+7) ^ M (xpj) = (pj(x)

for all j > k ^ t i , x e X . We will show that there is no Orlicz function equivalent to q> = ((pj) (for the definition of an equivalent function, see [ 1 J).

Suppose for a contrary that there are an Orlicz function N and constants

X

k, Ô > 0 , and a non-negative sequence (cj) such that Y cj < 00 with

j = i

(Pj (kx) — M(kxpj) < N(x) + Cj

for all j e N if N(x) ^ Ô. Since Cj -> 0 and M(kxpj) -> oo as j -> oo, we have N(x) = oo for x ^ O . Hence N is not an Orlicz function.

Let us remark additionally, that putting for example pj = 2j and M(x)

= |x|, condition (1) of Theorem 2 is not satisfied. In fact, the condition above 00

is not fulfilled iff sup Y e)’k = 00 f°r 1G N, where

k ^ i j = 1

e f = sup [ Ф)'к (( 1/2') x) - q>j (x)J.

q>j(x) < 1 / 2 * By the definition of Ф1,к, we have

el.'k ⁰ if ; c i + 1 ,

sup {<pk- i+J((l/2)x)-<pj(x)} if j ^ i + l.

,рЛх)^1/2*

Next, in this case there holds

j,k _ j 0 if к < 2i,

j’ ~ 1(1/2*) ((2V41) — 1) if к > 2i for j ^ / 4 -1. Hence

00 00

sup £ ef = SUP Z (1/2,)((2*/4,')-1) = 00 -

k ï t i j - 1 *5* 2 i j = i + 1

3. It is easy to calculate that if (pj) is a non-increasing sequence, <pj(x)

= M(xpj), then condition (1) of Theorem 2 is fulfilled and there is no Orlicz function equivalent to (p = (q>j). Also, if pj — l/2j and M(x) = |x|, then condition (1) of Theorem 1 does not hold.

References

[1] A. K a m in s k a , On comparison o f Orlicz spaces and Orlicz classes, Functiones et Approxim ate 11 (1981), 113-125.

[2 ] —, On some compactness criterion for Orlicz subspace Еф(й), Comment. Math. 22 (1981), 245-255.

[3] M. A. K r a s n o s i e l s k i t and Ya. B. R u tic k if, Convex functions and Orlicz spaces, Groningen 1961.

[4 ] J. L in d e n s t r a u s s and L. T z a fr ir i, Classical Banach spaces I, Springer-Verlag, Berlin-Heidelberg 1977.

[5] J. M u s ie la k , Modular approximation by a filtered family o f linear operators, Proceedings of the Conference on Function Analysis and Approximation, Oberwolfach 1980, 99-110.

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