Modular approximation by a filtered family of sublinear operators

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVII (1987)

Andrzej Kasperski (Gliwice)

Modular approximation by a filtered family of sublinear operators

1. Introduction. In this paper, (Q, Г, ц) will denote a cr-algebra Г of subsets of a nonempty set Q, fi a nonnegative, n-finite and complete measure on Q, and Ж is the space of all extended real-valued Г -measurable functions on Q, finite /a-almost everywhere: two functions equal /i-а.е. will be treated as the same element of Ж. Let g\ й х Ж - * [ 0, + oo] be a map satisfying the following conditions:

(1) g(t, x) is a modular in Ж for every teQ ,

(2) the function g(t, x) of variable t is Г -measurable for all х еЖ, (3) if x, у еЖ and |x(r)| |y(t)| a.e. in Q, then g(t, x) ^ g(t, y) a.e. in Ü.

Let us denote by X the set of functions x e Ж for which g(t, ax) —> 0 as a -> 0 a.e. in Q. Denoting

(4) e*{x) = $6{t, x)dfi,

n

we observe easily that ^s: X -* [0, + oo] is a modular in X. Let X Qs be the respective modular space, i.e.,

(5) X es = j x e l : gs(ax)->0 as x-»-0}.

It is known that if g(t, x) are convex, then

(6) ||x||es = inf {u > 0: gs(x/u) ^ 1]

is a homogeneous norm in X e .

All definitions and theorems connected with the idea of a modular space can be found in [1].

Throughout the paper, assumptions (1H3) and notations (4)—(6) will be in force.

2. A general theorem.

De f i n i t i o n 1. An operator А : Ж- * Ж will be called the q-sublinear

operator if for all x, y e SC and a, b e R

\lA{ax + by)](t)\ ^ |a[A (x)](l)|-H h[A (y)](l)| a.e. in Q.

Definition 2. An operator В: SC —> SC will be called the sublinear operator if for all x ,y e S C and a, b e R

[B(ax + byY\{t) ^ a[B(xÿ\(t) + b\_B{yy\(t) a.e. in Q.

Let V be an abstract nonempty set and let У be a filter of subsets of V Definition 3. A function g : V -> R tends to zero with respect to У, g(v)-*Q if for every £ > 0 there is a set У0е У such that \g(v)\ < s for all v e V 0.

Definition 4. A family T = (Tv)veV of nonlinear operators Tv: X e^ -+ X 6s will be called У -bounded if there exist positive numbers k lf k 2 and a function g: V -> R+ such that g(v) 0 and for all x, y e X Cs there is a set Ух>уе У for which

Qs(a(Tvx - T vy)) ^ kyQa(ak2( x - y ) ) + g{v) for all v e Vx y and every a > 0.

R e m a rk 1. If qs is convex, then the constant may be always taken equal to 1.

Theorem 1. Let T = (Tv)veV be а У -bounded family of q-sublinear operators Tv: X e -* X Qs and let S0 cz X Qs be such that for every x e S 0,

у

qs (a Tvx ) —>0 for all a > 0, let X%s be the Qs-closure in X Qs o f the set S o f all finite linear combinations of elements of the set S0.

Then for every x e X sQjh e r e exists b > 0 such that

Qs(bTvx)^> 0.

P ro o f. First, let us remark that the thesis holds for all x e S , since Л supposing x = ... +c„x„ with X;eS0 we have, writing c = £ |c,|

i= 1

П

Qs (b Tvx ) ^ X Qs (be Tv X;) 0.

i = 1

Now, let e > 0 be arbitrary and let x e X ^ be given. Then there exists b > 0 and an element s e S such that

qs(2bk2(x — s)) < and gs{2bTvs) Z 0,

where we may assume k l , k 2 ^ l . Let v eV xs, the set Vxs being chosen according to the definition of У -boundedness of ( Tv)vev corresponding to the

elements x, s. Then we have

Qs(bTvx) ^ Qs(2b(Tvx - T vs)) + gs(2bTvs)

** к 1gs (2 bk2 (x -s )) + g (v) + gs (2 8TV s)

^ U + g(v) + Qs(2-bTvs).

Now, let Vx, V2e i r be so that g(v) < e/2 for п еУ г and gs(2bTvs) < e/4 for

d gK2 . Taking F = n K2 n we obtain gs(bTvx) < £ for all ve V. Hence gs(bTvx ) ^ 0 .

Theorem 2. Let T = (Tv)veV be a 'V-bounded family of sublinear operators Tv: X Qs —> X Qs and let S 0 cz X 6s be such that for every x e S 0, gs(a(Tvx

— x)) 0 for all a > 0, let X^s be the gs-closure in X Q of the set S of all finite linear combinations of elements of the set So­

i f for every x e X 6s, [7i,x](0 ^ x(t) for all v e V a.e. in Q, then for every xeX% there exists a b > 0 such that

Qs(b{Tv x - x ) ) ^ > 0.

P ro o f. First, let us remark that the theorem holds for all x e S since П supposing x = c1 x 1 + ...+c„x„ with X;eS0 we have, writing c■= |c,|,

i = 1

n

Qs(b{Tvx - x ) ) ^ X & ( М ^ х г-Х ;))-^0 .

i= 1

Now, let £ > 0 be arbitrary and let x e X l be given. Then there exists a b > 0 and an element s e S such that

gs(3bk2( x - s)) < ^ - and Qs(3b(Tvs - s )) -^ 0,

where we may assume k 1, k 2 ^ l . Let v e V x>s, the set Vxs being chosen according to the definition of f "-boundedness of (Tv)veV corresponding to the elements x, s. Then we have

gs(b(Tvx - x ) ) ^ gs(3b(Tvx - T vs)) + gs (3b(Tvs - s ) ) + gs(3b(x-s))

^ gs(3b(x — s)) + k 1 gs(3bk2( x - s ) ) + gs(3b{Tvs - s ) ) + g(v)

^ 2kj gs(2bk2(x-s ))+ g (v ) + gs (3b(Tvs - s j ) .

Now, let Vu V2e V be such that g{v) < e/3 for d g^ and gs(3b(Tvs — s)) < e/3 for ve V2. Taking V — V1 n V 2 n Vx<s, we obtain gs(b(Tvx - x )) < £ for all ve V.

Hence gs(b(Tv x — x)) 0.

R e m a rk 2. If g{t, x) are convex and X* is the norm-closure in X Qs of the set of all finite linear combinations of elements of the set S0, where the

norm in X e is given by (6), then the thesis of Theorem 1 and Theorem 2 holds for every b > 0.

Let now I be a Banach space and

= [f: f : Q -* X and / is strongly measurable].

Let QsX{f) = &(ll/llx) and

X esx = f e and Qsx(af) - 0 as a-* 0].

Suppose F is a mapping from a subclass of J v into JT Then F is called a X- sublinear operator if it satisfies the following properties:

(i) If f = f \ + f2 and F f (i = 1, 2) are defined, then F f is defined.

(Ü) \\F{f1+ f2) h ^ \ \ F f 1\\x + \\Ff2\\x

(iii) For any scalar a, \\F(af)\\x = \a\\\Ff\\x .

Definition 4'. A family G = (Gv)veV of X-sublinear operators Gv: X PsX -* X &sX will be called iF-bounded if there exist positive numbers k {, k 2 and a function g : V -*■ R + such that g(v) -> 0 and for all / 1}/ 2 е 1 г there is a set

for which

e*x (a (Gv f i ~ Gv / 2)) ^ k x qsX (ak2 (Л - f 2)) + g (v) (

for all v eV f l j 2 and every a > 0.

Theorem V. Let G = (Gv)veV be a t -bounded family of X-sublinear operators Gv: X e —> X 6sX and let S0 cz X QsX be such that for every f e S 0, Qsx{aGvf ) X>0 for all a > 0. Let XgsA. be the QsX-closure in X 6sX of the set of all finite linear combinations o f elements of the set S 0. Then for every f there exists a b > 0 such that gsX(bGvx) -* 0.

The proof is quite analogous to that of Theorem 1 and we omit it.

R e m a rk 2'. If gsX is convex and X* is the norm closure in X QsX of the set of all finite linear combinations of elements of the set S0, where the norm in X 6sX is given by the formula \\f\\esX = ЦН/НхЦ^ then the thesis of Theorem V holds for every b > 0.

3. Application to the theory of approximatibn. Let P be an abstract nonempty set and let be a filter of subsets of P. Let X be normed linear space with the norm ||-||, it is known that every norm is a convex modular.

Let (X p)peP be a family of linear subspaces of X such that for every x e X and all p e P there exists spe X p such that

||x Sp|| = inf ||x - s ||.

seXp

Let us introduce a family of operators {U9p)peP by the formula U9Px = g inf ||x — s\\

se Xp

for every x eX and all p e P , where g e X .

It is easy to see that U9p are Y-sublinear operators on X for all p e P . In fact, let x, y e X and a e R ; we have

Щ (х + у) = I\g inf ||x + y - s |||| = y i l |x '+ y - s p>jeJ |

se Xp

< I to lK tk - ^ ll + lly -S p J ) = \\U9px\\ + \\U9py\\.

For а Ф 0 we have

\\U9p{ax)\\ = \\g inf ||a x - s |||| = \\g\| inf \\a(x-l)\\ = \a\\\Upx\\,

seX p IeX p

U9p( 0) = 0.

R e m a rk 3. If X = X(Q , Г, p) is the normed linear space of all extended real-valued, Г -measurable functions on Q, finite p-almost everywhere with the norm ||-||, then Up are g-sublinear operators on X for all p e P.

Proposition 1. I f ||gr|| = N, then for all x , y e X , and p e P , \\U9px - t / * y ||^ i V ||x - y ||.

P ro o f. Let x, у еХ and p e P , we have

\\U9px — U9py\\ = ||^(inf | | x - s | | - inf ||y -s||)||

seXp s e X p

^ N max |||x —sp>JC|| — ||у —sp>x||, | | x - s pJ - | | y - s pJ |}

< N \ \ x - y \ \ .

Ex a m p l e. Let now X = I?(Q, Г, p), where the measure p is <7-finite and atomless, <p is a locally integrable iV-function which satisfies the condition A2. Let ||i|e>0 be the Orlicz norm and ||-||e be the Luxemburg norm in X. Let 4' be complementary to tp in the sense of Young (see [3], p. 82, 83).

Let {et} be such that е^ЕИ and et e l f for i = 1, 2, ..., and J et ejdp n

= ôu , where

= (1 for i = j, i j ~ { 0 fori #7-

Let us denote: X n = span {cl5 e2, ..., enj, E„(x) = inf ||x — s||e>0, let

s e X „

Рп{х)еХп and ||x -p „ (x)||e>0 = En{x). Let / е Я \ f Ф 0.

Theorem 3. I f for every A cz Q, such that U{(xA)->Q as n-> oo,

8 — Prace Matematyczne 27.1

where Xa is the characteristic function o f the set A, then, for every x e l f , En (x) -* 0 as n-* oo. I f moreover, for every x e l f and every n e N there exists a„{x)eR such that p„(x) = p„-l {x) + a„(x)e„, po(x) = 0, then {e,} is the Schauder basis o f If.

P ro o f. From Theorem 1', Remark 2', Proposition 1 and condition A2 we obtain that U{,{x) -* 0 as n -* oo for every x e l f , so E„(x) -> 0 as n -> oo.

00 00

It is sufficient to prove now that if x =

£

Z

j=l /= i

for i — 1, 2, ... Let i be arbitrary and let s > i. We have

S

\at-bi\ = I = |J E ( a j - bj) ej eidii\

Q nj= 1

S

< Il

Z

j= i So a* = bt for i = l , 2 , . . .

References

[1] J. M u s ie la k , Modular spaces (in Polish), Poznan 1978.

[2] —, Modular approximation by a filtred family of linear operators, ISNM 60 Functional Analysis and Approximation 1981, Birkhâuser Verlag Basel, 99-110.

[3] —, Orlicz spaces and modular spaces, Springer Verlag, Berlin-Heidelberg-New York- Tokyo 1983.

INSTYTUT MATEMATYKI, POLITECHNIKA ALASKA

INSTITUTE OF MATHEMATICS, SILESIAN TECHNICAL UNIVERSITY GLIWICE