A N N A L ES S O C IE T A T IS M A T H E M A T IC A E PO LO N A E Series I : C O M M E N TA TIO N E S M A T H E M A T IC A E X X I I I (1983) R O C Z N IK I P O L S K IE G O T O W A R Z Y S T W A M A TEM A T Y C Z N E GO
Séria I : P R A C E M A T EM A T Y C Z N E X X I I I (1983)
D. J
acii(Szczecin) and W. O
rlicz(Poznan)
Some classes on Saks spaces
0. In this note, (T , 27, ju) will be a measure space, where 27 is a or-algebra of subsets of a non-empty set T and pt, a finite cr-additive measure on 27.
(Note, however, that our results remain valid when ц is <7-finite measure.) Moreover, pi is assumed to be non-atomic.
We denote by 8 the set of all real-valued 27-measurable functions on T which are ^-almost everywhere finite. I t is an F-space with an F-norm defined as usual, i.e., ||a?||s = j\x(t)\[l + \x(t)\)~1dfj,. The symbol %e will
T
denote the characteristic function of a set e e 27 and M a subspace of the space 8 of bounded measurable functions.
1. Let cp: <0, oo)-><0, oo) be a <p-function, i.e. cp is continuous non-de
creasing with <p(0) — 0, 9o{u) > 0 for 0 < и and lim<p(w) = oo. cp is said
u-+oo
to be r-convex, 0 < r < 1, if <p(au + ^v) < arcp(u) + @r<p{v) for a, /? ^ (P ar + j8r = l .
We say that a function cp satisfies the condition (Oj) if <p(u)lu -*0 as 0, and the condition ( oop if cp(u)lu-+ oo as u->oo. cp satisfies the con
dition (A2) for large и if cp(2n) = 0(cp(u)). For any <y-function cp satisfying conditions (op and ( oop we define a complementary function (in the Young sense) by the formula
cp* (v) — sup ( uv — у («)).
i о
The function cp* is a convex ^-function and satisfies conditions (ox) and ( °0])- For every x e S we put
I<p(x) — § cp[\x{t)\)dn, I{x) = fx {t)d [i.
t t
An Orlicz space is by definition:
L *v = {x e 8 : 19{Щ < oo for some A > 0}.
We will write
L * ,p — {x £ 8 : I^ ho) < oo for arbitrary A > 0}
(a subspace of finite elements). The sets L *<f, L *v are ^-spaces with the so-called generated norms ||-||v. If <p is r-convex, then
\\x\H = inf {e > 0 : I <p(xe~lir) < 1}
is an r-homogeneous norm in L *v and L *v.
In the case <p is convex (r = 1) and satisfies (oa), ( oof), we can define an equivalent norm to the norm ||*||* by the formula
IHIÎ = supl(|®| |y|)?
where the supremum is taken over the set {y e S: I^*(y) < 1}. If r = 1 we will write ||a?||° instead of \\x\\lr If cp(u) = up, p ^ 1, we rather use the sym
bol L p instead of L** and ||-|fp in the place of ||*||*.
1.1. Let X denote a real vector space with an r-homogeneous norm
||*|! and an L'-norm ||-||*. The set X s = { x e X : |[ж|| < 1} is called a Sales space if for the metric d(x, у) = ||a? — y\\* it is a complete space (cf. e.g.
[8 ], [9], [10] where ||*|| is assumed to be homogeneous). We will denote the Saks space by X s (co), со: ||*||, ||-||* or X s(||-||*), if the metric in X s is gen
erated by ||-j|*. Afunctional £ is called со-continuous if £ is defined and con
tinuous on X s ( со), со: ||-||, II* II*, £ is called linear in X 8( со) if £ is the restriction of a linear functional on X. We say that a Saks space X s.(co) has the BS- property (Banach-Steinhaus property) if the following condition holds:
i f (£») is a sequence o f linear co-continuous functionals and £n(x)->0 fo r every x e X s, then the set { £ „ : » e l } is equicontinuous at x = 0, in X s{co), i.e.
provided ||#J|*->0, xn e X s . ‘
2. Let ip be an arbitrary ^-function and cp an r-convex ^-function.
Let us write X = 1 / <рп Т *у>, X s = {x e X : ||a?||J < 1}. The set X s with the metric induced by the norm ||*||v, i.e. the set Х 8(||*||у) is a Saks space.
2 .1. The following conditions are equivalent:
(a) y(w) ^ le <p(lu) for и ^ u0 5 (b) X = L *<p-,
(c) The Sales space X s satisfies condition (n0):
W I ^ O implies ||a?Jv->0, xn e M.
The implication (a) => (b) is a consequence of a result of W. Matuszew- ska ([4], Th. 2.4, cf. also [5] ; use the Lebesgue measure there as a //-measure is inessential).
(b) => (c) and (c) =^(a) follow from Theorem 3.2, 3.2.1 of W. Matuszew-
ska [5].
Some classes on Saks spaces 43
2.2. Let ip and <p be as above. I f ip fulfils condition (A2) and
. . . f ( u )
(* ) ---> oo as u-> oo,
x ’ i p { c u ) ’
then L *9 c L*'% and the norms |!-[]s, ||*|[v are equivalent in X s.
The condition (*) implies 2.1 (a). Hence by 2.1 (b) we have L** c L *v.- Let xn e X s, IKJjs ->0 and let e > 0, d> 0 be arbitrary. Write
= {*: K ( t ) ! > ô},
% = {t- K (*) ! > М Щ
where u(k) is a number such that <p(u) ^ kip (eu) for и > и [к), к e X.
We have
(1) j > f у ( Ф п(1)\)dp,
an
{2) j ip(c\xn{t)\)d(j,^ f ... -f J ...
an a'rin e n an n e n
< ip[cu(k))pi(en) + ip(cô)pi(T) for n e N . Since к can be arbitrarily large, /л{еп)^-0 and then (1 ) and (2 ) imply I w{cxn)
— >0 and consequently ||$n||v-^-0, by condition (A2).
C
orollary, (a) I f ipx, ip2 are arbitrary ep-functions satisfying fo r large и condition (A2) and
lim w->oo
y(«) ip1 (c xu)
v <p(u)
o o ,
lim --- = oo,
«-*00 ^2(^2^) then the norms ||-||v,i and ||-||V2 are equivalent in X s.
((3) I f a (p-function <p satisfies condition (ooj), then L *<p a L 1 and the norms ||-||^ and Ц■ Цг are equivalent in X s.
2.2 .1 . Let (p by an r-convex (p-function, ip x, ip2 (p-functions satisfying con
dition {Af) for large u. I f the norms fj• ||V>1, Ц-Ц,^ are equivalent in X s, then the inequalities
( ~b) ^ ^2(vi(^ 0 “b tpilz'M'))
1( + + ) wAu) < +
hold fo r sufficiently large u.
Applying Theorem 3.2.1 [5] to the functions %% = щ -\-<p or Xi = Vz+V
we get the conclusion.
C oeollaey . Let a function <p satisfy condition (oox). I f the norm [[*j|v is equivalent to ||'||x (or |f-|J>s«) in X s, then ip(u) < Tcu-j-cpfu) fo r и > u0.
3. In the next theorem we will need the following well-known L emma . Let cpbe a convex (p-function satisfying conditions (ox) and ( oox).
Assume that S is a a-algebra o f Lebesgue measurable subsets o f the set T
= <«■, b}, (a, b fin ite). I f I ^ x J 1 for every n e N, then there exist a sub
sequence (xn.) and a function x such that
(1) I ( x n.xr)~+I(xxT) uniformly on T ;
(2 ) Iq>(®) < 1
(cf. Theorem 7 in [2], p. 55). (%x = * <a>T>.)
3.1. Under the assumptions of Lemma we define Xs = {x: ||a?||J then L *9 c L 1 and we can define a Б -norm on L *9 by the formula
(*) H®L = sup I Z(a?*T) I.
T e T
X s is compact in the norm Ц-Ц^ and then X e(H*||w) is a Saks space.
Obviously, the norm (*) is homogeneous and it is finite since by ( oox) we have L *9 <= L 1 and consequently
(**) M\w < M h for x e L * 9.
I t follows immediately from Lemma in 3 that the space X s (Ц-Ц^) is compact.
3.2. Let cp be a convex cp-function satisfying conditions (ox) and ( oox).
F or any (p-function y> the norm ||-||v is not weaker than |J * |[w in X s but the norm
|!*IU i8 strictly fin er than |]-||y.
The first part is a consequence of the inequality 3.1 (**). For the other part, let us take a sequence of orthogonal functions on T such that
b
suppess \xn (t) К m for n e N, where cp (m) (b — a )< 1 and j x2n (t) dt — d > 0 for
__ a
n e N. Since the set of functions yn = x jV d is an orthogonal system on (a, b), by Bessel’s inequality we get I(y nxr)-> 0 uniformly on T. Moreover, I<p(xn) < 1> therefore xn e X s and |[лУ|ц,->0. If for some subse
quence (xn.) then xn.-^ 0 which implies that j x2n.(t)dt->0, contrary to the % l Ъ definition of the sequence (xn). Consequently, lim inf ||а?и||у > 0.
4. Let cp be a convex cp-function satisfying conditions (ox), ( oox) and let X = L *9. A linear co-continuous functional in X s(co), со: Ц-||®, ||-j|x can be represented in the form
(*) i(x) = I(xy) for x e L * 9,
where
(**) y e L p ' .
Some classes on Saks spaces 45
Conversely, i f y satisfies condition (**), then the integral (*) is a linear o>-continuous functional in X s (co).
Let us take a linear functional rj on L *9 c L l which is continuous with respect to the norm Ц-H
j. The following inequality holds:
<1) \v(x)\ < ft 11®Hi for x e L * 9.
By the Hahn-Banach theorem, rj can be extended as a continuous linear functional rj to L 1. Applying the representation theorem describing the dual of L 1, we get
(2) 7](x) — I{xz) for x e L * 9,
where z e M. Let £ be a linear со-continuous functional in X s {co). By The
orem 4.2 in [1] there is a sequence of functionals of the form (2) (where instead of z we take zn e M) such that
(3) \I{xzn)-£ (x )\ ^ 0
uniformly on X s — {x: \\x\\cv < 1}. I t is clear that zn e L *9* and (3) implies j|2n —2m||“*->0 as n, m-+ oo. Moreover, the space L *9* is complete, thus there is an y e L *9* such that *
(4)
Since the linear subspace of all finite elements of L *9* is closed in L *9*
with respect to the norm ||-||°*, then y e L *9* and by (3) and (4) we have (*).
To prove the sufficiency, let us note that if y e L *9*, then the integral (*) exists for every x e L *9, and by Theorem 3.5 in [5] (cf. also [6 ]) the set M is dense in L *9* with respect to the norm ||-||°*. Hence there is a sequence zn e M such that \\zn — 2/||J«-*0. By Holder’s inequality we obtain
<5) \I(xzn) - I ( x y )\< |N|‘ |K - y If,..
Let us take xk, x0 e X s, \\xk — £P0||i->0 . Since I(x kzn)-> I(x0zn) for n e N , by (5) we have I( x ky)-+ I(x0y).
4.1. Let T — (a, b), X = L *9 and let <pbe a convex <p-function satisfying conditions (Oj), ( oof).
Every linear continuous functional in X s(co), со: ||-||*, ||-||ш has the representation 4 (*), where у fulfils condition 4 (**)•
The “necessary” part follows from 4 since the norm Ц-|]х is not weaker th£|n ll'IU-
Conversely, for every у satisfying condition 4 (**) the integral 4 (*) is a linear continuous functional in X s (co). To prove this, let us assume y e L *9* and take a sequence of step functions (yn) such that \yn{t) \ < \y{t)\, Уп{1)~>у(t) a.e. on T. Since we have
Ц\Х\ Ш Х е ) < Ц \ Х \ \У\Хе) < Ы Ц У Х е Щ * < e
provided ju(e) < <5, therefore, applying Egoroff’s theorem, we obtain
\I(xyn)-I(xy)\ -+ 0
uniformly on X s. Hence the functionals I(x y n) are continuous with respect to the norm Ц-Ц^; then, reasoning similarly as in the proof of (4), we can show the o>-continuity of I{xy).
Bern ark. I t is worth to note that the duals of the spaces
X(ll-lh) are not identical. (The representation of functionals from the duals of X{\\ Hj) is of the form 4(*) where у e M.)
5. Let cp be a (p-function satisfying the following conditions:
(a) <p is convex and fulfils conditions (ох), ( oox),
(b) lim --- = oo,
w->oo Cp(ll)(c) (p* satisfies condition (zl2) for large u.
Under these assumptions the Sake space X s{oo), to: ||-Ц®, ||-)|г has the B S -property.
B ern ark. There exist ^-functions which satisfy the above con
ditions a.e.
(p(u) = e u — u — 1 , cp*(v) = (1 + <y)log(l + v) — v.
Let £и(#)->0 for x e X s (co), where £n, n e N, are linear w-continuous functionals on X s (co). By 4 the functionals £n have the integral represen
tation 4(*), therefore
in (æ) = I for x e X s (со), yn £ L *9*.
Erom an Orlicz’s theorem in [7], p. 101-103 (where E is the cr-algebra of Lebesgue measurable sets) it follows that the functions (yn), n e N , have the following property:
There exists a constant Jc > 0 such that fo r every e > 0 there is ô > 0 such that
(1 ) I^ {kynxe) < e fo r n £ Ж i f p(e) < Ô.
Therefore if p(en)-+0 then I^ (k y aXen) - * 0, and ||yn* eJ| J.-*0 since <p*
satisfies condition (zJ2). I t follows that the norms [|yn||J* are absolutely equicon tinuous.
Let |fa?JJ < 1, p n||i->0. Applying Holder’s inequality we get
\1(ЯпУпХе)\ < Ы С <р\\Уп%е\\г,
\1 {ХпУп%е’)\ < \KXe'\\%\\lt n t * ,
