Inclusion theorems for classes of functions of generalized bounded variations

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIV (1984) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PR ACE MATEMATYCZNE XXIV (1984)

J. Ciem noczolo w sk i and W. Orlicz (Poznan)

Inclusion theorems for classes of functions of generalized bounded variations

Abstract. The paper is devoted to the spaces of functions of generalized bounded variation in the sense of Wiener-Young and Waterman. Authors aimed to give a methodical presentation of some inclusion theorems for the space Vv (see also [5], [6] which contain Theorems T, 2, with different proofs) and some theorems on relations between V* and W * .

0. The notion of p-th variation was first introduced by N. Wiener in [17].

Next, L. C. Young developed this idea in [18] and defined the ^-variation of a real-valued function of real argument. By (p we understand a continuous, non­

decreasing function <p{u): <0, oo) -> <0, oo) taking (p(u) — 0 for и = 0 only and tending to oo as u -^ o o . Any such (p will be referred to in this paper as a q>- function. These are the ones that gave rise to Orlicz spaces.

The number

i^ v (x) = sup X n i= 1

with supremum taken over all partitions n\ a = t0 < t 1 < ... < t„ = b of the interval {a, b} is called the cp-variation of the function x. Letter / is going to denote an interval I = <(a, /?> c= <a, by and we write x ( I ) = x(/?) —x(a). Y v (x) calculated over the interval / will be written Y q>(x, I). When Y ^ x ) < oo the function x is said to have bounded or finite (p-variation or to be o f q>-BV.

<p-BV functions are bounded. If for a sequence x„ there exists a constant к > 0 (not depending on n) such that Y <p(k(x„ — x)) 0, н —► oo, then x„ is uniformly convergent to x in (a, by. Throughout the paper it is assumed that x(a) = 0.

Variations of this kind were investigated in numerous settings such as stochastic processes (see [2]), convergence and absolute convergence of Fourier series ([13], [16]) and modular spaces (Musielak, Orlicz [10], [11]). Recently, in connection with Fourier series, D. Waterman introduced in [14] (see also [15]) the definition of Я-variation. Throughout the paper, X = (Я,) is going to denote a

QO strictly increasing sequence of positive numbers such that Я, - > х , £ 1Д

i = l

= oo.

The number

wA(x) = sup 2. - 1— ,

where supremum is taken over all collections (7{) of non-overlapping subintervals of <a, b>, is called À-variation of the function x (for the sequence À).

Infinite collections of subintervals (/,) can be replaced here by finite ones.

When н»я(х) < oo the function x is said to have bounded or finite À-variation or to be of Я-BV. This is equivalent to the definition given by D. Waterman in [15].

Я-BV functions are bounded.

1. For a ^-function (p the following conditions will be needed:

(0i) <p(u)

и as

(00i) (p{u)

--- ► 00 as и —► oo.

и

Function (p is said to satisfy condition A 2 (for small и) if there exist к > 0, u0 > 0 such that q>{2u) ^ k(p(u) for 0 < и ^ u0.

Let (p satisfy (OJ and (oo^. The function

(*) <P*(v) — sup ((mu — (p{u)): и ^ O) for v ^ 0 is called the function complementary to q>.

It is known that (p* is a convex (^-function satisfying (0^ and (oo i );(<?*)*

— (p if and only if tp is convex.

For arbitrary ^-functions satisfying (0^ and (oo^ there is the inequality ((p*)* ^ (p for и ^ 0. We call q> = ((p*)* the function associated with (p; (pis the largest convex function not exceeding (p. (*) implies the generalized Young’s inequality

uv ^ cp (u) -f- cp* (v) for all U, V > 0.

For every v > 0 there exists the smallest uv > 0 such that uvv = (p(uv) + (p*(v).

Besides, uv -> 0 as t?->0 ([1], [7]).

Two (^-functions (p, ф are called equivalent for small и when for some positive constants d1, d 2, k l , k2 > 0 and u0 > 0 the inequalities

(**) dl (p(k1u) ^ ф{и) ^ d2 (p(k2u)

hold for 0 ^ и ^ u0. This equivalence is denoted by tp ~ ф.

1.1. Suppose that (р,ф are ф-functions satisfying (Oj), (oOi):

Inclusion theorems for classes o f functions 183

(i) if ф{и) — a tp(bu), a, b > 0, then ф*(и) = a(p*(u/ab), (ii) if tp(u) ^ ф(и) for и çz Ug, then ф*(и) ^ <p*(u) for и ^ uS, (iii) if (p ~ ф, then (p* ~ ф* ([7]).

1.2. Consider the vector space V* = (x : t \ { k x ) < go for some constant k > 0). i \p{x) defined in V* is a modular in Musielak-Orlicz sense ([11]). If У ^(çx) -> 0 as ç -* 0, I ^(х) is said to satisfy condition Bl. A vector subspace of V* consisting of all elements for which Bl holds is called the modular space.

It can be given an F-norm generated by I \p:

||x||J = inf(e > 0: 1^(х/е) ^ г).

Moreover, when tp is convex, then there exists also a В-norm in the whole space

V *V4> ’

M i = inf (г > 0: У^(х/е) ^ 1),

equivalent to || • ||J, which now exists in the whole V*. We shall use the notation

= (*: ^ „ ( x ) < со).

13. For a given (^-function tp let us put OO Qv (a) = Z <PM)

i — 1

(a = (a,), b = (b,) always denote number sequences).

The set /J = (a : qv {ka) < oo for some constant k > 0) is a vector space with the modular q9. This modular satisfies the condition Bl, i.e., Q^ica) 0 as

£ ->0, so it generates an F-norm ||a||^ = inf(e > 0: ^ (a /e ) ^ e). When (p is convex one can define the Luxemburg norm IMI^ = inf(e > 0: Q9{a/e) ^ 1).

We shall use the following notations : l9 = (a : (a) < oo), l*m = [a e /* : a, \ 0), l*d = (u e /* : at \ 0 strictly, Z\ai = со). I™ and ldv are defined analogously.

1.4. Let us write W* = (x: wA(x) < oo). VKA* is a Banach space with the norm ||x||? = wA(x).

2. ([1], cf. Satz 1) If for any a e l r b e /J the series Z ai^« converges, then (*) uv ^ г((р(и) + ф ^)) for some r, u0 > 0, 0 ^ и ^ u0, 0 ^ v ^ u0.

Suppose (*) does not hold. So, by Hilfssatz 7 from [1] there exist sequences uf .->0, Vi ->0, щ ^ 0, Vf ^ 0 such that Z<P(Mi) < 00 » Z ^ ( y;) < 00 and Z u‘ v‘

= 00.

Let us take a decreasing rearrangement (vp.) of (t>,). By a simple construction we can find a decreasing sequence £,• -> 0, such that (в,- vp.) will be strictly decreasing and Z uPi £« \ = 00 • since Z up, e« \ ^ S4P ui Z et vPi> Z e« vPi diverges, so (£f vp.)e l ^ Z <p(uPi) < со, which leads to a contradiction.

2.1. I f the inclusion

00

(*) П Ç„ c K

n

holds, then there exists a natural N such that there is the inequality (**) ф(и) ^ dsup(q)l (u), (p2(u), (pN(u)) with some d > 0, u0 > 0,

and all 0 < и ^ u0 and conversely.

If (**) is fulfilled then, clearly, from (ul) e /<P2, . . . , (af) e /VJV, a{ ^ 0, follows

00 00

Y l 'xH ad < d ( £ ç>i(af) + ... + X <Ma«))

1 1

N

for almost all i, so f) Çn c K-

П

If condition (**) is not satisfied, then there exists a sequence un -> 0, Ф Ы ^ such that (Wj,) > 2"<p„(un), where <p„(u) = s u p ^ f u ) , <p2(u), ...

..., (p„{u)). Since ф„(и„) ^ 1/2" for n = 1, 2 ,..., one can choose integers kn > 0 in such a way that l/2"+1 ^ к„ф„(и„) < 2/2".

Let us construct a sequence (uj) consisting of all terms un, with each term repeated k„ times. For the n-th group of terms we have the estimations:

X ' <Ж ) ^ 2"k„ фп(u„) > i ; £ ' = *•.&(“■) < 2/ 2">

where means summation over the и-th group.

00 00

From the second inequality it follows £ <p„(w-) < 00 • Hence, £ <р„(и-)

i i

QO

< X , n = 1, 2 ,... and X ( Ж ) = oo. By a suitable rearrangement of (u-)

I uu

we are getting a sequence (bf), b{ \ 0 such that £ <p„(bf) < oo, « = 1 ,2 ,...

OO i

and J ij/{bi) = oo — a contradiction.

t

2 .2. / / r/ic inclusion

00

(*) Ç* <= U K„

n

holds, then

(**) <P/v(u) < dNij/(u) for some natural N , dN > 0, u0 > 0, 0 ^ и ^ m0, and conversely.

If (**) does not hold for any N, then there exist sequences (unr), « = 1 ,2 ,..., such that <jo„(«J ^ 1, ^ 2"+riJ/(unr) when n, r = 1, 2 ,... Hence unr -*■ 0, n, r -> oo.

Inclusion theorems for classes o f functions 185

Let us choose integers knr > 0 in such a way that the inequalities l/2"+r+1 < кпгф(ипг) ^ 2/2"+r hold for n, r = 1 ,2 ,...

Let us construct a sequence («,') consisting of all terms unr, with each term repeated knr times. We have the inequality with double summation

OO 00 J

I 'Ж ) ^ 2 X — <

i = 1 n ,r = 1 z

In (u'i) for the group of terms equal unr we have the inequality knr (p„(unr)

^ 2n+r кпгф(ипг) ^ j . So, for the respective group of terms ] Г (pn(u[) ^ j 00

and summing up all the groups we obtain £<p„(u-) = oo. Since

i

so, rearranging suitably (m-) we get a sequence (Ьг), bt \ 0 such that

00 00

£ «A(^i) < o°, X ФпФд = 00 for n = 1 ,2 ,... which leads to a contradiction.

i i

Sufficiency follows from 2.2.2.

These lemmas are analogous to the results in [8].

R e m a rk . In 2.2 (*) the set can be replaced by /$ if at least one (pn satisfies (Oi). Then <p„(bi) ^ b, for almost all i. There is £ fe, = oo, a contradiction, as in the proof above.

2.2.1. ly = l* iff (p(2u) ^ d(p(u), for d > 0, 0 ^ и ^ u0.

2.2.2. I™ ci / | iff the inequality

(*) ф(и) ^ dtp(ku) holds for some d, к, u0 > 0, 0 < и ^ u0.

When ф satisfies condition (0^, then c= Щ iff (*) holds.

GO

Set (pn(u) = ф{и/п). Then Ç cz l* implies Ç a (J lVn. An easy proof of П

sufficiency is omitted.

2.2.3. (a) l* а Щ iff 2.2.2 (*) holds, (b) Ç = Ç iff q> ~ ф.

2.3. Let (p convex satisfy (Oj) and (ooj). Let (b,), bt ^ 0 be a non-increasing sequence. I f for every a e I f the series

(1) converges,

then be Ip.

U

Set ф(и) = j* —j^-dt. is increasing so (p{u/2) ^ ф(и), ф{и) ^ tp{u), о

(p ~ ф and hence /*d = I f . This implies that £ а,Ь,- converges for a e l f . It is easy to show that for any sequence (a,) e I f series (1) converges. Indeed, suppose a1 ^ a2 ^ ... ^ at 0 and £ a, = oo and series (1) diverges. With the aid of a

simple construction one can find a decreasing sequence s„ -* 0, such that

£ Uj hi = -x , £ tij = со and putting aj, = we would get a contradiction to the assumption of our proposition. If £ af < oo, then (1) converges as (/?,-) is bounded. Repeating the reasoning from pp. 17-18 of [1] we conclude that there exists a constant к > 0 such that

(2) |£ аД-| < /с for и = 1 ,2 ,...

аеЩт for which дф(а) < 1.

Let us choose а,- in such a way that

(3) а ^ = ф(а1) + ф * ( ^ , i= 1 ,2 ,...

However, ф'(и) = q>(u)/u is strictly increasing in <0, oo), ф’(^) — Ь1, so the sequence (ôj) is non-increasing. The sequence , â 2, • • - , ân, 0, 0 ,... belongs to

n n

Щт so, if Q^(a) = Y , Ф(^д ^ I» then (2) and (3) imply £ ф*(Ь(/к) ^ 1 ; if

l i

n n

вф{$ > 1, then X (я./еДя)) ^ 1, (2) and (3) imply X Ф*(ЬЛ ) = 0. In consequence we have g^(h/k) < 1. From <p* ~ ф* there follows b e l

3. Let T denote an arbitrary sequence (r„), а = т0 < < т2 < ... < т„

< ... < b, т„-*Ь. V° (T) will stand for a class of functions in (a, b) such that x e V°(T) iff x(a) = 0, x(t) = 0 for t e (a, b ) \ T and x(t,) = ab where (a,) is an arbitrary element from s, the space of all sequences. The sequence a = (a,) will be said to correspond to x e V ° ( T ) , and conversely. V®*(T) will denote the vector space of elements from V°(T) such that their corresponding sequences (a,)e/J. V°m(T) is the set of functions from V°(T) for which я, \ 0.

3.1. I f a = (aj is the sequence corresponding to x e V ° ( T ) then the following inequalities hold:

(a) Qvia) ^ *%,(x), (b) ^ ( j x ) < 4Qv (a),

(c) iTv (x) ^ 4qv{a) for Oi \ 0.

(a) is obvious. To show (b) let us take a partition n: a = t0 < t x < ...

... < t n = b. Consider the following classes of subintervals: (i) /,. = (тк„ т,.), (ii) /, = <rk.,f> or Ij = where f e ( a , b ) \ T , (iii) / = < f',f" > , f ', Г"

e ( a , b } \ T . For subintervals of class (i) we have <p(||x(/f)|) ^ (p(\ak \) + + <Р(|я/,1) ог <р(1*(Л)1) ^ <p(|a*f|) for a( \ 0. For subintervals of class (ii) we have either <p(|x(J.-)|) = <p(\ak.\) or (p(\x(Ij)\) = (рЦаф for class (iii) <jf>(|x(/)|) = 0.

Intervals of class (i) are disjoint. If A is a set of different intervals from the

Inclusion theorems for classes o f functions 1X7

partition n which fall into class (i), then ф k[ implies Ф /• and summing over the intervals from A we get

2 > B l * № ) l K 2 f > » or I «р(|х(Л)1)< 2e M if a<4 °-

А Л

If В is a set of different intervals from n which belong to class (ii) then summing over В we get £ <p(|x(/,)|) < 2q9(a),

в

Finally, intervals of n belonging to class (iii) give £'< p(|x(/f)|) = 0.

Summing up over all intervals of n and considering n arbitrary, we obtain either (b) or (c).

3.2. Let a be a sequence corresponding to x. c) and Q^ia) in V®*(T) are modulars satisfying condition B l. The F norms || -\\9 , || -Ц^, generated by t and Qy respectively, are equivalent and

(*) . l | a H ^ I I < ^ 4 | | | < .

I f a{ \ 0, then in the inequality on the right-hand side \a can be replaced by ia .

By 3.1 (b) it follows, that if £n -»0, then ^ ( ^ „ x ) - > 0 , so Bl holds.

^ follows immediately from 3.1 (a). By 3.1 (b) if Qv (a/£) ^ e, then 1 ^ 4Q9 (a/e) ^ 4e, so \\2х\Ц ^ 4e, ||2x||J' < 4\\a\\v .

33. If x e V ° ( T ), a = (a,), at \ 0 is the sequence corresponding to x, then, for arbitrary X the following inequalities hold:

(*) f y < * я ( х ) < 3 £ у .

i A f i*

R e m a rk . The inequality on the left-hand side is valid for any x e V°(T), when a{ ^ 0.

Let us choose in (a, b) a sequence of non-overlapping intervals (/,) and consider the intervals classes mentioned in 3.1. For class (i) we have |x(/,-)| ^ ak., for class (ii) |x(/,)| = ak. or |x(/y)| = ax. , for class (iii) |x(/,)| = 0 .

Summation over intervals of class (i) gives

(1) ^{V W i ) \}

' X;

“ к; 00 ar. ^{x a}

< Г т + Е " г = Х л

*•> j * i к A k Pk

where ar. are all the terms of a different from ak. and put arbitrarily at Xj different

from all Я,-. So, (aPk) is a rearrangement of the sequence (a,). By a known theorem [3], [12],

f |x(/,)| a.

A i 2-, ^ •

i А,-

Summation over intervals of class (ii) gives x(Ii) I

(2) Z

, k

and by an analogous reasoning we get

i ^ " T V

3.4. f/|x(t)l ^ M for t e ( a , b}, q> satisfies A2, then there exists a constant

П

к > 0 such that if I — (J h n f — 0 , * # j, there is

l

П (x, /) < к ( Г 9 (x, I,) + (x, I 2)

+

+ r 9

Proof is based on the following remarks:

1° If (p satisfies A 2, sup|x(f)| ^ M, then iT 9(2x,/,) ^ k 'V ^ x , /,), where к depends on M only and is independent of the intervals.

2° If 1 = V и /", Г n /" = 0 , then

cp(\x(I)\) ^ ср(\х(Г)\ + \ х ( П \ ) ^ <р(2\х(П\) + <р(2\х(1")\).

Let Up denote the class of functions in <a, b) having the property: for any system of non-overlapping intervals f c <a ,b ), we have (cf) = (|х(/,)|)е/*.

It is easily seen that functions x e U ^ are bounded.

3.5. For an arbitrary (p-function c= U^, if (p satisfies A2, then cz Vv _ у *

r <p •

Proof of the first part of the theorem is omitted.

Let x e U 9, |x(f)| M . For an arbitrary sequence of non-overlapping intervals (/,), £<p(/c |х (/{)|) is convergent with some к > 0. Conditions A 2 and 2.2.1 imply 5 > (|x (/,)|) < oo. Suppose F~q,(x) = oo. Let us choose non­

overlapping intervals

k i

(1) / } , / 2, . . . , / ^ in such a way that £ <р(|х(/,-)|) ^ <p(2M)+l.

i = 1

kl ,

If the complement <a, b } \\ J I f is not empty, let

(2) l'i , I 2 , ..., Jj* denote intervals that compose this complement.

By the preceding lemma there would be тГ^(х, I 1) = 00 for at least one

Inclusion theorems for classes o f Junctions 189

interval / 1 of class (1) or (2). When / 1 belongs to the class (1), we remove it and for the remaining intervals of class (1) we have the estimation (p(\x(If)\) ^ 1.

In the case where l l belongs to the class (2), we let stand all intervals of class (1), so we get a collection of non-overlapping intervals denoted by T{, 7\, 7] , . . Tf ,

h

for which £ <p(|.x(//)|) ^ 1, and the interval 71 disjoint with them for which

l

*r v ( x , I 1) = zo.

Proceeding with / 1 analogously to (a, b), we get a collection of disjoint

l 2

intervals if , I f , ..., I f included in 11 and such that ]T ç>(|x(/?)|) ^ 2 and an

i = 1

interval I 2 disjoint, for which t f i f x , ! 2) = со. Applying the same procedure with I 2, and so on, we define by induction a sequence of systems of intervals (3) T \,Jr2, ... ,Tlf such that

(a) £ (р(\х{Щ ^ r, r = 1, 2 , . . . ,

/= i

(b) all the intervals of all the systems are disjoint with each other.

00

If we set all the systems of intervals in a sequence (/,), then ]T </>(|x(/,)|)

= x , a contradiction. So ^ v (x) < oo and x e l ^ . 1

Let Л denote some family of sequences X ; Лф is the family of all sequences X

= (Я,) such that (1 /Я ,)б /|а.

4. Theorem 1. The inclusion

00

(*) П Ущ с Уф

holds iff condition 2.1 (**) is satisfied.

N e c e s s ity . For some T let us consider the subspace I^.W(T). xeV£.m(T), хеУ ф т(Т) iff g (a) < x or дф(а) < со, respectively, where a, \ 0 is the sequence corresponding to x. So, now it is sufficient to apply 2.1.

S u ffic ie n c y . It is easy to see that by a suitable modification of the constant d, condition 2.1 (**) can be written in the form

(1) ф{и) ^ d((px (u) + . . . + (pm(u)) for 0 ^ и ^ u0,

m

where u0 is an arbitrary number. If x e П Vv.> then x is bounded and one can

l ‘

apply (1), putting |x(r)| ^ u0 for t e ( a , h ) . For arbitrary non-overlapping intervals / l 9/ 2, . . . , we get from (1)

I X M 7 -)I) ^ d & ( p 1(\x(Ii)\) + ... + f j (pm(\x(Ii)\j)

1 1 1

m

^ d (T fi1{ x )-f... + Y 'q>m{x)) so tfi{ x ) < со, f] V с

l

2 — Prace Matematyczne 24.2

4.1. Theorem 2. The inclusion

00

K <= U Kt holds iff condition 2.2 (**) is satisfied.

The proof is similar to the preceding one, with the aid of the spaces V f^fT ), Уфт(Т) and Lemma 2.2.

4.1.1. The inclusion

(*) V* c= V f

holds iff condition 2.2.2 (*) is satisfied,

(**) V f = V f iff (p ~ ф.

Ad (*). We use the preceding theorem, putting (pt = ip ^ u j , and a remark that the inclusion a V f implies (*) ([4], [10]).

4.2. Let q> convex satisfy (OJ, (oOi).

I f V f c W f, then (1 A )e /* f c /*,.

Let us consider a subspace Vf*{T) cz Vv . To every u g/J, a{ ^ 0, there corresponds a function x e V f* ( T ) so x e W f . By 3.3, is convergent.

From 2.3 we get (1 /A,-)g/** c /*„.

4.3. Theorem 3. Let tp convex satisfy (ОД (оох) and tp* condition A 2. A denotes a countable set o f sequences A", n = 1, 2 ,... There is no such A that

(*) V f = f ] W f = Wf .

ЛеЛ

Let A" = (A?), ||x||” = wA„(x). The set Wf is a B0_sPace with the norm N 1 = 1 1 N i;*

2" l + N I Л”

Suppose (*) holds. A( ) denotes the identical mapping A: W * -> V f. If

\\xk — x|| -> 0, IM(xfc) — y\\* -> 0, then у = A (x), because xk(t) -> x(t), xk(t) ->y(t) uniformly in (a , b ). By the closed graph theorem A (•) is continuous from W* to V f and consequently, for some <5 > 0 and some natural m, ||x||™„ < Ô for n

= 1, 2, ..., m there is ||x||J ^ s. Since ||-||J and ll'IIJ„ are homogeneous, for some constant к > 0, we have

(i) N l 2 < k ( N i r i + --. + N i r J .

Let us fix T. Let xr e V f* (T), where хг(т1) = 1 ,..., xr (ir) = 1, and xr (i,) = 0 for г, > тг. Obviously, i r (p{xrfi) = 2r(p(l/E),

M J = inf(e > 0: 2rç>(l/e) < 1), so ||xr||J = l/<jo_ x (l/2r).

Inclusion theorems for classes o f functions 191

On the other hand, ||хг|Гп ^ 2 r 1, n = 1 ,2 ,... and from (1) we get for x = xr i Ai

(2)

By 4.2 (1/A")e/**. Let us apply Holder’s inequality

(3)

where a = (0 ,0 ,..., 1, 1 ,...,1 ,0 ,0 ,...) , У — {1/Aj) for i = 1,1+1, and

l, 1+ 1, r oo /

If = 0 for the remaining i. By condition A2 the series £</>*( k j j J is convergent for every к > 0, so Y <p*[ k —. ] -» 0 and consequently ||hi|° -> 0 as l -> oo, for1

V Ц / <p'

any r > l. Let us notice that ЦаЦ^ = ---тг+г,---- м- By (3) we get

(4)

From (2) and (4)

<P-i(l/2(r-/))'

1 me.

V - 1 (l/2r) - i ( l / 2 ( r - / ) ) ^me.

However, </>_1(l/2r) < <p_ t (1/2(r — /)), so, when r -* oo, we get 1 < me, a contradiction as e -> 0.

43.1. Th e o r e m 3' (see [12]). Let V* = Vbe the space offunctions o f bounded variation. Л denotes a countable set o f A", n = 1, 2 ,... There is no such Л that relation 4.3 (*) holds.

If xr denotes the same function as in the preceding proof, by an analogous

/ r 1 r 1 \

reasoning we obtain 2r < 2 /c l£ ^ y + ... + £ — ). Since for sufficiently large /,

1 A i

i n 1 A i

r > l, Yj +] < (r — Oe for у = 1, 2 ,..., m, we have1 i = l A i

1 , / ' 1 ' 1 \ (r —Z)

which leads to a contradiction as first r -> oo and then e -> 0.

43.2. Th e o r e m 3". I f q> satisfies conditions (0 J, (oOi) and (p* condition A 2,

then

(*) V* = W*

does not hold.

In V f two norms are defined |H |^ , ||-||2 and the closed graph theorem implies that they are equivalent. In this way in V f*(T) the norm ||a||v is defined, equivalent to || -||J, where a is a sequence from l* corresponding to the element x of this space. This means that also in /* the norm || -||v is equivalent to the B- norm || • ||J\ By the known theorem ([9]) tp ~ ф, where ф is a convex (^-function satisfying (OJ, (oo^. However, У* = У^,ф* satisfies A 2 by tp* ~ ф*

(1.1 (iii)). Theorem 3 implies that (*) does not hold.

R e m a rk . It can analogously be shown that with the same assumption for (p,

П

ЯеЛ

4.4. Theorem 4. Let cp convex satisfy (OJ, (оох) and A2. Then

(*) V * =

П

Let (/,) be a sequence of non-overlapping intervals in (a, b ) rearranged in such a way that ^ ( / J l ^ |x (/2)| ^ ... If x e П h ? , then £ ——^ < oo

ЯеЛф * 1 Ai

for any ЛеЛ „. Lemma 2.3 implies (x (/())e/*, so x e U ^ . By 3.5, Uv = V f , consequently f | W f с= V*.

Now, let x e V f , A eA *. Since <p satisfies A2, we have T f (x) < oo and by Young’s inequality we obtain

« f W W '.)l)+f v * ( £ ) « П М + £ ч » * (£ ).

For some к > 0, £ tp* (k/A{) converges, so this inequality holds for any choice of (/,) which means that x e W * , V f cz f) W*.

кеЛу*

4.4.1. Theo rem 4' ([12]). V = Ç] W f, where Л denotes the set o f all A.

ЯеЛ

Inclusion V c= f) W f is obvious. Let x e f | W f and f — arbitrary non-

ЯеЛ ЯеЛ

overlapping intervals. For any Я,- s CO, 1 ( 1 Д ) = со, 1(|х(Л )|Д .) < oo and by simple considerations we obtain £ |x(/,)| < oo. By 3.5 (with cp(u) = и), xe V, in consequence П W f cz V.

ЯеЛ

Inclusion theorems for classes o f functions 193

4.5. Theorem 5. Let (p, ф satisfy (OJ, ( o o j. I f the condition

(*) ^v*= П

holds, then (p ~ ф*, where ф* is convex.

Let us consider the subspace V®* (T) a V f with fixed T. If a = (a,), a, ^ 0, denotes the sequence corresponding to x, such that £?v(a) < oo, then wA(x)

v

* by 33 for апу я with 1 Ф { Щ ) " °° ' By 2 we have * (p(и) + ф(t>) for 0 < и ^ u0, 0 ^ v ^ v0, r > 0. For a given v, let us choose uv such that

(1) uv- = (p{uv) + q>*

r

Since uv -> 0 as v -> 0, choosing Ô sufficiently small we have for 0 ^ v ^ Ô equality (1) and

(2) uv- < v ç>(m„) + ^ (ü), r

(3) (p*(u/r) ^ 1J/(u) for 0 ^ U ^ Ô.

oo

Choose aeZ" i.e. at \ 0, £ ф* (af) < oo and let À eA ^. From Young’s

^ i

OO 00 00

inequality we get к £ (a,A) < £ <A*A) + X <A(^A) < 00 for some /с > 0. This

i l l .

and 3.3 imply that x, whose corresponding sequence is a, is an element of W f c Fv*. Thus, by 3.1, a e /* .

Further, by 2.2.2, we get

(4) <p(u) ^ тф*(пи) for 0 ^ и ^ u1.

Application of 1.1 (i), 1.1 (ii), (3) and (4) yields 1 ( u \ 1 _ f u \

tp(ru) ^ ф*(и) ^ — (p - ^ — (p - for small u.

m \ n j m \ n j This means that <p ~ ф and ф* ~ (p, so tp ~ ф*.

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