up, p ^ 1 is well known in the theory of real functions, and a more general case of convex ^-functions also has been considered in papers [6] and [10]

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1977) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE X X (1977)

7j. Cy b e k t o w i c z and W. Ma t u s z e w s k a (Poznan)

Functions o f bounded generalized variations

This paper is devoted to a characterization of some real function spaces. We investigate the spaces of indefinite integrals, where the function under the sign of integral belongs to the class L9 or to the space L*v (Orlicz spaces). The case cp(u) = up, p ^ 1 is well known in the theory of real functions, and a more general case of convex ^-functions also has been considered in papers [6] and [10]. The known characterizations for cp (u) = up use the notion of Eiesz variation and the moduli of continuity. The same notions we can also use for convex ^-functions. The limit case cp(u) = u means the classical theorem of Hardy and Littlewood, which deals with functions equal almost everywhere to a function of finite variation.

One can find the classical proofs of Hardy-Littlewood theorem in [2]

or [12] for instance. W . Orlicz proposed different proofs in [8]. We apply, to some extent, his method of demonstration in our work.

The purpose of this paper is double. First of all we are interested in the systematic presentation of some fundament al facts concerning func­

tions mentioned in the title, which are known from the literature of the subject, but presented there in a less definite form or discussed only for the special case cp (u) = up, p > 1. Next, there is our intention to present the proofs with elementary tools of the real functions theory. In partic­

ular we avoid in the proofs, if possible, the theorem of differentiability almost everywhere of an absolute continuous function. We also do not use the Stieklov function or the Fejér’s means, applied very often in such situation by some authors. Of course, the integral representation of an absolutely continuous function occurs in the representation theorems, which we deal with. To keep ourselves on an elementary level, we can imagine that we obtain such integral representation by one of the elemen­

tary proofs of Badon-M kodyin theorem.

Following many authors we use the integral representation with / ' under the sign of integral but it is easy to see how to reformulate theorems for removing the explicite appearence of f under the sign of integral.

30 Z. C y b e rto w icz and W. M atuszew ska

1 . First of all in this section the notation will be given and some lemmas and theorems from the theory of Orliez spaces will be collected (see [3]).

The sym bols/.,#, ... от f(x), g( x) , ... will always denote the finite meas­

urable functions defined on the whole x-axis.

The phrase “function / ” will be used in a double sense. The first meaning of / is the class of measurable functions equivalent in the sense of equality almost everywhere, that is to say / will be a symbol of certain element of the space 8 of measurable functions. In the second meaning / will signify a function defined and finite for every x, i.e., an “ individually given” function (individual function).

In the following consideration we must distinguish carefully what meaning is given to a function in a concrete situation. For instance, the substitution of an individually given function of finite variation by another function different from it at only one point changes of course the total variation, but both functions mean the same element of 8.

If we denote the function f ( x + p ) , p Ф 0 by f p(x) a n d /, = /, we say that the function has the period p (function of the p-period).

Therefore, if the equation f p = f is interpreted as equality of the elements in 8, f(x-\-p) = f(x) means the identity for almost every x.

However, if / denotes an individually given function, then the equation f { x + p ) = f (x) must be satisfied for all x , in particular the values for x = 0, x = p are defined and /(0 ) — f(p)- In the following we always assume the functions to be ^-periodic (usually assuming p — l).

By sup / , sup* / (sometimes by sup f(x), sup* f(x)) we denote

{а,ЪУ (a ,ЬУ < a,b> {a,by

the supremum of a function / (individually given) over (a, by, and the essential supremum for f e S respectively. If <a , b > = < 0 , 1 > , then we write sup/, sup*/.

The symbol %e denotes the characteristic function of the set e.

The number gv, where gr is finite, denotes the right-hand side essen­

tial limit of a function / at the point x0, in other words gr has the property that sup* \gv— f(x)\->0 when ж->ж0+ . Instead of gT we shall often

< Х 0 ,Ж >

w rite /efiS(&<> + )• Analogously we define the left-hand side essential limit gl

= fess(xo — )î possesses the essential limit at the point x0 if / ess (xo + )

■ fcSB (X0 )•

1 .1 . By a <p-îunction we understand a non-decreasing continuous function in the interval <0, oo), such that <p{u) — 0 if and only if и — 0 and (p(u)->oо as u—>oo.

The following conditions will be needed:

(Oj) (p ('ll) I'M—^0 as îi—>-0 ,

(oox) (p(u)lu-+oo as u ^ o o .

Functions of bounded generalized variations 31

If a ^-function cp is convex and satisfies conditions (Oj) and ( oo1)?.

we can define the complementary function cp* by the formula cp*{v) — sup (uv — <p(u)).

11^0

It is known that cp* is also a convex ^-function satisfying conditions (ох) and (ooj). Besides {cp*)* — cp.

Let cp be a convex ^-function, satisfying (Oj) and (ocq), feS ,

1

-М Я = / <p[\f{p)\)àx.

0

By L*9 we denote the Orlicz space of measurable functions with the period ¹, i.e., such that the integral 7 , (A/) has a finite value for a certain A > 0.

It is known that L*9 is a linear space; if feL * 9, then + too (for an arbitrary Ji).

We can introduce in L*9 the following homogeneous norms :

\\f\\i<p) = inf{e > ⁰ : I ,( //e ) < 1} (Luxemburg’ s norm),

1

||/||, = sup I J f{x)g{x)dx I (Orlicz’s norm),

0

where the supremum is taken over all geL*9* for which I v*{g) < ¹. Both norms are equivalent, more precisely, the inequality

ll/llw < № < 211/11^ holds.

We also have for an arbitrary Ji, \\f{x + h)\\{(p) = \\f\\i(P), Wfix + Ji)]^

= II./II».

The following Holder’s inequality

1

(H) I f f{x) g (x) dx I < H/llç,\\д\\(<р*)

о

is satisfied.

Because ||/||(ç)) < 1 implies I v{f) < 1 and conversely (in (H) we may put (cp) instead of cp, cp* instead of (cp*)), from the facts mentioned above

1

follows that £ (/) = f f(x)g(x)dx is a linear functional continuous with

0

respect to the norm ||/||(,) , and its norm ||£||(,) equals ||</||,*. If we replace II/II(,p)? by ¹¹/¹¹, , then the norm of the functional, ||£||,, satisfies the in­

equality

UK < \\9b*< 2||f||„.

3 2 Z. C y b e rto w icz and W . M a tusze w ska

1.2. We call a function s (of period 1) the step function, if there are numbers ⁰ = u0 < ux < ... < un = ¹, cx, c.z, ..., cn, such that s (x) = ct

for хе<(и{_ х, %), i — 1 ,2 , n.

1.2.1. Let f be an integrable function on ( 0 ,1), let co1(d) be the integral modulus of continuity of f, i.e.,

i

C0i(<5) = sup f \Ahf{x)\dx, I fcK<50

where Ahf (x) = f ( x + h ) ~ f ( x ) . I f (S->0+, then to¹(<5)->0.

This lemma is well known and it can be proved in an elementary way by the theory of integral; namely the proof is an immediate conse­

quence of the fact that continuous functions are dense in L l( 0 , 1).

1.2.2. L e tf be an integrable function on <0, 1). Let u{ = ih,i = ⁰,¹ , . . . . .. , n, be a partition of the interval <0,1> with h — IJn. We define a step function

i r

Sh(x) = — I f(t)dt for u{_ x < x < Щ, i = 0 ,1 , . .. , n,

Щ- 1

■and extend it to a periodic function with the period 1. The following relation holds

.(*) J \f(x) — sh(x)\dx->0, as h^{- > 0} + .

о

For the proof we can apply the same method which is used by Uljanov in the proof of Lemma 2 ([11]). The following inequality holds:

1 f*

l f ( x ) ~ s h(x)l < — j \f(t)-f(x)\dt, щ_х ^ х < и { ,

ui - i

therefore

ui -j Щ ui

j \f(oo)-sh( x ) \ d x ^ j j ( J \f{t)-f(x)\di^dx

ui - 1 ui - l ui - i

and changing the order of integrals we have

о щ

J \ f { x ) - s h{x)\dx^ — j ( J \f{x + v)-f{x)\dxjdv +

Щ~ 1 ui - l ~ ui u i - 1 - »

^ Ui -vi_ 1 u{-v

+ ¥ / ^{( 1} If { x + v)-f{x)\dxjdv.

Functions of bounded generalized variations 33

It is easy to see that both integrals are equal. Hence

и j h U j — u

j \ f ( x ) - s h(x)\dx^ — J I J ^\f(æ + u ) —f(æ)\dxjdu

« i-i 0 « i - 1

h и -

< т / ( / * х + и) —f (х) [ dx^j du.

О ui —\

By addition of the integrals we obtain

1 2 h 1

f m x ) - s „ ( x ) \ d x ^ - j ( J I f ( x -f u) —f(x) I dxj dw < ² aq (&).

о o o

To finish the proof we apply 1.2.1.

1.3.(a) I f for a measurable fit notion f with the period 1 and for an arbi­

trary step function the inequality

i i

J^J s(x)f(x)dx I < ^о J ^\s(x)\dx

о о

holds, then

sup* \f(x)\ < c.

((3) I f for a measurable function f with the period ¹ and for an arbi­

trary step function the inequality i

j J s(x)f(x)dx j < csup* \s(x)\

0

holds, then

j i\f(x )\dx < ^c.

0

1.3.1. I f for a measurable function f with the period 1 and for an arbi­

trary step function s the inequality i

I / s{x)f{x)dx I < c||*||(v)

0

holds, thenfeL***, \\f\\v* ^ c. The norm equal to the snhallest constant c for which the above integral inequality is satisfied.

Lemmas 1.3.(a), ((3) are classical; we shall prove only 1.3.1. Let g be a bounded measurable function; from ¹.^{2 . 2} it follows that it is possible to approximate the function g by the step functions sn(x) almost every-

3 Roczniki PTM — Prace Matematyczne X X

34 Z. C y b e rto w icz and W. M atuszew ska

where, moreover, sn(x) are uniformly bounded. Using the definition of

||flf||w we verify that ||ej(v) ->|lflf||(v). So we obtain the inequality i

I f g{x)f{x)dx\ < c\\g\\{<P),

0

* when g and / are bounded.

The next step of the proof consists in the approximation of g and / by transcuted functions. Since the last inequality is true for all geL*<p1 the integral on the left-hand side is a linear continuous functional on the space L*v and consequently ||/||v* < c. The Holder inequality from 1.1 implies the second part of our statement.

1.3.2. (Generalization of F. Eiesz lemma). Let <p be a convex y-function satisfying conditions ( о х ), ( c x q ) and f be an individual function on < 0 , 1 ) . For a given partition n: u0 < u1 < u2 < ... < un = 1 write

о71 l/W-/K--i)l

Щ Щ—i

I f = super* < ^oo, t h e n the f u n c t i o n f i s a b s o l u t e l y c o n t i n u o u s

71

i n <0,1>, ^{a n d} _i

(*) £(*)(/) = f <p(\f'(x)\)fa}y

0 *

1

( * * ) сгяп ^ J <p(\f'{oo)\)dx

0 for any normal sequence of partition nn.

Conversely, i f f is an absolutely continuous function in <0 , 1 > and f <p(\f(x)\)dx < oo, then S{j ( f ) < oo (cf. [1], [6]).

0

The proof will be given in a few steps, bet S(9){f) < oo. Then we have

1° The function / is absolutely continuous. To prove this we take a system of disjoint intervals <%,%>.

Let r > S(<p)(f)le, cp(u)^ru when ti^ u * . If for a certain interval

<iq, и'{) the inequality

!/« • ) - Д щ (i) u

holds, then

\f(Ui)-f(u{)\

ut — Щ ^'0

Functions of bounded generalized variations 35

Therefore

^ \f(u'i) —f ( ui)\

where means the summation over intervals satisfying inequality (i).

By summation over intervals for which the inequality opposite to (i) holds, we get the estimation

! / « • ) - / ( % ) I < < (u'i- Щ ) < £,

if —ut) < ô = e/и*. Consequently |Дщ) —f ( u {)\ < 2e as —ut)< d.

2° Except for a countable set of points и there is

( + ) limç)

|Л^|-*-0

\f{u + h)—f(u)\ \

•h = ⁰,

and especially at the points ⁰ and ¹ there are the one-hand side limits ( +)•

In fact, let us assume that

(ü) lim sup ç»

1ЛК0

/ + ^{- f { Vj )}

\ Л •Л > e > 0 ,

i = 1 ,2 , . p, for different points vlt v27 ..., vp from (0,1). Then we may find p disjoint intervals uf) such that < p ( l — x х ( и { — щ ) ^ e/², and hence S ^ ( f ) > pe/². Therefore (ii) is fulfilled in at most ²S{<p)(f)le points and this implies ²°( + ).

Analogous arguments show the existence of the one-hand side limits ( + ) in the end points 0 ,1.

3° If 17 denotes a partition obtained from the partition n by addition of some new points to u{ , then а <rjj.

It is sufficient to observe that by adding a point w between and ut the inequality

(iii) <P \f(ui) —f ( ui-i)\

Ui ui-1 ^/

w - щ _j \f(w) —/(% _i)l

< (P --- ---

\Ui - U i_1 w — u

U j - W \f{Uj)-f{w)\

Щ — Щ-1 Ui — w

< <p

w — ui_ l (w — ui_ 1) + (p \f(Uj)-f(w)\

щ — w (Щ- W )

holds.

36 Z. C y b e rto w icz and W . M atuszew ska

Now we choose a sequence of normal partitions лп: u^l) < w(1n) < ...

... < u$n) — ¹ and a partition л in such a way that

% ) (Л ■e < ^ < P

i = l

1 f{Wj) - f { U j -i)

Ui Uj_i {Щ '4-1) = <7^t

According to l ° / i s continuous in <0,1). So we can assume ( + ) is satisfied at any of the points и{ .

For a sufficiently large n in the intervals of the partition лп there is at most one point and using 3°, we get the estimation

an ^ +

I <P !/(% )! -f{Vj)\

Ua — Vi

, Л , V ' / l / « - ) - / K - ) l \, ’

■»<)>

where (vif from the partition лп.

But for sufficiently small s u p ^ — v{) the two sums on the right-hand side of the last inequality are < e. This follows by application of 2° (+ ).

So we have

&W) ( f ) ~~ e ^ an ^ ann + 8 ^ 8(<p) ( f) + € >

therefore (o) lim аЛп = 8{v)(f).

П-УОО

и

To prove formulae (*) and (**) let us notice that f{u) — f f(t)dt-\-c

0

and by the Jensen inequality we have

<iv)

П

1 4i^{= 1}

l/(%)

Ui — uf j («<“ «<_ l)

» Uf

< \r(x )\dxi(ui - ui-i))(ui ~ ui - j

*=•1 м^{г-_ 1}

n Vf 1

< ^{J ?} J <p(\f{®)\)dx = J <p(\f'(x )\)dx -

i=l 0

v

Let g(v) = Js(t)dtf where s denotes a step function having constant

0

values in the intervals <^_1? v{), v0 = 0 < vx < ... < vn — 1. It is clear that

*•-1 ' * г_1 I 0

<▼)

Functions of bounded generalised variations 37

Let us take the partition n(h), in which v,i = ih, where Ji — 1/n,

1 p

sh(x) = — J f'(t)dt for u{_ x < x < ut . We have h H^{- 1}

Ujs

= i f Г Ш х =

ui — ui_ x h J Щ — и{_ х

ui —l

therefore according to (v),

i

(vi) am = f p(\sh{x)\)dx.

о

Xow, 1.2 . 2 implies that J\f'(x) — sh(x)\dx-+0 as or choosing

0

a sequence hn, we can assume sh ->f'(x) in measure (or almost every­

where).

Using the Fatou lemma, we get

i i

lim inf j <p(\shn{x)\)dx^ f <p(\f (x)\)dx.

n->oo 0 0

By 3° <тя(дп)- > ^ )(/), therefore (vi) implies S(q>)( f ) > f q>(\f'(x)\)dx,

0

which together with (iv) proves equality (*). Equality 3°(o), and (*,) give as a consequence (**).

1.3.3. Let p be a convex rp-function, f an individual and periodic function.

Let us construct for the translated function f ( x - f h ) the supremum S ^ ff) in the same way as 8^ ( f ) .

Then 8^ ( f ) = SM(f).

In fact, by 1.3.2

i i

®U(f) = f (p(\f(x + h)\)dx = f p(\f{x)\)dx,

\ 0 0

for f as an element of the space 8 has the period 1.

1.3.4. Let p be a convex function satisfying conditions ( o j, (o o j.

The following three statements are equivalent :

(cc) f is an absolutely continuous function and f eL*9\

(P) 8v{f) is finite, where 8w(f) denotes the supremum of the sums

П V

S ai U( ui) ~f(Ui_x)] taken over all partitions л and numbers ai such that

n

I > * ( K I )(ui - u i_ 1) < 1;

(ï) % ,(Я /) has a finite value for a certain X > 0.

38 Z. C y b e rto w icz and W. M atuszew ska

I f one of the statements is fulfilled, then

( + ) ll/X = « , ( / ) ,

{ + +) !l/'ll(«p) = inf {e > 0: S ^ ( f / e ) < 1}

(see [4], [5], [7], [8]).

To show the equivalence of the statements (a)o([3) we use the Young inequality uv < <p{u)-\-<p*(v) as u , v ^ 0. It follows from this inequality that

П

г = 1

1 f(Uj) ~/(Щ-l)l

Щ - Щ^{- 1} (ui ~ u i_ l) +

+ П

г=1

therefore Sv{f) < S((p){f) + ¹ .

Let S9(f) < oo. We choose a{ in such a manner that

«< [ / ( % ) - / ( % - 1)3

« ^ (K l )+<P

{ U i - U i . J S y i f ) ) ' The last equality implies

(i) «,[/(»,) -/(%-.)]/«»(/)

г = 1

V */, ,4/ N , V / f ( Ui ) ~ f ( Ui -1) \/я. X

= Zt’= 1 ’ ’ ( ¹» Л - **- >) + t = l² ^ ) ( *'” i- l) n

Let r = <p*(\ai\)(Ui — Щ-i)- Suppose that r > 1. Then 7 = ¹

*=i ' ' <=i

This means, by the definition of 8v{f), that

(Ü) _I a iU (Ui )- f( Uj -i)l

S, ( f )

which contradicts (i), except the case when /(% ) — = 0 for all i.

So we have r < 1. W e have again (ii) with 1 instead of r on the right- hand side.

Functions of bounded generalized variations 39

We see by (i) that

V / l / W - / K - i ) l

(Щ — Щ-i) < 1

and consequently fl(v) (//£ * (/)) ^ ¹, the function f is absolutely continuous, Equivalence of (a )o (y ) follows from 1.3.2, and, moreover, 1.3.2.(*) gives equality

from which ( + + ) follows at once.

i

By 1.3.1 И /'||p = sup J f (x)s{x)dx for all step functions, for which

1 0 ⁿ

t|s||(p.) < 1. But f f (x)s(x)dx = £ а{ [/(щ) — /(w ^ i)]; the inequality

0 , i —l

П

£<P*(\ai\)(ui — Щ-i) < ¹ is equivalent to ||s||(p*) < ¹, so formula ( + ) follows.

*=i

2. Let g be an individually given function with the period 1, which has a finite variation in < 0 ,1 ) in the classical sense. The change of a value of individual function at only one point changes completely its variation and so the variation is not invariant with respect to the relation of equality almost everywhere. To avoid such situation we shall introduce the notion of the essential variation, which is invariant with respect to equality almost overywhere. We remind that we always restrict ourselves to the finite­

valued functions.

Let e cz <⁰,¹), y(e) = ¹. We form П v a r j = sup £

<=i

where supremum is taken over all systems of numbers v1 < vz < ... < vn, such that vi e e.

We call inf{vare/ : /n(e) = 1} the essential variation of the fu n ction / in the interval <0,1>. We shall denote the essential variation in <0, 1>

by var* / (or var*/(a?)). Analogously we can define the essential variation in an arbitrary interval <a, /> ; in symbols var* / (or var*/(a?)) (cf. [8]).

<a,/3> <«,/S>

We shall prove the following lemmas:

2.1. Let g be a function defined everywhere in <0,1>, right-hand side continuous for ⁰ < x < ¹, and for which the limit lim g(x) = g( l — ) exists.

X - + 1 —

I f g(x) = f(x) almost everywhere, then var</ = var*/.

40 Z. C y b e rto w icz and W. M atuszew ska

Let e c: < 0 ,1 ), ju(e) = 1, let 0 = u0 < ux < ... < un = 1. We take the numbers щ < vQ < щ , % < <%»•••, w„_i < г?я_! < «„ < un = ¹, such that g(vt) = f ( v {) and c e. The equality

П

у \д(щ)-д(щ _i)l < vare/

г=1

is satisfied. Tending with % , v1, ..., г;и_х to u0, щ , ..., respectively and with vn to un = ¹, we get

ti

£ \9(щ) — 9(Щ-1)\ < varJ .

i = 1

Consequently var g < var*/.

Let v a r * /< oo, le t /(ж) = ^<7 (a?) for #<?a, where a t= < 0 ,1 ), //(a) = ¹, and varaf < oo. We take vi ea, v0 < vl < ... < vm in such a way that

m

var ^<7 ^ 2 l /K - )- /K - -i) l > v a r a/ —e. Consequently var^<7 > var*/. There­

in

fore var# = var*/. In the case when var*/ = ^{0 0} the proof is similar.

2.2. 7 / var*/ < 0 0, Йеге is a function f right-hand side continuous for ⁰ < x < ¹ and left-hand side continuous for x = ¹, which equals f almost everywhere in <(⁰,¹).

The definition of the essential variation implies that there is a set a c < 0 ,1 ), ju(a) = 1, for which vara/ = var*/. Indeed, it is sufficient to

OO

take sets en c. <0 ,1) , g{en) = 1, vare /->»var*/ and the set a = П en.

n ¹

We assert that if 0 < x < 1, then for an arbitrary sequence > x, vi e a, v ^ x , there exists the limit lim f(vf) = / * ( « ) . In fact, if it is not so, then

г-кх>

there exists a sequence {v j, such that v{ tends to x-\-, v^a, and |/(^) — I > £ for infinitely many different v{ , Vj. Clearly this contradicts vara/ < 0 0. Likewise for an arbitrary sequence {vi}1 v^a, , there exists lim f { v{) = /* ( ¹ —). From the above it follows that the limits

vi~* 1 —

f*(x), ^{/ * ( 1} — ) are defined independently of the choice of the sequence {y j, and it is easy to show /* is right-hand side continuous for ⁰ < x < ¹, lim /* (x) — / * ( 1 —). W e assert that except an at most countable set

X - + 1 —

of points, / is continuous for xe a with respect to the set a. Denote by ek a set of x e < 0 ,1 ), for which the oscillation of / over a is > 1 /Тс. Let х г, x %, ...

.. .j xm be a system of different points. We can define a system of disjoint intervals (щ , %> such that u{ , u'^a, |/(%) — f { u {)\ > 1/7. We see that m/k < vara/ < 00, hence the set ek is finite. Besides a countable set {Jek, f has the oscillation equal to zero with respect to a ; consequently for xed

= <(0,1) — [J ek, f is continuous with respect to a. Choosing xed, x < 1

Functions of bounded generalized variations 41

and vk > x, vkea in an appropriate way, we get f (vk)->f{x), f ( v k)->f*(æ}

as vk-^x, i.e., f{x) = f * ( x ) almost everywhere in <0 ,1).

Co r o l l a r y. 1 ) I f var*/, var*gr are finite, f(x) = g(x) almost every­

where, then f*{x) = g*{x) for 0 < x < 1, var* / = var*g — var/* = varg**

2) I f v a r * /< oo, then sup*|/| < oo.

<0,1>

2.3. Let var*/ be finite. The function /* defined in section 2.1 will be called the reduced function off, considering / as an element of the space $.

2.4. Let v a r * /< oo. There exist the limits: / ess(#o + ) — f * ( xo) for

0 < x0 < ¹ and f ess{x0~ ) = / * (x0 — ) for ⁰ < x0 < ¹.

Indeed, sin ce /* (x) = f{x) for almost every x, then sup* \f*(x0 — ) —

<ж,х0>

~ f * { x )I = sup* \f*(x0 — ) —f { x ) I, on the other hand sup* \f*(x0 — ) —

4>,я0> <ж,а;0>

~f *( x) ^{1—> 0} when x-+Xq — .

We reason similarly in the case of right-hand side limits.

Co r o l l a r y. Except for at most countable set f eee( x ~ ) —f eBB{x-\-) —f*{x).

2.5. Let v a r* /< ^{0 0}. If 0 < a < / ? < y < l , then var*/ = va r*/+ var*/ + |/ess( £ - ) - / eSs(£ + )l •

<а,У> <a,/3> <ЛГ>

We can repeat the proof of the existence of reduced function in the interval { 0 , 1 ) for an arbitrary closed interval contained in < 0 ,1 ). Taking into consideration the right-hand side continuity of reduced functions and that on the right end the value of reduced function equals its left- hand side limit, it can be easily seen that if /* is reduced function of the function / in <0 ,1), f*, f tj / 3 are reduced functions for intervals <a, />,

<£, У>, <a, У>, then f*(x) = f ' ( x ) for a < x < 0, / ? ( £ - ) = / * ( / - )

= /e »(0 - ) , / 2*(®) = / » for p < x < y , f * ( y ~ ) = f * ( y - ) , f № + ) =f*(P)r f*(x) = /* ( » ) for а < ж < у , / * ( 7 ) = f * { y — ).

Hence we haVe (counting common variations of individual functions.

/ ? , £ , £ )

var/3 * = var/* + v a r/* + |/3* ( /) - / 3* ( / - ) | .

<a,y> <<*,/?> </3,У>

Yet it is necessary to take into consideration that

/з № - /« e ( /S + ), Ж / ! - ) = Л*(Д-) = /.* ,(/» —), var*/ = v a r/f, var*/ = var/*, var*/ = var/*.

<а,/?> <а,й> <5,y> <|8,y> <«,У> <a,y>

2.6. Let var*/ < ^{0 0}. For every 0 < r < 1 there holds the equality (*) Var*/(tt + ®) = V a r * /+ i/ess( l + ) - / ess( l - ) | - | / ess(^ + ) -/e s s ( ^ -) U and except an at most countable set of values v there is

(**) var*/(a?+«) = var*/ + |/eS8( l + ) - / eSs(l ~)l •

4 2 Z. C y b e r t o w ic z and W . M a t u s z e w s k a

i

Let h(x) — f(x-\-v), 0 < x < 1, By virtue of 2.5 var* h = var* h +

<0,1> <0,I-w>

+ v a r * | ^ e s s ( ( l ~ f l ) + ) — Ü es s ( ( l — f l ) — ) j ,

^ e s s ( ( l - ' y ) + ) = / e « ( Ц - ) , & « * ( ( ! ~ / e s s ( l ) *

Since var* h = var* / ( « ) , var* A = var* f(u) = var/*(w), and

< 0 ,1 - 1 5 > <t? ,l> < 1 -1 ? , 1> < l , l + ® > < 0 ,v >

b y 2.5

var*/ = var*/+ var*/ + |/68в (■v + ) - / « (*> - ) I,

<0,1> <0,»> <»,!>

•equality (*) is valid. Since for all v from (0,1) except an at most count­

able set of values v, f csa(v-r) —f ess(v — ), (**) holds except an at most

•countable set of values v.

3. Lst fe L * <p. We shall introduce the following notations:

= sup \\xe(f(æ + h)-f(tB))\\v,

where e is a measurable set of finite measure;

cov(d) = 0)^(3, e) when e = <0 ,1) .

coplô) is so-called integral modulus of continuity of the function / in the metric of the space L*(p. If cp(u) = \u\, i.e., L*v is a space of inté­

grable functions, then we write a>x(d, e) instead of (ov(ô, e), co1(â) in place of cov(ô). Taking in L*<p the norm INI^) instead of the norm || - , we define a>(v)(â, e), a>l9>)(ô).

Now we give some properties of the integral moduli oj(g>)(3, e) and

<0)^(0, e) in the metric of L*<p space.

3.1.(a) I f (p satisfies (Д2)-condition, then

( + ) oj9(0, e) - > 0 as <5-^0 + for an arbitrary e of finite measure.

((3) <ofp(3, e) is a subadditive and non-decreasing function for ô > 0.

(y) cdv(ô2, e)/32 < 2co<p(ô1, e)/ô1 when ô2 > ^ > ⁰ (pseudomonotony of the quotient co^ô, e)/3).

(S) lim

<5—>-0 -j-

%(<*> e)

ô sup

0<Ô< 1

m<p(ô, e) Ô

(s) I f condition ( + ) is satisfied, then (ov{3, e) is a continuous function.

Ad (a). Under assumption (Д2) thq space L ** is separable and the set of continuous functions with period 1 is dense in L*v. Clearly relation (a) is true for continuous functions; hence by approximation of f e L **

by continuous functions we get ( ⁴- ) in the general case.

Functions of bounded generalised variations 43

Properties (y) and (8) are a consequence of subadditivity of ca(d, e) und these properties are known, but we give the proofs for completeness.

Let 0 < <5X < d2 and let n be a non-negative integer such that 2ndt

< d2 < 2n+l dx. The following inequalities are fulfilled

toyià2, e) < œ<p(2n+1ô1, e) < ^{2 W+1} о)у(^х, e) = g щ {дХ1 e)

àx *x

Let 0 < <5⁰ < 1, 0 < d < <5⁰ and let n denote a non-negative integer such that <50 = w<5-{-r((5), 0<r(<5) < 5. The inequality

<5 œ {ô,e) « (r(<5))

— --- < n --- --- --- —--- -

ô0 do d do

holds.

If <5->0, then m<3/<50->1, r(<$)->0. Suppose ( + ) is satisfied for e c <0,1).

Then so

^ liminf <0y(a,6) .

<50 ô->o+ <5 Prom the last inequality we conclude that

sup 0<<5⁰*S1 ^0

< lim inf

(5->0 +

>9{à, e)

< lim sup

<5—M)-f-

<M<3, e)

< sup

0<«50<1

Oy(^o> e) ào i.e., relation (⁸) is fulfilled.

If ( + ) does not hold, then by the monotony of cov(d) we have (ov(d) ->c > 0 when <5->0 + .

It is clear that in this case the expression on the right-hand side as well as on the left-hand side of formula (⁸) is equal to the infinity.

Ad (e). Prom subadditivity it follows that |co,,(<5) — w^d') |< ct)v(|<5 — <5'|);

hence the statement is a consequence of relation (+ ).

Let us remark that analogous properties (a)-(s) also hold for a>^(d, e),

<W<5)-

3.1.1. Let ⁹9, xp be convex <p-functions avid let a>(u) = <p(y)-i,{u)), where xp_x denotes the inverse function of ip (this function exists, because xp is strictly increasing).

I f co(u) is a convex function for 0 < u, co(l) ^ 1, then (oM(d, 6) < e)

for an arbitrary d > ⁰ and an measurable set e of measure < ¹.

44 Z. Cy b o rto w icz and W. M atuszew ska

According to a remark in [9] there is a number 0 < a < 1 such that и < aco (и) + (1 — a) for и ^ ⁰. Consequently we have

y>(u) < a<p(u) + { l - a ) , y)(\f(x + h)-f{x)\) < аср (|/(<в + Ц -/(ж)|) + ^{( 1} - a) , U и -/(ж)| > < „ [ J ^

J l Ite e ^ J W I L )/ J(®)ll(v)/ £ J '\ \\Xedhf{x)\\{v) where zJA/(æ) = /(а? + Л)— f (x ).

By definition of ||-|l(<p) we have the inequality

|ЫЛ/И)|и< \\Xe(A hf{x))\\

and consequently (*) holds.

dx+ ^{( 1} — a) < a -f ^{( 1 — a) = 1 ,}

3.2. Th e o r e m 1 (Hardy-Littlewood theorem). Suppose that for the integral modulus of continuity co1(<5) of an integrable function f, of period l r the condition

(*) lim

<5—>-0 ô - = Â < oc

holds. Then there is an individual function f* of bounded variation in <0, ¹>?

right-hand side continuous in <0 ,1), left-hand side continuous for x — 1, equal f(x) for almost every x, and such that

(**) A = var*/+ |/в8в(1 -)+ /е в в( 0 + )|

= v a r / * + l / * ( l- ) + / * ( 0 + )|.

Conversely, if f (x) is equal to a function of bounded variation almost everywhere in <⁰,¹), then condition (*) is fulfilled (cf. [²], [⁸]).

P r o o f. For an integrable fu n ction /w e construct a step function sn(x)r taking a partition of interval <0 , 1) by points u{ = ih, i = 0,1, . . . , nr where h — 1 jn and setting

1 r %

“ i - j J № № >

ui — 1

an Clearly,

1 U'n

— J ^f{t)dt,

un- 1

varsn

«u>

n—1

i= l

when < x < uit i = 1 , 2,..., n —1 ,

when un_j < ^x< un — 1 .

i=l

j f(t-\-h)dt— J f(t)dt

H- 1 ui - i

Functions of bounded generalized variations 45

1 / a>i(li)

J lf(t + h ) - f ( t ) \ d t ^ — — < Я.

0

Hence var sn < Я. By virtue of Lemma 1.2.2 sn tend to the function /

< 0 , 1>

in L 1. So there is a partial sequence sk (x), which tends to f(x) almost everywhere in <⁰,¹) ; in particular in a certain fixed point x0 the sequence sk (x0) is bounded. Since the variations of sk (x) are uniformly bounded, by the Helly’s lemma a partial sequence exists — let us denote it as above by skn{x) — such that for every же<⁰,¹), skn{x)->g(x), where g (ж) is a finite function and Я > limvarsj. > var g{x).

< o ,i > n <0,1>

So we have proved the existence of function of J:inite variation in

< 0 ,1 ), which is equal to f(x) almost everywhere and besides that Я

> vary (ж).

<0,1>

Using the definition of the essential variation, we see that var*/ < Я.

The translated function f(x-f-v) has the same modulus of continuity aq as the function f{x). Therefore var*/(cc + ,y) < Я.

By Lemma 2.6 there is v such that

<i) var*/ + |/es⁸( l + ) - / e s S( l - ) ! <

Assuming in addition that the function / is right-hand side continuous for 0 < x < 1, left-hand side continuous for x = 1, we get inequality (i) b y somewhat different method of proving.

The following formula will be useful in the proof

« V

j ( f (v + x) - f ( x ) ) dx — f (/(и + x) ~ f {x ) ) dx.

о 0

It follows from this formula that for arbitrary и', и", U, v" the following relation holds

u " v "

<ii) J (f(v" + x ) - f ( v ' + x))dx = f (f(u" + x ) - f ( u ' + x))dx.

и' V

Let us take the systems of numbers: ⁰ < u{ < u{ < ui+1 < ui+1

< ... < un < un < 1, 0 ■ = v,, < tq < ... < vm — 1. Moreover, let us take two systems of arbitrary real numbers: alt a2 ) . .. , an) blf b2i..., bm.

In the interval <0 , 1 ) we define step functions u(x) = a{ for x e ( u {, uf) and w(x) = bj for Х € ^ _ Х1 Vj}, j = ¹, ² , ..., m (the last interval is open) and equal ⁰ elsewhere. Next we extend them on the whole ж-axis to a func­

tion of period 1. By (ii) we get

1 m

(iii) J u(x) + x ) - f ( v j_1 + x)}^dx

0 j = l

1 n

= f w(x) ( ^ a i (f{u'i + x ) - f { u i + x))}dx.

о г = 1

46 Z. C y b e rto w ic z and W. M atuszew ska

Let us denote the left-hand side of this equality by and right- П

hand side by I v, e = [J ( u{, ufy.

i We have the inequality

i i

(iv) |6;-|| f ( f ( v j + x ) - f ( v j _ 1 + x ) ) d x I ⁼ \bj\ J \f(vj - v j _ 1 + x ) ~ f { x ) \ d x

о 0

< A ^j =1,2,..., ^{m .}

By (iii) we get

i

|Ir| < Asup*\u{x)\ J \ w ( x ) \ d x , 0

and by Lemma 1.3.(a) П

(v) sup* I J ? а < (/К + я?) - f ( U i + æ)) | < Asup* \u{x)\.

i = l

Assuming = ^{± 1} in all possible ways, we get the inequality

(vi) £ !/(% + + A,

t=i

valid for all x e ( 0 , l > besides a set of measure zero, which depends on the choise of the numbers u{, u'{.

Let us suppose that u{ give a partition of interval < 0 ,1 ), щ = ui+l for i = 0 ,1 , . .. , n — ¹. By (vi) we get

П ^—1

(^i') У \f(ui + x ) - f ( u i_ 1 + a})\ + \f{l + x)T-f(un_ l + x)\ < A.

i ti

Tending with x over a set a to 0, fi{a) = 1 and since var/ = var*/ and one-side limits of the / are equal to essential limits, we obtain

v a r 7 + l/eas(1 - ) - / e s s ( ° + ) I < A .

Now we are going to show that

v a r* /+ | /ess( l - ) - / esg(⁰ +)| ^ A.

Denote v a r * /+ ] /ess( l —) —/ e8a(0 + )l = k .

In (iii) we can assume /* for /. Since almost everywhere the inequality П

J ^ ai ( / ( % + ^ ) + x)) I < sup|tfq|var*/(w + a?) < sup* |w(a>)|

Functions of bounded generalized variations 47

holds, by (iii) we get

i

(vii) lljl < 7csup*\u(x)\ j \w(x)\dx.

о

Set vt == h, v0 = 0, 0 < h < d, v2 < v3 < ... < vn = 1 arbitrary, bt = 1 * b2 — bz — ... = bn = 0. By (yii) and by Lemma 1.3 we get

i

J \Ahf(x)\dx < kh < Ы.

0

So we have

(viii) ( <5) < &<5, i.e., X < fc, and (**) is proved.

To prove the second part of our statement suppose / is equal almost- everywhere to a function of finite variation; in other words, such that к is finite. The same arguments as above implies (viii).

3.3. Theorem 2. Let f e L * 9 and let <p be a convex <p-function satisfying1 conditions (ox) and ( cxq) and let

or equivalently

Xy = lim

<5—*-0

< OO,

\<p) lim

<5^{—>0 < CO.}

Then there is an absolutely continuous function f* defined in <0, l ) r almost everywhere equal f and such that f*' cL*4',

(*) ^К = llf ll„

(**) ^(<p) = II/* ll(?>}*

Conversely, when f (x) equals to an absolutely continuous function f*

almost everywhere in <0,1> and / * ' eL*9, then {*) and (**) holds (cf. [10],.

[^{1 2}]).

Let us remark that if Xv < oo, then, applying 3.1.1 to functions <p and tp = u/(p_1( 1), we get X < o o . We have the same value of X for all translated functions f(x-\-v). By Theorem 1 there exists for every v a func­

tion of bounded variation, which is right-hand side continuous in <⁰ ,¹} and left-hand side continuous at ¹ and in the following we choose the function f ( x + v) as the function f.

Let v, ⁰ < v < ¹, be a po _nt of continuity for the function /. Then the translated function f(x-\-v) is right-hand side continuous in <⁰, ¹} and continuous at 1. Next we shall take into consideration this translated function.