71 — 1^ Ш

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A N N A L ES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X V (1971)

B. T

a b e r s k i

(Poznan)

Some properties of M-variations

1. Introduction. We shall denote by M( u) , N{ u) and M x(u), N x(u) the pairs of non-negative continuous convex functions complementary in the sense of Young ([1], p. 16-20 or [ 6 ], p. 16). For the inverse func

tions the symbols М~г (v), N~l (v) etc. will be used.

Let f(t) be an arbitrary real function defined in a finite interval (a, b}. Consider partitions

P = {a = t 0 < t1 C . . . C t i < ti+l < ... < tn = b}

together with the sequences Ü = (u0, ux, .. ., un_x) of non-negative numbers щ such that

71 — 1

^ Ш < 1 •

i = 0

Putting Af(t{) = f ( t i+1)— we shall write VM (/; a, b) and **V*M(f; a, b) for the upper bounds of the sums**

n —1 n

— 1

У Ж ( И / ( У I) and Y м /(<4)|«4,

г= 0 г= 0

respectively. These quantities are called the first and the second M-va

riation of / in (a, by. Clearly, the ordinary variation

77— 1

V(Ji a, b) = sup У \Af{tt)I

p tr0

**with the factor N~l (l) majorises V*M{f; a, b). Moreover, by inequality of W. H. Young ([1], p. 24),**

V U f ; a , b ) < V M{ f ; a , b ) + 1.

.In this note we shall give some properties of Ж-variations. Most

of them concern the second Ж-variation. Our investigations completes

the results announced in [4].

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2. Basic properties. An argument similar to that of [ 6 ],p. 171, leads to **2.1. Let y = V*M(f ; a, b) be a positive number. Then,**

* Ум<ЛУ7а >ъ) < !•

The assumption F ^ (/; a t b) < oo implies **Ум№<>,Ь) = mV*M{f-,a,b)**

for any real number 1c. Further, as in [5], p. 191 we obtain

2 . 2 . Suppose that the functions f ( t ) , g( t ) are bounded in <a , b } f and set

**<p{t) = /(* ) + 0(t)> v(t) =f ( t ) 9( t ) i A -- su p |/(t)|, В = sup \g{t)\.**

a^Zt^b Then

**VM{<pya, b) < VM(f-, a, 6 ) + V*M{g-, a, b),** V* m **hP', a, b) < BVM (/; a, b) + AVM(g; a, b).**

Also, it is easily seen, 2.3. If a < c < b, then

**VMif', <>, b) < VM(f-, a, c)+ VM(f; c, b).**

2.4. I f a, < a < b < blt then

V m (I; «, &) < Ум(ff ai

7

bi).

2.5. I f Ж (и) < Жг(и) for ue( 0, A y 1 (l)), then **V U f i a , b ) ^ V* Mi(f-,a,b).**

Finally, we extend the corresponding propositions of [2], p. 36 and 38, to the first Ж-variation of 27t-periodic measurable f(t) defined for all t (cf. [3], p. 12).

2.6. (i) The assumption

sup M ( \ f ( t + h ) —f(t)\) = 0(h) 0 <£«S 2 tt

implies

(1) F M( / ; 0 , 2

tc

) < oo.

(ii) I f condition ( 1 ) holds, then 2 тг

/ ж ( 1 Л«+л) - Я«) | ) <в = о (ft)

as h 0

as й 0 .

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Proof. Case (i) is trivial. To prove (ii), we choose a positive integer к such that 27r/(fc+l) < \Ц < 2 л;/&. Considering 7 i >0, we have

2n к (v + l)7i

r= 0 vh

h к

where

< — I E h v =0 *

< 2h- VM{f; —h, 27Т + 2 Л-)

< 2 fr {4 F M( / ; 0 , 2it) + 3 M ( w ) } ,

w = 2 sup |/(t)| < oo.

0 « < 2 n

\dz

Further it is enough to observe that 2n

j = j M{ \ f { t +h) - f ( t ) \ ) dt .

о 0

3. Special theorems. A function N x(u) is said to be the strong majorant for N (u) in <0, Z> if, for every positive integer к and all integers r large enough (r > k),

rN(u) < N^rujk) whenever 0 < rujk < Z.

For example, N(u) ua

log и

(a > 1 , p > 0 , c > 1 )

possesses the strong majorant Жг( и ) = Ж ( и ) in <0,c/2>. If N{u) = и

lo g v

( / ? > 0 , o > l ) ,

the strong majorant, in the last interval, is of the form N^u) = и

log

(0 < р г < р ) .

Now, some generalizations of Marcinkiewicz’s results ([ 2 ], p. 38-40)

will be given.

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**3.1. Suppose that V*M{ f \ a , b ) < oo and that N {и) has the strong majorant N x(u) in the interval <0, V). Then, for any e > 0 and any xe( a, b), there is a a > 0 such that for every positive à < a,**

oc-\-ô,æ-{-a) < e .

Proof. Given a number e ^ 1 satisfying the inequality iVrf 1( 1 /c) < l, let us consider the modified pairs of functions

M(u) = c M ( u ) , N (u) —c N I ^— j

and

М г(и) — cMx(u), N x{u) — cNx I—j

coplementary in the sense of Young, too. The function N x{u) is the strong majorant for N (u) in < 0 , Y 1 _ 1 (l)); the inequality

**Yhif't «, &) < cV*M(f-, a, b) implies**

Vm(/; «» b) < оо.

First we shall prove that our thesis holds for the variation Ущ **instead of V*M . Supposing the contrary we could find, for an e > 0 and an x e f a , b } , a sequence of non-overlapping intervals (x-{- ôn, x-\- crny tending right-sidely to x such that**

Vh xifi я + д п, х + о п) > e (n = 1 , 2 , . . . ) . Let fc, r be some positive integers (r > h) for which ( 2 ) **~ Y > y h i f > < * i b ) les**

and rN{u) < N x(Xu) if 0 < Xu < N f l {1), where Я = rfk. Consider a parti

tion {x = x0 < xx < x2 < ... < xs = b} such that the pairs x1v, a?,- (v = 2r —1, 2r— 3 , . . . , 5 , 3 , 1 ) coincide with the end-points of r successive intervals <ж+ ôn, х + an} (n = 1, 2, ..., r). Choose some non-negative щ satisfying the condition

t v + l

£ S , W < 1 (V = l , 3 , 5 , . . . , 2 r - 3 , 2 r - l ) .

i = i v

Evidently, we may suppose that

t v + l

^ И/ ( ^ ) | я %> VwxU, Xu+x) ~ ^

i— jv

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Taking щ = 0 for other i, we have

s - i h h r

У в д < - | У а д » , ) + . . . + У s , w k i .

^—J г I— J —«/ J

г= 0 i = j x i = h r - \

Therefore,

S — 1

**^ж( / ; <*,b)> ^ И / M l %**

г= 0

? 2 ?4 h r

=

j

Œ + I + - - - + Z )

w

^ >

t

’

ï=j'i г' = ? ' 3 i = h r —l

which contradicts ( 2 ). Hence

Ущ(1; x + à, a? + a) < e ( « < # < & ) if cf is small enough and 0 < Ô < a.

Futher, in view of 2.1,

F ^ ( 0 /; °°+ à, x + a ) < 1 when 1/0 = F f^(/; oc+ ô, x + a ) . Consequently,

^м 1 ₁ (^/5ж+ <^ ж_Ь(Т) ^ — j FAf(0/;a?4-^J^ + o ,) < ---- (-1 •

c 1 c

Thus

Æ+^?Æ+ 0 ')^2TrM^(/?^+<5?Æ+or) < 2 e , and e being arbitrary, we get the desired assertion.

Clearly, the related inequality

x —о , x —ô) < e { a < x ^ b ) is valid, too.

3.2. Let f(t) be continuous at every point of the interval <a , b), Zetf F ^ (/; a —y, b + y) < oo for some у > 0 . Then, under the assumption on N (u) and N x(u) as ^n 3.1,

lim 7 ^ ( / ; x —h, x + h ) = 0

7i-->0-î-

uniformly in xe( a, ô).

Proof. Otherwise, there are an e > 0 and a certain x e ( a , b ) such that

(3) **V*Mi{f; x — a, x + a ) >** e

for arbitrary positive o. On the other hand, in view of 3.1, FlqC/j oc—a, x — <5)-j- oc+ ô, x + a ) < e/1

R oczn ik i PTM — P r a c e M a tem a ty czn e XV 10

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for sufficiently small a > 0 and <5e(0, a). Obviously, we may also sup

pose

№ ± Q)-f{æ)\ < g jy -6! ^ when 0 < ^ < cr.

Let {x— a = x0 < xx < ... < æk < x < xk+1 < ... < xm = x-j- a} be a partition of ( x — a, x-\- a}, and let %i be some positive numbers such,

that

m —1 m — 1

И / 0 *ч) 1 % > x —

0

, x + a ) —j .

i — 0 i = 0

Since

m —1 к — 1 m — l

£ = ( J £ + ^ ) M / ( ^ ) l % + l / ( % + i ) - / ( % ) l %

г = 0 i = 0 i = f c + l

**< ^ ( / ; ® * ) + ^ ( Z ; a w ® + f f ) + {l/(®*+i)—**

- /( * ) ! + l/(®)-/(®*)l}JVrr 1(i), we have

g g

^ ( / î ж+ <т)“ _ 2 <

This contradicts (3) (see also [2], p. 40).

References

[1] M. А. К р а сн о сел ь ск и й и Я. Б. Рутицкий, Выпуклые функции и простран

ства Орлича, Москва 1958.

[2] J. M a r c in k ie w ic z , Collected papers, Warszawa 1964.

[3] J. M u sie la k , O beswzglçdnej sbieznoéci szeregôw B ourlera pewnych funlccji prawie olcresowych, Zeszyty Naukowe Uniwersytetu im. A. Mickiewicza, mat.-chem., 1 (1957), p. 9-17.

[4] J. M u sie la k and W. O rlicz, On generalized variations (I), Studia Math. 18 (1959), p. 11-41.

[5] И. П. Н а т а н со н , Теория функций вещественной переменной, Москва-Ленин- град 1950.

[6] A. Z y g m u n d , Trigonometric series, I , Cambridge 1959.

IN ST IT U T E OF MATHEMATICS, A. MICKIEWICZ U N IV E R SIT Y IN POZNAN