Singular integrals depending on two parameters

(1)

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE VII (1962)

R.

Ta ber sk i

(Poznań)

Singular integrals depending on two parameters

1. Introduction. Let L2n be the class of all functions 27r-periodic and Lebesgue-integrable in the interval <—7c,7c>. Write L(_a,b} for the class of functions Lebesgue-integrable in <a, by. Denote, once for all, by K( t , £) a function defined for all t and £eE (where E is a given set of numbers), 2Tt-periodic, even, bounded and measurable with respect to t for every fixed £eE. Suppose that | 0 is an accumulation point of E.

It is easily observed that Fejer’s theorem concerning the convergence of singular integrals

7Г

'(1) U ( x , £ , f ) = j f ( t ) K ( t - x , ( ) d t ( f t L 2K)

----7T

can be extended to the convergence (x, S) -> (x0, | 0) on an arbitrary set of points of the plane. Namely,

1.1. If

7Г

(2) lim f K ( t , i ) d t = 1 (ŚeE),

— 7T

П

(3) J IK( t , 1)1 dt < 0 on E (C = const),

—П

(4) lim sup \K[t, f)| = 0 for д > 0 (д < те, £еЕ), ł->ło

then we have

(5) lim U(x, i , f ) = f(x0) ЦеЩ

(ж, i)— > (Xq, Iq)

at every x0 at which f is continuous (cf. [4], I, p. 89, [1], p. 418).

X

Delation (5) holds at the points of differentiability of f f(t)dt, under 5

the assumptions of Bomanovski’s theorem, if the convergence (x , I) ->

-> (x0, £0) is restricted to some set of points of the plane. Also a theorem of the Faddeev type and a result concerning the convergence of deriva

tives drU(x, i , f ) j dxr can be stated. The present note is devoted to these

problems.

(2)

1 7 4 R. T a b e r s k i

2. A generalization of Natanson’s lemma. We shall prove the following fundamental lemma (cf. [3], p. 243).

2.1. Let <p{t) be a function of bounded, variation in every interval

b

by (0 < у < b — a), such that f v(s)ds < oo, where

v(s) = var cp{t) {a < s < b) , v(b) — 0.

S t < b

Then, if

a + h

M = sup — f f(t)dt

0 < h t ą b —a h J

< o o (f e L ( a , b y),

the improper Lebesgue integral I = ff(t)<p(t)dt exists and a+

\i\ < m J [ * ( * ) + И & ) [ ] * > -

Proof. Write

F(t) = Jf( u)du (a <&),

a

P

Ia, n= f f(t)<p{t)dt (a < a < ft ^.b).

a

Integrating by parts, we obtain

fi /3

/ <p{l)dF(t) =<p(t)F(t)\f - f F(t)d?(t).

a a

Since

fi P

p

j J

F(t)dq>{t)\ < M

J

(t — a ) d [ —v(t)] = M^[a — t)v(t) j v{t)dt\,

a o . a

a

(a —a)\(p(a) — (p{b)\ ^ (a — a)v(a) < J v(t)dt,

a b

the integral fF(t)dy(t) exists and

ь ь

I f F{t)dcp{t)] < M f v(t)dt.

a+

a +

Our conclusion is now evident.

(3)

Singular integrals depending on two parameters 17 5

ь

R em ark. If ip(t) is such that / w(s)ds < oo, where

a

then

with

w{s) = var ip(t) (a < s < b), w{a) = 0,

a < < < s

b - b

I / f ( t ) y { t ) d t \ ^ Ж f [w(s)-\-\ip(a)\']ds,

a a

N = sup

0 < h < b—a

< O O .

3. Theorems of the Romanovski and the Faddeev type. Some results on the convergence of integrals (1) will now be given.

3.1. Suppose that the function K ( t, £) is non-negative and non-inereasing in t on <0, тс) for every £eE, that it satisfies condition (2) and that

(6)

Let

lim K( d, £) = 0 for 0 < д < тг (£ eF?).

(7) ^{lim —}

л.—> о h

J f ( t ) d t = f ( x 0) ( f €^2n) at some x0. Then integrals (1) tend to f(x0) as (x, £) ->

set Z in which the function

(x0, £0) on any plane

Ц х, I) = ( x - x 0) K ( 0, £) (£eE) is bounded (cf. [3], p. 245, [4], I, p. 101, (7.9)).

Proof. Of course, it is sufficient to show that T C

I(x, £) = f [f(t)—f(x0)]K(t — x, £ ) d t ^ 0

— 7T

as (x, £) -> (x0, £0) on Z.

We consider only the case — т с < x0 < 0 . By (7), given e > 0 there is a <5 > 0 such that

1 h

x 0 ± h

Xq

£ when 0 < h < 3.

Suppose that 6 <

tc

+ ^

o

, 0 < x0 — x < and write

Xq— d X 0 + 6 n

Ц о и , £ ) = ( ) ' + ] ' + j ) u m - f ( 4 № ( t - x , £ ) Ш = Ц + Ц + Ц . -•к ж0-<5 a'o+ó

(4)

176 E . T a b e r s k i

It is easy to see that

Xq— 0 7Г

|Za| J |/(«)-№ о)|< й < % № , ( ) / \ № - f ( x 0)\dt

— I t — 7t

and

I t I t

u,\ } \ m - n x 0)\m.

Xq -{- <5 — Tt

Hence, by (6),

lim l!

=

0

=

lim l3 as

(x, £) -> (x0, £0), £eE.

(x, o (x, $>

In view of 2.1,

Xq

\I2\ < £ f [ var К

(t — x, £ )+ K (x 0 — x — d,

£)]ds +

Xq- 8 x o ~ d < ł < s

x 0 +d

+ e f [ var

K ( t —x ,£ )-\-K (x 0—x -\-d ,£ ))d s

Xq S < < < Xq + 6

Tt

**< « [ /X (* ,f )< fo + 2 (^ -® ) X (0 ,f)] .**

— 7t

Therefore, if the points (я?,

£) eZ

are sufficiently near to

(x0, £0),

we have

\IZ\ < 2 г (ф + 1 ), where

Q =

sup

\(x — ж0)1Г(0, |) | ((x, £)eZ).

Thus, the proof is finished. Similarly, the following result can be obtained.

3.2.

Let K* (t, £) be. non-negative and non-increasing in t on

<0,

^tt) for

£eE, and'satisfy the same assumptions as K( t , £) in §

1,

above

1.1,

and let

\ K(t , 1)1 < K * ( t , I) (fe <0,7u>, U B ) .

Suppose that conditions

(2), (4)

for K( t , £) and condition

(3)

for K * (t, £) hold. Then the relation

limT Г \fit)—f{x0)\dt = 0 (feL2n)

h-> 0 h J

x0

implies

(5),

i.e.

7 t

lim

f f ( t ) K { t — x, £)dt = f ( x 0),

**(*,f)**

^{- n}

as (x^,I) -> (x0^{, £}^q⁾ on a plane set Z* in which the function A*(a?, f) = ( x - x o)K*(0, £) (£eE)

is bounded

(cf. [3], p. 246).

(5)

Singular integrals depending on two parameters 1 7 7

4. Differentiation of singular integrals. The following theorem hold.

4.1.

Let the function K( t ,

£)

and its derivatives dvK ( t , £)/dtv (v =

= 1, 2, . . . , r) be continuous with respect to t on ( — oo, oo) for every fixed

£eE. Suppose that conditions

(2), (3), (4)

together with

(8⁾

and

(9) lim sup

sup I sinrt

ę e is J

drK( t , £)

drK ( t , I)

dtr

dt < oo

dtr 0

for all 3 >

0 (<5 < тс, £e.Zy)

are satisfied. Denote by S a set of points (x, | ) (£eE) such that

drK( t , £)

(10)

\x — x0\vJ

sinr

vt

df

^{dt <}⁶

Y „

(v — 1, 2, . . . , r ) ,

ж0 fixed, where Cv are certain positive constants. Then, if the function f e L 2n possesses at x0 a finite derivative f (r\ x f ) , we have

(11) = f / m

( x , f ) dxr

**(.)**

drK{ t — x, £)

M

dt = f {r)(x0) as (x, £ ) —» (x0, £0) on S

(ef. [4], II, pp. 60-61, [1], pp. 419-420).

P roof. We construct a function Г

д(г) = ^ ап

sin” (t — a?0)

{ - u < x

0

< n )

n= 0

such that

**9т Ы = / <P,(*o) (P = 0 ,1 ,**

In this case, the coefficients an are certain linear combinations of the derivatives f {m)(x0) (m <

n),

e.g. a0 = f ( x 0), a1= / /(a?0), a2 = ^ f " ( x 0),

**а, = 1[/Ч * „ ) + / ' " « ] > «4 o ) + / , r (o)], «6 = i [ 9 / ( o ) + + 1 0 //"(^0) + / F(^o)] etc. II is easy to observe that**

oi(t)

= [/(£) —

**~#0O]/sinr(2 — #0) -*■ 0 as t -> x0 ([2], pp. 25-27).**

Obviously,

dr U(x, £, g)

dxr

u.

- i ) r f g ( t ) drK ( t — x , £)

—r

^{dt —}

7Г

■ 7Г

R oczn ik i PTM — P race M atem atyczn e VII 12

(6)

1 7 8 R . T a b e r s k i

Hence, by 1.1,

lim - _ g{r) ^ = j(r) ag (a?, f ) -> (a?0, £0) .

( X , S ) 0X

Since

dr U ( x , ę , f ) dr U ( x , ę , g ) dr U ( x , ę , f - g )

dxr dxr + dxr ~ ’

it is sufficient to show that the second term on the right-hand side of the last identity tends to zero as (x , |) -> (x0, £0) on /S'. But

H( x, £)

eter -1»' /

co(£)sinr(£ —a?9) drK ( t - x , £)

dt

and lim

o){t)

=

0.

Thus, for any e

> 0

there is a <5 >

0

( ó < min(7r —

x0,

<->x0

7г + а70)) such that

\H(x, |)|

-■0 r

/

sinr( t— x0) drK ( t — x, Ś)

d f

\ x0 — ® **

+ !( J +

J

Jco(0sinr(f — я?0) е1г (х, l ) + \I2{%, £) + I 9{x, £)|.

dt-\-

drK { t —x, £)

d f

dt

Evidently,

Ii(x, f) < /

S in r ( / + £I7— Ж0)

arir(<, f)

d f

dt

f sin rt ^{d rK ( t ,}^f)

d f dt +

7 Z

j

|sinr(ź +

x — x0) — s h ft I ^{d rK ( t ,}^f)

d f

dt

In view of (8), there is a constant C such that J x(£) < 0 ( $ e E ) .

Applying the formulas

an

— bn =

(a— b)(an~l -\-an~2bĄ- ...

-|-6n_1), sin a — &m(3 = 2 cos [(a-j-/?)/2] sin [(a — /?)/2], and taking into account (10), we can show that J 2{x, £) < P on 8, where P is a constant; whence I x( x, £) < (7 + P on 8.

(7)

Singular integrals depending on two parameters 1 7 9

Let \x — %0\ < \ d . Since y(t) = co(t)smr (t — xQ) is Lebesgue-integrable

m < — 7Г , 7T>,

Hence, by (9), lim l3(a?, £) = 0 as (%, |) (a?0, £„). Similarly, limZ2(#, £) =

= 0. The proof is completed.

Finally, we remark that from 1.1, 3.1 and 4.1 theorems 1, 3, 2 of [1], pp. 417-23, follow, respectively. An analogical result can be obtained for Cesaro’s integrals ([4], II, pp. 60-61).

[1] Gr. M. G olusin, Geometric theory of functions of complex variable (in Russian), Moscow-Leningrad 1952.

[2] Y. M atsuoka, Asymptotic formula for Vallee Poussins singular integrals, Sci. Reports of Kagoshima Univ., No. 9, 1960, pp. 25-34.

[3] I. P. N a ta n so n , Theory of functions of real variable (in Russian), Moscow- Leningrad 1950.

[4] A. Z ygm und, Trigonometric series, I, I I , Cambridge 1959.

*o+<5

7T

drK( t — x, I) d f

References

Singular integrals depending on two parameters

R.

(Poznań)

Singular integrals depending on two parameters

It is easily observed that Fejer’s theorem concerning the convergence of singular integrals

'(1) U ( x , £ , f ) = j f ( t ) K ( t - x , ( ) d t ( f t L 2K)

can be extended to the convergence (x, S) -> (x0, | 0) on an arbitrary set of points of the plane. Namely,

1.1. If

(2) lim f K ( t , i ) d t = 1 (ŚeE),

П

(3) J IK( t , 1)1 dt < 0 on E (C = const),

(4) lim sup \K[t, f)| = 0 for д > 0 (д < те, £еЕ), ł->ło

then we have

(5) lim U(x, i , f ) = f(x0) ЦеЩ

at every x0 at which f is continuous (cf. [4], I, p. 89, [1], p. 418).

Delation (5) holds at the points of differentiability of f f(t)dt, under 5

the assumptions of Bomanovski’s theorem, if the convergence (x , I) ->

-> (x0, £0) is restricted to some set of points of the plane. Also a theorem of the Faddeev type and a result concerning the convergence of deriva­

tives drU(x, i , f ) j dxr can be stated. The present note is devoted to these

problems.

2. A generalization of Natanson’s lemma. We shall prove the following fundamental lemma (cf. [3], p. 243).

2.1. Let <p{t) be a function of bounded, variation in every interval

by (0 < у < b — a), such that f v(s)ds < oo, where

v(s) = var cp{t) {a < s < b) , v(b) — 0.

Then, if

M = sup — f f(t)dt

the improper Lebesgue integral I = ff(t)<p(t)dt exists and a+

Proof. Write

F(t) = Jf( u)du (a <&),

P

Ia, n= f f(t)<p{t)dt (a < a < ft ^.b).

Integrating by parts, we obtain

fi /3

/ <p{l)dF(t) =<p(t)F(t)\f - f F(t)d?(t).

Since

fi P

F(t)dq>{t)\ < M

(t — a ) d [ —v(t)] = M^[a — t)v(t) j v{t)dt\,

(a —a)\(p(a) — (p{b)\ ^ (a — a)v(a) < J v(t)dt,

the integral fF(t)dy(t) exists and

ь ь

I f F{t)dcp{t)] < M f v(t)dt.

a+

Our conclusion is now evident.

ь

R em ark. If ip(t) is such that / w(s)ds < oo, where

then

with

w{s) = var ip(t) (a < s < b), w{a) = 0,

I / f ( t ) y { t ) d t \ ^ Ж f [w(s)-\-\ip(a)\']ds,

N = sup

3. Theorems of the Romanovski and the Faddeev type. Some results on the convergence of integrals (1) will now be given.

3.1. Suppose that the function K ( t, £) is non-negative and non-inereasing in t on <0, тс) for every £eE, that it satisfies condition (2) and that

Let

lim K( d, £) = 0 for 0 < д < тг (£ eF?).

(7) lim —

J f ( t ) d t = f ( x 0) ( f €^2n) at some x0. Then integrals (1) tend to f(x0) as (x, £) ->

set Z in which the function

(x0, £0) on any plane

Ц х, I) = ( x - x 0) K ( 0, £) (£eE) is bounded (cf. [3], p. 245, [4], I, p. 101, (7.9)).

Proof. Of course, it is sufficient to show that T C

I(x, £) = f [f(t)—f(x0)]K(t — x, £ ) d t ^ 0

as (x, £) -> (x0, £0) on Z.

We consider only the case — т с < x0 < 0 . By (7), given e > 0 there is a <5 > 0 such that

1 h

£ when 0 < h < 3.

Suppose that 6 <

+ ^

, 0 < x0 — x < and write

It is easy to see that

and

u,\ } \ m - n x 0)\m.

Hence, by (6),

lim l!

0

lim l3 as

(x, o (x, $>

In view of 2.1,

\I2\ < £ f [ var К

£)]ds +

-> (x0, £0) is restricted to some set of points of the plane. Also a theorem of the Faddeev type and a result concerning the convergence of deriva

(7) ^{lim —}

**< « [ /X (* ,f )< fo + 2 (^ -® ) X (0 ,f)] .**

**(*,f)**

**(.)**

**9т Ы = / <P,(*o) (P = 0 ,1 ,**

**а, = 1[/Ч * „ ) + / ' " « ] > «4 o ) + / , r (o)], «6 = i [ 9 / ( o ) + + 1 0 //"(^0) + / F(^o)] etc. II is easy to observe that**

**~#0O]/sinr(2 — #0) -*■ 0 as t -> x0 ([2], pp. 25-27).**