A theorem of the Romanovski type for double singular integrals

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXVIII (1989) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PR ACE MATEMATYCZNE XXVIII (1989)

St a n is l a w Si u d u t (Krakôw)

A theorem of the Romanovski type for double singular integrals

Abstract. In a paper by Taberski [2], among many other results, Theorem 6.2 (concerning the convergence of some double singular integrals on a given rectangle) is proved. It is the purpose of this note to prove an other theorem of this type.

1. Let a, b be positive real and let x0, y0 be real and P = < — a, a) x x ( — b,b}. Let X be a metric space and let £ be a given subset of X.

Suppose that £0 is an accumulation point of E.

Denote by К a real-valued function defined on R 2 x E satisfying the following conditions:

(1) K ^ O and K (-, -, Ç) is measurable for every £ e £ ;

(2) K(-,s,£) is even, 2a-periodic and non-increasing on (0, a) for all

£ eE, s e <0, b};

(3) K(t, -, £) is even, 2b-periodic and non-increasing on <0, by for all £ eE and t e <0, a );

a b

(4) lim f f K(t, s, Ç)dsdt = 1 (£e£);

«-*0-a-b

(5) lim K(ô, 0, £) = lim K{0, ô, £) = 0 for every Ô e(0, min (a, b)).

Denote b y '/a real-valued function defined on R2 satisfying the following conditions:

(6) / is Lebesgue integrable on the rectangle P;

(7) for each s é ( —b , b ) the function /( • , s) is 2a-periodic;

(8) for each t e ( — a, a ) the function f ( t , •) is 2b-periodic;

(9) the relation

j *o + A

lim - f f ( t , s ) d t = f ( x 0, s )

h->0 H ^{x q}

holds uniformly with respect to almost all s e ( — b,b}, and this relation holds for s = y0;

356 S t a n i s i a w Si udut

(10) the relation

i ?о + т

lim - j' f( t , s )d s = f ( t , y 0)

X~ * ° x У0

holds uniformly with respect to almost all f e < - a , a), and this relation holds for t = x 0.

For C > 0, let Zc denote the set

a

\(x, y, S)eR2 xE: |y0- y | f K ( t , 0 , i ) d t < C and

- a

b

|x0 —x| f K{0, s, f)ds <C).

- b

2. Theorem. Suppose that the functions K , f satisfy the conditions listed in Section 1.

Then, for any C > 0,

lim ] f f{ t, s)K(t — x, s - y , f)dsdt = / ( x 0, y0)

— a — b

as (x, y, f) ->(x0, y0> U) and (x, y, f ) e Z c.

P ro o f. Fix C > 0 and write

g(t, s ) = f ( t , s ) - f ( x 0, y0),

a b

W { x , y , f ) = f C g { t , s ) K { t - x , s - y , f ) d s d t .

— a — b

It is sufficient to show that W{x, y, f) ->0 as (x, y, £) -►(x0, y0, £0) and (x, y, f ) e Z c .

We may clearly assume that

(11) — a < x0 ^ 0, - b < y o ^ 0 , 0 <S <min{a + x0, b + y 0), 0 < |x0- x | <{ô and 0 < \y0 - y \ <i<5.

Let us divide the rectangle P into the rectangles Pi j (ci, Ci + j y x (dj, dj +1У

for 1 ^ i ^ 3 and 1 < 7 ^ 3, where c1 = —a, c2 = x 0 — S, c3 = x0 + S, c4 = a and dt = - b , d2 = y0- ô , d3 = y0 + ô, d4 = b.

If we write

f j = J.fôKL s ) K ( t - x , s - y , if) ds dt,

Theorem o f the Romanovski type 357

then W(x, y, £) = Y, hj and it is enough to show that

i . j = i

(12) / y ->0 as (x, y, <*) ->(x0, y0, £o) and (x, y, £)eZc, whenever 1 < г ^ 3 and 1 < j ^ 3.

We shall now prove (12) for the integrals / 13, 112 and / 22.

Let us consider first the integral 713. In view of (2), (3) and (11) we have

x 0 ~ ô b

| / , 3| « f I' \g (t ,s )\ d sd fK (i ô , iS , i)

- а Уо + д

<M \-K(±0, 0, 0 ,

a b

where \\g\| = f f |g(t, s)\dsdt.

— a — b

Hence, by (5), / 13 -*0 as (x, y, £) ->(x0, y0, C0)- Analogously, / n , I 31, / 33 tend to zero as (x, y, £) ->(x0, y0, £0).

Now we prove (12) for the integral / 12.

We begin with

(13) |/ i 2| = | J J { / (t, s) — f (x0, y0)}K(t — x, s - y , Ç)dsdt\

- a y Q - 0

X Q - à y Q + à

< | f f { f ( t , s ) - f ( t , y 0) } K { t - x , s - y , Ç ) d s d t \

- a y Q - 0

х0~дУ0 + â

+ .f .f \ f { t , y 0) - f ( x 0, y 0) \ K ( t - x , s - y , Z ) d s d t = A + B.

- a y ç ) - à

Applying (2) and (3), we obtain

x0-ô

(14) B ^ 2 6 - f \ f { t , y o) - f { x o, y o) \ K ( t - x , 0 , £ ) d t

- a

XQ-Ô

« 2ô-( f (|/(r, V’o)l +1/ ( ^0, Уо)!И<)КЙ,5, 0, c)

3

$ 26■( ( |

f(t, y0)\dt

By assumption (10), there exists т0 > 0 (т0 < b) such that

358 S t a n i s t a w Si udut

1 У0 + х0

- ! т° уо

f ( t , s)ds + 1 ^ | / ( f , y 0)|

for almost all t e ( — a, a}. Hence, by (14),

B ^ 2 S [ - \ \ f \ \ + 2a + 2a \f(x0, y0)| О, £);

T0 thus

(15) B -*0 as (x, y, £) ->(x0, y0, £o) (see (5)).

By virtue of (10) there exists, for every s > 0, a number Ô > 0 such that 1 Уо ±T

- f { f ( t , s ) - f ( t , y 0)\ds

T JL

^ £

for almost all t e ( — a , a ) when 0 <t^;<5. Hence, by Lemma 2.1 ([1], p. 174),

* 0 ~ ‘5 b

J' £ { J* K ( t - x , s, Ç)ds + 2\y0- y \ K ( t - x , 0, Ç)}dt

— a — b

a b a

< £ - { J ‘ J' K ( t - x , s, Ç)dsdt + 2\y0- y \ j K { t - x , 0, Ç)dt\

(see [1], p. 176, for the similar situation).

Therefore, if the points (x, y, £)eZ c are sufficiently near to (x0, y0, £o)>

we have

(16) Л ^ 2 г -(1 + С)

(see (4) and the definition of Zc).

In view of (13), (15) and (16) we have (12) for the integral / 12. For the integrals / 21, / 23, h i the proof of (12) is similar to the above one.

Let us pass to the integral I 22. Write x0 + <5 У0 + <*

|/2г1^ | f f {f ( t , s ) - f ( t , y 0) } K ( t - x , s - y , Ç ) d s d t \ x0~â yo~ô

y 0 + ô XQ + Ô

+ | j J { f ( t , y 0) - f { x 0, y 0) } K { t - x , s - y , Ç ) d t d s \

У 0 ~ 0 X Q ~ Ô

= Ai + A 2.

We estimate the integrals A 1 and A 2 as the integral A, i.e., we apply Lemma 2.1 ([1], p. 174) to each of them. Consequently, A t and A 2 tend to zero as (x, У, £) -*(xo> Уо» *so) and (x, y, £)eZ c . Hence (12) is true for the integral / 22. The above remark completes the proof of the theorem.

Theorem o f the Romanovski type 359

3. R e m a r k 1. Our theorem implies Taberski’s result ([1], p. 175).

P roof. Assume that f, К satisfies the assumptions of Taberski’s theorem and put / (t, s) = f (r), K (f, s, Ç) = К (t, £), a = n, b = Applying our theo­

rem, we are done.

R e ma r k 2. The request that both limits (9) and (10) are uniform cannot be omitted.

Counterexample. Let a = b = 1, E = (0, 1), Co = *o — Уо = ®- We define the functions f K ( -, -, Ç) (£e£) for t, se< — 1, 1) by:

f{t , s) = |t-s|/(t2 + s2) if t2 + s2 > 0, /(0 , 0) = 0;

( o i f ( ( , s ) 6 < - i , i > 2\ < - e . o 2-

Let f (t + k-2, s + l-2) = f (t, s), К (t+k-2, s + l-2, () = K(t, s, Ç) for all integers к, l.

The function К satisfies all assumptions of our theorem. The function / satisfies conditions (6), (7), (8), but not (9) and (10), because

lim - \ f ( t , s ) d t = 0 = /(0, s) 1 h but not uniformly a.e. on ( —1, 1), л-о h ô

l i m - f / ( f , s)ds — 0 = f ( t , 0) 1 T also not uniformly a.e. on ( —1, 1).

r -*0 T 0

It is easy to check that l l

lim f f f { t , s ) K ( t - x 0, s - y 0, Ç)dsdt = i l n 2 , J i J i

whence is not true that

f f f ( t , s)K(t — x, s - y , Qdsdt -> /( 0, 0) = 0 - i -T

as (x, y, Ç) -+(x0, y0, fo) and (x, y, ^)eZ c (observe that (x0, y0, £)eZ c for all £e£).

References

[1] R. T a b e r s k i, S in g u la r in te g r a ls d e p e n d in g o n tw o p a r a m e te r s , Roczniki PTM, Séria I, Prace Mat. 7 (1962), 173-179.

[2] —, O n d o u b le in te g r a ls a n d F o u r ie r s e r ie s , Ann. Polon. Math. 15 (1964), 97-115.