ROCZNIKI POL.SKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PR АСЕ МЛТЕМА TYCZNE XXVI (1986)
Stanfseaw Siudut (Krakow)
Some remarks on the singular integrals depending on two parameters
1. Introduction. Let L 2k be the class of all 2TT-periodic and Lebesgue- integrable functions in the interval ( — n, к}.
Denote, once for all, by K(t, ç) a function defined for all te R and ç e E (where £ is a given set of numbers), 2jc-periodic, even, non-negative and measurable with respect to t for every fixed ç gE. We assume that the function K(t, ç) is non-increasing in t on <0, n ) for every çeE.
Suppose that £0 is an accumulation point of E. We assume that the function К satisfies the conditions:
(1) lim f K ( t , c)dt = 1,
(2) Д lim K (<5, ç) = 0.
ôe(0,n) 4
Let us define, for any constants С > 0, P > 0, <5 > 0, 0 < e < 1 and any f e L 2n, the sets:
Z c = [(*, ç )e R xE: \ x - x o\K(0, ç) < Cj, Z c fi = l(x, c)g R x E : \ x - x 0\p K (0, ç) ^ C],
Z C)<5i£= {(x, Q e R x E : ( ] \ f ( t + x ) - f ( t + xo)\dty~eK ( 0 , ç ) ^ C } . In paper [3], p. 175, the following theorem was proved:
Theorem 1. I f f e L 2n and
x0 + h
lim-j- j f ( t ) d t = f { x 0)
h ^ o h J
xo
lim f i)dt = f ( x 0).
(*>s> -»(X0,Ç0)
(x.i)~Zç
at some x0, then
Let us put
U(x,i ,f) = } f(t)K(t-x,i)dt,
— я
K r = [(x, Ç)<=R2: ( x - x 0)2T (£ - £ 0)2 < r2\-
Let A be any subset of R x E and let (x0, ç0) be an accumulation point of A.
Suppose that assumptions of Theorem 1 holds.
Obviously, if relation
(P) V
V :
A n K r <= Z Cr > 0 C > 0
holds, then
(q) U (x, c,/) -» / (x0) as (x, c) -► (x0, Co) and (x, c)e A . If relation (p) does not hold, i.e., if
(p') Л A ( А п к г) \ г с Ф 0
r > О О 0
holds, then (q) is not true. We shall prove it in Section 2.
In other words, we shall prove that it is impossible to wider the sets Z c in Theorem 1, in essential way.
An analogical to Theorem 1 result we shall formulate and prove in Section 3.
Section 4 is devoted to some applications.
2. Impossibility of essential extension of the sets Zc. Let E = (0, я), Co = 0, x0 = 0 and
for fe< -c/ 2 ; ç/2>,
(t,Ç) j 0 for Г6 <-7T, K>\<-c7 2; c72>.
For t e ( — n, к) and any integer k we define K{t + 2kiz; c) = K(t, c).
The function K(t, ç) satisfies all the conditions of the introduction, and (3) Z c = {(x, c )e R x(0, я): М/с ^ C) .
Write
K r = [(x, c ) e R 2: x2 + c2 ^ r2}.
Let A cz R x E be a set such that
(4) A A ( A n K , ) \ Z c * 0 .
r >0 C > 0
(For example, A — R x E satisfies condition (4).)
We shall construct two sequences {x„{, of real numbers and a
sequence [Pn} of intervals such that 1° (*,,,
2° lim (x„, ç„) = (0, 0),
n~*oo
3° P„ = — xH + ZJ2\
4° P„ с (- Я , 7C), 5° ОфР
6° n ^ m = > P „ r \ P m = 0 (n, me N).
Construction. Let (Xj, £i)e{A n K 1) \Z ai when otl ^ 21 (from (4) it follows that this is possible) and P { = { х х — x 1+ ^ 1/2).
Applying (3), we get
Ci < il^il <4;
thus we obtain Г. 3°, 4°. 5°.
Let (x2, ^ ) e ( 4 n K lxiU2)\Ze2 when a2 ^ 22 (see (4)) and P2 = (x2- Ç 2/2;
*2 + C2/2).
Applying (3), we get
I I l l
C2 < " l - v 2| ^ ^ | x 2| < 23^1 < ? ;
thus we obtain 1°, 3°, 4°, 5°.
By the inequalities
l*2l < l * i l/2 and ç2 < i l * i l and Qi < { \ x l \ we get P, n P2 = 0.
Let (x„, O g(/1 n/CK l|/2)\Zan when a„ ^ 2" (see (4)) and P„ = (x„-ç„/2;
*п + £п/2).
We have the inequalities
(5) K -2I
22
< 2 ""11 and from (3) and (5) it follows that1 1 1 . |x,| 1
(6) Си < “ l*J ^ 2" l*»l ^ 2» +1 ^ ^ ^ 22" - 1 < 22n_ 1 ’ Thus we obtain 1°, 3°, 4°, 5° and
(7) P . - 1n P „ a= 0 .
We infer by (5) that the sequence {|x„|{ is decreasing; then (7) holds for any n e N (n ^ 2). Thus we obtain 6°.
Further, from (5), (6) we get 2°.
Remark. For sufficiently large a„ (n = 1,2,...) in the above construction we can obtain
(8)
and
(9 )
X 2v
n = 1
< X
1
W\\ I 0 as h 0.
Now, we construct a function f e L 2n as follows:
= for г ф Р п, t e < - к , тг>,
” “ I 1/х/|г-х„ + с„/2| for г е Р „
00
and /(f) = Z /„(f) for t e < - к , n } .
n= 1
For t e ( — к , к) and any integer к we define / ( f -h 2Acti) = /(f) (see 4°).
The function / is measurable (since /„ are measurable), non-negative (since /„
are non-negative) and 27[-periodic; moreover, by [2], p. 130, formula (1.13) and (8), we have
.f ( Z f n ( t ) ) d t = Z .f L i t )d t = Z 2 x/ç„ < X .
Therefore f e L 2n.
We observe that /(0) = 0 (see 5°) and
(10)
h limIh^oh }
0 Indeed,
0 ^
\ h \
f (t)dt 1 v- л
< — z 2
\h\
Hence, by (9) we get (10), thus / satisfies the assumptions of Theorem 1, where x0 = 0.
The hypothesis of Theorem 1 does not hold when (x, ç)eA and (x, ç)
(x0, ç0), because
f ( t ) K ( t - x „ , ç „ ) d t = f ( t - h x „ ) K ( t , çn)dt = — 1 I f { t + x„)dt
J J Çrt J
= T j f ( t ) d t = ^ - j f„{t)dt = -— 2 N/ ^ = —t== -»■ x
Cn J Qn J Сп - / ç„
я p
rn rn
as и -> x (see 2° and 6°).
Л
Then lim f f ( t ) K ( t - x , ç)dt is not equal to f ( x o) = 0.
(x,4) ->(х0,$0) -Л (x,s)e.4
3. A generalization of Theorem 1. Let us put Я
U{x, ç , / ) = J’ f { t ) K { t ~ x , ç)dt.
— П
Th eo r em 2. Z/- f e L 2n and
( П )
at some х0, then
lim -j- I U ( x 0 + t ) + f { x 0- t ) - 2 f ( x 0y\dt = 0 0
(12) L (x, ç , f ) - + f ( x 0) as (x, ç )- » (x 0, Со) ши! (x, ç )e Z Wi£.
//, in particular, f satisfies the assumptions of Theorem 1, then (13) L (x , ç , f ) - * f ( x 0) as (x, ç )- +( x0, ç0) ею/ (x, ç )e Z W t u Z C).
Proof. It is sufficient to prove (12) only. Indeed, we observe that if y, y0e R x E, T: R x E -> R,
lim T(y) — g = lim T{y),
y - * y о
ye.4 yeti
then lim T(y) = g.
У-+У0
yeA
Thus (12) implies (13).
To verify (12) it is sufficient to prove that Л
(14) U(x, { , / ) - / ( * „ ) J K{i, ( ) d t - > 0
— Я
as (x, £)->(x0, £0) and (x, ^ e Z Wi£.
Clearly,
Я
IU(x, ç ,/ )- / (x 0) j K { t, ç)dt\
— Я
Я
«|l/(x, i,/ )- l/ (x „ , с,/)\+\Щх0, ï , f ) - f ( x 0) f K (r, i ) d l \ = j t + j 2.
- Я
As we denote I ô = < —<5, ô}, then, by the definition of Z C<0<E, J , = \ ] (/ (f + x )-/ (r + x„))K(r, s: J |/(r + x )- / (f + x„)| K(r, ()dt
— я - я
« j|/(f + x)-/(/ + x0)|K(0, Qdt + f l/ (f- x )+ / (f + x0)| K ( f , 3 *
^ a: (0, ç) • ( j I f ( t + x) - f ( t + x0)| dt)1 ~e • ( { \f{t + x) - f i t + x0)| dtf +
i<5 h
+ K(Ô, ç)- J' \f(t + x ) - f ( t + x0)\dt
< - n,ny\Ijj a c ( f I f (r + X ) - f (f + JC0}| dtf + K 0 , 3 • 2• ||/||L2„ = : iv.
In virtue of [1], p. 4, (0.1.9) and (2), w -> 0 as (x, ç) -> (x0, ç0) and (x, ç )e Z Wi£ (e > 0). Hence J l -+0 as (x, £) (x0, ç0) and (x, c > Z w ,£.
We shall prove that J2^ 0 as (x, £)->(x0, ç0).
Since K ( -, 0 is even (and [1], p. 133; 3.2.5.), we have
J2 = |j f ( x 0 + t)K( t, ç) dt— { f ( x 0) K( t , c) dt\
— Я “ Я
= |J[/(*o + 0 + / (* o - 0 - 2 / (x o)]A:(r, ç)df|
о
< \jiLf(xo + t ) + f ( x0 — t) — 2f ( x 0) ] / C ( f , ç)dr| +
ô
+ |JC /(xo + 0 + / ( x 0 —0 —2 /(x0)]К (t, 3 * | = y, + r2 (0 < 5 < л).
’<5
Let be an arbitrary positive number. From assumption (11) it follows that there exists a positive number Ô < к such that
h
(15) 1
h [/ (*o + r) +/ (*o ~ t ) ~ 2 f (x0)] dt ^ £t if 0 < h ^ Ô.
The following inequalities
Гг « J(|/(x0 + O M / (X o -f)l + 2|/(x„H)K(<5, i)dl Ô
H K (S , £) • 2 • (||/||l2„ + l/(x0)|) hold. Therefore, in view of (2), Y2 -* 0 as ç -+ 0.
By (15) and a generalization of Natanson’s lemma ([3], p. 174; 2.1) for the functions f (x0 + t)-bf (x0 — r) — 2f(x0) and K ( t , ç ) we infer
ô S
Yj < e r J [ var K (t, ç) + K{S, ç)] ds ^ • J K (s, ç) ds ^ el
0 0
if ç is sufficiently near to ç0.
Since is an arbitrary positive number, the proof is complete.
Corollary. I f f e L 2n satisfies the assumptions of Theorem 2 and, in addition,
(16) V
V
: $ \f(t + x ) - f { t + x0)\dt = O ( \ x - x 0\*) a s x - > x 0,a > 0 <5e(0,n) - s
and 0 < P < a, then
U{x, £ ,/ )-> / (x 0) as (x, ç) —* (x0, Co) and (x, ç ) e Z c%p.
Proof. If x is sufficiently near to x0, then from (16) we have S\f(t + x ) - f ( t + . x 0)\dt ^ M \ x - x 0\x,
is whence
K(0, c ) ( \\f(t + x ) - f { t + x0)\dt)i ~^~lllx) is
< M pla • |x - xol^ К (0, £) ^ M pla • C for (x, £) e Z CtP.
In virtue of these inequalities
(17) (x , c ) e Z CJ implies (x ,ç )e Z c.MWui_Wa.
. Applying Theorem 2, we get
U{x, ç , f ) - * f ( x 0) as (x, £ )-»(x 0, Co) and (x, c )e Z c мР/л ôл thus from (17) it is still valid as (x, £)->(x0, ç0) and (x, c ) e Z c p .
4. Some applications.
A. Let, for x e ( — n, тг),
1
°№ = j i
if x < 0, if x = 0, if x > 0,
and, for fe ( — я, я> and any integer k, f { t + 2kn) = f{t).
The function f satisfies the assumptions of Corollary with у — 1 and .\'n = 0;
thus for the function К (f, Ç) in Section 2 and ç0 = 0 we obtain lim U (x, ç , f ) = f (xQ) = j ( P < U .
(jc,£)-(0,0) (x,4)eZc<p
We observe that the thesis of Theorem 1 does not hold in the above case because, for C = 1 and x = ç/2 (see (3)),
SI2 «
lim U(ç/2, ç , f ) = lim^ I f ( t + Ç/2)dt = \ i m\ ï f ( t ) d t = \im\-ç
£->0 S-OÇ J £-+oÇj
- ш о
= i Ф f (xo) == i-
B. Let, for x e ( —я, я>, \
s/x if 0 < x ^ я,
if II o'
i/xA if — я < x < 0, / (*) = 20
and, for t e( — я, я> and any integer k, f ( t + 2kn) = f ( t ) .
The function / satisfies the assumptions of Corollary with a = 1 and x0 = 0.
Thus
U(x, £,/)-►/ (x 0) = 0 as (x, ç) -* (0, f 0) and (x, c ) e Z c (/? < 1).
References
[1 ] P. L. B u tz e r , R. J. N e s s e l, Fourier Analysis and Approximation, Vol. I, Birkhauser Verlag, Basel und Stuttgart 1971.
[2 ] S. L o j a s i e w i c z , Wstyp do teorii funkcji rzeczywistych, P W N , Warszawa 1976.
[3 ] R. T a b e r ski, Singular integrals depending on two parameters, Prace Mat. 7 (1962), 173—
179.
