ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXX (1990)
Stanislaw Siudut (Krakôw)
On the Fatou type convergence of abstract singular integrals
Abstract. Some generalizations of Theorems 3.1, 3.2 of [5] are presented. The main results are contained in Theorems 1-4.
1. Introduction and statement of the results. Roman Taberski has proved in [5] two theorems on the pointwise convergence of singular integrals depending on two parameters (see [5], 3.1, 3.2). In these theorems the convergence of parameters is restricted to some subsets of the plane, i.e. the Fatou type convergence is discussed. The first of the parameters tends to a D-point or an L-point of a real-valued 27i-periodic function /, whereas the second one tends to an accumulation point of a given subset of R.
In this paper we extend Taberski’s results to the case of Banach space-valued function / and their D!l!- or L^-points (see the definition below).
The non-periodic case is also considered. Some applications of the obtained results to the norm-convergence of real-valued singular integrals are given.
We denote by G a real Banach space with the norm || ||. Let m be the Lebesgue measure on R. For 1 ^ p ^ oo, we define
& 2*(C) = { / I / : R ^ G is 27i-periodic and m-integrable on < — к, тс)}, i f2л = n(R), У P(G) = i f P{R, m; G), i f p = i f p(R) (see [2], p. 504 and Ch.
IV or [3], pp. 485, 486, 494).
Let 1 ^ p ^ oo and / e i f p(G) or / e i f 2lt(G).
Definition 1. A point x e R is called a D^-point [resp. an L^-point'] of / iff there are a, be G for which
л л
J(/(x + u) — a)du =o(h) as /г —> 0 — and §(f(x + u) — b)du = o(h) as h-* 0 +
o о
ft
[resp. j \\f(x + u) — a\\du = o(h) as h ^ 0 — о
ft
and j \\f(x + u) — b || du = o(h) as h -> 0 + ].
о
If x is a D*- or L*-point of/, then we write /_(х), f+(x) for a, b, respectively.
The point x is called a D-point [resp. an L-poinQ of / if it is a D^-point [resp.
an L*-point] of / and a = b. In this case we write /(x) for a.
Let £ be a non-empty subset of a metric space (X , @) and let £0 be an accumulation point of E. In the case X = R we also admit £0 = oo. In what follows Ç will always denote a parameter belonging to E.
Throughout this paper К, К * will denote real-valued functions defined on R x E and satisfying the conditions:
(1) for every ÇeE the functions K(-, f), K*(-, f) are even, bounded and measurable,
(2) IK{t, f)\ ^ K*(t, f) for all te R , ÇeE.
For С ^ 0, P > 0 we define
(3) Z c>/? = {(x, f e R x E : \ x - x ofK (0 , É)< C}, (4) Z t , = {(x, f e R x E : \ x - x 0f K*(0, Ç) ^ C}.
We now state the following generalizations of Theorems 3.1, 3.2 of [5]:
Th e o r e m 1. Suppose that for every ÇeE the function K(-, f) is 2n-periodic, even, non-negative and that
(5) K (-, is non-increasing on <0, к ) for every ÇeE, (6) lim K(ô, f) = 0 for every <5e(0, n),
Я
(7) lim J K(t, f)dt = 1,
€-€o -n
(8) 0 < fi < 1, C ^ 0, f e L2n(G) and x0 is a D^-point of f.
Then
(9) f K ( t - x , ( ) / № - > i { f - ( x 0) + / + ( x 0)}
— n
as (x, £)-*(x0, £0) and (x, £)eZ C;/}.
Furthermore, if x0 is a D-point of f then (9) also holds for /1=1.
Th e o r e m 2. Let K(-, if), K*(-, f) be 2n-periodic for every ÇeE. I f К, К * satisfy (1), (2) and:
(10) lim sup IK(t, f)\ = 0 for every <5g(0, я),
Я
(11) there is В > 0 such that j K*(t, f)dt ^ В for every ÇeE,
~ Я
(12) К satisfies (7) and K*(-, f) is non-increasing on <0, к ) for every ÇeE, (13) 0 < / ? < l , C ^ 0 , / e££2n(G) and x0 is an L^-point of f
then (9) holds with Z*,p instead of Z CtP. Furthermore, if x 0 is an L-point of f then (9) is also true for Z*,i (in place of Z Cjl).
Theorem 3. Suppose that for every ÇgE the function K(-, Ç) is even, non-negative and that
(5') K(-, Ç) is non-increasing on <0, oo) for every ÇgE, 00
(6') lim J K(t, Ç)dt = 0 for every <5e(0, oo),
«-«О г
00
(7') lim f K(t,Ç)dt = 1,
-oo
(8') 0 < fi < 1, С ^ О, 1 ^ p ^ oo, f g $ £p( G ) and x 0 is a D^-point of f.
Then 00
(9') J K ( t - x , a / ( t ) * - K / - ( * „ ) + / +(*o)}
— 00
as (x, Ç)-*(x0, £0) and (x, Ç)eZCtP.
Furthermore, if x 0 is a D-point of f then (9') also holds for f = 1.
Theorem 4. I f К, K* satisfy (1), (2) and § - œK*(t, Ç)dt ^ В for some В > 0 and every ÇgE and К satisfies (!'), К * satisfies (5') and (6'), 0 < f < 1, C ^ 0, 1 ^ p ^ oo, f G £FP(G), x 0 is an L^-point of f then the relation (9') holds with Z *,p. Furthermore, if x 0 is an L-point of f then (9') is also true for Z*, i (in place of Z c,i).
2. Proofs of Theorems.
Le m m a 1. I f К satisfies the assumptions of Theorem 1 and f is the In- periodic extension of the function
then
f e < - 7 c , 0 ) ,
t G (0 , я ) ,
(14) J K ( t - x , £)f(t)dt^>0 as (x, £)->(0, £0) and (x, Ç)gZ c,p, where C ^ 0, /?e(0, 1) are arbitrary fixed.
P ro o f. If — 7i < x < 0, then
J f ( t ) K ( t - x , Ç)dt = J f(s + x)K(s, Ç)ds
— л + |х| |x| n
= J K(s, Ç)ds— j K(s, <f)ds+ j K(s, Ç)ds = :a(x, f) , say.
- n - | x | it |x|
— n
Since K is even with respect to the first variable,
n |x |
a(x, f) = 2 J K(s, f)ds — 2 J K(s, f)ds
л — I j c| 0
and in view of the monotonicity of K(t, Ç) in t on <0, n) we obtain for C > 0
|a(x, £)| < 2 \x\ K (k— \x\, Ç) + 2 \x\ K ( 0 , f) ^ 4|x|K(0, 0 ^ 4C\x\l ~p according to the definition of Zc>/3. The last inequality proves (14) as (x, £)->(0, £0) and — я < л: < 0. Proceeding as before, we obtain (14) as (x, £)-*(0, £0) and 0 ^ x < 7t. Thus the proof is complete, because for C = 0 the relation (14) is obvious.
Le m m a 2. I f К satisfies the assumptions of Theorem l and C > 0, /?е(0, 1), a, beG and f is the In-periodic extension of the function
( t . = f a , t < E ( x 0 - n , x 0 ) ,
9 \b, te < x 0, х0 + я), then
П
(15) J K(t — x, Ç)f{t)dt-*j(a + b) as ( x ~ x 0, £)->(0, £0) and (x, f ) e Z c,p-
— П
P ro o f. Set U(x, £ ,/) = §n- nK{t — x, f)f{t)dt and observe that U(x, £ ,/)
— U(x — x 0, £ ,/(- + x0)), where f ( ’ + x 0)(t) = f ( t + x 0) for teR. Thus Lemma 1 yields
t / ( x - x 0, £ ,/(• + x0) - i ( a + b )) ^ 0 as (x —x0, ^)—»(0, £0) and |x — x ofK (0 , £) ^ C, which together with (7) proves (15).
Now we prove Theorem 1. Since the suitable generalizations of Natan- son’s lemma are shown in [4], the second part of Theorem 1 (concerning D-points) is an obvious consequence of the results of [5] (see [5], 3.1, 2.1 and Remark).
Define the functions , f 2 by
7 (t\ = S f№ ’ t e ( x 0- K , x 0), ~ f/+(*o)» t e ( x 0 — K, x0), 1 • \ f - ( x 0), t E<x0, x0 + 7l), 2 j/( t), te<X0, Х0 + я).
Denote by f t , f 2 the 27i-periodic extensions of / 15/ 2, resp. If we define
/3 = /1+ /2-/» then
U(x, Z , f ) = U { x , L f ^ f 2- h ) = Щх, £,Л )+ Щ х, Ç ,f2) - U { x , L h ) and applying the second part of Theorem 1 we obtain
U{x, Ç ' f J - t f - i x о), l/(x, £ , / 2)-*/+(*0)
as (x, f) —*(x0, f 0) and (x, f) e Z Ctl.
Applying Lemma 2 we get
U(x, £ , / 3H i( /- ( X o ) + /+(x0)) as (x> £o) and (x, Ç)eZc,is- It is sufficient to observe that ZCj/, <= Zca for |x — x0| < 1. Our conclusion is now evident. Similarly one proves Theorem 2.
P r o o f of T h e o re m 3. Using (5') and (6') we see that, for q ^ 1, <5 >0 fixed,
lim К ((5/2, £) = 0 and lim j K(t, Ç)qdt = 0.
Ç~*üo R \ ( - ô / 2 , ô / 2 )
This and the Holder inequality suffice to prove that
(16) lim j K(s — x,Ç){f{s)—f ( x 0)}ds = 0 with |x — x0| < <5/2.
4~*£o R \ ( x o — ô,xo + ô)
If x0 is a D-point of /, then proceeding just as in the proof of Theorem 3.1 in [5], we obtain
(17) lim J“ K { s - x , £ ) { f( s ) - f{ x 0)}ds = 0
(x,S)->(x0,4o) X o - 0 ( x . i ) e Z c . i
(for the abstract counterparts of Natanson’s lemma see [4]).
Now, the second part of Theorem 3 (i.e. (9') if x0 is a D-point of / and P = 1) is a consequence of (16), (17) and (7').
If x0 is a D^-point of f then the proof of (9') is a repetition of the proof of (9) (with the obvious modifications).
P r o o f of T h e o re m 4 (an outline). The relation (16) is easy to verify by using the triangle and* Holder’s inequalities. To establish (17) in the case where x0 is an L-point of /, we make use of Definition _ 1 and we apply the generalizations of Natanson’s lemma ([5], 2) to the quantity fêl-îK*{s — x, Ç)\\f(s)—f ( x 0)\\ds, which is not less than || f ô - ^ s - x , Ç) x { f( s) ~ f ( x o)}ds\\- The rest of the proof is similar to that of Theorem 1.
3. Some remarks and examples.
R em ark 1. In the first parts of Theorems 1-4 (concerning D or L^-points), the assumption /?е(0, 1) cannot be replaced by /?е(0, 1).
For example, consider the functions /, К defined by
/(0 ~ щХ(-1Д)(0> £) = 2^X(-s,$)(0>
where Хл denotes the characteristic function of. the set A c R, £e(0, 1). For
£0 = x0 = 0 we obtain l i m ^ oJV(0, £ ,/) = 0, but l i m ^ 0iV(£, £ ,f) = lim ^ 0 + Jo*(l/2£)dt = 1, although (0, Ç) and (£, Ç) belong to Z 1/2>1 (JV(x, £ ,/) denotes
the integral operator from (9')). Thus, the conclusion (9') of Theorem 3 does not hold.
Ex a m p l e 1. Let X(R) denote C(R) or B(R) with the sup or П norm, respectively, and let X 2n denote C2n or Щп with the sup or L%n norm, resp. ([1],
PP- 1, 2, 9).
Let gEX(R) and / : R-+X(R), f (t) = g(- 4-1). If 1 ^ p < oo then x 0 = 0 is a D-point and an L-point of the function / (this is a simple consequence of the Hôlder-Minkowski inequality and the continuity of f see [1], pp. 3,4).
Applying Theorems 3,4 (with C = 0) we obtain f K(t, Ç)g(- + t)dt^f{0) = g as
— 00
for every К satisfying the assumptions of Theorem 3 or Theorem 4. Thus, substituting the appropriate kernels К in the above relation, we obtain the following results of [1]: 3.1.18, 3.1.35, 3.1.41, i.e. we have the convergence of the singular integrals of Fejér, Gauss-Weierstrass and Cauchy-Poisson in the X(R) norm.
Analogously, in the periodic case one can obtain the following results of [1]: 1.2.26, 1.2.9, 1.3.4, i.e. the convergence of the singular integrals of Fejér, Abel-Poisson and Rogosinski in the X 2n norm.
We remark that it is possible to generalize many other theorems of [1] to the vector-valued case, for example 1.4.1, 1.4.2, 1.4.5, 3.2.1, 3.2.2, 3.2.4.
From the above considerations we conclude that the norm convergence of real-valued singular integrals is a consequence of the pointwise convergence of suitable vector-valued ones.
Finally, we observe that the result concerning differentiation of singular integrals ([5], 4.1) can also be extended to vector-valued functions f
References
[1] P. L. B u tz e r and R. J. N e s s e l, Fourier Analysis and Approximation, Vol. I, Birk- hâuser-Verlag, Basel and Stuttgart 1971.
[2] L. S c h w a r tz , Cours d ’analyse, Paris 1981.
[3] L. S c h w a r tz , Cours d ’analyse, Warszawa 1979 (in Polish).
[4] S. S iu d u t, Generalizations of Natanson’s lemma, Comment. Math. 29 (1990), 277-286.
[5] R. T a b e r s k i, Singular integrals depending on two parameters, ibid. 7 (1962), 173-179.
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