C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 67, N umb er 1, Pages 262–276 (2018) D O I: 10.1501/C om mua1_ 0000000848 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
NONLINEAR m SINGULAR INTEGRAL OPERATORS IN THE
FRAMEWORK OF FATOU TYPE WEIGHTED CONVERGENCE
GÜMRAH UYSAL
Abstract. In the present paper, we prove some theorems concerning Fatou type weighted pointwise convergence of nonlinear m singular integral opera-tors of the form:
T[m](f ; x) = Z R K t; m X k=1 ( 1)k 1 m k f (x + kt) ! dt;
where x 2 R; m 1 is a …nite natural number and 2 which is a non-empty set of non-negative indices, at a common m p Lebesgue point of f 2 Lp;'(R) (1 p < 1) and ': Here, ' : R ! R+ is a weight function en-dowed with some speci…c properties and Lp;'(R) is the space of all measurable functions for which 'f pis integrable on R:
1. Introduction
The approximation by singular integral operators is one of the highly studied and oldest topics of approximation theory. The researchers of this theory have investi-gated the limit behaviors of these type of operators while working on the problem of representing functions on some sets. The principal part of this survey belongs to approximation by linear singular integral operators on account of the comfort of the investigation. However, current problems of natural and applied sciences can not be interpreted by using only linear operator theory since nonlinear problems are included in the scope of those problems. Before giving brief information on nonlinear singular integrals, we …rst mention some of the studies and approxima-tion technics used therein which have come to the fore in the literature of linear singular integral operators.
Fourier series of the functions play an important role in the representation of functions at their characteristic points, such as point of continuity, d point and
Received by the editors: December 01, 2016, Accepted: May 26, 2017.
2010 Mathematics Subject Classi…cation. Primary 41A35, 41A25; Secondary 47G10, 45P05. Key words and phrases. Fatou type convergence; nonlinear m-singular integral; Lipschitz con-dition; nonlinear analysis.
c 2 0 1 8 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra . S é rie s A 1 . M a th e m a t ic s a n d S t a tis tic s .
Lebesgue point. Especially, the connection between the Fourier series of the func-tions and the following integral operator
L (f ; x) =
b
Z
a
f (t) K (t x) dt; x 2 ha; bi ; 2 ; (1.1)
where is a set of non-negative numbers with accumulation point 0; the symbol
ha; bi stands for closed, semi-closed or open arbitrary interval and K is the ker-nel function satisfying suitable conditions, is a well-known fact. Here, the kerker-nel function satis…es the singularity assumption lim ! 0K (0) = 1: For some
impor-tant studies concerning the pointwise convergence of the operators of type (1.1) in di¤erent settings, we refer the reader to [2, 6, 17, 22]. Also, we have to mention the studies [1, 17, 29] which contain many important results on weighted pointwise convergence of linear type of singular integral operators.
Now, we focus on Fatou type convergence, which is the heart of the matter, and give some related information. The great work by Fatou [9] gave an idea to the researchers such that the existence of one limit may indicate the existence of another one, and it was directly and indirectly used in the solutions of many approximation problems. The works directly based on this work with its theoretical background may be given as [8, 16].
Later on, Musielak [18] studied the nonlinear integral operators in the following setting:
Twf (y) =
Z
G
Kw(x y; f (x))dx; y 2 G; w 2 ; (1.2)
where G is a locally compact Abelian group equipped with Haar measure and is a non-empty index set with any topology. In this work, he used the Lipschitz condition for Kwwith respect to second variable. Therefore, the solution technics
developed for approximation problems for the linear case became applicable to nonlinear approximation problems. For some advanced studies about nonlinear integral operators in several settings, we refer the reader to [3, 4, 5, 11, 19, 20, 26].
Mamedov [17] obtained the following m singular integral operators
L[m](f ; x) = ( 1)m+1 Z R "m X k=1 ( 1)m k m k f (x + kt) # K (t)dt; (1.3)
where x 2 R; m 1 is a …nite natural number and 2 which is a non-empty set of non-negative indices, by using m th …nite di¤erence formulas. Here, the main concern is approximation of the m th derivatives of the integral of the functions pointwise by using the corresponding m singular integral operators. Later on, Karsli [15] studied the Fatou type convergence of nonlinear counterparts of the
operators of type (1.3) in the following form: T[m](f ; x) = Z R K t; m X k=1 ( 1)k 1 m k f (x + kt) ! dt; (1.4)
where x 2 R; m 1 is a …nite natural number and 2 which is a non-empty set of non-negative indices, at m p Lebesgue point of the functions f 2 Lp(R) (1 p < 1) ; where Lp(R) is the space of all measurable functions for which
jfjp is integrable on R: For some advanced studies concerning approximation by m singular integral operators in various settings, we refer the reader to [5, 13, 23]. As a continuation of the works [5, 15, 17], the current manuscript presents some results on the Fatou type weighted convergence of the operators of type (1.4). Our main concern is to prove that the family of the functions of type (1.4) converge to the functions f 2 Lp;'(R) (1 p < 1) ; where ' : R ! R+is a suitable weight function
satisfying submultiplication property, i.e., ' (t + x) ' (t) ' (x) ; 8t; x 2 R (for some speci…c weight functions satisfying this algebraic property see, for example, [12, 17]), and Lp;'(R) is the space of all measurable functions for which 'f
p
is integrable on R (see, e.g., [17]); at their m p Lebesgue points.
The paper is organized as follows: In Section 2, we introduce fundamental no-tions. In Section 3, we prove auxiliary results concerning existence and pointwise convergence of the operators of type (1.4). In Section 4, we present, as a main result, Fatou type convergence theorem for the indicated operators. In Section 5, we establish the rates of both pointwise and Fatou type convergences by using the results obtained in the previous two sections.
2. Preliminaries
De…nition 1. Let 1 p < 1 and 0 2 R+ be a …xed number. A point x0 2 R
characterized with the following relation lim h!0 1 (h) h Z 0 j mt g(x0)jpdt = 0; (2.1) where m t g(x0) = m X k=0 ( 1)m k m k g(x0+ kt);
is called m p Lebesgue point of locally p integrable function (i.e., a function whose p th power is Lebesgue integrable on arbitrary bounded subsets of R) g : R ! R. Here, : R ! R is an increasing and absolutely continuous function on 0 < h 0. Here, the relation( 2.1 ) also holds when the integral is taken from h
to 0.
Remark 1. De…nition 1 is similar to the characteristic point de…nition given and used in [15], i.e., the di¤ erence is the domain of the integration. On the other hand,
for some other generalized Lebesgue point characterizations, we refer the reader to [10, 23] and [30].
De…nition 2. (Class A') Let 1 p < 1 and be a empty set of
non-negative indices with accumulation point 0: Let ' : R ! R+ be a bounded weight
function on arbitrary bounded subsets of R such that
' (t + x) ' (t) ' (x) (2.2)
holds for every t; x 2 R. We will say that the family of the functions K : R R ! R, where K (#; u) is Lebesgue integrable on R for every u 2 R and 2 ; belongs to Class A' if the following conditions are satis…ed:
a) K (#; 0) = 0; for every # 2 R and 2 :
b) There exists a function L : R ! R which is integrable on R for each 2 such that the following inequality
jK (t; u) K (t; v)j L (t) ju vj holds for every t 2 R; u; v 2 R and 2 .
c) For every u 2 R; we have lim ! 0 Z R K t; m X k=1 ( 1)k 1 m k u ' (x0) '(x0+ kt) ! dt u = 0;
provided x02 R is a m p Lebesgue point of ':
d) lim ! 0 " sup jtj> 'kp(t)L (t) #
= 0; for every > 0 and k = 1; :::; m: e) lim ! 0 " R jtj> 'kp(t)L (t) dt #
= 0; for every > 0 and k = 1; :::; m:
f ) For a given number 1> 0; the function L (t) is non-decreasing on ( 1; 0]
and non-increasing on [0; 1) with respect to t; for any 2 :
g) 'kL
L1(R) Mk< 1; for every 2 and k = 0; 1; :::; m:
Throughout this article, we assume that K belongs to Class A':
Remark 2. The studies [3, 15, 17, 29], among others, are used as main reference works in the construction stage of Class A': For the Lipschitz condition (b); we
refer the reader to [3, 18, 19].
3. Auxiliary Results
Main results in this work are based on the following theorems. On the other hand, for the following type existence theorem, we refer the reader to [15].
Theorem 1. If f 2 Lp;'(R) (1 p < 1) ; then the operator T[m](f ) 2 Lp;'(R)
and the inequality T[m](f ) Lp;'(R) m X k=1 m k ' kL L1(R)kfkLp;'(R)
holds for every 2 .
Proof. We prove the theorem for the case 1 < p < 1: The proof for the case p = 1 is similar.
The norm in the space Lp;'(R) (see, e.g., [17]) is given by the following equality:
kfkLp;'(R)= 0 @Z R f (x) '(x) p dx 1 A 1 p :
By condition (a) ; we may write
T[m](f ) Lp;'(R) = 0 @Z R 1 'p(x) Z R K t; m X k=1 ( 1)k 1 m k f (x + kt) ! dt p dx 1 A 1 p 0 @Z R 1 'p(x) Z R L (t) m X k=1 ( 1)k 1 m k f (x + kt) dt p dx 1 A 1 p : Now, applying generalized Minkowski inequality (see, e.g., [25]), we have
T[m](f ) Lp;'(R) Z R 0 @Z R Lp(t) m X k=1 ( 1)k 1 m k f (x + kt) '(x) p dx 1 A 1 p dt = Z R L (t) 0 @Z R m X k=1 ( 1)k 1 m k f (x + kt) '(x + kt) '(x + kt) '(x) p dx 1 A 1 p dt: Using inequality (2.2), we can write
'(x + kt) '(x)'(kt); and since '(kt) = '((k 1 + 1)t) '((k 1) t)' (t) ; inductively, we set '(kt) 'k(t):
It follows that T[m](f ) Lp;'(R) Z R L (t) 0 @ Z R m X k=1 m k f (x + kt) '(x + kt) ' k(t) p dx 1 A 1 p dt m X k=1 m k Z R L (t) 'k(t)dt 0 @Z R f (u) '(u) p du 1 A 1 p = m X k=1 m k ' kL L1(R)kfkLp;'(R):
The desired result follows from condition (g). Thus the proof is completed. Now, we give a theorem concerning weighted pointwise convergence of the oper-ators of type (1.4).
Theorem 2. If x0 2 R is a common m p Lebesgue point of the functions
f 2 Lp;'(R) (1 p < 1) and '; then
lim
! 0
T[m](f ; x0) f (x0) = 0
provided that the function
Z
L (t) 0(jtj) dt; (3.1)
where 0 < < min f 0; 1g (for the de…nitions of the numbers 0 and 1; see
De…nition 1 and De…nition 2, respectively), is bounded as tends to 0on 1 .
Here, the set 1; which is the subset of (the de…nition of is given in De…nition
2), denotes the set of indices on which the function de…ned by (3.1) remains bounded for any …xed (0 < < min f 0; 1g) as tends to 0 on this set:
Proof. We prove the theorem for the case 1 < p < 1: The proof for the case p = 1 is similar. Let jI (x0)j = T[m](f ; x0) f (x0) and be a number such that
0 < < min f 0; 1g :
In view of (c) ; we may write
jI (x0)j = Z R K t; m X k=1 ( 1)k 1 m k f (x0+ kt) ! dt
+ Z R K t; m X k=1 ( 1)k 1 m k f (x0) ' (x0) '(x0+ kt) ! dt Z R K t; m X k=1 ( 1)k 1 m k f (x0) ' (x0) '(x0+ kt) ! dt f (x0) :
From above equality, and using (b) ; we may easily get jI (x0)j Z R m X k=1 ( 1)k 1 m k f (x0+ kt) '(x0+ kt) f (x0) ' (x0) '(x0+ kt) L (t) dt + Z R K t; m X k=1 ( 1)k 1 m k f (x0) ' (x0) '(x0+ kt) ! dt f (x0) :
Since whenever y; z being positive numbers the inequality (y + z)p 2p(yp+ zp)
holds (see, e.g., [21]), we have
jI (x0)jp 2p 0 @Z R m X k=1 ( 1)k 1 m k f (x0+ kt) '(x0+ kt) f (x0) ' (x0) '(x0+ kt) L (t) dt 1 A p +2p Z R K t; m X k=1 ( 1)k 1 m k f (x0) ' (x0) '(x0+ kt) ! dt f (x0) p = 2pI1+ 2pI2:
We can write the integral I1 as follows:
I1= 0 B @ 8 > < > : Z jtj> + Z 9>= > ; m X k=1 ( 1)k 1 m k f (x0+ kt) '(x0+ kt) f (x0) ' (x0) '(x0+ kt) L (t) dt 1 C A p :
Let 1p +1q = 1: Now, we apply Hölder’s inequality (see [21]) to the integral I1 as
follows: I1 M p q 0 8 > < > : Z jtj> + Z 9>= > ; m X k=1 ( 1)k 1 m k f (x0+ kt) '(x0+ kt) f (x0) ' (x0) '(x0+ kt) p L (t) dt = M p q 0 (I11+ I12) :
Let us show that I11! 0 as ! 0: The following inequality holds for I11: I11= Z jtj> m X k=1 ( 1)k 1 m k f (x0+ kt) '(x0+ kt) f (x0) ' (x0) '(x0+ kt) p L (t) dt 2p 0 B @ Z jtj> m X k=1 m k f (x0+ kt) '(x0+ kt) '(x0+ kt) 1 C A p L (t) dt +2p f (x0) ' (x0) p 0 B @ Z jtj> m X k=1 m k '(x0+ kt) 1 C A p L (t) dt: Applying (2.2) to the above integral, we have
I11 2p'p(x0) Z jtj> m X k=1 m k f (x0+ kt) '(x0+ kt) 'k(t) !p L (t) dt +2p f (x0) ' (x0) p 'p(x0) Z jtj> m X k=1 m k ' k(t) !p L (t) dt = 2p'p(x0)I111+ 2p f (x0) ' (x0) p 'p(x0)I112:
Expanding the summation inside the integral I111and using the inequality (y+z)p
2p(yp+ zp), we have I111 = Z jtj> m X k=1 m k f (x0+ kt) '(x0+ kt) 'k(t) !p L (t) dt 2mp m X k=1 m k p sup jtj> 'kp(t)L (t) kfkpL p;'(R): Similarly, we obtain I112 = Z jtj> m X k=1 m k ' k(t) !p L (t) dt 2mp Z jtj> m X k=1 m k p 'kp(t) ! L (t) dt:
Since m is …nite and using conditions (d) and (e) ; I111! 0 and I112! 0 as ! 0;
Now, we focus on I12: We have to show that I12 ! 0 as ! 0: Since ' is
bounded, 'k is bounded for each k: Set
D = max k supt ' k(t) m k=1 ; 0 < < min f 0; 1g :
Therefore, the following inequality holds: I12 'p(x0)Dp Z Xm k=0 ( 1)m k m k f (x0+ kt) '(x0+ kt) p L (t) dt = 'p(x0)Dp 8 < : 0 Z + Z 0 9 = ; m X k=0 ( 1)m k m k f (x0+ kt) '(x0+ kt) p L (t) dt = 'p(x0)DpfI121+ I122g :
For I121; let us de…ne
F (t) := 0 Z t m X k=0 ( 1)m k m k f (x0+ kv) '(x0+ kv) p dv:
According to De…nition 1, for every " > 0, there exists 2 > 0 such that the
inequality
F (t) " ( t) (3.2)
holds for every 2 satisfying 0 < 2 < min f 0; 1g : Using integration by parts
twice and (3.2), we have (for the similar situation, see [15, 17, 22]) jI121j " 0 Z 0( t) L (t) dt: Similarly, jI122j " Z 0 0(t) L (t) dt:
Combining above results, we have jI12j "Dp'p(x0)
Z
0(jtj) L (t) dt:
The remaining part follows from the arbitrariness of " and boundedness of Z
0(jtj) L (t) dt
4. Fatou Type Convergence
Throughout years many approximation theory researchers including Siudut [24], Carlsson [7] and Karsli [15] investigated the pointwise convergence of the linear and nonlinear singular integral operators by using Fatou type convergence investigation method, i.e., restriction of the pointwise convergence to some subsets of the plane. Because the sensitive analysis is obtained via this method. For further reading concerning this method, we refer the reader to [10, 14, 22, 27, 28]. Although the expression Fatou type convergence is not directly mentioned in some works (e.g., [22]), they are evaluated in this concept by some contemporary researchers.
In this section we will prove the Fatou type weighted pointwise convergence of the operators of type (1.4), i.e., the convergence will be restricted to a bounded planar subsets of R : For this purpose, we suppose that for every " > 0; there exists > 0 such that the function given as
(x; ) = m X k=1 m k Z jf(x + kt) f (x0+ kt)j L (t) dt;
where 0 < < min f 0; 1g ; is bounded on the set de…ned as
ZC; ;m= f(x; ) 2 R 1: ;m(x; ) < "Cg ;
where C is positive constant, as (x; ) tends to (x0; 0) :
Theorem 3. Suppose that the hypotheses of Theorem 2 hold. If x0 2 R is a
common m p Lebesgue point of the functions f 2 Lp;'(R) (1 p < 1) and
'; then
lim
(x; )!(x0; 0)
T[m](f ; x) f (x0) = 0
provided that (x; ) 2 ZC; ;m:
Proof. We prove the theorem for the case 1 < p < 1: The proof for the case p = 1 is similar. Let 0 < jx0 xj < 2 for a given 0 < < min f 0; 1g :
Now, set I (x) = T[m](f ; x) f (x0) : Let us write jI (x)j = Z R K t; m X k=1 ( 1)k 1 m k f (x + kt) ! dt f (x0) = Z R K t; m X k=1 ( 1)k 1 m k f (x + kt) ! dt Z R K t; m X k=1 ( 1)k 1 m k f (x0+ kt) ! dt + Z R K t; m X k=1 ( 1)k 1 m k f (x0+ kt) ! dt f (x0) :
From above equality, we deduce that
jI (x)jp 2p 0 @ Z R m X k=1 m k (f (x + kt) f (x0+ kt)) L (t) dt 1 A p +2p Z R K t; m X k=1 ( 1)k 1 m k f (x0+ kt) ! dt f (x0) p = 2p(I1+ I2) :
The following inequality holds for I1:
I1 = 0 B @ 8 > < > : Z jtj> + Z 9>= > ; m X k=1 m k jf(x + kt) f (x0+ kt)j L (t) dt 1 C A p 2p 0 B @ Z jtj> m X k=1 m k jf(x + kt) f (x0+ kt)j L (t) dt 1 C A p +2p 0 @Z m X k=1 m k jf(x + kt) f (x0+ kt)j L (t) dt 1 A p = 2p(I11+ I12) :
It is easy to see that I11 2p 0 B @'(x) Z jtj> m X k=1 m k f (x + kt) '(x + kt) ' k(t)L (t) dt 1 C A p +2p 0 B @'(x0) Z jtj> m X k=1 m k f (x0+ kt) '(x0+ kt) 'k(t)L (t) dt 1 C A p :
Following same strategy as in Theorem 2, we have I11 'p(x)2(m+1)p(M0) p q m X k=1 m k p sup jtj> 'kp(t)L (t) ! kfkpLp;'(R) +'p(x0)2(m+1)p(M0) p q m X k=1 m k p sup jtj> 'kp(t)L (t) ! kfkpLp;'(R):
Using the same method as in the proof of Theorem 2 and condition (d) ; we see that I11! 0 as ! 0: Clearly, by Theorem 2; I2! 0 as tends to 0: The result
follows from the hypothesis on the integral I12.
Thus the proof is completed.
5. Rate of Convergence
Theorem 4. Suppose that the hypotheses of Theorem 2 are satis…ed. Let
( ; ) = Z
L (t) 0(jtj) dt;
where 0 < < min f 0; 1g ; and the following conditions are satis…ed:
(i) ( ; ) ! 0 as ! 0 for some > 0:
(ii) For every > 0 and k = 1; :::; m; we have sup
jtj>
'kp(t)L (t) = o( ( ; ))
as ! 0:
(iii) For every > 0 and k = 1; :::; m; we have Z
jtj>
(iv) Letting ! 0; we have Z R K t; m X k=1 ( 1)k 1 m k f (x0) ' (x0) '(x0+ kt) ! dt f (x0) p = o( ( ; )): Then, at each common m p Lebesgue point of f 2 Lp;'(R) (1 p < 1)
and '; we have
T[m](f ; x0) f (x0) p
= o( ( ; )); as ! 0:
Proof. We prove the theorem for the case 1 < p < 1: The proof for the case p = 1 is similar. By the hypotheses of Theorem 2, we have
T[m](f ; x0) f (x0) p "2p'p(x0) (M0) p qDp Z 0(jtj) L (t) dt +2(m+2)p'p(x0) (M0) p q m X k=1 m k p sup jtj> 'kp(t)L (t) kfkpL p;'(R) +2(m+2)p'p(x0) (M0) p q f (x0) '(x0) p Z jtj> m X k=1 m k p 'kp(t) ! L (t) dt +2p Z R K t; m X k=1 ( 1)k 1 m k f (x0) ' (x0) '(x0+ kt) ! dt f (x0) p : The proof is completed by (i) (iv) :
Theorem 5. Suppose that the hypotheses of Theorem 3 are satis…ed. Let (x; ) = m X k=1 m k Z jf(x + kt) f (x0+ kt)j L (t) dt;
where 0 < < min f 0; 1g ; and the following conditions are satis…ed:
(i) (x; ) ! 0 as (x; ) ! (x0; 0) for some > 0:
(ii) For every > 0 and k = 1; :::; m; we have sup jtj> 'kp(t)L (t) = o ( (x; )) ; as ! 0: (iii) Letting (x; ) ! (x0; 0); Z R K t; m X k=1 ( 1)k 1 m k f (x0+ t) ! dt f (x0) p = o ( (x; )) :
Then, at each common m p Lebesgue point of f 2 Lp;'(R) (1 p < 1)
and '; we have T[m](f ; x) f (x0)
p
= o ( (x; )) ; as (x; ) ! (x0; 0):
Proof. We prove the theorem for the case 1 < p < 1: The proof for the case p = 1 is similar. Under the hypotheses of Theorem 3, we may write
T[m](f ; x) f (x0) p 'p(x)2(m+3)p(M0) p q m X k=1 m k p sup jtj> 'kp(t)L (t) ! kfkpLp;'(R) +'p(x0)2(m+3)p(M0) p q m X k=1 m k p sup jtj> 'kp(t)L (t) ! kfkpLp;'(R) +22p 0 @ m X k=1 m k Z jf(x + kt) f (x0+ kt)j L (t) dt 1 A p +2p Z R K t; m X k=1 ( 1)k 1 m k f (x0+ kt) ! dt f (x0) p : From conditions (i) (iii) ; the proof is completed.
6. Acknowledgements
The author presents her thanks and gratitudes to the unknown referees of the manuscript for their valuable comments and suggestions.
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