R E S E A R C H
Open Access
A note on rate of convergence of double
singular integral operators
Mine Menekse Yilmaz
1*, Gumrah Uysal
2and Ertan Ibikli
3*Correspondence: [email protected] 1Department of Mathematics, Faculty of Arts and Science, Gaziantep University, Gaziantep, Turkey
Full list of author information is available at the end of the article
Abstract
In this paper we prove the pointwise convergence and the rate of pointwise convergence for a family of singular integral operators with radial kernel in two-dimensional setting in the following form:Lλ(f;x,y) =
Df(s,t)Hλ(s–x,t–y)ds dt, (x,y)∈D, whereD=a,b × c,d(a,b × c,dis an arbitrary closed, semi-closed or open region inR2) and
λ
∈,
is a set of non-negative numbers with accumulation point
λ
0. Also we provide an example to support these theoretical results.MSC: Primary 41A35; secondary 41A25
Keywords: p-generalized Lebesgue point; radial kernel; pointwise convergence; rate of convergence
1 Introduction
Taberski [] studied the pointwise convergence of integrable functions and the approxi-mation properties of derivatives of integrable functions inL(–π,π) by a family of convo-lution type singular integral operators depending on two parameters of the form:
Uλ(f;x) =
π
–π
f(t)Kλ(t–x)dt, x∈(–π,π), ()
whereKλ(t) is the kernel satisfying suitable assumptions andλ∈(whereis a given set of non-negative numbers with accumulation pointλ).
Based on Taberski’s study [], Gadjiev [] investigated both the pointwise convergence theorems and the order of pointwise convergence theorems for operators type () at a gen-eralized Lebesgue point. Then Rydzewska [] conducted a similar study by changing the point to aμ-generalized Lebesgue point off ∈L(–π,π) instead of a generalized Lebesgue point.
Further, in [, ], Karsli and Ibikli extended the results of [, ], and [] by considering the more general integral operators defined by
Tλ(f;x) =
b
a
f(t)Kλ(t–x)dt, x∈ a,b,λ∈⊂R ()
for functions inLa,bwherea,bis an arbitrary interval inRsuch as [a,b], (a,b), [a,b) or (a,b].
In [, ], Karsli obtained the pointwise convergence theorems and the rate of pointwise convergence theorems for a family of nonlinear singular integral operators at aμ -gener-alized Lebesgue point and a gener-gener-alized Lebesgue point off ∈La,b, respectively.
In , Bardaro and Gori Cocchieri [] estimated the degree of pointwise conver-gence of Fejer-Type singular integrals at the generalized Lebesgue points of the functions
f ∈L(R). Also, Bardaro [] studied similar convergence results concerning moment type operators.
In [], Bardaro and Mantellini investigated the pointwise convergence of family of non-linear Mellin type convolution operators at Lebesgue points.
In paper [], Bardaroet al.obtained some approximation results concerning the point-wise convergence and the rate of pointpoint-wise convergence for non-convolution type linear operators at a Lebesgue point. In [], the same authors also obtained similar results for its nonlinear counterpart and then in [], they explored the pointwise convergence and the rate of pointwise convergence results for a family of Mellin type nonlinearm-singular integral operators atm-Lebesgue points off.
In this study, we also investigated the pointwise convergence and the rate of conver-gence of the operators similar to the studies above. However, this study considers the two-dimensional singular integral instead of one-two-dimensional integral similar to what Taberski [] investigated in . In [], Taberski explored the pointwise convergence of inte-grable functions inL(Q) by a three-parameter family of convolution type singular integral operators of the form
Uλ(f;x,y) =
Q
f(s,t)Kλ(s–x,t–y)ds dt, (x,y)∈Q, ()
whereQdenotes a given rectangle. Based on Taberski’s study [], Siudut [, ] obtained significant results relating the pointwise convergence of singular integrals by considering the operators of type ().
In [] Yilmazet al.studied the pointwise convergence of the singular integral operators in the following form:
Lλ(f;x,y) =
π
–π
π
–π
f(s,t)Hλ(s–x,t–y)ds dt, (x,y)∈ –π,π × –π,π ()
to the functionf(x,y) in the case (x,y,λ)→(x,y,λ) inL(–π,π × –π,π), where
–π,π × –π,πis a closed, semi-closed or open region inRand (x
,y) is a generalized Lebesgue point of the functionf∈L(–π,π × –π,π). In this study, the kernel function
Hλ(s,t) is chosen as a radial function.
Very recently, [] Serenbayet al.investigated the pointwise convergence of the operator of type () at aμ-generalized Lebesgue point.
The purpose of this paper is to investigate the pointwise convergence and the rate of convergence of the operators in the following form:
Lλ(f;x,y) =
D
f(s,t)Hλ(s–x,t–y)ds dt, (x,y)∈D, ()
whereD=a,b × c,d(a,b × c,dis an arbitrary closed, semi-closed or open region inR), at ap-generalized Lebesgue point off ∈L
is the collection of all measurable functionsffor which|f|pis integrable onD(≤p<∞), is a set of non-negative numbers with accumulation pointλand the kernel function
Hλ(s,t) is a radial function.
The paper is organized as follows: In Section , we introduce the fundamental defini-tions. In Section , we prove the existence of the operator of type (). In Section , we obtain two theorems concerning the pointwise convergence ofLλ(f;x,y) tof(x,y) when-ever (x,y) is ap-generalized Lebesgue point off in bounded region and unbounded re-gion. In Section , we establish the rate of convergence of operators of type () tof(x,y) as (x,y,λ) tends to (x,y,λ) and we conclude the paper with an example to support our results.
2 Preliminaries
In this section we introduce the main definitions used in this paper.
Definition . Letϕ(x,y) be a function defined in the rectangleD, let
P=
a=x<x<· · ·<xi<· · ·<xm<xm+=b,
c=y<y<· · ·<yj<· · ·<yn<yn+=d be a partition ofDand
ϕ(xi,yj) =ϕ(xi,yj) –ϕ(xi+,yj) –ϕ(xi,yj+) +ϕ(xi+,yj+).
Ifϕ(xi,yj)≥ (i= , , . . . ,m;j= , , . . . ,n) for any partition ofP, then it is said thatϕ(x,y)
satisfies the conditioninD[].
In other words, ifϕ(xi,yj)≥ for all partitions ofDthen it is said thatϕ(x,y) is
bi-monotonically increasing and ifϕ(xi,yj)≤ for all partitions ofD, then it is said that ϕ(x,y) is bimonotonically decreasing [].
Definition . A point (x,y)∈Dis called ap-generalized Lebesgue point of function
f ∈Lp(D) if
lim
(h,k)→(,)
hα+kα+
h
k
f(s+x,t+y) –f(x,y)pds dt
p =
(≤α< ; ≤p<∞).
Definition . A functionH∈L(R), is said to be radial if there exists a functionK:
R+
→Rsuch thatH(s,t) =K(
√
s+t) a.e. [].
Example . LetH:R→Ris given byH(s,t) =e–(s+t)and the corresponding function
K:R+
→RisK(z) =e–z
. Since the following equality holds for all (s,t)∈R:
e–(s+t)=e–(
√ s+t)
,
Definition .(Class A) LetHλ:R×→Rbe a radial functioni.e., there exists a functionKλ:R+×→Rsuch that the following equality holds for (s,t)∈Ra.e.
Hλ(s,t) :=Kλ
√
s+t ,
whereis a given set of non-negative numbers with accumulation pointλ.
We will say that the functionHλ(s,t) belongs to classA, if the following conditions are satisfied:
(a) Hλ(s,t) =Kλ(√s+t)is non-negative and integrable as a function of(s,t)onRfor each fixedλ∈.
(b) For fixed(x,y)∈D,Kλ(
x
+y)tends to infinity asλtends toλ. (c) lim(x,y,λ)→(x,y,λ)
RKλ(
(s–x)+ (t–y))ds dt= . (d) limλ→λ[supξ≤√s+tKλ(
√
s+t)] = ,∀ξ> . (e) limλ→λ[
ξ≤√s+tKλ(
√
s+t)ds dt] = ,∀ξ> .
(f ) Kλ((s–x)+ (t–y))is non-increasing with respect totonx
,dand non-decreasing onc,xand similarlyKλ(
(s–x)+ (t–y))is non-increasing with respect tosony,band non-decreasing ona,y, for anyλ∈and for fixed(x,y)∈D.
(g) KλL(R)≤M,∀λ∈.
Throughout this paper we suppose that the kernelHλ(s,t) belongs to classA.
3 Existence of the operator
Lemma . Let≤p<∞.If f ∈Lp(D),then the operator Lλ(f;x,y)defines a continuous
transformation over Lp(D).
Proof The proof of the casep= is quite similar to the proof in [].
We assume that <p<∞. By the linearity of the operatorLλ(f;x,y), it is sufficient to show that the following expression is bounded:
Lλ=sup
f=
Lλ(f,x,y)Lp(D)
fLp(D) .
We define a function such that
g(s,t) =
f(s,t), (s,t)∈D, , (s,t)∈R\D,
and then we rearrange and rewrite the norm as follows:
Lλ(f,x,y)L
p(D)=Lλ(g,x,y)Lp(D) =
D R
g(s,t)Hλ(s–x,t–y)ds dt
p
dx dy
p
=
D
Rg(s,t)Kλ
(s–x)+ (t–y) ds dt
p
dx dy
p
=
D R
g(s+x,t+y)Kλ
√
s+t ds dt
p
dx dy
By using the generalized Minkowsky inequality and by condition (g) of classA, we have
Thus the proof is completed.
4 Pointwise convergence
The following theorem gives a pointwise approximation of the integral operators type () to the functionf atp-generalized Lebesgue point off ∈Lp(D) wheneverDis an arbitrary
region inRthat is bounded, closed, semi-closed or open.
Theorem . Suppose that, as a function of (s,t), K((t–x)+ (s–y);λ)≥ on
on any set Z on which the function
SetIλ(x,y) :=|Lλ(f;x,y) –f(x,y)|. According to condition (c) of classA, we shall write
Using Hölder’s inequality for the termIwe have the following:
I+II≤
holds, by taking thepth power of both sides we have
The second part of the integral I∗ tends to one as (x,y,λ) tends to (x,y,λ) by
condi-For the integralIwe have the following inequality:
I=
δ, the following relation holds:
I≤pKλ
it is sufficient to show that the terms on the right hand side of the last inequality tends to zero as (x,y,λ)→(x,y,λ) onZ.
Let us consider first the integralI.
Let us define a new function as
Hence we can evaluate the integralI.
From [] (see Theorem ., p.), we can write the following:
I=
where (S) denotes the Stieltjes integral.
Two-dimensional integration by parts (see Theorem ., p. in []) gives us
x
After applying integration by parts toi,i, andiwe obtain the following result:
For the integralsI,I, andIthe proof is similar to the above one. Thus we obtain the following inequalities:
I≤–εp
x+δ
x
y
y–δ
Kλ
(s–x)+ (t–y) d(x
–s)p(α+)(y–t)p(α+)
,
I≤–εp
x
x–δ
y+δ
y
Kλ
(s–x)+ (t–y) d(s–x
)p(α+)(y–t)p(α+)
,
I≤εp
x+δ
x
y+δ
y
Kλ
(s–x)+ (t–y) d(x
–s)p(α+)(y–t)p(α+)
.
Collecting the estimatesI,I,I, andI, we have the following inequality:
I≤I+I+I+I =εp
x+δ
x–δ
y+δ
y–δ
Kλ
(s–x)+ (t–y) d(x
–s)p(α+)(y–t)p(α+). Therefore, if the points (x,y,λ)∈Zare sufficiently close to (x,y,λ), we have
I≤εC, where
C=sup x+δ
x–δ
y+δ
y–δ
Kλ
(s–x)+ (t–y) d(x
–s)p(α+)(y–t)p(α+): (x,y,λ)∈Z
.
Finally we consider the integralII∗. From condition (c), it is clear that
lim
(x,y,λ)→(x,y,λ)
II∗= .
Thus, the proof is finished.
Remark . For the casep= , the proof is quite similar to the proof in [].
The following theorem gives a pointwise approximation of the integral operators type () to the functionf atp-generalized Lebesgue point off ∈Lp(R) wheneverD=R.
Theorem . Suppose that the hypothesis of Theorem.is satisfied for D=R.If(x ,y)
is a p-generalized Lebesgue point of function f ∈Lp(R)then lim
(x,y,λ)→(x,y,λ)
Lλ(f;x,y) =f(x,y).
Proof Using the same strategy as in Theorem . we obtain
Lλ(f;x,y) –f(x,y)p
≤p
sup
δ≤√s+t
Kλ
√
t+s fp
+f(x,y)p
δ≤√s+tKλ
√
s+t ds dt
×
R
Kλ
(s–x)+ (t–y) ds dt
P q
+ pεp x+δ
x–δ
y+δ
y–δ
Kλ
(s–x)+ (t–y) d(x
–s)p(α+)(y–t)p(α+)
×
RKλ
(s–x)+ (t–y) ds dt
P q
+ pf(x,y)p
RKλ
(s–x)+ (t–y) ds dt–
p
.
By condition (e) of classAthe second term of the expression tends to zero asλtends toλ.
The remaining part of the proof is obvious by Theorem ..
5 Rate of convergence
In this section, we give a theorem concerning the rate of pointwise convergence.
Theorem . Suppose that the hypotheses of Theorem.and Theorem.are satisfied.
Let
(λ,δ,x,y) =
x+δ
x–δ
y+δ
y–δ
|x–s|p(α+)–|y–t|p(α+)–Kλ
(s–x)+ (t–y) ds dt
forδ> and the following assumptions are satisfied:
(i) (λ,δ,x,y)→as(x,y,λ)→(x,y,λ)for someδ> . (ii) For everyξ>
Kλ(ξ) =o(λ,δ,x,y)
as(x,y,λ)→(x,y,λ). (iii) For everyξ>
ξ≤√s+tKλ
√
s+t ds dt=o(λ,δ,x,y)
as(x,y,λ)→(x,y,λ).
Then at each p-generalized Lebesgue point of f ∈Lp(R)we have as(x,y,λ)→(x,y,λ)
Lλ(f;x,y) –f(x,y)=o
(λ,δ,x,y)p .
Proof In view of the equality of the Lebesgue-Stieltjes integrals we have
x+δ
x–δ
y+δ
y–δ
Kλ
(s–x)+ (t–y) d(x
–s)p(α+)(y–t)p(α+)
= (pα+p)
x+δ
x–δ
y+δ
y–δ
(x–s)p(α+)–(y–t)p(α+)–Kλ
By Theorem . and Theorem . we can write, forδ> ,
From (i)-(iii) and using classAconditions we have the desired result:
Lλ(f;x,y) –f(x,y)=o
λ satisfies the hypotheses of Theorem . and
Theorem ., see []. SinceKλ(√s+t) tends to infinity asλtends to zero at (s,t) = (, ), condition (b) of classAis satisfied.
In order to find for whichδ> condition (i) in Theorem . is satisfied, let(λ,δ,x,y)→ as (x,y,λ)→(, , ). Hence
lim
(x,y,λ)→(,,)(λ,δ,x,y) =
if and only ifδ=o(λ). Consequently the following equation holds:
(λ,δ,x,y) =
+δ
–δ
+δ
–δ
|s||t|
π λe
–((s–x)+(t–y)) λ ds dt
=
[,λ]×[,λ]|
s||t|
π λe
–((s–x)+(t–y)) λ ds dt.
From the above equation we see that
(λ,δ,x,y) =O(λ).
From [] it is clear that the conditions (ii) and (iii) in Theorem . are satisfied. Hence
Lλ(f;x,y) –f(x,y)=o
(λ,δ,x,y) =oλ .
Competing interests
The authors declare that they have no competing interests. Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details
1Department of Mathematics, Faculty of Arts and Science, Gaziantep University, Gaziantep, Turkey.2Department of Mathematics, Faculty of Science, Karabuk University, Karabuk, Turkey.3Department of Mathematics, Faculty of Science, Ankara University, Tandogan, Ankara, Turkey.
Acknowledgements
The authors thank the referees for their valuable comments and suggestions for the improvement of the manuscript.
Received: 22 July 2014 Accepted: 27 October 2014 Published:04 Nov 2014
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