- Open Access
A note on rate of convergence of double singular integral operators
© Yilmaz et al.; licensee Springer. 2014
Received: 22 July 2014
Accepted: 27 October 2014
Published: 4 November 2014
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: , , where ( is an arbitrary closed, semi-closed or open region in ) and , Λ is a set of non-negative numbers with accumulation point . Also we provide an example to support these theoretical results.
where is the kernel satisfying suitable assumptions and (where Λ is a given set of non-negative numbers with accumulation point ).
Based on Taberski’s study , Gadjiev  investigated both the pointwise convergence theorems and the order of pointwise convergence theorems for operators type (1) at a generalized Lebesgue point. Then Rydzewska  conducted a similar study by changing the point to a μ-generalized Lebesgue point of instead of a generalized Lebesgue point.
for functions in where is an arbitrary interval in ℝ such as , , or .
In [6, 7], Karsli obtained the pointwise convergence theorems and the rate of pointwise convergence theorems for a family of nonlinear singular integral operators at a μ-generalized Lebesgue point and a generalized Lebesgue point of , respectively.
In 1984, Bardaro and Gori Cocchieri  estimated the degree of pointwise convergence of Fejer-Type singular integrals at the generalized Lebesgue points of the functions . Also, Bardaro  studied similar convergence results concerning moment type operators.
In , Bardaro and Mantellini investigated the pointwise convergence of family of nonlinear Mellin type convolution operators at Lebesgue points.
In paper , Bardaro et al. obtained some approximation results concerning the pointwise convergence and the rate of pointwise 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 nonlinear m-singular integral operators at m-Lebesgue points of f.
where Q denotes a given rectangle. Based on Taberski’s study , Siudut [15, 16] obtained significant results relating the pointwise convergence of singular integrals by considering the operators of type (3).
to the function in the case in , where is a closed, semi-closed or open region in and is a generalized Lebesgue point of the function . In this study, the kernel function is chosen as a radial function.
Very recently,  Serenbay et al. investigated the pointwise convergence of the operator of type (4) at a μ-generalized Lebesgue point.
where ( is an arbitrary closed, semi-closed or open region in ), at a p-generalized Lebesgue point of as . Here is the collection of all measurable functions f for which is integrable on D (), Λ is a set of non-negative numbers with accumulation point and the kernel function is a radial function.
The paper is organized as follows: In Section 2, we introduce the fundamental definitions. In Section 3, we prove the existence of the operator of type (5). In Section 4, we obtain two theorems concerning the pointwise convergence of to whenever is a p-generalized Lebesgue point of f in bounded region and unbounded region. In Section 5, we establish the rate of convergence of operators of type (5) to as tends to and we conclude the paper with an example to support our results.
In this section we introduce the main definitions used in this paper.
If (; ) for any partition of P, then it is said that satisfies the condition Ω in D .
In other words, if for all partitions of D then it is said that is bimonotonically increasing and if for all partitions of D, then it is said that is bimonotonically decreasing .
Definition 2.3 A function , is said to be radial if there exists a function such that a.e. .
the function is a radial function.
Definition 2.4 (Class A)
where Λ is a given set of non-negative numbers with accumulation point .
is non-negative and integrable as a function of on for each fixed .
For fixed , tends to infinity as λ tends to .
is non-increasing with respect to t on and non-decreasing on and similarly is non-increasing with respect to s on and non-decreasing on , for any and for fixed .
Throughout this paper we suppose that the kernel belongs to class A.
3 Existence of the operator
Lemma 3.1 Let . If , then the operator defines a continuous transformation over .
Proof The proof of the case is quite similar to the proof in .
Thus the proof is completed. □
4 Pointwise convergence
The following theorem gives a pointwise approximation of the integral operators type (5) to the function f at p-generalized Lebesgue point of whenever D is an arbitrary region in that is bounded, closed, semi-closed or open.
where is the Lebesgue-Stieltjes measure with respect to , is bounded as tends to .
Proof Suppose that , , for all which satisfy and , for all which satisfy and .
By conditions (e) and (d) of class A, as .
hence by condition (d) of class A, whenever .
it is sufficient to show that the terms on the right hand side of the last inequality tends to zero as on Z.
Let us consider first the integral .
for all .
Hence we can evaluate the integral .
where denotes the Stieltjes integral.
Thus, the proof is finished. □
Remark 4.1 For the case , the proof is quite similar to the proof in .
The following theorem gives a pointwise approximation of the integral operators type (5) to the function f at p-generalized Lebesgue point of whenever .
By condition (e) of class A the second term of the expression tends to zero as λ tends to . The remaining part of the proof is obvious by Theorem 4.1. □
5 Rate of convergence
In this section, we give a theorem concerning the rate of pointwise convergence.
Theorem 5.1 Suppose that the hypotheses of Theorem 4.1 and Theorem 4.2 are satisfied.
as for some .
- (ii)For every
- (iii)For every
To verify that satisfies the hypotheses of Theorem 4.1 and Theorem 4.2, see . Since tends to infinity as λ tends to zero at , condition (b) of class A is satisfied.
The authors thank the referees for their valuable comments and suggestions for the improvement of the manuscript.
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