On the approximation by convolution type double singular integral operators

TL;DR

This paper establishes pointwise convergence and rates for a family of two-dimensional singular integral operators with flexible kernel functions, expanding the theoretical understanding of their approximation properties without restrictive kernel conditions.

Contribution

It proves convergence and rate results for convolution-type double singular integral operators with general kernels, not requiring evenness, positivity, or periodicity, which broadens previous theoretical frameworks.

Findings

Proved pointwise convergence of the operators.

Established rates of convergence under general conditions.

Provided an example illustrating the theoretical results.

Abstract

In this paper, we prove the pointwise convergence and the rate of pointwise convergence for a family of singular integral operators in two-dimensional setting in the following form: \begin{equation*} L_{\lambda }\left( f;x,y\right) =\underset{D}{\iint }f\left( t,s\right) K_{\lambda }\left( t-x,s-y\right) dsdt,\text{ }\left( x,y\right) \in D, \end{equation*} where $D=\left \langle a,b\right \rangle \times \left \langle c,d\right \rangle $ is an arbitrary closed, semi-closed or open rectangle in $\mathbb{R}^{2}$ and $% \lambda \in \Lambda ,$ $\Lambda $ is a set of non-negative indices with accumulation point $\lambda_{0}$. Also, we provide an example to support these theoretical results. In contrast to previous works, the kernel function $K_{\lambda }\left( t,s\right) $ does not have to be even, positive or 2$\pi -$periodic.

Equations

$$L_{\lambda }\left(f;x,y\right)=\underset{D}{\iint }f\left(t,s\right)K_{\lambda }\left(t-x,s-y\right)dsdt,\left(x,y\right)\in D,$$

$$L_{\lambda }\left(f;x\right)=\underset{-\pi }{\stackrel{\pi }{\int }}f\left(t\right)K_{\lambda }\left(t-x\right)dt,x\in \left(-\pi ,\pi \right),\lambda \in \Lambda ,$$

$$L_{\lambda }\left(f;x,y\right)=\underset{Q}{\iint }f\left(t,s\right)K_{\lambda }\left(t-x,s-y\right)dsdt,\left(x,y\right)\in Q,$$

$$L_{\lambda }\left(f;x,y\right)=\underset{D}{\iint }f\left(t,s\right)K_{\lambda }\left(t-x,s-y\right)dsdt,\left(x,y\right)\in D$$

$$\underset{\left(h,k\right)\to \left(0,0\right)}{\lim }\frac{1}{{\mu }_1(h){\mu }_2(k)}\stackrel{h}{\underset{0}{\int }}\stackrel{k}{\underset{0}{\int }}\left|f\left(t+x_0,s+y_0\right)-f\left(x_0,y_0\right)\right|dsdt=0,$$

$$V(g;[{\alpha }_1,{\alpha }_2]\times [{\beta }_1,{\beta }_2])$$

$${\Vert L_{\lambda }\Vert }_1=\underset{f\ne 0}{\sup }\frac{{\Vert L_{\lambda }(f;x,y)\Vert }_{L_1(D)}}{{\Vert f\Vert }_{L_1(D)}}$$

$$g(t,s)=\left\{\begin{array}[]{ccc}f(t,s),&if&(t,s)\in D,\\ 0,&if&(t,s)\in\mathbb{R}^{2}\backslash D.\end{array}\right.$$

$${\Vert L_{\lambda }(f;x,y)\Vert }_{L_1(D)}$$

$$\underset{\left(x,y,\lambda \right)\to \left(x_0,y_0,{\lambda }_0\right)}{\lim }{L}_{\lambda }\left(f;x,y\right)=f\left(x_0,y_0\right)$$

$$\stackrel{x_0+\delta }{\underset{x_0-\delta }{\int }}\stackrel{y_0+\delta }{\underset{y_0-\delta }{\int }}\left|K_{\lambda }(t-x,s-y)\right|{\rho }_1(\left|t-x_0\right|){\rho }_2(\left|s-y_0\right|)dsdt,$$

$$\left|K_{\lambda }\left(0,0\right)\right|{\mu }_1(\left|x-x_0\right|)\text{ and }\left|K_{\lambda }\left(0,0\right)\right|{\mu }_2(\left|y-y_0\right|)$$

$$g\left(t,s\right)=\left\{\begin{array}[]{c}f\left(t,s\right),\text{ \ \ \ \ \ \ }\left(t,s\right)\in D\text{,}\\ 0,\text{ \ \ \ \ \ \ }\left(t,s\right)\in\mathbb{R}^{2}\backslash D.\end{array}\right.$$

$$\stackrel{x_0}{\underset{x_0}{\int }}\stackrel{+\delta y_0}{\underset{y_0-\delta }{\int }}\left|f\left(t,s\right)-f\left(x_0,y_0\right)\right|dsdt<\epsilon {\mu }_1(h){\mu }_2(k)$$

$$I_{\lambda }(x,y)$$

$$=$$

$$I_3$$

$$I_1=\left\{\underset{D\backslash D_{22}}{\iint }+\underset{D_{22}}{\iint }\right\}\left|f\left(t,s\right)-f\left(x_0,y_0\right)\right|\left|K_{\lambda }\left(t-x,s-y\right)\right|dsdt$$

$$I_{D_{12}}$$

$$I_{11}\le \underset{(t,s)\in D\backslash D_{22}}{\sup }\left|K_{\lambda }\left(t-x,s-y\right)\right|\left({\Vert f\Vert }_{L_1\left(D\right)}+\left|f\left(x_0,y_0\right)\right|\left|b-a\right|\left|d-c\right|\right).$$

$$I_{12}$$

$$A_1\left(u,v\right)$$

$$A_2\left(u\right)$$

$$A_3\left(v\right)$$

$$I_{121}$$

$$i_1+i_2+i_3+i_4$$

$$I_{121}$$

$$I_{122}$$

$$I_{123}$$

$$I_{124}$$

$$I_{12}$$

Full text

[TABLE]

where $D=\left\langle a,b\right\rangle\times\left\langle c,d\right\rangle$ is an arbitrary closed, semi-closed or open rectangle in $\mathbb{R}^{2}$ and $\lambda\in\Lambda,$ $\Lambda$ is a set of non-negative numbers with accumulation point $\lambda_{0}$. Also, we provide an example to support these theoretical results. In contrast to previous works, the kernel function $K_{\lambda}\left(t,s\right)$ does not have to be even, positive or 2$\pi-$periodic.

1. Introduction

Taberski [7] studied the pointwise convergence of integrable functions and the approximation properties of derivatives of integrable functions in $L_{1}\left(-\pi,\pi\right)$ by a family of convolution type singular integral operators depending on two parameters in the following form:

[TABLE]

where $K_{\lambda}\left(t\right)$ is the kernel satisfying suitable assumptions and $\Lambda$ is a given set of non-negative numbers with accumulation point $\lambda_{0}.$

After Taberski’s study [7], Gadjiev [3] and Rydzewska [9] gave some approximation theorems concerning pointwise convergence and the order of pointwise convergence for operators of type (1.1) at a generalized Lebesgue point and a $\mu-$generalized Lebesgue point of $f\in L_{1}\left(-\pi,\pi\right)$, respectively. Later on, Bardaro [1] estimated the degree of pointwise convergence of general type singular integrals at generalized Lebesgue points of the functions $f\in L_{1}\left(\mathbb{R}\right)$.

In [8], Taberski explored the pointwise convergence of integrable functions in $L_{1}\left(Q\right)$ by a three-parameter family of convolution type singular integral operators of the form:

[TABLE]

where $Q=\left\langle-\pi,\pi\right\rangle\times\left\langle-\pi,\pi\right\rangle$ is a closed, semi-closed or open rectangle. In this work, the kernel function $K_{\lambda}$ was non-negative, even and $2\pi-$periodic with respect to both variables, seperately. Then, Rydzewska [10] extended this work by obtaining the rate of pointwise convergence at a $\mu-$generalized $d-$point. Here, note that all other assumptions on the indicated operators were same with [8]. In particular, for further studies on the convergence of double singular integral operators, we address the reader to [11, 12, 13, 4].

In this study, we also investigated the pointwise convergence and the rate of convergence of the operators similar to the studies above. In contrast to previous works, the kernel function $K_{\lambda}\left(t,s\right)$ does not have to be even, non-negative and 2$\pi-$periodic with respect to each variable.

The main contribution of this paper is to investigate the rate of pointwise convergence of the convolution type singular integral operators in the following form:

[TABLE]

where $D=\left\langle a,b\right\rangle\times\left\langle c,d\right\rangle$ is an arbitrary closed, semi-closed or open bounded rectangle in $\mathbb{R}^{2},$ to $f\in L_{1}\left(D\right),$ at a $\mu$-generalized Lebesgue point as $\left(x,y,\lambda\right)\rightarrow\left(x_{0},y_{0},\lambda_{0}\right).$ Here, $L_{1}\left(D\right)$ is the collection of all measurable functions $f$ for which $\left|f\right|$ is integrable on $D,$ $\Lambda\subset$ $\overset{-}{\mathbb{R}}$ is a set of non-negative indices with accumulation point $\lambda_{0}$.

The paper is organized as follows: In Section 2, we introduce the fundamental definitions. In Section 3, we prove the existence of the operators of type (1.3). In Section 4, we present two theorems concerning the pointwise convergence of $L_{\lambda}\left(f;x,y\right)$ to $f\left(x_{0},y_{0}\right)$ whenever $\left(x_{0},y_{0}\right)$ is a $\mu$-generalized Lebesgue point of $f$ , for the cases $D$ is bounded rectangle and $D=\mathbb{R}^{2}$. In Section 5, we establih the rate of convergence of operators of type (1.3) to $f\left(x_{0},y_{0}\right)$ as $\left(x,y,\lambda\right)$ tends to $\left(x_{0},y_{0},\lambda_{0}\right)$ 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.

The following definition is obtained by modifying the $\mu-$generalized Lebesgue point definition given in [10].

Definition 1**.**

A point $\left(x_{0},y_{0}\right)\in D$ is called a $\mu-$generalized Lebesgue point of function $f\in L_{1}\left(D\right)$ if

[TABLE]

where $\mu_{1}(h)=\underset{0}{\overset{h}{\mathop{\textstyle\int}}}\rho_{1}(t)dt>0,$ $0<h<\delta_{0}<\min\{b-a,d-c\}$ and $\rho_{1}(t)$ is an integrable and non-negative function on $\left[0,\delta_{0}\right]$ and similarly, $\mu_{2}(k)=\underset{0}{\overset{k}{\mathop{\textstyle\int}}}\rho_{2}(s)ds>0,$ $0<k<\delta_{0}<\min\{b-a,d-c\}$ and $\rho_{2}(s)$ is an integrable and non-negative function on $\left[0,\delta_{0}\right].$

Now, we define the new set of kernel functions by using some of the kernel conditions presented in [11].

Definition 2**.**

$\left(Class\text{ }A\right)$** We will say that the function $K_{\lambda}\left(t,s\right)$ belongs to class $A$, if the following criterions are fulfilled:**

**a: **

$K_{\lambda}\left(t,s\right)$ is defined and integrable as a function of $(t,s)$ on $\mathbb{R}^{2}$ for each fixed $\lambda\in\Lambda$ and $\left\|\left|K_{\lambda}\right|\right\|_{L_{1}(\mathbb{R}^{2})}\leq M,$ $\forall\lambda\in\Lambda.$

**b: **

For fixed $(t_{0},s_{0})\in D,$ $K_{\lambda}\left(t_{0},s_{0}\right)$ tends to infinity as $\lambda$ tends to $\lambda_{0}.$

**c: **

$\underset{\left(x,y,\lambda\right)\rightarrow\left(x_{0},y_{0},\lambda_{0}\right)}{\lim}\left|\underset{\mathbb{R}^{2}}{\mathop{\displaystyle\iint}}K_{\lambda}\left(t-x,s-y\right)dsdt-1\right|=0.$

**d: **

$\underset{\lambda\rightarrow\lambda_{0}}{\lim}\left|K_{\lambda}\left(\gamma,0\right)\right|=\underset{\lambda\rightarrow\lambda_{0}}{\lim}\left|K_{\lambda}\left(0,\gamma\right)\right|=0,\ \ \forall\gamma>0.$

**e: **

$\underset{\lambda\rightarrow\lambda_{0}}{\lim}\underset{\mathbb{R}^{2}\backslash\left\langle-\gamma,\gamma\right\rangle\times\left\langle-\gamma,\gamma\right\rangle}{\mathop{\displaystyle\iint}}\left|K_{\lambda}\left(t,s\right)\right|dsdt=0,\ \ \forall\gamma>0.$

**f: **

There exist the numbers $\delta_{1}>0$ and $\delta_{2}>0$ such that $\left|K_{\lambda}\left(t,s\right)\right|$ is monotonically increasing with respect to $t$ on $(-\delta_{1},0]$ and monotonically decreasing on $[0,\delta_{1})$ and similarly  $\left|K_{\lambda}\left(t,s\right)\right|$is monotonically increasing with respect to $s$ on $(-\delta_{2},0]$ and monotonically decreasing on $[0,\delta_{2})$ for any $\lambda\in\Lambda$. Similarly, $\left|K_{\lambda}\left(t,s\right)\right|$ is bimonotonically increasing with respect to $(t,s)$ on $[0,\delta_{1})\times[0,\delta_{2})$ and $(-\delta_{1},0]\times(-\delta_{2},0]$ and bimonotonically decreasing with respect to $(t,s)$ on $[0,\delta_{1})\times(\delta_{2},0]$ and $(-\delta_{1},0]\times[0,\delta_{2})$ for any $\lambda\in\Lambda$.

Remark 1**.**

If the function $g:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is bimonotonic on $[\alpha_{1},\alpha_{2}]\times[\beta_{1},\beta_{2}]\subset\mathbb{R}^{2},$ then the equality given by

[TABLE]

holds [8, 5].

Throughout this paper, we suppose that the kernel $K_{\lambda}\left(t,s\right)$* belongs to class $A$* and $\delta_{0}\leq\min\left\{\delta_{1},\delta_{2}\right\}.$

3. Existence of the operators

Lemma 1**.**

If $f\in L_{1}(D)$, then the operator $L_{\lambda}\left(f;x,y\right)$ defines a continuous transformation acting on $L_{1}(D\mathbb{)}$.

Proof.

Since $L_{\lambda}(f;x,y)$ is linear, it is sufficient to show that the expression given by

[TABLE]

is bounded.

Let $D=\left\langle a,b\right\rangle\times\left\langle c,d\right\rangle$ be an arbitrary closed, semi-closed or open bounded rectangle in $\mathbb{R}^{2}.$ The function $f$ is extended to $\mathbb{R}^{2}$ by defining $g$ such that

[TABLE]

Now, using Fubini’s Theorem [2], we have

[TABLE]

One may prove the assertion for the case $D=\mathbb{R}^{2}$ using similar method. The proof is completed.

4. Pointwise convergence

The following theorem gives a pointwise approximation of the integral operators of type (1.3) to the function $f$ at $\mu$-generalized Lebesgue point of $f\in L_{1}(D)$ whenever $D$ is an arbitrary rectangle in $\mathbb{R}^{2}$ such that bounded, closed, semi-closed or open.

Theorem 1**.**

If $\left(x_{0},y_{0}\right)$ is a $\mu$-generalized Lebesgue point of function $f\in L_{1}\left(D\right),$ then

[TABLE]

on any set $Z$ on which the functions

[TABLE]

and

[TABLE]

are bounded as $\left(x,y,\lambda\right)$ tends to $\left(x_{0},y_{0},\lambda_{0}\right)$. Here, the set $Z$ consists of the points $(x,y,\lambda)\in D\times\Lambda$ such that the functions given by (4.1) and (4.2) are bounded for all $\delta>0$ which satisfy $0<\delta<\delta_{0}.$

Proof.

The proof begins with the following compulsory assumptions: Suppose that $\left(x_{0},y_{0}\right)\in D,$ $0<x_{0}-x<\delta/2,$ for all $\delta>0$ which satisfies $x_{0}+\delta<b$ and $x_{0}-\delta>a$, $0<y_{0}-y<\delta/2$ for all $\delta>0$ which satisfies $y_{0}+\delta<d$ and $y_{0}-\delta>c$ whenever $0<\delta<\delta_{0}.$ Let us divide $D$ into the sets $D_{ij}=\left\langle c_{i},c_{i+1}\right\rangle\times\left\langle d_{j},d_{j+1}\right\rangle$ for $1\leq i\leq 3$ and $1\leq j\leq 3,$ where $c_{1}=a,$ $c_{2}=x_{0}-\delta,$ $c_{3}=x_{0}+\delta,$ $c_{4}=b$ and $d_{1}=c,$ $d_{2}=y_{0}-\delta,$ $d_{3}=y_{0}+\delta,$ $d_{4}=d.$ Recall the definition of the function $g(t,s)$ such that

[TABLE]

Let $\left(x_{0},y_{0}\right)\in D$ is a $\mu$-generalized Lebesgue point of function $f\in L_{1}\left(D\right).$ Therefore, for all given $\varepsilon>0$, there exists a $\delta>0$ such that for all $h,k$ satisfying $0<h,k\leq\delta,$ the inequality:

[TABLE]

holds.

Set $I_{\lambda}(x,y):=\left|L_{\lambda}\left(f;x,y\right)-f\left(x_{0},y_{0}\right)\right|$. According to condition (c) of class $A$, we shall write

[TABLE]

[TABLE]

It is easy to see that $I_{2}\rightarrow 0$ as $\lambda\rightarrow\lambda_{0}$ by condition (c) of class $A.$ On the other hand, since

[TABLE]

by condition (e) of class $A$ we have $I_{3}\rightarrow 0$ as $\left(x,y,\lambda\right)$ tends to* $\left(x_{0},y_{0},\lambda_{0}\right).$*

Obviously, the integral $I_{1}$ can be written as follows:

[TABLE]

We will show that $I_{11}\rightarrow 0$ as $\left(x,y,\lambda\right)$ tends to* $\left(x_{0},y_{0},\lambda_{0}\right)$ by following the similar steps as in [11]. *

Now, recall that the rectangle $D$ was splitted into nine rectangles at the beginning of the proof. Therefore, $I_{11}$ consists of eight integrals. Now, denote the integral which corresponds to $D_{12}$ by $I_{D_{12}}.$ Hence, by condition (f) of class $A,$ the following inequality

[TABLE]

holds. By condition (d) of class $A$, $I_{D_{12}}\rightarrow 0$ as $\left(x,y,\lambda\right)$ tends to* $\left(x_{0},y_{0},\lambda_{0}\right).$* The remaining seven integrals are evaluated by an analogous method and this part is omitted.

It follows that

[TABLE]

Consequently, $I_{11}\rightarrow 0$ as $\left(x,y,\lambda\right)$ tends to* $\left(x_{0},y_{0},\lambda_{0}\right).$*

Next, we prove that $I_{12}$ tends to zero as $\left(x,y,\lambda\right)$ tends to $\left(x_{0},y_{0},\lambda_{0}\right)$. Therefore, it is easy to see that the following inequality holds for $I_{12}$, that is

[TABLE]

Let us consider the integral $I_{121}.$ For the evaluations, we need to define the following variations:

[TABLE]

Taking above variations and (4.3) into account and applying bivariate integration by parts method (see, e.g., [8]) to last inequality, we have

[TABLE]

For the similar situation, we refer the reader to [8, 10].

Now, using Remark 1 and condition (f) of class $A,$ we get

[TABLE]

Hence, the following inequality holds for $I_{121}:$

[TABLE]

Analogous computations for $I_{122},$ $I_{123}$ and $I_{124}$ yield:

[TABLE]

[TABLE]

Hence the following inequality is obtained for $I_{12}:$

[TABLE]

or equivalently,

[TABLE]

The remaining part of the proof is obvious by the hyphotheses (4.1) and (4.2). Thus the proof is completed.

The following theorem gives a pointwise approximation of the integral operators of type (1.3) to the function $f$ at $\mu$-generalized Lebesgue point of $f\in L_{1}(\mathbb{R}^{2})$.

Theorem 2**.**

Suppose that the hypotheses of Theorem 1 are satisfied for $D=\mathbb{R}^{2}$. If $\left(x_{0},y_{0}\right)$ is a $\mu$-generalized Lebesgue point of $f\in L_{1}(\mathbb{R}^{2}),$ then

[TABLE]

Proof.

The proof of this theorem is quite similar to proof of preceding one and thus, it is omitted.

5. Rate of Convergence

In this section, we give a theorem concerning the rate of pointwise convergence of the operators of type (1.3).

Theorem 3**.**

Suppose that the hypotheses of Theorem 1 (or Theorem 2) are satisfied.

Let

[TABLE]

for $0<\delta<\delta_{0},$ and the following assumptions are satisfied:

**i: **

$\Delta(\lambda,\delta,x,y)\rightarrow 0$ as $(x,y,\lambda)\rightarrow(x_{0},y_{0},\lambda_{0})$ for some $\delta>0.$

**ii: **

For every $\gamma>0,$

[TABLE]

as $(x,y,\lambda)\rightarrow(x_{0},y_{0},\lambda_{0}).$

**iii: **

For every $\gamma>0,$

[TABLE]

as $(x,y,\lambda)\rightarrow(x_{0},y_{0},\lambda_{0}).$

**iv: **

As $(x,y,\lambda)\rightarrow(x_{0},y_{0},\lambda_{0}),$ we have

[TABLE]

Then, at each $\mu$-generalized Lebesgue point of $f\in L_{1}(D)$ we have

[TABLE]

as $(x,y,\lambda)\rightarrow(x_{0},y_{0},\lambda_{0}).$

Proof.

By the hypotheses of Theorem 1, we can write for $\delta>0$

[TABLE]

[TABLE]

From (i)-(iv) and using class $A$ conditions, we have the desired result for the case $D$ is bounded, that is

[TABLE]

One may prove the assertion for the case $D=\mathbb{R}^{2}$ following the same steps. Thus, the proof is completed.

Example 1**.**

In this example, we used two dimensional counterpart of the kernel function used in [6].

Let $\Lambda=[1,\infty),$ $\lambda_{0}=\infty$ and

[TABLE]

It is easy to check that given $K_{\lambda}(t,s)$ is from the class $A.$ Let $\mu_{1}(t)=t,$ $\mu_{2}(s)=s$.

Hence, we obtain

[TABLE]

In order to find for which $\delta>0$ the condition $(i)$ in Theorem 1 is satisfied, let $\Delta(\lambda,\delta,x,y)\rightarrow 0$ as $(x,y,\lambda)\rightarrow(x_{0},y_{0},\infty)$. Hence

[TABLE]

if and only if $\delta=o(1/\lambda).$ Consequently, the following equation

[TABLE]

holds for $\alpha\in(0,\infty).$ From above equation we see that

[TABLE]

By definition of $K_{\lambda}\left(t,s\right),$the conditions (ii) and (iii) of Theorem 5.1 are satisfied.

The last terms which must be considered are $K_{\lambda}\left(0,0\right)\mu_{2}(\left|y-y_{0}\right|)$ and $K_{\lambda}\left(0,0\right)\mu_{1}(\left|x-x_{0}\right|).$ Analyzing the following limits, we see that

[TABLE]

if and only if the rates of convergence of $\lambda^{2}$ $\rightarrow\infty$ and $x\rightarrow x_{0},$ $\lambda^{2}$ $\rightarrow\infty$ and $y\rightarrow y_{0}$ are equivalent.

Hence

[TABLE]