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A new approach to nonlinear singular integral operators depending on three parameters

Gumrah Uysal
• Corresponding author
• Department of Computer Technologies, Division of Technology of Information Security, Karabuk University, 78050, Karabuk, Turkey
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Published Online: 2016-11-11 | DOI: https://doi.org/10.1515/math-2016-0081

Abstract

In this paper, we present some theorems on weighted approximation by two dimensional nonlinear singular integral operators in the following form: $Tλ(f;x,y)=∬R2Kλ(t−x,s−y,f(t,s))dsdt,(x,y)∈R2,λ∈Λ,$ where Λ is a set of non-negative numbers with accumulation point λ0.

MSC 2010: 41A35; 41A25; 47G10

1 Introduction

Approximation by singular integral operators is one of the oldest topics of approximation theory. Here, the concept of singular integral operator refers to the integral operator whose kernel shows the behaviour of Dirac's δ function (for the properties of δ—function, see 1). Also, singular integral operators arise from the Fourier analysis of the functions. It is well known that Fourier analysis is one of the most useful tools of many branches of science. Therefore, indicated integral operators have various applications in many academic disciplines such as physics, engineering and medicine. In fact, magnetic resonance imaging, face recognition, differential equation solving and computer aided geometric design are some of the application areas in which the indicated operators are used. For mentioned applications, we refer the reader to [2, 3].

The convergence of various type linear integral operators have been examined at characteristic points such as continuity point, μ-generalized Lebesgue point, and so on, by many researchers throughout years: one parameter family of singular integral operators [4, 5], a sequence of singular integral operators with general Poisson type kernels 6, a family of singular integral operators depending on two parameters [79], a sequence of Gegenbauer singular integrals 10 and a sequence of msingular integral operators 11. One may consider the pointwise approximation of singular integral operators in weighted Lebesgue spaces as well as usual Lebesgue spaces. Therefore, for some advanced studies concerning weighted pointwise approximation by singular integral operators, we refer the reader to [1113].

Musielak 14 studied the convergence of convolution type nonlinear integral operators in the following form: $Tαf(y)=∫GKα(x−y,f(x))dx,y∈G,α∈Λ,$1 where G is a locally compact Abelian group equipped with Haar measure and Λ≠∅ is an index set with any topology, and he extended the concept of the singularity condition via replacing the linearity property of the integral operators by an assumption of Lipschitz condition for Kα with respect to second variable. Therefore, traditional solution technics became applicable to nonlinear problems by the aid of indicated Lipschitz condition. In 15, Musielak advanced his previous analysis 14 by obtaining significant results for generalized Orlicz spaces. After this important study, Swiderski and Wachnicki 16 investigated the pointwise convergence of the operators of type (1) at p Lebesgue points of functions $f\in {L}_{p}\left(-\pi ,\pi \right)\left(1\underset{_}{<}p<\mathrm{\infty }\right).$ For further results concerning the convergence of several types of nonlinear singular integral operators in different function spaces, the studies [17, 18] are strongly recommended.

In 19, Taberski studied the pointwise approximation of functions $f\in {L}_{1}\left(R\right)$ by convolution type two dimensional integral operators in the following form: $Vλ(f;x,y)=∫∫Rf(t,s)Kλ(t−x,s−y)dsdt,(x,y)∈R,$ where R denotes a given rectangle and Kλ(t,s) denotes a kernel satisfying suitable conditions with λ∈Λ, where Λ is a given set of non-negative numbers with accumulation point λ0. The earlier results concerning the operators of type (2) were obtained by Gahariya 20, i.e., the indicated operators were handled as a sequence of double integral operators in this work. The studies [2123], which are based on Taberski's study 19, are devoted to the study of pointwise convergence of the operators of type (2) on some planar sets consisting of characteristic points (x0,y0) of various types. Later, Musielak 24 investigated the conditions under which the two dimensional counterparts of the operators of type (2) are (σ,Iφ)–conservative, where σ is a modular defined on the space of functions which are Lebesgue measurable on arbitrary closed and bounded subset of ${\mathbb{R}}^{2}$, and I, is a Musielak-Orlicz modular. Recently, Karsli 25 obtained the convergence of convolution type linear singular integral operators depending on three parameters at μ-generalized Lebesgue points of the integrable functions. For some studies concerning double singular integral operators in several settings, we refer the reader to [2630]. On the other hand, for some other important works related to approximation by linear and nonlinear operators in several function spaces, we refer the reader to [3137].

Let $f\in {L}_{p}\left({\mathbb{R}}^{2}\right),$ is the space of all measurable functions $f:{\mathbb{R}}^{2}\to \mathbb{R}$ for which ${\left|\frac{f}{\phi }\right|}^{p}$ is integrable on ${\mathbb{R}}^{2}$. Here, $\phi :{\mathbb{R}}^{2}\to {\mathbb{R}}^{+}$ is aweight function satisfying suitable conditions. The norm formula for the space $f\in {L}_{1}\left(\mathbb{R}\right)$ (see, e.g., [11, 13]) is given by $||f||Lpφ(R2)=∬R2f(t,s)φ(t,s)pdsdt1p,1≤p<∞.$

The main aim of this paper is to investigate both the weighted pointwise convergence and the rate of weighted pointwise convergence of nonlinear double singular integral operators of the form as such: $Tλ(f;x,y)=∬R2Kλ(t−x,s−y,f(t,s)dsdt,(x,y)∈R2,λ∈Λ,$3 where Λ is a set of non-negative numbers with accumulation point λ0.

The paper is organized as follows: In Section 2, we give some preliminary concepts. In Section 3, main result is presented. In Section 4, the rate of pointwise convergence of the operators of type (3) is established.

2 Preliminaries

In this section, basic concepts used in this paper are introduced.

Let 1 ≤ p < ∞, and ${\delta }_{0},{\delta }_{1}\in {\mathbb{R}}^{+}$ it be fixed numbers. A point $\left({x}_{0},{y}_{0}\right)\in {\mathbb{R}}^{2}$ at which the following relations $limh→01μ1(h)∫x0x0+hg(t,y0)−g(x0;y0)pdt1p=0,$ and $limk→01μ2(k)∫Y0Y0+k|g(t,s)−g(t,y0)|pds1p=0,$ it hold uniformly with respect to almost every} $t\in \mathbb{R}$ is called a μ – p – generalized Lebesgue point of locally p– integrable function (i.e., a function whose p –th power is locally integrable) g: ${\mathbb{R}}^{2}\to \mathbb{R}$ Here, $\mathbb{R}\to \mathbb{R}$ is increasing and absolutely continuous on 0 < h ≤ δ0 and μ1(0) = 0 and also, ${\mu }_{2}:\mathbb{R}\to \mathbb{R}$ is increasing and absolutely continuous on 0 < k ≤ δ1 and μ2(0) = 0.

Basically Definition 2.1 is obtained by combining the characterization of the function μ(t) presented by Gadjiev 9 with the definition of d–point given by Siudut 21. Also, some diferent modifications are done according to our problem's needs, such as predispozing the definition to ${L}_{p}^{\phi }$ space. On the other hand, for some other μ-generalized Lebesgue point definitions, we reer the reader to [8, 25] and 28.

Let λ0 be an accumulation point of the non-negative set of numbers Λ or λ0 = ∞, and $\phi :{\mathbb{R}}^{2}\to {\mathbb{R}}^{+}$ be a locally bounded weightfunction such that the following inequality $φ(t+x,s+y)≤φ(t,s),(x,y)$6 holds for every $\left(t,s\right)\in {\mathbb{R}}^{2}$

A family (Kλ)λ∈Λ consisting of the functions ${\mathbb{R}}^{2}×\mathbb{R}\to \mathbb{R}$ is called class A, if the following conditions hold:

(a) ${K}_{\lambda }\left(t,s,0\right)=0\phantom{\rule{thinmathspace}{0ex}}\text{for every}\left(t,s\right)\in {\mathbb{R}}^{2}$ and for each λ∈Λ, and ${K}_{\lambda }\left(.,\phantom{\rule{thinmathspace}{0ex}}.,u\right)\in {L}_{1}\left({\mathbb{R}}^{2}\right)$ for every $u\in \mathbb{R}$ and for each λ∈Λ.

(b) There exists a family (Lλ)λ∈Λ consisting of the (globally) integrable functions L λ: ${\mathbb{R}}^{2}\to \mathbb{R}$ such that the Lipschitz inequality given by $|Kλ(t,s,u)−Kλ(t,s,v)|≤Lλ(t,s)|u−v|$ holds for every $\left(t,s\right)\in {\mathbb{R}}^{2},u,v\in \mathbb{R},$ and for each fixed λ∈Λ.

(c) $\underset{\left(x,y,\lambda \right)\to \left({x}_{0;}{y}_{0;}{\lambda }_{0}\right)}{lim}\left|\underset{{\mathbb{R}}^{2}}{\iint }{K}_{\lambda }\left(t-x,s-y,\frac{u}{\left({\mathrm{x}}_{0;}{y}_{0}\right)},\left(t,s\right)\right)dsdt-u\right|=0\phantom{\rule{thinmathspace}{0ex}}\text{for every}\phantom{\rule{thinmathspace}{0ex}}u\in \mathbb{R}$ and for any $\left({x}_{0},{y}_{0}\right)\in {\mathbb{R}}^{2}.$

(d) For every $\xi >0,\underset{\lambda \to {\lambda }_{0}}{lim}\left[su{p}_{\xi \le \sqrt{{t}^{2}+{s}^{2}}}\left[\phi \left(t,s\right){L}_{\lambda }\left(t,s\right)\right]dsdt\right]=0.$

(e)For every $\xi >0,\underset{\lambda \to {\lambda }_{0}}{lim}\left[{\iint }_{\xi ⩽\sqrt{{t}^{2}+{s}^{2}}}\left[\phi \left(t,s\right){L}_{\lambda }\left(t,s\right)\right]dsdt\right]=0.$

(f) $\parallel \phi {L}_{\lambda }{\parallel }_{{L}_{1}\left({\mathrm{R}}^{2}\right)}\le M<\mathrm{\infty }\phantom{\rule{thinmathspace}{0ex}}\text{for every}\phantom{\rule{thinmathspace}{0ex}}\lambda \in \mathrm{\Lambda }.$

(g) L λ(t,s) is non-increasing on [0,∞) and non-decreasing on (−∞,0] for each fixed λ∈Λ as a function of t. Similarly, L λ(t,s) is non-increasing on [0,∞) and non-decreasing on (−∞,0] for each fixed λ∈Λ as a function of s.

Throughout this paper the kernel function Kλ belongs to class A.

The studies [11, 13, 16, 21] and 18, among others, are used as main reerence works in the construction stage of class A. Therefore, we refer the reader to see the indicated works. On the other hand, we recommend the reader to compare the usage of the inequality (6) used in 11 with the current study Also, for the Lipschitz inequality included in the definition class $\mathcal{A}$, we reer the reader to see the works [14, 18].

Existence of the operators of type (3) is guaranteed by the conditions of class A, that is ${T}_{\lambda }\left(f;x,y\right)\in {L}_{p}^{\phi }\left({\mathbb{R}}^{2}\right)$ whenever $f\in {L}_{p}^{\phi }\left({\mathbb{R}}^{2}\right).$

A first example is the linear kernel. Let Λ be a set of non-negative numbers such that Λ = (0,;∞) with accumulation point λ0 = 0. Now, the definition of the function ${\mathbb{R}}^{2}×\mathbb{R}\to \mathbb{R}$ is as follows: $Kλ(t,s,u)=u4πλe−(t2+s2)4λ,u∈R.$ Since $|Kλ(t,s,u)−Kλ(t,s,v)|=14πλe−(t2+s2)4λ|u−v|=Lλ(t,s)|u−v|,$ one may easily observe that given function Kλ belongs to class A. For detailed analysis of the function Lλ(t,s), we recommend the reader to see 21.

Define the kernel function such that $Kλ(t,s,u)=λu2+sin⁡λu2;if(t,s)∈[−12λ;12λ]×[−12λ;12λ];0,if(t,s)∈R2∖[−12λ;12λ]×[−12λ;12λ];$ where λ∈ N and ∈0 = \∞. This kernel is the two dimensional analogue of the kernel given in 16.

It is easy to see that the conditions of class A are satisfied. Observe that one may take the desired linear kernel as $Lλ(t,s)=λ,if(t,s)∈−12λ,12λ×−12λ,12λ,0,ift,s∈R2,∖−12λ,12λ×−12λ,12λ.$

The appropriate weight functions, which are defined on ${\mathbb{R}}^{2}$, may be given by ${\phi }_{1}\left(t,s\right)={e}^{t+s}$ and ${\phi }_{2}\left(t,s\right)=\left(1+|t|\right)\left(1+|s|\right).$

3 Convergence at characteristic points

If $\left({x}_{0},{y}_{0}\right)\in {\mathbb{R}}^{2}$ is a common μ-generalized Lebesgue point of the functions $f\in {L}_{p}^{\phi }\left({\mathbb{R}}^{2}\right)$ $\left(1\le p<\mathrm{\infty }\right)$ and φ then $limx,y,λ→x0,y0,λ0Tλf;x,y=fx0,y0,$ on any set Z consisting of the points (x,y,λ) on which the functions $∫Y0−δY0+δ∫x0−δx0+δLλ(t−x,s−y)|{μ1(|t−x0|)}t′|dtds+2μ1(|x−x0|)∫Y0−δY0+δLλ(0,s−y)ds,$7 and $∫Y0−δxY0+δx∫0−δ0+δLλ(t−x,s−y)|{μ1(|t−x0|)}t′|dtds+2μ1(|x−x0|)∫Y0−δY0+δLλ(0,s−y)ds,$8 where 0 < δ < min {δ0, δ1, are bounded as (x,y,λ) tends to (x0,y00).

Let $0<|{x}_{0}-x|<\frac{\delta }{2}.$ Further, let $0<{y}_{0}-y<\frac{\delta }{2},$, and (x0,y0) be a common μ-generalized Lebesgue point of the functions $f\in {L}_{p}\left({}^{2}\right)\left(1p<\mathrm{\infty }\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}\phi .$ The proof of theorem will be given for the case 1 < p < ∞. The proof for the case p = 1 is similar. Now, set $I\left(x,y,\lambda \right)=|{T}_{\lambda }\left(f;x,y\right)-f\left({x}_{0},{y}_{0}\right)|.$ Using condition (c) , we obtain $I(x,y,λ)=∬R2Kλ(t−x,s−y,f(t,s))dsdt−f(x0,y0)=∬R2Kλ(t−x,s−y,f(t,s))dsdt−∬R2Kλ(t−x,s−y,f(x0,y0)φ(x0,y0)φ(t,s))dsdt+∬R2Kλ(t−x,s−y,f(x0,y0)φ(x0;y0)φ(t,s))dsdt−f(x0,y0)$

Using condition (b), it is easy to see that the following inequality holds: $I(x,y,λ)≤∬R2f(t,s)(t,s)−f(x0,y0)(x0,y0)φ(t,s)Lλ(t−x,s−y)dsdt+∬R2Kλ(t−x,s−y,f(x0,y0)(x0,y0)φ(t,s))dsdt−f(x0,y0).$

Since whenever m,n being positive numbers the inequality $\left(m+n{\right)}^{p}\le {2}^{p}\left({m}^{p}+{n}^{p}\right)$ holds (see, e.g., 38) we have $[I(x,y,λ)]p≤2p∬R2f(t,s)φ(t,s)−f(x0,y0)φ(x0,y0)φ(t,s)Lλ(t−x,s−y)dsdtp+2p∬R2Kλ(t−x,s−y,f(x0,y0)φ(x0,y0)φ(t,s))dsdt−f(x0,y0)p.$

Now, applying Hölder's inequality (see, e.g., 38) to the first integral of the resulting inequality, we have $[I(x,y,λ)]p≤2pβ(x,y,λ)∬R2f(t,s)φ(t,s)−f(x0,y0)φ(x0,y0)pφ(t,s)Lλ(t−x,s−y)dsdt+2p∬R2Kλ(t−x,s−y,f(x0,y0)φ(x0,y0)φ(t,s))dsdt−f(x0,y0)p,$ where $β(x,y,λ)=∬R2φ(t,s)Lλ(t−x,s−y)dsdtpq.$

Moreover, the following inequality holds: $[I(x,y,λ)]p≤2pβ(x,y,λ)∬R2∖Bδf(t,s)φ(t,s)−f(x0,y0)φ(x0,y0)pφ(t,s)Lλ(t−x,s−y)dsdt$ $+2pβ(x,y,λ)∬Bδ|f(t,s)φ(t,s)−f(x0,y0)φ(x0,y0)|pφ(t,s)Lλ(t−x,s−y)dsdt+2p|∬R2Kλ(t−x,s−y,f(x0,y0)φ(x0,y0)φ(t,s))dsdt−f(x0,y0)|p$ where $\left(t-{x}_{0}{\right)}^{2}+\left(s-{y}_{0}{\right)}^{2}<{\delta }^{2}\right\}.$

Now, applying the inequality given by $\left(m+n{\right)}^{p}\underset{_}{<}{2}^{p}\left({m}^{p}+{n}^{p}\right)$ once more to the right hand side of the resulting inequality, we obtain$[I(x,y,λ)]p<_22pβ(x,y,λ)|f(x0,y0)φ(x0,y0)|p∫R2∫∖Bδφ(t,s)Lλ(t−x,s−y)dsdt+22pβ(x,y,λ)∫R2∫∖Bδ|f(t,s)φ(t,s)|pφ(t,s)Lλ(t−x,s−y)dsdt+2pβ(x,y,λ)∫B∫δ|f(t,s)φ(t,s)−f(x0,y0)φ(x0,y0)|pφ(t,s)Lλ(t−x,s−y)dsdt+2p|∬R2Kλ(t−x,s−y,f(x0,y0)φ(x0,y0)φ(t,s))dsdt−f(x0,y0)|p$

Recalling the initial assumptions $0<|{y}_{0}-y|<\frac{\delta }{2}$, we may define the following set as follows:$Nδ={(x,y)∈R2:(x−x0)2+(y−y0)2<δ22}.$

Comparing geometric representations of the sets${B}_{\delta }$ gives the inclusion relation such that${\mathbb{R}}^{2}\mathrm{\setminus }{B}_{\delta }\subseteq {\mathbb{R}}^{2}\mathrm{\setminus }{A}_{\delta },$ where$Aδ={(t,s)∈Bδ:(t−x)2+(s−y)2<δ22,(x,y)∈Nδ}.$

In the light of these relations, we may write$[I(x,y,λ)]p<_22pφ(x,y)β(x,y,λ)|f(x0,y0)φ(x0,y0)|p∬R2∖Aδφ(t−x,s−y)Lλ(t−x,s−y)dsdt+22pφ(x,y)β(x,y,λ)sup(t,s)∈R2∖Aδ[φ(t−x,s−y)Lλ(t−x,s−y)]∥f∥Lpφ(R2)p+2pβ(x,y,λ)∬Bδ|f(t,s)φ(t,s)−f(x0,y0)φ(x0,y0)|pφ(t,s)Lλ(t−x,s−y)dsdts+2p|∬R2Kλ(t−x,s−y,f(x0,y0)φ(x0,y0)φ(t,s))dsdt−f(x0,y0)|p$

Rearranging and rewriting the last inequality, we obtain $[I(x,y,λ)]p<_22pφ(x,y)β(x,y,λ)|f(x0,y0)φ(x0,y0)|p∬u2+v2δ22φ(u,v)Lλ(u,v)dvdu$ $+22pφ(x,y)β(x,y,λ)supu2+v2⩾δ22⁡[φ(u,v)Lλ(u,v)]fLpφ(R2)p+2pβ(x,y,λ∬Bδf(t,s)φ(t,s)−f(x0,y0)φ(x0,y0)pφ(t,s)Lλ(t−x,s−y)dsdt+2p∬R2Kλ(t−x,s−y,f(x0,y0)φ(x0,y0)φ(t,s))dsdt−f(x0,y0)p=I1(x,y,λ)+I2(x,y,λ)+2pβ(x,y,λ)I3(x,y,λ)+I4(x,y,λ).$

Boundedness of the term $\left(x,y,\lambda \right)$ which follows from condition (f).On the other hand, ${I}_{4}\left(x,y,\lambda \right)\to 0$ by condition(c).Lastly, ${I}_{2}\left(x,y,\lambda \right)\to 0$ by conditions (d) and (e) , respectively.

Now, we may write the following inequality for the integral ${I}_{3}\left(x,y,\lambda \right)$ $I3(x,y,λ)=∬Bδf(t,s)φ(t,s)−f(x0,y0)φ(x0,y0)pφ(t,s)Lλ(t−x,s−y)dsdt<_sup(t,s)∈Qδφ(t,s)∫Q∫δf(t,s)φ(t,s)−f(x0,y0)φ(x0,y0)pLλ(t−x,s−y)dsdt=sup(t,s)∈Qδφ(t,s)I31(x,y,λ),$ where ${Q}_{\delta }=\left({x}_{0}-\delta ,{x}_{0}+\delta \right)×\left({y}_{0}-\delta ,{y}_{0}+\delta \right).$Observe that I31(x,y,λ) may be written in the following form:$I31(x,y,λ)=∬Qδf(t,s)φ(t,s)−f(x0,y0)φ(x0,y0)pLλ(t−x,s−y)dsdt=∫x0−δx0+δy∫y0−δy0+δf(t,s)φ(t,s)−f(x0,y0)φ(x0,y0)+f(t,y0)φ(t,y0)−f(t,y0)φ(t,y0)p×Lλ(t−x,s−y)dsdt.$

It is easy to see that the following inequality holds:$I31(x,y,λ)<_2p∫x0−δxo+δ∫y0−δy0+δf(t,s)φ(t,s)−f(t,y0)φ(t,y0)pLλ(t−x,s−y)dsdt+2p∫y0−δy0+δ∫xo+δxo+δf(t,y0)φ(t,y0)−f(x0,y0)φ(x0,y0)pLλ(t−x,s−y)dtds=2p∫x0−δx0+δiλ(x,y,t)dt+∫y0−δy0+δjλ(x,y,s)ds.$Now, we pass to the integral ${i}_{\lambda }\left(x,y,t\right)$.

This integral may be written in the following form: $iλ(x,y,t)=∫y0−δy0+δf(t,s)φ(t,s)−f(t,y0)φ(t,y0)pLλ(t−x,s−y)ds=∫y0−δy0+∫y0y0+δf(t,s)φ(t,s)−f(t,y0)φ(t,y0)pLλ(t−x,s−y)ds=iλ1(x,y,t)+iλ2(x,y,t).$

Now, we consider the integral${i}_{\lambda }^{1}\left(x,y,t\right)$ From relation (5), for every $ϵ>0$ there exists a corresponding number$\delta >0$ such that the expression $∫Y0−kY0f(t,s)φ(t,s)−f(t,y0)φ(t,y0)pds<ϵpμ2(k)$(9) holds for every $0

Define the function F(t,s) by $F(t,s)=∫sY0f(t,w)φ(t,w)−f(t,y0)φ(t,y0)pdw$10

From (10), we have $dsF(t,s)=−f(t,s)φ(t,s)−f(t,y0)φ(t,y0)pds$11

From (9) and (10), for every ssatisfying$0<{y}_{0}-s\underset{_}{<}\delta we have $|F(t,s)|<_ϵpμ2(y0−s)$12(12)for any fixed $t\in \mathbb{R}$ By virtue of (10) and (11), we have $iλ1(x,y,t)=(L)∫y0−δy0f(t,s)φ(t,s)−f(t,y0)φ(t,y0)pLλ(t−x,s−y)ds=(LS)∫y0−δy0Lλ(t−x,s−y)ds[−F(t,s)],$where (LS) denotes Lebesgue-Stieltjes integral.

Using integration by parts and applying (12), we have the following inequality:$|iλ1(x,y,t)|<_ϵpμ2(δ)Lλ(t−x,y0−δ−y)+ϵp∫y0−δy0μ2(y0−s)|dsLλ(t−x,s−y)|.$

It is easy to see that (for the similar situation, see [7, 9]) the following inequality holds:$iλ1(x,y,t)<_ϵpμ2(δ)Lλ(t−x,y0−δ−y)+ϵp∫y0−δy0μ2(y0−s−y)dsVy0−y−δs⁡(t−x,u).$

Applying integration by parts to the right hand side of the last inequality, we have the following expression:$|iλ1(x,y,t)|<_−ϵp∫y0−y−δy0−yVy0−y−δs⁡Lλ(t−x,u)μ2(y0−s−y)′sds.$

Remaining variational operations are evaluated using condition (g). Hence, we obtain $|iλ1(x,y,t)|<_ϵp∫y0−δY0Lλ(t−x,s−y){μ2(y0−s)}s′ds+2Lλ(t−x,0)μ2(y0−y)$13

Using preceding method, we can estimate the integral ${i}_{\lambda }^{2}\left(x,y,t\right)$as follows: $|iλ2(x,y,t)|<_ϵp∫y0y0+δLλ(t−x,s−y)|{μ2(s−y0)}s′|ds$14(14)Combining (13) and (14), we obtain$|iλ(x,y,t)|<_ϵp∫y0−δy0+δLλ(t−x,s−y)|{μ2(|s−y0|)}t′|ds+2μ2(|y−y0|)Lλ(t−x,0).$

Similar calculations for the integral ${j}_{\lambda }\left(x,y,s\right)$yield$|jλ(x,y,s)|<_ϵp∫x0−δxo+δLλ(t−x,s−y){μ1(|t−x0|)}t′|dt+2μ1(|x−x0)Lλ(0,s−y).$

Thus, we have$I31(x,y,λ)<_ϵp2p∫x0−δx0+δ∫y0−δy0+δLλ(t−x,s−y)|{μ2(|s−y0|)}t′|ds+2μ2(|y−y0|)Lλ(t−x,0)dt+ϵp2p∫y0−δy0+δ∫x0−δx0+δLλ(t−x,s−y)|{μ1(|t−x0|)}t′|dt+2μ1(|x−x0|)Lλ(0,s−y)ds.$

Since epsilon is arbitrary and Lλ is integrable with respect to eachvariable, the desired result follows from hypotheses (8)and (7), i.e.,$\left(x,y,\lambda \right)\to \left({x}_{0},{y}_{0},{\lambda }_{0}\right)$

Note that the same conclusion is obtained for the case $0 Thus the proof is completed.

4 Rate of pointwise convergence

it Suppose that the hypotheses of Theorem 3.1 are satisfied.

Let $△(x,y,λ,δ)=∫x0−δx0+δ△1(x,y,λ,δ,t)dt+∫y0−δy0+δ△2(x,y,λ,δ,s)ds,$ where $0<\delta $△1(x,y,λ,δ,t)=∫Y0−δY0+δLλ(t−x,s−y)|{μ2(|s−y0|)}t′|ds+2μ2(|y−y0|)Lλ(t−x,0),$ and $△2(x,y,λ,δ,s)=∫x0−δx0+δLλ(t−x,s−y)|{μ1(|t−x0|)}t′|dt+2μ1(|x−x0|)Lλ(0,s−y),$ and the following conditions are satisfied}

(i) $\mathrm{△}\left(x,y,\lambda ,\delta \right)\to 0as\left(x,y,\lambda \right)\phantom{\rule{thinmathspace}{0ex}}\text{tends to}\phantom{\rule{thinmathspace}{0ex}}\left({x}_{0},{y}_{0},{\lambda }_{0}\right)\phantom{\rule{thinmathspace}{0ex}}\text{for some}\phantom{\rule{thinmathspace}{0ex}}\delta >0.$

(ii) For every $\xi >0$ we have $\underset{\xi ⩽\sqrt{{t}^{2}+{s}^{2}}}{sup}$ $varphi\left(t,s\right){L}_{\lambda }\left(t,s\right)\right]=o\left(\mathrm{△}\left(x,y,\lambda ,\delta \right)\right)as\left(x,y,\lambda \right)tendsto\left({x}_{0},{y}_{0},{\lambda }_{0}\right)$

(iii) For every $u\in \mathbb{R}$we have $\left|{\iint }_{{\mathbb{R}}^{2}}{K}_{\lambda }\left(t-x,s-y,\frac{u}{\phi \left({x}_{0},{y}_{0}\right)}\phi \left(t,s\right)\right)dsdt-u\right|=o\left(\mathrm{△}\left(x,y,\lambda ,\delta \right)\right)$tends to $\left({x}_{0},{y}_{0},{\lambda }_{0}\right)$

(iv) For every$\xi >0$we have ${\iint }_{\xi \le \sqrt{{t}^{2}+{s}^{2}}}\phi \left(t,s\right){L}_{\lambda }\left(t,s\right)dsdt=o\left(\mathrm{△}\left(x,y,\lambda ,\delta \right)\right)\phantom{\rule{thinmathspace}{0ex}}\text{as}\phantom{\rule{thinmathspace}{0ex}}\left(x,y,\lambda \right)$tends to $\left({x}_{0},{y}_{0},{\lambda }_{0}\right).$

Then, at each common} \mu-generalized Lebesgue point of the functions $f\in {L}_{p}^{\phi }\left({\mathbb{R}}^{2}\right)\left(1\underset{_}{<}p<\mathrm{\infty }\right)$we have $|Tλ(f;x,y)−f(x0,y0)|=o(△(x,y,λ,δ)1p),$as (x,\ y,\ \lambda)tends to $\left({x}_{0},{y}_{0},{\lambda }_{0}\right)$

By the hypotheses of Theorem 3.1, we may write$|Tλ(f;x,y)−f(x0,y0)|p<_22pφ(x,y)β(x,y,λ)|f(x0,y0)φ(x0,y0)|p∬u2+v2⩾δ22φu2+v2(u,v)Lλ(u,v)dvdu+22pφ(x,y)β(x,y,λ)supu2+v2⩾δ22⁡φ(u,v)Lλ(u,v)]∥f∥Lpφ(R2)p+2pϵpβ(x,y,λ)sup(t,s)∈Qδφ(t,s)∫x0−δx0+δ△1(x,y,λ,δ,t)dt+2pϵpβ(x,y,λ)sup(t,s)∈Qδφ(t,s)∫y0−δy0+δ△2(x,yλ,δ,s)ds+2p∬R2Kλ(t−x,s−y,f(x0,y0)φ(x0,y0)φ(t,s))dsdt−f(x0,y0)p$

From (i)-(iv), and using class $\mathcal{A}$ conditions, the assertion follows. Thus, the proof is completed.

5 Conclusion

In this paper, the pointwise convergence of the convolution type nonlinear double singular integral operators depending on three parameters is investigated. In this work, we proved the theorems by using a specific weighted pointwise convergence method. Therefore, the main result is presented as Theorem 3.1. Also, by using main result, the rate of pointwise convergence of the indicated type operators is computed.

Acknowledgement

The author thanks to the unknown referees for their valuable comments and suggestions.

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Accepted: 2016-10-11

Published Online: 2016-11-11

Published in Print: 2016-01-01

Citation Information: Open Mathematics, Volume 14, Issue 1, Pages 897–907, ISSN (Online) 2391-5455,

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