In paper , Bardaro et al. obtained some approximation results concerning the point- wise convergence and the rate of pointwise convergence for non-convolutiontype 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-singularintegraloperators at m-Lebesgue points of f .
Bardaro, Karsli and Vinti  obtained some approximation results related to the point- wise convergence and the rate of pointwise convergence for non-convolutiontype linear op- erators at a Lebesgue point based on Bardaro and Mantellini’s study . After that, they got similar results for its nonlinear counterpart in  while in an another study , the pointwise convergence and the rate of pointwise convergence results for a family of Mellin type nonlinear m-singularintegraloperators at m-Lebesgue points of f were investigated. Almali  studied the problem of pointwise convergence of non-convolutiontype inte- gral operators at Lebesgue points of some classes of measurable functions.
of nonlinear singularintegraloperators [, ], a family of nonlinear m-singularintegraloperators , Fejer-Typesingular integrals , moment typeoperators , a family of nonlinear Mellin typeconvolutionoperators , nonlinear integraloperators with homo- geneous kernels  and a family of Mellin type nonlinear m-singularintegraloperators .
There were rather complete investigations on the method of solution for integral equa- tions of Cauchy type and integral equations of convolutiontype [1–5]. The solvability of a singularintegral equation (SIE) of Wiener–Hopf type with continuous coeﬃcients was considered in [6, 7]. For operators with Cauchy principal value integral and convolution, the conditions of their Noethericity were discussed in [8, 9]. Recently, Li [10–16] studied some classes of SIEs with convolution kernels and gave the Noether theory of solvability and the general solutions in the cases of normal type. It is well known that integral equa- tions of convolutiontype, mathematically, belong to an interesting subject in the theory of integral equations.
It is well known that singularintegral equations (SIEs) and integral equations of convo- lution type are two basic kinds of equations in the theory of integral equations. There have been many papers studying singularintegral equations and a relatively complete the- oretical system is almost formed (see, e.g., [1–6]). These equations play important roles in other subjects and practical applications, such as engineering mechanics, physics, frac- ture mechanics, and elastic mechanics. For operators containing both the Cauchy princi- pal value integral and convolution, Karapetiants-Samko  studied the conditions of their Noethericity in the more general case. In recent decades, many mathematicians studied some SIEs of convolutiontype. Litvinchuk  studied a class of Wiener-Hopf type inte- gral equations with convolution and Cauchy kernel and proved the solvability of the equa- tion. Giang-Tuan  studied the Noether theory of convolutiontype SIEs with constant coeﬃcients. Nakazi-Yamamoto  proposed a class of convolution SIEs with discontin- uous coeﬃcients and transformed the equations into a Riemann boundary value problem (RBVP) by Fourier transform, and given the general solutions of the equation. Later on, Li  discussed the SIEs with convolution kernels and periodicity, which can be trans- formed into a discrete jump problems by discrete Fourier transformation, and the solvable conditions and the explicit expressions of general solutions were obtained.
known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [–]). Hu and Yang (see ) proved a variant sharp estimate for the multilinear singularintegraloperators. In , Pérez and Trujillo- Gonzalez proved a sharp estimate for the multilinear commutator when b j ∈ Osc exp L rj (R n )
Recently, Wang et al. 3 obtained several inclusion relationships and integral- preserving properties associated with some subclasses involving the operator Q λ α,β , some sub- ordination and superordination results involving the operator are also derived. Furthermore, Sun et al. 4 investigated several other subordination and superordination results for the operator Q λ α,β .
Now as it is clear, the integral term on the right-hand side of (1.1) is at most weakly singular. Using this regularized formula we are going to solve some important ﬁrst and second kind Fredholm’s integral equations in which the kernels are singular. Before starting using (1.1), in the following we show its equivalent formulation on an interval (a,b).
The Hardy–Littlewood maximal function, fractional maximal function and frac- tional integrals are important technical tools in harmonic analysis, theory of functions and partial differential equations. On the real line, the Dunkl oper- ators are differential-difference operators associated with the reflection group Z 2 on R. In the works [1, 17, 24, 35] the maximal operator associated with
two methods are used to deal with higher-order boundary value problems. One approach is to transform the boundary value problems for k-regular functions and poly-harmonic functions into equivalent boundary value problems for regular functions in Cliﬀord anal- ysis by the Almansi type decomposition theorem . The other is to make use of higher- order integral representation formulas and a Cliﬀord algebra approach [, , ]. Obvi- ously, the ﬁrst method fails to solve a system of the fourth-order elliptic equation i.e., ( –κ )u = , coupled by the Riemann boundary conditions. Using the second method, we need to investigate factorizations of the fourth-order elliptic operator in the framework of a Cliﬀord algebra. Furthermore, we will construct higher-order kernels. The key idea is to choose an appropriate framework of the Cliﬀord algebra. A lot of boundary value prob- lems for some functions with the Cliﬀord algebra Cl(V n, ) (n ≥ ) have been studied; for
a multilinear commutator of the fractional integral operator. It is well known that multi- linear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [–, , , , , , ]). The purpose of this paper is to study the weighted boundedness properties for the multilinear operator.
As is well known, linear commutators are naturally appearing operators in harmonic analysis that have been extensively studied already. In general, the boundedness results of commutators in harmonic analysis can be used to characterize some important func- tion spaces such as BMO spaces, Lipschitz spaces, Besove spaces and so on (see [–]). Coifman et al.  applied the boundedness to some non-linear PDEs, which perfectly il- lustrate the intrinsic links between the theory of compensated compactness and the clas- sical tools of harmonic and real analysis. As for some other essential applications to PDEs such as characterizing pseudodiﬀerential operators, studying linear PDEs with measur- able coeﬃcients and the integrability theory of the Jacobians, interested researchers can
The final step of the construction is synthesis of an operator from the field of local representatives using the inverse covariant transform from Subsection 2.3. To this end we need to chose an invariant pairing on the group G, keeping the ¯ ax + b group as an archetypal example. For operators of local type the whole information is concentrated in the arbitrary small neighborhood of the subgroup G ⊂ G, cf. Cor. ¯ 18. Thus we select the Hardy-type functional (7) instead of the Haar one (6). Let dµ be the Haar measure on the group G. Then the following integral
The oscillation and variation for some families of operators have been studied by many authors on probability, ergodic theory, and harmonic analysis; see [–]. Recently, some authors [–] researched the weighted estimates of the oscillation and variation operators for the commutators of singular integrals.
Recently, Kajla  introduced a new sequence of summation-integraltypeoperators and established some approximation properties e.g. weighted approximation, asymptotic formula and error estimation in terms of modulus of smoothness. Very recently, Gupta and Agrawal  proposed the integral modification of the operators (1) by taking weights of Beta basis functions as follows:
The integraloperators, in particular convolutionoperators have already been studied extensively over the last few decades. For more detail about composition operators, integraloperators, convolutionoperators, composite integraloperators and composite convolutionoperators we refer to Singh and Manhas , Halmos and Sunder (,), Stepanov (, ), Gupta and Komal  and Gupta (, , ). Whitley  established the Lyubic's  conjecture and generalized it to Volterra composition operators on L p [0,1]. This paper broadens the approach that was taken into account in the papers of Gupta ( , ).
Recently, a class of integral transforms that is related to the generalized convolution (1.11) has been introduced and investigated in . In this paper, we will consider a class of in- tegral transform which has a connection with the generalized convolution (1.13), namely, the transforms of the form