# Singular integral operators of convolution type

## Top PDF Singular integral operators of convolution type:

### A note on rate of convergence of double singular integral operators

In paper [], Bardaro et al. obtained some approximation results concerning the point- wise 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 .

### Convergence of Singular Integral Operators in Weighted Lebesgue Spaces

Bardaro, Karsli and Vinti [5] obtained some approximation results related to the point- wise convergence and the rate of pointwise convergence for non-convolution type linear op- erators at a Lebesgue point based on Bardaro and Mantellini’s study [4]. After that, they got similar results for its nonlinear counterpart in [6] while in an another study [7], 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 were investigated. Almali [1] studied the problem of pointwise convergence of non-convolution type inte- gral operators at Lebesgue points of some classes of measurable functions.

### A study on pointwise approximation by double singular integral operators

of nonlinear singular integral operators [, ], a family of nonlinear m-singular integral operators [], Fejer-Type singular integrals [], moment type operators [], a family of nonlinear Mellin type convolution operators [], nonlinear integral operators with homo- geneous kernels [] and a family of Mellin type nonlinear m-singular integral operators [].

### Solvability of some classes of singular integral equations of convolution type via Riemann–Hilbert problem

There were rather complete investigations on the method of solution for integral equa- tions of Cauchy type and integral equations of convolution type [1–5]. The solvability of a singular integral 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 convolution type, mathematically, belong to an interesting subject in the theory of integral equations.

### Some classes of singular integral equations of convolution type in the class of exponentially increasing functions

It is well known that singular integral 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 singular integral 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 [7] studied the conditions of their Noethericity in the more general case. In recent decades, many mathematicians studied some SIEs of convolution type. Litvinchuk [8] 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 [9] studied the Noether theory of convolution type SIEs with constant coeﬃcients. Nakazi-Yamamoto [10] 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 [11] 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.

### A sharp inequality for multilinear singular integral operators with non smooth kernels

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 singular integral operators. In [], Pérez and Trujillo- Gonzalez proved a sharp estimate for the multilinear commutator when b j ∈ Osc exp L rj (R n )

### Some Properties of Certain Class of Integral Operators

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 .

### A Regularization of Fredholm type singular integral equations

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).

### On an extension of singular integrals along manifolds of finite type

Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Inventiones Mathematicae 84 1986, no.. Pan, L p estimates for singular integrals associated to [r]

### ON THE BOUNDEDNESS OF DUNKL-TYPE FRACTIONAL INTEGRAL OPERATOR IN THE GENERALIZED DUNKL-TYPE MORREY SPACES

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

### A fourth-order elliptic Riemann type problem in \(\mathbb{R}^{3}\)

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

### Inequality estimates for the boundedness of multilinear singular and fractional integral operators

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.

### Lipschitz estimates for commutators of singular integral operators associated with the sections

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

### Operator Covariant Transform and Local Principle

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

### Oscillation and variation inequalities for the multilinear singular integrals related to Lipschitz functions

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.

### Direct Estimates for Certain Integral Type Operators

Recently, Kajla [15] introduced a new sequence of summation-integral type operators 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 [11] proposed the integral modification of the operators (1) by taking weights of Beta basis functions as follows:

### Vol 8, No 8 (2017)

The integral operators, in particular convolution operators have already been studied extensively over the last few decades. For more detail about composition operators, integral operators, convolution operators, composite integral operators and composite convolution operators we refer to Singh and Manhas [11], Halmos and Sunder ([5],[6]), Stepanov ([9], [10]), Gupta and Komal [1] and Gupta ([2], [3], [4]). Whitley [12] established the Lyubic's [7] 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 ( [2], [3]).

### Weighted norm inequalities for multilinear Calderón Zygmund operators in generalized Morrey spaces

Abstract In this paper, the authors study the boundedness of multilinear Calderón-Zygmund singular integral operators and their commutators in generalized Morrey spaces.. MSC: 42B20 Keyw[r]

### On the initial value problem for a partial differential equation with operator coefficients

Singular Integral Operators and Cauchy’s Problem for Some Partial Differential Equations With Operator Coefficients, Transaction of the Science Centre, Alexandria University Vol.. Lnear [r]

### Integral Transforms of Fourier Cosine and Sine Generalized Convolution Type

Recently, a class of integral transforms that is related to the generalized convolution (1.11) has been introduced and investigated in [12]. 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