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 .

13 Read more

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.

Show more
13 Read more

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** [].

10 Read more

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.

Show more
19 Read more

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.

Show more
14 Read more

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 )

14 Read more

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 .

10 Read more

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).

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]

16 Read more

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

20 Read more

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

Show more
19 Read more

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.

15 Read more

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

Show more
12 Read more

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**

11 Read more

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.

14 Read more

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:

18 Read more

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]).

Show more
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]

10 Read more

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]

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

11 Read more