Some properties of singular integral operators over an open curve Dedicated to the 85th birthday of Prof.Lu Jinake

In this paper, the stability properties, the endpoint behavior and the invertible relations of Cauchy-type singular integral operators over an open curve are discussed. If the endpoints of the curve are not special, this type of operators are proved to be stable. At the endpoints, either the singularity or smoothness of the operators are exactly described. And the function sets or spaces on which the operators are invertible as well as the corresponding inverted operators are given. Meanwhile, some applications for the solution of Cauchy-type singular integral equations are illustrated.     

Some properties of singular integral operators over an open curve

In this paper, the stability properties, the endpoint behavior and the invertible relations of Cauchy-type singular integral operators over an open curve are discussed. If the endpoints of the curve are not special, this type of operators are proved to be stable. At the endpoints, either the singularity or smoothness of the operators are exactly described. And the function sets or spaces on which the operators are invertible as well as the corresponding inverted operators are given. Meanwhile, some applications for the solution of Cauchy-type singular integral equations are illustrated. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10471048)     

Boundary properties for several singular integral operators in real Clifford analysis

On the basis of the Cauchy integral formulas for regular and biregular functions, we define some Cauchy-type singular integral operators. Then we discuss the Hlder continuous property of some singular integral operators with one integral variable. Then we divide a singular integral operator with two variables into three parts and prove its Hlder continuous property on the boundary.     

Factorization of singular integral operators with a Carleman backward shift: The case of bounded measurable coefficients

In this paper, we generalize our recent results concerning scalar singular integral operators with a Carleman backward shift, allowing more general coefficients, bounded measurable functions on the unit circle. Our aim is to obtain an operator factorization for singular integral operators with a backward shift and bounded measurable coefficients, from which such Fredholm characteristics as the kernel and the cokernel can be described. The main tool is the factorization of matrix functions. In the course of the analysis performed, we obtain several useful representations, which allow us to characterize completely the set of invertible operators in that class, thus providing explicit examples of such operators Dedicated to Professor A. Ferreira dos Santos on the occasion of his seventieth birthday     

On the Spectrum of Invertible Semi-hyponormal Operators

It is known that for a semi-hyponormal operator, the spectrum of the operator is equal to the union of the spectra of the general polar symbols of the operator. The original proof of this theorem involves the so-called singular integral model. The purpose of this paper is to give a different proof of the same theorem for the case of invertible semi-hyponormal operators without using the singular integral model.      

Singular Integral Operators with Fixed Singularities on Weighted Lebesgue Spaces

The paper is devoted to study of singular integral operators with fixed singularities at endpoints of contours on weighted Lebesgue spaces with general Muckenhoupt weights. Compactness of certain integral operators with fixed singularities is established. The membership of singular integral operators with fixed singularities to Banach algebras of singular integral operators on weighted Lebesgue spaces with slowly oscillating Muckenhoupt weights is proved on the basis of Balakrishnans formula from the theory of strongly continuous semi-groups of closed linear operators. Symbol calculus for such operators, Fredholm criteria and index formulas are obtained.     

Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations

Our main interest is an analog of a Cauchy-type integral for the theory of the Moisil-Theodoresco system of differential equations in the case of a piecewise-Lyapunov surface of integration. The topics of the paper concern theorems that cover basic properties of this Cauchy-type integral: the Sokhotskii-Plemelj theorem for it as well as a necessary and sufficient condition for the possibility of extending a given Hölder function from such a surface up to a solution of the Moisil-Theodoresco system of partial differential equations in a domain. A formula for the square of a singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between the theory of the Moisil-Theodoresco system of partial differential equations and some versions of quaternionic analysis.     

Invertible linear relations generated by an integral equation with Nevanlinna measure

We define families of maximal and minimal relations generated by integral equations with Nevanlinna operator measure and non-selfadjoint operator measure. We prove that if a restriction of a maximal relation is continuously invertible, then the inverse operator is integral. We study the case when the convergence of non-selfadjoint operator measures implies the convergence of the corresponding integral operators inverse to restrictions of maximal relations, and establish a sufficient condition for the validity of this implication. The obtained results are applicable to the study of differential equations with singular potentials.     

A note on a singular integral equation arising in water wave scattering

A quick method of solution of a singular integral equationof the first kind involving both logarithmic singularity aswell as Cauchy-type singularity is explained.     

Invertibility theory for Toeplitz plus Hankel operators and singular integral operators with flip

It is well known that a Toeplitz operator is invertible if and only if its symbols admits a canonical Wiener-Hopf factorization, where the factors satisfy certain conditions. A similar result holds also for singular integral operators. More generally, the dimension of the kernel and cokernel of Toeplitz or singular integral operators which and Fredholm operators can be expressed in terms of the partial indices of an associated Wiener-Hopf factorization problem.In this paper we establish corresponding results for Toeplitz plus Hankel operators and singular integral operators with flip under the assumption that the generating functions are sufficiently smooth (e.g., Hölder continuous). We are led to a slightly different factorization problem, in which pairs , instead of the partial indices appear. These pairs provide the relevant information about the dimension of the kernel and cokernel and thus answer the invertibility problem.     

On strongly elliptic singular integral operators with piecewise continuous coefficients

The concept of strongly elliptic operators is one of the main tools for approximating the solution of boundary integral equations by finite element methods (see [4-10]). In the present paper necessary and sufficient conditions for the strong ellipticity of singular integral operators with piecewise continuous matrix coefficients on a closed or open Ljapunov curve are obtained.     

Some properties of the Cauchy-type integral for the time-harmonic Maxwell equations

We study the analog of the Cauchy-type integral for the theory of time-harmonic electromagnetic fields in case of a piece-wise Liapunov surface of integration and we prove the Sokhotski-Plemelj theorem for it as well as the necessary and sufficient condition for the possibility to extend a given pair of vector fields from such a surface up to a solution of the time-harmonic Maxwell equations in a domain. Formula for the square of the singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between time-harmonic solutions of the Maxwell equations and some versions of quaternionic analysis.     

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part I: Shape Differentiability of Pseudo-homogeneous Boundary Integral Operators

In this paper we study the shape differentiability properties of a class of boundary integral operators and of potentials with weakly singular pseudo-homogeneous kernels acting between classical Sobolev spaces, with respect to smooth deformations of the boundary. We prove that the boundary integral operators are infinitely differentiable without loss of regularity. The potential operators are infinitely shape differentiable away from the boundary, whereas their derivatives lose regularity near the boundary. We study the shape differentiability of surface differential operators. The shape differentiability properties of the usual strongly singular or hypersingular boundary integral operators of interest in acoustic, elastodynamic or electromagnetic potential theory can then be established by expressing them in terms of integral operators with weakly singular kernels and of surface differential operators.     

Clifford分析中Cauchy型奇异积分算子的一些性质

First,we give a module estimation of the singular integral with a differential element.Then by proving the existences of Cauchy principal value we obtain the transformation formula of the Cauchy-type singular integrals with a parameter.     

Differentiation of Singular Integrals with Piecewise-Continuous Density and Boundary Values of Derivatives of a Cauchy-Type Integral

We establish sufficient conditions for the differentiability of a singular Cauchy integral with piecewise-continuous density. Formulas for the nth-order derivatives of a singular Cauchy integral and for the boundary values of the nth-order derivatives of a Cauchy-type integral are obtained.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 222–229, February, 2005.     

Some Properties of Cauchy-Type Singular Integrals in Clifford Analysis

First,we give a module estimation of the singular integral with a differential element.Then by proving the existences of Cauchy principal value we obtain the transformation formula of the Cauchy-type singular integrals with a parameter.     

刚性线夹杂与弹性圆夹杂的相互作用

本文将刚性线夹杂与弹性圆夹杂的相互作用，归为解一个标准的柯西型奇异积分方程，获得了刚性夹杂端点的应力强度因子及夹杂的界面应力．     

On homotopy properties and indices of families of singular and bisingular operators with piecewise-continuous coefficients

We present the stable homotopy classification of families of Fredholm and invertible singular integral operators with piecewise-continuous coeffcients in weighted L _p -spaces with Khvedelidze weights. We obtain a new topological formula for the calculation of the indices of families. These results allowed the authors to investigate homotopy properties of bisingular operators.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol.9, Suzdal Conference-3, 2003.     

Koch曲线上的齐次Riemann边值问题

当L为典型的分形曲线一Koch曲线时,提出了Riemann边值问题,但在一般情况下,在Koch曲线上所做的Cauchy型积分无意义.当对已知函数G(z),g(z)增加一定的解析条件,同时利用一列Cauchy型积分的极限函数,对定义在Koch曲线上的齐次Riemann边值问题进行了讨论,并得到与经典解析函数边值问题相类似的结果.     

裂纹与弹性夹杂的相互影响*

本文利用无限域上单根弹性夹杂和单根裂纹产生的位移和应力,将裂纹与弹性夹杂的相互影响问题归为解一组柯西型奇异积分方程,然后用此对夹杂分枝裂纹解答的奇性性态作了理论分析,并求得了振荡奇性界面应力场,对于不相交的夹杂裂纹问题,具体计算了端点的应力强度因子及夹杂上的界面应力,结果令人满意。