A Class of Singular Integral Equation of Convolution Type with csc(τ-θ) Kernel

In this paper, we propose and discuss a class of singular integral equation of convolution type with csc(τ- θ) kernel in class L2[-π, π]. Using discrete Fourier transform and the lemma, this kind of equations is transformed to discrete system of equations, and then we obtain the solvable conditions and the explicit solutions in class L2[-π, π].     

Integral Solution Operators for the Cauchy－Riemann Equations on an Open Set with Piecewise C~1－Boundary

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Principal Value and Its Applications for One Type of Cauchy—Fantappiè Integral and the System of Equations

Acta Mathematica Sinica, English Series - It is known that principal value plays vital roles in the study of singular integral. In this paper, one type of Cauchy—Fantappiè integral with...     

Simplest Singular Points of 1–forms Invariant with Respect to an Action of Group of Order Three

Singular points that cannot be removed by small deformations of 1–forms and invariant with respect to the action of a cyclic group of order 3 are classified. It is proved that for ?₃–invariant 1–forms the equivariant index of a singular point, as an element of the representation ring of the group, coincides with the class of the representation on the space of germs of forms of highest order factored by the subspace of forms divisible by the given 1–form.     

Unicity of the Integrated Density of States for Relativistic Schrödinger Operators with Regular Magnetic Fields and Singular Electric Potentials

We show coincidence of the two definitions of the integrated density of states (IDS) for a class of relativistic Schrödinger operators with magnetic fields and scalar potentials introduced in Iftimie et al. (Publ Res Inst Math Sci 43(3):585–623, 2007; Topics in applied mathematics and mathematical physics, Editura Academiei Române, 2008), the first one relying on the eigenvalue counting function of operators induced on open bounded sets with Dirichlet boundary conditions, the other one involving the spectral projections of the operator defined on the entire space. In this way one generalizes the results of Doi et al. (Math Z 237:335–371, 2001) and Iftimie (Publ Res Inst Math Sci 41(2):307–327, 2005) for non-relativistic operators. The proofs needs the magnetic pseudodifferential calculus developed in Iftimie et al. (Publ Res Inst Math Sci 43(3):585–623, 2007), as well as a Feynman-Kac-Itô formula for Lévy processes (Ichinose and Tamura, Commun Math Phys 105(2):239–257, 1986; Iftimie et al. Topics in applied mathematics and mathematical physics, Editura Academiei Române, 2008). In addition, in case when both the magnetic field and the scalar potential are periodic, one also proves the existence of the IDS.     

Estimation of the Density of Hypoelliptic Diffusion Processes with Application to an Extended Itô's Formula

We prove a uniform bound for the density, p _t (x), of the solution at time t(0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p _t (x). These results are applied to extend the Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al. ⁽⁵⁾     

Hölder estimates for solutions of the Cauchy problem for the porous medium equation with external forces

We study the interior Hölder regularity problem for weak solutions of the porous medium equation with external forces. Since the porous medium equation is the typical example of degenerate parabolic equations, Hölder regularity is a delicate matter and does not follow by classical methods. Caffrelli-Friedman, and Caffarelli-Vazquez-Wolansky showed Hölder regularity for the model equation without external forces. DiBenedetto and Friedman showed the Hölder continuity of weak solutions with some integrability conditions of the external forces but they did not obtain the quantitative estimates. The quantitative estimates are important for studying the perturbation problem of the porous medium equation. We obtain the scale invariant Hölder estimates for weak solutions of the porous medium equations with the external forces. As a particular case, we recover the well known Hölder estimates for the linear heat equation.     

Singular Integral Operators with Cauchy Kernel in Fractional Spaces

We single out the Besov spaces that embed into the class of continuous functions and enjoy the Fredholm theory of linear singular integral equations with Cauchy kernel. We give basic results of this theory in the class of continuous (rather than Holder continuous) functions in terms of Besov spaces. Alongside elliptic operators we consider violations of ellipticity: the degeneration of the symbol of an operator at finitely many points.     

一类奇异积分和Cauchy型积分关于积分曲线的稳定性

本文讨论了当任意给定的f（τ,t）在某个区域E内属于H类时,奇异积分在封闭或开口光滑曲线E发生光滑扰动时的稳定性,并给出了相应的误差估计．作为应用,我们还讨论了当（t）在E内属于H类时,Cauchy型积分,在封闭光滑曲线E发生光滑扰动时的稳定性及误差估计．     

Stability of Cauchy-type Singular Integral Equations over an Interval

About the Cauchy type of singular integral equations over an interval, the paper presents three main results: the unified normalization form, the stability property on the smooth function space C^m,, and the pointwise error estimate of the solution for their perturbed equations. As the applications of the main results, the approximate methods for the equation are mentioned. Mathematics Subject Classification (2000) 45E05.     

一类对偶积分方程组正则化为Cauchy奇异积分方程组解法

本文基于Mellin变换法求解复杂更一般形式的对偶积分方程组．通过积分变换，由实数域化成复数域上的方程组，引入未知函数的积分变换，移动积分路径，应用Cauchy积分定理，实现退耦正则化为Cauchy奇异积分方程组，由此给出一般性解，并严格证明了对偶积分方程组退耦正则化为Cauchy奇异积分方程组与原对偶积分方程组等价性，以及对偶积分方程组解的存在性和唯一性．给出的解法和理论解，作为求解复杂对偶积分方程组一种有效解法，可供求解复杂的数学、物理、力学中的混合边值问题应用．     

On the Cauchy Principal Value of the Khenkin-Ramirez Singular Integral in Strictly Pseudoconvex Domains of ℂ <Superscript>n</Superscript>

We show that in the multidimensional case (unlike the complex plane) the Cauchy principal value of the Khenkin-Ramirez singular integral in strictly pseudoconvex domains is equal to the limit value of this integral inside the domain.Original Russian Text Copyright © 2005 Kytmanov A. M. and Myslivets S. G.The first author was supported by a grant of the President of the Russian Federation and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-1212.2003.1); the second author was supported by the Krasnoyarsk Region Science Foundation (Grant 12F0063C).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 625–633, May–June, 2005.     

cauchy核奇异积分方程关于积分曲线的稳定性

当E为复平面上的有界连通区域,所有已知函数在E上满足Holder条件,光滑封闭曲线гсE时,借助广义逆,讨论了正则型Cauchy核奇异积分方程在г发生某种光滑扰动时的稳定性问题,给出了相应的误差估计,并建立了收敛性定理.     

一类具有准齐次核的Hilbert型奇异重积分算子的范数及应用

引入准齐次函数的概念,研究了具有准齐次核的一类Hilbert型奇异重积分算子的范数问题,并讨论其应用.     

复双球垒域上奇异积分的Cauchy主值

利用一种新的挖法定义复双球垒域上的立体角系数,得到奇异积分的Cauchy主值的存在性.推广了复超球上的奇异积分理论.     

BMO与弱核条件下的Calderón-Zygmund算子交换子的有界性

本文证明了BMO与弱核条件下Calderón-Zygmund奇异积分算子的交换子 [B,T]f=BTf-T(Bf)的L2有界性,这里B ∈BMO,T对增生的b1,b2,满足 T(b1)∈BMO,T*(b2)∈BMO,b2Tb1∈WBP．     

BMO与弱核条件下的Calderón-Zygmund算子交换子的有界性

本文证明了BMO与弱核条件下Calderón-Zygmund奇异积分算子的交换子 [B,T]f=BTf-T(Bf)的L2有界性,这里B ∈BMO,T对增生的b1,b2,满足 T(b1)∈BMO,T*(b2)∈BMO,b2Tb1∈WBP．