Some results and problems on commutators

In this paper, the author introduces some late results and puts forward a few problems on commutators of many important operators in harmonic analysis, included the Bochner-Riesz operator below the critical index, the strongly singular integral operator, the pseudo-differential operator, a class of convolution operators with oscillatory kernel, the Marcinkiewicz integral operator, and the fractional integral operator with rough kernel.     

Singular integral operators in a weighted L2-space

Cauchy singular integral operators are characterized as operators in a weighted L²-space. The integral operator arises from a singular integral equation with variable coefficients. An appropriate weight function associated with the singular integral operator is constructed, and the set of polynomials orthogonal with respect to this weight function is defined. The action of the operator on polynomial sets is studied, and the definition of the operator is extended to a weighted L²-space. In this space, the operator is shown to be bounded, and, in some cases, isometric. Formulas are developed for the composition of the singular integral operator and its one sided inverse.     

Multilinear integral operators and mean oscillation

In this paper, the boundedness properties for some multilinear operators related to certain integral operators from Lebesgue spaces to Orlicz spaces are obtained. The operators include Calderón—Zygmund singular integral operator, fractional integral operator, Littlewood—Paley operator and Marcinkiewicz operator.     

Properties of an integral operator preserved under pointwise multiplication of its kernel by a function

Let K be a kernel that determines an integral operator on some space of functions, and let H be a function. This paper investigates conditions under which certain properties of the integral operator determined by K (especially compactness properties) also hold for the integral operator determined by HK.     

On the solvability of a complete two-dimensional hypersingular integral equation

We study the solvability of a complete two-dimensional linear hypersingular integral equation that contains a hypersingular integral operator in which the integral is understood in the sense of Hadamard finite value as well as an integral operator in which the integral is understood in the sense of principal value, an integral operator with a weakly singular kernel, and an integral-free term. We consider smooth solutions in the class of functions that have Hölder continuous derivatives outside a neighborhood of the boundary. We prove the Fredholm alternative and estimate the norm of the solution in a special metric.     

About one system of integral equations in kinetic theory

One system of integral convolution equations is considered on a half-line with an noninvertible matrix integral operator whose symbol has a fourth-order zero. The application of a special factorization method makes it possible to distinguish noninvertible factors in the original noninvertible operator and reduce the system to a new system with a nonsingular integral operator. The structural theorem of the existence of a solution to the original system is proved.     

On the univalence of an integral operator

In this work the author introduces a general integral operator and determines conditions for the univalence of this integral operator.     

与一类奇异积分算子相关的Toeplitz型算子的有界性

对与具有一般核的分数次奇异积分算子相关的Toeplitz型算子,本文证明了其sharp极大函数不等式,作为应用,得到了该算子在Lebesgue空间,Morrey空间和Triebel-Lizorkin空间的有界性.     

On new fractional integral inequalities using Marichev–Saigo–Maeda operator

Generalizations of fractional integral inequalities were introduced by many authors. The aim of our investigation is to establish some new fractional integral inequalities using Marichev–Saigo–Maeda (MSM) fractional integral operator for convex function. Further, we obtain some more fractional integral inequalities of Grüss type using MSM operator.     

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 integral representations of linear operators

In this paper we describe families of those bounded linear operators that are simultaneously unitarily equivalent to integral operators with smooth Carleman kernels. The singleton case of the main result implies that every integral operator is unitarily equivalent to an integral operator having smooth Carleman kernel.     

广义算子半群与广义分布参数系统的适定性

首先,针对广义分布参数系统的求解问题,提出了由Hilbert空间中有界线性算子所引导的广义算子半群和广义积分半群;其次,讨论了广义预解算子的性质、广义算子半群与广义积分半群的性质;最后,研究了广义分布参数系统的适定性问题.     

Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-Line

1.IntroductionInthispaPer,westudynumericalsolutionstointegralequationsofthesecondkinddefinedonthehalfline.Morepreciselyweconsidertheequationy(t)+Iooa(t,s)y(s)ds=g(t),OS相似文献   

Feynman operator calculus: The constructive theory

In this paper, we survey progress on the Feynman operator calculus and path integral. We first develop an operator version of the Henstock-Kurzweil integral, construct the operator calculus and extend the Hille-Yosida theory. This shows that our approach is a natural extension of operator theory to the time-ordered setting. As an application, we unify the theory of time-dependent parabolic and hyperbolic evolution equations. Our theory is then reformulated as a sum over paths, providing a completely rigorous foundation for the Feynman path integral. Using our disentanglement approach, we extend the Trotter-Kato theory.     

Integral representations of irregular root functions of loaded second-order differential operators

We consider a second-order differential operator on an interval of the real line with integral boundary conditions. We show how to construct the adjoint operator. The differential operation of the adjoint operator can be loaded, and the domain of that operator can contain functions that, together with their derivatives, have jump discontinuities at countably many points. For the root functions of the adjoint operator, we obtain integral representations, in particular, a mean-value formula.     

A Hilbert-Schmidt integral operator and the Weil distribution

In this paper, a positive operator is given. It is shown that the product of this positive operator and the convolution operator is a trace class Hilbert-Schmidt integral operator and has nonnegative eigenvalues. A formula is given for the trace of this product operator. It seems that this product operator is the closest trace class integral operator which has nonnegative eigenvalues and is related to the Weil distribution in the context of Connes’ program for the Riemann hypothesis. A relation is given between the trace of the product operator and the Weil distribution.

     

Banach空间中一类非线性Volterra型积分方程整体解的存在性

利用一个新的不动点定理在较弱的条件下考虑Banach空间中一类非线性Voherra型积分方程整体解的存在性．由于非线性项中含有非线性积分算子,相对于线性积分算子,文章所得结论推广并丰富了已有文献的一些结果．     

Clifford分析中一类高阶奇异积分算子关于区域摄动的稳定性

定义了Clifford分析中一类高阶奇异Teodorescu算子,利用几个重要的不等式研究了这类算子关于积分区域的边界摄动的稳定性.     

修正的isotonic算子的不动点与Mann迭代

在论文的第一部分, 研究与讨论了isotonic函数的Cauchy型奇异积分算子$T$的 H\"{o}lder连续性及$\|T[f]\|_{\alpha}$与$\|f\|_{\alpha}$间的关系. 在第二部分, 引入了isotonic函数的修正的Cauchy型奇异积分算子$T'$, 首先利用压缩不动点定理证明此修正的Cauchy型奇异积分算子$T'$存在唯一不动点, 然后给出其不动点的Mann迭代序列并证明此序列强收敛于$T'$算子的不动点.     

An exponentially convergent algorithm for nonlinear differential equations in Banach spaces

An exponentially convergent approximation to the solution of a nonlinear first order differential equation with an operator coefficient in Banach space is proposed. The algorithm is based on an equivalent Volterra integral equation including the operator exponential generated by the operator coefficient. The operator exponential is represented by a Dunford-Cauchy integral along a hyperbola enveloping the spectrum of the operator coefficient, and then the integrals involved are approximated using the Chebyshev interpolation and an appropriate Sinc quadrature. Numerical examples are given which confirm theoretical results.