多线性奇异积分算子的一些新进展（英文）

多线性奇异积分算子产生于调和分析、偏微分方程以及遍历论等领域的各类问题.本文将综述多线性奇异积分算子的一些最新进展,主要包括多线性Calderón-Zygmund算子、多线性H?rmander乘子以及一些相关问题.     

带非光滑核的多线性奇异积分算子的有界性

该文对带非光滑核的多线性奇异积分算子建立了sharp函数估计,作为应用,得到了该多线性奇异积分算子的L~p(1p∞)范数不等式.     

多线性分数次积分算子交换子的双权弱型不等式

多线性分数次积分算子定义为利用分数次Orlicz极大算子和sharp函数,得到了多线性分数次积分算子交换子的双权弱(p,p)型不等式成立的充分条件.     

乘积广义局部Morrey空间上由多线性分数次积分算子生成的多重次线性算子和交换子（英文）

本文在调和分析中大多数算子都满足的一般尺度条件下,得到了乘积广义局部Morrey空间上由多线性分数次积分算子生成的特定多重次线性算子的有界性.还证明了由局部Campanato函数和多线性分数次积分算子生成的多线性算子的交换子在乘积广义局部Morrey空间上有界.     

极大多线性奇异积分算子的一些点态估计

建立了联系极大多线性奇异积分算子与某些古典极大算子的两个点态估计, 这些点态估计隐含着极大多线性奇异积分算子的重排估计和BLO(Rⁿ)估计.     

度量测度空间上多线性分数次积分算子的加权模不等式

设(X, d,μ)为满足几何双倍条件的度量测度空间.本文建立了(X, d,μ)上多线性分数次积分算子以及多线性分数次极大函数的弱型加权模不等式.作为应用,本文还建立了(X, d,μ)上多线性分数次积分算子在Morrey空间上的弱型加权模不等式.     

几类多线性算子在B_σ-Morrey空间上的有界性

研究了多线性Hardy-Littlewood极大算子,多线性分数次极大算子,多线性奇异积分和分数次积分算子在Bσ-Morrey空间上的有界性.     

加权耦合型空间上的多线性算子（英文）

本文证明了,如果满足特定点态估计的多线性算子T和它的多线性交换子、迭代交换子分别在乘积加权Lebesgue空间上有界,那么它们也在加权耦合型空间上有界.作为应用,我们说明了多线性Littlewood-Paley函数、具有卷积或非卷积核的多线性Marcinkiewicz积分和它们的线性交换子和迭代交换子均在乘积加权耦合型空间上有界.引入耦合型Campanato空间后,我们得到了多线性分数次积分算子是从耦合型空间到耦合型Campanato空间上有界的.我们的结果对于线性的分数次积分算子也是新的.     

强奇异积分算子的多线性交换子的Sharp极大函数估计和连续性

本文对由强奇异积分算子和加权Lipschitz函数生成的多线性交换子证明了其sharp极大函数估计,作为应用,得到了该多线性交换子的连续性.     

Calderon-Zygmund核的多线性振荡奇异积分算子的Lp-有界性

研究关于Calderon-Zygmund标准核的多线性振荡奇异积分算子,证明了这类算子的Lp-有界性.     

Integral Equation Method for the First and Second Problems of the Stokes System

Using the integral equation method we study solutions of boundary value problems for the Stokes system in Sobolev space H ¹(G) in a bounded Lipschitz domain G with connected boundary. A solution of the second problem with the boundary condition $\partial {\bf u}/\partial {\bf n} -p{\bf n}={\bf g}$ is studied both by the indirect and the direct boundary integral equation method. It is shown that we can obtain a solution of the corresponding integral equation using the successive approximation method. Nevertheless, the integral equation is not uniquely solvable. To overcome this problem we modify this integral equation. We obtain a uniquely solvable integral equation on the boundary of the domain. If the second problem for the Stokes system is solvable then the solution of the modified integral equation is a solution of the original integral equation. Moreover, the modified integral equation has a form f?+?S f?=?g, where S is a contractive operator. So, the modified integral equation can be solved by the successive approximation. Then we study the first problem for the Stokes system by the direct integral equation method. We obtain an integral equation with an unknown ${\bf g}=\partial {\bf u}/\partial {\bf n} -p{\bf n}$ . But this integral equation is not uniquely solvable. We construct another uniquely solvable integral equation such that the solution of the new eqution is a solution of the original integral equation provided the first problem has a solution. Moreover, the new integral equation has a form ${\bf g}+\tilde S{\bf g}={\bf f}$ , where $\tilde S$ is a contractive operator, and we can solve it by the successive approximation.     

On a class of optimum problems in structural design

The paper deals with the existence of solutions for a class of optimal design problems. The notion of relaxation of an integral functional with respect toG-convergence is introduced, and a general integral representation theorem is obtained for the relaxed functional. For a particular class of functionals, this integral representation is computed explicitly.This work has been realized in a National Research Project in Mathematics supported by the Ministero della Pubblica Istruzione (Italy).     

Numerical solution of some classes of integral equations using Bernstein polynomials

This paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using Bernstein polynomials as basis. The integral equations considered are Fredholm integral equations of second kind, a simple hypersingular integral equation and a hypersingular integral equation of second kind. The method is explained with illustrative examples. Also, the convergence of the method is established rigorously for each class of integral equations considered here.     

多线性奇异积分在Hardy空间的有界性

通过相关的分数次积分算子的性质,证明了一类多线性算子在Hardy空间上的(Hp,Lp)有界性,得到一种简明的方法.     

滞后型泛函微分方程的有界变差解

本文研究了一类滞后型泛函微分方程的有界变差解.利用Henstock-Kurzweil积分与Schauder不动点定理,在Henstock-Kurzweil积分下,得到了这类滞后型泛函微分方程有界变差解的存在性定理,推广了一些相关的结果.     

Smooth functional derivatives in Feynman path integrals by time slicing approximation

We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, functional differentiation, translation and real linear transformation. The integration by parts and Taylor's expansion formula with respect to functional differentiation holds in Feynman path integral. Feynman path integral is invariant under translation and orthogonal transformation. The interchange of the order with Riemann-Stieltjes integrals, the interchange of the order with a limit, the semiclassical approximation and the fundamental theorem of calculus in Feynman path integral stay valid as well as N. Kumano-go [Bull. Sci. Math. 128 (3) (2004) 197-251].     

Non-absolutely convergent integrals with respect to distributions

We define an integral of a function with respect to a distribution. In case that the underlying distribution is just the Lebesgue measure, the definition leads to a new non-absolutely convergent integral which is wider than the Denjoy–Perron integral. We present a version of the Gauss–Green theorem where the new integral is used for both interior and boundary terms. As a by-product, we characterize the predual Sobolev space \(W^{-1,1}\) .     

Weighted boundedness of some integral operators on weighted <Emphasis Type="Italic">λ</Emphasis>- central Morrey space

In this paper, the authors prove the weighted boundedness of singular integral and fractional integral with a rough kernel on the weighted λ-central Morrey space. Moreover, the weighted estimate for commutators of singular integral with a rough kernel on the weighted λ-central Morrey space is also given.     

SOLUTION OF A CLASS INTEGRALEQUATIONS

1TwoTheoremsonmultipleintegrationTheorem1.1Supposethatf(t)isarbitrarycontinuousfunctiondefinedonRandh(fl,'')fi)isarbitrarycontinuousfunctiondefinedonm(1Sp0}ProofWeshallproveitbymathematicalinduction…     

A new <Emphasis Type="Italic">q</Emphasis>-Selberg integral,Schur functions,and Young books

Recently, Kim and Oh expressed the Selberg integral in terms of the number of Young books which are a generalization of standard Young tableaux of shifted staircase shape. In this paper the generating function for Young books according to major index statistic is considered. It is shown that this generating function can be written as a Jackson integral which gives a new q-Selberg integral. It is also shown that the new q-Selberg integral has an expression in terms of Schur functions.