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Harmonic Analysis and its Applications

谐波分析及其应用

基本信息

批准号：
08454022
负责人：
IGARI Satoru
金额：
$ 4.86万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1996
资助国家：
日本
起止时间：
1996 至 1998
项目状态：
已结题

项目摘要

The problem of the harmonic analysis is often reduced to the invitational of translation invariant operators. As a useful real analysis method of such operators, there is the singular integral theory. However, we pay attention to the important translation invariant operators which aren't satisfied with the category of that theory. There, the Kakeya maximal function plays an important role an auxiliary function. Igari defines a maximal function which has a special base, and gave an estimate which gives the best possible estimate for radial functions, and also a partially Bourgain's result.Arai studied, form the viewpoint of the harmony analysis, the elliptic partial differential operators which degenerate in all points in the boundary, such as operators in pseudo-convex domain of the Stein manifolds and in the manifolds with theta-structure. He gets a basic result in the harmony analysis of such operators and is preparing a paper on it. Also, he found he fact which has a possibility to solve the pointwise Fatou problem for functions of several complex variables.It is studied for a long time the uniqueness of the solution of the Navier-Strokes equation, but it is still an unsolved major theme. Masuda introduced a quite new way which is of real variable method for this problem, and found a beginning of solving the problem. This result was reported in the international conference held in Verona. Also, he solved the conjecture on the classification of the minimal surfaces of constant curvature in two dimensional complex space from by the collaboration with K.Kenmotsu..Saito developed an important tool for the study of the monotone complete C*-algebra by giving an elementary proof for a regularity of the Rickart C*-algebra. He also gets a result about the completely additive measures on von Neumann algebras.

调和分析的问题常常归结为对平移不变算子的邀请。奇异积分理论是分析此类算子的一种实用方法。然而，我们注意到一些重要的平移不变算子并不满足于该理论的范畴。其中，Kakeya极大函数作为辅助函数起着重要的作用。Igari定义了具有特殊基的极大函数，给出了径向函数的最佳估计，部分地得到了Bourgain的结果。Arai从调和分析的角度，研究了在边界上所有点都退化的椭圆型偏微分算子，如Stein流形的伪凸域上的算子和具有结构的流形上的算子。他在这些算子的和声分析中得到了一个基本结果，并准备就此发表一篇论文。同时，他还发现了有可能求解多复变函数的逐点法图问题的事实。长期以来，人们对Navier-Strokes方程解的唯一性进行了研究，但这仍然是一个未解决的重大课题。增田为这一问题引入了一种全新的方法——实变量法，为解决这一问题开辟了一个开端。在维罗纳举行的国际会议上报告了这一结果。此外，他还与kenmotsu合作解决了二维复空间中常曲率极小曲面的分类猜想，并通过对ricart C*-代数的正则性的初等证明，为单调完全C*-代数的研究提供了重要的工具。他也得到了关于冯·诺依曼代数上的完全加性测度的结果。