Comprehensive Study of Real Analysis and Functional Analysis

实分析与泛函分析综合研究

• 批准号: 11304009
• 负责人: MIYACHI Akihiko
• 金额: $ 17.09万
• 依托单位: Tokyo Woman's Christian University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A).
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11304009/
• 关键词: Singular integral; Hardy space; Navier-Stokes equation; Phase transition; Dirac equation; Pfaffian; p-hyponormal operator; chaos theory; Hardy空間; Navier-Stokes方程式; 相転移; ホロノミック系; Turnbull等式; Putnam不等式; Jaynes-Cummingsモデル

The research is extended to the following 5 areas : (1) Harmonic analysis by real variable methods ; (2) Partial differential equations by functional- analysis method ; (3) Operator algebras and function algebras ; (4) Represen- tation theory ; (5) Quantum logics and function spaces.Our main results are the following. In the area (1) : we obtained real variable characterization for the weighted Hardy spaces on a domain and for the Herz-type Hardy spaces ; proved boundedness of several singular integral operators with nonregular kernels ; and got Hardy space estimate for the mul- tilinear operator which is defined as a sum of products of singular integrals. In (2) : we got strong unique continuation theorem for the Maxwell equation, the Dirac equation, and for the Schrodinger equations ; studied propagation of singularities for the Schrodinger equations ; used Hardy space estimate to study the Stokes operator ; proved time-global existence of solutions to the Cauchy problem for the Navier-Stokes equation with nondecaying initial data ; studied the properties of the attractor of the nonlinear dynamical systoms related to the phase-transition phenomena ; obtained several algorithms for computing the Grothendieck residues, de Rham cohomology of algebraic varieties, and D-modules. In (3) : we gave a generalization of Putnam's inequality ; got the classification of actions of factors to operator algebras ; constructed the index theory for C^* algebras ; and solved the single generator problem on the Bohr group. In (4) : we got Hardy space theory on rank 1 semisimple Lie group ; and got summation formula for Pfaffians and its applications. In (5) : we gave a solution to an NP complete problem as an application of the quantum algorithm and the chaos theory ; studied several properties of p-hyponormal operators.

研究内容扩展到以下5个领域：（1）实变量方法调和分析； (2) 泛函分析法偏微分方程； (3)算子代数和函数代数； (4) 表示论； (5)量子逻辑和函数空间。我们的主要结果如下。在区域（1）中：我们获得了域上的加权Hardy空间和Herz型Hardy空间的实变量表征；证明了几个具有非正则核的奇异积分算子的有界性；并得到了多重线性算子的 Hardy 空间估计，其定义为奇异积分的乘积之和。在(2)中：我们得到了麦克斯韦方程、狄拉克方程和薛定谔方程的强唯一连续定理；研究了薛定谔方程的奇点传播；利用Hardy空间估计研究Stokes算子；证明了具有非衰减初始数据的纳维-斯托克斯方程的柯西问题解的时间全局存在性；研究了与相变现象相关的非线性动力系统吸引子的性质；获得了计算 Grothendieck 留数、代数簇的 de Rham 上同调和 D 模的多种算法。在(3)中：我们给出了普特南不等式的推广；得到因子对算子代数作用的分类；构建了 C^* 代数的指数理论；并解决了玻尔群上的单发电机问题。在(4)中：我们得到了关于1阶半单李群的Hardy空间理论；得到了Pfaffians的求和公式及其应用。在(5)中：我们给出了一个NP完全问题的解决方案，作为量子算法和混沌理论的应用；研究了 p-次正规算子的几个性质。

项目成果

Masanori Ohya: "NP problem in quantum algorithm"Open Systems and Information Dynamics. 7. 33-39 (2000)

Masanori Ohya：“量子算法中的NP问题”开放系统和信息动力学。

儀我美一: "非線形偏微分方程式-解の漸近挙動と自己相似解"共立出版. (1999)

Yoshikazu Giga：“非线性偏微分方程 - 解和自相似解的渐近行为”Kyoritsu Shuppan (1999)。

Toshinori Oaku: "A localization algorithm for D-modules"Journal of Symlolic computation. 29. 721-728 (2000)

Toshinori Oaku：“D 模块的定位算法”Symlolic 计算杂志。

Akihiko Miyachi: "Remarks on Herz-type Hardy spaces"Acta Math.Sinica (New Series). 17(掲載予定). (2000)

宫地明彦：“关于赫兹型哈代空间的评论”数学学报（新丛书）17（待出版）。

Akihiko Miyachi: "Hardy space estimates for the product of singular integrals."Canadian Journal of Mathematics. 52. 381-411 (2000)

Akihiko Miyachi：“奇异积分乘积的哈代空间估计。”加拿大数学杂志。