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.

该研究扩展到以下5个领域：（1）通过实际变量方法进行谐波分析； （2）通过功能分析方法进行部分微分方程； （3）算子代数和功能代数； （4）代表理论； （5）量子逻辑和功能空间。我们的主要结果如下。在区域（1）中：我们获得了域上的加权强壮空间和HERZ型耐寒空间的实际变量表征；几个奇异整体操作员与非规范内核的证明有限。并获得了定义为奇异积分产物之和的Multinear操作员的Hardy空间估计。在（2）中：对于Maxwell方程，Dirac方程和Schrodinger方程，我们获得了强烈的独特延续定理。研究了Schrodinger方程的奇异点的传播；使用Hardy空间估算来研究Stokes操作员；事实证明，对于Navier-Stokes方程的Cauchy问题的解决方案证明了具有无原始数据的库奇问题的解决方案；研究了与相变现象相关的非线性动力学收缩期的吸引子的性能；获得了几种用于计算Grothendieck残基，代数品种的DE RHAM协同学和D模块的算法。在（3）中：我们对普特南的不平等进行了概括；将因素的行为分类为操作员代数；构建了C^*代数的索引理论；并在Bohr组上解决了单发器问题。在（4）中：我们对等级1半谎言群体获得了强硬的空间理论；并获得了Pfaffians及其应用的总和。在（5）中：我们为NP完全问题提供了解决方案，作为量子算法和混乱理论的应用。研究了p-螺旋正常运算符的几种特性。

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。