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.

研究扩展到以下5个领域：（1）用真实的变量方法进行调和分析;（2）用泛函分析方法进行偏微分方程;（3）算子代数和函数代数;（4）表示理论;（5）量子逻辑和函数空间。我们的主要结果如下。区域（1）：得到了整环上加权哈代空间和Herz型哈代空间的真实的变量特征，证明了几个非正则核奇异积分算子的有界性，并得到了定义为奇异积分乘积之和的穆尔线性算子的哈代空间估计.在（2）中：我们得到了麦克斯韦方程、Dirac方程和Schrodinger方程的强唯一延拓定理，研究了Schrodinger方程的奇性传播，利用哈代空间估计研究了Stokes算子，证明了具有非衰减初值的Navier-Stokes方程Cauchy问题解的时间整体存在性，得到了非衰减初值的Navier-Stokes方程Cauchy问题解的时间整体存在性。研究了与相变现象有关的非线性动力系统的吸引子的性质，得到了计算Grothendieck留数、代数簇的de Rham上同调和D-模的几种算法。在（3）中：给出了Putnam不等式的一个推广，得到了因子对算子代数作用的分类，构造了C^* 代数的指标理论，解决了Bohr群上的单生成元问题。在（4）中，我们得到了秩为1的半单李群上的哈代空间理论，并得到了Pfrons的求和公式及其应用。在（5）中：作为量子算法和混沌理论的应用，我们给出了一个NP完全问题的解，研究了p-亚正规算子的若干性质。

项目成果

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）。

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。

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。

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)。

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。