Comprehensive Study of Real Analysis and Functional Analysis

本文的研究扩展到以下5个方面：(1)实变量法谐波分析；(2)用泛函分析法求解偏微分方程；(3)算子代数与函数代数；(4)表征理论；(5)量子逻辑和函数空间。我们的主要结果如下。在区域(1)中，我们得到了域上加权Hardy空间和herz型Hardy空间的实变量表征；证明了几种非正则核奇异积分算子的有界性得到了定义为奇异积分积和的多线性算子的Hardy空间估计。式(2)：得到了麦克斯韦方程、狄拉克方程和薛定谔方程的强唯一延拓定理；研究了薛定谔方程奇点的传播；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Navier-Stokes方程Cauchy问题解的时间全局存在性研究了非线性动力系统中与相变现象相关的吸引子的性质；获得了计算Grothendieck残数、代数变种的de Rham上同调和d模的几种算法。在(3)中：我们给出了Putnam不等式的推广；得到了因子对算子代数作用的分类；构造了C^*代数的指标论；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；得到了普氏级数的求和公式及其应用。在(5)中，我们给出了应用量子算法和混沌理论求解NP完全问题的方法；研究了p-次正规算子的几个性质。

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.