Cooperative Researches on Real Analysis and Functional Analysis

1.在实分析领域，关于欧几里德区域上的调和分析得到了各种结果。其中，详细地发展了H^P空间的一个理论。建立了谐波分析的新发展。后者与微分几何、偏微分方程组和随机过程有很深的关系。在函数空间的应用领域，量子信息论获得了新的方面，并应用于不可逆过程。研究了多端口网络连接中的矩阵不等式问题。得到了发展方程领域的各种结果，并将其应用于偏微分方程组。在表象理论和调和分析领域，取得了显著的成果。研究了齐次约化型空间中的不连续群。实现了无限维李群的构造。阐明了半单李群上齐次特征方程的解的维度不变性。在泛函分析和偏微分方程领域，发展方程、拟微分算子、非线性抛物型方程、特征值反问题、Hilbert空间中的发展方程、超函数和薛定谔方程都得到了成功的发展。在算子代数和函数代数领域，发展了C^*-代数的无界导子理论，并将其应用于统计力学。详细分析了算子代数、量子群、无界算子代数和函数代数的结构。

1. In the area of real analysis, various results pertaining to harmonic analysis on Euclidean domains were obtained. Among others, a theory of H^P space was developed in detail. A new development of Harmonic analysis was established. The latter was deeply related to differential geometry, partial differential equations and stochastic processes.2. In the area of applications of function spaces, new aspects of quantum information theory were obtained and applied to irreversible processes. Matrix inequalities in multiport network connections were studied. Various results in the field of evolution equations were obtained and applied to partial differential equations.3. In the area of representation theory and harmonic analysis, remarkable results were obtained. Discontinuous group in a homogeneous space of reductive type was studied. A construction of infinite-dimensional Lie groups was realized. Invariance of dimension of solutions to homogeneous characteristi cequations on semisimple Lie groups was clarified.4. In the area of functional analysis and partial differential equations, evolution equations, pseudo-differential operators, nonlinear parabolic equations, inverse eigenvalue problems, evolution equations in Hilbert spaces, hyperfunctions and Schrodinger equations were successfully developed.5. In the area of operator algebras and function algebras, the theory of unbounded derivations in C^*-algebras was developed and applied to statistical mechanics. Structure of operator algebras, quantum groups, structure of unbounded operator algebras, and function algebras were analyzed in detail.