Research on Applications of Real and Functional Analysis

我们研究了实分析和泛函分析的各个领域，并获得了许多与偏微分方程、数学物理、概率论和微分几何密切相关的结果。下面是一些重要的结果。偏微分方程的应用。非线性Schrodinger方程，Vlasov-Maxwell方程，若干非线性特征值问题，伪微分算子有界性的精确充分条件，超函数框架下无限阶一般边值问题的微分算子，二次微定位。算子代数。2 .因子代数的Jones指数，Banach函数代数的Baum-Connes猜想泛函微积分的研究，2参数brown鞅的Hardy空间。函数空间。3 .道格拉斯代数与内函数，parreui - widom型定域，可测函数空间上的凸规划，赋范空间中的正交性，Banach空间中的非线性演化算子及其在具体问题中的应用。真正的分析。多元函数的傅里叶变换，奇异积分算子，群的调和分析，负曲率流形的调和分析。李群的表示。半简单群，半简单对称空间，对称空间上的傅里叶变换，可解群的单项式表示，d模理论，无限维群的表示在物理中的应用。

We investigated various fields in real analysis and functional analysis, and have obtained many results which are closely related to partial differential equations, mathematical physics, probability theory and differential geometry. Some of important results are as follows.1. Application to Partial Differential Equations.Nonlinear Schrodinger equation, Vlasov-Maxwell equation, some nonlinear eigenvalue problems, precise sufficient conditions for boundenness of pseudo-differential operators, differential operators of infinite order general boundary value problems in the framework of hyperfunctions, 2nd microlocalization.2. Operator Algebra.Jones' index of factor algebra, investigation on Baum-Connes conjecture functional calculus for Banach function algebra, Hardy spaces of 2-parameter Brownian martingales.3. Function Spaces.Douglas algebra and inner functions, domain of Parreu-Widom type, convex programming on spaces of measurable functions, orthogonality in normed spaces, nonlinear evolution operators in Banach spaces and their applications to concrete problems.4. Real Analysis.Fourier transform for functions of several variables, singular integral operators, harmonic analysis on groups, harmonic anslysis on manifolds of negative curvature.5. Representation of Lie Groups.Semi-simple groups, Semi-simple symmetric spaces, Fourier transform on symmetric spaces, monomial representations of solvable group, theory of D-modules, applications of representation of infinite dimentional groups to physics.