Research on Applications of Real and Functional Analysis

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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.