Functional Analytic Methods for Heat Green Operators

Dr. Kazuaki Taira was awarded a Doctor of Science degree by the University of Tokyo (1976) and a Doctorat d'État degree by Université de Paris-Sud (Orsay) (1978), where he had studied on a French government scholarship (1976-1978). He was also a member of the Institute for Advanced Study (Princeton) (1980-1981), an associate professor at the University of Tsukuba (1981-1995), and a professor at Hiroshima University (1995-1998). In 1998, he accepted an offer from the University of Tsukuba to teach there again as a professor.  He was also a part-time professor at Waseda University (2009-2017). His current research interests are in three interrelated subjects in analysis: semigroups, elliptic boundary value problems and Markov processes.

- Introduction and Main Results.- Part I Functional Analytic Approach to Heat Operators.- Preliminaries from Manifold Theory.- Finite Double-Norm and Trace-Class Operators on a Hilbert Space.- Hilbert-Schmidt and Trace-Class Operators on a Manifold.- Analytic Semigroups via Dunford Integrals.- Part II Pseudo-Differential Operators and Elliptic Boundary Value Problems.- Lp Theory of Pseudo-Differential Operators.- Elliptic Boundary Value Problems on a Manifold.- Lp Theory of Elliptic Boundary Value Problems.- Boutet de Monvel Calculus.- Pseudo-Differential Operator Approach to Agmon's Method.- Part III Analytic Semigroup Approach to Heat Operators.- Generation Theorem for Analytic Semigroups via Agmon's Method.- Hypoelliptic Robin Problems via Boutet de Monvel Calculus.- Distribution Kernel of Analytic Semigroups in the Hypoelliptic Case.- Part IV The Fundamental Solution for the Cauchy Problem.- The Cauchy Problem for the Heat Operator on Rn.- Fundamental Solution Operator for the Cauchy Problem on a Manifold.- Part V Symbolic Calculus for Dirichlet, Neumann and Regular Robin Problems.- Symbolic Calculus near the Boundary.- Analytic Version of Weyl Bases for the Heat Kernel on the Half Axis.- Symbolic Calculus for Boundary Value Problems on a Manifold.- Part VI Transport Equations and Trace Formulas on the Half Space.- Transport Equations via the Weyl-Hörmander Calculus.- Several Trace Formulas for Auxiliary Operators on the Half Space.- Part VII Heat Kernel Asymptotics for Dirichlet, Neumann and Regular Robin Problems.- Heat Kernel Asymptotics for the Dirichlet Problem.- Heat Kernel Asymptotics for the Neumann Problem.- Heat Kernel Asymptotics for the Regular Robin Problem.- Heat Kernel Asymptotics for the Generalized Regular Robin Problem.- Part VIII Heat Kernel Asymptotics for the Hypoelliptic Robin Problem.- Heat Kernel Asymptotics for the Hypoelliptic Robin Problem.- Examples in the Plane R2.- Concluding Remarks.