- 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.