Hörmander Operators

Description

Hörmander operators are a class of linear second order partial differential operators with nonnegative characteristic form and smooth coefficients, which are usually degenerate elliptic-parabolic, but nevertheless hypoelliptic, that is highly regularizing. The study of these operators began with the 1967 fundamental paper by Lars Hörmander and is intimately connected to the geometry of vector fields. Motivations for the study of Hörmander operators come for instance from Kolmogorov-Fokker-Planck equations arising from modeling physical systems governed by stochastic equations and the geometric theory of several complex variables. The aim of this book is to give a systematic exposition of a relevant part of the theory of Hörmander operators and vector fields, together with the necessary background and prerequisites. The book is intended for self-study, or as a reference book, and can be useful to both younger and senior researchers, already working in this area or aiming to approach it. Dedication Contents Foreword Introduction 1. Basic geometry of vector fields 2. Function spaces defined by systems of vector fields 3. Homogeneous groups in ℝᴺ 4. Hypoellipticity of sublaplacians on Carnot groups 5. Hypoellipticity of general Hörmander operators 6. Fundamental solutions of Hörmander operators 7. Real analysis and singular integrals in locally doubling metric spaces 8. Sobolev and Hölder estimates for Hörmander operators on groups 9. More geometry of vector fields: metric balls and equivalent distances 10. Lifting and approximation 11. Sobolev and Hölder estimates for general Hörmander operators 12. Nonvariational operators constructed with Hörmander vector fields Appendix: A Short summary of distribution theory Bibliography Index