Hormander 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 Hormander and is intimately connected to the geometry of vector fields.
Motivations for the study of Hormander 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 Hormander 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.
Contents:
- Foreword
- Introduction
- Basic Geometry of Vector Fields
- Function Spaces Defined by Systems of Vector Fields
- Homogeneous Groups in ?N
- Hypoellipticity of Sublaplacians on Carnot Groups
- Hypoellipticity of General Hormander Operators
- Fundamental Solutions of Hormander Operators
- Real Analysis and Singular Integrals in Locally Doubling Metric Spaces
- Sobolev and Holder Estimates for Hormander Operators on Groups
- More Geometry of Vector Fields: Metric Balls and Equivalent Distances
- Lifting and Approximation
- Sobolev and Holder Estimates for General Hormander Operators
- Nonvariational Operators Constructed with Hormander Vector Fields
- Appendix A: Short Summary of Distribution Theory
- Bibliography
- Index
Readership: Advanced undergraduate and graduate students, researchers in the fields of mathematical analysis and partial differential equations; academic libraries.
Key Features:
- It is the first book entirely devoted to the study of general Hormander operators and vector fields
- The book contains detailed proofs of the main classical results in the subject
- The exposition is self-contained, and the necessary prerequisites are presented with a language which anybody working in Mathematical Analysis and PDEs should find comfortable
- The book contains a thorough introduction of the tools which are necessary to do active research in the field of Hormander operators
- The book includes some results due to the authors about nonvariational operators structured on Hormander vector fields
