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.Key Features: oIt is the first book entirely devoted to the study of general Hörmander operators and vector fieldsoThe book contains detailed proofs of the main classical results in the subjectoThe exposition is self-contained, and the necessary prerequisites are presented with a language which anybody working in Mathematical Analysis and PDEs should find comfortableoThe book contains a thorough introduction of the tools which are necessary to do active research in the field of Hörmander operatorsoThe book includes some results due to the authors about nonvariational operators structured on Hörmander vector fields

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