Title An invitation to hypoelliptic operators and Hörmander's vector fields / Marco Bramanti

Published Cham : Springer, 2014

Table of Contents
1.	Hormander's Operators: What they are					1
1.1.	The Context of Distribution Theory					1
1.2.	Local Solvability					2
1.3.	Hypoellipticity					3
1.3.1.	Hypoelliptic Operators with Constant Coefficients					4
1.3.2.	Hypoelliptic Operators with Variable Coefficients					5
1.3.3.	An Unsatisfactory Situation					6
1.3.4.	A Turning Point: Hormander 1967, Acta Mathematica					7
1.3.5.	Subelliptic Estimates					13
	References					13
2.	Hormander's Operators: Why they are Studied					15
2.1.	First Motivation: Kolmogorov-Fokker-Planck Equations					15
2.1.1.	Brownian Motion and Langevin's Equation					15
2.1.2.	Wiener Process and Gaussian White Noise					16
2.1.3.	Stochastic Differential Equations					17
2.1.4.	Kolmogorov and Fokker-Planck Equations					18
2.1.5.	Examples of Kolmogorov-Fokker-Planck Equations Arising from Applications of Stochastic Models					21
2.2.	Second Motivation: PDEs Arising in the Theory of Several Complex Variables					26
2.2.1.	Background on the Cauchy-Riemann Complex					26
2.2.2.	The Neumann Problem					29
2.2.3.	The Tangential Cauchy-Riemann Complex and the Kohn Laplacian					30
2.2.4.	The Kohn Laplacian on the Heisenberg Group					31
	References					34
3.	A Priori Estimates in Sobolev Spaces for Hormander's Operators?					37
3.1.	What are the "Natural" a Priori-Estimates to be Proved for Hormander's Operators?					37
3.2.	The Sublaplacian on the Heisenberg Group					39
3.2.1.	The Classical Laplacian					39
3.2.2.	Geometry of the Sublaplacian					41
3.2.3.	Fundamental Solution of the Sublaplacian					43
3.2.4.	What we can do with a Good Fundamental Solution					46
3.2.5.	Singular Integrals in Spaces of Homogeneous Type					50
3.2.6.	LP Estimates for the Sublaplacian and the Kohn- Laplacian on the Heisenberg Group					53
3.3.	Hormander's Operators on Homogeneous Groups					54
3.3.1.	Homogeneous Groups					54
3.3.2.	Homogeneous Lie Algebras					57
3.3.3.	Hormander's Operators on Homogeneous Groups					59
3.3.4.	Homogeneous Fundamental Solutions and LP Estimates					61
3.3.5.	Higher Order Estimates					63
3.3.6.	Some Classes of Examples of Homogeneous Groups and Corresponding Hormander's Operators					65
3.4.	General Hormander's Operators					69
3.4.1.	The Problem, and How to Approach It					69
3.4.2.	Lifting					73
3.4.3.	Approximation with Left Invariant Vector Fields					74
3.4.4.	Parametrix and LP Estimates					77
3.4.5.	Singular Integral Estimates					81
3.5.	Some Final Comments on the Quest of a-Priori Estimates in Sobolev Spaces					83
3.5.1.	Local Versus Global Estimates					83
3.5.2.	Levels of Generality					84
	References					85
4.	Geometry of Hormander's Vector Fields					87
4.1.	Connectivity, and Some of its Meanings					87
4.1.1.	Exponential of a Vector Field, and How to Move Along the Direction of a Commutator					87
4.1.2.	Rashevski-Chow's Connectivity Theorem					90
4.1.3.	Caratheodory Foundations of Thermodynamics and Inaccessibility					90
4.1.4.	Connectivity, Controllability, and Nonholonomy					92
4.1.5.	Propagation of Maxima					97
4.2.	Metric Balls Induced by Systems of Vector Fields					98
4.2.1.	Motivation					98
4.2.2.	Distance Induced by a System of Hormander's Vector Fields					100
4.2.3.	Volume of Metric Balls					101
4.2.4.	The Control Distance					104
4.2.5.	Relation Between Lifted and Unlifted Balls					107
4.2.6.	Estimates on the Fundamental Solution					109
4.3.	Heat Kernels and Gaussian Estimates					110
4.4.	Poincare's Inequality, and Some of its Consequences					111
4.5.	Carnot-Caratheodory Spaces					113
4.6.	Franchi---Lanconelli Operators with Diagonal Vector Fields					114
	References					115
5.	Beyond Hormander's Operators					119
5.1.	Kolmogorov---Fokker---Planck Equations with Linear Drift					119
5.1.1.	The Class of Operators Introduced by Lanconelli---Polidoro					119
5.1.2.	Developments of the Theory of Homogeneous Operators of Lanconelli---Polidoro Type					126
5.1.3.	Developments of the Theory of Nonhomogeneous Operators of Lanconelli---Polidoro Type					127
5.2.	Nonlinear Equations Coming from the Theory of Several Complex Variables					129
5.2.1.	Regularity Theory for the Levi Equation and the Study of "Nonlinear Vector Fields"					129
5.2.2.	Levi---Monge---Ampere Equations and Nonvariational Operators Structured on Hormander's Vector Fields					131
5.3.	Nonvariational Operators Structured on Hormander's Vector Fields					132
5.3.1.	LP Estimates for Nonvariational Operators Structured on Hormander's Vector Fields					133
5.3.2.	Gaussian Estimates for Nonvariational Operators Structured on Hormander's Vector Fields					138
5.4.	Nonsmooth Hormander's Vector Fields					140
5.4.1.	Motivation and History of the Problem					140
5.4.2.	Some Results from the Theory of Nonsmooth Hormander's Vector Fields and Operators					142
	References					147

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Description 1 online resource (xi, 150 pages) : illustrations Series SpringerBriefs in Mathematics, 2191-8198 SpringerBriefs in mathematics, 2191-8198 Contents Hörmander's operators: what they are -- Hörmander's operators: why they are studied -- A priori estimates in Sobolev spaces -- Geometry of Hörmander's vector fields -- Beyond Hörmander's operators Summary Hörmander's operators are an important class of linear elliptic-parabolic degenerate partial differential operators with smooth coefficients, which have been intensively studied since the late 1960s and are still an active field of research. This text provides the reader with a general overview of the field, with its motivations and problems, some of its fundamental results, and some recent lines of development Bibliography Includes bibliographical references Notes Online resource; title from PDF title page (SpringerLink, viewed November 25, 2013) Subject Hypoelliptic operators. Mathematics. Mathematics mathematics. applied mathematics. SCIENCE -- Life Sciences -- Evolution. Mathematics Hypoelliptic operators Form Electronic book ISBN 9783319020877 3319020870

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