Table Of ContentSPRINGER BRIEFS IN MATHEMATICS
Marco Bramanti
An Invitation
to Hypoelliptic
Operators and
Hörmander‘s
Vector Fields
123
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Marco Bramanti
An Invitation to Hypoelliptic
Operators and Hörmander’s
Vector Fields
123
MarcoBramanti
Dipartimentodi Matematica
Politecnico diMilano
Milan
Italy
ISSN 2191-8198 ISSN 2191-8201 (electronic)
ISBN 978-3-319-02086-0 ISBN 978-3-319-02087-7 (eBook)
DOI 10.1007/978-3-319-02087-7
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Preface
Hörmander’s operators are an important class of linear elliptic-parabolic degen-
erate 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 is based on the notes that I wrote for a short course held at the
Department of Mathematics of Politecnico di Milano in February–March 2012.
The audience consisted of a group of colleagues and Ph.D. students working on
PDEs. In accordance with the style of that course, this text is written for people
whoareinterestedinbeginningtodoresearchinthisarea,orarejustcuriousabout
it, and look for a general overview of this field of research, with its motivations
andproblems,someofitsfundamentalresults,andatleastsomeofitsrecentlines
of development. In this text proofs are almost completely absent, which is an
uncommon feature for a mathematical booklet, and therefore needs some justifi-
cation, which will appear from the rest of this introduction.
Let me first sketch the plan of these notes. Chapters 1 and 2 deal, in two
different ways, with motivations for studying Hörmander’s operators. (Actually,
the quest ofmotivation is aleitmotiv ofthesenotes). Chapter 1 explains the PDE
context at the origin of the concept of Hörmander’s operators, while Chap. 2
focuses on therelevanceinotherareas ofpure orapplied mathematicsofspecific
classes of partial differential operators which are actually of Hörmander’s type.
Chapters 3 and 4 discuss some fundamental ideas and results in this area, dating
back to the 1970s and 1980s, which everyone interested in doing research, or
studying contemporary research papers about Hörmander’s operators, needs to
know.Morespecifically,Chap.3,perhapsthetechnicalcoreofthistext,dealswith
the theme of a-priori estimates, in the suitable Sobolev spaces, for Hörmander’s
operators. This involves the concept of homogeneous groups, the construction of
fundamental solutions, the use of abstract singular integral theories, and the
developmentofsuitablealgebraicanddifferentialgeometrictools.Chapter4deals
with the geometry of the vector fields which appear in the definition of Hör-
mander’s operators, the concept of distance induced by a system of vector fields,
andrelatedproblems.Actually,thestudyofgeometryofHörmander’svectorfields
is nowadays an independent field of research with respect to the study of second
order PDEs ofHörmander’stype.These notesare mainlyfocussedonthesecond
theme, with a touch on the first one mainly as a tool for the study of the second
one.However,inChap.4Ihave atleasttried togive someofthemotivations for
v
vi Preface
thestudyofthegeometryofHörmander’svectorfields,besidesitsapplicationsto
PDEs. Finally, Chap. 5 presents an overview of some of the developments of the
theory of Hörmander’s operators in the 1990s and 2000s. Here the choice of the
topics particularly reflects my personal interests. As we will see, the evolution of
this area has mainly consisted of the study of classes of operators which are no
longer of Hörmander type, strictly speaking, but are structured on Hörmander’s
vector fields, in several senses, and also contain some nonsmooth ingredients,
which poses new problems. Again, I have always tried to give some motivations
for the study of the particular classes of operators which are considered.
Let me now try to justify the style of this text. Motivation for the study of
Hörmander’s operators and vector fields involve, among other areas, systems of
stochastic differential equations, the theory offunctions of several complex vari-
ables, geometric control theory and nonholonomic mechanics: a wide range of
subjects, which is impossible to give account of in a few pages. The fundamental
ideasandresultswhichstillnowconstitutethebasictoolstoworkonHörmander’s
operators appeared in the literature in a small number of very important papers,
whichareverytechnical,long,andoftennoteasytoread.Adetailedexpositionof
the contents of those papers, in the style of a graduate course, would require
severalhundredsofpages.Ithinkthatbothforapersonwhoisjustcurious about
thisareaandforonewhowantstobegindoingactiveresearchinit,acquiringfrom
the very beginning a general picture of the landscape can be of great help. Doing
thisinalimitedtimerequiresavoidingproofs.Thestudyofproofsandtechniques
is a, clearly unavoidable, second step which a person will take, starting with the
specific techniques involved in the specific problem he/she wants to attack.
Any comment on this book will be appreciated. Please, write to:
marco.bramanti@polimi.it
Acknowledgments
I wish to thank all the people who attended the short course at the Department of
Mathematics of Politecnico di Milano for their interest and for stimulating par-
ticipation to the lessons. I am also grateful to some collegues who read this text
and made useful comments and suggestions: Marco Peloso, Ermanno Lanconelli,
and Sergio Polidoro.
vii
Contents
1 Hörmander’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: Hörmander 1967,
Acta Mathematica. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.5 Subelliptic Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . 13
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Hörmander’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 o-Neumann Problem. . . . . . . . . . . . . . . . . . . . . . . 29
2.2.3 The Tangential Cauchy-Riemann Complex
and the Kohn Laplacian h . . . . . . . . . . . . . . . . . . . . . 30
b
2.2.4 The Kohn Laplacian on the Heisenberg Group. . . . . . . . 31
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 A Priori Estimates in Sobolev Spaces for Hörmander’s
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 What are the ‘‘Natural’’ a Priori-Estimates to be Proved
for Hörmander’s Operators? . . . . . . . . . . . . . . . . . . . . . . . . . . 37
ix
x Contents
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 Hörmander’s Operators on Homogeneous Groups . . . . . . . . . . . 54
3.3.1 Homogeneous Groups . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 Homogeneous Lie Algebras . . . . . . . . . . . . . . . . . . . . . 57
3.3.3 Hörmander’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 Hörmander’s Operators. . . . . 65
3.4 General Hörmander’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 Hörmander’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 Carathéodory 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 Hörmander’s
Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2.3 Volume of Metric Balls. . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.4 The Control Distance. . . . . . . . . . . . . . . . . . . . . . . . . . 104
Contents xi
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 Poincaré’s Inequality, and Some of its Consequences . . . . . . . . 111
4.5 Carnot-Carathéodory Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.6 Franchi–Lanconelli Operators with Diagonal Vector Fields . . . . 114
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5 Beyond Hörmander’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–Ampère Equations and Nonvariational
Operators Structured on Hörmander’s Vector Fields . . . . 131
5.3 Nonvariational Operators Structured on Hörmander’s
Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3.1 Lp Estimates for Nonvariational Operators Structured
on Hörmander’s Vector Fields . . . . . . . . . . . . . . . . . . . 133
5.3.2 Gaussian Estimates for Nonvariational Operators
Structured on Hörmander’s Vector Fields . . . . . . . . . . . 138
5.4 Nonsmooth Hörmander’s Vector Fields . . . . . . . . . . . . . . . . . . 140
5.4.1 Motivation and History of the Problem . . . . . . . . . . . . . 140
5.4.2 Some Results from the Theory of Nonsmooth
Hörmander’s Vector Fields and Operators . . . . . . . . . . . 142
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
