Analysis and Geometry on Carnot-Carathéodory Spaces Alexander Nagel

264 Pages·2008·1.3 MB·English

Analysis and Geometry
on
Carnot-Carath´eodory Spaces
Alexander Nagel

Contents
Preface iv
Part 1. Introduction 1
Chapter 1. Spaces of Homogeneous Type: Definitions and Examples 2
1. Spaces of homogeneous type 3
1.1. Pseudometrics and doubling measures 4
1.2. Covering Lemmas 6
1.3. The Hardy-Littlewood maximal operator 11
1.4. The Cald´eron-Zygmund Decomposition 13
2. The Laplace operator and Euclidean Analysis 15
2.1. Symmetries of the Laplace operator 16
2.2. The Newtonian Potential 17
2.3. The role of Euclidean geometry 19
2.4. Hardy-Littlewood-Sobolev estimates 21
2.5. Lipschitz Estimates 23
2.6. L2-estimates 28
2.7. L1-estimates 31
3. The heat operator and a non-isotropic metric on Rn+1 33
3.1. Symmetry properties 33
3.2. The initial value problem and a fundamental solution 34
3.3. Parabolic convergence 38
4. Operators on the Heisenberg group 39
4.1. The Siegal upper half space and its boundary 39
4.2. Dilations and translations on H 41
n
4.3. The sub-Laplacian and its geometry 42
4.4. The space H2(Hn) and the Szeg¨o Projection 45
4.5. Weighted spaces of entire functions 46
5. The Grushin plane 49
5.1. The geometry 50
5.2. Fundamental solution for L 50
5.3. Hermite functions 51
Chapter 2. VECTOR FIELDS 53
1. Vector fields and commutators 54
1.1. Germs of functions, tangent vectors, and vector fields 54
1.2. Commutators 55
1.3. Vector fields of finite type 56
1.4. Derivatives of determinants 58
2. Vector fields and smooth mappings 59
iii
CONTENTS iv
2.1. The differential acting on tangent vectors, vector fields, and
commutators 60
2.2. Action on determinants 61
3. Integral curves and the exponential maps 61
3.1. Integral curves 62
3.2. The exponential mapping 63
3.3. Exponential maps of several vector fields 65
4. Composition of flows 67
4.1. Commuting vector fields 67
4.2. Non-commuting vector fields 71
5. Normal coordinates 74
5.1. The definition of Θ 74
x
5.2. The action of dΘ on tangent vectors 75
x
6. Submanifolds and tangent vectors 76
6.1. Submanifolds 76
6.2. Tangential vector fields 77
6.3. The Frobenius theorem 77
7. Normal forms for vector fields 80
7.1. The case of a single vector field 80
7.2. The case of two vector fields 81
8. Vector fields and Derivations 81
8.1. Factorization 82
8.2. Derivations 83
Chapter 3. CARNOT-CARATHE´ODORY METRICS 84
1. Construction of Carnot-Carath´eodory metrics 85
1.1. Homogeneous norms 85
1.2. Control systems 86
1.3. Carnot-Carath´eodory metrics 88
1.4. Vector fields of finite type 92
2. Examples of metrics and operators 94
2.1. Isotropic Euclidean space 94
2.2. Non-isotropic Euclidean space 95
2.3. The Heisenberg group 96
2.4. The Grushin plane 101
3. Local structure of Carnot-Carath´eodory metrics 101
3.1. Notation 102
3.2. Exponential Balls 103
3.3. (x,δ,η)-dominance 105
3.4. The local structure of Carnot-Carath´eodory balls 105
4. Proof of the main theorem 107
4.1. Algebraic preliminaries 107
4.2. Estimates at x using (x,δ,κ)-dominance 111
4.3. Estimates on exponential balls 112
4.4. Some topological remarks 115
4.5. Proof of Theorem ??, part (??) 116
4.6. Proof of Theorem ??, part (??) 120
4.7. Proof of Theorem ??, part (??) 122
4.8. Proof of Theorem ??, part (??) 124
CONTENTS v
5. Smooth Metrics 125
5.1. Differentiable metrics 125
5.2. Scaled bump functions 129
Chapter 4. Subelliptic estimates and hypoellipticity 130
1. Introduction 130
2. Subelliptic estimates 130
2.1. Statement of the main theorem 130
2.2. Hypoellipticity of L 132
2.3. Commentary on the argument 132
3. Estimates for (cid:80)pj=1(cid:12)(cid:12)(cid:12)(cid:12)Xjϕ(cid:12)(cid:12)(cid:12)(cid:12)2s 133
3.1. Integration by parts 134
3.2. The basic L2-identity and L2-inequality 135
3.3. Commutators 136
3.4. The passage from L2 to Hs 141
4. Estimates for smooth functions 144
4.1. Introduction 144
4.2. The space of sub-elliptic multipliers 145
4.3. The main theorem on sub-elliptic multipliers 147
5. The theorem for distributions 153
5.1. Mollifiers 153
5.2. The main result 154
6. Examples of vector fields Y satisfying hypothesis (H-2) 154
0
Chapter 5. Estimates for fundamental solutions 160
1. Introduction 160
2. Unbounded operators on Hilbert space 161
2.1. Closed, densely defined operators and their adjoints 162
2.2. The spectrum of an operator 165
2.3. Self-adjoint operators 166
2.4. The spectral theorem for self-adjoint operators 167
3. The initial value problem for the heat operator ∂ +L 168
t x
3.1. The Friedrich’s construction 168
3.2. The heat semigroup {e−tL} 171
4. The heat kernel and heat equation 175
4.1. The heat kernel 175
4.2. The heat equation on R×Ω 178
5. Pointwise estimates for H 180
5.1. Scaling maps 180
5.2. A bound for the scaled initial value problem 180
5.3. Pointwise estimates for H 182
5.4. Action of e−tL on bump functions 184
Chapter 6. Non-isotropic smoothing operators 185
1. Definitions and properties of NIS operators 185
1.1. Definition 185
1.2. Elementary properties 186
Chapter 7. Algebras 191
1. Associative Algebras 191
CONTENTS vi
1.1. Free Associative Algebras 191
1.2. Algebras of formal series 193
2. Lie Algebras 195
2.1. Iterated Commutators 196
3. Universal Enveloping Algebras 198
3.1. Construction of U(L) 198
3.2. The Poincar´e-Birkhoff-Witt Theorem 199
4. Free Lie Algebras 202
4.1. The construction 202
4.2. Characterizations of Lie elements 204
5. The Campbell-Baker-Hausdorff formula 207
6. Hall Bases for Free Lie Algebras 208
6.1. Hall Sets 209
6.2. Two Examples 209
6.3. Proof of Theorem ?? 210
Chapter 8. The ∂ and ∂-Neumann Problems 212
1. Differential forms and the d and ∂-operators 212
1.1. Real differential forms 212
1.2. Notation 213
1.3. Differential forms with coordinates 214
1.4. The exterior derivative d 215
1.5. Complex coordinates and holomorphic functions 217
1.6. Splitting the complexified tangent space 218
1.7. Complex differential forms 219
1.8. The ∂-operator 220
1.9. The ∂-problem 222
2. The ∂-Neumann problem 223
2.1. A finite dimensional analogue 223
2.2. Hilbert spaces 224
2.3. Notation 229
2.4. Computation of ∂ and its formal adjoint 232
2.5. The basic identity 233
2.6. Further estimates 238
2.7. Hilbert spaces 239
2.8. The Neumann operator 242
Chapter 9. Appendix on background material 243
1. Distributions, Fourier transforms, and Fundamental Solutions 243
1.1. Notation 243
1.2. Spaces of smooth functions 243
1.3. Spaces of distributions 244
1.4. The Fourier Transform 245
1.5. Fundamental solutions and parametrices 246
Chapter 10. Pseudodifferential operators 248
1. Functions, distributions, and Fourier transforms 248
1.1. Spaces of smooth functions 248
1.2. Spaces of distributions 249
CONTENTS vii
2. Pseudodifferential Operators 251
2.1. Results 252
Bibliography 254
Preface
The subject matter of this book lies at the interface of the fields of harmonic
analysis, complex analysis, and linear partial differential equations, and has been
at the center of considerable research effort since at least the late 1960’s. Some
aspects of this work are presented in monographs and texts. Any brief list would
have to include:
• G.B. Folland and J.J. Kohn’s monograph, The Neumann Problem for the
Cauchy-Riemann Complex [FK72], on the ∂-Neumann problem;
• L. H¨ormander’s books, An Introduction to Complex Analysis in Several
Variables [H¨66] and Notions of Convexity [H¨94], on estimates for the
∂-problem;
• L. H¨ormander’s encyclopedic The Analysis of Linear Partial Differential
Operators I - IV[H¨85]whichcontainsmaterialonhypoellipticityofsums
of squares of vector fields;
• E.M. Stein’s short monograph Boundary Behavior of Holomorphic Func-
tions of Several Complex Variables [Ste72];
• E.M. Stein’s definitive classic Harmonic Analysis: Real-Variables Meth-
ods, Orthogonality, and Oscillatory Integrals [Ste93];
• The article by M. Gromov in the collection Sub-Riemannian Geomerty
[BR96] by A. Bella¨ıche and J.-J. Risler;
• TherecentbookbyS.-C.ChenandM.-C.ShawPartialDifferentialEqua-
tions in Several Complex Variables [CS01].
Despite this list of references, many of the developments that have occurred
in this subject remain largely unchronicled except in the original papers. Also,
many of these results require general techniques that are not fully discussed in the
earlier texts. This situation makes it difficult for a student to start work in this
area. The most recent papers refer to older papers, which in turn cite earlier work,
and a student often becomes discouraged at the prospect of trying to navigate this
seemingly infinite regress.
Thusthisbookhastwoobjectives. Thefirstistoprovideanaccessiblereference
for material and techniques from the subject of ‘control’ or Carnot-Carath´eodory
metrics. The second is to provide a coherent account of some applications of these
techniques to problems in complex analysis, harmonic analysis, and linear partial
differential equations. These two objectives are inseparable. One needs the general
theoryofthegeometryofCarnot-Carath´eodorymetricsinordertodealwithcertain
kinds of problems in complex and harmonic analysis, but the theory by itself is
essentially indigestible unless leavened with interesting problems and examples.
viii
PREFACE ix
Part I of this book provides an introduction to the geometry associated to
certain families of vector fields, and to analytic results about a related class of
integral of operators. The basic geometric structures go by various names such as
control metrics, Carnot-Carath´eodory spaces, or sub-Riemannian manifolds. The
associatedanalyticobjects,inthisgeneralcontext,areoftenknownasnon-isotropic
smoothing (NIS) operators. These concepts arose in part through attempts to
present a more unified description of classical results, and from the need for more
flexible tools to deal with new problems and phenomena arising in complex and
harmonic analysis and linear partial differential equations.
Part 1
Introduction