Table Of ContentHörmander
Operators
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Hörmander
Operators
Marco Bramanti
Politecnico di Milano, Italy
Luca Brandolini
Università degli Studi di Bergamo, Italy
World Scientific
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HÖRMANDER OPERATORS
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Contents
Foreword xi
Introduction xiii
0.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
0.2 Scope and structure of the book . . . . . . . . . . . . . . . . . . . xviii
0.3 Why study H¨ormander operators? . . . . . . . . . . . . . . . . . xxii
1. Basic geometry of vector fields 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Exponentials and commutators of vector fields. . . . . . . . . . . 2
1.3 Lie algebras, H¨ormander’s condition, H¨ormander operators. . . . 8
1.4 The control distance . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 The weighted control distance . . . . . . . . . . . . . . . . . . . . 21
1.6 Connectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7 Other properties related to connectivity . . . . . . . . . . . . . . 35
1.8 Maximum principles for degenerate elliptic operators . . . . . . . 37
1.9 Propagation of maxima and strong maximum principle for sum of
squares operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.10 Propagation of maxima for operators with drift . . . . . . . . . . 48
1.11 Some examples of explicit computations with the control distance 56
1.12 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2. Function spaces defined by systems of vector fields 67
2.1 Sobolev spaces induced by vector fields . . . . . . . . . . . . . . . 67
2.2 H¨older spaces induced by H¨ormander vector fields. . . . . . . . . 80
2.3 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3. Homogeneous groups in RN 93
3.1 Homogeneous groups . . . . . . . . . . . . . . . . . . . . . . . . . 94
vii
viii Ho¨rmander operators
3.2 Homogeneous Lie algebras of invariant vector fields on a
homogeneous group . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.3 Exponential maps on a homogeneous group . . . . . . . . . . . . 116
3.4 Convolution and mollifiers on a homogeneous group. . . . . . . . 118
3.5 Homogeneous stratified Lie groups and Lie algebras, and their
control distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.6 Connectivity matters and Poincar´e inequality on stratified groups 128
3.7 Weak solutions to Dirichlet problems for divergence form
equations structured on vector fields . . . . . . . . . . . . . . . . 132
3.8 Homogeneous stratified Lie algebras and Lie groups of type II . . 135
3.9 Distributions on homogeneous groups. . . . . . . . . . . . . . . . 141
3.10 Examples of homogeneous groups and homogeneous H¨ormander
operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.11 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4. Hypoellipticity of sublaplacians on Carnot groups 153
4.1 Introduction, statement of the main results and strategy of the
proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.2 Notation and preliminary facts about Sobolev spaces and finite
differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.3 Regularity estimates for the canonical sublaplacian . . . . . . . . 163
4.4 Hypoellipticity of the canonical sublaplacian . . . . . . . . . . . . 174
4.5 General sublaplacians and uniform estimates . . . . . . . . . . . 184
4.6 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5. Hypoellipticity of general H¨ormander operators 191
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.2 The Fourier transform on the Schwartz space S(Rn) and on
tempered distributions . . . . . . . . . . . . . . . . . . . . . . . . 192
5.3 Fractional order Sobolev spaces . . . . . . . . . . . . . . . . . . . 195
5.4 Some classes of operators on S(Rn) . . . . . . . . . . . . . . . . . 199
5.5 Subelliptic estimates . . . . . . . . . . . . . . . . . . . . . . . . . 217
5.6 Localized subelliptic estimate . . . . . . . . . . . . . . . . . . . . 229
5.7 Hypoellipticity of H¨ormander operators . . . . . . . . . . . . . . 232
5.8 Uniform subelliptic estimates . . . . . . . . . . . . . . . . . . . . 238
5.9 Some applications of the subelliptic estimates . . . . . . . . . . . 242
5.10 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
6. Fundamental solutions of H¨ormander operators 247
6.1 Fundamental solutions and solvability of general H¨ormander
operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
6.2 Homogeneous H¨ormander operators . . . . . . . . . . . . . . . . . 254
Contents ix
6.3 Existence of a global homogeneous fundamental solution and
uniform estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
6.4 Properties of the global fundamental solution . . . . . . . . . . . 268
6.5 Some explicit examples of fundamental solutions on homogeneous
groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
6.6 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
7. Real analysis and singular integrals in locally doubling metric spaces 291
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
7.2 Locally doubling metric spaces . . . . . . . . . . . . . . . . . . . 295
7.3 Localized kernels of singular and fractional integrals . . . . . . . 298
7.4 Singular and fractional integrals on H¨older spaces . . . . . . . . . 301
7.5 L2 continuity of singular integrals via continuity on Cα . . . . . 309
7.6 Local maximal function and fractional integrals on Lp spaces . . 312
7.7 Calder´on-Zygmund theory in locally doubling metric spaces . . . 318
7.8 Integral characterization of H¨older continuity . . . . . . . . . . . 327
7.9 Some geometric results . . . . . . . . . . . . . . . . . . . . . . . . 331
7.10 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
8. Sobolev and H¨older estimates for H¨ormander operators on groups 337
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
8.2 Homogeneous kernels on G, fractional integrals and Sobolev
embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
8.3 Singular integrals associated to homogeneous kernels of type 0 . . 354
8.4 Global Sobolev estimates. . . . . . . . . . . . . . . . . . . . . . . 360
8.5 Local Sobolev estimates . . . . . . . . . . . . . . . . . . . . . . . 370
8.6 H¨older estimates for solutions of Lu=f . . . . . . . . . . . . . . 378
8.7 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
9. More geometry of vector fields: metric balls and equivalent distances 399
9.1 Introduction and statement of the main results . . . . . . . . . . 399
9.2 Dependence of the constants . . . . . . . . . . . . . . . . . . . . . 406
9.3 The Baker-Campbell-Hausdorff formula . . . . . . . . . . . . . . 408
9.4 Suboptimal bases and their properties . . . . . . . . . . . . . . . 420
9.5 Structure of metric balls . . . . . . . . . . . . . . . . . . . . . . . 437
9.6 Local equivalence of the distances d,d∗ . . . . . . . . . . . . . . . 458
9.7 Segment properties and the global doubling condition . . . . . . 462
9.8 Proof of the BCH formula for formal series and other consequences 466
9.9 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
10. Lifting and approximation 479
10.1 Motivation and statement of the main results . . . . . . . . . . . 479
