Part I. Elements of Analysis of Stratified Groups
1. Stratified Groups and Sub-Laplacians ........................ 3
1.1. Vector Fields in  N: Exponential Maps and Lie
Algebras ............................................... 3
1.1.1. Vector Fields in N ............................. 3
1.1.2. Integral Curves ................................. 6
1.1.3. Exponentials of Vector Fields ................... 8
1.1.4. Lie Brackets of Vector Fields in N ............ 10
1.2. Lie Groups on N ...................................... 13
1.2.1. The Lie Algebra of a Lie Group on N ........... 13
1.2.2. The Jacobian Basis ............................. 19
1.2.3. The (Jacobian) Total Gradient .................. 22
1.2.4. The Exponential Map of a Lie Group on N ....... 23
1.3. Homogeneous Lie Groups on N .......................... 31
1.3.1. δλ-homogeneous Functions and Differential
Operators ...................................... 32
1.3.2. The Composition Law of a Homogeneous Lie
Group .......................................... 38
1.3.3. The Lie Algebra of a Homogeneous Lie Group
on N .......................................... 44
1.3.4. The Exponential Map of a Homogeneous Lie
Group .......................................... 48
1.4. Homogeneous Carnot Groups ............................. 56
1.5. The Sub-Laplacians on a Homogeneous Carnot Group ...... 62
1.5.1. The Horizontal  -gradient ...................... 68
1.6. Exercises of Chapter 1 ................................ 73
2. Abstract Lie Groups and Carnot Groups ...................... 87
2.1. Abstract Lie Groups ................................... 87
2.1.1. Differentiable Manifolds ....................... 87
2.1.2. Tangent Vectors ................................ 91
2.1.3. Differentials .................................. 95
2.1.4. Vector Fields .................................. 97
2.1.5. Commutators. φ-relatedness .................... 102
2.1.6. Abstract Lie Groups ........................... 106
2.1.7. Left Invariant Vector Fields and the Lie
Algebra ....................................... 107
2.1.8. Homomorphisms ................................. 112
2.1.9. The Exponential Map ........................... 116
2.2. Carnot Groups ........................................ 121
2.2.1. Some Properties of the Stratification of a Carnot
Group ......................................... 125
2.2.2. Some General Results on Nilpotent Lie
Groups ........................................ 128
2.2.3. Abstract and Homogeneous Carnot Groups ........ 130
2.2.4. More Properties of the Lie Algebra ............ 138
2.2.5. Sub-Laplacians of a Stratified Group .......... 144
2.3. Exercises of Chapter 2 ............................... 147
3. Carnot Groups of Step Two ................................. 155
3.1. The Heisenberg-Weyl Group ............................ 155
3.2. Homogeneous Carnot Groups of Step Two ................ 158
3.3. Free Step-two Homogeneous Groups ..................... 163
3.4. Change of Basis ...................................... 165
3.5. The Exponential Map of a Step-two Homogeneous
Group ................................................ 166
3.6. Prototype Groups of Heisenberg Type .................. 169
3.7. H-groups (in the Sense of Metivier) .................. 173
3.8. Exercises of Chapter 3 ............................... 177
4. Examples of Carnot Groups ................................. 183
4.1. A Primer of Examples of Carnot Groups ................ 183
4.1.1. Euclidean Group ............................... 183
4.1.2. Carnot Groups with Homogeneous
Dimension Q ≤ 3 ............................... 184
4.1.3. B-groups ...................................... 184
4.1.4. K-type Groups ................................. 186
4.1.5. Sum of Carnot Groups .......................... 190
4.2. From a Set of Vector Fields to a Stratified Group .... 191
4.3. Further Examples ..................................... 198
4.3.1. The Vector Fields ∂1, ∂2 + х1∂3 ................ 198
4.3.2. Classical and Kohn Laplacians ................. 200
4.3.3. Bony-type Sub-Laplacians ...................... 202
4.3.4. Kolmogorov-type Sub-Laplacians ................ 204
4.3.5. Sub-Laplacians Arising in Control Theory ...... 205
4.3.6. Filiform Carnot Groups ........................ 207
4.4. Fields not Satisfying One of the Hypotheses (H0), (H1),
(H2) ................................................. 210
4.4.1. Fields not Satisfying Hypothesis (H0) ......... 210
4.4.2. Fields not Satisfying Hypothesis (H1) ......... 212
4.4.3. Fields not Satisfying Hypothesis (H2) ......... 215
4.5. Exercises of Chapter 4 ............................... 215
5. The Fundamental Solution for a Sub-Laplacian and
Applications .............................................. 227
5.1. Homogeneous Norms .................................... 229
5.2. Control Distances or Carnot-Caratheodory Distances ... 232
5.3. The Fundamental Solution ............................. 236
5.3.1. The Fundamental Solution in the Abstract
Setting ....................................... 244
5.4. -gauges and -radial Functions ...................... 246
5.5. Gauge Functions and Surface Mean Value Theorem ....... 251
5.6. Superposition of Average Operators ................... 257
5.7. Harnack Inequalities ................................. 262
5.8. Liouville-type Theorems .............................. 269
5.8.1. Asymptotic Liouville-type Theorems ............ 274
5.9. Sobolev-Stein Embedding Inequality ................... 276
5.10. Analytic Hypoellipticity of Sub-Laplacians .......... 280
5.11. Harmonic Approximation .............................. 287
5.12. An Integral Representation Formula for Г ............ 291
5.13. Appendix A. Maximum Principles ...................... 293
5.13.1. A Decomposition Theorem for -harmonic
Functions ................................... 303
5.14. Appendix B. The Improved Pseudo-triangle
Inequality .......................................... 306
5.15. Appendix C. Existence of Geodesies .................. 309
5.16. Exercises of Chapter 5 .............................. 319
Part II Elements of Potential Theory for Sub-Laplacians
6. Abstract Harmonic Spaces .................................. 337
6.1. Preliminaries ........................................ 338
6.2. Sheafs of Functions. Harmonic Sheafs ................. 340
6.2.1. Harmonic Measures and Hyperharmonic
Functions ..................................... 341
6.2.2. Directed Families of Functions ................ 342
6.3. Harmonic Spaces ...................................... 345
6.3.1. Directed Families of Harmonic and Hyperharmonic
Functions ..................................... 347
6.4.  -hyperharmonic Functions. Minimum Principle ......... 348
6.5. Subharmonic and Superharmonic Functions. Perron
Families ............................................. 353
6.6. Harmonic Majorants and Minorants ..................... 358
6.7. The Perron-Wiener-Brelot Operator .................... 359
6.8.  -harmonic Spaces: Wiener Resolutivity Theorem ....... 363
6.8.1. Appendix: The Stone-Weierstrass Theorem ....... 366
6.9.  -harmonic Measures for Relatively Compact Open
Sets ................................................. 367
6.10. *-harmonic Spaces: Bouligand's Theorem ............. 370
6.11. Reduced Functions and Balayage ...................... 375
6.12. Exercises of Chapter 6 .............................. 378
7. The -harmonic Space ...................................... 381
7.1. The -harmonic Space ................................. 381
7.2. Some Basic Definitions and Selecta of Properties ..... 388
7.3. Exercises of Chapter 7 ............................... 392
8. -subharmonic Functions ................................... 397
8.1. Sub-mean Functions ................................... 397
8.2. Some Characterizations of -subharmonic Functions .... 401
8.3. Continuous Convex Functions on  ..................... 411
8.4. Exercises of Chapter 8 ............................... 422
9. Representation Theorems ................................... 425
9.1. -Green Function for -regular Domains ............... 425
9.2. -Green Function for General Domains ................. 427
9.3. Potentials of Radon Measures ......................... 432
9.3.1. Potentials Related to the Average Operators ... 435
9.4. Riesz Representation Theorems for -subharmonic
Functions ............................................ 441
9.5. The Poisson-Jensen Formula ........................... 445
9.6. Bounded-above -subharmonic Functions in ........... 451
9.7. Smoothing of -subharmonic Functions ................. 455
9.8. Isolated Singularities—Bocher-type Theorems .......... 458
9.8.1. An Application of Bocher's Theorem ............ 462
9.9. Exercises of Chapter 9 ............................... 463
10. Maximum Principle on Unbounded Domains .................... 473
10.1. MP Sets and -thinness at Infinity .................. 473
10.2. q-sets and the Maximum Principle .................... 477
10.3. The Maximum Principle on Unbounded Domains .......... 482
10.4. The Proof of Lemma 10.2.3 ........................... 483
10.5. Exercises of Chapter 10 ............................. 487
11. -capacity, -polar Sets and Applications ................. 489
11.1. The Continuity Principle for -potentials ........... 489
11.2. -polar Sets ........................................ 491
11.3. The Maria-Frostman Domination Principle ............. 494
11.4. -energy and -equilibrium Potentials ............... 497
11.5. -balayage and -capacity ........................... 500
11.6. The Fundamental Convergence Theorem ................. 510
11.7. The Extended Poisson-Jensen Formula ................. 514
11.8. Further Results. A Miscellanea ...................... 519
11.9. Further Reading and the Quasi-continuity Property ... 527
11.10. Exercises of Chapter 11 ............................ 533
12. -thinness and -fine Topology ............................ 537
12.1. The -fine Topology: A More Intrinsic Tool .......... 537
12.2. -thinness at a Point ............................... 538
12.3. -thinness and -regularity ......................... 542
12.3.1. Functions Peaking at a Point ................ 542
12.3.2. -thinness and -regularity ................. 544
12.4. Wiener's Criterion for Sub-Laplacians ............... 547
12.4.1. A Technical Lemma ........................... 547
12.4.2. Wiener's Criterion for .................... 550
12.5. Exercises of Chapter 12 ............................. 553
13. d-Hausdorff Measure and -capacity ........................ 557
13.1. d-Hausdorff Measure and Dimension ................... 557
13.2. d-Hausdorff Measure and -capacity .................. 561
13.3. New Phenomena Concerning the d-Hausdorff
Dimension ........................................... 569
13.4. Exercises of Chapter 13 ............................. 572
Part III Further Topics on Carnot Groups
14. Some Remarks on Free Lie Algebras ......................... 577
14.1. Free Lie Algebras and Free Lie Groups ............... 577
14.2. A Canonical Way to Construct Free Carnot Groups ..... 584
14.2.1. The Campbell-Hausdorff Composition ◊ ........ 584
14.2.2. A Canonical Way to Construct Free Carnot
Groups ...................................... 586
14.3. Exercises of Chapter 14 ............................. 589
15. More on the Campbell-Hausdorff Formula .................... 593
15.1. The Campbell-Hausdorff Formula for Stratified
Fields .............................................. 593
15.2. The Campbell-Hausdorff Formula for Formal Power
Series-1 ............................................ 599
15.3. The Campbell-Hausdorff Formula for Formal Power
Series-2 ............................................ 605
15.4. The Campbell-Hausdorff Formula for Smooth Vector
Fields .............................................. 610
15.5. Exercises of Chapter 15 ............................. 616
16. Families of Difleomorphic Sub-Laplacians .................. 621
16.1. An Isomorphism Turning ∑i,jai,jXiXj into ΔG ........... 622
16.2. Examples and Counter-examples ....................... 628
16.3. Canonical or Non-canonical? ......................... 637
16.3.1. An Example .................................. 641
16.4. Further Reading: An Application to PDE's ............ 644
16.5. Exercises of Chapter 16 ............................. 645
17. Lifting of Carnot Groups .................................. 649
17.1. Lifting to Free Carnot Groups ....................... 649
17.2. An Example of Lifting ............................... 659
17.3. An Example of Application to PDE's .................. 661
17.4. Folland's Lifting of Homogeneous Vector Fields ...... 666
17.4.1. The Hypotheses on the Vector Fields ......... 669
17.5. Exercises of Chapter 17 ............................. 676
18. Groups of Heisenberg Type ................................. 681
18.1. Heisenberg-type Groups .............................. 681
18.2. A Direct Characterization of H-type Groups .......... 686
18.3. The Fundamental Solution on H-type Groups ........... 695
18.4. H-type Groups of Iwasawa-type ....................... 702
18.5. The H-inversion and the H-Kelvin Transform .......... 704
18.6. Exercises of Chapter 18 ............................. 709
19. The Caratheodory-Chow-Rashevsky Theorem ................... 715
19.1. The Caratheodory-Chow-Rashevsky Theorem for
Stratified Vector Fields ............................ 715
19.2. An Application of Caratheodory-Chow-Rashevsky
Theorem ............................................. 727
19.3. Exercises of Chapter 19 ............................. 730
20. Taylor Formula on Carnot Groups ........................... 733
20.1. Polynomials and Derivatives on Homogeneous Carnot
Groups .............................................. 734
20.1.1. Polynomial Functions on ................... 734
20.1.2. Derivatives on ............................ 736
20.2. Taylor Polynomials on Homogeneous Carnot Groups ..... 741
20.3. Taylor Formula on Homogeneous Carnot Groups ......... 746
20.3.1. Stratified Taylor Formula with Peano
Remainder ................................... 751
20.3.2. Stratified Taylor Formula with Integral
Remainder ................................... 754
20.4. Exercises of Chapter 20 ............................. 766
References .................................................... 773
Index of the Basic Notation ................................... 789
Index ......................................................... 795
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