Geometric Function Theory and Non-linear Analysis

Front Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Preface......Page 7
CONTENTS......Page 10
1 Introduction and overview ......Page 18
1.1 The planar theory ......Page 21
1.2 n-Dimensional quasiconformal mappings ......Page 30
1.3 The Liouville theorem ......Page 33
1.4 Higher integrability ......Page 34
1.5 Stability and rigidity phenomena ......Page 35
1.6 Quasiconformal structures on manifolds ......Page 36
1.7 Nevanlinna theory ......Page 40
1.9 Singular integral operators ......Page 42
1.11 Quasiconformal groups, semigroups and dynamics ......Page 44
1.12 Continuum mechanics and non-linear elasticity ......Page 46
1.13 Mostow rigidity ......Page 48
2.1 The Cauchy-Riemann system ......Page 49
2.2 The Mobius group ......Page 51
2.4 Curvature ......Page 53
2.5 Computing the Jacobian ......Page 56
2.6 Conclusions ......Page 57
2.7 Further aspects ......Page 58
3.1 Mapping classes ......Page 60
3.2 Harnack inequalities ......Page 62
3.3 A stability function ......Page 64
3.4 Passing Harnack inequalities on to Mt ......Page 65
3.5 Local injectivity ......Page 67
4.1 Schwartz distributions ......Page 70
4.3 Mollification ......Page 74
4.4 Lebesgue points ......Page 75
4.5 Pointwise coincidence of Sobolev functions ......Page 76
4.6 Alternative characterizations ......Page 77
4.7 Cross product of gradient fields ......Page 80
4.8 The adjoint differential ......Page 82
4.9 Subharmonic distributions ......Page 84
4.10 Embedding theorems ......Page 85
4.11 Duals and compact embeddings ......Page 90
4.12 Orlicz-Sobolev spaces ......Page 91
4.13 Hardy spaces and BMO ......Page 97
5.1 Introduction ......Page 102
5.2 Second-order estimates ......Page 104
5.3 Identities ......Page 107
5.4 Second-order equations ......Page 110
5.5 Continuity of the Jacobian ......Page 112
5.6 A formula for the Jacobian ......Page 114
5.7 Concluding arguments ......Page 115
6 Mappings of finite distortion ......Page 116
6.1 Differentiability ......Page 117
6.2 Integrability of the Jacobian ......Page 121
6.3 Absolute continuity ......Page 122
6.4 Distortion functions ......Page 125
6.5.1 Radial stretchings ......Page 129
6.5.2 Winding maps ......Page 132
6.5.3 Cones and cylinders ......Page 135
6.5.4 The Zorich exponential map ......Page 136
6.5.5 A regularity example ......Page 139
6.5.6 Squeezing the Sierpinski sponge ......Page 143
6.5.7 Releasing the sponge ......Page 151
7 Continuity ......Page 155
7.1 Distributional Jacobians ......Page 157
7.2 The Ll integrability of the Jacobian ......Page 160
7.3 Weakly monotone functions ......Page 165
7.4 Oscillation in a ball ......Page 167
7.5 Modulus of continuity ......Page 169
7.6 Exponentially integrable outer distortion ......Page 173
7.7 Holder estimates ......Page 177
7.8 Fundamental LP-inequality for the Jacobian ......Page 180
7.8.1 A class of Orlicz functions ......Page 181
7.8.2 Another proof of Corollary 7.2.1 ......Page 183
8.1 Distributional Jacobians revisited ......Page 186
8.2 Weak convergence of Jacobians ......Page 189
8.3 Maximal inequalities ......Page 192
8.4 Improving the degree of integrability ......Page 193
8.5 Weak limits and orientation ......Page 198
8.6 L log L integrability ......Page 202
8.7 A limit theorem ......Page 203
8.8 Polyconvex functions ......Page 204
8.8.1 Null Lagrangians ......Page 205
8.8.2 Polyconvexity of distortion functions ......Page 207
8.9 Biting convergence ......Page 208
8.10 Lower semicontinuity of the distortion ......Page 210
8.11 The failure of lower semicontinuity ......Page 214
8.12 Bounded distortion ......Page 217
8.13 Local injectivity revisited ......Page 218
8.14 Compactness for exponentially integrable distortion ......Page 222
9.1 The 1-covectors ......Page 225
9.2 The wedge product ......Page 226
9.4 The pullback ......Page 228
9.5 Matrix representations ......Page 229
9.6 Inner products ......Page 230
9.7 The volume element ......Page 233
9.8 Hodge duality ......Page 234
9.9 Hadamard-Schwarz inequality ......Page 237
9.10 Submultiplicity of the distortion ......Page 238
10.1 Differential forms in R" ......Page 239
10.2 Pullback of differential forms ......Page 245
10.3 Integration by parts ......Page 246
10.4 Orlicz-Sobolev spaces of differential forms ......Page 249
10.5 The Hodge decomposition ......Page 251
10.6 The Hodge decomposition in R" ......Page 253
11.1 The Beltrami equation ......Page 257
11.2 A fundamental example ......Page 261
11.2.1 The construction ......Page 262
11.3 Liouville-type theorem ......Page 267
11.4 The principal solution ......Page 268
11.5 Stoilow factorization ......Page 270
11.6 Failure of factorization ......Page 272
11.7 Solutions for integrable distortion ......Page 274
11.8 Distortion in the exponential class ......Page 276
11.8.1 An example ......Page 278
11.8.2 Statement of results ......Page 279
11.9.1 An example ......Page 281
11.9.2 Statement of results ......Page 282
11.9.3 Further generalities ......Page 284
11.10 Preliminaries ......Page 285
11.10.1 Results from harmonic analysis ......Page 286
11.10.2 Existence for exponentially integrable distortion ......Page 287
11.10.3 Uniqueness ......Page 293
11.10.4 Critical exponents ......Page 295
11.10.5 Existence for subexponentially integrable distortion ......Page 297
11.11 Global solutions ......Page 301
11.12 Holomorphic dependence ......Page 306
11.13 Examples and non-uniqueness ......Page 309
11.14 Compactness ......Page 316
11.15 Removable singularities ......Page 317
11.16 Final comments ......Page 318
12.1 Singular integral operators ......Page 320
12.2 Fourier multipliers ......Page 325
12.3 Trivial extension of a scalar operator ......Page 329
12.4 Extension to C" ......Page 330
12.5 The real method of rotation ......Page 332
12.6 The complex method of rotation ......Page 333
12.7 Polarization ......Page 336
12.8 The tensor product of Riesz transforms ......Page 338
12.9 Dirac operators and the Hilbert transform on forms ......Page 340
12.10 The LP-norms of the Hilbert transform on forms ......Page 347
12.11 Further estimates ......Page 349
12.12 Interpolation ......Page 350
13.1 Non-linear commutators ......Page 354
13.2 The complex method of interpolation ......Page 357
13.3 Jacobians and wedge products revisited ......Page 360
13.4 The H'-theory of wedge products ......Page 362
13.5 An L log L inequality ......Page 364
13.6 Estimates beyond the natural exponent ......Page 367
13.7 Proof of the fundamental inequality for Jacobians ......Page 369
14 The Gehring lemma ......Page 371
14.1 A covering lemma ......Page 373
14.2 Calderdn-Zygmund decomposition ......Page 374
14.3 Gehring's lemma in Orlicz spaces ......Page 376
14.4 Caccioppoli's inequality ......Page 380
14.5 The order of zeros ......Page 384
15.1 Equations in the plane ......Page 387
15.2 Absolute minima of variational integrals ......Page 392
15.3 Conformal mappings ......Page 397
15.4 Equations at the level of exterior algebra ......Page 403
15.5 Even dimensions ......Page 408
15.6 Signature operators ......Page 410
15.7 Four dimensions ......Page 415
16 Topological properties of mappings of bounded distortion ......Page 418
16.1 The energy integrand ......Page 419
16.2 The Dirichlet problem ......Page 422
16.3 The A-harmonic equation ......Page 423
16.5 The comparison principle ......Page 427
16.6 The polar set ......Page 428
16.7 Sets of zero conformal capacity ......Page 431
16.8 Qualitative analysis near polar points ......Page 433
16.9 Local injectivity of smooth mappings ......Page 436
16.10 The Jacobian is non-vanishing ......Page 439
16.11 Analytic degree theory ......Page 440
16.12 Openness and discreteness for mappings of bounded distortion ......Page 443
16.13 Further generalities ......Page 444
16.14 An update ......Page 445
17.1 Painleve's theorem in the plane ......Page 448
17.2 Hausdorff dimension and capacity ......Page 449
17.3 Removability of singularities ......Page 451
17.4 Distortion of dimension ......Page 454
18 Even dimensions ......Page 457
18.1 The Beltrami operator ......Page 458
18.2 Integrability theorems in even dimensions ......Page 460
18.3 Mappings with exponentially integrable distortion ......Page 463
18.4 The L^2 inverse of I-\nu S ......Page 466
18.5 Wl"n-regularity ......Page 469
18.6 Singularities ......Page 477
18.7 An example ......Page 478
19 Picard and Montel theorems in space ......Page 484
19.2 Serrin's theorem and Harnack functions ......Page 485
19.3 Estimates in H(R^n) ......Page 486
19.4 Harnack inequalities near zeros ......Page 489
19.5 Collections of Harnack functions ......Page 492
19.6 Proof of Rickman's theorem ......Page 494
19.7 Normal families ......Page 497
19.8 Montel's theorem in space ......Page 500
19.9 Further generalizations ......Page 501
20.1 The space S(n) ......Page 503
20.2 Conformal structures ......Page 506
20.3 The smallest ball ......Page 508
21 Uniformly quasiregular mappings ......Page 510
21.1 A first uniqueness result ......Page 511
21.2 First examples ......Page 513
21.3 Fatou and Julia sets ......Page 516
21.4 Lattes-type examples ......Page 518
21.5 Invariant conformal structures ......Page 522
22 Quasiconformal groups ......Page 527
22.1 Convergence properties ......Page 528
22.2 The elementary quasiconformal groups ......Page 530
22.3 Non-elementary quasiconformal groups ......Page 534
22.4 The triple space ......Page 536
22.5 Conjugacy results ......Page 537
22.6 Hilbert-Smith conjecture ......Page 541
22.7 Remarks ......Page 544
23.1 Uniqueness ......Page 545
23.2 Proof of Theorem 23.1.1 ......Page 546
23.3 Remarks ......Page 547
Bibliography ......Page 548
Index ......Page 564
Back Cover......Page 570