Geometric Function Theory and Non-linear Analysis

This book provides a survey of recent developments in the field of non-linear analysis and the geometry of mappings. Sobolev mappings, quasiconformal mappings, or deformations, between subsets of Euclidean space, or manifolds or more general geometric objects may arise as the solutions to certain optimisation problems in the calculus of variations or in non-linear elasticity, as the solutions to differential equations (particularly in conformal geometry), as local co-ordinates on a manifold or as geometric realisations of abstract isomorphisms between spaces such as those that arise in dynamical systems (for instance in holomorphic dynamics and Kleinian groups). In each case the regularity and geometric properties of these mappings and related non-linear quantities such as Jacobians, tells something about the problems and the spaces under consideration. The applications studied include aspects of harmonic analysis, elliptic PDE theory, differential geometry, the calculus of variations as well as complex dynamics and other areas. Indeed it is the strong interactions between these areas and the geometry of mappings that underscores and motivates the authors' work. Much recent work is included. Even in the classical setting of the Beltrami equation or measurable Riemann mapping theorem, which plays a central role in holomorphic dynamics, Teichmuller theory and low dimensional topology and geometry, the authors present precise results in the degenerate elliptic setting. The governing equations of non-linear elasticity and quasiconformal geometry are studied intensively in the degenerate elliptic setting, and there are suggestions for potential applications for researchers in other areas 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 A survey of recent developments in the field of non-linear analysis and the geometry of mappings. Sobolev mappings, quasiconformal mappings, or deformations, between subsets of Euclidean space, or manifolds or more general geometric objects may arise as the solutions to certain optimization problems in the calculus of variations or in non-linear elasticity, as the solutions to differential equations (particularly in conformal geometry), as local co-ordinates on a manifold or as geometric realizations of abstract isomorphisms between spaces such as those that arise in dynamical systems (for instance in holomorphic dynamics and Kleinian groups). In each case the regularity and geometric properties of these mappings and related non-linear quantities such as Jacobians, tells something about the problems and the spaces under consideration. Iwaniec (math, Syracuse U.) and Martin (math, U. of Auckland) explain recent developments in the geometry of mappings, related to functions or deformations between subsets of the Euclidean n-space Rn and more generally between manifolds or other geometric objects. Material on mappings intersects with aspects of differential geometry, topology, partial differential equations, harmonic analysis, and the calculus of variations. Chapters cover topics such as conformal mappings, stability of the Mobius group, Sobolev theory and function spaces, the Liouville theorem, even dimensions, Picard and Montel theorems in space, uniformly quasiregular mappings, and quasiconformal groups. Annotation c. Book News, Inc., Portland, OR (booknews.com) This unique book explores the connections between the geometry of mappings and many important areas of modern mathematics such as Harmonic and non-linear Analysis, the theory of Partial Differential Equations, Conformal Geometry and Topology. Much of the book is new. It aims to provide students and researchers in many areas with a comprehensive and up to date account and an overview of the subject as a whole.