Monographs in Mathematics 102 Methods of Geometric Analysis in Extension and Trace Problems Volume 1 Bearbeitet von Alexander Brudnyi, Prof. Yuri Brudnyi Technion R&D Foundation Ltd 1. Auflage 2011. Buch. xxiii, 560 S. Hardcover ISBN 978 3 0348 0208 6 Format (B x L): 15,5 x 23,5 cm Gewicht: 1039 g Weitere Fachgebiete > Mathematik > Mathematische Analysis Zu Leseprobe schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.
Contents Preface Basic Terms and Notation xi xvii I Classical Extension-Trace Theorems and Related Results 1 1 Continuous and Lipschitz Functions 5 Continuous Functions............................ 5 1.1 Notation and definitions........................ 6 1.2 Extension and trace problems: formulations and examples..... 7 1.2.1 Example: Continuous functions................ 9 1.2.2 Example: Uniformly continuous functions.......... 9 1.2.3 Example: Continuously differentiable functions on R n... 10 1.2.4 Example: BMO and Sobolev spaces............. 11 1.3 Continuous selections.......................... 12 1.4 Simultaneous continuous extensions.................. 14 1.5 Extensions of continuous maps acting between metric spaces.... 16 1.6 Absolute metric retracts........................ 17 Lipschitz Functions.............................. 20 1.7 Notation and definitions........................ 20 1.8 Trace and extension problems for Lipschitz functions........ 22 1.9 Lipschitz selection problem...................... 23 1.9.1 Counterexample........................ 23 1.9.2 Combinatorial geometric selection results.......... 25 1.10 Extensions preserving Lipschitz constants.............. 27 1.10.1 Banach-valued Lipschitz functions.............. 27 1.10.2 Extension and the intersection property of balls....... 30 1.10.3 Proof of Theorem 1.26..................... 33 1.10.4 Lipschitz maps acting in spaces of constant curvature... 33 1.11 Lipschitz extensions.......................... 42 1.12 Simultaneous Lipschitz extensions.................. 53
vi Contents 1.13 Simultaneous Lipschitz selection problem.............. 55 Comments................................... 56 Appendices.................................. 61 A Topological dimension and continuous extensions of maps into S n. 61 B Helly s topological theorem...................... 64 B.1 The Classical Helly theorem and related results....... 64 B.2 Cohomology theory a computational aspect........ 67 B.3 Helly s topological theorem.................. 71 C Sperner s lemma and its consequences................ 73 D Contractions of n-spheres....................... 78 2 Smooth Functions on Subsets of R n 83 2.1 Classical function spaces: notation and definitions.......... 84 2.1.1 Differentiable functions.................... 84 2.1.2 k-jets.............................. 86 2.1.3 Lipschitz functions of higher order.............. 87 2.1.4 Extension and trace problems for classical function spaces. 92 2.2 Whitney s extension theorem..................... 93 2.3 Divided differences, local approximation and differentiability.... 101 2.4 Trace and extension problems for univariate C k functions..... 120 2.4.1 Whitney s theorem....................... 120 2.4.2 Reformulation of Whitney s theorem............. 129 2.4.3 Finiteness and linearity.................... 130 2.4.4 Basic conjectures........................ 133 2.5 Restricted main problem for some classes of domains in R n.... 134 2.5.1 Quasiconvex domains..................... 134 2.5.2 Lipschitz domains....................... 149 2.6 Sobolev spaces: selected trace and extension results......... 155 2.6.1 P. Jones theorem and related results............. 155 2.6.2 Peetre s nonexistence theorem................. 160 Comments................................... 165 Appendices.................................. 170 E Difference identities.......................... 170 E.1 Difference identities...................... 170 E.2 Marchaud s identity...................... 176 F Local polynomial approximation and moduli of continuity..... 178 F.1 Degree of local polynomial approximation.......... 178 F.2 Whitney s constants...................... 186 F.3 Conjectures........................... 188 G Local inequalities for polynomials................... 188
Contents vii II Topics in Geometry of and Analysis on Metric Spaces 197 3 Topics in Metric Space Theory 201 3.1 Principal concepts and related facts................. 201 3.1.1 Pseudometrics, metrics and quasimetrics........... 201 3.1.2 Metric and quasimetric spaces................ 203 3.1.3 Paracompactness and continuous partitions of unity.... 212 3.1.4 Compact and precompact metric spaces........... 215 3.1.5 Proper metric spaces...................... 219 3.1.6 Doubling metric spaces.................... 222 3.1.7 Metric length structure.................... 226 3.1.8 Basic metric constructions................... 237 3.2 Measures on metric spaces....................... 252 3.2.1 Measure theory......................... 252 3.2.2 Integration........................... 254 3.2.3 Measurable selections..................... 255 3.2.4 Hausdorff measures....................... 256 3.2.5 Doubling measures....................... 263 3.2.6 Families of pointwise doubling measures........... 266 3.3 Basic classes of metric spaces..................... 274 3.3.1 Ultrametric spaces....................... 274 3.3.2 Spaces of bounded geometry................. 278 3.3.3 Riemannian manifolds as metric spaces........... 280 3.3.4 Gromov hyperbolic spaces................... 286 3.3.5 Sub-Riemannian manifolds.................. 291 3.3.6 Metric graphs.......................... 292 3.3.7 Metric groups.......................... 301 Comments................................... 312 4 Selected Topics in Analysis on Metric Spaces 317 4.1 Dvoretsky type theorem for finite metric spaces........... 318 4.2 Covering metric invariants....................... 326 4.2.1 Metric dimension........................ 326 4.2.2 Hausdorff dimension...................... 329 4.2.3 Hausdorff dimension of doubling metric spaces....... 334 4.2.4 Nagata dimension....................... 341 4.3 Existence of doubling measures.................... 357 4.3.1 Finite metric spaces...................... 358 4.3.2 Compact metric spaces.................... 367 4.3.3 Complete metric spaces.................... 369 4.3.4 Dyn kin conjecture....................... 369 4.3.5 Concluding remarks...................... 371 4.4 Space of balls.............................. 372 4.4.1 B(M) as a length space.................... 373
viii Contents 4.4.2 B(R n ) as a space of pointwise homogeneous type...... 379 4.4.3 Generalized hyperbolic spaces H n+1 ω............. 384 4.5 Differentiability of Lipschitz functions................ 386 4.5.1 Lipschitz functions on R n................... 386 4.5.2 Lipschitz functions on metric spaces............. 389 4.6 Lipschitz spaces............................. 394 4.6.1 Modulus of continuity..................... 394 4.6.2 Real interpolation of Lipschitz spaces............ 397 4.6.3 Duality theorem........................ 405 Comments................................... 413 5 Lipschitz Embedding and Selections 417 5.1 Embedding of metric spaces into the space forms.......... 418 5.1.1 Finite metric spaces...................... 418 5.1.2 Infinite metric trees...................... 423 5.1.3 Doubling metric spaces.................... 438 5.1.4 Gromov hyperbolic spaces................... 447 5.2 Roughly similar embeddings of Gromov hyperbolic spaces..... 454 5.2.1 Coarse Geometry, a survey.................. 454 5.2.2 Coarse geometry of H n.................... 458 5.2.3 The Bonk Schramm theorem................. 461 5.3 Lipschitz selections........................... 470 5.3.1 Barycenter and Steiner selectors............... 470 5.3.2 Helly type result: a conjecture................ 476 5.3.3 A Sylvester type selection result............... 483 5.4 Simultaneous Lipschitz selections................... 493 5.4.1 The problem.......................... 493 5.4.2 Formulation of the main theorem............... 494 5.4.3 Auxiliary results........................ 495 5.4.4 Proof of Theorem 5.66..................... 497 5.4.5 Proof of Proposition 5.68................... 512 5.4.6 Proof of Proposition 5.69................... 519 Comments................................... 523 Bibliography 527 Index 557
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