Methods of Geometric Analysis in Extension and Trace Problems

Transcription

1 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 Buch. xxiii, 560 S. Hardcover ISBN 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.

2 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 Notation and definitions Extension and trace problems: formulations and examples Example: Continuous functions Example: Uniformly continuous functions Example: Continuously differentiable functions on R n Example: BMO and Sobolev spaces Continuous selections Simultaneous continuous extensions Extensions of continuous maps acting between metric spaces Absolute metric retracts Lipschitz Functions Notation and definitions Trace and extension problems for Lipschitz functions Lipschitz selection problem Counterexample Combinatorial geometric selection results Extensions preserving Lipschitz constants Banach-valued Lipschitz functions Extension and the intersection property of balls Proof of Theorem Lipschitz maps acting in spaces of constant curvature Lipschitz extensions Simultaneous Lipschitz extensions

3 vi Contents 1.13 Simultaneous Lipschitz selection problem Comments Appendices A Topological dimension and continuous extensions of maps into S n. 61 B Helly s topological theorem B.1 The Classical Helly theorem and related results B.2 Cohomology theory a computational aspect B.3 Helly s topological theorem C Sperner s lemma and its consequences D Contractions of n-spheres Smooth Functions on Subsets of R n Classical function spaces: notation and definitions Differentiable functions k-jets Lipschitz functions of higher order Extension and trace problems for classical function spaces Whitney s extension theorem Divided differences, local approximation and differentiability Trace and extension problems for univariate C k functions Whitney s theorem Reformulation of Whitney s theorem Finiteness and linearity Basic conjectures Restricted main problem for some classes of domains in R n Quasiconvex domains Lipschitz domains Sobolev spaces: selected trace and extension results P. Jones theorem and related results Peetre s nonexistence theorem Comments Appendices E Difference identities E.1 Difference identities E.2 Marchaud s identity F Local polynomial approximation and moduli of continuity F.1 Degree of local polynomial approximation F.2 Whitney s constants F.3 Conjectures G Local inequalities for polynomials

4 Contents vii II Topics in Geometry of and Analysis on Metric Spaces Topics in Metric Space Theory Principal concepts and related facts Pseudometrics, metrics and quasimetrics Metric and quasimetric spaces Paracompactness and continuous partitions of unity Compact and precompact metric spaces Proper metric spaces Doubling metric spaces Metric length structure Basic metric constructions Measures on metric spaces Measure theory Integration Measurable selections Hausdorff measures Doubling measures Families of pointwise doubling measures Basic classes of metric spaces Ultrametric spaces Spaces of bounded geometry Riemannian manifolds as metric spaces Gromov hyperbolic spaces Sub-Riemannian manifolds Metric graphs Metric groups Comments Selected Topics in Analysis on Metric Spaces Dvoretsky type theorem for finite metric spaces Covering metric invariants Metric dimension Hausdorff dimension Hausdorff dimension of doubling metric spaces Nagata dimension Existence of doubling measures Finite metric spaces Compact metric spaces Complete metric spaces Dyn kin conjecture Concluding remarks Space of balls B(M) as a length space

5 viii Contents B(R n ) as a space of pointwise homogeneous type Generalized hyperbolic spaces H n+1 ω Differentiability of Lipschitz functions Lipschitz functions on R n Lipschitz functions on metric spaces Lipschitz spaces Modulus of continuity Real interpolation of Lipschitz spaces Duality theorem Comments Lipschitz Embedding and Selections Embedding of metric spaces into the space forms Finite metric spaces Infinite metric trees Doubling metric spaces Gromov hyperbolic spaces Roughly similar embeddings of Gromov hyperbolic spaces Coarse Geometry, a survey Coarse geometry of H n The Bonk Schramm theorem Lipschitz selections Barycenter and Steiner selectors Helly type result: a conjecture A Sylvester type selection result Simultaneous Lipschitz selections The problem Formulation of the main theorem Auxiliary results Proof of Theorem Proof of Proposition Proof of Proposition Comments Bibliography 527 Index 557

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