Methods of Geometric Analysis in Extension and Trace Problems Volume 1 [Hardcover]

The book presents a comprehensive exposition of extension results for maps between different geometric objects and of extension-trace results for smooth functions on subsets with no a priori differential structure (Whitney problems). The account covers development of the area from the initial classical works of the first half of the 20th century to the flourishing period of the last decade. Seemingly very specific these problems have been from the very beginning a powerful source of ideas, concepts and methods that essentially influenced and in some cases even transformed considerable areas of analysis. Aside from the material linked by the aforementioned problems the book also is unified by geometric analysis approach used in the proofs of basic results. This requires a variety of geometric tools from convex and combinatorial geometry to geometry of metric space theory to Riemannian and coarse geometry and more. The necessary facts are presented mostly with detailed proofs to make the book accessible to a wide audience.

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Preface.- Basic Terms and Notation.- Part 1. Classical Extension-Trace Theorems and Related Results.- Chapter 1. Continuous and Lipschitz Functions.- Chapter 2. Smooth Functions on Subsets of Rn.- Part 2. Topics in Geometry of and Analysis on Metric Spaces.- Chapter 3. Topics in Metric Space Theory.- Chapter 4. Selected Topics in Analysis on Metric Spaces.- Chapter 5. Lipschitz Embedding and Selections.- Bibliography.- Index.

Book under review is enormous in scope and containsmost of the old and current results on extension problems. Many of the theoremsappear here for the first time in book form. The book is self-contained, andthe detailed arguments make it accessible to a wide audience, l_