Lectures on Analysis on Metric Spaces

Juha Heinonen

ISBN 10: 1461265258 / ISBN 13: 9781461265252

Editore: Springer New York, Springer New York Sep 2012, 2012

Riguardo questo articolo

Neuware -Analysis in spaces with no a priori smooth structure has progressed to include concepts from the first order calculus. In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in rather general settings; abstract Sobolev space theories have been instrumental in this development. The purpose of this book is to communicate some of the recent work in the area while preparing the reader to study more substantial, related articles. The material can be roughly divided into three different types: classical, standard but sometimes with a new twist, and recent. The author first studies basic covering theorems and their applications to analysis in metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is relatively recent and appears for the first time in book format. There are plenty of exercises. The book is well suited for self-study, or as a text in a graduate course or seminar. The material is relevant to anyone who is interested in analysis and geometry in nonsmooth settings.Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg 156 pp. Englisch.

Codice articolo 9781461265252

Dati bibliografici

• Titolo: Lectures on Analysis on Metric Spaces
• Editore: Springer New York, Springer New York Sep 2012
• Data di pubblicazione: 2012
• Lingua: Inglese
• ISBN 10: 1461265258
• ISBN 13: 9781461265252
• Rilegatura: Taschenbuch
• Condizione: Neu

Descrizione articolo

Riassunto

Analysis is an outgrowth of calculus which considers functions defined in Euclidean spaces which have derivatives. The theory of distributions and Sobolev spaces allowed mathematicians to extend these ideas to functions which were no longer differentiable. Recent developments in complex analysis have permitted researchers to replace Euclidean spaces by general metric spaces. The author is a great expositor and gives a first taste of this exciting new area.

Contenuti

1. Covering Theorems.- 2. Maximal Functions.- 3. Sobolev Spaces.- 4. Poincaré Inequality.- 5. Sobolev Spaces on Metric Spaces.- 6. Lipschitz Functions.- 7. Modulus of a Curve Family, Capacity, and Upper Gradients.- 8. Loewner Spaces.- 9. Loewner Spaces and Poincaré Inequalities.- 10. Quasisymmetric Maps: Basic Theory I.- 11. Quasisymmetric Maps: Basic Theory II.- 12. Quasisymmetric Embeddings of Metric Spaces in Euclidean Space.- 13. Existence of Doubling Measures.- 14. Doubling Measures and Quasisymmetric Maps.- 15. Conformal Gauges.- References.