Juha Heinonen

Sobolev Spaces on Metric Measure Spaces (eBook, PDF)

An Approach Based on Upper Gradients

• Format: PDF

Produktbeschreibung

Analysis on metric spaces emerged in the 1990s as an independent research field providing a unified treatment of first-order analysis in diverse and potentially nonsmooth settings. Based on the fundamental concept of upper gradient, the notion of a Sobolev function was formulated in the setting of metric measure spaces supporting a Poincare inequality. This coherent treatment from first principles is an ideal introduction to the subject for graduate students and a useful reference for experts. It presents the foundations of the theory of such first-order Sobolev spaces, then explores geometric implications of the critical Poincare inequality, and indicates numerous examples of spaces satisfying this axiom. A distinguishing feature of the book is its focus on vector-valued Sobolev spaces. The final chapters include proofs of several landmark theorems, including Cheeger's stability theorem for Poincare inequalities under Gromov-Hausdorff convergence, and the Keith-Zhong self-improvement theorem for Poincare inequalities.

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Produktdetails

• Verlag: Cambridge University Press
• Erscheinungstermin: 5. Februar 2015
• Englisch
• ISBN-13: 9781316237250
• Artikelnr.: 53011821

Autorenporträt

Juha Heinonen (1960-2007) was Professor of Mathematics at the University of Michigan. His principal areas of research interest included quasiconformal mappings, nonlinear potential theory, and analysis on metric spaces. He was the author of over 60 research articles, including several posthumously, and two textbooks. A member of the Finnish Academy of Science and Letters, Heinonen received the Excellence in Research Award from the University of Michigan in 1997 and gave an invited lecture at the International Congress of Mathematicians in Beijing in 2002.

Inhaltsangabe

Preface
1. Introduction
2. Review of basic functional analysis
3. Lebesgue theory of Banach space-valued functions
4. Lipschitz functions and embeddings
5. Path integrals and modulus
6. Upper gradients
7. Sobolev spaces
8. Poincaré inequalities
9. Consequences of Poincaré inequalities
10. Other definitions of Sobolev-type spaces
11. Gromov-Hausdorff convergence and Poincaré inequalities
12. Self-improvement of Poincaré inequalities
13. An Introduction to Cheeger's differentiation theory
14. Examples, applications and further research directions
References
Notation index
Subject index.