Sobolev Spaces on Metric Measure Spaces
An Approach Based on Upper Gradients
Nonfiction, Science & Nature, Mathematics, Functional Analysis, Algebra
| Author: | Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson | ISBN: | 9781316235362 |
| Publisher: | Cambridge University Press | Publication: | February 5, 2015 |
| Imprint: | Cambridge University Press | Language: | English |
| Author: | Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson |
| ISBN: | 9781316235362 |
| Publisher: | Cambridge University Press |
| Publication: | February 5, 2015 |
| Imprint: | Cambridge University Press |
| Language: | English |
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 Poincaré 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 Poincaré 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 Poincaré inequalities under Gromov–Hausdorff convergence, and the Keith–Zhong self-improvement theorem for Poincaré inequalities.
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 Poincaré 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 Poincaré 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 Poincaré inequalities under Gromov–Hausdorff convergence, and the Keith–Zhong self-improvement theorem for Poincaré inequalities.
