Heinonen J., Koskela P., Shanmugalingam N., Tyson J.T. Sobolev Spaces on Metric Measure Spaces. An Approach Based on Upper Gradients

Cambridge University Press, 2015. — 448 p. — (New Mathematical Monographs: 27).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.A clear introduction to a burgeoning research area
The extensive bibliography introduces readers to the rapidly expanding literature
Readers need only a graduate-level knowledge in analysisReview of basic functional analysis
Lebesgue theory of Banach space-valued functions
Lipschitz functions and embeddings
Path integrals and modulus
Upper gradients
Sobolev spaces
Poincaré inequalities
Consequences of Poincaré inequalities
Other definitions of Sobolev-type spaces
Gromov–Hausdorff convergence and Poincaré inequalities
Self-improvement of Poincaré inequalities
An Introduction to Cheeger's differentiation theory
Examples, applications and further research directionsNotation index
Subject index.