Sobolev Spaces on Metric Measure Spaces

This book provides a thorough introduction to Sobolev spaces, ideal for both graduate students and experts, highlighting foundational principles and significant theorems.

The field of analysis on metric spaces has experienced significant growth since the 1990s, evolving into a distinct area of research. Sobolev Spaces on Metric Measure Spaces is authored by some of the pioneers in this domain, offering a comprehensive exploration of the subject, starting from fundamental principles. This text serves as an excellent introduction for graduate students while also acting as a valuable reference for seasoned experts in the field.

At the core of Sobolev Spaces on Metric Measure Spaces is the concept of upper gradients, which paved the way for defining Sobolev functions within the framework of metric measure spaces that uphold a Poincaré inequality. The book meticulously outlines the foundational aspects of first-order Sobolev spaces and delves into the geometric ramifications of the critical Poincaré inequality. Furthermore, it presents a variety of examples illustrating spaces that conform to this essential axiom.

A notable aspect of this work is its emphasis on vector-valued Sobolev spaces, which distinguishes it from other texts. The concluding chapters contain proofs of several significant theorems, including Cheeger's stability theorem regarding Poincaré inequalities under Gromov–Hausdorff convergence and the Keith–Zhong self-improvement theorem for Poincaré inequalities. Overall, this book is not only a well-structured introduction but also a rich resource for advanced study in the field.