Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients

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.

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.

Pekka Koskela is Professor of Mathematics at the University of Jyväskylä, Finland. He works in Sobolev mappings and in the associated nonlinear analysis, and he has authored over 140 publications. He gave invited lectures at the European Congress of Mathematics in Barcelona in 2000 and at the International Congress of Mathematicians in Hyderabad in 2010. Koskela is a member of the Finnish Academy of Science and Letters. He received the Väisälä Award in 2001 and the Magnus Ehrnrooth Foundation prize in 2012.

Nageswari Shanmugalingam is Professor of Mathematics at the University of Cincinnati. Her research interests include analysis in metric measure spaces, potential theory, functions of bounded variation and quasiminimal surfaces in metric setting. The foundational part of the structure of Sobolev spaces in metric setting was developed by her in her PhD thesis in 1999, and she has also contributed to the development of potential theory in metric setting. Her research contributions were recognized by the College of Arts and Sciences at the University of Cincinnati with a McMicken Dean's Award in 2008.

Jeremy T. Tyson is Professor of Mathematics at the University of Illinois, Urbana-Champaign, working in analysis in metric spaces, geometric function theory and sub-Riemannian geometry. He has authored over 40 research articles and co-authored two other books. Tyson has received awards for teaching from the University of Illinois at both the departmental and college level. He is a Fellow of the American Mathematical Society.