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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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