Hajłasz, Piotr Sobolev spaces on metric-measure spaces. (English) Zbl 1048.46033 Auscher, Pascal (ed.) et al., Heat kernels and analysis on manifolds, graphs, and metric spaces. Lecture notes from a quarter program on heat kernels, random walks, and analysis on manifolds and graphs, April 16–July 13, 2002, Paris, France. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3383-9/pbk). Contemp. Math. 338, 173-218 (2003). This is a survey paper, but it includes some new results. Various possible extensions of the classical theory of the first order Sobolev spaces to the setting of metric spaces equipped with a Borel measure are discussed. The paper is well written and may serve as an introduction to the subject. Most of the results discussed in the paper are proved. Some of them are proved here for the first time or have new proofs. Motivation and the historical background are given. The following subjects are discussed: rectifiable curves in metric spaces, doubling measures, path families and their moduli, upper gradient and abstract derivatives, different definitions of Sobolev spaces on metric spaces and relations between them. Bibliographical remarks are made. The last include applications of the theory to quasi-conformal mappings, non-linear subelliptic operators, analysis on graphs and fractals.For the entire collection see [Zbl 1029.00030]. Reviewer: Leszek Skrzypczak (Poznań) Cited in 1 ReviewCited in 161 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis Keywords:Sobolev spaces; metric spaces; Poincaré inequality PDFBibTeX XMLCite \textit{P. Hajłasz}, Contemp. Math. 338, 173--218 (2003; Zbl 1048.46033)