{"product_id":"9781316235362","title":"Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients","description":"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.","brand":"Cambridge University Press","offers":[{"title":"Default Title","offer_id":47119344271600,"sku":"9781316235362","price":64.49,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0737\/7593\/9824\/files\/9781316235362_p0.jpg?v=1763707490","url":"https:\/\/shop-qa.barnesandnoble.com\/products\/9781316235362","provider":"Barnes \u0026 Noble (DEV)","version":"1.0","type":"link"}