Lectures on Analysis on Metric Spaces (Universitext) by Juha Heinonen (PDF)

Description

The purpose of this book is to communicate some of the recent advances in this field while preparing the reader for more advanced study. The material can be roughly divided into three different types: classical, standard but sometimes with a new twist, and recent. The author first studies basic covering theorems and their applications to analysis in metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is recent and appears for the first time in book format.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is intended as an introduction to some modern topics that now comprise the field of “analysis on metric spaces”. This is an active research area, closely interacting with many other fields such as fractal geometry, complex analysis, sub-Riemannian geometry, geometric group theory, and more. The primary topics covered are metric Sobolev spaces, quasisymmetric maps, and metric measure spaces.This book is a polished version of Heinonen’s notes from a class he taught at Michigan. That means it offers a brief, broad sweep of the field, picking out some highlights and helping a newcomer orient themself without attempting to be comprehensive. The style aims at being elegant and concise–sometimes at the expense of precision and detail. This book alone is probably insufficient for one to really grasp the material. Fortunately Heinonen includes a fairly extensive bibiliography with some 200 references; hence this book can be seen as a springboard for more in-depth study, especially by reading the original papers.The difficulty of the book really depends on your mathematical maturity and any prior acquaintance with the subject matter. In my case, this is the first book my advisor (actually a student of Heinonen) had me read. Some of the topics required some motivation on his part, and maybe some supplemental reading, before they really made sense.I don’t know of any book that is equivalent, though a few similar ones exist. The closest would be “Topics in Analysis on Metric Spaces” by Ambrosio and Tilli, which covers similar material but more from the vantage point of geometric measure theory (whereas Heinonen’s book comes from the tradition of quasiconformal mappings). Perhaps one would also consider “A Course in Metric Geometry” by Burago, Burago, and Ivanov, though their focus is on Alexandrov geometry.

⭐this book may well save the universe

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