Lectures On Analysis On Metric Spaces
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Lectures on Analysis on Metric Spaces
Author | : Juha Heinonen |
Publsiher | : Springer Science & Business Media |
Total Pages | : 149 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 9781461301318 |
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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.
Introduction to the Analysis of Metric Spaces
Author | : John R. Giles |
Publsiher | : Cambridge University Press |
Total Pages | : 276 |
Release | : 1987-09-03 |
Genre | : Mathematics |
ISBN | : 0521359287 |
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This is an introduction to the analysis of metric and normed linear spaces for undergraduate students in mathematics. Assuming a basic knowledge of real analysis and linear algebra, the student is exposed to the axiomatic method in analysis and is shown its power in exploiting the structure of fundamental analysis, which underlies a variety of applications. An example is the link between normed linear spaces and linear algebra; finite dimensional spaces are discussed early. The treatment progresses from the concrete to the abstract: thus metric spaces are studied in some detail before general topology is begun, though topological properties of metric spaces are explored in the book. Graded exercises are provided at the end of each section; in each set the earlier exercises are designed to assist in the detection of the structural properties in concrete examples while the later ones are more conceptually sophisticated.
Lectures on analysis in metric spaces
Author | : Luigi Ambrosio,Francesco Serra Cassano |
Publsiher | : Edizioni della Normale |
Total Pages | : 0 |
Release | : 2001-10-01 |
Genre | : Mathematics |
ISBN | : 8876422552 |
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This book contains the notes of an international summer school on Analysis in Metric Spaces. The contributions are the following: T. Coulhon, Random walks and geometry on infinite graphs; G. David, Uniform rectifiability and quasiminimal sets; P. Koskela, Upper gradients and Poincaré inequalities; S. Semmes, Derivatives and difference quotients for Lipschitz or Sobolev functions on various spaces; R. L. Wheeden, Some weighted Poincaré estimates in spaces of homogenous type.
Analysis and Geometry of Metric Measure Spaces
Author | : Galia Devora Dafni,Robert John McCann,Alina Stancu |
Publsiher | : American Mathematical Soc. |
Total Pages | : 241 |
Release | : 2013 |
Genre | : Mathematics |
ISBN | : 9780821894187 |
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Contains lecture notes from most of the courses presented at the 50th anniversary edition of the Seminaire de Mathematiques Superieure in Montreal. This 2011 summer school was devoted to the analysis and geometry of metric measure spaces, and featured much interplay between this subject and the emergent topic of optimal transportation.
New Trends on Analysis and Geometry in Metric Spaces
Author | : Fabrice Baudoin,Séverine Rigot,Giuseppe Savaré,Nageswari Shanmugalingam |
Publsiher | : Springer Nature |
Total Pages | : 312 |
Release | : 2022-02-04 |
Genre | : Mathematics |
ISBN | : 9783030841416 |
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This book includes four courses on geometric measure theory, the calculus of variations, partial differential equations, and differential geometry. Authored by leading experts in their fields, the lectures present different approaches to research topics with the common background of a relevant underlying, usually non-Riemannian, geometric structure. In particular, the topics covered concern differentiation and functions of bounded variation in metric spaces, Sobolev spaces, and differential geometry in the so-called Carnot–Carathéodory spaces. The text is based on lectures presented at the 10th School on "Analysis and Geometry in Metric Spaces" held in Levico Terme (TN), Italy, in collaboration with the University of Trento, Fondazione Bruno Kessler and CIME, Italy. The book is addressed to both graduate students and researchers.
Topics on Analysis in Metric Spaces
Author | : Luigi Ambrosio,Paolo Tilli |
Publsiher | : Oxford University Press, USA |
Total Pages | : 148 |
Release | : 2004 |
Genre | : Mathematics |
ISBN | : 0198529384 |
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This book presents the main mathematical prerequisites for analysis in metric spaces. It covers abstract measure theory, Hausdorff measures, Lipschitz functions, covering theorums, lower semicontinuity of the one-dimensional Hausdorff measure, Sobolev spaces of maps between metric spaces, and Gromov-Hausdorff theory, all developed ina general metric setting. The existence of geodesics (and more generally of minimal Steiner connections) is discussed on general metric spaces and as an application of the Gromov-Hausdorff theory, even in some cases when the ambient space is not locally compact. A brief and very general description of the theory of integration with respect to non-decreasing set functions is presented following the Di Giorgi method of using the 'cavalieri' formula as the definition of the integral. Based on lecture notes from Scuola Normale, this book presents the main mathematical prerequisites for analysis in metric spaces. Supplemented with exercises of varying difficulty it is ideal for a graduate-level short course for applied mathematicians and engineers.
Sobolev Spaces on Metric Measure Spaces
Author | : Juha Heinonen,Pekka Koskela,Nageswari Shanmugalingam,Jeremy T. Tyson |
Publsiher | : Cambridge University Press |
Total Pages | : 447 |
Release | : 2015-02-05 |
Genre | : Mathematics |
ISBN | : 9781107092341 |
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This coherent treatment from first principles is an ideal introduction for graduate students and a useful reference for experts.
Geometry and Analysis of Metric Spaces via Weighted Partitions
Author | : Jun Kigami |
Publsiher | : Springer |
Total Pages | : 164 |
Release | : 2020-11-17 |
Genre | : Mathematics |
ISBN | : 3030541533 |
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The aim of these lecture notes is to propose a systematic framework for geometry and analysis on metric spaces. The central notion is a partition (an iterated decomposition) of a compact metric space. Via a partition, a compact metric space is associated with an infinite graph whose boundary is the original space. Metrics and measures on the space are then studied from an integrated point of view as weights of the partition. In the course of the text: It is shown that a weight corresponds to a metric if and only if the associated weighted graph is Gromov hyperbolic. Various relations between metrics and measures such as bilipschitz equivalence, quasisymmetry, Ahlfors regularity, and the volume doubling property are translated to relations between weights. In particular, it is shown that the volume doubling property between a metric and a measure corresponds to a quasisymmetry between two metrics in the language of weights. The Ahlfors regular conformal dimension of a compact metric space is characterized as the critical index of p-energies associated with the partition and the weight function corresponding to the metric. These notes should interest researchers and PhD students working in conformal geometry, analysis on metric spaces, and related areas.