Topics on Analysis in Metric Spaces

Topics on Analysis in Metric Spaces

by Luigi Ambrosio, Paolo Tilli
     
 

ISBN-10: 0198529384

ISBN-13: 9780198529385

Pub. Date: 02/28/2004

Publisher: Oxford University Press, USA

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

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Overview

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.

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Product Details

ISBN-13:
9780198529385
Publisher:
Oxford University Press, USA
Publication date:
02/28/2004
Series:
Oxford Lecture Series in Mathematics and Its Applications Series
Edition description:
New Edition
Pages:
144
Product dimensions:
9.30(w) x 6.30(h) x 0.50(d)

Table of Contents

1Some preliminaries on measure theory1
1.1Outer measures1
1.2Signed and vector measures9
1.3Weak convergence of measures14
2Hausdorff measures and covering theorems in metric spaces19
2.1Hausdorff measures19
2.2Covering theorems23
2.3Relationships between Hausdorff and Lebesgue measures28
2.4Densities30
3Lipschitz functions in metric spaces35
3.1Definition and general properties35
3.2Lipschitz functions of several real variables39
3.3The area formula44
3.4The one-dimensional area formula45
4The geodesic problem and Gromov-Hausdorff convergence53
4.1Metric derivative and geodesics in metric spaces54
4.2Reparametrization59
4.3Existence of geodesics61
4.4The intrinsic formulation64
4.5Gromov-Hausdorff convergence of metric spaces79
5Sobolev spaces in a metric framework89
5.1Definition of metric Sobolev spaces89
5.2Doubling measures and maximal operators92
5.3Equivalence between classical and metric Sobolev spaces98
5.4Poincare and Sobolev inequalities101
6A quick overview on the theory of integration113
6.1The Archimedean integral114
6.2Integration with respect to nondecreasing set functions114
6.3Integral of extended real-valued functions121
Bibliography125
Index131

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