Topics on Analysis in Metric Spaces

by Ambrosio, Luigi; Tilli, Paolo

• ISBN13: 9780198529385
• ISBN10: 0198529384
• Format: Hardcover
• Copyright: 2004-02-26
• Publisher: Oxford University Press

Summary

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 sp

Author Biography

Paolo Tilli is a Researcher in mathematical analysis at the Scuola Normale Superiore, Pisa.

Table of Contents

1 Some preliminaries on measure theory	1	(18)
1.1 Outer measures	1	(8)
1.2 Signed and vector measures	9	(5)
1.3 Weak convergence of measures	14	(3)
Exercises	17	(2)
2 Hausdorff measures and covering theorems in metric spaces	19	(16)
2.1 Hausdorff measures	19	(3)
Exercises	22	(1)
2.2 Covering theorems	23	(5)
2.3 Relationships between Hausdorff and Lebesgue measures	28	(2)
2.4 Densities	30	(2)
Exercises	32	(3)
3 Lipschitz functions In metric spaces	35	(18)
3.1 Definition and general properties	35	(4)
Exercises	39	(1)
3.2 Lipschitz functions of several real variables	39	(4)
Exercise	43	(1)
3.3 The area formula	44	(1)
3.4 The one-dimensional area formula	45	(6)
Exercises	51	(2)
4 The geodesic problem and Gromov-Hausdorff convergence	53	(36)
4.1 Metric derivative and geodesics in metric spaces	54	(4)
Exercises	58	(1)
4.2 Reparametrization	59	(2)
4.3 Existence of geodesics	61	(2)
Exercises	63	(1)
4.4 The intrinsic formulation	64	(15)
Exercises	79	(1)
4.5 Gromov-Hausdorff convergence of metric spaces	79	(7)
Exercises	86	(3)
5 Sobolev spaces in a metric framework	89	(24)
5.1 Definition of metric Sobolev spaces	89	(3)
5.2 Doubling measures and maximal operators	92	(6)
Exercises	98	(1)
5.3 Equivalence between classical and metric Sobolev spaces	98	(2)
Exercise	100	(1)
5.4 Poincare and Sobolev inequalities	101	(9)
Exercises	110	(3)
6 A quick overview on the theory of integration	113	(12)
6.1 The Archimedean integral	114	(1)
6.2 Integration with respect to nondecreasing set functions	114	(7)
6.3 Integral of extended real-valued functions	121	(4)
Bibliography	125	(6)
Index	131