Analysis of and on uniformly rectifiable sets. (English) Zbl 0832.42008

David, Guy; Semmes, Stephen

Mathematical Surveys and Monographs. 38. Providence, RI: American Mathematical Society (AMS). xii, 356 p. (1993).

This book is a mixture of geometric measure theory and harmonic analysis which deals mainly with the quantitative notion of the rectifiability of sets in \(R^n\), so-called uniform rectifiability, by some techniques in harmonic analysis, such as singular integrals, Littlewood-Paley theory (square function estimates), Carleson measure, Corona decomposition, etc. The two authors of the book did a systematic investigation on the analysis of functions that live in a \(d\)-dimensional set in \(R^n\), on the behavior of certain linear operators that act on these functions (for example, variants of the Cauchy integral operators), and on the related geometric properties of the involved set. They devoted a series of publications, including a book “Singular integrals and rectifiable sets in \(R^n\). Au-delà des graphes lipschitziens” [Astérisque 193 (1991; Zbl 0743.49018)] to the topic. This book is a summing up and a remarkable development of these researches. It divides into 4 parts. Part 1 (about 50 pages) gives the background information and the summary of the main results. Part 2 and Part 3 give the proofs of the main results (various characterizations of the uniform rectifiability). Part 4 provides an alternative approach by giving new and more direct proofs.

Reviewer: Long Ruilin (Beijing)

MSC:

42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
49Q15	Geometric measure and integration theory, integral and normal currents in optimization
28A75	Length, area, volume, other geometric measure theory
42-02	Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
49-02	Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
30C65	Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30G35	Functions of hypercomplex variables and generalized variables
30C85	Capacity and harmonic measure in the complex plane

Keywords:

Lipschitz graph; geometric measure theory; harmonic analysis; rectifiability of sets; singular integrals; Carleson measure

Citations:

Zbl 0743.49018