Rectifiability, analytic capacity, and singular integrals
type: Item
instance of: scholarly article
title: Rectifiability, analytic capacity, and singular integrals
zbMATH Open document ID: 0917.28003
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/232510
full work available at URL: Mattila.MAN.html
review text: A survey of the remarkable progress made in the field in question in recent time. In a clear and vivid manner, the author discusses the material around the following ``principles'': 1) The analytic capacity of a one-dimensional compact subset \(E\) of the complex plane \(\mathbb{C}\) is zero if and only if \(E\) is purely unrectifiable. 2) The analytic capacity of a one-dimensional compact subset \(E\) of \(\mathbb{C}\) is positive if and only if the Cauchy integral operator is \(L^2\)-bounded on a large part of \(E\). 3) Singular integrals behave nicely on an \(m\)-dimensional subset \(E\) of \(\mathbb{R}^n\) if and only if \(E\) is rectifiable in a sense. Table of contents: 1. Analytic capacity; finite length. 2. Analytic capacity; infinite length. 3. Singular integrals on regular sets. 4. Existence of principal values. 5. Calderón-Zygmund theory on nonhomogeneous spaces. Section 1 is mainly devoted to the David theorem saying that principle 1) becomes a true statement if the one-dimensional Hausdorff measure of \(E\) is finite. The key elements of the proof are outlined. The rest of the material is presented without proofs.
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 31A05
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 30E20
zbMATH DE Number: 1184300
zbMATH Keywords: rectifiable set
zbMATH Keywords: Menger curvature
zbMATH Keywords: singular integrals
zbMATH Keywords: analytic capacity
zbMATH Keywords: Cauchy integral
zbMATH Keywords: Calderón-Zygmund theory
zbMATH Keywords: Hausdorff measure
author: Pertti Mattila
reviewed by: S. V. Kislyakov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1184300