Rectifiability, analytic capacity, and singular integrals
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.
scientific article; zbMATH DE number 1184300