The Cauchy integral, analytic capacity and subsets of quasicircles
The paper revolves around a theme that suggests a connection between the \(L^2\)-boundedness of the Cauchy integral operator \({\mathcal C}\) and the geometric properties of sets \(K \subseteq \mathbb{C}\). In this direction, G. David, P. Jones and S. Semmes have raised several questions, and the following problems are still open: Problem 1: Suppose \(K\) is \(AD\)-regular and \({\mathcal C}\) is bounded on \(L^2 (K)\). Is it true that \(K\) is contained in an \(AD\)-regular curve? Problem 2: Suppose \(K\) has a finite one-dimensional Hausdorff measure, \(\Lambda (K) < \infty\). Is it true that \(K\) has a positive analytic capacity, \(\gamma (K) > 0\), if and only if there exists a rectifiable curve \(\Gamma\) such that \(\Lambda (K \cap \Gamma) > 0\)? \textit{G. David} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 17, 157-189 (1984; Zbl 0537.42016)] has shown that Problem 1 is true when \(K\) is connected. Later \textit{G. David} and \textit{A. Semmes} [Montrogue: Societé Mathematique de France Astérisque. 193, 145 p. (1991; Zbl 0537.42016)] have also proved that Problem 1 is true when \({\mathcal C}\) is replaced by all singular integrals with odd kernels. In this paper, the author provides a partial answer to Problems 1 and 2 by showing that if \(K\) is \(AD\)-regular and sufficiently flat, and \({\mathcal C}\) is bounded on \(L^2 (K)\), then \(K\) is a subset of a chord-arc curve. He also proves that if \(K\) is \(AD\)-regular and locally flat with a positive analytic capacity, then \(K\) contains a subset of a rectifiable curve of positive length. Finally, the author provides a characterization of subset \(K\) of some quasicircles in terms of a geometric quantity \(\beta (K)\) introduced by \textit{P. W. Jones} [Lect. Notes Math. 1384, 24-68 (1987; Zbl 0675.30029)].
scientific article; zbMATH DE number 759241