A counterexample in conformal welding concerning chord-arc curves
type: Item
instance of: scholarly article
title: A counterexample in conformal welding concerning chord-arc curves
zbMATH Open document ID: 0602.30011
DOI: 10.1007/BF02384395
author: Stephen Semmes
published in: Arkiv för Matematik
publication date: 1986-01-01T00:00:00Z
review text: Let \(\Gamma\) be a Jordan curve in the plane that passes through \(\infty\), and let \(\Phi_+\) and \(\Phi_-\) be conformal maps of the upper and lower half-planes onto the complementary regions of \(\Gamma\). We can assume that \(\Phi_{\pm}(\infty)=\infty\), so that \(h=\Phi_+^{- 1}\circ (\Phi_-|_{{\mathbb{R}}})\) defines a homeomorphism of \({\mathbb{R}}\) onto itself. The reverse process through which \(\Gamma\) is obtained from h is called conformal welding, and the problem is to understand how the geometry of \(\Gamma\) is reflected in h. It is well known that \(\Gamma\) is a quasicircle if and only if h satisfies a doubling condition. If \(\Gamma\) is a chord-arc curve, then h' lies in the Muckenhoupt class of weights \(A_{\infty}\). \textit{D. Jerison} and \textit{C. E. Kenig} [Math. Scand. 50, 221-247 (1982; Zbl 0509.30025)] asked whether the converse is true. A small constant result holds, as was shown by \textit{G. David} [Ann. Inst. Fourier 32, No.3, 227-239 (1982; Zbl 0464.30028)]. However, the global result fails: in the paper under review, a curve is constructed that is not locally rectifiable, whose h satisfies \(c\leq h'\leq C\).
Mathematics Subject Classification ID: 30C35
Mathematics Subject Classification ID: 30C85
zbMATH DE Number: 3971126
zbMATH Keywords: conformal welding
zbMATH Keywords: chord-arc curve
MaRDI profile type: Publication
cites work: Quasiconformal reflections
cites work: Q5512337
cites work: Weighted norm inequalities for maximal functions and singular integrals
cites work: Courbes corde-arc et espaces de Hardy généralisés
cites work: Hardy spaces, $A_\infty$, and singular integrals on chord-arc domains
Wikidata QID: Q124965859
description: scientific article; zbMATH DE number 3971126