A counterexample in conformal welding concerning chord-arc curves
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 \(c\leq h'\leq C\).
scientific article; zbMATH DE number 3971126