Remarks on the regularity of quasislits

Abstract

A quasislit is the image of a vertical line segment \([0, iy]\), \(y > 0\), under a quasiconformal homeomorphism of the upper half-plane fixing \(\infty\). Quasislits correspond precisely to curves generated by the Loewner equation with a driving function in the Lip-\(\frac{1}{2}\) class. It is known that a quasislit is contained in a cone depending only on its Loewner driving function Lip-\(\frac{1}{2}\) seminorm, \(\sigma\). In this note we use the Loewner equation to give quantitative estimates on the opening angle of this cone in the maximal range \(\sigma <4\). The estimate is shown to be sharp for small \(\sigma\). As consequences, we derive explicit Hölder exponents for \(\sigma < 4\) as well as estimates on winding rates. We also relate quantitatively the Lip-\(\frac{1}{2}\) seminorm with the quasiconformal dilatation and discuss the optimal regularity of quasislits achievable through reparametrization.