Remarks on the regularity of quasislits - Schoug, Lukas; Shekhar, Atul; Viklund, Fredrik (Annales Fennici Mathematici, 2021)
A quasislit is the image of a vertical line segment \([0, iy]\), \(y > 0\), under a quasiconformal homeomorphism of the upper half-plane fixing \(∞\). Quasislits correspond precisely to curves generated by the Loewner equation with a driving function in the Lip-\(12\) class. It is known that a quasislit is contained in a cone depending only on its Loewner driving function Lip-\(12\) seminorm, \(σ\). In this note we use the Loewner equation to give quantitative estimates on the opening angle of this cone in the maximal range \(σ <4\). The estimate is shown to be sharp for small \(σ\). As consequences, we derive explicit Hölder exponents for \(σ < 4\) as well as estimates on winding rates. We also relate quantitatively the Lip-\(12\) seminorm with the quasiconformal dilatation and discuss the optimal regularity of quasislits achievable through reparametrization.