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 $\mathrm{\infty }$. 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.