Remarks on the regularity of quasislits

Lukas Schoug, Atul Shekhar, Fredrik Viklund
2021 Annales Fennici Mathematici  
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-1 2 class. It is known that a quasislit is contained in a cone depending only on its Loewner driving function Lip-1 2 seminorm, σ. In this note we use the Loewner equation to give quantitative estimates on the opening angle of this cone in the
more » ... l 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-1 2 seminorm with the quasiconformal dilatation and discuss the optimal regularity of quasislits achievable through reparametrization.
doi:10.5186/aasfm.2021.4623 fatcat:46dmkc6vpjbgvbwpq6qphffs64