Hausdorff measure of quasicircles

TL;DR

This paper establishes a precise measure-theoretic property of K-quasicircles, showing their Hausdorff measure scales predictably and deriving sharp integrability results for the Riemann map derivative.

Contribution

It proves a new upper bound on the Hausdorff measure of quasicircles and derives sharp integrability conditions for the Riemann map derivative, extending previous dimension results.

Findings

Hausdorff measure of quasicircles scales as r^{1+k^2}

Derived sharp weak-integrability of the Riemann map derivative

Confirmed and extended Smirnov's dimension bound

Abstract

S. Smirnov proved recently that the Hausdorff dimension of any K-quasicircle is at most 1+k^2, where k=(K-1)/(K+1). In this paper we show that if $\Gamma$ is such a quasicircle, then $H^{1+k^2}(B(x,r)\cap \Gamma)\leq C(k) r^{1+k^2}$ for all x in \C and r>0, where H^s stands for the s-Haudorff measure. On a related note we derive a sharp weak-integrability of the derivative of the Riemann map of a quasidisk.