Quasiconformal distortion of Hausdorff measures and quasicircles

In this talk I will explain some new results in connection with quasiconformal distortion of Hausdorff measures. One of the results, in collaboration with Prause and Uriarte-Tuero asserts that if $G$ is a K-quasicercle, then $H^{1+h^2}(G \cap B(x,r)) \leq C r^{1+h^2}$ for any ball $B(x,r)$, where $H$ denotes the Hausdorff measure and $h=(K-1)/(K+1)$. I will also talk about other results concerning the distortion of Hausdorff measure of general sets, partially based on a work from Astala, Clop, Tolsa, Uriarte-Tuero and Verdera.