Hausdorff measure of quasicircles

S. Smirnov (2010) [10] proved recently that the Hausdorff dimension of any K-quasicircle is at most 1+k2, where k=(K−1)/(K+1). In this paper we show that if Γ is such a quasicircle, thenH1+k2(B(x,r)∩Γ)⩽C(k)r1+k2for allx∈C,r>0, where Hs stands for the s-Hausdorff measure. On a related note we derive a sharp weak-integrability of the derivative of the Riemann map of a quasidisk.