Hausdorff measure of quasicircles

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 $H^{1+k^2}(B(x,r)\cap \Gamma )\le C(k)r^{1+k^2}$ 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.

15 pages

References in corpus (2)

• Quasisymmetric distortion spectrum
• Quasiconformal distortion of Hausdorff measures

Cited by in corpus (3)

• Asymptotic variance of the Beurling transform
• Quasisymmetry and rectifiability of quasispheres
• Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps