Hausdorff measure of quasicircles

Details Hausdorff measure of quasicircles Hausdorff measure of quasicircles

2009-12-17

arxiv.org/abs/0912.3365v1

Adv. Math. 229 (2) 2012, 1313-1328

Mathematics

Complex Variables

15 pages

Scientific paper

10.1016/j.aim.2011.11.001

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.

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