Hausdorff measure of quasicircles
Copyright policy )Hausdorff measure of quasicircles
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 $Γ$ is such a quasicircle, then $H^{1+k^2}(B(x,r)\cap Γ)\leq C(k) r^{1+k^2}$ 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.
15 pages
- Michigan State University United States
- University of Missouri United States
- Autonomous University of Barcelona Spain
- University of Helsinki Finland
- Centre de Recerca Matemàtica Spain
Mathematics(all), Mathematics - Complex Variables, Aplicacions quasi-conformes, Quasiconformal mappings in the complex plane, 30C62, 517 - Anàlisi, Hausdorff measure, quasicircle, quasiconformal mappings in the plane, Hausdorff and packing measures, FOS: Mathematics, Complex Variables (math.CV), Hausdorff, Mesures de
Mathematics(all), Mathematics - Complex Variables, Aplicacions quasi-conformes, Quasiconformal mappings in the complex plane, 30C62, 517 - Anàlisi, Hausdorff measure, quasicircle, quasiconformal mappings in the plane, Hausdorff and packing measures, FOS: Mathematics, Complex Variables (math.CV), Hausdorff, Mesures de
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