Distortion of area and dimension under quasiconformal mappings in the plane.

Abstract

We find the exact estimates for the distortion of area and Hausdorff dimension under a K-quasiconformal mapping of the complex plane. This solves also the problem of finding the precise bound p(K) of the exponents p such that for each planar K-quasiconformal mapping f the Jacobian Jf is locally p-integrable; it follows that p(K) = 2K/(K - 1). Further consequences include among others the regularity and removability results for quasiregular mappings and sharp estimates for the complex Hilbert transform.