Optimal regularity for planar mappings of finite distortion

Abstract

Let $f:Ω\to\IR^2$ be a mapping of finite distortion, where $Ω\subset\IR^2 .$ Assume that the distortion function $K(x,f)$ satisfies $e^{K(\cdot ,f)}\in L_{loc}^p(\Omega )$ for some $p>0.$ We establish optimal regularity and area distortion estimates for $f$. Especially, we prove that $|Df{|}^2{\log }^{\beta -1}(e+|Df|)\in L_{loc}^1(\Omega )$ for every $\beta <p.$ This answers positively well known conjectures due to Iwaniec and Martin \cite{IMbook} and to Iwaniec, Koskela and Martin \cite{IKM}.

22 pages, formula (3) has been corrected