Optimal regularity for planar mappings of finite distortion

Abstract: Let $f:\Omega\to\IR^2$ be a mapping of finite distortion, where $\Omega\subset\IR^2.$ Assume that the distortion function $K(x,f)$ satisfies $e^{K(\cdot, f)}\in L^p_{loc}(\Omega)$ 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^1_{loc}(\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}.