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 $e^{K(\cdot, f)}\in L^p_{loc}(Ω)$ for some $p>0.$ We establish optimal regularity and area distortion estimates for $f$. Especially, we prove that $|Df|^2 \log^{β-1}(e + |Df|) \in L^1_{loc}(Ω) $ for every $β<p.$ 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