ACL homeomorphisms and linear dilatation (Q2781254)

For a domain \(D\) in \(\mathbb R^n\), (\(n\geq 1\)) and for a homeomorphism \(f:D\to \mathbb R^n\), let \(H(x,f)\) denote the linear dilatation of \(f\) at \(x\in D\). It is well-known that if \(H(x,f)\) is uniformly bounded in \(D\), then \(f\) is a quisiconformal mapping. The authors study the consequences of the condition that \(H(x,f)<\infty\) for each \(x\in D\setminus S\) under various assumptions on the exceptional set \(S\). The main result states that if \(S\) has \(\sigma\)-finite \((n-1)\)-dimensional Hausdorff measure and if \(H(x,f) \in L_{\text{loc}}^s(D)\) for \(s>n/(n-1)\), then \(f\) is absolutely continuous on lines (ACL). Another result is the following: If \(S\) has \(n\)-dimensional Lebesgue measure zero, then \(f\) is a.e. differentiable. Integrability properties of \(f^\prime\) are also proved.