Absolute continuity on lines of hyper \((\alpha, Q)\)-homeomorphisms (Q2901737)

A homeomorphism \(f:D\rightarrow {\mathbb R}^n\), \(n\geq 2,\) is said to be a hyper \((\alpha, Q)\)-homeomorphism, if the modulus of every family of surfaces of order \(\alpha\) has an upper integral estimate depending on \(Q\) under \(f\). It is proved that every hyper \((\alpha, Q)\)-homeomorphism \(f:D\rightarrow {\mathbb R}^n\) is absolutely continuous on lines provided that \(Q\in L_{\text{loc}}^1(D)\) and \(\alpha> n-1\). As a consequence, such a mapping \(f\) has partial derivatives almost everywhere.