On the theory of the mappings that quasiconformal in the mean on Riemann manifolds (Q2879962)

The present paper is devoted to the investigation of mappings \(f:D\rightarrow D_*\) where the domains \(D\) and \(D_*\) belong to some Riemann manifolds \({\mathbb M}^n\) and \(\mathbb M_*^n\), respectively, and \(\dim{\mathbb M}^n=\dim \mathbb M_*^n=n\). The so-called \(Q\)-homeomorphisms which are more general than quasiconformal mappings are studied in the paper. The main result is the following. Let \(g\) and \(g_{*}\) be the metric tensors of \(D\) and \(D_*\), respectively, and assume that the isoperimetrical dimension \(\mu\) of \(\left({\mathbb M}^n, g\right)\) is bigger than 1. Suppose also that \(D\) is locally connected at the boundary and that the boundary of \(D_*\) is strongly accessible. Then every \(Q\)-homeomorphism \(f:D\rightarrow D_*\) admits a continuous extension \(\overline{f}:\overline{D}\rightarrow \overline{D_*}\) provided that \(\int\limits_{D}\Phi(Q(x))dm(x)\) holds for some non-negative convex, non-decreasing function \(\Phi:[0, \infty]\rightarrow [0, \infty]\) satisfying the requirement \(\int\limits_{\delta_0}^{\infty}\frac{d\tau}{\tau\left[\Phi^{-1}(\tau)\right]^{\frac{1}{n-1}}} =\infty\) for some \(\delta_0>\Phi(0)\).