Equicontinuity of mean quasiconformal mappings

We establish the equicontinuity and normality of the families $\mathfrak R^\Phi$ of ring $Q(x)$-homeomorphisms with integral-type restrictions $\int\Phi(Q(x))\,dm(x)<\infty$ on a domain $D\subset\mathbb R^n$ with $n\ge2$. The resulting conditions on $\Phi$ are not only sufficient but also necessary for the equicontinuity and normality of these families of mappings. We give some applications of these results to the Sobolev classes $W^{1,n}_\mathrm{loc}$.