Equicontinuity of mappings quasiconformal in the mean

It is stated equicontinuity and normality of families \(\mathfrak{R}^{\Phi}\) of the so-called ring \(Q(x)\)-homeomorphisms with integral constraints of the type \(\int \Phi (Q(x)) dm(x) < \infty\) in a domain \(D \subset \mathbf{R}^n\), \(n \ge 2\). It is shown that the found conditions on the function \(\Phi\) are not only sufficient but also necessary for equicontinuity and normality of such families of mappings. It is also given applications of these results to families of mappings in the Sobolev class \(W_{\text{loc}}^{1,n}\).