On equicontinuity of one family of space mappings with unbounded characteristic

In the present paper, some class of space mappings satisfying geometric estimates with respect to some outer measure, that is conformal modulus of families of curves, is studied. It is proved the equicontinuity of above classes in a closure of a domain provided that the majorant corresponding to a distortion of families of curves has a finite mean oscillation at every point, or satisfies some other conditions.
Let $D$ be a domain in ${\Bbb R}^n,$ $n\ge 2,$ and $f:D\rightarrow {\Bbb R}^n$ be a continuous mapping. Set $\overline{{\Bbb R}^n}={\Bbb R}^n\cup\{\infty\},$ let $m$ be the Lebesgue measure in ${\Bbb R}^n,$ and $M$ be the conformal modulus of families of curves. Given a domain $D$ and two sets $E$ and $F$ in ${\overline{{\Bbb R}^n}},$ $n\ge 2,$ $\Gamma (E,F,D)$ denotes the family of all paths $\gamma:[a,b]\rightarrow {\overline{{\Bbb R}^n}}$ which join $E$ and $F$ in $D$, i.e., $\gamma(a)\in E,$ $\gamma(b)\in F$ and $\gamma(t)\in D$ for $a<t<b.$ Denote by $S(x_0,r_1)$ and $S(x_0,r_2)$ the corresponding boundaries of the spherical ring $A(x_0,r_1,r_2) = \{ x\in{\Bbb R}^n : r_1<|x-x_0|<r_2\}$ and let $S_i=S(x_0, r_i),$ $i=1,2.$ Given a (Lebesgue) measurable function $Q: D \rightarrow [0,\infty]$, a mapping $f:D\rightarrow {\Bbb R}^n$ is called ring $Q$–mapping at a point $x_0\in D$ if
\begin{equation} M\left(f(\Gamma(S_1, S_2, A(x_0,r_1,r_2)))\right) \le \int\limits_{A(x_0,r_1,r_2)} Q(x)\cdot \eta^n(|x-x_0|)dm(x)\tag{1} \end{equation}
for $0<r_1<r_2< r_0={\rm dist}(x_0,\partial D),$ and for every Lebesgue measurable function $\eta: (r_1,r_2)\rightarrow [0,\infty ]$ such that $\int\limits_{r_1}^{r_2}\eta(r)dr\ge 1.$ By analogy, given a Lebesgue measurable function $Q:{\Bbb R}^n\rightarrow [0, \infty],$ $Q(x)\equiv 0$ for every $x\not\in D,$ we say that a mapping $f:D\rightarrow \overline{{\Bbb R}^n}$ is a ring $Q$-mapping at $x_0\in \overline{D},$ $x_0\ne \infty,$ if for every $r_0=r(x_0)$ and $A=A(r_1,r_2,x_0)$ the relation (1) holds for every continua $E_1\subset \overline{B(x_0, r_1)}\cap D$ and $E_2\subset \left(\overline{{\Bbb R}^n}\setminus B(x_0, r_2)\right)\cap D.$
Note that analytic functions ($n=2$) are ring $Q$-mappings with $Q\equiv 1,$ ant that the so-called mappings with bounded distortion are ring $Q$-mappings with $Q\le K=const.$ We say that a function $\varphi:D\rightarrow {\Bbb R} $ has finite mean oscillation at a point $x_0 \in {D} $ if $\limsup\limits_{\varepsilon\rightarrow 0}\frac{1}{\Omega_n\cdot \varepsilon^n} \int\limits_{B(x_0, \,\varepsilon)} |\varphi(x)-\widetilde{\varphi_{\varepsilon}}|\,dm(x)< \infty$ where $\widetilde{\varphi_{\varepsilon}}= \frac{1}{\Omega_n\cdot \varepsilon^n}\int\limits_{B( x_0, \,\varepsilon)} \varphi(x)\, dm(x)\,.$ We say that a boundary $\partial D$ of $D$ is strongly accessible at $x_0\in \partial D$ if, for every neighborhood $U$ of $x_0$ there exists a compactum $E\subset D,$ a neighborhood $V\subset U$ of $x_0$ and a number $\delta >0$ such that $M(\Gamma(E,F, D))\ge \delta$ for every continua $F$ in $D,$ $F\cap\partial U\ne\varnothing\ne F\cap\partial V.$ It is known that, in particular, all convex bounded domains have strongly accessible boundaries.
Given domains $D,$ $D^{\,\prime}\subset {\Bbb R}^n,$ $z_1, z_2\in D,$ $z_1\ne z_2,$ $z_1^{\prime},$ $z_2^{\prime}\in D^{\prime}$ and Lebesgue measurable function $Q(x): {\Bbb R}^n\rightarrow [0, \infty]$ obeying $Q(x)\equiv 0$ for $x\not\in D,$ denote $\frak{R}_{z_1, z_2, z_1^{\,\prime}, z_2^{\,\prime}, Q}(D, D^{\,\prime})$ a family of all ring $Q$-homeomorphisms $f:D\rightarrow D^{\,\prime}$ satisfying to (1) in $\overline{D},$ $f(D)=D^{\,\prime},$ such that $f(z_1)=z_1^{\prime},\quad f(z_2)=z_2^{\prime}\,.$ Given a Lebesgue measurable function $Q\colon{\Bbb R}^n\rightarrow[0, \infty]$ and $x_0\in{\Bbb R}^n,$ $q_{x_0}(r)$ is integral mean value of $Q$ under sphere $S(x_0, r).$ Denote $q_{x_0}^*(r)$ a mean integral value of
$ Q^*(x)=\begin{cases} Q(x), & Q(x)\ge 1,\\ 1, & Q(x)<1 \end{cases} $ under the sphere $S(x_0, r).$ Now we have the following.
Theorem. Let a domain $D$ be locally connected at all boundary points, $\partial D^{\,\prime}$ is strongly accessible, and $Q(x)$ satisfies for every $x_0\in\overline D$ at least one of the following conditions: 1) $Q(x)\in FMO(x_0)$; 2) $q_{x_0}(r)=O([\log\frac{1}{r}]^{n-1})$ at $r\rightarrow 0$; 3) for some $\delta(x_0)>0,$ $\int\limits_{0}^{\delta(x_0)}\frac{dt}{t{q_{x_0}^*}^{1/(n-1)}(t)}=\infty. $ Then every $f\in \frak{R}_{z_1, z_2, z_1^{\,\prime}, z_2^{\,\prime}, Q}(D, D^{\,\prime})$ has a continuous extension $\overline f\colon\overline D\rightarrow\overline {D^{\,\prime}}$, moreover, a family $\overline{\frak{R}_{z_1, z_2, z_1^{\,\prime}, z_2^{\,\prime}, Q}(D, D^{\,\prime})}$ which consists of all extended mappings mentioned above, is equicontinuous (normal) in $\overline D$.