# Sevost'yanov E. A.

### On the equicontinuity of families of mappings in the case of variable domains

Sevost'yanov E. A., Skvortsov S. A.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 938-951

UDC 517.5

We study the problem of local behavior of maps in the closure of a domain in the Euclidean space. The equicontinuity of
families of these mappings is established in the case where the mapped domain is not fixed. We separately consider the
domains with bad and good boundaries, as well as the homeomorphisms and maps with branching.

### On the local behavior of Sobolev classes on two-dimensional Riemannian manifolds

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 663-676

UDC 517.9

We study open discrete maps of two-dimensional Riemannian manifolds from the Sobolev class. For these mappings,
we obtain the lower estimates of distortions of the moduli of the families of curves. As a consequence, we establish the
equicontinuity of Sobolev classes at interior points of the domain.

### On the lower estimate of the distortion of distance for one class of mappings

Markish A. A., Salimov R. R., Sevost'yanov E. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1553-1562

We study the behavior of one class of mappings with finite distortion in a neighborhood of the origin. Under certain conditions imposed on the characteristic of quasiconformality, we establish a lower estimate for the distortion of distance under mappings of the indicated kind.

### On the equicontinuity of one family of inverse mappings in terms of prime ends

Salimov R. R., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1264-1273

For a class of mappings satisfying upper modular estimates with respect to families of curves, we study the behavior of the corresponding inverse mappings. In the terms of prime ends, we prove that the families of these homeomorphisms are equicontinuous (normal) in the closure of a given domain.

### On convergence of mappings in metric spaces with direct and inverse modulus conditions

Sevost'yanov E. A., Skvortsov S. A.

Ukr. Mat. Zh. - 2018. - 70, № 7. - pp. 952-967

For mappings in metric spaces satisfying one inequality with respect to the modulus of families of curves, we establish the property of lightness of the limit mapping. It is shown that the uniform limit of these mappings is a light mapping, whenever the function responsible for the distortion of the families of curves, is of finite mean oscillation at every point. In addition, for one class of homeomorphisms of metric spaces, we prove theorems on the equicontinuity of the families of inverse mappings.

### On equicontinuity of homeomorphisms of the Orlicz and Orlicz – Sobolev classes in the closure of a domain

Petrov E. A., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2017. - 69, № 11. - pp. 1564-1576

We investigate the behavior of homeomorphisms of the Orlicz – Sobolev classes in the closure of a domain. The theorems on equicontinuity of the indicated classes are obtained in terms of the prime ends of regular domains. In particular, it is shown that indicated classes are equicontinuous in domains with certain restrictions imposed on their boundaries provided that the corresponding inner dilatation of order p has a majorant of finite mean oscillation at every point. We also prove theorems on the (pointwise) equicontinuity of the analyzed classes in the case of locally connected boundaries.

### On the absolute continuity of mappings distorting the moduli of cylinders

Salimov R. R., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 860-864

We consider the mappings satisfying one modular inequality with respect to cylinders in the space. The distortion of modulus is majorized by an integral depending on a certain locally integrable function. We also prove a result on the absolute continuity of the analyzed mappings on lines.

### On the equicontinuity of mappings with branching in the closure of the domain

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 273-279

We study the problem of local behavior of mappings f : D \rightarrow R^n,\; n \geq 2,$ in $D$. Under certain conditions imposed on a measurable function $Q(x), Q : D \rightarrow [0,\infty ]$, and the boundaries of $D$ and $D\prime = f(D)$, we show that a family of open discrete mappings $f : D \rightarrow R^n$ with a characteristic of quasiconformality $Q(x)$ is equicontinuous in $D$.

### On the local behavior of open discrete mappings of the Orlicz – Sobolev classes

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1259-1272

The paper is devoted to the study of mappings with unbounded characteristic of quasiconformality and, in particular, of mappings with finite distortion extensively studied in recent years. We obtain theorems on equicontinuity of families of mappings that belong to the Orlicz–Sobolev class for $n \geq 3$, and have finite distortion. To do this, we also investigate some auxiliary classes of mappings, namely, we study the relationship between the so-called lower $Q$-mappings and some inequalities of the capacity type.

### On the removability of isolated singularities of Orlicz – Sobolev classes with branching

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 683-693

The local behavior of closed-open discrete mappings of the Orlicz – Sobolev classes in $R^n,\; n \geq 3$, is investigated. It is proved that the indicated mappings have continuous extensions to an isolated boundary point $x_0$ of the domain $D \setminus \{ x0\}$, whenever the $n - 1$ degree of its inner dilatation has FMO (finite mean oscillation) at this point and, in addition, the limit sets of $f$ at $x_0$ and $\partial D$ are disjoint. Another sufficient condition for the possibility of continuous extension can be formulated as a condition of divergence of a certain integral.

### Normality of the Orlicz - Sobolev classes

Ryazanov V. I., Salimov R. R., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 106-116

We establish a series of new criteria of equicontinuity and, hence, normality of the mappings of Orlicz – Sobolev classes in terms of inner dilatations.

### Analog of the Montel Theorem for Mappings of the Sobolev Class with Finite Distortion

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 829-837

We study the classes of mappings with unbounded characteristic of quasiconformality and obtain a result on the normal families of open discrete mappings $f : D → ℂ \backslash \{a, b\}$ from the class $W\{\text{loc}^{1,1}$ with finite distortion that do not take at least two fixed values $a 6 ≠ b$ in $ℂ$ whose maximal dilatation has a majorant of finite mean oscillation at every point. This result is an analog of the well-known Montel theorem for analytic functions and is true, in particular, for the so-called $Q$-mappings.

### On the Radius of Injectivity for Generalized Quasiisometries in the Spaces of Dimension Higher Than Two

Gol'berg A. L., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 174-184

We consider a class of local homeomorphisms more general than the mappings with bounded distortion. Under these homeomorphisms, the growth of the *p*-module (*n-*1 *< p ≤ n*) of the families of curves is controlled by an integral containing an admissible metric and a measurable function *Q.* It is shown that, under generic conditions imposed on the majorant *Q,* this class has a positive radius of injectivity (and, hence, a ball in which every mapping is homeomorphic). Moreover, one of the conditions imposed on *Q* is not only sufficient but also necessary for existence of a radius of injectivity.

### On Equicontinuous Families of Mappings Without Values in Variable Sets

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 361–370

The present paper is devoted to the study of the classes of mappings with unbounded characteristics of quasiconformality. We prove sufficient conditions for the equicontinuity of the families of these mappings that do not take values from a set *E* provided that a real-valued characteristic *c*(*E*) of these mappings has a lower bound of the form *c*(*E*) ≥ \( \delta \) , \( \delta \) \( \epsilon \) ℝ.

### On the Orlicz–Sobolev Classes and Mappings with Bounded Dirichlet Integral

Ryazanov V. I., Salimov R. R., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1254–1265

It is shown that homeomorphisms *f* in \( {{\mathbb{R}}^n} \) , *n* ≥ 2, with finite Iwaniec distortion of the Orlicz–Sobolev classes *W* ^{1,φ} _{loc} under the Calderon condition on the function φ and, in particular, the Sobolev classes *W* ^{1,φ} _{loc}, *p* > *n* - 1, are differentiable almost everywhere and have the Luzin (*N*) -property on almost all hyperplanes. This enables us to prove that the corresponding inverse homeomorphisms belong to the class of mappings with bounded Dirichlet integral and establish the equicontinuity and normality of the families of inverse mappings.

### On the boundary behavior of open discrete mappings with unbounded characteristic

Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 855-859

We study the problem of extension of mappings $f : D → R^n,\; n ≥ 2$, to the boundary of a domain $D$. Under certain conditions imposed on a measurable function $Q(x),\; Q: D → [0, ∞]$, and the boundaries of the domains $D$ and $D' = f(D)$, we show that an open discrete mapping $f : D → R^n,\; n ≥ 2$, with quasiconformality characteristic $Q(x)$ can be extended to the boundary $\partial D$ by continuity. The obtained statements extend the corresponding Srebro’s result to mappings with bounded distortion.

### Analogs of the Ikoma?Schwartz lemma and Liouville theorem for mappings with unbounded characteristic

Salimov R. R., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1368-1380

In the present paper, we obtain results on the local behavior of open discrete mappings $f:\;D \rightarrow \mathbb{R}^n, \quad n \geq 2,$, that satisfy certain conditions related to the distortion of capacities of condensers. It is shown that, in an infinitesimal neighborhood of zero, the indicated mapping cannot grow faster than an integral of a special type that corresponds to the distortion of the capacity under this mapping, which is an analog of the well-known growth estimate of Ikoma proved for quasiconformal mappings of the unit ball into itself and of the classical Schwartz lemma for analytic functions. For mappings of the indicated type, we also obtain an analogue of the well-known Liouville theorem for analytic functions.

### On the openness and discreteness of mappings with unbounded characteristic of quasiconformality

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1128-1134

The paper is devoted to the investigation of the topological properties of space mappings. It is shown that sense-preserving mappings $f : D \rightarrow \overline{\mathbb{R}^n}$ in a domain $D \subset \mathbb{R}^n$, n ≥ 2, which are more general than mappings with bounded distortion, are open and discrete if a function Q corresponding to the control of the distortion of families of curves under these mappings has slow growth in the domain f(D), e.g., if Q has finite mean oscillation at an arbitrary point $y0 \in f(D)$.

### On some properties of generalized quasiisometries with unbounded characteristic

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 385-398

We consider a family of the open discrete mappings $f:\; D \rightarrow \overline{\mathbb{R}^n}$ that distort in a special way the $p$ -modulus of families of curves connecting the components of spherical condenser in a domain $D$ in $\mathbb{R}^n$, $p > n — 1,\;\; p < n$, and omitting a set of positive $p$-capacity. We establish that this family is normal provided that some function realizing the control of the considered distortion of curve family has a finite mean oscillation at every point or only logarithmic singularities of the order, which is not larger than $n − 1$. We prove that, under these conditions, an isolated singularity $x_0 \in D$ of the mapping $f : D \ \{x_0\} \rightarrow \overline{\mathbb{R}^n}$ is removable and, moreover, the extended mapping is open and discrete. As applications we obtain analogs of the known Liouville and Sokhotski – Weierstrass theorems.

### On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 69-79

For the open discrete mappings *f*: *D* \ {*b*} → **R**^{3} of the domain *D* ⊂ **R**^{3} satisfying relatively general geometric conditions in *D* \ {*b*} and having the essential singularity *b* ∈ **R**^{3}, we prove the following
statement.
Let *y*_{0} belong to **R**^{3} \ *f* (*D* \ {*b*}) and let the inner dilatation *K*_{I } (*x*, *f*) and the outer dilatation
K_{Ο }(*x*, *f*) of the mapping *f* at a point *x* satisfy certain conditions.
Denote by *B _{f }* the set of branch points of

*f*. Then for an arbitrary neighborhood

*V*of the point

*y*

_{0}, a set

*V*∩

*f*(

*B*) cannot be contained in the set

_{f }*A*such that

*g*(

*A*) =

*I*, where

*I*= {

*t*∈

**R**: |

*t*| < 1} and

*g*:

*U*→

**R**

^{n}is a quasiconformal mapping of the domain

*U*⊂

**R**

^{n}such that

*A*⊂

*U*.

### Estimation of dilatations for mappings more general than quasiregular mappings

Salimov R. R., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1531–1537

We consider the so-called ring $Q$-mappings, which are natural generalizations of quasiregular mappings in a sense of Väisälä’s geometric definition of moduli. It is shown that, under the condition of nondegeneracy of these mappings, their inner dilatation is majorized by a function $Q(x)$ to within a constant depending solely on the dimension of the space.

### On the sets of branch points of mappings more general than quasiregular

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 215–230

It is shown that if a point $x_0 ∊ ℝ^n, \; n ≥ 3$, is an essential isolated singularity of an open discrete $Q$-mapping $f : D → \overline{ℝ^n}, B_f$ is the set of branch points of $f$ in $D$; and a point $z_0 ∊ \overline{ℝ^n}$ is an asymptotic limit of $f$ at the point $x_0$; then, for any neighborhood $U$ containing the point $x_0$; the point $z_0 ∊ \overline{f(B_f ∩ U)}$ provided that the function $Q$ has either a finite mean oscillation at the point $x_0$ or a logarithmic singularity whose order does not exceed $n − 1$: Moreover, for $n ≥ 2$; under the indicated conditions imposed on the function $Q$; every point of the set $\overline{ℝ^n}\ f(D)$ is an asymptotic limit of $f$ at the point $x_0$. For $n ≥ 3$, the following relation is true: $\overline{ℝ^n}∖f(D) ⊂\overline{f(B_f ∩ U)}$. In addition, if $∞ ∉ f(D)$, then the set $f B_f$ is infinite and $x_0 ∈ \overline{B_f}$.

### On the integral characterization of some generalized quasiregular mappings and the significance of the conditions of divergence of integrals in the geometric theory of functions

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1367-1380

The paper deals with the theory of space mappings. For a generalization of quasiregular mappings important for the investigation of the Sobolev and other known classes of mappings, we propose a simple condition satisfied by all mappings of this kind and only by these mappings. On the basis of conditions of divergence of the integrals, we establish sufficient conditions for the normality of the families of the analyzed classes of mappings and solve the problem of removing isolated singularities. Some applications of the obtained results to mappings from the Sobolev class are discussed.

### Generalization of one Poletskii lemma to classes of space mappings

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 969-975

The paper is devoted to investigations in the field of space mappings. We prove that open discrete mappings $f ∈ W^{1,n}_{\text{loc}}$ such that their outer dilatation $K_O (x, f)$ belongs to $L^{n−1}_{\text{loc}}$ and the measure of the set $B_f$ of branching points of $f$ is equal to zero have finite length distortion. In other words, the images of almost all curves $γ$ in the domain $D$ under the considered mappings $f : D → ℝ^n,\;n ≥ 2$, are locally rectifiable, $f$ possesses the $(N)$-property with respect to length on $γ$, and, furthermore, the $(N)$-property also holds in the inverse direction for liftings of curves. The results obtained generalize the well-known Poletskii lemma proved for quasiregular mappings.

### On one modulus inequality for mappings with finite length distortion

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 680-688

The Väisälä inequality, which is well known in the theory of quasilinear mappings, is extended to the class of mappings with finite length distortion.

### Removal of singularities and analogs of the Sokhotskii–Weierstrass theorem for Q-mappings

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 116-126

We prove that an open discrete *Q*-mapping \( f:D \to \overline {{\mathbb{R}^n}} \) has a continuous extension to an isolated boundary point if the function *Q*(*x*) has finite mean oscillation or logarithmic singularities of order at most *n* – 1 at this point. Moreover, the extended mapping is open and discrete and is a *Q*-mapping. As a corollary, we obtain an analog of the well-known Sokhotskii–Weierstrass theorem on *Q*-mappings. In particular, we prove that an open discrete *Q*-mapping takes any value infinitely many times in the neighborhood of an essential singularity, except, possibly, for a certain set of capacity zero.

### On the normality of families of space mappings with branching

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1389–1400

We study space mappings with branching that satisfy modulus inequalities. For classes of these mappings, we obtain several sufficient conditions for the normality of families.