# Salimov R. R.

### Lower bounds for the volume of the image of a ball

Klishchuk B. A., Salimov R. R.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 774-785

UDC 517.5

We consider ring $Q$-homeomorphisms with respect to $p$-modulus in
the space $\Bbb R^{n}$ as $p>n$.
We obtain a lower bound for the volume of the image of a ball under these mappings.
We solve the extremal problems of minimization of functionals of the volume of the image of a ball and the area of the image of a sphere.

### 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 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.

### 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.

### 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.

### One Property of Ring *Q*-Homeomorphisms With Respect to a *p*-Module

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 728–733

We establish sufficient conditions for a ring *Q*-homeomorphisms in \( {{\mathbb{R}}^n} \) , *n* ≥ 2, with respect to a *p*-module with *n* − 1 < *p* < *n* to have the finite Lipschitz property. We also construct an example of the ring *Q*-homeomorphism with respect to a *p*-module at a fixed point which does not have the finite Lipschitz property.

### 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.

### Asymptotic behavior of generalized quasiisometries at a point

Kovtonyuk D. A., Salimov R. R.

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 481-488

We consider $Q$-homeomorphisms with respect to the $p$-modulus. An estimate for a measure of a ball image is obtained under such mappings and the asymptotic behavior at zero is investigated.

### 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 order of growth of ring $Q$-homeomorphisms at infinity

Salimov R. R., Smolovaya E. S.

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 829 – 836

For ring homeomorphisms $f : ℝn → ℝn$ , we establish the order of growth at infinity.

### Local behavior of Q-homeomorphisms in Loewner spaces

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1378–1388

We study the problem of the elimination of isolated singularities for so-called *Q*-homeomorphisms in Loewner spaces. We formulate several conditions for a function *Q*(*x*) under which every *Q*-homeomorphism admits a continuous extension to an isolated singular point. We also consider the problem of the homeomorphicity of the extension obtained. The results are applied to Riemannian manifolds and Carnot groups.

### On the boundary behavior of imbeddings of metric spaces into a Euclidean space

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1068–1074

We investigate the boundary behavior of so-called *Q*-homeomorphisms with respect to a measure in some metric spaces. We formulate a series of conditions for the function *Q*(*x*) and the boundary of the domain under which any *Q*-homeomorphism with respect to a measure admits a continuous extension to a boundary point.