On isolated singularities of mappings with inverse moduli inequalities

TL;DR

This paper investigates the boundary behavior of open discrete mappings satisfying inverse modulus inequalities, proving they can be continuously extended to isolated boundary points under certain integrability conditions without additional topological constraints.

Contribution

It establishes conditions under which mappings with inverse modulus inequalities extend continuously to isolated boundary points, relaxing previous topological assumptions.

Findings

Mappings extend continuously to isolated boundary points under integrability conditions.

No topological restrictions are needed for the mappings in the main results.

Extension holds even when the majorant has finite averages over infinitesimal spheres.

Abstract

We consider open discrete mappings that satisfy the modulus condition of the inverse Poletsky inequality type. We study the case when the majorant in it is integrable, or more generally, has finite averages over infinitesimal spheres. We proved that such mappings have a continuous extension to an isolated point of the boundary of some domain without any a priori requirements on the corresponding mapped domain for integrable majorants. In the case of majorants, integrable on spheres, we require only the boundedness of the mapped domain. We do not require any other topological conditions on the mappings.