Equicontinuity and normality of mappings with integrally bounded $p$-moduli

TL;DR

This paper investigates the equicontinuity and normality of certain open discrete mappings in alculus, controlled by integrals involving a measurable function Q, and establishes conditions under which the family of such mappings is normal.

Contribution

It introduces new conditions on the measurable function Q ensuring the normality of families of open discrete mappings with controlled p-moduli.

Findings

Established normality criteria for mappings with integrally bounded p-moduli.

Extended understanding of the behavior of discrete open mappings in alculus.

Provided conditions on Q for equicontinuity of mapping families.

Abstract

We consider the generic discrete open mappings in ${\mathbb R}^n$ under which the perturbation of extremal lengths of curve collections is controlled integrally via $\int Q(x)\eta^p(|x-x_0|) dm(x)$ with $n-1<p<n$, where $Q$ is a measurable function on ${\mathbb R}^n$ and $\int\limits_{r_1}^{r_2} \eta(r) dr \ge 1$ for any $\eta$ on a given interval $[r_1,r_2].$ We proved that the family of all open discrete mappings of above type is normal under appropriate restrictions on the majorant $Q.$