On the inverse $K_I$-inequality for one class of mappings

TL;DR

This paper investigates a class of differentiable mappings with specific measure-theoretic properties, establishing a lower bound for their Poletsky-type distortion, which advances understanding of their geometric behavior.

Contribution

It proves a new inverse $K_I$-inequality for a class of mappings with measure-theoretic properties, extending previous results in geometric function theory.

Findings

Mappings satisfy the lower bound for Poletsky-type distortion

Mappings possess the $N$-Luzin and $N^{-1}$-properties on spheres

Image of zero Jacobian set has zero Lebesgue measure

Abstract

We study mappings differentiable almost everywhere, possessing the $N$-Luzin property, the $ N^{\,-1}$-property on the spheres with respect to the $(n-1)$-dimensional Hausdorff measure and such that the image of the set where its Jacobian equals to zero has a zero Lebesgue measure. It is proved that such mappings satisfy the lower bound for the Poletsky-type distortion in their domain of definition.