An inequality on the mass of image of rectifiable chain under chain map

TL;DR

This paper establishes an inequality relating the mass of the image of a rectifiable chain under a Lipschitz-induced chain map to the integral of the Jacobian of the map over the chain's support, within a normed abelian group setting.

Contribution

It introduces a new inequality controlling the mass of the image of rectifiable chains under Lipschitz maps using Jacobian integrals in a normed abelian group context.

Findings

Mass of image controlled by Jacobian integral

Inequality holds for rectifiable chains in normed abelian groups

Provides a new tool for geometric measure theory analysis

Abstract

For a complete normed abelian group $G$, we show that the mass of image of a rectifiable $G$-chain $S$ under chain map $f_{\sharp}$ induced by Lipschitz map $f$ is controlled by the integral of Jacobi of $f$ restricted on the support of $S$ with respect to associated radon mesure $\mu_S$.