# Weak BLD mappings and Hausdorff measure

## TL;DR

This paper establishes a relationship between Hausdorff measures of Lipschitz images under weak BLD mappings between quasiconvex metric spaces, extending previous Euclidean space results.

## Contribution

It generalizes earlier Euclidean space results to mappings between quasiconvex metric spaces, linking Hausdorff measure properties under weak BLD mappings.

## Key findings

- Hausdorff measure zero preservation under weak BLD mappings
- Extension of Euclidean results to quasiconvex metric spaces
- Characterization of Lipschitz images in metric spaces

## Abstract

We prove that if $\Phi:X\to Y$ a mapping of weak bounded length distortion from a quasiconvex and complete metric space $X$ to any metric space $Y$, then for any Lipschitz mapping $f:\mathbb{R}^k\supset E\to X$ we have that ${\mathcal H}^k(f(E))=0$ in $X$ if and only if ${\mathcal H}^k(\Phi(f(E)))=0$ in $Y$. This generalizes an earlier result of Haj\l{}asz and Malekzadeh where the target space $Y$ was a Euclidean space $Y=\mathbb{R}^N$.