Weak BLD mappings and Hausdorff measure

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 Hajłasz and Malekzadeh where the target space $Y$ was a Euclidean space $Y={\mathbb{R}}^N$.