Lipschitz mappings, metric differentiability, and factorization through metric trees

Abstract

Given a Lipschitz map 𝑓 from a cube into a metric space, we find several equivalent conditions for 𝑓 to have a Lipschitz factorization through a metric tree. As an application, we prove a recent conjecture of David and Schul. The techniques developed for the proof of the factorization result yield several other new and seemingly unrelated results. We prove that if 𝑓 is a Lipschitz mapping from an open set in ℝ 𝑛 onto a metric space 𝑋, then the topological dimension of 𝑋 equals 𝑛 if and only if 𝑋 has positive 𝑛-dimensional Hausdorff measure. We also prove an area formula for length-preserving maps between metric spaces, which gives, in particular, a new formula for integration on countably rectifiable sets in the Heisenberg group.

Lower bounds on mapping content and quantitative factorization through trees

“…DMS‐1763973. The authors thank Behnam Esmayli and Piotr Hajłasz for comments, and for sharing an early draft of [6].…”

“…Recently, Esmayli–Hajłasz [6] showed that a mapping with $$\mathcal {H}^{2,m}_\infty$$ equal to zero must factor through a tree, answering the qualitative cases of Question 1.3 and, equivalently, Conjecture 1.6. (This is the equivalence “(e) $$\Leftrightarrow$$ (a)” in [6, Theorem 1.1], combined with their Remark 1.2.) Theorem Let $$Q_0=[0,1]^{2+m}$$ and $$f\colon Q_0 \rightarrow Y$$ be a Lipschitz map into a metric space.…”