Bochner Partial Derivatives, Cheeger-Kleiner Differentiability, and Non-Embedding

TL;DR

This paper introduces Cheeger fractals within Poincaré inequality spaces, demonstrating their non-embeddability into Banach spaces under certain differentiability conditions, extending prior results in metric geometry.

Contribution

It defines Cheeger fractals and proves their non-embeddability into Banach spaces with integrable Bochner derivatives, generalizing previous work on metric space embeddings.

Findings

Cheeger fractals include the sub-Riemannian Heisenberg group.

No bi-Lipschitz embedding into Banach spaces preserves certain differentiability properties.

Extends results of Creutz and Evseev on non-embedding conditions.

Abstract

Among all Poincar\'e inequality spaces, we define the class of Cheeger fractals, which includes the sub-Riemannian Heisenberg group. We show that there is no bi-Lipschitz embedding $\iota$ of any Cheeger fractal $X$ into any Banach space $V$ with the following property: there exists a bounded Euclidean domain $\Omega$ such that for any Lipschitz mapping $f \colon \Omega \to X$, the Bochner partial derivatives of $\iota \circ f$ exist and are integrable. This extends and provides context for an important related result of Creutz and Evseev.