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

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.