Tensorization of $p$-weak differentiable structures

TL;DR

This paper investigates the tensorization properties of $p$-weak differentiable structures, demonstrating that products of such structures retain their properties and providing partial results on Sobolev space tensorization, especially when involving PI spaces.

Contribution

It proves that the product of $p$-weak charts is a $p$-weak chart and advances the tensorization problem for Sobolev spaces, including cases with PI spaces.

Findings

Product of $p$-weak charts is a $p$-weak chart

Product spaces with $p$-weak structures also admit such structures

Tensorization holds when one factor is a PI space

Abstract

We consider $p$-weak differentiable structures that were recently introduced by the first and last named authors, and prove that the product of $p$-weak charts is a $p$-weak chart. This implies that the product of two spaces with a $p$-weak differentiable structure also admits a $p$-weak differentiable structure. We make partial progress on the tensorization problem of Sobolev spaces by showing an isometric embedding result. Further, we establish tensorization when one of the factors is PI.