A step towards the tensorization of Sobolev spaces

TL;DR

This paper demonstrates that Sobolev spaces on product metric spaces can be tensorized under certain conditions, extending the understanding of Sobolev space behavior in complex geometric settings.

Contribution

It establishes the tensorization property of Sobolev spaces on Cartesian and warped products of metric spaces with minimal assumptions.

Findings

Sobolev spaces tensorize on product spaces with doubling and Poincaré conditions

Tensorization holds for both Cartesian and warped products

Extends previous results to more general metric space settings

Abstract

We prove that Sobolev spaces on Cartesian and warped products of metric spaces tensorize, only requiring that one of the factors is a doubling space supporting a Poincar\'e inequality.