A step towards the tensorization of Sobolev spaces
Silvia Ghinassi, Vikram Giri, Elisa Negrini
TL;DR
The paper addresses the tensorization problem for first-order Sobolev spaces on Cartesian and warped product spaces of metric measure spaces. It proves that under the assumption that one factor is doubling and supports a $(2,2)$-Poincaré inequality, the Beppo-Levi space ${\mathsf{BL}}(X,Y)$ coincides with $W^{1,2}(X\times Y)$ and the product weak upper gradient is controlled by the factor gradients via $|Df|_{{\mathsf{BL}}} \le |Df| \le C_0|Df|_{{\mathsf{BL}}}$, with a parallel result for warped products using ${\mathsf{BL}}_w$ and $|Df|_{X\times_w Y}$. The authors provide a largely self-contained proof built on Lipschitz density, a dyadic partition of $Y$, and a decomposition into factorwise gradients, extending Gigli and Han’s interval case to general Cartesian and warped products. These results advance the understanding of Sobolev spaces on product spaces under mild geometric hypotheses, with implications for analysis on spaces with lower curvature bounds as in the RCD framework.
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 Poincaré inequality.