# Tensorization of quasi-Hilbertian Sobolev spaces

### Sylvester Eriksson-Bique

Research Unit of Mathematical Sciences, Oulu; University of Jyvaskyla, Finland### Tapio Rajala

University of Jyväskylä, Finland### Elefterios Soultanis

Radboud University, Nijmegen, Netherlands

## Abstract

The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space $X×Y$ can be determined from its factors. We show that two natural descriptions of the Sobolev space from the literature coincide, $W_{1,2}(X×Y)=J_{1,2}(X,Y)$, thus settling the tensorization problem for Sobolev spaces in the case $p=2$, when $X$ and $Y$ are *infinitesimally quasi-Hilbertian*, i.e., the Sobolev space $W_{1,2}$ admits an equivalent renorming by a Dirichlet form. This class includes in particular metric measure spaces $X,Y$ of finite Hausdorff dimension as well as infinitesimally Hilbertian spaces.

More generally, for $p∈(1,∞)$ we obtain the norm-one inclusion $∥f∥_{J_{1,p}(X,Y)}≤∥f∥_{W_{1,p}(X×Y)}$ and show that the norms agree on the algebraic tensor product

When $p=2$ and $X$ and $Y$ are infinitesimally quasi-Hilbertian, standard Dirichlet forms theory yields the density of $W_{1,2}(X)⊗W_{1,2}(Y)$ in $J_{1,2}(X,Y)$, thus implying the equality of the spaces. Our approach raises the question of the density of $W_{1,p}(X)⊗W_{1,p}(Y)$ in $J_{1,p}(X,Y)$ in the general case.

## Cite this article

Sylvester Eriksson-Bique, Tapio Rajala, Elefterios Soultanis, Tensorization of quasi-Hilbertian Sobolev spaces. Rev. Mat. Iberoam. 40 (2024), no. 2, pp. 565–580

DOI 10.4171/RMI/1433