The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space can be determined from its factors. We show that two natural descriptions of the Sobolev space from the literature coincide, , thus settling the tensorization problem for Sobolev spaces in the case , when and are infinitesimally quasi-Hilbertian, i.e., the Sobolev space admits an equivalent renorming by a Dirichlet form. This class includes in particular metric measure spaces of finite Hausdorff dimension as well as infinitesimally Hilbertian spaces.
More generally, for we obtain the norm-one inclusion and show that the norms agree on the algebraic tensor product
When and and are infinitesimally quasi-Hilbertian, standard Dirichlet forms theory yields the density of in , thus implying the equality of the spaces. Our approach raises the question of the density of in in the general case.
Cite this article
Sylvester Eriksson-Bique, Tapio Rajala, Elefterios Soultanis, Tensorization of quasi-Hilbertian Sobolev spaces. Rev. Mat. Iberoam. (2023), published online firstDOI 10.4171/RMI/1433