Tensorization of quasi-Hilbertian Sobolev spaces

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 / D J 1;2.X; Y /, thus settling the tensorization problem for Sobolev spaces in the case p D 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 2 .1;1/ we obtain the norm-one inclusion kf kJ1;p.X;Y / kf kW 1;p.XY / and show that the norms agree on the algebraic tensor product W 1;p.X / ˝ W 1;p.Y / W 1;p.X Y /: When p D 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.