Tensorization of quasi-Hilbertian Sobolev spaces

Abstract

The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space X\times 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\times 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\in (1,\infty) we obtain the norm-one inclusion \|f\|_{J^{1,p}(X,Y)}\le \|f\|_{W^{1,p}(X\times Y)} and show that the norms agree on the algebraic tensor product W^{1,p}(X)\otimes W^{1,p}(Y)\subset W^{1,p}(X\times Y). When p=2 and X and Y are infinitesimally quasi-Hilbertian, standard Dirichlet forms theory yields the density of W^{1,2}(X)\otimes 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)\otimes W^{1,p}(Y) in J^{1,p}(X,Y) in the general case.