Tensorization of quasi-Hilbertian Sobolev spaces

Abstract

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

\[W^{1,p}(X)⊗W^{1,p}(Y)⊂W^{1,p}(X×Y).\]

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.