Tensorization of Cheeger energies, the space $H^{1,1}$ and the area formula for graphs

Abstract

First we study in detail the tensorization properties of weak gradients in metric measure spaces $(X,d,m)$. Then, we compare potentially different notions of Sobolev space $H^{1,1}(X,d,m)$ and of weak gradient with exponent 1. Eventually we apply these results to compare the area functional $\int\sqrt{1+|\nabla f|_w^2}\,dm$ with the perimeter of the subgraph of $f$, in the same spirit as the classical theory.