Tensorization of Cheeger energies, the space H1,1H1,1 and the area formula for graphs

First we study in detail the tensorization properties of weak gradients in metric measure spaces (X,d,m)(X,d,m). Then, we compare potentially different notions of the Sobolev space H1,1(X,d,m)H1,1(X,d,m) and of weak gradient with exponent 1. Eventually we apply these results to compare the area functional ∫1+|∇f|w2dm with the perimeter of the subgraph of f, in the same spirit as the classical theory.