Local clustering of the non-zero set of functions in W1,1(E)

We extend to the p=1 case a measure theoretic result previously proved by DiBenedetto and Vespri for functions that belong to u∈ W1,p(K_ρ(x_0)) where K_ρ(x_0)) is a N -dimensional cube of edge ρ centered at x_0 . It basically states that if the set where u is bounded away from zero occupies a sizable portion of K_ρ , then the set where u is positive clusters about at least one point of K_ρ .