BMO-type seminorms generating Sobolev functions

Abstract

In the recent literature certain BMO-type seminorms provide characterizations of Sobolev functions. In the same order of ideas, we obtain the norm of the gradient of a function in Lp(Ω), where Ω⊂Rn, n>1 and p>1, as limit of BMO-type seminorms involving families of pairwise disjoint sets with all orientations, the sets being not necessarily cubes or tessellation cells. An analogous result is obtained when rotations are not allowed.