Localization of nonlocal gradients in various topologies

In this paper, we study nonlocal gradients and their relationship to classical gradients. As the nonlocality vanishes we demonstrate the convergence of nonlocal gradients to their local analogue for Sobolev and BV functions. As a consequence of these localizations we give new characterizations of the Sobolev and BV spaces that are in the same spirit of Bourgain, Brezis, and Mironsecu’s 2001 characterization. Integral functional of the nonlocal gradient with proper growth are shown to converge to corresponding functional of classical gradients both pointwise and in the sense of Γ-convergence.