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 (Optimal control and partial differential equations (a volume in honour of A. Benssoussan's 60th birthday). IOS Press, Amsterdam, pp. 439-455. 2001) characterization. Integral functionals of the nonlocal gradient with proper growth are shown to converge to a corresponding functional of the classical gradient both pointwise and in the sense of -convergence.