Some new characterizations of Sobolev spaces

AbstractIn this paper, we present some new characterizations of Sobolev spaces. Here is a typical result. Let g∈Lp(RN), 1<p<+∞; we prove that g∈W1,p(RN) if and only ifsup0<δ<1∫RN∫RN|g(x)−g(y)|>δδp|x−y|N+pdxdy<+∞. Moreover,limδ→0∫RN∫RN|g(x)−g(y)|>δδp|x−y|N+pdxdy=1pKN,p∫RN|∇g(x)|pdx,∀g∈W1,p(RN), where KN,p is defined by (12).This result is somewhat related to a characterization of Sobolev spaces due to J. Bourgain, H. Brezis, P. Mironescu (see [J. Bourgain, H. Brezis, P. Mironescu, Another look at Sobolev spaces, in: J.L. Menaldi, E. Rofman, A. Sulem (Eds.), Optimal Control and Partial Differential Equations, A Volume in Honour of A. Bensoussan's 60th Birthday, IOS Press, 2001, pp. 439–455]). However, the precise connection is not transparent