Some new characterizations of Sobolev spaces

In this paper, we present some new characterizations of Sobolev spaces. Here is a typical result. Let g ∈ Lp(RN), 1δ δp |x − y|N+p dx dy <+∞. Moreover, lim δ→0 RN RN |g(x)−g(y)|>δ δp |x − y|N+p dx dy = 1 p KN,p RN ∇g(x) p dx, ∀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. © 2006 Elsevier Inc. All rights reserved.