Further characterizations of Sobolev spaces

Let (Fn)n∈N be a sequence of non-decreasing functions from [0,+∞) into [0,+∞). Under some suitable hypotheses on (Fn)n∈N, we prove that if g ∈ Lp(RN ), 1 < p < +∞, satisfies lim inf n→∞ ∫ RN ∫ RN Fn(|g(x)− g(y)|) |x − y|N+p dx dy < +∞, then g ∈ W1,p(RN ) and moreover lim n→∞ ∫ RN ∫ RN Fn(|g(x)− g(y)|) |x − y|N+p dx dy = KN,p ∫ RN |∇g(x)| dx, whereKN,p is a positive constant depending only onN and p. This extends some results in J. Bourgain and H.-M. Nguyen [A new characterization of Sobolev spaces, C. R. Math. Acad. Sci. Paris 343, 75–80 (2006)] and H.-M. Nguyen [Some new characterizations of Sobolev spaces, J. Funct. Anal. 237, 689–720 (2006)]. We also present some partial results concerning the case p = 1 and various open problems.