Further characterizations of Sobolev spaces
Let be a sequence of non-decreasing functions from into . Under some suitable hypotheses of , we will prove that if , , satisfies , then and moreover , where is a positive constant depending only on and . This extends some results in J. Bourgain and H-M. Nguyen [A new characterization of Sobolev spaces, C. R. Acad Sci. Paris, Ser. I 343 (2006) 75-80] and H-M. Nguyen [Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689-720]. We also present some partial results...