A Marcinkiewicz integral type characterization of the Sobolev space

In this paper we present a new characterization of the Sobolev space W^1,p , 1 < p < ∞ which is a higher dimensional version of a result of Waterman [32]. We also provide a new and simplified proof of a recent result of Alabern, Mateu, and Verdera [2]. Finally, we generalize the results to the case of weighted Sobolev spaces with respect to a Muckenhoupt weight.