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In this paper we present a new characterization of the Sobolev space$W^{1,p}$, $1<p<\infty$ which is a higher dimensional version of a result ofWaterman. We also provide a new and simplified proof of a recent result ofAlabern, Mateu and Verdera. Finally, we generalize the results to the case ofweighted Sobolev spaces with respect to a Muckenhoupt weight.

A Marcinkiewicz integral type characterization of the Sobolev space