  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 of
Waterman. We also provide a new and simplified proof of a recent result of
Alabern, Mateu and Verdera. Finally, we generalize the results to the case of
weighted Sobolev spaces with respect to a Muckenhoupt weight.
