A characterization of ${\dot{W}}^{1,p}({\mathbb{R}}^d)$

For $1<p<\mathrm{\infty }$ we give a characterization of the Sobolev space $\dot W^{1,p}(\mathbb R^d)$ in terms of the oscillations of a function on balls of varying centers and radii. Our work is motivated both by the study of trace ideal properties of commutators with singular integral operators and by work of Nguyen and by Brezis, Van Schaftingen and Yung on derivative-free characterizations of Sobolev spaces.