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

We improve an earlier characterization of the Sobolev space $\dot W^{1,p}(\mathbb R^d)$, $1<p<\infty$, in terms of the oscillations of a function on balls of varying centers and radii. In contrast to earlier work, finiteness of the supremum is replaced by the formally weaker assumption of finiteness