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

We improve an earlier characterization of the Sobolev space ${\dot{W}}^{1,p}({\mathbb{R}}^d)$, $1<p<\mathrm{\infty }$, 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 of the liminf, thereby answering a question by Brezis.