Pointwise Characterization of Sobolev Classes

We prove that a function $f$ is in the Sobolev class $W_{\mathrm {loc}}^{m,p}(\mathbb R^n)$ or $W^{m,p}(Q)$ for some cube $Q\subset \mathbb R^n$ if and only if the formal $(m-1)$-Taylor remainder $R^{m-1}f(x,y)$ of $f$ satisfies the pointwise inequality $|R^{m-1}f(x,y)|\le |x-y|^m [a(x)+a(y)]$ for some $a\in L^p(Q)$ outside a set $N\subset Q$ of null Lebesgue measure. This is analogous to H. Whitney's Taylor remainder condition characterizing the traces of smooth functions on closed subsets of $\mathbb R^n$.