Pointwise characterization of Sobolev classes

We prove that a function f is in the Sobolev class W _loc ^m,p (ℝ ⁿ ) or W ^m,p (Q) for some cube Q ⊂ 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)| ≤ |x − y| ^m a(x) + a(y)] for some a ε L ^p (Q) outside a set N ⊂ 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 ℝ ⁿ . Dedicated to S.M. Nikol’skiĭ on the occasion of his 100th birthday The main results and ideas of this paper were presented in the plenary lecture of the author at the International Conference and Workshop Function Spaces, Approximation Theory and Nonlinear Analysis dedicated to the centennial of Sergei Mikhailovich Nikol’skii, Moscow, May 24–28, 2005.