Maximally singular Sobolev functions

It is known that for any Sobolev function in the space Wm,p(RN), p⩾1, mp⩽N, where m is a nonnegative integer, the set of its singular points has Hausdorff dimension at most N−mp. We show that for p>1 this bound can be achieved. This is done by constructing a maximally singular Sobolev function in Wm,p(RN), that is, such that Hausdorff's dimension of its singular set is equal to N−mp. An analogous result holds also for Bessel potential spaces Lα,p(RN), provided αp0, and p>1. The existence of maximally singular Sobolev functions has been announced in [Chaos Solitons Fractals 21 (2004), p. 1287].