engleski

Maximally singular Sobolev functions

It is known that for any Sobolev function in the space W_m, p(R^N), 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 W_m, p(R^N), 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 L_α, p(R^N), provided αp<N, α>0, and p>1.

Sobolev function; Bessel potential; fractal set; Minkowski content; Sierpinski carpet

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