Maximally singular functions in Besov spaces

Assuming that 0 < alpha p < N, p, q is an element of (1,infinity), we construct a class of functions in the Besov space B-alpha(p,q)(R-N) such that the Hausdorff dimension of their singular set is equal to N - alpha p. We show that these functions are maximally singular, that is, the Hausdorff dimension of the singular set of any other Besov function in B-alpha(p,q)(R-N) is <= N - alpha p. Similar results are obtained for Lizorkin-Triebel spaces F-alpha(p,q)(R-N) and for the Hardy space H-1(R-N). Some open problems are listed.