Maximally singular functions in Besov spaces

Abstract.

Assuming that 0 <  α p <  N, p, q ∈(1,∞), we construct a class of functions in the Besov space $B^{{p,q}}_{\alpha } \,({\user2{\mathbb{R}}}^{N} )$ such that the Hausdorff dimension of their singular set is equal to N − α p. We show that these functions are maximally singular, that is, the Hausdorff dimension of the singular set of any other Besov function in $B^{{p,q}}_{\alpha } \,({\user2{\mathbb{R}}}^{N} )$ is ≦ N − α p. Similar results are obtained for Lizorkin-Triebel spaces $F^{{p,q}}_{\alpha } \,({\user2{\mathbb{R}}}^{N} )$ and for the Hardy space $H^{1} \,({\user2{\mathbb{R}}}^{N} )$. Some open problems are listed.