Maximally singular functions in Besov spaces

Assuming that 0 <  α p <  N, p, q ∈(1,∞), we construct a class of functions in the Besov space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{{p,q}}_{\alpha } \,({\user2{\mathbb{R}}}^{N} )$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{{p,q}}_{\alpha } \,({\user2{\mathbb{R}}}^{N} )$$\end{document} is ≦ N − α p. Similar results are obtained for Lizorkin-Triebel spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^{{p,q}}_{\alpha } \,({\user2{\mathbb{R}}}^{N} )$$\end{document} and for the Hardy space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{1} \,({\user2{\mathbb{R}}}^{N} )$$\end{document}. Some open problems are listed.