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Maximally singular functions in Besov spaces

Authors:	Darko Zubrinić

Institution:	(1) Faculty of Electrical Engineering and Computing, Department of Applied Mathematics, University of Zagreb, Unska 3, 10000 Zagreb, Croatia

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. Received: 5 July 2005; revised: 18 October 2005

Keywords:	46E30
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