Abstract
One describes closed subsets of the circle, having the property that the restrictions, to it of analytic functions with H1 derivatives can be characterized only by their typical smoothness. The spaces of the traces of such functions on thin sets are described in intrinsic terms. It is proved that on any closed subset of the circumference, of dimension less than 1, there exists a nonzero measure satisfying the doubling condition. The spaces of traces of functions with H1 derivative have a description of the Besov type with respect to this measure.
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