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Let$E$ç$S$^{1}be a set with Minkowski dimension$d(E)1$. We consider the Hardy-Littlewood maximal function, the Hilbert transform and the maximal Hilbert transform along the directions of$E$. The main result of this paper shows that these operators are bounded on$L$_{$rad$}^{$p$}(R^{2}) for$p>1+d(E)$and unbounded when$p<1+d(E)$. We also give some end-point results.

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Perturbations by analytic discs along generating CR-submanifolds of C^{n}are considered. In the case all partial indices of a closed path$p$in a generating CR-fibration {$M$(ξ)}_{ξ∈∂$D$}are greater or equal to −1 we can completely parametrize all small holomorphic perturbations of the path$p$along the fibration {$M$(ξ)}_{ξ∈∂$D$}. In this case we also study the geometry of perturbations by analytic discs and their relation to the conormal bundle of the fibration.

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