Perturbations by analytic discs along generating CR-submanifolds of Cⁿ 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.

LetEçS¹ 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 ${}_{rad}^{p}$ (R²) for p>1+d(E) and unbounded when p<1+d(E). We also give some end-point results.

Weighted weak type estimates are proved for some maximal operators on the weighted Hardy spaces H ${}_{ω}^{p}$ (0 < p < 1, ω ∈A₁) (0<p<1, ω∞A₁); in particular, weighted weak type endpoint estimates are obtained for the maximal operators arising from the Bochner-Riesz means and the spherical means on H ${}_{ω}^{p}$ .

We introduce and study the Jarník limit set ℐ_σ of a geometrically finite Kleinian group with parabolic elements. The set ℐ_σ is the dynamical equivalent of the classical set of well approximable limit points. By generalizing the method of Jarník in the theory of Diophantine approximations, we estimate the dimension of ℐ_σ with respect to the Patterson measure. In the case in which the exponent of convergence of the group does not exceed the maximal rank of the parabolic fixed points, and hence in particular for all finitely generated Fuchsian groups, it is shown that this leads to a complete description of ℐ_σ in terms of Hausdorff dimension. For the remaining case, we derive some estimates for the Hausdorff dimension and the packing dimension of ℐ_σ.