Weighted weak type estimates are proved for some maximal operators on the weighted Hardy spaces$H$_{ω}^{$p$}(0 <$p$< 1, ω ∈$A$_{1}) (0<p<1, ω∞A_{1}); 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 present a homogeneous real analytic hypersurface in C^{3}, two-nondegenerate, uniformly Levi degenerate of rank one, with a seven-dimensional CR automorphism group such that the isotropy group of each point is two-dimensional and commutative. The classical tube Γ_{C}over the two-dimensional real cone in R^{3}is also homogeneous and has a seven-dimensional CR automorphism group. However, our example is$not$biholomorphic to the tube over the real cone, because the two-dimensional isotropy groups of Γ_{C}are, in contrast, noncommutative.

We investigate local properties of the Green function of the complement of a compact set$E$υ[0,1] with respect to the extended complex plane. We demonstrate, that if the Green function satisfies the 1/2-Hölder condition locally at the origin, then the density of$E$at 0, in terms of logarithmic capacity, is the same as that of the whole interval [0, 1]..

Laplace operators on metric graphs are considered. It is proven that for compact graphs the spectrum of the Laplace operator determines the total length, the number of connected components, and the Euler characteristic. For a class of non-compact graphs the same characteristics are determined by the scattering data consisting of the scattering matrix and the discrete eigenvalues.

For an odd prime p, we prove the p-part of the Birch and Swinnerton-Dyer formula for semistable elliptic curves of analytic rank one.