We prove sharp weighted weak type (1,1) estimates for rough singular integral operators on homogeneous groups. Similar results are shown for singular integrals on $\mathbb{R}^{2}$ with the generalized homogeneity.

For two-sided topological Markov chains, we show that the set of points for which the two-sided Birkhoff averages of a continuous function diverge is residual. We also show that the set of points for which the Birkhoff averages have a given set of accumulation points other than a singleton is residual. A nontrivial consequence of our results is that the set of points for which the local entropies of an invariant measure on a locally maximal hyperbolic set does not exist is residual. This strongly contrasts to the Shannon–McMillan–Breiman theorem in the context of ergodic theory, which says that local entropies exist on a full measure set.

In this paper we prove a monotonicity formula for the integral of the mean curvature for complete and proper hypersurfaces of the hyperbolic space and, as consequences, we obtain a lower bound for the integral of the mean curvature and that the integral of the mean curvature is infinity.

We introduce and study the notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and Păun. We define what it means for such a metric to be positively and negatively curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the curvature Θ^{$h$}as a matrix of currents. We then proceed to show that such metrics can be regularised in such a way that the corresponding curvature tensors converge weakly to Θ^{$h$}. Finally we define what it means for$h$to be strictly negatively curved in the sense of Nakano and show that it is possible to regularise such metrics with a sequence of smooth, strictly Nakano negative metrics.

We show that the standard graded cover of the well-known category $\mathcal{O}$ of Bernstein–Gelfand–Gelfand can be characterized by its compatibility with the action of the center of the enveloping algebra.