Tubular neighborhoods in the sub-Riemannian Heisenberg groups

In the present paper we consider the Carnot–Carathéodory distance

On CR volume growth estimate in a complete pseudohermitian 3-manifold

In this paper, we first derive the CR Reilly's formula and a CR Ricatti equation for sub-Laplacian of the Carnot–Carathéodory distance in a complete pseudohermitian 3-manifold. As a consequence, we obtain the CR volume growth estimate in a complete pseudohermitian 3-manifold under a lower bound of pseudohermitian curvature tensors. This is a generalization of Nagel, Stein and Wainger's volume growth estimate for the Heisenberg ball in the standard Heisenberg group.

Differences of Composition Operators on the Space of Bounded Analytic Functions in the Polydisc

This paper gives some estimates of the essential norm for the difference of composition operators induced byφandψacting on the space,H∞(Dn), of bounded analytic functions on the unit polydiscDn, whereφandψare holomorphic self-maps ofDn. As a consequence, one obtains conditions in terms of the Carathéodory distance onDnthat characterizes those pairs of holomorphic self-maps of the polydisc for which the difference of two composition operators onH∞(Dn)is compact.

MASS TRANSPORTATION IN GROUPS OF TYPE H

We give a solution of the optimal transport problem in groups of type H when the cost function is the square of either the Carnot–Carathéodory or Korányi distance. This generalizes results previously proved for the Heisenberg groups. We use the same strategy that the one which was developed in that special case together with slightly refined technicalities that essentially reflect the fact that the center of the group can be of dimension larger than one. For each distance we prove existence, uniqueness and give a characterization of the optimal transport. In the case of the Carnot–Carathéodory distance we also prove that the optimal transport arises as the limit of the optimal transports in natural Riemannian approximations.