Hardy-Type Inequalities for the Carnot-Caratheodory Distance in the Heisenberg Group

摘要

In this paper we study Hardy inequalities in the Heisenberg group Hn, with respect to the Carnot-Caratheodory distance delta from the origin. We firstly show that, letting Q be the homogenous dimension, the optimal constant in the (unweighted) Hardy inequality is strictly smaller than n2=(Q-2)2/4. Then, we prove that, independently of n, the Heisenberg group does not support a radial Hardy inequality, i.e., a Hardy inequality where the gradient term is replaced by its projection along backward difference H delta. This is in stark contrast with the Euclidean case, where the radial Hardy inequality is equivalent to the standard one, and has the same constant. Motivated by these results, we consider Hardy inequalities for non-radial directions, i.e., directions tangent to the Carnot-Caratheodory balls. In particular, we show that the associated constant is bounded on homogeneous cones C sigma with base sigma subset of S2n, even when sigma degenerates to a point. This is a genuinely sub-Riemannian behavior, as such constant is well known to explode for homogeneous cones in the Euclidean space.