Hardy–Sobolev Type Inequalities with Sharp Constants in Carnot–Carathéodory Spaces

Abstract

We prove a generalization with sharp constants of a classical inequality due to Hardy to Carnot groups of arbitrary step, or more general Carnot–Caratheodory spaces associated with a system of vector fields of Hormander type. Under a suitable additional assumption (see Eq. 1.6 below) we are able to extend such result to the nonlinear case \(p\not= 2\). We also obtain a sharp inequality of Hardy–Sobolev type.