Rearrangements in Carnot Groups

In this paper we extend the notion of rearrangement of nonnegative functions to the setting of Carnot groups. We define rearrangement with respect to a given family of anisotropic balls B-r, or equivalently with respect to a gauge x and prove basic regularity properties of this construction. If u is a bounded nonnegative real function with compact support, we denote by u* its rearrangement. Then, the radial function u* is of bounded variation. In addition, if u is continuous then u* is continuous, and if u belongs to the horizontal Sobolev space Wh where p 1.