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 W_h^1,p , then D_hu^ ⋆(x)| D_h( x )| is in L p . Moreover, we found a generalization of the inequality of Pólya and Szegö ∫| D_hu^ ⋆|^p| D_h( x )|^p dx ≤ C ∫| D_hu|^p dx, where p ≥ 1.