Rearrangements in Carnot Groups

TL;DR

This paper extends the concept of function rearrangement to Carnot groups, establishing regularity properties and inequalities analogous to classical results in Euclidean spaces.

Contribution

It introduces a new rearrangement framework in Carnot groups and proves regularity and inequality results generalizing classical Euclidean theorems.

Findings

Rearrangement with respect to anisotropic balls in Carnot groups is well-defined.

Rearranged functions are of bounded variation.

Continuity and Sobolev inequalities are generalized to this setting.

Abstract

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, we found a generalization of the inequality of Polya and Szeg\"o.