Approximations of Sobolev norms in Carnot groups

TL;DR

This paper extends Sobolev space concepts and Poincaré inequalities from Euclidean spaces to Carnot groups, establishing equivalences of seminorms and exploring self-improving properties in this non-Euclidean setting.

Contribution

It generalizes Sobolev seminorms and Poincaré inequalities to Carnot groups, providing new equivalences and constructive proofs in this non-Euclidean context.

Findings

Seminorms in Carnot groups are equivalent to intrinsic gradient norms.

A Poincaré-type inequality is established using one-dimensional estimates.

Self-improving properties are analyzed for specific cases.

Abstract

This paper deals with a notion of Sobolev space $W^{1,p}$ introduced by J.Bourgain, H.Brezis and P.Mironescu by means of a seminorm involving local averages of finite differences. This seminorm was subsequently used by A.Ponce to obtain a Poincar\'e-type inequality. The main results that we present are a generalization of these two works to a non-Euclidean setting, namely that of Carnot groups. We show that the seminorm expressd in terms of the intrinsic distance is equivalent to the $L^p$ norm of the intrinsic gradient, and provide a Poincar\'e-type inequality on Carnot groups by means of a constructive approach which relies on one-dimensional estimates. Self-improving properties are also studied for some cases of interest.