Basic properties of nonsmooth Hormander's vector fields and Poincare's inequality

TL;DR

This paper extends fundamental properties and inequalities related to Hormander's vector fields to a nonsmooth setting, enabling analysis of associated differential operators under weaker regularity assumptions.

Contribution

It generalizes key geometric and analytic results for Hormander's vector fields from smooth to nonsmooth coefficients, including distance properties, doubling condition, connectivity, and Poincare's inequality.

Findings

Established basic properties of the induced distance in nonsmooth context

Proved doubling condition and Chow's connectivity theorem for nonsmooth vector fields

Extended Poincare's inequality to nonsmooth Hormander's vector fields

Abstract

We consider a family of vector fields defined in some bounded domain of R^p, and we assume that they satisfy Hormander's rank condition of some step r, and that their coefficients have r-1 continuous derivatives. We extend to this nonsmooth context some results which are well-known for smooth Hormander's vector fields, namely: some basic properties of the distance induced by the vector fields, the doubling condition, Chow's connectivity theorem, and, under the stronger assumption that the coefficients belong to C^{r-1,1}, Poincare's inequality. By known results, these facts also imply a Sobolev embedding. All these tools allow to draw some consequences about second order differential operators modeled on these nonsmooth Hormander's vector fields.