On the lifting and approximation theorem for nonsmooth vector fields

TL;DR

This paper extends Rothschild-Stein's lifting and approximation theorem to nonsmooth Hörmander vector fields with Hölder continuous commutators, including cases with mixed weights, advancing the understanding of sub-Riemannian geometry in less regular settings.

Contribution

It generalizes the lifting and approximation theorem to nonsmooth vector fields with Hölder continuous commutators, covering cases with mixed weights.

Findings

Established a version of Rothschild-Stein's theorem for nonsmooth vector fields.

Explicitly covered cases with vector fields of different weights.

Provided new tools for analysis in sub-Riemannian geometry with low regularity.

Abstract

We prove a version of Rothschild-Stein's theorem of lifting and approximation and some related results in the context of nonsmooth Hormander's vector fields for which the highest order commutators are only Holder continuous. The theory explicitly covers the case of one vector field having weight two while the others have weight one.