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

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.

60 pages, LaTeX; Section 6 added and Section 7 (6 in the previous version) changed. Some references added

References in corpus (2)

• Nonsmooth Hormander vector fields and their control balls
• Geometry of Carnot--Carathéodory Spaces, Differentiability and Coarea Formula

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