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

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

2008-09-17

arxiv.org/abs/0809.2872v2

Mathematics

Analysis of PDEs

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

Scientific paper

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.

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