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arXiv · 0809.2872

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

Marco Bramanti·Luca Brandolini·Marco Pedroni
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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.

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Marco Bramanti, Luca Brandolini, Marco Pedroni. 2009-04-08. Basic properties of nonsmooth Hormander's vector fields and Poincare's inequality. https://doi.org/10.1515/form.2011.133

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