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arXiv · 1701.08831

Jacobian determinant inequality on corank 1 Carnot groups with applications

Zoltán M. Balogh·Alexandru Kristály·Kinga Sipos
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Abstract

We establish a weighted pointwise Jacobian determinant inequality on corank 1 Carnot groups related to optimal mass transportation akin to the work of Cordero-Erausquin, McCann and Schmuckenschläger. In this setting, the presence of abnormal geodesics does not allow the application of the general sub-Riemannian optimal mass transportation theory developed by Figalli and Rifford and we need to work with a weaker notion of Jacobian determinant. Nevertheless, our result achieves a transition between Euclidean and sub-Riemannian structures, corresponding to the mass transportation along abnormal and strictly normal geodesics, respectively. The weights appearing in our expression are distortion coefficients that reflect the delicate sub-Riemannian structure of our space. As applications, entropy, Brunn-Minkowski and Borell-Brascamp-Lieb inequalities are established on Carnot groups.

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Zoltán M. Balogh, Alexandru Kristály, Kinga Sipos. 2019-07-28. Jacobian determinant inequality on corank 1 Carnot groups with applications. https://arxiv.org/abs/1701.08831

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