Jacobian determinant inequality on corank 1 Carnot groups with applications

Balogh, Zoltán M.; Kristály, Alexandru; Sipos, Kinga (2019). Jacobian determinant inequality on corank 1 Carnot groups with applications. Journal of functional analysis, 277(12), p. 108293. Elsevier 10.1016/j.jfa.2019.108293

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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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