Jacobian determinant inequality on corank 1 Carnot groups with applications

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 su...

Full description

Saved in:
Bibliographic Details
Published in:Journal of functional analysis 2019-12, Vol.277 (12), p.108293, Article 108293
Main Authors: Balogh, Zoltán M., Kristály, Alexandru, Sipos, Kinga
Format: Article
Language:English
Subjects:
Abnormal and normal geodesics
Brunn-Minkowski inequality
Carnot group
Optimal mass transportation
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2019.108293