Jacobian determinant inequality on corank 1 Carnot groups with applications

Zoltán M. Balogh, Alexandru Kristály, Kinga Sipos

Research output: Contribution to journalArticle

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.

Original languageEnglish
Article number108293
JournalJournal of Functional Analysis
Volume277
Issue number12
DOIs
Publication statusPublished - Dec 15 2019

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Carnot Group
Determinant
Geodesic
Euclidean
Strictly
Entropy
Coefficient

Keywords

  • Abnormal and normal geodesics
  • Brunn-Minkowski inequality
  • Carnot group
  • Optimal mass transportation

ASJC Scopus subject areas

  • Analysis

Cite this

Jacobian determinant inequality on corank 1 Carnot groups with applications. / Balogh, Zoltán M.; Kristály, Alexandru; Sipos, Kinga.

In: Journal of Functional Analysis, Vol. 277, No. 12, 108293, 15.12.2019.

Research output: Contribution to journalArticle

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