  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 Schmuckenschl\"ager. 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.
