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. The weights appearing in our expression are distortion coefficients that reflect the delicate sub-Riemannian structure of our space including the presence of abnormal geodesics. Our inequality interpolates in some sense between Euclidean and sub-Riemannian structures, corresponding to the mass transportation along abnormal and strictly normal geodesics, respectively. As applications, entropy, Brunn-Minkowski and Borell-Brascamp-Lieb inequalities are established.
Keywords: Carnot group; Jacobian determinant inequality; optimal mass transportation; abnormal and normal geodesics; entropy inequality; Brunn-Minkowski inequality; Borell-Brascamp-Lieb inequality.
MSC: 53C17, 35R03, 49Q20.
As a general framework of our results, let be a suitably regular geodesic metric measure space with topological dimension where the theory of optimal mass transportation can be successfully developed. Examples for such spaces include Riemannian and Finsler manifolds, see McCann  and Ohta , the Heisenberg group , see Ambrosio and Rigot , or even more general sub-Riemannian structures with ’well-behaved’ cut locus, see Figalli and Rifford . Let and be two probability measures on which are absolutely continuous w.r.t. the reference measure m, and let be the unique displacement interpolation measure joining and throughout the so-called -intermediate optimal transport map . Roughly speaking, for fixed, the Jacobian determinant inequality reads as
Here, and are interpreted as the Radon-Nikodym derivatives of and of w.r.t. the reference measure while is the distortion coefficient which encodes information on the geometric structure of the space . Expressions of can be calculated in terms of the Jacobian of the exponential map or estimated in terms of a curvature condition. The expression can be given as a function of or its derivatives.
The Jacobian determinant inequality (1.1) in the above general form has been considered first in the setting of complete Riemannian manifolds (endowed with the natural Riemaniann distance and volume form) in the pioneering work of Cordero-Erausquin, McCann and Schmuckenschläger . This result constituted the starting point of an extensive study of the geometry of metric measure spaces, while relation (1.1) became an equivalent formulation of the famous curvature-dimension condition , due to Lott and Villani , and Sturm [16, 17], where is replaced by explicit expressions , being the lower bound of the Ricci curvatures in the Riemannian setting. Namely, is given by
and is precisely the Riemannian distance .
Juillet  proved that the Lott-Sturm-Villani curvature-dimension condition does not hold for any pair of parameters on the Heisenberg group (endowed with its usual Carnot-Carathéodory metric and -measure), which is the simplest sub-Riemannian structure. Accordingly, there were strong doubts on the validity of a sub-Riemannian version of the Jacobian determinant inequality in the sub-Riemannian context. However, by using a natural Riemannian approximation of the Heisenberg group as in Ambrosio and Rigot , the authors of the present paper proved (1.1) on , see [3, 4], where the Heisenberg distortion coefficient is defined by
and is the ’vertical’ derivative of at the point .
In the present paper we prove a Jacobian determinant inequality on corank 1 Carnot groups where the sub-Riemannian geometry is more complicated than the one of the model Heisenberg group due to the presence of abnormal geodesics and the ’anisotropic’ structure of the cut locus. Our method is different from the one in [3, 4] as we obtain the Jacobian determinant inequality by an intrinsic approach, without using a Riemannian approximation. As in [3, 4], we apply our Jacobian determinant inequality to establish various functional and geometric inequalities in the present setting including entropy, Brunn-Minkowski and Borell-Brascamp-Lieb inequalities. These results should open up the way to considering the above inequalities in a broader context outside the realm of -type conditions by replacing the coefficients by expressions that are suitable for sub-Riemannian geometries.
In order to present our main result, let us fix some notation. We denote by a dimensional corank 1 Carnot group with its Lie algebra , where dim and dim. The operation on (in exponential coordinates on ) can be given by
where , , and is a real skew-symmetric matrix. Let be the neutral element in The layers and are generated by the left-invariant vector fields
Moreover, By the spectral theorem for skew-symmetric matrices one can consider the diagonalized representation of given by
where and is the square null-matrix.
For further use, let us introduce the functions given by
To define the distortion coefficient, we introduce the set
where , and let be the closure of . The distortion coefficient on the Carnot group is defined by
Let us consider two compactly supported probability measures and on which are absolutely continuous w.r.t. . Since the distribution on the corank 1 Carnot group is two-generating, there exists a unique map realizing the optimal transportation between the measures and w.r.t. the cost function , see Figalli and Rifford [8, Proposition 4.2 and Theorem 3.2]; this map can be defined -a.e. through a -concave function as
Hereafter, is the Carnot-Carathéodory metric on and the sets and denote the moving and static sets of the transportation, respectively; see Section 2 for details. For fixed, we also introduce the -interpolant optimal transport map as
Our main result reads as follows.
(Jacobian determinant inequality on Carnot groups) Let be a dimensional corank 1 Carnot group, and assume that and are two compactly supported Borel probability measures on , both absolutely continuous w.r.t. . Let be fixed, be the unique optimal transport map transporting to associated to the cost function and its -interpolant map. Then the following Jacobian determinant inequality holds
where is given by
Let us notice that if , we have that
Furthermore, monotonicity properties of the functions and show that
Therefore, the measure contraction property MCP proved by Rizzi  is formally a consequence of (1.7). Notice, however that we use Rizzi’s result to prove the absolute continuity of the interpolant measure (see Proposition 2.5), needed in the proof of the Jacobian determinant inequality.
In our next remark we consider the situation when is the -dimensional Heisenberg group. In this case we have and for every . In this case no abnormal geodesics appear and the Carnot distortion coefficient reduces to the Heisenberg distortion coefficient which is nothing but relation (1.2) (introduced in ). Consequently, most of the results of  will be covered in the present work.
Let us notice furthermore, that in general corank 1 Carnot groups, the coefficients and depend not only on the parameter (as in the Heisenberg group) but also on , , showing a more anisotropic character of the present geometric setting as compared to the Heisenberg group. As we shall see later, and can be obtained by differentiating w.r.t. the horizontal vector fields from the distribution and the vertical vector field , respectively (see Lemma 2.2 below).
Our final remark is of technical nature, but the details will be clear by reading the proof of Theorem 1.1. In this proof, we shall distinguish the cases when the mass is transported along abnormal and strictly normal geodesics, respectively. On one hand, when the mass transport is realized along abnormal geodesics, it turns out that the Jacobian determinant inequality reduces to an Euclidean-type determinant inequality thus the distortion coefficient can be as in the Euclidean framework. We notice that in this case the full Jacobian matrix of might not exist; however, since the matrix has a triangular structure, the Jacobian can be reduced to two parts of the diagonal which are well defined and inequality (1.7) makes sense. Furthermore, the triangular structure of the Jacobi matrix will allow us to perform the necessary changes of variable in order to provide important applications (see e.g. the entropy and Borell-Brascamp-Lieb inequalities via a suitable Monge-Ampère equation). On the other hand, once the mass transport is along strictly normal geodesics, the distortion coefficient encodes information on the genuine sub-Riemannan character of the Carnot group obtained by a careful analysis of the Jacobian for the exponential map. It could also happen that a positive part of the mass is transported along abnormal geodesics while the complementary mass is transported by strictly normal geodesics, so different formulas for will be used in the same instance of the mass transportation; such a scenario will be presented in Example 3.1 (see also Figure 2).
The organization of the paper is as follows. The proof of Theorem 1.1 will be provided in Section 3 after a self-contained presentation of the needed technical details in Section 2, i.e., properties of the Carnot-Carathéodory metric , exponential map and its Jacobian, the cut locus, and the optimal mass transportation on corank 1 Carnot groups. We emphasize that the optimal mass transportation developed by Figalli and Rifford  for large classes of sub-Riemannian manifolds cannot be directly applied since the squared distance function is not necessarily locally semiconcave outside of the diagonal of which is crucial in  (e.g. the regularity of optimal mass transport maps and , or the validity of the Monge-Ampère equation). Section 4 is devoted to applications, i.e., by the Jacobian determinant inequality we shall derive entropy inequalities, the Brunn-Minkovski inequality and the Borell-Brascamp-Lieb inequality on corank 1 Carnot groups.
Acknowledgements. We express our gratitude to Luca Rizzi for motivating conversations about the subject of this paper. A. Kristály is grateful to the Mathematisches Institute of Bern for the warm hospitality where this work has been developed.
2.1. Carnot-Carathéodory metric, geodesics and their Jacobians on Carnot groups
A horizontal curve on is an absolutely continuous curve for which there exist measurable functions () such that
The length of this curve is
The classical Chow-Rashewsky theorem assures that any two points from the Carnot group can be joined by a horizontal curve. Thus we can equip the Carnot group with its natural Carnot-Carathéodory metric by
where are arbitrarily fixed.
Let be the neutral element in . The left invariance of the vector fields in the distribution is inherited by the distance , thus
Following Rizzi , the explicit form of the minimal geodesics on a general corank 1 Carnot group can be described as follows.
(Rizzi ) On a corank 1 Carnot group the geodesic starting from , with direction
has the following equation
when . When , the geodesic is
Hereafter, denotes the unit matrix.
We notice that once has a non-trivial kernel, there are abnormal geodesics on which appear precisely when their equations are given by
for every . Note that each abnormal geodesic can be described uniquely by a normal geodesic as well, namely, letting and ; thus the map in (2.2) may define an abnormal geodesic.
We recall from Rizzi  that the Jacobian determinant of the exponential map is
By left-invariance, the minimal geodesics on starting from an arbitrary point are represented by , where the two covectors and can be identified. Moreover, since for every the left-translation , is a volume-preserving map, it follows that
Given and assume that for some . Then , where is given by
From now on, we assume the matrix has the diagonal representation given in (1.4).
We notice that is not a fat distribution whenever the kernel of is non-trivial. Indeed, in this case we have for every and . However, is two-generating, i.e.,
For simplicity of notation, we reorganize the vector fields in as
We split the distribution on into two types of vector fields; namely, and . This splitting gives the following trivial representation of the distance function :
(Pythagorean rule) For every , we have
where is the Euclidean metric in while is the Carnot-Carathéodory distance on w.r.t. to the distribution inherited from the original sub-Riemannian structure.
Proof. By the left-invariance of the metric , we have
2.2. Cut locus
Let us consider the set
Rizzi [15, Lemma 16] proved that is precisely the injectivity domain of parameters associated to geodesics joining the origin to points of . We know that all points in the corank 1 Carnot group can be reached by a minimal normal geodesic; namely, for every there exists a parametrization in the closure of , i.e.,
which defines a minimal normal geodesic joining and .
The cut locus of the origin in is
The set in the above representation corresponds to the image of abnormal geodesics while the latter set contains the conjugate points to , see (2.4). By left-invariance, the cut locus of the point is
Note that is closed and for every moreover, by (2.6) it follows that if and only if .
Fix and let . If then we have
for every ,
Proof. By exploring the left-invariance, it is enough to consider the case when . Let us introduce the auxiliary functions defined by
We consider the case when the case can be obtained by a limiting procedure. Since and the cut locus is closed, there exists a small neighborhood of such that . Let be arbitrarily fixed. By (2.1) (for ) we have that
Thus, one has
Note that ; thus, the latter relations give at once that
A direct consequence of Lemma 2.2 is:
Fix such that . If , then
Proof. Let for some . According to Lemma 2.2, we have that
Thus, by relation (2.6) it follows that
which concludes the proof.
Our next proposition states that for fixed the function fails to be locally semiconvex at the points that are in the cut locus of .
Let be such that where satisfies and the number being the multiplicity of , i.e., . Then