Geometric inequalities on Heisenberg groups

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb H^n$$\end{document}. Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott–Villani and Sturm and also a geodesic version of the Borell–Brascamp–Lieb inequality akin to the one obtained by Cordero-Erausquin, McCann and Schmuckenschläger. The latter statement implies sub-Riemannian versions of the geodesic Prékopa–Leindler and Brunn–Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb H^n$$\end{document} developed by Ambrosio and Rigot. These results refute a general point of view, according to which no geometric inequalities can be derived by optimal mass transportation on singular spaces.


General background and motivation
Due to the seminal papers by Lott and Villani [27] and Sturm [38,39], metric measure spaces with generalized lower Ricci curvature bounds support various geometric and functional inequalities including Borell-Brascamp-Lieb, Brunn-Minkowski, Bishop-Gromov inequalities. A basic assumption for these results is the famous curvature-dimension condition CD(K , N ) which -in the case of a Riemannian manifold M, represents the lower bound K ∈ R for the Ricci curvature on M and the upper bound N ∈ R for the dimension of M, respectively. It is a fundamental question whether the method used in [27,38,39], based on optimal mass transportation works in the setting of singular spaces with no apriori lower curvature bounds. A large class of such spaces are the sub-Riemannian geometric structures or Carnot-Carathéodory geometries, see Gromov [20].
During the last decade considerable effort has been made to establish geometric and functional inequalities on sub-Riemannian spaces. The quest for Borell-Brascamp-Lieb and Brunn-Minkowski type inequalities became a hard nut to crack even on simplest sub-Riemannian setting such as the Heisenberg group H n endowed with the usual Carnot-Carathéodory metric d CC and L 2n+1 -measure. One of the reasons for this is that although there is a good first order Riemannian approximation (in the pointed Gromov-Hausdorff sense) of the sub-Riemannian metric structure of the Heisenberg group H n , there is no uniform lower bound on the Ricci curvature in these approximations (see e.g. Capogna et al. [11,Section 2.4.2]); indeed, at every point of H n there is a Ricci curvature whose limit is −∞ in the Riemannian approximation. The lack of uniform lower Ricci bounds prevents a straightforward extension of the Riemannian Borell-Brascamp-Lieb and Brunn-Minkowski inequalities of Cordero-Erausquin et al. [12] to the setting of the Heisenberg group. Another serious warning is attributed to Juillet [21] who proved that both the Brunn-Minkowski inequality and the curvature-dimension condition CD(K , N ) fail on (H n , d CC , L 2n+1 ) for every choice of K and N .
These facts tacitly established the view according to which there are no entropy-convexity and Borell-Brascamp-Lieb type inequalities on singular spaces such as the Heisenberg groups. The purpose of this paper is to deny this paradigm. Indeed, we show that the method of optimal mass transportation is powerful enough to yield good results even in the absence of lower curvature bounds. By using convergence results for optimal transport maps in the Riemannian approximation of H n due to Ambrosio and Rigot [2] we are able to introduce the correct sub-Riemannian geometric quantities which can replace the lower curvature bounds and can be successfully used to establish geodesic Borell-Brascamp-Lieb, Prékopa-Leindler, Brunn-Minkowski and entropy inequalities on the Heisenberg group H n . The main statements from the papers of Figalli and Juillet [15] and Juillet [21] will appear as special cases of our results.
Before stating our results we shortly recall the aforementioned geometric inequalities of Borell-Brascamp-Lieb and the curvature dimension condition of Lott-Sturm-Villani and indicate their behavior in the sub-Riemannian setting of Heisenberg groups.

An overview of geometric inequalities
The classical Borell-Brascamp-Lieb inequality in R n states that for any fixed s ∈ (0, 1), p ≥ − where A and B are positive and finite measure subsets of R n , and L n denotes the ndimensional Lebesgue measure. For a comprehensive survey on geometric inequalities in R n and their applications to isoperimetric problems, sharp Sobolev inequalities and convex geometry, we refer to Gardner [19].
In his Ph.D. Thesis, McCann [29, Appendix D] (see also [30]) presented an optimal mass transportation approach to Prékopa-Leindler, Brunn-Minkowski and Brascamp-Lieb inequalities in the Euclidean setting. This pioneering idea led to the extension of a geodesic version of the Borell-Brascamp-Lieb inequality on complete Riemannian manifolds via optimal mass transportation, established by Cordero-Erausquin et al. [12]. Closely related to the Borell-Brascamp-Lieb inequalities on Riemannian manifolds is the convexity of the entropy functional [12]. The latter fact served as the starting point of the work of Lott and Villani [27] and Sturm [38,39] who initiated independently the synthetic study of Ricci curvature on metric measure spaces by introducing the curvature-dimension condition CD(K , N ) for K ∈ R and N ≥ 1. Their approach is based on the effect of the curvature of the space encoded in the reference distortion coefficients where s ∈ (0, 1), see e.g. Sturm [39] and Villani [41]. To be more precise, let (M, d, m) be a metric measure space, K ∈ R and N ≥ 1 be fixed, P 2 (M, d) be the usual Wasserstein space, and Ent N (·|m) : P 2 (M, d) → R be the Rényi entropy functional given by where ρ is the density function of It turns out that a Riemannian (resp. Finsler) manifold (M, d, m) satisfies the condition CD(K , N ) if and only if the Ricci curvature on M is not smaller than K and the dimension of M is not greater than N , where d is the natural metric on M and m is the canonical Riemannian (resp. Busemann-Hausdorff) measure on M, see Sturm [39] and Ohta [33]. Coming back to the Borell-Brascamp-Lieb inequality in curved spaces, e.g., when (M, d, m) is a complete N -dimensional Riemannian manifold, we have to replace the convex combination (1 − s)x + sy in (1.1) by the set of s-intermediate points Z s (x, y) between x and y w.r.t. the Riemannian metric d on M defined by With this notation, we can state the result of Cordero-Erausquin et al. [12] (see also Bacher [3]), as the Borell-Brascamp-Lieb inequality BBL(K , N ) on (M, d, m) which holds if and only if for all s ∈ (0, 1), We would like to emphasize the fact that in [12] the main ingredient is provided by a weighted Jacobian determinant inequality satisfied by the optimal transport interpolant map.
It turns out, even in the more general setting of non-branching geodesic metric spaces, that both CD(K , N ) and BBL(K , N ) imply the geodesic Brunn-Minkowski inequality BM(K , N ), see Bacher [3], i.e., if (M, d, m) is such a space, for Borel sets A, B ⊂ M with m(A) = 0 = m(B) and s ∈ (0, 1), As we already pointed out, Juillet [21] proved that the Brunn-Minkowski inequality BM(K , N ) fails on (H n , d CC , L 2n+1 ) for every choice of K and N ; therefore, both CD(K , N ) and BBL(K , N ) fail too. In fact, a closer investigation shows that the failure of these inequalities on H n is not surprising: indeed, the distortion coefficient τ K ,N s is a 'pure Riemannian' object coming from the behavior of Jacobi fields along geodesics in Riemannian space forms. More quantitatively, since certain Ricci curvatures tend to −∞ in the Riemannian approximation of the first Heisenberg group H 1 (see Capogna, Danielli, Pauls and Tyson [11,Section 2.4.2]) and lim K →−∞ τ K ,N s (θ ) = 0 for every s ∈ (0, 1) and θ > 0, some Riemannian quantities blow up and they fail to capture the subtle sub-Riemannian metric structure of the Heisenberg group. In particular, assumption (1.3) in BBL(K , N ) degenerates to an impossible condition.
On the other hand, there is a positive effect in the Riemannian approximation (see [11,Section 2.4.2]) that would be unfair to conceal. It turns out namely, that the two remaining Ricci curvatures in H 1 will blow up to +∞ in the Riemannian approximation scheme. This can be interpreted as a sign of hope for a certain cancellation that could save the day at the end. This will be indeed the case: appropriate geodesic versions of Borell-Brascamp-Lieb and Brunn-Minkowski inequalities still hold on the Heisenberg group as we show in the sequel.

Statement of main results
According to Gromov [20], the Heisenberg group H n with its sub-Riemannian, or Carnot-Carathéodory metric, can be seen as the simplest prototype of a singular space. In this paper we shall use a model of H n that is identified with its Lie algebra R 2n+1 C n × R via canonical exponential coordinates. At this point we just recall the bare minimum that is needed of the metric structure of H n in order to state our results. In the next section we present a more detailed exposition of the Heisenberg geometry, its Riemannian approximation and the connection between their optimal mass transportation maps. We denote a point in H n by x = (ξ, η, t) = (ζ, t), where ξ = (ξ 1 , . . . , ξ n ) ∈ R n , η = (η 1 , . . . , η n ) ∈ R n , t ∈ R, and we identify the pair (ξ, η) with ζ ∈ C n having coordinates ζ j = ξ j + iη j for all j = 1, . . . , n. The correspondence with its Lie algebra through the exponential coordinates induces the group law where Im denotes the imaginary part of a complex number and ζ, ζ = n j=1 ζ j ζ j is the Hermitian inner product. In these coordinates the neutral element of H n is 0 H n = (0 C n , 0) and the inverse element of (ζ, t) is (−ζ, −t). Note that x = (ξ, η, t) = (ζ, t) form a real coordinate system for H n and the system of vector fields given as differential operators forms a basis for the left invariant vector fields of H n . The vectors X j , Y j , j ∈ {1, . . . , n} form the basis of the horizontal bundle and we denote by d CC the associated Carnot-Carathéodory metric.
Following the notations of Ambrosio and Rigot [2] and Juillet [21], we parametrize the sub-Riemannian geodesics starting from the origin as follows. For every (χ, θ ) ∈ C n × R we consider the curve γ χ,θ : [0, 1] → H n defined by For the parameters (χ, θ ) ∈ (C n \{0 C n }) × [−2π, 2π], the paths γ χ,θ are length-minimizing non-constant geodesics in H n joining 0 H n and γ χ,θ (1). If θ ∈ (−2π, 2π) then it follows that the geodesics connecting 0 H n and γ χ,θ (1) = 0 H n are unique, while for θ ∈ {−2π, 2π} the uniqueness fails. Let In analogy to τ K ,N s we introduce for s ∈ (0, 1) the Heisenberg distortion coefficients τ n s : The function θ → τ n s (θ ) is increasing on [0, 2π] (cf. Lemma 2.1), in particular τ n s (θ ) → +∞ as θ → 2π; and also: For s ∈ (0, 1), we introduce the notatioñ A rough comparison of the Riemannian and Heisenberg distortion coefficients is in order. First of all, both quantities τ K ,N s and τ n s encode the effect of the curvature in geometric inequalities. Moreover, both of them depend on the dimension of the space, as indicated by the parameter N in the Riemannian case and n in the Heisenberg case. However, by τ K ,N s there is an explicit dependence of the lower bound of the Ricci curvature K , while in the expression of τ n s no such dependence shows up. Let us recall that in case of R n the elegant proof of the Borell-Brascamp-Lieb inequality by the method of optimal mass transportation, see e.g. Villani [40,41] is based on the concavity of det(·) 1 n defined on the set of n ×n-dimensional real symmetric positive semidefinite matrices. In a similar fashion, Cordero-Erausquin, McCann and Schmuckenschläger derive the Borell-Brascamp-Lieb inequality on Riemannian manifolds by the optimal mass transportation approach from a concavity-type property of det(·) 1 n as well, which holds for the n × ndimensional matrices, obtained as Jacobians of the map Here ϕ M is a c = d 2 2 -concave map defined on the complete Riemannian manifold (M, g), d is the Riemannian metric, and exp M and ∇ M denote the exponential map and Riemannian gradient on (M, g). Here, the concavity is for the Jacobian matrices s → Jac Here Let μ 0 and μ 1 be two compactly supported probability measures on H n that are absolutely continuous w.r.t. L 2n+1 . According to Ambrosio and Rigot [2], there exists a unique optimal transport map ψ : H n → H n transporting μ 0 to μ 1 associated to the cost function d 2
Note that the maps ψ and ψ s are essentially injective thus their inverse functions ψ −1 and ψ −1 s are well defined μ 1 -a.e. and μ s -a.e., respectively, see Figalli and Rifford [16,Theorem 3.7] and Figalli and Juillet [15, p. 136]. If ψ(x) is not in the Heisenberg cut-locus of x ∈ H n (i.e., x −1 · ψ(x) / ∈ L * , which happens μ 0 -a.e.) and ψ(x) = x, there exists a unique 'angle' is Borel measurable on H n . Our main result can now be stated as follows. Theorem 1.1 (Jacobian determinant inequality on H n ) Let s ∈ (0, 1) and assume that μ 0 and μ 1 are two compactly supported, Borel probability measures, both absolutely continuous w.r.t. L 2n+1 on H n . Let ψ : H n → H n be the unique optimal transport map transporting μ 0 to μ 1 associated to the cost function d 2 CC 2 and ψ s its interpolant map. Then the following Jacobian determinant inequality holds: If ρ 0 , ρ 1 and ρ s are the density functions of the measures μ 0 , μ 1 and μ s = (ψ s ) # μ 0 w.r.t. to L 2n+1 , respectively, the Monge-Ampère equations (1.10) show the equivalence of (1.9) to (1.11) It turns out that a version of Theorem 1.1 holds even in the case when only μ 0 is required to be absolutely continuous. In this case we consider only the first term on the right hand side of (1.11). Inequality (1.7) shows that where ρ = dμ dL 2n+1 is the density of μ. Our entropy inequality is stated as follows: Theorem 1.2 General entropy inequality on H n ) Let s ∈ (0, 1) and assume that μ 0 and μ 1 are two compactly supported, Borel probability measures, both absolutely continuous w.r.t. L 2n+1 on H n with densities ρ 0 and ρ 1 , respectively. Let ψ : H n → H n be the unique optimal transport map transporting μ 0 to μ 1 associated to the cost function d 2 CC 2 and ψ s its interpolant map. If μ s = (ψ s ) # μ 0 is the interpolant measure between μ 0 and μ 1 , and U : [0, ∞) → R is a function such that U (0) = 0 and t → t 2n+1 U 1 t 2n+1 is non-increasing and convex, the following entropy inequality holds: Inequality (1.7), Theorem 1.2 and the assumptions made for U give the uniform entropy estimate (see also Corollary 3.2): Various relevant choices of admissible functions U : [0, ∞) → R will be presented in the sequel. In particular, Theorem 1.2 provides an curvature-dimension condition on the metric measure space (H n , d CC , L 2n+1 ) for the choice of (1.12) Then the following inequality holds: Consequences of Theorem 1.3 are uniformly weighted and non-weighted Borell-Brascamp-Lieb inequalities on H n which are stated in Corollaries 3.5 and 3.6, respectively. As particular cases we obtain Prékopa-Leindler-type inequalities on H n , stated in Corollaries 3.7-3.9.
Let us emphasize the difference between the Riemannian and sub-Riemannian versions of the entropy and Borell-Brascamp-Lieb inequalites. In the Riemannian case, we notice the appearance of the distance function in the expression of τ K ,N s (d(x, y)). The explanation of this phenomenon is that in the Riemannian case the effect of the curvature accumulates in dependence of the distance between x and y in a controlled way, estimated by the lower bound K of the Ricci curvature. In contrast to this fact, in the sub-Riemannian framework the argument θ(x, y) appearing in the weight τ n s (θ (x, y)) is not a distance but a quantity measuring the deviation from the horizontality of the points x and y, respectively. Thus, in the Heisenberg case the effect of positive curvature occurs along geodesics between points that are situated in a more vertical position with respect to each other. On the other hand an effect of negative curvature is manifested between points that are in a relative 'horizontal position' to each other. The size of the angle θ(x, y) measures the 'degree of verticality' of the relative positions of x and y which contributes to the curvature.
The geodesic Brunn-Minkowski inequality on the Heisenberg group H n will be a consequence of Theorem 1.3. For two nonempty measurable sets A, B ⊂ H n we introduce the quantity where the sets A 0 and B 0 are nonempty, full measure subsets of A and B, respectively.

Theorem 1.4 (Weighted Brunn-Minkowski inequality on H n ) Let s ∈ (0, 1) and A and B be two nonempty measurable sets of H n . Then the following geodesic Brunn-Minkowski inequality holds:
Here we consider the outer Lebesgue measure whenever Z s (A, B) is not measurable, and the convention +∞·0 = 0 for the right hand side of (1.13). The latter case may happen e.g. when The value A,B represents a typical Heisenberg quantity indicating a lower bound of the deviation of an essentially horizontal position of the sets A and B. An intuitive description of the role of weights τ n 1−s ( A,B ) and τ n s ( A,B ) in (1.13) will be given in Sect. 4. By Theorem 1.4 we deduce several forms of the Brunn-Minkowski inequality, see Corollary 4.2. Moreover, the weighted Brunn-Minkowski inequality implies the measure contraction property MCP(0, 2n + 3) on H n proved by Juillet [21,Theorem 2.3], see also Corollary 4.1, namely, for every s ∈ (0, 1), x ∈ H n and nonempty measurable set E ⊂ H n , Our proofs are based on techniques of optimal mass transportation and Riemannian approximation of the sub-Riemannian structure. We use extensively the machinery developed by Cordero-Erausquin et al. [12] on Riemannian manifolds and the results of Ambrosio and Rigot [2] and Juillet [21] on H n . In our approach we can avoid the blow-up of the Ricci curvature to −∞ by not considering limits of the expressions of τ K ,N s . Instead of this, we apply the limiting procedure to the coefficients expressed in terms of volume distortions. It turns out that one can directly calculate these volume distortion coefficients in terms of Jacobians of exponential maps in the Riemannian approximation. These quantities behave in a much better way under the limit, avoiding blow-up phenomena. The calculations are based on an explicit parametrization of the Heisenberg group and the approximating Riemannian manifolds by an appropriate set of spherical coordinates that are based on a fibration of the space by geodesics.
The paper is organized as follows. In the second section we present a series of preparatory lemmata obtaining the Jacobian representations of the volume distortion coefficients in the Riemannian approximation of the Heisenberg group and we discuss their limiting behaviour.
In the third section we present the proof of our main results, i.e., the Jacobian determinant inequality, various entropy inequalities and Borell-Brascamp-Lieb inequalities.
The forth section is devoted to geometric aspects of the Brunn-Minkowski inequality. In the last section we indicate further perspectives related to this research. The results of this paper have been announced in [4].

Volume distortion coefficients in H n
The left translation l x : H n → H n by the element x ∈ H n is given by l x (y) = x · y for all y ∈ H n . One can observe that l x is affine, associated to a matrix with determinant 1. Therefore the Lebesgue measure of R 2n+1 will be the Haar measure on H n (uniquely defined up to a positive multiplicative constant).
For λ > 0 define the nonisotropic dilation ρ λ : thus the homogeneity dimension of the Lebesgue measure L 2n+1 is 2n + 2 on H n .
In order to equip the Heisenberg group with the Carnot-Carathéodory metric we consider the basis of the space of the horizontal left invariant vector fields {X 1 , . . . , X n , Y 1 , . . . Y n }. A horizontal curve is an absolutely continuous curve γ : [0, r ] → H n for which there exist measurable functions h j : [0, r ] → R ( j = 1, . . . , 2n) such that The length of this curve is The classical Chow-Rashewsky theorem assures that any two points from the Heisenberg group can be joined by a horizontal curve, thus it makes sense to define the distance of two points as the infimum of lengths of all horizontal curves connecting the points, i.e., for every x, y ∈ H n and λ > 0.
We recall the curve γ χ,θ introduced in (1.5). One can observe that for every x ∈ H n \L, there exists a unique minimal geodesic γ χ,θ joining 0 H n and x, where L = {0 C n } × R is the center of H n . In the sequel, following Juillet [21], we consider the diffeomorphism (2.1) Moreover, by (1.6) and (2.2), we have for every θ ∈ [0, 2π) (and Proof Let s ∈ (0, 1) be fixed and consider the functions f i,s : (0, π) → R, i ∈ {1, 2}, given by Note that both functions f i,s are positive on (0, π), i ∈ {1, 2}. First, for every t ∈ (0, π) one has where we use the Mittag-Leffler expansion of the cotangent function Therefore, f 1,s is increasing on (0, π). In a similar way, we have that Thus, f 2,s is also increasing on (0, π). Since the claim follows.
The following property gives a formula for the Heisenberg volume distortion coefficient in terms of the Jacobian Jac( s ).
Because of the homogeneities of d CC and L 2n+1 , we have .
Since x −1 · y / ∈ L, we have that B(x −1 · y, r ) ∩ L = ∅ for r small enough, thus the map s • −1 1 realizes a diffeomorphism between the sets B(x −1 · y, r ) and Z s (0 H n , B(x −1 · y, r )). This constitutes the basis for the following change of variable By the continuity of the integrand in the latter expression, the volume derivative of , which gives precisely the claim.
(ii) At first glance, this property seems to be just the symmetric version of (i). Note however that thus we need the explicit form of the geodesic from 0 H n to −x −1 · y in terms of (χ, θ ). A direct computation based on (2.1) shows that Therefore, the minimal geodesic joining 0 H n and −x −1 · y is given by the curve s → s −χe −iθ , −θ , s ∈ [0, 1]. Now, it remains to apply (i) with the corresponding modifications, obtaining which concludes the proof.
For further use (see Corollary 2.1 Let s ∈ (0, 1) and x, y ∈ H n such that x = y. The following properties hold: Moreover, we have for every s ∈ (0, 1) and x, y ∈ H n that

Similar relations hold for v 1−s (y, x) by replacing s by (1 − s).
Proof (i) Directly follows by Proposition 2.1 and relation (2.2).
where c 1 > 0 is a constant which depends on t > 0 (but not on r > 0). To check inequality (2.5) we may replace the ball B(x −1 · y, r ) in the Carnot-Carathéodory metric d CC by the ball in the Korányi metric d K (introduced as the gauge metric in [2]). Since the two metrics are bi-Lipschitz equivalent, it is enough to check (2.5) for the Korányi ball; for simplicity, we keep the same notation.
Recalling that r < √ t 2 , by inequality (2.7) we obtain that Combining this estimate with inequality (2.6), it yields that sin θ w 2 ≤ r |θ w | 2|χ w | ≤ r π |χ w | ≤ r 8π 3t , proving inequality (2.5). Note that θ w is close to 2π whenever r is very small. Therefore, by continuity reasons, since , t > 0 and n ∈ N. Consequently, by relation (2.2) one has for every w ∈ B(x −1 · y, r )\L that Since the map s • −1 1 is a diffeomorphism between the sets B(x −1 · y, r )\L and Z s (0 H n , B(x −1 · y, r )\L), a similar argument as in the proof of Proposition 2.1 gives By the latter estimate and (2.5) we have The fact that v s (x, y) = +∞ for x −1 · y ∈ L * encompasses another typical sub-Riemannian feature of the Heisenberg group H n showing that on 'vertical directions' the curvature blows up even in small scales (i.e., when x and y are arbitrary close to each other), described by the behavior of the Heisenberg volume distortion coefficient. This phenomenon shows another aspect of the singular space structure of the Heisenberg group H n .

Volume distortion coefficients in the Riemannian approximation M ε of H n
We introduce specific Riemannian manifolds in order to approximate the Heisenberg group H n , following Ambrosio and Rigot [2] and Juillet [21,23]. For every ε > 0, let M ε = R 2n+1 be equipped with the usual Euclidean topology and with the Riemannian structure where the orthonormal basis of the metric tensor g ε at the point x = (ξ, η, t) is given by the vectors (written as differential operators):  γ (s)), . . . , X n (γ (s)), Y 1 (γ (s)), . . . , Y n (γ (s)), T ε (γ (s)). One can check thaṫ γ j (s) is equal with the j-th cartesian coordinate ofγ (s), j = 1, . . . , 2n, anḋ Note that (M ε , d ε ) is complete and the distance d ε inherits the left invariance of the vector fields X 1 , . . . , X n , Y 1 , . . . , Y n , T ε , similarly as in the Heisenberg group H n . Moreover, one can observe that d ε is decreasing w.r.t. ε > 0 and due to Juillet [23] for a fixed c > 0 constant, For ε > 0 fixed, we recall from Ambrosio and Rigot [2] that the ε-geodesic γ ε : [0, 1] → M ε with starting point 0 H n and initial vector (2.11) Using the complex notation C n × R for the Heisenberg group H n , we can write the expression of the ε-geodesics γ ε explicitly as where With these notations, let ε s (χ ε , θ ε ) = γ ε (s).
For further use, let cut ε (x) be the cut-locus of x ∈ M ε in the Riemannian manifold (M ε , g ε ).
In particular, we can simplify the computations by setting χ ε 0 = (0, . . . , 0, |χ ε |); in this way the above matrix has several zeros, and its determinant is the product of n − 1 identical determinants corresponding to the matrix The rest of the computation is straightforward.
For a fixed s ∈ [0, 1] and (x, Following Cordero-Erausquin et al. [12], we consider the volume distortion coefficient in when s ∈ (0, 1]. Note that v ε 1 (x, y) = 1 for every x, y ∈ H n . Moreover, the local behavior of geodesic balls shows that v ε s (x, x) = 1 for every s ∈ (0, 1) and x ∈ H n . The following statement provides an expression for the volume distortion coefficient in terms of the Jacobian Jac( ε s ).
Proof Since y / ∈ cut ε (x) and cut ε (x) is closed, there exists r > 0 small enough such that B ε (y, r ) ∩ cut ε (x) = ∅. In particular, the point x and every element from B ε (y, r ) can be joined by a unique minimal ε-geodesic and Z ε s (x, z) is a singleton for every z ∈ B ε (y, r ). By the left-translation (valid also on the (2n +1)−dimensional Riemannian manifold (M ε , g ε )), 1 · y, r ) .
Because of the asymptotic behaviour of the volume of small balls in the Riemannian geometry (see Gallot, Hulin and Lafontaine [18] 1 · y, r ) .
In the last step we used dm ε = 1 ε dL 2n+1 . The rest of the proof goes in the same way as in case of Proposition 2.1(i); see also Cordero-Erausquin et al. [12].

Optimal mass transportation on H n and M ε
Let us fix two functions f, g : H n → [0, ∞) and assume that Let μ 0 = f L 2n+1 and μ 1 = gL 2n+1 . By the theory of optimal mass transportation on H n for c = d 2 CC /2, see Ambrosio and Rigot [2, Theorem 5.1], there exists a unique optimal transport map from μ 0 to μ 1 which is induced by the map for some c-concave and locally Lipschitz map ϕ, where 1 comes from (2.1). In fact, according to Figalli and Rifford [16], there exists a Borel set C 0 ⊂ supp f of null L 2n+1 -measure such that for every x ∈ supp f \C 0 , there exists a unique minimizing geodesic from x to ψ(x); this geodesic is represented by The sets M ψ = {x ∈ H n : ψ(x) = x} and S ψ = {x ∈ H n : ψ(x) = x} correspond to the moving and static sets of the transport map ψ, respectively. On the Riemannian manifold (M ε , g ε ), we may consider the unique optimal transport map ψ ε from μ ε

The existence and uniqueness of such a map is provided by McCann [31, Theorem 3.2]. This map is defined by a
see Ambrosio and Rigot [2, p. 292]. Note that we may always assume that ϕ ε (0 H n ) = 0. Due to Cordero-Erausquin et al. [12,Theorem 4.2], there exists a Borel set C ε ⊂ supp f of null m ε -measure such that ψ ε (x) / ∈ cut ε (x) for every x ∈ supp f \C ε . Now we consider the interpolant map ψ ε Using again a left-translation, we equivalently have With the above notations we summarize the results in this section, establishing a bridge between notions in H n and M ε which will be crucial in the proof of our main theorems:

Proposition 2.3
There exists a sequence {ε k } k∈N ⊂ (0, 1) converging to 0 and a full μ 0measure set D ⊂ H n such that f is positive on D and for every x ∈ D we have: x) for every s ∈ (0, 1). Proof of Proposition 2.3. Let us start with an arbitrary sequence {ε k } k∈N of positive numbers such that lim k→∞ ε k = 0 and C = C 0 ∪ ∪ k∈N C ε k , where C 0 and C ε k are the sets with null L 2n+1 -measure coming from the previous construction, i.e., there is a unique minimizing geodesic from x to ψ(x) and ψ ε k (x) / ∈ cut ε k (x) for every x ∈ supp f \C. We define D = {x ∈ H n : f (x) > 0}\C. Notice that D has full μ 0 -measure by its definition. It is clear that every volume distortion coefficient appearing in (ii) and (iii) is well-defined for every x ∈ D. The set D from the claim will be obtained in the course of the proof by subsequently discarding null measure sets several times from D. In order to simplify the notation we shall keep the notation D for the sets that are obtained in this way. Similarly, we shall keep the notation for {ε k } k∈N when we pass to a subsequence.

Remark 2.2 Note that the limiting value of the distortion coefficients in the
Accordingly, by Ambrosio and Rigot [2, Theorem 6.2] we have that In the proof of (i) we shall distinguish two cases. Let s ∈ (0, 1] be fixed. Case 1: the moving set M ψ . By using [2, Theorem 6.11] of Ambrosio and Rigot, up to the removal of a null measure set and up to passing to a subsequence we have (2.19) where ϕ ε k and ϕ are the c ε k -concave and c-concave functions appearing in (2.15) and (2.18).
Due to the form of w ε k (x) ∈ T 0 H n M ε k from (2.18), we introduce the complex vector-field The limits in (2.19) imply that for a.e. x ∈ D ∩ M ψ we have Case 2: the static set S ψ . From the representation (2.16) we have that ψ s (x) = x for any x ∈ S ψ . Clearly, we only need to consider values of ε k for which ψ ε k (x) = x. Again, by [2, Theorem 6.2] of Ambrosio and Rigot, lim k→∞ ψ ε k (x) = ψ(x) = x for a.e. x ∈ D ∩ S ψ . According to (2.17) the point ψ ε k s (x) lies on the ε k -geodesic connecting x and ψ ε k (x). The latter limit and the estimate (2.10) imply the following chain of inequalities so lim k→∞ ψ ε k s (x) = x, which ends the proof of (i). To prove inequality (ii) we distinguish again two cases.

Remark 2.3
In the second case of the above proof (i.e., x ∈ S ψ ) we could expect a better lower bound than s 2 for v ε k s (x, ψ ε k (x)) as k → ∞ since no explicit presence of Heisenberg volume distortion coefficient is expected. However, in the general case s 2 is the optimal bound. Indeed, since x ∈ S ψ we first notice that |χ ε k | → 0 as k → ∞. Thus, if θ ε k → 0 and we assume that

Jacobian determinant inequality on H n
In this subsection we shall prove our Jacobian determinant inequality on H n as the key result of the paper.
Proof of Theorem 1.1. We shall consider the sequence {ε k } k∈N ⊂ (0, 1] such that lim k→∞ ε k = 0 and the statement of Proposition 2.3 holds. Let (M ε k , g ε k ) be the Riemannian manifolds approximating H n , k ∈ N. Let us consider the measures μ 0 = ρ 0 L 2n+1 ,  [12,Lemma 5.3], the maps ψ, ψ s and ψ ε k s are essentially injective on D, respectively. Consequently, there is a set D 0 ⊂ D of null L 2n+1 -measure such that the maps ψ, ψ s and ψ ε k s (k ∈ N) are injective on D\D 0 ; for simplicity, we keep the notation D for D\D 0 . Let μ s = (ψ s ) # μ 0 and μ ε k s = (ψ ε k s ) # μ 0 be the push-forward measures on H n and M ε k , and ρ s and ε k ρ ε k s be their density functions w.r.t. to the measures L 2n+1 and m ε k , respectively.
Let A i ⊂ H n be the support of the measures μ i , i ∈ {0, 1}. On account of (2.10), definition (2.14) and the compactness of the sets A 0 and A 1 , one has for every x ∈ D that y).
Since by (2.10) we have that Clearly, A 0 , A 1 ⊂ B(0, R). Thus, it is enough to take m = L 2n+1 | B(0,R) as the reference measure.
The proof is based on the Jacobian determinant inequality from [12, Lemma 6.1] on M ε k , i.e., for every x ∈ D, The technical difficulty is that we cannot simply pass to a point-wise limit in the latter inequality because we do not have an almost everywhere convergence of Jacobians. To overcome this issue we aim to prove a weak version of the inequality by multiplying by a continuous test function and integrating. As we shall see in the sequel, this trick allows the process of passing to the limit and we can obtain an integral version of the Jacobian inequality which in turn gives us the desired point-wise inequality almost everywhere.
To carry out the aforementioned program, we combine the above Jacobian determinant inequality with the Monge-Ampère equations on M ε k , namely, Let us fix an arbitrary non-negative test function h ∈ C c (H n ) with support in B(0, R); for simplicity of notation, let S = supp(h). Multiplying (3.4) by h(ψ ε k s (x)) ≥ 0, an integration on D w.r.t. the measure μ 0 = ρ 0 m gives where Note that by Fatou's lemma, the continuity of h and Proposition 2.3, we have By the Monge-Ampère equations (1.10) and (3.3), it turns out that for every k ∈ N we have ψ ε k (D) = ψ(D) = supp(μ 1 ) (up to a null measure set). Therefore, by performing a change of variables y = ψ ε k (x) in the integrand R k s,2 , we obtain by (3.3) that Taking the lower limit as k → ∞, Fatou's lemma, the continuity of h and Proposition 2.3 imply that Changing back the variable y = ψ(x), it follows by (1.10) that By Corollary 2.1, relations (1.6), (2.4) and (1.8), we observe that for every x ∈ D, . Therefore, by the estimates (3.6) and (3.7) we obtain In the sequel, we shall prove that Let us notice first that μ ε k s μ s as k → ∞. Indeed, let ϕ : H n → R be a continuous test function with support in B(0, R). By the definition of interpolant measures μ where all integrals are over B(0, R). Since μ 0 is compactly supported and lim k→∞ ψ ε k s (x) = ψ s (x) for μ 0 -a.e. x (cf. Proposition 2.3), the Lebesgue dominated convergence theorem implies ϕ(ψ ε k s (x))dμ 0 (x) → ϕ(ψ s (x))dμ 0 (x) as k → ∞. Combined the latter limit with (3.10) the claim follows, i.e., ϕ(y)dμ ε k s (y) → ϕ(y)dμ s (y) as k → ∞. In partic- ε k = ρ ε k s and dμ s dm = ρ s , the latter limit implies that In what follows we need an inequality version of this weak convergence result valid for upper semicontinuous functions. We shall formulate the result as the following:

) be a bounded, upper semicontinous function. Then the following inequality holds:
lim sup k→∞ ϕ(y)ρ ε k s (y)dm(y) ≤ ϕ(y)ρ s (y)dm(y). (3.11) To prove the claim let us notice that by the definition of densities and push-forwards of measures the inequality (3.11) is equivalent to By the upper semicontinuity of ϕ and from the fact that lim k→∞ ψ ε k Let M > 0 be an upper bound of ϕ, i.e., 0 ≤ ϕ ≤ M. For an arbitrarily fixed δ > 0 we shall prove that there exists k δ ∈ N such that for k ≥ k δ , Since δ > 0 is arbitrarily small, the claim (3.12) would follow from (3.14). In order to show (3.14) let us introduce for all l ∈ N the set S l δ := {x ∈ D : ϕ(ψ ε k s (x)) ≤ ϕ(ψ s (x)) + δ for all k ≥ l}.
Note that S l δ ⊆ S l+1 δ for all l ∈ N and ∪ l S l δ = D; the latter property follows by (3.13). Since D is a full μ 0 -measure set it follows that for δ > 0 there exists k δ ∈ N such that for k ≥ k δ we have μ 0 (S k δ ) ≥ 1 − δ. This implies that for every k ≥ k δ we have the estimates concluding the proof of the claim.
We resume now the proof of the theorem. Since ρ s ∈ L 1 (dm), there exists a decreasing sequence of non-negative lower semicontinuous functions {ρ i s } i∈N approximating ρ s from above. More precisely, we have that ρ i s ≥ ρ s and ρ i s → ρ s in L 1 (dm) as i → ∞. Replacing ρ i s by ρ i s + 1 i if necessary, we can even assume that ρ i s > ρ s . In particular, ρ i s is strictly positive and lower semicontinuous. This implies that (ρ i s ) − 1 2n+1 is positive, bounded from above and upper semicontinuous for every i ∈ N. We introduce the sequence of functions defined by Note that ρ ε k ,i s > 0 on D. To continue the proof of the theorem we notice that the injectivity of the function ψ ε k s on D, relation (3.3) and a change of variable y = ψ ε k s (x) give that The sub-unitary triangle inequality (i.e., |a + b| α ≤ |a| α + |b| α for a, b ∈ R and α ≤ 1), and the convexity of the function t → −t 1− 1 2n+1 , t > 0 imply the following chain of inequalities Let δ > 0 be arbitrarily fixed. On one hand, by Hölder's inequality and the fact that ρ i s → ρ s in L 1 (dm) as i → ∞, it follows the existence of i δ ∈ N such that for every i ≥ i δ , On the other hand, since y → ϕ(y) = h(y)(ρ i δ s (y)) − 1 2n+1 is positive, bounded from above and upper semicontinuous, by (3.11) we find k δ ∈ N such that 2n 2n Summing up the above estimates, for every k ≥ k δ we have Thus, the arbitrariness of δ > 0 implies that h(y) (ρ s (y)) 1− 1 2n+1 dm(y). Now, the injectivity of the map D x → ψ s (x), a change of variable y = ψ s (x) in the right hand side of the latter estimate, and the Monge-Ampère equation (3.16) give the inequality in (3.9). Combining the estimates (3.8) and (3.9), we obtain Applying the change of variables y = ψ s (x) and (3.16) we obtain Observe that the function on the left side of the above estimate that multiplies h is ρ Since we are considering only positive functions it follows that the function on the right side multiplying h is also in L 1 (dm). We shall use the well-known fact that convolutions with mollifiers converge point-wise almost everywhere to the function values for functions in L 1 (dm).
Since the test function h ≥ 0 is arbitrary, it can play the role of convolution kernels. From here we can conclude that the latter integral inequality implies the point-wise inequality: for a.e. y ∈ ψ s (D). Composing with ψ s the above estimate, it yields a.e. x ∈ D. (3.17) By the Monge-Ampère equations (3.16) and ρ 0 (x) = ρ 1 (ψ(x))Jac(ψ)(x), x ∈ D, we obtain the inequality a.e. x ∈ D, which concludes the proof.

Remark 3.1
Observe that the Jacobian identity on the Riemannian manifolds M ε k (cf. [12,Lemma 6.1]) that constitutes the starting point of the proof of our determinant inequality holds also in the case when μ 1 is not necessarily absolutely continuous w.r.t. the L 2n+1measure. In this case our arguments are based on the inequality that we obtain by canceling the second term of the right side, namely Now we can perform the same steps as in the proof of Theorem 1.1 by obtaining a.e. x ∈ D, (3.18) or equivalently (Jac(ψ s )(x)) ≥ τ n 1−s (θ x ) 2n+1 a.e. x ∈ D. (3.19) A direct consequence of (3.18) and (1.7) is the main estimate from the paper of Figalli and Juillet [15] formulated and refined in the following statement: Remark 3.2 A closer inspection of inequality (1.9) from Theorem 1.1 shows that it can be improved in the presence of a positive measure set of static points. Indeed, if x is a static point of ψ than it follows that it will be a static point for ψ s (x) as well. Considering density points of the static set, (i.e. discarding a null set if necessary) we obtain that both Jacobians Jac(ψ s )(x) = Jac(ψ)(x) = 1 on a full measure of stationary points. This implies that relation (1.9) holds with τ n s (θ x ) = s and τ n 1−s (θ x ) = 1 − s. Based on this observation it is natural to define a new, optimal transport based Heisenberg distortion coefficientτ n s,ψ which depends directly on x ∈ H n rather than on the angle θ x . If s ∈ (0, 1), we considerτ (3.20) With this notation, under the assumptions of Theorem 1.1, the following improved version of (1.9) holds:

Entropy inequalities on H n
As a first application of the Jacobian determinant inequality we prove several entropy inequalities on H n .
Proof of Theorem 1.2. We shall keep the notations from the proof of Theorem 1.1. Since the function t → t 2n+1 U (t −(2n+1) ) is non-increasing, relation (3.17) implies that for a.e. x ∈ D we have .
Recalling relation sτ n s = τ n s , the right hand side of the above inequality can be written as ⎛ By using the convexity of t → t 2n+1 U (t −(2n+1) ), the latter term can be estimated from above by Summing up, for a.e. x ∈ D we have 2n+1 .
Let U S (t) = t log t for t > 0 and U S (0) = 0; Corollary 3.2 implies the following convexity-type property of the Shannon entropy s → Ent U S (μ s |m) on H n : Corollary 3.4 (Uniform Shannon entropy inequality on H n ) Under the same assumptions as in Theorem 1.1, the following entropy inequality holds: and state a corresponding version of Theorem 1.2 with respect to these coefficients.

Borell-Brascamp-Lieb and Prékopa-Leindler inequalities on H n
In this subsection we prove various Borell-Brascamp-Lieb and Prékopa-Leindler inequalities on H n by showing another powerful application of the Jacobian determinant inequality. . Thus, we may assume that both integrals are positive. The proof is divided into three parts.
Step 1. We first consider the particular case when the functions f, g are compactly supported and normalized, i.e., for some N ≥ 1, where · denotes the Heisenberg group law. It turned out that (4.1) fails for the homogeneous dimension N = 2n + 2, see Monti [32]; moreover, it fails even for all N > 2n + 1 as shown by Juillet [21]. However, inequality (4.1) holds for the topological dimension N = 2n + 1, see [26].
In this subsection we shall present several geodesic Brunn-Minkowski inequalities on H n and discuss their geometric features.
Proof of Theorem 1.4. We have nothing to prove when both sets have zero L 2n+1 -measure.
Let A, B ⊂ H n be two nonempty measurable sets such that at least one of them has positive L 2n+1 -measure. We first claim that A,B < 2π. To check this we recall that where the sets A 0 and B 0 are nonempty, full measure subsets of A and B, respectively. Arguing by contradiction, if A,B = 2π, it follows that up to a set of null L 2n+1 -measure, we have for every (x, y) ∈ A × B that In particular, up to a set of null L 2n+1 -measure, A −1 ·B ⊂ {0 C n }×R, thus L 2n+1 (A −1 ·B) = 0. Therefore, the multiplicative Brunn-Minkowski inequality (4.1) for N = 2n + 1 gives that Due to the fact that A,B < 2π, we have 0 < c s 1 , c s 2 < +∞. We now distinguish two cases.  (Z s (A, B) We consider the first sub-case; the second one is treated in a similar way.
By the first part of the proof, we have that A,B < 2π. By setting μ 0 = L 2n+1 | A L 2n+1 (A) and μ 1 = δ x the point-mass associated to a point x ∈ B, the Jacobian determinant inequality (3.19) can be explored in order to obtain L 2n+1 (Z s (A, B) where we used that A,{x} ≥ A,B .
Proof The first inequality is nothing but the weighted Brunn-Minkowski inequality for A = {x} and B = E (see also Case 2 in the proof of Theorem 1.4). As τ n s ≥ s 2n+3 2n+1 , the proof is complete.
The geodesic Brunn-Minkowski inequality carries more information on the sub-Riemannian geometry of the Heisenberg group. To illustrate this aspect, we give the geometric interpretation of the expression A,B appearing in Theorem 1.4 for sets A, B ⊂ H n with positive measure and of the Heisenberg distortion coefficients τ n 1−s ( A,B ) and τ n s ( A,B ) that appear as weights in the Brunn-Minkowski inequality.
We say that A and B are essentially horizontal if there exist full measure subsets A 0 ⊂ A and B 0 ⊂ B such that for every x 0 ∈ A 0 there exists y 0 ∈ B 0 ∩ H x 0 , where H x 0 = y = (ζ, t) ∈ H n : t = t 0 + 2Im ζ 0 , ζ denotes the horizontal plane at x 0 = (ζ 0 , t 0 ). In such a case, for some χ 0 ∈ C n we have x −1 0 · y 0 = (χ 0 , 0) = 1 (χ 0 , 0), i.e., A,B = 0. We now turn our attention to the case when the sets A and B are not essentially horizontal to each other. Bellow we indicate an example showing that in such a case, the Heisenberg distortion coefficients τ n 1−s ( A,B ) and τ n s ( A,B ) can even take arbitrarily large values. To be more precise, let s ∈ (0, 1) and consider the CC-balls A r = B((0 C n , t 1 ), r ) and B r = B((0 C n , t 2 ), r ) in H n for sufficiently small values of r > 0 and t 1 = t 2 . Clearly, the sets A r and B r are horizontally far from each other, i.e., B r ∩ H x 0 = ∅ for every x 0 ∈ A r .
The geodesics joining the elements of A r and B r largely deviate from the t-axis and Z s (A r , B r ) becomes a large set w.r.t. A r and B r ; see Fig. 1 for n = 1. More precisely, we have as r → 0, which is in a perfect concordance with the competition of the two sides of (1.13).

Remark 4.3
We notice that instead of A,B in the weighted Brunn-Minkowski inequality, we can use a better quantity depending on the optimal mass transport A,ψ = sup where the set A 0 is a nonempty, full measure subset of A and ψ is the optimal transport map resulting from the context. Sinceˆ A,ψ ≥ A,B and τ n s is increasing, one has τ n s ˆ A,ψ ≥ τ n s ( A,B ). In this way, one can slightly improve the Brunn-Minkowski inequality (1.13). A further improvement can be obtained by replacing τ n s byτ n s from (3.20).

Concluding remarks and further questions
The purpose of this final section is to indicate open research problems that are closely related to our results and can be considered as starting points of further investigations. Let us mention first that there have been several different approaches to functional inequalities for sub-Riemannian geometries. One such possibility was initiated by Baudoin, Bonnefont and Garofalo [7] via the Bakry-Émery carré du champ operator by introducing an analytic curvature-dimension inequality on sub-Riemannian manifolds. A challenging may have a pathological behavior, see e.g. Figalli and Rifford [16,§5.8,p. 145], and the geodesics in the Riemannian approximants may converge to singular geodesics.
After posting the first version of the present work to the mathematical community, followup works have been obtained by establishing intrinsic geometric inequalities on corank 1 Carnot groups (by Balogh et al. [5]) and on ideal sub-Riemannian manifolds (by Barilari and Rizzi [6]) by different methods than ours. Naturally, the Heisenberg distortion coefficient τ n s introduced in the present paper and those from the latter works coincide on H n . This confirms the efficiency of the approximation arguments in suitable sub-Riemannian geometric contexts. In addition, as C. Villani suggested in [42, p. 43], the results in the present paper (together with those from [5,6]) motivate the so-called "grande unification" of geometric inequalities appearing in Riemannian, Finslerian and sub-Riemannian geometries.