Geometric inequalities on Heisenberg groups

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group $\mathbb H^n$. 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 Schmuckenschl\"ager. The latter statement implies sub-Riemannian versions of the geodesic Pr\'ekopa-Leindler and Brunn-Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of $\mathbb H^n$ 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.

and Sturm [Stu06a,Stu06b], 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 [LV09], [Stu06a,Stu06b], 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 [Gro96]. 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, Danielli, Pauls and Tyson [CDPT07, 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, McCann and Schmuckenschläger [CEMS01] to the setting of the Heisenberg group. Another serious warning is attributed to Juillet [Jui09] who proved that both the Brunn-Minkowski inequality and the curvaturedimension 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 Riemannian approximation of H n 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 [FJ08] and Juillet [Jui09] 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.
1.2. An overview of geometric inequalities. The classical Borell-Brascamp-Lieb inequality in R n states that for any fixed s ∈ (0, 1), p ≥ − 1 n and integrable functions f, g, h : R n → [0, ∞) which satisfy h((1 − s)x + sy) ≥ M p s (f (x), g(y)) for all x, y ∈ R n , (1.1) one has Here and in the sequel, for every s ∈ (0, 1), p ∈ R ∪ {±∞} and a, b ≥ 0, we consider the p-mean 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 [Gar02].
In his Ph.D. Thesis, McCann [McC94, Appendix D] (see also [McC97]) 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, McCann and Schmuckenschläger [CEMS01]. Closely related to the Borell-Brascamp-Lieb inequalities on Riemannian manifolds is the convexity of the entropy functional [CEMS01]. The latter fact served as the starting point of the work of Lott and Villani [LV09] and Sturm [Stu06a,Stu06b] 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 [Stu06b] and Villani [Vil09]. 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 µ w.r.t. m, and N ≥ N. The metric measure space (M, d, m) satisfies the curvature-dimension condition CD(K, N ) for K ∈ R and N ≥ 1 if and only if for every µ 0 , µ 1 ∈ P 2 (M, d) there exists an optimal coupling q of µ 0 = ρ 0 m and µ 1 = ρ 1 m and a geodesic Γ : [0, 1] → P 2 (M, d) joining µ 0 and µ 1 such that for all s ∈ [0, 1] and N ≥ N , 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 [Stu06b] and Ohta [Oht09].
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, McCann and Schmuckenschläger [CEMS01] (see also Bacher [Bac10]), as the Borell-Brascamp-Lieb inequality BBL(K, N ) on (M, d, m) which holds if and only if for all s ∈ (0, 1), p ≥ − 1 N and integ- We would like to emphasize the fact that in [CEMS01] 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 [Bac10], 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), Here Z s (A, B) is the set of s-intermediate points between the elements of the sets A and B w.r.t. the metric d, defined by Z s (A, B) = (x,y)∈A×B Z s (x, y), and As we already pointed out, Juillet [Jui09] 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 [CDPT07, 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 [CDPT07, 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.
1.3. Statement of main results. According to Gromov [Gro96], 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.
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 [Vil09,Vil03] 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 × n-dimensional matrices, obtained as Jacobians of the map x → exp M x (−s∇ M ϕ M (x)). 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(ψ M s )(x), where ψ M s is the interpolant map defined for µ 0 -a.e. x ∈ M as to the Riemannian metric d, and ψ M : M → M is the optimal transport map between the absolutely continuous probability measures µ 0 and µ 1 defined on M minimizing the transportation cost w.r.t. the quadratic cost function d 2 2 . Our first result is an appropriate version of the Jacobian determinant inequality on the Heisenberg group. In order to formulate the precise statement we need to introduce some more notations.
Let s ∈ (0, 1). Hereafter, Z s (A, B) denotes the s-intermediate set associated to the nonempty sets A, B ⊂ H n w.r.t. the Carnot-Carathéodory metric d CC . Note that (H n , d CC ) is a geodesic metric space, thus Z s (x, y) = ∅ for every x, y ∈ H n .
The first application of Theorem 1.1 is an entropy inequality. In order to formulate the result, we recall that for a function U : [0, ∞) → R one defines the U -entropy of an absolutely continuous measure µ w.r.t. L 2n+1 on H n as 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 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 intrinsic curvature-dimension condition on the metric measure space (H n , d CC , L 2n+1 ) for the choice of U R (t) = −t 1− 1 2n+1 , see Corollary 3.3. Further consequences of Theorem 1.2 are also presented for the Shannon entropy in Corollary 3.4.
Another consequence of Theorem 1.1 is the following Borell-Brascamp-Lieb inequality: (1.11) 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 (1.12). The latter case may happen e.g.
The value Θ 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 (Θ A,B ) and τ n s (Θ A,B ) in (1.12) will be given in Section 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 [Jui09, 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, McCann and Schmuckenschläger [CEMS01] on Riemannian manifolds and the results of Ambrosio and Rigot [AR04] and Juillet [Jui09] 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.
Acknowledgements. The authors wish to express their gratitude to Luigi Ambrosio, Nicolas Juillet, Pierre Pansu, Ludovic Rifford, Séverine Rigot and Jeremy Tyson for helpful conversations on various topics related to this paper.

Preliminary results
2.1. 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 thaṫ 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.
Note that both functions f i,s are positive on (0, π), i ∈ {1, 2}. First, for every t ∈ (0, π) one has f 1, 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.
Let s ∈ (0, 1) and x, y ∈ H n be such that x = y. If B(y, r) = {w ∈ H n : d CC (y, w) < r} is the open CC-ball with center y ∈ H n and radius r > 0, we introduce the Heisenberg volume distortion coefficient , sr)) .
The following property gives a formula for the Heisenberg volume distortion coefficient in terms of the Jacobian Jac(Γ s ).
Since x −1 · y / ∈ L, we have that B(x −1 · y, r) ∩ L = ∅ for r small enough, thus the map Γ s • Γ −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 Z s (0 H n , ·) at the point , which gives precisely the claim.
(ii) At first glance, this property seems to be just the symmetric version of (i). Note 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 → which concludes the proof.
For further use (see Proposition 2.3), we consider Corollary 2.1. Let s ∈ (0, 1) and x, y ∈ H n such that x = y. The following properties hold: then v s (x, y) = +∞. 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).
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 [AR04]). 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.

Because of the asymptotic behaviour of the volume of small balls in the Riemannian geometry (see Gallot, Hulin and Lafontaine [GHL87]), we have
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, McCann and Schmuckenschläger [CEMS01].
(ii) Taking into account that v ε 1−s (y, x) = v ε 1−s (0 H n , y −1 · x) = v ε 1−s (0 H n , −x −1 · y) and the ε-geodesic joining 0 H n and −x −1 ·y is given by the curve s → Γ ε s −χ ε e −iθ ε , −θ ε , s ∈ [0, 1], a similar argument works as in Proposition 2.1 (ii). 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 [AR04, 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 Γ 1 comes from (2.1). In fact, according to Figalli and Rifford [FR10], there exists a Borel set C 0 ⊂ suppf of null L 2n+1 -measure such that for every x ∈ suppf \ C 0 , there exists a unique minimizing geodesic from x to ψ(x); this geodesic is represented by (2.16) 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 0 = (εf )m ε to µ ε 1 = (εg)m ε . The existence and uniqueness of such a map is provided by McCann [McC01, Theorem 3.2]. This map is defined by a c ε = (d ε ) 2 /2-concave function ϕ ε via see Ambrosio and Rigot [AR04,p. 292]. Note that we may always assume that ϕ ε (0 H n ) = 0. Due to Cordero-Erausquin, McCann and Schmuckenschläger [CEMS01, Theorem 4.2], there exists a Borel set C ε ⊂ suppf of null m ε -measure such that ψ ε (x) / ∈ cut ε (x) for every x ∈ suppf \C ε . Now we consider the interpolant map (2.17) Using again a left-translation, we equivalently have (2.18) 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).
Remark 2.2. Note that the limiting value of the distortion coefficients in the Riemannian approximation (i.e., (ii) and (iii)) are not the Heisenberg volume distortion coefficients v s (x, y). The appropriate limits are given by v 0 s (x, y), see (2.3). 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 ∈ suppf \ 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.
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 [AR04, 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 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 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 [AR04, Theorem 6.2] of Ambrosio and Rigot, lim 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 which ends the proof of (i).
To prove inequality (ii) we distinguish again two cases.

Proof of main results
3.1. 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 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 Since by (2.10) we have that ) + ε k cπ, the estimate (3.1) assures the existence of R > 0 such that the ball B(0, R) contains the supports of the measures µ s = (ψ s ) # µ 0 and µ ε k 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 [CEMS01, 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.
Note that S l δ ⊆ S l+1 δ for all l ∈ N and ∪ l S l δ = D; the latter property follows by (3.10). 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 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 h(y) (ρ ε k s (y)) 1− 1 2n+1 dm(y).
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 (3.8) we find k δ ∈ N such that 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). (3.12) Since supp(ρ s ) ⊆ ψ s (D), by (3.12) we have that 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 give the inequality in (3.6). Combining the estimates (3.5) and (3.6), we obtain Applying the change of variables y = ψ s (x) and (3.13) we obtain Observe that the function on the left side of the above estimate that multiplies h is s which is in L 1 (dm). 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.(3.14) By the Monge-Ampère equations (3.13) 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. [CEMS01, 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+1 -measure. 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, (3.15) or equivalently (Jac(ψ s )(x)) ≥ τ n 1−s (θ x ) 2n+1 a.e. x ∈ D. (3.16) A direct consequence of (3.15) and (1.7) is the main estimate from the paper of Figalli and Juillet [FJ08] formulated and refined in the following statement: ρ 0 (ψ −1 s (y)) ≤ 1 (1 − s) 2n+3 ρ 0 (ψ −1 s (y)) for µ s -a.e. y ∈ H n .
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 stationary points. Indeed, if x is a stationary point of ψ than it follows that it will be a stationary point for ψ s (x) as well.
(3.18) 3.2. 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.
x ∈ D we have .
Applying directly Theorem 1.1 to U R (t) = −t 1− 1 2n+1 one has Corollary 3.3. (Rényi entropy inequality on H n ) Under the same assumptions as in Theorem 1.1, the following entropy inequality holds: Let U S (t) = t log t for t > 0 and U S (0) = 0; Corollary 3.2 implies the following convexitytype 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: Remark 3.3. The positive concave function w(s) = −2 log((1 − s) 1−s s s ) compensates the lack of convexity of s → Ent U S (µ s |m), s ∈ (0, 1). Notice also that we have 0 < w(s) ≤ log 4 = w 1 2 for every s ∈ (0, 1), and lim s→0 w(s) = lim s→1 w(s) = 0. Remark 3.4. Based on Remark 3.2 we can also define optimal transport based coefficientŝ τ n s,ψ asτ and state a corresponding version of Theorem 1.2 with respect to these coefficients.
3.3. 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.
Proof of Theorem 1.3. We assume that hypotheses of Theorem 1.3 are fulfilled. Let s ∈ (0, 1) and p ≥ − 1 2n+1 . Note that if either H n f = 0 or H n g = 0, the conclusion follows due to our convention concerning the p-mean M p s . 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., (3.20) Let us keep the notations from the proof of Theorem 1.1, by identifying the density functions ρ 0 and ρ 1 of the measures µ 0 and µ 1 with f and g, respectively. Since the Jacobian determinant inequality is equivalent to (1.10), we have that for a.e. x ∈ D. (3.21) Choosing y = ψ(x) in hypothesis (1.11) for points x ∈ D we obtain: (3.22) Since p ≥ − 1 2n+1 , the monotonicity of the p-mean, relationτ n s = s −1 τ n s and inequalities (3.21), (3.22) imply that g(ψ(x)) (τ n s (θ x )) 2n+1 ≥ ρ s (ψ s (x)) for a.e. x ∈ D. Since µ s = (ψ s ) # µ 0 , an integration and change of variables give that Step 2. We assume that the functions f, g are compactly supported and 0 < for every a, b, c, d ≥ 0, s ∈ (0, 1) and p, q ∈ R such that p + q ≥ 0 with η = pq p+q when p and q are not both zero, and η = 0 if p = q = 0.

Geometric aspects of Brunn-Minkowski inequalities on H n
We first notice that different versions of the Brunn-Minkowski inequality have been studied earlier in the setting of the Heisenberg group. In particular, Leonardi and Masnou [LM05] considered the multiplicative Brunn-Minkowski inequality on H n , i.e., if A, B ⊂ H n are compact sets, then 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 [Mon03]; moreover, it fails even for all N > 2n + 1 as shown by Juillet [Jui09]. However, inequality (4.1) holds for the topological dimension N = 2n + 1, see [LM05].
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 Θ 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 Θ 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 which implies that L 2n+1 (A) = L 2n+1 (B) = 0, a contradiction. Fix s ∈ (0, 1) and let n 1−s (θ(y, x)) and c s 2 = sup where A 0 and B 0 are nonempty, full measure subsets of A and B. Since the function θ → τ n s (θ) is increasing on [0, 2π), cf. Lemma 2.1, it turns out that c s 1 =τ n 1−s (Θ A,B ) and c s 2 =τ n s (Θ A,B ). Due to the fact that Θ A,B < 2π, we have 0 < c s 1 , c s 2 < +∞. We now distinguish two cases. Case 2: L 2n+1 (A) = 0 = L 2n+1 (B) or L 2n+1 (A) = 0 = L 2n+1 (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 Θ 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 (3.16) can be explored in order to obtain L 2n+1 (Z s (A, B)) ≥ L 2n+1 (Z s (A, {x})) = L 2n+1 (∪ y∈A Z s (y, x)) ≥ L 2n+1 (ψ s (A)) = A Jac(ψ s )(y)dL 2n+1 (y) where we used that Θ A,{x} ≥ Θ A,B .
The arguments in Theorem 1.4 put the measure contraction property MCP(0, 2n + 3) of Juillet [Jui09, Theorem 2.3] into the right perspective. In particular, it explains the appearance of the somewhat mysterious value 2n + 3 of the exponent: 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 Θ 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 (Θ A,B ) and τ n s (Θ 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) = Γ 1 (χ 0 , 0), i.e., Θ 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 (Θ A,B ) and τ n s (Θ 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 H n , r) and B r = B((0 C n , 1), r) in H n for sufficiently small values of r > 0. 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 Figure 1 for n = 1. More precisely, we have Proposition 4.1. Let A r = B(0 H n , r), B r = B((0 C n , 1), r) and s ∈ (0, 1). Then (i) L 2n+1 (Z s (A r , B r )) = ω(r 3 ) as r → 0; 1 (ii) 2π − Θ Ar,Br = O(r) as r → 0; (iii) τ n s (Θ Ar,Br ) = ω r 1−2n 1+2n as r → 0.
In particular, Proposition 4.1 implies that L 2n+1 (Z s (A r , B r )) L 2n+1 (A r ) → +∞ as r → 0; this is the reason why the weights τ n 1−s (Θ Ar,Br ) and τ n s (Θ Ar,Br ) appear in the Brunn-Minkowski inequality (1.12) in order to compensate the size of Z s (A r , B r ) w.r.t. A r and B r . Quantitatively, the left hand side of (1.12) is L 2n+1 (Z s (A r , B r )) 1 2n+1 = ω r as r → 0, which is in a perfect concordance with the competition of the two sides of (1.12).
Remark 4.3. We notice that instead of Θ A,B in the weighted Brunn-Minkowski inequality, we can use a better quantity depending on the optimal mass transport where the set A 0 is a nonempty, full measure subset of A and ψ is the optimal transport map resulting from the context. SinceΘ A,ψ ≥ Θ A,B and τ n s is increasing, one has τ n s Θ A,ψ ≥ τ n s (Θ A,B ). In this way, one can slightly improve the Brunn-Minkowski inequality (1.12). A further improvement can be obtained by replacing τ n s byτ n s from (3.17).