Dario Prandi

Luca Rizzi
email: luca.rizzi@univ-grenoble-alpes.fr

AND Marcello Seri
email: m.seri@ucl.ac.uk

A SUB-RIEMANNIAN SANTAL Ó FORMULA WITH APPLICATIONS TO ISOPERIMETRIC INEQUALITIES AND FIRST DIRICHLET EIGENVALUE OF HYPOELLIPTIC OPERATORS

Keywords: Classification. 53C17, 53C65, 35P15, 57R30, 35R03, 53C65

A sub-Riemannian Santaló formula with applications to isoperimetric inequalities and first Dirichlet eigenvalue of hypoelliptic operators

Introduction and results

Let (M, g) be a compact connected Riemannian manifold with boundary ∂M . Santaló formula [START_REF] Chavel | Riemannian geometry[END_REF][START_REF] Santaló | Integral geometry and geometric probability[END_REF] is a classical result in integral geometry that describes the Liouville measure µ of the unit tangent bundle U M in terms of the geodesic flow φ t : U M → U M . Namely, for any measurable function F : U M → R we have (1)

U M F µ = ∂M U + q ∂M (v) 0 F (φ t (v))dt g(v, n q )η q (v) σ(q),
where σ is the surface form on ∂M induced by the inward pointing normal vector n, η q is the Riemannian spherical measure on U q M , U + q ∂M is the set of inward pointing unit vectors at q ∈ ∂M and (v) is the exit length of the geodesic with initial vector v. Finally, U M ⊆ U M is the visible set, i.e. the set of unit vectors that can be reached via the geodesic flow starting from points on ∂M .

In the Riemannian setting, (1) allows to deduce some very general isoperimetric inequalities and Dirichlet eigenvalues estimates for the Laplace-Beltrami operator as showed by Croke in the celebrated papers [START_REF] Croke | Some isoperimetric inequalities and eigenvalue estimates[END_REF][START_REF] Croke | A sharp four-dimensional isoperimetric inequality[END_REF][START_REF] Croke | A lower bound for λ1 on manifolds with boundary[END_REF].

The extension of (1) to the sub-Riemannian setting and its consequences are not straightforward for a number of reasons. Firstly, in sub-Riemannian geometry the geodesic flow is replaced by a degenerate Hamiltonian flow on the cotangent bundle. Moreover, the unit cotangent bundle (the set of covectors with unit norm) is not compact, but rather has the topology of an infinite cylinder. Finally, in sub-Riemannian geometry there is not a clear agreement on which is the "canonical" volume, generalizing the Riemannian measure. Another aspect to consider is the presence of characteristic points on the boundary.

In this paper we extend (1) to the most general class of sub-Riemannian structures for which Santaló formula makes sense. As an application we deduce Hardy-like inequalities, sharp universal estimates on the first Dirichlet eigenvalue of the sub-Laplacian and sharp isoperimetric-type inequalities.

To our best knowledge, a sub-Riemannian version of (1) appeared only in [START_REF] Pansu | Une inégalité isopérimétrique sur le groupe de Heisenberg[END_REF] for the three-dimensional Heisenberg group, and more recently in [START_REF] Montefalcone | Some relations among volume, intrinsic perimeter and one-dimensional restrictions of BV functions in Carnot groups[END_REF] for Carnot groups, where the natural global coordinates allow for explicit computations. As far as other sub-Riemannian structures are concerned, Santaló formula is an unexplored technique with potential applications to different settings, including CR (Cauchy-Riemann) and QC (quaternionic contact) geometry, Riemannian foliations, and Carnot groups.

1.1. Setting and examples. Let (N, D, g) be a sub-Riemannian manifold of dimension n, where D ⊆ T N is a distribution that satisfies the bracket-generating condition and g is a smooth metric on D. Smooth sections X ∈ Γ(D) are called horizontal. We consider a compact n-dimensional submanifold M ⊂ N with boundary ∂M = ∅.

If (N, D, g) is Riemannian, we equip it with its Riemannian volume ω R . In the genuinely sub-Riemannian case we fix any smooth volume form ω on M (or a density if M is not orientable). In any case, the surface measure σ = ι n ω on ∂M is given by the contraction with the horizontal unit normal n to ∂M . For what concerns the regularity of the boundary, we assume only that ∂M is piecewise C 1,1 . (See Remark 7 for the Lipschitz case.)

A central role is played by sub-Riemannian geodesics, i.e., curves tangent to D that locally minimize the sub-Riemannian distance between endpoints. In this setting, the geodesic flow 1 is a natural Hamiltonian flow φ t : T * M → T * M on the cotangent bundle, induced by the Hamiltonian function H ∈ C ∞ (T * M ). The latter is a non-negative, degenerate, quadratic form on the fibers of T * M that contains all the information on the sub-Riemannian structure. Length-parametrized geodesics are characterized by an initial covector λ in the unit cotangent bundle U * M = {λ ∈ T * M | 2H(λ) = 1}.

A key ingredient for most of our results is the following reduction procedure. Fix a transverse sub-bundle V ⊂ T M such that T M = D ⊕ V. We define the reduced cotangent bundle T * M r as the set of covectors annihilating V. On T * M r we define a reduced Liouville volume Θ r , which depends on the choice of the volume ω on M . These must satisfy the following stability hypotheses:

(H1) The bundle T * M r is invariant under the Hamiltonian flow φ t ; (H2) The reduced Liouville volume is invariant, i.e. L H Θ r = 0. This allows to replace the non-compact U * M with a compact slice U * M r := U * M ∩T * M r , equipped with an invariant measure (see Section 4.3). These hypotheses are verified for:

• any Riemannian structure, equipped with the Riemannian volume;

• any sub-Riemannian structure associated with a Riemannian foliation with totally geodesic leaves, equipped with the Riemannian volume. These includes contact, CR, QC structures with transverse symmetries, and also some non-equiregular structures as the Martinet one on R 3 . See Section 5.2; • any left-invariant sub-Riemannian structure on a Carnot group 2 , equipped with the Haar volume, see Section 5.1. An interesting example, coming from CR geometry, is the complex Hopf fibration (CHF)

S 1 → S 2d+1 p - → CP d , d ≥ 1,
1 Abnormal geodesics are allowed, but strictly abnormal ones, not given by the Hamiltonian flow on the cotangent bundle, do not play any role in our construction. 2 We stress that Carnot groups are not Riemannian foliations if their step is > 2. where D := (ker p * ) ⊥ is the orthogonal complement of the kernel of the differential of the Hopf map w.r.t. the round metric on S 2d+1 , and the sub-Riemannian metric g is the restriction to D of the round one. Another interesting structure, coming from QC geometry and with corank 3, is the quaternionic Hopf fibration (QHF)

S 3 → S 4d+3 p - → HP d , d ≥ 1,
where HP d is the quaternionic projective space of real dimension 4d and the sub-Riemannian structure on S 4d+3 is defined similarly to its complex version.

1.2. Sub-Riemannian Santaló formulas. Consider a sub-Riemannian geodesic γ(t) with initial covector λ ∈ U * M . The exit length (λ) ∈ [0, +∞) is the length after which γ leaves M by crossing ∂M . Similarly, ˜ (λ) is the minimum between (λ) and the cut length c(λ). That is, after length ˜ (λ) the geodesic either loses optimality or leaves M .

The visible unit cotangent bundle U M ⊂ U * M is the set of unit covectors λ such that (-λ) < +∞. (See Fig. 1.) Analogously, the optimally visible unit cotangent bundle Ũ M is the set of unit covectors such that ˜ (-λ) < +∞.

For any non-characteristic point q ∈ ∂M , we have a well defined inner pointing unit horizontal vector n q ∈ D q , and U + q ∂M ⊂ U * q M is the set of initial covectors of geodesics that, for positive time, are directed toward the interior of M .

As anticipated, we do not consider all the length-parametrized geodesics, i.e. all initial

covectors λ ∈ U * q M S k-1 × R n-k , but a reduced subset U * q M r S k-1 .
In the following the suffix r always denotes the intersection with the reduced unit cotangent bundle U * M r . We stress the critical fact that U * M r is compact, while U * M never is, except in the Riemannian setting where the reduction procedure is trivial. With these basic definitions at hand, we are ready to state the sub-Riemannian Santaló formulas.

Theorem 1 (Reduced Santaló formulas). The visible set U M r and the optimally visible set Ũ M r are measurable. For any measurable function

F : U * M r → R we have U M r F µ r = ∂M U + q ∂M r (λ) 0 F (φ t (λ))dt λ, n q η r q (λ) σ(q), (2) Ũ M r F µ r = ∂M U + q ∂M r ˜ (λ) 0 F (φ t (λ))dt λ, n q η r q (λ) σ(q). (3)
In ( 2)-(3), µ r is a reduced invariant Liouville measure on U * M r , η r q is an appropriate smooth measure on the fibers U * q M r and λ, • denotes the action of covectors on vectors. Indeed both include the Riemannian case, where the reduction procedure is trivial and U * M U M since the Hamiltonian is not degenerate. Remark 1. Hypotheses (H1) and (H2) are essential for the reduction procedure. An unreduced version of Theorem 1 holds for any volume ω and with no other assumptions but the Lipschitz regularity of ∂M (see Theorem 16 and Remark 7). However, the consequences we present do not hold a priori, as their proofs rely on the summability of certain functions on U * M r , generally false on the non-compact U * M . 1.3. Hardy-type inequalities. For any f ∈ C ∞ (M ), let ∇ H f ∈ Γ(D) be the horizontal gradient: the horizontal direction of steepest increase of f . It is defined via the identity ( 4)

g(∇ H f, X) = df (X), ∀X ∈ Γ(D).
Consider all length-parametrized sub-Riemannian geodesic passing through a point q ∈ M , with covector λ ∈ U * q M . Set L(λ) := (λ) + (-λ); this is the length of the maximal geodesic that passes through q with covector λ.

Proposition 2 (Hardy-like inequalities). For any

f ∈ C ∞ 0 (M ) it holds M |∇ H f | 2 ω ≥ kπ 2 |S k-1 | M f 2 R 2 ω, ( 5
) M |∇ H f | 2 ω ≥ k 4|S k-1 | M f 2 r 2 ω, ( 6
)
where k = rank D and r, R :

M → R are: 1 R 2 (q) := U * q M r 1 L 2 η r q , 1 r 2 (q) := U * q M r 1 2 η r q , ∀q ∈ M.
We observe that r is the harmonic mean distance from the boundary defined in [START_REF] Davies | A review of Hardy inequalities[END_REF]. One can also consider the following generalization of Proposition 2 for L p (M, ω) norms.

Proposition 3 (p-Hardy-like inequality). Let p > 1 and f

∈ C ∞ 0 (M ). Then M |∇ H f | p ω ≥ π p p C p,k M |f | p R p ω, (7) M |∇ H f | p ω ≥ p -1 p p C p,k M |f | p r p ω, ( 8
)
where k = rank D, the constants π p and C p,k are

π p = 2π(p -1) 1/p p sin(π/p) , C p,k = k |S k-1 | √ π Γ( k+p 2 ) 2Γ( 1+p 2 )Γ( k 2 + 1)
,

and r p , R p : M → R are 1 R p (q) := U * q M r 1 L p η r q , 1 r p (q) := U * q M r 1 p η r q , ∀q ∈ M.
1.4. Lower bound for the first Dirichlet eigenvalue. For any given smooth volume ω, a fundamental operator in sub-Riemannian geometry is the sub-Laplacian ∆ ω , playing the role of the Laplace-Beltrami operator in Riemannian geometry. Under the bracketgenerating condition, this is an hypoelliptic operator on L 2 (M, ω). Its principal symbol is (twice) the Hamiltonian, thus the Dirichlet spectrum of -∆ ω on the compact manifold M is positive and discrete. We denote it

0 < λ 1 (M ) ≤ λ 2 (M ) ≤ . . . .
As a consequence of Proposition 2 and the min-max principle, we obtain a universal lower bound for the first Dirichlet eigenvalue λ 1 (M ) on the given domain. Here by universal we mean an estimate not requiring any assumption on curvature or capacity.

Proposition 4 (Universal spectral lower bound). Let L = sup λ∈U * M r L(λ) be the length of the longest reduced geodesic contained in M . Then, letting k = rank D,

(9) λ 1 (M ) ≥ kπ 2 L 2 ,
where we set the r.h.s. to 0 if L = +∞.

Remark 2. In ( 9), L cannot be replaced by the sub-Riemannian diameter, as M might contain very long (non-minimizing) geodesics, for example closed ones, and L = +∞. See Appendix B for more details.

In the Riemannian case, as noted by Croke, we attain equality in [START_REF] Baudoin | Poincaré, and isoperimetric inequalities for subelliptic diffusion operators satisfying a generalized curvature dimension inequality[END_REF] when M is the hemisphere of the Riemannian round sphere. We prove the following extension to the sub-Riemannian setting. all equipped with the Riemannian volume of the corresponding round sphere. In all these cases, L = π and λ 1 (M ) = d, 2d or 4d, respectively. Moreover, the associated eigenfunction is Ψ = cos(δ), where δ is the Riemannian distance from the north pole. Remark 3. The Riemannian volume of the sub-Riemannian Hopf fibrations coincides, up to a constant factor, with their Popp volume [START_REF] Barilari | A formula for Popp's volume in sub-Riemannian geometry[END_REF][START_REF] Montgomery | A tour of subriemannian geometries, their geodesics and applications[END_REF], an intrinsic smooth measure in sub-Riemannian geometry. This is proved for 3-Sasakian structures (including the QHF) in [START_REF] Rizzi | Sub-Riemannian Ricci curvatures and universal diameter bounds for 3-Sasakian structures[END_REF]Prop. 34] and can be proved exactly in the same way for Sasakian structures (including the CHF) using the explicit formula for Popp volume of [START_REF] Barilari | A formula for Popp's volume in sub-Riemannian geometry[END_REF]. For the case (i) ∆ ω is the Laplace-Beltrami operator. For the cases (ii) and (iii) ∆ ω is the standard sub-Laplacian of CR and QC geometry, respectively.

In principle, L can be computed when the reduced geodesic flow is explicit. This is the case for Carnot groups, where reduced geodesics passing through the origin are simply straight lines (they fill a k-plane for rank k Carnot groups). It turns out that, in this case, L = diam H (M ) (the horizontal diameter, that is the diameter of the set M measured through left-translations of the aforementioned straight lines). Thus [START_REF] Baudoin | Poincaré, and isoperimetric inequalities for subelliptic diffusion operators satisfying a generalized curvature dimension inequality[END_REF] gives an easily computable lower bound for the first Dirichlet eigenvalue in terms of purely metric quantities. Corollary 6. Let M be a compact n-dimensional submanifold with piecewise C 1,1 boundary of a Carnot group of rank k, with the Haar volume. Then, [START_REF] Baudoin | The subelliptic heat kernel on the CR sphere[END_REF] λ

1 (M ) ≥ kπ 2 diam H (M ) 2 ,
where diam H (M ) denotes the horizontal diameter of M .

In particular, if M is the metric ball of radius R, we obtain λ 1 (M ) ≥ kπ 2 /(2R) 2 . Clearly [START_REF] Baudoin | The subelliptic heat kernel on the CR sphere[END_REF] is not sharp, as one can check easily in the Euclidean case. 1.5. Isoperimetric-type inequalities. In this section we relate the sub-Riemannian area and perimeter of M with some of its geometric properties. Since M is compact, the sub-Riemannian diameter diam(M ) can be characterized as the length of the longest optimal geodesic contained in M . Analogously, the reduced sub-Riemannian diameter diam r (M ) is the length of the longest reduced optimal geodesic contained in M . Indeed diam r (M ) ≤ diam(M ).

Consider all reduced geodesics passing through q ∈ M with covector λ. Some of them originate from the boundary ∂M , that is (-λ) < +∞; others do not, i.e. (-λ) = +∞. The relative ratio of the lengths of these two types of geodesics (w.r.t. an appropriate measure on U * q M r ) is called the visibility angle ϑ q ∈ [0, 1] at q (see Definition 6). Roughly speaking, if ϑ q = 1 then any geodesic passing through q will hit the boundary and, on the opposite, if it is equal to 0 then q is not visible from the boundary (see Fig. 2). Similarly, we define the optimal visibility angle θ q by replacing (-λ) with ˜ (-λ). Finally, the least visibility angle is ϑ := inf q∈M ϑ q , and similarly for the least optimal visibility angle θ := inf q∈M θ q .

Proposition 7 (Isoperimetric-type inequalities). Let := sup{ (λ) | λ ∈ U * q M r , q ∈ ∂M } be the length of the longest reduced geodesic contained in M starting from the boundary ∂M . Then [START_REF] Baudoin | The subelliptic heat kernels of the quaternionic Hopf fibration[END_REF] σ(∂M )

ω(M ) ≥ C ϑ and σ(∂M ) ω(M ) ≥ C θ diam r (M ) ,
where

C = 2π|S k-1 |/|S k | and we set the r.h.s. to 0 if = +∞.
The equality in [START_REF] Baudoin | The subelliptic heat kernels of the quaternionic Hopf fibration[END_REF] holds for the hemisphere of the Riemannian round sphere, as pointed out in [START_REF] Croke | Some isoperimetric inequalities and eigenvalue estimates[END_REF]. We have the following generalization to the sub-Riemannian setting. where ω is the Riemannian volume of the corresponding round sphere. In all these cases ϑ = θ = 1 and = diam r (M ) = π.

We can apply Proposition 7 to Carnot groups equipped with the Haar measure. In this case ϑ = θ = 1 and = diam r (M ) = diam H (M ). Moreover, ω is the Lebesgue volume of R n and σ is the associated perimeter measure of geometric measure theory [START_REF] Capogna | An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem[END_REF].

Corollary 9.

Let M be a compact n-dimensional submanifold with piecewise C 1,1 boundary of a Carnot group of rank k, with the Haar volume. Then,

σ(∂M ) ω(M ) ≥ 2π|S k-1 | |S k | diam H (M )
,

where diam H (M ) is the horizontal diameter of the Carnot group.

This inequality is not sharp even in the Euclidean case, but it is very easy to compute the horizontal diameter for explicit domains. For example, if M is the sub-Riemannian metric ball of radius R, then diam H (M ) = 2R.

1.6. Remark on change of volume. Fix a sub-Riemannian structure (N, D, g), a compact set M with piecewise C 1,1 boundary and a complement V such that (H1) holds. Now assume that, for some choice of volume form ω, also (H2) is satisfied, so that we can carry on with the reduction procedure and all our results hold. One can derive the analogous of Propositions 2, 3, 4, 7 for any other volume ω = e ϕ ω, with ϕ ∈ C ∞ (M ). In all these results, it is sufficient to multiply the r.h.s. of the inequalities by the volumetric constant 0 < α ≤ 1 defined as α := min e ϕ max e ϕ , and indeed replace ω with ω = e ϕ ω in Propositions 2 and 3, σ with σ = e ϕ σ in Proposition 7, and the sub-Laplacian ∆ ω with ∆ ω = ∆ ω + dϕ, ∇ H • in Proposition 4. Analogously, one can deal with the Corollaries 6 and 9 about Carnot groups.

This remark allows, for example, to obtain results for (sub-)Riemannian weighted measures. This is particularly interesting in the genuinely sub-Riemannian setting since, in some cases, the volume satisfying (H2) might not coincide with the intrinsic Popp one. 1.7. Remark on rigidity. The sharpness results of Propositions 5 and 8 hold for hemispheres of (sub-)Riemannian structures associated with Riemannian submersions of the sphere with totally geodesic fibers, which have been completely classified in [START_REF] Escobales | Riemannian submersions with totally geodesic fibers[END_REF]. The only case which is not covered in these propositions is the so-called octonionic Hopf fibration (OHF) S 7 → S 15 → OP 1 , which to our best knowledge has not yet been studied from the sub-Riemannian point of view, and for which explicit expressions for the sub-Laplacian do not appear in the literature. It is however likely that the sharpness results of Propositions 5 and 8 hold also for the hemisphere S 15 + of the sub-Riemannian OHF. Finally, concerning the universal lower bound of Proposition 4, Croke proved the following rigidity result in the Riemannian case (see [START_REF] Croke | Some isoperimetric inequalities and eigenvalue estimates[END_REF]Thm. 16]). As we already remarked, the lower bound ( 9) is non-trivial if and only if all geodesics starting from points of M hit the boundary at some finite time (i.e. ϑ = 1). If, furthermore, every such geodesic minimizes distance up to the point of intersection with the boundary (i.e. θ = 1), then we have equality in [START_REF] Baudoin | Poincaré, and isoperimetric inequalities for subelliptic diffusion operators satisfying a generalized curvature dimension inequality[END_REF] if and only if M is an hemisphere of the round sphere. See also [START_REF] Croke | A lower bound for λ1 on manifolds with boundary[END_REF] for a more general rigidity result. The following question is thus natural.

Open question. Are the hemispheres of the CHF, QHF, and possibly OHF, the only domains on compact sub-Riemannian manifolds tamed by a foliation with totally geodesic leaves (see Section 5.2) where the lower bound of Proposition 4 is attained? 1.8. Afterwords and further developments. Despite its broad range of applications in Riemannian geometry and its Finsler generalizations [START_REF] Yuan | Some formulas of Santaló type in Finsler geometry and its applications[END_REF], only a few works used Santaló formula in the hypoelliptic setting, all of them in the specific case of Carnot groups [START_REF] Montefalcone | Some relations among volume, intrinsic perimeter and one-dimensional restrictions of BV functions in Carnot groups[END_REF][START_REF] Pansu | Une inégalité isopérimétrique sur le groupe de Heisenberg[END_REF] or 3D Sasakian structures [START_REF] Chanillo | Isoperimetric inequalities & volume comparison theorems on CR manifolds[END_REF]. It is interesting to notice that, in [START_REF] Pansu | Une inégalité isopérimétrique sur le groupe de Heisenberg[END_REF], Pansu was able to use Santaló formula in pairs with minimal surfaces to eliminate the diameter term in Corollary 9 and obtain his celebrated isoperimetric inequality. In our general setting, this is something worth investigating.

The study of spectral properties of hypoelliptic operators is an active area of research. Many results are available for the complete spectrum of the sub-Laplacian on closed manifolds (with no boundary conditions). We recall [START_REF] Baudoin | The subelliptic heat kernel on SU(2): representations, asymptotics and gradient bounds[END_REF][START_REF] Baudoin | The subelliptic heat kernel on the CR sphere[END_REF][START_REF] Baudoin | The subelliptic heat kernels of the quaternionic Hopf fibration[END_REF] for the case of SU(2), CHF and QHF. Furthermore, in [START_REF] De Verdière | Spectral asymptotics for sub-Riemannian Laplacians. I: quantum ergodicity and quantum limits in the 3D contact case[END_REF], one can find the spectrum of the "flat Heisenberg case" (a compact quotient of the Heisenberg group) together with quantum ergodicity results for 3D contact sub-Riemannian structures. Lower bounds for the first (non-zero) eigenvalue of the sub-Laplacian on closed foliation, under curvature-like assumptions, appeared in [START_REF] Baudoin | Poincaré, and isoperimetric inequalities for subelliptic diffusion operators satisfying a generalized curvature dimension inequality[END_REF] (see also [START_REF] Baudoin | Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations[END_REF] for a more general statement).

Concerning the Dirichlet spectrum on Riemannian manifolds with boundary, a classical reference is [START_REF] Chavel | Eigenvalues in Riemannian geometry[END_REF]. In the sub-Riemannian setting, we are aware of results for the sum of Dirichlet eigenvalues [START_REF] Strichartz | Estimates for sums of eigenvalues for domains in homogeneous spaces[END_REF] by Strichartz and related spectral inequalities [START_REF] Hansson | Sharp spectral inequalities for the Heisenberg Laplacian[END_REF] by Hannson and Laptev, both for the case of the Heisenberg group. To our best knowledge, Proposition 4 is the first sharp universal lower bound for the first Dirichlet eigenvalue in the sub-Riemannian setting and in particular for non-Carnot structures.

The study of Hardy's inequalities, already in the Euclidean setting, ranges across the last century and continues to the present day (see [START_REF] Barbatis | On the Hardy Constant of Some Non-convex Planar Domains[END_REF][START_REF] Brezis | Hardy's inequalities revisited[END_REF][START_REF] Ekholm | Hardy inequalities for p-Laplacians with Robin boundary conditions[END_REF] and references therein). The sub-Riemannian case is more recent, for an account of the known result we mention the works for Carnot groups of Capogna, Danielli and Garofalo (see e.g. [START_REF] Capogna | An isoperimetric inequality and the geometric Sobolev embedding for vector fields[END_REF][START_REF] Danielli | Hardy-Sobolev type inequalities with sharp constants in Carnot-Carathéodory spaces[END_REF]).

Poincaré inequalities are strictly connected to Hardy's ones. On this subject the literature is again huge, we already mentioned the works of Croke and Derdzinski concerning the Riemannian case [START_REF] Croke | Some isoperimetric inequalities and eigenvalue estimates[END_REF][START_REF] Croke | A sharp four-dimensional isoperimetric inequality[END_REF][START_REF] Croke | A lower bound for λ1 on manifolds with boundary[END_REF]. Finally, see [START_REF] Ivanov | The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds[END_REF] for results on CR and QC manifolds under Ricci curvature assumptions in the spirit of the Lichnerowicz-Obata theorem.

In this paper we focused mostly on foliations, where our results are sharp. For Carnot groups, Corollaries 6 and 9 appeared in [START_REF] Montefalcone | Some relations among volume, intrinsic perimeter and one-dimensional restrictions of BV functions in Carnot groups[END_REF] and are not sharp. Let us consider for simplicity the 3D Heisenberg group, with coordinates (x, y, z) ∈ R 3 . A relevant class of domains for the Dirichlet eigenvalues problem are the "Heisenberg cubes" [0, ε] × [0, ε] × [0, ε 2 ], obtained by non-homogeneous dilation of the unit cube [0, 1] 3 . These represent a fundamental domain for the quotient H 3 /εΓ of the 3D Heisenberg group H 3 by the (dilation of the) integer Heisenberg subgroup Γ (a lattice). This is the basic example of nilmanifold, (we thank R. Montgomery for pointing out this example). For these fundamental domains, the first Dirichlet eigenvalue is unknown. However, we mention that for any Carnot group the reduction technique developed here can be further improved leading to a λ 1 estimate for cubes, via the technique sketched in Appendix B.) 1.9. Structure of the paper. In Section 2 we recall some basic definitions about sub-Riemannian geometry and sub-Laplacians. In Section 3 we introduce some preliminary constructions concerning integration on vector bundles that we need for the reduction procedure. In Section 4 we prove the main result of the paper, namely the reduced Santaló formula. Section 5 is devoted to examples, and contains the general class of structures where our construction can be carried out. Finally, in Section 6 we apply the reduced Santaló formula to prove Poincaré, Hardy, and isoperimetric-type inequalities.

Sub-Riemannian geometry

We give here only the essential ingredients for our analysis; for more details see [START_REF] Agrachev | Introduction to Riemannian and Sub-Riemannian geometry from Hamiltonian viewpoint[END_REF][START_REF] Montgomery | A tour of subriemannian geometries, their geodesics and applications[END_REF][START_REF] Rifford | Sub-Riemannian geometry and optimal transport[END_REF]. A sub-Riemannian manifold is a triple (M, D, g), where M is a smooth, connected manifold of dimension n ≥ 3, D is a vector distribution of constant rank k ≤ n and g is a smooth metric on D. We assume that the distribution is bracket-generating, that is

span{[X i 1 , [X i 2 , [. . . , [X i m-1 , X im ]]]] | m ≥ 1} q = T q M, ∀q ∈ M,
for some (and thus any) set X 1 , . . . , X k ∈ Γ(D) of local generators for D.

A horizontal curve γ : [0, T ] → R is a Lipschitz continuous path such that γ(t) ∈ D γ(t) for almost any t. Horizontal curves have a well defined length

(γ) = T 0 g( γ(t), γ(t))dt.
Furthermore, the sub-Riemannian distance is defined by:

d(x, y) = inf{ (γ) | γ(0) = x, γ(T ) = y, γ horizontal}.
By the Chow-Rashevskii theorem, under the bracket-generating condition, d is finite and continuous. Sub-Riemannian geometry includes the Riemannian one, when D = T M .

2.1. Sub-Riemannian geodesic flow. Sub-Riemannian geodesics are horizontal curves that locally minimize the length between their endpoints. Let π : T * M → M be the cotangent bundle. The sub-Riemannian Hamiltonian H :

T * M → R is H(λ) := 1 2 k i=1 λ, X i 2 , λ ∈ T * M, where X 1 , . . . , X k ∈ Γ(D)
U * M = {λ ∈ T * M | 2H(λ) = 1}.
It is a fiber bundle with fiber

U * q M = S k-1 × R n-k . For λ ∈ U * q M , the curve γ λ (t) is a length-parametrized geodesic with length (γ| [t 1 ,t 2 ] ) = t 2 -t 1 .
Remark 4. There is also another class of minimizing curves, called abnormal, that might not follow the Hamiltonian dynamic of [START_REF] Brezis | Hardy's inequalities revisited[END_REF]. Abnormal geodesics do not exist in Riemannian geometry, and they are all trivial curves in some basic but popular classes of sub-Riemannian structures (e.g. fat ones). Our construction takes in account only the normal sub-Riemannian geodesic flow, hence abnormal geodesics are allowed, but ignored. Some hard open problems in sub-Riemannian geometry are related to abnormal geodesics [1,[START_REF] Montgomery | A tour of subriemannian geometries, their geodesics and applications[END_REF][START_REF] Le Donne | Sard property for the endpoint map on some Carnot groups[END_REF].

2.2. The intrinsic sub-Laplacian. Let (M, D, g) be a compact sub-Riemannian manifold with piecewise C 1,1 boundary ∂M , and ω ∈ Λ n M be any smooth volume form (or a density, if M is not orientable). We define the Dirichlet energy functional as

E(f ) = M 2H(df ) ω, f ∈ C ∞ 0 (M ).
The Dirichlet energy functional induces the operator -∆ ω on L 2 (M, ω). Its Friedrichs extension is a non-negative self-adjoint operator on L 2 (M, ω) that we call the Dirichlet sub-Laplacian. Its domain is the space H 1 0 (M ), the closure in the H 1 (M ) norm of the space C ∞ 0 (M ) of smooth functions that vanish on ∂M . Since ∇ H f 2 = 2H(df ), for smooth functions we have

∆ ω f = div ω (∇ H f ), ∀f ∈ C ∞ 0 (M )
, where the divergence is computed w.r.t. ω, and ∇ H is the horizontal gradient defined by [START_REF] Balogh | Size of tangencies to non-involutive distributions[END_REF]. The spectrum of -∆ ω is discrete and positive,

0 < λ 1 (M ) ≤ λ 2 (M ) ≤ . . . → +∞.
In particular, by the min-max principle we have [START_REF] Capogna | An isoperimetric inequality and the geometric Sobolev embedding for vector fields[END_REF] λ

1 (M ) = inf E(f ) f ∈ C ∞ 0 (M ), M |f | 2 ω = 1 .

Preliminary constructions

We discuss some preliminary constructions concerning integration on vector bundles that we need for the reduction procedure. In this section π : E → M is a rank k vector bundle on an n dimensional manifold M . For simplicity we assume M to be oriented and E to be oriented (as a vector bundle). If not, the results below remain true replacing volumes with densities. We use coordinates

x on O ⊂ M and (p, x) ∈ R k × R n on U = π -1 (O) such that the fibers are E q 0 = {(p, x 0 ) | p ∈ R k }. In a compact notation we write, in coordinates, dp = dp 1 ∧ . . . ∧ dp k and dx = dx 1 ∧ . . . ∧ dx n .
3.1. Vertical volume forms. Consider the fibers E q ⊂ E as embedded submanifolds of dimension k. For each λ ∈ E q , let Λ k (T λ E q ) be the space of alternating multi-linear functions on T λ E q . The space

Λ k v (E) := λ∈E Λ k (T λ E π(λ) ) defines a rank 1 vector bundle Π : Λ k v (E) → E, such that Π(η) = λ if η ∈ Λ k (T λ E π(λ) ). To see this, choose coordinates (p, x) ∈ R k × R n on U = π -1 (O) such that the fibers are E q 0 = {(p, x 0 ) | p ∈ R k }. Thus the vectors ∂ p 1 , . . . , ∂ p k tangent to the fibers E q are well defined. The map Ψ : Π -1 (U ) → U × R, defined by Ψ(η) = (Π(η), η(∂ p 1 , . . . , ∂ p k )
) is a bijection. Suppose that (U , p , x ) is another chart, and similarly Ψ :

Π -1 (U ) → U × R. Then, on Π -1 (U ∩ U ) × R we have Ψ • Ψ -1 (λ, α) = (λ, det(∂q /∂q)).
∈ Γ(Λ k v (E)
) is called a vertical volume form on E. In particular, the restriction ν q := ν| Eq of a vertical volume form defines a measure on each fiber E q .

Lemma 10 (Disintegration 1). Fix a volume form Ω ∈ Λ n+k (E) and a volume form ω ∈ Λ n (M ) on the base space. Then there exists a unique vertical volume form ν ∈ Λ k v (E) such that, for any measurable set D ⊆ E and measurable f :

D → R, ( 14
) D f Ω = π(D ) D q f q ν q ω(q), f q := f | Eq , D q := E q ∩ D . If, in coordinates, Ω = Ω(p, x)dp ∧ dx and ω = ω(x)dx, then ν| (p,x) = Ω(p, x) ω(x) dp.
Proof. The last formula does not depend on the choice of coordinates (p, x) on E. So we can use this as a definition for ν. Moreover, in coordinates,

Ω| (p,x) = Ω(p, x)dp ∧ dx = Ω(p, x) ω(x) dp ∧ (ω(x)dx).
Both uniqueness and ( 14) follow from the definition of integration on manifolds and Fubini theorem.

Vertical surface forms. Let

E ⊂ E be a corank 1 sub-bundle of π : E → M .
That is, a submanifold E ⊂ E such that π| E : E → M is a bundle, and the fibers E q := π -1 (q) ∩ E ⊂ E q are diffeomorphic to a smooth hypersurface C ⊂ R k . As a matter of fact, we will only consider the cases in which C is a cylinder or a sphere. Fix a smooth volume form Ω ∈ Λ n+k (E). The Euler vector field is the generator of homogenous dilations on the fibers λ → e α λ, for all α ∈ R. In coordinates (p, x) on E we have e = n i=1 p i ∂ p i . If e is transverse to E we induce a volume form on E by µ := ι e Ω. In this setting, a volume form µ ∈ Λ n+k-1 (E ) is called a surface form. For any vertical volume form ν ∈ Λ k v (E), we define a measure on the fibers E q as η q = ι e ν| Eq . With an abuse of language, we will refer to such measures as vertical surface forms.

Lemma 11 (Disintegration 2). Fix a surface form µ = ι e Ω ∈ Λ n+k-1 (E ) and a volume form ω ∈ Λ n (M ) on the base space. For any measurable set D ⊆ E and measurable f :

D → R, ( 15
) D f µ = π(D) Dq f q η q ω(q), f q := f | E q , D q := E q ∩ D.
Here, η q = ι e ν| Eq and ν is the vertical volume form on E defined in Lemma 10.

Proof. Choose coordinates (p, x) on E. As in the proof of Lemma 10

µ| (p,x) = ι e Ω| (p,x) = Ω(p, x) ι e dp ∧ dx + (-1) k dp ∧ ι e dx = Ω(p, x)ι e dp ∧ dx = Ω(p, x) ω(x) ι e dp ∧ (ω(x)dx).
Thus [START_REF] Chanillo | Isoperimetric inequalities & volume comparison theorems on CR manifolds[END_REF] holds with η| (p,x) = Ω(p,x) ω(x) ι e dp. This, together with the local expression of ν in Lemma 10, yields η = ι e ν.

Example 1 (The unit cotangent bundle). We apply the above constructions to E = T * M and E = U * M . In this case E q = U * q M are diffeomorphic to cylinders (or spheres, in the Riemannian case). Moreover, we set Ω = Θ, the Liouville volume form, and µ = ι e Θ, the Liouville surface form. 3 One can check that Θ = dµ.

Let ν ∈ Λ n v (T * M ) and η = ι e ν as in Lemmas 10 and 11. In canonical coordinates,

Θ = dp ∧ dx. Then, if ω = ω(x)dx, ν = 1 ω(x) dp and η = 1 ω(x) n i=1 (-1) i-1 p i dp 1 ∧ . . . ∧ dp i ∧ . . . ∧ dp n . Choose coordinates x around q 0 ∈ M such that ∂ x 1 | q 0 , . . . , ∂ x k | q 0 is
an orthonormal basis for the sub-Riemannian distribution D q 0 . In the associated canonical coordinates we have

U * q 0 M = {(p, x 0 ) ∈ R 2n | p 2 1 + . . . + p 2 k = 1} S k-1 × R n-k .
In this chart, η q 0 is the (n -1)-volume form of the above cylinder times 1/ω(x 0 ).

Remark 5. This construction gives a canonical way to define a measure on U * M and its fibers in the general sub-Riemannian case, depending only on the choice of the volume ω on the manifold M . It turns out that this measure is also invariant under the Hamiltonian flow. Notice though that in the sub-Riemannian setting, fibers have infinite volume.

3.3.

Invariance. Here we focus on the case of interest where E ⊆ T * M is a rank k vector sub-bundle and E ⊂ E is a corank 1 sub-bundle as defined in Section 3.2. We stress that E is not necessarily a vector sub-bundle, but typically its fibers are cylinders or spheres.

Recall that the sub-Riemannian geodesic flow φ t : T * M → T * M is the Hamiltonian flow of H : T * M → R. Moreover, in our picture, M ⊂ N is a compact submanifold with boundary ∂M of a larger manifold N , with dim M = dim N = n.

Definition 2. A sub-bundle E ⊆ T * M is invariant if φ t (λ) ∈ E for all λ ∈ E and t such that φ t (λ) ∈ T * M is defined. A volume form Ω ∈ Λ n+k (E) is invariant if L H Ω = 0.
Our definition includes the case of interest for Santaló formula, where sub-Riemannian geodesics may cross ∂M = ∅. In other words, E is invariant if the only way to escape from E through the Hamiltonian flow is by crossing the boundary π -1 (∂M ). Moreover, if Ω is an invariant volume on an invariant sub-bundle E, then φ * t Ω = Ω. Lemma 12 (Invariant induced measures). Let E ⊆ T * M be an invariant vector bundle with an invariant volume Ω. Let E ⊂ E be a corank 1 invariant sub-bundle. Let e be a vector field transverse to E and µ = ι e Ω the induced surface form on E . Then µ is invariant if and only if [ H, e] is tangent to E . Let q ∈ ∂M such that the tangent space is well defined. We say that q is a characteristic point if D q ⊆ T q ∂M . If q is non-characteristic, the horizontal normal at q is the unique inward pointing unit vector n q ∈ D q orthogonal to T q ∂M ∩D q . If q ∈ ∂M is characteristic, we set n q = 0. We call C(∂M ) the set of characteristic points. The size of C(∂M ) has been studied in [START_REF] Derridj | Sur un théorème de traces[END_REF][START_REF] Balogh | Size of tangencies to non-involutive distributions[END_REF] under various regularity assumptions on ∂M . We give a self-contained proof of the negligibilty of C(∂M ), which we need in the following. The C 1,1 regularity assumption cannot be weakened to C 1,α , with 0 < α < 1, as shown in [START_REF] Balogh | Size of characteristic sets and functions with prescribed gradient[END_REF]Thm. 1.4].

In

Proposition 13.

Let ∂M be piecewise C 1,1 . Then, the set of characteristic points C(∂M ) has zero measure in ∂M .

Proof. Without loss of generality we assume that N = R n and that locally ∂M is the

graph of a C 1,1 function f : R n-1 → R. Let also u(x, z) = z -f (x), so that locally ∂M = {(x, z) ∈ R n-1 × R | u(x, z) = 0}.
Let à ⊂ R n-1 be measurable with positive measure, and let A = {(x, f (x)) | x ∈ Ã} ⊆ ∂M . We claim that if X, Y are smooth vector fields (not necessarily horizontal), tangent to ∂M a.e. on A, then also [X, Y ] is tangent to ∂M a.e. on A. Notice that X is tangent to ∂M a.e. on A if and only if X(u)(x, f (x)) = 0 for a.e. x ∈ Ã. Consider the Lipschitz function ξ(x) := X(u)(x, f (x)). As a consequence of coarea formula [START_REF] Evans | Measure theory and fine properties of functions[END_REF], we have

à |∇ξ(x)| dx = R H n-2 ( à ∩ ξ -1 (t)) dt = 0,
where |∇ξ| is the norm of the Euclidean gradient of ξ : R n-1 → R, and H n-2 is the Hausdorff measure. In particular, ∇ξ = 0 a.e. on Ã. Since Y is tangent to ∂M a.e. on A, the above identity yields that Y (X(u)) = 0 a.e. on A. A similar argument shows that also

Y (X(u)) = 0 a.e. on A. Since [X, Y ](u)(x, f (x)) = X(Y (u))(x, f (x)) -Y (X(u))(x, f (x))
for a.e. x ∈ R n-1 , we have that [X, Y ] is tangent to ∂M a.e. on A, as claimed.

Assume by contradiction that C(∂M ) has positive measure. In particular, applying the above claim to any pair X, Y ∈ Γ(D), and A = C(∂M ), we obtain that [X, Y ] is tangent to ∂M a.e. on C(∂M ). Since [X, Y ] ∈ Γ(T M ), we can apply the claim a finite number of times, obtaining that any iterated Lie bracket of elements of Γ(D) is tangent to ∂M a.e. on C(∂M ). This contradicts the bracket-generating assumption. 4.2. (Sub-)Riemannian Santaló formula. For any covector λ ∈ U * q M , the exit length (λ) is the first time t ≥ 0 at which the corresponding geodesic γ λ (t) = π • φ t (λ) leaves M crossing its boundary, while ˜ (λ) is the smallest between the exit and the cut length along γ λ (t). Namely

(λ) = sup{t ≥ 0 | γ λ (t) ∈ M }, ˜ (λ) = sup{t ≤ (λ) | γ λ | [0,t] is minimizing}.
We also introduce the following subsets of the unit cotangent bundle π : U * M → M :

U + ∂M = {λ ∈ U * M | ∂M | λ, n > 0} , U M = {λ ∈ U * M | (-λ) < +∞}, Ũ M = {λ ∈ U M | ˜ (-λ) = (-λ)}.
Some comments are in order. The set U + ∂M consists of the unit covectors λ ∈ π -1 (∂M ) such that the associated geodesic enters the set M for arbitrary small t > 0. The visible set U M is the set of covectors that can be reached in finite time starting from π -1 (∂M ) and following the geodesic flow. If we restrict to covectors that can be reached optimally in finite time, we obtain the optimally visible set Ũ M (see Fig. 1).

Lemma 14. The cut-length c : U * M → (0, +∞] is upper semicontinuous (and hence measurable). Moreover, if any couple of distinct points in M can be joined by a minimizing non-abnormal geodesic, c is continuous.

Proof. The result follows as in [17, Thm. III.2.1]. We stress that the key part of the proof of the second statement is the fact that, in absence of non-trivial abnormal minimizers, a point is in the cut locus of another if and only if (i) it is conjugate along some minimizing geodesic or (ii) there exist two distinct minimizing geodesics joining them.

Lemma 15. The exit length : U + ∂M → (0, +∞] is lower semicontinuous (and hence measurable). Moreover, ˜ :

U + ∂M → (0, +∞] is measurable. Proof. Let λ 0 ∈ U + ∂M . Consider a sequence λ n such that lim inf λ→λ 0 (λ) = lim n (λ n ).
Then, the trajectories γ n (t) = π • φ t (λ n ) for t ∈ [0, (λ n )] converge uniformly as n → +∞ to the trajectory γ 0 (t) = φ t (λ 0 ) for t ∈ [0, δ] where δ = lim n (λ n ). Moreover, by continuity of ∂M and the fact that γ n ( (λ n )) ∈ ∂M , it follows that γ 0 (δ) ∈ ∂M . This proves that δ ≥ (λ 0 ), proving the first part of the statement.

To complete the proof, observe that ˜ = min{ , c}, which are measurable by the previous claim and Lemma 14.

Fix a volume form ω on M (or density, if M is not orientable). In any case, ω and σ := ι n ω induce positive measures on M and ∂M , respectively. According to Lemmas 10 and 11, these induce measures ν q and η q = ι e ν q on T * q M and U * q M , respectively. Proof. Let A ⊂ [0, +∞) × U + ∂M be the set of pairs (t, λ) such that 0 < t < (λ). By Lemma 15 it follows that A is measurable. Let also Z = π -1 (∂M ) ⊂ U M which has zero measure in U * M . Define φ : A → U M \ Z as φ(t, λ) = φ t (λ). This is a smooth diffeomorphism, whose inverse is φ -1 ( λ) = ( (-λ), -φ (-λ) (-λ)). In particular, U M is measurable. Then, using Lemma 17 (see below), and Fubini theorem, we have ( 18)

Theorem 16 (Santaló formulas). The visible set U M and the optimally visible set Ũ M are measurable. Moreover, for any measurable function

F : U * M → R we have U M F µ = ∂M U + q ∂M (λ) 0 F (φ t (λ))dt λ, n q η q (λ) σ(q), (16) Ũ M F µ = ∂M U + q ∂M ˜ (λ) 0 F (φ t (λ))dt λ, n q η q (λ) σ(q). ( 17
U M F µ = φ(A) F µ = A (F • φ) φ * µ = = ∂M U + q ∂M (λ) 0 F (φ t (λ))dt λ, n q η q (λ) σ(q),
which proves [START_REF] Chavel | Eigenvalues in Riemannian geometry[END_REF]. Analogously, with à = {(t, λ) | 0 < t < ˜ (λ)} and Z = Z ∪ {φ ˜ (λ) (λ) | λ ∈ U + ∂M } the map φ : à → Ũ M \ Z is a diffeomorphism with the same inverse. Then, the same computations as [START_REF] De Verdière | Spectral asymptotics for sub-Riemannian Laplacians. I: quantum ergodicity and quantum limits in the 3D contact case[END_REF] replacing A with à and Z with Z yield [START_REF] Chavel | Riemannian geometry[END_REF].

Lemma 17. The following local identity of elements of

Λ 2n-1 (R × U + ∂M ) holds φ * µ| (t,λ) = λ, n q dt ∧ σ ∧ η, λ ∈ U + ∂M,
where, in canonical coordinates (p, x) on

T * M η = ι e ν, ν = 1 ω(x) dp, σ = ι n ω, ω = ω(x)dx. Proof. For any (t, λ) ∈ R × U + ∂M let {∂ t , v 1 , . . . , v 2n-2 } be a set of independent vectors in T (R × U + ∂M ) = T R ⊕ T U + ∂M . Observe that φ * µ = dt ∧ (ι ∂t φ * µ). Then, ι ∂t φ * µ(v 1 , . . . , v 2n-2 ) = µ| φ(t,λ) d (t,λ) φ ∂ t , d (t,λ) φ v 1 , . . . , d (t,λ) φ v 2n-2 .
Notice that, (a)

d (t,λ) φ ∂ t = (d λ φ t ) H, this is in fact just H| φt(λ) , (b) d (t,λ) φ v i = (d λ φ t )v i for any i = 1, . . . , 2n -2. Hence it follows that (19) ι ∂t φ * µ = ι H φ * t µ = ι H µ,
where in the last passage we used the invariance of µ (see the discussion below Lemma 12). By Lemma 11 and its proof (in particular see Example 1) locally µ = η ∧ ω. By the properties of the interior product, [START_REF] Croke | A sharp four-dimensional isoperimetric inequality[END_REF] ι

H µ = (ι H η) ∧ ω + (ι H ω) ∧ η.
The first term on the r.h.s. vanishes: as a 2n -2 form, its value at a point λ ∈ U + ∂M is completely determined by its action on 2n -2 independent vectors of T λ U + ∂M . We can choose coordinates such that ∂M = {x n = 0}. Then a basis of T λ U + ∂M is given by ∂ x 1 , . . . , ∂ x n-1 and a set of n -

1 vectors v i = n j=1 v j i ∂ p j in T λ U + π(λ) ∂M .
Since ι H η is a n -2 form, then ω necessarily acts on at least one v i , and vanishes. Now, notice that ( 21)

ι H ω| λ (•) = ω| π(λ) (π * H, π * •) = λ, n π(λ) ω| π(λ) (n π(λ) , π * •) = λ, n π(λ) σ| λ (•).
Putting together [START_REF] Croke | Some isoperimetric inequalities and eigenvalue estimates[END_REF], [START_REF] Croke | A sharp four-dimensional isoperimetric inequality[END_REF], and ( 21) completes the proof of the statement.

Reduced Santaló formula.

The following reduction procedure replaces the noncompact set U M in Theorem 16 with a compact subset that we now describe.

To carry out this procedure we fix a transverse sub-bundle V ⊂ T M such that T M = D⊕V. We assume that V is the orthogonal complement of D w.r.t. to a Riemannian metric g such that g| D coincides with the sub-Riemannian one and the associated Riemannian volume coincides with ω. In the Riemannian case, where V is trivial, this forces ω = ω R , the Riemannian volume. In the genuinely sub-Riemannian case there is no loss of generality since this assumption is satisfied for any choice of ω. Definition 3. The reduced cotangent bundle is the rank k vector bundle π : T * M r → M of covectors that annihilate the vertical directions:

T * M r := {λ ∈ T * M | λ, v = 0 for all v ∈ V} . The reduced unit cotangent bundle is U * M r := U * M ∩ T * M r .
Observe that U * M r is a corank 1 sub-bundle of T * M r , whose fibers are spheres S k-1 . If T * M r is invariant in the sense of Definition 2, we can apply the construction of Section 3.3. The Liouville volume Θ on T * M induces a volume on T * M r as follows.

Let X 1 , . . . , X k and Z 1 , . . . , Z n-k be local orthonormal frames for D and V, respectively. Let u i (λ) := λ, X i ) and v j (λ) := λ, Z j smooth functions on T * M . Thus

T * M r = {λ ∈ T * M | v 1 (λ) = . . . = v n-k (λ) = 0}.
For all q ∈ M where the fields are defined, (u, v) : T * q M → R n are smooth coordinates on the fiber and hence ∂ u 1 , . . . , ∂ u k ,∂ v 1 , . . . , ∂ v n-k are vectors on T λ (T * q M ) ⊂ T λ (T * M ) for all λ ∈ π -1 (q). In particular, the vector fields ∂ v 1 , . . . , ∂ v n-k are transverse to T * M r , hence we give the following definition.

Definition 4. The reduced Liouville volume

Θ r ∈ Λ n+k (T * M r ) is Θ r λ := Θ λ (. . . , . . . , . . . k vectors , ∂ v 1 , . . . , ∂ v n-k , . . . , . . . , . . . n vectors ), ∀λ ∈ T * M r .
The above definition of Θ r does not depend on the choice of the local orthonormal frame {X 1 , . . . , X k , Z 1 , . . . , Z n-k } and Riemannian metric g| V on the complement, as long as its Riemannian volume remains the fixed one, ω. In fact, let X , Z be a different frame for a different Riemannian metric g | V . Then4 , X = RX and

Z = SX + T Z for R ∈ SO(k), T ∈ SL(n -k) and S ∈ M(k, n). One can check that ∂ v = S∂ u + T ∂ v and that Θ r = Θ(. . . , ∂ v , . . .) = Θ(. . . , T ∂ v , . . .) = (Θ r )
, where both frames are defined.

Assumptions for reduction. We assume the following hypotheses:

(H1) The bundle

T * M r ⊆ T * M is invariant. (H2)
The reduced Liouville volume is invariant, i.e. L H Θ r = 0.

Remark 8. Assumption (H1) depends only on V, while (H2) depends also on ω (since Θ r does). In the Riemannian case, with ω = ω R , both are trivially satisfied.

Under these assumptions U * M r = U * M ∩ T * M r is an invariant corank 1 sub-bundle of T * M r . Moreover, µ r = ι e Θ r is an invariant surface form on U * M r . This follows from Lemma 12 observing that [ H, e] = -H is tangent to U * M r . As in Section 3.1, the volume Θ r ∈ Λ n+k (T * M r ) induces a vertical volume ν r q on the fibers T * q M r and a vertical surface form η r q = ι e ν q on U * q M r . As a consequence of Lemmas 10 and 11 the latter has the following explicit expression, whose proof is straightforward.

Lemma 18 (Explicit reduced vertical measure). Let q 0 ∈ M and fix a set of canonical coordinates (p, x) such that q 0 has coordinates x 0 and

• {∂ x 1 , . . . , ∂ x k } q 0 is an orthonormal basis of D q 0 , • {∂ x k+1 , . . . , ∂ xn } q 0 is an orthonormal basis of V q 0 . In these coordinates ω| x 0 = dx| q 0 . Then ν r q 0 = vol R k and η r q 0 = vol S k-1 . In particular,

U * q 0 M r η r q 0 = |S k-1 |, ∀q 0 ∈ M, where |S k-1 | denotes the Lebesgue measure of S k-1 and vol R k , vol S k-1 denote the Euclidean volume forms of R k and S k-1 .
We now state the reduced Santaló formulas. The sets U + ∂M r , U M r , and Ũ M r are defined from their unreduced counterparts by taking the intersection with T * M r .

Theorem 19 (Reduced Santaló formulas). The visible set U M r and the optimally visible set Ũ M r are measurable. For any measurable function

F : U * M r → R we have U M r F µ r = ∂M U + q ∂M r (λ) 0 F (φ t (λ))dt λ, n q η r q (λ) σ(q), ( 22
) Ũ M r F µ r = ∂M U + q ∂M r ˜ (λ) 0 F (φ t (λ))dt λ, n q η r q (λ) σ(q).
Proof. The proof follows the same steps as the one of Theorem 16 replacing the invariant sub-bundles, volumes, and surface forms with their reduced counterparts.

Remark 9. Let H sR be the sub-Riemannian Hamiltonian and H R be the Riemannian Hamiltonian of the Riemannian extension. The two Hamiltonians are (locally on T * M )

H R = 1 2   k i=1 u 2 i + n-k j=1 v 2 j   , H sR = 1 2 k i=1 u 2 i .
Let φ sR t = e t H sR and φ R t = e t H R be their Hamiltonian flows. Since

T * M r = {λ | v 1 (λ) = . . . = v n-k (λ) = 0},

by assumption (H1) we have

H sR = H R , and φ sR t = φ R t on T * M r .
In particular, the sub-Riemannian geodesics with initial covector λ ∈ U * M r are also geodesics of the Riemannian extension and viceversa.

Examples

Carnot groups.

A Carnot group (G, ) of step m is a connected, simply connected Lie group of dimension n, such that its Lie algebra g = T e G admits a nilpotent stratification of step m, that is

g = g 1 ⊕ . . . ⊕ g m , with [g 1 , g j ] = g 1+j , ∀1 ≤ j ≤ m, g m = {0}, g m+1 = {0}.
Let D be the left-invariant distribution generated by g 1 , and consider any left-invariant sub-Riemannian structure on G induced by a scalar product on g 1 . We identify G R n with a polynomial product law by choosing a basis for g as follows. Recall that the group exponential map, exp G : g → G, associates with V ∈ g the element γ (1), where γ : [0, 1] → G is the unique integral line starting from γ(0) = 0 of the left invariant vector field associated with V . Since G is simply connected and g is nilpotent, exp G is a smooth diffeomorphism.

Let d j := dim g j . Indeed d 1 = k. Let {X j i }, for j = 1, . . . , m and i = 1, . . . , d j be an adapted basis, that is g j = span{X j 1 , . . . , X j d j }. In exponential coordinates we identify

(x 1 , . . . , x m ) exp G   m j=1 d j i=1 x j i X j i   , x j ∈ R d j .
The identity e ∈ G is the point (0, . . . , 0) ∈ R n and, by the Baker-Cambpell-Hausdorff formula the group law is a polynomial expression in the coordinates (x 1 , . . . , x m ). Finally,

X j i = ∂ ∂x j i 0 , so that D| e {(x, 0, . . . , 0) | x ∈ R k } and D q = L q * D| e ,
where L q * is the differential of the left-translation L q (p) := q p. We equip G with the Lebesgue volume of R n , which is a left-invariant Haar measure. In order to apply the reduction procedure of Section 4.3, let V be the left-invariant distribution generated by V| e := g 2 ⊕ . . . ⊕ g m , and consider any left-invariant scalar product g| V on V. Thus, up to a renormalization, g = g| D ⊕ g| V is a left-invariant Riemannian extension such that T M = D ⊕ V is an orthogonal direct sum and its Riemannian volume coincides with the Lebesgue one.

Proposition 20. Any Carnot group satisfies assumptions (H1) and (H2).

Proof. Let X 1 , . . . , X k ∈ Γ(D) and Z 1 , . . . , Z n-k ∈ Γ(V) be a global frame of left-invariant orthonormal vector fields. Let u i (λ) := λ, X i and v j (λ) := λ, Z j be smooth functions on T * G. We have the following expressions for the Poisson brackets

{u i , v j } = k i=1 n-k =1 d ij v , i = 1, . . . , k, j = 1, . . . , n -k,
for some constants d ij . We stress that the above expression does not depend on the u i 's, as a consequence of the graded structure. Denoting the derivative along the integral curves of H with a dot, we have

vj = {H, v j } = k i=1 u i {u i , v j } = k i=1 n-k =1 u i d ij v .
Thus, any integral line of H starting from λ ∈ T * M r = {v 1 = . . . = v n-k = 0} remains in T * M r and the latter is invariant.

To prove the invariance of Θ r , consider, for any fixed left-invariant X ∈ Γ(D), the adjoint map ad X : V| e → V| e , given by ad X (Z) = [X, Z]| e . This map is well defined (as a consequence of the graded structure) and nilpotent. In particular Trace(ad X ) = 0. Thus, we obtain from an explicit computation (see Appendix A)

L H Θ r = -   k i=1 n-k j=1 u i d j ij   Θ r = - k i=1 u i Trace(ad X i ) Θ r = 0.
Proposition 21 (Characterization of reduced geodesics for Carnot groups). The geodesics γ λ (t) with initial covector λ ∈ T * q M r are obtained by left-translation of straight lines, that is, in exponential coordinates,

γ λ (t) = q (ut, 0, . . . , 0), u ∈ R k . Proof. Let X 1 , . . . , X k ∈ Γ(D) and Z 1 , . . . , Z n-k ∈ Γ(V)
be a global frame of left-invariant orthonormal vector fields. Let u i (λ) := λ, X i and v j (λ) := λ, Z j be smooth functions on

T * G. Let u ∈ R k . The extremal λ(t) = φ t (λ)
, with initial covector λ = (q, u, 0) satisfies v ≡ 0 by Proposition 20 and, as a consequence of the graded structure,

ui = {H, u i } = k j=1 u j {u j , u i } = k j=1 n-k =1 u j c ji v = 0,
In particular λ(t) = (q(t), u, 0). Moreover the geodesic γ λ (t

) = π(λ(t)) satisfies γλ (t) = k i=1 u i X i (γ λ (t)). Since the u i 's are constants, γ λ (t) is an integral curve of k i=1 u i X i starting from q. Then L -1 q γ λ (t) is an integral curve of k i=1 u i L -1 q * X i = k i=1 u i X i starting from the identity. By definition of exponential coordinates γ λ (t) = q exp G t k i=1 u i X i q (ut, 0).
Remark 10. In the case of a step 2 Carnot group, the group law is linear when written in exponential coordinates. In fact, for a fixed left-invariant basis X 1 , . . . , X k ∈ Γ(D) and

Z 1 , . . . , Z n-k ∈ Γ(V) it holds [X i , X j ] = n-k =1 c ij Z , c ij ∈ R.
By the Baker-Campbell-Hausdorff formula, (x, z) (x , z ) = (x + x, z + z + f (x, x )), where

f (x, x ) = 1 2 k i,j=1 x i c ij x j , = 1, . . . , n -k.
As a consequence, the geodesics γ λ (t) with initial covector λ ∈ T * q M r span the set q D| e . The latter is not an hyperplane, in general, when q = e and the step m > 2.

Example 2 (Heisenberg group). The (2d + 1)-dimensional Heisenberg group H 2d+1 is the sub-Riemannian structure on R 2d+1 where (D, g) is given by the following set of global orthonormal fields

X i := ∂ x i - 1 2 2d i=1 J ij x j ∂ z , J = 0 I n -I n 0 , i = 1, . . . , 2d, written in coordinates (x, z) ∈ R 2d × R. The distribution is bracket-generating, as [X i , X j ] = J ij ∂ z .
These fields generate a stratified Lie algebra, nilpotent of step 2, with

g 1 = span{X 1 , . . . , X 2d }, g 2 = span{∂ z }.
There is a unique connected, simply connected Lie group G such that g = g 1 ⊕ g 2 is its Lie algebra of left-invariant vector fields. The group exponential map exp G : g → G is a smooth diffeomorphism and then we identify G = R 2d+1 with the polynomial product law

(x, z) (x , z ) = x + x , z + z + 1 2 x • Jx .
Notice that X 1 , . . . , X 2d (and ∂ z ) are left-invariant.

To carry on the reduction, we consider the Riemannian extension g such that ∂ z is a unit vector orthogonal to D. The geodesics associated with λ ∈ U * M r and starting from q reach the whole Euclidean plane q {z = 0} (the left-translation of R 2d ⊂ R 2d+1 ). At q = (x, z) this is the plane orthogonal to the vector 1 2 Jx, 1 w.r.t. the Euclidean metric. 5.2. Riemannian foliations with bundle like metric. Roughly speaking, a Riemannian foliation has bundle like metric if locally it is a Riemannian submersion w.r.t. the projection along the leaves. Definition 5. Let M be a smooth and connected n-dimensional Riemannian manifold. A k-codimensional foliation F on M is said to be Riemannian with bundle like metric if there exists a maximal collection of pairs

{(U α , π α ), α ∈ I} of open subsets U α of M and submersions π α : U α → U 0 α ⊂ R k such that • {U α } α∈I is a covering of M • If U α ∩ U β = ∅, there exists a local diffeomorphism Ψ αβ : R k → R k such that π α = Ψ αβ π β on U α ∩ U β • the maps π α : U α → U 0
α are Riemannian submersions when U 0 α are endowed with a given Riemannian metric On each U α , the preimages π -1 α (x 0 ) for fixed x 0 ∈ U 0 α are codimension k embedded submanifolds, called the plaques of the foliation. These submanifolds form maximal connected injectively immersed submanifolds called the leaves of the foliation. The foliation is totally geodesic if its leaves are totally geodesic submanifolds [START_REF] Tondeur | Foliations on Riemannian manifolds[END_REF].

To any Riemannian foliation with bundle-like metric we associate the splitting T M = D ⊕ V, where V is the bundle of vectors tangent to the leaves of the foliation and D is its orthogonal complement (we call V the bundle of vertical directions, and its sections vertical vector fields). If D is bracket-generating, then (D, g| D ) is indeed a sub-Riemannian structure on M that we refer to as tamed by a foliation and we assume to be equipped with the corresponding Riemannian volume.

We say that a vector field X ∈ Γ(T M ) is basic if, locally on any U α , it is π α -related with some vector X 0 on U 0 α . If X ∈ Γ(T M ) is basic, and V ∈ Γ(V) is vertical, then the Lie bracket [X, V ] is vertical. In this setting we consider a local orthonormal frame Z 1 , . . . , Z n-k ∈ Γ(V) of vertical vector fields and a local orthonormal frame of basic vector fields X 1 , . . . , X k ∈ Γ(D) for the distribution. The structural functions are defined as

[X i , X j ] = k =1 b ij X + n-k =1 c ij Z , [X i , Z j ] = n-k =1 d ij Z , [Z i , Z j ] = n-k =1 e ij Z .
The totally geodesic assumption is equivalent to the fact that any basic horizontal vector field X generates a vertical isometry, that is

(23) (L X g)(Z, W ) = 0, ∀Z, W ∈ Γ(V) ⇐⇒ d ij = -d j i . Proposition 22.
Any sub-Riemannian structure tamed by a foliation with totally geodesic leaves satisfies assumptions (H1) and (H2).

Proof. Locally, T * M r is the zero-locus of the functions v i (λ) = λ, Z i for some family {Z j } n-k j=1 of generators of V. Thus, denoting the derivative along the integral curves of H with a dot, we have

vj = {H, v i } = k i=1 u i {u i , v j } = k i=1 n-k =1 u i d ij v = 0, on T * M r .
This readily implies the invariance of T * M r . To prove the invariance of Θ r , we obtain from an explicit computation (see Appendix A)

L H (Θ r ) = - k i=1 n-k =1 u i d i Θ r = 0,
where, in the last step, we used the totally geodesic assumption [START_REF] Davies | A review of Hardy inequalities[END_REF].

X i = Xi . Let Z 1 , . . . , Z n-k ∈ Γ(V) be a local orthonormal frame for V. Indeed [X i , X j ] = k =1 b ij X + n-k =1 c ij Z , b ij , c ij ∈ C ∞ (M ).
Since the X i 's are basic, the functions b ij ∈ C ∞ (M ) are constant along the fibers of the submersion and descend to well defined functions in C ∞ ( M ). Moreover

[ Xi , Xj ] = k =1 b ij X . Sub-Riemannian extremals λ(t) ∈ U * M r satisfy (24) v j (t) ≡ 0, uj (t) = k i, =1 u i (t)b ij u (t), γλ (t) = k i=1 u i (t)X i (γ λ (t)),
where the structural functions b ij = b ij (γ λ (t)) are computed along the sub-Riemannian geodesic. On the other hand, Riemannian extremals λ(t) ∈ U M satisfy ( 25)

uj (t) = k i, =1 ūi (t)b ij ū (t), γλ (t) = k i=1 ūi (t) Xi (γ λ (t)),
where ūi : T * M → R are the smooth functions u i ( λ) = λ, Xi , for i = 1, . . . , k and are computed along the extremal. The statement follows by observing that the projections γλ = π •γ λ of sub-Riemannian extremals satisfy [START_REF] Ekholm | Hardy inequalities for p-Laplacians with Robin boundary conditions[END_REF] with ūi (t) = u i (t). Viceversa, for any Riemannian geodesic γλ on M , its horizontal lift γ λ on M satisfies [START_REF] Derridj | Sur un théorème de traces[END_REF] with u i (t) = ūi (t) and v j ≡ 0. Hence, the quotient space S 2d+1 /S 1 CP d (the complex projective space) has a unique Riemannian structure (the Fubini-Study metric) such that the projection p(z 0 , . . . , z d ) = [z 0 : . . . : z d ] is a Riemannian submersion. The fibration S 1 → S 2d+1 p -→ CP d is called the complex Hopf fibration. In real coordinates z j = x j + iy j on C d+1 , the vertical distribution V = ker p * is generated by the restriction to S 2d+1 of the unit vector field

ξ = d j=0 (x j ∂ y j -y j ∂ x j ).
The orthogonal complement D := V ⊥ with the restriction g| D of the round metric define the standard sub-Riemannian structure on the complex Hopf fibrations. In real coordinates, as subspaces of R 2d+2 , the hemisphere and its boundary are

M = S 2d+1 + := d i=0 x 2 i + y 2 i = 1 | x 0 ≥ 0 , ∂M = d i=0 x 2 i + y 2 i = 1 | x 0 = 0 .
A different set of coordinates we will use is the following

(ϑ, w 1 , . . . , w d ) → e iϑ 1 + |w| 2 , w 1 e iϑ 1 + |w| 2 , . . . , w d e iϑ 1 + |w| 2 ,
where ϑ ∈ (-π, π) and w = (w 1 , . . . , w d ) ∈ C d . In particular (w 1 , . . . , w d ) are inohomgeneous coordinates for CP d given by w j = z j /z 0 and ϑ is the fiber coordinate. The north pole corresponds to ϑ = 0 and w = 0. The hemisphere is characterized by ϑ ∈ [-π 2 , π 2 ] and its boundary by cos(ϑ) = 0.

Example 4 (Quaternionic Hopf fibrations). Let H be the field of quaternions. If q = x + iy + jz + kw, with x, y, z, w ∈ R, the quaternionic norm is

q = x 2 + y 2 + z 2 + w 2 .
Consider the sphere S 4d+3 as a subset of the quaternionic space H d , S 4d+3 = {(q 0 , q 1 , . . . , q d ) ∈ H d+1 | q = 1}, equipped with the standard round metric. The left multiplication by unit quaternions S 3 = {q ∈ H | |q| = 1} gives an isometric action of SU(2) on S 4d+3 . The quotient space S 4d+3 /S 3 HP d (the quaternionic projective space) has a unique Riemannian structure such that the projection p(q 0 , . . . , q d ) = [q 0 : . . . : q d ] is a Riemannian submersion. The fibration S 3 → S 4d+3 p -→ HP d is the quaternionic Hopf fibration. In real coordinates q j = x j + iy j + jz j + kw j on H d+1 , the vertical distribution V = ker p * is generated by

ξ I = d i=0 y i ∂ x i -x i ∂ y i + w i ∂ z i -z i ∂ w i , ξ J = d i=0 z i ∂ x i -w i ∂ y i -x i ∂ z i + y i ∂ w i , ξ K = d i=0 w i ∂ x i + z i ∂ y i -y i ∂ z i -x i ∂ w i .
The orthogonal complement D := V ⊥ with the restriction g| D of the round metric define the standard sub-Riemannian structure on the quaternionic Hopf fibrations.

In real coordinates, the hemisphere M = S 4d+4 + ⊂ R 4d+4 and its boundary are

M = d i=0 x 2 i + y 2 i + z 2 i + w 2 i = 1 | x 0 ≥ 0 , ∂M = d i=0 x 2 i + y 2 i + z 2 i + w 2 i = 1 | x 0 = 0 .
A different set of coordinates we will use is the following

(ϑ 1 , ϑ 2 , ϑ 3 , w 1 , . . . , w d ) → e iϑ 1 +jϑ 2 +kϑ 3 1 + |w| 2 , w 1 e iϑ 1 +jϑ 2 +kϑ 3 1 + |w| 2 , . . . , w d e iϑ 1 +jϑ 2 +kϑ 3 1 + |w| 2 ,
where

|ϑ| 2 = ϑ 2 1 + ϑ 2 2 + ϑ 2 3 < π 2 and w = (w 1 , . . . , w d ) ∈ H d .
In particular (w 1 , . . . , w d ) are inohomgeneous coordinates for HP d given by w j = q -1 0 q j and ϑ 1 , ϑ 2 , ϑ 3 are local coordinates on SU [START_REF] Agrachev | Introduction to Riemannian and Sub-Riemannian geometry from Hamiltonian viewpoint[END_REF]. The north pole corresponds to ϑ 1 = ϑ 2 = ϑ 3 = 0 and w = 0. The hemisphere is characterized by |ϑ| ≤ π/2 and its boundary by cos |ϑ| = 0.

Applications

In this section we present the proofs of the applications of the reduced Santaló formula presented in Sections 1.3, 1.4 and 1.5.

Hardy-type inequalities. It is well known that for all

f ∈ C ∞ 0 ([0, a]) one has (26) a 0 f (t) 2 dt ≥ π 2 a 2 a 0 f (t)
L(λ) := (-φ (-λ) (-λ)), if (-λ) < +∞, +∞, otherwise.
Proof of Proposition 2. Choose coordinates x around a fixed q ∈ M as in Lemma 18, and let (p, x) be the associated canonical coordinates on T * M . Let Q be a quadratic form on T * q M r . In particular Q(λ) = k i,j=1 p i Q ij p j , where λ = (p 1 , . . . , p k ). By Lemma 18 we have

U * q M r Q(λ)η r q 0 (λ) = S k-1 k i,j=1 Q ij p i p j dvol S k-1 (p) = |S k-1 | k Trace(Q),
where we performed the standard integral of a quadratic form on S

k-1 . Choosing Q(λ) = λ, ∇ H f (q) 2 , then Trace(Q) = |∇ H f (q)| 2 .
Thus, for any point q ∈ M we have

(27) |S k-1 | k |∇ H f (q)| 2 = U * q M r λ, ∇ H f (q) 2 η r q (λ), ∀f ∈ C ∞ (M ).
Using the reduced Santaló formula [START_REF] Danielli | Hardy-Sobolev type inequalities with sharp constants in Carnot-Carathéodory spaces[END_REF],

|S k-1 | k M |∇ H f (q)| 2 ω(q) = M U * q M r λ, ∇ H f (q) 2 η r q (λ) ω(q) = U * M r λ, ∇ H f 2 µ r ≥ U M r λ, ∇ H f 2 µ r = ∂M U + q ∂M r (λ) 0 φ t (λ), ∇ H f 2 dt λ, n q η r q (λ) σ(q). Consider the subset D = U + ∂M r ∩ { < +∞}. Let f λ (t) := f (π • φ t (λ)). For λ ∈ D we have f λ (0) = f λ ( (λ))
= 0 and the one-dimensional Poincaré inequality ( 26) gives ( 28)

(λ) 0 φ t (λ), ∇ H f 2 dt = (λ) 0 f λ (t) 2 dt ≥ π 2 2 (λ) (λ) 0 f λ (t) 2 dt.
Indeed we can replace with L, which is φ t -invariant. Then

|S k-1 | k M |∇ H f (q)| 2 ω(q) ≥ π 2 ∂M Dq (λ) 0 f λ (t) 2 L(λ) 2 dt λ, n q η r q (λ) σ(q).
Since on U + q ∂M \ D q the function 1/L(λ) 2 = 0, we can replace D q with U + q ∂M . Using again Santaló formula to restore the integral on U M r , we obtain

M |∇ H f (q)| 2 ω(q) ≥ kπ 2 |S k-1 | U M r (π * f ) 2 L 2 µ r = kπ 2 |S k-1 | M U * q M r 1 L 2 η r q f (q) 2 ω(q).
The second equality follows by Lemma 11. This concludes the proof of (5).

To prove [START_REF] Barilari | A formula for Popp's volume in sub-Riemannian geometry[END_REF] we replace Poincaré inequality with Hardy in (28):

(λ) 0 φ t (λ), ∇ H f 2 dt = (λ) 0 f λ (t) 2 dt ≥ 1 4 (λ) 0 f λ (t) 2 min{t, (λ) -t} 2 dt ≥ 1 4 (λ) 0 f λ (t) 2 (φ t (λ)) 2 dt,
where we used the fact that if λ ∈ U + ∂M then (φ t (λ)) = (λ) -t. We then proceed as in the previous case without replacing with L.

Proof of Proposition 3. The result is obtained by mimicking the proof of Proposition 2. Observe that, for any f ∈ C ∞ 0 (M ) and q ∈ M , we have

U * q M r | λ, ∇ H f (q) | p η r q (λ) = |∇ H f (q)| p U * q M r λ, ∇ H f (q) |∇ H f (q)| p η r q (λ) = 2|∇ H f (q)| p S k-1 ∩{p 1 >0} p p 1 dvol S k-1 (p) = |∇ H f (q)| p C -1 p,k
, where we used coordinates as in Lemma 18, the rotational invariance of the measure and

C p,k := Γ( k+p 2 ) 2Γ( 1+p 2 )π (k-1)/2 = k |S k-1 | √ π Γ( k+p 2 ) 2Γ( 1+p 2 )Γ( k 2 + 1)
.

Using the above in place of [START_REF] Evans | Measure theory and fine properties of functions[END_REF], by Santaló formula ( 22) we obtain

C -1 p,k M |∇ H f | p ω(q) ≥ ∂M U + q ∂M r (λ) 0
| φ t (λ), ∇ H f | p dt λ, n q η r q (λ) σ(q).

To prove [START_REF] Baudoin | Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations[END_REF] we proceed as in the proof of Proposition 2 replacing the step [START_REF] Hansson | Sharp spectral inequalities for the Heisenberg Laplacian[END_REF]

M |∇ H f | 2 ω ≥ kπ 2 L 2 M f 2 ω.
By the min-max principle [START_REF] Capogna | An isoperimetric inequality and the geometric Sobolev embedding for vector fields[END_REF], whenever any f ∈ C ∞ 0 (M ) such that M f 2 ω = 1, we have

λ 1 (M ) ≥ M |∇ H f | 2 ω ≥ kπ 2 L 2 .
Proof of Proposition 5. Fix a north pole q 0 and the hemisphere M whose center is q 0 . By Remark 9, in all three cases, the reduced sub-Riemannian geodesics are a subset of the Riemannian ones (great circles). In particular L = π and Proposition 4 gives In particular ∆Φ = ∆Φ = (-4d)Φ.

6.2. Isoperimetric inequalities. We define some quantities that we already introduced. Definition 6. The visibility angle at q ∈ M and the optimal visibility angle are ϑ q := η r q (U q M r ) η r q (U q M r ) , θ q := η r q ( Ũ q M r ) η r q (U q M r )

.

The least visibility angle and the least optimal visibility angle are

ϑ = inf q∈M ϑ q , θ = inf q∈M θ q .
Notice that ϑ q , θ q , ϑ , θ ∈ [0, 1] and do not depend on the choice of the volume ω. Clearly diam r (M ) ≤ diam(M ).

Proof of Proposition 7. The proof follows as in [START_REF] Chavel | Riemannian geometry[END_REF][START_REF] Croke | Some isoperimetric inequalities and eigenvalue estimates[END_REF], considering F = 1 in [START_REF] Danielli | Hardy-Sobolev type inequalities with sharp constants in Carnot-Carathéodory spaces[END_REF]. The l.h.s. is estimated from below using the disintegration of µ r given in Lemma 11. For the estimate of the r.h.s. we only observe that, by Lemma 18 we have

U + q ∂M r λ, n q η r q (λ) = S k-1 ∩{p 1 >0} p 1 dvol S k-1 (p) = |S k | 2π .
Proof of Proposition 8. For all these structures, all the inequalities in the proof of Proposition 7 are equalities, hence the sharpness follows. Anyway, here we perform the explicit computation for the hemisphere of the sub-Riemannian complex Hopf fibration; the remaining case of the quaternionic Hopf fibration can be checked following the same steps. We use the notation of Example 3, and real coordinates. Let q = (0, y 0 , . . . , x d , y d ) ∈ ∂M . The sub-Riemannian normal n q is the unique inward pointing unit vector in D q orthogonal to D q ∩ T q ∂M . Indeed, T q ∂M is the orthogonal complement to ∂ x 0 w.r.t. to the Riemannian round metric, while D q is the orthogonal complement to ξ. Thus, n q = αξ + β∂ x 0 . The condition n q ∈ D and the normalization imply

n q = 1 1 -g(∂ x 0 , ξ) 2 (∂ x 0 -g(∂ x 0 , ξ)ξ) .
Using the explicit expression for ξ we obtain

n q = 1 -y 2 0 ∂ x 0 mod T q ∂M.
Notice that C(∂M ) = {y 2 0 = 1} ∩ ∂M . Due to the factor ρ(y 0 ) := 1 -y 2 0 , the sub-Riemannian surface measure σ is different from the Riemannian one in cylindrical coordinates σ R = ι ∂x 0 ω = ρ 2d-2 dy 0 dvol S 2d-1 . In particular and we set T * M r = V ⊥ = span{dϑ}. The restriction of g to D induces a sub-Riemannian structure (which does not satisfy the bracket-generating condition, but this is inconsequential here). The Dirichlet energy of (M, D, g) is not greater than the one of the original Riemannian structure. Hence a lower bound for the first eigenvalue of the sub-Laplacian of (M, D, g| D ) yields a lower bound for the Laplace-Beltrami operator on M . Geodesics with λ ∈ U * M r cross the spherical band longitudinally, and L(λ) = 2ε. Both (H1) and (H2) are verified, and thus we obtain from Proposition 4 the sharp estimate

σ(∂M ) = ∂M ρ(y 0 ) σ R = |S 2d-2 |
λ 1 (M ) ≥ π 2 (2ε) 2 .
This construction highlights the fact that the reduction procedure can be used in both the Riemannian and sub-Riemannian case to improve estimates such as the one of Proposition 4, when the geometry of the problem is quite explicit. The general philosophy is that the smaller is the set of reduced geodesics, the better is the bound in Proposition 4.

Figure 1 .

 1 Figure 1. Exit length (left) and visible set (right). Covectors are represented as hyperplanes, the arrow shows the direction of propagation of the associated geodesic for positive time.

Proposition 5 (

 5 Sharpness of the eigenvalue lower bound). In Proposition 4, in the following cases we have equality, for all d ≥ 1: (i) the hemispheres S d + of the Riemannian round sphere S d ; (ii) the hemispheres S 2d+1 + of the sub-Riemannian complex Hopf fibration S 2d+1 ; (iii) the hemispheres S 4d+3 + of the sub-Riemannian quaternionic Hopf fibration S 4d+3 ;

Figure 2 .

 2 Figure 2. Visibility angle on a 2D Riemannian manifold. Only the geodesics with tangent vector in the dashed slice go to ∂M .

Proposition 8 (

 8 Sharpness of isoperimetric inequalities). In Proposition 7, in the following cases we have equality, for all d ≥ 1: (i) the hemispheres S d + of the Riemannian round sphere S d ; (ii) the hemispheres S 2d+1 + of the sub-Riemannian complex Hopf fibration S 2d+1 ; (iii) the hemispheres S 4d+3 + of the sub-Riemannian quaternionic Hopf fibration S 4d+3 ;

  Finally, we apply the vector bundle construction Lemma [32, Lemma 5.5]. Definition 1. A smooth, strictly positive section ν

3

  Let ϑ ∈ Λ 1 (T * M ) be the tautological form ϑ(λ) := π * (λ). The Liouville invariant volume Θ ∈ Λ 2n (T * M ) is Θ := (-1) n(n-1) 2 dϑ ∧ . . . ∧ dϑ. In canonical coordinates (p, x) on T * M we have Θ = dp ∧ dx.

1 .

 1 Example 1, E = T * M and E = U * M are clearly invariant; in particular H is tangent to E . By Liouville theorem, Ω = Θ is invariant for any Hamiltonian flow Moreover, if the Hamiltonian H is homogeneous of degree d (on fibers), one checks that [ H, e] = -(d -1) H and Lemma 12 yields the invariance of the Liouville surface measure µ = ι e Θ. In particular this holds in Riemannian and sub-Riemannian geometry, with d = 2. 4. Santaló formula 4.Assumptions on the boundary. Let (N, D, g) be a smooth connected sub-Riemannian manifold, of dimension n, without boundary. We focus on a compact n-dimensional submanifold M with piecewise C 1,1 boundary ∂M .

) Remark 6 .

 6 Even if M is compact and hence ˜ < +∞, in general Ũ M U M . Fur- thermore, if < +∞ (that is, all geodesics reach the boundary of M in finite time), then U M = U * M . Thus, our statement of Santalò formula contains [17, Thm. VII.4.1]. Remark 7. If ∂M is only Lipschitz and C(∂M ) has positive measure, the above Santaló formulas still hold by removing on the left hand side from U M and Ũ M the set {φ t (λ) | π(λ) ∈ C(∂M ) and t ≥ 0}. Nothing changes on the right hand side as σ(C(∂M )) = 0, since σ = ι n ω and n vanishes on C(∂M ) by definition.

Example 3 (

 3 Complex Hopf fibrations). Consider the odd dimensional spheres S 2d+1 S 2d+1 = {(z 0 , z 1 , . . . , z d ) ∈ C d+1 | z = 1}, equipped with the standard round metric. The unit complex numbers S 1 = {z ∈ C | |z| = 1} give an isometric action of U(1) on S 2d+1 by z → e iϑ z, z ∈ S 2d+1 , ϑ ∈ (-π, π].

λ 1 ( 1 +

 11 M ) ≥ k, where k = d for the Riemannian sphere, k = 2d for the CHF and k = 4d for the QHF. In all cases, uniqueness of Φ = cos(δ) ∈ C ∞ 0 (M ) follows as in the Riemannian case from the min-max principle [16, Corollary 2, p. 20]. To complete the proof, we show that Φ is an eigenfunction of the positive (sub-)Laplacian with eigenvalue d, 2d, 4d, respectively. In the Riemannian case this is well known. For the CHF we use coordinates (ϑ, w) of Example 3. Then, Φ = x 0 = cos(ϑ) 1 + |w| 2 = cos(ϑ) cos(r), where we have set tan(r) = |w|. In [10, Proposition 2.3] the authors show that for a function depending only on ϑ and r the action of the sub-Laplacian reduces to the action of its cylindrical part, given by ∆ = ∂ 2 r + ((2d -1) cot(r) -tan(r))∂ r + tan 2 (r)∂ 2 ϑ . In particular ∆Φ = ∆Φ = (-2d)Φ. For the QHF we use coordinates (ϑ 1 , ϑ 2 , ϑ 3 , w) of Example 4. Using the expression e iϑ 1 +jϑ 2 +kϑ 3 = cos(η) + (iϑ 1 + jϑ 2 + kϑ 3 |w| 2 = cos(η) cos(r), where we have set tan(r) = |w|. In [11, Definition 2.1 and Proposition 2.2] the authors show that, for a function depending only on η and r, the action of the sub-Laplacian reduces to the action of its cylindrical part, given by ∆ = ∂ 2 r + ((4d -1) cot(r) -3 tan(r))∂ r + tan 2 (r) ∂ 2 η + 2 cot(η)∂ η .

Definition 7 .

 7 The sub-Riemannian diameter and reduced diameter are:diam(M ) := sup{d(x, y) | x, y ∈ M } = sup{ ˜ (λ) | λ ∈ U * M }, diam r (M ) := sup{ ˜ (λ) | λ ∈ U * M r }.

1 - 1 ρ

 11 (y 0 ) 2d-1 dy 0 = |S 2d+1 | |S 2d | .Moreover ω(M ) = |S 2d+1 |/2. By Remark 9, the reduced geodesics γ λ with λ ∈ U * M r are a subset of Riemannian geodesics hence ϑ = θ = 1 and = diam r (M ) = π.

  5.2.1. Riemannian submersions.A Riemannian submersion π : (M, g) → ( M , ḡ) is trivially a Riemannian foliation with bundle-like metric. Let M be a sub-Riemannian manifold tamed by a Riemannian submersion π : M → M . We have the following characterization. Let M be a sub-Riemannian manifold tamed by a Riemannian submersion π : M → M . Then γ λ : [0, T ] → M is a sub-Riemannian geodesic associated with λ ∈ U * M r if and only if it is the lift of a Riemannian geodesic γλ := π • γ λ of M .Proof. Let X1 , . . . , Xk ∈ Γ(T M ) be a local orthonormal frame for ( M , ḡ). Let X 1 , . . . , X k ∈ Γ(D) the corresponding local orthonormal frame of basic vector fields on M , such that π *

	Proposition 23.

  with the L p Poincaré inequality[START_REF] Lindqvist | A nonlinear eigenvalue problem[END_REF] Sec. 5.3] We proceed similarly for the proof of (8), replacing[START_REF] Hansson | Sharp spectral inequalities for the Heisenberg Laplacian[END_REF] with the L p Hardy's inequalities[START_REF] Hardy | Inequalities. Cambridge Mathematical Library[END_REF] Thm. 327] Proof of Proposition 4. With L := sup λ∈U * M r L(λ), the Hardy inequality (5) can be further simplified into

			0	a	|f (t)| p dt ≥	π p a	p	0	a	|f (t)| p dt.
	0	a	|f (t)| p dt ≥	p -1 p	p	0	a	|f (t)| p d(t) p dt.

For simplicity, assume that D is orientable as a vector bundle and that X1 . . . , X k is an oriented frame.

Acknowledgments

We are grateful to the organizers of the Trimester Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds, whose stimulating atmosphere fostered this collaboration, and the Institut Henri Poincaré, Paris, where most of this research has been carried out. We also thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for the support and hospitality during the programme Periodic and Ergodic Spectral Problems. The last named author is grateful for the support and hospitality of the Erwin Schrödinger Institute of Mathematical Physics in Vienna during the programme Modern theory of wave equations, where he stayed when this article was completed. We thank F. Baudoin and J. Wang for useful discussions about the spectrum of the sub-Laplacian on the quaternionic Hopf fibration and L. Parnovski for useful comments on the spectral theoretic part. We thank Z. Balogh for useful discussions on the negligibility of the set of characteristic points. This research has been supported by the European Research Council, ERC StG 2009 "GeCoMethods", contract n. 239748, by the iCODE institute (research project of the Idex Paris-Saclay), and by the SMAI project "BOUM". The first and second authors were partially supported by the Grant ANR-15-CE40-0018 of the ANR. This research benefited from the support of the "FMJH Program Gaspard Monge in optimization and operation research" and from the support to this program from EDF.

Appendix A. Lie derivative of the reduced Liouville volume Lemma 24. In the notation of Section 4.3, we have

Proof. For any -tuple w = (w 1 , . . . , w ) of vector fields and -form α, we denote

for any -form α and -uple w, we obtain

where, in the last step, we used that L H Θ = 0. Now observe that, for j = 1, . . . , n -k,

Plugging [START_REF] Ivanov | The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds[END_REF] in [START_REF] Hardy | Inequalities. Cambridge Mathematical Library[END_REF], and using complete skew-symmetry, we obtain the statement.

Appendix B. Improving estimates through reduction

We sketch a strategy to improve the lower bound for the (sub-)Laplacian of Proposition 4, when the latter is trivial, i.e. when L = +∞. Similar considerations hold also for isoperimetric-type inequalities.

Let M be a (sub-)Riemannian manifold with boundary ∂M , and assume that a reduced bundle U * M r has been found in such a way that the reduced Santaló formula, and all its consequences, hold. Assume that there exists a reduced geodesic that never hits the boundary of M . This happens if there exists a covector λ ∈ U * M r such that L(λ) = +∞. This phenomenon occurs already in the Riemannian case, where no reduction is required, and in particular if M contains closed Riemannian geodesics. For example, consider the small rotationally symmetric neighborhood

of the equator ϑ = π/2 of the two dimensional round sphere, equipped with the standard measure. In this cases, the lower bound of Proposition 4 for the Dirichlet spectrum on M is trivial, since L = sup λ∈U * M L(λ) = +∞. Nevertheless the reduction procedure can be still applied to circumvent this problem, as we now sketch for the spherical band above.

The idea is to define a set of reduced geodesics by considering only those which are normal to the boundary ∂M . More precisely, we let D := span{∂ ϑ }, V := span{∂ φ },