MEASURE CONTRACTION PROPERTIES FOR TWO-STEP ANALYTIC SUB-RIEMANNIAN STRUCTURES AND LIPSCHITZ CARNOT GROUPS

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Zeinab Badreddine & Ludovic Rifford

Measure contraction properties for two-step analytic sub-Riemannian structures and Lipschitz Carnot groups Tome 70, n^o6 (2020), p. 2303-2330.

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MEASURE CONTRACTION PROPERTIES FOR TWO-STEP ANALYTIC SUB-RIEMANNIAN STRUCTURES AND LIPSCHITZ CARNOT GROUPS

by Zeinab BADREDDINE & Ludovic RIFFORD (*)

Abstract. —We prove that two-step analytic sub-Riemannian structures on a compact analytic manifold equipped with a smooth measure and Lipschitz Carnot groups satisfy measure contraction properties.

Résumé. —On démontre que les structures sous-riemanniennes analytiques com- pactes de pas 2 munies de mesures lisses et les groupes de Carnot dits Lipschitz vérifient des propriétés de contraction de la mesure.

1. Introduction

The aim of this paper is to provide new examples of sub-Riemannian structures satisfying measure contraction properties. LetM be a smooth manifold of dimension n >3 equipped with a sub-Riemannian structure (∆, g) of rankm < n, whose geodesic distanced_SRis supposed to be com- plete. We refer the reader to Appendix B for the notations used throughout the paper. As in the previous paper of the second author on the same sub- ject [25], we restrict our attention to the notion of measure contraction properties in metric measured spaces with negligeable cut loci (ifA⊂M is a Borel set thenLⁿ(A) = 0 means that A has vanishingn-dimensional Lebesgue measures in charts):

Definition 1.1. — We say that the sub-Riemannian structure (∆, g) onM has negligeable cut loci if for everyx∈M, there is a measurable set C(x)⊂M with

Lⁿ(C(x)) = 0,

Keywords:sub-Riemannian geometry, Measure Contraction Properties.

2020Mathematics Subject Classification:53C23, 22E25, 58C15, 58J60.

(*) Both authors are supported by the ANR project SRGI “Sub-Riemannian Geometry and Interactions”, ANR-15-CE40-0018.

and a measurable mapγ_x : (M \ C(x))×[0,1]−→M such that for every y∈M\ C(x)the curve

s∈[0,1]7−→γ_x(s, y) is the unique minimizing geodesic fromxtoy.

Measure contraction properties consist in comparing the contraction of volumes along minimizing geodesics from a given point with what hap- pens in classical model spaces of Riemannian geometry. We recall that for every K ∈ R, the comparison function s_K : [0,+∞) → [0,+∞) (sK : [0, π/√

K)→[0,+∞) ifK >0) is defined by

s_K(t) :=











sin(√

√Kt)

K ifK >0

t ifK= 0

sinh(√

√ −Kt)

−K ifK <0.

In our setting, the following definition is equivalent to the notion of measure contraction property introduced by Ohta in [23] for more general measured metric spaces (see also [32]).

Definition 1.2. — Let (∆, g) be a sub-Riemannian structure on M with negligeable cut loci,µa measure absolutely continuous with respect to LⁿandK∈R, N >1be fixed. We say that(∆, g)equipped withµsatisfies MCP(K, N) if for every x∈ M and every measurable setA ⊂M \ C(x) (provided thatA⊂B_SR(x, πp

N−1/K)ifK >0) with0< µ(A)<∞, µ(As)>

Z

A

s

"

sK sdSR(x, z)/√ N−1 s_K d_SR(x, z)/√

N−1

#^N−1

dµ(z) ∀s∈[0,1], whereA_sis thes-interpolation ofAfromxdefined by

As:={γx(s, y)|y∈A\ C(x)} ∀s∈[0,1].

In particular,(∆, g)equipped withµsatisfiesMCP(0, N)if for everyx∈M and every measurable setA⊂M\ C(x)with0< µ(A)<∞,

µ(As)>s^Nµ(A) ∀s∈[0,1].

To our knowledge, the first study of measure contraction properties in the sub-Riemannian setting has been performed by Juillet in his thesis. In [16], Juillet proved that then-th Heisenberg group Hⁿ (with n>1) equipped with its sub-Riemannian distance and the Lebesgue measureL²ⁿ⁺¹(in this case the ambient space isR²ⁿ⁺¹) satisfies MCP(0,2n+ 3). This result is sharp for two reasons. First, Juillet proved thatHⁿ does not satisfy any

other stronger notion of “Ricci curvature bounded from below” in met- ric measured spaces such as for example the so-called curvature dimension property (see [20, 31, 32, 33]). Secondly, Juillet showed that 2n+ 3 is the optimal dimension for whichHⁿsatisfies MCP(0, N), there is noN <2n+3 such thatHⁿ (equipped withdSRandL²ⁿ⁺¹) satisfies MCP(0, N). Juillet’s Theorem, which settled the case of the simplest sub-Riemannian structures, paved the way to the study of measure contraction properties for more general sub-Riemannian structures. In [6], Agrachev and Lee investigated the case of sub-Riemannian structures associated with contact distribu- tions in dimension 3. In [18, 19], Lee and Lee, Li and Zelenko studied the particular case of Sasakian manifolds. In [25], the second author proved that any ideal Carnot group satisfies MCP(0, N) for some N >1 (it has been shown later by Rizzi [29] that a Carnot group is ideal if and only if it is fat). In [29], Rizzi showed that any co-rank 1 Carnot group of di- mension k+ 1 (equipped with the sub-Riemannian distance and a left- invariant measure) satisfies MCP(0, k+ 3). Finally, more recently, Barilari and Rizzi [10] proved thatH-type Carnot groups of rankkand dimensionn satisfy MCP(0, k+ 3(n−k)). The purpose of the present paper is to pursue the qualitative approach initiated by the second author in [25]. We aim to show that some assumptions on the sub-Riemannian structure insure that the sub-Riemannian distance enjoys some properties which guarantee that some measure contraction property of the form MCP(0, N) is satisfied for someN >1 (in factN has to be greater or equal to the geodesic dimension of the sub-Riemannian structure as introduced by Rizzi [28]). Our approach is purely qualitative, we do not compute any curvature type quantity in order to find the best exponents. Our results are concerned with two-step analytic sub-Riemannian structures and Lipschitz Carnot groups.

Given a (real) analytic manifoldM, we say that (∆, g) is analytic if both

∆ andg are analytic onM. Moreover, we recall that a distribution ∆, or a sub-Riemannian structure (∆, g), istwo-step if

[∆,∆](x) :={[X, Y](x)|X, Y smooth sections of ∆}=TxM ∀x∈M.

A measure onM is calledsmoothif it is locally defined by a positive smooth density times the Lebesgue measureLⁿ, our first result is the following:

Theorem 1.3. — Every two-step analytic sub-Riemannian structure on a compact analytic manifold equipped with a smooth measure satisfies MCP(0, N)for someN >0.

In the case of Carnot groups which are, as Lie groups equipped with left- invariant sub-Riemannian structures, analytic manifolds with analytic sub- Riemannian structures, the homogeneity allows us to extended the above result to left-invariant Lipschitz distributions.

Following [26], we say that a sub-Riemannian structure (∆, g) or a Carnot group whose first layer ∆ is equipped with a left-invariant metric, isLips- chitzif it is complete and the associated geodesic distancedSR:M×M → Ris locally Lipschitz outside of the diagonalD={(x, y)∈M×M|x=y}.

Examples of Lipschitz sub-Riemannian structures include two-step distri- butions and more generally medium-fat distributions. A distribution ∆ (or a sub-Riemannian structure with distribution ∆ or a Carnot group whose first layer ∆ is equipped with a left-invariant metric) is calledmedium-fat if, for everyx∈M and every smooth sectionX of ∆ withX(x)6= 0, there holds

(1.1) T_xM = ∆(x) + [∆,∆](x) +

X,[∆,∆]

(x), where

X,[∆,∆]

(x) :=

X,[Y, Z]

(x)

Y, Z smooth sections of ∆ . The notion of medium-fat distribution has been introduced by Agrachev and Sarychev in [7]. Of course, in the case of a Carnot group the property of being medium-fat depends only on the properties of its Lie algebra. Our second result is the following:

Theorem 1.4. — Any Lipschitz Carnot group whose first layer is equipped with a left-invariant metric and equipped with Haar measure satisfiesMCP(0, N)for someN >0.

The proofs of Theorem 1.3 and 1.4 are based on the fact that squared sub-Riemannian pointed distancesdSR(x,·)² satisfy a certain property of horizontal semiconcavity. Note that for the moment, we are only able to prove this property in the analytic case under an assumption of compact- ness of length minimizers. In particular, we don’t know if the assumption of analyticity is really necessary. The property of horizontal semiconcav- ity together with the lipschitzness ofdSR(x,·)² allows us to give an upper bound for divergence of horizontal gradients off^xwhich implies the desired measure contraction property.

It is worth to notice that, thanks to a seminal result by Cavaletti and Huesmann [13], measure contraction properties are strongly connected with the well-posedness of the Monge problem for quadratic geodesic distances.

We refer the interested reader to [9, 8] for further details.

We recall that all the notations used throughout the paper are listed in Appendix B. The material required for the proof of the two theorems above is worked out in Section 2. The proofs of Theorems 1.3 and 1.4 are respectively given in Sections 3 and 4.

Acknowledgement

The authors are grateful to the referee for useful remarks and for point- ing out a gap in the initial proof of Proposition 12. The authors are also indebted to Adam Parusinski for fruitful discussions and the reference to the paper by Denef and Van den Dries [14].

2. Preliminaries

Throughout all this section, (∆, g) denotes a complete sub-Riemannian structure onM of rankm < n.

2.1. The minimizing Sard conjecture

The minimizing Sard conjecture is concerned with the size of points that can be reached from a given point by singular minimizing geodesics.

Following [27], givenx∈M, we set S_∆,min^x g:=n

γ(1)

γ∈W_∆^1,2([0,1], M), γ sing.,d_SR(x, γ(1))²= energy_g(γ)o . Note that for everyx∈M, the setS_∆,min^x gis closed and containsx(because m < n). Let us introduce the following definition.

Definition 2.1. — We say that (∆, g) satisfies the minimizing Sard conjecture atx∈M if the set S_∆,min^x g has Lebesgue measure zero inM. We say that it satisfies the minimizing Sard conjecture if this property holds for anyx∈M.

It is not known if all complete sub-Riemannian structures satisfy the minimizing Sard conjecture (see [2, 27]). The best general result is due to Agrachev who proved in [1] that all closed setsS_∆,min^x g have empty interior.

As the next result shows, the minimizing Sard conjecture is related to regularity properties of pointed distance functions. Following Agrachev [1], we call smooth point of the function y 7→ dSR(x, y) (for a fixed x ∈ M)

anyy∈M for which there is p∈T_x^∗M which is not a critical point of the exponential mapping exp_x and such that the projectionγx,p of the normal extremal ψ : [0,1] → T^∗M starting at (x, p) is the unique minimizing geodesic from x to y = γx,p(1). By Agrachev’s Theorem, the set Ox of smooth points is always open and dense inM. The following holds:

Proposition 2.2. — Letx∈M be fixed, the following properties are equivalent:

(1) the structure(∆, g)satisfies the minimizing Sard conjecture atx∈M, (2) the functiony7→dSR(x, y)is differentiable almost everywhere inM, (3) the set of smooth pointsO_xis an open set with full measure inM. Furthermore, the functiony7→dSR(x, y)is smooth onOx and ifM and (∆, g)are analytic, then the setO_xis geodesically star-shaped atx, that is (2.1) γ(s, y)∈ Ox ∀s∈(0,1], ∀y∈ Ox,

whereγx(·, y)∈W_∆^1,2([0,1], M)is the unique minimizing geodesic fromx toy.

Proof of Proposition 2.2. — Letx ∈ M be fixed. The part (3) ⇒ (2) is immediate. Let us prove that (2)⇒(1). By assumption the set of dif- ferentiability D of f := dSR(x,·) has full measure in M. Recall that for every y ∈ D, there is a unique minimizing geodesic from xto y which is given by the projection of the normal extremalψ: [0,1]→T^∗M such that ψ(1) = (y, dSR(x, y)dyf) (see [26, Lemma 2.15 p. 54]). By Sard’s Theorem, the set S of exp_x(p) with p ∈ T_x^∗M critical has Lebesgue measure zero in M. Therefore, the set D\S has full measure and for every y ∈D\S there is a unique minimizing geodesic fromx to y and it is not singular, which shows that y does not belong to S_∆,min^x g. Let us now show that (1)⇒ (3). By definition ofS_∆,min^x g, for every y /∈ S_∆,min^x g all minimizing horizontal paths betweenxandy are not singular. So repeating the proof of [26, Theorem 3.14 p. 98] (see also [11]), we can show that the function f : y 7→ d_SR(x, y) is locally semiconcave and so locally Lipschitz on the open setU :=M \ S_∆,min^x g. Thus for every compact setK⊂U, there is a compact setPK ⊂T_x^∗M such that for everyy∈K, there isp∈ PK with exp_x(p) =y and H(x, p) =dSR(x, y)²/2 (in other wordsγx,p: [0,1]→M is a minimizing geodesic fromxto y). By Sard’s Theorem, the setSK of exp_x(p) withp∈ PK critical is a closed set of Lebesgue measure zero. For every positive integer k, set (here the diameter of the convex set d⁺_yf is taken with respect to some geodesic distance onT^∗M)

Σ^k(f) :=

y∈U

diam(d⁺_yf)>1/k .

By local semiconcavity off in U, each set Σ^k(f) is a closed set inU with Lebesgue measure zero (see [12, Proposition 4.1.3 p. 79]). We claim that

S_K⁰ :=K∩ [

k>0

Σ^k(f)⊂ K∩ [

k>0

Σ^k(f)

!

∪S_K.

As a matter of fact, ify∈Kbelongs to∪k>0Σ^k(f)\ ∪k>0Σ^k(f), thend⁺_yf is a singleton and there is a sequence{yl}l converging toy such that all d⁺_y

lf have dimension at least one and tend to d⁺_yf. This implies that the covectorp such that exp_x(p) = y and H(x, p) = dSR(x, y)²/2 is critical, which shows thaty belongs toS_K. By construction, every point inK\S_K⁰ is a smooth point. We conclude easily.

It remains to prove the second part. The smoothness off :y7→dSR(x, y) is an easy consequence of the inverse function theorem. As a matter of fact, we can show easily that for everyy∈ Oxsuch thaty= exp_x(p) with H(x, p) =d_SR(x, y)²/2 andp∈T_x^∗M non-critical, there is a neighborhood U ofy in Oxsuch that

f(z)²= 2H(x,exp_x(z)⁻¹) ∀z∈U,

where exp⁻¹_x denotes a local inverse of the exponential mapping from a neighborhood ofptoU. To prove (2.1), we argue by contradiction. Assume that there arex∈M,y∈ O_xands∈(0,1) such thatz:=γ(s, y)∈ O_xand letp∈ T_x^∗M be such thaty =γx,p(1) = exp_x(p). We have z =γx,p(s) = exp_x(sp) so there are two possibilities: Either there are two distinct mini- mizing geodesics fromxtozor there is only one minimizing geodesic from xto y and sp is a critical point of the exponential mapping. In the first case, we infer the existence of two distinct minimizing geodesics fromxto y, which contradicts the smoothness of y. In the second case, we deduce thatz=γ_x,p(s) is conjugate tox. But by real-analyticityγcannot contain non-trivial singular subsegments (this is easy to prove by characterization of singular curves, see e.g. [26, Proposition 1.11 p. 21]) and the presence of a conjugate timesin (0,1) implies thatγis no longer minimizing for times greater thans(see [3, Theorem 8.48]) which gives a contradiction.

Remark 2.3. — By Proposition 2.2, any (complete) sub-Riemannian structure satisfying the minimizing Sard conjecture has negligeable cut loci.

Remark 2.4. — As pointed out by the referee, the set Ox could be re- placed by the set A_x ⊂ O_x of ample points from x. This set is defined as the set of y ∈ Ox where the unique minimizing geodesic from xto y is ample, that is whose growth vector saturates the tangent space. It can be shown to be open with full measure and geodesically star-shaped in the

smooth case, we refer the interested reader to the monograph [4] for further details. In our case, since we need the analyticity to prove other results, we prefer to work with the simplerO_x which is geodesically star-shaped in the analytic case.

2.2. Two characterizations forMCP(0, N)

The following result was implicit in the previous paper [25] of the second author (it is also the case in [17, p. 5] and [24, Section 6.2]). The mea- sure contraction property MCP(0, N) is equivalent to some upper bound on the divergence of the horizontal gradient of the squared pointed sub- Riemannian distance. This result holds at least whenever the horizontal gradient is well-defined and the setsOxare geodesically star-shaped.

Proposition 2.5. — Assume that(∆, g)satisfies the minimizing Sard conjecture and that all its setsOx are geodesically star-shaped, and letµ be a smooth measure on M and N > 0 be fixed. Then (∆, g) equipped withµsatisfiesMCP(0, N)if and only if

(2.2) div_y^µ ∇^hf^x

6N ∀y∈ Ox, ∀x∈M,

wheref^x:M →Ris the function defined byf^x(y) :=dSR(x, y)²/2.

Proof. — Let x ∈ M be fixed, the vector field Z := −∇^hf^x is well- defined and smooth on Ox. Moreover by assumption, every solution of

˙

y(t) = Z(y(t)) with y(0) ∈ Ox remains in Ox for all t > 0, we denote by{ϕt}t>0 the flow of Z on Ox. For every y ∈ Ox, the function θ : t ∈ [0,+∞)7→d_SR(ϕ_t(y), y) satisfies

θ(0) = 0 and θ(t) = length^g ϕ_[0,t](y)

= Z t

0

|Z(ϕ_s(y)|ds.

So that, for allt>0,

θ(t) =˙ |Z(ϕ_t(y)|=d_SR(x, ϕ_t(y)) =d_SR(x, y)−d_SR(y, ϕ_t(y))

=d_SR(x, y)−θ(t), which yields

θ(t) =dSR(x, y) 1−e^−t

∀t>0.

Consequently, ifA⊂ Oxis a Borel set ands∈(0,1], then we have As={γx(s, y)|y∈A}=ϕt(A) with t=−ln(s).

Let us now assume that (2.2) is satisfied. By definition of div^µZ, for every x∈M and any measurable setA ⊂ Ox, we have for every t>0 (see for example, see [30, Proposition B.1]),

µ(ϕ_t(A)) = Z

A

exp Z t

0

div^µ_ϕ

s(y)(Z) ds

dµ(y), which by (2.2) implies withs=e^−t,

µ(As) =µ(ϕt(A))>

Z

A

exp (−N t) dµ(y) =s^Nµ(A).

This shows that (2.2) implies MCP(0, N). Conversely, if (∆, g) equipped with µ satisfies MCP(0, N) then for every x ∈ M and every small ball Bδ(y)⊂ Ox(say a Riemannian ball with respect to the Riemannian exten- siong), we have

µ(ϕt(Bδ(y))) = Z

B_δ(y)

exp Z t

0

div^µ_ϕ

s(y)(Z) ds

dµ(y)

>e^{−N t}µ(Bδ(y)) ∀t>0.

For everyt>0, lettingδgo to 0 yields exp

Z t 0

div^µ_ϕ

s(y)(Z) ds

>e^{−N t}.

We infer (2.2) by dividing bytand lettingt go to 0.

In the case of Carnot groups, the invariance of the divergence of ∇^hf^x by dilation allows us to characterize MCP(0, N) in term of a control on the divergence over a compact set not containing the origin.

Proposition 2.6. — Let G be a Carnot group whose first layer is equipped with a left-invariant metric satisfying the minimizing Sard conjec- ture andN >0 fixed. Then the metric space(G, dSR)with Haar measure µsatisfiesMCP(0, N)if and only if

(2.3) div_y^µ ∇^hf⁰

6N ∀y∈ O₀∩S_SR(0,1),

wheref⁰:M →Ris the function defined byf⁰(y) :=d_SR(0, y)²/2.

Proof. — Since Carnot groups are indeed analytic, by the second part of Proposition 2.2 and Proposition 2.5, it is sufficient to show that (2.3) is equivalent to

(2.4) div_y^µ ∇^hf⁰

6N ∀y∈ O0.

Recall that by taking a set of exponential coordinates (x₁, . . . , x_n), we can identify Gwith its Lie algebrag'Rⁿ and indeed consider that we work

with the Lebesgue measure inRⁿ and that the sub-Riemannian structure is globally parametrized by an orthonormal family of analytic vector fields X¹, . . . , Xⁿ inRⁿ satisfying

(2.5) Xⁱ(δλ(x)) =λ⁻¹δλ Xⁱ(x)

∀x∈Rⁿ, ∀i= 1, . . . , n, where {δλ}λ>0 is a family of dilations defined as (d1, . . . , dn are positive integers)

δ_λ(x₁, . . . , x_n) = λ^d1x₁, λ^d2x₂, . . . , λ^dnx_n

∀x∈Rⁿ.

By the homogeneity property, we havedSR(0, δλ(x)) =λ dSR(0, x) for all x∈Rⁿ andλ >0. Then we have

(2.6) f⁰(δ_λ(x)) =λ²f⁰(x) and d_δλ(x)f⁰◦δ_λ=λ²d_xf⁰

∀x∈Rⁿ, ∀λ >0.

Recall that the horizontal gradient∇^hf⁰is given by

∇^h_xf⁰=

m

X

i=1

Xⁱ·f⁰

(x)Xⁱ(x) ∀x∈Rⁿ.

Therefore, by (2.5)–(2.6), we infer that for everyx∈Rⁿ andλ >0,

∇^h_δλ(x)f⁰=

m

X

i=1

d_δλ(x)f⁰ Xⁱ(δ_λ(x))

Xⁱ(δ_λ(x))

=λ⁻²

m

X

i=1

dδ_λ(x)f⁰ δλ Xⁱ(x)

δλ Xⁱ(x)

=

m

X

i=1

dxf⁰ Xⁱ(x)

δλ Xⁱ(x)

=δλ ∇^h_xf⁰ . We deduce that

div_δ^µ

λ(x) ∇^hf⁰

= div_x^µ ∇^hf⁰

∀x∈Rⁿ,∀λ >0,

which shows that (2.3) and (2.4) are equivalent and concludes the proof.

2.3. Nearly horizontally semiconcave functions

Recall that without loss of generality, we can assume that the metric g over ∆ is the restriction of a global Riemannian metric on M. This metric allows us to define theC²-norms of functions fromR^mtoM. In the following statement, (e1, . . . , em) stands for the canonical basis inR^m.

Definition 2.7. — LetC >0andU be an open subset ofM, a function f :U →Ris said to beC-nearly horizontally semiconcave with respect to (∆, g)if for everyy∈U, there are an open neighborhoodV^y of0inR^m, a functionϕ^y:V^y⊂R^m→U of classC²and a functionψ^y:V^y ⊂R^m→R of classC² such that

ϕ^y(0) =y, ψ^y(0) =f(y), f(ϕ^y(v))6ψ^y(v) ∀v∈V^y, (2.7)

{d0ϕ^y(e₁), ..., d₀ϕ^y(e_m)}is an orthonormal family of vectors in∆(y), (2.8)

and

kϕ^yk_C2,kψ^yk_C2 6C, (2.9)

wherekϕ^yk_C2,kψ^yk_C2 denote theC²-norms ofϕ^y andψ^y.

Remark 2.8. — We refer the reader to [21] for an other notion of hori- zontal semiconcavity of interest that has been investigated by Montanari and Morbidelli in the framework of Carnot groups.

If m were equal to n that is if we were in the Riemannian case, the above definition would coincide with the classical definitions of semiconcave functions (see [12, 26]). Here, in the casem < n, the definition tells that at each point, there is a support function from above of classC²which bounds the function along aC² submanifold which is tangent to the distribution.

This type of mild horizontal semiconcavity will allow us, at least in certain cases, to bound the divergence of the horizontal gradient of squared pointed sub-Riemannian distance functions.

Before stating the main result of this section, we recall that a minimizing geodesicγ: [0,1]→M fromxtoyis called normal if it is the projection of a normal extremal, that is a trajectoryψ: [0,1]→T^∗M of the Hamiltonian vector field associated with (∆, g). We refer the reader to Appendix B for more details. Here is our result (|p|^∗ =|p|^∗_x for every (x, p)∈T^∗M is the dual norm associated withg, in particular for any compact setK⊂M and A >0, the set of (x, p) inT^∗M withx∈Kand|p|^∗6Ais a compact set):

Proposition 2.9. — Assume that M and (∆, g) are analytic and let K be a compact subset of M, U ⊂ M a relatively compact open set of M, and A > 0 satisfying the following property: For every x∈K, every y ∈ U and every minimizing geodesic γ : [0,1] → M from x to y, there is p ∈ T_x^∗M with |p|^∗ 6 A such that γ is the projection of the normal extremal ψ : [0,1] → T^∗M starting at (x, p). Then there is C > 0 such that for everyx∈K, the functiony7→d_SR(x, y)² isC-nearly horizontally semiconcave inU.

Proof of Proposition 2.9. — Let K be a compact set of M, U be a relatively compact open set of M andx∈ K fixed, let us first show how to construct functionsϕ^y¯, ψ^y¯ of classC² satisfying (2.7)–(2.8) fory ∈U. Pick a minimizing geodesic γ : [0,1] → M from x to y = γ(1). There is an open neighborhood Uγ_¯ of γ([0,1]) and a family Fγ_¯ of m analytic vector fields X_γ¹_¯, . . . , X_¯γ^m in M such that for every z ∈ U¯γ the family {X_γ¯¹(z), . . . , X_γ^m_¯ (z)} is orthonormal with respect to g and parametrize ∆ (that is Span{X_γ¹_¯(z), . . . , X_¯γ^m(z)} = ∆(z)) and for every z ∈ M \ U_¯γ, X_¯γ¹(z), . . . , X_¯γ^m(z) belongs to ∆(z). Consider the End-Point mapping from xin time 1 associated with the familyF_¯γ={X_γ¹_¯, . . . , X_¯γ^m}, it is defined by

E^x,1_F^¯

¯

γ :u∈L²([0,1];R^m) 7−→ γ_u^Fγ¯(1)∈M, whereγu^F¯γ(1) : [0,1]→M is the solution to the Cauchy problem (2.10) γ(t) =˙

m

X

i=1

ui(t)X_¯γⁱ(γ(t)) for a.e. t∈[0,1], γ(0) =x.

Note that taking the vector fieldsX_γ¹_¯, . . . , X_¯γ^m equal to zero outside of an neighborhood ofU_¯γ, we may assume without loss of generality thatE_F^x,1

¯ γ is well-defined onL²([0,1];R^m). So we assume from now on thatX_¯γ¹, . . . , X_γ^m_¯ are analytic on U_γ¯ and smooth on M. Recall that the function E_F^x,1

¯ γ is smooth and satisfies (see [26, Proposition 1.10 p. 19])

(2.11) ∆ (y)⊂Im

du¯γE^x,1_F^¯

¯ γ

,

where u¯γ is the unique control u ∈ L²([0,1],R^m) such that γu^Fγ¯ = γ.

Therefore, there arev¹_¯γ, . . . , v_¯γ^m∈L²([0,1],R^m) such that (2.12) du_¯γE_F^x,1¯

¯ γ vⁱ_¯γ

=X_γ¯ⁱ(y) ∀i= 1, . . . , m.

Defineϕ¯γ:R^m→M by ϕγ¯(v) :=E_F^¯x,1

¯ γ uγ¯+

m

X

i=1

vivⁱ_¯γ

!

∀v= (v1, . . . , vm)∈R^m. By construction,ϕ_¯γ is smooth and satisfies

ϕ¯γ(0) =E^x,1_F^¯

¯

γ (uγ¯) =γ(1) =y, and

d₀ϕ_γ¯(e_i) =d_uγ¯E_F^¯x,1

¯ γ v_γⁱ_¯

=X_γⁱ_¯(y) ∀i= 1, . . . , m.

Moreover, for everyv∈R^msuch that the solution to (2.10) associated with the controluγ¯+Pm

i=1vivⁱ_¯γ remains inU¯γ, we have dSR(x, ϕγ¯(v))²6

u¯γ+

m

X

i=1

viv_γⁱ_¯

2

L²

=:ψ¯γ(v).

By construction,ϕ¯γ andψγ¯ are smooth, defined in a neighborhood of 0∈ R^m and satisfy (2.7)–(2.8). It remains to show that the C² norms of ϕ^γ¯ andψ^γ¯ can be taken to be uniformly bounded, this is the purpose of the next lemma.

Lemma 2.10. — There are neighborhoodsU¯xand Uy¯respectively ofx and y in M, a neighborhood Uγ¯ of u_γ¯ in L²([0,1],R^m)and C_¯γ > 0 such that for everyx∈K∩Ux¯, y ∈ U∩Uy¯ and every control ux,y associated with a minimizing geodesicγ_x,y : [0,1]→M from xto y with u_x,y ∈ U_¯γ, there arev_u¹_x,y, . . . , v_u^m_x,y ∈L²([0,1],R^m)such that

(2.13) du_x,yE_F^x,1

¯ γ

v_uⁱ_x,y

=X_γⁱ_¯(y) ∀i= 1, . . . , m and

(2.14)

v_uⁱ

x,y

_L26C_¯γ ∀i= 1, . . . , m.

Proof of Lemma 2.10. — Note that if we prove for eachi∈ {1, . . . , m}

the existence of neighborhoodsU_xⁱ, U_yⁱ and U_γ¯ⁱ such that (2.13)–(2.14) are satisfied for i, then the result follows by taking the intersections of the neighborhoodsU_¯xⁱ, U_y¯ⁱ and U_γ¯ⁱ. So, let us fix i in {1, . . . , m}. By taking a chart on a neighborhood ofγ([0,1]) (that we still denote byUγ¯) we may assume that the restriction of E^x,1_F^¯

¯

γ to a neighborhood of u_γ¯ is valued in Rⁿ and doing a change of coordinates (y₁, . . . , y_n) in a neighborhood U_y¯⁰ we may also assume that there are analytic vector fieldsY¹, . . . , Y^m such that

Y¹(y) =Xⁱ(y) =∂y₁, Y^j(y) =∂y_j +

n

X

l=m+1

a^j_l(y)∂y_l

∀j= 2, . . . , m, ∀y∈U_y¯⁰ and

∆(z) = Span

Y¹(z), . . . , Y^m(z) ∀z∈U¯γ.

Observe that by construction, there are analytic mappings f_k^j on U¯γ for j∈ {1, . . . , m} andk∈ {1, . . . , m} such that

Y^j(z) =

m

X

k=1

f_k^j(z)X^k(z) ∀z∈Uγ_¯,∀j = 1, . . . , m.

As a consequence, ifγ: [0,1]→U_¯γ is solution to (2.15) γ(t) =˙

m

X

j=1

wj(t)Y^j(γ(t)) for a.e.t∈[0,1],

for somew∈L²([0,1],R^m), then there holds for a.e.t∈[0,1],

˙ γ(t) =

m

X

j=1

wj(t)

m

X

k=1

f_k^j(γ(t))X^j(γ(t))

=

m

X

k=1





m

X

j=1

wj(t)f_k^j(γ(t))



X^k(γ(t)).

This shows that if we denote by F the family {Y¹, . . . , Y^m}, then any horizontal curve parametrized byF can be parametrized byF¯γ as above.

On the other hand, any parametrization with respect to Fγ¯ leads to a parametrization with respect to F. Consequently, there is a control w ∈ L²([0,1),R^m) such that the solution of (2.15) starting atx is equal toγ and there are neighborhoodsU_¯x⁰ ofxandWofwtogether with a mapping G:U_x¯⁰× W →L²([0,1],R^m) of classC¹such that

E_F^x,1(w) =E_F^x,1

¯

γ(G(x, w)) ∀x∈U_¯x⁰,∀w∈ W,

where E_F^x,1 denotes the End-Point mapping from x in time 1 associated with the familyF. Therefore, in order to prove our result it is sufficient to prove that there isC >0 such that for everyxinKclose tox, everyyinU close toy and anyγminimizing geodesic fromxtoy close toγ associated to the controlw∈ W with respect to F, there is ω ∈L²([0,1],R^m) such that

(2.16) dwE_F^x,1(ω) =Xⁱ(y) =∂y1 and kωk_L2 6C.

Letx∈U_x¯⁰andw∈ W be fixed, the differential ofE_F^x,1 atwis given by (see [26, Remark 1.5 p. 15])

dwE_F^x,1(v) =S(1) Z 1

0

S(t)⁻¹B(t)v(t) dt ∀v∈L²([0,1],R^m), whereS: [0,1]→Mn(R) is solution to the Cauchy problem

S(t) =˙ A(t)S(t) for a.e.t∈[0,1], S(0) =In

and where the matrices A(t) ∈ M_n(R), B(t) ∈ M_n,m(R) are defined by (note thatJ_Y1 =J_Xi = 0)

A(t) :=

m

X

j=2

w_j(t)J_Yj(γ(t)) a.e.t∈[0,1], B(t) = Y¹(γ(t)), . . . , Y^m(γ(t))

∀t∈[0,1].

By construction ofY¹, . . . , Y^m, there isτ⁰∈(0,1) (which does not depend uponx, ybut only uponx, y) such that for almost everyt∈[1−τ⁰,1] the matrixA(t) has the form

A(t) =























0 0 · · · 0 0 · · · 0 0

... ... 0

0

0

α(t) β(t) δ(t)





















 ,

whereα(t) is inR^n−m,β(t) = (β(t)k,l) is a (n−m)×(m−1) matrix and δ(t) is a (n−m) square matrix. Therefore, as the solution of the Cauchy problem S(t) = ˜˙˜ S(t)A(1−t),S(0) =˜ In, the first column of the matrix S(t) :=˜ S(1)S(1−t)⁻¹ witht∈[0,1] has the form

(2.17)







˜ s1(t)

... s˜_n(t)





 with

(˜s1(t) = 1

˜

sj(t) = 0 ∀j= 2, . . . , m ∀t∈[0, τ₀] and the column vector ˜s(t) with coordinates (˜sm+1(t), . . . ,s˜n(t)) is solution of

(2.18) s(t) =˙˜ D(t)α(1−t) ∀t∈[0, τ0], s(0) = 0˜ withD(t) the (n−m) square matrices satisfying

(2.19) D(t) =˙ D(t)δ(1−t) ∀t∈[0, τ₀], D(0) =I_n−m.

Thus, a way to solvedwE_F^x,1(ω) =∂y₁ is to takeω ∈L²([0,1],R^m) of the form

(2.20) ω(t) = (ω1(1−t),0, . . . ,0) a.e. t∈[0,1]

with

(2.21) Supp(ω1)⊂[0, τ⁰], Z 1

0

ω1(t) dt= 1 and

Z 1 0

ω1(t) ˜sl(t) dt= 0 ∀l=m+ 1, . . . , n.

Remember now that by assumption, M and (∆, g) are analytic and for everyx∈K andy∈U all minimizing geodesics joiningxtoy are normal with initial covector bounded byA. Letp∈ T_¯x^∗M be such that γ =γx,¯p¯

where for any (x, p)∈T^∗M,γ_x,p: [0,1]→M denotes the projection of the normal extremalψx,p : [0,1]→T^∗M starting at (x, p) (see Appendix B).

Then, the desired result will follow if we show that there are a neighborhood T of (x, p) in T^∗M and C > 0 such that for every (x, p) ∈ T, the point xbelongs to U_x¯⁰, the curve γx,p([0,1]) is contained in U¯γ, the associated control w = w^x,p belongs to W and there is ω = ω^x,p ∈ L²([0,1],R^m) satisfying (2.20)–(2.21) (with the function ˜s= ˜s^x,p= (˜sm+1(t), . . . ,s˜n(t)) : [0,1]→R^n−mgiven by the first column of ˜S(t) = ˜S^x,p(t) :=S(1)S(1−t)⁻¹ associated withγx,p and wx,p which satisfies (2.17)–(2.18) with α=α^x,p andδ=δ^x,p associated withγx,p andwx,p as well) such that

(2.22) kω^x,pk_L2 6C.

We need the following lemma whose proof is postponed to Appendix A, the result does not hold in the smooth case.

Lemma 2.11. — Let K ⊂R^l be a compact set and h: [0,1]× K →R be an analytic mapping such thath(0, κ) = 0for allκ∈ K. Then there are τ >0as small as desired andν∈(0,1) such that

(2.23)

Z τ 0

h(t, κ) dt ²

6ν τ Z τ

0

h(t, κ)²dt ∀κ∈ K.

Let T be a compact neighborhood of (x, p) in T^∗M such that for every (x, p) ∈ T, the point x belongs to U_x¯⁰, the curve γx,p([0,1]) is contained in U_γ¯ and the associated control w =w^x,p belongs to W. Then we note that since ˜s1, . . . ,˜sn−m are the coordinates of a vector solution of a ordi- nary differential equation with analytic coefficients (see (2.17)–(2.19)), the mapping

(t, x, p, λ)∈[0,1]× T ×[−1,1]^n−m7−→h(t, x, p, λ) :=

n−m

X

k=1

λks˜^x,p_k (t)

is analytic. Therefore, by Lemma 2.11, there areτ∈(0, τ⁰) andν ∈(0,1) such that

(2.24) Z τ

0

h(t, x, p, λ) dt 2

6ντ Z τ

0

h(t, x, p, λ)²dt

∀(x, p)∈ T, ∀λ∈[−1,1]^n−m. Define I ∈ L²([0,1],R) by I(t) := 1_[0,τ](t) for all t ∈ [0,1] and for every (x, p) ∈ T denote by P ∈ L²([0,1],R) the orthogonal projection of I in L²([0,1],R) over the vector space

V^x,p:=

f ∈L²([0,1],R)

hf,1_[0,τ]˜s^x,p_l i_L2 = 0,∀l=m+ 1, . . . , n . Let (x, p)∈ T be fixed, then there are Λ^x,p₁ , . . . ,Λ^x,p_n−m∈Rsuch that

P =I+

n−m

X

k=1

Λk1_[0,τ]s˜^x,p_m+k =:I+J satisfies by definition

hP,1_[0,τ]˜s^x,p_l i_L2 = 0 ∀l=m+ 1, . . . , n−m and

Z 1 0

1_[0,τ](t)P(t)(t) dt=hP, Ii_L2 =kPk²_L2

=kIk²_L2+kJk²_L2+ 2hI, Ji_L2

>kIk²_L2+kJk²_L2−2√

νkIk_L2kJk_L2

> 1−√ ν

kIk²_L2 =τ 1−√ ν

, where we used that consideringksuch that|Λ¯k|= max{|Λ₁|, . . . ,|Λ_n−m|}

thanks to (2.24) yields (note that the inequality below holds obviously in the case Λk¯= 0 so we may assume that Λ¯k 6= 0)

(hI, Ji_L2)²= Z τ

0 n−m

X

k=1

Λks˜^x,p_m+k(t) dt

!²

= Λ²_k¯ Z τ

0 n−m

X

k=1

Λk

Λ¯k

˜

s^x,p_m+k(t) dt

!²

6Λ²k¯ντ Z τ

0 n−m

X

k=1

Λ_k Λ¯k

˜s^x,p_m+k(t)

!² dt

=νkIk²_L2kJk²_L2.

In conclusion, we deduce that for every (x, p)∈ T, the functionω=ω^x,p ∈ L²([0,1],R^m) of the form (2.20) withω1=ω₁^x,p given by

ω^x,p₁ (t) :=1_[0,τ](t)P(t) kPk²_L2

∀t∈[0,1]

satisfies (2.21) and

kω₁^x,pk_L26 1 pτ(1−√

ν).

The proof of Lemma 2.10 is complete.

We conclude easily by compactness ofK andU.

3. Proof of Theorem 1.3

LetM be an analytic compact manifold, (∆, g) a two-step analytic sub- Riemannian structure andµa smooth measure onM. The following result, due to Agrachev and Lee [5] (see also [26]), is a consequence of the fact that ∆ is two-step (and the compactness ofM and the fact that an upper bound on the gradient off^xatygives an upper bound on a normal extremal joiningxto y, see [26, Lemma 2.15 p. 54]). We refer the reader to [5, 26]

for the proof. In fact, it is worth mentioning that Agrachev and Lee prove that a sub-Riemannian structure is two-step if and only if d_SR is locally Lipschitz in charts.

Lemma 3.1. — The functiond²_SR:M ×M →Ris locally Lipschitz in charts. In particular, there isL > 0 such that|∇yf^x|6L for all x∈ M andy∈ Ox, where∇yf^xstands for the gradient off^xatywith respect to the global Riemannian metricg. Furthermore, there isA >0such that for everyx, y∈M and every minimizing geodesicγ: [0,1]→M from xto y, there isp∈T_x^∗M with|p|^∗6Asuch thatγis the projection of the normal extremalψ: [0,1]→T^∗M starting at(x, p).

By the above lemma and Proposition 2.9, there is C >0 such that for every x∈M the functionf^x :y 7→ dSR(x, y)²/2 is C-nearly horizontally semiconcave inM.

Lemma 3.2. — There isB >0such that for everyx∈M the following property holds: for every y ∈ Ox, there is a neighborhood U^y ⊂ Ox of y along with an orthonormal family of smooth vector fields X¹, . . . , X^m which parametrize∆inU^y such that

(3.1)

Xⁱ

_C1 6B ∀i= 1, . . . , m, and

(3.2)

Xⁱ·(Xⁱ·f^x)

(z)6B|∇zf^x|+B ∀z∈U^y,∀i= 1, . . . , m, where∇_zf^x stands for the gradient of f^x at z with respect to the global Riemannian metricg.