The tangent space in sub-Finsler geometry and applications
Mattia Magnabosco Tommaso Rossi





















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The tangent space in sub-Finsler geometry and applications
     Mattia Magnabosco[Mathematical Institute, University of Oxford. E-mail:  mailto:mattia.magnabosco@maths.ox.ac.ukmattia.magnabosco@maths.ox.ac.uk]  and  
Tommaso Rossi[Laboratoire Jacques-Louis Lions, Sorbonne Université. E-mail: mailto:tommaso.rossi@inria.frtommaso.rossi@inria.fr] 
    April 29, 2024
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In this paper, we study the tangent space to a manifold in the measured Gromov-Hausdorff sense. We prove that the metric tangent is described by the nilpotent approximation, generalizing the result. Additionally, we study the blow-up of a measure on a manifold. We identify the new notion of bounded measure which ensures that, in the limit, the blow-up is a scalar multiple of the Lebesgue measure. Our results have applications in the study of the Lott-Sturm-Villani curvature-dimension condition in sub-Finsler manifolds. In particular, we show the failure of the  condition in equiregular manifolds with growth vector (2,3), equipped with a bounded measure. 



Keywords: sub-Finsler geometry, Gromov-Hausdorff tangent space, Carnot groups. 
 

 AMS Mathematics Subject Classifications 2020: 53C17, 53C23, 49J52.















§ INTRODUCTION


In this paper, we study the tangent space to a manifold in the measured Gromov-Hausdorff sense. Sub-Finsler geometry is a generalization of Finsler and sub-Riemannian geome­try, where, given a smooth manifold M, a smoothly varying norm is defined only on a set of preferred directions, called the distribution. 
Under the so-called Hörmander condition, the length-minimization procedure among admissible curves (namely those curves that are tangent to the distribution) yields a well-defined distance on M, called distance. 

For a general metric measure space (,,) (i.e. a complete and separable metric space, equipped with a locally finite Borel measure), a metric measure tangent space at a point x∈ is obtained as a limit of blow-ups, with respect to the measured Gromov-Hausdorff topology. Loosely speaking, a metric measure tangent captures the infinitesimal geometry of (,,) around the base point x. In a Riemannian manifold (M,g), the metric measure tangent space at a point x∈ M coincides with T_xM, equipped with the scalar product g_x and with the Lebesgue measure. In a manifold, identifying the tangent space becomes less trivial, as the infinitesimal model is instead a quotient of a Carnot group. Recall that a Carnot group is a connected and simply connected Lie group with stratified Lie algebra.

    
    
    
    
In <cit.>, it is shown that, the metric tangent to a complete manifold M at a point x∈ M is identified by its nilpotent approximation at x. The nilpotent approximation can be regarded as a non-holonomic first-order approximation of the structure at x and, in turn, is isometric to a quotient of a Carnot group (where a scalar product is defined on the first layer of its Lie algebra). 

Our main theorem extends these classical results to the setting, including also the measure in the convergence. More precisely, we characterize the metric measure tangent space of any manifold, equipped with a bounded measure, namely a Borel measure which, in coordinates, is absolutely continuous with respect to the Lebesgue measure, with density bounded from above and below by strictly positive constants, see Definition <ref>.


     Let M be a complete manifold, equipped with a bounded measure . Then, for -a.e. q∈ M, there is a unique metric measure tangent space of M at q and it is its nilpotent approximation equipped with (a multiple of) the Lebesgue measure. 
 

The nilpotent approximation of a manifold is isometric to a quotient of a Carnot group (where a norm is defined on the first layer of its Lie algebra). On the one hand, for the metric side of Theorem <ref>, we follow an argument analogous to the one developed in <cit.> for manifolds, adapting it to the setting. In particular, this requires to prove the continuity of the distance with respect to the structure, see Theorem <ref>. On the other hand, for identifying the blow-up of the bounded measure  in Theorem <ref>, we leverage its Lebesgue points, in the sense of Definition <ref>. In detail, we show that -almost every point is Lebesgue and, at those points, the blow-up is a scalar multiple of the Lebesgue measure of the correct dimension. Along the way, we also show that on equiregular manifolds, a measure is bounded if and only if it is locally Ahlfors regular. We mention that the metric side of our result is stated, without proof, for equiregular manifolds in <cit.>.

Theorem <ref> has several applications to the study of the so called curvature-dimension (in short ) condition in manifolds. The latter was pioneered by Sturm <cit.> and Lott-Villani <cit.> and is a synthetic notion of curvature-dimension bounds, formulated for metric measure spaces. In particular, the (K,N) condition asks a suitable (K,N)-convexity property of the Rényi entropy functionals in the Wasserstein space (see <cit.> for the precise definition) and represents a lower bound on the (Ricci) curvature by K∈ and an upper bound on the dimension by N∈ (1,∞). We remark that the  condition is a consistent curvature-dimension bound, as in the Riemannian setting it is equivalent to a lower bound on the Bakry-Émery Ricci curvature tensor. 

It has been recently proved through successive contributions <cit.> that a similar relation between curvature and  condition does not hold in the setting. Indeed, every complete manifold, equipped with a positive smooth measure, cannot satisfy the (K,N) condition for any choice of the parameters K and N. In other recent contributions, the same result has been proved for the Heisenberg group <cit.> and for smooth manifolds <cit.>. In this paper, we generalize some of these results in two directions. First of all, we relax the assumptions on the measure by allowing bounded reference measures. Second of all, we prove the failure of the  condition for equiregular manifolds with growth vector (2,3). We refer to Section <ref> for the precise statements of the results.



 §.§ Acknowledgments
 
M.M. acknowledges support from the Royal Society through the Newton International Fellowship (award number: NIF\R1\231659). T.R. acknowledges support from the ANR-DFG project “CoRoMo” (ANR-22-CE92-0077-01). 




§ PRELIMINARIES


Let M be a smooth manifold of dimension n and let k∈. A structure on M is a couple (ξ,), such that: 

    
  * ξ: E→ TM is a morphism of vector bundles, where E is a normed vector bundle over M, meaning that, for every q∈ M, the fiber E_q is a Banach space with norm _q. In addition, the dependence q↦_q is smooth.
    
  * The set of horizontal vector fields, defined as 
    
    := {ξ∘σ  : σ∈Γ(E) }⊂Γ(TM),

    is a bracket-generating family of vector fields (or it satisfies the Hörmander condtion), namely, setting
    
    Lie_q():={X(q) :  X∈span{[X_1,…,[X_j-1,X_j]] :  X_i∈,j∈}},    ∀ q∈ M,

    it holds that Lie_q()=T_qM, for every q∈ M.

We say that (M,ξ,), or simply M if there is no ambiguity, is a manifold. 


    Every structure (ξ,) is equivalent to a free one, meaning that we can (and will) always assume that E=M×^k, for some k∈. See <cit.> for details. However, differently from the setting, we can not ensure that the norm _q on the fiber E_q={q}×^k is independent of q.    


At every point q∈ M we define the distribution at q as 

    _q := {ξ(q,w)   :  w ∈^k }={X(q) : X∈}⊂ T_qM.

This is a vector subspace of T_qM whose dimension is called rank (of the distribution) and denoted by r(q):=_q≤ n. Moreover, the distribution can be described by a family of horizontal vector fields: letting {e_i}_i=1, …, k be the standard basis of ^k, the generating family is the set {X_i}_i=1, …, k, where 

    X_i(q) := ξ(q, e_i)     ∀ q∈ M,   for i= 1, … ,k.

Then, according to (<ref>), _q= span{X_1(q), …, X_k(q)}. On the distribution we define the induced norm as

    v_q := inf{w_q   :  v = ξ(q,w) }    for every v∈_q.

Since the infimum is actually a minimum, the function _q is a norm on _q, so that (_q, _q) is a normed space. Moreover, the dependence on the base point M∋ q↦_q is smooth.

A continuous curve γ: [0,1]→ M is admissible if its velocity γ̇(t) is defined almost everywhere and there exists a function u=(u_1,…,u_k)∈ L^2([0,1];^k) such that

    γ̇(t)=∑_i=1^k u_i(t)X_i(γ(t)),    for a.e. t∈ [0,1].

The function u is called control. Furthermore, given an admissible curve γ, there exists u̅=(u̅_1,…, u̅_k): [0,1]→^k such that 

    γ̇(t) = ∑_i=1^k u̅_i(t) X_i(γ(t)),
        and    γ̇(t)_γ(t) = u̅(t)_γ(t),    for a.e. t∈[0,1].

The function u̅ is called minimal control, and the map t↦u̅(t)_γ(t) belongs to L^∞([0,1]), cf. <cit.>. We define the length of an admissible curve as

    ℓ(γ):=∫_0^1 γ̇(t)_γ(t) t∈[0,∞).

Finally, for every pair of points q_0,q_1∈ M, we define the distance between them as

    (q_0,q_1)= inf{ℓ(γ)  : γ admissible, γ(0)=q_0  and γ(1)=q_1}.

Since every norm on ^k is equivalent to the standard scalar product on ^k, it follows that the structure on M induces a distance which is locally equivalent to . Namely, denoting by _SR the induced distance, for every compact K⊂ M there exist constants C>c>0 (depending on K) such that

    c _SR≤≤ C_SR,    on K× K.

Thus, as a consequence of the classical Filippov and Chow–Rashevskii theorems in sub-Rieman­nian geometry, 
we obtain the following result.


    Let M be a manifold. The distance is finite, continuous on M× M and the induced topology is the manifold one. In particular, M is locally compact.












We say that a manifold (M,ξ, ) is complete if the metric space (M,) is complete. 

















§ CONTINUITY OF THE DISTANCE WITH RESPECT TO THE SUB-FINSLER STRUCTURE


In this section we prove the continuity of the sub-Finser distance  with respect to the sub-Finsler structure, following the same lines of the case, cf. <cit.>. Consider a manifold M with M=n. Given k∈, let 𝒮ℱ_k be the space of free and complete structures on M of bundle rank k. We endow 𝒮ℱ_k with the C^0-topology in the following way. Call B_1 the Euclidean ball of radius 1 in ^k and define the distance 𝔇 between norms on ^k as
 
    𝔇 (_1, _2) := sup_ x ∈ B_1| x_1 - x_2| .

 Moreover, using Whitney theorem, we embed M in ^N for some N∈, and we define the norm 𝔑 on linear functions g:^k →^N as
 
    𝔑 (g) = sup_ x ∈ B_1g(x)_^N,

 where _^N denotes the Euclidean norm on ^N. Then, given (ξ,), (ξ',')∈𝒮ℱ_k and a compact set K⊂ M, we introduce the seminorm
 
    (ξ,)-(ξ',')_K = sup_q∈ K[ 𝔇 (_q, '_q) + 𝔑(ξ(q, ·)- ξ' (q,·)) ],

 where we consider T_qM embedded in ^N. Finally, to define the C_0-topology on 𝒮ℱ_k, let {K_ℓ}_ℓ∈ be an increasing family of compact sets exhausting M and, for every (ξ,)∈𝒮ℱ_k, consider the topology generated by the countable local basis of neighborhoods of (ξ,) given by: 
  
    U_(ξ,|·|),ℓ := {(ξ',') ∈𝒮ℱ_k : (ξ',')-(ξ,)_K_ℓ≤1/ℓ, ℓ∈}.

 
 For every (ξ,)∈𝒮ℱ_k, we denote by _(ξ,|·|) the associated distance on M.

 
    Let M be a n-dimensional smooth manifold. Given two points q_0,q_1 ∈  M, the function 
     
    dist_q_0,q_1: 𝒮ℱ_k →,     dist_q_0,q_1(ξ,) = _(ξ,|·|)(q_0,q_1)

is continuous with respect to the C^0-topology on 𝒮ℱ_k.
 

 
     We prove the lower semicontinuity first, and then the upper semicontinuity. We follow the strategy of <cit.>, adapting it to the setting. 

    (i) Fix (ξ,) ∈𝒮ℱ_k and r<_(ξ,|·|)(q_0,q_1). Given any ϵ>0, we are going to show the existence of δ(ϵ)>0 and a compact set K such that, for every (ξ',') ∈𝒮ℱ_k,
    
    (ξ,)-(ξ',')_K < δ(ϵ)     _(ξ',|·|')(q_0,q_1)>(1-ϵ)r.

     Let B̅(q_0,r) be the closed ball centered at q_0 of radius r with respect to the distance _(ξ,|·|). As B̅(q_0,r) is compact by completeness and q_1 ∉B̅(q_0,r), we can find a relatively compact open neighbourhood O of B̅(q_0,r) such that q_1 ∉ O. Let K be a compact set containing O. By compactness, we can find a constant C>0 such that 
    
    ξ (q,v) -  ξ (q',v)_^N≤ C q-q'_^N    for every q,q'∈ K,  v∈ B_1.

    Then, for a control u∈ L^∞([0,1],^k), given any (ξ',')∈𝒮ℱ_k, we consider the curves γ,γ':[0,1]→ M solving the equations q̇= ξ(q,u) and q̇=ξ'(q,u) respectively, with initial conditions γ(0)=γ'(0)=q_0. Moreover, we consider the quantity
    
    T_u := sup{ t∈ [0,1]   : γ(s),γ'(s)∈ O  for every s∈[0,t] },

    and the function 
     
    _u:[0,1]→,     _u(t):= γ(t)-γ'(t)_^N.

     Then, as O⊂ K, for every t≤ T_u, we obtain the estimate
     
    _u(t)    ≤∫_0^t ξ (γ(s),u(s)) -  ξ' (γ'(s),u(s))_^N s 
       ≤∫_0^t ξ (γ(s),u(s)) -  ξ (γ'(s),u(s))_^N s 
          + ∫_0^t ξ (γ'(s),u(s)) -  ξ' (γ'(s),u(s))_^N s
       ≤ C∫_0^t _u(s) s+ (ξ,)-(ξ',')_K ∫_0^t u(s)_^k s.

    Now, take δ(ϵ)>0 sufficiently small so that there exist two constants A,B>0 for which, for every (ξ',')∈𝒮ℱ_k with (ξ,)-(ξ',')_K < δ(ϵ), it holds that 
    
    '_q ≥1/A_^k,  ∀ q∈ K     and    𝔇 (_q, _q') ≤ B q-q'_^N,  ∀ q,q'∈ K.

    Then, consider any control u∈ L^∞([0,1],^k) such that ∫_0^1 |u(s)|'_γ'(s) s ≤ (1-ϵ)r and observe that, for every t≤ T_u,
    
    ∫_0^t u(s)_^k s ≤ A ∫_0^t |u(s)|'_γ'(s) s ≤ A (1-ϵ)r .

    Let ρ>0 be such that ρ<r ϵ/1+ AB(1-ϵ) r and ρ < inf{q-q'_^N  :q∈B̅(q_0,r),   q'∈ O^c}. Up to taking a smaller δ(ϵ), we can assume that
    
    δ (ϵ) < ρ/Ae^C(1-ϵ)r.

    Keeping in mind (<ref>), Gronwall's lemma allows us to deduce that, for every (ξ',')∈𝒮ℱ_k with (ξ,)-(ξ',')_K < δ(ϵ) and every t≤ T_u,
    
    _u(t)≤ e^C (ξ,)-(ξ',')_K  ∫_0^t u(s)_^k s < ρ,

    where, in the last inequality, we used (<ref>) and (<ref>). In order to conclude the proof, we need to check that, for every (ξ',')∈𝒮ℱ_k with (ξ,)-(ξ',')_K < δ(ϵ), we have T_u=1. Suppose by contradiction this is not the case. On the one hand, by continuity of γ and γ', this means that either γ(T_u)∉ O or γ'(T_u)∉ O. On the other hand, we have the following estimate
    
    ∫_0^T_u   |u(s)|_γ(s) s ≤∫_0^T_u |u(s)|_γ'(s) s + ∫_0^T_u| |u(s)|_γ(s) - |u(s)|_γ'(s)|  s 
         (<ref>)  ≤∫_0^T_u |u(s)|'_γ'(s) s + ∫_0^T_u| |u(s)|_γ'(s) - |u(s)|'_γ'(s)|  s + B ∫_0^T_uu(s)_^k_u(s) s 
         (<ref>)  ≤∫_0^T_u |u(s)|'_γ'(s) s + (ξ,)-(ξ',')_K ∫_0^T_uu(s)_^k s + B ρ∫_0^T_uu(s)_^k s
       (<ref>)-(<ref>)≤ (1-ϵ)r + [1+ AB(1-ϵ) r ]ρ≤ r,

      where the last inequality follows by our choice of ρ. This proves that γ(T_u) ∈B̅(q_0,r) ⊂ O, and, in addition, by our choice of ρ, the estimate (<ref>) ensures that γ'(T_u)∈ O. Thus, we obtain the desired contradiction. In particular, we have proved that for every 
      
      (ξ',')∈𝒮ℱ_k with (ξ,)-(ξ',')_K < δ(ϵ) and every control u∈ L^∞([0,1],^k) such that ∫_0^1 |u(s)|'_γ'(s)≤ (1-ϵ)r, we have γ'(1)∈ O, and therefore 
     
    {q∈ M : _(ξ',|·|')(q_0,q) ≤ (1-ϵ)r}⊂ O.

     Thus, if (ξ,)-(ξ',')_K < δ(ϵ), then _(ξ',|·|')(q_0,q_1)> (1-ϵ)r, proving (<ref>).
     

     (ii) To prove the upper semicontinuity, we fix any r>_(ξ,|·|)(q_0,q_1) and show that _(ξ',|·|')(q_0,q_1) < r for every (ξ',')∈𝒮ℱ_k sufficiently C^0-close to (ξ,)∈𝒮ℱ_k. 
     
     In the following, for every u∈ L^∞([0,1],^k) and q we will denote by γ_ξ^u,q_0:[0,1]→ M the curve solving q̇= ξ(q,u), with initial condition q(0)=q_0. Consider a control u∈ L^∞([0,1],^k) such that γ_ξ^u,q_0(1)=q_1 and 
     
    ∫_0^1 |u(s)|_γ_ξ^u,q_0(s) s= r'<r.
 
     Given any ϵ>0, we apply <cit.> with V= B^^n(0,ϵ), finding ŝ∈ B^^n(0,ϵ) and a choice of n horizontal vector fields X_i_1, …, X_i_n (as defined in (<ref>)) such that the map
     
    ϕ: ^n → M,      ϕ (s_1, …, s_n) := e^-ŝ_1 X_i_1∘⋯∘ e^-ŝ_n X_i_n∘ e^ s_n X_i_n∘⋯∘ e^ s_1 X_i_1(q_1)

     is a local diffeomorphism on a neighborhood U⊂ B^^n(0,ϵ) of ŝ. Moreover, up to taking a smaller ϵ, we can assume that the sum of the lengths of all curves (computed with respect to (ξ,)) involved in the definition of ϕ on U is less that r-r'/3.  With this we mean that, for every s∈ U, defining the control ũ as      
     
    ũ:[0,1] →^k,      t ↦ũ(t):= 
                 s_j e_i_j   if t∈[j-1/2n,j/2n),  for  j=1, …,n, 
    
                 - ŝ_j e_i_j   if t∈[n+j-1/2n,n+j/2n),  for  j=1, …,n,

     it holds that
     
    ∫_0^1 |ũ (t)|_γ_ξ^ũ,q_1(t) t < r-r'/3.
 
     We remark that the control ũ depends on s, but we prefer not to explicit this dependence, to ease the notation. 
       

     Given any (ξ',')∈𝒮ℱ_k, consider the map
     
    ϕ': ^n → M,      ϕ' (s_1, …, s_n) := e^-ŝ_1 X'_i_1∘⋯∘ e^-ŝ_n X'_i_n∘ e^ s_n X'_i_n∘⋯∘ e^ s_1 X'_i_1(q'_1),

     where q'_1:=γ_ξ'^u,q_0(1). Now, we want to prove that, if (ξ',') is sufficiently C^0-close to (ξ,), then the map ϕ' is uniformly close to ϕ on U. To this purpose, fix ρ>0. With an analogous argument to the one of point (i), we can prove that, for every (ξ',')∈𝒮ℱ_k sufficiently C^0-close to (ξ,), we have 
     
    q'_1-q_1_^N < ρ/2    and    _(ξ',|·|')(q_0,q'_1)  < r' + 1/3 (r-r').

     
    Observe that, according to (<ref>), we know that the curve γ_ξ^ũ,q_1 is entirely contained in the closed ball B̅(q_1,r-r'/3) (with respect to the distance _(ξ,|·|)). Take an open neighborhood O of B̅(q_1,r-r'/3) and a compact set K containing O. With the same argument of point (i), we can prove that if ρ and (ξ,)-(ξ',')_K are sufficiently small, then γ_ξ'^ũ,q'_1 is entirely contained in O and 
    
    ϕ' (s)- ϕ (s)_^N = γ_ξ'^ũ,q'_1(1) - γ_ξ^ũ,q_1(1)_^N < ρ,

    
    for every s∈ U. Recall indeed that ũ is defined in terms of s, but all the estimates can be obtained uniformly in s as they only depend on the estimate (<ref>). 
    
    
    We have then showed that ϕ' is uniformly close to ϕ on U, if (ξ',') is sufficiently C^0-close to (ξ,). Moreover, with an argument similar to the one developed in (<ref>) and keeping in mind (<ref>), we can prove that, up to taking (ξ',') further C^0-closer to (ξ,), we have that
    
    _(ξ',|·|')(q'_1,ϕ'(s)) ≤∫_0^1 |ũ (t)|'_γ_ξ'^ũ,q'_1(t) s < 2/3 (r-r') ,

    for every s∈ U.

    Now, we can apply <cit.> with F=ϕ'|_U and Φ= ϕ|_U, deducing that, if (ξ',') is sufficiently C^0-close to (ξ,), then there exists s'∈ U such that ϕ'(s')=ϕ(ŝ)=q_1. Finally, the combination of (<ref>) and (<ref>) allows us to we conclude that
    
    _(ξ',|·|')(q_0,q_1) ≤_(ξ',|·|')(q_0,q'_1) + _(ξ',|·|')(q'_1,q_1) ≤  r' + 1/3 (r-r') + 2/3 (r-r') < r,

    concluding the proof.
     
    


     
    
     
     
     
     
 
 


§ THE NILPOTENT APPROXIMATION OF A SUB-FINSLER MANIFOLD



We introduce the notion of nilpotent approximation of a sub-Finsler manifold, see <cit.> for details. This construction is analogous to the nilpotent approximation of a manifold, taking into account the necessary adaptations to include norms on the distribution. 



  
The flag of the distribution.
Let M be an n-dimensional manifold with distribution . We define the flag of  as the sequence of subsheafs ^j⊂ TM such that

    ^1=,    ^j+1=^j+[,^j],    ∀ j≥ 1,

with the convention that ^0={0}. Under the Hörmander condition, the flag of the distribution defines an exhaustion of T_qM, for any point q∈ M, i.e. there exists s(q)∈ such that:

    {0}=^0_q⊂^1_q⊂…⊂^s(q)-1_q⊊^s(q)_q=T_qM.

The number s(q) is called degree of nonholonomy at q. We set n_j(q) = ^j_q, for any j≥ 0, then the collection of s(q) integers 

    (n_1(q),…,n_s(q)(q))

is called growth vector at q, and we have n_s(q)(q)=n= M. Associated with the growth vector, we can define the weights w_i(q) of  at q: for every i∈{1,…,n}, we set 

    w_i(q):=j,    if and only if     n_j-1(q)+1≤ i≤ n_j(q).

A point q∈ M is said to be regular if the growth vector is constant in a neighborhood of q, and singular otherwise. The structure on M is said to be equiregular if all points of M are regular. In this case, the weights are constant as well on M. Finally, given any q∈ M, we define the homogeneous dimension of M at q as 

    𝒬(q) = ∑_i=1^s(q)
     i ( n_i(q)-n_i-1(q) ) =∑_i=1^n w_i(q).

We recall that, if q is regular, then 𝒬(q) coincides with the Hausdorff dimension of (M,) at q, cf. <cit.>. Moreover, 𝒬(q)>n, for any q∈ M such that _q⊊ T_qM. 



  
Privileged coordinates.
Let M be a manifold with generating family {X_i}_i=1,…,k and f be a smooth function at q∈ M. We call nonholonomic derivative of order ℓ∈ of f at q the quantity

    X_j_1⋯ X_j_ℓf(q),

for any family of indexes {j_1,…,j_ℓ}⊂{1,…,k}. Then, the nonholonomic order of f at q is 

    ord_q(f)=min{ℓ∈ : ∃{j_1,…,j_ℓ}⊂{1,…,k} s.t. X_j_1⋯ X_j_ℓf(q)≠ 0}.







Let M be a n-dimensional manifold and q∈ M. A system of local coordinates (z_1,…,z_n) centered at q is said to be privileged at q if 

    ord_q(z_j) = w_j(q),    ∀  j=1,…,n.



We remark that privileged coordinates always exist, see <cit.> for the construction.













  
Nilpotent approximation.
Let M be a manifold and let q∈ M with weights as in (<ref>). Consider ψ^q=(z_1,…,z_n) U→ V a chart of privileged coordinates centered at q, where U⊂ M is a relatively compact neighborhood of q and V⊂^n is a neighborhood of 0. Then, for any ε∈, we can define the dilation at q as 

    δ_ε^n→^n;    δ_ε
    (z)=(ε^w_1(q)z_1,…,ε^w_n(q)z_n).
 
Using such dilations, we can obtain the nilpotent (or first-order) approximation of the generating family {X_i}_i=1,…,k. Indeed, for any i=1…,k we set Y_i:=ψ^q_*X_i and define the vector field 

    Y_i^ε:=εδ_1/ε*(Y_i),    ∀ ε>0.

Observe that Y_i^ε is defined on the set δ_1/ε(V) and, when ε→ 0, this set tends to cover the whole ^n. Then, according to <cit.>, for every i=1…,k the family {Y_i^ε}_ε>0 admits a limit in the C^∞-topology of ^n as ε→ 0.
In particular, for every i=1…,k, we can define the vector field 

    X_i^q :=lim_ε→0 Y_i^ε∈Γ(T^n).






The family {X_1^q,…,X_k^q} of vector fields on ^n generates a nilpotent Lie algebra of step s(q)=w_n(q) and satisfies the Hörmander condition.


This theorem is well-known in the setting, cf.  <cit.>. Being the result purely algebraic, it carries over to the setting without modifications. Recall that a Lie algebra is said to be nilpotent of step s if s is the smallest integer such that all the brackets of length greater than s are zero. 

According to Theorem <ref>, we can then define the nilpotent approximation of (M, ξ, ) at q.



Let M be a manifold and let q∈ M. Consider the family {X_1^q,…,X_k^q} on ^n, provided by  Theorem <ref>. The nilpotent approximation of M at the point q is the structure (ξ^q, _q) on ^n where

    ξ^q: ^n×^k→^2n,    ξ^q(z,e_i):=(z,X_i^q(z)),   for i=1,…,k.
 


In the following, we denote by ^q and by ^q, the distribution and the distance associated to the structure (ξ^q, _q) on ^n, respectively.


    By construction, the vector fields X_i^q are homogeneous of degree -1 with respect to δ_ε, namely 
    
    X_i^q=ε(δ_1/ε)_*X_i^q,     ∀ ε>0,  for i=1,…,k.

    Moreover, by definition, the norm on ^q is given by 
    
    v^q_z:=inf{|u|_q:ξ^q(z,u)=∑_i=1^k u_i X_i^q(z)=v},    ∀ v∈^q_z, ∀ z∈^n,

    and it is 1-homogeneous with respect to δ_ε, meaning that δ_ε^*·^q_z=ε·^q_z. Thus, the distance ^q of the nilpotent approximation is 1-homogeneous with respect to δ_ε. In addition, since (ξ^q, _q) induces a locally compact topology, by homogeneity, (^n,^q) is complete.



















    The nilpotent approximation of M at a point q is isometric to the quotient of a Carnot group G^q/H^q, for some normal subgroup H^q G^q, equipped with a left-invariant norm. If q is regular, then H^q={0} and the nilpotent approximation is isometric to a Carnot group.    


The proof of the previous theorem follows the blueprint of <cit.>. The main difference lies of course in the norm, which is left-invariant, since the generating family is so (cf. (<ref>)), and thus the metric structure of the nilpotent approximation is indeed the one of a (quotient of a) Carnot group. 




§ THE METRIC MEASURE TANGENT TO A MANIFOLD


In the present section, we combine our previous results to show that the nilpotent approximation of a complete manifold M at q, equipped with (a scalar multiple of) the Lebesgue measure, represents its metric measure tangent. To prove this, we need to introduce an approximating family of structures. 






































 























































































































































































 









 









 

 













 

 











 §.§ The approximating family of structures

Let M be a complete manifold and let q∈ M. Fix a chart of privileged coordinates centered at q, (z_1,…,z_n):U→ V⊂^n. With a slight abuse of notation, we identify M with ^n and work in coordinates. Let r̅>0 such that B^(0,4r̅) ⊂ V and take two cut-off functions φ, ψ∈ C^∞(^n,[0,1]) such that 

    φ(x)= 
            1    if x∈ B^(0,3 r̅),
    
            0    if x∈^n ∖ B^(0,4 r̅);
            ψ(x)= 
            0    if x∈ B^(0,r̅),
    
            1    if x∈^n ∖ B^(0,2 r̅).








For every ε>0, we define the structure (ξ^ε, ^ε) on ^n as 

    ξ^ε   : ^n×^2k→^2n,    ξ(z,e_i):=
            (z,φ(δ_ε(z))· Y_i^ε(z))    for i=1,… k, 
    
            (z,ψ (δ_ε(z))·X^q_i)    for i=k+1,… 2k,
        
    ^ε   :^n×^2k→,          |u|^ε_z := |(u_1,…,u_k)|_δ_ε(z)+ |(u_k+1,…,u_2k)|_q,

where the vector fields Y_i^ε are defined in (<ref>). Note that _δ_ε(z) is defined only for z∈δ_1/ε(V), however, as for i=1,…,k we have ξ(z,e_i)=0 for every z∈^n ∖δ_1/ε(B^(0,4 r̅)), the value of the norm ^ε on the first k coordinates outside B^(0,4 r̅) is irrelevant.  


For every ε>0, denote by ^ε the distribution associated with the structure (ξ^ε, ^ε). Then, ^ε satisfies the Hörmander condition. Indeed, this is obvious for every z∈^n ∖δ_1/ε(B^(0,2 r̅)), since at those points ^q_z⊂^ε_z and Theorem <ref> holds. While, for z ∈δ_1/ε(B̅_3r̅(0)) this is an immediate consequence of the definition of Y_i^ε, keeping in mind that the family of vector fields {Y_i}_i=1,…,k satisfies the Hörmander condition.


By the previous remark, (ξ^ε, ^ε) defines a structure on ^n. We denote by ^ε the distance associated with (ξ^ε, ^ε).



    For every ε,ρ>0, we have the following homogeneity properties: 
    
    δ_ρ^*^ε=ρ ^ρε,    ρ(δ_1/ρ)_*ξ^ε(·,e_i)=ξ^ρε(·,e_i)  for i=1,…,2k.




    Note that, by definition of Y_i^ε, cf. (<ref>), we have the following: 
    
    ρ(δ_1/ρ)_*Y_i^ε=Y_i^ρε.
    
    Thus, the homogeneity of the vector fields is a consequence of (<ref>) and (<ref>). In addition, by our definition of norm, for every z∈^n and u∈^2k, we have
    
    
    
    
    
    
    |u|^ε_δ_ρ(z) =|(u_1,…,u_k)|_δ_ρε(z)+|(u_k+1,…,u_2k)|_q=|u|^ρε_z.

    Finally, the homogeneity of the distance is a routine computation combining the homogeneity of the structure with (<ref>).




    There exists a neighborhood O of 0 such that for every ε>0 it holds that 
    
    ^ϵ(δ_1/ε(x),δ_1/ε(y))= 1/ε (x, y),     for every x, y ∈ O.




    By the homogeneity property of the distance of Lemma <ref>, taking ρ=1/ε, we obtain: 
    
    ^ϵ(δ_1/ε(x),δ_1/ε(y))= 1/ε^1 (x, y),     for every x, y ∈^n.

    Now, let O:=B^(0,r̅/3) and we claim that  and ^1 coincide on O. First of all, as in B^(0,r̅) the structures (ξ,) and (ξ^1,^1) are equivalent, we observe that 
    
    B^(0,r̅)⊂ B^^1(0,r̅).
 
    Second of all, also the converse inclusion holds. Indeed, let x∈ B^^1(0,r̅), η:[0,1]→ M be a geodesic realizing ^1(0,x) and assume by contradiction that there exists t∈ [0,1] such that η_t∉ B^(0,r̅). We set t_0:=inf{t∈ [0,1]:η_t∉ B^(0,r̅)} and then, 
    
    
    
    
    r̅>^1(0,x)=ℓ_(ξ^1, |·|^1)(η)>ℓ_(ξ^1, |·|^1)(η_|[0,t_0]) = ℓ_(ξ, |·|)(η_|[0,t_0])≥r̅,

    giving a contradiction. Therefore, (<ref>) is actually an identity. Finally, note that any geodesic γ, with respect to the structure (ξ,) connecting two points x,y∈ O is contained in B^(0,r̅), since we have
    
    (0, γ(t)) ≤(0, x) + t (x,y) ≤r̅/3 + 2 t r̅/3≤r̅.

    Analogously, the same holds for any geodesic with respect to the structure (ξ^1, |·|^1), connecting x and y in O. We conclude that,  and ^1 must coincide on O.
    




    For every ε>0, (ξ^ε, ^ε) is a complete structure on ^n.



    By homogeneity, it is enough to prove completeness for ε=1. In particular, it is sufficient to show that B̅^^1(0,R) is compact for every R>0 sufficiently large. To this aim, let R̅>0 such that B^(0,4 r̅) ⊂ B^^q(0,R̅). Then, as the structure (ξ^1,^1) coincides with (ξ^q,^q), outside B^(0,4 r̅), an argument analogous to the one leading to (<ref>) implies that:  
    
    B̅^^1(0,R) ⊂B̅^^q(0,R̅+R).

    This concludes the proof as (^n,^q) is complete, cf. Remark <ref>.


The following proposition is the analogous of <cit.> and can be proved starting from the result.


    For every compact set K ⊂^n there exist constants ε_0, C >0 such that 
    
    ^ε(x,y) ≤ C |x-y|^1/s(q)    for every ε∈ (0,ε_0)  and every x,y∈ K,

    where s(q) is the degree of nonholonomy of M at q.



    Consider the (and thus sub-Finsler) structure (ξ^ε, _^2k) and denote by ^ε the associated distance.
    Reasoning as in the proof of Corollary <ref>, we know that, for every r>0, every geodesic with respect to the structure (ξ^ε, _^2k), joining two points in B^^ε(0,r), is contained in B^^ε(0,3r). On the other hand, since the function z ↦_z is smooth, there exists a constant C_0 such that 
    
    _z ≤ C_0 _^k    for every z ∈ B^(0,r̅).

     Then, given any compact set K, fix R̅ such that K⊂ B^^ε(0,R̅) and take ε_0 sufficiently small such that B^^ε(0,3R̅) ⊂δ_1/ε_0(B^(0,r̅)). As a consequence, observe that for every ε<ε_0 we have
     
    |u|^ε_z = |(u_1,…,u_k)|_δ_ε(z)+ |(u_k+1,…,u_2k)|_q≤ 2 C_0 |u|_^2k,

      for every z∈ B^^ε(0,3R̅) and every u∈^2k.
      
      Therefore, we can deduce that 
      
    ^ε (x,y) ≤ 2C_0  ^ε(x,y),     for every ε∈ (0,ε_0)  and every x,y∈ K.

     Up to taking a smaller ε_0, <cit.> guarantees the existence of a constant C_1>0 such that 
    
    ^ε(x,y) ≤ C_1 |x-y|^1/s(q)    for every ε∈ (0,ε_0)  and every x,y∈ K.

    Putting together (<ref>) and (<ref>), we obtain (<ref>).



In the next proposition we prove the convergence of the structures {(ξ^ε, ^ε)}_ε>0 to (ξ^q, _q) as ε→ 0. As a first step, we have to interpret (ξ^q, _q) as a structure of rank 2k. Therefore, we introduce the structure (ξ^0, ^0) 










    ξ^0   : ^n×^2k→^2n,    ξ^0(z,e_i):= ξ^q(z,e_i)   for i=1,…,k 
    
            (z,0)    for i=k+1,…,2k
        ,
    ^0    :^n×^2k→,            |u|^0_z := |(u_1,…,u_k)|_q+ |(u_k+1,…,u_2k)|_q.

Note that (ξ^0, ^0) is equivalent to (ξ^q, _q), hence it induces the same distance ^q.




    The family of structures {(ξ^ε, ^ε)}_ε>0 converges as ε→ 0 to the structure (ξ^0, ^0), with respect to the C^0-topology on 𝒮ℱ_2k, cf. Section <ref>.



    To prove the thesis, we fix any compact set K ⊂^n and show that 
    
    (ξ^0,^0)-(ξ^ε,^ε)_K → 0     as ε→ 0.

    We start by proving that 
    
    sup_z∈ K𝔇(_z^0, ^ε_z) → 0     as ε→ 0.

    To this aim, given any ϱ>0, we take an open neighborhood O of q such that 𝔇 (_q, _z) < ϱ for every z∈ O (the existence of O is guaranteed by the smoothness of the function z↦_z). Now, for every ε>0 so small that δ_ε(K) ⊂ O, we have:
    
    𝔇(_z^0, ^ε_z) = 𝔇(_q, _δ_ε(z)) < ϱ,     for every z ∈ K.

    As ϱ is arbitrary, this proves (<ref>). Then, we proceed to show that
    
    sup_z∈ K𝔑(ξ^0(z, ·)- ξ^ε (z,·)) → 0     as ε→ 0.

    As the maps ξ^0(z, ·) and ξ^ε (z,·) are linear, it is sufficient to prove that for every i=1,…,2k
    
    sup_z∈ K𝔑(ξ^0(z,e_i)- ξ^ε (z,e_i)) → 0     as ε→ 0.

    However, this is an immediate consequence of the C^∞-convergence (<ref>) and of the existence of ε_0>0 such that
    
    K⊂δ_1/ε(B^(0,r̅)),    ∀ ε<ε_0.

    The thesis follows from combining (<ref>) and (<ref>), cf. (<ref>).



    The family of distances {^ε}_ε>0 converges uniformly on compact sets of ^n ×^n to the distance ^q, as ε→ 0.



    The combination of Theorem <ref> and Proposition <ref> yields the pointwise convergence of {^ε}_ε>0 to ^q, as ε→ 0. However, according to Proposition <ref>, the sequence {^ε}_ε>0 is equicontinuous when restricted to any compact subset of ^n ×^n. Therefore, the thesis follows from the Arzelà-Ascoli theorem. 


A first important consequence of the previous theorem is the Ball-Box theorem. Its proof is analogous to the one, cf. <cit.>. 



    Let M be a complete manifold, let q∈ M and let ψ^q:U→^n be a chart of privileged coordinates at q. Then, there exist constants C_q≥ 1 and r_q>0 such that
    
    Box^q(1/C_qr) ⊂ψ^q(B^(q,r))⊂Box^q(C_qr),    ∀  r∈ (0,r_q)

    where Box^q(r):={(z_1,…,z_n) : |z_i|≤ r^w_i(q), 1 ≤ i ≤ n}.




    If M is equiregular, the constants in the previous theorem can be chosen uniformly on compact sets. Namely, for every K⊂ M compact, the constants C_q and r_q of Corollary <ref> can be taken to be independent of q∈ K, cf. <cit.>.





 §.§ Bounded measures on a sub-Finsler manifold




















































































    

















In this section, we describe a class of measures whose blow-up is the Lebesgue measure almost-everywhere, up to a constant. 



    Let M be a n-dimensional manifold, equipped with a Borel measure . We say that  is a bounded measure if, for every coordinate chart ψ:U→^n, ψ_#≪^n and 
    
    ψ_#=ρ^n,    with     c≤ρ≤ C  ^n-a.e.,

    for some positive constants C≥ c>0, depending on the chart.





    Let M be a complete n-dimensional sub-Finsler manifold, equipped with a bound­ed measure . Then, for every q∈ M, there exists C_q≥ 1 and r_q>0 such that
    
    C_q^-1r^𝒬(q)≤(B(q,r))≤ C_q r^𝒬(q),    ∀ q∈ M, r∈ (0,r_q),
 
    where 𝒬(q) is the homogeneous dimension of M at q. 



    Let q∈ M and let ψ^q:U→^n be a set of privileged coordinates centered at q and consider ψ^q_#. On the one hand, by assumptions, there exists a measurable function ρ:^n→_+ satisfying (<ref>). Then, we easily see that, for r>0 sufficiently small,
    
    c  (ψ^q)_#^-1^n(B(q,r)) ≤(B(q,r))≤ C (ψ^q)_#^-1^n(B(q,r)).

    On the other hand, from the Ball-Box theorem, cf. Corollary <ref>, we deduce that there exist constants C̃_q and r_q∈ (0, U), such that
    
    C̃_q^-1 r^𝒬(q)≤ (ψ^q)^-1_#^n(B(q,r))≤C̃_qr^𝒬(q),    ∀ r≤ r_q.

    The combination of the previous inequality and (<ref>) concludes the proof.





    Let M be a manifold, equipped with a bounded measure . Then, we say that q∈ M is a Lebesgue point for  if there exists a coordinate chart ψ:U→^n, defined on a neighborhood U of q, such that
    
    lim_r→ 01/r^𝒬(q)∫_ψ(B(q,r))|ρ-ρ(ψ(q))|^n=0,

    where ψ_#=ρ^n and 𝒬(q) is the homogeneous dimension of M at q. 


















  


    Let M be a complete manifold, equipped with a bounded measure . Then, the definition of Lebesgue point for  is independent on the choice of ψ and -a.e. point of M is a Lebesgue point for , with strictly positive density.



    Let ψ:U⊂ M→^n be a coordinate chart, with U relatively compact. First of all, we show that ^n  satisfies the asymptotic doubling property on (ψ(U),(ψ^-1)^*), namely 
    
    lim sup_r→ 0^n(ψ(B(q,2r)))/^n(ψ(B(q,r)))<∞,

    for all q∈ U. Indeed, by the Ball-Box theorem, cf. Corollary <ref>, there exist constants C_q≥ 1 and r_q>0 such that 
    
    1/C_qr^𝒬(q)≤^n(ψ^q(B(q,r)))≤ C_q r^𝒬(q),    ∀ r<r_q,

    where ψ^q:V_q⊂ U→^n is a chart of privileged coordinates at q. Thus, for every r< r_q, we have: 
    
    ^n(ψ(B(q,2r)))   =∫_ψ^q(B(q,2r))| d(ψ∘ (ψ^q)^-1)|^n
       ≤sup_ψ^q(V_q)| d(ψ∘ (ψ^q)^-1)| ^n(ψ^q(B(q,2r)))
       ≤sup_ψ^q(V_q)| d(ψ∘ (ψ^q)^-1)|   C_q(2r) ^𝒬(q)
       ≤sup_ψ^q(V_q)| d(ψ∘ (ψ^q)^-1)|  C_q^2 2^𝒬(q)^n(ψ^q(B(q,r)))
       ≤sup_ψ^q(V_q)| d(ψ∘ (ψ^q)^-1)| sup_ψ(U)| d(ψ^q∘ψ^-1)|  C_q^2 2^𝒬(q)^n(ψ(B(q,r))).

    This shows that condition (<ref>) is fulfilled.
    Therefore, Lebesgue differentiation theorem holds, cf. <cit.>, and we deduce that, for ^n-a.e. x∈ψ(U), we have
    
    lim_r→ 01/^n(ψ(B(p,r)))∫_ψ(B(p,r))|ρ-ρ(x)|^n=lim_r→ 01/r^𝒬(p)∫_ψ(B(p,r))|ρ-ρ(x)|^n=0,

    having set p:=ψ^-1(x). From this, we immediately see that, if the limit (<ref>) is independent on the coordinate chart, -a.e. point of M is a Lebesgue point for . 
    
    Second of all, let φ:V→^n be another coordinate chart containing q. Then, φ_#=η^n and we need to check that (<ref>) implies that
    
    lim_r→ 01/r^𝒬(q)∫_φ(B(q,r))|η-η(φ(q))|^n=0.

    Consider the smooth change of coordinates given by ψ∘φ^-1:^n→^n and observe that, by definition, we have
    
    (ψ∘φ^-1)_#(η^n)=ρ^n,

    which implies that η=ρ∘ψ∘φ^-1, ^n-a.e.. Hence, using the change of variables x↦ψ∘φ^-1(x) in (<ref>), we obtain: 
    
    1/r^𝒬(q)∫_φ(B(q,r))|η-ρ(ψ(q))|^n=1/r^𝒬(q)∫_ψ(B(q,r))|ρ-ρ(ψ(q))|·| d(φ∘ψ^-1)|^n.

    Since φ∘ψ^-1 is smooth, we easily see that the right-hand side converges to 0, as a consequence of (<ref>). Recalling that η=ρ∘ψ∘φ^-1, ^n-a.e., we conclude the proof of the claim, possibly choosing a different representative of η.  


When M is an equiregular manifold, Lemma <ref> can be improved to a complete characterization of bounded measures, in terms of the notion of 𝒬-Ahlfors regularity.



    Let (,,) be a metric measure space and let 𝒬>1. We say that  is locally 𝒬-Ahlfors regular if, for every compact K⊂, there exists C≥ 1 and r_0∈ (0,], depending on K, such that
    
    C^-1r^𝒬≤(B(p,r))≤ C r^𝒬,    ∀ p∈ K, r∈ (0,r_0).






    Let M be a complete equiregular manifold of homogeneous dimension 𝒬. Then, a measure  on M is bounded if and only if it is locally 𝒬-Ahlfors regular. 



    The only if part of the statement is the content of Lemma <ref>, taking into account that all the constants can be chosen locally uniformly since M is equiregular. To prove the converse statement, let ψ:U⊂ M→^n be a chart and define μ:=ψ_# and ν:=^nψ(U). On the one hand, from the proof of Lemma <ref>, we deduce that ν is asymptotically doubling and thus (ψ(U),(ψ^-1)^*,ν) satisfies Vitali's covering lemma, cf. <cit.>. On the other hand, from the upper bound of the uniform Ball-Box theorem, cf. Remark <ref>, we obtain the estimate: 
    
    D_νμ(p):=lim sup_r→ 0μ(ψ(B(p,r)))/ν(ψ(B(p,r)))≤ C,    ∀ p∈ U,

    for some positive constant C depending only on U. From (<ref>), a classical application of Vitali covering's theorem ensures that for every open set A⊂ψ(U), μ(A)≤ Cν(A). Finally, the monotone class theorem allows to deduce that 
    
    μ(E)≤ Cν(E),    ∀ E⊂ψ(U) Borel.

    This implies that μ≪ν and the density is bounded by C. To prove that ν≪μ with a bounded density we repeat the same argument, observing that also μ is asympotically doubling, thanks to the 𝒬-Ahlfors regularity, and using the lower bound of the uniform Ball-Box theorem.





 §.§ Convergence to the metric measure tangent





In this section, we conclude the proof of Theorem <ref>. Recall that a pointed metric measure space is a metric measure space with a distinguished point. For such class of spaces, a natural notion of convergence is the pointed measured Gromov-Hausdorff convergence, see <cit.>. 



     Let (_n, _n,_n, x_n), n∈∪{∞} be a collection of pointed metric measure spaces. Then, we say that (_n, _n,_n, x_n) converges to (_∞, _∞,_∞, x_∞) in the pointed measured Gromov-Hausdorff sense (pmGH for short) if for any ε,R > 0 there exists N(ε,R)∈ such that for all n>N(ε,R) there exists a Borel map f^ε,R_n : B^_n(x_n,R)→_∞ such that
     
         
  (i) f^ε,R_n(x_n)=x_∞,
         
  (ii) sup_x,y ∈ B^_n(x_n,R) |_n(x,y)- _∞(f^ε,R_n(x), f^ε,R_n(y))|≤ε,
         
  (iii) the ε-neighbourhood of f^ε,R_n (B^_n(x_n,R)) contains B^_∞(x_∞,R-ε),
         
  (iv) (f^ε,R_n)_#(_nB^_n(x_n,R)) weakly converges to B^_∞(x_∞,R) as n→∞, for a.e. R > 0.
     
     
  

 The notion of metric measured tangents for a general metric measure space can be given according to the notion of pmGH-convergence. Let (,,) be a metric measure space, let x∈() and r∈(0,1). Define 

    _r^x:= (∫_B(x,r)1-1/r(·,x))^-1.


 
    Let (,,) be a metric measure space, let x∈() and r∈(0,1). and consider the rescaled and normalized pointed metric measure space (, _r,^x_r,x), where _r:=r^-1 and _r^x is defined in (<ref>). A pointed metric measure space ( Y,_Y, 𝔫, y) is called a tangent to (,,) at x if there exists a sequence of radii r_i→ 0^+ so that 
    (, _r_i,^x_r_i,x) → ( Y,_Y, 𝔫, y) as i→∞ in the pointed measured Gromov-Hausdorff topology. We denote the collection of all the tangents of (,,) at x by Tan(,,,x).    
 

We are now in position to prove our main result, namely Theorem <ref>. 

 
 
     Let M be a complete manifold, equipped with a bounded measure . Then, for -a.e. q∈ M, we have 
     
    Tan(M,,,q)={(M^q,^q,m(q)^n,0)},

     where (M^q,^q) is the nilpotent approximation of M at q and is isometric to a quotient Carnot group and m(q)>0 is defined by 
     
    m(q):=(∫_B̂^q(0,1)(1-^q)^n)^-1.
 
 

 
     Let q∈ M be a Lebesgue point for  and let  ψ^q:U→^n be a chart of privileged coordinates at q, with U open and relatively compact. Fix ε,R>0 and choose r_0=r_0(ε,R) sufficiently small to ensure that B^_r(q,R)=B^(q,rR)⊂ U for every r<r_0. In this way, we can work in coordinates and, for ease of notation, we can identify M with ^n and q with the origin (we also drop the superscript q of the nilpotent approximation). Then, define 
     
    f^ε,R_r : B^_r(0,R)→^n;     f^ε,R_r(p):=δ_r^-1(p),

     where δ_r:^n→^n is the dilation defined in (<ref>). Note that f^ε,R_r(0)=0 and therefore item (i) of Definition <ref> is verified. Possibly taking a smaller r_0, we may assume that B^(0,r_0R)⊂ O, where O is the set identified by Corollary <ref>. Moreover, as a consequence of Corollary <ref>, we have 
     
    δ_1/r(B^(0,rR))=B^^r(0,R),    ∀  r≤ r_0,

    From Theorem <ref>, we immediately deduce that δ_1/r(B^(0,rR))→ B^(0,R) as r→ 0 as sets, proving that also item (iii) of Definition <ref> is verified. Combining this convergence with Corollary <ref> and Theorem <ref>, we obtain
    
    sup_p,p' ∈ B^_r(0,R)|_r(p,p')- (f^ε,R_r(p), f^ε,R_r(p'))|    =sup_p,p' ∈ B^(0,rR)|1/r(p,p')-(δ_1/r(p),δ_1/r(p'))|
       =sup_p,p' ∈ B^(0,rR)|^r(δ_1/r(p),δ_1/r(p'))- (δ_1/r(p),δ_1/r(p'))|
       =sup_x,x' ∈δ_1/r(B^(0,rR))|^r(x,x')- (x,x')| 0.

     Hence, we have proved that, up to possibly choosing a smaller r_0, item (ii) of Definition <ref> is verified. Therefore we are left to check item (iv). 
     
     First of all,  is a bounded measure, hence ψ^q_#=ρ^n for a measurable function ρ with 0<c≤ρ≤ C. Thus, since 0∈^n is a Lebesgue point for , after the change of variables x↦δ^-1_r(x), we have 
     
    0=lim_r→ 01/r^𝒬(q)∫_B^(0,r)|ρ-ρ(0)|^n=lim_r→ 0∫_δ^-1_r(B^(0,r))|ρ∘δ_r-ρ(0)|^n,

     where ρ(0) is strictly positive and finite. Second of all, for every φ∈ L^∞(^n), we claim that 
     
    lim_r→ 0∫_δ^-1_r(B^(0,rR))φ(ρ∘δ_r)^n=ρ(0)∫_ B^(0,R)φ^n.

     Indeed, recall that δ^-1_r(B^(0,rR))→ B^(0,R) as r→ 0, as sets and we can compute: 
    
    |∫_δ^-1_r(B^(0,rR))φ(ρ∘δ_r)^n-ρ(0)∫_ B^(0,R)φ^n|
            
    ≤∫_δ^-1_r(B^(0,rR))|φ(ρ∘δ_r)-ρ(0)φ|^n+ρ(0)|∫_δ^-1_r(B^(0,rR))φ^n-∫_B^(0,R)φ^n|,

    which converges to 0 as a consequence of (<ref>).
    We are in position to prove the weak convergence of the sequence (f^ε,R_r)_#_r^0B^_r(0,R) to the tangent measure m(q)^nB^(0,R).
    Let φ∈ C_c(^n), then by definition of _r^0, we have
     
    ∫_^nφ (f^ε,R_r)_#(_r^0B^_r(0,R))   =(∫_B^(0,r)1-1/r(·,0))^-1∫_B^_r(0,R)(φ∘δ_r^-1)ρ^n 
       =(∫_B^(0,r)1-_r(·,0))^-1r^𝒬(q)∫_δ_r^-1(B^(0,rR))φ(ρ∘δ_r)^n,

     having performed a change of variable x↦δ_r(x). For the renormalization term, using once again (<ref>) and the convergence of the sets, we have  
 
 
 
 
 
 
    ∫_B^(0,r)1-_r(·,0)   =r^𝒬(q)∫_δ^-1_r(B^(0,r)) (1-_r(δ_r(·),0))ρ∘δ_r^n 
       =r^𝒬(q)∫_δ^-1_r(B^(0,r)) (1-^r(·,0))ρ∘δ_r^n r→0∼
     m(q)^-1 ρ(0) r^𝒬(q),

     where m(q)>0 is defined in (<ref>). Hence, combining the convergence (<ref>) with (<ref>), we may deduce that 
     
    lim_r→0∫_^nφ (f^ε,R_r)_#(_r^0B^_r(0,R))   =m(q)∫_B^(0,R)φ^n=m(q)∫_^nφ(^nB^(0,R)),

     concluding the proof of item (iv).
 





§ APPLICATIONS TO THE CURVATURE-DIMENSION CONDITION



Recall that the Heisenberg group  is the only Carnot group of topological dimension 3 and Hausdorff dimension 4. A Heisenberg group is the Heisenberg group, equipped with a norm :^2→_+ on its first layer. We refer the reader to <cit.> for a precise definition. 


    Let (M,ξ,) be a complete equiregular manifold with growth vector (2,3), equipped with a locally 4-Ahlfors regular measure . Then, for -a.e. q∈ M, we have 
     
    Tan(M,,,q)={(,^q,m(q)^3,0)},

     where  is the Heisenberg group, ^q is the distance on  induced by _q:^2→_+, and m(q)>0 is defined by (<ref>). 



    Applying Theorem <ref>, we deduce that, at -almost every point q∈ M, the nilpotent approximation (M^q,^q) of M, equipped with the measure m(q)^3 and with distinguished point 0, is the unique (pointed) metric measure tangent at q. Since q is a regular point, Theorem <ref> implies that (M^q,^q) is a Carnot group of (topological) dimension 3 and thus it must be the Heisenberg group, where ^q is defined in Definition <ref>.    
 




    Let (M,ξ,) be a complete equiregular manifold with growth vector (2,3), equipped with a locally 4-Ahlfors regular measure . Then, (M,,) does not satisfy the (K,N) condition for every K>0 and N∈ (1,∞).



    Arguing by contradiction, we assume that (M,,) satisfies the (K,N) condition for some K>0 and N∈ (1,∞).
    The scaling property of the  condition (see <cit.>) guarantees that the metric measure space (, _r,^x_r) satisfies the (r^2 K, N) condition. Moreover, as the  condition is stable with respect to the pointed measured Gromov-Hausdorff convergence (see <cit.>), Proposition <ref> allows us to conclude that 
    
    (,^q,m(q)^3,0)   is a (0,N) space.

    Finally, Theorem 1.6 in <cit.> gives the desired contradiction.



    
    In certain situations, the strategy used in the previous proof can be refined to show the failure of the measure contraction property (K,N) (cf. <cit.> for a definition). Indeed, under the same assumptions of Proposition <ref>, if there exists a positive -measure set where _q is neither C^1 nor strongly convex, then Theorem 1.1 of <cit.> implies that (K,N) fails for every K∈ and every N∈ (1,∞). 



    Propositions <ref> and <ref> apply in the case M is a 3D-contact manifold (see <cit.> for a definition), since any 3D-contact manifold is equiregular with growth vector (2,3).


The following corollary is a specification of Proposition <ref> in the case of the Heisenberg group. We believe it is worth emphasize it, in light of <cit.>.


    
    Let  be a Heisenberg group, equipped with a locally 4-Ahlfors regular measure . Then, (,,) does not satisfy the (K,N) condition for every K>0 and N∈ (1,∞).


Finally, we extend the known results to bounded measures.


    Let M be a complete manifold with M=n, equipped with a bounded measure . Assume there exists an -positive set 𝒮⊂ M such that r(q)<n, for every q∈𝒮. Then, (M,,) does not satisfy the (K,N) condition for every K>0 and N∈ (1,∞).



    Assume by contradiction that (M,,) satisfies the (K,N) condition for some K>0 and N∈ (1,∞). Since 𝒮 has positive measure, we can find q∈𝒮 which is a Lebesgue point for , cf. Lemma <ref>. By Theorem <ref>, the unique (pointed) metric measure tangent of M at q is a manifold (M^q,^q), equipped with the measure m(q)^n and with distinguished point 0. 
    Arguing as in the proof of Proposition <ref>, we conclude that (M^q,^q, m(q)^n) is a (0,N) space. This is in contradiction with <cit.>, since r(0)<n. 



alpha 


