Universal Infinitesimal Hilbertianity of Sub-Riemannian Manifolds

We prove that sub-Riemannian manifolds are infinitesimally Hilbertian (i.e., the associated Sobolev space is Hilbert) when equipped with an arbitrary Radon measure. The result follows from an embedding of metric derivations into the space of square-integrable sections of the horizontal bundle, which we obtain on all weighted sub-Finsler manifolds. As an intermediate tool, of independent interest, we show that any sub-Finsler distance can be monotonically approximated from below by Finsler ones. All the results are obtained in the general setting of possibly rank-varying structures.


Introduction
General overview.In the last two decades, weakly differentiable functions over metric measure spaces have been extensively studied and have played a fundamental role in the development of abstract calculus in the nonsmooth setting (see, e.g., [15,12,11]).The definition of Sobolev space we adopt in this paper is the one introduced in [9], which is equivalent to the notions proposed in [7,20,5].At this level of generality, however, Sobolev calculus might not be fully satisfactory from a functional-analytic viewpoint.For instance, not only the Sobolev space can fail to be Hilbert (consider the Euclidean space endowed with the L ∞ -norm and the Lebesgue measure), but it can be also non-reflexive (as shown in [2,Proposition 7.8]).In view of this, the class of infinitesimally Hilbertian metric measure spaces (i.e., whose associated Sobolev space is Hilbert) is particularly relevant.These spaces enjoy nice features, among which the strong density of boundedly-supported Lipschitz functions in the Sobolev space (as proven in [4]); we refer to the introduction of [18] for an account of the several benefits of working within this class of spaces.
A strictly related concept is that of universally infinitesimally Hilbertian metric space, that is to say, a metric space that is infinitesimally Hilbertian with respect to whichever Radon measure.The interest in this property is mainly motivated by the study of metric structures that are important from a geometric perspective, but do not carry any 'canonical' measure (such as sub-Riemannian manifolds that are not equiregular).The purpose of this paper is to prove the following claim: All sub-Riemannian manifolds are universally infinitesimally Hilbertian.
The goal will be achieved by building an isometric embedding of the 'analytic' space of derivations over any weighted sub-Finsler manifold (which provide us with a synthetic notion of vector field, linked to the Sobolev calculus) into the 'geometric' space of sections of the horizontal bundle.The abstract differential structure of the space under consideration and the behaviour of its (purely metric) tangent spaces are -a priori -unrelated, thus the role of the above-mentioned embedding result is to bridge this gap, showing that Sobolev functions are suitable to capture the fiberwise Hilbertianity of the horizontal bundle.As an intermediate tool, of independent interest, we prove that a sub-Finsler distance can be monotonically approximated from below by Finsler distances.
Outline of the paper.We consider a (generalised) sub-Finsler manifold (M, E, σ, ψ).This means that M is a smooth connected manifold, while E is a smooth vector bundle over M equipped with a continuous metric σ : E → [0, +∞) (as in Definition 3.7) and ψ : E → TM is a bundle morphism; moreover, a Hörmander-like condition is required to hold, cf.Definition 4.1.Whenever it holds that for every x ∈ M the norm σ| Ex on the fiber E x is induced by a scalar product that smoothly depends on x, we say that (M, E, σ, ψ) is a (generalised) sub-Riemannian manifold.This notion of sub-Riemannian manifold is the most general one that we have in the literature (see, e.g., [1]).
The horizontal bundle HM is obtained by 'patching together' the horizontal fibers D x := ψ(E x ), which form a continuous distribution on M (in the sense of Theorem 3.2).We then define a generalised metric ρ : TM → [0, +∞] over the tangent bundle (cf.Definition 3.4 for this term) as ρ(x, v) = v x := inf σ(u) u ∈ E x , (x, v) = ψ(u) , for every (x, v) ∈ TM.
Observe that the finiteness domain of ρ(x, •) coincides with the horizontal fiber D x for every x ∈ M. The space M can be made into a metric space by considering the Carnot-Carathéodory distance: given any two points x, y ∈ M, we define d CC (x, y) as the length of the shortest path among all horizontal curves (i.e., tangent to HM) joining x and y.Here, the length of a horizontal curve is computed with respect to the generalised metric ρ.See Definition 4.3 for the details.
Let us now fix a non-negative Radon measure µ on (M, d CC ), say that µ is finite (for simplicity).We may consider two (completely different in nature) notions of vector field over (M, d CC , µ): • The space Der 2,2 (M; µ) of L 2 -derivations having divergence in L 2 (in the sense of [9]).
These are linear functionals acting on Lipschitz functions and taking values into the space of Borel functions over M, that satisfy a suitable Leibniz rule and a locality property.The Sobolev space W 1,2 (M, d CC , µ) is then obtained in duality with Der 2,2 (M; µ), as described in Definition 2.4.The whole Section 2 is devoted to the key results about L 2 -derivations.• The space L 2 (HM; µ) of 2-integrable sections of the horizontal bundle; see Definition 4.7.
Whenever M is a sub-Riemannian manifold, the elements of L 2 (HM; µ) satisfy a pointwise parallelogram rule (thanks to geometric considerations, see Remark 4.8).Nevertheless, it is not clear -a priori -how to deduce from this information that the metric measure space (M, d CC , µ) is infinitesimally Hilbertian.
The main result of the paper aims at providing a relation between Der 2,2 (M; µ) and L 2 (HM; µ): the former space is isometrically embeddable into the latter one.The precise statement is: Theorem 1.1 (Embedding theorem).Let (M, E, σ, ψ) be a sub-Finsler manifold with d CC complete.Let µ be a finite, non-negative Borel measure on (M, d CC ).Then there exists a unique linear operator I : Der 2,2 (M; µ) → L 2 (HM; µ) such that As a consequence, sub-Riemannian manifolds are universally infinitesimally Hilbertian: Theorem 1.2 (Infinitesimal Hilbertianity of sub-Riemannian manifolds).Let (M, E, σ, ψ) be a sub-Riemannian manifold with d CC complete.Let µ be a non-negative Radon measure on (M, d CC ).Then the metric measure space (M, d CC , µ) is infinitesimally Hilbertian.
The proof of the embedding result (Theorem 1.1) builds upon the following key ingredients: a) The Carnot-Carathéodory distance d CC can be written as pointwise limit of an increasing sequence of Finsler distances; cf.Theorem 5.1.This property follows from the results we develop in Section 3, where we show that the sub-Finsler metric ρ (or, more generally, any generalised metric as in Definition 3.4) can be approximated from below by Finsler ones.This technical statement can be achieved by exploiting the lower semicontinuity of ρ, as done in Lemma 3.8.b) The pointwise norm of a given derivation can be recovered by just considering its evaluation at smooth 1-Lipschitz functions.More precisely, we can find a sequence (f n ) n ⊆ C 1 c (M) of 1-Lipschitz functions (with respect to d CC ) such that the identity |b| = sup n b(f n ) holds µ-a.e. for every b ∈ Der 2,2 (M; µ).This representation formula is obtained by combining item a) above with an approximation result for Finsler manifolds proven in [18].c) Any derivation b ∈ Der 2,2 (M; µ) can be represented by a suitable measure π on the space of continuous curves in M, as granted by the metric version [19] of Smirnov's superposition principle for normal 1-currents; see Theorem 2.8.The presence of such a measure π is an essential tool in the construction of the embedding map I : Der 2,2 (M; µ) → L 2 (HM; µ), which preserves the pointwise norm of all vector fields as a consequence of item b).
Comparison with previous works.The results of the present paper enrich the list of metric spaces that are known to be universally infnitesimally Hilbertian, which previously consisted of: i) Euclidean spaces [13], ii) Riemannian manifolds [18], iii) Carnot groups [18], iv) locally CAT(κ)-spaces [10].
Let us now briefly comment on the main differences and analogies between the technique we exploit here and the previous approaches.To the best of our knowledge, the strategy proposed in [13,18] does not carry over to the framework of sub-Riemannian manifolds.In the classes of spaces i), ii), and iii), a fact which plays a fundamental role is that any Lipschitz function (with respect to the relevant distance) can be approximated by smooth ones having the same Lipschitz constant; it seems that this property, achieved by a convolution argument, cannot be generalised to sub-Riemannian manifolds, the problem being to keep the Lipschitz constant under control.
However, a different approach has been developed in [10] in order to overcome the lack of smoothness of the spaces in iv).The proof in the sub-Riemannian case is inspired by the ideas introduced in [10]: indeed, the universal infinitesimal Hilbertianity of CAT spaces stems -similarly to what described above -from an embedding result, which in turn relies upon Smirnov's superposition principle and a representation formula for the pointwise norm of derivations.While the former is available on any metric measure space, the latter requires an ad hoc argument for the sub-Riemannian setting.This makes a significant difference with [10]: on CAT spaces, distance functions from given points are 1-Lipschitz and everywhere have some form of differentiability, thus they are suitable candidates for the representation formula; on sub-Riemannian manifolds, on the contrary, this is no longer true, whence we need to find an alternative way to show that there is plenty of smooth 1-Lipschitz functions that are µ-a.e.differentiable (where µ is an arbitrary measure).Most of the present paper is actually dedicated to addressing this last point.Once the representation formula is at disposal, the proof of the embedding result closely follows along the lines of [10,Theorem 6.2].
We call LIP(X) or LIP d (X) the space of real-valued Lipschitz functions on (X, d), while LIP bs (X) or LIP d bs (X) stand for the set of elements of LIP(X) with bounded support.The (global) Lipschitz constant of f ∈ LIP(X) is denoted by Lip(f ) or Lip d (f ), while the functions lip(f ) : X → [0, +∞) and lip a (f ) : X → [0, +∞) are defined as whenever x ∈ X is an accumulation point, and lip(f )(x) = lip a (f )(x) := 0 elsewhere.We say that lip(f ) and lip a (f ) are the local Lipschitz constant and the asymptotic Lipschitz constant of the function f , respectively.The vector space of all (equivalence classes up to m-a.e.equality of) real-valued Borel functions on X is denoted by L 0 (m).
A derivation on (X, d, m) is a linear map b : LIP bs (X) → L 0 (m) with these two properties:  The space Der 2,2 (X; m) is a module over the commutative ring LIP bs (X) and is a Banach space when endowed with the norm Der 2,2 (X; Lemma 2.2.Let (X, d, m) be a metric measure space and b ∈ Der 2,2 (X; m).Let (f n ) n ⊆ LIP bs (X) be a sequence with sup n Lip(f n ) < +∞ that pointwise converges to some limit f ∈ LIP bs (X).Then Proof.See item (1) of [10,Lemma 5.4].
It will be convenient to work with the following representation formula for the pointwise norm of the elements of Der 2,2 (X; m).Proposition 2.3.Let (X, d, m) be a metric measure space.Let b ∈ Der 2,2 (X; m) be given.Fix a countable dense set (x k ) k ⊆ X.For any j, k ∈ N, let η jk : X → [0, 1−1/j] be a boundedly-supported Lipschitz function such that η jk = 1 − 1/j on B j (x k ) and Lip(η jk ) ≤ 1/j 2 .Then it holds that Proof.It follows from [10,Proposition 5.5].
By duality with Der 2,2 (X; m), it is possible to introduce a notion of Sobolev space W 1,2 (X, d, m).
Definition 2.4 (Sobolev space).Let (X, d, m) be a metric measure space.Then we say that a function f ∈ L 2 (m) belongs to the Sobolev space W 1,2 (X, d, m) provided there exists a continuous morphism L f : Der 2,2 (X; m) → L 1 (m) of LIP bs (X)-modules such that The map L f is uniquely determined.Furthermore, there exists a function G ∈ L 2 (m) such that for every b ∈ Der 2,2 (X; m).
The minimal such function G (in the m-a.e.sense) is called 2-weak gradient of f and is denoted by ) is a Banach space when equipped with the norm for every f ∈ W Given any function f ∈ W 1,2 (X, d, m), we define the element L f ∈ B as Then it holds that the map W The following definition -which has been introduced in [12] -plays a key role in this paper.
The following result provides a sufficient condition for the infinitesimal Hilbertianity to hold.
Proposition 2.7.Let (X, d, m) be a metric measure space.Suppose that  d, m) is a Hilbert space, as required.
Finally, we conclude the subsection by reporting the following consequence of the metric version of Smirnov's superposition principle, which has been proven by E. Paolini and E. Stepanov in [19].
Theorem 2.8 (Superposition principle).Let (X, d, m) be a metric measure space with m finite.Let b ∈ Der 2,2 (X; m).Then there exists a finite, non-negative Borel measure π on C [0, 1], X , concentrated on the set of non-constant Lipschitz curves on X having constant speed, such that for every f, g ∈ LIP bs (X).

Monotone approximation of generalised metrics
3.1.Set-up and auxiliary results.We begin with some classical definitions.A norm n defined on a finite-dimensional vector space V is said to be smooth provided it is of class C ∞ on V \ {0}.
In addition, we say that n is strongly convex if the Hessian matrix of n 2 at any vector v ∈ V \ {0} is positive definite.With the notation W ≤ V we intend that W is a vector subspace of V .
By smooth manifold we shall always mean a connected differentiable manifold of class C ∞ .Given a smooth manifold M and a smooth vector bundle (E, π) over M, we say that a function is continuous, it is smooth on the complement of the zero section, and F | Ex is a strongly convex norm on the fiber E x := π −1 (x) for every x ∈ M.
By Finsler metric on M we mean a Finsler metric F over the tangent bundle TM.In this case, we also say that the couple (M, F ) is a Finsler manifold.(In the literature, (M, F ) is often referred to as a reversible Finsler manifold; cf., for instance, the monograph [6].)Definition 3.1 (Generalised norm).Let V be a vector space.Then a function n : V → [0, +∞] is said to be a generalised norm if there exists a vector subspace Theorem 3.2 (Definition of continuous distribution).Let M be a smooth manifold and let (E, π) be a smooth vector bundle over M. Let {V x } x∈M be a family of vector spaces such that V x ≤ E x for all x ∈ M. Then the following conditions are equivalent: i) Given x ∈ M and v ∈ V x, there exists a continuous section v of E, defined on some neighbourhood U of x, such that v(x) = v and v(x) ∈ V x for every x ∈ U .ii) Given x ∈ M, there exist finitely many continuous sections v 1 , . . ., v k of E, defined on some neighbourhood U of x, such that V x = span v 1 (x), . . ., v k (x) for every x ∈ U .iii) Given x ∈ M, there exist a neighbourhood U of x, a smooth vector bundle Ẽ over U , and a continuous vector bundle morphism ψ : Ẽ → E| U , such that V x = ψ( Ẽx ) for all x ∈ U .
If the above conditions are satisfied, we say that {V x } x∈M is a continuous distribution (of possibly varying rank) over M.Moreover, we can assume that k and the rank of Ẽ are at most d 2 2•5 n −1 , where d is the rank of E and n is the dimension of M.
Proof.The real novelty of the theorem is the implication i) =⇒ ii).i) =⇒ ii) Suppose item i) holds.Given a point x ∈ M, we can choose an open set U ′ ⊆ R n containing 0 and a map ϕ : U ′ → M satisfying ϕ(0) = x that is a homeomorphism with its image.
Possibly shrinking U ′ , we can assume there exists a Finsler metric F over E| U , where U := ϕ(U ′ ).Fix any radius λ > 0 such that Bλ (0) ⊆ U ′ and call K := ϕ Bλ (0) ⊆ U .For any i = 1, . . ., d we set In order to prove ii), it would be enough to find some finite families F 1 ⊆ . . .⊆ F d of continuous sections of E| K such that for any i = 1, . . ., d it holds that We build F 1 , . . ., F d via a recursive argument.Suppose to have already defined F 1 , . . ., F i for some i < d.Notice that item i) grants that the function M ∋ x → dim V x is lower semicontinuous, thus C i is a compact set.For any j ∈ N we define the compact set K j ⊆ K as Observe that j∈N K j = K \ C i and that Kj ∩ Kj ′ = ∅ for all j, j ′ ∈ N such that |j − j ′ | is even.Let j ∈ N be fixed.For any x ∈ K j , we choose a vector wx ∈ V x such that F ( wx ) = 1 and dim F i (x)+R wx ≥ i+1.By item i), we can find a neighbourhood W x ⊆ U of x and a continuous section w x of E| Wx , such that w x (x) = wx and w x (z) ∈ V z for all z ∈ W x .Possibly shrinking W x , we can further assume that 0 < F w x (z) ≤ 2 and dim F i (z) + R w x (z) ≥ i + 1 hold for every point z ∈ W x .By compactness of K j , we can thus find an open covering W 1 , . . ., W m ⊆ U of K j and continuous sections w 1 , . . ., w m of E| W1 , . . ., E| Wm , respectively, such that 0 < F w ι (x) ≤ 2 and dim F i (x) + R w ι (x) ≥ i + 1 for every ι = 1, . . ., m and x ∈ W ι .By Lebesgue's number lemma, there exists r > 0 such that any ball in R n of radius r centered at ϕ −1 (K j ) is entirely contained in one of the sets ϕ −1 (W 1 ), . . ., ϕ −1 (W m ).Choose a maximal r-separated subset S of ϕ −1 (K j ), i.e., S is maximal among all subsets satisfying |p − q| ≥ r for every p, q ∈ S with p = q.Note that S is a finite set by compactness of ϕ −1 (K j ).For any p ∈ S, call G p := ϕ B r (p) and pick ι(p) ∈ {1, . . ., m} such that G p ⊆ W ι(p) .By definition of S, it holds that K j ⊆ p∈S G p .Moreover, given any p ∈ S we have that the balls B r/2 (q) : q ∈ S \ {p}, |p − q| < 2r are pairwise disjoint and contained in B 5r/2 (p)\ B r/2 (p), whence accordingly # q ∈ S \ {p} : |p− q| < 2r ≤ 5 n − 1 for all p ∈ S. Therefore, we can take a partition S = S 1 ∪ . . .∪ S 5 n with the property that G p ∩ G q = ∅ whenever ℓ = 1, . . ., 5 n and p, q ∈ S ℓ satisfy p = q.Given ℓ = 1, . . ., 5 n and p ∈ S ℓ , we can pick a continuous function Notice that F v jα (x) ≤ 2 • 5 n for every α ∈ {−1, 1} 5 n −1 and x ∈ K j .For any j ∈ N we fix a continuous functions η j : K → [0, 1] such that η j = 0 on K \ Kj and η j > 0 on Kj .Then we set Therefore, it follows from the construction that the family F i+1 := F i ∪F ′ i+1 of continuous sections of E| K satisfies dim F i+1 (x) ≥ i + 1 for every x ∈ K, as required.Observe also that #F ′ i ≤ 2 2•5 n −1 for all i = 1, . . ., d, thus the cardinality of F := F d does not exceed d 2 2•5 n −1 .This proves ii).ii) =⇒ iii) Suppose item ii) holds.Given a point x ∈ M, pick a neighbourhood U of x and some continuous sections v 1 , . . ., v k of E| U such that V x = span v 1 (x), . . ., v k (x) for every x ∈ U .Let us define Ẽ := U × R k and the continuous vector bundle morphism ψ : Ẽ → E| U as Therefore, we conclude that ψ( Ẽx ) = span v 1 (x), . . ., v k (x) = V x for all x ∈ U , thus proving iii).iii) =⇒ i) Suppose item iii) holds.Fix x ∈ M and v ∈ V x.There exist a smooth vector bundle Ẽ over some neighbourhood U ′ of x and a continuous vector bundle morphism ψ : Ẽ → E| U ′ such that V x = ψ( Ẽx ) for all x ∈ U ′ .Choose any w ∈ Ẽx for which ψ( w) = v.Then we can find a neighbourhood U ⊆ U ′ of x and a continuous section w of Ẽ| U such that w(x) = w.Now let us define v(x) := ψ w(x) for every x ∈ U .Therefore, it holds that v is a continuous section of E| U such that v(x) = v and v(x) ∈ V x for all x ∈ U , thus proving i).
Remark 3.3.As already observed during the proof of Theorem 3.2, the function M ∋ x → dim V x is lower semicontinuous whenever {V x } x∈M is a continuous distribution over M. Definition 3.4 (Generalised metric).Let M be a smooth manifold, (E, π) a smooth vector bundle over M. Then a generalised metric over E is a lower semicontinuous function ρ : E → [0, +∞] having the following properties: ii) The family D(ρ x )} x∈M is a continuous distribution.
In the case E = TM, we just say that ρ is a generalised metric on M.
Let us fix some notation: given any vector subspace V ≤ R d , we denote by V ⊥ its orthogonal complement (with respect to the Euclidean norm).We denote by S d−1 the Euclidean unit sphere in R d , i.e., the set of all points (3.1) Then there exists a norm n on R d such that the following properties are satisfied: thus completing the proof of the statement.
In the following results, we shall consider the trivial bundle M × R d over M. Given any x ∈ M, a vector subspace of the fiber of M × R d at x is of the form {x} × V , for some vector subspace V ≤ R d .For simplicity, we will always implicitly identify {x} × V with the vector space V itself.Lemma 3.6.Let M be a smooth manifold and ρ a generalised metric over M × R d .Fix x ∈ M and any norm • on R d .Call V x := D(ρ x ) for every x ∈ M and k := dim V x.Then for any ε > 0 there exists a neighbourhood U of x such that where the Hausdorff distance d H is computed with respect to the norm • .
Proof.Since {V x } x∈M is a continuous distribution, we can find a neighbourhood U ′ of x and some continuous maps v 1 , . . ., v k ′ : U ′ → R d such that V x = span v 1 (x), . . ., v k ′ (x) for all x ∈ U ′ .Up to relabelling, we can assume that v 1 (x), . . ., v k (x) constitute a basis of V x.Then there is a neighbourhood U ⊆ U ′ of x such that v 1 (x), . . ., v k (x) are linearly independent for all x ∈ U .Define W x := span v 1 (x), . . ., v k (x) for every point x ∈ U .Let us apply a Gram-Schmidt orthogonalisation process to the vector fields v 1 , . . ., v k , with respect to the Euclidean norm | • |: for every x ∈ U .Therefore, the resulting continuous maps w 1 , . . ., w k : U → R d satisfy w i (x) • w j (x) = δ ij for every i, j = 1, . . ., k and x ∈ U, W x = span w 1 (x), . . ., w k (x) for every x ∈ U.
Fix any C > 0 such that v ≤ C |v| for all v ∈ R d .Possibly shrinking U , we can assume that ≤ sup for every x ∈ U .The statement follows by noticing that Definition 3.7 (Continuous metric).Let M be a smooth manifold.Let (E, π) be a smooth vector bundle over M. Then a continuous metric σ over E is a continuous function σ : E → [0, +∞) such that σ| Ex is a norm on the fiber E x for every point x ∈ M.
Lemma 3.8.Let M be a smooth manifold, ρ a generalised metric over M × R d .Call V x := D(ρ x ) for every x ∈ M. Fix x ∈ M and two constants ε, λ > 0. Let σ be a continuous metric over M × R d such that σ(x, v) < ρ(x, v) for every x ∈ M and v ∈ R d \ {0}.Then there exist a smooth, strongly convex norm n on R d and a neighbourhood U of x such that the following properties hold: Proof.We divide the proof into several steps: Step 1. Lemma 3.5 grants the existence of a norm n ′ on R d such that (3.4) Given that M ∋ x → dim V x is lower semicontinuous, we can find a neighbourhood U ′ of x such that dim V x is the minimum of the function Step 2. The function we deduce from (3.5) that there exists a neighbourhood , which together with (3.6) imply that ) − ε by the first line of (3.4).
Step 3. In light of (3.7), there exists a constant δ ′ > 0 with (λ + 1 Choose any smooth norm • on R d such that v − n ′′ (v) ≤ δ ′ /2 holds for all v ∈ S d−1 , whose existence follows, e.g., from [14,Theorem 103].Then let us finally define the sought norm n as Clearly, it is a smooth and strongly convex norm by construction.Moreover, it can be immediately checked that n satisfies Accordingly, by combining (3.8) with (3.9) we obtain that (3.10) Step 4. Observe that (3.9) and the third line of (3.4) give that for some neighbourhood U ⊆ U ′′ of x we have Therefore, item iii) is verified (recall the last claim in Step 1).Finally, we deduce from (3.10) that also items i) and ii) hold, thus concluding the proof of the statement.

The approximation result.
Let M be a smooth manifold and let ρ be a generalised metric over M × R d .Calling V x := D(ρ x ) for every x ∈ M, it holds that M ∋ x → dim V x is a lower semicontinuous function (recall Remark 3.3), thus for any x ∈ M there exists r x > 0 such that dim V x = min dim V y y ∈ B rx (x) .

Let us define
G n := x ∈ M r x ≥ 1/n for every n ∈ N. (3.12) Observe that for any point x ∈ M there exists n ∈ N such that x ∈ n≥n G n .
Proposition 3.9.Let M be a smooth manifold.Let d be any distance on M that induces the manifold topology.Fix a generalised metric ρ over M × R d .Then there exists a sequence (F n ) n of Finsler metrics over M × R d such that the following properties are satisfied: , where V x := D(ρ x ) for all x ∈ M and the set G n is defined as in (3.12).c) For any n ∈ N there exists a countable set S n ⊆ M such that where the Hausdorff distance d H is computed with respect to the norm Proof.We recursively define the Finsler metrics F n : M × R d → [0, +∞).Suppose to have already defined F 0 , . . ., F n−1 for some n ∈ N, where F 0 := 0. By using Lemma 3.6, Lemma 3.8, and the paracompactness of M, we can find a family (U n i , z n i , n n i ) : i ∈ N such that: i) {U n i } i∈N is a locally finite, open covering of M, and diam(U n i ) < 1/n for every i ∈ N.
ii) z n i ∈ U n i and dim V z n i = min{dim V x : x ∈ U n i } for every i ∈ N. iii) Given any i ∈ N, we have that n n i is a smooth, strongly convex norm on R d that satisfies where d H is intended with respect to Since each norm n n i is smooth and strongly convex, it can be readily checked that F n is a Finsler metric over M × R d .Let us then conclude by verifying that F n satisfies the desired properties: a) It follows from iii) that Indeed, we know that r x ≥ 1/n by definition of G n , whence it holds U n i ⊆ B rx (x) by i) and accordingly dim V x = dim V z n i by ii).This shows that x ∈ D n i , thus proving the above claim.Fix v ∈ V ⊥ x ∩ S d−1 .Therefore, we deduce from the previous claim and v) that Moreover, as in the proof of item b) we deduce that x ∈ D n j .Therefore, we know from item vi) , as diam(U n j ) < 1/n by i).This gives the statement.
Lemma 3.10.M, ρ, and (F n ) n be as in Proposition 3.9.Then it holds that Proof.It clearly suffices to prove that F n (x, v) ր ρ(x, v) for any fixed x ∈ M and v ∈ S d−1 .Case 1. Assume v ∈ V x := D(ρ x ).We argue by contradiction: suppose there is t > 0 such that Choose any n ∈ N such that 1/n < min{r, t/4} and x ∈ n≥n G n , with G n defined as in (3.12).Fix any n ≥ n.Item d) of Proposition 3.9 grants the existence of a point z ∈ S n ∩ B 1/n (x) and a ) of Proposition 3.9 ensures that F n (z, w z ) ≥ ρ(z, w z ) − 1/n.All in all, we conclude that which is in contradiction with (3.13).This proves that , and β > 0 for which v = v ′ + β w.Fix any n ∈ N with x ∈ n≥n G n .Then item b) of Proposition 3.9 yields F n (x, w) ≥ n for all n ≥ n.Taking into account item a) of Proposition 3.9, this gives Therefore, the statement is proven.Theorem 3.11 (Approximation of generalised metrics).Let M be a smooth manifold.Let ρ be a generalised metric on M. Then there exists a sequence (F n ) n of Finsler metrics on M such that Given any i ∈ N, we can apply Proposition 3.9 and Lemma 3.10 to obtain a sequence ( It can be readily checked that (F n ) n is a sequence of Finsler metrics on M satisfying (3.15).
Remark 3.12.Under some additional assumptions, we can actually improve the statement of Theorem 3.11: if we further suppose that ρ x | D(ρx) is a Hilbert norm for every x ∈ M, then there exists a sequence (g n ) n of Riemannian metrics on M such that This fact can be proven by slightly modifying (actually, simplifying) the arguments we discussed in the present section.More precisely, it is sufficient to notice that in this case the norm n defined in (3.Given a smooth manifold M, we denote by Vec(M) the space of all smooth vector fields on M.Moreover, we define the map Der : It is well-known that Der is a Borel map.For any v, w ∈ Vec(M), we denote by [v, w] ∈ Vec(M) the Lie brackets of v and w.Given any subset F of Vec(M), we define the space Lie(F ) ⊆ Vec(M) as the Lie algebra generated by the family F , i.e., We set Lie x (F ) := v(x) : v ∈ Lie(F ) ≤ T x M for every x ∈ M. We say that the family F satisfies the Hörmander condition provided Lie x (F ) = T x M holds for every x ∈ M.
Definition 4.1 (Sub-Finsler manifold).Let M be a smooth manifold.Then a triple (E, σ, ψ) is said to be a sub-Finsler structure on M provided the following properties hold: i) E is a smooth vector bundle over M, ii) σ is a continuous metric over E, iii) ψ : E → TM is a morphism of smooth vector bundles such that the family D of smooth horizontal vector fields on M, which is defined as satisfies the Hörmander condition.
The quadruple (M, E, σ, ψ) is said to be a generalised sub-Finsler manifold (or just a sub-Finsler manifold, for brevity).If (E x , σ| Ex ) is a Hilbert space for every x ∈ M and the family of squared norms (σ| Ex ) 2 smoothly depends on x, then (M, E, σ, ψ) is called a generalised sub-Riemannian manifold (or just a sub-Riemannian manifold).
The family D of smooth horizontal vector fields is a finitely-generated module over C ∞ (M).The continuous distribution {D x } x∈M associated with (M, E, σ, ψ) is defined as We say that r(x) := dim D x ≤ dim M is the rank of the sub-Finsler structure (E, σ, ψ) at x ∈ M.
Given any point x ∈ M and any vector v ∈ D x , we define the quantity v x as Therefore, it holds that • x is a norm on D x .Furthermore, if (E, σ, ψ) is a sub-Riemannian structure on M, then each norm • x is induced by some scalar product •, • x .
Definition 4.2 (Horizontal curve).Let (M, E, σ, ψ) be a sub-Finsler manifold.Let γ : [0, 1] → M be a continuous curve such that for any t ∈ [0, 1] there exist δ > 0 and a chart (U, φ) of M such that γ(I) ⊆ U and φ • γ| I : I → R dim M is Lipschitz, where we set I := ( t − δ, t + δ) ∩ [0, 1].Then the curve γ is said to be horizontal provided there is an L ∞ -section u of the pullback bundle γ * E -i.e., a map [0, 1] ∋ t → u(t) ∈ E γt that is measurable and essentially bounded -such that The sub-Finsler length of the curve γ is defined as ℓ CC (γ) := The metric space (M, d CC ) is complete if and only if Br (x) is compact for all x ∈ M and r > 0.
Proof.First of all, observe that x → D(ρ x ) = D x is a continuous distribution by Theorem 3.2.
With this said, we only have to prove that the function ρ is lower semicontinuous.To this aim, let us fix a sequence ( Without loss of generality, we can assume that lim n ρ(x n , v n ) < +∞.For any n ∈ N choose an element Therefore, it holds that σ(u n k ) k∈N is bounded, thus there exists u ∞ ∈ E x∞ such that (possibly passing to a not relabelled subsequence) we have ) by continuity of ψ.Consequently, by using the continuity of σ and the definition of ρ x∞ we conclude that which proves the claim (4.5).Hence, the statement is finally achieved.
A vector field v : M → TM is said to be a section of HM provided v(x) ∈ D x for every x ∈ M. We say that a section v of HM is Borel provided it is Borel measurable as a map from M to TM.
It immediately follows from Lemma 4.6 that x ∈ R is a Borel function, for every Borel section v of HM.
The space of Borel sections of HM is a vector space with respect to the usual pointwise operations.
Definition 4.7 (The space L 2 (HM; µ)).Let (M, E, σ, ψ) be a sub-Finsler manifold.Let µ be a non-negative Borel measure on (M, d CC ).Then we define the space L 2 (HM; µ) as the set of (equivalence classes up to µ-a.e.equality of ) all Borel sections v of the horizontal bundle HM such that M ∋ x → v(x) x ∈ R belongs to L 2 (µ).The space L 2 (HM; µ) is an L ∞ (µ)-module with respect to the natural pointwise operations, thus in particular it is a vector space.
Given that each space D x , • x is Hilbert, we readily deduce that v(x) + w(x) holds for µ-a.e.x ∈ M, for every v, w ∈ L 2 (HM; µ).
Given any smooth function f ∈ C ∞ (M), we denote by df its differential, which is a smooth section of the cotangent bundle T * M. Then the horizontal differential d H f of f is defined as Lemma 4.9.Let (M, E, σ, ψ) be a given sub-Finsler manifold.Then there exists a countable family of functions Proof.Call n := dim M. By Lindelöf lemma we know that there exists an open covering (Ω j ) j∈N of M with the following property: for every j ∈ N there exist some functions x for every x ∈ Ω j .Therefore, the countable family of functions C := j∈N V j fulfills the required properties.
5. Main result: infinitesimal Hilbertianity of sub-Riemannian manifolds 5.1.Derivations on weighted sub-Finsler manifolds.The aim of this subsection is to provide an alternative to the representation formula (2.1) for the pointwise norm of a derivation (with divergence) over a weighted sub-Finsler manifold M. We would like to express the pointwise norm of a derivation b as the essential supremum of the functions b(f ), where f varies in a countable family of 1-Lipschitz smooth functions.Given that the distance functions d CC (•, x) from fixed points x ∈ M are not smooth (thus in particular not almost everywhere differentiable with respect to an arbitrary measure on M), a new representation formula is needed.
The following result states that any Carnot-Carathéodory distance can be (monotonically) approximated by distances associated to suitable Finsler metrics.A word on notation: given a Finsler metric F on a manifold M, we denote by d F the induced distance on M. Theorem 5.1.Let (M, E, σ, ψ) be a sub-Finsler manifold.Then there exists a sequence (F n ) n of Finsler metrics on M such that d Fn (x, y) ր d CC (x, y) holds for every x, y ∈ M.
Proof.Define ρ as in (4.4) and consider a sequence (F n ) n of approximating Finsler metrics as in Theorem 3.11.Let x, y ∈ M be fixed.Let γ : [0, 1] → M be a curve joining x and y that is Lipschitz when read in charts (i.e., as in Definition 4.2).Calling ℓ Fn the length functional associated to F n , it holds that ℓ Fn (γ) ≤ ℓ Fn+1 (γ) ≤ ℓ CC (γ) for every n ∈ N, thus by taking the infimum over γ we deduce that d Fn (x, y) ≤ d Fn+1 (x, y) ≤ d CC (x, y).Given any n ∈ N, by definition of d Fn we find a constant-speed Lipschitz curve γ n : [0, 1] → M, where the target is endowed with d Fn , such that Fix n ∈ N. The above considerations yield This shows that (γ i ) i≥n is an equiLipschitz family of curves (with respect to d Fn ).By combining Arzelà-Ascoli theorem with a diagonalisation argument, we thus obtain a curve γ : [0, 1] → M, which is Lipschitz with respect to each distance d Fn , such that (up to a not relabelled subsequence) d Fn (γ i t , γ t ) = 0 for every n ∈ N. (5.2) Since ℓ Fn is lower semicontinuous under uniform convergence of curves, we deduce from (5.2) that ≤ lim i→∞ d Fi (x, y) for every n ∈ N.
(5.3) Therefore, by using the monotone convergence theorem we obtain that Although not strictly needed for our purposes, let us point out an immediate well-known consequence (already proven in [16]) of the previous theorem.
Corollary 5.2.Let (M, E, σ, ψ) be a sub-Riemannian manifold.Then there exists a sequence (g n ) n of Riemannian metrics on M such that d gn (x, y) ր d CC (x, y) holds for every x, y ∈ M.
Proof.It follows from Theorem 5.1 by taking Remark 3.12 into account.
We shall also need the ensuing approximation result for real-valued Lipschitz functions that are defined on a Finsler manifold.Lemma 5.3.Let (M, F ) be a Finsler manifold.Let f ∈ LIP bs (M) be given.Then there exists a sequence Proof.We call C := max M |f |.We know (for instance, from [18, Theorem 2.6]) that for any n ∈ N there exists a function thus accordingly f n → f uniformly on M, as required.
We are now in a position to prove a representation formula for the pointwise norm of derivations on weighted sub-Finsler manifolds, by combining the above two results with Proposition 2. Proof.Fix a dense sequence (x k ) k ⊆ M. Theorem 5.1 grants the existence of a sequence (F i ) i of Finsler metrics on M such that d Fi ր d CC pointwise on M × M. Choose a family {η jk } j,k∈N of cut-off functions with these properties: given j, k ∈ N, we have that η jk : M → [0, 1 − 1/j] is a boundedly-supported Lipschitz function (with respect to d F1 ) such that η jk = 1 − 1/j on B dCC j (x k ) and Lip dF 1 (η jk ) ≤ 1/j 2 .Observe that for any i, j, k ∈ N it holds that Therefore, Lemma 5.3 guarantees the existence of a function 5.2.Embedding theorem and its consequences.This subsection is devoted to our main result, namely Theorem 1.1, which states that the space of derivations Der 2,2 (M; µ) associated with a weighted sub-Finsler manifold M can be isometrically embedded into the space L 2 (HM; µ) of all 'geometric' 2-integrable sections of the horizontal bundle HM.For the reader's convenience, we also recall here the statement.Proof.We divide the proof into several steps: Borel regularity.We aim to prove that any section I(b) of HM satisfying (5.5) is (equivalent to) a Borel section.Given any x ∈ M, we can find an open neighbourhood Ω of x and some functions f  (5.9) Given that π is concentrated on non-constant Lipschitz curves having constant speed, we also have that γt = 0 for π-a.e.(γ, t), or equivalently that ρ • Der > 0 in the π-a.e.sense.Hence Finally, we conclude by expounding how to deduce from Theorem 5.5 that all sub-Riemannian manifolds are universally infinitesimally Hilbertian.This is the content of the following result, which has been already stated in Theorem 1.2.Remark 5.7.Given a sub-Finsler manifold (M, E, σ, ψ) equipped with a non-negative Radon measure µ, it is not necessarily true that W 1,2 (M, d CC , µ) is a Hilbert space.Nevertheless, we can still deduce from Theorem 5.5 that W 1,2 (M, d CC , µ) is reflexive, as we are going to explain.
First of all, it can be readily checked that L 2 (HM; µ) is a reflexive Banach space if endowed with the norm L 2 (HM; µ) ∋ v → v(x) 2 x dµ(x) 1 /2 .Calling B the dual of Der 2,2 (M; µ), • 2 , where the norm • 2 is defined as in Remark 2.5, we deduce from Theorem 5.5 that B is a reflexive Banach space.Consequently, the product space L 2 (µ) × B is reflexive as well.Define L f ∈ B for every f ∈ W 1,2 (M, d CC , µ) as in (2.2).Observe that Remark 2.5 grants that the linear operator is an isometry.Therefore, we can finally conclude that W 1,2 (M, d CC , µ) is reflexive.

a)
Leibniz rule.The identity b(f g) = f b(g) + g b(f ) holds for every f, g ∈ LIP bs (X).b) Weak locality.There exists a function G ∈ L 0 (m) such that b(f ) ≤ G lip a (f ) is satisfied in the m-a.e.sense for every f ∈ LIP bs (X).The pointwise norm |b| := ess sup b(f ) f ∈ LIP bs (X), Lip(f ) ≤ 1 is the minimal function (in the m-a.e.sense) that can be chosen as G in item b) above.

Definition 2 . 1 (
The space Der 2,2 (X; m)).Let (X, d, m) be a metric measure space.Then we denote by Der 2,2 (X; m) the space of all derivations b on (X, d, m) such that |b| ∈ L 2 (m) and whose distributional divergence can be represented as a function in L 2 (m), i.e., there exists a (uniquely determined) function div(b) ∈ L 2 (m) such that b(f ) dm = − f div(b) dm for every f ∈ LIP bs (X).
while x • y stands for the Euclidean scalar product between x ∈ R d and y ∈ R d .Finally, given a metric space (X, d) and two compact non-empty sets A, B ⊆ X, we shall denote by d H (A, B) the Hausdorff distance between A and B, i.e., d H (A, B) := max sup x∈A inf y∈B d(x, y), sup y∈B inf x∈A d(x, y) .

. 13 )
Fix any distance d on M that induces the manifold topology.Since the function ρ is lower semicontinuous by definition, we can find r > 0 such that ρ(y, w) ≥ ρ(x, v) − t 2 for every (y, w) ∈ M × R d with d(x, y), |v − w| < r.(3.14)

4 . 4 . 1 .
2) is induced by a scalar product, and to omit Step 3 from the proof of Lemma 3.8 (just defining n := n ′′ ).Another proof of this result can be found in [16, Proof of Corollary 1.5].Sub-Finsler manifolds Definitions and main properties.In this subsection we recall the notion of sub-Finsler manifold and its main properties.The following material is taken from[17, Section 2.3].
Structure of the horizontal bundle.Let (M, E, σ, ψ) be a sub-Finsler manifold, whose associated distribution is denoted by {D x } x∈M .Then we define the horizontal bundle HM as Then γ is d CC -Lipschitz if and only if it is horizontal.Moreover, in such case it holds that γt γt = lim h→0 d CC (γ t+h , γ t ) |h| for a.e.t ∈ [0, 1].4.2.x∈M D x .