Hausdorff volume in non equiregular sub-Riemannian manifolds

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Hausdorff volume in non equiregular sub-Riemannian manifolds

Roberta Ghezzi, Frédéric Jean

To cite this version:

Roberta Ghezzi, Frédéric Jean. Hausdorff volume in non equiregular sub-Riemannian mani- folds. Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2015, 126, pp.345-377.

�10.1016/j.na.2015.06.011�. �hal-01107470v2�

Hausdorff volume in non equiregular sub-Riemannian manifolds

R. Ghezzi

, F. Jean

June 25, 2015

Abstract

In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume.

Then we study the regular part, show that it is not commensurable with the smooth volume, and give conditions under which it is a Radon measure. We finally give a complete characterization of the singular part. We illustrate our results and techniques on numerous examples and cases (e.g. to generic sub-Riemannian structures).

Contents

1 Introduction 2

2 Definitions 4

2.1 Hausdorff and smooth volumes . . . 4

2.2 Sub-Riemannian manifolds . . . 5

3 Lebesgue decomposition of the Hausdorff volume 8 3.1 Equiregular case . . . 8

3.2 Non equiregular case . . . 9

3.3 Stratification assumption . . . 11

4 Local integrability of the Radon-Nikodym derivative 13 4.1 Computation in local coordinates . . . 13

4.2 Sufficient conditions . . . 15

4.2.1 Conditions on nonholonomic orders . . . 16

4.2.2 Conditions on usual orders . . . 18

5 Hausdorff measures on submanifolds 18 5.1 Algebraic structure associated with equisingular submanifolds . . . 19

5.2 Hausdorff volume of equisingular submanifolds . . . 21

5.3 Weak equivalent ofq7→µˆ^q(TqN∩Bbq) . . . 23

∗Institut de Math´ematiques de Bourgogne UBFC, 9 Avenue Alain Savary BP47870 21078 Dijon Cedex France roberta.ghezzi@u-bourgogne.fr

†828 Boulevard des Mar´echaux 91762 Palaiseau, France, and Team GECO, INRIA Saclay – ˆIle-de-France, frederic.jean@ensta-paristech.fr

‡This work was partially supported by the European project AdG ERC “GeMeThNES”, grant agreement number 246923 (see alsogemethnes.sns.it); by iCODE (Institute for Control and Decision), research project of the IDEX Paris- Saclay; and by the Commission of the European Communities under the 7th Framework Programme Marie Curie Initial Training Network (FP7-PEOPLE-2010-ITN), project SADCO, contract number 264735.

6 Applications 25 6.1 Generic smooth case . . . 25 6.2 Examples . . . 26

A Appendix 28

1 Introduction

The present work is motivated by the analysis of intrinsic volumes in sub-Riemannian geometry. Here a sub-Riemannian manifold is a triplet (M,D, g), where M is a smooth manifold,Da Lie-bracket generating distribution onM andga Riemannian metric onD(note that our framework will permit us to consider rank- varying distributions as well). As in Riemannian geometry, one defines the length of absolutely continuous paths which are almost everywhere tangent toDby integrating theg-norm of their tangent vectors. Then, the sub-Riemannian distancedis defined as the infimum of length of paths between two given points. Since Dis Lie-bracket generating, for every pointp∈M there existsr(p)∈Nsuch that

{0}=D⁰p⊂ Dp¹⊂ · · · ⊂ Dp^r(p)=TpM, (1) where Dpⁱ ={X(p)|X ∈ Dⁱ} and Dⁱ ⊂Vec(M) is the submodule defined recursively byD¹ =D,Dⁱ⁺¹= Dⁱ+ [D,Dⁱ]. The sub-Riemannian manifold is equiregular if the dimensions dimDⁱp do not depend onp.

Intrinsic measures on sub-Riemannian manifolds are those which are associated with the sub-Riemannian structure. There are essentially two ways to build such measures: either using the metric structure defined by the sub-Riemannian distance which provides Hausdorff and spherical Hausdorff measures, or by means of the algebraic structure associated with the distribution which allows to construct the so called Popp’s measure (see [23]) on equiregular manifolds.

Intrinsic measures have been widely studied in the equiregular case, where the algebraic structure is well understood (in Carnot groups [4, 7, 10] and on equiregular manifolds [11]). The relevance of the study of intrinsic volumes, e.g., top-dimensional Hausdorff measures, is due to their use in PDE’s analysis. For instance, to generalize the Laplace-Beltrami operator in sub-Riemannian geometry, one needs a (sufficiently smooth) intrinsic volume: this motivates the analysis of regularity of the Hausdorff volume in [1]. We also mention the recent work [8] where smoothness of intrinsic volumes is needed to apply some nice PDE’s proof ideas. In the non equiregular case, no study of intrinsic volumes exists so far. We refer the reader to [16]

for a survey of many facts and interesting questions. Hausdorff measures are also studied in [13, 19] along curves.

The main aim of this paper is the analysis of the Hausdorff volume in non equiregular sub-Riemannian manifolds.

For convenience, let us first recall the equiregular case. Let (M,D, g) be an equiregular sub-Riemannian manifold. The Hausdorff dimension of M can be algebraically computed in terms of the flag (1) of the distribution by dim_HM =Q, where

Q=

∑r i=1

i(dimDpⁱ −dimDⁱp⁻¹),

see [22]. The Hausdorff volume, denoted by volH, is by definition the top-dimensional spherical Hausdorff measure S^dimHM. Assume M to be oriented. A natural way to understand the behavior of volH is to compare it with a smooth volume µ on M, i.e., a measure defined on open sets by µ(A) = ´

Aω, where ω∈ΛⁿM is a positively oriented non degeneraten-form. The equiregular assumption implies that volH and any smooth volumeµare mutually absolutely continuous. Moreover the Radon-Nikodym derivative of volH

with respect toµat a pointp, denoted by ^dvol_dµ^H(p), can be computed explicitly by the formula (see [1])

rlim→0

volH(B(p, r))

µ(B(p, r)) = 2^Q ˆ

µ^p(Bbp), (2)

where B(p, r) is the sub-Riemannian ball centered at p of radius r, Bbp is the unit ball in the nilpotent approximation atpand ˆµ^p is a measure obtained through a blow-up procedure ofµatp. As a consequence,

dvol_H

dµ is continuous onM and hence locally bounded and locally bounded away from zero onM. With the language of [22, 23], this implies that vol_Handµare commensurable and, in particular, that vol_H is a Radon measure, i.e., vol_H(K)<∞for every compact setK. Therefore, when the manifold is equiregular, vol_H is well understood in the sense that it behaves essentially as a smooth volume. Nevertheless, further regularity of ^dvol_dµ^H is not granted see [1].

In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold. A pointpis called regular if the growth vectorq7→(dimD¹q, . . . ,dimD^r(q)q ) is constant in a neighborhood ofp, otherwise p is called singular. The natural assumption under which we perform our work is that the manifold is stratified by equisingular submanifolds, where both the growth vector of the distribution and the growth vector of the distribution restricted to the submanifold are constant. These submanifolds were introduced in [16] and, thanks to their simple Lie algebraic structure, they constitute the fundamental block that allows us to carry through our investigation.

When the setSof singular points is not empty, the Hausdorff dimension ofM is obviously the maximum between the Hausdorff dimension ofS and the Hausdorff dimension of the setR=M\Sof regular points.

The first question is whether volH is absolutely continuous with respect to a smooth volumeµ. It turns out that this may not be the case and vol_H may have a singular part. More precisely, under the assumption thatS isµ-negligible, then vol_H admits the following Lebesgue decomposition

vol_H= vol_HxR+vol_HxS,

that is, volHxR≪ µ and volHxS⊥µ (see Corollary 3.9). As a consequence, volH is absolutely continuous with respect to a smooth volumeµif and only if the Hausdorff dimension of the singular set is smaller than the Hausdorff dimension of the set of regular points.

The next problem is to analyze the behavior of the absolutely continuous part volHxR(which is nontrivial if and only if dimHR ≥ dimHS). Thanks to (2), this amounts to study the function q 7→ µˆ^q(Bbq) near singular points. To this aim, we show that the asymptotics of this function is characterized by the one of the determinants of adapted bases at regular points (see Proposition 3.12). As a direct consequence, we deduce that ^dvol_dµ^HxR(q) blows up when q approaches the singular set (see Proposition 3.15). In particular, ^dvol_dµ^HxR is not essentially bounded near the singular set, that is, unlike the equiregular case, µ and vol_HxR are no longer commensurable. Going further in the regularity analysis, we find out that vol_HxRmay even fail to be locally integrable with respect toµand therefore vol_HxR may fail to be a Radon measure. More precisely, we exhibit a sufficient condition involving the nonholonomic order of ^dvol_dµ^HxR and the algebraic structure of the distribution at a singular point for non-integrability of vol_HxR (see Proposition 4.4), and another one involving the usual order of functions and the codimension of the singular set (see Proposition 4.10). For instance, these conditions are satisfied when the Hausdorff dimension ofRis not greater than the Hausdorff dimension ofS or whenS topologically splits M (see Corollaries 4.6 and 4.11). We also exhibit a sufficient condition for the integrability of volHxR (see Proposition 4.9), but there is a gap between integrability and non-integrability conditions. In Figure 1 we summarize the relations between the Hausdorff volume and µ in all cases.

For generic sub-Riemannian manifolds, we apply our technique and we characterize the integrability of the absolutely continuous part by comparing the dimension of the manifold to dimensions of free Lie algebras (see Proposition 6.1).

As for the singular part vol_HxS, the stratification assumption permits to focus on each equisingular submanifold N ⊂S. For an equisingular submanifold N ⊂S, we give an algebraic characterization of the Hausdorff dimension ofN and we compare the Hausdorff volume with a smooth volume on the submanifold (see Theorem 5.3). We show actually that on the restricted metric space (N, d|N) the situation is very similar to the one in equiregular manifolds: the Hausdorff volume is absolutely continuous with respect to any smooth volume onN and we have an expression of the Radon-Nikodym derivative in terms of the nilpotent approximation. Results of this part of the paper have been announced in [14].

Prop. 4.9

S̸=∅ QS< QR

volH not Radon volH≪µ volH≪µ

volH, µ volH Radon

commensurable

volH̸≪µ volHxR, µ

volH⊥µ

S=∅ S̸=∅ S̸=∅

QS=QR QS> QR

Prop. 4.4 and 4.10

Cor. 4.6

volH not Radon

not commensurable not commensurable

volH, µ volH Radon

Figure 1: Summary of relations between volH and a smooth volume µ on an oriented stratified sub- Riemannian manifold (QS, QR denote the Hausdorff dimensions ofSandR, respectively)

The structure of the paper is the following. In Section 2 we briefly recall Hausdorff measures on metric spaces and the basic concepts in sub-Riemannian geometry. Section 3 is devoted to the analysis of the Hausdorff volume in the equiregular case and of the Lebesgue decomposition of the Hausdorff volume in the non equiregular case. In Section 4 we perform the study of regularity of the Radon-Nikodym derivative and we provide sufficient conditions for integrability and non-integrability. Then, Section 5 deals with equisingular submanifolds by providing first a complete study of the algebraic and metric structure of such submanifolds, and then an analysis of the properties of the Hausdorff volume. Finally, in Section 6 we apply the methods of Section 4 to the study of the generic smooth case and we list some examples. We end with an appendix containing the proof of a technical result, Proposition A.1 which is a uniform Ball-Box Theorem on equisingular submanifolds and which is a key ingredient to study the behavior of ^dvol_dµ^HxR at singular points.

2 Definitions

2.1 Hausdorff and smooth volumes

Let us first recall some basic facts on Hausdorff measures. Let (M, d) be a metric space. We denote by diamS the diameter of a setS⊂M. Letα≥0 be a real number. For every setE⊂M, theα-dimensional Hausdorff measure H^α ofE is defined asH^α(E) = lim_ϵ→0+H^αϵ(E), where

H^αϵ(E) = inf {_∞

∑

i=1

(diamSi)^α : E⊂∪^∞

i=1

Si, Si nonempty set, diamSi≤ϵ }

,

and theα-dimensional spherical Hausdorff measure is defined asS^α(E) = lim_ϵ→0+Sϵ^α(E), where Sϵ^α(E) = inf

{_∞

∑

i=1

(diamSi)^α : E⊂

∪∞ i=1

Si, Si is a ball, diamSi≤ϵ }

.

For every setE⊂M, the non-negative number

D= sup{α≥0| H^α(E) =∞}= inf{α≥0| H^α(E) = 0}

is called theHausdorff dimension ofE. Notice thatH^D(E) may be 0,>0, or∞. By construction, for every subsetS⊂M,

H^α(S)≤ S^α(S)≤2^αH^α(S), (3) hence the Hausdorff dimension can be defined equivalently using spherical measures. In the sequel we will callHausdorff volume the spherical Hausdorff measureS^dimHM and we will denote this measure by volH.

Given a subset N ⊂M, we consider the metric space (N, d|N). Denoting byH^αN andSN^α the Hausdorff and spherical Hausdorff measures in this space, by definition we have

H^αxN(E) := H^α(E∩N) =H^αN(E∩N),

S^αxN(E) := S^α(E∩N)≤ SN^α(E∩N). (4) Notice that the inequality (4) is strict in general, as coverings in the definition of SN^α are made with sets B which satisfy B =B(p, ρ)∩N with p∈ N, whereas coverings in the definition of S^αxN include sets of the type B(p, ρ)∩N withp /∈N (S^αxN is very similar to centered Hausdorff measures, see [9]). Moreover, thanks to (3), for everyS⊂N, there holds H^αN(S)≤ SN^α(S)≤2^αH^αN(S). Hence

H^α(S)≤ SN^α(S)≤2^αH^α(S),

and SN^α is absolutely continuous with respect to H^αxN. We will denote by vol^N_H the Hausdorff volume SN^dimHN.

WhenM is an oriented manifold we can introduce another kind of volume. We say that a measure µis asmooth volumeonM if there exists a positively-oriented non degeneraten-formω∈ΛⁿM onM (i.e. ω is a volume form) such that, for every Borel setE⊂M,µ(E) =´

Eω.

Finally we will say that two measures µ, ν on M are commensurable if µ, ν are mutually absolutely continuous, i.e.,µ≪ν andν ≪µ, and if both Radon-Nikodym derivatives ^dµ_dν and ^dν_dµ exist and are locally essentially bounded. When it is the case, for every compact setK there existsC >0 such that

1

CµxK≤νxK≤CµxK.

In particular, thanks to (3),H^αandS^α are mutually absolutely continuous and commensurable.

2.2 Sub-Riemannian manifolds

Usually, a sub-Riemannian manifold is a triplet (M,∆, g^R), where M is a smooth (i.e., C^∞) manifold, ∆ is a subbundle of T M of rank m < dimM and g^R is a Riemannian metric on ∆. Using g^R, the length of horizontal curves, i.e., absolutely continuous curves which are almost everywhere tangent to ∆, is well- defined. When ∆ is Lie bracket generating, the mapd: M×M →R defined as the infimum of length of horizontal curves between two given points is a continuous distance (Rashevsky-Chow Theorem), and it is called sub-Riemannian distance.

In this paper we study sub-Riemannian manifolds with singularities. Thus it is natural to work in a larger setting, where the mapq7→∆q itself may have singularities. This leads us to the following generalized definition [2, 6].

Definition 2.1. A sub-Riemannianstructure on a manifoldM is a triplet (U,⟨·,·⟩, f) where (U,⟨·,·⟩) is a Euclidean vector bundle overM (i.e., a vector bundleπU:U→M equipped with a smoothly-varying scalar product q 7→ ⟨·,·⟩q on the fibre Uq) and f is a morphism of vector bundles f : U → T M, i.e. a smooth map linear on fibers and such that, for every u∈ U, π(f(u)) = πU(u), where π: T M →M is the usual projection.

Let (U,⟨·,·⟩, f) be a sub-Riemannian structure onM. We define the submoduleD ⊂Vec(M) as

D={f◦σ|σsmooth section ofU}, (5)

and for q∈M we setDq ={X(q)|X ∈ D} ⊂TqM. ClearlyDq =f(Uq). The length of a tangent vector v∈ Dq is defined as

gq(v) := inf{⟨u, u⟩q|f(u) =v, u∈Uq}. (6) An absolutely continuous curve γ: [a, b]→M is horizontal if ˙γ(t)∈ Dγ(t) for almost everyt. IfDis Lie bracket generating, that is

∀q∈M LieqD=TqM, (7)

then the mapd:M×M →Rdefined as the infimum of length of horizontal curves between two given points is a continuous distance as in the classic case. In this paper, all sub-Riemannian manifolds are assumed to satisfy the Lie bracket generating condition (7).

Remark 2.2. Definition 2.1 includes the following cases.

• Classic sub-Riemannian structures: in this caseU is a subbundle ofT M andf is the inclusion. With the notations used at the beginning of this section, this amounts to takeU= ∆, and⟨·,·⟩=g^R. Then the moduleDcoincides with the module of smooth sections of the subbundle ∆ and Dq has constant dimension. Moreover, for everyq∈M andv∈ Dq,gq(v) =g_q^R(v, v).

• Sub-Riemannian structures associated with a family of vector fields X1, . . . , Xm: in this case U = M ×R^m is the trivial bundle of rank m over M, ⟨·,·⟩ is the Euclidean scalar product on R^m, and f :M×R^m→M is defined asf(q, u) =∑m

i=1uiXi(q). HereDis the module generated byX1, . . . Xm

andg is given by

g_q(v) = inf { _m

∑

i=1

u²_i v=

∑m i=1

u_iX_i(q) }

.

Let (U,⟨·,·⟩, f) be a sub-Riemannian structure on a manifoldM, andD,gthe corresponding module and quadratic form as defined in (5) and (6). In analogy with the classic sub-Riemannian case and to simplify notations, in the sequel we will refer to the sub-Riemannian manifold as the triplet (M,D, g). This is justified since all the constructions and definitions below rely only onDandg.

Given i≥1, define recursively the submoduleDⁱ⊂Vec(M) by D¹=D, Dⁱ⁺¹=Dⁱ+ [D,Dⁱ].

Fixp∈M and setDpⁱ ={X(p)|X ∈ Dⁱ}. The Lie-bracket generating assumption implies that there exists an integerr(p) such that

{0}=D⁰p⊂ Dp¹⊂ · · · ⊂ Dp^r(p)=T_pM. (8) The sequence of subspaces (8) is called theflag ofD atp. Setni(p) = dimDⁱp and

Q(p) =

r(p)∑

i=1

i(n_i(p)−n_i−1(p)). (9)

This integer will play a crucial role for determining the Hausdorff dimension of (M, d). To writeQ(p) in a different way, we define theweightsof the flag (8) atpas the integersw1(p), . . . , wn(p) such thatwi(p) =s if dimD^sp⁻¹< i≤dimD^sp. ThenQ(p) =∑n

i=1wi(p).

We say that a point p is regular if, for every i, ni(q) is constant as q varies in a neighborhood of p.

Otherwise, the point is said to besingular. The sub-Riemannian manifold is calledequiregularif every point is regular.

When the dimensionsni(q) are constant onM, the moduleDⁱcoincides with the module of vector fields {X ∈Vec(M)|X(q)∈ Dⁱq ∀q∈M},

i.e., vector fields that are tangent to the distribution q 7→ Dⁱq. Yet the identification between the module Dⁱ and the mapq7→ Dqⁱ is no more meaningful when the dimension ofDⁱq varies as a function ofq(see the discussion in [6, page 48]). Indeed, in the rank-varying case, a vector field tangent to Dⁱq at every q∈ M may fail¹ to be in the moduleDⁱ. Definition 2.1 allows to take account of structures where the dimensions ni(q) (and in particularn1(q)) may vary.

1For instance, onM =R, take the moduleD ⊂VecRgenerated byX(x) =x²∂x. Then the vector fieldY(x) =x∂x is clearly tangent to the distributionx7→ Dxbut does not belong toD.

Example 2.3 (Grushin plane). LetM =R²,U=R²×R² endowed with the canonical Euclidean structure, andfbe the morphism defined as follows. Ifσ1, σ2is a global orthonormal basis onU, we setf(σ1(x, y)) =∂x

andf(σ₂(x, y)) =x∂_y. ThenD(x,y)is two dimensional for everyx̸= 0, whereas dimD(0,y)= 1.

Remark 2.4. At a regular point p, the equalityni(p) =ni+1(p) implies that the local distribution q7→ Dqⁱ

is involutive, and so thatni(p) = nj(p) for any j ≥i. From the Lie bracket generating assumption (7) we deduceni(p)< ni+1(p) fori < r(p), which in turn implies Q(p)≤n².

Given any sub-Riemannian manifold (M,D, g) there always exist a (possibly very big) integer m and vector fieldsX1, . . . , Xmsuch thatDis globally generated byX1, . . . , Xmand

g_q(v) = inf {_m

∑

i=1

u²_i v=

∑m i=1

u_iX_i(q) }

for every q∈M, v∈T_qM.

We call such a familyX1, . . . , Xma(global) generating family for the sub-Riemannian structure (D, g). The existence of a generating family is a consequence of [2, Corollary 3.16]. For an alternative proof see also [27].

Consider a generating family X1, . . . , Xm for (D, g). A multi-index I of length |I| = j ≥ 1 is an element of {1, . . . , m}^j. With any multi-index I = (i₁, . . . , i_j) is associated an iterated Lie bracketX_I = [X_i1,[X_i2, . . . ,[X_ij−1, X_ij]. . .] (we setX_I =X_i1 ifj = 1). The set of vector fields{X_I | |I| ≤j} generates the module D^j. As a consequence, if the values ofX_I1, . . . , X_In at p ∈M are linearly independent, then

∑

i|I_i| ≥Q(p).

LetY be a vector field. We define thelength of Y by

ℓ(Y) = min{i∈N|Y ∈ Dⁱ}.

In particular,ℓ(X_I)≤ |I|. By an adapted basisto the flag (8) at p, we meannvector fields Y₁, . . . , Y_n such that their values atpsatisfy

Dpⁱ = span{Yj(p)|ℓ(Yj)≤i}, ∀i= 1, . . . , r(p).

In particular, ∑n

i=1ℓ(Y_i) = Q(p). As a consequence, a family of Lie brackets X_I1, . . . , X_In such that X_I1(p), . . . , X_In(p) are linearly independent is an adapted basis to the flag (8) atpif and only if ∑

i|I_i|= Q(p).

Leth:U →Rbe a continuous function on a neighborhood ofp. Thenonholonomic order ofhatpis ordp(h) := inf{s∈(0,+∞)|h(q) =O(d(p, q)^s)}.

Ifhis a smooth function, then ord_p(h) admits an algebraic characterization, namely,

ordp(h) = min{s∈N| ∃i1, . . . , is∈ {1. . . , m}such that (Xi₁Xi₂· · ·Xi_sh)(p)̸= 0}. A smooth functionhis calledprivilegedatpif

ordp(h) = max{s∈N|dh(D^sp/D^sp⁻¹)̸= 0},

where Dp^s/Dp^s−1 is a vector subspace such that Dp^s = D^sp⁻¹⊕(Dp^s/Dp^s−1). We say that coordinates φ = (x₁, . . . , x_n) :U →Rⁿareprivileged atpif they are centered atpand linearly adapted (that is the coordinate vector fields are an adapted basis to the flag atp), and if every coordinate functionx_i is privileged atp.

Note that in this definition of privileged coordinates, the coordinate functions are not ordered by in- creasing nonholonomic order. However the set{ord_p(x_i), i= 1, . . . , n}coincides with{w₁(p), . . . , w_n(p)}, so we can relabel the weights in such a way that ord_p(x_i) =w_i(p). We then say that the weights arelabeled according to the coordinatesφ.

For every pointp∈M the metric tangent cone² to (M, d) atpexists and is isometric to (TpM,dbp) where dbp is the sub-Riemannian distance associated with the nilpotent approximation of the structure atp (see

2in Gromov’s sense, see [15]

[6, Definition 5.15] for the definition of nilpotent approximation). This was shown in [22] for equiregular sub-Riemannian manifolds and in [6] for the general case. Isometries between the metric tangent cone and (T_pM,db_p) are given by privileged coordinates atp.

By construction, the metric tangent cone is endowed with a family of dilationsδ_λ with respect to which the distancedb_p is homogeneous. Ifφ:U →Rⁿ is a system of privileged coordinates atpthen dilations are given by

δ_λ(x₁, . . . , x_n) = (λ^w1(p)x₁, . . . , λ^wn(p)x_n), wherewi(p) = ordp(xi).

We refer the reader to [2, 6, 23] for a primer in sub-Riemannian geometry.

3 Lebesgue decomposition of the Hausdorff volume

In this section we investigate the relation between the Hausdorff volume and a smooth volume on a sub- Riemannian manifold. We first recall the equiregular case, where the situation is well understood, then we consider the case where singular points are present. We write the Lebesgue decomposition of the Hausdorff volume with respect to the smooth volume and we start the analysis of both the absolutely continuous part and the singular part.

3.1 Equiregular case

Assume (M,D, g) is an equiregular connected sub-Riemannian manifold. Thenq7→Q(q) is constant on M (see (9)) and we denote byQits constant value. Moreover at every pointp∈M the metric tangent cone to (M, d), which is isometric to (TpM,dbp), has a structure of a Carnot group. AssumeM is also oriented and letµbe a smooth volume onM. The associated volume form ωinduces canonically a left-invariant volume form ˆωponTpM. We denote by ˆµ^p the smooth volume onTpM defined by ˆωp. To clarify this construction, we refer the reader to Proposition 5.1 in Section 5.

Theorem 3.1. Let (M,D, g)be an equiregular connected oriented sub-Riemannian manifold and let µbe a smooth volume onM. Then

(i) dimHM =QandvolH =S^Q; (ii) vol_H is a Radon measure onM; (iii) volH ≪µandµ≪volH;

(iv) the Radon-Nikodym derivative ^dvol_dµ^H(p) coincides with the densitylimϵ→0vol_H(B(p,ϵ))

µ(B(p,ϵ)) , whose value is lim

ϵ→0

vol_H(B(p, ϵ))

µ(B(p, ϵ)) = 2^Q ˆ

µ^p(Bb_p), ∀p∈M, (10)

where Bb_p denotes the ball centered at0 and of radius1 in T_pM with respect todb_p.

Remark 3.2. As we will see in Theorem 5.3 below, one can interpret the constant 2 in (10) as the diameter of the ball Bbp with respect to the distance dbp. Since the nilpotent approximation is a Carnot group here, we then have 2^Q= diam_db

p(Bbp)^Q=S_d^Q_b

p

(Bbp) (hereS_d^Q_b

p

denotes the spherical Hausdorff measure inTpM with respect to the distancedbp), which gives a clear interpretation to (10).

Theorem 3.1 is a special case of Theorem 5.3 proved in Section 5. Note that all the statements in Theorem 3.1 are well-known, but they have been proved only fragmentarily in several references: properties (i) and (iii) are stated in [22] (for a rigorous proof see [23]), property (iv) in [1]. Up to the authors’

knowledge, property (ii) has never been stated as is and it is a consequence of basic covering arguments in geometric measure theory. In particular, this property is needed to apply the differentiation theorem for Radon measures [26, Theorem 4.7 p.24] and ensures that ^dvol_dµ^H(p) coincides with lim_ϵ→0^vol_µ(B(p,ϵ))^H(B(p,ϵ)).

Remark 3.3. Properties (i), (ii), (iii) hold true if we replace volH byH^dimHM. In particular, applying the differentiation theorem for Radon measures [26, Theorem 4.7 p.24] we get that (i) the limit limϵ→0H^Q(B(p,ϵ)) µ(B(p,ϵ))

exists µ-almost everywhere and it coincides with the Radon-Nikodym derivative of H^Q with respect to µ.

Nevertheless, we do not have an explicit representation of such limit as we have for the spherical Hausdorff case in (10).

A first consequence of Theorem 3.1 is that the Radon-Nikodym derivative of vol_H with respect to µ is continuous on M, see [1, Corollary 2 and Section 4.1]. This is due to the fact that both the nilpotent approximation atqand the tangent measure ˆµ^q depend smoothly onq. Studying higher regularity of ^dvol_dµ^H is the main subject in [1], to which we refer the interested reader.

A further consequence is thatµand volH are commensurable (see the last remark in [22]). This follows directly by the fact that q 7→ µˆ^q(Bbq) is positive and continuous on M, so that ^dvol_dµ^H(q) and its inverse

dµ

dvolH(q) are locally bounded.

In conclusion, when the manifold is equiregular, the Hausdorff volume essentially behaves as a smooth measure. This fails when singular points are present, as we see in the next section.

3.2 Non equiregular case

Assume (M,D, g) is an oriented sub-Riemannian manifold andµis a smooth volume onM. The manifold is split into the disjoint union of two sets

M =R∪S,

where R,S denote respectively the set of regular and singular points. Since the functions q7→ dimDⁱq are lower semi-continuous, Ris an open and dense subset of M and conversely S is a closed subset of empty interior.

We will assume in this paper that µ(S) = 0. This assumption is satisfied for a very wide class of sub- Riemannian manifolds: for instance for analytic sub-Riemannian manifolds, for generic sub-Riemannian structures on a given manifold (see Section 6.1), or whenq7→ Dq is a distribution of corank 1 [25, Proposi- tion 3.3]. Nevertheless one can build up sub-Riemannian manifolds whereµ(S)̸= 0 as in the next example.

Example 3.4. Let S ⊂ R⁴ be any closed set having empty interior and positive Lebesgue measure. By Whitney’s Extension Theorem (see for instance [21, Proposition A.8]), there exists a function f ∈ C^∞(R⁴) such thatS =f⁻¹(0). Consider the sub-Riemannian structure on R⁴ for which the following vector fields are a generating family,

X₁=∂₁, X₂=∂₂+x₁∂₃+x²₁

2 ∂₄, X₃=f(x)∂₄.

One easily checks that the singular set coincides withS and thus it is of positive measure.

We are interested in properties of vol_H = S^dimHM, hence we first compute dim_HM. Clearly, the Hausdorff dimension ofM is max{dim_HR,dim_HS}. Then, using property (i) in Theorem 3.1 the Hausdorff dimension of the regular set can be computed algebraically as

dim_HR= sup{dim_HO, Oconnected component ofR}= max

q∈R Q(q).

This follows from the fact thatRis a countable union of open connected components, and by the inequality Q(q)≤n² at everyq∈R(see Remark 2.4). To remind its algebraic characterization, we denote byQ_R the number maxq∈RQ(q) (which equals dimHR). We present in Section 5 an analogous method to compute the Hausdorff dimension ofSunder some stratification assumptions.

Applying Theorem 3.1 to every connected component ofRwe obtain the following result.

Proposition 3.5. Let (M,D, g)be an oriented sub-Riemannian manifold and let µbe a smooth volume on M. Assume µ(S) = 0. Then S^QRxR≪µ, the Radon-Nikodym derivative of S^QRxR with respect to µ exists µ-almost everywhere, and it coincides with the density limϵ→0S^QR(B(q,ϵ))

µ(B(q,ϵ)) for everyq∈R.

Moreover, if O is a connected component of Rsuch that dimHO=QR, then dS^QRxR

dµ (q) = 2^QR ˆ

µ^q(Bbq) for everyq∈O. (11)

Remark 3.6. The measure S^QRxR may not be a Radon measure on M (see Section 4 below). Thus the existence of a Radon-Nikodym derivative and the fact that it coincides with the density are not consequences ofS^QRxR≪µ.

Remark 3.7. In the statement above we can replaceS^QR by volHxR. Indeed, if dimHS> QR, then volH = S^dimHS. As a consequence, volHxR≡0 and volHxR(B(q, ϵ)) = 0, for everyq∈R, so that the density becomes trivial. On the other hand, if dim_HS≤Q_R then vol_HxR=S^QRxR, that is, the two measures coincides.

Remark 3.8. When there is a connected componentO⊂Rwith dimHO < QR, the density limϵ↓0S^QR(B(q,ϵ)) µ(B(q,ϵ))

vanishes at every pointq∈O and so ^dSQ_dµ^RxR ≡onO.

Proof of Proposition 3.5. Apply Theorem 3.1 to each connected component of R. Then, by property (ii), S^QR, as a measure on the metric space (R, d|R), is Radon, that is,S^QR is finite on compact subsets contained in R. Moreover, by property (iii), S^QR, as a measure on R, is absolutely continuous with respect to µ.

Obviously,µis also a Radon measure on M (and thus onR) which is also locally doubling (see [24]). Thus we can apply the differentiation theorem for Radon measures (see for instance [26, Theorem 4.7 p.24] with X =R,µ₂= vol_H andµ₁=µ) and deduce that, onR, the Radon-Nikodym derivative ofS^QR with respect toµcoincides with the density lim_ϵ→0^S_µ(B(q,ϵ))^QR(B(q,ϵ)). In other words, for every Borel setE⊂R,

S^QR(E) = ˆ

E

lim

ϵ→0

S^QR(B(q, ϵ)) µ(B(q, ϵ)) dµ(q).

SinceRis open andµ(M\R) = 0, we deduce that for every Borel setE ⊂M, S^QRxR(E) =

ˆ

E∩R ϵlim→0

S^QR(B(q, ϵ)) µ(B(q, ϵ)) dµ(q), which gives the conclusion.

Since Ris open and Hausdorff measures are Borel regular,RandS areµ- and vol_H-measurable. Hence for every setE⊂M

vol_H(E) = vol_H(E∩R) + vol_H(E∩S),

or equivalently, vol_H = vol_HxR+vol_HxS. Moreover, sinceµ(S) = 0, vol_HxS is concentrated onSwhereasµis concentrated onR. Thereforeµand vol_HxS are mutually singular. We thus get directly the following fact.

Corollary 3.9. Let(M,D, g)be an oriented sub-Riemannian manifold and letµbe a smooth volume onM. Assume µ(S) = 0. Then µand volHxS are mutually singular and the Lebesgue decomposition of volH with respect to µis

volH= volHxR+volHxS.

As a consequence, (i) if dimHR<dimHS thenvolH andµ are mutually singular; (ii) ifdimHR>dimHS thenvolH ≪µ.

Corollary 3.9 provides the Lebesgue decomposition of volH with respect to a smooth measureµ. In the sequel we are interested in studying the absolutely continuous part and the singular part of vol_H. Note that the only case where vol_HxR̸≡0 is when dim_HR≥dim_HS. In this case vol_HxR=S^QRxR. Thus the latter is the measure we study in Section 4.

3.3 Stratification assumption

To go further in the characterization of vol_H, the first question is how to compute the Hausdorff dimension dimHM and how to relate dimHM to algebraic properties of the sub-Riemannian structure.

Recall that to compute dimHRone simply considersRas the disjoint union of open sets where the growth vector (n1(q), . . . , n_r(q)(q)) is constant and then compute the Hausdorff dimension of each componentO⊂R by the algebraic formula

dimHO=

r(q)∑

i=1

i(ni(q)−ni−1(q)).

This idea can be carried out on the whole manifold, provided thatM can be stratified by suitable submani- folds. This motivates the following definition.

Let N ⊂M be a smooth connected submanifold and q∈N. Theflag at q of D restricted to N is the sequence of subspaces

{0} ⊂(Dq¹∩TqN)⊂ · · · ⊂(Dq^r(q)∩TqN) =TqN. (12) Set

n^N_i (q) = dim(Dⁱq∩T_qN) and Q_N(q) =∑r(q)

i=1i(n^N_i (q)−n^N_i−1(q)). (13) We say thatN isequisingular³ if

(i) for everyi, the dimensionni(q) is constant as qvaries inN; (ii) for everyi, the dimensionn^N_i (q) is constant asqvaries in N.

In this case, we denote byQN the constant value ofQN(q),q∈N. We will see in Theorem 5.3 thatQN is actually the Hausdorff dimension ofN.

Remark 3.10. If N is an open connected submanifold of M then N is equisingular if and only if it is equiregular, that is, condition (i) is equivalent to condition (ii). For submanifolds N of dimension smaller than dimM being equisingular implies thatN is contained in the singular set (see Example 3.14 below for a sub-Riemannian structure where the singular set is itself an equisingular submanifold).

Definition 3.11. We say that the sub-Riemannian manifold (M,D, g) isstratified by equisingular subman- ifolds if there exists a locally finite stratificationS=∪iS_i where everyS_i is an equisingular submanifold.

In the rest of the paper we will make the assumption that (M,D, g) is stratified by equisingular subman- ifolds. The interest of such an assumption has already been raised by Gromov [16, 1.3.A]. It holds true in particular for generic smooth sub-Riemannian manifolds and for analytic ones. It obviously impliesµ(S) = 0 for any smooth measure µ, hence Corollary 3.9 and all the analysis of the previous section are valid. To shorten the notations we gather all the hypotheses we need on the sub-Riemannian manifold in the following assumption:

(A) The sub-Riemannian manifold (M,D, g) is stratified by equisingular submanifolds, M is oriented, and Q(q) =QR at every pointq∈R.

As a first consequence of this assumption, we obtain dim_HS = sup_idim_HS_i and vol_HxS=∑

ivol_HxSi. Thus, to characterize vol_HxS, it suffices to understand what happens on each stratum. This will be done Section 5 where we develop the analysis of Hausdorff volumes on equisingular submanifolds started in [14].

As concerns the regular part of the Hausdorff volume, note that all connected componentsO of Rhave the same Hausdorff dimension QR, which implies that (11) holds for every q∈ R. Also, it makes sense to study S^QRxR rather than volHxR. Indeed, if dimHS ≤QR then volH =S^QR. Conversely, if dimHS> QR

then the absolutely continuous part volHxR vanishes identically, butS^QRxR still satisfies Proposition 3.5.

3Note that in our previous paper [14, Definition 1], these submanifolds are calledstrongly equiregular.

We will focus our analysis on the regularity properties of the derivative ^dS_dµ^QRxR as they translate directly into properties of the measure S^QRxR. In particular, S^QRxR is commensurable toµif and only if ^dSQ_dµ^RxR is locally essentially bounded and bounded away for zero andS^QRxR is a Radon measure if and only if ^dSQ_dµ^RxR is locally integrable with respect toµ. To carry out the analysis of this Radon-Nikodym derivative the main tool is the proposition below.

For simplicity, we introduce the following notations. For two real numbers t andt^′ we writet ≍C t^′ if t^′/C ≤ t ≤ Ct^′, where C > 0 is a constant. Moreover, given a family of vector fields X₁, . . . , X_m and a n-tupleI = (I₁, . . . , I_n) of multi-indices we denote by X_I then-tuple of vector fields (X_I1, . . . , X_In) (where XI_j is the Lie bracket corresponding to the multi-indexIj).

Proposition 3.12. Assume (A) and consider a generating family X1, . . . , Xm for the sub-Riemannian structure. For every compact subset K⊂M, there exists a constant C >0such that

dS^QRxR

dµ (q)≍C

1

ν(q) ∀q∈K∩R, where

ν(q) =√∑

I∈F

ω(XI)(q)² and F= {

I= (I1, . . . , In)

∑

i

|Ii|=QR

} .

The proof is postponed to Section 5.3 . Actually, since ^dSQ_dµ^RxR(q) = ^2QR

ˆ

µ^q(Bbq) for q∈R, Proposition 3.12 follows from a particular instance of Proposition 5.7, see also Remark 5.8.

Remark 3.13. Since the family F is finite, one easily computes the nonholonomic order of the function q7→ν(q) atp∈Sas

ordpν = min

I∈Fordp(ω(XI)).

Moreover, in the statement of Proposition 3.12,ν can be replaced by

¯

ν(q) = max

I∈F|ω(XI)(q)|.

Example 3.14 (the Martinet space). Let us computeν on a specific example. Consider the sub-Riemannian structure onR³given by the generating family

X1=∂1, X2=∂2+x²₁ 2 ∂3.

We chooseω =dx1∧dx2∧dx3, that is, the canonical volume form on R³. The growth vector is equal to (2,2,3) on the plane {x₁= 0}, and it is (2,3) elsewhere. As a consequence, Scoincides with {x₁ = 0} and is equisingular, sincen^S₁(0, x₂, x₃) = n^S₂(0, x₂, x₃) = 1, n^S₃(0, x₂, x₃) = 2. At regular points we have Q_R = 4 and the only adapted basis at regular points is (X₁, X₂,[X₁, X₂]). Then

ν(x1, x2, x3)≍Cν¯(x1, x2, x3) =|x1|.

By Proposition 3.12, onRthe Radon-Nikodym derivative ^dS_dµ^QRxR is locally equivalent to 1/ν. Henceforth it is sufficient to study essential boundedness and integrability of 1/ν. It turns out that this function is not locally essentially bounded around the singular set.

Proposition 3.15. Under assumption (A), for everyp∈S and every neighborhoodU ⊂M ofp, essupdS^QRxR

dµ

U = +∞. As a consequence,S^QRxR andµare not commensurable when S̸=∅.