T := { z ∈M :Φ ( z ) ,..., Φ ( z ) ∈ T M} define U ⊂ R N,m ≥ M N C U Let Φ ,..., Φ (with k ≥ 1 )bevectorfieldsofclass C inanopenset 1. Introduction SilvanoDelladio INVOLUTIVITYDEGREEOFADISTRIBUTIONATSUPERDENSITYPOINTSOFITSTANGENCIES

ARCHIVUM MATHEMATICUM (BRNO) Tomus 57 (2021), 195–219

INVOLUTIVITY DEGREE OF A DISTRIBUTION AT SUPERDENSITY POINTS OF ITS TANGENCIES

Silvano Delladio

Abstract.

Let Φ1, . . . ,Φk+1(withk≥1) be vector fields of classC^kin an open set U⊂R^N+m, letMbe aN-dimensionalC^ksubmanifold ofUand define

T :={z∈ M: Φ1(z), . . . ,Φk+1(z)∈TzM}

whereTzMis the tangent space toMatz. Then we expect the following property, which is obvious in the special case whenz0 is an interior point (relative toM) ofT:

Ifz0∈ Mis a(N+k)-density point (relative toM) ofT then all the iterated Lie brackets of order less or equal tok

Φi1(z0),[Φi1,Φi2](z0),[[Φi1,Φi2],Φi3](z0), . . . (h, ih≤k+1) belong toTz₀M.

Such a property has been proved in [9] fork= 1 and its proof in the case k= 2 is the main purpose of the present paper. The following corollary follows at once:

LetDbe aC² distribution of rankN on an open setU⊂R^N+m andMbe aN-dimensionalC² submanifold of U. Moreover let z0 ∈ M be a (N+ 2)-density point of the tangency set {z ∈ M |TzM=D(z)}. ThenDmust be2-involutive atz0, i.e., for every family{Xj}^N_j=1of classC² in a neighborhoodV ⊂U ofz0

which generatesDone has

Xi₁(z0),[Xi₁, Xi₂](z0),[[Xi₁, Xi₂], Xi₃](z0)∈Tz0M for all1≤i1, i2, i3≤N.

1. Introduction

Let Φ₁, . . . ,Φ_k+1 (with k ≥ 1) be vector fields of class C^k in an open set U ⊂R^N+m(withN, m≥1), let Mbe aN-dimensionalC^k submanifold ofU and define

T :={z∈ M: Φ1(z), . . . ,Φk+1(z)∈TzM}

2020Mathematics Subject Classification: primary 28Axx; secondary 58A30, 58C35, 58A17.

Key words and phrases: tangency set, distributions, superdensity, integral manifold, Frobenius theorem.

Received March 22, 2021. Editor J. Slovák.

DOI: 10.5817/AM2021-4-195

whereT_zMis the tangent space toMatz. One has the following obvious property:

If z₀ is an interior point (relative toM) of T, then all the iterated Lie brackets of order less or equal to k

Φi₁(z₀),[Φi₁,Φi₂](z₀),[[Φi₁,Φi₂],Φi₃](z₀), . . . (h, ih≤k+ 1) belong toTz₀M.

With reference to this property, we are interested in understanding whether it remains true when we substitute the hypothesis thatz0 is an internal point of T with the assumption that it is a point of sufficiently high density ofT. In this regard, for the convenience of the reader, we recall that z0∈ M is said to be a (N+h)-density point of a setE ⊂ M(relative to M) ifh≥0 and

H^N(B_M(z₀, r)\ E) =o(r^N+h) (asr→0+)

whereB_M(z0, r)⊂ Mis the metric ball of radiusrcentered atz0, compare Section 2 below. According to this definition, roughly speaking, we can say that: the larger h, the higher the concentration ofEatz0(and the morez0will resemble an interior point ofE). Observe that, in the special case whenm= 0 and M=R^N, the point z0 is a N-density point of E if and only if it is a point of Lebesgue density ofE, that is L^N(B_RN(z0, r)∩E)/L^N(B_RN(z0, r))→1 (asr→0+).

In [9] we have proved the following result answering “yes” to the question above, whenk= 1.

Theorem 1.1([9]).Given an open setU ⊂R^N+m, considerΦ1,Φ2∈C¹(U,R^N+m) and aN-dimensional C¹ submanifoldM ofU. Moreover define

T :=

z∈ M: Φ₁(z),Φ₂(z)∈T_zM

and assume that z0∈ M is a (N+ 1)-density point ofT (relative toM). Then one hasΦ1(z0),Φ2(z0),[Φ1,Φ2](z0)∈Tz₀M.

The main purpose of this work is to prove that also fork= 2 the answer to the previous question is affirmative. More precisely one has

Theorem 1.2. Given an open setU ⊂R^N+m, considerΦ1,Φ2,Φ3∈C²(U,R^N+m) and aN-dimensional C² submanifoldM ofU. Moreover define

T :=

z∈ M: Φ1(z),Φ2(z),Φ3(z)∈TzM

and assume that z₀∈ M is a (N+ 2)-density point ofT (relative toM). Then one has Φ_i1(z₀),[Φ_i1,Φ_i2](z₀),[[Φ_i1,Φ_i2],Φ_i3](z₀)∈T_z0Mfor all 1≤i₁, i₂, i₃≤3.

Now, in order to better understand the continuation of this introduction, we will recall some definitions and some well-known facts. First of all, let N, m, k be positive integers and recall that a C^k distribution of rank N on an open set U ⊂R^N+mis a mapDassigning aN-dimensional vector subspaceD(z) ofR^N+m to each pointz∈U and satisfying the following property: Ifz∈U then there exist a neighborhood V^(z)⊂U of zand a family {X_i^(z)}^N_i=1⊂C^k(V^(z),R^N+m) which generatesDinV^(z), i.e., such that{X₁^(z)(z⁰), . . . , X_N^(z)(z⁰)}is a basis of D(z⁰) for

all z⁰ ∈ V^(z). The distribution D is said to be k-involutive at z ∈ U if all the iterated Lie brackets of order less or equal tok

X_i^(z)

1 (z),[X_i^(z)

1 , X_i^(z)

2 ](z),[[X_i^(z)

1 , X_i^(z)

2 ], X_i^(z)

3 ](z), . . . (withi_h≤N andh≤k+ 1) belong to D(z). Such a definition does not depend on the choice of the family {X_i^(z)}^N_i=1, compare Proposition 6.1 below. In the special case whenk= 1 we will omit the prefix, i.e., we will simply say “involutive” instead of “1-involutive”(that makes this definition consistent with the classical one, compare [12, Definition 2.11.5]).

Let Dbe a C¹ distribution of rankN on an open setU ⊂R^N+mand letMbe aN-dimensionalC¹submanifold ofU. Then, according to a celebrated theorem by Frobenius (see [12, Section 2.11]), the distributionDis involutive at every point of U if and only if the following integrability property is verified:For all z0∈U there exists aC¹ submanifoldMofU such thatz0∈ Mand the tangency set ofMwith respect toDcoincides withM, namelyτ(M,D) :={z∈ M : TzM=D(z)}=M.

The size of the tangency with respect to a noninvolutive distribution has been the subject of recent investigations in the field of sub-Riemannian geometry. The follo- wing list collects some of the results produced by this research activity. They describe the “integrability degree” of noninvolutiveC¹distributionsD, mainly by providing upper bounds for dimH(τ(M,D)) asMvaries among all theN-dimensionalC² submanifolds of U (where dim_H denotes the Hausdorff dimension).

(1) LetHH^kbe the horizontal subbundle of the tangent bundle to the Heisenberg groupH^k, that is the distribution of rank 2konR^2k+1 generated by the vector fields

(x₁, . . . , x2k+1)7→ ∂

∂x_i + 2xk+i

∂

∂x_2k+1 (i= 1, . . . , k) (x₁, . . . , x_2k+1)7→ ∂

∂xk+i

−2x_i ∂

∂x2k+1

(i= 1, . . . , k). This distribution is noninvolutive everywhere and one has

(1.1) dimH τ(M, HH^k)

≤k

for every (2k)-dimensionalC² submanifoldMofR^2k+1 (see [1, Theorem 1.2], [2, Example 6.5], [7, Corollary 4.1]).

(2) An explicit estimate of the number

sup{dim_H(τ(M,D)) : MisC²-smooth}

is provided by [2, Theorem 1.3] in terms of the involutiveness degree ofD. An elementary proof, based on the implicit function theorem, can be found in [6].

In [2, Example 6.5], already mentioned above, this result is used to prove the inequality (1.1).

(3) IfDis of classC^∞ and fulfils the Hörmander noninvolutiveness condition (see [2, Definition 4.1]), then one has

sup{dimH(τ(M,D)) : MisC²-smooth} ≤N−1

compare [2, Theorem 4.5]. The well-known result by Derridj [10, Theorem 1]

follows immediately from this property.

(4) Roughly speaking, theC¹smooth submanifoldsMare expected to produce much larger tangencies (with respect toD) than those produced byC²smooth submanifolds. In fact, even if there are no points at whichD is involutive, it can well be that a C¹ smooth M exists such that H^N(τ(M,D)) > 0.

According to [2, Proposition 8.2], this is true for a large class of distributions includingHH^k and there are good reasons to believe that it is true in general (compare [2, Problem 8.3]).

(5) IfDis a C¹ distribution of rankN on an open set U ⊂R^N+mandM is aN-dimensional C¹ submanifold ofU, then Dmust be involutive at each pointz0∈ Mwhich is a (N+ 1)-density point of τ(M,D) (relative toM) [8, Corollary 5.1]. In other words: despite (4), ifDis not involutive at a point z0∈ M then there is no N-dimensionalC¹ submanifoldMof U such that z0∈ Mandz0 is a (N+ 1)-density point (relative toM) ofτ(M,D).

As we observed in [9], the result mentioned in (5) follows at once from Theorem 1.1. Analogously, the following corollary follows immediately from Theorem 1.2.

Corollary 1.1. IfDis aC²distribution of rankN on an open setU ⊂R^N+mand M is aN-dimensionalC² submanifold ofU, thenDmust be 2-involutive at each pointz₀∈ Mwhich is a(N+ 2)-density point ofτ(M,D)(relative toM). In other words: if Dis not2-involutive at a pointz₀∈ M then there is noN-dimensional C² submanifold M of U such that z₀ ∈ M and z₀ is a (N + 2)-density point (relative to M) ofτ(M,D).

When we started working on Theorem 1.2, our belief that it could be valid was rather weak, while (by virtue of Theorem 1.1 and Theorem 1.2) we are now firmly convinced that the following conjecture is true and will be the subject of future work.

Conjecture 1.1. Given an open set U ⊂ R^N+m, consider Φ1, . . . ,Φk+1 ∈ C^k(U,R^N+m)and aN-dimensionalC^k submanifoldMofU (withk≥1). Moreo- ver define

T :=

z∈ M: Φ₁(z), . . . ,Φk+1(z)∈TzM .

and assume that z₀∈ M is a(N+k)-density point ofT (relative toM). Then all the iterated Lie brackets of order less or equal tok

Φi₁(z0),[Φi₁,Φi₂](z0),[[Φi₁,Φi₂],Φi₃](z0), . . . (h, ih≤k+ 1) belong toT_z0M.

We conclude by observing that, just as Corollary 1.1 followed at once from Theorem 1.2, this property follows immediately for allk≥1 such that Conjecture 1.1 holds:

If D is aC^k distribution of rank N on an open setU ⊂R^N+m andMis aN-dimensionalC^k submanifold ofU, thenDmust be k-involutive at each pointz₀∈ Mwhich is a (N+k)-density point

ofτ(M,D) (relative toM). In other words: ifDis notk-involutive at a pointz₀∈ Mthen there is noN-dimensionalC^k submanifold M of U such that z₀ ∈ M and z₀ is a (N +k)-density point (relative toM) ofτ(M,D).

2. General notation and preliminaries

We will have to deal with maps fromR^N toR^m. The standard basis ofR^N+mand the corresponding coordinates are denoted by e1, . . . , eN+m and (x1, . . . , xN, y1, . . . , ym), respectively. We may also writeR^Nx in place ofR^N andR^my in place of R^m. If U is an open subset ofR^Nx ×R^my andG∈C¹(U,R^k), then DxGandDyG denote the Jacobian matrix ofGwith respect toxand the Jacobian matrix ofG with respect toy, respectively, that is

DxG:=∂G

∂x1

. . .

∂G

∂xN

, DyG:=∂G

∂y1

. . .

∂G

∂ym

.

In general, the Jacobian matrix of anyC¹ vector fieldF is denoted byDF. The Hessian matrix of anyC²function f is denoted byD²f, while D²_ijf stands for the (i, j)-entry of D²f. Theh^th-order derivative of aC^h function of one variableg is indicated withg^(h). For simplicity, we define

D1:= ∂

∂x1

, . . . , DN := ∂

∂xN

, DN+1:= ∂

∂y1

, . . . , DN+m:= ∂

∂ym

. Forα= (α1, . . . , αN)∈N^N, define

|α|:=α1+· · ·+αN, Dα:= ∂^|α|

∂x^α₁¹· · ·∂x^α_N^N.

The Euclidean norms involved throughout this paper are all denoted byk · k. The constants depending only onp, q, . . .are indicated byC(p, q, . . .). LetU be an open subset ofR^N+m. Ifk≥1 and

H = (H1, . . . , HN+m), K= (K1, . . . , KN+m)∈C^k(U,R^N+m) then we recall that the Lie bracket product ofH, K is the vector field

[H, K] = ([H, K]1, . . . ,[H, K]_N+m)∈C^k−1(U,R^N+m) where

(2.1) [H, K]j :=

N+m

X

i=1

(HiDiKj−KiDiHj), j= 1, . . . , N+m

compare [12, Remark 2.4.5]. Recall that the Lie bracket product is anti-symmetric, bilinear and verifies the following identity

(2.2) [f H, gK] =f(H·Dg)K−g(K·Df)H+f g[H, K] (f, g∈C^k(U)) compare [4, Chapter 1, Theorem 4.2]. If k ≥ 1 and X := {X1, . . . , Xp} ⊂ C^k(U,R^N+m), then we state the following inductive definition ofh^th-order iterated

Lie brackets of the vector fieldsX_i, with 0≤h≤k and 1≤i₁, . . . , i_h+1≤p:

Λ^X_(i

1,...,i_h+1):=

(Xi₁ if h= 0 h

Λ^X_(i

1,...,i_h), Xi_h+1

i

if 1≤h≤k

e.g. Λ^X₍₁₎ = X1, Λ^X_(1,2) = [X1, X2], Λ^X_(1,2,1) = [[X1, X2], X1] (provided k ≥ 2).

Observe that

(2.3) Λ^X_(i

1,...,ih+1)∈C^k−h(U,R^N+m).

Let L^X_h(z) be the vector space spanned by the family of theh^th-order iterated Lie brackets (of the vector fields Xi) atz∈U, namely

L^X_h(z) := spann

Λ^X_{(i1,...,ih+1)}(z) : 1≤i1, . . . , ih+1≤po for allz∈U and 0≤h≤k.

Let M be a N-dimensional C¹ submanifold of R^N+m and let d denote the distance defined on each connected component ofMby taking the infimum over the joining paths (compare [3, Section 1.6]). Then for z₀∈ Mandr >0 we define

B_M(z₀, r) :={z∈ M^(z0)|d(z, z₀)< r}

whereM^(z0) is the connected component of Mcontaining z0. Recall that for r small enough exp_z0 mapsBT_z0M(0, r) diffeomorphically onto a neighborhood ofz0

and one has

exp_x0 BT_z0M(0, r)

=B_M(z0, r)

compare [3, Theorem 1.6 and Corollary 1.1]. In the special case whenm= 0 and M=R^N the distancedreduces to the usual Euclidean distance and we denote B_RN(z0, r) simply byBr(z0).

The Lebesgue outer measure onR^N and theN-dimensional Hausdorff measure onR^N+mwill be denoted byL^N andH^N, respectively.

A point x ∈ R^N is said to be a (N +k)-density point of E ⊂ R^N (where k∈[0,+∞)) if

L^N(B_r(x)\E) =o(r^N+k) (as r→0+).

The set of all (N+k)-density points of Eis denoted byE^(N+k). Analogously, if Mis aN-dimensionalC¹ submanifold ofR^N+mandz₀∈ M, then we say thatz₀ is a (N+k)-density point ofE ⊂ M(relative toM) if

H^N(B_M(z₀, r)\ E) =o(r^N+k) (as r→0+).

The set of all (N+k)-density points ofE (relative toM) is denoted by E^(N+k). Observe that

E^(N+k)⊂ E^(N+h)

for allh∈[0, k]. In particular, ifkis a positive integer, one has (2.4) E^(N+k)⊂ E^(N+k−1)⊂ · · · ⊂ E^(N).

By [11, 3.2.46] and the area formula [11, Theorem 3.2.3] one can prove that C¹ embeddings preserve density-degree, namely the following property holds [8, Proposition 3.3].

Proposition 2.1. Let Mbe aN-dimensionalC¹ submanifold of R^N+m, letΩbe an open subset of R^N and letF : Ω→R^N+m be an injective immersion of class C¹ such that F(Ω)⊂ M. Moreover let E be a subset of Ωand letx₀∈Ω. Then (fork≥0) one has

L^N(B_r(x₀)\E) =o(r^N+k) (as r→0+) if and only if

H^N(B_M(F(x₀), r)\F(E)) =o(r^N+k) (as r→0+). In particular,x₀∈E^(N+k) if and only ifF(x₀)∈F(E)^(N+k).

We conclude this section with a remark which will be very useful below.

Remark 2.1. LetH = (H1, . . . , HN+m) be a vector field of classC¹in an open setU ⊂R^Nx ×R^my . Moreover let Ω be an open subset ofR^Nx andf = (f1, . . . , fm)∈ C¹(Ω,R^my). Denote by Γ the graph of f, that is Γ :=F(Ω) where

F: Ω→R^Nx ×R^my , F(x) := x, f(x)

and assume that Γ⊂U. Givenx∈Ω, obviously one has thatH(F(x))∈T_F(x)Γ if and only ifH(F(x))∈Im(DF). Recalling that

DF = I

Df

we get at once the following property:H(F(x))∈T_F(x)Γ if and only if

(2.5) H# F(x)

=Df(x)H_∗ F(x) where we have defined

H∗:= (H1, . . . , HN), H#:= (HN+1, . . . , HN+m).

Moreover, ifK= (K1, . . . , KN+m) is another vector field of classC¹ inU and if one hasH(F(x)), K(F(x))∈T_F(x)Γ for a certainx∈Ω, then

(2.6) DH F(x)

K F(x)

=D(H◦F)(x)K_∗ F(x) compare [9, Lemma 4.1].

3. Some localization properties at a superdensity point Consider a function g∈C^k(R), withk≥1, such that

0≤g≤1, g|_(−∞,0]≡1, g|_[1,+∞)≡0 and, forρ∈(0,1), define

ψρ(x) :=gkxk −ρ 1−ρ

, x∈R^N. Observe that

ψρ|_B

ρ(0) ≡1, ψρ|_RN\B1(0)≡0.

Proposition 3.1. For all α∈N^N\ {0} one has

Dαψρ(x) =

|α|

X

h=1

(1−ρ)^−hg^(h)kxk −ρ 1−ρ

X

{β1,...,β_h}∈Ph(α)

Dβ₁kxk · · ·Dβ_hkxk where

Ph(α) :=n

{β1, . . . , βh} : βi∈N^N \ {0},

h

X

i=1

βi=|α|o .

Proof. The statement is obvious if|α|= 1. Then letkbe a positive integer and assume that the identity holds whenever|α| ≤k. We have to prove that it continues to be true for any α ∈N^N such that |α| = k+ 1. To this aim, without loss of generality, we can suppose thatα1≥1. If define

ε1:= (1,0, . . . ,0), ε2:= (0,1, . . . ,0), . . . , εN := (0, . . . ,0, N) then one has|α−ε1|=k, hence (by assumption)

D_α−ε1ψ_ρ(x) =

k

X

h=1

(1−ρ)^−hg^(h)kxk −ρ 1−ρ

X

{β₁,...,β_h}∈P_h(α−ε₁)

D_β1kxk · · ·D_βhkxk. Thus

Dαψρ(x) =Dε₁(D_α−ε1ψρ) (x)

=

k

X

h=1

(1−ρ)^−h−1g^(h+1)kxk −ρ 1−ρ

D_ε1kxk X

{β1,...,β_h}∈Ph(α−ε1)

D_β1kxk · · ·D_βhkxk

+

k

X

h=1

(1−ρ)^−hg^(h)kxk −ρ 1−ρ

X

{β1,...,β_h}∈Ph(α−ε1)

Dβ₁+ε₁kxkDβ₂kxk · · ·Dβ_hkxk +Dβ₁kxkDβ₂+ε₁kxk · · ·Dβ_hkxk+· · ·+Dβ₁kxk · · ·Dβ_h−1kxkDβ_h+ε₁kxk

= (1−ρ)⁻¹g⁽¹⁾kxk −ρ 1−ρ

D_αkxk+

k

X

h=2

(1−ρ)^−hg^(h)kxk −ρ 1−ρ

×

X

{β1,...,βh−1}∈Ph−1(α−ε1)

Dε₁kxkDβ₁kxk · · ·Dβh−1kxk

+ X

{β1,...,βh}∈Ph(α−ε1)

Dβ₁+ε₁kxkDβ₂kxk · · ·Dβ_hkxk

+D_β1kxkDβ2+ε1kxk · · ·D_βhkxk+· · ·+D_β1kxk · · ·D_βh−1kxkDβh+ε1kxk

+ (1−ρ)^−k−1g^(k+1)kxk −ρ 1−ρ

Dε₁kxkD^α_ε1¹⁻¹kxkD^α_ε2²kxk · · ·D^α_εN^Nkxk

hence the conclusion follows immediately.

Remark 3.1. By a completely standard argument (e.g. by induction) one can easily prove that for allα∈N^N one has

Dαkxk= pα(x)

kxk^2|α|−1 (x6= 0)

wherepαis a homogeneous polynomial of degree|α|whose coefficients depend only onα. It follows that

max

x∈B1(0)\Bρ(0)

Dαkxk

≤ C(α) ρ^2|α|−1. Corollary 3.1. Let x0∈R^N,r >0,ρ∈(1/2,1)and define

ϕ_ρ,r(x) :=ψ_ρx−x₀ r

, x∈R^N. Then, for allα∈N^N, one has

kDαϕρ,rk_∞≤ C(α) (1−ρ)^|α|r^|α|.

Proof. The statement is obvious for |α|= 0, so we can assume|α| ≥1. Observe that

D_αϕ_ρ,r(x) =r^−|α|(D_αψ_ρ)x−x₀ r

, x∈R^N. Hence, by Proposition 3.1 and Remark 3.1, we get

kDαϕρ,rk_∞=r^−|α|kDαψρk_∞

=r^−|α| max

x∈B1(0)\Bρ(0)

|D_αψ_ρ(x)|

≤r^−|α|

|α|

X

h=1

kg^(h)k_∞ (1−ρ)^h

X

{β₁,...,β_h}∈P_h(α)

C(β₁). . . C(β_h) ρ^2|β1|−1. . . ρ^2|βh|−1

≤C(α) r^|α|

|α|

X

h=1

1

(1−ρ)^hρ^2|α|−h.

The conclusion follows by observing that if 1/2< ρ <1, then one has (1−ρ)^hρ^2|α|−h≥(1−ρ)^|α|

2^2|α| (for h= 1, . . . ,|α|).

Proposition 3.2. LetΘ be a continuous function defined in a neighborhood of x0∈R^N. Assume that for allρ∈(1/2,1) one has

(3.1)

Z

B_r(x₀)

Θ(x)ϕ_ρ,r(x)dx=o(r^N) asr→0+. ThenΘ(x0) = 0.

Proof. As in [9, Proposition 3.1].

Proposition 3.3. Let E be a measurable subset of R^N and x₀ ∈ E^(N+k), with k≥1. Moreover letΘ andΛbe a couple of continuous real valued functions defined in a neighborhood ofx₀ such that Θ|_E∩Br(x0)= Λ|_E∩Br(x0) (for rsmall enough).

If ρ∈(1/2,1), then one has Z

Br(x0)

ΘDαϕρ,rdx= Z

Br(x0)

ΛDαϕρ,rdx+o(r^N) (as r→0+) for all α∈Nⁿ such that |α| ≤k.

Proof. Let α∈Nⁿ be such that|α| ≤k and observe that (since the integral is linear) it will be enough to prove the statement for Λ≡0. Then

Z

B_r(x₀)

ΘDαϕρ,rdx =

Z

B_r(x₀)\E

ΘDαϕρ,rdx

≤ sup

B_r(x₀)

|Θ|

kDαϕρ,rk_∞L^N(Br(x0)\E).

We conclude by Corollary 3.1 and recalling thatx₀∈E^(N+k). Definition 3.1. Let Θ and Λ be a couple of real valued functions, each one defined and summable in a neighborhood of x₀∈R^N, such that

Z

Br(x0)

Θ(x)ϕρ,r(x)dx= Z

Br(x0)

Λ(x)ϕρ,r(x)dx+o(r^N) (as r→0+) for allρ∈(1/2,1). Then we write Θ^x∼⁰Λ and say that Θ and Λ are equivalent at x0.

Remark 3.2. It is trivial to verify that^x∼⁰ is actually an equivalence relation on the family of real valued functions defined and summable in a neighborhood ofx₀. Proposition 3.4. Let Θ and Λ be a couple of real valued functions, each one defined and continuous in a neighborhood of x₀∈R^N, such that Θ^x∼⁰Λ. Moreover, letg be a function of class C¹ in a neighborhood ofx₀. Then one hasgΘ^x∼⁰gΛ.

Proof. Forrsufficiently small andρ∈(0,1), one has

Z

B_r(x₀)

g(x)Θ(x)ϕ_ρ,r(x)dx− Z

B_r(x₀)

g(x)Λ(x)ϕ_ρ,r(x)dx

= Z

B_r(x₀)

[g(x)−g(x₀)]Θ(x)ϕ_ρ,r(x)dx− Z

B_r(x₀)

[g(x)−g(x₀)]Λ(x)ϕ_ρ,r(x)dx +g(x₀)

Z

B_r(x₀)

Θ(x)ϕ_ρ,r(x)dx− Z

B_r(x₀)

Λ(x)ϕ_ρ,r(x)dx

≤C(N) sup

B_r(x₀)

|g−g(x₀)|

sup

B_r(x₀)

|Θ|+ sup

B_r(x₀)

|Λ|

r^N +|g(x₀)|o(r^N). Moreover, for all x∈Br(x₀), one has

g(x)−g(x0) = Z 1

0

Dg(x0+t(x−x0))·(x−x0)dt

hence

sup

Br(x0)

|g−g(x₀)| ≤ sup

Br(x0)

kDgk r . It follows that

Z

Br(x0)

g(x)Θ(x)ϕρ,r(x)dx− Z

Br(x0)

g(x)Λ(x)ϕρ,r(x)dx=o(r^N) (as r→0+). From Proposition 3.3 and the integration by parts formula it follows at once the following result.

Theorem 3.1. Let E be a measurable subset of R^N and x0 ∈ E^(N+k), with k≥1. Moreover let ΘandΛbe a couple of real valued functions of classC^k in a neighborhood of x₀ such thatΘ|_E∩Br(x₀)= Λ|_E∩Br(x₀) (forr small enough). Then one has

DαΘ^x∼⁰DαΛ (henceDαΘ(x0) =DαΛ(x0), by Proposition 3.2) for all α∈Nⁿ such that |α| ≤k.

4. The proof of Theorem 1.2 (main result)

This section is devoted to the proof Theorem 1.2. It is an example of how Theorem 3.1 can serve to extend to (N +k)-density points a property which is known to hold at interior points. Actually we will prove the following result which is trivially equivalent to Theorem 1.2, by Theorem 1.1 and (2.4) (withk= 2 and E =T). We state it by using a subscript-free notation that will produce shorter formulas.

Theorem 4.1. Let H, K, L be three vector fields of class C² in an open set U ⊂R^N+m. Moreover let Mbe aN-dimensionalC² submanifold ofU and define

T :=

z∈ M:H(z), K(z), L(z)∈T_zM .

Ifz0∈ Mis a (N+ 2)-density point ofT (relative toM) then[[H, K], L](z0)∈ Tz₀M.

Proof. Since Mis locally the graph of aC² function, we can assume that there exist an open set Ω⊂R^Nx andf = (f1, . . . , fm)∈C²(Ω,R^my) such that

z₀∈Γ :={(x, f(x)) : x∈Ω} ⊂ M. Define F∈C²(Ω,R^N+m) by

F(x) := x, f(x)

, x∈Ω. Moreover let

x0:=F⁻¹(z0), T :=F⁻¹(T) and observe that

x0∈Ω∩T^(N+2) (hence alsox0∈Ω∩T^(N+1))

by Proposition 2.1. The following notation will be useful: ifAandB are functions defined in Ω such thatA|T =B|T, then we write A=^TB.

If for allh= 1, . . . , mdefineDh∈C(Ω) as

D_h(x) := Df(x)[[H, K], L]_∗(F(x))−[[H, K], L]_#(F(x))

·e_N+h then, by Remark 2.1, we have to prove that

(4.1) Dh(x0) = 0 (h= 1, . . . , m).

From now on the argument is a very long and technical computation, divided into steps, whose hardest details are collected in the next section.

Step 1.First of all, observe that [[H, K], L]j =

N+m

X

i=1

([H, K]iDiLj−LiDi[H, K]j)

=

N+m

X

i,l=1

H_lD_lK_iD_iL_j−K_lD_lH_iD_iL_j−L_iD_i(H_lD_lK_j−K_lD_lH_j)

= [(DK)H]·DLj−[(DH)K]·DLj−[(D²Kj)H]·L+

−[(DH)L]·DKj+ [(D²Hj)K]·L+ [(DK)L]·DHj

by (2.1). Hence we get (compare Section 5.1)

(4.2) Dh^x∼⁰Gh(H, K, L)− Gh(K, H, L) where

(4.3)

G_h(H, K, L) := [D(K◦F)(H_∗◦F)]·X^N

p=1

[(DL_p)◦F]D_pf_h−[(DL_N+h)◦F]

+ [D(K◦F)(L∗◦F)]·X^N

p=1

[(DHp)◦F]Dpfh−[(DHN+h)◦F]

−

N

X

p=1

Dpfh [(D²Kp)◦F](H◦F)

·(L◦F) + [(D²KN+h)◦F](H◦F)

·(L◦F). Thus we are reduced to prove that

G_h(H, K, L)(x₀) =G_h(K, H, L)(x₀) (h= 1, . . . , m).

Step 2.Forl= 1, . . . , N+mthe following identity holds (compare Section 5.2) [(D²Kl)◦F](H◦F)

·(L◦F)^x∼⁰Al(H, K, L)

−[D(H◦F)(L_∗◦F)]·[(DK_l)◦F] (4.4)

where

Al(H, K, L) :=D[D(K_l◦F)·(H∗◦F)]·(L∗◦F).

Step 3.From (4.3) and (4.4), we obtain

Gh(H, K, L)^x∼⁰[D(K◦F)(H_∗◦F)]·X^N

p=1

[(DLp)◦F]Dpfh−[(DLN+h)◦F]

+

N

X

p=1

D_pf_h[D(K◦F)(L_∗◦F)]·[(DH_p)◦F]+

−

N

X

p=1

DpfhAp(H, K, L) +

N

X

p=1

Dpfh[D(H◦F)(L∗◦F)]·[(DKp)◦F]

+AN+h(H, K, L)−[D(H◦F)(L_∗◦F)]·[(DKN+h)◦F]

−[D(K◦F)(L_∗◦F)]·[(DHN+h)◦F]

that is

Gh(H, K, L)^x∼⁰Bh(H, K, L) +Ch(H, K, L) +S_h⁽¹⁾(H, K, L) +S_h⁽²⁾(H, K, L) (4.5)

where

Bh(H, K, L) :=AN+h(H, K, L)−

N

X

p=1

DpfhAp(H, K, L)

Ch(H, K, L) := [D(K◦F)(H_∗◦F)]·X^N

p=1

[(DLp)◦F]Dpfh−[(DLN+h)◦F]

S_h⁽¹⁾(H, K, L) :=−[D(H◦F)(L∗◦F)]·[(DKN+h)◦F]

−[D(K◦F)(L_∗◦F)]·[(DHN+h)◦F] S_h⁽²⁾(H, K, L) :=

N

X

p=1

D_pf_h[D(K◦F)(L_∗◦F)]·[(DHp)◦F]

+

N

X

p=1

Dpfh[D(H◦F)(L∗◦F)]·[(DKp)◦F].

Observe that S_h⁽¹⁾(H, K, L) and S_h⁽²⁾(H, K, L) are symmetric with respect to the couple (H, K), that is

S_h⁽¹⁾(H, K, L) =S_h⁽¹⁾(K, H, L), S_h⁽²⁾(H, K, L) =S_h⁽²⁾(K, H, L). Step 4.One has (compare Section 5.3)

(4.6) Ch(H, K, L)^x∼⁰Fh(H, K, L) +S_h⁽³⁾(H, K, L)

where

Fh(H, K, L) :=−

N

X

i=1

[D(Ki◦F)·(H∗◦F)][Di(Dfh)·(L∗◦F)]

S_h⁽³⁾(H, K, L) :=

m

X

q=1

[D²f_q(K_∗◦F)]·(H_∗◦F)

× [(DN+qL_∗)◦F]·Dfh−[(DN+qLN+h)◦F]

. Observe thatS_h⁽³⁾(H, K, L) is symmetric with respect to the couple (H, K).

Step 5.One has (compare Section 5.4)

Bh(H, K, L)+Fh(H, K, L)^x∼⁰div [D(KN+h◦F)·(H_∗◦F)](L_∗◦F)

−divX^N

i=1

D_if_h[D(K_i◦F)·(H_∗◦F)](L_∗◦F) +S_h⁽⁴⁾(H, K, L)

(4.7)

where

S_h⁽⁴⁾(H, K, L) :=−[D²fh(K_∗◦F)]·(H_∗◦F) div(L_∗◦F)

which is symmetric with respect to (H, K). From (4.5), (4.6) and (4.7) we obtain Gh(H, K, L)^x∼⁰div [D(KN+h◦F)·(H∗◦F)](L∗◦F)

−divX^N

i=1

Difh[D(Ki◦F)·(H_∗◦F)](L_∗◦F) +Sh(H, K, L)

(4.8)

where

Sh(H, K, L) :=

4

X

l=1

S_h^(l)(H, K, L). Step 6.One has (compare Section 5.5)

Z

Br(x0)

ϕρ,rdiv [D(KN+h◦F)·(H_∗◦F)](L_∗◦F) dx

− Z

B_r(x₀)

ϕ_ρ,rdivX^N

i=1

D_if_h[D(K_i◦F)·(H∗◦F)](L∗◦F) dx

=σh(H, K, L) +o(r^N) (4.9)

as r→0+, where

σh(H, K, L) :=− Z

Br(x0)

[D²fh(K_∗◦F)]·(H_∗◦F)[(L_∗◦F)·Dϕρ,r]dx .

Step 7 (the conclusion).One has Z

Br(x0)

ϕρ,rGh(H, K, L)dx= Z

Br(x0)

ϕρ,rSh(H, K, L)dx +σh(H, K, L) +o(r^N) (4.10)

as r→0+, by (4.8) and (4.9). Since the right hand side of (4.10) is symmetric with respect to the couple (H, K), we obtain

Gh(H, K, L)^x∼⁰Gh(K, H, L) that is

Dh^x∼⁰0

by (4.2). Finally, the identity (4.1) follows from Proposition 3.2.

Remark 4.1. Actually in [9] we have not proved Theorem 1.1, but the following

“step-1” analogous of Theorem 4.1, which is however trivially equivalent to Theorem 1.1.

Theorem 4.2. letH, K be two vector fields of classC¹ in an open setU ⊂R^N+m. Moreover letM be aN-dimensionalC¹ submanifold ofU and define

T :=

z∈ M:H(z), K(z)∈TzM .

Ifz0∈ Mis a(N+ 1)-density point ofT (relative toM) then[H, K](z0)∈Tz₀M.

It is now quite clear that Theorem 4.1 and Theorem 4.2 strongly support the following conjecture, which is equivalent to Conjecture 1.1, by (2.4) (withE=T).

Conjecture 4.1. Consider an open setU ⊂R^N+m, a N-dimensionalC^k subma- nifold M of U, a familyΦ :={Φ₁, . . . ,Φ_k+1} ⊂C^k(U,R^N+m)(with k≥1) and define

T :=

z∈ M: Φ1(z), . . . ,Φk+1(z)∈TzM .

Ifz0∈ Mis a(N+k)-density point ofT (relative to M) thenΛ^Φ(1,...,k+1)(z0)⊂ T_z0M.

5. Collection of the computational details needed to prove Theorem 4.1

5.1. Proof of (4.2). One has Dh=

N

X

p=1

Dpfh[[H, K], L]p◦F−[[H, K], L]N+h◦F

=

N

X

p=1

[(DK)◦F](H◦F)

·[(DLp)◦F]Dpfh

−

N

X

p=1

[(DH)◦F](K◦F)

·[(DLp)◦F]D_pf_h