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

**INVOLUTIVITY DEGREE OF A DISTRIBUTION**
**AT SUPERDENSITY POINTS OF ITS TANGENCIES**

Silvano Delladio

Abstract.

Let Φ1*, . . . ,*Φ*k+1*(with*k*≥1) be vector fields of class*C** ^{k}*in an open set

*U*⊂R

*, letMbe a*

^{N+m}*N*-dimensional

*C*

*submanifold of*

^{k}*U*and define

T :={z∈ M: Φ1(z), . . . ,Φ*k+1*(z)∈*T**z*M}

where*T**z*Mis the tangent space toMat*z. Then we expect the following*
property, which is obvious in the special case when*z*0 is an interior point
(relative toM) ofT:

*If**z*0∈ M*is a*(N+*k)-density point (relative to*M) ofT *then*
*all the iterated Lie brackets of order less or equal to**k*

Φ*i*1(z0),[Φ*i*1*,*Φ*i*2](z0),[[Φ*i*1*,*Φ*i*2],Φ*i*3](z0), . . . (h, i*h*≤*k+1)*
*belong to**T**z*_{0}M.

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

*Let*D*be a**C*^{2} *distribution of rank**N* *on an open set**U*⊂R^{N+m}*and*M*be a**N-dimensional**C*^{2} *submanifold of* *U. Moreover let*
*z*0 ∈ M *be a* (N+ 2)-density point of the tangency set {z ∈
M |*T**z*M=D(z)}. ThenD*must be*2-involutive at*z*0*, i.e., for*
*every family*{X*j*}^{N}_{j=1}*of class**C*^{2} *in a neighborhood**V* ⊂*U* *of**z*0

*which generates*D*one has*

*X**i*_{1}(z0),[X*i*_{1}*, X**i*_{2}](z0),[[X*i*_{1}*, X**i*_{2}], X*i*_{3}](z0)∈*T**z*0M
*for all*1≤*i*1*, i*2*, i*3≤*N.*

1. Introduction

Let Φ_{1}*, . . . ,*Φ* _{k+1}* (with

*k*≥ 1) be vector fields of class

*C*

*in an open set*

^{k}*U*⊂R

*(with*

^{N+m}*N, m*≥1), let Mbe a

*N-dimensionalC*

*submanifold of*

^{k}*U*and define

T :={z∈ M: Φ1(z), . . . ,Φ*k+1*(z)∈*T**z*M}

2020*Mathematics 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

where*T** _{z}*Mis the tangent space toMat

*z. One has the following obvious property:*

*If* *z*_{0} *is an interior point (relative to*M) of T*, then all the iterated Lie brackets of*
*order less or equal to* *k*

Φ*i*_{1}(z_{0}),[Φ*i*_{1}*,*Φ*i*_{2}](z_{0}),[[Φ*i*_{1}*,*Φ*i*_{2}],Φ*i*_{3}](z_{0}), . . . (h, i*h*≤*k*+ 1)
*belong toT**z*_{0}M.

With reference to this property, we are interested in understanding whether
it remains true when we substitute the hypothesis that*z*0 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 *z*0∈ M is said to be a
(N+*h)-density point of a set*E ⊂ M(relative to M) if*h*≥0 and

H* ^{N}*(B

_{M}(z

_{0}

*, r)*\ E) =

*o(r*

*) (as*

^{N+h}*r*→0+)

where*B*_{M}(z0*, r)*⊂ Mis the metric ball of radius*r*centered at*z*0, compare Section
2 below. According to this definition, roughly speaking, we can say that: the larger
*h, the higher the concentration of*Eat*z*0(and the more*z*0will resemble an interior
point ofE). Observe that, in the special case when*m*= 0 and M=R* ^{N}*, the point

*z*0 is a

*N-density point of*E if and only if it is a point of Lebesgue density ofE, that is L

*(B*

^{N}_{R}

*N*(z0

*, r)*∩

*E)/L*

*(B*

^{N}_{R}

*N*(z0

*, r))*→1 (as

*r*→0+).

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

**Theorem 1.1**([9]).*Given an open setU* ⊂R^{N}^{+m}*, consider*Φ1*,*Φ2∈*C*^{1}(U,R^{N}^{+m})
*and aN-dimensional* *C*^{1} *submanifold*M *ofU. Moreover define*

T :=

*z*∈ M: Φ_{1}(z),Φ_{2}(z)∈*T** _{z}*M

*and assume that* *z*0∈ M *is a* (N+ 1)-density point ofT *(relative to*M). Then
*one has*Φ1(z0),Φ2(z0),[Φ1*,*Φ2](z0)∈*T**z*_{0}M.

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

**Theorem 1.2.** *Given an open setU* ⊂R^{N}^{+m}*, consider*Φ1*,*Φ2*,*Φ3∈*C*^{2}(U,R* ^{N+m}*)

*and aN-dimensional*

*C*

^{2}

*submanifold*M

*ofU. Moreover define*

T :=

*z*∈ M: Φ1(z),Φ2(z),Φ3(z)∈*T**z*M

*and assume that* *z*_{0}∈ M *is a* (N+ 2)-density point ofT *(relative to*M). Then
*one has* Φ_{i}_{1}(z_{0}),[Φ_{i}_{1}*,*Φ_{i}_{2}](z_{0}),[[Φ_{i}_{1}*,*Φ_{i}_{2}],Φ_{i}_{3}](z_{0})∈*T*_{z}_{0}M*for all* 1≤*i*_{1}*, i*_{2}*, i*_{3}≤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

*is a mapDassigning a*

^{N+m}*N-dimensional vector subspace*D(z) ofR

*to each point*

^{N+m}*z*∈

*U*and satisfying the following property: If

*z*∈

*U*then there exist a neighborhood

*V*

^{(z)}⊂

*U*of

*z*and a family {X

_{i}^{(z)}}

^{N}*⊂*

_{i=1}*C*

*(V*

^{k}^{(z)}

*,*R

^{N}^{+m}) which generatesDin

*V*

^{(z)}, i.e., such that{X

_{1}

^{(z)}(z

^{0}), . . . , X

_{N}^{(z)}(z

^{0})}is a basis of D(z

^{0}) for

all *z*^{0} ∈ *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 to*k*

*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), . . . (with*i** _{h}*≤

*N*and

*h*≤

*k*+ 1) belong to D(z). Such a definition does not depend on the choice of the family {X

_{i}^{(z)}}

^{N}*, compare Proposition 6.1 below. In the special case when*

_{i=1}*k*= 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 *C*^{1} distribution of rank*N* on an open set*U* ⊂R^{N}^{+m}and letMbe
a*N*-dimensional*C*^{1}submanifold of*U*. 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* *z*0∈*U* *there*
*exists aC*^{1} *submanifold*M*ofU* *such thatz*0∈ M*and the tangency set of*M*with*
*respect to*D*coincides with*M, namely*τ(M,*D) :={z∈ M : *T**z*M=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 noninvolutive*C*^{1}distributionsD, mainly by providing
upper bounds for dim*H*(τ(M,D)) asMvaries among all the*N*-dimensional*C*^{2}
submanifolds of *U* (where dim* _{H}* denotes the Hausdorff dimension).

(1) Let*H*H* ^{k}*be the horizontal subbundle of the tangent bundle to the Heisenberg
groupH

*, that is the distribution of rank 2konR*

^{k}^{2k+1}generated by the vector fields

(x_{1}*, . . . , x*2k+1)7→ *∂*

*∂x** _{i}* + 2x

*k+i*

*∂*

*∂x*_{2k+1} (i= 1, . . . , k)
(x_{1}*, . . . , x*_{2k+1})7→ *∂*

*∂x**k+i*

−2x_{i}*∂*

*∂x*2k+1

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

(1.1) dim*H* *τ(M, H*H* ^{k}*)

≤*k*

for every (2k)-dimensional*C*^{2} 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)) : Mis

*C*

^{2}-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 class*C*^{∞} and fulfils the Hörmander noninvolutiveness condition
(see [2, Definition 4.1]), then one has

sup{dim*H*(τ(M,D)) : Mis*C*^{2}-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, the*C*^{1}smooth submanifoldsMare expected to produce
much larger tangencies (with respect toD) than those produced by*C*^{2}smooth
submanifolds. In fact, even if there are no points at whichD is involutive,
it can well be that a *C*^{1} 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
including*H*H* ^{k}* and there are good reasons to believe that it is true in general
(compare [2, Problem 8.3]).

(5) IfDis a *C*^{1} distribution of rank*N* on an open set *U* ⊂R^{N}^{+m}andM is
a*N*-dimensional *C*^{1} submanifold of*U*, then Dmust be involutive at each
point*z*0∈ 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
*z*0∈ M then there is no *N-dimensionalC*^{1} submanifoldMof *U* such that
*z*0∈ Mand*z*0 is a (N+ 1)-density point (relative toM) of*τ(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.** *If*D*is aC*^{2}*distribution of rankN* *on an open setU* ⊂R^{N}^{+m}*and*
M *is aN-dimensionalC*^{2} *submanifold ofU, then*D*must be* 2-involutive at each
*pointz*_{0}∈ M*which is a*(N+ 2)-density point of*τ(M,*D)*(relative to*M). In other
*words: if* D*is not*2-involutive at a point*z*_{0}∈ M *then there is noN-dimensional*
*C*^{2} *submanifold* M *of* *U* *such that* *z*_{0} ∈ M *and* *z*_{0} *is a* (N + 2)-density point
*(relative to* M) of*τ(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*, . . . ,*Φ*k+1* ∈
*C** ^{k}*(U,R

*)*

^{N+m}*and aN-dimensionalC*

^{k}*submanifold*M

*ofU*

*(withk*≥1). Moreo-

*ver define*

T :=

*z*∈ M: Φ_{1}(z), . . . ,Φ*k+1*(z)∈*T**z*M *.*

*and assume that* *z*_{0}∈ M *is a*(N+*k)-density point of*T *(relative to*M). Then all
*the iterated Lie brackets of order less or equal tok*

Φ*i*_{1}(z0),[Φ*i*_{1}*,*Φ*i*_{2}](z0),[[Φ*i*_{1}*,*Φ*i*_{2}],Φ*i*_{3}](z0), . . . (h, i*h*≤*k*+ 1)
*belong toT*_{z}_{0}M.

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

If D is a*C** ^{k}* distribution of rank

*N*on an open set

*U*⊂R

*andMis a*

^{N+m}*N*-dimensional

*C*

*submanifold of*

^{k}*U*, thenDmust be

*k-involutive at each pointz*

_{0}∈ Mwhich is a (N+

*k)-density point*

of*τ*(M,D) (relative toM). In other words: ifDis not*k-involutive*
at a point*z*_{0}∈ Mthen there is no*N*-dimensional*C** ^{k}* submanifold
M of

*U*such that

*z*

_{0}∈ M and

*z*

_{0}is a (N +

*k)-density point*(relative toM) of

*τ(M,*D).

2. General notation and preliminaries

We will have to deal with maps fromR* ^{N}* toR

*. The standard basis ofR*

^{m}

^{N}^{+m}and the corresponding coordinates are denoted by

*e*1

*, . . . , e*

*N*+m and (x1

*, . . . , x*

*N*

*, y*1

*,*

*. . . , y*

*m*), respectively. We may also writeR

^{N}*x*in place ofR

*andR*

^{N}

^{m}*y*in place of R

*. If*

^{m}*U*is an open subset ofR

^{N}*x*×R

^{m}*y*and

*G*∈

*C*

^{1}(U,R

*), then*

^{k}*D*

*x*

*G*and

*D*

*y*

*G*denote the Jacobian matrix of

*G*with respect to

*x*and the Jacobian matrix of

*G*with respect to

*y, respectively, that is*

*D**x**G*:=*∂G*

*∂x*1

*. . .*

*∂G*

*∂x**N*

*,* *D**y**G*:=*∂G*

*∂y*1

*. . .*

*∂G*

*∂y**m*

*.*

In general, the Jacobian matrix of any*C*^{1} vector field*F* is denoted by*DF*. The
Hessian matrix of any*C*^{2}function *f* is denoted by*D*^{2}*f*, while *D*^{2}_{ij}*f* stands for the
(i, j)-entry of *D*^{2}*f*. The*h*^{th}-order derivative of a*C** ^{h}* function of one variable

*g*is indicated with

*g*

^{(h)}. For simplicity, we define

*D*1:= *∂*

*∂x*1

*, . . . , D**N* := *∂*

*∂x**N*

*, D**N*+1:= *∂*

*∂y*1

*, . . . , D**N+m*:= *∂*

*∂y**m*

*.*
For*α*= (α1*, . . . , α**N*)∈N* ^{N}*, define

|α|:=*α*1+· · ·+*α**N**,* *D**α*:= *∂*^{|α|}

*∂x*^{α}_{1}^{1}· · ·*∂x*^{α}_{N}^{N}*.*

The Euclidean norms involved throughout this paper are all denoted byk · k. The
constants depending only on*p, q, . . .*are indicated by*C(p, q, . . .). LetU* be an open
subset ofR^{N}^{+m}. If*k*≥1 and

*H* = (H1*, . . . , H**N*+m), K= (K1*, . . . , K**N*+m)∈*C** ^{k}*(U,R

^{N}^{+m}) then we recall that the Lie bracket product of

*H, 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*

(H*i**D**i**K**j*−*K**i**D**i**H**j*), *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

*, . . . , X*

*p*} ⊂

*C*

*(U,R*

^{k}*), then we state the following inductive definition of*

^{N+m}*h*

^{th}-order iterated

Lie brackets of the vector fields*X** _{i}*, with 0≤

*h*≤

*k*and 1≤

*i*

_{1}

*, . . . , i*

*≤*

_{h+1}*p:*

Λ^{X}_{(i}

1*,...,i** _{h+1}*):=

(*X**i*_{1} if *h*= 0
h

Λ^{X}_{(i}

1*,...,i** _{h}*)

*, X*

*i*

_{h+1}i

if 1≤*h*≤*k*

e.g. Λ^{X}_{(1)} = *X*1, Λ^{X}_{(1,2)} = [X1*, X*2], Λ^{X}_{(1,2,1)} = [[X1*, X*2], X1] (provided *k* ≥ 2).

Observe that

(2.3) Λ^{X}_{(i}

1*,...,i**h+1*)∈*C** ^{k−h}*(U,R

^{N}^{+m}).

Let L^{X}* _{h}*(z) be the vector space spanned by the family of the

*h*

^{th}-order iterated Lie brackets (of the vector fields

*X*

*i*) at

*z*∈

*U*, namely

L^{X}* _{h}*(z) := spann

Λ^{X}_{(i}_{1}_{,...,i}_{h+1}_{)}(z) : 1≤*i*1*, . . . , i**h+1*≤*p*o
for all*z*∈*U* and 0≤*h*≤*k.*

Let M be a *N-dimensional* *C*^{1} 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

*z*

_{0}∈ Mand

*r >*0 we define

*B*_{M}(z_{0}*, r) :=*{z∈ M^{(z}^{0}^{)}|*d(z, z*_{0})*< r}*

where*M*^{(z}^{0}^{)} is the connected component of Mcontaining *z*0. Recall that for *r*
small enough exp_{z}_{0} maps*B**T*_{z}_{0}M(0, r) diffeomorphically onto a neighborhood of*z*0

and one has

exp_{x}_{0} *B**T*_{z}_{0}M(0, r)

=*B*_{M}(z0*, r)*

compare [3, Theorem 1.6 and Corollary 1.1]. In the special case when*m*= 0 and
M=R* ^{N}* the distance

*d*reduces to the usual Euclidean distance and we denote

*B*

_{R}

*N*(z0

*, r) simply byB*

*r*(z0).

The Lebesgue outer measure onR* ^{N}* and the

*N*-dimensional Hausdorff measure onR

*will be denoted byL*

^{N+m}*andH*

^{N}*, respectively.*

^{N}A point *x* ∈ R* ^{N}* is said to be a (N +

*k)-density point of*

*E*⊂ R

*(where*

^{N}*k*∈[0,+∞)) if

L* ^{N}*(B

*(x)\*

_{r}*E) =o(r*

*) (as*

^{N+k}*r*→0+)

*.*

The set of all (N+*k)-density points of* *E*is denoted by*E*^{(N+k)}. Analogously, if
Mis a*N*-dimensional*C*^{1} submanifold ofR^{N}^{+m}and*z*_{0}∈ M, then we say that*z*_{0}
is a (N+*k)-density point of*E ⊂ M(relative toM) if

H* ^{N}*(B

_{M}(z

_{0}

*, r)*\ E) =

*o(r*

^{N}^{+k}) (as

*r*→0+).

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

E^{(N}^{+k)}⊂ E^{(N+h)}

for all*h*∈[0, k]. In particular, if*k*is 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
*C*^{1} embeddings preserve density-degree, namely the following property holds [8,
Proposition 3.3].

**Proposition 2.1.** *Let* M*be aN-dimensionalC*^{1} *submanifold of* R^{N}^{+m}*, let*Ω*be*
*an open subset of* R^{N}*and letF* : Ω→R^{N+m}*be an injective immersion of class*
*C*^{1} *such that* *F*(Ω)⊂ M. Moreover let *E* *be a subset of* Ω*and letx*_{0}∈Ω. Then
*(fork*≥0) one has

L* ^{N}*(B

*(x*

_{r}_{0})\

*E) =o(r*

^{N}^{+k}) (as

*r*→0+)

*if and only if*

H* ^{N}*(B

_{M}(F(x

_{0}), r)\

*F*(E)) =

*o(r*

*) (as*

^{N+k}*r*→0+)

*.*

*In particular,x*

_{0}∈

*E*

^{(N+k)}

*if and only ifF*(x

_{0})∈

*F*(E)

^{(N+k)}

*.*

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

**Remark 2.1.** Let*H* = (H1*, . . . , H**N+m*) be a vector field of class*C*^{1}in an open
set*U* ⊂R^{N}*x* ×R^{m}*y* . Moreover let Ω be an open subset ofR^{N}*x* and*f* = (f1*, . . . , f**m*)∈
*C*^{1}(Ω,R^{m}*y*). Denote by Γ the graph of *f, that is Γ :=F*(Ω) where

*F*: Ω→R^{N}*x* ×R^{m}*y* *,* *F*(x) := *x, f(x)*

and assume that Γ⊂*U*. Given*x*∈Ω, obviously one has that*H*(F(x))∈*T** _{F(x)}*Γ if
and only if

*H(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*, . . . , H**N*)*,* *H*#:= (H*N*+1*, . . . , H**N*+m)*.*

Moreover, if*K*= (K1*, . . . , K**N+m*) is another vector field of class*C*^{1} in*U* and if
one has*H*(F(x)), K(F(x))∈*T*_{F}_{(x)}Γ for a certain*x*∈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), with

*k*≥1, such that

0≤*g*≤1, *g|*_{(−∞,0]}≡1, *g|*_{[1,+∞)}≡0
and, for*ρ*∈(0,1), define

*ψ**ρ*(x) :=*g*kxk −*ρ*
1−*ρ*

*,* *x*∈R^{N}*.*
Observe that

*ψ**ρ*|_{B}

*ρ*(0) ≡1*,* *ψ**ρ*|_{R}*N*\B1(0)≡0*.*

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

*one has*

*D**α**ψ**ρ*(x) =

|α|

X

*h=1*

(1−*ρ)*^{−h}*g*^{(h)}kxk −*ρ*
1−*ρ*

X

{β1*,...,β** _{h}*}∈P

*h*(α)

*D**β*_{1}kxk · · ·*D**β** _{h}*kxk

*where*

P*h*(α) :=n

{β1*, . . . , β**h*} : *β**i*∈N* ^{N}* \ {0},

*h*

X

*i=1*

*β**i*=|α|o
*.*

**Proof.** The statement is obvious if|α|= 1. Then let*k*be 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 that

*α*1≥1. If define

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

*D*_{α−ε}_{1}*ψ** _{ρ}*(x) =

*k*

X

*h=1*

(1−*ρ)*^{−h}*g*^{(h)}kxk −*ρ*
1−*ρ*

X

{β_{1}*,...,β** _{h}*}∈P

*(α−ε*

_{h}_{1})

*D*_{β}_{1}kxk · · ·*D*_{β}* _{h}*kxk

*.*Thus

*D**α**ψ**ρ*(x) =*D**ε*_{1}(D_{α−ε}_{1}*ψ**ρ*) (x)

=

*k*

X

*h=1*

(1−*ρ)*^{−h−1}*g*^{(h+1)}kxk −*ρ*
1−*ρ*

*D*_{ε}_{1}kxk X

{β1*,...,β** _{h}*}∈P

*h*(α−ε1)

*D*_{β}_{1}kxk · · ·*D*_{β}* _{h}*kxk

+

*k*

X

*h=1*

(1−*ρ)*^{−h}*g*^{(h)}kxk −*ρ*
1−*ρ*

X

{β1*,...,β** _{h}*}∈P

*h*(α−ε1)

*D**β*_{1}+ε_{1}kxkD*β*_{2}kxk · · ·*D**β** _{h}*kxk
+

*D*

*β*

_{1}kxkD

*β*

_{2}+ε

_{1}kxk · · ·

*D*

*β*

*kxk+· · ·+*

_{h}*D*

*β*

_{1}kxk · · ·

*D*

*β*

*kxkD*

_{h−1}*β*

*+ε*

_{h}_{1}kxk

= (1−*ρ)*^{−1}*g*^{(1)}kxk −*ρ*
1−*ρ*

*D** _{α}*kxk+

*k*

X

*h=2*

(1−*ρ)*^{−h}*g*^{(h)}kxk −*ρ*
1−*ρ*

×

X

{β1*,...,β**h−1*}∈P*h−1*(α−ε1)

*D**ε*_{1}kxkD*β*_{1}kxk · · ·*D**β**h−1*kxk

+ X

{β1*,...,β**h*}∈P*h*(α−ε1)

*D**β*_{1}+ε_{1}kxkD*β*_{2}kxk · · ·*D**β** _{h}*kxk

+*D*_{β}_{1}kxkD*β*2+ε1kxk · · ·*D*_{β}* _{h}*kxk+· · ·+

*D*

_{β}_{1}kxk · · ·

*D*

_{β}*kxkD*

_{h−1}*β*

*h*+ε1kxk

+ (1−*ρ)*^{−k−1}*g*^{(k+1)}kxk −*ρ*
1−*ρ*

*D**ε*_{1}kxkD^{α}_{ε}_{1}^{1}^{−1}kxkD^{α}_{ε}_{2}^{2}kxk · · ·*D*^{α}_{ε}_{N}* ^{N}*kxk

hence the conclusion follows immediately.

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

*D**α*kxk= *p**α*(x)

kxk^{2|α|−1} (x6= 0)

where*p**α*is a homogeneous polynomial of degree|α|whose coefficients depend only
on*α. It follows that*

max

*x∈B*1(0)\B*ρ*(0)

D*α*kxk

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

*ϕ** _{ρ,r}*(x) :=

*ψ*

_{ρ}*x*−

*x*

_{0}

*r*

*,* *x*∈R^{N}*.*
*Then, for allα*∈N^{N}*, one has*

kD*α**ϕ**ρ,r*k_{∞}≤ *C(α)*
(1−*ρ)*^{|α|}*r*^{|α|}*.*

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

*D*_{α}*ϕ** _{ρ,r}*(x) =

*r*

^{−|α|}(D

_{α}*ψ*

*)*

_{ρ}*x*−

*x*

_{0}

*r*

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

kD*α**ϕ**ρ,r*k_{∞}=*r*^{−|α|}kD*α**ψ**ρ*k_{∞}

=*r*^{−|α|} max

*x∈B*1(0)\B*ρ*(0)

|D_{α}*ψ** _{ρ}*(x)|

≤*r*^{−|α|}

|α|

X

*h=1*

kg^{(h)}k_{∞}
(1−*ρ)*^{h}

X

{β_{1}*,...,β** _{h}*}∈P

*(α)*

_{h}*C(β*_{1})*. . . C(β** _{h}*)

*ρ*

^{2|β}

^{1}

^{|−1}

*. . . ρ*

^{2|β}

^{h}^{|−1}

≤*C(α)*
*r*^{|α|}

|α|

X

*h=1*

1

(1−*ρ)*^{h}*ρ*^{2|α|−h}*.*

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

2^{2|α|} (for *h*= 1, . . . ,|α|)*.*

**Proposition 3.2.** *Let*Θ *be a continuous function defined in a neighborhood of*
*x*0∈R^{N}*. Assume that for allρ*∈(1/2,1) *one has*

(3.1)

Z

*B** _{r}*(x

_{0})

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* *x*_{0} ∈ *E*^{(N+k)}*, with*
*k*≥1. Moreover letΘ *and*Λ*be a couple of continuous real valued functions defined*
*in a neighborhood ofx*_{0} *such that* Θ|_{E∩B}_{r}_{(x}_{0}_{)}= Λ|_{E∩B}_{r}_{(x}_{0}_{)} *(for* *rsmall enough).*

*If* *ρ*∈(1/2,1), then one has
Z

*B**r*(x0)

ΘD*α**ϕ**ρ,r**dx*=
Z

*B**r*(x0)

ΛD*α**ϕ**ρ,r**dx*+*o(r** ^{N}*) (as

*r*→0+)

*for all*

*α*∈N

^{n}*such that*|α| ≤

*k.*

**Proof.** Let *α*∈N* ^{n}* 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}*(x

_{0})

ΘD*α**ϕ**ρ,r**dx*
=

Z

*B** _{r}*(x

_{0})\E

ΘD*α**ϕ**ρ,r**dx*

≤ sup

*B** _{r}*(x

_{0})

|Θ|

kD*α**ϕ**ρ,r*k_{∞}L* ^{N}*(B

*r*(x0)\

*E).*

We conclude by Corollary 3.1 and recalling that*x*_{0}∈*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 *x*_{0}∈R* ^{N}*, such that

Z

*B**r*(x0)

Θ(x)ϕ*ρ,r*(x)*dx*=
Z

*B**r*(x0)

Λ(x)ϕ*ρ,r*(x)*dx*+*o(r** ^{N}*) (as

*r*→0+) for all

*ρ*∈(1/2,1). Then we write Θ

*∼*

^{x}^{0}Λ and say that Θ and Λ are equivalent at

*x*0.

**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 of

*x*

_{0}.

**Proposition 3.4.**

*Let*Θ

*and*Λ

*be a couple of real valued functions, each one*

*defined and continuous in a neighborhood of*

*x*

_{0}∈R

^{N}*, such that*Θ

*∼*

^{x}^{0}Λ. Moreover,

*letg*

*be a function of class*

*C*

^{1}

*in a neighborhood ofx*

_{0}

*. Then one hasgΘ*

*∼*

^{x}^{0}

*gΛ.*

**Proof.** For*r*sufficiently small and*ρ*∈(0,1), one has

Z

*B** _{r}*(x

_{0})

*g(x)Θ(x)ϕ** _{ρ,r}*(x)

*dx*− Z

*B** _{r}*(x

_{0})

*g(x)Λ(x)ϕ** _{ρ,r}*(x)

*dx*

= Z

*B** _{r}*(x

_{0})

[g(x)−*g(x*_{0})]Θ(x)ϕ* _{ρ,r}*(x)

*dx*− Z

*B** _{r}*(x

_{0})

[g(x)−*g(x*_{0})]Λ(x)ϕ* _{ρ,r}*(x)

*dx*+

*g(x*

_{0})

Z

*B** _{r}*(x

_{0})

Θ(x)ϕ* _{ρ,r}*(x)

*dx*− Z

*B** _{r}*(x

_{0})

Λ(x)ϕ* _{ρ,r}*(x)

*dx*

≤*C(N*) sup

*B** _{r}*(x

_{0})

|g−*g(x*_{0})|

sup

*B** _{r}*(x

_{0})

|Θ|+ sup

*B** _{r}*(x

_{0})

|Λ|

*r** ^{N}* +|g(x

_{0})|

*o(r*

*)*

^{N}*.*Moreover, for all

*x*∈

*B*

*r*(x

_{0}), one has

*g(x)*−*g(x*0) =
Z 1

0

*Dg(x*0+*t(x*−*x*0))·(x−*x*0)*dt*

hence

sup

*B**r*(x0)

|g−*g(x*_{0})| ≤
sup

*B**r*(x0)

kDgk
*r .*
It follows that

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*=*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* *x*0 ∈ *E*^{(N}^{+k)}*, with*
*k*≥1. Moreover let Θ*and*Λ*be a couple of real valued functions of classC*^{k}*in a*
*neighborhood of* *x*_{0} *such that*Θ|_{E∩B}* _{r}*(x

_{0})= Λ|

_{E∩B}*(x*

_{r}_{0})

*(forr*

*small enough). Then*

*one has*

*D**α*Θ* ^{x}*∼

^{0}

*D*

*α*Λ (hence

*D*

*α*Θ(x0) =

*D*

*α*Λ(x0), by Proposition 3.2)

*for all*

*α*∈N

^{n}*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) (with*k*= 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* *C*^{2} *in an open set*
*U* ⊂R^{N+m}*. Moreover let* M*be aN-dimensionalC*^{2} *submanifold ofU* *and define*

T :=

*z*∈ M:*H(z), K(z), L(z)*∈*T** _{z}*M

*.*

*Ifz*0∈ M*is a* (N+ 2)-density point ofT *(relative to*M) then[[H, K], L](z0)∈
*T**z*_{0}M.

**Proof.** Since Mis locally the graph of a*C*^{2} function, we can assume that there
exist an open set Ω⊂R^{N}*x* and*f* = (f1*, . . . , f**m*)∈*C*^{2}(Ω,R^{m}*y*) such that

*z*_{0}∈Γ :={(x, f(x)) : *x*∈Ω} ⊂ M*.*
Define *F*∈*C*^{2}(Ω,R^{N}^{+m}) by

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

*,* *x*∈Ω*.*
Moreover let

*x*0:=*F*^{−1}(z0)*,* *T* :=*F*^{−1}(T)
and observe that

*x*0∈Ω∩*T*^{(N+2)} (hence also*x*0∈Ω∩*T*^{(N+1)})

by Proposition 2.1. The following notation will be useful: if*A*and*B* are functions
defined in Ω such that*A|**T* =*B|**T*, then we write *A*=^{T}*B.*

If for all*h*= 1, . . . , mdefineD*h*∈*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) D*h*(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]*i**D**i**L**j*−*L**i**D**i*[H, K]*j*)

=

*N*+m

X

*i,l=1*

*H*_{l}*D*_{l}*K*_{i}*D*_{i}*L** _{j}*−

*K*

_{l}*D*

_{l}*H*

_{i}*D*

_{i}*L*

*−*

_{j}*L*

_{i}*D*

*(H*

_{i}

_{l}*D*

_{l}*K*

*−*

_{j}*K*

_{l}*D*

_{l}*H*

*)*

_{j}= [(DK)H]·*DL**j*−[(DH)K]·*DL**j*−[(D^{2}*K**j*)H]·*L+*

−[(DH)L]·*DK**j*+ [(D^{2}*H**j*)K]·*L*+ [(DK)L]·*DH**j*

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

(4.2) D*h** ^{x}*∼

^{0}G

*h*(H, K, L)− G

*h*(K, H, L) where

(4.3)

G* _{h}*(H, K, L) := [D(K◦F)(H

_{∗}◦F)]·X

^{N}*p=1*

[(DL* _{p}*)◦F]D

_{p}*f*

*−[(DL*

_{h}

_{N}_{+h})◦F]

+ [D(K◦F)(L∗◦F)]·X^{N}

*p=1*

[(DH*p*)◦F]D*p**f**h*−[(DH*N*+h)◦F]

−

*N*

X

*p=1*

*D**p**f**h* [(D^{2}*K**p*)◦F](H◦F)

·(L◦F)
+ [(D^{2}*K**N*+h)◦F](H◦F)

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

G* _{h}*(H, K, L)(x

_{0}) =G

*(K, H, L)(x*

_{h}_{0}) (h= 1, . . . , m)

*.*

**Step 2.**For*l*= 1, . . . , N+*m*the following identity holds (compare Section 5.2)
[(D^{2}*K**l*)◦F](H◦F)

·(L◦F)* ^{x}*∼

^{0}A

*l*(H, K, L)

−[D(H◦F)(L_{∗}◦F)]·[(DK* _{l}*)◦F]
(4.4)

where

A*l*(H, K, L) :=*D[D(K** _{l}*◦F)·(H∗◦F)]·(L∗◦F).

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

G*h*(H, K, L)* ^{x}*∼

^{0}[D(K◦F)(H

_{∗}◦F)]·X

^{N}*p=1*

[(DL*p*)◦F]D*p**f**h*−[(DL*N*+h)◦F]

+

*N*

X

*p=1*

*D*_{p}*f** _{h}*[D(K◦F)(L

_{∗}◦F)]·[(DH

*)◦F]+*

_{p}−

*N*

X

*p=1*

*D**p**f**h*A*p*(H, K, L) +

*N*

X

*p=1*

*D**p**f**h*[D(H◦F)(L∗◦F)]·[(DK*p*)◦F]

+A*N*+h(H, K, L)−[D(H◦F)(L_{∗}◦F)]·[(DK*N*+h)◦F]

−[D(K◦F)(L_{∗}◦F)]·[(DH*N+h*)◦F]

that is

G*h*(H, K, L)* ^{x}*∼

^{0}B

*h*(H, K, L) +C

*h*(H, K, L) +S

_{h}^{(1)}(H, K, L) +S

_{h}^{(2)}(H, K, L) (4.5)

where

B*h*(H, K, L) :=A*N+h*(H, K, L)−

*N*

X

*p=1*

*D**p**f**h*A*p*(H, K, L)

C*h*(H, K, L) := [D(K◦F)(H_{∗}◦F)]·X^{N}

*p=1*

[(DL*p*)◦F]D*p**f**h*−[(DL*N+h*)◦F]

S_{h}^{(1)}(H, K, L) :=−[D(H◦F)(L∗◦F)]·[(DK*N*+h)◦F]

−[D(K◦F)(L_{∗}◦F)]·[(DH*N*+h)◦F]
S_{h}^{(2)}(H, K, L) :=

*N*

X

*p=1*

*D*_{p}*f** _{h}*[D(K◦F)(L

_{∗}◦F)]·[(DH

*p*)◦F]

+

*N*

X

*p=1*

*D**p**f**h*[D(H◦F)(L∗◦F)]·[(DK*p*)◦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) C*h*(H, K, L)* ^{x}*∼

^{0}F

*h*(H, K, L) +S

_{h}^{(3)}(H, K, L)

where

F*h*(H, K, L) :=−

*N*

X

*i=1*

[D(K*i*◦F)·(H∗◦F)][D*i*(Df*h*)·(L∗◦F)]

S_{h}^{(3)}(H, K, L) :=

*m*

X

*q=1*

[D^{2}*f** _{q}*(K

_{∗}◦F)]·(H

_{∗}◦F)

× [(D*N+q**L*_{∗})◦F]·Df*h*−[(D*N*+q*L**N+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)

B*h*(H, K, L)+F*h*(H, K, L)* ^{x}*∼

^{0}div [D(K

*N+h*◦F)·(H

_{∗}◦F)](L

_{∗}◦F)

−divX^{N}

*i=1*

*D*_{i}*f** _{h}*[D(K

*◦F)·(H*

_{i}_{∗}◦F)](L

_{∗}◦F) +S

_{h}^{(4)}(H, K, L)

(4.7)

where

S_{h}^{(4)}(H, K, L) :=−[D^{2}*f**h*(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
G*h*(H, K, L)* ^{x}*∼

^{0}div [D(K

*N*+h◦F)·(H∗◦F)](L∗◦F)

−divX^{N}

*i=1*

*D**i**f**h*[D(K*i*◦F)·(H_{∗}◦F)](L_{∗}◦F)
+S*h*(H, K, L)

(4.8)

where

S*h*(H, K, L) :=

4

X

*l=1*

S_{h}^{(l)}(H, K, L)*.*
**Step 6.**One has (compare Section 5.5)

Z

*B**r*(x0)

*ϕ**ρ,r*div [D(K*N+h*◦F)·(H_{∗}◦F)](L_{∗}◦F)
*dx*

− Z

*B** _{r}*(x

_{0})

*ϕ** _{ρ,r}*divX

^{N}*i=1*

*D*_{i}*f** _{h}*[D(K

*◦F)·(H∗◦F)](L∗◦F)*

_{i}*dx*

=*σ**h*(H, K, L) +*o(r** ^{N}*)
(4.9)

as *r*→0+, where

*σ**h*(H, K, L) :=−
Z

*B**r*(x0)

[D^{2}*f**h*(K_{∗}◦F)]·(H_{∗}◦F)[(L_{∗}◦F)·Dϕ*ρ,r*]*dx .*

**Step 7 (the conclusion).**One has
Z

*B**r*(x0)

*ϕ**ρ,r*G*h*(H, K, L)*dx*=
Z

*B**r*(x0)

*ϕ**ρ,r*S*h*(H, K, L)*dx*
+*σ**h*(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

G*h*(H, K, L)* ^{x}*∼

^{0}G

*h*(K, H, L) that is

D*h** ^{x}*∼

^{0}0

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 classC*^{1} *in an open setU* ⊂R^{N}^{+m}*.*
*Moreover let*M *be aN-dimensionalC*^{1} *submanifold ofU* *and define*

T :=

*z*∈ M:*H(z), K(z)*∈*T**z*M *.*

*Ifz*0∈ M*is a*(N+ 1)-density point ofT *(relative to*M) then[H, K](z0)∈*T**z*_{0}M.

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 family*Φ :={Φ_{1}*, . . . ,*Φ* _{k+1}*} ⊂

*C*

*(U,R*

^{k}*)*

^{N+m}*(with*

*k*≥1) and

*define*

T :=

*z*∈ M: Φ1(z), . . . ,Φ*k+1*(z)∈*T**z*M *.*

*Ifz*0∈ M*is a*(N+*k)-density point of*T *(relative to* M) thenΛ^{Φ}(1,...,k+1)(z0)⊂
*T*_{z}_{0}M.

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

5.1. **Proof of** (4.2). One has
D*h*=

*N*

X

*p=1*

*D**p**f**h*[[H, K], L]*p*◦F−[[H, K], L]*N*+h◦F

=

*N*

X

*p=1*

[(DK)◦F](H◦F)

·[(DL*p*)◦F]D*p**f**h*

−

*N*

X

*p=1*

[(DH)◦F](K◦F)

·[(DL*p*)◦F]D_{p}*f*_{h}