Almost exponential maps and integrability results for a class of horizontally regular vector fields

0.37 MB·English

by  Daniele Morbidelli

Almost exponential maps and integrability results
for a class of horizontally regular vector fields ∗
Annamaria Montanari Daniele Morbidelli
2
1
0 Abstract
2
n We consider a family H := {X1,...,Xm} of C1 vector fields in Rn and we dis-
a cuss the associated H-orbits. Namely, we assume that our vector fields belong to a
J horizontal regularity class and we require that a suitable s-involutivity assumption

5 holds. Then we show that any H-orbit O is a C1 immersed submanifolds and it is an
2 integralsubmanifoldofthedistributiongeneratedbythefamilyofallcommutatorsup

to length s. Our main tool is a class of almost exponential maps of which we discuss
]
G carefully some precise first order expansions.
D
.
h
Contents
t
a
m
1 Introduction and main results 1
[ 2 Preliminaries 4

1 3 Approximate exponentials and regularity of As orbits 8
3.1 Geometric properties of orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
v
3.2 Derivatives of almost exponentialmaps and regularity of orbits. . . . . . . . . . . . . 13
8
2 A Appendix 20
2 References 21
5
.
1 1. Introduction and main results
0
2
1 In this paper we discuss the integrability of distributions defined by families of vector
: fields under a higher order horizontal regularity hypothesis and assuming an involutivity
v
i conditionoforders ∈ N. Thecentraltoolweexploitisgivenbyaclassofalmostexponential
X
maps which we will analyze in details assuming only low regularity on the coefficients of
r
a the vector fields.
To start the discussion, fix a family H = {X ,...,X } of at least Lipschitz-continuous
1 m
vector fields. For any x ∈Rn define the Sussmann’s orbit, or leaf
OHx := {et1Xj1 ···etpXjpx :p ∈ N,J := (j1,...,jp) ∈ {1,...,m}p,t ∈ ΩJ,x}, (1.1)
where for fixed x ∈ Rn we denote by Ω ⊂ Rp the open neighborhood of the origin where
J,x
the map t 7→ et1Xj1 ···etpXjpx is well defined. We equip the leaf OHx with the topology τd
defined by the Franchi–Lanconelli distance d; see (2.1).
∗
2010 MathematicsSubjectClassification. Primary53C17; Secondary53C12. KeywordsandPhrases:
Integrabledistribution, Carnot–Carathéodory distance.
1
A.MontanariandD.Morbidelli Almostexponential mapsandintegrability
Ourpurposeis todescribearegularity class oforder s ≥ 2andas-involutivity assump-
tion that ensure that each orbit O is a integral manifold of the distribution generated
H
by the family P := P := {Y ,...,Y } of all nested commutators of length at most s
s 1 q
constructed from the original family H. To give coordinates on O we shall use the follow-
ing almost exponential maps. Fix s ≥ 2 and denote by P the aforementioned family of
commutators. Assign to each Y the length ℓ ≤ s, just its order. Then, let
j j
E (h) := exp (h Y )···exp (h Y )x, (1.2)
I,x ap 1 i1 ap p ip
where I = (i ,...,i ) is a multiindex which fixes p commutators Y ,...,Y ∈ P, h ∈ Rp
1 p i1 ip
belongs to a neighborhood of the origin and p ∈ {1,...,n} is suitable. See (2.15) for
the definition of the approximate exponential exp . We shall use the maps in (1.2) to
ap
construct charts, developing a higher order, nonsmooth, quantitative extension of some
ideas appearing in a paper by Lobry; see [Lob70]; see Theorem 3.5 and Remarks 3.6
and 3.7 below.
Here is a description of our regularity class. Let H = {X ,...,X } and let s ≥ 2.
1 m
Assume that X =: f ·∇ ∈ C1 for all j (here and hereafter C1 refers to Euclidean
j j Euc Euc
regularity). Assume also that for each p ≤ s and j ,...,j ∈ {1,...,m}, all derivatives
1 p
♯ ♯
X ···X f exist and are locally Lipschitz-continuous functions with respect to dis-
j1 jp−1 jp
tance d associated to the vector fields. Here, following [MM12a], we denote by X♯f the
Lie derivative along the vector field X of the scalar function f. Moreover we require that
for any commutator Y =: g ·∇ ∈ P, all maps of the form g ◦E are continuous for all
j j j I,x
p ∈ {1,...,n}, I = (i ,...,i ) and x ∈Rn. 1
1 p
Furthermore, we require the following s-involutivity condition. For any X ∈ H and
j
♯
for any Y ∈ P with maximal length ℓ = s, at any x ∈ Ω where the derivative X g (x)
k k j k
exists one can write for suitable bi = bi(x)
q
(ad Y ) :=(X♯g (x)−Y f (x))·∇ = biY with bi locally bounded. (1.3)
Xj k x j k k j i,x
i=1
X
The class of vector fields satisfying all those assumptions will be denoted by A ; see Def-
s
inition 2.5, where a more precise formulation of this assumption is described. Note that
in the smooth case we have ad Y = [X ,Y ] and ultimately (1.3) is equivalent to the
Xj k j k
Hermann condition [Her62]
[Y ,Y ] = ckY , with ck ∈ L∞, (1.4)
i j ij k ij loc
1≤k≤q
X
which ensures that any Sussmann’s orbit O of the family of commutators P is a integral
P
manifold of the distribution generated by P. If furthermore s = 1, then P = H and (1.4)
and (1.3) are the same. Note that the appearance of operators of the form ad Y is
Xj k
very natural in the framework of our almost exponential maps; see the non-commutative
calculus formulas discussed in [MM12a, Section 3].
Here is the statement of our result.
1Thisconditioniswidelyensuredforinstanceassoonasweassumethatg iscontinuousintheEuclidean
j
topology, or at least in the Sussmann’sorbit topology definedon O bythefamily H; see [Sus73].
2
A.MontanariandD.Morbidelli Almostexponential mapsandintegrability
Theorem 1.1. Let H = {X ,...,X } be a family of vector fields of class A . Then, for
1 m s
any x ∈ Rn, the orbit O := Ox0 with the topology τ is a C1 immersed submanifold of Rn
0 H d
with tangent space T O = P for all y ∈ O.
y y
Note that this result does not follow from standard ones, because the commutators Y
j
are not assumed to be C1 in the Euclidean sense. In Example 3.14 we exhibit a family of
vector fields where our theorem apply, but classical results do not. See also Remark 3.15
for some further comments. Furthermore, let us mention that if s = 1, i.e. H = P, then
Theorem 1.1 is a consequence of the Frobenius Theorem for singular C1 distributions (it
is well known to experts that in such case one can prove that orbits are even C2 smooth).
Note that if s = 1, in [MM11a] we proved a singular Frobenius-type theorem assuming
only Lipschitz-continuity of the involved vector fields, generalizing part of Rampazzo’s
results [Ram07] to singular distributions; in fact, in [MM11a], orbits are C1,1.
On a technical level, the main tool we discuss is the approximate exponential E
I,x
in(1.2). Introducethenotationp := dimP := dimspan{Y (x),...,Y (x)}forallx ∈ Rn.
x x 1 q
Fix x, take p := p commutators Y ,...,Y , which are linearly independent at x and
x i1 ip
construct the map E, defined in (1.2). Then, under the hypotheses of Theorem 1.1, we
shall show that if the family H satisfies condition A , then E is a C1 , full rank map
s Euc
in a neighborhood of the origin 0 ∈ Rp, whose derivative enjoys the following remarkable
expansion
s q
E∗(∂hk)= Yik(E(h))+ ajk(h)Yj(E(h))+ ωki(x,h)Yi(E(h)). (1.5)
ℓj=Xℓik+1 Xi=1
The functions aj and ωi have a very precise rate of convergence to 0, as h → 0 which will
k k
bespecifiedin (3.22)and (3.23). Notethat an expansion of E (∂ )can beobtained either
∗ hk
with the Campbell–Hausdorff formula in the smooth case (see [Mor00] or [VSCC92]), or
in nonsmooth situations with the techniques of [MM12b]. However, the expansions in the
mentioned papers contain some remainders appearing either as formal series, or in integral
j
form. Here we are able to express such reminders via the pointwise terms ω , improving
k
all previous results. Note also that we are improving the mentioned papers both from a
regularity standpoint andbecauseherewedonotassumetheHörmandercondition. Atthe
authors’ knowledge, expansion (1.5)with precise estimates on aj and ωi is new even inthe
k k
smooth case. As a final remark, observe that Theorem 3.11 contains an explicit detailed
proofofthefactthatthemapE isC1 smooth,avoidinganyuseoftheCampbell–Hausdorff
formula. Note that, even if the vector fields are smooth, such maps are not much more
than C1; see Remark 3.12-(ii).
Theusefulinformationonecanextractfrom(1.5)isthatE (∂ ) ∈ P (notethatwe
∗ hk E(h)
are interested to situations where the inclusion P ⊂ Rn is strict); see Theorem 3.11 for
E(h)
aprecise statement. Observethat, ifO ⊂ Rp isasmallopensetcontaining theorigin, then
E(O) is a C1 submanifold of Rn and (1.5) shows that T E(O) ⊆ P for all h. This
E(h) E(h)
is the starting point to prove that Ox is a integral manifold of the distribution generated
H
byP. AnotherfactweneedtoproveisthatthedimensionofP := span{Y (y) :1 ≤ j ≤ q}
y j
is constant if y belongs to a fixed orbit Ox. This is obtained by means of a nonsmooth
H
quantitative curvilinear version of the original Hermann’s argument inspired to the work
of Nagel, Stein and Wainger [NSW85] and Street [Str11].
3
A.MontanariandD.Morbidelli Almostexponential mapsandintegrability
To conclude this introduction, we give some references and motivations to study our
almost exponential maps E. Such maps appear in [NSW85], and were used by the authors
to show equivalence between different control distances; see also [VSCC92]. More recently
theyhaverevealedtobeausefultooltostudyPoincaréinequalities(see[LM00]),subelliptic
Sobolev spaces (see [Dan91,Mor00,CRTN01,MM12b]), and geometric theory of Carnot–
Carathéodory spaces (see [MM02,FF03,Vit12]). Finally, note that the precise expansion
(1.5) will be a fundamental tool in the companion paper [MM11b], where we shall prove
a Poincaré inequality on orbits for a family of vector fields satisfying an integrability
condition.
2. Preliminaries
Vector fields and the control distance. Consider a family of vector fields H =
{X ,...,X } and assume that X ∈ C1 (Rn) for all j. Here and later C1 means
1 m j Euc Euc
C1 in the Euclidean sense. Write X =: f ·∇, where f : Rn → Rn. The vector field X ,
j j j j
evaluated at a point x ∈ Rn, will be denoted by X or X (x). All the vector fields in this
j,x j
paper are always defined on the whole space Rn.
Define the Franchi–Lanconelli distance [FL83]
d(x,y) := inf r >0 : y = et1Z1···etµZµx for some µ ∈N
(2.1)
n
where |t | ≤ 1 with Z ∈ rH .
j j
X o
Here and hereafter we let rH := {rX ,...,rX } and ±rH := {±rX ,...,±rX }. The
1 m 1 m
topology associated with d will be denoted with τ . We denote instead by d the standard
d cc
Carnot–Carathéodory or control distance (see Feffermann–Phong [FP83] and Nagel–Stein–
Wainger [NSW85]). In the present paper we shall make a prevalent use of the distance d.
It is well known that τ is (possibly strictly) stronger than the topology τ | received by
d Euc O
O from Rn. See [BCH08, Chapter 3] and [AS04, Example 5.5].
Inview of the mentioned examples, we need touse the broad definition of submanifold;
see [Che46,KN96]. Below, if Σ ⊂ Rn, we denote by τ | the induced topology.
Euc Σ
Definition 2.1 (Immersed submanifold). Let Σ ⊂ Rn and let τ ⊇ τ | bea topology on
Euc Σ
Σ. We say that Σ is a Ck submanifold if Σ is connected and for all x ∈ Σ there is Ω ∈ τ,
open neighborhood of x such that Ω is a Ck graph. If moreover τ = τ | then we say
Euc Σ
that Σ is an embedded submanifold.
Horizontal regularity classes. Here we define our notion of horizontal regularity in
terms of the distance d. Note that we do not use the control distance d .
cc
Definition 2.2. Let H := {X ,,...,X } be a family of vector fields, X ∈ C1 . Let
1 m j Euc
d be their distance (2.1) Let g : Rn → R. We say that g is d-continuous, and we write
g ∈ C0(Rn), if for all x ∈ Rn, we have g(y) → g(x), as d(y,x) → 0. We say that
H
g :Rn → R is H-Lipschitz or d-Lipschitz in A ⊂ Rn if
|g(x)−g(y)|
Lip (g;A) := sup < ∞.
H d(x,y)
x,y∈A,x=6 y
4
A.MontanariandD.Morbidelli Almostexponential mapsandintegrability
We say that g ∈ CH1(Rn) if the derivative Xj♯g(x) := limt→0(f(etXjx) − f(x))/t is a d-
continuous function for any j = 1,...,m. We say that g ∈ Ck(Rn) if all the derivatives
H
♯ ♯
X ···X g are d-continuous for p ≤ k and j ,...,j ∈ {1,...,m}. If all the derivatives
j1 jp 1 p
♯ ♯
X ···X g are d-Lipschitz on each Ω bounded set in the Euclidean metric, then we say
j1 jk
thatg ∈ Ck,1 (Rn). Finally, denotetheusualEuclideanLipschitzconstantofg onA⊂ Rn
H,loc
by Lip (g;A).
Euc
We will usually deal with vector fields which are of class at least C1 ∩Cs−1,1, where
Euc H,loc
s ≥1 is a suitable integer. Inthis caseit turns out that commutators up tothe order s can
be defined; see Definition 2.3. In the companion paper [MM12a] we study several issues
related with this definition.
Definitions of commutator. Our purpose now is to show that, given a family H of
vector fields with X ∈ Cs−1,1∩C1 , then commutators can be defined up to length s.
j H,loc Euc
For any ℓ ∈ N, denote by W := {w ···w : w ∈ {1,...,m}} the words of length
ℓ 1 ℓ j
|w| := ℓ in the alphabet 1,2,...,m. Let also S be the group of permutations of ℓ letters.
ℓ
Then for all ℓ ≥1, there are functions π :S → {−1,0,1} such that
ℓ ℓ
[A ,[A ,...[A ,A ]]...]= π (σ)A A ···A , (2.2)
w1 w2 wℓ−1 wℓ ℓ σ1(w) σ2(w) σℓ(w)
σX∈Sℓ
for all A ,...,A : V → V linear operators on a vector space V. See [MM12a] for a more
1 m
formal definition and an in-depth discussion.
We are now ready to define commutators for vector fields in our regularity classes.
Definition 2.3 (Definitions of commutator). Given a family H = {X ,...,X } of vector
1 m
fields of class Cs−1,1 ∩ C1 , define for any function ψ ∈ C1 the operator X♯ψ(x) :=
H,loc Euc H j
L ψ(x), the Lie derivative. Let also X ψ(x) := f (x)·∇ψ(x) where ψ ∈C1 . Moreover,
Xj j j Euc
let
f := π (σ) X ···X f for all w with |w| ≤ s,
w ℓ σ1(w) σℓ−1(w) σℓ(w)
σX∈Sℓ (cid:0) (cid:1)
X ψ := [X ,,...,[X ,X ]]ψ := f ·∇ψ for all ψ ∈ C1 |w| ≤ s,
w w1 wℓ−1 wℓ w Euc
X♯ψ := π (σ)X♯ ···X♯ X♯ ψ for all ψ ∈ Cℓ |w| ≤ s−1.
w ℓ σ1(w) σℓ−1(w) σℓ(w) H
σX∈Sℓ
Finally, for any j ∈ {1,...,m} and w with 1 ≤ |w| ≤ s, let
ad X ψ := (X♯f −f ·∇f )·∇ψ = (X♯f −X f )·∇ψ for all ψ ∈C1 . (2.3)
Xj w j w w j j w w j Euc
Non-nested commutators are precisely defined in [MM12a].
Remark 2.4. • Let Z ∈ ±H. If |w| ≤ s−1, then there are no problems in defining
ad X . More precisely, in [MM12a] we show that ad X = [Z,X ]. If instead
Z w Z w w
|w| = s, then the function t 7→ f (etZx) is Euclidean Lipschitz. In particular it is
w
differentiable for a.e. t. In other words, for any fixed x ∈ Rn, the limit df (etZx) =:
dt w
Z♯f (etZx) exists for a.e. t close to 0. Therefore the pointwise derivative Z♯f (y)
w w
exists for almost all y ∈ Rn and ultimately ad X is defined almost everywhere.
Z w
5
A.MontanariandD.Morbidelli Almostexponential mapsandintegrability
♯
• Both our definitions of commutator, X and X are well posed from an algebraic
w w
point of view, i.e. they satisfy antisymmetry and the Jacobi identity; see [MM12a].
♯
• In [MM12a] we will also recognize that the first order operator X agrees with X
w w
against functions ψ ∈ Cs−1,1∩C1 as soon as |w| ≤ s−1.
H,loc Euc
The integrability class A .
s
Definition 2.5 (Vector fields of class A ). Let H = {X ,...,X } be a family in the
s 1 m
regularity class C1 ∩Cs−1,1. We say that the family H belongs to the class A if, fixed
Euc H,loc s
an open bounded set Ω ⊂ Rn, there is C > 1 such that the following holds: for any
0
Z ∈ ±H, for any word w with |w| = s, for each x ∈ Ω and for a.e. t ∈ [−C−1,C−1], there
0 0
are coefficients bv ∈ R such that
ad X (etZx) = buX (etZx) with (2.4)
Z w u
1≤|u|≤s
X
|bu| ≤ C for all u with 1≤ |u| ≤ s; (2.5)
0
finally assume that if 1 ≤ |w| ≤ s, for all p ∈ {1,...,n}, for any I ∈ I(p,q), x ∈ Rn, we
have at any h∗ where E is defined
I,x
f (E (h)) −→ f (E (h∗)) as h → h∗. (2.6)
w I,x w I,x
Remark 2.6. • Assumption (2.6) will be used only once, in (3.25), but it is essential
in order to ensure that the almost exponential maps we define later are actually
C1 smooth. It is easy to check that assumption (2.6) is satisfied as soon as f :
Euc w
(O ,τ ) → Riscontinuous, whereτ denotestheSussmann’sorbittopologydefined
H H H
by the family H, see [Sus73]. Note that at this stage assumption (2.6) is not ensured
by the d-Lipschitz continuity of f .
w
• Conditions (2.4) and (2.5) scale nicely. Namely, letting for all r ≤ 1, Z = rZ,
X = r|w|X with |w| = s, we have
w w
e
e adZeXw(x) = buXu(x) where |bu| ≤ C0r ≤ C0 for all u. (2.7)
1≤|u|≤s
X
e e e e
• Let H bea family of vector fields inthe class C1 ∩Cs−1,1 satisfyingthe Hörmander
Euc H,loc
bracket-generating condition of step s and assume that each f with |w| ≤ s is
w
continuous in the Euclidean sense. Then H satisfies A . The constant C in (2.5)
s 0
depends also on a positive lower bound on inf |Λ (x,1)|, see (2.13). This case is
Ω n
discussed in [MM12a, Section 4].
• The pathological vector fields X = ∂ and X = e−1/x12∂ , in spite of their C∞
1 x1 2 x2
smoothness, do not satisfy (2.5) for any s ∈ N.
Let Ω ⊂ Rn be a fixed open set, bounded in the Euclidean metric. Given a family H
0
of vector fields of class C1 ∩Cs−1,1, introduce the constant
Euc H,loc
m
♯ ♯
L : = sup |f |+|∇f |+ |X ···X f |
0 j1 j1 j1 jp−1 jp
j1,.X..,js=1n Ω0 (cid:16) Xp≤s (cid:17) (2.8)
♯ ♯
+Lip (X ···X f ;Ω ) .
H j1 js−1 js 0
o
6
A.MontanariandD.Morbidelli Almostexponential mapsandintegrability
We shall always choose points x ∈ Ω ⋐ Ω and we fix a constant t > 0 small enough to
0 0
ensure that
eτ1Z1···eτNZNx ∈Ω if x ∈ Ω, Z ∈ H, |τ |≤ t and N ≤ N , (2.9)
0 j j 0 0
where N is a suitable constant which depends on the data n,m and s.
0
Proposition 2.7 (measurability). Let H be a family of class A . Let |w| = s and let
s
Z ∈ ±H, Then for any x ∈ Ω we can write
ad X (etZx) = bv(t)X (etZx) for a.e. t ∈ (−t ,t ), (2.10)
Z w v 0 0
1≤|v|≤s
X
where the functions t 7→ bv(t) are measurable and for a.e. t we have |bv(t)| ≤ C , where
0
C denotes the constant in (2.5).
0
Proof. The statement can be proved arguing as in [MM12a, Proposition 4.1].
Wedge products and η-maximality conditions. Following [Str11], denote by P :=
{Y ,...,Y } = {X : 1 ≤ |w| ≤ s} the family of commutators of length at most s.
1 q w
Let ℓ ≤ s be the length of Y and write Y =: g · ∇. Define for any p,µ ∈ N, with
j j j j
1 ≤ p ≤ µ, I(p,µ) := {I = (i ,...,i ) : 1 ≤ i < i < ··· < i ≤ µ}. For each x ∈ Rn
1 p 1 2 p
define p := dimspan{Y : 1 ≤ j ≤ q}. Obviousely, p ≤ min{n,q}. Then for any
x j,x x
p ∈ {1,...,min{n,q}}, let
Y := Y ∧···∧Y ∈ T Rn ∼ Rn for all I ∈ I(p,q),
I,x i1,x ip,x p x p
and, for all K ∈ I(p,n) and I ∈ I(p,q)V V
YK(x) := dxK(Y ,...,Y )(x) := det(gkβ) . (2.11)
I i1 ip iα α,β=1,...,p
Here we let dxK := dxk1 ∧···∧dxkp for any K = (k ,...,k ) ∈ I(p,n).
1 p
The family e := e ∧ ··· ∧ e , where K ∈ I(p,n), gives an othonormal basis of
K k1 kp
Rn, i.e. he ,e i = δ for all K,H. Then we have the orthogonal decomposition
p K H K,H
Y (x) = YK(x)e ∈ Rn, so that the number
VI K J K p
P |Y (x)| :=V YK(x)2 1/2 = |Y (x)∧···∧Y (x)|
I I i1 ip
K∈I(p,n)
(cid:0) X (cid:1)
gives the p-dimensional volume of the parallelepiped generated by Y (x),...,Y (x).
i1 ip
LetI = (i ,...,i ) ∈ I(p,q)suchthat|Y | =6 0. Considerthelinearsystem p ξkY =
1 p I k=1 ik
W, for some W ∈ span{Y ,...,Y }. The Cramer’s rule gives the unique solution
i1 ip P
hY ,ιk(W)Y i
ξk = I I for each k = 1,...,p, (2.12)
|Y |2
I
where we let ιk Y := ιk(W)Y := Y ∧W ∧Y .
W I I (i1,...,ik−1) (ik+1,...,ip)
Let r > 0. Given J ∈ I(p,q), let ℓ(J) := ℓ +···+ℓ . Introduce the vector-valued
j1 jp
function
Λ (x,r) := YK(x)rℓ(J) =: YK(x) , (2.13)
p J J∈I(p,q),K∈I(p,n) J J∈I(p,q),K∈I(p,n)
where we adopt th(cid:0)e tilde nota(cid:1)tion Yk = rℓkYk an(cid:0)deits ob(cid:1)vious generalization for wedge
products. Note that |Λ (x,r)|2 = r2ℓ(I)|Y (x)|2.
p I∈I(p,q) I
e
P
7
A.MontanariandD.Morbidelli Almostexponential mapsandintegrability
Definition 2.8 (η-maximality). Let x ∈ Rn, let I ∈ I(p ,q) and η ∈ (0,1). We say that
x
(I,x,r) is η-maximal if |Y (x)|rℓ(I) > η max |Y (x)|rℓ(J).
I J
J∈I(px,q)
Notethat, if(I,x,r)isacandidate tobeη-maximal withI ∈ I(p,q),thenbydefinition
it must be p = p = dimspan{Y (x) : 1 ≤ j ≤ q}.
x j
Approximate exponentials of commutators. Let w ,...,w ∈ {1,...,m}. Given
1 ℓ
τ > 0, we define, as in [NSW85,Mor00] and [MM12b],
C (X ):= exp(τX ),
τ w1 w1
C (X ,X ):= exp(−τX )exp(−τX )exp(τX )exp(τX ),
τ w1 w2 w2 w1 w2 w1
.
.
.
C (X ,...,X ):= C (X ,...,X )−1exp(−τX )C (X ,...,X )exp(τX ).
τ w1 wℓ τ w2 wℓ w1 τ w2 wℓ w1
(2.14)
Then let
C (X ,...,X ), if t ≥ 0,
etXw1w2...wℓ := exp (tX ):= t1/ℓ w1 wℓ (2.15)
ap ap w1w2...wℓ (C|t|1/ℓ(Xw1,...,Xwℓ)−1, if t < 0.
By standard ODE theory, there is t depending on ℓ,Ω, Ω , sup|f | and sup|∇f | such
0 0 j j
that exp (tX )x ∈ Ω for any x ∈ Ω and |t| ≤ t . Define, given I = (i ,...,i ) ∈
∗ w1w2...wℓ 0 0 1 p
{1,...,q}p, x ∈ Ω and h ∈ Rp, with |h| ≤ C−1
E (h) := exp (h Y )···exp (h Y )(x)
I,x ap 1 i1 ap p ip
(2.16)
h I := j=m1a,..x.,p|hj|1/ℓij and QI(r):= {h ∈Rp : khkI < r}.
(cid:13) (cid:13)
(cid:13) (cid:13)
Gronwall’s inequality. We shall refer several times to the following standard fact: for
all a ≥ 0, b > 0, T > 0 and f continuous on [0,T],
t a
0≤ f(t)≤ at+b f(τ)dτ ∀ t ∈ [0,T] ⇒ f(t)≤ (ebt −1) ∀t ∈ [0,T]. (2.17)
b
Z0
3. Approximate exponentials and regularity of A orbits
s
Let H = {X ,...,X } be a family of A vector fields in Rn. The main purpose of this
1 m s
section is to prove that any H-orbit O with the topology τ generated by the distance d
H d
is a C1 integral manifold of the distribution generated by P. Recall our usual notation
P := {Y :1 ≤ j ≤ q}, P := span{Y :1 ≤ j ≤ q} and p := dimP .
j x j,x x x
3.1. Geometric properties of orbits
In this subsection we look at the properties of orbits O for vector fields of class A .
H s
First we study how the geometric determinants YK change along a given orbit O . The
J H
argument we use is known, see for instance [TW03,MM12b] and especially [Str11]. How-
ever, we need to address some issues which appeaer due to our low regularity assumptions.
Ultimately, we will show that the positive integer p is constant as x ∈ O .
x H
8
A.MontanariandD.Morbidelli Almostexponential mapsandintegrability
Below we shall use the following notation: given r > 0, we let Yj = rℓjYj =: gj ·∇
and Z = rZ, if Z ∈ ±H. Let also YK := rℓ(J)YK, where the notation for YK has been
J J J
introduced in (2.11). e e
e e
Lemma 3.1. Let H be a family of vector fields of class A . Let p ∈ {1,...,q ∧n}. Let
s
x ∈ Ω and r > 0 so that B (x,r ) ⊂ Ω . Let J ∈ I(p,q), K ∈ I(p,n), r ∈ (0,r ] and
0 d 0 0 0
e
Z ∈ ±rH. Then the function [−1,1] ∋ t 7→ YK(etZx) is Lipschitz continuous and there is
J
C > 1 depending on C and L in (2.5) and (2.8) such that
0 0
e e
d e e
YK(etZx) ≤ C|Λ (etZx,r)| for a.e. t ∈ (−1,1).
dt J p
(cid:12) (cid:12)
Proof. Denote γ :(cid:12)(cid:12)= eetZex and le(cid:12)(cid:12)t t,τ ∈ (−1,1). Then
t
|YK(γ )−YK(γ )| = dxK(...,Y (γ ),Y (γ )−Y (γ ),Y (γ ),...)
J τ J t jα+1 t jα τ jα t jα+1 t
(cid:12)1≤Xα≤p (cid:12)
(cid:12) (cid:12)
e e ≤ C(cid:12) |τ −t|, e e e e (cid:12)
where C depends on L in (2.8). Then t 7→ YK(γ )belongs to Lip (−1,1). The estimate
0 J t Euc
for the Lipschitz constant here is quite rough and it can be refined through a computation
of the derivative. Indeed, we claim that forea.e. t ∈ (−1,1) we have
d
YK(γ )= dxK(...,Y ,[Z,Y ],Y ,...,Y )(γ )
dt J t jα−1 jα jα+1 jp t
1≤α≤p
X
e ℓjα≤s−1 e e e e e
+ bβ(γ )dxK(...,Y ,Y ,Y ,...,Y )(γ )
α t jα−1 β jα+1 jp t
(3.1)
1≤α≤p1≤β≤q
X X
ℓjα=s e e e e
+ ∂ fkβdx(k1,...,kβ−1,γ,kβ+1,...,kp)(Y ,...,Y )(γ )
γ j1 jp t
1≤γ≤n1≤β≤p
X X
=: (A)+(B)+(Ce), e e
where we wrote Z = f ·∇ ∈ C1 and bβ are measurable functions with |bβ| ≤ C . To
Euc α α 0
prove (3.1), observe that, if ℓ(Y ) ≤ s−1, then t 7→ Y (γ ) is C1 (−1,1) and
jα jα t Euc
e e
lim Yjα(γτ)−Yjα(γt) = Z♯g (γ )·∇ = [Z,Y ](γ )+eY f(γ )·∇ for all t ∈ [−1,1].
τ→t τ −t jα t jα t jα t
e e
Note that here we used [MeMe12a, Theoreme3.1e] to claimethaet adZeYjα = [Z,Yjα]. If instead
ℓ(Y )= s, then for almost any t we have
jα
e e e
Y (γ )−Y (γ )
τli→mt jα ττ −tjα t = Z♯gjα(γt)·∇ = adZeYjα(γt)+Yjαf(γt)·∇
e e q (3.2)
= e ebβ(t)Y (γ )+Y ef(γ )·∇.e e
α β t jα t
β=1
X
e e e
9
A.MontanariandD.Morbidelli Almostexponential mapsandintegrability
In the first equality we used the definition of ad. Here Y f := g ·∇f, is well defined. In
jα jα
the second line we used Proposition 2.7. The term Y f, in view of Lemma A.1 gives the
jα
third line of (3.1). e e e e
Next we estimate each line of (3.1), starting withe(Ae).
|(A)| ≤ dxK(...,Y (γ ),[Z,Y ](γ ),Y (γ ),...) ≤ C|Λ (γ ,r)|,
jα−1 t jα t jα+1 t p t
(cid:12) (cid:12)
for all t ∈ [−1,1](cid:12). Estimateeis correcteeveen if Λp(eγt,r) = 0. To(cid:12) estimate (B), recall that
β
|b | ≤ C. Then, for all t ∈[−1,1],
α
|(B)| ≤ dxK(...,Y ,Y ,Y ,...) ≤ C|Λ (γ ,r)|.
jα−1 β jα+1 p t
1≤α≤p1≤β≤q
X X (cid:12) (cid:12)
(cid:12) e e e (cid:12)
Finally the estimate of (C) is easy and takes the form
|(C)| ≤ sup |∇f| max |YK(γ )| ≤ C|Λ (γ ,r)| if |t| ≤ 1.
J t p t
Bd(x,r) K∈I(p,n)
e e
The previous lemma immediately implies the following proposition.
Proposition 3.2. Let H be a family in the regularity class A . Let x ∈ Ω, let r ≤ r ,
s 0
where r is small enough so that B (x,r ) ⊂ Ω . Let γ(t) := γ be a piecewise integral
0 d 0 0 t
curve of ±rH with γ(0) =x. Let p ∈ {1,...,q∧n}. Then we have
Λ (γ(t),r)−Λ (x,r) ≤ |Λ (x,r)|(eCt −1) for all t ∈ [0,1]. (3.3)
p p p
(cid:12) (cid:12)
In particular,(cid:12)if p = px and (I,x,r)(cid:12)is η-maximal, then
Ct
|Y (γ(t))−Y (x)| ≤ |Y (x)| for all J ∈ I(p,q) t ∈ [0,1]. (3.4)
J J I
η
Finally, if x,yebelong to thee same orbite, then p = p .
x y
Remark 3.3. As a consequence of the proposition and of the Cramer’s rule (2.12), if
(I,x,r) is η-maximal, then (I,y,r) is C−1η-maximal for all y ∈ B (x,C−1ηr) and we may
d
write for all such y and for any j ∈ {1,...,q}
p bk
j
Y = Y , (3.5)
j,y η ik,y
k=1
X
e e
where |bk|≤ C.
j
Remark 3.4. Proposition 3.2 shows that the oscillation of determinants Λ on a ball is
p
controlled in terms of the value of Λ at the center of the ball. It is not true that the
p
oscillation of a single vector field on a ball can be controlled by its value at the center of
the ball. For instance, we can take the vector fields X = ∂ and Y = y∂ +x∂ . Look
x y x
at the ball B((0,y),r), where 0 < y ≪ r. Note that (r,y) belongs to such ball, but the
oscillation |Y(0,y)−Y(r,y)| ∼ r can not be controlled with the value |Y(0,y)| = |y|.
10