Nonsmooth Hormander vector fields and their control balls

NONSMOOTH H ¨ ORMANDER VECTOR FIELDS AND THEIR CON TR OL BALLS ANNAMARIA MONT ANARI AND D ANIELE MORBIDELLI Abstract. W e prov e a ball -b o x theorem for nonsmo oth H¨ ormander vect or ﬁelds of step s ≥ 2 . 1. Introduction In this pap er we give a self-contained pro o f of a ball-b ox theor em fo r a family { X 1 , . . . , X m } of nonsmoo th vector ﬁelds satisfying the H¨ ormander condition. This is the third paper , after [M] and [MM], where we inv estiga te ideas of the classica l article by Nagel Stein and W ainger [NSW]. Our purp o s e is to prove a ball-b ox theorem using only elementary analysis tech- niques and at the same time t o relax as m uch as possible the regular it y assumptions on the vector ﬁelds. Roughly spea k ing, our results hold as soon a s th e comm utators inv olved in the H¨ ormander condition are Lipsc hitz c o ntin uous. Moreover, our proof do es not rely on alg ebraic tools , lik e formal series and the Campbell–Ha us dorﬀ for- m ula. T o descr ib e our work, w e recall th e basic ideas of [NSW]. Notation and language are mo r e pr ecisely des c rib ed in Sec tion 2 . An y control ball B ( x, r ) a sso ciated w ith a family { X 1 , . . . , X m } of H¨ ormander vector ﬁelds in R n satisﬁes, for x b elonging to some compact set K and small ra dius r < r 0 , the double inclusion (1.1) Φ x ( Q ( C − 1 r )) ⊂ B ( x, r ) ⊂ Φ x ( Q ( C r )) . Here, the map Φ x is an exp onential of the form (1.2) Φ x ( h ) = ex p( h 1 U 1 + · · · + h n U n )( x ) , where the vector ﬁelds U 1 , . . . , U n are suitable commutators of leng ths d 1 , . . . , d n and Q ( r ) = { h ∈ R n : max j | h j | 1 /d j < r } . Usually , (1.1) is referred to as a b al l-b ox inclusion. A control on the Jacobia n matrix of Φ x gives a n es timate o f the measur e of the ball and ultimately it provides the doubling pro per ty . A remark a ble a chiev ement in [NSW] co ncerns the choice of the vector ﬁelds U j which guara nt ee inclusio ns (1.1) for a given co ntrol ball B ( x, r ), see also the discussion in [Ste, p. 44 0]. Enumerate as Y 1 , . . . , Y q all c o mmu tators o f length at most s a nd let ℓ i be the leng th o f Y i . If the H¨ ormander condition of step s is fulﬁlled, then the vector ﬁelds Y i span R n at an y point. Given a m ulti-index I = ( i 1 , . . . , i n ) ∈ { 1 , . . . , q } n =: S a nd its corr esp onding n − tuple Y i 1 , . . . , Y i n of commutators, let (1.3) λ I ( x ) = det( Y i 1 , . . . , Y i n )( x ) and ℓ ( I ) = ℓ i 1 + · · · + ℓ i n . 2010 Mathematics Subje ct Classiﬁc ation. Primary 53C17; Secondary 35R03. 1 2 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI In [NSW], the a uthors prove the following fact: given a ball B ( x, r ), inclusion (1.1) holds with U 1 = Y i 1 , . . . , U n = Y i n if the n − tuple I ∈ S satisﬁes the η -maximality condition (1.4) | λ I ( x ) | r ℓ ( I ) > η max K ∈ S | λ K ( x ) | r ℓ ( K ) , where η ∈ (0 , 1) is gr eater than so me a bsolute cons tant . Although the choice of the n -tuple I may dep end on b oth the p oint and the radius, the constant C is uniform in x ∈ K and r ∈ (0 , r 0 ). In [M] the seco nd a uthor prov ed that (1.1) a lso holds if o ne changes the map Φ x with the almost ex p onential map (1.5) E x ( h ) = ex p ∗ ( h 1 U 1 ) ◦ · · · ◦ exp ∗ ( h n U n )( x ) , where h j 7→ e xp ∗ ( h j U j ) is the appr oximate exp onential of the c o mm utator U j , whose main feature is that it can b e factorized as a suitable comp osition of ex- po nentials o f the or ig inal vector ﬁelds X 1 , . . . , X m . See (2.3) for the deﬁnition of exp ∗ . Lanconelli and the seco nd a uthor in [LM] pr ov e d that, if inclusion (1 .5), with per tinent estimates for the Jacobian o f E x are known, then the Poincar´ e inequality follows (see [J ] for the or ig inal pro of ). It is worth to obser ve no w that all the res ults in [NSW] and [M] are proved for C M vector ﬁelds, where M is m uch larg e r than the step s . This can be seen b y carefully reading the pro ofs of Lemmas 2.10 a nd 2.13 in [NSW]. In [T W, Section 4], T ao and W righ t g ave a new pro of of the ba ll-b ox theo rem with a diﬀerent appro a ch, based on Gronwall’s ineq ua lity . The author s in [TW] use sca ling ma ps of the form Φ x,r ( t ) := ex p( t 1 r d 1 U 1 + · · · + t n r d n U n ) x , which are naturally deﬁned on a b ox | t | ≤ ε 0 , where ε 0 > 0 is a sma ll constant indep endent of x and r , see the discus s ion in Subsection 5.2. The arguments in [TW] do not rely on the Campb ell–Hausdo rﬀ formula. 1 Moreov er, although the statement is phr a sed for C ∞ vector ﬁelds, one can see that their results hold under the assumption that the vector ﬁelds hav e a C M smo othness, with M = 2 s for vector ﬁelds of step s . See Remark 5.10 for a more detailed discussion. In [MM ] we started to work in low regular ity hypo theses and we obtained a ball-b ox theo rem and the Poincar´ e inequality for L ips chit z con tinuous vector ﬁelds of step tw o with L ipschitz contin uo us commutators. W e used the maps (1.5), but several a sp ects o f the work [MM] are p eculia r o f the step tw o situation and until now it was not c le ar ho w to genera liz e those results to higher step vector ﬁelds. Recently , Brama n ti, Brandolini and Pedroni [BBP] have proved a doubling prop- erty and the Poincar´ e inequality for nonsmo oth H¨ ormander vector ﬁelds with an algebraic metho d. Infor ma lly sp eaking , they truncate the T aylor series of the coef- ﬁcient s of the vector ﬁelds a nd then they apply to the p olyno mial approximations the results in [NSW, LM] and [M ]. The pap er [BBP] also involv es a study o f the almost exp onential maps in (1.5). The results in [BBP] and in the present pap er were obtained indep endently and sim ultaneously . In this pap er we complete the result in [MM], na mely we prove a ball-b ox the- orem for general vector ﬁelds of ar bitrary s tep s , requiring ba sically that all the commutators inv olved in the H¨ ormander condition are Lipschitz contin uous. Our precise hypothes e s are s ta ted in Deﬁnition 2.1. W e improv e all pr e vious res ults in term of reg ularity , see Remark 5.1 0. As in [MM], w e use the a lmost expo nential 1 The metho ds of [TW] hav e b een further exploited in a very recent paper by Street [Str]. NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 3 maps (1.5 ), but we need to provide a very detailed study of such functions in the higher step case. The scheme of the pro of of o ur theorem is basically the Nagel, Stein and W ainge r ’s one, but there ar e some new to ols that should be emphasized. Namely , we obtain some non commutativ e calculus formulas develop ed in order to show that, given a commutator Y , the deriv a tive d dt exp ∗ ( tY ) can be precise ly wr itten a s a ﬁnite sum of hig her order commutators plus an inte gr al r emainder . This is done in Section 3. These results a re a pplied in Section 5 to the a lmost exp o ne ntial maps E in (1.5). Our main structure theorem is Theorem 5.8. As in [MM], par t of our computations will b e given for smo oth vector ﬁelds, namely the standard Euclidean regular ization X σ j of the vector ﬁelds X j . W e will keep everywhere under control all constants in order to make sure that they ar e s table as σ go es to 0. It is w ell known (see [LM, MM]) that the doubling prop erty and the Poincar´ e inequality follow immediately from Theorem 5.8. Observe a lso that our ball- box theorem can be useful in a ll situations where in tegra ls of the form R | f ( x ) − f ( y ) | w ( x, y ) dxdy nee d to b e estimated, for some w eight w . See for example [M] or [MoM]. As an application, in P rop osition 6 .2 we pr ov e a sub elliptic H¨ orma nder– t yp e estimate for nonsmo o th vector ﬁelds. W e b elieve that the results in Section 3 may b e useful in other, related, situations. Concerning the mac hinary dev elop ed in Section 3, it is w orth to men tio n the pap ers [RaS, Ra S2], where non co mmutative ca lculus fo rmulas are used in the pro of of a nonsmo oth v ersion of Chow’s The o rem for vector ﬁelds of step tw o. Geometric a nalysis for nonsmo oth vector ﬁelds star ted in the 80s with the pa- per s by F ranchi and Lanconelli [FL1, FL2], who prov ed the Poincar ´ e ineq uality for diagonal vector ﬁelds in R n of the for m X j = λ j ( x ) ∂ j , j = 1 , . . . , n . In the diago na l case completely diﬀerent techniques a re av ailable. In the rece nt pap er b y Sa wyer and Wheeden [SW], which proba bly contains the b est res ults to date o n diagonal vector ﬁelds, the re ader can ﬁnd a ric h bibliography on the sub ject. Plan of the pap er. In Sectio n 2 we intro duce notatio n. In Section 3 we pr ove our noncommutativ e calculus formulas a nd in Section 4 we prove a stabilit y prop erty of the “a lmost-maximality” condition (1.4). These to o ls are applied in Subsec tio n 5.1 to the maps E . In subsection 5.2 we brieﬂy discus s the “s caled version” of our maps E . Subsection 5.3 co nt ains the ball-b ox theorem. In Sectio n 6 we show some examples. Finally , Section 7 co ntains the smooth approximation result for the original vector ﬁelds. Ac knowledgmen t. W e wish to express our gra titude to Ermanno L a nconelli, for his contin uous advice, encoura gement and interest in our work, pas t a nd prese nt. W e dedica te this paper to him with admiration. 2. Preliminaries and not a tion W e consider v e c tor ﬁelds X 1 , . . . , X m in R n . F or a ny ℓ ∈ N we deﬁne a wor d w = j 1 . . . j ℓ to b e an y ﬁnite or dered collection of ℓ letters, j k ∈ { 1 , . . . , m } , and we int ro duce the no tation X w = [ X j 1 , · · · , [ X j ℓ − 1 , X j ℓ ]] for commutators. Let | w | := ℓ be the length of X w . W e assume the H¨ ormander condition of step s , i.e. that { X w ( x ) : | w | ≤ s } generate all R n at any p oint x ∈ R n . Sometimes it will b e useful to have a diﬀerent notation b etw een a vector ﬁeld in R n and its a sso ciated vector function. In these situations we will write X w = f w · ∇ = P n k =1 f k w ∂ k . W e will also 4 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI enum erate as Y 1 , . . . , Y q all the co mm utators X w with leng th | w | ≤ s and denote by ℓ i or ℓ ( Y i ) the length of Y i . W e iden tify a n o rdered n –tuple o f c ommut ators Y i 1 , . . . , Y i n by the index I = ( i 1 , . . . , i n ) ∈ S := { 1 , . . . , q } n . F or x, y ∈ R n , denote b y d ( x, y ) the control dista nc e , that is the inﬁm um of the r > 0 such that there is a Lipschitz pa th γ : [0 , 1 ] → R n with γ (0) = x , γ (1) = y and ˙ γ = P m j =1 b j X j ( γ ), for a.e. t ∈ [0 , 1]. The measur able functions b j m ust satisfy | b j ( t ) | ≤ r for almost any t . Corresp o nding ba lls will be indica ted as B ( x, r ). Denote also by  ( x, y ) the inﬁmum of the r > 0 such tha t there is a Lipschitz contin uous path γ : [0 , 1 ] → R n with γ (0) = x , γ (1) = y a nd γ s atisﬁes for a.e. t ∈ [0 , 1 ], ˙ γ = P q i =1 c i Y i ( γ ) for suitable mea surable functions c j with | c j ( t ) | ≤ r ℓ ( Y j ) . Corresp o nding balls will b e de no ted by B  ( x, r ). The deﬁnition of  is meaningful as so on as the v ector ﬁelds Y j are at least contin uous. Deﬁnition 2.1 (V ector ﬁelds o f class A s ) . Let X 1 , . . . , X m be vector ﬁelds in R n and let s ≥ 2. W e say that the vector ﬁelds X j are of class A s if they are of class C s − 2 , 1 lo c ( R n ) and for any word w with | w | = s − 1, and for every j, k ∈ { 1 , . . . , m } , (1) the deriv ative X k f w exists and it is contin uous; (2) the distributional deriv a tive X j ( X k f w ) exists and (2.1) X j ( X k f w ) ∈ L ∞ lo c ( R n ) . Recall that X j ∈ C s − 2 , 1 lo c means that all the E uclidean deriv atives of order at most s − 2 of the functions f 1 , . . . , f m are lo cally L ips chit z contin uous. In particular , all the commut ators X w , with | w | ≤ s − 1 ar e lo cally Lipschit z con tinuous in the Euclidean sense and by item (1) all commutators X w of length | w | = s ar e p o int wise deﬁned. If w e knew that d deﬁnes the Euclidean to p o logy , condition (2) would equiv alent to the fact that X w is lo cally d -Lipschitz, if | w | = s , see [GN, FSSC]. Let { X 1 , . . . , X m } b e in the class A s and assume that they satisfy the H¨ ormander condition of step s . Fix once for a ll a pair o f b ounded connected op en sets Ω ′ ⊂⊂ Ω and denote K = Ω ′ . W e denote by D Euclidean deriv atives. If D = ∂ j 1 · · · ∂ j p for some j 1 , . . . , j p ∈ { 1 , . . . , n } , then | D | := p indica tes the order of D . It is understo o d that a deriv ative of o r der 0 is the iden tity . Introduce the p ositive constan t (2.2) L : = max 1 ≤ j ≤ m 0 ≤| D |≤ s − 2 sup Ω | D f j | + max j =1 ,...,m | D | = s − 1 ess sup Ω | D f j | + max k,j =1 ,.. .,m, | w | = s − 1 ess sup Ω | X k X j f w | . Remark 2.2. W e will pro ve in Section 5 a ball-b ox theorem for v ector ﬁelds of step s in the class A s . This impr ov es b oth the results in [TW] a nd [BBP] in term of re gularity . Indeed, in [TW] a C M regular ity , with M = 2 s must b e assumed (see Remar k 5.1 0). In [BBP] the authors assume that the v ector ﬁelds b elong to the Euclidea n Lipschitz space C s − 1 , 1 lo c ( R n ), which req uires the bo undedness o n the Euclidean gr adient ∇ f w of any commutator f w of length s , while we only need to control only the “hor izontal” gradient of f w . NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 5 Appro xi mate com m utators. F or vector ﬁelds X j 1 , . . . , X j ℓ , and for τ > 0 , w e deﬁne, as in [NSW], [M] and [MM], C τ ( X j 1 ) := ex p( τ X j 1 ) , C τ ( X j 1 , X j 2 ) := ex p( − τ X j 2 ) exp( − τ X j 1 ) exp( τ X j 2 ) exp( τ X j 1 ) , . . . C τ ( X j 1 , . . . , X j ℓ ) := C τ ( X j 2 , . . . , X j ℓ ) − 1 exp( − τ X j 1 ) C τ ( X j 2 , . . . , X j ℓ ) exp( τ X j 1 ) . Then let (2.3) e tX j 1 j 2 ...j ℓ ∗ := exp ∗ ( tX j 1 j 2 ...j ℓ ) := ( C t 1 /ℓ ( X j 1 , . . . , X j ℓ ) , if t > 0 , C | t | 1 /ℓ ( X j 1 , . . . , X j ℓ ) − 1 , if t < 0 . By standard ODE theory , there is t 0 depe nding on ℓ, K , Ω, sup | f j | and ess sup |∇ f j | such that ex p ∗ ( tX j 1 j 2 ...j ℓ ) x is well deﬁned for any x ∈ K and | t | ≤ t 0 . The a p- proximate commutators C t are quite natural (indeed, they make a n appea rance in the original pap er [4 ]). Assuming that the vector ﬁelds a re s mo oth and using the Campb ell–Hausdorﬀ formula, we ha ve the forma l expa nsion C τ ( X j 1 , . . . , X j ℓ ) = ex p  τ ℓ X j 1 j 2 ...j ℓ + ∞ X k = ℓ +1 τ k R k  , where R k denotes a linear combination o f commutators of length k . See [NSW, Lemma 2 .2 1]. A study of these maps in the smo o th case based on this for m ula is carried out in [M]. Deﬁne, given I = ( i 1 , . . . , i n ) ∈ S , x ∈ K and h ∈ R n , with | h | ≤ C − 1 (2.4) E I ,x ( h ) := E I ( x, h ) := exp ∗ ( h 1 Y i 1 ) · · · exp ∗ ( h n Y i n )( x ) , k h k I := max j =1 ,...,n | h j | 1 /ℓ i j , Q I ( r ) := { h ∈ R n : k h k I < r } Λ( x, r ) := ma x K ∈ S | λ K ( x ) | r ℓ ( K ) , where ℓ ( K ) = ℓ k 1 + · · · + ℓ k n , the determinants λ K are deﬁned in (1.3), and we have (2.5) ν := inf x ∈ Ω Λ( x, 1) > 0 . The low er b ound (2.5) will app ear many times in the fo llowing sectio ns. All the constants in our main theorem will dep end o n ν in (2.5) and on L in (2.2). In order to refer to the crucial condition (1.4), we giv e the follo wing deﬁnition Deﬁnition 2.3 ( η − max imal triple) . Let η ∈ ]0 , 1[, I ∈ S , x ∈ R n and r > 0. W e say that ( I , x, r ) is η − maximal, if w e ha ve | λ I ( x ) | r ℓ ( I ) > η Λ( x, r ) . Regularized v ector ﬁelds . Here we describe our pro c edure of smo othing of the vector ﬁelds X j of step s . F or for any function f , let f ( σ ) ( x ) = R f ( x − σy ) ϕ ( y ) dy , where ϕ ∈ C ∞ 0 is a standard nonnega tive averaging kernel supp or ted in the unit ball. Deﬁne (2.6) X σ j := n X k =1 ( f k j ) ( σ ) ∂ k and X σ j 1 ...j ℓ := [ X σ j 1 , · · · , [ X σ j ℓ − 1 , X σ j ℓ ]] =: n X k =1 ( f k j 1 ...j ℓ ) σ ∂ k , 6 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI for any word j 1 . . . j ℓ , with 2 ≤ ℓ ≤ s . (Obser ve that f σ w 6 = f ( σ ) w , if | w | > 1. See Section 7) Then: Prop ositio n 2. 4. L et X 1 , . . . , X m b e ve ctor ﬁelds in the class A s . Then the fol- lowing hold. (1) F or any ℓ = 1 , . . . , s , for any wor d w of lenght | w | ≤ ℓ , (2.7) X σ w → X w , as σ → 0 , uniformly on K . In p articular, for any m ulti-index I = ( i 1 , . . . , i n ) ∈ S , we have λ σ I := det( Y σ i 1 , . . . , Y σ i n ) − → λ I , uniformly on K , as σ → 0 . (2) Ther e is σ 0 > 0 such that, if | w | = s and k = 1 , . . . , m , then (2.8) sup 0 <σ<σ 0 sup x ∈ K | X σ k f σ w | ≤ C , with C dep ending on L in (2.2) . (3) Ther e is r 0 dep ending on K, Ω and the c onstant in (2.2) su ch that the fol lowing holds. L et x ∈ K , r < r 0 and b ∈ L ∞ ([0 , 1] , R m ) with k b j k L ∞ ≤ r for all j . Then ther e is a unique ϕ ∈ L ip([0 , 1] , R n ) , a.e. solution of ˙ ϕ = P j b j X j ( ϕ ) , with ϕ (0) = x . D enote also by ϕ σ ∈ Lip([0 , 1] , R n ) , the a.e. solution of the ˙ ϕ σ = P j b j X σ j ( ϕ σ ) , with ϕ σ (0) = x . Then (2.9) ϕ σ (1) → ϕ (1) , as σ → 0 , uniformly in x ∈ K . As a c onse quenc e, for any I ∈ S , uniformly in x ∈ K , | h | ≤ C − 1 , (2.10) E σ I ( x, h ) := exp ∗ ( h 1 Y σ i 1 ) · · · exp ∗ ( h n Y σ i n ) → E I ( x, h ) . Pr o of. The pro ofs of items 1 and 2 a re given in details in Section 7. Item 3 follows from standard prop erties of ODE.  Remark 2.5. The approximation result contained in Prop ositio n 2.4 is crucial for o ur subsequent arguments. Note that the class A s requires a co ntrol o n the Euclidean gr adients of all co mm utators of length strictly les s than s . How ever, it is natural to conjecture that a c ontrol only along the horizontal directions could be suﬃcien t to ensure our ma in structure theorem in Section 5. Unfortunately , it seems quite diﬃcult to get an approximation theorem a s Pr op osition 2.4 for a more general class than A s . On the other side, w ork ing without molliﬁed vector ﬁelds seems to rise some non trivial new issues which we plan to face in a further study . Some mo re notation. Our notation for consta nts are the following: C , C 0 de- note large a bsolute constants, ε 0 , r 0 , t 0 , C − 1 or C − 1 0 denote p ositive small absolute constants. “Abso lute consta nt s” may dep end on the dimension n , the n umber m of the ﬁelds, their step s , the constan t L in (2.2) and p o ssibly the co ns tant ν in (2.5). W e also use the nota tion ε η (or C η ) to denote a sma ll (or a la rge) cons ta nt depe nding a lso on η . The co ns tants σ 0 or e σ a pp e aring in the regula r izing para meter σ may also depend on the E uclidean contin uity mo duli of the v ector ﬁelds f w , with | w | = s, whic h ar e not included in L. Comp osition of functions are shortened a s follows: f g stands for f ◦ g . The notation u is alwa ys used for functions of the for m exp( t 1 Z 1 ) · · · exp( t ν Z ν ) for some t j ∈ R , ν ≥ 1, Z j ∈ { X 1 , . . . , X m } . NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 7 3. Appro xima te exponential s of commut a tors The main result of this section is Theorem 3.6 in Subsection 3.3, where we prov e an exac t form ula for the deriv ative d dt u ( e tX w ∗ ( x )), wher e X w is a commut ator of length | w | ≤ s , while e ∗ is the approximate ex p o nent ial deﬁned in (2.3). All this sectio n is written for smo oth v ector ﬁelds, namely the molliﬁed X σ j , but a ll constants are app earing in our computatio ns are stable as σ go es to 0. W e drop everywhere in this section the sup ersc r ipt σ . W e will s how that (3.1) d dt e tX w ∗ ( x ) = X w ( e tX w ∗ ( x )) + higher order commutators + integral remainder. The integral rema inder is rather complica ted, but we do not need its e x act form. In order to understand what w e need to co mpute the deriv ative in (3.1), let us try to calculate for example the der iv ative d dt u ( e tX e tY x ), where X , Y ∈ {± X 1 , . . . , ± X m } and u denotes the iden tit y function in R n . Since X and Y ar e C 1 , we hav e d dt u ( e tX e tY x ) = ( X u )( e tX e tY x ) + Y ( ue tX )( e tY x ) . In order to compare the terms in the right-hand side, we ma y write Y ( ue tX )( e tY x ) = Y u ( e tX e tY x ) + Z t 0 d dτ Y ( ue τ X )( e − τ X e tX e tY x ) dτ . Lemma 3.1 b elow shows that the deriv ative inside the integral can b e wr itten in an exact fo rm in term o f the co mmu tator of X and Y . The purp os e of the following Subsection 3 .1 is to es tablish a forma lism to study in a pr e c ise way more g eneral, related, integral expre ssions. Lemma 3. 1. L et Z, X b e smo oth ve ct or ﬁ elds. Then, (3.2) d dt Z ( ue − tX )( e tX y ) = [ X , Z ]( ue − tX )( e tX y ) . Pr o of. The lemma is kno wn but we provide a pr o of for completeness. Obser ve ﬁrs t that d dt Z ( ue − tX )( e tX x ) = d dτ Z ( ue − tX )( e τ X x )    τ = t + d dτ Z ( ue − τ X )( e tX x )    τ = t =: (1) + (2) . Obviously , (1) = X Z ( ue − tX )( e tX x ). W r ite now (2) as follows d dt Z ( ue − tX )( ξ )   ξ = e tX x = d dt Z j ( ξ ) ∂ ξ j ( ue − tX )( ξ )   ξ = e tX x = Z j ( ξ ) ∂ ξ j d dt ( ue − tX )( ξ )   ξ = e tX x The pro of of formula (3.2) will b e co nc luded as so o n as w e prov e that (3.3) d dt ( ue − tX )( ξ ) = − X ( ue − tX )( ξ ) . 8 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI T o prov e (3.3), start from the ident ity u ( η ) = u ( e − tX e tX η ), for small t . Diﬀeren ti- ating, 0 = d dt ( u ( e − tX e tX η )) = d dτ ( u ( e − tX e τ X η )) | τ = t + d dτ ( u ( e − τ X e tX η )) | τ = t = Z ( ue − tX )( e tX η ) + d dτ ( u ( e − τ X e tX η )) | τ = t . Then, (3.3) is prov e d by le tting e tX η = ξ .  3.1. Notation for in tegral remainders. Let λ ∈ N , p ∈ { 2 , . . . , s + 1 } . W e denote, for y ∈ K , and t ∈ [0 , t 0 ], t 0 small enough, (3.4) O p ( t λ , u, y ) = N X i =1 Z t 0 ω i ( t, τ ) X w i ( uϕ − 1 i e − τ Z i )( e τ Z i ϕ i y ) dτ , where N is a suita ble integer a nd u is the identit y map or u = exp( tY 1 ) · · · exp( tY µ ) , for some in teger µ and suitable vector ﬁelds Y j ∈ {± X 1 , . . . , ± X m } . Here X w i actually stands for a molliﬁed X σ w i , but we drop the sup er s cript for simplicity . T o describ e the generic term of the sum ab ov e, w e drop the dependenc e on i : (3.5) ( R ) := Z t 0 ω ( t, τ ) X w ( uϕ − 1 e − τ X )( e τ X ϕy ) dτ . Here X w is a commutator o f length | w | = p and X ∈ { ± X j } . Moreover, for a ny t < t 0 , the function ω ( t, τ ) is a polyno mial, homog eneous of degr ee λ − 1 in all v ariables ( t, τ ), such that ω ( t, τ ) > 0 if 0 < τ < t. Thus (3.6) Z t 0 ω ( t, τ ) dτ = bt λ for any t > 0 , for a suitable constant b ∈ R . The map ϕ is the iden tity map or ϕ = exp( tZ 1 ) · · · exp( tZ ν ) for some ν ∈ N , where Z j ∈ {± X 1 , . . . , ± X m } . Remark 3.2. All the num b ers N , µ, ν, b, app earing in the computations of this pap er will b e b ounded by abso lute constants. In o rder to ex pla in how this formalism works, we give the main pr op erties o f o ur int egra l r emainders. Prop ositio n 3.3. A r emainder of t he form (3.4) satisﬁes for every α ∈ N (3.7) t α O p ( t λ , u, y ) = O p ( t α + λ , u, y ) for al l y ∈ K t ∈ [0 , t 0 ] . Mor e over, for p ≤ s + 1 , (3.8) | O p ( t λ , u, y ) | ≤ C t λ for al l y ∈ K t ∈ [0 , t 0 ] , wher e t 0 and C dep end on the c onstant L in (2.2) and on the numb ers N , µ, ν, b app e aring in the sum (3.4) . F urthermor e, if ℓ ( Z ) = 1 and p ≤ s + 1 , (3.9) O p ( t λ , ue tZ , y ) = O p ( t λ , u, e tZ y ) . Final ly, if p ≤ s , we may write, for suitable c onst ants c w , | w | = p, (3.10) O p ( t λ , u, y ) = X | w | = p c w t λ X w u ( y ) + O p +1 ( t λ +1 , u, y ) . NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 9 Pr o of. The pro of of (3 .7) and (3.9) ar e ra ther eas y and we leav e them to the rea der. So w e start with the pro of of (3 .8 ). A typical ter m in O p ( t λ , u, y ) has the form (3.11) Z t 0 ω ( t, τ ) Y ( uϕ − 1 e − τ Z )( e τ Z ϕy ) dτ , with ℓ ( Y ) = p ≤ s + 1. Thus, b y P r op osition 2.4, we hav e   Y ( uϕ − 1 e − τ Z )( e τ Z ϕy )   ≤ C (observe that we need (2.8), if p = s + 1 ). Therefore, (3.8) follows fro m the prop erty (3 .6) of ω . Finally we establish the key pr op erty (3.10). Star t fro m the generic term of O p ( t λ , u, y ) in (3.1 1), where we intro duce the notation g k := e tZ k · · · e tZ ν , for k = 1 , . . . , ν and g ν +1 denotes the identit y map. Recall also that ℓ ( Y ) ≤ s . Therefore , we get Z t 0 ω ( t, τ ) Y ( ue − tZ ν · · · e − tZ 1 e − τ X )  e τ X e tZ 1 · · · e tZ ν y  dτ = Z t 0 ω ( t, τ ) Y ( ug − 1 1 e − τ X )  e τ X g 1 y  dτ = Z t 0 ω ( t, τ ) Y u ( y ) dτ + Z t 0 ω ( t, τ ) n Y ( ug − 1 1 e − τ X )  e τ X g 1 y  − Y u ( y ) o dτ = bY u ( y ) t λ + Z t 0 ω ( t, τ ) n Y ( ug − 1 1 e − τ X )  e τ X g 1 y  − Y ( ug − 1 1 )  g 1 y  o dτ + ν X k =1 Z t 0 ω ( t, τ ) n Y ( ug − 1 k )  g k y  − Y ( ug − 1 k +1 )  g k +1 y  o dτ . Recall that Y has length p ≤ s . The pe n ultimate term ca n be written a s Z t 0 ω ( t, τ ) n Y ( ug − 1 1 e − τ X )  e τ X g 1 y  − Y u ( ug − 1 1 )  g 1 y  o dτ = Z t 0 dτ ω ( t, τ ) Z τ 0 dσ d dσ Y ( ug − 1 1 e − σX )  e σX g 1 y ) = Z t 0 dσ n Z t σ ω ( t, τ ) dτ o [ X , Y ]( ug − 1 1 e − σX )  e σX g 1 y ) . Observe that, as require d, the function e ω ( t, σ ) := R t σ ω ( t, τ ) dτ sa tisﬁes Z t 0 e ω ( t, σ ) dσ = Z t 0 dτ ω ( t, τ ) Z τ 0 dσ = Z t 0 dτ τ ω ( t, τ ) = e bt λ +1 , bec ause ( t, τ ) 7→ τ ω ( t, τ ) is homogeneo us o f degree λ . The k − th term in the sum has the for m Z t 0 dτ ω ( t, τ ) Z t 0 dσ d dσ Y ( ug − 1 k +1 e − σZ k )  e σZ k g k +1 y  = Z t 0 dσ e ω ( t, σ )[ Z k , Y ]( ug − 1 k +1 e − σZ k )  e σZ k g k +1 y  , where e ω ( t, σ ) := R t 0 ω ( t, τ ) dτ = b t λ has the co rrect form. The pro o f is concluded.  10 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI 3.2. Higher order non commuta tiv e calculus form ul as. In order to prove Theorem 3 .5, we ﬁrst need to iterate for m ula (3.2). Start from smo oth vector ﬁelds X := X σ j of leng th one and Z := X w of leng th ℓ ( Z ) := | w | . Diﬀerentiating ident ity (3.2) w e get, by the T aylor formula Z ( ue − tX )( e tX y ) = r X k =0 t k k ! ad k X Z u ( y ) + Z t 0 ( t − τ ) r r ! ad r +1 X Z ( ue − τ X )( e τ X y ) dτ , where we intro duced the notatio n: ad X Z = [ X , Z ], ad 2 X Z = [ X , [ X , Z ]], etcetera. In other w ords, (3.12) Z ( ue tX )( y ) − Z u ( e tX y ) = r X k =1 t k k ! ad k − X Z u ( e tX y ) + Z t 0 ( t − τ ) r r ! ad r +1 − X Z ( ue τ X )( e − τ X e tX y ) dτ = r X k =1 t k k ! ad k − X Z u ( e tX y ) + O r +1+ ℓ ( Z ) ( t r +1 , u, e tX y ) . If we take r = s − ℓ ( Z ), w e may write (3.13) Z ( ue tX )( y ) − Z u ( e tX y ) = s − ℓ ( Z ) X k =1 t k k ! ad k − X Y u ( e tX y ) + O s +1 ( t s − ℓ ( Z )+1 , u, e tX y ) . In view of (3.8), this order of expansion is the highest which ensures that the remainder can be estimated with C t s − ℓ ( Z )+1 , with a control on C in term of the constant in (2.2), as so o n as y ∈ K and | t | ≤ C − 1 . Next, we see k for a family of higher o rder formulas, in which we change e tX with an a pproximate expo nential exp ∗ ( tX w ). The co eﬃcients of the expansio n (3.1 2) are all explicit but we do not nee d such an accur acy in the higher o r der formulae. T o explain what suﬃces for our purpo ses, s tart with the case of co mm utators of length tw o. Let C t = C t ( X, Y ) = e − tY e − tX e tY e tX , where X := X σ j and Y := X σ k are molliﬁed vector ﬁelds with length o ne. Let Z := X σ v be a smo o th commutator with length ℓ ( Z ) := | v | . Assume ﬁrst that ℓ ( Z ) = s . Then, iterating (3 .13) we can write ( F 2 , 1 ) Z ( uC t )( x ) − Z u ( C t x ) = O s +1 ( t, u, C t x ) . If instead ℓ ( Z ) = s − 1 , then so me elementary computations based on (3.12) give ( F 2 , 2 ) Z ( uC t )( x ) − Z u ( C t x ) = X k 1 + k 2 + k 3 + k 4 =1 t k 1 + ··· k 4 k 1 ! · · · k 4 ! ad k 4 Y ad k 3 X ad k 2 − Y ad k 1 − X Z u ( C t x ) + O 2+ ℓ ( Z ) ( t 2 , u, C t x ) = O 2+ ℓ ( Z ) ( t 2 , u, C t x ) = O s +1 ( t 2 , u, C t x ) . NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 11 Next, if ℓ ( Z ) = s − 2 , (this can happ en only if s ≥ 3), then we must expand more. Namely , we ha ve ( F 2 , 3 ) Z ( uC t )( x ) − Z u ( C t x ) = 2 X k 1 + k 2 + k 3 + k 4 =1 t k 1 + ··· k 4 k 1 ! · · · k 4 ! ad k 4 Y ad k 3 X ad k 2 − Y ad k 1 − X Z u ( C t x ) + O 3+ ℓ ( Z ) ( t 3 , u, C t x ) = t 2 [ Z, [ X , Y ]] u ( C t x ) + O 3+ ℓ ( Z ) ( t 3 , u, C t x ) = t 2 [ Z, [ X , Y ]] u ( C t x ) + O s +1 ( t 3 , u, C t x ) . Finally , if ℓ ( Z ) ≤ s − 3 (this requires at least s ≥ 4), we must expand even more: ( F 2 , 4 ) Z ( uC t )( x ) − Z u ( C t x ) = 3 X k 1 + k 2 + k 3 + k 4 =1 t k 1 + ··· k 4 k 1 ! · · · k 4 ! ad k 4 Y ad k 3 X ad k 2 − Y ad k 1 − X Z u ( C t x ) + O 4+ ℓ ( Z ) ( t 4 , u, C t x ) = t 2 [ Z, [ X , Y ]] u ( C t x ) + t 3 n 1 2 ad 2 Y ad X Z u ( C t x ) − 1 2 ad Y ad 2 X Z u ( C t x ) − 1 2 ad 2 X ad Y Z u ( C t x ) + 1 2 ad X ad 2 Y Z u ( C t x ) − ad Y ad X ad Y Z u ( C t x ) + a d X ad Y ad X Z u ( C t x ) o + O 4+ ℓ ( Z ) ( t 4 , u, C t x ) . Observe that if ℓ ( Z ) = s − 3 , then O 4+ ℓ ( Z ) ( t 4 , u, C t x ) = O s +1 ( t 4 , u, C t x ). If instead ℓ ( Z ) < s − 3, then we can expand up to the or der O s +1 ( t s +1 − ℓ ( Z ) , u, C t x ) by means of (3.1 0). W e have s tarted to put tags of the form ( F k,λ ) in our formulae. The num b er k indicates the length o f the co mmutator we are appr oximating, while the num b er λ denotes the power of t which con trols the remainder. Note that in ( F 2 , 4 ), the cur ly bracket changes sign if we exchange X with Y . Brieﬂy , we can write Z ( uC t )( x ) = Z u ( C t x ) + t 2 [ Z, [ X , Y ]] u ( C t x ) + t 3 X | w | =3+ ℓ ( Z ) c w X w u ( C t x ) + O 4+ ℓ ( Z ) ( t 4 , u, C t x ) . for all x ∈ K , t ∈ [0 , C − 1 ], where the co eﬃcients c w are determined in ( F 2 , 4 ). The corres p o nding formula for C − 1 t ( X, Y ) is Z ( uC − 1 t )( x ) = Z u ( C − 1 t x ) − t 2 [ Z, [ X , Y ]] u ( C − 1 t x ) + t 3 X | w | =3+ ℓ ( Z ) e c w X w u ( C − 1 t x ) + O 4+ ℓ ( Z ) ( t 4 , u, C − 1 t x ) , where, since C − 1 t ( X, Y ) = C t ( Y , X ), the coeﬃcie nts e c w are obtained again from ( F 2 , 4 ), by changing X a nd Y . W e are not in terested in the explicit knowledge of all the co eﬃcients c w and e c w . W e only need to observe the following r e mark able cancellation prop erty: X | w | =3+ ℓ ( Z ) ( c w + e c w ) X w ( x ) = 0 for a ny x ∈ K . 12 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI Next we generalize form ulae ( F 2 ,λ ) ab ov e . The ge ne r al statemen t we prov e tells that this ca nce llation p ersists when the length of the commutator w e are appr oxi- mating with C t is three or more. Theorem 3.4. F or any ℓ ∈ { 2 , . . . , s } , x ∈ K , t ∈ [0 , C − 1 ] , the fol lowing family ( F ℓ, 1 , F ℓ, 2 , . . . , F ℓ,s ) of formulas holds. F or mulas F ℓ,k . F or any C t = C t ( X w 1 , . . . , X w ℓ ) , k = 1 , . . . , ℓ and for any c ommu- tator Z su ch that ℓ ( Z ) + k ≤ s + 1 , we have Z ( uC t )( x ) − Z u ( C t x ) = O k + ℓ ( Z ) ( t k , u, C t x ) Z ( uC − 1 t )( y ) − Z u ( C − 1 t y ) = O k + ℓ ( Z ) ( t k , u, C − 1 t x ) . F or mula F ℓ,ℓ +1 . L et ℓ ≥ 2 b e s uch t hat ℓ + 1 ≤ s . Then, for any C t ( X w 1 , . . . , X w ℓ ) and Z such that ℓ + 1 + ℓ ( Z ) ≤ s + 1 , Z ( uC t )( x ) − Z u ( C t x ) = t ℓ [ Z, X w ] u ( C t x ) + O ℓ +1+ ℓ ( Z ) ( t ℓ +1 , u, C t x ) , Z ( uC − 1 t )( y ) − Z u ( C − 1 t y ) = − t ℓ [ Z, X w ] u ( C − 1 t x ) + O ℓ +1+ ℓ ( W ) ( t ℓ +1 , u, C − 1 t x ) . F or mula F ℓ,ℓ +2 . If s ≥ 4 , let ℓ ≥ 2 b e su ch that ℓ + 2 ≤ s . Then, for any C t ( X w 1 , . . . , X w ℓ ) and Z such that ℓ + 2 + ℓ ( Z ) ≤ s + 1 , ther e ar e numb ers c v , e c v , with | v | = ℓ + ℓ ( Z ) + 1 , su ch t hat (3.14) Z ( uC t )( x ) − Z u ( C t x ) = t ℓ [ Z, X w ] u ( C t x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 c v X v u ( C t x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C t x ) Z ( uC − 1 t )( x ) − Z u ( C − 1 t x ) = − t ℓ [ Z, X w ] u ( C − 1 t x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 e c v X v u ( C − 1 t x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C − 1 t x ) . Cancellation pr op erty . If s ≥ 4 , let ℓ ≥ 2 b e su ch that ℓ +2 ≤ s . If formulae F ℓ, 1 . . . , F ℓ,ℓ +2 hold, then, for any C t ( X w 1 , . . . , X w ℓ ) and Z such that ℓ + 2 + ℓ ( Z ) ≤ s + 1 , the c o eﬃcients c w , e c w in (3.14) s atisfy (3.15) X | w | = ℓ + ℓ ( Z )+1 ( c w + e c w ) X w ( x ) = 0 for any x ∈ K. F or mulae F ℓ,r , with ℓ + 3 ≤ r ≤ s . L et s ≥ 5 and assume t hat ℓ ≥ 2 and r ar e such that ℓ + 3 ≤ r ≤ s . Then, for any C t ( X w 1 , . . . , X w ℓ ) and Z with r + ℓ ( Z ) ≤ s + 1 , ther e ar e c v , e c v such that Z ( uC t )( x ) − Z u ( C t x ) = t ℓ [ Z, X w ] u ( C t x ) + r − 1+ ℓ ( Z ) X | v | = ℓ + ℓ ( Z )+1 t | v | − ℓ ( Z ) c v X v u ( C t x ) + O r + ℓ ( Z ) ( t r , u, C t x ) , Z ( uC − 1 t )( x ) − Z u ( C − 1 t x ) = − t ℓ [ Z, X w ] u ( C − 1 t x ) + r − 1+ ℓ ( Z ) X | v | = ℓ + ℓ ( Z )+1 t | v | − ℓ ( Z ) e c v X v u ( C − 1 t x ) + O r + ℓ ( Z ) ( t r , u, C − 1 t x ) . Observe aga in that in the form ula F ℓ,k , ℓ is the length of the comm utator which deﬁnes C t , while k is the degree of the p ower o f t whic h cont rols the remainder. NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 13 Pr o of of The or em 3.4. If ℓ = 2, w e hav e alr eady prov ed the statement. See for- m ulae ( F 2 , 1 ), ( F 2 , 2 ), ( F 2 , 3 ) and ( F 2 , 4 ), p. 11, and recall pr op erty (3.10) of the remainders. The pro o f will be acc omplished in t wo s teps. Step 1. Let s ≥ 4 and ℓ ≥ 2 be such that ℓ +2 ≤ s . Assume that F ℓ, 1 , F ℓ, 2 , . . . , F ℓ,ℓ +2 hold. Then the cancellation (3.15) ho lds fo r an y C t ( X j 1 , . . . , X j ℓ ) and W s uch that ℓ + 2 + ℓ ( W ) ≤ s + 1. Step 2. Assume tha t for some ℓ ≥ 2, all formulae F ℓ,k hold, for k = 1 , . . . , s . Then formula F ℓ +1 ,k holds, for any k = 1 , . . . , s . Pr o of of Step 1. Let C t = C t ( X w 1 , . . . , X w ℓ ) and Z such that ℓ ( Z ) + ℓ + 2 ≤ s + 1. Applying t wice formula F ℓ,ℓ +2 , we obtain, (3.16) Z u ( x ) = Z ( uC − 1 t C t )( x ) = Z ( uC − 1 t )( C t x ) + t ℓ [ Z, X w ]( uC − 1 t )( C t x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 c v X v ( uC − 1 t )( C t x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , uC − 1 t , C t x ) = Z u ( x ) − t ℓ [ Z, X w ] u ( x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 e c v X v u ( x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, x ) + t ℓ [ Z, X w ]( uC − 1 t )( C t x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 c v X v ( uC − 1 t )( C t x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , uC − 1 t , C t x ) . Observe ﬁrs t that pro per ty (3.9) g ives O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , uC − 1 t , C t x ) = O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, x ) . Later on, we will tacitly use such prop e rty ma n y times. Rec all that ℓ ≥ 2 . By means of F ℓ, 2 and F ℓ, 1 , resp e ctively , w e obtain [ Z, X w ]( uC − 1 t )( C t x ) = [ Z , X w ] u ( x ) + O 2+ ℓ ( Z )+ ℓ ( t 2 , u, x ) and X v ( uC − 1 t )( C t x ) = X v u ( x ) + O 2+ ℓ + ℓ ( Z ) ( t, u, x ) . Inserting this information in to (3.1 6) gives, after algebraic simpliﬁcations 0 = t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 e c v X v u ( x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, x ) + t ℓ O 2+ ℓ + ℓ ( Z ) ( t 2 , u, x ) + t ℓ +1  X | v | = ℓ + ℓ ( Z )+1 c v X v u ( x ) + O ℓ +2+ ℓ ( Z ) ( t, u, x )  . T o conc lude the pro o f, recall (3.7), divide b y t ℓ +1 and let t → 0. Pr o of of Step 2. Let ℓ + 2 ≤ s . W e prov e formula F ℓ +1 ,ℓ +2 , which is the mos t signiﬁcant among a ll formulae F ℓ +1 , 1 , . . . , F ℓ +1 ,s . I ndee d, once F ℓ +1 ,ℓ +2 is prov ed, if ℓ + 3 ≤ s , then for mu lae F ℓ +1 ,ℓ +3 , . . . , F ℓ +1 ,s follow ea sily from F ℓ +1 ,ℓ +2 and from prop erty (3 .10). On the other side, the lower order formulae F ℓ +1 ,k with k < ℓ + 2 are ea sier (just truncate at the corre ct order all the expa nsions in the pro of b elow). 14 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI T o star t, recall that we are ass uming that F ℓ, 1 , . . . , F ℓ,s hold. Let, for t > 0 (3.17) C t : = C t ( X w 1 , . . . , X w ℓ ) and C 0 t : = C t ( X, X w 1 , . . . , X w ℓ ) = C − 1 t e − tX C t e tX , where X = X w 0 . Let Z b e a co mm utator with ℓ ( Z ) + ℓ + 2 ≤ s + 1. In the subs e q uent form ulae, we expand everywhere up to a remainder of the fo r m O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ). By (3 .12), Z ( uC 0 t )( x ) = Z ( uC − 1 t e − tX C t )( e tX x ) + t [ − X , Z ]( uC − 1 t e − tX C t )( e tX x ) + ℓ +1 X k =2 t k k ! ad k − X Z ( uC − 1 t e − tX C t )( e tX x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , uC − 1 t e − tX C t , e tX x ) =: ( A ) + ( B ) + ( C ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) , where we also used (3.9). Next we use F ℓ,ℓ +2 in ( A ) . ( A ) = Z ( uC − 1 t e − tX )( C t e tX x ) + t ℓ [ Z, X w ]( uC − 1 t e − tX )( C t e tX x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 c v X v ( uC − 1 t e − tX )( C t e tX x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) =: ( A 1 ) + ( A 2 ) + ( A 3 ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) . W e ﬁr st treat ( A 1 ). By (3.12), (3.18) ( A 1 ) = Z ( uC − 1 t )( e − tX C t e tX x ) + t [ X , Z ]( uC − 1 t )( e − tX C t e tX x ) + ℓ +1 X k =2 t k k ! ad k X Z ( uC − 1 t )( e − tX C t e tX x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) . Consider now the v ar ious terms in ( A 1 ). Firs t use F ℓ,ℓ +2 to get Z ( uC − 1 t )( e − tX C t e tX x ) = Z u ( C 0 t x ) − t ℓ [ Z, X w ] u ( C 0 t x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 e c v X v u ( C 0 t x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) . Moreov er, b y F ℓ,ℓ +1 we g et t [ X , Z ]( uC − 1 t )( e − tX C t e tX x ) = t n [ X , Z ] u ( C 0 t x ) − t ℓ [[ X , Z ] , X w ] u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +1 , u, C 0 t x ) o . Finally , we use F ℓ,ℓ +2 − k in the k − th ter m of the sum in (3.18). Obser ve that ℓ + 2 − k ∈ { 1 , . . . , ℓ } so that w e use only remainders. t k k ! ad k X Z ( uC − 1 t )( e − tX C t e tX x ) = t k k ! n ad k X Z u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 − k , u, C 0 t x ) o . NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 15 Therefore ( A 1 ) = Z u ( C 0 t x ) − t ℓ [ Z, X w ] u ( C 0 t x ) + t ℓ +1 X | v | = ℓ +1+ ℓ ( Z ) e c v X v u ( C 0 t x ) + t [ X , Z ] u ( C 0 t x ) − t ℓ +1 [[ X , Z ] , X w ] u ( C 0 t x ) + ℓ +1 X k =2 t k k ! ad k X Z u ( C 0 t x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) . Next we consider ( A 2 ). F orm ula (3.12) gives ( A 2 ) = t ℓ [ Z, X w ]( uC − 1 t )( e − tX C t e tX x ) + t ℓ +1 [ X , [ Z, X w ]]( uC − 1 t )( e − tX C t e tX x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . Since ℓ ≥ 2 , form ulas F ℓ, 2 and F ℓ, 1 give re s pe c tively t ℓ [ Z, X w ]( uC − 1 t )( e − tX C t e tX x ) = t ℓ [ Z, X w ] u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) , t ℓ +1 [ X , [ Z, X w ]]( uC − 1 t )( e − tX C t e tX x ) = t ℓ +1 [ X , [ Z, X w ]] u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) , so that ( A 2 ) = t ℓ [ Z, X w ] u ( C 0 t x ) + t ℓ +1 [ X , [ Z, X w ]] u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . T o handle ( A 3 ), observe that a rep ea ted application of (3.12) gives ( A 3 ) = t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 c v X v u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . Next we study ( B ). Start with formula F ℓ,ℓ +1 : ( B ) = t [ − X , Z ]( uC − 1 t e − tX )( C t e tX x ) + t ℓ +1 [[ − X , Z ] , X w ]( uC − 1 t e − tX )( C t e tX x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) = t [ − X , Z ]( uC − 1 t e − tX )( C t e tX x ) + t ℓ +1 [[ − X , Z ] , X w ] u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) =: ( B 1 ) + ( B 2 ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) , W e ﬁr st consider ( B 1 ). In view of (3 .12), we o btain ( B 1 ) = t [ − X , Z ]( uC − 1 t )( e − tX C t e tX x ) − t ℓ X k =1 t k k ! ad k X [ X , Z ]( uC − 1 t )( e − tX C t e tX x ) + tO ℓ + ℓ ( Z ) +2 ( t ℓ +1 , u, C 0 t x ) . But b y F ℓ,ℓ +1 we g et t [ − X , Z ]( uC − 1 t )( e − tX C t e tX x ) = t [ − X , Z ] u ( C 0 t x ) − t ℓ +1 [[ − X , Z ] , X w ] u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) , while for an y k = 1 , . . . , ℓ , formula F ℓ,ℓ +1 − k gives t t k k ! ad k X ([ X, Z ])( uC − 1 t )( e − tX C t e tX x ) = t k +1 k ! ad k +1 X Z u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . 16 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI Therefore ( B 1 ) = t ℓ +1 [[ X , Z ] , X w ] u ( C 0 t x ) − ℓ X k =0 t k +1 k ! ad k +1 X Z u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . Observe that t ℓ +1 [[ X , Z ] , X w ] u ( C 0 t x ) = − ( B 2 ). Finally we consider ( C ). In the k − th term of the sum us e form ula F ℓ,ℓ +2 − k . Then ( C ) = ℓ +1 X k =2 t k k ! ad k − X Z ( uC − 1 t e − tX )( C t e tX x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) = by (3.12) = ℓ +1 X k =2 t k k ! n ℓ +1 − k X h =0 t h h ! ad h X ad k − X Z ( uC − 1 t )( e − tX C t e tX x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 − k , u, C 0 t x ) o + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) = ℓ +1 X k =2 ℓ +1 − k X h =0 t k + h k ! h ! ( − 1) k ad k + h X Z u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . Collecting together all the previous computations and making so me simpliﬁca- tions (in particular w e need here the cancellatio n pro per ty (3 .15)), w e get Z ( uC 0 t )( x ) = ( A 1 ) + ( A 2 ) + ( A 3 ) + ( B 1 ) + ( B 2 ) + ( C ) = Z u ( C 0 t x ) + t ℓ +1  − [[ X , Z ] , X w ] u ( C 0 t x ) + [ X , [ Z , X w ]] u ( C 0 t x )  + ℓ +1 X k =1 t k k ! ad k X Z u ( C 0 t x ) − ℓ X k =0 t k +1 k ! ad k +1 X Z u ( C 0 t x ) + ℓ +1 X k =2 ℓ +1 − k X h =0 t k + h k ! h ! ( − 1) k ad k + h X Z u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) =: Z u ( C 0 t x ) + t ℓ +1 {· · · } + (1) + (2) + (3) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . The Jacobi ident ity gives t ℓ +1 {· · · } = t ℓ +1 [ Z, [ X , X w ]] , which is the desir ed term. Ultimately w e need to consider all the ter ms with sums. C ha nging k and h in (2), w e may wr ite (2) + (3) = ℓ +1 X k =1 ℓ +1 − k X h =0 ( − 1) k t k + h k ! h ! ad k + h X Z u ( C 0 t x ) and (1) + Z u ( C 0 t x ) = ℓ +1 X h =0 t h h ! ad h X Z u ( C 0 t x ) . Therefore, (1) + (2) + (3) + Z u ( C 0 t x ) = ℓ +1 X k =0 ℓ +1 − k X h =0 ( − 1) k t k + h k ! h ! ad k + h X Z u ( C 0 t x ) = ℓ +1 X s =0  X k + h = s k,h ≥ 0 ( − 1) k k ! h !  t s ad s X Z u ( C 0 t x ) = Z u ( C 0 t x ) , NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 17 bec ause X k + h = s k,h ≥ 0 ( − 1) k k ! h ! = 0 for all s ≥ 1. The pro o f of Step 2 and of Theorem 3.4 is concluded.  3.3. Deriv ativ es of approx imate exp one ntials. Her e we give the formula fo r the deriv ative of a n approximate exp onential. All the subsection is written for the molliﬁed v ector ﬁelds X σ j , but we drop everywhere the supesr cript. Theorem 3.5. Ther e is t 0 > 0 such that, for any ℓ ∈ { 2 , . . . , s } , w = ( w 1 , . . . , w ℓ ) , letting C t = C t ( X w 1 , · · · , X w ℓ ) , ther e ar e c onstants a w , e a w such t hat, for any x ∈ K , t ∈ [0 , t 0 ] , (3.19) d dt u ( C t x ) = ℓt ℓ − 1 X w u ( C t x ) + s X | v | = ℓ +1 a v t | v | − 1 X v u ( C t x ) + O s +1 ( t s , u, C t x ) , (3.20) d dt u ( C − 1 t x ) = − ℓt ℓ − 1 X w u ( C − 1 t x ) + s X | v | = ℓ +1 e a v t | v | − 1 X v u ( C − 1 t x ) + O s +1 ( t s , u, C − 1 t x ) . wher e, if ℓ = s , the su m is empty, while, if 2 ≤ ℓ < s , we have t he c anc el lation (3.21) X | w | = ℓ +1  a w + e a w  X w ( x ) = 0 for al l x ∈ K . F ro m Theorem 3.5 it is v ery easy to obtain the following result: Theorem 3 .6. F or any c ommutator X w with length | w | = ℓ ≤ s , we have, for x ∈ K and t ∈ [ − t 0 , t 0 ] , (3.22) d dt u ( e tX w ∗ ( x )) = X w u ( e tX w ∗ ( x )) + s X | v | = ℓ +1 α v ( t ) X v u  e tX w ∗ ( x )  + O s +1  | t | ( s +1 − ℓ ) /ℓ , u, e tX w ∗ ( x )  , wher e the sum is empty if ℓ = s , α w ( t ) = ℓ − 1 a v t ( | v | /ℓ ) − 1) , if t > 0 and α v ( t ) = − ℓ − 1 e a v | t | ( | v | /ℓ ) − 1) if t < 0 . In p articular, the map ( t, x ) 7− → e tX w ∗ ( x ) is of class C 1 on ( − t 0 , t 0 ) × Ω ′ . Example 5.7 shows that, even if the vector ﬁelds ar e smo oth, then the map exp ∗ ( tX w ) is at most C 1 ,α for some α < 1 . Pr o of of The or em 3.6. F ormula (3.2 2) follows immediately fro m (3.19), (3.20) and the deﬁnition (2.3) of e ∗ . W e only need to show now that the map is C 1 in b oth v ariables t, x . Recall that the vector ﬁelds X σ j are smo o th and in particular C 1 . By classical ODE theory , s ee [Ha, Chap. 5], an y map of the for m ( τ 1 , . . . , τ ν , x ) 7→ e τ 1 X i 1 · · · e τ ν X i ν x is C 1 if the τ j ’s b elong to so me neig hborho o d of the o rigin and x ∈ Ω ′ . This implies that for any comm utator X w , the map ∇ x exp ∗ ( tX w ) x is contin uous o n ( t, x ) ∈ I × Ω ′ , while d dt exp ∗ ( tX w ) x is contin uous in ( t, x ) ∈  I \ { 0 }  × Ω ′ . 18 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI Next we prov e tha t d dt exp ∗ ( tX w ) x exists and it is c ontin uous also at a ll points of the form (0 , x ). Obser ve ﬁr st that formula (3.2 2) gives (3.23) lim t → 0 d dt exp ∗ ( tX w ) x = 0 uniformly in x ∈ Ω ′ . Now, (3.23) a nd l’Hˆ opital’s rule imply that d dt exp ∗ ( tX w ) x   t =0 = 0 , for all x ∈ Ω ′ . Finally , the uniformit y of the limit ensures that the map ( t, x ) 7→ d dt exp ∗ ( tX w ) x is actually contin uo us in I × Ω ′ .  Pr o of of The or em 3.5. W e divide the pro of in tw o steps. Step 1. W e ﬁr st prove that, if (3.19) and (3.20) hold for so me w with ℓ := | w | ∈ { 2 , . . . , s − 1 } , then the cancellation formula (3.2 1) must hold. Fix such a w and start from the ident ity d dt u ( C − 1 t C t x ) = 0. (3.24) 0 = d ds u ( C − 1 s C t x )    s = t + d ds ( uC − 1 t )( C s x )    s = t = − ℓ t ℓ − 1 X w u ( x ) + X | v | = ℓ +1 e a v t ℓ X v u ( x ) + O ℓ +2 ( t ℓ +1 , u, x ) + ℓt ℓ − 1 X w ( uC − 1 t )( C t x ) + X | v | = ℓ +1 a v t ℓ X v ( uC − 1 t )( C t x ) + O ℓ +2 ( t ℓ +1 , uC − 1 t , C t x ) . But, since ℓ ≥ 2, formula F ℓ, 2 shows that t ℓ − 1  X w ( uC − 1 t )( C t x ) − X w u ( x )  = t ℓ − 1 O 2+ | w | ( t 2 , u, x ) = O ℓ +2 ( t ℓ +1 , u, x ) , while F ℓ, 1 gives for a n y v with | v | = ℓ + 1, t ℓ { X v ( uC − 1 t )( C t x ) − X v u ( x ) } = t ℓ O 1+ | v | ( t, u, x ) = O ℓ +2 ( t ℓ +1 , u, x ) . Divide (3.24) by t ℓ and let t → 0 to get (3.21). Step 1 is concluded. Step 2. W e prove by an induction argument, tha t, if Theo rem 3.5 holds for s ome ℓ ∈ { 2 , . . . , s − 1 } , then it holds for ℓ + 1. T o show the result for ℓ = 2, it suﬃces to follow the pro of b elow, taking into account that formulas (3.1 9) and (3.20) ar e trivial, if ℓ = 1. W e use the notation in (3.17) for C t and C 0 t . In view of (3.10) a nd of the already accomplished Step 1, it suﬃces to prov e that (3.25) d dt u ( C 0 t x ) = ( ℓ + 1 ) t ℓ [ X , X w ] u ( C 0 t x ) + O ℓ +2 ( t ℓ +1 , u, C 0 t x ) and d dt u (( C 0 t ) − 1 x ) = − ( ℓ + 1) t ℓ [ X , X w ] u (( C 0 t ) − 1 x ) + O ℓ +2 ( t ℓ +1 , u, ( C 0 t ) − 1 x ) . W e pr ov e only the ﬁrst line of (3.2 5). The latter is similar. W e know that d dt ( u ( C t x )) = ℓt ℓ − 1 X w u ( C t x ) + t ℓ X | v | = ℓ +1 a v X v u ( C t x ) + O ℓ +2 ( t ℓ +1 , u, C t x ) d dt u ( C − 1 t x ) = − ℓt ℓ − 1 X w u ( C − 1 t x ) + t ℓ X | v | = ℓ +1 e a v X v u ( C − 1 t x ) + O ℓ +2 ( t ℓ +1 , u, C − 1 t x ) , (3.26) NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 19 with the remark a ble ca ncellation (3.21). Obse r ve that a v = e a v = 0, if ℓ = 1. Next, d dt  u ( C 0 t x )  = d dt  u ( C − 1 t e − tX C t e tX x )  = X  uC − 1 t e − tX C t  ( e tX x ) + d ds ( uC − 1 t e − tX )( C s e tX x )    s = t − X  uC − 1 t  ( e − tX C t e tX x ) + d ds u ( C − 1 s e − tX C t e tX x )    s = t =: A 1 + A 2 + A 3 + A 4 . First we study A 1 + A 3 , by (3 .12) and F ℓ,ℓ +1 . A 1 + A 3 = X  uC − 1 t e − tX C t  ( e tX x ) − X  uC − 1 t e − tX  ( C t e tX x ) = (b y F ℓ,ℓ +1 ) = t ℓ [ X , X w ]( uC − 1 t e − tX )( C t e tX x ) + O ℓ +2 ( t ℓ +1 , uC − 1 t e tX , C t e tX x ) = t ℓ [ X , X w ] u ( C 0 t x ) + O ℓ +2 ( t ℓ +1 , u, C 0 t x ) . Next we study A 2 + A 4 , b y means of (3 .26). A 2 + A 4 = ℓt ℓ − 1 X w ( uC − 1 t e − tX )( C t e tX x ) + t ℓ X | v | = ℓ +1 a v X v ( uC − 1 t e − tX )( C t e tX x ) − ℓt ℓ − 1 X w u ( C 0 t x ) + t ℓ X | v | = ℓ +1 ˜ a v X v u ( C 0 t x ) + O ℓ +2 ( t ℓ +1 , u, C 0 t x ) = ℓt ℓ − 1 n X w ( uC − 1 t )( e − tX C t e tX x ) + t [ X , X w ]( uC − 1 t )( e − tX C t e tX x ) + O ℓ +2 ( t 2 , u, C 0 t x ) o + t ℓ X | v | = ℓ +1 a v  X v u ( C 0 t x ) + O ℓ +2 ( t, u, C 0 t x )  − ℓt ℓ − 1 X w u ( C 0 t x ) + t ℓ X | v | = ℓ +1 ˜ a v X v u ( C 0 t x ) + O ℓ +2 ( t ℓ +1 , u, C 0 t x ) . Now obser ve that b y formula F ℓ, 2 we hav e, if ℓ ≥ 2, t ℓ − 1 n X w ( uC − 1 t )( e − tX C t e tX x ) − X w u ( C 0 t x ) o = O ℓ +2 ( t ℓ +1 , u, C 0 t x ) , while, if ℓ = 1 the left-ha nd side v anishes identically . Th us, cancella tion (3.21) g ives A 2 + A 4 = ℓt ℓ [ X , X w ] u ( C 0 t x )+ O ℓ +2 ( t ℓ +1 , u, C 0 t x ) and ultimately A 1 + A 2 + A 3 + A 4 = ( ℓ + 1 ) t ℓ [ X , X w ] u ( C 0 t x ) + O ℓ +2 ( t ℓ +1 , u, C 0 t x ) . The pro of is concluded.  4. Persistence of maximal ity conditions on bal ls Here we establish a key prop er t y o f stability of the η − max imality co ndition. The argument, as in [TW], is bas ed on Gron wall’s inequality . Theorem 4.1. L et X 1 , . . . , X m b e ve ctor ﬁelds in A s . Then, ther e ar e r 0 > 0 and ε 0 > 0 dep ending on the c onstants L and ν in (2.5) and (2.2) such that, if for some η ∈ ]0 , 1[ , x ∈ K and r < r 0 , the triple ( I , x, r ) is η − maximal, then for any y ∈ B ( x, η ε 0 r ) , we have t he estimates (4.1) | λ I ( y ) − λ I ( x ) | ≤ 1 2 | λ I ( x ) | , (4.2) | λ I ( y ) | r ℓ ( I ) > C − 1 η Λ( y , r ) . 20 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI T o prove Theorem 4.1 we need the following easy lemma. Lemma 4.2. Ther e is C > 0 dep en ding on L and ν such t hat, given y ∈ Ω and z ∈ R n , the line ar system P q i =1 Y i ( y ) ξ i = z has a solution ξ ∈ R q such that | ξ | ≤ C | z | . Pr o of. T ake y ∈ Ω and cho ose ( k 1 , . . . , k n ) ∈ S such that det ( Y k 1 ( y ) , . . . , Y k n ( y )) ≥ ν . Let A := ( Y k 1 ( y ) , . . . , Y k n ( y )). Thus,   A − 1   ≤ C / | det( A ) | ≤ C /ν , where C depe nds on L . The lemma is ea s ily prov e d by studying the s ystem Aξ = z with ξ = ( ξ k 1 , . . . , ξ k n ) ∈ R n .  Pr o of of The or em 4.1. Obs erve that if ( I , x, r ) is η -maximal, then there is e σ > 0 which may also dep end o n I , x, r , such that ( I , x, r ) is η -maxima l for the molliﬁed X σ j for all σ ≤ e σ . Therefore, we will give the pr o of for smo oth vector ﬁelds (without writing a ny sup ers cript). The nonsmo oth case will follow by pa ssing to the limit as σ → 0 a nd taking int o account tha t all constants ar e s table. Let J ∈ S a nd let λ J ( x ) := det[ Y j 1 ( x ) , . . . , Y j n ( x )]. Let X b e a vector ﬁeld o f length one. Recall the following form ula (see [NSW, Lemma 2.6]): X λ J = (div X ) λ J + n X k =1 det( . . . , Y j k − 1 , [ X , Y j k ] , Y j k +1 , . . . ) = (div X ) λ J + X k ≤ n, ℓ j k 0 such that, given I ∈ S , then, for any j = 1 , . . . , n , σ ≤ σ 0 , x ∈ K and h ∈ Q I ( r 0 ) , the C 1 map E σ I ,x satisﬁes (5.1) ∂ ∂ h j E σ I ,x ( h ) = U σ j ( E σ I ,x ( h )) + s X | w | = d j +1 a w j ( h ) X σ w ( E σ I ,x ( h )) + ω σ j ( x, h ) , 22 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI wher e the sum is empty if d j = ℓ ( U j ) = s and t he fol lo wing estimates hold: (5.2) | ω σ j ( x, h ) | ≤ C k h k s +1 − d j I for any x ∈ K h ∈ Q I ( r 0 ) σ ≤ σ 0 , (5.3) | a w j ( h ) | ≤ C k h k | I |− d j I for al l h ∈ Q I ( r 0 ) | w | = d j + 1 , . . . , s. Theorem 5.1 holds without assuming η - maximality . If the triple ( I , x, r ) is η − maximal, w e have mor e. T o state the r esult, ﬁx once for all a dimensional constant χ > 0 such tha t (5.4) det( I n + A ) ∈ h 1 2 , 2 i for all A ∈ R n × n with norm | A | ≤ χ. Theorem 5. 2. L et r 0 , σ 0 > 0 as in The or em 5.1. Given an η − maximal triple ( I , x, r ) for the ve ctor ﬁelds X i , with x ∈ K , r < r 0 and σ ≤ σ 0 , then, for any h ∈ Q I ( ε 0 η r ) , j = 1 , . . . , n , we may write (5.5) ∂ ∂ h j E σ I ,x ( h ) = U σ j ( E σ I ,x ( h )) + n X k =1 ( b k j ) σ ( x, h ) U σ k ( E σ I ,x ( h )) , wher e, (5.6) | ( b k j ) σ ( x, h ) | ≤ C η k h k I r r d k − d j ≤ χr d k − d j for al l h ∈ Q I ( η ε 0 r ) . Remark 5.3. Estimate (5 .6) and the res ults on Section 4 imply that, under the hypotheses of Theorem 5.2, we hav e | λ σ I ( x ) | ≤ C 1 | λ σ I ( E σ I ,x ( h )) | ≤ C 2     det ∂ ∂ h E σ I ,x ( h )     ≤ C 3 | λ σ I ( x ) | for all h ∈ Q I ( ε 0 η r ) . Pr o of of The or em 5.1. Without loss of ge ne r ality we ma y work in R 2 . W e drop everywhere the sup erscr ipt σ . Then E I ( x, h ) = e h 1 U 1 ∗ e h 2 U 2 ∗ x. Deno te by u the ident ity function in R n . W e ﬁr st lo ok at ∂ /∂ h 1 . Theore m 3.6 with X w = U 1 and t = h 1 gives: ∂ ∂ h 1 u  e h 1 U 1 ∗ e h 2 U 2 ∗ x  = U 1 u ( E I ,x ( h )) + s X | v | = d 1 +1 α v ( h 1 ) X v u ( E I ,x ( h )) + O s +1  | h 1 | ( s +1 − d 1 ) /d 1 , u, E I ,x ( h )  , where we know that | α v ( h 1 ) | ≤ C | h 1 | ( | v |− d 1 ) /d 1 and   O s +1  | h 1 | ( s +1 − d 1 ) /d 1 , u, E I ,x ( h )    ≤ C | h 1 | ( s +1 − d 1 ) /d 1 . Thu s, since | h 1 | 1 /d 1 ≤ k h k I , we hav e prov ed (5 .10) and (5.11) for j = 1. Next we lo ok at the v ariable h 2 . Theore m 3.6 gives (5.7) ∂ ∂ h 2 u  e h 1 U 1 ∗ e h 2 U 2 ∗ x  = U 2  ue h 1 U 1 ∗  e h 2 U 2 ∗ x  + s X | v | = d 2 +1 α v ( h 2 ) X v  ue h 1 U 1 ∗  e h 2 U 2 ∗ x  + O s +1  h ( s +1 − d 2 ) /d 2 2 , ue h 1 U 1 ∗ , e h 2 U 2 ∗ x  , where we know that | α v ( h 2 ) | ≤ C | h 2 | ( | v |− d 2 ) /d 2 ≤ C k h k | v | − d 2 I and    O s +1  h ( s +1 − d 2 ) /d 2 2 , ue h 1 U 1 ∗ , e h 2 U 2 ∗ x     ≤ C | h 2 | ( s +1 − d 2 ) /d 2 ≤ C k h k s +1 − d 2 I . NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 23 Now, a rep eated application of for m ula (3.12) gives (5.8) U 2  ue h 1 U 1 ∗  e h 2 U 2 ∗ x  = U 2 u  E I ,x ( h )  + s − d 2 X α 1 + ··· + α ν =1 C α h ( α 1 + ··· + α ν ) /d 1 1 ad α 1 Z 1 · · · ad α ν Z ν U 2 u  E I ,x ( h )  + O s +1  h ( s +1 − d 2 ) /d 1 1 , u, E I ,x ( h )  , where w e denoted brieﬂy e h 1 U 1 ∗ = e − h 1 /d 1 1 Z 1 · · · e − h 1 /d 1 1 Z ν , where ν is suitable, h 1 > 0 and Z j ∈ {± X 1 , · · · ± X m } . If h 1 < 0 the computation is a na logue. T o conclude the pr o of it suﬃces to write a ll the ter ms X v ( ue h 1 U 1 ∗ )( e h 2 U 2 ∗ x ) in (5.7) in the form X v u  E I ,x ( h )  plus an appropria te rema inder. The ar gument is the same used in equation (5.8) and we leave it to the reader .  Pr o of of The or em 5.2. The pro of r e lie s o n Coro llary 4 .3. W e drop everywhere the sup erscript σ . If ( I , x, r ) is η − maximal, then (4.2 ) gives | λ I ( E I ,x ( h )) | r ℓ ( I ) ≥ C − 1 η Λ( E I ,x ( h ) , r ), as so on as h ∈ Q I ( ε 0 η r ). W rite brieﬂy E instea d of E I ,x ( h ) . Lo oking at the right-hand side of (5.9), we need to study , for any word w of length | w | = ℓ , with ℓ = d j + 1 , . . . , s , the linear system a w j ( h ) X w ( E ) = P n k =1 b k j U k ( E ) and we must show tha t the solution b k j satisﬁes (5.6), if k h k I ≤ ε 0 η r . By Corollary 4.3 write X w ( E ) = P p k w U k ( E ), where | p k w | ≤ C η r d k −| w | . Thus | b k j | = | a w j p k w | ≤ C k h k | w |− d j I C η r d k −| w | ≤  k h k I r  | w |− d j C η r d k − d j . Here w e also used (5.11). This gives the estimate of the ter ms in the sum in (5.9). Next we lo o k at the the r emainder ω j . Fix j = 1 , . . . , n . W e know that | ω j | ≤ C k h k s +1 − d j I and we wan t to write ω j = P k b k j U k ( E ) with estimate (5 .6). It is conv e nient to multiply by r d j . Let r d j ω j =: θ ∈ R n and ξ k = r d j b k j . Thus it suﬃces to sho w that w e can write θ = P k ξ k U k ( E ), where ξ k satisﬁes the estimate | ξ k | ≤ C η k h k r r d k . W e know that | θ | = | r d j ω j | ≤ C k h k s +1 − d j I r d j = C  k h k I r  s +1 − d j r s +1 . T o estimate ξ k , we follow a t wo steps argument: Step 1. W rite, b y Lemma 4 .2, θ = P q i =1 µ i Y i ( E ) , for some µ ∈ R q satisfying | µ | ≤ C | θ | ≤ C  k h k r  s +1 − d j r s +1 . Step 2. F or any i = 1 , . . . , q write Y i ( E ) = P n k =1 λ k i U k ( E ). This can b e done in a unique w ay a nd estimate | λ k i | ≤ C η r d k − ℓ ( Y i ) holds, b y Corolla ry 4.3. Collecting Step 1 and Step 2, we conclude that | ξ k | =    q X i =1 µ i λ k i    ≤ C  k h k r  s +1 − d j r s +1 · C η r d k − ℓ ( Y i ) ≤ C η k h k r r d k , as required. This ends the pro o f.  Next we pass to the limit a s σ → 0 in b oth Theor ems 5 .1 and 5.2. 24 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI Theorem 5. 4. If ( I , x, r ) is η − maximal for some x ∈ K , r ≤ r 0 , then the map E I ,x   Q I ( ε 0 ηr ) is lo c al ly biLipschitz in t he Euclide an sense and satisﬁ es for a.e. h , (5.9) ∂ ∂ h j E I ,x ( h ) = U j ( E I ,x ( h )) + s X | w | = d j +1 a w j ( h ) X w ( E I ,x ( h )) + ω j ( x, h ) = U j ( E I ,x ( h )) + n X k =1 b k j ( x, h ) U k ( E I ,x ( h )) , wher e the sum is empty if d j = ℓ ( U j ) = s and otherwise the fol lowing estimates hold: (5.10) | ω j ( x, h ) | ≤ C k h k s +1 − d j I for al l x ∈ K h ∈ Q I ( r 0 ) , (5.11) | a w j ( h ) | ≤ C k h k | w |− d j I if | w | = d j + 1 , . . . , s and h ∈ Q I ( r 0 ) , and (5.12) | b k j ( x, h ) | ≤ C η k h k I r r d k − d j ≤ χr d k − d j for al l h ∈ Q I ( ε 0 η r ) . Remark 5.5. If s ≥ 3, then vector ﬁelds o f the c lass A s are C 1 . Then, as discussed in the b eginning of the pro of of Theor em 3 .6, the map E I ,x is a ctually C 1 smo oth. This is not ensured if s = 2 . Pr o of of The or em 5.4. Lo o k ﬁrst at the C 1 map E σ = E σ I ,x deﬁned on Q I ( r 0 ). Denote by E its po int wise limit as σ → 0. By Theore m, 5.1, the map E σ satisﬁes for any σ < σ 0 , k h k I ≤ r 0 , (5.13) ∂ ∂ h j E σ ( h ) = U σ j ( E σ ( h )) + s X | w | = d j +1 a w j ( h ) X σ w ( E σ ( h )) + ω σ j ( h ) , where a w j do not depe nd on σ , while | ω σ j ( h ) | ≤ C k h k s +1 − d j I , uniformly in σ ≤ σ 0 . Let E σ k be a sequence weakly conv erging to E in W 1 , 2 . Ther efore, by (5 .1 3), the remainder ω σ k j has a weak limit in L 2 . Denote it by ω j . Standar d prop erties of weak conv ergence ensure that | ω j ( h ) | ≤ C 0 k h k s +1 − d j I for a .e. h . There fo re, we hav e proved the ﬁr st line o f (5.9) a nd estimates (5.10) and (5.11). T o prov e the second line and (5 .1 2), it s uﬃces to rep eat the a rgument of Theor em 5.2, tak ing into account that the main ingredient ther e, namely Cor o llary 4.3, holds fo r nonsmo oth vector ﬁelds in A s . Now we hav e to prove the lo cal injectivit y of E . Let σ be small eno ugh to ensure that ( I , x, r ) is η -maximal for the vector ﬁelds X σ j . In view o f Theor e m 5.2, we can write dE σ ( h ) = U σ ( E σ ( h ))( I n + B σ ( h )), where U σ = [ U σ 1 , . . . , U σ n ], and the ent ries of the ma tr ix B satisfy | ( b k j ) σ | ≤ C r d k − d j , by (5.5). Fix now h 0 ∈ Q I ( ε 0 η r ), where ε 0 η comes fr o m Theor em 5.2. W e will sho w that E σ is lo ca lly one-to- one around h 0 , with a s ta ble co ercivity estimate as σ → 0. By Pr op osition 2.4 and by the con tinu ity of the v ector ﬁelds U j , we may claim that for any δ > 0 there is  > 0 such that | U σ j ( ξ ) − U σ j ( ξ ′ ) | < δ as s o on as ξ , ξ ′ ∈ K , | ξ − ξ ′ | <  a nd σ <  . Recall also that E σ is Lipschitz contin uous, unifor mly in σ , s ee (5.1 3). Then, for any δ > 0 there is  > 0 such that B Eucl ( h 0 ,  ) ⊂ Q I ( ε 0 η r ), and, if | h − h 0 | ≤  and σ <  , then | U σ ( E σ ( h )) − U σ ( E σ ( h 0 )) | ≤ δ . NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 25 T a ke h, h ′ ∈ B Eucl ( h 0 , δ ). By integrating on the path γ ( t ) = h ′ + t ( h − h ′ ), we hav e | E σ ( h ) − E σ ( h ′ ) | =    Z 1 0 U σ ( E σ ( γ ))( I + B σ ( γ ))( h − h ′ ) dt    ≥    Z 1 0 U σ ( E σ ( h 0 ))( I + B σ ( γ ))( h − h ′ ) dt    −    Z 1 0  U σ ( E σ ( γ )) − U σ ( E σ ( h 0 ))  ( I + B σ ( γ ))( h − h ′ ) dt    T o estimate from b elow the ﬁrst line recall the easy inequality | Ax | ≥ C − 1 | d et A | | A | n − 1 | x | , for all A ∈ R n × n . The po in twise estimate | ( b k j ) σ | ≤ χr d k − d j gives | R 1 0 ( b k j ) σ ( γ )) dt | ≤ χr d k − d j . Thus (5.4) gives    det Z 1 0 ( I + B σ ( γ )) dt    =    det  I + Z 1 0 B σ ( γ ) dt     ≥ 1 2 . Observe als o that | I + B σ ( γ ) | ≤ C r 1 − s . Moreover, in view of Remark 5.3, it must be | det U σ ( E σ ( h 0 )) | ≥ C − 1 | λ I ( x ) | , for small σ . This suﬃces to es tima te from be low the ﬁr st line. T o get an estimate of the second line we need a gain the inequality | I + B σ ( γ ) | ≤ C r 1 − s . Even tually we get | E σ ( h ) − E σ ( h ′ ) | ≥ { C − 1 0 | λ I ( x ) | r ( n − 1)( s − 1) − C 0 r 1 − s δ }| h − h ′ | , for any σ <  and | h − h ′ | <  . The pro of is concluded a s so on as w e cho ose δ = δ ( I , x, r ) small enough and let σ → 0 . This argument shows that the map is lo cally biLipschitz, as desired.  5.2. Pullbac k of v ector ﬁelds through scaling maps. Given an η -maxima l triple ( I , x, r ), for vector ﬁelds of the class A s we c an deﬁne, as in [TW], the “scaling map” (5.14) Φ I ,x,r ( t ) = exp  n X j =1 t j r ℓi j Y i j  x, for small | t | . T he dilation δ I r ( t ) := ( t 1 r ℓ i 1 , . . . , t n r ℓ i n ) mak es the na tur al domain o f Φ I ,x,r independent o f r . Obse rve the prop erty k δ I r t k I = r k t k I . It turns o ut that, if b X k ( k = 1 , . . . , m ) denotes the pullback of rX k under Φ I ,x,r , then b X 1 , . . . , b X m satisfy the H¨ ormander condition in an uniform way . This fact enables the authors in [TW] to give several simpliﬁca tions to the arguments in [NSW]. W e can also co nsider the scaling ma p a sso ciated with our exp onentials. Namely , (5.15) S I ,x,r ( t ) := exp ∗ ( t 1 r ℓ i 1 Y i 1 ) · · · exp ∗ ( t n r ℓ i n Y i n ) = E I ,x ( δ I r t ) , It will b e proved in Subsection 5.3 that, if ( I , x, r ) is η -ma ximal, then S is one-to- one on the set {k t k I ≤ ε 0 η } . If we ass ume that the orig inal vector ﬁelds are of class C 1 , s e e Remark 5.5, thus, w e ma y deﬁne, for all i ∈ { 1 , . . . , q } the vector ﬁelds b Y j := S − 1 ∗ ( r ℓ i Y i ) . Theorem 5.4 thus b ecomes Prop ositio n 5. 6. L et X 1 , . . . , X m b e ve ctor ﬁ elds in A s . L et ( I , x, r ) b e an η − maximal triple and let S := S I ,x,r b e the asso ciate d sc aling map. Then S   Q I ( ε 0 η ) is a lo c al ly 26 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI biLipschi tz map and for a.e. t ∈ Q I ( ε 0 η ) we may write (5.16) S ∗ ( ∂ t j ) = r ℓ i j Y i j ( S ( t )) + n X k =1 b b k j r ℓ i k Y i k ( S ( t )) , wher e the functions b b k j satisfy (5.17) | b b k j | ≤ C η k t k I for a.e. t ∈ Q I ( ε 0 η ) . Mor e over, if S is C 1 and we write b Y i j = ∂ t j + P n k =1 a k j ( t ) ∂ t k , t hen (5.18) | a k j ( t ) | ≤ C η k t k I for al l t ∈ Q I ( ε 0 η ) . Pr o of. F ormula (5.16) is just Theorem 5.4. The pro of o f (5.18) is a conse quence o f (5.17) and of the following elemen tary fact: given a s q uare matrix B ∈ R n × n with norm | B | ≤ 1 2 , w e ma y write ( I n + B ) − 1 = I n + A , and | A | =   P k ≥ 1 ( − B ) k   ≤ 2 | B | .  In the fr amework of our a lmost exp onential maps, es tima te (5.18) is sharp, even for smo oth vector ﬁelds. T he b etter estimate Y i j ( t ) = ∂ j + P k a k j ( t ) ∂ k with | a k j ( t ) | ≤ C | t | , obtained in [TW] fo r maps of the for m (5.14), g e nerically fails for S , as the following example shows. Example 5. 7 . Let X 1 = ∂ 1 , X 2 = a ( x 1 ) ∂ 2 with a ( s ) = s + s 2 , or any smo oth function with a (0) = 0 and a ′ (0) 6 = 0 6 = a ′′ (0). A computation shows that exp ∗ ( h [ X 1 , X 2 ])( x 1 , x 2 ) =  x 1 , x 2 +  a ( x 1 + | h | 1 / 2 ) − a ( x 1 )  | h | − 1 / 2 h  . Therefore, at ( x 1 , x 2 ) = (0 , 0), for small r , we must cho ose the maximal pa ir of commutators X 1 , [ X 1 , X 2 ] and w e hav e S ( t 1 , t 2 ) = ex p ∗ ( t 1 rX 1 ) exp ∗ ( t 2 r 2 [ X 1 , X 2 ])(0 , 0) =  t 1 r , a ( r | t 2 | 1 / 2 ) | t 2 | − 1 / 2 t 2 r  . =  t 1 r , t 2 r 2 + | t 2 | 1 / 2 t 2 r 3  . Therefore, b X 1 = ∂ t 1 , b X 2 = t 1 + r t 2 1 1 + 3 2 r | t 2 | 1 / 2 ∂ t 2 , \ [ X 1 , X 2 ] = 1 + 2 rt 1 1 + 3 2 r | t 2 | 1 / 2 ∂ t 2 . Clearly the form ula \ [ X 1 , X 2 ] = ∂ t 2 + O ( | t | ) ca nnot hold, but (5.18) holds. O bserve also tha t, writing \ [ X 1 , X 2 ] = b f (1 , 2) · ∇ , we have sup t ∈ U   b X 2 b f (1 , 2)   ≃ sup t ∈ U | t 2 | − 1 / 2 = + ∞ , for a ny neighbo rho o d U of the or igin. Therefore, the vector ﬁelds b X 1 , b X 2 do not even b elong to the class A 2 . 5.3. Ball-b ox theorem . Her e we g ive o ur main re s ult. W e keep the notation fro m Subsection 5.1. Theorem 5.8. L et X 1 , . . . , X m b e H¨ ormander ve ct or ﬁelds of step s in t he class A s . Ther e ar e r 0 , e r 0 , C 0 > 0 , and for al l η ∈ (0 , 1 ) ther e ar e ε η , C η > 0 such that: (A) if ( I , x, r ) is η − maximal for some x ∈ K , r ≤ r 0 , t hen, for any ε ≤ ε η , we have (5.19) E I ,x ( Q I ( εr )) ⊃ B  ( x, C − 1 η ε s r ); NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 27 (B) if ( I , x, r ) is η − maximal for some x ∈ K , r ≤ e r 0 , then the map E I ,x is one-to-one on the set Q I ( ε η r ) . Remark 5.9. Obser ve that in the right-hand s ide of inclusio n (5.19) w e use the distance d  . Therefore, a standard consequence o f (5.19) is the well known prop erty B ( x, r ) ⊃ B E ( x, C − 1 r s ), for any x ∈ K , r < r 0 . See [FP]. Remark 5.10. In the pap er [TW] the a utho r s use the exp onential maps in (1.2). If the vector ﬁelds hav e step s , then their metho d require s that the commutators of length 2 s ar e at least contin uous. (Here, we specia lize [TW] to the case ε = 1 and we do not dis cuss the higher r e gularity estimate [TW, Eq. (2.1)].) This app ears in the pro o f o f (22) and (23) of [T W , P r op osition 4.1 ]. Indeed in equation (29), the commutator [ X w , X w j ] m ust b e written as a linear combination of commutators X w ′ , where for algebraic r easons it must b e | w ′ | = | w | + | w k | . If | w | = | w k | = s , then co mm utators of degree 2 s app ear. A similar issue app ears for [ Y w i , Y w j ] a t the beg inning of p. 619. Remark 5.11. The rea son wh y we in tro duce tw o diﬀeren t constants r 0 and e r 0 is that C 0 , ε 0 and r 0 depe nd only on L and ν in (2.2) and (2.5) (together with universal co nstants, like m, n and s ). The co nstants ε η and C η depe nd on ν , L and η als o. W e do not hav e a cont rol of e r 0 (whic h app ear s only in the injectivity statement) in ter m of L and ν . This is a delicate q uestion b ecause o f the cov ering argument implicitly contained in [NSW, p. 1 32] a nd des crib ed in [M, p. 230]. B e low we provide a constructive pr o cedure to provide a low er b ound for e r 0 in term of the functions λ I . See p. 31. This can be of s ome in ter est in view of applica tions of our results to nonlinear problems. Remark 5.12. T he pr o of o f the injectivity re s ult would b e consider ably simpliﬁed if we could pr ove (uniformly in x ∈ K , r < r 0 ) an eq uiv alence b etw een the ba lls and their conv ex hulls, i.e. co B ( x, r ) ⊂ B ( x, C r ) , which is rea sonable for diag onal vector ﬁelds (see [SW, Remark 5]) or a “ c ontractabilit y” prop er ty of the ball B ( x, r ) inside B ( x, C r ). See [Sem, Deﬁnition 1.7]. Unfortunately , in spite of their reasona ble asp ect, b oth these conditions seem quite diﬃcult to prove in our s ituation. It seems a lso that the clever ar gument in [TW, p. 622] can no t be adapted to o ur almost exp onential maps. In the proo f of inclusion (5.1 9), w e follow the argument in [NSW , M]. Befor e giving the pro o f, we need to show that some constants in the pr o of actua lly dep end only on L and ν in (2.2) and (2 .5). Ba sically , what we ne e d is co ntained in Co rollary 4.3 and in the following Lemma. Se e [NSW, p. 129]. Lemma 5.13. Assu me that ( I , x, r ) is η − maximal for ve ctor ﬁ elds X j in A s , x ∈ K , r ≤ r 0 . L et e σ > 0 b e such that ( I , x, r ) is η -m aximal for t he mol liﬁe d X σ j for al l σ ≤ e σ . L et U ⊂ Q I ( ε η r ) , wher e ε η c omes fr om The or em 5.4 , and assu me that a C 1 diﬀe omorphism ψ = ( ψ 1 , . . . , ψ n ) : E σ ( U ) → U satisﬁes ψ ( E σ ( h )) = h, for any h ∈ U. Then we have t he estimate | U σ j ψ k ( E σ ( h )) | ≤ C r d k − d j , for al l h ∈ U , wher e C is indep en dent of σ . Pr o of. It is co nv enient to work with the map S σ ( t ) := E σ ( δ r t ), so that ϕ := δ 1 /r ψ satisﬁes t = ϕ ( S σ ( t )), for all t ∈ V := δ 1 /r U . The chain rule g ives dϕ ( S σ ( t )) dS σ ( t ) = I , for all t ∈ V . But, by (5.16) we hav e dS σ ( t ) = [ r d 1 U σ 1 ( S σ ) , . . . , r d n U σ n ( S σ )]( I + 28 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI b B σ ( t )), where | b B σ ( t ) | ≤ C η k t k I , if k t k I ≤ ε 0 η . There fore w e ma y write dϕ ( S σ )[ r d 1 U 1 ( S σ ) , . . . , r d n U n ( S σ )] = I + A σ , where, a s in the pro of of P rop osition 5.6, | A σ ( t ) | ≤ 2 | b B σ ( t ) | . This implies that | r d j U σ j ϕ k ( S σ ( t )) | ≤ C and ultimately that | r d j − d k U σ j ψ k | ≤ C , as desired.  Pr o of of The or em 5.8, (A). Since the vector ﬁelds Y j are not E uclidean Lipschitz contin uous, if ℓ j = s , we do not know whether or not a ny p o int in a  -ba ll o f the Y j can b e a pproximated b y p oints in the a nalogous ball of the molliﬁed Y σ j . In orde r to av o id this problem, observe the inclusion B  ( x, r ) ⊂ B e  ( x, C r ) where C is absolute and the distance e  is deﬁned using the family { Y j : ℓ j ≤ s − 1 , ∂ k : k = 1 , . . . , n } , where w e a ssign to the vector ﬁelds ∂ k maximal w eight s . Therefo r e, we will prov e the inclusion using the distance e  , whic h is deﬁned b y Lipschitz v ector ﬁelds. Let ( I , x, r ) be a η − maximal triple for the original vector ﬁelds X j and let e σ b e as in Lemma 5.13. Let y ∈ B e  ( x, C − 1 η ε s r ), where ε ≤ ε η , and ε η comes from statement (A), while C η will b e discussed b elow. Thus, y = γ (1), where ˙ γ = P ℓ j ≤ s − 1 b j Y j ( γ ) + P n i =1 ˜ b j ∂ i ( γ ) a.e. on [0 , 1] , with | b j ( t ) | ≤ ( C − 1 η ε s r ) ℓ ( Y j ) and | ˜ b i ( t ) | ≤ ( C − 1 η ε s r ) s for a.e . t ∈ [0 , 1]. L et a ls o y σ ∈ B e  ( x, C − 1 η ε s r ) b e an approximating family , y σ = γ σ (1), wher e ˙ γ σ = P ℓ j ≤ s − 1 b j Y σ j ( γ σ ) + P n i =1 ˜ b j ∂ i ( γ σ ) a.e. on [0 , 1] . Observe that y σ → y , a s σ → 0 . Claim. If C η is la r ge enoug h, then for an y σ ≤ e σ there is a lifting map θ σ ( t ) , t ∈ [0 , 1 ], with θ σ (0) = 0 and suc h that (5.20) E σ ( θ σ ( t )) = γ σ ( t ) and k θ σ ( t ) k I < εr for all t ∈ [0 , 1] . Once the claim is prov ed, the surjectivit y statement follows. T o prove the cla im the key es tima te w e need is the following. Let U ⊂ Q I ( ε η r ), σ ≤ e σ and assume that a C 1 − diﬀeomorphism ψ = ( ψ 1 , . . . , ψ n ) satisﬁes lo ca lly ψ ( E σ ( h )) = h , for all h ∈ U , where, fo r some t ∈ [0 , 1], E σ ( U ) is a neighborho o d of γ σ ( t ). Then, for µ = 1 , . . . , n and for all τ clos e to t (5.21)    d dτ ψ µ ( γ σ ( τ ))    =    X ℓ j ≤ s − 1 b j ( τ ) Y σ j ψ µ ( γ σ ( τ )) + n X i =1 ˜ b i ( τ ) ∂ j ψ µ ( γ σ ( τ ))    =    X ℓ j ≤ s b j ( τ ) n X k =1 a k j ( γ σ ( τ )) U σ k ψ µ ( γ σ ( τ )) + n X i =1 ˜ b i ( τ ) n X k =1 ˜ a k i ( γ σ ( τ )) U σ k ψ µ ( γ σ ( τ ))    ≤ X j,k C ( C − 1 η ε s r ) ℓ ( Y j ) · C η r d k − ℓ ( Y j ) · C r d µ − d k + X i,k C ( C − 1 η ε s r ) s · C η r d k − s · C r d µ − d k ≤ C C − 1 η η ε s r d µ ≤ C C − 1 η η ( εr ) d µ . NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 29 The constant C η will be chosen b elow, while C dep ends on L, ν , in force of Cor ollary 4.3 and Lemma 5.13. W e used the estimate ∂ i = ˜ a k i U σ k with   ˜ a k i   ≤ C η r d k − s , which follows from Lemma 4.2 and Coro llary 4 .3. With estimate (5.21) in hands we can prove the cla im along the lines of [M, p. 228]. Here is a sketc h of the argument. Step 1. If C η is large enough, then, if θ σ ( t ) satisﬁes E ( θ σ ( t )) = γ σ ( t ) on [0 , ¯ t ], for some ¯ t ≤ 1, then k θ σ ( t ) k I < 1 2 εr , for any t ≤ ¯ t . T o prov e Step 1, a ssume by contradiction that the statement is fals e. There is e t ≤ ¯ t suc h that k θ σ ( t ) k I < 1 2 εr for all t < e t and k θ σ ( e t ) k I = 1 2 εr . Then for some µ ∈ { 1 , . . . , n } , w e ha ve (5.22)  1 2 εr  d µ = | θ σ µ ( e t ) | ≤ C C − 1 η η ( εr ) d µ . This estimate c a n b e obtained writing lo ca lly θ σ ( t ) = ψ ( γ σ ( t )) and using (5 .2 1). If we choose C η large enough to ensure that C C − 1 η η < ( 1 2 ) s , then (5 .22) c a n not hold and w e hav e a contradiction. This ends the pro of of Step 1. Step 2. Ther e exists a path θ σ on [0 , 1] satisfying (5.20). The pro o f of Step 2 can be done a s in [M, p. 2 29] by a v ery class ical argument, inv olving an upper bo und “ of Hada mard type ” k dE σ ( θ σ ( t )) − 1 k ≤ C , which ho lds uniformly in t . The pro of of the statement (A) is concluded.  Before pr oving part ( B ) of Theor em 5.8, w e need the following rough injectivit y statement. Lemma 5.14. L et x ∈ K and I such that λ I ( x ) 6 = 0 . Then the function E I ,x is one-to-one on the set Q I ( C − 1 | λ I ( x ) | ) . Pr o of. Observe ﬁrst that for all j = 1 , . . . , n and small σ , we have (5.23)    ∂ ∂ h j E σ ( h ) − U σ j ( x )    ≤    ∂ ∂ h j E σ ( h ) − U σ j ( E σ ( h ))    + | U σ j ( E ( h )) − U σ j ( x ) | ≤ C k h k I , by estimates (5.2), (5.3) and the d -Lipschitz contin uity o f U σ j . Fix h, h ′ ∈ Q I ( C − 1 | λ I ( x ) | ) and let γ ( t ) = h ′ + t ( h − h ′ ). Then | E σ ( h ) − E σ ( h ′ ) | =    Z 1 0 dE σ ( γ )( h − h ′ ) dt    ≥ | dE σ (0)( h − h ′ ) | −    Z 1 0  dE σ ( γ ) − dE σ (0)  dt ( h − h ′ )    ≥  C − 1 | λ σ I ( x ) | − C max { k h k I , k h ′ k I }  | h − h ′ | . by (5.23) and b ecause dE σ (0) = U σ ( x ) = [ U σ 1 ( x ) , . . . , U σ n ( x )] has determinant λ σ I ( x ). The pro of is concluded by letting σ → 0.  As announced in Remark 5.11, we provide a constructive pro cedur e for the “ in- jectivit y ra dius” e r 0 in Theor em 5.8 in term of the functions λ I . Co mpare [M, p. 229-230 ]. 30 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI Denote by D 1 , . . . , D p all the v alues attained b y ℓ ( I ), as I ∈ S . Ass ume that D 1 < · · · < D p and in tro duce the notation: (5.24) X I | λ I ( x ) | r ℓ ( I ) = p X j =1 r D j X ℓ ( I )= D j | λ I ( x ) | =: p X j =1 r D j µ j ( x ) , where µ j is deﬁned by (5.24). Let Σ 1 := K and, for all k = 2 , . . . , p , Σ k := { x ∈ K : µ j ( x ) = 0 for any j = 1 , . . . , k − 1 } . Observe that Σ 1 = K ⊇ Σ 2 ⊇ · · · ⊇ Σ p . Let x ∈ K . T ake j ( x ) = min { j ∈ { 1 , . . . , p } : µ j ( x ) 6 = 0 } . Then choose I x ∈ S such that | λ I x ( x ) | = max ℓ ( J )= D j ( x ) | λ J ( x ) | is maximal. Therefore, we have | λ I x ( x ) | ≃ µ j ( x ) ( x ) , through absolute constants. F ro m the construction ab ov e we get the following prop ositio n. Prop ositio n 5.15. Ther e is C > 1 such that, lett ing r x := C − 1 | λ I x ( x ) | for al l x ∈ K , then: (1) we have (5.25) | λ I x ( y ) | r ℓ ( I x ) x > C − 1 Λ( y , r x ) for al l y ∈ B ( x , ε 0 r x ); (2) the m ap h 7→ E I x ( y , h ) is one-to-one on t he set Q I x ( r x ) , for any y ∈ B ( x, ε 0 r x ) . Observe that Pro po sition 5.15 is far fro m what we need, becaus e it may b e inf K r x = 0, (for exa mple this happ ens in the elementary situation X 1 = ∂ 1 , X 2 = x 1 ∂ 2 .) Pr o of. W e ﬁrst prov e (1) for y = x . Namely we show that (5.26) | λ I x ( x ) | r ℓ ( I x ) ≥ | λ J ( x ) | r ℓ ( J ) for all J ∈ S r ∈ [0 , r x ] , where r x = C − 1 | λ I x ( x ) | , as req uired. L e t J ∈ S . If λ J ( x ) = 0, then (5.2 6) holds for a ll r > 0. If instead λ J ( x ) 6 = 0, by the choice of I x it must b e ℓ ( J ) = ℓ ( I x ) o r ℓ ( J ) > ℓ ( I x ). If ℓ ( J ) = ℓ ( I x ), then (5.26) holds for any r > 0, bec ause | λ I x ( x ) | is maximal, by the constr uction above. If ℓ ( J ) > ℓ ( I x ), then | λ J ( x ) | r ℓ ( J ) ≤ | λ I x ( x ) | r ℓ ( I x ) ⇐ C r ℓ ( J ) − ℓ ( I x ) ≤ | λ I x ( x ) | ⇐ r ≤ C − 1 | λ I x ( x ) | . Thu s (5.26) holds for any r ≤ r x , where r x has the required form. The pro of of (1) for y 6 = x follows fro m Theorem 4.1. Finally , to prove (2) observe that, in view of Lemma 5 .14, the map h 7→ E I x ( y , h ) is one-to -one o n Q I x ( C − 1 | λ I x ( y ) | ). But Theo rem 4.1, in par ticular (4.1) show that, if d ( x, y ) ≤ ε 0 r x , then | λ I x ( y ) | and | λ I x ( x ) | are co mpa rable. This co ncludes the pro of.  Pr o of of The or em 5.8 , (C). Let p 1 ≤ p b e the lar gest integer such that Σ p 1 6 = ∅ . Then deﬁne the “injectivit y ra dius” (5.27) r ( p 1 ) := min x ∈ Σ p 1 r x = min x ∈ Σ p 1 C − 1 | λ I x ( x ) | ≥ C − 1 min x ∈ Σ p 1 µ p 1 ( x ) > 0 . Denote also Ω p 1 = [ x ∈ Σ p 1 Ω ′ ∩ B ( x, r ( p 1 ) ) , where the op en set Ω ′ was in tro duced befor e (2.2). Recall that all metric ba lls B ( x, r ) ar e op en, by the already accomplished Theorem (5.8), part (A). Then, by NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 31 Prop ositio n 5 .15, for an y y ∈ Ω p 1 there is x ∈ Σ p 1 such that the map h 7→ E I x ( y , h ) is one- to-one on Q I x ( ε 0 r x ) a nd ( I x , y , r x ) is C − 1 -maximal. Recall that r x ≥ r ( p 1 ) , on Σ p 1 . Next let p 2 < p 1 be the lar gest n umber such that K p 2 := Σ p 2 \ Ω p 1 6 = ∅ . Then, let r ( p 2 ) := min x ∈ Σ p 2 \ Ω p 1 r x ≥ C − 1 min x ∈ Σ p 2 \ Ω p 1 µ p 2 ( x ) > 0 . W e may claim that for any y ∈ Ω p 2 := S x ∈ K p 2 Ω ′ ∩ B ( x, r ( p 2 ) ) , there is x ∈ K p 2 such that the map h 7→ E I x ( y , h ) is one - to-one on the set Q I x ( ε 0 r x ) and ( I x , y , r x ) is C − 1 -maximal. Iterating the argument, and letting e r 0 = min { r ( p k ) } we conclude that for any x ∈ K there is a n -tuple I 0 = I 0 ( x ), and  0 =  0 ( x ) ≥ e r 0 such that E I 0 ( x, · ) is one-to-one on the s e t Q I 0 ( ε 0  0 ) and ( I 0 , x,  0 ) is C − 1 -maximal. Clear ly , I 0 can be diﬀerent from I x . This is the starting p oint for the pro of of the injectivity statement, Theorem 5.8, item (C). F ro m now on I , x ∈ K and r < e r 0 are ﬁxed and ( I , x, r ) is η − maximal, a s in the hypothesys of (C). Let I 0 and  0 be the n -tuple a nd the injectivity r adius as so ciated with x b y the argument a b ov e. Recall that  0 ≥ e r 0 . Arguing as in [M, p. 230 ], see also [NSW, p. 1 33], we may ﬁnd a sequence o f n − tuples I = I N , I N − 1 , . . . , I 1 , I 0 and corres po ding num b er s 0 ≤  N +1 <  N < · · · <  0 , with  0 ≥ e r 0 , r ∈ [  N +1 ,  N ] such that for any j = 0 , 1 , . . . , N − 1, (5.28) | λ I j ( x ) |  ℓ ( I j ) ≥ η Λ ( x,  ) , ∀  ∈ [  j +1 ,  j ] . In o rder to show that E I = E I N is one-to-one on the s e t Q I ( ε η r ), for some ε η > 0, we start by showing that E I 1 is one-to -one on the set Q I 1 ( ε ′ η  1 ) , for a suitable ε ′ η . What we know is that E I 0 is one- to-one on the set Q I 0 (  0 ). W e also know that (5 .28) holds for j = 0 , 1 and  =  1 . Therefore, applying twice (5.19), we hav e (5.29) E I 1 ( Q I 1 ( ε η  1 )) ⊇ E I 0 ( Q I 0 ( C − 1 η  1 )) ⊇ E I 1  Q I 1 ( ε ′ η  1 )  . Assume by contradiction that E I 1 ( h ) = E I 1 ( h ′ ) = y for some h, h ′ ∈ Q I 1 ( ε ′ η  1 ). Let r ( t ) = h ′ + t ( h − h ′ ), t ∈ [0 , 1] be the line segment connecting h and h ′ . Let also γ ( t ) = E I 1 ( r ( t )). Since E I 0 is one-to-one (actually a C 1 diﬀeomorphism on its image ), we may contract γ to a p oint just by letting q ( λ, t ) = E I 0  λE − 1 I 0 ( y ) + (1 − λ ) E − 1 I 0 ( γ ( t ))  , where ( λ, t ) ∈ [0 , 1] × [0 , 1] . Observe tha t q is contin uous on [0 , 1] 2 , and q ( λ, t ) ∈ Q I 1 ( ε η  1 ), by (5.2 9). Moreover q (0 , t ) = γ ( t ) = E I 1 ( r ( t )) a nd q (1 , t ) = y , for any t ∈ [0 , 1]. By standard pro p e r ties of lo cal diﬀeomorphisms we may c la im that there is a contin uous lifting p : [0 , 1] 2 → Q I 1 ( ε η  1 ) such that E I 1 ( p ( λ, t )) = q ( λ, t ) and p (0 , t ) = r ( t ) for all λ and t ∈ [0 , 1]. Next observe that b oth the maps λ 7→ E I 1 ( p ( λ, 1)) a nd λ 7→ E I 1 ( p ( λ, 0)) are constants on [0 , 1]. Therefore, since E I 1 is a lo cal diﬀeomorphism, b oth λ 7→ p ( λ, 0) and λ 7→ p ( λ, 1) m ust b e constant. In par ticular p (1 , 1) = p (0 , 1) = h ′ and p (1 , 0) = p (0 , 0) = h . Finally observe that the path t 7→ p (1 , t ) must b e constant, b ecause E I 1 ( p (1 , t )) = y for all t ∈ [0 , 1]. Ther efore we conclude that h = h ′ . Then we hav e proved that E I 1 is o ne-to-one on Q I 1 ( ε ′ η  1 ). Iterating the argument at most N times, we get the pro o f of statemen t (C) of Theorem 5.8. 32 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI 6. Examples Example 6.1 (Levi v ector ﬁelds) . In or de r to illustrate the previous pr o cedure to ﬁnd ˜ r 0 we exhibit the following three-step example. In R 3 consider the vector ﬁelds X 1 = ∂ x 1 + a 1 ∂ x 3 and X 2 = ∂ x 2 + a 2 ∂ x 3 . Assume that the vector ﬁelds belo ng to the class A 3 . Let us deﬁne f = X 1 a 2 − X 2 a 1 . Mor ov er assume that | f | + | X 1 f | + | X 2 f | 6 = 0 at every p oint of the closure of a bo unded set Ω ⊃ K = Ω ′ . Assume also that f has so me zero inside K . This condition na turally arises in the regular ity theory for graphs of the fo rm { ( z 1 , z 2 ) ∈ C 2 : Im( z 2 ) = ϕ ( z 1 , ¯ z 1 , Re( z 2 )) } having s ome ﬁrst order zeros. See [CM], where the smoothness o f C 2 ,α graphs with prescrib ed smo o th Levi curv ature is prov ed. In this situa tio n we ha ve n = 3 , m = 2 , s = 3 and Y 1 = X 1 , Y 2 = X 2 , Y 3 = [ X 1 , X 2 ] = f ∂ x 3 , Y 4 = [ X 1 , [ X 1 , X 2 ]] = ( X 1 f − f ∂ x 3 a 1 ) ∂ x 3 , Y 5 = [ X 2 , [ X 1 , X 2 ]] = ( X 2 f − f ∂ x 3 a 2 ) ∂ x 3 . Th us, q = 5 a nd λ (1 , 2 , 3) = f , d (1 , 2 , 3) = 4 , λ (1 , 2 , 4) = X 1 f − f ∂ x 3 a 1 , d (1 , 2 , 4) = 5 , λ (1 , 2 , 5) = X 2 f − f ∂ x 3 a 2 , d (1 , 2 , 5) = 5 . Let us put D 1 = 4 , D 2 = 5 and, b y (5.24), µ 1 = | f | , µ 2 = | X 1 f − f ∂ x 3 a 1 | + | X 2 f − f ∂ x 3 a 2 | . In this situation Σ 1 = K, Σ 2 = { x ∈ K : µ 1 ( x ) = 0 } = { x ∈ K : f ( x ) = 0 } . Hence, r (2) = min x ∈ Σ 2 r x = min x ∈ Σ 2 max {| X 1 f ( x ) | , | X 2 f ( x ) |} > 0 . Let Ω 2 = ∪ x ∈ Σ 2 Ω ′ ∩ B ( x, r (2) ), with ¯ Ω ′ = K , and let K 1 = Σ 1 \ Ω 2 . Since K 1 ⊆ { x ∈ K : f ( x ) 6 = 0 } , if K 1 6 = ∅ then r (1) = min x ∈ K 1 r x = min x ∈ K 1 | f ( x ) | > 0 . Finally , if K 1 6 = ∅ then ˜ r 0 = min { r (1) , r (2) } , while if K 1 = ∅ then ˜ r 0 = r (2) . In next exa mple we show a s ub elliptic-type estimate for nonsmo oth vector ﬁelds. The argument of the pro of b elow is due to E rmanno Lanconelli (unpublished). Prop ositio n 6.2 (H¨ ormander–type estimate [H]) . L et X 1 , . . . , X m b e a family of ve ctor ﬁ elds of step s and in the class A s . Then, given Ω ′ ⊂⊂ Ω , and ε ∈ ]0 , 1 /s [ , ther e is e r 0 and C > 0 such that su ch that, for any f ∈ C 1 (Ω) , (6.1) [ f ] 2 ε := Z Ω ′ × Ω ′ , d ( x,y ) ≤ e r 0 | f ( x ) − f ( y ) | 2 | x − y | n +2 ε dxdy ≤ C Z Ω X j | X j f ( y ) | 2 dy . Pr o of. W e just sk etch the pro of, leaving so me details to the rea der. F or an y I ∈ S , let Ω I := { x ∈ Ω : I 0 ( x ) = I } , w he r e I 0 ( x ) comes from the pro of of Theo rem 5.8, together with  0 =  0 ( x ) ≥ e r 0 , see the discussio n b e fore equatio n (5.28). If x ∈ Ω I , we have B ( x,  0 ) ⊂ E I ( x, Q I ( C  0 )), where the biLipschitz map E I satisﬁes C − 1 ≤ | det dE I ( x, h ) | ≤ C , for a.e. h ∈ Q I ( C  0 ). Thus, [ f ] 2 ε = Z Ω ′ × Ω ′ d ( x,y ) ≤ e r 0 | f ( x ) − f ( y ) | 2 | x − y | n +2 ε dxdy ≤ X I ∈S Z Ω I dx Z d ( x,y ) ≤  0 dy | f ( x ) − f ( y ) | 2 | x − y | n +2 ε ≤ C X I Z Ω I dx Z Q I ( C  0 ) dh | f ( x ) − f ( E I ( x, h )) | 2 | x − E I ( x, h ) | n +2 ε . Now observe that, arguing as in the pro of o f Lemma 5.1 4, we hav e | E I ( x, h ) − x | ≥ C − 1 | h | , if k h k I ≤ C  0 . Let δ 0 = max x ∈ K  0 ( x ). Next we follow the argument in NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 33 [LM]. W rite E I ( x, h ) = γ I ( x, h, T ( h )), wher e γ I ( x, h, t ), t ∈ [0 , T ( h )] is a c ont r ol function , with the prop erties describ ed in [LM]. There fo re [ f ] 2 ε = X I Z Ω I dx Z Q I ( C δ 0 ) dh | f ( x ) − f ( E I ( x, h )) | 2 | h | n +2 ε ≤ C Z Q I ( C δ 0 ) dh | h | n +2 ε Z Ω I dx    Z T ( h ) 0 dt | X f ( γ I ( x, h, t )) |    2 ≤ C Z Q I ( C δ 0 ) dh | h | n +2 ε n Z T ( h ) 0 dt  Z Ω I dx | X f ( γ I ( x, h, t )) | 2  1 / 2 o 2 ≤ C Z Q I ( C δ 0 ) dh | h | n +2 ε T ( h ) 2 k X f k 2 L 2 (Ω) ≤ C k X f | 2 L 2 (Ω) , bec ause x 7→ γ I ( x, h, t ) is a change of v ariable, b y e s timate T ( h ) ≤ k h k I ≤ | h | 1 /s and the strict inequality ε < 1 /s .  The b order line inequa lit y k f k 1 /s ≤ C k X f k L 2 , which can no t be obtained with the a r gument ab ove, was pr ov ed in the smo oth case by Rothschild and Stein [Ro S]. 7. Proof of Proposition 2. 4 Here we pr ov e Pro p osition 2.4. By deﬁnition, (2.1) mea ns that for a ll j, k ∈ { 1 , . . . , m } and | w | ≤ s − 1 there is a b ounded function X j ( X k f w ) such that for a ny test function ψ ∈ C ∞ c ( R n ), (7.1) Z ( X k f w )( X j ψ ) = − Z { X j ( X k f w ) + div( X j ) X k f w } ψ . If D = ∂ j 1 · · · ∂ j p for some j 1 , . . . , j p ∈ { 1 , . . . , n } is an E uclidean deriv ative, denote by | D | = p its order . It is unders to o d that a deriv ative of o rder 0 is the iden tity . The ﬁrst item of Pro p o sition 2 .4 is a consequence of the following lemma: Lemma 7.1. L et X 1 , . . . , X m b e ve ctor ﬁelds in A s . Then for any wor d w with | w | ≤ s and for any Euclide an derivative D of or der | D | = p ∈ { 0 , . . . , s − | w |} , we have (7.2) sup K    D f σ w − ( D f w ) ( σ )    ≤ C σ. Note that, the case p = 0 of (7.2) provides the pro of of item 1 of Pr op osition 2.4. Observe also that, if | w | = s , then we ha ve | f w − f σ w | ≤ | f w − ( f w ) σ | + | f σ w − ( f w ) σ | . Lemma 7.1 gives the estimates of the second term. The ﬁrst one is estimated by means of the cont inuit y mo dulus of f w , which is not included in L in (2.2). Pr o of of L emma 7.1. W e arg ue b y induction on | w | . If | w | = 1 , then the left hand side of (7.2) v anishes. Assume that for some ℓ ∈ { 1 , . . . , s − 1 } , (7.2) holds for any word w of length ℓ and for eac h D with | D | ≤ s − ℓ . Let v = kw b e a word of length | k w | = ℓ + 1 . W e must s how that for a n y Euclidea n deriv ative D of order 0 ≤ | D | ≤ s − | v | , (7 .2) holds. W e have f v = X k f w − X w f k and f σ v = X σ k f σ w − X σ w f σ k . W e ﬁrst prov e (7.2) whe n the o rder of D satisﬁes 1 ≤ | D | ≤ s − | v | = s − ℓ − 1, 34 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI which can o c c ur only if ℓ ≤ s − 2 (in particular this implies s ≥ 3). The easier case is when D is the iden tity opera tor and it will be proven b elow. D f σ v − ( D f v ) ( σ ) = D  X σ k f σ w − X σ w f σ k  −  D ( X k f w − X w f k )  ( σ ) = D  X σ k f σ w  − ( D X k f w ) ( σ ) − { D ( X σ w f σ k ) −  D X w f k  ( σ ) } =: ( A ) − ( B ) . Omitting summation sign on α = 1 , . . . , n , w e may write ( A ) = ( D f α k ) ( σ )  ∂ α f σ w − ( ∂ α f w ) ( σ )  +  ( D f α k ) ( σ ) ( ∂ α f w ) ( σ ) −  ( D f α k ) ∂ α f w  ( σ )  + ( f α k ) ( σ )  D ∂ α f σ w − ( D ∂ α f w ) ( σ )  +  ( f α k ) ( σ ) ( D ∂ α f w ) ( σ ) − ( f α k D ∂ α f w  ( σ )  =: ( A 1 ) + ( A 2 ) + ( A 3 ) + ( A 4 ) . The e s timate | ( A 1 ) | + | ( A 3 ) | ≤ C σ follo ws from the induction as sumption. T o estimate ( A 4 ) observe that | ( A 4 ) | =    Z  ( f α k ) ( σ ) ( x ) − f α k ( x − σ y )  D ∂ α f w ( x + σ y ) ϕ ( y ) dy    ≤ C σ, bec ause f k is Lipschitz, while D ∂ α f w ∈ L ∞ lo c . Indeed, since | w | = ℓ , f w ∈ W s − ℓ, ∞ . Moreov er, D has leng th at most s − ℓ − 1 so that D∂ α has leng ht a t most s − ℓ . The estimate of ( A 2 ) is analog ous to that of A 4 . J ust recall that D f α k is Lipschitz and ∂ α f w is b ounded. Next we estimate ( B ). ( B ) = D (( f α w ) σ ∂ α f σ k ) − ( D ( f α w ∂ α f k )) ( σ ) = { D ( f α w ) σ − ( D f α w ) ( σ ) } ∂ α f σ k +  ( D f α w ) ( σ ) ∂ α f σ k − (( D f α w ) ∂ α f k ) ( σ )  + ( f α w ) σ D ∂ α f σ k − ( f α w ( D ∂ α f k )) ( σ ) =: ( B 1 ) + ( B 2 ) + ( B 3 ) . The term ( B 1 ) can be estimated by the inductive assumption. Mor eov er, | ( B 2 ) | =    Z  ∂ α f σ k ( x ) − ∂ α f k ( x − σ y )  D f α w ( x − σ y ) ϕ ( y ) dy    ≤ C σ, bec ause ∂ α f k is Lipschitz and D f α w ∈ L ∞ lo c . Finally | ( B 3 ) | =    Z  ( f α w ) ( σ ) ( x ) − f α w ( x − σ y )  D ∂ α f k ( x − σ y ) ϕ ( y ) dy    ≤ C σ. Indeed, since | w | ≤ s − 2, f α w is loc a lly Lipsc hitz. Moreov er, since the length o f the deriv ative D ∂ α is at most s − 1 and f k ∈ W s − 1 , ∞ lo c , we hav e D∂ α f k ∈ L ∞ . Next we lo ok at the case wher e D has length zer o, i.e. D is the identit y o p er ator. W e hav e to estima te, for v w ith | v | ≤ s , the diﬀerence f σ v − ( f v ) ( σ ) . W rite v = k w , where k ∈ { 1 , . . . , m } . Thus f σ v − ( f v ) ( σ ) = X σ k f σ w − X σ w f σ k − ( X k f w ) ( σ ) + ( X w f k ) ( σ ) = ( f α k ) ( σ ) { ∂ α f σ w − ( ∂ α f w ) ( σ ) } +  ( f α k ) ( σ ) ( ∂ α f w ) ( σ ) − ( f α k ∂ α f w ) ( σ )  − { ( f α w ) σ ∂ α f σ k − ( f α w ∂ α f k ) ( σ ) } = ( S 1 ) + ( S 2 ) + ( S 3 ) . NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 35 Now ( S 1 ) can be estimated by the inductive assumption. Moreover, | ( S 2 ) | =    Z { ( f α k ) ( σ ) ( x ) − f α k ( x − σ y ) } ∂ α f w ( x − σ y ) ϕ ( y ) dy    ≤ C σ, bec ause f α k is lo cally Lipschitz contin uous a nd ∂ α f w is lo cally b ounded, sinc e | w | ≤ s − 1 . Fina lly | ( S 3 ) | =    Z  ( f α w ) σ ( x ) − f α w ( x − σ y )  ∂ α f k ( x − σ y ) ϕ ( y ) dy    ≤ C σ, bec ause f w is Lipschitz and ∂ α f k is b ounded. This co ncludes the pro of of the the ﬁrst item of Prop os ition 2 .4.  Pr o of of Pr op osition 2.4, item 2. W e need to show that, for a ny j, k ∈ { 1 , . . . , m } , | w | = s − 1 we hav e the estimate | X σ j X σ k f σ w | ≤ C , unifor mly in x ∈ K and σ ≤ σ 0 . W rite X σ j X σ k f σ w = X σ j ( X k f w ) ( σ ) + X σ j  X σ k f σ w − ( X k f w ) ( σ )  =: M + N . Now, letting ϕ σ ( ξ ) = σ − n ϕ ( ξ /σ ), we hav e M ( x ) = ( f α j ) ( σ ) ( x ) ∂ x α Z X k f w ( x − σ y ) ϕ ( y ) dy = − Z ( f α j ) σ ( x ) X k f w ( z ) ∂ z α ( ϕ σ ( x − z )) dz = − Z f α j ( z ) X k f w ( z ) ∂ z α ( ϕ σ ( x − z )) dz + Z  ( f α j ) σ ( x ) − f α j ( z )  X k f w ( z ) ( ∂ α ϕ ) σ ( x − z ) σ dz . The ﬁrst line can b e estimated integrating by parts by means of (7.1). The es timate of the second line follows fro m the Lipsc hitz contin uity o f the functions f i . Next we control N . N ( x ) = ( f α j ) ( σ ) ( x ) ∂ x α n X σ k Z f w ( x − σ y ) ϕ ( y ) dy − Z ( X k f w )( x − σ y ) ϕ ( y ) dy o = ( f α j ) ( σ ) ( x ) ∂ x α n Z ( f β k ) σ ( x ) ∂ β f w ( z ) ϕ σ ( x − z ) dz − Z f β k ( z ) ∂ β f w ( z ) ϕ σ ( x − z ) dz o = ( f α j ) ( σ ) ( x ) Z n ( ∂ α f β k ) σ ( x ) ∂ β f w ( z ) ϕ σ ( x − z ) + [( f β k ) σ ( x ) − f β k ( z )] ∂ β f w ( z ) 1 σ ( ∂ α ϕ ) σ ( x − z ) o dz The estimate is co ncluded, b ecause ∂ β f w is b ounded, while | ( f β k ) σ ( x ) − f β k ( z ) | ≤ C σ .  References [BBP] M. Bramanti, L. Br andolini , M. Pedroni, B asi c prop erties of nonsmo oth H¨ ormander’s v ector ﬁelds and Po incar´ e’s inequality , preprint [CM] G. Citti, A. Mont anari, Regularity prop erties of solutions of a class of ell iptic-parab ol ic nonlinear Levi t yp e equations, T rans. Amer. Math. So c. 3 54 (200 2), no. 7, 2819–2848. 36 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI [FP] C. F eﬀerman, D. H. Phong, Sub elliptic eigenv alue problems. Conference on harmonic analysis in honor of Antoni Zygm und, V ol. I, II (Chicago, Ill . , 1981), 590–606, W adsworth Math. Ser., W adsw orth, Belmont, CA, 1983. [FL1] B. F ranc hi, E. Lanconelli, Une m´ etrique associ´ ee ` a une classe d’op ´ erateurs ell iptiques d ´ eg ´ en´ er´ es, (F renc h) [A metric associ ated wi th a class of degenerate ell iptic op erators] Conference on l inear partial and pseudo diﬀerential op erators (T orino, 1982). Rend. Sem. Mat. Univ. Politec . T orino 1983, Sp ecial Issue, 105–114 (1984). [FL2] B. F ranc hi, E. Lanconelli, H¨ ol der regulari t y theorem for a class of linear non uniformly elliptic operators with measurable coeﬃcients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (198 3), 523–541. [FSSC] B. F ranc hi, R. Serapioni, F. Serra Cassano, Approximation and imbedding theorems for we ighte d Sob olev spaces associated with Lipschitz contin uous vect or ﬁelds, Boll. Un. Mat. Ital. B (7) 11 (1997), 83–117. [GN] N Garofalo, D . M. Nhieu, Lipsch itz contin ui ty , global smo oth approximations and ex- tension theorems for Sobol ev functions i n Carnot-Carath ´ eodory spaces, J. Anal. Math. 74 (199 8), 67–97. [Ha] P . Hartman, Ordinary diﬀeren tial equa tions, John Wiley & Sons, Inc., New Y ork- London- Sydney 1964. [H] L. H¨ ormander, Hyp o elliptic second order diﬀerential equations, Acta Math. 1 19 (1967), 147–171. [J] D. Jerison, The Po incar´ e inequality for ve ctor ﬁelds satisfying the H¨ ormander condition, Duk e Math. J. 53 (1986) , No. 2, 503–523. [LM] E. Lanconelli, D. Mor bidelli, On the Poincar ´ e inequalit y for vecto r ﬁelds, Ar k. Mat. 3 8 (2000), 327–342. [MM] A. Monta nari, D. Morbidelli, Balls deﬁned by nonsmooth vec tor ﬁelds and the Poincar ´ e inequalit y , Ann. Inst. F ourier (Grenoble) 54 (2004), 431–452. [MoM] R . Monti, D. Morbidelli, T race theorems for ve ctor ﬁelds, M ath. Z. 239 (2002), 747–776. [M] D. Morbidelli, F ractional Sob olev norms and str ucture of Carnot–Carath ´ eodory balls for H¨ ormander vect or ﬁelds, Studia Math. 139 (2000), 213–244. [NSW] A . Nagel, E. M . Stein, S. W ainger, B al ls and metrics deﬁned by v ector ﬁelds I: basic properties, Acta M ath, 155 (1985), 103–147. [RaS] F. Rampazzo, H. J. Sussmann, Set –v alued diﬀeren tial and a nonsmooth version of Cho w’s theorem, Proceedings of the 40th IEEE Conference on Decision and Con trol; Orlando, Flori da, 2001. [RaS2] F. Rampazzo, H. J. Sussmann, Commut ators of ﬂow maps of nonsmo oth vect or ﬁelds, J. Diﬀerential Equations 23 2 (20 07), 134–175. [RoS] L. P . Rothsc hi ld, E. M . Stein, Hyp o elliptic di ﬀeren tial operators and nilpotent groups, Acta Math. 137 (1976) , 247–320. [SW] E. T. Sawy er , R. L. Whee den, H¨ older con tinuit y of weak solutions to sub elliptic equations with rough co eﬃcients, Mem. Amer. Math So c. 180 (2006). [Sem] S. Semmes, Finding curves on general spaces through quantitativ e top ology , with appli- cations to Sob olev and Poinc ar´ e inequalities, Selecta M ath. (N.S.) 2 (1996), 155–295. [Ste] E. M . Stein, Some geometrical concepts arisi ng in harmonic analysi s , GAF A 2000 (T el Aviv, 1999). Geom. F unct. Anal. 2000, Sp ecial V olume, Part I, 434–453. [Str] B. Street, Multi-parameter Carnot-Carath ´ eodory ball s and the theorem of F rob enius, preprint [TW] T. T ao, J. W right , L p improving b ounds for av erages along curve s, J. Amer. M ath. Soc. 16 (200 3), 605–638. Dip ar timento di Ma tema tica, Universit ` a di Bologn a (IT AL Y) E-mail addr ess : montanar@dm. unibo.it, morbidel@dm.uni bo.it NONSMOOTH H ¨ ORMANDER VECTOR FIELDS AND THEIR CON TR OL BALLS ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI Abstract. W e pro ve a ball -b o x theorem for nonsmo oth H¨ ormander vecto r ﬁelds of step s ≥ 2 . Contents 1. Introduction 1 2. Preliminar ies and no tation 4 Approximate co mm utators 5 Regularized vector ﬁelds . 6 Some more notation. 7 3. Approximate exp onentials of commutators 7 3.1. Notation for integral r emainders 8 3.2. Higher order non commutativ e calculus formulas 10 3.3. Deriv atives of approximate exp onentials 17 4. Persistence o f maximalit y conditions on balls 20 5. Ball-b ox theor em 22 5.1. Deriv atives of almost expo nent ial maps 22 5.2. Pullback o f v ector ﬁelds through scaling maps 25 5.3. Ball-b ox theorem 27 6. Examples 32 7. Pro o f of P rop osition 2.4 33 References 36 1. Introduction In this pap er we give a self-contained pro o f of a ball-b ox theor em fo r a family { X 1 , . . . , X m } of nonsmo o th vector ﬁelds satisfying the H¨ ormander co ndition. This is the third pa p er , after [M] and [MM], where we inv estigate idea s o f the cla ssical article by Nagel Stein and W ainger [NSW]. Our purp o s e is to prov e a ball- b ox theorem using o nly elementary a nalysis tech- niques and at the same time to relax as much as p oss ible the regular ity assumptions on the vector ﬁelds. Roughly sp eaking , o ur res ults ho ld a s so o n as the commutators inv olved in the H¨ ormander condition are Lipschitz contin uous. Moreov er, our pro o f do es not rely on alg ebraic to ols, lik e formal series and the Campbell–Haus dorﬀ for- m ula. 2010 Mathematics Subje ct Classiﬁc ation. Primary 53C17; Secondary 35R03. 1 2 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI T o describ e our work, we recall the basic ideas of [NSW]. Notation and lang ua ge are mo r e pr ecisely des c rib ed in Sec tion 2 . An y control ba ll B ( x, r ) a sso ciated with a family { X 1 , . . . , X m } of H¨ ormander vector ﬁelds in R n satisﬁes, for x b elonging to some compact set K and small ra dius r < r 0 , the double inclusion (1.1) Φ x ( Q ( C − 1 r )) ⊂ B ( x, r ) ⊂ Φ x ( Q ( C r )) . Here, the map Φ x is an exp onential of the form (1.2) Φ x ( h ) = ex p( h 1 U 1 + · · · + h n U n )( x ) , where the vector ﬁelds U 1 , . . . , U n are suitable commutators of leng ths d 1 , . . . , d n and Q ( r ) = { h ∈ R n : ma x j | h j | 1 /d j < r } . Usually , (1.1) is referred to as a b al l-b ox inclusion. A c o ntrol o n the Jacobian matr ix of Φ x gives a n es timate o f the measur e of the ball and ultimately it provides the doubling pro per ty . A remark a ble a chiev ement in [NSW] co ncerns the choice of the vector ﬁelds U j which guara nt ee inclusio ns (1.1) for a given co ntrol ball B ( x, r ), see also the discussion in [Ste, p. 44 0]. Enumerate as Y 1 , . . . , Y q all c o mmu tators o f length at most s a nd let ℓ i be the leng th o f Y i . If the H¨ ormander condition of step s is fulﬁlled, then the vector ﬁelds Y i span R n at an y point. Given a m ulti-index I = ( i 1 , . . . , i n ) ∈ { 1 , . . . , q } n =: S a nd its corr esp onding n − tuple Y i 1 , . . . , Y i n of commutators, let (1.3) λ I ( x ) = det( Y i 1 , . . . , Y i n )( x ) and ℓ ( I ) = ℓ i 1 + · · · + ℓ i n . In [NSW], the a uthors prove the following fact: given a ball B ( x, r ), inclusion (1.1) holds with U 1 = Y i 1 , . . . , U n = Y i n if the n − tuple I ∈ S satisﬁes the η -maximality condition (1.4) | λ I ( x ) | r ℓ ( I ) > η max K ∈ S | λ K ( x ) | r ℓ ( K ) , where η ∈ (0 , 1) is gr eater than so me a bsolute cons tant . Although the choice of the n -tuple I may dep end on b oth the p oint and the radius, the constant C is uniform in x ∈ K and r ∈ (0 , r 0 ). In [M] the seco nd a uthor prov ed that (1.1) a lso holds if o ne changes the map Φ x with the almost ex p onential map (1.5) E x ( h ) = ex p ∗ ( h 1 U 1 ) ◦ · · · ◦ exp ∗ ( h n U n )( x ) , where h j 7→ e xp ∗ ( h j U j ) is the appr oximate exp onential of the c o mm utator U j , whose main feature is that it can b e factorized as a suitable comp osition of ex- po nentials o f the or ig inal vector ﬁelds X 1 , . . . , X m . See (2.3) for the deﬁnition of exp ∗ . Lanconelli and the seco nd a uthor in [LM] pr ov e d that, if inclusion (1 .5), with per tinent estimates for the Jacobian o f E x are known, then the Poincar´ e inequality follows (see [J ] for the or ig inal pro of ). It is worth to obser ve no w that all the res ults in [NSW] and [M] are proved for C M vector ﬁelds, where M is m uch larg e r than the step s . This can b e seen b y carefully reading the pro ofs of Lemmas 2.10 a nd 2.13 in [NSW]. In [T W, Section 4], T ao and W righ t g ave a new pro of of the ba ll-b ox theo rem with a diﬀerent appro a ch, based on Gronwall’s ineq ua lity . The author s in [TW] use sca ling ma ps of the form Φ x,r ( t ) := ex p( t 1 r d 1 U 1 + · · · + t n r d n U n ) x , which are naturally deﬁned on a b ox | t | ≤ ε 0 , where ε 0 > 0 is a sma ll constant indep endent of x and r , see the discus s ion in Subsection 5.2. The arguments in [TW] do not rely on NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 3 the Campb ell–Hausdo rﬀ formula. 1 Moreov er, although the statement is phr a sed for C ∞ vector ﬁelds, one can see that their results hold under the assumption that the vector ﬁelds hav e a C M smo othness, with M = 2 s for vector ﬁelds of step s . See Remark 5.10 for a more detailed discussion. In [MM ] we started to work in low regular ity hypo theses and we obtained a ball-b ox theo rem and the Poincar´ e inequality for L ips chit z con tinuous vector ﬁelds of step tw o with L ipschitz contin uo us commutators. W e used the maps (1.5), but several a sp ects o f the work [MM] are p eculia r o f the step tw o situation and until now it was not c le ar ho w to genera liz e those results to higher step vector ﬁelds. Recently , Brama n ti, Brandolini and Pedroni [BBP] have proved a doubling prop- erty and the Poincar´ e inequality for nonsmo oth H¨ ormander vector ﬁelds with an algebraic metho d. Infor ma lly sp eaking , they truncate the T aylor series of the coef- ﬁcient s of the vector ﬁelds a nd then they apply to the p olyno mial approximations the results in [NSW, LM] and [M ]. The pa p er [BBP] also in volv es a study o f the almost exp onential maps in (1.5). The results in [BBP] and in the present pap er were obtained indep endently and sim ultaneously . In this pap er we complete the result in [MM], na mely we prove a ball-b ox the- orem for general vector ﬁelds of ar bitrary s tep s , requiring ba sically that all the commutators inv olved in the H¨ ormander condition are Lipschitz contin uous. Our precise hypothes e s are s ta ted in Deﬁnition 2.1. W e improv e all pr e vious res ults in term of reg ularity , see Remark 5.1 0. As in [MM], w e use the a lmost expo nential maps (1.5 ), but we need to provide a very detailed study of such functions in the higher step case. The scheme of the pro of of o ur theorem is basically the Nagel, Stein and W ainge r ’s one, but there ar e some new to ols that should be emphasized. Namely , we obtain some non commutativ e calculus formulas develop ed in order to show that, given a commutator Y , the deriv a tive d dt exp ∗ ( tY ) can be precise ly wr itten a s a ﬁnite sum of hig her order commutators plus an inte gr al r emainder . This is done in Section 3. These results a re a pplied in Section 5 to the a lmost exp o ne ntial maps E in (1.5). Our main structure theorem is Theorem 5.8. As in [MM], par t of our computations will b e given for smo oth vector ﬁelds, namely the standard Euclidean regular ization X σ j of the vector ﬁelds X j . W e will keep everywhere under control all constants in order to make sure that they ar e s table as σ go es to 0. It is w ell known (see [LM, MM]) that the doubling prop erty and the Poincar´ e inequality follow immediately from Theorem 5.8. Observe a lso that our ball- box theorem can be useful in a ll situations where in tegra ls of the form R | f ( x ) − f ( y ) | w ( x, y ) dxdy nee d to b e estimated, for some w eight w . See for example [M] or [MoM]. As an application, in P rop osition 6 .2 we pr ov e a sub elliptic H¨ orma nder– t yp e estimate for nonsmo o th vector ﬁelds. W e b elieve that the results in Section 3 may b e useful in other, related, situations. Concerning the mac hinary dev elop ed in Section 3, it is w orth to men tio n the pap ers [RaS, Ra S2], where non co mmutative ca lculus fo rmulas are used in the pro of of a nonsmo oth v ersion of Chow’s The o rem for vector ﬁelds of step tw o. Geometric a nalysis for nonsmo oth vector ﬁelds star ted in the 80s with the pa- per s by F ranchi and Lanconelli [FL1, FL2], who prov ed the Poincar ´ e ineq uality for diagonal vector ﬁelds in R n of the for m X j = λ j ( x ) ∂ j , j = 1 , . . . , n . In the diago na l case completely diﬀerent techniques a re av ailable. In the rece nt pap er b y Sa wyer 1 The metho ds of [TW] hav e b een further exploited in a very recent paper by Street [Str]. 4 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI and Wheeden [SW], which proba bly contains the b est res ults to date o n diagonal vector ﬁelds, the re ader can ﬁnd a ric h bibliography on the sub ject. Plan of the pap er. In Sectio n 2 we intro duce notatio n. In Section 3 we pr ove our noncommutativ e calculus formulas a nd in Section 4 we prove a stabilit y prop erty of the “a lmost-maximality” condition (1.4). These to o ls are applied in Subsec tio n 5.1 to the maps E . In subsection 5.2 we brieﬂy discus s the “s caled version” of our maps E . Subsection 5.3 co nt ains the ball-b ox theorem. In Sectio n 6 we show some examples. Fina lly , Sectio n 7 con tains the smo oth approximation result fo r the original vector ﬁelds. Ac knowledgmen t. W e wish to express our gra titude to Ermanno L a nconelli, for his contin uous advice, encoura gement and interest in our work, pas t a nd prese nt. W e dedica te this paper to him with admiration. 2. Preliminaries and not a tion W e consider v e c tor ﬁelds X 1 , . . . , X m in R n . F or a ny ℓ ∈ N we deﬁne a wor d w = j 1 . . . j ℓ to b e an y ﬁnite or dered collection of ℓ letters, j k ∈ { 1 , . . . , m } , and we int ro duce the no tation X w = [ X j 1 , · · · , [ X j ℓ − 1 , X j ℓ ]] for commutators. Let | w | := ℓ be the length of X w . W e assume the H¨ ormander condition of step s , i.e. that { X w ( x ) : | w | ≤ s } generate all R n at any p oint x ∈ R n . Sometimes it will b e useful to have a diﬀerent notation b etw een a vector ﬁeld in R n and its a sso ciated vector function. In these situations we will write X w = f w · ∇ = P n k =1 f k w ∂ k . W e will also enum erate as Y 1 , . . . , Y q all the co mm utators X w with leng th | w | ≤ s and denote by ℓ i or ℓ ( Y i ) the length of Y i . W e iden tify a n o rdered n –tuple o f c ommut ators Y i 1 , . . . , Y i n by the index I = ( i 1 , . . . , i n ) ∈ S := { 1 , . . . , q } n . F or x, y ∈ R n , denote b y d ( x, y ) the control dista nc e , that is the inﬁm um of the r > 0 such that there is a Lipschitz pa th γ : [0 , 1 ] → R n with γ (0) = x , γ (1) = y and ˙ γ = P m j =1 b j X j ( γ ), for a.e. t ∈ [0 , 1]. The measur able functions b j m ust satisfy | b j ( t ) | ≤ r for almost any t . Corresp o nding ba lls will be indica ted as B ( x, r ). Denote also by  ( x, y ) the inﬁmum of the r > 0 such tha t there is a Lipschitz contin uous path γ : [0 , 1 ] → R n with γ (0) = x , γ (1) = y a nd γ s atisﬁes for a.e. t ∈ [0 , 1 ], ˙ γ = P q i =1 c i Y i ( γ ) for suitable mea surable functions c j with | c j ( t ) | ≤ r ℓ ( Y j ) . Corresp o nding balls will b e de no ted by B  ( x, r ). The deﬁnition of  is meaningful as so on as the v ector ﬁelds Y j are at least contin uous. Deﬁnition 2.1 (V ector ﬁelds o f class A s ) . Let X 1 , . . . , X m be vector ﬁelds in R n and let s ≥ 2. W e say that the vector ﬁelds X j are of class A s if they are of class C s − 2 , 1 lo c ( R n ) and for any word w with | w | = s − 1, and for every j, k ∈ { 1 , . . . , m } , (1) the deriv ative X k f w exists and it is contin uous; (2) the distributional deriv a tive X j ( X k f w ) exists and (2.1) X j ( X k f w ) ∈ L ∞ lo c ( R n ) . Recall that X j ∈ C s − 2 , 1 lo c means that all the E uclidean deriv atives of order at most s − 2 of the functions f 1 , . . . , f m are lo cally L ips chit z contin uous. In particular , all the commut ators X w , with | w | ≤ s − 1 ar e lo cally Lipschit z con tinuous in the Euclidean sense and by item (1) all commutators X w of length | w | = s ar e p o int wise deﬁned. If we knew that d deﬁnes the E uclidean top olo gy , condition (2) w ould equiv alent to the fact that X w is lo cally d -Lipschitz, if | w | = s , see [GN, FSSC]. NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 5 Let { X 1 , . . . , X m } b e in the class A s and assume that they satisfy the H¨ o rmander condition of step s . Fix once for a ll a pair o f b ounded connected op en sets Ω ′ ⊂⊂ Ω and denote K = Ω ′ . W e denote by D Euclidean deriv atives. If D = ∂ j 1 · · · ∂ j p for some j 1 , . . . , j p ∈ { 1 , . . . , n } , then | D | := p indica tes the order of D . It is understo o d that a deriv ative of o r der 0 is the iden tity . Introduce the p ositive constan t (2.2) L : = max 1 ≤ j ≤ m 0 ≤| D |≤ s − 2 sup Ω | D f j | + max j =1 ,...,m | D | = s − 1 ess sup Ω | D f j | + max k,j =1 ,.. .,m, | w | = s − 1 ess sup Ω | X k X j f w | . Remark 2.2. W e will pro ve in Section 5 a ball-b ox theorem for v ector ﬁelds of step s in the class A s . This impr ov es b oth the results in [TW] a nd [BBP] in term of re gularity . Indeed, in [TW] a C M regular ity , with M = 2 s must b e assumed (see Remar k 5.1 0). In [BBP] the authors assume that the v ector ﬁelds b elong to the Euclidea n Lipschitz space C s − 1 , 1 lo c ( R n ), which req uires the bo undedness o n the Euclidean gr adient ∇ f w of any commutator f w of length s , while we only need to control only the “hor izontal” gradient of f w . Appro xi mate com m utators. F or vector ﬁelds X j 1 , . . . , X j ℓ , and for τ > 0 , w e deﬁne, as in [NSW], [M] and [MM], C τ ( X j 1 ) := ex p( τ X j 1 ) , C τ ( X j 1 , X j 2 ) := ex p( − τ X j 2 ) exp( − τ X j 1 ) exp( τ X j 2 ) exp( τ X j 1 ) , . . . C τ ( X j 1 , . . . , X j ℓ ) := C τ ( X j 2 , . . . , X j ℓ ) − 1 exp( − τ X j 1 ) C τ ( X j 2 , . . . , X j ℓ ) exp( τ X j 1 ) . Then let (2.3) e tX j 1 j 2 ...j ℓ ∗ := exp ∗ ( tX j 1 j 2 ...j ℓ ) := ( C t 1 /ℓ ( X j 1 , . . . , X j ℓ ) , if t > 0 , C | t | 1 /ℓ ( X j 1 , . . . , X j ℓ ) − 1 , if t < 0 . By standard ODE theory , there is t 0 depe nding on ℓ, K , Ω, sup | f j | and ess sup |∇ f j | such that exp ∗ ( tX j 1 j 2 ...j ℓ ) x is w ell deﬁned for an y x ∈ K and | t | ≤ t 0 . The appr ox- imate co mmutators C t are quite natural (indeed, they make an app eara nce in the original pap er [NSW ]). Assuming tha t the vector ﬁelds are smo oth and using the Campb ell–Hausdorﬀ formula, we ha ve the forma l expa nsion C τ ( X j 1 , . . . , X j ℓ ) = ex p  τ ℓ X j 1 j 2 ...j ℓ + ∞ X k = ℓ +1 τ k R k  , where R k denotes a linear combination o f commutators of length k . See [NSW, Lemma 2 .2 1]. A study of these maps in the smo o th case based on this for m ula is carried out in [M]. Deﬁne, given I = ( i 1 , . . . , i n ) ∈ S , x ∈ K and h ∈ R n , with | h | ≤ C − 1 (2.4) E I ,x ( h ) := E I ( x, h ) := exp ∗ ( h 1 Y i 1 ) · · · exp ∗ ( h n Y i n )( x ) , k h k I := max j =1 ,...,n | h j | 1 /ℓ i j , Q I ( r ) := { h ∈ R n : k h k I < r } Λ( x, r ) := ma x K ∈ S | λ K ( x ) | r ℓ ( K ) , 6 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI where ℓ ( K ) = ℓ k 1 + · · · + ℓ k n , the determinants λ K are deﬁned in (1.3), and we have (2.5) ν := inf x ∈ Ω Λ( x, 1) > 0 . The low er b ound (2.5) will app ear many times in the fo llowing sectio ns. All the constants in our main theorem will dep end o n ν in (2.5) and on L in (2.2). In order to refer to the crucial condition (1.4), we giv e the follo wing deﬁnition Deﬁnition 2.3 ( η − max imal triple) . Let η ∈ ]0 , 1[, I ∈ S , x ∈ R n and r > 0. W e say that ( I , x, r ) is η − maximal, if w e ha ve | λ I ( x ) | r ℓ ( I ) > η Λ( x, r ) . Regularized v ector ﬁelds . Here we describe our pro c edure of smo othing of the vector ﬁelds X j of step s . F or for any function f , let f ( σ ) ( x ) = R f ( x − σy ) ϕ ( y ) dy , where ϕ ∈ C ∞ 0 is a standard nonnega tive averaging kernel supp or ted in the unit ball. Deﬁne (2.6) X σ j := n X k =1 ( f k j ) ( σ ) ∂ k and X σ j 1 ...j ℓ := [ X σ j 1 , · · · , [ X σ j ℓ − 1 , X σ j ℓ ]] =: n X k =1 ( f k j 1 ...j ℓ ) σ ∂ k , for any word j 1 . . . j ℓ , with 2 ≤ ℓ ≤ s . (Obser ve that f σ w 6 = f ( σ ) w , if | w | > 1. See Section 7) Then: Prop ositio n 2. 4. L et X 1 , . . . , X m b e ve ctor ﬁelds in the class A s . Then the fol- lowing hold. (1) F or any ℓ = 1 , . . . , s , for any wor d w of lenght | w | ≤ ℓ , (2.7) X σ w → X w , as σ → 0 , uniformly on K . In p articular, for any m ulti-index I = ( i 1 , . . . , i n ) ∈ S , we have λ σ I := det( Y σ i 1 , . . . , Y σ i n ) − → λ I , uniformly on K , as σ → 0 . (2) Ther e is σ 0 > 0 such that, if | w | = s and k = 1 , . . . , m , then (2.8) sup 0 <σ<σ 0 sup x ∈ K | X σ k f σ w | ≤ C , with C dep ending on L in (2.2) . (3) Ther e is r 0 dep ending on K, Ω and the c onstant in (2.2) su ch that the fol lowing holds. L et x ∈ K , r < r 0 and b ∈ L ∞ ([0 , 1] , R m ) with k b j k L ∞ ≤ r for all j . Then ther e is a unique ϕ ∈ L ip([0 , 1] , R n ) , a.e. solution of ˙ ϕ = P j b j X j ( ϕ ) , with ϕ (0) = x . D enote also by ϕ σ ∈ Lip([0 , 1] , R n ) , the a.e. solution of the ˙ ϕ σ = P j b j X σ j ( ϕ σ ) , with ϕ σ (0) = x . Then (2.9) ϕ σ (1) → ϕ (1) , as σ → 0 , uniformly in x ∈ K . As a c onse quenc e, for any I ∈ S , uniformly in x ∈ K , | h | ≤ C − 1 , (2.10) E σ I ( x, h ) := exp ∗ ( h 1 Y σ i 1 ) · · · exp ∗ ( h n Y σ i n ) → E I ( x, h ) . Pr o of. The pro ofs of items 1 and 2 a re given in details in Section 7. Item 3 follows from standard prop erties of ODE.  NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 7 Remark 2.5. The approximation result contained in Prop ositio n 2.4 is crucial for o ur subsequent arguments. Note that the class A s requires a co ntrol o n the Euclidean gr adients of all co mm utators of length strictly les s than s . How ever, it is natural to conjecture that a c ontrol only along the horizontal directions could be suﬃcien t to ensure our ma in structure theorem in Section 5. Unfortunately , it seems quite diﬃcult to get an approximation theorem a s Pr op osition 2.4 for a more general class than A s . On the other side, w ork ing without molliﬁed vector ﬁelds seems to rise some non trivial new issues which we plan to face in a further study . Some mo re notation. Our notation for consta nts are the following: C , C 0 de- note large a bsolute constants, ε 0 , r 0 , t 0 , C − 1 or C − 1 0 denote p ositive small absolute constants. “Abso lute consta nt s” may dep end on the dimension n , the n umber m of the ﬁelds, their step s , the constan t L in (2.2) and p o ssibly the co ns tant ν in (2.5). W e also use the nota tion ε η (or C η ) to denote a sma ll (or a la rge) cons ta nt depe nding a lso on η . The co ns tants σ 0 or e σ a pp e aring in the regula r izing para meter σ may also depend on the E uclidean contin uity mo duli of the v ector ﬁelds f w , with | w | = s, whic h ar e not included in L. Comp osition of functions are shortened a s follows: f g stands for f ◦ g . The notation u is alwa ys used for functions of the for m exp( t 1 Z 1 ) · · · exp( t ν Z ν ) for some t j ∈ R , ν ≥ 1, Z j ∈ { X 1 , . . . , X m } . 3. Appro xima te exponential s of commut a tors The main result of this section is Theorem 3.6 in Subsection 3.3, where we prov e an exac t form ula for the deriv ative d dt u ( e tX w ∗ ( x )), wher e X w is a commut ator of length | w | ≤ s , while e ∗ is the approximate ex p o nent ial deﬁned in (2.3). All this sectio n is written for smo oth v ector ﬁelds, namely the molliﬁed X σ j , but a ll constants are app earing in our computatio ns are stable as σ go es to 0. W e drop everywhere in this section the sup ersc r ipt σ . W e will s how that (3.1) d dt e tX w ∗ ( x ) = X w ( e tX w ∗ ( x )) + higher order commutators + integral remainder. The integral rema inder is rather complica ted, but we do not need its e x act form. In order to understand what w e need to co mpute the deriv ative in (3.1), let us try to calculate for example the der iv ative d dt u ( e tX e tY x ), where X , Y ∈ {± X 1 , . . . , ± X m } and u denotes the iden tit y function in R n . Since X and Y ar e C 1 , we hav e d dt u ( e tX e tY x ) = ( X u )( e tX e tY x ) + Y ( ue tX )( e tY x ) . In order to compare the terms in the right-hand side, we ma y write Y ( ue tX )( e tY x ) = Y u ( e tX e tY x ) + Z t 0 d dτ Y ( ue τ X )( e − τ X e tX e tY x ) dτ . Lemma 3.1 b elow shows that the deriv ative inside the integral can b e wr itten in an exact fo rm in term o f the co mmu tator of X and Y . The purp os e of the following Subsection 3 .1 is to es tablish a forma lism to study in a pr e c ise way more g eneral, related, integral expre ssions. Lemma 3. 1. L et Z, X b e smo oth ve ct or ﬁ elds. Then, (3.2) d dt Z ( ue − tX )( e tX y ) = [ X , Z ]( ue − tX )( e tX y ) . 8 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI Pr o of. The lemma is kno wn but we provide a pr o of for completeness. Obser ve ﬁrs t that d dt Z ( ue − tX )( e tX x ) = d dτ Z ( ue − tX )( e τ X x )    τ = t + d dτ Z ( ue − τ X )( e tX x )    τ = t =: (1) + (2) . Obviously , (1) = X Z ( ue − tX )( e tX x ). W r ite now (2) as follows d dt Z ( ue − tX )( ξ )   ξ = e tX x = d dt Z j ( ξ ) ∂ ξ j ( ue − tX )( ξ )   ξ = e tX x = Z j ( ξ ) ∂ ξ j d dt ( ue − tX )( ξ )   ξ = e tX x The pro of of formula (3.2) will b e co nc luded as so o n as w e prov e that (3.3) d dt ( ue − tX )( ξ ) = − X ( ue − tX )( ξ ) . T o prov e (3.3), start from the ident ity u ( η ) = u ( e − tX e tX η ), for small t . Diﬀeren ti- ating, 0 = d dt ( u ( e − tX e tX η )) = d dτ ( u ( e − tX e τ X η )) | τ = t + d dτ ( u ( e − τ X e tX η )) | τ = t = Z ( ue − tX )( e tX η ) + d dτ ( u ( e − τ X e tX η )) | τ = t . Then, (3.3) is prov e d by le tting e tX η = ξ .  3.1. Notation for in tegral remainders. Let λ ∈ N , p ∈ { 2 , . . . , s + 1 } . W e denote, for y ∈ K , and t ∈ [0 , t 0 ], t 0 small enough, (3.4) O p ( t λ , u, y ) = N X i =1 Z t 0 ω i ( t, τ ) X w i ( uϕ − 1 i e − τ Z i )( e τ Z i ϕ i y ) dτ , where N is a suita ble integer a nd u is the identit y map or u = exp( tY 1 ) · · · exp( tY µ ) , for some in teger µ and suitable vector ﬁelds Y j ∈ {± X 1 , . . . , ± X m } . Here X w i actually stands for a molliﬁed X σ w i , but we drop the sup er s cript for simplicity . T o describ e the generic term of the sum ab ov e, w e drop the dependenc e on i : (3.5) ( R ) := Z t 0 ω ( t, τ ) X w ( uϕ − 1 e − τ X )( e τ X ϕy ) dτ . Here X w is a commutator o f length | w | = p and X ∈ { ± X j } . Moreover, for a ny t < t 0 , the function ω ( t, τ ) is a polyno mial, homog eneous of degr ee λ − 1 in all v ariables ( t, τ ), such that ω ( t, τ ) > 0 if 0 < τ < t. Thus (3.6) Z t 0 ω ( t, τ ) dτ = bt λ for any t > 0 , for a suitable constant b ∈ R . The map ϕ is the iden tity map or ϕ = exp( tZ 1 ) · · · exp( tZ ν ) for some ν ∈ N , where Z j ∈ {± X 1 , . . . , ± X m } . Remark 3.2. All the num b ers N , µ, ν, b, app earing in the computations of this pap er will b e b ounded by abso lute constants. In o rder to ex pla in how this formalism works, we give the main pr op erties o f o ur int egra l r emainders. NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 9 Prop ositio n 3.3. A r emainder of t he form (3.4) satisﬁes for every α ∈ N (3.7) t α O p ( t λ , u, y ) = O p ( t α + λ , u, y ) for al l y ∈ K t ∈ [0 , t 0 ] . Mor e over, for p ≤ s + 1 , (3.8) | O p ( t λ , u, y ) | ≤ C t λ for al l y ∈ K t ∈ [0 , t 0 ] , wher e t 0 and C dep end on the c onstant L in (2.2) and on the numb ers N , µ, ν, b app e aring in the sum (3.4) . F urthermor e, if ℓ ( Z ) = 1 and p ≤ s + 1 , (3.9) O p ( t λ , ue tZ , y ) = O p ( t λ , u, e tZ y ) . Final ly, if p ≤ s , we may write, for suitable c onst ants c w , | w | = p, (3.10) O p ( t λ , u, y ) = X | w | = p c w t λ X w u ( y ) + O p +1 ( t λ +1 , u, y ) . Pr o of. The pro of of (3 .7) and (3.9) ar e ra ther eas y and we leav e them to the rea der. So w e start with the pro of of (3 .8 ). A typical ter m in O p ( t λ , u, y ) has the form (3.11) Z t 0 ω ( t, τ ) Y ( uϕ − 1 e − τ Z )( e τ Z ϕy ) dτ , with ℓ ( Y ) = p ≤ s + 1. Thus, b y P r op osition 2.4, we hav e   Y ( uϕ − 1 e − τ Z )( e τ Z ϕy )   ≤ C (observe that we need (2.8), if p = s + 1 ). Therefore, (3.8) follows fro m the prop erty (3 .6) of ω . Finally we establish the key pr op erty (3.10). Star t fro m the generic term of O p ( t λ , u, y ) in (3.1 1), where we intro duce the notation g k := e tZ k · · · e tZ ν , for k = 1 , . . . , ν and g ν +1 denotes the identit y map. Recall also that ℓ ( Y ) ≤ s . Therefore , we get Z t 0 ω ( t, τ ) Y ( ue − tZ ν · · · e − tZ 1 e − τ X )  e τ X e tZ 1 · · · e tZ ν y  dτ = Z t 0 ω ( t, τ ) Y ( ug − 1 1 e − τ X )  e τ X g 1 y  dτ = Z t 0 ω ( t, τ ) Y u ( y ) dτ + Z t 0 ω ( t, τ ) n Y ( ug − 1 1 e − τ X )  e τ X g 1 y  − Y u ( y ) o dτ = bY u ( y ) t λ + Z t 0 ω ( t, τ ) n Y ( ug − 1 1 e − τ X )  e τ X g 1 y  − Y ( ug − 1 1 )  g 1 y  o dτ + ν X k =1 Z t 0 ω ( t, τ ) n Y ( ug − 1 k )  g k y  − Y ( ug − 1 k +1 )  g k +1 y  o dτ . Recall that Y has length p ≤ s . The pe n ultimate term ca n be written a s Z t 0 ω ( t, τ ) n Y ( ug − 1 1 e − τ X )  e τ X g 1 y  − Y u ( ug − 1 1 )  g 1 y  o dτ = Z t 0 dτ ω ( t, τ ) Z τ 0 dσ d dσ Y ( ug − 1 1 e − σX )  e σX g 1 y ) = Z t 0 dσ n Z t σ ω ( t, τ ) dτ o [ X , Y ]( ug − 1 1 e − σX )  e σX g 1 y ) . 10 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI Observe that, as require d, the function e ω ( t, σ ) := R t σ ω ( t, τ ) dτ sa tisﬁes Z t 0 e ω ( t, σ ) dσ = Z t 0 dτ ω ( t, τ ) Z τ 0 dσ = Z t 0 dτ τ ω ( t, τ ) = e bt λ +1 , bec ause ( t, τ ) 7→ τ ω ( t, τ ) is homogeneo us o f degree λ . The k − th term in the sum has the for m Z t 0 dτ ω ( t, τ ) Z t 0 dσ d dσ Y ( ug − 1 k +1 e − σZ k )  e σZ k g k +1 y  = Z t 0 dσ e ω ( t, σ )[ Z k , Y ]( ug − 1 k +1 e − σZ k )  e σZ k g k +1 y  , where e ω ( t, σ ) := R t 0 ω ( t, τ ) dτ = b t λ has the co rrect form. The pro o f is concluded.  3.2. Higher order non commuta tiv e calculus form ul as. In order to prove Theorem 3 .5, we ﬁrst need to iterate for m ula (3.2). Start from smo oth vector ﬁelds X := X σ j of leng th one and Z := X w of leng th ℓ ( Z ) := | w | . Diﬀerentiating ident ity (3.2) w e get, by the T aylor formula Z ( ue − tX )( e tX y ) = r X k =0 t k k ! ad k X Z u ( y ) + Z t 0 ( t − τ ) r r ! ad r +1 X Z ( ue − τ X )( e τ X y ) dτ , where we intro duced the notatio n: ad X Z = [ X , Z ], ad 2 X Z = [ X , [ X , Z ]], etcetera. In other w ords, (3.12) Z ( ue tX )( y ) − Z u ( e tX y ) = r X k =1 t k k ! ad k − X Z u ( e tX y ) + Z t 0 ( t − τ ) r r ! ad r +1 − X Z ( ue τ X )( e − τ X e tX y ) dτ = r X k =1 t k k ! ad k − X Z u ( e tX y ) + O r +1+ ℓ ( Z ) ( t r +1 , u, e tX y ) . If we take r = s − ℓ ( Z ), w e may write (3.13) Z ( ue tX )( y ) − Z u ( e tX y ) = s − ℓ ( Z ) X k =1 t k k ! ad k − X Y u ( e tX y ) + O s +1 ( t s − ℓ ( Z )+1 , u, e tX y ) . In view of (3.8), this order of expansion is the highest which ensures that the remainder can be estimated with C t s − ℓ ( Z )+1 , with a control on C in term of the constant in (2.2), as so o n as y ∈ K and | t | ≤ C − 1 . Next, we see k for a family of higher o rder formulas, in which we change e tX with an a pproximate expo nential exp ∗ ( tX w ). The co eﬃcients of the expansio n (3.1 2) are all explicit but we do not nee d such an accur acy in the higher o r der formulae. T o explain what suﬃces for our purpo ses, s tart with the case of co mm utators of length tw o. Let C t = C t ( X, Y ) = e − tY e − tX e tY e tX , where X := X σ j and Y := X σ k are molliﬁed vector ﬁelds with length o ne. Let Z := X σ v be a smo o th commutator with length ℓ ( Z ) := | v | . Assume ﬁrst that ℓ ( Z ) = s . Then, iterating (3 .13) we can write ( F 2 , 1 ) Z ( uC t )( x ) − Z u ( C t x ) = O s +1 ( t, u, C t x ) . NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 11 If instead ℓ ( Z ) = s − 1 , then so me elementary computations based on (3.12) give ( F 2 , 2 ) Z ( uC t )( x ) − Z u ( C t x ) = X k 1 + k 2 + k 3 + k 4 =1 t k 1 + ··· k 4 k 1 ! · · · k 4 ! ad k 4 Y ad k 3 X ad k 2 − Y ad k 1 − X Z u ( C t x ) + O 2+ ℓ ( Z ) ( t 2 , u, C t x ) = O 2+ ℓ ( Z ) ( t 2 , u, C t x ) = O s +1 ( t 2 , u, C t x ) . Next, if ℓ ( Z ) = s − 2 , (this can happ en only if s ≥ 3), then we must expand more. Namely , we ha ve ( F 2 , 3 ) Z ( uC t )( x ) − Z u ( C t x ) = 2 X k 1 + k 2 + k 3 + k 4 =1 t k 1 + ··· k 4 k 1 ! · · · k 4 ! ad k 4 Y ad k 3 X ad k 2 − Y ad k 1 − X Z u ( C t x ) + O 3+ ℓ ( Z ) ( t 3 , u, C t x ) = t 2 [ Z, [ X , Y ]] u ( C t x ) + O 3+ ℓ ( Z ) ( t 3 , u, C t x ) = t 2 [ Z, [ X , Y ]] u ( C t x ) + O s +1 ( t 3 , u, C t x ) . Finally , if ℓ ( Z ) ≤ s − 3 (this requires at least s ≥ 4), we must expand even more: ( F 2 , 4 ) Z ( uC t )( x ) − Z u ( C t x ) = 3 X k 1 + k 2 + k 3 + k 4 =1 t k 1 + ··· k 4 k 1 ! · · · k 4 ! ad k 4 Y ad k 3 X ad k 2 − Y ad k 1 − X Z u ( C t x ) + O 4+ ℓ ( Z ) ( t 4 , u, C t x ) = t 2 [ Z, [ X , Y ]] u ( C t x ) + t 3 n 1 2 ad 2 Y ad X Z u ( C t x ) − 1 2 ad Y ad 2 X Z u ( C t x ) − 1 2 ad 2 X ad Y Z u ( C t x ) + 1 2 ad X ad 2 Y Z u ( C t x ) − ad Y ad X ad Y Z u ( C t x ) + a d X ad Y ad X Z u ( C t x ) o + O 4+ ℓ ( Z ) ( t 4 , u, C t x ) . Observe that if ℓ ( Z ) = s − 3 , then O 4+ ℓ ( Z ) ( t 4 , u, C t x ) = O s +1 ( t 4 , u, C t x ). If instead ℓ ( Z ) < s − 3, then we can expand up to the or der O s +1 ( t s +1 − ℓ ( Z ) , u, C t x ) by means of (3.1 0). W e have s tarted to put tags of the form ( F k,λ ) in our formulae. The num b er k indicates the length o f the co mmutator we are appr oximating, while the num b er λ denotes the power of t which con trols the remainder. Note that in ( F 2 , 4 ), the cur ly bracket changes sign if we exchange X with Y . Brieﬂy , we can write Z ( uC t )( x ) = Z u ( C t x ) + t 2 [ Z, [ X , Y ]] u ( C t x ) + t 3 X | w | =3+ ℓ ( Z ) c w X w u ( C t x ) + O 4+ ℓ ( Z ) ( t 4 , u, C t x ) . 12 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI for all x ∈ K , t ∈ [0 , C − 1 ], where the co eﬃcients c w are determined in ( F 2 , 4 ). The corres p o nding formula for C − 1 t ( X, Y ) is Z ( uC − 1 t )( x ) = Z u ( C − 1 t x ) − t 2 [ Z, [ X , Y ]] u ( C − 1 t x ) + t 3 X | w | =3+ ℓ ( Z ) e c w X w u ( C − 1 t x ) + O 4+ ℓ ( Z ) ( t 4 , u, C − 1 t x ) , where, since C − 1 t ( X, Y ) = C t ( Y , X ), the coeﬃcie nts e c w are obtained again from ( F 2 , 4 ), by changing X a nd Y . W e are not in terested in the explicit knowledge of all the co eﬃcients c w and e c w . W e only need to observe the following r e mark able cancellation prop erty: X | w | =3+ ℓ ( Z ) ( c w + e c w ) X w ( x ) = 0 for a ny x ∈ K . Next we generalize form ulae ( F 2 ,λ ) ab ov e . The ge ne r al statemen t we prov e tells that this ca nce llation p ersists when the length of the commutator w e are appr oxi- mating with C t is three or more. Theorem 3.4. F or any ℓ ∈ { 2 , . . . , s } , x ∈ K , t ∈ [0 , C − 1 ] , the fol lowing family ( F ℓ, 1 , F ℓ, 2 , . . . , F ℓ,s ) of formulas holds. F or mulas F ℓ,k . F or any C t = C t ( X w 1 , . . . , X w ℓ ) , k = 1 , . . . , ℓ and for any c ommu- tator Z su ch that ℓ ( Z ) + k ≤ s + 1 , we have Z ( uC t )( x ) − Z u ( C t x ) = O k + ℓ ( Z ) ( t k , u, C t x ) Z ( uC − 1 t )( y ) − Z u ( C − 1 t y ) = O k + ℓ ( Z ) ( t k , u, C − 1 t x ) . F or mula F ℓ,ℓ +1 . L et ℓ ≥ 2 b e s uch t hat ℓ + 1 ≤ s . Then, for any C t ( X w 1 , . . . , X w ℓ ) and Z such that ℓ + 1 + ℓ ( Z ) ≤ s + 1 , Z ( uC t )( x ) − Z u ( C t x ) = t ℓ [ Z, X w ] u ( C t x ) + O ℓ +1+ ℓ ( Z ) ( t ℓ +1 , u, C t x ) , Z ( uC − 1 t )( y ) − Z u ( C − 1 t y ) = − t ℓ [ Z, X w ] u ( C − 1 t x ) + O ℓ +1+ ℓ ( Z ) ( t ℓ +1 , u, C − 1 t x ) . F or mula F ℓ,ℓ +2 . If s ≥ 4 , let ℓ ≥ 2 b e su ch that ℓ + 2 ≤ s . Then, for any C t ( X w 1 , . . . , X w ℓ ) and Z such that ℓ + 2 + ℓ ( Z ) ≤ s + 1 , ther e ar e numb ers c v , e c v , with | v | = ℓ + ℓ ( Z ) + 1 , su ch t hat (3.14) Z ( uC t )( x ) − Z u ( C t x ) = t ℓ [ Z, X w ] u ( C t x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 c v X v u ( C t x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C t x ) Z ( uC − 1 t )( x ) − Z u ( C − 1 t x ) = − t ℓ [ Z, X w ] u ( C − 1 t x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 e c v X v u ( C − 1 t x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C − 1 t x ) . Cancellation pr op erty . If s ≥ 4 , let ℓ ≥ 2 b e su ch that ℓ +2 ≤ s . If formulae F ℓ, 1 . . . , F ℓ,ℓ +2 hold, then, for any C t ( X w 1 , . . . , X w ℓ ) and Z such that ℓ + 2 + ℓ ( Z ) ≤ s + 1 , the c o eﬃcients c w , e c w in (3.14) s atisfy (3.15) X | w | = ℓ + ℓ ( Z )+1 ( c w + e c w ) X w ( x ) = 0 for any x ∈ K. NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 13 F or mulae F ℓ,r , with ℓ + 3 ≤ r ≤ s . L et s ≥ 5 and assume t hat ℓ ≥ 2 and r ar e such that ℓ + 3 ≤ r ≤ s . Then, for any C t ( X w 1 , . . . , X w ℓ ) and Z with r + ℓ ( Z ) ≤ s + 1 , ther e ar e c v , e c v such that Z ( uC t )( x ) − Z u ( C t x ) = t ℓ [ Z, X w ] u ( C t x ) + r − 1+ ℓ ( Z ) X | v | = ℓ + ℓ ( Z )+1 t | v | − ℓ ( Z ) c v X v u ( C t x ) + O r + ℓ ( Z ) ( t r , u, C t x ) , Z ( uC − 1 t )( x ) − Z u ( C − 1 t x ) = − t ℓ [ Z, X w ] u ( C − 1 t x ) + r − 1+ ℓ ( Z ) X | v | = ℓ + ℓ ( Z )+1 t | v | − ℓ ( Z ) e c v X v u ( C − 1 t x ) + O r + ℓ ( Z ) ( t r , u, C − 1 t x ) . Observe aga in that in the form ula F ℓ,k , ℓ is the length of the comm utator which deﬁnes C t , while k is the degree of the p ower o f t whic h cont rols the remainder. Pr o of of The or em 3.4. If ℓ = 2, w e hav e alr eady prov ed the statement. See for- m ulae ( F 2 , 1 ), ( F 2 , 2 ), ( F 2 , 3 ) and ( F 2 , 4 ), p. 11, and recall pr op erty (3.10) of the remainders. The pro o f will be acc omplished in t wo s teps. Step 1. Let s ≥ 4 and ℓ ≥ 2 be such that ℓ +2 ≤ s . Assume that F ℓ, 1 , F ℓ, 2 , . . . , F ℓ,ℓ +2 hold. Then the cancellation (3.15) ho lds fo r an y C t ( X j 1 , . . . , X j ℓ ) and W s uch that ℓ + 2 + ℓ ( W ) ≤ s + 1. Step 2. Assume tha t for some ℓ ≥ 2, all formulae F ℓ,k hold, for k = 1 , . . . , s . Then formula F ℓ +1 ,k holds, for any k = 1 , . . . , s . Pr o of of Step 1. Let C t = C t ( X w 1 , . . . , X w ℓ ) and Z such that ℓ ( Z ) + ℓ + 2 ≤ s + 1. Applying t wice formula F ℓ,ℓ +2 , we obtain, (3.16) Z u ( x ) = Z ( uC − 1 t C t )( x ) = Z ( uC − 1 t )( C t x ) + t ℓ [ Z, X w ]( uC − 1 t )( C t x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 c v X v ( uC − 1 t )( C t x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , uC − 1 t , C t x ) = Z u ( x ) − t ℓ [ Z, X w ] u ( x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 e c v X v u ( x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, x ) + t ℓ [ Z, X w ]( uC − 1 t )( C t x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 c v X v ( uC − 1 t )( C t x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , uC − 1 t , C t x ) . Observe ﬁrs t that pro per ty (3.9) g ives O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , uC − 1 t , C t x ) = O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, x ) . Later on, we will tacitly use such prop e rty ma n y times. Rec all that ℓ ≥ 2 . By means of F ℓ, 2 and F ℓ, 1 , resp e ctively , w e obtain [ Z, X w ]( uC − 1 t )( C t x ) = [ Z , X w ] u ( x ) + O 2+ ℓ ( Z )+ ℓ ( t 2 , u, x ) and X v ( uC − 1 t )( C t x ) = X v u ( x ) + O 2+ ℓ + ℓ ( Z ) ( t, u, x ) . 14 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI Inserting this information in to (3.1 6) gives, after algebraic simpliﬁcations 0 = t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 e c v X v u ( x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, x ) + t ℓ O 2+ ℓ + ℓ ( Z ) ( t 2 , u, x ) + t ℓ +1  X | v | = ℓ + ℓ ( Z )+1 c v X v u ( x ) + O ℓ +2+ ℓ ( Z ) ( t, u, x )  . T o conc lude the pro o f, recall (3.7), divide b y t ℓ +1 and let t → 0. Pr o of of Step 2. Let ℓ + 2 ≤ s . W e prov e formula F ℓ +1 ,ℓ +2 , which is the mos t signiﬁcant among a ll formulae F ℓ +1 , 1 , . . . , F ℓ +1 ,s . I ndee d, once F ℓ +1 ,ℓ +2 is prov ed, if ℓ + 3 ≤ s , then for mu lae F ℓ +1 ,ℓ +3 , . . . , F ℓ +1 ,s follow ea sily from F ℓ +1 ,ℓ +2 and from prop erty (3 .10). On the other side, the lower order formulae F ℓ +1 ,k with k < ℓ + 2 are ea sier (just truncate at the corre ct order all the expa nsions in the pro of b elow). T o star t, recall that we are ass uming that F ℓ, 1 , . . . , F ℓ,s hold. Let, for t > 0 (3.17) C t : = C t ( X w 1 , . . . , X w ℓ ) and C 0 t : = C t ( X, X w 1 , . . . , X w ℓ ) = C − 1 t e − tX C t e tX , where X = X w 0 . Let Z b e a co mm utator with ℓ ( Z ) + ℓ + 2 ≤ s + 1. In the subs e q uent form ulae, we expand everywhere up to a remainder of the fo r m O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ). By (3 .12), Z ( uC 0 t )( x ) = Z ( uC − 1 t e − tX C t )( e tX x ) + t [ − X , Z ]( uC − 1 t e − tX C t )( e tX x ) + ℓ +1 X k =2 t k k ! ad k − X Z ( uC − 1 t e − tX C t )( e tX x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , uC − 1 t e − tX C t , e tX x ) =: ( A ) + ( B ) + ( C ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) , where we also used (3.9). Next we use F ℓ,ℓ +2 in ( A ) . ( A ) = Z ( uC − 1 t e − tX )( C t e tX x ) + t ℓ [ Z, X w ]( uC − 1 t e − tX )( C t e tX x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 c v X v ( uC − 1 t e − tX )( C t e tX x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) =: ( A 1 ) + ( A 2 ) + ( A 3 ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) . W e ﬁr st treat ( A 1 ). By (3.12), (3.18) ( A 1 ) = Z ( uC − 1 t )( e − tX C t e tX x ) + t [ X , Z ]( uC − 1 t )( e − tX C t e tX x ) + ℓ +1 X k =2 t k k ! ad k X Z ( uC − 1 t )( e − tX C t e tX x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) . Consider now the v ar ious terms in ( A 1 ). Firs t use F ℓ,ℓ +2 to get Z ( uC − 1 t )( e − tX C t e tX x ) = Z u ( C 0 t x ) − t ℓ [ Z, X w ] u ( C 0 t x ) + t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 e c v X v u ( C 0 t x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) . NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 15 Moreov er, b y F ℓ,ℓ +1 we g et t [ X , Z ]( uC − 1 t )( e − tX C t e tX x ) = t n [ X , Z ] u ( C 0 t x ) − t ℓ [[ X , Z ] , X w ] u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +1 , u, C 0 t x ) o . Finally , we use F ℓ,ℓ +2 − k in the k − th ter m of the sum in (3.18). Obser ve that ℓ + 2 − k ∈ { 1 , . . . , ℓ } so that w e use only remainders. t k k ! ad k X Z ( uC − 1 t )( e − tX C t e tX x ) = t k k ! n ad k X Z u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 − k , u, C 0 t x ) o . Therefore ( A 1 ) = Z u ( C 0 t x ) − t ℓ [ Z, X w ] u ( C 0 t x ) + t ℓ +1 X | v | = ℓ +1+ ℓ ( Z ) e c v X v u ( C 0 t x ) + t [ X , Z ] u ( C 0 t x ) − t ℓ +1 [[ X , Z ] , X w ] u ( C 0 t x ) + ℓ +1 X k =2 t k k ! ad k X Z u ( C 0 t x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) . Next we consider ( A 2 ). F orm ula (3.12) gives ( A 2 ) = t ℓ [ Z, X w ]( uC − 1 t )( e − tX C t e tX x ) + t ℓ +1 [ X , [ Z, X w ]]( uC − 1 t )( e − tX C t e tX x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . Since ℓ ≥ 2 , form ulas F ℓ, 2 and F ℓ, 1 give re s pe c tively t ℓ [ Z, X w ]( uC − 1 t )( e − tX C t e tX x ) = t ℓ [ Z, X w ] u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) , t ℓ +1 [ X , [ Z, X w ]]( uC − 1 t )( e − tX C t e tX x ) = t ℓ +1 [ X , [ Z, X w ]] u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) , so that ( A 2 ) = t ℓ [ Z, X w ] u ( C 0 t x ) + t ℓ +1 [ X , [ Z, X w ]] u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . T o handle ( A 3 ), observe that a rep ea ted application of (3.12) gives ( A 3 ) = t ℓ +1 X | v | = ℓ + ℓ ( Z )+1 c v X v u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . Next we study ( B ). Start with formula F ℓ,ℓ +1 : ( B ) = t [ − X , Z ]( uC − 1 t e − tX )( C t e tX x ) + t ℓ +1 [[ − X , Z ] , X w ]( uC − 1 t e − tX )( C t e tX x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) = t [ − X , Z ]( uC − 1 t e − tX )( C t e tX x ) + t ℓ +1 [[ − X , Z ] , X w ] u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) =: ( B 1 ) + ( B 2 ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) , W e ﬁr st consider ( B 1 ). In view of (3 .12), we o btain ( B 1 ) = t [ − X , Z ]( uC − 1 t )( e − tX C t e tX x ) − t ℓ X k =1 t k k ! ad k X [ X , Z ]( uC − 1 t )( e − tX C t e tX x ) + tO ℓ + ℓ ( Z ) +2 ( t ℓ +1 , u, C 0 t x ) . 16 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI But b y F ℓ,ℓ +1 we g et t [ − X , Z ]( uC − 1 t )( e − tX C t e tX x ) = t [ − X , Z ] u ( C 0 t x ) − t ℓ +1 [[ − X , Z ] , X w ] u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) , while for an y k = 1 , . . . , ℓ , formula F ℓ,ℓ +1 − k gives t t k k ! ad k X ([ X, Z ])( uC − 1 t )( e − tX C t e tX x ) = t k +1 k ! ad k +1 X Z u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . Therefore ( B 1 ) = t ℓ +1 [[ X , Z ] , X w ] u ( C 0 t x ) − ℓ X k =0 t k +1 k ! ad k +1 X Z u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . Observe that t ℓ +1 [[ X , Z ] , X w ] u ( C 0 t x ) = − ( B 2 ). Finally we consider ( C ). In the k − th term of the sum us e form ula F ℓ,ℓ +2 − k . Then ( C ) = ℓ +1 X k =2 t k k ! ad k − X Z ( uC − 1 t e − tX )( C t e tX x ) + O ℓ +2+ ℓ ( Z ) ( t ℓ +2 , u, C 0 t x ) = by (3.12) = ℓ +1 X k =2 t k k ! n ℓ +1 − k X h =0 t h h ! ad h X ad k − X Z ( uC − 1 t )( e − tX C t e tX x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 − k , u, C 0 t x ) o + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) = ℓ +1 X k =2 ℓ +1 − k X h =0 t k + h k ! h ! ( − 1) k ad k + h X Z u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . Collecting together all the previous computations and making so me simpliﬁca- tions (in particular w e need here the cancellatio n pro per ty (3 .15)), w e get Z ( uC 0 t )( x ) = ( A 1 ) + ( A 2 ) + ( A 3 ) + ( B 1 ) + ( B 2 ) + ( C ) = Z u ( C 0 t x ) + t ℓ +1  − [[ X , Z ] , X w ] u ( C 0 t x ) + [ X , [ Z , X w ]] u ( C 0 t x )  + ℓ +1 X k =1 t k k ! ad k X Z u ( C 0 t x ) − ℓ X k =0 t k +1 k ! ad k +1 X Z u ( C 0 t x ) + ℓ +1 X k =2 ℓ +1 − k X h =0 t k + h k ! h ! ( − 1) k ad k + h X Z u ( C 0 t x ) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) =: Z u ( C 0 t x ) + t ℓ +1 {· · · } + (1) + (2) + (3) + O ℓ + ℓ ( Z ) +2 ( t ℓ +2 , u, C 0 t x ) . The Jacobi ident ity gives t ℓ +1 {· · · } = t ℓ +1 [ Z, [ X , X w ]] , which is the desir ed term. NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 17 Ultimately w e need to consider all the ter ms with sums. C ha nging k and h in (2), w e may wr ite (2) + (3) = ℓ +1 X k =1 ℓ +1 − k X h =0 ( − 1) k t k + h k ! h ! ad k + h X Z u ( C 0 t x ) and (1) + Z u ( C 0 t x ) = ℓ +1 X h =0 t h h ! ad h X Z u ( C 0 t x ) . Therefore, (1) + (2) + (3) + Z u ( C 0 t x ) = ℓ +1 X k =0 ℓ +1 − k X h =0 ( − 1) k t k + h k ! h ! ad k + h X Z u ( C 0 t x ) = ℓ +1 X s =0  X k + h = s k,h ≥ 0 ( − 1) k k ! h !  t s ad s X Z u ( C 0 t x ) = Z u ( C 0 t x ) , bec ause X k + h = s k,h ≥ 0 ( − 1) k k ! h ! = 0 for all s ≥ 1. The pro o f of Step 2 a nd o f Theorem 3.4 is concluded.  3.3. Deriv ativ es of approx imate exp one ntials. Her e we give the formula fo r the deriv ative of a n approximate exp onential. All the subsection is written for the molliﬁed v ector ﬁelds X σ j , but we drop everywhere the supesr cript. Theorem 3.5. Ther e is t 0 > 0 such that, for any ℓ ∈ { 2 , . . . , s } , w = ( w 1 , . . . , w ℓ ) , letting C t = C t ( X w 1 , · · · , X w ℓ ) , ther e ar e c onstants a w , e a w such t hat, for any x ∈ K , t ∈ [0 , t 0 ] , (3.19) d dt u ( C t x ) = ℓt ℓ − 1 X w u ( C t x ) + s X | v | = ℓ +1 a v t | v | − 1 X v u ( C t x ) + O s +1 ( t s , u, C t x ) , (3.20) d dt u ( C − 1 t x ) = − ℓt ℓ − 1 X w u ( C − 1 t x ) + s X | v | = ℓ +1 e a v t | v | − 1 X v u ( C − 1 t x ) + O s +1 ( t s , u, C − 1 t x ) . wher e, if ℓ = s , the su m is empty, while, if 2 ≤ ℓ < s , we have t he c anc el lation (3.21) X | w | = ℓ +1  a w + e a w  X w ( x ) = 0 for al l x ∈ K . F ro m Theorem 3.5 it is v ery easy to obtain the following result: Theorem 3 .6. F or any c ommutator X w with length | w | = ℓ ≤ s , we have, for x ∈ K and t ∈ [ − t 0 , t 0 ] , (3.22) d dt u ( e tX w ∗ ( x )) = X w u ( e tX w ∗ ( x )) + s X | v | = ℓ +1 α v ( t ) X v u  e tX w ∗ ( x )  + O s +1  | t | ( s +1 − ℓ ) /ℓ , u, e tX w ∗ ( x )  , 18 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI wher e the sum is empty if ℓ = s , α v ( t ) = ℓ − 1 a v t ( | v | /ℓ ) − 1) , if t > 0 and α v ( t ) = − ℓ − 1 e a v | t | ( | v | /ℓ ) − 1) if t < 0 . In p articular, the map ( t, x ) 7− → e tX w ∗ ( x ) is of class C 1 on ( − t 0 , t 0 ) × Ω ′ . Example 5.7 shows that, even if the vector ﬁelds ar e smo oth, then the map exp ∗ ( tX w ) is at most C 1 ,α for some α < 1 . Pr o of of The or em 3.6. F ormula (3.2 2) follows immediately fro m (3.19), (3.20) and the deﬁnition (2.3) of e ∗ . W e only need to show now that the map is C 1 in b oth v ariables t, x . Recall that the vector ﬁelds X σ j are smo o th and in particular C 1 . By classical ODE theory , s ee [Ha, Chap. 5], an y map of the for m ( τ 1 , . . . , τ ν , x ) 7→ e τ 1 X i 1 · · · e τ ν X i ν x is C 1 if the τ j ’s b elong to so me neig hborho o d of the o rigin and x ∈ Ω ′ . This implies that for any comm utator X w , the map ∇ x exp ∗ ( tX w ) x is contin uous o n ( t, x ) ∈ I × Ω ′ , while d dt exp ∗ ( tX w ) x is contin uous in ( t, x ) ∈  I \ { 0 }  × Ω ′ . Next we prov e tha t d dt exp ∗ ( tX w ) x exists and it is c ontin uous also at a ll points of the form (0 , x ). Obser ve ﬁr st that formula (3.2 2) gives (3.23) lim t → 0 d dt exp ∗ ( tX w ) x = X w ( x ) uniformly in x ∈ Ω ′ . Now, (3.23) and l’Hˆ opital’s rule imply that d dt exp ∗ ( tX w ) x   t =0 = X w ( x ) , for all x ∈ Ω ′ . Finally , the uniformity of the limit e ns ures that the map ( t, x ) 7→ d dt exp ∗ ( tX w ) x is actually contin uous in I × Ω ′ .  Pr o of of The or em 3.5. W e divide the pro of in tw o steps. Step 1. W e ﬁrs t prove that, if (3.1 9) and (3.20) hold for so me w with ℓ := | w | ∈ { 2 , . . . , s − 1 } , then the cancellation formula (3.2 1) must hold. Fix such a w and start from the ident ity d dt u ( C − 1 t C t x ) = 0. (3.24) 0 = d ds u ( C − 1 s C t x )    s = t + d ds ( uC − 1 t )( C s x )    s = t = − ℓ t ℓ − 1 X w u ( x ) + X | v | = ℓ +1 e a v t ℓ X v u ( x ) + O ℓ +2 ( t ℓ +1 , u, x ) + ℓt ℓ − 1 X w ( uC − 1 t )( C t x ) + X | v | = ℓ +1 a v t ℓ X v ( uC − 1 t )( C t x ) + O ℓ +2 ( t ℓ +1 , uC − 1 t , C t x ) . But, since ℓ ≥ 2, formula F ℓ, 2 shows that t ℓ − 1  X w ( uC − 1 t )( C t x ) − X w u ( x )  = t ℓ − 1 O 2+ | w | ( t 2 , u, x ) = O ℓ +2 ( t ℓ +1 , u, x ) , while F ℓ, 1 gives for a n y v with | v | = ℓ + 1, t ℓ { X v ( uC − 1 t )( C t x ) − X v u ( x ) } = t ℓ O 1+ | v | ( t, u, x ) = O ℓ +2 ( t ℓ +1 , u, x ) . Divide (3.24) by t ℓ and let t → 0 to get (3.21). Step 1 is concluded. Step 2. W e prove by an induction argument, tha t, if Theo rem 3.5 holds for s ome ℓ ∈ { 2 , . . . , s − 1 } , then it holds for ℓ + 1. T o show the result for ℓ = 2, it suﬃces to follow the pro of b elow, taking into account that formulas (3.1 9) and (3.20) ar e trivial, if ℓ = 1. W e use the notation in (3.17) for C t and C 0 t . In view of (3.10) a nd NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 19 of the already accomplished Step 1, it suﬃces to prov e that (3.25) d dt u ( C 0 t x ) = ( ℓ + 1 ) t ℓ [ X , X w ] u ( C 0 t x ) + O ℓ +2 ( t ℓ +1 , u, C 0 t x ) and d dt u (( C 0 t ) − 1 x ) = − ( ℓ + 1) t ℓ [ X , X w ] u (( C 0 t ) − 1 x ) + O ℓ +2 ( t ℓ +1 , u, ( C 0 t ) − 1 x ) . W e pr ov e only the ﬁrst line of (3.2 5). The latter is similar. W e know that d dt ( u ( C t x )) = ℓt ℓ − 1 X w u ( C t x ) + t ℓ X | v | = ℓ +1 a v X v u ( C t x ) + O ℓ +2 ( t ℓ +1 , u, C t x ) d dt u ( C − 1 t x ) = − ℓt ℓ − 1 X w u ( C − 1 t x ) + t ℓ X | v | = ℓ +1 e a v X v u ( C − 1 t x ) + O ℓ +2 ( t ℓ +1 , u, C − 1 t x ) , (3.26) with the remark a ble ca ncellation (3.21). Obse r ve that a v = e a v = 0, if ℓ = 1. Next, d dt  u ( C 0 t x )  = d dt  u ( C − 1 t e − tX C t e tX x )  = X  uC − 1 t e − tX C t  ( e tX x ) + d ds ( uC − 1 t e − tX )( C s e tX x )    s = t − X  uC − 1 t  ( e − tX C t e tX x ) + d ds u ( C − 1 s e − tX C t e tX x )    s = t =: A 1 + A 2 + A 3 + A 4 . First we study A 1 + A 3 , by (3 .12) and F ℓ,ℓ +1 . A 1 + A 3 = X  uC − 1 t e − tX C t  ( e tX x ) − X  uC − 1 t e − tX  ( C t e tX x ) = (b y F ℓ,ℓ +1 ) = t ℓ [ X , X w ]( uC − 1 t e − tX )( C t e tX x ) + O ℓ +2 ( t ℓ +1 , uC − 1 t e tX , C t e tX x ) = t ℓ [ X , X w ] u ( C 0 t x ) + O ℓ +2 ( t ℓ +1 , u, C 0 t x ) . Next we study A 2 + A 4 , b y means of (3 .26). A 2 + A 4 = ℓt ℓ − 1 X w ( uC − 1 t e − tX )( C t e tX x ) + t ℓ X | v | = ℓ +1 a v X v ( uC − 1 t e − tX )( C t e tX x ) − ℓt ℓ − 1 X w u ( C 0 t x ) + t ℓ X | v | = ℓ +1 ˜ a v X v u ( C 0 t x ) + O ℓ +2 ( t ℓ +1 , u, C 0 t x ) = ℓt ℓ − 1 n X w ( uC − 1 t )( e − tX C t e tX x ) + t [ X , X w ]( uC − 1 t )( e − tX C t e tX x ) + O ℓ +2 ( t 2 , u, C 0 t x ) o + t ℓ X | v | = ℓ +1 a v  X v u ( C 0 t x ) + O ℓ +2 ( t, u, C 0 t x )  − ℓt ℓ − 1 X w u ( C 0 t x ) + t ℓ X | v | = ℓ +1 ˜ a v X v u ( C 0 t x ) + O ℓ +2 ( t ℓ +1 , u, C 0 t x ) . Now obser ve that b y formula F ℓ, 2 we hav e, if ℓ ≥ 2, t ℓ − 1 n X w ( uC − 1 t )( e − tX C t e tX x ) − X w u ( C 0 t x ) o = O ℓ +2 ( t ℓ +1 , u, C 0 t x ) , 20 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI while, if ℓ = 1 the left-ha nd side v anishes identically . Th us, cancella tion (3.21) g ives A 2 + A 4 = ℓt ℓ [ X , X w ] u ( C 0 t x )+ O ℓ +2 ( t ℓ +1 , u, C 0 t x ) and ultimately A 1 + A 2 + A 3 + A 4 = ( ℓ + 1 ) t ℓ [ X , X w ] u ( C 0 t x ) + O ℓ +2 ( t ℓ +1 , u, C 0 t x ) . The pro of is concluded.  4. Persistence of maximal ity conditions on bal ls Here we establish a key prop er t y o f stability of the η − max imality co ndition. The argument, as in [TW], is bas ed on Gron wall’s inequality . Theorem 4.1. L et X 1 , . . . , X m b e ve ctor ﬁelds in A s . Then, ther e ar e r 0 > 0 and ε 0 > 0 dep ending on the c onstants L and ν in (2.5) and (2.2) such that, if for some η ∈ ]0 , 1[ , x ∈ K and r < r 0 , the triple ( I , x, r ) is η − maximal, then for any y ∈ B ( x, η ε 0 r ) , we have t he estimates (4.1) | λ I ( y ) − λ I ( x ) | ≤ 1 2 | λ I ( x ) | , (4.2) | λ I ( y ) | r ℓ ( I ) > C − 1 η Λ( y , r ) . T o prove Theorem 4.1 we need the following easy lemma. Lemma 4.2. Ther e is C > 0 dep en ding on L and ν such t hat, given y ∈ Ω and z ∈ R n , the line ar system P q i =1 Y i ( y ) ξ i = z has a solution ξ ∈ R q such that | ξ | ≤ C | z | . Pr o of. T ake y ∈ Ω and cho ose ( k 1 , . . . , k n ) ∈ S such that det ( Y k 1 ( y ) , . . . , Y k n ( y )) ≥ ν . Let A := ( Y k 1 ( y ) , . . . , Y k n ( y )). Thus,   A − 1   ≤ C / | det( A ) | ≤ C /ν , where C depe nds on L . The lemma is ea s ily prov e d by studying the s ystem Aξ = z with ξ = ( ξ k 1 , . . . , ξ k n ) ∈ R n .  Pr o of of The or em 4.1. Obs erve that if ( I , x, r ) is η -maximal, then there is e σ > 0 which may also dep end o n I , x, r , such that ( I , x, r ) is η -maxima l for the molliﬁed X σ j for all σ ≤ e σ . Therefore, we will give the pr o of for smo oth vector ﬁelds (without writing a ny sup ers cript). The nonsmo oth case will follow by pa ssing to the limit as σ → 0 a nd taking int o account tha t all constants ar e s table. Let J ∈ S a nd let λ J ( x ) := det[ Y j 1 ( x ) , . . . , Y j n ( x )]. Let X b e a vector ﬁeld o f length one. Recall the following form ula (see [NSW, Lemma 2.6]): X λ J = (div X ) λ J + n X k =1 det( . . . , Y j k − 1 , [ X , Y j k ] , Y j k +1 , . . . ) = (div X ) λ J + X k ≤ n, ℓ j k 0 such that, given I ∈ S , then, for any j = 1 , . . . , n , σ ≤ σ 0 , x ∈ K and h ∈ Q I ( r 0 ) , the C 1 map E σ I ,x satisﬁes (5.1) ∂ ∂ h j E σ I ,x ( h ) = U σ j ( E σ I ,x ( h )) + s X | w | = d j +1 a w j ( h ) X σ w ( E σ I ,x ( h )) + ω σ j ( x, h ) , wher e the sum is empty if d j = ℓ ( U j ) = s and t he fol lo wing estimates hold: (5.2) | ω σ j ( x, h ) | ≤ C k h k s +1 − d j I for any x ∈ K h ∈ Q I ( r 0 ) σ ≤ σ 0 , (5.3) | a w j ( h ) | ≤ C k h k | w |− d j I for al l h ∈ Q I ( r 0 ) | w | = d j + 1 , . . . , s. Theorem 5.1 holds without assuming η - maximality . If the triple ( I , x, r ) is η − maximal, w e have mor e. T o state the r esult, ﬁx once for all a dimensional constant χ > 0 such tha t (5.4) det( I n + A ) ∈ h 1 2 , 2 i for all A ∈ R n × n with norm | A | ≤ χ. Theorem 5. 2. L et r 0 , σ 0 > 0 as in The or em 5.1. Given an η − maximal triple ( I , x, r ) for the ve ctor ﬁelds X i , with x ∈ K , r < r 0 and σ ≤ σ 0 , then, for any h ∈ Q I ( ε 0 η r ) , j = 1 , . . . , n , we may write (5.5) ∂ ∂ h j E σ I ,x ( h ) = U σ j ( E σ I ,x ( h )) + n X k =1 ( b k j ) σ ( x, h ) U σ k ( E σ I ,x ( h )) , wher e, (5.6) | ( b k j ) σ ( x, h ) | ≤ C η k h k I r r d k − d j ≤ χr d k − d j for al l h ∈ Q I ( η ε 0 r ) . Remark 5.3. Estimate (5 .6) and the res ults on Section 4 imply that, under the hypotheses of Theorem 5.2, we hav e | λ σ I ( x ) | ≤ C 1 | λ σ I ( E σ I ,x ( h )) | ≤ C 2     det ∂ ∂ h E σ I ,x ( h )     ≤ C 3 | λ σ I ( x ) | for all h ∈ Q I ( ε 0 η r ) . Pr o of of The or em 5.1. Without loss of ge ne r ality we ma y work in R 2 . W e drop everywhere the sup erscr ipt σ . Then E I ( x, h ) = e h 1 U 1 ∗ e h 2 U 2 ∗ x. Deno te by u the ident ity function in R n . W e ﬁr st lo ok at ∂ /∂ h 1 . Theore m 3.6 with X w = U 1 and t = h 1 gives: ∂ ∂ h 1 u  e h 1 U 1 ∗ e h 2 U 2 ∗ x  = U 1 u ( E I ,x ( h )) + s X | v | = d 1 +1 α v ( h 1 ) X v u ( E I ,x ( h )) + O s +1  | h 1 | ( s +1 − d 1 ) /d 1 , u, E I ,x ( h )  , NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 23 where we know that | α v ( h 1 ) | ≤ C | h 1 | ( | v |− d 1 ) /d 1 and   O s +1  | h 1 | ( s +1 − d 1 ) /d 1 , u, E I ,x ( h )    ≤ C | h 1 | ( s +1 − d 1 ) /d 1 . Thu s, since | h 1 | 1 /d 1 ≤ k h k I , we hav e prov ed (5 .10) and (5.11) for j = 1. Next we lo ok at the v ariable h 2 . Theore m 3.6 gives (5.7) ∂ ∂ h 2 u  e h 1 U 1 ∗ e h 2 U 2 ∗ x  = U 2  ue h 1 U 1 ∗  e h 2 U 2 ∗ x  + s X | v | = d 2 +1 α v ( h 2 ) X v  ue h 1 U 1 ∗  e h 2 U 2 ∗ x  + O s +1  h ( s +1 − d 2 ) /d 2 2 , ue h 1 U 1 ∗ , e h 2 U 2 ∗ x  , where we know that | α v ( h 2 ) | ≤ C | h 2 | ( | v |− d 2 ) /d 2 ≤ C k h k | v | − d 2 I and    O s +1  h ( s +1 − d 2 ) /d 2 2 , ue h 1 U 1 ∗ , e h 2 U 2 ∗ x     ≤ C | h 2 | ( s +1 − d 2 ) /d 2 ≤ C k h k s +1 − d 2 I . Now, a rep eated application of for m ula (3.12) gives (5.8) U 2  ue h 1 U 1 ∗  e h 2 U 2 ∗ x  = U 2 u  E I ,x ( h )  + s − d 2 X α 1 + ··· + α ν =1 C α h ( α 1 + ··· + α ν ) /d 1 1 ad α 1 Z 1 · · · ad α ν Z ν U 2 u  E I ,x ( h )  + O s +1  h ( s +1 − d 2 ) /d 1 1 , u, E I ,x ( h )  , where w e denoted brieﬂy e h 1 U 1 ∗ = e − h 1 /d 1 1 Z 1 · · · e − h 1 /d 1 1 Z ν , where ν is suitable, h 1 > 0 and Z j ∈ {± X 1 , · · · ± X m } . If h 1 < 0 the computation is a na logue. T o conclude the pr o of it suﬃces to write a ll the ter ms X v ( ue h 1 U 1 ∗ )( e h 2 U 2 ∗ x ) in (5.7) in the form X v u  E I ,x ( h )  plus an appropria te rema inder. The ar gument is the same used in equation (5.8) and we leave it to the reader .  Pr o of of The or em 5.2. The pro of r e lie s o n Coro llary 4 .3. W e drop everywhere the sup erscript σ . If ( I , x, r ) is η − maximal, then (4.2 ) gives | λ I ( E I ,x ( h )) | r ℓ ( I ) ≥ C − 1 η Λ( E I ,x ( h ) , r ), as so on as h ∈ Q I ( ε 0 η r ). W rite brieﬂy E instea d of E I ,x ( h ) . Lo oking at the right-hand side of (5.1), we need to study , for any word w of length | w | = ℓ , with ℓ = d j + 1 , . . . , s , the linear system a w j ( h ) X w ( E ) = P n k =1 b k j U k ( E ) and we must show tha t the solution b k j satisﬁes (5.6), if k h k I ≤ ε 0 η r . By Corollary 4.3 write X w ( E ) = P p k w U k ( E ), where | p k w | ≤ C η r d k −| w | . Thus | b k j | = | a w j p k w | ≤ C k h k | w |− d j I C η r d k −| w | ≤  k h k I r  | w |− d j C η r d k − d j . Here w e also used (5.3). This gives the estimate of the terms in the sum in (5.1). Next we lo o k at the the r emainder ω j . Fix j = 1 , . . . , n . W e know that | ω j | ≤ C k h k s +1 − d j I and we wan t to write ω j = P k b k j U k ( E ) with estimate (5 .6). It is conv e nient to multiply by r d j . Let r d j ω j =: θ ∈ R n and ξ k = r d j b k j . Thus it suﬃces to sho w that w e can write θ = P k ξ k U k ( E ), where ξ k satisﬁes the estimate | ξ k | ≤ C η k h k r r d k . W e know that | θ | = | r d j ω j | ≤ C k h k s +1 − d j I r d j = C  k h k I r  s +1 − d j r s +1 . 24 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI T o estimate ξ k , we follow a t wo steps argument: Step 1. W rite, b y Lemma 4 .2, θ = P q i =1 µ i Y i ( E ) , for some µ ∈ R q satisfying | µ | ≤ C | θ | ≤ C  k h k r  s +1 − d j r s +1 . Step 2. F or any i = 1 , . . . , q write Y i ( E ) = P n k =1 λ k i U k ( E ). This can b e done in a unique w ay a nd estimate | λ k i | ≤ C η r d k − ℓ ( Y i ) holds, b y Corolla ry 4.3. Collecting Step 1 and Step 2, we conclude that | ξ k | =    q X i =1 µ i λ k i    ≤ C  k h k r  s +1 − d j r s +1 · C η r d k − ℓ ( Y i ) ≤ C η k h k r r d k , as required. This ends the pro o f.  Next we pass to the limit a s σ → 0 in b oth Theor ems 5 .1 and 5.2. Theorem 5. 4. If ( I , x, r ) is η − maximal for some x ∈ K , r ≤ r 0 , then the map E I ,x   Q I ( ε 0 ηr ) is lo c al ly biLipschitz in t he Euclide an sense and satisﬁ es for a.e. h , (5.9) ∂ ∂ h j E I ,x ( h ) = U j ( E I ,x ( h )) + s X | w | = d j +1 a w j ( h ) X w ( E I ,x ( h )) + ω j ( x, h ) = U j ( E I ,x ( h )) + n X k =1 b k j ( x, h ) U k ( E I ,x ( h )) , wher e the sum is empty if d j = ℓ ( U j ) = s and otherwise the fol lowing estimates hold: (5.10) | ω j ( x, h ) | ≤ C k h k s +1 − d j I for al l x ∈ K h ∈ Q I ( r 0 ) , (5.11) | a w j ( h ) | ≤ C k h k | w |− d j I if | w | = d j + 1 , . . . , s and h ∈ Q I ( r 0 ) , and (5.12) | b k j ( x, h ) | ≤ C η k h k I r r d k − d j ≤ χr d k − d j for al l h ∈ Q I ( ε 0 η r ) . Remark 5.5. If s ≥ 3, then vector ﬁelds o f the c lass A s are C 1 . Then, as discussed in the b eginning of the pro of of Theor em 3 .6, the map E I ,x is a ctually C 1 smo oth. This is not ensured if s = 2 . Pr o of of The or em 5.4. Lo o k ﬁrst at the C 1 map E σ = E σ I ,x deﬁned on Q I ( r 0 ). Denote by E its po int wise limit as σ → 0. By Theore m, 5.1, the map E σ satisﬁes for any σ < σ 0 , k h k I ≤ r 0 , (5.13) ∂ ∂ h j E σ ( h ) = U σ j ( E σ ( h )) + s X | w | = d j +1 a w j ( h ) X σ w ( E σ ( h )) + ω σ j ( h ) , where a w j do not depe nd on σ , while | ω σ j ( h ) | ≤ C k h k s +1 − d j I , uniformly in σ ≤ σ 0 . Let E σ k be a sequence weakly conv erging to E in W 1 , 2 . Ther efore, by (5 .1 3), the remainder ω σ k j has a weak limit in L 2 . Denote it by ω j . Standar d prop erties of weak conv ergence ensure that | ω j ( h ) | ≤ C 0 k h k s +1 − d j I for a .e. h . There fo re, we hav e proved the ﬁr st line o f (5.9) a nd estimates (5.10) and (5.11). T o prov e the second line and (5 .1 2), it s uﬃces to rep eat the a rgument of Theor em 5.2, tak ing into NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 25 account that the main ingredient ther e, namely Cor o llary 4.3, holds fo r nonsmo oth vector ﬁelds in A s . Now we hav e to prove the lo cal injectivit y of E . Let σ be small eno ugh to ensure that ( I , x, r ) is η -maximal for the vector ﬁelds X σ j . In view o f Theor e m 5.2, we can write dE σ ( h ) = U σ ( E σ ( h ))( I n + B σ ( h )), where U σ = [ U σ 1 , . . . , U σ n ], and the ent ries of the ma tr ix B satisfy | ( b k j ) σ | ≤ C r d k − d j , by (5.5). Fix now h 0 ∈ Q I ( ε 0 η r ), where ε 0 η comes fr o m Theor em 5.2. W e will sho w that E σ is lo ca lly one-to- one around h 0 , with a s ta ble co ercivity estimate as σ → 0. By Pr op osition 2.4 and by the con tinu ity of the v ector ﬁelds U j , we may claim that for any δ > 0 there is  > 0 such that | U σ j ( ξ ) − U σ j ( ξ ′ ) | < δ as s o on as ξ , ξ ′ ∈ K , | ξ − ξ ′ | <  a nd σ <  . Recall also that E σ is Lipschitz contin uous, unifor mly in σ , s ee (5.1 3). Then, for any δ > 0 there is  > 0 such that B Eucl ( h 0 ,  ) ⊂ Q I ( ε 0 η r ), and, if | h − h 0 | ≤  and σ <  , then | U σ ( E σ ( h )) − U σ ( E σ ( h 0 )) | ≤ δ . T a ke h, h ′ ∈ B Eucl ( h 0 , δ ). By integrating on the path γ ( t ) = h ′ + t ( h − h ′ ), we hav e | E σ ( h ) − E σ ( h ′ ) | =    Z 1 0 U σ ( E σ ( γ ))( I + B σ ( γ ))( h − h ′ ) dt    ≥    Z 1 0 U σ ( E σ ( h 0 ))( I + B σ ( γ ))( h − h ′ ) dt    −    Z 1 0  U σ ( E σ ( γ )) − U σ ( E σ ( h 0 ))  ( I + B σ ( γ ))( h − h ′ ) dt    . T o estimate from b elow the ﬁrst line recall the easy inequality | Ax | ≥ C − 1 | d et A | | A | n − 1 | x | , for all A ∈ R n × n . The po in twise estimate | ( b k j ) σ | ≤ χr d k − d j gives | R 1 0 ( b k j ) σ ( γ )) dt | ≤ χr d k − d j . Thus (5.4) gives    det Z 1 0 ( I + B σ ( γ )) dt    =    det  I + Z 1 0 B σ ( γ ) dt     ≥ 1 2 . Observe als o that | I + B σ ( γ ) | ≤ C r 1 − s . Moreover, in view of Remark 5.3, it must be | det U σ ( E σ ( h 0 )) | ≥ C − 1 | λ I ( x ) | , for small σ . This suﬃces to es tima te from be low the ﬁr st line. T o get an estimate of the second line we need a gain the inequality | I + B σ ( γ ) | ≤ C r 1 − s . Even tually we get | E σ ( h ) − E σ ( h ′ ) | ≥ { C − 1 0 | λ I ( x ) | r ( n − 1)( s − 1) − C 0 r 1 − s δ }| h − h ′ | , for any σ <  and | h − h ′ | <  . The pro of is concluded a s so on as w e cho ose δ = δ ( I , x, r ) small enough and let σ → 0 . This argument shows that the map is lo cally biLipschitz, as desired.  5.2. Pullbac k of v ector ﬁelds through scaling maps. Given an η -maxima l triple ( I , x, r ), for vector ﬁelds of the class A s we c an deﬁne, as in [TW], the “scaling map” (5.14) Φ I ,x,r ( t ) = exp  n X j =1 t j r ℓi j Y i j  x, for small | t | . T he dilation δ I r ( t ) := ( t 1 r ℓ i 1 , . . . , t n r ℓ i n ) mak es the na tur al domain o f Φ I ,x,r independent o f r . Obse rve the prop erty k δ I r t k I = r k t k I . It turns o ut that, if b X k ( k = 1 , . . . , m ) denotes the pullback of rX k under Φ I ,x,r , then b X 1 , . . . , b X m 26 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI satisfy the H¨ ormander condition in an uniform way . This fact enables the authors in [TW] to give several simpliﬁca tions to the arguments in [NSW]. W e can also co nsider the scaling ma p a sso ciated with our exp onentials. Namely , (5.15) S I ,x,r ( t ) := exp ∗ ( t 1 r ℓ i 1 Y i 1 ) · · · exp ∗ ( t n r ℓ i n Y i n ) = E I ,x ( δ I r t ) , It will b e proved in Subsection 5.3 that, if ( I , x, r ) is η -ma ximal, then S is one-to- one on the set {k t k I ≤ ε 0 η } . If we ass ume that the orig inal vector ﬁelds are of class C 1 , s e e Remark 5.5, thus, w e ma y deﬁne, for all i ∈ { 1 , . . . , q } the vector ﬁelds b Y j := S − 1 ∗ ( r ℓ i Y i ) . Theorem 5.4 thus b ecomes Prop ositio n 5. 6. L et X 1 , . . . , X m b e ve ctor ﬁ elds in A s . L et ( I , x, r ) b e an η − maximal triple and let S := S I ,x,r b e the asso ciate d sc aling map. Then S   Q I ( ε 0 η ) is a lo c al ly biLipschi tz map and for a.e. t ∈ Q I ( ε 0 η ) we may write (5.16) S ∗ ( ∂ t j ) = r ℓ i j Y i j ( S ( t )) + n X k =1 b b k j r ℓ i k Y i k ( S ( t )) , wher e the functions b b k j satisfy (5.17) | b b k j | ≤ C η k t k I for a.e. t ∈ Q I ( ε 0 η ) . Mor e over, if S is C 1 and we write b Y i j = ∂ t j + P n k =1 a k j ( t ) ∂ t k , t hen (5.18) | a k j ( t ) | ≤ C η k t k I for al l t ∈ Q I ( ε 0 η ) . Pr o of. F ormula (5.16) is just Theorem 5.4. The pro of o f (5.18) is a conse quence o f (5.17) and of the following elemen tary fact: given a s q uare matrix B ∈ R n × n with norm | B | ≤ 1 2 , w e ma y write ( I n + B ) − 1 = I n + A , and | A | =   P k ≥ 1 ( − B ) k   ≤ 2 | B | .  In the fr amework of our a lmost exp onential maps, es tima te (5.18) is sharp, even for smo oth vector ﬁelds. T he b etter estimate Y i j ( t ) = ∂ j + P k a k j ( t ) ∂ k with | a k j ( t ) | ≤ C | t | , obtained in [TW] fo r maps of the for m (5.14), g e nerically fails for S , as the following example shows. Example 5. 7 . Let X 1 = ∂ 1 , X 2 = a ( x 1 ) ∂ 2 with a ( s ) = s + s 2 , or any smo oth function with a (0) = 0 and a ′ (0) 6 = 0 6 = a ′′ (0). A computation shows that exp ∗ ( h [ X 1 , X 2 ])( x 1 , x 2 ) =  x 1 , x 2 +  a ( x 1 + | h | 1 / 2 ) − a ( x 1 )  | h | − 1 / 2 h  . Therefore, at ( x 1 , x 2 ) = (0 , 0), for small r , we must cho ose the maximal pa ir of commutators X 1 , [ X 1 , X 2 ] and w e hav e S ( t 1 , t 2 ) = ex p ∗ ( t 1 rX 1 ) exp ∗ ( t 2 r 2 [ X 1 , X 2 ])(0 , 0) =  t 1 r , a ( r | t 2 | 1 / 2 ) | t 2 | − 1 / 2 t 2 r  . =  t 1 r , t 2 r 2 + | t 2 | 1 / 2 t 2 r 3  . Therefore, b X 1 = ∂ t 1 , b X 2 = t 1 + r t 2 1 1 + 3 2 r | t 2 | 1 / 2 ∂ t 2 , \ [ X 1 , X 2 ] = 1 + 2 rt 1 1 + 3 2 r | t 2 | 1 / 2 ∂ t 2 . NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 27 Clearly the form ula \ [ X 1 , X 2 ] = ∂ t 2 + O ( | t | ) ca nnot hold, but (5.18) holds. O bserve also tha t, writing \ [ X 1 , X 2 ] = b f (1 , 2) · ∇ , we have sup t ∈ U   b X 2 b f (1 , 2)   ≃ sup t ∈ U | t 2 | − 1 / 2 = + ∞ , for a ny neighbo rho o d U of the or igin. Therefore, the vector ﬁelds b X 1 , b X 2 do not even b elong to the class A 2 . 5.3. Ball-b ox theorem . Her e we g ive o ur main re s ult. W e keep the notation fro m Subsection 5.1. Theorem 5.8. L et X 1 , . . . , X m b e H¨ ormander ve ct or ﬁelds of step s in t he class A s . Ther e ar e r 0 , e r 0 , C 0 > 0 , and for al l η ∈ (0 , 1 ) ther e ar e ε η , C η > 0 such that: (A) if ( I , x, r ) is η − maximal for some x ∈ K , r ≤ r 0 , t hen, for any ε ≤ ε η , we have (5.19) E I ,x ( Q I ( εr )) ⊃ B  ( x, C − 1 η ε s r ); (B) if ( I , x, r ) is η − maximal for some x ∈ K , r ≤ e r 0 , then the map E I ,x is one-to-one on the set Q I ( ε η r ) . Remark 5.9. Obser ve that in the right-hand s ide of inclusio n (5.19) w e use the distance d  . Therefore, a standard consequence o f (5.19) is the well known prop erty B ( x, r ) ⊃ B E ( x, C − 1 r s ), for any x ∈ K , r < r 0 . See [FP]. Remark 5.10. In the pap er [TW] the a utho r s use the exp onential maps in (1.2). If the vector ﬁelds hav e step s , then their metho d require s that the commutators of length 2 s ar e at least contin uous. (Here, we specia lize [TW] to the case ε = 1 and we do not dis cuss the higher r e gularity estimate [TW, Eq. (2.1)].) This app ears in the pro o f o f (22) and (23) of [T W , P r op osition 4.1 ]. Indeed in equation (29), the commutator [ X w , X w j ] m ust b e written as a linear combination of commutators X w ′ , where for algebraic r easons it must b e | w ′ | = | w | + | w k | . If | w | = | w k | = s , then co mm utators of degree 2 s app ear. A similar issue app ears for [ Y w i , Y w j ] a t the beg inning of p. 619. Remark 5.11. The rea son wh y we in tro duce tw o diﬀeren t constants r 0 and e r 0 is that C 0 , ε 0 and r 0 depe nd only on L and ν in (2.2) and (2.5) (together with universal co nstants, like m, n and s ). The co nstants ε η and C η depe nd on ν , L and η als o. W e do not hav e a cont rol of e r 0 (whic h app ear s only in the injectivity statement) in ter m of L and ν . This is a delicate q uestion b ecause o f the cov ering argument implicitly contained in [NSW, p. 1 32] a nd des crib ed in [M, p. 230]. B e low we provide a constructive pr o cedure to provide a low er b ound for e r 0 in term of the functions λ I . See p. 31. This can be of s ome in ter est in view of applica tions of our results to nonlinear problems. Remark 5.12. T he pr o of o f the injectivity re s ult would b e consider ably simpliﬁed if we could pr ove (uniformly in x ∈ K , r < r 0 ) an eq uiv alence b etw een the ba lls and their conv ex hulls, i.e. co B ( x, r ) ⊂ B ( x, C r ) , which is rea sonable for diag onal vector ﬁelds (see [SW, Remark 5]) or a “ c ontractabilit y” prop er ty of the ball B ( x, r ) inside B ( x, C r ). See [Sem, Deﬁnition 1.7]. Unfortunately , in spite of their reasona ble asp ect, b oth these conditions seem quite diﬃcult to prove in our s ituation. It seems a lso that the clever ar gument in [TW, p. 622] can no t be adapted to o ur almost exp onential maps. In the proo f of inclusion (5.1 9), w e follow the argument in [NSW , M]. Befor e giving the pro o f, we need to show that some constants in the pr o of actua lly dep end 28 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI only on L and ν in (2.2) and (2 .5). Ba sically , what we ne e d is co ntained in Co rollary 4.3 and in the following Lemma. Se e [NSW, p. 129]. Lemma 5.13. Assu me that ( I , x, r ) is η − maximal for ve ctor ﬁ elds X j in A s , x ∈ K , r ≤ r 0 . L et e σ > 0 b e such that ( I , x, r ) is η -m aximal for t he mol liﬁe d X σ j for al l σ ≤ e σ . L et U ⊂ Q I ( ε η r ) , wher e ε η c omes fr om The or em 5.4 , and assu me that a C 1 diﬀe omorphism ψ = ( ψ 1 , . . . , ψ n ) : E σ ( U ) → U satisﬁes ψ ( E σ ( h )) = h, for any h ∈ U. Then we have t he estimate | U σ j ψ k ( E σ ( h )) | ≤ C r d k − d j , for al l h ∈ U , wher e C is indep en dent of σ . Pr o of. It is co nv enient to work with the map S σ ( t ) := E σ ( δ r t ), so that ϕ := δ 1 /r ψ satisﬁes t = ϕ ( S σ ( t )), for all t ∈ V := δ 1 /r U . The chain rule g ives dϕ ( S σ ( t )) dS σ ( t ) = I , for all t ∈ V . But, by (5.16) we hav e dS σ ( t ) = [ r d 1 U σ 1 ( S σ ) , . . . , r d n U σ n ( S σ )]( I + b B σ ( t )), where | b B σ ( t ) | ≤ C η k t k I , if k t k I ≤ ε 0 η . There fore w e ma y write dϕ ( S σ )[ r d 1 U 1 ( S σ ) , . . . , r d n U n ( S σ )] = I + A σ , where, a s in the pro of of P rop osition 5.6, | A σ ( t ) | ≤ 2 | b B σ ( t ) | . This implies that | r d j U σ j ϕ k ( S σ ( t )) | ≤ C and ultimately that | r d j − d k U σ j ψ k | ≤ C , as desired.  Pr o of of The or em 5.8, (A). Since the vector ﬁelds Y j are not E uclidean Lipschitz contin uous, if ℓ j = s , we do not know whether or not a ny p o int in a  -ba ll o f the Y j can b e a pproximated b y p oints in the a nalogous ball of the molliﬁed Y σ j . In orde r to av o id this problem, observe the inclusion B  ( x, r ) ⊂ B e  ( x, C r ) where C is absolute and the distance e  is deﬁned using the family { Y j : ℓ j ≤ s − 1 , ∂ k : k = 1 , . . . , n } , where w e a ssign to the vector ﬁelds ∂ k maximal w eight s . Therefo r e, we will prov e the inclusion using the distance e  , whic h is deﬁned b y Lipschitz v ector ﬁelds. Let ( I , x, r ) be a η − maximal triple for the original vector ﬁelds X j and let e σ b e as in Lemma 5.13. Let y ∈ B e  ( x, C − 1 η ε s r ), where ε ≤ ε η , and ε η comes from statement (A), while C η will b e discussed b elow. Thus, y = γ (1), where ˙ γ = P ℓ j ≤ s − 1 b j Y j ( γ ) + P n i =1 ˜ b j ∂ i ( γ ) a.e. on [0 , 1] , with | b j ( t ) | ≤ ( C − 1 η ε s r ) ℓ ( Y j ) and | ˜ b i ( t ) | ≤ ( C − 1 η ε s r ) s for a.e . t ∈ [0 , 1]. L et a ls o y σ ∈ B e  ( x, C − 1 η ε s r ) b e an approximating family , y σ = γ σ (1), wher e ˙ γ σ = P ℓ j ≤ s − 1 b j Y σ j ( γ σ ) + P n i =1 ˜ b j ∂ i ( γ σ ) a.e. on [0 , 1] . Observe that y σ → y , a s σ → 0 . Claim. If C η is la r ge enoug h, then for an y σ ≤ e σ there is a lifting map θ σ ( t ) , t ∈ [0 , 1 ], with θ σ (0) = 0 and suc h that (5.20) E σ ( θ σ ( t )) = γ σ ( t ) and k θ σ ( t ) k I < εr for all t ∈ [0 , 1] . Once the claim is prov ed, the surjectivit y statement follows. T o prove the cla im the key es tima te w e need is the following. Let U ⊂ Q I ( ε η r ), σ ≤ e σ and assume that a C 1 − diﬀeomorphism ψ = ( ψ 1 , . . . , ψ n ) satisﬁes lo ca lly ψ ( E σ ( h )) = h , for all h ∈ U , where, fo r some t ∈ [0 , 1], E σ ( U ) is a neighborho o d NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 29 of γ σ ( t ). Then, for µ = 1 , . . . , n and for all τ clos e to t (5.21)    d dτ ψ µ ( γ σ ( τ ))    =    X ℓ j ≤ s − 1 b j ( τ ) Y σ j ψ µ ( γ σ ( τ )) + n X i =1 ˜ b i ( τ ) ∂ j ψ µ ( γ σ ( τ ))    =    X ℓ j ≤ s b j ( τ ) n X k =1 a k j ( γ σ ( τ )) U σ k ψ µ ( γ σ ( τ )) + n X i =1 ˜ b i ( τ ) n X k =1 ˜ a k i ( γ σ ( τ )) U σ k ψ µ ( γ σ ( τ ))    ≤ X j,k C ( C − 1 η ε s r ) ℓ ( Y j ) · C η r d k − ℓ ( Y j ) · C r d µ − d k + X i,k C ( C − 1 η ε s r ) s · C η r d k − s · C r d µ − d k ≤ C C − 1 η η ε s r d µ ≤ C C − 1 η η ( εr ) d µ . The constant C η will be chosen b elow, while C dep ends on L, ν , in force of Cor ollary 4.3 and Lemma 5.13. W e used the estimate ∂ i = ˜ a k i U σ k with   ˜ a k i   ≤ C η r d k − s , which follows from Lemma 4.2 and Coro llary 4 .3. With estimate (5.21) in hands we can prove the cla im along the lines of [M, p. 228]. Here is a sketc h of the argument. Step 1. If C η is large enough, then, if θ σ ( t ) satisﬁes E ( θ σ ( t )) = γ σ ( t ) on [0 , ¯ t ], for some ¯ t ≤ 1, then k θ σ ( t ) k I < 1 2 εr , for any t ≤ ¯ t . T o prov e Step 1, a ssume by contradiction that the statement is fals e. There is e t ≤ ¯ t suc h that k θ σ ( t ) k I < 1 2 εr for all t < e t and k θ σ ( e t ) k I = 1 2 εr . Then for some µ ∈ { 1 , . . . , n } , w e ha ve (5.22)  1 2 εr  d µ = | θ σ µ ( e t ) | ≤ C C − 1 η η ( εr ) d µ . This estimate c a n b e obtained writing lo ca lly θ σ ( t ) = ψ ( γ σ ( t )) and using (5 .2 1). If we choose C η large enough to ensure that C C − 1 η η < ( 1 2 ) s , then (5 .22) c a n not hold and w e hav e a contradiction. This ends the pro of of Step 1. Step 2. Ther e exists a path θ σ on [0 , 1] satisfying (5.20). The pro o f of Step 2 can be done a s in [M, p. 2 29] by a v ery class ical argument, inv olving an upper bo und “ of Hada mard type ” k dE σ ( θ σ ( t )) − 1 k ≤ C , which ho lds uniformly in t . The pro of of the statement (A) is concluded.  Before pr oving part ( B ) of Theor em 5.8, w e need the following rough injectivit y statement. Lemma 5.14. L et x ∈ K and I such that λ I ( x ) 6 = 0 . Then the function E I ,x is one-to-one on the set Q I ( C − 1 | λ I ( x ) | ) . 30 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI Pr o of. Observe ﬁrst that for all j = 1 , . . . , n and small σ , we have (5.23)    ∂ ∂ h j E σ ( h ) − U σ j ( x )    ≤    ∂ ∂ h j E σ ( h ) − U σ j ( E σ ( h ))    + | U σ j ( E ( h )) − U σ j ( x ) | ≤ C k h k I , by estimates (5.2), (5.3) and the d -Lipschitz contin uity o f U σ j . Fix h, h ′ ∈ Q I ( C − 1 | λ I ( x ) | ) and let γ ( t ) = h ′ + t ( h − h ′ ). Then | E σ ( h ) − E σ ( h ′ ) | =    Z 1 0 dE σ ( γ )( h − h ′ ) dt    ≥ | dE σ (0)( h − h ′ ) | −    Z 1 0  dE σ ( γ ) − dE σ (0)  dt ( h − h ′ )    ≥  C − 1 | λ σ I ( x ) | − C max { k h k I , k h ′ k I }  | h − h ′ | . by (5.23) and b ecause dE σ (0) = U σ ( x ) = [ U σ 1 ( x ) , . . . , U σ n ( x )] has determinant λ σ I ( x ). The pro of is concluded by letting σ → 0.  As announced in Remark 5.11, we provide a constructive pro cedur e for the “ in- jectivit y ra dius” e r 0 in Theor em 5.8 in term of the functions λ I . Co mpare [M, p. 229-230 ]. Denote by D 1 , . . . , D p all the v alues attained b y ℓ ( I ), as I ∈ S . Ass ume that D 1 < · · · < D p and in tro duce the notation: (5.24) X I | λ I ( x ) | r ℓ ( I ) = p X j =1 r D j X ℓ ( I )= D j | λ I ( x ) | =: p X j =1 r D j µ j ( x ) , where µ j is deﬁned by (5.24). Let Σ 1 := K and, for all k = 2 , . . . , p , Σ k := { x ∈ K : µ j ( x ) = 0 for any j = 1 , . . . , k − 1 } . Observe that Σ 1 = K ⊇ Σ 2 ⊇ · · · ⊇ Σ p . Let x ∈ K . T ake j ( x ) = min { j ∈ { 1 , . . . , p } : µ j ( x ) 6 = 0 } . Then choose I x ∈ S such that | λ I x ( x ) | = max ℓ ( J )= D j ( x ) | λ J ( x ) | is maximal. Therefore, we have | λ I x ( x ) | ≃ µ j ( x ) ( x ) , through absolute constants. F ro m the construction ab ov e we get the following prop ositio n. Prop ositio n 5.15. Ther e is C > 1 such that, lett ing r x := C − 1 | λ I x ( x ) | for al l x ∈ K , then: (1) we have (5.25) | λ I x ( y ) | r ℓ ( I x ) x > C − 1 Λ( y , r x ) for al l y ∈ B ( x , ε 0 r x ); (2) the m ap h 7→ E I x ( y , h ) is one-to-one on t he set Q I x ( r x ) , for any y ∈ B ( x, ε 0 r x ) . Observe that Pro po sition 5.15 is far fro m what we need, becaus e it may b e inf K r x = 0, (for exa mple this happ ens in the elementary situation X 1 = ∂ 1 , X 2 = x 1 ∂ 2 .) Pr o of. W e ﬁrst prov e (1) for y = x . Namely we show that (5.26) | λ I x ( x ) | r ℓ ( I x ) ≥ | λ J ( x ) | r ℓ ( J ) for all J ∈ S r ∈ [0 , r x ] , NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 31 where r x = C − 1 | λ I x ( x ) | , as req uired. L e t J ∈ S . If λ J ( x ) = 0, then (5.2 6) holds for a ll r > 0. If instead λ J ( x ) 6 = 0, by the choice of I x it must b e ℓ ( J ) = ℓ ( I x ) o r ℓ ( J ) > ℓ ( I x ). If ℓ ( J ) = ℓ ( I x ), then (5.26) holds for any r > 0, bec ause | λ I x ( x ) | is maximal, by the constr uction above. If ℓ ( J ) > ℓ ( I x ), then | λ J ( x ) | r ℓ ( J ) ≤ | λ I x ( x ) | r ℓ ( I x ) ⇐ C r ℓ ( J ) − ℓ ( I x ) ≤ | λ I x ( x ) | ⇐ r ≤ C − 1 | λ I x ( x ) | . Thu s (5.26) holds for any r ≤ r x , where r x has the required form. The pro of of (1) for y 6 = x follows fro m Theorem 4.1. Finally , to prove (2) observe that, in view of Lemma 5 .14, the map h 7→ E I x ( y , h ) is one-to -one o n Q I x ( C − 1 | λ I x ( y ) | ). But Theo rem 4.1, in par ticular (4.1) show that, if d ( x, y ) ≤ ε 0 r x , then | λ I x ( y ) | and | λ I x ( x ) | are co mpa rable. This co ncludes the pro of.  Pr o of of The or em 5.8 , (C). Let p 1 ≤ p b e the lar gest integer such that Σ p 1 6 = ∅ . Then deﬁne the “injectivit y ra dius” (5.27) r ( p 1 ) := min x ∈ Σ p 1 r x = min x ∈ Σ p 1 C − 1 | λ I x ( x ) | ≥ C − 1 min x ∈ Σ p 1 µ p 1 ( x ) > 0 . Denote also Ω p 1 = [ x ∈ Σ p 1 Ω ′ ∩ B ( x, r ( p 1 ) ) , where the op en set Ω ′ was in tro duced befor e (2.2). Recall that all metric ba lls B ( x, r ) are open, by the a lready a c c omplished Theo rem 5 .8, part (A). Then, by Prop ositio n 5 .15, for an y y ∈ Ω p 1 there is x ∈ Σ p 1 such that the map h 7→ E I x ( y , h ) is one- to-one on Q I x ( ε 0 r x ) a nd ( I x , y , r x ) is C − 1 -maximal. Recall that r x ≥ r ( p 1 ) , on Σ p 1 . Next let p 2 < p 1 be the lar gest n umber such that K p 2 := Σ p 2 \ Ω p 1 6 = ∅ . Then, let r ( p 2 ) := min x ∈ Σ p 2 \ Ω p 1 r x ≥ C − 1 min x ∈ Σ p 2 \ Ω p 1 µ p 2 ( x ) > 0 . W e may claim that for any y ∈ Ω p 2 := S x ∈ K p 2 Ω ′ ∩ B ( x, r ( p 2 ) ) , there is x ∈ K p 2 such that the map h 7→ E I x ( y , h ) is one - to-one on the set Q I x ( ε 0 r x ) and ( I x , y , r x ) is C − 1 -maximal. Iterating the argument, and letting e r 0 = min { r ( p k ) } we conclude that for any x ∈ K there is a n -tuple I 0 = I 0 ( x ), and  0 =  0 ( x ) ≥ e r 0 such that E I 0 ( x, · ) is one-to-one on the s e t Q I 0 ( ε 0  0 ) and ( I 0 , x,  0 ) is C − 1 -maximal. Clear ly , I 0 can be diﬀerent from I x . This is the starting p oint for the pro of of the injectivity statement, Theorem 5.8, item (B). F ro m now on I , x ∈ K and r < e r 0 are ﬁxed and ( I , x, r ) is η − maximal, a s in the hypothesis of (B). Let I 0 and  0 be the n -tuple and the injectivity radius as so ciated with x b y the argument a b ov e. Recall that  0 ≥ e r 0 . Arguing as in [M, p. 230 ], see also [NSW, p. 1 33], we may ﬁnd a sequence o f n − tuples I = I N , I N − 1 , . . . , I 1 , I 0 and corres po ding num b er s 0 ≤  N +1 <  N < · · · <  0 , with  0 ≥ e r 0 , r ∈ [  N +1 ,  N ] such that for any j = 0 , 1 , . . . , N − 1, (5.28) | λ I j ( x ) |  ℓ ( I j ) ≥ η Λ ( x,  ) , ∀  ∈ [  j +1 ,  j ] . In o rder to show that E I = E I N is one-to-one on the s e t Q I ( ε η r ), for some ε η > 0, we start by showing that E I 1 is one-to -one on the set Q I 1 ( ε ′ η  1 ) , for a suitable ε ′ η . What we know is that E I 0 is one- to-one on the set Q I 0 (  0 ). W e also 32 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI know that (5 .28) holds for j = 0 , 1 and  =  1 . Therefore, applying twice (5.19), we hav e (5.29) E I 1 ( Q I 1 ( ε η  1 )) ⊇ E I 0 ( Q I 0 ( C − 1 η  1 )) ⊇ E I 1  Q I 1 ( ε ′ η  1 )  . Assume by contradiction that E I 1 ( h ) = E I 1 ( h ′ ) = y for some h, h ′ ∈ Q I 1 ( ε ′ η  1 ). Let r ( t ) = h ′ + t ( h − h ′ ), t ∈ [0 , 1] be the line segment connecting h and h ′ . Let also γ ( t ) = E I 1 ( r ( t )). Since E I 0 is one-to-one (actually a C 1 diﬀeomorphism on its image ), we may contract γ to a p oint just by letting q ( λ, t ) = E I 0  λE − 1 I 0 ( y ) + (1 − λ ) E − 1 I 0 ( γ ( t ))  , where ( λ, t ) ∈ [0 , 1] × [0 , 1] . Observe tha t q is contin uous on [0 , 1] 2 , and q ( λ, t ) ∈ Q I 1 ( ε η  1 ), by (5.2 9). Moreover q (0 , t ) = γ ( t ) = E I 1 ( r ( t )) a nd q (1 , t ) = y , for any t ∈ [0 , 1]. By standard pro p e r ties of lo cal diﬀeomorphisms we may c la im that there is a contin uous lifting p : [0 , 1] 2 → Q I 1 ( ε η  1 ) such that E I 1 ( p ( λ, t )) = q ( λ, t ) and p (0 , t ) = r ( t ) for all λ and t ∈ [0 , 1]. Next observe that b oth the maps λ 7→ E I 1 ( p ( λ, 1)) a nd λ 7→ E I 1 ( p ( λ, 0)) are constants on [0 , 1]. Therefore, since E I 1 is a lo cal diﬀeomorphism, b oth λ 7→ p ( λ, 0) and λ 7→ p ( λ, 1) m ust b e constant. In par ticular p (1 , 1) = p (0 , 1) = h ′ and p (1 , 0) = p (0 , 0) = h . Finally observe that the path t 7→ p (1 , t ) must b e constant, b ecause E I 1 ( p (1 , t )) = y for all t ∈ [0 , 1]. Ther efore we conclude that h = h ′ . Then we hav e proved that E I 1 is o ne-to-one on Q I 1 ( ε ′ η  1 ). Iterating the argument at most N times, we get the pro o f of statemen t (B) of Theorem 5.8. 6. Examples Example 6.1 (Levi v ector ﬁelds) . In or de r to illustrate the previous pr o cedure to ﬁnd ˜ r 0 we exhibit the following three-step example. In R 3 consider the vector ﬁelds X 1 = ∂ x 1 + a 1 ∂ x 3 and X 2 = ∂ x 2 + a 2 ∂ x 3 . Assume that the vector ﬁelds belo ng to the class A 3 . Let us deﬁne f = X 1 a 2 − X 2 a 1 . Mor ov er assume that | f | + | X 1 f | + | X 2 f | 6 = 0 at every p oint of the closure of a bo unded set Ω ⊃ K = Ω ′ . Assume also that f has so me zero inside K . This condition na turally arises in the regular ity theory for graphs of the fo rm { ( z 1 , z 2 ) ∈ C 2 : Im( z 2 ) = ϕ ( z 1 , ¯ z 1 , Re( z 2 )) } having s ome ﬁrst order zeros. See [CM], where the smoothness o f C 2 ,α graphs with prescrib ed smo o th Levi curv ature is prov ed. In this situa tio n we ha ve n = 3 , m = 2 , s = 3 and Y 1 = X 1 , Y 2 = X 2 , Y 3 = [ X 1 , X 2 ] = f ∂ x 3 , Y 4 = [ X 1 , [ X 1 , X 2 ]] = ( X 1 f − f ∂ x 3 a 1 ) ∂ x 3 , Y 5 = [ X 2 , [ X 1 , X 2 ]] = ( X 2 f − f ∂ x 3 a 2 ) ∂ x 3 . Th us, q = 5 a nd λ (1 , 2 , 3) = f , d (1 , 2 , 3) = 4 , λ (1 , 2 , 4) = X 1 f − f ∂ x 3 a 1 , d (1 , 2 , 4) = 5 , λ (1 , 2 , 5) = X 2 f − f ∂ x 3 a 2 , d (1 , 2 , 5) = 5 . Let us put D 1 = 4 , D 2 = 5 and, b y (5.24), µ 1 = | f | , µ 2 = | X 1 f − f ∂ x 3 a 1 | + | X 2 f − f ∂ x 3 a 2 | . In this situation Σ 1 = K, Σ 2 = { x ∈ K : µ 1 ( x ) = 0 } = { x ∈ K : f ( x ) = 0 } . Hence, r (2) = min x ∈ Σ 2 r x = min x ∈ Σ 2 max {| X 1 f ( x ) | , | X 2 f ( x ) |} > 0 . Let Ω 2 = ∪ x ∈ Σ 2 Ω ′ ∩ B ( x, r (2) ), with ¯ Ω ′ = K , and let K 1 = Σ 1 \ Ω 2 . Since K 1 ⊆ { x ∈ K : f ( x ) 6 = 0 } , if K 1 6 = ∅ then r (1) = min x ∈ K 1 r x = min x ∈ K 1 | f ( x ) | > 0 . Finally , if K 1 6 = ∅ then ˜ r 0 = min { r (1) , r (2) } , while if K 1 = ∅ then ˜ r 0 = r (2) . In next exa mple we show a s ub elliptic-type estimate for nonsmo oth vector ﬁelds. The argument of the pro of b elow is due to E rmanno Lanconelli (unpublished). NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 33 Prop ositio n 6.2 (H¨ ormander–type estimate [H]) . L et X 1 , . . . , X m b e a family of ve ctor ﬁ elds of step s and in the class A s . Then, given Ω ′ ⊂⊂ Ω , and ε ∈ ]0 , 1 /s [ , ther e is e r 0 and C > 0 such that su ch that, for any f ∈ C 1 (Ω) , (6.1) [ f ] 2 ε := Z Ω ′ × Ω ′ , d ( x,y ) ≤ e r 0 | f ( x ) − f ( y ) | 2 | x − y | n +2 ε dxdy ≤ C Z Ω X j | X j f ( y ) | 2 dy . Pr o of. W e just sk etch the pro of, leaving so me details to the rea der. F or an y I ∈ S , let Ω I := { x ∈ Ω : I 0 ( x ) = I } , w he r e I 0 ( x ) comes from the pro of of Theo rem 5.8, together with  0 =  0 ( x ) ≥ e r 0 , see the discussio n b e fore equatio n (5.28). If x ∈ Ω I , we have B ( x,  0 ) ⊂ E I ( x, Q I ( C  0 )), where the biLipschitz map E I satisﬁes C − 1 ≤ | det dE I ( x, h ) | ≤ C , for a.e. h ∈ Q I ( C  0 ). Thus, [ f ] 2 ε = Z Ω ′ × Ω ′ d ( x,y ) ≤ e r 0 | f ( x ) − f ( y ) | 2 | x − y | n +2 ε dxdy ≤ X I ∈S Z Ω I dx Z d ( x,y ) ≤  0 dy | f ( x ) − f ( y ) | 2 | x − y | n +2 ε ≤ C X I Z Ω I dx Z Q I ( C  0 ) dh | f ( x ) − f ( E I ( x, h )) | 2 | x − E I ( x, h ) | n +2 ε . Now observe that, arguing as in the pro of o f Lemma 5.1 4, we hav e | E I ( x, h ) − x | ≥ C − 1 | h | , if k h k I ≤ C  0 . Let δ 0 = max x ∈ K  0 ( x ). Next we follow the argument in [LM]. W rite E I ( x, h ) = γ I ( x, h, T ( h )), wher e γ I ( x, h, t ), t ∈ [0 , T ( h )] is a c ont r ol function , with the prop erties describ ed in [LM]. There fo re [ f ] 2 ε = X I Z Ω I dx Z Q I ( C δ 0 ) dh | f ( x ) − f ( E I ( x, h )) | 2 | h | n +2 ε ≤ C Z Q I ( C δ 0 ) dh | h | n +2 ε Z Ω I dx    Z T ( h ) 0 dt | X f ( γ I ( x, h, t )) |    2 ≤ C Z Q I ( C δ 0 ) dh | h | n +2 ε n Z T ( h ) 0 dt  Z Ω I dx | X f ( γ I ( x, h, t )) | 2  1 / 2 o 2 ≤ C Z Q I ( C δ 0 ) dh | h | n +2 ε T ( h ) 2 k X f k 2 L 2 (Ω) ≤ C k X f | 2 L 2 (Ω) , bec ause x 7→ γ I ( x, h, t ) is a change of v ariable, b y e s timate T ( h ) ≤ k h k I ≤ | h | 1 /s and the strict inequality ε < 1 /s .  The b order line inequa lit y k f k 1 /s ≤ C k X f k L 2 , which can no t be obtained with the a r gument ab ove, was pr ov ed in the smo oth case by Rothschild and Stein [Ro S]. 7. Proof of Proposition 2. 4 Here we pr ov e Pro p osition 2.4. By deﬁnition, (2.1) mea ns that for a ll j, k ∈ { 1 , . . . , m } and | w | ≤ s − 1 there is a b ounded function X j ( X k f w ) such that for a ny test function ψ ∈ C ∞ c ( R n ), (7.1) Z ( X k f w )( X j ψ ) = − Z { X j ( X k f w ) + div( X j ) X k f w } ψ . If D = ∂ j 1 · · · ∂ j p for some j 1 , . . . , j p ∈ { 1 , . . . , n } is an E uclidean deriv ative, denote by | D | = p its order . It is unders to o d that a deriv ative of o rder 0 is the iden tity . The ﬁrst item of Pro p o sition 2 .4 is a consequence of the following lemma: 34 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI Lemma 7.1. L et X 1 , . . . , X m b e ve ctor ﬁelds in A s . Then for any wor d w with | w | ≤ s and for any Euclide an derivative D of or der | D | = p ∈ { 0 , . . . , s − | w |} , we have (7.2) sup K    D f σ w − ( D f w ) ( σ )    ≤ C σ. Note that, the case p = 0 of (7.2) provides the pro of of item 1 of Pr op osition 2.4. Observe also that, if | w | = s , then we ha ve | f w − f σ w | ≤ | f w − ( f w ) σ | + | f σ w − ( f w ) σ | . Lemma 7.1 gives the estimates of the second term. The ﬁrst one is estimated by means of the cont inuit y mo dulus of f w , which is not included in L in (2.2). Pr o of of L emma 7.1. W e arg ue b y induction on | w | . If | w | = 1 , then the left hand side of (7.2) v anishes. Assume that for some ℓ ∈ { 1 , . . . , s − 1 } , (7.2) holds for any word w of length ℓ and for eac h D with | D | ≤ s − ℓ . Let v = kw b e a word of length | k w | = ℓ + 1 . W e must s how that for a n y Euclidea n deriv ative D of order 0 ≤ | D | ≤ s − | v | , (7 .2) holds. W e have f v = X k f w − X w f k and f σ v = X σ k f σ w − X σ w f σ k . W e ﬁrst prov e (7.2) whe n the o rder of D satisﬁes 1 ≤ | D | ≤ s − | v | = s − ℓ − 1, which can o c c ur only if ℓ ≤ s − 2 (in particular this implies s ≥ 3). The easier case is when D is the iden tity opera tor and it will be proven b elow. D f σ v − ( D f v ) ( σ ) = D  X σ k f σ w − X σ w f σ k  −  D ( X k f w − X w f k )  ( σ ) = D  X σ k f σ w  − ( D X k f w ) ( σ ) − { D ( X σ w f σ k ) −  D X w f k  ( σ ) } =: ( A ) − ( B ) . Omitting summation sign on α = 1 , . . . , n , w e may write ( A ) = ( D f α k ) ( σ )  ∂ α f σ w − ( ∂ α f w ) ( σ )  +  ( D f α k ) ( σ ) ( ∂ α f w ) ( σ ) −  ( D f α k ) ∂ α f w  ( σ )  + ( f α k ) ( σ )  D ∂ α f σ w − ( D ∂ α f w ) ( σ )  +  ( f α k ) ( σ ) ( D ∂ α f w ) ( σ ) − ( f α k D ∂ α f w  ( σ )  =: ( A 1 ) + ( A 2 ) + ( A 3 ) + ( A 4 ) . The e s timate | ( A 1 ) | + | ( A 3 ) | ≤ C σ follo ws from the induction as sumption. T o estimate ( A 4 ) observe that | ( A 4 ) | =    Z  ( f α k ) ( σ ) ( x ) − f α k ( x − σ y )  D ∂ α f w ( x + σ y ) ϕ ( y ) dy    ≤ C σ, bec ause f k is Lipschitz, while D ∂ α f w ∈ L ∞ lo c . Indeed, since | w | = ℓ , f w ∈ W s − ℓ, ∞ . Moreov er, D has leng th at most s − ℓ − 1 so that D∂ α has leng ht a t most s − ℓ . The estimate of ( A 2 ) is analog ous to that of A 4 . J ust recall that D f α k is Lipschitz and ∂ α f w is b ounded. Next we estimate ( B ). ( B ) = D (( f α w ) σ ∂ α f σ k ) − ( D ( f α w ∂ α f k )) ( σ ) = { D ( f α w ) σ − ( D f α w ) ( σ ) } ∂ α f σ k +  ( D f α w ) ( σ ) ∂ α f σ k − (( D f α w ) ∂ α f k ) ( σ )  + ( f α w ) σ D ∂ α f σ k − ( f α w ( D ∂ α f k )) ( σ ) =: ( B 1 ) + ( B 2 ) + ( B 3 ) . The term ( B 1 ) can be estimated by the inductive assumption. Mor eov er, | ( B 2 ) | =    Z  ∂ α f σ k ( x ) − ∂ α f k ( x − σ y )  D f α w ( x − σ y ) ϕ ( y ) dy    ≤ C σ, NONSMOOTH H ¨ ORMANDER VE CTOR FIELDS 35 bec ause ∂ α f k is Lipschitz and D f α w ∈ L ∞ lo c . Finally | ( B 3 ) | =    Z  ( f α w ) ( σ ) ( x ) − f α w ( x − σ y )  D ∂ α f k ( x − σ y ) ϕ ( y ) dy    ≤ C σ. Indeed, since | w | ≤ s − 2, f α w is loc a lly Lipsc hitz. Moreov er, since the length o f the deriv ative D ∂ α is at most s − 1 and f k ∈ W s − 1 , ∞ lo c , we hav e D∂ α f k ∈ L ∞ . Next we lo ok at the case wher e D has length zer o, i.e. D is the identit y o p er ator. W e hav e to estima te, for v w ith | v | ≤ s , the diﬀerence f σ v − ( f v ) ( σ ) . W rite v = k w , where k ∈ { 1 , . . . , m } . Thus f σ v − ( f v ) ( σ ) = X σ k f σ w − X σ w f σ k − ( X k f w ) ( σ ) + ( X w f k ) ( σ ) = ( f α k ) ( σ ) { ∂ α f σ w − ( ∂ α f w ) ( σ ) } +  ( f α k ) ( σ ) ( ∂ α f w ) ( σ ) − ( f α k ∂ α f w ) ( σ )  − { ( f α w ) σ ∂ α f σ k − ( f α w ∂ α f k ) ( σ ) } = ( S 1 ) + ( S 2 ) + ( S 3 ) . Now ( S 1 ) can be estimated by the inductive assumption. Moreover, | ( S 2 ) | =    Z { ( f α k ) ( σ ) ( x ) − f α k ( x − σ y ) } ∂ α f w ( x − σ y ) ϕ ( y ) dy    ≤ C σ, bec ause f α k is lo cally Lipschitz contin uous a nd ∂ α f w is lo cally b ounded, sinc e | w | ≤ s − 1 . Fina lly | ( S 3 ) | =    Z  ( f α w ) σ ( x ) − f α w ( x − σ y )  ∂ α f k ( x − σ y ) ϕ ( y ) dy    ≤ C σ, bec ause f w is Lipschitz and ∂ α f k is b ounded. This co ncludes the pro of of the the ﬁrst item of Prop os ition 2 .4.  Pr o of of Pr op osition 2.4, item 2. W e need to show that, for a ny j, k ∈ { 1 , . . . , m } , | w | = s − 1 we hav e the estimate | X σ j X σ k f σ w | ≤ C , unifor mly in x ∈ K and σ ≤ σ 0 . W rite X σ j X σ k f σ w = X σ j ( X k f w ) ( σ ) + X σ j  X σ k f σ w − ( X k f w ) ( σ )  =: M + N . Now, letting ϕ σ ( ξ ) = σ − n ϕ ( ξ /σ ), we hav e M ( x ) = ( f α j ) ( σ ) ( x ) ∂ x α Z X k f w ( x − σ y ) ϕ ( y ) dy = − Z ( f α j ) σ ( x ) X k f w ( z ) ∂ z α ( ϕ σ ( x − z )) dz = − Z f α j ( z ) X k f w ( z ) ∂ z α ( ϕ σ ( x − z )) dz + Z  ( f α j ) σ ( x ) − f α j ( z )  X k f w ( z ) ( ∂ α ϕ ) σ ( x − z ) σ dz . The ﬁrst line can b e estimated integrating by parts by means of (7.1). The es timate of the second line follows fro m the Lipsc hitz contin uity o f the functions f i . 36 ANNAMARIA MONT ANARI AND DANIELE MORBIDELLI Next we control N . N ( x ) = ( f α j ) ( σ ) ( x ) ∂ x α n X σ k Z f w ( x − σ y ) ϕ ( y ) dy − Z ( X k f w )( x − σ y ) ϕ ( y ) dy o = ( f α j ) ( σ ) ( x ) ∂ x α n Z ( f β k ) σ ( x ) ∂ β f w ( z ) ϕ σ ( x − z ) dz − Z f β k ( z ) ∂ β f w ( z ) ϕ σ ( x − z ) dz o = ( f α j ) ( σ ) ( x ) Z n ( ∂ α f β k ) σ ( x ) ∂ β f w ( z ) ϕ σ ( x − z ) + [( f β k ) σ ( x ) − f β k ( z )] ∂ β f w ( z ) 1 σ ( ∂ α ϕ ) σ ( x − z ) o dz The estimate is co ncluded, b ecause ∂ β f w is b ounded, while | ( f β k ) σ ( x ) − f β k ( z ) | ≤ C σ .  References [BBP] M. Bramanti, L. Br andolini , M. Pedroni, B asi c prop erties of nonsmo oth H¨ ormander’s v ector ﬁelds and Po incar´ e’s inequality , preprint [CM] G. Citti, A. Mont anari, Regularity prop erties of solutions of a class of ell iptic-parab ol ic nonlinear Levi t yp e equations, T rans. Amer. Math. 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