The Heisenberg group and Pansu s Theorem

Transcription

1 The Heisenberg group and Pansu s Theorem July 31, 2009 Absrac The goal of hese noes is o inroduce he reader o he Heisenberg group wih is Carno- Carahéodory meric and o Pansu s differeniaion heorem. As hey are very similar, we will firs sudy Rademacher s heorem abou Lipschiz maps and hen see how he same echnique can be applied in he more complex seing of he Heisenberg group. Conens 1 The Heisenberg group Coninuous Heisenberg group Using he ools from differenial geomery Naive definiion Dilaaions and Derivaives Discree Heisenberg group Bilipschiz maps Rademacher s heorem Pansu s heorem Proof of Pansu s heorem 10 1

2 1. The Heisenberg group 2 1 The Heisenberg group 1.1 Coninuous Heisenberg group [1.A] Definiion (Real Heisenberg group) The Heisenberg group H 3 (R) is he group of 3 3 upper riangular marices of he form 1 X Z 0 1 Y, where X, Y, Z R The Heisenberg group is mainly sudied for he srange and remarkable properies of a paricular meric, called Carno-Carahéodory or subriemannian meric. The idea behind his meric is fairly simple, bu slighly counerinuiive : a classical (Riemannian) meric is defined as he lengh of he minimal pah beween wo given poins, over all possible pahs on he manifold, or here on he group. The CC meric will be defined almos he same way, bu he infimum will no be aken over all he possible pahs, bu only over some special pahs, which will be chosen angen o a given disribuion. A rigorous definiion of his meric requires ools from differenial geomery, wih which he reader migh no be familiar. In his case, he should sraigh away jump o he naive definiion, which should give him an idea of how i works wihou asphyxiaing him wih oo many new conceps Using he ools from differenial geomery We can firs noice ha he firs definiion of he Heisenberg group is equivalen o he following one [1.B] Definiion (Real Heisenberg group 2) The Heisenberg group is he simply conneced Lie group whose Lie algebra is R 3 (ξ, η, ζ) wih [ξ, η] = ζ, all he oher brackes being zero. ξ and η will span wha we will call he horizonal disribuion, and he idea of he Carno-Carahéodory meric is o jus consider curves angen o his disribuion, which are called horizonal curves. [1.C] Definiion (Carno-Carahéodory meric) Le H be he disribuion (i.e. a vecor subbundle T H 3 of he angen bundle of H 3 ) spanned by lef ranslaion of {ξ, η}. A curve on H 3 is called horizonal if i is angen o H. An arbirary norm on T H 3 being chosen, he CC meric on H 3 is hen defined by, for p, q H 3 d CC (p, q) = inf{lengh of he smooh horizonal curves ha connec A and B}, his disance being infinie if here is no such curve. This definiion needs an explanaion. I may seem counerinuiive o ry o connec wo poins wih only horizonal curves. The same definiion on R 3 would give no resuls : he disance beween wo poins would be he Euclidean one if hey re boh on he same horizonal plane, and infinie if hey re no. The main difference is ha, unlike he R 3 case, he horizonal disribuion we have defined on H 3 is clearly no a foliaion. Recall ha he Frobenius heorem saes ha a subbundle H of he angen bundle is inegrable if, for any wo vecor fields X, Y on H, he Lie bracke [X, Y ] akes values in H as well. This is obviously no he case here, and he behavior of he Heisenberg group is bes described by he following heorem by Chow, which is he saring poin of Carno Carahéodory geomery [1.D] Definiion (Bracke generaing disribuion) Given a collecion {X a } of vecor fields, form is Lie Hull, he collecion of all vecor fields {X a, [X b, X c ], [X a, [X b, X c ]]...} generaed by Lie brackes of he X a. We say ha he collecion {X a } is bracke generaing if his Lie hull spans he whole angen bundle. A disribuion H T Q is called bracke generaing if any local horizonal frame {X a } for he disribuion is bracke generaing.

3 1. The Heisenberg group 3 [1.E] Theorem (Chow) If a disribuion H T Q is bracke generaing hen he se of poins ha can be conneced o A Q by a horizonal pah is he componen of Q conaining A. As he horizonal disribuion we have defined on he Heisenberg group is clearly bracke generaing, his heorem explains why on his paricular example, he Carno Carahéodory meric is always finie Naive definiion We will see how he CC meric works by sudying i in comparison wih he usual Euclidean meric. In he usual R 3 space, one can always connec wo disinc poins wih he sraigh line joining hem. However, if one ries o connec wo arbirary poins wih horizonal lines, one is doomed o fail, as he z-coordinae says consan on horizonal lines. This is because he R 3 group law is exremely regular and symeric : i is (x, y, z), (x, y, z ) (x + x, y + y, z + z ). Hence, he horizonal plane, which is spanned by ((1, 0, 0), (0, 1, 0)), is exremely regular oo, and maches he inuiive idea of a plane. Le us see wha happens wih he Heisenberg group now. The main difference is ha he group law is no as regular, and is wised : ((X, Y, Z), (X, Y, Z )) (X + X, Y + Y, Z + Z + XY ); in paricular, if X, Y 0, (X, Y, 0) + (X, Y, 0) has a nonzero z-coordinae. If we define he horizonal disribuion (we use his erm raher han plane, because i is no one) as he one spanned by ((1, 0, 0), (0, 1, 0)), his shows why you can connec any arbirary pair of poins using only horizonal curves, i.e. curves angen o he horizonal disribuion! The CC meric is hen defined fairly easily, as he lengh of he minimal horizonal pah beween wo poins. The scepic reader migh appreciae some picures These picures, couresy of Parick Masso, show wha he horizonal disribuion looks like, in caresian and cylindrical coordinaes. Horizonal curves are curves which always say angen o he lile planes drawn on he picure. The wis can be easily seen on boh picures : he horizonal planes are no horizonal. This is why horizonal curves can reach he enire group. To illusrae his, he las picure, couresy of Pierre Pansu, shows how o connec x = (0, 0, 0) and x = (0, 0, z) wih horizonal curves.

4 1. The Heisenberg group 4 A good way o familiarize oneself wih he CC meric is o compare he firs picure wih his one o check ha he used curves are indeed horizonal Dilaaions and Derivaives The Heisenberg group can be provided wih a dilaaion δ, which is he analog of a classical homoheic ransformaion δ λ : x λx in he Euclidean R n space. We define δ (X, Y, Z) = (X, Y, 2 Z), so ha [δ (ξ), δ (η)] = δ (ζ) and d CC (δ (x), δ (y)) = (d CC (x, y)). This allows us o define a derivaive on he f(x) Heisenberg group : as he classical derivaive a 0 is defined by lim 0 = lim λ 0 (δ λ ) 1 (f(δ λ x)), one can define a derivaive on he Heisenberg group : df (g)(h) = lim 0 (δ 1 )((F (g)) 1 F (gδ h)). As in he Euclidean case, where we expec he derivaive o be linear, Pansu s derivaive needs o be an homomorphism. Pansu s heorem, which will be proved in he hird par and is he main resul of his lecure, is ha Lipschiz maps (for he Carno-Carahéodory meric) are differeniable almos everywhere on he Heisenberg group. 1.2 Discree Heisenberg group We now inroduce he discree Heisenberg group, which is he same as he real one excep ha he enries are now aken in Z. [1.F] Definiion (Discree Heisenberg group) The Heisenberg group H 3 (Z) is he group of 3 3 upper riangular marices of he form 1 X Z 0 1 Y, where X, Y, Z Z From is definiion, i is easily seen ha he discree Heisenberg group is generaed by and The main ool o sudy meric properies of discree groups finiely generaed is called he Cayley graph.

5 2. Bilipschiz maps 5 [1.G] Definiion (Cayley Graph) Suppose ha G is a group and S is a generaing se. The Cayley graph Γ = Γ(G, S) is a colored direced graph consruced as follows. Each elemen g of G is assigned a verex : he verex se V (Γ) of Γ is idenified wih G. Each generaor s of S is assigned a color c s. For any g G, s S, he verices corresponding o he elemens g and gs are joined by a direced edge of colour c s. Thus he edge se E(Γ) consiss of pairs of he form (g, gs), wih s S providing he color. To see wha a Cayley graph looks like, here is he example of he Cayley graph of he free group wih wo generaors. A graph is always naurally equipped wih a disance : he graph disance which is he disance of he shores pah beween wo edges. A remarkable fac abou he Heisenberg group is ha if he norm on he angen bundle is he l 1 norm, he disances defined by d CC H3(Z) and he graph disance are he same. This follows from he following observaion, which is fundamenal in he geomery of he Carno- Carahéodory meric : geodesics for he Carno-Carahéodory meric correspond o soluions of Dido s problem 1 in R 2. Le us explain wha his means : I is easily shown ha for an arbirary curve on he Heisenberg group, being angen o he horizonal disribuion is equivalen o saisfying dz xdy = 0. Hence we have z() = 0 x(s)y (s)ds which, by Green-Riemann (or Sokes) s formula is, up o a consan, exacly he area in R 2 beween he curve and he y axis. As he soluions o Dido s problem in R 2 for he l 1 norm are well known (hese are arcs of squares wih sides parallel o he axes), i follows easily ha he graph and CC disances are exacly he same. 2 Bilipschiz maps 2.1 Rademacher s heorem In his secion we will prove he following heorem, iniially saed by Rademacher, which is he analog of Pansu s heorem in he usual R n Euclidean space. 1 Dido s problem is he following varian of he isoperimeric problem : you are given a segmen [AB] and a consan L, and you need o find he curve wih lengh L beween A and B which maximizes he area beween he segmen and he curve. If he norm is Euclidean, he soluion is obviously a circle

6 2. Bilipschiz maps 6 [2.A] Theorem (Rademacher) Le n, m (N ) 2 and f : R n R m a C-Lipschiz map. Then f is differeniable almos everywhere. The proof of his heorem is spli ino wo pars. The firs one shows ha one can suppose m = 1, and he second par is he proof of he case n = 1, for which here is a more general heorem, firs saed by Lebesgue, which applies here because Lipschiz maps have bounded variaion. There are numerous proofs for boh hose resuls, and we have ried o find he easies ones, hese come from [GT98]. However, he proof of [2.A] which we presen here is prey hard o generalize o Heisenberg groups, his is why anoher one will be used in he hird par, and for he course iself. We chose o keep his one in he noes because i is he mos direc one and i gives a good grasp on how his works, while everyhing is kep hidden behind Lusin and Egorov heorems in he oher one. [2.B] Theorem (Lebesgue) Le f : [a, b] R be a bounded variaion map. Then f is differeniable almos everywhere. Proof of [2.A]. Le us prove he firs par of he heorem, i.e. prove i under he assumpion ha i is rue if n = 1. One can suppose ha m = 1, and ha s wha we do, and we wrie <, > for he Euclidean scalar produc, and S for he corresponding uni sphere. Two facs can be immediaely deduced from his. f(x) exiss for almos all x R n. If e S, f(x+e) f(x) has, for almos all x R n, a limi when 0, ha we will name L x (e). We claim ha for a fixed e S, for almos all x R n, L x (e) =< f(x), e >. Indeed, we will show ha for every map ϕ C wih compac suppor, (L R n x (e) < f(x), e >)ϕ(x)dx = 0, which proves he announced resul. The dominaed convergence heorem gives easily, as ϕ has compac suppor, f(x + e) f(x) ϕ(x)dx 0 L x (e)ϕ(x)dx (2.i) R n We also have R n R n f(x + e) f(x) ϕ(x)dx = R n ϕ(x e) ϕ(x) f(x)dx The dominaed convergence heorem shows ha his ends o R n < ϕ(x), e > f(x)dx as 0. By inegraion by pars, we also have (legiimae because he parial derivaives are defined almos everywhere). (2.ii) And 2.i, 2.ii, 2.iii show ha : R n < ϕ(x), e > f(x)dx = n e i i=1 = R n n e i i=1 ϕ x i f(x)dx R n f x i (x)ϕ(x)dx = < f(x), e > ϕ(x)dx R n R n (L x (e) < f(x), e >)ϕ(x)dx = 0 (2.iii)

7 2. Bilipschiz maps 7 which is wha we waned. Le A = {x R n, f(x)exiss}. We know ha µ(r n \A) = 0, and for a fixed e S, x A, f(x + e) f(x) 0 < f(x), e > (2.iv) We sill have o show ha x A, 2.iv is rue for all e S, and his uniformly in e. To prove his, le (e i ) i 1 be a dense sequence in S, and le B = {x A, i 1, f(x + e i) f(x) 0 < f(x), e i >} We have µ(r n \B) = 0. Bu if x B, he maps Φ : e S f(x+e) f(x) are all C-Lipschiz. Hence he convergence Φ (e i ) < f(x), e i > shows he uniform convergence of Φ o < f(x), e >, and hus he differeniabiliy almos everywhere of f. Before proving Lebesgue s heorem, we firs have o recall some properies of bounded variaion funcions. [2.C] Definiion (Bounded variaion funcions) A funcion f : I R is said o have bounded variaion if i saisfies one of he following equivalen condiions : 1. The oal variaion of f defined by is finie. n 1 V a,b (f) = sup{ f(x i+1 ) f(x i ), a x 1 < x 2 <.. < x n b} i=1 2. The graph of f is recifiable. 3. f is he he difference of wo nondecreasing funcions. The equivalence beween 1. and 2. is rivial, is clear, as a nondecreasing funcion has bounded variaion, is easy if we noice ha x V a,x (f) is nondecreasing on [a, b], and x V a,x (f) f(x) oo. Condiion 3. shows ha such a funcion is coninuous excep on a counable se. We will firs need he following lemma, which is a weak version of Viali s lemma. [2.D] Lemma Le (I n =]x n r n, x n + r n [) 1 n N be N inervals of R. Then here exiss a subse J [ 1, N ] such ha he (I j ) j J are pairwise disjoin and N n=1 I n j J ]x j 3r j, x j + 3r j [ which implies N µ( I n ) 3 µ(i j ) n=1 j J Proof of [2.D]. Le us sor he r n in a decreasing sequence R 1..R k. Take hen J 1 [ 1, N ] maximal such ha he (I j ) j J1 don inersec and have radius R 1. Then J 2 [ 1, N ] maximal such ha he (I j ) j J1 J 2 don inersec and such ha he (I j ) j J2 have radius R 2. Go on unil J k is defined, and hen define J = k i=1 J i. Then J verifies he given properies : he I j are pairwise disjoin by definiion, and he

8 2. Bilipschiz maps 8 oher propery follows immediaely from he following easy fac : If ]x r, x + r[ ]y s, y + s[ wih r s, hen ]x r, x + r[ ]y 3s, y + 3s[. We now have all he ools needed o prove Lebesgue s heorem. Proof of [2.B]. We will firs need some noaions. Wrie D for he he se (a mos counable) of disconinuiy poins of f. For x ]a, b[\d, wrie and define he following ses : f + (x) = lim sup y x f(y) f(x), f f(y) f(x) (x) = lim inf y x y x y x A + = {x ]a, b[\d, f + (x) = + }, A = {x ]a, b[\d, f (x) = } B = {x ]a, b[\d, f + (x) > f (x)} We will show ha µ(a + ) = µ(a ) = µ(b) = 0, which will prove Lebesgue s heorem. Le us begin wih A + and A. Suppose ad absurdum ha µ(a + ) > 0 and le K be an arbirary big consan. For x A +, here exiss I x =]a x, b x [ included in ]a, b[ and conaining x such ha f(b x ) f(a x ) K(b x a x ). R being separable, one can exrac a counable covering (I xn ) n N ou of I x, and here exis N 0 such ha N µ( I xn ) n=0 I xn ) 1 2 µ( n N By applying he lemma o his family, we now have a finie se F such ha : 1. The (I x ) x F are pairwise disjoin. 2. x F µ(i x) 1 6 µ( x ]a,b[ ). Hence (f(b x ) f(a x )) K 6 µ(a+ ) x F and V (f) K 6 µ(a+ ). K being arbirary, his conradics he fac ha f has bounded variaion. Suppose now ha µ(b) > 0. Then here exis (α, β) Q + Q such ha he se C = {x B, f + (x) > β + αandf (x) < β α} has a posiive measure (because Q is counable, dense in R, and a measure is counably addiive). By adding an affine funcion, one can suppose ha β = 0 Here begins he echnical par of he proof. For a bounded variaion map g, wrie l(g) for he lengh of is graph. Le F be an arbirary finie subse in [a, b] conaining a and b and a F he funcion inerpolaing affinely f a he poins of F. We will build anoher finie subse G [a, b] such ha l(a G ) l(a F ) + µ(c) 6 ( 1 + α 2 1) This will be absurd because by ieraing he same process wih G insead of F and so on, his shows ha he graph of f is no recifiable. Le us build his subse G. For x C\F, choose I x =]a x, b x [ included in ]a, b[ and conaining x such ha I x F = and { f(bx ) f(a x ) α(b x a x ) if a F is nondecreasing on I x. f(b x ) f(a x ) α(b x a x ) oherwise. The same echnique as he one used for A + gives us a finie subse E of C\F such ha he (I x ) x E are pairwise disjoin and µ(i x ) µ(c). We now jus have o show ha G = E F verifies he 6 x E announced inequaliy.

9 2. Bilipschiz maps 9 Insead of a complicaed rigorous proof, we will use he following picure, corresponding o he case where a F is always nondecreasing, he reader will have no rouble considering he general case which works he same way. Wih he noaions of he picure, AD is he graph of a F, ABD is he graph of a G, and we have l(a G ) AB + BD, hus Bu l(a F ) = AD AC + CD = (AC AB) + (AB + CD) This gives exacly he resul we announced. 2.2 Pansu s heorem (AC AB) + (AB + BD) AC AB + l(a G ) AC AB = (1 1 + α 2 )AC = (1 1 + α 2 )µ(c) Le us sae Pansu s heorem [Pan89], which is our main resul. I has a more general seing wih he Carno groups, bu for he sake of simpliciy we will only use see is special case wih he Heisenberg group. [2.E] Theorem (Pansu s heorem) Le f be a Lipschiz map from he Heisenberg group o anoher one or he Euclidean R n space. Then f is differeniable almos everywhere. The proof of his heorem will be delayed unil he 3rd par, bu we can already sae a relaively easy, bu essenial, corollary, iniially saed in [Sem96]. [2.F] Corollary (Semmes) A map f : (H 3, d CC ) (R n, d eucl ) is never bilipschiz. Proof of [2.F]. If f was bilipschiz, i would be differeniable almos everywhere, and as he definiion of he derivaive scales in he naural way, he differenial is bilipschiz oo. This gives a conradicion, because any homomorphism from he Heisenberg group o he Euclidean R n mus have a kernel which is a leas 1-dimensional (all commuaors mus be mapped o 0) and hence canno be bilipschiz. Such a corollary shows ha here is a fundamenal inerplay beween differeniaion heorems and bilipschiz embeddings, which can be used in boh direcions. Indeed, o see if we can find differenaion heorems abou maps beween a saring space X and an arbirary space Y, a good idea is o look for bilipschiz maps : heir exisence will prove ha differeniaion fails for such meric spaces. A good

10 3. Proof of Pansu s heorem 10 example of his echnique is he case Y = L (X). In his case, here is he classical Kuraowsky embedding : x d(x, ) d(x 0, ) where x 0 is a basepoin. I is easily seen ha i is an isomery, and ha shows ha here canno be any relevan differeniaion heorem from an arbirary space (e.g. H 3 ) and L (H). Anoher classical example is he case X = [0; 1] and Y = L 1 (R) : one can use he moving indicaor embedding : f : χ [0;] where χ [0;] is he indicaor funcion of [0; ]. I is easily shown ha his embedding is an isomery, and one can even show direcly ha his embedding is no differeniable almos everywhere, because he difference quoien ends o a Dirac δ, which is no in L 1. This coninuous group provides herefore a good example of a meric space which is hard o embed, bu i is a very absrac one which doesn apply o he heoreical compuer science problemaics we are ineresed in : embedding algorihms such as he one from Linial, London and Rabinovich are used on graphs, which are discree and finie meric spaces, and no on coninuous Lie groups. However, he following corollary shows ha he same conclusion holds for he discree Heisenberg group, giving a much handier and visual example of hard-o-embed meric spaces. [2.G] Corollary (Discree version) A map f : (H 3 (Z), d graph ) (R n, d eucl ) is never bilipschiz. Proof of [2.G]. Le f : H 3 (Z) R n be a L-bilipschiz map and wrie δ for he dilaaion on H 3 and he homoheic dilaaion on R n. Define f = δ f δ 1 : δ (H 3 (Z)) R n, i is clearly a family of L-bilipschiz maps, and he saring space can be rivially embedded in H 3 (R). As he L consan is uniform, he family is equiconinuous, and a varian of Ascoli s heorem (he usual heorem canno be applied here as he domains change wih he maps, bu he same demonsraion can be applied o our seing) shows ha here exiss a subsequence of his family which converges o f 0 : H 3 (R) R n. f 0 being L-bilipschiz oo. As we have seen, his is absurd, and herefore he firs map f canno exis. The las proof is relaively easy bu is main drawback is ha i can be applied when he end space is no locally compac (because Ascoli s heorem can no be applied in such spaces), which will be he case in he oher courses (for example L p and especially L 1 spaces). However, here is anoher consrucion, based on ulralimis and asympoic cones, which yields he same resul : one can associae a map on he real group o a map on he discree group, wih he same Lipschiz consan. Therefore, any resul on he real Heisenberg group, which is easier o manipulae due o he numerous ools we have for Lie groups, can be exended o he discree Heisenberg group. 3 Proof of Pansu s heorem As in he proof of Rademacher s heorem, he proof is spli in wo pars. The firs sep is o reduce he problem o a one dimensional problem, he second one is o prove he differeniabiliy almos everywhere of recifiable curves. However, he second par is no needed in his course. Indeed, using only he firs one and Lebesgue s heorem, we shows ha every Lipschiz map f : H 3 (R) R n is differeniable almos everywhere, which is he resul we need. The second par is useful o consider he case where he range is anoher Heisenberg group or a Carno group (i.e. a group on which one can define a Carno- Carahéodory meric he same way ha we did for he Heisenberg group), bu i is of no use for compuer science problems, and we will herefore omi his par.

11 3. Proof of Pansu s heorem 11 [3.A] Proposiion Le f : O R n be a Lipschiz map where O is an open subse of H. Le a, b H be such ha for almos all x H, he limis df (x)(a) = lim 0 δ 1 (F (x)) 1 F (xδ a) df (x)(b) = lim 0 δ 1 (F (x)) 1 F (xδ b) exis. Then for almos all c of he form x = δ u (a)δ u (b) and for almos all x H, he limi df (x)(c) = lim (δ 1 )(F (x)) 1 F (xδ c) 0 exiss and is δ u df (x)(a)δ u df (x)(b). Proof of [3.A]. Le us wrie M for he Haar mesure on H (which exiss because H is a locally compac group), i coincides wih he Lebesgue measure in he case of he Heisenberg group. We firs noe ha if df (x)(a) exiss, hen df (x)(δ a) exiss for all so we can suppose c = ab wih d(1, a) = 1. The following facs resul from he classical Egorov and Lusin heorems : There exiss, for all τ > 0 a closed subspace F H such ha M(N\F ) < τ and 1. df (x)(a), df (x)(b) exis for all x F. 2. x df (x)(b) is coninuous on F. 3. (δ 1 )(F (x)) 1 F (xδ b) ends o df (x)(b), uniformly for x F. I would suffice o show ha xδ a F. Indeed, we have δ 1 (F (x)) 1 F (xδ c) = (1)(2)(3) where ends o df (x)(a) because of 1. (1) = δ 1 (F (x)) 1 F (xδ 1 a) ends o 1 because of 3. and (2) = δ 1 (F (xδ 1 a)) 1 F (xδ 1 aδ b)(df (xδ a)(b)) (3) = df (x(δ ) 1 a)(b) ends o df (x)(b) because of 2. However, his is no always rue. Bu if x is a densiy poin of F, i.e. if M(B(x, r)\f ) M(B(x, r)) ends o 0 wih r, hen here is a poin of F very close o xδ b. Se λ = d(xδ 1 a, F ) and xδ 1 a a poin where his disance is reached. As x is a densiy poin, we see easily ha d(xδ 1 a, xδ 1 a ) ends o 0 as ends o 0, and hus a a. We have hen δ 1 (F (x) 1 F (xδ c)) = (1)(2)(3)(4)(5) where ends o df (x)(a) because of 1. (1) = δ 1 (F (x)) 1 F (xδ a) (2) = δ 1 (F (xδ a)) 1 F (xδ a ), (3) = δ 1 (F (xδ a ) 1 F (xδ a δ b))(df (xδ a )(b)) 1

12 REFERENCES 12 ends o 1 because of 3. ends o df (x)(b) because of 2. and (4) = df (xδ a )(b) (5) = δ 1 (F (xδ a δ b) 1 F (xδ aδ b)) If f is M-Lipschiz, we have d((2), 1) = 1 d(f (xδ a), F (xδ a )) M 1 d(δ a, δ a ) = Md(a, a ) which ends o 0. And d((5), 1) = 1 d(f (xδ a δ b, F (xδ aδ b))) M 1 d(δ (ab), δ (a b)) = Md(ab, a b) which ends o 0 as ends o 0. This shows he exisence of he limi of (δ 1 )((F (x)) 1 F (xδ c)) for x a densiy poin of F, i.e. almos everywhere in F. [3.B] Corollary (Reducion o dimension 1) Le f be a bilipschiz map from O open subse of H o R n. If every curve in H s f(x exp(sv)) (where v is a horizonal vecor in he Lie algebra) is differeniable almos everywhere, hen f is differeniable almos everywhere and a df (x)(a) is a group homomorphism. Proof of [3.B]. We jus have o verify he uniform convergence for a fixed x of f (a) = (δ 1 )((F (x)) 1 F (xδ a)) o df (x)(a). One easily shows ha H as a group is generaed by he subses (δ s exp(ξ)) s R+ and (δ s exp(η)) s R+ (his follows from he fac ha he Lie algebra of H is generaed by he wo horizonal vecors ξ and η). As he speed of convergence of f o df (x) only depends on sup a j and he speed of f (v j ) o d F (x)(v j ), his is he uniform convergence we wan. Wih he resuls menioned in he second par, his heorem is a powerful ool o show he absence of bilipschiz embeddings from H 3 o R n. As he finie dimensional case is no he only case of ineres, we migh wonder if he same proof can be generalized. In fac, he same proof can easily be exended o arbirary Hilber spaces, or more generally o spaces Y wih he Radon-Nikodym propery, which saes ha every Lipschiz map f : R Y is differeniable almos everywhere. However, as L 1 doesn have he Radon-Nikodym propery he non-embeddabiliy resul does no follow from Pansu s heorem. References [GT98] S. Gonnord and N. Tosel. Calcul différeniel pour l agrégaion. Ellipses, [Mon02] R. Mongomery. A Tour of Subriemannian Geomeries, Their Geodesics, and Applicaions, volume 91 of Mahemaical Surveys and Monographs. American Mahemaical Sociey, 2002.

13 REFERENCES 13 [Pan89] P. Pansu. Mériques de Carno-Carahéodory e quasiisoméries des espaces symériques de rang un. Ann. of Mah., [Sem96] S. Semmes. On he nonexisence of bilipschiz paramerizaions and geomeric problems abou a weighs. Revisa Maemáica Iberoamericana, 1996.