A variant of Gromov's problem on Hölder equivalence of Carnot groups
aa r X i v : . [ m a t h . M G ] J u l A variant of Gromov’s problem on H¨older equivalence of Carnotgroups
Derek Jung ∗ Department of MathematicsUniversity of Illinois at Urbana-Champaign1409 West Green St.Urbana, IL 61801 [email protected]
October 17, 2018
Abstract
It is unknown if there exists a locally α -H¨older homeomorphism f : R → H for any <α ≤ , although the identity map R → H is locally -H¨older. More generally, Gromov asked:Given k and a Carnot group G , for which α does there exist a locally α -H¨older homeomorphism f : R k → G ? Here, we equip a Carnot group G with the Carnot-Carath´eodory metric. In2014, Balogh, Haj lasz, and Wildrick considered a variant of this problem. These authors provedthat if k > n , there does not exist an injective, ( +)-H¨older mapping f : R k → H n that isalso locally Lipschitz as a mapping into R n +1 . For their proof, they use the fact that H n ispurely k -unrectifiable for k > n . In this paper, we will extend their result from the Heisenberggroup to model filiform groups and Carnot groups of step at most three. We will now requirethat the Carnot group is purely k -unrectifiable. The main key to our proof will be showingthat ( +)-H¨older maps f : R k → G that are locally Lipschitz into Euclidean space, are weaklycontact. Proving weak contactness in these two settings requires understanding the relationshipbetween the algebraic and metric structures of the Carnot group. We will use coordinates ofthe first and second kind for Carnot groups. Contents J k ( R )
104 Result for Carnot groups of step at most three 155 Future work 21 ∗ Supported by U.S. Department of Education GAANN fellowship P200A150319.
Key Words and Phrases: sub-Riemannian geometry, Carnot groups, H¨older mappings, geometric measure theory, jetspaces, Gromov conjecture, unrectifiability
Primary 53C17, 22E25; Secondary 53C23, 26B35, 49Q15, 58A20 Introduction
A Lie algebra g is said to have an r -step stratification if g = g ⊕ g ⊕ · · · ⊕ g r , where g ⊆ g is a subspace, g j +1 = [ g , g j ] for all j = 1 , . . . , r −
1, and [ g , g r ] = 0. A Carnotgroup is a connected, simply-connected, nilpotent Lie group with a stratified Lie algebra. If theLie algebra of a Carnot group G admits an r -step stratification, then we will say G is step r . EachCarnot group can be identified with a Euclidean space equipped with a metric structure and agroup operation arising from its Lie algebra structure.It is natural to ask the following general question:When are two Carnot groups equivalent?In [11], Pansu proved that two Carnot groups are biLipschitz homeomorphic if and only if theyare isomorphic. With the problem of biLipschitz equivalence somewhat well-understood, we can goon to ask when two Carnot groups are H¨older equivalent.In [7], Gromov considered the problem of H¨older equivalence of Carnot groups: If a Carnotgroup G is identified with R n equipped with a group operation, for which α does there exist alocally α -H¨older homeomorphism f : R n → G ? If such α exist, what is the supremum of the set ofsuch α ? Here, we do not require any regularity of f − beyond continuity.Before we discuss past work on this problem, we will comment on the notation that will be usedthroughout this paper. We will simply write R n to denote Euclidean space equipped with additionand the standard Euclidean metric. We will write ( R n , · ) to denote a Carnot group equipped withcoordinates of the first or second kind and with the Carnot-Carath´eodory metric. When we equip aCarnot group with coordinates of the first or second kind, it is implied that we are taking coordinateswith respect to a basis compatible with the stratification of its Lie algebra. We will introduce thesetwo systems of coordinates and the Carnot-Carath´eodory metric for Carnot groups in section 2. Insection 3, we will discuss coordinates of the second kind for a class of jet spaces: the model filiformgroups. We will begin section 4 by looking at the geometry of Carnot groups of step at most three.Nagel, Stein, and Wenger [10, Proposition 1.1] proved the existence of α as above: Proposition 1.1.
Let ( R n , · ) be a step r Carnot group. Then id : R n → ( R n , · ) is locally r -H¨olderand id : ( R n , · ) → R n is locally Lipschitz. On the other hand, Gromov [7, Section 4] used an isoperimetric inequality for Carnot groups[16] to prove that if there exists a locally α -H¨older homeomorphism f : R n → ( R n , · ), then α ≤ n − Q − . Here, Q denotes the Hausdorff dimension of ( R n , · ) with respect to its cc-metric.Beyond these results, little is known about this problem. For example, in the case of the firstHeisenberg group, the supremum of α for which there exists a locally α -H¨older homeomorphism f : R → H is only known to lie between 1 / / C ,α ( X ; Y ) of α -H¨older maps f : X → Y . Definition 1.2.
Fix metric spaces (
X, d X ) , ( Y, d Y ) and α >
0. We say a map f : X → Y is ofclass C ,α + ( X ; Y ) if there exists a homeomorphism β : [0 , ∞ ) → [0 , ∞ ) such that(1.1) d Y ( f ( a ) , f ( b )) ≤ d X ( a, b ) α β ( d X ( a, b )) for all a, b ∈ X. We will sometimes simply write C ,α + if the domain and target are clear.2 emark 1.3. Suppose
X, Y are metric spaces with X bounded. It is easy to check that C ,η ( X ; Y ) ⊆ C ,α + ( X ; Y ) ⊆ C ,α ( X ; Y ) . whenever 0 < α < η . Thus, C ,α + ( X ; Y ) can thought of as a right limit of H¨older spaces.For certain models of model filiform groups and Carnot groups of small step, we will prove thatthere do not exist ( α +)-H¨older equivalences for α ≥ /
2. Before stating our paper’s two mainresults, we make the following definition.
Definition 1.4.
A Carnot group ( R n , · ) is said to be purely k -unrectifiable if for every A ⊆ R k and Lipschitz map f : A → ( R n , · ), we have H kcc ( f ( A )) = 0 . Here, we endow ( R n , · ) with the Carnot-Carath´eodory metric to be described in subsection 2.2.Ambrosio and Kirchheim proved that H is purely k -unrectifiable for k = 2 , , k -unrectifiable if and only if itshorizontal layer does not contain a Lie subalgebra of dimension k [9, Theorem 1.1]. In particular, H n is purely k -unrectifiable for all k > n . In 2014, Balogh, Haj lasz, and Wildrick provided adifferent proof of this last result by using approximate derivatives and a weak contact condition [2,Theorem 1.1]. In the process, they prove that a Lipschitz mapping of an open subset of R k , k > n ,into H n has an approximate derivative that is horizontal almost everywhere.Motivated by Gromov’s H¨older equivalence problem, Balogh, Haj lasz, and Wildrick go on toprove that one cannot embed R k , k > n , into H n via a sufficiently regular ( α +)-H¨older mapping.More specifically, they prove that if k > n and Ω ⊆ R k is open, then there is no injective mappingof class C , + (Ω , H n ) that is locally Lipschitz as a mapping into R n +1 [2, Theorem 1.11]. Themain key to their proof is showing that if such a map existed, then it would have to be horizontalalmost everywhere. Notice that Remark 1.3 combined with the identity map id : R → H beinglocally -H¨older suggest that this result is sharp except for the extra local Lipschitz assumption.In this paper, we will extend the result in the previous paragraph to more general Carnotgroups, specifically model filiform groups and Carnot groups of step at most three. The modelfiliform groups can be realized as the class of jet spaces J k ( R ). In these groups, there are fewnontrivial bracket relations relative to the step. For Carnot groups of small step, the Baker-Campbell-Hausdorff has a simple form; this allows one to describe the structure (e.g., left-invariantvector fields and contact forms) of the Carnot group in coordinates and perform computations. TheLie algebraic properties of these two classes of Carnot group make them ideal settings to generalizethe result from the previous paragraph. The proofs for these Carnot groups will again boil downto showing the almost everywhere horizontality of certain C , + mappings into these groups.The standard basis { e ( k ) , e k , . . . , e } of Lie ( J k ( R )) is such that [ e j , e ( k ) ] = e j − , j ≥
1, are theonly nontrivial bracket relations. We will equip J k ( R ) with coordinates of the first and secondkind with respect to this basis. For example, J ( R ) is isomorphic to H . This will be discussedfurther in subsection 3.1. It is implied that J k ( R ) is equipped with either one of the two systemsof coordinates in the following result, the first of our two main theorems. Theorem 1.5.
Fix α ≥ and positive integers n, k with n > . Let Ω be an open subset of R n .Then there is no injective mapping in the class C ,α + (Ω; J k ( R )) that is also locally Lipschitz whenconsidered as a map into R k +2 .
3e will prove this result in the case α = , and the cases for α > will follow from the fact C ,α + (Ω; J k ( R )) ⊂ C , + (Ω; J k ( R )) . The identity map R k +2 → J k ( R ) is locally k +1 -H¨older. From the Heisenberg case, one mayexpect for it to be unknown whether there exist locally α -H¨older, injective maps f : R n → J k ( R )for α > k +1 . However, we will give an example of a locally -H¨older, injective map f : R → J k ( R )that is locally Lipschitz as a map into R k +2 (Example 3.5). Comparing with Remark 1.3, thissuggests that our result is sharp, at least in the case n = 2.We will first prove Theorem 1.5 for when J k ( R ) is equipped with coordinates of the second kind.We will then prove at end of the subsection 2.4 that this implies the theorem holds for first kindcoordinates as well. We will use Warhurst’s model for jet spaces equipped with coordinates of thesecond kind (see [17, Section 3]). Rigot, Wenger, and Young have used this model to investigateextendability of Lipschitz maps into jet spaces [18, 13].For the next result, we can choose coordinates with respect to any basis compatible with thestratification of g , but the metric on ( R n , · ) will be induced by this choice. Theorem 1.6.
Fix α ≥ and an open subset Ω ⊆ R k . Suppose ( R n , · ) is a Carnot group ofstep at most three that is purely k -unrectifiable. Then there is no injective mapping in the class C ,α + (Ω; ( R n , · )) that is also locally Lipschitz when considered as a map into R n . As for J k ( R ), we will only explicitly prove this for α = . These two theorems will be provenin a similar fashion, implied by the following result: Proposition 1.7.
Fix an open subset Ω ⊆ R k . Let ( R n , · ) be a Carnot group that is purely k -unrectifiable. Then there is no injective mapping f : Ω → ( R n , · ) that is weakly contact and locallyLipschitz when considered as a map into R n . Thus, to prove Theorems 1.5 and 1.6, it suffices to show that if a map in C , + (Ω , ( R n , · )) islocally Lipschitz as a map into R n , then it is weakly contact. We prove this for the class of modelfiliform jet spaces, J k ( R ), in Proposition 3.3, for step 2 Carnot groups in Lemma 4.1, and for step3 Carnot groups in Lemma 4.3. We will discuss weakly contact maps further in subsection 2.3.Proposition 3.3 follows from considering the group structure on J k ( R ), specifically Lemma 3.1.The proofs of Lemmas 4.1 and 4.3 are a bit technical and requires one to carefully work withgroup structures, bounding terms via the Ball-Box Theorem (Theorem 2.2) and the modulatinghomeomorphism. It is expected that Theorem 1.5 and 1.6 should generalize to all Carnot groupsif one attains a better understanding of the group structure arising from coordinates of the firstkind. We will discuss this more at the end of this paper. In this section, we will review the basics of Carnot groups, discussing two systems of coordinates,the Carnot-Carath´eodory metric, and weakly contact maps.For some r , the Lie algebra g of a Carnot group G admits an r -step stratification: g = g ⊕ g ⊕ · · · ⊕ g r , where g ⊆ g is a subspace, g j +1 = [ g , g j ] for all j = 1 , . . . , r −
1, and [ g , g r ] = 0. We write[ g , g j ] to denote the subspace generated by commutators of elements of g with elements of g j ,4nd similarly with [ g , g r ]. The subspaces g j are commonly referred to as the layers of g , with g referred to as the horizontal layer . We define the step of G to be r , and this is well-defined [3,Proposition 2.2.8]. Throughout this paper, we will implicitly fix a stratification for each Carnotgroup. In other words, we will view the stratification of g as data of a Carnot group G .After combining bases of the subspaces g j to obtain a basis of g , we can define an inner product g = h· , ·i on g by declaring the combined basis to be orthonormal. Thus, we say that a basis B = { X , . . . , X n } of g is compatible with the stratification of g if { X h j − +1 , . . . , X h j } is a basis of g j for each j , where h j = P ji =1 dim( g i ). As we discuss coordinates of the first andsecond kind, it will be implied that coordinates are being taken with respect to a basis compatiblewith the stratification of g . While choosing different bases may technically result in different groupstructures, we will see that the resulting Carnot groups are all isomorphic to G . For Carnot groups, the exponential map exp : g → G is a diffeomorphism [5, Page 13]. Hence wecan define ⋆ : g × g → g by X ⋆ Y = exp − (exp( X ) exp( Y )) . The Baker-Campbell-Hausdorff formula gives us an explicit formula for
X ⋆ Y : X ⋆ Y = X n> ( − n +1 n X
X ⋆ Y up to order 3 is given by X + Y + 12 [ X, Y ] + 112 ([ X, [ X, Y ]] + [ Y, [ Y, X ]]) . Set n equal to the topological dimension of G , and let B ⊂ g be a basis compatible with thestratification of g . We can identify g with R n via coordinates of B , and then ⋆ on g translates intoan operation on R n . With a slight abuse of notation, we will also denote this operation on R n by ⋆ .Then ( R n , ⋆ ) is a Carnot group isomorphic to G via exp [3, Proposition 2.2.22]. We say that ( R n , ⋆ )is a normal model of the first kind of G and that ( R n , ⋆ ) is G equipped with coordinates ofthe first kind with respect to B . Observe that if G is of step r , each coordinate of X ⋆ Y is apolynomial of homogeneous degree at most r in the coordinates of X and Y . Let ( R n , · ) be a Carnot group, and set m j = dim( g j ) for each j . Fix a basis B = { X , . . . , X m } for the horizontal layer g . The horizontal bundle of ( R n , · ) is defined fiberwise by H p ( R n , · ) := span { X p , . . . , X m p } . H p ( R n , · ) = dL p H ( R n , · ) . Declaring ( B ) p to be orthonormal, we obtain an inner product on each fiber H p ( R n , · ).Recall that we just write R n to denote Euclidean space equipped with the standard Euclideanmetric. Definition 2.1.
We say a path γ : [ a, b ] → ( R n , · ) is horizontal if it is absolutely continuous as amap into R n and γ ′ ( t ) ∈ H γ ( t ) ( R n , · ) for a.e. t ∈ [ a, b ] . We define the length of a horizontal path to be l H ( γ ) := Z ba | γ ′ ( t ) | dt. Here, | γ ′ ( t ) | := q h γ ′ ( t ) , γ ′ ( t ) i γ ( t ) whenever γ is differentiable at t with γ ′ ( t ) ∈ H γ ( t ) ( R n , · ).Note the length of a horizontal path is finite (see 3.35, [6]).A theorem by Chow [4] states that ( R n , · ) is horizontally path-connected. This enables us todefine the Carnot-Carath´eodory metric on ( R n , · ): d cc ( x, y ) := inf γ :[ a,b ] → ( R n , · ) { l H ( γ ) : γ is horizontal , γ ( a ) = x, γ ( b ) = y } . Another common name for this metric is cc-metric . It is well-known that the Carnot-Carath´eodorymetric defines a geodesic metric on ( R n , · ), i.e., for every x , y ∈ ( R n , · ), there exists a horizontalpath γ connecting x to y with d cc ( x, y ) = l H ( γ ) [3, Theorem 5.15.5].Suppose ( R n , · ) is step r . From the previous two sections, a point x ∈ ( R n , · ) is of the form( ~x , ~x . . . , ~x r ), where each ~x j lies in R m j and corresponds to the coefficients of the elements of g j .For each ǫ >
0, we can define a dilation δ ǫ : ( R n , · ) → ( R n , · ) by δ ǫ ( ~x , ~x , . . . , ~x r ) := ( ǫ ~x , ǫ ~x , . . . , ǫ r ~x r ) . The Carnot-Carath´eodory metric is left-invariant and one-homogeneous with respect to these dila-tions:For all ǫ > x, y, z ∈ ( R n , · ), • d cc ( z · x, z · y ) = d cc ( x, y ) • d cc ( δ ǫ ( x ) , δ ǫ ( y )) = ǫ · d cc ( x, y ) . One may wonder how the Carnot-Carath´eodory metric on ( R n , · ) relates to the standard Eu-clidean metric on R n . From Proposition 1.1, ( R n , · ) and R n have the same topologies. Furthermore,Proposition 1.1 (combined with left-invariance and homogeneity) implies the following version ofthe Ball-Box Theorem: Theorem 2.2. (Ball-Box Theorem) Suppose ( R n , · ) is a step r Carnot group. For ǫ > and p ∈ ( R n , · ) , define Box ( ǫ ) := r Y j =1 [ − ǫ j , ǫ j ] m j nd B cc ( p, ǫ ) := { q ∈ ( R n , · ) : d cc ( p, q ) ≤ ǫ } . There exists
C > such that for all ǫ > and p ∈ ( R n , · ) , B cc ( p, ǫ/C ) ⊆ p · Box ( ǫ ) ⊆ B cc ( p, Cǫ ) . We obtain an important corollary which allows us to estimate the cc-metric:
Corollary 2.3.
Suppose ( R n , · ) is a step r Carnot group. There exists
C > such that for all p = ( a , . . . , a m , a , . . . , a rm r ) ∈ ( R n , · ) , C · d cc (0 , p ) ≤ max {| a jk | /j : 1 ≤ j ≤ r, ≤ k ≤ m j } ≤ Cd cc (0 , p ) . Fix an open set Ω ⊆ R k and a Carnot group ( R n , · ). If f : Ω → ( R n , · ) is Lipschitz, f is locallyLipschitz as a map into R n by Proposition 1.1. By Rademacher’s Theorem, then f is differentiablealmost everywhere in Ω. We say a locally Lipschitz map f : Ω → R n is weakly contact ifim df x ⊂ H f ( x ) ( R n , · ) for H k − almost every x ∈ Ω . Here, we write df x to denote the differential or total derivative of f at x . Observe that byTheorem 9.18 of [14], if f is differentiable at x ∈ Ω, thenim df x ⊂ H f ( x ) ( R n , · ) ⇐⇒ ∂ i f ( x ) ∈ H f ( x ) ( R n , · ) for all i = 1 , . . . , k. Balogh, Haj lasz, and Wildrick proved for the n th Heisenberg group H n that if a Lipschitz map f : [0 , k → R n +1 is weakly contact, then it is actually Lipschitz as a map into H n [2, Proposition8.2]. Their proof easily converts into a statement for all Carnot groups. To keep this paper asself-contained as possible, we will repeat the argument here. Proposition 2.4.
Let k be a positive integer. If f : [0 , k → R n is Lipschitz and weakly contact,then f : [0 , k → ( R n , · ) is Lipschitz.Proof. Fix a weakly contact map f : [0 , k → R n that is L -Lipschitz. Fubini’s Theorem impliesthe restriction of f to almost every line segment parallel to a coordinate axis is horizontal. Onbounded sets, the lengths with respect to the sub-Riemannian metrics and to the Euclidean metricsare equivalent for horizontal vectors. As f [0 , k is bounded and the Euclidean speed of f is boundedby L on line segments, it follows that the restriction of f on almost every line segment parallelto a coordinate axis is CL -Lipschitz as a map into ( R n , · ). Hence the restriction of f on each line segment parallel to a coordinate axis is CL -Lipschitz as a map into ( R n , · ), and the resultfollows.This enables us to prove Proposition 1.7, a result fundamental to our paper. The proof ofTheorem 1.11 in [2] for the Heisenberg group translates into a result for all Carnot groups. Proof of Proposition 1.7.
Assume that there is an injective map f : Ω → ( R n , · ) that is locallyLipschitz as a map into R n . Restricting f , we may assume Ω is a closed cube and f is Lipschitz asa map into R n . If f is weakly contact, f : Ω → ( R n , · ) is Lipschitz, which implies H k ( R n , · ) ( f (Ω)) = 0.As the identity map from ( R n , · ) to R n is locally Lipschitz (by Proposition 1.1), H k R n ( f (Ω)) = 0. Itfollows from Theorem 8.15 of [8] that the topological dimension of f (Ω) is at most k −
1. Since f | Ω is a homeomorphism, f (Ω) is of the same topological dimension as Ω, which is a contradiction.7he main theorems of this paper thus reduce to showing locally Lipschitz maps f into R n thatare of class C , + (Ω , ( R n , · )), are weakly contact. Balogh, Haj lasz, and Wildrick proved this forthe Heisenberg group [2, Proposition 8.1]. In this paper, we will prove it for models of jet spacesand models of Carnot groups of step at most three. Suppose G is a Carnot group with stratification g = g ⊕ · · · ⊕ g r . We define the family of dilations { d ǫ } ǫ> to be the collection of isomorphism of g induced by d ǫ ( X j ) = ǫ j X j , X j ∈ g j . Each d ǫ is a Lie group automorphism of ( g , ⋆ ) [3, Remark 1.3.32], i.e.,(2.1) d ǫ ( X ⋆ Y ) = ( d ǫ ( X )) ⋆ ( d ǫ ( Y )) for all X, Y ∈ g . These dilations on g are also commonly notated as δ ǫ , but we will not do so here to avoid confusionwith the dilations on G .As the exponential map exp G : g → G is a diffeomorphism, this induces a family of dilations δ ǫ on G :(2.2) δ ǫ := exp G ◦ d ǫ ◦ exp − G . This aligns with our earlier definition of dilations in subsection 2.2.Suppose H is a Carnot group isomorphic to G , with stratification h = h ⊕ · · · ⊕ h r . A Lie group isomorphism ϕ : G → H induces a Lie algebra isomorphism ϕ ∗ : g → h that satisfiesthe following identity:(2.3) exp H ◦ ϕ ∗ = ϕ ◦ exp G . We say that a Lie group isomorphism ϕ : G → H commutes with dilation if ϕ ( δ Gǫ g ) = δ Hǫ ϕ ( g ) for all g ∈ G, ǫ > , where δ Gǫ , δ Hǫ denote the dilations on G , H , respectively. If we say that a Lie algebra isomorphism f : g → h commutes with dilation if f ( d Gǫ X ) = d Hǫ f ( X ) for all X ∈ g , ǫ > , it is easy to check using (2.2) and (2.3) that an isomorphism ϕ : G → H commutes with dilationsif and only if ϕ ∗ : g → h commutes with dilations. Example 2.5.
Let G be a Carnot group. Suppose B ⊂ g is a basis compatible with the stratifi-cation of g . Let ( R n , ⊙ ) and ( R n , ⋆ ) be G equipped with coordinates of the second and first kind,respectively, with respect to B . Then ( R n , ⊙ ) is isomorphic to ( R n , ⋆ ) via exp − ◦ Φ and coordinates.Moreover, this isomorphism commutes with dilations.We say that an isomorphism ϕ : G → H is strata-preserving if ϕ ∗ ( g j ) = h j for all j = 1 , . . . , r. Note that ϕ is strata-preserving if and only if ϕ − is strata-preserving.The next result follows from the use of dilations:8 emma 2.6. Let G , H be isomorphic Carnot groups. An isomorphism ϕ : G → H commutes withdilations if and only if ϕ is strata-preserving. In fact, if we say that an isomorphism ϕ : G → H is contact if ϕ ∗ ( g ) = h , it’s easy to checkfrom the stratifications of g and h that ϕ is a contact map if and only if it is strata-preserving.We will show weakly contact mappings are invariant under isomorphisms that commute withdilations. We first prove that such isomorphisms are biLipschitz. Proposition 2.7.
Let ϕ : ( R n , · ) → ( R n , ∗ ) be an isomorphism between Carnot groups, that com-mutes with dilations. Then ϕ is biLipschitz, i.e., there exists a constant C such that C d ( R n , · ) cc ( g, h ) ≤ d ( R n , ∗ ) cc ( ϕ ( g ) , ϕ ( h )) ≤ Cd ( R n , · ) cc ( g, h ) for all g, h ∈ ( R n , · ) . Proof. As ϕ commutes with dilations and the cc-metrics on ( R n , · ) and ( R n , ∗ ) are one-homogeneous,it suffices to show ϕ is biLipschitz when restricted to B cc ( e, { X , . . . , X m } , { Y , . . . , Y m } be left-invariant frames for H ( R n , · ), H ( R n , ∗ ), respectively.For each g ∈ ( R n , · ), define the linear isomorphism S g : H ϕ ( g ) ( R n , ∗ ) → H ϕ ( g ) ( R n , ∗ ) induced by( ϕ ∗ X j ) ϕ ( g ) Y jϕ ( g ) . The function g
7→ || S g || is continuous, and hence, is bounded on B cc ( e, C . This implies for all g ∈ B cc ( e,
2) and v ∈ H g ( R n , · ), we have | dϕ g ( v ) | ϕ ( g ) ≤ C | v | g . It thenfollows from Lemma 2.6 that d ( R n , ∗ ) cc ( ϕ ( g ) , ϕ ( h )) ≤ Cd ( R n , · ) cc ( g, h )for all g, h ∈ B cc ( e, ϕ − , the lemma follows.It follows from the chain rule that weak contactness is preserved by strata-preserving isomor-phisms. Corollary 2.8.
Fix Ω ⊆ R k an open subset. Let ϕ : ( R n , · ) → ( R n , ∗ ) be an isomorphism betweenCarnot groups, that commutes with dilations. If f : Ω → ( R n , · ) is locally Lipschitz and weaklycontact, then ϕ ◦ f : Ω → ( R n , ∗ ) is also locally Lipschitz and weakly contact. Now that we have defined dilations on G and g , we can introduce coordinates of the second kind,another model for Carnot groups. The Carnot group that we obtain via this construction will beisomorphic to the coordinates of the first kind model we described in subsection 2.1. We will firststate a result that will allow us to define our other model. Theorem 2.9. ([15, Theorem 2.10.1]) Let G be a Lie group with Lie algebra g . Suppose g isthe direct sum of linear subspaces h , . . . , h s . Then there are open neighborhoods B i of in h i ( ≤ i ≤ s ) and U of in G , such that the map Ψ : ( Z , . . . , Z s ) exp Z · · · exp Z s is an analytic diffeomorphism of B × · · · × B s onto U . Fix a basis B = { X , . . . , X n } of g compatible with the stratification, and define Φ : g → G byΦ( a X + · · · + a n X n ) = exp( a X ) · · · exp( a n X n ) .
9y Theorem 2.9, the restriction Φ | V : V → U is a diffeomorphism for some open neighborhoods V ⊂ g of 0 and U ⊂ G of e . After noticing Φ( a X + · · · + a n X n ) = exp( a X ⋆ · · · ⋆ a n X n ) , itfollows from (2.1) and (2.2) that Φ is a global diffeomorphism.We can then define ⊙ : g × g → g by X ⊙ Y = Φ − (Φ( X )Φ( Y )) . As in subsection 2.1, we can identify g with R n and define a corresponding operation ⊙ on R n ,with a slight abuse of notation. We say that ( R n , ⊙ ) is a normal model of the second kind of g , and ( R n , ⊙ ) is G equipped with coordinates of the second kind with respect to B .Identifying ( R n , ⋆ ) with g via the same basis, observe that exp − ◦ Φ : ( R n , ⊙ ) → ( R n , ⋆ ) is a Liegroup isomorphism. In particular, ( R n , ⊙ ) is isomorphic to G . It then follows from Corollary 2.8that it suffices to prove each of Theorems 1.5 and 1.6 for a single system of coordinates. J k ( R ) J k ( R ) as Carnot groups We will only do our discussion in this section for jet spaces J k ( R ) = J k ( R , R ) (for k ≥
1) to makethings clearer. Similar constructions can be used to define more general jet spaces J k ( R m , R n ) (see[17, Section 4]). The results in this paper concerning model filiform groups translate into resultsfor general jet spaces, and I will note the more general results.Given f, g ∈ C k ( R ), we say f is equivalent to g at x ∈ R , and write f ∼ x g , if their k th -orderTaylor polynomials agree at x . Define J k ( R ) = [ x ∈ R C k ( R ) / ∼ x , and observe we have global coordinates on J k ( R ) by J k ( R ) ∋ [ f ] ∼ x ( x, u k , . . . , u ) ∈ R k +2 , where u j := f ( k ) ( x ).The horizontal bundle HJ k ( R ) is defined pointwise by H p J k ( R ) = { v ∈ T p J k ( R ) | ω i ( v ) = 0 , i = 0 , . . . , k − } , where ω i := du i − u i +1 dx. In coordinates, HJ k ( R ) is a 2-dimensional tangent distribution on J k ( R ) with global frame { X ( k ) , ddu k } ,where X ( k ) = ∂∂x + u k ∂∂u k − + · · · + u ∂∂u . The nontrivial bracket relations are (cid:20) ∂∂u j , X ( k ) (cid:21) = ∂∂u j − , j = 1 , . . . , k. It follows that
Lie ( J k ( R )) = HJ k ( R ) ⊕ span (cid:26) ∂∂u k − (cid:27) ⊕ · · · ⊕ span (cid:26) ∂∂u (cid:27)
10s a ( k + 1)-step stratified Lie algebra.One can use coordinates of the second kind to turn J k ( R ) into a Carnot group with the followinggroup operation: ( x, u k , . . . , u ) ⊙ ( y, v k , . . . , v ) = ( z, w k , . . . , w ) , where z = x + y , w k = u k + v k , and w s = u s + v s + k X j = s +1 u j y j − s ( j − s )! , s = 0 , . . . , k − x, u k , . . . , u ) ∈ J k ( R ), it is easy to show(( x, u k , . . . , u ) − ) s = − k X j = s ( − x ) j − s ( j − s )! u j , s = 0 , . . . , k. J k ( R ) In this section, we will prove a horizontality condition for J k ( R ), from which Theorem 1.5 will follow.We begin with a crucial lemma concerning the group structure of J k ( R ), similar to Corollary 1.3.18of [3]. Lemma 3.1.
For ( x, u k , . . . , u ) , ( y, v k , . . . , v ) ∈ J k ( R ) , (( x, u k , . . . , u ) − ⊙ ( y, v k , . . . , v )) = v − u − k X j =1 u j j ! ( y − x ) j . Proof.
Recall (( x, u k , . . . , u ) − ) s = − k X j = s ( − x ) j − s ( j − s )! u j , s = 0 , . . . , k, and the last coordinate of ( x, u k , . . . , u ) ⊙ ( y, v k , . . . , v ) is(( x, u k , . . . , u ) ⊙ ( y, v k , . . . , v )) = v + k X s =0 y s s ! u s . Thus, (( x, u k , . . . , u ) − ⊙ ( y, v k , . . . , v )) = v − k X s =0 n X j = s y s s ! · ( − x ) j − s ( j − s )! · u j = v − k X j =0 j X s =0 (cid:18) js (cid:19) y s ( − x ) j − s · u j j != v − k X j =0 j ! · ( y − x ) j u j , where the last equality comes from the Binomial Theorem.11 emark 3.2. The same reasoning using the Multinomial Theorem gives us the following general-ization for all jet spaces:Fix positive integers k, m, n . Let the notation for J k ( R m , R n ) be as in Warhurst (see [17,Subsection 4.4]), and equip J k ( R m , R n ) with the group operation arising from coordinates of thesecond kind (see [17, subsection 4.4]). Given ( x, u ( k ) ) , ( y, v ( k ) ) ∈ J k ( R m , R n ),(( x, u ( k ) ) − ⊙ ( y, v ( k ) )) l = v l − X I ∈ ˜ I ( m ) u lI I ! ( y − x ) I , l = 1 , . . . n. Here, for I = ( i , . . . , i m ) ∈ ˜ I ( m ) and z = ( z , . . . , z m ) ∈ R m , we define I ! = i ! · · · i m ! and z I = z i · · · z i m m . Proposition 3.3.
Let k, n be positive integers with Ω ⊆ R n an open set. Suppose that f =( f x , f u k , . . . , f u ) : Ω → J k ( R ) is of class C , + . If the component f x is differentiable at a point p ∈ Ω , then the components f u k − , f u k − , . . . , f u are also differentiable at p with df u j p = f u j +1 ( p ) df xp for all j = 0 , . . . , k − . In particular, if f u k is also differentiable at p , then the image of df p liesin the horizontal space H f ( p ) J k ( R ) .Proof. We prove this result by induction on k ≥
1. Below, p is a point in Ω.Let f = ( f x , f u , f u ) : Ω → J ( R ) be given of class C , + . Choose a map β for f satisfying(1.1). By Lemma 3.1,( f ( p ) − f ( p )) = f u ( p ) − f u ( p ) − f u ( p )( f x ( p ) − f x ( p )) . Thus by Corollary 2.3, there exists
C > | f u ( p ) − f u ( p ) − f u ( p )( f x ( p ) − f x ( p )) | / ≤ Cd cc ( f ( p ) , f ( p )) ≤ Cβ ( | p − p | ) · | p − p | / . We have | f u ( p ) − f u ( p ) − f u ( p ) df xp ( p − p ) |≤ C β ( | p − p | ) · | p − p | + | f u ( p )( f x ( p ) − f x ( p )) − f u ( p ) df xp ( p − p ) | = o ( | p − p | ) , where we used the differentiability of f x at p for the last equality.Suppose we have proven the result up to k . Let f = ( f x , f u k +1 , . . . , f u ) : Ω → J k +1 ( R ) begiven of class C , + with f x differentiable at p . Let ˜ β be a map satisfying (1.1) for f . Define theprojection π : J k +1 ( R ) → J k ( R ) by π ( x, u k +1 , . . . , u ) = ( x, u k +1 , . . . , u ) . As π maps horizontal curves to horizontal curves of the same length, it’s not hard to see that π isa contraction. This implies π ◦ f = ( f x , f u k +1 , . . . , f u ) is of class C , + (Ω , J k ( R )). By induction, f u k , . . . , f u are differentiable at p with df u j p = f u j +1 ( p ) df xp j = 1 , . . . , k .It remains to show f u is also differentiable at p with df u p = f u ( p ) df xp . Lemma 3.1 and Corollary 2.3 combine to imply (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f u ( p ) − f u ( p ) − k +1 X j =1 f u j ( p ) j ! ( f x ( p ) − f x ( p )) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / ( k +1) ≤ C ˜ β ( | p − p | ) · | p − p | / . Moreover, as f x is differentiable at p , f x ( p ) − f x ( p ) = O ( | p − p | ) , and hence | f x ( p ) − f x ( p ) | j = o ( | p − p | ) for all j ≥ . It follows | f u ( p ) − f u ( p ) − f u ( p ) df xp ( p − p ) |≤ C k +1 ˜ β k +1 ( | p − p | ) · | p − p | k +12 + | f u ( p )( f x ( p ) − f x ( p )) − f u ( p ) df xp ( x − x ) | + k +1 X j =2 (cid:12)(cid:12)(cid:12)(cid:12) f u j ( p ) j ! ( f x ( p ) − f x ( p )) (cid:12)(cid:12)(cid:12)(cid:12) j = o ( | x − x | ) . This proves f u is differentiable at p with df u p = f u ( p ) df xp , and the proposition follows. Remark 3.4.
In the above proof, we needed f to lie in C , + (Ω , J k ( R )) in order to ensure f u k − wasdifferentiable at the point with the desired form. To prove the differentiability of the componentsof f corresponding to higher layers, one can assume lower regularity. In fact, the above proof showsthe following:Assume Ω ⊆ R n is open and j ≥
2. Suppose f = ( f x , f u k , . . . , f u ) : Ω → J k ( R ) is of class C , j + . If f x is differentiable at a point p ∈ Ω, then f u k +1 − j , f u k − j , . . . , f u are also differentiableat p with df u l p = f u l +1 ( p ) df xp , l = k + 1 − j, . . . , . Before we prove Theorem 1.5, we will give an example of a locally -H¨older map f : R → J k ( R )that is Lipschitz as a map into R k +2 . Comparing with Remark 1.3, this suggests that our result issharp in the case n = 2. 13 xample 3.5. Define f : R → J k ( R ) by f ( x, y ) = (0 , x, y, , . . . , . Then f is Lipschitz (in fact, is an isometry) as a map into R k +2 .To show f is locally -H¨older, first note in J k ( R ),(0 , − x , − y , , . . . , ⊙ (0 , x , y , , . . . ,
0) = (0 , x − x , y − y , , . . . , . By Corollary 2.3, there exists a constant C such that d cc ( f ( x , y ) , f ( x , y )) ≤ C max {| x − x | , | y − y | / } for all ( x , y ) , ( x , y ) ∈ R . By considering cases, one can then show d cc ( f ( x , y ) , f ( x , y )) ≤ √ M C | ( x , y ) − ( x , y ) | / whenever ( x , y ) , ( x , y ) ∈ [ − M, M ] with M > Proof of Theorem 1.5.
Fix positive integers n, k with n ≥
2. Suppose f : Ω → J k ( R ) is of class C , + and is locally Lipschitz as a map into R k +2 . By Rademacher’s Theorem, each of the compo-nents of f is differentiable almost everywhere, and in particular, f x is differentiable almost every-where. Proposition 3.3 then implies that f is weakly contact. Since J k ( R ) is purely n -unrectifiable[9, Theorem 1.1], Theorem 1.5 in the case of second kind coordinates follows from Proposition2.4. The discussion at the end of subsection 2.5 then proves the result for coordinates of the firstkind. Remark 3.6.
Observe that J k ( R m , R n ) is purely j -unrectifiable if j > (cid:0) m + k − k (cid:1) [9, Theorem 1.1].Hence, from Remark 3.2, one can use similar reasoning to show the following generalization:Fix a jet space J k ( R m , R n ) and equip it with the group structure from Subsection 4.4 of[17]. Suppose j > (cid:0) m + k − k (cid:1) and Ω is an open subset of R j . If N is the topological dimensionof J k ( R m , R n ), there is no injective mapping in the class C , + (Ω; J k ( R m , R n )) that is also locallyLipschitz when considered as a map into R N . Remark 3.7.
Theorem 1.5 has an easier proof if we assume n < (cid:16) ( k +1)( k +2)2 (cid:17) . Making thisassumption, suppose that f : Ω → J k ( R ) is injective and of class C , + . Let B ( x, r ) be an openball with B ( x, r ) ⊆ Ω. Then the restriction f | B ( x,r ) is injective and of class C , + ( B ( x, r )) , J k ( R )).Since B ( x, r ) is bounded, it follows that f | B ( x,r ) is a -H¨older homeomorphism. In particular, f ( B ( x, r )) is open in J k ( R ), which impliesdim Hau f ( B ( x, r )) = dim Hau J k ( R ) = 1 + ( k + 1)( k + 2)2 . But as f is -H¨older, dim Hau f ( B ( x, r )) ≤ · dim Hau B ( x, r ) = 2 n, which is a contradiction. 14 Result for Carnot groups of step at most three
In this section, we will consider the geometry of Carnot groups of step two and equip these groupswith coordinates of the first kind.Fix a step two Carnot group G with Lie algebra g . Writing g = g ⊕ g , let d , . . . , d r be a basisfor g and e , . . . , e s be a basis for g . We can write[ d i , d j ] = s X k =1 α ijk e k for some structural constants α ijk , with all other bracket relations trivial. By antisymmetry, α ijk = − α jik for all i, j, and k . In fact, Bonfiglioli, Lanconelli, and Ugozzoni prove that there exists aCarnot group of step two with these bracket relations if and only if the skew-symmetric matrices( α ijk ), k = 1 , . . . , s , are linearly independent [3, Proposition 3.2.1].Using the procedure described in subsection 2.1, we can identify G with R r + s equipped withthe following multiplication via coordinates of the first kind:( A , . . . , A r , B , . . . , B s ) ⋆ ( a , . . . , a r , b , . . . , b s ) = ( A , . . . , A r , B , . . . , B s ) , where A i = A i + a i , B k = B k + b k + 12 X ≤ i
Lie ( R r + s , ⋆ ) = h X i i ≤ i ≤ r ⊕ h Y k i ≤ k ≤ s [3, Remark 1.4.8]. In fact, it is easy to check that the linear map ϕ : Lie ( R r + s , ⋆ ) → g induced by X i d i , Y k e k is a Lie algebra isomorphism.The contact forms, satisfying H ( R r + s , ⋆ ) = s \ k =1 ker ω k , are given by ω k := dB k − r X i =1 X ji α ijk A j dA i . In other words, if v ∈ T p ( R r + s , ⋆ ), then v ∈ H p ( R r + s , ⋆ ) ⇐⇒ ω kp ( v ) = 0 for all k = 1 , . . . , s. .2 Geometry of step three Carnot groups Let G be a step three Carnot group. Let d , . . . , d r be a basis for g , e , . . . , e s a basis for g , and f , . . . , f t a basis for g . We write [ d i , d j ] = s X k =1 α ijk e k [ d i , e k ] = t X m =1 β ikm f m with all other bracket relations trivial.As in the step two case, we can identify G with R r + s + t equipped with the following operationvia coordinates of the first kind:( A i , B k , C m ) ⋆ ( a i , b k , c m ) = ( A i , B k , C m ) , where A i = A i + a i B k = B k + b k + 12 X i
16t is clear that { X i } i ∪{ Y k } k ∪{ Z m } m forms a basis for Lie ( R n , ⋆ ). Moreover, we have the expectedstep three stratification of Lie ( R n , ⋆ ) [3, Remark 1.4.8]:(4.2) Lie ( R r + s + t , ⋆ ) = h X i i ≤ i ≤ r ⊕ h Y k i ≤ k ≤ s ⊕ h Z m i ≤ m ≤ t In fact, one can show using the Jacobi identity that the linear map ϕ : Lie ( R n , ⋆ ) → g induced by X i d i , Y k e k , Z m f m is a Lie algebra isomorphism.The contact forms are given by ω k := dB k − r X i =1 X ji α ijk A j dA i ω m := dC m − r X i =1 − s X j =1 B j β ijm + 112 r X l =1 s X k =1 A l ( X ji α ijk A j ) β lkm dA i . We have H ( R r + s + t , ⋆ ) = s \ k =1 ker ω k ∩ t \ m =1 ker ω m , so that a tangent vector v lies in H p ( R r + s + t , ⋆ ) if and only if ( ω k ) p ( v ) = ( ω m ) p ( v ) = 0 for all k and m . In this subsection, we will consider step two Carnot groups G using the notation from subsection4.1. Recall that we identity G with R r + s equipped with an operation arising from coordinates ofthe first kind: ( A i , B k ) ⋆ ( a i , b k ) = ( A i , B k ) , where A i = A i + a i , B k = B k + b k + 12 X ≤ i
Fix a step two Carnot group G and an open set Ω ⊆ R n . Let f = ( f A , . . . , f A r , f B , . . . , f B s ) :Ω → G be of class C , + , where f A , . . . , f A r are the horizontal components of f . If each f A i isdifferentiable at a point x ∈ Ω , then f is differentiable at x with the image of df x contained in H f ( x ) G . roof. We need to show for all k , the component f B k is differentiable at x with df B k x = 12 r X i =1 X ji α ijk f A j ( x ) df A i x . Fix k . By Corollary 2.3, there exists a constant C such that(4.3) | f B k ( x ) − f B k ( x ) − X ≤ i Lemma 4.1 was proven in the case G = H n by Balogh, Haj lasz, and Wildrick [2,Proposition 8.1]. The proof of Lemma 4.1 above was directly obtained from their proof by takinginto account structural constants. In this section, we will prove Theorem 1.6 for step three Carnot groups using similar reasoning asin subsection 4.3. We begin by reviewing notation:Let G be a step 3 Carnot group. We identify G with R r + s + t equipped with an operation arisingfrom coordinates of the first kind:( A i , B k , C m ) ⋆ ( a i , b k , c m ) = ( A i , B k , C m )where A i = A i + a i , B k = B k + b k + 12 X i Lemma 4.3. Fix a step three Carnot group G and an open set Ω ⊆ R n . Let f = ( f A , . . . , f A r , f B , . . . , f B s , f C , . . . , f C t ) :Ω → G be given of class C , + , where f A , . . . , f A r are the horizontal components of f . If each f A i is differentiable at a point x ∈ Ω , then f is differentiable at x with the image of df x lying in H f ( x ) G .Proof. By the proof of Lemma 4.1, each component f B k is differentiable at x with df B k x = 12 X i X ji α ijk f A j ( x ) df A i x . It remains to show that each component f C m is differentiable at x with df C m x = X i − X j β ijm f B j ( x ) + 112 X l,k f A l ( x ) X ji α ijk f A j ( x ) β lkm df A i x . Fix m . Choose β so that (1.1) holds. By the calculations in (4.1) and Corollary 2.3, we have | f C m ( x ) − f C m ( x ) − X i,j β ijm ( f A i ( x ) f B j ( x ) − f B j ( x ) f A i ( x )) + 112 X l,k X i Let ( R n , · ) be a Carnot group and Ω ⊆ R k an open subset. Suppose f : Ω → ( R n , · ) is of class C , + (Ω , ( R n , · )) . If each of the components of f are differentiable at a point x ∈ Ω , then theimage of df x lies in H f ( x ) ( R n , · ) . Acknowledgements. The author deeply thanks Jeremy Tyson for bringing this problem to hisattention and for many hours of discussion on the content and presentation of this paper. Theauthor also thanks Ben Warhurst for explaining why the map Φ that is used to define coordinatesof the second kind for Carnot groups, is a diffeomorphism. Finally, we are grateful to the refereefor their very careful reading of the paper and for suggesting which results should be cited ratherthan reproven. References [1] Luigi Ambrosio and Bernd Kirchheim. 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