Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups
Gioacchino Antonelli, Daniela Di Donato, Sebastiano Don, Enrico Le Donne
aa r X i v : . [ m a t h . M G ] M a y CHARACTERIZATIONS OF UNIFORMLY DIFFERENTIABLECO-HORIZONTAL INTRINSIC GRAPHS IN CARNOT GROUPS
GIOACCHINO ANTONELLI, DANIELA DI DONATO,SEBASTIANO DON AND ENRICO LE DONNE
Abstract.
In arbitrary Carnot groups we study intrinsic graphs of maps with horizontaltarget. These graphs are C regular exactly when the map is uniformly intrinsically dif-ferentiable. Our first main result characterizes the uniformly intrinsic differentiability bymeans of Hölder properties along the projections of left-invariant vector fields on the graph.We strengthen the result in step-2 Carnot groups for intrinsic real-valued maps by onlyrequiring horizontal regularity. We remark that such a refinement is not possible already inthe easiest step-3 group.As a by-product of independent interest, in every Carnot group we prove an area-formulafor uniformly intrinsically differentiable real-valued maps. We also explicitly write the areaelement in terms of the intrinsic derivatives of the map. Contents
0. Notation 31. Introduction 31.1. An historical account of the notion of C -surface in Carnot groups 31.2. Main theorems 51.3. Structure of the paper 92. Preliminaries 102.1. Carnot groups 102.2. Little Hölder continuous functions 122.3. Intrinsic surfaces, Intrinsically Lipschitz and Intrinsically differentiable functions 133. Intrinsic projected vector fields on subgroups 193.1. Definition of D ϕ and main properties 203.2. Invariance properties of D ϕ D ϕ ϕ along integral curves of D ϕ to regularity of ϕ Date : 26th May 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Carnot groups, intrinsically C surfaces, co-horizontal surfaces, area formula,intrinsically differentiable functions, little Hölder functions, broad solutions.G.A., D.D.D., S.D. and E.L.D. are partially supported by the Academy of Finland (grant 288501 ‘ Geo-metry of subRiemannian groups ’ and by grant 322898 ‘
Sub-Riemannian Geometry via Metric-geometry andLie-group Theory ’) and by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘
Geometryof Metric Groups ’). .4. Main theorem 515. Some applications 525.1. A different proof of the propagation of broad* regularity in H n , with n ≥ . Notation ( ϕ, e ϕ ) , ( γ, e γ ) , . . . (cid:3) is the representation of e (cid:3) in exponential coordinates. SeeDefinition 2.3. L g , R g Left-translation and right-translation by g ∈ G . deg Holonomic degree. See Section 2. k·k , k·k G Homogeneous norm on G . See (7). V Horizontal bundle of G . See (5). π V Projection on the horizontal bundle. See (6). ( W , L ) Complementary subgroups. See Definition 2.1. π W ( g ) , g W Projection of g ∈ G onto the homogeneous subgroup W , giventhe splitting G = W · L . See (8). h α ( U ; R k ) R k -valued α -little Hölder functions defined on U ⊆ R n . SeeDefinition 2.6. e Φ( e U ) , graph( e ϕ ) Intrinsic graph of e ϕ : e U ⊆ W → L , given the splitting G = W · L .See Definition 2.8. e ϕ q : e U q ⊆ W → L Intrinsic q -translation, with q ∈ G , of the function e ϕ : e U ⊆ W → L , given the splitting G = W · L . See Definition 2.9. d P f Pansu differential of the C function f . See Definition 2.15. d ϕ ϕ a , d ϕ ϕ ( a ) : W → L Intrinsic differential of the function e ϕ : e U ⊆ W → L , at a point a ∈ e U , given the splitting G = W · L . See Definition 2.17. (U)ID( e U , W ; L ) Set of (uniformly) intrinsically differentiable functions e ϕ : e U ⊆ W → L , given a splitting G = W · L . See Definition 2.17. ∇ ϕ ϕ a , ∇ ϕ ϕ ( a ) When L is horizontal, the map, identified with a k × ( m − k ) -matrix, in Lin(Lie( W ) ∩ V , Lie( L )) , corresponding to the intrinsicdifferential d ϕ ϕ a . See Definition 2.20 and Remark 2.21. ∇ ϕj ϕ, ∇ ϕ ϕ ( X j ) The j -th component, either a number or a vector, of the intrinsicgradient ∇ ϕ ϕ . ∇ H f = ( ∇ L f | ∇ W f ) Pansu differential of f ∈ C ( e V ; R k ) in coordinates adapted tothe splitting G = W · L . See Definition 2.26. D ϕW Intrinsic projected vector field on e U ⊆ W , relative to the vectorfield W ∈ Lie( W ) , and to the function e ϕ : e U ⊆ W → L , giventhe splitting G = W · L . See Definition 3.1. D ϕj Intrinsic projected vector field when W = X j , with X j ∈ Lie( W ) .See Proposition 3.9. D ϕ (cid:3) ϕ The vector field D ϕ (cid:3) acting on the function ϕ . L n n -dimensional Lebesgue measure. S n n -dimensional spherical Hausdorff measure.1. Introduction
An historical account of the notion of C -surface in Carnot groups. In theselast twenty years there has been an increasing interest in a fine study of parametrized in-trinsically regular surfaces in sub-Riemannian settings. The search for a good such notionwas motivated by a negative result obtained in [AK99]. Indeed, in the reference the authorsshow that the sub-Riemannian Heisenberg group H is not k -rectifiable in Federer’s sense[Fed69], for every k ≥ . notion of intrinsic C regular surface was firstly introduced and studied in [FSSC01],and then in [FSSC03a] in arbitrary Carnot groups G . Initially, the authors only took C -hypersurfaces into account. A first step toward a general definition of C -surfaces in ar-bitrary codimensions was done in [FSSC07, Definition 3.1, Definition 3.2] in the setting ofHeisenberg groups H n . Then a general notion of ( G , M ) regular surface, where G and M areCarnot groups, was proposed by Magnani in [Mag06, Definition 3.5]. According to the latterdefinition, a ( G , M ) regular surface is locally the zero level set of an M -valued C -functiondefined on an open subset of G and whose intrinsic Pansu differential d P f is surjective.A first natural question one could try to answer is whether it is possible to (locally) writea C -surface as an intrinsic graph of a function. An intrinsic graph in a Carnot group G isthe set of points of the form p · ϕ ( p ) , given a function ϕ : U ⊆ W → L , where W and L arehomogeneous and complementary subgroups, namely G = W · L , and W ∩ L = { e } . Theanswer to this question is affirmative for C -hypersurfaces. Moreover the graphing functionis intrinsically Lipschitz according to the definition of [FSSC06, FS16], while it is in generalneither Euclidean Lipschitz nor Lipschitz with respect to any sub-Riemannian distance, see[FSSC06, Example 3.3 & Proposition 3.4].A more general implicit function theorem was proved by Magnani in [Mag13, Theorem 1.4].This theorem holds for arbitrary ( G , M ) regular surfaces with the additional property that Ker(d P f ( x )) has a complementary subgroup in G , where x is the point around which wewant to parametrize the surface. From [Mag13, Eq. (1.8)] it follows that this parametrizationis intrinsically Lipschitz. The validity of the implicit function theorem leads the way to avery general definition of ( G , M ) regular sets for G , where M is just a homogeneous group,given in [Mag13, Definition 10.2]. We will not deal with objects at this level of generality,but we refer the interested reader to [Mag13, Sections 10,11,12]. The class of intrinsicallyregular surfaces is also studied in [JNGV20], where area and coarea formulae are proved. Foran alternative proof of the implicit function theorem, one can also see [JNGV20, Section 2.5].We will mainly deal with co-horizontal C -surfaces, that have been studied in [Koz15,DD18, Cor19], see Definition 2.27. Definition 1.1 (Co-horizontal C -surface) . Let G be a Carnot group, and let k ∈ N . Wesay that Σ ⊂ G is a co-horizontal C -surface of codimension k if, for any p ∈ Σ , there exista neighborhood U of p and a map f ∈ C ( U ; R k ) such that Σ ∩ U = { g ∈ U : f ( g ) = 0 } , and the Pansu differential d P f ( p ): G → R k is surjective.If, morevoer, the subgroup Ker(d P f ( p )) admits a complementary subgroup, that is ho-rizontal, we say that Σ is a co-horizontal C -surface with complemented tangents . We call Ker(d P f ( p )) the homogeneous tangent space to Σ at p . If the previous holds, the homogen-eous subgroup at p is independent of the choice of f , see [Mag13, Theorem 1.7]. Uniform intrinsic differentiability of the parametrizing function . A fine study onthe regularity of the parametrizing function of a C -surface has been initiated in [ASCV06]in the setting of Heisenberg groups H n , for the class of C -hypersurfaces. For this study inarbitrary CC -spaces, see also [CM06]. In [ASCV06], the authors introduced the notion ofuniform intrinsic differentiability. In this paper we abbreviate “intrinsically differentiable”and “uniformly intrinsically differentiable” with ID and UID, respectively.From the analytic viewpoint, the notion of (U)ID is defined in a translation invariant way,mimicking the Euclidean notion of derivative. For the sake of exposition, we here recall the efinition of ID at the identity. Then, building on this definition, one can define the notionof (U)ID at any point by means of translations, see Definition 2.9, and Definition 2.17. Inthe following, k·k is a homogeneous norm on G . Definition 1.2 (Intrinsic differentiability) . Let G be a Carnot group, with identity e , anda splitting G = W · L . Let e ∈ U ⊆ W be a relatively open subset, and ϕ : U ⊆ W → L afunction with ϕ ( e ) = e .We say that ϕ is intrinsically differentiable at e if there exists an intrinsically linear map d ϕ ϕ e : W → L such that lim ̺ → (cid:18) sup (cid:26) k d ϕ ϕ e ( b ) − · ϕ ( b ) kk b k : b ∈ U, < k b k < ̺ (cid:27)(cid:19) = 0 , where we say that a function is intrinsically linear if its intrinsic graph is a homogeneoussubgroup. The function d ϕ ϕ e is called intrinsic differential of ϕ at e .Building upon the implicit function theorem, the authors in [ASCV06] prove that in H n thegraphing map ϕ for a C -hypersurface is UID. The idea behind this implication is the follow-ing: a function f ∈ C not only has continuous derivatives, but also its horizontal gradient ∇ H f uniformly approximates f at first order, see [Mag13, Theorem 1.2], and [JNGV20,Proposition 2.4]. This notion is often referred to as strict differentiability. This fact hasa strong analogy with the Euclidean setting. Indeed, in the Euclidean framework, a func-tion f with continuous partial derivatives is Fréchet-differentiable, and the proof relies ona use of a mean value inequality, that is exactly what one finds in [Mag13, Theorem 1.2],and [ASCV06, Lemma 4.2]. Eventually, the uniform differentiability of f translates into theuniform intrinsic differentiability of ϕ .The fact that the graphing function is UID was proved in the case of co-horizontal C -surfaces in H n [AS09], and more in general for co-horizontal C -surfaces with complementedtangents in any Carnot groups, in [DD18]. The inverse implication, i.e., the fact that thegraph of a UID function is a C -surface, was firstly shown to be true in [ASCV06, AS09] inthe setting of H n , and lately generalized in [DD18] for arbitrary Carnot groups G to functionswith horizontal target, see our Proposition 2.28 for a precise statement. Notice that the lackof generality in the statement, namely, the fact that one restricts the target to be horizontal,is due to the fact that a generalized version of Whitney’s extension theorem is not known tobe true.1.2. Main theorems. Definitions & statements . For the notation we refer to Section 2.Given an arbitrary Carnot group G of step s , with layers V i , with ≤ i ≤ s , we fix a splitting G = W · L , where L is horizontal. We denote by k·k a (fixed) homogeneous norm on G .We first aim at characterizing the uniform intrinsic differentiability of a map ϕ : U ⊆ W → L defined on a relatively open subset U ⊆ W . This is done by means of a correctLipschitz/Hölder property of ϕ along the integral curves of projected vector fields on U ⊆ W ,that we define here. See also Definition 3.1. Definition 1.3 (Projected vector fields) . Let U be a relatively open subset of W . Given acontinuous function ϕ : U ⊆ W → L , for every W ∈ Lie( W ) , we take D ϕW as the continuousvector field on U defined by(1) D ϕW ( p ) := (d π W ) p · ϕ ( p ) W p · ϕ ( p ) . ∀ p ∈ U, here π W is the projection on W associated to the splitting G = W · L . Notice that if ϕ is C , then D ϕ is a C vector field.We define the notion of broad* regularity, mimicking the definition in [ASCV06, BSC10b],and we then introduce the notion of vertically broad* hölder regularity, see Definition 3.24,and Definition 4.3, respectively. Definition 1.4 (Broad* regularity) . Let G be a Carnot group, with splitting G = W · L ,and L horizontal. Let U be a relatively open subset in W , ϕ : U ⊆ W → L be a continuousfunction, and let ω : U ⊆ W → Lin(Lie( W ) ∩ V ; L ) be a continuous function with values inthe space of linear maps.We say that D ϕ ϕ = ω in the broad* sense on U if the following holds: for every a ∈ U ,there exist a neighborhood U a ⋐ U of a and T > such that, for every a ∈ U a and every W ∈ Lie( W ) ∩ V with k W k≤ , there exists an integral curve γ : [ − T, T ] → U of D ϕW suchthat γ (0) = a and ϕ ( γ ( s )) − · ϕ ( γ ( t )) = Z ts ω ( γ ( r ))( W ) d r, ∀ t, s ∈ [ − T, T ] . Definition 1.5 (Vertically broad* hölder regularity) . Let G be a Carnot group, with splitting G = W · L , and L horizontal. Let U be a relatively open subset in W , and let ϕ : U ⊆ W → L be a continuous function.We say that ϕ is vertically broad* hölder on U if the following holds: for every a ∈ U there exist a neighborhood U a ⋐ U of a and T > such that for every a ∈ U a and every W ∈ Lie( W ) ∩ V d , with d > and k W k≤ , there exists an integral curve γ : [ − T, T ] → U of D ϕW such that γ (0) = a and lim ̺ → (cid:18) sup (cid:26) k ϕ ( γ ( s )) − · ϕ ( γ ( t )) k| t − s | /d : t, s ∈ [ − T, T ] , | t − s |≤ ̺ (cid:27)(cid:19) = 0 , and the limit is uniform on a ∈ U a and on W ∈ Lie( W ) ∩ V d , with d > , and k W k≤ .We next state our first main result in a free-coordinate fashion. We refer the readerto Theorem 4.17 for a coordinate-dependent, though equivalent, statement. We remarkthat the equivalence (a) ⇔ (b) of the forthcoming Theorem 1.6 is not in the statement ofTheorem 4.17, but it is an outcome of Proposition 2.28, and the implicit function theorem[Mag13, Theorem 1.4]. Theorem 1.6.
Let G be a Carnot group with splitting G = W · L , and L horizontal. Let U be a relatively open subset in W , and let ϕ : U ⊆ W → L be a continuous function. Thefollowing facts are equivalent. (a) graph( ϕ ) is a co-horizontal C -surface with homogeneous tangent spaces complemen-ted by L (see Definition 1.1). (b) ϕ is uniformly intrinsically differentiable on U (see Definition 1.2). (c) ϕ is vertically broad* hölder on U (see Definition 1.5), and there exists a continu-ous function ω : U → Lin(Lie( W ) ∩ V ; L ) , such that, for every a ∈ U , there exist δ > and a family of functions { ϕ ε ∈ C ( B ( a, δ ); L ) : ε ∈ (0 , } such that, for every W ∈ Lie( W ) ∩ V , one has (2) lim ε → ϕ ε ( p ) = ϕ ( p ) and lim ε → ( D ϕ ε W ϕ ε )( p ) = ω ( p )( W ) uniformly in p ∈ B ( a, δ ) . d) ϕ is vertically broad* hölder on U (see Definition 1.5) and there exists a continu-ous function ω : U → Lin(Lie( W ) ∩ V ; L ) such that D ϕ ϕ = ω in the broad* sense on U (see Definition 1.4). Our second main result is an improvement of Theorem 1.6 for step-2 Carnot groups, inthe case L is one-dimensional. Again, we here give a coordinate-free statement. We referthe reader to Theorem 6.17 for a coordinate-dependent, though equivalent, statement. Westress that in the forthcoming theorem we are removing the vertically broad* hölder condition of Theorem 1.6, and we work with L that is one-dimensional . We also stressthat in general in the statement of Theorem 1.6 one cannot remove the vertically broad*hölder condition, see Remark 4.18, and [Koz15, Example 4.5.1] for a counterexample in theeasiest step-3 group, namely the Engel group. Theorem 1.7.
Let G be a Carnot group of step with a splitting G = W · L , and L one-dimensional horizontal . Let U be a relatively open subset in W , and let ϕ : U ⊆ W → L be a continuous function. The following facts are equivalent. (a) graph( ϕ ) is a C -hypersurface with homogeneous tangent spaces complemented by L (see Definition 1.1). (b) ϕ is uniformly intrinsically differentiable on U (see Definition 1.2). (c) There exists a continuous map ω : U → Lin(Lie( W ) ∩ V ; L ) , such that, for every a ∈ U , there exist δ > and a family of functions { ϕ ε ∈ C ( B ( a, δ ); L ) : ε ∈ (0 , } such that, for every W ∈ Lie( W ) ∩ V , one has lim ε → ϕ ε ( p ) = ϕ ( p ) and lim ε → ( D ϕ ε W ϕ ε )( p ) = ω ( p )( W ) uniformly in p ∈ B ( a, δ ) . (d) There exists a continuous function ω : U → Lin(Lie( W ) ∩ V ; L ) such that D ϕ ϕ = ω in the broad* sense on U (see Definition 1.4). The last main result that we state is an area-formula for uniformly differentiable intrinsicreal-valued maps, which we believe has its independent interest. We here give a coordinate-free statement. We refer the reader to Proposition 4.10, and Remark 4.11 for a coordinate-dependent statement.
Proposition 1.8.
Let G be a Carnot group of homogeneous dimension Q with a splitting G = W · L , with L horizontal and one-dimensional. Let h· , ·i denote a scalar product onthe first layer, and let d be a homogeneous and left-invariant distance on G . Consider U a relatively open subset in W , and a uniformly intrinsically differentiable function ϕ : U ⊆ W → L .Then, the subgraph E ϕ of ϕ has locally finite G -perimeter in U · L and its G -perimetermeasure | ∂E ϕ | G is given by (3) | ∂E ϕ | G ( V ) = Z Φ − ( V ) p | d ϕ ϕ | d S Q − W , for every Borel set V ⊆ U · L , where Φ denotes the graph map of ϕ , d ϕ ϕ is the intrinsic differential of ϕ , and S Q − W is the spherical Hausdorff measure of dimension Q − restricted to W .Moreover, the reduced boundary of E ϕ coincides with graph( ϕ ) , it is a C -hypersurface,and there exists a positive measurable function β on graph( ϕ ) , only depending on the tangent The G -perimeter is defined with respect to the scalar product h· , ·i . o the hypersurface and on the homogeneous distance d , such that (4) Z V β d S Q − graph( ϕ ) = Z Φ − ( V ) p | d ϕ ϕ | d S Q − W , for every Borel set V ⊆ U · L . Comments on the statements . We point out that in Definition 1.4, and Definition 1.5we give coordinate-free definitions of the broad* conditions, while in Definition 3.24, andDefinition 4.3, we will give apparently weaker definitions, choosing an adapted basis. Nev-ertheless the broad* condition and the vertically broad* hölder condition, when coupledtogether, are independent on the choice of the basis, see Remark 3.26, and Remark 4.4.We comment on the statement of Theorem 1.6, and we refer the reader to the introductionof Section 4 for a more detailed discussion. We notice that (a) ⇔ (d) is [Koz15, Theorem4.3.1]. In [Koz15] the proof of this latter fact is heavily based on the characterization of co-horizontal C -surfaces in terms of uniform convergence to Hausdorff tangents, see [Koz15,Theorem 3.1.12]. We give a self-contained different proof of this equivalence, with analyticflavor. Namely, we first show (b) ⇔ (d) in Theorem 1.6, whose proof is based on ideas comingfrom [ASCV06] and [DD18], and thus, as a corollary, we eventually recover [Koz15, Theorem4.3.1] by using the latter equivalence and (a) ⇔ (b).The approximating condition in (2) of item (c) of Theorem 1.6 could be interpreted as aweak formulation of the equality D ϕ ϕ = ω on U , and it is the one that was first proposedand studied in [ASCV06], see Remark 4.14 for a detailed discussion about this condition.Indeed, in [ASCV06], in the case G = H n and L one-dimensional, the equivalence (b) ⇔ (c)of Theorem 1.6 has been proved, even in the stronger version obtained by removing thevertically broad* hölder regularity, see [ASCV06, Theorem 5.1]. We stress that the fact that D ϕ ϕ = ω holds in the sense of distributions on U , that is part of [ASCV06, Theorem 5.1],follows in general from the argument of item (c) of our Proposition 4.10.We comment on the statement of Theorem 1.7, and we refer the reader to the introductionof Section 6 for a more detailed discussion on the idea behind the proof. First, we noticethat the main difference between this statement and the one in Theorem 1.6 is in the factthat, in all the equivalences, we are able to drop the vertically broad* hölder regularity on ϕ . The equivalence (b) ⇔ (c) of Theorem 1.7 results in being a generalization of [ASCV06,Theorem 5.1] to all step-2 Carnot groups.The equivalence (b) ⇔ (d) generalizes the result in [BSC10b, Theorem 1.2], if one also takesitem (d) of Proposition 4.10 into account, in order to explicitly write the intrinsic normal of graph( ϕ ) in terms of the intrinsic derivatives of ϕ .We comment on the statement of Proposition 1.8. First of all, let us notice that (4) simplycomes from (3) and the general area formula in [Mag17], see Remark 4.11 for details. Letus notice that for some particular choices of the homogeneous distance on G , i.e., when it isvertically symmetric, the function β is constant, thus simplifying (4), see again Remark 4.11.We also mention that in orthonormal coordinates we can explicitly compute the intrinsicnormal, see item (d) of Proposition 4.10, thus generalizing the formulas already proved inHeisenberg groups and in Carnot groups of step 2 in [ASCV06, DD18].We finally remark that a formula in the spirit of (4) has been recently obtained in [CM20]for parametrized co-horizontal C -surfaces with complemented tangents of arbitrary codi-mensions in H n (see [CM20, Theorem 4.2]), building on an upper blow-up thorem, see[CM20, Theorem 1.1]. Very recently, in [JNGV20, Theorem 1.1], a general area formula for -surfaces has been proved in great generality. In [JNGV20] the area element is left impli-cit, depending only on the tangent to the surface and on the homogeneous distance on thegroup. We stress that, with our formula (4), we explicitly write the area element in terms ofthe intrinsic derivatives of ϕ , in the case the target is one-dimensional. For an area-formulafor intrinsically Lipschitz functions in H n , one can also see [CMPSC14]. Geometric characterizations of intrinsic differentiability . The notion of (U)ID has ageometric meaning. Indeed, given U ⊆ W open, a function ϕ : U ⊆ W → L is ID at w ∈ U if and only if the Hausdorff tangent to graph( ϕ ) at w · ϕ ( w ) is a homogeneous subgroupthat is complementary to L , see Remark 2.24.We stress that, at least in the case L is horizontal, the previous convergence to the tangentis uniform if and only if ϕ is UID, see [Koz15, Theorem 3.1.1], and [Koz15, Theorem 3.1.12].The proof of these statements are rather involved and based on the so-called four conesTheorem, see [BK14]. We remark that we will not use this particular uniformity resultthroughout the paper.We point out that the mere existence of Hausdorff tangents for C regular surfaces -without any information on the uniformity of convergence - has been proved in great gener-ality also in [Mag13, Theorem 1.7], and in [JNGV20, Lemma 2.14].Now a natural question can be raised. Is it true that an ID function ϕ with con-tinuous intrinsic gradient d ϕ ϕ is UID? Taking the geometric interpretation into account,the question can be reformulated, at least in the category of co-horizontal C -surfaces: isit true that, if a co-horizontal graph( ϕ ) has continuously varying Hausdorff tan-gents, then it is a co-horizontal C -surface ? If true, this would be the counterpart ofan already known result in the Euclidean setting that goes back to the beginning of twenti-eth century. We refer the reader to [BNG14, Proposition 2.1] and references therein for anhistorical account of the problem.The answer to the previous question is affirmative in Heisenberg groups H n , see [SC16,Theorem 4.95], and [Cor19, Theorem 1.4]. In this paper we obtain a new result in thisdirection. We prove that the answer is affirmative also for hypersurfaces in every step-2Carnot group, see (b) ⇒ (a) in Theorem 6.17, thus generalizing [SC16, Theorem 4.95].We give also a partial affirmative answer for arbitrary Carnot groups, by requiring theadditional hypothesis of the vertically broad* hölder condition on ϕ , see Corollary 4.7. Thisweaker implication might not be so satisfactory. Indeed, the intrinsic differentiability (evenif it is not uniform) by itself already implies a /d -little Hölder continuity on integral curvesof the vector fields D ϕW , with W ∈ Lie( W ) ∩ V d . Nevertheless, this little Hölder continuity,a priori, might not be uniform, see Proposition 3.19, Remark 4.8, and Example 2.7.1.3. Structure of the paper.
In Section 2 we introduce the common terminology andnotation we use throughout the paper. We introduce Carnot groups, little Hölder functions,intrinsic submanifolds, intrinsically Lipschitz functions, (uniformly) intrinsically differenti-able functions and we describe their basic properties and relations.In Section 3 we introduce the projected vector fields and we study their basic properties:in particular, we show some invariance properties that will be crucial in the proof of themain theorems. We also show how the (uniformly) intrinsic differentiability affects metricproperties along integral curves of the projected vector fields.In Section 4 we prove the main results Theorem 1.6 and Proposition 1.8 we discussed above. n Section 5 we construct examples and apply our results also to obtain different proofs ofparticular cases of theorems already contained in the literature. We refer the reader to thebeginning of Section 5 for a more detailed discussion.In Section 6 we give the proof of Theorem 1.7. Acknowledgments . We are grateful to Katrin Fässler for several discussions around thetopic of the paper. We warmly thank Raul Serapioni for several stimulating discussions andfor having shared and discussed with us the examples in Example 2.7 and Remark 5.4. Wealso thank Sebastiano Nicolussi Golo for stimulating discussions, in particular regarding thelast part of Remark 2.24. We finally thank Francesco Serra Cassano and Davide Vittone forhaving discussed with us the content of Remark 4.14.2.
Preliminaries
Carnot groups.
A Carnot group G is a connected and simply connected Lie group,whose Lie algebra g is stratified. Namely, there exist subspaces V , . . . , V s of the Lie algebra g such that g = V ⊕ . . . ⊕ V s , [ V , V j ] = V j +1 ∀ j = 1 , . . . , s − , [ V , V s ] = { } . The integer s is called step of the group G , while m := dim( V ) is called rank of G . We call n := dim G the topological dimension of G . We denote by e the identity element of G .It is well known that the exponential map exp: g → G is a diffeomorphism. We call(5) V := exp( V ) , the horizontal bundle of G . We write(6) π V := exp ◦ π V ◦ exp − , to denote the projection on the horizontal bundle V , where π V is the linear projection in g onto V .Every Carnot group has a one-parameter family of dilations that we denote by { δ λ : λ > } .These dilations act on g as ( δ λ ) | Vi = λ i (id) | Vi , ∀ λ > , ∀ ≤ i ≤ s, and are extended linearly. We will indicate with δ λ both the dilations on g and the groupautomorphisms corresponding to them via the exponential map.We fix a scalar product h· , ·i in V , that can be extended left-invariantly on the horizontalbundle V = exp( V ) , and a homogeneous norm k·k on G . We recall that k·k is a homogeneousnorm on G if k g k = 0 if and only if g = 0 , k δ λ g k = λ k g k , ∀ λ > , ∀ g ∈ G , k g k = k g − k , ∀ g ∈ G . (7)Sometimes we will also call the homogeneous norm k·k G . We also fix on G a left-invariant δ λ -homogeneous distance d and we denote by B ( g, r ) (respectively B ( g, r ) ) the open (re-spectively closed) balls of center g ∈ G and radius r > according to this distance. We nextgive the definition of complementary subgroups. efinition 2.1 (Complementary subgroups) . Given a Carnot group G , we say that twosubgroups W and L are complementary subgroups in G if they are homogeneous , i.e., closedunder the action of δ λ for every λ > , G = W · L and W ∩ L = { e } .We say that the subgroup L is horizontal and k -dimensional , if there exist linearly in-dependent X , . . . , X k ∈ V such that L = exp(span { X , . . . , X k } ) . Given W and L twocomplementary subgroups, we denote the projection maps from G onto W and onto L by π W and π L , respectively. Defining g W := π W g and g L := π L g for any g ∈ G , one has(8) g = ( π W g ) · ( π L g ) = g W · g L , and, whenever W is normal (for example this is true when L is horizontal), we have ( g · h ) L = g L · h L , ( g · h ) W = g W · (cid:0) g L · h W · ( g L ) − (cid:1) , ∀ g, h ∈ G . Remark . If W and L are complementary subgroups of G and L is one-dimensional, thenit is easy to see that L is horizontal. For the sake of clarity, we will always write L horizontaland one-dimensional even if one-dimensional is technically sufficient.Let us set m := 0 and m j := dim V j for any j = 1 , . . . , s . We recall that m = m . Let usdefine n := 0 , and n j := P jℓ =1 m ℓ . The ordered set ( X , . . . , X n ) is an adapted basis for g ifthe following facts hold.(i) The vector fields X n j +1 , . . . , X n j +1 are chosen among the iterated commutators oforder j of the vector fields X , . . . , X m , for every j = 0 , . . . , s − .(ii) The set { X n j +1 , . . . , X n j +1 } is a basis for V j +1 for every j = 0 , . . . , s − .If we fix an adapted basis ( X , . . . , X n ) , and ℓ ∈ { , . . . , n } , we define the holonomic degreeof ℓ to be the unique j ∗ ∈ { , . . . , s } such that n j ∗ − + 1 ≤ ℓ ≤ n j ∗ . We denote deg ℓ := j ∗ and we also say that j ∗ is the holonomic degree of X ℓ , i.e., deg( X ℓ ) := j ∗ . Definition 2.3 (Exponential coordinates) . Let G be a Carnot group of dimension n and let ( X , . . . , X n ) be an adapted basis of its Lie algebra. We define the exponential coordinatesof the first kind associated with ( X , . . . , X n ) by the map F : R n → G defined by F ( x , . . . , x n ) := exp ( x X + . . . + x n X n ) . It is well known that F is a diffeomorphism from R n to G . We will often need to considermaps in exponential coordinates. To avoid inconvenient notation we will use the followingconventions. • If e U ⊂ G , then U := F − ( e U ) . • If U ⊆ R n , then e U := F ( U ) . • If W and L are complementary subgroups of G and L is horizontal and k -dimensional,we may assume that L = exp(span { X , . . . , X k } ) and W = exp(span { X k +1 , . . . , X n } ) for an adapted basis ( X , . . . , X n ) . Therefore F is one-to-one from R k × { R n − k } onto L and also from { R k } × R n − k onto W . • If e U ⊆ W and e ϕ : e U → L is a function, then ϕ : U ⊆ R n − k → R k denotes thecomposition of e ϕ with F , namely ϕ := F − ◦ e ϕ ◦ F . • If ϕ : U ⊆ R n − k → R k is a function, then we denote by e ϕ : e U ⊆ W → L the mapdefined by e ϕ := F ◦ ϕ ◦ F − . • If p ∈ G and j = 1 , . . . , s , then p j ∈ R m j is the vector of the coordinates of p in the j th layer, namely p j := ( F − ( p ) n j − +1 , . . . , F − ( p ) n j ) . • If p ∈ G and j = 1 , . . . , s , then k p j k m j denotes the Euclidean norm of p j in R m j . t is well known that all the homogeneous norms on G are bi-Lipschitz equivalent. Thus,when it will be convenient in the proofs, we work with the anisotropic norm that in expo-nential coordinates reads as k ( x , . . . , x n ) k G = n X ℓ =1 | x ℓ | / deg ℓ . We remark that a slight variation of the previous homogeneous norm gives rise to a homogen-eous norm that induces a left-invariant homogeneous distance, see [FSSC03a, Theorem 5.1].We recall that the homogeneous degree of the monomial x a · . . . · x a n n in exponential co-ordinates, is P nℓ =1 a ℓ · deg ℓ .For the expression of the operation on the group G in exponential coordinates we refer to[FSSC03a, Proposition 2.1]. In the following result we point out a useful estimate for thenorm of the conjugate. Proposition 2.4 ([FS16, Lemma 3.12]) . There exist P = ( P , . . . , P s ): G × G → R m ×· · · × R m s such that, for every p, q ∈ G , one has (9) F − ( p − qp ) = F − ( q ) + P ( p, q ) , where P = 0 and, for each i = 2 , . . . s , P i is a vector valued δ λ -homogeneous polynomial ofdegree i . Moreover, for any bounded set B ⊂ G , there exists C := C ( B, G ) > such that |P i ( p, q ) |≤ C ( k q k m + · · · + k q i − k m i − ) , for every p, q ∈ B and every i = 2 , . . . , s .Remark . With a little abuse of notation, (9) will be always written as(10) p − qp = q + P ( p, q ) , where the identification of G with R n has to be understood via exponential coordinates.Notice, however, that if one chooses a different diffeomorphism between R n and G , such asexponential coordinates of the second kind or of mixed type, the polynomial P , the constant C and the components p j have to be changed accordingly.2.2. Little Hölder continuous functions.
We introduce and discuss the notion of α -littleHölder continuous function. Definition 2.6 (little Hölder functions, [Lun95]) . Let U ⊆ R n be an open set. We denoteby h α ( U ; R k ) the set of all α -little Hölder continuous functions of order < α < , i.e., theset of maps ϕ ∈ C ( U ; R k ) satisfying(11) lim r → sup ( | ϕ ( b ′ ) − ϕ ( b ) || b ′ − b | α : b, b ′ ∈ U , < | b ′ − b | < r )! = 0 . We also define h α loc ( U ; R k ) the set of all functions ϕ ∈ C ( U ; R k ) such that ϕ ∈ h α ( U ′ ; R k ) forany open set U ′ ⋐ U .The following example is, in some sense, “pathological”. As it will be clear during thepaper, it gives a flavor of the difference between intrinsically differentiable functions anduniformly intrinsically differentiable functions, see Remark 4.8. We thank R. Serapioni forhaving shared this example with us. xample . We are going to construct a real-valued function ϕ : R → R such that ϕ ∈ h / ( R \ { } ) , ϕ / ∈ h / ( R ) , but still it holds(12) lim x → | ϕ ( x ) − ϕ (0) || x | / = 0 . Let us first notice that, for n ≥ , the intervals I n := [1 /n − /n , /n + 1 /n ] are mutuallydisjoint. Let us define, for n ≥ , the functions ϕ n : R → R as(13) ϕ n ( x ) := ( n (cid:12)(cid:12) x − n (cid:12)(cid:12) if x ∈ I n , otherwise.Notice that for each n ≥ , the function ϕ n is globally Lipschitz. Define ϕ : R → R as ϕ ( x ) := | x |· + ∞ Y n =2 ϕ n ( x ) . Notice that, being I n pairwise disjoint for n ≥ , the infinite product is well-defined, since,if x / ∈ I n , then ϕ n ( x ) = 1 . Moreover, being each ϕ n globally Lipschitz, we get that ϕ ∈ Lip loc ( R \ { } ) and thus ϕ ∈ h / ( R \ { } ) . We now prove that ϕ / ∈ h / ( R ) . In particular thiswill follow from the fact that, for every compact neighborhood U of the origin, ϕ / ∈ h / ( U ) .Indeed, by definition of ϕ , we get the following equalities(14) (cid:12)(cid:12) ϕ (cid:0) n + n (cid:1) − ϕ (cid:0) n (cid:1)(cid:12)(cid:12)(cid:0) n (cid:1) / = n + n n / = n + 1 n / , ∀ n ≥ . Thus, if U is an arbitrary compact neighborhood of 0, we get that, for every n large enough,one has [1 /n, /n + 1 /n ] ⊂ U , and thus (14) implies that (11) cannot hold, because /n =(1 /n + 1 /n ) − /n → but ( n + 1) / ( n / ) → + ∞ as n → + ∞ . Thus ϕ h / ( U ) .Finally, by definition of ϕ , we get that, for x = 0 , | ϕ ( x ) − ϕ (0) || x | / = | x | / · + ∞ Y n =2 ϕ n ( x ) , and thus (12) holds, because Q + ∞ n =2 ϕ n is bounded.We remark that, by a little modification of this example, one can replace / with any < α < .2.3. Intrinsic surfaces, Intrinsically Lipschitz and Intrinsically differentiable func-tions.
In this section we recall the notion of intrinsic graph of a function, and see whathappens to the defining map if we translate the graph. Then we recall the definitions ofintrinsically Lipschitz and intrinsically differentiable maps. Finally we discuss the notion ofco-horizontal C -surface. Definition 2.8 (Intrinsic graph of a function) . Given W and L two complementary sub-groups in G , and e ϕ : e U ⊆ W → L a function, we denote e Φ( e U ) = graph( e ϕ ) := { e Φ( w ) := w · e ϕ ( w ) : w ∈ e U } . Definition 2.9 (Intrinsic translation of a function) . Given W and L two complementarysubgroups of a Carnot group G and a map e ϕ : e U ⊆ W → L , we define, for every q ∈ G , e U q := { a ∈ W : π W ( q − · a ) ∈ e U } , nd e ϕ q : e U q ⊆ W → L by setting(15) e ϕ q ( a ) := (cid:0) π L ( q − · a ) (cid:1) − · e ϕ (cid:0) π W ( q − · a ) (cid:1) . Proposition 2.10.
Let W and L be two complementary subgroups of a Carnot group G andlet e ϕ : e U ⊆ W → L be a function. Then, for every q ∈ G , the following facts hold. (a) graph( e ϕ q ) = q · graph( e ϕ ) ; (b) ( e ϕ q ) q − = e ϕ ; (c) If W is normal, then e U q = q W · (cid:16) q L · e U · q − L (cid:17) and e ϕ q ( a ) = q L · e ϕ ( q − L · q − W · a · q L ) , for any a ∈ e U q ; (d) If q = e ϕ ( a ) − · a − for some a ∈ e U , then e ϕ q ( e ) = e. Proof.
The proof of (a), directly follows from (15), which yields(16) a · e ϕ q ( a ) = q · π W ( q − · a ) · e ϕ (cid:0) π W ( q − · a ) (cid:1) , ∀ a ∈ e U q . To prove (b), it is enough to apply twice (15). For the proof of (c), decompose q = q W · q L .Then, for every a ∈ e U q , q − · a = ( q − L · q − W · a · q L ) · q − L , and whenever W is normal one gets(17) π L ( q − · a ) = q − L , π W ( q − · a ) = q − L · q − W · a · q L . As a consequence we get e U q = q W · (cid:16) q L · e U · q − L (cid:17) and, using again (15), we obtain (c).To prove (d), it is enough to evaluate (16) in a = e and q = e ϕ ( a ) − · a − . (cid:3) We introduce the notion of intrinsically Lipschitz function and state some properties. See[FS16, Section 3].
Definition 2.11 (Intrinsic Cone) . Let W and L be two complementary subgroups of aCarnot group G . The intrinsic cone C W , L ( q, α ) of basis W and axis L , centered at q and ofopening α ≥ , is defined by C W , L ( q, α ) := q · { p ∈ G : k p W k≤ α k p L k} . Definition 2.12 (Intrinsically Lipschitz function) . Let W and L be complementary sub-groups of a Carnot group G . We say that a function e ϕ : e U ⊆ W → L is intrinsically L -Lipschitz in e U , with L > , if C W , L ( p, L − ) ∩ graph ( e ϕ ) = { p } , ∀ p ∈ graph ( e ϕ ) . Proposition 2.13 ([FS16, Theorem 3.2 & Proposition 3.3]) . Let W and L be two comple-mentary subgroups in a Carnot group G and let e ϕ : e U ⊆ W → L be a function. Then thefollowing facts are equivalent. (a) e ϕ is intrinsically L -Lipschitz in e U ; (b) k π L ( p − · q ) k≤ L k π W ( p − · q ) k for every p, q ∈ graph( e ϕ ) ; (c) for any a ∈ e U , setting q := e ϕ ( a ) − · a − , one has k e ϕ q ( b ) k≤ L k b k for every b ∈ e U q ,where e ϕ q and e U q are defined in Definition 2.9. oreover, for every a ∈ e U , setting q := e ϕ ( a ) − · a − , one has that e ϕ is intrinsically L -Lipschitzin e U if and only if e ϕ q is intrinsically L -Lipschitz in e U q . We now define the notion of intrinsically linear function, intrinsically differentiable func-tion and uniformly intrinsically differentiable function. General properties are studied in[FMS14], see for example [FMS14, Proposition 3.1.3 & Proposition 3.1.6]. For the forthcom-ing definitions and properties of intrinsically differentiable functions we follow also [DD18].The notion of intrinsic differentiability was first given in [FSSC06, Definition 4.4] and thenfirst studied in [ASCV06], see [ASCV06, Definition 1.1]. In this last reference the notion ofintrinsic differentiability is given in terms of the graph distance. We here give a slightlydifferent definition of intrinsic differentiability that is indeed equivalent to ours, by [SC16,Proposition 4.76], when W is a normal subgroup, that will always be in our case. Definition 2.14 (Intrinsically linear function) . Let W and L be complementary subgroupsin G . Then ℓ : W → L is intrinsically linear if graph( ℓ ) is a homogeneous subgroup of G . Definition 2.15 (Pansu differentiability) . Let G and G ′ be two Carnot groups endowedwith left-invariant homogeneous distances d G and d G ′ and let Ω ⊆ G be an open set. Afunction f : G → G ′ is said to be Pansu differentiable at a point p ∈ Ω if there exists aCarnot homomorphism L : G → G ′ , i.e., a group homomorphism that commutes with thedilations δ λ , such that lim x → p d G ′ ( f ( p ) − f ( x ) , L ( p − x )) d G ( x, p ) = 0 . The map L is uniquely determined, whenever it exists, and it is called the Pansu differential of f at p and it is denoted by d P f ( p ) . Definition 2.16 ( C -function) . Let Ω ⊆ G be an open subset of a Carnot group G . A map f : Ω → R k is said to be of class C if it is Pansu differentiable and the Pansu differential d P f : G → R k is continuous. We denote by C (Ω; R k ) the set of R k -valued functions of class C in Ω . Definition 2.17 ((Uniformly) intrinsic differentiability) . Let W and L be complementarysubgroups of a Carnot group G and let e ϕ : e U ⊆ W → L be a function with e U open in W . For a ∈ e U , let p := e ϕ ( a ) − · a − and denote by e ϕ p : e U p ⊆ W → L the shifted function definedin Definition 2.9.We say that e ϕ is intrinsically differentiable at a if the shifted function e ϕ p is intrinsicallydifferentiable at e , i.e., if there is an intrinsically linear map d ϕ ϕ a : W → L such that(18) lim r → (cid:18) sup (cid:26) k d ϕ ϕ a ( b ) − · e ϕ p ( b ) kk b k : b ∈ e U p , < k b k < r (cid:27)(cid:19) = 0 . The function d ϕ ϕ a , sometimes denoted also by d ϕ ϕ ( a ) , is called intrinsic differential of e ϕ at a , and we say that e ϕ is intrinsically differentiable if it is intrinsically differentiable at anypoint a ∈ e U . We also denote by ID( e U , W ; L ) the set of intrinsically differentiable functions e ϕ : e U ⊆ W → L .We say that e ϕ is uniformly intrinsically differentiable at a if, setting p a := e ϕ ( a ) − · a − for any a ∈ e U , we have(19) lim r → (cid:18) sup (cid:26) k d ϕ ϕ a ( b ) − · e ϕ p a ( b ) kk b k : a ∈ e U ∩ B ( a , r ) , b ∈ e U p a ∩ B ( a , r ) , a = b (cid:27)(cid:19) = 0 . e say that e ϕ is uniformly intrinsically differentiable on e U if it is uniformly intrinsicallydifferentiable at any a ∈ e U . We finally denote by UID( e U , W ; L ) the set of uniformlyintrinsically differentiable functions e ϕ : e U ⊆ W → L . Remark . In the papers [FSSC06, Ser17], the authorsintroduce and study the following two notions, giving characterizations for intrinsicallyLipschitz continuity, see [Ser17, Proposition 3.11 & Theorem 3.21]. For a continuous func-tion e ϕ : e U ⊆ W → L , defined on e U open, the intrinsic difference quotients of e ϕ at the point w ∈ e U in the direction Y ∈ Lie( W ) at time t > , are defined as ∇ Y e ϕ ( w, t ) := δ /t e ϕ p ( δ t exp Y ) , for every t > , where p := e ϕ ( w ) − · w − , and whenever δ t exp Y is in e U p . The intrinsicdirectional derivative of e ϕ at w ∈ e U in the direction Y ∈ Lie( W ) is defined by D Y e ϕ ( w ) := lim t → ∇ Y e ϕ ( w, t ) , whenever the limit exists. In analogy with Euclidean Calculus, we notice that, if e ϕ ∈ ID( e U , W ; L ) , then it admits intrinsic directional derivatives at any w ∈ e U along any Y ∈ Lie( W ) , and, moreover, one has D Y e ϕ ( w ) = d ϕ ϕ ( w )(exp Y ) , for any w ∈ e U and every Y ∈ Lie( W ) . Indeed, this is a consequence of the following identity k d ϕ ϕ ( w )(exp Y ) − δ /t e ϕ p ( δ t exp Y ) k = k (d ϕ ϕ ( w )( δ t exp Y )) − e ϕ p ( δ t exp Y ) k t , that simply comes from the fact both the norm k·k and d ϕ ϕ are δ λ -homogeneous. Then fromthe previous equality and (18) with a = w , and b = δ t exp Y , we get the sought claim taking t → . Proposition 2.19 ([DD18, Proposition 3.4]) . Let W and L be two complementary subgroupsof a Carnot group G with L horizontal and k -dimensional and let ℓ : W → L be an intrinsicallylinear function. Then ℓ only depends on the horizontal components of the elements in W ,namely on W := W ∩ V , where V = exp( V ) . In particular, if π V denotes the projectionfrom G to V , see (6) , one has ℓ ( a ) = ℓ ( π V a ) , ∀ a ∈ W . As a consequence, exp − ◦ ℓ ◦ exp | Lie( W ) ∩ V : Lie( W ) ∩ V → Lie( L ) is linear, and there exists aconstant C := C ( ℓ ) > such that (20) k ℓ ( a ) k≤ C k π V a k , ∀ a ∈ W . Definition 2.20 (Intrinsic gradient) . Let W and L be two complementary subgroups of aCarnot group G with L horizontal and k -dimensional, let e U ⊆ W be open, and let e ϕ : e U → L be intrinsically differentiable at a ∈ e U . By Proposition 2.19, the map exp − ◦ (d ϕ ϕ a ) ◦ exp | Lie( W ) ∩ V is linear and thus there exists a linear map ∇ ϕ ϕ a ∈ Lin(Lie( W ) ∩ V ; Lie( L )) such that d ϕ ϕ a (exp W ) = exp ( ∇ ϕ ϕ a ( W )) , ∀ W ∈ Lie( W ) ∩ V . Remark . Assume ( X , . . . , X n ) is an ad-apted basis of the Lie algebra g such that L = span { X , . . . , X k } and W = span { X k +1 , . . . , X n } and identify W and L with R n − k and R k , respectively, through exponential coordinates as xplained in Definition 2.3. Then, by Definition 2.20, with a little abuse of notation, we geta k × ( m − k ) matrix ∇ ϕ ϕ a such that, in coordinates, one has d ϕ ϕ a ( a ) = (cid:0) ∇ ϕ ϕ a ( a k +1 , . . . , a m ) T , , . . . , (cid:1) , ∀ a = ( a k +1 , . . . , a n ) ∈ W ≡ R n − k . The following proposition gives us a more explicit way to write the definition of functionsin
ID( e U , W ; L ) and in UID( e U , W ; L ) , whenever L is horizontal. Proposition 2.22 ([DD18, Proposition 3.5]) . Let W and L be complementary subgroups ina Carnot group G , with L horizontal, and let e ϕ : e U ⊆ W → L with e U open in W . Then thefollowing facts hold. (a) e ϕ is intrinsically differentiable at e a ∈ e U if and only if (21) lim r → (cid:18) sup (cid:26) | ϕ ( b ) − ϕ ( a ) − ∇ ϕ ϕ a ( a − · b ) |k e ϕ ( a ) − a − · b e ϕ ( a ) k : b ∈ U, < k a − · b k < r (cid:27)(cid:19) = 0 . (b) e ϕ is uniformly intrinsically differentiable at e a if and only if (22) lim r → (cid:18) sup (cid:26) | ϕ ( b ) − ϕ ( a ) − ∇ ϕ ϕ a ( a − · b ) |k e ϕ ( a ) − a − b e ϕ ( a ) k : a, b ∈ B ( a , r ) ∩ U, a = b (cid:27)(cid:19) = 0 . Remark . We notice that in (21) and (22) there is a little abuse of notation, for thesake of simplicity. First we are identifying L with R k in order to write the differences in thenumerators, and moreover we write ∇ ϕ ϕ a ( a − · b ) but we mean ∇ ϕ ϕ a ( π V exp − ( e a − · exp b )) . Remark . Let us collect the followingobservations about Definition 2.17.(i) If e ϕ is intrinsically differentiable at a ∈ e U , there is a unique intrinsically linear function d ϕ ϕ a satisfying (18). Moreover e ϕ is continuous at a , see [FMS14, Theorem 3.2.8 andProposition 3.2.3].(ii) The notion of intrinsic differentiability is invariant under group translations. Moreprecisely, let a, b be in e U and let p := e ϕ ( a ) − · a − and q := e ϕ ( b ) − · b − . Then e ϕ isintrinsically differentiable at a if and only if e ϕ q − p = ( e ϕ p ) q − is intrinsically differentiable at b , see ([FMS14, Remark 3.2.2]).(iii) The analytic definition of intrinsic differentiability has an equivalent geometric for-mulation. Indeed, the intrinsic differentiability at one point is equivalent to the existence ofa tangent subgroup to the graph, see [FSSC11, Theorem 4.15] for the proof in the case ofHeisenberg groups H n . If we have e ϕ : e U ⊆ W → L , and w ∈ e U , we say that a homogeneoussubgroup T of G is a tangent subgroup to graph ( e ϕ ) at w · e ϕ ( w ) if the following facts hold.(i) T is a complementary subgroup of L ;(ii) In any compact subset of G , the limit lim λ →∞ δ λ (cid:0) ( w · e ϕ ( w )) − · graph ( e ϕ ) (cid:1) = T holds in the sense of Hausdorff convergence.In the introduction of [FMS14] the authors say that e ϕ is intrinsically differentiable at w if andonly if graph ( e ϕ ) has a tangent subgroup T in w · e ϕ ( w ) and in this case T = graph (d ϕ ϕ w ) .The complete proof can be given building on [FMS14, Theorem 3.2.8], that shows one part ofthe statement, and generalizing [FSSC11, Theorem 4.15], that holds verbatim in the contextof arbitrary Carnot groups. We thank Sebastiano Nicolussi Golo for having shared with ussome notes containing a detailed proof of the previously discussed statement. roposition 2.25 ([DD18, Proposition 3.7]) . Let W and L be complementary subgroupsof a Carnot group G with L horizontal and k -dimensional, let e U ⊆ W be open and let e ϕ ∈ UID( e U , W ; L ) . Then the following facts hold. (a) e ϕ is intrinsically Lipschitz continuous on every relatively compact subset of e U . (b) the function a
7→ ∇ ϕ ϕ a is continuous from e U to the space of matrices R k × ( m − k ) . Here ∇ ϕ ϕ is the intrinsic gradient, see Definition 2.20. Definition 2.26 ( ∇ W , ∇ L ) . Let W and L be two complementary subgroups of a Carnotgroup G , with L horizontal and k -dimensional and let f ∈ C ( e U ; R k ) . Consider an adaptedbasis ( X , . . . , X n ) of the Lie algebra g such that L = exp(span { X , . . . , X k } ) and W = exp(span { X k +1 , . . . , X n } ) . Then, we define ∇ L f and ∇ W f by setting ∇ L f := X f (1) . . . X k f (1) ... . . . ... X f ( k ) . . . X k f ( k ) , ∇ W f := X k +1 f (1) . . . X m f (1) ... . . . ... X k +1 f ( k ) . . . X m f ( k ) . In particular, one has that, in exponential coordinates, ∇ H f = ( ∇ L f | ∇ W f ) .We recall the notion of co-horizontal C -surface of arbitrary codimension, see [Koz15,Definition 3.3.4]. We stress that we changed the terminology with respect to [Koz15,Definition 3.3.4] . What he calls co-Abelian surface, for us is a co-horizontal surface .For a very general definition of C -surface, we refer the reader to [Mag06, Definition 3.1],[Mag13, Definition 10.2] and to [JNGV20, Section 2.5]. Definition 2.27 (co-horizontal C -surface) . Let G be a Carnot group of rank m and let ≤ k ≤ m . We say that Σ ⊂ G is a co-horizontal C -surface of codimension k if, for any p ∈ Σ , there exist a neighborhood e U of p and a map f ∈ C ( e U ; R k ) such that(23) Σ ∩ e U = { g ∈ e U : f ( g ) = 0 } , and the Pansu differential d P f ( p ): G → R k of f is surjective.We say that Σ is a codimension k co-horizontal C -surface with complemented tangents if, in addition, given a representation around p as in (23), the homogeneous subgroup Ker(d P f ( p )) admits a horizontal complement (of dimension k ). In this case, we call Ker(d P f ( p )) the homogeneous tangent space to Σ at p . This homogeneous subgroup at p is independentof the choice of f , see [Mag13, Theorem 1.7].We remark that, if Σ ⊆ G is a co-horizontal C -surface with complemented tangents,then one can use the implicit function Theorem, see [FSSC03b, Theorem 2.1] for the one-codimensional case, and see [Mag13, Theorem 1.4] for the more general statement, to locallyrepresent the surface as a graph of a function e ϕ : e U ⊆ W := Ker(d P f ( p )) → L , with W and L complementary subgroups.The following proposition follows from [DD18, Theorem 4.1 & Theorem 4.6] and relateslevel sets of R k -valued C -functions, and ultimately co-horizontal C -surfaces with comple-mented tangents, with uniformly intrinsically differentiable functions. Proposition 2.28 ([DD18, Theorem 4.1 & Theorem 4.6]) . Let W and L be two comple-mentary subgroups of a Carnot group G , with L horizontal and k -dimensional, take e U ⊆ W pen and e ϕ ∈ UID( e U , W ; L ) . Then, for every a ∈ e U , there exist a neighborhood e V of a · e ϕ ( a ) in G , and f ∈ C ( e V ; R k ) , such that e Φ( e U ) ∩ e V = { g ∈ e V : f ( g ) = 0 } , and, for every g ∈ e V , the Pansu differential d P f ( g ) | L : L → R k is bijective. As a con-sequence graph ( e ϕ ) is a co-horizontal C -surface of codimension k , with tangents comple-mented by L . Moreover, if ( X , . . . , X n ) is an adapted basis of the Lie algebra g such that L = exp(span { X , . . . , X k } ) and W = exp(span { X k +1 , . . . , X n } ) , then det ∇ L f = 0 and, inexponential coordinates, one has (24) ∇ ϕ ϕ ( a ) = − (cid:16) ∇ L f ( e Φ( a )) (cid:17) − ∇ W f ( e Φ( a )) , ∀ a ∈ e U .
For the definition of ∇ ϕ ϕ, ∇ W and ∇ L we refer to Definition 2.20 and Definition 2.26.On the other hand, if ≤ k ≤ m and Σ is a codimension k co-horizontal C -surfacewith complemented tangents, then, for every p ∈ Σ , there exist two complementary subgroups W and L of G with L horizontal and k -dimensional, a neighborhood e V ⊆ G of p and e ϕ ∈ UID( e U , W ; L ) , with e U = π W ( e V ) , such that Σ ∩ e V = graph ( e ϕ ) . Remark . Notice that, in the setting of Proposition 2.28, in the case k = 1 , one mayassume X f = 0 on e V , and, in coordinates, formula (24) reads as(25) ∇ ϕ ϕ ( a ) = − (cid:18) X fX f , . . . , X m fX f (cid:19) ◦ Φ( a ) , ∀ a ∈ e U .
Remark C -surfaces) . From the previous Proposition 2.28 andRemark 2.24 it directly follows that every co-horizontal C -surface with complemented tan-gents has Hausdorff tangent everywhere. For a proof of this property in a more generalcontext one can see [Mag13, Theorem 1.7], or [JNGV20, Lemma 2.14, point (iii)]. Thisconvergence is moreover locally uniform: we will not use this information, but this comesfrom [Koz15, Theorem 3.1.1].3. Intrinsic projected vector fields on subgroups
In this section we mainly deal with complementary subgroups W and L of a Carnot group G along with a continuous map e ϕ : e U ⊆ W → L , where e U is open in W .In Section 3.1 we shall define, for some W ∈ Lie( W ) , the projected vector field D ϕW on W by taking the projection on W of W restricted the graph e Φ( e U ) of e ϕ (see Definition 3.1), andwe discuss some basic properties of these vector fields. We give explicit formulas for thesevector fields in Heisenberg groups H n , in Carnot groups of step 2, and in the Engel group E (see Example 3.4, Example 3.6, and Example 3.8, respectively). In Proposition 3.9 weshow an explicit expression of such vector fields in exponential coordinates. The definition ofthe projected vector fields appeared first in [Koz15], see Remark 3.2. In [Koz15] the authorgives equivalent conditions for e Φ( e U ) to be an intrinsically Lipschitz graph (respectively aco-horizontal C -surfaces with complemented tangents) in terms of Hölder properties of theintegral curves of the vector fields D ϕW , see [Koz15, Theorem 4.2.16] (respectively [Koz15,Theorem 4.3.1]). Within our context we recover these results by using invariance propertiesof such vector fields, see the introduction to Section 4. n Section 3.2 we prove some invariance properties of the projected vector fields withrespect to the intrinsic translations (see Definition 2.9) of e ϕ . In particular we write how thevector field D ϕ q changes with respect to D ϕ and how the integral curves of D ϕ q change withrespect to the integral curves of D ϕ , see Lemma 3.12 and Lemma 3.13 when L is horizontal,and Remark 3.14 for the general case in which W is normal. These invariance properties willbe crucial for the proof of the results in Section 4.In Section 3.3 we recall that if e ϕ is intrinsically Lipschitz, then e ϕ ◦ e γ is /j -Hölder whenever e γ is an integral curve of D ϕW with deg W = j , see Proposition 3.17. We stress that thisproperty was already known from [Koz15, Theorem 4.2.16]. We improve this result when e ϕ is more regular. Namely if e ϕ is intrinsically differentiable, then e ϕ ◦ e γ is Euclidean differentiablewhenever e γ is an integral curve of D ϕW with deg W = 1 , while if deg W > we obtain a pointwise little Hölder continuity of e ϕ ◦ e γ , see Proposition 3.19. This pointwise little Höldercontinuity improves to a uniform little Hölder continuity if e ϕ is uniformly intrinsicallydifferentiable, see Proposition 3.21, and Proposition 3.22 for a more refined conclusion.In Section 3.4 we recall the notion of broad* solution to the system D ϕ ϕ = ω , with acontinuous datum ω . The study of the relation between being a broad* solution to the system D ϕ ϕ = ω with a continuous ω and the intrinsic regularity of graph( e ϕ ) was first initiated, in H n for L one-dimensional in [ASCV06, Section 5], and then continued in [BSC10a, BSC10b,BCSC14]. For the case G = H n and L horizontal and k -dimensional see also [Cor19] and forthe general case of Carnot groups of step 2 and L one-dimensional, see [DD18, DD19]. InProposition 3.27 we give a sufficient condition for the map ϕ to be a broad* solution to thesystem D ϕ ϕ = ω , with ω continuous. This condition is the intrinsic differentiability plus thecontinuity of the intrinsic gradient.3.1. Definition of D ϕ and main properties. In this subsection we define the projectedvector fields D ϕW and state some of their properties. Definition 3.1 (Projected vector fields) . Given two complementary subgroups W and L ina Carnot group G , and a continuous function e ϕ : e U ⊆ W → L defined on an open set e U of W , we define, for every W ∈ Lie( W ) , the continuous projected vector field D ϕW , by setting(26) ( D ϕW ) | w ( f ) := W | w · e ϕ ( w ) ( f ◦ π W ) , for all w ∈ e U and all f ∈ C ∞ ( W ) . When W is an element X j of an adapted basis ( X , . . . , X n ) we also denote D ϕj := D ϕX j . Remark . The Definition 3.1 is well posed since the projection π W is polynomial and hence C ∞ for every arbitrary splitting. Notice that if e ϕ is C ∞ then D ϕW is a vector field with C ∞ coefficients.Definition 3.1 has been given in [Koz15, Definition 4.2.12] and it has been studied in thecase W is a homogeneous normal subgroup and, more specifically, when L is horizontal and k -dimensional. We refer to the discussion in the introduction of Section 3. From now on W denotes a homogeneous normal subgroup of G . Remark . Notice that (26) is equivalent to(27) ( D ϕW ) | w = d( π W ) e Φ( w ) ( W | e Φ( w ) ) , hat is, D ϕW is the push-forward of the vector field W towards the map ( π W ) | e Φ( e U ) : e Φ( e U ) → e U .Thus, as already observed in [Koz15, Equation 4.4], one has(28) ( D ϕW ) | w = dd t | t =0 π W ( e Φ( w ) · exp( tW )) = dd t | t =0 w · e ϕ ( w ) · exp( tW ) · e ϕ ( w ) − == dd t | t =0 L w ◦ L e ϕ ( w ) ◦ R e ϕ ( w ) − (exp( tW )) == d( L w ) e ◦ d( L e ϕ ( w ) ) e ϕ ( w ) − ◦ d( R e ϕ ( w ) − ) e ( W | e ) = d( L w ) e ◦ Ad e ϕ ( w ) ( W | e ) . For the previous computation, we used the definition of the differential and, in the secondequality, the fact that π W ( e Φ( w ) · exp( tW )) = π W ( w · e ϕ ( w ) · exp( tW ) · e ϕ ( w ) − · e ϕ ( w )) == w · e ϕ ( w ) · exp( tW ) · e ϕ ( w ) − , for every w ∈ W and W ∈ Lie( W ) , where the last equality holds since W is normal. Example . Consider the Heisenberg group H n , with an adapted basis ( X , . . . , X n +1 ) of its Lie algebra such that the only nonvanishingrelations are [ X i , X n + i ] = X n +1 , for every ≤ i ≤ n . Fix ≤ k ≤ n , identify H n with R n +1 by means of exponential coordinates associated with ( X , . . . , X n +1 ) and define W := { x = · · · = x k = 0 } , and L := { x k +1 = · · · = x n +1 = 0 } . Then, for a continuous e ϕ : e U ⊆ W → L , with e U open, one can compute in exponential coordinates ( D ϕX j ) | w = ( X j ) | w , k + 1 ≤ j ≤ n ∨ k + n + 1 ≤ j ≤ n, ( D ϕX n + i ) | w = ( ∂ x n + i ) | w + ϕ ( i ) ( w )( ∂ x n +1 ) | w , i = 1 , . . . , k. (29)(30) ( D ϕX n +1 ) | w = ( ∂ x n +1 ) | w . for every w ∈ U , where ϕ denotes the composition of e ϕ with the exponential coordinatesand ϕ ( i ) is its i -th component of ϕ . Notice that we do not have the first condition in case k = n . Remark . For the computations of Example 3.4, we refer to [Koz15, Section 4.4.2], wherethe constant is slightly different from ours because of the fact that the author considers themodel of H n with relations [ X i , X n + i ] = − X n +1 , for every ≤ i ≤ n . The expression ofthe projected vector fields in case L is k -dimensional is also in [Cor19, Definition 3.6].It is by now well known, from the papers [ASCV06], [BSC10a], [BSC10b], and [BCSC14],that in every Heisenberg group H n , in case L is one-dimensional, the intrinsic regularityof graph( e ϕ ) depends on the regularity of the vector field D ϕ applied to ϕ , i.e., D ϕ ϕ :=( D ϕX ϕ, . . . , D ϕX n ϕ ) , which has to be considered in the sense of distributions. For the fullresults we refer to [SC16, Theorems 4.90 & 4.92]. In particular graph( e ϕ ) is an intrinsicallyLipschitz graph (respectively a C -hypersurface) if and only if D ϕ ϕ = ω , in the distributionalsense, for some ω ∈ L ∞ ( U ) (respectively ω ∈ C ( U ) ).A step towards obtaining analogous results in H n , in case L has higher dimension and ω is continuous, has been recently done by Corni in [Cor19]. In particular, the author provesthat, if L is horizontal k -dimensional, the set graph( e ϕ ) is a co-horizontal C -surface if andonly if ϕ is a broad* solution to D ϕ ϕ = ω for some ω ∈ C ( U ) . We shall recall the definitionof broad* in Definition 3.24. xample . Consider a Carnot group G of rank m and step 2 with an adapted basis ( X , . . . , X n ) . For ≤ s, ℓ ≤ m and m +1 ≤ i ≤ n ,let us define the structure constants c iℓs by means of the relation [ X ℓ , X s ] =: P ni = m +1 c iℓs X i .Identify G with R n by means of exponential coordinates and take W := { x = 0 } , and L := { x = · · · = x n = 0 } . Then, if e ϕ : e U ⊆ W → L is continuous, with e U open, by explicitcomputations one has in exponential coordinates(31) ( D ϕX j ) | w = ( X j ) | w + n X i = m +1 c i j ϕ ( w )( ∂ x i ) | w , for j = 2 , . . . , m ; (32) ( D ϕX j ) | w = ( X j ) | w = ( ∂ x j ) | w , for j = m + 1 , . . . , n, for every w ∈ U , where ϕ denotes the composition of e ϕ with the exponential coordinates. Remark . For the expression of the projected vector fields in Example 3.6 we refer alsoto [DD18, Definition 5.2]. In the papers [DD18] and [DD19] the author started to generalizethe results already proved in the Heisenberg groups H n (see Remark 3.5) to Carnot groupsof step 2, in case L is one-dimensional.In particular, in [DD18], in the setting of Carnot groups of step 2, the author dealswith the characterization of maps e ϕ such that graph( e ϕ ) is a C -hypersurface. In [DD18,Theorem 5.8], the author recovers partially the result in [ASCV06, Theorem 5.1], thus makingthe first step through the complete characterization in step 2 Carnot groups analogous tothe one discussed in Remark 3.5 for the Heisenberg group. We stress that in this paper ofours we generalize [ASCV06, Theorem 5.1] to all step-2 Carnot groups, thus improving also[DD18, Theorem 5.8], see Theorem 6.17.In [DD19, Theorem 7.1 & 7.2], in the setting of Carnot groups of step 2, with L one-dimensional, the author recovers [BCSC14, Theorem 1.1] with an additional assumption: graph( e ϕ ) is intrinsically Lipschitz if and only if D ϕ ϕ = ω in the sense of distribution for some ω ∈ L ∞ ( U ) and ϕ is locally / -Hölder along the vertical coordinates . We expectthat the techniques of Section 6 can be used to drop the previous additional assumptionon the locally 1/2-Hölder continuity along the vertical coordinates. This will be subject offurther investigations. Example . Consider the Engel group E , whichis the Carnot group of topological dimension whose Lie algebra e admits an adapted basis ( X , X , X , X ) such that e := span { X , X } ⊕ span { X } ⊕ span { X } , where the only nonvanishing relations are [ X , X ] = X , [ X , X ] = X . We identify E with R by means of exponential coordinates, and we define W := { x = 0 } , and L := { x = x = x = 0 } . Then, by explicit computations that can be found in [Koz15, Section 4.4.1], we getthat, given a continuous function e ϕ : e U ⊆ W → L , with e U open, the projected vector fieldson W are(33) D ϕX = ∂ x + ϕ∂ x + ϕ ∂ x , D ϕX = ∂ x + ϕ∂ x , D ϕX = ∂ x . In [Koz15, Setion 4.4.1], one can find the computations of the projected vector fieldsalso for the pair of complementary subgroups ( { x = 0 } , { x = x = x = 0 } ) . It is worthmentioning that in [Koz15, Section 4.5] there are some counterexamples, for the Engel group with the splitting discussed here, to some of the statements, discussed in Remark 3.5, andRemark 3.7, that holds for Carnot groups of step 2. For one of these examples, see alsoRemark 4.18.We now prove a proposition about the general form of the projected vector fields, in anarbitrary Carnot group G , in exponential coordinates. The proposition below can be foundin [Koz15, Proposition 4.1.15], but we however write the proof for the sake of completeness. Proposition 3.9 (Projected vector fields in coordinates) . Let W and L be complementarysubgroups of a Carnot group G such that L is horizontal and k -dimensional, and let e ϕ : e U ⊆ W → L be a continuous function on an open set e U . Fix an adapted basis ( X , . . . , X n ) of theLie algebra g such that W = exp (span { X k +1 , . . . , X n } ) and L = exp (span { X , . . . , X k } ) .If we identify G with R n by means of exponential coordinates associated with ( X , . . . , X n ) ,then the vector fields D ϕj := D ϕX j defined in (26) have the following expression: (34) D ϕj | (0 ,..., ,xk +1 ,...,xn ) = ∂ x j + n X i = n deg j +1 P ji ( ϕ (1) , . . . , ϕ ( k ) , x k +1 , . . . , x n deg i − ) ∂ x i , ∀ j = k + 1 , . . . , n, where ϕ is the composition of e ϕ with the exponential map, ϕ ( i ) = ϕ ( i ) ( x k +1 , . . . , x n ) denotesthe i -th component of ϕ , and P ji is a polynomial of homogeneous degree deg i − deg j , withthe convention that the degree of the ϕ ( i ) components in the polynomial is 1. For the notation n (cid:3) and deg , see the discussion before Definition 2.3.Proof. Since L is horizontal, then W is normal and (28) holds. Now the result followsfrom (28) and the following general fact that holds for arbitrary Carnot groups. Let us fix,in exponential coordinates, x ∈ G ≡ R n . The differential of the left (respectively right)translation evaluated at a point x ′ ∈ G ≡ R n , that we denote by d( L x ) x ′ (respectively d( R x ) x ′ ), is a matrix with identity ( m ℓ × m ℓ ) -blocks on the diagonal, with ≤ ℓ ≤ s , andmoreover the element of position ij is a polynomial in the coordinates of x, x ′ of homogeneousdegree deg i − deg j , if deg i > deg j . Instead if deg i < deg j the element of position ij is zero.This last statement about the structure of the differential of left and right translations followsby the explicit expression of the product in coordinates, see [FSSC03a, Proposition 2.1]. (cid:3) Remark . Notice that, for notational purposes, Proposition 3.9 is stated just for L horizontal but an expression similar to (34) holds also in the more general case in which W is normal, see also [Koz15, Proposition 4.1.15]. The difference is that the zeros shouldbe put in the components of L , which are not necessarily all in the first layer, and the ϕ ( i ) components in the polynomial are not necessarily of degree 1. Lemma 3.11 (Locally connectible with projected vector fields) . Let W and L be comple-mentary subgroups of G with W normal, and let e ϕ : e U ⊆ W → L be a continuous functionon an open set e U . Then, for every e U ′ ⋐ e U and every e w ∈ e U ′ , there exists a neighborhood e V ⋐ e U ′ of e w such that, for every e v, e v ′ ∈ e V there exists a path, entirely contained in e U ′ ,connecting e v to e v ′ , made of a finite concatenation of integral curves for the vector fields D ϕW ,for W ∈ Lie( W ) .Proof. We give the proof in the case L is horizontal and k -dimensional. The same proof canbe given in the general case in which W is normal taking Remark 3.10 into account. We fixan adapted basis ( X , . . . , X n ) of the Lie algebra g , such that W = exp( span { X k +1 , . . . , X n } ) nd L = exp( span { X , . . . , X k } ) . We denote by ϕ : U ⊆ R n − k → R k the composition of e ϕ with the exponential coordinates and we set D ϕj := D ϕX j for every j = k + 1 , . . . , n . Usingthe particular form of D ϕj , see (34), we shall prove that U is locally connectible by means ofintegral curves of the vector fields D ϕj , with j = k + 1 , . . . , n . Indeed, if we fix w ∈ U , and j ∈ { k + 1 , . . . , n } , Peano’s Theorem [Hal80, Theorem 1.1] and the estimate for the existingtime [Hal80, Corollary 1.1] imply that, for every open neighborhood U ′ ⋐ U of w , there exist < α := α ( U ′ , ϕ ) and an open neighborhood V ′′ := V ′′ ( U ′ , ϕ ) of w with V ′′ ⋐ U ′ , such that,for every v ∈ V ′′ , there exists at least one integral curve γ of D ϕj starting from v , defined in [ − α, α ] , and such that γ ([ − α, α ]) ⊆ U ′ .We can iterate the argument for each j , eventually considering a smaller neighborhood V ′′ and a smaller time at each stage of the iteration. Thus we obtain a neighborhood V ′ ⋐ U ′ of w and a time δ > such that, starting from an arbitrary point in V ′ , there exists aconcatenation of n − k integral curves of D ϕj , with j = k + 1 , . . . , n , with interval of definitioncontaining [ − δ, δ ] , and this concatenation is supported in U ′ . Eventually, up to reducing V ′ ,we deduce that for every U ′ ⋐ U containing w there exists a neighborhood V ⋐ V ′ ⋐ U ′ of w such that, for every v, v ′ ∈ V , there exists at least one path from v to v ′ , made of integralcurves of the vector fields D ϕj , with j = k + 1 , . . . , n , entirely contained in U ′ . We remarkthat this can be done taking into account that any integral curve of D ϕj is a line along the j -th coordinate, and adjusting one coordinate at time. When we adjust one coordinate, wedo not have the check the previous ones because of the triangular form of the vector fields D ϕj , see (34). (cid:3) Invariance properties of D ϕ . Now we prove some invariance properties of the vectorfields D ϕ under the operation of translating graphs that we have introduced in Definition 2.9. Lemma 3.12.
Let W and L be two complementary subgroups of a Carnot group G , with L k -dimensional and horizontal and let e ϕ : e U ⊆ W → L be a continuous function defined on e U open. Take W ∈ Lie( W ) , let e f : e U ⊆ W → L be a C ∞ function and let us denote D ϕ := D ϕW .Fix q ∈ G and denote by e ϕ q and e f q the translated functions defined as in (15) with domain e U q .Denote by f, ϕ : U ⊆ R n − k → R k and f q , ϕ q : U q ⊆ R n − k → R k the composition of e f , e ϕ, e f q , e ϕ q with the exponential coordinates, respectively. Then, for every w ∈ U q , the following equalityholds in exponential coordinates (35) D ϕ q | w ( f q ) = D ϕ | π W ( q − · w ) ( f ) , where D ϕ ( f ) , for a vector valued f , stands for the vector ( D ϕ ( f (1) ) , . . . , D ϕ ( f ( k ) )) .Proof. We stress a little abuse of notation throughout the proof, for the sake of simplicity.We exploit the identifications as in Definition 2.3 without explicitly write the exponentialmap F or F − .By (26) and (16), if we fix w ∈ U q and set g := π W ( q − · w ) · e ϕ ( π W ( q − · w )) , we get(36) D ϕ q | w ( f q ) = W | w · e ϕq ( w ) ( f q ◦ π W ) = W | q · g ( f q ◦ π W ) = W | g ( f q ◦ π W ◦ L q ) , where in the last equality we used the fact that W is left-invariant, namely W | q · g = d( L q ) g ( W | g ) .By definition of g and the definition of D ϕ , see (26), one has(37) D ϕ | π W ( q − · w ) ( f ) = W | g ( f ◦ π W ) . hus, taking (36) and (37) into account, we are left to show that(38) W | g ( f q ◦ π W ◦ L q ) = W | g ( f ◦ π W ) . Indeed, if a ∈ G , we have(39) e f q ◦ π W ◦ L q ( a ) = e f q ◦ π W ( q · a ) = e f q ◦ π W (cid:0)(cid:0) q W · q L · a W · q − L (cid:1) · q L · a L (cid:1) = e f q ( q W · q L · a W · q − L ) = q L · e f ( a W ) = q L · e f ◦ π W ( a ) , where in the third equality we used that W is a normal subgroup, and in the fourth equalitywe used (c) of Proposition 2.10. Then, the functions e f q ◦ π W ◦ L q and e f ◦ π W differ only by a lefttranslation of the element q L . Thus, in exponential coordinates, they are R k -valued functionsthat differ by the fixed Euclidean translation of the R k -vector corresponding to q L . This lastobservation comes from the fact that, in exponential coordinates, the operation of the grouprestricted to L is the Euclidean sum, being L horizontal, see [FSSC03a, Proposition 2.1].Finally, (38) holds true by the fact that, component by component, we are differentiatingalong a vector field two functions that differ by a fixed constant. (cid:3) Lemma 3.13.
Consider the setting of the Lemma 3.12 above, let
T > and let e γ : [0 , T ] → e U be a C regular solution of the Cauchy problem (40) (e γ ′ ( t ) = D ϕ ◦ e γ ( t ) , e γ (0) = w. Then for every q ∈ G there exists a unique C map e γ q : [0 , T ] → e U q such that (41) π W ( q − · e γ q ( t )) = e γ ( t ) , ∀ t ∈ [0 , T ] . In addition, e γ q is a solution of the Cauchy problem (42) (e γ ′ q ( t ) = D ϕ q ◦ e γ q ( t ) , e γ q (0) = q W · q L · w · q − L . Moreover, if there exists a continuous function ω : U ⊆ R n − k → R k such that ϕ ( γ ( t )) − ϕ ( γ (0)) = Z t ω ( γ ( s )) d s, ∀ t ∈ [0 , T ] , then, there exists a continuous function ¯ ω q : U q ⊆ R n − k → R k such that ϕ q ( γ q ( t )) − ϕ q ( γ q (0)) = Z t ¯ ω q ( γ q ( s )) d s, ∀ t ∈ [0 , T ] . Proof.
We stress a little abuse of notation throughout the proof, for the sake of simplicity.We exploit the identifications as in Definition 2.3 without explicitly write the exponentialmap F or F − .For every q ∈ G , we define e γ q ( t ) := q W · q L · e γ ( t ) · q − L , ∀ t ∈ [0 , T ] . Then e γ q takes values in e U q , see item (c) of Proposition 2.10, and e γ q (0) = q W · q L · w · q − L .Moreover, one also has π W ( q − · e γ q ( t )) = π W ( q − L · q − W · q W · q L · e γ ( t ) · q − L ) = e γ ( t ) , ∀ t ∈ [0 , T ] . oreover, if we impose (41), the uniqueness of e γ q is guaranteed by the second equation of(17), and the equivalence(43) q − L · q − W · e γ q ( t ) · q L = e γ ( t ) ⇔ e γ q ( t ) = q W · q L · e γ ( t ) · q − L , ∀ t ∈ [0 , T ] . Now we want to check that e γ q is a solution to the Cauchy problem (42). We work inexponential coordinates. For every vector-valued function f : U q ⊆ R n − k → R k of class C ∞ ,we have that, for every t ∈ [0 , T ] , it holds, in exponential coordinates, the following chain ofequalities:(44) D ϕ q | γq ( t ) ( f ) = D ϕ q | γq ( t ) (( f q − ) q ) = D ϕ | γ ( t ) ( f q − )= γ ′ ( t )( f q − ) = dd t ( f q − ◦ γ ( t ))= dd t ( f q − ◦ π W ◦ L q − ( γ q ( t )))= dd t (cid:0) ( q − ) L · f ◦ π W ( γ q ( t )) (cid:1) = dd t (cid:0) ( q − ) L · f ◦ γ q ( t )) (cid:1) = dd t ( f ◦ γ q ( t )) = γ ′ q ( t )( f ) , where in the first equality we used item (b) of Proposition 2.10, and in the second onewe used (35) and the fact that π W ( q − · e γ q ( t )) = e γ ( t ) . In the fifth equality we used thecoordinate version of π W ( q − · e γ q ( t )) = e γ ( t ) with a little abuse of notation, and in the sixthequality we used the coordinate version of (39) with q − in place of q . Finally, in the eighthequality we used the fact that, being L horizontal, the product with a fixed element of L , in exponential coordinates, is a Euclidean translation and hence it does not affect thetime derivative. This comes from the explicit expression of the product in coordinates, see[FSSC03a, Proposition 2.1]. Thus the proof of (42) is finished by (44).By a further inspection, following the equalities starting from the right hand side of thesecond line to the right hand side of the fourth line of (44), we have proved, only byexploiting the fact that W is normal , that if e U is open then for every function e G : e U ⊆ W → L , for every q ∈ G , and every t ∈ [0 , T ] , it holds(45) e G ◦ e γ ( t ) = ( q − ) L · e G q ◦ e γ q ( t ) . Since L is horizontal, in exponential coordinates this equality reads as(46) G ◦ γ ( t ) = G q ◦ γ q ( t ) + ( q − ) L . Assume there exists a continuous map ω : U ⊆ R n − k → R k as in the statement. Then, bycomposing with exponential coordinates we get a continuous e ω : e U ⊆ W → L . We thendefine e ω q as in (15) and we set ω q : U q ⊆ R n − k → R k to be the composition of e ω q with theexponential coordinates. We are then in a position to define ω q by setting(47) ω q ( x ) := ω q ( x ) + ( q − ) L , ∀ x ∈ U q . Then, we have, for every t ∈ [0 , T ] ,(48) ϕ q ( γ q ( t )) − ϕ q ( γ q (0)) = ϕ ( γ ( t )) − ϕ ( γ (0)) = Z t ω ( γ ( s )) d s == Z t (cid:0) ω q ( γ q ( s )) + ( q − ) L (cid:1) d s = Z t ¯ ω q ( γ q ( s )) d s, here in the first and in the third equality we used (46). This completes the proof. (cid:3) Remark . The curve e γ q defined in (43) solves (42) also in the general case when W isnormal, but one should run more involved computations, because the invariance property(35) might not be true. We presented the invariance in (35) in the specific case L is horizontaland k -dimensional because it will be frequently used in the last theorems of this section andin Section 4.We give a sketch of the proof in the general case. For a reference, one can also readthe first item of [Koz15, Proposition 4.2.15]. Given q ∈ G , define the map σ q on W by σ q ( w ) := q W · q L · w · ( q L ) − . By (17), we get that σ q − ( w ) = π W ( q − · w ) . Then, in case L ishorizontal, the invariance property proved in (35) reads as D ϕ q | w = D ϕ ◦ σ q − ( w ) . In case L isnot horizontal, using the properties stated in Proposition 2.10, one can prove that D ϕ q | w = d( σ q ) σ q − ( w ) ( D ϕ ◦ σ q − ( w )) , for every w ∈ e U and q ∈ G . One can hence use the previous equality and the definition of e γ q as in (43) to show that, if a curve e γ satisfies (40), then e γ q satisfies (42).3.3. Metric properties of integral curves of D ϕ . In this subsection we study how theintrinsically Lipschitz regularity of e ϕ affects the metric regularity of the integral curves ofthe vector fields D ϕ (see Proposition 3.17). We also see how the conclusions obtained can beimproved when we assume e ϕ to be intrinsically differentiable (Proposition 3.19) or uniformlyintrinsically differentiable (see Proposition 3.21 and Proposition 3.22).The following lemma is essentially the implication (1) ⇒ (3) of [Koz15, Theorem 4.2.16].For the reader convenience, and for some benefit toward Remark 3.18, we give the proofhere, without going through all the precise estimates, and claiming no originality. Lemma 3.15.
Let W and L be two complementary subgroups of a Carnot group G , with W normal, let e U ⊆ W be an open neighborhood of the identity e and let e ϕ : e U → L be a functionsuch that e ϕ ( e ) = e . Assume there exists a constant C > with (49) k e ϕ ( w ) k G ≤ C k w k G , ∀ w ∈ e U .
Then for every integer d ≥ , every W ∈ Lie( W ) ∩ V d , and every integral curve e γ : [0 , T ] → e U of D ϕW starting from e , there exists C ′ depending only on C and W , such that (50) k e γ ( t ) k G ≤ C ′ t /d , k e ϕ ( e γ ( t )) k G ≤ C ′ t /d , ∀ t ∈ [0 , T ] . Proof.
We give the proof in case L is k -dimensional and horizontal, just for notational pur-poses. Then, taking Remark 3.10 into account, the proof can be given in the general case.We recall that, from the fact that all the homogeneous norms are equivalent, we may fix k ( x , . . . , x n ) k G := P ni =1 | x i | / deg i . Then, in the case L is horizontal, k e ϕ ◦ e γ k G = | ϕ ◦ γ | .We fix an adapted basis ( X , . . . , X n ) of the Lie algebra g such that W = X j for some j ∈ { k +1 , . . . , n } . Then deg j = d and, according to Proposition 3.9, we have, in exponentialcoordinates adapted to this basis, that D ϕj := D ϕX j = D ϕW writes as(51) D ϕj | (0 ,..., ,x k +1 ,...,x n ) = ∂ x j + n X i = n d +1 P ji ( ϕ (1) , . . . , ϕ ( k ) , x k +1 , . . . , x n deg i − ) ∂ x i , for some polynomials P ji of homogeneous degree deg i − d . By the use of triangle inequalityand Young’s inequality, we get that for every i = n d +1 , . . . , n there exists a constant C ,i > epending only on the polynomial P ji , and a constant C ,i > that depends only on C ,i and on n , such that(52) | P ji ( ϕ (1) , . . . , ϕ ( k ) , x k +1 , . . . , x n deg i − ) | ≤ C ,i k (0 , . . . , , ϕ (1) , . . . , ϕ ( k ) , x k +1 , . . . , x n deg i − ) k deg i − d G ≤ C ,i ( k (0 , . . . , , ϕ (1) , . . . , ϕ ( k ) , , . . . , k deg i − d G + k (0 , . . . , , x k +1 , . . . , x n ) k deg i − d G ) . Fix t ∈ [0 , T ] and define m γ ( t ) := max s ∈ [0 ,t ] k e γ ( s ) k G , m ϕ ( t ) := max s ∈ [0 ,t ] | ϕ ( γ ( s )) | . Then, by the fact that e γ is an integral curve of D ϕj and the particular form of D ϕj in (51),we get that γ j ( t ) = t for every t ∈ [0 , T ] , and γ ℓ ( t ) ≡ for every t ∈ [0 , T ] and every ℓ = j with ≤ ℓ ≤ n d . The estimate (52) implies that, for every i ≥ n d + 1 , one has | γ i ( s ) | ≤ Z s (cid:12)(cid:12) P ji (cid:0) ϕ (1) ( γ ( r )) , . . . , ϕ ( k ) ( γ ( r )) , γ k +1 ( r ) , . . . , γ n deg i − ( r ) (cid:1)(cid:12)(cid:12) d r ≤ tC ,i (cid:0) ( m γ ( t )) deg i − d + ( m ϕ ( t )) deg i − d (cid:1) ≤ tC ,i ( m γ ( t )) deg i − d , ∀ s ∈ [0 , t ] , (53)where in the last inequality we used m ϕ ( t ) ≤ Cm γ ( t ) for every t ∈ [0 , T ] , that comes fromthe hypothesis (49), and where C ,i depends only on C ,i and C . Thus, from (53) and thefact that the only other nonzero component of γ is γ j ( t ) = t , we get k e γ ( s ) k G ≤ C t /d + n X i = m d +1 t / deg i ( m γ ( t )) − d/ deg i ! ≤ nC max i ∈{ n d +1 ,...,n } (cid:8) t /d , t / deg i ( m γ ( t )) − d/ deg i (cid:9) , for every s ∈ [0 , t ] , where C depends only on all the constants C ,i , with i ≥ m d + 1 .Maximizing the previous inequality with respect to s ∈ [0 , t ] , gives m γ ( t ) ≤ nC max i ∈{ n d +1 ,...,n } (cid:8) t /d , t / deg i ( m γ ( t )) − d/ deg i (cid:9) , for every t ∈ [0 , T ] . As a consequence of this inequality we get m γ ( t ) ≤ C t /d for all t ∈ [0 , T ] , for a constant C depending only on C and on G . We have thus proved k e γ ( t ) k G ≤ C t /d , ∀ t ∈ [0 , T ] . To conclude the proof, it is enough to use (49) and choose C ′ := max { C , CC } . Finally,taking into account all the dependencies of the constants, the constant C ′ only dependson C and on the coefficients of D ϕj in coordinates, and thus ultimately on the vector field W ∈ Lie( W ) . (cid:3) Remark . Condition (49) in Lemma 3.15 can be deduced as soon as e ϕ : e U ⊆ W → L is intrinsically Lipschitz at e , see the item (c) of Proposition 2.13. We will give a generalstatement in this direction in the forthcoming Proposition 3.17, that is a restatement of theimplication (1) ⇒ (2)&(3) of [Koz15, Theorem 4.2.16].Notice that in Lemma 3.15 we only exploited the particular triangular form of D ϕj , see(34). The same result as in Lemma 3.15 holds if we take the integral curves, startingfrom e , of any vector field of the same form as the right hand side of (51), satisfying the omogeneity conditions on the polynomials P ji given in Proposition 3.9. On the contrary,for the forthcoming Proposition 3.17, we need that we are dealing precisely with the vectorfields D ϕj in order to use the invariance properties in Lemma 3.12, and Lemma 3.13. Proposition 3.17.
Let W and L be two complementary subgroups of a Carnot group G , with W normal, let e U be an open subset of W , and let e ϕ : e U → L be an intrinsically L -Lipschitzfunction with Lipschitz constant L > .Then for every integer d ≥ , every W ∈ Lie( W ) ∩ V d , and every integral curve e γ : [0 , T ] → e U of D ϕW , there exists a constant C ′ > depending only on L and W such that (54) k e ϕ ( e γ ( s )) − · e γ − ( s ) · e γ ( t ) · e ϕ ( e γ ( s )) k G ≤ C ′ | t − s | /d , for ≤ s < t ≤ T ; (55) k e ϕ ( e γ ( s )) − · e ϕ ( e γ ( t )) k G ≤ C ′ | t − s | /d , for ≤ s < t ≤ T. Proof.
Fix s ∈ [0 , T ] and define q := e ϕ ( e γ ( s )) − · e γ ( s ) − . By exploiting item (d) of Propos-ition 2.10, item (c) of Proposition 2.13, and the fact that e ϕ is intrinsically L -Lipschitz, weget that e ϕ q ( e ) = e and k e ϕ q ( w ) k G ≤ L k w k G for every w ∈ e U q . We now apply Lemma 3.13 toget that the curve e γ q defined by e γ q ( t ) := e ϕ ( e γ ( s )) − · e γ − ( s ) · e γ ( t ) · e ϕ ( e γ ( s )) is an integral curveof the vector field D ϕ q W . Notice that this curve takes values in e U q as noticed in Lemma 3.13.We stress that Lemma 3.13 is stated only for L horizontal, but it also holds in case W isnormal, see Remark 3.14.Define e γ + q ( · ) := e γ q ( · + s ) . Since e γ + q (0) = e , we are in a position to apply Lemma 3.15 tothe function e ϕ q and to the curve e γ + q . Evaluating the first inequality of (50) at time t − s we get (54). Finally, by (45) - that holds in the general case in which W is normal - and byevaluating the second inequality of (50) at time t − s , we get (55). (cid:3) Remark . A simple modification of theproof of Proposition 3.17 provides a general argument for the second part of [ALD19, Pro-position 6.6], that was proved only in the setting of Carnot groups of step 2 and in case L is 1-dimensional. The generalization reads as follows. Fix two complementary subgroups W and L of a Carnot group G , with W normal, and an intrinsically Lipschitz function e ϕ : e U ⊆ W → L , where e U is open. Consider a solution e γ : I → e U of(56) e γ ′ ( t ) = m X j = k +1 a j ( t )( D ϕX j ) | e γ ( t ) , for some controls a j ( t ) of class L ∞ ( I ) , where { X k +1 , . . . , X m } is a basis of Lie( W ) ∩ V . Thenthe curve e ϕ ◦ e γ is Lipschitz.To prove this last statement one first proves the analogous of Lemma 3.15 for curvessatisfying (56). The estimates are done in the same way but the constant C ′ also dependson a uniform bound of the controls a j ( · ) in L ∞ ( I ) . In order to conclude, one can run thesame argument of Proposition 3.17, taking into account that the invariance property shownin Lemma 3.13 also holds for curves satisfying (56) with exactly the same proof. For thesketch of the proof of Lemma 3.13 in the general case when W is normal we refer the readerto Remark 3.14.The forthcoming Proposition 3.19 is, to our knowledge, new. It tells us what are themetric properties of the integral curves e γ of D ϕ whenever e ϕ ∈ ID( e U , W ; L ) . The counterpartof Proposition 3.19 in the setting of the Heisenberg group H n is already known: for the case n which L is one-dimensional, the proof follows from the argument of [SC16, Theorem 4.95, (3) ⇒ (2) ], while for the case in which L is k -dimensional, the proof is in [Cor19, Propos-ition 4.6]. A weaker version of this proposition, which also requires e ϕ ◦ e γ to be C , hasappeared in [ASCV06, Proposition 3.7] in the Heisenberg group, for L one-dimensional, andin [DD18, Proposition 5.6] in Carnot groups of step 2, for L one-dimensional. Proposition 3.19.
Let W and L be two complementary subgroups in a Carnot group G ,with L horizontal and k-dimensional, let e U be an open set in W and let e ϕ : e U → L be anintrinsically differentiable function at w ∈ e U . Then the following facts hold. (i) For every W ∈ Lie( W ) ∩ V and every integral curve e γ : [0 , T ] → e U of D ϕW startingfrom w , the composition e ϕ ◦ e γ is differentiable at 0 and (57) dd t | t =0 ( e ϕ ◦ e γ )( t ) = d ϕ ϕ ( w )(exp W ) . (ii) For every d > , W ∈ Lie( W ) ∩ V d and every integral curve e γ : [0 , T ] → e U of D ϕW starting from w , the following holds: (58) lim t → k e ϕ ( w ) − · e ϕ ( e γ ( t )) k G t /d = 0 . Proof.
Notice that, since L is horizontal, the homogeneous norm k·k G restricted to L isequivalent to the Euclidean norm of the exponential coordinates. First of all, by item (i) ofRemark 2.24, e ϕ is continuous at w . If we define q := e ϕ ( w ) − · w − we get, from item (d) ofProposition 2.10, that e ϕ q ( e ) = e and, from item (ii) of Remark 2.24, that e ϕ q is intrinsicallydifferentiable at e with d ϕ q ϕ q ( e ) = d ϕ ϕ ( w ) . By (21), (20) and the triangle inequality, forevery e V ⋐ e U q containing e , there exists a constant C such that | ϕ q ( w ) |≤ C k w k G for every w ∈ e V .(i) Fix W ∈ Lie( W ) ∩ V and e γ as in the assumption. By the first part of Lemma 3.13,there exists e γ q : [0 , T ] → e U q ⊆ W such that e γ q is an integral curve of D ϕ q W starting from e .Then, since | ϕ q ( w ) |≤ C k w k G for every w ∈ e V , and since for sufficiently small times t > itholds e γ q ([0 , t ]) ⊆ e V , we are in the setting of Lemma 3.15. Thus, from the first inequality in(50), we can write lim sup t → k e γ q ( t ) k G t < + ∞ . Fix an adapted basis ( X , . . . , X n ) of the Lie algebra g such that L = exp(span { X , . . . , X k } ) and W = exp(span { X k +1 , . . . , X n } ) . We use the notation of Definition 2.3 and Defini-tion 2.20. By using the previous inequality we get that there is a constant ¯ C > such that,for every small enough t ∈ [0 , T ] ,(59) | ϕ q ( γ q ( t )) − ϕ q ( γ q (0)) − t ∇ ϕ q ϕ q ( e )( W ) | t ≤ ¯ C | ϕ q ( γ q ( t )) − ϕ q ( γ q (0)) − t ∇ ϕ q ϕ q ( e )( W ) |k e γ q ( t ) k G . Notice that ϕ q ( γ q (0)) = ϕ q (0) = 0 . Moreover, using the particular form of the projectedvector fields in (34) and the fact that W ∈ V , it is easy to see that π V ( e γ q ( t )) = exp( tW ) forall t ∈ [0 , T ] . By exploiting the fact that the intrinsic differential is linear on the horizontalcomponents (see Proposition 2.19), we get that for all t ∈ [0 , T ] t ∇ ϕ q ϕ q ( e )( W ) = ∇ ϕ q ϕ q ( e )( tW ) = exp − (d ϕ q ϕ q ( e )(exp( tW ))) . et us conclude the proof. The intrinsic differentiability of e ϕ q at e provides (21). Thus, byexploiting π V ( e γ q ( t )) = exp( tW ) for all t ∈ [0 , T ] , the previous equality, and the fact that theintrinsic differential depends only on the projection on V (see Proposition 2.19), the righthand side of (59) goes to zero as t → . Thus, also the left hand side goes to zero as t → and this means that dd t | t =0 ( ϕ q ◦ γ q )( t ) = ∇ ϕ q ϕ q ( e )( W ) . By using (46), and since ∇ ϕ q ϕ q ( e ) = ∇ ϕ ϕ ( w ) , we get (57). This concludes the proof of (i).(ii) Assume W ∈ Lie( W ) ∩ V d with d > , and e γ as in the assumption. We proceed withthe same argument as in (i). Then, following the lines of the proof in item (i) and by thefirst inequality in (50), we obtain that there exists ¯ C > such that for sufficiently small t ∈ [0 , T ] (60) | ϕ q ( γ q ( t )) − ϕ q ( γ q (0)) | t /d ≤ ¯ C | ϕ q ( γ q ( t )) − ϕ q ( γ q (0)) |k e γ q ( t ) k G . Since W ∈ V d with d > and ϕ q (0) = 0 , the projection of every integral curve of D ϕ q W ,starting from 0, on the horizontal bundle is zero. This follows by exploiting the particularform of D ϕ q in coordinates, see (34). Then the intrinsic differentiability of e ϕ q at e jointlywith (60), the fact that the projection of e γ q on V is zero, and that the intrinsic gradient islinear on V (see Proposition 2.19) yields, with the same reasoning as before, lim t → | ϕ q ( γ q ( t )) − ϕ q ( γ q (0)) | t /d = 0 . Then, by using (46) we conclude (58) and thus the proof. (cid:3)
Remark . For the ease of notation, we considered in Proposition 3.19 only intervals [0 , T ] ,and thus we got conclusions only on the right limits and the right derivatives. The sameproof provides the same conclusion on the full limit, or the full derivative, whenever theinterval is centered at the origin.Now we want to deduce metric properties of e ϕ when we know that it is UID. The followingproposition shows that any uniformly intrinsically differentiable function e ϕ is s -little Höldercontinuous on any Carnot group of step s , when read in exponential coordinates. It is ageneralization of [ASCV06, Proposition 4.4]. Proposition 3.21.
Let W and L be two complementary subgroups of a Carnot group G with L horizontal and k-dimensional, and let e U ⊆ W be an open set. If e ϕ ∈ UID( e U , W ; L ) , thensuch a function read in exponential coordinates is in h /s loc ( U ; R k ) , that is ϕ ∈ C ( U ; R k ) andfor all U ′ ⋐ U one has (61) lim r → (cid:18) sup (cid:26) | ϕ ( b ) − ϕ ( a ) || b − a | /s : a, b ∈ U ′ , < | b − a | < r (cid:27)(cid:19) = 0 . Proof.
We fix an adapted basis ( X , . . . , X n ) such that L = exp(span { X , . . . , X k } ) and W =exp(span { X k +1 , . . . , X n } ) . We use the convention in Definition 2.3 and Proposition 2.22,taking into account the little abuse of notation as in Remark 2.23. In these coordinates, upto bi-Lipschitz equivalence, we can suppose to work with the anisotropic norm. If a ∈ U ,we denote by a , . . . , a s the vector of components of a in each layer, so a j ∈ R m j for every = 1 , . . . , s and a = ( a , . . . , a s ) . For any a ∈ U and r > we set ρ a ( r ) := sup (cid:26) | ϕ ( b ) − ϕ ( a ) − ∇ ϕ ϕ a ( a − b ) |k e ϕ ( a ) − a − b e ϕ ( a ) k : a, b ∈ B ( a , r ) ∩ U, a = b (cid:27) . Assuming e ϕ ∈ UID( e U , W ; L ) , we have by (22)(62) lim r → ρ a ( r ) = 0 , for every a ∈ U . Fix U ′ ⋐ U with a ∈ U ′ . From Proposition 2.19, since the intrinsic-ally linear function d ϕ ϕ a depends only on the variables on the first layer of W , and it ishomogeneous, we can find a constant C > depending on a for which |∇ ϕ ϕ a ( a − b ) |≤ C | b − a | , ∀ a, b ∈ R n , and, consequently, we have(63) |∇ ϕ ϕ a ( a − b ) || b − a | /s ≤ Cr − /s , for all a, b ∈ R n with < | a − b | < r < . By a consequence of Proposition 2.4, see [FS16,Corollary 3.13], we have with a little abuse of notation e ϕ ( a ) − a − b e ϕ ( a ) = b − a + P ( e ϕ ( a ) , a − b ) , for every a, b ∈ R n , where P ( e ϕ ( a ) , a − b ) := ( P ( e ϕ ( a ) , a − b ) , . . . , P s ( e ϕ ( a ) , a − b )) , with P ( e ϕ ( a ) , a − b ) = 0 . Moreover, for each i = 2 , . . . , s , there is C i > depending only U ′ and e ϕ such that |P i ( e ϕ ( a ) , a − b ) |≤ C i (cid:0) | b − a | + · · · + | b i − − a i − | (cid:1) , for all a, b ∈ U ′ . Hence there exists C ′ > depending only on C i and on the group G suchthat k e ϕ ( a ) − a − b e ϕ ( a ) k| b − a | /s ≤ C ′ , ∀ a, b ∈ U ′ with < | a − b | < . Finally, by the last inequality together with (63), we get(64) | ϕ ( b ) − ϕ ( a ) || b − a | /s ≤ | ϕ ( b ) − ϕ ( a ) − ∇ ϕ ϕ a ( a − b ) |k e ϕ ( a ) − a − b e ϕ ( a ) k k e ϕ ( a ) − a − b e ϕ ( a ) k| b − a | /s + |∇ ϕ ϕ a ( a − b ) || b − a | /s ≤ C ′ ρ a ( r ) + Cr − /s , for all a ∈ U and all a, b ∈ U ′ ∩ B ( a , r ) with < | a − b | < r < . We stress that, ultimately, C ′ depends only U ′ and e ϕ , while C depends only on a .We conclude the proof by contradiction. Assume we can find U ′ ⋐ U , two sequences ( a h ) and ( b h ) in U ′ , and an infinitesimal sequence ( r h ) of positive numbers such that < | a h − b h | < r h and | ϕ ( b h ) − ϕ ( a h ) || b h − a h | /s > M, for some M > . Since U ′ is compact, we can assume that, up to passing to subsequences,both ( a h ) and ( b h ) converge to some a ∈ U ′ . By (64) we would find some M ′ > such thatthat ρ a ( r h ) > M ′ , or arbitrarily large h ∈ N , a contradiction to (62). (cid:3) The previous proposition tells us what is the regularity of ϕ in all the exponential coordin-ates, in case it is UID. Actually, we can refine Proposition 3.21 by improving the property(58). We stress that the forthcoming proposition would also follow from the implication(1) ⇒ (2) of [Koz15, Theorem 4.3.1] but, up to our knowledge, the proof presented here isnew. Indeed, in (1) ⇒ (2) of [Koz15, Theorem 4.3.1] it is proved that if the intrinsic graphof ϕ is a co-horizontal C -surface with complemented tangents, then (65) holds. Then thefollowing Proposition 3.22 would be a consequence of that implication and Proposition 2.28.Instead, we here give a direct proof within our context. In conclusion we obtain, in a differentway, the implication (1) ⇒ (2) of [Koz15, Theorem 4.3.1] by making use of Proposition 3.22and Proposition 2.28. Proposition 3.22.
Let W and L be two complementary subgroups of a Carnot group G , with L horizontal and k -dimensional. Let e U ⊆ W be open and e ϕ ∈ UID( e U , W ; L ) . Fix an adaptedbasis ( X , . . . , X n ) in which W = exp(span { X k +1 , . . . , X n } ) , L = exp(span { X , . . . , X k } ) and let e V ⋐ e U . Then (65) lim ̺ → (cid:18) sup (cid:26) | ϕ ( γ ( t )) − ϕ ( γ ( s )) || t − s | / deg j : j = m + 1 , . . . , n, γ ′ = D ϕX j ◦ γ, γ ⊆ V, < | t − s |≤ ̺ (cid:27)(cid:19) = 0 . Proof.
We use the convention in Definition 2.3 and Proposition 2.22, taking into account thelittle abuse of notation as in Remark 2.23. By item (a) of Proposition 2.25 we have that e ϕ is intrinsically Lipschitz on e V ⋐ e U . We denote by C the constant for which e ϕ is intrinsically C -Lipschitz in e V .Fix w ∈ e V . Let us take m + 1 ≤ j ≤ n , and an integral curve e γ : I → e V ⊆ W of D ϕX j .Without loss of generality we may assume that the curve is defined on I = [0 , T ] , with T > possibly depending on the curve. By the particular form of D ϕj in coordinates, see (34), andthe fact that j ≥ m + 1 , we have that the projection of e γ on V is constant. Then, since byProposition 2.19 ∇ ϕ ϕ w depends only on the projection on V and it is linear, we have, forall t, s ∈ [0 , T ] ,(66) | ϕ ( γ ( t )) − ϕ ( γ ( s )) − ∇ ϕ ϕ w ( γ ( s ) − · γ ( t )) |k e ϕ ( e γ ( s )) − · e γ ( s ) − · e γ ( t ) · e ϕ ( e γ ( s )) k = | ϕ ( γ ( t )) − ϕ ( γ ( s )) |k e ϕ ( e γ ( s )) − · e γ ( s ) − · e γ ( t ) · e ϕ ( e γ ( s )) k . Since e ϕ is intrinsically C -Lipschitz in e V , by (54) there exists a constant C j > dependingonly on j, C and the adapted basis such that(67) k e ϕ ( e γ ( s )) − · e γ ( s ) − · e γ ( t ) · e ϕ ( e γ ( s )) k≤ C j | t − s | / deg j , ∀ ≤ s < t ≤ T. In particular, we can find a constant C ′ > depending only on C and the adapted basissuch that for every j = m + 1 , . . . , n , for every integral curve e γ : [0 , T ] → e V of D ϕX j and every ≤ s < t ≤ T we have(68) | ϕ ( γ ( t )) − ϕ ( γ ( s )) |k e ϕ ( e γ ( s )) − · e γ ( s ) − · e γ ( t ) · ϕ ( e γ ( s )) k ≥ | ϕ ( γ ( t )) − ϕ ( γ ( s )) | C ′ | t − s | / deg j . Combining (68) and (66) we get(69) | ϕ ( γ ( t )) − ϕ ( γ ( s )) || t − s | / deg j ≤ C ′ | ϕ ( γ ( t )) − ϕ ( γ ( s )) − ∇ ϕ ϕ w ( γ ( s ) − · γ ( t )) |k e ϕ ( e γ ( s )) − · e γ ( s ) − · e γ ( t ) · e ϕ ( e γ ( s )) k , or every j = m + 1 , . . . , n , every integral curve e γ : [0 , T ] → e V of D ϕX j , every ≤ t < s ≤ T and every w ∈ e V . If | t − s |≤ ̺ , by the estimate in Proposition 2.4 and (67), we get(70) k e γ ( s ) − · e γ ( t ) k≤ C ( ̺ ) , for every j = m + 1 , . . . , n and every integral curve e γ : [0 , T ] → e V of D ϕX j , where C ( ̺ ) is acontinuous increasing function such that lim ̺ → C ( ̺ ) = 0 , depending on e ϕ and independenton the choices of j and e γ .Assume by contradiction that (65) is false. Then there exist ε > , a sequence of integralcurves e γ ℓ : [0 , T ℓ ] → e V of D ϕX iℓ , for some i ℓ ∈ { m + 1 , . . . , n } , and sequences of times ≤ t ℓ
In this subsection we give the notion of broad* solution to thesystem D ϕ ϕ = ω , with ω continuous. Eventually we show that an intrinsically differentiablefunction ϕ with continuous intrinsic gradient ∇ ϕ ϕ (see Definition 2.20) is a broad* solutionto D ϕ ϕ = ∇ ϕ ϕ .Let W and L be complementary subgroups of a Carnot group G , with L horizontaland k -dimensional. Let e U be an open subset of W and let e ϕ : e U ⊆ W → L be a con-tinuous function. We fix an adapted basis ( X , . . . , X n ) of the Lie algebra g , such that W = exp( span { X k +1 , . . . , X n } ) and L = exp( span { X , . . . , X k } ) . We give the notion ofbroad* solution of the system(72) D ϕX k +1 ϕ (1) . . . D ϕX m ϕ (1) ... . . . ... D ϕX k +1 ϕ ( k ) . . . D ϕX m ϕ ( k ) = ω k +1 . . . ω m ... . . . ... ω k k +1 . . . ω k m , here ω := ( ω ℓj ) : U → R k × ( m − k ) , with ℓ ∈ { , . . . , k } , j ∈ { k + 1 , . . . , m } , is a continuousmatrix valued function, and where we refer to the notation introduced in Definition 2.3. Definition 3.24 (Broad* and broad solutions) . Let W and L be complementary subgroupsof a Carnot group G , with L horizontal and k -dimensional. Let e U ⊆ W be open and let e ϕ : e U → L be a continuous function. Consider an adapted basis ( X , . . . , X n ) of the Liealgebra g such that L = exp(span { X , . . . , X k } ) and W = exp(span { X k +1 , . . . , X n } ) . Let ω := ( ω ℓj ) : U → R k × ( m − k ) be a continuous matrix valued function with ℓ ∈ { , . . . , k } and j ∈ { k + 1 , . . . , m } . We say that ϕ =: ( ϕ (1) , . . . , ϕ ( k ) ) ∈ C ( U ; R k ) is a broad* solution of D ϕ ϕ = ω in U if for every a ∈ U there exist < δ < δ and m − k maps E ϕj : B ( a , δ ) × [ − δ , δ ] → B ( a , δ ) for j = k + 1 , . . . , m , where the balls are considered restricted to U , satisfying the following two properties.(a) For every a ∈ B ( a , δ ) and every j = k + 1 , . . . , m , the map E ϕj ( a ) := E ϕj ( a, · ) is C regular and it is a solution of the Cauchy problem ( ˙ γ = D ϕj ◦ γγ (0) = a, in the interval [ − δ , δ ] , where the vector field D ϕj := D ϕX j is defined in (26).(b) For every a ∈ B ( a , δ ) , for every t ∈ [ − δ , δ ] , every j = k + 1 , . . . , m and every ℓ = 1 , . . . , k one has ϕ ( ℓ ) ( E ϕj ( a, t )) − ϕ ( ℓ ) ( a ) = Z t ω ℓj ( E ϕj ( a, s )) d s. We say that D ϕ ϕ = ω in the broad sense on U if for every W ∈ Lie( W ) ∩ V and every γ : I → U integral curve of D ϕW , it holds that dd s | s = t ( ϕ ◦ γ )( s ) = ω ( W )( γ ( t )) , ∀ t ∈ I, where by ω ( W ) we mean the matrix ω applied to the ( m − k ) -vector W . Remark . We stress that, in the setting of Definition 3.24, if e ϕ ∈ C ( e U ) then D ϕ ϕ = ∇ ϕ ϕ both pointwise and in the broad sense on U . First e ϕ ∈ UID( e U , W ; L ) by [DD18,Theorem 4.9], because e ϕ ∈ C ( e U ) . Then we can consider the intrinsic gradient ∇ ϕ ϕ as inDefinition 2.20, which is continuous, see Proposition 2.25. Thus the claim is an outcomeof point (i) of Proposition 3.19 and point (c) of Proposition 4.10, that becomes a pointwiseequality if e ϕ is C ( e U ) , see the proof of Proposition 4.10. Remark . Let us notice that the definition given in Definition 3.24 is a priori susceptibleto the choice of an adapted basis. Nevertheless, when it is coupled with the vertically broad*hölder condition in the same basis, see Definition 4.3, it is independent of this choice. Thisis an outcome of Theorem 4.17. Indeed, from (d) ⇒ (a) of Theorem 4.17, it follows that thebroad* condition and the vertically broad* hölder condition on a fixed basis imply that e ϕ isUID. Thus, from item (i) of Proposition 3.19, we get that the broad* condition is satisfiedfor every other basis. Finally, from Proposition 3.22, we get that also the vertically broad*hölder condition holds in every other basis.With the above reasoning, we remark that we can conclude something stronger: if thebroad* condition and the vertically broad* hölder condition hold on a fixed basis, then they old uniformly on the choice of W ∈ Lie( W ) with bounded norm, see Definition 1.4, andDefinition 1.5.The following result is already known in the Heisenberg groups H n : in case L is one-dimensional, it is proved in [SC16, (3) ⇒ (2) & Theorem 4.95], while in case L is k -dimensionalit is proved in [Cor19, Theorem 1.4, (iii) ⇒ (ii)]. We here generalize it to arbitrary Carnotgroups, in the case L is horizontal and k -dimensional. Proposition 3.27.
Let W and L be complementary subgroups of a Carnot group G , with L horizontal and k -dimensional, and consider an adapted basis of the Lie algebra g suchthat W = exp(span { X k +1 , . . . , X n } ) and L = exp(span { X , . . . , X k } ) . Let e U be an opensubset of W , and e ϕ ∈ ID( e U , W ; L ) be such that d ϕ ϕ is continuous on e U . Denote by ∇ ϕ ϕ the k × ( m − k ) matrix that represents d ϕ ϕ in coordinates, see Definition 2.20.Then, we have that (73) dd t | t = t ( ϕ ( ℓ ) ◦ γ )( t ) = ∇ ϕℓj ϕ ( γ ( t )) , for every j = k + 1 , . . . , m , every integral curve e γ : I → e U of D ϕj := D ϕX j , every ℓ = 1 , . . . , k ,and every t ∈ I . In particular the function ϕ is a broad solution, and thus also a broad*solution, of the system D ϕ ϕ = ∇ ϕ ϕ .Proof. Equation (73) directly follows from (57) seen in coordinates. Then, from (73) and thefact that ∇ ϕ ϕ is continuous by hypothesis, the second part of the thesis follows. (cid:3) Main Theorems in arbitrary Carnot groups
In this section we prove Theorem 4.17, that is Theorem 1.6 in the introduction. We dealwith an arbitrary Carnot group G along with a continuous function e ϕ : e U ⊆ W → L , where W and L are complementary subgroups of G , with L horizontal and k -dimensional, and e U is an open subset of W .In Section 4.1, we study how Hölder properties of e ϕ along integral curves of the vectorfields D ϕ as defined in (26) affect the intrinsic regularity of the function e ϕ . The main resultof this section is a converse of Proposition 3.22: if D ϕ ϕ = ω holds in the broad* sense (seeDefinition 3.24) and there is, locally around every point, a family of curves satisfying thelittle Hölder regularity condition (65) (we shall call this property vertically broad* hölderregularity, see Definition 4.3), then e ϕ is uniformly intrinsically differentiable. For the fullstatement, see Proposition 4.5. We notice that, taking Remark 3.23 into account, the latterproposition generalizes [ASCV06, Theorem 5.7], which deals with the case G = H n and L one-dimensional, [DD18, Theorem 5.8, (4) ⇒ (2)], which is proved in case G has step 2 and L is one-dimensional, and [Cor19, Theorem 5.3] that solves the problem for G = H n with L horizontal and k -dimensional. We remark that, also in these cases, we obtain slightly strongerresults, requiring just a locally 1/2-little Hölder regularity in the vertical components.Proposition 4.5 could also be obtained by a combination of (2) ⇒ (1) of [Koz15, The-orem 4.3.1] and Proposition 2.28, which is proved in [DD18]. The idea of (2) ⇒ (1) in [Koz15,Theorem 4.3.1] is to show that the Hölder conditions on e ϕ along a family of integral curves,that is the assumptions of Proposition 4.5, imply that the intrinsic graph of e ϕ is a co-horizontal C -surface with complemented tangents. To prove this latter fact the authoruses a characterization of co-horizontal C -surfaces by means of uniform Hausdorff conver-gence to tangents - see [Koz15, Theorem 3.1.12] - that is in turn based on the so-called four ones Theorem, see [BK14, Theorem 1.2]. With the independent proof we give in Propos-ition 4.5, with more analytic flavor, we stress we can indirectly obtain (2) ⇒ (1) of [Koz15,Theorem 4.3.1] by making use of Proposition 4.5 and Proposition 2.28. We also obtained(1) ⇒ (2) of [Koz15, Theorem 4.3.1], see the discussion before Proposition 3.22.Our proof of Proposition 4.5 requires Proposition 4.1, that is stated only for L horizontaland k -dimensional. The Proposition 4.1 is a converse of Lemma 3.15, i.e., it can be roughlyread in the following way: the uniform Hölder regularity of the curves e ϕ ◦ e γ , where e γ is anintegral curve of the vector field D ϕ , implies the intrinsically Lipschitz regularity of e ϕ . Wegive a proof of Proposition 4.1 as we crucially need it for the proof of Proposition 4.5, but weremark that a more general statement can be given, in case W is normal, with a proof that isvery similar to the one of Proposition 4.1. Notice that the general statement with W normalcan be found in (3) ⇒ (1) of [Koz15, Theorem 4.2.16]. For more details, see Remark 4.2.As a by-product, we obtain some analytical results, that can have their own independentinterests. The first one is given by Corollary 4.9 and it states that broad* regularity impliesbroad regularity. Roughly speaking, having a function that is Hölder regular on a precisefamily of integral curves implies the Hölder regularity on every integral curve. Then,in Corollary 4.7, we prove that every intrinsically differentiable function that is verticallybroad* hölder (see Definition 4.3), and that has a continuous intrinsic gradient, is uniformlyintrinsically differentiable. We do not know at present whether the assumption on thevertically broad* hölder regularity can be dropped in Corollary 4.7, see also Remark 4.8. Weexpect that the hypothesis on the vertically broad* hölder regularity in Corollary 4.7 canbe dropped in general, see also the paragraph Geometric characterizations of intrinsicdifferentiability in the introduction. From the results proved in [BSC10b] and [Cor19],we know that the assumption on the vertically broad* hölder regularity in Corollary 4.7 isnot necessary in the case of the Heisenberg groups H n , with L horizontal k -dimensional, seealso the introduction to Section 5. We stress we obtain that we can remove the assumptionon the vertically broad* hölder regularity in Corollary 4.7 also in the case of step-2 Carnotgroups with L one-dimensional, see Section 6.In Section 4.2 we focus on the case in which L is one-dimensional and we prove Propos-ition 4.10. We present an area formula that represents the perimeter of the subgraph of auniformly intrinsically differentiable function ϕ in terms of the density p |∇ ϕ ϕ | . Formore details about the area formula and a representation involving the Hausdorff measures,we refer the reader to Remark 4.11. In Proposition 4.10 we also prove that, whenever thetarget L is one-dimensional, every uniformly intrinsically differentiable function ϕ is a dis-tributional solution of the system D ϕ ϕ = ∇ ϕ ϕ and, in Corollary 4.12, we deduce that if D ϕ ϕ = ω holds in the broad* sense with a continuous datum ω and ϕ is vertically broad*hölder, then D ϕ ϕ = ω in the sense of distributions. We do not know, in general, if in Corol-lary 4.12 we can remove the assumption on the vertically broad* hölder regularity. In fact,one can remove the hypothesis on the vertically broad* hölder regularity in Corollary 4.12 inHeisenberg groups, and it is a consequence of the results in [BSC10b]. We stress that thanksto the results obtained in Section 6, we drop the assumption on the vertically broad* hölderregularity in Corollary 4.12 also in the case of step-2 Carnot groups.It is interesting to investigate the converse implication: if one has D ϕ ϕ = ω in the senseof distributions with a continuous function ω , is it true that D ϕ ϕ = ω in the broad* sense?This is actually the case in the Heisenberg groups, see [BSC10a], and the techniques usedin Section 6 seem a good tool to address this implication in arbitrary step-2 Carnot groups. e will not address this issue in this paper and it will be the target of further investigations.It is however interesting to notice that, in some examples besides the step-2 case and forparticular ϕ , one can obtain that if D ϕ ϕ = ω holds in the sense of distributions with acontinuous function ω , then D ϕ ϕ = ω in the broad* sense. We will not discuss this issue inthe paper, but we refer the reader to [ABC16].In Section 4.3 we come back to the general case in which the target L is horizontal andnot necessarily one-dimensional. We prove that if ϕ is locally approximable with a sequenceof smooth functions whose intrinsic derivatives converge to a continuous function ω , then D ϕ ϕ = ω in the broad*, see Proposition 4.15. This notion of local approximability has beenfirst introduced and studied in [ASCV06], see also Remark 4.14. We exploit this result toprove that every uniformly intrinsically differentiable function ϕ always solves D ϕ ϕ = ∇ ϕ ϕ in the broad* sense.In Section 4.4, we combine some of the previous results together to prove our main theoremTheorem 4.17, which is Theorem 1.6 in the introduction. Notice that our result provides thegeneralization to all Carnot groups, and to any possible horizontal and k -dimensional target L , of [DD18, Theorem 5.8]. We stress that Theorem 4.17 will be strengthened in Section 6dropping the hypothesis on the vertical broad* hölder regularity in the setting of Carnotgroups of step 2. We stress that, in general, the assumption on the vertical broad* hölderregularity cannot be dropped in Theorem 4.17, see Remark 4.18 for a counterexample in theeasiest step-3 group, namely the Engel group.4.1. From regularity of ϕ along integral curves of D ϕ to regularity of ϕ . In thissubsection we show how the Hölder regularity of ϕ along integral curves of D ϕ affects theintrinsic regularity of e ϕ . Proposition 4.1.
Let W and L be complementary subgroups of a Carnot group G , with L horizontal and k -dimensional, let e U be open, and let e ϕ : e U ⊆ W → L be a continuous functionwith e ∈ e U and e ϕ ( e ) = e .Let ( X , . . . , X n ) be an adapted basis of the Lie algebra g such that L = exp(span { X , . . . , X k } ) and W = exp(span { X k +1 , . . . , X n } ) . Denote by D ϕj := D ϕX j , for every j = k + 1 , . . . , n . Let L > .Fix v ∈ U and consider a concatenation of curves γ k +1 , . . . , γ n in U connecting to v suchthat γ j : I j → U is an integral curve of D ϕj for j = k + 1 , . . . , n . Assume that the function ϕ ◦ γ j is j -Hölder continuous on I j , for j = k + 1 , . . . , n , with Hölder constant L . Then,there exists C > only depending on L, W , L and the adapted basis ( X , . . . , X n ) such that (74) | ϕ ◦ γ j ( t ) |≤ C k v k G , ∀ t ∈ I j , ∀ j = k + 1 , . . . , n, (75) | γ ( i ) j ( t ) |≤ C k v k deg i G , ∀ t ∈ I j , ∀ i, j = k + 1 , . . . , n, where γ ( i ) j denotes the i -th component of γ j in exponential coordinates.Proof. Up to bi-Lipschitz equivalence, we can prove the result choosing the anisotropic norm k x k G := P nℓ =1 | x ℓ | / deg ℓ . For the sake of readability, we fix k = 1 and assume that all thelayers V i of the algebra g , with i ≥ , are 1-dimensional, so that deg i = i − for every i = 2 , . . . , n . The proof in the general case only requires a typographical effort to deal withmultiple components in each layer and the fact that we have more zeros in the componentsof the first layer. e work in exponential coordinates so that v = (0 , v , . . . , v n ) , and denote the extremalpoints of the concatenation of γ , . . . , γ n by the following chain(76) e = (0 , , . . . , → γ (0 , v , r , , . . . , r n, ) → γ . . .. . . → γ j (0 , v , . . . , v j , r j +1 ,j , . . . , r n,j ) → γ j +1 . . .. . . → γ n v = (0 , v , . . . , v n ) . In the previous chain, we used the triangular form of D ϕj given in (34), so that the flow along D ϕj does not affect the coordinates with index less than j . Notice also that, without lossof generality we can assume I j ⊆ [ −| v j − r j,j − | , | v j − r j,j − | ] , with the convention r , := 0 .Indeed, again by using (34), in order to correct the error r j,j − in the j -th component, wehave to flow for a time v j − r j,j − along D ϕj .We prove (74) and (75) by induction on j . When we say that a constant depends on W and L we mean that also depends on the chosen adapted basis ( X , . . . , X n ) .By assumption, the curve ϕ ◦ γ is L -Lipschitz on I and ϕ ◦ γ (0) = 0 . Consequently, wehave that(77) | ϕ ◦ γ ( t ) | = | ϕ ◦ γ ( t ) − ϕ ◦ γ (0) |≤ L | t |≤ L | v |≤ L k v k G , ∀ t ∈ I . Therefore (74) is proved for j = 2 .Next we shall prove (75) for j = 2 by means of induction on i . For i = 2 , we have γ (2)2 ( t ) = t and then | γ (2)2 ( t ) | = | t |≤ | v |≤ k v k G , ∀ t ∈ I . Assume now that for some i ≥ , there is a constant C ,i > such that(78) | γ ( i )2 ( t ) |≤ C ,i k v k deg i G = C ,i k v k i − G , ∀ t ∈ I , ∀ i = 2 , . . . , i . We want to prove that there exists C ,i +1 > such that | γ ( i +1)2 ( t ) |≤ C ,i +1 k v k deg( i +1) G = C ,i +1 k v k i G , ∀ t ∈ I , where C ,i +1 only depends on i , L, C ,i , W and L . By using the particular triangular formof D ϕ in exponential coordinates, see (34), we get that(79) | γ ( i +1)2 ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z t P i +1 ( ϕ ◦ γ ( s ) , γ (2)2 ( s ) , . . . , γ ( i )2 ( s )) d s (cid:12)(cid:12)(cid:12)(cid:12) , ∀ t ∈ I , for some polynomial P i +1 of homogeneous degree deg( i + 1) − deg 2 = i − . Then, from(77) and (78), we deduce there exists C ,i +1 depending on i , L , C ,i , W and L such that (cid:12)(cid:12)(cid:12) P i +1 ( ϕ ◦ γ ( t ) , γ (2)2 ( t ) , . . . , γ ( i )2 ( t )) (cid:12)(cid:12)(cid:12) ≤ C ,i +1 k v k i − G , ∀ t ∈ I , and thus, by using this inequality in (79), and since | t |≤ | v |≤ k v k G , we get | γ ( i +1)2 ( t ) |≤ C ,i +1 | t |k v k i − G ≤ C ,i +1 k v k i G , ∀ t ∈ I . To conclude the proof of (74) and (75) in the case j = 2 , it is enough to set C :=max i =2 ,...,n C ,i , where C , = max { L, } .Assume now that for some ≤ j ≤ n and some constant C j > one has(80) | ϕ ◦ γ j ( t ) |≤ C j k v k G , ∀ t ∈ I j , ∀ j = 2 , . . . , j , (81) | γ ( i ) j ( t ) |≤ C j k v k deg i G , ∀ t ∈ I j , ∀ i ≥ , ∀ j = 2 , . . . , j . e want to find C j +1 > only depending on L, j , C j , W and L such that(82) | ϕ ◦ γ j +1 ( t ) |≤ C j +1 k v k G , ∀ t ∈ I j +1 , (83) | γ ( i ) j +1 ( t ) |≤ C j +1 k v k deg i G , ∀ t ∈ I j +1 , ∀ i = 2 , . . . , n. To prove (82), we develop the following inequalities:(84) | ϕ ◦ γ j +1 ( t ) | ≤ | ϕ ◦ γ j +1 ( t ) − ϕ ◦ γ j +1 (0) | + | ϕ ◦ γ j +1 (0) |≤ L | t | / deg( j +1) + | ϕ ◦ γ j +1 (0) |≤ L | v j +1 − r j +1 ,j | /j + C j k v k G ≤ L | v j +1 | /j + L | r j +1 ,j | /j + C j k v k G ≤ L k v k G + LC j j k v k G + C j k v k G = e C j +1 k v k G , ∀ t ∈ I j +1 , where in the second inequality we used the fact that ϕ ◦ γ j +1 is Hölder of constant L ; inthe third inequality we used the definition of I j +1 for the first term, while the estimate onthe second term comes from (80) and the fact that γ j +1 (0) is the endpoint of γ j ; the fifthinequality is a consequence of (81), inequality | v j +1 | /j ≤ k v k G , and the fact that r j +1 ,j isthe endpoint of γ ( j +1) j .We are left to prove (83). First we notice that, by (34), for ≤ i ≤ j we have γ ( i ) j +1 ( t ) ≡ v i ≤ k v k deg i G for all t ∈ I j +1 . Using again (34), for i = j + 1 we have that γ ( j +1) j +1 ( t ) = r j +1 ,j + t . Then, since | t |≤ | v j +1 − r j +1 ,j | , for every t ∈ I j +1 , arguing similarly to (84),one can obtain | γ ( j +1) j +1 ( t ) |≤ | r j +1 ,j | + | v j +1 |≤ (2 C j + 1) k v k j G . Setting C j +1 ,j +1 = max { C j + 1 , } , we get | γ ( i ) j +1 ( t ) |≤ C j +1 ,j +1 k v k deg i G , ∀ t ∈ I j +1 , ∀ i = 2 , . . . , j + 1 . Let us now proceed by induction on i and assume there exists i ∈ { j + 1 , . . . , n } and aconstant C j +1 ,i > such that(85) | γ ( i ) j +1 ( t ) |≤ C j +1 ,i k v k deg i G = C j +1 ,i k v k i − G , ∀ t ∈ I j +1 , ∀ i = 2 , . . . , i . We want to find C j +1 ,i +1 > only depending on L, C j , C j +1 ,i , W and L such that | γ ( i +1) j +1 ( t ) |≤ C j +1 ,i +1 k v k deg( i +1) G = C j +1 ,i +1 k v k i G , ∀ t ∈ I j +1 . By using again (34), we get that, for every t ∈ I j +1 , one has(86) | γ ( i +1) j +1 ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12) r i +1 ,j + Z t P j +1 i +1 ( ϕ ◦ γ j +1 ( s ) , γ (2) j +1 ( s ) , . . . , γ ( i ) j +1 ( s )) d s (cid:12)(cid:12)(cid:12)(cid:12) ≤ | r i +1 ,j | + Z t (cid:12)(cid:12)(cid:12) P j +1 i +1 ( ϕ ◦ γ j +1 ( s ) , γ (2) j +1 ( s ) , . . . , γ ( i ) j +1 ( s )) (cid:12)(cid:12)(cid:12) d s, for a polynomial P j +1 i +1 of homogeneous degree deg( i + 1) − deg( j + 1) = i − j . From(84) and (85), we deduce that there exists a constant M > depending only on j , i , L, W and L (it indeed depends on the coefficients of the polynomial, the constant e C j +1 , and theinduction constant C j +1 ,i ) such that (cid:12)(cid:12)(cid:12) P j +1 i +1 ( ϕ ◦ γ j +1 ( t ) , γ (2) j +1 ( t ) , . . . , γ ( i ) j +1 ( t )) (cid:12)(cid:12)(cid:12) ≤ M k v k i − j G , ∀ t ∈ I j +1 , nd thus from (86) we get(87) | γ ( i +1) j +1 ( t ) |≤ | r i +1 ,j | + M | t |k v k i − j G ≤ | r i +1 ,j | + M | v j +1 − r j +1 ,j |k v k i − j G , for every t ∈ I j +1 . Notice that r i +1 ,j is the endpoint of γ ( i +1) j , as well as r j +1 ,j is theendpoint of γ ( j +1) j . Thus, by (81), we get that | r i +1 ,j |≤ C j k v k deg( i +1) G = C j k v k i G , and | v j +1 − r j +1 ,j |≤ (1 + C j ) k v k deg( j +1) G = (1 + C j ) k v k j G . Replacing these last two equalities in (87), we get | γ ( i +1) j +1 ( t ) |≤ C j +1 ,i +1 k v k i G , ∀ t ∈ I j +1 , for the constant C j +1 ,i +1 := C j + M (1+ C j ) , which only depends on j , i , L, W , and L . In-equalities (82) and (83) are completed by choosing C j +1 := max { max i = j +1 ,...,n C j +1 ,i , e C j +1 } .To conclude the proof, it is enough to set C := max j =2 ,...,n C j . (cid:3) Remark . The careful reader could have noticedthat the scheme of the proof Proposition 4.1 above can be adapted to prove the very samestatement, in the more general case in which W is normal. We will not use the conclusionsof this remark in what follows, but we nonetheless give a sketch of the proof of this fact.Indeed, also in the case in which W is normal, taking Remark 3.10 into account, the vectorfields D ϕj , for k + 1 ≤ j ≤ n , in exponential coordinates, have a triangular form analogousto Proposition 3.9. Then, if we adopt the same notation and setting as the statement ofProposition 4.1 and if we assume that the curves e ϕ ◦ e γ j are j -Hölder with respect to thenorm k·k G , the same double-induction argument of the proof of Proposition 4.1 implies that,for any k + 1 ≤ j ≤ n , and any t ∈ I j , one has k e ϕ ◦ e γ j ( t ) k G ≤ C k v k G , and k e γ j ( t ) k G ≤ C k v k G ,instead of (74), and (75),respectively.Thus, by evaluating the general form of (74) at j = n and time t corresponding to theendpoint of I n , we get k e ϕ ( v ) k G ≤ C k v k G , with a constant C only depending on L , W , L and the basis adapted to the splitting. Then, if we assume that the bound L on the Hölderconstant of e ϕ ◦ e γ j is uniform with respect to the choice of the integral curve γ j of D ϕj , with k + 1 ≤ j ≤ n , we get, by exploiting the fact that W is locally D ϕ -connectible according toLemma 3.11, that(88) k e ϕ ( v ) k G ≤ C k v k G , for every v ∈ e U ′ ⋐ e U , where the constant C only depends on L , W , L and the chosen basisadapted to the splitting.Finally, if we do not necessarily assume e ϕ ( e ) = e as in the statement of Proposition 4.1,but we still assume that the / deg j -Hölder constant of e ϕ ◦ e γ j is uniformly bounded withrespect to the choice of the integral curves e γ j , the same translation argument as in thebeginning of the proof of Proposition 3.17, joined with the conclusion (88) and the thirdpoint of Proposition 2.13, implies that e ϕ is intrinsically Lipschitz on e U ′ ⋐ e U . This laststatement is the local converse of Proposition 3.17.We finally remark that the improved result we described here is the implication (3) ⇒ (1)of [Koz15, Theorem 4.2.16]. e now exploit Proposition 4.1 to show a criterion to prove that a function is uniformlyintrinsically differentiable in an arbitrary Carnot group. To do so we introduce the definitionof the vertically broad* hölder property in a fixed adapted basis. Definition 4.3 (Vertically broad* hölder and vertically broad hölder condition) . Let W and L be two complementary subgroups of a Carnot group G with L horizontal and k -dimensional, and fix an adapted basis ( X , . . . , X n ) of the Lie algebra g such that W =exp(span { X k +1 , . . . , X n } ) and L = exp(span { X , . . . , X k } ) . Let us fix e U ⊆ W an opensubset and let us denote D ϕj := D ϕX j as defined in (26). We say that a continuous function e ϕ : e U → L is vertically broad* hölder if for every a in U there exist δ a > and neighborhoods U ′ a ⋐ U a ⋐ U of a such that for every a ∈ U ′ a and every j = m + 1 , . . . , n one can find a C regular solution E ϕj ( a ): [ − δ a , δ a ] → U a of the Cauchy problem ( ˙ γ = D ϕj ◦ γγ (0) = a such that(89) lim ̺ → (cid:18) sup (cid:26) | ϕ ( E ϕj ( a, t )) − ϕ ( E ϕj ( a, s )) || t − s | / deg j : j = m + 1 , . . . , n, a ∈ U ′ a , < | t − s |≤ ̺ (cid:27)(cid:19) = 0 . We moreover say that e ϕ is vertically broad hölder if for every V ⋐ U one has lim ̺ → (cid:18) sup (cid:26) | ϕ ( γ ( t )) − ϕ ( γ ( s )) || t − s | / deg j : j = m + 1 , . . . , n, ˙ γ = D ϕj ◦ γ, γ ⊆ V, < | t − s |≤ ̺ (cid:27)(cid:19) = 0 . Remark . Notice that Definition 4.3 is a priori susceptible to the choice of a basis adaptedto the splitting. However, when it is coupled with the broad* condition in the same basis,see Definition 3.24, it is independent on this choice. See Remark 3.26 for details.
Proposition 4.5.
Let W and L be complementary subgroups of a Carnot group G , with L horizontal and k -dimensional, and fix an adapted basis ( X , . . . , X n ) of the Lie algebra g suchthat W = exp(span { X k +1 , . . . , X n } ) and L = exp(span { X , . . . , X k } ) . Let e U ⊆ W be open,let e ϕ : e U → L be a vertically broad* hölder map and assume ω : U ⊆ R n − k → R k × ( m − k ) is acontinuous function such that D ϕ ϕ = ω in the broad* sense on U . Then e ϕ ∈ UID( e U , W ; L ) .Moreover ∇ ϕ ϕ = ω , where ∇ ϕ ϕ is the intrinsic gradient defined in Definition 2.20.Proof. Up to bi-Lipschitz equivalence, we can prove the result choosing the anisotropic norm k x k G := P nℓ =1 | x ℓ | / deg ℓ . For the sake of readability, we give the proof in the case k = 1 . Theproof for a larger k only requires a typographical effort due to the fact that ϕ has more thanone component.Fix a ∈ U . According to the definition of vertically broad* hölder, we find δ a > ,neighborhoods U ′ a ⋐ U a ⋐ U of a and C maps E ϕj ( a ): [ − δ a , δ a ] → U a satisfying theconditions of Definition 4.3. Define for every ̺ > sufficiently small the quantity(90) f ( ̺ ) := sup (cid:26) | ϕ ( E ϕj ( a, t )) − ϕ ( E ϕj ( a, s )) || t − s | / deg j : j = m + 1 , . . . , n, a ∈ U ′ a , < | t − s |≤ ̺ (cid:27) , which by assumption converges to as ̺ → . Throughout the proof, we will often write E ϕj instead of E ϕj ( a ) , where the dependence on the starting point has to be understood fora suitable a ∈ U ′ a . e claim that ϕ is UID at a and ∇ ϕ ϕ a ( b ) = ω ( a ) · b , for every b ∈ R n − , where b ∈ R m − denotes the projection of b onto the first ( m − components, and where ω ( a ) · b := P mi =2 ω i ( a ) · b i denotes the usual scalar product on R m − .By (22), we just need to prove that(91) lim ̺ → (cid:18) sup (cid:26) | ϕ ( b ) − ϕ ( a ) − ω ( a ) · ( b − a ) |k e ϕ ( a ) − a − b e ϕ ( a ) k : a ∈ B ( a , ̺ ) , k a − b k < ̺ (cid:27)(cid:19) = 0 . Since D ϕ ϕ = ω in the broad* sense, by Definition 3.24, we can find neighborhoods U ′′ ⋐ U ′ ⋐ U ′ a ⋐ U of a and δ > such that, for every j = 2 , . . . , m , one has E ϕj ( U ′′ × [ − δ, δ ]) ⊆ U ′ . We can improve this observation using the triangular form of D ϕj , see (34), and arguingas in Lemma 3.11. Indeed, the sets U ′′ and U ′ can be chosen small enough such that, forevery a, b ∈ U ′′ , there exists a path connecting a to b , entirely contained in U ′ , made first bya concatenation of the maps E ϕ , . . . , E ϕm (defined accordingly to Definition 3.24) and then bya concatenation E ϕm +1 , . . . , E ϕn of integral curves of D ϕm +1 , . . . , D ϕn provided by the verticallybroad* hölder condition.Let use improve this conclusion. We know from Lemma 3.13 that if e γ is an integralcurve of D ϕj , then e γ q := q · e γ · ( q L ) − is an integral curve of D ϕ q j , see (42). Then, by possiblytaking a smaller U ′′ , we can suppose without loss of generality that there exist neighborhoods V ′′ ⋐ V ′ of , such that, for every a ∈ U ′′ and every b ′ ∈ V ′′ , there exists a path connecting to b ′ , entirely contained in V ′ made of q -translations of exponential maps. Indeed, if a, b ∈ U ′′ , it is enough to set q := e ϕ ( a ) − a − and build the concatenation of q -translatedof the maps ( E ϕ ) q , . . . , ( E ϕm ) q , that are integral curves of D ϕ q , . . . , D ϕ q m , respectively, byLemma 3.13, and then chain it with the q -translated curves ( E ϕm +1 ) q , . . . , ( E ϕn ) q , that areintegral curves of D ϕ q m +1 , . . . , D ϕ q n , respectively. By construction, this concatenation connects to b ′ = e ϕ ( a ) − a − b e ϕ ( a ) . It suffices then to take a small enough V ′′ ⊆ [ { e ϕ ( a ) − a − b e ϕ ( a ) : a, b ∈ U ′′ } . Moreover, by (d) of Proposition 2.10, we get ϕ q (0) = 0 and by (c) of Proposition 2.10, onealso has ϕ q ( b ′ ) = ϕ q ( b ′ ) − ϕ q (0) = ϕ ( b ) − ϕ ( a ) . Notice also that ( e ϕ ( a ) − a − b e ϕ ( a )) = b − a ,so that we have(92) | ϕ ( b ) − ϕ ( a ) − ω ( a ) · ( b − a ) |k e ϕ ( a ) − a − b e ϕ ( a ) k = | ϕ q ( b ′ ) − ϕ q (0) − ω ( a ) · ( b ′ ) |k b ′ k , ∀ a, b ∈ U ′′ . For any a, b ∈ U ′′ , we hence consider the concatenation ( E ϕ ) q , . . . , ( E ϕn ) q of integral curvesof D ϕ q , . . . , D ϕ q n entirely lying in V ′ , constructed as above, that connects to b ′ . Similarlyto (76), we denote the concatenation by the following chain(93) ¯ b ′ := (0 , , . . . , → ( E ϕ ) q ¯ b ′ := (0 , b ′ , r , , . . . , r n, ) → ( E ϕ ) q . . .. . . → ( E ϕj ) q ¯ b ′ j := (0 , b ′ , . . . , b ′ j , r j +1 ,j , . . . , r n,j ) → ( E ϕj +1 ) q . . .. . . → ( E ϕn ) q ¯ b ′ n := b ′ = (0 , b ′ , . . . , b ′ n ) , where each ( E ϕj ) q is defined on I j ⊆ [ −| b ′ j − r j,j − | , | b ′ j − r j,j − | ] , with the convention r , := 0 .Since D ϕ ϕ = ω in the broad* sense, by Lemma 3.13 and in particular (47) and (48), we getthat(94) ( ϕ q ◦ ( E ϕj ( a )) q ) ′ ( t ) = ( ω j ) q (( E ϕj ( a )) q ( t )) = ω j ( E ϕj (( a, t ))) , or all j = 2 , . . . , m , all t ∈ I j and all a ∈ U ′′ ; where the first equality follows from (48) andthe second one by the definition of ( ω j ) q , see (46) and (43). For every a, b ∈ U ′′ , we can nowperform the following estimates, which we subsequently explain:(95) | ϕ q ( b ′ ) − ϕ q (0) − ω ( a ) · ( b ′ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X j =2 (cid:0) ϕ q (¯ b ′ j ) − ϕ q (¯ b ′ j − ) − ω j ( a ) b ′ j (cid:1) + n X j = m +1 (cid:0) ϕ q (¯ b ′ j ) − ϕ q (¯ b ′ j − ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X j =2 (cid:0) ω j ( b ∗ j ) − ω j ( a ) (cid:1) b ′ j + n X j = m +1 (cid:0) ϕ q (¯ b ′ j ) − ϕ q (¯ b ′ j − ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup j =2 ,...,m | ω j ( b ∗ j ) − ω j ( a ) |k b ′ k + f (cid:18) sup j = m +1 ,...,n | b ′ j − r j,j − | (cid:19) n X j = m +1 | b ′ j − r j,j − | / deg j . In the second equality we used (94) and, for every j = 2 , . . . , m , the point b ∗ j is on E ϕj ( I j ) and satisfies the conditions of Lagrange’s Theorem. In the third inequality we estimated thefirst term with the supremum norm and the second term by exploiting the definition of f in(90), but replacing in (90) ϕ ◦ E ϕj with ϕ q ◦ ( E ϕj ) q . Indeed, by (46), the map ϕ q ◦ ( E ϕj ) q is aEuclidean translation (in the horizontal coordinates) of ϕ ◦ E ϕj .Notice that, by continuity and with simple estimates coming from Proposition 2.4, bydefinition of b ′ and b ∗ j one has lim ̺ → (cid:0) sup {k b ′ k : a, b ∈ U ′′ , k a − b k≤ ̺ } (cid:1) = 0 , and lim ̺ → (cid:0) sup {k a − b ∗ j k : a, b ∈ U ′′ , k a − b k≤ ̺ } (cid:1) = 0 , ∀ j = 2 , . . . , n, where we implicitly mean that b ∗ j is also a function of the concatenation and the limit isuniform also with respect to that choice. Moreover, the concatenation in (93) uniformlycollapses to as ̺ → , since by continuity one has lim ̺ → (cid:0) sup {| b ′ j − r j,j − | : a, b ∈ U ′′ , k a − b k≤ ̺ } (cid:1) = 0 , ∀ j = 2 , . . . , n, where again, the uniformity has to be understood also in the choice of the concatenation. Wecan then find two continuous functions α , α : (0 , + ∞ ) → (0 , + ∞ ) such that lim ̺ → α ( ̺ ) =lim ̺ → α ( ̺ ) = 0 and, combining (95) and (92), one has(96) | ϕ ( b ) − ϕ ( a ) − ω ( a ) · ( b − a ) |k e ϕ ( a ) − a − b e ϕ ( a ) k ≤ sup j =2 ,...,m k ω j − ω j ( a ) k L ∞ ( B ( a ,α ( ̺ ))) + f ( α ( ̺ )) n X j = m +1 | b ′ j − r j,j − | / deg j k b ′ k , for every sufficiently small ̺ > and every a ∈ B ( a , ̺ ) and b ∈ U ′′ such that k a − b k < ̺ .We claim that we are in a position to apply Proposition 4.1 to the function ϕ q and to thecurves E ϕ q j that connect 0 to b ′ . Indeed, for j = 2 , . . . , m , the uniform bound on the Lipschitzconstant of the map ϕ q ◦ ( E ϕj ) q follows from (94) and the fact that ω is continuous, whilefor j ≥ m + 1 we use the fact that, since f ( ̺ ) → as ̺ → , the maps ϕ q ◦ ( E ϕj ) q are in h / deg j ( I j ) with a uniform bound on the Hölder constant. In particular, by (75), we get that or every ̺ sufficiently small, there exists C ′ > depending on the uniform bound of ω on U ′ such that for every j = 2 , . . . , n | r j,j − | / deg j ≤ C ′ k b ′ k , and thus from (96) we get | ϕ ( b ) − ϕ ( a ) − ω ( a ) · ( b − a ) |k e ϕ ( a ) − a − b e ϕ ( a ) k ≤ sup j =2 ,...,m k ω j − ω j ( a ) k L ∞ ( B ( a ,α ( ̺ ))) + f ( α ( ̺ ))( n − m )(1+ C ′ ) . By the continuity of each ω j , the proof follows by letting ̺ → . (cid:3) For the forthcoming corollaries we recall for the reader’s benefit that we are going to usethe notation in Definition 2.3.
Corollary 4.6.
Let W and L be two complementary subgroups of a Carnot group G , with L horizontal and k -dimensional and let ( X , . . . , X n ) be an adapted basis of the Lie algebra g such that W = exp(span { X k +1 , . . . , X n } ) and L = exp(span { X , . . . , X k } ) . Let e U ⊆ W beopen and let e ϕ : e U → L and ω : U ⊆ R n − k → R k × ( m − k ) be two continuous functions. Assumethat e ϕ is vertically broad* hölder and assume D ϕ ϕ = ω holds in the broad* sense on U .Then the graph of e ϕ is a co-horizontal C -surface with tangents complemented by L .Proof. It is enough to combine Proposition 4.5 and Proposition 2.28. (cid:3)
Corollary 4.7.
Let W and L be two complementary subgroups of a Carnot group G , with L horizontal and k -dimensional and let ( X , . . . , X n ) be an adapted basis of the Lie algebra g such that W = exp(span { X k +1 , . . . , X n } ) and L = exp(span { X , . . . , X k } ) . Let e U ⊆ W beopen and let e ϕ ∈ ID( e U , W ; L ) be a vertically broad* hölder map and assume d ϕ ϕ is continuouson e U . Then e ϕ ∈ UID( e U , W ; L ) .Proof. It is enough to combine Proposition 3.27 and Proposition 4.5. (cid:3)
Remark . We do not know whether, in arbitrary Carnot groups, one can drop the conditionof vertically broad* hölder regularity in Corollary 4.7. This is the case for step-2 Carnotgroups, with L horizontal and one-dimensional, see Corollary 6.13. Corollary 4.9.
Let W and L be two complementary subgroups of a Carnot group G , with L horizontal and k -dimensional and let ( X , . . . , X n ) be an adapted basis of the Lie algebra g such that W = exp(span { X k +1 , . . . , X n } ) and L = exp(span { X , . . . , X k } ) . Let e U ⊆ W beopen and let e ϕ : e U → L and ω : U ⊆ R n − k → R k × ( m − k ) be two continuous functions. Assume e ϕ is vertically broad* hölder and assume D ϕ ϕ = ω holds in the broad* sense on U . Then e ϕ is vertically broad hölder and D ϕ ϕ = ω holds in the broad sense on U .Proof. It is enough to combine Proposition 4.5, Proposition 3.22 and Proposition 3.27. (cid:3)
Area formula for codimension-one graphs in terms of intrinsic derivatives.
We prove here that, if L is horizontal and one-dimensional, and e ϕ ∈ UID( e U , W ; L ) , with e U open, then D ϕ ϕ = ∇ ϕ ϕ holds in the sense of distributions . For a precise definitionof the distribution D ϕ ϕ the interested reader may soon read the first lines of the proof ofProposition 4.10. We also provide an area formula for graph( e ϕ ) in this case. For the case G = H n , this formula was already obtained in [ASCV06, Proposition 2.22 & Remark 2.23].Recall that we are going to use the notation given in Definition 2.3. roposition 4.10. Let W and L be two complementary subgroups of a Carnot group G with L horizontal and one-dimensional, and let ( X , . . . , X n ) be an adapted basis of the Liealgebra g such that L = exp(span { X } ) and W = exp(span { X , . . . , X n } ) . Let e U ⊆ W beopen and consider e ϕ ∈ UID( e U , W ; L ) . Then the following facts hold. (a) For every j = 2 , . . . , m , the distribution D ϕj ϕ := D ϕX j ϕ is well-defined on U . (b) For every a ∈ U , there exist δ > and a family of functions { ϕ ε ∈ C ( B ( a, δ )) : ε ∈ (0 , } such that lim ε → ϕ ε = ϕ and lim ε → D ϕ ε j ϕ ε = ∇ ϕ ϕ ( X j ) in L ∞ ( B ( a, δ )) , for every j = 2 , . . . , m . (c) The equation (97) D ϕj ϕ = ∇ ϕ ϕ ( X j ) =: ∇ ϕj ϕ, holds in the distributional sense on U , for every j = 2 , . . . , m . Here ∇ ϕ ϕ is theintrinsic gradient of ϕ , see Definition 2.20. (d) The subgraph of ϕ defined by E ϕ := { w · exp( tX ) : w ∈ U, t < ϕ ( w ) } has locally finite G -perimeter in U · exp( R X ) and its G -perimeter measure | ∂E ϕ | G is given by (98) | ∂E ϕ | G ( V ) = Z Φ − ( V ) q |∇ ϕ ϕ | m − d L n − , for every Borel set V ⊆ U · exp ( R X ) , where Φ: U → R n is the graph function of e ϕ composed with the exponential coordinates, and with a little abuse of notation wewrote U · exp( R X ) meaning the set e U · exp( R X ) embedded in R n through exponentialcoordinates.Moreover, the set graph ( e ϕ ) has a unit normal given, up to a sign, by (99) ν E ϕ = − p |∇ ϕ ϕ | m − , ∇ ϕ ϕ p |∇ ϕ ϕ | m − , . . . , ∇ ϕm ϕ p |∇ ϕ ϕ | m − ! ∈ R m . Proof.
Notice that the fact that D ϕj ϕ is a well-defined distribution on U is a consequenceof Proposition 3.9 and the fact that L is one-dimensional. Indeed, in coordinates, we getthat the vector field D ϕj is the sum of terms g ( x ) ϕ h ∂ x i , for some polynomial function g of the coordinates of W , some integer h ≥ , and some i = 2 , . . . , n . Thus in order todefine the distribution D ϕj ϕ , we only need to define g ( x ) ϕ h ∂ x i ϕ := g ( x ) h +1 ∂ x i ϕ h +1 . Since g ( x ) h +1 ∂ x i ϕ h +1 is well-defined in the sense of distributions, because ϕ is continuous, (a) isproved.We now show that equality (97) holds pointwise if e ϕ : e U ⊆ W → L is of class C in e U . Incase e ϕ ∈ C ( e U ) , then, by [DD18, Theorem 4.9], one has e ϕ ∈ UID( e U , W ; L ) .We can assume without loss of generality that e ∈ e U and e ϕ ( e ) = e . Indeed, if we wantto prove identity (97) in a ∈ U , we may consider e ϕ p with p := e ϕ ( a ) − · a − , and use theinvariance properties given by Lemma 3.12 and Remark 2.24, and then notice that e ϕ p ( e ) = e .By Proposition 2.28, since e ϕ is UID , the set graph( e ϕ ) is a C -hypersurface and thereforethere exist a neighborhood e V of e in G and a function f ∈ C ( e V ) such that(100) graph( e ϕ ) ∩ e V = { p ∈ G : f ( p ) = 0 } ∩ e V and ∇ G f = 0 on e V . Here we take the usual definition of the horizontal perimeter with respect to the orthonormal basis ( X , . . . , X m ) , see [FSSC03a, Definition 2.18]. hen, for every sufficiently small ε > and for every j = 2 , . . . , n , one has f (cid:16) exp( εX j ) e ϕ (exp( εX j )) (cid:17) = 0 . Therefore, with a little abuse of notation, one can differentiate with respect to ε to get ε | ε =0 f (cid:16) exp( εX j ) e ϕ (exp( εX j )) (cid:17) = dd ε | ε =0 f (cid:16) e ϕ (exp( εX j )) (cid:17) + dd ε | ε =0 f (cid:16) exp( εX j ) (cid:17) = X f | e (cid:18) dd ε | ε =0 ϕ ( εX j ) (cid:19) + X j f | e = X f | e (cid:18) dd ε | ε =0 ( ϕ ◦ π W )( εX j ) (cid:19) + X j f | e = ( X f ) | e D ϕj ϕ | e +( X j f ) | e , (101)where we used the fact that G = W · L and exploited the fact that e ϕ takes values in L = exp(span { X } ) . The last equality follows by using the definition of D ϕj acting on ϕ , see(26). The claim is then obtained by (101) and (25).To prove (b), we use some ideas of [FSSC03b, Theorem 2.1] to show that, for any a ∈ U ,there exist δ > and a family of functions { ϕ ε ∈ C ( U ) : 0 < ε < } , such that ϕ ε → ϕ, and D ϕ ε j ϕ ε → ∇ ϕ ϕ ( X j ) , ∀ j = 2 , . . . , m, uniformly in the Euclidean ball B e ( a, δ ) ⋐ U , as ε → .By Proposition 2.28, we can find a neighborhood e V of a · e ϕ ( a ) and f ∈ C H ( e V ) satisfying (100).Let δ > be such that, setting B := B e ( a, δ ) , one has e B · e ϕ ( e B ) ⊆ e V . Then, up to reducing δ and e V and a regularization argument analogous to [FSSC03b, Step 1 of Theorem 2.1], wecan construct a family { f ε ∈ C ( e V ) : 0 < ε < } such that(102) lim ε → sup e V | X j f ε − X j f | ! = 0 , ∀ j = 1 , . . . , m. Since e ϕ takes values in exp(span { X } ) , by Proposition 2.28, we can assume without loss ofgenerality that X f > on e V , since X f = 0 and we are free to possibly exchange f with − f . By (102), we can find ε > such that X f ε > on e V , for every ε ∈ (0 , ε ) . For anysuch ε > , by the Euclidean implicit function theorem, we can find e ϕ ε , defined on e B , suchthat graph( e ϕ ε ) ∩ e B · e ϕ ε ( e B ) = { p ∈ G : f ε ( p ) = 0 } ∩ e B · e ϕ ε ( e B ) . Moreover, since f ε is smooth, then also e ϕ ε is smooth. In particular, for every b ∈ e B , one has f ε ( b · e ϕ ε ( b )) = 0 , ∀ ε ∈ (0 , ε ) . From [FSSC03b, Step 3 of Theorem 1.2] we deduce that(103) lim ε → (cid:18) sup B | ϕ ε − ϕ | (cid:19) = 0 . enote by Φ ε : B → R n the graph function of e ϕ ε composed with the exponential coordinates.Since e ϕ ε is of class C , by using the pointwise version of (97) for C functions and (25), wededuce that(104) D ϕ ε j ϕ ε ( x ) = − X j f ε X f ε ◦ Φ ε ( x ) , ∀ j = 2 , . . . , m, ∀ x ∈ B. Then, by (104), (102), (25) and (103), we conclude that, for any j = 2 , . . . , m , the family D ϕ ε j ϕ ε converges uniformly on B to − X j fX f ◦ Φ = ∇ ϕ ϕ ( X j ) , as ε → .The proof of (c) follows directly from (b) and the particular form of the distribution D ϕj ϕ we discussed at the beginning of this proof. In fact from the convergence proved in (b),we know that, for any a ∈ U , there exists δ > such that D ϕj ϕ = ∇ ϕ ϕ ( X j ) on B ( a, δ ) inthe sense of distributions, for every j = 2 , . . . , m . It is then enough to consider a locallyfinite countable open sub-covering { B ( a h , δ h ) : h ∈ N } of U and build a partition of unitysubordinate to it. The fact that D ϕj ϕ = ∇ ϕ ϕ ( X j ) holds in the sense of distributions on U isa consequence of the local identity, the linearity of distributions, and a standard argumentusing the partition of unity.To prove (d), we first notice that, by [FSSC03b, Theorem 2.1] and Proposition 2.28, weknow that, for every p ∈ graph( e ϕ ) there exists a neighborhood e V ′ of p and a function f ∈ C H ( e V ′ ) , with X f > on e V ′ , representing graph( e ϕ ) as non critical level set and suchthat | ∂E ϕ | G ( V ) = Z Φ − ( V ) |∇ G f | X f ◦ Φ d L n − holds for every Borel set V ⊆ V ′ . Now by (25), we can write(105) | ∂E | G ( V ) = Z Φ − ( V ) q |∇ ϕ ϕ | m − d L n − , for every Borel set V ⊆ V ′ . Since the right hand side of (105) does not depend on the choice of f , by a covering argumentwe can extend it to every Borel subset V of U · exp( R X ) .The explicit expression of the unit normal in (99) comes from the fact that the graph of e ϕ is locally the zero-level set of f . Thus, the unit normal of graph( e ϕ ) is in the direction of ∇ H f , and taking (25) into account, one has ∇ ϕj ϕ = − X j fX f ◦ Φ for every j = 2 , . . . , m , andthen we conclude by normalization. (cid:3) Remark . For what concerns the area formula in arbitrary Carnot groups, in [Mag17,Theorem 1.2], the author proves that(106) P G ( E ) = β ( d, ν E ) S Q − F E, for any set E of finite perimeter with C -rectifiable reduced boundary F E . The density β is explicitly computed in [Mag17, Theorem 3.2] and depends on the metric d and on thenormal ν E of E that is defined in the sense of Geometric Measure Theory. Moreover, in casethe distance d is vertically symmetric, β is a constant that only depends on the group G and on the metric d , see [Mag17, Theorem 6.3]. We finally notice that every Carnot groupadmits a metric d that is vertically symmetric, see [FSSC03a, Theorem 5.1]. For a surveyon the area formula in Carnot groups, we refer the reader to [SC16, Section 4].Notice that if e ϕ is in UID( e U , W ; L ) , the set graph( e ϕ ) is a C -hypersurface by Propos-ition 2.28. Then, by definition of C -rectifiability and by [FSSC03b, Theorem 2.1], the ubgraph E ϕ of e ϕ has a C -rectifiable reduced boundary F E ϕ = graph( e ϕ ) . Thus we are in aposition to apply Proposition 4.10 and [Mag17, Theorem 1.2], and in particular to compare(98) with (106) in order to obtain the explicit representation(107) Z e V ∩ graph( e ϕ ) β ( d, ν E ϕ ) d S Q − = Z Φ − ( V ) q |∇ ϕ ϕ | m − d L n − , for every Borel set e V ⊆ e U · exp( R X ) .A general area formula for a C -surface Σ , valid in an arbitrary Carnot groups, has beenvery recently obtained in [JNGV20, Theorem 1.1]. If we call α the Hausdorff dimensionof Σ , and we suppose Σ = graph( e ϕ ) , this formula allows for a representation of S α Σ asan integral on e U of a properly defined area element with respect to S α W , see [JNGV20,Lemma 3.2]. According to the previous equation (107), we thus get that for an arbitraryCarnot group G , and in case L is one-dimensional, the area element of [JNGV20, The-orem 1.1] is, up to the function β , explicitly written in terms of the intrinsic gradient of e ϕ . Eventually, by the recent work [CM20], in particular [CM20, Eq. (43) after Theorem 4.2],we get that on H n equipped with a vertically symmetric distance, in case L is horizontal and k -dimensional, one can explicitly write the area element of [JNGV20, Theorem 1.1] in termsof the intrinsic gradient ∇ ϕ ϕ . Corollary 4.12.
Let W and L be two complementary subgroups of a Carnot group G , with L horizontal and one-dimensional, and let ( X , . . . , X n ) be an adapted basis of the Lie algebra g such that W = exp(span { X , . . . , X n } ) and L = exp(span { X } ) . Let e U ⊆ W be open, andlet e ϕ : e U → L and ω : U ⊆ R n − → R m − be two continuous functions. Assume e ϕ is verticallybroad* hölder and assume that D ϕ ϕ = ω in the broad* sense on U . Then D ϕ ϕ = ω in thesense of distributions and ω = ∇ ϕ ϕ .Proof. It is enough to combine Proposition 4.5 and item (c) of Proposition 4.10. (cid:3)
Relations between intrinsic differentiability and local approximability.
Remark . Point (b) of Proposition 4.10 can actually be generalized to the case L hori-zontal and k -dimensional. The computation for the k -dimensional case is similar but oneshould pay attention to the fact that if f =: ( f (1) , . . . , f ( k ) ) ∈ C ( e V ; R k ) is a vector valuedmap, its horizontal gradient is a ( k × m ) -dimensional matrix (see Definition 2.26). Usinga regularization argument that is similar to [FSSC03b, Step 1 of Theorem 2.1] as in theproof of Proposition 4.10, we can find a family of functions { f ε ∈ C ( e V ; R k ) : 0 < ε < } ,such that each component X j f ( i ) ε converges uniformly to X j f ( i ) for every i = 1 , . . . , k and j = 1 , . . . , m as ε → , and such that, for every ε ∈ (0 , , the associated matrix ∇ L f ε defined in Definition 2.26 has det ∇ L f ε = 0 on e V .Then in order to prove point (b) of Proposition 4.10 in this general case, we take theprevious changes into account, the straightforward changes in (101), and we run the samecomputations of the proof of Proposition 4.10 by using (24) instead of (25) at the end of theargument. Remark . The statement (i) ⇒ (ii) of [ASCV06, Theorem 5.1] claims that, in the Heis-enberg groups H n , if L is one-dimensional and ϕ ∈ UID( U, W ; L ) , then there exists ω ∈ C ( U ; R n − ) such that ∇ ϕ ϕ = ω in the sense of distributions and there exists a family ϕ ε ∈ C ( U ) : ε ∈ (0 , } such that ϕ ε → ϕ and ∇ ϕ ε ϕ ε → ω uniformly on any compactsubset K ⊆ U .However, the implication (5.17) ⇒ (5.19) in its proof is imprecise. In sight of this, onecould replace point (ii) with a local version of it in which the approximating functions { ϕ ε } depend on the point a and the convergence is uniform in a neighborhood B ( a, δ ) , see[ASCV06, Proposition 4.6]. This does not affect the validity of the proofs, that only referto [ASCV06, Lemma 5.6], which holds true with the weakened approximation assumptions,since it has a local statement. Indeed, the proof of (ii) ⇒ (i) of [ASCV06, Theorem 5.1] justneeds the local approximation. To the best of our knowledge, all the references to [ASCV06,Theorem 5.1] just require the local approximation: see point (3) [DD18, Theorem 5.8],point (3) of [DD19, Theorem 8.2], [Cor19, Theorem 1.3], [Cor19, Proposition 4.4], [BSC10b,Theorem 2.7], [BSC10a, Theorem 1.1 & Corollary 1.4], [BCSC14, Theorem 2.7].Anyway, the original formulation of (i) ⇒ (ii) of [ASCV06, Theorem 5.1] can be fixed withthe approximation argument exploited in the proof of [MV12, Theorem 1.2] and we expectthis to hold true in all Carnot groups of step 2. We warmly thank Francesco Serra Cassanoand Davide Vittone for precious suggestions. Proposition 4.15.
Let W and L be two complementary subgroups in a Carnot group G , with L k -dimensional and horizontal. Let ( X , . . . , X n ) be an adapted basis of the Lie algebra g such that L = exp(span { X , . . . , X k } ) and W = exp(span { X k +1 , . . . , X n } ) . Let e U ⊆ W bean open set and let e ϕ : e U → L be a continuous function.Assume there exists ω ∈ C ( U ; R k × ( m − k ) ) such that, for every a ∈ U , there exist r > , anda family of functions { ϕ ε ∈ C ( B ( a, r ); R k ) : 0 < ε < } satisfying lim ε → ϕ ε = ϕ in L ∞ ( B ( a, r )) and lim ε → D ϕ ε ϕ ε = ω in L ∞ ( B ( a, r ); R k × ( m − k ) ) . Then ϕ is a broad* solution in U of the system D ϕ ϕ = ω .Proof. The proof closely follows the argument used in [ASCV06, Lemma 5.6]. For simplicity,we consider the case k = 1 . The case k ≥ can be reached with the same proof and sometypographical effort.Let a ∈ U and fix j ∈ { , . . . , m } and ε > . Since ϕ ε ∈ C ( B ( a, r )) , we can find <δ ( ε ) < δ < r and a map E ϕ ε j : B ( a, δ ( ε )) × [ − δ ( ε ) , δ ( ε )] → B ( a, δ ) such that E ϕ ε j ( b, · ) isthe unique solution of the Cauchy problem (cid:26) γ ′ = D ϕ ε j ◦ γ,γ (0) = b, in the interval [ − δ ( ε ) , δ ( ε )] , for every b ∈ B ( a, δ ( ε )) . By Peano’s estimate on the exist-ence time for solutions of ordinary differential equations (see e.g. [Mus05, Theorem 1]) wecan choose δ ( ε ) = C/ k P j ( x, ϕ ε ) k L ∞ ( B ( a,δ )) , with C only depending on δ and with P j apolynomial function of the coordinates x of W and the components of ϕ ε . This polynomialfunction depends only on the structure of D ϕj , see (34). In particular, since ϕ ε is converginguniformly on B ( a, δ ) , then δ ( ε ) has a positive lower bound M independent on ε and weare going to verify the conditions of Definition 3.24 with δ ≤ M .Since ϕ ε are bounded on B ( a, δ ) uniformly in ε > and D ϕ ε j are vector fields with poly-nomial coefficients in ϕ ε and, possibly, in some of the coordinates, the functions E ϕ ε j areequi-Lipschitz with respect to ε on the compact set B ( a, δ ) × [ − δ , δ ] . By Arzelá-Ascoli heorem, we can therefore find an infinitesimal sequence ( ε h ) in (0 , such that E ϕ εh j con-verges to some continuous function E ϕj uniformly on B ( a, δ ) × [ − δ , δ ] . By definition of E ϕ εh j , one has E ϕ εh j ( b, t ) = b + Z t D ϕ εh j (cid:0) E ϕ εh j ( b, s ) (cid:1) d s and ϕ ε h (cid:0) E ϕ εh j ( b, t ) (cid:1) − ϕ ε h (cid:0) E ϕ εh j ( b, (cid:1) = Z t D ϕ εh j ϕ ε h (cid:0) E ϕ εh j ( b, s ) (cid:1) d s, for every b ∈ B ( a, δ ) and every t ∈ [ − δ , δ ] .By letting h → ∞ and using that all the involved convergences are uniform, we get E ϕj ( b, t ) = b + Z t D ϕj (cid:0) E ϕj ( b, s ) (cid:1) d s, and ϕ ( E ϕj ( b, t )) − ϕ ( E ϕj ( b, Z t ω j ( E ϕj ( b, s )) d s, for every b ∈ B ( a, δ ) and every t ∈ [ − δ , δ ] , which are the conditions we were looking forto make D ϕ ϕ = ω hold in the broad* sense. (cid:3) Corollary 4.16.
Let W and L be two complementary subgroups in a Carnot group G , with L horizontal and k-dimensional. Let ( X , . . . , X n ) be an adapted basis of the Lie algebra g such that L = exp(span { X , . . . , X k } ) and W = exp(span { X k +1 , . . . , X n } ) . Let e U ⊆ W beopen and let e ϕ ∈ UID( e U , W ; L ) . Then, there exists ω ∈ C ( U ; R k × ( m − k ) ) such that ϕ is abroad* solution in U of the system D ϕ ϕ = ω . Moreover, ω = ∇ ϕ ϕ .Proof. It is enough to choose ω = ∇ ϕ ϕ , which is continuous taking Proposition 2.25 intoaccount, since e ϕ ∈ UID( e U , W ; L ) . The proof follows by combining item (b) of Propos-ition 4.10, which also holds in case L is k -dimensional, see Remark 4.13, together withProposition 4.15. (cid:3) Main theorem.
Now we are in a position to give the following theorem, which is ageneralization of [DD18, Theorem 5.8] to all Carnot groups.
Theorem 4.17.
Let W and L be two complementary subgroups in a Carnot group G ,with L horizontal and k-dimensional. Let e U ⊆ W be open and let e ϕ : e U ⊆ W → L be a continuous function. Fix a graded basis ( X , . . . , X n ) of the Lie algebra g such that L = exp(span { X , . . . , X k } ) and W = exp(span { X k +1 , . . . , X n } ) . Then, the following factsare equivalent. (a) e ϕ ∈ UID( e U , W ; L ) ; (b) ϕ is vertically broad* hölder and there exists ω ∈ C ( U ; R k × ( m − k ) ) such that, for every a ∈ U , there exist δ > and a family of functions { ϕ ε ∈ C ( B ( a, δ ); R k ) : ε ∈ (0 , } such that lim ε → ϕ ε = ϕ and lim ε → D ϕ ε j ϕ ε = ω j in L ∞ ( B ( a, δ ); R k ) , for every j = k + 1 , . . . , m ; (c) ϕ is vertically broad hölder and there exists ω ∈ C ( U ; R k × ( m − k ) ) such that D ϕ ϕ = ω in the broad sense. d) ϕ is vertically broad* hölder and there exists ω ∈ C ( U ; R k × ( m − k ) ) such that D ϕ ϕ = ω in the broad* sense.Moreover, if any of the previous holds, then ω = ∇ ϕ ϕ .Proof. (a) ⇒ (b) follows by combining Proposition 3.22, item (b) of Proposition 4.10 in the k -dimensional case following the lines of Remark 4.13. (b) ⇒ (d) follows from Proposition 4.15.(d) ⇒ (a) follows from Proposition 4.5. (a) ⇒ (c) follows from Proposition 3.22, Proposi-tion 3.27 and the continuity of ω follows from Proposition 2.25. The implication (c) ⇒ (d) istrivial. (cid:3) Remark . Notice that in [Koz15, Example 4.5.1], in the setting of Example 3.8, theauthor constructs a function e ϕ : e U ⊆ W → L such that D ϕ ϕ = − in the broad* sense but e ϕ is not vertically broad* hölder, see Definition 4.3. Taking Theorem 4.17 into account, thismeans that e ϕ cannot be UID. This also means that, in general, in Theorem 4.17, one cannotdrop the assumption on the vertically broad* hölder regularity in arbitrary Carnot groups.We will show that this is possible for step 2 Carnot groups, in Section 6. For more examplesrelated to this topic, we refer the reader to [Koz15, Section 4.5].5. Some applications
Let us begin with an observation that will motivate the first part of this section. Considerthe first Heisenberg group H with Lie algebra g := span { X, Y } ⊕ span { Z } , and the onlynontrivial relation [ X, Y ] = Z . By a direct application of the Baker-Campbell-Hausdorffformula one gets exp X exp Y exp( − X ) exp( − Y ) = exp Z . By exploiting this formula, inProposition 5.1 below, we give an alternative proof, in the case of H n with n ≥ , of[BSC10b, Theorem 3.2]. The argument we use is different because we prove that being abroad* solution with a continuous datum implies being / -little Hölder continuous alongvertical coordinates (see also Remark 3.23) that is actually simpler than proving 1/2-littleHölder continuity in all the coordinates as in [BSC10b, Theorem 3.2]. Nevertheless, thisis sufficient for applying Proposition 4.5. Similarly, in Proposition 5.2, we obtain the sameresult of [BSC10b, Theorem 1.2], by making use of Proposition 5.1 and Proposition 4.5. Weremark that our argument is different by the one used in [BSC10b] but, on the contrary, itdoes not work for n = 1 .Recently, in [Cor19, Proposition 4.10], the author has proved a generalization of [BSC10b,Theorem 3.2] that holds for arbitrary complemented subgroups in H n , and this statement isone of the key step in order to get the main theorem [Cor19, Theorem 1.4]. By using our reas-oning, we are also able to recover [Cor19, Proposition 4.10], and thus [Cor19, Theorem 1.4],in the case H n = W · L with L horizontal and k -dimensional, with k < n . However, ourargument does not apply to the remaining case k = n , while the argument in the referencedoes.Even if these results already appeared in the literature, we think it is worth writingthem down with these different proofs. Indeed, the proof of Proposition 5.1 provides auseful “toolkit” for the forthcoming theorems: in fact, a similar idea to the one exploited inProposition 5.1 will be used in Theorem 6.6 to prove the analogous version of Proposition 5.1in the setting of free Carnot groups of step 2, when L is one-dimensional. This will be thekey step to obtain the analogous Proposition 5.2 for Carnot groups of step 2 with L one-dimensional. We refer the reader to the introduction of Section 6. n Example 5.3 below we give a class of nontrivial examples of UID functions in the Engelgroup. We build them by making use of Theorem 4.17. This class of examples is inspired by[BV10, Eq. (3.1)]. Moreover, a slight modification of [BV10, Eq. (3.1)], gives rise to functionswhose intrinsic graphs are both of class C and of class C , but they possess a characteristicpoint, see Remark 5.4. We thank R. Serapioni for having discussed this example with us.5.1. A different proof of the propagation of broad* regularity in H n , with n ≥ .Proposition 5.1. Let W and L be two complementary subgroups of H n , with n ≥ see Example . such that L is horizontal and k -dimensional with k < n , and let e ϕ : e U ⊆ W → L be a continuous function, with e U open.Then there exists ( X , . . . , X n +1 ) an adapted basis of the Lie algebra such that L =exp(span { X , . . . , X k } ) , W = exp(span { X k +1 , . . . , X n +1 } ) , and such that the only nonvan-ishing bracket relations are given by [ X i , X n + i ] = X n +1 for every ≤ i ≤ n .Moreover, if there exists a continuous function ω : U ⊆ R n +1 − k → R k × (2 n − k ) such that D ϕ ϕ = ω in the broad* sense, then e ϕ is vertically broad* holder. Namely, since we are in H n (see Remark 3.23), for every a ∈ U there exists a neighborhood U a ⋐ U of a such that (108) lim ̺ → (cid:18) sup (cid:26) | ϕ ( ξ, z ) − ϕ ( ξ, z ) || z − z | / : ( ξ, z ) ∈ U a , ( ξ, z ) ∈ U a , < | z − z | < ̺ (cid:27)(cid:19) = 0 , where we mean that ξ ∈ U a ∩ { z = 0 } ⊆ R n − k and z , z ∈ R are the (2 n + 1) th coordinatesof ( ξ, z ) and ( ξ, z ) when seen as elements of H n .Proof. The existence of the basis as in the first part of the statement comes from [FSSC07,Lemma 3.26]. Fix a ∈ U and find δ > and sufficiently small neighborhoods V ′ a ⋐ V a ⋐ U of a such that, for every a ∈ V ′ a , and every j = k + 1 , . . . , n , we have the existence ofintegral curves E ϕj ( a ): [ − δ, δ ] → V a satisfying the conditions of Definition 3.24. Denote by β : [0 , + ∞ ) → [0 , + ∞ ) a modulus of uniform continuity for ω on V a .We fix a neighborhood U a ⋐ V ′ a of a , ̺ > and points ( ξ, z ) , ( ξ, z ) ∈ U a such that | z − z | < ̺ . The set U a has to be chosen small enough: this will be clear during the proof.We are going to prove that, for every sufficiently small ̺ , we have(109) | ϕ ( ξ, z ) − ϕ ( ξ, z ) || z − z | / ≤ kβ ( α ( ̺ )) , for a continuous function α with α (0) = 0 that only depends on the norms of ϕ and ω on V a . From this fact (108) would follow, concluding the proof.Recall that, by (29) and since k < n , one has D ϕk +1 = X k +1 and D ϕn + k +1 = X n + k +1 .Assume without loss of generality that z > z , and set t := ( z − z ) / . We exploit therelation [ D ϕk +1 , D ϕn + k +1 ] = X n +1 and the Baker-Campbell-Hausdorff formula to conclude thatwe can join ( ξ, z ) and ( ξ, z ) by means of a concatenation of integral curves of D ϕk +1 = X k +1 and D ϕn + k +1 = X n + k +1 . In particular a := ( ξ, z ) → E ϕk +1 ( a ) a := E ϕk +1 ( a, t ) → E ϕn + k +1 ( a ) a := E ϕn + k +1 ( a , t ) → E ϕk +1 ( a ) a := E ϕk +1 ( a , − t ) → E ϕn + k +1 ( a ) a := ( ξ, z ) = E ϕn + k +1 ( a , − t ) , (110)and if ̺ is sufficiently small with respect to δ (for example ̺ < δ ) we know that the integrallines in (110) are defined in [ − t , t ] and all the points a , a , a ∈ V ′ a . Notice that it is recisely here that we have to take U a small enough to guarantee that the points defined in(110) are in V ′ a . This can be done since we are taking the concatenation of integral curvesliving up to a time whose norm is bounded above by t < ̺ / .We set ϕ =: ( ϕ (1) , . . . , ϕ ( k ) ) in coordinates. From Definition 3.24 we get that, for every a ∈ V ′ a , we have(111) dd s | s = t ϕ ( ℓ ) ( E ϕj ( a, s )) = ω ℓj ( E ϕj ( a, t )) , ∀ j = k +1 , . . . , n +1 , ∀ ℓ = 1 , . . . , k, ∀ t ∈ [ − δ, δ ] . Thus, by using the points defined in (110), Lagrange’s theorem and the triangle inequality,we get | ϕ ( ξ, z ) − ϕ ( ξ, z ) || z − z | / = | ( ϕ ( a ) − ϕ ( a )) + ( ϕ ( a ) − ϕ ( a )) + ( ϕ ( a ) − ϕ ( a )) + ( ϕ ( a ) − ϕ ( a )) || z − z | / ≤ k X ℓ =1 | z − z | / · (cid:12)(cid:12)(cid:12) (cid:0) ϕ ( ℓ ) ( a ) − ϕ ( ℓ ) ( a ) (cid:1) + (cid:0) ϕ ( ℓ ) ( a ) − ϕ ( ℓ ) ( a ) (cid:1) ++ (cid:0) ϕ ( ℓ ) ( a ) − ϕ ( ℓ ) ( a ) (cid:1) + (cid:0) ϕ ( ℓ ) ( a ) − ϕ ( ℓ ) ( a ) (cid:1) (cid:12)(cid:12)(cid:12) = k X ℓ =1 |− t ω ℓ,k +1 ( b ,ℓ ) − t ω ℓ,n + k +1 ( b ,ℓ ) + t ω ℓ,k +1 ( b ,ℓ ) + t ω ℓ,n + k +1 ( b ,ℓ ) | t ≤ k X ℓ =1 ( | ω ℓ,k +1 ( b ,ℓ ) − ω ℓ,k +1 ( b ,ℓ ) | + | ω ℓ,n + k +1 ( b ,ℓ ) − ω ℓ,n + k +1 ( b ,ℓ ) | ) ≤ k X ℓ =1 ( β ( | b ,ℓ − b ,ℓ | ) + β ( | b ,ℓ − b ,ℓ | )) , (112)where, for every ℓ = 1 , . . . , k , the points b ,ℓ , b ,ℓ , b ,ℓ , b ,ℓ are respectively chosen accordingto Lagrange’s Theorem, that can be applied in view of (111), on the images of the integralcurves E ϕk +1 ( a ) , E ϕn + k +1 ( a ) , E ϕk +1 ( a ) , E ϕn + k +1 ( a ) used in (110). By simple estimates relyingon the triangle inequality, we get a constant C > only depending on V a such that forevery ℓ = 1 , . . . , k , the estimates | b ,ℓ − b ,ℓ |≤ C t , and | b ,ℓ − b ,ℓ |≤ C t hold. Since t = | z − z | / , and | z − z | < ̺ , from (112) we thus get (109) with α ( ̺ ) := C ̺ / . (cid:3) Proposition 5.2.
In the setting of Proposition 5.1, it holds D ϕ ϕ = ω in the broad* sense ⇒ e ϕ ∈ UID( e U , W ; L ) . Proof.
It is a direct consequence of Proposition 5.1 and Proposition 4.5. Indeed, taking intoaccount (30) of Example 3.4, the integral curves of D ϕ n +1 are vertical lines and, therefore,condition (108) of Proposition 5.1 implies that ϕ is vertically broad* hölder. (cid:3) Examples of uniformly intrinsically differentiable functions.
We show a classof nontrivial examples of
UID functions in the Engel group, see Example 3.8.
Example . Consider the Engel group E , with the splitting E = W · L described in Ex-ample 3.8. We show that the function(113) ϕ α (0 , x , x , x ) := x α χ { x ≥ } (0 , x , x , x ) , roduces e ϕ α ∈ UID( W ; L ) for any α > / . We first claim that D ϕ α X ϕ α = α x α − χ { x ≥ } inthe broad* sense. Indeed, for any ε ∈ (0 , , the functions ( ϕ α ) ε := ε / χ { x < } + ( x α + ε ) / χ { x ≥ } are globally C and lim ε → ( ϕ α ) ε = ϕ α in L ∞ loc ( R ) , and D ( ϕ α ) ε X ( ϕ α ) ε = α x α − χ { x ≥ } , ∀ ε ∈ (0 , , where the last equality comes from the particular form of D ϕX in (33). By applying Propos-ition 4.15, we get the claim.We claim now that ϕ is vertically broad* holder, see Definition 4.3. Indeed, since theintegral curves of D ϕ α X are the vertical lines along direction x , see (33), and since α > / ,we get that ϕ α is locally uniformly / -little Hölder continuous along these curves. We areleft to prove that, locally around any a := (0 , x , x , x ) , condition (89) is satisfied for theintegral curves of D ϕ α X , whose expression is in (33).According to the sign of x we identify three families of integral curves of the vectorfield D ϕ α X starting from a . If x < , the only integral curve of D ϕ α X starting from a is γ ( t ) = (0 , x , x + t, x ) , and it is well-defined for every time t . If x = 0 , we have that anintegral curve of D ϕ α X , existing for all times t , starting from a is given by γ ( t ) = (cid:0) , x , x + t, (1 − α ) / (1 − α ) t / (1 − α ) χ { t ≥ } ( t ) (cid:1) , while, if x > , we have that an integral curve of D ϕ α X starting from a is given by γ ( t ) = , x , x + t, (1 − α ) / (1 − α ) (cid:18) t + ( x ) − α − α (cid:19) / (1 − α ) ! , defined for t ∈ (cid:16) − x − α − α , + ∞ (cid:17) . By exploiting this explicit choice of integral curves, we noticethat γ ( t ) , in the three cases, is constant or it is of order t / (1 − α ) on the last component.Thus, we get that ϕ α is 1/2-little Hölder continuous along these integral curves, because α/ (1 − α ) > / for any α > / , and this happens locally uniformly. From this, weconclude that ϕ α is vertically broad* holder. Now we conclude by applying (d) ⇒ (a) ofTheorem 4.17.Notice that, if α = 1 / , the function ϕ α is not UID in every neighborhood of the origin.Indeed, by Theorem 4.17, if ϕ α is UID, then it is vertically broad* hölder and so 1/3-littleHölder continuous along the integral curves of the vector field D ϕ α X . This means that ϕ α should be / -little Hölder continuous in the last coordinate, but this is not true when α = 1 / .We remark here that, in the case of the Heisenberg group H , a slight modification of thistypes of examples gives rise to a C -hypersurface, that is also C Euclidean, but it has 0 asa characteristic point, see Remark 5.4.
Remark . Consider the first Heisenberg group H identified with R by means of expo-nential coordinates, and consider the splitting H = W · L described in Example 3.4, with L one-dimensional. Define ϕ : W ≡ R → L ≡ R by setting ϕ (0 , x , x ) := sgn( x ) | x | / . ince ϕ (0 , x , x ) = | x | / ∈ C , by [ASCV06, Corollary 5.11], then e ϕ ∈ UID( W ; L ) and,consequently, its graph is a C -hypersurface. Moreover, in coordinates, one has(114) (0 , x , x ) · ( ϕ (0 , x , x ) , ,
0) = (cid:18) sgn( x ) | x | / , x , x −
12 sgn( x ) x | x | / (cid:19) . The surface Σ parametrized by (114) is the union of two surfaces Σ , Σ given by Σ := (cid:26) ( x , x , x ) ∈ R : x ≥ , x − x / + 12 x x = 0 (cid:27) , Σ := (cid:26) ( x , x , x ) ∈ R : x ≤ , x + ( − x ) / + 12 x x = 0 (cid:27) , (115)that are glued along the x -axis { ( x , x , x ) ∈ R : x = x = 0 } . We thus get that, for any x ∈ int(Σ ) , one has T x Σ = (cid:16) x − x / , x , (cid:17) ⊥ with respect to the standard Euclideanscalar product, while for any x ∈ int(Σ ) , one has T x Σ = (cid:0) x − ( − x ) / , x , (cid:1) ⊥ withrespect to the standard Euclidean scalar product. From these explicit expressions, Σ and Σ glue together in C regular way along the x -axis, and thus Σ is also C -Euclidean surface.Moreover, from the previous expressions, we get that T Σ = (0 , , ⊥ = { x = 0 } = V andhence is a characteristic point for Σ .We remark here that a similar example appeared in [FSSC07, Remark 3.8]. In that casethe intrinsic graph is C regular but not C regular.6. Carnot groups of step 2
As already underlined in Remark 4.18, a co-horizontal C -surface cannot be always char-acterized only by its horizontal geometry. This is however possible inside Carnot groups ofstep 2. Indeed, in this section, we show that the assumption on the vertical broad* hölderregularity of Theorem 4.17 can be dropped when G is a Carnot group of step 2 and L is ahorizontal one-dimensional subgroup of G . In particular, as main result of this section, weget Theorem 6.17, which is Theorem 1.7 in the introduction.We describe the strategy of the proof of Theorem 6.17. The key idea is to first show thatthe implication(116) D ϕ ϕ = ω broad* ⇒ ϕ vertically broad* hölderholds in free Carnot groups of step 2, for a continuous ω . This is done in two main steps.First, we explicitly write the intrinsic vector fields D ϕ and we notice that the nonlinearitygiven by ϕ only shows up in one vertical coordinate for each vector field, see (120). Second,by using the structure of the vector fields D ϕ , one can propagate the broad* regularity fromthe horizontal components of ϕ to the little Hölder regularity along vertical components byusing a geometric trick: if [ X, Y ] = Z , and ϕ is C on the integral curves of X and Y , weexpect ϕ to be / -little Hölder continuous on the integral curves of Z .More precisely, the first step allows us to obtain the / -little Hölder regularity on thevertical coordinates affected by the nonlinearity by means of an adaptation of [ASCV06,Theorem 5.8]. The second step allows us to obtain the / -little Hölder regularity on allthe remaining vertical coordinates. Notice that the efforts made to prove (d) ⇒ (a) in The-orem 4.17, asking for just the regularity along the integral curves of D ϕ (i.e., the verticallybroad* regularity), are here payed back from a crucial simplification of this proof. Indeed, o conclude the proof of Theorem 6.17 in the case of free Carnot groups of step 2, namelyTheorem 6.6, we just use (116) and apply (d) ⇒ (a) of Theorem 4.17. This can be done afterhaving noticed that on Carnot groups of step 2 the vertically broad* hölder regularity readsas the locally 1/2-little Hölder continuity along vertical coordinates, see also Remark 3.23.Finally, to conclude the proof of the difficult implication (e) ⇒ (a) of Theorem 6.17, weuse (116) together with the fact that, on a Carnot group G of step 2, the broad* conditionlifts to the free Carnot group F of step 2 and of the same rank of G , see Proposition 6.10,while having a vertically broad* hölder property is naturally transferred from F to G , seeProposition 6.11. The resulting strategy presents some similarity to [LDPS19].We point out that (a) ⇔ (c) of Theorem 6.17 is a generalization to all step 2 Carnot groupsof [ASCV06, Theorem 5.1] and (a) ⇔ (e) is a generalization of [BSC10b, Theorem 1.2]. Werefer the reader to the introduction for a more detailed discussion on the literature.In the current section, without loss of generality, we will always work in coordinatesand there will be no distinction between (cid:3) and e (cid:3) . See Remark 6.4 for details on theidentifications.6.1. Regularity results for broad* solutions in free Carnot groups of step 2.
Free-nilpotent Lie algebras can be defined as follows (see Definition 14.1.1 in [BLU07]).
Definition 6.1 (Free-nilpotent Lie algebras of step 2) . Let m ≥ be integer. We say that f m, is the free-nilpotent Lie algebra of step 2 with m generators X , . . . , X m if the followingfacts hold.(i) f m, is a Lie algebra generated by the elements X , . . . , X m (i.e., f m, is the smallestLie algebra containing { X , . . . , X m } );(ii) f m, is nilpotent of step (i.e., nested Lie brackets of length are always );(iii) for every nilpotent Lie algebra g of step and for every map Ψ: { X , . . . , X m } → g ,there exists a unique homomorphism of Lie algebras Ψ : f m, → g that extends Ψ . Definition 6.2 (Free Carnot groups of step 2) . A free Carnot group of step 2 is a Carnotgroup whose Lie algebra is isomorphic to a free-nilpotent Lie algebra f m, for some m ≥ .In this case, the horizontal layer of the free Carnot group is isomorphic to the linear span ofthe generators of the Lie algebra f m, . Remark . We give an explicitrepresentation of free Carnot groups of step . Fix an integer m ≥ and denote by n := m + m ( m − . In R n denote the coordinates by x j , for ≤ j ≤ m , and by y ℓs , for ≤ s < ℓ ≤ m .Let ∂ j and ∂ ℓs denote the standard basis vectors in this coordinate system. We define n linearly independent vector fields on R n by setting:(117) X j := ∂ j + 12 X j<ℓ ≤ m x ℓ ∂ ℓj − X ≤ ℓ Let W and L be two complementary subgroups of the free Carnot group F of step 2 with L horizontal and one-dimensional. Let V ⊆ W be open and ψ : V → L be acontinuous function and assume that ψ is a broad* solution of the system D ψ ψ = ω in V ,for some ω ∈ C ( V ; R m − ) , with respect to the basis chosen in Remark 6.3 and Remark 6.4.Then ψ is vertically broad* hölder.Proof. Working in the coordinates described in Remark 6.3 and Remark 6.4, we can assumewithout loss of generality that W and L are defined as in (119). If m = 2 , then F = H ,and therefore the statement would follow from [BSC10b, Theorem 1.2]. We therefore assumethat m > .We prove the following stronger fact, from which (121) follows. For each a ∈ V there aresufficiently small neighborhoods I ⋐ I ′ ⋐ V of a such that, for every ≤ s < ℓ ≤ m , onecan find a continuous and increasing function α ℓs : (0 , + ∞ ) → [0 , + ∞ ) only depending on I ′ , k ψ k L ∞ ( I ′ ) , k ω k L ∞ ( I ′ ) and on the modulus of continuity of ω on I ′ with the property that(122) lim ̺ → α ℓs ( ̺ ) = 0 , and | ψ ( ξ, η ) − ψ ( ξ, y ) || η ℓs − y ℓs | / ≤ α ℓs ( ̺ ) , (123)for every ( ξ, η ) , ( ξ, y ) ∈ I such that η kτ = y kτ for every ( k, τ ) = ( ℓ, s ) and < | y ℓs − η ℓs |≤ ̺ .Fix a ∈ V . Since ψ is a broad* solution of D ψ ψ = ω in V , there exist < δ < δ and afamily of maps E ψj : B ( a , δ ) × [ − δ , δ ] → B ( a , δ ) , for j = 2 , . . . , m , such that the conditions of Definition 3.24 are satisfied. Define I ′ := B ( a , δ ) and I := B ( a , δ ) , and set M := k ψ k L ∞ ( I ′ ) . Let also β be an increasing modulusof uniform continuity of ω on I ′ . We are going to prove (123) with α ℓs defined by(124) α ℓs ( ̺ ) := ( δ ( ̺ ) , if ( ℓ, s ) = ( j, and j = 2 , . . . , m,G ℓs ( ̺ ) , otherwise,where G ℓs will be determined later in (138) and(125) δ ( ̺ ) := max (cid:26) ̺ / , q β ( C̺ / ) (cid:27) , for some constant C := C ( j ) > such that(126) | η j − y j | +2( | ξ | + M ) | η j − y j | / ≤ C | η j − y j | / or any ( ξ, η ) , ( ξ, y ) ∈ I ′ with η kτ = y kτ for every couple ( k, τ ) = ( j, . First step. If a = ( x, y ) ∈ I , by using (120), we have that for any ≤ j ≤ m and any t ∈ [ − δ , δ ] , it holds(127) E ψj ( a, t ) = ( x , . . . , x j − , x j + t, x j +1 , . . . , x m , y ( t )) , where y ℓs ( t ) = y j − R t ψ ( E ψj ( a, r )) d r, if ( ℓ, s ) = ( j, , tx ℓ + y ℓj , if s = j, and j < ℓ ≤ m, − tx s + y js , if ℓ = j, and < s < j,y ℓs , otherwise,and consequently, since E ψj are the maps provided by Definition 3.24, t y j ( t ) is a solutionof the Cauchy problem ¨ y j ( t ) = dd t h − ψ ( E ψj ( a, t )) i = − ω j ( E ψj ( a, t )) , t ∈ [ − δ , δ ] ,y j (0) = y j , ˙ y j (0) = − ψ ( a ) . As a consequence of (127) one gets(128) max r ∈ [ −| t | , | t | ] | ˙ y ( r ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j<ℓ ≤ m x ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Fix j = 2 , . . . , m and assume a = ( ξ, η ) , ˆ a = ( ξ, y ) ∈ I with η kτ = y kτ forall couples ( k, τ ) = ( j, . We will need to possibly shrink I in a way that will be clearthroughout the proof. We aim to show that(129) | ψ ( a ) − ψ (ˆ a ) || η j − y j | / ≤ α j ( | η j − y j | ) , where, according to (124), α j ( ̺ ) = 3 δ ( ̺ ) . This would imply (123) for ( ℓ, s ) = ( j, .Set δ := δ ( | η j − y j | ) and suppose (129) is not true, namely(130) | ψ ( a ) − ψ (ˆ a ) || η j − y j | / > δ. Let E ψj ( a, · ) and E ψj (ˆ a, · ) be the integral curves of D ψj given by the broad* condition. By thefirst step they satisfy E ψj ( a, t ) = ( ξ , . . . , ξ j − , ξ j + t, ξ j +1 , . . . , ξ m , η ( t )) , and E ψj (ˆ a, t ) = ( ξ , . . . , ξ j − , ξ j + t, ξ j +1 , . . . , ξ m , y ( t )) . We claim we can find t ∗ ∈ [ − δ , δ ] such that η j ( t ∗ ) = y j ( t ∗ ) , with ψ ( E ψj ( a, t ∗ )) = ψ ( E ψj (ˆ a, t ∗ )) . This will lead to a contradiction, by the fact that the equality η j ( t ∗ ) = y j ( t ∗ ) would also imply η ( t ∗ ) = y ( t ∗ ) . ithout loss of generality, assume that the initial data satisfy η j > y j . By the firststep of this proof, for every t ∈ [ − δ , δ ] , one has η j ( t ) − y j ( t ) − ( η j − y j ) = Z t " ˙ η j (0) − ˙ y j (0) + Z r ′ (¨ η j ( r ) − ¨ y j ( r )) d r d r ′ = − t ( ψ ( a ) − ψ (ˆ a )) − Z t Z r ′ (cid:16) ω j ( E ψj ( a, r )) − ω j ( E ψj (ˆ a, r )) (cid:17) d r d r ′ . Using (128) and the fundamental theorem of Calculus, one gets a constant C > onlydepending on C such that(131) | E ψj ( a, r ) − E ψj (ˆ a, r ) | ≤ | η − y | + | r | (cid:18) max r ∈ [ −| t | , | t | ] | ˙ η ( r ) | + max r ∈ [ −| t | , | t | ] | ˙ y ( r ) | (cid:19) ≤ | η − y | + | t | (cid:18) max r ∈ [ −| t | , | t | ] | ˙ η ( r ) | + max r ∈ [ −| t | , | t | ] | ˙ y ( r ) | (cid:19) ≤ C ( | η j − y j | +2 | t | ( | ξ | + M )) , for every r ∈ [ −| t | , | t | ] and t ∈ [ − δ , δ ] . Hence, we obtain that(132) η j ( t ) − y j ( t ) − ( η j − y j ) ≤ − t ( ψ ( a ) − ψ (ˆ a )) + t max r ∈ [ −| t | , | t | ] β ( | E ψj ( a, r ) − E ψj (ˆ a, r ) | ) ≤ − t ( ψ ( a ) − ψ (ˆ a )) + t β ( C ( | η j − y j | +2 | t | ( | ξ | + M ))) , for every t ∈ [ − δ , δ ] . Up to redefining β , we can replace without loss of generality β ( C ̺ ) with β ( ̺ ) . Now, by (130) we know that(133) − | ψ ( a ) − ψ (ˆ a ) | < − δ | η j − y j | / . Without loss of generality, up to restricting I , we can assume that(134) δ ≥ | y j − η j | / ≥ | y j − η j | / /δ, where the second inequality directly follows from (125). If ψ ( a ) = ψ (ˆ a ) , then (129) wouldbe trivial. We study two cases: ψ ( a ) − ψ (ˆ a ) < or ψ ( a ) − ψ (ˆ a ) > . If ψ ( a ) − ψ (ˆ a ) < ,set t := − | η j − y j | / δ (otherwise we can choose t := | η j − y j | / δ ) and evaluate (132) in t = t .Combining it with (126), (133), (134), the definition of δ , and the fact that β is increasing,we obtain (in both cases)(135) η j ( t ) − y j ( t ) ≤ η j − y j + | η j − y j | / −| ψ ( a ) − ψ (ˆ a ) | δ ++ 1 δ | η j − y j | β (cid:18) | η j − y j | +2( | ξ | + M ) | η j − y j | / δ (cid:19) ≤ η j − y j − | η j − y j | + | η j − y j | β (cid:0) C | η j − y j | / (cid:1) δ ≤ η j − y j − | η j − y j | + | η j − y j |≤ −| η j − y j | < . If ψ ( a ) − ψ (ˆ a ) > we can define t ∗ := sup { r ∈ [0 , δ ] : η j ( s ) − y j ( s ) > , ∀ s ∈ [0 , r ] } , since the set { r ∈ [0 , δ ] : η j ( s ) − y j ( s ) > , ∀ s ∈ [0 , r ] } is not empty; indeed, recall that weassumed without loss of generality that η j > y j and therefore η j (0) − y j (0) = η j − y j > . oreover < t ∗ < t ≤ δ , and recalling that η kτ = y kτ except for ( k, τ ) = ( j, , one has,by (127) that(136) η j ( t ∗ ) = y j ( t ∗ ) ,η ℓj ( t ∗ ) = y ℓj ( t ∗ ) = 12 t ∗ ξ ℓ + η ℓj , for j < ℓ ≤ m,η js ( t ∗ ) = y js ( t ∗ ) = − t ∗ ξ s + η js , for < s < j,η ℓs ( t ∗ ) = y ℓs ( t ∗ ) = η ℓs , otherwise.Hence, by definition of E ψj ( a, · ) and E ψj (ˆ a, · ) in (127), one gets E ψj ( a, t ∗ ) = E ψj (ˆ a, t ∗ ) andtherefore(137) ψ ( E ψj ( a, t ∗ )) = ψ ( E ψj (ˆ a, t ∗ )) . In the case ψ ( a ) − ψ (ˆ a ) < , we consider t = − | η j − y j | / δ and define t ∗ := inf { r ∈ [ − δ , 0] : η j ( s ) − y j ( s ) > , ∀ s ∈ [ r, } . Then − δ ≤ t < t ∗ < and, also in this case, (136) and(137) are satisfied.We now show that this leads to a contradiction. In case ψ ( a ) − ψ (ˆ a ) > , by using theproperties of E ψj , (131), (133), (126), the definition of β and the fact that t ∗ < t , we deduce − ( ψ ( E ψj ( a, t ∗ )) − ψ ( E ψj (ˆ a, t ∗ ))) = − ( ψ ( a ) − ψ (ˆ a )) − Z t ∗ (cid:16) ω j ( E ψj ( a, r )) − ω j ( E ψj (ˆ a, r )) (cid:17) d r ≤ − η j − y j ) / δ + | t ∗ | max r ∈ [0 ,t ∗ ] β (cid:16) | E ψj ( a, r ) − E ψj (ˆ a, r ) | (cid:17) ≤ − η j − y j ) / δ + | t ∗ | β ( | η j − y j | +2 | t ∗ | ( | ξ | + M )) ≤ − η j − y j ) / δ + | t ∗ | β (cid:18) | η j − y j | +2( | ξ | + M ) | η j − y j | / δ (cid:19) ≤ − η j − y j ) / δ + ( η j − y j ) / δ β (cid:0) C ( η j − y j ) / (cid:1) δ ≤ ( − η j − y j ) / δ < . Similarly, if ψ ( a ) − ψ (ˆ a ) < , then ψ ( E ψj ( a, t ∗ )) − ψ ( E ψj (ˆ a, t ∗ )) = ψ ( a ) − ψ (ˆ a ) + Z t ∗ (cid:16) ω j ( E ψj ( a, r )) − ω j ( E ψj (ˆ a, r )) (cid:17) d r ≤ − η j − y j ) / δ + | t ∗ | max r ∈ [ − t ∗ , β (cid:16) | E ψj ( a, r ) − E ψj (ˆ a, r ) | (cid:17) ,< . Hence, in both cases we have ψ ( E ψj ( a, t ∗ )) = ψ ( E ψj (ˆ a, t ∗ )) that is in contradiction with (137),so (129) follows. Third step. Fix ℓ, s with < s < ℓ ≤ m , denote by M := k ω k L ∞ ( I ′ ) and define(138) G ℓs ( ̺ ) := 2 p M α ℓ (4 M ̺ ) + 2 p M α s (4 M ̺ ) + 2 β ( C ̺ / ) , where α ℓ and α s are defined as in (124) for j = ℓ and j = s , respectively, β is an increasingmodulus of uniform continuity of ω on I ′ and C > is a suitable constant, only dependingon the supremum norm of ω on I ′ , that will be determined later. e want to show that(139) | ψ ( a ) − ψ (ˆ a ) || η ℓs − y ℓs | / ≤ G ℓs ( ̺ ) , for every sufficiently small ̺ > , every a = ( ξ, η ) , ˆ a = ( ξ, y ) ∈ I , such that η kτ = y kτ forevery ( k, τ ) = ( ℓ, s ) and < | η ℓs − y ℓs |≤ ̺ . Denote by T ℓs := | η ℓs − y ℓs | / . We will need topossibly shrink I in a way that will be clear from the proof. Rough idea of the proof. We build a concatenation of integral curves of the vector fields D ψℓ and D ψs that joins ˆ a and a suitable point a that that can be connected to a with twovertical lines on which we can use the result of the second step of this proof. We start from ˆ a and follow the integral curve of D ψℓ for a time T ℓs and we follow the integral curve of D ψs for the same time T ℓs ; finally, we repeat the same procedure but for time − T ℓs . At the endof this process, we obtain a point with three different vertical components with respect to ˆ a : two increments are given by the non linear terms − ψ∂ ℓ and − ψ∂ s coming respectivelyfrom D ψℓ and from D ψs and one increment is given by the commutator [ X ℓ , X s ] (which is Y ℓs ). In particular, whenever η ℓs − y ℓs > , the ( ℓ, s ) -component becomes equal to the ( ℓ, s ) -component of a , that is η ℓs . Vice versa, if η ℓs − y ℓs < , one has to replace the times ± T ℓs with ∓ T ℓs . In the end, we complete the proof by using the estimate of the second step ofthis proof applied to a and a . The desired estimates come by using Lagrange’s Theorem.Assume η ℓs − y ℓs > . Then we construct the following chain of points. ˆ a → E ψℓ (ˆ a ) a := E ψℓ (ˆ a, T ℓs ) → E ψs ( a ) a := E ψs ( a , T ℓs ) → E ψℓ ( a ) a := E ψℓ ( a , − T ℓs ) → E ψs ( a ) a := E ψs ( a , − T ℓs ) , (140)where we recall that E ψℓ and E ψs are the integral curves of the vector fields D ψℓ and D ψs ,respectively, given by the fact that D ψ ψ = ω in the broad* sense. In particular, E ψℓ and E ψs satisfy (127).In case η ℓs − y ℓs < we repeat the same construction by replacing ± T ℓs with ∓ T ℓs . Inboth cases, we have that a =: ( ξ, y ( a ) ) is such that y ( a ) ℓs = η ℓs . Indeed, if η ℓs − y ℓs > , then y ( a ) kτ = η ℓs , if ( k, τ ) = ( ℓ, s ) ,η ℓ − Z T ℓs ψ ( E ψℓ (ˆ a, t )) d t + Z T ℓs ψ ( E ψℓ ( a , t )) d t, if ( k, τ ) = ( ℓ, ,η s − Z T ℓs ψ ( E ψs ( a , t )) d t + Z T ℓs ψ ( E ψs ( a , t )) d t, if ( k, τ ) = ( s, ,y kτ , otherwise.We can assume that a , a , a , a ∈ I . Indeed, this can be done because for a sufficientlysmall ̺ only depending on the supremum norms of ϕ and ω on I ′ , we can possibly reduce I to some I so that all the curves as in (140) starting in I , and living for times boundedabove by ̺ , lie inside I .Let a = ( ξ, y ( a ) ) be a point that has the same components of a , except for position ( ℓ, for which y ( a ) ℓ = y ( a ) ℓ . As remarked above, we can assume without loss of generality thatalso a ∈ I . Moreover, we can estimate | ψ ( a ) − ψ (ˆ a ) | as follows:(141) | ψ ( a ) − ψ (ˆ a ) | ≤ | ψ ( a ) − ψ ( a ) | + | ψ ( a ) − ψ ( a ) | + | ψ ( a ) − ψ (ˆ a ) | . e start by considering | ψ ( a ) − ψ ( a ) | . Evaluating (129) for j = ℓ , we get that | ψ ( a ) − ψ ( a ) | ≤ | η ℓ − y ( a ) ℓ | / α ℓ ( | η ℓ − y ( a ) ℓ | ) , and we also notice that | η ℓ − y ( a ) ℓ | = (cid:12)(cid:12)(cid:12)(cid:12)Z T ℓs ψ ( E ψℓ (ˆ a, t )) d t − Z T ℓs ψ ( E ψℓ ( a , t )) d t (cid:12)(cid:12)(cid:12)(cid:12) . Recalling that M = k ω k L ∞ ( I ′ ) , we aim to show that(142) (cid:12)(cid:12)(cid:12)(cid:12)Z T ℓs ψ ( E ψℓ (ˆ a, t )) d t − Z T ℓs ψ ( E ψℓ ( a , t )) d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ M T ℓs , that would imply(143) | ψ ( a ) − ψ ( a ) | ≤ p M T ℓs α ℓ ( | η ℓ − y ( a ) ℓ | ) ≤ p M T ℓs α ℓ (4 M T ℓs ) . We first observe that (cid:12)(cid:12)(cid:12)(cid:12)Z T ℓs ψ ( E ψℓ (ˆ a, t )) d t − Z T ℓs ψ ( E ψℓ ( a , t )) d t (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z T ℓs (cid:16) ψ ( E ψℓ (ˆ a, t )) − ψ (ˆ a ) (cid:17) d t − Z T ℓs (cid:16) ψ ( E ψℓ ( a , t )) − ψ ( a ) (cid:17) d t + T ℓs (( ψ (ˆ a ) − ψ ( a ))) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T ℓs (cid:12)(cid:12)(cid:12) ψ ( E ψℓ (ˆ a, t )) − ψ (ˆ a ) (cid:12)(cid:12)(cid:12) d t + Z T ℓs (cid:12)(cid:12)(cid:12) ψ ( E ψℓ ( a , t )) − ψ ( a ) (cid:12)(cid:12)(cid:12) d t + T ℓs | ψ (ˆ a ) − ψ ( a ) | . Recalling that for every t in the interval of definition of the curve E ψℓ , one has dd s | s = t ψ ( E ψℓ ( A, s )) = ω ℓ ( E ψℓ ( A, t )) for A = ˆ a, a , by exploiting the fundamental theorem ofCalculus the previous estimate then yields (cid:12)(cid:12)(cid:12)(cid:12)Z T ℓs ψ ( E ψℓ (ˆ a, t )) d t − Z T ℓs ψ ( E ψℓ ( a , t )) d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ M T ℓs + T ℓs | ψ (ˆ a ) − ψ ( a ) | . By Lagrange’s Theorem one also gets | ψ (ˆ a ) − ψ ( a ) |≤ | ψ (ˆ a ) − ψ ( a ) | + | ψ ( a ) − ψ ( a ) | = T ℓs | ω ℓ ( b ∗ ) | + T ℓs | ω s ( b ∗ ) |≤ M T ℓs , where b ∗ and b ∗ are two points on E ψℓ (ˆ a, [0 , T ℓs ]) and E ψs ( a , [0 , T ℓs ]) , respectively. Hence,combining together the last two estimates, one obtains (142) and, consequently, also (143)holds.Now we consider | ψ ( a ) − ψ ( a ) | . Since | a − a | = | y ( a ) s − y ( a ) s | , analogously to the previouscase, one obtains(144) | ψ ( a ) − ψ ( a ) | ≤ p M T ℓs α s ( | y ( a ) s − y ( a ) s | ) ≤ p M T ℓs α s (4 M T ℓs ) , where α s is defined as in (124) and | y ( a ) s − y ( a ) s | = (cid:12)(cid:12)(cid:12)(cid:12)Z T ℓs ψ ( E ψs ( a , t )) d t − Z T ℓs ψ ( E ψs ( a , t )) d t (cid:12)(cid:12)(cid:12)(cid:12) . ventually, we estimate ψ ( a ) − ψ (ˆ a ) in the following way: | ψ ( a ) − ψ (ˆ a ) | ≤ | ( ψ ( a ) − ψ ( a )) + ( ψ ( a ) − ψ ( a )) + ( ψ ( a ) − ψ ( a )) + ( ψ ( a ) − ψ (ˆ a )) | = |− T ℓs ω s ( a ∗ ) − T ℓs ω ℓ ( a ∗ ) + T ℓs ω s ( a ∗ ) + T ℓs ω ℓ ( a ∗ ) |≤ T ℓs ( | ω s ( a ∗ ) − ω s ( a ∗ ) | + | ω ℓ ( a ∗ ) − ω ℓ ( a ∗ ) | ) ≤ T ℓs ( β ( | a ∗ − a ∗ | ) + β ( | a ∗ − a ∗ | )) , where a ∗ ∈ E ψℓ (ˆ a, [0 , T ℓs ]) , a ∗ ∈ E ψs ( a , [0 , T ℓs ]) , a ∗ ∈ E ψℓ ( a , [ − T ℓs , and a ∗ ∈ E ψs ( a , [ − T ℓs , are chosen to fulfill the conditions of Lagrange’s Theorem. By simple estimates relying onthe triangle inequality, we get a constant C > , only depending on the supremum norm of ω on I ′ , such that | a ∗ − a ∗ |≤ C T ℓs , and | a ∗ − a ∗ |≤ C T ℓs . Hence(145) | ψ ( a ) − ψ (ˆ a ) | ≤ T ℓs ( β ( | a ∗ − a ∗ | ) + β ( | a ∗ − a ∗ | )) ≤ T ℓs β ( C T ℓs ) . By combining (141) with (143), (144) and (145) and recalling that T ℓs = | η ℓs − y ℓs | / and | η ℓs − y ℓs | < ̺ , we thus get (139). Indeed, one has ψ ( a ) − ψ (ˆ a ) | η ℓs − y ℓs | / ≤ √ M T ℓs α ℓ (4 M T ℓs ) + 2 √ M T ℓs α s (4 M T ℓs ) + 2 T ℓs β ( C T ℓs ) | η ℓs − y ℓs | / = 2 p M (cid:16) α ℓ (4 M | η ℓs − y ℓs | ) + α s (4 M | η ℓs − y ℓs | ) (cid:17) + 2 β ( C | η ℓs − y ℓs | / ) ≤ G ℓs ( ̺ ) . Finally, since G ℓs is defined as sum of continuous maps that are at , it follows that, if ̺ → , then G ℓs ( ̺ ) → for which we get (123) also for all ( ℓ, s ) with s = 1 and the proof iscomplete. (cid:3) Regularity results for broad* solutions in Carnot groups of step 2. In thissection we see how to generalize Theorem 6.6, valid for free Carnot groups of step , to anyCarnot group of step 2. We adapt some techniques already exploited in [LDPS19]. Moreprecisely, in Proposition 6.10 we prove that the broad* condition lifts from G to the freegroup F with same step and rank of G . In Proposition 6.11 we show that the verticallybroad* hölder regularity on F implies the vertically broad* hölder regularity on G . Thesetwo facts will put us in a position to prove Theorem 6.17 by exploiting Theorem 4.17 andTheorem 6.6.We here introduce Carnot groups of step 2 and we refer the reader to [BLU07, Chapter 3].We denote with m the rank of G and we identify G with ( R m + h , · ) . If q ∈ G , we write q = ( x, y ) meaning that x ∈ R m and y ∈ R h . The group operation · between two elements q = ( x, y ) and q ′ = ( x ′ , y ′ ) is given by(146) q · q ′ = (cid:18) x + x ′ , y + y ′ − hB x, x ′ i (cid:19) , where hB x, x ′ i := ( hB (1) x, x ′ i , . . . , hB ( h ) x, x ′ i ) and B ( i ) are linearly independent and skew-symmetric matrices in R m × m , for i = 1 , . . . , h . or any i = 1 , . . . , h and any j, ℓ = 1 , . . . , m , denote by ( B ( i ) ) jℓ =: ( b ( i ) jℓ ) , and define m + h linearly independent left-invariant vector fields by setting X ′ j ( p ) := ∂ x j − h X i =1 m X ℓ =1 b ( i ) jℓ x ℓ ∂ y i , for j = 1 , . . . , m,Y ′ i ( p ) := ∂ y i , for i = 1 , . . . , h. The ordered set ( X ′ , . . . , X ′ m , Y ′ , . . . , Y ′ h ) is an adapted basis of the Lie algebra g of G . Usingthe skew-symmetry of B , it easy to see that(147) [ X ′ j , X ′ ℓ ] = h X i =1 b ( i ) jℓ Y ′ i , and [ X ′ j , Y ′ i ] = 0 , ∀ j, ℓ = 1 , . . . , m and ∀ i = 1 , . . . , h. Remark . When G is a free Carnot group of step with coordinate representation definedas in Remark 6.3, we denote the matrices of the beginning of Section 6.2 with B ( i ) =: B ( ℓ,s ) with ≤ s < ℓ ≤ m . The composition law (146) also tells us that B ( ℓ,s ) has entry inposition ( ℓ, s ) , − in position ( s, ℓ ) and elsewhere.Since the space of skew-symmetric m -dimensional matrices has dimension m ( m − , in anyCarnot group G of step 2, the dimensions m of the horizontal layer and h of the verticallayer are related by the inequality h ≤ m ( m − , and G is free if and only if h = m ( m − . From now on G is a Carnot group of rank m and step , with the coordinaterepresentation previously discussed, and F is the free Carnot group of step 2 andrank m with the coordinate representation as in Remark 6.3 and Remark 6.4 .We denote by ( X , . . . , X m ) a basis of the first layer of the Lie algebra of F . By definitionof free Lie algebra, there exists a Lie group surjective homomorphism π : F → G such that π ∗ ( X ℓ ) = X ′ ℓ for any ℓ = 1 , . . . , m (see e.g., [LDPS19, Section 6]).If we consider on F and G the Carnot-Carathéodory metrics d F and d G respectively, themap π preserves the length of horizontal curves, so it is Lipschitz with Lip( π ) = 1 . Thefollowing lemma is well-known. We refer the reader to [LDPS19, Lemma 6.1] for a proof. Lemma 6.8. For any p ∈ F and any q ′ ∈ G , there exists p ′ ∈ π − ( q ′ ) such that d F ( p, p ′ ) = d G ( π ( p ) , q ′ ) . Recall that the dimension of F is n = m + m ( m − . From the definition of π , we noticethat it preserves the horizontal coordinates, namely, for any ( x, y ) ∈ R n , there exists y ∗ ∈ R h such that(148) π ( x, y ) = ( x, y ∗ ) . We denote by W G and L G two complementary subgroups of G with L G horizontal andone-dimensional. Similarly to Remark 6.4, by means of exponential coordinates we identifythem with R m + h − and R by setting(149) L G := { ( x , . . . , 0) : x ∈ R } , W G := { (0 , x , . . . , x m , y , . . . , y h ) : x i , y k ∈ R for i = 2 , . . . , m, k = 1 , . . . h } . emark . Let G be a Carnot group of step 2 and W G , L G be the complementary subgroupsof G defined as in (149). Then, according to Example 3.6 the projected vector fields relativeto a continuous ϕ : U ⊆ W G → L G , with U open, are given by(150) D ϕj = ∂ x j − h X i =1 b ( i ) j ϕ + 12 m X k =2 x k b ( i ) jk ! ∂ y i = X ′ j | U − h X i =1 b ( i ) j ϕY ′ i | U , for j = 2 , . . . , m,D ϕi = ∂ y i = Y ′ i | U , for i = 1 , . . . , h. Now let W F and L F be the complementary subgroups of F defined as in (119). Then π | LF : L F → L G is an isomorphism and more precisely, with our identification, we can assumeit is the identity, see (148). Moreover, by (148), it follows that π | WF : W F → W G is onto.Since π is a Lie group homomorphism, its differential is a Lie algebra homomorphism.Hence, for any ≤ s < ℓ ≤ m , one also has π ∗ ( Y ℓs ) = π ∗ ([ X ℓ , X s ]) = [ π ∗ ( X ℓ ) , π ∗ ( X s )] = [ X ′ ℓ , X ′ s ] = h X i =1 b ( i ) ℓs Y ′ i , where we used (118), (147) and π ∗ ( X j ) = X ′ j . We can therefore write the following formula π ( x , . . . , x m , y , . . . , y m ( m − ) = ( x , . . . , x m , y ∗ , . . . , y ∗ h ) , where y ∗ i = X ≤ s<ℓ ≤ m b ( i ) ℓs y ℓs , ∀ i = 1 , . . . , h. (151) Proposition 6.10. Let G be a Carnot group of step 2 and let W G and L G be two com-plementary subgroups of G , with L G horizontal and one-dimensional and choose coordinatessuch that (149) is satisfied. Let F be the free Carnot group of step 2, rank m and let W F and L F be the complementary subgroups of F satisfying the identification (119) .Let U ⊆ W G be an open set and denote by V ⊆ W F the open set defined by V := π − ( U ) .Let ϕ : U → L G be a continuous map and let ψ : V → L F be the map defined as ψ := π − ◦ ϕ ◦ π | V . Assume there exists ω ∈ C ( U ; R m − ) such that D ϕ ϕ = ω in the broad* sense.Then ψ is a broad* solution in V of the system D ψ ψ = ω ◦ π .Proof. Fix j = 2 , . . . , m . By (120), we have(152) D ψX j = ∂ x j − ψ∂ j + 12 X j Let G be a Carnot group of step 2 and let W G and L G be two com-plementary subgroups of G , with L G horizontal and one-dimensional and choose coordinatessuch that (149) is satisfied. Let F be the free Carnot group of step 2, rank m and let W F and L F be the complementary subgroups of F satisfying the identification (119) . Let U ⊆ W G bean open set and denote by V ⊆ W F the open set defined by V := π − ( U ) . Let ϕ : U → L G bea continuous map and let ψ : V → L F be the map defined as ψ := π − ◦ ϕ ◦ π | V . Then, if ψ is vertically broad* hölder, also ϕ is vertically broad* hölder.Proof. We observe that in the case we are dealing with, the vertically broad* hölder conditionis equivalent to the locally 1/2-little Hölder continuity along vertical coordinates, see thediscussion before the statement of Theorem 6.6, that holds verbatim for arbitrary Carnotgroups of step 2. Fix b ∈ U and let a ∈ π − ( b ) . Since ψ is vertically broad* hölder,there exist two neighborhoods V ′ and V ′′ of a with V ′ ⋐ V ′′ ⋐ V and an increasing map α : (0 , + ∞ ) → [0 , + ∞ ) only depending on V ′′ , such that lim ̺ → α ( ̺ ) = 0 and(159) | ψ ( ξ, η ) − ψ ( ξ, y ) || η ℓs − y ℓs | / ≤ α ( ̺ ) for every ( ℓ, s ) such that ≤ s < ℓ ≤ m , every sufficiently small ̺ > and every ( ξ, η ) , ( ξ, y ) ∈ V ′ with η kτ = y kτ for every ( k, τ ) = ( ℓ, s ) and < | η ℓs − y ℓs |≤ ̺ . et U ′ := π ( V ′ ) so that b ∈ U ′ and clearly U ′ ⋐ U. We aim to prove that there existsan increasing function β : (0 , + ∞ ) → [0 , + ∞ ) only depending on U ′′ := π ( V ′′ ) such that lim ̺ → β ( ̺ ) = 0 and(160) | ϕ ( ξ, η ) − ϕ ( ξ, y ) || η i − y i | / ≤ β ( ̺ ) , for every i = 1 , . . . , h , every sufficiently small ̺ > and every ( ξ, η ) , ( ξ, y ) ∈ U ′ with y k = y k for every k = i and < | η i − y i |≤ ̺ . Fix i = 1 , . . . , h and ̺ > sufficiently small andconsider b = ( ξ, η ) ˆ b = ( ξ, y ) in U ′ such that η k = y k for all k = i and < | y i − η i |≤ ̺ .Applying Lemma 6.8 to the points b and b we find a = ( ξ, η ∗ ) ∈ π − ( b ) such that d G ( b , b ) = d F ( a , a ) and, since π is continuous, we can also assume, up to possibly reducing U ′ , that a ∈ V ′ . Applying again Lemma 6.8 to the points b and ˆ b , we find ˆ a = ( ξ, y ∗ ) ∈ π − (ˆ b ) ∩ V ′ such that d G ( b, ˆ b ) = d F ( a, ˆ a ) . Since the horizontal components of the points b and ˆ b are equal and the norm induced by the distance d G is equivalent to the anisotropic normon G , we have that d G ( b, ˆ b ) is equivalent to | η i − y i | / . Similarly, notice that a, ˆ a have thesame horizontal components and by the fact that the norm induced by d F is equivalent tothe anisotropic norm on F , it follows that | η ∗ − y ∗ | / is equivalent to d F ( a, ˆ a ) . In particular,we can find a geometric constant C > such that | η ∗ − y ∗ | / ≤ C | η i − y i | / .We can then make the following estimates: | ϕ ( ξ, η ) − ϕ ( ξ, y ) || η i − y i | / = | ψ ( ξ, η ∗ ) − ψ ( ξ, y ∗ ) || η ∗ − y ∗ | / | η ∗ − y ∗ | / | η i − y i | / ≤ C | ψ ( ξ, η ∗ ) − ψ ( ξ, y ∗ , η ∗ , . . . , η ∗ m ( m − ) || η ∗ − y ∗ | / + . . . · · · + | ψ ( ξ, y ∗ , . . . , y ∗ m ( m − , η ∗ m ( m − ) − ψ ( ξ, y ∗ ) || η ∗ m ( m − − y ∗ m ( m − | / ! ≤ C (cid:0) α ( | η ∗ − y ∗ | ) + · · · + α ( | η ∗ m ( m − − y ∗ m ( m − | ) (cid:1) , where we used (159) and assumed without loss of generality that all the considered pointsin the chain belong to V ′ . Observe that, for any ( ℓ, s ) such that ≤ s < l ≤ m , one has | η ∗ ℓs − y ∗ ℓs |≤ | η ∗ − y ∗ |≤ C | η i − y i |≤ C ̺ , and then we can define β ( ̺ ) := C m ( m − α ( C ̺ ) . The previous computations have shown that | ϕ ( ξ, η ) − ϕ ( ξ, y ) || η i − y i | / ≤ β ( ̺ ) , and hence (160) holds, completing the proof. (cid:3) Theorem 6.12. Let W and L be complementary subgroups of a Carnot group G of step 2with L horizontal and one-dimensional and choose coordinates such that (149) is satisfied.Let U ⊆ W be an open set and let ϕ : U → L and ω : U → R m − be two continuous functionssuch that D ϕ ϕ = ω in the broad* sense. Then ϕ is vertically broad* hölder.Proof. Let F be the free Carnot group of step 2, rank m and let W F and L F be the comple-mentary subgroups of F satisfying the identification (119). By Proposition 6.10, we know hat ψ := π − ◦ ϕ ◦ π is a broad* solution of D ψ ψ = ω ◦ π in V = π − ( U ) . Then, byTheorem 6.6, ψ is vertically broad* hölder and finally, by using Proposition 6.11, we obtainthe thesis. (cid:3) We state here some corollaries of the previous results. Corollary 6.13. Let W and L be complementary subgroups of a Carnot group G of step2 with L horizontal and one-dimensional and choose coordinates on G such that (149) issatisfied. Let U ⊆ W be an open set and let ϕ : U → L and ω : U → R m − be two continuousfunctions such that D ϕ ϕ = ω in the broad* sense on U . Then ϕ ∈ UID( U, W ; L ) .Proof. It is enough to combine Proposition 4.5 and Theorem 6.12. (cid:3) Corollary 6.14. Let W and L be complementary subgroups of a Carnot group G of step2 with L horizontal and one-dimensional and choose coordinates on G such that (149) issatisfied. Let U ⊆ W be an open set and let ϕ : U → L and ω : U → R m − be two continuousfunctions such that D ϕ ϕ = ω in the broad* sense on U . Then, the intrinsic graph of ϕ is asurface of class C .Proof. It is enough to combine Corollary 4.6 and Theorem 6.12. (cid:3) Corollary 6.15. Let W and L be complementary subgroups of a Carnot group G of step2 with L horizontal and one-dimensional and choose coordinates on G such that (149) issatisfied. Let U ⊆ W be an open set, and let ϕ : U ⊆ W → L and ω : U → R m − be twocontinuous functions. Assume that D ϕ ϕ = ω in the broad* sense on U . Then D ϕ ϕ = ω inthe sense of distributions on U .Proof. It is enough to combine Theorem 6.12 and Corollary 4.12. (cid:3) Remark . The converse implication of Corollary 6.15 is, up to now, only known forHeisenberg groups, see [BSC10a]. This implication in the general step-2 case will be asubject of further investigations by means of the techniques exploited in this section.6.3. Main theorem in Carnot groups of step 2. Now we are in a position to give thefollowing theorem (stated in Theorem 1.7), which shows that the assumption on the verticallybroad* hölder regularity in Theorem 4.17 can be dropped if we are inside a Carnot group ofstep 2 and L is one-dimensional. We use the same conventions as in Theorem 4.17, followingthe notation of Definition 2.3. Theorem 6.17. Let W and L be complementary subgroups of a Carnot group G of step 2with L horizontal and one-dimensional. Let e U ⊆ W be an open set and let e ϕ : e U → L be acontinuous function. Then the following conditions are equivalent: (a) e ϕ ∈ UID( e U , W ; L ) ; (b) e ϕ ∈ ID( e U , W ; L ) and d ϕ ϕ is continuous on e U ; (c) there exists ω ∈ C ( U ; R m − ) such that, for every a ∈ U , there exist δ > and afamily of functions { ϕ ε ∈ C ( B ( a, δ )) : ε ∈ (0 , } such that lim ε → ϕ ε = ϕ and lim ε → D ϕ ε j ϕ ε = ω j in L ∞ ( B ( a, δ )) , for every j = 2 , . . . , m ; (d) there exists ω ∈ C ( U ; R m − ) such that D ϕ ϕ = ω in the broad sense on U ; (e) there exists ω ∈ C ( U ; R m − ) such that D ϕ ϕ = ω in the broad* sense on U with thechoice of coordinates of (149) . roof. (a) ⇒ (b) is trivial, by item (b) of Proposition 2.25 and (b) ⇒ (a) follows from Propos-ition 3.27, Theorem 6.12 and Corollary 4.7.(a) ⇒ (c) follows from (a) ⇒ (b) of Theorem 4.17 and (c) ⇒ (a) follows by combining Pro-position 4.15 and Corollary 6.13.(d) ⇒ (e) is trivial, by Definition 3.24. (e) ⇒ (a) follows from Corollary 6.13. (a) ⇒ (d)follows from (a) ⇒ (c) of Theorem 4.17. 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E-mail address : [email protected] Enrico Le Donne: Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo5, 56127 Pisa, Italy, &, University of Jyväskylä, Department of Mathematics and Statistics,P.O. Box (MaD), FI-40014, Finland E-mail address : [email protected]@unipi.it only depends on the topological dimensionof F . Second step.