Distributional solutions of Burgers' type equations for intrinsic graphs in Carnot groups of step 2
aa r X i v : . [ m a t h . M G ] A ug DISTRIBUTIONAL SOLUTIONS OF BURGERS’ TYPE EQUATIONSFOR INTRINSIC GRAPHS IN CARNOT GROUPS OF STEP 2
GIOACCHINO ANTONELLI, DANIELA DI DONATO AND SEBASTIANO DON
Abstract.
We prove that in arbitrary Carnot groups G of step 2, with a splitting G = W · L with L one-dimensional, the graph of a continuous function ϕ : U ⊆ W → L is C -regularprecisely when ϕ satisfies, in the distributional sense, a Burgers’ type system D ϕ ϕ = ω ,with a continuous ω . We stress that this equivalence does not hold already in the easieststep-3 Carnot group, namely the Engel group.As a tool for the proof we show that a continuous distributional solution ϕ to a Burgers’type system D ϕ ϕ = ω , with ω continuous, is actually a broad solution to D ϕ ϕ = ω . As aby-product of independent interest we obtain that all the continuous distributional solutionsto D ϕ ϕ = ω , with ω continuous, enjoy / -little Hölder regularity along vertical directions. Contents
1. Introduction 12. Preliminaries 52.1. Carnot groups 52.2. Carnot groups of step 2 G in exponential coordinates. 62.3. Free Carnot groups of step 2 F in exponential coordinates. 72.4. Projection from F to G . 82.5. Projected vector fields in Carnot groups of step 2 82.6. Invariance properties of projected vector fields 102.7. Broad, broad* and distributional solutions of D ϕ ϕ = ω Introduction
Due to the multitude of applications, sub-Riemannian geometry has attracted a lot ofattention in the mathematical community in the recent years. A sub-Riemannian manifoldis a generalization of Riemannian manifold for which the metric is induced by a smooth scalar
Date : 4th August 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Carnot groups, step-2 Carnot groups, intrinsically C -surfaces, broad solutions,Burgers’ equation, distributional solutions to non-linear first order PDEs.D.D.D., S.D. are partially supported by the Academy of Finland (grant 288501 ‘ Geometry of subRieman-nian groups ’ and by grant 322898 ‘
Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory ’).G.A., D.D.D., S.D. are partially supported by the European Research Council (ERC Starting Grant 713998GeoMeG ‘
Geometry of Metric Groups ’). roduct only defined on a sub-bundle of the tangent bundle. The infinitesimal model of asub-Riemannian manifold, namely the class of its Gromov-Hausdorff tangents, is representedby the class of (quotients of) Carnot groups [SC16, LD17]. Carnot groups are connectedand simply connected Lie groups G whose Lie algebra g admits a stratification, namely adecomposition into nontrivial complementary linear subspaces V , . . . , V s such that g = V ⊕ . . . ⊕ V s , [ V j , V ] = V j +1 , for j = 1 , . . . , s − , [ V s , V ] = { } , where [ V j , V ] denotes the subspace of g generated by the commutators [ X, Y ] with X ∈ V j and Y ∈ V . Carnot groups have been studied from very different point of views such asDifferential Geometry [CDPT07], Subelliptic Differential Equations [BLU07, Fol73, Fol75,SC84], Complex Analysis [SS03] and Neuroimaging [CMS04].Concerning Geometric Measure Theory in the setting of Carnot groups, one of the moststudied problems in the past twenty years is represented by the rectifiability problem: is itpossible to cover the boundary of a finite perimeter set with a countable union of C -regularsurfaces? The answer to this question is affirmative in the Euclidean case and it was studiedin [DG54, DG55] via a blow-up analysis. The proof of De Giorgi has then been adapted inthe framework of step-2 Carnot groups in [FSSC01, FSSC03] and then generalized to theso-called Carnot groups of type ⋆ in [Mar14], see also the recent [LDM20]. When dealingwith Carnot groups of step 3 or higher, only partial results concerning this question areavailable in the literature. One of the main difficulty is represented by the fact that it is notknown in general if C rectifiability is equivalent to a Lipschitz-type rectifiability. ConcerningHeisenberg groups, see [Vit20] for a Rademacher-type theorem for intrinsic Lipschitz graphsof any codimension. Different notions of rectifiability have also been recently investigated,see [ALD20, DLDMV19].The rectifiability problem represents an example that underlines the importance of a fineunderstanding of intrinsic surfaces inside Carnot groups. The study of different notionsof surfaces in Carnot groups has been quite extensive in the recent years and we mention[FSSC07] for a definition of regular submanifold in the Heisenberg groups, [FMS14, FS16]for intrinsic Lipschitz graphs and their connection to C -hypersurfaces, [Mag19] for a notionof non-horizontal transversal submanifold and [Mag13, JNGV20] for a notion of C -surfacewith Carnot group target, but the list is far from being complete.We focus our attention on codimension-one intrinsic graphs. A codimension-one intrinsicgraph Γ inside a Carnot group G comes with a couple of homogeneous and complementarysubgroups W and L with L one-dimensional, see Section 2, and a map ϕ : U ⊆ W → L such that Γ = { x ∈ G : x = w · ϕ ( w ) , w ∈ U } . It turns out that the regularity of thegraph Γ is strictly related to the regularity of ϕ and its intrinsic gradient ∇ ϕ ϕ , see Section 2.As a geometric pointwise approach, we just say that ϕ is intrinsically differentiable if itsgraph has a homogeneous subgroup as blow-up. However, one can define some differentnotions of regularity that rely on some ϕ -dependent operators D ϕW whenever W ∈ Lie( W ) ,see Definition 2.6. If an adapted basis of the Lie algebra ( X , . . . , X n ) is fixed and is suchthat L := exp(span { X } ) and W := exp(span { X , . . . , X n } ) , then D ϕ is the vector valuedoperator ( D ϕX , . . . , D ϕX m ) =: ( D ϕ , . . . , D ϕm ) . The regularity of Γ is related to the validity ofthe equation D ϕ ϕ = ω in an open subset U ⊆ W , for some ω : U → R m − , which can beunderstood in different ways. We briefly present some of them here. istributional sense . Since L is one-dimensional, D ϕ ϕ is a well-defined distribution,see the last part of Definition 2.13. Thus we could interpret D ϕ ϕ = ω in the distri-butional sense. Broad* sense . For every j = 2 , . . . , m and every point a ∈ U , there exists a C integralcurve of D ϕX j starting from a for which the Fundamental Theorem of Calculus withderivative ω holds, see Definition 2.13. Broad sense . For every j = 2 , . . . , m and every point a ∈ U , all the integral curvesof D ϕX j starting from a are such that the Fundamental Theorem of Calculus withderivative ω holds, see Definition 2.13. Approximate sense . For every a ∈ U , there exist δ > and a family { ϕ ε ∈ C ( B ( a, δ )) : ε ∈ (0 , } such that ϕ ε → ϕ and D ϕ ε j ϕ ε → ω j uniformly on B ( a, δ ) as ε goes to zero.When G has step 2 and L is one-dimensional, the following theorem holds, see [ADDDLD20,Theorem 6.17] for a proof and [ADDDLD20, Theorem 1.7] for an equivalent and coordinate-independent statement. Notice that the statement of the result below needs a choice ofcoordinates as explained in Section 2.2, see also (5). We also refer the reader to the pre-liminary section of [ADDDLD20] for the notion used in the statement below that are nottreated in the current paper. Theorem 1.1 ([ADDDLD20, Theorem 6.17]) . Let G be a Carnot group of step 2 and rank m , and let W and L be two complementary subgroups of G , with L horizontal and one-dimensional. Let U ⊆ W be an open set, and let ϕ : U → L be a continuous function. Thenthe following conditions are equivalent (a) graph( ϕ ) is a C -hypersurface with tangents complemented by L ; (b) ϕ is uniformly intrinsically differentiable on U ; (c) ϕ is intrinsically differentiable on U and its intrinsic gradient is continuous; (d) 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 ; (e) there exists ω ∈ C ( U ; R m − ) such that D ϕ ϕ = ω in the broad sense on U ; (f) there exists ω ∈ C ( U ; R m − ) such that D ϕ ϕ = ω in the broad* sense on U .Moreover if any of the previous holds, ω is the intrinsic gradient of ϕ . The main result of the current paper is given by the following implication(1) D ϕ ϕ = ω in the sense of distributions ⇒ D ϕ ϕ = ω in the broad* sense , in every Carnot group G of step 2 and for every continuous ϕ : U ⊆ W → L , with U open,and ω ∈ C ( U ; R m − ) with L one-dimensional, see Theorem 4.1. This result allows us toimprove Theorem 1.1 adding a seventh equivalent condition to the list above :(g) there exists ω ∈ C ( U ; R m − ) such that D ϕ ϕ = ω holds in the distributional sense on U . To complete the chain of implication one also needs (a) ⇒ (g) and this follows from [ADDDLD20,Proposition 4.10]. tem (g) allows us to complete the chain of implications of Theorem 1.1 in the setting ofstep-2 Carnot groups generalizing the results scattered in [ASCV06, BSC10a, BSC10b] wherethe authors study the same problem in the Heisenberg groups, and [DD20a, DD20b] wherepartial generalizations of the results in [ASCV06, BSC10a, BSC10b] are obtained in the caseof step-2 Carnot groups.The strategy of the proof of (1) goes as follows. Given a Carnot group G of step 2,we consider the free Carnot group F with step 2 and the same rank of G , see Section 2.3for the precise choice of identifications. We show in Proposition 3.3 that if D ϕ ϕ = ω indistributional sense inside G with some continuous ω ∈ C ( U ; R m − ) , then also D ψ ψ = ω ◦ π in distributional sense in F , where ψ := π − ◦ ϕ ◦ π , and π : F → G is the projection. Then,we prove Proposition 3.2 that tells us that D ψ ψ = ω in distributional sense in F with ω ∈ C ( U ; R m − ) implies that D ψ ψ = ω in the broad* sense, which is exactly implication(1) in the setting of free Carnot groups of step 2. Finally, we prove in Proposition 3.4 that D ψ ψ = ω ◦ π in the broad* sense in F implies D ϕ ϕ = ω in the broad* sense in G . Theglobal strategy of lifting the problem to the free Carnot groups resembles the one used in[ADDDLD20, Section 6] and [LDPS19].The main difficulty arises in the proof of Proposition 3.2 where we have to combine thedimensional reduction given by Lemma 3.1 and the translation invariance of Proposition 2.12to reduce ourselves to the Burgers’ equation of the first Heisenberg group, and then applythe arguments used for this case in [Daf06, Eqq. (3.4) and (3.5)] and [BSC10a, Step 1, proofof Theorem 1.2]. We point out that this argument is essentially different by the one used in[BSC10a]. One of the reasons for this is that the distributional equation D ϕ ϕ = ω in arbitraryCarnot groups of step 2 has a significantly different structure compared to the one in theHeisenberg groups. For example, consider a Carnot group of dimension 5, step 2 and rank 3with Lie algebra g = span { X , X , X , X , X } , horizontal layer V := span { X , X , X } andwhere the only nonvanishing commutators are given by [ X , X ] = X + X and [ X , X ] = X − X . Define, in exponential coordinates, W := { x = 0 } and L := { x = x = x = x =0 } . Then, given a continuous ϕ : U ⊆ W → L on an open set U , the operators D ϕj := D ϕX j for j = 2 , have the following form (see [ADDDLD20, Example 3.6]) D ϕ = ∂ + ϕ∂ + ϕ∂ ,D ϕ = ∂ + ϕ∂ − ϕ∂ , which show a nonlinearity in two vertical directions, instead of only one as in the Heisenberggroups. We remark that Proposition 3.2 and Theorem 4.1 have also an interesting PDE point ofview which allows to see the problem independently of the Carnot group structure. Indeed,the result can be read to obtain the following regularity result. Assume that the Burgers’type system D ϕ ϕ = ω holds in the distributional sense for a continuous map ϕ and with thecontinuous datum ω . Then, from each single equation of the system, we infer the followingproperty: for every j = 2 , . . . , m , ϕ is (uniformly) Lipschitz continuous on all the integralcurves of the operator D ϕj . In addition, the Fundamental Theorem of Calculus with derivative ω holds on some particular local family of integral curves of D ϕj , namely the broad* conditionholds, and then also the broad condition holds, see (f) ⇒ (e) of Theorem 4.1. Moreover, when Clearly this double nonlinearity can be removed by considering the Lie algebra automorphism such that Ψ( X ) = X , Ψ( X ) = X + X , Ψ( X ) = X − X . This is basically our idea of properly lifting step-2Carnot groups to free Carnot groups with the same rank. e consider all the equations together, we obtain a remarkable piece of information: ϕ is / -little Hölder continuous on the vertical coordinates, see Theorem 4.7.We remark that Theorem 1.1 complemented with (g) is optimal in step-2 Carnot groupsfor the following reason. Already in the Engel group, which is the easiest step-3 Carnotgroup, we can find a continuous map ϕ that solves D ϕ ϕ = ω in the sense of distributionsfor a constant ω whose graph is not uniformly intrinsically differentiable (UID). We howevernotice that we do not know at present if implication (1) holds in Carnot groups of higherstep, see Remark 4.5.We briefly describe the situation in which ω is less regular. In the paper [BCSC15], theauthors show that, in Heisenberg groups, D ϕ ϕ = ω holds in the sense of distributions forsome ω ∈ L ∞ ( U ; R m − ) if and only if ϕ is intrinsically Lipschitz. The validity of (1) with ω ∈ L ∞ ( U ; R m − ) in the setting of step-2 Carnot groups would open to a slightly modifiedversion of Theorem 1.1 where ω ∈ L ∞ ( U ; R m − ) and (a) is replaced by(a’) graph ( ϕ ) is intrinsically Lipschitz for the splitting given by W and L .Indeed, having D ϕ ϕ = ω in the broad* sense with ω ∈ L ∞ ( U ; R m − ) would imply that ϕ is / -Hölder continuous along vertical directions. This topic is out of the aims of this paperand will be target of future investigations.We notice here that if a generalization of the a priori estimate [MV12, Lemma 3.1] wouldhold in any step-2 Carnot group, then we could improve Theorem 1.1 replacing (d) with(d’) There exists ω ∈ C ( U ; R m − ) and a family of functions { ϕ ε ∈ C ( U ) : ε ∈ (0 , } such that, for every compact set K ⊆ U and every j = 2 , . . . , m , one has lim ε → ϕ ε = ϕ and lim ε → D ϕ ε j ϕ ε = ω j in L ∞ ( K ) . We refer the reader to [ADDDLD20, Remark 4.14] for a discussion of the literature and ofthe difference between item (d) and item (d’). We also remark that a smooth approximationthat does not involve the intrinsic gradient holds in any Carnot group for intrinsic Lipschitzgraphs, see [Vit20, Theorem 1.6].Intrinsic surfaces of higher codimensions have been studied in the Heisenberg groups in[Cor20, CM20]. For what concerns the approach via distributional solutions, finding a mean-ing of the distributional system D ϕ ϕ = ω in higher codimension is still open. The maindifficulty comes from the fact that it is not known how to give meaning to mixed terms ofthe form ϕ i ∂ x ϕ j . This was already noticed in [Koz15, Remark 4.3.2]. A weak formulationthat goes in this direction is collected in [MST18], where the authors relate zero-level sets ofmaps in C ,α H ( H ; R ) with curves that satisfies certain “Level Set Differential Equations”, see[MST18, Theorem 5.6]. 2. Preliminaries
Carnot groups.
We give a very brief introduction on Carnot groups. We refer thereader to e.g. [BLU07, SC16, LD17] for a comprehensive introduction to Carnot groups. ACarnot group G is a connected and simply connected Lie group, whose Lie algebra g isstratified. Namely, there exist subspaces V , . . . , V s of the Lie algebra g such that g = V ⊕ . . . ⊕ V s , [ V j , V ] = V j +1 ∀ j = 1 , . . . , s − , [ V s , V ] = { } . The integer s is called step of the group G , while m := dim( V ) is called rank of G . We set n := dim( G ) to be the topological dimension of G . We equivalently denote by e or theidentity element of the group G . very Carnot group has a one-parameter family of dilations that we denote by { δ λ : λ > } defined as the unique linear maps on g such that δ λ ( X ) = λ j X , for every X ∈ V j . We denoteby δ λ both the dilations on G and on g , with the usual identification given by the exponentialmap exp: g → G which is a diffeomorphism. We fix a homogeneous norm k·k on G , namelysuch that k δ λ x k = λ k x k for every λ > and x ∈ G , k xy k≤ k x k + k y k for every x, y ∈ G , k x k = k x − k for every x ∈ G , and k x k = 0 if and only if x = e . The norm k·k induces aleft-invariant homogeneous distance and we denote with B ( a, r ) the open ball of center a andradius r > according to this distance. We stress that on a Carnot group a homogeneousnorm always exists, and every two left-invariant homogeneous distances are bi-Lipschitzequivalent. Definition 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 inde-pendent X , . . . , X k ∈ V such that L = exp(span { X , . . . , X k } ) . Given two complementarysubgroups W and L , 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(2) g = ( π W g ) · ( π L g ) = g W · g L . 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. Notice also that, if W and L are complementary subgroups and L is horizontal, then W is a normal subgroup of G . Definition 2.3 (Adapted basis) . Denote by n j := P ji =1 dim( V i ) , for j = 1 , . . . , s and n := 0 .We say that a basis ( X , . . . , X n ) of g is adapted if the following facts hold • For every j = 1 , . . . , s , the set { X n j − +1 , . . . , X n j } is a basis for V j . • For any j = 1 , . . . , s , the vectors X n j − +1 , . . . , X n j are chosen among the iteratedcommutators of length j − of the vectors X , . . . , X m . Definition 2.4 (Exponential coordinates) . Let G be a Carnot group of dimension n and let ( X , . . . , X n ) be an adapted basis of its Lie algebra. The exponential coordinates of the firstkind associated with ( X , . . . , X n ) are given by the one-to-one correspondence R n ↔ G ( x , . . . , x n ) ↔ exp ( x X + . . . + x n X n ) . It is well known that this defines a diffeomorphism from R n to G that allows us to identify G with R n .2.2. Carnot groups of step 2 G in exponential coordinates. We here introduce Carnotgroups of step 2 in exponential coordinates. We adopt as a general reference [BLU07,Chapter 3], but the interested reader could also read the beginning of [ADDDLD20, Subsec-tion 6.2]. In this subsection G will always be an arbitrary Carnot group of step 2.We denote with m the rank of G and we identify G with ( R m + h , · ) by means of exponentialcoordinates associated with an adapted basis ( X ′ , . . . , X ′ m , Y ′ , . . . , Y ′ h ) of the Lie algebra g . n this coordinates, we will identify any point q ∈ G with q ≡ ( x , . . . , x m , y ∗ , . . . , y ∗ h ) . Thegroup operation · between two elements q = ( x, y ∗ ) and q ′ = ( x ′ , ( y ∗ ) ′ ) is given by(3) 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 . For any i = 1 , . . . , h and any j, ℓ = 1 , . . . , m ,we set ( B ( i ) ) jℓ =: ( b ( i ) jℓ ) , and it is standard to observe that we can write 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. We stress that the operation in (3) is precisely the one obtained by means of the Baker-Campbell-Hausdorff formula in exponential coordinates of the first kind associated with theadapted basis ( X ′ , . . . , X ′ m , Y ′ , . . . , Y ′ h ) . We also stress that(4) [ 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, so that it is clear that b ( i ) jℓ , with i = 1 , . . . , h , and ≤ j, ℓ ≤ m , are the so-called structurecoefficients .In the sequel we denote by W G and L G two arbitrary complementary subgroups of G with L G horizontal and one-dimensional. Up to choosing a proper adapted basis of the Lie algebra g , we may suppose that L G = exp(span { X } ) . Thus, by means of exponential coordinateswe can identify W G and L G with R m + h − and R , respectively, as follows(5) 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 } . Free Carnot groups of step 2 F in exponential coordinates. We here introducefree Carnot groups of step 2 in exponential coordinates. We adopt as a general reference[BLU07, Chapter 3], but the interested reader could also read the beginning of [ADDDLD20,Subsection 6.1]. In this subsection F will always denote a free Carnot group of step 2and rank m . Recall that the topological dimension of F is m + m ( m − and denote by ( X , . . . , X m , Y , . . . , Y m ( m − ) an adapted basis of the Lie algebra of F such that [ X ℓ , X s ] = Y ℓs for every ≤ s < ℓ ≤ m .If we set n := m + m ( m − , we can identify F with R n by means of exponential coordinatesassociated with the adapted basis ( X , . . . , X m , Y , . . . , Y m ( m − ) . In this coordinates, wewill identify any point q ∈ F with q ≡ ( x , . . . , x m , y , . . . , y m ( m − ) . It is readily seen that,in such coordinates, we have(6) X j = ∂ x j + 12 X j<ℓ ≤ m x ℓ ∂ y ℓj − X ≤ ℓ We recall here the definitionof projected vector fields [ADDDLD20, Definition 3.1]. Definition 2.6 (Projected vector fields) . Given two complementary subgroups W and L ina Carnot group G , and a continuous function ϕ : U ⊆ W → L defined on an open set U of W , we define, for every W ∈ Lie( W ) , the continuous projected vector field D ϕW , by setting(11) ( D ϕW ) | w ( f ) := W | w · ϕ ( w ) ( f ◦ π W ) , or all w ∈ U and all f ∈ C ∞ ( W ) . When W is an element X j of an adapted basis ( X , . . . , X n ) we also write D ϕj := D ϕX j .Let us fix G a Carnot group of step 2 and rank m along with two complementary subgroups W G and L G such that L G is horizontal and one-dimensional. Assume we have chosen abasis in such a way that (5) is satisfied. Take F the free step-2 Carnot group of rank m and introduce W F and L F as in Remark 2.5. In this subsection we work in exponentialcoordinates and we use the identifications and the coordinate representations discussed inSection 2.2, Section 2.3, and Remark 2.5. We recall that from [ADDDLD20, Example 3.6 &Remark 6.9] the projected vector fields relative to a continuous function ϕ : U ⊆ W G → L G ,with U open, are given by(12) 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. In addition, in the particular case of the free Carnot group F , given V ⊆ W F an open set,and given a continuous map ψ : V ⊆ W F → L F , the projected vector fields are given by(13) D ψj = ∂ x j − ψ∂ y j + 12 X j<ℓ ≤ m x ℓ ∂ y ℓj − X 0) = ( x , . . . , x m , y ∗ , . . . , y ∗ h ) . or every q ∈ G , we define the map(16) P q : W G → W G w π W G ( q · w ) . Set q = ( q , . . . , q m , q m +1 , . . . , q m + h ) ∈ G and w = ( w := 0 , w , . . . , w m , w m +1 , . . . , w m + h ) ∈ W G . By using (15) and (3), one has, being i = 1 , . . . , h , that the following chain of equalitiesholds(17) P q ( w ) = π W G ( q · w )= (cid:16) , q + w , . . . , q m + w m , . . . , q m + i + w m + i + 12 m X j =1 m X ℓ =2 b ( i ) jℓ q j w ℓ − m X j =1 b ( i ) j ( q j + w j ) q , . . . (cid:17) = (cid:16) , q + w , . . . , q m + w m , . . . ,. . . , q m + i + w m + i + 12 m X j =1 m X ℓ =2 b ( i ) jℓ q j w ℓ + 12 m X ℓ =2 b ( i )1 ℓ w ℓ q − m X ℓ =2 b ( i ) ℓ q ℓ q , . . . (cid:17) = (cid:16) , q + w , . . . , q m + w m , . . . ,. . . , q m + i + w m + i + 12 m X ℓ =2 w ℓ m X j =1 b ( i ) jℓ q j + b ( i )1 ℓ q ! − m X ℓ =2 b ( i ) ℓ q ℓ q , . . . (cid:17) , where we used the fact that the first component of w is zero and that B ( i ) is skew-symmetricand therefore b ( i )11 = 0 . If we see P q as a map from R m + h − to R m + h − , the differential of P q at a point w ∈ W is identified with a ( m + h − × ( m + h − matrix with the followingcomponents(18) (d P q )( w ) ii = 1 , ∀ i = 1 , . . . , m + h − , (d P q )( w ) m + i − ,ℓ − = m X j =2 b ( i ) jℓ q j + b ( i )1 ℓ q ! , ∀ i = 1 , . . . , h ; ℓ = 2 , . . . , m, (d P q )( w ) jℓ = 0 , otherwise.In particular, det(d P q )( w ) = 1 for any w ∈ W .2.6. Invariance properties of projected vector fields. We collect here some invarianceproperties that we will use later on. We introduce the operation of q -translation of a function. Definition 2.8 (Intrinsic graph of a function) . Given two complementary subgroups W and L of a Carnot group G , and a function ϕ : U ⊆ W → L , we define the graph of ϕ by setting graph( ϕ ) := { Φ( w ) := w · ϕ ( w ) : w ∈ U } = Φ( U ) . Definition 2.9 (Intrinsic translation of a function) . Given two complementary subgroups W and L of a Carnot group G and a map ϕ : U ⊆ W → L , we define, for every q ∈ G , U q := { a ∈ W : π W ( q − · a ) ∈ U } , and ϕ q : U q ⊆ W → L by setting(19) ϕ q ( a ) := (cid:0) π L ( q − · a ) (cid:1) − · ϕ (cid:0) π W ( q − · a ) (cid:1) . otice that U q = P q ( U ) , where P q is defined as in (16). This easily comes from the fact thatfor every q ∈ G P q ◦ P q − = Id W , see e.g. the proof of Proposition 2.12.The following results can be found in [ADDDLD20, Proposition 2.10] and [ADDDLD20,Lemma 3.13, and Equations (45)-(46)], respectively. Proposition 2.10 ([ADDDLD20, Proposition 2.10]) . Let W and L be two complementarysubgroups of a Carnot group G and let ϕ : U ⊆ W → L be a function. Then, for every q ∈ G ,the following facts hold. (a) graph( ϕ q ) = q · graph( ϕ ) ; (b) ( ϕ q ) q − = ϕ ; (c) If W is normal, then U q = q W · ( q L · U · ( q L ) − ) and ϕ q ( a ) = q L · ϕ (( q L ) − · q − W · a · q L ) , for any a ∈ U q ; (d) If q = ϕ ( a ) − · a − for some a ∈ U , then ϕ q ( e ) = e. Lemma 2.11 ([ADDDLD20, Lemma 3.13, and Equations (45)-(46)]) . Let W and L be twocomplementary subgroups of a Carnot group G , with L k -dimensional and horizontal and let ϕ : U ⊆ W → L be a continuous function defined on U open. Take W ∈ Lie( W ) , and let usdenote D ϕ := D ϕW . Let T > , w ∈ W , and let γ : [0 , T ] → U be a C -regular solution of theCauchy problem (20) ( γ ′ ( t ) = D ϕ ◦ γ ( t ) ,γ (0) = w. Then for every q ∈ G there exists a unique C map γ q : [0 , T ] → U q such that (21) π W ( q − · γ q ( t )) = γ ( t ) , ∀ t ∈ [0 , T ] . In addition, γ q is a solution of the Cauchy problem (22) ( γ ′ q ( t ) = D ϕ q ◦ γ q ( t ) ,γ q (0) = q W · q L · w · ( q L ) − . Moreover the following equality holds (23) ϕ q ( γ q (0)) − · ϕ q ( γ q ( t )) = ϕ ( γ (0)) − · ϕ ( γ ( t )) , ∀ t ∈ [0 , T ] . In the following proposition we prove the invariance of being a distributional solution withrespect to q -translation. Proposition 2.12. Let W and L be two complementary subgroups of a step-2 Carnot group G with L one-dimensional. Let Ω be an open set in W and let ω ∈ L (Ω) . Let us choosecoordinates on G as explained in Section 2.2, see also (5) . Assume that for some ℓ = 2 , . . . , m the map ϕ : U → L is a distributional solution of the equation D ϕℓ ϕ = ω on U . Then, forevery q ∈ G , the map ϕ q defined in Definition 2.9 is a distributional solution of D ϕ q ℓ ϕ q = ω ◦ P q − , on the open set P q (Ω) . roof. By item (c) of Proposition 2.10, we know that in exponential coordinates ϕ q ( w ) = q + ϕ ( P q − ( w )) , where P q − is defined in (16). Indeed, since W is normal, the followingequality holds(24) P q − ( w ) = π W ( q − · w ) = π W (( q L ) − · ( q W ) − · w · q L · ( q L ) − ) = q − · w · q L . Moreover we claim P q − = P − q , for all q ∈ G . Indeed, since W is normal, the followingequality holds(25) P q ( w ) = π W ( q · w ) = π W ( q · w · q − · q W · q L ) = q · w · q − · q W = q · w · ( q L ) − , and hence it is clear by comparing (24) and (25) that P q ◦ P q − = P q − ◦ P q = Id W , for all q ∈ G . Moreover notice that from item (c) of Proposition 2.10 and (25) we get that(26) P q (Ω) = Ω q , for all q ∈ G . For every ξ ∈ C ∞ c ( P q (Ω)) , using (12) we can write the action of the distribution D ϕ q ℓ ϕ q on ξ , where we mean that the coordinates are w = ( x , . . . , x m , y ∗ , . . . , y ∗ h ) ∈ W , as follows(27) h D ϕ q ℓ ϕ q , ξ i = Z P q (Ω) − ϕ q ∂ξ∂x ℓ + 12 h X i =1 b ( i ) ℓ ϕ q ∂ξ∂y ∗ i + 12 h X i =1 m X j =2 x j b ( i ) ℓj ϕ q ∂ξ∂y ∗ i ! d w = Z P q (Ω) − ( q + ϕ ◦ P q − ) ∂ξ∂x ℓ + 12 h X i =1 b ( i ) ℓ ( q + ϕ ◦ P q − ) ∂ξ∂y ∗ i ! d w + Z P q (Ω) h X i =1 m X j =2 x j b ( i ) ℓj ( q + ϕ ◦ P q − ) ∂ξ∂y ∗ n + i ! d w = Z P q (Ω) − ϕ ◦ P q − ∂ξ∂x ℓ + h X i =1 b ( i ) ℓ (cid:20) 12 ( ϕ ◦ P q − ) + q ϕ ◦ P q − (cid:21) ∂ξ∂y ∗ i ! d w + Z P q (Ω) h X i =1 m X j =2 x j b ( i ) ℓj ( ϕ ◦ P q − ) ∂ξ∂y ∗ i ! d w, where in the third equality we used the fact that ξ has compact support in P q (Ω) and q doesnot depend on w . Taking into account that, by Remark 2.7, det(d P q ) = 1 everywhere on W ,we perform in (27) the change of variable w ′ = P q − ( w ) . Thus, recalling that P q ◦ P q − = Id W ,and by exploiting (17), we obtain the following equality(28) h D ϕ q ℓ ϕ q , ξ i = Z Ω − ϕ ∂ξ∂x ℓ ◦ P q + h X i =1 b ( i ) ℓ (cid:20) ϕ + q ϕ (cid:21) ∂ξ∂y ∗ i ◦ P q ! d w + Z Ω h X i =1 m X j =2 ( x j + q j ) b ( i ) ℓj ϕ ∂ξ∂y ∗ i ◦ P q ! d w. We can now use (18) to compute the derivatives of ξ ◦ P q as follows(29) ∂∂x ℓ ( ξ ◦ P q ) = ∂ξ∂x ℓ ◦ P q + h X i =1 m X j =2 b ( i ) jℓ q j + b ( i )1 ℓ q ! ∂ξ∂y ∗ i ◦ P q , ∀ ℓ = 2 , . . . , m,∂∂y ∗ i ( ξ ◦ P q ) = ∂ξ∂y ∗ i ◦ P q , ∀ i = 1 , . . . , h. y using (29) into (28) we get h D ϕ q ℓ ϕ q , ξ i = Z Ω − ϕ ∂ ( ξ ◦ P q ) ∂x ℓ − h X i =1 m X j =2 b ( i ) jℓ q j + b ( i )1 ℓ q ! ∂ ( ξ ◦ P q ) ∂y ∗ i ! d w + Z Ω h X i =1 b ( i ) ℓ (cid:20) ϕ + q ϕ (cid:21) + 12 m X j =2 ( x j + q j ) b ( i ) ℓj ϕ ! ∂ ( ξ ◦ P q ) ∂y ∗ i d w = Z Ω − ϕ ∂ ( ξ ◦ P q ) ∂x ℓ + 12 h X i =1 b ( i ) ℓ ϕ ∂ ( ξ ◦ P q ) ∂y ∗ i + 12 h X i =1 m X j =2 x j b ( i ) ℓj ϕ ∂ ( ξ ◦ P q ) ∂y ∗ i ! d w = h D ϕℓ ϕ, ξ ◦ P q i , where, in order to write the second equality, we used that the matrices B are skew-symmetric.Now, exploiting the last identity, the assumption and the fact that for every ξ ∈ C ∞ c ( P q (Ω)) we have ξ ◦ P q ∈ C ∞ c (Ω) , we conclude h D ϕ q ℓ ϕ q , ξ i = h D ϕℓ ϕ, ξ ◦ P q i = Z Ω ω ( ξ ◦ P q ) d w = Z P q (Ω) ( ω ◦ P q − ) ξ d w, where in the last equality we changed the variables and exploited the fact that det(d P ( q )) =1 . By the arbitrariness of ξ ∈ C ∞ c ( P q ( ω )) , the proof is complete. (cid:3) Broad, broad* and distributional solutions of D ϕ ϕ = ω . We recall the followingdefinition as a particular case of [ADDDLD20, Section 3.4, Definition 3.24]. For discussionsabout the dependence of the definition of broad* regularity on the chosen adapted basis, werefer the reader to [ADDDLD20, Remark 3.26 & Remark 4.4]. Definition 2.13 (Broad*, broad and distributional solutions) . Let W and L be comple-mentary subgroups of a Carnot group G , with L one-dimensional. Let U ⊆ W be openand let ϕ : U → L be a continuous function. Consider an adapted basis ( X , . . . , X n ) ofthe Lie algebra of G such that L = exp(span { X } ) and W = exp(span { X , . . . , X n } ) . Let ω := ( ω j ) j =2 ,...,m : U → R m − be a continuous function. Up to identifying L with R by meansof exponential coordinates, we say that ϕ ∈ C ( U ) is a broad* solution of D ϕ ϕ = ω in U if for every a ∈ U there exist < δ < δ such that B ( a , δ ) ∩ W ⊆ U and there exist m − maps E ϕj : ( B ( a , δ ) ∩ W ) × [ − δ , δ ] → B ( a , δ ) ∩ W for j = 2 , . . . , m , satisfying thefollowing two properties.(a) For every a ∈ B ( a , δ ) ∩ W and every j = 2 , . . . , m , the map E ϕj ( a ) := E ϕj ( a, · ) is C -regular and it is a solution of the Cauchy problem ( ˙ γ = D ϕj ◦ γ,γ (0) = a, in the interval [ − δ , δ ] , where the vector field D ϕj := D ϕX j is defined in (11).(b) For every a ∈ B ( a , δ ) ∩ W , for every t ∈ [ − δ , δ ] , and every j = 2 , . . . , m one has ϕ ( E ϕj ( a, t )) − ϕ ( a ) = Z t ω j ( E ϕj ( a, s )) d s. oreover, we say that D ϕ ϕ = ω in the broad sense on U if for every W ∈ Lie( W ) ∩ V andevery γ : I → U integral curve of D ϕW , it holds that dd s | s = t ( ϕ ◦ γ )( s ) = h ω ( γ ( t )) , W i , ∀ t ∈ I, where by h ω, W i is the standard scalar product on R m − in exponential coordinates.Finally, let us notice that, for every j = 2 , . . . m , D ϕj is a continuous vector field with coef-ficients that might depend polinomially on ϕ and on some of the coordinates, see Section 2.5for the case in which G is of step 2 and [ADDDLD20, Proposition 3.9] for the general case.We say that D ϕ ϕ = ω holds in the distributional sense on U if for every j = 2 , . . . , m onehas D ϕj ϕ = ω j in the distributional sense. Notice that the distribution ( D ϕj ) ϕ is well definedsince the coefficients of D ϕj just contain polynomial terms in ϕ and terms depending on thecoordinates, see also [ADDDLD20, Item (a) of Proposition 4.10].3. Reduction of the main theorems to free Carnot groups of step 2 In this section we analyze the link between distributional and broad* solutions to D ϕ ϕ = ω with a continuous datum ω . We first show that a distributional solution of D ϕ ϕ = ω with a continuous datum ω is a broad* solution inside free Carnot groups F of step 2, seeProposition 3.2 (b). In this proof, a crucial role is played by the particular structure ofthe projected vector fields inside free Carnot groups of step 2, which produces Burgers’type operators in higher dimensions, see (13). Indeed, combining the invariance result inProposition 2.12 and the dimensional reduction of Lemma 3.1, we can reduce ourselves todeal with Burgers’ distributional equation with continuous datum on the first Heisenberggroup H , and then exploit the arguments used by Dafermos in [Daf06] and by Bigolin andSerra Cassano in [BSC10a].Secondly, by the explicit expression of the projection π from F to a Carnot group G of step 2, we prove that being a distributional solution to D ϕ ϕ = ω on G lifts to F , seeProposition 3.3. Finally, Proposition 3.4 states that the notion of broad* solution is preservedby π , i.e., a broad* solution on F becomes a broad* solution on G . The resulting strategyresembles the one used in [LDPS19]. Lemma 3.1. Let n , n ∈ N and let Ω be an open set in R n + n + k . Let f , f , . . . , f n ∈ C (Ω) and assume that, for every ϕ ∈ C ∞ c (Ω) , one has Z Z Z n X i =1 f i ( x, y, z ) ∂ϕ∂x i ( x, y, z ) + f ( x, y, z ) ϕ ( x, y, z ) ! d x d y d z = 0 , where ( x, y, z ) ∈ R n × R n × R k . Then, for every z ∈ R k such that Ω := { ( x, y ) ∈ R n × R n :( x, y, z ) ∈ Ω } is nonempty, and any ϕ ∈ C ∞ c (Ω ) , one has Z Z n X i =1 f i ( x, y, z ) ∂ϕ∂x i ( x, y ) + f ( x, y, z ) ϕ ( x, y ) ! d x d y = 0 . Proof. By translation invariance, we can assume without loss of generality that z = 0 . Upto iterate the argument k times, we can also assume without loss of generality that k = 1 .Fix b ϕ := b ϕ ( x, y ) be such that supt( b ϕ ) ⊆ Ω . Choose ε > small enough and consider, forany ε ∈ (0 , ε ] , the map ϕ ε ( x, y, z ) := ε ϕ ε ( z ) b ϕ ( x, y ) with supt( ϕ ε ) ⊆ [ − ε − ε , ε + ε ] , ϕ ε ≥ nd ϕ ε = 1 on [ − ε, ε ] , and such that supt( b ϕ ) × supt( ϕ ε ) ⊆ Ω . Then, by the hypothesis andFubini’s Theorem we may write(30) ε Z ε + ε − ε − ε ϕ ε ( z ) Z Z n X i =1 f i ( x, y, z ) ∂ b ϕ∂x i ( x, y ) + f ( x, y, z ) b ϕ ( x, y ) d x d y ! d z = 0 . Notice that the function F ( z ) := Z Z n X i =1 f i ( x, y, z ) ∂ b ϕ∂x i ( x, y ) + f ( x, y, z ) b ϕ ( x, y ) d x d y, is continuous on [ − ε − ε , ε + ε ] . We can then decompose the left-hand side of (30) in thefollowing way(31) ε Z ε + ε − ε − ε ϕ ε ( z ) F ( z ) d z = 12 ε Z ε − ε F ( z ) d z + 12 ε Z [ − ε − ε ,ε + ε ] \ [ − ε,ε ] ϕ ε ( z ) F ( z ) d z. Since ϕ ε ≤ , we have (cid:12)(cid:12)(cid:12)(cid:12) ε Z [ − ε − ε ,ε + ε ] \ [ − ε,ε ] ϕ ε ( z ) F ( z ) d z (cid:12)(cid:12)(cid:12)(cid:12) ≤ M ε ε = M ε, where M is the maximum of | F | in [ − ε − ε , ε + ε ] . Letting ε → in (31), the thesisfollows by means of Lebesgue’s Theorem. (cid:3) Proposition 3.2. Let F be a free Carnot group of step 2, rank m and topological dimension n , and let W and L be two complementary subgroups of F such that L is one-dimensional.Let Ω be an open subset of W and ψ : Ω → L be a continuous function. Choose an adaptedbasis and exponential coordinates on F as in Section 2.3, see also (10) . Assume there exists ω ∈ C (Ω; R m − ) such that D ψ ψ = ω holds in the distributional sense on Ω . Then, thefollowing facts hold. (a) For every j = 2 , . . . , m and for every integral curve γ : [0 , T ] → Ω of D ψj , the map ψ ◦ γ : [0 , T ] → L is Lipschitz and the Lipschitz constant only depends on j and ω . (b) D ψ ψ = ω holds in the broad* sense on Ω .Proof. Preliminary dimensional reduction . Fix j = 2 , . . . , m . Assume ∈ Ω , ψ (0) = 0 and D ψj ψ = ω j in the sense of distributions on Ω . Taking (13) into account, this amounts tosaying that(32) Z − ψ∂ x j ϕ + ψ ∂ y j ϕ − X j<ℓ ≤ m x ℓ ψ∂ y ℓj ϕ + 12 X Let G be a Carnot group of step 2, rank m and topological dimension m + h , and let W G and L G be two complementary subgroups of G , with L G one-dimensional.Let F be the free Carnot group of step 2, and rank m , and choose coordinates on G and F asexplained in Remark 2.5. Denote with W F and L F the complementary subgroups of F with L F one-dimensional such that π ( W F ) = W G and π ( L F ) = L G , see (10) . Let U be an openset in W G and denote by V ⊆ W F the open set defined by V := π − ( U ) . Let ϕ : U → L G be acontinuous 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 holds in the distributional sense on U . Then, ψ is a distributional solution to D ψ ψ = ω ◦ π on V .Proof. Fix j = 2 , . . . , m . Let us identify any element in G with ( x, y ∗ ) where x ∈ R m and y ∗ ∈ R h , and let us identify any element in F with ( x, y ) where x ∈ R m and y ∈ R m ( m − / .Then, taking (12) into account, we have that(42) Z U − ϕ ∂ξ∂x j + 12 h X i =1 b ( i ) j ϕ ∂ξ∂y ∗ i + 12 h X i =1 m X ℓ =2 x ℓ b ( i ) jℓ ϕ ∂ξ∂y ∗ i ! d x d y ∗ = Z U ω j ξ d x d y ∗ , for every ξ ∈ C ∞ c ( U ) . Hence, by exploiting (13), we would like to show that(43) h D ψj ψ, e ξ i := Z V − ψ ∂ e ξ∂x j + 12 ψ ∂ e ξ∂y j − X j<ℓ ≤ m x ℓ ψ ∂ e ξ∂y ℓj + 12 X Let G be a Carnot group of step 2, rank m and topological dimension m + h and let W G and L G be two complementary subgroups of G , with L G one-dimensional. Let F bethe free Carnot group of step 2, rank m and topological dimension n , and choose coordinateson G and F as explained in Remark 2.5. Denote by W F and L F the complementary subgroupsof F with L F one-dimensional such that π ( W F ) = W G and π ( L F ) = L G , see (10) . Let U bean open set in W G and denote with 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 ψ ψ = ω ◦ π holds in the broad* sense on V .Then, ϕ is a broad* solution to D ϕ ϕ = ω on U .Proof. In order to give the proof of the statement we first show the following intermediateresult: for every j = 2 , . . . , m , every point q := (0 , x , . . . , x m , y , . . . , y m ( m − ) ∈ V , andevery integral curve γ : [0 , T ] → V of D ψj starting from q we have that π ◦ γ : [0 , T ] → U is anintegral curve of D ϕj starting from π ( q ) =: (0 , x , . . . , x m , y ∗ , . . . , y ∗ h ) , see (8). Moreover westress that from (9) we have y ∗ i = P ≤ s<ℓ ≤ m b ( i ) ℓs y ℓs , for all i = 1 , . . . , h .Take an integral curve γ : [0 , T ] → U of D ψj starting from q . Then, the components of γ satisfy the system of ODEs in (14). From the explicit expression of the projection in (9), wecan write the components of π ◦ γ as a linear combination of the components of γ . Then,exploiting the ODEs in (14), taking the derivatives of those linear expressions, and by usingthe definition of ψ in terms of ϕ in the statement, one simply obtains that π ◦ γ : [0 , T ] → U is an integral curve of D ϕj starting from π ( q ) .In order to conclude, notice that, from the relation between ψ and ϕ in the statement, weobtain the following equivalence(47) ϕ ( π ◦ γ ( t )) − ϕ ( π ◦ γ (0)) = Z t ω j ( π ◦ γ ( s )) d s ⇔ ψ ( γ ( t )) − ψ ( γ (0)) = Z t ( ω j ◦ π )( γ ( s )) d s, for every integral curve γ : [0 , T ] → V of D ψj , with j = 2 , . . . , m , and every t ∈ [0 , T ] . Thus,from the previous observation on the projection of the integral curves and the equivalence(47), we get the thesis by taking the definition of broad* solution in Definition 2.13 intoaccount. (cid:3) . Main theorems We are ready to prove the main theorem of this paper, by making use of the invarianceresults proved in Section 3. The following theorem is a converse of [ADDDLD20, Corollary6.15]. Theorem 4.1. Let G be a Carnot group of step 2 and rank m , and let W and L be twocomplementary subgroups of G , with L horizontal and one-dimensional. Let U ⊆ W be anopen set, and let ϕ : U → L be a continuous function. Choose coordinates on G as explainedin Section 2.2, see also (5) . Assume there exists ω ∈ C ( U ; R m − ) such that D ϕ ϕ = ω holdsin the distributional sense on U . Then D ϕ ϕ = ω holds in the broad* sense on U .Proof. It directly follows by joining together Proposition 3.3, Proposition 3.2, and Proposi-tion 3.4. (cid:3) By making use of the previous theorem and [ADDDLD20, Theorem 6.17] we obtain thefollowing characterization of C -hypersurfaces in Carnot groups of step 2. For the notion ofintrinsic differentiabilty we refer the reader to [ADDDLD20, Definition 2.17], while for thenotion of intrinsic gradient we refer the reader to [ADDDLD20, Definition 2.20 & Remark2.21]. For the definition of C -hypersurface we refer the reader to [FSSC03, Definition 1.6].For a detailed account on this notion we refer the reader to the introduction of [ADDDLD20]and in particular to [ADDDLD20, Definition 2.27] for the definition of co-horizontal C -regular surfaces with complemented tangents. Theorem 4.2. Let G be a Carnot group of step 2 and rank m , and let W and L be twocomplementary subgroups of G , with L horizontal and one-dimensional. Let U ⊆ W be anopen set and let ϕ : U → L be a continuous function. Choose coordinates on G as explainedin Section 2.2, see also (5) . Then the following conditions are equivalent: (a) graph( ϕ ) is a C -hypersurface with tangents complemented by L ; (b) ϕ is uniformly intrinsically differentiable on U ; (c) ϕ is intrinsically differentiable on U and its intrinsic gradient is continuous; (d) 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 ; (e) there exists ω ∈ C ( U ; R m − ) such that D ϕ ϕ = ω holds in the broad sense on U ; (f) there exists ω ∈ C ( U ; R m − ) such that D ϕ ϕ = ω holds in the broad* sense on U ; (g) there exists ω ∈ C ( U ; R m − ) such that D ϕ ϕ = ω holds in the distributional sense on U .Moreover, if any of the previous holds, ω is the intrinsic gradient of ϕ .Proof. The equivalence between (a),(b),(c),(d),(e), and (f) follows form [ADDDLD20, The-orem 6.17]. The implication (g) ⇒ (f) follows from Theorem 4.1. The implication (b) ⇒ (g)follows from [ADDDLD20, Item (c) of Proposition 4.10]. (cid:3) Remark . We stress that if any of the hypotheses inTheorem 4.2 is satisfied, we can write the intrinsic normal to graph( ϕ ) and an area formulafor graph( ϕ ) explicitely in terms of the intrinsic gradient ω , see [ADDDLD20, Item (d) ofProposition 4.10 and Remark 4.11]. emark D ϕ ϕ = ω ) . The approximatingsequence in item (d) of Theorem 4.2 is a priori dependent on the point a ∈ U we choose. Thisis true because in order to obtain [ADDDLD20, Theorem 6.17], from which Theorem 4.2follows, we use [ADDDLD20, Item (b) of Proposition 4.10], in which the approximatingsequence is constructed in a way that is a priori dependent on the point a ∈ U . Neverthelessthe upgrade of such approximation from a local one on balls to an approximation on arbitrarycompact sets, with sequences of functions that are not dependent on the compact set itself,is very likely to be true in the setting of Carnot groups of step 2 by exploiting the sametechnique explained in [ADDDLD20, Remark 4.14] and based on [MV12]. Since this topicdoes not fit in this paper we will not treat it here, and it will subject of further investigations. Remark . Consider the Engel group E , i.e., the Carnot group whose Lie algebra e admits an adapted basis ( X , X , X , X ) suchthat e := span { X , X } ⊕ span { X } ⊕ span { X } , where [ X , X ] = X , and [ X , X ] = X . We identify E with R by means of exponentialcoordinates, and we define the couple of homogeneous complementary subgroups W := { x =0 } , and L := { x = x = x = 0 } in such coordinates. Then, by explicit computations thatcan be found in [Koz15, Section 4.4.1], we get that, for a continuous function ϕ : U ⊆ W → L ,with U open, the projected vector fields on W are(48) D ϕX = ∂ x + ϕ∂ x + ϕ ∂ x , D ϕX = ∂ x + ϕ∂ x , D ϕX = ∂ x . Thus, if we consider the function ϕ (0 , x , x , x ) := x / on W , we get that D ϕX ϕ = ∂ x ( ϕ ) = in the distributional sense on W . On the other hand ϕ : W → L is not uniformlyintrinsically differentiable, since it is not / -little Hölder continuous along the coordinate x , see Definition 4.6, while for a function to be uniformly intrinsically differentiable this is anecessary condition, see [ADDDLD20, Example 5.3] and [ADDDLD20, (a) ⇒ (c) of Theorem4.17]. Then we conclude that the chain of equivalences of Theorem 4.2 cannot be extendedalready in the easiest step-3 Carnot group.Nevertheless we do not know whether Theorem 4.1 holds in some cases beyond the settingof step-2 Carnot groups. In particular we do not know whether Theorem 4.1 holds in theEngel group with the splitting previously discussed. Interesting develpoments in the directionof studying whether distributional solutions to Burgers’ type equations with non-convexfluxes are also broad solutions are given in [ABC16] and [ABC].We conclude with the following Hölder property that happens to be a consequence of ϕ being a distributional solution to D ϕ ϕ = ω with a continuous datum ω . For the purpose,we here recall the definition of little Hölder continuity. Definition 4.6 (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(49) lim r → sup ( | ϕ ( b ′ ) − ϕ ( b ) || b ′ − b | α : b, b ′ ∈ U , < | b ′ − b | < r )! = 0 . Theorem 4.7. Let G be a Carnot group of step 2 and rank m , and let W and L be twocomplementary subgroups of G , with L horizontal and one-dimensional. Let U ⊆ W be an pen set and let ϕ : U → L be a continuous function. Choose coordinates on G as explainedin Section 2.2, see also (5) . If one of the items of Theorem 4.2 holds, then ϕ is / -littleHölder continuous along the vertical coordinates.Proof. It is a consequence of Theorem 4.2 and [ADDDLD20, Remark 3.23 & Theorem 6.12]. (cid:3) References [ABC] G. Alberti, S. Bianchini, and L. Caravenna. Eulerian, lagrangian and broad continuous solu-tions to a balance law with non convex flux II. preprint SISSA 32/2016/MATE .[ABC16] G. Alberti, S. Bianchini, and L. Caravenna. Eulerian, Lagrangian and broad continuous solu-tions to a balance law with non-convex flux I. J. Differential Equations , 261(8):4298–4337,2016.[ADDDLD20] Gioacchino Antonelli, Daniela Di Donato, Sebastiano Don, and Enrico Le Donne. Charac-terizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups. 2020.Preprint, available at https://arxiv.org/abs/2005.11390 .[ALD20] Gioacchino Antonelli and Enrico Le Donne. Pauls rectifiable and purely Pauls unrectifiablesmooth hypersurfaces. Nonlinear Anal. , 200:111983, 30, 2020.[ASCV06] Luigi Ambrosio, Francesco Serra Cassano, and Davide Vittone. Intrinsic regular hypersurfacesin Heisenberg groups. J. Geom. Anal. , 16(2):187–232, 2006.[BCSC15] F. Bigolin, L. Caravenna, and F. Serra Cassano. Intrinsic Lipschitz graphs in Heisenberggroups and continuous solutions of a balance equation. Ann. Inst. H. Poincaré Anal. NonLinéaire , 32(5):925–963, 2015.[BLU07] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni. Stratified Lie groups and potential theory fortheir sub-Laplacians . Springer Monographs in Mathematics. Springer, Berlin, 2007.[BSC10a] Francesco Bigolin and Francesco Serra Cassano. Distributional solutions of Burgers’ equationand intrinsic regular graphs in Heisenberg groups. J. Math. Anal. Appl. , 366(2):561–568, 2010.[BSC10b] Francesco Bigolin and Francesco Serra Cassano. Intrinsic regular graphs in Heisenberg groupsvs. weak solutions of non-linear first-order PDEs. Adv. Calc. Var. , 3(1):69–97, 2010.[CDPT07] Luca Capogna, Donatella Danielli, Scott D. Pauls, and Jeremy T. Tyson. An introduction tothe Heisenberg group and the sub-Riemannian isoperimetric problem , volume 259 of Progressin Mathematics . Birkhäuser Verlag, Basel, 2007.[CM20] Francesca Corni and Valentino Magnani. Area formula for regular submani-folds of low codimension in Heisenberg groups. 2020. Preprint, available at https://arxiv.org/abs/2002.01433 .[CMS04] Giovanna Citti, Maria Manfredini, and Alessandro Sarti. Neuronal oscillations in the visualcortex: Γ -convergence to the Riemannian Mumford-Shah functional. SIAM J. Math. Anal. ,35(6):1394–1419, 2004.[Cor20] Francesca Corni. 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FMS14] Bruno Franchi, Marco Marchi, and Raul Serapioni. Differentiability and approximate differ-entiability for intrinsic lipschitz functions in carnot groups and a rademarcher theorem. Anal.Geom. Metr. Spaces , 2(3):258–281, 2014.[Fol73] G. B. Folland. A fundamental solution for a subelliptic operator. Bull. Amer. Math. Soc. ,79:373–376, 1973.[Fol75] G. B. Folland. Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. ,13(2):161–207, 1975.[FS16] Bruno Franchi and Raul Paolo Serapioni. Intrinsic Lipschitz graphs within Carnot groups. J.Geom. Anal. , 26(3):1946–1994, 2016.[FSSC01] Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano. Rectifiability and perimeter inthe Heisenberg group. Math. Ann. , 321(3):479–531, 2001.[FSSC03] Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano. On the structure of finite peri-meter sets in step 2 Carnot groups. The Journal of Geometric Analysis , 13(3):421–466, 2003.[FSSC07] Bruno Franchi, Raul Serapioni, and Francesco Serra Cassano. Regular submanifolds, graphsand area formula in Heisenberg groups. Adv. Math. , 211(1):152–203, 2007.[JNGV20] Antoine Julia, Sebastiano Nicolussi Golo, and Davide Vittone. Area of in-trinsic graphs and coarea formula in Carnot groups. 2020. Preprint, available at https://arxiv.org/abs/1811.05457 .[Koz15] Artem Kozhevnikov. Propriétés métriques des ensembles de niveau des applications différen-tiables sur les groupes de carnot. PhD Thesis, Université Paris Sud - Paris XI , 2015.[LD17] Enrico Le Donne. A primer on Carnot groups: homogenous groups, Carnot-Carathéodoryspaces, and regularity of their isometries. Anal. Geom. Metr. Spaces , 5:116–137, 2017.[LDM20] Enrico Le Donne and Terhi Moisala. Semigenerated step-3 Carnot algebrasand applications to sub-Riemannian perimeter. 2020. Preprint, available at https://arxiv.org/abs/2004.08619 .[LDPS19] Enrico Le Donne, Andrea Pinamonti, and Gareth Speight. Universal differentiability sets andmaximal directional derivatives in Carnot groups. J. Math. Pures Appl. (9) , 121:83–112, 2019.[Lun95] Alessandra Lunardi. Analytic semigroups and optimal regularity in parabolic problems . ModernBirkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 1995. [2013 reprint of the 1995original] [MR1329547].[Mag13] Valentino Magnani. Towards differential calculus in stratified groups. J. Aust. Math. Soc. ,95(1):76–128, 2013.[Mag19] Valentino Magnani. Towards a theory of area in homogeneous groups. Calc. Var. PartialDifferential Equations , 58(3):Paper No. 91, 39, 2019.[Mar14] Marco Marchi. Regularity of sets with constant intrinsic normal in a class of Carnot groups. Ann. Inst. Fourier (Grenoble) , 64(2):429–455, 2014.[MST18] Valentino Magnani, Eugene Stepanov, and Dario Trevisan. A rough calculus approach to levelsets in the Heisenberg group. J. Lond. Math. Soc. (2) , 97(3):495–522, 2018.[MV12] Roberto Monti and Davide Vittone. Sets with finite H -perimeter and controlled normal. Math.Z. , 270(1-2):351–367, 2012.[SC84] Antonio Sánchez-Calle. Fundamental solutions and geometry of the sum of squares of vectorfields. Invent. Math. , 78(1):143–160, 1984.[SC16] Francesco Serra Cassano. Some topics of geometric measure theory in Carnot groups. In Geometry, analysis and dynamics on sub-Riemannian manifolds. Vol. 1 , EMS Ser. Lect.Math., pages 1–121. Eur. Math. Soc., Zürich, 2016.[SS03] Elias M. Stein and Rami Shakarchi. Complex analysis , volume 2 of Princeton Lectures inAnalysis . Princeton University Press, Princeton, NJ, 2003.[Vit20] Davide Vittone. Lipschitz graphs and currents in Heisenberg groups. 2020. Preprint, availableat https://arxiv.org/abs/2007.14286 . Gioacchino Antonelli: Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa,Italy. E-mail address : [email protected] aniela Di Donato: Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI–40014, University of Jyväskylä, Finland. E-mail address : [email protected] Sebastiano Don: Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI–40014,University of Jyväskylä, Finland. E-mail address : [email protected]@jyu.fi
suchthat | ψ ( γ ( T )) − ψ ( γ (0)) | ≤ CT, whenever γ : [0 , T ] → Ω is an integral curve of D ψj with γ (0) = w , where w is any point in Ω .Fix ≤ j ≤ m . We first assume ∈ Ω with ψ (0) = 0 and we consider an integral curve γ : [0 , T ] → Ω of D ψj such that γ (0) = 0 . Taking (13) and (14) into account, we can explicitlywrite all the components of γ ( t ) as follows: γ j ( t ) = t,γ j ( t ) = − R t b ψ ( τ, γ j ( τ )) d τ,γ i ( t ) = 0 , ∀ i = 1 , . . . , m, i = j,γ ℓs ( t ) = 0 , ∀ ( ℓ, s ) with ≤ s < ℓ ≤ m and ( ℓ, s ) = ( j, , where b ψ is defined in (35). We can thus define b γ : [0 , T ] → R by letting b γ ( t ) := ( t, γ j ( t )) .By the same choice of test functions given in [Daf06, Eq. (2.5) and (2.6)], from (34) we can erive(36) Z γ j ( T ) γ j ( T ) − ε b ψ ( T, x ) d x − Z − ε b ψ (0 , x ) d x − Z T Z γ j ( t ) γ j ( t ) − ε b ω j ( t, x ) d x d t = − Z T (cid:16) b ψ ( t, γ j ( t ) − ε ) − b ψ ( t, γ j ( t )) (cid:17) d t, for every sufficiently small ε > , see [Daf06, Eq. (3.4)] with the choice g = b ω j , u = b ψ , σ = 0 , τ = T , ξ = γ j and the change of sign of the right hand side with respect to the referencecomes by the fact that in our case f ( u ) = − u instead of f ( u ) = u . Since the right handside of (36) is negative, we can write Z γ j ( T ) γ j ( T ) − ε b ψ ( T, x ) d x − Z − ε b ψ (0 , x ) d x ≤ Z T Z γ j ( t ) γ j ( t ) − ε b ω j ( t, x ) d x d t ≤ ε k ω j k L ∞ (Ω) T. Dividing both sides by ε and letting ε → , by the continuity of ψ we get(37) b ψ ( T, γ j ( T )) − b ψ (0 , ≤ k ω j k L ∞ (Ω) T. Similarly, by mimicking [Daf06, Eq. (3.5)] we can write for every sufficiently small ε > theequation Z γ j ( T )+ εγ j ( T ) b ψ ( T, x ) d x − Z ε b ψ (0 , x ) d x − Z T Z γ j ( t )+ εγ j ( t ) b ω j ( t, x ) d x d t = 12 Z T (cid:16) b ψ ( t, γ j ( t ) + ε ) − b ψ ( t, γ j ( t )) (cid:17) d t. Noticing that the right hand side is positive one gets Z γ j ( T )+ εγ j ( T ) b ψ ( T, x ) d x − Z ε b ψ (0 , x ) d x ≥ Z T Z γ j ( t )+ εγ j ( t ) b ω j ( t, x ) d x d t ≥ − ε k ω j k L ∞ (Ω) T. Dividing both sides by ε and letting ε → we get(38) b ψ ( T, γ j ( T )) − b ψ (0 , ≥ −k ω j k L ∞ (Ω) T. Combining (37) and (38) we finally obtain | b ψ ( T, γ j ( T )) − b ψ (0 , | = | b ψ ( b γ ( T )) − b ψ ( b γ (0)) | = | ψ ◦ γ ( T ) − ψ ◦ γ (0) |≤ k ω j k L ∞ (Ω) T, for any integral curve γ : [0 , T ] → Ω of D ψj with γ (0) = 0 .For the general case, assume w ∈ Ω and let γ : [0 , T ] → Ω be an integral curve of D ψj with γ (0) = w . Setting q := ( w · ψ ( w )) − , by Lemma 2.11 and in particular (23), there exists anintegral curve γ q : [0 , T ] → P q (Ω) of D ψ q j such that γ q (0) = 0 and(39) ψ q ( γ q ( t )) − ψ q ( γ q (0)) = ψ ( γ ( t )) − ψ ( γ (0)) , ∀ t ∈ [0 , T ] . We also know by Proposition 2.12 that D ψ q j ψ q = ω j ◦ P q − in the distributional sense in P q (Ω) .Since w ∈ Ω , then ∈ P q (Ω) and ψ q (0) = 0 , see items (c) and (d) of Proposition 2.10. Wecan therefore run the same argument used in the preliminary dimensional reduction and thefirst part of (a) to ψ q , P q (Ω) , γ q and ω j ◦ P q − , to get that | ψ q ◦ γ q ( T ) − ψ q ◦ γ q (0) |≤ k ω j ◦ P q − k L ∞ ( P q (Ω)) T. he proof of (a) is complete if we use (39) and we observe that the Lipschitz constant isuniform by the fact that k ω j k L ∞ (Ω) = k ω j ◦ P q − k L ∞ ( P q (Ω)) . (b) Fix a ∈ Ω , ≤ j ≤ m and let δ > be such that B ( a , δ ) ∩ W ⊆ Ω . Up to reducing δ ,recalling the explicit expression of P q in (16), we can assume that for every w ∈ B ( a , δ ) ∩ W one has B (0 , δ ) ∩ W ⊆ P q (Ω) where, as before, q := ( w · ψ ( w )) − .Let w ∈ B ( a , δ ) ∩ W . From the fact that D ψj ψ = ω j in the distributional sense on Ω , we conclude that D ψ q j ψ q = ω j ◦ P q − in the distributional sense on P q (Ω) , where q :=( w · ψ ( w )) − , see Proposition 2.12. Moreover ∈ P q (Ω) and ψ q (0) = 0 , see items (c) and(d) of Proposition 2.10. Thus from the preliminary result on the reduction of dimension,see (34) and (35), we conclude that D c ψ q j c ψ q = \ ω j ◦ P q − holds in the distributional sense on \ P q (Ω) . Here we recall that by D c ψ q j we mean the classical Burgers’ operator ∂ j − c ψ q ∂ j onthe open subset \ P q (Ω) of R := { ( x j , y j ) : x j , y j ∈ R } . Then we exploit this informationand the argument in [BSC10a, Step 1 of proof of Theorem 1.2] to find < δ < δ and a C -smooth integral curve b γ : [ − δ , δ ] → B (0 , δ ) ∩ \ P q (Ω) of D c ψ q j such that b γ (0) = 0 and(40) c ψ q ( b γ ( t )) − c ψ q ( b γ (0)) = Z t ( \ ω j ◦ P q − ) ( b γ ( s )) d s, ∀ t ∈ [ − δ , δ ] . Moreover, by the same argument used in [BSC10a, Step 1 of proof of Theorem 1.2], wecan choose δ := min { δ / , δ / (2 M q ) } , where M q := sup B (0 , δ ) ∩ \ P q (Ω) | c ψ q | . In particular, if w ∈ B ( a , δ ) ∩ W and δ is small enough, M q has a uniform bound depending on thesupremum of ψ in some a priori fixed neighborhood of a , since ψ q is explicit in terms of q ,see item (c) of Proposition 2.10. As a consequence, up to eventually reducing and fixing δ , δ has a positive lower bound independent of q = ( w · ψ ( w )) − , when we allow w to run in B ( a , δ ) ∩ W . We still denote this lower bound with δ .Recalling (14) and the first part of this proof, we can write b γ ( t ) = ( t, γ j ( t )) for some γ j : [ − δ , δ ] → R . For any w ∈ B ( a , δ ) ∩ W we can hence define a w -dependent γ : [ − δ , δ ] → B (0 , δ ) ∩ W ⊆ P q (Ω) by letting(41) γ ( t ) := (0 , . . . , , t, , . . . , , γ j ( t ) , , . . . , , ∀ t ∈ [ − δ , δ ] . Then, since γ (0) = 0 , from the particular expression of D ψ q j , see (13) and (14), and by thefact that b γ is an integral curve of D b ψ q j , we get that γ is an integral curve of D ψ q j , and from(41), (40), and (35) the following equality holds ψ q ( γ ( t )) − ψ q ( γ (0)) = Z t ( ω j ◦ P q − )( γ ( s )) d s, ∀ t ∈ [ − δ , δ ] . Thanks to Lemma 2.11 and to item (b) of Proposition 2.10, we can translate the integralcurve γ to an integral curve γ q − : [ − δ , δ ] → Ω of D ψj with γ q − (0) = w , such that, exploiting(23), the following equality holds ψ q ( γ ( t )) − ψ q ( γ (0)) = ψ ( γ q − ( t )) − ψ ( γ q − (0)) = Z t ω j ( γ q − ( s )) d s, ∀ t ∈ [ − δ , δ ] , where the last equality is true since P q − ◦ γ = γ q − , see (21) and (16). Thus we have shownthat if we fix a ∈ Ω , j = 2 , . . . , m , and δ sufficiently small, we can find δ only depending on , ω , and δ such that for every w ∈ B ( a , δ ) there exists an integral curve γ : [ − δ , δ ] → Ω of D ψj such that γ (0) = w and ψ ( γ ( t )) − ψ ( γ (0)) = Z t ω j ( γ ( s )) d s, ∀ t ∈ [ − δ , δ ] . By the continuity of ψ this suffices to conclude that D ψ ψ = ω in the broad* sense on Ω , seeDefinition 2.13. (cid:3) Proposition 3.3.