Distributional Solutions of Burgers’ type Equations for Intrinsic Graphs in Carnot Groups of Step 2

We prove that in arbitrary Carnot groups G of step 2, with a splitting G = W · L with L one-dimensional, the intrinsic graph of a continuous function ϕ : U ⊆ W → L is C 1H -regular precisely 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 easiest step-3 Carnot group, namely the Engel group. We notice that our results generalize previous works by Ambrosio-Serra Cassano-Vittone and Bigolin-Serra Cassano in the setting of Heisenberg groups. 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 a by-product of independent interest we obtain that all the continuous distributional solutions to D ϕ ϕ = ω , with ω continuous, enjoy 1 / 2-little H¨older regularity along vertical directions.

product only defined on a sub-bundle of the tangent bundle.The infinitesimal model of a sub-Riemannian manifold, namely the class of its Gromov-Hausdorff tangents, is represented by the class of (quotients of) Carnot groups [SC16,LD17].Carnot groups are connected and simply connected Lie groups G whose Lie algebra g admits a stratification, namely a decomposition into nontrivial complementary linear subspaces V 1 , . . ., V s such that where [V j , V 1 ] denotes the subspace of g generated by the commutators [X, Y ] with X ∈ V j and Y ∈ V 1 .Carnot groups have been studied from very different point of views such as Differential 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 most studied problems in the past twenty years is represented by the rectifiability problem: is it possible to cover the boundary of a finite perimeter set with a countable union of C 1 -regular surfaces?The answer to this question is affirmative in the Euclidean case and it was studied in [DG54,DG55] via a blow-up analysis.The proof of De Giorgi has then been adapted in the framework of step-2 Carnot groups in [FSSC01,FSSC03] and then generalized to the so-called Carnot groups of type ⋆ in [Mar14], see also the recent [LDM20].When dealing with Carnot groups of step 3 or higher, only partial results concerning this question are available in the literature.One of the main difficulty is represented by the fact that it is not known in general if C 1 rectifiability is equivalent to a Lipschitz-type rectifiability.Concerning Heisenberg groups, see [Vit20] for a Rademacher-type theorem for intrinsic Lipschitz graphs of 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 fine understanding of intrinsic surfaces inside Carnot groups.The study of different notions of 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 1 -hypersurfaces, [Mag19] for a notion of non-horizontal transversal submanifold and [Mag13,JNGV20] for a notion of C 1 -surface with Carnot group target, but the list is far from being complete.
We focus our attention on codimension-one intrinsic graphs.A codimension-one intrinsic graph Γ inside a Carnot group G comes with a couple of homogeneous and complementary subgroups 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 the graph 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 its graph has a homogeneous subgroup as blow-up.However, one can define some different notions of regularity that rely on some ϕ-dependent operators D ϕ W whenever W ∈ Lie(W), see Definition 2.6.If an adapted basis of the Lie algebra (X 1 , . . ., X n ) is fixed and is such that L := exp(span{X 1 }) and W := exp(span{X 2 , . . ., X n }), then D ϕ is the vector valued operator (D ϕ X 2 , . . ., D ϕ Xm ) =: (D ϕ 2 , . . ., D ϕ m ).The regularity of Γ is related to the validity of the equation D ϕ ϕ = ω in an open subset U ⊆ W, for some ω: U → R m−1 , which can be understood in different ways.We briefly present some of them here.
Distributional 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 distributional sense.Broad* sense.For every j = 2, . . ., m and every point a ∈ U, there exists a C1 integral curve of D ϕ X j starting from a for which the Fundamental Theorem of Calculus with derivative ω holds, see Definition 2.13.Broad sense.For every j = 2, . . ., m and every point a ∈ U, all the integral curves of D ϕ X j starting from a are such that the Fundamental Theorem of Calculus with derivative ω holds, see Definition 2.13.Approximate sense.For every a ∈ U, there exist δ > 0 and a family {ϕ ε ∈ C 1 (B(a, δ)) : ε ∈ (0, 1)} such that ϕ ε → ϕ and D ϕε 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 coordinateindependent statement.Notice that the statement of the result below needs a choice of coordinates as explained in Section 2.2, see also (5).We also refer the reader to the preliminary section of [ADDDLD20] for the notion used in the statement below that are not treated in the current paper.Item (g) allows us to complete the chain of implications of Theorem 1.1 in the setting of step-2 Carnot groups generalizing the results scattered in [ASCV06,BSC10a,BSC10b] where the authors study the same problem in the Heisenberg groups, and [DD20a,DD20b] where partial generalizations of the results in [ASCV06,BSC10a,BSC10b] are obtained in the case of 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 2.3 for the precise choice of identifications.We show in Proposition 3.3 that if D ϕ ϕ = ω in distributional sense inside G with some continuous ω ∈ C(U; R m−1 ), then also D ψ ψ = ω • π in distributional sense in F, where ψ := π −1 • 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−1 ) 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 The global 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 the dimensional reduction given by Lemma 3.1 and the translation invariance of Proposition 2.12 to reduce ourselves to the Burgers' equation of the first Heisenberg group, and then apply the arguments used for this case in [Daf06,Eqq. (3.4) and (3.5)] and [BSC10a, Step 1, proof of 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 arbitrary Carnot groups of step 2 has a significantly different structure compared to the one in the Heisenberg groups.For example, consider a Carnot group of dimension 5, step 2 and rank 3 with Lie algebra g = span{X 1 , X 2 , X 3 , X 4 , X 5 }, horizontal layer V 1 := span{X 1 , X 2 , X 3 } and where the only nonvanishing commutators are given by [X 1 , X 2 ] = X 4 + X 5 and [X 1 , X 3 ] = X 4 − X 5 .Define, in exponential coordinates, W := {x 1 = 0} and L := {x 2 = x 3 = x 4 = x 5 = 0}.Then, given a continuous ϕ: U ⊆ W → L on an open set U, the operators D ϕ j := D ϕ X j for j = 2, 3 have the following form (see [ADDDLD20, Example 3.6]) , which show a nonlinearity in two vertical directions, instead of only one as in the Heisenberg groups. 2  We remark that Proposition 3.2 and Theorem 4.1 have also an interesting PDE point of view 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 the continuous datum ω.Then, from each single equation of the system, we infer the following property: for every j = 2, . . ., m, ϕ is (uniformly) Lipschitz continuous on all the integral curves of the operator D ϕ j .In addition, the Fundamental Theorem of Calculus with derivative ω holds on some particular local family of integral curves of D ϕ j , namely the broad* condition holds, and then also the broad condition holds, see (f)⇒(e) of Theorem 4.1.Moreover, when we consider all the equations together, we obtain a remarkable piece of information: ϕ is 1/2-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 groups for the following reason.Already in the Engel group, which is the easiest step-3 Carnot group, we can find a continuous map ϕ that solves D ϕ ϕ = ω in the sense of distributions for a constant ω whose graph is not uniformly intrinsically differentiable (UID).We however notice that we do not know at present if implication (1) holds in Carnot groups of higher step, see Remark 4.5.
We briefly describe the situation in which ω is less regular.In the paper [BCSC15], the authors show that, in Heisenberg groups, D ϕ ϕ = ω holds in the sense of distributions for some ω ∈ L ∞ (U; R m−1 ) if and only if ϕ is intrinsically Lipschitz.The validity of (1) with ω ∈ L ∞ (U; R m−1 ) in the setting of step-2 Carnot groups would open to a slightly modified version of Theorem 1.1 where ω ∈ L ∞ (U; R m−1 ) 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−1 ) would imply that ϕ is 1/2-Hölder continuous along vertical directions.This topic is out of the aims of this paper and will be target of future investigations.
We notice here that if a generalization of the a priori estimate [MV12, Lemma 3.1] would hold in any step-2 Carnot group, then we could improve Theorem 1.1 replacing (d) with (d') There exists ω ∈ C(U; R m−1 ) and a family of functions {ϕ ε ∈ C 1 (U) : ε ∈ (0, 1)} such that, for every compact set K ⊆ U and every j = 2, . . ., m, one has We refer the reader to [ADDDLD20, Remark 4.14] for a discussion of the literature and of the difference between item (d) and item (d').We also remark that a smooth approximation that does not involve the intrinsic gradient holds in any Carnot group for intrinsic Lipschitz graphs, 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 meaning of the distributional system D ϕ ϕ = ω in higher codimension is still open.The main difficulty comes from the fact that it is not known how to give meaning to mixed terms of the form ϕ i ∂ x ϕ j .This was already noticed in [Koz15,Remark 4.3.2].A weak formulation that goes in this direction is collected in [MST18], where the authors relate zero-level sets of maps in C 1,α H (H; R 2 ) with curves that satisfies certain "Level Set Differential Equations", see [MST18, Theorem 5.6].

Preliminaries
2.1.Carnot groups.We give a very brief introduction on Carnot groups.We refer the reader to e.g.[BLU07, SC16, LD17] for a comprehensive introduction to 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 1 , . . ., V s of the Lie algebra g such that The integer s is called step of the group G, while m := dim(V 1 ) is called rank of G.We set n := dim(G) to be the topological dimension of G.We equivalently denote by e or 0 the identity element of the group G.
Every Carnot group has a one-parameter family of dilations that we denote by {δ λ : λ > 0} defined as the unique linear maps on g such that δ λ (X) = λ j X, for every X ∈ V j .We denote by δ λ both the dilations on G and on g, with the usual identification given by the exponential map exp: g → G which is a diffeomorphism.We fix a homogeneous norm • on G, namely such that δ λ x = λ x for every λ > 0 and x ∈ G, xy ≤ x + y for every x, y ∈ G, x = x −1 for every x ∈ G, and x = 0 if and only if x = e.The norm • induces a left-invariant homogeneous distance and we denote with B(a, r) the open ball of center a and radius r > 0 according to this distance.We stress that on a Carnot group a homogeneous norm always exists, and every two left-invariant homogeneous distances are bi-Lipschitz equivalent.
Definition 2.1 (Complementary subgroups).Given a Carnot group G, we say that two subgroups W and L are complementary subgroups in G if they are homogeneous, i.e., closed under the action of δ λ for every λ > 0, We say that the subgroup L is horizontal and k-dimensional if there exist linearly independent X 1 , . . ., X k ∈ V 1 such that L = exp(span{X 1 , . . ., X k }).Given two complementary subgroups 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 Remark 2.2.If W and L are complementary subgroups of G and L is one-dimensional, then it is easy to see that L is horizontal.For the sake of clarity, we will always write L horizontal and 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 2.4 (Exponential coordinates).Let G be a Carnot group of dimension n and let (X 1 , . . ., X n ) be an adapted basis of its Lie algebra.The exponential coordinates of the first kind associated with (X 1 , . . ., X n ) are given by the one-to-one correspondence 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 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 6.2].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 exponential coordinates associated with an adapted basis In this coordinates, we will identify any point q ∈ G with q ≡ (x 1 , . . ., x m , y * 1 , . . ., y * h ).The group operation • between two elements q = (x, y * ) and q ′ = (x ′ , (y * ) ′ ) is given by where Bx, x ′ := ( B (1) x, x ′ , . . ., B (h) x, x ′ ) and B (i) are linearly independent and skewsymmetric 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 jℓ ), and it is standard to observe that we can write 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 the adapted basis jℓ , with i = 1, . . ., h, and 1 ≤ j, ℓ ≤ m, are the so-called structure coefficients.
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 1 }).Thus, by means of exponential coordinates we can identify W G and L G with R m+h−1 and R, respectively, as follows (5)

2.3.
Free Carnot groups of step 2 F in exponential coordinates.We here introduce free 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 2 and rank m.Recall that the topological dimension of

2
, we can identify F with R n by means of exponential coordinates associated with the adapted basis (X 1 , . . ., X m , Y 21 , . . ., Y m(m−1) ).In this coordinates, we will identify any point q ∈ F with q ≡ (x 1 , . . ., x m , y 1 , . . ., y m(m−1) ).It is readily seen that, in such coordinates, we have (6) Moreover, for any q = (x, y) and q ′ = (x ′ , y ′ ) in F, the product q • q ′ is given by the Baker-Campbell-Hausdorff formula, and yields 2.4.Projection from F to G. Fix a Carnot group G of step 2 and rank m as in Section 2.2 and let F be a free Carnot group of rank m and step s.By definition of free Carnot groups, there exists a Lie group surjective homomorphism π: F → G such that (7) π * (X ℓ ) = X ′ ℓ , for any ℓ = 1, . . ., m (see e.g.[LDPS19, Section 6]).We identify F and G with R n and R m+h , respectively, by means of exponential coordinates as explained above and in Section 2.2 and Section 2.3.From (7), jointly with the very definition of exponential coordinates, we notice that for any (x, y) ∈ R n , where x ∈ R m and y ∈ R m(m−1)/2 , there exists Since π is a Lie group homomorphism, its differential is a Lie algebra homomorphism.Hence, for any 1 ≤ s < ℓ ≤ m, we have that where we used (4), (7), and the fact that for 1 ≤ s < ℓ ≤ m one has [X ℓ , X s ] = Y ℓs .We can therefore write the following formula ℓs y ℓs , ∀i = 1, . . ., h. (9) Remark 2.5 (Main identification).Given a Carnot group G of step 2 and rank m, and a free Carnot group F of step 2 and rank m we work in the coordinates of Section 2.2 and Section 2.3.Let W G and L G be two complementary subgroups of G, with L G one-dimensional.Up to a proper choice of an adapted basis, we can assume we are working in a basis in which (5) holds.Thus, taking into account (8), we are in a position to lift W G and L G to two complementary subgroups W F and In this way we have the following identification (10) for all w ∈ U and all f ∈ C ∞ (W).When W is an element X j of an adapted basis (X 1 , . . ., X n ) we also write D ϕ j := D ϕ X 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 a basis 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 exponential coordinates and we use the identifications and the coordinate representations discussed in Section 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) 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) Then, see also [ADDDLD20, Remark 6.5], for each j = 2, . . ., m, every integral curve satisfying the following equations ( 14) ẏj1 (t) = −ψ(x 2 , . . ., x j−1 , x j + t, x j+1 , . . ., x m , y(t)), where the horizontal components of γ j (0) are (0, x 2 , . . ., x m ).
2.6.Invariance properties of projected vector fields.We collect here some invariance properties that we will use later on.We introduce the operation of q-translation of a function.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, and ϕ q : U q ⊆ W → L by setting Notice that U q = P q (U), where P q is defined as in ( 16).This easily comes from the fact that for every q ∈ G P q • P q −1 = 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 complementary subgroups 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); Then for every q ∈ G there exists a unique C 1 map γ q : [0, T ] → U q such that In addition, γ q is a solution of the Cauchy problem Moreover the following equality holds In the following proposition we prove the invariance of being a distributional solution with respect 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 1 loc (Ω).Let us choose coordinates 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 ϕ ℓ ϕ = ω on U.Then, for every q ∈ G, the map ϕ q defined in Definition 2.9 is a distributional solution of D ϕq ℓ ϕ q = ω • P q −1 , on the open set P q (Ω).
Proof.By item (c) of Proposition 2.10, we know that in exponential coordinates ϕ q (w) = q 1 + ϕ(P q −1 (w)), where P q −1 is defined in (16).Indeed, since W is normal, the following equality holds (24) Moreover we claim P q −1 = P −1 q , for all q ∈ G. Indeed, since W is normal, the following equality holds (25) and hence it is clear by comparing ( 24) and (25) that P q • P q −1 = P q −1 • 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 ϕq ℓ ϕ q on ξ, where we mean that the coordinates are w = (x 2 , . . ., x m , y * 1 , . . ., y * h ) ∈ W, as follows where in the third equality we used the fact that ξ has compact support in P q (Ω) and q 1 does not depend on w.Taking into account that, by Remark 2.7, det(dP q ) = 1 everywhere on W, we perform in (27) the change of variable w ′ = P q −1 (w).Thus, recalling that P q •P q −1 = Id W , and by exploiting (17), we obtain the following equality (28) We can now use (18) to compute the derivatives of ξ • P q as follows (29) By using (29) into (28) we get 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 where in the last equality we changed the variables and exploited the fact that det(dP (q)) = 1.By the arbitrariness of ξ ∈ C ∞ c (P q (ω)), the proof is complete.Let ω := (ω j ) j=2,...,m : U → R m−1 be a continuous function.Up to identifying L with R by means of exponential coordinates, we say that ϕ ∈ C(U) is a broad* solution of D ϕ ϕ = ω in U if for every a 0 ∈ U there exist 0 < δ 2 < δ 1 such that B(a 0 , δ 1 ) ∩ W ⊆ U and there exist m − 1 maps E ϕ j : (B(a 0 , δ 2 ) ∩ W) × [−δ 2 , δ 2 ] → B(a 0 , δ 1 ) ∩ W for j = 2, . . ., m, satisfying the following two properties.
(a) For every a ∈ B(a 0 , δ 2 ) ∩ W and every j = 2, . . ., m, the map E ϕ j (a) := E ϕ j (a, •) is C 1 -regular and it is a solution of the Cauchy problem where the vector field D ϕ j := D ϕ X j is defined in (11).(b) For every a ∈ B(a 0 , δ 2 ) ∩ W, for every t ∈ [−δ 2 , δ 2 ], and every j = 2, . . ., m one has Moreover, we say that D ϕ ϕ = ω in the broad sense on U if for every W ∈ Lie(W) ∩ V 1 and every γ: where by ω, W is the standard scalar product on R m−1 in exponential coordinates.Finally, let us notice that, for every j = 2, . . .m, D ϕ j is a continuous vector field with coefficients that might depend polinomially on ϕ and on some of the coordinates, see Section 2.5 for 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 one has D ϕ j ϕ = ω j in the distributional sense.Notice that the distribution (D ϕ j )ϕ is well defined since the coefficients of D ϕ j just contain polynomial terms in ϕ and terms depending on the coordinates, see also [ADDDLD20, Item (a) of Proposition 4.10].

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, see Proposition 3.2 (b).In this proof, a crucial role is played by the particular structure of the 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 in Proposition 2.12 and the dimensional reduction of Lemma 3.1, we can reduce ourselves to deal with Burgers' distributional equation with continuous datum on the first Heisenberg group H 1 , and then exploit the arguments used by Dafermos in [Daf06] and by Bigolin and Serra 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, see Proposition 3.3.Finally, Proposition 3.4 states that the notion of broad* solution is preserved by π, i.e., a broad* solution on F becomes a broad* solution on G.The resulting strategy resembles the one used in [LDPS19].
Lemma 3.1.Let n 1 , n 2 ∈ N and let Ω be an open set in R n 1 +n 2 +k .Let f 0 , f 1 , . . ., f n 1 ∈ C(Ω) and assume that, for every ϕ ∈ C ∞ c (Ω), one has Proof.By translation invariance, we can assume without loss of generality that z 0 = 0. Up to iterate the argument k times, we can also assume without loss of generality that k = 1.Fix ϕ := ϕ(x, y) be such that supt( ϕ) ⊆ Ω 0 .Choose ε 0 > 0 small enough and consider, for any ε ∈ (0, ε 0 ], the map ϕ ε (x, y, z) and such that supt( ϕ) × supt(ϕ ε 0 0 ) ⊆ Ω.Then, by the hypothesis and Fubini's Theorem we may write (30) 1 2ε Notice that the function . We can then decompose the left-hand side of (30) in the following way 31), the thesis follows by means of Lebesgue's Theorem.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 adapted basis and exponential coordinates on F as in Section 2.3, see also (10).Assume there exists ω ∈ C(Ω; R m−1 ) such that D ψ ψ = ω holds in the distributional sense on Ω.Then, the following facts hold.
Observe that the dimensional reduction shown above can be performed in the same way to the distributional equation D ψq ψ q = ω • P q −1 in the set P q (Ω), where for any w ∈ Ω we have set q = (w • ψ(w)) −1 and the map ψ q is defined in Definition 2.9.Indeed in this case, if w ∈ Ω, then by item (c) of Proposition 2.10 we have 0 ∈ P q (Ω) and, thanks to item (d) of Proposition 2.10, ψ q (0) = 0.
By the same choice of test functions given in [Daf06, Eq. (2.5) and (2.6)], from (34) we can derive (36) for every sufficiently small ε > 0, see [Daf06, Eq. (3.4)] with the choice g = ω j , u = ψ, σ = 0, τ = T , ξ = γ j1 and the change of sign of the right hand side with respect to the reference comes by the fact that in our case f (u) = − 1 2 u 2 instead of f (u) = 1 2 u 2 .Since the right hand side of (36) is negative, we can write Dividing both sides by ε and letting ε → 0, by the continuity of ψ we get Similarly, by mimicking [Daf06, Eq. (3.5)] we can write for every sufficiently small ε > 0 the equation Noticing that the right hand side is positive one gets Dividing both sides by ε and letting ε → 0 we get (38) ψ(T, γ j1 (T )) − ψ(0, 0) ≥ − ω j L ∞ (Ω) T.
Then, since γ(0) = 0, from the particular expression of D ψq j , see ( 13) and ( 14), and by the fact that γ is an integral curve of D ψq j , we get that γ is an integral curve of D ψq j , and from (41), (40), and (35) the following equality holds Thanks to Lemma 2.11 and to item (b) of Proposition 2.10, we can translate the integral curve γ to an integral curve γ q −1 : [−δ 2 , δ 2 ] → Ω of D ψ j with γ q −1 (0) = w, such that, exploiting (23), the following equality holds where the last equality is true since P q −1 • γ = γ q −1 , see (21) and ( 16).Thus we have shown that if we fix a 0 ∈ Ω, j = 2, . . ., m, and δ 1 sufficiently small, we can find δ 2 only depending on j, ω, and δ 1 such that for every w ∈ B(a 0 , δ 1 ) there exists an integral curve γ: By the continuity of ψ this suffices to conclude that D ψ ψ = ω in the broad* sense on Ω, see Definition 2.13.
Assume there exists ω ∈ C(U; R m−1 ) 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−1)/2 .Then, taking (12) into account, we have that for every ξ ∈ C ∞ c (U).Hence, by exploiting (13), we would like to show that (43) , and then, by using the chain rule in order to write the partial derivatives of ξ with respect to ξ, we can write (45) Since, by construction, the projection of M(V ) onto the variables (x * , y * ) is precisely π(V ) = U, we get that ξ(•, ŷ * 0 ) ∈ C ∞ c (U) for every ŷ * 0 ∈ R m(m−1)/2−h , and thus by using (42), Remark 4.4 (Approximation of a distributional solutions to D ϕ ϕ = ω).The approximating sequence in item (d) of Theorem 4.2 is a priori dependent on the point a ∈ U we choose.This is true because in order to obtain [ADDDLD20, Theorem 6.17], from which Theorem 4.2 follows, we use [ADDDLD20, Item (b) of Proposition 4.10], in which the approximating sequence is constructed in a way that is a priori dependent on the point a ∈ U. Nevertheless the upgrade of such approximation from a local one on balls to an approximation on arbitrary compact 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 same technique explained in [ADDDLD20, Remark 4.14] and based on [MV12].Since this topic does not fit in this paper we will not treat it here, and it will subject of further investigations.
Remark 4.5 (Counterexample to Theorem 4.2 on the Engel group).Consider the Engel group E, i.e., the Carnot group whose Lie algebra e admits an adapted basis (X 1 , X 2 , X 3 , X 4 ) such that e := span{X 1 , X 2 } ⊕ span{X 3 } ⊕ span{X 4 }, where [X 1 , X 2 ] = X 3 , and [X 1 , X 3 ] = X 4 .We identify E with R 4 by means of exponential coordinates, and we define the couple of homogeneous complementary subgroups W := {x 1 = 0}, and L := {x 2 = x 3 = x 4 = 0} in such coordinates.Then, by explicit computations that can 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) Thus, if we consider the function ϕ(0, x 2 , x 3 , x 4 ) := x 1/3 4 on W, we get that D ϕ X 2 ϕ = 1 6 ∂ x 4 (ϕ 3 ) = 1 6 in the distributional sense on W. On the other hand ϕ: W → L is not uniformly intrinsically differentiable, since it is not 1/3-little Hölder continuous along the coordinate x 4 , see Definition 4.6, while for a function to be uniformly intrinsically differentiable this is a necessary condition, see [ADDDLD20, Example 5.3] and [ADDDLD20, (a)⇒(c) of Theorem 4.17].Then we conclude that the chain of equivalences of Theorem 4.2 cannot be extended already in the easiest step-3 Carnot group.
Nevertheless we do not know whether Theorem 4.1 holds in some cases beyond the setting of step-2 Carnot groups.In particular we do not know whether Theorem 4.1 holds in the Engel group with the splitting previously discussed.Interesting develpoments in the direction of studying whether distributional solutions to Burgers' type equations with non-convex fluxes 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.
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 onedimensional.Let U ⊆ W be an open set, and let ϕ: U → L be a continuous function.Then the following conditions are equivalent (a) graph(ϕ) is a C 1 H -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−1 ) such that, for every a ∈ U, there exist δ > 0 and a family of functions {ϕ ε ∈ C 1 (B(a, δ)) : ε ∈ (0, 1)} such that lim ε→0 ϕ ε = ϕ, and lim ε→0 D ϕε j ϕ ε = ω j in L ∞ (B(a, δ)), for every j = 2, . . ., m; (e) there exists ω ∈ C(U; R m−1 ) such that D ϕ ϕ = ω in the broad sense on U; (f) there exists ω ∈ C(U; R m−1 ) 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−1 ) with L one-dimensional, see Theorem 4.1.This result allows us to improve Theorem 1.1 adding a seventh equivalent condition to the list above 1 : (g) there exists ω ∈ C(U; R m−1 ) such that D ϕ ϕ = ω holds in the distributional sense on U.

2. 5 .
Projected vector fields in Carnot groups of step 2. We recall here the definition of projected vector fields [ADDDLD20, Definition 3.1].Definition 2.6 (Projected vector fields).Given two complementary subgroups W and L in a 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 ϕ W , by setting (11)

Lemma 2 .
11 ([ADDDLD20, Lemma 3.13, and Equations (45)-(46)]).Let W and L be two complementary 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 us denote D ϕ := D ϕ W .Let T > 0, w ∈ W, and let γ: [0, T ] → U be a C 1 -regular solution of the Cauchy problem

2. 7 .
Broad, broad* and distributional solutions of D ϕ ϕ = ω.We recall the following definition as a particular case of [ADDDLD20, Section 3.4, Definition 3.24].For discussions about the dependence of the definition of broad* regularity on the chosen adapted basis, we refer the reader to [ADDDLD20, Remark 3.26 & Remark 4.4].Definition 2.13 (Broad*, broad and distributional solutions).Let W and L be complementary subgroups of a Carnot group G, with L one-dimensional.Let U ⊆ W be open and let ϕ: U → L be a continuous function.Consider an adapted basis (X 1 , . . ., X n ) of the Lie algebra of G such that L = exp(span{X 1 }) and W = exp(span{X 2 , . . ., X n }).

Proposition 3. 3 .
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 as explained 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 open set in W G and denote by V ⊆ W F the open set defined by V := π −1 (U).Let ϕ: U → L G be a continuous map and let ψ: V → L F be the map defined as
Definition 4.6 (little Hölder functions,[Lun95]).Let U ⊆ R n be an open set.We denote by h α (U; R k ) the set of all α-little Hölder continuous functions of order 0 < α < 1, i.e., the set of maps ϕ ∈ C(U; R k ) satisfying (49)lim r→0 sup |ϕ(b ′ ) − ϕ(b)| |b ′ − b| α : b, b ′ ∈ U , 0 < |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 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.Choose coordinates on G as explained in Section 2.2, see also (5).If one of the items of Theorem 4.2 holds, then ϕ is 1/2-little Hölder continuous along the vertical coordinates.Proof.It is a consequence of Theorem 4.2 and [ADDDLD20, Remark 3.23 & Theorem 6.12].