We stress that, in the setting of <ref>, if $\widetilde\phi \in C^1(\widetilde U)$ then $D^{\phi}\phi=\nabla^{\phi}\phi$ both pointwise and in the broad sense on $U$. First $\widetilde\phi\in {\rm UID}(\widetilde U,\mathbb W;\mathbb L)$ by <cit.>, because $\widetilde\phi\in C^1(\widetilde U)$. Then we can consider the intrinsic gradient $\nabla^{\phi}\phi$ as in <ref>, which is continuous, see <ref>. Thus the claim is an outcome of point (i) of <ref> and point (c) of <ref>, that becomes a pointwise equality if $\widetilde\phi$ is $C^1(\widetilde U)$, see the proof of <ref>.

Let us notice that the definition given in <ref> is a priori susceptible to the choice of an adapted basis. Nevertheless, when it is coupled with the vertically broad* hölder condition in the same basis, see <ref>, it is independent of this choice. This is an outcome of <ref>. Indeed, from (d)$\Rightarrow$(a) of <ref>, it follows that the broad* condition and the vertically broad* hölder condition on a fixed basis imply that $\widetilde\phi$ is UID. Thus, from item (i) of <ref>, we get that the broad* condition is satisfied for every other basis. Finally, from <ref>, we get that also the vertically broad* hölder condition holds in every other basis.

With the above reasoning, we remark that we can conclude something stronger: if the broad* condition and the vertically broad* hölder condition hold on a fixed basis, then they hold uniformly on the choice of $W\in {\rm Lie}(\mathbb W)$ with bounded norm, see <ref>, and <ref>.

The following result is already known in the Heisenberg groups $\mathbb H^n$: in case $\mathbb L$ is one-dimensional, it is  proved in <cit.>, while in case $\mathbb L$ is $k$-dimensional it is proved in <cit.>. We here generalize it to arbitrary Carnot groups, in the case $\mathbb L$ is horizontal and $k$-dimensional.

Let $\mathbb W$ and $\mathbb L$ be complementary subgroups of a Carnot group $\mathbb G$, with $\mathbb L$ horizontal and $k$-dimensional, and consider an adapted basis of the Lie algebra $\mathfrak g$ such that $\mathbb W=\exp(\mathrm{span}\{X_{k+1},\dots,X_n\})$ and $\mathbb L=\exp(\mathrm{span}\{X_{1},\dots,X_k\})$. Let $\widetilde U$ be an open subset of $\mathbb W$, and $\widetilde\phi\in {\rm ID}(\widetilde U,\mathbb W;\mathbb L)$ be such that $\de^{\phi}\!\phi$ is continuous on $\widetilde U$. Denote by $\nabla^{\phi}\phi$ the $k\times (m-k)$ matrix that represents $\de^{\phi}\!\phi$ in coordinates, see <ref>.

Then, we have that

\begin{equation}\label{eqn:Semplice}
\frac{\de}{\de t}_{|_{t=t_0}}(\phi^{(\ell)}\circ\gamma)(t)=\nabla^{\phi}_{\ell j}\phi(\gamma(t_0)),
\end{equation}

for every $j=k+1,\dots,m$, every integral curve $\widetilde\gamma\colon I\to\widetilde U$  of $D^{\phi}_j\coloneqq D^{\phi}_{X_j}$, every $\ell=1,\dots,k$, and every $t_0\in I$. In particular the function $\phi$ is a broad solution, and thus also a broad* solution, of the system $D^{\phi}\phi=\nabla^{\phi}\phi$.

Equation (<ref>) directly follows from (<ref>) seen in coordinates. Then, from (<ref>) and the fact that $\nabla^{\phi}\phi$ is continuous by hypothesis, the second part of the thesis follows.

§ MAIN THEOREMS IN ARBITRARY CARNOT GROUPS

In this section we prove <ref>, that is <ref> in the introduction. We deal with an arbitrary Carnot group $\mathbb G$ along with a continuous function $\widetilde{\phi}\colon\widetilde U\subseteq\mathbb W\to\mathbb L$, where $\mathbb W$ and $\mathbb L$ are complementary subgroups of $\mathbb G$, with $\mathbb L$ horizontal and $k$-dimensional, and $\widetilde U$ is an open subset of $\mathbb W$.

In <ref>, we study how Hölder properties of $\widetilde\phi$ along integral curves of the vector fields $D^{\phi}$ as defined in (<ref>) affect the intrinsic regularity of the function $\widetilde \phi$. The main result of this section is a converse of <ref>: if $D^{\phi}\phi=\omega$ holds in the broad* sense (see <ref>) and there is, locally around every point, a family of curves satisfying the little Hölder regularity condition (<ref>) (we shall call this property vertically broad* hölder regularity, see <ref>), then $\widetilde \phi$ is uniformly intrinsically differentiable. For the full statement, see <ref>. We notice that, taking <ref> into account, the latter proposition generalizes <cit.>, which deals with the case  $\G=\mathbb H^n$ and $\mathbb L$ one-dimensional, <cit.>, which is proved in case $\G$ has step 2 and $\mathbb L$ is one-dimensional, and <cit.> that solves the problem for $\G=\mathbb H^n$ with $\mathbb L$ horizontal and $k$-dimensional. We remark that, also in these cases, we obtain slightly stronger results, requiring just a locally 1/2-little Hölder regularity in the vertical components.

<ref> could also be obtained by a combination of (2)$\Rightarrow$(1) of <cit.> and <ref>, which is proved in [26]. The idea of (2)$\Rightarrow$(1) in <cit.> is to show that the Hölder conditions on $\widetilde\phi$ along a family of integral curves, that is the assumptions of <ref>, imply that the intrinsic graph of $\widetilde \phi$ is a co-horizontal $C^1_{\rm H}$-surface with complemented tangents. To prove this latter fact the author uses a characterization of co-horizontal $C^1_{\rm H}$-surfaces by means of uniform Hausdorff convergence to tangents - see <cit.> - that is in turn based on the so-called four cones Theorem, see <cit.>. With the independent proof we give in <ref>, with more analytic flavor, we stress we can indirectly obtain (2)$\Rightarrow$(1) of <cit.> by making use of <ref> and <ref>. We also obtained (1)$\Rightarrow$(2) of <cit.>, see the discussion before <ref>.

Our proof of <ref> requires <ref>, that is stated  only for $\mathbb L$ horizontal and $k$-dimensional. The <ref> is a converse of <ref>, i.e., it can be roughly read in the following way: the uniform Hölder regularity of the curves $\widetilde\phi\circ \widetilde\gamma$, where $\widetilde\gamma$ is an integral curve of the vector field $D^\phi$, implies the intrinsically Lipschitz regularity of $\widetilde\phi$. We give a proof of <ref> as we crucially need it for the proof of <ref>, but we remark that a more general statement can be given, in case $\mathbb W$ is normal, with a proof that is very similar to the one of <ref>. Notice that the general statement with $\mathbb W$ normal can be found in (3)$\Rightarrow$(1) of <cit.>. For more details, see <ref>.

As a by-product, we obtain some analytical results, that can have their own independent interests. The first one is given by <ref> and it states that broad* regularity implies broad regularity. Roughly speaking, having a function that is Hölder regular on a precise family of integral curves implies the Hölder regularity on every integral curve. Then, in <ref>, we prove that every intrinsically differentiable function that is vertically broad* hölder (see <ref>), and that has a continuous intrinsic gradient, is uniformly intrinsically differentiable. We do not know at present whether the assumption on the vertically broad* hölder regularity can be dropped in <ref>, see also <ref>. We expect that the hypothesis on the vertically broad* hölder regularity in <ref> can be dropped in general, see also the paragraph Geometric characterizations of intrinsic differentiability in the introduction. From the results proved in [16] and [23],  we know that the assumption on the vertically broad* hölder regularity in <ref> is not necessary in the case of the Heisenberg groups $\mathbb H^n$, with $\mathbb L$ horizontal $k$-dimensional, see also the introduction to <ref>. We stress we obtain that we can remove the assumption on the vertically broad* hölder regularity in <ref> also in the case of step-2 Carnot groups with $\mathbb L$ one-dimensional, see <ref>.

In <ref> we focus on the case in which $\mathbb L$ is one-dimensional and we prove <ref>. We present an area formula that represents the perimeter of the subgraph of a uniformly intrinsically differentiable function $\phi$ in terms of the density $\sqrt{1+|\nabla^{\phi}\phi|^2}$. For more details about the area formula and a representation involving the Hausdorff measures, we refer the reader to <ref>. In <ref> we also prove that, whenever the target $\mathbb L$ is one-dimensional, every uniformly intrinsically differentiable function $\phi$ is a distributional solution of the system $D^{\phi}\phi=\nabla^{\phi}\phi$ and, in <ref>, we deduce that if $D^{\phi}\phi=\omega$ holds in the broad* sense with a continuous datum $\omega$ and $\phi$ is vertically broad* hölder, then $D^{\phi}\phi=\omega$ in the sense of distributions. We do not know, in general, if in <ref> we can remove the assumption on the vertically broad* hölder regularity. In fact, one can remove the hypothesis on the vertically broad* hölder regularity in <ref> in Heisenberg groups, and it is a consequence of the results in [16]. We stress that thanks to the results obtained in <ref>, we drop the assumption on the vertically broad* hölder regularity in <ref> also in the case of step-2 Carnot groups.

It is interesting to investigate the converse implication: if one has $D^{\phi}\phi=\omega$ in the sense of distributions with a continuous function $\omega$, is it true that $D^{\phi}\phi=\omega$ in the broad* sense? This is actually the case in the Heisenberg groups, see [15], and the techniques used in <ref> seem a good tool to address this implication in arbitrary step-2 Carnot groups. We will not address this issue in this paper and it will be the target of further investigations. It is however interesting to notice that, in some examples besides the step-2 case and for particular $\phi$, one can obtain that if $D^{\phi}\phi=\omega$ holds in the sense of distributions with a continuous function $\omega$, then $D^{\phi}\phi=\omega$ in the broad* sense. We will not discuss this issue in the paper, but we refer the reader to [1].

In <ref> we come back to the general case in which the target $\mathbb L$ is horizontal and not necessarily one-dimensional. We prove that if $\phi$ is locally approximable with a sequence of smooth functions whose intrinsic derivatives converge to a continuous function $\omega$, then $D^{\phi}\phi=\omega$ in the broad*, see <ref>. This notion of local approximability has been first introduced and studied in [5], see also <ref>. We exploit this result to prove that every uniformly intrinsically differentiable function $\phi$ always solves $D^{\phi}\phi=\nabla^{\phi}\phi$ in the broad* sense.

In <ref>, we combine some of the previous results together to prove our main theorem <ref>, which is <ref> in the introduction. Notice that our result provides the generalization to all Carnot groups, and to any possible horizontal and $k$-dimensional target $\mathbb L$, of <cit.>. We stress that <ref> will be strengthened in <ref> dropping the hypothesis on the vertical broad* hölder regularity in the setting of Carnot groups of step 2.
We stress that, in general, the assumption on the vertical broad* hölder regularity cannot be dropped in <ref>, see <ref> for a counterexample in the easiest step-3 group, namely the Engel group.

§.§ From regularity of $\phi$ along integral curves of $D^{\phi}$ to regularity of $\phi$

In this subsection we show how the Hölder regularity of $\phi$ along integral curves of $D^{\phi}$ affects the intrinsic regularity of $\widetilde\phi$.

Let $\mathbb W$ and $\mathbb L$ be complementary subgroups of a Carnot group $\G$, with $\mathbb L$ horizontal and $k$-dimensional, let $\widetilde U$ be open, and let $\widetilde{\phi}\colon\widetilde{U}\subseteq \mathbb W\to\mathbb L$ be a continuous function with $e\in\widetilde{U}$ and $\widetilde{\phi}(e)=e$.

Let $(X_1,\dots,X_n)$ be an adapted basis of the Lie algebra $\mathfrak g$ such that $\mathbb L=\exp({\rm span}\{X_{1},\dots,X_k\})$ and $\mathbb W=\exp({\rm span}\{X_{k+1},\dots,X_n\})$. Denote by
$D^{\phi}_j\coloneqq D^{\phi}_{X_j}$, for every $j=k+1,\dots,n$. Let $L>0$.

Fix $v\in U$ and consider a concatenation of curves $\gamma_{k+1},\dots,\gamma_n$ in $U$ connecting $0$ to $v$ such that $\gamma_j\colon I_j\to U$ is an integral curve of $D^\phi_j$ for $j=k+1,\dots, n$. Assume that the function $\phi\circ \gamma_j$ is $\frac{1}{\mathrm{deg}j}$-Hölder continuous on $I_j$, for $j=k+1,\dots,n$, with Hölder constant $L$.
Then, there exists $C>0$ only depending on $L, \mathbb W$, $\mathbb L$ and the adapted basis $(X_1,\dots, X_n)$ such that

\begin{equation}\label{eqn:ThesisOfConverseLipschitz1}
|\phi\circ \gamma_j(t)|\leq C\|v\|_{\mathbb G}, \qquad \forall t\in I_j, \quad  \forall j=k+1,\dots, n,
\end{equation}

\begin{equation}\label{eqn:ThesisOfConverseLipschitz2}
\;\;\;|\gamma_j^{(i)}(t)| \leq C\|v\|_{\mathbb G}^{\deg i}, \qquad\; \forall t\in I_j, \quad \forall i,j=k+1,\dots,n,
\end{equation}

where $\gamma^{(i)}_j$ denotes the $i$-th component of $\gamma_j$ in exponential coordinates.

Up to bi-Lipschitz equivalence, we can prove the result choosing the anisotropic norm $\|x\|_{\mathbb G}\coloneqq \sum_{\ell=1}^n |x_{\ell}|^{1/\deg \ell}$. For the sake of readability, we fix $k=1$ and assume that all the layers $V_i$ of the algebra $\mathfrak g$, with $i\geq 2$, are 1-dimensional, so that $\deg i=i-1$ for every $i=2,\dots, n$. The proof in the general case only requires a typographical effort to deal with multiple components in each layer and the fact that we have more zeros in the components of the first layer.

We work in exponential coordinates so that $v=( 0,v_{2},\dots,v_n)$, and denote the extremal points of the concatenation of $\gamma_2,\dots, \gamma_n$ by the following chain

\begin{equation}\label{eqn:Concatenation}
\begin{aligned}
e=(0,0,\dots,0)&\rightarrow_{\gamma_{2}}( 0,v_2,r_{3,2},\dots,r_{n,2})\rightarrow_{\gamma_{3}}\dots  \\\dots
&\rightarrow_{\gamma_j}( 0,v_{2},\dots,v_j,r_{j+1,j},\dots,r_{n,j})\rightarrow_{\gamma_{j+1}}\dots \\
\dots &\rightarrow_{\gamma_{n}} v=( 0,v_{2},\dots,v_n).
\end{aligned}
\end{equation}

In the previous chain, we used the triangular form of $D^\phi_j$ given in (<ref>), so that the flow along $D^\phi_j$ does not affect the coordinates with index less than $j$. Notice also that, without loss of generality we can assume $I_j\subseteq[-|v_j-r_{j,j-1}|,|v_j-r_{j,j-1}|]$, with the convention $r_{2,1}\coloneqq 0$. Indeed, again by using (<ref>), in order to correct the error $r_{j,j-1}$ in the $j$-th component, we have to flow for a time $v_j-r_{j,j-1}$ along $D^{\phi}_j$.

We prove (<ref>) and (<ref>) by induction on $j$. When we say that a constant depends on $\mathbb W$ and  $\mathbb L$ we mean that also depends on the chosen adapted basis $(X_1,\dots, X_n)$.

By assumption, the curve $\phi\circ\gamma_2$ is $L$-Lipschitz on $I_2$  and $\phi\circ\gamma_2(0)=0$. Consequently, we have that

\begin{equation}\label{eqn:FirstInduction}
|\phi\circ\gamma_2(t)|=|\phi\circ\gamma_2(t)-\phi\circ\gamma_2(0)|\leq L|t|\leq L|v_2| \leq L\|v\|_{\mathbb G}, \qquad \forall t\in I_2.
\end{equation}

Therefore (<ref>) is proved for $j=2$.

Next we shall prove (<ref>) for $j=2$ by means of induction on $i$. For $i=2$, we have $\gamma_2^{(2)}(t)=t$ and then

\[
|\gamma_2^{(2)}(t)|=|t|\leq |v_2|\leq \|v\|_{\mathbb G}, \qquad \forall t\in I_2.
\]

Assume now that for some $i_0\geq 2$, there is a constant $C_{2,i_0}>0$ such that

\begin{equation}\label{eqn:ipotesiinduttiva1}
|\gamma_2^{(i)}(t)|\leq C_{2,i_0}\|v\|_{\mathbb G}^{\deg i}= C_{2,i_0}\|v\|_{\mathbb G}^{i-1},\quad \forall t\in I_2,\quad \forall i=2,\dots,i_0.
\end{equation}

We want to prove that there exists $C_{2,i_0+1}>0$ such that

\[
|\gamma_2^{(i_0+1)}(t)|\leq C_{2,i_0+1}\|v\|_{\mathbb G}^{\deg (i_0+1)} =C_{2,i_0+1}\|v\|_{\mathbb G}^{i_0}, \quad \forall t\in I_2,
\]

where $C_{2,i_0+1}$ only depends on  $ i_0, L,C_{2,i_0}, \mathbb W$ and $\mathbb L$. By using the particular triangular form of $D_2^{\phi}$ in exponential coordinates, see (<ref>), we get that

\begin{equation}\label{eqn:SecondInduction}
|\gamma_2^{(i_0+1)}(t)|=\left|\int_0^t P_{i_0+1}^2(\phi\circ\gamma_2(s),\gamma_2^{(2)}(s),\dots,\gamma_2^{(i_0)}(s))\de\!s\right|,\quad \forall t\in I_2,
\end{equation}

for some polynomial $P^2_{i_0+1}$ of homogeneous degree $\deg(i_0+1)-\deg 2=i_0-1$. Then, from (<ref>) and (<ref>), we deduce there exists $ C_{2,i_0+1}$ depending on $i_0$, $L$, $C_{2,i_0}$ ,$\mathbb W$ and $\mathbb L$ such that

\[
\left|P_{i_0+1}^2(\phi\circ\gamma_2(t),\gamma_2^{(2)}(t),\dots,\gamma_2^{(i_0)}(t))\right| \leq C_{2,i_0+1}\|v\|_{\mathbb G}^{i_0-1}, \qquad \forall t\in I_2,
\]

and thus, by using this inequality in (<ref>), and since $|t|\leq |v_2|\leq \|v\|_{\mathbb G}$, we get

\[
|\gamma_2^{(i_0+1)}(t)|\leq C_{2,i_0+1}|t|\|v\|_{\mathbb G}^{i_0-1}\leq  C_{2,i_0+1}\|v\|_{\mathbb G}^{i_0}, \qquad \forall t\in I_2.
\]

To conclude the proof  of (<ref>) and (<ref>) in the case $j=2$, it is enough to set $C_2\coloneqq \max_{i=2,\dots,n} C_{2,i}$, where $C_{2,2}=\max\{L,1\}$.

Assume now that for some $2\leq j_0\leq n$  and some constant $C_{j_0}>0$ one has

\begin{equation}\label{eqn:ipotesiinduttivaj1}
|\phi\circ \gamma_{j}(t)|\leq C_{j_0}\|v\|_{\mathbb G}, \qquad \forall t\in I_{j},\quad \forall j=2,\dots,j_0, \\
\end{equation}

\begin{equation}\label{eqn:ipotesiinduttivaj2}
|\gamma_{j}^{(i)}(t)| \leq C_{j_0}\|v\|_{\mathbb G}^{\deg i}, \qquad \forall t\in I_{j}, \quad \forall i\geq 2,\quad \forall j=2,\dots, j_0.
\end{equation}

We want to find $C_{j_0+1}>0$ only depending on $L, j_0, C_{j_0}, \mathbb W$ and $\mathbb L$ such that

\begin{equation}\label{eqn:ThirdInduction1}
|\phi\circ \gamma_{j_0+1}(t)|\leq C_{j_0+1}\|v\|_{\mathbb G}, \qquad \forall t\in I_{j_0+1},
\end{equation}

\begin{equation}\label{eqn:ThirdInduction2}
|\gamma_{j_0+1}^{(i)}(t)| \leq C_{j_0+1}\|v\|_{\mathbb G}^{\deg i}, \qquad \forall t\in I_{j_0+1}, \quad \forall i= 2,\dots, n.
\end{equation}

To prove (<ref>), we develop the following inequalities:

\begin{equation}\label{eqn:CentralEstimate}
\begin{aligned}
|\phi\circ \gamma_{j_0+1}(t)|&\leq |\phi\circ \gamma_{j_0+1}(t)-\phi\circ \gamma_{j_0+1}(0)|+|\phi\circ \gamma_{j_0+1}(0)|\leq L|t|^{1/\deg(j_0+1)}+|\phi\circ \gamma_{j_0+1}(0)| \\
&\leq L|v_{j_0+1}-r_{j_0+1,j_0}|^{1/j_0}+C_{j_0}\|v\|_{\mathbb G}\leq L|v_{j_0+1}|^{1/j_0}+L|r_{j_0+1,j_0}|^{1/j_0}+C_{j_0}\|v\|_{\mathbb G} \\
&\leq L\|v\|_\G+LC_{j_0}^{\frac 1{j_0}}\|v\|_\G+C_{j_0}\|v\|_\G=\widetilde C_{j_0+1}\|v\|_\G, \qquad \forall t\in I_{j_0+1},
\end{aligned}
\end{equation}

where in the second inequality we used the fact that $\phi\circ\gamma_{j_0+1}$ is Hölder of constant $L$; in the third inequality we used the definition of $I_{j_0+1}$ for the first term, while the estimate on the second term  comes from (<ref>) and the fact that $\gamma_{j_0+1}(0)$ is the endpoint of $\gamma_{j_0}$; the fifth inequality is a consequence of (<ref>), inequality $|v_{j_0+1}|^{1/j_0}\leq \|v\|_{\mathbb G}$, and the fact that $r_{j_0+1,j_0}$ is the endpoint of $\gamma_{j_0}^{(j_0+1)}$.

We are left to prove (<ref>). First we notice that, by (<ref>), for $2\leq i\leq j_0$ we have $\gamma_{j_0+1}^{(i)}(t)\equiv v_i\leq \|v\|^{\deg i}_\G$ for all $t\in I_{j_0+1}$. Using again (<ref>), for $i=j_0+1$ we have that $\gamma_{j_0+1}^{(j_0+1)}(t)=r_{j_0+1,j_0}+t$. Then, since  $|t|\leq |v_{j_0+1}-r_{j_0+1,j_0}|$, for every $t\in I_{j_0+1}$, arguing similarly to (<ref>), one can obtain

\[
|\gamma_{j_0+1}^{(j_0+1)}(t)|\leq 2|r_{j_0+1,j_0}|+|v_{j_0+1}|\leq (2C_{j_0}+1)\|v\|_\G^{j_0}.
\]

Setting $C_{j_0+1,j_0+1}=\max\{2C_{j_0}+1,1\}$, we get

\[
|\gamma_{j_0+1}^{(i)}(t)| \leq C_{j_0+1,j_0+1}\|v\|_{\mathbb G}^{\deg i},\qquad \forall t\in I_{j_0+1},\quad \forall i=2,\dots,j_0+1.
\]

Let us now proceed by induction on $i$ and assume there exists $i_0\in\{j_0+1,\dots,n\}$ and a constant $C_{j_0+1,i_0}>0$ such that

\begin{equation}\label{eqn:induzionenellinduzione}
|\gamma_{j_0+1}^{(i)}(t)|\leq  C_{j_0+1,i_0}\|v\|_{\mathbb G}^{\deg i}=C_{j_0+1,i_0}\|v\|_{\mathbb G}^{i-1}, \quad \forall t\in I_{j_0+1}, \quad \forall i=2,\dots,i_0.
\end{equation}

We want to find $C_{j_0+1,i_0+1}>0$ only depending on $L, C_{j_0}, C_{j_0+1,i_0}, \mathbb W$ and $\mathbb L$ such that

\[
|\gamma_{j_0+1}^{(i_0+1)}(t)|\leq  C_{j_0+1,i_0+1}\|v\|_{\mathbb G}^{\deg (i_0+1)} = C_{j_0+1,i_0+1}\|v\|_{\mathbb G}^{i_0},\quad \forall t\in I_{j_0+1}.
\]

By using again (<ref>), we get that, for every $t\in I_{j_0+1}$, one has

\begin{equation}\label{eqn:SecondInduction2}
\begin{aligned}
|\gamma_{j_0+1}^{(i_0+1)}(t)|&=\left|r_{i_0+1,j_0}+\int_0^t P_{i_0+1}^{j_0+1}(\phi\circ\gamma_{j_0+1}(s),\gamma_{j_0+1}^{(2)}(s),\dots,\gamma_{j_0+1}^{(i_0)}(s))\de\!s\right| \\
&\leq |r_{i_0+1,j_0}|+\int_0^t\left|P_{i_0+1}^{j_0+1}(\phi\circ\gamma_{j_0+1}(s),\gamma_{j_0+1}^{(2)}(s),\dots,\gamma_{j_0+1}^{(i_0)}(s))\right|\de\!s,
\end{aligned}
\end{equation}

for a polynomial $P^{j_0+1}_{i_0+1}$ of homogeneous degree $\deg(i_0+1)-\deg(j_0+1)=i_0-j_0$. From (<ref>) and (<ref>), we deduce that there exists a constant $M>0$ depending only on $j_0, i_0, L, \mathbb W$ and $\mathbb L$ (it indeed depends on the coefficients of the polynomial, the constant $\widetilde C_{j_0+1}$, and the induction constant $C_{j_0+1,i_0}$) such that

\[
\left|P_{i_0+1}^{j_0+1}(\phi\circ\gamma_{j_0+1}(t),\gamma_{j_0+1}^{(2)}(t),\dots,\gamma_{j_0+1}^{(i_0)}(t))\right| \leq M\|v\|_{\mathbb G}^{i_0-j_0}, \qquad \forall t\in I_{j_0+1},
\]

and thus from (<ref>) we get

\begin{equation}\label{eqn:FourthInduction}
|\gamma_{j_0+1}^{(i_0+1)}(t)|\leq |r_{i_0+1,j_0}|+M|t|\|v\|_{\mathbb G}^{i_0-j_0}\leq |r_{i_0+1,j_0}|+M|v_{j_0+1}-r_{j_0+1,j_0}|\|v\|_{\mathbb G}^{i_0-j_0},
\end{equation}

for every $t\in I_{j_0+1}$. Notice that $r_{i_0+1,j_0}$ is the endpoint of $\gamma_{j_0}^{(i_0+1)}$, as well as $r_{j_0+1,j_0}$ is the endpoint of $\gamma_{j_0}^{(j_0+1)}$. Thus, by (<ref>), we get that

\[
|r_{i_0+1,j_0}|\leq C_{j_0}\|v\|^{\deg(i_0+1)}_{\mathbb G}=C_{j_0}\|v\|^{i_0}_{\mathbb G},
\]

\[
|v_{j_0+1}-r_{j_0+1,j_0}|\leq (1+C_{j_0})\|v\|_{\mathbb G}^{\deg(j_0+1)}=(1+C_{j_0})\|v\|_{\mathbb G}^{j_0}.
\]

Replacing these last two equalities in (<ref>), we get

\[
|\gamma_{j_0+1}^{(i_0+1)}(t)|\leq  C_{j_0+1,i_0+1}\|v\|^{i_0}_{\mathbb G}, \qquad \forall t\in I_{j_0+1},
\]

for the constant $ C_{j_0+1,i_0+1}\coloneqq C_{j_0}+M(1+C_{j_0})$, which only depends on $j_0, i_0, L, \mathbb W$, and $\mathbb L$. Inequalities (<ref>) and (<ref>) are completed by choosing $C_{j_0+1}\coloneqq\max\{ \max_{i=j_0+1,\dots,n} C_{j_0+1,i}, \widetilde C_{j_0+1} \}$. To conclude the proof, it is enough to set $C\coloneqq \max _{j=2,\dots, n} C_j$.

The careful reader could have noticed that the scheme of the proof  <ref> above can be adapted to prove the very same statement, in the more general case in which $\mathbb W$ is normal. We will not use the conclusions of this remark in what follows, but we nonetheless give a sketch of the proof of this fact.

Indeed, also in the case in which $\mathbb W$ is normal, taking <ref> into account, the vector fields $D^{\phi}_{j}$, for $k+1\leq j\leq n$, in exponential coordinates, have a triangular form analogous to <ref>. Then, if we adopt the same notation and setting as the statement of <ref> and if we assume that the curves $\widetilde\phi\circ\widetilde\gamma_j$ are $\frac{1}{\deg j}$-Hölder with respect to the norm $\|\cdot\|_{\mathbb G}$, the same double-induction argument of the proof of <ref> implies that, for any $k+1\leq j\leq n$, and any $t\in I_j$, one has $\|\widetilde\phi\circ\widetilde\gamma_j(t)\|_{\mathbb G} \leq\! C\| v\|_{\mathbb G}$, and $\|\widetilde\gamma_j(t)\|_{\mathbb G}\leq\! C\|v\|_{\mathbb G}$, instead of (<ref>), and (<ref>),respectively.

Thus, by evaluating the general form of (<ref>) at $j=n$ and  time $t$ corresponding to the endpoint of $I_n$, we get $\|\widetilde\phi(v)\|_{\mathbb G}\leq C\|v\|_{\mathbb G}$, with a constant $C$ only depending on $L$, $\mathbb W$, $\mathbb L$ and the basis adapted to the splitting. Then, if we assume that the bound $L$ on the Hölder constant of $\widetilde\phi\circ\widetilde\gamma_j$ is uniform with respect to the choice of the integral curve $\gamma_j$ of $D^{\phi}_j$, with $k+1\leq j\leq n$, we get, by exploiting the fact that $\sW$ is locally $D^\phi$-connectible according to <ref>, that

\begin{equation}\label{eqn:ConverseLipschitzWNormal}
\|\widetilde{\phi}(v)\|_{\mathbb G}\leq C\|v\|_{\mathbb G},
\end{equation}

for every $v\in\widetilde U'\Subset\widetilde U$, where the constant $C$ only depends on $L$, $\mathbb W$, $\mathbb L$ and the chosen basis adapted to the splitting.

Finally, if we do not necessarily assume $\widetilde{\phi}(e)=e$ as in the statement of <ref>, but we still assume that the $1/\deg j$-Hölder constant of $\widetilde\phi\circ\widetilde\gamma_j$ is uniformly bounded with respect to the choice of the integral curves $\widetilde\gamma_j$, the same translation argument as in the beginning of the proof of <ref>, joined with the conclusion (<ref>) and the third point of <ref>, implies that $\widetilde\phi$ is intrinsically Lipschitz on $\widetilde U'\Subset\widetilde U$. This last statement is the local converse of <ref>.

We finally remark that the improved result we described here is the implication (3)$\Rightarrow$(1) of <cit.>.

We now exploit <ref> to show a criterion to prove that a function is uniformly intrinsically differentiable in an arbitrary Carnot group. To do so we introduce the definition of the vertically broad* hölder property in a fixed adapted basis.

[Vertically broad* hölder and vertically broad hölder condition]
Let $\mathbb W$ and $\mathbb L$ be two complementary subgroups of a Carnot group $\G$ with $\mathbb L$ horizontal and $k$-dimensional, and fix an adapted basis $(X_1,\dots,X_n)$ of the Lie algebra $\mathfrak g$ such that $\mathbb W=\exp({\rm span}\{X_{k+1},\dots,X_n\})$ and $\mathbb L=\exp({\rm span}\{X_{1},\dots,X_k\})$. Let us fix $\widetilde U\subseteq \mathbb W$ an open subset and let us denote $D^{\phi}_j\coloneqq D^{\phi}_{X_j}$ as defined in (<ref>). We say that a continuous function $\widetilde \phi\colon\widetilde U\to \mathbb L$ is vertically broad* hölder if for every $a_0$ in $U$ there exist $\delta_{a_0}>0$ and neighborhoods $U'_{a_0}\Subset U_{a_0}\Subset U$ of $a_0$ such that for every $a\in U'_{a_0}$ and every $j=m+1,\dots, n$ one can find a $C^1$ regular solution $E_j^\phi(a)\colon[-\delta_{a_0},\delta_{a_0}]\to U_{a_0}$ of the Cauchy problem

\[
\begin{cases}
\dot\gamma=D^\phi_j\circ\gamma& \\
\gamma(0)=a&
\end{cases}
\]

such that

\begin{equation}\label{eqn:vertcicallybroad*holder}
\lim_{\varrho\to0}\left(\sup\left\{\frac{|\phi(E_j^\phi(a,t))-\phi(E_j^\phi(a,s))|}{|t-s|^{1/{\mathrm{deg}j}}}: j=m+1,\dots,n, a\in U'_{a_0}, 0<|t-s|\leq\varrho \right\}\right) =0.
\end{equation}

We moreover say that $\widetilde\phi$ is vertically broad hölder if for every $V\Subset U$ one has

\begin{equation*}
\lim_{\varrho\to0}\left(\sup\left\{\frac{|\phi(\gamma(t))-\phi(\gamma(s))|}{|t-s|^{1/{\deg j}}}: j=m+1,\dots,n,\,\dot\gamma=D^{\phi}_{j}\circ\gamma, \gamma\subseteq V, 0<|t-s|\leq\varrho \right\}\right) =0.
\end{equation*}

Notice that <ref> is a priori susceptible to the choice of a basis adapted to the splitting. However, when it is coupled with the broad* condition in the same basis, see <ref>, it is independent on this choice. See <ref> for details.

Let $\mathbb W$ and $\mathbb L$ be complementary subgroups of a Carnot group $\mathbb G$, with $\mathbb L$ horizontal and $k$-dimensional, and fix an adapted basis $(X_1,\dots, X_n)$ of the Lie algebra $\mathfrak g$ such that $\mathbb W=\exp({\rm span}\{X_{k+1},\dots,X_n\})$ and $\mathbb L=\exp({\rm span}\{X_{1},\dots,X_k\})$. Let $\widetilde U\subseteq \mathbb W$ be open, let $\widetilde{\phi}\colon\widetilde{U}\to\mathbb L$ be a vertically broad* hölder map and assume $\omega\colon U\subseteq \mathbb R^{n-k}\to\mathbb R^{k\times(m-k)}$ is a continuous function such that $D^{\phi}\phi=\omega$ in the broad* sense on $U$.
Then $\widetilde{\phi}\in \mathrm{UID}(\widetilde{U}, \mathbb W; \mathbb L)$. Moreover $\nabla^{\phi}\phi=\omega$, where $\nabla^{\phi}\phi$ is the intrinsic gradient defined in <ref>.

Up to bi-Lipschitz equivalence, we can prove the result choosing the anisotropic norm $\|x\|_{\mathbb G}\coloneqq \sum_{\ell=1}^n |x_{\ell}|^{1/\deg \ell}$. For the sake of readability, we give the proof in the case $k=1$. The proof for a larger $k$ only requires a typographical effort due to the fact that $\phi$ has more than one component.

Fix $a_0\in U$. According to the definition of vertically broad* hölder, we find $\delta_{a_0}>0$, neighborhoods $U'_{a_0}\Subset U_{a_0}\Subset U$ of $a_0$ and $C^1$ maps $E^\phi_j(a)\colon[-\delta_{a_0},\delta_{a_0}]\to U_{a_0}$ satisfying the conditions of <ref>. Define for every $\varrho>0$ sufficiently small the quantity

\begin{equation}\label{eqn:DefinitionOff}
f(\varrho)\coloneqq\sup\left\{\frac{|\phi(E_j^\phi(a,t))-\phi(E_j^\phi(a,s))|}{|t-s|^{1/{\mathrm{deg}j}}}: j=m+1,\dots,n, a\in U'_{a_0}, 0<|t-s|\leq\varrho \right\},
\end{equation}

which by assumption converges to $0$ as $\varrho\to0$. Throughout the proof, we will often write $E_j^\phi$ instead of $E_j^\phi(a)$, where the dependence on the starting point has to be understood for a suitable $a\in U'_{a_0}$.

We claim that $\phi$ is UID at $a_0$ and $\nabla^{\phi}\phi _{a_0}(b)=\omega(a_0)\cdot b^1$, for every $b\in \mathbb R^{n-1}$, where $b^1\in \mathbb R^{m-1}$ denotes the projection of $b$ onto the first $(m-1)$ components, and where $\omega(a_0)\cdot b^1\coloneqq\sum_{i=2}^{m} \omega_i(a_0)\cdot b^1_i$ denotes the usual scalar product on $\mathbb R^{m-1}$.

By (<ref>), we just need to prove that

\begin{equation}\label{eqn:EquationToShow}
\lim_{\varrho\to 0}\left(\sup\left\{\frac{|\phi(b)-\phi(a)-\omega(a_0)\cdot(b^1-a^1)|}{\|\widetilde{\phi}(a)^{-1}a^{-1}b\,\widetilde{\phi}(a)\|}: a\in B(a_0,\varrho), \|a^{-1}b\|<\varrho\right\}\right)= 0.
\end{equation}

Since $D^{\phi}\phi=\omega$ in the broad* sense, by <ref>, we can find neighborhoods $ U''\Subset U' \Subset U'_{a_0}\Subset U$ of $a_0$ and $\delta>0$ such that, for every $j=2,\dots,m$, one has $E_j^\phi(U''\times[-\delta,\delta])\subseteq U'$. We can improve this observation using the triangular form of $D^{\phi}_j$, see (<ref>), and arguing as in <ref>. Indeed, the
sets $ U''$ and $U'$ can be chosen small enough such that, for every $a,b\in U''$, there exists a path connecting $a$ to $b$, entirely contained in $U'$, made first by a concatenation of the maps $E_2^\phi,\dots,E_m^\phi$ (defined accordingly to <ref>) and then by a concatenation $E^\phi_{m+1},\dots,E^\phi_n$ of integral curves of $D^{\phi}_{m+1},\dots,D^{\phi}_n$ provided by the vertically broad* hölder condition.

Let use improve this conclusion. We know from <ref> that if $\widetilde{\gamma}$ is an integral curve of $D^{\phi}_j$, then $\widetilde{\gamma}_q\coloneqq q\cdot \widetilde{\gamma}\cdot (q_{\mathbb L})^{-1}$ is an integral curve of $D^{\phi_q}_j$, see (<ref>). Then, by possibly taking a smaller $U''$, we can suppose without loss of generality that there exist neighborhoods $ V''\Subset V'$  of $0$, such that, for every $a\in U''$ and every $b'\in  V''$, there exists a path connecting $0$ to $b'$, entirely contained in $V'$ made of $q$-translations of exponential maps. Indeed, if $a, b\in U''$, it is enough to set $q\coloneqq\widetilde{\phi}(a)^{-1}a^{-1}$ and build the concatenation of $q$-translated of the maps $(E_2^\phi)_q,\dots,(E_m^\phi)_q$, that are integral curves of $D^{\phi_q}_2,\dots,D^{\phi_q}_m$, respectively, by <ref>, and then chain it with the $q$-translated curves $(E^\phi_{m+1})_q,\dots,(E^\phi_n)_q$, that are integral curves of $D^{\phi_q}_{m+1},\dots,D^{\phi_q}_n$, respectively. By construction,  this concatenation connects $0$ to $b'=\widetilde\phi(a)^{-1}a^{-1}b\widetilde \phi(a)$. It suffices then to take a small enough
\[V''\subseteq\bigcup\{\widetilde\phi(a)^{-1}a^{-1}b\,\widetilde \phi(a): a, b\in U''\}.\]

Moreover, by (d) of <ref>, we get $\phi_q(0)=0$ and by (c) of <ref>, one also has $\phi_q(b')=\phi_q(b')-\phi_q(0)=\phi(b)-\phi(a)$.

Notice also that $\left(\widetilde{\phi}(a)^{-1}a^{-1}b\,\widetilde{\phi}(a)\right)^1 = b^1-a^1$, so that we have

\begin{equation}\label{eqn:KeyEstimate2}
\frac{|\phi(b)-\phi(a)-\omega(a_0)\cdot(b^1-a^1)|}{\|\widetilde{\phi}(a)^{-1}a^{-1}b\,\widetilde{\phi}(a)\|} = \frac{|\phi_q(b')-\phi_q(0)-\omega(a_0)\cdot (b')^1|}{\|b'\|}, \quad \forall a,b\in U''.
\end{equation}

For any $a, b\in U''$, we hence consider the concatenation $(E^\phi_2)_q,\dots,(E^\phi_n)_q$ of integral curves of $D^{\phi_q}_2,\dots,D^{\phi_q}_n$ entirely lying in $V'$, constructed as above, that connects $0$ to $b'$. Similarly to (<ref>), we denote the concatenation by the following chain

\begin{equation}\label{eqn:ConcatenationInTheorem}
\begin{aligned}
\bar b'_1\coloneqq( 0,0,\dots,0)&\rightarrow_{(E^\phi_{2})_q}\bar b'_{2}\coloneqq( 0,b'_2,r_{3,2},\dots,r_{n,2})\rightarrow_{(E^\phi_{3})_q}\dots  \\\dots
&\rightarrow_{(E^\phi_j)_q} \bar b'_j\coloneqq( 0,b'_2,\dots,b'_j,r_{j+1,j},\dots,r_{n,j})\rightarrow_{(E^\phi_{j+1})_q}\dots \\
\dots &\rightarrow_{(E^\phi_{n})_q}\bar b'_n\coloneqq b'=(0,b'_2,\dots,b'_n),
\end{aligned}
\end{equation}

where each $(E^\phi_j)_q$ is defined on  $I_j\subseteq[-|b_j'-r_{j,j-1}|,|b_j'-r_{j,j-1}|]$, with the convention  $r_{2,1}\coloneqq 0$. Since $D^{\phi}\phi=\omega$ in the broad* sense, by <ref> and in particular (<ref>) and (<ref>), we get that

\begin{equation}\label{eqn:FundamentalOfCalculus}
(\phi_q\circ(E_j^\phi(a))_q)'(t)=(\overline \omega_j)_q((E^\phi_j(a))_q(t))=\omega_j(E^\phi_j((a,t))),
\end{equation}

for all $j=2,\dots,m$, all $t\in I_j$ and all $a\in U''$; where the first equality follows from (<ref>) and the second one by the definition of $(\overline \omega_j)_q$, see (<ref>) and (<ref>). For every $a, b \in U''$, we can now perform the following estimates, which we subsequently explain:

\begin{equation}\label{eqn:KeyEstimate}
\begin{aligned}
|\phi_q(b') -&\phi_q(0)-\omega(a_0)\cdot (b')^1|\\ &=\left|\sum_{j=2}^{m}\left(\phi_q(\bar b'_j)-\phi_q(\bar b'_{j-1})-\omega_j(a_0)b'_j\right)+\sum_{j=m+1}^n \left(\phi_q(\bar b'_j)-\phi_q(\bar b'_{j-1})\right)\right| \\
&=\left|\sum_{j=2}^{m}\left(\omega_j(b_j^*)-\omega_j(a_0)\right)b'_j+\sum_{j=m+1}^n \left(\phi_q(\bar b'_j)-\phi_q(\bar b'_{j-1})\right)\right|  \\
\leq \sup_{j=2,\dots,m}|&\omega_j(b_j^*)-\omega_j(a_0)|\|b'\|+ f\left(\sup_{j=m+1,\dots,n}| b'_j-r_{j,j-1}|\right)\sum_{j=m+1}^n|b'_j-r_{j,j-1}|^{1/\deg j}.
\end{aligned}
\end{equation}

In the second equality we used (<ref>) and, for every $j=2,\dots,m$, the point $b_j^*$ is on $E_j^\phi(I_j)$ and satisfies the conditions of Lagrange's Theorem. In the third inequality we estimated the first term with the supremum norm and the second term by exploiting the definition of $f$ in (<ref>), but replacing in (<ref>) $\phi\circ E^{\phi}_j$ with $\phi_q\circ(E^\phi_j)_q$. Indeed, by (<ref>), the map $\phi_q\circ (E^\phi_j)_q$ is a Euclidean translation (in the horizontal coordinates) of $\phi\circ E^\phi_j$.

Notice that, by continuity and with simple estimates coming from <ref>, by definition of $b'$ and $b^*_j$ one has

\[
\begin{aligned}
\lim_{\varrho\to0}& \left(\sup\{\|b'\|: a,b\in U'', \|a^{-1}b\|\leq \varrho\}\right)=0, \qquad\text{and}\\
\lim_{\varrho\to0}&\left(\sup\{\|a_0^{-1}b^*_j\|: a,b\in U'', \|a^{-1}b\|\leq \varrho\}\right)=0, \quad\forall j=2,\dots, n,
\end{aligned}
\]

where we implicitly mean that $b^*_j$ is also a function of the concatenation and the limit is uniform also with respect to that choice.
Moreover, the concatenation in (<ref>) uniformly collapses to $0$ as $\varrho\to 0$, since by continuity one has

\[
\lim_{\varrho\to0}\left(\sup\{|b'_j-r_{j,j-1}|: a,b\in U'', \|a^{-1}b\|\leq \varrho\}\right)=0, \qquad \forall j=2,\dots,n,
\]

where again, the uniformity has to be understood also in the choice of the concatenation.
We can then find two continuous functions $\alpha_1,\alpha_2\colon (0,+\infty)\to (0,+\infty)$ such that $\lim_{\varrho\to 0} \alpha_1(\varrho)=\lim_{\varrho\to0}\alpha_2(\varrho)=0$ and, combining (<ref>) and (<ref>), one has

\begin{equation}\label{eqn:Final}
\begin{aligned}
\frac{|\phi(b)-\phi(a)-\omega(a_0)\cdot(b^1-a^1)|}{\|\widetilde{\phi}(a)^{-1}a^{-1}b\,\widetilde{\phi}(a)\|}&\leq \sup_{j=2,\dots,m}\|\omega_j-\omega_j(a_0)\|_{L^\infty(B(a_0,\alpha_1(\varrho)))}\\&\hphantom{=}+f(\alpha_2(\varrho))\sum_{j=m+1}^n\frac{|b'_j-r_{j,j-1}|^{1/\deg j}}{\|b'\|},
\end{aligned}
\end{equation}

for every sufficiently small $\varrho>0$ and every $a\in B(a_0,\varrho)$ and $b\in U''$ such that $\|a^{-1}b\|<\varrho$.
We claim that we are in a position to apply <ref> to the function $\phi_q$ and to the curves $E^{\phi_q}_j$ that connect 0 to $b'$. Indeed, for $j=2,\dots,m$, the uniform bound on the Lipschitz constant of the map $\phi_q\circ(E^\phi_j)_q$ follows from (<ref>) and the fact that $\omega$ is continuous, while for $j\geq m+1$ we use the fact that, since $f(\varrho)\to0$ as $\varrho\to0$, the maps $\phi_q\circ(E^\phi_j)_q$ are in $h^{1/\deg j}(I_j)$ with a uniform bound on the Hölder constant. In particular, by (<ref>), we get that for every $\varrho$ sufficiently small, there exists $C'>0$ depending on the uniform bound of $\omega$ on $U'$ such that for every $j=2,\dots,n$

\[
|r_{j,j-1}|^{1/{\rm deg}j}\leq C'\|b'\|,
\]

and thus from (<ref>) we get

\begin{equation*}
\frac{|\phi(b)-\phi(a)-\omega(a_0)\cdot(b^1-a^1)|}{\|\widetilde{\phi}(a)^{-1}a^{-1}b\,\widetilde{\phi}(a)\|} \leq \sup_{j=2,\dots,m}\|\omega_j-\omega_j(a_0)\|_{L^\infty(B(a_0,\alpha_1(\varrho)))}+f(\alpha_2(\varrho))(n-m)(1+C').
\end{equation*}

By the continuity of each $\omega_j$, the proof follows by letting $\varrho\to0$.

For the forthcoming corollaries we recall for the reader's benefit that we are going to use the notation in <ref>.

Let $\mathbb W$ and $\mathbb L$ be two complementary subgroups of a Carnot group $\G$, with $\mathbb L$ horizontal and $k$-dimensional and let $(X_1,\dots,X_n)$ be an adapted basis of the Lie algebra $\mathfrak g$ such that $\mathbb W=\exp({\rm span}\{X_{k+1},\dots,X_n\})$ and $\mathbb L=\exp({\rm span}\{X_{1},\dots,X_k\})$. Let $\widetilde U\subseteq \mathbb W$ be open and let $\widetilde \phi\colon\widetilde U\to \mathbb L$ and $\omega\colon U\subseteq \R^{n-k}\to\R^{k\times(m-k)}$ be two continuous functions. Assume that $\widetilde\phi$ is vertically broad* hölder and assume $D^\phi\phi=\omega$ holds in the broad* sense on $U$. Then the graph of $\widetilde\phi$ is a co-horizontal $C^1_{\rm H}$-surface with tangents complemented by $\mathbb L$.

It is enough to combine <ref> and <ref>.

Let $\mathbb W$ and $\mathbb L$ be two complementary subgroups of a Carnot group $\G$, with $\mathbb L$ horizontal and $k$-dimensional and let $(X_1,\dots,X_n)$ be an adapted basis of the Lie algebra $\mathfrak g$ such that $\mathbb W=\exp({\rm span}\{X_{k+1},\dots,X_n\})$ and $\mathbb L=\exp({\rm span}\{X_{1},\dots,X_k\})$. Let $\widetilde U\subseteq \mathbb W$ be open and let $\widetilde \phi\in \mathrm{ID}(\widetilde U,\mathbb W;\mathbb L)$ be a vertically broad* hölder map and assume $\de ^\phi\!\phi$ is continuous on $\widetilde U$. Then $\widetilde\phi\in \mathrm{UID}(\widetilde U, \mathbb W;\mathbb L)$.

It is enough to combine <ref> and <ref>.

We do not know whether, in arbitrary Carnot groups, one can drop the condition of vertically broad* hölder regularity in <ref>. This is the case for step-2 Carnot groups, with $\mathbb L$ horizontal and one-dimensional, see <ref>.

Let $\mathbb W$ and $\mathbb L$ be two complementary subgroups of a Carnot group $\G$, with $\mathbb L$ horizontal and $k$-dimensional and let $(X_1,\dots,X_n)$ be an adapted basis of the Lie algebra $\mathfrak g$ such that $\mathbb W=\exp({\rm span}\{X_{k+1},\dots,X_n\})$ and $\mathbb L=\exp({\rm span}\{X_{1},\dots,X_k\})$. Let $\widetilde U\subseteq \mathbb W$ be open and let $\widetilde \phi\colon\widetilde U\to \mathbb L$ and $\omega\colon U\subseteq \R^{n-k}\to\R^{k\times(m-k)}$ be two continuous functions. Assume $\widetilde\phi$ is vertically broad* hölder and assume $D^\phi \phi=\omega$ holds in the broad* sense on $U$. Then $\widetilde\phi$ is vertically broad hölder and $D^\phi\phi=\omega$ holds in the broad sense on $U$.

It is enough to combine <ref>, <ref> and <ref>.

§.§ Area formula for codimension-one graphs in terms of intrinsic derivatives

We prove here that, if $\mathbb L$ is horizontal and one-dimensional, and $\widetilde\phi\in {\rm UID}(\widetilde U,\mathbb W;\mathbb L)$, with $\widetilde U$ open, then $D^{\phi}\phi=\nabla^{\phi}\phi$ holds in the sense of distributions. For a precise definition of the distribution $D^{\phi}\phi$ the interested reader may soon read the first lines of the proof of <ref>. We also provide an area formula for ${\rm graph}(\widetilde\phi)$ in this case. For the case $\mathbb G= \mathbb H^n$, this formula was already obtained in <cit.>. Recall that we are going to use the notation given in <ref>.

Let $\W$ and $\mathbb L$  be two complementary subgroups of a Carnot group $\G$ with $\mathbb L$ horizontal and one-dimensional, and let $(X_1,\dots, X_n)$ be an adapted basis of the Lie algebra $\mathfrak g$ such that $\mathbb L=\exp({\rm span}\{X_1\})$ and $\mathbb W=\exp({\rm span}\{X_2,\dots,X_n\})$. Let $\widetilde U \subseteq \mathbb W$ be open and consider $\widetilde \phi\in \mathrm{UID}(\widetilde U, \mathbb W; \mathbb L)$. Then the following facts hold.

(a) For every $j=2,\dots,m$, the distribution $D_j^{\phi}\phi\coloneqq D_{X_j}^{\phi}\phi$ is well-defined on $U$.

(b) For every $a\in U$, there exist $\delta>0$ and a family of functions $\{\phi_\eps\in C^1(B(a,\delta)):\eps\in (0,1)\}$ such that

\[
\lim_{\eps\to0}\phi_\eps=\phi \quad\text{and}\quad\lim_{\eps\to0}D_j^{\phi_\eps}\phi_\eps=\nabla^\phi\phi(X_j)  \quad\text{in $L^\infty(B(a,\delta))$},
\]

for every  $j=2,\dots,m$.

(c) The equation

\begin{equation}\label{eqn:DistributionalEquation}
D_j^{\phi}\phi = \nabla^{\phi}\phi(X_j)\eqqcolon\nabla^\phi_j\phi,
\end{equation}

holds in the distributional sense on $U$, for every $j=2,\dots,m$. Here $\nabla^{\phi}\phi$ is the intrinsic gradient of $\phi$, see <ref>.

(d) The subgraph of $\phi$ defined by $ E_\phi\coloneqq\{ w\cdot\exp(tX_1): w\in U, t<\phi(w)\}$ has locally finite $\G$-perimeter[Here we take the usual definition of the horizontal perimeter with respect to the orthonormal basis $(X_1,\dots,X_m)$, see <cit.>.] in $U\cdot \exp(\R X_1)$ and its $\G$-perimeter measure $|\partial E_\phi|_{\G}$ is given by

\begin{equation}\label{eqn:PerimeterSubgraph}
|\partial E_\phi|_{\G}(V) =\int _{\Phi^{-1}( V)} \sqrt{1+  |\nabla^{ \phi} \phi|_{m-1}^2} \,  \de\mathscr{L}^{n-1},
\end{equation}

for every Borel set $ V\subseteq U\cdot \exp{(\R X_1)} $, where $\Phi\colon U\to \R^n$ is the graph function of $\widetilde \phi$ composed with the exponential coordinates, and with a little abuse of notation we wrote $U\cdot \exp(\R X_1)$ meaning the set $\widetilde U\cdot \exp(\R X_1)$ embedded in $\R^n$ through exponential coordinates.

Moreover, the set $\mbox{graph}(\widetilde\phi)$ has a unit normal given, up to a sign, by

\begin{equation}\label{eqn:Normal}
\nu_{E_{\phi}} = \left(-\frac{1}{\sqrt{1+  |\nabla^{ \phi} \phi|_{m-1}^2}},\frac{\nabla_2^{\phi}\phi}{\sqrt{1+  |\nabla^{ \phi} \phi|_{m-1}^2}},\dots,\frac{\nabla_m^{\phi}\phi}{\sqrt{1+  |\nabla^{ \phi} \phi|_{m-1}^2}}\right)\in \R^m.
\end{equation}

Notice that the fact that $D^{\phi}_j\phi$ is a well-defined distribution on $U$ is a consequence of <ref> and the fact that $\mathbb L$ is one-dimensional. Indeed, in coordinates, we get that the vector field $D^{\phi}_j$ is the sum of terms $g(x)\phi^h\partial_{x_i}$, for some polynomial function $g$ of the coordinates of $\mathbb W$, some integer $h\geq 0$, and some $i=2,\dots,n$. Thus in order to define the distribution $D^{\phi}_j\phi$, we only need to define $g(x)\phi^h\partial_{x_i}\phi\coloneqq g(x)\frac{1}{h+1}\partial_{x_i}\phi^{h+1}$. Since $g(x)\frac{1}{h+1}\partial_{x_i}\phi^{h+1}$ is well-defined in the sense of distributions, because $\phi$ is continuous, (a) is proved.

We now show that equality (<ref>) holds pointwise if $\widetilde{\phi}\colon\widetilde{U}\subseteq \mathbb W\to\mathbb L$ is of class $C^1$ in $\widetilde{U}$. In case $\widetilde \phi\in C^1(\widetilde U)$, then, by <cit.>, one has $\widetilde{\phi}\in \mathrm{UID}(\widetilde U, \mathbb W; \mathbb L)$.
We can assume without loss of generality that $e\in\widetilde{U}$ and $\widetilde{\phi}(e)=e$. Indeed, if we want to prove identity (<ref>) in $a\in U$, we may consider $\widetilde{\phi}_p$ with $p\coloneqq\widetilde{\phi}(a)^{-1}\cdot  a^{-1}$, and use the invariance properties given by <ref> and <ref>, and then notice that $\widetilde{\phi}_p(e)=e$.
By <ref>, since $\widetilde \phi$ is ${\rm UID}$, the set ${\rm graph}(\widetilde{\phi})$ is a $C^1_{\rm H}$-hypersurface and therefore there exist a neighborhood $\widetilde V$ of $e$ in $\G$ and a function $f\in C^1_{\rm H}(\widetilde V)$ such that

\begin{equation}\label{eqn:proprietaC1H}
\mathrm{graph}(\widetilde\phi)\cap \widetilde V=\{p\in \G: f(p)=0\}\cap \widetilde V \quad \mbox{and} \quad \nabla_\G f\neq 0 \text{ on $\widetilde V$}.
\end{equation}

Then, for every sufficiently small $\eps>0$ and for every $j=2,\dots,n$, one has

\[
f\Big(\exp (\eps X_j) \widetilde{\phi} (\exp (\eps X_j)) \Big)=0.
\]

Therefore, with a little abuse of notation, one can differentiate with respect to $\eps$ to get

\begin{equation}\label{eqn:DeriveF}
\begin{split}
0& = \frac \de {\de\eps }_{|_{\eps=0}} f\Big(\exp (\eps X_j) \widetilde{\phi} (\exp (\eps X_j)) \Big)\\
&= \frac \de {\de\eps }_{|_{\eps=0}} f\Big(\widetilde{\phi} (\exp (\eps X_j)) \Big) + \frac \de {\de\eps }_{|_{\eps =0}} f\Big(\exp (\eps X_j)  \Big) \\
&= X_1 f_{|_e}\left(\frac \de {\de\eps }_{|_{\eps =0}} \phi  (\eps X_j) \right) + X_jf_{|_e}\\
&= X_1 f_{|_e}\left(\frac \de {\de\eps }_{|_{\eps =0}} ( \phi \circ \pi_{\mathbb W})(\eps X_j) \right) + X_jf|_e\\
&= (X_1 f)_{|_e} D_j^\phi\phi|_e  + (X_jf)_{|_e},
\end{split}
\end{equation}

where we used the fact that $\G=\mathbb W\cdot\mathbb L$ and exploited the fact that $\widetilde{\phi}$ takes values in $\mathbb L=\exp({\rm span}\{X_1\})$. The last equality follows by using the definition of $D^{\phi}_j$ acting on $\phi$, see (<ref>). The claim is then obtained by (<ref>) and (<ref>).

To prove (b), we use some ideas of <cit.> to show that, for any $a\in U$, there exist $\delta>0$ and a family of functions $\{\phi_{\eps}\in C^1(U): 0<\eps<1\}$, such that

\[
\phi_{\eps}\to\phi, \qquad {\rm and} \quad  D_j^{\phi_{\eps}}\phi_{\eps} \to \nabla^{\phi}\phi(X_j), \quad \forall j=2,\dots,m,
\]

uniformly in the Euclidean ball $B_{\rm e}(a,\delta)\Subset U$, as $\eps\to 0$.

By <ref>, we can find a neighborhood $\widetilde V$ of $a\cdot \widetilde \phi(a)$ and $f\in C^1_H(\widetilde V)$ satisfying (<ref>). Let $\delta>0$  be such that, setting  $B\coloneqq B_{\rm e}(a,\delta)$, one has $\widetilde{B}\cdot\widetilde{\phi}(\widetilde{B})\subseteq \widetilde V$. Then, up to reducing $\delta$ and $\widetilde V$ and a regularization argument analogous to <cit.>, we can construct a family  $\{f_{\eps}\in C^1(\widetilde V): 0<\varepsilon<1\}$ such that

\begin{equation}\label{eqn:UniformConvergenceXj}
\lim_{\eps\to0}\left(\sup_{\widetilde V}|X_jf_{\eps}- X_jf|\right)=0, \qquad \forall j=1,\dots,m.
\end{equation}

Since $\widetilde{\phi}$ takes values in $\exp({\rm span}\{X_1\})$, by <ref>, we can assume without loss of generality that $X_1f>0$ on $\widetilde V$, since $X_1f\neq 0$ and we are free to possibly exchange $f$ with $-f$. By (<ref>), we can find $\varepsilon_0>0$ such that $X_1f_{\eps}>0$ on $\widetilde V$, for every $\eps\in (0,\eps_0)$. For any such $\eps>0$, by the Euclidean implicit function theorem, we can find $\widetilde \phi_{\eps}$, defined on $\widetilde B$, such that

\[
\mathrm{graph}(\widetilde \phi_\eps)\cap \widetilde{B}\cdot\widetilde{\phi}_\eps(\widetilde B)=\{p\in \G: f_\eps(p)=0\}\cap\widetilde{B}\cdot\widetilde{\phi}_\eps(\widetilde B).
\]

Moreover, since $f_\eps$ is smooth, then also $\widetilde \phi_\eps$ is smooth. In particular, for every $b\in \widetilde{B}$, one has

\[
f_{\eps}(b\cdot\widetilde{\phi}_{\eps}(b))=0, \qquad \forall \eps\in (0,\varepsilon_0).
\]

From <cit.> we deduce that

\begin{equation}\label{eqn:UniformConvergencePhi}
\lim_{\eps\to 0}\left(\sup_B|\phi_{\eps}-\phi|\right)=0.
\end{equation}

Denote by $\Phi_{\eps}\colon B\to \R^n$ the graph function of $\widetilde\phi_\eps$ composed with the exponential coordinates. Since $\widetilde{\phi}_{\eps}$ is of class $C^1$, by using the pointwise version of (<ref>) for $C^1$ functions and (<ref>), we deduce that

\begin{equation}\label{eqn:nonnevica}
D^{\phi_{\eps}}_j\phi_{\eps}(x)=-\frac{X_jf_{\eps}}{X_1f_{\eps}}\circ \Phi_{\eps}(x), \qquad \forall j=2,\dots,m, \quad \forall x\in B.
\end{equation}

Then, by (<ref>), (<ref>), (<ref>) and (<ref>), we conclude that, for any $j=2,\dots, m$, the family $D^{\phi_{\eps}}_j\phi_{\eps}$ converges uniformly on $B$ to $-\frac{X_jf}{X_1f}\circ\Phi=\nabla^{\phi}\phi(X_j)$, as $\varepsilon \to 0$.

The proof of (c) follows directly from (b) and the particular form of the distribution $D^{\phi}_j\phi$ we discussed at the beginning of this proof. In fact from the convergence proved in (b), we know that, for any $a\in U$, there exists $\delta>0$ such that $D_j^\phi \phi=\nabla^\phi\phi(X_j)$ on $B(a, \delta)$ in the sense of distributions, for every $j=2,\dots, m$. It is then enough to consider a locally finite countable open sub-covering $\{B(a_h,\delta_h):h\in \mathbb N\}$ of $U$ and build a partition of unity subordinate to it. The fact that $D^\phi_j\phi=\nabla^\phi \phi(X_j)$ holds in the sense of distributions on $U$ is a consequence of the local identity, the linearity of distributions, and a standard argument using the partition of unity.

To prove (d), we first notice that, by <cit.> and <ref>, we know that, for every $p\in \mathrm{graph}(\widetilde \phi)$ there exists a neighborhood $\widetilde V'$ of $p$ and a function $ f\in C_H^1(\widetilde V')$, with $X_1 f>0$ on $\widetilde V'$, representing $\mathrm{graph}(\widetilde \phi)$ as non critical level set and such that

\[
|\partial  E_\phi|_\G ( V) = \int _{\Phi^{-1}(V)} \frac{ |\nabla_\G  f|}{X_1 f}\circ \Phi \, \de \mathscr{L}^{n-1}
\]

holds for every Borel set $ V\subseteq  V'$. Now by (<ref>), we can write

\begin{equation}\label{eq:arealocale}
|\partial  E|_\G ( V) = \int _{\Phi^{-1}(V)} \sqrt{1+|\nabla^\phi \phi|^2_{m-1}} \de \mathscr{L}^{n-1}, \quad\text{for every Borel set }\quad V \subseteq  V'.
\end{equation}

Since the right hand side of (<ref>) does not depend on the choice of $ f$, by a covering argument we can extend it to every Borel subset $V$ of $ U\cdot \exp(\R X_1)$.

The explicit expression of the unit normal in (<ref>) comes from the fact that the graph of $\widetilde\phi$ is locally the zero-level set of $ f$. Thus, the unit normal of ${\rm graph}(\widetilde\phi)$ is in the direction of $\nabla_{\mathbb H} f$, and taking  (<ref>) into account, one has $\nabla_j^{\phi}\phi=-\frac{X_j f}{X_1 f}\circ\Phi$ for every $j=2,\dots,m$, and then we conclude by normalization.

For what concerns the area formula in arbitrary Carnot groups, in <cit.>, the author proves that

\begin{equation}\label{eqn:AreaFormulaMagnani}
P_\G(E)=\beta(d,\nu_E)\mathscr S^{Q-1}\res \mathcal FE,
\end{equation}

for any set $E$ of finite perimeter with $C^1_{\rm H}$-rectifiable reduced boundary $\mathcal FE$. The density $\beta$ is explicitly computed in <cit.> and depends on the metric $d$ and on the normal $\nu_E$ of $E$ that is defined in the sense of Geometric Measure Theory. Moreover, in case the distance $d$ is vertically symmetric, $\beta$ is a constant that only depends on the group $\G$ and on the metric $d$, see <cit.>. We finally notice that every Carnot group admits a metric $d$ that is vertically symmetric, see <cit.>.
For a survey on the area formula in Carnot groups, we refer the reader to <cit.>.

Notice that if $\widetilde{\phi}$ is in ${\rm UID}(\widetilde U,\mathbb W;\mathbb L)$, the set ${\rm graph}(\widetilde\phi)$ is a $C^1_{\rm H}$-hypersurface by <ref>. Then, by definition of $C^1_{\rm H}$-rectifiability and by <cit.>, the subgraph $E_{\phi}$ of $\widetilde\phi$ has a $C^1_{\rm H}$-rectifiable reduced boundary $\mathcal{F}E_{\phi}={\rm graph}(\widetilde\phi)$. Thus we are in a position to apply <ref> and <cit.>, and in particular to compare (<ref>) with (<ref>) in order to obtain the explicit representation

\begin{equation}\label{eqn:AreaFormula2}
\int_{\widetilde V\cap {\rm graph}(\widetilde \phi)} \beta(d,\nu_{E_\phi})\de \mathscr S^{Q-1} = \int_{\Phi^{-1}(V)} \sqrt{1+|\nabla^{\phi}\phi|_{m-1}^2}\de\mathscr L^{n-1},
\end{equation}

for every Borel set $\widetilde V\subseteq \widetilde U\cdot \exp(\mathbb R X_1)$.

A general area formula for a $C^1_{\rm H}$-surface $\Sigma$, valid in an arbitrary Carnot groups, has been very recently obtained in <cit.>. If we call $\alpha$ the Hausdorff dimension of $\Sigma$, and we suppose $\Sigma={\rm graph}(\widetilde\phi)$, this formula allows for a representation of $\mathscr S^\alpha\res \Sigma$ as an integral on $\widetilde U$ of a properly defined area element with respect to $\mathscr S^{\alpha}\res \mathbb W$, see <cit.>. According to the previous equation (<ref>), we thus get that for an arbitrary Carnot group $\mathbb G$, and in case $\mathbb L$ is one-dimensional, the area element of <cit.> is, up to the function $\beta$, explicitly written in terms of the intrinsic gradient of $\widetilde \phi$.

Eventually, by the recent work [24], in particular <cit.>, we get that on $\mathbb H^n$ equipped with a vertically symmetric distance, in case $\mathbb L$ is horizontal and $k$-dimensional, one can explicitly write the area element of <cit.> in terms of the intrinsic gradient $\nabla^{\phi}\phi$.

Let $\mathbb W$ and $\mathbb L$ be two complementary subgroups of a Carnot group $\G$, with $\mathbb L$ horizontal and one-dimensional, and let $(X_1,\dots, X_n)$ be an adapted basis of the Lie algebra $\mathfrak g$ such that $\mathbb W=\exp({\rm span}\{X_{2},\dots,X_n\})$ and $\mathbb L=\exp({\rm span}\{X_{1}\})$. Let $\widetilde U\subseteq \mathbb W$ be open, and let $\widetilde \phi\colon\widetilde U\to \mathbb L$ and $\omega\colon U\subseteq \R^{n-1}\to\R^{m-1}$ be two continuous functions. Assume $\widetilde\phi$ is vertically broad* hölder and assume that $D^\phi \phi=\omega$ in the broad* sense on $U$. Then $D^\phi\phi=\omega$ in the sense of distributions and $\omega=\nabla^\phi\phi$.

It is enough to combine <ref> and item (c) of <ref>.

§.§ Relations between intrinsic differentiability and local approximability

Point (b) of <ref> can actually be generalized to the case $\mathbb L$ horizontal and $k$-dimensional. The computation for the $k$-dimensional case is similar but one should pay attention to the fact that  if $f\eqqcolon(f^{(1)},\dots, f^{(k)})\in C^1_{\rm H}(\widetilde V;\R^k)$ is a vector valued map, its horizontal gradient is a $(k \times m)$-dimensional matrix (see <ref>). Using a regularization argument that is similar to <cit.> as in the proof of <ref>, we can find a family of functions $\{f_{\eps}\in C^1(\widetilde V;\R^k):0<\eps<1\}$, such that each component $X_jf^{(i)}_{\eps}$ converges uniformly to $X_jf^{(i)}$ for every $i=1,\dots, k$ and $j=1,\dots, m$ as $\eps\to 0$, and such that, for every $\eps\in (0,1)$, the associated matrix $\nabla_{\mathbb L}f_\eps$ defined in <ref> has det$\nabla_{\mathbb L}f_\eps \ne 0$ on $\widetilde V$.

Then in order to prove point (b) of <ref> in this general case, we take the previous changes into account, the straightforward changes in (<ref>), and we run the same computations of the proof of <ref> by using (<ref>) instead of (<ref>) at the end of the argument.

The statement (i)$\Rightarrow$(ii) of <cit.>
claims that, in the Heisenberg groups $\mathbb H^n$, if $\sL$ is one-dimensional and $\phi\in {\rm UID}(U,\sW;\sL)$, then there exists $\omega\in C(U;\R^{2n-1})$ such that $\nabla ^\phi\phi=\omega$ in the sense of distributions and there exists a family $\{\phi_\eps\in C^1(U): \eps\in (0,1)\}$ such that $\phi_\eps\to \phi$ and $\nabla^{\phi_\eps}\phi_\eps\to \omega$ uniformly on any compact subset $K\subseteq U$.

However, the implication (5.17)$\Rightarrow$(5.19) in its proof is imprecise. In sight of this, one could replace point (ii) with a local version of it in which the approximating functions $\{\phi_{\eps}\}$ depend on the point $a$ and the convergence is uniform in a neighborhood $B(a,\delta)$, see <cit.>. This does not affect the validity of the proofs, that only refer to <cit.>, which holds true with the  weakened approximation assumptions, since it has a local statement. Indeed, the proof of (ii)$\Rightarrow$(i) of <cit.> just needs the local approximation. To the best of our knowledge, all the references to <cit.> just require the local approximation: see point (3) <cit.>, point (3) of <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>.

Anyway, the original formulation of (i)$\Rightarrow$(ii) of <cit.> can be fixed with the approximation argument exploited in the proof of <cit.> and we expect this to hold true in all Carnot groups of step 2. We warmly thank Francesco Serra Cassano and Davide Vittone for precious suggestions.

Let $\mathbb W$ and $\mathbb L$ be two complementary subgroups in a Carnot group $\mathbb G$, with $\mathbb L$ $k$-dimensional and horizontal. Let $(X_1,\dots, X_n)$
be an adapted basis of the Lie algebra $\mathfrak g$
such that $\mathbb L=\exp({\rm span}\{X_1,\dots,X_k\})$ and
$\mathbb W=\exp({\rm span}\{X_{k+1},\dots,X_n\})$. Let $\widetilde U\subseteq \mathbb W$ be an open set and let $\widetilde{\phi}\colon\widetilde{U}\to\mathbb L$ be a continuous function.

Assume there exists $\omega \in C(U;\R^{k\times(m-k)})$ such that, for every $a\in U$, there exist $r>0$,  and a family of functions $\{\phi_\eps \in C^1(B(a,r);\R^k): 0<\eps<1\}$ satisfying

\begin{equation*}
\lim_{\eps\to0}\phi _\eps=\phi  \quad\text{in $L^\infty(B(a,r))$}\quad  \text{and} \quad  \lim_{\eps\to0}D^{{\phi _\eps}} {\phi _\eps}=\omega \quad\text{in $L^\infty(B(a,r);\R^{k\times(m-k)})$}.
\end{equation*}

Then $\phi $ is a broad* solution in $U$ of the system $ D^{{\phi}} {\phi} = \omega $.

The proof closely follows the argument used in <cit.>. For simplicity, we consider the case $k=1$. The case $k\geq 2$ can be reached with the same proof and some typographical effort.

Let $a\in U$ and fix $j\in \{2,\dots , m\}$ and $\eps >0$.
Since $\phi_\eps\in C^1(B(a,r))$, we can find $0<\delta_2(\eps)<\delta_1<r$ and a map $E^{\phi_\eps}_j \colon \overline{B(a, \delta_2 (\eps))}\times[-\delta _2(\eps),\delta _2(\eps)] \to  \overline{B(a,\delta_1)}$ such that $E^{\phi_{\eps}}_j(b,\cdot)$ is the unique solution of the Cauchy problem

\begin{equation*}
\left\{
\begin{array}{l}
\gamma '=D^{\phi _\eps} _{j} \circ\gamma, \\
\gamma (0)= b,
\end{array}
\right.
\end{equation*}

in the interval $[-\delta_2(\eps),\delta_2(\eps)]$, for every $b\in \overline{B(a,\delta_2(\eps))}$.
By Peano's estimate on the existence time for solutions of ordinary differential equations (see e.g. <cit.>) we can choose $\delta_2 (\eps)=C/\| P_j(x,\phi _\eps) \|_{L^\infty ( \overline{B(a, \delta_1 ) })} $, with $C$ only depending on $\delta _1$ and with $P_j$ a polynomial function of the coordinates $x$ of $\mathbb W$ and the components of $\phi_{\eps}$. This polynomial function depends only on the structure of $D^\phi_j$, see (<ref>). In particular, since $\phi_\eps$ is converging uniformly on $B(a,\delta_1)$, then $\delta_2(\eps)$ has a positive lower bound $M$ independent on $\eps$ and we are going to verify the conditions of <ref> with $\delta_2\leq M$.
Since $\phi_\eps$ are bounded on $B(a,\delta_1)$ uniformly in $\varepsilon>0$ and $D^{\phi_\eps}_j$ are vector fields with polynomial coefficients in $\phi_\eps$ and, possibly, in some of the coordinates, the functions $E^{\phi_\eps}_j$ are equi-Lipschitz with respect to $\varepsilon$ on the compact set $\overline{B(a, \delta_2)}\times[-\delta_2,\delta_2]$. By Arzelá-Ascoli Theorem, we can therefore find an infinitesimal sequence $(\eps _h)$ in $(0,1)$ such that $E^{\phi_{\eps _h}}_j$ converges to some continuous function $E_j^\phi $ uniformly on $\overline{B(a, \delta_2)}\times[-\delta_2,\delta_2]$. By definition of $E^{\phi_{\eps_h}}_j$, one has

\begin{equation*}
E^{\phi_{\eps _h}}_j(b,t)= b + \int_0^t D^{\phi _{ \eps_h} }_{j} \left(E^{ \phi_{\eps_h}}_j(b,s)\right)\de\! s \qquad \text{and}
\end{equation*}

\begin{equation*}
\phi _{\eps_h}\left(E_j^{\phi_{\eps_h}}(b,t)\right) - \phi _{\eps_h}\left(E_j^{\phi_{ \eps_h}}(b,0)\right)= \int_0^t  D^{\phi _{ \eps_h} }_{j} \phi _{ \eps_h}\left(E_j^{\phi_{\eps_h}}(b,s)\right)\de \!s,
\end{equation*}

for every $b\in \overline{B(a,\delta_2)}$ and every $t\in [-\delta_2,\delta_2]$.
By letting $h\to \infty$ and using that all the involved convergences are uniform, we get

\begin{equation*}
E _j^\phi(b,t)= b+ \int_0^t D^\phi _{j}\left({E_j^\phi(b,s)}\right)\de \!s,\qquad \text{and}
\end{equation*}

\begin{equation*}
\quad\phi(E_j^\phi(b,t))-\phi(E_j^\phi(b,0))= \int_0^t \omega _{j} (E _j^\phi(b,s)) \de\! s,
\end{equation*}

for every $b\in \overline{B(a,\delta_2)}$ and every $t\in [-\delta_2,\delta_2]$, which are the conditions we were looking for to make $D^\phi\phi=\omega$ hold in the broad* sense.

Let $\mathbb W$ and $\mathbb L$ be two complementary subgroups in a Carnot group $\mathbb G$, with $\mathbb L$ horizontal and k-dimensional. Let $(X_1,\dots, X_n)$
be an adapted basis of the Lie algebra $\mathfrak g$
such that $\mathbb L=\exp({\rm span}\{X_1,\dots,X_k\})$ and
$\mathbb W=\exp({\rm span}\{X_{k+1},\dots,X_n\})$. Let $\widetilde U\subseteq \mathbb W$ be open and let  $\widetilde{\phi}\in \mathrm{UID}(\widetilde{U},\mathbb W;\mathbb L)$.
Then, there exists  $\omega \in C(U ; \R^{k\times (m-k)})$ such that $\phi $ is a broad* solution in $U$ of the system $ D^{{\phi}} {\phi} = \omega $. Moreover, $\omega=\nabla^\phi\phi$.

It is enough to choose $\omega=\nabla^\phi\phi$, which is continuous taking <ref> into account, since $\widetilde{\phi}\in \mathrm{UID}(\widetilde{U},\mathbb W;\mathbb L)$. The proof follows by combining item (b) of <ref>, which also holds in case $\mathbb L$ is $k$-dimensional, see <ref>, together with <ref>.

§.§ Main theorem

Now we are in a position to give the following theorem, which is a generalization of <cit.> to all Carnot groups.

Let $\mathbb W$ and $\mathbb L$ be two complementary subgroups in a Carnot group $\mathbb G$, with $\mathbb L$ horizontal and k-dimensional. Let $\widetilde U\subseteq \mathbb W$ be open and let  $\widetilde{\phi}\colon\widetilde{U}\subseteq\mathbb W\to\mathbb L$ be a continuous function. Fix a graded basis $(X_1,\dots,X_n)$  of the Lie algebra $\mathfrak g$ such that $\mathbb L=\exp({\rm span}\{X_1,\dots,X_k\})$ and $\mathbb W=\exp({\rm span}\{X_{k+1},\dots,X_n\})$.  Then, the following facts are equivalent.

(a) $\widetilde\phi\in {\rm UID}(\widetilde U, \mathbb W; \mathbb L)$;

(b) $\phi$ is vertically broad* hölder and there exists $\omega\in C(U;\R^{k\times(m-k)})$ such that, for every $a\in U$, there exist $\delta>0$ and a family of functions $\{\phi_\eps\in C^1(B(a,\delta);\R^k):\eps\in (0,1)\}$ such that

\[
\lim_{\eps\to0}\phi_\eps=\phi \quad\text{and}\quad\lim_{\eps\to0}D_j^{\phi_\eps}\phi_\eps=\omega_j  \quad\text{in $L^\infty(B(a,\delta);\R^k)$},
\]

for every  $j=k+1,\dots,m$;

(c) $\phi$ is vertically broad hölder and there exists $\omega\in C(U;\R^{k\times(m-k)})$ such that $D^\phi \phi=\omega$ in the broad sense.

(d) $\phi$ is vertically broad* hölder and there exists $\omega\in C(U;\R^{k\times(m-k)})$ such that $D^\phi \phi=\omega$ in the broad* sense.

Moreover, if any of the previous holds, then $\omega=\nabla^\phi\phi$.

(a)$\Rightarrow$(b) follows by combining <ref>, item (b) of <ref> in the $k$-dimensional case following the lines of <ref>. (b)$\Rightarrow$(d) follows from <ref>.
(d)$\Rightarrow$(a) follows from <ref>. (a)$\Rightarrow$(c) follows from <ref>, <ref> and the continuity of $\omega$ follows from <ref>. The implication (c)$\Rightarrow$(d) is trivial.

Notice that in <cit.>, in the setting of <ref>, the author constructs a function $\widetilde\phi\colon\widetilde U\subseteq \mathbb W\to\mathbb L$ such that $D^{\phi}\phi=-1$ in the broad* sense but $\widetilde\phi$ is not vertically broad* hölder, see <ref>. Taking <ref> into account, this means that $\widetilde\phi$ cannot be UID. This also means that, in general, in <ref>, one cannot drop the assumption on the vertically broad* hölder regularity in arbitrary Carnot groups. We will show that this is possible for step 2 Carnot groups, in <ref>. For more examples related to this topic, we refer the reader to <cit.>.

§ SOME APPLICATIONS

Let us begin with an observation that will motivate the first part of this section. Consider the first Heisenberg group $\mathbb H^1$ with Lie algebra $\mathfrak g\coloneqq {\rm span}\{X,Y\}\oplus{\rm span}\{Z\}$, and the only nontrivial relation $[X,Y]=Z$. By a direct application of the Baker-Campbell-Hausdorff formula one gets $\exp X\exp Y\exp (-X)\exp (-Y)=\exp Z$.
By exploiting this formula, in <ref> below, we give an alternative proof, in the case of $\mathbb H^n$ with $n\geq 2$, of <cit.>.
The argument we use is different because we prove that being a broad* solution with a continuous datum implies being $1/2$-little Hölder continuous along vertical coordinates (see also <ref>) that is actually simpler than proving 1/2-little Hölder continuity in all the coordinates as in <cit.>. Nevertheless, this is sufficient for applying <ref>.  Similarly, in <ref>, we obtain the same result of <cit.>, by making use of  <ref> and <ref>. We remark that our argument is different by the one used in [16] but, on the contrary, it does not work for $n=1$.

Recently, in <cit.>, the author has proved a generalization of <cit.> that holds for arbitrary complemented subgroups in $\mathbb H^n$, and this statement is one of the key step in order to get the main theorem <cit.>. By using our reasoning, we are also able to recover <cit.>, and thus <cit.>, in the case $\mathbb H^n=\mathbb W\cdot \mathbb L$ with $\mathbb L$ horizontal and $k$-dimensional, with $k<n$. However, our argument does not apply to the remaining case $k=n$, while the argument in the reference does.

Even if these results already appeared in the literature, we think it is worth writing them down with these different proofs. Indeed,  the proof of <ref> provides a useful “toolkit” for the forthcoming theorems: in fact, a similar idea to the one exploited in <ref> will be used in <ref> to prove the analogous version of <ref> in the setting of free Carnot groups of step 2, when $\mathbb L$ is one-dimensional. This will be the key step  to obtain the analogous <ref> for Carnot groups of step 2 with $\mathbb L$ one-dimensional. We refer the reader to the introduction of <ref>.

In <ref> below we give a class of nontrivial examples of ${\rm UID}$ functions in the Engel group. We build them by making use of  <ref>. This class of examples is inspired by <cit.>. Moreover, a slight modification of <cit.>, gives rise to functions whose intrinsic graphs are both of class $C^1_{\rm H}$ and of class $C^1$, but they possess a characteristic point, see <ref>. We thank R. Serapioni for having discussed this example with us.

§.§ A different proof of the propagation of broad* regularity in $\mathbb H^n$, with $n\geq 2$

Let $\mathbb W$ and $\mathbb L$ be two complementary subgroups of $\mathbb H^n$, with $n\geq2\, ($see $\cref{example:Heisenberg})$ such that $\mathbb L$ is horizontal and $k$-dimensional with $k< n$, and let $\widetilde\phi\colon\widetilde U \subseteq \mathbb W\to\mathbb L$ be a continuous function, with $\widetilde U$ open.

Then there exists $(X_1,\dots,X_{2n+1})$ an adapted basis of the Lie algebra such that $\mathbb L=\exp({\rm span}\{X_1,\dots,X_k\})$, $\mathbb W=\exp({\rm span}\{X_{k+1},\dots,X_{2n+1}\})$, and such that the only nonvanishing bracket relations are given by $[X_i,X_{n+i}]=X_{2n+1}$ for every $1\leq i\leq n$.

Moreover, if there exists a continuous function $\omega\colon U\subseteq \mathbb R^{2n+1-k}\to\mathbb R^{k\times (2n-k)}$ such that $D^{\phi}\phi=\omega$ in the broad* sense, then  $\widetilde\phi$ is vertically broad* holder. Namely, since we are in $\mathbb H^n$ (see <ref>), for every $a_0 \in U$ there exists a neighborhood $U_{a_0}\Subset U$ of $a_0$ such that

\begin{equation}\label{eqn:ThesisBroad*}
\lim_{\varrho\to 0}\left(\sup\left\{\frac{|\phi(\xi,z_1)-\phi(\xi,z_2)|}{|z_1-z_2|^{1/2}}: (\xi,z_1)\in U_{a_0}, (\xi,z_2)\in U_{a_0}, 0<|z_1-z_2|<\varrho\right\}\right)=0,
\end{equation}

where we mean that $\xi\in  U_{a_0}\cap\{z=0\}\subseteq  \R^{2n-k}$ and $z_1,z_2\in \R$ are the $(2n+1)^{th}$ coordinates of $(\xi, z_1)$ and $(\xi,z_2)$ when seen as elements of $\mathbb H^n$.

The existence of the basis as in the first part of the statement comes from  <cit.>. Fix $a_0\in U$ and find $\delta>0$ and sufficiently small neighborhoods $V'_{a_0} \Subset V_{a_0}\Subset U$ of $a_0$ such that, for every $a\in V'_{a_0}$, and every $j=k+1,\dots,2n$, we have the existence of integral curves $E^{\phi}_j(a)\colon[-\delta,\delta]\to V_{a_0}$ satisfying the conditions of <ref>. Denote by $\beta\colon[0,+\infty)\to[0,+\infty)$ a modulus of uniform continuity for $\omega$ on $\overline{V_{a_0}}$.

We fix a neighborhood $U_{a_0}\Subset V'_{a_0}$ of $a_0$, $\varrho>0$ and points $(\xi,z_1),(\xi, z_2)\in U_{a_0}$ such that $|z_1-z_2|<\varrho$. The set $U_{a_0}$ has to be chosen small enough: this will be clear during the proof. We are going to prove that, for every sufficiently small $\varrho$, we have

\begin{equation}\label{eqn:FinalToObtain}
\frac{|\phi(\xi,z_1)-\phi(\xi,z_2)|}{|z_1-z_2|^{1/2}} \leq 2k\beta(\alpha(\varrho)),
\end{equation}

for a continuous function $\alpha$ with $\alpha(0)=0$ that only depends on the norms of $\phi$ and $\omega$ on $V_{a_0}$. From this fact (<ref>) would follow, concluding the proof.

Recall that, by (<ref>) and since $k<n$, one has $D^{\phi}_{k+1}=X_{k+1}$ and $D^{\phi}_{n+k+1}=X_{n+k+1}$. Assume without loss of generality that $z_2>z_1$, and set $t_0\coloneqq (z_2-z_1)^{1/2}$. We exploit the relation $[D^{\phi}_{k+1},D^{\phi}_{n+k+1}]=X_{2n+1}$ and the Baker-Campbell-Hausdorff formula to conclude that we can join $(\xi,z_1)$ and $(\xi,z_2)$ by means of a concatenation of integral curves of $D^{\phi}_{k+1}=X_{k+1}$ and $D^{\phi}_{n+k+1}=X_{n+k+1}$. In particular

\begin{equation}\label{eqn:ConcatenationOfCurve2}
\begin{split}
a\coloneqq (\xi,z_1)&\to_{E^{\phi}_{k+1}(a)} \, a_1\coloneqq E^{\phi}_{k+1}(a,t_0) \to_{E^{\phi}_{n+k+1}(a_1)} \, a_2\coloneqq E^{\phi}_{n+k+1}(a_1,t_0) \\
&\to_{E^{\phi}_{k+1}(a_2)} \, a_3\coloneqq E^{\phi}_{k+1}(a_2,-t_0) \to_{E^{\phi}_{n+k+1}(a_3)} \,  a_4\coloneqq (\xi,z_2)= E^{\phi}_{n+k+1}(a_3,-t_0),
\end{split}
\end{equation}

and if $\varrho$ is sufficiently small with respect to $\delta$ (for example $\varrho<\delta^2$) we know that the integral lines in (<ref>) are defined in $[-t_0,t_0]$ and all the points $a_1, a_2, a_3\in V'_{a_0}$. Notice that it is precisely here that we have to take $U_{a_0}$ small enough to guarantee that the points defined in (<ref>) are in $V'_{a_0}$. This can be done since we are taking the concatenation of integral curves living up to a time whose norm is bounded above by $t_0<\varrho^{1/2}$.

We set $\phi=:(\phi^{(1)},\dots,\phi^{(k)})$ in coordinates. From <ref> we get that, for every $a\in V'_{a_0}$, we have

\begin{equation}\label{eqn:USELGRANGE}
\frac{\de}{\de \! s}{|_{s=t}}\phi^{(\ell)}(E^{\phi}_{j}(a,s))=\omega_{\ell j}(E^{\phi}_{j}(a,t)), \qquad \forall j=k+1,\dots,2n+1,\quad \forall \ell=1,\dots,k, \quad \forall t\in [-\delta,\delta].
\end{equation}

Thus, by using the points defined in (<ref>), Lagrange's theorem and the triangle inequality, we get

\begin{equation}\label{eqn:ComplEstimate}
\begin{split}
\frac{|\phi(\xi,z_1)-\phi(\xi,z_2)|}{|z_1-z_2|^{1/2}} &=
\frac{\left|\left(\phi(a)-\phi(a_1)\right)+\left(\phi(a_1)-\phi(a_2)\right)+\left(\phi(a_2)-\phi(a_3)\right)+\left(\phi(a_3)-\phi(a_4)\right)\right|}{|z_1-z_2|^{1/2}} \\
&\leq \sum_{\ell=1}^k \frac{1}{|z_1-z_2|^{1/2}}\cdot  \Big|\left(\phi^{(\ell)}(a)-\phi^{(\ell)}(a_1)\right)+\left(\phi^{(\ell)}(a_1)-\phi^{(\ell)}(a_2)\right)+ \\
&= \sum_{\ell=1}^k \frac{|-t_0\omega_{\ell,k+1}(b_{1,\ell})-t_0\omega_{\ell,n+k+1}(b_{2,\ell})+t_0\omega_{\ell,k+1}(b_{3,\ell})+t_0\omega_{\ell,n+k+1}(b_{4,\ell})|}{t_0} \\
&\leq \sum_{\ell=1}^k \left(|\omega_{\ell,k+1}(b_{3,\ell})-\omega_{\ell,k+1}(b_{1,\ell})|+|\omega_{\ell,n+k+1}(b_{4,\ell})-\omega_{\ell,n+k+1}(b_{2,\ell})|\right) \\
&\leq \sum_{\ell=1}^k\left(\beta(|b_{3,\ell}-b_{1,\ell}|)+\beta(|b_{4,\ell}-b_{2,\ell}|)\right),
\end{split}
\end{equation}

where, for every $\ell=1,\dots,k$, the points $b_{1,\ell},b_{2,\ell},b_{3,\ell},b_{4,\ell}$ are respectively chosen according to Lagrange's Theorem, that can be applied in view of (<ref>), on the images of the integral curves $E^{\phi}_{k+1}(a)$, $E^{\phi}_{n+k+1}(a_1)$, $E^{\phi}_{k+1}(a_2)$, $E^{\phi}_{n+k+1}(a_3)$ used in (<ref>). By simple estimates relying on the triangle inequality, we get a constant $C_0>0$ only depending on $V_{a_0}$ such that for every $\ell=1,\dots,k$, the estimates $|b_{3,\ell}-b_{1,\ell}|\leq C_0t_0$, and $|b_{4,\ell}-b_{2,\ell}|\leq C_0t_0$ hold. Since $t_0=|z_1-z_2|^{1/2}$, and $|z_1-z_2|< \varrho$, from (<ref>) we thus get (<ref>) with $\alpha(\varrho)\coloneqq C_0\varrho^{1/2}$.

In the setting of <ref>, it holds

D^{\phi}\phi=\omega \quad \mbox{{\rm in the broad* sense}} \Rightarrow \widetilde{\phi} \in{\rm UID}(\widetilde U,\mathbb W;\mathbb L).

It is a direct consequence of <ref> and <ref>. Indeed, taking into account (<ref>) of <ref>, the integral curves of $D^{\phi}_{2n+1}$ are vertical lines and, therefore, condition (<ref>) of <ref> implies that $\phi$ is vertically broad* hölder.

§.§ Examples of uniformly intrinsically differentiable functions

We show a class of nontrivial examples of ${\rm UID}$ functions in the Engel group, see <ref>.

Consider the Engel group $\mathbb E$, with the splitting $\mathbb E=\mathbb W\cdot\mathbb L$ described in <ref>. We show that the function

\begin{equation}
\phi_{\alpha}(0,x_2,x_3,x_4)\coloneqq x_4^{\alpha}\chi_{\{x_4\geq 0\}}(0,x_2,x_3,x_4),
\end{equation}

produces $\widetilde\phi_\alpha\in {\rm UID}(\mathbb W;\mathbb L)$ for any $\alpha> 1/3$. We first claim that $D^{\phi_{\alpha}}_{X_2}\phi_{\alpha}=\frac{\alpha}{2}x_4^{3\alpha-1}\chi_{\{x_4\geq 0\}}$ in the broad* sense.
Indeed, for any $\eps\in(0,1)$, the functions
\[(\phi_{\alpha})_{\eps}\coloneqq \eps^{1/3}\chi_{\{x_4<0\}}+ (x_4^{3\alpha}+\eps)^{1/3}\chi_{\{x_4\geq 0\}}\]
are globally $C^1$ and

\[
\lim_{\eps\to0}(\phi_{\alpha})_{\eps}=\phi_{\alpha} \quad \mbox{ in } L^\infty_{\rm loc}(\R^3), \quad \text{ and }\quad D^{(\phi_{\alpha})_{\eps}}_{X_2}(\phi_{\alpha})_{\eps}=\frac{\alpha}{2}x_4^{3\alpha-1}\chi_{\{x_4\geq 0\}}, \quad \forall \eps\in(0,1),
\]

where the last equality comes from the particular form of $D_{X_2}^{\phi}$ in (<ref>).
By applying <ref>, we get the claim.

We claim now that $\phi$ is vertically broad* holder, see <ref>. Indeed, since the integral curves of $D_{X_4}^{\phi_{\alpha}}$ are the vertical lines along direction $x_4$, see (<ref>), and since $\alpha>1/3$, we get that $\phi_{\alpha}$ is locally uniformly $1/3$-little Hölder continuous along these curves. We are left to prove that, locally around any $ a \coloneqq (0, x_2, x_3, x_4)$, condition (<ref>) is satisfied for the integral curves of $D_{X_3}^{\phi_{\alpha}}$, whose expression is in (<ref>).

According to the sign of $x_4$ we identify three families of integral curves of the vector field $D^{\phi_{\alpha}}_{X_3}$ starting from $a$. If $x_4<0$, the only integral curve of $D_{X_3}^{\phi_{\alpha}}$ starting from $a$ is $\gamma(t)= (0,x_2,x_3+t, x_4)$, and it is well-defined for every time $t$. If $ x_4=0$, we have that an integral curve of $D^{\phi_{\alpha}}_{X_3}$, existing for all times $t$, starting from $ a$ is given by
\[\gamma(t)=\left(0,x_2, x_3+t, (1-\alpha)^{1/(1-\alpha)}t^{1/(1-\alpha)}\chi_{\{t\geq 0\}}(t)\right),\]
while, if $ x_4>0$, we have that an integral curve of $D^{\phi_{\alpha}}_{X_3}$ starting from $ a$ is  given by
\[\gamma(t)=\left(0, x_2, x_3+t, (1-\alpha)^{1/(1-\alpha)}\left(t+\frac{( x_4)^{1-\alpha}}{1-\alpha}\right)^{1/(1-\alpha)}\right),\]
defined for $t\in \left(-\frac{ x_4^{1-\alpha}}{1-\alpha},+\infty\right)$. By exploiting this explicit choice of integral curves, we notice that $\gamma(t)$, in the three cases, is constant or
it is of order $t^{1/(1-\alpha)}$ on the last component. Thus, we get that $\phi_{\alpha}$ is 1/2-little Hölder continuous along these integral curves, because $\alpha/(1-\alpha)>1/2$ for any $\alpha>1/3$, and this happens locally uniformly. From this, we conclude that $\phi_{\alpha}$ is vertically broad* holder.
Now we conclude by applying (d)$\Rightarrow$(a) of <ref>.

Notice that, if $\alpha=1/3$, the function $\phi_{\alpha}$ is not UID in every neighborhood of the origin. Indeed, by <ref>, if  $\phi_{\alpha}$ is UID, then it is vertically broad* hölder and so  1/3-little Hölder continuous along the integral curves of the vector field $D^{\phi_{\alpha}}_{X_4}$. This means that $\phi_{\alpha}$ should be  $1/3$-little Hölder continuous in the last coordinate, but this is not true when $\alpha=1/3$.

We remark here that, in the case of the Heisenberg group $\mathbb H^1$, a slight modification of this types of examples gives rise to a $C^1_{\rm H}$-hypersurface, that is also $C^1$ Euclidean, but it has 0 as a characteristic point, see <ref>.

Consider the first Heisenberg group $\mathbb H^1$ identified with $\mathbb R^3$ by means of exponential coordinates, and consider the splitting $\mathbb H^1=\mathbb W\cdot\mathbb L$ described in <ref>, with $\mathbb L$ one-dimensional. Define $\phi\colon\mathbb W\equiv \mathbb R^2\to \sL\equiv\R$ by setting $\phi(0,x_2,x_3)\coloneqq {\rm sgn}(x_3)|x_3|^{2/3}$.

Since $\phi ^2(0,x_2,x_3)=|x_3|^{4/3}\in C^1$, by <cit.>, then $\widetilde\phi\in{\rm UID}(\mathbb W;\mathbb L)$ and, consequently, its graph is a $C^1_{\rm H}$-hypersurface. Moreover, in coordinates, one has

\begin{equation}\label{eqn:Parametrized}
(0,x_2,x_3)\cdot(\phi(0,x_2,x_3),0,0)=\left({\rm sgn}(x_3)|x_3|^{2/3},x_2,x_3-\frac{1}{2}{\rm sgn}(x_3)x_2|x_3|^{2/3}\right).
\end{equation}

The surface $\Sigma$ parametrized by (<ref>) is the union of two surfaces $\Sigma_1,\Sigma_2$ given by

\begin{equation}
\begin{split}
\Sigma_1&\coloneqq \left\{(x_1,x_2,x_3)\in \R^3:x_1\geq 0, x_3-x_1^{3/2}+\frac{1}{2}x_1x_2=0\right\}, \\ \Sigma_2&\coloneqq \left\{(x_1,x_2,x_3)\in \R^3:x_1\leq 0, x_3+(-x_1)^{3/2}+\frac{1}{2}x_1x_2=0\right\},
\end{split}
\end{equation}

that are glued along the $x_2$-axis $\{(x_1,x_2,x_3)\in \R^3: x_1=x_3=0\}$. We thus get that, for any $x\in {\rm int}(\Sigma_1)$, one has $T_x\Sigma_1=\left(\frac{1}{2}x_2-\frac{3}{2}x_1^{1/2},\frac{1}{2}x_1,1\right)^{\perp}$ with respect to the standard Euclidean scalar product, while for any $x\in {\rm int}(\Sigma_2)$, one has $T_x\Sigma_2=\left(\frac{1}{2}x_2-\frac{3}{2}(-x_1)^{1/2},\frac{1}{2}x_1,1\right)^{\perp}$ with respect to the standard Euclidean scalar product. From these explicit expressions, $\Sigma_1$ and $\Sigma_2$ glue together in $C^1$ regular way along the $x_2$-axis, and thus $\Sigma$ is also $C^1$-Euclidean surface. Moreover, from the previous expressions, we get that $T_0\Sigma = (0,0,1)^{\perp}=\{x_3=0\}=V_1$ and hence $0$ is a characteristic point for $\Sigma$.

We remark here that a similar example appeared in <cit.>. In that case the intrinsic graph is $C^1_{\rm H}$ regular but not $C^1$ regular.

§ CARNOT GROUPS OF STEP 2

As already underlined in <ref>, a co-horizontal $C^1_{\mathrm H}$-surface cannot be always characterized only by its horizontal geometry. This is however possible inside Carnot groups of step 2. Indeed, in this section, we show that the assumption on the vertical broad* hölder regularity of <ref> can be dropped when $\G$ is a Carnot group of step 2 and $\mathbb{L}$ is a horizontal one-dimensional subgroup of $\G$. In particular, as main result of this section, we get <ref>, which is <ref> in the introduction.

We describe the strategy of the proof of <ref>. The key idea is to first show that the implication

\begin{equation}\label{eqn:Propagation}
D^{\phi}\phi=\omega \; \mbox{broad*}\; \Rightarrow\; \phi \; \mbox{vertically broad* h\"older}
\end{equation}

holds in free Carnot groups of step 2, for a continuous $\omega$. This is done in two main steps. First, we explicitly write the intrinsic vector fields $D^{\phi}$ and we notice that the  nonlinearity given by $\phi$ only shows up in one vertical coordinate for each vector field, see (<ref>). Second, by using the structure of the vector fields $D^{\phi}$, one can propagate the broad* regularity from the horizontal components  of $\phi$ to the little Hölder regularity along vertical components by using a geometric trick: if $[X,Y]=Z$, and $\phi$ is $C^1$ on the integral curves of $X$ and $Y$, we expect $\phi$ to be $1/2$-little Hölder continuous on the integral curves of $Z$.

More precisely, the first step allows us to obtain the $1/2$-little Hölder regularity on the vertical coordinates affected by the nonlinearity by means of an adaptation of <cit.>. The second step allows us to obtain the $1/2$-little Hölder  regularity on all the remaining vertical coordinates. Notice that the efforts made to prove (d)$\Rightarrow$(a) in <ref>,  asking for just the regularity along the integral curves of $D^{\phi}$ (i.e., the vertically broad* regularity), are here payed back from a crucial simplification of this proof. Indeed, to conclude the proof of <ref> in the case of free Carnot groups of step 2, namely <ref>, we just use (<ref>) and apply (d)$\Rightarrow$(a) of <ref>. This can be done after having noticed that on Carnot groups of step 2 the vertically broad* hölder regularity reads as the locally 1/2-little Hölder continuity along vertical coordinates, see also <ref>.

Finally, to conclude the proof of the difficult implication (e)$\Rightarrow$(a) of <ref>, we use (<ref>) together with the fact that, on a Carnot group $\mathbb G$ of step 2, the broad* condition lifts to the free Carnot group $\mathbb F$ of step 2 and of the same rank of $\mathbb G$, see <ref>, while having a vertically broad* hölder property is naturally transferred  from $\mathbb F$ to $\mathbb G$, see <ref>. The resulting strategy presents some similarity to [50].

We point out that (a)$\Leftrightarrow$(c) of <ref> is a generalization to all step 2 Carnot groups of <cit.> and (a)$\Leftrightarrow$(e) is a generalization of <cit.>. We refer the reader to the introduction for a more detailed discussion on the literature.

In the current section, without loss of generality, we will always work in coordinates and there will be no distinction between $\square$ and $\widetilde\square$. See <ref> for details on the identifications.

§.§ Regularity results for broad* solutions in free Carnot groups of step 2

Free-nilpotent Lie algebras can be defined as follows (see Definition 14.1.1 in [18]).
[Free-nilpotent Lie algebras of step 2]
Let $m\geq 2$ be integer. We say that $\mathfrak{f}_{m,2}$ is the free-nilpotent Lie algebra of step 2 with $m$ generators $X_1,\dots , X_m$ if the following facts hold.

(i) $\mathfrak{f}_{m,2}$ is a Lie algebra generated by the elements $X_1,\dots , X_m$ (i.e., $\mathfrak{f}_{m,2}$ is the smallest Lie algebra containing $\{X_1,\dots , X_m\}$);

(ii) $\mathfrak{f}_{m,2}$ is nilpotent of step $2$ (i.e., nested Lie brackets of length $3$ are always $0$);

(iii) for every nilpotent Lie algebra $\mathfrak g$ of step $2$ and for every map $\Psi \colon \{X_1,\dots , X_m \}\to \mathfrak g$, there exists a unique homomorphism of Lie algebras $\overline{\Psi}: \mathfrak{f}_{m,2} \to \mathfrak g$ that extends $\Psi$.

[Free Carnot groups of step 2]
A free Carnot group of step 2 is a Carnot group whose Lie algebra is isomorphic to a
free-nilpotent Lie algebra $\mathfrak{f}_{m,2}$  for some $m\geq 2$. In this case, the horizontal layer of
the free Carnot group is isomorphic to the linear span of the generators of the Lie algebra $\mathfrak{f}_{m,2}$.

We give an explicit representation of free Carnot groups of step $2$. Fix an integer $m\geq 2$ and denote by $n\coloneqq m+\frac{m(m-1)}{2}$. In $\R^n$ denote the coordinates by $x_j$, for $1\leq j\leq m$, and by $y_{\ell s}$, for
$1 \leq s< \ell \leq m$. Let $\partial _j$ and $\partial _{\ell s}$ denote the standard basis vectors in this coordinate system.
We define $n$ linearly independent vector fields on $\R^n$ by setting:

\begin{equation}\label{eqn:VectorFieldsFree}
\begin{aligned}
X_j &\coloneqq \partial _j+ \frac 12 \sum_{ j<\ell\leq m } x_\ell \partial _{\ell j} -\frac 12 \sum_{ 1\leq \ell<j} x_\ell \partial _{j\ell}, \quad \mbox{ if } 1\leq j\leq m,\\
Y_{\ell s}&\coloneqq \partial _{\ell s}, \hphantom{\frac 12 \sum_{ j<\ell\leq m } x_\ell \partial _{\ell j} -\frac 12 \sum_{ 1\leq \ell<j} x_\ell \partial _{j\ell}} \qquad  \mbox{ if } 1\leq s<\ell\leq m.
\end{aligned}
\end{equation}

Let  $\mathbb{F} \coloneqq (\R^{m+\frac{m(m-1)}{2}}, \cdot)$ be the coordinate representation of the step 2 Carnot group with $m$ generators whose Lie algebra is generated by the vector fields in (<ref>). Then $\mathbb F$ is free and its Carnot structure is given by

\begin{equation*}
V_1\coloneqq\mbox{span} \{ X_j \,:\, 1\leq j\leq m\}\quad \mbox{and} \quad V_2\coloneqq\mbox{span} \{ Y_{\ell s} \,:\, 1\leq s<\ell \leq m\}.
\end{equation*}

Moreover, for any $p$ and $q\in \mathbb F$, the product $p\cdot q$  is given by the Baker-Campbell-Hausdorff formula, and yields

\begin{equation*}
\begin{aligned}
(p\cdot q)_j &=p_j+q_j,\quad \quad \qquad \qquad \qquad \,\,\,\,\, \text{ if } 1\leq j\leq m,\\
(p\cdot q)_{\ell s} &=p_{\ell s}+q_{\ell s} +\frac 1 2 (p_\ell q_s-q_\ell p_s), \quad \text{ if } 1\leq s<\ell\leq m.\\
\end{aligned}
\end{equation*}

It is easily verified that for $1\leq s<\ell\leq m$ and $1\leq j\leq m$, one has

\begin{equation}\label{commutatorifree}
[X_\ell , X_s] = Y_{\ell s} \quad \mbox{and} \quad  [X_j , Y_{\ell s}] = 0.
\end{equation}

Let $\mathbb{F}$ be a free Carnot groups of step $2$ and rank $m$.
To keep the notation simpler,
throughout this section, we identify without loss of generality $\mathbb F$ with $\R^n$ with $n\coloneqq m+\frac{m(m+1)}{2}$ by means of the coordinates described in <ref>. Moreover, given two complementary subgroups $\W$ and $\mathbb L$ with $\mathbb L$ horizontal and one-dimensional, we identify them in the following way:

\begin{equation}\label{5.2.0}
\begin{aligned}
\mathbb L & \coloneqq\left\{ (x_1,\dots , x_n)\in \R^n : x_2=\dots=x_n=0 \right\},\\
\W & \coloneqq\left\{ (x_1,\dots, x_n)\in\R^n : x_1=0 \right\}.
\end{aligned}
\end{equation}

Therefore, there are natural identifications between $ \R^{n-1}$ and $\W$ and  between $\R$ and $\sL$. We stress that, given an arbitrary free Carnot group of step 2 and complementary subgroups $\sW$ and $\sL$, with $\sL$ horizontal and one-dimensional, we can always choose coordinates satisfying (<ref>) and such that the identifications of <ref> are satisfied.

Let $\mathbb{F}$ be a free Carnot group of step $2$ represented in coordinates as in <ref> and let $\W$ and $\mathbb L$ be complementary subgroups of $\mathbb{F}$ as in (<ref>). Given $V\subseteq \mathbb W$ an open set, and given a continuous map $\psi\colon V\subseteq\sW\to \sL$, according to <ref>, the projected vector fields are given by

\begin{equation}\label{operatoriproiettatiinF}
\begin{aligned}
D_j^\psi  = \partial _{j}-\psi \partial _{j1} + \frac 1 2 \sum_{  j<\ell\leq m } x_\ell\partial _{\ell j} -  \frac 1 2 \sum_{1<s <j} x_s \partial _{js}=  {X_j}_{| V}-\psi  {Y_{j1}}_{|V},  \quad &\mbox{ for } j=2,\dots, m,\\
D^\psi_{\ell s}  = \partial _{\ell s}  =  {Y_{\ell s}}_{\vert \W},  \hphantom{\frac 1 2 \sum_{  j<\ell\leq m } x_\ell\partial _{\ell j} -  \frac 1 2 \sum_{1<s <j} x_s \partial _{js}=  {X_j}_{\vert \W}-\psi  {Y_{j1}}_{|V}}\quad &\mbox{ for }  1\leq s<\ell\leq m.
\end{aligned}
\end{equation}

Then, for each $j=2,\dots,m$, every integral curve $\gamma_j\colon I\to \mathbb R^{n-1}$ of $D^{\phi}_j$ has vertical components $y\coloneqq(y_{\ell s})_{1\leq s<\ell\leq m}\colon I\to \mathbb R^{\frac{m(m-1)}2}$  satisfying the following equations

\begin{equation*}
\begin{aligned}
\dot y _{j1} (t)&=-\psi (x_2,\dots ,x_{j-1}, x_j+t,x_{j+1},\dots ,x_m, y(t)), \\
\dot y _{\ell j} (t)&= \frac 12 x_\ell , \quad \quad  \qquad \mbox{ if }  j<\ell\leq m,\\
\dot y _{js} (t)&=  -\frac 12 x_s, \quad  \,\qquad \mbox{ if }  1<s <j, \\
\dot y_{\ell s} (t)&= 0,\quad \qquad  \quad \,\quad  \mbox{ otherwise,}
\end{aligned}
\end{equation*}

where the horizontal components of $\gamma_j(0)$ are $(x_1,\dots, x_m)$.

We now prove that, given an open set $V\subseteq \sW$ and a continuous function $\psi\colon V\to \sL$, a broad* solution of $D^\psi \psi =\omega $ for some continuous $\omega$ is also vertically broad* hölder (see <ref>). In particular, since the vector field $D^\psi_{\ell s}$ satisfies $D^\psi_{\ell s}=\partial _{{\ell s}}$ for every $1\leq s<\ell\leq m$ (see (<ref>)), then (<ref>) is equivalent to the following condition: for every $a_0$ in $V$, there exists a neighborhood $V'$ of $a_0$ with $V'\Subset  V$ such that for every  $\ell$ and $s$ with $1\leq  s< \ell\leq m$, one has

\begin{equation}\label{eq:equazioneprimadella66}
\lim_{ \varrho  \to0} \left(\sup\left\{\frac{|\psi(\xi,\eta)-\psi(\xi, y)|}{|\eta_{ \ell  s}-  y_{ l s}|^{1/2}}\right\}\right) =0,
\end{equation}

where the supremum is taken on all the couples $(\xi,\eta), (\xi,y)\in V'$ such that $\eta_{k\tau}=y_{k\tau}$ for any $(k,\tau)\neq (\ell,s)$ and $0<|\eta_{\ell s}-y_{\ell s}|\leq \varrho$.

In the first part of the proof  of <ref> below, we use techniques that are similar to the ones exploited in <cit.> in the context of Heisenberg groups. The third step of the proof is new.

Let $\W$ and $\mathbb{L}$ be two complementary subgroups of the free Carnot group $\mathbb{F}$ of step 2 with $\mathbb L$ horizontal and one-dimensional. Let $ V \subseteq \W$ be open and $\psi \colon{V} \to \mathbb L$ be a continuous function and assume that $\psi$ is a broad* solution of the system $D^{\psi } \psi =\omega$ in $V$, for some $\omega\in C(V;\mathbb R^{m-1})$, with respect to the basis chosen in <ref> and <ref>.
Then $\psi$ is vertically broad* hölder.

Working in the coordinates described in <ref> and <ref>, we can assume without loss of generality that $\sW$ and $\sL$ are defined as in (<ref>). If $m=2$, then $\mathbb{F}=\mathbb{H}^1$, and therefore the statement would follow from <cit.>.
We therefore assume that $m>2$.

We prove the following stronger fact, from which (<ref>) follows. For each $a_0\in V$ there are sufficiently small neighborhoods $I\Subset  I'\Subset V$ of $a_0$ such that, for every $1\leq s<\ell\leq m$, one can find a continuous and increasing function  $\alpha _{ \ell s} \colon(0,+\infty )\to [0,+\infty )$ only depending on $I'$, $\|\psi \|_{L^\infty (I')}$, $\| \omega \|_{L^\infty (I')}$ and on the modulus of continuity of $\omega$ on $I'$ with the property that

\begin{equation}\label{limitealpha.2}
\lim_{\varrho \to 0}\alpha  _{ \ell  s}(\varrho )=0,
\end{equation}

\begin{equation}\label{alpha88.2}
\begin{split}
\frac{|\psi(\xi,\eta)-\psi(\xi, y)|}{|\eta_{ \ell  s}-  y_{ \ell  s}|^{1/2}}\leq \alpha  _{  \ell  s} (\varrho ),
\end{split}
\end{equation}

for every $(\xi,\eta), (\xi, y)\in I$ such that  $\eta_{ k\tau} =y_{ k\tau}$ for every $(k,\tau)\ne ( \ell, s)$ and $0< |y_{\ell  s}-  \eta_{ \ell s}|\leq \varrho$.

Fix $a_0\in V$. Since $\psi $ is a broad* solution of $D^{\psi } \psi =\omega$ in $V$, there exist $0<\delta _2<\delta  _1$ and a family of maps

\begin{equation*}
E_j^\psi\colon\overline{B(a_0, \delta _2) }\times[-\delta _2, \delta _2]  \to   \overline{B(a_0, \delta _1) },
\end{equation*}

for $j=2,\dots ,m$,  such that the conditions of <ref> are satisfied.
Define $I'\coloneqq \overline{B(a_0, \delta _1) }$ and $I\coloneqq \overline{B(a_0, \delta _2) },$ and set
$ M_1\coloneqq\| \psi \|_{L^\infty (I' )}$. Let also $\beta $ be an increasing modulus of uniform continuity of $\omega$ on $I' $. We are going to prove (<ref>) with $\alpha _{ \ell s}$ defined by

\begin{equation}\label{Ndefialfa0.2}
\alpha _{ \ell s}(\varrho ) \coloneqq
\begin{cases}
3 \delta (  \varrho ),   &  \mbox{ if } (\ell, s)=(j,1) \mbox{ and } j=2,\dots, m,\\
G_{ \ell s} (\varrho ), & \mbox{ otherwise,}
\end{cases}
\end{equation}

where $G_{ \ell s}$ will be determined later in (<ref>) and

\begin{equation}\label{eq:definizionedelta}
\delta ( \varrho)\coloneqq\max \left\{  \varrho^{1/4}, \sqrt{\beta(C  \varrho^{1/4})} \right\},
\end{equation}

for some constant $C\coloneqq C(j) >0$ such that
\begin{equation}\label{eq:costanteC}
|\eta_{j1}- y_{j1}| + 2(|\xi|+M_1) |\eta_{j1}- y_{j1}|^{1/4} \leq C |\eta_{j1}- y_{j1}|^{1/4}
\end{equation}

for any $(\xi,\eta), (\xi, y) \in  I'$ with $\eta_{k \tau}=  y_{k\tau}$ for every couple $(k,\tau)\neq(j,1)$.

First step. If $a=(x,y)\in I$, by using (<ref>), we have that for any $2\leq j\leq m$ and any $t \in [-\delta_2,\delta_2]$, it holds

\begin{equation}\label{eq:sistemadiCauchyy}
\begin{aligned}
E_j^\psi(a,t) &=(x_2,\dots ,x_{j-1}, x_j+t,x_{j+1},\dots ,x_m, y(t)), \qquad \text{where}\\
y_{\ell s}(t) &=
\begin{cases}
y_{j1}- \int_{0}^t \psi (E_j^\psi(a,r)) \, \de \! r, & \mbox{if } (\ell,s)=(j,1),\\
\frac{1}{2} t x_\ell  +y_{\ell j},   & \mbox{if }s=j,\text{ and }  j<\ell\leq m,\\
-\frac{1}{2}tx_s   +y_{j s}, &\mbox{if } \ell=j,\text{ and }  1<s <j, \\
y_{\ell s}, &  \mbox{otherwise,}
\end{cases}
\end{aligned}
\end{equation}

and consequently, since $E^{\psi}_j$ are the maps provided by <ref>,
$t\mapsto y_{j1}(t)$ is a solution of the Cauchy problem

\begin{equation*}\label{prcauchy.2}
\left\{
\begin{array}{l}
\ddot y_{j1}(t)=\displaystyle\frac{ \de }{ \de \! t} \Big[ -\psi (E_j^\psi(a,t))\, \Big] = - \omega_j(E_j^\psi(a,t)),\qquad   t\in [-\delta _2,\delta _2],\\
\\
y_{j1}(0)=y_{j1}, \\
\\
\dot y_{j1}(0) = - \psi (a).
\end{array}
\right.
\end{equation*}

As a consequence of (<ref>) one gets

\begin{equation}\label{eq:stimaderivata}
\begin{aligned}
\max_{ r \in [-|t|, |t|] } | \dot y (r)| & \leq  \frac{1}{2}  \left|   \sum_{  j<\ell\leq m } x_\ell \right| +  \frac{1}{2}\left|\sum_{ 1<s<j} x_s \right| +\max_{ r  \in [-|t|, |t|]} \left|  \psi (E_j^\psi(a,r)) \right|  \leq C_1(|x|+M_1),\\
\end{aligned}
\end{equation}

for every $t\in [-\delta_2,\delta_2]$, where the constant $C_1>0$ only depends on the topological dimension of $\mathbb{F}$.

Second step.  Fix $j=2,\dots , m$ and assume $a=(\xi,\eta), \hat a=(\xi, y) \in I $ with $\eta_{k\tau}= y_{k\tau}$ for all couples $(k,\tau)\neq(j,1)$. We will need to possibly shrink $I$ in a way that will be clear throughout the proof.
We aim to show that

\begin{equation}\label{finaleCC.2}
\frac{|\psi (a)-\psi  (\hat a)|}{|\eta_{j1}- y_{j1}|^{1/2}  }  \leq \alpha _{j1} (|\eta_{j1}- y_{j1}| ),
\end{equation}

where, according to (<ref>), $\alpha _{j1}(\varrho) = 3\delta(\varrho) $. This would imply (<ref>) for $( \ell, s )=(j,1)$.

Set $\delta \coloneqq\delta( |\eta_{j1}- y_{j1}| )$ and suppose (<ref>) is not true, namely

\begin{equation}\label{finaleCCassurdo}
\frac{|\psi (a)-\psi  (\hat a)|}{|\eta_{j1}- y_{j1}|^{1/2} }  > 3\delta .
\end{equation}

Let $E_j^\psi(a, \cdot )$ and $E_j^\psi( \hat a,\cdot) $ be the integral curves of $D^\psi_j$ given by the broad* condition. By the first step they satisfy

\[
E_j^\psi(a,t)=(\xi_2,\dots, \xi_{j-1},\xi_j+t,\xi_{j+1},\dots ,\xi_m,\eta(t)), \]

\[
E_j^\psi(\hat a,t)=(\xi_2,\dots ,\xi_{j-1},\xi_j+t,\xi_{j+1},\dots ,\xi_m, y(t)).
\]

We claim we can find $t^*\in [-\delta _2,\delta_2]$ such that $\eta_{j1}(t^*)= y_{j1} (t^* )$, with $\psi (E_j^\psi(a,t^*) ) \ne  \psi (E_j^\psi(\hat a,t^*))$. This will lead to a contradiction, by the fact that the equality $\eta_{j1}(t^*)= y_{j1} (t^* )$ would also imply $\eta(t^*)=y(t^*)$.

Without loss of generality, assume that the initial data satisfy $\eta_{j1}> y_{j1}$.  By the first step of this proof, for every $t\in [-\delta _2,\delta _2]$, one has

\[
\begin{aligned}
\eta_{j1}(t)- y_{j1}(t) - (\eta_{j1}- y_{j1})
&=\int _{0}^t \left[ \dot \eta_{j1}(0) -\dot y_{j1} (0)+\int _{0}^{r'} (\ddot \eta_{j1} (r) - \ddot { y }_{j1} (r))\,  \de \! r   \right] \de \! r'  \\
& = - t(\psi (a)-\psi (\hat a))- \int _{0}^t\int _{0}^{r'} \left( \omega_j (E_j^\psi(a,r)) -\omega_j (E_j^\psi(\hat a,r))\right)\,  \de \! r   \de \! r' .\\
\end{aligned}
\]

Using (<ref>) and the fundamental theorem of Calculus, one gets a constant $C_2>0$ only depending on $C_1$ such that

\begin{equation}\label{eq240318}
\begin{aligned}
|E_j^\psi(a,r) - E_j^\psi(\hat a,r)| & \leq  \vert \eta-y\vert+ |r|\left(\max_{r  \in [-|t|, |t|]} | \dot \eta (r)| + \max_{ r \in [-|t|, |t|] } | \dot {y} (r)| \right)\\
& \leq \vert\eta-y\vert   + |t|\left(\max_{ r  \in [-|t|, |t|]} | \dot \eta(r)| + \max_{ r  \in [-|t|, |t|]} | \dot {y}(r)| \right)\\
& \leq  C_2\left(|\eta_{j1}-y_{j1}|+2|t|(|\xi|+M_1 )\right),
\end{aligned}
\end{equation}

for every $r \in [-|t|, |t|]$ and $t\in [-\delta_2,\delta_2]$. Hence, we obtain that

\begin{equation}\label{5.24}
\begin{aligned}
\eta_{j1}(t)- y_{j1}(t) - (\eta_{j1}- y_{j1})& \leq  - t(\psi (a)-\psi (\hat a))+ t^2 \max_{r  \in [-|t|, |t|] }\beta \big(\vert E_j^\psi(a,r) - E_j^\psi(\hat a,r)\vert\big)  \\
& \leq - t(\psi (a)-\psi (\hat a))+
t^2 \beta \left(  C_2(|\eta_{j1}- y_{j1}|  +2|t|(|\xi|+M_1 )) \right),
\end{aligned}
\end{equation}

for every $t\in [-\delta_2,\delta_2]$. Up to redefining $\beta$, we can replace without loss of generality $\beta(C_2\varrho)$ with $\beta(\varrho)$. Now, by (<ref>) we know that

\begin{equation}\label{phi1}
- |\psi (a)-\psi (\hat a)|  < -3 \delta  |\eta_{j1}- y_{j1}|^{1/2}.
\end{equation}

Without loss of generality, up to restricting $I$, we can assume that

\begin{equation}\label{eq:restrizioneI}
\delta_2 \geq |y_{j1}-\eta_{j1}|^{1/4} \geq |y_{j1}-\eta_{j1}|^{1/2}/\delta,
\end{equation}

where the second inequality directly follows from (<ref>).
If $\psi(a)=\psi(\hat a)$, then (<ref>) would be trivial. We study two cases: $\psi(a)-\psi(\hat a)<0$ or $\psi(a)-\psi (\hat a)>0$. If $\psi(a)-\psi(\hat a)<0$, set $t_0\coloneqq - \frac{|\eta_{j1}- y_{j1}|^{1/2}}{\delta }$ (otherwise we can choose $t_0\coloneqq\frac{|\eta_{j1}- y_{j1}|^{1/2} }{\delta }$) and evaluate $\eqref{5.24}$ in $t=t_0$. Combining it with (<ref>), (<ref>), (<ref>), the definition of $\delta$, and the fact that $\beta$ is increasing, we obtain (in both cases)

\begin{equation}\label{5.24bis}
\begin{aligned}
\eta_{j1}(t_0)- y_{j1}(t_0) & \leq \eta_{j1}- y_{j1} + |\eta_{j1}- y_{j1}|^{1/2} \frac{-|\psi (a)-\psi (\hat a)|}{\delta } +\\
& \hspace{0,5 cm} + \frac{1}{\delta ^2}|\eta_{j1}- y_{j1}|  \beta  \left(  |\eta_{j1}- y_{j1}| + 2(|\xi|+M_1) \frac{|\eta_{j1}- y_{j1}|^{1/2} }{\delta} \right)    \\
& \leq \eta_{j1}- y_{j1}  - 3 |\eta_{j1}- y_{j1}| + |\eta_{j1}- y_{j1}| \frac{ \beta  \left( C |\eta_{j1}- y_{j1}|^{1/4}\right)}{\delta ^2}    \\
& \leq \eta_{j1}-y_{j1} - 3|\eta_{j1}- y_{j1}|+|\eta_{j1}- y_{j1}| \leq -|\eta_{j1}- y_{j1}|<0.
\end{aligned}
\end{equation}

If $ \psi (a)-\psi (\hat a)>0$  we can define

\begin{equation*}
t^*\coloneqq \sup\{ \, r\in [0, \delta _2] \, :\,\eta_{j1}(s)- y_{j1}(s) >0,\, \forall s\in [0,r]\},
\end{equation*}

since the set $\{ \, r\in [0, \delta _2] \, :\,\eta_{j1}(s)- y_{j1}(s) >0,\, \forall s\in [0,r]\}$ is not empty; indeed, recall that we assumed without loss of generality that $\eta_{j1}>y_{j1}$ and therefore $\eta_{j1}(0)- y_{j1}(0)=\eta_{j1}- y_{j1}>0$. Moreover $0<t^*<t_0 \leq\delta _2$, and recalling that $\eta_{k\tau}=y_{k\tau}$ except for $(k,\tau)=(j,1)$, one has, by (<ref>) that

\begin{equation}\label{3.44big.2}
\begin{aligned}
\eta_{j1}(t^*) & = y_{j1}(t^*), \\
\eta_{\ell j}(t^*)& = y_{\ell j}(t^*)= \frac{1}{2}t^* \xi_\ell  +\eta_{\ell j}, \quad \qquad \quad \mbox{ for }  j<\ell\leq m,\\
\eta_{j s}(t^*) & = y_{j s}(t^*) =-\frac{1}{2}t^* \xi_s   +\eta_{j s}, \,\quad \quad \quad  \mbox{ for }  1<s <j, \\ \eta_{\ell s}(t^*)& =y_{\ell s}(t^*) =\eta_{\ell s},  \qquad  \qquad   \qquad \quad  \,\, \,\,  \mbox{ otherwise.}
\end{aligned}
\end{equation}

Hence, by definition of $E_j^\psi(a,\cdot )$ and $E_j^\psi(\hat a,\cdot)$ in (<ref>), one gets $E_j^\psi(a,t^*)=E_j^\psi(\hat a, t^*)$ and therefore

\begin{equation}\label{legame3.2}
\begin{aligned}
\psi (E_j^\psi(a,t^*) ) =  \psi (E_j^\psi(\hat a,t^*)).
\end{aligned}
\end{equation}

In the case $ \psi (a)-\psi (\hat a)<0$, we consider $t_0= -\frac{|\eta_{j1}- y_{j1}|^{1/2}}{\delta }$ and define $t^*\coloneqq\inf \{  r\in [-\delta _2,0]: \eta_{j1}(s)- y_{j1}(s)>0,\, \forall s\in [r,0] \}$. Then  $-\delta _2\leq t_0<t^*<0$ and, also in this case, (<ref>) and (<ref>) are satisfied.

We now show that this leads to a contradiction. In case $\psi (a)-\psi (\hat a)>0$, by using the properties of $E^\psi_j$, (<ref>), (<ref>), (<ref>), the definition of $\beta$  and the fact that $t^*< t_0$, we deduce

\begin{equation*}
\begin{aligned}
-(\psi (E_j^\psi(a,t^*) )  -  &\psi (E_j^\psi(\hat a,t^*)))= -(\psi (a)-\psi (\hat a))- \int _{0}^{t^*} \left( \omega_j (E_j^\psi(a,r)) -\omega_j (E_j^\psi(\hat a,r))\right)\,  \de  \! r  \\
& \leq   -3 (\eta_{j1}-y_{j1})^{1/2}\delta + |t^*| \,\max_{r  \in [0, t^*] }  \beta \left(\vert E_j^\psi(a,r) - E_j^\psi(\hat a,r) \vert \right)  \\
& \leq  -3 (\eta_{j1}- y_{j1})^{1/2}\delta +  |t^*| \beta  \left( |\eta_{j1}- y_{j1}|+ 2|t^*|(|\xi|+M_1) \right) \\
& \leq -3 (\eta_{j1}- y_{j1})^{1/2}\delta  + |t^*| \, \beta  \left( |\eta_{j1}- y_{j1}|+ 2(|\xi|+M_1) \frac{|\eta_{j1}- y_{j1}|^{1/2}}{\delta} \right)    \\
& \leq -3 (\eta_{j1}- y_{j1})^{1/2}\delta  +  (\eta_{j1}- y_{j1})^{1/2} \delta \, \frac{\beta  \left(C(\eta_{j1}- y_{j1})^{1/4} \right)}{\delta ^2}    \\
& \leq (-3+1)(\eta_{j1}- y_{j1})^{1/2} \delta   <0.
\end{aligned}
\end{equation*}
Similarly, if $\psi (a)-\psi (\hat a)<0$, then

\begin{equation*}
\begin{aligned}
\psi (E_j^\psi(a,t^*) )  -  \psi (E_j^\psi(\hat a,t^*))&= \psi (a)-\psi (\hat a)+ \int _{0}^{t^*} \left( \omega_j (E_j^\psi(a,r)) -\omega_j (E_j^\psi(\hat a,r))\right) \de  \! r  \\
& \leq   -3 (\eta_{j1}- y_{j1})^{1/2}\delta + |t^*| \,\max_{r  \in [-t^*, 0] }  \beta \left(\vert E_j^\psi(a,r) - E_j^\psi(\hat a,r)\vert \right),  \\
\end{aligned}
\end{equation*}

Hence, in both cases we have $\psi (E_j^\psi(a,t^*) ) \ne  \psi ( E_j^\psi(\hat a,t^*))$ that is in contradiction with (<ref>), so $\eqref{finaleCC.2}$ follows.

Third step.
Fix $ \ell, s$ with $1< s < \ell \leq m $,  denote by  $M_2\coloneqq\|\omega\|_{L^\infty (I' )}$ and define

\begin{equation}\label{defiGls}
G_{\ell s}(\varrho)\coloneqq 2\sqrt {M_2} \alpha _{\ell 1} (4M_2 \varrho ) + 2\sqrt {M_2}  \alpha _{ s 1} (4M_2 \varrho )+2 \beta (C_0 \varrho^{1/2} ),
\end{equation}

where $\alpha_{\ell 1}$ and $\alpha_{s 1} $ are defined as in (<ref>) for $j=\ell$ and $j= s$, respectively, $\beta$ is an increasing modulus of uniform continuity of $\omega$ on $I' $ and $C_0>0$ is a suitable constant, only depending on the supremum norm of $\omega$ on $I'$, that will be determined later.

We want to show that

\begin{equation}\label{equVERIFICA.2}
\frac{|\psi(a)-\psi(\hat a)|}{|\eta_{\ell  s}-  y_{ \ell s}|^{1/2}}\leq G_{\ell s}(\varrho),
\end{equation}

for every sufficiently small $\varrho>0$, every $a=(\xi,\eta), \hat a= (\xi, y)\in I$, such that $\eta_{ k \tau} = y_{ k\tau}$ for every $(k,\tau)\ne (\ell, s)$ and $0< |\eta_{\ell  s}-  y_{ \ell s}| \leq   \varrho$. Denote by $T_{\ell s}\coloneqq|\eta_{\ell s}- y_{\ell  s}|^{1/2}$. We will need to possibly shrink $I$ in a way that will be clear from the proof.

Rough idea of the proof. We build a concatenation of integral curves of the vector fields $D^\psi_{\ell}$ and $D^\psi_{s}$ that joins $\hat a$ and a suitable point $a_4$ that that can be connected to $a$ with two vertical lines on which we can use the result of the second step of this proof. We start from $\hat a$ and follow the integral curve of $D^\psi_{\ell}$ for a time $T_{\ell s}$ and we follow the integral curve of $D^\psi_{s}$ for the same time $T_{ \ell s}$; finally, we repeat the same procedure but for time $-T_{ \ell s}$.
At the end of this process, we obtain a point with three different vertical components with respect to $\hat a$: two increments are given by the non linear terms $-\psi \partial _{\ell 1}$ and $-\psi \partial _{ s1}  $ coming respectively from $D^\psi_{\ell}$ and from $D^\psi_{ s}$ and one increment is given by the commutator $[X_{\ell}, X_{s}]$ (which is $Y_{\ell s}$). In particular,  whenever $\eta_{\ell s}-  y_{\ell s}>0$, the $(\ell, s)$-component becomes equal to the $(\ell, s)$-component of $a$, that is $\eta_{\ell s}$. Vice versa, if $\eta_{\ell s}-y_{\ell s}<0$, one has to replace the times $\pm T_{ \ell s}$ with $\mp T_{\ell s}$.
In the end, we complete the proof by using the estimate of the second step of this proof applied to $a_4$ and $a$. The desired estimates come by using Lagrange's Theorem.

Assume $\eta_{\ell s} -y_{\ell s}>0$. Then we construct the following chain of points.

\begin{equation}\label{aggiustamentieq}
\begin{split}
\hat a &\to_{E_{\ell}^\psi (\hat a)}\, a_1\coloneqq E_{\ell}^\psi (\hat a, T_{\ell s})  \to_{E_{s}^\psi ( a_{1}) } \, a_2\coloneqq E_{ s}^\psi ( a_{1}, T_{\ell  s}) \\
&\to_{E_{\ell}^\psi ( a_{2}) } \, a_3\coloneqq  E_{\ell}^\psi ( a_2 , -T_{\ell s}) \to_{E_{s}^\psi ( a_{3})} \,  a_4\coloneqq E_{ s}^\psi ( a_{3}, -T_{\ell  s}), \\
\end{split}
\end{equation}

where we recall that $E_{\ell}^\psi$ and $ E_{s}^\psi $ are the integral curves of the vector fields $D^\psi_{\ell}$ and $D^\psi_{s}$, respectively, given by the fact that $D^{\psi } \psi =\omega$ in the broad* sense. In particular, $E_{\ell}^\psi $ and $ E_{s}^\psi $ satisfy (<ref>).

In case $\eta_{\ell s} - y_{\ell s} <0$ we repeat the same construction by replacing $\pm T_{\ell s}$ with $\mp T_{\ell s}$. In both cases, we have that $a_{4}\eqqcolon(\xi, y^{(a_{4})})$ is such that $y^{(a_{4})}_{\ell s} =\eta_{\ell s}$. Indeed, if $\eta_{\ell s} -y_{\ell s}>0$, then

\begin{equation*}
y^{(a_{4})}_{k\tau} =
\begin{cases}
\eta_{\ell s}, & \mbox{if }  (k,\tau)=(\ell, s), \\
\displaystyle\eta_{\ell 1}-\int_{0}^{T_{\ell s}} \psi ( E_{\ell}^\psi ( \hat a , t))  \de \!  t+ \int_{0}^{T_{\ell s}} \psi (E_{\ell}^\psi ( a_2 , t)) \de  \! t, & \mbox{if }  (k,\tau)=(\ell,1),\\
\displaystyle\eta_{ s1} -\int_{0}^{T_{\ell s}} \psi (E_{ s}^\psi ( a_{1}, t))  \de  \! t+ \int_{0}^{ T_{\ell s}} \psi (E_{ s}^\psi ( a_{3}, t))  \de  \! t, &  \mbox{if }  (k,\tau)=(s,1),\\
y_{k\tau}, &\mbox{otherwise.}
\end{cases}
\end{equation*}

We can assume that $a_1,a_2, a_3, a_4\in I$. Indeed, this can be done because for a sufficiently small $\varrho$ only depending on the supremum norms of $\phi$ and $\omega$ on $I'$, we can possibly reduce $I$ to some $I_0$ so that all the curves as in (<ref>) starting in $I_0$, and living for times bounded above by $\varrho$, lie inside $I$.

Let $a_5=(\xi,y^{(a_{5})})$ be a point that has the same components of $a$, except for position $(\ell, 1)$ for which $y^{(a_{5})}_{\ell1}=y^{(a_{4})}_{\ell1}$. As remarked above, we can assume without loss of generality that also $a_5 \in I$.  Moreover, we can estimate $|\psi(a)-\psi(\hat a)|$ as follows:

\begin{equation}\label{stima2903.6}
\begin{aligned}
\left|\psi (a)-\psi (\hat a) \right|  & \leq   \left|\psi (a)-\psi (a_5) \right|   + \left|\psi (a_5)-\psi ( a_4) \right|  +   \left|\psi (a_4)-\psi ( \hat a)\right|.
\end{aligned}
\end{equation}

We start by considering $|\psi(a)-\psi( a_5)|$. Evaluating (<ref>) for $j=\ell$, we get that

\begin{equation*}
\begin{aligned}
\left|\psi (a)-\psi (a_5) \right|  & \leq  |\eta_{\ell 1}-y^{(a_{5})}_{\ell 1}|^{1/2}  \alpha _{\ell 1} ( |\eta_{\ell 1}-y^{(a_{5})}_{\ell1}|), \\
\end{aligned}
\end{equation*}

and we also notice that

\begin{equation*}
|\eta_{\ell 1}-y^{(a_{5})}_{\ell 1}|= \left|\int_{0}^{ T_{\ell s}} \psi ( E_{\ell}^\psi ( \hat a , t))  \de \!  t  -\int_{0}^{ T_{\ell s}} \psi ( E_{\ell }^\psi ( a _2, t))  \de  \!  t \right|.
\end{equation*}

Recalling that $M_2=\|\omega\|_{L^\infty(I')}$, we aim to show that

\begin{equation}\label{stima2903.1}
\begin{aligned}
\left|\int_{0}^{ T_{\ell s}} \psi ( E_{\ell}^\psi ( \hat a , t))  \de  \! t  -\int_{0}^{ T_{\ell s}} \psi ( E_{\ell}^\psi ( a _2, t)) \de  \!  t \right|  \leq 4M_2 T_{\ell s}^2, \\
\end{aligned}
\end{equation}

that would imply

\begin{equation}\label{stima2903.2}
\begin{aligned}
\left|\psi (a)-\psi (a_5) \right|  & \leq  2\sqrt {M_2} T_{\ell  s}  \alpha _{\ell 1} ( |\eta_{\ell 1}-y^{(a_{5})}_{\ell1}|) \leq  2\sqrt{M_2} T_{\ell s}  \alpha _{\ell 1} ( 4M_2 T_{\ell s}^2). \\
\end{aligned}
\end{equation}

We first observe that

\begin{equation*}
\begin{aligned}
& \left|\int_{0}^{ T_{\ell  s}} \psi (E_{\ell}^\psi ( \hat a , t)) \de  \!  t  -\int_{0}^{ T_{\ell s}} \psi ( E_{\ell}^\psi ( a_2 , t)) \de  \!  t \right| \\& \quad = \left|\int_{0}^{ T_{\ell s}} \left(\psi ( E_{\ell}^\psi (\hat a , t)) -\psi (\hat a)\right)  \de \!   t  -\int_{0}^{ T_{\ell s}} \left(\psi (E_{\ell}^\psi ( a_2 , t)) -\psi (a_2)\right) \de  \!  t  +T_{\ell s} \left((\psi (\hat a)-\psi (a_2))\right) \right|  \\
&\quad \leq \int_{0}^{ T_{\ell s}}  \left| \psi (E_{\ell}^\psi (\hat a , t)) -\psi (\hat a)\right| \de  \!  t  + \int_{0}^{ T_{\ell s}}  \left| \psi ( E_{\ell}^\psi ( a _2, t)) -\psi (a_2) \right| \de  \!  t  +T_{\ell s} |\psi (\hat a)-\psi (a_2)|. \\
\end{aligned}
\end{equation*}

Recalling that for every $t$ in the interval of definition of the curve $E^{\psi}_{\ell}$, one has
$\frac{\de }{\de \! s}_{|s=t} \psi (E_{\ell}^\psi ( A , s)) = \omega_{\ell} ( E_{\ell}^\psi ( A , t))$ for $A=\hat a, a_2$, by exploiting the fundamental theorem of Calculus the previous estimate then yields

\begin{equation*}
\begin{aligned}
& \left|\int_{0}^{ T_{\ell s}} \psi ( E_{\ell}^\psi (\hat a , t)) \de  \!  t  -\int_{0}^{ T_{\ell s}} \psi ( E_{\ell}^\psi ( a_2 , t))  \de  \!  t \right| \leq 2M_2 T^2_{\ell s} +T_{\ell s} |\psi (\hat a)-\psi (a_2)|. \\
\end{aligned}
\end{equation*}

By Lagrange's Theorem one also gets

\begin{equation*}
\begin{aligned}
|\psi (\hat a)-\psi (a_2)| \leq  |\psi (\hat a)-\psi (a_1)|+ |\psi (a_1)-\psi (a_2)| =T_{\ell s} |\omega_{\ell } (b^*_0)|+ T_{\ell s} |\omega_{s} (b^*_1)| \leq 2M_2T_{\ell s},   \\
\end{aligned}
\end{equation*}

where $b^*_0$ and $b^*_1$ are two points on $E_{\ell}^\psi ( \hat a , [0,T_{\ell s}])$ and $E_{s}^\psi ( a _1, [0,T_{\ell s}])$, respectively. Hence, combining together the last two estimates, one obtains (<ref>) and, consequently, also (<ref>) holds.

Now we consider $|\psi(a_5)-\psi(a_4)|$. Since  $|a_5-a_4|= |y^{(a_{5})}_{s1}-y^{(a_{4})}_{ s1}|$, analogously to the previous case, one obtains

\begin{equation}\label{stima2903.3}
\left|\psi (a_5)-\psi (a_4) \right|   \leq  2\sqrt{M_2} T_{\ell s}  \alpha _{ s 1} ( |y^{(a_{5})}_{s1}-y^{(a_{4})}_{s1}|)  \leq  2\sqrt{M_2} T_{\ell s}  \alpha _{s 1} ( 4{M_2} T_{\ell s}^2),
\end{equation}

where  $\alpha _{ s 1}$ is defined as in (<ref>) and

\begin{equation*}
|y^{(a_{5})}_{s1}-y^{(a_{4})}_{s1}| = \left|\int_{0}^{T_{\ell s}} \psi (E_{s}^\psi (a _1, t)) \de \!  t- \int_{0}^{ T_{\ell s}} \psi (E_{ s}^\psi ( a _3, t))  \de \!  t \right|.
\end{equation*}

Eventually, we estimate $\psi (a_4)-\psi ( \hat a)$ in the following way:

\begin{equation*}
\begin{aligned}
\left|\psi (a_4)-\psi ( \hat a)\right| & \leq    \left|(\psi (a_4)-\psi ( a_3))+ (\psi (a_3)-\psi ( a_2))+(\psi (a_2)-\psi ( a_1))+(\psi (a_1)-\psi ( \hat a)) \right|  \\
&=| -T_{\ell s}  \omega_{ s } (a^*_3)-T_{\ell s}  \omega_{\ell } (a^*_2) + T_{\ell s}  \omega_{ s } (a^*_1)+T_{\ell  s}  \omega_{\ell } (a^*_0)| \\
& \leq T_{\ell s} (|\omega_{ s } (a^*_1)-  \omega_{ s } (a^*_3)| +|\omega_{\ell } (a^*_2)-  \omega_{\ell } (a^*_0)|)\\
& \leq T_{\ell  s}\left( \beta (|a^*_1- a^*_3|)+ \beta (|a^*_2- a^*_0|) \right),
\end{aligned}
\end{equation*}

where  $a^*_0 \in E_{\ell}^\psi (\hat a, [0,T_{\ell s}]), a^*_1\in E_{ s}^\psi (a_1, [0,T_{\ell s}]), a^*_2 \in E_{\ell}^\psi (a_2,[-T_{\ell s} , 0])$ and $a^*_3\in E_{ s}^\psi (a_3, [-T_{\ell s}, 0])$ are chosen to fulfill the conditions of Lagrange's Theorem. By simple estimates relying on the triangle inequality, we get a constant $C_0>0$, only depending on the supremum norm of $\omega$ on $I'$, such that $|a^*_1- a^*_3|\leq C_0T_{\ell s} $, and $|a^*_2- a^*_0|\leq C_0 T_{\ell s}$. Hence

\begin{equation}\label{stima2903.4}
\begin{aligned}
\left|\psi (a_4)-\psi ( \hat a)\right|    & \leq T_{\ell s}\left( \beta (|a^*_1- a^*_3|)+ \beta (|a^*_2- a^*_0|) \right) \leq 2  T_{\ell s} \beta (C_0 T_{\ell s}).
\end{aligned}
\end{equation}

By combining (<ref>) with (<ref>), (<ref>) and (<ref>) and recalling that $T_{\ell s}= |\eta_{\ell s}- y_{ \ell  s}|^{1/2}$ and $|\eta_{\ell s}- y_{\ell s}|< \varrho$, we thus get (<ref>). Indeed, one has

\[
\begin{aligned}
\frac{\psi(a)-\psi(\hat a)}{|\eta_{\ell s}-y_{\ell s}|^{1/2}}&\leq \frac{2\sqrt {M_2}T_{\ell s} \alpha_{\ell 1}(4M_2 T_{\ell s}^2)+2\sqrt {M_2}T_{\ell s} \alpha_{s 1}(4M_2 T_{\ell s}^2)+2T_{\ell s}\beta(C_0T_{\ell s})}{|\eta_{\ell s}- y_{\ell s}|^{1/2}}\\
&= 2\sqrt{M_2}\Big(\alpha_{\ell 1}(4M_2|\eta_{\ell s}-y_{\ell s}|)+\alpha_{s 1}(4M_2|\eta_{\ell s}-y_{\ell s}|)\Big)+ 2\beta(C_0|\eta_{\ell s}- y_{\ell s}|^{1/2})\\
&\leq G_{\ell s}(\varrho).
\end{aligned}
\]

Finally, since $G_{\ell s}$ is defined as sum of continuous maps that are $0$ at $0$, it follows that, if $\varrho \to 0$, then $G_{\ell  s} (\varrho)\to 0 $ for which we get (<ref>) also for all $( \ell, s )$ with $s\ne 1$ and the proof is complete.

§.§ Regularity results for broad* solutions in Carnot groups of step 2

In this section we see how to generalize <ref>, valid for free Carnot groups of step $2$, to any Carnot group of step 2. We adapt some techniques already exploited in [50]. More precisely, in <ref> we prove that the broad* condition lifts from $\mathbb G$ to the free group $\mathbb F$ with same step and rank of $\G$. In <ref> we show that the vertically broad* hölder regularity on $\mathbb F$ implies the vertically broad* hölder regularity on $\mathbb G$. These two facts will put us in a position to prove <ref> by exploiting <ref> and <ref>.

We here introduce Carnot groups of step 2 and we refer the reader to <cit.>.
We denote with $m$ the rank of $\G$ and we identify $\mathbb G$ with $(\R^{m+h}, \cdot )$. If $q\in \G$, we write $q=(x,y)$ meaning that $x\in \R^m$ and $y\in \R^h$. The group operation $\cdot$ between two elements $q=(x,y)$ and $q'=(x',y')$ is given by

\begin{equation}\label{5.1.0}
q\cdot q'= \left(x+ x',y+ y'-\frac 1 2\langle \mathcal{B} x,  x' \rangle \right),
\end{equation}

where $\langle \mathcal{B}x,x' \rangle \coloneqq (\langle \mathcal{B}^{(1)}x,x' \rangle, \dots , \langle \mathcal{B}^{(h)} x, x' \rangle)$ and $\mathcal B^{(i)}$ are linearly independent and skew-symmetric matrices in $\R^{m\times m}$, for $i=1,\dots, h$.

For any $i=1,\dots, h$ and any $j,\ell=1,\dots,m$, denote by $(\mathcal{B}^{(i)})_{j\ell}\eqqcolon (b_{j\ell}^{(i)})$, and define $m+h$ linearly independent left-invariant vector fields by setting

\begin{equation*}
\begin{aligned}
X'_j (p) & \coloneqq \partial _{x_j}  -\frac{1}{2 } \sum_ {i=1 }^{h} \sum_ {\ell=1 }^{m} b_ {j\ell}^{(i)} x_\ell  \,\partial _{y_i},  \quad \mbox{ for } j=1,\dots ,m,\\
Y'_i(p)  & \coloneqq \partial _{y_i }, \,\qquad \qquad \qquad \qquad \qquad \mbox{ for } i=1,\dots , h.
\end{aligned}
\end{equation*}

The ordered set $(X'_1,\dots,X'_m,Y'_1,\dots,Y'_h)$ is an adapted basis of the Lie algebra $\mathfrak g$ of $\G$. Using the skew-symmetry of $\mathcal B$, it easy to see that

\begin{equation}\label{commutatoripasso2}
[X'_j, X'_\ell]= \sum _{i=1}^h b_{j\ell}^{(i)} Y'_i,  \quad \mbox{and} \quad  [X'_j , Y'_{i}] = 0,\quad \forall j,\ell=1,\dots, m \;\text{ and }\; \forall i=1,\dots,h.
\end{equation}

When $\G$ is a free Carnot group of step $2$ with coordinate representation defined as in <ref>, we denote the matrices of the beginning of <ref> with $\mathcal B^{(i)}\eqqcolon\mathcal{B}^{(\ell,s)}$ with $1\leq s< \ell\leq m$. The composition law $\eqref{5.1.0}$ also tells us that $\mathcal{B}^{(\ell,s)}$ has entry $1$ in  position $(\ell,s)$, $-1$ in  position $(s,\ell)$ and $0$ elsewhere.

Since the space of skew-symmetric $m$-dimensional matrices has dimension $\frac{m(m-1)}{2}$, in any Carnot group $\G$ of step 2, the dimensions $m$ of the horizontal layer and $h$ of the vertical layer are related by the inequality

\[
h \leq \frac{m(m-1)}{2},
\]

and $\G$ is free if and only if $h=\frac{m(m-1)}{2}$.

From now on $\G$ is a Carnot group of rank $m$ and step $2$, with the coordinate representation previously discussed, and $\mathbb{F}$ is the free Carnot group of step 2 and rank $m$ with the coordinate representation as in <ref> and <ref>. We denote by $(X_1,\dots, X_m)$ a basis of the first layer of the Lie algebra of $\mathbb F$. By definition of free Lie algebra, there exists a Lie group surjective homomorphism $\pi \colon \mathbb F \to \G$ such that $\pi _*(X_\ell)=X'_\ell$ for any $\ell=1,\dots ,m$ (see e.g., <cit.>).

If we consider on $\mathbb{F}$ and $\G$ the Carnot-Carathéodory metrics $d_{\mathbb{F}}$ and $d_{\G}$ respectively, the map $\pi$  preserves the length of horizontal curves, so it is Lipschitz with
${\rm Lip}(\pi) = 1$. The following lemma is well-known. We refer the reader to <cit.> for a proof.

For any $p\in \mathbb{F}$ and any $ q' \in \G$, there exists $ p' \in \pi^{-1}(q')$ such that
\[d_{\mathbb{F}} (p, p') = d_{\G} (\pi(p), q').\]

Recall that the dimension of $\mathbb F$ is $n=m+\frac{m(m-1)}{2}$. From the definition of $\pi $, we notice that it preserves the horizontal coordinates, namely, for any $(x,y)\in \R^{n}$, there exists $y^*\in \R^h$ such that

\begin{equation}\label{proiezione}
\pi(x,y)= (x,y^*).
\end{equation}

We denote by $\W_\G$ and $\mathbb L_\G$ two complementary subgroups of $\G$ with $\mathbb L_\G$ horizontal and one-dimensional. Similarly to <ref>, by means of exponential coordinates we identify them with $\R^{m+h-1}$ and $\R$ by setting

\begin{equation}\label{5.2.0.1}
\begin{aligned}
\mathbb L_\G & \coloneqq\{ (x_1,0\dots , 0) \,:\, x_1\in\R \},\\
\W_\G & \coloneqq\{ (0,x_2, \dots,x_m,y_1,\dots, y_h) \,:\, x_i, y_k \in\R \mbox{ for }i=2,\dots, m,\, k=1,\dots h \}.
\end{aligned}
\end{equation}

Let $ \G$  be a Carnot group of step 2 and $\W _\G$, $\mathbb L_\G$ be the complementary subgroups of $\mathbb{G}$ defined as in (<ref>). Then, according to <ref> the projected vector fields relative to a continuous $\phi\colon U\subseteq \sW_\G\to\sL_\G$, with $U$ open, are given by

\begin{equation}\label{operatoriproiettatiinG}
\begin{aligned}
D_{j}^\phi & = \partial _{x_j}-  \sum_{ i=1 }^{h  } \left(  b_{j1}^{(i)} \phi  +\frac 1 2 \sum_{ k=2 }^{m  } x_k b_{jk}^{(i)}\right) \partial _{y_i} =  {X'_j}_{\vert U}-  \sum_{i=1}^hb_{j1}^{(i)}  \phi  {Y'_{i}}_{\vert U},  \quad \mbox{ for } j=2,\dots, m,\\
D^\phi_{i} & = \partial_{y_i}  =  {Y'_{i}}_{\vert U},  \qquad \mbox{ for }  i=1,\dots, h.
\end{aligned}
\end{equation}

Now let $\W_{ \mathbb{F}}$ and $\mathbb{L}_{ \mathbb{F}}$ be the complementary subgroups of $\mathbb{F}$ defined as in (<ref>).  Then $\pi_{|_{ \mathbb L _{ \mathbb{F}} }} \colon\mathbb L _{ \mathbb{F}}  \to \mathbb L_\G$ is an isomorphism and more precisely, with our identification, we can assume it is the identity, see (<ref>).  Moreover, by (<ref>), it follows that $\pi_{|_{ \mathbb W_{ \mathbb{F}} }} \colon\mathbb W_{ \mathbb{F}}  \to \mathbb W_\G$ is onto.

Since $\pi$ is a Lie group homomorphism, its differential is a Lie algebra homomorphism. Hence, for any $1\leq s<\ell\leq m$, one also has

\begin{equation*}
\pi_* (Y_{\ell s})=\pi_*  ([X_\ell, X_s])=[\pi_* (X_\ell),\pi_*  (X_s)]=[X'_\ell, X'_s] = \sum _{i=1}^h b_{\ell s}^{(i)} Y'_i,
\end{equation*}

where we used (<ref>), (<ref>) and $\pi_*(X_j)=X'_j$. We can therefore write the following formula

\begin{equation}\label{definizioneesplicitaP}
\begin{split}
& \pi(x_1,\dots , x_m, y_{21},\dots , y_{m(m-1)})  =(x_1,\dots , x_m, y^*_1,\dots , y^*_h), \quad \text{where}\\
& y^*_i  =  \sum _{1\leq s<\ell \leq m}  b_{\ell s}^{(i)} y_{\ell s}, \quad \forall i=1,\dots, h.\\
\end{split}
\end{equation}

Let $\mathbb{G}$ be a Carnot group of step 2 and let $\mathbb W_\G$ and $\mathbb L_\G$ be two complementary subgroups of $\mathbb G$, with $\mathbb L _\G$ horizontal and one-dimensional and choose coordinates such that (<ref>) is satisfied. Let $\mathbb F$ be the free Carnot group of step 2, rank $m$ and let $\mathbb W_{\mathbb F}$ and $\mathbb L_{\mathbb F}$ be the complementary subgroups of $\mathbb F$ satisfying the identification (<ref>).

Let $ U\subseteq \sW _\G$   be an open set and denote  by ${V} \subseteq\mathbb W_{\mathbb F}$ the open set defined by $  V\coloneqq\pi^{-1}( U)$. Let ${\phi}\colon{U} \to\mathbb L_\G$ be a continuous map and let $ \psi \colon{V} \to\mathbb L_{\mathbb F} $ be the map defined as

\[
{\psi} \coloneqq \pi^{-1} \circ {\phi}  \circ \pi _{|_{ V}}.
\]

Assume there exists $\omega\in C(U; \R^{m-1})$ such that $D^\phi \phi=\omega$ in the broad* sense.

Then $\psi $ is a broad* solution in $V$ of the system $ D^{{\psi}} {\psi} = \omega \circ \pi$.

Fix $j=2,\dots , m$. By (<ref>), we have

\begin{equation}\label{formula3}
\begin{aligned}
D_{X_j}^\psi & = \partial _{x_j}-\psi \partial _{j1} + \frac 1 2 \sum_{  j<l\leq m } x_\ell\partial _{\ell j} -  \frac 1 2 \sum_{1<s <j} x_s \partial _{js},\\
\end{aligned}
\end{equation}

and by (<ref>), it follows

\begin{equation}\label{formula4}
\begin{aligned}
D_{X'_j}^\phi & = \partial _{x_j}-  \sum_{ i=1 }^{h  } \left(  b_{j1}^{(i)} \phi  +\frac 1 2 \sum_{ k=2 }^{m  } x_k b_{jk}^{(i)}\right) \partial _{y_i}.\\
\end{aligned}
\end{equation}

Let $ a = (0, x_2,\dots , x_m, \eta_{21},\dots , \eta_{m(m-1)}) \in V$, and denote with $ b\coloneqq  \pi (a)$ the point in $\mathbb W_\G$ with coordinates $ b =(0, x_2,\dots,  x_m, y^*_1,\dots , y^*_h)$. Let $\delta >0$ and let $ \gamma_j \colon[-\delta , \delta ] \to U$ be an arbitrary integral curve of $D_{X'_j}^\phi$ starting from $ b$ such that

\begin{equation}\label{eqn:PhiBroadStep2}
\phi (\gamma _j (t))-\phi ( b)=\int_0^t \omega_{j}(\gamma_j(s))\de\!s ,  \quad \forall t\in [- \delta, \delta ].
\end{equation}

Recall that by (<ref>), there is an explicit relation between the coordinates of $ a$ and the coordinates of $b$. We are going to prove that we can lift $\gamma_j$ to an integral curve $\zeta_j$ of $D^{\psi}_{X_j}$ starting from $ a$, defined on $[-\delta,\delta]$ and with values in $V$, that satisfies

\begin{equation*}
\psi (\zeta _j (t))-\psi ( a)=\int_0^t (\omega_{j} \circ \pi ) (\zeta _j (s))\,\de\!s ,  \quad \forall t\in [- \delta, \delta ].
\end{equation*}

Indeed, let $\zeta _j \colon[-\delta , \delta ] \to \mathbb W_{\mathbb F}$ be defined as

\begin{equation}\label{definizione di eta}
\begin{split}
\zeta _j(t) &=(0, x_2,\dots , x_{j-1}, x_j+t,  x_{j+1},\dots , x_m,\eta_{21}(t),\dots, \eta_{m(m-1)}(t)), \quad \text{where}\\
\eta_{\ell s}(t)& = \begin{cases}
\displaystyle \eta_{j1}- \int_{0}^t \phi (\gamma _j(r))  \de \! r,  & \mbox{for } (\ell,s)=(j,1),\\
\displaystyle\frac{1}{2} t x_\ell  + \eta_{\ell j},   &\mbox{for }s=j\,\text{ and }\,  j<\ell\leq m,\\
\displaystyle-\frac{1}{2}t x_s   + \eta_{js},  &\mbox{for } \ell=j\,\text{ and }\, 1<s <j, \\
\eta_{\ell s},& \mbox{otherwise,}
\end{cases}
\end{split}
\end{equation}

for $t\in [-\delta , \delta]$. By definition, one immediately gets $ \zeta _j(0) = a$. We are left to prove the following facts.

(i) $ \pi  \circ \zeta _j = \gamma _j$;

(ii)  $\zeta _j \colon[-\delta , \delta ] \to V$ is  an integral curve of $ D_{X_j}^\psi$;

(iii) For every $t\in [-\delta, \delta]$ one has
\begin{equation*}
\psi (\zeta _j (t))-\psi ( a)=\int_0^t (\omega_{j} \circ\pi) (\zeta _j (s))\de\!s.
\end{equation*}

(i). By (<ref>), and the fact that $\gamma _j (0)= b=(0, x_2,\dots, x_m, y^*_1,\dots , y^*_h)$, we can explicitly write $ \gamma _j$, exploiting the fact that it is an integral curve of $D^{\phi}_{X_j'}$, and get

\begin{equation}\label{defitildegammaj}
\begin{split}
\gamma _j(t) &=(0, x_2,\dots , x_{j-1}, x_j+t, x_{j+1},\dots ,  x_m, y^*_1(t),\dots, y^*_h(t)), \quad \text{where}\\
y^*_{i}(t)& =  y^*_{i}  -\frac 1 2 t\left( \sum_{ k=2 }^{m  } x_k b_{jk}^{(i)} \right) -b_{j1}^{(i)}  \int_{0}^t \phi (\gamma _j(r)) \de \! r,  \qquad \forall i=1,\dots, h,\text{ and }\,\forall t\in[-\delta , \delta].\\
\end{split}
\end{equation}

Using the explicit formula of $\pi$ given in (<ref>), we obtain that

\begin{equation}\label{defigamma.1}
\begin{split}
& \pi (\zeta _j(t )) =(0, x_2,\dots , x_{j-1}, x_j+t,  x_{j+1},\dots , x_m, \eta^*_{ 1}(t),\dots, \eta^*_{ h}(t)), \quad \text{where}\\
& \eta^*_{i} (t) =  \sum _{1\leq s<\ell \leq m}  b_{\ell s}^{(i)} \eta_{\ell s}(t ), \qquad \forall i=1,\dots, h,\text{ and }\,\forall t\in[-\delta , \delta].\\
\end{split}
\end{equation}

In particular, by (<ref>), we have that

\begin{equation*}
\begin{split}
\eta^*_{i} (t)&  = b_{j1}^{(i)}\left( \eta_{j1}- \int_{0}^t \phi (\gamma _j(r))\de \! r\right) +  \sum _{j<\ell\leq m } b_{\ell j}^{(i)}\left( \frac{1}{2} t x_\ell  +  \eta_{\ell j} \right)   \\
&\quad +  \sum _{1<s <j } b_{js}^{(i)} \left( -\frac{1}{2}t x_s   + \eta_{j s}\right) +\sum _{\substack{1\leq s<\ell \leq m \\ s\ne j, \ell\ne j}}  b_{\ell s}^{(i)} \eta_{\ell s},
\end{split}
\end{equation*}

for every $i=1,\dots, h$ and $t\in [-\delta , \delta]$.  Using again (<ref>), we also notice that $ y^*_i  =  \sum _{1\leq s<\ell \leq m}  b_{\ell s}^{(i)}\eta_{\ell s}$, and, by the skew-symmetry of $\mathcal{B}^{(i)}$, we deduce that

\begin{equation*}
\begin{split}
\eta^*_i (t) &  = y^*_{i}  -\frac 1 2 t\left( \sum_{ k=2 }^{m  } x_k b_{jk}^{(i)} \right) -b_{j1}^{(i)}  \int_{0}^t \phi (\gamma _j(r)) \de \! r, \qquad \forall i=1,\dots, h \text{ and }\,\forall t\in [-\delta , \delta].  \\
\end{split}
\end{equation*}

Comparing (<ref>), (<ref>) and the last equality, we get $\pi \circ \zeta _j = \gamma _j$, as desired.

(ii). We now check that $\zeta_j$ is an integral curve of $ D_{X_j}^\psi$. Recalling (<ref>), and by the definition of the components of $\zeta_j$ in (<ref>), it suffices to check its $(j,1)$-coordinate, since for all the others is trivial. Observe that, from (i), $(\phi \circ \gamma _j) (t) = (\phi \circ \pi\circ \zeta_j )(t)= (\psi \circ \zeta _j)(t), $ for every $t\in [-\delta , \delta]$, and so

\[\eta_{j1} (t ) = \eta_{j1}- \int_{0}^t \phi (\gamma _j(r)) \de \! r= \eta_{j1}- \int_{0}^t \psi (\zeta _j(r)) \, \de \! r, \qquad \forall t\in [-\delta , \delta],\]
as desired. Notice that we have used that $\pi_{|_{\mathbb L_{\mathbb F}}}$ identifies $\mathbb L_{\mathbb F}$ with $\mathbb L_{\mathbb G}$.

(iii). We first notice that, since ${U} \supseteq \gamma _j ([-\delta , \delta ]) =( \pi  \circ \zeta _j)  ([-\delta , \delta ])$ and $ V=\pi^{-1}( U)$, then also  $\zeta _j ([-\delta , \delta ])\subseteq {V}$.

Using  $\pi  \circ \zeta _j = \gamma _j$ and the fact that $\gamma_j$ satisfies (<ref>), we finally obtain

\begin{equation}\label{equaintegrale}
\psi (\zeta _j (t))-\psi (a)=\int_0^t (\omega_{j} \circ \pi ) (\zeta _j (s))\de\!s ,  \qquad  \forall t\in [- \delta, \delta ].
\end{equation}