# On some intrinsic differentiability properties for Absolutely Continuous
functions between Carnot groups and the Area formula

Andrea Pinamonti  Dipartimento di Matematica
Universitá di Trento
Via Sommarive, 14, 38123 Povo TN, Italy<EMAIL_ADDRESS>,
Francesco Serra Cassano  Dipartimento di Matematica
Universitá di Trento
Via Sommarive, 14, 38123 Povo TN, Italy<EMAIL_ADDRESS>and
Kilian Zambanini  Dipartimento di Matematica
Universitá di Trento
Via Sommarive, 14, 38123 Povo TN, Italy<EMAIL_ADDRESS>
###### Abstract.

We discuss $Q-$absolutely continuous functions between Carnot groups,
following Malý’s definition for maps of several variables ([54]). Such maps
enjoy nice regularity properties, like continuity, Pansu differentiability
a.e., weak differentiability and an Area formula. Furthermore, we extend
Stein’s result concerning the sharp condition for continuity and
differentiability a.e. of a Sobolev map in terms of the integrability of the
weak gradient: more precisely, we prove that a Sobolev map between Carnot
groups with horizontal gradient of its sections uniformly bounded in $L^{Q,1}$
admits a representative which is $Q$-absolutely continuous.

In Memory of Jan Malý

††Keywords: Carnot groups, Sobolev spaces, Pansu differentiability, Lorentz
space, $Q-$absolutely continuous functions, Area formula.

##  1\. Introduction

In this paper we study Sobolev regularity, continuity and Pansu
differentiability for maps $f:\Omega\subset\mathbb{G}_{1}\to\mathbb{G}_{2}$
satisfying suitable intrinsic notions of absolute continuity or bounded
variation, called $Q$-absolute continuity and bounded $Q$-variation, where
$\mathbb{G}_{1}$ and $\mathbb{G}_{2}$ are Carnot groups and $\Omega$ is an
open set in $\mathbb{G}_{1}$. In particular, we will apply such a study for
extending in this setting an analogous of the remarkable Euclidean Stein’s
result [72] concerning the sharp condition providing continuity and
differentiability a.e. in terms of integrability of the weak gradient, thus
paving the way to the study of the gradient regularity of solutions to the
sub-Laplace Poisson equation, see e.g. [43]. Moreover we will prove an area
formula for $Q$-absolutely continuous maps between Carnot group.

The definition and main properties of a Carnot group are collected in section
2. Let us quickly recall here that a Carnot group $\mathbb{G}$ is a simply
connected nilpotent Lie group which can be represented as
$\mathbb{G}\equiv\,(\mathbb{R}^{n},\cdot)$ equipped with:

  * •

a family of intrinsic dilations
$(\delta_{\lambda})_{\lambda>0}\colon\mathbb{G}\to\mathbb{G}$, which are
automorphisms of $\mathbb{G}$;

  * •

a subbundle $H\mathbb{G}$ of the tangent bundle $T\mathbb{G}$, called
horizontal subbundle, which generates by commutation the whole $T\mathbb{G}$;

  * •

a Carnot-Carathéodory (or, also, control) metric $d_{c}$ associated to the
horizontal subbundle $H\mathbb{G}$, which is left-invariant and homogeneous
with respect to the intrinsic dilations.

The Haar measure of $\mathbb{G}$ turns out to be the $n$-dimensional Lebesgue
measure $\mathcal{L}^{n}$ on $\mathbb{R}^{n}$ and the metric (Hausdorff)
dimension of $(\mathbb{G},d_{c})$ coincides with its homogeneous dimension
$Q$. If $\mathbb{G}$ is commutative, then $\mathbb{G}\equiv\mathbb{R}^{n}$ as
Euclidean vector space; if $\mathbb{G}$ is not commutative, then $Q>\,n$. In
particular, we will see that the homogeneous dimension $Q$ will play the same
role of the topological dimension $n$ in the Euclidean setting.

A Carnot group is a sub-Riemannian structure, where several classical
differential notions and results of the Euclidean/Riemannian geometry do not
apply (see[30, 31]). Thus new intrinsic notions are needed in this setting,
such as, for instance, the ones of differentiable and Sobolev regular function
between Carnot groups (see Definitions 2.5 and 2.17, respectively).
Nevertheless, from the intrinsic point of view, some important analogous
results to Euclidean setting still hold for maps between Carnot groups, such
as a Rademacher differentiability result due to Pansu (see Theorem 2.6) and a
Sobolev-Morrey embedding theorem (see, Theorem 2.20).

Let us stress that the intrinsic notion of differentiability for mappings
between Carnot groups, and the related results, cannot be always recovered
within the theory of differentiation for mappings between metric spaces (see
Section 7 ).

The notions of $n$-dimensional absolutely continuous function and
$n$-variation for a map $f:\,\Omega\subset\mathbb{R}^{n}\to\mathbb{R}^{m}$
were initially introduced by Malý in [54]. They have been studied in several
papers (see, for instance, [15, 36, 37, 39]). An historical account as far as
the concept of absolute continuity for functions can be found in [54] and the
references therein.

These notions can also be introduced in a metric measure setting, thus it is
immediate to extend them to maps between Carnot groups. In the following the
balls involved are the ones with respect to the CC distance $d_{c}$.

######  Definition 1.1.

Let $\mathbb{G}_{1}=\,(\mathbb{R}^{n},\cdot,d_{1})$ and
$\mathbb{G}_{2}=\,(\mathbb{R}^{m},*,d_{2})$ be two Carnot groups, let $Q$ be
the homogeneous dimension of $\mathbb{G}_{1}$ (see (1)) and let
$f:\,\Omega\subset\mathbb{G}_{1}\to\mathbb{G}_{2}$, with $\Omega$ open set in
$\mathbb{G}_{1}$.

  * (i)

$f$ is said to be $Q$-absolutely continuous if for each $\varepsilon>0$ there
exists $\delta=\,\delta(\varepsilon)>\,0$ such that, for each finite disjoint
family of open balls $U_{1},\dots,U_{k}$ contained in $\Omega$,

$\text{if }\sum_{i=1}^{k}\mathcal{L}^{n}(U_{i})<\delta\text{ then
}\sum_{i=1}^{k}\left({\rm osc}_{U_{i}}f\right)^{Q}<\,\varepsilon$

where

${\rm osc}_{U}f:=\,{\rm diam}_{\mathbb{G}_{2}}(f(U))\,.$

  * (ii)

The $Q$-variation of $f$ is defined as

$V_{Q}(f,\Omega):=\,\sup\left\\{\sum_{i=1}^{k}\left({\rm
osc}_{U_{i}}f\right)^{Q}:\\{U_{1},\dots,U_{k}\\}\text{ disjoint family of
balls in }\Omega\,\right\\}\,.$

We say that $f$ has bounded $Q$-variation if $V_{Q}(f,\Omega)<\,\infty$.

We will denote by $BV^{Q}(\Omega,\mathbb{G}_{2})$ the class of functions
$f:\,\Omega\subset\mathbb{G}_{1}\to\mathbb{G}_{2}$ with bounded $Q$-variation
and with $AC^{Q}(\Omega,\mathbb{G}_{2})$ the class of $Q$-absolutely
functions. We use the notation $BV_{\rm loc}^{Q}(\Omega,\mathbb{G}_{2})$ for
functions which are locally of bounded $Q$-variation and $AC_{\rm
loc}^{Q}(\Omega,\mathbb{G}_{2})$ for the family of functions which are locally
$Q$-absolutely continuous. It actually holds that
$AC^{Q}(\Omega.\mathbb{G}_{2})\subset BV^{Q}(\Omega,\mathbb{G}_{2})$ if
$\Omega\subset\mathbb{G}_{1}$ is a bounded open set (see Proposition 3.1).

The study of absolutely continuous functions has been extended from the
Euclidean framework to the one of metric measures spaces: firstly, the study
of real-valued functions defined on measure metric spaces (see, for instance,
[63, 64]), and then the one of metric space-valued maps (see, for instance,
[79, 80, 55, 44]). We are going to compare later our results with previous
ones.

Note that also in this context the homogeneous dimension $Q$ of the Carnot
group $\mathbb{G}_{1}$ plays the same role of the Euclidean dimension. Indeed,
we will see that, roughly speaking, the class
$AC_{\rm{loc}}^{Q}(\Omega,\mathbb{G}_{2})$ locally lies between the class of
Sobolev maps which admit weak horizontal gradient with summability $Q$, that
is the class $W_{\rm loc}^{1,Q}(\Omega,\mathbb{G}_{2})$ (see Definition 2.17),
and the union over $p>Q$ of the spaces $W_{\rm
loc}^{1,p}(\Omega,\mathbb{G}_{2})$. Moreover it shares good properties with
the maps in $W_{\rm loc}^{1,p}(\Omega,\mathbb{G}_{2})$ with $p>Q$.

In section 3 we begin our study by stressing some simple classes of functions
contained in $AC^{Q}(\Omega,\mathbb{G}_{2})$. In particular we will see that,
if $f$ satisfies the so-called (RR) condition, then $f\in
AC^{Q}(\Omega,\mathbb{G}_{2})$ and we will show that, if $f$ is a Sobolev map
with exponent $p>Q$, that is $f\in W^{1,p}(\Omega,\mathbb{G}_{2})$, then $f\in
AC^{Q}(\Omega,\mathbb{G}_{2})$ (see Theorem 3.6). Moreover we will prove that
maps with finite $Q$-variation between Carnot groups, that is maps in
$BV^{Q}(\Omega,\mathbb{G}_{2})$, are Pansu-differentiable a.e. and admit weak
horizontal gradient (see Theorems 3.7 and 3.10). As a consequence, since
$AC^{Q}_{\rm loc}(\Omega,\mathbb{G}_{2})\subset BV^{Q}_{\rm
loc}(\Omega,\mathbb{G}_{2})$, also functions in
$AC^{Q}(\Omega,\mathbb{G}_{2})$ share the same properties.

In section 4 we will prove that a Sobolev function
$f:\Omega\subset\mathbb{G}_{1}\to\mathbb{G}_{2}$ admitting weak gradient in
the Lorenz space $L^{Q,1}(\Omega)$ admits a representative in
$AC^{Q}(\Omega,\mathbb{G}_{2})$ (Theorem 4.2), extending in this way Stein’s
result on Euclidean spaces. Moreover we will discuss this result also for maps
in a suitable class of Orlicz-Sobolev spaces between Carnot groups (see
Appendix 6 and in particular Theorem 6.7).

In section 5 we will show an area formula for functions in
$AC^{Q}(\Omega,\mathbb{G}_{2})$ (see Theorem 5.7 and Corollary 5.8), which
extends the one proved in [50] for Lipschitz functions.

Eventually in Appendix 7 we will collect, for sake of completeness, some
relevant notions and results about the differentiability and Sobolev
regularity of metric spaces-valued maps defined on a metric measure space, in
order to compare our results with the previuos ones.

Acknowledgments. The authors are members of Gruppo Nazionale per l’Analisi
Matematica, la Probabilità e le loro Applicazioni (GNAMPA), of the Istituto
Nazionale di Alta Matematica (INdAM) and they are partially supported by the
PRIN project: MUR-PRIN 2022 Project Regularity problems in Sub-Riemannian
structures, Project code: 2022F4F2LH. The authors would like to thank B.
Franchi and R. Serapioni for very useful discussions on the topics of the
present paper.

##  2\. Preliminaries

###  2.1. Carnot groups

A Carnot group $(\mathbb{G},\cdot)$ is a simply connected and nilpotent Lie
group whose Lie algebra $\mathfrak{g}$ of left-invariant vector fields has
dimension $n$ and admits a stratification of step $s\in\mathbb{N}$, that is

$\mathfrak{g}=V_{1}\oplus V_{2}\oplus\cdots\oplus V_{s},$

where the vector spaces $V_{1},\dots,V_{s}\subset\mathfrak{g}$ satisfy

$V_{i}=[V_{1},V_{i-1}]\quad\text{for}\
i=2,\ldots,s,\quad[V_{1},V_{s}]=\\{0\\}.$

We set $m_{i}=\dim(V_{i})$ and $h_{i}=\sum_{j=1}^{i}m_{j}$ for $i=1,\dots,s$.
We fix an _adapted basis_ of $\mathfrak{g}$, i.e. a basis $X_{1},\dots,X_{n}$
such that

$X_{h_{i-1}+1},\dots,X_{h_{i}}\ \text{is a basis of}\ V_{i},\quad
i=1,\dots,s.$

The subspace $V_{1}$ of $\mathfrak{g}$ is also called horizontal layer, while
$X_{1},\dots,X_{m_{1}}$ are the generating vector fields of the group. We
endow the algebra $\mathfrak{g}$ with the left-invariant Riemannian metric
$\langle{\cdot,\cdot}\rangle$ that makes the basis $X_{1},\dots,X_{n}$
orthonormal. Exploiting the exponential identification
$p=\exp\left(\sum_{i=1}^{n}p_{i}X_{i}\right)$, we can identify $\mathbb{G}$
with $\mathbb{R}^{n}$, endowed with the group law determined by the
Campbell–Hausdorff formula. In particular, the identity $e\in\mathbb{G}$
corresponds to $0\in\mathbb{R}^{n}$ and the _inversion map_ becomes
$\iota(p)=p^{-1}=-p$ for any $p\in\mathbb{G}$. Moreover, it is not restrictive
to assume that $X_{i}(0)=\mathrm{e}_{i}$ for any $i=1,\dots,n$. Therefore, by
left-invariance, we get

$X_{i}(p)=d\tau_{p}\mathrm{e}_{i},\quad i=1,\dots,n,$

where $\tau_{p}\colon\mathbb{G}\to\mathbb{G}$ is the _left-translation_ by
$p\in\mathbb{G}$, i.e. $\tau_{p}(q)=p\cdot q$ for any $q\in\mathbb{G}$.
The subbundle of the tangent bundle $T\mathbb{G}$ that is spanned by the
vector fields $X_{1},\dots,X_{m_{1}}$ plays a particularly important role in
the theory and it is called the horizontal bundle $H\mathbb{G}$; the fibers of
$H\mathbb{G}$ are

$H\mathbb{G}_{x}=\mbox{span }\\{X_{1}(x),\dots,X_{m_{1}}(x)\\},\qquad
x\in\mathbb{G}.$

For any $i=1,\dots,n$, the _degree_ $d(i)\in\\{1,\dots,s\\}$ of the basis
vector field $X_{i}$ is $d(i)=j$ if and only if $X_{i}\in V_{j}$. The group
_dilations_ $(\delta_{\lambda})_{\lambda\geq 0}\colon\mathbb{G}\to\mathbb{G}$
are hence given by

$\delta_{\lambda}(p)=\delta_{\lambda}(p_{1},\dots,p_{n})=(\lambda
p_{1},\dots,\lambda^{d(i)}p_{i},\dots,\lambda^{s}p_{n})\quad\text{for all}\
p\in\mathbb{G}.$

The Haar measure of the group $\mathbb{G}$ coincides with the $n$-dimensional
Lebesgue measure $\mathcal{L}^{n}$. The homogeneity property
$\mathcal{L}^{n}(\delta_{\lambda}(E))=\lambda^{Q}\mathcal{L}^{n}(E)$ holds for
any measurable set $E\subset\mathbb{G}$, where

(1) $Q:=\sum_{i=1}^{s}i\dim(V_{i})\in\mathbb{N}$

is the _homogeneous dimension_ of $\mathbb{G}$. For notational convenience, we
will use also the shorthand $|E|=\mathcal{L}^{n}(E)$.
In the following proposition (see [25]) we collect some fairly elementary
properties of the group operation.

######  Proposition 2.1.

The group product has the form

(2) $x\cdot y=x+y+\mathcal{Q}(x,y),\qquad\forall x,y\in\mathbb{R}^{n}$

where
$\mathcal{Q}=(\mathcal{Q}_{1},\dots,\mathcal{Q}_{n}):\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$
and each $\mathcal{Q}_{i}$ is a homogeneous polynomial of degree $d(i)$ i.e.

$\mathcal{Q}_{i}(\delta_{\lambda}x,\delta_{\lambda}y)=\lambda^{d(i)}\mathcal{Q}_{i}(x,y),\quad\forall
x,y\in\mathbb{G}.$

Moreover $\mathcal{Q}$ is anti-symmetric, which means

$\mathcal{Q}_{j}(x,y)=-\mathcal{Q}_{j}(-y,-x),$

and also

$\displaystyle\mathcal{Q}_{1}(x,y)=...=\mathcal{Q}_{m_{1}}(x,y)=0,$ (3)
$\displaystyle\mathcal{Q}_{j}(x,0)=\mathcal{Q}_{j}(0,y)=0,$
$\displaystyle\mathcal{Q}_{j}(x,x)=\mathcal{Q}_{j}(x,-x)=\mathcal{Q}_{j}(-x,x)=0,\quad\text{for}\;\;m_{1}<j\leq
n,$
$\displaystyle\mathcal{Q}_{j}(x,y)=\mathcal{Q}_{j}(x_{1},\dots,x_{h_{i-1}},y_{1},\dots,y_{h_{i-1}}),\quad\text{if}\quad
1<i\leq s\quad\text{and}\quad j\leq h_{i}.$

######  Definition 2.2.

An absolutely continuous curve $\gamma\colon[a,b]\to\mathbb{G}$ is horizontal
if there exist $u_{1},\dots,u_{m_{1}}\in L^{1}[a,b]$ such that

$\gamma^{\prime}(t)=\sum_{j=1}^{m_{1}}u_{j}(t)X_{j}(\gamma(t))\quad\mbox{for
a.e.}\ t\in[a,b].$

We define the _horizontal length_ of a horizontal curve $\gamma$ by
$L(\gamma)=\int_{a}^{b}|u(t)|dt$, where $u=(u_{1},\dots,u_{m_{1}})$ and
$|\cdot|$ denotes the Euclidean norm on $\mathbb{R}^{m_{1}}$.

Chow-Rashevskii Theorem asserts that any two points of $\mathbb{G}$ can be
connected by horizontal curves [9, Theorem 9.1.3]. Actually, any two points
can be connected by a geodesic, despite such a curve can be not unique [58,
Theorem 1.4.4]. This allows us to define the _Carnot-Carathéodory distance (CC
distance)_ as follows.

######  Definition 2.3.

The _CC distance_ between $x,y\in\mathbb{G}$ is

$d_{c}(x,y):=\inf\\{L(\gamma)|\,\gamma\colon[0,1]\to\mathbb{G}\mbox{
horizontal joining }x\mbox{ to }y\\}.$

This CC metric induces the Euclidean topology and it is an invariant distance,
since it satisfies $d_{c}(zx,zy)=d_{c}(x,y)$ (left translation invariance) and
$d_{c}(\delta_{r}(x),\delta_{r}(y))=rd_{c}(x,y)$ (homogeneity) for
$x,y,z\in\mathbb{G}$ and $r>0$. Every invariant distance $d$ induces a
homogeneous norm, simply defining $\|x\|:=d(x,0)$. We denote the CC open ball
with center $x$ and radius $r$ by $U_{r}(x)$ or by $U(x,r)$ and in particular
we use the shorthand $U_{1}=U(0,1)$. When convenient, we will consider the
closed CC ball of center $x$ and radius $r$, denoted by $B(x,r)$. It can be
proved that $\mathcal{L}^{n}(U_{r}(x))=r^{Q}\mathcal{L}^{n}(U_{1}(0))$, where
$Q$ is the homogeneous dimension of $\mathbb{G}$ (see (1)). Therefore $Q$
turns out to be the Hausdorff dimension of the metric measure space
$(\mathbb{G},d_{c},\mathcal{L}^{n})$. In particular note that, if the step of
$\mathbb{G}$ is $s\geq\,2$, then $Q>\,n$, which means that the Hausdorff
metric dimension of $\mathbb{G}$ is larger than its topological dimension $n$.

Following [25, Thm. 5.1], we also introduce the left-invariant and homogeneous
distance $d_{\infty}(p,q)=d_{\infty}(q^{-1}\cdot p,0)$ for $p,q\in\mathbb{G}$,
where, identifying
$\mathbb{G}=\mathbb{R}^{n}=\mathbb{R}^{m_{1}}\times\dots\times\mathbb{R}^{m_{s}}$
and letting
$\pi_{\mathbb{R}^{m_{i}}}\colon\mathbb{R}^{n}\to\mathbb{R}^{m_{i}}$ to be the
canonical projection for $i=1,\dots,s$,

(4)
$d_{\infty}(p,0)=\max\\{\varepsilon_{i}|\pi_{\mathbb{R}^{m_{i}}}(p)|_{\mathbb{R}^{m_{i}}}^{1/i}:i=1,\dots,s\\}\quad\text{for
all}\ p\in\mathbb{G},$

with constants $\varepsilon_{1}=1$ and $\varepsilon_{i}\in(0,1)$ for all
$i=2,\dots,s$ depending on the structure of $\mathbb{G}$. We use the shorthand
$\|p\|_{\infty}=d_{\infty}(p,0)$ for $p\in\mathbb{G}$. Consequently, for
$p\in\mathbb{G}$ and $r>0$, we define the open and closed balls with respect
to this metric as

$U_{\infty}(p,r)=\\{q\in\mathbb{G}:d_{\infty}(q,p)<r\\},\qquad
B_{\infty}(p,r)=\\{q\in\mathbb{G}:d_{\infty}(q,p)\leq r\\}.$

###  2.2. Differential Calculus on Carnot Groups

This section is taken from [25] and [50, Chapter 3]. We refer to [52] for a
generalization of the content of this section. The notion of _P-
differentiability_ for functions acting between Carnot groups was introduced
by Pansu in [60].

######  Definition 2.4.

Let $\mathbb{G}_{1},\mathbb{G}_{2}$ be Carnot groups, with homogeneous norms
$\|\cdot\|_{1}$, $\|\cdot\|_{2}$ and dilations $\delta^{1}_{\lambda}$,
$\delta^{2}_{\lambda}$. We say that $L:\mathbb{G}_{1}\to\mathbb{G}_{2}$ is _H-
linear_ , or is a _homogeneous homomorphism_ , if $L$ is a group homomorphism
such that

$L(\delta^{1}_{\lambda}g)=\delta^{2}_{\lambda}L(g),\quad\text{for all
}g\in\mathbb{G}_{1}\text{ and }\lambda>0.$

######  Definition 2.5.

Let $\Omega\subset\mathbb{G}_{1}$ be open. We say that
$f:\Omega\to\mathbb{G}_{2}$ is _P-differentiable_ at $g_{0}\in\Omega$ if there
is a $H$-linear function
$d_{P}f(g_{0}):\mathbb{G}_{1}\rightarrow\mathbb{G}_{2}$ such that

$\|\left(d_{P}f(g_{0})(g_{0}^{-1}\cdot g)\right)^{-1}\cdot f(g_{0})^{-1}\cdot
f(g)\|_{2}=o\big{(}\|g_{0}^{-1}\cdot g\|_{1}\big{)},\quad\text{as
$\|g_{0}^{-1}\cdot g\|_{1}\to 0$}$

where $o(t)/t\to 0$ as $t\to 0^{+}$. The $H$-linear map $d_{P}f(g_{0})$ is
called the _Pansu’s differential_ of $f$ at $g_{0}$.

The fundamental result where P-differentiability applies is the so called
Pansu-Rademacher theorem for Lipschitz functions between Carnot groups, see
[60] and [53, 61] for a different proof.

######  Theorem 2.6 (Pansu [60]).

Let $\mathbb{G}_{1}=(\mathbb{R}^{n},\cdot)$ and
$\mathbb{G}_{2}=(\mathbb{R}^{m},*)$ be two Carnot groups equipped with two CC
metrics denoted, respectively, by $d_{1}$ and $d_{2}$. Let
$f:\,\Omega\subset\mathbb{G}_{1}\to\mathbb{G}_{2}$ be a Lipschitz continuous
function with $\Omega$ an open set. Then $f$ is P-differentiable at
$\mathcal{L}^{n}$-a.e. in $\Omega$ .

A version of the classical Stepanov’s theorem [73] holds for maps between
Carnot groups, which extends Pansu-Rademacher theorem (see [77, Theorem 3.1]).

######  Theorem 2.7 (Stepanov’s theorem for Carnot groups).

Let $\mathbb{G}_{1}=(\mathbb{R}^{n},\cdot)$ and
$\mathbb{G}_{2}=(\mathbb{R}^{m},*)$ be two Carnot groups equipped with two CC
metrics denoted, respectively, by $d_{1}$ and $d_{2}$. Let
$f:\,\Omega\subset\mathbb{G}_{1}\to\mathbb{G}_{2}$ and assume that

(5) ${\rm Lip}(f)(p):=\,\limsup_{q\to
p}\frac{d_{2}(f(q),f(p))}{d_{1}(q,p)}<\infty\text{ for $\mathcal{L}^{n}$-a.e.
}p\in\Omega.$

Then $f$ is P-differentiable at $\mathcal{L}^{n}$-a.e. $p\in\Omega$ .

Let $X_{1},\dots,X_{m_{1}}$ be a generating family of vector fields. For
$j=1,\dots,m_{1}$, we say that $f:\mathbb{G}\to\mathbb{R}$ is differentiable
along $X_{j}$ at a point $x_{0}$ if the map $\lambda\mapsto
f(\tau_{x_{0}}(\delta_{\lambda}e_{j}))$ is differentiable at $\lambda=0$,
where $e_{j}$ is the $j$-th vector of the canonical basis of $\mathbb{R}^{n}$.
In other words, $f$ is differentiable along $X_{j}$ at $x_{0}$ if there exists

$\lim_{t\to 0}\frac{f(\exp(tX_{j})(x_{0}))-f(x_{0})}{t}=X_{j}f(x_{0}).$

For any function $f:\mathbb{G}\to\mathbb{R}$ for which the partial derivatives
$X_{j}f$ exist, we define the horizontal gradient of $f$, denoted by
$\nabla_{\mathbb{G}}f$, as

$\nabla_{\mathbb{G}}f:=\sum_{i=1}^{m_{1}}(X_{i}f)X_{i}.$

We will represent the horizontal gradient in coordinates
$\nabla_{\mathbb{G}}f=(X_{1}f,\dots,X_{m_{1}}f).$

Finally, if $x=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\equiv\mathbb{G}$ and
$x_{0}\in\mathbb{G}$ are given, we consider the projection of $x$ onto
$H\mathbb{G}_{x_{0}}$:

(6) $\pi_{x_{0}}(x)=\sum_{j=1}^{m_{1}}x_{j}X_{j}(x_{0}).$

The following proposition is proved in [59, Remark 3.3].

######  Proposition 2.8.

If $f:\Omega\subset\mathbb{G}\to\mathbb{R}$ is Pansu-differentiable at
$x_{0}$, then it is differentiable along $X_{j}$ at $x_{0}$ for
$j=1,\dots,m_{1}$, and for any $v\in\mathbb{G}$ the following formula holds:

(7)
$d_{P}f(x_{0})(v)=\langle{\nabla_{\mathbb{G}}f(x_{0})},{\pi_{x_{0}}(v)}\rangle_{x_{0}}.$

In particular, $d_{P}f(x_{0})(e_{j})=X_{j}f(x_{0})$ for all $j=1,\dots m_{1}$.

We conclude this section by recalling the notion of convolution in Carnot
groups; we address the interested reader to [70] for the proof.

######  Lemma 2.9.

Let $\mathbb{G}=(\mathbb{R}^{n},\cdot)$ be a Carnot group and let
$\varrho\in{\mathbf{C}}^{\infty}(\mathbb{R}^{n})$ be such that
$0\leq\varrho\leq 1$, $\int_{\mathbb{R}^{n}}\varrho\,d\mathcal{L}^{n}=1$,
${\mathrm{spt}}\varrho\subset U_{1}$ and
$\varrho(x^{-1})=\varrho(-x)=\varrho(x)$ for all $x\in\mathbb{R}^{n}$. Let us
denote

(8)
$\varrho_{\varepsilon}(x):=\varepsilon^{-Q}\varrho\left(\delta_{1/\varepsilon}(x)\right)\qquad
x\in\mathbb{R}^{n}\,;$ (9)
$(f\star\varrho_{\varepsilon})(x):=\int_{\mathbb{R}^{n}}\varrho_{\varepsilon}(x\cdot\,y^{-1})\,f(y)\,d\mathcal{L}^{n}(y)=\int_{\mathbb{R}^{n}}\varrho_{\varepsilon}(y)\,f(y^{-1}\cdot\,x)\,d\mathcal{L}^{n}(y)$

Then

  * (i)

if $f\in L^{p}(\mathbb{R}^{n}),\ 1\leq p<\infty$, then
$f\star\varrho_{\varepsilon}\in{\mathbf{C}}^{\infty}(\mathbb{R}^{n})$ and
$f\star\varrho_{\varepsilon}\to f$ in $L^{p}(\mathbb{R}^{n})$ as
$\varepsilon\to 0$;

  * (ii)

${\mathrm{spt}}\ f\star\varrho_{\varepsilon}\subset{\mathrm{spt}}\ f\cdot
B(0,\varepsilon)$;

  * (iii)

$X_{j}(\varphi\star\varrho_{\varepsilon})=X_{j}\varphi\star\varrho_{\varepsilon}$
for any $\varphi\in{\mathbf{C}}_{c}^{\infty}(\mathbb{R}^{n})$ and each
$j=1,\dots,m_{1}$;

  * (iv)

$\int_{\mathbb{R}^{n}}(f\star\varrho_{\varepsilon})\,g\,d\mathcal{L}^{n}=\int_{\mathbb{R}^{n}}(g\star\varrho_{\varepsilon})\,f\,d\mathcal{L}^{n}$
for every $f\in L^{\infty}(\mathbb{R}^{n})$, $g\in L^{1}(\mathbb{R}^{n})$;

  * (v)

if $f\in L^{\infty}(\mathbb{R}^{n})\cap{\mathbf{C}}^{0}(\Omega)$ for a
suitable open set $\Omega\subset\mathbb{R}^{n}$ then
$f\star\varrho_{\varepsilon}\to f$ uniformly on compact subsets of $\Omega$ as
$\varepsilon\to 0$.

###  2.3. Lorentz space

Let $\Omega\subset\mathbb{G}\equiv\mathbb{R}^{n}$ be an open set, let $Q$
denote the homogeneous dimension of $\mathbb{G}$ and let
$A:=\,\mathcal{L}^{n}(\Omega)$. If $g:\Omega\to\bar{\mathbb{R}}$ is a
measurable function, $\mathcal{L}^{n}$-a.e. finite, we define its distribution
function $\lambda_{g}:[0,\infty)\to[0,\infty]$ as

$\lambda_{g}(s):=\,\mathcal{L}^{n}\left(\left\\{x\in\Omega:\,|g(x)|>\,s\right\\}\right)$

and the nonincreasing rearrangement $g^{*}:\,[0,\infty)\to[0,\infty]$ of $g$
as

$g^{*}(t):=\,\inf\left\\{s\geq\,0:\,\lambda_{g}(s)\leq\,t\right\\}\,.$

It is well-known that $\lambda_{g}$ and $g^{*}$ are nonincreasing and right-
continuous. Moreover $g^{*}$ is the nonincreasing function which is
equimeasurable with $|g|$, namely

$\mathcal{L}^{n}(\\{x\in\Omega:|g(x)|>s\\})=\mathcal{L}^{1}(\\{t\in\mathbb{R}^{+}:g^{*}(t)>s\\})\qquad\text{for
all }s>0.$

The Lorentz space $L^{Q,1}(\Omega)$ is defined as the class of all measurable
functions $g:\Omega\to\bar{\mathbb{R}}$, $\mathcal{L}^{n}$-a.e. finite, for
which the quasi-norm

$\|g\|_{L^{Q,1}(\Omega)}:=\,\int_{0}^{A}t^{\frac{1-Q}{Q}}\,g^{*}(t)\,dt$

is finite. Note that it is not a norm since the triangle inequality may fail.
A systematic study of Lorentz spaces is carried out in [71] whereas a simpler
recent account can be found in [39].
Let us recall some properties of the Lorentz space $L^{Q,1}(\Omega)$ whose
proof can be found in [8] and [39].

######  Proposition 2.10.

1

  * (i)

For any measurable function $g:\Omega\to\mathbb{R}$,

$\|g\|_{L^{Q,1}(\Omega)}=\,Q\int_{0}^{\infty}\lambda_{g}(s)^{1/Q}\,ds\,.$

  * (ii)

$L^{Q,1}(\Omega)\subset L^{Q}(\Omega)$; moreover, if $|\Omega|<\,\infty$, it
holds that $L^{p}(\Omega)\subset L^{Q,1}(\Omega)$ for each $p>Q$.

  * (iii)

If $|g_{1}|\leq\,|g_{2}|$ a.e. in $\Omega$, then

$\|g_{1}\|_{L^{Q,1}(\Omega)}\leq\,\|g_{2}\|_{L^{Q,1}(\Omega)}\,.$

Notice that, in view of point (ii) of the previous Proposition, the Lorentz
space $L^{Q,1}(\Omega)$ is, locally, an intermediate space between
$L^{Q}(\Omega)$ and $L^{p}(\Omega)$ for all $p>Q$.
Let us also recall the following remarkable characterization of
$L^{Q,1}(\Omega)$ in terms of Orlicz integrability conditions, contained in
[39, Corollary 2.4], which we will need later.

######  Theorem 2.11.

Let $g$ be a measurable function on $\Omega$. Then the following properties
are equivalent:

  * (i)

$g\in L^{Q,1}(\Omega)$.

  * (ii)

There is a positive nonincreasing function $\varphi\in L^{1/Q}((0,\infty))$
such that

(10) $\int_{\Omega}F_{\varphi}(|g|)\,dx<\,\infty\,,$

where

(11) $F_{\varphi}(s):=\begin{cases}s\,\varphi(s)^{\frac{1}{Q}-1}&\text{ if
}s>\,0\\\ 0&\text{ if }s=\,0.\end{cases}$

###  2.4. Sobolev spaces on Carnot groups.

In this section we introduce and study $L^{p}$ and Sobolev spaces for maps
between Carnot groups.

####  2.4.1. Sobolev functions from a Carnot group to $\mathbb{R}$

Sobolev spaces for scalar functions defined on a Carnot group $\mathbb{G}$ are
constructed in the usual way by considering distributional derivatives along
the basis $X_{1},\dots,X_{m_{1}}$ of the horizontal layer of $\mathbb{G}$: a
function $g\in L^{1}_{loc}(\Omega)$ is the weak (or distributional) derivative
of $f$ along $X_{i}$ if

$\int_{\Omega}g\varphi\,dx=-\int_{\Omega}fX_{i}\varphi\,dx\qquad\text{for all
}\varphi\in C^{\infty}_{c}(\Omega).$

As usual we will write $X_{i}f$ also to denote the weak derivative of $f$
along $X_{i}$ and $\nabla_{\mathbb{G}}f=(X_{1}f,\dots,X_{m_{1}}f)$ for the
weak horizontal gradient.

######  Definition 2.12.

Let $\Omega$ be an open subset of $\mathbb{G}\equiv\mathbb{R}^{n}$ and $1\leq
p\leq\infty$. We define the (horizontal) Sobolev space
$W^{1,p}_{\mathbb{G}}(\Omega)$ as the space of all the functions with finite
norm

(12) $\|u\|_{W^{1,p}_{\mathbb{G}}}:=\|u\|_{p}+\|\nabla_{\mathbb{G}}u\|_{p},$

where $\nabla_{\mathbb{G}}u$ denotes the horizontal distributional gradient of
$u$.

The following result is well-known, see e.g. [71].

######  Proposition 2.13.

For any $1\leq p\leq\infty$,
$(W^{1,p}_{\mathbb{G}}(\Omega),\|\cdot\|_{W^{1,p}_{\mathbb{G}}})$ is a Banach
space, reflexive if $1<p<\infty$. Moreover, $W^{1,2}_{\mathbb{G}}(\Omega)$ is
a Hilbert space.

Another way to define the $W^{1,p}_{\mathbb{G}}$ with $1\leq p<\infty$ is to
take the closure of $C^{\infty}$ functions w.r.t the norm (12). As in the
Euclidean case, the two approaches are equivalent. This was obtained
independently in [23] and [28]. However, the method goes back to the classical
result by Friedrichs [27]. The result can be stated as follows.

######  Theorem 2.14.

Let $1\leq p<\infty$. We have

$\mathbf{C}^{\infty}(\Omega)\cap W^{1,p}_{\mathbb{G}}(\Omega)\mbox{ is dense
in }W^{1,p}_{\mathbb{G}}(\Omega).$

If in addition $\partial\Omega$ is a smooth manifold, then

$\mathbf{C}^{\infty}(\overline{\Omega})\mbox{ is dense in
}W^{1,p}_{\mathbb{G}}(\Omega).$

We state now an interesting result which will be useful in the proof of
Theorem 3.10.

######  Proposition 2.15.

Let $f:\,\Omega\subset\mathbb{G}\to\mathbb{R}$. Suppose that

  * (i)

$f$ is P-differentiable a.e. in $\Omega$;

  * (ii)

$f\in W^{1,1}_{\mathbb{G},\rm loc}(\Omega)$.

Then

$\exists\lim_{t\to
0}\frac{f\left(\exp(tX_{i})(x)\right)-f(x)}{t}=\,X_{i}f(x)\text{ for a.e.
}x\in\Omega\,$

for each $i=1,\dots,m_{1}$, where $X_{i}f$ denotes the distributional
derivative of $f$ along the vector field $X_{i}$.

###### Proof.

Let us fix $i=1,\dots,m_{1}$ and $\varphi\in C^{\infty}_{c}(\Omega)$.
If $x\in\Omega$ is a point of P-differentiability for $f$ then, by Proposition
2.8, $f$ is differentiable along $X_{i}$ at $x$ and

$\exists\lim_{t\to
0}\frac{f\left(\exp(tX_{i})(x)\right)-f(x)}{t}=\,X_{i}f(x),$

where in this case $X_{i}f$ stands for the classical derivative of $f$ along
$X_{i}$.
We have

$\displaystyle\int_{\Omega}\lim_{t\to
0}\frac{f\left(\exp(tX_{i})(x)\right)-f(x)}{t}\varphi(x)\,dx$
$\displaystyle=\lim_{t\to
0}\int_{\Omega}\frac{f\left(\exp(tX_{i})(x)\right)-f(x)}{t}\varphi(x)\,dx$
$\displaystyle=\lim_{t\to
0}\int_{\Omega}\frac{\varphi\left(\exp(-tX_{i})(x)\right)-\varphi(x)}{t}f(x)\,dx$
$\displaystyle=-\int_{\Omega}X_{i}\varphi(x)f(x)\,dx$
$\displaystyle=\int_{\Omega}X_{i}f(x)\varphi(x)\,dx$

and the conclusion follows by the fundamental Lemma of the Calculus of
Variations. ∎

We conclude this section by recalling a well-known estimate on the Riesz
potential that will be helpful later on. The proof easily follows using the
Poincaré inequality in Carnot groups [35], [21] and [26] and we omit it.

######  Theorem 2.16.

Let $U=U(x,r)$ and let $f:\mathbb{G}\to\mathbb{R}$ be a continuously
differentiable function. Then there exists $C>0$, independent of $U$ and $f$,
such that

(13) $|f(x)-f_{U}|\leq
C\int_{U}\frac{|\nabla_{\mathbb{G}}f(y)|}{\;\;\;\;d(x,y)^{Q-1}}\;dy$

for all $x\in U$, where $f_{U}:={\mathchoice{\kern 4.30554pt\vbox{\hrule
height=0.4pt,width=6.0pt,depth=0.0pt}\kern-9.7pt}{\kern 4.30554pt\vbox{\hrule
height=0.4pt,width=4.3pt,depth=0.0pt}\kern-7.0pt}{}{}}\int_{U}f\,dx$.
Moreover, if $f\in W^{1,1}_{\mathbb{G},\rm loc}(U)$, then (13) holds for a.e.
$x\in U$ and $\nabla_{\mathbb{G}}f(y)$ denotes the weak horizontal gradient of
$f$ at $y$.

####  2.4.2. Sobolev functions between Carnot groups.

We are now going to introduce the notions of $L^{p}$ functions and Sobolev
functions with values in a Carnot group. These concepts actually work for
mappings between general metric spaces and can be introduced in several
different ways (see Section 7). A possible approach relies on upper gradients
(this is for instance the case discussed in [44]), while here we use the
definition by sections of mappings. These notions go back to Ambrosio [3] and
Reshetnyak [65] for functions defined on open sets of $\mathbb{R}^{n}$ or a
Riemannian manifold and have been extended by Vodop’yanov [76] to the setting
of Carnot groups. A recent account containing different approaches to this
argument is due to Kleiner, Müller and Xie [41, 42].

######  Definition 2.17.

Let $\mathbb{G}_{1}=(\mathbb{R}^{n},\cdot,d_{1})$ and
$\mathbb{G}_{2}=(\mathbb{R}^{m},\star,d_{2})$ be two Carnot groups and let
$\Omega\subset\mathbb{G}_{1}$ be an open set.

  * (i)

We say that a map $f:\,\Omega\to\mathbb{G}_{2}$ belongs to
$L^{p}(\Omega,\mathbb{G}_{2})$, and we will write $f\in
L^{p}(\Omega,\mathbb{G}_{2})$, if
$f:\,\Omega\subset\mathbb{R}^{n}\to\mathbb{R}^{m}$ is measurable and the map
$\mathbb{R}^{n}\ni x\mapsto d_{2}(0,f(x))$ is in $L^{p}(\Omega)$.

  * (ii)

We say that a map $f:\,\Omega\to\mathbb{G}_{2}$ belongs to the Sobolev space
$W^{1,p}(\Omega,\mathbb{G}_{2})$, and we will write $f\in
W^{1,p}(\Omega,\mathbb{G}_{2})$, if

($\bullet$) $f\in L^{p}(\Omega,\mathbb{G}_{2})$;

($\bullet$) for all $z\in\mathbb{G}_{2}$ the functions
$u_{z}:\,\Omega\to\mathbb{R}$

(14) $u_{z}(x):=\,d_{2}(z,f(x))\text{ if }x\in\Omega\,,$

are in $W^{1,p}_{\mathbb{G}_{1},\rm loc}(\Omega)$;

($\bullet$) there exists a nonnegative function $g\in L^{p}(\Omega)$
(independent of $z$) such that

(15) $|\nabla_{\mathbb{G}_{1}}u_{z}|\leq\,g\text{ a.e. in $\Omega$, for each
$z\in\mathbb{G}_{2}$}\,.$

In the following we will refer to a function $u_{z}$ as a section of $f$.

As usual, we will denote by $W^{1,p}_{\rm loc}(\Omega,\mathbb{G}_{2})$ the set
of functions $f:\Omega\to\mathbb{G}_{2}$ such that $f\in
W^{1,p}(\Omega^{\prime},\mathbb{G}_{2})$ for each
$\Omega^{\prime}\subset\subset\Omega$.

######  Remark 2.18.

The stability condition requested in (15) with respect to the family of
$1$-Lipschitz maps $\varphi:\,\mathbb{G}_{2}\to\mathbb{R}$,
$\varphi(y):=\,d_{2}(z,y)$, can be extended to all Lipschitz maps between
$\mathbb{G}_{2}$ and $\mathbb{R}$. Indeed the following characterization holds
(see [76, Proposition 1] and [41, Proposition 2.31], see also Lemma 7.4):
$f\in W^{1,p}(\Omega,\mathbb{G}_{2})$ if and only if there exists a
nonnegative function $g\in L^{p}(\Omega)$ such that, for each Lipschitz
function $\varphi:\,\mathbb{G}_{2}\to\mathbb{R}$, the composition map
$\varphi\circ f:\Omega\to\mathbb{R}$ is in $W^{1,p}_{\mathbb{G}_{1},\rm
loc}(\Omega)$ and $|\nabla_{\mathbb{G}_{1}}(\varphi\circ f)|\leq\,{\rm
Lip}(\varphi)\,g$ a.e. in $\Omega$.

Notice that, exploiting this remark, it is easy to see that Definition 2.17
does not depend on the specific distance $d_{2}$ on $\mathbb{G}_{2}$ which
appears in (14). In fact, suppose $\tilde{d}_{2}$ is another invariant
distance on $\mathbb{G}_{2}$. Then $d_{2}$ and $\tilde{d}_{2}$ determine the
same Sobolev spaces. Indeed, since $d_{2}$ and $\tilde{d}_{2}$ are both
invariant, they are equivalent as distances (see [9, Corollary 5.1.5]), namely
there exists a constant $L>1$ such that

(16) $L^{-1}d_{2}(x,y)\leq\tilde{d}_{2}(x,y)\leq Ld_{2}(x,y)$

for every $x,y\in\mathbb{G}_{2}$. Suppose now $f\in
W^{1,p}(\Omega,\mathbb{G}_{2})$ according to the distance $d_{2}$. Then

  * •

The map $\mathbb{R}^{n}\ni x\mapsto\tilde{d}_{2}(0,f(x))$ is in
$L^{p}(\Omega)$ by (16);

  * •

For every $z\in\mathbb{G}_{2}$, the map
$\varphi_{z}:(\mathbb{G}_{2},d_{2})\to(\mathbb{R},|\cdot|)$ defined by
$\varphi_{z}(y):=\tilde{d}_{2}(z,y)$ is $L$-Lipschitz since

$|\varphi_{z}(x)-\varphi_{z}(y)|=|\tilde{d}_{2}(z,x)-\tilde{d}_{2}(z,y)|\leq\tilde{d}_{2}(x,y)\leq
Ld_{2}(x,y).$

Hence we can apply Remark 2.18 and deduce that the composition
$\tilde{d}_{2}(z,f(\cdot))=\varphi_{z}\circ f\in W^{1,p}_{\mathbb{G}_{1},\rm
loc}(\Omega)$;

  * •

Again by the previous considerations and Remark 2.18, we conclude that
$|\nabla_{\mathbb{G}_{1}}(\varphi_{z}\circ f)|\leq Lg$ a.e. in $\Omega$ for
each $z\in\mathbb{G}_{2}$.

Thus $f\in W^{1,p}(\Omega,\mathbb{G}_{2})$ with respect to $\tilde{d}_{2}$:
this shows that Definition 2.17 does not depend on the specific invariant
distance on $\mathbb{G}_{2}$.

P-differentiability and Morrey embedding theorem for real-valued Sobolev
spaces on Carnot groups when $p>Q$ (see, for instance, [59] and [28, Theorem
1.11]) can be extended to Sobolev maps between Carnot groups. Here $Q$ always
denotes the homogeneous dimension of $\mathbb{G}_{1}$.

######  Theorem 2.19 (P-differentiability with $p>Q$, [78, Corollary 2]).

Let $f\in W^{1,p}(\Omega,\mathbb{G}_{2})$ with $p>\,Q$. Then $f$ is
P-differentiable a.e. in $\Omega$.

The following theorem can be found in [42, Lemma 2.7].

######  Theorem 2.20 (Morrey-Sobolev embedding with $p>Q$).

Let $f:U(x,r)\subset\mathbb{G}_{1}\to\mathbb{G}_{2}$ and $p>\,Q$. There is a
positive constant $C_{p}$ (depending on $p$) such that, if $f\in
W^{1,p}(U(x,r),\mathbb{G}_{2})$ and $g$ is as in (15), then $f$ has a
continuous representative and for each $y\in U(x,r)$ it satisfies

(17)
$d_{2}(f(x),f(y))\leq\,C_{p}\,r^{1-\frac{Q}{p}}\,\left(\int_{U(x,r)}g^{p}\,d\mathcal{L}^{n}\right)^{1/p}.$

##  3\. Absolutely continuous functions between Carnot groups and some
properties

Let us collect in this section some interesting properties of $Q-$absolutely
continuous functions and functions of bounded $Q-$variation.
Let us begin to observe that, on bounded open sets (with finite measure would
suffice), the class of $Q$-absolutely functions is contained in the one of
bounded $Q$-variation functions.

######  Proposition 3.1.

Let $\Omega\subset\mathbb{G}_{1}$ be a bounded open set. Then
$AC^{Q}(\Omega,\mathbb{G}_{2})\subset BV^{Q}(\Omega,\mathbb{G}_{2})$.

###### Proof.

The strategy of the proof is inspired by [44, Lemma 2.16], where the authors
prove a local version of this result for abstract metric measure spaces. Let’s
denote by $A:=\mathcal{L}^{n}(\Omega)<+\infty$ and let $f\in
AC^{Q}(\Omega,\mathbb{G}_{2})$. Fix $\varepsilon=1$: by $Q-$absolute
continuity, there exists $\delta>0$ such that, for each finite disjoint family
of open balls $U_{1},\dots,U_{k}$ contained in $\Omega$,

(18) $\text{if }\sum_{i=1}^{k}\mathcal{L}^{n}(U_{i})<2\delta\,\text{ then
}\,\sum_{i=1}^{k}\left({\rm osc}_{U_{i}}f\right)^{Q}<\,1.$

Let $n\in\mathbb{N}$ be sufficiently large so that $A<n\delta$. Let now
$\mathcal{F}=\\{U_{1},\dots,U_{k}\\}$ be a disjoint family of balls in
$\Omega$ and write $\mathcal{F}=\mathcal{F}_{1}\cup\mathcal{F}_{2}$, where

$\mathcal{F}_{1}=\\{U_{j}\in\mathcal{F}:\mathcal{L}^{n}(U_{j})\geq\delta\\},$
$\mathcal{F}_{2}=\\{U_{j}\in\mathcal{F}:\mathcal{L}^{n}(U_{j})<\delta\\}.$

Notice that the elements in $\mathcal{F}_{1}$ are at most $n$ and their radii
are equibounded. For any $U=U(x,r)\in\mathcal{F}_{1}$ and any $y\in U$
consider a geodesic joining $x$ and $y$. On such a curve pick points
$x=x_{0},\,x_{1},\dots,\,x_{N-1},\,x_{N}=y$ such that $x_{i}$ and $x_{i+1}$
live in a common small ball (contained in $U$) of measure less than $2\delta$.
Note that it is possible to find $N$ independent of $y$ and $U$ since all the
elements in $\mathcal{F}_{1}$ have equibounded radii. By (18) we can now
estimate

$\begin{split}d_{2}(f(x),f(y))^{Q}&\leq(d_{2}(f(x),f(x_{1}))+\dots+d_{2}(f(x_{N-1}),f(y)))^{Q}\\\
&\leq C(Q)(d_{2}(f(x),f(x_{1}))^{Q}+\dots+d_{2}(f(x_{N-1}),f(y))^{Q})\leq
C(Q)N:=K.\end{split}$

By the triangular inequality the same estimate holds (with a greater constant
$\widetilde{K}$) for any $y,z\in U$. We conclude that
$\sum_{U\in\mathcal{F}_{1}}(\text{osc}_{U}f)^{Q}\leq n\widetilde{K}.$
Collect instead the elements of $\mathcal{F}_{2}$ into at most $n$ sets
$\mathcal{F}_{2,1},\dots\mathcal{F}_{2,n}$ with
$\sum_{U\in\mathcal{F}_{2,j}}\mathcal{L}^{n}(U)<2\delta$, so that by (18)

$\sum_{U\in\mathcal{F}_{2}}(\text{osc}_{U}f)^{Q}=\sum_{j=1}^{n}\sum_{U\in\mathcal{F}_{2,j}}(\text{osc}_{U}f)^{Q}<n$

Hence $V_{Q}(f,\Omega)<n(\widetilde{K}+1)$ and the thesis follows. ∎

It is clear that a $Q$-absolutely continuous function is actually continuous;
here we provide a first example of a class of $Q$-absolutely continuous maps.

######  Remark 3.2.

Let $f:\,\Omega\subset\mathbb{G}_{1}\to\mathbb{G}_{2}$ be Lipschitz
continuous. Then $f$ turns out to be $Q$-absolutely continuous. Indeed,

${\rm osc}_{U_{r}(x)}(f):={\rm
diam}_{\mathbb{G}_{2}}(f(U_{r}(x)))\leq\mathrm{Lip}(f)\,{\rm
diam}_{\mathbb{G}_{1}}(U_{r}(x))$

for each $x\in\mathbb{G}_{1}$ and $r>\,0$. Using the previous inequality we
immediately get that there exists $L_{1}>\,0$ such that

(19) ${\rm osc}_{U}(f)^{Q}\leq(2\,\mathrm{Lip}(f))^{Q}\,r^{Q}=L_{1}\,|U|$

for each (open) CC open ball $U\subset\mathbb{G}_{1}$ and the conclusion
follows.

######  Definition 3.3.

A function $f:\,\Omega\subset\mathbb{G}_{1}\to\mathbb{G}_{2}$ satisfies the
condition (RR) if there is a nonnegative function $w\in L^{1}(\Omega)$, called
weight, such that

(20) ${\rm osc}_{U_{r}(x)}(f)^{Q}\leq\,\int_{U_{r}(x)}w\,dy$

for every ball $U_{r}(x)\subset\Omega$, being $Q$ the homogeneous dimension of
$\mathbb{G}_{1}$.

######  Remark 3.4.

The acronym (RR) stands for Rado and Reichelderfer [62], who introduced a
similar condition for proving an area formula in $\mathbb{R}^{n}$. The
condition (RR), as stated in Definition 3.3, is a generalization to Carnot
groups of the one due to Malý [54].

######  Remark 3.5.

It is immediate to check that if $f:\Omega\to\mathbb{G}_{2}$ satisfies the
condition (RR), then $f\in AC^{Q}(\Omega,\mathbb{G}_{2})$, by the absolute
continuity of the Lebesgue integral. The viceversa is known in Euclidean
spaces for functions $f\in AC^{n}(\Omega,\mathbb{R}^{m})$ [15, Thm. 9] and it
is still open for maps between Carnot groups.

As observed in Remark 3.2, Lipschitz functions are $Q-$absolutely continuous.
Also Sobolev functions with a high order of summability share the same
property. This result was shown in [54, Thm. 4.1] for the Euclidean case and
the proof can be carried out in the same way by exploiting Morrey-Sobolev
embedding Theorem 2.20. Here we mention an alternative approach which relies
on an equivalent definition of Sobolev spaces through upper gradients.

######  Theorem 3.6.

Let $\Omega\subset\mathbb{G}_{1}$ be an open bounded set and let $f\in
W^{1,p}(\Omega,\mathbb{G}_{2})$ with $p>Q$, where $Q$ is the homogeneous
dimension of $\mathbb{G}_{1}$. Then, for a suitable representative, $f\in
AC^{Q}(\Omega,\mathbb{G}_{2})$.

###### Proof.

By the following Corollary 7.6, we can assume $f\in
N^{1,p}(\Omega:\mathbb{G}_{2})$, where $N^{1,p}(\Omega:\mathbb{G}_{2})$
denotes the Newtonian Sobolev space between $\Omega$ and $\mathbb{G}_{2}$ (see
Appendix 7). Applying [44, Theorem 1.4] we immediately get the conclusion. ∎

We are now interested in studying some differentiability properties of
absolutely continuous maps between Carnot groups. We start with the following

######  Theorem 3.7 (Differentiability a.e.).

Let $f\in BV^{Q}(\Omega,\mathbb{G}_{2})$. Then

  * (i)

$f$ is P-differentiable for a.e. $p\in\Omega$;

  * (ii)

${\rm Lip}(f)\in L^{Q}(\Omega)$, where ${\rm Lip}(f)$ is the function defined
in (5).

######  Remark 3.8.

Note that Pansu’s differentiability for maps in
$BV^{Q}(\Omega,\mathbb{G}_{2})$ does not follow from differentiability result
à la Cheeger-Kleiner proved in [44, Proposition 1.3] for maps between metric
spaces, of which the target metric space is biLipschitz embedded into a Banach
space satisfying the Radon- Nikodym property (RNP). Indeed it is well-known
that Carnot groups may not be biLipschitz embedded into a Banach space
satisfying (RNP) (see Appendix 7.1)

###### Proof.

We can follow the same proof of [54, Thm. 3.3] with suitable changes. For the
sake of completeness, we collect here the main steps. Let
$g:\Omega\to\mathbb{R}$ be a nonnegative upper semicontinuous function such
that

$0\leq g(x)\leq[Lip(f)(x)]^{Q}\qquad\text{for all }x\in\Omega.$

For any $x\in\\{g>0\\}$, there exists a unique integer $k=k(x)$ such that
$2^{k}\leq g(x)<2^{k+1}$. Furthermore, since $g$ is upper semicontinuous,
there exists $0<r=r(x)<1$ such that $U(x,5r)\subset\\{g<2^{k+1}\\}$. In
addition, from the fact that $g(x)\leq[Lip(f)(x)]^{Q}$, we can assume, up to
reduce $r$, that

(21) $2^{k}r^{Q}\leq g(x)r^{Q}<2(\underset{U(x,r)}{\text{osc}}f)^{Q}.$

Applying Vitali’s covering Lemma, we can extract a disjoint (hence at most
countable) family $\\{U(x_{i},r_{i})\\}_{i}\subset\\{U(x,r(x))\\}$ of balls
such that

$\\{g>0\\}\subset\bigcup_{x\in\\{g>0\\}}U(x,r(x))\subset\bigcup_{i}U(x_{i},5r_{i}).$

We define, for all $k\in\mathbb{Z},$ $I_{k}:=\\{i:k(x_{i})\geq k\\}$ and
$E_{k}:=\\{x\in\Omega:2^{k}\leq g(x)<2^{k+1}\\}$.
Notice that $E_{k}\subset\bigcup_{i\in I_{k}}U(x_{i},5r_{i})$. In fact, let
$x\in E_{k}$. In particular $g(x)>0$, hence $x\in U(x_{i},5r_{i})$ for some
$i$. If it were $i\not\in I_{k}$, then $k(x_{i})<k$, which implies $x\in
U(x_{i},5r_{i})\subset\\{g<2^{k}\\}$, contradicting $x\in E_{k}$. Thus we get

$\begin{split}\mathcal{L}^{n}(E_{k})&\leq\sum_{i\in
I_{k}}\mathcal{L}^{n}(U(x_{i},5r_{i}))\leq C\sum_{i\in I_{k}}r_{i}^{Q}\\\
&\stackrel{{\scriptstyle(\ref{esti})}}{{\leq}}C\sum_{i\in
I_{k}}2^{-k(x_{i})}(\underset{U(x_{i},r_{i})}{\text{osc}}f)^{Q}.\end{split}$

Finally we use this estimate to bound

$\begin{split}\int_{\Omega}g(x)\,dx&\leq\sum_{k=-\infty}^{+\infty}\int_{E_{k}}g(x)\,dx\leq\sum_{k=-\infty}^{+\infty}2^{k+1}\mathcal{L}^{n}(E_{k})\\\
&\leq C\sum_{k=-\infty}^{+\infty}\sum_{i\in
I_{k}}2^{k+1-k(x_{i})}(\underset{U(x_{i},r_{i})}{\text{osc}}f)^{Q}\\\
&=C\sum_{i=1}^{+\infty}\sum_{k\leq
k(x_{i})}2^{k+1-k(x_{i})}(\underset{U(x_{i},r_{i})}{\text{osc}}f)^{Q}\\\
&=4C\sum_{i=1}^{+\infty}(\underset{U(x_{i},r_{i})}{\text{osc}}f)^{Q}\leq
CV_{Q}(f,\Omega),\end{split}$

where we exploited the properties of the $Q-$variation and the fact that
$\sum_{k\leq k(x_{i})}2^{k}=2^{k(x_{i})+1}$. It follows that

$\int_{\Omega}[Lip(f)(x)]^{Q}\,dx=\sup\Big{\\{}\int_{\Omega}g(x)\,dx:g\text{
u.s.c., }0\leq g\leq Lip(f)^{Q}\Big{\\}}\leq CV_{Q}(f,\Omega).$

Thus (ii) follows and also that ${\rm Lip}(f)(x)<\infty$ for a.e.
$x\in\Omega$. From Theorem 2.7, (i) follows, too. ∎

We will now focus our attention to the case when
$\mathbb{G}_{2}=\,(\mathbb{R}^{m},+)$. Note that in this case, since
$\mathbb{R}^{m}$ is abelian, a map
$f=\,(f_{1},\dots,f_{m}):\,\Omega\subset\mathbb{G}\to\mathbb{R}^{m}$ is
$Q$-absolutely continuous (respectively of finite $Q$-variation) if and only
if each component $f_{i}:\,\Omega\subset\mathbb{G}\to\mathbb{R}$ is
$Q$-absolutely continuous (respectively of finite $Q$-variation), for each
$i=1,\dots,m$. Thus we can reduce the study to real-valued functions defined
on an open set of a Carnot group. We will denote

$BV_{\mathbb{G}}^{Q}(\Omega):=\,BV^{Q}(\Omega,\mathbb{R}),\,AC_{\mathbb{G}}^{Q}(\Omega):=\,AC^{Q}(\Omega,\mathbb{R})\,,$
$BV_{\mathbb{G};\rm loc}^{Q}(\Omega):=\,BV_{\rm
loc}^{Q}(\Omega,\mathbb{R}),\,AC_{\mathbb{G};\rm loc}^{Q}(\Omega):=\,AC_{\rm
loc}^{Q}(\Omega,\mathbb{R})\,.$

The following theorem can be proved mutatis mutandis as its Euclidean
counterpart, and the proof relies on standard arguments. For this reason we
omit it.

######  Theorem 3.9.

If $f\in L^{Q}(\Omega)\cap BV_{\mathbb{G}}^{Q}(\Omega)$ let us denote

$\|f\|_{Q,BV}:=\,\|f\|_{L^{Q}(\Omega)}+V_{Q}(f,\Omega)^{1/Q}.$

Then $(L^{Q}(\Omega)\cap BV_{\mathbb{G}}^{Q}(\Omega),\|\cdot\|_{Q,BV})$ is a
Banach space. Moreover $L^{Q}(\Omega)\cap AC_{\mathbb{G}}^{Q}(\Omega)$ is a
closed linear subspace of $L^{Q}(\Omega)\cap BV_{\mathbb{G}}^{Q}(\Omega)$.

Theorem 3.7 states that $BV^{Q}$-functions are P-differentiable pointwise
almost everywhere. We will now prove that such maps enjoy additional
differentiability properties, namely they are weakly differentiable according
to Definition 2.17. More precisely,

######  Theorem 3.10 (Weak differentiability).

If $Q>\,1$, then

$BV^{Q}_{\rm loc}(\Omega,\mathbb{G}_{2})\subset W^{1,Q}_{\rm
loc}(\Omega,\mathbb{G}_{2}).$

######  Remark 3.11.

Note that from [44, Theorem 1.2] it follows that $BV^{Q}_{\rm
loc}(\Omega,\mathbb{G}_{2})\subset D^{1,Q}_{\rm loc}(\Omega:\mathbb{G}_{2})$,
where $D^{1,Q}_{\rm loc}(\Omega:\mathbb{G}_{2})$ denotes the (local) Dirichlet
class of maps between metric spaces $(\Omega,d_{1},\mathcal{L}^{n})$ and
$(\mathbb{G}_{2},d_{2})$ according to [34, Chap. 7] (see Appendix 7.2). It is
also straightforward from definition that $BV^{Q}_{\rm
loc}(\Omega,\mathbb{G}_{2})\subset L^{\infty}_{\rm
loc}(\Omega,\mathbb{G}_{2})$. By definition, $D^{1,Q}_{\rm
loc}(\Omega:\mathbb{G}_{2})\cap L^{Q}_{\rm
loc}(\Omega,\mathbb{G}_{2})=\,N^{1,Q}_{\rm loc}(\Omega:\mathbb{G}_{2})$, where
$N^{1,Q}_{\rm loc}(\Omega:\mathbb{G}_{2})$ denotes the (local) Newtonian
Sobolev space of maps between the previous metric spaces. On the other hand,
from Corollary 7.6, it follows that $N^{1,Q}_{\rm
loc}(\Omega:\mathbb{G}_{2})=\,W^{1,Q}_{\rm loc}(\Omega,\mathbb{G}_{2})$.
Therefore Theorem 3.10 can be shown also by this strategy.

Here we propose an alternative proof of Theorem 3.10 with respect to the one
in [44, Theorem 1.2], by using the intrinsic convolution in Carnot groups and
in the same spirit of [54, Theorem 3.2]. In order to prove Theorem 3.10 we
need the following technical lemma. Point (i) establishes a uniform estimate
on the distance between the images of integer points after a right-
translation. Point (ii) provides a way to cover $\mathbb{G}$ with arbitrarily
small balls centered at suitable dilations of the integer lattice.

######  Lemma 3.12.

  * (i)

For every $R_{0}>0$, it holds that

(22) $\sigma_{0}:=\inf\\{d_{c}\left(\xi\cdot p,\eta\cdot
p\right):\,\xi,\eta\in\mathbb{Z}^{n},\,\xi\neq\eta,\,p\in
B(0,R_{0})\\}>\,0\,.$

  * (ii)

Given a positive integer $k$, let ${\bf Z}_{k}$ be the set of all points
$z\in\mathbb{G}\equiv\mathbb{R}^{n}$ such that $\delta_{k}(z)$ has integer
coordinates, that is,

(23) ${\bf Z}_{k}:=\delta_{1/k}(\mathbb{Z}^{n})\,.$

Then there exists a positive constant $\sigma_{1}>0$ such that

$\cup_{z\in{\bf Z}_{k}}U(z,\sigma_{1}/k)=\,\mathbb{G}\,.$

###### Proof.

(i) Since $d_{c}$ and $d_{\infty}$ are equivalent distances, it is sufficient
to show that

(24) $\sigma^{*}_{0}:=\inf\\{d_{\infty}\left(\xi\cdot p,\eta\cdot
p\right):\,\xi,\eta\in\mathbb{Z}^{n},\,\xi\neq\eta,\,p\in
B_{\infty}(0,R_{0})\\}>\,0\,.$

By contradiction, if $\sigma^{*}_{0}=\,0$, there exist three sequences
$(\xi_{h})_{h},\,(\eta_{h})_{h}$ in $\mathbb{Z}^{n}$, $(p_{h})_{h}$ in
$B_{\infty}(0,R_{0})$,with $\xi_{h}\neq\eta_{h}$ and
$\lim_{h\to\infty}d_{\infty}\left(\xi_{h}\cdot p_{h},\eta_{h}\cdot
p_{h}\right)=\,\lim_{h\to\infty}\|p_{h}^{-1}\cdot\eta_{h}^{-1}\cdot\xi_{h}\cdot
p_{h}\|_{\infty}=\,0$. Let $x_{h}:=\eta_{h}^{-1}\cdot\xi_{h}$. We observe that
$(x_{h})_{h}$ is bounded in $(\mathbb{R}^{n},d_{\infty})$. In fact, for all
$h$,

$\begin{split}\|x_{h}\|_{\infty}&=\|p_{h}\cdot p_{h}^{-1}\cdot x_{h}\cdot
p_{h}\cdot p_{h}^{-1}\|_{\infty}\\\ &\leq\|p_{h}\|_{\infty}+\|p_{h}^{-1}\cdot
x_{h}\cdot p_{h}\|_{\infty}+\|p_{h}^{-1}\|_{\infty}\\\ &\leq
2R_{0}+\|p_{h}^{-1}\cdot x_{h}\cdot p_{h}\|_{\infty}.\end{split}$

Hence, up to a subsequence, we can assume that $p_{h}\to p_{*}$ and $x_{h}\to
x_{*}$ as $h\to\infty$. This implies, by the continuity of the norms and of
the group operation, that

(25) $0=\lim_{h\to\infty}\|p_{h}^{-1}\cdot x_{h}\cdot
p_{h}\|_{\infty}=\|p_{*}^{-1}\cdot x_{*}\cdot p_{*}\|_{\infty},$ (26)
$\|x_{*}\|_{\mathbb{R}^{n}}=\lim_{h\to\infty}\|\eta_{h}^{-1}\cdot\xi_{h}\|_{\mathbb{R}^{n}}.$

From (25) it follows that $x_{*}=0$.
Let us now show that

(27) $\|\eta^{-1}\cdot\xi\|_{\mathbb{R}^{n}}\geq\,1\text{ for each
}\xi,\,\eta\in\mathbb{Z}^{n}\text{, with }\xi\neq\eta\,,$

thus a contradiction will follow from (26). Let us represent now
$\xi,\,\eta\in\mathbb{Z}^{n}$ by means of the layer representation

$\xi=[\xi^{1},\xi^{2},\dots,\xi^{s}]\text{ and
}\eta=[\eta^{1},\eta^{2},\dots,\eta^{s}]\,,$

with $\xi^{i},\,\eta^{i}\in\mathbb{Z}^{m_{i}}$ ($i=1,\dots,s$) and, since
$\xi\neq\eta$, we can assume that $\xi^{i_{0}}\neq\eta^{i_{0}}$ for a suitable
$i_{0}$. Here we recall that $\mathfrak{g}=V_{1}\oplus\dots\oplus V_{s}$ and
$m_{i}=\dim(V_{i})$. Meanwhile, we can represent the element
$\eta^{-1}\cdot\xi$ as

$\eta^{-1}\cdot\xi=[(\eta^{-1}\cdot\xi)^{1},(\eta^{-1}\cdot\xi)^{2},\dots,(\eta^{-1}\cdot\xi)^{s}]\,,$

$(\eta^{-1}\cdot\xi)^{1}=\xi^{1}-\eta^{1},\,(\eta^{-1}\cdot\xi)^{i}=\xi^{i}-\eta^{i}+\mathcal{Q}^{i}(-\eta^{1},\dots,-\eta^{i-1},\xi^{1},\dots,\xi^{i-1})$

with $i=2,\dots,s$ and
$\mathcal{Q}^{i}:=(\mathcal{Q}_{h_{i-1}+1},\dots,\mathcal{Q}_{h_{i}})$. It is
clear that if $\xi^{1}-\eta^{1}\neq 0$, then
$\|\eta^{-1}\cdot\xi\|_{\mathbb{R}^{n}}\geq\,\|\xi^{1}-\eta^{1}\|_{\mathbb{R}^{m_{1}}}\geq\,1$
and (27) holds. If $\xi^{1}-\eta^{1}=\,0$, then, by the properties of
$\mathcal{Q}_{i}$’ s (see (3))
$\mathcal{Q}^{2}(-\eta^{1},\xi^{1})=\,\mathcal{Q}^{2}(-\xi^{1},\xi^{1})=\,0$,
which implies
$\|\eta^{-1}\cdot\xi\|_{\mathbb{R}^{n}}\geq\,\|(\eta^{-1}\cdot\xi)^{2}\|_{\mathbb{R}^{m_{2}}}=\,\|\xi^{2}-\eta^{2}\|_{\mathbb{R}^{m_{2}}}$.
Thus if $\xi^{2}-\eta^{2}\neq\,0$, (27) holds. Otherwise, in the baddest case,
it could happen that
$\xi^{1}-\eta^{1}=\,\dots=\,\xi^{i_{0}-1}-\eta^{i_{0}-1}=\,0$. This implies
that
$\mathcal{Q}^{i_{0}}(-\eta^{1},\dots,-\eta^{i_{0}-1},\xi^{1},\dots,\xi^{i_{0}-1})=\,\mathcal{Q}^{i_{0}}(-\xi^{1},\dots,-\xi^{i_{0}-1},\xi^{1},\dots,\xi^{i_{0}-1})=\,0$.
Thus
$\|\eta^{-1}\cdot\xi\|_{\mathbb{R}^{n}}\geq\,\|(\eta^{-1}\cdot\xi)^{i_{0}}\|_{\mathbb{R}^{m_{i_{0}}}}=\,\|\xi^{i_{0}}-\eta^{i_{0}}\|_{\mathbb{R}^{m_{i_{0}}}}\geq\,1$
and (27) holds.

(ii) It is well-known that it is possible to construct a pavage in a Carnot
group (see, for instance, [20, Sect. 2.3]). More precisely, if
$Y:=\\{x=(x_{1},\dots,x_{n}):\,-1/2<\,x_{i}\leq 1/2\\}$ denote the unit cube
in $\mathbb{R}^{n}$ centered at the origin, denote

$Q^{\varepsilon}_{\xi}:=\,\delta_{\varepsilon}(\tau_{\xi}(Y))\text{ if
}\varepsilon>\,0,\,\xi\in\mathbb{Z}^{n}\,.$

Then, it holds that, for each $\varepsilon>\,0$, the family of sets
$\\{Q^{\varepsilon}_{\xi}:\,\xi\in\mathbb{Z}^{n}\\}$ is a pavage of
$\mathbb{G}=(\mathbb{R}^{n},\cdot)$, that is, the sets are disjoint and

(28) $\mathbb{R}^{n}=\,\cup_{\xi\in\mathbb{Z}^{n}}Q^{\varepsilon}_{\xi}\,.$

It is easy to see that there exist $\sigma^{*}_{1},\sigma_{1}>\,0$ such that

(29) $Y\subset U_{\infty}(0,\sigma^{*}_{1})\subset U(0,\sigma_{1})\,.$

Thus, choosing $\varepsilon=1/k$, from (28) and (29), we get the desired
conclusion. ∎

###### Proof of Theorem 3.10.

1st step. Let us first show our conclusion in the case
$\mathbb{G}_{1}\equiv\mathbb{G}$ and $\mathbb{G}_{2}=\,\mathbb{R}$. Proof’s
strategy will be the same as [54, Theorem 3.2]. We will point out in the
following the main changes and differences which need in the setting of Carnot
groups. First assume that $f\in BV^{Q}_{\mathbb{G}}(\mathbb{R}^{n})\cap
L^{Q}(\mathbb{R}^{n})$. Let $\sigma_{1}$ be as in (ii) of the previous lemma
and find $\sigma_{0}$ as in (i) for $R_{0}=2\sigma_{1}$. This means that, for
each $y\in B(0,2\sigma_{1})$,

$\sigma_{0}\leq\inf\\{d_{c}(\xi\cdot y,\eta\cdot
y):\xi,\eta\in\mathbb{Z}^{n},\xi\neq\eta\\}.$

This also implies that, for every integer $k$ and every $y\in
B(0,2\sigma_{1}/k)$,

$\frac{\sigma_{0}}{k}\leq\inf\\{d_{c}(p\cdot y,q\cdot
y):p,q\in\mathbf{Z}_{k},p\neq q\\}.$

Hence, for every $y\in B(0,2\sigma_{1}/k)$, the family of CC balls
$\\{U(z\cdot y,\sigma_{0}/3k):z\in\mathbf{Z}_{k}\\}$ is disjoint. In addition,
up to increase $\sigma_{1}$, we can assume $\sigma_{1}\geq\sigma_{0}$.
Let $(\varrho_{k})_{k}$ be the sequence of mollifiers defined in (8), where
$\varepsilon=1/k$, $\varrho_{k}\equiv\varrho_{1/k}$ and suppose that
${\mathrm{spt}}(\varrho)\subset B(0,\sigma_{0}/6)$. Observe that it holds that

(30)
$\nabla_{\mathbb{G}}\varrho_{k}(x)=\,k^{Q+1}\,\nabla_{\mathbb{G}}\varrho(\delta_{k}(x))\text{
for each }x\in\mathbb{R}^{n},\,k\in\mathbb{N}\,,$ (31)
$\int_{\mathbb{R}^{n}}X_{i}\varrho_{k}(r_{x}(y))\,dy=\,0\text{ for each
}x\in\mathbb{R}^{n},\,i=1,\dots,m_{1}\,,$ (32)
$\begin{split}\nabla_{\mathbb{G}}(g\star\varrho_{k})(x)&=\,\int_{\mathbb{R}^{n}}\nabla_{\mathbb{G}}\varrho_{k}(r_{x}(y^{-1}))\,g(y)\,dy\\\
&=\,\int_{\mathbb{R}^{n}}\nabla_{\mathbb{G}}\varrho_{k}(w^{-1})\,g(\ell_{x}(w))\,dw\\\
&\text{ for each }g\in L^{1}_{\rm
loc}(\mathbb{R}^{n}),\,x\in\mathbb{R}^{n}\,.\end{split}$

Indeed (30) follows by the definition of $\varrho_{k}$ and homogeneity of
intrinsic gradient $\nabla_{\mathbb{G}}:=(X_{1},\dots,X_{m_{1}})$;(31) follows
by the properties of $\varrho_{k}$ , the structure of the horizontal vector
fields $X_{i}$ and the right-invariance of $n$-dimensional Lebesgue measure
$\mathcal{L}^{n}$;(32) follows by the definition of convolution and by the
left invariance of the horizontal vector fields.

Thus , given $y\in B(0,2\sigma_{1}/k)$, by (31),(32) and the fact that the
family $\\{U(z\cdot y,\sigma_{0}/3k):z\in\mathbf{Z}_{k}\\}$ is disjoint,

$\begin{split}&\sum_{z\in{\bf Z}_{k}}\int_{U(z\cdot
y,\sigma_{0}/6k)}|\nabla_{\mathbb{G}}(f\star\varrho_{k})(x)|^{Q}\,dx=\\\
&=\,\sum_{z\in{\bf Z}_{k}}\int_{U(z\cdot
y,\sigma_{0}/6k)}\left|\int_{\mathbb{R}^{n}}\nabla_{\mathbb{G}}\varrho_{k}(w^{-1})\,f(\tau_{x}(w))\,dw\right|^{Q}\,dx\\\
&=\,\sum_{z\in{\bf Z}_{k}}\int_{U(z\cdot
y,\sigma_{0}/6k)}\left|\int_{\mathbb{R}^{n}}\nabla_{\mathbb{G}}\varrho_{k}(w^{-1})\,(f(\tau_{x}(w))-f(x))\,dw\right|^{Q}\,dx\\\
&=\,\sum_{z\in{\bf Z}_{k}}\int_{U(z\cdot
y,\sigma_{0}/6k)}\left|k^{Q+1}\,\int_{\mathbb{R}^{n}}\nabla_{\mathbb{G}}\varrho(\delta_{k}(w^{-1}))\,(f(\tau_{x}(w))-f(x))\,dw\right|^{Q}\,dx\\\
&=\,\sum_{z\in{\bf Z}_{k}}\int_{U(z\cdot
y,\sigma_{0}/6k)}\left|k^{Q+1}\,\int_{B(0,\sigma_{0}/6k)}\nabla_{\mathbb{G}}\varrho(\delta_{k}(w^{-1}))\,(f(\tau_{x}(w))-f(x))\,dw\right|^{Q}\,dx\\\
&\leq\,c_{1}\,\sum_{z\in{\bf Z}_{k}}({\rm osc}_{U(z\cdot
y,\sigma_{0}/3k)}f)^{Q}\leq\,c_{1}\,V_{Q}(f,\mathbb{R}^{n})\,.\end{split}$

By integrating the previous inequality with respect to $y\in
B(0,2\sigma_{1}/k)$, it follows that there exists a positive constant
$c_{2}>\,0$ such that

(33) $\begin{split}&k^{Q}\,\int_{B(0,2\sigma_{1}/k)}\left(\sum_{z\in{\bf
Z}_{k}}\int_{U(z\cdot
y,\sigma_{0}/6k)}|\nabla_{\mathbb{G}}(f\star\varrho_{k})(x)|^{Q}\,dx\right)\,dy\\\
&\leq\,c_{2}\,V_{Q}(f,\mathbb{R}^{n})\text{ for each positive integer
}k\,.\end{split}$

Notice also that, writing
$g(x)=|\nabla_{\mathbb{G}}(f\star\varrho_{k})(x)|^{Q}$, we get, for each
$z\in\mathbb{G},$

$\begin{split}\int_{U(z,\sigma_{1}/k)}g(x)\,dx&=\int_{U(0,\sigma_{1}/k)}g(z\cdot
x)\,dx\\\
&=c_{3}k^{Q}\int_{U(0,\sigma_{0}/6k)}\left(\int_{U(0,\sigma_{1}/k)}g(z\cdot
x)\,dx\right)dy\\\
&=c_{3}k^{Q}\int_{U(0,\sigma_{0}/6k)}\left(\int_{U(0,\sigma_{1}/k)\cdot
y^{-1}}g(z\cdot x\cdot y)\,dx\right)dy\\\ &\leq
c_{3}k^{Q}\int_{U(0,\sigma_{0}/6k)}\left(\int_{B(0,2\sigma_{1}/k)}g(z\cdot
x\cdot y)\,dx\right)dy\\\
&=c_{3}k^{Q}\int_{B(0,2\sigma_{1}/k)}\left(\int_{U(0,\sigma_{0}/6k)}g(z\cdot
x\cdot y)\,dy\right)dx\\\
&=c_{3}k^{Q}\int_{B(0,2\sigma_{1}/k)}\left(\int_{U(z\cdot
x,\sigma_{0}/6k)}g(y)\,dy\right)dx.\end{split}$

In these passages we performed some changes of variables (which are allowed
being the $n$-dimensional Lebesgue measure both left and right invariant), we
noticed that $U(0,\sigma_{1}/k)\cdot y^{-1}\subset B(0,2\sigma_{1}/k)$ for
every $y\in U(0,\sigma_{0}/6k)$ and applied Fubini-Tonelli theorem.
Let us now notice that, by applying Lemma 3.12 (ii) and the previous
estimates, it follows that

(34)
$\begin{split}&\int_{\mathbb{R}^{n}}|\nabla_{\mathbb{G}}(f\star\varrho_{k})(x)|^{Q}\,dx\leq\,\sum_{z\in{\bf
Z}_{k}}\int_{U(z,\sigma_{1}/k)}|\nabla_{\mathbb{G}}(f\star\varrho_{k})(x)|^{Q}\,dx\\\
&\leq\,c_{3}\,k^{Q}\,\int_{B(0,2\sigma_{1}/k)}\left(\sum_{z\in{\bf
Z}_{k}}\int_{U(z\cdot
y,\sigma_{0}/6k)}|\nabla_{\mathbb{G}}(f\star\varrho_{k})(x)|^{Q}\,dx\right)\,dy\,.\end{split}$

From (33) and (34), we get that there exists $c_{4}>\,0$ such that

(35)
$\int_{\mathbb{R}^{n}}|\nabla_{\mathbb{G}}(f\star\varrho_{k})(x)|^{Q}\,dx\leq\,c_{4}\,V_{Q}(f,\mathbb{R}^{n})<\,\infty,\,\forall\,k\in\mathbb{N}\,.$

Since $W^{1,Q}_{\mathbb{G}}(\mathbb{R}^{n})$ is a reflexive space and
$f\star\varrho_{k}\to f$ in $L^{Q}(\mathbb{R}^{n})$, as $k\to\infty$, we can
infer that $f\in W^{1,Q}_{\mathbb{G}}(\mathbb{R}^{n})$. In the general case
$f\in BV^{Q}_{\mathbb{G};\rm loc}(\Omega)$, it is sufficient to show that
$f\,\varphi\in W_{\mathbb{G}}^{1,Q}(\Omega)$ for each
$\varphi\in{\mathbf{C}}^{\infty}_{c}(\Omega)$ which is a cutoff function in a
small neighbourhood of a point. Using zero-extension, since $f\,\varphi\in
BV^{Q}_{\mathbb{G}}(\mathbb{R}^{n})\cap L^{Q}(\mathbb{R}^{n})$, by the
previous argument, $f\,\varphi\in W_{\mathbb{G}}^{1,Q}(\mathbb{R}^{n})$ and we
are done.

2nd step. Let us now show the conclusion in the general case. We assume $f\in
BV^{Q}_{\rm loc}(\Omega,\mathbb{G}_{2})$ and prove that $f\in
W^{1,Q}(\Omega^{\prime},\mathbb{G}_{2})$ for every
$\Omega^{\prime}\subset\subset\Omega.$ Let $u_{z}:\,\Omega\to\mathbb{R}$ be
the function in (14). Then, by the triangle inequality for distances, it
follows that

${\rm osc}_{U_{d_{1}}(x,r)}(u_{z})\leq\,{\rm osc}_{U_{d_{1}}(x,r)}(f)$

for each ball $U_{d_{1}}(x,r)\subset\Omega$ and $z\in\mathbb{G}_{2}$. In
particular, $u_{z}\in BV^{Q}_{\mathbb{G};\rm loc}(\Omega)$ and

$V_{Q}(u_{z},\Omega)\leq\,V_{Q}(f,\Omega)\text{ for each
}z\in\mathbb{G}_{2}\,.$

From Theorem 3.7 and the previous step, $u_{z}:\,\Omega\to\mathbb{R}$ is
P-differentiable a.e. and $u_{z}\in W_{\mathbb{G}_{1},\rm loc}^{1,Q}(\Omega)$
(implying $u_{z}\in W^{1,Q}_{\mathbb{G}_{1}}(\Omega^{\prime})$). Thus, by
Proposition 2.15,

$\begin{split}\left|X_{i}u_{z}(x)\right|&=\,\left|\lim_{t\to
0}\frac{u_{z}\left(\exp(tX_{i})(x)\right)-u_{z}(x)}{t}\right|\\\ &\leq\,{\rm
Lip}(u_{z})(x)\leq\,\,{\rm Lip}(f)(x)\end{split}$

for a.e. $x\in\Omega$ and for each $z\in\mathbb{G}_{2}$. Since
$\Omega^{\prime}\subset\subset\Omega$ and $f\in BV^{Q}_{\rm
loc}(\Omega,\mathbb{G}_{2})$, by Theorem 3.7 it follows that ${\rm Lip}(f)\in
L^{Q}(\Omega^{\prime})$. To conclude notice that $f\in
L^{Q}(\Omega^{\prime},\mathbb{G}_{2})$: indeed $f\in BV^{Q}_{\rm
loc}(\Omega,\mathbb{G}_{2})$, so it is locally bounded and
$\Omega^{\prime}\subset\subset\Omega$. Hence $f\in
W^{1,Q}(\Omega^{\prime},\mathbb{G}_{2})$ according to Definition 2.17.
∎

##  4\. Sobolev maps between Carnot groups with horizontal gradient of
sections in $L^{Q,1}$

In this section we deal with the sharp condition on the summability of the
weak gradient of a Sobolev function which ensures continuity and
differentiability almost everywhere. It is well known that these properties
hold for maps in the Sobolev space $W^{1,p}_{\rm loc}(\mathbb{R}^{n})$
provided that $p>n$ (in the limit case $p=n$ the answer is negative, see also
Remark 4.3). Such a result has been extended to the case of Carnot groups (see
Theorem 2.19 and Theorem 3.6). In his article [72], Elias M. Stein sharpened
the previous result on Euclidean spaces: if a weakly differentiable function
admits weak gradient in the Lorentz space $L^{n,1}(\mathbb{R}^{n})$, then it
is continuous and differentiable a.e. This is a remarkable improvement since,
by Proposition 2.10, $L^{n,1}$ is locally an intermediate space between
$L^{n}$ and $L^{p}$ (for all $p>n$). The approach of Stein makes use of the
machinery of harmonic analysis, while later Kauhanen, Koskela and Malý [39]
obtained an alternative proof exploiting tools of real analysis and, in
particular, the already discussed notion of $n-$absolutely continuous
functions. Next we will follow this last strategy in order to discuss Stein’s
result for maps between Carnot groups.
Continuity and differentiability have been already studied also for real
valued Sobolev functions on metric measure spaces with upper gradient in the
Lorentz space $L^{Q,1}$, where $Q$ denotes the Hausdorff dimension of the
metric space (see, for instance, [64, 63, 81, 79, 80]). Here we are going to
prove that a Sobolev map $f:\,\Omega\subset\mathbb{G}_{1}\to\mathbb{G}_{2}$,
with horizontal gradients of its sections $u_{z}:\,\Omega\to\mathbb{R}$
uniformly bounded in $L^{Q,1}(\Omega)$ (see (14)), actually satisfies the
condition (RR), if $Q$ denotes the homogeneous dimension of $\mathbb{G}_{1}$.

Let us first recall an estimate of the Riesz potential in Carnot groups in
terms of the Orlicz conditions, which extends the Euclidean one first
introduced in [39, Theorem 3.1].

######  Theorem 4.1.

Let $\mathbb{G}\equiv\mathbb{R}^{n}$ be a Carnot group, $d$ the associated CC
distance, $Q$ its homogeneous dimension and $\Omega\subset\mathbb{G}$. Let $g$
be a nonnegative measurable function on $\Omega$ and let $\varphi$ be a
nonincreasing positive function on $(0,\infty)$. If $F_{\varphi}(s)$ is
defined as in (11), then for any $z\in\Omega$ and any measurable set
$E\subset\Omega$, we have the inequality

$\left(\int_{E}d(x,z)^{1-Q}g(x)\,dx\right)^{Q}\leq\,C_{Q}\,\left(\int_{0}^{\infty}\varphi^{\frac{1}{Q}}ds\right)^{Q-1}\int_{E}F_{\varphi}(g)\,dx$

where $C_{Q}:=\,2^{Q}\,\left(|\partial
U(0,1)|_{\mathbb{G}}(\mathbb{G})\right)^{Q-1}$ and $|\partial
U(0,1)|_{\mathbb{G}}(\mathbb{G})$ denotes the $\mathbb{G}$-perimeter of the CC
open ball $U(0,1)$.

###### Proof.

The proof can be verbatim carried out as in [39, Theorem 3.1], by replacing
the Euclidean dimension $n$ with the homogeneous dimension $Q$, the Euclidean
norm with the homogeneous norm induced by $d$, and by using the integration
formula in polar coordinates in Carnot groups (see, for instance, [59,
Corollary 4.6]). ∎

######  Theorem 4.2.

Let $\mathbb{G}_{1}\equiv\mathbb{R}^{n}$,
$\mathbb{G}_{2}\equiv\mathbb{R}^{m}$, $\Omega\subset\mathbb{G}_{1}$ be an open
set and let $Q$ denote the homogeneous dimension of $\mathbb{G}_{1}$. Assume
that $f\in W^{1,1}_{\rm loc}(\Omega,\mathbb{G}_{2})$ and the function $g$ in
(15) belongs to $L^{Q,1}(\Omega)$. Then, a suitable representative of $f$ is
continuous and satisfies the condition (RR). In particular $f\in
AC^{Q}(\Omega,\mathbb{G}_{2})$ and then it is P-differentiable a.e. in
$\Omega$.

###### Proof.

1st step. First assume $\mathbb{G}_{1}=\mathbb{G}$,
$\mathbb{G}_{2}=\mathbb{R}$ with $f\in W^{1,1}_{\mathbb{G},\rm loc}(\Omega)$
and $|\nabla_{\mathbb{G}}f|\in L^{Q,1}(\Omega)$. We stress that this case was
already studied in the general case of measure metric spaces (see, for
instance, [79, Thm. 6.6]) by extending the strategy exploited in [39]. However
we are going to repeat this strategy since we will use it in the second step,
which is still open to our knowledge.
First of all notice that we can apply Theorem 2.11 inferring that, since
$|\nabla_{\mathbb{G}}f|\in L^{Q,1}(\Omega)$, there exists a nonnegative
nonincreasing function $\varphi:(0,+\infty)\to[0,+\infty)$ such that

(36) $\int_{0}^{+\infty}\varphi^{\frac{1}{Q}}(s)\,ds<+\infty,$ (37)
$\int_{\Omega}F_{\varphi}(|\nabla_{\mathbb{G}}f(x)|)\,dx<+\infty.$

Let’s apply now Theorem 2.16, finding a negligible set $N$ such that

(38) $|f(z)-f_{U}|\leq
C\int_{U}\frac{|\nabla_{\mathbb{G}}f(y)|}{d(z,y)^{Q-1}}\,dy$

for each CC open ball $U\subset\Omega$ and $z\in U\setminus N$. This implies
that, for every CC open ball $U\subset\Omega$,

(39) $(\text{osc}_{U\setminus N}f)^{Q}\leq\int_{U}w(x)\,dx,$

where

$w(x)=2^{Q}C^{Q}C_{Q}\left(\int_{0}^{+\infty}\varphi^{\frac{1}{Q}}(s)\,ds\right)^{Q-1}F_{\varphi}(|\nabla_{\mathbb{G}}f(x)|).$

In order to prove (39) we can assume the left-hand side to be positive, and
choose $0\leq c<\text{osc}_{U\setminus N}f$. By definition of oscillation and
the triangular inequality, there exists $z\in U\setminus N$ such that $c\leq
2|f(z)-f_{U}|$. Combining this fact with (38) and Theorem 4.1, we infer that

$\begin{split}c^{Q}&\leq
2^{Q}C^{Q}\left(\int_{U}\frac{|\nabla_{\mathbb{G}}f(y)|}{d(z,y)^{Q-1}}\,dy\right)^{Q}\\\
&\leq
2^{Q}C^{Q}C_{Q}\,\left(\int_{0}^{\infty}\varphi^{\frac{1}{Q}}ds\right)^{Q-1}\int_{U}F_{\varphi}(|\nabla_{\mathbb{G}}f(x)|)\,dx\\\
&=\int_{U}w(x)\,dx.\end{split}$

By arbitrariness of $0\leq c<$osc${}_{U\setminus N}u$, we conclude that (39)
is satisfied. As a consequence, $f$ is locally uniformly continuous on
$\Omega\setminus N$ and can be extended to a continuous function $\tilde{f}$
on the whole space $\Omega$ which satisfies the (RR) condition with weight
$w$.

2nd step. Let us now prove the general case. For given
$z\in\mathbb{G}_{2}\equiv\mathbb{R}^{m}$ let $u_{z}:\,\Omega\to\mathbb{R}$
denote a section of $f$, that is, the function defined in (14). By definition
$u_{z}\in\ W^{1,1}_{\mathbb{G}_{1},\rm loc}(\Omega)$ and satisfies (15) with
$g\in L^{Q,1}(\Omega)$. Thus, by the 1st step, there exists a continuous
function $\tilde{u}_{z}:\,\Omega\to\mathbb{R}$ and a negligible set
$N_{z}\subset\Omega$, found as in (38), such that

(40) $\tilde{u}_{z}(x)=\,u_{z}(x)=\,d_{2}(z,f(x))\text{ for each
}x\in\Omega\setminus N_{z}\,.$

Now we proceed again as in the 1st step: since $g\in L^{Q,1}(\Omega)$, we know
that there exists a nonnegative nonincreasing function
$\varphi:(0,+\infty)\to[0,+\infty)$ such that

(41) $\int_{0}^{+\infty}\varphi^{\frac{1}{Q}}(s)\,ds<+\infty,$ (42)
$\int_{\Omega}F_{\varphi}(g(x))\,dx<+\infty.$

We want to show that

(43) $(\text{osc}_{U\setminus
N_{z}}\tilde{u}_{z})^{Q}\leq\widetilde{C}\left(\int_{0}^{+\infty}\varphi^{\frac{1}{Q}}(s)\,ds\right)^{Q-1}\int_{U}F_{\varphi}(g(x))\,dx$

for each CC open ball $U\subset\Omega$ and each $z\in\mathbb{G}_{2}$. If the
left-hand side is 0, clearly there is nothing to prove. Otherwise pick any
$0\leq c<\text{osc}_{U\setminus N_{z}}\tilde{u}_{z}$. This implies that there
exists $x\in U\setminus N_{z}$ such that $c\leq
2|\tilde{u}_{z}(x)-(\tilde{u}_{z})_{U}|$. Hence, using Theorem 4.1,

$\begin{split}c^{Q}&\,\leq\,\,2^{Q}|\tilde{u}_{z}(x)-(\tilde{u}_{z})_{U}|^{Q}\stackrel{{\scriptstyle(\ref{rieszformula})}}{{\leq}}2^{Q}C^{Q}\left(\int_{U}\frac{|\nabla_{\mathbb{G}_{1}}\tilde{u}_{z}(y)|}{d_{1}(x,y)^{Q-1}}\,dy\right)^{Q}\\\
&\,\leq\,\,2^{Q}C^{Q}\left(\int_{U}\frac{g(y)}{d_{1}(x,y)^{Q-1}}\,dy\right)^{Q}\\\
&\,\leq
2^{Q}C^{Q}C_{Q}\left(\int_{0}^{+\infty}\varphi^{\frac{1}{Q}}(s)\,ds\right)^{Q-1}\int_{U}F_{\varphi}(g(x))\,dx.\end{split}$

Since this estimate holds for every $0\leq c<\text{osc}_{U\setminus
N_{z}}\tilde{u}_{z}$, we get (43).
From the fact that $\tilde{u}_{z}$ is continuous on $\Omega$, the same
estimate holds replacing osc${}_{U\setminus N_{z}}\tilde{u}_{z}$ with
osc${}_{U}\,\tilde{u}_{z}$. Let $N=\,\cup_{z\in\mathbb{Q}^{m}}N_{z}$. Then $N$
is still negligible and, by (40), it follows that

(44) $\begin{split}&|d_{2}(z,f(x))-d_{2}(z,f(y))|^{Q}\leq\,({\rm
osc}_{U}\tilde{u}_{z})^{Q}\\\
&\leq\,\widetilde{C}\left(\int_{0}^{+\infty}\varphi^{\frac{1}{Q}}(s)\,ds\right)^{Q-1}\int_{U}F_{\varphi}(g(x))\,dx\end{split}$

for each $x,\,y\in U\setminus N$ and $z\in\mathbb{Q}^{m}$, for each CC open
ball $U\subset\Omega$. Now observe that, by the triangle inequality and
density of $\mathbb{Q}^{m}$ in $\mathbb{G}_{2}\equiv\mathbb{R}^{m}$, it
follows that

(45)
$\sup_{z\in\mathbb{Q}^{m}}|d_{2}(z,f(x))-d_{2}(z,f(y))|=\,d_{2}(f(x),f(y))$

for each $x,\,y\in U\setminus N$. Thus, from (44) and (45), it follows that

(46)
$d_{2}(f(x),f(y))^{Q}\leq\widetilde{C}\left(\int_{0}^{+\infty}\varphi^{\frac{1}{Q}}(s)\,ds\right)^{Q-1}\int_{U}F_{\varphi}(g(x))\,dx,$

for each $x,\,y\in U\setminus N$ and each CC open ball $U\subset\Omega$. This
implies that $f:\,\Omega\setminus N\to\mathbb{G}_{2}$ is locally uniformly
continuous and, since $\Omega\setminus N$ is dense in $\Omega$, it can be
extended to a (unique) continuous function $f:\Omega\to\mathbb{G}_{2}$.
Moreover, from (46), it follows also that

$({\rm
osc}_{U}f)^{Q}\leq\,\widetilde{C}\left(\int_{0}^{+\infty}\varphi^{\frac{1}{Q}}(s)\,ds\right)^{Q-1}\int_{U}F_{\varphi}(g(x))\,dx,$

for each CC open ball $U\subset\Omega$. This means that $f$ satisfies the
condition (RR) with weight

$w(x)=\widetilde{C}\left(\int_{0}^{+\infty}\varphi^{\frac{1}{Q}}(s)\,ds\right)^{Q-1}F_{\varphi}(g(x)).\qed$

######  Remark 4.3.

Let us recall that Theorem 4.2 is sharp already in the Euclidean case. Indeed
, Stein [72] pointed out that, if $\nabla f\notin L^{n,1}$, then both
continuity and differentiability of $f$ may fail. For instance, an explicit
example is given by the function $u(x):=\log(\log(1/|x|))$ on $B(0,1/3)$
extended in a smooth way on $\mathbb{R}^{n}$ with compact support. By a direct
computation it is easy to prove that $u\in W^{1,n}(\mathbb{R}^{n})$ and
moreover $|\nabla u|\in L^{n,q}$ for every $q>1$ (see [71] for the definition
of general Lorentz spaces). If $\\{q_{k}\\}_{k\in\mathbb{N}}$ is an
enumeration of the rational points in $\mathbb{R}^{n}$, we set
$f_{k}(x)=u(x-q_{k})$ and define

$f(x)=\sum_{k=0}^{+\infty}2^{-k}f_{k}(x).$

Then such a map $f$, and its weak gradient, have the same summability
properties of $u$, but it is essentially unbounded (thus discontinuous and not
differentiable) around any point. Hence Theorem 4.2 is sharp also in the
refined class of Lorentz spaces.

######  Remark 4.4.

Note that Theorem 4.2 is not invertible, in the following sense: even though
$\Omega$ is a bounded open set of an Euclidean space $\mathbb{R}^{n}$, it is
well known that a function $f\in AC^{n}(\Omega)$ need not admit weak gradient
$\nabla f$ in $L^{n,1}(\Omega)$ (see, for instance, [37, Theorem 3.7]).

##  5\. Area formula and change of variables for absolutely continuous
functions between Carnot groups

An important property of Lipschitz functions between Euclidean spaces is the
so called Area formula, relating the Hausdorff measure of the image of a
Lipschitz map (counted with multiplicity) to the integral of the Jacobian on
the domain. Such a formula holds also in the setting of Carnot groups (Theorem
5.3); we will extend this result to the case of absolutely continuous
functions (see Theorem 5.7).

###  5.1. Area formula between Carnot groups

The content of this section mainly relies on [50, Chapter 4].

An important tool in Euclidean geometric measure theory is the so-called _area
formula_ for Lipschitz maps, which turns to be true also for Lipschitz maps
between Carnot groups.

######  Definition 5.1.

[H-Jacobian] Given two Carnot groups $(\mathbb{G}_{i},d_{i})$, $i=1,2$,
endowed with an invariant distance $d_{i}$, let
$L:\,\mathbb{G}_{1}\to\mathbb{G}_{2}$ be a $H$\- linear map. Let $Q$ denote
the homogeneous dimension of $\mathbb{G}_{1}$. The _horizontal jacobian_ of
$L$ is defined by

$J_{Q}(L):=\frac{\mathcal{H}_{d_{2}}^{Q}(L(B_{1}))}{\mathcal{H}_{d_{1}}^{Q}(B_{1})}$

where $B_{1}$ denotes the unit open ball in $(\mathbb{G}_{1},d_{1})$ and for
$i=1,2$, $\mathcal{H}_{d_{i}}^{Q}$ denotes the Hausdorff measure w.r.t.
$d_{i}$.

######  Remark 5.2.

In the previous definition it is possible to replace $B_{1}$ with any other
measurable subset of $\mathbb{G}_{1}$ of positive finite measure, by the
$H$-linearity of the map $L$.

The following Theorem is taken from [50, Theorem 4.3.4].

######  Theorem 5.3.

(Area formula for Lipschitz maps) Let
$f:\,E\subset(\mathbb{G}_{1},d_{1})\to(\mathbb{G}_{2},d_{2})$ be a Lipschitz
continuous function with $E$ measurable set. Then

$\int_{E}J_{Q}(d_{P}f(x))\,d\mathcal{H}^{Q}_{d_{1}}(x)=\,\int_{\mathbb{G}_{2}}N(f,E,y)\,d\mathcal{H}^{Q}_{d_{2}}(y)$

where $N(f,E,\cdot)$ denotes the multiplicity function of $f$ relatively to
$E$, that is $N(f,E,y)=\\#\\{f^{-1}(y)\cap
E\\}\in\mathbb{N}\cup\\{+\infty\\}$.

######  Remark 5.4.

The original definition of P-differentiability (cf. Definition 2.5) requires
$E$ to be an open subset of a Carnot group $\mathbb{G}_{1}$. However, it is
possible to extend this notion to the set of density points of a general
measurable set. In particular, the Pansu’s differential, if it exists, is
uniquely determined at a density point of $E$. Moreover, a generalization of
Theorem 2.6 holds: it has been proved by Magnani in [50] that a Lipschitz map
$f:E\to\mathbb{G}_{2}$ is $\mathcal{H}^{Q}_{d_{1}}$ a.e. P-differentiable,
where $E\subset\mathbb{G}_{1}$ is measurable, while $Q$ and $d_{1}$ denote the
homogeneous dimension and the CC distance of $\mathbb{G}_{1}$ respectively.

Aim of this chapter is to extend Theorem 5.3 to the case of maps in
$AC^{Q}(\Omega,\mathbb{G}_{2})$, which are not necessarily Lipschitz
continuous. The following useful result is proved in [44, Lemma 2.12] in the
general setting of metric measure spaces.

######  Proposition 5.5 (Lusin’s (N) condition for $AC^{Q}$).

Let $\mathbb{G}_{1}\equiv\mathbb{R}^{n}$ and
$\mathbb{G}_{2}\equiv\mathbb{R}^{m}$ be Carnot groups and let
$\Omega\subset\mathbb{G}_{1}$ be an open set. Let $f\in
AC^{Q}(\Omega,\mathbb{G}_{2})$, where $Q$ denotes the homogeneous dimension of
$\mathbb{G}_{1}$. Then, for every measurable subset $E\subset\Omega$, it holds

$\mathcal{L}^{n}(E)=0\;\Longrightarrow\;\mathcal{H}^{Q}(f(E))=0.$

The next result is inspired by Theorem 3.1.9 in [18] and adapted to our
setting.

######  Theorem 5.6.

Let $\Omega\subset\mathbb{G}_{1}\equiv\mathbb{R}^{n}$ be an open set and
$A\subset\Omega$ be measurable. If
$f:\Omega\to\mathbb{G}_{2}\equiv\mathbb{R}^{m}$ has the property that

$\limsup_{y\to x}\frac{d_{2}(f(y),f(x))}{d_{1}(y,x)}<+\infty\qquad\text{for
all $x\in A$,}$

then there exists a countable measurable partition $\\{A_{k}\\}_{k}$ of $A$
such that $f_{\upharpoonright A_{k}}$ is Lipschitz continuous.

###### Proof.

For every $j\in\mathbb{N}$, we define

(47) $C_{j}:=\\{z\in\Omega:d_{2}(f(y),f(z))\leq jd_{1}(y,z)\quad\text{for
every }y\in U_{d_{1}}(z,1/j)\\}.$

It follows that $A$ is contained in $\cup_{j}C_{j}$. Indeed, let $a\in A,$ so
that

$\limsup_{y\to a}\frac{d_{2}(f(y),f(a))}{d_{1}(y,a)}<+\infty.$

This implies that there exists $j\in\mathbb{N}$ and $\bar{r}>0$ sufficiently
small such that $\frac{d_{2}(f(y),f(a))}{d_{1}(y,a)}<j$ for every $y\in
U_{d_{1}}(a,\bar{r})$. Hence, up to increase $j$, we deduce that the previous
inequality holds for every $y\in U_{d_{1}}(a,\frac{1}{j})$, which means that
$a\in C_{j}$.
Notice that each $C_{j}$ is closed in $\Omega$. In fact, let $x_{n}\in C_{j}$
which converge to $x\in\Omega$ as $n\to\infty$ and let $y\in
U_{d_{1}}(x,\frac{1}{j})$. If $n\in\mathbb{N}$ is chosen sufficiently large,
we can assume $\\{x,y\\}\subset U_{d_{1}}(x_{n},1/j)$. Hence

$\begin{split}d_{2}(f(y),f(x))&\leq
d_{2}(f(y),f(x_{n}))+d_{2}(f(x_{n}),f(x))\\\ &\leq
jd_{1}(y,x_{n})+jd_{1}(x_{n},x)\end{split}$

and, passing to the limit as $n\to\infty$, we get $d_{2}(f(y),f(x))\leq
jd_{1}(y,x)$, so that $x\in C_{j}$.
Exploiting the Lindelöf property of $\mathbb{G}_{1}\equiv\mathbb{R}^{n}$, we
can write $C_{j}$ as a countable union of closed sets $C_{j}^{i}$, for
$i\in\mathbb{N}$, each of which of diameter less than $\frac{1}{j}$. If we
relabel $\\{C_{k}\\}_{k}=\\{C_{j}^{i}\\}_{j,i}$, it follows that
$f_{\upharpoonright C_{k}}$ is Lipschitz by (47). To conclude, define
$A_{k}:=C_{k}\cap A$ which are still measurable and notice that we can suppose
the sets $A_{k}$ to be disjoint (hence a partition of $A$) by the usual
differences procedure. ∎

We now combine Theorem 5.6 and Theorem 5.3 to get an Area formula for
absolutely continuous maps between Carnot groups.

######  Theorem 5.7.

Let $\Omega\subset\mathbb{G}_{1}$ be open and $f\in
AC^{Q}(\Omega,\mathbb{G}_{2})$. For every measurable subset $E$ of $\Omega$,
we have

(48)
$\int_{E}J_{Q}(d_{P}f(x))\,d\mathcal{H}^{Q}_{d_{1}}(x)=\int_{\mathbb{G}_{2}}N(f,E,y)\,d\mathcal{H}^{Q}_{d_{2}}(y)$

where $J_{Q}$ is the Jacobian introduced in Definition 5.1, while
$N(f,E,\cdot)$ denotes the multiplicity function of $f$ relatively to $E$.

###### Proof.

We note that, by Theorem 3.7, Lip$(f)$ is finite almost everywhere, which
means

$\text{Lip}(f)(x):=\limsup_{y\to
x}\frac{d_{2}(f(y),f(x))}{d_{1}(y,x)}<+\infty\quad
for\,\,\mathcal{L}^{n}-a.e.\,x\in\Omega.$

We can therefore apply Theorem 5.6 and write $\Omega$ as a disjoint union of
measurable subsets

$\Omega=A_{0}\cup\bigcup_{k=1}^{+\infty}A_{k},$

with $\mathcal{L}^{n}(A_{0})=0$ and $f_{\upharpoonright A_{k}}$ Lipschitz.
Moreover we know, again by Theorem 3.7, that $f$ is P-differentiable almost
everywhere. In the negligible set where $f$ does not satisfy this condition,
we can define $d_{P}f(x)$ to be for instance the zero map.
Let now $E$ be a measurable subset of $\Omega$ and denote $E_{k}:=E\cap
A_{k}$. Applying Theorem 5.3 we get, for every $k\geq 1$,

$\int_{E_{k}}J_{Q}(d_{P}f_{\upharpoonright
E_{k}}(x))\,d\mathcal{H}^{Q}_{d_{1}}(x)=\int_{\mathbb{G}_{2}}N(f,E_{k},y)\,d\mathcal{H}^{Q}_{d_{2}}(y).$

Moreover this identity trivially holds for $k=0$ in view of Lemma 5.5. Note
that, if $x$ is a density point of $E_{k}$ and $f:\Omega\to\mathbb{G}_{2}$ is
P-differentiable at $x\in E_{k}$, then also $f_{\upharpoonright E_{k}}$ is
P-differentiable at $x$ in the sense of Remark 5.4 and the two differentials
coincide. Hence, summing over $k\in\mathbb{N}$ we find, by a simple
application of monotone convergence,

$\begin{split}&\int_{E}J_{Q}(d_{P}f(x))\,d\mathcal{H}^{Q}_{d_{1}}(x)=\sum_{k=0}^{+\infty}\int_{E_{k}}J_{Q}(d_{P}f_{\upharpoonright
E_{k}}(x))\,d\mathcal{H}^{Q}_{d_{1}}(x)\\\
&=\sum_{k=0}^{+\infty}\int_{\mathbb{G}_{2}}N(f,E_{k},y)\,d\mathcal{H}^{Q}_{d_{2}}(y)=\int_{\mathbb{G}_{2}}N(f,E,y)\,d\mathcal{H}^{Q}_{d_{2}}(y).\qed\end{split}$

Through standard arguments of approximation by simple functions, it is easy to
extend formula (48) to a more general form:

######  Corollary 5.8.

If $f\in AC^{Q}(\Omega,\mathbb{G}_{2})$, then for any measurable subset $E$ of
$\Omega$ and each summable function $u:E\to\mathbb{R}$, it holds

$\int_{E}u(x)J_{Q}(d_{P}f(x))\,d\mathcal{H}^{Q}_{d_{1}}(x)=\int_{\mathbb{G}_{2}}\sum_{x\in
f^{-1}(y)}u(x)\,d\mathcal{H}^{Q}_{d_{2}}(y).$

As a straightforward consequence of Theorem 4.2 and Theorem 5.7, we get the
following

######  Corollary 5.9.

Let $\Omega\subset\mathbb{G}_{1}$ be an open set and let $f\in W^{1,1}_{\rm
loc}(\Omega,\mathbb{G}_{2})$ such that the function $g$ in (15) belongs to
$L^{Q,1}(\Omega)$. Then a suitable representative of $f$ satisfies

$\int_{E}J_{Q}(d_{P}f(x))\,d\mathcal{H}^{Q}_{d_{1}}(x)=\int_{\mathbb{G}_{2}}N(f,E,y)\,d\mathcal{H}^{Q}_{d_{2}}(y)$

for every measurable subset $E\subset\Omega$. Moreover, for every
$u:E\to\mathbb{R}$ summable,

$\int_{E}u(x)J_{Q}(d_{P}f(x))\,d\mathcal{H}^{Q}_{d_{1}}(x)=\int_{\mathbb{G}_{2}}\sum_{x\in
f^{-1}(y)}u(x)\,d\mathcal{H}^{Q}_{d_{2}}(y).$

##  6\. Appendix: Orlicz-Sobolev spaces on Carnot groups

Orlicz spaces arise naturally in many applications in real, harmonic or
functional analysis. Such spaces were introduced by the Polish mathematician
Władysław Orlicz in 1932 and are generalizations of the classical $L^{p}$
spaces. Indeed, the role played by the function $t^{p}$ is replaced by more
general convex maps, called $N$-functions.
We recall now the main definitions and basic properties of Orlicz spaces: see
for instance [1],[45] or [49] for an exhaustive treatment.

######  Definition 6.1.

A map $A:[0,+\infty)\to[0,+\infty)$ is called an N-function if it is convex,
vanishes only at $0$ and has the following limit properties:

$\lim_{t\to
0^{+}}\frac{A(t)}{t}=0,\qquad\lim_{t\to\infty}\frac{A(t)}{t}=+\infty.$

It is easy to verify that a map satisfying these properties is actually
continuous on $[0,+\infty)$ and strictly increasing. Given an $N$-function
$A:[0,+\infty)\to[0,+\infty)$, its Young conjugate $\widetilde{A}$ is defined
as the $N$-function

$\widetilde{A}(t):=\sup_{s>0}\,\\{st-A(s)\\}.$

Examples of $N$-functions and associated conjugates are for instance

$\begin{split}A(t)=\frac{t^{p}}{p},&\qquad\widetilde{A}(t)=\frac{t^{p^{\prime}}}{p^{\prime}}\qquad\left(1<p<+\infty,\quad
p^{\prime}=\frac{p}{p-1}\right);\\\
A(t)=e^{t}-&t-1,\qquad\widetilde{A}(t)=(1+t)\log(1+t)-t.\end{split}$

######  Definition 6.2.

Let $\Omega\subset\mathbb{R}^{n}$ be measurable and let $A$ be an
$N-$function. For every measurable map $u:\Omega\to\mathbb{R}$, we define its
Luxemburg norm as

$\|u\|_{L^{A}(\Omega)}:=\inf\left\\{\lambda>0:\int_{\Omega}A\left(\frac{|u(x)|}{\lambda}\right)\,dx\leq
1\right\\}.$

The Orlicz space associated to the $N$\- function $A$ is defined as

$L^{A}(\Omega):=\\{u:\Omega\to\mathbb{R}\text{
measurable}:\|u\|_{L^{A}(\Omega)}<+\infty\\}.$

Equipped with $\|\cdot\|_{L^{A}(\Omega)}$, the Orlicz space $L^{A}(\Omega)$ is
a Banach space. Moreover, an interesting relation between an $N-$function and
its Young conjugate is expressed by the following Hölder-type inequality:

$\int_{\Omega}u(x)v(x)\,dx\leq
2\|u\|_{L^{A}(\Omega)}\|v\|_{L^{\widetilde{A}}(\Omega)}.$

Notice that, when considering $A(t):=t^{p}$, for $1<p<+\infty$, we get that
$L^{A}(\Omega)$ coincides with the classical $L^{p}(\Omega)$ space. Moreover
in this case we get
$\|\cdot\|_{L^{A}(\Omega)}=p^{-1/p}\|\cdot\|_{L^{p}(\Omega)}$.

In order to establish an inclusion relation similar to the classical one for
Lebesgue spaces on domain with finite measure, we introduce two new concepts:
if $A$ and $B$ are $N$-functions, we will say that $B$ dominates $A$ globally
if there exists a positive constant $k$ such that $A(t)\leq B(kt)$ for all
$t\geq 0$. If this estimate holds only for large values of $t$, we will say
that $B$ dominates $A$ near infinity. An interesting result is that the
embedding $L^{B}(\Omega)\hookrightarrow L^{A}(\Omega)$ holds if and only if
either $B$ dominates $A$ globally or $B$ dominates $A$ near infinity and
$|\Omega|<+\infty.$ Hence functions which are globally equivalent, i.e. each
one dominates the other globally, determine the same Orlicz space (and clearly
the same holds if we require that they are equivalent near infinity and
$|\Omega|<+\infty$).
Note that, if $1<p<q<+\infty$, then the map $B(t):=t^{q}$ dominates the
function $A(t):=t^{p}$ near infinity, so that $L^{q}(\Omega)\subset
L^{p}(\Omega)$ if $\Omega$ has finite measure.

In the same spirit of Sobolev space, it is now straightforward to define
Orlicz-Sobolev functions by means of Orlicz integrability conditions on the
weak derivatives. We give this definition directly in the general setting of
Carnot groups:

######  Definition 6.3.

Let $\Omega\subset\mathbb{G}\equiv\mathbb{R}^{n}$ be open and let $A$ be an
$N$-function. We define the Orlicz-Sobolev space

$W^{1,A}_{\mathbb{G}}(\Omega):=\\{u\in L^{A}(\Omega):\exists X_{i}u\in
L^{A}(\Omega)\quad i=1,\dots,m_{1}\\},$

where $X_{i}u$ are the distributional derivatives of $u$ computed along the
generating vector fields of the group. We also set

$\|u\|_{W^{1,A}_{\mathbb{G}}(\Omega)}:=\|u\|_{L^{A}(\Omega)}+\|\nabla_{\mathbb{G}}u\|_{L^{A}(\Omega)}.$

Equipped with such a norm, it is easy to see that
$(W^{1,A}_{\mathbb{G}}(\Omega),\|\cdot\|_{W^{1,A}_{\mathbb{G}}(\Omega)})$ is a
Banach space. If $\mathbb{G}$ is the Euclidean $\mathbb{R}^{n}$, we will use
the notation $W^{1,A}(\Omega)$.

The classical embedding theorems for Sobolev spaces have been extended also to
Orlicz-Sobolev spaces in the Euclidean setting. In particular, the condition
$p>n$ has to be replaced by a more general requirement on the growth of the
$N$-function which induces the space. In [1] or [16] it has been proven that,
if $\Omega\subset\mathbb{R}^{n}$ is sufficiently regular and $A$ is an
$N$-function, then $W^{1,A}(\Omega)$ is embedded in the space of continuous
and bounded maps provided that the condition

(49) $\int_{1}^{+\infty}\frac{A^{-1}(t)}{t^{1+1/n}}\,dt<+\infty$

is satisfied. In the case of $A(t)=t^{p}$, we recover the classical embedding
theorem for $p>n$. However, as remarked in [13], this condition is not sharp,
since it can be improved by requiring

(50)
$\int_{1}^{+\infty}\frac{\widetilde{A}(t)}{t^{1+n^{\prime}}}\,dt<+\infty,$

which is a weaker assumption than (49) (see [74]). Indeed, G. Talenti proved
in the same work that (50) is a sufficient condition for boundedness (see also
[14]), while A. Cianchi showed in [13] that it is also the sharp condition for
continuity. Moreover the same author proved in [2] that this condition is also
the correct one to ensure differentiability almost everywhere.
The strategy of Cianchi relies on rearrangement tools, while in the following
we will provide an alternative way to extend this result in the setting of
Carnot groups, following the approach introduced by Kauhanen, Koskela and Malý
in [39], in the same spirit of Theorem 4.2.
In order to reach this result, we need first to extend Definition 6.3 to the
case of group-valued maps, inspired by Definition 2.17. This generalization is
not straightforward as for Euclidean spaces (in which it suffices to work in
components), since a Carnot group cannot be seen in general as direct product
of subgroups.

######  Definition 6.4.

Let $\mathbb{G}_{1}\equiv\mathbb{R}^{n},\mathbb{G}_{2}\equiv\mathbb{R}^{m}$ be
Carnot groups. Let $\Omega\subset\mathbb{G}_{1}$ be open and let $A$ be an
$N$-function.

  1. (i)

We say that a function $u:\Omega\to\mathbb{G}_{2}$ belongs to the space
$L^{A}(\Omega,\mathbb{G}_{2})$ if
$u:\Omega\subset\mathbb{R}^{n}\to\mathbb{R}^{m}$ is measurable and the map
$u_{0}:\Omega\to\mathbb{R}$ defined by $u_{0}(x)=d_{2}(0,u(x))$ is in
$L^{A}(\Omega)$;

  2. (ii)

We say that a function $u:\Omega\to\mathbb{G}_{2}$ belongs to the Orlicz-
Sobolev space $W^{1,A}(\Omega,\mathbb{G}_{2})$ if

     * •

$u\in L^{A}(\Omega,\mathbb{G}_{2})$;

     * •

for all $z\in\mathbb{G}_{2}$, the functions $u_{z}:\Omega\to\mathbb{R}$
defined by

(51) $u_{z}(x):=d_{2}(z,u(x))\quad\text{if }x\in\Omega$

are in $W^{1,A}_{\mathbb{G}_{1},\rm loc}(\Omega)$ (according to Definition
6.3);

     * •

There exists a nonnegative function $g\in L^{A}(\Omega)$ (which does not
depend on $z$) such that

$|\nabla_{\mathbb{G}_{1}}u_{z}|\leq g\quad\text{a.e. in }\Omega,\;\;\text{for
each }z\in\mathbb{G}_{2}.$

We provide now the natural extension of Theorem 3.6 and Theorem 4.2 to the
case of Orlicz-Sobolev maps: under a sufficient growth condition on $A$,
functions in the space $W^{1,A}(\Omega,\mathbb{G}_{2})$ are $Q$-absolutely
continuous and so enjoy nice regularity properties.

The following Lemma represents the key step in order to apply the strategy
proposed by the authors in [39] also to the case of Orlicz-Sobolev spaces.

######  Lemma 6.5.

Let $\Omega\subset\mathbb{G}\equiv\mathbb{R}^{n}$ be open with
$\mathcal{L}^{n}(\Omega)<+\infty$. Let $A$ be an N-function satisfying

(52)
$\int_{1}^{+\infty}\left(\frac{t}{A(t)}\right)^{\frac{1}{Q-1}}\,dt<+\infty,$

and let $g$ be a nonnegative function in the space $L^{A}(\Omega)$. Then there
exists a nonincreasing function $\varphi:(0,+\infty)\to(0,+\infty)$ such that

(53) $\int_{\Omega}F_{\varphi}(g(x))\,dx<+\infty$

and

(54) $\int_{0}^{+\infty}\varphi^{1/Q}(t)\,dt<+\infty.$

###### Proof.

Notice first that we can assume that

(55) $\int_{0}^{1}\left(\frac{t}{A(t)}\right)^{\frac{1}{Q-1}}\,dt<+\infty,$

since it is always possible to replace $A$ with another $N-$function which is
equivalent near infinity (so that it determines the same Orlicz space) and
decays sufficiently slow to 0.
Since $g\in L^{A}(\Omega)$, there exists for sure $\bar{\lambda}>0$ such that
$\int_{\Omega}A(\bar{\lambda}g(x))\,dx<+\infty$. Define
$\varphi:(0,+\infty)\to(0,+\infty)$ as

$\varphi(t):=\left(\frac{A(\bar{\lambda}t)}{t}\right)^{\frac{Q}{1-Q}}.$

By definition $\varphi$ is positive and also nonincreasing. In fact, by
convexity of $A$, we know that $A(\lambda t_{1}+(1-\lambda)t_{2})\leq\lambda
A(t_{1})+(1-\lambda)A(t_{2})$ for every
$\lambda\in(0,1),\,t_{1},t_{2}\in[0,+\infty).$ Consider $0<t<s$ and apply this
inequality with $t_{1}=\bar{\lambda}s,\,t_{2}=0,\,\lambda=\frac{t}{s}$. We get
$A(\bar{\lambda}t)\leq\frac{t}{s}A(\bar{\lambda}s)$ and, by simple
rearrangements, $\varphi(t)\geq\varphi(s)$. Moreover we can calculate

$\begin{split}\int_{\Omega}F_{\varphi}(g(x))\,dx&=\int_{\\{g>0\\}}g(x)\left[\left(\frac{A(\bar{\lambda}g(x))}{g(x)}\right)^{\frac{Q}{1-Q}}\right]^{\frac{1-Q}{Q}}\,dx\\\
&=\int_{\Omega}A(\bar{\lambda}g(x))\,dx<+\infty\end{split}$

and also, exploiting (52) and (55),

$\int_{0}^{+\infty}\varphi^{1/Q}(t)\,dt=\int_{0}^{+\infty}\left(\frac{A(\bar{\lambda}t)}{t}\right)^{\frac{1}{1-Q}}dt=\bar{\lambda}^{\frac{Q}{1-Q}}\int_{0}^{+\infty}\left(\frac{t}{A(t)}\right)^{\frac{1}{Q-1}}\,dt<+\infty.\qed$

######  Remark 6.6.

It is proved in [14] that condition (52) is indeed equivalent to the
following:

(56)
$\int_{0}^{+\infty}\frac{\widetilde{A}(t)}{t^{1+Q^{\prime}}}\,dt<+\infty,$

which clearly extends (50) in the case of Carnot groups of homogeneous
dimension $Q$.

Under condition (52), or equivalently (56), functions in the space
$W^{1,A}(\Omega,\mathbb{G}_{2})$ are $Q-$absolutely continuous. Actually, only
an Orlicz bound on the weak horizontal gradient of the sections is needed.
This is expressed by the following result.

######  Theorem 6.7.

Let $\mathbb{G}_{1}\equiv\mathbb{R}^{n}$, $\mathbb{G}_{2}\equiv\mathbb{R}^{m}$
be Carnot groups and let $\Omega\subset\mathbb{G}_{1}$ open. Suppose also that
$\mathcal{L}^{n}(\Omega)<+\infty$ and denote by $Q$ the homogeneous dimension
of $\mathbb{G}_{1}$. Let $A$ be an N-function satisfying (52). If $f\in
W^{1,1}_{\rm loc}(\Omega,\mathbb{G}_{2})$ and there exists a function $g\in
L^{A}(\Omega)$ such that

$|\nabla_{\mathbb{G}}u_{z}|\leq g\qquad\text{for all }z\in\mathbb{G}_{2},$

where $u_{z}$ are defined as in (51), then there exists a representative of
$f$ which is continuous and satisfies the (RR) condition. In particular $f\in
AC^{Q}(\Omega,\mathbb{G}_{2})$ and is P-differentiable a.e. in $\Omega$.

###### Proof.

The proof of this result can be achieved using the same strategy exploited in
Theorem 4.2. In particular the existence of a nonincreasing function
$\varphi:(0,+\infty)\to(0,+\infty)$ satisfying properties (53) and (54) is
ensured by Lemma 6.5 instead of Theorem 2.11. The other steps of the proof are
unchanged. ∎

######  Remark 6.8.

Equivalently one can observe that, combining Lemma 6.5 and the
characterization given by Theorem 2.11, a function $g$ as in the assumptions
belongs actually to the Lorentz space $L^{Q,1}(\Omega)$ and then apply Theorem
4.2.

We deduce that, under condition (52), Orlicz-Sobolev maps between Carnot
groups are actually in $AC^{Q}(\Omega,\mathbb{G}_{2})$: hence they share
properties like continuity, $P$-differentiability a.e. (Theorem 3.7) and Area
formula (Theorem 5.7 and Corollary 5.8).

##  7\. Appendix: Differentiation for maps between metric spaces and metric
space-valued Sobolev maps

###  7.1. Differentiation for maps between metric spaces:

In this section we will assume that the metric measure space
$X\equiv(X,d_{X},\mu)$ admits a $p$-Poincaré inquality with $p\geq\,1$, the
ambient measure $\mu$ is doubling and $Y\equiv(Y,d_{Y})$ is a (separable)
metric space.
Let us recall that a general differentiation theory has been carried out for
maps $f:X\to Y$, whenever $Y$ can be isometrically embedded into a Banach
space $V$ satisfying the Radon-Nikodym property (RNP) (see [11, 12]). We
recall that a Banach space $V$ sarisfies (RNP) if each Lipschitz function
$f:\,\mathbb{R}\to V$ is differentiable a.e. The first pioneering result is
due to Cheeger[10] when $Y=\,\mathbb{R}$, who was able to endow the metric
measure space $X$ by a so-called Cheeger differentiable structure and proved a
Rademacher differentiability theorem for Lipschitz function in this context.
This result was remarkably extended by Cheeger and Kleiner to maps with target
space $Y=V$ satisfying (RNP). Then, whenever the target space $Y$ can be bi-
Lipschitz embedded into a Banach space $V$ satisfying (RNP), a differentiation
theory can be carried out also for maps $f:\Omega\subset X\to Y\subset V$ and
one can get a Stepanov’s type theorem (see, for instance, [80]). Moreover
metric space-valued Sobolev functions can be introduced by means of the notion
of upper gradient in this setting as in [34, Chap. 7]).
This approach may not work, as concern the differentiability issue, if the
target space is a Carnot group. Indeed, as showed in [11, Theorem 6.1], the
simplest Carnot group $\mathbb{H}^{1}$ does not admit a bi-Lipschitz embedding
into any Banach space satisfying (RNP).

###  7.2. Metric space-valued Sobolev maps:

As far as the notion of metric space-valued Sobolev mapping is concerned,
different equivalent approaches can be introduced. Let us point out here two
ones related to our arguments: the approach by embedding of the target space
and the one by sections of mappings. From now on we can weaken the assumptions
on the involved metric spaces. More precisely, we assume that
$X\equiv(X,d_{X},\mu)$ is a metric measure space with ambient measure $\mu$
finite on bounded sets, that is $\mu(E)<\,\infty$ for each bounded $E\subset
X$, and $Y\equiv(Y,d_{Y},y_{0})$ is a pointed complete and separable metric
space. Moreover, we will denote by $B(x,r)$ and $U(x,r)$ the closed and open
ball of $X$ centered at $x$ with radius $r$, respectively. Approach by
embedding of the target metric space: This approach extends the one due to
Shanmugalingam [69] who introduced the notion of Newtonian Sobolev space
$N^{1,p}(X)$ for real-valued functions defined on $X$ and it was carried out
in [34, Chap. 7]. Let us briefly recall here the main notions and results. We
first introduce vector-valued Sobolev functions on metric spaces. If $V$ is a
Banach space and $1\leq\,p<\infty$, we say that a mapping $f:X\to V$ belongs
to the Dirichlet space $D^{1,p}(X:V)$ if $f$ is measurable and it possesses a
$p$-integrable $p$-weak upper gradient (see [34, Section 6.2]). When
$V=\mathbb{R}$, we simply denote $D^{1,p}(X):=\,D^{1,p}(X:\mathbb{R})$. It is
clear that $D^{1,p}(X:V)$ is a vector space.
We denote as Sobolev space of $V$-valued functions on $X$ the space
$N^{1,p}(X:V):=\,D^{1,p}(X:V)\cap L^{p}(X,V)$ and $N^{1,p}(X:V)$ turns out to
be a Banach space with respect to the norm

$\|u\|_{N^{1,p}(X:V)}:=\,\|u\|_{L^{p}(X,V)}+\|\varrho_{u}\|_{L^{p}(X)},$

where $\varrho_{u}$ denotes the minimal $p$-weak upper gradient of $u$
guaranteed by [34, Chap. 6 and Theorem 6.3.20].
Consider now the Kuratowski isometric embedding of $Y$ into
$V:=\,\ell^{\infty}(Y)$, with respect to the base point $y_{0}$. More
precisely, let $\ell^{\infty}(Y)$ denote the Banach space of bounded functions
from $Y$ to $\mathbb{R}$, equipped with the norm

$\|\varphi\|_{\ell^{\infty}(Y)}:=\,\sup_{y\in Y}|\varphi(y)|\quad\text{ if
}\varphi\in\ell^{\infty}(Y)\,.$

Then the mapping

$\begin{split}Y\ni y\mapsto\,&\varphi_{y}\in\ell^{\infty}(Y)\\\
&\varphi_{y}(z):=\,d(z,y)-d(y_{0},z)\quad z\in Y\end{split}$

is an isometric embedding (see [34, Theorem 4.1]). According to [34], we can
introduce the Sobolev and Dirichlet classes of mappings between metric spaces
$X$ to $Y$ (depending on the base point $y_{0}$) as follows:

$N^{1,p}(X:Y):=\left\\{f\in N^{1,p}(X:\ell^{\infty}(Y)):f(x)\in
Y\,\mu\text{-a.e. }x\in X\right\\}$

and similarly

$D^{1,p}(X:Y):=\left\\{f\in D^{1,p}(X:\ell^{\infty}(Y)):f(x)\in
Y\,\mu\text{-a.e. }x\in X\right\\}\,.$

Moreover we denote as $N_{\rm loc}^{1,p}(X:Y)$ ($D_{\rm loc}^{1,p}(X:Y)$,
respectively) the class of all maps $f:\,X\to\ell^{\infty}(Y)$ such that each
$x\in X$ has a small radius $r>\,0$ for which $f|_{U(x,r)}\in
N^{1,p}(U(x,r):Y)$ ($f|_{U(x,r)}\in D^{1,p}(U(x,r):Y)$, respectively).
Eventually we denote as $L^{p}_{\rm loc}(X,\mu)$ the class of measurable maps
$f\to\bar{\mathbb{R}}$ such that $\int_{E}|f|^{p}\,d\mu<\infty$ for each
bounded set $E\subset X$. Let us recall the following characterizations of
$N^{1,p}(X:Y)$ and $D^{1,p}(X:Y)$.

######  Theorem 7.1.

([34, Theorem 7.1.20]) Let $V$ be a Banach space and let $f:\,X\to V$ be a
measurable function. The the following conditions are equivalent:

  * (i)

$f$ has a $\mu$-representative in the Dirichlet class $D^{1,p}(X:V)$;

  * (ii)

there esists a $p$-integrable Borel function $\varrho:\,X\to[0,\infty]$ with
the following property: for each $1$-Lipschitz function
$\varphi:V\to\mathbb{R}$ there esists a $\mu$-representative $f_{\varphi}$ of
the function $\varphi\circ f$ in $D^{1,p}(X)$ so that the minimal upper
gradient $\varrho_{f_{\varphi}}$ of $f_{\varphi}$ satisfies
$\varrho_{f_{\varphi}}\leq\,\varrho$ $\mu$-a.e. in $X$.

######  Theorem 7.2.

([34, Proposition 7.1.38]) A map $f:X\to Y$ is in $D^{1,p}(X:Y)$ if and only
if there is a non-negative $p$-integrable Borel function $\varrho$ on $X$ such
that whenever $\gamma:\,[0,\ell(\gamma)]\to X$ is a non constant rectifiable
curve in $X$

$d_{Y}(f\left(\gamma(\ell(\gamma))\right),f\left(\gamma(0)\right))\leq\,\int_{\gamma}\varrho\,ds\,,$

where $\ell(\gamma)$ denotes the length of $\gamma$ and

$\int_{\gamma}\varrho\,ds:=\,\int_{0}^{\ell(\gamma)}\varrho(\gamma(t))\,dt\,.$

Moreover, if $\mu(X)$ is finite, then $f\in N^{1,p}(X:Y)$ if and only if $f\in
D^{1,p}(X:Y)$ and the function $X\ni x\mapsto d_{Y}(y_{0},f(x))$ is
$p$-integrable for some (and hence every ) $y_{0}\in Y$.

Theorem 7.2 shows that there is an intrinsic way of determining membership in
$D^{1,p}(X:Y)$ and $N_{\rm loc}^{1,p}(X:Y)$ that is independent of the
embedding of $Y$ into a Banach space. Instead, membership in $N^{1,p}(X:Y)$
(as well as in $L^{p}(X:Y))$ depends on the specific embedding and, in
particular, on the base point $y_{0}$.
Approach by sections of mappings:  It is an extension of the one exploited in
Definition 2.17, first introduced by Ambrosio [3], for the case of $BV$
functions, and then by Reshetnyak [65], for the case of Sobolev functions.
More precisely, first one provides the notion of a suitable real-valued
Sobolev space $W^{1,p}(X)$ and, for each $u\in W^{1,p}(X)$ a notion of modulus
of the “weak gradient” $|\nabla_{X}u|$ is introduced. For instance, by using
the previous Newtonian Sobolev space $N^{1,p}(X)$, the minimal $p$-weak upper
gradient $\varrho_{u}$ plays the role of $|\nabla_{X}u|$ (see, for instance,
[5, 6]). In the following we assume that $W^{1,p}(X)=\,N^{1,p}(X)$. Then,
given a mapping $f:X\to Y$, we say that $f\in W^{1,p}(X,Y)$ if

  * •

the map $X\ni x\mapsto d_{Y}(y_{0},f(x))$ belongs to $L^{p}(X,\mu)$;

  * •

for each $y\in Y$, the section $u_{y}:\,X\to\mathbb{R}$,

(57) $u_{y}(x):=\,d_{Y}(y,f(x))\text{ if }x\in X\text{ belongs to
$D^{1,p}(X)$}$

and there exists $g\in L^{p}(X,\mu)$ such that

(58) $\varrho_{u_{y}}\leq g\,\,\mu\text{-a.e. in $X$, for each $y\in Y$}\,,$

denoting by $\varrho_{u}$ the minimal $p$-weak upper gradient of $u\in
D^{1,p}(X)$. A function $g$ as in (58) will be called a $p$-weak upper
gradient of $f$.

As usual, let us denote by $W^{1,p}_{\rm loc}(X,Y)$ the class of all maps
$f:\,X\to Y$ such that each $x\in X$ has a bounded open neighborhood $U_{x}$
for which $f|_{U_{x}}\in W^{1,p}(U_{x},Y)$.

As a consequence of Theorem 7.1, for fixed $y_{0}$, we get the following
coincidence of spaces:

######  Theorem 7.3.

Let $1<\,p<\,\infty$. Then $W^{1,p}(X,Y)=\,N^{1,p}(X:Y)$.

Before the proof of Theorem 7.3, we will introduce a preliminary result, which
is an extension of [41, Proposition 2.31] to a metric space.

######  Lemma 7.4.

Let $1<\,p<\,\infty$, $f\in W^{1,p}(X,Y)$ and let $\varphi:\,Y\to\mathbb{R}$
be Lipschitz. Then

  * (i)

$X\ni x\mapsto\varphi(f(x))-\varphi(y_{0})$ is in $L^{p}(X,\mu)$.

  * (ii)

If $f_{\varphi}:=\,\varphi\circ f$, then $f_{\varphi}-\varphi(y_{0})\in
N^{1,p}(X)$ and $f_{\varphi}$ admits a minimal $p$-weak upper gradient
$\varrho_{f_{\varphi}}\leq L\,g$ $\mu$-a.e. in $X$, being $g$ the function in
(58) and $L:=\,Lip(\varphi)$.

###### Proof.

(i) By definition of $W^{1,p}(X,Y)$, the map $X\ni x\mapsto d_{Y}(y_{0},f(x))$
is in $L^{p}(X,\mu)$. Being $\varphi$ Lipschitz,

$|\varphi(f(x))-\varphi(y_{0})|\leq\,L\,d_{Y}(y_{0},f(x))\text{ for each }x\in
X\,,$

and we get the desired conclusion.
(ii) Let $Z:=\,\\{z_{i}:\,i\in\mathbb{N}\\}\subset Y$ be a dense set. Let us
denote

$\tilde{f}_{\varphi}:=\,f_{\varphi}-\varphi(y_{0}):\,X\to\mathbb{R}\text{ and
}u_{i}:=\varphi(z_{i})+L\,u_{z_{i}}:\,X\to\mathbb{R}\,,$

where $u_{z}$ is the section of $f$ defined in (57). Note that we can
represent $f_{\varphi}$ as

(59) $\varphi(f(x))=\,\inf_{i\in\mathbb{N}}u_{i}(x)\text{ for each }x\in X\,.$

Indeed, being $\varphi$ Lipschitz,

$|\varphi(f(x))-\varphi(z_{i})|\leq L\,d_{Y}(z_{i},f(x)).$

This means that

$\varphi(f(x))\leq\varphi(z_{i})+L\,d_{Y}(z_{i},f(x))=\varphi(z_{i})+L\,u_{z_{i}}(x)=u_{i}(x).$

Moreover, consider a sequence $(i_{k})_{k\in\mathbb{N}}$ such that
$z_{i_{k}}\to f(x)$. We get

$\begin{split}|u_{i_{k}}(x)-\varphi(f(x))|&=|\varphi(z_{i_{k}})+L\,d_{Y}(z_{i_{k}},f(x))-\varphi(f(x))|\\\
&\leq|\varphi(z_{i_{k}})-\varphi(f(x))|+|L\,d_{Y}(z_{i_{k}},f(x))|\leq
2L\,d_{Y}(z_{i_{k}},f(x))\to 0,\end{split}$

which implies (59). Thus, from (59), it also follows that

(60)
$\tilde{f}_{\varphi}(x)=\,\varphi(f(x))-\varphi(y_{0})=\,\inf_{i\in\mathbb{N}}(\tilde{u}_{i}(x))\,.$

where $\tilde{u}_{i}:=\,u_{i}-\varphi(y_{0})$. Note also that
$\tilde{u}_{i}\in D^{1,p}(X)\cap L^{p}_{\rm loc}(X,\mu)$ with

(61) $\varrho_{\tilde{u}_{i}}\leq Lg\,\mu\text{-a.e. in $X$ for each
$i\in\mathbb{N}$}$

where $\varrho_{\tilde{u}_{i}}$ denotes the minimal $p$-weak upper gradient of
$\tilde{u}_{i}$ and $g$ a $p$-weak upper gradient of $f$, that is a function
satisfying in (58). Let $v_{j}:=\,\inf_{1\leq i\leq j}\tilde{u}_{i}$. From
[34, Proposition 7.1.8]), $D^{1,p}(X)$ is a lattice, that is $v_{j}\in
D^{1,p}(X)$ for each $j$ and

(62) $\varrho_{v_{j}}\leq L\,g\quad\mu\text{-a.e. in $X$, for each $j$}\,.$

Now observe that, since

(63) $\tilde{f}_{\varphi}\leq\,v_{j}\leq\,v_{1}\quad\text{for each $j$}\,,$

from (60) and (63), by applying Lebesgue dominated convergence theorem, it
follows that

(64) $v_{j}\to\tilde{f}_{\varphi}\text{ in }L^{p}_{\rm loc}(X,\mu)\text{ as
$j\to\infty$.}$

Let $x_{0}\in X$ and $\Omega_{n}:=\,U(x_{0},n)$, if $n\in\mathbb{N}$. Then
$(\Omega_{n})_{n}$ is an increasing sequence of open balls of $X$ and
$X=\cup_{n=1}^{\infty}\Omega_{n}$. From (64) and (62), by the locality of
Sobolev functions in $N^{1,p}(X)$ (see [34, Lemma 7.3.22]), there exists

(65) $\hat{f}\in N^{1,p}(X)\text{ such that
}\hat{f}=\,\tilde{f}_{\varphi}\,\mu\text{-a.e. in }X$

and $\hat{f}$ admits a $p$-weak upper gradient $\varrho_{\infty}$ in $X$
satisfying

(66) $\varrho_{\infty}\leq\,L\,g\quad\mu\text{-a.e. in }X.$

By (65) and (66), it follows that $\tilde{f}_{\varphi}$ admits a
$\mu$-representative $\hat{f}\in N^{1,p}(X)$ with minimal $p$-weak upper
gradient $\varrho_{\hat{f}}\leq\,L\,g$ $\mu$-a.e. in $X$. Thus we get the
desired conclusion. ∎

###### Proof of Theorem 7.3.

Let us first show the inclusion

$N^{1,p}(X:Y)\subset W^{1,p}(X,Y)\,.$

Let $f\in N^{1,p}(X:Y)$. From Theorem 7.2,

(67) $f\in N^{1,p}(X:Y)\subset\,D^{1,p}(X:Y)\subset
D^{1,p}(X:\ell^{\infty}(Y))\,,$

and the map

$X\ni x\mapsto d_{Y}(y_{0},f(x))\text{ is in }L^{p}(X,\mu)\,.$

For a given $y\in Y$, let $\varphi_{y}:Y\to\mathbb{R}$ be the function,
$\varphi_{y}(z):=\,d_{Y}(y,z)$. Then by applying Theorem 7.1 (ii)

(68) $f_{y}:=\,\varphi_{y}\circ f\in D^{1,p}(X)\text{ and
}\varrho_{f_{y}}\leq\varrho_{f}\,\text{$\mu$- a.e. in }X\text{, for each }y\in
Y\,.$

Thus, combining (67) and (68), it follows that $f\in W^{1,p}(X,Y)$.
Let us now show the inclusion

$W^{1,p}(X,Y)\subset N^{1,p}(X:Y)\,.$

Let $f\in W^{1,p}(X,Y)$. By definition, we can can assume $X\ni x\mapsto
d_{Y}(y_{0},f(x))$ is in $L^{p}(X,\mu)$. By applying Lemma 7.4 and Theorem 7.1
(ii), we can conclude that there exists a $\mu$-representative of $f$ such
that $f\in N^{1,p}(X:Y)$. ∎

We are going to apply Theorem 7.3 in the setting of Carnot groups. In
particular, for maps between Carnot groups, we have chosen to fix
$y_{0}=0_{\mathbb{G}_{2}}$ (see Definition 2.17). Let us first recall the
coincidence between the Newtonian Sobolev space and the horizontal Sobolev
space of real valued functions, which is well-known (see, for instance, [41,
Proposition C.12]).

######  Theorem 7.5.

Let $1<\,p<\,\infty$ and $\Omega\subset\mathbb{G}$ be an open set of a Carnot
group $\mathbb{G}\equiv\mathbb{R}^{n}$. Then

$N_{\mathbb{G}}^{1,p}(\Omega)=\,W_{\mathbb{G}}^{1,p}(\Omega)$

where $N_{\mathbb{G}}^{1,p}(\Omega)$ and $W_{\mathbb{G}}^{1,p}(\Omega)$ denote
the Newtonian Sobolev space defined before between metric spaces
$X=\,(\Omega,d_{c},\mathcal{L}^{n})$ and $Y=\mathbb{R}$ and the horizontal
Sobolev space defined in Definition 2.12, respectively.

As an immediate consequence of Theorems 7.3 and 7.5 and Definition 2.17, the
following result follows (see also [41, Proposition C.13])

######  Corollary 7.6.

Let $\Omega\subset\mathbb{G}$ be an open set of a Carnot group
$\mathbb{G}\equiv\mathbb{R}^{n}$ and let $Y\equiv(Y,d_{Y},y_{0})$ be a pointed
complete separable metric space. Then

$N^{1,p}(\Omega:Y)=\,W^{1,p}(\Omega,Y)$

where $N^{1,p}(\Omega:Y)$ and $W^{1,p}(\Omega,Y)$ denote the Newtonian space
between metric space $X=\,(\Omega,d_{c},\mathcal{L}^{n})$ and $Y$, and the
metric space-valued Sobolev space defined by sections introduced before,
respectively.

## References

  * [1] R.A. Adams, J.J.F. Fournier, Sobolev Spaces. Elsevier Science, 2003.
  * [2] A.Alberico, A.Cianchi, Differentiability properties of Orlicz-Sobolev functions. Ark. Mat. 43 (2005),1–28.
  * [3] L.Ambrosio, Metric space valued functions of bounded variation. Ann. Scuola Norm. Sup. Pisa Cl. Sci.,4 (17), (1990), 439–478.
  * [4] L.Ambrosio, Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces. Adv. in Math.,159, (2001), 51–67.
  * [5] L.Ambrosio, R.Ghezzi, Sobolev and bounded variation functions on metric measures spaces. Geometry,Analysis and Dynamics, vol. II,EMS Lectures Notes Series,pp. 63, 2016.
  * [6] L.Ambrosio, T.Ikonen, D.Lucić, E.Pasqualetto, Metric Sobolev spaces I: equivalence of definitions. ArXiv 2404.11190v1,2024.
  * [7] Z.M. Balogh, K.Rogovin, T. Zürcher, The Stepanov differentiability theorem in metric measure spaces. J. Geom. Anal. 14 (2004), no. 3, 405–422.
  * [8] C.Bennett, R.Sharpley, Interpolation of operators, volume 129 of Pure and Applied Mathematics. Academic Press Inc, Boston (1988).
  * [9] A.Bonfiglioli, E.Lanconelli, F.Uguzzoni, Stratified Lie Groups and Potential Theory for their Sub–Laplacians. Springer Monographs in Mathematics, Springer-Verlag Berlin Heidelberg, New York (2007).
  * [10] J.Cheeger, Differentiability of Lipschitz functions on metric measure spaces. Geometry and Functional Analysis, 9, (1999), 428–517.
  * [11] J.Cheeger, B.Kleiner. On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces. Nankai Tracts Math., vol. 11, pp 129-152. World Sci. Publ., Hackensack, 2006.
  * [12] J.Cheeger, B.Kleiner. Differentiability of Lipschitz functions from metric measure spaces to Banach spaces with Radon-Nikodym property. Geom. Funct. Anal. 19 (2009), 1017–1027.
  * [13] A.Cianchi, Continuity properties of functions from Orlicz-Sobolev spaces and embedding theorems, Annali della Scuola Normale Superiore di Pisa, 1996.
  * [14] A.Cianchi, Optimal Orlicz-Sobolev embeddings, Revista Matematica Iberoamericana, 2004.
  * [15] M.Csörnyei, Absolutely continuous functions of Rado, Reichelderfer and Malý. J. Math. Anal. Appl., 252 (2000), 147–166.
  * [16] T.K. Donaldson and N.S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems 1971\.
  * [17] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, BocaRaton, 1992 .
  * [18] H.Federer, Geometric Measure Theory. Springer, 1969.
  * [19] G.B. Folland, E.M. Stein, Hardy spaces on homogeneous groups. Princeton University Press, 1982.
  * [20] B.Franchi, C.E. Gutiérrez,T.Van Nguyen, Homogenization and Convergence of Correctors in Carnot groups. Comm. Partial Differential Equations, 30 (2005), 1817–1841.
  * [21] B.Franchi, G.Lu, R.L. Wheeden, A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type. Internat. Math. Res. Notices, 1 (1996), 1–14.
  * [22] B.Franchi, R.Serapioni, F.Serra Cassano, Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields. Houston Journal of Mathematics, 22, 4, (1996), 859–889.
  * [23] B.Franchi, R.Serapioni, F.Serra Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields. Boll. Un. Mat. Ital.,(7) 11-B (1997), 83–117.
  * [24] B.Franchi, R.Serapioni, F.Serra Cassano, Rectifiability and perimeter in the Heisenberg group. Math Ann 321, (2001), 479–531.
  * [25] B.Franchi, R.Serapioni, F.Serra Cassano, On the Structure of Finite Perimeter Sets in Step 2 Carnot Groups. The Journal of Geometric Analysis. 13 (2003), no 3, 421–466.
  * [26] B.Franchi, R.L. Wheeden, Some remarks about Poincaré type inequalities and representation formulas in metric spaces of homogeneous type. J. of Inequal. Appl. 3 (1999), 65–89.
  * [27] O.K. Friedrichs, , The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc. 55 (1944), 132–151.
  * [28] N.Garofalo, D.M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math., 49 (1996), 1081–1144.
  * [29] N. Gigli, A. Tyulenev, Korevar-Scoen’s energy on strongly rectifiable spaces. Calc. Var. Partial Differential Equations 60 (2021), no. 6,Paper No. 235, 54pp.
  * [30] M.Gromov, Carnot-Carathéodory spaces seen from within. In Subriemannian Geometry, Progress in Mathematics, 144. ed. by A.Bellaiche and J.Risler, Birkhauser Verlag, Basel, 1996.
  * [31] M.Gromov, Metric structures for Riemannian and Non Riemannian Spaces. Progress in Mathematics, 152, Birkhauser Verlag, Boston, (1999).
  * [32] P.Hajłasz, Sobolev spaces on an arbitrary metric space. Potential Anal., 5 (1996), no. 4, 403–415.
  * [33] P.Hajłasz, P.Koskela, Sobolev met Poincare., Memoirs of the American Mathematical Society, 688, (2000).
  * [34] J.Heinonen, P.Koskela, N.Shanmugalingam, and J.T. Tyson, Sobolev spaces on metric measure spaces. New Mathematical Monographs, vol. 27, Cambridge University Press, Cambridge, 2015.
  * [35] D.Jerison, The Poincaré inequality for vector fields satisfying Hörmander condition. Duke Math. J., 53 (1986), 503-523.
  * [36] S.Hencl, On the notions of absolute continuity for functions of several variables. Fund. Math., 173 (2002), 175–189.
  * [37] S.Hencl, Notes on absolutely continuous functions of several variables. Real Anal. Exch. 30 (1) (2004/2005), 59–74.
  * [38] S.Hencl, J.Malý, Absolutely continuous functions of several variables and diffeomorphisms. Centr. European J. Math. 4 (2003), 690–705.
  * [39] J.Kauhanen, P.Koskela and J.Malý, On functions with derivatives in a Lorentz space. Manuscripta Math. 100 (1999), 87–101.
  * [40] B.Kirchheim, F.Serra Cassano, Rectifiability and parameterizations of intrinsically regular surfaces in the Heisenberg group. Ann. Scuola Norm. Sup. Pisa, Cl.Sc. (5) III, (2004), 871–896.
  * [41] B.Kleiner, S.Müller and X. Xie, Pansu pullback and exterior differentiation for Sobolev maps on Carnot groups. ArXiv:2007.092741v2 [math. DG] 3 Dec 2021.
  * [42] B.Kleiner, S.Müller and X. Xie, Pansu pullback and rigidity of mappings between Carnot groups. ArXiv:2004.06694v2 [math. DG] 18 Dec 2021.
  * [43] T. Kuusi, G. Mingione, Guide to nonlinear potential estimates. Bull. Math. Sci. 4(1), 1–82 (2014).
  * [44] P.Lahti and X.Zhou,Absolutely continuous mappings on doubling metric measure spaces. manuscripta math. 173, 1–21 (2024).
  * [45] M.A. Krasnosel’skii, Ya.B. Rutickii, Convex Functions and Orlicz spaces, 1961.
  * [46] E.Le Donne, A metric characterization of Carnot groups., To appear on PAMS.
  * [47] E.Le Donne, S.Rigot Besicovitch covering property for homogeneous distances in Heisenberg groups. Preprint, 2014.
  * [48] G.Lu, R.L. Wheeden, An optimal representation formula for Carnot-Carathéodory vector fields. Bull. London Math. Soc. 30 (1998), no. 6, 578–584.
  * [49] W.A.J. Luxemburg, Banach function spaces, Getal en Figuur. 6. Assen: Van Gorcum & Comp. 1955.
  * [50] V.Magnani, Elements of Geometric Measure Theory on sub-Riemannian Groups, PhD Thesis, Scuola Normale Superiore,Edizioni della Normale, Pisa, 2002.
  * [51] V.Magnani, An area formula in metric spaces. Colloq. Math. 124 (2011), 275–283.
  * [52] V.Magnani, Towards differential calculus in stratified groups. J. Aust. Math. Soc. 95 (2013),No.1 , 76–128.
  * [53] V.Magnani, A.Pinamonti, G.Speight, Porosity and Differentiability of Lipschitz Maps from Stratified Groups to Banach Homogeneous Groups. Annali di Matematica Pura ed Applicata 199, (2020), 1197–1220.
  * [54] J.Malý, Absolutely Continuous Functions of Several Variables. J. Math. Anal. Appl. 231 (1999), 492–508.
  * [55] N.Marola, W.P. Ziemer, Aspects of area formula by way of Luzin, Radó and Reichelderfer on metric measure spaces. J. Math. Soc. Japan 67 (2015), no. 2, 561–579.
  * [56] J.Mitchell, On Carnot-Carathèodory metrics. J.Differ. Geom. 21, (1985), 35–45.
  * [57] R.Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications. Mathematical Surveys and Monographs 91, American Mathematical Society, Providence, RI, 2002.
  * [58] R.Monti, Distances, Boundaries and Surface Measures in Carnot-Carathéodory Spaces. PhD Thesis, University of Trento, 2001.
  * [59] R.Monti, F.Serra Cassano, Surface measures in Carnot-Carathéodory spaces. Calc. Var. Partial Diff. Eq. 13, (2001), 339–376.
  * [60] P.Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. of Math., 129, (1989), 1–60.
  * [61] A.Pinamonti, G.Speight, Porosity, Differentiability and Pansu’s Theorem. J. Geom. Anal., 27(3),(2017), 2055–-2080
  * [62] T.Rado, P.V. Reichelderfer, Continuous Transformations in Analysis. Springer- Verlag, New York, 1955.
  * [63] A.Ranjbar-Motlagh, An embedding theorem for Sobolev type functions with gradients in a Lorentz space. Studia Math. 19 (2009), no.1 ,1–9.
  * [64] A.S. Romanov, Absolute continuity of the Sobolev type functions on metric measure spaces. Sibirsk. Mat. Zh. 49 (2008), no. 5, 1147–1156 .
  * [65] Yu.G. Reshetnyak, Sobolev classes of functions with values in a metric space. Sibirsk. Mat. Zh. 38 (1997), no. 3, 657–675.
  * [66] S.Rigot, Counterexample to the Besicovitch covering property for some Carnot groups equipped with their Carnot-Carathéodory metric. Math.Z. 248 (2004), no. 4,827–848.
  * [67] E.Sawyer, R.L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Amer.J.Math., 114, (1992), 813–874.
  * [68] S.Semmes, On the non existence of bilipschitz parametrization and geometric problems about $A_{\infty}$-weights. Revista Matematica Iberoamericana. 12, (1996), 337–410.
  * [69] N. Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev.Mat. Iberoamericana 16 (2000), 243–279.
  * [70] E.M. Stein, Singular Integrals and Differential Properties of Functions. Princeton University Press, Princeton, N J, 1970.
  * [71] E.M. Stein, G.Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, N.J., 1971.
  * [72] E.M. Stein, Editor’s note: the differentiability of functions in $\mathbb{R}^{n}$. Ann. of Math. 113 (1981), 383–385.
  * [73] W.Stepanoff , Ueber totale Differenzierbarkeit.Math. Ann., 90 (1923), 318–320.
  * [74] G.Talenti, An embedding theorem, Partial differential equations and the calculus of variations. Essays in Honor of Ennio De Giorgi, 1989.
  * [75] S.K. Vodopyanov, Monotone functions and quasiconformal mappings on Carnot groups. Sibirsk. Mat. Zh. 37 (1996), no. 6, 1269–1295.
  * [76] S.K. Vodopyanov, Mappings with bounded distortion and with finite distortion on Carnot groups. Sibirsk.Mat. Zh. 40 (1999), no. 4, 764–804.
  * [77] S.K. Vodop’yanov, ${\mathcal{P}}$-differentiability on Carnot groups in different topologies and related topics. Proc.on Analysis and Geometry,603–670, Sobolev Institute Press, Novosibirsk, (2000)
  * [78] S.K. Vodop’yanov, Differentiability of maps of Carnot groups of Sobolev classes. Sb. Math. 194 (2003), 857–877.
  * [79] K.Wildrick, T.Zürcher, Space filling with metric measure spaces. Math. Z. 270 (2012), 103–131.
  * [80] K.Wildrick, T.Zürcher, Sharp Differentiability Results for the Lower Local Lipschitz Constant and Applications to Non-embedding.  J. Geom. Anal. 25 (2015), 2590–2616.
  * [81] K.Wildrick, T.Zürcher, Mapping with an upper gradient in a Lorentz space. Preprint
  * [82] W.P. Ziemer, Weakly Differentiable Functions, Springer, New York, 1989\.