# Pansu pullback and rigidity of mappings between Carnot groups

Bruce Kleiner  ,  Stefan Müller  and  Xiangdong Xie

###### Abstract.

We consider mappings $f:G\supset U\rightarrow G^{\prime}$ where $G$ and
$G^{\prime}$ are Carnot groups. We prove a number of new structural results
for Sobolev (in particular quasisymmetric) mappings, establishing (partial)
rigidity or (partial) regularity theorems, depending on the context. In
particular, we prove the quasisymmetric rigidity conjecture for Carnot groups
which are not rigid in the sense of Ottazzi-Warhurst.

BK was supported by NSF grants DMS-1405899 and DMS-1711556 and a Simons
Collaboration grant.

SM has been supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through the Hausdorff Center for Mathematics (GZ EXC 59
and 2047/1, Projekt-ID 390685813) and the collaborative research centre The
mathematics of emerging effects (CRC 1060, Projekt-ID 211504053). This work
was initiated during a sabbatical of SM at the Courant Institute and SM would
like to thank R.V. Kohn and the Courant Institute members and staff for their
hospitality and a very inspiring atmosphere.

XX has been supported by Simons Foundation grant #315130.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Differential forms on Carnot groups
  4. 4 The pullback theorem
  5. 5 Center of mass and mollification in Carnot groups
  6. 6 Pansu pullback and mollification
  7. 7 Rigidity of products
  8. 8 Complexifed Carnot groups
  9. 9 Quasisymmetric rigidity of nonrigid Carnot groups
  10. 10 Structure of nonrigid Carnot groups
  11. 11 Rigidity of product-quotients
  12. 12 Indecomposable product-quotients with $\dim K=1$
  13. 13 Indecomposable product-quotients with $\dim K\geq 2$
  14. 14 Product quotients of complex Heisenberg groups and higher real Heisenberg groups

##  1\. Introduction

### Background and overview

Quasiconformal mappings between domains in $\mathbb{R}^{n}$ have been heavily
studied since the 1930s. They initially arose in Teichmuller theory, and over
time found applications in a number of areas of mathematics, including
Kleinian groups, complex dynamics, PDE, rigidity of lattices, and geometric
group theory. Quasiconformal mappings in non-Euclidean settings first appeared
around 1970 in the work of Mostow on rigidity of lattices in Lie groups, and
the topic gained momentum in the 80s and 90s due to the convergence of
developments from several directions, including Gromov’s notion of
hyperbolicity, fundamental work of Pansu on quasi-isometries of symmetric
spaces, advances in the analytical theory of quasiconformal homeomorphisms,
and the explosion of interest in analysis on metric spaces (see for instance
[Gro87, Pan89, IM93, MM95, HK98, Ast98, Bon06, Kle06, Hei07]). A new
phenomenon in the non-Euclidean setting is rigidity [Pan89]: in some
situations quasiconformal mappings turn out to be very restricted, in sharp
contrast to mappings in $\mathbb{R}^{n}$, which are flexible and come in
infinite dimensional families. A heuristic explanation for rigidity is that in
the case of smooth mappings quasiconformality is equivalent to a first order
system which is formally overdetermined except in very special cases. However,
the limited regularity of quasiconformal homeomorphisms has posed a major
obstacle to converting this heuristic into theorems: since quasiconformal
homeomorphisms are not a priori smooth, the standard technique for exploiting
overdetermined conditions – repeated differentiation – is not applicable. For
this reason, rigidity is only known in some special cases where the regularity
problem has been successfully overcome or circumvented [Pan89, Cap99, CC06,
Xie15]. Regularity issues aside, the general picture remains unclear even for
smooth quasiconformal homeomorphisms, because the precise implications of the
condition depend in a subtle way on the structure of the Carnot group $G$; see
below for further discussion.

In this paper we establish a number of rigidity (or partial rigidity)
theorems. A key technical result is a generalization to Carnot groups of the
fact that in Euclidean space, pullback of differential forms by Sobolev
mappings commutes with exterior differentiation, under suitable assumptions on
the Sobolev exponent [Res89]. We consider Sobolev (in particular
quasiconformal) mappings $f:G\supset U\rightarrow U^{\prime}\subset
G^{\prime}$ between open subsets of Carnot groups, and define the Pansu
pullback $f_{P}^{*}\omega$ of a smooth differential form
$\omega\in\Omega^{*}(U^{\prime})$ using the (approximate) Pansu differential.
Although Pansu pullback does not commute with exterior differentiation in
general, even for smooth Sobolev maps, it turns out that a partial analog does
hold: for some closed forms $\omega$, certain components of the distributional
exterior derivative of $f_{P}^{*}\omega$ are zero. In this paper we use this
result to exclude certain types of oscillatory behavior of the Pansu
differential and obtain rigidity results. Further applications to geometric
mapping theory will be given elsewhere.

Although the initial motivation for this work arose from geometric mapping
theory and geometric group theory, in spirit the phenomena and methods have
much in common with the analytical side of geometric mapping theory, and with
the literature on rigidity and oscillatory solutions to PDEs, see for instance
[Nas54, Tar79, Mur81, Gro86, Sch93, DM99, M9̈9, Mv03, DLS09, DLS17, Ise18,
BV19].

### Statement of results

We begin by setting notation and briefly recalling a few facts about Carnot
groups; see Section 2 for more detail.

Let $G$ be a Carnot group with Lie algebra $\mathfrak{g}$, grading
$\mathfrak{g}=\oplus_{j\geq 1}^{s}V_{j}$, and dilation group
$\\{\delta_{r}:G\rightarrow G\\}_{r\in(0,\infty)}$. Without explicit mention,
in what follows all Carnot groups will be equipped with Haar measure and a
Carnot-Caratheodory metric denoted generically by $d_{CC}$. We recall that if
$f:G\supset U\rightarrow G^{\prime}$ is a Sobolev mapping between Carnot
groups, where $U$ is open, then $f$ has a well-defined approximate Pansu
differential $D_{P}f(x):G\rightarrow G^{\prime}$ for a.e. $x\in U$; this is a
graded group homomorphism, which we often conflate with the associated
homomorphism of graded Lie algebras
$D_{P}f(x):\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$. If $f:G\supset
U\rightarrow U^{\prime}\subset G$ is a quasiconformal homeomorphism, then
$f\in W^{1,p}_{\operatorname{loc}}$ for some $p$ strictly larger than the
Hausdorff (homogeneous) dimension of $G$, and $D_{P}f(x)$ is an isomorphism
for a.e. $x\in U$.

We first discuss mappings and product structure. We remark that the problem of
determining to what extent a mapping $X_{1}\times X_{2}\rightarrow Y_{1}\times
Y_{2}$ must respect the product structure has arisen in various places in
mathematics [Mau81, Whi40, Fox47, Fou71, J5́7]; our original motivation was
the product rigidity theorem in geometric group theory [KKL98].

We observe that bilipschitz homeomorphisms need not respect product structure
in Carnot groups: if $G$ is a Carnot group then for any nonconstant Lipschitz
map $\alpha:G\rightarrow\mathbb{R}^{n}$ the shear
$(x,y)\mapsto(x,y+\alpha(x))$ defines a bilipschitz homeomorphism
$G\times\mathbb{R}^{n}\rightarrow G\times\mathbb{R}^{n}$ which does not
respect the product structure. Another type of example comes from products of
the form $G_{1}\times G_{2}$, where $G_{1}\simeq K$, $G_{2}\simeq K\times K$
for some Carnot group $K$: the map
$(x_{1},(x_{2},x_{3}))\mapsto(x_{3},(x_{1},x_{2}))$ does not respect the given
product structure. Our first result asserts that nondegenerate Sobolev
mappings between products must respect the product structure once these two
types of examples are excluded. Let $\\{G_{i}\\}_{1\leq i\leq n}$,
$\\{G_{j}^{\prime}\\}_{1\leq j\leq n^{\prime}}$ be collections of Carnot
groups, where each $G_{i}$, $G_{j}^{\prime}$ is nonabelian and does not admit
a nontrivial decomposition as a product of Carnot groups. Let
$G:=\prod_{i}G_{i}$, $G^{\prime}:=\prod_{j}G^{\prime}_{j}$.

######  Theorem 1.1.

Let $\nu$ denote the homogeneous dimension of $G$. Suppose for some $p>\nu$
that $f:G\supset U\rightarrow G^{\prime}$ is a
$W^{1,p}_{\operatorname{loc}}$-mapping, $U=\prod_{i}U_{i}$ is a product of
connected open sets $U_{i}\subset G_{i}$, and the Pansu differential
$D_{P}f(x)$ is an isomorphism for a.e. $x\in G$. Then $f$ is a product of
mappings, i.e. $n=n^{\prime}$, and for some permutation
$\sigma:\\{1,\ldots,n\\}\rightarrow\\{1,\ldots,n\\}$ there are mappings
$\\{f_{\sigma(i)}:U_{i}\rightarrow G_{\sigma(i)}^{\prime}\\}_{1\leq i\leq n}$
such that

(1.2)
$f(x_{1},\ldots,x_{n})=(f_{1}(x_{\sigma^{-1}(1)}),\ldots,f_{n}(x_{\sigma^{-1}(n)}))$

for every $(x_{1},\ldots,x_{n})\in U$. In particular, any quasiconformal
homeomorphism $G\rightarrow G^{\prime}$ is a product of homeomorphisms.

By means of a blow-down argument, we use the product rigidity theorem above to
see that quasi-isometries between products must asymptotically preserve
product structure.

Let $\\{G_{i}\\}_{1\leq i\leq n}$, $\\{G_{j}^{\prime}\\}_{1\leq j\leq
n^{\prime}}$ be as above, and suppose $\\{\hat{G}_{i}\\}_{1\leq i\leq n}$,
$\\{\hat{G}_{j}^{\prime}\\}_{1\leq j\leq n^{\prime}}$ are collections where:

  * •

Each $\hat{G}_{i}$, $\hat{G}^{\prime}_{j}$ is either a simply-connected
nilpotent Lie group with a left invariant Riemannian metric, or a finitely
generated nilpotent group equipped with a word metric.

  * •

For every $1\leq i\leq n$, $1\leq j\leq n^{\prime}$ the asymptotic cone of
$\hat{G}_{i}$, $\hat{G}_{j}^{\prime}$ is bilipschitz homeomorphic to $G_{i}$,
$G_{j}^{\prime}$, respectively [Pan83].

Set $\hat{G}:=\prod_{i}\hat{G}_{i}$,
$\hat{G}^{\prime}=\prod_{j}G_{j}^{\prime}$, and equip $\hat{G}$,
$\hat{G}^{\prime}$ with the $\ell^{2}$-distance functions,

$d^{2}_{\hat{G}}(x,x^{\prime}):={\sum_{i}d_{G_{i}}^{2}(\pi_{i}(x),\pi_{i}(x^{\prime}))}\,,\quad
d^{2}_{\hat{G}^{\prime}}(x,x^{\prime}):={\sum_{j}d_{G_{j}^{\prime}}^{2}(\pi_{i}(x),\pi_{i}(x^{\prime}))}\,.$

If $1\leq i\leq n$, we say that two points
$x=(x_{1},\ldots,x_{n}),\,x^{\prime}=(x_{1}^{\prime},\ldots,x_{n}^{\prime})\in
G$ form an $i$-pair if $x_{j}=x_{j}^{\prime}$ for every $j\neq i$. The next
result says that an $i$-pair $x,x^{\prime}\in\hat{G}$ with $d(x,x^{\prime})$
large is nearly mapped to a $j$-pair, where $j$ depends only on $i$.

######  Theorem 1.3.

For every $L\geq 1$, $A<\infty$, there is a function
$\varepsilon=\varepsilon_{L,A}:[0,\infty)\rightarrow(0,1]$ with
$\lim_{R\rightarrow\infty}\varepsilon(R)=0$ with the following property.

For every $(L,A)$-quasi-isometry $\Phi:\hat{G}\rightarrow\hat{G}^{\prime}$,
there is a bijection
$\sigma:\\{1,\ldots,n\\}\rightarrow\\{1,\ldots,n^{\prime}\\}$, such that for
every $R\in[0,\infty)$, $1\leq i\leq n$, and every $i$-pair
$x,x^{\prime}\in\hat{G}$ with $d(x,x^{\prime})\geq R$, we have

(1.4)
$\frac{d(\pi_{j}(\Phi(x)),\pi_{j}(\Phi(x^{\prime})))}{d(\Phi(x),\Phi(x^{\prime}))}\in\begin{cases}(1-\varepsilon,1]\quad\text{if}\quad
j=\sigma(i)\\\ [0,\varepsilon)\quad\text{if}\quad
j\neq\sigma(i)\,.\end{cases}$

One may compare Theorem 1.3 with the rigidity theorem for quasi-isometries
between products of Gromov hyperbolic spaces [KKL98]; in that case the quasi-
isometry is at finite sup distance from a product of quasi-isometries, modulo
reindexing factors. This stronger assertion is false in the setting of Theorem
1.3, because of central shears. Nonetheless, Theorem 1.3 is still strong
enough to imply that the factors are quasi-isometric:

######  Corollary 1.5.

If $\hat{G}$, $\hat{G}^{\prime}$, and $\Phi$ are as in Theorem 1.3, then
modulo reindexing, $\hat{G}_{i}$ is quasi-isometric to $\hat{G}_{i}^{\prime}$.

We recall that the quasi-isometry classification of nilpotent groups has
remained a major open problem in geometric group theory since Pansu’s work
[Pan89, Sha04]; Corollary 1.5 shows that this reduces to the case of groups
which are indecomposable (in an appropriate sense).

Our next result is a regularity theorem for nondegenerate Sobolev maps between
certain complexified Carnot groups. For simplicity we only state the result
for complex Heisenberg groups here, and refer the reader to Section 8 for the
general case.

######  Theorem 1.6.

Let $\mathbb{H}_{n}^{\mathbb{C}}$ denote the complexification of the $n^{th}$
Heisenberg group $\mathbb{H}_{n}$, and let
$U\subset\mathbb{H}_{n}^{\mathbb{C}}$ be a connected open set. If
$f:U\rightarrow\mathbb{H}_{n}^{\mathbb{C}}$ belongs to
$W^{1,p}_{\operatorname{loc}}$ for some $p>4n+2$, and the Pansu differential
$D_{P}f(x)$ is an isomorphism for a.e. $x\in U$, then $f$ is holomorphic or
antiholomorphic.

######  Corollary 1.7.

Any quasiconformal homeomorphism $\mathbb{H}_{n}^{\mathbb{C}}\supset
U\rightarrow U^{\prime}\subset\mathbb{H}_{n}^{\mathbb{C}}$ between connected
open subsets is either holomorphic or antiholomorphic.

In the case of $C^{2}$ diffeomorphisms Corollary 1.7 is due to Reimann-Ricci
[RR00]. We also show that global quasiconformal homeomorphisms are rigid:

######  Theorem 1.8.

The group of quasiconformal homeomorphisms
$\mathbb{H}_{n}^{\mathbb{C}}\rightarrow\mathbb{H}_{n}^{\mathbb{C}}$ is
generated by left translations, complex graded automorphisms, and complex
conjugation.

In contrast to global quasiconformal homeomorphisms, locally defined
quasiconformal homeomorphisms $\mathbb{H}_{n}^{\mathbb{C}}\supset U\rightarrow
U^{\prime}\subset\mathbb{H}_{n}^{\mathbb{C}}$ are quite flexible, and come in
infinite dimensional families, see Section 8.

We now shift our attention to quasiconformal homeomorphisms $G\supset
U\rightarrow U^{\prime}\subset G$ for a general Carnot group $G$. For a smooth
diffeomorphism, being locally quasiconformal is equivalent to being a contact
diffeomorphism, i.e. preservation of the subbundle of the tangent bundle $TG$
defined by the first layer $V_{1}\subset\mathfrak{g}$. A parameter count
indicates that the contact condition is formally overdetermined unless $G$ is
isomorphic to the Engel group or to $\mathbb{R}^{k}\times\mathbb{H}_{\ell}$
for some $k,\ell$, and hence one expects some form of rigidity. However, the
analytical character of the condition is quite different for different groups:

  * •

When $G=\mathbb{H}\times\mathbb{H}$ contact diffeomorphisms must be products
(locally) but otherwise are quite flexible [CR03].

  * •

When $G=\mathbb{H}_{n}^{\mathbb{C}}$ the contact condition is locally
flexible, but still “hypoelliptic”, i.e. contact diffeomorphisms are
holomorphic or antiholomorphic.

  * •

When $G$ is an $H$-type group with center of dimension at least $3$, (e.g. one
of the Carnot groups studied by Pansu) or a free Carnot group of step $s\geq
3$, then the smooth contact embeddings $G\supset U\rightarrow G$ form a finite
dimensional family when $U$ is a connected open subset [Pan89, Rei01, War07].

Following Ottazzi-Warhurst [OW11], for $1\leq k\leq\infty$ we say that a
Carnot group $G$ is $C^{k}$-rigid if the space of $C^{k}$ contact embeddings
$G\supset U\rightarrow G$ forms a finite dimensional family for every
connected open subset $U\subset G$. Building on the theory of overdetermined
systems [GQS67, Spe69], Ottazzi-Warhurst [OW11] gave several different
characterizations of $C^{\infty}$-rigid groups, and proved the following
regularity theorem:

######  Theorem 1.9 ( [OW11]).

If $G$ is a $C^{\infty}$-rigid Carnot group, then any $C^{2}$-contact
diffeomorphism $G\supset U\rightarrow U^{\prime}\subset G$ is $C^{\infty}$.

In view of the theorem, a group is $C^{k}$-rigid for $k\geq 2$ if and only if
it is $C^{\infty}$-rigid, so we will call such groups rigid. This leads
naturally to the following (cf. [OW11, p.2]):

######  Regularity Conjecture 1.10.

If $G$ is a rigid Carnot group, then any quasiconformal homeomorphism
$G\supset U\rightarrow U^{\prime}\subset G$ is $C^{\infty}$.

The conjecture is known for $H$-type groups with center of dimension at least
$3$ by using subelliptic regularity [Cap99, CC06]; it follows from Theorem 1.1
that the Regularity Conjecture holds for a product of Carnot groups
$\prod_{i}G_{i}$ if it holds for all of the factors $\\{G_{i}\\}$. We will
present further results on rigid groups elsewhere [KMX], and focus here on the
case of nonrigid groups. Our main result for such groups is that one always
has partial rigidity, apart from some exceptional cases:

######  Theorem 1.11 (See Section 9 for definitions).

If $G$ is a nonrigid Carnot group with homogeneous dimension $\nu$, then one
of the following holds:

  1. (1)

$G$ is isomorphic to $\mathbb{R}^{n}$ or to a real or complex Heisenberg group
$\mathbb{H}_{n}$, $\mathbb{H}_{n}^{\mathbb{C}}$ for some $n\geq 1$.

  2. (2)

There is a closed horizontally generated subgroup $\\{e\\}\subsetneq
H\subsetneq G$, and a finite group $A$ of graded automorphisms of $G$ with the
following properties:

     * •

For every $p>\nu$, if $U\subset G$ is a connected open subset and $f:G\supset
U\rightarrow G$ is a $W^{1,p}_{\operatorname{loc}}$-mapping such that
$\det(D_{P}f)$ has essentially constant sign, then for some $\Phi\in A$ the
composition $\Phi\circ f$ preserves the coset foliation of $H$. In particular,
the conclusion holds for quasiconformal homeomorphisms.

     * •

The Lie algebra of $H$ is generated by a linear subspace $\\{0\\}\subsetneq
W\subsetneq V_{1}$ with $[W,V_{j}]=\\{0\\}$ for all $j\geq 2$.

Thus, apart from exceptional cases quasiconformal homeomorphisms preserve a
foliation, up to post-composition with a graded automorphism. This suggests
that for the (nonexceptional) nonrigid cases there may be a more detailed
description of quasiconformal homeomorphisms along the lines of [Xie15] for
model filiform groups. To our knowledge such a description is not known even
for smooth contact diffeomorphisms, either locally or globally.

Another aspect of quasiconformal regularity/flexibility has to do with the
Sobolev exponent, i.e. higher integrability of the derivative. Quasiconformal
homeomorphisms $f:G\supset U\rightarrow U^{\prime}\subset G$ are always in
$W^{1,p}_{\operatorname{loc}}$ for some $p$ strictly larger than the
homogeneous dimension of $G$, where $p$ depends on $G$ and the quasiconformal
distortion of $f$ [HK98]. However, except for $\mathbb{R}^{n}$ and the
Heisenberg groups $\mathbb{H}_{n}$, all known examples of quasiconformal
homeomorphisms are in $W^{1,\infty}_{\operatorname{loc}}$, i.e. are locally
bilipschitz. This motivated the following:

######  Conjecture 1.12 (Xie).

If $G$ is a Carnot group other than $\mathbb{R}^{n}$ or $\mathbb{H}_{n}$ for
some $n$, then every quasiconformal homeomorphism $f:G\supset U\rightarrow
U^{\prime}\subset G$ is locally bilipschitz; moreover, if $U=G$ then $f$ is
bilipschitz.

Using Theorem 1.11 and a variation on [SX12, Xie13b, LDX16] we show:

######  Theorem 1.13.

Conjecture 1.12 holds for nonrigid Carnot groups.

Note that for rigid groups Conjecture 1.12 would follow from the Regularity
Conjecture and [CO15].

Our next result, which is a key ingredient in the rigidity theorems above,
concerns the interaction between Sobolev mappings and differential forms. We
recall that pullback of smooth $k$-forms by a
$W^{1,k+1}_{\operatorname{loc}}$-mapping $f:\mathbb{R}^{n}\supset
U\rightarrow\mathbb{R}^{m}$ commutes with the exterior derivative, i.e.

(1.14) $df^{*}\omega=f^{*}d\omega$

for all $\omega\in\Omega^{k}(\mathbb{R}^{m})$; here the exterior derivative on
the left hand side is interpreted distributionally. To our knowledge this was
first observed by Reshetnyak, who used it in his approach to quasiconformal
and quasiregular mappings [Res89]; it has many other applications to geometric
mapping theory in Euclidean space. Now let $f:G\supset U\rightarrow
G^{\prime}$ be a $W^{1,p}_{\operatorname{loc}}$ mapping where $G$,
$G^{\prime}$ are Carnot groups and $U$ is an open subset. Since $f$ need not
be a Sobolev mapping with respect to the Euclidean metrics (1.14) is not
applicable. On the other hand we may define the Pansu pullback of a smooth
$k$-form $\omega\in\Omega^{k}(U^{\prime})$ to be the $k$-form with measurable
coefficients $f_{P}^{*}\omega$ given by

$f_{P}^{*}\omega(p)=(D_{P}f(p))^{*}\omega(f(p))\,;$

here we are viewing the Pansu differential as a graded homomorphism
$D_{P}f(p):\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$, and identifying
tangent spaces of $G$, $G^{\prime}$ with their respective Lie algebras via
left translation. It turns out that unlike ordinary pullback, Pansu pullback
does not commute with the exterior derivative. For instance, the equation
$df_{P}^{*}\omega=f_{P}^{*}d\omega$ holds for a smooth contact diffeomorphism
$f:\mathbb{H}\rightarrow\mathbb{H}$ of the Heisenberg group and every smooth
form $\omega$ only if $f$ preserves the foliation of $\mathbb{H}$ by cosets of
the center. Nonetheless, there is a partial generalization of (1.14) to Carnot
groups. To state this, we will need the notion of the weight of a differential
form; the weight $\operatorname{wt}(\alpha)$ of a differential form $\alpha$
is defined using its decomposition with respect to the diagonalizable action
of the Carnot scaling on $\Lambda^{*}\mathfrak{g}$, see Section 3.3.

######  Theorem 1.15 (Pullback theorem).

Let $G$, $G^{\prime}$ be Carnot groups where $G$ has dimension $N$ and
homogeneous dimension $\nu$. Let $f:G\supset U\rightarrow U^{\prime}\subset
G^{\prime}$ be a $W^{1,p}_{\operatorname{loc}}$-mapping between open subsets,
for some $p>\nu$. Suppose $\omega\in\Omega^{k}(U^{\prime})$ and
$\eta\in\Omega^{N-k-1}_{c}(U)$ are smooth forms such that $\omega$ is closed
and $\operatorname{wt}(\omega)+\operatorname{wt}(d\eta)\leq-\nu$. Then

(1.16) $\int_{U}f_{P}^{*}\omega\wedge d\eta=0\,.$

In particular, if $\zeta\in\Omega^{N-k-1}(U)$ is closed and
$\operatorname{wt}(\omega)+\operatorname{wt}(\zeta)\leq-\nu+1$, then the
distributional exterior derivative $df_{P}^{*}\omega$ satisfies

$\zeta\wedge df_{P}^{*}\omega=0\,.$

Theorem 1.15 is a consequence of an approximation theorem (see Theorem 1.19
below), which is the key technical result in this paper.

Although it is not used here, we mention one more consequence of Theorem 1.19;
we will discuss further applications elsewhere. For every open subset $U$ of a
Carnot group, let $I(U)$ denote the differential ideal generated by the
$1$-forms $\theta\in\Omega^{1}(U)$ which vanish on the horizontal subbundle
$V_{1}\subset TU$, i.e.

$I(U)=\operatorname{span}\\{\alpha\wedge\theta,\;\beta\wedge
d\theta\mid\alpha,\,\beta\in\Omega^{*}(U)\,,\;\theta\in\Omega^{1}(U),\;\theta\mbox{\Large$|$\normalsize}_{V_{1}}\equiv
0\\}\,.$

Let $J(U)$ be the annihilator of $I$ in the algebra of differential forms:

$J(U):=\\{\beta\in\Omega^{*}(U)\mid\beta\wedge\theta=0\;\;\text{for
every}\;\;\theta\in I(U)\\}\,.$

######  Theorem 1.17.

Let $G$ be a Carnot group with homogeneous dimension $\nu$, and suppose
$f:G\supset U\rightarrow U^{\prime}\subset G$ is a
$W^{1,p}_{\operatorname{loc}}$-mapping for some $p>\nu$. Then for every
$\omega\in J(U^{\prime})$ we have

$d(f_{P}^{*}\omega)=f_{P}^{*}d\omega\,,$

where the left hand side refers to the distributional exterior derivative.

Theorem 1.17 was shown by Vodopyanov when $G$ is a $2$-step Carnot group, and
$\omega$ has codegree $1$, i.e. $\omega\in\Omega^{\dim G-1}$ [Vod07]; see also
[Dai99] for related work. Their work was motivated by applications to
quasiregular mappings.

### Discussion of proofs

We begin with Theorem 1.1, and for simplicity we take
$U=G=G^{\prime}=\mathbb{H}\times\mathbb{H}$.

Consider a $W^{1,p}_{\operatorname{loc}}$ mapping
$f:\mathbb{H}\times\mathbb{H}\rightarrow\mathbb{H}\times\mathbb{H}$ for some
$p>8$, such that the Pansu differential $D_{P}f(x)$ is an isomorphism for a.e.
$x\in\mathbb{H}\times\mathbb{H}$. Since the Pansu differential
$D_{P}f(x):\mathbb{H}\times\mathbb{H}\rightarrow\mathbb{H}\times\mathbb{H}$ is
a graded isomorphism of Carnot groups, it is easy to see that $D_{P}f(x)$ must
respect the direct sum decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{h}$, i.e. there is a permutation
$\sigma_{x}:\\{1,2\\}\rightarrow\\{1,2\\}$ such that

(1.18)
$D_{P}f(x)(x_{1},x_{2})=(h_{1}(x_{\sigma_{x}^{-1}(1)}),h_{2}(x_{\sigma_{x}^{-1}(2)}))$

for some automorphisms $h_{1},h_{2}:\mathfrak{h}\rightarrow\mathfrak{h}$. If
the permutation $\sigma_{x}$ is independent of $x$, then one may readily
integrate (1.18) to deduce that $f$ itself is a product as in (1.2). If
$D_{P}f(x)$ varied continuously with $x$, it would follow immediately that the
permutation $\sigma_{p}$ is constant since it takes values in a discrete set;
however $D_{P}f(x)$ varies only measurably with $x$. To appreciate the
delicacy of this issue, consider the Lipschitz folding map
$F:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ defined by

$F(p)=\begin{cases}p\quad\text{if}\quad p\in H_{+}\\\
r_{\ell}(p)\quad\text{if}\quad p\in H_{-}\,,\end{cases}$

where $H_{\pm}\subset\mathbb{R}^{2}$ are the two halfplanes bounded by the
line $\ell:=\\{(x,x)|x\in\mathbb{R}\\}$, and
$r_{\ell}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is reflection across the
line $\ell$. The derivative $DF(x):\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$
preserves the direct sum decomposition
$\mathbb{R}^{2}=\mathbb{R}\oplus\mathbb{R}$ for a.e. $x\in\mathbb{R}^{2}$, but
$F$ is not a product mapping. We are able to exclude this kind of behavior in
the permutation $\sigma_{x}$ from (1.18) using the Pullback Theorem and a
short argument with differential forms.111Our first proof of the product
rigidity theorem used currents and the pullback theorem. We thank Mario Bonk
for the simpler dual argument using forms, which he suggested to us after
hearing the original proof.

Although the details are different, the starting point of the proof of Theorem
1.8 is also the exclusion of oscillatory behavior. If
$f:U\rightarrow\mathbb{H}_{n}^{\mathbb{C}}$ is as in the theorem, then the
Pansu differential
$D_{P}f(x):\mathfrak{h}_{n}^{\mathbb{C}}\rightarrow\mathfrak{h}_{n}^{\mathbb{C}}$
is a graded automorphism for a.e. $x\in U$ by assumption, and is therefore
either $J$-linear or $J$-antilinear (see [RR00] or Lemma 10.6). Using the
pullback theorem one can show that the map $f$ has to make a global choice
between these two possibilities. The remainder of the proof is based on more
standard PDE techniques, see Section 8.

We now give some indication of the proof of Theorem 1.11.

Let $G$ be a nonrigid Carnot group with graded Lie algebra $\mathfrak{g}$. By
[OW11, Theorem 1], the first layer $V_{1}^{\mathbb{C}}$ of the
complexification $\mathfrak{g}^{\mathbb{C}}$ contains a nonzero element $Z$
such that $[Z,\mathfrak{g}^{\mathbb{C}}]$ is a subspace of dimension at most
$1$. The first phase of the proof of Theorem 1.11 is to work out the algebraic
implications of this, which leads to the following trichotomy222The trichotomy
is a variation on unpublished work of the third author [Xie13a].:

  1. (a)

$G$ is isomorphic to $\mathbb{R}^{n}$, $\mathbb{H}_{n}$, or
$\mathbb{H}_{n}^{\mathbb{C}}$ for some $n\geq 1$.

  2. (b)

There is a linear subspace $\\{0\\}\subsetneq W\subsetneq V_{1}$ which is
invariant under graded automorphisms of $\mathfrak{g}$, such that
$[W,V_{j}]=\\{0\\}$ for all $j\geq 2$.

  3. (c)

$G$ is a product quotient: it is of the form $G=\tilde{G}/\exp(K)$ where
$\tilde{G}=\prod_{j=1}^{n}\tilde{G}_{j}$ and the $\tilde{G}_{j}$s are copies
of $\mathbb{H}_{m}$ or $\mathbb{H}_{m}^{\mathbb{C}}$ for some $m\geq 1$, and
$K\subset\tilde{\mathfrak{g}}$ is a linear subspace of the second layer
$\tilde{V}_{2}\subset\tilde{\mathfrak{g}}$ which satisfies several conditions,
see Definition 10.21.

In case (a) assertion (1) of Theorem 1.11 holds, so we are done. In case (b),
letting $\mathfrak{h}\subset\mathfrak{g}$ be the Lie subalgebra generated by
$W$, and $H:=\exp(\mathfrak{h})$, the
$\operatorname{Aut}(\mathfrak{g})$-invariance of $W$ implies that the Pansu
differential of $f$ respects the coset foliation of $H$, and then (2) follows
by an integration argument. In case (c), the main obstacle is that there are
automorphisms $\Phi:\tilde{G}/\exp(K)=G\rightarrow G=\tilde{G}/\exp(K)$
induced by an automorphism $\tilde{\Phi}:\tilde{G}\rightarrow\tilde{G}$ which
permutes the factors of $\tilde{G}=\prod_{j}\tilde{G}_{j}$, and one has to
show that the Pansu differentials of the map $f:U\rightarrow G$ induce a
constant permutation of the factors. A special case is when $K=\\{0\\}$, i.e.
$G=\tilde{G}$ is a product, and then the constancy of the permutation was
addressed in Theorem 1.1. The treatment of case (c) in general breaks down
into three main subcases depending on the dimension of $K$ and the type of the
factors $\tilde{G}_{j}$. The arguments are lengthy and hard to summarize
briefly; we refer the reader to Section 11 and to the beginning of Sections
12-14 for more explanation.

We now turn to Theorems 1.15 and 1.17.

The proof of (1.14) in the Euclidean case follows from a straightforward
mollification argument, so it is natural to try to use mollification to prove
analogous results in the Carnot group setting. The standard approach to
mollifying a map $f:G\rightarrow G^{\prime}$ would be as follows. Let
$\sigma_{1}$ be a symmetric compactly supported smooth probability measure on
$G$. For $\rho\in(0,\infty)$, $x\in G$, let
$\sigma_{\rho}:=(\delta_{\rho})_{*}\sigma_{1}$ be the pushforward of
$\sigma_{1}$ under the Carnot dilatation $\delta_{\rho}$, and
$\sigma_{x,\rho}:=(\ell_{x})_{*}\sigma_{\rho}$ be the pushforward of
$\sigma_{\rho}$ under left translation by $x$. One defines the mollified
mapping $f_{\rho}:G\rightarrow G^{\prime}$ by letting $f_{\rho}(x)$ to be the
average of the pushforward measure $f_{*}(\sigma_{x,\rho})$, where the average
is defined by treating $G^{\prime}$ as a linear space, for instance by
identifying $G^{\prime}$ with its Lie algebra via the exponential map.
Unfortunately, such a mollification introduces uncontrollable error terms, a
fact which was pointed out in [Vod07] (however, see [Dai99] for a proof of
some analytical results using standard mollification in the special case of
the Heisenberg group). Instead of using standard averaging to define
$f_{\rho}(x)$ we use a center of mass construction due to Buser-Karcher which
is compatible with left translation and Carnot dilation [BK81]. This avoids
both the uncontrollable error terms and the limitation to forms of codegree
$1$ faced by the method of [Vod07]. The main consequence is the following
approximation theorem, which is the central technical result in this paper:

######  Theorem 1.19.

Let $G$, $G^{\prime}$ be Carnot groups where $G$ has dimension $N$ and
homogeneous dimension $\nu$. Let $f:G\supset U\rightarrow U^{\prime}\subset
G^{\prime}$ be a $W^{1,p}_{\operatorname{loc}}$-mapping between open subsets,
for some $p>\nu$. Then there is a family of smooth mappings
$\\{f_{\rho}:U\rightarrow G^{\prime}\\}_{\rho>0}$ with the following
properties:

  * •

$f_{\rho}\rightarrow f$ locally uniformly.

  * •

If $\omega\in\Omega^{k}(U^{\prime})$, $\eta\in\Omega^{N-k}_{c}(U)$ are smooth
differential forms whose weights satisfy
$\operatorname{wt}(\omega)+\operatorname{wt}(\eta)\leq-\nu$, then

(1.20) $\int_{U}f_{P}^{*}\omega\wedge\eta=\lim_{\rho\rightarrow
0}\int_{U}f_{\rho}^{*}\omega\wedge\eta\,.$

Theorem 1.15 follows immediately from Theorem 1.19 by Stokes’ theorem.

Acknowledgements. We would like to thank Mario Bonk for suggesting a rigidity
argument using differential forms, which simplified an earlier proof based on
currents. We would also like to thank Misha Kapovich for pointing out that the
center of mass construction we were using had been discovered long ago by
Buser-Karcher.

##  2\. Preliminaries

We collect some background material and conventions.

###  2.1. Carnot groups

We recommend the survey [LD17] for a discussion of the material here.

Without explicit mention will equip all Lie groups with Haar measure. As is
customary, we will conflate two standard definitions of the Lie algebra of a
Lie group $G$ – the tangent space at the identity $T_{e}G$ and the space of
left invariant vector fields; likewise we will conflate a linear subspace of
$\mathfrak{g}$ with the corresponding left invariant subbundle of the tangent
bundle $TG$. We will typically use the same notation for a Lie group
homomorphism $G\rightarrow G^{\prime}$ and the induced homomorphism of Lie
algebras $\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$.

######  Definition 2.1.

A Carnot group is a simply connected Lie group $G$ together with a
decomposition of its Lie algebra $\mathfrak{g}$ into linear subspaces

$\mathfrak{g}=V_{1}\oplus\ldots\oplus V_{s}$

such that:

  * •

$[V_{i},V_{j}]\subset V_{i+j}$ for $1\leq i,j\leq s$.

  * •

The horizontal space (or first layer) $V_{1}$ generates $\mathfrak{g}$.

A graded homomorphism of Carnot groups
$(G,\mathfrak{g}=\oplus_{j=1}^{s}V_{j})$,
$(G^{\prime},\mathfrak{g}^{\prime}=\oplus_{k=1}^{s^{\prime}}V_{k}^{\prime})$
is a Lie group homomorphism $G\rightarrow G^{\prime}$ such that the induced
homomorphism of Lie algebras $\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$
carries $V_{j}$ into $V^{\prime}_{j}$ for all $1\leq j\leq s$. We will use
$\operatorname{Aut}(G)$ to denote the group of graded automorphisms of a
Carnot group.

Properties of a Carnot group $(G,\mathfrak{g}=\oplus_{j}V_{j})$:

  * •

$G$ is nilpotent, and $V_{j+1}=[V_{1},V_{j}]$ for all $1\leq j\leq s$.

  * •

There is a $1$-parameter group of Carnot group automorphisms
$\\{\delta_{r}:G\rightarrow G\\}$ such that $\delta_{r}$ scales $V_{j}$ by the
factor $r^{j}$. We refer to these automorphisms as (Carnot) dilations.

  * •

Without explicit mention, we will assume that the first layer
$V_{1}\subset\mathfrak{g}$ is equipped with an inner product, and extend this
to a left-invariant Riemannian metric on the left-invariant subbundle of $TG$
determined by $V_{1}$.

  * •

The Carnot-Caratheodory distance $d_{CC}$ between two points $x_{0},x_{1}\in
G$ is the infimal length of a $C^{1}$-path $\gamma:[0,1]\rightarrow G$ with
$\gamma(i)=x_{i}$, $\gamma^{\prime}(t)\in V_{1}$ for all $t\in[0,1]$, and the
length is taken with respect to the given Riemannian metric on $V_{1}$.

  * •

The topology induced by the metric $d_{CC}$ is the same as the topology coming
from $G$ as a manifold.

  * •

$d_{CC}$ is invariant under left translation, and scaled by dilations:

$d_{CC}(\delta_{r}(x),\delta_{r}(x^{\prime}))=rd_{CC}(x,x^{\prime})\,.$

  * •

The homogeneous dimension of $G$ is $\nu:=\sum_{j}j\dim V_{j}$; this is the
Hausdorff dimension of $(G,d_{CC})$, and there is a constant $C\neq 0$ such
that for every $x\in G$, $r\in[0,\infty)$, the $\nu$-dimensional Hausdorff
measure of the $r$-ball satisfies:

$\mathcal{H}^{\nu}(B(x,r))=Cr^{\nu}\,.$

######  Definition 2.2.

The $n^{th}$ real Heisenberg group $\mathbb{H}_{n}$ is the Carnot group with
graded Lie algebra $\mathfrak{h}_{n}=V_{1}\oplus V_{2}$, where
$V_{1}=\mathbb{R}^{2n}$, $V_{2}=\mathbb{R}$, and the bracket $X,Y\in V_{1}$ is
given by $[X,Y]=\omega(X,Y)\in V_{2}$; here $\omega=\sum_{i}dx_{2i-1}\wedge
dx_{2i}$ is the standard symplectic form on $\mathbb{R}^{2n}$.

The $n^{th}$ complex Heisenberg group $\mathbb{H}_{n}^{\mathbb{C}}$ is the
complexification of $\mathbb{H}_{n}$; its graded Lie algebra is
$\mathfrak{h}_{n}^{\mathbb{C}}:=\mathfrak{h}_{n}\otimes\mathbb{C}$; more
concretely, it has the grading
$\mathfrak{h}_{n}^{\mathbb{C}}=V_{1}^{\mathbb{C}}\oplus V_{2}^{\mathbb{C}}$,
where $V_{1}^{\mathbb{C}}=V_{1}\otimes\mathbb{C}=\mathbb{C}^{2n}$,
$V_{2}^{\mathbb{C}}=V_{2}\otimes\mathbb{R}=\mathbb{C}$, and for $X,Y\in
V_{1}^{\mathbb{C}}$ the bracket is given by
$[X,Y]=\omega^{\mathbb{C}}(X,Y)\in\mathbb{C}=V_{2}^{\mathbb{C}}$, and
$\omega^{\mathbb{C}}$ is the extension of $\omega$ to a $\mathbb{C}$-bilinear
form $V_{1}^{\mathbb{C}}\times V_{1}^{\mathbb{C}}\rightarrow\mathbb{C}$.

###  2.2. Sobolev mappings

Starting from the mid-90s the theory of Sobolev spaces in metric measure
spaces advanced rapidly, so that basic results were established in great
generality. However in this paper we only need results in the Carnot group
case, where they were (almost all) already known, and in most cases
technically simpler to establish. We refer the reader to [Pan89, KR95, MM95,
Hei01, HaK00, Vod07].

Let $G$, $G^{\prime}$ be Carnot groups, where $G$ has homogeneous dimension
$\nu$ and grading $\mathfrak{g}=\oplus_{j}V_{j}$.

######  Definition 2.3.

If $1\leq q\leq\infty$, $U\subset G$ is open, then
$W^{1,q}_{\operatorname{loc}}(U)$ is the set of elements $u\in
L^{q}_{\operatorname{loc}}(U)$ such that for every horizontal left invariant
vector field $X\in V_{1}$, distributional directional derivative $Xu$ belongs
to $L^{q}_{\operatorname{loc}}(U)$. We let $|D_{h}u|\in
L^{q}_{\operatorname{loc}}(U)$ be the pointwise supremum of the directional
derivatives $Xu$, where $X\in V_{1}$ and $|X|=1$.

We remark that a variety of other definitions of Sobolev spaces have been
developed in Hajlasz, Shanmugalingam, Cheeger, and they are all equivalent to
Definition 2.3 in the case of Carnot groups equipped with Haar measure.

######  Lemma 2.4.

[HaK00, Section 11 and Theorem 5.1] There are constants $\Lambda>1$,
$C_{q}<\infty$ such that for every $x\in G$, $r>0$, $u\in W^{1,q}(B(x,\Lambda
r))$

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B(x,r)}|u-\bar{u}|\leq
Cr\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B(x,\Lambda
r)}|D_{h}u|^{q}d\mu\right)^{\frac{1}{q}}$

where $\bar{u}:=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B(x,r)}u\,d\mu$. If $q>\nu$, then $u$ has a
continuous representative, and it satisfies

$\sup_{y\in B(x,r)}|u(y)-\bar{u}|\leq
C^{\prime}r\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B(x,\Lambda
r)}|D_{h}u|^{q}d\mu\right)^{\frac{1}{q}}$

where $C^{\prime}=C^{\prime}(q)$.

######  Definition 2.5.

Suppose $1\leq q\leq\infty$ and $U\subset G$ is open. Then a mapping
$f:U\rightarrow G^{\prime}$ is in $W^{1,q}_{\operatorname{loc}}$ if there
exists $\rho\in L^{q}_{\operatorname{loc}}(U)$ such that for every $z\in
G^{\prime}$, the composition $d_{z}\circ f:U\rightarrow\mathbb{R}$ belongs to
$W^{1,q}_{\operatorname{loc}}(U)$ and its horizontal differential satisfies
$|D_{h}(d_{z}\circ f)|\leq\rho$ almost everywhere. Here
$d_{z}:=d_{CC}(z,\cdot)$ denotes the distance function from $z$.

In this paper we will consider $W^{1,q}_{\operatorname{loc}}$-mappings
$f:U\rightarrow G^{\prime}$ where $q>\nu$; by Lemma 2.4 these are continuous.

Fix a mapping $f:G\supset U\rightarrow G^{\prime}$, where $U\subset G$ is
open. For all $x\in U$, $0<r<\infty$, define

$f_{x}=\ell_{f(x)^{-1}}\circ f\circ\ell_{x}\,,\qquad
f_{x,r}=\delta_{r^{-1}}\circ f_{x}\circ\delta_{r}\,,$

where $\ell_{h}$, $\delta_{r}$ denote left translation by $h$ and Carnot
dilation, respectively.

######  Definition 2.6.

$f$ is Pansu differentiable at $x\in U$ if there is a graded group
homomorphism $\Phi:G\rightarrow G^{\prime}$ such that

$f_{x,r}\longrightarrow\Phi$

uniformly on compact sets as $r\rightarrow 0$. We refer to $\Phi$, as well as
the associated homomorphism of graded Lie algebras
$\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$ as the Pansu differential of
$f$ at $x$. We use $D_{P}f(x)$ to denote both objects $D_{P}f(x):G\rightarrow
G^{\prime}$, $D_{P}f(x):\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$.

######  Theorem 2.7.

If $f:U\rightarrow G^{\prime}$ is a $W^{1,q}_{\operatorname{loc}}$ mapping for
some $q>\nu$, then $f$ is Pansu differentiable almost everywhere, and there is
a universal constant $C$ such that for every $z\in G^{\prime}$, we have

$|D_{h}(d_{z}\circ f)|(x)\leq C\|D_{P}f(x)|_{V_{1}}\|$

where $D_{P}f(x)|_{V_{1}}:V_{1}\rightarrow V_{1}^{\prime}$ is the restriction
of the Pansu differential
$D_{P}f(x):\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$ to the first layer.

######  Lemma 2.8.

There is a constant $\Lambda\geq 1$ such that if $q>\nu$ and $f:B(x,\Lambda
r)\rightarrow G^{\prime}$ belongs to $W^{1,q}$, then for all $y\in B(x,r)$ we
have

$d_{CC}(f(y),f(x))\lesssim r\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B(x,\Lambda
r)}\|D_{P}f|_{V_{1}}\|^{q}\,d\mu\right)^{\frac{1}{q}}\,.$

###### Proof.

Since $d_{CC}(f(y),f(x))=(d_{f(x)}\circ f)(y)$, the lemma follows from Lemma
2.4 and Theorem 2.7. ∎

###  2.3. Quasiconformal and quasisymmetric mappings

We let $G$, $G^{\prime}$ denote Carnot groups as before, where $G$ has
homogeneous dimension $\nu>1$.

######  Definition 2.9.

Let $f:X\rightarrow Y$ be a homeomorphism between metric spaces, and
$\eta:[0,\infty)\rightarrow[0,\infty)$ be a homeomorphism. For $x\in X$, let

$H_{f}(x):=\limsup_{r\rightarrow 0}\frac{\sup\\{d(f(x),f(y))\;|\;d(x,y)\leq
r\\}}{\inf\\{d(f(x),f(y))\;|\;d(x,y)\geq r\\}}\,,$

and $H_{f}:=\sup_{x}H_{f}(x)$. Then $f$ quasiconformal if $H_{f}<\infty$, and
$f$ is $\eta$-quasisymmetric if for any distinct points $p,x,y\in X$,

$\frac{d(f(p),f(x))}{d(f(p),f(y))}\leq\eta\left(\frac{d(p,x)}{d(p,y)}\right)\,.$

We will need the following properties of quasiconformal and quasisymmetric
homeomorphisms:

######  Theorem 2.10 (See [HK95, HK98]).

Let $f:G\supset U\rightarrow U^{\prime}\subset G^{\prime}$ be a homeomorphism.
Then

  1. (1)

If $f$ is quasiconformal, then it is locally $\eta$-quasisymmetric for some
$\eta=\eta(H_{f})$, i.e. for every $x\in U$, there is an $r=r(x)>0$ such that
$f\mbox{\Large$|$\normalsize}_{B(x,r)}:B(x,r)\rightarrow f(B(x,r))$ is an
$\eta$-quasisymmetric homeomorphism.

  2. (2)

Every global quasiconformal homeomorphism $f:G\rightarrow G^{\prime}$ is
$\eta=\eta(H_{f})$-quasisymmetric.

  3. (3)

If $f$ is quasiconformal, then $f\in W^{1,p}_{\operatorname{loc}}$ for some
$p=p(H_{f})>\nu$.

  4. (4)

If $f$ is quasiconformal, then the sign of the determinant of $D_{P}f(x)$ is
locally constant, and agrees with the local degree of $f$.

###### Proof.

For (1) and (2) see Remark 1.8(b) and Theorem 1.7 in [HK95].

(3). This is based on Gehring’s lemma, and is treated in Section 7 of [HK98].
However, in the Carnot group case, one may equally well use the
differentiability and absolute continuity results from [Pan89] or [MM95] and
Gehring’s lemma instead.

(4). Since $f$ is a homeomorphism, so is $f_{x,r}$ for every $r\in(0,\infty)$.
If $x$ is a point of Pansu differentiability, then $f_{x,r}\rightarrow
D_{P}f(x)$ uniformly on compact sets, and therefore its local degree near
$e\in G$ is constant for small $r$, and coincide with the local degree of
$D_{P}f(x)$, which is the sign of the determinant of
$D_{P}f(x):\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$. This implies the
local constancy of the sign of $\det(D_{P}f(x))$.

∎

###  2.4. Miscellaneous

The following Lemma is the distributional version of the statement: “$g$
constant along a line implies $h\circ g$ constant on this line”.

We recall that the flow $\tau_{s}$ of a left invariant vector field $X$ is
given by right translation by the $1$-parameter group generated by $X$, i.e.
$\tau_{s}(p)=p\exp(sX)$ see [Lee09], Thm. 5.74 or [Mic08, 4.17]. We also
recall that Carnot groups are unimodular, since they are nilpotent; see, for
example, [Mic08, Corollary 14.3]

######  Lemma 2.11.

Let $G$ be a unimodular Lie group with bi-invariant Haar measure $\mu$,
$U\subset G$ be an open subset, $g\in L^{1}_{loc}(U;\mathbb{R}^{l})$. For
every $X\in\mathfrak{g}$, if $\tau_{s}:=r_{\exp sX}$ denotes $1$-parameter
group of right translations generated by $X$, then:

  1. (1)

We have

(2.12) $\int_{U}g\,X\varphi\,d\mu=0\quad\forall\varphi\in C_{c}^{\infty}(U)$

if and only if for every compact set $V\subset U$ there exists an
$\varepsilon>0$ such that

(2.13) $g\circ\tau_{s}=g$ a.e. on $V$ $\forall
s\in(-\varepsilon,\varepsilon)$.

  2. (2)

Let $h:\mathbb{R}^{l}\to\mathbb{R}$ be a Borel function with $|h(t)|\leq
C_{1}|t|+C_{2}$. Then (2.12) implies that

(2.14) $\int_{U}(h\circ g)\,X\varphi\,d\mu=0\quad\forall\varphi\in
C_{c}^{\infty}(U).$

###### Proof.

The main point is that (2.12) is equivalent to

(2.15)
$\int_{U}g\,(\varphi\circ\tau_{s}-\varphi)\,d\mu=0\quad\forall\varphi\in
C_{c}^{\infty}(U),s\in(-\varepsilon_{\varphi},\varepsilon_{\varphi})$

where $\varepsilon_{\varphi}$ depends on the support of $\varphi$, $U$ and
$X$. Since $\mu$ is bi-invariant (2.15) is equivalent to

$\int_{U}(g\circ\tau_{-s}-g)\,\varphi\,d\mu=0\quad\forall\varphi\in
C_{c}^{\infty}(U),s\in(-\varepsilon_{\varphi},\varepsilon_{\varphi})$

or to $g\circ\tau_{s}=g$ a.e. This concludes the proof of the first assertion.

To prove the second assertion note that the condition $h$ is Borel ensures
that $h\circ g$ is measurable. The growth condition implies that $h\circ g\in
L^{1}_{loc}(U)$. Now apply $h$ to both sides of (2.13) and use the equivalence
of (2.13) and (2.12). ∎

Now let $G$, $G^{\prime}$ be Carnot groups, where $G$ has homogeneous
dimension $\nu$. Let $\mathfrak{h}\subset\mathfrak{g}$,
$\mathfrak{h}^{\prime}\subset\mathfrak{g}^{\prime}$ be Lie subalgebras, where
$\mathfrak{h}$ is horizontally generated (i.e. $\mathfrak{h}$ is generated by
$\mathfrak{h}\cap V_{1}$), and let $H:=\exp(\mathfrak{h})$,
$H^{\prime}:=\exp(\mathfrak{h}^{\prime})$. Note that $H$, $H^{\prime}$ are
closed subgroups since the exponential maps are diffeomorphisms.

######  Lemma 2.16.

There exists $\Lambda\in[1,\infty)$ such that if $x\in G$, $r\in(0,\infty)$,
$f:B(x,\Lambda r)\rightarrow G^{\prime}$ is a
$W^{1,p}_{\operatorname{loc}}$-mapping for some $p>\nu$, and the Pansu
differential satisfies $D_{P}f(x)(\mathfrak{h})\subset\mathfrak{h}^{\prime}$,
then for every $y\in G$ the image of $yH\cap B(x,r)$ under $f$ is contained in
a single left coset of $H^{\prime}$.

###### Proof.

This is a slight variation on [Xie13b, Proposition 3.4].

Since $\mathfrak{h}$ is horizontally generated, it is invariant under the
dilation $\delta_{r}:G\rightarrow G$ for every $r\in(0,\infty)$. Therefore
there is a direct sum decomposition
$\mathfrak{h}=\oplus_{j}V_{j}^{\mathfrak{h}}$ where
$V_{j}^{\mathfrak{h}}:=\mathfrak{h}\cap V_{j}$, and the pair
$(H,\mathfrak{h}=\oplus_{j}V_{j}^{\mathfrak{h}})$ is a Carnot group. We may
equip the first layer $V_{1}^{\mathfrak{h}}$ with the restriction of the inner
product on $V_{1}$, and let $d_{CC}^{H}$ be the corresponding Carnot-
Caratheodory metric on $H$.

Note that there is a constant $C_{1}$ such that

$d_{CC}(x,x^{\prime})\leq d_{CC}^{H}(x,x^{\prime})\leq
C_{1}d_{CC}(x,x^{\prime})\,;$

the first inequality follows directly from the definitions, while the second
follows for pairs $(x,x^{\prime})=(e,y)$ with $d_{CC}(e,y)=1$ by compactness,
and for general pairs by applying translation and scaling.

Let $\Lambda:=2C_{1}\Lambda_{0}+1$, where $\Lambda_{0}$ is the constant given
by Lemma 2.17. Suppose $x$, $r$, and $f$ are as in the statement of Lemma
2.16; by rescaling and left translation we may assume that $x=e$ and $r=1$.
Arguing as in [Xie13b, Proposition 3.4], for every path $[0,1]\rightarrow
B(e,\Lambda)$ with constant horizontal velocity, the image $f(\gamma([0,1]))$
is contained in a single coset of $H^{\prime}\subset G^{\prime}$.

Choose $y\in G$, and $x,x^{\prime}\in yH\cap B(e,1)$. Then
$d_{CC}(x,x^{\prime})<2$, so $d_{CC}^{H}(x,x^{\prime})<2C_{1}$. Therefore by
Lemma 2.17, for every $\varepsilon>0$ there is a piecewise horizontal path
$\gamma:[0,1]\rightarrow B(e,\Lambda)$ with $\gamma(0)=x$ and $\gamma(1)\in
B(x^{\prime},\varepsilon)$; since $f(\gamma([0,1]))$ lies in a single coset of
$H^{\prime}$ and $\varepsilon$ is arbitrary, by continuity it follows that $x$
and $x^{\prime}$ belong to the same coset of $H^{\prime}$. ∎

######  Lemma 2.17.

Let $H$ be a Carnot group. Then there exists $\Lambda_{0}>1$ with the
following property.

Let $\Gamma$ be the collection of paths $\gamma:[0,1]\rightarrow H$ such that
$\gamma(0)=e$ and $\gamma^{\prime}$ is piecewise constant and horizontal, i.e.
there exists a partition $0=t_{0}<\ldots<t_{n}=1$, and for every $1\leq j\leq
n$ elements $X_{j}\in V_{1}$, $h_{j}\in H$, such that

$\gamma(t)=h_{j}\exp(ta_{j}X_{j})\quad\text{for all}\quad t\in[t_{j-1},t_{j}]$

for all $1\leq j\leq k$.

Then the set

$\\{\gamma(1)\mid\gamma\in\Gamma\,,\;\gamma([0,1])\subset
B(e,\Lambda_{0})\\}\cap B(e,1)$

is dense in $B(e,1)$.

###### Proof.

Let $\hat{H}$ be the closure of the subgroup generated by the collection of
horizontal $1$-parameter subgroups. Then $\hat{H}$ is a Lie subgroup, and its
Lie algebra $\hat{\mathfrak{h}}$ contains $V_{1}$, so $\hat{H}=H$. It follows
that the collection $\\{\gamma(1)\mid\gamma\in\Gamma\\}$ is dense in $H$.

Now choose a finite collection
$\Gamma_{1}=\\{\gamma_{1},\ldots,\gamma_{n}\\}\subset\Gamma$ such that the
endpoints are $\frac{1}{2}$-dense in $B(e,1)$, i.e. for every $x\in B(e,1)$ we
have $d(x,\\{\gamma_{j}(1)\\}_{1\leq j\leq n}\\})<\frac{1}{2}$. Then for some
$\Lambda_{1}$ and every $1\leq j\leq n$ the image $\gamma_{j}([0,1])$ is
contained in the ball $B(e,\Lambda_{1})$.

Pick $y\in B(e,1)$. By induction, there is a sequence $y_{0}=e,y_{1},\ldots$
and for every $j\geq 1$ an element $i_{j}\in\\{1,\ldots,n\\}$ such that
$d(y_{j},y)<2^{-j}$ for all $j\geq 0$, and the path
$\ell_{y_{j}}\circ\delta_{2^{-j}}\circ\gamma_{i_{j}}$ joins $y_{j}$ to
$y_{j+1}$. By concatenating, for every $j$ we obtain a path $\gamma\in\Gamma$
where $\gamma(1)=y_{j}$ and $\gamma([0,1])\subset B(e,\Lambda_{0})$, where
$\Lambda_{0}:=2\Lambda_{1}$. ∎

##  3\. Differential forms on Carnot groups

The purpose of this section is to review some (mostly) standard material for
the convenience of the reader, and also to discuss some special features of
the de Rham complex on Carnot groups; this sets the stage for the Pullback
Theorem in the next section, which asserts that the distributional exterior
derivative commutes with Pansu pullback, but only if the form which is pulled
back and the test form satisfy certain restriction on their weights.

In Subsection 3.1 we begin by reviewing basic facts from multilinear algebra,
in particular the properties of the interior product. Then in Subsection 3.2
we recall some formulae for the exterior derivative of interior products of
the volume form in a unimodular Lie group. This will be useful to obtain
interesting low codegree test forms for the application of the Pullback
Theorem. We also recall that exterior differentiation, viewed as a linear map
on $\Lambda^{*}\mathfrak{g}$, commutes with pullback by Lie algebra
homomorphisms.

In Subsection 3.3 we then focus on Carnot groups and define two key notions:
the weight of of a differential from (which reflects the action of the
dilation map $\delta_{r}$) and the Pansu pullback.

###  3.1. Exterior algebra and interior products

Here we recall some basic facts about interior products. This will be useful
for constructing closed left-invariant forms of low codegree that will be
useful in the application of the Pullback Theorem. In this subsection we work
purely in the setting of multilinear algebra. Exterior differentiation of
forms will be considered in the next subsection.

Given a finite-dimensional vector space $V$ over $\mathbb{R}$ we denote the
space of $m$-vectors by $\Lambda_{m}V$ and the space of alternating $m$-linear
forms by $\Lambda^{m}V$. The elements of $\Lambda^{m}V$ can be identified with
linear functionals on $\Lambda_{m}V$ (or $m$-covectors).

The interior product of a vector $X\in V$ and a $p$-form
$\alpha\in\Lambda^{p}V$ is the $(p-1)$-form $i_{X}\alpha$ given by

(3.1) $i_{X}\alpha(X_{2},\ldots,X_{p})=\alpha(X,X_{2},\ldots,X_{p}).$

Using the dual pairing of vectors and covectors this can be rewritten as

$i_{X}\alpha(Z)=\alpha(X\wedge Z).$

For $q\leq p$, $X\in\Lambda_{q}V$ and $\alpha\in\Lambda^{p}V$ one defines
$i_{X}\alpha\in\Lambda^{p-q}V$ by

(3.2) $(i_{X}\alpha)(Z)=\alpha(X\wedge Z)\quad\forall Z\in\Lambda_{p-q}V.$

Then for $X\in\Lambda_{q}$ and $Y\in\Lambda_{q^{\prime}}$

$(i_{X}i_{Y}\alpha)(Z)=i_{Y}\alpha(X\wedge Z)=\alpha(Y\wedge X\wedge Z)$

and thus

(3.3) $i_{X}\circ i_{Y}=i_{Y\wedge X}\quad\forall
X\in\Lambda_{q},\,\,Y\in\Lambda_{q^{\prime}}$

The interior product satisfies the graded Leibniz rule

(3.4)
$i_{X}(\beta\wedge\gamma)=i_{X}\beta\wedge\gamma+(-1)^{\deg\beta}\beta\wedge
i_{X}\gamma\quad\forall X\in V.$

If $\omega$ is a top-degree form on $V$, then

(3.5) $\alpha\wedge
i_{X}\omega=i_{X}\alpha\wedge\omega=\alpha(X)\omega\,,\quad\forall\alpha\in\Lambda^{p}V\,,\quad
X\in\Lambda_{p}V$

This can easily be checked using the standard basis and dual basis and
verifying the assertion for $X$ and $\beta$ being simple vectors in terms of
theses basis vectors.

We finally briefly discuss the interior product on quotient spaces. Let
$\tilde{V}$ be a finite-dimensional vector space over $\mathbb{R}$, let
$K\subset\tilde{V}$ be a subspace and let $V=\tilde{V}/K$. We say that
$\beta\in\Lambda^{p}\tilde{V}$ annihilates $K$ if
$\beta(X_{1},\ldots,X_{p})=0$ whenever at least one of the $X_{i}$ satisfies
$X_{i}\in K$. If $\tilde{\alpha}\in\Lambda^{p}\tilde{V}$ annihilates $K$ then
it gives rise to a unique form $\alpha\in\Lambda^{p}V$ defined by

(3.6)
$\alpha(X_{1}+K,\ldots,X_{p}+K)=\tilde{\alpha}(X_{1},\ldots,X_{p})\quad\forall
X_{i}\in\tilde{V}.$

Conversely for every $\alpha\in\Lambda^{p}V$ there exists a unique lift
$\tilde{\alpha}\in\Lambda^{p}\tilde{V}$ which annihilates $K$ and satisfies
(3.6). Consider such $\alpha$ and $\tilde{\alpha}$ as well as $X\in\tilde{V}$.
Then

$\widetilde{i_{X+K}\alpha}=i_{X}\tilde{\alpha}.$

To simplify the notation we sometimes use the notation $i_{X}\alpha$ for the
form $i_{X+K}\alpha$ and identify $i_{X}\alpha$ and $i_{X}\tilde{\alpha}$.

###  3.2. Exterior derivatives of left-invariant forms with low codegree

In this subsection we recall some useful formulae for exterior derivatives of
forms on unimodular Lie groups, particularly for low codegree forms defined
using interior products of vectors with the volume form. Recall that Carnot
groups are unimodular; we will only use the results in this section in the
case of Carnot groups, but for the sake of logical clarity we have formulated
them for unimodular groups, because they hold in such generality.

Let $G$ be a Lie group. We identify the corresponding algebra $\mathfrak{g}$
with the space of left invariant vectorfields on $G$, i.e., with the left
invariant sections in the tangential bundle $TG$. Similarly we identify the
space of $1$-forms $\Lambda^{1}\mathfrak{g}$ with the space of left-invariant
sections of the cotangent bundle $T^{*}G$ and the spaces of $p$-vectors in
$\mathfrak{g}$ and $p$-forms on $\mathfrak{g}$ with the spaces of left-
invariant sections of the corresponding tensor bundles.

For a vectorfield $X$ let $\mathcal{L}_{X}$ denote the Lie derivative with
respect to $X$. We will use Cartan’s formula

(3.7) $\mathcal{L}_{X}\alpha=d(i_{X}\alpha)+i_{X}d\alpha$

as well as the relation

(3.8) $i_{[X,Y]}=[\mathcal{L}_{X},i_{Y}],$

see, e.g., [Mic08, Theorem 9.9].

######  Lemma 3.9.

Let $G$ be a unimodular group and let $\omega$ be a bi-invariant volume form
on $G$. Then

(3.10) $\displaystyle d(i_{X}\omega)$ $\displaystyle=$
$\displaystyle\mathcal{L}_{X}\omega=0\quad\forall X\in\mathfrak{g},$ (3.11)
$\displaystyle d(i_{X}i_{Y}\omega)$ $\displaystyle=$ $\displaystyle
i_{[X,Y]}\omega\quad\forall X,Y\in\mathfrak{g},$ (3.12) $\displaystyle
d(i_{X}i_{Y}i_{Z}\omega)$ $\displaystyle=$ $\displaystyle
i_{A}\omega\quad\forall X,Y,Z\in\mathfrak{g},\quad\hbox{where}$ (3.13)
$\displaystyle A$ $\displaystyle=$ $\displaystyle
X\wedge[Y,Z]+Y\wedge[Z,X]+Z\wedge[X,Y].$

Finally if

(3.14)
$X,X^{\prime},X^{\prime\prime},Z\in\mathfrak{g}\quad\hbox{and}\quad[X,Z]=[X^{\prime},Z]=[X^{\prime\prime},Z]=0$

then

(3.15)
$d(i_{Z}i_{X^{\prime\prime}}i_{X^{\prime}}i_{X}\omega)=-i_{Z}d(i_{X^{\prime\prime}}i_{X^{\prime}}i_{X}\omega)$

###### Proof.

Since $\omega$ has top degree it is closed and thus Cartan’s formula gives

(3.16) $d(i_{X}\omega)=\mathcal{L}_{X}\omega\quad\forall X\in\mathfrak{g}.$

Since the flow of a left-invariant vector field is given by right translation
and since $\omega$ is also right-invariant we have $\mathcal{L}_{X}\omega=0$.

To show (3.11) we use Cartan’s formula and (3.10) to get

$\displaystyle d(i_{X}i_{Y}\omega)$ $\displaystyle=$
$\displaystyle\mathcal{L}_{X}(i_{Y}\omega)-i_{X}d(i_{Y}\omega)=\mathcal{L}_{X}(i_{Y}\omega)$
$\displaystyle=$
$\displaystyle\mathcal{L}_{X}(i_{Y}\omega)-i_{Y}(\mathcal{L}_{X}\omega)\underset{\eqref{eq:commutator_Lie_derivative}}{=}i_{[X,Y]}\omega.$

To prove (3.12) we start again from Cartan’s formula

$\displaystyle d(i_{X}i_{Y}i_{Z}\omega)$ $\displaystyle=$
$\displaystyle\mathcal{L}_{X}(i_{Y}i_{Z}\omega)-i_{X}d(i_{Y}i_{Z}\omega)$
$\displaystyle=$
$\displaystyle[\mathcal{L}_{X},i_{Y}]i_{Z}\omega+i_{Y}\mathcal{L}_{X}i_{Z}\omega-
i_{X}i_{[Y,Z]}\omega$ $\displaystyle=$ $\displaystyle
i_{[X,Y]}i_{Z}\omega+i_{Y}[\mathcal{L}_{X},i_{Z}]\omega+i_{Y}i_{Z}\underbrace{\mathcal{L}_{X}\omega}_{=0}-i_{X}i_{[Y,Z]}\omega.$

Now the formula for $A$ follows from (3.8) and (3.3).

To prove (3.15) we use that $Z$ commutes with every vector in $\mathfrak{g}$.
Thus successive application of (3.8) and (3.10) give

$\mathcal{L}_{Z}(i_{X^{\prime\prime}}i_{X^{\prime}}i_{X}\omega)=i_{X^{\prime\prime}}\mathcal{L}_{Z}(i_{X^{\prime}}i_{X}\omega)=\ldots=i_{X^{\prime\prime}}i_{X^{\prime}}i_{X}\mathcal{L}_{Z}\omega=0.$

Now (3.15) follows from Cartan’s formula (3.7). ∎

The exterior derivative of a left-invariant form is left-invariant since left
translation is smooth and hence commutes with exterior differentiation. Thus
the exterior derivative induces a linear map on $\Lambda^{*}\mathfrak{g}$ via

(3.17) $d\alpha:=(d\tilde{\alpha})(e).$

where $\tilde{\alpha}$ is the left-invariant extension of $\alpha$.

######  Lemma 3.18.

Let $G$ be a Lie group and let $\alpha\in\Lambda^{k}\mathfrak{g}$. Then

(3.19) $d\alpha(X_{0},\ldots,X_{k})=\sum_{0\leq i<j\leq
k}(-1)^{i+j}\alpha([X_{i},X_{j}],X_{0},\ldots,\widehat{X}_{i},\ldots,\widehat{X}_{j},\ldots,X_{k}).$

###### Proof.

See [Mic08, Lemma 14.14]. ∎

For $1$-forms this reduces to

(3.20) $d\alpha(X_{0},X_{1})=-\alpha([X_{0},X_{1}]).$

Let $\Phi\in\operatorname{Aut}(\mathfrak{g})$. We define the pull-back of
$\alpha\in\Lambda^{k}\mathfrak{g}$ by
$(\Phi^{*}\alpha)(X_{1},\ldots,X_{k}):=\omega(\Phi(X_{1}),\ldots,\Phi(X_{k}))$
for $X_{i}\in\mathfrak{g}$. Then (3.19) implies that

(3.21) $d\Phi^{*}\omega=\Phi^{*}d\omega.$

Note that if $\Phi$ is the tangent map of group homomorphism, then the above
definition of pullback agrees with the usual one, i.e. one first extends
$\alpha\in\Lambda^{k}\mathfrak{g}$ to a left-invariant form and applies the
usual pullback operation.

###  3.3. Differentials forms, scaling, and weights

In this subsection we define a weight for multivectors and differential forms.
The weight is determined by the action of the dilation map $\delta_{r}$. In
particular the weight of a vector $X$ in the $j$-th layer $V_{j}$ of
$\mathfrak{g}$ is $j$ and the weight behaves naturally under wedge product,
vector-form duality and pull-back by graded isomorphims.

The importance of the weight stems for the fact that our main technical tool,
the Pullback Theorem (Theorem 4.1), holds only under a restriction on the
weights of the forms involved. This is a key difference between the pullback
by the Pansu differential which is crucial in our setting and the usual
pullback of forms by smooth maps.

Let $G$ be a Carnot group of step $s$ with Lie algebra $\mathfrak{g}$. We
retain the notation from before:

  * •

$N$ and $\nu$ are the dimensions and homogeneous dimensions of $G$
respectively.

  * •

$\delta_{r}:G\rightarrow G$ is dilation by $r$; we use the same notation for
the dilation on $\mathfrak{g}$.

  * •

$\\{X_{i}\\}_{1\leq i\leq N}$, $\\{\theta_{i}\\}_{1\leq i\leq N}$ are dual
graded bases of $\mathfrak{g}$ and one-forms
$\mathfrak{g}^{*}=\Lambda^{1}\mathfrak{g}$, i.e., each $X_{i}$ lies in one of
the layers $V_{1},\ldots,V_{s}$.

  * •

If $I=\\{i_{1},\ldots,i_{k}\\}\subset\\{1,\ldots,N\\}$, then
$\theta_{I}:=\theta_{i_{1}}\wedge\ldots\wedge\theta_{i_{k}}$.

As usual, we will identify $X_{i}$ and $\theta_{I}$ with the corresponding
left-invariant vector fields and forms on $G$ when convenient.

We first define weights on the graded algebras of multi-vectors
$\Lambda_{*}\mathfrak{g}=\oplus_{k=1}^{N}\Lambda_{k}\mathfrak{g}$ and
alternating forms
$\Lambda^{*}\mathfrak{g}=\oplus_{k=1}^{N}\Lambda^{k}\mathfrak{g}$.

######  Definition 3.22.

A $k$-form $\alpha\in\Lambda^{k}\mathfrak{g}$ or $k$-vector
$\xi\in\Lambda_{k}(\mathfrak{g})$ is homogeneous with weight $w$ if
$(\delta_{r})_{*}\alpha=r^{w}\,\alpha$ or $(\delta_{r})_{*}\xi=r^{w}\,\xi$,
respectively, for all $r\in(0,\infty)$.

Note that the weight of a non-vanishing homogeneous $k$-vector or $k$-form is
uniquely defined. We denote it by $\operatorname{wt}(\xi)$ and
$\operatorname{wt}(\alpha)$. Note also that the zero form and the zero vector
are homogeneous with any weight. Hence they do not have a well-defined weight.

######  Remark 3.23.

Most of the facts about weights and homogeneity discussed below follow the
behavior of diagonalizable actions under dualizing, tensor products, and
contraction. However, we give a calculation based treatment.

If we denote the elements of a homogeneous basis of
$\mathfrak{g}=\oplus_{i=1}^{s}V_{i}$ by $X_{i,j}$ with $X_{i,j}\in V_{i}$,
$1\leq i\leq s$ and $1\leq j\leq\dim V_{i}$, then
$\operatorname{wt}(X_{i,j})=i$. If $\theta_{i,j}$ denotes the dual basis of
one forms then $\operatorname{wt}(\theta_{i,j})=-i$. By direct inspection we
see that for a multiindex $I=((i_{1},j_{1}),\ldots,(i_{k},j_{k}))$ with
$(i_{\ell},j_{\ell})\neq(i_{m},j_{m})$ for $\ell\neq m$ we have

(3.24)
$\operatorname{wt}(\theta_{I})=-\sum_{\ell=1}^{k}i_{\ell},\quad\operatorname{wt}(X_{I})=\sum_{\ell=1}^{k}i_{\ell}.$

It follows from (3.24) that every $k$-form
$\alpha\in\Lambda^{k}(\mathfrak{g})$ has a canonical decomposition

$\alpha=\sum_{-\nu\leq j\leq-k}\alpha_{j}$

where $\alpha_{j}$ zero or a homogeneous $k$-form of weight $j$. Likewise
every $k$-vector $X$ has a decomposition

$X=\sum_{k\leq j\leq\nu}X_{j}$

where $X_{j}$ is homogeneous of weight $j$ or zero. By definition, the weight
of a $k$-form or $k$-vector is the maximal weight of its homogeneous
nonvanishing summands.

######  Lemma 3.25.

  1. (1)

The sets $\\{\alpha\in\Lambda^{k}\mathfrak{g}:\alpha\neq
0,\operatorname{weight}(\alpha)\leq w\\}\cup\\{0\\}$ and
$\\{X\in\Lambda_{k}\mathfrak{g}:\xi\neq 0,\operatorname{weight}(X)\leq
w\\}\cup\\{0\\}$ are vector spaces.

  2. (2)

If $\alpha,\beta\in\Lambda^{*}\mathfrak{g}$ and
$X,Y\in\Lambda_{*}\mathfrak{g}$ are homogeneous, then so are
$\alpha\wedge\beta$, $X\wedge Y$, and the interior product $i_{X}\alpha$.
Moreover

$\displaystyle\operatorname{weight}(\alpha\wedge\beta)=$
$\displaystyle\operatorname{weight}(\alpha)+\operatorname{weight}(\beta)$
$\displaystyle\operatorname{weight}(X\wedge Y)=$
$\displaystyle\operatorname{weight}(X)+\operatorname{weight}(Y)\,,$
$\displaystyle\operatorname{weight}(i_{X}\alpha)=$
$\displaystyle\operatorname{weight}(\alpha)+\operatorname{weight}(X)$

as long as the form or vector on the left hand side does not vanish.

  3. (3)

If $\gamma\in\Lambda^{k}\mathfrak{g}^{\prime}$ is homogeneous and
$\Phi:\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$ is a graded algebra
homomorphism, then $\Phi^{*}\gamma$ is homogeneous with
$\operatorname{weight}(\Phi^{*}\beta)=\operatorname{weight}(\beta)$ as long as
$\Phi^{*}\beta\neq 0$.

  4. (4)

If $\Phi:\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$ is a graded
homomorphism, $\alpha,\beta\in\Lambda^{*}\mathfrak{g}$,
$\gamma\in\Lambda^{*}\mathfrak{g}^{\prime}$,

(3.26) $\displaystyle\operatorname{weight}(\alpha\wedge\beta)\leq$
$\displaystyle\operatorname{weight}(\alpha)+\operatorname{weight}(\beta)$
(3.27) $\displaystyle\operatorname{weight}(d\alpha)\leq$
$\displaystyle\operatorname{weight}(\alpha)\quad\forall\alpha\in\Lambda^{*}\mathfrak{g},$
(3.28) $\displaystyle\operatorname{weight}(\Phi^{*}\gamma)\leq$
$\displaystyle\operatorname{weight}(\gamma)\quad\hbox{$\forall\gamma\in\Lambda^{*}\mathfrak{g}^{\prime}$}$

whenever the forms on the left hand side do not vanish. Moreover if $\Phi$ is
an isomorphism and $\gamma\neq 0$ then

(3.29)
$\displaystyle\quad\operatorname{weight}(\Phi^{*}\gamma)=\operatorname{weight}(\gamma)$

  5. (5)

If $\alpha\in\Lambda^{k}\mathfrak{g}$ then

$\displaystyle\alpha\neq
0\quad\Longrightarrow\quad\operatorname{weight}(\alpha)\geq-k+N-\nu$

  6. (6)

If $\alpha\in\Lambda^{*}\mathfrak{g}$ is homogeneous and $d\alpha\neq 0$ then

$\displaystyle\operatorname{weight}(d\alpha)=\operatorname{weight}(\alpha)$

In assertion (6) of the lemma, the action of the exterior derivative on
$\Lambda^{*}\mathfrak{g}$ is defined by extending the elements of
$\Lambda^{*}\mathfrak{g}$ to left-invariant forms, see (3.19).

###### Proof.

Assertion (1) follows directly from the definition of the weight. The first
two identities in (2) also follow immediately from the definition and the fact
that pullback und pushward under $\delta_{r}$ commute with the wedge product.
The third identity follows from a short calculation which starts from (3.2)
and uses that
$(\delta_{r})_{*}i_{X}\alpha(Z)=\delta_{r^{-1}}^{*}i_{X}\alpha(Z)=(i_{X}\alpha)((\delta_{r^{-1}})_{*}Z)$.

To show (3) we use that $(\delta_{r})_{*}=\delta_{r^{-1}}^{*}$ and that graded
algebra homomorphisms commute with the dilation map.

Assertion (4) follows from assertions (2) and (3) and the definition of the
weight.

Assertion (5) is clear for $k=N$ since the volume form has weight $-\nu$. Let
$k<N$ and assume that $\alpha\in\Lambda^{k}\mathfrak{g}$ with $\alpha\neq 0$
and $\operatorname{weight}(\alpha)<-\nu+N-k$. Every one-form has weight
$\leq-1$. Thus by (3.26) we get
$\operatorname{weight}(\alpha\wedge\beta)<-\nu$ for every $k$-form $\beta$.
Hence $\alpha\wedge\beta=0$ for all $k$-forms $\beta$. This implies $\alpha=0$
and yields a contradiction.

Finally (6) follows from (3.19) as well as the identities
$(\delta_{r})_{*}=\delta_{r^{-1}}^{*}$ and
$(\delta_{r^{-1}})_{*}[X_{i},X_{j}]=[(\delta_{r^{-1}})_{*}X_{i},(\delta_{r^{-1}})_{*}X_{j}]$.

∎

We now consider differential forms $\alpha$ defined on an open subset $U$ of a
Carnot group $G$. We denote the differential forms on $U$ by $\Omega^{*}(U)$
and the of compactly supported forms by $\Omega^{*}_{c}(U)$.

Identifying the left-invariant differential forms with
$\Lambda^{*}\mathfrak{g}$ we can write a $k$-form $\alpha\in\Omega^{k}(U)$ as

$\displaystyle\alpha=\sum_{I}a_{I}\theta_{I}\quad\hbox{where}\quad
a_{I}:U\to\mathbb{R}$

where the sum runs over ordered multiindices of length $k$. The functions
$a_{I}$ are determined uniquely. We say that $\alpha$ is a measurable (or
distributional, continuous, smooth, …) form if the coefficient functions
$a_{I}$ are measurable (or distributions, continuous, smooth, …). We say that
$\alpha$ vanishes a.e. if all functions $a_{I}$ vanish a.e. For fixed $x\in U$
we can view $\alpha(x)=\sum_{I}a_{I}(x)\theta_{I}$ as an element of
$\Lambda^{k}\mathfrak{g}$ and we define its weight as above.

######  Definition 3.30.

If $\alpha$ does not vanish a.e. in $U$ we define

(3.31) $\operatorname{weight}(\alpha):=\operatorname{esssup}_{x\in
U:\alpha(x)\neq 0}\operatorname{weight}(\alpha(x))$ (3.32)

If $E\subset U^{\prime}$ is measurable and the restriction of $\alpha$ to $E$
does not vanish a.e. then we define

(3.33) $\operatorname{weight}(\alpha_{|E}):=\operatorname{esssup}_{x\in
E:\alpha(x)\neq 0}\operatorname{weight}(\alpha(x))$

It follows from (3.26) that

(3.34)
$\operatorname{wt}(\alpha\wedge\beta)\leq\operatorname{wt}(\alpha)+\operatorname{wt}(\beta)\quad\forall\alpha,\beta\in\Omega^{*}(U)$

whenever $\alpha\wedge\beta$ does not vanish a.e.

We now come to a key object in our analysis, the Pansu pullback of a
differential form under map which is a.e. Pansu differentiable. Let $G$ and
$G^{\prime}$ be Carnot groups, where $G$ has dimension $N$ and homogeneous
dimension $\nu$. Suppose $f:G\supset U\rightarrow U^{\prime}\subset
G^{\prime}$ a $W^{1,p}_{\operatorname{loc}}$ mapping between open sets for
some $p>\nu$. Then $f$ is Pansu differentiable at almost every $x\in U$. The
Pansu differential $f_{P}(x)$ can be viewed as a graded homomorphism from $G$
to $G^{\prime}$ or from $\mathfrak{g}$ to $\mathfrak{g}^{\prime}$ (here we
identify a graded group homomorphism with its tangent map).

######  Definition 3.35.

Let $\alpha$ be a continuous differential form on $U^{\prime}$. The Pansu
pullback $f_{P}^{*}\alpha$ is defined by

(3.36) $(f_{P}^{*}\alpha)(x)=\sum_{I}(a_{I}\circ
f)(x)\,\,(f_{P}(x))^{*}\theta_{I}$

######  Lemma 3.37.

Let $f:G\supset U\rightarrow U^{\prime}\subset G^{\prime}$ and $\alpha$ be as
above. If $f_{P}^{*}\alpha$ does not vanish a.e. in $U$ then

(3.38)
$\operatorname{weight}(f_{P}^{*}\alpha)\leq\operatorname{weight}(\alpha).$

Assume in addition that the Pansu differential $f_{P}(x)$ is an isomorphism
for a.e. $x\in U$ and that $\alpha$ does not vanish a.e. in $f(U)$. Then
$f_{P}^{*}(\alpha)$ does not vanish a.e. in $U$ and

(3.39)
$\operatorname{weight}(f_{P}^{*}\alpha)=\operatorname{weight}(\alpha_{|f(U)}).$

###### Proof.

The inequality (3.38) follows from (3.28), while (3.39) follows from (3.29) ∎

##  4\. The pullback theorem

The key technical result in this paper is:

######  Theorem 4.1 (Pullback theorem).

Let $G$ and $G^{\prime}$ be Carnot groups, where $G$ has dimension $N$ and
homogeneous dimension $\nu$. Suppose $f:G\supset U\rightarrow
U^{\prime}\subset G^{\prime}$ a $W^{1,p}_{\operatorname{loc}}$ mapping between
open sets for some $p>\nu$, and $\omega\in\Omega^{k}(U^{\prime})$ is a
continuous closed form, $\eta\in\Omega^{N-k-1}_{c}(U)$ is smooth, and
$\operatorname{wt}(\omega)+\operatorname{wt}(d\eta)\leq-\nu$. Then

$\int_{U}f_{P}^{*}\omega\wedge d\eta=0\,.$

Note that by the Sobolev embedding theorem for Carnot groups, a
$W^{1,p}_{\operatorname{loc}}$ mapping is continuous when $p>\nu$, see Lemma
2.8. We will prove this result by a new approximation of $f$ by smooth maps
$f_{\rho}$ which respects the structure of the Carnot group. The approximation
is introduced in Section 5 and the proof of Theorem 4.1 is carried out in
Section 6.

We will usually use the following special case of Theorem 4.1:

######  Theorem 4.2 (Pullback Theorem (special case)).

Let $G,G^{\prime},U,U^{\prime}$ and $f$ be as in Theorem 4.1. Suppose
$\varphi\in C_{c}^{\infty}(U)$ and that $\alpha$ and $\beta$ are closed left-
invariant forms which satisfy

(4.3)
$\deg\alpha+\deg\beta=N-1\quad\hbox{and}\quad\operatorname{wt}(\alpha)+\operatorname{wt}(\beta)\leq-\nu+1.$

Then

(4.4) $\int_{U}f_{P}^{*}(\alpha)\wedge d(\varphi\beta)=0$

###### Proof.

Letting $\eta=\varphi\beta$, we have $d\eta=d\varphi\wedge\beta$. Thus
$\operatorname{wt}(d\eta)\leq\operatorname{wt}(\beta)-1$. Hence the assertion
follows from Theorem 4.1. ∎

For a closed form $\beta$ we have $d(\varphi\beta)=d\varphi\wedge\beta$. Thus
the identity (4.4) implies that the codegree $1$ form
$f_{P}^{*}(\alpha)\wedge\beta$ is distributionally closed, i.e. its
distributional exterior derivative vanishes. Letting $\alpha$ and $\beta$
vary, we obtain conditions on the distribution derivatives of certain wedge
powers of the Pansu differential $D_{P}f(x)$. We will study these conditions
more systematically in a subsequent paper; our focus here will be on making
judicious choices of forms to deduce rigidity for mappings.

Interestingly, Theorem 4.2 yields no information on derivatives of the Pansu
differential if the form $\alpha$ or $\beta$ is exact. More precisely we have
the following result:

######  Lemma 4.5.

Assume that $f:U\subset G\to G^{\prime}$ is Pansu differentiable a.e. in $U$.
Let $\alpha$ and $\beta$ be closed left-invariant forms which satisfy

(4.6)
$\deg\alpha+\deg\beta=N-1\quad\hbox{and}\quad\operatorname{wt}(\alpha)+\operatorname{wt}(\beta)\leq-\nu+1.$

and assume that there exists a left-invariant form $\gamma$ such that
$\alpha=d\gamma$ (respectively $\beta=d\gamma$). Then

(4.7) $f_{P}^{*}(\alpha)\wedge\beta=0\quad\hbox{a.e.}$

###### Proof.

In view of the definition of the Pansu pullback it suffices to show the
following pointwise statement for $\alpha\in\Lambda^{*}\mathfrak{g}^{\prime}$,
$\beta\in\Lambda^{*}\mathfrak{g}$. If (4.6) holds, if $\alpha=d\gamma$ or
$\beta=d\gamma$ and if $\Phi:\mathfrak{g}\to\mathfrak{g}^{\prime}$ is a graded
algebra homomorphism then

(4.8) $\Phi^{*}\alpha\wedge\beta=0.$

Here we use that exterior differentiation of left-invariant forms induces a
linear operation on $\Lambda^{*}\mathfrak{g}$ (and on
$\Lambda^{*}\mathfrak{g}^{\prime}$), see (3.19).

Now assume first that $\alpha=d\gamma$. Then $\deg\gamma=\deg\alpha-1$. We
claim that we may assume that
$\operatorname{wt}(\gamma)=\operatorname{wt}(\alpha)$. To see this write
$\gamma$ as a sum of homogeneous forms of weight $s$

$\gamma=\sum_{s=-\nu}^{-\deg\gamma}\gamma_{s}.$

By Lemma 3.25 (6) we have $d\gamma_{s}=0$ or
$\operatorname{wt}(d\gamma_{s})=s$. By the definition of
$\operatorname{wt}(\alpha)$ we thus must have $d\gamma_{s}=0$ if
$s>\operatorname{wt}(\alpha)$ and $\gamma_{\operatorname{wt}(\alpha)}\neq 0$.
Hence $d\gamma=d\tilde{\gamma}$ where
$\tilde{\gamma}=\sum_{s=-\nu}^{\operatorname{wt}(\alpha)}\gamma_{-s}$ and
$\operatorname{wt}(\tilde{\gamma})=\operatorname{wt}(\alpha)$.

Since pull-back by $\Phi$ commutes with $d$ (see (3.21)) and since $d\beta=0$

$\Phi^{*}(\alpha)\wedge\beta=d\Phi^{*}(\gamma)\wedge\beta=d(\Phi^{*}(\gamma)\wedge\beta).$

The form $\Phi^{*}(\gamma)\wedge\beta$ has degree $N-2$ and weight $\leq\nu-1$
and hence vanishes by Lemma 3.25 (5). This proves (4.8) if $\alpha=d\gamma$.

If $\beta=d\gamma$ then $\deg\gamma=\deg\beta-1$ and we may assume that
$\operatorname{wt}(\gamma)=\operatorname{wt}(\beta)$. Now using again (3.21)
and the fact that $\alpha$ is closed we get $d\Phi^{*}\alpha=0$. Thus

$\Phi^{*}\alpha\wedge\beta=\Phi^{*}\alpha\wedge d\gamma=\pm
d(\Phi^{*}\alpha\wedge\gamma).$

Now by (3.26) and (3.28) the form $\Phi^{*}\alpha\wedge\gamma$ has weight
$\leq-\nu+1$. Since this form has degree $N-2$ it must vanish. ∎

##  5\. Center of mass and mollification in Carnot groups

In this section we introduce a mollification procedure which is based on
special features of Carnot groups. This will be used in the proof of the
pullback theorem in the next section. This is a variation on the treatment in
[BK81], adjusted to the Carnot group case.

Let $G$ and $G^{\prime}$ be Carnot groups with respective Lie algebras
$\mathfrak{g}$, $\mathfrak{g}^{\prime}$, topological dimensions $N$ and
$N^{\prime}$, horizontal dimensions $n$ and $n^{\prime}$, and homogeneous
dimensions $\nu$ and $\nu^{\prime}$.

###  5.1. Center of mass in Carnot groups

We define a center-of-mass for a compactly supported probability measure in a
Carnot group; this behaves well with respect to left translation and
rescaling.

Recall that if $G$ is a Carnot group, then the exponential map
$\exp:\mathfrak{g}\rightarrow G$ is a diffeomorphism; we let
$\log:G\rightarrow\mathfrak{g}$ denote the inverse diffeomorphism. Recall that
$\exp$ (and hence also $\log$) are $\operatorname{Aut}(G)$-equivariant, where
$\operatorname{Aut}(G)$ acts on $G$ in the tautological way, and on
$\mathfrak{g}$ by the derivative action, i.e. for
$\phi\in\operatorname{Aut}(G)$ we have $\exp(D\phi(v))=\phi(\exp(v))$. For
$x\in G$, let $\log_{x}=\log\circ\ell_{x^{-1}}$, where $\ell_{g}$ is the left
translation by $g$.

Suppose $\nu$ is a compactly supported Borel probability measure on $G$. We
define a function $C_{\nu}:G\rightarrow\mathfrak{g}$ by integration:

$C_{\nu}(x)=\int_{\mathfrak{g}}\operatorname{id}_{\mathfrak{g}}\,(d(\log_{x})_{*}\nu)=\int_{G}\log_{x}\,d\nu=\int_{G}\log_{x}(y)\;d\nu(y)\,,$

where we are viewing $\log_{x}$ as a vector-valued function in the latter two
integrals.

######  Lemma 5.1.

For any compactly support Borel probability measure $\nu$:

  1. (1)

$C_{\nu}$ is smooth.

  2. (2)

$C_{\nu}$ is a local diffeomorphism.

  3. (3)

$C_{\nu}$ is proper.

  4. (4)

$C_{\nu}:G\rightarrow\mathfrak{g}$ is a diffeomorphism.

  5. (5)

If $\nu$ is inversion symmetric, $I_{*}\nu=\nu$, then $C_{\nu}(e)=0$.

In particular, $C_{\nu}^{-1}(0)$ contains a unique element, which we denote by
$\operatorname{cent}(\nu)$.

###### Proof.

(1). Since $\log_{x}(y)=\log\circ\ell_{x^{-1}}(y)=\log\circ r_{y}\circ I(x)$
where $r_{y}$ is right translation by $y$ and $I(g)=g^{-1}$, the function
$x\mapsto\log_{x}(y)$ has derivatives bounded locally (in $x$) where the
bounds depend on $y$; Since $\nu$ is a compactly supported probability
measure, this implies that $C_{\nu}$ is smooth. (In fact it is real analytic.)

(2). Unwinding definitions, one gets that for any automorphism $\phi$,

(5.2) $C_{\phi_{*}\nu}(\phi(x))=D\phi(C_{\nu}(x))\,.$

Therefore by applying a strongly contracting automorphism to $\nu$ if
necessary, we may assume without loss of generality that
$\operatorname{spt}(\nu)$ and $x$ are both close to $e\in G$. Now in
exponential coordinates, the left translations are $C^{1}$ close to Euclidean
translations, so (2) follows.

(3). Pick $R$ such that $\operatorname{spt}(\nu)\subset B(e,R)$. For $x\in
G\setminus\\{e\\}$, let $r=d_{CC}(x^{-1},e)=d_{CC}(x,e)$, and
$\phi=\delta_{r^{-1}}$. Since $\phi$ multiplies distances by $r^{-1}$, when
$d_{CC}(x,e)$ is large, the measure $\phi_{*}\nu$ is concentrated near $e$,
and $(\ell_{\phi(x^{-1})})_{*}\phi_{*}\nu$ is concentrated near
$\phi(x^{-1})$, which lies in the $d_{CC}$-unit sphere $S(e,1)\subset G$.
Hence

$C_{\phi_{*}\nu}(\phi(x))=\int_{\mathfrak{g}}\operatorname{id}_{\mathfrak{g}}\,\log_{*}(\ell_{\phi(x^{-1})})_{*}\phi_{*}\nu$

lies close to $\log(\phi(x^{-1}))\in\log(S(e,1))$. Therefore

$C_{\nu}(x)=(D\phi)^{-1}(C_{\phi_{*}\nu}(\phi(x))$

tend to infinity as $x\rightarrow\infty$. This proves that $C_{\nu}$ is
proper.

(4). Since $C_{\nu}$ is a proper map and is a local diffeomorphism, it is a
covering map. It is a diffeomorphism since $\mathfrak{g}$ is simply connected
and $G$ is connected.

(5). Since inversion $I:G\rightarrow G$ satisfies $\log\circ I=-\log$, we have

$C_{\nu}(e)=\int_{\mathfrak{g}}\operatorname{id}_{\mathfrak{g}}\,\log_{*}\nu=\int_{\mathfrak{g}}\operatorname{id}_{\mathfrak{g}}\,\log_{*}I_{*}\nu$
$=-\int_{\mathfrak{g}}\operatorname{id}_{\mathfrak{g}}\,\log_{*}\nu=-C_{\nu}(e)\,,$

forcing $C_{\nu}(e)=0$. ∎

######  Lemma 5.3 (Properties of the center-of-mass).

  1. (1)

If $g\in G$, $\Phi\in\operatorname{Aut}(G)$, then

$\operatorname{cent}((\Phi\circ\ell_{g})_{*}\nu)=(\Phi\circ\ell_{g})(\operatorname{cent}\nu)\,.$

  2. (2)

There is a constant $C$ such that if $\operatorname{spt}(\nu)\subset B(x,R)$,
then $\operatorname{cent}(\nu)\subset B(x,CR)$.

###### Proof.

(1). If suffices to show the results for push-forward by $l_{g}$ and by $\Phi$
separately. The result for push-forward by $\ell_{g}$ follows from the
identity

$\displaystyle C_{(\ell_{g})_{*}\nu}(gx)=$
$\displaystyle\int_{\mathfrak{g}}\operatorname{id}_{\mathfrak{g}}\,\log_{*}(\ell_{x^{-1}})_{*}(\ell_{g^{-1}})_{*}(\ell_{g})_{*}\nu$
$\displaystyle=$
$\displaystyle\int_{\mathfrak{g}}\operatorname{id}_{\mathfrak{g}}\,\log_{*}\circ(\ell_{x^{-1}})_{*}\nu=C_{\nu}(x)\,.$

For $\Phi\in\operatorname{Aut}(G)$ we can use (5.2) to get

$C_{\Phi_{*}\nu}(\Phi(\operatorname{cent}\nu))=D\Phi(C_{\nu}(\operatorname{cent}\nu))=D\Phi(0)=0$

and thus $\operatorname{cent}\Phi_{*}\nu=\Phi(\operatorname{cent}\nu)$.

(2). By part (1) it suffices to show the assertion for $x=0$. If the assertion
does not hold for $x=0$ then there exist probability measure $\mu_{k}$
supported in $B(0,R_{k})$ such that
$r_{k}:=d(\operatorname{cent}\mu_{k},e)>kR_{k}$. Applying part (1) with
$\Phi=\delta_{r_{k}^{-1}}$ we obtain probability measures $\nu_{k}$ supported
in $B(0,\frac{1}{k})$ such that $d(\operatorname{cent}\nu_{k},e)=1$. Passing
to a subsequence, not relabelled, we may assume
$x_{k}:=\operatorname{cent}\nu_{k}\to x_{\infty}$ with $d(x_{\infty},e)=1$. By
the definition of the center of mass and the uniform continuity of the map
$(x,y)\mapsto\log(x^{-1}y)$ on compact subsets we get

$0=C_{\nu_{k}}(x_{k})=\int_{G}\log(x_{k}^{-1}y)d\nu_{k}(y)\to\log x_{\infty}.$

Thus $x_{\infty}=e$, a contradiction. ∎

###  5.2. Mollifying maps between Carnot groups

Let $\sigma_{1}$ be a smooth probability measure on $G$ with
$\operatorname{spt}(\sigma_{1})\subset B(e,1)$. We also assume that
$\sigma_{1}$ is symmetric under inversion: $I_{*}\sigma_{1}=\sigma_{1}$, where
$I(x)=x^{-1}$. For $x\in G$, $\rho\in(0,\infty)$, let $\sigma_{\rho}$,
$\sigma_{x}$, and $\sigma_{x,\rho}$ be the pushforwards of $\sigma_{1}$ under
the the corresponding Carnot scaling and left translation:

$\sigma_{\rho}=(\delta_{\rho})_{*}\sigma_{1}\,,\quad\sigma_{x}=(\ell_{x})_{*}\sigma_{1}\,,\quad\sigma_{x,\rho}=(\ell_{x}\circ\delta_{\rho})_{*}\sigma_{1}=(\ell_{x})_{*}(\sigma_{\rho})\,.$

Let $U\subset G$ be open. Let $G^{\prime}$ be another Carnot group and let
$f:U\to G^{\prime}$ be continuous. For $\rho>0$ set

(5.4) $U_{\rho}:=\\{x:B(x,\rho)\subset U\\}$

For $x\in U_{1}$ we have
$\operatorname{spt}(\sigma_{x})=x+\operatorname{spt}(\sigma)\subset U$. Thus
we may define $f_{1}:U_{1}\rightarrow G^{\prime}$ by

$f_{1}(x)=\operatorname{cent}(f_{*}(\sigma_{x}))\,,$

and $f_{\rho}:U_{\rho}\rightarrow G^{\prime}$ by

(5.5) $f_{\rho}=\delta_{\rho}\circ(\delta_{\rho^{-1}}\circ
f\circ\delta_{\rho})_{1}\circ\delta_{\rho^{-1}}\,.$

######  Lemma 5.6.

  1. (1)

For all $\rho\in(0,\infty)$,

$\delta_{\rho^{-1}}\circ f_{\rho}\circ\delta_{\rho}=(\delta_{\rho^{-1}}\circ
f\circ\delta_{\rho})_{1}\,.$

  2. (2)

If $x\in U_{1}$ and $f(B(x,1))\subset B(f(x),R)$, then the derivatives of
$\delta_{R^{-1}}\circ f_{1}$ are controlled near $x$, i.e.
$\|D^{i}(\delta_{R^{-1}}\circ f_{1})(x)\|<C=C(i,R)$. In particular,
$Df_{1}(x)=D(\delta_{R}\circ(\delta_{R^{-1}}\circ f_{1})(x)=D\delta_{R}\circ
T$ where $\|T\|<C^{\prime}$ and $C^{\prime}=C^{\prime}(G,G^{\prime})$ is
independent of $R$.

  3. (3)

If $f:G\rightarrow G^{\prime}$ is a group homomorphism, then $f_{1}=f$.

  4. (4)

If $\\{f_{k}:U\rightarrow G^{\prime}\\}$ is a sequence of continuous maps, and
$f_{k}\rightarrow f_{\infty}$ in $C^{0}_{loc}(U)$, then the sequence of
mollified maps $\\{(f_{k})_{1}\\}$ converges in $C^{j}_{loc}(U_{1})$ to
$(f_{\infty})_{1}$, for all $j$.

  5. (5)

We have $f_{\rho}\to f$ in $C_{loc}(U)$ as $\rho\to 0$.

###### Proof.

(1). This is immediate from the definition.

(2). Note that $\delta_{R^{-1}}\circ f_{1}=(\delta_{R^{-1}}\circ f)_{1}$, so
we are reduced to the case when $R=1$.

Since the derivative estimates are unchanged if we pre or post-compose $f$
with left translations, we may assume without loss of generality that
$f(e)=e$, and that we want to estimate the derivative of $f_{1}$ at $e$. So we
are interested in $f_{1}(z)=\operatorname{cent}(f_{*}\sigma_{z})$ for $z$
close to $e\in G$, which is determined implicitly by the equation

$C_{f_{*}\sigma_{z}}(\operatorname{cent}(f_{1}(z))=0\,.$

Using that $\operatorname{spt}(\nu_{z})\subset B(0,1)$ for $z$ close to $e$
and that $f$ is defined on $B(0,1)$ we get

$\displaystyle C_{f_{*}\sigma_{z}}(x)=$
$\displaystyle\int_{G^{\prime}}\log_{x}\,d(f_{*}\sigma_{z})$ $\displaystyle=$
$\displaystyle\int_{B(0,1)}(\log_{x}\circ f)\,d\sigma_{z}$ $\displaystyle=$
$\displaystyle\int_{B(0,1)}(\log_{x}\circ f)\,d((\ell_{z})_{*}\sigma_{1})$
$\displaystyle=$ $\displaystyle\int_{B(0,1)}(\log\circ\ell_{x^{-1}}\circ
f)(y)(\alpha\circ\ell_{z^{-1}})(y)\,d\mu(y)\,,$

where $\mu$ is the left-invariant volume form on $G$, $\sigma_{1}=\alpha\,\mu$
and $x\approx f_{1}(z)$.

Since $z\approx e$, we get that $d_{CC}(f_{1}(z),e)<C_{1}$. Also, we may
assume that
$\operatorname{spt}((\ell_{z})_{*}\sigma_{1})=\operatorname{spt}(\alpha\circ\ell_{z^{-1}})\subset
B(e,1)$. Thus when $y$ is fixed, the integrand is a smooth function in $x$ and
$z$ with derivatives bounded uniformly independent of
$y\in\operatorname{spt}(\alpha\circ\ell_{z^{-1}})$.

(3). For $y\in G$,

$\displaystyle f_{1}(y)=$
$\displaystyle\operatorname{cent}(f_{*}\sigma_{y})=\operatorname{cent}(f_{*}(\ell_{y})_{*}\sigma_{1})$
$\displaystyle=$
$\displaystyle\operatorname{cent}((\ell_{f(y)})_{*}(f_{*}\sigma_{1}))=\ell_{f(y)}(\operatorname{cent}(f_{*}\sigma_{1}))$
$\displaystyle=$ $\displaystyle f(y)\operatorname{cent}(f_{*}\sigma_{1})=f(y)$

since $f_{*}\sigma_{1}$ is symmetric.

(4). If this were false, there would be a compact subset $Y\subset U_{1}$, an
integer $j$, $\varepsilon>0$, and a subsequence $\\{f_{k_{i}}\\}$ such that

(5.7)
$d_{C^{j}}((f_{k_{i}})_{1}\mbox{\Large$|$\normalsize}_{Y},(f_{\infty})_{1}\mbox{\Large$|$\normalsize}_{Y})\geq\varepsilon$

for all $i$. By (3), all derivatives of $(f_{k})_{1}$ are uniformly bounded on
compact subsets of $U_{1}$. Therefore a subsequence converges in
$C^{j}_{loc}(U_{1})$, to a limit $g_{\infty}$. To obtain a contradiction with
(5.7) it only remains to show that $g_{\infty}=(f_{\infty})_{1}$ in $U_{1}$.
This identity follows from the convergence $f_{k}\stackrel{{\scriptstyle
C^{0}_{loc}(U)}}{{\rightarrow}}f_{\infty}$ by passing to the limit in the
defining equation for $(f_{k})_{1}(x)$ namely

$0=C_{(f_{k})_{*}\sigma_{x}}\big{(}(f_{k})_{1}(x)\big{)}=\int_{U}(\log\circ\ell_{((f_{k})_{1}(x))^{-1}}\circ
f_{k})(y)\,\,d\sigma_{x}(y).$

To pass to the limit one uses that for $x\in U_{1}$ the measure
$\sigma_{x}=(\ell_{x})_{*}\sigma_{1}$ has compact support in $U$.

(5). Unwinding definitions and using Lemma 5.3(1) for dilations we get
$f_{\rho}(x)=\operatorname{cent}f_{*}(\sigma_{x,\rho})$. Let $K\subset U$ be
compact. Then there exists a $\rho^{\prime}>0$ such that $f$ is uniformly
continuous in a $\rho^{\prime}$-neighbourhood of $K$. Let $\omega$ denote a
modulus of continuity of $f$ restricted to that neighbourhood. Then for
$\rho\in(0,\rho^{\prime})$ we have $\operatorname{spt}\sigma_{x,\rho}\subset
B(x,\rho)$ and $\operatorname{spt}f_{*}\sigma_{x,\rho}\subset
B(f(x),\omega(\rho))$. Hence by Lemma 5.3(2) we have $f_{\rho}(x)\in
B(f(x),C\omega(\rho))$ for all $x\in K$. Thus $f_{\rho}$ converges to $f$,
uniformly in $K$. ∎

##  6\. Pansu pullback and mollification

The main result of this section is Theorem 6.1 below, which implies Theorem
4.1 as an easy corollary.

######  Theorem 6.1 (Main approximation theorem).

Let $G$, $G^{\prime}$, and $f:U\rightarrow U^{\prime}$ be as in Theorem 4.1.
Suppose $\omega\in\Omega^{k}(U^{\prime})$, $\eta\in\Omega^{N-k}_{c}(U)$ are
forms with continuous coefficients, such that
$\operatorname{wt}(\omega)+\operatorname{wt}(\eta)\leq-\nu$. Then

(6.2) $\int_{U}f_{P}^{*}\omega\wedge\eta=\lim_{\rho\rightarrow
0}\int_{U}f_{\rho}^{*}\omega\wedge\eta\,,$

where $f_{\rho}$ is the mollified map defined in (5.5).

###### Proof of Theorem 4.1 using Theorem 6.1.

If $\omega$ is smooth, then

$\displaystyle\int_{U}f_{P}^{*}\omega\wedge d\eta=$
$\displaystyle\lim_{\rho\rightarrow 0}\int_{U}f_{\rho}^{*}\omega\wedge d\eta$
$\displaystyle=$ $\displaystyle\lim_{\rho\rightarrow
0}(-1)^{\deg\omega}\int_{U}d(f_{\rho}^{*}\omega\wedge\eta)=0$

by Stokes’ theorem. The general case follows by approximating $\omega$ by
smooth forms. ∎

We now turn to the proof of the Main Approximation Theorem

The strategy of the proof is as follows. We show convergence of the integrals
with the help of the Dominated Convergence Theorem. To verify pointwise
convergence $f_{\rho}^{*}\omega\wedge\eta\rightarrow f_{P}^{*}\omega\wedge
d\eta$, by simple algebraic manipulations we first express
$f_{\rho}^{*}\omega\wedge\eta(x)$ in terms of the rescaled, re-centered, and
smoothed map $(\delta_{\rho^{-1}}\circ f_{x}\circ\delta_{\rho})_{1}$ where
$f_{x}=\ell_{f(x)^{-1}}\circ f\circ\ell_{x}$. Then we use the definition of
the Pansu derivative and the properties of the center of mass smoothing to see
that $(\delta_{\rho^{-1}}\circ f_{x}\circ\delta_{\rho})_{1}$ converges in
$C^{\infty}_{\operatorname{loc}}$ to the Pansu differential $D_{P}f(x)$ when
$x$ is a point of Pansu differentiability. This gives pointwise convergence of
the integrands.

######  Lemma 6.3.

Let $U\subset G$ be open and let $f:U\to G^{\prime}$ be continuous. Suppose
$\alpha\in\Omega^{k}(G^{\prime})$, $\gamma\in\Omega^{N-k}(G)$ are left
invariant forms of weight $w_{\alpha}$ and $w_{\gamma}$, respectively.

  1. (1)

For every $x\in U_{\rho}$,

$(f_{\rho}^{*}\alpha\wedge\gamma)(x)=\rho^{-(\nu+w_{\alpha}+w_{\gamma})}(h_{1}^{*}\alpha\wedge\gamma)(\delta_{\rho^{-1}}(x))\,,$

where $h=\delta_{\rho^{-1}}\circ f\circ\delta_{\rho}$.

  2. (2)

For every $x\in U_{\rho}$,

$(f_{\rho}^{*}\alpha\wedge\gamma)(x)=\rho^{-(\nu+w_{\alpha}+w_{\gamma})}\big{(}(\delta_{\rho^{-1}}\circ
f_{x}\circ\delta_{\rho})_{1}^{*}\alpha\wedge\gamma\big{)}(e)\,,$

where $f_{x}=\ell_{f(x)^{-1}}\circ f\circ\ell_{x}$.

  3. (3)

If $x\in U_{\rho}$ and $f(B(x,\rho))\subset B(f(x),C\rho)$, then

$\|(f_{\rho}^{*}\alpha)\wedge\gamma)(x)\|\leq
C^{\prime}\,C^{-w_{\alpha}}\rho^{-(\nu+w_{\alpha}+w_{\gamma})}\|\alpha\|\,\|\gamma\|.$

###### Proof.

(1). Note that
$\\{z:B(z,1)\subset\delta_{\rho^{-1}}U\\}=\delta_{\rho^{-1}}U_{\delta}$ and
thus

$h:\delta_{\rho^{-1}}U\to G^{\prime},\quad
h_{1}:\delta_{\rho^{-1}}U_{\delta}\to G^{\prime}.$

For $x\in U_{\rho}$ we have

$\displaystyle(f_{\rho}^{*}(\alpha)\wedge\gamma)(x)=$
$\displaystyle\big{(}(\delta_{\rho}\circ
h_{1}\circ\delta_{\rho^{-1}})^{*}\alpha\wedge\gamma\big{)}(x)$
$\displaystyle=$
$\displaystyle(\delta_{\rho^{-1}}^{*}h_{1}^{*}\delta_{\rho}^{*}\alpha\,\wedge\,\delta_{\rho^{-1}}^{*}\delta_{\rho}^{*}\gamma)(x)$
$\displaystyle=$
$\displaystyle\rho^{-(w_{\alpha}+w_{\gamma})}(\delta_{\rho^{-1}}^{*}h_{1}^{*}\alpha\wedge\delta_{\rho^{-1}}^{*}\gamma)(x)$
$\displaystyle=$
$\displaystyle\rho^{-(w_{\alpha}+w_{\gamma})}\big{(}\delta_{\rho^{-1}}^{*}(h_{1}^{*}\alpha\wedge\gamma)\big{)}(x)$
$\displaystyle=$
$\displaystyle\rho^{-(\nu+w_{\alpha}+w_{\gamma})}(h_{1}^{*}\alpha\wedge\gamma)(\delta_{\rho^{-1}}x)$

In the last step we used that $h_{1}^{*}\alpha\wedge\gamma$ is a multiple of
the volume form, which has weight $-\nu$.

(2). With $h$ as in (1),

$\displaystyle h=$ $\displaystyle\delta_{\rho^{-1}}\circ f\circ\delta_{\rho}$
$\displaystyle=$
$\displaystyle(\delta_{\rho^{-1}}\circ\ell_{f(x)}\circ\delta_{\rho})\circ\delta_{\rho^{-1}}\circ\ell_{f(x)^{-1}}\circ
f\circ\ell_{x}\circ\delta_{\rho}\circ(\delta_{\rho^{-1}}\circ\ell_{x^{-1}}\circ\delta_{\rho})$
$\displaystyle=$
$\displaystyle\ell_{\delta_{\rho^{-1}}f(x)}\circ\delta_{\rho^{-1}}\circ
f_{x}\circ\delta_{\rho}\circ\ell_{\delta_{\rho^{-1}}x^{-1}}$

and so

$h_{1}=\ell_{\delta_{\rho^{-1}}f(x)}\circ(\delta_{\rho^{-1}}\circ
f_{x}\circ\delta_{\rho})_{1}\circ\ell_{\delta_{\rho^{-1}}x^{-1}}\,.$

Since $\alpha$ and $\gamma$ are left invariant we have for $x\in U_{\rho}$

$\displaystyle(h_{1}^{*}\alpha\wedge\gamma)(\delta_{\rho^{-1}}(x))$ (6.4)
$\displaystyle=$
$\displaystyle\ell_{\delta_{\rho^{-1}}x^{-1}}^{*}[(\delta_{\rho^{-1}}\circ
f_{x}\circ\delta_{\rho})_{1}^{*}\alpha\wedge\gamma](\delta_{\rho^{-1}}(x))$
$\displaystyle=$ $\displaystyle[(\delta_{\rho^{-1}}\circ
f_{x}\circ\delta_{\rho})_{1}^{*}\alpha\wedge\gamma](e).$

Combining (1) with (6) gives (2).

(3). By our assumptions we have
$B(\delta_{\rho^{-1}}(x),1)\subset\delta_{\rho^{-1}}U$ and
$h(B(\delta_{\rho^{-1}}(x),1))\subset B(h(\delta_{\rho^{-1}}(x)),C)$, so
$\|D(\delta_{C^{-1}}\circ h)(\delta_{\rho^{-1}}(x))\|\leq C^{\prime}$ by Lemma
5.6(2). Using (1) and
$\delta_{C}^{*}\alpha=(\delta_{C^{-1}})_{*}\alpha=C^{-w_{\alpha}}\alpha$ we
get

$\displaystyle\|(f_{\rho}^{*}\alpha)\wedge\gamma)(x))\|$ $\displaystyle=$
$\displaystyle\|\rho^{-(\nu+w_{\alpha}+w_{\gamma})}(\delta_{C}\circ(\delta_{C^{-1}}\circ
h_{1}))^{*}\alpha\wedge\gamma)(\delta_{\rho^{-1}}(x))\|$ $\displaystyle=$
$\displaystyle
C^{-w_{\alpha}}\rho^{-(\nu+w_{\alpha}+w_{\gamma})}\|(\delta_{C^{-1}}\circ
h_{1})^{*}\alpha\wedge\gamma)(\delta_{\rho^{-1}}(x))\|$ $\displaystyle\leq$
$\displaystyle
C^{\prime}C^{-w_{\alpha}}\rho^{-(\nu+w_{\alpha}+w_{\gamma})}\|\alpha\|\,\|\gamma\|.$

∎

###### Proof of Theorem 6.1.

It suffices to show the assertion if $\omega=a\alpha$ and $\eta=u\gamma$ where
$\alpha$ and $\gamma$ are left invariant forms in $\Omega^{k}(G)$ and
$\Omega^{N-k}(G)$, respectively, with
$\operatorname{wt}(\alpha)+\operatorname{wt}(\gamma)\leq-\nu$ and where $a$
and $u$ are continuous functions and $u$ has compact support in $U$. Replacing
$U$ by an open set $V$ with $\operatorname{spt}u\subset V\subset\subset U$ if
necessary we may assume without loss of generality that

(6.5) $f\in W^{1,p}(U;G^{\prime}).$

In order to show that

$\lim_{\rho\rightarrow
0}\int_{U}f_{\rho}^{*}\omega\wedge\eta=\int_{U}f_{P}^{*}\omega\wedge\eta$

we will use the Dominated Convergence Theorem.

We first verify pointwise convergence of the integrands in
$\operatorname{spt}u\subset U$. Since $\operatorname{spt}u$ is compact there
exists a $\rho_{0}>0$ such that $\operatorname{spt}u\subset U_{\rho}$ for all
$\rho\in(0,\rho_{0})$. Moreover it follows from Lemma 5.6(5) that $f_{\rho}\to
f$ in $C^{0}(\operatorname{spt}u)$. Since $f(\operatorname{spt}u)$ is a
compact subset of $U^{\prime}$ we may assume that
$f_{\rho}(\operatorname{spt}u)\subset U^{\prime}$ for all
$\rho\in(0,\rho_{0})$ by making $\rho_{0}$ smaller if needed. In the following
we will always assume $\rho\in(0,\rho_{0})$. Then the assertions in Lemma 6.3
hold for all $x\in\operatorname{spt}u$. Assertion (2) of that lemma gives for
$x\in\operatorname{spt}u$

$\displaystyle f_{\rho}^{*}(\omega)\wedge\eta(x)$ $\displaystyle=$
$\displaystyle(a\circ f_{\rho})(x)\,u(x)\,(f_{\rho}^{*}\alpha\wedge\gamma)(x)$
$\displaystyle=$ $\displaystyle(a\circ
f_{\rho})(x)\,u(x)\,\rho^{-(\nu+w_{\alpha}+w_{\xi})}\big{(}(\delta_{\rho^{-1}}\circ
f_{x}\circ\delta_{\rho})_{1}^{*}\alpha\wedge\gamma\big{)}(e)\,.$

If $x$ is a point of differentiability of $f$, then $\delta_{\rho^{-1}}\circ
f_{x}\circ\delta_{\rho}\stackrel{{\scriptstyle
C^{0}_{loc}}}{{\longrightarrow}}D_{P}f$. (Recall that we are using the
notation $D_{P}f(x)$ to denote a graded Lie algebra homomorphism
$\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$ and a homomorphism of Carnot
groups $G\rightarrow G^{\prime}$, depending on the context.) So by Lemma 5.6
parts (4) and (3) we get $D(\delta_{\rho^{-1}}\circ
f_{x}\circ\delta_{\rho})_{1}(e)\rightarrow D_{P}f(x)$ as $\rho\rightarrow 0$.
If $w_{\alpha}+w_{\gamma}=-\nu$, then

(6.6) $\displaystyle(f_{\rho}^{*}(\omega)\wedge\eta)(x)\rightarrow$
$\displaystyle(a\circ
f_{\rho})(x)\,u(x)\,((D_{P}f(x))^{*}\alpha)(x)\wedge\gamma(x)$
$\displaystyle=$ $\displaystyle(f_{P}^{*}\omega\wedge\eta)(x)$

so we have pointwise convergence in this case. If
$w_{\alpha}+w_{\gamma}<-\nu$, then
$(f_{\rho}^{*}(\omega)\wedge\eta)(x)\rightarrow 0$ as $\rho\rightarrow 0$,
while

$(f_{P}^{*}\omega\wedge\eta)(x)=(a\circ
f)(x)\,u(x)\,\,((D_{P}f)(x)^{*}\alpha)(x)\wedge\gamma(x).$

By Lemma 3.25 parts (3) and (1) the last expression is a form of weight
strictly less than $\nu$ and hence vanishes. Thus we have $\mu$-a.e. pointwise
convergence of the integrand.

To find an integrable majorant pick $\nu<q<p$. Let $\psi=\|D_{h}f\|^{q}$ in
$U$ and extend $\psi$ by zero to $G\setminus U$. By (6.5) we have $\psi\in
L^{\frac{p}{q}}(G)$. Let $M\psi$ be the maximal function:

$M\psi(x)=\sup_{r>0}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B(x,r)}\psi\,d\mu\,.$

Then it is standard (see [Hei01, Chapter 2]) that $M\psi\in
L^{\frac{p}{q}}(G)$, and thus $(M\psi)^{-\frac{w_{\alpha}}{q}}\in
L^{1}(\operatorname{spt}u)$.

Let $x\in\operatorname{spt}u$. Let $\Lambda$ be the constant in Lemma 2.8. If
$\rho<\Lambda^{-1}\rho_{0}$ we have $B(x,\Lambda\rho)\subset U$ and using
Lemma 2.8 we get

$f(B(x,\rho))\subset B(f(x),C_{x}\rho)$

where

$C_{x}\lesssim(M\psi)^{\frac{1}{q}}(x)\,.$

Now by Lemma 6.3(3) we have for all $x\in\operatorname{spt}u$

$\displaystyle\|f_{\rho}^{*}(\omega)\wedge\eta(x)\|$ $\displaystyle\leq$
$\displaystyle\|a\|_{\infty}\,\|u\|_{\infty}\,\|(f_{\rho}^{*}\alpha\wedge\gamma)(x)\|$
$\displaystyle\leq$ $\displaystyle
C^{\prime}C_{x}^{-w_{\alpha}}\rho^{-(\nu+w_{\alpha}+w_{\gamma})}\|a\|_{\infty}\,\|u\|_{\infty}\,\|\alpha\|\,\|\gamma\|$
$\displaystyle\lesssim$ $\displaystyle(M\psi)^{\frac{-w_{\alpha}}{q}}(x)\,.$

Since $f_{\rho}^{*}(\omega)\wedge\eta$ vanishes on
$G\setminus\operatorname{spt}u$ this shows that for
$\rho<\Lambda^{-1}\rho_{0}$ the integrands $\|f_{\rho}^{*}\omega\wedge\eta\|$
are dominated by a fixed $L^{1}$ function.

∎

##  7\. Rigidity of products

Let $\\{G_{i}\\}_{i\in I}$, $\\{G_{j}^{\prime}\\}_{j\in I^{\prime}}$ be finite
collections of Carnot groups, where each $G_{i}$, $G_{j}^{\prime}$ is
nonabelian and does not admit a nontrivial decomposition as a product of
Carnot groups. Let $G:=\prod_{i\in I}G_{i}$, $G^{\prime}:=\prod_{j\in
I^{\prime}}G^{\prime}_{j}$, and $\mathfrak{g}$, $\mathfrak{g}^{\prime}$ be the
graded Lie algebras.

######  Theorem 7.1.

If $U\subset G$ is open and $f:U\rightarrow G^{\prime}$ is a Sobolev mapping
such that such that

  1. (1)

$f\in W^{1,p}_{\operatorname{loc}}$ for $p>\nu$.

  2. (2)

$D_{P}f(x)$ is an isomorphism for $\mu$-a.e. $x\in V$.

Then up to reindexing the collections $\\{G_{i}\\}_{i\in I}$,
$\\{G^{\prime}_{j}\\}_{j\in I^{\prime}}$ are the same up to isomorphism, and
$f$ is locally a product of homeomorphisms, i.e. after shrinking $U$ if
necessary, there is a bijection $\sigma:I\rightarrow I^{\prime}$ such that the
composition $\pi_{\sigma(i)}\circ f:U\rightarrow G_{\sigma(i)}^{\prime}$
factors through the projection $\pi_{i}:U\rightarrow G_{i}$. In particular, if
$U=\prod_{i}U_{i}$ where each $U_{i}$ is connected then $f$ is a product, and
if $U=G$ and $f$ is a quasiconformal/quasisymmetric homeomorphism, then $f$ is
a product of homeomorphisms.

As an application of Theorem 7.1, we show that any quasi-isometry between
products of simply-connected nilpotent Lie groups must asymptotically respect
the product structure.

Let $\\{\hat{G}_{i}\\}_{i\in I}$, $\\{\hat{G}_{j}^{\prime}\\}_{j\in
I^{\prime}}$ be collections where each $\hat{G}_{i}$, $\hat{G}^{\prime}_{j}$
is either a simply-connected nilpotent Lie group with a left invariant
Riemannian metric, or a finitely generated nilpotent group equipped with a
word metric. By [Pan83] the associated collections of asymptotic cones
$\\{G_{i}\\}_{i\in I}$, $\\{G^{\prime}_{j}\\}_{j\in I^{\prime}}$ are Carnot
groups equipped with Carnot-Caratheodory metrics, up to bilipschitz
homeomorphism. We assume that every $G_{i}$, $G_{j}^{\prime}$ is nonabelian
and indecomposable as a Carnot group.

Set $\hat{G}:=\prod_{i}\hat{G}_{i}$,
$\hat{G}^{\prime}=\prod_{j}G_{j}^{\prime}$, and equip $\hat{G}$,
$\hat{G}^{\prime}$ with the $\ell^{2}$-distance functions

$d^{2}_{\hat{G}}(x,x^{\prime}):={\sum_{i}d_{G_{i}}^{2}(\pi_{i}(x),\pi_{i}(x^{\prime}))}\,,\quad
d^{2}_{\hat{G}^{\prime}}(x,x^{\prime}):={\sum_{j}d_{G_{j}^{\prime}}^{2}(\pi_{i}(x),\pi_{i}(x^{\prime}))}\,.$

If $i\in I$, we say that $x,x^{\prime}\in G$ are an $i$-pair if
$\pi_{j}(x)=\pi_{j}(x^{\prime})$ for every $j\neq i$.

######  Theorem 7.2.

For every $L\geq 1$, $A<\infty$, there is a function
$\varepsilon=\varepsilon(L,A):[0,\infty)\rightarrow(0,1]$ with
$\lim_{R\rightarrow\infty}\varepsilon(R)=0$ with the following property.

For every $(L,A)$-quasi-isometry $\Phi:\hat{G}\rightarrow\hat{G}^{\prime}$,
there is a bijection $\sigma:I\rightarrow I^{\prime}$, such that for every
$R\in[0,\infty)$, $i\in I$, and every $i$-pair $x,x^{\prime}\in\hat{G}$ with
$d(x,x^{\prime})\geq R$, we have

(7.3)
$\frac{d(\pi_{j}(\Phi(x)),\pi_{j}(\Phi(x^{\prime})))}{d(\Phi(x),\Phi(x^{\prime}))}\in\begin{cases}(1-\varepsilon,1]\quad\text{if}\quad
j=\sigma(i)\\\ [0,\varepsilon)\quad\text{if}\quad
j\neq\sigma(i)\,.\end{cases}$

######  Corollary 7.4.

If $\hat{G}$, $\hat{G}^{\prime}$, and $\Phi$ are as in Theorem 7.2, then
modulo reindexing, $\hat{G}_{i}$ is quasi-isometric to $\hat{G}_{i}^{\prime}$.

###  7.1. Proof of Theorem 7.1

We first treat a special case which exhibits the essential ideas, and then
explain how to modify the argument to handle the general case.

###### Proof of Theorem 7.1 when $G=G^{\prime}=\mathbb{H}\times\mathbb{H}$.

Since $G=G_{1}\times G_{2}$ with $G_{i}\simeq\mathbb{H}$, we have a graded
decomposition $\mathfrak{g}=\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$, so the
grading $\mathfrak{g}=V_{1}\oplus V_{2}$ is of the form
$V_{i}=\oplus_{j}V_{ij}$ where $i,j\in\\{1,2\\}$ and
$V_{ij}:=V_{i}\cap\mathfrak{g}_{j}$.

Let $f:G\supset U\rightarrow U^{\prime}\subset G$ be as in the statement of
Theorem 7.1. We may assume without loss of generality that $U=U_{1}\times
U_{2}$, where $U_{1}$, $U_{2}$ are connected open sets.

Step 1: Any graded automorphism $\phi:\mathfrak{g}\rightarrow\mathfrak{g}$
preserves the decomposition $\mathfrak{g}=\oplus_{i}\mathfrak{g}_{i}$, i.e.
there is a permutation $\sigma:\\{1,2\\}\rightarrow\\{1,2\\}$ such that
$\phi(\mathfrak{g}_{i})=\mathfrak{g}_{\sigma(i)}$.

Pick $j\in\\{1,2\\}$. Then $[V_{1j},V_{1}]=V_{2j}$, and since $\phi$ is a
graded automorphism, it follows that the image $\phi(V_{1j})$ is a
$2$-dimensional subspace of $V_{1}$ such that $[\phi(V_{1j}),V_{1}]\subset
V_{2}$ is $1$-dimensional. Hence the projection $\pi_{k}(\phi(V_{1,j}))\subset
V_{1k}$ can be nontrivial for at most one $k\in\\{1,2\\}$, which means that
$\phi(V_{1j})=V_{1\sigma(j)}$ for some $\sigma(j)\in\\{1,2\\}$. Since $\phi$
is an automorphism the map $\sigma:\\{1,2\\}\rightarrow\\{1,2\\}$ is a
bijection.

Step 2: There is a permutation $\sigma$ such that for a.e. $x\in U$, the Pansu
derivative $Df(x):\mathfrak{g}\rightarrow\mathfrak{g}$ satisfies
$Df(x)(\mathfrak{g}_{i})=\mathfrak{g}_{\sigma(i)}$.

Let $X_{1},\ldots,X_{6}$ be a graded basis for $\mathfrak{g}$ such that
$[X_{1},X_{2}]=-X_{3}$, $[X_{4},X_{5}]=-X_{6}$,
$V_{11}=\operatorname{span}\\{X_{1},X_{2}\\}$,
$V_{12}=\operatorname{span}\\{X_{4},X_{5}\\}$. Now let
$\theta_{1},\ldots,\theta_{6}$ be the dual basis, and we adopt the notation
from Subsection 3.3:
$\theta_{J}=\theta_{j_{1}}\wedge\ldots\wedge\theta_{j_{k}}$ if
$J=\\{j_{1}<\ldots<j_{k}\\}$.

Since the Pansu derivative $Df(x)$ is a well-defined measurably varying graded
automorphism of $\mathfrak{g}$ for a.e. $x\in U$, by Step 1 we may define a
measurable mapping $\sigma:U\rightarrow S_{2}$ such that

(7.5) $Df(x)(\mathfrak{g}_{i})=\mathfrak{g}_{\sigma(x)(i)}$

for a.e. $x\in U$. It follows that
$f_{P}^{*}\theta_{123}=a_{123}\theta_{123}+a_{456}\theta_{456}$ where
$a_{123},a_{456}\in L^{1}_{\operatorname{loc}}$ and

(7.6) $\displaystyle a_{456}(x)$
$\displaystyle=0\;\;\text{when}\;\;\sigma(x)=\operatorname{id}$ $\displaystyle
a_{123}(x)$ $\displaystyle=0\;\;\text{when}\;\;\sigma\neq\operatorname{id}\,.$

Pick $\psi\in C^{\infty}_{c}(U)$. Then $d(\psi\theta_{56})=\sum_{1\leq i\leq
4}(X_{i}\psi)\theta_{i56}$ so $\operatorname{wt}(d(\psi\theta_{56}))\leq-4$.
Since $\operatorname{wt}(\theta_{123})=-4$, by Theorem 4.1 we have

$\displaystyle 0=\int_{U}f_{P}^{*}\theta_{123}\wedge d(\psi\theta_{56})$
$\displaystyle=\int_{U}(a_{123}\theta_{123}+a_{456}\theta_{456})\wedge
d(\psi\theta_{56})$
$\displaystyle=\int_{U}a_{123}(X_{4}\psi)\operatorname{vol}_{G}\,.$

Since this holds for all $\psi$, we get that $X_{4}a_{123}=0$ as a
distribution. Similarly $X_{5}a_{123}=X_{1}a_{456}=X_{2}a_{456}$. This forces
$X_{6}a_{123}=X_{3}a_{456}=0$ since $X_{3}=-[X_{1},X_{2}]$ and
$X_{6}=-[X_{4},X_{5}]$. In view of the fact that $U_{1}$, $U_{2}$ are
connected, these distributional equations imply that, up to null sets,
$a_{123}$, $a_{456}$ will be nonzero on subsets of the form $A_{123}\times
U_{2}$, and $U_{1}\times A_{456}$, respectively. If both subsets had positive
measure, then both $a_{123}$, $a_{456}$ would both be nonzero a.e. in
$A_{123}\times A_{456}$, which contradicts (7.6). Therefore, after modifying
$\sigma$ on a set of measure zero, it will be constant in $U$ as desired.

Step 3: $f$ is a product of mappings. We may assume without loss of generality
that $\sigma=\operatorname{id}$.

By Fubini’s theorem, for a.e. $y\in U_{2}$, the map $f_{y}:U\rightarrow G$
defined by $f_{y}(x)=f(x,y)$ is in $W^{1,p}_{\operatorname{loc}}(U_{1})$, and
by the chain rule its Pansu derivative satisfies
$(D_{P}f)(x)(\mathfrak{g}_{1})\subset\mathfrak{g}_{1}$ for a.e. $x\in U_{1}$.
Hence for a.e. $y\in U_{2}$ we have $D_{P}(\pi_{2}\circ f_{y})(x)=0$ for a.e.
$x\in U_{1}$, and since $U_{1}$ is connected it follows that $\pi_{2}\circ
f_{y}$ is constant. Because $p>\nu$ we know that $f$ is continuous, and
therefore $\pi_{2}\circ f_{y}$ is constant for every $y\in U_{2}$, i.e.
$\pi_{2}\circ f(x,y)$ depends only on $y$. Similarly $(\pi_{1}\circ f)(x,y)$
depends only on $x$. Thus $f$ is a product of mappings. ∎

We now return to the general case of Theorem 7.1.

The generalization of Step 1 of the above proof is the following result from
[Xie13b, Prop. 2.5]. We include a short proof here.

######  Lemma 7.7.

Suppose $\mathfrak{g}=\oplus_{i\in I}\mathfrak{g}_{i}$,
$\mathfrak{g}^{\prime}=\oplus_{j\in I^{\prime}}\mathfrak{g}_{j}^{\prime}$
where every $\mathfrak{g}_{i}$, and $\mathfrak{g}^{\prime}_{j}$ is nonabelian
and does not admit a nontrivial decomposition as a direct sum of graded
ideals. Then any graded isomorphism
$\phi:\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$ is a product of graded
isomorphisms, i.e. there is a bijection $\sigma:I\rightarrow I^{\prime}$ and
for every $i\in I$ there exists a graded isomorphism
$\phi_{i}:\mathfrak{g}_{i}\rightarrow\mathfrak{g}^{\prime}_{\sigma(i)}$ such
that for all $i\in I$ we have $\pi_{\sigma(i)}\circ\phi=\phi_{i}\circ\pi_{i}$.

###### Proof.

For subsets $S_{1},S_{2}$ of $\mathfrak{g}$ or $\mathfrak{g}^{\prime}$, we let
$[S_{1},S_{2}]=\operatorname{span}(\\{[s_{1},s_{2}]\mid s_{i}\in S_{i}\\})$,
and $\mathcal{I}(S_{1})$ be the ideal generated by $S_{1}$. Note that the
center of $\mathfrak{g}$ intersects $V_{1}$ trivially, since every
$\mathfrak{g}_{i}$ is nonabelian and has no nontrivial decomposition as a
direct sum of graded ideals; similarly the center of $\mathfrak{g}^{\prime}$
intersects $V_{1}^{\prime}$ trivially.

We define a linear subspace $W\subset V_{1}\subset\mathcal{H}$ to be bracket
maximal if for every $Z\subset V_{1}$ with $Z\supset W$ and
$[Z,V_{1}]=[W,V_{1}]$ we have $Z=W$. Note that

$[Z,V_{1}]=[\oplus_{i\in I}\pi_{i}(Z),V_{1}],$

so if $W$ is bracket maximal then $W=\oplus_{i\in I}\pi_{i}(W)$. A similar
definition and remark applies in $\mathfrak{g}^{\prime}$.

For $i\in I$, let $V_{1i}=V_{1}\cap\mathfrak{g}_{i}$. Then $V_{1i}$ is bracket
maximal for all $i\in I$, since $V_{1i}\subsetneq Z\subset V_{1}$ implies that
$\pi_{j}(Z)\neq\\{0\\}$ for some $j\neq i$, and hence
$[\pi_{j}(Z),V_{1j}]\neq\\{0\\}$ since the center of $\mathfrak{g}_{j}$
intersects $V_{1j}$ trivially.

Since $\Phi$ is a graded isomorphism, it follows that $\Phi(V_{1i})\subset
V_{1}^{\prime}$ is bracket maximal, and
$\mathfrak{g}^{\prime}=\oplus_{i}\mathcal{I}(\Phi(V_{1i}))$. But since
$\Phi(V_{1i})$ is bracket maximal we have

$\Phi(V_{1i})=\oplus_{j\in I^{\prime}}(\pi_{j}(\Phi(V_{1i})))\,,$

giving

$\displaystyle\mathfrak{g}^{\prime}=\oplus_{i\in I}\mathcal{I}(\Phi(V_{1i}))=$
$\displaystyle\oplus_{i}\oplus_{j}\mathcal{I}(\pi_{j}(\Phi(V_{1i})))$
$\displaystyle=$
$\displaystyle\oplus_{j}\oplus_{i}\mathcal{I}(\pi_{j}(\Phi(V_{1i})))\,.$

Since $\mathfrak{g}^{\prime}_{j}$ does not admit a nontrival decomposition as
a direct sum of graded ideals, it follows that for every $j\in I^{\prime}$,
the ideal $\mathcal{I}(\pi_{j}(\Phi(V_{1i})))$ is $=\mathfrak{g}^{\prime}_{i}$
for precisely one $\tau(j)\in I$, and $=\\{0\\}$ for the rest. Since $\phi$ is
a graded isomorphism, $\tau$ is the inverse of a bijection
$\sigma:I\rightarrow I^{\prime}$, and the lemma follows. ∎

Let $f:G\supset U\rightarrow G$ be as in the statement of Theorem 7.1, where
$U=\prod_{i}U_{i}$ and the $U_{i}$s are connected. By Lemma 7.7, we may assume
without loss of generality that $I=I^{\prime}$ and
$\mathfrak{g}_{i}=\mathfrak{g}_{i}^{\prime}$ for all $i\in I$, and so there is
a measurable function $\sigma:U\rightarrow\operatorname{Perm}(I)$ such that
$Df(x)(\mathfrak{g}_{i})=\mathfrak{g}_{\sigma(x)(i)}$ for a.e. $x\in U$.

For $i\in I$ let $\theta_{i}$ denote the pullback
$\pi_{i}^{*}\operatorname{vol}_{G_{i}}$ where $\operatorname{vol}_{G_{i}}$ is
a volume form on the factor $G_{i}$. Pick $i_{0}\in I$, and let $I_{0}\subset
I$ be the indices $j\in I$ for which
$\mathfrak{g}_{j}\simeq\mathfrak{g}_{i_{0}}$. Then

$f_{P}^{*}\theta_{i_{0}}=\sum_{j\in I_{0}}a_{j}\theta_{j}\,,$

where $a_{j}\in L^{1}_{\operatorname{loc}}(U)$. For $j,j^{\prime}\in I_{0}$
and $X\in V_{1}\cap\mathfrak{g}_{j^{\prime}}$, let $\omega:=(\Lambda_{i\neq
j,j^{\prime}}\theta_{i})\wedge i_{X}\theta_{j^{\prime}}$. Taking $\psi\in
C^{\infty}_{c}(U)$, we apply Theorem 4.1 to get

$0=\int_{U}f_{P}^{*}\theta_{i_{0}}\wedge
d(\psi\omega)=\pm\int_{U}a_{j}(X\psi)\wedge_{i\in I}\theta_{i}$

so $Xa_{j}=0$ distributionally. Since $V_{1}\cap\mathfrak{g}_{j^{\prime}}$
generates $\mathfrak{g}_{j^{\prime}}$ as a Lie algebra, we get that $Za_{j}=0$
for all $Z\in\mathfrak{g}_{j^{\prime}}$. Arguing as before, we get that there
is a $\sigma(j)\in I_{0}$ such that $a_{j}\neq 0$ and $a_{k}=0$ a.e. in $U$.
The index $i_{0}$ was arbitrary, so we conclude that there is a permuation
$\sigma\in\operatorname{Perm}(I)$ such that
$Df(x)(\mathfrak{g}_{i})=\mathfrak{g}_{\sigma(i)}$ for a.e. $x\in U$. The last
step of the proof is similar to the special case.

###  7.2. Proof of Theorem 7.2

We begin with the following geometric result.

######  Lemma 7.8.

Let $X$ be a metric space, where one of the following holds:

  1. (a)

$X$ is isometric to a simply connected nilpotent Lie group $H$ with a left
invariant Riemannian metric, where $H\not\simeq\mathbb{R}$.

  2. (b)

$X$ is the Cayley graph of a finitely generated nilpotent group $H_{0}$, where
$H_{0}$ is not virtually cyclic.

Then there exist $\underline{R}<\infty$, $\lambda>0$ such that:

  1. (1)

Then for every $p,x_{1},x_{2}\in X$ such that
$R:=d(p,x_{1})\geq\underline{R}$, $x_{2}\in A(p,R,2R)$, there is a path
$\gamma:[0,1]\rightarrow X\setminus B(x,\lambda R)$ from $x_{1}$ to $x_{2}$.

  2. (2)

Let $P_{r}:=\\{(x,x^{\prime})\in X\times X\mid d(x,x^{\prime})\geq r\\}$ for
$r\in[0,\infty)$. Then for all $R\geq\underline{R}$, the subset $P_{R}\subset
P_{\lambda R}$ is contained in a single path component of $P_{\lambda R}$.

###### Proof.

(1). Suppose (1) is false. Then there are sequences $\\{p_{k}\\}$,
$\\{x_{1,k}\\}$, $\\{x_{2,k}\\}$ in $X$ such that
$R_{k}:=d(p_{k},x_{1,k})\rightarrow\infty$, $x_{2,k}\in
A(p_{k},R_{k},2R_{k})$, but there is no path from $x_{1,k}$ to $x_{2,k}$ in
$X\setminus B(p_{k},\frac{1}{k}R_{k})$. After passing to a subsequence, the
tuples $(X,R_{k}^{-1}d_{X},p_{k};x_{1,k},x_{2,k})$ will pointed Gromov-
Hausdorff converge to a tuple
$(X_{\infty},d_{\infty},p_{\infty};x_{1,\infty},x_{2,\infty})$ where
$(X_{\infty},d_{\infty})$ is bilipschitz homeomorphic to a Carnot group
[Pan83]. Since in case (a) $H\not\simeq\mathbb{R}$ and in case (b) $H_{0}$ is
not virtually cyclic, we know that $X_{\infty}$ is homeomorphic to
$\mathbb{R}^{n}$ for some $n\geq 2$. Therefore there is a path
$\gamma:[0,1]\rightarrow X_{\infty}\setminus B(p_{\infty},3a)$ from
$x_{1,\infty}$ to $x_{2,\infty}$ for some $a>0$. Subdividing $\gamma$, we
obtain a sequence of points

$x_{1,\infty}=z_{0},\ldots,z_{n}=x_{2,\infty}\subset X_{\infty}\setminus
B(p_{\infty},3a)$

with $d(z_{j-1},z_{j})<a$ for all $1\leq j\leq n$. Hence for large $k$ there
exist

$x_{1,k}=z_{0,k},\ldots,z_{n,k}=x_{2,k}\subset X\setminus B(p_{k},2aR_{k})$

such that $d(z_{j-1,k},z_{j,k})<aR_{k}$ for all $1\leq j\leq n$. Joining
$z_{j-1,k}$ to $z_{j,k}$ by a geodesic $\gamma_{j,k}$ and concatenating the
$\gamma_{j,k}$s, we get a contradiction.

(2). Iterating (1), we may reduce to the case when
$d(x,x^{\prime})=d(y,y^{\prime})=10d(x,y)$. Then applying (1) twice more, we
get that $(x,x^{\prime})$, $(x,y^{\prime})$ and $(y,y^{\prime})$ lie in the
same path component of $P_{\lambda R}$.

∎

Now let $\\{\hat{G}_{i}\\}_{i\in I}$, $\\{\hat{G}^{\prime}_{j}\\}_{j\in
I^{\prime}}$ be as in Theorem 7.2.

Our first step is to show that the image of an $i$-pair under a quasi-isometry
is approximately a $j$-pair for some $j=j(i,x,x^{\prime})$, provided
$d(x,x^{\prime})$ is large.

######  Lemma 7.9.

For every $L\geq 1$, $A<\infty$, $\varepsilon>0$, there is an
$R=R(L,A,\varepsilon)<\infty$ such that if
$\Phi:\hat{G}\rightarrow\hat{G}^{\prime}$ is an $(L,A)$-quasi-isometry, $i\in
I$, and $x,x^{\prime}\in G$ is an $i$-pair such that $d(x,x^{\prime})\geq R$,
then for every $j\in I^{\prime}$ we have

$\frac{d(\pi_{j}(\Phi(x)),\pi_{j}(\Phi(x^{\prime})))}{d(\Phi(x),\Phi(x^{\prime}))}\in[0,\varepsilon)\cup(1-\varepsilon,1]\,.$

###### Proof.

If not, then for some $L$, $A$, $\varepsilon$, there is a sequence of
$(L,A)$-quasi-isometries $\\{\Phi_{k}:\hat{G}\rightarrow\hat{G}^{\prime}\\}$,
and for some $i\in I$ a sequence of $i$-pairs
$\\{x_{k},y_{k}\\}\subset\hat{G}$ such that
$d(x_{k},x_{k}^{\prime})\rightarrow\infty$, and for every $k$

(7.10)
$\frac{d(\pi_{j}(\Phi_{k}(x_{k})),\pi_{j}(\Phi_{k}(x^{\prime}_{k})))}{d(\Phi_{k}(x_{k}),\Phi_{k}(x^{\prime}_{k}))}\in[\varepsilon,1-\varepsilon]\,.$

By pre/postcomposing with translations, we may assume that $x_{k}=e$ and
$\Phi_{k}(x_{k})=e$ for all $k$. Letting $R_{k}:=d(x_{k},y_{k})$, the maps
$\Phi_{k}$ induce $(L,R_{k}^{-1}A)$-quasi-isometries
$(\hat{G},R_{k}^{-1}d_{\hat{G}})\rightarrow(\hat{G}^{\prime},R_{k}^{-1}d_{\hat{G}^{\prime}})$.
Extracting a subsequential limit, we obtain an $L$-bilipschitz homeomorphism
$\Phi_{\omega}:G_{\omega}=\prod_{i}G_{i}\rightarrow\prod_{j}G_{j}^{\prime}=G_{\omega}^{\prime}$,
and by (7.10) there is an $i$-pair $x_{\omega},y_{\omega}\in G_{\omega}$ whose
image under $\Phi_{\omega}$ is not a $j$-pair for any $j$. This is a
contradiction, since $\\{G_{i}\\}_{i\in I}$, $\\{G_{j}^{\prime}\\}_{j\in
I^{\prime}}$ satisfy the assumptions of Theorem 7.1 and so $\Phi_{\omega}$ is
a product mapping. ∎

We now fix an $(L,A)$-quasi-isometry
$\Phi:\hat{G}\rightarrow\hat{G}^{\prime}$.

For $i\in I$, $R\in[0,\infty)$, let $P_{i,R}$ be the collection of $i$-pairs
$x,x^{\prime}\in G$ with $d(x,x^{\prime})\geq R$. Let $R_{0}:=2L^{-1}A$. Since
$\Phi$ is an $(L,A)$-quasi-isometry, the mapping
$\Phi_{ij}:P_{i,R_{0}}\rightarrow[0,1]$ given by

$\Phi_{ij}(x,x^{\prime}):=\frac{d(\pi_{j}(\Phi(x)),\pi_{j}(\Phi(x^{\prime})))}{d(\Phi(x),\Phi(x^{\prime}))}$

is well-defined. Let $R_{1}=\max(R_{0},R(L,A,\frac{1}{4}))$ where
$R(L,A,\frac{1}{4})$ is the constant from Lemma 7.9. By that lemma we have a
well-defined function

$\bar{\Phi}_{ij}:P_{i,R}\rightarrow\\{0,1\\}$

where

$\bar{\Phi}_{ij}(x,x^{\prime})=\begin{cases}1\quad\text{if}\quad\Phi_{ij}(x,x^{\prime})\in(\frac{3}{4},1]\\\
0\quad\text{if}\quad\Phi_{ij}(x,x^{\prime})\in[0,\frac{1}{4})\,.\end{cases}$

######  Lemma 7.11.

When $R\geq R_{2}=R_{2}(L,A)$ then
$\bar{\Phi}_{ij}(x_{1},x^{\prime}_{1})=\bar{\Phi}(x_{2},x_{2}^{\prime})$ for
every pair $(x_{1},x_{1}^{\prime}),(x_{2},x_{2}^{\prime})\in P_{i,R}$ with
$d(x_{1},x_{2}),d(x_{1}^{\prime},x_{2}^{\prime})\leq 1$.

###### Proof.

We have

$d(\Phi(x_{1}),\Phi(x_{2})),d(\Phi(x_{1}^{\prime}),\Phi(x_{2}^{\prime}))\leq
L+A\,,$

which implies

$\displaystyle|\Phi_{ij}(x_{1},x_{1}^{\prime})-\Phi_{ij}(x_{2},x_{2}^{\prime})|\leq\frac{2(L+A)}{L(R-1)-2A}$

which is $<\frac{1}{4}$ when $R>R_{2}=R_{2}(L,A)$. The distance between the
intervals $[0,\frac{1}{4})$ and $(\frac{3}{4},1]$ is $>\frac{1}{4}$, and
therefore
$\bar{\Phi}_{ij}(x_{1},x^{\prime}_{1})=\bar{\Phi}(x_{2},x_{2}^{\prime})$ as
claimed. ∎

Combining Lemma 7.11 with Lemma 7.8, we get that for $R_{3}=R_{3}(L,A)>R_{2}$,
for every $i\in I$, the subset $P_{i,R_{3}}\subset P_{i,R_{2}}$ lies in a
single path component, and hence $\bar{\Phi}_{ij}$ is constant on
$P_{i,R_{3}}$. Applying Lemma 7.9 to an $i$-pair $(x,x^{\prime})$ with
$d(x,x^{\prime})$ large, we see that for every $i$ there is a $\sigma(i)\in
I^{\prime}$ such that in $P_{i,R_{3}}$ we have $\bar{\Phi}_{i\sigma(i)}\equiv
1$ and $\bar{\Phi}_{ij}\equiv 0$ for $j\in I^{\prime}\setminus\sigma(i)$. If
$\Phi^{\prime}:\hat{G}^{\prime}\rightarrow\hat{G}$ is an
$(L^{\prime},A^{\prime})$-quasi-inverse of $\Phi$, where
$L^{\prime}=L^{\prime}(L,A)$, $A^{\prime}=A^{\prime}(L,A)$, then we argue
similarly to define $\Phi^{\prime}_{ij}$, $\bar{\Phi}^{\prime}_{ij}$,
$R_{3}^{\prime}$, and $\sigma^{\prime}:I^{\prime}\rightarrow I$. Again taking
an $i$-pair $(x,x^{\prime})$ with $d(x,x^{\prime})$ large, we can find a
$\sigma(i)$-pair $(y,y^{\prime})\subset P^{\prime}_{\sigma(i),R_{3}^{\prime}}$
with $y:=\Phi(x)$, $d(y^{\prime},\Phi(x^{\prime}))\ll
d(\Phi(x),\Phi(x^{\prime}))$, and applying Lemma 7.9 we get that
$\sigma^{\prime}(\sigma(i))=i$. Similarly
$\sigma\circ\sigma^{\prime}=\operatorname{id}_{I^{\prime}}$, so $\sigma$ is a
bijection. Now (7.3) follows from Lemma 7.9.

###  7.3. Proof of Corollary 7.4

Let $\Phi$, $\sigma$, etc be as in Theorem 7.2. Let
$\Phi^{\prime}:\hat{G}^{\prime}\rightarrow\hat{G}$ be a quasi-inverse of
$\Phi$, i.e. $\Phi^{\prime}$ is an $(L,A_{1})$-quasi-isometry such that
$d(\Phi^{\prime}\circ\Phi,\operatorname{id}_{\hat{G}})$,
$d(\Phi\circ\Phi^{\prime},\operatorname{id}_{\hat{G}^{\prime}})<A_{1}$, where
$A_{1}=A_{1}(L,A)$.

Choose $i\in I$ and for every $j\neq i$ pick some $x_{j}\in\hat{G}_{j}$. Now
define $\alpha:\hat{G}_{i}\rightarrow\hat{G}$ by $\pi_{j}(\alpha(x))=x_{j}$
for $j\neq i$, and $\pi_{i}(\alpha(x))=x$, and let
$\phi:\hat{G}_{i}\rightarrow\hat{G}^{\prime}_{\sigma(i)}$ be the composition
$\phi:=\pi_{\sigma(i)}\circ\Phi\circ\alpha$. It follows from (7.3) that $\phi$
is an $(L,A_{2})$ quasi-isometric embedding for $A_{2}=A_{2}(L,A)$. We will
show that $\phi$ is a quasi-isometry.

Pick $y_{0}^{\prime}\in\hat{G}^{\prime}_{\sigma(i)}$. Now let
$y_{1}\in\hat{G}_{i}$ satisfy $d(\phi(y_{1}),y_{0}^{\prime})\leq 2\rho$ where
$\rho:=\inf_{x\in\hat{G}_{i}}d(\phi(y),y_{0}^{\prime})$. Let
$x_{0}^{\prime}\in\hat{G}^{\prime}$ be the point with
$\pi_{\sigma(i)}(x_{0}^{\prime})=y_{0}^{\prime}$,
$\pi_{j}(x_{0}^{\prime})=\pi_{j}(\Phi(y_{1}))$ for all $j\in
I^{\prime}\setminus\\{\sigma(i)\\}$. Let
$y_{2}:=\pi_{i}(\Phi^{\prime}(x_{0}^{\prime}))$. Now

$\displaystyle\rho$ $\displaystyle\leq d(\phi(y_{2}),y_{0}^{\prime})\leq
d(\Phi\circ\alpha(y_{2}),x_{0}^{\prime})$ $\displaystyle\leq
d(\Phi\circ\alpha(y_{2}),\Phi(\Phi^{\prime}(x_{0}^{\prime})))+d(\Phi(\Phi^{\prime}(x_{0})),x_{0}^{\prime})$
$\displaystyle\leq Ld(\alpha(y_{2}),\Phi^{\prime}(x_{0}^{\prime}))+A+A_{1}\,.$

Applying Theorem 7.2 to $\Phi^{\prime}$, it follows from (7.3) that

$\rho^{-1}\,d(\Phi^{\prime}(x_{0}^{\prime}),\alpha(y_{2}))\leq\varepsilon_{1}(\rho)\,,$

where $\varepsilon_{1}=\varepsilon_{1}(L,A)$, and
$\varepsilon_{1}(t)\rightarrow 0$ as $t\rightarrow 0$. Hence $\rho\leq
A_{3}=A_{3}(L,A)$. Thus $\phi$ is an $(L,A_{4})$-quasi-isometry. The index
$i\in I$ was arbitrary, so the corollary follows.

##  8\. Complexifed Carnot groups

In this section we study mappings between complexified Carnot groups, the
complex Heisenberg groups being prime examples. In Subsection 8.1 we discuss
local results, in Subsection 8.3 global rigidity, and in Subsection 8.4 we
give examples of mappings exhibiting flexibility.

Recall that the complexification $V_{\mathbb{C}}$ of a real vector space $V$
is the tensor product $V\otimes_{\mathbb{R}}\mathbb{C}$, and that
complexification has obvious compatibilities with tensor/wedge products and
complex conjugation:

$\displaystyle(V\otimes_{\mathbb{R}}W)_{\mathbb{C}}$
$\displaystyle=(V\otimes_{\mathbb{R}}W)\otimes_{\mathbb{R}}\mathbb{C}\simeq(V_{\mathbb{C}})\otimes_{\mathbb{C}}(W_{\mathbb{C}})$
$\displaystyle\overline{v\otimes w}$
$\displaystyle=\overline{v}\otimes\overline{w}\,,\qquad v\in
V_{\mathbb{C}}\,,w\in W_{\mathbb{C}}\,.$

If $M$ is a manifold, then may complexify the algebra $\Omega^{*}(M)$ of
differential forms, and then pullback and exterior derivative are
$\mathbb{C}$-linear and real (i.e. they commute with complex conjugation).
Similarly for Pansu pullback, since it is based on an algebra homomorphism of
exterior algebras.

###  8.1. Local results

Let $H$ be a Carnot group of topogical dimension $N$ and homogeneous dimension
$\nu$. Let $\mathfrak{h}$ be the corresponding Carnot algebra. By
$\mathfrak{g}=\mathfrak{h}^{\mathbb{C}}$ we denote the complexification of
$\mathfrak{h}$. It follows from the Baker-Campbell-Hausdorff formula that
$\mathfrak{g}$ is the Lie algebra of a complex Lie group $G$. In the following
we view $G$ as a real Carnot group of topological dimension $2N$ and
homogeneous dimension $2\nu$ and we denote complex multiplication by $J$. Our
main result is:

######  Theorem 8.1.

Suppose $U\subset G$ is a connected open set, and $f:U\rightarrow G$ is a map
in the Sobolev space $W^{1,p}(U,G)$ with $p>2\nu$ such that for a.e. $x\in U$
the Pansu derivative $D_{P}f(x)$ is either a $J$-linear isomorphism or a
$J$-antilinear isomorphism. Then $f$ is holomorphic or antiholomorphic (with
respect to the complex structure $J$).

Since graded isomorphisms
$\mathfrak{h}_{m}^{\mathbb{C}}\rightarrow\mathfrak{h}_{m}^{\mathbb{C}}$ of the
complex $m^{th}$ Heisenberg Lie algebra are either $J$-linear or
$J$-antilinear (Lemma 10.6), Theorem 8.1 applies if $G$ is a complex
Heisenberg group and $f$ is quasisymmetric, or more generally, when its Pansu
derivative is an isomorphism almost everywhere.

The assumption that the Pansu derivative is an isomorphism a.e. cannot be
dropped. After the end of the proof we give an example of a Lipschitz map $f$
from the complexified first Heisenberg group $\mathbb{H}^{\mathbb{C}}$ to
itself for which the Pansu differential is a.e. $J$-linear or $J$-antilinear,
but which is holomorphic nor antiholomorphic.

######  Corollary 8.2.

Let $f:G\supset U\rightarrow G$ be as in Theorem 8.1, where $U=B(p,r)$ for
some $p\in G$, $r>0$. If $f$ is an $\eta$-quasisymmetric homeomorphism, then:

  * •

$f$ is either $J$-biholomorphic or $J$-antibiholomorphic.

  * •

Modulo post-composition with a Carnot dilation, the restriction of $f$ to the
subball $B(p,\frac{r}{K})$ is $K$-bilipschitz, where $K=K(\eta)$.

######  Lemma 8.3.

Let $G$ be a complexified Carnot group and suppose that $f:G\supset B(p,r)\to
B(p^{\prime},\rho)\subset G$ is a holomorphic contact map. Then the
restriction of $f$ to the ball $B(p,r/4)$ is $K_{G}\frac{\rho}{r}$-Lipschitz
where $K_{G}$ depends only on the group $G$.

###### Proof.

Since dilation and left translation are holomorphic contact maps it suffices
to show the result for $p=p^{\prime}=e$ and $r=\rho=1$. Note that the ball
$B(e,1)$ is contained in an Euclidean ball $B^{\rm eucl}(e,R_{G})$ and that
the left-invariant norm of horizontal vectorfields on $B(e,1)$ is comparable
to the Euclidean norm.

By compactness there exist a $r_{G}>0$ such for every point in
$q\in\overline{B}(e,\frac{3}{4})$ we have $B^{\rm eucl}(q,r_{G})\subset
B(e,1)$.

Thus by standard estimates for holomorphic maps $f:B^{\rm eucl}(q,r_{G})\to
B^{\rm eucl}(e,R_{G})$ we get

$|Df(q)X|_{\rm eucl}\leq C_{N}R_{G}r_{G}^{-1}|X|_{\rm eucl}\quad\forall q\in
B(e,\frac{3}{4})$

where $N$ is the topological dimension of $G$. Since the left-invariant norm
and the Euclidean norm of horizontal vectorfields on $B(e,\frac{3}{4})$ are
equivalent we get

$|DfX|\leq C_{G}R_{G}r_{G}^{-1}|X|$

for all horizontal vectorfields $X$ in $B(e,\frac{3}{4})$. Finally we use that
any two points $q,q^{\prime}$ in $B(e,\frac{1}{4})$ can be connected by a
horizontal curve of length $<\frac{1}{2}$. In particular this curves stays
inside $B(0,\frac{3}{4})$. Thus $d(\tilde{f}(q),\tilde{f}(q^{\prime}))\leq
C_{G}R_{G}r_{G}^{-1}d(q,q^{\prime})$. ∎

###### Proof of Corollary 8.2.

After composing $f$ with complex conjugation if necessary, we may assume by
Theorem 8.1 that $f$ is holomorphic. Since $f$ is a holomorphic homeomorphism,
its Jacobian is nonzero away from a complex analytic subvariety $Z\subset U$,
so $f(Z)\subset$ has measure zero. Hence $f^{-1}$ is locally a holomorphic
diffeomorphism on $f(U)\setminus f(Z)$. Since $f^{-1}$ is
$\eta$-quasisymmetric, by Theorem 8.1 it is holomorphic, and hence both $f$
and $f^{-1}$ are biholomorphic; furthermore both maps are contact
diffeomorphisms because they are Pansu differentiable almost everywhere.

Let $M=\eta(1)$. Since $f$ is $\eta$-quasisymmetric we may assume after
postcomposing with a dilation that $B(p^{\prime},M^{-1}r)\subset
f(B(p,r/2))\subset B(p^{\prime},r)$ where $p^{\prime}=f(p)$. Thus by Lemma 8.3
the restriction of $f$ to $B(p,r/8)$ is $2K_{G}$-Lipschitz. Let $g$ be the
restriction of $f^{-1}$ to $B(p^{\prime},M^{-1}r)$. Then by Lemma 8.3 the
restriction of $g$ to $B(p^{\prime},M^{-1}r/4)$ is $MK_{G}$-Lipschitz.
Moreover $f(B(p,K_{G}^{-1}M^{-1}r/8))$ is contained in
$B(p^{\prime},M^{-1}r/4)$. Thus the assertion holds with $K=8MK_{G}$. ∎

###### Proof of Theorem 8.1.

The main point in the proof is to show that the pullback of the top degree
holomorphic form cannot oscillate between a holomorphic and anti-holomorphic
form.

For simplicity, we begin by proving the result for the complexified first
Heisenberg group. The general case is very similar, although it requires a few
minor modifications which we discuss at the end. We first set up some
notation. Let $\mathfrak{g}$ be the first complex Heisenberg algebra. Since we
will shortly be complexifying a second time, we recall the convention to
denote the complex multiplication on $\mathfrak{g}$ by
$J:\mathfrak{g}\rightarrow\mathfrak{g}$. Let $\\{X_{j},Y_{j}=JX_{j}\\}_{1\leq
j\leq 3}$ be the basis for which $[X_{1},X_{2}]=-X_{3}$, so

$[X_{1},X_{2}]=-X_{3}\,,\quad[Y_{1},X_{2}]=-Y_{3}$
$[X_{1},Y_{2}]=-Y_{3}\,,\quad[Y_{1},Y_{2}]=J^{2}(-X_{3})=X_{3}\,.$

and all other brackets of basis vectors are zero. Let
$\\{\alpha_{j},\beta_{j}\\}_{1\leq j\leq 3}$ be the dual basis. Then

$d\alpha_{3}=\alpha_{1}\wedge\alpha_{2}-\beta_{1}\wedge\beta_{2}\,,\quad
d\beta_{3}=\beta_{1}\wedge\alpha_{2}+\alpha_{1}\wedge\beta_{2}\,.$

Now we complexify $\mathfrak{g}$, the exterior algebra, and differential
forms.

Considering the $(1,0)$-forms

$\displaystyle\zeta_{j}=\alpha_{j}+i\beta_{j}\,,\quad\bar{\zeta}_{j}=\alpha_{j}-i\beta_{j}$

and the Wirtinger vector fields

$\displaystyle
Z_{j}=\frac{1}{2}(X_{j}-iY_{j})\quad\bar{Z}_{j}=\frac{1}{2}(X_{j}+iY_{j}).$

one gets:

$\displaystyle\zeta_{j}(Z_{k})$