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\datereceived{2018-03-08}
\daterevised{2018-11-09}
\dateaccepted{2018-12-04}
\editor{X. Caruso}
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\title[\texorpdfstring{$\ell^{q,p}$}{ell(q,p)} cohomology of Carnot groups]{On the
\texorpdfstring{$\ell^{q,p}$}{ell(q,p)} cohomology of Carnot groups}
\alttitle{Sur la cohomologie \texorpdfstring{$\ell^{q,p}$}{ell(q,p)} des groupes de Carnot}
\author[\initial{P.} \lastname{Pansu}]{\firstname{Pierre} \lastname{Pansu}}
\address{Laboratoire de Math\'ematiques d'Orsay\\
Universit\'e Paris-Sud, CNRS\\
Universit\'e Paris-Saclay, 91405 Orsay (France)}
\email{pierre.pansu@math.u-psud.fr}
\author[\initial{M.} \lastname{Rumin}]{\firstname{Michel} \lastname{Rumin}}
\address{Laboratoire de Math\'ematiques d'Orsay\\
Universit\'e Paris-Sud, CNRS\\
Universit\'e Paris-Saclay, 91405 Orsay (France)}
\email{michel.rumin@math.u-psud.fr}
\begin{abstract}
We study the simplicial $\ell^{q,p}$ cohomology of Carnot groups $G$. We show vanishing and non-vanishing results depending of the range of the $(p,q)$ gap with respect to the weight gaps in the Lie algebra cohomology of $G$.
\end{abstract}
\begin{altabstract}
On \'etudie la cohomologie $\ell^{q,p}$ des groupes de Carnot. On d\'emontre qu'elle s'annule (ou non) suivant la position des exposants $(p,q)$ par rapport aux poids pr\'esents dans la cohomologie de l'alg\`ebre de Lie.
\end{altabstract}
\subjclass{20F65, 22E30, 32V20, 58J40, 35R03, 20J06, 57T10, 47B06, 43A85}
\keywords{$\ell^{q,p}$ cohomology, Carnot group, global analysis on manifolds}
\thanks{Both authors supported in part by MAnET Marie Curie Initial Training Network. P.P. supported by Agence Nationale de la Recherche, ANR-15-CE40-0018.}
\begin{document}
\enlargethispage{12pt}
\maketitle
\section{Introduction}
\subsection{\texorpdfstring{$\ell^{q,p}$}{ell(q,p)} cohomology}
Let $T$ be a countable simplicial complex. Given $1\leq p\leq q\leq +\infty$, the $\ell^{q,p}$ cohomology of $T$ is the quotient of the space of $\ell^p$ simplicial cocycles by the image of $\ell^q$ simplicial cochains by the coboundary $\dd$,
\[
\ell^{q,p}H^k(T)=(\ell^p C^k(T)\cap \ker \dd)/\dd(\ell^q C^{k-1}(T)) \cap \ell^p C^k(T).
\]
It is a quasiisometry invariant of bounded geometry simplicial complexes whose usual cohomology vanishes in a uniform manner, see~\cite{Bourdon-Pajot,Elek,Genton,Gromov,Pansu2}. Riemannian manifolds $M$ with bounded geometry admit quasiisometrically homotopy equivalent simplicial complexes (a construction is provided below, in Section~\ref{sec:leray}). Uniform vanishing of cohomology passes through. Therefore one can take the $\ell^{q,p}$ cohomology of any such complex as a definition of the $\ell^{q,p}$ cohomology of $M$.
$\ell^{q,p}$ cohomology has an analytic flavor. Its vanishing is equivalent to a \emph{discrete Poincar\'e inequality}: every $\ell^p$ $k$-cocycle $\kappa$ admits an $\ell^q$ primitive, i.e. a $(k-1)$-cochain $\lambda$ such that $\dd\lambda=\kappa$ and
\begin{equation}\label{eq:1}
\|\lambda\|_q \leq C \|\kappa\|_p.
\end{equation}
However, one may also think of $\ell^{q,p}$ cohomology as a (large scale) topological invariant. It has been useful in several contexts, mainly for the class of hyperbolic groups where the relevant value of $q$ is $q=p$, see~\cite{Bourdon-Kleiner,Bourdon-Pajot, Cornulier-Tessera,Drutu-Mackay} for instance. For this class of groups, $\ell^{p,p}$ cohomology is often Hausdorff and can sometimes be viewed as a function space of objects (functions, forms,\dots) living on the ideal boundary, opening the way to analysis on the ideal boundary. The intervals where $\ell^{p,p}$ cohomology vanishes (resp. is Hausdorff) provide numerical quasiisometry invariants of hyperbolic groups and negatively curved manifolds. In Riemannian geometry, negative curvature pinching seems to have a direct relation with $\ell^{p,p}$ cohomology, see~\cite{Pansu4}.
It is interesting to study a class of spaces where values of $q\not=p$ play a significant role. The goal of the present paper is to compute $\ell^{q,p}$ cohomology, to some extent, for certain Carnot groups. Even the case of abelian groups is not straightforward. A more remote goal would be to cover the whole family of groups of polynomial growth, the quasiisometry classification of which is still open. In this class, $\ell^{q,p}$ cohomology is expected to be either $0$ or nonHausdorff. Indeed, a result of Gromov, \cite[8.C]{Gromov}, states that for $p>1$, the reduced $\ell^{p,p}$ cohomology of a group with infinite center must vanish. Nevertheless, the intervals of $p$ and $q$ where $\ell^{q,p}$ cohomology vanishes could be exploited. Unfortunately, our present results have no direct consequences for the quasiisometry classification problem, since the quasiisometry classification of Carnot groups is known.
Before stating a general result, we shall describe a few examples, and indicate ideas along the way. A more thorough explanation of the methods used appears in Section~\ref{sec:method}.
\subsection{The Euclidean Sobolev inequality}
In Euclidean space $\R^n$, the Sobolev inequality states that there exists a constant $C=C(n)$ such that for every smooth compactly supported function $u$,
\begin{equation}\label{eq:2}
\|u\|_q \leq C \|\du\|_p,
\end{equation}
provided $p\geq 1$ and
\begin{equation*}
\frac{1}{p}-\frac{1}{q}=\frac{1}{n}.
\end{equation*}
Here is a sophisticated proof, valid if $p>1$. Let $\dd^*$ denote the adjoint of $\dd$, a differential operator from $1$-forms to functions. Let $\Delta=\dd^*\dd$ be the Laplacian. Then $\Delta$ admits a pseudodifferential inverse $\Delta^{-1}$ which commutes with $\dd$. Let $K=\Delta^{-1}\dd^*$. Then $\dK=1$, hence $v=K\du$ satisfies $\du=\dv$. Calderon--Zygmund theory shows that if $p>1$ and $\frac{1}{p}-\frac{1}{q}=\frac{1}{n}$, $K$ is bounded from $L^p$ to $L^q$. Hence $v=u$ and $\|u\|_q=\|K\du\|_q\leq C\,\|\du\|_p$.
This argument proves a slightly different statement, the \emph{continuous Poincar\'e inequality}: under the same assumption on $p>1$ and $q$, there exists a constant $C=C(n)$ such that for every smooth function $u$ such that $\du\in L^p$, there exists $c_u\in\R$ such that
\begin{equation}\label{eq:3}
\|u-c_u\|_q \leq C \|\du\|_p.
\end{equation}
This is strongly reminiscent of the discrete Poincar\'e inequality~\eqref{eq:1} in degree $k=1$.
Conversely, if $\frac{1}{p}-\frac{1}{q}\not=\frac{1}{n}$, sequences of functions yielding counterexamples to~\eqref{eq:2} can be constructed from functions which are homogeneous under the $1$-parameter group of homotheties $s\mapsto h_s=e^{s}\Id$ of $\R^n$. If $\frac{1}{p}-\frac{1}{q}>\frac{1}{n}$, examples exist with compact support in a fixed ball: the inequality fails for local reasons. If $\frac{1}{p}-\frac{1}{q}<\frac{1}{n}$, examples exist which satisfy a uniform bound on any finite number of derivatives: the inequality fails for large scale reasons. This dichotomy indicates that~\eqref{eq:2} is not equivalent to~\eqref{eq:1}. The continuous analogue of~\eqref{eq:1} amounts to requiring that~\eqref{eq:2} holds with Sobolev norms involving many derivatives instead of mere $L^p$ norms, see Theorem~\ref{prop:leray}.
Once the discretization dictionary is established, validity of~\eqref{eq:2} for $\frac{1}{p}-\frac{1}{q}=\frac{1}{n}$ implies validity of~\eqref{eq:1}. Since $\ell^{q'}\subset \ell^{q}$ for $q'\leq q$, the validity of~\eqref{eq:1} for $\frac{1}{p}-\frac{1}{q}\geq\frac{1}{n}$ follows automatically.
\subsection{Higher degree forms on abelian groups}
As we have just seen, inequality~\eqref{eq:2} is related to cohomology in degree~$1$. In higher degrees, let us define the continuous Poincar\'e inequality as follows: there exists a constant $C$ such that every closed differential $k$-form $\omega$ admits a primitive, i.e. a differential $k-1$-form $\phi$ such that $\dphi=\omega$ and
\begin{equation}\label{eq:4}
\|\phi\|_q \leq C \|\omega\|_p,
\end{equation}
where $\dd$ now denotes the exterior differential. For which exponents does such an inequality hold? Homogeneity under homotheties imposes a restriction. If $\omega$ is a differential $k$-form on $\R^n$, then
\begin{equation}\label{eq:5}
\|h_s^*\omega\|_p=e^{s(\frac{n}{p}-k)}\|\omega\|_p.
\end{equation}
Therefore in order for inequality~\eqref{eq:4} to hold for every closed $k$-form $\omega$ on $\R^n$, one needs that $\frac{n}{p}-k=\frac{n}{q}-(k-1)$, i.e.
\begin{equation*}
\frac{1}{p}-\frac{1}{q}=\frac{1}{n},
\end{equation*}
like in degree~$1$.
The argument based on properties of the scalar Laplacian extends to other degrees. To prove~\eqref{eq:4}, a homotopy $K$ is constructed. This is an operator on differential forms that decreases the degree by $1$ unit and satisfies $1=\dK+K\dd$. If $\dd\omega=0$, then $\omega=\dd(K\omega)$. Our favorite choice is $K=\dd^*\Delta^{-1}$, where $\dd^*$ is the $L^2$ adjoint of $\dd$, $\Delta= \dd\dd^*+\dd^*\dd$ and $\Delta^{-1}$ is a pseudodifferential inverse of $\Delta$, which commutes with $\dd$. Calderon--Zygmund theory shows again that if $p>1$ and $\frac{1}{p}-\frac{1}{q}=\frac{1}{n}$ is bounded from $L^p$ to $L^q$.
We shall see that this argument can be adapted to Carnot groups, with substantial changes.
\subsection{The Carnot Sobolev inequality}
A \emph{homogeneous group} is a Lie group $G$ quipped with a $1$-parameter group $s\mapsto h_s$ of expanding automorphisms. Expanding means that the eigenvalues of the derivation $D$ generating the $1$-parameter group are real and positive. When the eigenspace relative to eigenvalue $1$ generates the Lie algebra $\mathfrak{g}$, one speaks of a \emph{Carnot group}.
The derivation defines gradations, called \emph{weight} gradations, on $\mathfrak{g}$ and $\Lambda^\cdot\mathfrak{g}^*$. The trace of $D$,
\begin{equation*}
Q=\Trace(D)
\end{equation*}
is called the \emph{homogeneous dimension} of $(G,\{h_s\})$.
\begin{exem}
For abelian groups $G$, the derivation $D$ is the identity, all $k$-forms have weight $k$, $Q=\dim(G)$.
For Heisenberg groups, $D$ has two eigenvalues, $1$ of even multiplicity and $2$ of multiplicity $1$. For $1\leq k\leq \dim(G)-1$, $\Lambda^k\mathfrak{g}^*$ splits into two eigenspaces with weights $k$ and $k+1$, $Q=\dim(G)+1$.
\end{exem}
When $G$ is not abelian, $1$-forms come in several different weights $1,2,\dots,r$ where $r$ is the step of $G$. Assume that an inequality of the form~\eqref{eq:4},
\begin{equation*}
\|u\|_q \leq C \|\du\|_p
\end{equation*}
holds for all compactly supported functions $u$ on $G$. Splitting $\du=\dd_1 u+\dots +\dd_r u$ into weights and applying the obvious generalization of Formula~\eqref{eq:5}, we get, for every $s\in\R$,
\begin{equation*}
e^{s\frac{Q}{q}} \|u\|_q \leq C' \left(e^{s(\frac{Q}{p}-1)}\|\dd_1 u\|_p+\dots +e^{s(\frac{Q}{p}-r)}\|\dd_r u\|_p\right).
\end{equation*}
Letting $s$ tend to $+\infty$, we find two necessary conditions:
\begin{equation}\label{eq:6}
\frac{1}{p}-\frac{1}{q}=\frac{1}{Q} \quad \text{and}\quad \|u\|_q \leq C' \|\dd_1 u\|_p.
\end{equation}
It turns out that such an inequality indeed holds, see~\cite{FollandStein} for $p>1$ and~\cite{VCSC} for $p=1$.
\subsection{Higher degree forms on Carnot groups}
For higher degrees, one can similarly reduce the range of weights occurring in each degree. The \emph{contracted complex} is a resolution of the map $\dd_1$ from $0$ to $1$-forms along $\mathfrak{g}_1$. It is homotopy equivalent to de Rham's complex. The weights of forms occurring in the contracted complex are the same which occur in Lie algebra cohomology. This is an important step: if, in two consecutive degrees, Lie algebra cohomology exists in only one weight, then the differential of the contracted complex is homogeneous, and a variant of the method based on inverting the Laplacian leads to sharp intervals for vanishing and nonvanishing of $\ell^{q,p}$-cohomology. This happens in all degrees for Heisenberg groups, for instance. This approach has been pursued in~\cite{BFP}.
In this paper, we shall obtain results in the general case, using a different method. The main trick, stated in~\cite{Rumin_CRAS_1999}, is a homogeneous (pseudodifferential) version of the exterior differential. Inverting the corresponding Laplacian leads to a homotopy $K$ which is bounded on certain function spaces of Sobolev type where each weight component of a form is differentiated as many times as its weight. A Poincar\'e inequality for functions, similar to~\eqref{eq:6}, allows to get back to usual Lebesgue spaces, but with losses caused by lack of homogeneity. More details on the method will be given after the statement of our main theorem.
\subsection{Main result}
Our main result relates the vanishing and non-vanishing of $\ell^{q,p}$ cohomology of a Carnot group $G$ to the weight gaps in its Lie algebra cohomology $H^*(\mathfrak{g})$ in two consecutive degrees. Recall that this cohomology can be seen as the cohomology of translation invariant forms on $G$. It is graded by degree and weight,
\[
H^\cdot(\mathfrak{g})=\bigoplus_{k,w}H^{k,w}(\mathfrak{g}).
\]
\begin{defi}
For $k=0,\dots,\dim(\mathfrak{g})$, let $w_{\min}(k)$ (resp. $w_{\max}(k)$) be the smallest (resp. the largest) weight $w$ such that $H^{k,w}(\mathfrak{g})\not=0$.
\end{defi}
For instance, on $G=\R^n$, all invariant forms are closed, therefore $H^*(\mathfrak{g}) = \Lambda^* \mathfrak{g}$, and $w_{\min}(k) =w_{\max}(k) = k$ for $k=0, \dots, n$.
\begin{theo}\label{1}
Let $G$ be a Carnot group of dimension $n$ and of homogeneous dimension $Q$. Let $k=1,\dots,n$. Denote by
\[
\delta N_{\max}(k)=w_{\max}(k)-w_{\min}(k-1),\quad \delta N_{\min}(k)=\max\{1,w_{\min}(k)-w_{\max}(k-1)\}.
\]
Let $p$ and $q$ be real numbers.
\begin{enumerate}%[label=(\roman*)]
\item \label{1.3.1} If
\[
10$. Then $P$ is bounded from $W^{h,p}\Omega^{k}(U)$ to $W^{h -n-1,q}\Omega^{k-1}(U)$, provided $p\geq 1$, $q\geq 1$, $n+1 \leq h \leq \ell-1$. Its norm depends on $K$, $\ell$, $n$, $R$ and $r$ only.
\end{prop}
\begin{proof}
The assumptions on $R$ guarantee that minimizing geodesics between points at distance $n$ is required and $L^q$ is replaced with $C^0$). If $p\geq 1$, this implies that $W^{n+1,p}\subset C^0$, hence $W^{h,p}\subset C^{h-n-1}$. Obviously, $C^{h-n-1}\subset W^{h-n-1,q}$.
Since curvature and its derivatives are bounded up to order $\ell$, the Riemannian exponential map and its inverse are $C^{\ell-1}$-bounded, hence $P_y$ is bounded on $C^h$ for $h \leq \ell-1$. If furthermore $h \geq n+1$, the embeddings
\[
W^{h,p}\subset C^{h-n-1}\subset W^{h-n-1,q}
\]
hold on $U$ with bounds depending on $K$, $R$ and $r$ only, hence $P$ maps $W^{h,p}$ to $W^{h-n-1,q}$, with uniform bounds.
\end{proof}
\section{\texorpdfstring{$\ell^{q,p}$}{ell(q,p)} cohomology and Sobolev \texorpdfstring{$L^{q,p}$}{L(q,p)} cohomology}\label{sec:leray}
\begin{defi}
Let $M$ be a Riemannian manifold. Let $h, h'\in\N$. The
\emph{Sobolev $L^{q,p}$ cohomology} is
\[
L_{h',h}^{q,p}H^{\cdot}(M)=\{\text{closed forms in }W^{h,p}\} / \dd(\{\text{forms in }W^{h',q}\}) \cap W^{h,p}.
\]
\end{defi}
\begin{rema}\label{rem:decreasing}
On a bounded geometry $n$-manifold, this is nonincreasing in $q$ in the following sense. Let $q'\geq q$ and $h'\geq n/q + 1$. Then $L_{h',h}^{q,p}H^{k}(M)$ surjects onto $L_{h'-1 - n/q,h}^{q',p}H^{k}(M)$, see Remark~\ref{whpdecr}.
\end{rema}
\begin{theo}\label{prop:leray}
Let $1\leq p\leq q\leq \infty$. Let $M$ be a Riemannian manifold of $C^\ell$-bounded geometry, with $\ell > n(n+1)$. There exists a simplicial complex $T$, admitting a quasiisometric homotopy equivalence to $M$, with the following property. For every integers $h,h'$ such that $n(n+1) \leq h,h'\leq \ell -1$, there exists an isomorphism between the $\ell^{q,p}$ cohomology of $T$ and the Sobolev $L^{q,p}$ cohomology $L_{h',h}^{q,p}H^{\cdot}(M)$.
\end{theo}
The proof is a careful inspection of Leray's acyclic covering theorem.
First construct a simplicial complex $T$ quasiisometric to $M$. Up to rescaling, one can assume that sectional curvature is $\leq 1/n^2$ and injectivity radius is $>2n$. Pick a maximal $1/2$-separated subset $\{x_i\}$ of $M$. Let $B_i$ be the covering by closed unit balls centered on this set. Let $T$ denote the nerve of this covering. Let $U_i=B(x_i,3)$. Note that if $x_{i_0},\dots,x_{i_j}$ span a $j$-simplex of $T$, then the intersection
\[
U_{{i_0}\dots{i_j}} := \bigcap_{m=0}^{j}U_{i_m}
\]
is contained in a ball of radius 3 and contains a concentric ball of radius 1.
Pick once and for all a smooth cut-off function with support in $[-1,1]$, compose it with distance to points $x_i$ and convert the obtained collection of functions into a partition of unity $\chi_i$ by dividing by the sum.
Define a bicomplex $C^{j,k}=$ skew-symmetric maps associating to $j+1$-tuples $(i_0,\dots, i_j)$ differential $k$-forms on $j+1$-fold intersections $U_{i_0}\cap\dots\cap U_{i_j}$. It is convenient to extend the notation to
\[
C^{-1,k} =\Omega^k(M),\quad C^{j,-1}=C^j(T),\quad C^{j,\cdot}=C^{\cdot,j}=0 \text{ if } j<-1.
\]
The two commuting complexes are $\dd:C^{j,k}\to C^{j,k+1}$ and the simplicial coboundary $\delta: C^{j,k}\to C^{j+1,k}$ defined by
\[
\delta(\phi)_{i_0 \dots i_{j+1}}= \phi_{i_0 \dots i_{j}}-\phi_{i_0
\dots i_{j-1}j_{j+1}} +\dots+(-1)^{j+1}\phi_{i_1 \dots i_{j+1}},
\]
restricted to $U_{i_0}\cap\dots\cap U_{i_{j+1}}$. By convention, $\dd:C^{j,-1}\to C^{j,0}$ maps scalar $j$-cochains to skew-symmetric maps to functions on intersections which are constant. Also, $\delta:C^{-1,k}\to C^{0,k}$ maps a globally defined differential form to the collection of its restrictions to open sets $U_i$. We define the differential to be zero in all other cases.
\[
\xymatrix %@!=1pc
{ & C^{-1,0}\ar[d]_{\delta} & C^{-1,1}\ar[d]_{\delta} & C^{-1,2}\ar[d]_{\delta} & C^{-1,3}\ar[d]_{\delta} &\dots \\ C^{0,-1} \ar[r]^{d} & C^{0,0}
\ar[r]^{d}\ar[d]_{\delta} & C^{0,1}\ar[r]^{d}\ar[d]_{\delta} & C^{0,2}\ar[r]^{d}\ar[d]_{\delta} & C^{0,3}\ar[r]^{d}\ar[d]_{\delta} &\dots
\\
C^{1,-1} \ar[r]^{d} & C^{1,0} \ar[r]^{d}\ar[d]_{\delta} & C^{1,1}
\ar[r]^{d}\ar[d]_{\delta} & C^{1,2} \ar[r]^{d}\ar[d]_{\delta} & C^{1,3} \ar[r]^{d}\ar[d]_{\delta} &\dots
\\
C^{2,-1} \ar[r]^{d} & C^{2,0} \ar[r]^{d}\ar[d]_{\delta} & C^{2,1}
\ar[r]^{d}\ar[d]_{\delta} & C^{2,2} \ar[r]^{d}\ar[d]_{\delta} & C^{2,3} \ar[r]^{d}\ar[d]_{\delta} &\dots
\\
\dots &\dots &\dots &\dots &\dots &\dots }
\]
The coboundary $\delta$ is inverted by the operator
\[
\epsilon:C^{j,k}\to C^{j-1,k}
\]
defined by
\[
\epsilon(\phi)_{i_0 \dots i_{j-1}}=\sum_{m}\chi_m \phi_{m i_0 \dots i_{j-1}}.
\]
By an inverse, we mean that $\delta\epsilon+\epsilon\delta=1$. This identity persists in all nonnegative bidegrees provided $\epsilon : C^{0,k} \to C^{-1,k}$ is defined by $\epsilon (\phi) = \sum_m \chi_m \phi_m$ and $\epsilon=0$ on $C^{j,k}$, $j<0$.
\[
\xymatrix %@!=1pc
{ & C^{-1,0} & C^{-1,1} & \textcolor{red}{C^{-1,2}} & C^{-1,3} &\dots
\\
C^{0,-1} & C^{0,0} \ar[l]^{P}\ar[u]^{\epsilon} & C^{0,1}\ar[l]^{P}\ar[u]^{\epsilon} & C^{0,2} & C^{0,3}\ar[l]^{P}\ar[u]^{\epsilon} &\dots\ar[l]^{P}
\\
C^{1,-1} & C^{1,0} \ar[l]^{P}\ar[u]^{\epsilon} & C^{1,1} & C^{1,2} \ar[l]^{P}\ar[u]^{\epsilon} & C^{1,3} \ar[l]^{P}\ar[u]^{\epsilon} &\dots\ar[l]^{P}
\\
\textcolor{red}{C^{2,-1}} \ar@{=>>} '[r]^-{\textcolor{red}{d}} '[ru]^{\textcolor{red}{\epsilon}} '[rru]^{\textcolor{red}{d}} '[rruu]^{\textcolor{red}{\epsilon}} '[rrruu]^{\textcolor{red}{d}} '[rrruuu]^{\textcolor{red}{\epsilon}}& C^{2,0} & C^{2,1}
\ar[l]^{P}\ar[u]^{\epsilon} & C^{2,2} \ar[l]^{P}\ar[u]^{\epsilon} & C^{2,3} \ar[l]^{P}\ar[u]^{\epsilon} &\dots\ar[l]^{P}
\\
\dots &\dots\ar[u]^{\epsilon} &\dots\ar[u]^{\epsilon} &\dots\ar[u]^{\epsilon} &\dots\ar[u]^{\epsilon} &\dots }
\]
Consider the maps
\[
\Phi_j = (\epsilon \dd)^{j+1} : C^j(T)=C^{j,-1} \to C^{-1,j}=\Omega^j(M).
\]
($\Phi_2$ is featured on the diagram). By definition, given a cochain $\kappa$, $\Phi_j(\kappa)$ is a local linear expression of the constants defining $\kappa$, multiplied by polynomials of the $\chi_i$ and their differential. Therefore $\Phi_j(\kappa)$ is $C^\infty$ and belongs to $W^{h,q}(\Omega^j(M))$ for all $q \geq p\geq 1$ if $\kappa \in \ell^p(C^j(T))$.
Since
\[
(\epsilon \dd)\delta=\epsilon\delta \dd=(1-\delta\epsilon)\dd=\dd-\delta\epsilon \dd,
\]
for $j\geq 0$, on $C^{j,-1}$,
\begin{align*}
(\epsilon \dd)^{j+2}\delta &=(\epsilon \dd)^{j+1} (\dd-\delta\epsilon \dd)= -(\epsilon \dd)^{j+1} \delta(\epsilon \dd) \\
&=\dots\\
&=(-1)^{j+1}(\epsilon \dd)\delta(\epsilon \dd)^{j+1}=(-1)^{j+1}(\dd-\delta\epsilon \dd) (\epsilon \dd)^{j+1}\\
&=(-1)^{j+1}\dd(\epsilon \dd)^{j+1}-(-1)^{j+1}\delta(\epsilon \dd)^{j+2}\\
&=(-1)^{j+1}\dd(\epsilon \dd)^{j+1}.
\end{align*}
Indeed, $(\epsilon \dd)^{j+2}(C^{j,-1})\subset C^{-2,j+1}=\{0\}$. In other words,
\[
\Phi_{j+1} \circ \delta=(-1)^{j+1}\dd\circ\Phi_j.
\]
%\medskip
We now proceed in the opposite direction to produce a cohomological inverse of $\Phi$. Let us mark each intersection $U_{i_0 \dots i_j}:=U_{i_0}\cap\dots\cap U_{i_j}$ with the point $y=x_{i_0}$. Proposition~\ref{local} provides us with an operator $P_{i_0\dots i_j}$ on usual $k$-forms on $U_{i_0 \dots i_j}$. Putting them together yields an operator $P:C^{j,k}\to C^{j,k-1}$ such that $1=\dd P+P\dd$. Furthermore, $P$ is bounded from $W^{\cdot,p}$ to $W^{\cdot-n-1,q}$.
Exchanging the formal roles of $(\delta, \epsilon)$ and $(\dd,P)$, we define
\[
\Psi_k = (P\delta)^{k+1} :\Omega^k(M)=C^{-1,k}\to C^{k,-1}=C^k(T),
\]
As above, one checks easily that $\Psi_{k+1} \circ \dd = (-1)^{k+1}\delta \circ \Psi_k$.
Observe that the maps $\Psi_j \circ \Phi_j$, $j=0,1,\dots$, put together form a morphism of the complex $C^{\cdot,-1}=C^\cdot(T)$ (i.e. they commute with $\delta$). We next show that it is homotopic to the identity. Let us prove by induction on $i$ that, on $C^{\cdot,-1}$,
\begin{equation}\label{eq:7}
(P \delta)^i (\epsilon \dd)^i = 1 - R_i \delta - \delta R_{i-1}\,,
\end{equation}
with $R_0= 0$ and $R_i = \sum_{k=0}^{i-1} (-1)^k (P \delta)^k P (\epsilon \dd)^{k+1}$. This implies the result for $\Psi_j \circ \Phi_j$.
\begin{proof}
For $i=1$, one has on $C^{\cdot,-1}$
\begin{align*}
(P \delta) (\epsilon \dd) & = P(1 - \epsilon \delta)\dd = P\dd - P\epsilon \dd \delta \\
& = 1 - (P\epsilon \dd) \delta,
\end{align*}
since $P\dd = 1$ on $C^{\cdot,-1}$. Assuming~\eqref{eq:7} for $i$, one writes
\begin{align}
(P\delta)^{i+1} (\epsilon \dd)^{i+1} & = (P \delta)^i P\delta
\epsilon \dd (\epsilon \dd)^i = (P \delta)^i P(1 - \epsilon
\delta) \dd (\epsilon \dd)^i \nonumber \\
& = (P
\delta)^i (1 - \dd P - P \epsilon \delta \dd) (\epsilon \dd)^i
\nonumber \\
& = (P \delta)^i (\epsilon \dd)^i - (P \delta)^i \dd P (\epsilon \dd)^i - (P \delta)^i P (\epsilon \dd) \delta (\epsilon \dd)^i.\label{eq:8}
\end{align}
About the second term in~\eqref{eq:8}, one finds that $\delta(P\delta) \dd = - \delta \dd (P\delta)$, hence by induction, one can push the isolated $\dd$ term to the left,
\[
(P \delta)^i d = (-1)^{i-1} P\delta \dd (P\delta)^{i-1} = (-1)^{i-1} P\dd \delta (P\delta)^{i-1} = (-1)^{i-1}\delta (P\delta)^{i-1},
\]
when the image lies within $C^{\cdot,-1}$, as it does in~\eqref{eq:8}.
For the third term in~\eqref{eq:8}, one sees that $(\epsilon \dd) \delta (\epsilon \dd) = (\epsilon \dd)(1 - \epsilon \delta) \dd = - (\epsilon \dd)^2 \delta$, so that one can push the isolated $\delta$ term to the right,
\[
(P \delta)^i P (\epsilon \dd) \delta (\epsilon \dd)^i = (-1)^i (P \delta)^i P (\epsilon \dd)^{i+1} \delta \,.
\]
Gathering in~\eqref{eq:8} gives
\[
(P\delta)^{i+1} (\epsilon \dd)^{i+1} = 1 - (R_i + (-1)^i (P\delta)^i P (\epsilon \dd)^{i+1}) \delta - \delta (R_{i-1} + (-1)^{i-1} (P\delta)^{i-1} P (\epsilon \dd)^i),
\]
that proves~\eqref{eq:7}.
\end{proof}
Similarly, $\Phi\circ\Psi$ is homotopic to the identity on the complex $(C^{-1,\cdot},\dd)$, $\Phi\circ\Psi=1-\dd R'-R'\dd$.
Finally, let us examine how Sobolev norms behave under the class of endomorphisms we are using. Maps from cochains to differential forms, i.e. $\epsilon$, $\Phi$ are bounded from $\ell^p$ to $W^{\ell-1,p}$. Maps from differential forms, i.e. $P$, $\Psi$, loose derivatives (but this is harmless since the final outputs are scalar cochains) but are bounded from $W^{h,p}$ to $\ell^p$, $h\leq \ell-1$. One merely needs $h$ large enough to be able to apply $P$ $n$ times, whence the assumption $h\geq n(n+1)$. Maps from cochains to cochains, e.g. $R$, are bounded on $\ell^p$, maps from differential forms to differential forms, e.g. $R'$, are bounded from $W^{h',p}$ to $W^{h,p}$ for every $h'\geq n(n+1)$ such that $h'\leq \ell-1$.
If $q\geq p$, the $\ell^q$-norm is controlled by the $\ell^p$-norm, hence $R$ is bounded from $\ell^p$ to $\ell^q$. It is also true that $R'$ is bounded from $W^{h'-1,p}$ to $W^{h-1,q}$. Indeed, it is made of bricks which map differential forms to cochains or cochains to differential forms, so no loss on derivatives affects the final differentiability. For the same reason, one can gain local integrability from $L^p_{loc}$ to $L^q_{loc}$ without restriction on $p$ and $q$ but $p$, $q\geq 1$.
It follows that $\Phi$ and $\Psi$ induce isomorphisms between the $\ell^{q,p}$ cohomology of $T$ and the Sobolev $L^{q,p}$ cohomology.
\section{De Rham complex and graduation on Carnot groups}\label{sec:dc_complex}
From now on, we will work on Carnot Lie groups. These are nilpotent Lie groups $G$ such that their Lie algebra $\mathfrak{g}$ splits into a direct sum
\[
\mathfrak{g} = \mathfrak{g}_1 \oplus \mathfrak{g}_2 \oplus \dots
\oplus\mathfrak{g}_r \quad \mathrm{satisfying} \quad [\mathfrak{g}_1,
\mathfrak{g}_i] = \mathfrak{g}_{i+1} \ \mathrm{for} \ 1\leq i \leq r-1.
\]
The weight $w = i$ on $\mathfrak{g}_i$ induces a family of dilations $\delta_t = t^w$ on $\mathfrak{g} \simeq G$.
In turn, the tangent bundle $TG$ splits into left invariant sub-bundles $H_1 \oplus \dots \oplus H_r$ with $H_i = \mathfrak{g}_i$ at the origin. Finally, differential forms decompose through their weight $\Omega^k G = \oplus_w \Omega^{k,w} G$ with $\Omega^{k_1} H^*_{w_1}\wedge \dots \wedge \Omega^{k_i}H^*_{w_i}$ of weight $w = k_1 w_1 + \dots + k_i w_i$. De Rham differential $\dd$ itself splits into
\[
\dd = \dd_0 + \dd_1 + \dots + \dd_r,
\]
with $\dd_i$ increasing weight by $i$. Indeed, this is clear on functions where $\dd_0 = 0$ and $\dd_i = d $ along $H_i$. This extends to forms, using $\dd(f\alpha) = \dd f \wedge \alpha + f \dalpha$ and observing that for left invariant forms $\alpha$ and left invariant vectors $X_i$, Cartan's formula reads
\begin{align*}
\dalpha (X_1,\dots, X_{k+1}) & = \sum_{1\leq i < j\leq k+1} (-1)^{i+j}
\alpha ([X_i,X_j], \dots, \widehat{X}_{i,j},
\dots X_{k+1}) \\
& = \dd_0 \alpha (X_1,\dots, X_{k+1}).
\end{align*}
Hence $\dd= \dd_0$ is a weight preserving algebraic (zero order) operator on invariant forms, and $\ker \dd_0 / \im \dd_0 = H^*(\mathfrak{g})$ is the Lie algebra cohomology of $G$. Note also that from these formulas, $\dd_i$ is a homogeneous differential operator of degree~$i$ and increases the weight by $i$. Like $\dd$, it is homogeneous of degree~$0$ through $h^*_\lambda$, since $\dd h^*_\lambda = h^*_\lambda \dd$.
%\medskip
This algebraic $\dd_0$ allows to split and contract de Rham complex on a smaller subcomplex, as we now briefly describe. This was shown in~\cite{Rumin_CRAS_1999} and~\cite{Rumin_TSG} in the more general setting of Carnot--Caratheodory manifolds. More details may be found there.
Pick an invariant metric so that the $\mathfrak{g}_i$ are orthogonal to each others, and let $\delta_0 = \dd_0^*$ and $\dd_0^{-1}$ be the partial inverse of $\dd_0$ such that $\ker \dd_0^{-1} = \ker \delta_0$, $\dd_0^{-1} \dd_0 = \Pi_{\ker \delta_0}$ and $\dd_0 \dd_0^{-1} = \Pi_{\im \dd_0}$.
Let $E_0 = \ker \dd_0 \cap \ker \delta_0 \simeq \Omega^* H^*(\mathfrak{g})$. Iterating the homotopy $r = 1 - \dd_0^{-1}\dd - \dd \dd_0 ^{-1}$ one can show the following results, stated here in the particular case of Carnot groups.
\let\oldpointir\pointir
\renewcommand{\pointir}{}
\begin{theo}[{\cite[Theorem~1]{Rumin_CRAS_1999}}]\label{thm:dc-complex}
\ %\ \\*[-2.6em]
\begin{enumerate}
\item The de Rham complex on $G$ splits as the direct sum of two sub-complexes $E \oplus F$, where $E = \ker \dd_0^{-1} \cap \ker \dd \dd_0^{-1}$ and $F= \im \dd_0^{-1} + \im (\dd\dd_0^{-1}) $.
\item The retractions $r^k$ stabilize to $\Pi_E$ the projection on $E$ along $F$. $\Pi_E$ is a homotopy equivalence of the form $\Pi_E = 1 - R\dd - \dd R$ where $R$ is a differential operator.
\item One has $\Pi_E \Pi_{E_0} \Pi_E = \Pi_E$ and $\Pi_{E_0} \Pi_E \Pi_{E_0} = \Pi_{E_0}$ so that the complex $(E,\dd)$ is conjugated through $\Pi_{E_0}$ to the complex $(E_0, \dd_c)$ with $\dd_c = \Pi_{E_0} \dd \Pi_E \Pi_{E_0}$.
\end{enumerate}
\end{theo}
\let\pointir\oldpointir
This shows in particular that the de Rham complex, $(E,\dd)$ and $(E_0, \dd_c)$ are homotopically equivalent complexes on smooth forms. For convenience in the sequel, we will refer to $(E_0,\dd_c)$ as the \emph{contracted de Rham complex} (also known as Rumin complex) and sections of $E_0$ as \emph{contracted forms}, since they have a restricted set of components with respect to usual ones.
We shall now describe its analytical properties we will use.
\section{Inverting \texorpdfstring{$\dd_c$}{dc} and \texorpdfstring{$\dd$}{d} on \texorpdfstring{$G$}{G}}\label{sec:inverting-d-d_c}
De Rham and contracted de Rham complexes are not homogeneous as differential operators, but are indeed invariant under the dilations $\delta_t$ taking into account the weight of forms. This leads to a notion of sub-ellipticity in a graded sense, called C-C ellipticity in~\cite{Rumin_CRAS_1999, Rumin_TSG}, that we now describe.
Let $\nabla = \dd_1$ the differential along $H = H_1$. Extend it on all forms using $\nabla (f \alpha) = (\nabla f) \alpha$ for left invariant forms $\alpha$ on $G$. Kohn's Laplacian $\Delta_H = \nabla^* \nabla$ is hypoelliptic since $H$ is bracket generating on Carnot groups, and positive self-adjoint on $L^2$. Let then
\[
|\nabla| = \Delta_H^{1/2}
\]
denotes its square root. Following~\cite[Section~3]{Folland_1975} or~\cite{CGGP}, it is a homogeneous first order pseudodifferential operator on $G$ in the sense that its distributional kernel, acting by group convolution, is homogeneous and smooth away from the origin. It possesses an inverse $|\nabla|^{-1}$, which is also a homogeneous pseudodifferential operator of order $-1$ in this calculus. Actually according to~\cite[Theorem~3.15]{Folland_1975}, it belongs to a whole analytic family of pseudodifferential operators $|\nabla|^\alpha$ of order $\alpha \in \mathbb{C}$. Note that kernels of these homogeneous pseudodifferential operators may contain logarithmic terms, when the order is an integer $\leq -Q$. We refer to~\cite{CGGP} and~\cite[Section~1]{Folland_1975} for more details and properties of this calculus.
%\smallskip
A particularly useful test function space for these operators is given by the space of Schwartz functions all of whose polynomial moments vanish,
\begin{equation}\label{eq:9}
\calS_0=\{f\in\calS\,;\,\langle f,P\rangle=0 \text{ for every polynomial }P\},
\end{equation}
where $\mathcal{S}$ denotes the Schwartz space of $G$ and $\langle f,P\rangle=\int_Gf(x)P(x)\,\dx$. Unlike more usual test functions spaces as $C^\infty_c$ or $\mathcal{S}$, this space $\calS_0$ is stable under the action of pseudodifferential operators of any order in the calculus, see~\cite[Proposition~2.2]{CGGP}, so that they can be composed on it. In particular by~\cite[Theorem~3.15]{Folland_1975}, for every $\alpha$, $\beta \in \mathbb{C}$,
\begin{equation}\label{eq:10}
|\nabla|^\alpha |\nabla|^{\beta} = |\nabla|^{\alpha+\beta} \
\mathrm{on} \ \calS_0 \,.
\end{equation}
We shall prove in Proposition~\ref{prop:density-s_0} that $\calS_0$ is dense in all Sobolev spaces $W^{h,p}$ with $h \in \N$ and $1< p< +\infty$, but we shall work mainly in $\calS_0$ in this section.
%\medskip
Now let $N=w$ on forms of weight $w$. Consider the operator $|\nabla|^N$, preserving the degree and weight of forms, and acting componentwise on $\calS_0$ in a left-invariant frame.
From the previous discussion, $\dd^{\nabla} = |\nabla|^{-N} \dd |\nabla|^N$ and $\dd_c^{\nabla} = |\nabla|^{-N} \dd_c |\nabla|^N$ are both homogeneous pseudodifferential operators of (differential) order $0$. Indeed, as observed in Section~\ref{sec:dc_complex}, $\dd$ splits into $\dd=\sum_i \dd_i$ where $\dd_i$ is a differential operator of horizontal order $i$ which increases weight by $i$. On forms of weight $w$, $|\nabla|^N$ has differential order $w$, $\dd_i |\nabla|^N$ has order $w+i$ and maps to forms of weight $w+i$, on which $ |\nabla|^{-N}$ has order $-(w+i)$, hence $|\nabla|^{-N} \dd |\nabla|^N$ has order $0$. The same argument applies to $\dd_c$.
Viewed in this Sobolev scale, these complexes become invertible in the pseudodifferential calculus. Let
\[
\Delta^{\nabla } = \dd^{\nabla} (\dd^{\nabla})^* + (\dd^{\nabla})^* \dd^\nabla
\quad \mathrm{and} \quad \Delta_c^{\nabla} = \dd_c^{\nabla} (\dd_c^{\nabla})^* + (\dd_c^{\nabla})^* \dd_c^\nabla,
\]
not to be confused with the non homogeneous $\dd_c^{\nabla} (\dd_c^*)^{\nabla} + (\dd_c^*)^{\nabla} \dd_c^{\nabla} = |\nabla|^{-N} (\dd_c \dd_c^* + \dd_c^* \dd_c) |\nabla|^N$.
\begin{theo}[{\cite[Theorem~3]{Rumin_CRAS_1999}, \cite[Theorem~5.2]{Rumin_TSG}}]\label{thm:left_inverse}
The Laplacians $\Delta^{\nabla} $ and $\Delta_c^{\nabla } $ have left inverses $Q^{\nabla}$ and $Q_c^{\nabla}$, which are zero order homogeneous pseudodifferential operators.
\end{theo}
By~\cite[Theorem~6.2]{CGGP}, this amounts to show that these Laplacians satisfy Rockland's injectivity criterion. This means that their symbols are injective on smooth vectors of any non trivial irreducible unitary representation of $G$.
%\medskip
This leads to a global homotopy for $\dd_c$ on $G$. Indeed following~\cite[Proposition~1.9]{Folland_1975}, homogeneous pseudodifferential operators of order zero on $G$, such as $\dd_c^{\nabla}$, $\Delta_c^\nabla$ and $Q_c^{\nabla}$, are bounded on all $L^p(G)$ spaces for $1< p< \infty$. Therefore the positive self-adjoint $\Delta_c^{\nabla }$ on $L^2(G)$ is bounded from below since $Q_c^{\nabla} \Delta_c^{\nabla} = 1$. Hence, it is invertible in $L^2(G)$ and $Q_c^{\nabla} = (\Delta_c^{\nabla})^{-1}$ is the inverse of $\Delta_c^{\nabla}$.
Since $(\Delta_c^{\nabla})^{-1}$ commute with $\dd_c^{\nabla}$, the zero order homogeneous pseudodifferential operator $K_c^\nabla = (\dd_c^{\nabla})^* (\Delta_c^{\nabla})^{-1} $ is a global homotopy,
\[
1 = \dd_c^\nabla K_c^\nabla + K_c^\nabla \dd_c^\nabla.
\]
Let us set
\[
K_c = |\nabla|^{N}K_c^\nabla|\nabla|^{-N}
\]
on $\calS_0$, in order that
\[
1 = \dd_c K_c + K_c \dd_c.
\]
Since $K_c^\nabla$ is bounded from $L^p$ to $L^p$ for $10$ it holds on $\Omega_{[a,b]} \cap \mathcal{S}$ that
\[
C \| \ \|_{W^{h,q}_c} \leq \sum_{m=a+ \mu}^{b+\mu + h} \| \
\|_{|\nabla|, N-m,p} \,.
\]
\end{enumerate}
\end{prop}
\begin{proofc}
\eqref{6.1.1}\pointir By definition $a\leq N= w \leq b$ on $\Omega_{[a,b]}$, hence
\[
\sum_{m=b}^{a+h} \|\ \|_{|\nabla|, N-m, p} = \sum_{m=b}^{a+h}
\||\nabla|^{m-N}\ \|_p \leq \sum_{m=N}^{N+h}
\||\nabla|^{m-N} \ \|_p = \sum_{k=0}^h
\||\nabla|^k \ \|_p \,.
\]
One has also
\[
\sum_{m=a}^{b+h} \|\ \|_{|\nabla|, N-m, p} = \sum_{m=a}^{b+h}
\||\nabla|^{m-N}\ \|_p \geq \sum_{m=N}^{N+h}
\||\nabla|^{m-N} \ \|_p = \sum_{k=0}^h
\||\nabla|^k \ \|_p \,.
\]
%\smallskip
\eqref{6.1.2}\pointir Since $|\nabla|^{-\mu}$ is a homogeneous pseudodifferential operator of order $-\mu$ and $\mu0$, the series $f_{\mathbf{m},\eps\mathbf{t}}$ converges in $W^{F,p}$ and for $\eps \leq 1$
\[
\|f_{\mathbf{m},\eps \mathbf{t}}\|_{W^{F,p}}\leq
\eps^{Q/p'}|m_0|N^{F,p}_0 + \eps \sum_{\alpha\not=0} |m_\alpha|t_\alpha^{w(\alpha)+Q/p'} N^{F,p}_\alpha.
\]
Therefore, as $\eps$ tends to $0$, $f_{\mathbf{m},\eps\mathbf{t}}$ tends to $0$ in $W^{F,p}$ (if $p=1$, one must assume that $m_0=0$).
Given $f\in W^{F,p}$ (assume furthermore that $\langle f,1 \rangle=0$ if $p=1$ or that $f\in C_0^h(G)$ if $p=\infty$), approximate $f$ with an element $g\in\calS$ (resp. such that $\langle g,1 \rangle=0$ if $p=1$). Set $m_\alpha=\langle x^\alpha,g \rangle$. Pick $\mathbf{t}$ satisfying the above smallness assumptions with respect to $\mathbf{m}$. Then $g-f_{\mathbf{m},\eps\mathbf{t}}\in \calS_0$ and $f_{\mathbf{m},\eps\mathbf{t}}$ tends to $0$ in $W^{F,p}$, thus $f$ belongs to the closure of $\calS_0$.
Finally, $\|f\|_{W_c^{h,p}}\leq \|f\|_{W^{F,p}}$ for a suitable finite set $F$. Indeed, $G$ admits a basis of left-invariant vectorfields $X_i$ of the form
\begin{equation}\label{eq:13}
X_i=\frac{\partial}{\partial x_i}+\sum_{j>i}P_{i,j}
\frac{\partial}{\partial x_j},
\end{equation}
where $P_{i,j}$ is a $\delta_t$-homogeneous of weight $w(P_{i,j})0$. Pick a real number $\lambda$ in the dense set given in Definition~\ref{dfi:wsG} and such that
\begin{equation}\label{eq:18}
w-\frac{T}{p}-\frac{T}{2}\epsilon \leq \lambda < w-\frac{T}{p}.
\end{equation}
Then $\lambda'=-\lambda$ satisfies
\begin{align}
\lambda'-w'+\frac{T}{q'} & = -\lambda - w' + \frac{T}{q'} \nonumber \\
& \leq -w + \frac{T}{p} + \frac{T\epsilon}{2} -w' + \frac{T}{q'} = - \frac{T\epsilon}{2} < 0. \label{eq:19}
\end{align}
By definition, there exist differential $k-1$ and $n-k$-forms $\alpha$ and $\alpha'$, homogeneous of degrees $\lambda$ and $\lambda'$, such that $\dalpha$ and $\dalpha'$ have weights $\geq w$ and $\geq w'$. Then $\dalpha \wedge \alpha'$ is homogeneous of degree~$0$. Using the notations of Section~\ref{sec:homog-diff-forms}, for all $j$, $\dalpha\wedge \chi_j\alpha'= h_{-j}^*(\dalpha\wedge \chi_1
\alpha')$, hence
\[
\int_{G} \dalpha \wedge \chi_j\alpha'=\int_{G} \dalpha \wedge
\chi_1\alpha'=I(\dalpha,\alpha')\int_{\R}\chi(t)\,\dt\not=0.
\]
Since $\dalpha$ is homogeneous of degree $\lambda$ and has weight $\geq w$, it belongs to $L^p$ (away from a neighborhood of the origin) by~\eqref{eq:16} and~\eqref{eq:18}. Furthermore, derivatives along left invariant vector fields decrease homogeneity. Hence all such derivatives of $\dalpha$ belong to $L^p$. After smoothing $\alpha$ near the origin, we get a closed form $\omega$ on $G$ that coincides with $\dalpha$ on $\{\rho \geq 1\}$, and which belongs to $W^{h,p}$ for all $h$. Set
\[
\omega'_j=\chi_j \alpha'.
\]
Then $\int_{G}\omega\wedge \omega'_j$ does not depend on $j$.
Assume by contradiction that $\omega=\dphi$ where $\phi\in W^{h,q}$. In particular, $\phi\in L^q$. Since $\omega'_j$ are compactly supported, Stokes theorem applies and
\begin{align*}
\left|\int_{G}\omega\wedge \omega'_j\right| &=\left|\int_{G}\dphi\wedge \omega'_j\right|\\
&=\left|\int_{G}\phi\wedge \domega'_j\right|\\
&\leq\|\phi\|_{q}\|\domega'_j\|_{q'}
\end{align*}
which tends to $0$, as $\alpha'$ and $\dalpha' \in L^q(\{\rho \geq 1\}$) by~\eqref{eq:16} and~\eqref{eq:19}, contradiction. We conclude that $[\omega]\not=0$ in the Sobolev $L^{q,p}$ cohomology of $G$. According to Proposition~\ref{prop:leray}, this implies that the $\ell^{q,p}$ cohomology of $G$ does not vanish.
\end{proof}
\subsection{Lower bounds on \texorpdfstring{$ws_G$}{wsG}}
We give here two lower bounds on $ws_G$. Combined with Theorem~\ref{thm:nonzero}, they complete the proof of Theorem~\ref{1}$\MK$\eqref{1.3.2} in the wider setting of homogeneous groups. We start with a lemma on the contracted complex.
\begin{lemm}\label{lem:existence_formes}
Let $G$ be a homogeneous Lie group of dimension $n$, let $k=1,\dots,n$. Then for an open dense set of real numbers $\lambda$, there exist smooth non $\dd_c$-closed contracted $k-1$-forms on $G \setminus \{1\}$ which are homogeneous of degree $\lambda$.
\end{lemm}
\begin{proof}
The differential $\dd_c \not\equiv 0$ on $E_0^{k-1}$, since the complex $(\dd_c, E_0^*)$ is a resolution on $G$. Then, by the Stone--Weierstrass approximation theorem, their exist non $\dd_c$-closed contracted forms with homogeneous polynomial components in an invariant basis. Pick one term $P \alpha_0$ with $\alpha_0\in E_0^{k-1}$ invariant, and a non constant homogeneous polynomial $P$ such that $\dd_c (P\alpha_0)\not=0$. Up to changing $P$ into $-P$, pick $x_0 \in G$ such that $\dd_c(P\alpha_0) (x_0) \not=0$ and $P(x_0) >0$. Consider the map
\[
F \ :\ \lambda \in \mathbb{C} \mapsto \dd_c (P^\lambda \alpha_0) (x_0).
\]
Since $\dd_c$ is a differential operator, $F$ is analytic. Since $F(1) \not=0$, one has $F(\lambda) \not= 0$ except for a set of isolated values of $\lambda$. Let $\chi$ be a smooth homogeneous function on $G \setminus \{1\}$ of degree~$0$ with support in $\{P> 0\}$ and $\chi=1$ around $x_0$. Then $\alpha = \chi P^\lambda \alpha_0$ is a smooth non $\dd_c$-closed homogeneous contracted form on $G\setminus \{1\}$ of degree $w(\alpha)= \lambda w(P) + w(\alpha_0)$.
\end{proof}
\begin{prop}\label{lower}
Let $G$ be a homogeneous Lie group of dimension $n$. For all $k=1,\dots,n$,
\[
ws_G(k)\geq \max\{1, w_{\min}(k)-w_{\max}(k-1)\}.
\]
\end{prop}
\begin{proof}
By Lemma~\ref{lem:existence_formes}, pick a non $\dd_c$-closed contracted $k-1$-form $\alpha$, homogeneous of degree $\lambda$. Assume that $\dd_c\alpha$ has weight $\geq w$ and no better (i.e. its weight $w$ component $(\dd_c \alpha)_w$ does not vanish identically). Pick a smooth contracted $n-k$-form $\alpha'$ of weight $T-w$, homogeneous of degree $-\lambda$ and such that $I(\dd_c\alpha,\alpha')\not=0$. For instance $\alpha'= \rho^{-2\lambda+2w-T} * (\dd_c\alpha)_w$ will do. Set $\alpha_E = \Pi_E \alpha$ and $\alpha_E'= \Pi_E \alpha'$.
By construction (see Theorem~\ref{thm:dc-complex}), $\Pi_E = \Pi_{E_0}+ D$ where $D$ strictly increases the weight. Hence $\dalpha_E - \dd_c \alpha= \Pi_E \dd_c \alpha - \dd_c\alpha$ has weight $\geq w +1$, and $\alpha'_E - \alpha'$ has weight $\geq T-w + 1$. Therefore $\dalpha_E \wedge \alpha'_E - \dd_c \alpha \wedge \alpha'$ has weight $\geq T+1$, thus vanishes. Then it holds that
\[
I(\dalpha_E, \alpha'_E) = I(\dd_c \alpha, \alpha') \not=0 \,.
\]
Consider now the weight of $\dalpha'_E$. By construction, $E \subset \ker \dd_0$, so that $\dd$ strictly increases the weight on $E$, see Section~\ref{sec:dc_complex}. Therefore
\[
w(\dalpha'_E) \geq w(\alpha'_E) + 1 = w(\alpha')+ 1 = T - w(\dd_c
\alpha) + 1 = T -w(\dalpha_E) + 1,
\]
hence $ws_G(k) \geq w(\dalpha_E) + w (\dalpha'_E) - T \geq 1 $ as needed. One has also that
\[
w(\dalpha'_E) = w (\dd_c \alpha') \geq w_{\min}(n-k+1) = T - w_{\max}(k-1),
\]
by Hodge $*$-duality, see proof of~\eqref{eq:15}, while
\[
w(\dalpha_E) = w (\dd_c \alpha) \geq w_{\min}(k) \,.
\]
This gives $ws_G(k) \geq w(\dalpha_E) + w (\dalpha'_E) - T \geq w_{\min}(k) - w_{\max}(k-1)$.
\end{proof}
\subsection{An example: Engel's group}\label{sec:engels-group}
We illustrate the non-vanishing results on the Engel group $E^4$.
It has a 4-dimensional Lie algebra with basis $X,Y,Z,T$ and nonzero brackets $[X,Y]=Z$ and $[X,Z]=T$. One finds, see e.g.~\cite[Section~2.3]{Rumin_TSG}, that
\[
H^1(\mathfrak{g}) \simeq \mathrm{span}(\theta_X, \theta_Y) \
\mathrm{and} \ H^2(\mathfrak{g}) \simeq \mathrm{span}(\theta_Y \wedge
\theta_Z, \theta_X \wedge \theta_T) \,.
\]
The following table gives the values of $\delta N_{\max}$ and $\delta N_{\min}$ for $E^4$ with respect to its standard Carnot weight : $w(X)= w(Y) =1$, $w(Z)= 2$ and $w(T) = 3$. One has $Q=7$ and
\[
\begin{tabular}{|l|c|c|c|c|}
\hline k & 1 & 2 & 3 & 4 \\
\hline\hline $w_{\max}(k)$ & 1 & 4 & 6 & 7 \\
\hline $w_{\min}(k)$ & 1 & 3 & 6 & 7 \\
\hline $\delta N_{\max}(k)$ & 1 & 3 & 3 & 1 \\
\hline $\delta N_{\min}(k)$ & 1 & 2 & 2 & 1 \\
\hline
\end{tabular}
\]
We see that Theorem~\ref{1} is sharp in degrees $1$ and $4$. However, there are gaps in degrees $2$ and $3$. In particular, $H^{2,q,p}(E^4)$ vanishes when $\frac{1}{p}- \frac{1}{q} \geq \frac{3}{7}$ and does not when $\frac{1}{p}- \frac{1}{q} < \frac{2}{7}$, provided $1