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\newcommand{\divg}{\mathrm{div}_\G\,} \newcommand{\divh}{\mathrm{div}_{\he{}}\,} \newcommand{\e}{\mathrm{Euc}} \newcommand{\Hom}{\mathrm{Hom}\,} \newcommand{\hatcov}[1]{{\bigwedge\nolimits^{#1}{\hat{\mfrak g}}}} \usepackage[usenames]{color} \newcommand\cyan{\textcolor{cyan}} \newcommand\blue{\textcolor{blue}} %\newcommand\red{\textcolor{red}} \newcommand{\red}[1]{{\color{red}{#1}}} \begin{document} %\today %\tableofcontents \title[Poincar\'e and Sobolev inequalities for differential forms ] {Poincar\'e and Sobolev inequalities for differential forms in Heisenberg groups \\ and contact manifolds} \author[Annalisa Baldi, Bruno Franchi, Pierre Pansu]{ Annalisa Baldi\\ Bruno Franchi\\ Pierre Pansu } \begin{abstract} In this paper, we prove contact Poincar\'e and Sobolev inequalities in Heisenberg groups $\he n$, where the word ``contact'' is meant to stress that de Rham's exterior differential is replaced by the exterior differential of the so-called Rumin's complex $(E_0^\bullet,d_c)$, that recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of $\he n$. In addition, we construct smoothing operators for differential forms on sub-Riemannian contact manifolds with bounded geometry, that act trivially on cohomology. For instance, this allows to replace a closed form, up to adding a controlled exact form, with a much more regular differential form. \end{abstract} \keywords{Heisenberg groups, differential forms, Sobolev-Poincar\'e inequalities, contact manifolds, homotopy formula} \subjclass{58A10, 35R03, 26D15, 43A80, 53D10 46E35} \maketitle \tableofcontents \section{Introduction} \subsection{Sobolev and Poincar\'e inequalities for differential forms}\label{intro 1} {The Sobolev inequality in $\R^n$ states that, if $u$ is a compactly supported function, then \begin{eqnarray*} \|u\|_q \leq C_{p,q,n}\|du\|_p \end{eqnarray*} whenever \begin{eqnarray*} 1\leq p,q< +\infty,\quad \frac{1}{p}-\frac{1}{q}=\frac{1}{n}, \end{eqnarray*} where $du$ is the differential of $u$ (that is a 1-form). } A local version, for functions supported in the unit ball, holds under the weaker assumption \begin{eqnarray*} 1\leq p,q< +\infty,\quad \frac{1}{p}-\frac{1}{q}\leq\frac{1}{n}. \end{eqnarray*} Poincar\'e's inequality is a variant for functions $u$ defined, but not necessarily compactly supported, in the unit ball $B$. It states that there exists a real number $c_u$ such that \begin{eqnarray*} \|u-c_u\|_q \leq C_{p,q,n}\|du\|_p. \end{eqnarray*} Alternatively, for a given exact $1$-form $\omega$ on $B$, there exists a function $u$ on $B$ such that $du=\omega$ on $B$, and such that \begin{eqnarray*} \|u\|_q \leq C_{p,q,n}\|\omega\|_p. \end{eqnarray*} This suggests the following generalization for higher degree differential forms in Riemannian manifolds. Let $M$ be a Riemannian manifold, with or without boundary. We say that a \emph{global} Poincar\'e inequality holds on $M$, if there exists a positive constant $C=C(M,p,q)$ such that for every exact $h$-form $\omega$ on $M$, belonging to $L^p$, there exists a $(h-1)$-form $\phi$ such that $d\phi=\omega$ and \begin{eqnarray*} \|\phi\|_q \leq C\,\|\omega\|_p. \end{eqnarray*} Shortly, we shall say that Poincar\'e$_{p,q}(h)$ holds. \medskip A \emph{global} Sobolev inequality holds on $M$, if for every exact compactly supported $h$-form $\omega$ on $M$, belonging to $L^p$, there exists a compactly supported $(h-1)$-form $\phi$ such that $d\phi=\omega$ and \begin{eqnarray*} \|\phi\|_q \leq C\,\|\omega\|_p. \end{eqnarray*} Again, we shall say that Sobolev$_{p,q}(h)$ holds. In both statements, the assumption that given forms are exact is there to separate the topological problem (whether a given closed form is exact) from the analytical one (whether a primitive can be upgraded to one which satisfies estimates). For bounded convex domains, the { global} Poincar\'e and Sobolev inequalities hold for $1
0$ and large enough $\lambda\geq 1$, there exists a constant $C=C(M,p,q,r,\lambda)$ such that for every $x\in M$ and every exact $h$-form $\omega$ on $B(x,\lambda r)$, belonging to $L^p$, there exists a $(h-1)$-form $\phi$ on $B(x,r)$ such that $d\phi=\omega$ on $B(x,r)$ and \begin{eqnarray}\label{int poinc} \|\phi\|_{L^q(B(x,r))} \leq C\,\|\omega\|_{L^p(B(x,\lambda r))}. \end{eqnarray} By \emph{interior Sobolev inequalities}, we mean that, if $\omega$ is supported in $B(x,r)$, then there exists $\phi$ supported in $B(x,\lambda r)$ such that $d\phi = \omega$ and \begin{eqnarray}\label{int sob} \|\phi\|_{L^q(B(x,\lambda r))} \leq C\,\|\omega\|_{L^p(B(x, r))}. \end{eqnarray} It turns out that in several situations, the loss on domain is harmless. This is the case for $L^{q,p}$-cohomological applications, see \cite{Pcup}. Let us comment on the terminology. Due to the loss on domain, inequality \eqref{int poinc} provides no information on the behaviour of differential forms near the boundary of their domain of definition, this is why we speak of an interior Poincar\'e inequality. \subsection{Contact manifolds}\label{contact introduction} A contact structure on an odd-dimensional manifold $M$ is a smooth distribution of hyperplanes $H$ which is maximally nonintegrable in the following sense: if $\theta$ is a locally defined smooth 1-form such that $H=\mathrm{ker}(\theta)$, then $d\theta$ restricts to a non-degenerate 2-form on $H$, i.e. if $2n+1$ is the dimension of $M$, then $\theta\wedge(d\theta)^n\neq 0 $ on $M$ (see \cite{mcduff_salamon}, Proposition 3.41). A contact manifold $(M,H)$ is the data of a smooth manifold $M$ and a contact structure $H$ on $M$. Contact diffeomorphisms (also called contactomomorphisms: see Definition \ref{uggioso}) are contact structure preserving diffeomorphisms between contact manifolds. %^* The prototype of a contact manifold is the Heisenberg group $\he n$, the simply connected Lie group whose Lie algebra is the central extension \begin{equation}\label{strat intro} \mathfrak{h}=\mathfrak{h}_1\oplus\mathfrak{h}_2,\quad\mbox{with $\mathfrak{h}_2=\R=Z(\mathfrak{h})$,} \end{equation} with bracket $\mathfrak{h}_1\otimes\mathfrak{h}_1\to\mathfrak{h}_2=\R$ being a non-degenerate skew-symmetric 2-form. The contact structure is obtained by left-translating $\mathfrak{h}_1$. According to a theorem by Darboux, every contact manifold is locally contactomorphic to $\he n$. The Heisenberg Lie algebra admits a one parameter group of automorphisms $\delta_t$, \begin{eqnarray*} \delta_t=t\textrm{ on }\mathfrak{h}_1,\quad \delta_t=t^2 \textrm{ on }\mathfrak{h}_2, \end{eqnarray*} which are {counterpart of the usual Euclidean dilations in $\rn n$. }Thus, differential forms on $\mathfrak{h}$ split into 2 eigenspaces under $\delta_t$, therefore de Rham complex lacks scale invariance under these anisotropic dilations. A substitute for de Rham's complex, that recovers scale invariance under $\delta_t$ has been defined by M. Rumin, \cite{rumin_jdg}. It makes sense for arbitrary contact manifolds $(M,H)$ and it is invariant under contactomorphisms. Let $h=0,\ldots,2n+1$. Rumin's substitute for smooth differential forms of degree $h$ are the smooth sections of a vector bundle $E_0^h$. If $h\leq n$, $E_0^h$ is a subbundle of $\Lambda^h H^*$. If $h\geq n$, $E_0^h$ is a subbundle of $\Lambda^h H^*\otimes (TM/H)$. Rumin's substitute for de Rham's exterior differential is a linear differential operator $d_c$ from sections of $E_0^h$ to sections of $E_0^{h+1}$ such that $d_c^2=0$. \medskip {\bf We stress that the operator $d_c$ has order $2$ when $h=n$ and order $1$ otherwise. } \medskip This phenomenon will be a major issue in the proofs of our results and will affect the choice of the exponents $p,q$ in our inequalities The data of $(M,H)$ equipped with a scalar product $g$, defined on sub-bundle $H$ only, is called a \emph{sub-Riemannian} contact manifold and we shall write $(M,H,g)$. The scalar product on $H$ determines a choice of a local contact form $\theta$, hence a norm on the line bundle $TM/H$. Therefore $E_0^h$ are endowed with a scalar product. Using $\theta\wedge(d\theta)^n$ as a volume form, one gets $L^p$-norms on spaces of smooth Rumin differential forms. In any sub-Riemannian contact manifold $(M,H,g)$ we can define a sub-Rieman\-nian distance $d_M$ (see e.g. \cite{montgomery}) inducing on $M$ the same topology of $M$ as a manifold. In particular, Heisenberg groups $\he n$ can be viewed as sub-Riemannian contact manifolds. If we choose on the contact sub-bundle of $\he n$ a left-invariant metric, it turns out that the associated sub-Riemanian metric is also left-invariant. {\ It is customary to call this distance in $\he n$ a {\it Carnot-Carath\'eodory distance}.} Poincar\'e and Sobolev inequalities for differential forms make sense on contact sub-Riemannian manifolds: merely replace the exterior differential $d$ with $d_c$. All left-invariant sub-Riemannian metrics on Heisenberg group are bi-Lipschitz equivalent, hence we may refer to sub-Riemannian Heisenberg group without referring to a specific left-invariant metric: if a Poincar\'e inequality holds for some left-invariant metric, it holds for all of them. On the other hand, in absence of symmetry assumptions, large scale behaviors of sub-Riemannian contact manifolds are diverse. {Examples illustrating this phenomenon will be given in Section \ref{diverse}.} \subsection{Results on Poincar\'e and Sobolev inequalities} In this paper, we prove {global} $\he{}$-$\mathrm{Poincar\acute{e}}$ and $\he{}$-$\mathrm{Sobolev}$ inequalities and interior $\he{}$-$\mathrm{Poincar\acute{e}}$ and $\he{}$-$\mathrm{Sobolev}$ inequalities in Heisenberg groups, where the prefix $\he{}$ is meant to stress that the exterior differential is replaced with Rumin's exterior differential $d_c$. The range of parameters differs slightly from the Euclidean case, due to the fact that $d_c$ has order 2 in middle dimension. Let $h\in \{0,\ldots,2n+1\}$. We say that assumption $E(h,p,q,n)$ holds if $1
1$, the global estimates of Theorem \ref{strongglobal} are more or less straightforward
consequences of the $L^p - L^q$ continuity of singular integrals of potential type (see Section \ref{Global homotopy operators intro}
below).
}
\end{remark}
Here is a simple consequence of these results. Combining both theorems with results from \cite{Pcup}, we get
\begin{cor}
Under assumption $E(h,p,q,n)$, the $\ell^{q,p}$-cohomology in degree $h$ of $\he n$ vanishes.
\end{cor}
Our third result is the construction of a smoothing homotopy on general contact manifolds. Under a bounded geometry assumption, uniform estimates can be given
(precise
definitions of bounded geometry contact manifolds, as well as of associated Sobolev spaces $W^{j,p}$,
will be given in Section \ref{sobolev contact}).
\begin{theorem}\label{1.5}
Let $k\geq 3$ be an integer. Let $(M,H,g)$ be a $2n+1$-dimensional sub-Riemannian contact manifold of bounded $C^k$-geometry.
Under assumption $I(h,p,q,n)$, there exist operators $S_M$ and $T_M$ on $h$-forms on $M$ which are bounded from $W^{j,p}$ to $W^{j,q}$ for all $0\leq j\leq k-1$, and such that
\begin{equation}\label{hom intro}
1=S_M+d_c T_M +T_Md_c.
\end{equation}
Furthermore, $S_M$ and $T_M$ are bounded from $W^{j-1,p}$ to $W^{j,p}$ if $j\geq 1$ (resp. from $W^{j-2,p}$ to $W^{j,p}$ if $j\geq 2$ and degree $h=n+1$).
\end{theorem}
We stress that the ``approximate homotopy formula'' \eqref{hom intro} has no consequences for the cohomology of $M$.
The iteration of the process yields an operator $S_M$ which is bounded from $L^{p}$ to $W^{k-1,q}$, and still acts trivially on cohomology. {For instance, it is possible to replace a closed form with a much more regular differential form (up to adding a controlled exact form).}
{
\subsection{State of the art}\label{State of the art}
This paper is part of a larger project aimed to prove $(p,q)$-Poincar\'e and Sobolev inequalities in Heisenberg groups
when $1\le p < q\le \infty$.
Thus it seems convenient to point out the different cases we have to deal with. Let us restrict ourselves for a while to
Euclidean spaces $\rn n$ and Heisenberg groups $\he n$. The first fundamental distinction is the following:
\begin{itemize}
\item[i)] global inequalities (i.e. inequalities on all the space $\rn n$ or $\he n$);
\item[ii)] interior inequalities (for instance on Carnot-Carath\'eodory balls).
\end{itemize}
For each one of the above geometric assumptions we must distinguish between
\begin{itemize}
\item[iii)] the case $p=1$;
\item[iv)] the case $p>1$.
\end{itemize}
In the scalar case, $(p,q)$-Poincar\'e and Sobolev inequalities are well understood
both in Euclidean spaces and in Heisenberg groups for all $p\ge 1$.
Consider now differential forms of higher degree.
For the case $p=1$, \emph{global} inequalities in $\rn n$ (Gagliardo-Nirenberg inequalities for differential forms)
have been proved by Bourgain \& Brezis (\cite{BB2007})
and Lanzani \& Stein (\cite{LS}) via a suitable identity for closed differential forms
and relying on careful estimates for divergence-free vector fields. Thanks to the counterpart
of this identity proved
by Chanillo \& van Schaftingen in homogeneous groups (\cite{CvS2009}), similar global
inequalites for differential forms in $\he n$
were proved in \cite{BFP}. We stress that in \cite{BFP} algebra plays an important role precisely in the proof of
the identities for closed forms, therefore apart from Heisenberg groups,
only a handful of more general nilpotent groups have been treated, \cite{BFTr}.
Interior inequalities when $p=1$ use the estimate of \cite{BFP} combined with an approximate
homotopy formula introduced in the present paper, but require a new different argument to control the
commutator between Rumin's exterior differential (or de Rham's exterior differential in $\rn n$) and multiplication by a cut-off function.
These inequalities are proved for Heisenberg groups in \cite{BFP3} and in \cite{BFP4} for Euclidean spaces.
Notice that in the Heisenberg group case, one more algebraic obstacle shows up, averages of $L^1$ forms, see \cite{PT}.
Consider now the case $p>1$. In the Euclidean setting, \emph{interior} Poincar\'e inequalities for $p>1$ are proved in \cite{IL}.
However, the arguments of \cite{IL} do not extend to Heisenberg groups.
Thus, the core of the present paper is the proof of
interior Poincar\'e and Sobolev inequalities in $\he n$ when $p>1$.
Indeed, as we shall point out later (see Remark \ref{p.10}),
when $p>1$ global inequalities in $\he n$ (as well as in $\rn n$) are more or less straightforward.
On the contrary, interior inequalities require
a different more sophisticated argument (see Section \ref{Local homotopy operators intro} for a gist
of our proof). At the same time, the techniques introduced in the present paper differ substantially from those
of \cite{BFP} for global inequalities for $p=1$.
The case when $q=\infty$ can be obtained by duality, and this will appear in \cite{BFP5}. We refer also to \cite{BFP_Catania}
for endpoint inequalities in Orlicz spaces.
For more general sub-Riemannian spaces, the strategy is to reduce to large scale invariants (see section \ref{diverse}).
For this, one must pass via interior inequalities and a global smoothing procedure, like in Theorems \ref{local intro} and \ref{1.5}.
In particular, in the present paper and in \cite{BFP3} we deal with a special class of sub-Riemannian manifolds,
the sub-Riemannian contact manifold of bounded $C^k$-geometry as in Definition \ref{contact bis}.
}
\subsection{Open questions}\label{questions}
Keeping in mind the analogous inequalities in the scalar case, the following (still open) questions
naturally arise.
\begin{itemize}
\item[1.] {Do Poincar\'e and Sobolev inequalities hold without loss of the domain
for some family of specific domains as, e.g., for metric balls associated with a left-invariant
homogeneous distance?
}
\item[2.] {Since Heisenberg groups provide the simplest non-commutative instance
of arbitrary Carnot groups (connected, simply connected stratified nilpotent groups: see \cite{pansu_annals}), the following
question naturally arises: how much of these results do extend to more general Carnot groups?
}
\end{itemize}
{
Let us make a few comments about the previous questions.
\begin{itemize}
\item[1.] When dealing with scalar functions it is possible to obtain $\he{}$-$\mathrm{Poincar\acute{e}}_{p,q}$
inequalities on Carnot-Carath\'eodory balls without loss on the domain and the argument
relies on the so-called Boman chain condition (see, e.g. \cite{FGuW}, \cite{FLW}). However, it is not clear at all how to extend
this technique to differential forms.
\item[2.] The argument used in this paper relies on an appropriate approximate homotopy formula (see point (2) in Section
\ref{Local homotopy operators intro} below). It is reasonable to expect that the construction of the
approximate homotopy operator could be generalized to more general Carnot groups using the construction
carried out in \cite{BFTT} and \cite{PR} to prove a compensated compactness result (see formul\ae\ (37) and (38) in \cite{BFTT}).
However, for Carnot groups we expect only unsharp estimates, due to the crucial role of a fundamental solution
of a 0-order Laplace operator mixing up components of forms of different homogeneity.
Further comments related to this question can be found in Remark \ref{quaternionic} below where specific
examples in more general Carnot groups are given.
\end{itemize}
}
Let us give now a sketch of the proofs.
\subsection{Global homotopy operators}\label{Global homotopy operators intro}
The most efficient way to prove a Poincar\'e inequality is to find a homotopy between identity and 0 on the complex of differential forms, i.e. a linear operator $K$ that raises the degree by 1 and satisfies
\begin{eqnarray*}
I=dK+Kd.
\end{eqnarray*}
More generally, we shall deal with homotopies between identity and other operators $P$, i.e. of the form
\begin{eqnarray*}
I-P=dK+Kd.
\end{eqnarray*}
In Euclidean space, the Laplacian provides us such a homotopy. Write $\Delta=d\delta+\delta d$. Denote by $\Delta^{-1}$ the operator of convolution with the fundamental solution of the Laplacian. Then $\Delta^{-1}$ commutes with $d$ and its adjoint $\delta$, hence $K_{\mathrm{Euc}}=\delta \Delta^{-1}$ satisfies $I=dK_{\mathrm{Euc}}+K_{\mathrm{Euc}} d$ on globally defined $L^p$ differential forms. Furthermore, $K_{\mathrm{Euc}}$ is bounded $L^p\to L^q$ provided $\frac{1}{p}-\frac{1}{q}=\frac{1}{n}$. This proves the {global} Poincar\'e$_{p,q}(h)$ inequality for Euclidean space.
{Rumin defines a Laplacian $\Delta_c$ by $\Delta_c=d_c\delta_c+\delta_c d_c$ when both $d_c$ and $\delta_c$ are first order horizontal differential operators, and by $\Delta_c=(d_c\delta_c)^2+\delta_c d_c$ or $\Delta_c=d_c\delta_c+(\delta_c d_c)^2$ near middle dimension (i.e. when $h=n$ or $h=n+1$, respectively), when one of them has order 2. This leads to a homotopy of the form $K_0=\delta_c \Delta_c^{-1}$ or $K_0=\delta_c d_c\delta_c \Delta_c^{-1}$ depending on degree. Again, $K_0$ is a singular integral of potential type associated with a homogeneous kernel and therefore
is bounded from $L^p$ to $L^q$ under assumption $E(h,p,q,n)$ (see \cite{folland} or \cite{folland_stein} for the continuity of Riesz potentials
in homogeneous groups). This proves the {global} $\he{}$-Poincar\'e$_{p,q}(h)$ inequality for Heisenberg group, Theorem \ref{strongglobal}.
}
\subsection{Local homotopy operators}\label{Local homotopy operators intro}
We pass to interior estimates.
%local results.
In Euclidean space, Poincar\'e's Lemma asserts that every closed form on a ball is exact. We need a quantitative version of this statement. The standard proof of Poincar\'e's Lemma relies on a homotopy operator which depends on the choice of an origin. Averaging over origins yields a bounded operator $K_{\mathrm{Euc}}:L^p\to L^q$, as was observed by Iwaniec and Lutoborski, \cite{IL}. This proves the {global} Euclidean Poincar\'e$_{p,q}(h)$ inequality for convex Euclidean domains. A support preserving variant $J_{\mathrm{Euc}}:L^p\to L^q$ appears in Mitrea-Mitrea-Monniaux, \cite{mitrea_mitrea_monniaux} and this proves the {global} Euclidean Sobolev$_{p,q}$ inequality for bounded convex Euclidean domains. Incidentally, since for balls constants do not depend on the radius of the ball, this reproves the {global} Euclidean Sobolev$_{p,q}$ inequality for Euclidean spaces.
In this paper a sub-Riemannian counterpart is obtained using the homotopy equivalence of de Rham's and Rumin's complexes. Since this homotopy is a differential operator, a preliminary smoothing operation is needed. This is obtained by localizing (multiplying the kernel with cut-offs) the global homotopy $K_0$ provided by the inverse of Rumin's (modified) Laplacian.
Hence the proof goes as follows (see Section \ref{poincare}):
\begin{enumerate}
\item Show that the inverse $K_0$ of Rumin's modified Laplacian on all of $\he n$ is given by a homogeneous kernel $k_0$. Deduce bounds $L^p\to W^{1,q}$,
where $q,p$ are as above. Conclude that $K_0$ is an exact homotopy for globally defined $L^p$ forms.
Basically, this step does not contain any new idea, relying only on the estimates of the fundamental solution of Rumin's modified Laplacian (see \cite{BFT3}) and on classical estimates
for convolution kernels in homogeneous groups (see \cite{folland}, \cite{folland_stein}).
\item Take a smooth cut-off function $\psi$, $\psi\equiv 1$ in a neighborhood of the origin, and split $k_0=\psi k_0+(1-\psi)k_0$, so that $\psi k_0$ has small support
near the origin and $(1-\psi)k_0$ is smooth. Denote by $T$ the convolution operator associated with the kernel $\psi k_0$, and by
$K_{\mathrm{smooth}}$ the convolution operator associated with the kernel { $(1-\psi) k_0$}. It turns out that $T$ is a homotopy on balls
(with a loss on domain) between the identity $I$ and the operator $S:=d_c K_{\mathrm{smooth}}+K_{\mathrm{smooth}}d_c$ (which is smoothing),
i.e. $I-S = d_cT + Td_c$. The operator $S$ provides the required local smoothing operator.
\item Compose Iwaniec \& Lutoborski's averaged Poincar\'e homotopy for the de Rham complex and Rumin's homotopy, and apply the result to smoothed forms. This proves an interior Poincar\'e inequality in Heisenberg groups. Replacing Iwaniec \& Lutoborski's homotopy with Mitrea, Mitrea \& Monniaux's homotopy leads to an interior Sobolev inequality
in Heisenberg groups.
\end{enumerate}
\subsection{Global smoothing}
Now we piece together local homotopy operators into globally defined smoothing operators. Let $k\ge 3$. Let $(M,H,g)$ be a bounded $C^k$-geometry sub-Riemannian contact manifold. Pick a uniform covering by equal radius balls. Let $\chi_j$ be a partition of unity subordinate to this covering. Let $\phi_j$ be the corresponding charts from the unit Heisenberg ball. Let $S_j$ and $T_j$ denote the smoothing and homotopy operators associated with $\phi_j$ { using the pull-back operator}. Set
\begin{eqnarray*}
T=\sum_j T_j \chi_j ,\quad S=\sum_j S_j \chi_j +T_j[\chi_j,d_c].
\end{eqnarray*}
When $d_c$ is first order, the commutator $[\chi_j,d_c]$ is an order 0 differential operator, hence $T_j[\chi_j,d_c]$ gains 1 derivative. When $d_c$ is second order, $[\chi_j,d_c]$ is a first order differential operator. It turns out that precisely in this case, $T_j$ gains 2 derivatives, hence $T_j[\chi_j,d_c]$ gains 1 derivative in this case as well.
The details are discussed in Section \ref{final}.
\subsection{Structure of the paper}\label{structure} In Section \ref{Rumin} we collect basic results
about Heisenberg groups $\he n$ and differential forms in $\he n$. Successively, we remind the notion of Rumin's complex for
Heisenberg groups as well as for general contact manifolds, providing explicit examples in low dimensions.
In Section \ref{kernels} we present a list of general results for Folland-Stein homogeneous kernels, and,
in particular, for matrix-valued kernels associated with Rumin's homogeneous Laplacian in $\he n$.
Section \ref{function spaces} is devoted to theory of Folland-Stein Sobolev spaces in Heisenberg groups and
in sub-Riemannian contact manifolds with bounded geometry. In particular, in Section \ref{sobolev contact}
we precise the notion and the properties of manifolds with bounded geometry. Section \ref{poincare} is the core of the paper,
containing an approximate homotopy formulae (i.e. and homotopy formula with a smoothing error term)
and Poincar\'e and Sobolev inequalities for differential forms in $\he n$.
Then, in Section \ref{final} we are able to prove a similar approximate homotopy formula for sub-Riemannian contact
manifolds with bounded geometry. The error term is a regularizing operator with ``maximal regularity''.
Finally, Section \ref{diverse} contains a few examples of contact manifolds with bounded geometry, and a brief discussion
of the $\ell^{q,p}$ cohomology.
\section{Heisenberg groups and Rumin's complex $(E_0^\bullet,d_c)$}
\label{Rumin}
\subsection{Differential forms on Heisenberg groups}
We denote by $\he n$ the $n(2n+1)$-dimensional Heisenberg
group, identified with $\rn {2n+1}$ through exponential
coordinates. A point $p\in \he n$ is denoted by
$p=(x,y,t)$, with both $x,y\in\rn{n}$
and $t\in\R$.
If $p$ and
$p'\in \he n$, the group operation is defined by
\begin{equation*}
p\cdot p'=(x+x', y+y', t+t' + \frac12 \sum_{j=1}^n(x_j y_{j}'- y_{j} x_{j}')).
\end{equation*}
{Notice that $\he n$ can be equivalently identified with $\mathbb C\times \mathbb R$
endowed with the group operation
$$
(z,t)\cdot (\zeta,\tau): = (z+\zeta, t+\tau + \frac12\,Im\,(z\bar{\zeta})).
$$
}
The unit element of $\he n$ is the origin, that will be denote by $e$.
For
any $q\in\he n$, the {\it (left) translation} $\tau_q:\he n\to\he n$ is defined
as $$ p\mapsto\tau_q p:=q\cdot p. $$
For a general review on Heisenberg groups and their properties, we
refer to \cite{Stein}, \cite{GromovCC} and to \cite{VarSalCou}.
We limit ourselves to fix some notations, following \cite{FSSC_advances}.
{ First we notice that Heisenberg groups are smooth manifolds (and therefore
are Lie groups). In particular, the pull-back of differential forms is well
defined as follows (see, e.g. \cite{GHL}, Proposition 1.106);
\begin{definition} If $\;\mc U,\mc V$ are open subsets of $\he n$, and $f: \mc U\to
\mc V$ is a
diffeomorphism, then for any differential form $\alpha$ of degree $h$,
we denote by $f^\sharp \alpha$ the pull-back form in $\mc
\mc U$ defined by
$$
(f^\sharp \alpha)(p) (v_1,\dots,v_h):= \alpha(f(p)) (df(p)v_1,\dots,df(p)v_h)
$$
for any $h$-tuple $(v_1,\dots,v_h)$ of tangent vectors at $p$.
\end{definition}
}
The Heisenberg group $\he n$ can be endowed with the homogeneous
norm (Cygan-Kor\'anyi norm): if $p=(x,y,t)\in \he n$, then we set
\begin{equation}\label{gauge}
\varrho (p)=\big((x^2+y^2)^2+t^2\big)^{1/4},
\end{equation}
and we define the gauge distance (a true distance, see
\cite{Stein}, p.\,638), that is left invariant i.e. $d(\tau_q p,\tau_q p')=d(p,p' )$ for all $p,p'\in\he n$)
as
\begin{equation}\label{def_distance}
d(p,q):=\varrho ({p^{-1}\cdot q}).
\end{equation}
{Notice that $d$ is equivalent to the Carnot-Carath\'eodory distance on $\he n$ (see, e.g., \cite{BLU}, Corollary 5.1.5).}
Finally, the balls for the metric $d$ are the so-called Cygan-Kor\'anyi balls
\begin{equation}\label{koranyi}
B(p,r):=\{q \in \he n; \; d(p,q)< r\}.
\end{equation}
Notice that Cygan-Kor\'anyi balls are convex smooth sets.
{A straightforward computation shows that, if $ \rho(p) <1$, then
\begin{equation}\label{c0}
|p| \le \rho(p) \le |p|^{1/2}.
\end{equation}
}
It is well known that the topological dimension of $\he n$ is $2n+1$,
since as a smooth manifold it coincides with $\R^{2n+1}$, whereas
the Hausdorff dimension of $(\he n,d)$ is $Q:=2n+2$
(the so called \emph{homogeneous dimension} of $\he n$).
We denote by $\mfrak h$
the Lie algebra of the left
invariant vector fields of $\he n$. The standard basis of $\mfrak
h$ is given, for $i=1,\dots,n$, by
\begin{equation*}
X_i := \partial_{x_i}-\frac12 y_i \partial_{t},\quad Y_i :=
\partial_{y_i}+\frac12 x_i \partial_{t},\quad T :=
\partial_{t}.
\end{equation*}
The only non-trivial commutation relations are $
[X_{j},Y_{j}] = T $, for $j=1,\dots,n.$
The {\it horizontal subspace} $\mfrak h_1$ is the subspace of
$\mfrak h$ spanned by $X_1,\dots,X_n$ and $Y_1,\dots,Y_n$:
${ \mfrak h_1:=\mathrm{span}\,\left\{X_1,\dots,X_n,Y_1,\dots,Y_n\right\}\,.}$
\noindent Coherently, from now on, we refer to $X_1,\dots,X_n,Y_1,\dots,Y_n$
(identified with first order differential operators) as
the {\it horizontal derivatives}. Denoting by $\mfrak h_2$ the linear span of $T$, the $2$-step
stratification of $\mfrak h$ is expressed by
\begin{equation*}
\mfrak h=\mfrak h_1\oplus \mfrak h_2.
\end{equation*}
\bigskip
{
The stratification of the Lie algebra $\mfrak h$ induces a family of non-isotropic dilations
$\delta_\lambda: \he n\to\he n$, $\lambda>0$ as follows: if
$p=(x,y,t)\in \he n$, then
\begin{equation}\label{dilations}
\delta_\lambda (x,y,t) = (\lambda x, \lambda y, \lambda^2 t).
\end{equation}
}
The vector space $ \mfrak h$ can be
endowed with an inner product, indicated by
$\scalp{\cdot}{\cdot}{} $, making
$X_1,\dots,X_n$, $Y_1,\dots,Y_n$ and $ T$ orthonormal.
Throughout this paper, we write also
\begin{equation}\label{campi W}
W_i:=X_i, \quad W_{i+n}:= Y_i\quad { \mathrm{and} } \quad W_{2n+1}:= T, \quad \text
{for }i =1, \dots, n.
\end{equation}
The dual space of $\mfrak h$ is denoted by $\covH 1$. The basis of
$\covH 1$, dual to the basis $\{X_1,\dots , Y_n,T\}$, is the family of
covectors $\{dx_1,\dots, dx_{n},dy_1,\dots, dy_n,\theta\}$ where
\begin{equation}\label{theta}
\theta
:= dt - \frac12 \sum_{j=1}^n (x_jdy_j-y_jdx_j)
\end{equation}
is called the {\it contact
form} in $\he n$.
We denote by $\scalp{\cdot}{\cdot}{} $ the
inner product in $\covH 1$ that makes $(dx_1,\dots, dy_{n},\theta )$
an orthonormal basis.
Coherently with the previous notation \eqref{campi W},
we set
\begin{equation*}
\omega_i:=dx_i, \quad \omega_{i+n}:= dy_i \quad { \mathrm{and} }\quad \omega_{2n+1}:= \theta, \quad \text
{for }i =1, \dots, n.
\end{equation*}
{
We put
$ \vetH 0 := \covH 0 =\R $
and, for $1\leq h \leq 2n+1$,
\begin{equation*}
\begin{split}
\covH h& :=\mathrm {span}\{ \omega_{i_1}\wedge\dots \wedge \omega_{i_h}:
1\leq i_1< \dots< i_h\leq 2n+1\}.
\end{split}
\end{equation*}
In the sequel we shall denote by $\Theta^h$ the basis of $ \covH h$ defined by
$$
\Theta^h:= \{ \omega_{i_1}\wedge\dots \wedge \omega_{i_h}:
1\leq i_1< \dots< i_h\leq 2n+1\}.
$$
To avoid cumbersome notations, if $I:=({i_1},\dots,{i_h})$, we write
$$
\omega_I := \omega_{i_1}\wedge\dots \wedge \omega_{i_h}.
$$
The inner product $\scal{\cdot}{\cdot}$ on $ \covH 1$ yields naturally an inner product
$\scal{\cdot}{\cdot}$ on $ \covH h$
making $\Theta^h$ an orthonormal basis.
The volume $(2n+1)$-form $ \omega_1\wedge\cdots\wedge \omega_{ 2n+1}$
will be also
written as $dV$.
Throughout this paper, the elements of $\cov h$ are identified with \emph{left invariant} differential forms
of degree $h$ on $\he n$.
\begin{definition}\label{left} A $h$-form $\alpha$ on $\he n$ is said left invariant if
$$\tau_q^\#\alpha
=\alpha\qquad\mbox{for any $q\in\he n$.}
$$
\end{definition}
The same construction can be performed starting from the vector
subspace $\mfrak h_1\subset \mfrak h$,
obtaining the {\it horizontal $h$-covectors}
\begin{equation*}
\begin{split}
\covh h& :=\mathrm {span}\{ \omega_{i_1}\wedge\dots \wedge \omega_{i_h}:
1\leq i_1< \dots< i_h\leq 2n\}.
\end{split}
\end{equation*}
It is easy to see that
$$
\Theta^h_0 := \Theta^h \cap \covh h
$$
provides an orthonormal
basis of $ \covh h$.
Keeping in mind that the Lie algebra $\mathfrak h$ can be identified with the
tangent space to $\he n$ at $x=e$ (see, e.g. \cite{GHL}, Proposition 1.72),
starting from $\cov h$ we can define by left translation a fiber bundle
over $\he n$ that we can still denote by $\cov h$. We can think of $h$-forms as sections of
$\cov h$. We denote by $\Omega^h$ the
vector space of all smooth $h$-forms.
\bigskip
We already pointed out in Section \ref{contact introduction} that the stratification
of the Lie algebra $\mfrak h$ yields a lack of homogeneity of de Rham's exterior differential
with respect to group dilations $\delta_\lambda$. Thus, to keep into account the different degrees
of homogeneity of the covectors when they vanish on different layers of the
stratification, we introduce the notion of {\sl weight} of a covector as follows.
}
\begin{definition}\label{weight} If $\eta\neq 0$, $\eta\in \covh 1$,
we say that $\eta$ has \emph{weight $1$}, and we write
$w(\eta)=1$. If $\eta = \theta$, we say $w(\eta)= 2$.
More generally, if
$\eta\in \covH h$, { $\eta\neq 0$, }we say that $\eta$ has \emph {pure weight} $p$ if $\eta$ is
a linear combination of covectors $\omega_{i_1}\wedge\cdots\wedge\omega_{i_h}$
with $w(\omega_{i_1})+\cdots + w(\omega_{ i_h})=p$.
\end{definition}
{ Notice that, if $\eta,\zeta \in \covH h$ and $w(\eta)\neq w(\zeta)$, then
$\scal{\eta}{\zeta}=0$ (see \cite{BFTT}, Remark 2.4). We notice also that
$w(d\theta) = w(\theta)$.
We stress that generic covectors may fail to have a pure weight: it is enough to
consider $\he 1$ and the covector $dx_1+\theta\in \covH{1}$. However, the
following result holds
(see \cite{BFTT}, formula (16)):
\begin{equation}\label{dec weights}
\covH h = \covw {h}{h}\oplus \covw {h}{h+1} = \covh h\oplus \Big(\covh {h-1}\Big)\wedge \theta,
\end{equation}
where $\covw {h}{p}$ denotes the linear span of the $h$-covectors of weight $p$.
By our previous remark, the decomposition \eqref{dec weights} is orthogonal.
In addition, since the elements of the basis $\Theta^h$ have pure weights, a basis of
$ \covw {h}{p}$ is given by $\Theta^{h,p}:=\Theta^h\cap \covw {h}{p}$
(such a basis is usually called an adapted basis).
{We notice that, according to \eqref{dec weights}, the weight of a $h$-form
is either $h$ or $h+1$ and there are no forms of weight $h+2$, since there
is only one 1-form of weight 2. Something analogous can be possible for instance in
$\he n\times \mathbb R$, but it fails to be possible already in the case of general step 2 groups
with higher dimensional center (see also Remark \ref{quaternionic} below).
}
As above, starting from $\covw {h}{p}$, we can define by left translation a fiber bundle
over $\he n$ that we can still denote by $\covw {h}{p}$.
Thus, if we denote by $\Omega^{h,p} $ the vector space of all
smooth $h$--forms in $\he n$ of weight $p$, i.e. the space of all
smooth sections of $\covw {h}{p}$, we have
\begin{equation}\label{deco forms}
\Omega^h = \Omega^{h,h}\oplus\Omega^{h,h+1} .
\end{equation}
}
\subsection{Rumin's complex on Heisenberg groups}\label{rumin heisenberg}
{ Let us give a short introduction to Rumin's complex. For a more detailed presentation we
refer to Rumin's papers \cite{rumin_grenoble}. Here we follow the presentation of \cite{BFTT}.
}
The exterior differential $d$ does not preserve weights. It splits into
\begin{eqnarray*}
d=d_0+d_1+d_2
\end{eqnarray*}
where $d_0$ preserves weight, $d_1$ increases weight by 1 unit and $d_2$ increases weight by 2 units.
{
More explicitly,
let $\alpha\in \Omega^{h}$ be a (say) smooth form
of pure weight $h$. We can write
$$
\alpha= \sum_{\omega_I\in\Theta^{h}_0}\alpha_{I}\, \omega_I
,\quad
\mbox{with } \alpha_I \in \mc C^\infty (\he n).
$$
Then
$$
d\alpha= \sum_{\omega_I\in\Theta^{h}_0}\sum_{j=1}^{2n} (W_j\alpha_{I})\, \omega_j\wedge\omega_I +
\sum_{\omega_I\in\Theta^{h}_0} (T \alpha_{I})\, \theta\wedge \omega_I = d_1\alpha + d_2\alpha,
$$
and $d_0\alpha =0$. On the other hand, if $\alpha\in \Omega^{h,h+1}$ has pure weight $h+1$, then
$$
\alpha = \sum_{\omega_J\in\Theta^{h-1}_0}\alpha_{J}\, \theta\wedge\omega_J,
$$
and
$$
d\alpha= \sum_{\omega_J\in\Theta^{h}_0}\alpha_J\,d\theta\wedge\omega_J + \sum_{\omega_J\in\Theta^{h}_0}\sum_{j=1}^{2n} (W_j\alpha_{J})\, \omega_j\wedge\theta\wedge\omega_I
=d_0\alpha+d_1\alpha,
$$
and $d_2\alpha=0$.
It is crucial to notice that $d_0$ is an algebraic operator, in the sense that
for any real-valued $f\in\mc C^\infty (\he n)$ we have
$$
d_0(f\alpha)= f d_0\alpha,
$$
so that its action can be identified at any point with the action of a linear
operator from $\cov h$ to $\cov {h+1}$ (that we denote again by $d_0$).
Following M. Rumin (\cite{rumin_grenoble}, \cite{rumin_cras}) we give the following definition:
\begin{definition}\label{E0}
If $0\le h\le 2n+1$, keeping in mind that $\cov h$ is endowed with a canonical
inner product, we set
$$
E_0^h:= \ker d_0\cap (\mathrm{Im}\; d_0)^{\perp}.
$$
Straightforwardly, $E_0^h$ inherits from $\cov h$ the
inner product.
\end{definition}
As above, $E_0^\bullet$ defines by left translation a fibre bundle over $\he n$,
that we still denote by $E_0^\bullet$. To avoid cumbersome notations,
we denote also by $E_0^\bullet$ the space of sections of this fibre bundle.
Let $L: \cov h \to \cov{h+2}$ the Lefschetz operator defined by
\begin{equation}\label{lefs}
L\, \xi = d\theta\wedge\xi.
\end{equation}
Then the spaces $E_0^\bullet$ can be defined explicitly as follows:
\begin{theorem}[see \cite{rumin_jdg}, \cite{rumin_gafa}] \label{rumin in H} We have:
\begin{itemize}
\item[i)] $E_0^1= \covh{1}$;
\item[ii)] if $2\le h\le n$, then $E_0^h= \covh{h}\cap \big(\covh{h-2}\wedge d\theta\big)^\perp$
(i.e. $E_0^h$ is the space of the so-called \emph{primitive covectors} of $\covh h$);
\item[iii)] if $n< h\le 2n+1$, then $E_0^h = \{\alpha = \beta\wedge\theta, \; \beta\in \covh{h-1},
\; \gamma\wedge d\theta =0\} = \theta\wedge\ker L$;
\item[iv)] if $1 0$ such that}
$$
\| u\ast K\|_{L^p(\he n)} \le C \| u\|_{L^p(\he n)}.
$$
\end{itemize}
\end{theorem}
\begin{proof}
For statements i) and iii), we refer to \cite{folland}, Propositions 1.11 and 1.9. As for ii),
if $p\ge Q/\alpha$, we choose $1<\tilde p n$, together with
Remark 4.1 in \cite{IL}.
As for iii), the statement can be proved similarly to i), noticing that $K = \Pi_{E_0}\Pi_E K_{\mathrm{Euc}}$ on forms of degree $n+1$.
Finally, $\mathrm{supp}\, J \omega
\subset B$ since both $\Pi_E$ and $\Pi_{E_0}$ preserve the support.
\end{proof}
The operators $K$ and $J$ provide a local homotopy in Rumin's complex, but fail to yield the
Sobolev and Poincar\'e inequalities we are looking for, since, because of the presence
of the projection operator $\Pi_E$ (that on forms of low degree is a first order differential
operator) they loose regularity as is stated in Lemma \ref{senza nome}, ii) above. In order
to build ``good'' local homotopy operators with the desired gain of regularity, we have
to combine them with homotopy operators which, though not local, in fact provide the ``good''
gain of regularity.
\begin{proposition}\label{homotopy formulas} If $\alpha\in \mc D(\he n, E_0^h)$
for $p>1$ and $h= 1,\dots,2n$, then the following homotopy formulas hold:
there exist operators $K_1$, $\tilde K_1$ and $K_2$, $\tilde K_1$ acting on
$ \mc D(\he n, E_0^\bullet)$
such that
\begin{itemize}
\item if $h\neq n, n+1$, then $\alpha = d_c K_1 \alpha + \tilde K_1d_c \alpha $, where $K_1$ and $\tilde K_1$ are associated with kernels $k_1, \tilde k_1$ of type 1;
\item if $h = n$, then $\alpha = d_c K_1 \alpha + \tilde K_2 d_c\alpha$, where $K_1$ and $\tilde K_2$ are associated with kernels $k_1, \tilde k_2$ of type 1 and 2,
respectively;
\item if $h = n+1 $, then $\alpha = d_c K_2 \alpha + \tilde K_1 d_c\alpha$, where $K_2$ and $\tilde K_1$ are associated with kernels $k_2, \tilde k_1$ of type 2
and $1$, respectively.
\end{itemize}
\end{proposition}
\begin{proof}
Suppose $h\neq n-1,n,n+1$. By Lemma \ref{comm}, we have:
\begin{equation*}\begin{split}
\alpha &=
\Delta_{\mathbb H,h} \Delta^{-1}_{\mathbb H,h}\alpha = d_c (\delta_c \Delta^{-1}_{\mathbb H,h})\alpha
+ \delta_c (d_c \Delta^{-1}_{\mathbb H,h})\alpha
\\&
=d_c (\delta_c \Delta^{-1}_{\mathbb H,h})\alpha +
(\delta_c \Delta^{-1}_{\mathbb H,h+1})d_c \alpha.
\end{split}\end{equation*}
where $\delta_c \Delta^{-1}_{\mathbb H,h}$ and $\delta_c \Delta^{-1}_{\mathbb H,h+1}$ are
associated with a kernel of type $1$ (by Proposition \ref{kernel}
and Theorem \ref{global solution}).
Analogously, if $h=n-1$
\begin{equation*}\begin{split}
\alpha &=
\Delta_{\mathbb H,n-1} \Delta^{-1}_{\mathbb H,n-1}\alpha = d_c (\delta_c \Delta^{-1}_{\mathbb H,n-1})\alpha
+ \delta_c (d_c \Delta^{-1}_{\mathbb H,n-1})\alpha
\\&
=d_c (\delta_c \Delta^{-1}_{\mathbb H,n-1})\alpha +
(\delta_c d_c\delta_c\Delta^{-1}_{\mathbb H, n} )d_c \alpha.
\end{split}\end{equation*}
Again $\delta_c \Delta^{-1}_{\mathbb H,n-1}$ and $\delta_c d_c\delta_c\Delta^{-1}_{\mathbb H, n}$
are
associated with kernels of type $1$.
Take now $h=n$. Then
\begin{equation*}\begin{split}
\alpha &= \Delta_{\mathbb H,n} \Delta^{-1}_{\mathbb H,n}\alpha = (d_c \delta_c )^2 \Delta^{-1}_{\mathbb H,n} \alpha
+ \delta_c (d_c \Delta^{-1}_{\mathbb H,n})\alpha
\\&
= d_c (\delta_c d_c \delta_c \Delta^{-1}_{\mathbb H,n})\alpha +
\delta_c \Delta^{-1}_{\mathbb H,n+1}d_c \alpha
\end{split}\end{equation*}
where $\delta_c d_c \delta_c \Delta^{-1}_{\mathbb H,n}$ and $\delta_c \Delta^{-1}_{\mathbb H,n+1}$ are associated with a kernel of type $1$ and $2$,
respectively).
Finally, take $h=n+1$. Then
\begin{equation*}\begin{split}
\alpha &= \Delta_{\mathbb H,n+1} \Delta^{-1}_{\mathbb H,n+1}\alpha = d_c \delta_c \Delta^{-1}_{\mathbb H,n+1} \alpha
+ (\delta_c d_c )^2 \Delta^{-1}_{\mathbb H,n+1} \alpha
\\&
= d_c \delta_c \Delta^{-1}_{\mathbb H,n+1} \alpha +
\delta_c \Delta^{-1}_{\mathbb H,n+2} d_c\alpha
\end{split}\end{equation*}
where $\delta_c \Delta^{-1}_{\mathbb H,n+1} $ and $\delta_c \Delta^{-1}_{\mathbb H,n+2}$ associated with kernels of type $2$ and $1$, respectively.
\end{proof}
The $L^p-L^q$ continuity properties of convolution operators associated with Folland's kernels yields the following
{global} $\he{}$-$\mathrm{Poincar\acute{e}}_{p,q}(h)$ inequality in $\he n$ (the {global} $\he{}$-$\mathrm{Sobolev}_{p,q}(h)$
is obtained in Corollary \ref{strong sobolev}).
\begin{corollary}\label{strong poincare}
Take $1\le h\le 2n+1$. Suppose $1 1$, be concentric balls of $\he{n}$. If $1\le h\le 2n+1$, there exist operators $T$ and $\tilde T$ from
$C^\infty(B', E_0^h)$ to $C^\infty(B, E_0^{h-1})$ and $S$ from
$C^\infty(B', E_0^h)$ to $C^\infty(B, E_0^h)$ satisfying
\begin{equation}\label{approx homotopy tilde}
d_c T+ \tilde Td_c + S=I\qquad\mbox{on $B$.}
\end{equation}
\end{theorem}
\begin{proof} Suppose first $h\neq n,n+1$. We consider a cut-off function $\psi_R$ supported in a $R$-neighborhood
of the origin, such that $\psi_R\equiv 1$ near the origin. With the notations of Proposition \ref{homotopy formulas}, we can write
\begin{equation}\label{morti}
k_1=k_1\psi_R + (1-\psi_R)k_1\qquad\mbox{and}\qquad \tilde k_1=\tilde k_1\psi_R + (1-\psi_R)\tilde k_1,
\end{equation}
where
\begin{equation}\label{23nov}
k_1=:(k_1)_{\ell,\lambda}\qquad\mbox{and}\qquad \tilde k_1=:(\tilde k_1)_{\ell,\lambda}
\end{equation}
are the matrix-valued kernels associated with the operators $\delta_c \Delta_{\mathbb H, h}$ and
$\delta_c \Delta_{\mathbb H, h+1}$, respectively, as shown in the proof of Proposition \ref{homotopy formulas}.
Let us denote by $K_{1,R}$, $\tilde K_{1,R}$ the convolution operators associated with
$\psi_R k_1$, $\psi_R \tilde k_1$, respectively.
Let us fix two balls $B_0$, $B_1$ with
\begin{equation}\label{varie B}
B\Subset B_0 \Subset B_1\Subset B',
\end{equation}
and a cut-off function $\chi \in \mc D(B_1)$,
$\chi\equiv 1$ on $B_0$. If $\alpha \in C^\infty(B', E_0^\bullet)$, we set $\alpha_0= \chi\alpha$, continued by zero outside $B_1$.
%Keeping in mind \eqref{convolution by parts} and Proposition \ref{kernel},
We have:
\begin{equation}\label{sept 9 eq:1}
\alpha_0 = d_c K_{1,R} \alpha_0 + \tilde K_{1,R}d_c \alpha_0 + S_0\alpha_0,
\end{equation}
where $S_0$ is
\begin{equation}\label{santi}
S_0\alpha_0 := d_c( \alpha_0\ast (1-\psi_R)k_1 ) + d_c\alpha_0 \ast(1-\psi_R)\tilde k_1.
\end{equation}
We set
\begin{equation}\label{we set}
T\alpha := K_{1,R} \alpha_0, \qquad \tilde T d_c\alpha:= \tilde K_{1,R}d_c \alpha_0, \qquad S\alpha:= S_0\alpha_0.
\end{equation}
We notice that, provided $R>0$ is small enough, the definition of $T$ and $\tilde T$
does not depend on the continuation of $\alpha$ outside $B_0$.
By \eqref{sept 9 eq:1} we have
$$
\alpha = d_c T\alpha + \tilde Td_c \alpha + S\alpha \qquad\mbox{in $B$}.
$$
If $h=n$ we can carry out the same construction, replacing $\tilde k_1$ by $\tilde k_2$ (keep in mind that $\tilde k_2$
is a kernel of type 2, again by Proposition \ref{homotopy formulas}). Analogously, if $h=n+1$ we can carry out the same construction, replacing $k_1$ by $ k_2$ (again
a kernel of type 2).
\end{proof}
Later on, we need the following remark:
\begin{remark} \label{supp T} By construction, if $\mathrm{supp}\, \alpha \subset B$ then $\mathrm{supp}\, T \alpha$ is contained in
a $R$-neighborhood of $B$ and then is contained in $B_0$ provided $R 1$, be concentric balls of $\he{n}$, and let $1\le h\le 2n+1$.
Then the operator $S$ defined in \eqref{we set}
is a smoothing operator. In particular, for any $m, s\in \mathbb Z$, $m 0$, then the map $x\to f(x):= \tau_{p} \delta_{t} (x)$
maps $B(e,\rho)$ into $B(p,t\rho)$ for $\rho>0$. Therefore, by Proposition \ref{pull},
from the previous theorem for balls of fixed radius, we obtain the following result for general balls.
\begin{theorem} Take $1\le h\le 2n+1$. Suppose $1 0$ in the definition of the kernel of $T$ has
been chosen in such a way that the $R$-neighborhood of $\phi_j^{-1}(\mathrm{supp}\;\chi_j) \subset B(e,1)$.
In particular $v_j - d_cT v_j - Td_cv_j $ is supported in $B(e,1)$ and therefore also $Sv_j$ is supported
in $B(e,1)$.
In particular, $ (\phi_j^{-1})^\# \big(
d_cTv_j + Td_cv_j + Sv_j
\big)$ is supported in $\phi_j (B(e,1))$ so that it can be continued by zero on $M$.
Thus
\begin{equation*}\begin{split}
u &= \sum_j (\phi_j^{-1})^\# \big(
d_cT v_j + Td_cv_j + S v_j
\big)
\\&=
d_c \sum_j \ (\phi_j^{-1})^\#
T \phi_j^\#(\chi_j u )
\\&+
\sum_j ( (\phi_j^{-1})^\#
T \phi_j^\#\chi_j ) d_c u
-
\sum_j (\phi_j^{-1})^\#
T \phi_j^\#([\chi_j ,d_c] u )
\\&+
\sum_j ( (\phi_j^{-1})^\# (S\phi_j^\# \chi_j )u.
\end{split}\end{equation*}
We set
\begin{equation}\label{T}
{ \bf{ T}}u:= \sum_j (\phi_j^{-1})^\#
T \phi_j^\# (\chi_j u)
\end{equation}
and
\begin{equation}\label{S}
{\bf{S} }u:= \sum_j (\phi_j^{-1})^\# S\phi_j^\# ( \chi_j u) -
\sum_j (\phi_j^{-1})^\#
T \phi_j^\#([\chi_j ,d_c] u).
\end{equation}
\begin{lemma}\label{homotopy manifold lemma}
Let $(M,H,g)$ be a bounded $C^k$-geometry sub-Riemannian contact manifold with $k\ge 3$. If $2\le\ell\le k-1$ and ${\bf T}$
and ${ {\bf S}}$ are
defined in \eqref{T} and \eqref{S}, then the following homotopy formula holds:
\begin{equation}\label{homotopy M0}
I= d_c {\bf T}+ {\bf T}d_c + { {\bf S}}.
\end{equation}
In particular, ${ {\bf S}d_c = d_c{\bf S}}$.
In addition, if $1\leq h \le 2n+1$, the following maps are continuous:
\begin{itemize}
\item[i)] $ {\bf T}: W^{-1,p}(M,E_0^{h+1}) \to L^p(M,E_0^{h})$ if $h\neq n$, whereas $ {\bf T}: W^{-2,p}(M,E_0^{n+1}) \to L^p(M,E_0^{n})$;
\item[ii)] ${\bf T}: L^p (M,E_0^{h}) \to W^{1,p}(M,E_0^{h-1})$ if $h\neq n+1$, whereas $ {\bf T}: L^p(M,E_0^{n+1}) \to W^{2,p}(M,E_0^{n})$;
\item[iii)] if $1\le\ell\le k$, then ${ {\bf S}}: W^{\ell-1,p}(M, E_0^h) \longrightarrow W^{\ell,p}(M, E_0^h)$.
\end{itemize}
\end{lemma}
\begin{proof} First of all, we notice that, if $\alpha$ is supported in $\phi_j(B(e,\lambda))$, then, by Definition
\ref{contact bis} the norms
$$
\|\alpha\|_{W^{m,p}(M,E_0^\bullet)} \qquad \mbox{and}\qquad \|\phi_j^\#\alpha\|_{W^{m,p}(\he n,E_0^\bullet)}
$$
are equivalent for $-k\le m\le k$, with equivalence constants independent of $j$. Thus, assertions i) and ii)
follow straightforwardly from Theorem \ref{smoothing2}.
To get iii) we only need to note that
the operators $ (\phi_j^{-1})^\#
T \phi_j^\#[\chi_j ,d_c]$ are bounded $W^{\ell-1,p}(M, E_0^\bullet)\to W^{\ell,p}(M, E_0^\bullet)$ in every degree.
Indeed, by Proposition \ref{memory}, the differential operator $\phi_j^\#[\chi_j ,d_c](\phi_j^{-1})^\#$ in $\he n$ has order 1 if $h=n$, and
order 0 if $h\neq n$. Since the kernel of $ T$ can be estimated by kernel of type 2 if $T$
acts on forms of degree $h=n$, and of type 1 if it
acts on forms of degree $h \neq n$, the assertion follows straightforwardly.
Summing up in $j$ and keeping into account that the sum is locally finite, we obtain:
\begin{equation*}\begin{split}
\| \sum_j &(\phi_j^{-1})^\#
T \phi_j^\#[\chi_j ,d_c] u\|_{W^{\ell,p}(M, E_0^\bullet)}
\le
\sum_j\| (\phi_j^{-1})^\#
T \phi_j^\#[\chi_j ,d_c]u\|_{W^{\ell,p}(\phi_j (B(e,1)), E_0^\bullet)}
\\&
\le
C \sum_j\|
T \phi_j^\#[\chi_j ,d_c]u\|_{W^{\ell,p}(B(e,1), E_0^\bullet) }
\le
C \sum_j
\| \phi_j^\# u\|_{W^{\ell-1,p}(B(e,1), E_0^\bullet)}
\\&
\le
C \| u\|_{W^{\ell-1,p}(M, E_0^\bullet)}.
\end{split}\end{equation*}
\end{proof}
Now the following global homotopy formula holds in $M$.
\begin{theorem}\label{homotopy manifold}
Let $(M,H,g)$ be a bounded $C^k$-geometry sub-Riemannian contact manifold, $k\ge 3$. Then
\begin{equation}\label{homotopy M}
I= d_c T_M+ T_Md_c + S_M,
\end{equation}
where
$$
T_M:= \big(\sum_{i=0}^{k-1}{\bf{ S}}^i\big){\bf{ T}}, \qquad S_M:= {\bf{ S}}^{k},
$$
and ${\bf{ T}}$
and ${\bf{ S}}$ are
defined in \eqref{T} and \eqref{S}.
Moreover
\begin{equation}\label{S commuta su M}
{ d_c S_Mu = S_M d_c u,}
\end{equation}
and, if $1\le h\le 2n+1$, the following maps are continuous:
\begin{itemize}
\item[i)] $ T_M: W^{-1,p}(M,E_0^{h+1}) \to L^p(M,E_0^{h})$ if $h\neq n$, whereas $ T_M: W^{-2,p}(M,E_0^{n+1}) \to L^p(M,E_0^{n})$;
\item[ii)] $T_M: L^p (M,E_0^{h}) \to W^{1,p}(M,E_0^{h-1})$ if $h\neq n+1$, whereas $ T_M:L^p(M,E_0^{n+1}) \to W^{2,p}(M,E_0^{n})$;
\item[iii)] $S_M:L^p(M,E_0^h)\to { W^{k-1,p}(M,E_0^{h})}$.
\end{itemize}
\end{theorem}
\begin{proof}
By \eqref{S commuta su M},
\begin{equation*}\begin{split}
d_c T_M &+ T_Md_c + S_M
\\&
= d_c \big(\sum_{i=0}^{k-1}S^i\big)T + \big(\sum_{i=0}^{k-1}S^i\big)\tilde Td_c
+ S^k
\\&
=\sum_{i=0}^{k-1}S^i \big(d_cT + Td_c\big)
+S^k
\\&
=\sum_{i=0}^{k-1}S^i \big(I-S)
+S^k =I.
\end{split}\end{equation*}
Then statements i), ii) and iii) follow straightforwardly from i), ii) and iii) of
Lemma \ref{homotopy manifold lemma}.
\end{proof}
\section{Large scale geometry of contact sub-Riemannian manifolds}
\label{diverse}
Theorems \ref{local intro} and \ref{1.5} are the key to proving that the validity of {global} Poincar\'e inequalities is equivalent to vanishing of $\ell^{q,p}$ cohomology, a large scale invariant of metric spaces. This equivalence will be established in \cite{Pcup}. By large scale invariant, we mean preserved, under uniform local assumptions, by quasiisometries, i.e. maps $f$ between metric spaces which satisfy
$$
-C +\frac{1}{L}d(x,x') \leq d(f(x),f(x'))\leq Ld(x,x')+C,
$$
for suitable positive constants $L$ and $C$.
Avoiding the general metric definition of $\ell^{q,p}$ cohomology, let us give a construction valid for bounded geometry Riemannian manifolds with uniform vanishing of cohomology (the cohomology of an $R'$-ball dies when restricted to a concentric $R$-ball, where the radius $R'$ depends only on the radius $R$). First, one defines the $\ell^{q,p}$ cohomology of a simplicial complex: it is the quotient of the space of $\ell^p$ simplicial cocycles by the image of $\ell^q$ simplicial cochains by the coboundary operator. One shows that $\ell^{q,p}$ cohomology is a quasiisometry invariant of simplicial complexes with bounded geometry (i.e. bounded number of simplices through a vertex) and uniform vanishing of cohomology. Then one observes that every bounded geometry Riemannian manifold is quasiisometric to such a simplicial complex.
Under similar boundedness and uniformity assumptions, one can show (\cite{Pcup}) that various locally acyclic complexes can be used to compute $\ell^{q,p}$ cohomology. For contact sub-Riemannian manifolds, one can use either the exterior differential or Rumin's differential. As alluded to above, the building blocks are interior estimates and global smoothing, i.e. Theorems \ref{local intro} and \ref{1.5} and their Riemannian analogues. It follows that a global Poincar\'e inequality holds if and only if a global $\he{}$-Poincar\'e inequality holds.
Using the Riemannian Hodge Laplacian, D. M\"uller, M. Peloso and F. Ricci prove a Poincar\'e inequality Poincar\'e$_{2,q}$ for the exterior differential on the Riemannian Heisenberg group (\cite{MPR}, Lemma 11.2), under the assumption $\frac{1}{2}-\frac{1}{q}=\frac{1}{2n+1}$. Therefore, their result combined with \cite{Pcup} provides an alternative proof of part of Corollary 1.4 above. We note that in degree $h=n+1$, they miss the sharp exponent, given by our condition $E(n+1,2,q,n)$.
The advantage of Rumin's Laplacian over its Riemannian sibling is its scale invariance. This allows to apply the theory of singular integral operators, to treat $\ell^{q,p}$ cohomology for all $p$ and to get the sharp exponent in degree $h=n+1$. The drawback of Rumin's complex is that interior Poincar\'e inequalities become hard.
\subsection{Three-dimensional Lie groups}
There are four $3$-dimensional Lie algebras which cannot be generated by a pair of vectors: the abelian Lie algebra $\R^3$, $\mathfrak{dil}(2)$, the direct sum $\mathfrak{dil}(1)\oplus\R$, where $\mathfrak{dil}(n)$ denotes the Lie algebra of the group of dilations and translations of $\R^n$, and the solvable unimodular Lie algebra $\mathfrak{sol}$. The Lie groups corresponding to other $3$-dimensional Lie algebras admit left-invariant contact structures. All left-invariant sub-Riemannian metrics have bounded geometry, so Theorem \ref{1.5} applies. When simply connected, they satisfy all uniform local assumptions required for identification of $\he{}$-$\mathrm{Poincar\acute{e}}_{p,q}$ inequality with vanishing of $\ell^{q,p}$ cohomology and its quasiisometry invariance. Here are examples.
Heisenberg group $\he{1}$ is covered by Theorem \ref{strongglobal}. Note that the corresponding facts about $\ell^{q,p}$ cohomology are new.
$\widetilde{M_1}:=\widetilde{\text{Mot}(E^2)}$, the universal covering of the group of planar Euclidean motions, is quasiisometric to Euclidean $3$-space $E^3$. Its $\ell^{q,p}$ cohomology vanishes if and only if $\frac{1}{p}-\frac{1}{q}\geq\frac{1}{3}$ (this is the Euclidean analogue of Theorem \ref{strongglobal}). Therefore, assuming $1 1$, see \cite{PP07}. Since $PSL(2,\R)$ acts isometrically and simply transitively on hyperbolic plane $H^2$, it is quasiisometric to $H^2$. Since the $\ell^{p,p}$-cohomology of $H^2$ in degree $1$ is Hausdorff and nonzero, the K\"unneth formula of \cite{GKS} applies, and the $\ell^{p,p}$-cohomology in degree $2$ of the product does not vanish, because the $\ell^{p,p}$-cohomology in degree $1$ of the line does not vanish. We conclude that, assuming $10$ such that}
$$
\| u\ast K\|_{L^q(\he n)} \le C \| u\|_{L^p(\he n)}
$$
for all $u\in L^p(\he n)$.
\item[ii)] If $p\ge Q/\alpha$ and $B, B' \subset \he n$ are fixed balls with $B\subset B'$, then
for any $q\ge p$ { there exists $C=C(B,B',p,q,\alpha)>0$}
$$
\| u\ast K\|_{L^q(B')} \le C \| u\|_{L^p({ B})}
$$
for all $u\in L^p(\he n)$ with $\supp u\subset B$.
\item[iii)] If $K$ is a kernel of type 0 and $1
0$.
Then, if $f\in \mc D(\he n)$ and $R$ is an homogeneous
polynomial of degree $\ell\ge 0$ in the horizontal derivatives,
we have
$$
R( f\ast g)(p)= O(|p|^{\mu-Q-\ell})\quad\mbox{as }p\to\infty.
$$
In addition, let $g$ be
a smooth function in $\he n\setminus\{0\}$
{ satisfying} the logarithmic estimate
$$|g(p)|\le C(1+|\ln|p|| ), $$ and suppose
its horizontal derivatives are kernels of { type} $Q-1$
with respect to group dilations.
Then, if $f\in \mc D(\he n)$ and $R$ is an homogeneous
polynomial of degree $\ell\ge 0$ in the horizontal derivatives,
we have
\begin{eqnarray*}
R( f\ast g)(p)&=&O(|p|^{-\ell})\quad \mbox{as }p\to\infty
\quad\mbox{ if $\ell>0$}; \\
R( f\ast g)(p)&=&O(\ln|p| )\quad \mbox{as }p\to\infty
\quad\mbox{ if $\ell=0$}.
\end{eqnarray*}
\end{lemma}
{ \begin{proof}
The first part of the lemma is a particular instance of Lemma 6.4 in \cite{folland_stein}. As for the second part, we can repeat the same argument.
Indeed, the first statement follows straightforwardly from the first part of the lemma, since, by the Poincar\'e--Birkhoff--Witt theorem, we can write
$$
R( f\ast g) = \sum_j R_\ell' ( f\ast W_j g)\,,
$$
where the differential operators $R_j'$ have homogeneous degree $\ell-1$. Finally, the last statement can be proved as follows: suppose
$\supp f\subset B(0,M)$, $M>1$, and take $|p|>2M$. Then, keeping in mind that $1
0$ such that, for every $x\in M$,
there exists a contactomorphism (i.e. a diffeomorphism preserving the contact forms) $\phi_x : B(e,1)\to M$ { that satisfies}
\begin{enumerate}
\item $B(x,r)\subset\phi_x(B(e,1))$;
\item $\phi_x$ is $C$-bi-Lipschitz, i.e.
\begin{equation}\label{bilip}
\frac1Cd(p,q)\le d_M(\phi_x (p), \phi_x(q)) \le Cd(p,q)\qquad \mbox{for all $p,q\in B(e,1)$};
\end{equation}
\item coordinate changes $\phi_x\circ\phi_y^{-1}$ and their first $k$ derivatives with respect to unit left-invariant horizontal vector fields are bounded by $C$.
\end{enumerate}
\end{dfi}
\begin{remark}Compact sub-Riemannian contact manifolds have bounded geometry. More examples arise from covering spaces of such compact manifolds.
Note that every orientable compact $3$-manifold admits a contact structure (\cite{Martinet}), it can be equipped with sub-Riemannian structures,
its universal covering space is usually noncompact. This leads to a large variety of non-compact bounded geometry sub-Riemannian contact $3$-manifolds.
\end{remark}
The following covering lemma is basically \cite{mattila}, Theorem 1.2.
\begin{lemma}\label{covering} Let $(M,H,g)$ be a bounded $C^k$-geometry sub-Riemannian contact manifold,
where $k$ is a positive integer. Then there
exists $\rho>0$ (depending only on the radius $r$ of Definition \ref{contact bis}) and an at most countable covering $\{ B(x_j, \rho)\}$ of $M$ such that
\begin{itemize}
\item[i)] each ball $B(x_j, \rho)$ is contained in the image of one of the contact charts of Definition \ref{contact bis};
\item[ii)] $B(x_j, \frac15\rho)\cap B(x_i, \frac15\rho)=\emptyset$ if $i\neq j$;
\item[iii)] the covering is uniformly locally finite. Even more, there exists a $N=N(M)\in \mathbb N$ such that
for each ball $B(x,\rho)$
$$
\#\{k\in \mathbb N \mbox{ such that } B(x_k,\rho)\cap B(x,\rho)\neq\emptyset\} \le N.
$$
In addition, if $B(x_k,\rho)\cap B(x,\rho)\neq\emptyset$, then $B(x_k,\rho)\subset B(x,r)$, where $B(x,r)$ has
been defined in Definition \ref{contact bis}-(2)).
\end{itemize}
\end{lemma}
\begin{proof} First we notice that $M$ is separable. Indeed, let $x\in M$ be fixed.
With the notations of Definition \ref{contact bis},
if we set $\phi_x(B(0,1)):= \mc U_x$ then $\{\mc U_x, x\in M\}$ is an open covering of $M$.
Let now $\{\mc V_{x_j}, j\in \mathbb N\}$ be a countable refinement of $\{\mc U_x, x\in M\}$ (see \cite{warner}, Lemma 1.9).
For any $j\in \mathbb N$, let $S_j$ be a countable dense subset of $\phi_{x_j}^{-1}(\mc V_{x_j})$; then
$\phi_{x_j}(S_j)$ is a countable dense subset of
$\mc V_{x_j}$. Thus $\Sigma:= \cup_j \phi_{x_j}(S_j)$ is a countable dense subset of $M$.
Let now $\rho \in (0,r/2)$ be fixed. Then, by \cite{mattila}, Theorem 1,2, there exists a family of disjoint balls $\{B({x_\alpha}, \frac {\rho }{5})\}$
such that $\{B({x_\alpha}, \rho )\}$ is an open covering of $M$. We prove now that we can extract a countable sub-family
$\{B({x_{\alpha_j}}, \rho )\} =: \{B({x_j}, \rho )\} $ which is still an open covering of $M$. Indeed, for any $y\in\Sigma$,
let us prove that $\#\{\alpha\; \mbox{such that}\; y \in B({x_{\alpha}}, \rho ) \} \le N$,
where $N$ is a geometric constant. If $y\in B({x_{\alpha}}, \rho ) \cap B({x_{\beta}}, \rho )$, then $d_M(x_\alpha,x_\beta)<2\rho $.
In addition, then $B({x_{\alpha}}, \rho )$ and $B({x_{\beta}}, \rho )$ are contained in $\phi_y(B(e,1))$ since $2\rho < r$. From
now on we assume $\rho>0$ is fixed with $3\rho
p$. First take $m=0$.
Again, let us fix two balls $B_0$, $B_1$ with
$B\Subset B_0 \Subset B_1\Subset B'$.
If $\chi \in \mc D(B_1)$ is cut-off function such that
$\chi\equiv 1$ on $B_0$, we set $\alpha_0 = \chi\alpha$.
Keeping the notations of the proof of Theorem \ref{smoothing1}, it is easy to check that $S\alpha$ can be written as
(see \eqref{santi})
\begin{equation}\label{santi2}
S\alpha = S_0\alpha_0 := \alpha_0\ast d_c (1-\psi_R)k_1 ) \pm \alpha_0 \ast \ccheck d_c\ccheck(1-\psi_R)\tilde k_1.
\end{equation}
Thus, if $\alpha= \sum_j \alpha_j \xi_j^h$, then each entry of $S\alpha$ is a sum of terms of the form
$$
(\chi\alpha_j)\ast \kappa,
$$
where $\kappa $ ia a smooth kernel.
Thus we are lead to estimate the $L^q$-norms in $B$ of a sum of terms of the form
$$
(\chi\alpha_j)\ast W^J\kappa = (\chi\alpha_j)\ast \mathbf 1_{2B'} W^J\kappa\qquad\mbox{with $|J|=s$,}
$$
and the assertion follows by classical Hausdorff-Young inequality (see \cite{folland}, Proposition 1.10 ), since
the kernel $\mathbf 1_{2B'} W^J\kappa$ belongs to all $L^r$, $ r\ge 1$. Therefore $S$ is bounded from
$L^{p}(B',E_0^h)$ to $W^{s,q}(B,E_0^h)$.
Clearly, this yields the continuity of $S$ from $W^{m,p}(B',E_0^h)$ to $W^{s,q}(B,E_0^h)$ for $m\ge 0$.
The proof in the case $m<0$ can be carried out by a duality argument akin to the one we used in the
proof of Theorem \ref{smoothing2}.
\end{proof}
\begin{remark}\label{tenda} Apparently, in previous theorem, two different homotopy operators $T$ and $\tilde T$
appear. In fact, they coincide when acting on form of the same degree.
More precisely, in Proposition \ref{homotopy formulas} the homotopy formulas involve four operators
$K_1, \tilde K_1, K_2, \tilde K_2$, where the notation is meant to distinguish operators acting on $d_c\alpha$ (the operators
with tilde) from those on which the differential acts (the operators
without tilde), whereas the lower index 1 or 2 denotes the type of the associated kernels. Alternatively, a different notation could be used:
if $\alpha\in \mc D(\he n, E_0^h)$ we can write
$$
\alpha = d_c K_h + \tilde K_{h+1}d_c\alpha,
$$
where the tilde has the same previous meaning, whereas the lower index refers now to the degree of the forms on which
the operator acts.
It is important to notice that
$$
K_{h+1} = \tilde K_{h+1}, \qquad h=1,\dots, 2n.
$$
Indeed, take $h