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\title{$\ell^{q,p}$ cohomology of certain Carnot groups}
\author{Pierre Pansu, Univ. Paris-Sud. Joint with A. Baldi, B. Franchi, M. Rumin}
\date{February 22nd, 2018}


\mode<presentation>{\usetheme{Warsaw}}

\begin{document}

\begin{frame}
\titlepage
\end{frame}

\section{$\ell^{q,p}$ cohomology}

\subsection{Motivation}

\begin{frame}

\begin{defi}
$X$ simplicial complex. Cochains are functions on simplices. When is an $\ell^p$ cocycle the coboundary of an $\ell^q$ cochain? Set
\begin{eqnarray*}
\ell^{q,p}H^k(X)=\{\ell^p \,k\textrm{-cocycles}\}/d\{\ell^q\,k-1\textrm{-cochains}\}.
\end{eqnarray*}
\end{defi}
$X$ finite: topological invariant. $X$ infinite: quasi-isometry invariant. Well defined for discrete groups, gives rise to numerical invariants of discrete groups...\pause

\bigskip 

\textbf{Example}. $X=$ subdivided line. Then all $\ell^{q,p}H^0(X)=0$, $\ell^{\infty,1}H^1(X)=0$, all other $\ell^{q,p}H^1(X)\not=0$.\pause

\bigskip

\textbf{Example}. $X=$ tesselated plane. Then $\ell^{q,p}H^1(X)=0$ if
$\displaystyle\frac{1}{p}-\frac{1}{q}\geq \frac{1}{2}$.
Indeed, Sobolev inequality allows to handle the case of finitely supported cocycles. It states that, for a smooth compactly supported function $u$ on the plane, if $p<2$,
\begin{eqnarray*}
\|u\|_q \leq C\,\|du\|_p.
\end{eqnarray*}

\textbf{Questions}. Handle infinitely supported cocycles ? Pass from discrete to continuous and backward ? 

\end{frame}


\subsection{Euclidean Poincar\'e inequality}

\begin{frame}

\begin{defi}
$X$ Riemannian manifold. Set
\begin{eqnarray*}
L^{q,p}H^k(X)=\{L^p \,\textrm{closed }k\textrm{-forms}\}/d\{L^q\,k-1\textrm{-forms }\omega\textrm{ such that }d\omega\in L^p\}.
\end{eqnarray*}
\end{defi}

\textbf{Questions}. Compute it. If $X$ is triangulated, does $L^{q,p}H^k(X)=\ell^{q,p}H^k(X)$? \pause

\bigskip

\textbf{Example}. $X=\R^n$. Then $L^{q,p}H^k(X)=0$ if $1< p\leq q<\infty$ and $\displaystyle\frac{1}{p}-\frac{1}{q}= \frac{1}{n}$.

\smallskip

\textbf{Proof}. Let $\Delta=d^*d+dd^*$. Then $\Delta$ has a pseudo-differential inverse which commutes with $d$. $T=d^*\Delta^{-1}$ has a homogeneous kernel of degree $1-n$, hence is bounded $L^p\to L^q$ provided $\frac{1}{p}-\frac{1}{q}= \frac{1}{n}$ (\textcolor{red}{Calderon-Zygmund 1952}). Finally $1=dT+Td$.\pause

\bigskip

\textbf{Example}. $X=$ ball in $\R^n$. Then $L^{q,p}H^k(X)=0$ if $1< p\leq q<\infty$  and $\displaystyle\frac{1}{p}-\frac{1}{q}\leq \frac{1}{n}$.

\smallskip

\textbf{Proof \textcolor{red}{(Iwaniec-Lutoborsky 1993)}}. Cartan's homotopy formula provides a homotopy $T$, $1=dT+Td$, which has a homogeneous kernel of degree $1-n$. It does not require forms to be globally defined. Hölder $\Rightarrow$ $q$ can be lessened. Works for convex sets.

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\subsection{Leray's acyclic covering theorem revisited}

\begin{frame}

\begin{thm}[\textcolor{red}{Leray, circa 1946}]
Vanishing of $L^{q,p}H^\cdot$ of all simplices suffices to prove that $L^{q,p}H^\cdot=\ell^{q,p}H^\cdot$ for bounded geometry triangulated manifolds.
\end{thm}
Thus Iwaniec-Lutoborsky $\implies$ $\ell^{q,p}H^\cdot=L^{q,p}H^\cdot$ if $\displaystyle\frac{1}{p}-\frac{1}{q}\leq \frac{1}{n}$.
\pause

%\medskip

%\textbf{Example}. $(U_i)$ covering by stars of vertices. $\omega$ closed 1 form on $X$. $\omega_{|U_i}=du_i$, $u_i-u_j=\kappa_{ij}$ is constant on simplex $U_i\cap U_j$, it is a 1-cocycle of the triangulation. Conversely, pick partition of unity $\chi_i$. Given cocycle $\kappa$, set $u_i=\sum_j \chi_j\kappa_{ij}$. Then $du_i-du_j=0$ on $U_i\cap U_j$, hence defines a closed 1-form.\pause

\medskip

\begin{prop}[Rumin]
Proposition persists in the form $\ell^{q,p}H^\cdot=L_\infty^{q,p}H^\cdot$, for all $1\leq p,q\leq\infty$, where $L_\infty^p=\{\text{forms all of whose derivatives are in } L^p\}$.
\end{prop}
"Loss on differentiability is allowed". Useful since no restriction on exponent $q$.\pause

\medskip

\begin{prop}[Pansu]
Proposition merely requires even weaker analytic information: it suffices that a closed form on ball of radius 2 have a primitive on unit ball with controlled norms.
\end{prop}
"Loss on domain is allowed". Useful to prove quasiisometry invariance.

\end{frame}

\section{Carnot group case}

\subsection{Vanishing result}

\begin{frame}

Need homogeneous Laplacian. Use homogeneous groups. Special case : Carnot groups $G$, $\mathfrak{g}=\mathfrak{g}_1 \oplus \cdots \oplus \mathfrak{g}_s$, $\delta_t=t^i$ on $\mathfrak{g}_i$.\pause

\bigskip

\textbf{Kohn's Laplacian}. For a function $u$, let $d_R u=du_{|\mathfrak{g}_1}$ and let $\Delta u=d_R^* d_R$. This homogeneous of degree $2$ under Carnot dilations $\delta_t$. \pause

\bigskip

In general, there is no differential homogeneous Laplacian on forms, since left-invariant forms split under $\delta_t$ into several weight spaces. 

\medskip

\textbf{Example}. On the Heisenberg group, two weights on $k$-forms, $k$ and $k+1$, since $\Lambda^k\mathfrak{g}^*=\Lambda^k\mathfrak{g}_1^*\oplus\Lambda^{k-1}\mathfrak{g}_1^*\otimes\mathfrak{g}_2^*$.

\end{frame}


\subsection{Role of anisotropic pseudodifferential calculus}

\begin{frame}

\textbf{A pseudodifferential homogeneous Laplacian}. Let $|\nabla|=\Delta^{1/2}$. Let $|\nabla|^N$ be the operator acting componentwise which is $|\nabla|^w$ on forms of weight $w$. Then $d^\nabla:=|\nabla|^{-N} d |\nabla|^N$ is pseudodifferential of order $0$, so is its Laplacian $\Delta^\nabla=(d^\nabla)^*d^\nabla+d^\nabla(d^\nabla)^*$. Both are homogeneous.\pause

\bigskip

$\Delta^\nabla$ admits a pseudodifferential inverse (\textcolor{red}{Helffer-Nourrigat 1979}, \textcolor{red}{Christ-Geller-Glowacki-Polin 1992}), hence $d^\nabla$ admits a homotopy $K^\nabla:=(d^\nabla)^*(\Delta^\nabla)^{-1}$, $1=d^\nabla K^\nabla+K^\nabla d^\nabla$, hence a homogeneous homotopy $K:=|\nabla|^{N} K^\nabla |\nabla|^{-N}$ for $d$.\pause

\bigskip

$K^\nabla$ is bounded on $L^p$ (\textcolor{red}{Folland 1975}), thus $K$ is bounded on
$$L^{N,p}:=\{\alpha\,;\,|\nabla|^{-N}\alpha\in L^p\},$$
and on $L^{N-m,p}$ for every constant $m$.

\end{frame}

\begin{frame}

On functions, 
$$L_\infty^p=\bigcap_{m=0}^\infty L^{N-m,p}.$$ 
On the space of forms whose weight lies between $a$ and $b$, $\Omega_{a}^{b}$,
$$\Omega_{a}^{b}\cap\bigcap_{m=a}^{\infty}L^{N-m,p}\subset \Omega_{a}^{b}\cap L_\infty^{p}\subset \bigcap_{m=b}^{\infty}L^{N-m,p}.$$\pause

\medskip

$|\nabla|^{-\mu}$ is bounded from $L^p$ to $L^q$ if $\frac{1}{p} - \frac{1}{q} = \frac{\mu}{Q}$, $Q=\sum i\, \mathrm{dim}(\mathfrak{g}_i)$ (\textcolor{red}{Folland 1975}). Whence the graded Poincar\'e inequality
$$L^{N-m,p}\subset L^{N-m+\mu,q}.$$\pause

Assume that $K(\Omega_{a}^{b})\subset \Omega_{a'}^{b'}$. If $\mu=b-a'$, then 
$$K(L_\infty^{p}\cap\Omega_{a}^{b})\subset \Omega_{a'}^{b'}\cap\bigcap_{m=b}^{\infty}L^{N-m,p}
\subset\Omega_{a'}^{b'}\cap\bigcap_{m=b}^{\infty}L^{N-m+\mu,q}=\Omega_{a'}^{b'}\cap\bigcap_{m=a'}^{\infty}L^{N-m,q}\subset L_\infty^{q},$$

\end{frame}

\subsection{Rumin's complex}

\begin{frame}

\begin{prop}
$\ell^{q,p}H^k(G)=0$ if $1<p,q<\infty$, $\frac{1}{p}-\frac{1}{q}\geq\frac{b-a'}{Q}$, where $b$ is the maximal weight in degree $k$, $a'$ is the minimal weight in degree $k-1$.
\end{prop}

\textbf{Example}. For Heisenberg group, $b=k+1$, $a'=k-1$, $b-a'=2$. Unsharp.\pause

\bigskip

\textbf{Strategy}. Replace $d$ with a subcomplex that uses less weights: Rumin's complex (1994, 1999).

\medskip

Forms on $G$ split into several weights under dilations. Let $d_0$ be the weight 0 (algebraic) part of $d$. Pick complements of $\mathrm{ker}(d_0)$ and $\mathrm{im}(d_0)$ and define $d_0^{-1}$. Then powers of $1-d_0^{-1}d-dd_0^{-1}$ stabilize to a projector $\Pi_E$ onto a subcomplex $E$. $\Pi_0 =1-d_0^{-1}d_0-d_0d_0^{-1}$ projects to a subspace $E_R$ of forms where less weights occur. Set
\begin{eqnarray*}
d_R=\Pi_0 \circ d\circ \Pi_E.
\end{eqnarray*}
Then \textcolor{red}{Rumin's complex} $(E_R,d_R)$ is homotopic to de Rham's complex.

\end{frame}

\begin{frame}

\textbf{Example}. For the $3$-dimensional Heisenberg group, 

\begin{itemize}
  \item $E_R^1$ consists of horizontal 1-forms, $d_R:E_R^0\to E_R^1$ is the horizontal gradient.

  \item $E_R^2$ consists of vertical 2-forms. $\Pi_E$ extends a horizontal 1-form $\alpha$ in such a way that $d\Pi_E\alpha$ is vertical (unique choice). Hence $d_R\alpha=d\Pi_E\alpha$ involves second derivatives.

\end{itemize}

In general, for the $2m+1$-dimensional Heisenberg group, $E_R$ has one weight in each degree, $w=k$ if $k\leq m$, $w=k+1$ if $k\geq m+1$, and $d_R:E_R^m\to E_R^{m+1}$ has order $2$.\pause

\begin{thm}[Pansu-Rumin]
Let $G$ be a Carnot group of homogeneous dimension $Q$. Let $[a,b]$ be the scope of weights in Rumin $k$-forms, let $[a',b']$ be the scope of weights in Rumin $k-1$-forms. \begin{enumerate}
  \item $\ell^{q,p}H^k(G)=0$ provided $1<p,q<\infty$ and
$$\frac{1}{p}-\frac{1}{q}\geq\frac{b-a'}{Q}.$$
  \item $\ell^{q,p}H^k(G)\not=0$ if $1\leq p,q\leq\infty$, $\frac{1}{p}-\frac{1}{q}<\frac{\max\{1,b'-a\}}{Q}$.
\end{enumerate}
\end{thm}

\textbf{Example}. For Heisenberg groups, $b-a'=b'-a=1$, except in middle dimension where $b-a'=b'-a=2$.

\medskip

\textbf{Example}. For all Carnot groups, $b-a'=b'-a=1$ in degrees $1$ and $n=\mathrm{dim}(G)$.

\end{frame}

\section{Local Poincar\'e inequality}

\subsection{Smoothing homotopy}

\begin{frame}

\begin{thm}[Baldi-Franchi-Pansu]
Closed $L^p$ forms defined on the Heisenberg 2-ball have $d_R$-primitives on the unit ball which are $L^q$, provided $1<p,q<\infty$ and $\frac{1}{p}-\frac{1}{q}\leq\frac{1}{2m+2}$ (resp. $\frac{2}{2m+2}$ in degree $m+1$).
\end{thm}\pause

\textbf{Proof}.
No subRiemannian Cartan homotopy formula.

\smallskip

Instead, one uses Rumin's homotopy $\Pi_E$ followed by Iwaniec-Lutoborsky's Euclidian homotopy. Since $\Pi_E$ is differential, a preliminary smoothing homotopy is needed. \pause

\medskip

Fix a subRiemannian metric in order to define adjoints. $\Delta_R:=d_R^*d_R+d_Rd_R^*$ is replaced with $\Delta_R:=(d_R^*d_R)^2+d_Rd_R^*$ ou $d_R^*d_R+(d_Rd_R^*)^2$ in degrees $m$ and $m+1$. Then $\Delta_R$ is maximally hypoelliptic, thus $T_R=d_R^{*}\Delta_R^{-1}$ has a smooth and homogeneous kernel.\pause

\medskip

Let $K_R$ be the kernel of $T_R$. Write $K_R=K^1+K^2$ where $K_1$ has small support and $K_2$ is smooth. Then $T_R=T^1+T^2$,
\begin{eqnarray*}
1=d_R T^1+T^1d_R+S
\end{eqnarray*}
where $S$ is smoothing. $T^1$ and therefore $S$ map forms defined on ball of radius 2 to forms defined on unit ball. $T^1$ maps $L^p$ to $L^q$ like $T_R$. $S$ wins the derivatives that $\Pi_E$ looses.

\end{frame}

\subsection{Result}

\begin{frame}

\begin{cor}[Baldi-Franchi-Pansu]
Rumin's complex can be used to compute $\ell^{q,p}$-cohomology of contact $2m+1$-manifolds with bounded geometry, provided $1<p,q<\infty$ and $\frac{1}{p}-\frac{1}{q}\leq\frac{1}{2m+2}$ (resp. $\frac{2}{2m+2}$ in degree $m+1$): $\ell^{q,p}H^\cdot=L^{q,p}H^\cdot(d_c)$. Also, the complex $(E_R,d_R)$ admits a smoothing homotopy.
\end{cor}

\bigskip

\textbf{Question}. Cases when $p=1$ or $q=\infty$? Work in progress with Baldi and Franchi.

\end{frame}

\end{document}

%\section{Cas où \texorpdfstring{$p=1$}{} ou \texorpdfstring{$q=\infty$}{}}

%\subsection{Inégalité de Poincaré locale \texorpdfstring{$L^1$}{}}

\begin{frame}

L'inégalité de Sobolev $\|u\|_\infty\leq C\,\|du\|_n$ sur $\R^n$ n'est pas vraie (invariance conforme). \pause

\medskip

Par dualité, l'inégalité de Poincaré $L^1$ sur les $n$-formes sur $\R^n$ n'est pas vraie : trouver une primitive $L^{n/n-1}$ pour une $n$-forme $L^1$ d'intégrale nulle n'est pas toujours possible. Autrement dit $L^{n/n-1,1}H^n(\R^n)\not=0$.\pause

\medskip

En revanche, l'inégalité de Sobolev $\|u\|_{n/n-1}\leq C\,\|du\|_1$ est vraie. Quid des autres degrés ?

\begin{thm}[\textcolor{red}{Bourgain-Brezis 2003, 2006}]
Dans $\R^n$, si $1\leq k\leq n-1$, $d^*\Delta^{-1}$ est borné des $k$-formes \textbf{fermées} $L^1$ vers $L^{n/n-1}$. En particulier $L^{n/n-1,1}H^k(\R^n)=0$.
\end{thm}\pause

\begin{thm}[Baldi-Franchi-Pansu]
L'inégalité de Poincaré locale pour $d_R$, lorsque $p=1$ et $q=Q/Q-1$ (resp. $q=Q/Q-2$ en degré $m+1$) est vraie en tout degré $<2m+1$ pour le groupe d'Heisenberg.
\end{thm}

Repose sur un procédé initié par \textcolor{red}{Lanzani-Stein 2005} dans $\R^n$, une alternative à Bourgain-Brezis due à \textcolor{red}{Van Schaftingen 2004} et son extension aux groupes de Carnot due à \textcolor{red}{Chanillo-Van Schaftingen 2009}.

\end{frame}

\subsection{Inégalité de Poincaré globale \texorpdfstring{$L^1$}{}}

\begin{frame}

\begin{thm}[Baldi-Franchi-Pansu]
On suppose que $k<n$, $q=Q/Q-1$ (resp. $q=Q/Q-2$ en degré $m+1$). Pour une variété de contact à géométrie bornée de dimension $n=2m+1$, $\ell^{q,1}H^k=L^{q,1}H^k$ calculé avec $d_R$. Pour le groupe d'Heisenberg, $\ell^{q,1}H^k$ est de dimension finie.
\end{thm}
En fait, probablement de dimension $0$.

\end{frame}


\end{document}