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\title{Poincaré inequalities for differential forms on Heisenberg groups}
\author{Pierre Pansu, Univ. Paris-Sud. Joint with A. Baldi, B. Franchi, Bologna}
\date{October 17th, 2017}


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

\begin{document}

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

\section{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 $n-1$, hence is bounded $L^p\to L^q$ provided $\frac{1}{p}-\frac{1}{q}= \frac{1}{n}$ (\textcolor{red}{Calderon-Zygmund}). 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)}}. Poincar\'e's homotopy formula provides a homotopy $T$, $1=dT+Td$, which has a homogeneous kernel of degree $n-1$. It does not require forms to be globally defined. Hölder $\Rightarrow$ $q$ can be lessened. Works for convex sets.

\end{frame}

\subsection{Leray's acyclic covering theorem revisited}

\begin{frame}

\begin{prop}[Leray]
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{prop}

\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{thm}[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{thm}
"Loss on domain is allowed".

\end{frame}

\section{Sub-Riemannian case}

\subsection{Rumin's complex}

\begin{frame}

To handle Carnot groups $G$, need homogeneous Laplacian: Rumin?

\smallskip

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 the de Rham complex.\pause

\bigskip

\textbf{Example}. Heisenberg group $\mathbb{H}^1$. 

\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}

\end{frame}

\begin{frame}

More generally, for Heisenberg group $\mathbb{H}^n$, $E_R$ has exactly one weight in each degree, hence $d_R$ is homogeneous under Heisenberg dilations. It has order 1, except in degree $n$ where it has order 2. 

\medskip

One can make choices in a contact invariant manner: $(E_R,d_R)$ is invariantly defined for contact manifolds.

\medskip

In presence of a sub-Riemannian metric, adjoints are defined. Let $\Delta_R:=d_R^*d_R+d_Rd_R^*$ be replaced with $\Delta_R:=(d_R^*d_R)^2+d_Rd_R^*$ or $d_R^*d_R+(d_Rd_R^*)^2$ in degrees $n$ and $n+1$. 

Then $\Delta_R$ is maximally hypoelliptic, hence $d_R^{*}\Delta_R^{-1}$ has a smooth (away from the origin), homogeneous kernel.

\begin{prop}[\textcolor{red}{Coifman-Weiss, Koranyi-Vagi 1971}]
$T_R:=d_R^{*}\Delta_R^{-1}$ is bounded $L^p$ to $L^q$ provided $1<p<q<\infty$ and $\frac{1}{p}-\frac{1}{q}=\frac{1}{2n+2}$ (resp. $\frac{2}{2n+2}$ in degree $n+1$).
\end{prop}
Again, $1=d_R T+Td_R$.

\end{frame}

\subsection{Heisenberg Poincar\'e inequality}

\begin{frame}

Need local version.

\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 $\frac{1}{p}-\frac{1}{q}\leq\frac{1}{2n+2}$ (resp. $\frac{2}{2n+2}$ in degree $n+1$).
\end{thm}\pause

\textbf{Proof}.
The sub-Riemannian analogue of Poincar\'e's homotopy formula is not compatible with Rumin's construction.

\medskip

Instead, use Rumin's homotopy $\Pi_E$ and then apply Iwaniec-Lutoborsky. Since $\Pi_E$ is differential, this requires a preliminary smoothing homotopy. 

\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{Back to $\ell^{p,q}$-cohomology}

\begin{frame}

Leray's method requires control on $\|d_R(\chi\omega)\|_p$ in terms of $\|\omega\|_p$ and $\|d_R\omega\|_p$. This fails when $d_R$ is second order. The way around this is

\begin{thm}[Baldi-Franchi-Pansu]
Let $1<p,q<\infty$ and $\frac{1}{p}-\frac{1}{q}\leq\frac{1}{2n+2}$ (resp. $\frac{2}{2n+2}$ in degree $n+1$). On contact manifolds with $C^{k+1}$-bounded geometry, global smoothing homotopies $1=d_RT+Td_R+S$ are defined, where $T:W^{s,p}\to W^{s+1,q}$ and $S:W^{s,p}\to W^{s+k,q}$, $-k\leq s\leq 0$.
\end{thm}

\begin{cor}
Rumin's complex can be used to compute $\ell^{q,p}$-cohomology of contact manifolds with bounded geometry for this range of $p,q$.
\end{cor}

\begin{cor}
$\ell^{q,p}H^\cdot(\mathbb{H}^n)=0$ provided $1<p<q<\infty$ and $\frac{1}{p}-\frac{1}{q}\geq\frac{1}{2n+2}$ (resp. $\frac{2}{2n+2}$ in degree $n+1$).
\end{cor}

\end{frame}

\section{Questions}

\begin{frame}

\textbf{Questions}.

\medskip

\begin{enumerate}
\item Sobolev inequality (i.e. for compactly supported forms)? Yes.
  \item Sharpness? Probably yes.
  \item Cases when $p=1$ and $q=\infty$? Work in progress with Baldi and Franchi.
  \item Other Carnot groups? Work in progress with Rumin, but sharp intervals are rarely attained.
\end{enumerate}

\end{frame}

\end{document}
