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\title{$L^1$ analysis and related geometry}
\author{Pierre Pansu, joint with A. Baldi, B. Franchi and F. Tripaldi}
\date{September 17th, 2021}

\usepackage[T1]{fontenc}

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

\begin{document}

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

\begin{frame}

\begin{center}
\textbf{A few milestones in Bruno Franchi's achievements}
\end{center}

\begin{itemize}
  \item 1983: Franchi and Lanconelli extend de Giorgi's H\"older regularity of solutions of linear PDEs. Ellipticity needs be measured with respect to a metric naturally associated with the operator. Birth of \emph{Analysis on Metric Spaces}.
  \item 1999: Franchi, Hajlasz and Koskela define Sobolev spaces on metric spaces. Consolidation of Analysis on Metric Spaces.
  \item 2001: Franchi, Serapioni and Serra Cassano inaugurate \emph{Geometric Measure Theory} on Carnot groups. Their first rectifiability result is crucial in Cheeger, Kleiner and Naor's work on the Goemans-Linial conjecture.
  \item 2011: Franchi, Serapioni and Serra Cassano introduce intrinsic Lipschitz graphs, as candidates for models of rectifiable sets. These play a key role in Naor and Young's final answer to Goemans-Linial's conjecture.
  \item 2009-2021: Franchi and coauthors extend the theory from functions to differential forms, guided by the idea of currents.
\end{itemize}
\end{frame}

\section{Motivation}

\subsection{Sobolev and \texorpdfstring{$\ell^{q,p}$} cohomology}


\begin{frame}

\textbf{The Euclidean Sobolev inequality}

Let $1\leq p<n$, $\displaystyle\frac{1}{p}-\frac{1}{q}= \frac{1}{n}$. If $u$ is a smooth, compactly supported function on $\R^n$, then
$$
\|u\|_q\leq C(n,p,q)\|du\|_p.\hskip2cm (Sobol_{p,q})
$$\pause

\bigskip

The most important $(Sobol_{p,q})$, is the one with $p=1$,
\begin{itemize}
  \item It implies all the others.
  \item It is equivalent to the classical isoperimetric inequality.
\end{itemize}\pause

\bigskip

\begin{minipage}{5cm}

$(Sobol_{p,q})$ has a discrete version on $\Z^n$, $n\geq 2$ : every finitely supported $1$-cocycle $c$ admits a primitive $u$ such that 
$$
\|u\|_q\leq C'(n,p,q)\|c\|_p.
$$

\end{minipage}\hskip2cm
\begin{minipage}{3cm}

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\end{minipage}\pause

\bigskip

We study a higher dimensional generalization, which has a topological flavour.
\end{frame}


\begin{frame}

\begin{defi}
$X$ simplicial complex. Is every $\ell^p$ cocycle the coboundary of a $\ell^q$ cochain? 
\begin{eqnarray*}
\ell^{q,p}H^k(X)=\{ \ell^p\,k\text{-cocycles }\}/d\{\ell^q\,k-1\text{-cochains }\}.
\end{eqnarray*}
\end{defi}
If $X$ is finite, it is a topological invariant. If $X$ is infinite, it is a quasiisometry invariant. This gives rise to numerical invariants of discrete groups.\pause

\bigskip 

\textbf{Example}. $X=$ the line tiled with equal intervals. Then $\ell^{q,p}H^0(X)=0$ except $\ell^{q,\infty}H^0(X)=\R$. In degre $1$,  $\ell^{\infty,1}H^1(X)=0$, all other $\ell^{q,p}H^1(X)\not=0$.\pause

\bigskip

\textbf{Example}. $X=$ plane tiled with equal triangles. Then $\ell^{q,p}H^1(X)=0$ if
$\displaystyle\frac{1}{p}-\frac{1}{q}\geq \frac{1}{2}$.
For finitely supported $1$-cocycles, this is the discrete Sobolev inequality. 

\medskip

Two steps, 
\begin{itemize}
  \item one passes from the discrete to the continuous setting (requires analysis: interior estimates on balls),
  \item in the continuous setting, global estimates.
\end{itemize}

\end{frame}

\subsection{Continuous version}

\begin{frame}

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

\textbf{Property}. If $X$ has a bounded geometry triangulation, $L^{q,p}H^k(X)=\ell^{q,p}H^k(X)$.\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 pseudodifferential inverse which commutes with $d$. $T=d^*\Delta^{-1}$ has a kernel which is homogeneous  of degree $1-n$, hence is bounded $L^p\to L^q$ as soon as $\frac{1}{p}-\frac{1}{q}= \frac{1}{n}$ (\textcolor{red}{Calderon-Zygmund 1952}). Finally, $1=dT+Td$.\pause

\bigskip

\textbf{Case where $p=1$}. On $n$-forms, $d^*\Delta^{-1}$ is not bounded from $L^1$ to $L^{n/(n-1)}$.

Otherwise, by duality, Sobolev inequality $(Sobol_{\infty,n})$ would hold. But is does not: if $n\geq 2$, $\R^n$ is $n$-parabolic.

\end{frame}

\section{\texorpdfstring{$L^1$}-analysis}

\subsection{Degree $n-1$}

\begin{frame}

\begin{thm}[\textcolor{red}{Bourgain-Brezis-Mironescu 2004}]
Let $\omega$ be a \textbf{closed} compactly supported $n-1$-form on $\R^n$. Then for all $1$-forms $\alpha$ on $\R^n$ such that $\nabla\alpha\in L^n$,
$$
|\int_{\R^n} \alpha\wedge \omega|\leq C\,\|\omega\|_1\|\nabla\alpha\|_n.
$$
\end{thm}\pause

\begin{cor}
If $\phi$ is compactly supported and satisfies $d^*\phi=0$ and $d\phi=\omega$, then $\|\phi\|_{n/(n-1)}\leq C\,\|\omega\|_1$.
\end{cor}\pause

The \emph{averages} of a $L^1$ $k$-form $\omega$ are the integrals $\int_{\R^n}\beta\wedge\omega$, where $\beta$ has constant coefficients.

\begin{cor}
$d^*\Delta^{-1}:L^1\to L^{n/(n-1)}$ is bounded on $n-1$-forms which are \textbf{closed and of vanishing averages}.
\end{cor}

\end{frame}

\subsection{Proof of BBM's inequality}

\begin{frame}

\textbf{Proof of BBM's inequality} [after \textcolor{red}{van Schaftingen}] \pause 

\medskip

Let us approximate $\omega$ with forms which are Poincar\'e-dual to loops $\gamma$ and let us prove that
$$
|\int_{\gamma} \alpha|\leq C\,\ell(\gamma)\|\nabla\alpha\|_n,
$$
for $\alpha=u\,dx_1$. 

\vskip4cm

\end{frame}

\begin{frame}

\begin{minipage}{4cm}

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\end{minipage}\begin{minipage}{6cm}

Use the projection along parallel hyperplanes $\Pi=\{x_1=t\}$. \pause

\medskip

Since  $u\in C^{(n-1)/n}(\Pi)$ and $\gamma\cap\Pi=\sum_{i=1}^{m} z_i-w_i$, by Morrey-Sobolev,
$$
|u(z_i)-u(w_i)|\leq C\,\|\nabla\alpha\|_{L^n(\Pi)}|z_i-w_i|^{1/n}.
$$\pause
By H\"older,
$$
\sum_{i=1}^{m}|u(z_i)-u(w_i)|\leq  C\,\|\nabla\alpha\|_{L^n(\Pi)}\ell(\gamma)^{1/n}m^{(n-1)/n}.
$$
We integrate with respect to $x_1$, then apply H\"older, we get an estimate of $\int_{\gamma} \alpha$ by $\|\nabla\alpha\|_n$ and $\int_\R m(x_1)\,dx_1\leq \ell(\gamma)$. 
\end{minipage}

\end{frame}

\section{Case of Heisenberg group}

\subsection{A Heisenberg avatar of BBM's inequality}

\begin{frame}

The \emph{Heisenberg group} $\HH^n$ has Lie algebra $\mathfrak{g}=\mathfrak{g}_1\oplus\mathfrak{g}_2$, where $\mathfrak{g}_1=\R^{2n}$ et $[\cdot,\cdot]:\mathfrak{g}_1\to\mathfrak{g}_2=\R$ is symplectic. By left-translation, $\mathfrak{g}_1$ defines a distribution, hence a subRiemannian metric. Vectors or curves tangent to it are called \emph{horizontal}. Curves of finite length must be horizontal. For a function $u$, $\nabla_\HH u$ denotes the \emph{horizontal gradient}.

\medskip \pause

Following \textcolor{red}{Chanillo-van Schaftingen 2009}, let us prove that for every horizontal loop $\gamma$ in Heisenberg group $\HH^n$ and every horizontal $1$-form $\alpha$,
$$
|\int_{\gamma} \alpha|\leq C\,\ell(\gamma)\|\nabla_\HH\alpha\|_Q,
$$
for $Q=2n+2$. It suffices to treat $\alpha=u\,dx_1$. \pause

\medskip

Caveat: in a hyperplane $\Pi_t=\{x_1=t\}$, if $n=1$, the horizontal direction defines a foliation by parallel lines, so no Morrey-Sobolev embedding. \pause

\medskip

Trick: for every $\lambda>0$, there exists a smooth function $u_\lambda$ on $\HH^n$ such that
\begin{align*}
\|u-u_\lambda\|_{L^\infty(\Pi_t)} &\leq C\,\lambda^{1/Q}M(t),\\
\|\nabla_\HH u_\lambda\|_{L^\infty(\HH^1)} &\leq C\,\lambda^{(1/Q)-1}M(t),
\end{align*}
where $M$ is the maximal function of $t\mapsto \|(\nabla_\HH u)_{|\Pi_t}\|_Q$ (based on \textcolor{red}{Jerison 1986}).

\end{frame}

\begin{frame}

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\end{minipage}\begin{minipage}{6.5cm}

We use the projection along parallel hyperplanes $\Pi_t=\{x_1=t\}$. 
We write $\gamma\cap\Pi_t=\sum_{i=1}^{m(t)} z_i(t)-w_i(t)$.
We set $\lambda(t)=\frac{\ell(\gamma)}{m(t)}$. \pause

\medskip

We estimate
\begin{align*}
|\sum_{i=1}^{m}((u-u_\lambda)(z_i)-&(u-u_\lambda)(w_i))|
\leq 2mC\,\lambda^{1/Q}M\\
&\leq 2C\ell(\gamma)^{1/Q}m^{1-1/Q}M,
\end{align*}\pause
and
\begin{align*}
|\sum_{i=1}^{m} (u_\lambda(z_i)-&u_\lambda(w_i))|\leq C\,\sum_{i=1}^{m}d(z_i,w_i)\lambda^{(1/Q)-1}M\\
&\leq  C\,\ell(\gamma)\ell(\gamma)^{(1/Q)-1}m^{1-1/Q}M\\
&=C\,\ell(\gamma)^{1/Q}m^{1-1/Q}M.
\end{align*}
\pause
We integrate with respect to $x_1$, then we apply H\"older, we get an estimate of $\int_{\gamma} \alpha$ by $\ell(\gamma)$ times $\|M\|_Q\leq C\|\nabla_\HH\alpha\|_Q$. 

\end{minipage}

\end{frame}

\subsection{Results}
\begin{frame}

\begin{thm}[Baldi-Franchi-Pansu-Tripaldi]
Let $Q=2n+2$. Then $\ell^{q,p}H^k(\HH^n)=0$ in the following cases:
\begin{itemize}
  \item If $k\not=n$, $1\leq p\leq Q$, $\frac{1}{p}-\frac{1}{q}\geq\frac{1}{Q}$.
  \item If $k=n+1$, $1\leq p\leq \frac{Q}{2}$, $\frac{1}{p}-\frac{1}{q}\geq\frac{2}{Q}$.
\end{itemize}
except in two cases: (i) $k=2n+1$ and $p=1$, and (ii) $k=1$ and $p=Q$ and $q=\infty$.
\end{thm}

\bigskip

\textbf{Remark} (Pansu-Rumin). The converse is true: $\ell^{q,p}H^k(\HH^n)\not=0$ in all other cases.\pause

\medskip

\begin{cor}
Quasiconformal classification of $3$-dimensional subRiemannian Lie groups: $\HH^1$ and $\widetilde{Mot(\R^2)}$ are distinguished by $L^{4,2}H^2$ (but not by $L^{\infty,4}H^1$ or $L^{2,1}H^3$).
\end{cor}

This completes the classification in the simply connected case.

\begin{ques}
Let $G$ be the (subRiemannian) quotient of $\HH^1$ by a cyclic subgroup of its center. Is $G$ quasiconformal to the group $Mot(\R^2)$ of planar Euclidean motions?
\end{ques}

\end{frame}

\end{document}

%\subsection{Proof}

\begin{frame}

The Heisenberg group $\HH^n$ possesses automorphisms $\delta_\epsilon:=\epsilon^i$ on $\mathfrak{g}_i$ which make it \emph{selfsimilar}.\pause

\bigskip

\begin{enumerate}

  \item The Laplacian on forms $dd^*+d^*d$ is not homogeneous under $\delta_\epsilon$. In \textcolor{red}{1994, Rumin} introduced a complex $d_R$ homotopic to the complex of differential forms and homogeneous under $\delta_\epsilon$. The corresponding Laplacian has order $2$ in degrees $k\not=n$, $n+1$ and order $4$ in degrees $k=n$ and $n+1$.
  
  \item If $p>1$, the theory of singular integrals extends to homogeneous groups, \textcolor{red}{Folland 1975}.

\item The generalisation to homogeneous groups of Bourgain-Brezis-Mironescu's inequality is due to \textcolor{red}{Chanillo-Van Schaftingen 2008}.

\item The averages of an $L^1$ $d_R$-closed form $\omega$ are the integrals $\int_{\HH^n}\beta\wedge\omega$, where $\beta$ is a left-invariant Rumin form.

Averages pass to the quotient $L^{Q/(Q-1),1}H^k(\HH^n)\to\R$ (resp. $L^{Q/(Q-2),1}H^k(\HH^n)\to\R$ in degree $k=n+1$). 

\textcolor{red}{Tripaldi} shows that averages of all $d_R$-closed $L^1$ forms of degrees $<2n+1$ vanish.

\end{enumerate}

\end{frame}

%\subsection{Averages}

\begin{frame}

\begin{prop}
Let $\omega$ be a closed $L^1$ $n-1$-form on $\R^n$. Then $\omega$ has vanishing averages, i.e. for every constant coefficients $1$-form $\beta$,
$\int_{\R^n}\beta\wedge\omega=0$.
\end{prop}

Indeed, $\beta=d\gamma$ where $\gamma$ is a polynomial of degree $1$.
\begin{align*}
\int\chi_r\beta\wedge\omega&=-\int\gamma d\chi_r\wedge\omega+\int d(\chi_r \gamma\omega)\\
&=-\int_{B(2r)\setminus B(r)}\gamma d\chi_r\wedge\omega\\
&\leq C\,\int_{B(2r)\setminus B(r)}|\omega|
\end{align*}
tends to $0$. Thus $\int\beta\wedge\omega=0$.\pause

\begin{cor}[\textcolor{red}{Bourgain-Brezis 2007}]
$L^{n/(n-1),1}H^{n-1}(\R^n)=0$.
\end{cor}

\end{frame}

%\subsection{Other degrees}

\begin{frame}

\begin{prop}[\textcolor{red}{Lanzani-Stein 2005}]
The Bourgain-Brezis-Mironescu inequality for closed compactly supported forms $\omega$
$$
|\int_{\R^n} \alpha\wedge \omega|\leq C\,\|\omega\|_1\|\nabla\alpha\|_n.
$$
holds for all degrees $k\leq n-1$ (but not in degree $n$).
\end{prop}\pause

Indeed, each component of $\omega$ is a component of a closed compactly supported $n-1$-form: $\omega\wedge dx_{i_1}\wedge\cdots \wedge dx_{i_{n-1-h}}$.\pause

\begin{cor}
$L^{n/(n-1),1}H^{k}(\R^n)=0$ for all $k=0,\ldots,n-1$. However,
$L^{n/(n-1),1}H^{n}(\R^n)$ is not Hausdorff.
\end{cor}

\end{frame}



\end{document}

A.-P. Calderon, and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171-225.
