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\begin{document}




\begin{center}
{\bf ISOPERIMETRIC TYPE INEQUALITIES FOR\\
DIFFERENTIAL FORMS ON MANIFOLDS }
\end{center}

\medskip

\begin{center}
{ Flavia Giannetti * Antonia Passarelli di 
Napoli *}
\end{center}

\medskip


\begin{center}
{\small  *Dipartimento di Matematica e Applicazioni "R. Caccioppoli"}

{\small Universit\`{a} di Napoli ``Federico II", via Cintia - 80126 
Napoli}

{\footnotesize e-mail:giannett@unina.it\qquad 
antonia.passarelli@unina.it}


\end{center}
\vskip2cm
\bigskip
\bigskip

\noindent {\footnotesize  ABSTRACT. Let $X$ be a smooth oriented Riemannian $n$-manifold without 
  boundary and $(\Phi,\Psi)\in {\cal L}^p(\wedge ^l X)\times
{\cal L}^r(\wedge ^{n-l} X)$ ,$\frac {1}{p}+\frac {1}{r}=1+\frac {1}{n}$ , be a  pair 
of closed differential forms. We prove an isoperimetric type 
inequality for such differential forms under suitable assumptions. As 
an application we derive H\"older continuity for solutions of Hodge 
systems.
}

\bigskip

\noindent {\footnotesize {\bf AMS Classifications.}  58A10; 58A14 ; 
58J99.}

\bigskip

\noindent {\footnotesize {\bf Key words.} {\it Wedge Products; 
Isoperimetric Inequality;
H\"older continuity.}}
\bigskip
\bigskip
\bigskip


\vskip2cm

\section
{Introduction}
The familiar geometric form of the isoperimetric inequality 
reads as
$$n^{n-1}\omega_{n-1}|U|^{n-1}\leqslant |\partial U|^{n}$$
where $|U|$ stands for the volume of a region $U\subset \RR^{n}$, 
while $|\partial U|$ is its $(n-1)$-dimensional surface area.  Now, 
if $f:R\to U$ is a smooth diffeomorphism of a ``regular'' domain 
$Q\subset \RR^{n}$
onto $U$ then $|U|=\int_{Q}{\cal J}(x,f)dx$, where ${\cal J}(x,f)$ is 
the Jacobian determinant of $Df$, while $|\partial 
U|$ is dominated by $\int_{\partial 
Q}|Df|^{n-1}dx$ or $\int_{\partial 
Q}|D^{\sharp}f|dx$. Thus we obtain the following integral form of the 
isoperimetric inequalities
\begin{equation}\label{isopuno}
    \Bigg|\int_{Q}{\cal J}(x,f)dx\Bigg|\leqslant 
C(n)\Bigg(\int_{\partial 
Q}|Df|^{n-1}dx\Bigg)^{\frac{n}{n-1}}
\end{equation}
and
\begin{equation}\label{isopdue}
    \Bigg|\int_{Q}{\cal J}(x,f)dx\Bigg|\leqslant 
C(n)\Bigg(\int_{\partial 
Q}|D^{\sharp}\!f|dx\Bigg)^{\frac{n}{n-1}},
\end{equation}
valid also for all smooth mappings $f:Q\to \RR^{n}$, not necessarily 
diffeomorphisms.
Reshetnyak  $\cite{[13]}$ established sharp H\"older continuity for  mappings of 
bounded distortion  extending  ideas of Morrey \cite{[7]}. This 
required  the isoperimetric inequality (\ref{isopdue}) for 
mappings in the Sobolev class $W^{1,n}$. This inequality has later been explored 
 in  nonlinear elasticity \cite{[8]},\cite{[9]}.
Let us note that given a mapping $f:\RR^{n}\to\RR^{n}$ in 
$W^{1,n}$, we have two vector fields associated with $f$. The first 
one is curl free while the second is divergence free. These are, 
respectively
$$E=\nabla f^{1}\qquad{\rm and}\qquad  B=(B^{1},\dots, B^{n})$$ where
$$B^{j}=(-1)^{j+1}\frac{\partial (f^{2},\dots,f^{n})}{\partial 
(x^{1},\dots,x^{j-1},x^{j+1},\dots,x^{n})}$$
Their scalar product $\langle B, E\rangle$, which equals exactly the Jacobian 
determinant of $Df$, is integrable for $f\in 
W^{1,n}(\RR^{n})$.

Now, any pair $(B,E)$ of vector fields with div $B$=0 and curl $E$=0 
will be 
 referred to as {\it div-curl couple}. By analogy to the previous 
 discussion the scalar product $J=\langle B, 
 E\rangle$ will be called the Jacobian of $(B,E)$.
 
 Recently  \cite{[1]} an isoperimetric inequality of 
 type (\ref{isopuno}) for div-curl couples $(B,E)\in L^{p}\times 
 L^{q}(\RR^{n}\times\RR^{n})$ has been proved, under the 
 assumption that $p,q$ are H\"older conjugate exponents. This makes 
 their Jacobian integrable.
 
 As a  consequence the authors derived a H\"older 
 continuity property  of $p$-harmonic functions.
 
 
 Actually,  local integrability of the Jacobian of  
 an orientation preserving  mapping $f$ was established  under 
 somewhat weaker assumption, i.e.
 $f\in W^{1,\frac {n^{2}}{n+1}}(\RR^{n})$ and
 $$\liminf_{t\to\infty}t^{{n+1\over n}}\int_{|Df|>t}|Df(x)|^{\frac 
 {n^{2}}{n+1}}dx=0\,.$$
 Later  \cite{[11]}    an isoperimetric inequality of the type
 (\ref{isopdue})  for such  mappings $f$  has also been proved.
 
 In this spirit one might try to obtain an  isoperimetric type inequality  for  a 
 wedge product of closed differential forms on a smooth Riemannian manifold 
 analogous to that in \cite{[1]}. However, attempts to weaken the assumptions on the 
 integrability of the forms fail when the wedge product 
 doesn't obey  the rule of integration by parts. Actually,  this 
 problem translates into the  question how the
 distributional  wedge 
 product  relates to the pointwise product.
 
 In this paper we were able to identify these two
  quantities. Then we 
  proved  an isoperimetric type inequality  under assumptions similar to 
  those in \cite{[3]}.

  Let $X$ be a smooth oriented Riemannian $n$-manifold without 
  boundary.
 A  pair of closed differential   forms 
 $\Phi\in {\cal L}^p(\wedge ^l X)\cap$ ker $d$ and $\Psi\in
{\cal L}^r(\wedge ^{n-l} X)\cap$ ker $d$, where  $1< p,r<\infty$ 
satisfy
Sobolev's relation $\frac {1}{p}+\frac {1}{r}=1+\frac {1}{n}$,  will be 
called   an {\it admissible pair} 
 if $\Phi\wedge 
\Psi \ge 0$\footnote{Throughout this text we fix an orientation 
of $X$, that is a nonvanishing $n$-form $dx\in {\cal C}^{\infty}(\wedge 
^n X)$. We then say that an $n$-form $\Theta$ on $X$ is nonnegative 
if $\Theta=\lambda dx$ for some nonnegative function $\lambda$.} and
\begin{equation}\label{acca}
\liminf_{t\to\infty}t^{{1\over n}}\int_{H>t}H(x)dx=0
\end{equation}
where $H=|\Phi|^{p}+|\Psi|^{r}$.    

The reader may wish to notice that this equation is automatic if 
$H\in {\cal L}^{n+1\over n}( X)$. However, there are many interesting 
cases in which (\ref{acca}) holds for $H\not \in {\cal L}^{n+1\over n}( X)$

 Our main result is contained in the following 

 
\medskip
\begin{thm}\label{dis} Let $(\Phi, 
\Psi)$ be an admissible pair.
   Given $x\in X$, for almost every ball ${\bf B}={\bf B}(x,\rho)$ in $X$ 
   we have
   \begin{equation}\label{tesi}
\int_{\bf B}\Phi\wedge \Psi<\!\!< \Big(\int_{\partial{\bf 
B}}|\Phi|^{p}d{\cal H}^{n-1}\Big)^{\frac{1}{p}} 
\Big(\int_{\partial{\bf 
B}}|\Psi|^{s}d{\cal H}^{n-1}\Big)^{\frac{1}{s}}
\end{equation}
provided $ 1< s=\frac{p(n-1)}{np-n+1}$ and $0<\rho\le R_{X}$, 
where $R_{X}$ depends only on $X$.
\end{thm}


The symbol $<\!\!<$ means that inequality (\ref{tesi}) holds with a 
constant depending on $X$.

\noindent As a first consequence we obtain an isoperimetric inequality of the type 
(\ref{isopuno})  when the target manifold has a
nontrivial cohomology ${\cal H}^{l}( Y)$, for some $1< l<n-1$. Secondly, 
as in \cite{[1]}, we derive a  H\"older 
 continuity property for solutions of Hodge systems.

\medskip

\section
{Preliminaries}

{\bf 2.1 Differential forms}

\bigskip

Our ambient space is an oriented compact (without boundary) smooth 
Riemannian manifold $X$ of dimension $n> 2$.

\noindent Fix an atlas 
$\cal A$ consisting of a finite number of coordinate charts 
$(\Omega_{i}, \vartheta_{i})$, where $\vartheta_{i}:\Omega_{i} \to \RR^{n}$ is a $\cal 
C^{\infty}$- diffeomorphism of the open region $\Omega_{i}\subset X$ onto 
$\RR^{n}$ and choose a ${\cal C}^{\infty}$- partition of unity 
$\{\chi_{i}\in {\cal C}_{0}^{\infty}(\Omega_{i}):\,\, \sum \chi_{i}=1\}$ 
subordinate to $\cal A$.

\noindent Introduce a radius $R=R_X$ such that all concentric 
balls ${\bf B}(x,\rho)\subset {\bf B}(x,3\rho)\subset X$, with $\rho\le R$ lay 
in the same coordinate region $\Omega_{i}$ of the fixed atlas  $\cal A$. 
Such balls share many properties of the Euclidean balls.

Let us denote by ${\cal C}^{\infty}(\wedge ^l X)$, $ 1\le l \le n=dim X$, the space of smooth $l$-forms on $X$. 
Consider the exterior derivative 
$$d:{\cal 
C}^{\infty}(\wedge ^l 
X)\to {\cal 
C}^{\infty}(\wedge ^{l+1} 
X)$$
and its formal adjoint 
$$d^{*}=(-1)^{nl+1}*d*:{\cal 
C}^{\infty}(\wedge ^{l+1} 
X)\to {\cal 
C}^{\infty}(\wedge ^l
X)$$
where $*:{\cal 
C}^{\infty}(\wedge ^l 
X)\to {\cal 
C}^{\infty}(\wedge ^{n-l} 
X)$ denotes the Hodge star duality operator. The duality between $d$ 
and $d^{*}$ is emphasized in the integration by parts 
$$ \int_{X}\langle d\varphi, \psi\rangle =\int_{X}\langle \varphi, 
d^{*}\psi\rangle$$
for $\varphi\in {\cal 
C}^{\infty}(\wedge ^l 
X)$ and $\psi\in {\cal 
C}^{\infty}(\wedge ^{l+1} 
X)$.

We denote  by ${\cal L}^p(\wedge ^l 
X)$ the space of all measurables $l$-forms $\omega$ for which 
$$||\omega||_{p}=\Big(\int_{X}|\omega|^{p}\Big)^{1\over p}<+\infty 
\qquad {\rm for}\,\,\, 1\le p<+\infty$$
and
$$||\omega||_{\infty}={\rm ess}\sup |\omega|\qquad {\rm for}\,\,\,  p=\infty$$

In order to define the Sobolev space ${\cal W}^{1,p}(\wedge ^l 
X)$, $1\le p\le +\infty$, we decompose an $l$-form $\omega$ as 
$\omega=\sum \omega_{i}$ where $\omega_{i}=\chi_{i}\omega$ and 
consider the pullback of $\omega_{i}$ via the mapping 
$\vartheta_{i}^{-1}:\RR^{n}\to \Omega_{i}$
$$\omega_{i}^{\sharp}(x)=\sum_{I}\omega_{i}^{I}(x)dx^{I}\qquad x\in 
\theta_{i}(\Omega_{i})\subset \RR^{n} $$
The sum is taken over all ordered $l$-tuples $I=(i_{1},\ldots, 
i_{l})$ and $dx^{I}=dx_{i_{1}}\wedge\ldots\wedge  
dx_{i_{l}}$. If each function $\omega_{i}^{I}$ has generalized partial 
derivative $\partial\omega_{i}^{I}\in {\cal L}^p$ then the Sobolev 
norm of $\omega$ is defined by
$$||\omega||_{1,p}=\sum_{i} ||\omega_{i}||_{1,p}$$


A differential form $\varphi \in {\cal L}^p(\wedge ^l 
X)$ is said to be closed (in the sense of distributions), in writing 
$d\varphi=0$, if $$ \int_{X}\langle \varphi, d^{*}\psi\rangle =0$$ for 
every test form $\psi\in {\cal 
C}^{\infty}(\wedge ^{l+1} 
X)$.
Similarly we define the coclosed forms, $ d^{*}\psi=0$. Finally
a form $h\in {\cal L}^p(\wedge ^l 
X)$ which is both closed and coclosed is called  a harmonic field of 
degree $l$. Note that the space of all harmonic fields of degree $l$ 
is finite dimensional and consists of ${\cal 
C}^{\infty}$-smooth forms.

We shall use the following Hodge decomposition of a differential form $\omega \in {\cal L}^p(\wedge ^l 
X)$  (see \cite{[6]}):
\begin{thm}\label{hodge}
    For $1<p<\infty$, and $1\le l\le n$ every $\omega \in {\cal L}^p(\wedge ^l 
X)$ can be decomposed as
$$\omega = d\alpha+d^{*}\beta +h$$
where $\alpha,\beta$ are in the Sobolev classes ${\cal W}^{1,p}(\wedge 
^{l-1} 
X)$ , ${\cal W}^{1,p}(\wedge 
^{l+1} 
X)$ respectively and $h $ is a harmonic field of degree $l$.
\end{thm}

\bigskip

\noindent {\bf 2.2 Weakly differentiable mappings}

\bigskip


Our applications will be concerned with weakly differentiable 
mappings $f:X\to Y$ where $X$ and $Y$ are  oriented smooth 
Riemannian manifolds  of dimension $n\ge 2$ without boundary. 

\noindent In order to define the 
Sobolev class ${\cal W}^{1,p}(X,Y)$ we appeal to the well known 
Theorem of J. Nash (\cite{[10]}) which ensures that $Y$ can be ${\cal 
C}^{\infty}$-isometrically imbedded in the euclidean space $\RR^{n}$.

\noindent Namely, we say that $f: X\to \RR^{n}$ lies in 
${\cal W}^{1,p}(X,Y)$ if each coordinate function $f^{i}:X\to \RR$ 
belongs to the  usual Sobolev space ${\cal W}^{1,p}(X)$ and $f(x)\in Y$ 
for almost every $x\in X$.

To every ${\cal 
C}^{\infty}$-smooth $l$-form $\omega$ in the target  space there 
corresponds its pullback $f^{\sharp}\!\omega$ via $f$. The pointwise 
estimate $$ |f^{\sharp}\!\omega|<\!\!< |Df|^{l}$$
relates the degree of integrability of $f^{\sharp}\!\omega$ to the one 
of $Df$.

\noindent Here and in what follows the symbol $<\!\!<$ means that the inequality 
holds with a constant depending  on the manifold $X$.

\noindent Note that the commutation rule 
$$ f^{\sharp}\circ d=d\circ f^{\sharp}$$
holds provided $f:X \to Y$ belongs to $C^{\infty}$.

Given a function $h\in {\cal L}^p(X)$  define the maximal function 
of $h$ as  
$${\bf M}_{p}h(x)=\sup\Big\{\Big(\medint_{\bf 
B}|h(x)|^{p}dx\Big)^{1\over p};\,\, x\in {\bf 
B}\subset X\Big\}$$
where the supremun runs over all balls ${\bf 
B}\subset X$ containing $x$.

\noindent As in the Euclidean setting, the following weak type estimate holds 
(see \cite{[4]})
\medskip
\begin{prop}\label{propmax}
    For every $h\in {\cal L}^p(X)$  we have
\begin{equation}\label{massimale}
\int_{{\bf M}_{p}h>2t}dx<\!\!< {1\over t^{p}}\int_{|h|>t}|h(x)|^{p}dx
\end{equation}
for all $t>0$.
\end{prop}

\noindent As a consequence of the weak type estimate at (\ref{massimale}) we 
obtain
\medskip
\begin{cor}\label{cormax}
 Let $\{h_{j}\}$ converges to $h$ in ${\cal L}^p(X)$, then
\begin{equation}
    \lim_{j \to \infty} \int_{M_{p}h_{j}>\lambda} dx= \int_{M_{p}h>\lambda} dx
\end{equation}
for every $\lambda>0$. Moreover $\{M_{p}h_{j}\}$ contains a 
subsequence converging to $M_{p}h$ almost everywhere.
\end{cor}

\bigskip
\section
{Weak wedge products}
\bigskip
Our intention here is to identify the distributional wedge product of 
closed forms with their pointwise product. As a matter of fact, a 
significant part of this section comes from \cite{[4]}.

\noindent Consider closed differential forms $\Phi\in {\cal L}^p(\wedge ^l 
X)\cap$ ker $d$ and $\Psi\in
{\cal L}^r(\wedge ^k X)\cap$ ker $d$, where $1\le k,l<n,\, k+l=n$ and 
$1< p,r<\infty$. First
assume that $p$ is H\"older conjugate to $r$. Thus $\Phi\wedge \Psi$ 
is integrable and, therefore,
defines a Schwartz distribution of order zero
\begin{equation}
(\Phi\wedge \Psi)[\eta]=\int_X \eta (\Phi\wedge \Psi)\qquad\qquad 
\mbox{for}\,\, \eta \,\in {\cal
C}^\infty (X) 
\end{equation}
But we can do much better if we use the following Hodge decomposition, see 
Theorem \ref{hodge}, 
\begin{equation}
\Phi= d\varphi+\theta
\end{equation}
where the exact component $d\varphi\in {\cal L}^p(\wedge ^l X)$ and the 
harmonic field $\theta\in {\cal C}^\infty(\wedge ^l X)$.
Then, using integration by parts, we obtain  the following formula 
\begin{eqnarray}
(\Phi\wedge \Psi)[\eta]& = &\int_X \eta (d\varphi\wedge \Psi)+\int_X 
\eta 
(\theta\wedge \Psi)\cr\cr
&  =& \int_X \eta 
(\theta\wedge \Psi)- \int_X d\eta \wedge(\varphi\wedge \Psi)
\end{eqnarray}
This latter integral converges whenever $ \varphi\wedge \Psi$ is 
integrable. Assume now that $1< p,r<\infty$ are Sobolev conjugate 
exponents, that is 
$$ \frac{1}{p}+\frac{1}{r}=1+\frac{1}{n}$$
This condition forces that one of the exponents is less than $n$, for 
instance $1< p<n$. Then, by Sobolev imbedding Theorem, we find 
that $\varphi\in {\cal L}^{\frac{np}{n-p}}(\wedge ^{l-1} X)$, where the 
exponent $\frac{np}{n-p}$ is exactly H\"older conjugate to $r$  so 
that $\varphi\wedge \Psi \in {\cal L}^1(\wedge ^{n-1} X)$.

In this way we are justified  to give the  definition of the {\it distributional 
wedge 
product}.
\begin{defn} 
 Let $\Phi\in {\cal L}^p(\wedge ^l X)\cap$ ker $d$ and $\Psi\in
{\cal L}^r(\wedge ^{n-l} X)\cap$ ker $d$, where  $1< p,r<\infty$ are
Sobolev conjugate exponents. The distribution $\Phi\wedge \Psi$ 
operates on a test function $\eta\in {\cal C}^\infty( X)$ by the 
rule
\begin{equation}\label{limit}
    (\Phi\wedge \Psi)[\eta]=\lim_{j\to \infty}\int_X \eta 
(\Phi_{j}\wedge 
\Psi_{j})
\end{equation}
where $\Phi_{j}\in {\cal C}^\infty(\wedge ^l X)$ and $\Psi_{j}\in 
{\cal 
C}^\infty(\wedge  ^{n-l} X)$ are closed forms converging to $\Phi$ and 
$\Psi$ in ${\cal L}^p(\wedge ^l X)$ and $
{\cal L}^r(\wedge  ^{n-l} X)$, respectively.
\end{defn}
\medskip




\bigskip
We shall use  the following two Lemmas.

\medskip
\begin{lem}\label{lemma1}
Let $\Phi\in {\cal C}^\infty(\wedge ^l X)$  and $\Psi\in
{\cal C}^\infty(\wedge  ^{n-l} X)$ be closed differential forms and let  
$1< 
p,r<\infty$ be
Sobolev conjugate exponents. Then for every nonempty open set 
$\Omega\subset X$ and every test function $\eta\in {\cal 
C}^\infty(X)$ we have
\begin{equation}
    \Bigg|\medint_{\Omega}\eta (\Phi\wedge 
\Psi)\Bigg|<\!\!<\Vert\eta\Vert_{{\cal 
C}^\infty(\Omega)}\Vert{\bf M}_{p}\Phi\Vert_{{\cal 
L}^{\infty}(X\setminus\Omega)}\Vert{\bf 
M}_{r}\Psi\Vert_{{\cal L}^{\infty}(X\setminus\Omega)}
\end{equation}
\end{lem}


\bigskip

\begin{lem}\label{lemma2}
Let $\Phi\in {\cal C}^\infty(\wedge ^l X)$  and $\Psi\in
{\cal C}^\infty(\wedge  ^{n-l} X)$ be closed differential forms, let  
$1< 
p,r<\infty$ be
Sobolev conjugate exponents and let $\lambda >0$. Then, setting 
$H=|\Phi|^{p}+|\Psi|^{r}$, we have
\begin{equation}\label{distr}
    \Bigg|(\Phi\wedge \Psi)[\eta]-\int_{X\setminus\Omega}\eta (\Phi\wedge 
    \Psi)\Bigg|<\!\!<\lambda^{1+{1\over n}}|\Omega|\,\Vert\eta\Vert_{{\cal 
C}^\infty(X)}
\end{equation}
where $\Omega=\{x\in X:\,\,  {\bf M}H(x)>\lambda\}$.
\end{lem}

\medskip

\medskip
\noindent {\it Proof:}\quad The left hand side of the inequality 
(\ref{distr}) reduces to 
\begin{equation}
\Bigg|\int_{\Omega}\eta(\Phi\wedge \Psi)\Bigg|
\end{equation}

Using Lemma \ref{lemma1} and the definition of $\Omega$ we get
\begin{eqnarray}
\Bigg|\int_{\Omega}\eta(\Phi\wedge 
\Psi)\Bigg|&<\!\!<&|\Omega|\,\Vert\eta\Vert_{{\cal 
C}^\infty(X)}\Vert{\bf M}_{p}\Phi\Vert_{{\cal L}^{\infty}(X\setminus\Omega)}\Vert{\bf 
M}_{r}\Psi\Vert_{{\cal L}^{\infty}(X\setminus\Omega)}\cr\cr
&<\!\!<&|\Omega|\,\Vert\eta\Vert_{{\cal 
C}^\infty(X)}\Vert{\bf M}H\Vert^{1\over p}_{{\cal 
L}^{\infty}(X\setminus\Omega)}\Vert{\bf 
M}H\Vert^{1\over r}_{{\cal L}^{\infty}(X\setminus\Omega)}\cr\cr
&<\!\!<&|\Omega|\,\Vert\eta\Vert_{{\cal 
C}^\infty(X)}\lambda^{1\over p}\lambda^{1\over 
r}=|\Omega|\,\Vert\eta\Vert_{{\cal 
C}^\infty(X)}\lambda^{1+{1\over n}}
\end{eqnarray}

Now, we are in position to prove 

\medskip

\begin{thm}
Let $(\Phi,\Psi)$  be an admissible pair of differential forms. Then there are measurable sets 
$X_{1}\subset X_{2}\ldots \subset X$ whose union is $X$ and 
$\Phi\wedge 
\Psi$ is integrable over each $X_{j}$, such that 
\begin{equation}
(\Phi\wedge \Psi)[\eta]=\lim_{j\to \infty}\int_{X_{j}}\eta(\Phi\wedge 
\Psi)
\end{equation}
for every $\eta\in {\cal 
C}^\infty(X)$.
\end{thm}

\medskip

\medskip
\noindent {\it Proof:}\quad Observe that $\Phi$ and $\Psi$ satisfy 
inequality (\ref{distr}) for all but a countable set of the 
parameters $\lambda>0$. Indeed, there exist sequences of closed forms 
$\Phi_{j}\in {\cal 
C}^\infty(\wedge ^l X)$  and $\Psi_{j}\in
{\cal C}^\infty(\wedge  ^{n-l} X)$ converging to $\Phi$ and 
$\Psi$ in ${\cal L}^p(\wedge ^l X)$ and $
{\cal L}^r(\wedge  ^{n-l} X)$, respectively. We need only to justify  a 
passage to the limit in the inequalities
\begin{equation}
    \Bigg|(\Phi_{j}\wedge \Psi_{j})[\eta]-\int_{{\bf M}H_{j}\le 
\lambda}\eta 
    (\Phi_{j}\wedge 
    \Psi_{j})\Bigg|<\!\!<\lambda^{1+{1\over n}}||\eta||_{{\cal 
C}^\infty(X)}\int_{{\bf M}H_{j}> \lambda}dx 
\end{equation}
Letting $j\to\infty$ the integrals in the right hand side converge 
to $\int_{{\bf M}H> \lambda}dx $, by Corollary \ref{cormax}. Of course, 
$(\Phi_{j}\wedge 
\Psi_{j})[\eta]$ converges to $(\Phi\wedge \Psi)[\eta]$ by the 
definition of the distributional wedge product. 

Moreover, since ${\bf 
M}H_{j}$ converges to ${\bf 
M}H$ a.e. in X, the characteristic functions $\Xi_{j}$ of the sets $\{x\in 
X:\,\, {\bf 
M}H_{j}\le \lambda\} $ converge almost 
everywhere to the characteristic function $\Xi$ of the set $\{x\in 
X:\,\, {\bf 
M}H\le \lambda\} $. Therefore if we write the
 integral in the left hand side as
$$ \int_{X}\eta 
    (\Phi_{j}\wedge 
    \Psi_{j})\Xi_{j}$$
and observe that the integrands are uniformly bounded by $\lambda^{1+{1\over 
n}}||\eta||_{{\cal 
C}^\infty(X)}$, we can pass to the limit thank to the Dominated 
Convergence Theorem.   

Let us consider a sequence $t_{j}$ increasing to infinity such that 
\begin{equation}\label{assu}
\liminf_{j\to\infty}t_{j}^{{1\over n}}\int_{H>t_{j}}H(x)dx=0
\end{equation}
and define the sets $X_{j}$ as
$$X_{j}=\{x\in X:\,\, {\bf 
M}H\le 2t_{j}\} $$
We make use of the estimate in Lemma \ref{lemma2} with 
$\lambda=2t_{j}$
\begin{eqnarray}
\Bigg|(\Phi\wedge \Psi)[\eta]-\int_{X_{j}}\eta 
    (\Phi\wedge 
    \Psi)\Bigg|&<\!\!<&\Vert\eta\Vert_{{\cal 
C}^\infty(X)} t_{j}^{1+{1\over n}}\int_{{\bf M}H> 2t_{j}}dx\cr\cr
&<\!\!<&\Vert\eta\Vert_{{\cal 
C}^\infty(X)} t_{j}^{{1\over n}}\int_{H> t_{j}}H(x)dx
\end{eqnarray}
The latter inequality follows by Proposition \ref{propmax}.
Letting $t_{j}$ go to $\infty$, we conclude using  
(\ref{assu}).

\bigskip
Using a monotone convergence argument, from previous Theorem we obtain

\medskip

\begin{thm}
Let $(\Phi,\Psi)$  be an admissible pair of differential forms. Then 
\begin{equation}
(\Phi\wedge \Psi)[\eta]=\int_{X}\eta(\Phi\wedge 
\Psi)
\end{equation}
for every $\eta\in {\cal 
C}^\infty(X)$.
\end{thm}




\medskip

Fix a nonnegative $\eta\in {\cal 
C}^\infty(\RR^{n})$ supported in the closed unit ball  and having 
integral 
1. We shall denote by   $\eta_{t}$, $t>0$, the usual sequence of 
mollifiers, i.e.  $\eta_{t}(x)=t^{-n}\eta(t^{-1}x)$. The convolution 
between $\Phi\wedge 
\Psi $ and $\eta_{t}$ is defined by the rule
\begin{equation}\label{conv}
(\Phi\wedge \Psi)\star\eta_{t}(x)=\sum_{\Omega_{i}\in{\cal 
A}}\chi_{i}(x)\int_{\Omega_{i}}\eta_{t}(\vartheta_{i}(x)-\vartheta_{i}(z))J(z,\vartheta_{i}(x))\Phi\wedge \Psi(z)
\end{equation}
where $\{\chi_{i}, i\in I\}$ is the  partition of unity subordinate 
to the fixed atlas ${\cal 
A}$ consisting of the coordinate charts $(\Omega_{i},\vartheta_{i})$.

Note that $(\Phi\wedge \Psi)\star\eta_{t}$ converges to $\Phi\wedge 
\Psi$ in the sense of distributions, as $t\to 0$.

Then if $(\Phi,\Psi)$  is an admissible pair of differential forms we 
have
\begin{equation}\label{conv}
(\Phi\wedge \Psi)(x)=\lim_{t\to 0}(\Phi\wedge \Psi)\star\eta_{t}(x)
\end{equation}

A simple limiting argument implies  the following 
rule of integration by parts for admissible pairs of differential forms 
\begin{equation}\label{parts}
\int_X \eta (\Phi\wedge \Psi)=\int_X 
d\eta \wedge
(\varphi\wedge \Psi)+\int_X 
\eta 
(h\wedge \Psi)
\end{equation}
where $\Phi=d\varphi+ h$.



\bigskip
\bigskip

\section{Isoperimetric type inequalities}

\bigskip
\bigskip

In the Euclidean case it is possible to derive  isoperimetric 
type inequalities  from estimates of the  
div-curl  product $\langle  B,E\rangle$ under suitable assumptions on the 
degree of
integrability of $B$ and $E$ (see \cite{[1]}). In our context this idea 
translates in Theorem \ref{dis}, whose proof is the following


\medskip

\noindent {\it Proof:}\quad Since ${\bf B}$  has 
trivial cohomology, then every closed form $\Phi$ is also exact on ${\bf 
B}$, i.e. there exists a $(l-1)$-form 
$\varphi$ such that $d\varphi=\Phi$.

\noindent Moreover, from the assumption on $s$ we
infer that $1< p<n-1$. Then there exists a closed $(l-1)$-form 
$\varphi_{0}$ such that $\varphi-\varphi_{0}\in 
{\cal W}^{1,p^{*}}(\Lambda^{l-1}X)$ and the following Sobolev-Poincar\'e inequality 
holds
\begin{equation}
\Big( \int_{\partial {\bf 
B}}|\varphi-\varphi_{o}|^{p^{*}}d{\cal H}^{n-1}\Big)^{1\over 
p^{*}}<\!\!<\Big( \int_{\partial {\bf 
B}}|d\varphi|^{p}d{\cal H}^{n-1}\Big)^{1\over p}
\end{equation}
 where $ p^{*}=\frac{(n-1)p}{n-1-p}$ and ${\cal H}^{n-1}$ 
denotes the $(n-1)$-dimensional  measure on $\partial {\bf 
B}$.


By the rule of integration by parts at (\ref{parts})  and the 
assumption $\Psi\in$ ker $d$ we have
\begin{eqnarray}
\int_{\bf B}\Phi\wedge \Psi&=& \int_{\bf 
B}d(\varphi-\varphi_{o})\wedge 
\Psi\cr\cr
&=& \int_{\bf 
B}d[(\varphi-\varphi_{o}) 
\Psi]\cr\cr
&= &\int_{\partial {\bf B}}(\varphi-\varphi_{o}) 
\Psi 
\end{eqnarray}

Now, since $ \frac{1}{p^{*}}+\frac{1}{s}=1$, by H\"older's inequality  
we conclude
\begin{eqnarray}
\int_{\bf B}\Phi\wedge \Psi &<\!\!< &\int_{\partial {\bf 
B}}|\varphi-\varphi_{o}|\, 
|\Psi|\,d{\cal H}^{n-1}\cr\cr
&<\!\!< &\Big(\int_{\partial {\bf 
B}}|\varphi-\varphi_{o}|^{p^{*}}d{\cal H}^{n-1}\Big)^{1\over p^{*}} 
\Big(\int_{\partial {\bf 
B}}|\Psi|^{s}d{\cal H}^{n-1}\Big)^{1\over s} \cr\cr
&<\!\!< &\Big(\int_{\partial {\bf 
B}}|\Phi|^{p}d{\cal H}^{n-1}\Big)^{1\over p} 
\Big(\int_{\partial {\bf 
B}}|\Psi|^{s}d{\cal H}^{n-1}\Big)^{1\over s}
\end{eqnarray}



\bigskip

\bigskip

One can easily deduce that 
\begin{cor}\label{isop}
 Let $(\Phi,\Psi)$ be an admissible pair. Given $x\in X$, for almost every  
 ball  ${\bf B}={\bf B}(x,\rho)$ in $X$ we have
\begin{equation}\label{lavagna}
\int_{\bf B}\Phi\wedge \Psi<\!\!< \Bigg(\int_{\partial{\bf 
B}}\Big(|\Phi|^{p}+|\Psi|^{s}\Big) d{\cal H}^{n-1}\Bigg)^{\frac{n}{n-1}} 
\end{equation}
provided $ 1< s=\frac{p(n-1)}{np-n+1}$ and $0<\rho\le R_{X}$.
\end{cor}




\bigskip

\bigskip

Next we fix $\omega \in {\cal 
C}^\infty(\wedge^{n} Y)$ and a Sobolev mapping $f\in {\cal W}^{1,\frac {n^{2}}{n+1}}(X,Y)$.
The pointwise Jacobian ${\cal J}_{\omega}(x,f)$ is defined by the 
rule 
$${\cal J}_{\omega}(x,f)dx=f^{\sharp}\omega$$
Suppose that the target manifold $Y$ has one 
nontrivial cohomology ${\cal H}^{l}( Y)$,  $1<l<n-1$, then every $n$-form $\omega \in {\cal 
C}^\infty(\wedge^{n} Y)$ decomposes into wedge products of closed 
forms of degree smaller than $n$, namely
$$\omega= \sum_{i=1}^{N} \a_{i} \wedge \b_{i}\qquad {\rm 
deg}\alpha_{i}> 1,\qquad {\rm 
deg}\beta_{i}> 1 $$
These forms, once pulled back via the mapping $f:X \to Y$, bring us to 
analogous decomposition of $f^{\sharp}\omega$:
$${\cal J}_{\omega}(x,f)dx=f^{\sharp}\omega= \sum_{i=1}^{N} f^{\sharp} \a_{i} \wedge f^{\sharp} \b_{i}$$
see \cite{[4]}.
Therefore
 we obtain the following variant of Theorem \ref{dis} 
 reminiscent to the classical isoperimetric inequality (\ref{isopuno}).

\medskip

\begin{cor}\label{discor}
 Let $f\in {\cal W}^{1,\frac {n^{2}}{n+1}}(X,Y)$ be an orientation 
 preserving mapping. If 
  $$\liminf_{t\to \infty}t^{\frac {n+1}{n}}\int_{|Df|>t}|Df|^{\frac {n^{2}}{n+1}}=0$$
  then 
\begin{equation}
\int_{\bf B}{\cal J}_{\omega}(x,f)dx<\!\!< \Bigg(\int_{\partial{\bf 
B}}|Df|^{n-1}d{\cal H}^{n-1}\Bigg)^{\frac{n}{n-1}} 
\end{equation}
for almost every ball  ${\bf B}={\bf B}(x,\rho)$ in $X$ with $0<\rho<R_{X}$.
\end{cor}

\medskip

\noindent {\it Proof:}\qquad We need only consider one wedge product 
$\omega= \a \wedge \b$ with $\a \in {\cal 
C}^\infty(\wedge^{l} X)\cap$ ker $d$ and $\b \in {\cal 
C}^\infty(\wedge^{n-l} X)\cap$ ker $d$.
Taking $f^{\sharp}\a$ and $f^{\sharp}\b$ for  the closed forms $\Phi$ and $\Psi$  
in Theorem \ref{dis} and  using the pointwise 
inequalities
$$|f^{\sharp}\alpha|<\!\!< |Df|^{l}\qquad\qquad |f^{\sharp}\b|<\!\!< 
|Df|^{n-l}$$
 we get
\begin{eqnarray*}
\int_{\bf B}{\cal J}_{\omega}(x,f)dx &<\!\!<& \Big(\int_{\partial{\bf 
B}}|f^{\sharp}\alpha|^{p}d{\cal H}^{n-1}\Big)^{\frac{1}{p}} 
\Big(\int_{\partial{\bf 
B}}|f^{\sharp}\b|^{s}d{\cal H}^{n-1}\Big)^{\frac{1}{s}}\cr\cr
&<\!\!<& \Big(\int_{\partial{\bf 
B}}|Df|^{pl}d{\cal H}^{n-1}\Big)^{\frac{1}{p}} \Big(\int_{\partial{\bf 
B}}|Df|^{s(n-l)}d{\cal H}^{n-1}\Big)^{\frac{1}{s}}
\end{eqnarray*}
In order to conclude it is  sufficient to 
choose $ p=\frac{n-1}{l}$ and $s=\frac{n-1}{n-l}$ in previous  inequality. 
Note that this choice is legitimate thank to assumption $1<l<n-1$.

\bigskip
Take $n=2l$ and denote by $D^{l\times l}f$ the matrix of all $l\times 
l$-subdeterminants of $Df$. If ${\cal H}^{l}( Y)\not=0$, the pointwise inequalities 
$$|f^{\sharp}\alpha|<\!\!< |D^{l\times l}f|\qquad\qquad |f^{\sharp}\b|<\!\!< 
|D^{l\times l}f|$$
hold.
The following estimate,  reminiscent to 
the isoperimetric type inequality (\ref{isopdue}), can be obtained 
arguing as in the previous corollary.

\medskip

\begin{cor}\label{scor}
 Let $f\in {\cal W}^{1,\frac {n^{2}}{n+1}}(X,Y)$ be an orientation 
 preserving mapping. If 
  $$\liminf_{t\to \infty}t^{\frac {n+1}{n}}\int_{|Df|>t}|Df|^{\frac {n^{2}}{n+1}}=0$$
  then 
\begin{equation}
\int_{\bf B}{\cal J}_{\omega}(x,f)dx<\!\!< \Bigg(\int_{\partial{\bf 
B}}|D^{\frac{n}{2}\times \frac{n}{2}}f|^{2(n-1)\over n}d{\cal H}^{n-1}\Bigg)^{\frac{n}{n-1}} 
\end{equation}
for almost every ball  ${\bf B}={\bf B}(x,\rho)$ in $X$ with $0<\rho<R_{X}$.
\end{cor}


\bigskip

\bigskip



\section{Some applications}



\bigskip
\bigskip

As an application of the isoperimetric type inequality proved in 
Theorem \ref{dis}, we give a Morrey's type  estimate  
 for the wedge product of some special closed forms. Namely we 
have the following

\medskip
\begin{thm}\label{is}
 Let $(\Phi,\Psi)$ be a pair of closed differential forms in
 ${\cal L}^q(\wedge ^l X)\times
{\cal L}^{q'}(\wedge ^{n-l} X)$, where  $1< q,q'<\infty$ are H\"older conjugate 
exponents. If
 there exists a constant $K>1$ such that
\begin{equation}\label{distorsione}
    |\Phi|^{q}+|\Psi|^{q'}\le K (\Phi \wedge\Psi)
\end{equation}
then 
\begin{equation}
\int_{{\bf B}(0,\rho)}\Phi\wedge \Psi\le 
\Big(\frac{\rho}{R}\Big)^{\frac{1}{c_{n}K}} \int_{{\bf 
B}(0,R)}\Phi\wedge \Psi
\end{equation}
for all  $0<\rho<R =R_{X}$.
\end{thm}

\medskip
\noindent  Remark that inequality (\ref{distorsione}) means that the 
nonnegative function $\lambda$ such that the $n$-form $\Phi\wedge 
\Psi=\lambda dx$ satisfies  $|\Phi|^{q}+|\Psi|^{q'}\le K \lambda$.

\medskip

\noindent {\it Proof:}\quad  Observe that the exponents 
$p=\frac{q(n-1)}{n}$ and $s=\frac{q'(n-1)}{n}$ are related by the 
equality $ s=\frac{p(n-1)}{np-n+1}$.
The isoperimetric type inequality  
(\ref{lavagna}) combined with assumption (\ref{distorsione}) gives
\begin{eqnarray}
\int_{{\bf B}(0,\rho)}\Phi\wedge \Psi & \le  & c_{n} 
\Bigg(\int_{\partial {\bf B}(0,\rho)} 
\Big(|\Phi|^{p}+|\Psi|^{s}\Big) d{\cal H}^{n-1}\Bigg)^{\frac{n}{n-1}}\cr\cr
& \le  & c_{n} K \Bigg(\int_{\partial {\bf B}(0,\rho)} 
(\Phi\wedge \Psi)^{\frac{n-1}{n}}\Bigg)^{\frac{n}{n-1}}\cr\cr
& \le  &  c_{n}K \Bigg(\int_{\partial {\bf 
B}(0,\rho)}dx\Bigg)^{\frac{1}{n-1}} 
\int_{\partial {\bf B}(0,\rho)}\Phi\wedge \Psi\cr\cr
& \le  &  c_{n}K \rho \int_{\partial {\bf B}(0,\rho)}\Phi\wedge \Psi
\end{eqnarray}
Setting $$h(\rho)=\int_{{\bf B}(0,\rho)}\Phi\wedge \Psi$$
a routine calculation shows that 
$$h(\rho)\le \Big(\frac{\rho}{R}\Big)^{\frac{1}{c_{n}K}} h(R)$$
for $\rho\le R$, which concludes the proof.

\bigskip

Now we deduce the following  H\"older continuity result. 

\medskip
\begin{thm}\label{appli}
  Let $(\Phi,\Psi)$ be a pair of closed differential forms in
 ${\cal L}^q(\wedge ^l X)\times
{\cal L}^{q'}(\wedge ^{n-l} X)$, where  $1< q,q'<\infty$ are H\"older conjugate 
exponents. If
 there exists a constant $K>1$ such that
\begin{equation}
    |\Phi|^{q}+|\Psi|^{q'}\le K (\Phi\wedge\Psi)
\end{equation}
then the (l-1)-form $\varphi$ such that $d\varphi=\Phi$  is H\"older 
continuous, provided $K<\frac{C_{n}}{n-q}$.
\end{thm}    

\medskip

\noindent {\it Proof}\qquad  Inequality  (\ref{distorsione})  yields
$$|d\varphi|^{q}\le K (\Phi\wedge\Psi)$$
Then Theorem \ref{is} gives
\begin{eqnarray}
\int_{{\bf B}(0,\rho)}|d\varphi|^{q} d{\cal H}^{n} & \le  & c_{n} 
K\int_{ {\bf B}(0,\rho)} 
\Phi\wedge \Psi\cr\cr
& \le  & c_{n} K \Big(\frac{\rho}{R}\Big)^{\frac{1}{c_{n}K}}
\int_{{\bf B}(0,R)}\Phi\wedge \Psi
\cr\cr & =  & c \rho^{\frac{1}{c_{n}K}}
\end{eqnarray}
for all  $0<\rho<R =R_{X}$.
Therefore $\varphi$  is H\"older continuous of exponent 
$\gamma=1+\frac{1}{c_{n}K 
q}-\frac{n}{q}$.  

\bigskip

Now, following the lead of the classical $q$-harmonic operator, 
$1<q<\infty$, we consider a bundle map $${\cal H}_{q}: \wedge^{l} X\to 
\wedge^{l} X $$
such that
$$|{\cal H}_{q}(\xi)|\le M|\xi|^{q-1}$$
$$\langle{\cal H}_{q}(\xi),\xi\rangle \ge M^{-1}|\xi|^{q}$$
for every l-form $\xi$ on $X$, where $M\ge 1$ is a constant.
Consider a pair $(\Phi,\Psi)\in {\cal L}^q(\wedge^{l} X)\times {\cal 
L}^{q'}(\wedge^{l} X) $ such that $d\Phi=0$, $d^{*}\Psi=0$ and coupled by
\begin{equation}\label{Hodge}
    \Psi={\cal H}_{q}(\Phi)
\end{equation}    
We refer to (\ref{Hodge}) as a Hodge system. Existence and uniqueness 
 of solutions to (\ref{Hodge}) has been studied in \cite{[5]}, 
 \cite{[6]}.
 Applying  the operator $d^{*}$ to (\ref{Hodge}) we obtain the second 
 order elliptic equation 
 $$d^{*}{\cal H}_{q}(\Phi)=0$$
One can easily check that the $l$-form $\Phi$ and the $(n-l)$-form $\Psi$  
satisfy inequality (\ref{distorsione}). From Theorem \ref{appli} it 
follows that 
the $(l-1)$-form $\varphi$ such that $d\varphi=\Phi$ is H\"older continuous.

\bigskip
\newpage


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\end{document}