Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

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\def\RR{{\Bbb R}}
\def\CC{{\Bbb C}}
\def\QQ{{\Bbb Q}}
\def\NN{{\Bbb N}}
\def\ZZ{{\Bbb Z}}
\def\II{{\Bbb I}}

\def\bw{\bigwedge\nolimits}
\def\ra{\rightarrow}
\def\btu{\bigtriangleup}			%{\Box}
\def\dbar{\overline{\partial}}
\def\bvp{boundary value problem}
\def\dom{\mbox{dom}\,}
\def\K{{\cal K}}
\def\vn{\vec{n}}
\def\baro{\overline{\Omega}}
\def\o{\Omega}



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\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}{EXAMPLE}
\newtheorem{remark}[theorem]{REMARK}
\newtheorem{proposition}[theorem]{Proposition}

\begin{document}
\title[Hodge Theory in the Sobolev
Topology]{
%\large 
Hodge Theory in the Sobolev Topology
for the de Rham Complex
on a Smoothly Bounded Domain
in Euclidean Space} 

\subjclass{35J55; 35S15, 35N15, 58A14, 58G05}
\author[L. Fontana]{Luigi Fontana}
\address{Dipartimento di Matematica\\ Via Saldini 50\\
Universit\`a di Milano\\ 
20133 Milano (Italy)}
\email{fontana@@vmimat.mat.unimi.it}
\author[S. G. Krantz]{Steven G. Krantz}
\address{Department of Mathematics\\ Washington University\\ St.
Louis, MO 63130 
(U.S.A.)}
\email{sk@@math.wustl.edu}
\thanks{Second author supported in part by the
National Science Foundation}
\author[M. M. Peloso]{Marco M. Peloso}
\address{Dipartimento di Matematica\\ Politecnico di Torino\\ 10129 
Torino (Italy)}
\email{peloso@@polito.it}
\thanks{Third author supported in part
by the Consiglio Nazionale delle Ricerche} 

\keywords{Hodge theory, 
de Rham complex, $\dbar$-Neumann complex,
elliptic estimates,
subelliptic estimates,
pseudodifferential boundary problems} 
%\maketitle 

\def\currentvolume{1}
\def\currentissue{3}

\date{July 29, 1995}

\communicated_by{Robert Lazarsfeld}

\setcounter{page}{103}

\begin{abstract}  The Hodge theory of the de Rham complex
in the setting of the Sobolev topology is studied.  As a result,
a new elliptic boundary value problem is obtained. Next, the Hodge 
theory of the $\dbar$-Neumann problem in the Sobolev
topology is studied.  A new $\dbar$-Neumann boundary condition is
obtained, and the corresponding subelliptic estimate derived.
\end{abstract} 

\maketitle 

\vskip1cm

The classical Hodge theory on a domain in $\Omega \subseteq \RR^{N+1}$ 
 (or, 
more generally, on a real manifold) is based on the complex
$$
\bw^0 \stackrel{d}{\ra} \bw^1 \stackrel{d}{\ra} 
      \bw^2 \stackrel{d}{\ra} \cdots
$$
In the topology of $L^2(\Omega)$, one can calculate both the domain
of existence and the actual form of $d^*$, the Hilbert space
adjoint of $d$.  It turns out that $d^*$ equals $d'$, the formal
adjoint of $d$ 
{\em on the domain of $d^*$}.  See [FOK] for details.

Of course these facts are well known.  They lead to the study
of the second order, self-adjoint operator
$$
\Box \equiv d \, d^* + d^* \, d .
$$
Said operator $\Box$ makes sense precisely on those
forms $\phi$ that lie both in the domain of $d$ and in the domain
of $d^*$.  Calculated on a domain or region $\Omega$ 
in space, the operator
$\Box$ turns out to be the (negative of the) ordinary Laplacian
$\btu$.
The Hodge theory of $\Box$, which by today's standards
is rather straightforward, shows that $\Box$ has
closed range in $L^2(\Omega)$.  The orthogonal complement of
the range is of course the kernel of the adjoint of 
$\Box$ (which is nothing other than $\Box$
itself).  The {\em Neumann operator} $N$ for the $d$-complex
is a right inverse for $\Box$.
  
It turns out that the operator $N$ is, essentially, a
pseudodifferential operator of degree $-2$.  The regularity theory
for the operator $\Box$, and also for the operator $d$, may
be read off from the mapping properties of $N$.   
However, because the operator $d$ has a large kernel, one must
choose a solution to the
equation $du = f$ carefully.

The form
\begin{equation}\label{**}
u_c \equiv d^* N f        
\end{equation}
will always be a solution to $d u = f$ (so long as $f$ is orthogonal
to the harmonic space) and it follows from formal
considerations that $u_c \perp \hbox{Ker}\, d$.  We call
$u_c$ the {\em canonical solution} to $du=f$.  In many
applications, $u_c$ is the ``good'' solution that we seek.

It is by now well known (see [SWE]) that $u_c$ satisfies standard
elliptic regularity estimates for a first order (uniformly) elliptic
linear partial differential operator:  measured in either the
Sobolev topology, the Lipschitz topology, 
or in fact in any Triebel-Lizorkin space topology,
the solution $u_c$ exhibits a gain in smoothness of order 1
when compared to the smoothness of the data.  

The purpose of the present work is to develop the Hodge theory for
the $d$ operator in a {\em Sobolev space} $W^1$ topology.  While 
the general $W^s$-case, $s=1,2,\dots$ is of
considerable intrinsic interest, the case of the topology of $W^1$
is of particular interest in geometric applications.  This work on
the $d$ operator
also serves as a step towards working out the theory of the
$\overline{\partial}$-Neumann problem in a Sobolev space topology. 
We shall spend the remainder of this announcement doing two things:  
{\bf (1)}  summarizing the key features of the Hodge theory for $d$
in  the
Sobolev space $W^1$ topology, and {\bf (2)}  describing progress
on our program to study the $\dbar$-Neumann problem in the
Sobolev space $W^1$ topology.  


\section{The Formalism of the de Rham Complex in the Sobolev
Topology} 

We consider the complex
$$
\bw^0 \stackrel{d}{\ra} \bw^1 \stackrel{d}{\ra} 
      \bw^2 \stackrel{d}{\ra} \cdots .
$$
This is the same complex as considered above but now, when we
turn our attention to the operator $\Box \equiv d d^* + d^* d$,
we consider the adjoints in the $W^s$ topology.

Fix now a smoothly bounded domain $\Omega \subseteq \RR^{N+1}$
(or the half space $\RR^{N+1}_+$).

Following notation of [FOK], we let $\bw^{r}_0(\overline{\Omega})$  
denote  those
$r$-forms with $C_0^\infty(\RR^{N+1})$ coefficients and such that 
the intersection of the support of the form with 
$\overline{\Omega}$ is
{\em compact in $\overline{\Omega}$}.

\begin{proposition} 
With $\Omega$ as above, and with $q = 0,1,2, \dots, N$, we have
$$
\mbox{\rm dom}\, d^* \bigcap \, \bw_0^{q+1} (\overline{\Omega}) = 
 \left \{ \phi \in \bw_0^{q+1}(\overline{\Omega}) : 
 \nabla_{\vec{n}} \phi \, \RightContraction \, \vec{n} \biggr |_{b\Omega}
 = 0   \right \} .
$$
\end{proposition}

Here we denote by $\nabla_{\vec{n}} \phi$ the covariant derivative 
in the normal direction of
the form $\phi$, and by $\RightContraction$ the standard  contraction operator.

\begin{proposition} 
Let $\Omega$ be a smoothly bounded domain.
Then, on $\dom d^*$,
$$
d^* = d' + \K,
$$
where $d'$ is the formal adjoint of $d$, and 
$\K$ is an operator sending $(q+1)$-forms to $q$-forms.
The components of ${\cal K}\phi$ 
are solutions  of the following \bvp 
$$
\left \{  \begin{array}{lll}
\displaystyle{(-\btu+I)(\K\phi)_I} & =0  & \hbox{on }  \Omega \\
\displaystyle{\frac{\partial }{\partial n} (\K\phi)_I }
	& = T_2\phi\RightContraction \vec{n} & \hbox{on }  b\Omega ,
          \end{array}
\right. 
$$
where $T_2$ is a second order tangential differential operator on
forms whose top order terms equal those of the Laplacian
on the boundary. 
\end{proposition}

\begin{definition} 
For a smoothly bounded domain $\Omega$ we set
$$
G_\Omega =\K d+d\K  .
$$
\end{definition}
\noindent Notice that now $\Box \equiv dd^* +d^* d = -\btu + G_\Omega$.
\smallskip 

In the language just introduced, our \bvp\ 
becomes
\begin{equation}\label{dagger}
\left \{   \begin{array}{lcl}
\displaystyle{ (-\btu+G_\Omega) \phi= \alpha }& 
					\hbox{on} & \Omega \\
\phi\in\dom d^*  & & \\
d\phi\in\dom d^*  & &   
           \end{array}  .
\right.               
\end{equation}

We are now ready to state our results about existence and regularity
of the \bvp.

\begin{theorem}\label{MAIN-THM-DMN} 
Let $\Omega$ be a smoothly
bounded domain.
Consider the \bvp\ (\ref{dagger}).
Let $s>1/2$. 
Then there exists a finite dimensional subspace 
(the harmonic space) ${\cal H}_q$ 
of $\bw^q (\overline{\Omega})$
and a constant $c=c_s >0$ such
that if 
$\alpha\in
W^s_q (\Omega)$
is orthogonal (in the
$W^1$-sense) to ${\cal H}_q$, then
the \bvp\ (\ref{dagger})
has a unique solution $\phi$ orthogonal to ${\cal H}_q$ 
in the $W^1$ topology such that
$$
\| \phi \|_{s+2} \le c \|  \alpha\|_s .
$$
\end{theorem}

\noindent {\bf Remark.} If $s<1$ then, by saying that $\alpha$ is
orthogonal in the
$W^1$-sense, we mean that $\alpha$ is the $W^s$-limit of smooth forms 
that are orthogonal in $W^1$ to ${\cal H}_q$.

Finally we have
\begin{theorem}  
Let $\Omega$ be a smoothly bounded domain in $\RR^{N+1}$.  
Let $W^1_q (\Omega)$ 
denote the 1-Sobolev space of $q$-forms.  Then we have the
strong orthogonal decomposition
$$
W^1_q = dd^* (W^1_q) \bigoplus d^* d(W^1_q) \bigoplus {\cal H}_q 
 \, , 
$$
where ${\cal H}_q$ is a finite dimensional subspace.
\end{theorem}

What is novel in our treatment of the Hodge theory for the de Rham
operator in the Sobolev topology 
is the presence 
of the operator ${\cal K}$ when mediating between $d'$ and $d^*$.
The contribution of ${\cal K}$ to the \bvp\
is non-trivial and its study occupies a large
portion  of our work.

The analysis of $d^*$ in the Sobolev topology fits rather naturally
into an abstract theory of pseudodifferential operators of transmission type
as developed by Boutet de Monvel in [BDM1-BDM3].  Details are
provided in our paper [FKP1].

\section{The $\dbar$-Neumann Problem}

As previously indicated, the main goal of this work is to develop
a new approach to the $\dbar$-Neumann problem.  We have used our
work on the Hodge theory of the de Rham operator in the Sobolev
topology as a stepping stone in this program.  We now outline
the steps of our theory that have been developed thus far for
the $\dbar$-Neumann problem.  Full details will appear in [FKP2].

In analogy with what occurred above, we let $\Omega$ be a smoothly
bounded domain in $\CC^n$.  We consider the complex
$$
\bw^{0,0} \stackrel{\dbar}{\ra} \bw^{0,1} \stackrel{\dbar}{\ra} 
      \bw^{0,2} \stackrel{\dbar}{\ra} \cdots
$$
All notation is analogous to that stated for the $d$- complex; we
shall not repeat all the definitions.
When we calculate adjoints, it shall be in the $W^1(\Omega)$
topology.

Let $\rho$ be a defining function for $\Omega$ and set
$$
N = \sum_{j=1}^n \frac{\partial \rho}{\partial \bar{z}_j} 
                    \frac{\partial}{\partial z_j} .
$$
Then we set
$$
\frac{\partial}{\partial n} \equiv N + \overline{N} .
$$

\begin{proposition} 
Let $\phi \in \bw^{0,1}(\overline{\Omega})$.  Then 
$\phi \in \dom  \dbar^*$
if and only if 
$$
\frac{\partial}{\partial n} \bigl ( \phi \RightContraction \overline{N} 
\bigr )   \biggr |_{b\Omega} = 0 ,
$$
that is, if and only if
$$
\sum_{j} \frac{\partial}{\partial n} \left ( \phi_j 
\frac{\partial \rho}{\partial z_j} \right ) \biggr |_{b\Omega}
\equiv \sum_j \frac{\partial \rho}{\partial z_j} 
\frac{\partial \phi_j}{\partial n} \biggr |_{b\Omega}
 = 0 .
$$
\end{proposition}

\begin{proposition} 
Let $\phi \in \bw^{0,1} (\overline{\Omega}) \cap \dom \dbar{}^*$.
Then
$$
\dbar{}^* \phi = \vartheta \phi + \K \phi ,
$$
where $\vartheta$ is the formal adjoint of
the $\dbar$ operator and
$\K$ is the function which is the solution of the following
boundary value problem:
$$
\left \{ \begin{array}{lll}
\displaystyle{
\bigl ( - \bigtriangleup + I \bigr ) (\K \phi) } 
& \displaystyle{=  0 } & \mbox{on} \ \o \medskip \\
\displaystyle{
\frac{\partial}{\partial n} \bigl ( \K \phi \bigr ) }
& \displaystyle{ =
\sum_{k=1}^n \biggl[ 2 \cdot \sum_{p=1}^n \biggl(
Y_p^* \bigl[ (Y_p \phi_k ) \frac{\partial \rho}{\partial z_k}
 \bigr]} & \\
& \qquad \qquad \qquad
+  \overline{Y}_p^* \bigl[ (\overline{Y}_p \phi_k ) 
 \frac{\partial \rho}{\partial z_k} \bigr] \biggr)
         + \phi_k \frac{\partial \rho}{\partial z_k} \biggr]  
   &  \mbox{on} \ b\o  
         \end{array} \ .
\right. 
$$
Here $Y_p$ denotes the tangential component of 
$\partial/\partial z_p$, and $Y_p^*$ is its formal adjoint.
\end{proposition}

Now recall that, in the classical $L^2$ theory (see [FOK]), a major
point of the analysis is that the boundary value problem is not
elliptic: the classical coercive estimate does not hold.  
One obtains
a substitute estimate from below for the quadratic form $Q$ 
that
is defined to be essentially the polarization of $\Box$.  A similar, but more
complicated, circumstance now obtains when we work in the Sobolev
$W^1$ topology.

If $\phi, \psi \in \bw^{0,1}(\baro) \cap \dom \dbar^*$, then we set
$$
Q(\phi,\psi) = \langle \dbar \phi, \dbar \psi \rangle_1 +
       \langle \dbar{}^* \phi, \dbar{}^* \psi \rangle_1 +
            \langle \phi, \psi \rangle_1 .
$$
We also set
$$
E(\phi)^2 \equiv  \sum_{j,k} \biggl\| 
\frac{\partial \phi_j}{\partial \bar{z}_k} \biggr\|_1^2 
+ \| \phi\|_1^2 + \|\phi\|_{W^1 (b\Omega)}^2 
+ \|\phi\, \RightContraction \, \bar N \|_{W^{3/2}(b\Omega)}^2 .
$$
We let ${\cal L}$ denote the Levi form.  Then our basic estimate
is this:

\begin{theorem} 
There is a constant $C_0 > 0$ such that, for all 
$\phi \in \bw^{0,1}(\baro) \cap \dom \dbar{}^*$, we have
\begin{eqnarray*}
Q(\phi, \phi) 
& \geq & C_0 \Biggl \{ 
\sum_{j,k} \biggl \| \frac{\partial \phi_j}{\partial \bar{z}_k} 
\biggr \|_1^2 
	+ \| \K \phi\|_1^2 + \| \phi \|_1^2 
	+ \int_{b\Omega} {\cal L}(\phi, \phi) \\
&  & \qquad + 2 \sum_p \int_{b\Omega} {\cal L}\biggl (
          \frac{\partial \phi}{\partial z_p}, 
          \frac{\partial \phi}{\partial z_p} \biggr )
      + 2 \sum_p \int_{b\Omega} {\cal L} \biggl (
    \frac{\partial \phi}{\partial \bar{z}_p} , 
    \frac{\partial \phi}{\partial \bar{z}_p} \biggr ) \Biggr \} .
\end{eqnarray*}
As a consequence, if $\Omega$ is a smoothly bounded strongly 
pseudoconvex domain, then there exists $C_1 >0$ such that
$$
\frac{1}{C_1} E(\phi) \le  Q(\phi,\phi) \le C_1 E(\phi) .
$$
\end{theorem}

Of course this theorem makes it clear (especially in view of the 
well known calculations in [FOK]) how the Levi form---in particular
how the property of strong pseudoconvexity---will come into play in further
developments of the theory.  The inequality in Theorem 2.3 will
enable us to prove subelliptic estimates for the canonical
solution to the $\dbar$ problem in the Sobolev inner product.
In particular, our work produces a complete existence and 
regularity theory for a new canonical solution to the
$\dbar$-Neumann problem.  Other canonical solutions, besides that
of Kohn, have been explored in [PHO].

A complete explication of the $\dbar$-Neumann problem in the $W^s$ 
topology, together with applications, will appear in [FKP2].
     
\begin{thebibliography}{ABCDE}
 
\bibitem[BDM1]{BDM1}  L. Boutet de Monvel, Comportement
d'un op\'{e}rator pseudo-diff\'{e}rential sur une
vari\'{e}t\'{e} \`{a} bord. I. {\em Journal d'Anal.
Math.} 17(1966), 241-253.
 
\bibitem[BDM2]{BDM2}  L. Boutet de Monvel, {\em ibid},
254-304.
 
\bibitem[BDM3]{BDM3}  L. Boutet de Monvel, Boundary
problems for pseudo-differential operators, {\em Acta
Math.} 126(1971), 11-51.
 
\bibitem[FOK]{FOK} G. B. Folland and J. J. Kohn,
{\em The Neumann Problem for the Cauchy-Riemann Complex},
Princeton University Press, Princeton, 1972.

\bibitem[FKP1]{FKP1}  L. Fontana, S. G. Krantz, M. M. Peloso,
Hodge theory in the Sobolev topology for the de Rham complex,
preprint.

\bibitem[FKP2]{FKP2}  L. Fontana, S. G. Krantz, M. M. Peloso,
The $\dbar$-Neumann problem in the Sobolev topology, in progress.

\bibitem[KOH]{KOH} J. J. Kohn, Harmonic integrals on
strongly pseudoconvex manifolds I, {\em Ann. Math.}
78(1963), 112-148; II, ibid. 79(1964), 450-472.

\bibitem[PHO]{PHO} D. H. Phong, thesis, Princeton
University, 1977.
 
\bibitem[SWE]{SWE}  R. Sweeney, The $d$-Neumann
problem, {\em Acta Math.} 120(1968), 224-277.

\end{thebibliography} 

\end{document}