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\theoremstyle{definition}
\newtheorem*{definition1}{Step I}
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\begin{document}

\title[ON BOJARSKI'S INDEX FORMULA]
{On Bojarski's index formula for nonsmooth interfaces}
\author{Marius Mitrea}
\address{Department of Mathematics,
University of Missouri-Columbia,
Columbia, MO 65211}
\email{marius@math.missouri.edu}
\subjclass{Primary 58G10, 42B20; Secondary 34L40, 30D55}
\thanks{Partially supported by NSF}
\issueinfo{5}{06}{}{1999}
\dateposted{April 6, 1999}
\pagespan{40}{46}
\PII{S 1079-6762(99)00060-8}
\def\copyrightyear{1999}
\copyrightinfo{1999}{American Mathematical Society}
\commby{Stuart Antman}
\date{December 02, 1998}
\begin{abstract}
Let $D$ be a Dirac type operator on a compact manifold ${\mathcal{M}}$
and let $\Sigma $ be a Lipschitz submanifold of codimension one 
partitioning
${\mathcal{M}}$ into two Lipschitz domains $\Omega _{\pm }$. Also, let
${\mathcal{H}}^{p}_{\pm }(\Sigma ,D)$ be the traces on $\Sigma $ of the
($L^{p}$-style) Hardy spaces associated with $D$ in $\Omega _{\pm }$.
Then $({\mathcal{H}}^{p}_{-}(\Sigma ,D),{\mathcal{H}}^{p}_{+}(\Sigma 
,D))$ is
a Fredholm pair of subspaces for $L^{p}(\Sigma )$ (in Kato's sense) whose
index is the same as the index of the Dirac operator $D$ considered
on the whole manifold ${\mathcal{M}}$.
\end{abstract}
\maketitle

\section{Introduction and statement of results}Let $\Sigma $ be 
a rectifiable curve in ${\mathbb{R}}^{2}\equiv {\mathbb{C}}$
which divides the plane into two domains, denoted by $\Omega _{\pm }$.
Corresponding to these, introduce the Hardy spaces on $\Sigma $,
\begin{equation}\label{eq:1.1}{\mathcal{H}}^{p}_{\pm }(\Sigma ):=\{u|_{\Sigma 
};\,\bar {\partial }u=0\,\,
\text{in}\,\,\Omega _{\pm },\,\,{\mathcal{N}}(u)\in L^{p}(\Sigma )\},
\qquad 1m:=\text{dim}\,{\mathcal{M}}$.
Our aim is to investigate the analogue of \eqref{eq:1.2} when $\Sigma $
is a Lipschitz submanifold of codimension one of ${\mathcal{M}}$ and
$D:{\mathcal{E}}\to {\mathcal{F}}$ is a first-order, elliptic 
differential
operator whose coefficients have a limited amount of smoothness.
Specifically, we assume that
\begin{equation}\label{eq:1.3}\text{the top coefficients of}\,\,D\,\,\text{belong 
to}\,\,H^{2,r},
\end{equation}
\begin{equation}\label{eq:1.4}\text{the coefficients of the zero order part 
of}\,\,D\,\,
\text{belong to}\,\,H^{1,r},
\end{equation}
for some $r>m$. Under the current assumptions, $D^{*}$, the adjoint of 
$D$,
also satisfies \eqref{eq:1.3}--\eqref{eq:1.4}.

The major analytical hypothesis we make on the differential operator
is that
\begin{equation}\label{eq:1.5}D\,\,\text{and}\,\,D^{*}\,\,\text{have the unique 
continuation
property}.
\end{equation}
Recall that $D$ is said to have the unique continuation property
if the following holds: $u\in H^{1,2}({\mathcal{M}},{\mathcal{E}})$
and $Du=0$ implies that either $u\equiv 0$ or
$\text{supp}\,u={\mathcal{M}}$. The hypothesis \eqref{eq:1.5} is natural
inasmuch as it is automatically satisfied (for $r=\infty $) by
Dirac type operators; cf. \cite{Ar}, \cite{Co}.

Consider now a Lipschitz subdomain $\Omega $ of ${\mathcal{M}}$ and
set $\Sigma :=\partial \Omega $, $\Omega _{+}:=\Omega $,
$\Omega _{-}:={\mathcal{M}}\setminus \bar {\Omega }$. Also, for 
$10$. Hence, the integral operator $K:=C_{+}-C_{-}$ is compact
from $L^{p}(\Sigma ,{\mathcal{E}})$ into itself, $10.
\end{equation}
These are proved by implementing a symbol decomposition to the
effect that $D=D^{\#}+D^{b}+B$ with
\begin{equation}\label{eq:2.16}D^{\#}\in OPC^{\infty }S^{1}_{1,\delta },\quad 
D^{b}\in OPC^{1}S^{1-\delta }_{1,\delta },\quad B \in L^{\infty 
}({\mathcal{M}},\text{Hom}({\mathcal{E}},{\mathcal{F}})),
\end{equation}
for some $0<\delta <1$. Now \eqref{eq:2.14}--\eqref{eq:2.15} follow from the existence
of a parametrix for $D^{\#}$ and mapping properties for
pseudodifferential operators whose symbols have a limited amount
of smoothness. See, e.g., \cite{Ta} for references and a general
discussion of such issues.

Now, if $f\in {\mathcal{H}}^{p}_{-}(\Sigma ,D)\cap {\mathcal{H}}^{p}_{+
}(\Sigma ,D)$,
then there exist (unique, by \eqref{eq:2.6}) functions
$u_{\pm }\in {\mathcal{H}}^{p}(\Omega _{\pm },D)$ so that
$u_{+}|_{\Sigma }=f=u_{-}|_{\Sigma }$. Set $u:=u_{-}$ in $\Omega _{-}$ 
and
$u:=u_{+}$ in $\Omega _{+}$, $u\in L^{p}({\mathcal{M}},{\mathcal{E}})$ 
so that
$u\in \text{Ker}\,\left (D:H^{1,p}({\mathcal{M}},{\mathcal{E}})\to 
L^{p}({\mathcal{M}},{\mathcal{F}})\right )$. Consequently, the (just 
defined)
assignment $f\mapsto u$ is linear, well defined and, by \eqref{eq:2.6},
one-to-one from ${\mathcal{H}}^{p}_{-}(\Sigma ,D)\cap 
{\mathcal{H}}^{p}_{+}(\Sigma ,D)$
into $\text{Ker}\,\left (D:H^{1,p}({\mathcal{M}},{\mathcal{E}})\to 
L^{p}({\mathcal{M}},{\mathcal{F}})\right )$.
Invoking \eqref{eq:2.15}, it is clear that this is also onto. Hence,
\begin{equation}\label{eq:2.17}\text{dim}\left ({\mathcal{H}}^{p}_{-}(\Sigma ,D)
\cap {\mathcal{H}}^{p}_{+}(\Sigma ,D)\right )
=\text{dim}\,\text{Ker}\left (D:H^{1,p}({\mathcal{M}},{\mathcal{E}})
\to L^{p}({\mathcal{M}},{\mathcal{F}})\right )<\infty .
\end{equation}
At this point, the first part in Theorem \ref{thm:1} follows. There remain
\eqref{eq:1.9} and \eqref{eq:1.10}, which require refining further the analysis carried out
so far. This makes the object of the next couple of steps.
\end{definition3}
\begin{definition4} Let $[...]^{\circ }$ stand for the annihilator of
$[...]$ under the pairing $(u,v)=\int _{\Sigma }\langle u,v^{c}\rangle 
\,dS$.
We shall prove that the mapping
\begin{equation}\label{eq:2.18}\Phi :{\mathcal{H}}^{p}_{\pm }(\Sigma ,D^{*})\to \left 
[{\mathcal{H}}^{q}_{\pm }(\Sigma ,D)\right ]^{\circ },\,\,
\Phi (u):=i\sigma (D^{*};\nu )u,\quad 1/p+1/q=1,
\end{equation}
is an isomorphism for any $1