\documentclass[12pt]{amsart}
\headheight=8pt     \topmargin=0pt
\textheight=624pt   \textwidth=432pt
\oddsidemargin=18pt \evensidemargin=18pt
\usepackage{amssymb,amscd}
\usepackage{enumerate}
%\usepackage{showkeys}
%\usepackage[arrow,matrix]{xy}
%\CompileMatrices
\setcounter{tocdepth}{1}

%
% HAVE A NICE ``DATE:''
%
%\let\labelc\label\renewcommand\label[1]{\mar{#1}\labelc{#1}}
\setlength{\marginparwidth}{1.12in}
%\newcommand{\mar}[1]{{\marginpar{\textsf{#1}}}}
%\newcommand\datver[1]{\def\datverp%
% {\par\boxed{\boxed{\text{Version: #1; Run: \today}}}}}
%\datver{3.5; Revised: Jan 21, 2004 by B.}
%
%\newcommand{\mnote}[1]{\marginpar{\tiny\em #1}}
%
%
% OPERATORS:
%
\newcommand{\ie}{{\em i.e., }}
\newcommand{\Cal}{\mathcal}
\newcommand{\mfk}{\mathfrak}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\CI}{\mathcal{C}^\infty}
\newcommand{\CIc}{{\mathcal C}^{\infty}_{\text{c}}}
\newcommand{\pa}{\partial}
\newcommand\<{\langle}
\renewcommand\>{\rangle}

\newcommand{\maC}{\mathcal C}
\newcommand{\maD}{\mathcal D}
\newcommand{\maE}{\mathcal E}
\newcommand{\maF}{\mathcal F}
\newcommand{\maH}{\mathcal H}
\newcommand{\maI}{\mathfrak I}
\newcommand{\maL}{\mathcal L}
\newcommand{\maP}{\mathcal P}
\newcommand{\maR}{\mathcal R}
\newcommand{\maO}{\mathcal O}
\newcommand{\maW}{\mathcal W}
\newcommand{\maX}{\mathcal X}

\newcommand\mW{\maW}
\newcommand\mL{\maL}
\newcommand\mH{\maH}
\newcommand\mP{\maP}

\newcommand{\CC}{\mathbb C}
\newcommand{\NN}{\mathbb N}
\newcommand{\PP}{\mathbb P}
\newcommand{\RR}{\mathbb R}
\newcommand{\Sz}{\mathbb S}
\newcommand{\ZZ}{\mathbb Z}

\newcommand{\scal}{\operatorname{scal}}
\newcommand{\dvol}{\mathop{{\rm dvol}}}
\let\ep\epsilon

%deleted

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% new counter for enumerate %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\theenumi}{\alph{enumi}}
\renewcommand{\labelenumi}{\rm {({\theenumi})}}
\renewcommand{\labelenumii}{(\roman{enumii})}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%
%%%% PSEUDODIFFERENTIAL OPERATORS:
%%%%
\newcommand{\PS}[1]{\Psi^{#1}(\GR; E)}
\newcommand{\tPS}[1]{\Psi^{#1}(\GR)}
\newcommand{\tPSI}[1]{\Psi^{#1}(M,\GR)}
\newcommand{\PSI}[1]{\Psi^{#1}(M,\GR; E)}
\newcommand{\PSEF}[1]{\Psi^{#1}(\GR; E,F)}
\newcommand{\Dir}{{\not \!\!D}} %%% old: {D\!\!\!\!\!\!/}
\newcommand{\Cliff}{{\rm Cliff}}
\newcommand{\Lie}{\operatorname{Lie}}
\newcommand{\fgg}{\mathfrak g}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                              %
%%  MACROS FOR THIS MANUSCRIPT  %
%%                              %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% NEW FOR THIS MANUSCRIPT
%%
%% Lipschitz and Sobolev spaces
%%
\newcommand\comment[1]{}
\newcommand\Wkp{W^{k,p}}
\newcommand\WkpM{W^{k,p}(M_0)}
\newcommand\WkpO{W^{k,p}(\Omega_0)}
\newcommand\Wkinf{W^{k, \infty}}
\newcommand\WkinfM{W^{k, \infty}(M_0)}
\newcommand\WkinfO{W^{k, \infty}(\Omega)}
\newcommand\Cka{C^{k, \alpha}}
\newcommand\CkaM{C^{k, \alpha}(M_0)}
\newcommand\CkaO{C^{k, \alpha}(\Omega)}

\newcommand\smb{\nu^A}
\newcommand{\VV}{\mathcal V}
\newcommand{\WW}{\mathcal W}
\newcommand{\Diff}[1]{{\rm Diff}(#1)}
\newcommand{\DiffV}[1]{{\rm Diff}^{#1}_{\VV}}
\newcommand{\cl}{\rm cl}
\newcommand{\lzmn}{L^{2}(M_{0})}
\newcommand{\maHa}[1]{H^{#1}(M_{0};T)}
\newcommand{\maHk}{\maHa{k}}
\newcommand{\maHn}{\maHa{0}}
\newcommand{\maHs}{\maHa{s}}
\newcommand{\maHu}{\maHa{\infty}}
\newcommand{\nzn}{\NN_{0}}
\newcommand{\hTa}[1]{\maH_{#1}(T)}
\newcommand{\hTk}{\hTa{k}}
\newcommand{\hTn}{\hTa{0}}
\newcommand{\hTs}{\hTa{s}}
\newcommand{\vol}{\operatorname{vol}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand\Kond[2]{{\mathcal K}^{#1}_{#2}}
%\newcommand\normK[3]{\|#1\|_{\Kond{#2}{#3}}}

% from the submanifolds section
\def\II{{\rm II}}
\newcommand\codim{\operatorname{codim}}


%
% THEOREM TYPE ENVIRONMENTS:
%
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notations}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\newtheorem{examples}[theorem]{Examples}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}



%\dedicatory\datverp

\title[Sobolev spaces]{Sobolev spaces
on Lie manifolds and polyhedral domains  }

\author[B. Ammann]{Bernd Ammann} \address{Bernd Ammann, 
Math. Science Res. Inst.,
17 Gauss Way, Berkeley, CA 94720-5070, USA, 
{\rm http://www.math.uni-hamburg.de/home/ammann}}

\author[A. Ionescu]{Alexandru D. Ionescu} \address{Alexandru D. Ionescu,
University of Wisconsin, Department of Mathematics, Madison, WI 53705, USA}
\email{ionescu@math.wisc.edu}

\author[V. Nistor]{Victor Nistor} \address{Victor Nistor, Pennsylvania State
University, Math. Dept., University Park, PA 16802, USA}
\email{nistor@math.psu.edu}

\thanks{Ammann was partially supported by MSRI's NSF grant DMS-9810361.
Ionescu was supported in part by NSF Grant No. 0100021. Nistor
was partially supported by NSF Grants DMS 0200808 and DMS 0209497.
Manuscripts available from {\bf
http:{\scriptsize//}www.math.psu.edu{\scriptsize/}nistor{\scriptsize/}}
.}

\begin{abstract}
We study Sobolev spaces on Lie manifolds, which we define as a
class of manifolds described by vector fields (see Definition
\ref{def.unif.str}).  The class of Lie manifolds includes the
Euclidean spaces $\RR^n$, asymptotically flat manifolds,
conformally compact manifolds, and manifolds with cylindrical and
polycylindrical ends. As in the classical case of $\RR^n$, we
define Sobolev spaces using derivatives, powers of the Laplacian,
or a suitable class of partitions of unity. We extend the basic
results about Sobolev spaces on Euclidean spaces to the setting of
Lie manifolds. These results include the definition of the trace
map, a characterization of its range, the extension theorem, the
density of smooth functions, and interpolation properties. One of
the main motivations is that, in the examples we have studied so
far, the totally-characteristic Sobolev spaces on polyhedral
domains identify with Sobolev spaces on suitable Lie manifolds
with boundary. The analysis we develop may be useful for solving
certain types of non-linear partial differential equations on
non-compact manifolds that appear, for instance, in Einstein's
constraint equations. We also sketch two applications, one to the
Yamabe functional and one to the regularity of boundary value
problems on polyhedral domains.
\end{abstract}


\maketitle \tableofcontents

\section*{Introduction}


Function spaces play a central role in Analysis and are often used
in practical applications of mathematics. In many of these
applications the domains are not smooth, which has lead to work
under Lipschitz-type conditions, as in \cite{JerisonKenig,
MitreaTaylor, Necas, Verchota} etc. These papers have extended
many classical results on function spaces to Lipschitz domains,
but have also revealed the limitations of the standard Sobolev
spaces. For example, the usual regularity theorems for solutions
of elliptic differential equations on smooth domains \cite{Evans,
Taylor1} do not hold on Lipschitz domains. See  \cite{Costabel,
Dauge, Grisvard1, Grisvard2} in addition to the papers quoted
above.

To explain the question of regularity, denote by $\Delta$ the
Laplace operator and consider the following example. Let $\PP$ be
a polygon in the plane and let $u \in H^1(\PP)$ be a solution of
the Poisson problem $\Delta u = f \in \CI(\PP)$ with Dirichlet
boundary conditions (\ie $u = 0$ on $\pa \PP$). One can show
\cite{Grisvard1, Grisvard2} that there exists a constant $s_0$,
explicitly determined in terms of the angles of $\PP$, such that
$u \in H^s(\PP)$ for any $f \in \CI(\PP)$ and any $s < s_0$, but
not better. That is, $u \not \in H^{s_0}(\PP)$ for a suitable
choice of $f \in \CI(\PP)$. This is in sharp contrast with the
case of smooth boundary, in which case $u$ is smooth whenever $f$
is smooth. A deep study of these issues in the setting of
Lipschitz domains can be found in the papers of Jerison and Kenig
\cite{JerisonKenig} and Mitrea and Taylor \cite{MitreaTaylor}, and
in the other papers quoted there.

The loss of regularity in the Poisson problem mentioned above can
be avoided, however, if one considers a different class of Sobolev
spaces on the polygon $\PP$ \cite{BNZ, BNZ2, MitreaNistor, NP}.
These Sobolev spaces, sometimes denoted $H_b^m(\PP)$ are the so
called {\em totally characteristic Sobolev spaces} and were used
by many researchers, see \cite{BabuGuo, BabuPA, Costabel, Dauge,
Kondratiev, Mazya, meaps, MelroseMendoza, NP, Schulze98} and the
references therein.
%% V3.3
The definition of the spaces $H_b^m(\PP)$ uses the distance
function $\rho(x)$ from $x \in \PP$ to the vertices of $\PP$:
\begin{equation}\label{def.t.c.a0}
    H^m_b(\PP) := \{ u \in L^2_{loc}(\PP), \
    \rho^{|\alpha|-1} D^{\alpha} u \in L^2(\PP), \ \
    |\alpha| \le m \}.
\end{equation}


It is one of the purposes of this paper to study totally
characteristic Sobolev spaces on  polyhedral domains and to extend
some of the main results in the theory of classical Sobolev spaces
to this setting. This will help clarify the role of the totally
characteristic Sobolev spaces in the study of boundary value
problems on polyhedral domains, in particular. More specific
results in three dimension will be included in \cite{BNZ2}.

Our approach to Sobolev spaces on polyhedral domains is to reduce
their study to that of Sobolev spaces on certain non-compact
manifolds with boundary. These non-compact manifolds are obtained
from our polyhedral domain by conformally changing the metric with
a factor that blows up at the faces of codimension $\ge 2$. The
resulting non-compact manifolds are ``Lie manifolds with
boundary,'' (See Definition \ref{def.unif.str} and Subsection
\ref{ssec.PD} for definitions.)

Lie manifolds were first introduced informally in \cite{meicm,
MelroseScattering}. Their definition was formalized in
\cite{aln1}, where several simple but basic properties of these
manifolds were proved in a general setting. (Lie manifolds were
called ``manifolds with a Lie structure at infinity'' in that
paper.) In addition to the non-compact manifolds that arise from
polyhedral domains, other examples of Lie manifolds include the
Euclidean spaces $\RR^n$, manifolds that are Euclidean at
infinity, conformally compact manifolds, manifolds with
cylindrical and polycylindrical ends, and asymptotically
hyperbolic manifolds. These classes of non-compact manifolds are
relevant in many problems in Mathematical Physics and
Computational Sciences, such as domain decomposition methods and
the Finite Element Method, quasi-linear parabolic equations,
Yamabe's problem, Einstein's equations, and the positive mass
theorem. Classes of Sobolev spaces on non-compact manifolds have
been studied in many papers, of which we mention only a few
\cite{anderson:p03, akutagawa.botvinnik, JeffresLoya,
KlainermanSel, Lauter, LauterMoroianu, Maniccia,
MelroseScattering, MaMeAsian, Mazzeo, SchroheSI, SchroheFC,
TataruH} in addition to the works mentioned before.

A large part of the technical material in this paper is devoted to
the study of Sobolev spaces on Lie manifolds, with or without
boundary. The first three sections of the paper are more
elementary and we have attempted to make them essentially
self-contained. We begin in Section \ref{sec.LI} with a review of
the definition of a structural Lie algebra of vector fields $\VV$
on a manifold with corners $M$. This Lie algebra of vector fields
will provide the derivatives appearing in the definition of the
Sobolev spaces. Then we define Lie manifolds.
The interior $M_0$ of $M$ is by definition a {\em Lie manifold}.
It turns out that $M_0$ carries a complete metric $g$, unique up
to Lipschitz equivalence (some authors use the term
``quasi-isometric'' to describe two Lipschitz equivalent metrics).

In Section \ref{sec.LI} we define the Sobolev spaces $W^{s,
p}(M_0)$ on a Lie manifold $M_0$, where the superscripts have the
following possible ranges: either $s \in \ZZ_+$ and $1 \le p \le
\infty$ or $s \in \RR$ and $1 < p < \infty$. The main goal of this
paper is to study the spaces $W^{s, p}(M_0)$. We first define the
spaces $W^{s, p}(M_0)$, $s \in \ZZ_+$ and $1 \le p \le \infty$, by
differentiating with respect to vector fields in $\VV$. This
definition is in the spirit of the definition of Sobolev spaces on
$\RR^n$. Then we prove two alternative definitions of these
Sobolev spaces, either using a suitable class of partitions of
unity (as in \cite{Shubin, StrichartzM} for example), or as the
domains of the powers of the Laplace operator (for $p = 2$). We
also consider these spaces on open subsets $\Omega_0 \subset M_0$.
The spaces $W^{s, p}(M_0)$, for $s \in \RR$, $1 < p < \infty$ are
defined by interpolation and duality or, alternatively, using
partitions of unity. In Section \ref{sec.MB}, we discuss domains
$\Omega_0$ whose boundary $\pa \Omega_0$ is a (smooth) Lie
submanifold of $M_0$.

We extend several of the classical results on Sobolev spaces to
the setting of the spaces $W^{s, p}(M_0)$. These results include
the density of smooth, compactly supported functions, the
Gagliardo-Nirenberg-Sobolev inequalities, the extension theorem,
the trace theorem, the characterization of the range of the trace
map in Hilbert space case ($p = 2$), and the Rellich-Kondrachov
compactness theorem. Some of these results follow from the
analogous results for manifolds with bounded geometry. We conclude
the first three, more elementary sections, with an application to
the regularity of boundary value problems on polyhedral domains,
Theorem \ref{thm.reg1}.

The last three sections are slightly less elementary. In
particular, we no longer attempt to make them self-contained. Some
applications to geometry (the Yamabe problem) are given in
Section~\ref{sec.appl}. In Section \ref{sec.SM} we discuss the
geometric results needed on Lie submanifolds, most importantly,
the global tubular neighborhood theorem. Finally, in Section
\ref{sec.PO} we discuss the continuity of pseudodifferential
operators acting on the spaces $W^{s, p}(M_0)$. The class of
pseudodifferential operators that we consider is $\Psi_{1, 0,
\VV}^{\infty}(M_0)$. This algebra was introduced in \cite{aln2} as
a quantization of the algebra of differential operators
$\DiffV{*}(M)$ generated by the structural Lie algebra of vector
fields $\VV$ defining the Lie structure on $M_0$, thus solving a
conjecture from \cite{meicm}. In particular, we obtain a
description of the spaces $W^{s, p}(M_0)$, $s \ge 0$, $1 < p <
\infty$, as the domain of any elliptic operator $P\in \Psi_{1, 0,
\VV}^s(M_0)$.

We include few concrete examples of manifolds with a Lie structure
at infinity besides those needed to treat polyhedral domains. The
reader can find more examples in \cite{LN1} or in \cite{aln1}, for
example.


%% V3.3 I added Dahmen
\subsubsection*{Acknowledgements:}\ We would like to thank W.
Dahmen,  R. Lauter, R. Strichartz, and A. Vasy for useful
discussions. The first named author wants to thank MSRI, Berkeley, CA 
for its hospitality. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Lie manifolds}
\label{sec.LI}

As mentioned already in the Introduction, our approach to the
study of totally-characteristic Sobolev spaces on polyhedral
domains is based on their relation to Sobolev spaces on Lie
manifolds with boundary. For the convenience of the reader, we
recall the definition of a Lie manifold and some basic results.
This Section is to a large extent based on \cite{aln1}. (Note that
what we are calling here ``Lie manifolds'' were called ``manifolds
with a Lie structure at infinity'' in \cite{aln1}.)

We shall treat Lie manifolds as well as Lie submanifolds (of Lie
manifolds) in Section \ref{sec.MB}.


\subsection{Definition}\ We need to recall first manifolds with
corners. By definition, every point $p$ in a manifold with corners
$M$ has a coordinate neighborhood diffeomorphic to $[0,\infty)^k
\times \RR^{n-k}$ such that the transition functions are smooth up
to the boundary.

We write $M_0$ for the interior of $M$, and $\partial M=
M\smallsetminus M_0$ for the boundary, \ie $\partial M$ is the
union of all boundary faces of dimension $0$ to $n-1$. In the
sequel, by a {\em manifold} we shall always understand a
$C^\infty$-manifold {\em possibly with corners}, whereas a {\em
smooth manifold} is a $C^\infty$-manifold {\em without corners}.

As we shall see below, a Lie manifold is described by a Lie
algebra of vector fields satisfying certain conditions. We now
discuss some of these conditions.


\begin{definition}\label{def.str.LA}\
A subspace $\VV\subseteq \Gamma(M, TM)$ of the Lie algebra of all
smooth vector fields on $M$ is said to be a {\em structural Lie
algebra of vector fields on $M$} provided that the following
conditions are satisfied:
\begin{enumerate}[(i)]
\item\ $\VV$ is closed under the Lie bracket of vector fields;
\item\ every $V\in\VV $ is tangent to all hyperfaces of $M$;
\item\ $\CI(M)\VV = \VV$; and
\item\ each point $p\in M$ has a neighborhood $U_p$ such that
\begin{equation*}
    \VV_{U_p} := \{X|_{\overline{U}_p}\,|X\in \VV\} \simeq
    C^\infty(\overline{U}_p)^k,
\end{equation*}
that is, $\VV_{U_p}$ is a free $C^\infty(\overline{U}_p)$-module
of dimension $k$, for some $k$.
\end{enumerate}
\end{definition}

\noindent The condition (iv) in the definition above can be
reformulated as follows:
\begin{enumerate}[(i')]
\setcounter{enumi}3
\item For every
$p \in M$ there exist a neighborhood $U_p \subset M$ of $p$ and
vector fields $X_1, X_2, \ldots, X_k \in \VV$ with the property
that for any $Y \in \VV$, there exist functions $f_1, \ldots, f_k
\in \CI(M)$, uniquely determined on $U_p$, such that
\begin{equation}\label{eq.local.basis}
    Y = \sum_{j = 1}^k f_j X_j \quad \text{on} \; U_p.
\end{equation}
\end{enumerate}

Here are some examples of structural Lie algebras of vector
fields. If $F \subset TM$ is a sub-bundle of the tangent bundle of
a smooth manifold (so $M$ has no boundary) such that $\VV_F :=
\Gamma(M, F)$ is closed under the Lie bracket, then $\VV_F$ is a
structural Lie algebra of vector fields. Another example arises
from manifolds with boundary and is related to the totally
characteristic Sobolev spaces defined on an angle, as in the
introduction. More precisely, let $M$ be a manifold with boundary
and let $\VV_b$ be the space of vector fields on $M$ {\em tangent}
to the boundary of $M$. Then $\VV_b$ is a structural Lie algebra
of vector fields. See \cite{me81, meaps} and Subsection
\ref{ssec.PD}.


\begin{definition}\label{def.unif.str}\
A {\em Lie structure at infinity} on a smooth manifold $M_0$ is a
pair $(M, \VV)$, where $M$ is a compact manifold, possibly with
corners, and $\VV \subset \Gamma(M, TM)$ is a structural Lie
algebra of vector fields on $M$ with the following properties:
\begin{enumerate}[(i)]
\item\ $M_0 = M \smallsetminus \pa M$, the interior of $M$, and
\item\ If $p \in M_0$, then any local basis of $\VV$ in a
neigborhood of $p$ is also a local basis of the tangent space to
$M_0$. (In particular, the constant $k$ of Equation
\eqref{eq.local.basis} equals the dimension of $M_0$.)
\end{enumerate}
\end{definition}


A {\em manifold with a Lie structure at infinity} (or, simply, a
{\em Lie manifold}) is a manifold $M_0$ together with a Lie
structure at infinity $(M, \VV)$ on $M_0$. We shall sometimes
denote a Lie manifold as above by $(M_0, M, \VV)$, or, simply, by
$(M, \VV)$, because $M_0$ is determined as the interior of $M$.

We include only a few examples of Lie manifolds. The reader can
find more examples in \cite{MelroseScattering}, from where these
examples were borrowed or in \cite{LN1, aln1}.

\begin{examples}\label{ex1}\ \\[-5mm]
\begin{enumerate}
\item Take $\VV_b$ to be the set of all vector fields tangent to
all faces of a manifold with corners $M$. Then $(M, \VV_b)$ is a
Lie manifold. We shall say following Melrose's terminology that
$M_0 = M \smallsetminus \pa M$ is endowed with the $b$-structure
at infinity.
\item Take $\VV_0$ to be the set of all vector fields vanishing on
all faces of a manifold with corners $M$. Then $(M, \VV_0)$ is a
Lie manifold. We shall say following Melrose's terminology that
$M_0 = M \smallsetminus \pa M$ is endowed with the zero-structure
at infinity.
\end{enumerate}
\end{examples}


\begin{remark}\label{rem.A}\
Let us observe, that Conditions (iii) and (iv) of Definition
\ref{def.str.LA} are equivalent to the condition that $\VV$ be a
projective $\CI(M)$-module. Thus, by the Serre-Swan theorem
\cite{Karoubi}, there exists a vector bundle $A \to M$, unique up
to isomorphism, such that $\VV = \Gamma(M, A)$. Since $\VV$
consists of vector fields, that is $\VV \subset \Gamma(M, TM)$, we
also obtain a natural vector bundle morphism $\varrho : A \to M$,
called the {\em anchor map}. The Condition (ii) of Definition
\ref{def.unif.str} is then equivalent to the fact that $\varrho$
is an isomorphism $A\vert_{M_0} \simeq TM_0$ on $M_0$. We will
take this isomorphism to be an identification, and thus we can say
that $A$ is {\em an extension} of $TM_0$ to $M$ (that is, $TM_0
\subset A$).
\end{remark}


\subsection{Riemannian metric\label{ssec.Rm}}
Let $(M_0, M,\VV)$ be a Lie manifold. By definition, {\em a
Riemannian metric on $M_0$ compatible with the Lie structure at
infinity $(M, \VV)$} is a metric $g$ such that for any $p \in M$,
we can choose the basis $X_1,\dots,X_k$ in
Definition~\ref{def.str.LA}, (iv') and \eqref{eq.local.basis} to
be orthonormal with respect to this metric everywhere on $U_p$.
(Note that this condition is a restriction only for $p \in \pa M
:= M \smallsetminus M_0$.) Alternatively, we will also say that
$(M_{0},g_{0})$ is a {\em Riemannian Lie manifold}. Any Lie
manifold carries a compatible Riemannian metric, and any two
compatible metrics are bi-Lipschitz to each others.

\begin{remark}\ Using the language of Remark \ref{rem.A}, $g$ is a
compatible metric on $M_0$ if, and only if, there exists a metric
on the vector bundle $A \to M$ which restricts to $g$ on $TM_0
\subset A$.
\end{remark}

The geometry of a Riemannian manifold $(M_{0},g_{0})$ with a Lie
structure $(M,\VV)$ at infinity has been studied in \cite{aln1}. For
instance, $(M_{0},g_{0})$ is necessarily of infinite volume and
complete.  Moreover, all the covariant derivatives of the
Riemannian curvature tensor are bounded. Under additional mild
assumptions, we also know that the injectivity radius is bounded
from below by a positive constant, \ie\ $(M_{0},g_{0})$ is of
bounded geometry. (A {\em manifold with bounded geometry} is a
Riemannian manifold with positive injectivity radius and with
bounded covariant derivatives of the curvature tensor, see
\cite{Shubin} and references therein).

On a Riemannian Lie manifold $(M_0, M, \VV)$, the exponential map
$\exp_p : TM_0 \to M_0$ is well-defined for all $p \in M_0$ and
extends to a differentiable map $\exp_p : A_p \to M$ depending
smoothly on $p \in M$. A convenient way to introduce the
exponential map is via the geodesic spray, as done in \cite{aln1}.
Similarly, any vector field $X \in \VV = \Gamma(M, A)$ is
integrable and will map any (connected) face of $M$ to itself. The
resulting diffeomorphism of $M_0$ will be denoted $\psi_X$.

{\em We assume from now on that $r_{\mathrm{inj}}(M_0)$, the
injectivity radius of $(M_{0}, g_{0})$, is positive.}


\subsection{$\VV$-differential operators}
We are especially interested in the analysis of the differential
operators generated using only derivatives in $\VV$. Let
$\DiffV{*}(M)$ be the algebra of differential operators on $M$
generated by multiplication with functions in $\CI(M)$ and by
differentiation with vector fields $X \in \VV$. The space of order
$m$ differential operators in $\DiffV{*}(M)$ will be denoted
$\DiffV{m}(M )$. A differential operator in $\DiffV{*}(M)$ will be
called a $\VV$-differential operator.

We can define $\VV$-differential operators acting between sections
of smooth vector bundles $E, F \rightarrow M$, $E, F \subset M
\times \CC^N$ by
\begin{equation}\label{vectorvalued}
    \DiffV{*}(M ;E,F) := e_F M_N(\DiffV{*}(M )) e_E\,,
\end{equation}
where $e_E, e_F \in M_N(\CI(M))$ are the projections onto $E$ and,
respectively, $F$. It follows that $\DiffV{*}(M; E,E) = :
\DiffV{*}(M; E)$ is an algebra. It is also closed under taking
adjoints of operators in $L^2(M_0)$, where the volume form is
defined using a compatible metric $g$ on $M_0$.


\subsection{Lie manifolds with boundary}
One of the main motivation for this work is to study Sobolev
spaces on polyhedral domains. We shall do that by reducing their
study to that of Sobolev spaces on ``Lie manifolds with
boundary,'' a class of manifolds with boundary that we introduce
below.

%% V3.3 See the next two-three paragraphs
To understand the following constructions, let us take a closer
look at the the local structure of the Sobolev space $H^m_b(\PP)$
associated to a polygon $\PP$ (recall \eqref{def.t.c.a0}).
Consider $\Omega := \{ (r,\theta)\,|\, 0<r<r_0, 0 < \theta <
\alpha \}$, which models an angle of $\PP$. Then the {\em totally
characteristic Sobolev spaces} associated to $\Omega$,
$H_b^m(\Omega)$, can alternatively be described as
\begin{equation}\label{def.t.c.a}
    H^m_b(\Omega) := \{ u \in L^2_{loc}(\Omega), \
    r^{-1}(r\pa_r)^i \pa_\theta^ju \in L^2(\Omega), \ \
    i + j \le m \}.
\end{equation}
An important point of the above definition is that first the angle
$\Omega$ was desingularized and then a different basis of vector
fields on the desingularization was used instead of the standard
basis in the definition of the usual Sobolev spaces. For example,
in the case above of the angle $\Omega$, the basis $r\pa_r$ and
$\pa_\theta$ was used instead of the usual basis $\pa_x$ and
$\pa_y$. This underscores the importance of vector fields in our
approach, which owes to the work of several authors. See
\cite{Cordes, CO1, CO2, meicm, MelroseScattering, Mazya, Mazzeo,
NP, SchroheSI, Schulze98, Shubin, Stein, StrichartzM, StrichartzB}
and the references therein.

Let $N \subset M$ be a submanifold with corners of codimension one
of $M$ (see Section \ref{sec.SM}). Recall that this implies that
$N$ is transverse to all faces of $M$. We shall say that $N$ is a
{\em regular} submanifold of $(M, \VV)$ if we can choose a tubular
neighborhood $V$ of $N_0 := N \smallsetminus \pa N = N \cap M_0$
in $M_0$ and a compatible metric $g$ on $M_0$ that restricts to a
product-type metric on $V \simeq (\pa N_0) \times (-\varepsilon_0,
\varepsilon_0)$. (In Section \ref{sec.SM}, we shall show that
every tame
submanifold of codimension one
is regular; in turn, this will give an easy, geometric criterion
to decide when a codimension one submanifold of $M$ is regular.)

Let $\Omega \subset M$ be an open subset. We say that $\Omega$ is
a {\em Lie domain} in $M$ if and only if $\pa \Omega = \pa
{\overline{\Omega}}$ and $\pa \Omega$ is a regular submanifold of
$M$. Let $\Omega_0 = \Omega \cap M_0$. Then $\pa \Omega_0 = (\pa
\Omega) \cap M_0$ is a smooth submanifold of codimension one of
$M_0$.

\begin{definition}\label{def.man.wb}\
A {\em Lie manifold with boundary} is a triple $(\Omega_0,
\Omega', \VV')$, where $\Omega_0$ is a smooth manifold with
boundary, $\Omega'$ is a compact manifold with corners containing
$\Omega_0$ as an open subset, and $\VV'$ is a Lie algebra of
vector fields on $\Omega'$ with the property that there exists a
Lie manifold $(M_0, M, \VV)$, a Lie domain $\Omega \subset M$ in
$M$ and a diffeomorphism $\phi : \Omega' \to \Omega$ such that
$\phi(\Omega_0) = \Omega \cap M_0$ and $\phi^*(\VV\vert_{\Omega})
= \VV'$.
\end{definition}

Note that if $(\Omega_0, \Omega, \VV)$ is a Lie manifold with
boundary, then $\Omega_0$ is determined by $(\Omega, \VV)$, so we
can write $(\Omega, \VV)$ instead of $(\Omega_0, \Omega, \VV)$.


\subsection{Polyhedral domains\label{ssec.PD}}
We now discuss in an example the relation between polyhedral
domains and Lie manifolds with boundary. Let us consider
\begin{equation*}
    \Delta_n := \{(x_1, \ldots, x_n) \in \RR^n, x_j
    \ge 0, \sum x_j \le 1\},
\end{equation*}
the unit simplex in $\RR^n$. To $\Delta_n$ we now associate a Lie
manifold with boundary $(\Sigma(\Delta_n), \VV(\Delta_n))$,
together with a ``desingularization'' map
\begin{equation*}
    \kappa_n : \Sigma(\Delta_n) \to \Delta_n,
\end{equation*}
satisfying $\kappa_n (\pa \Sigma(\Delta_n)) \subset \pa\Delta_n$.
Let $S \subset \Delta_n$ be the set of points not belonging to a
face of codimension $\ge 2$. The map $\kappa_n$ will turn out to
be a bijection between $\kappa_n^{-1}(S)$ and $S$. We shall
proceed by induction as follows.

Let $\PP$ be the angle $\{ 0 \le \theta \le \alpha \}$, a closed
subset of $\RR^2$, where $(r, \theta)$ are the polar coordinates
in the plane. We define its canonical desingularization by
$\Sigma(\PP) := [0,\infty) \times [0, \alpha]$, which maps
surjectively to $\PP$ by $\kappa(r, \theta) =  (r\cos \theta,
r\sin \theta)$. The Lie algebra of vector fields on $\Sigma(\PP)$
is given by
\begin{equation}
    \{X \in \Gamma(T\Sigma(\PP)),\; X(0, \theta)\
    \text{ tangent to } \{0\} \times [0, \alpha]\, \}
\end{equation}
We can realize $\Sigma(\PP)$ as a Lie domain in the manifold with
boundary $[0, \infty) \times S^1$, by realizing $[0, \alpha]$ as a
subset of the unit circle $S^1$. On $[0, \infty) \times S^1$ we
then consider all vector fields tangent to the boundary.

The desingularization $\Sigma(\Delta_2)$ of the unit triangle
$\Delta_2$ is obtained as follows. First we desingularize each
angle with the opposite face removed. Then we glue these
desingularizations using the desingularization map $\kappa$. This
yields, up to a diffeomorphism a hexagon. Let $e_1, e_2, e_3$ be
the three non-intersecting edges of this hexagon corresponding to
the vertices of $\Delta_2$.  These faces are the ones that, under
the desingularization map will go to the three vertices of the
triangle. Then $\VV(\Delta_2)$ consists of vector fields tangent
to the faces $e_1, e_2, e_3$.

Assume now that $(\Sigma(\Delta_n), \VV(\Delta_n), \kappa_n)$ were
constructed, and let us construct $(\Sigma(\Delta_{n+1}),
\VV(\Delta_{n+1}), \kappa_{n+1})$. We first construct an analogous
desingularization of
\begin{equation*}
    C\Delta_n := \{(x_0, x_1, \ldots, x_n)
    \in \RR^{n+1}, x_j \ge 0\}.
\end{equation*}
The space $C\Delta_n$ is the ``cone'' over $\Delta_n$. We shall
use next the alternative description of $\Delta_n$ as
\begin{equation*}
    \Delta_n := \{(y_0, y_1, \ldots, y_n) \in
    \RR^{n+1}, y_j \ge 0, \sum y_j = 1\}.
\end{equation*}
The desingularization of $C\Delta_n$ is, by definition, $[0,
\infty) \times \Sigma(\Delta_n)$, with desingularization map
$\kappa(r, z) = r \kappa_n(z)$. The Lie algebra of vector fields
on this desingularization is
\begin{equation*}
    \VV = \{(X, Y) \in \Gamma(T[0, \infty)) \times
    \Gamma(T\Sigma(\Delta_n)), \, X(0) = 0 \text{
    and } Y \in \VV(\Delta_n)\}.
\end{equation*}
In other words, if we split the tangent space to $[0, \infty)
\times \Sigma(\Delta_n)$ into the direct sum of the vector bundles
tangent to the two factors, then on the first component we get
vector fields tangent to the boundary (``tangent to the boundary''
means ``vanishing at the boundary'' in this case of $[0, \infty)$)
and on the second component we simply get vectors in the
structural Lie algebra of vector fields $\VV(\Sigma(\Delta_n))$
corresponding to the desingularization of $\Delta_n$.

The desingularization $\Sigma(\Delta_{n+1})$ of $\Delta_{n+1}$ is
obtained by gluing the desingularizations of the cones
corresponding to each of the vertices, these cones being obtained
by removing the opposite face to the given vertex, one vertex at a
time.

This rather complicated construction is justified by the following
simple proposition.


\begin{proposition}\label{prop.tc}\
Let $h$ be the standard euclidean metric on $\Delta_n$. Let
$\rho(x)$ be the distance from the point $x$ to the set of points
belonging to a face of codimension $\ge 2$. Then $\rho^{-2}h$ is
Lipschitz equivalent to any compatible metric $g$ on the interior
of $\Sigma(\Delta_n)$, that is, there exists $C > 0$ such that
\begin{equation*}
    C^{-1}g(\xi) \le \rho^{-2}h(\xi) \le C g(\xi)
\end{equation*}
for any tangent vector $\xi \in T\Delta_n$ tangent to an interior
point of $\Delta_n$. In particular,
\begin{equation*}
    C^{-n/2}d\vol_g \le \rho^{-n}d\vol_h \le C^{n/2}
    d\vol_g.
\end{equation*}
\end{proposition}


\begin{proof}\ By induction. For $n = 2$, this follows right away
from the definition of totally characteristic Sobolev spaces on a
triangle. For the induction step, denote by $h_n$ the metric on
$\Delta_n$. We can cover $\Delta_n$ with open sets $V_j$, such
that the closure $\overline{V}_j$ contains exactly one vertex of
$\Delta_n$. It is enough to prove our statement on
$\overline{V}_j$, for all $j$. This will change the function
$\rho$ on $\overline{V}_j$, but the metric $\rho^{-2}h_n$ will be
in the same Lipschitz equivalence class, because the function
$\rho$ changes only by a factor that is bounded from above and
bounded from zero (\ie it takes values in a compact interval of
$(0, \infty)$).

We then consider the same construction on $C\Delta_{n-1}$, the
cone over the simplex $\Delta_{n-1}$. Note that the function
$\rho$ does not change by considering the bigger set
$C\Delta_{n-1}$ instead of $\overline{V}_j$. Let $r$ denote the
distance to the vertex of the cone $C\Delta_{n-1}$. After a
suitable translation, we can (and will) assume that the vertex of
this cone is the origin. The metric $r^{-2}h_n$ makes the interior
of $C\Delta_{n-1}$ isometric to the interior of $\Delta_{n-1}
\times \RR$ with the product metric $\tilde{h}_{n-1} \times
(dt)^2$, $t \in \RR$, where $\tilde{h}_{n-1}$ is the metric on
$\Delta_{n-1}$ obtained by mapping $\Delta_{n-1}$ to the unit
sphere via the map $x \to x/\|x\|$. This metric is Lipschitz
equivalent to $h_{n-1}$.

Let $\tilde{\rho}(x)$ be the distance from $x \in C\Delta_{n-1}$
to the subset of points belonging to a face of codimension $\ge 2$
in the metric $h_{n-1} \times (dt)^2$. The induction hypothesis is
that $\tilde{\rho}^{-2} (h_{n-1} \times (dt)^2)$ is Lipschitz
equivalent to $g$. Hence
\begin{equation*}
    (r \tilde{\rho})^{-2} h_n =
    \tilde{\rho}^{-2} \tilde{h}_{n-1} \times (dt)^2
\end{equation*}
is Lipschitz equivalent to $g$. To prove our result, it is enough
then to show that $f(x) := r(x) \tilde{\rho}(x)/\rho(x)$ is
bounded from above and bounded from zero. Let us observe that
$\rho(tx) = t \rho(x)$, $r(tx) = tr(x)$, and $\tilde{\rho}(tx) =
\tilde{\rho}(x)$, for all $t \in (0, \infty)$ for which this makes
sense. Consequently, $f(tx) = f(x)$, whenever both sides are
defined. To prove that $f$ takes on values in a compact interval
contained in $(0, \infty)$, it is therefore enough to do that for
$\|x\|$ constant.  The result hence follows from the fact that $f$
is continuous $\neq 0$ on the set $\{\|x\| = c > 0 \} \cap
C\Delta_{n-1}$.
\end{proof}


For further reference, let us record there a consequence of the
proof of the above proposition.


\begin{corollary}\label{cor.sm.rho}\ We use the notation of
Proposition \ref{prop.tc}. There exists $f \in C(\Delta_n)$, $f$
smooth and $\neq 0$ on the interior of $\Delta_n$ such that
$f/\rho$ takes values in  a compact interval of $(0, \infty)$.
Moreover, $f^{-2}h$ is a compatible metric on the interior of
$\Delta_n$ for any such $f$.
\end{corollary}



\section{Sobolev spaces\label{sec.SS}}


In this section we discuss the Sobolev spaces on $M_0$ from an
elementary point of view, that is, without using
pseudodifferential operators. Our treatment is standard, following
\cite{Evans, Hebey2, Hebey1}. See also \cite{Aubin}. Some of these
elementary results simply follow from the fact that $M_0$ has
bounded geometry whenever its injectivity radius is positive.
These results include the density of smooth, compactly supported
functions, the identification of the $L^2$--Sobolev spaces with
the domains of suitable powers of $1 + \Delta$, and the
Gagliardo--Nirenberg--Sobolev Embedding theorem. We include these
results here for completeness and further references.
\smallskip

{\bf Conventions.}\ {\em Throughout the rest of this paper, $(M_0,
M, \VV)$ will be a fixed Lie manifold. We also fix a compatible
metric $g$ on $M_0$ (\ie, a metric compatible with the Lie
structure at infinity on $M_0$, see Subsection \ref{ssec.Rm}). By
$\Omega$ we shall denote an open subset of $M$ and
$\Omega_0=\Omega\cap M_0$. The letters $C$ and $c$ will be used to
denote possibly different constants that may depend only on
$(M_0,g)$ and its Lie structure at infinity $(M,\VV)$.}
\smallskip

We shall denote the volume form (or measure) on $M_0$ associated
to $g$ by $d\vol_g(x)$ or simply by $dx$, when there is no danger
of confusion. Also, we shall denote by $L^p(\Omega_0)$ the
resulting $L^p$-space on $\Omega_0$ (\ie defined with respect to
the volume form $dx$). These spaces are independent of the choice
of the compatible metric $g$ on $M_0$, but their norms, denoted by
$\|\; \cdot \;\|_{L^p}$, do depend upon this choice, although this
is not reflected in the notation. Also, we shall use the fixed
metric $g$ on $M_0$ to trivialize all density bundles. Then the
space $\maD'(\Omega_0)$ of distributions on $\Omega_0$ is defined,
as usual, as the dual of $\CIc(\Omega_0)$. The spaces
$L^p(\Omega_0)$ identify with spaces of distributions on
$\Omega_0$ via the pairing
\begin{equation*}
    \langle u, \phi \rangle = \int_{\Omega_0} u(x) \phi(x)dx,\,
    \quad \,\text{where }\, \phi \in \CIc(\Omega_0)
    \,\text{ and }\, u \in L^p(\Omega_0).
\end{equation*}


\subsection{Definition of Sobolev spaces using vector fields and
connections} Anticipating, let us mention that we will define the
Sobolev spaces $W^{s,p}(\Omega_0)$ in the following two cases:
$s\in \mathbb{Z}_+$, $1\leq p\leq \infty$, and arbitrary open sets
$\Omega_0$ or $s\in \RR$, $1<p<\infty$, and $\Omega_0=M_0$. In
fact, we will give several definitions and then show their
equivalence. The first definition is in terms of the Levi-Civita
connection $\nabla$ on $TM_0$. We shall denote also by $\nabla$
the induced connections on tensors (\ie on tensor products of
$TM_0$ and $T^*M_0$).


\begin{definition}[$\nabla$-definition of Sobolev
spaces]\label{def.Sobolev}\ Let $k \in \ZZ_+$, then the Sobolev
space $W^{k, p}(\Omega_0)$ is the space of distributions $u$ on
$\Omega_0 \subset M_0$ such that
\begin{equation}\label{eq.norm.Wkp'}
    \|u\|_{\nabla, \Wkp}^p : = \sum_{l = 1}^k \int_{\Omega_0}
    |\nabla^l u(x)|^pdx < \infty \,, \quad 1 \le p < \infty.
\end{equation}
For $p = \infty$ we change this definition in the obvious way,
namely we require that,
\begin{equation}\label{eq.norm.Wkinf'}
    \|u\|_{\nabla, \Wkinf} : = \sup |\nabla^l u(x)|
    < \infty \,, \quad 0 \le l \le k.
\end{equation}
\end{definition}

Let $p = 2$. When $\overline{\Omega} = \Sigma(\Delta_n)$, see
Subsection \ref{ssec.PD}, we can regard $\overline{\Omega}$ as a
subset of its double, which is a Lie manifold. This gives then the
totally characteristic Sobolev spaces on $\Delta_n$, denoted
$H^{k}_b(\Delta_n)$. Let $\rho(x)$ be the distance to the set of
points belonging to a face of codimension $\ge 2$ of $\Delta_n$,
as in Proposition~\ref{prop.tc}. Then
\begin{multline}
    H_b^k(\Delta_n) = W^{k, 2}(\Sigma(\Delta_n)) =
    \{u, \int_{\Delta_n} |\pa^\alpha u|^2
    \rho^{-n+2|\alpha|}d\vol_h
    < \infty, \quad |\alpha| \le k \} \\
    = \{u, \rho^{|\alpha|-n/2}\pa^\alpha u
    \in L^2(\Delta_n) = L^2(\Delta_n, dx),
    \quad |\alpha| \le k \}.
\end{multline}

We introduce an alternative definition of Sobolev spaces.


\begin{definition}[vector fields definition of Sobolev spaces]
Let again $k\in \ZZ_+$. Choose a finite set of vector fields
$\maX$ such that $\CI(M) \maX = \VV$. This condition is equivalent
to the fact that the set $\{X(p), X \in \maX\}$ generates $A_p$
linearly, for any $p \in M$. Then the system $\maX$ provides us
with the norm
\begin{equation}\label{eq.norm.Wkp}
    \|u\|_{\maX, \Wkp}^p : = \sum \|X_{1} X_{2} \ldots X_{l} u
    \|_{L^p}^p\,,\quad 1 \le p < \infty,
\end{equation}
the sum being over all possible choices of $0 \le l \le k$ and all
possible choices of not necessarily distinct vector fields $X_1,
X_2, \ldots, X_l \in \maX$. For $p = \infty$, we change this
definition in the obvious way:
\begin{equation}\label{eq.norm.Wkinf}
    \|u\|_{\maX, \Wkinf} : = \max \|X_{1} X_{2} \ldots X_{l} u
    \|_{L^\infty}\,,
\end{equation}
the maximum being taken over the same family of vector fields.
\end{definition}

In particular,
\begin{equation}
    W^{k, p}(\Omega_0) = \{u \in L^p(\Omega_0), Pu \in
    L^p(\Omega_0), \text{ for all } P \in \DiffV{k}(M) \}
\end{equation}

The following proposition shows that the second definition yields
equivalent norms.


\begin{proposition}\label{prop.Banach}\ The norms
$\|\;\cdot\;\|_{\maX, \Wkp}$  and $\|\;\cdot\;\|_{\nabla, \Wkp}$
are equivalent for any choice of the compatible metric $g$ on
$M_0$ and any choice of a system of the finite set $\maX$ such
that $\CI(M)\maX = \VV$. The spaces $W^{k,p}(\Omega_0)$ are
complete Banach spaces in the resulting topology. Moreover,
$W^{k,2}(\Omega_0)$ is a Hilbert space.
\end{proposition}


\begin{proof}\
As all compatible metrics $g$ are bi-Lipschitz to each others, the
equivalence classes of the $\|\;\cdot\;\|_{\maX, \Wkp}$-norms are
independent of the choice of $g$. We will show that for any choice
$\maX$ and $g$, $\|\;\cdot\;\|_{\maX, \Wkp}$  and
$\|\;\cdot\;\|_{\nabla, \Wkp}$ are equivalent. It is clear that
then the equivalence class of $\|\;\cdot\;\|_{\maX, \Wkp}$ is
independent of the choice of $\maX$, and the equivalence class of
$\|\;\cdot\;\|_{\nabla, \Wkp}$ is independent of the choice of
$g$.

We argue by induction in $k$. The equivalence is clear for $k=0$.
We assume now that the $W^{l,p}$-norms are already equivalent for
$l=0,\dots,k-1$. Observe that if $X,Y\in \VV$, then the Koszul
formula implies $\nabla_XY\in\VV$ \cite{aln1}. To simplify
notation, we define inductively $\maX^{0}:=\maX$, and
$\maX^{i+1}=\maX^i\cup \{\nabla_X Y\,|\, X,Y\in \maX^i\}$.

By definition any $V\in \Gamma(T^*M^{\otimes k})$ satisfies
$(\nabla \nabla V)(X,Y)=\nabla_X \nabla_Y V - \nabla_{\nabla_X
Y}V.$ This implies for $X_1,\ldots, X_k \in \maX$
\begin{equation*}
    (\underbrace{\nabla\dots\nabla f}_{k \mbox{-times}})
    (X_1, \dots, X_k) = X_1\dots X_k f + \sum_{l=0}^{k-1}
    \sum_{Y_j\in \maX^{k-l}}
    a_{Y_1,\ldots,Y_l} Y_1\ldots Y_l\,f,
\end{equation*}
for appropriate choices of $a_{Y_1,\ldots,Y_l}\in \ZZ_+$. Hence,
\begin{equation*}
    \|(\underbrace{\nabla\dots\nabla
    f}_{k\mbox{-times}})\|_{L^p}\leq
    C \sum \|\nabla\dots\nabla f(X_1,\ldots,X_k)\|_{L^p} \leq
    C \|f\|_{\maX,W^{k,p}}.
\end{equation*}
By induction, we know that $\|Y_1,\ldots,Y_l f \|_{L^p}\leq C
\|f\|_{\nabla,W^{l,p}}$ for $Y_i\in \maX^{k-l}$, $0\leq l\leq
k-1$, and hence
\begin{multline*}
    \|X_1\ldots X_k f\|_{L^p}\leq
    \underbrace{\|\nabla\dots\nabla f\|_{L^p}
    \|X_1\|_{L^\infty} \cdots
    \|X_k\|_{L^\infty}}_{\leq C \|f\|_{\nabla, W^{k,p}}} \\
    + \underbrace{\sum_{l=0}^{k-1}\sum_{Y_1,\ldots,Y_l\in
    \maX^{k-l}} a_{Y_1,\ldots,Y_l} Y_1\ldots Y_l \, f}_{
    \leq C\|f\|_{\nabla,W^{k-1},p}}.
\end{multline*}
This implies the equivalence of the norms.

The proof of completeness is standard, see for example
\cite{Evans, Taylor3}.
\end{proof}


We shall also need the following simple observation.


\begin{lemma}\label{lemma.res}\
Let $\Omega ' \subset \Omega \subset M$ be open subsets, $\Omega_0
= \Omega \cap M_0$, and $\Omega_0' = \Omega' \cap M_0$. The
restriction then defines continuous operators $W^{s,p}(\Omega_0)
\to W^{s, p}(\Omega'_0)$. If the various choices ($\maX,g,x_j$)
are done in the same way on $\Omega$ and $\Omega'$, then the
restriction operator has norm $1$.
\end{lemma}


\subsection{Definition of Sobolev spaces using partitions of unity}
Yet another description of the spaces
$W^{k,p}(\Omega_0)$ can be obtained by using suitable partitions
of unity as in \cite[Lemma 1.3]{Shubin}, whose definition we now
recall. See also \cite{CGT, StrichartzM, Roe}.


\begin{lemma}\label{lemma.Shubin}\ For any
$0 < \epsilon < r_{\mathrm{inj}}(M_0)/6$ there is a sequence of
points $\{x_j\} \subset M_0$, and a partition of unity $\phi_j \in
\CIc(M_0)$ with the following properties:
\begin{enumerate}[\rm (i)]
\item $\supp(\phi_j) \subset B(x_j, 2\epsilon)$;
\item $\|\nabla^k \phi_j\|_{L^\infty(M_0)}\le C_{k,\epsilon}$,
with $C_{k,\epsilon}$ independent of $j$; and
\item the sets $B(x_j, \epsilon/2)$ are disjoint, the
sets $B(x_j, \epsilon)$ form a covering of $M_0$, and the sets
$B(x_j, 4\epsilon)$ form a covering of $M_0$ of finite
multiplicity, \ie $$\sup_{y\in M_0}\# \{x_j\,|\,y\in B(x_j,
4\epsilon)\}<\infty.$$
\end{enumerate}
\end{lemma}


Fix $\epsilon\in(0,r_{\mathrm{inj}}(M_0)/6)$. Let $\psi_j
:B(x_j,4\epsilon) \to B_{\RR^n}(0,4\epsilon)$ normal coordinates
around $x_j$, \ie a composition of the exponential maps
$\exp_{x_j}:T_{x_j} M_0 \to M_0$ and by some isometries
$T_{x_j}M_0 \simeq \RR^n$. The uniform bounds on the Riemann
tensor $R$ and its derivatives $\nabla^k R$ imply uniform bounds
on
\begin{equation*}
    \nabla^k d \exp_{x_j}:B_{T_{x_j}M_0}(0,4\epsilon)\to
    \RR^{n(k+1)}\otimes TM,
\end{equation*}
which simply means that all derivatives of $\psi_j$ are uniformly
bounded.

\begin{proposition}\label{prop.alt.desc}\
Let $\phi_i$ and $\psi_i$ be as in the two paragraphs above. Let
$U_j=\psi_j(\Omega_0 \cap B(x_j,2\epsilon)) \subset\RR^n$. We
define
\begin{equation*}
    \nu_{k,\infty}(u) := \sup_j \|(\phi_j u)
    \circ \psi_j^{-1}\|_{W^{k,\infty}(U_j)}
\end{equation*}
and, for $1\leq p<\infty$,
\begin{equation*}
    \nu_{k,p}(u)^p := \sum_j \|(\phi_j u) \circ
    \psi_j^{-1}\|_{W^{k,p}(U_j)}^p.
\end{equation*}
Then $u \in W^{k, p}(\Omega_0)$ if, and only if, $\nu_{k,p}(u) <
\infty$. Moreover, $\nu_{k, p}(u)$ defines an equivalent norm on
$W^{k, p}(\Omega_0)$.
\end{proposition}


\begin{proof}\ We shall assume $p < \infty$, for simplicity of
notation. The case $p = \infty$ is completely similar. Consider
then $\mu(u)^p = \sum_j \|\phi_j u \|_{W^{k,p}(\Omega_0)}^p$. Then
there exists $C_{k,\varepsilon} > 0$ such that
\begin{equation}\label{eq.mu.equiv}
    C_{k,\varepsilon}^{-1}\|u\|_{W^{k, p}(\Omega_0)} \le
    \mu(u) \le C_{k,\varepsilon}\|u\|_{W^{k, p}(\Omega_0)},
\end{equation}
for all $u \in W^{k, p}(\Omega_0)$, by Lemma \ref{lemma.Shubin}
(\ie the norms are equivalent). The fact that all derivatives of
$\exp_{x_j}$ are bounded uniformly in $j$ further shows that $\mu$
and $\nu_{k,p}$ are also equivalent.
\end{proof}


The proposition gives rise to a third, equivalent definition of
Sobolev spaces. This definition was inspired from \cite{Shubin,
StrichartzM}, and can be used to define the spaces $W^{s,
p}(\Omega_0)$, for any $s \in \RR$, $1 < p < \infty$, and
$\Omega_0=M_0$. The cases $p = 1$ or $p = \infty$ are more
delicate and we shall not discuss them here.

Recall that the spaces $W^{s, p}(\RR^n)$, $s \in \RR$, $ 1 < p <
\infty$ are defined using the powers of $1 + \Delta$, see
\cite[Chapter V]{Stein} or \cite[Section 13.6]{Taylor3}.


\begin{definition}[Partition of unity
definition of Sobolev spaces]\label{def.sreal}
Let $s\in \RR$, and $1<p<\infty$. Then we define
\begin{equation}\label{sreal}
    \|u\|_{W^{s,p}(M_0)}^p:=\sum_j\|(\phi_j u) \circ
\psi_j^{-1}\|_{W^{s,p}(\RR^n)}^p,\,\,\,\,\,1<p<\infty.
\end{equation}
\end{definition}


By Proposition
\ref{prop.alt.desc} this norm is equivalent to our previous norm
on $W^{k,p}(M_0)$ when $k$ is a nonnegative integer.


\begin{proposition}\label{prop.dense}\ The space $\CIc(M_0)$ is
dense in $W^{k,p}(M_0)$, for $1 < p < \infty$ and $s\in\RR$, or
$1\leq p<\infty$ and $s=k\in\mathbb{Z}_+$.
\end{proposition}


\begin{proof}\ For $k \in \ZZ_+$, the result is true for any
manifold with bounded geometry, see \cite[Theorem 2]{Aubin} or
\cite[Theorem 2.8]{Hebey2}, or \cite{Hebey1}. For $\Omega_0 =
M_0$, $s \in \RR$, and $1 < p < \infty$, the definition of the
norm on $W^{s, p}(M_0)$ allows us to reduce right away the proof
to the case of $\RR^n$, by ignoring enough terms in the sum
defining the norm \eqref{sreal}. (We also use a cut-off function
$0 \le \chi \le 1$, $\chi \in \CIc(B_{\RR^n}(0, 4\epsilon))$,
$\chi = 1$ on $B_{\RR^n}(0, 4\epsilon)$.)
\end{proof}


We now give a characterization of the spaces $W^{s, p}(M_0)$ using
interpolation. Let $\widetilde W^{-k, p}(M_0)$ be the set of
distributions on $M_0$ that extend by continuity to linear
functionals on $W^{k, q}(M_0)$, $p^{-1} + q^{-1} = 1$, using
Proposition \ref{prop.dense}. That is, $\widetilde W^{-k, p}(M_0)$
be the set of distributions on $M_0$ that define continuous linear
functionals on $W^{k, q}(M_0)$, $p^{-1} + q^{-1} = 1$. We let
\begin{equation*}
    \widetilde W^{\theta k, k, p}(M_0) :=
    [\widetilde W^{0, p}(M_0), W^{k, p}(M_0)]_{\theta}\,,
    \quad 0 \le \theta \le 1\,,
\end{equation*}
be the complex interpolation spaces. Similarly, we define
\begin{equation*}
    \widetilde W^{-\theta k, k, p}(M_0) = [\widetilde W^{0,
    p}(M_0), W^{-k,p}(M_0)]_{\theta}.
\end{equation*}
(See \cite{BerghLofstrom} or \cite[Chapter 4]{Taylor1} for the
definition of the complex interpolation spaces.)

The following proposition is an analogue of Proposition
\ref{prop.alt.desc}. Its main role is to give an intrinsic
definition of the spaces $W^{s, p}(M_0)$, a definition that is
independent of choices.


\begin{proposition}\label{prop.alt.desc1}\
Let $1 < p < \infty$ and $k > |s|$. Then we have a topological
equality $\widetilde W^{s, k , p}(M_0) = W^{s, p}(M_0)$. In
particular, the spaces $W^{s, p}(M_0)$, $s \in \RR$, do not depend
on the choice of the covering $B(x_j, \epsilon)$ and of the
subordinated partition of unity and we have
\begin{equation*}
    [W^{s, p}(M_0), W^{0 , p}(M_0)]_{\theta} = W^{\theta s,
    p}(M_0)\,,\quad 0 \le \theta \le 1\,.
\end{equation*}
Moreover, the pairing between functions and distributions defines
an isomorpism $W^{s, p}(M_0)^* \simeq W^{-s, q}(M_0)$, where $1/p
+ 1/q = 1$.
\end{proposition}


\begin{proof}\
This proposition is known if $M_0 = \RR^n$ with the usual metric
\cite{Taylor3}[Equation (6.5), page 23]. In particular, $\tilde
W^{s,p}(\RR^n) = W^{s, p}(\RR^n)$. As in the proof of Proposition
\ref{prop.alt.desc} one shows that the quantity
\begin{equation}\label{eq.nu.sp}
    \nu_{s, p}(u)^p := \sum_j\|(\phi_j u) \circ
    \psi_j^{-1}\|_{\tilde W^{s,p}(\RR^n)}^p,
\end{equation}
is equivalent to the norm on $\tilde W^{s, p}(M_0)$. This implies
$\tilde W^{s,p}(M_0) = W^{s, p}(M_0)$.

Choose $k$ large. Then we have
\begin{multline*}
    [W^{s, p}(M_0), W^{0, p}(M_0)]_{\theta} = [W^{s, k, p}(M_0), W^{0,
    k, p}(M_0)]_{\theta}\\ = W^{\theta s, k, p}(M_0) = W^{\theta s,
    p}(M_0).
\end{multline*}
The last part follows from the compatibility of interpolation with
taking duals. This completes the proof.
\end{proof}

The above proposition provides us with several corollaries. First,
from the interpolation properties of the spaces $W^{s, p}(M_0)$, we
obtain the following corollary.

\begin{corollary}\label{cor.n.mult}\
Let $\phi \in W^{k, \infty}(M_0)$ and $p \in (1, \infty)$. Then
multiplication by $\phi$ defines a bounded operator on
$W^{s,p}(M_0)$ of norm at most $C_k\|\phi\|_{W^{k, \infty}(M_0)}$
where $k \ge |s|$, $k \in \ZZ_+$. Similarly, any
differential operator $P \in \DiffV{m}(M)$ defines continuous maps
$P : W^{s, p}(M_0) \to W^{s-m, p}(M_0)$.
\end{corollary}

\begin{proof}\
For $s \in \ZZ_+$, this follows from the definition of the norm on
$W^{k, \infty}(M_0)$ and from the definition of $\DiffV{m}(M)$ as the
linear span of differential operators of the form $fX_1 \ldots X_k$,
($f \in \CI(M) \subset W^{k, \infty}$, $X_j \in \VV$, and $0
\le k \le m$), and from the definition of the spaces
$W^{k,p}(\Omega_0)$.

For $s \le m$, the follows by duality. For the other values of $s$,
the result follows by interpolation.
\end{proof}


Next, recall that an isomorphism $\phi: M \to M$ of the Lie
manifolds $(M_0, M, \VV)$ and $(M_0', M', \VV')$ is a
diffeomorphism such that $\phi_*(\VV) = \VV'$. We then have the
following invariance property of the Sobolev spaces that we have
introduced.


\begin{corollary}\label{cor.invariance}\
Let $\phi : M \to M'$ be an isomorphism of Lie manifolds,
$\Omega_0 \subset M_0$ be an open subset and $\Omega' =
\phi(\Omega)$. Let $p \in [1, \infty]$ if $s \in \ZZ_+$, and $p
\in (1, \infty)$ if $s \not \in \ZZ_+$. Then $f \to f \circ \phi$
extends to an isomorphism $\phi^* : W^{s, p}(\Omega') \to W^{s,
p}(\Omega)$ of Banach spaces.
\end{corollary}


\begin{proof}\
For $k \in \ZZ_+$, this follows right away from definitions and
Proposition \ref{prop.Banach}. For $-k \in \ZZ_+$, this follows by
duality, Proposition \eqref{prop.alt.desc1}. For the other values
of $s$, the result follows from the same proposition, by
interpolation.
\end{proof}

Recall now that $M_0$ is complete \cite{aln1}. Hence the Laplace
operator $\Delta = \nabla^*\nabla$ is essentially self-adjoint on
$\CIc(M_{0})$ by \cite{gaffney, roel, strichartz}. We shall define
then $(1 + \Delta)^{s/2}$ using the spectral theorem.

\begin{proposition}\label{prop.domain}\ The space
$H^s(M_0) := W^{s, 2}(M_0)$, $s \ge 0$, identifies with the domain
of $(1 + \Delta)^{s/2}$, if we endow the latter with the graph
topology.
\end{proposition}


\begin{proof}\ For $s \in \ZZ_+$, the result is true for
any manifold of bounded geometry, by \cite[Proposition 3]{Aubin}.
For $s \in \RR$, the result follows from interpolation, because
the interpolation spaces are compatible with powers of operators,
see the chapter on Sobolev spaces in Taylor's book \cite{Taylor1}.
\end{proof}

The well known Gagliardo--Nirenberg--Sobolev inequality
\cite{Aubin, Evans, GilbargTrudinger, Hebey2} holds also in our
setting.


\begin{proposition}\label{prop.Sobolev}\ Denote by $n$ the dimension
of $M_0$. Assume that $1/p = 1/q - m/n$, $1 < q \leq p < \infty$,
where $m\geq 0$. Then $W^{s, q}(M_0)$ is continuously embedded in
$W^{s-m, p}(M_0)$.
\end{proposition}


\begin{proof}\
If $s$ and $m$ are integers, $s\geq m\geq 0$, the statement of the
proposition is true for manifolds with bounded geometry,
\cite[Theorem 7]{Aubin} or \cite[Corollary 3.1.9]{Hebey2}. By
duality (see Proposition \ref{prop.alt.desc1}), we obtain the same
result when $s \le 0$, $s\in \ZZ$. Then, for integer $s,m$,
$0<s<m$ we obtain the corresponding embedding by composition
$W^{s,q}(M_0)\to W^{0,r}(M_0)\to W^{s-m,p}(M_0)$, with
$1/r=1/q-s/n$. This proves the result for integral values of $s$.
For non-integral values of $s$, the result follows by
interpolation using again Proposition~\ref{prop.alt.desc1}.
\end{proof}


The Rellich-Kondrachov's theorem on the compactness of the
embeddings of Proposition \ref{prop.Sobolev} for $1/p > 1/q -m/n$
is true if $M_0$ is compact \cite[Theorem 9]{Aubin}. This happens
precisely when $M = M_0$, which is a trivial case of a manifold
with a Lie structure at infinity. On the other hand, it is easily
seen (and well known) that these compactness cannot be true for
$M_0$ non-compact. We will nevertheless restore this by using
Sobolev spaces with weights in the next section, see
Theorem~\ref{theorem.Kondrachov}.


\section{Manifolds with boundary\label{sec.MB}}


We continue to assume that $(M_0, M, \VV)$ is Lie manifold. Let $N
\subset M$ be a submanifold with corners of codimension one of
$M$. Recall that this implies that $N$ is transverse to all faces
of $M$. Also, recall that $N$ is a {\em regular} submanifold of
$(M, \VV)$ if we can choose a tubular neighborhood $V$ of $N_0 :=
N \smallsetminus \pa N = N \cap M_0$ in $M_0$ and a compatible
metric $g$ on $M_0$ that restricts to a product-type metric on $V
\simeq (\pa N_0) \times (-\varepsilon_0, \varepsilon_0)$ (see
Proposition~\ref{prop.prod.struct}). In Section \ref{sec.SM}, we
shall show that every tame manifold is regular; in turn, this will
give an easy, geometric criterion to decide when a codimension one
submanifold of $M$ is regular.

Let $\Omega \subset M$ be an open subset. Recall that $\Omega$ is
a  Lie domain in $M$ if, and only if, $\pa \Omega =
\pa{\overline{\Omega}}$ and $\pa \Omega$ is a regular submanifold
of $M$. Let $\Omega_0 = \Omega \cap M_0$. Then $\pa\Omega_0 :=
(\pa \Omega) \cap M_0$ is a smooth submanifold of codimension one
of $M_0$. We have the following analogue of the classical
extension theorem.


\begin{theorem}\label{theorem.ext}\
There exists a linear operator $E$ mapping measurable functions on
$\Omega_0$ to measurable functions on $M_0$ with the properties:
\begin{enumerate}[\rm (i)]
\item\ $E$ maps $W^{k,p}(\Omega_0)$ continuously into $\WkpM$ for
every $p\in[1,\infty]$ and every integer $k\geq 0$, and
\item\ $E u \vert_{\Omega_0} = u$.
\end{enumerate}
\end{theorem}


\begin{proof}\
Since $\pa\Omega_0$ is a regular submanifold we can fix a
compatible metric $g$ on $M_0$ and a tubular neighborhood $V_0$ of
$\pa\Omega_0$ such that $V_0 \simeq (\pa\Omega_0) \times
(-\varepsilon_0, \varepsilon_0)$, $\varepsilon_0>0$. Let
$\varepsilon=\min(\varepsilon_0,r_{\mathrm{inj}}(M_0))/20$, where
$r_{\mathrm{inj}}(M_0)>0$ is the injectivity radius of $M_0$. By
Zorn's lemma and the fact that $M_0$ has bounded geometry we can
choose a maximal, countable set of disjoint balls
$B(x_i,\varepsilon)$, $i\in I$. Since this family of balls is
maximal we have $M_0=\cup_iB(x_i,2\varepsilon)$. For each $i$ we
fix a smooth function $\eta_i$ supported in $B(x_i,3\varepsilon)$
and equal to $1$ in $B(x_i,2\varepsilon)$. This can be done easily
in local coordinates around the point $x_i$; since the metric $g$
is induced by a metric $g$ on $A$ we may also assume that all
derivatives of order up to $k$ of $\eta_i$ are bounded by a
constant $C_{k,\varepsilon}$ independent of $i$. By replacing
$\eta_i$ with $\eta_i/\sqrt{\sum_{j}\eta_j^2}$, we can further
assume that $\sum_i \eta_i^2 = 1$.

Following \cite[Ch. 6]{Stein} we also define two smooth cutoff
functions adapted to the set $\Omega_0$. We start with a function
$\psi:\RR\to[0,1]$ which is equal to $1$ on $[-3,3]$ and which has
support in $[-6,6]$

Let $\varphi = (\varphi_1,\varphi_2)$ denote the isomorphism
between $V_0$ and $\pa\Omega_0 \times
(-\varepsilon_0,\varepsilon_0)$, where $\varphi_1 : V_0 \to
\pa\Omega_0$ and $\varphi_2 :
V_0\to(-\varepsilon_0,\varepsilon_0)$. We define
\begin{equation*}
\Lambda_+(x):=
\begin{cases}
    \ \ 0 & \text{if }\; x\in M_0\setminus V_0\\ \
    \psi(\varphi_2(x)/\varepsilon) & \text{if }\; x\in V_0,
\end{cases}
\end{equation*}
and $\Lambda_-(x):=1-\Lambda_+(x)$. Clearly $\Lambda_+$ and
$\Lambda_-$ are smooth functions on $M_0$ and $\Lambda_+(x) +
\Lambda_-(x)= 1$. Obviously, $\Lambda_+$ is supported in a
neighborhood of $\pa\Omega_0$ and $\Lambda_-$ is supported in the
complement of a neighborhood of $\pa\Omega_0$.


Let $\pa\Omega_0 = A_1 \cup A_2 \cup \ldots$ denote the
decomposition of $\pa\Omega_0$ into connected components. Let $V_0
= B_1 \cup B_2 \cup \ldots$ denote the corresponding decomposition
of $V_0$ into connected components, namely, $B_j =
\varphi^{-1}(A_j \times (-\varepsilon_0, \varepsilon_0))$. Since
$\pa\Omega_0=\pa{\overline{\Omega}_0}$, we have
$\varphi(\Omega_0\cap B_j)=A_j\times(-\varepsilon_0,0)$ or
$\varphi(\Omega_0\cap B_j)=A_j\times(0,\varepsilon_0)$. Thus, if
necessary, we may change the sign of $\varphi$ on some of the
connected components of $V_0$ in such a way that
\begin{equation*}
    \varphi(\Omega_0\cap V_0) = \pa\Omega_0 \times
    (0,\varepsilon_0).
\end{equation*}
Let $\psi_0$ denote a fixed smooth function, $\psi_0 :
\RR\to[0,1]$, $\psi_0(t)=1$ if $t\geq-\varepsilon$ and
$\psi_0(t)=0$ if $t\leq-2\varepsilon$, and let
\begin{equation*}
\Lambda_0(x)=
\begin{cases}
\ \ 1 & \text{if }\; x \in \Omega_0 \setminus V_0\\ \ \ 0 &
\text{if }\; x \in M_0 \setminus (\Omega_0 \cup V_0) \\ \
\psi_0(\varphi_2(x)) & \text{if }\; x \in V_0.
\end{cases}
\end{equation*}


We look now at the points $x_i$ defined in the first paragraph of
the proof. Let $J_1=\{i\in I:d(x_i,\pa\Omega_0)\leq
10\varepsilon\}$ and $J_2=\{i\in
I:d(x_i\pa\Omega_0)>10\varepsilon\}$. For every point $x_i$, $i\in
J_1$, there is a point $y_i\in\pa\Omega_0$ with the property that
$B(x_i,4\varepsilon)\subset B(y_i,15\varepsilon)$. Let
$B_{\pa\Omega_0}(y_i,15\varepsilon)$ denote the ball in
$\pa\Omega_0$ of center $y_i$ and radius $15\varepsilon$ (with
respect to the induced metric on $\pa\Omega_0$). Let $h_i :
B_{\pa\Omega_0}(y_i,15\varepsilon) \to B_{\RR^{n-1}}(0,
15\varepsilon)$ denote the normal system of coordinates around the
point $y_i$. Finally let $g_i : B_{\RR^{n-1}}(0, 15\varepsilon)
\times (-15\varepsilon, 15\varepsilon) \to V_0$ denote the map
$g_i(v,t) = \varphi^{-1}(h_i^{-1}(v),t)$.

Let $E_{\RR^n}$ denote the extension operator that maps
$W^{k,p}(\RR^n_+)$ to $W^{k,p}(\RR^n)$ continuously, where
$\RR^n_+$ denotes the half-space $\{x:x_n > 0\}$. Clearly,
$E_{\RR^n}u\vert_{\RR^n_+} = u$. The existence of this extension
operator is a classical fact, for instance, see \cite[Chapter
6]{Stein}. For any $u\in W^{k,p}(\Omega_0)$ and $i\in J_1$ the
function $(\eta_iu)\circ g_i$ is well defined on $\RR^n_+$ simply
by setting it equal to $0$ outside the set
$B_{\RR^{n-1}}(0,15\varepsilon)\times(0, 15\varepsilon)$. Clearly,
$(\eta_iu)\circ g_i\in W^{k,p}(\RR^n_+)$. We define the extension
$Eu$ by the formula
\begin{equation}\label{extension}
    Eu(x) = \Lambda_0(x) \Lambda_-(x)u(x) + \Lambda_0(x)
    \Lambda_+(x) \sum_{i\in J_1} \eta_i(x)
    E_{\RR^n}[(\eta_iu)\circ g_i](g_i^{-1}x)\,.
\end{equation}
Notice that
\begin{equation}\label{verify}
    \sum_{i\in J_1}\eta_i^2(x) = 1 \;
    \text{ in }\mathrm{supp}\, \Lambda_+.
\end{equation}
Indeed, since $M_0 = \cup_{i\in I}B(x_i, 2\varepsilon)$, we have
$\sum_{i\in I} \eta_i^2(x) \geq 1$ for any $x \in M_0$. Also,
$\eta_i(x)\equiv 0$ in $\mathrm{supp}\,\Lambda_+$ if $i\in J_2$,
thus \eqref{verify} follows. This shows that $Eu$ in
\eqref{extension} is well-defined. Clearly, by the formula,
$Eu\vert_{\Omega_0}=u$. It remains to verify that
\begin{equation*}
    \|Eu\|_{W^{k,p}(M_0)}\leq C_k\|u\|_{W^{k,p}(\Omega_0)}.
\end{equation*}
This follows as in \cite{Stein} using \eqref{verify}, the fact
that the extension $E_{\RR^n}$ satisfies the same bound, and the
definition of the Sobolev spaces using partitions of unity
(Proposition \ref{prop.alt.desc}).
\end{proof}


\begin{theorem}\label{theorem.dense}\ The space
$\CIc(\overline{\Omega}_0)$, where the closure is defined in
$M_0$, is dense in $W^{k,p}(\Omega_0)$, for $1 \le p < \infty$.
\end{theorem}


\begin{proof} For any $u\in W^{k,p}(\Omega_0)$ let $Eu$
denote its extension from Theorem \ref{theorem.ext}, $Eu\in
W^{k,p}(M_0)$. By Proposition \ref{prop.dense}, there is a
sequence of functions $f_j\in C_c^{\infty}(M_0)$ with the property
that
\begin{equation*}
\lim_{j\to\infty}f_j=Eu\text{ in }W^{k,p}(M_0).
\end{equation*}
Thus $\lim_{j\to\infty}f_j\vert_{{\Omega_0}}=u$ in
$W^{k,p}(\Omega_0)$, as desired.
\end{proof}


\begin{theorem}\label{theorem.trace.I}\ The restriction map
$\CIc(\Omega_0) \to \CIc(\pa\Omega_0)$ extends to a continuous
map $T : W^{k,p}(\Omega_0) \to W^{k-1, p}(\pa\Omega_0)$, for $1
\le p \leq\infty$.
\end{theorem}


\begin{proof}\ To simplify the notation assume $1 \leq p < \infty$.
We shall assume that the compatible metric on $M_0$ restricts to a
product type metric on $V_0$, our distinguished tubular
neighborhood of $\pa\Omega_0$.

We use the definitions of the Sobolev spaces using partitions of
unity, Proposition \ref{prop.alt.desc} and Lemma
\ref{lemma.Shubin} with $\varepsilon = \min(\epsilon_0,
r_{\mathrm{inj}}(M_0))/10$. Let $B(x_j, 2\varepsilon)$ denote the
balls in the cover of $X$ in Lemma \ref{lemma.Shubin} and
$1=\sum_j\phi_j$ the corresponding partition of unity. Then
$\widetilde{\phi}_j = \phi_j\vert_{\pa\Omega_0}$ form a partition
of unity on $\pa\Omega_0$. Clearly,
\begin{equation*}
    r_{\mathrm{inj}}(\pa\Omega_0) \geq
    r_{\mathrm{inj}}(M_0)>0.
\end{equation*}

Start with a function $u\in W^{k,p}(\Omega_0)$ and let
$u_j=(u\phi_j)\circ\psi_j^{-1}$, $u_j\in
W^{k,p}(\psi_j(\Omega_0\cap B(x_j,4\varepsilon)))$. In addition
$u_j\equiv 0$ outside the set $\psi_j(\Omega_0\cap
B(x_j,2\varepsilon))$. If
$B(x_j,4\varepsilon)\cap\pa\Omega_0=\emptyset$ let
$\widetilde{T}(u_j)=0$. Otherwise notice that
$B(x_j,4\varepsilon)$ is included in $V_0$, the tubular
neighborhood of $\pa\Omega_0$, thus the set
$\psi_j(\pa\Omega_0\cap B(x_j,4\varepsilon))$ is the intersection
of a hyperplane and the ball $B_{\RR^n}(0,4\varepsilon)$. We can
then let $\widetilde{T}(u_j)$ denote the Euclidean restriction of
$u_j$ to $\psi_j(\pa\Omega_0\cap B(x_j,4\varepsilon))$ (see
\cite[Section 5.5]{Evans}). Clearly $\widetilde{T}(u_j)$ is
supported in $\psi_j(\pa\Omega_0\cap B(x_j,2\varepsilon))$ and
\begin{equation*}
    \|\widetilde{T}(u_j) \circ \widetilde{\psi}_j\|_{W^{k-1,p}(\pa\Omega_0)}
    \leq C\|u_j\|_{W^{k,p}(\psi_j(\Omega_0\cap B(x_j,4\varepsilon)))},
\end{equation*}
where $\widetilde{\psi}_j=\psi_j\vert_{\Omega_0}$ and the constant
$C$ is independent of $j$ (recall that $\psi_j(\pa\Omega_0\cap
B(x_j,4\varepsilon))$ is the intersection of a hyperplane and the
ball $B_{\RR^n}(0,4\varepsilon)$). Let
\begin{equation*}
Tu=\sum_j\widetilde{T}(u_j)\circ\widetilde{\psi}_j.
\end{equation*}
Since the sum is uniformly locally finite we have
\begin{equation*}
\begin{split}
    \|Tu\|_{W^{k-1,p}(\pa\Omega_0)}^p&\leq
    C\sum_j\|\widetilde{T}(u_j)\circ
    \widetilde{\psi}_j\|_{W^{k-1,p}(\pa\Omega_0)}^p\\
    &\leq C\sum_j\|u_j\|_{W^{k,p}(\psi_j(\Omega_0\cap
    B(x_j,4\varepsilon)))}^p\leq C\|u\|_{W^{k,p(\Omega_0)}}^p,
\end{split}
\end{equation*}
with constants $C$ independent of $u$. The fact
$Tu\vert_{\mathcal{C}_c^\infty(\Omega_0)}$ is indeed the
restriction operator follows immediately from the definition.
\end{proof}

We shall see that if $p=2$, we get a surjective map $W^{s,
2}(\Omega_0) \to W^{s-1/2, 2}(\pa\Omega_0)$ (Theorem
\ref{theorem.res}).


\begin{theorem}\label{theorem.zero}\ The closure of
$\CIc(\Omega_0)$ in $\WkpO$ is the intersection of the kernels of
$T \circ \pa_\nu^j$, $0 \le j \le k-1$.
\end{theorem}


\begin{proof}\
The proof is reduced to the Euclidean case \cite{Adams} following
the same pattern of reasoning as in the previous theorem.
\end{proof}


The Gagliardo--Nirenberg--Sobolev theorem holds also for manifolds
with boundary.


\begin{theorem}\label{theorem.Sobolev}\ Denote by $n$ the dimension
of $M$ and let $\Omega \subset M$ be a Lie domain in $M$. Assume
that $1/p = 1/q - m/n > 0$, $1 \le q < \infty$, where $m \le k$ is
an integer. Then $W^{k, q}(\Omega_0)$ is continuously embedded in
$W^{k-m, p}(\Omega_0)$.
\end{theorem}


\begin{proof}\  This can be proved using Proposition
\ref{prop.Sobolev} and Theorem \ref{theorem.ext}. Indeed, denote
by
\begin{equation*}
    j : W^{k, q}(M_0)  \to  W^{k-m, p}(M_0)
\end{equation*}
the continuous inclusion of Proposition \ref{prop.Sobolev}. Also,
denote by $r$ the restriction maps $\WkpM \to \WkpO$. Then the
maps
\begin{equation*}
    W^{k, q}(\Omega_0) \stackrel{E}{\longrightarrow} W^{k, q}(M_0)
    \stackrel{j}{\longrightarrow} W^{k-m, p}(M_0)
    \stackrel{r}{\longrightarrow} W^{k-m, p}(\Omega_0)
\end{equation*}
are well defined and continuous. Their composition is the
inclusion of $W^{k, q}(\Omega_0)$ into $W^{k-m, p}(\Omega_0)$.
This completes the proof.
\end{proof}


For the proof of a variant of Rellich--Kondrachov's compactness
theorem, we shall need Sobolev spaces with weights. Let $a_H > 0$
be a parameter associated to each hyperface (\ie face of
codimension one) of $M$. Fix for any hyperface $H \subset M$ a
defining function $\rho_H$, that is a function $\rho_H \ge 0$ such that
$H = \{\rho_H = 0\}$ and $d\rho_H \not = 0$ on $H$. Let
\begin{equation}\label{eq.def.rho}
    \rho = \prod \rho_H^{a_H},
\end{equation}
the product being taken over all hyperfaces of $M$.
%% V3.4 Two more sentences.
The function $\rho$ will be called an {\em admissible weight}.
(The function $\rho$ considered in the beginning of this paper is,
in fact, an admissible weight, so there is no conflict in the
notation.) We define then
\begin{equation}\label{eq.def.weights}
    \rho^s \WkpO := \{\rho^s u, u \in \WkpO\},
\end{equation}
with the norm $\|\rho^su\|_{\rho^s \WkpO} := \|u\|_{\WkpO}$.


\begin{theorem}\label{theorem.Kondrachov}\
Denote by $n$ the dimension of $M$ and let $\Omega \subset M$ be a
Lie domain, $\Omega_0 = \Omega \cap M_0$. Let $s < s'$ be real
parameters. Assume that $1/p > 1/q - m/n > 0$, $1 \le q < \infty$,
where $m \le k$ is an integer. Then $\rho^sW^{k, q}(\Omega_0)$ is
compactly embedded in $\rho^{s'}W^{k-m, p}(\Omega_0)$.
\end{theorem}


\begin{proof}\ The same argument as that in the proof of Theorem
\ref{theorem.Sobolev} allows us to assume that $\Omega_0 = M_0$.
The norms are chosen such that $\WkpO \ni u \to \rho^s u \in
\rho^s\WkpO$ is an isometry. Thus, it is enough to prove that
$\rho^s : W^{k, q}(\Omega_0) \to W^{k - m, p}(\Omega_0)$, $s > 0$,
is a compact operator.

For any defining function $\rho_H$ and any $X \in \VV$, we have that
$X(\rho_H)$ vanishes on $H$, since $X$ is tangent to $H$. Recall now
the function $\rho$ defined in Equation \eqref{eq.def.rho}. We obtain
that $X(\rho^s) = \rho^s f_X$, for some $f_X \in \CI(M)$. Then, by
induction, $X_1 X_2 \ldots X_k (\rho^s) = \rho^s g$, for some $g \in
\CI(M)$.

Let $\chi \in \CI([0, \infty)$ be equal to $0$ on $[0, 1/2]$,
equal to $1$ on $[1, \infty)$, and non-negative everywhere. Define
$\phi_\epsilon = \chi(\epsilon^{-1} \rho^s)$. Then
\begin{equation*}
    \| X_1 X_2 \ldots X_k \big(\rho^s \phi_\epsilon
    - \rho^s\big) \|_{L^\infty} \to 0\,, \quad \text{as }\;
    \epsilon \to 0,
\end{equation*}
for any $X_1, X_2, \ldots, X_k \in \VV$. Corollary
\ref{cor.n.mult} then shows that $\rho^s \phi_\epsilon \to \rho^s$ in
the norm of bounded operators on $W^{s, p}(\Omega_0)$. But
multiplication by $\rho^s\phi_\epsilon$ is a compact operator, by the
Rellich-Kondrachov's theorem for compact manifolds with boundary
\cite[Theorem 9]{Aubin}. This completes the proof.
\end{proof}


We end with the following generalization of the classical
restriction theorem for the Hilbertian Sobolev spaces $H^s(M_0) :=
W^{s, 2}(M_0)$. The tame submanifolds are defined in the next
section.


\begin{theorem}\label{theorem.res}\
Let $N_0 \subset M_0$ be a tame submanifold of codimension $k$ of
the Lie manifold $(M_0, M, \VV)$. Restriction of smooth functions
extends to a bounded, surjective map
\begin{equation*}
    H^{s}(M_0) \to H^{s - k/2}(N_0),
\end{equation*}
for any $s > k/2$. In particular, $H^{s}(\Omega_0) \to H^{s -
1/2}(\pa\Omega_0)$ is continuous and surjective.
\end{theorem}


\begin{proof}\
Let $B \to N$ be the vector bundle defining the Lie structure at
infinity $(N, B)$ on $N_0$ and $A \to M$ be the vector bundle
defining the Lie structure at infinity $(M, A)$ on $M_0$. (See
Section \ref{sec.SM} for further explanation of this notation.)
The existence of tubular neighborhoods, Theorem
\ref{thm.tub.neigh}, and a partition of unity argument, allows us
to assume that $M = N \times S^1$ and that $A = B \times TS^1$
(external product). Since the Sobolev spaces $H^s(M_0)$ and
$H^{s-1/2}(N_0)$ do not depend on the metric on $A$ and $B$, we
can assume that the circle $S^1$ is given the invariant metric
making it of length $2\pi$ and that $M_0$ is given the product
metric.  The rest of the proof now is independent of the way we
have arrived at the product metric on $M_0$.

Then $\Delta_{M_0} = \Delta_{N_0} + \Delta_{S^1}$ and
$\Delta_{S^1}= -\pa^2/\pa \theta^2$ has spectrum $\{4\pi^2n^2\}$,
$n \in \ZZ$. We can decompose $L^2(N_0 \times S^1)$ according to
the eigenvalues $n \in \ZZ$ of $-\frac{1}{2\pi\imath}\pa_\theta$:
\begin{equation*}
    L^2(N_0 \times S^1) \simeq \oplus_{n \in \ZZ} L^2(N_0 \times
    S^1)_n \simeq \oplus_{n \in \ZZ} L^2(N_0),
\end{equation*}
where the isomorphism $L^2(N_0 \times S^1)_n \simeq \oplus_{n \in
\ZZ} L^2(N_0)$ is obtained by restricting to $N_0 = N_0 \times
\{1\}$, $1 \in S^1$.

To prove our theorem, it is enough to check that, if $\xi_n \in
L^2(N_0)$ is a sequence such that
\begin{equation}\label{eq.sum.n}
    \sum_n \|( 1 + n^2 + \Delta_{N_0})^{s/2}\xi_n\|^2 < \infty,
\end{equation}
then $\sum (1 + \Delta_{N_0})^{s/2-1/4}\xi_n$ is convergent.

Let $C = 1 + \int_\RR (1 + t^2)^{-s}dt$ and assume that each
$\xi_n$ is in the spectral subspace of $\Delta_{N_0}$
corresponding to $[m,m+1) \subset \RR_+$.  Then $$
    (1 + m^2)^{s - 1/2} \Bigl( \sum_n \| \xi_n \| \Bigr )^2 \le C
    \sum_n \| (1 + n^2 + m^2)^{s/2} \xi_n\|^2.
$$ Since the constant $C$ is independent of $m$ and the spectral
spaces of $\Delta_{N_0}$ corresponding to $[m,m+1) \subset \RR$
give an orthogonal direct sum decomposition of $L^2(N_0)$, this
checks Equation \eqref{eq.sum.n} and completes the proof.
\end{proof}


We conclude this section with an application to the regularity of
boundary value problems. Applications of this result to the
regularity of boundary value problems on polyhedra as well as more
details will be included in \cite{BNZ2}.

Let us introduce some notation first that will be also useful in
the following, especially in Section~\ref{sec.PO}.

Let $\exp : TM_{0} \longrightarrow M_{0} \times M_{0}$ be given by
$\exp(v) := (x, \exp_{x}(v)),$, $v \in T_xM_0$. If $E$ is a real
vector bundle with a metric, we shall denote by $(E)_r$ the set of
all vectors $v$ of $E$ with $|v|< r$. Let $(M_0^2)_r := \{ (x, y),
x, y \in M_0,\, d(x, y) < r\}$. Then the exponential map defines a
diffeomorphism $\exp :(TM_{0})_{r} \to (M_0^2)_r$. We shall also
need the function $\rho$ defined in Equation \eqref{eq.def.rho}
and the weighted Sobolev spaces $\rho^s \WkpO := \{\rho^s u, u \in
\WkpO\}$ introduced in Equation \ref{eq.def.weights}.


\begin{theorem}\label{thm.reg1}\
Let $\Omega_0 \subset M_0$ be a Lie domain in a Lie manifold
$(M_0, M, \VV)$. Let $P \in \DiffV{2}(M)$ be an order $2$ elliptic
operator on $M_0$ generated by $\VV$. Let $u \in \rho^{s}W^{1,
p}(\Omega_0)$ be such that $Pu \in \rho^{s}W^{t, p}(\Omega_0)$, $s
\in \RR$, $t \in \ZZ$, $1 < p < \infty$, and $u\vert_{\pa
\Omega_0} = 0$. Then $u \in \rho^{s}W^{t + 2, p}(\Omega_0)$.
\end{theorem}


\begin{proof}\
Note that locally, this is a well known statement. In particular,
$\phi u \in W^{t + 2, p}(\Omega_0)$, for any $\phi \in \CIc(M_0)$.
The result will follow then if we prove that
\begin{equation}\label{eq.ell.est0}
    \|u\|_{\rho^{s}W^{t + 2, p}(M_0)} \le
    C(\|Pu\|_{\rho^{s}W^{t, p}(M_0)} +
    \|u\|_{\rho^{s}W^{1, p}(M_0)} )
\end{equation}
for any $u \in W^{t + 2, p}_{\mathrm{loc}}(\Omega_0)$.

Let $r < r_{\mathrm{inj}}(M_0)$ and let $\exp : (TM_0)_{r} \to
(M_0^2)_{r}$ be the exponential map.  The statement is trivially
true for $t \le - 1$, so we will assume $t \ge 0$ in what follows.
Also, we will assume first that $s = 0$. The general case will be
reduced to this one at the end. Assume first that $\Omega_0 =
M_0$.

Let $P_x$ be the differential operators on defined on $B_{T_xM_0}(0,
r)$ obtained from $P$ by the local diffeomorphism $\exp :
B_{T_xM_0}(0, r) \to M_0$. We claim that there exists a constant $C >
0$, independent of $x \in M_0$ such that
\begin{equation}\label{eq.ell.reg}
    \|u\|_{W^{t+2, p}(T_xM_0)}^p \le C \big( \|P_xu\|_{W^{t,
    p}(T_xM_0)}^p + \|u\|_{W^{1, p}(T_xM_0)}^p \big),
\end{equation}
for any function $u \in \CIc(B_{T_xM_0}(0, r))$. This is seen as
follows. We can find a constant $C_x>0$ with this property for any
$x \in M_0$ by the ellipticity of $P_x$ using \cite{Taylor1}.
Choose $C_x$ to be the least such constant. Let $\pi : A \to M$ be
the extension of the tangent bundle of $M_0$, see Remark
\ref{rem.A} and let $A_x = \pi^{-1}(x)$. The family $P_x$, $x \in
M_0$, extends to a family $P_x$, $x \in M$, that is smooth in $x$.
The smoothness of the family $P_x$ in $x \in M$ shows that $C_x$
is lower semi-continuous. Since $M$ is compact, $C_x$ will attain
its minimum, which therefore must be positive. Let $C$ be that
minimum value.

Let now $\phi_j$ be the partition of unity and $\psi_j$ be the
diffeomorphisms appearing in Equation \eqref{eq.nu.sp}, for some
$0 < \epsilon < r/6$. In particular, the partition of unity
$\phi_j$ satisfies the conditions of Lemma \ref{lemma.Shubin},
which implies that $\supp(\phi_j) \subset B(x_j, 2\epsilon)$ and
the sets $B(x_j, 4\epsilon)$ form a covering of $M_0$ of finite
multiplicity.  Let $\eta_j = 1$ on the support of $\phi_j$,
$\supp(\eta_j) \subset B(x_j, 4\epsilon)$. We then have
\begin{multline*}
    \nu_{t + 2, p}(u)^p :=  \sum_j \|(\phi_j u) \circ
    \psi_j^{-1}\|_{W^{t+2,p}(\RR^n)}^p \\ \le C \sum_j \Big( \|P_x
    (\phi_j u)\|_{W^{t, p}(T_xM_0)}^p + \|\phi_j u\|_{W^{1,
    p}(T_xM_0)}^p \Big) \\ \le C \sum_j \Big( \| \phi_j P_x u\|_{W^{t,
    p}(T_xM_0)}^p + \|[P_x, \phi_j] u \|_{W^{t, p}(T_xM_0)}^p +
    \|\phi_j u\|_{W^{1, p}(T_xM_0)}^p \Big) \\ \le C \sum_j \Big( \|
    \phi_j P_x u\|_{W^{t, p}(T_xM_0)}^p + \| \eta_j u \|_{W^{t+1,
    p}(T_xM_0)}^p + \|\phi_j u\|_{W^{1, p}(T_xM_0)}^p \Big) \\ \le C
    \big( \nu_{t, p}(P u)^p + \nu_{t+1}(u)^p \big).
\end{multline*}
The equivalence of the norm $\nu_{s, p}$ with the standard norm on
$W^{s, p}(M_0)$ (Propositions \ref{prop.alt.desc} and
\ref{prop.alt.desc1}) shows that $\|u\|_{W^{t + 2, p}(M_0)} \le C(
\|Pu\|_{W^{t, p}(M_0)} + \|u\|_{W^{t + 1, p}(M_0)} )$, for any
$t$. This is known to imply
\begin{equation}\label{eq.ell.est}\
    \|u\|_{W^{t + 2, p}(M_0)} \le C( \|Pu\|_{W^{t, p}(M_0)} +
    \|u\|_{W^{1, p}(M_0)} )
\end{equation}
by a boot-strap procedure, for any $t \ge -1$. This proves our
statement if $s = 0$ and $\Omega_0 = M_0$.

The case $\Omega_0$ arbitrary follows in exactly the same way, but
using a product type metric in a neighborhood of $\pa \Omega_0$
and the analogue of Equation \eqref{eq.ell.reg} for a half-space,
%% V3.4
which shows that Equation \eqref{eq.ell.est0} continues to hold
for $M_0$ replaced with $\Omega_0$.

The case $s$ arbitrary is obtained by applying Equation
\eqref{eq.ell.est} to the elliptic operator $\rho^{-s} P \rho^{s}
\in \DiffV{2}(M)$ and to the function $\rho^{-s}u \in W^{k,
p}(\Omega_0)$, which then gives Equation \eqref{eq.ell.est0}
righaway.
\end{proof}


%% V3.4 I added the following standard corollary
For $p = 2$, by combining the above theorem (including its proof,
especially Equation \eqref{eq.ell.est0}) with Theorem
\ref{theorem.res}, we obtain the following corollary.


\begin{corollary}\label{cor.reg1}\
We keep the assumptions of Theorem \ref{thm.reg1}. Let $u \in
\rho^{s}H^{1}(\Omega_0)$ be such that $Pu \in
\rho^{s}H^{t}(\Omega_0)$ and $u \vert_{\pa \Omega_0} \in
\rho^{s}H^{t + 3/2}(\Omega_0)$, $s \in \RR$, $t \in \ZZ$. Then $u
\in \rho^{s}H^{t + 2}(\Omega_0)$ and
\begin{equation}\label{eq.ell.estB}
    \|u\|_{\rho^{s}H^{t + 2}(\Omega_0)} \le
    C(\|Pu\|_{\rho^{s}H^{t}(\Omega_0)} +
    \|u\|_{\rho^{s}H^{1}(\Omega_0)} +
    \|u\vert_{\pa \Omega_0}\|_{\rho^{s}H^{1}(\Omega_0)}).
\end{equation}
\end{corollary}


\begin{proof}\ For $u\vert_{\pa \Omega_0} = 0$, the result follows
from Equation \eqref{eq.ell.est0}, with $M_0$ replaced by
$\Omega_0$, which is proved as explained in the proof of Theorem
\ref{thm.reg1}. In general, choose $v \in H^{t+2}(\Omega_0)$ such
that $v\vert_{\pa \Omega_0} = u\vert_{\pa \Omega_0}$, which is
possible by Theorem \ref{theorem.res}. Then we use our result for
$u-v$.
\end{proof}







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Yamabe functional on Lie manifolds\label{sec.appl}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



As an example of the consequences and applications of the analysis
developped in the preceeding sections, we will show that the
Yamabe functional on a Lie manifold is bounded from below.

As before, let $(M_0,g)$ be a Riemannian Lie manifold, with
$(M,\VV)$ the Lie structure at infinity. Recall that the scalar
curvature $\scal_g(p)$ at $p\in M_0$ is defined as
$\scal_g(p)=\sum_{i,j=1}^n \<R(e_i,e_j)e_j,e_i\>$, where $e_1,
e_2, \ldots,e_n$ denotes an orthonormal basis of $T_p M_0$.


We define the Yamabe functional for functions $u\in \CIc(M_0)$
\begin{equation*}
    Y(u)=\inf \frac{\int_M (4\,\frac{n-1}{n-2} |du|^2 +
    u^2 \scal_g )\,\dvol_g}{\|u\|^2_{L^p(M_0,g)}},
\end{equation*}
where $p=2n/(n-2)$. The infimum is a conformal invariant. It is
called the \emph{Yamabe constant} of $(M_0, g)$.
%% The literature does not agree whether this should be the
%% Yamabe invariant or Yamabe constant.

The geometric problem of finding a minimizer of $Y$ is called the
Yamabe problem. The interest in this problem is due to the
following observation. If a minimizer $u$ exists, then $|u|$ is an
everywhere strictly positive minimizer of $Y$. Then, the Euler
Lagrange equation in this case is equivalent to the statement that
the metric $|u|^{4/(n-2)}g$ is a metric of constant scalar
curvature.

Intensive investigations from 1960 to 1987 (see
\cite{lee.parker:87} for a nice overview, and \cite{trudinger:68},
\cite{aubin:76}, \cite{schoen:84, schoen.yau:79a, schoen.yau:81}
for original literature) lead to the celebrated result that a
minimizer of $Y$ always exists on compact manifolds.


\begin{remark}\
The original motivation for studying the Yamabe functional comes
from the following observation (essentially due to Einstein and
Hilbert): We view $Y$ as a functional on the space of all metrics
$g$ on $M$, and positive functions $u$ on $M$; then $(g,u)$ a
stationary point of $Y$ (with respect to compactly supported
perturbations of $g$ and $u$) if and only if $u^{4/(n-2)}g$ is an
Einstein manifold (see e.g.\ \cite{besse:87}). Yamabe's idea
\cite{yamabe:60} was to find such stationary points via a minimax
procedure. As a first step one mimimzes $u$ for fixed $g$, as
described above. Then, as a second step one takes the maximum over
all metrics $g$. However, Yamabe's program did not succeed as
Yamabe was not aware of analytical difficulties in his approach.
As indicated above, on compact manifolds, the first step in his
program (\ie the solution of the Yamabe problem) could be
repaired. However, the second step cannot be repaired, there are
obstructions to the existence of an Einstein metric.
\end{remark}

For various reasons, one is interested in finding minimizers of
$Y$ on a large class of non-compact, but complete manifolds. E.g.\
to derive glueing formulas that describe the behavior of the
Yamabe constant under taking a connected sum, it is helpful to
find minimizers of $Y$ on manifolds with cylindrical ends
\cite{akutagawa.botvinnik}.

The following proposition is a preliminary step in the solution of
the Yamabe problem on Lie manifolds.

\begin{proposition}
The Yamabe functional extends to a continuous functional on
$W^{1,2}(M_0)$, and it is bounded from below.
\end{proposition}


\begin{proof}
Because of bounded geometry
\begin{equation*}
    \int_{M_0} 4\,\frac{n-1}{n-2} |du|^2 + {\rm scal}_g
    u^2\leq C \|u\|^2_{W^{1,2}(M_0)}.
\end{equation*}
Theorem~\ref{theorem.Sobolev} tells us that $W^{1,2}(M_0)$ embeds
into $L^{\frac{2n}{n-2}}(M_0)$, hence the functional extends to
$W^{1,2}(M_0)$, and the constant from the embedding yields a lower
bound for the Yamabe functional.
\end{proof}

It would be interesting to obtain a criterion under which the
infimum of he functional is attained on general non-compact Lie
manifolds.

Another important case, in which the Yamabe constant on a Lie
manifold is of central importance, is general relativity theory. A
preliminary step to obtaining solutions to the Einstein equations
on a space-time is to obtain solutions to the Einstein constraint
equations on a asymptotically euclidean or hyperbolic Riemannian
manifold $M$ of dimension $3$, embedded as a space-like
hypersurface into a space-time. The constraint equation is the
compatibility condition at the metric and at the second
fundamental form on $M$ in order that it can be extended to a
solution of an Einstein metric on a space-time.

The ``conformal-cmc method'' is a method to obtain solutions to
these Einstein constraint equations. After having chosen a
divergence free symmetric $(0,2)$-tensor $\sigma$ on $M$ and an
additional real constant $\tau$ in the space-time, each solution
the \emph{Lichnerowicz} equation
\begin{equation*}
    8\Delta u + {\rm scal} \cdot u -|\sigma|^2 u^{-7}=0
\end{equation*}
gives rise to a solution of the Einstein constraint equations with
mean curvature $\tau$. Whether the Lichnerowicz equation has
solutions depends on the Yamabe constant of $M$
\cite{isenberg-flat, isenberg-compact, isenberg-hyperbolic,
maxwell:p03}.

String theory suggests to search for solutions to the Einstein
constraint equations on $9$, $10$ and $25$ dimensional manifolds
with other kind of asymptotic at infinity.

\begin{remark}
Most of the results of this paper, in particular
Corollary~\ref{cor.n.mult}, Proposition~\ref{prop.Sobolev}, and
Theorem~\ref{theorem.Kondrachov}, still hold if one replaces
functions by sections of a vector bundle, if this vector bundle
extends to $M$ in the sense of vector bundles with metric and
connection.

For the special case that $\pa M=\emptyset$ (\ie $M_0$ compact)
these statements are applied to sections of the spinor bundle in
\cite{ammann.habil,ammann.comin}. This proves the existence of a
maximizer of the functional
\begin{equation*}
    {\mathcal F}(\phi) = \frac{\int_M \< D\phi,\phi \>}
    {\|D\phi\|_{L^q(M)}^2}, \quad q = 2n/(n+1),
\end{equation*}
where $\phi$ is a section of the spinor bundle, and $D$ is the
Dirac operator. The functional is bounded from above  because of
the boundedness of the Sobolev embedding
\begin{equation*}
    W^{1,2}(M_0)\to W^{1/2,q}(M_0).
\end{equation*}
Maximizers of ${\mathcal F}$ satisfy $D\phi =
c|\phi|^{2/(n+1)}\phi$. If dim $M=2$, then the spinorial
Weierstrass representation tells us that these solutions represent
constant mean curvature surfaces in $\RR^3$ and $S^3$.

Extensions of this functional to non-compact Lie manifolds are the
object of current research.
\end{remark}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Submanifolds\label{sec.SM}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In this section we introduce various classes of submanifolds of a
Lie manifold. Some of these classes were already used in the
previous sections.


\subsection{General submanifolds}
We first introduce the most general class of  submanifolds of a
Lie manifold.

We first fix some notation. Let $(M_0, M, A)$ and  $(N_0, N, B)$
be Lie manifolds. We know that there exist vector bundles $A \to
M$ and $B \to N$ such that $\VV \simeq \Gamma(M, A)$ and $\WW
\simeq \Gamma(N, B)$, see Remark \ref{rem.A}. We can assume that
$\VV = \Gamma(M, A)$ and $\WW = \Gamma(N, B)$.


\begin{definition}\label{def.submanif}\
Let $(M_0, M, A)$ be a Lie manifold. Then $(N_0, N, B)$ is called
a {\em submanifold} of $(M_0, M, A)$ if
\begin{enumerate}[{\rm (i)}]
\item $N$ is a closed submanifold of $M$ (possibly with corners, no
transversality at the boundary required),
\item\label{sp2} $N_0 \subset M_0$, $\partial N \subset \partial M$,
and $B \subset A\vert_{N}$.
\item $(N_0, N, B)$ is a Lie manifold,
\item \label{c.BA.rel}the Lie structures at infinity satisfy the
compatibility condition
\begin{equation*}
    \Gamma(N; B) = \{X\in \Gamma(N;A|_N)\,|\,
    \rho\circ X \in \Gamma(N;TN)\}
\end{equation*}
\end{enumerate}
\end{definition}


We now make three simple observations.


\begin{remark}\ An alternative form of Condition
\eqref{c.BA.rel} of the above definition is
\begin{equation}\label{eq.alt.f}
    \Gamma(N; B) = \{X\vert_N\,|\,  X \in \Gamma(M,A)\mbox{ and }
    X\vert_N \text{ tangent to } N\}.
\end{equation}
\end{remark}


\begin{remark}\ Equation \ref{eq.alt.f} shows that there exists a
natural vector bundle morphism $f : B \to A$. Since $B_x = T_xN_0
\subset T_xM_0 = A_x$ for $x \in N_0$, the map $f$ is injective
above $N_0$. The assumption \ref{sp2} of our definition implies
that $f$ is injective everywhere.
\end{remark}


We have the following simple corollary that justifies Condition
\eqref{c.BA.rel} of Definition \ref{def.submanif}.


\begin{corollary}\label{cor.res.met}\
Let $g_0$ be a metric on $M_0$ compatible with the Lie structure
at infinity on $M_0$. Then the restriction of $g$ to $N_0$ is
compatible with the Lie structure at infinity on $N_0$.
\end{corollary}


\begin{proof}\ Let $g$ be a metric on $A$ whose restriction to
$TM_0$ defines the metric $g$. Then $g$ restricts to a metric $h$
on $B$, which in turn defines a metric $h_0$ on $N_0$. By
definition, $h_0$ is the restriction of $g$ to $N_0$.
\end{proof}


We thus see that any submanifold (in the sense of the above
definition) of a Riemannian Lie manifold is itself a Riemannian
Lie manifold.



\subsection{Second fundamental form}
We define the \emph{$A$-normal bundle} of the sub-manifold $(N_0,
N, B)$ of $(M_0, M, A)$ as $\smb = (A|_N)/B$ which is a bundle
over $N$. Then the anchor map $\varrho$ defines a map $\rho^\nu:
\smb \to (TM|_N)/TN$ which is an isomorphism over $N_0$.

We denote the Levi-Civita-connection on $A$ by $\nabla^A$ and the
Levi-Civita connection on $B$ by $\nabla^B$ \cite{aln1}. Let
$X,Y\in \VV = \Gamma(M, A)$ such that $X\vert_{N_0}, Y\vert_{N_0}
\in TN_0$. Then $\nabla^A_X Y|_N$ depends only on $X\vert_{N},
Y\vert_{N} \in \WW = \Gamma(N, B)$. Furthermore the Koszul formula
implies that $\nabla^B_X Y$ is the tangential part of $\nabla^A_X
Y|_N$. The normal part gives rise to the \emph{second fundamental
form}
\begin{equation*}
    \II:B\times B\to \smb, \quad \II(X,Y)=
    \nabla^A_X Y - \nabla^B_X Y.
\end{equation*}
The Levi-Civita connections $\nabla^A$ and $\nabla^B$ are torsion
free, and hence $\II(X,Y)-\II(Y,X)=[X,Y]-[Y,X]=0$ is symmetric. A
direct computation reveals also that $\II(X,Y)$ is tensorial in
$X$, and hence, because of the symmetry, it is also tensorial in
$Y$. (``Tensorial'' here means $\II(fX, Y) = f\II(X, Y)$.) It then
follows from the compactness of $N$ that
\begin{equation*}
    \|\II_p(X_p,Y_p)\|\leq C \|X_p\|\,\|Y_p\|,
\end{equation*}
with a constant $C$ independent of $p \in N$. Clearly, on the
interior $N_0\subset M_0$ the second fundamental form coincides
with the classical second fundamental form.


\begin{corollary}
Let $(N_0,N,B)$ be a submanifold of $(M_0,M,A)$ with a compatible
metric. Then the (classical) second fundamental form of $N_0$ in
$M_0$ is uniformly bounded.
\end{corollary}


\subsection{Tame submanifolds} We now introduce tame manifolds.
Our main interest in tame manifolds is the tubular neighborhood
theorem, Theorem \ref{thm.tub.neigh}, which asserts that a tame
submanifold of a Lie manifold has a tubular neighborhood in a
strong sense. In particular, we will obtain that a tame
submanifold of codimension one is regular. This is interesting
because being tame is an algebraic condition that can be easily
verified by looking at the structural Lie algebras of vector
fields. On the other hand, being a regular submanifold is an
analytic condition on the metric that is very difficult to check
directly.


\begin{definition}\label{def.submanif.ii}\
A submanifold $(N_0,N,B)$ of a Lie manifold $(M_0,M,A)$ is called
{\em tame} if $T_pN$ and $\varrho(A_p)$ span $T_pM$ for all $p \in
\partial N$.
\end{definition}


As a consequence of these properties the anchor map $\varrho$
defines an isomorphism from $A_p/B_p$ to $T_pM/T_pN$ for any $p\in
N$. In particular, the anchor map $\rho$ maps $B^\perp$, the
orthogonal complement of $B$ in $A$, injectively into
$\varrho(A)\subset TM$. For any boundary face $F$ and $p\in F$ we
have $\rho(A_p)\subset T_pF$. Hence, $N$ is transversal to $F$,
\ie for any $p\in N\cap F$, the space $T_pM$ is spanned by $T_pN$
and $T_pF$. As a consequence, $N \cap F$ is a submanifold of $F$
of codimension $\dim M -\dim N$. The codimension of $N \cap F$ in
$F$ is therefore independent of $F$, in particular independent of
the dimension of $F$.


\begin{examples}\ \\[-5mm]
\begin{enumerate}[{\rm (1)}]
\item Let $M$ be any compact manifold (without boundary).
Fix a $p\in M$. Let $(N_0,N,B)$ be a manifold with a Lie structure
at infinity. Then $(N_0\times \{p\},N\times \{p\},B)$ is a tame
submanifold of $(N_0\times M,N\times M,B\times TM)$.
\item If $\partial N\neq \emptyset$, the diagonal
$N$ is a submanifold of $N\times N$, but not a tame submanifold.
\item Let $N$ be a submanifold with corners of $M$ (so $N$ is
transverse to all faces of $M$). We endow these manifolds with the
$b$-structure at infinity $\VV_b$ (see Example \ref{ex1} (i)).
Then $(N, \VV_b)$ is a tame Lie submanifold of $(M, \VV_b)$.
\end{enumerate}
\end{examples}


We now prove the main theorem of this section.


\begin{theorem}[Global tubular neighborhood theorem]
\label{thm.tub.neigh}  Let $(N_0,N,B)$ be a tame submanifold of
the Lie manifold $(M_0,M,A)$. For $\epsilon>0$, let $(\nu)_\epsilon$
be the set of all vectors normal to $N$ of length smaller than
$\epsilon$. If $\epsilon > 0$ is sufficiently small, then the normal
exponential map $\exp^\nu$ defines a diffeomorphism from
$(\nu)_\epsilon$ to an open neighborhood $V_\epsilon$ of~$N$ in
$M$. Moreover, $dist(\exp^\nu(X),N)=|X|$ for $|X|<\ep$.
\end{theorem}


\begin{proof}\
Recall from \cite{aln1} that the exponential map $\exp: TM_0 \to M_0$
extends to a map $\exp: A \to M$. The definition of the normal
exponential function $\exp^\nu$ is obtained by identifying the
quotient bundle $\nu^A$ with $B^{\perp}$, as discussed earlier. This
gives
\begin{equation*}
    \exp^\nu : (\nu)_\epsilon \to M.
\end{equation*}
The differential $d\exp^\nu$ at $0_p\in \nu_p$, $p\in N$ is the
identity, hence any point $p\in N$ has a neighborhood $U(p)$ and
$\tau_p>0$ such that
\begin{equation}\label{map.res}
    \exp^\nu:(\nu)_{\tau_p}|_{U_p}\to M
\end{equation}
is a diffeomorphism onto its image. By compactness $\tau_p\geq
\tau>0$. Hence, $\exp^\nu$ is a local diffeomorphism of $(\nu)_\tau$
to a neighborhood of~$N$ in $M$. It remains to show that it is
injective for small $\epsilon\in(0,\tau)$.

Let us assume now that there is no $\epsilon>0$ such that the
proposition holds. Then there are sequences $X_i,Y_i\in \nu$, $i
\in \NN$, $X_i\neq Y_i$ such that $\exp^\nu X_i=\exp^\nu Y_i$ with
$|X_i|,|Y_i|\to 0$ for $i \to \infty$. After taking a subsequence
we can assume that the basepoints $p_i$ of $X_i$ converge to
$p_\infty$ and the basepoints $q_i$ of $Y_i$ converge to
$q_\infty$. As the distance in $M$ of $p_i$ and $q_i$ converges to
$0$, we conclude that $p_\infty=q_\infty$. However, $\exp^\nu$ is
a diffeomorphism from $(\nu)_{\tau}|_{U(p_\infty)}$ into a
neighborhood of $U(p_\infty)$. Hence, we see that $X_i=Y_i$ for
large $i$, which  contradicts the assumptions.
\end{proof}


We now prove that every tame, codimension one Lie submanifold is
regular.


\begin{proposition}\label{prop.prod.struct}\
Let $(N_0,N,B)$ be a tame submanifold of codimension one of
$(M_0,M,A)$. Fix a diffeomorphism
\begin{equation*}
    \exp^\nu:(\nu)_\epsilon\cong N\times (-\epsilon,\epsilon)\to
    \{x\,|\,d(x,N)<\epsilon\}:=U_\epsilon
\end{equation*}
as in Theorem \ref{thm.tub.neigh}. Then $M_0$ carries a compatible
metric $g$ such that $(\exp^\nu)^*g$ is a product metric, \ie
$(\exp^\nu)^*g = g_N + dt^2$ on $N \times
(-\epsilon/2,\epsilon/2)$.
\end{proposition}


\begin{proof}\
Choose any compatible metric $g_1$ on $M_0$. Let $g_2$ be a metric
on $U_\epsilon$ such that $(\exp^\nu)^*g_2 = g_1|_N + dt^2$ on
$N\times (-\epsilon, \epsilon)$. Let $d(x) := dist(x, N)$. Then
\begin{equation*}
    g =\chi\circ d g_1 +(1-\chi\circ d) g_2,
\end{equation*}
has the desired properties, where the cut-off function $\chi : \RR
\to [0,1]$ is $1$ on $(-\epsilon/2, \epsilon/2)$ and has support
in $(-\epsilon,\epsilon)$, and satisfies $\chi(-t)=\chi(t)$.
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Pseudodifferential operators\label{sec.PO}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


We now recall the definition of pseudodifferential operators on
$M_0$ generated by a Lie structure at infinity $(M, \VV)$ on
$M_0$.

%------------------------------
\subsection{Definition}
%------------------------------

We fix in what follows a compatible Riemannian metric
$g$ on $M_0$ (that is, a metric coming by restriction from a
metric on the bundle $A \to M$ extending $TM_0$), see Section
\ref{sec.LI}. In order to simplify our discussion below, we shall
use the metric $g$ to trivialize all density bundles on $M$.
Recall that $M_0$ with the induced metric is complete \cite{aln1}.

Let $\exp_x : T_x M_0 \to M_0$ be the exponential map, which is
everywhere defined because $M_0$ is complete. We let
\begin{equation}\label{eq.Phi}
    \Phi : TM_{0} \longrightarrow M_{0} \times M_{0}, \quad
    \Phi(v) := (x, \exp_{x}(-v)), \; v \in T_xM_0,
\end{equation}

If $E$ is a real vector bundle with a metric, we shall denote by
$(E)_r$ the set of all vectors $v$ of $E$ with $|v|< r$. Let
$(M_0^2)_r := \{ (x, y), x, y \in M_0,\, d(x, y) < r\}$. Then the
map $\Phi$ of Equation \eqref{eq.Phi} restricts to a
diffeomorphism $\Phi :(TM_{0})_{r} \to (M_0^2)_r$, for any $0 < r
< r_{\mathrm{inj}}(M_0)$, where $r_{\mathrm{inj}}(M_0)$ is the
injectivity radius of $M_0$, which was assumed to be positive. The
inverse of $\Phi$ is of the form
\begin{equation*}
    (M_0^2)_{r} \ni (x,y) \longmapsto
    (x,\tau(x,y))\in (TM_{0})_{r}\,.
\end{equation*}

We shall denote by $S^m_{1,0}(E)$ the space of symbols of order $m$
and type $(1,0)$ on $E$ (in H\"ormander's sense) and by $S^m_{cl}(E)$
the space of classical symbols of order $m$ on $E$ \cite{hoe67, NS,
Taylor, Taylor2}. See \cite{aln2} for a review of these spaces of
symbols in our framework.

Let $\chi \in \CI(A^*)$ be a smooth function that is equal to $1$
on $(A^*)_{r}$ and is equal to $0$ outside $(A^*)_{2r}$, for some
$r < r_{\mathrm{inj}}(M_0)/3$. Then, following \cite{aln2}, we
define
\begin{equation*}
    q(a)u(x) = (2\pi )^{-n} \int_{T^{*}M_{0}} e^{i\tau(x,y)\cdot\eta}
    \chi(x,\tau(x,y)) a(x,\eta)u(y)\, d \eta \,dy \,.
\end{equation*}
This integral is an oscillatory integral with respect to the
symplectic measure on $T^*M_0$ \cite{hor3}. Alternatively, we consider
the measures on $M_0$ and on $T^*_xM_0$ defined by some choice of a
metric on $A$ and we integrate first in the fibers $T^*_xM_0$ and then
on $M_0$. The map $\sigma_{tot} : S_{1,0}^{m}(A^{*}) \to
\Psi^{m}(M_{0})/\Psi^{-\infty}(M_{0})$,
\begin{equation*}
    \sigma_{tot}(a) := q(a) + \Psi^{-\infty}(M_{0})
\end{equation*}
is independent of the choice of the function $\chi \in \CIc((A)_{r})$
\cite{aln2}.

We now enlarge the class of order $-\infty$ operators that we
consider. Any $X \in \Gamma(A)$ generates a global flow $\Psi_{X}
: \RR\times M\rightarrow M$ because $X$ is tangent to all boundary
faces of $M$ and $M$ is compact.  Evaluation at $t = 1$ yields a
diffeomorphism
\begin{equation}\label{eq.def.psiX}
    \psi_X := \Psi_{X}(1,\cdot) : M \rightarrow M.
\end{equation}

We now define the pseudodifferential calculus on $M_0$ that we
will consider following \cite{aln2}.


\begin{definition}\label{def.psi.MA}\
Fix $0< r < r_{\mathrm{inj}}(M_0)$ and $\chi \in \CIc((A)_{r})$
such that $\chi = 1$ in a neighborhood of $M \subseteq A$.  For $m
\in \RR$, the space $\Psi_{1,0,\VV}^{m}(M_{0})$ of {\em
pseudodifferential operators generated by the Lie structure at
infinity $(M, \VV)$} is defined to be the linear space of
operators $\CIc(M_0) \rightarrow \CIc(M_{0})$ generated by $q(a)$,
$a \in S_{1,0}^m(A^*)$, and $q(b)\psi_{X_1}\ldots \psi_{X_k}$, $b
\in S^{-\infty}(A^*)$ and $X_j \in \Gamma(A)$, $\forall j$.

Similarly, the space $\Psi_{cl,\VV}^{m}(M_{0})$ of {\em classical
pseudodifferential operators generated by the Lie structure at
infinity $(M, \VV)$} is obtained by using classical symbols $a$ in
the construction above.
\end{definition}


We have that $\Psi_{cl,\VV}^{-\infty}(M_{0}) =
\Psi_{1,0,\VV}^{-\infty}(M_{0}) =: \Psi_{\VV}^{-\infty}(M_{0})$
(we dropped some subscripts).


%----------------------------
\subsection{Properties}
%----------------------------

 We now review some properties of the
operators in $\Psi^{m}_{1, 0, \VV}(M_{0})$ and $\Psi^{m}_{cl,
\VV}(M_{0})$ from \cite{aln2}. These properties will be used
below. Let $\Psi^{\infty}_{1, 0, \VV}(M_{0}) = \cup_{m \in \ZZ}
\Psi^{m}_{1, 0, \VV}(M_{0})$ and $\Psi^{\infty}_{cl, \VV}(M_{0}) =
\cup_{m \in \ZZ} \Psi^{m}_{cl, \VV}(M_{0})$.

First of all, each operator $P \in \Psi^{m}_{1, 0, \VV}(M_{0})$
defines continuous maps $\CIc(M_{0}) \rightarrow \CI(M_{0})$, and
$\CI(M) \rightarrow \CI(M)$, still denoted by $P$. An operator $P
\in \Psi^{m}_{1, 0, \VV}(M_{0})$ has a distribution kernel $k_P$
in the space $I^m(M_0 \times M_0, M_0)$ of distributions on $M_0
\times M_0$ that are conormal of order $m$ to the diagonal, by
\cite{hor3}. If $P = q(a)$, then $k_P$ has support in $(M_0 \times
M_0)_r$. If we extend the exponential map $(TM_0)_r \to M_0 \times
M_0$ to a map $A \to M$, then the distribution kernel of $P =
q(a)$ is the restriction of a distribution, also denoted $k_P$ in
$I^m(A, M)$.

If $\maP$ denotes the space of polynomial symbols on $A^*$ and
$\Diff{M_0}$ denotes the algebra of differential operators on
$M_0$, then
\begin{equation}
    \Psi^{\infty}_{1, 0, \VV}(M_{0}) \cap \Diff{M_0}
    = \DiffV{\infty}{(M)} = q(\maP).
\end{equation}

The spaces $\Psi_{1, 0, \VV}^m(M_{0})$ and $\Psi_{1, 0,
\VV}^m(M_{0})$ are independent of the choice of the metric on $A$
and the function $\chi$ used to define it, but depend, in general,
on the Lie structure at infinity $(M,A)$ on $M_0$. They are also
closed under multiplication, which is a quite non-trivial fact.


\begin{theorem}\label{thm.alg}\
The spaces $\Psi_{1,0, \VV}^{\infty}(M_{0})$ and $\Psi_{1,0,
\VV}^{\infty}(M_{0})$ are filtered algebras that are closed under
adjoints.
\end{theorem}


For $\Psi_{1,0, \VV}^m(M_{0})$, the meaning of the above theorem
is that
\begin{equation*}
    \Psi_{1,0, \VV}^m(M_{0}) \Psi_{1, 0, \VV}^{m'}(M_{\VV})
    \subseteq \Psi_{1, 0, \VV}^{m + m'}(M_{0}) \text{ and }
    \big(\Psi_{1,0, \VV}^m(M_{0})\big)^* = \Psi_{1,0,
    \VV}^m(M_{0})
\end{equation*}
for all $m,m' \in \CC \cup \{-\infty\}.$


The usual properties of the principal symbol remain true.


\begin{proposition}\label{prop.princ.symb}\
The principal symbol establishes isomorphisms
\begin{equation}\label{eq.def.princ}
    \sigma^{(m)} : \Psi_{1,0,\VV}^{m}(M_{0}) /
    \Psi_{1,0,\VV}^{m-1}(M_{0}) \to S^m_{1,0}(A^*) /
    S^{m-1}_{1,0}(A^*)
\end{equation}
and
\begin{equation}\label{eq.def.princ'}
    \sigma^{(m)} : \Psi_{cl,\VV}^{m}(M_{0}) /
    \Psi_{cl,\VV}^{m-1}(M_{0}) \to S^m_{cl}(A^*) /
    S^{m-1}_{cl}(A^*).
\end{equation}
Moreover, $\sigma^{(m)}(q(a)) = a + S^{m-1}_{1, 0}(A^*)$ for any
$a \in S^{m}_{1, 0}(A^*)$ and $\sigma^{(m+m')}(PQ) =
\sigma^{(m)}(P)\sigma^{(m')}(Q)$, for any $P \in
\Psi_{1,0,\VV}^{m}(M_{0})$ and $Q \in \Psi_{1,0,\VV}^{m'}(M_{0})$.
\end{proposition}


We shall need also the following result.


\begin{proposition} \label{prop.auto}\
Let $x$ be a defining function of some hyperface of $M$. Then
$\rho^{s} \Psi_{1,0, \VV}^m(M_{0}) \rho^{-s} = \Psi_{1, 0,
\VV}^m(M_{0})$ and $\rho^{s} \Psi_{cl, \VV}^m(M_{0}) \rho^{-s} =
\Psi_{cl, \VV}^m(M_{0})$ for any $s \in \CC$.
\end{proposition}


%----------------------------------------
\subsection{Continuity on $W^{s,p}(M_0)$}
%----------------------------------------


The preparations above will allow us to prove the continuity of the
operators $P \in \Psi_{1, 0, \VV}^m(M_{0})$ between suitable Sobolev
spaces. This is the main result of this section.  Some of the ideas
and constructions in the proof below have already been used in
\ref{thm.reg1}, which the reader may find convenient to review first.


\begin{theorem}\label{thm.Wkp.cont}\
Let $P \in \Psi_{1,0, \VV}^m(M_{0})$ and $p \in (0,\infty)$. Then
$P$ maps $\rho^rW^{s, p}(M_0)$ continuously to $\rho^rW^{s-m,
p}(M_0)$ for any $r, s \in \RR$.
\end{theorem}


\begin{proof}\
We have that $P$ maps $\rho^rW^{s, p}(M_0)$ continuously to
$\rho^rW^{s-m, p}(M_0)$ if, and only if, $\rho^{-r}P\rho^r$ maps
$W^{s, p}(M_0)$ continuously to $W^{s-m, p}(M_0)$. By Proposition
\ref{prop.auto} it is therefore enough to check our result for $r =
0$.

We shall first prove our result if the Schwartz kernel of $P$ has
support close enough to the diagonal. To this end, let us choose
$\epsilon < r_{\mathrm{inj}}(M_0)/9$ and assume that the distribution
kernel of $P$ is supported in the set $(M_0^2)_{\epsilon} := \{(x, y),
d(x, y) < \epsilon\} \subset M_0^2$. This is possible by choosing the
function $\chi$ used to define the spaces $\Psi_{1, 0, \VV}^m(M_0)$ to
have support in the set $(M_0^2)_{\epsilon}$. There will be no loss of
generality then to assume that $P = q(a)$.

Then choose a smooth function $\eta : [0, \infty) \to [0, 1]$,
$\eta(t) = 1$ if $t \le 6\epsilon$, $\eta(t) = 0$ if $t \le
7\epsilon$. Let $\psi_x : B(x, 8\epsilon) \to B_{T_xM_0}(0,
8\epsilon)$ denote the normal system of coordinates induced by the
exponential maps $\exp_x : T_x M_0 \to M_0$. Denote $\pi : A \to M$ be
the natural (vector bundle) projection and
\begin{equation}
    B := A \times_{M}A := \{(\xi_1, \xi_2) \in A \times A, \pi(\xi_1)
    = \pi(\xi_2) \},
\end{equation}
which defines a vector bundle $B \to M$. In the language of vector
bundles, $B := A \oplus A$. For any $x \in M_0$, let $\eta_x$ denote
the function $\eta \circ \exp_x$, and consider the operator $\eta_x P
\eta_x$ on $B(x, 13\epsilon)$. The diffeomorphism $\psi_x$ then will
map this operator to an operator $P_x$ on $B_{T_xM_0}(0,
8\epsilon)$. Then $P_x$ maps continuously $W^{s, p}(T_xM_0) \to
W^{s-m, p}(T_xM_0)$, by the continuity of pseudodifferential operators
on $\RR^n$ \cite[XIII, \S 5]{Taylor3} or \cite{Taylor, StrichartzB}.

The distribution kernel $k_x$ of $P_x$ is a distribution with compact
support on
\begin{equation*}
    T_xM_0 \times T_xM_0 = A_x \times A_x = B_x
\end{equation*}
If $P = q(a) \in \Psi_{1, 0, \VV}^m(M_0)$, then the distributions
$k_x$ can be determined in terms of the distribution $k_P \in I^m(A,
M)$ associated to $P$. This shows that the distributions $k_x$ extend
to a smooth family of distributions on the fibers of $B \to M$. From
this, it follows that the family of operators $P_x : W^{s, p}(A_x) \to
W^{s - m, p}(A_x)$, $x \in M_0$, extends to a family of operators
defined for $x \in M$ (recall that $A_x = T_xM_0$ if $x \in
M_0$). This extension is obtained by extending the distribution
kernels. In particular, the resulting family $P_x$ will depend
smoothly on $x \in M$. Since $M$ is compact, we obtain, in particular,
that the norms of the operators $P_x$ are uniformly bounded for $x \in
M_0$.

By abuse of notation, we shall denote by $P_x : W^{s, p}(M_0) \to
W^{s-m, p}(M_0)$ the induced family of pseudodifferential operators,
and we note that it will still be a smooth family that is uniformly
bounded in norm. Note that it is possible to extend $P_x$ to an
operator on $M_0$ because its distribution kernel has compact support.

Then choose the sequence of points $\{x_j\} \subset M_0$ and a
partition of unity $\phi_j \in \CIc(M_0)$ as in Lemma
\ref{lemma.Shubin}. In particular, $\phi_j$ will have support in
$B(x_j, 2\epsilon)$. Also, let $\psi_j : B(x_j,4\epsilon) \to
B_{\RR^n}(0,4\epsilon)$ denote the normal system of coordinates
induced by the exponential maps $\exp_x : T_x M_0 \to M_0$ and some
fixed isometries $T_xM_0 \simeq \RR^n$. Then all derivatives of
$\psi_j \circ \psi_k^{-1}$ are bounded on their domain of
definition, with a bound that may depend on $\epsilon$ but does not
depend on $j$ and $k$ \cite{CGT, Shubin}.

Let
\begin{equation*}
    \nu_{s,p}(u)^p := \sum_j \|(\phi_j u) \circ
    \psi_j^{-1}\|_{W^{s,p}(\RR^n)}^p.
\end{equation*}
be one of the several equivalent norms defining the topology on $W^{s,
p}(M_0)$ (see Proposition \ref{prop.alt.desc1} and Equation
\eqref{sreal}. It is enough to prove that
\begin{multline}\label{mult.cont}
    \nu_{s,p}(Pu)^p := \sum_j \|(\phi_j Pu) \circ
    \psi_j^{-1}\|_{W^{s,p}(\RR^n)}^p \\ \le C \sum_j \|(\phi_j u)
    \circ \psi_j^{-1}\|_{W^{s,p}(\RR^n)}^p =: C\nu_{s,p}(u)^p,
\end{multline}
for some constant $C$ independent of $u$.

We now prove this statement. Indeed, for the reasons explained below,
we have the following inequalities.
\begin{multline*}
    \sum_j \|(\phi_j Pu) \circ \psi_j^{-1}\|_{W^{s,p}(\RR^n)}^p \le
    C \sum_{j, k} \|(\phi_j P \phi_k u ) \circ
    \psi_j^{-1}\|_{W^{s,p}(\RR^n)}^p \\ = C \sum_{j, k} \|(\phi_j
    P_{x_j} \phi_k u ) \circ \psi_j^{-1}\|_{W^{s,p}(\RR^n)}^p \le C
    \sum_{j, k} \|(\phi_j \phi_k u ) \circ
    \psi_j^{-1}\|_{W^{s,p}(\RR^n)}^p \\ \le C \sum_{j} \|(\phi_j u )
    \circ \psi_j^{-1}\|_{W^{s,p}(\RR^n)}^p = C \nu_{s,p}(u)^p.
\end{multline*}
Above, the first and last inequalities are due to the fact that the
family $\phi_j$ is uniformly locally finite, that is, there exists a
constant $N$ such that at any given point $x$, at most $N$ of the
functions $\phi_j(x)$ are different from zero. The first equality is
due to the support assumptions on $\phi_j$, $\phi_k$, and
$P_{x_j}$. Finally, the second inequality is due to the fact that the
operators $P_{x_j}$ are continuous, with norms bounded by a constant
independent of $j$, as explained above. We have therefore proved that
$P = q(a) \in \Psi_{1, 0, \VV}^m(M_0)$ defines a bounded operator
$W^{s, p}(M_0) \to W^{s-m, p}(M_0)$, provided that the Schwartz kernel
of $P$ has support in a set of the $(M_0^2)_{\epsilon}$, for $\epsilon
< r_{\mathrm{inj}}(M_0)/9$.

Assume now that $P \in \Psi_{\VV}^{-\infty}(M_{0})$. We shall check
that $P$ is bounded as a map $W^{2k, p}(M_0) \to W^{-2k, p}(M_0)$. For
$k = 0$, this follows from the fact that the Schwartz kernel of $P$ is
given by a smooth function $k(x, y)$ such that $\int_{M_0}|k(x,
y)|d\vol_g(x)$ and $\int_{M_0}|k(x, y)|d\vol_g(y)$ are uniformly
bounded in $x$ and $y$. For the other values of $k$, it is enough to
prove that the bilinear form
\begin{equation*}
    W^{2k, p}(M_0) \times W^{2k, p}(M_0) \ni (u, v) \to \<Pu, v\> \in
    \CC
\end{equation*}
is continuous. Choose $Q$ a parametrix of $\Delta^k$ and let $R = 1 -
Q\Delta^k$ be as above. Let $R' = 1 - \Delta^kQ \in
\Psi_{\VV}^{-\infty}(M_{0}).$ Then
\begin{equation*}
    \<Pu, v\> = \<(Q P Q)\Delta^{k} u, \Delta^{k}v\> + \<(Q P R) u,
    \Delta^{k}v\> + \<(R' P Q)\Delta^{k} u, v\> + \<(R' P R) u, v\>,
\end{equation*}
which is continuous since $QPQ, Q P R, R' P Q$, and $R' P R$ are in
$\Psi_{\VV}^{-\infty}(M_{0})$ and hence they are continuous on on
$L^p(M_0)$ and because $\Delta^k : W^{2k, p}(M_0) \to L^p(M_0)$ is
continuous.

Since any $P \in \Psi_{1, 0, \VV}^{m}(M_{0})$ can be written $P = P_1
+ P_2$ with $P_2 \in \Psi_{\VV}^{-\infty}(M_{0})$ and $P_1 = q(a) \in
\Psi_{1, 0, \VV}^{m}(M_{0})$ with support arbitrarily close to the
diagonal in $M_0$, the result follows.
\end{proof}


As in \cite{StrichartzM}[Proposition 1.8], we obtain the following
characterization of Sobolev spaces.


\begin{theorem}\label{theorem.three}\
Let $s \in \RR_+$ and $p \in (1, \infty)$. We have that $u \in W^{s,
p}(M_0)$ if, and only if, $u \in L^p(M_0)$ and $Pu \in L^p(M_0)$ for
any $P \in \Psi_{1, 0, \VV}^s(M_0)$. The norm $u \to \|u\|_{L^p(M_0)}
+ \|Pu\|_{L^p(M_0)}$ is equivalent to the original norm on
$W^{s,p}(M_0)$ for any elliptic $P \in \Psi_{1, 0, \VV}^s(M_0)$.

Similarly, the space $W^{-s, p}(M_0)$ is the quotient of $L^p(M_0)
\oplus L^p(M_0)$ with respect to the map $(u, v) \to u + Pv$.
\end{theorem}


\begin{proof}\
Clearly, if $u \in W^{s, p}(M_0)$, then $Pu, u \in L^p(M_0)$. Let
us prove the converse.


Assume $Pu, u \in L^p(M_0)$. Let $Q \in \Psi_{1, 0,
\VV}^{-s}(M_0)$ be a parametrix of $P$ and let $R, R' \in
\Psi_{\VV}^{-\infty}(M_0)$ be defined by $R := 1 - QP$ and $R' = 1
- PQ$.  Then $u = QPu + Ru$. Since both $Q, R : L^p(M_0) \to W^{s,
p}(M_0)$ are defined and bounded, $u \in W^{s, p}(M_0)$ and
$\|u\|_{W^{s, p}(M_0)} \le C \big( \|u\|_{L^p(M_0)} +
\|Pu\|_{L^p(M_0)} \big )$. This proves the first part.

To prove the second part, we observe that the mapping
\begin{equation*}
    W^{s, q}(M_0) \ni u \to (u, Pu) \in L^q(M_0)
    \oplus L^q(M_0), \qquad q^{-1} + p^{-1} = 1,
\end{equation*}
is an isomorphism onto its image. The result then follows by
duality using the Hahn-Banach theorem.
\end{proof}


We conclude our paper with the sketch of two regularity results on
solutions of elliptic equations. We formulate the first result
only for order two operators with Dirichlet boundary conditions,
in order to avoid a discussion of regular boundary conditions
\cite{Taylor1} in our setting.

The proof of the following result is a standard application of the
previous ideas. Recall the Sobolev spaces with weights $\rho^sW^{s,
p}(\Omega_0)$ introduced in Equation \eqref{eq.def.weights}.


\begin{theorem}\label{thm.reg2}\
Let $P \in \DiffV{m}(M)$ be an order $m$ elliptic operator on
$M_0$ generated by $\VV$. Let $u \in \rho^{s}W^{r, p}(M_0)$ be
such that $Pu \in \rho^{s}W^{t, p}(M_0)$, $s, r, t \in \RR$, $1 <
p < \infty$. Then $u \in \rho^{s}W^{t + m, p}(M_0)$.
\end{theorem}


\begin{proof}\ Let $Q \in \Psi_{\VV}^{-\infty}(M_0)$
be a parametrix of $P$. Then $R = I - QP \in
\Psi_{\VV}^{-\infty}(M_0)$. This gives $u = Q(Pu) + Ru$. But
$Q(Pu) \in \rho^{s}W^{t + m, p}(M_0)$, by Theorem
\ref{thm.Wkp.cont}, because $Pu \in \rho^{s}W^{t, p}(M_0)$.
Similarly, $Ru \in \rho^{s}W^{t + m, p}(M_0)$. This completes the
proof.
\end{proof}


Note that the above theorem was already proved in the case $t \in
\ZZ$ and $m = 2$, using more elementary methods, as part of
Theorem \ref{thm.reg1}. The proof here is much shorter, however,
which attests to the power of pseudodifferential operator algebra
techniques.


%\bibliographystyle{siam}
%\bibliography{sobolev}

\def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\begin{thebibliography}{10}

\bibitem{Adams}
{\sc R.~A. Adams}, {\em Sobolev spaces}, Pure and Applied Mathematics, Vol. 65,
  Academic Press, New York-London, 1975.

\bibitem{akutagawa.botvinnik}
{\sc K.~Akutagawa and B.~Botvinnik}, {\em Yamabe metrics on cylindrical
  manifolds}, Geom. Funct. Anal., 13 (2003), pp.~259--333.

\bibitem{ammann.comin}
{\sc B.~Ammann}, {\em The smallest {D}irac eigenvalue in a spin-conformal class
  and cmc-immersions}, {P}reprint, 2003.
\newblock {A}r{X}iv math.DG/0309061.

\bibitem{ammann.habil}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em A Variational
  Problem in Conformal Spin Geometry}, Habilitationsschrift, Universit\"at
  Hamburg, 2003.

\bibitem{aln1}
{\sc B.~Ammann, R.~Lauter, and V.~Nistor}, {\em On the {R}iemannian geometry of
  manifolds with a {L}ie structure at infinity}.
\newblock to appear in Int. J. Math. {\&} Math. Sc.

\bibitem{aln2}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Pseudodifferential
  operators on manifolds with a {L}ie structure at infinity}.
\newblock Preprint, December 2002.

\bibitem{anderson:p03}
{\sc M.~Anderson}, {\em Dehn filling and {E}instein metrics in higher
  dimensions}.
\newblock Preprint ArXiv math.DG/0303260.

\bibitem{Aubin}
{\sc T.~Aubin}, {\em Espaces de {S}obolev sur les vari\'et\'es riemanniennes},
  Bull. Sc. Math., 100 (1970), pp.~149--173.

\bibitem{aubin:76}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em {{\'E}quations
  diff{\'e}rentielles non lin{\'e}aires et probl{\`e}me de Yamabe concernant la
  courbure scalaire.}}, J. Math. Pur. Appl., IX. Ser., 55 (1976), pp.~269--296.

\bibitem{BabuPA}
{\sc I.~Babu{\v{s}}ka, T.~von Petersdorff, and B.~Andersson}, {\em Numerical
  treatment of vertex singularities and intensity factors for mixed boundary
  value problems for the {L}aplace equation in {${\bf R}\sp 3$}}, SIAM J.
  Numer. Anal., 31 (1994), pp.~1265--1288.

\bibitem{BNZ2}
{\sc C.~Bacuta, V.~Nistor, and L.~Zikatanov}, {\em Boundary value on polyhedra
  (tentative title)}.
\newblock work in progress.

\bibitem{BNZ}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em A note on improving
  the rate of convergence of `high order finite elements' on polygons}.
\newblock ESI Preprint no. 1280, 2003.

\bibitem{BerghLofstrom}
{\sc J.~Bergh and J.~L{\"o}fstr{\"o}m}, {\em Interpolation spaces. An
  introduction}, Springer-Verlag, Berlin, 1976.
\newblock Grundlehren der Mathematischen Wissenschaften, No. 223.

\bibitem{besse:87}
{\sc A.~L. Besse}, {\em Einstein manifolds}, vol.~10 of Ergebnisse der
  Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1987.

\bibitem{CGT}
{\sc M.~Cheeger, J.~Gromov and M.~Taylor}, {\em Finite propagation speed,
  kernel estimates for functions of the {L}aplace operator, and the geometry of
  complete {R}iemannian manifolds}, J. Differential Geom., 17 (1982),
  pp.~15--53.

\bibitem{isenberg-flat}
{\sc Y.~Choquet-Bruhat, J.~Isenberg, and J.~W. York, Jr.}, {\em Einstein
  constraints on asymptotically {E}uclidean manifolds}, Phys. Rev. D (3), 61
  (2000), pp.~084034, 20.

\bibitem{Cordes}
{\sc H.~Cordes}, {\em Spectral theory of linear differential operators and
  comparison algebras}, London Mathematical Society, Lecture Notes Series 76,
  Cambridge University Press, Cambridge - London - New York, 1987.

\bibitem{CO2}
{\sc H.~Cordes and R.~McOwen}, {\em The {$C\sp*$}-algebra of a singular
  elliptic problem on a noncompact {R}iemannian manifold}, Math. Z., 153
  (1977), pp.~101--116.

\bibitem{CO1}
{\sc H.~O. Cordes and R.~C. McOwen}, {\em Remarks on singular elliptic theory
  for complete {R}iemannian manifolds}, Pacific J. Math., 70 (1977),
  pp.~133--141.

\bibitem{Costabel}
{\sc M.~Costabel}, {\em Boundary integral operators on curved polygons}, Ann.
  Mat. Pura Appl. (4), 133 (1983), pp.~305--326.

\bibitem{Dauge}
{\sc M.~Dauge}, {\em Elliptic boundary value problems on corner domains},
  vol.~1341 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988.
\newblock Smoothness and asymptotics of solutions.

\bibitem{Evans}
{\sc L.~C. Evans}, {\em Partial differential equations}, vol.~19 of Graduate
  Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.

\bibitem{gaffney}
{\sc M.~P. Gaffney}, {\em The harmonic operator for exterior differential
  forms}, Proc. Nat. Acad. Sci. U. S. A., 37 (1951), pp.~48--50.

\bibitem{GilbargTrudinger}
{\sc D.~Gilbarg and N.~S. Trudinger}, {\em Elliptic partial differential
  equations of second order}, vol.~224 of Grundlehren der Mathematischen
  Wissenschaften, Springer-Verlag, Berlin, 1977.

\bibitem{Grisvard1}
{\sc P.~Grisvard}, {\em Elliptic problems in nonsmooth domains}, vol.~24 of
  Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program),
  Boston, MA, 1985.

\bibitem{Grisvard2}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Singularities in
  boundary value problems}, vol.~22 of Research in Applied Mathematics, Masson,
  Paris, 1992.

\bibitem{BabuGuo}
{\sc B.~Guo and I.~Babu{\v{s}}ka}, {\em Regularity of the solutions for
  elliptic problems on nonsmooth domains in {$\bold R\sp 3$}. {I}. {C}ountably
  normed spaces on polyhedral domains}, Proc. Roy. Soc. Edinburgh Sect. A, 127
  (1997), pp.~77--126.

\bibitem{Hebey2}
{\sc E.~Hebey}, {\em Sobolev spaces on {R}iemannian manifolds}, vol.~1635 of
  Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1996.

\bibitem{Hebey1}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Nonlinear analysis
  on manifolds: {S}obolev spaces and inequalities}, vol.~5 of Courant Lecture
  Notes in Mathematics, New York University Courant Institute of Mathematical
  Sciences, New York, 1999.

\bibitem{hoe67}
{\sc L.~H\"ormander}, {\em Pseudo-differential operators and hypoelliptic
  equations}, vol.~X of Proc.\ Symp.\ in Pure Math.\ -- Singular Integrals,
  Amer.\ Math.\ Soc., Providence, Rhode Island, 1967, pp.~138--183.

\bibitem{hor3}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em The analysis of
  linear partial differential operators, {\rm vol.\ 3}. {P}seudo-differential
  operators}, vol.~274 of Grundlehren der Mathematischen Wissenschaften,
  Springer-Verlag, Berlin - Heidelberg - New York, 1985.

\bibitem{isenberg-compact}
{\sc J.~Isenberg}, {\em Constant mean curvature solutions of the {E}instein
  constraint equations on closed manifolds}, Classical Quantum Gravity, 12
  (1995), pp.~2249--2274.

\bibitem{isenberg-hyperbolic}
{\sc J.~Isenberg and J.~Park}, {\em Asymptotically hyperbolic non-constant mean
  curvature solutions of the {E}instein constraint equations}, Classical
  Quantum Gravity, 14 (1997), pp.~A189--A201.
\newblock Geometry and physics.

\bibitem{JeffresLoya}
{\sc T.~Jeffres and P.~Loya}, {\em Regularity of solutions of the heat equation
  on a cone}, Int. Math. Res. Not.,  (2003), pp.~161--178.

\bibitem{JerisonKenig}
{\sc D.~Jerison and C.~E. Kenig}, {\em The inhomogeneous {D}irichlet problem in
  {L}ipschitz domains}, J. Funct. Anal., 130 (1995), pp.~161--219.

\bibitem{Karoubi}
{\sc M.~Karoubi}, {\em Homologie cyclique et {K}-theorie}, Asterisque, 149
  (1987), pp.~1--147.

\bibitem{KlainermanSel}
{\sc S.~Klainerman and S.~Selberg}, {\em Bilinear estimates and applications to
  nonlinear wave equations}, Commun. Contemp. Math., 4 (2002), pp.~223--295.

\bibitem{Kondratiev}
{\sc V.~A. Kondrat{\cprime}ev}, {\em Boundary value problems for elliptic
  equations in domains with conical or angular points}, Transl. Moscow Math.
  Soc., 16 (1967), pp.~227--313.

\bibitem{Lauter}
{\sc R.~Lauter}, {\em Pseudodifferential analysis on conformally compact
  spaces}, Mem. Amer. Math. Soc., 163 (2003).

\bibitem{LauterMoroianu}
{\sc R.~Lauter and S.~Moroianu}, {\em Homology of pseudodifferential operators
  on manifolds with fibered cusps}, Trans. Amer. Math. Soc., 355 (2003),
  pp.~3009--3046.

\bibitem{LN1}
{\sc R.~Lauter and V.~Nistor}, {\em Analysis of geometric operators on open
  manifolds: a groupoid approach}, in Quantization of Singular Symplectic
  Quotients, N.~Landsman, M.~Pflaum, and M.~Schlichenmaier, eds., vol.~198 of
  Progress in Mathematics, Birkh{\"a}user, Basel - Boston - Berlin, 2001,
  pp.~181--229.

\bibitem{lee.parker:87}
{\sc J.~M. Lee and T.~H. Parker}, {\em {The Yamabe problem.}}, Bull. Am. Math.
  Soc., New Ser., 17 (1987), pp.~37--91.

\bibitem{Maniccia}
{\sc L.~Maniccia and P.~Panarese}, {\em Eigenvalue asymptotics for a class of
  md-elliptic {$\psi$}do's on manifolds with cylindrical exits}, Ann. Mat. Pura
  Appl. (4), 181 (2002), pp.~283--308.

\bibitem{maxwell:p03}
{\sc D.~Maxwell}, {\em Solutions of the {E}instein constraint equations with
  apparent horizon boundary}, preprint, {Arxiv: \tt gr-qc/0307117}, 2003.

\bibitem{Mazya}
{\sc V.~Maz{\cprime}ya, S.~Nazarov, and B.~Plamenevskij}, {\em Asymptotic
  theory of elliptic boundary value problems in singularly perturbed domains.
  {V}ol. {I}}, vol.~111 of Operator Theory: Advances and Applications,
  Birkh\"auser Verlag, Basel, 2000.
\newblock Translated from the German by Georg Heinig and Christian Posthoff.

\bibitem{Mazzeo}
{\sc R.~Mazzeo}, {\em {Elliptic theory of differential edge operators. I.}},
  Commun. Partial Differ. Equations, 16 (1991), pp.~1615--1664.

\bibitem{MaMeAsian}
{\sc R.~Mazzeo and R.~B. Melrose}, {\em Pseudodifferential operators on
  manifolds with fibred boundaries}, Asian J. Math., 2 (1998), pp.~833--866.

\bibitem{me81}
{\sc R.~B. Melrose}, {\em Transformation of boundary value problems}, Acta
  Math., 147 (1981), pp.~149--236.

\bibitem{meicm}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Pseudodifferential
  operators, corners and singular limits}, in Proceeding of the International
  Congress of Mathematicians, Kyoto, Berlin - Heidelberg - New York, 1990,
  Springer-Verlag, pp.~217--234.

\bibitem{meaps}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em {The
  Atiyah-Patodi-Singer index theorem.}}, {Research Notes in Mathematics
  (Boston, Mass.). 4. Wellesley, MA: A. K. Peters, Ltd.. xiv, 377 p. }, 1993.

\bibitem{MelroseScattering}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Geometric scattering
  theory}, Stanford Lectures, Cambridge University Press, Cambridge, 1995.

\bibitem{MelroseMendoza}
{\sc R.~B. Melrose and G.~Mendoza}, {\em Elliptic operators of totally
  characteristic type}.
\newblock MSRI Preprint.

\bibitem{MitreaNistor}
{\sc M.~Mitrea and V.~Nistor}, {\em Boundary layer potentials on manifolds with
  cylindrical ends}.
\newblock ESI Preprint no.~1244, 2002.

\bibitem{MitreaTaylor}
{\sc M.~Mitrea and M.~Taylor}, {\em Boundary layer methods for {L}ipschitz
  domains in {R}iemannian manifolds}, J. Funct. Anal., 163 (1999),
  pp.~181--251.

\bibitem{NS}
{\sc A.~Nagel and E.~Stein}, {\em Lectures on pseudodifferential operators:
  regularity theorems and applications to nonelliptic problems}, vol.~24 of
  Mathematical Notes, Princeton University Press, Princeton, N.J., 1979.

\bibitem{NP}
{\sc S.~Nazarov and B.~Plamenevsky}, {\em Elliptic problems in domains with
  piecewise smooth boundaries}, vol.~13 of de Gruyter Expositions in
  Mathematics, Walter de Gruyter \& Co., Berlin, 1994.

\bibitem{Necas}
{\sc J.~Ne{\v{c}}as}, {\em Les m\'ethodes directes en th\'eorie des \'equations
  elliptiques}, Masson et Cie, \'Editeurs, Paris, 1967.

\bibitem{Roe}
{\sc J.~Roe}, {\em An index theorem on open manifolds. {I}, {II}}, J.
  Differential Geom., 27 (1988), pp.~87--113, 115--136.

\bibitem{roel}
{\sc W.~Roelcke}, {\em \"{U}ber den {L}aplace-{O}perator auf {R}iemannschen
  {M}annigfaltigkeiten mit diskontinuierlichen {G}ruppen}, Math. Nachr., 21
  (1960), pp.~131--149.

\bibitem{schoen:84}
{\sc R.~Schoen}, {\em {Conformal deformation of a Riemannian metric to constant
  scalar curvature}}, J. Diff. Geom., 20 (1984), pp.~479--495.

\bibitem{schoen.yau:79a}
{\sc R.~Schoen and S.-T. Yau}, {\em {On the proof of the positive mass
  conjecture in general relativity.}}, Comm. Math. Phys., 65 (1979),
  pp.~45--76.

\bibitem{schoen.yau:81}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em {Proof of the
  positive mass theorem. II.}}, Comm. Math. Phys., 79 (1981), pp.~231--260.

\bibitem{SchroheSI}
{\sc E.~Schrohe}, {\em {Spectral invariance, ellipticity, and the Fredholm
  property for pseudodifferential operators on weighted Sobolev spaces.}}, Ann.
  Global Anal. Geom., 10 (1992), pp.~237--254.

\bibitem{SchroheFC}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Fr\'echet algebra
  techniques for boundary value problems: {F}redholm criteria and functional
  calculus via spectral invariance}, Math.\ Nachr., 199 (1999), pp.~145--185.

\bibitem{Schulze98}
{\sc B.~W. Schulze}, {\em {Boundary value problems and singular
  pseudo-differential operators.}}, {Wiley-Interscience Series in Pure and
  Applied Mathematics. Chichester: John Wiley \& Sons.}, 1998.

\bibitem{Shubin}
{\sc M.~A. Shubin}, {\em Spectral theory of elliptic operators on noncompact
  manifolds}, Ast\'erisque, 207/5 (1992), pp.~35--108.
\newblock M\'ethodes semi-classiques, Vol.\ 1 (Nantes, 1991).

\bibitem{Stein}
{\sc E.~Stein}, {\em Singular integrals and differentiability properties of
  functions}, Princeton Mathematical Series, No. 30, Princeton University
  Press, Princeton, N.J., 1970.

\bibitem{StrichartzM}
{\sc R.~Strichartz}, {\em Multipliers on fractional {S}obolev spaces}, J. Math.
  Mech., 16 (1967), pp.~1031--1060.

\bibitem{StrichartzB}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Invariant
  pseudo-differential operators on a {L}ie group}, Ann. Scuola Norm. Sup. Pisa
  (3), 26 (1972), pp.~587--611.

\bibitem{strichartz}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Analysis of the
  {L}aplacian on the complete {R}iemannian manifold}, J. Funct. Anal., 52
  (1983), pp.~48--79.

\bibitem{TataruH}
{\sc D.~Tataru}, {\em Strichartz estimates in the hyperbolic space and global
  existence for the semilinear wave equation}, Trans. Amer. Math. Soc., 353
  (2001), pp.~795--807.

\bibitem{Taylor}
{\sc M.~Taylor}, {\em Pseudodifferential operators}, vol.~34 of Princeton
  Mathematical Series, Princeton University Press, Princeton, N.J., 1981.

\bibitem{Taylor1}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Partial differential
  equations I, Basic theory}, vol.~115 of Applied Mathematical Sciences,
  Springer-Verlag, New York, 1995.

\bibitem{Taylor2}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Partial differential
  equations II, Qualitative studies of linear equations}, vol.~116 of Applied
  Mathematical Sciences, Springer-Verlag, New York, 1996.

\bibitem{Taylor3}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Partial differential
  equations III, Nonlinear equations}, vol.~117 of Applied Mathematical
  Sciences, Springer-Verlag, New York, 1997.

\bibitem{trudinger:68}
{\sc N.~S. Trudinger}, {\em {Remarks concerning the conformal deformation of
  Riemannian structures on compact manifolds}}, Ann. Sc. Norm. Super. Pisa,
  Sci. Fis. Mat., III. Ser., 22 (1968), pp.~265--274.

\bibitem{Verchota}
{\sc G.~Verchota}, {\em Layer potentials and regularity for the {D}irichlet
  problem for {L}aplace's equation in {L}ipschitz domains}, J. Funct. Anal., 59
  (1984), pp.~572--611.

\bibitem{yamabe:60}
{\sc H.~Yamabe}, {\em {On a deformation of Riemannian structures on compact
  manifolds.}}, Osaka Math. J., 12 (1960), pp.~21--37.

\end{thebibliography}


\end{document}












