baaux application˙regularity-arxiv-v2a.baauxbaaux

# A regularity result for the bound states of $N$-body Schrödinger operators:
Blow-ups and Lie manifolds

Bernd Ammann  B. A., Fakultät für Mathematik, Universität Regensburg, 93040
Regensburg, Germany<EMAIL_ADDRESS>,  Jérémy
Mougel  J. M., Mathematisches Institut Georg-August-Universität Göttingen,
37083 Göttingen, Germany<EMAIL_ADDRESS>and  Victor Nistor
V. N., Université de Lorraine, CNRS, IECL, F-57000 Metz, France and Inst.
Math. Romanian Acad. PO BOX 1-764, 014700 Bucharest Romania
<EMAIL_ADDRESS>
###### Abstract.

We prove regularity estimates in weighted Sobolev spaces for the
$L^{2}$-eigenfunctions of Schrödinger type operators whose potentials have
inverse square singularities and uniform radial limits at infinity. In
particular, the usual $N$-body Hamiltonians with Coulomb-type singular
potentials are covered by our result: in that case, the weight is
$\delta_{\mathcal{F}}(x)\coloneqq\min\\{d(x,\mathchoice{{\bigcup}\,\mathcal{F}}{\medmath{\textstyle\bigcup}\,\mathcal{F}}{\medmath{\scriptstyle\bigcup}\mathcal{F}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{F}}),1\\}$,
where
$d(x,\mathchoice{{\bigcup}\,\mathcal{F}}{\medmath{\textstyle\bigcup}\,\mathcal{F}}{\medmath{\scriptstyle\bigcup}\mathcal{F}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{F}})$
is the usual euclidean distance to the union
$\mathchoice{{\bigcup}\,\mathcal{F}}{\medmath{\textstyle\bigcup}\,\mathcal{F}}{\medmath{\scriptstyle\bigcup}\mathcal{F}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{F}}$
of the set of collision planes $\mathcal{F}$. The proof is based on blow-ups
of manifolds with corners and Lie manifolds. More precisely, we start with the
radial compactification $\overline{X}$ of the underlying space $X$ and we
first blow-up the spheres $\SS_{Y}\subset\SS_{X}$ at infinity of the collision
planes $Y\in\mathcal{F}$ to obtain the Georgescu-Vasy compactification. Then
we blow-up the collision planes $\mathcal{F}$. We carefully investigate how
the Lie manifold structure and the associated data (metric, Sobolev spaces,
differential operators) change with each blow-up. Our method applies also to
higher order differential operators, to certain classes of pseudodifferential
operators, and to matrices of scalar operators.

B.A. has been partially supported by SPP 2026 (Geometry at infinity) and the
SFB 1085 (Higher Invariants), both funded by the DFG (German Science
Foundation). V.N. has been partially supported by ANR-14-CE25-0012-01
(SINGSTAR) funded by ANR (French Science Foundation).
AMS Subject classification (2010): 35J10 (Primary), 35B65, 35J47, 58J05
(Secondary)

###### Contents

  1. 1 Introduction
    1. 1.1 Basic notation and constructions
    2. 1.2 Statement of the main result and comments
    3. 1.3 Contents of the paper
  2. 2 Preliminaries
    1. 2.1 A few basic notations and definitions
    2. 2.2 Manifolds with corners and their submanifolds
    3. 2.3 Submanifolds of manifolds with corners
    4. 2.4 Clean intersections
  3. 3 Blow-ups
    1. 3.1 The blow-up along a closed p-submanifold
    2. 3.2 Iterated blow-ups
    3. 3.3 Admissible orders
    4. 3.4 A blow-up point of view on the spherical compactification
  4. 4 Distance functions and Sobolev spaces for blown-up spaces
    1. 4.1 Smoothed distance function to a p-submanifold
    2. 4.2 Smoothed distance functions to a family
    3. 4.3 Blow-ups and Lie manifolds
  5. 5 The Lie manifolds associated to the generalized $N$-body problem
    1. 5.1 Semilattices and blow-ups for the $N$-body problem
    2. 5.2 A lifting lemma and smoothed distance functions
    3. 5.3 More on smooth distance functions on blow-ups
  6. 6 Regularity results for eigenfunctions
  7. A The equivalence of $\rho$ and $\delta_{\mathcal{F}}$
  8. B Group actions on compactifications of vector spaces
  9. C A splitting lemma
  10. D More about clean intersections
  11. E Notation

##  1\. Introduction

We obtain regularity estimates for the eigenfunctions of a class of elliptic
differential operators with singular coefficients. Our results cover
Schrödinger-type operators whose potentials have inverse square type
singularities and uniform radial limits at infinity. Weaker singularities,
such as Coulomb-type singularities, are included in our results; in
particular, the “usual” $N$-body Hamiltonians [7, 11, 13] are covered by our
results. Our results are obtained by combining and extending the results of
our previous papers [1] and [6], which will be two basic references in what
follows.

###  1.1. Basic notation and constructions

Here is first a minimum of notation needed to state our results for case of
Coulomb-type singularities, Theorem 1.1 below. This notation was explained in
more detail in [6]. Let us fix from now on a _finite-dimensional,_ euclidean
(real) vector space $X$. We shall let $\SS_{X}$ denote its _sphere at
infinity_ (more precisely: we define $\SS_{X}$ as the set of rays in $X$
emanating form $0$) and, then, we let $\overline{X}=X\sqcup\SS_{X}$, denote
the spherical compactification of $X$, as described, for instance, in [6,
Section 5.1] (see also [36, 37, 44, 45]). Note that here $\sqcup$ denotes
disjoint union as sets, but $\overline{X}$ also carries a differential
structure that makes $\overline{X}$ diffeomorphic to a closed disk. In
particular, we have a diffeomorphism

(1) $\overline{X}\smallsetminus\\{0\\}\simeq\SS_{X}\times(0,\infty]\,.$

Moreover, the euclidean metric on $X$ defines a natural diffeomorphism from
$\SS_{X}$ to the unit sphere in $X$.

Let us assume throughout the paper that $\mathcal{F}$ is a finite, non-empty
set of linear subspaces of $X$. We need to define various distance functions
in terms of $\mathcal{F}$. First, by
$\operatorname{dist}(x,y)=\operatorname{dist}_{\mathrm{eucl}}(x,y)$, we shall
denote the euclidean distance on $X$ and, for any subset $Z\subset X$, we let

(2) $d_{Z}(x)\ \coloneqq\ \operatorname{dist}(x,Z)\ \coloneqq\ \inf_{z\in
Z}|x-z|$

denote the euclidean distance from $x$ to $Z$. Let then
$\mathchoice{{\bigcup}\,\mathcal{F}}{\medmath{\textstyle\bigcup}\,\mathcal{F}}{\medmath{\scriptstyle\bigcup}\mathcal{F}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{F}}\coloneqq\bigcup_{Y\in\mathcal{F}}Y$
denote the union of the elements of $\mathcal{F}$ and

(3) $\delta_{\mathcal{F}}(x)\ \coloneqq\
\min_{Y\in\mathcal{F}}\\{d_{Y}(x),1\\}\ =\
\min\\{\operatorname{dist}(x,\mathchoice{{\bigcup}\,\mathcal{F}}{\medmath{\textstyle\bigcup}\,\mathcal{F}}{\medmath{\scriptstyle\bigcup}\mathcal{F}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{F}}),1\\}\,.$

This function will be the weight used in the definition of the weighted
Sobolev spaces that appear (sometimes only implicitly) in the statements of
our results.

We shall assume throughout the paper, as in, for instance, [9, 11], that
$\mathcal{F}$ is stable under intersections. More precisely, let $2^{A}$
denote the set of all subsets of some set $A$. We shall say that
$\mathcal{S}\subset 2^{A}$ is a _semilattice_ 111In the literature, what we
call a semilattice is often called a _meet-semilattice_. if
$A,B\in\mathcal{S}$ implies $A\cap B\in\mathcal{S}$. Thus, from now on, we
shall assume that $\mathcal{F}$ is a _finite semilattice_ of linear subspaces
of $X$ such that $X\notin\mathcal{F}$ but $\\{0\\}\in\mathcal{F}$.
(Mathematically the assumption that $X\notin\mathcal{F}$ is not a very
important assumption, but it allows us to simplify the statement of Theorem
1.1, for instance.)

If $Y$ is a linear subspace of $X$, its closure $\overline{Y}$ in
$\overline{X}$ coincides with the spherical compactification of $Y$, so there
is no danger of confusion. As in [6], to the semilattice $\mathcal{F}$ we will
associate the semilattices of _spherical compactifications_ and, respectively,
_spheres at infinity_ of subspaces in $\mathcal{F}$:

(4) $\overline{\mathcal{F}}\ \coloneqq\ \\{\,\overline{Y}\,|\
Y\in\mathcal{F}\,\\}\ \ \mbox{ and }\ \ \SS_{\mathcal{F}}\ \coloneqq\
\\{\,\mathbb{S}_{Y}\,|\ Y\in\mathcal{F}\,\\}\,.$

Finally, we can now introduce the main spaces considered in our main results
as iterated blow-ups:

(5) $X_{\mathrm{GV}}\ \coloneqq\ [\overline{X}:\SS_{\mathcal{F}}]\ \ \mbox{
and }\ \ X_{\mathcal{F}}\ \coloneqq\
[\overline{X}:\SS_{\mathcal{F}}\cup\overline{\mathcal{F}}]\,.$

The notions of blow-up and iterated blow-up will be recalled in Section 3. We
will see in 5.1.1 that
$X_{\mathcal{F}}\simeq[X_{\mathrm{GV}}:\widehat{\mathcal{F}}]$, where
$\widehat{\mathcal{F}}$ denotes the lift of $\overline{\mathcal{F}}$ to
$X_{\mathrm{GV}}$, see Subsection 5.1 for details.

###  1.2. Statement of the main result and comments

The following result for eigenfunctions is formulated, for simplicity, just
for the usual Coulomb type singularities (see Theorems 6.1 and 6.6 for more
general versions). There is no loss of generality to assume
$X=\mathbb{R}^{n}$, and we shall do that when convenient, for instance, in the
next theorem. Let $\partial_{i}:=\frac{\partial}{\partial x^{i}}$ and
$\partial^{\alpha}:=\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}\ldots\partial_{n}^{\alpha_{n}}$,
as usual. The Laplacian will always have the “analytic” sign convention, i.e.
$\Delta=\sum_{i}\partial_{i}^{2}$.

######  Theorem 1.1.

Let $\mathcal{F}\not\kern 1.0pt\ni X$ be a finite semilattice of linear
subspaces of $X\coloneqq\mathbb{R}^{n}$, $\\{0\\}\in\mathcal{F}$. Let
$X_{\mathcal{F}}\ \coloneqq\
[\overline{X}:\SS_{\mathcal{F}}\cup\overline{\mathcal{F}}]$ and let
$d_{Y}(x)\coloneqq\operatorname{dist}(x,Y)$ and
$\delta_{\mathcal{F}}(x)\coloneqq\min\\{\operatorname{dist}(x,\mathchoice{{\bigcup}\,\mathcal{F}}{\medmath{\textstyle\bigcup}\,\mathcal{F}}{\medmath{\scriptstyle\bigcup}\mathcal{F}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{F}}),1\\}$
be the distance functions introduced in the previous subsection. For every
$Y\in\mathcal{F}\cup\\{X\\}$, let
$a_{Y}\in{\mathcal{C}}^{\infty}(X_{\mathcal{F}})$ and

(6) $V(x)\ \coloneqq\ \sum_{Y\in\mathcal{F}}a_{Y}(x)d_{Y}(x)^{-1}+a_{X}(x)\,.$

Let us assume that $u\in L^{2}(X)$ satisfies $(\Delta+V)u=\lambda u$ in
distribution sense on
$X\smallsetminus\mathchoice{{\bigcup}\,\mathcal{F}}{\medmath{\textstyle\bigcup}\,\mathcal{F}}{\medmath{\scriptstyle\bigcup}\mathcal{F}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{F}}$,
for some $\lambda\in\mathbb{C}$. Then, for all multi-indices
$\alpha\in\mathbb{N}^{n}$, we have

$\delta_{\mathcal{F}}^{|\alpha|}\partial^{\alpha}u\in L^{2}(X)\,.$

Define
$\rho(x)\coloneqq\operatorname{dist}_{\overline{g}}(x,\mathchoice{{\bigcup}\,\mathcal{F}}{\medmath{\textstyle\bigcup}\,\mathcal{F}}{\medmath{\scriptstyle\bigcup}\mathcal{F}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{F}})$,
where $\overline{g}$ is a (true) metric on $X_{\mathrm{GV}}$ (that is, a
metric on the compact manifold with corners $X_{\mathrm{GV}}$ that is smooth
up to the boundary). Then we can use $\rho$ instead of $d_{\mathcal{F}}$, see
the Appendix A. That is, for $u$ as in the theorem above and for all multi-
indices $\alpha\in\mathbb{N}^{n}$, we have

$\rho^{|\alpha|}\partial^{\alpha}u\in L^{2}(X)\,.$

Theorem 1.1 is, as we have already mentioned, a particular case of Theorem 6.6
(which, in turn, follows from Theorem 6.1). In particular, that theorem covers
also the case of inverse square potentials, which have been studied recently
in [12, 14, 17, 31, 32, 39] and in many other papers. We also consider higher
order uniformly strongly elliptic operators. This result generalizes a similar
regularity result in [1]. Our results generalize immediately to systems.
Regularity results for the eigenfunctions of Schrödinger operators have been
obtained in many papers. See, for instance, [1, 10, 15, 16, 18, 19, 20, 21,
22, 23, 26, 29, 33, 42, 43, 46, 47] and the references therein. Many methods
in this article may be generalized to non-linear equations, e.g. to semi-
linear equations of the form

$\Delta u+V|u|^{s}u=\lambda u$

with $V$ as in (6), $\lambda\in\mathbb{C}$ and $0<s\leq 2/(n-2)$, see Remark
6.7 for details.

###  1.3. Contents of the paper

The paper is essentially self-contained and consists of two parts. The first
part contains mostly background material, but presented in a novel way. It
consists of sections 2 and 3. In Section 2, we review manifolds with corners,
their submanifolds, clean intersections of submanifolds, and some other basic
concepts needed in the paper. We also recall in this section some results from
[6] and from some earlier papers, including [1, 36, 40]. In Section 3 we
recall and study the blow-ups and their iteration with respect to one or more
suitable subsets. In particular, we extend the definition of an _admissible
order_ to a $k$-tuple of closed subsets of $M$ and explain that the iterated
blow-up with respect to a clean semilattice with an admissible order is
defined and independent of the chosen admissible order [1, 6]. We also explain
that the action of a Lie group on $(M,\mathcal{S})$ extends to an action on
the iterated blow-up $[M:\mathcal{S}]$. The second part of the paper begins
with Section 4, where we study metric aspects of the iterated blow-ups. We
start by introducing the concept of a “smoothed distance function” to a
p-submanifold $P\subset M$, which is a function that behaves like the distance
to $P$ close to $P$, but is smooth outside $P$. We study then the smoothed
distance function to a blow-up suitable $k$-tuple, in general, and to a clean
semilattice, in particular. Then we recall how the metrics, the Sobolev
spaces, and the differential operators change if one conformally changes the
metric using a smoothed distance function. We apply these results to blow-ups
(including the iterated ones). We distinguish here the case of a blow-up along
a submanifold contained in the boundary and the case of a manifold not
contained in the boundary. The main technical result of this section is the
behavior of the smoothed distance functions when performing iterated blow-ups,
Proposition 4.15. We also recall here the regularity result for the natural
elliptic operators on Lie manifolds [3], which will then provide the
regularity result of this paper. (The main work of this paper is to position
ourselves to be able to use the regularity theorem for Lie manifolds.) In
Section 5, we introduce the relevant semilattices and Lie manifolds needed to
deal with the $N$-body problem and we particularize the constructions and
results of Section 4 to this setting. Section 6 explains how to obtain our
regularity result, Theorem 1.1 and some of its generalizations from the
results of the other sections. Finally, the paper contains ??? appendices. In
Appendix A, we prove the equivalence of the functions $\rho$ and
$\delta_{\mathcal{F}}$ used in Theorem 1.1 and right after. (In fact, they are
both equivalent to $\rho_{\mathcal{F}}$, the smoothed distance function to
$\mathcal{F}$, which is the function that is actually used in the proofs, but
is more difficult to define.) Appendices B, C and D contain technical details
used in the article. Appendix E summarizes some notation.

Acknowledgment: V.N. thanks Chris Kottke for some useful references. The
authors thank Daniel Grieser for useful discussions about iterated blowups and
lifts of submanifolds.

##  2\. Preliminaries

We now recall a few concepts and results needed to understand the main results
of this paper. For the concepts and results not recalled here, we refer to [1]
or [6], on which this paper is heavily based.

###  2.1. A few basic notations and definitions

We need to complete the notation introduced in the Introduction as follows. We
let

(7) $\mathbb{R}_{k}^{n}\ \coloneqq\ [0,\infty)^{k}\times\mathbb{R}^{n-k}\mbox{
and }\ \SS_{k}^{n-1}\ \coloneqq\ \SS^{n-1}\cap\mathbb{R}_{k}^{n}\,,$

where $\mathbb{S}^{n-1}$ is the unit sphere of $\mathbb{R}^{n}$ for the
euclidean norm. The space $\SS_{k}^{n-1}$ will be called a _(generalized)
orthant of the sphere_ , for instance, $\SS_{1}^{n-1}$ is a half-sphere. It
will be used in order to help us understand the smooth structure on the
spherical compactification $\overline{X}$. More precisely, let us assume that
$X=\mathbb{R}^{n}$ and let us consider the bijective map
$\Theta_{n}:\overline{X}\to\mathbb{S}^{n}_{1}$

(8) $\begin{gathered}\begin{cases}\ \Theta_{n}(x)\ \coloneqq\
\frac{1}{\sqrt{1+\|x\|^{2}}}(1,x)\in\mathbb{S}^{n}_{1}&\mbox{ if }x\in X\,,\\\
\ \Theta_{n}(\mathbb{R}_{+}v)\ \coloneqq\
\frac{1}{\|v\|}(0,v)\in\mathbb{S}^{n}_{1}&\mbox{ if
}\mathbb{R}_{+}v\in\mathbb{S}_{X}\,.\end{cases}\end{gathered}$

and its inverse $\Theta^{-1}_{n}:\mathbb{S}^{n}_{1}\to\overline{X}$

(9)
$\Theta^{-1}_{n}(y_{0},y_{1},\ldots,y_{n})\longmapsto\begin{cases}\frac{1}{y_{0}}(y_{1},\ldots,y_{n})\in\mathbb{R}^{n}&\text{if
}y_{0}\neq 0\\\ \mathbb{R}_{+}(y_{1},\ldots,y_{n})\in\SS_{X}&\text{if
}y_{0}=0\end{cases}\,.$

We define the smooth structure on $\overline{X}$ by the requirement that the
map $\Theta_{n}$ be a diffeormphism as in [36, 40, 44]. For general $X$, we
define the smooth structure on $\overline{X}$ using a linear isomorphism
$X\simeq\mathbb{R}^{n}$, for suitable $n$.

######  Remark 2.1.

Let us assume that $Y=\mathbb{R}^{p}\subset\mathbb{R}^{n}=X$. Then we have the
following commutative diagram of smooth embeddings

(10)
$\begin{matrix}Y&\hookrightarrow&\overline{Y}&\stackrel{{\scriptstyle\Theta_{p}}}{{\longrightarrow}}&\mathbb{S}^{p}_{1}\\\
\mathrel{\rotatebox[origin={c}]{-90.0}{$\hookrightarrow$}}&&\mathrel{\rotatebox[origin={c}]{-90.0}{$\hookrightarrow$}}&&\mathrel{\rotatebox[origin={c}]{-90.0}{$\hookrightarrow$}}\\\
X&\hookrightarrow&\overline{X}&\stackrel{{\scriptstyle\Theta_{n}}}{{\longrightarrow}}&\mathbb{S}^{n}_{1}\,\end{matrix},$

which is a basic functoriality property of the spherical compactification. In
particular, this shows that the closure of $Y$ in $\overline{X}$ can be
identified as a differential manifold with $\overline{Y}$, the spherical
compactification of $Y$, so there is not danger of confusion.

Let $|A|$ denote the number of elements of a set $A$. We continue with several
basic definitions.

######  Definition 2.2.

Let $I\subset\\{1,\ldots,n\\}$ and

(11) $L_{I,k}^{n}\ \coloneqq\
\\{\,x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}_{k}\ \coloneqq\
[0,\infty)^{k}\times\mathbb{R}^{n-k}\mid x_{i}=0\ \mbox{ if }\ i\in I\,\\}\,.$

Then $b_{I,k}^{n}\coloneqq|I\cap\\{1,\ldots,k\\}|$ will be called the
_boundary depth_ of $L_{I,k}^{n}$ in $\mathbb{R}_{k}^{n}$ and
$b_{x}\coloneqq|\\{i\in\\{1,\ldots,k\\}\mid x_{i}=0\\}|$ will be called the
_boundary depth_ of $x=(x_{1},\ldots,x_{n})$ in $\mathbb{R}_{k}^{n}$. Clearly,
$c\coloneqq|I|$ is the _codimension_ of $L_{I}$ in $\mathbb{R}_{k}^{n}$ and
$d\coloneqq n-c$ is its _dimension_.

Obviously $b_{I,k}^{n}\coloneqq\min\\{b_{x}\mid x\in L_{I,k}^{n}\\}$. The role
of the sets $L_{I,k}^{n}$ is to serve as models for p-submanifolds [40,
Definition 1.7.4]. A p-submanifold is a suitable submanifold of a manifold
with corners, a concept recalled next. The letter “p” comes from the fact that
for such a p-submanifold and any of its points, a local chart can be found
around that point such that the submanifold is a factor of a _product_.

###  2.2. Manifolds with corners and their submanifolds

Let us now summarize a few basic definitions related to manifolds with corners
[4, 34, 36, 35, 37, 41]. Recall that a _manifold with corners_ $M$ of
dimension $n$ is a topological space locally modeled on
$\mathbb{R}^{n}_{k}\coloneqq[0,\infty)^{k}\times\mathbb{R}^{n-k}$ with smooth
transition functions. The spaces $\mathbb{R}^{n}_{k}$ and $\SS^{n}_{k}$
introduced in (7) provide simple examples of manifolds with corners. Given a
point $x$ in a manifold with corners $M$, the _(boundary) depth_ of $x$ in $M$
is defined locally using Definition 2.2. More precisely, the boundary depth of
$x$ in $M$ is the number of non-negative coordinate functions vanishing at $p$
in any local coordinate chart at $p$. It is the least $k$ such that, for all
$x\in M$, there exists a chart $\phi:U\to\mathbb{R}_{k}^{n}$ defined on an
open neighborhood of $x$ in $M$. For further reference, let us formalize the
following concept in a definition.

######  Definition 2.3.

The _boundary depth_ of $P\subset M$ is the minimum over all boundary depths
of points $x\in P$.

Let $(M)_{k}$ be the set of points of $M$ of boundary depth $k$. Its connected
components are called the _open_ boundary faces of codimension (or boundary
depth) $k$ of $M$. A _boundary face_ of boundary depth $k$ is the closure of
an open boundary face of boundary depth $k$. A boundary face $H$ of
codimension one of $M$ will be called a _boundary hyperface_ of $M$. The union
of the boundary hyperfaces $H$ of $M$ is denoted $\partial M$; it is the
_boundary_ of $M$. It is the set of points of $M$ of boundary depth $\geq 1$.
In particular, a subset $A\subset M$ has boundary depth 0 if, and only if, it
is not contained in $\partial M$, the boundary of $M$.

The smooth functions on a manifold with corners $M$ are defined as on their
counterparts without corners. This is conveniently done by embedding $M$ into
a smooth manifold without corners $\widetilde{M}$, called an _enlargement_ of
$M$. This enlargement can also be used to define the tangent spaces of $M$.

######  Definition 2.4.

Let $H$ be a hyperface of $M$ or a union of hyperfaces of $M$. A _boundary
defining function_ of $H$ (in $M$) is a function $0\leq
x_{H}\in{\mathcal{C}}^{\infty}(M)$ such that $H=x_{H}^{-1}(0)$ and $dx_{H}\neq
0$ on $H$.

######  Remark 2.5.

The example of the teardrop domain, see Figure 1, shows that not all
hypersurfaces have a boundary defining function. However, each (connected)
boundary face $F$ of codimension $k$ can _locally_ (i.e. , in a sufficiently
small neighborhood $U$ of any given point $x$) be represented as

$F\cap U\ =\ \\{x_{1}=x_{2}=\ldots=x_{k}=0\\}\,,$

where $x_{j}$ are boundary defining functions in $U$ of the hyperfaces of $U$
containing $F\cap U$. Then $k$ is the boundary depth of $F$.

Figure 1. A teardrop domain as a subset in $\mathbb{R}^{2}$.

In order to define the blow-up, we need the concept of an _inward pointing
normal bundle_ of a p-submanifold $P$ of $M$. We first recall the concept of
an inward pointing tangent space to a manifold with corners $M$, a concept
used also in [3, 4].

######  Definition 2.6.

Let $M$ be a manifold with corners. We let the _inward pointing tangent space_
$T_{x}^{+}M$ to be defined as the set of derivatives

$T_{x}^{+}M\ \coloneqq\ \\{\gamma^{\prime}(0)\mid\gamma:[0,\epsilon)\to
M\text{ smooth},\quad\gamma(0)=x\\}\,.$

Its elements are called _inward-pointing tangent vectors_ to $T_{x}M$.

We stress that, according to this definition, a tangent vector to $M$ that is
tangent to the boundary will automatically be an inward pointing tangent
vector.

###  2.3. Submanifolds of manifolds with corners

The following concept introduced in [40] will play a crucial role in what
follows. (Recall that $|A|$ denotes the number of elements of a set $A$.)

######  Definition 2.7.

A subset $P$ of a manifold with corners $M$ is a _p-submanifold_ if, for every
$x\in P$, there exists a chart $\phi:U\to\mathbb{R}_{k}^{n}$, with $U$ an open
neighborhood of $x$ in $M$, and $I\subset\\{1,2,\ldots,n\\}$ such that

$\phi(P\cap U)\ =\ L_{I,k}^{n}\cap\phi(U),$

with $L_{I,k}^{n}$ as defined in Equation (11). The number $n-|I|$
(respectively, $|I|$, respectively, $|I\cap\\{1,\ldots,k\\}|$) will be called
the _dimension_ (respectively, the _codimension_ of $P$ at $x$, respectively,
the _boundary depth_ of $P$ at $x$). We allow p-submanifolds $Y$ of non-
constant dimension. We define $\dim(Y)$ as the maximum of the dimensions of
the connected components of $Y$ and $\dim(\emptyset)=-\infty$.

In particular, $\SS^{n}_{k}$ is a closed p-submanifold of
$S^{n+n^{\prime}}_{k+k^{\prime}}$ of codimension $n^{\prime}$ and boundary
depth $k^{\prime}$.

######  Definition 2.8.

Let $P\subset M$ be a p-submanifold of the manifold with corners $M$.

  1. (i)

The quotient bundle $N^{M}P\coloneqq TM|_{P}/TP$ is called the _normal bundle
of $P$ in $M$_.

  2. (ii)

The image $N^{M}_{+}P$ of $T^{+}M|_{P}$ in $N^{M}P$ is called the _inward
pointing normal fiber bundle of $P$ in $M$._ One can show that, in a
neighborhood of $x\in P$, it is a fiber bundle with fiber
$\mathbb{R}^{n^{\prime}}_{k^{\prime}}$ where $n^{\prime}$ is the codimension
of $P$ in $M$ at $x$ and where $k^{\prime}$ is the boundary depth of $P$ in
$M$.

  3. (iii)

The set $\SS(N^{M}_{+}P)$ of rays (emanating from $0$) in $N^{M}_{+}P$ is
called the _inward pointing spherical normal bundle of $P$ in $M$._ It is a
fiber bundle with fibers $\SS^{n^{\prime}-1}_{k^{\prime}}$, where $n^{\prime}$
and $k^{\prime}$ are as above.

######  Remark 2.9.

A (positive definite) scalar product on $TM$ induces a scalar product on
$N^{M}P$. The choice of a metric on $M$ will thus define a natural
diffeomorphism from $\SS(N^{M}_{+}P)$ to the set of unit vectors in
$N^{M}_{+}P$.

Besides p-submanifolds, we shall also need _weak submanifolds,_ which are
really just plain submanifolds (without any other qualification or condition).
We now recall (and slightly reformulate) the definition of a weak submanifold
from [6]; the reformulation provides a definition equivalent to [6, Definition
LABEL:schrSpace-def.weak-submanifold], but avoids to introduce the concept of
“submanifolds in Melrose’s sense,” [6, Definition LABEL:schrSpace-
def.submanifold-gen] and its simplified version [6, Definition
LABEL:schrSpace-def.weak-s.0]

######  Definition 2.10.

A subset $S$ of a manifold with corners $M$ is a _weak submanifold of $M$_ if,
for every $p\in S$, there are

  * •

natural numbers $k=k_{p},m=m_{p}\in\\{0,\ldots,n\\}$ and
$\ell=\ell_{p}\in\\{0,\ldots,m\\}$,

  * •

a chart $\phi=\phi_{p}:U\xrightarrow{\sim}V\subset\mathbb{R}^{n}_{k}$ on $M$
with $V$ open in $\mathbb{R}^{n}_{k}$,

  * •

a diffeomorphism $\psi=\psi_{p}:\widetilde{V}\xrightarrow{\sim}W$ with
$\widetilde{V}$ and $W$ open in $\mathbb{R}^{n}$

such that

  1. (1)

$p\in U$,

  2. (2)

$V=\widetilde{V}\cap\mathbb{R}^{n}_{k}$,

  3. (3)

$\psi\bigl{(}\phi(S\cap U)\bigr{)}=W\cap\mathbb{R}^{m}_{\ell}$.

If $S$ is a weak submanifold of a manifold with corners $M$, then
$\\{\psi_{p}\circ\phi_{p}\mid p\in S\\}$ provides an atlas for a manifold with
corners structure on $S$ such that the inclusion map $S\hookrightarrow M$ is
an injective immersion and a homeomorphism onto its image. In conclusion, a
weak submanifold of $M$ is a manifold with corners on its own. Conversely, if
$S$ is an abstract manifold with corners, together with an injective immersion
$\iota:S\to M$ that defines a homeomorphism from $S$ to $\iota(S)$, then [6,
Proposition LABEL:schrSpace-prop.smfd.crit] states that $S$ is weak
submanifold of $M$. Moreover, any p-submanifold of $M$ is a weak submanifold
of $M$.

The number $m_{p}$ is the dimension of $S$ at $p$ and thus locally constant.
(Again we do not require the function $\dim_{p}(S)$ to be constant on $S$.)
The number $\ell_{p}$ is the (boundary) depth of $p$ in $S$, and might by
smaller, larger or equal to $k_{p}$, the boundary depth of $p$ in $M$.

Note that Definition 2.10 is weaker than the notion of a submanifold in
Melrose’s sense [6, Definition LABEL:schrSpace-def.submanifold-gen], which
explain the word “weak” in the above definition.

###  2.4. Clean intersections

In order to study the iterate the blow-up construction in the next secton, we
will need _clean intersections,_ which we recall next. For simplicity, we
discuss only the case of p-submanifolds.

######  Definition 2.11.

Let $P$ and $Q$ be two p-submanifolds of a manifold with corners $M$ such that
$P\cap Q$ is also a p-submanifold of $M$. We say that $P$ and $Q$ _intersect
cleanly_ or that _they have a clean intersection_ if, for every $x\in P\cap
Q$, we have $T_{x}(P\cap Q)=T_{x}P\cap T_{x}Q$.

For example, if $P$ is a p-submanifold of $M$ and $F$ is a boundary face of
$M$, then $F$ and $P$ have a clean intersection. This is not, however, the
case, in general, if $P$ is just a _weak submanifold_ of $M$. This explains,
in particular, the need to consider p-submanifolds. We next recall how to
extend the definition of clean intersection to semilattices of p-submanifolds.

######  Definition 2.12.

Let $\mathcal{S}$ be a finite semilattice of p-submanifolds of $M$. We call
$\mathcal{S}$ a _clean semilattice of p-submanifolds (of $M$)_ if any
$P,Q\in\mathcal{S}$ intersect cleanly.

######  Remark 2.13.

If $\mathcal{S}$ is a clean semilattice of p-submanifolds then, for all
$P_{1},\ldots,P_{k}\in\mathcal{S}$ and all $x\in\bigcap_{j=1}^{k}P_{j}$, we
obtain that

(12) $T_{x}\Big{(}\bigcap_{j=1}^{k}P_{j}\Big{)}\ =\
\bigcap_{j=1}^{k}T_{x}P_{j}\,.$

In particular, our definition of clean semilattices coincides with Definition
2.7 in [1]. This property does not hold anymore, if $\mathcal{S}$ is not a
semilattice, as seen in the next example.

######  Example 2.14.

Assume that $\mathcal{S}$ is a set of p-submanifolds with $k$ elements, such
that each pair $\\{P_{i},P_{j}\\}\subset\mathcal{S}$ intersections cleanly,
then we cannot conclude, in general, that we have the property expressed in
Equation (12) for $k\geq 3$. Indeed, let us consider the family consisting of
the following three surfaces $P_{1}$, $P_{2}$ and $P_{3}$ in $\mathbb{R}^{3}$
(i.e. $2$-dimensional submanifolds of $\mathbb{R}^{3}$):

$\begin{gathered}P_{1}\ \coloneqq\ \left\\{(x,y,z)^{T}\in\mathbb{R}^{3}\mid
y=0\right\\},\qquad P_{2}\ \coloneqq\ \left\\{(x,y,z)^{T}\in\mathbb{R}^{3}\mid
z=0\right\\}\,,\\\ P_{3}\ \coloneqq\ \left\\{(x,y,z)^{T}\in\mathbb{R}^{3}\mid
y+z=x^{2}\right\\}\,.\end{gathered}$

Each pair intersects cleanly in a $1$-dimensional submanifold. However
$P_{1}\cap P_{2}\cap P_{3}=\\{0\\}$, but $T_{0}P_{1}\cap T_{0}P_{2}\cap
T_{0}P_{3}=\mathbb{R}(1,0,0)^{T}$. Thus the triple $(P_{1},P_{2},P_{3})$ does
not intersect cleanly.

The following Lemma will provide the needed examples of pairs of cleanly
intersecting p-submanifolds.

######  Lemma 2.15.

Let $Y,Z$ be two linear subspaces of $X$. Then all subsets of
$\\{\overline{Y},\overline{Z},\SS_{Y},\SS_{Z}\\}$ intersect cleanly.

The proof is straightforward but involves many cases, so we relegate it to
Appendix D, see Corollary D.2.

##  3\. Blow-ups

In this section we gather the needed results on blow-ups, following the
approach of [6]. Most of the results in this section were discussed in a
similar form in the literature, see [1, 6, 26, 27, 36, 40] for more details.
In particular, the basic construction of the blow-up along a p-submanifold
coincides with the one in these references. However, the lifting (or pull-
back) of submanifolds differs in certain cases from the one used in, for
instance, [28, 36, 40]. As a consequence, our notion of iterated blow-up
exhibits some subtle differences to the iterated blow-up discussed in the
aforementioned papers. Our slightly different approach to the iterated blow-up
has better locality and functorial properties, in particular it is compatible
with restriction to open subsets, and thus avoids discussing additional
special cases. However, our modified approach requires additional attention
when citing existing literature: for example [28, Lemma 7.2 (b)] does not hold
using our definitions. As a consequence, the current section will be written
in a self-contained way, both in order to provide reliable foundations and for
better readability. The subtle difference in the approaches, however will
finally have no effect in our applications: in these applications, our
iterated blow-up spaces coincide with the iterated blow-up spaces in the work
of Melrose and in the related work cited above.

###  3.1. The blow-up along a closed p-submanifold

We now recall the definition of the blow-up $[M:P]$ of a manifold with corners
$M$ along a closed p-submanifold $P$ of $M$. See [6, 36] for further details
and references. We begin by specifying the underlying set of the blow-up
$[M:P]$. Its topology and smooth structure will be defined shortly after that.
If $A$ and $B$ are disjoint, we sometimes denote $A\sqcup B\ \coloneqq\ A\cup
B$ their union.

######  Definition 3.1.

[see [6, Definition 3.1]] Let $M$ be a manifold with corners and $P$ be a
_closed_ p-submanifold of $M$. We let $\SS(N^{M}_{+}P)$ denote the inward
pointing spherical normal bundle of $P$ in $M$ (Definition 2.8). Then, as a
set, the _blow-up of $M$ along $P$_ is the disjoint union

$[M:P]\ \coloneqq\ (M\smallsetminus P)\,\sqcup\,\SS(N^{M}_{+}P)\,.$

The blow-down map $\beta=\beta_{M,P}:[M:P]\to M$ is defined as the identity
map on $M\smallsetminus P$ and as the fiber bundle projection
$\SS(N^{M}_{+}P)\to P$ on the complement.

If $P\subset\partial M$ in the above definition, we shall say that $[M:P]$ is
a _boundary blow-up_. If no component of $P$ is contained in the boundary of
$M$, then we shall say that $[M:P]$ is an _interior blow-up_. See [6, 36, 40],
for instance, for the definition of the topology and smooth structure on the
blow-up $[M:P]$. Let us say, nevertheless, that the smooth structure on
$[M:P]$ is defined using euclidean model spaces and is such that it induces
the given smooth structures on $M\smallsetminus P$ and on $\SS(N^{M}_{+}P)$.
We also remark that the topology on $[M:P]$ is such that $M$ has the quotient
topology with respect to $\beta_{M,P}$. The following extreme cases of blow-
ups deserve a special treatment.

######  Remark 3.2.

If $P$ is a union of connected components of $M$, then $[M:P]=M\smallsetminus
P$. In particular, we have $[M:\emptyset]=M$ and $[M:M]=\emptyset$.

We shall need the following Proposition from [6]. In that proposition a _lift_
denotes a smooth map $j^{\beta}$ that yields a commutative diagram

$\textstyle{[Q:P\cap
Q]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\beta}}$$\scriptstyle{\beta_{Q,P\cap
Q}}$$\textstyle{[M:P]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{M,P}}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{M.}$

######  Proposition 3.3 ( [6, Proposition LABEL:schrSpace-prop.beta.m1]).

Let $P$ and $Q$ be closed p-submanifolds of $M$ intersecting cleanly. Then the
inclusion $j:Q\to M$ lifts uniquely to a smooth map

$\underbrace{(Q\smallsetminus(P\cap Q))\,\sqcup\,\SS(N^{Q}_{+}(P\cap
Q))}_{\displaystyle[Q:P\cap Q]\ \coloneqq\
}\xrightarrow{\;\;j^{\beta}\;\;}\underbrace{(M\smallsetminus
P)\,\sqcup\,\SS(N^{M}_{+}P)}_{\displaystyle[M:P]\ \coloneqq\ }.$

This map is an injective immersion, a homeomorphism onto its image, and the
image of $j^{\beta}$ is a p-submanifold. Moreover

$\overline{\beta_{M,P}^{-1}(Q\smallsetminus P)}\ =\ j^{\beta}([Q:P\cap Q]).$

In view of the above proposition, we shall write $[Q:P\cap Q]\subset[M:P]$, by
abuse of notation.

We have the following useful factorization lemma due (essentially) to Kottke
[36]. It deals with a particular, but important case of the iterated blow-up.
We shall need to recall the formulation of the first factorization Lemma from
[6], i.e. [6, Lemma LABEL:schrSpace-lemma.factorization1], which deals with
the setting $Q\subset P\subset M$ (see also [1]).

######  Lemma 3.4.

Let us assume that $Q$ is a p-submanifold of $P$ and that $P$ is a
p-submanifold of $M$. Then $Q$ is a p-submanifold of $M$. Moreover, there
exists a smooth, canonical map

$\zeta_{M,Q,P}:[M:Q,P]\ \coloneqq\ \bigl{[}[M:Q]:[P:Q]\bigr{]}\ \to\ [M:P]$

that restricts to the identity on $M\smallsetminus P$.

The following factorization lemma is similar in spirit, only easier (see [1,
6, 36]).

######  Lemma 3.5.

Let us assume that $P$ and $Q$ are closed, _disjoint_ p-submanifolds of $M$.
We have $\beta_{M,Q}^{*}(P)=P$ and similarly for $P$ and $Q$ switched. With
these identifications, we then have
$\bigl{[}[M:Q]:P\bigr{]}\simeq\bigl{[}[M:P]:Q\bigr{]}$ canonically, and hence
there exists a smooth, canonical map

$\zeta_{M,Q,P}:\bigl{[}[M:Q]:P\bigr{]}\ \to\ [M:P]$

that restricts to the identity on $M\smallsetminus(P\cup Q)$.

###  3.2. Iterated blow-ups

In the applications of the blow-up, we typically have to blow-up several
subsets. This subsection deals with some of the intricancies of this
procedure. The first thing to discuss is the _pull-back_ of a subset in $M$ to
the blow-up $[M:P]$ following [6, 36, 37].

######  Definition 3.6.

Let $P$ be a p-submanifold of $M$ and $Q$ be a closed subset of $M$. The
_pull-back_ or _lifting_ $\beta^{\ast}_{M,P}(Q)$ of $Q$ to $[M:P]$ is defined
by

(13) $\beta^{\ast}_{M,P}(Q)\ \coloneqq\ \overline{\beta^{-1}(Q\smallsetminus
P)}$

In the case $Q\subset P$, our definition of $\beta^{\ast}_{M,P}(Q)$ is
different from the one given by Melrose in [40, Chapter 5, Section 7], see [6,
Remark LABEL:schrSpace-betaMPQremark] for details.

In order to introduce the iterated blow-up with respect to an ordered family
($n$-tuple) of closed subsets, we first introduce the families with respect to
which we can define the blow-up. We found it convenient to allow repetitions
in these families. The following is Definition LABEL:schrSpace-def.iterated.bu
in [6] and Definition 2.9 in [1], but see also [36] and [40].

######  Definition 3.7.

Let $M$ be a manifolds with corners and let
$\mathcal{P}\coloneqq(P_{i})_{i=1}^{k}$ be a $k$-tuple of closed subsets of
$M$, $k\geq 1$. If $P_{1}$ is a _closed p-submanifold_ of $M$ and $k>1$, we
define the _pull-back_ of $\mathcal{P}$ to be the $(k-1)$-tuple

$\mathcal{P}^{\prime}\ \coloneqq\
\beta_{M,P_{1}}^{*}(\mathcal{P}\smallsetminus\\{P_{1}\\})\ \coloneqq\
\bigl{(}\beta_{M,P_{1}}^{*}(P_{i})\bigr{)}_{i=2}^{k}\,.$

Then, by induction on $k$, we say that $\mathcal{P}$ is _blow-up-suitable (in
$M$)_ if:

  1. (i)

$P_{1}$ is a _closed p-submanifold_ of $M$ and

  2. (ii)

if $k>1$, the pull-back $\mathcal{P}^{\prime}$ is blow-up-suitable in
$[M:P_{1}]$.

Of course, $k=|\mathcal{P}|$. If $k=0$, that is, if $\mathcal{P}=\emptyset$,
then we also say that $\mathcal{P}$ is blow-up-suitable in $M$.

In what follows, we will often use the pull-back $\mathcal{P}^{\prime}$ of
various $k$-tuples $\mathcal{P}$. We now define the iterated blow-up with
respect to blow-up-suitable families.

######  Definition 3.8.

We use the notation of Definition 3.7, in particular,
$\mathcal{P}\coloneqq(P_{i})_{i=1}^{k}$ is a $k$-tuple of closed subsets of
$M$ and
$\mathcal{P}^{\prime}\coloneqq\beta_{M,P_{1}}^{*}(\mathcal{P}\smallsetminus\\{P_{1}\\})\coloneqq\bigl{(}\beta_{M,P_{1}}^{*}(P_{i})\bigr{)}_{i=2}^{k}$
(the pull-back of $\mathcal{P}$). If $\mathcal{P}$ is blow-up-suitable, then
we define by induction the _iterated blow-up_ $[M:\mathcal{P}]$ by

$[M:\mathcal{P}]\ =\ [M:(P_{i})_{i=1}^{k}]\ \coloneqq\ \begin{cases}\
[M:P_{1}]&\mbox{ if }\ k=1\,,\\\
\,\big{[}[M:P_{1}]:\mathcal{P}^{\prime}\big{]}&\mbox{ if }\ k>1\,.\end{cases}$

The blow-down map $\beta_{M,\mathcal{P}}:[M:\mathcal{P}]\to M$ is defined by
induction on $k$ as the composition of the blow-down maps
$[M:\mathcal{P}]:=[[M:P_{1}]:\mathcal{P}^{\prime}]\to[M:P_{1}]\to M$.

It is not difficult to see, by induction, that Definition 3.8 makes sense.
This is the point of introducing Definition 3.7. We shall also use
$[M:P_{1},P_{2},\ldots,P_{k}]$ as an alternative notation for
$[M:\mathcal{P}]$. Let $\mathcal{E}:=\emptyset$, (which is regarded as a
$k$-tuple with $k=0$) and define $[M:\mathcal{E}]=M$ and
$\mathcal{P}^{\prime}=\mathcal{E}$ for if $\mathcal{P}$ has only one element.
Then the relation
$[M:\mathcal{P}]=\big{[}[M:P_{1}]:\mathcal{P}^{\prime}\big{]}$ remains true
also for $k=1$. This may be useful for proofs by induction, for instance in
the proof of Lemma 4.10.

######  Remark 3.9.

Of course, a $k$-tuple $\mathcal{P}$ _without_ repetitions is the same thing
as a _linearly ordered set_. We want the $\mathcal{P}$ in the definition of
the iterated blow-up $[M:\mathcal{P}]$ to be a $k$-tuple rather than a
linearly ordered finite set of p-submanifolds. That is, we want to _allow
repetitions_ in $\mathcal{P}$. The reason for this choice is that, even if
$\mathcal{P}$ does not have repetitions, its pull-back
$\mathcal{P}^{\prime}\coloneqq\bigl{(}\beta^{*}(P_{i})\bigr{)}_{i=2}^{k}$
might have repetitions. An example is provided by $\mathcal{P}=(A,B,A\cup B)$,
where $A$ and $B$ are disjoint closed p-submanifolds of $M$, in which case
$\mathcal{P}^{\prime}=(B,B)$. We shall often consider semilattices with an
additional total order (other than the order given by inclusion).

In the next remark we will explain how to eliminate repetitions. For this
purpose, the following proposition is a useful technical result. Recall that
$\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}\coloneqq\bigcup_{P\in\mathcal{P}}P$.

######  Proposition 3.10.

We use the notations of Definition 3.8 and we assume that $\mathcal{P}$ is
blow-up-suitable (and hence that $[M:\mathcal{P}]$ is defined). Then

  1. (i)

$M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}\subset[M:\mathcal{P}]$
and $\beta=\operatorname{\mathrm{id}}$ on
$M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}$
and

  2. (ii)

$M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}$
is open and dense in $[M:\mathcal{P}]$.

###### Proof.

The first part follows from the definitions of the blow-up and of the iterated
blow-up. To prove the second part, we proceed by induction on $k$, the number
of elements of $\mathcal{P}$. The case $k=1$ follows from the definition (and
was discussed also in [6]). For the induction step, we notice that

$M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}\
=\ (M\smallsetminus
P_{1})\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}^{\prime}\subset[M:P_{1}]\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}^{\prime}$

is dense in
$[M:P_{1}]\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}^{\prime}$
by the case $k=1$, since
$\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}^{\prime}$
is closed. Since
$[M:P_{1}]\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}^{\prime}$
is dense in
$[M:\mathcal{P}]\coloneqq\big{[}[M:P_{1}]:\mathcal{P}^{\prime}\big{]}$ by the
case $k-1$ (the induction hypothesis), the result follows. ∎

Given two blow-ups of $M$, we shall say that they are _canonically
diffeomorphic_ if there exists a diffeomorphism that is the identity outside
the sets that are blown up. In view of the above proposition, this entails a
uniqueness property for the canonical diffeomorphisms. We now explain how we
can eliminate the repetitions in $\mathcal{P}$.

######  Remark 3.11.

We use the notation of Definition 3.8 and the results of Proposition 3.10. Let
$\mathcal{P}_{\mathrm{red}}$ be obtained from $\mathcal{P}$ by removing
repetitions by keeping only the _first_ appearence of a p-submanifold. Thus,
if $\mathcal{P}=(\emptyset,A,B,A)$, then
$\mathcal{P}_{\mathrm{red}}=(\emptyset,A,B)$. Then, $[M:\mathcal{P}]$ is
defined if, and only if, $[M:\mathcal{P}_{\mathrm{red}}]$ is defined.
Moreover, these two iterated blow-ups are canonically diffeomorphic (when
defined), in the sense that:

  1. (i)

$M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}=M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}_{\mathrm{red}}$;

  2. (ii)

$M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}$
is dense in $[M:\mathcal{P}]$ and
$M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}_{\mathrm{red}}$
is dense in $[M:\mathcal{P}_{\mathrm{red}}]$, by Proposition 3.10; and

  3. (iii)

the identity map
$M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}\to
M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}_{\mathrm{red}}$
extends to a diffeomorphism
$[M:\mathcal{P}]\to[M:\mathcal{P}_{\mathrm{red}}]$, which is unique by the
density properties of ii.

In addition to removing repetitions, we could as well remove entries
$P_{j}=\emptyset$ from $\mathcal{P}$ with the same effect. However, we found
it convenient for exposition purposes to do exactly the opposite, that is, to
usually assume $\emptyset$ to be the first entry of $\mathcal{P}$, especially
if $\mathcal{P}$ is a semilattice. We have the following properties.

######  Remark 3.12.

We use the notation of Definition 3.8. In what follows, the pull-back
operation $\mathcal{P}\mapsto\mathcal{P}^{\prime}$ introduced in

  1. (i)

We have
$[M:\mathcal{P}]\coloneqq[[M:P_{1}],\mathcal{P}^{\prime}]=[[M:P_{1}]:\mathcal{P}^{\prime}_{\mathrm{red}}]$,
which may be useful if one wants to deal only with _reduced_ tuples (which is
the same as linearly ordered finite sets).

  2. (ii)

Let us introduced the _iterated pull-backs_ of $\mathcal{P}$ by
$\mathcal{P}^{(1)}\coloneqq\mathcal{P}^{\prime}$ and
$\mathcal{P}^{(k+1)}\coloneqq(\mathcal{P}^{(k)})^{\prime}.$ Then

$[M:\mathcal{P}]\ =\ [[M:P_{1},P_{2},\ldots,P_{k}]:\mathcal{P}^{(k+1)}]\,.$

  3. (iii)

$(\mathcal{P}^{\prime}_{\mathrm{red}})^{\prime}_{\mathrm{red}}=\mathcal{P}^{\prime\prime}_{\mathrm{red}}$
(where ′ always commes before red, meaning that
$\mathcal{P}^{\prime}_{\mathrm{red}}\coloneqq(\mathcal{P}^{\prime})_{\mathrm{red}}$).

  4. (iv)

A special situation arises if $P_{1}$, the first element of $\mathcal{P}$, is
$P_{1}=\emptyset$, which is, in fact, the norm when dealing with semilattices.
Then, in our definition of the blow-up, the first step, the blow-up with
respect to $P_{1}$ is trivial (it does not change our sets, except that it
removes $P_{1}$ from the list). In particular,
$\mathcal{P}=(\emptyset,\mathcal{P}^{\prime})$. It is the next blow-up that
may be interesting. For us, it will then be useful to introduce the following
notation

$\widetilde{\mathcal{P}}\ \coloneqq\ (\emptyset,\mathcal{P}^{\prime\prime})\
\coloneqq\
(\emptyset,\beta_{M:P_{2}}^{*}(P_{3}),\beta_{M:P_{2}}^{*}(P_{4}),\ldots,\beta_{M:P_{2}}^{*}(P_{k}))\,.$

(Note that $\beta_{M:P_{2}}^{*}(P_{2})=\emptyset$.) We then have the relation

$[M:\mathcal{P}]\ =\ [M:\mathcal{P}^{\prime}]\ =\
[[M:P_{2}]:\mathcal{P}^{\prime\prime}]\ =\
[[M:P_{2}]:\widetilde{\mathcal{P}}]\ =\
[[M:P_{2}]:\widetilde{\mathcal{P}}_{\mathrm{red}}]\,.$

We also notice that, if $\mathcal{P}_{\mathrm{red}}$ is a clean semilattice,
then
$\widetilde{\mathcal{P}}_{\mathrm{red}}=(\widetilde{\mathcal{P}_{\mathrm{red}}})_{\mathrm{red}}$
is also a clean semilattice; here the semilattice property is obvious and the
cleanness was proved in [1, Theorem 2.8]. (In this context we define again
$\widetilde{\mathcal{P}}_{\mathrm{red}}\coloneqq(\widetilde{\mathcal{P}})_{\mathrm{red}}$,
that is, the “tilde” comes before “reduced”.)

  5. (v)

To wrap up the list of needed properties, let us notice that

$(\emptyset,\mathcal{P})^{\prime}=\mathcal{P}\ \mbox{ and }\
[M:(\emptyset,\mathcal{P})]=[M:\mathcal{P}]\,.$

In particular, if we define $\mathcal{P}_{0}:=\mathcal{P}^{(0)}:=\mathcal{P}$,
$\mathcal{P}_{k+1}:=\widetilde{\mathcal{P}}_{k}$ and
$\mathcal{P}^{(k+1)}=(\mathcal{P}^{(k)})^{\prime}$, then
$\mathcal{P}_{k}=(\emptyset,\mathcal{P}^{(k+1)})$ for $k\geq 0$. Hence the
iteration of the tilde operation can be expressed in terms of the iteration
with respect to the prime operation and the addition of the emptyset.

The following results generalizes to the iterated blow-up [6, Lemma
LABEL:schrSpace-lemma.product] on the compatibility of the blow-up with
products.

######  Proposition 3.13.

Let $M$ and $M_{1}$ be two manifolds with corners and
$\mathcal{P}=(P_{1},P_{2},\ldots,P_{k})$ be a blow-up-suitable (in $M$)
$k$-tuple of closed subsets of $M$. Then $\mathcal{P}\times
M_{1}\coloneqq(P_{1}\times M_{1},P_{2}\times M_{1},\ldots,P_{k}\times M_{1})$
is a blow-up-suitable (in $M\times M_{1})$ $k$-tuple of closed subsets of
$M\times M_{1}$ and there exists a canonical diffeomorphism $[M\times
M_{1}:\mathcal{P}\times M_{1}]\simeq[M:\mathcal{P}]\times M_{1}$ such that the
following diagram commutes:

(14) $\begin{CD}[M\times M_{1}:\mathcal{P}\times
M_{1}]@>{\simeq}>{}>[M:\mathcal{P}]\times M_{1}\\\ @V{\beta_{M\times
M_{1},\mathcal{P}\times
M_{1}}}V{}V@V{}V{\beta_{M,\mathcal{P}}\times\operatorname{\mathrm{id}}}V\\\
M\times M_{1}@>{\operatorname{\mathrm{id}}}>{}>M\times M_{1}\,.\end{CD}$

###### Proof.

If $k=1$ (that is, $\mathcal{P}$ consists of a single set), then the result
was proved in [1] (it can be found also in [6]). In general, it follows by
induction, using again the result from [1] (the case $k=1$) and using also
that $\beta^{*}(P)\times M_{1}=\beta^{*}(P\times M_{1})$, where $\beta$ is an
appropriate blow-down map. ∎

###  3.3. Admissible orders

A natural questions when we do iterated blow-ups of a manifold with corners
$M$ along an ordered family $\mathcal{P}=(P_{0},P_{1},\ldots,P_{k})$, is to
decide how the order of the blow-up influences the final space [36]. In
particular, a related question is whether the iterated blow-up is defined for
a given order. The aim of this subsection is to recall the results of [1, 6,
36] that give a positive answer to these questions if “admissible orders” are
used. Before stating the main result from [6], we first introduce admissible
orders and graph blow-ups, which will be needed for the statement of the
theorem.

If, in the definition of a blow-up-suitable $k$-tuple $\mathcal{P}$, we
further require $P_{1}$ to be minimal for inclusion, we obtain the notion of
an “admissible” $k$-tuple. Let us state this explicitly.

######  Definition 3.14.

Let $M$ be a manifolds with corners and let
$\mathcal{P}\coloneqq(P_{i})_{i=1}^{k}$ be a $k$-tuple of closed subsets of
$M$, $k\geq 1$. By induction on $k$, we say that $\mathcal{P}$ is _admissible
(in $M$)_ if:

  1. (i)

$P_{1}$ is a _closed p-submanifold_ of $M$,

  2. (ii)

there is no $i>1$ such that $P_{i}\subsetneqq P_{1}$, and

  3. (iii)

if $k>1$, $\mathcal{P}^{\prime}$ is admissible in $[M:P_{1}]$, where
$\mathcal{P}^{\prime}=(\beta_{M,P_{1}}^{*}(P_{j}))_{j=2}^{k}$ is the pull-back
of $\mathcal{P}$ as before (in particular, $\beta^{*}_{M,P_{1}}$ is as defined
in (13)).

Of course, $k\geq|\mathcal{P}|$. If $k=0$, that is, if
$\mathcal{P}=\emptyset$, then we also say that $\mathcal{P}$ is admissible in
$M$.

This definition of and admissible $k$-tuple is more general than the one in
[6].

######  Remark 3.15.

We notice the following

  1. (1)

A $k$-tuple with an admissible order is, in particular, also blow-up-suitable.

  2. (2)

$\mathcal{P}$ is admissible if, and only if, $(\emptyset,\mathcal{P})$ is
admissible.

  3. (3)

Assume the $(k+1)$-tuple $\mathcal{S}=(P_{0},P_{1},\ldots,P_{k})$ is a clean
semilattice, $P_{0}=\emptyset$. Then
$\widetilde{\mathcal{S}}\coloneqq\bigl{(}\emptyset,\beta_{M,P_{1}}^{*}(P_{2}),\beta_{M,P_{1}}^{*}(P_{3}),...,\beta_{M,P_{1}}^{*}(P_{k})\bigr{)}$
is also a clean semilattice by the results of [1], see Remark 3.12 (iv).

Moreover, $\mathcal{S}$ (with the indicated order) is admissible if, and only
if, $\widetilde{\mathcal{S}}$ (with the induced order) is admissible. This
explains why we sometimes consider $\widetilde{\mathcal{S}}$ and do not work
exclusively with the iterated pull-backs
$\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime},\ldots,\mathcal{S}^{(j)}$.

We now consider a different type of blow-up with respect to a $k$-tuple of
p-submanifolds that is immediately seen not to depend on the choice of the
order. To define it, we introduce the _multi-diagnonal map_. For a semi-
lattice $\mathcal{S}$ endowed with an admissible order, we let $U\coloneqq
M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{S}}{\medmath{\textstyle\bigcup}\,\mathcal{S}}{\medmath{\scriptstyle\bigcup}\mathcal{S}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{S}}$,
which is a dense subset of $[M:\mathcal{S}]$. The multi-diagonal map is the
map

(15) $\delta:U\to\prod_{P\in\mathcal{S}}[M:P],\quad x\mapsto(x,x,\ldots,x).$

######  Definition 3.16.

Let $\mathcal{P}=(P_{i})$ be a $k$-tuple of closed p-submanifolds of the
manifold with corners $M$ and let $\delta$ be the multi-diagonal map defined
in (15). Then the _graph blow-up_ $\\{M:\mathcal{P}\\}$ of $M$ along
$\mathcal{P}$ is defined by

$\\{M:\mathcal{P}\\}\ \coloneqq\
\overline{\delta(M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}})}\
=\ \overline{\\{(x,x,\ldots,x)\,|\ x\in
M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}\\}}\
\subset\ \prod_{i\in I}[M:P_{i}]\,.$

For the graph blow-up, we can also remove the repetitions.

######  Remark 3.17.

The graph blow-up $\\{M:\mathcal{P}\\}$ is a weak submanifold of
$Q\coloneqq\prod_{i\in I}[M:P_{i}]$. The notion of a “weak submanifold” was
introduced in [6, Subsec. 2.3.1], however the only aspect we need to know
about it here is that, if one restricts the sheaf of smooth functions on $Q$
to $\\{M:\mathcal{P}\\}$, then this restriction defines the structure of a
manifold with corners on $\\{M:\mathcal{P}\\}$ which is compatible with the
topology induced from $Q$.

Theorem LABEL:schrSpace-thm.main1 of [6] shows that, if $\mathcal{S}$ is a
semilattice endowed with an admissible order, then the iterated blow-up and
the graph blow-ups of $M$ with respect to $\mathcal{S}$ are canonically
diffeomorphic. The following statement combines the statement of that theorem
with part of Remark LABEL:schrSpace-rem.size-order and with Theorem
LABEL:schrSpace-thm.cor.main1 of that paper.

######  Theorem 3.18.

Let $\mathcal{S}\ni\emptyset$ be a clean semilattice of closed p-submanifolds
of $M$ with an admissible order on its elements (Definition 3.14). Then the
iterated blow-up $[M:\mathcal{S}]$ is defined and $\\{M:\mathcal{S}\\}$ is a
manifold with corners that is canonically diffeomorphic to $[M:\mathcal{S}]$.
If $G$ is a Lie group acting smoothly on $M$ such that $G$ maps $\mathcal{S}$
to itself, then $G$ acts by diffeomorphisms on $[M:\mathcal{S}]$.

In the theorem “canonically diffeomorphic” means that the multi-diagonal map
defined in (15) has a unique smooth extension which provides this
diffeomorphism.

The iterated blow-up does not depend on the order of the sets (up to a
canonical diffeomorphism). Hence, an immediate consequence of the last theorem
is that the iterated blow-up $[M:\mathcal{S}]$ does not depend on the choice
of the _admissible_ order on $\mathcal{S}$.

The reader may wonder, at this time, whether admissible orders exists on a
given $k$-tuple. Note that, at this time, it is not clear even that a blow-up-
suitable order exists on a given $k$-tuple of closed p-submanifolds of $M$.
The following proposition gives a positive answer to this question if our
$k$-tuple is a _clean semilattice_. This result and the previous theorem
motivates the use of semilattices in our work.

######  Remark 3.19.

Let $\mathcal{S}$ be a finite, clean semilattice of closed p-submanifolds of
$M$, $\emptyset\in\mathcal{S}$. Let us assume construct an order
$(P_{0},P_{1},\ldots,P_{k})$ on the elements of
$\mathcal{S}=\\{P_{0},P_{1},\ldots,P_{k}\\}$ by choosing $P_{j}\in\mathcal{S}$
by induction on $0\leq j\leq k=|\mathcal{S}|-1$ as follows.

  1. (i)

Let us choose an arbitrary initial order on the elements of $\mathcal{S}$, to
be able to talk about pull-backs. With each choice of $P_{j}$, we modifiy this
order by moving $P_{j}$ on the $(j+1)$-position. (So, after choosing $P_{j}$,
the first $(j+1)$-elements of the modified order will be the chosen elements
$(P_{0},P_{1},\ldots,P_{j})$.)

  2. (ii)

We begin by choosing $P_{0}\coloneqq\emptyset$, which implies that the pull-
back $\mathcal{S}^{\prime}$ is given by
$\mathcal{S}^{\prime}=\mathcal{S}\smallsetminus\\{\emptyset\\}$ (after which
we change the order on $\mathcal{S}$ according to the previous point).

  3. (iii)

We let $P_{1}$ to be an arbitrary minimal element of $\mathcal{S}^{\prime}$
for inclusion.

  4. (iv)

Let

$\widetilde{\mathcal{S}}\coloneqq(\emptyset,\mathcal{S}^{\prime\prime})\coloneqq\big{(}\emptyset,\beta_{M,P_{1}}^{*}(\mathcal{S}\smallsetminus\\{\emptyset,P_{1}\\})\big{)}=\beta_{M,P_{1}}^{*}(\mathcal{S}\smallsetminus\\{\emptyset\\})\,.$

Then the elements of $\widetilde{\mathcal{S}}$ form again a clean semilattice
of closed p-submanifolds of $M$ [1, Theorem 2.8] and we choose
$P_{2}\in\mathcal{S}\smallsetminus\\{P_{0},P_{1}\\}$ such that the lift
$\beta_{M,P_{1}}^{*}(P_{2})$ of $P_{2}$ is minimal element of
$\mathcal{S}^{\prime\prime}$ for inclusion. In particular,
$\beta_{M,P_{1}}^{*}(P_{2})$ will be a closed p-submanifold of $[M:P_{1}]$ by
the aforementioned result in [1].

  5. (v)

We then iterate this construction with $\widetilde{\mathcal{S}}$ in place of
$\mathcal{S}$. More precisely, recall from Remark 3.12(v) that
$\mathcal{S}_{j}:=\widetilde{\mathcal{S}}_{j-1}=(\emptyset,\mathcal{S}^{(j+1)})$,
where $\mathcal{S}^{(j+1)}=(\mathcal{S}^{(j)})^{\prime}$ and
$\mathcal{S}_{0}=\mathcal{S}^{(0)}=\mathcal{S}$. Assume
$P_{0},P_{1},\ldots,P_{j}$ were chosen. Then we choose
$P_{j+1}\in\mathcal{S}\smallsetminus\\{P_{0},P_{1},\ldots,P_{j}\\}$, $j+1\leq
k$, to correspond to a minimal element of $\mathcal{S}^{(j+1)}$ for inclusion.
This satisfies the desired condition that the lift of $P_{j+1}$ in
$\mathcal{S}^{(j+1)}$ be a p-submanifold.

######  Proposition 3.20.

We use the notation of Remark 3.19. First, the procedure of that remark is
well defined in the sense that it yields an order $(P_{0},P_{1},\ldots,P_{k})$
on the elements of $\mathcal{S}$. Most importantly, the resulting order is
admissible. Let $\emptyset\neq Y\in\mathcal{S}$. With suitable choices in the
procedure of that remark, we obtain an admissible order on $\mathcal{S}$ such
that all elements that preceede $Y$ in this order are contained in $Y$.

###### Proof.

The procedure of Remark 3.19 is well defined since, at each step, the
resulting tuples
$\widetilde{\mathcal{S}}:=(\emptyset,\mathcal{S}^{\prime\prime})$ and
$(\emptyset,\mathcal{S}^{(j+1)}):=\widetilde{(\emptyset,\mathcal{S}^{(j)})}$,
$k\geq 1$, are clean semilattices, by [1], so the procedure stops only after
we have chosen $P_{k}$ (i.e. after having ordered all elements). (See Remark
3.12(v) for the notation.) The resulting order on $\mathcal{S}$ is admissible,
by the definition of an admissible order and by induction on the number of
elements of $\mathcal{S}$. Finally, given $Y\in\mathcal{S}$, we are going to
successively choose the sets $P_{k}$ to be either $Y$ or a set contained in
$Y$, if possible (that is, unless $Y$ has already been chosen). This procedure
will yield the desired admissible order on $\mathcal{S}$. ∎

###  3.4. A blow-up point of view on the spherical compactification

This subsection is not needed for the main results of the article. We will
show the following proposition which might add a helpful perspective for
future research. In particular, it shows that if we blow-up at infinity the
one-point compactification $X_{\infty}$ of a finite-dimensional vector space
$X$ obtained by stereographic projection, then we obtain a compactification
canonically diffeomorphic to the spherical compactification $\overline{X}$.

For introducing the one-point compactification $X_{\infty}$, we use the scalar
product
$\langle\mathchoice{\,\raisebox{1.30005pt}{\scalebox{0.5}{$\bullet$}}\,}{\raisebox{1.30005pt}{\scalebox{0.5}{\,$\bullet$\,}}}{\,\raisebox{0.70007pt}{\scalebox{0.4}{$\bullet$}}\,}{\,\raisebox{0.59998pt}{\scalebox{0.35}{$\bullet$}}\,},\mathchoice{\,\raisebox{1.30005pt}{\scalebox{0.5}{$\bullet$}}\,}{\raisebox{1.30005pt}{\scalebox{0.5}{\,$\bullet$\,}}}{\,\raisebox{0.70007pt}{\scalebox{0.4}{$\bullet$}}\,}{\,\raisebox{0.59998pt}{\scalebox{0.35}{$\bullet$}}\,}\rangle$
on $X$, and we consider the (unit) sphere $\SS_{\mathbb{R}\times X}$ in
$\mathbb{R}\times X$. We define the _south pole_
$S\coloneqq(-1,0)\in\SS_{\mathbb{R}\times X}$. The _stereographic projection_
is the map

$\displaystyle\sigma\colon\SS_{\mathbb{R}\times X}\smallsetminus\\{S\\}$
$\displaystyle\to$ $\displaystyle X,$ $\displaystyle\mathbb{R}\times
X\ni\begin{pmatrix}\cos\theta\\\ \sin(\theta)\cdot y\end{pmatrix}$
$\displaystyle\mapsto$
$\displaystyle\tan\Bigl{(}\frac{\theta}{2}\Bigr{)}\,\cdot\,y,\qquad\theta\in(0,\pi],\quad
y\in\SS_{X}\,.$

After formally defining $\infty\coloneqq\sigma(S)$, there is a unique topology
and a unique smooth structure on $X_{\infty}\coloneqq X\cup\\{\infty\\}$ such
that $\sigma$ is a diffeomorphism. This manifold $X_{\infty}$ is called the
_one-point-compactification_ of X.

$X$$\mathbb{R}$$\SS_{\mathbb{R}\times
X}$$S$$\scriptscriptstyle\alpha$$\scriptscriptstyle\alpha$$\scriptscriptstyle
2\alpha$ Figure 2. One-point and disc compactification. The stereographic
projection $\sigma$ maps the blue dot to the red dot, and the map $\Theta_{X}$
maps the red dot to the green dot.

We also consider a new map $\Theta_{X}$, which is a slightly modified version
of the map $\Theta_{n}$ from (8). For that purpose, let

$\begin{gathered}\SS_{+}^{\prime}:=\bigl{\\{}x+S\mid x\in\SS_{[0,\infty)\times
X}\bigr{\\}}=\bigl{\\{}x-(1,0)\mid x=(x^{0},x^{\prime}),\;\|x\|=1,\;x^{0}\geq
0\bigr{\\}}\,,\\\
\Theta_{X}:\overline{X}\to\SS_{+}^{\prime},\qquad\begin{cases}\ \Theta_{X}(x)\
\coloneqq\ \frac{1}{\sqrt{1+\langle
x,x\rangle}}(0,x)\in\SS_{+}^{\prime}&\mbox{ if }x\in X\,,\\\ \
\Theta_{X}(\mathbb{R}_{+}v)\ \coloneqq\
\frac{1}{\|v\|}(-1,v)\in\SS_{+}^{\prime}&\mbox{ if
}\mathbb{R}_{+}v\in\mathbb{S}_{X}\,.\end{cases}\end{gathered}$

We write $\SS_{X}\coloneqq\overline{X}\smallsetminus X$ for the sphere at
infinity of $X$, as usual.

######  Lemma 3.21.

Let $\Psi\colon\overline{X}\to X_{\infty}$ be the map with
$\Psi|_{X}=\operatorname{\mathrm{id}}_{X}$ and $\Psi(\SS_{X})=\\{\infty\\}$.
This is a smooth map from the spherical compactification $\overline{X}$ to the
one-point-compactification $X_{\infty}$, extending the identity
$\operatorname{\mathrm{id}}_{X}$. There is a diffemorphism
$\Phi:\overline{X}\to\bigl{[}X_{\infty}:\\{\infty\\}\bigr{]}$ extending
$\operatorname{\mathrm{id}}_{X}$, and thus $\Psi\circ\Phi^{-1}$ is the blow-
down map $\beta_{X_{\infty},\\{\infty\\}}$.

###### Proof.

We define
$\psi\coloneqq\sigma^{-1}\circ\Theta_{X}^{-1}\colon\SS_{+}^{\prime}\to\SS_{\mathbb{R}\times
X}$. Then $\Psi=\sigma\circ\psi\circ\Theta_{X}$. By construction, $\sigma$ and
$\Theta_{X}$ are diffeomorphisms. To prove the lemma, one thus has to show
that $\psi$ is that there is a diffeomorphism
$\phi\colon\SS_{+}^{\prime}\to\bigl{[}\SS_{\mathbb{R}\times
X}:\\{S\\}\bigr{]}$ such that $\psi\circ\phi^{-1}=\beta_{\SS_{\mathbb{R}\times
X},\\{S\\}}$. The construction of such a $\phi$ is an easy exercise, if one
uses the following formula for $\psi$, which is apparent in view of Figure 2,
setting $\theta\coloneqq 2\alpha$ above.

$\psi\Bigl{(}\bigl{(}(\cos\alpha)-1,(\sin\alpha)y\bigr{)}\Bigr{)}\ =\
\bigl{(}(\cos 2\alpha),(\sin
2\alpha)y\bigr{)},\qquad\alpha\in[0,\pi/2],\;Y\in\SS_{X}\,.$

The function $\cos\alpha$ is a boundary defining function for
$\SS_{+}^{\prime}$ and a smoothed distance function for $S$ in
$\SS_{\mathbb{R}\times X}$. This completes the proof. ∎

##  4\. Distance functions and Sobolev spaces for blown-up spaces

We now investigate how several geometric quantities (metrics, distance
functions, Sobolev spaces, natural differential operators, …) change when
performing a blow-up. The manifolds $M$ we consider have a complete metric in
the interior $M_{0}$ described precisely in terms of Lie manifolds, Definition
4.20. One important case will be that when $M_{0}$ is a euclidean vector space
$X$ with its spherical compactification $M=\overline{X}$, which was described
in the Introduction. Some other times, $M$ will be a blow-up of
$\overline{X}$.

We distinguish here the case of a blow-up along a submanifold _contained in
the boundary_ (the easy case) and the case of a manifold _not contained in the
boundary_ (the difficult case, but treated already in [1] and in other
papers). In the first case, the _boundary case,_ we will addtionally require
that our vector fields are tangent to the submanifold contained in the
boundary. In that case, the metric in the interior remains the same, only the
compactification is altered, which makes many investigations much easier. In
the second case, the _interior case,_ the metric will be changed conformally
by multiplication with $r^{-2}$, where $r$ is a “smoothed distance function,”
(a concept that will be defined in this section). The main technical result of
this section is the behavior of “smoothed distance functions” when performing
iterated blow-ups, Proposition 4.15. We also recall in this section the needed
regularity result on Lie manifolds from [3].

###  4.1. Smoothed distance function to a p-submanifold

Let $M$ denote a manifold with corners, as before. We assume that
$\mathcal{P}\subset 2^{M}$ – equipped with a suitable ordering – is a blow-up-
suitable $k$-tuple of subsets of $M$ (so that $[M:\mathcal{P}]$ is defined).
As always, we shall write

$\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}\
\coloneqq\ \bigcup\limits_{P\in\mathcal{P}}P\,.$

We shall need the following simple concept of “equivalent functions” on
$[M:\mathcal{P}]$.

######  Definition 4.1.

Let $M$ be a manifold with corners, let $\mathcal{P}$ be a blow-up-suitable
$k$-tuple of p-submanifolds of $M$ (so, in particular, $[M:\mathcal{P}]$ is
defined), and let $k\in\mathbb{N}\cup\\{0,\infty\\}$. Assume that we have two
continuous functions $f_{i}:M\to[0,\infty)$, $i=0,1$, such that the functions
$f_{i}$ are $\mathcal{C}^{k}$ on $M\smallsetminus\bigcup\mathcal{P}$ and
nowhere vanishing. We shall say that $f_{0}$ and $f_{1}$ are
_$\mathcal{C}^{k}$ -equivalent (on $[M:\mathcal{P}]$)_ or that _$f_{0}$ is
$\mathcal{C}^{k}$-equivalent (on $[M:\mathcal{P}]$) to $f_{1}$_ if

$\frac{f_{1}\mathchoice{\lower 3.24005pt\hbox{\big{|}}\lower
6.84009pt\hbox{$\scriptstyle M\smallsetminus\bigcup\mathcal{P}$}}{\lower
2.88007pt\hbox{\big{|}}\lower 5.75995pt\hbox{$\scriptstyle
M\smallsetminus\bigcup\mathcal{P}$}}{\lower 2.52008pt\hbox{|}\lower
5.03998pt\hbox{$\scriptscriptstyle M\smallsetminus\bigcup\mathcal{P}$}}{\lower
2.15991pt\hbox{|}\lower 4.68pt\hbox{$\scriptscriptstyle
M\smallsetminus\bigcup\mathcal{P}$}}}{f_{0}\mathchoice{\lower
3.24005pt\hbox{\big{|}}\lower 6.84009pt\hbox{$\scriptstyle
M\smallsetminus\bigcup\mathcal{P}$}}{\lower 2.88007pt\hbox{\big{|}}\lower
5.75995pt\hbox{$\scriptstyle M\smallsetminus\bigcup\mathcal{P}$}}{\lower
2.52008pt\hbox{|}\lower 5.03998pt\hbox{$\scriptscriptstyle
M\smallsetminus\bigcup\mathcal{P}$}}{\lower 2.15991pt\hbox{|}\lower
4.68pt\hbox{$\scriptscriptstyle M\smallsetminus\bigcup\mathcal{P}$}}}$

extends to a nowhere vanishing $\mathcal{C}^{k}$ function on
$[M:\mathcal{P}]$. In the case $k=\infty$, we shall say that $f_{0}$ is
_smoothly equivalent_ or just _equivalent_ to $f_{1}$ and write $f_{0}\sim
f_{1}$. In the case $k=0$, we say $f_{0}$ and $f_{1}$ are _continuously
equivalent._

The family $\mathcal{P}$ as well as the space $[M:\mathcal{P}]$, which we
assumed to be defined, will sometimes be understood, so we may occasionally
not mention them. Recall that two functions $f_{1},f_{2}:U\to[0,\infty)$ are
called _Lipschitz equivalent_ if there exists $C>0$ such that

(16) $C^{-1}f_{1}\leq f_{2}\leq Cf_{1}\,.$

Two functions $f_{1},f_{2}:M\to[0,\infty)$ are called _locally Lipschitz
equivalent_ if any point in $M$ has an open neighborhood $U$ such that
$f_{1}|_{U}$ and $f_{2}|_{U}$ are Lipschitz equivalent.

######  Remark 4.2.

Let us record a few easy consequences of the definiton:

  1. (i)

Clearly, the $\mathcal{C}^{k}$-equivalence of functions on $[M:\mathcal{P}]$
is an equivalence relation.

  2. (ii)

Since
$M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}$
is dense in $[M:\mathcal{P}]$ (Proposition 3.10) the extension of the quotient
$\dfrac{f_{1}|_{M\smallsetminus\bigcup\mathcal{P}}}{f_{0}|_{M\smallsetminus\bigcup\mathcal{P}}}$
to $[M:\mathcal{P}]$ by continuity is unique (when it exists).

  3. (iii)

Obviously $\mathcal{C}^{k}$-equivalence implies
$\mathcal{C}^{\ell}$-equivalence for $\ell\leq k$.

  4. (iv)

If two functions $f_{i}:M\to[0,\infty)$ as above are continuously equivalent,
then they are locally Lipschitz equivalent. As a consequence, for $M$ compact,
$\mathcal{C}^{0}$-equivalence implies Lipschitz equivalence.

  5. (v)

If $H$ is a boundary hyperface of $M$ with at least one defining function,
then any two defining functions of $H$ will be equivalent on $[M:H]\simeq M$.

Let us now discuss the type of metrics that we will use. Let $M$ be a manifold
with corners. In the following, we shall make use of two kinds of metrics on
$M$. The first kind of metrics will consist of what we are calling “true
Riemannian metrics” on $M$. A _true Riemannian metric_ on $M$ is, by
definition, nothing but a smooth, fiberwise positive definite symmetric
section of $T^{*}M\otimes T^{*}M\to M$ (the usual kind of metrics on $M$). The
second kind of metrics will consist of the so-called _compatible metrics,_
which will be defined below, Definition 4.20, and whose definition requires
some additional data (a structural Lie algebra of vector fields on $M$).
Compatible metrics are Riemannian metrics on $M_{0}\coloneqq
M\smallsetminus\partial M$ and do not extend to a true metric on $M$, unless
$\partial M=\emptyset$.

Let $P$ be a closed p-submanifold of $M$. Recall that $\beta_{M,P}:[M:P]\to M$
denotes the blow-down map (see Definition 3.1).

######  Definition 4.3.

Let $M$ be a manifold with corners and $P\subset M$ be a closed p-submanifold.
A function $r_{P}:M\to[0,\infty)$ will be called _a smoothed distance function
to $P$ (in $M$)_ if its lift $\beta_{M,P}^{*}(r_{P})\coloneqq
r_{P}\circ\beta_{M,P}$ is a boundary defining function (Definition 2.4) for
$\SS N_{+}^{M}P=\beta_{M,P}^{-1}(P)$, the hyperface (or union of hyperfaces)
of $[M:P]$ obtained by blowing up $M$ along $P$.

We also remark that a smoothed distance function to $P$ in $M$ is continuous
since it lifts to a continuous function on $[M:P]$ and $M$ has the quotient
topology. The following remark explains the name “smoothed distance function
to $P$” in $M$ for $r_{P}$.

######  Remark 4.4.

The function $f:M\to[0,1]$ is a smoothed distance function to $P$ in $M$ if,
and only if, it satisfies the following conditions:

  1. (i)

it is continuous on $M$ and smooth on $M\smallsetminus P$,

  2. (ii)

$f^{-1}(\\{0\\})=P$, and most importantly,

  3. (iii)

there is a neighborhood $V$ of $P$ such that $f$ is (smoothly) equivalent on
$[V:P]$ to the distance to $P$ with respect to some suitable true metric on
$M$.

We omit details here, as this fact will not be used in our article.

######  Remark 4.5.

In general, a smoothed distance to $P$ in $M$ function $r_{P}$ will _not_ be
smooth on $M$; in fact, smoothness on $M$ will only hold if $P$ is empty, or a
union of connected components of $M$. However, it will be continuous on $M$
and smooth on $[M:P]$. This is one of the main reasons for considering the
blow-up $[M:P]$.

######  Remark 4.6.

Given a closed p-submanifold $P\subset M$, we see that a smoothed distance to
$P$ in $M$ (Definition 4.3) is _not uniquely determined_. However, any two
such smoothed distance functions are equivalent on $[M:P]$, and hence they are
equivalent on any other iterated blow-up $[M:\mathcal{P}]$ as well, as long as
$P\in\mathcal{P}$. This is, in fact, our motivation for introducing the notion
of equivalence of such functions. For this reason, we will sometimes talk
about _the_ smoothed distance to $P$, although we will really mean the
equivalence class of the smoothed distances to $P$ in $M$.

######  Remark 4.7.

In the case that $M=\overline{X}$ is the spherical compactificaction of some
euclidean vector space $X$ as explained in Subsection 1.1 and if $Y$ is a
linear subspace of $X$, we may consider the euclidean distance function
$d_{Y}:=\min\bigl{\\{}\|x-y\|\mid y\in Y\bigr{\\}}$ introduced in Equation
(2). If $r_{\overline{Y}}:\overline{X}\to[0,\infty)$ is a _smoothed_ distance
function to $\overline{Y}$, then the function $r_{\overline{Y}}$ is clearly
not Lipschitz equivalent to $d_{Y}$, as the first one is bounded and the
second one is not. One can also show that $r_{\overline{Y}}$ is also not bi-
Lipschitz to $\arctan\circ d_{Y}$, as these two functions behave differently
close to $\SS_{X}$. The precise behavior of $r_{\overline{Y}}$ is that it is
continuously equivalent and Lipschitz equivalent to
$x\mapsto\arctan\bigl{(}(\|x\|^{2}+1)^{-1/2}d_{Y}(x)\bigr{)}$. Furthermore,
$r_{\overline{Y}}$ is continuously equivalent and thus locally Lipschitz
equivalent to $x\mapsto(\|x\|^{2}+1)^{-1/2}d_{Y}(x)$.

We now consider as in [6] and [36] pairs $(P,Q)$ of closed p-submanifolds of
$M$. As in those papers, we need to consider the cases $Q\subset P$ and $Q\cap
P=\emptyset$. We begin with the first case.

######  Lemma 4.8.

Let $P\subset M$ be a closed p-submanifold of the manifold with corners $M$,
let $Q\subset P$ be a closed p-submanifold of $P$, and let $r_{P}$
(respectively, $r_{Q}$) be a smoothed distance function to $P$ (respectively,
to $Q$) in $M$. Then

$(r_{Q}^{-1}r_{P})\mathchoice{\lower 3.24005pt\hbox{\big{|}}\lower
6.84009pt\hbox{$\scriptstyle M\smallsetminus P$}}{\lower
2.88007pt\hbox{\big{|}}\lower 5.75995pt\hbox{$\scriptstyle M\smallsetminus
P$}}{\lower 2.52008pt\hbox{|}\lower 5.03998pt\hbox{$\scriptscriptstyle
M\smallsetminus P$}}{\lower 2.15991pt\hbox{|}\lower
4.68pt\hbox{$\scriptscriptstyle M\smallsetminus P$}}$

extends to a smoothed distance function to
$\beta_{M,Q}^{*}(P)\coloneqq\overline{\beta_{M,Q}^{-1}(P\smallsetminus Q)}$ in
$[M:Q]$.

Of course, in the above lemma, we have
$\beta_{M,Q}^{*}(P)\coloneqq\overline{\beta_{M,Q}^{-1}(P\smallsetminus
Q)}\simeq[P:Q]$ see Proposition 3.3. Here $\simeq$ denotes the existence of a
diffeomorphism extending the identity on $P\smallsetminus Q$. Also, for the
simplicity of the presentation, we may and will assume in the following that
$\beta_{M,Q}^{*}(P)=[P:Q]$.

######  Remark 4.9.

Note that the assumptions in the lemma allow the case when, for all $q\in Q$,
we have $\dim_{q}(Q)=\dim_{q}(P)$. In this case
$\beta_{M,Q}^{*}(P)=P\smallsetminus Q$ is a union of some connected components
of $P$, and thus for any $q\in Q$ we have
$(\beta_{M,Q})^{-1}(q)\cap[P:Q]=\emptyset$. If $U$ is an open neighborhood of
$q$, then any positive function defined on $\beta_{M,Q}^{-1}(U)$ is thus a
smoothed distance function to $[P:Q]\cap\beta_{M,Q}^{-1}(U)=\emptyset$.

On the other hand, in this special case, $r_{Q}^{-1}r_{P}$, defined on
$U\smallsetminus\\{q\\}$, where $U$ is an open neighborhood of $q$ with $U\cap
P=Q$, extends to a positive smooth function on
$\beta_{[M:Q],[P:Q]}^{*}(\beta_{M,Q}^{*}(U))=\beta_{M,Q}^{*}(U)$. The latter
statement follows as $r_{P}|U$ is a smooth distance function to $Q$ in $U$ and
using Remark 4.5. These argument provide a proof of the lemma in this
exceptional case.

This special case is also included in the following proof, if we use the
convention $\SS^{-1}=\emptyset$ (here $\SS^{-1}$ is the unit sphere in
$\mathbb{R}^{0}$).

###### Proof of Lemma 4.8.

We need to prove that the function $r_{Q}^{-1}r_{P}:M\smallsetminus
P\to(0,\infty)$, extends to a smooth function on

$[M:Q,P]\ \coloneqq\ \bigl{[}[M:Q]:[P:Q]\bigr{]}$

and that this extension of $r_{Q}^{-1}r_{P}$ is a defining function for the
hyperface

$P^{\prime}\ \coloneqq\ \SS\bigl{(}N^{[M:Q]}_{+}[P:Q]\bigr{)}\ =\
\beta_{[M:Q],[P:Q]}^{-1}([P:Q])$

of $[M:Q,P]$. Let

$\beta\ \coloneqq\ \beta_{M,Q,P}\ \coloneqq\
\beta_{M,Q}\circ\beta_{[M:Q],[P:Q]}:[M:Q,P]\to M$

be the blow-down map and $z\in P^{\prime}$ (see Definition 3.8). Then
$\beta(z)\in P$.

Recall that by the definition of $r_{P}$, we have that $r_{P}\circ\beta_{M,P}$
is a boundary defining function
$r_{P}\circ\beta_{M,P}\colon[M:P]\to[0,\infty)$ of
$\SS(N^{M}_{+}P)=\beta_{M,P}^{-1}(P)\subset[M:P]$ as a hyperface of $[M:P]$.
The map $[M:Q,P]\to[M:P]$, defined by Lemma 3.4 is a diffemorphism outside the
preimage of $Q$. Thus, if $\beta(z)\notin Q$, the function
$r_{Q}^{-1}r_{P}:M\smallsetminus P=[M:Q]\smallsetminus[P:Q]\to[0,\infty)$
extends locally – i.e. in a neighborhood of $z$ – to a defining function for
$P^{\prime}\subset[M:Q,P]$, since $r_{Q}>0$ at and near $z$ and since $r_{P}$
is a defining function of $\SS(N^{M}_{P})$, the pull-back of $P$ in $[M:P]$,
as we have just explained.

On the other hand, if $\beta(z)\in Q$, we use (again) the fact that our
problem is local. Thus by choosing a suitable chart around $\beta(z)$, we can
reduce the lemma to the special case

$\begin{cases}\
M=\mathbb{R}_{\ell}^{k}\times\mathbb{R}_{\ell^{\prime}}^{k^{\prime}}\times\mathbb{R}_{\ell^{\prime\prime}}^{k^{\prime\prime}}\ni(x,x^{\prime},x^{\prime\prime})\\\
\
P=\mathbb{R}_{\ell}^{k}\times\mathbb{R}_{\ell^{\prime}}^{k^{\prime}}\times\\{0\\}\ni(x,x^{\prime},0)\\\
\ Q=\mathbb{R}_{\ell}^{k}\times\\{0\\}\times\\{0\\}\ni(x,0,0)\\\ \
\beta(z)=(0,0,0)\,.\end{cases}$

Assume first that $\ell^{\prime}=\ell^{\prime\prime}=0$. In this very special
case we have

$\displaystyle[M:Q]$ $\displaystyle=$
$\displaystyle\mathbb{R}_{\ell}^{k}\times\SS^{k^{\prime}+k^{\prime\prime}-1}\times[0,\infty)\ni(x,\xi,r),$
$\displaystyle{}[P:Q]$ $\displaystyle=$
$\displaystyle\mathbb{R}_{\ell}^{k}\times\SS^{k^{\prime}-1}\times[0,\infty),$

where $r=\sqrt{(x^{\prime})^{2}+(x^{\prime\prime})^{2}}$ and
$\xi=(x^{\prime},x^{\prime\prime})/r$ away from $Q$. Furthermore, the
inclusion $[P:Q]\subset[M:Q]$ is given by the inclusion of
$\SS^{k^{\prime}-1}\subset\SS^{k^{\prime}+k^{\prime\prime}-1}$ on the first
$k^{\prime}$ components of $\SS^{k^{\prime}+k^{\prime\prime}-1}$ and by the
identity on the other factors (that is, on $\mathbb{R}_{\ell}^{k}$ and on
$[0,\infty)$). For $\xi\in\SS^{k^{\prime}+k^{\prime\prime}-1}$ let
$\theta(\xi)$ be the length of the shortest geodesic from $\xi$ to
$\SS^{k^{\prime}-1}$, unless $k^{\prime}=0$. In the case $k^{\prime}=0$ we set
$\theta\equiv\pi/2$, thus $\sin\circ\theta\equiv 1$. Then

$\hat{r}\ \coloneqq\
\sin\circ\,\theta\colon\SS^{k^{\prime}+k^{\prime\prime}-1}\to[0,1]$

is a smoothed distance function to $\SS^{k^{\prime}-1}$ in
$\SS^{k^{\prime}+k^{\prime\prime}-1}$, thus – by pullback – $\hat{r}$ is also
a smoothed distance function to $[P:Q]$ in $[M:Q]$. Smoothed distance
functions $r_{Q}$ and $r_{P}$ (for $Q$ and $P$ in $M$) are then given by

$r_{Q}(x,\eta,r)\ =\ \sqrt{\|x^{\prime}\|^{2}+\|x^{\prime\prime}\|^{2}}\ =\ r\
\text{ and }\ r_{P}\ =\ \|x^{\prime\prime}\|\ =\ r\sin(\theta(\xi))\,.$

Since, obviously, $\hat{r}=r_{Q}^{-1}r_{P}$, we obtain the desired statement
if $\ell^{\prime}=\ell^{\prime\prime}=0$. The general case follows by
replacing $\SS^{k^{\prime}-1}$ with $\SS_{\ell^{\prime}}^{k^{\prime}-1}$ and
$\SS^{k^{\prime}+k^{\prime\prime}-1}$ with
$\SS^{k^{\prime}+k^{\prime\prime}-1}\cap(\mathbb{R}_{\ell^{\prime}}^{k^{\prime}}\times\mathbb{R}_{\ell^{\prime\prime}}^{k^{\prime\prime}})$.
∎

We now turn to the second case, that when $P$ and $Q$ are disjoint. For later
use, we prove a more general statement.

######  Lemma 4.10.

Let $M$ be a compact manifold with corners, let $P\subset M$ be a closed
p-submanifold and $\mathcal{Q}=(Q_{1},Q_{2},\ldots,Q_{k})$ be a blow-up-
suitable $k$-tuple of closed subsets of $M$ _disjoint_ from $P$. Let
$\beta:[M:\mathcal{Q}]\to M$ be the blow-down map and let $r_{P}$ be a
smoothed distance to $P$ (in $M$). Let
$\widehat{P}:=\beta^{-1}(P)=\beta^{*}(P)$. Then $r_{P}\circ\beta$ is a
smoothed distance function to $\widehat{P}$ in $[M:\mathcal{Q}]$.

###### Proof.

Let us prove our result by induction on $k$. For $k=0$ (i.e. for
$\mathcal{Q}=\emptyset$) there is nothing to prove according to our
conventions for the blow-up with respect to an empty family. Let us write
$Q=Q_{1}$, for the simplicity of the notation. Let us prove our result for
$k=1$. The blow-down $[M:Q,P]\to[M:P]$ induces a diffeomorphism
$[M:Q,P]\smallsetminus\beta^{-1}(Q)\to[M\smallsetminus Q:P]$ (see Lemma 3.5).
The function $r_{P}\circ\beta_{M,Q}$ is a smoothed distance function to
$\beta^{-1}(P)$ on $M\smallsetminus Q\subset[M:Q]\textasciiacute$ (since it
coincides with $r_{P}$ there). Moreover, $r_{P}\circ\beta_{M,Q}$ is smooth
everywhere on the iterated blow-up $[M:Q,P]$ and vanishes only on the preimage
of $P$ (in particular, it is $>0$ on the preimage of $Q$, which is disjoint
from the preimage of $P$). Hence $r_{P}\circ\beta_{M,Q}$ is a smoothed
distance function to $\beta_{M,Q}^{*}(P)=\beta_{M,Q}^{-1}(P)$ in $[M:Q]$. The
induction step is always to perform an additional blow-up, so it reduces to
the case $k=1$ just proved. ∎

###  4.2. Smoothed distance functions to a family

We need to extend the definition of the smoothed distance function to
families. (The reader may want to review Definition 3.8 at this point.)

######  Definition 4.11.

Let $\mathcal{P}\coloneqq(P_{i})_{i=1}^{k}$, $P_{i}\subset M$ be a blow-up-
suitable $k$-tuple (that is, an ordered family of closed subsets of $M$ such
that the iterated blow-up $[M:P_{1},P_{2},\ldots,P_{k}]$ is defined,
Definition 3.7). Let $\beta:[M:P_{1}]\to M$ be the blow-down map and
$\mathcal{P}^{\prime}\coloneqq\bigl{(}\beta^{*}(P_{2}),\beta^{*}(P_{3}),\ldots,\beta^{*}(P_{k})\bigr{)}$,
which, we recall, is a family of closed p-submanifolds of $[M:P_{1}]$. We then
define _a smoothed distance function_ $\rho_{\mathcal{P}}$ to $\mathcal{P}$ in
$M$ to be a function $\rho_{\mathcal{P}}:M\to[0,\infty)$ given by induction on
$k$ by the formula

$\rho_{\mathcal{P}}(x)\ \coloneqq\ \begin{cases}\ \ r_{P_{1}}(x)&\mbox{ if
}k=1\\\ \ r_{P_{1}}(x)\rho_{\mathcal{P}^{\prime}}(y)&\mbox{ if }k>1\ \mbox{
and }y\in\beta^{-1}(\\{x\\})\,,\end{cases}$

where $r_{P_{1}}:M\to[0,\infty)$ is a smoothed distance function to $P_{1}$ in
$M$ (Definition 4.3) and $\rho_{\mathcal{P}^{\prime}}$ is a smoothed distance
function to $\mathcal{P}^{\prime}$ in $[M:P_{1}]$. (We note that the last
expression is well-defined since, for $x\notin P_{1}$, $y$ is unique whereas,
for $x\in P_{1}$, we have $r_{P_{1}}(x)=0$.)

Again, we use the notation of Definition 3.8 (which is, in turn, the same as
the one in [6, Definition LABEL:schrSpace-def.iterated.bu]). We have then,
similarly to [6, Remark LABEL:schrSpace-rem.cond.ex].

######  Remark 4.12.

We have that $\rho_{\mathcal{P}}$ is continuous on $M$ and that
$\rho_{\mathcal{P}}^{-1}(\\{0\\})=\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}=\bigcup_{j=1}^{k}P_{j}$.
Let $\gamma_{0}=\operatorname{\mathrm{id}}$,
$\gamma_{1}\coloneqq\beta^{*}_{1}$ and
$\gamma_{j}\coloneqq\beta^{*}_{j}\circ\gamma_{j-1}=\beta^{*}_{j}\circ...\circ\beta^{*}_{1}$,
where

$\beta_{k}\ \coloneqq\
\beta_{[M:P_{1},\ldots,P_{k-1}],[P_{k}:P_{1},\ldots,P_{k-1}]}:[M:P_{1},\ldots,P_{k}]\to[M:P_{1},\ldots,P_{k-1}]\,.$

If,
$x\notin\mathchoice{{\bigcup}\,\mathcal{P}}{\medmath{\textstyle\bigcup}\,\mathcal{P}}{\medmath{\scriptstyle\bigcup}\mathcal{P}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{P}}$,
then

$\rho_{\mathcal{P}}(x)\ =\
r_{\gamma_{0}^{*}(P_{1})}(x)r_{\gamma_{1}^{*}(P_{2})}(x)\ldots
r_{\gamma_{k-1}^{*}(P_{k})}(x)\,.$

In particular, it follows that the pull-back of $\rho_{\mathcal{P}}$ to
$[M:\mathcal{P}]$ is smooth, or equivalently, suppressing all pull-backs from
the notation,
$\rho_{\mathcal{P}}\in\mathcal{C}^{\infty}\bigl{(}[M:\mathcal{P}]\bigr{)}$.
However note, that in contrast to the non-iterated case,
$\rho_{\mathcal{P}}\in\mathcal{C}^{\infty}\bigl{(}[M:\mathcal{P}]\bigr{)}$ is
no longer a boundary defining function for some boundary hypersurface of
$[M:\mathcal{P}]$.

######  Remark 4.13.

As in Remark 4.6, a smoothed distance to $\mathcal{P}$ function
$\rho_{\mathcal{P}}$ (see Definition 4.11) is not unique, but it is unique _up
to equivalence_ on $[M:\mathcal{P}]$. Indeed, this follows from Remark 4.12
since all the factors $r_{\gamma_{k-1}^{*}(P_{k})}$ are uniquely determined up
to equivalence on $[M:\mathcal{P}]$ and the equivalence is compatible with
products and preserved when we increase $\mathcal{P}$.

The following result extends to smoothed distance functions the compatibility
with products of Proposition 3.13.

######  Proposition 4.14.

Let $M$ and $M_{1}$ be two manifolds with corners and
$\mathcal{P}=(P_{i})_{i=1}^{k}$ be a blow-up-suitable $k$-tuple of subsets of
$M$. Then

$\mathcal{P}\times M_{1}\coloneqq\left(P_{j}\times M_{1}\right)_{j=1}^{k}$

is a blow-up-suitable $k$-tuple of subsets of $M\times M_{1}$. Let
$r_{\mathcal{P}}$ be a smoothed distance to $\mathcal{P}$ in $M$ and
$p:M\times M_{1}\to M$ the projection. Then $r_{\mathcal{P}}\circ p$ is a
smoothed distance function to $\mathcal{P}\times M_{1}$ in $M\times M_{1}$.

###### Proof.

If $H$ is a hyperface of $M$ and $r_{H}$ is a defining function for $H$ in
$M$, then $r_{H}\circ p$ is a defining function for the hyperface $H\times
M_{1}$ in $M\times M_{1}$. The result follows from definitions by induction on
$k$ using repeatedly this observation. Indeed, there is nothing to prove if
$k=0$. The case $k=1$ follows from Proposition 3.13 (used also for $k=1$) and
the observation about hyperfaces. The induction step is completely similar to
the case $k=1$ and it reduces to that one, as in the proof of Proposition
3.13, using that proposition, the relation $\beta^{*}(P\times
M_{1})=\beta^{*}(P)\times M_{1}$, and the definition of smoothed distance
functions. ∎

From now on, we shall assume that our $k$-tuple is a clean semilattice of
closed p-submanifolds of $M$, endowed with an admissible order. Also, from now
on we shall denote our $k$-tuple with $\mathcal{S}$ instead of $\mathcal{P}$
in order to stress that it is stable for intersections (i.e. that
$\mathcal{S}$ is a semilattice). (Recall that we can choose any admissible
order on $\mathcal{S}$.) The following proposition will play a crucial role in
our application to $N$-body type problems.

######  Proposition 4.15.

Let $\mathcal{S}$ be a clean semilattice of closed p-submanifolds of a
connected manifold with corners $M$ and $\emptyset\neq Y\in\mathcal{S}$. Let
$r_{Y}$ be a smoothed distance function to $Y$ in $M$ (Definition 4.3) and
$\rho_{\mathcal{S}}$ be a smoothed distance function to $\mathcal{S}$ in $M$
(Definition 4.11). Then the function
$\rho_{\mathcal{S}}/r_{Y}:M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{S}}{\medmath{\textstyle\bigcup}\,\mathcal{S}}{\medmath{\scriptstyle\bigcup}\mathcal{S}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{S}}\to(0,\infty)$
extends to a smooth function on $[M:\mathcal{S}]$.

Note that in the statement of the proposition, it is important that $r_{Y}$ is
a smoothed distance in $M$ and not in some blow-up of $M$. The proposition
would be a trivial consequence of the recursive definition of
$\rho_{\mathcal{S}}$ if we replaced $r_{Y}$ by a smooth distance function in
some suitable blow-up of $M$.

###### Proof.

There is no loss of generality to assume that $\emptyset\in\mathcal{S}$, so we
will do so for brevity. (The case $\emptyset\notin\mathcal{S}$ is completely
similar.) We shall prove the statement by induction on the number of elements
of $\mathcal{S}$.

If $\mathcal{S}$ has only one non-trivial element, that is, if
$\mathcal{S}=(\emptyset,Y)$, then we have $\rho_{\mathcal{S}}\simeq r_{Y}$, by
the definition of $\rho_{\mathcal{S}}$, as explained in Remarks 4.6 and 4.13.

Now, for the induction step, let us arrange
$\mathcal{S}=(P_{0}=\emptyset,P_{1},P_{2},\ldots,P_{k})$ in an admissible
order. Let us blow up along $P_{1}\in\mathcal{S}$ (which is a minimal element
of $\mathcal{S}\smallsetminus\\{\emptyset\\}$, by the definition of an
admissible order). We use the notation of Definition 4.11, in particular
$\mathcal{S}^{\prime}$ consists of the lifts to $[M:P_{1}]$ of the
p-submanifolds $P_{2},\cdots,P_{k}$. If $Y=P_{1}$, then the result follows
from the formula for $\rho_{\mathcal{S}}$ in Remark 4.12. Let us assume
therefore that $Y\neq P_{1}$. Since $P_{1}$ is a minimal (non-empty) element
of $\mathcal{S}$, we have two possibilities: either $P_{1}\subset Y$ or
$P_{1}\cap Y=\emptyset$.

Let us assume that we are in the case $P_{1}\subset Y$. We choose then as a
smoothed distance for $[Y:P_{1}]$ in $[M:P_{1}]$ the function
$r_{[Y:P_{1}]}^{\prime}\coloneqq r_{Y}/r_{P_{1}}$, which is possible in view
of Lemma 4.8. By the induction hypothesis, we know that
$(r_{Y}/r_{P_{1}})^{-1}\rho_{\mathcal{S}^{\prime}}$ extends to a smooth
function on $\bigl{[}[M:P_{1}]:\mathcal{S}^{\prime}\bigr{]}$.

Since $[[M:P_{1}]:\mathcal{S}^{\prime}]=[M:\mathcal{S}]$, by the definition of
the iterated blow-up, by suppressing pull-backs in notation we obtain that

(17) $r_{Y}^{-1}\rho_{\mathcal{S}}\ =\
r_{Y}^{-1}r_{P_{1}}\rho_{\mathcal{S}^{\prime}}\ =\
(r_{Y}/r_{P_{1}})^{-1}\rho_{\mathcal{S}^{\prime}}\in{\mathcal{C}}^{\infty}([M:\mathcal{S}])\,.$

This equation should be understood in the sense of functions
$M\smallsetminus\bigcup\mathcal{S}\to\mathbb{R}_{+}$ that extend to smooth
functions on $[M:\mathcal{S}]$. Thus, we have obtained the desired relation,
so the proof is complete in the case $P_{1}\subset Y$.

Let us assume now that $P_{1}\cap Y=\emptyset$. The proof is then an immediate
application of Lemma 4.10 for $k=1$. ∎

The following remark recalls [1, Lemma 3.17].

######  Remark 4.16.

Recall that in this and in the following sections, $\mathcal{S}$ is a clean
semilattice of closed p-submanifolds of $M$. Let
$\rho(x)\coloneqq\operatorname{dist}_{\overline{g}}(x,\mathcal{S})\coloneqq\operatorname{dist}_{\overline{g}}(x,\mathchoice{{\bigcup}\,\mathcal{S}}{\medmath{\textstyle\bigcup}\,\mathcal{S}}{\medmath{\scriptstyle\bigcup}\mathcal{S}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{S}})$
be the distance to $\mathcal{S}$ in some _true_ metric $\overline{g}$ on $M$.
(That is, a metric on $TM$ that extends smoothly to the boundary of $M$,
unlike a “compatible metric,” a concept that will be introduced shortly and
which is really only a metric on $M\smallsetminus\partial M$ with some
additional properties.) One can show that the functions $\rho_{\mathcal{S}}$
and $\rho$ are continuously equivalent. A special case of this statement which
is in fact the only case needed in the present article was proved in [1, Lemma
3.17] based on preliminary work in [8]. In [1] it is additionally assumed that
all p-submanifolds in $\mathcal{S}$ are _interior submanifolds_. By
definition, an interior submanifold is a p-submanifold of positive boundary
depth, i.e. no component is contained in $\partial M$. However, a proof may be
extended to the full statement claimed above; the result extends to
p-submanifolds that are not interior, either by adapting the proofs or by
embedding the manifold $M$ into a Riemannian manifold $N$ without corners and
boundary, such that all boundary hypersurfaces of $M$ are totally geodesic.
The function $\rho$, however is not smooth on
$M\smallsetminus\mathchoice{{\bigcup}\,\mathcal{S}}{\medmath{\textstyle\bigcup}\,\mathcal{S}}{\medmath{\scriptstyle\bigcup}\mathcal{S}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{S}}$,
in general. This justifies the notations $\rho$ and $\rho_{\mathcal{S}}$,
since they are very similar in scope. See Lemma A.2 for an extension of this
remark.

###  4.3. Blow-ups and Lie manifolds

We now look at a class of differential operators that will “desingularize” the
$N$-body Hamiltonian (with Coulomb or inverse square potentials). Let thus $M$
be a manifold with corners and
$\mathcal{V}\subset{\mathcal{C}}^{\infty}(M;TM)$ be a subspace of smooth
vector fields on $M$ that is stable for the Lie bracket and for multiplication
with functions in ${\mathcal{C}}^{\infty}(M)$. We then let
$\operatorname{Diff}_{\mathcal{V}}^{m}(M)$ denote the space of differential
operators of order $\leq m$ on $M$ generated by ${\mathcal{C}}^{\infty}(M)$
and by $\mathcal{V}$ and

(18) $\operatorname{Diff}_{\mathcal{V}}(M)\ \coloneqq\
\bigcup_{m\in\mathbb{N}_{0}}\operatorname{Diff}_{\mathcal{V}}^{m}(M)\ =\
\Bigl{\\{}\,\sum aV_{1}V_{2}\ldots V_{k}\,|\
a\in{\mathcal{C}}^{\infty}(M),V_{j}\in\mathcal{V}\,\Bigr{\\}}\,.$

Furthermore, we will assume that $(M,\mathcal{V}$) is a “Lie manifold,” a
concept from [4] that we now recall.

######  Definition 4.17.

Let $M$ be a smooth, compact manifold with corners and
$\mathcal{V}\subset{\mathcal{C}}^{\infty}(M;TM)$ be a subspace of smooth
vector fields on $M$. In addition to the relations
${\mathcal{C}}^{\infty}(M)\mathcal{V}=\mathcal{V}$ and
$[\mathcal{V},\mathcal{V}]\subset\mathcal{V}$, let us also assume that:

  1. (i)

${\mathcal{C}}^{\infty}_{\text{c}}(M\smallsetminus\partial
M;TM)\subset\mathcal{V}$;

  2. (ii)

all vector fields in $\mathcal{V}$ are tangent to the boundary; and

  3. (iii)

$\mathcal{V}$ has a ${\mathcal{C}}^{\infty}(M)$-local basis near each point.

Then we shall say that $(M;\mathcal{V})$ is a _Lie manifold_ ; the space
$\mathcal{V}$ will be called its associated _structural Lie algebra of vector
fields_.

######  Remark 4.18.

The condition (iii) in the above definition means that any $p\in M$ has a
closed neighborhood $U$ such that the restriction of $\mathcal{V}$ to $U$ is a
free ${\mathcal{C}}^{\infty}(U)$-module. An even better way of expressing
condition (iii) is to assume the existence of a vector bundle $A\to M$
together a vector bundle morphism $\varrho:A\to TM$ over
$\operatorname{\mathrm{id}}_{M}$ with $\varrho(\Gamma(A))=\mathcal{V}$. The
morphism $\varrho$ is called an _anchor_ map, see [4, 38] and the references
therein. Then Condition (i) is equivalent to saying that $\varrho$ restricts
to a vector bundle isomorphism on $M_{0}$ $\ \coloneqq\
M\smallsetminus\partial M$. Similarly, Condition (ii) is equivalent to saying
that, for any $p\in\partial M$ of boundary depth $1$, we have
$\varrho(A_{p})\subset T_{p}(\partial M)\subsetneq T_{p}M$. This condition
then implies analogous tangency conditions for points of boundary depth $\geq
2$.

The following simple and basic example(s) will play a crucial role in our
applications to the $N$-body problem.

######  Example 4.19.

We continue to assume that $X$ is a finite-dimensional real vector space and
and that $\overline{X}$ is its spherical compactification. We shall take
$\overline{X}$ for our ambient manifold (thus $\overline{X}$ plays the role of
the manifold typically called $M$ so far). Then $X$ identifies with the space
of constant vector fields on itself. Corollary B.3 tells us that these vector
fields extend to vector fields on $\overline{X}$ and that these extended
vector fields vanish on $\SS_{X}$. We shall consider then the structural Lie
algebra of vector fields
$\mathcal{V}\coloneqq{\mathcal{C}}^{\infty}(\overline{X})X$. Then the pair
$(\overline{X};\mathcal{V})$ is a Lie manifold (Definition 4.17). Indeed,
$\mathcal{V}$ is a free ${\mathcal{C}}^{\infty}(\overline{X})$–module since
any basis of $X$, viewed as a vector space, also provides a basis of
$\mathcal{V}$ as a ${\mathcal{C}}^{\infty}(\overline{X})$-module. The
resulting algebra of differential operators
$\operatorname{Diff}_{\mathcal{V}}(\overline{X})$ is the algebra of
differential operators on $X$ with coefficients in
${\mathcal{C}}^{\infty}(\overline{X})$.

To a Lie manifold $(M;\mathcal{V})$ there is canonically associated a class of
metrics on the interior of $M$, called _$\mathcal{V}$ -compatible metrics_ on
$M_{0}=M\smallsetminus\partial M$, as follows:

######  Definition 4.20.

Let $(M,\mathcal{V})$ be a Lie manifold, $n=\dim(X)$. A _$\mathcal{V}$
-compatible metric on $M$_ is defined as a Riemannian metric on $M_{0}$ with
the following property: each $x\in M$ has a closed neighborhood $U_{x}$ such
that the restriction of $\mathcal{V}$ to $U_{x}$ has a
${\mathcal{C}}^{\infty}(U_{x})$-basis $(v_{1},v_{2},\ldots,v_{n})$ that is
orthonormal in the sense

$\forall y\in M_{0}\cap U_{x}:\;(v_{1},v_{2},\ldots,v_{n})\mbox{ is an
orthonormal basis of}(T_{y}M,g_{y}).$

(Of course, this is a true restriction on $g$ and $\mathcal{V}$ only at the
boundary of $M$).

A $\mathcal{V}$-compatible metric on $M$ will never be a true metric on $M$
since the induced metric on $TM_{0}$ will not extend continuously to $M$. For
instance, for Example 4.19 (and the related Example 5.2 relevant for the
$N$-body problem), any euclidean metric on $X$ will be a
$\mathcal{V}$-compatible metric on $M$.

######  Remark 4.21.

An alternative way to define a _$\mathcal{V}$ -compatible_ is as follows. We
briefly recall this, as it will be used in Appendix A. See [4] for a proof of
the equivalence of the definitions. Let $(M,\mathcal{V})$ be a Lie manifold
with anchor map $\varrho:A\to TM$ as defined above. A compatible metric may be
defined as a metric on $A$, which is by definition a symmetric positive
definite section $\gamma\in\Gamma(A^{*}\otimes A^{*})$. The anchor map
$\varrho$ induces a map $\rho_{*}=\rho\otimes\rho\colon A\otimes A\to
TM\otimes TM$, and on $M_{0}$ the inverse of the anchor map, namely
$\left(\varrho|_{M_{0}}\right)^{-1}$, may be dualized to a pull-back map
$\left(\left(\varrho|_{M_{0}}\right)^{-1}\right)^{*}\colon A^{*}|_{M_{0}}\to
T^{*}M_{0}$. Then
$g\coloneqq\left(\left(\varrho|_{M_{0}}\right)^{-1}\right)^{*}\otimes\left(\left(\varrho|_{M_{0}}\right)^{-1}\right)^{*}(\gamma)$
is a Riemannian metric on $M_{0}$. A Riemannian metric on $M_{0}$ is
_$\mathcal{V}$ -compatible_ if, and only if it arises this way.

If $B\in V^{*}\otimes V^{*}$ is a symmetric bilinear form on a finite-
dimensional vector space $V$, then we denote $\flat_{B}:V\to V^{*}$, $X\mapsto
B(X,\mathchoice{\,\raisebox{1.30005pt}{\scalebox{0.5}{$\bullet$}}\,}{\raisebox{1.30005pt}{\scalebox{0.5}{\,$\bullet$\,}}}{\,\raisebox{0.70007pt}{\scalebox{0.4}{$\bullet$}}\,}{\,\raisebox{0.59998pt}{\scalebox{0.35}{$\bullet$}}\,})$
the associated canonical homorphism. If $B$ is non-degenerate, $\flat_{B}$ is
invertible and one defines $\sharp_{B}\coloneqq\flat_{B}^{-1}$. In this case
we can define $\prescript{\lower
1.95pt\hbox{\reflectbox{$\scriptstyle\sharp$}\kern-0.8pt}}{}{B^{\sharp}}\coloneqq
B(\sharp_{B}(\mathchoice{\,\raisebox{1.30005pt}{\scalebox{0.5}{$\bullet$}}\,}{\raisebox{1.30005pt}{\scalebox{0.5}{\,$\bullet$\,}}}{\,\raisebox{0.70007pt}{\scalebox{0.4}{$\bullet$}}\,}{\,\raisebox{0.59998pt}{\scalebox{0.35}{$\bullet$}}\,}),\sharp_{B}(\mathchoice{\,\raisebox{1.30005pt}{\scalebox{0.5}{$\bullet$}}\,}{\raisebox{1.30005pt}{\scalebox{0.5}{\,$\bullet$\,}}}{\,\raisebox{0.70007pt}{\scalebox{0.4}{$\bullet$}}\,}{\,\raisebox{0.59998pt}{\scalebox{0.35}{$\bullet$}}\,}))\in
V\otimes V$ whose matrix with respect to some basis of $V$ is the inverse of
the matrix of $B$ with respect to its dual basis. Now, if $\gamma$ is a metric
on the vector bundle $A$, we map apply this construction for any $q\in M$ to
$\gamma|_{q}\in A_{q}^{*}\otimes A_{q}^{*}$, and we obtain a smooth section
$\prescript{\lower
1.95pt\hbox{\reflectbox{$\scriptstyle\sharp$}\kern-0.8pt}}{}{\gamma^{\sharp}}\in\Gamma(A\otimes
A)$. Then $G\coloneqq\bigl{(}\rho\otimes\rho\bigr{)}(\prescript{\lower
1.95pt\hbox{\reflectbox{$\scriptstyle\sharp$}\kern-0.8pt}}{}{\gamma^{\sharp}})\in\Gamma(TM\otimes
TM)$ is a well-defined smooth tensor. One can easily show that $G$ is the
unique continuous extension of $\prescript{\lower
1.95pt\hbox{\reflectbox{$\scriptstyle\sharp$}\kern-0.8pt}}{}{g^{\sharp}}$ for
the $g$ defined above. We will use this description to proof in Appendix A for
a true metric $\overline{g}$ and a compatible metric $g$ we have, see Lemma
A.2, a constant $C$ with $\overline{g}\leq Cg$ in the sense of symmetric
forms, i.e. $Cg-\overline{g}$ is positive semi-definite.

Any two $\mathcal{V}$-compatible metrics $g$ and $\tilde{g}$ are Lipschitz
equivalent, i.e. there exists a constant $C>0$ with $C^{-1}\tilde{g}\leq g\leq
C\tilde{g}$, again in the sense of symmetric forms. Thus the volume forms are
Lipschitz equivalent as well, and hence the definition of the space
$L^{p}(M,g)=L^{p}(M;\mathcal{V})$ only depends on $\mathcal{V}$, and not on
the metric. This allow us to define the associated _Sobolev spaces_ on $X$ (or
$M$) as

(19) $W^{k,p}(M;\mathcal{V})\ =\ \bigl{\\{}u\,\bigm{|}\;V_{1}\ldots V_{j}u\in
L^{p}(M;\mathcal{V}),\,\forall
V_{1},\ldots,V_{j}\in\mathcal{V},\,j\in\\{0,1,\ldots,k\\}\bigr{\\}}\,$

where $k\in\mathbb{N}_{0}$ and $1\leq p\leq\infty$. Obviously, they do not
depend on the choice of $\mathcal{V}$-compatible metric. These Sobolev spaces
coincide with the standard Soblev spaces on the complete Riemannian manifold
associated to any compatible metric.

The main reason we are interested in Lie manifolds is because there is also a
notion of an associated pseudodifferential calculus [5], a notion of
ellipticity in $\operatorname{Diff}_{\mathcal{V}}(M)$, and, most importantly
for this paper, an elliptic regularity result [3]. Explicitly, an operator in
$\operatorname{Diff}_{\mathcal{V}}^{m}(M)$ is _elliptic_ in
$\operatorname{Diff}_{\mathcal{V}}^{m}(M)$ if its principal symbol is
uniformly elliptic in one (equivalently, any) $\mathcal{V}$-compatible metric.

The following theorem is the special case $s=0$ and $m\in\mathbb{Z}_{+}$ of
[3, Theorem 8.7].

######  Theorem 4.22 (Ammann-Ionescu-Nistor).

Let $(M;\mathcal{V})$ be a Lie manifold. Let $k,j\in\mathbb{Z}$,
$m\in\mathbb{Z}_{+}$, $1<p<\infty$, and
$P\in\operatorname{Diff}_{\mathcal{V}}^{m}(M)$ be elliptic. Let $u\in
W^{k,p}(M;\mathcal{V})$ be such that $Pu\in W^{j,p}(M;\mathcal{V})$. Then
$u\in W^{j+m,p}(M;\mathcal{V})$.

Let us mention also that for in Example 4.19 (and the related Example 5.2
relevant for the $N$-body problem), an operator
$P\in\operatorname{Diff}_{\mathcal{V}}(M)$ is elliptic in that algebra if, and
only if, it is uniformly elliptic (in the usual, euclidean metric of $X$). In
particular, the above regularity theorem (Theorem 6.1) can also be obtained
from the regularity result in [2].

Let us now provide an alternative, more general method to obtain a Lie
manifold structure on the Georgescu-Vasy space $X_{\mathrm{GV}}$. (See the
Introduction or Section 5 (22) for the definition of $X_{\mathrm{GV}}$.)

######  Remark 4.23.

Let $(M;\mathcal{V})$ be a Lie manifold and $Y\subset M$ be a closed
p-submanifold. We assume $Y\subset\partial M$, i.e. $Y$ is everywhere of
positive boundary depth. We also assume that all $V\in\mathcal{V}$ satisfy
$V|_{Y}\in\Gamma(TY)$. The latter condition implies (due to a straightforward
modification of [1, Proposition 3.2]) that any $V\in\mathcal{V}$ has a _lift_
$\tilde{V}$, that is, a vector field on $[M:Y]$ such that

$\textstyle{[M:Y]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{V}}$$\scriptstyle{\beta_{M,Y}}$$\textstyle{T[M:Y]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\beta_{M,Y}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{TM.}$

commutes. As a consequence the inclusion
$\mathcal{V}\hookrightarrow\Gamma(TM)$ “lifts” to a map
$\mathcal{V}\hookrightarrow\Gamma(T[M:Y])$, both maps are injective Lie
algebra homomorphisms and $\mathcal{C}^{\infty}(M)$-linear. We obtain a Lie
manifold structure on $[M:Y]$ by taking as structural Lie algebra of vector
fields

(20) $\mathcal{W}_{\mathrm{bdry}}\ \coloneqq\
{\mathcal{C}}^{\infty}([M:Y]\,\mathcal{V}\,.$

Let us record here how the various quantities associated to the Lie manifold
$(M,\mathcal{V})$ change when going to the blow-up Lie manifold
$([M:Y],\mathcal{W}_{\mathrm{bdry}})$:

  * •

The interior smooth manifold is the same: $M_{0}=M\smallsetminus\partial
M=[M:Y]\smallsetminus\partial[M:Y]$.

  * •

If a metric on $M_{0}$ is $\mathcal{V}$-compatible, then it is also
$\mathcal{W}_{\mathrm{bdry}}$-compatible.

  * •

$W^{k,p}(M;\mathcal{V})=W^{k,p}([M:Y];\mathcal{W}_{\mathrm{bdry}})$.

  * •

$\operatorname{Diff}_{\mathcal{W}_{\mathrm{bdry}}}([M:Y])={\mathcal{C}}^{\infty}([M:Y])\operatorname{Diff}_{\mathcal{V}}(M)$.

The last equality is based on the fact that the lift of $\mathcal{V}$ acts on
${\mathcal{C}}^{\infty}([M:Y])$ by derivations.

By iterating this construction, we can obtain similarly a Lie manifold
structure on $[M:\mathcal{Y}]$, where $\mathcal{Y}$ is a clean semilattice of
p-submanifolds of $\partial M$. Indeed, let us assume that $\mathcal{V}$ is
tangent to each manifold $Y\subset\mathcal{Y}$, then $\mathcal{V}$ will lift
to all intermediate blow-ups leading to $[M:\mathcal{Y}]$. At each step, by
density, the lifting of $\mathcal{V}$ will be tangent to the manifold with
respect to which we blow-up.

######  Example 4.24.

Recall from the Introduction that $\mathcal{F}$ is a finite semi-lattice of
linear subspaces of $X$ such that $\\{0\\}\in\mathcal{F}$ and
$X\not\in\mathcal{F}$. The Lie manifold structure on $X_{\mathrm{GV}}$ will be
obtained starting from $\overline{X}$ by taking
$\mathcal{Y}\coloneqq\SS_{\mathcal{F}}\coloneqq\\{\SS_{Y}\mid\,Y\in\mathcal{F}\\}$.
Since the action of $X$ on the boundary $\SS_{X}$ of $\overline{X}$ is
trivial, see Remark 4.19 and Appendix C, the vector fields in
$\mathcal{V}\coloneqq\\{fV\mid f\in{\mathcal{C}}^{\infty}(\overline{X}),V\in
X\\}$ are tangent to all the submanifolds of $\SS_{\mathcal{F}}$.

The situation described in the last remark changes, however, dramatically if
$Y$ has boundary depth $0$, i.e. if no connected component of $Y$ is contained
of the boundary. In that case, there is, in principle, more work to be done,
but that has already been completed to a large extent in [1]. Let us summarize
some required results from that article.

######  Remark 4.25.

We use the notation of Remark 4.23. Unlike there, howevever, we shall
consider, this time, the case when $Y$ is connected and has boundary depth $0$
(that is, it is not contained in the boundary $\partial M$ of $M$). Let
$r_{Y}$ be a smoothed distance function to $Y$, Definition 4.3. We then obtain
a Lie manifold structure on $[M:Y]$ as in [1] by taking as structural Lie
algebra of vector fields

(21) $\mathcal{W}_{\mathrm{int}}\ \coloneqq\
r_{Y}{\mathcal{C}}^{\infty}([M:Y])\mathcal{V}\,.$

See [1, Lemma 3.8 and Theorem 3.10] where it was proved that all vector fields
in $r_{Y}\mathcal{V}$ lift canonically to smooth vector fields on $[M:Y]$.
(This justifies the additional factor $r_{Y}$, since without it, that would
not cover our case.) Let us record here how the various quantities associated
to the Lie manifold $(M,\mathcal{V})$ change when going to the blow-up Lie
manifold $([M:Y],\mathcal{W}_{\mathrm{int}})$:

  * •

The interior smooth manifold of the blow up is, this time:

$M_{1}\ \coloneqq\ [M:Y]\smallsetminus\partial[M:Y]\ =\
M\smallsetminus(\partial M\cup Y)\,.$

  * •

If $g$ is a $\mathcal{V}$-compatible metric on $M$, then $\tilde{g}\coloneqq
r_{Y}^{-2}\beta_{M,Y}^{*}g$ is a $\mathcal{W}_{\mathrm{int}}$-compatible
metric on $[M:Y]$.

  * •

The volume forms for $g$ and $\tilde{g}$ are related by the formula
$\operatorname{dvol}\nolimits^{\tilde{g}}=r_{Y}^{-n}\beta_{[M:Y]}^{*}\operatorname{dvol}\nolimits^{g}$,
$n=\dim(M)$. As a consequence we have

$u\in
L^{p}([M:Y];\mathcal{W}_{\mathrm{int}})\;\Longleftrightarrow\;r_{Y}^{-n/p}u\in
L^{p}(M;\mathcal{V})$

  * •

The Sobolev spaces defined by $\mathcal{W}_{\mathrm{int}}$ are “weighted
Sobolev spaces” in the old metric (compare to Equation (19) and notice the
factor $r_{Y}^{j}$):

$\displaystyle W^{k,p}\bigl{(}[M:Y],\mathcal{W}_{\mathrm{int}}\bigr{)}$
$\displaystyle=$ $\displaystyle\Bigl{\\{}u\,\big{|}\;r_{Y}^{j}V_{1}\ldots
V_{j}u\in L^{p}\bigl{(}[M:Y];\mathcal{W}_{\mathrm{int}}\bigr{)},\,\forall
V_{1},\ldots,V_{j}\in\mathcal{V},$
$\displaystyle\phantom{\Bigl{\\{}u\mid}j\in\\{0,1,\ldots,k\\}\Bigr{\\}}\,$

  * •

Let $m\in\mathbb{Z}_{+}$, then
$\operatorname{Diff}_{\mathcal{W}_{\mathrm{int}}}^{m}([M:Y])$ is the linear
span of differential monomials of the form $r_{Y}^{j}aV_{1}V_{2}\ldots V_{j}$,
with $a\in{\mathcal{C}}^{\infty}([M:Y])$,
$V_{1},V_{2},\ldots,V_{j}\in\mathcal{V}$, and $0\leq j\leq m$. Moreover, if
$D\in\operatorname{Diff}_{\mathcal{V}}^{m}(M)$, then
$r_{Y}^{m}D\in\operatorname{Diff}_{\mathcal{W}_{\mathrm{int}}}^{m}([M:Y])$
and, if $D$ is elliptic in $\operatorname{Diff}_{\mathcal{V}}^{m}(M)$, then
$r_{Y}^{m}D$ is elliptic in
$\operatorname{Diff}_{\mathcal{W}_{\mathrm{int}}}^{m}([M:Y])$.

The last two statements are obtained using also that
$V(r_{Y})\in{\mathcal{C}}^{\infty}([M:Y])$ for all $V\in\mathcal{V}$. (We note
in passing that $V(r_{Y})\notin{\mathcal{C}}^{\infty}(M)$.)

As in Remark 4.23, we can iterate the construction of Remark 4.25, to obtain
similarly a Lie manifold structure on $[M:\mathcal{Y}]$, where $\mathcal{Y}$
is a clean semilattice (i.e. a cleanly intersecting semilattice) of
p-submanifolds of $M$. All the resulting objects on $[M:\mathcal{Y}]$ will be
independent on the admissible order chosen on $\mathcal{Y}$.

##  5\. The Lie manifolds associated to the generalized $N$-body problem

We now investigate how the constructions of the previous section particularize
to the case of $N$-body problems. Let us recall our standing conventions that
$X$ is a finite dimensional, real vector space and that $\mathcal{F}$ is a
finite semilattice of _linear_ subspaces of $X$ satisfying
$\\{0\\}\in\mathcal{F}$ and $X\notin\mathcal{F}$. (We stress, however, that
the assumptions $\\{0\\}\in\mathcal{F}$ and $X\notin\mathcal{F}$ are not
essential, since the general case of a finite semilattice $\mathcal{F}$ easily
reduces to this case.)

###  5.1. Semilattices and blow-ups for the $N$-body problem

We begin by introducing the semilattices that we will use for the study of the
$N$-body problem. These semilattices will then be used to introduce the
associated blow-ups and Lie manifolds.

####  5.1.1. The semilattices and the blow-ups

We denote by $\\{0\\}$ the vector space consisting of just $0\in X$. We agree
that $\SS_{\\{0\\}}=\emptyset$, and hence $\overline{\\{0\\}}=\\{0\\}$. As in
[1, 6], we shall consider the semilattices
$\overline{\mathcal{F}}\coloneqq\\{\overline{Y}\mid Y\in\mathcal{F}\\}$ and
$\SS_{\mathcal{F}}\coloneqq\\{\SS_{Y}\mid Y\in\mathcal{F}\\}$, recalled
earlier in Equation (4). Note that $\emptyset\in\SS_{\mathcal{F}}$ because
$0\in\mathcal{F}$.

######  Proposition 5.1.

The sets $\overline{\mathcal{F}}$, $\SS_{\mathcal{F}}$,
$\SS_{\mathcal{F}}\cup\overline{\mathcal{F}}$ are _clean semilattices_ of
p-submanifolds of $\overline{X}$.

###### Proof.

In view of Remark 2.13, this result is obtained by a direct application of
Lemma 2.15. ∎

We fix an admissible order on $\SS_{\mathcal{F}}$, but the constructions below
do not depend on this choice. For instance, the _Georgescu-Vasy space_

(22) $X_{\mathrm{GV}}\ \coloneqq\ [\overline{X}:\SS_{\mathcal{F}}]$

does not depend on the choice of an admissible order on $\SS_{\mathcal{F}}$.
(This space is the same as the one introduced in [6] and in Equation (5).) We
now introduce the Lie manifold structure on $X_{\mathrm{GV}}$.

######  Example 5.2.

Let $X$ act on $\overline{X}$ by translations, as in Example 4.19. Let also
$\mathcal{S}$ be a finite, clean semilattice of closed submanifolds of
$\SS_{X}=\overline{X}\smallsetminus X$. The Lie group action of $X$ on
$\overline{X}$ acts trivially on $\SS_{X}$, thus it preserves $\mathcal{S}$,
and therefore we obtain an action of $X$ on the blown-up space
$[\overline{X}:\mathcal{S}]$ (where the blow-up is defined using any
admissible order on $\mathcal{S}$). As explained in Lemma B.4, this implies
that any constant vector field on $X$ extends to a smooth vector field on
$M\coloneqq[\overline{X}:\mathcal{S}]$ that is _tangent_ to the boundary. Let
$\mathcal{W}_{\mathrm{bdry}}\coloneqq{\mathcal{C}}^{\infty}(M)X$.

As in Example 4.19, then $(M;\mathcal{W}_{\mathrm{bdry}})$ is a Lie manifold
(Definition 4.17). Indeed, again, $\mathcal{W}_{\mathrm{bdry}}$ is a free
${\mathcal{C}}^{\infty}(M)$–module, a basis being given by extensions of the
canonical basis, viewed as constant vector fields. In this case, the resulting
algebra of differential operators
$\operatorname{Diff}_{\mathcal{W}_{\mathrm{bdry}}}(M)$ is the algebra of
differential operators on $X$ with coefficients in
${\mathcal{C}}^{\infty}(M)$. So the blow-up of $\overline{X}$ by $\mathcal{S}$
leads to a larger algebra of differential operators than the one associated to
the Lie algebroid $(\overline{X},\mathcal{V})$,
$\mathcal{V}\coloneqq{\mathcal{C}}^{\infty}(\overline{X})X$, considered in
Example 4.19. The class of differential operators to which our regularity
results apply, however, is even larger than
$\operatorname{Diff}_{\mathcal{W}_{\mathrm{bdry}}}(M)$ (it allows also for
interior blow-ups).

For us, the relevant choice is $\mathcal{S}=\SS_{\mathcal{F}}$, which yields
$[\overline{X}:\SS_{\mathcal{F}}]=:X_{\mathrm{GV}}.$ (Note that
$X_{\mathrm{GV}}$ was defined using only boundary blow-ups.) We thus have
obtained a _Lie manifold structure on the Georgescu-Vasy space
$X_{\mathrm{GV}}$._ This class of examples appears implicitly (but
prominently) in Georgescu’s work [24, 25].

We now proceed to further blow-up $X_{\mathrm{GV}}$ with respect to
$\overline{\mathcal{F}}$, i.e. the lifts of elements of $\mathcal{F}$.

######  Lemma 5.3.

Let us arrange both $\SS_{\mathcal{F}}$ and $\overline{\mathcal{F}}$ in
admissible orders (see Definition 3.14). We then use the resulting order to
obtain an order relation on $\SS_{\mathcal{F}}\cup\overline{\mathcal{F}}$ in
which all the elements of $\SS_{\mathcal{F}}$ preceed the ones of
$\overline{\mathcal{F}}$. Then the resulting order on the union
$\SS_{\mathcal{F}}\cup\overline{\mathcal{F}}$ is an admissible order.

###### Proof.

Let $\overline{Y}\in\overline{\mathcal{F}}$, where $Y\in\mathcal{F}$. Let also
$\mathcal{P}\subset\SS_{\mathcal{F}}$, $\mathcal{P}\neq\SS_{\mathcal{F}}$, and
and $\beta:[\overline{X}:\mathcal{P}]\to\overline{X}$ be the blow-down map.
Then $\beta^{*}(\overline{Y})$ will contain $Y$ and hence cannot be a minimal
element of
$(\SS_{\mathcal{F}}\cup\overline{\mathcal{F}})\smallsetminus\mathcal{P}$ since
it is not contained in any element of
$\SS_{\mathcal{F}}\smallsetminus\mathcal{P}$. ∎

We shall need for the following result the following notation for the lifts of
$\mathcal{F}$ to $X_{\mathrm{GV}}$. Let
$\beta_{\mathrm{GV}}:X_{\mathrm{GV}}\coloneqq[\overline{X}:\SS_{\mathcal{F}}]\to\overline{X}$
be the blow-down map. If $Y\in\mathcal{F}$, we let

(23) $\begin{gathered}\widehat{Y}\ \coloneqq\
\beta_{\mathrm{GV}}^{*}\overline{Y}\subset X_{\mathrm{GV}}\quad\mbox{ and}\\\
\widehat{\mathcal{F}}\ \coloneqq\ \\{\widehat{Y}\mid\,\ Y\in\mathcal{F}\\}\ =\
\beta_{\mathrm{GV}}^{*}\overline{\mathcal{F}}\,.\end{gathered}$

(In general, later on, we shall denote by $\widehat{Y}$ the lift of
$\overline{Y}$ to any intermediate blow-up between
$X_{\mathrm{GV}}:=[X:\SS_{\mathcal{F}}]$ and
$[\overline{X}:\SS_{\mathcal{F}}\cup\overline{\mathcal{F}}]$.)

The independence of the blow-up on the admissible order [6] (a result reminded
in Theorem 3.18), gives the following

######  Corollary 5.4.

We have canonical diffeomorphisms

$X_{\mathcal{F}}\ \coloneqq\
[\overline{X}:\SS_{\mathcal{F}}\cup\overline{\mathcal{F}}]\,\simeq\,\bigl{[}[\overline{X}:\SS_{\mathcal{F}}]:\widehat{\mathcal{F}}\bigr{]}\
=\ [X_{\mathrm{GV}}:\widehat{\mathcal{F}}]\,,$

where $\widehat{\mathcal{F}}$ is the lift of $\overline{\mathcal{F}}$ to
$[\overline{X}:\SS_{\mathcal{F}}]$.

######  Remark 5.5.

Let us assume that we have an order
$\mathcal{F}=(Y_{0}=\\{0\\},Y_{1},\ldots,Y_{k})$ on the linear subspaces of
$\mathcal{F}$ such that $Y_{i}\subsetneq Y_{j}\ \Rightarrow i<j$ (that is, we
have a “size-order” on $\mathcal{F}$, in the sense of Kottke [36]). Then this
is an admissible order on $\mathcal{F}$ and the induced orders on
$\SS_{\mathcal{F}}$, $\overline{\mathcal{F}}$, and $\widehat{\mathcal{F}}$ are
again admissible.

So $X_{\mathcal{F}}$ is obtained from $X_{\mathrm{GV}}$ using only _interior_
blow-ups. By contrast, in [1], the process of blowing up along the collision
planes $\overline{\mathcal{F}}$ started from $\overline{X}$ to arrive at
$[\overline{X}:\overline{\mathcal{F}}]$. This had the disadvantage that the
smooth functions on $\overline{X/Y}$ do not lift to smooth functions on
$[\overline{X}:\overline{\mathcal{F}}]$. Our paper thus fixes this issue from
[1] and provides uniform estimates at infinity.

####  5.1.2. Boundary blow-ups and the data on $X_{\mathrm{GV}}$

Let us now see how the constructions of Lie manifolds and of their geometric
objects of the previous section particularize to our setting, taking into
account the two types of blow-ups: boundary blow-up (along a p-submanifold
contained in the boundary) and interior blow-up (along a p-submanifold not
contained in the boundary). We begin with the blow-up of $\overline{X}$ with
respect to $\SS_{\mathcal{F}}$, which, we recall, is obtained using a sequence
of boundary blow-ups. We will let $\mathcal{W}_{\mathrm{eucl}}$ denote the
structural Lie algebra of vector fields on $\overline{X}$.

######  Remark 5.6.

We will now define a Lie manifold structure on $\overline{X}$, as well as on
any of the intermediate blow-ups leading all the way up to
$X_{\mathrm{GV}}\coloneqq[\overline{X}:\SS_{\mathcal{F}}]$. At first, the Lie
manifold structure on $\overline{X}$ is defined by the action of $X$ on itself
and on $\overline{X}$ as described in Example 4.19, namely, the structural Lie
algebra of vector fields $\mathcal{W}_{\mathrm{eucl}}$ on $\overline{X}$ is
$\mathcal{W}_{\mathrm{eucl}}\coloneqq{\mathcal{C}}^{\infty}(\overline{X})X$.
We have already described in [6] and in then in the Introduction of this paper
how to get from $\overline{X}$ to $X_{\mathrm{GV}}$ by iterated blow-ups. On
any of these intermediate blow-ups $\widehat{X}$ Theorem 3.18 defines an
action of $X$ (as a Lie group) on $\widehat{X}$ extending smoothly the
translation action. Deriving this Lie group action yields a Lie algebra action
of $X$ on $\widehat{X}$, given by a linear map $E:X\to\Gamma(\widehat{X})$,
which in fact provides smooth extensions of all constant vector fields on $X$
to the compactification $\widehat{X}$. In general $E(X)$ will not vanish at
the boundary, but it will be tangent to the boundary, see below for further
discussion.

On all these blow-ups leading to the Georgescu-Vasy space $X_{\mathrm{GV}}$,
the euclidean metric will be compatible with the Lie manifold structure on the
(intermediate) blow-ups, since we are performing only boundary blow-ups.

An alternative way to obtain the Lie manifold structure on all the iterated
blow-ups $\widehat{X}$, including $X_{\mathrm{GV}}$ is to use Remark 4.23 to
define the Lie manifold structure inductively, starting from the Lie manifold
structure on $\overline{X}$. It is easy to check by induction over the
iterated blow-ups that this construction yields the same Lie manifold
structures as above.

We stress that, as a consequence of Remark 4.23, the resulting Sobolev spaces
on all the blow-ups between $\overline{X}$ and $X_{\mathrm{GV}}$ will be the
same. Namely, they are the _usual Sobolev spaces_ on $X$. Recall that we
identify $X$ with constant vector fields on itself. The algebra of
differential operators, however, will change with each blow-up (along some
$\SS_{Y}$, $Y\in\mathcal{F}$), eventually leading on
$[\overline{X}:\SS_{\mathcal{F}}]$ to the structural Lie algebra of vector
fields
$\mathcal{W}_{\mathrm{bdry}}\coloneqq{\mathcal{C}}^{\infty}([\overline{X}:\SS_{\mathcal{F}}])X$.
Therefore,

$\operatorname{Diff}_{\mathcal{W}_{\mathrm{bdry}}}(X_{\mathrm{GV}})\ =\
{\mathcal{C}}^{\infty}(X_{\mathrm{GV}})\mathbb{C}[X]\,,$

where $\mathbb{C}[X]$ denotes the algebra of polynomials on $X$ and where a
monomial of degree $d$ is interpreted as a differential operator of degree $d$
with constant coefficients.

####  5.1.3. Interior blow-ups

As we have seen already, the situation becomes more complicated once we start
performing blow-ups along (lifts of) elements of $\overline{\mathcal{F}}$.

######  Remark 5.7.

We use the notation of Example 5.2 and Lemma 5.3. In particular, let
$\widehat{\mathcal{F}}$ be the lifting of $\overline{\mathcal{F}}$ to
$X_{\mathrm{GV}}$. Let $\rho_{\widehat{\mathcal{F}}}$ be a smoothed distance
function to $\widehat{\mathcal{F}}$ in
$X_{\mathrm{GV}}\coloneqq[\overline{X}:\SS_{\mathcal{F}}]$. By iterating
Remark 4.25, we obtain a Lie manifold structure on

$X_{\mathcal{F}}\ \coloneqq\
[\overline{X}:\SS_{\mathcal{F}}\cup\overline{\mathcal{F}}]\ =\
\bigl{[}[\overline{X}:\SS_{\mathcal{F}}]:\widehat{\mathcal{F}}\bigr{]}\ =\
[X_{\mathrm{GV}}:\widehat{\mathcal{F}}]$

(see Corollary 5.4) by taking

$\mathcal{W}_{\mathcal{F}}\ \coloneqq\
\rho_{\widehat{\mathcal{F}}}\,{\mathcal{C}}^{\infty}(X_{\mathcal{F}})\mathcal{W}_{\mathrm{int}}\
=\
\rho_{\widehat{\mathcal{F}}}\,{\mathcal{C}}^{\infty}(X_{\mathcal{F}})\mathcal{W}_{\mathrm{eucl}}\
=\ \rho_{\widehat{\mathcal{F}}}\,{\mathcal{C}}^{\infty}(X_{\mathcal{F}})X\,.$

Let us record here what are the various quantities associated to the Lie
manifold $(X_{\mathcal{F}},\mathcal{W}_{\mathcal{F}})$, again by iterating
Remark 4.25 (starting from $M\coloneqq X_{\mathrm{GV}}$ and successively
blowing up with respect to the lifts of $Y\in\mathcal{F}$):

  * •

The interior smooth manifold of the blow-up is this time:

$X_{\mathcal{F}}\smallsetminus\partial(X_{\mathcal{F}})\ =\
X\smallsetminus\mathchoice{{\bigcup}\,\mathcal{F}}{\medmath{\textstyle\bigcup}\,\mathcal{F}}{\medmath{\scriptstyle\bigcup}\mathcal{F}}{\medmath{\scriptscriptstyle\bigcup}\mathcal{F}}\,.$

  * •

If $g$ is a $\mathcal{W}_{\mathrm{eucl}}$-compatible metric on
$X_{\mathrm{GV}}$, then
$\tilde{g}\coloneqq\rho_{\widehat{\mathcal{F}}}^{-2}\beta^{*}g$ is a
$\mathcal{W}_{\mathcal{F}}$-compatible metric on
$X_{\mathcal{F}}\coloneqq[X_{\mathrm{GV}}:\widehat{\mathcal{F}}]$. Here
$\beta:X_{\mathcal{F}}\to X_{\mathrm{GV}}$ is the associated blow-down map.

  * •

The volume forms for $g$ and $\tilde{g}$ are related by the formula
$\operatorname{dvol}\nolimits^{\tilde{g}}=\rho_{\widehat{\mathcal{F}}}^{-n}\beta^{*}\operatorname{dvol}\nolimits^{g}$,
$n=\dim(M)=\dim(X)$. As a consequence, we have

$u\in
L^{p}(X_{\mathcal{F}};\mathcal{W}_{\mathcal{F}})\;\Longleftrightarrow\;\rho_{\widehat{\mathcal{F}}}^{-n/p}u\in
L^{p}(X_{\mathrm{GV}};\mathcal{W}_{\mathrm{int}})\;\Longleftrightarrow\;\rho_{\widehat{\mathcal{F}}}^{-n/p}u\in
L^{p}(X;\operatorname{dvol}\nolimits_{g})\,.$

  * •

The Sobolev spaces defined by $\mathcal{W}_{\mathcal{F}}$ are “weighted
Sobolev spaces” in the old metric $g$ (compare to Equation (19) and notice the
factor $\rho_{\widehat{\mathcal{F}}}^{j}$):

$W^{k,p}\bigl{(}X_{\mathcal{F}},\mathcal{W}_{\mathcal{F}}\bigr{)}\ =\
\Bigl{\\{}u\,\bigm{|}\;\rho_{\widehat{\mathcal{F}}}^{j}v_{1}\ldots v_{j}u\in
L^{p}\bigl{(}X_{\mathcal{F}};\mathcal{W}_{\mathcal{F}}\bigr{)}\,,\ \forall
v_{1},\ldots,v_{j}\in X,\,0\leq j\leq k\,\Bigr{\\}}\\\
=\Bigl{\\{}u\,\bigm{|}\;\rho_{\widehat{\mathcal{F}}}^{j}V_{1}\ldots V_{j}u\in
L^{p}\bigl{(}X_{\mathcal{F}};\mathcal{W}_{\mathcal{F}}\bigr{)}\,,\ \forall
V_{1},\ldots,V_{j}\in\mathcal{W}_{\mathrm{int}},\,0\leq j\leq
k\,\Bigr{\\}}\,.$

  * •

$\operatorname{Diff}_{\mathcal{W}_{\mathcal{F}}}^{m}(X_{\mathcal{F}})$ is the
linear span of differential monomials of the form
$a\rho_{\widehat{\mathcal{F}}}^{j}V_{1}V_{2}\ldots V_{j}$, with
$a\in{\mathcal{C}}^{\infty}(X_{\mathcal{F}})$,
$V_{1},V_{2},\ldots,V_{j}\in\mathcal{W}_{\mathrm{eucl}}$, $0\leq j\leq m$.

In order to obtain the last statement, we use, in particular, that
$\rho_{\widehat{\mathcal{F}}}$ is the product of the smoothed distance
functions to the blow-ups of $\widehat{Y}$, where
$\widehat{Y}\in\widehat{\mathcal{F}}$, see Remark 4.12. It is important to
keep in mind that $\rho_{\widehat{\mathcal{F}}}$ is a smooth distance
functions in $X_{\mathrm{GV}}$ and that it is a product of smoothed distance
functions in blow-ups of $X_{\mathrm{GV}}$; the statements would be different
for smoothed distance functions in $\overline{X}$ and the corresponding blow-
ups thereof.

###  5.2. A lifting lemma and smoothed distance functions

We continue to let $X$ be a finite-dimensional euclidean vector space. Let $Y$
be a linear subspace of $X$. In this subsection, we prove a technical lemma (a
lifting lemma) and apply it to the study of smoothed distance functions,
leading to some results that will be used in the next subsection. Recall that
the projection map $\pi_{X/Y}:X\to X/Y$ extends canonically to a smooth map
$\psi_{Y}:[\overline{X}:\SS_{Y}]\to\overline{X/Y}$, see [6, Proposition
LABEL:schrSpace-prop.X/Y] and the proof of [36, Theorem 4.1]. See also [40].

######  Lemma 5.8.

There exists a diffeomorphism

(24) $\Psi_{Y}\ =\
(\psi_{Y},\chi):[\overline{X}:\SS_{Y}]\to\overline{X/Y}\times\overline{Y}$

such that $\chi(y)=y$ if $y\in Y$ and
$\beta_{\overline{X},\SS_{Y}}\circ\Psi_{Y}^{-1}$ restricts on
$\overline{X/Y}\times\SS_{Y}$ to the projection
$\overline{X/Y}\times\SS_{Y}\to\SS_{Y}$.

###### Proof.

This follows from [6, Lemma LABEL:schrSpace-blow.sphere] for $k=1$ and
$k^{\prime}=0$, see Proposition C.2 in Appendix C for details. ∎

######  Remark 5.9.

In order to compare the above well-known lemma to the literature, we mention
that it says that the map $\psi_{Y}$ is the projection maps of a trivial
fibration with fiber $\overline{Y}$. In this context a smooth map $f:A\to B$
between manifolds with corners is the projection of a trivial fibration with
fiber $C$, if and only if, $f$ is the first component of a diffeomorphism
$A\to B\times C$. This is a special case of a fibration in the sense of [40,
Subsection 2.4], a concept whose definition will not be recalled here, as it