# Semiclassical analysis of a nonlocal boundary value problem related to
magnitude

Heiko Gimperlein, Magnus Goffeng, Nikoletta Louca  Heiko Gimperlein
Leopold-Franzens-Universität Innsbruck
Technikerstraße 13
6020 Innsbruck
Austria

Magnus Goffeng,
Centre for Mathematical Sciences
University of Lund
Box 118, SE-221 00 Lund
Sweden

Nikoletta Louca
Maxwell Institute for Mathematical Sciences and
Department of Mathematics, Heriot-Watt University
Edinburgh EH14 4AS
United Kingdom
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study a Dirichlet boundary problem related to the fractional Laplacian in a
manifold. Its variational formulation arises in the study of magnitude, an
invariant of compact metric spaces given by the reciprocal of the ground state
energy. Using recent techniques developed for pseudodifferential boundary
problems we discuss the structure of the solution operator and resulting
properties of the magnitude. In a semiclassical limit we obtain an asymptotic
expansion of the magnitude in terms of curvature invariants of the manifold
and the boundary, similar to the invariants arising in short-time expansions
for heat kernels.

###### Key words and phrases:

47F10 (primary), 35R11, 35S15, 58J40, 31C45 (secondary)

##  1\. Introduction

The analysis of boundary problems for nonlocal operators has attracted much
interest in recent years, including Dirichlet and Neumann problems for
fractional Laplacians. In this article we initiate the semiclassical analysis
of related boundary problems motivated by applications to the Leinster-
Willerton conjecture for the magnitude invariant of compact metric spaces.

To be specific, we consider the integral equation with parameter $R>0$

(1) $\int_{X}\mathrm{e}^{-R\mathrm{d}(x,y)}u(y)\mathrm{d}y=f(x).$

Here $(X,\mathrm{d})$ is a compact metric space, and we focus on when $X$ is a
manifold with boundary and $\mathrm{d}$ is a distance function satisfying
additional regularity assumptions. Already when $X\subseteq\mathbb{R}^{2}$ is
the unit disc, close to nothing was known for the solutions to (1). We shall
prove in this paper that for $X$ a compact $n$-dimensional manifold with
boundary, Equation (1) is well posed for $f$ in the Sobolev space
$\overline{H}^{(n+1)/2}(X)$ and admits a unique solution
$u_{R}\in\dot{H}^{-(n+1)/2}(X)$ for sufficiently large $R\gg 0$ (for notation,
see page 4). We relate the integral equation (1) to a pseudodifferential
boundary value problem which is elliptic with parameter.

Our main results concern structural properties of solutions to Equation (1)
such as asymptotic behavior as $R\to\infty$ and meromorphic extensions in the
parameter $R$ to sectors $\Gamma\subseteq\mathbb{C}$. The methods for
pseudodifferential boundary value problems that we use date far back, see the
work of Gregory Eskin [9] and Lars Hörmander [23], but have in recent years
seen much development in work of Gerd Grubb [16, 17, 18, 19, 20].

The solution to (1) for the right hand side $f=1$ enters in the so called
magnitude function of $(X,\mathrm{d})$, studied extensively in for instance
[2, 11, 12, 27, 28, 30, 33, 34, 44, 45]. The empirical properties of the
solution to (1) have recently found applications in data science, leading to
precise conjectures for its structural properties [6, 7].

The case $f=1$ can be considered as a minimizing problem relating to the
ground state energy

$\displaystyle\mathcal{E}(R;X,\mathrm{d}):=$
$\displaystyle\inf\left\\{\int_{X}\int_{X}\mathrm{e}^{-R\mathrm{d}(x,y)}\mathrm{d}\mu(x)\mathrm{d}\mu(y):\;\mu\;\mbox{a
signed Borel measure with $\mu(X)=1$}\right\\}=$ $\displaystyle=$
$\displaystyle\inf\left\\{\sum_{x,y\in
X}\mathrm{e}^{-R\mathrm{d}(x,y)}c(x)c(y):\;\mbox{$c:X\to\mathbb{R}$ has finite
support and $\sum_{x\in X}c(x)=1$}\right\\}.$

More precisely, if $R$ is such that $(X,R\mathrm{d})$ is positive definite
(i.e. the matrix $(\mathrm{e}^{-R\mathrm{d}(x,y)})_{x,y\in F}$ is positive
definite for any finite $F\subseteq X$), then by [34] a solution $u_{R}$ to
Equation (1) with $f=1$ satisfies

$\int_{X}u_{R}(x)\mathrm{d}x=\frac{1}{\mathcal{E}(R;X,\mathrm{d})}.$

Let us digest on the problem of finding $\mathcal{E}(R;X,\mathrm{d})$ and
studying its semiclassical limit, as it has been broadly studied in various
mathematical communities. The ground state energy
$\mathcal{E}(R;X,\mathrm{d})$ is that of a signed distribution of finitely
many particles on $X$ where a particle in $x$ interacts with a particle in $y$
under the potential $\mathrm{e}^{-R\mathrm{d}(x,y)}$. As such, the scaling
parameter $R>0$ should be thought of as an order parameter with $R\to\infty$
corresponding to a semiclassical limit. The non-locality of Equation (1) and
the ground state energy $\mathcal{E}(R;X,\mathrm{d})$ makes the problem of
explicit computation an impossibility, however in the semiclassical limit the
problem localizes and is asymptotically described in terms of geometric
invariants. Related problems concerning ground state energies with nonlocal
interaction potentials arise in the mean field description of interacting
particle systems, such as [10]. Specifically for log gases the dependence of
the ground state energy on the geometry has been of interest [47]. In complex
geometry, Berman has studied similar minimization problems from which
geometric structures emerged [3, 4]. Related operators also appear in image
processing [1].

The integral equation (1) is, as mentioned above, related to magnitude – an
invariant that has been extensively studied since it was introduced by Tom
Leinster [27]. We presume no prerequisites from the reader on magnitude in
this paper, but for the convenience of the reader we summarize the
implications to magnitude here and expand on this relation in the follow up
paper [13]. From its category-theoretic origin, magnitude has found unexpected
applications from algebraic topology [15, 29, 41] and applied category theory
[8, 36] to data science [6, 7] and mathematical biology [26].

For a metric space $(X,\mathrm{d})$ this invariant leads to a function
$\mathcal{M}_{X}:(0,\infty)\to\mathbb{R}\cup\\{\infty\\}$. When the metric
space $(X,R\mathrm{d})$ is positive definite, and in particular for compact
sets $X\subset\mathbb{R}^{n}$ [33], $\mathcal{M}_{X}(R)$ is defined as
$\mathcal{M}_{X}(R)=\int_{X}u_{R}(x)\mathrm{d}x$, where $u_{R}$ satisfies the
magnitude equation

(2) $\int_{X}\mathrm{e}^{-R\mathrm{d}(x,y)}u(y)\mathrm{d}y=1.$

The work [34] provides an abstract Hilbert space framework in which to pose
this equation.

In the case of compact sets $X\subset\mathbb{R}^{n}$, Meckes [34] gives an
interpretation of the magnitude in potential theoretic terms, as a generalized
capacity:

(3)
$\mathcal{M}_{X}(R)=\frac{1}{Rn!\omega_{n}}\inf\left\\{\|(R^{2}+\Delta)^{(n+1)/4}h_{R}\|_{L^{2}(\mathbb{R}^{n})}^{2}:h_{R}\in
H^{(n+1)/2}(\mathbb{R}^{n}),\ h_{R}=1\ \text{on}\ X\right\\}\ .$

Here $\omega_{n}$ denotes the volume of the unit ball in $\mathbb{R}^{n}$. The
minimizer of (3) is attained at a function $h_{R}\in
H^{(n+1)/2}(\mathbb{R}^{n})$ solving the non-local exterior problem

(4) $\begin{cases}(R^{2}+\Delta)^{(n+1)/2}h_{R}=0,\;&\mbox{in
$\mathbb{R}^{n}\setminus X$},\\\ h_{R}=1,\;&\mbox{in $X$}.\end{cases}$

If $n$ is odd, this was studied as a boundary value problem for the five-
dimensional unit ball in [2], and extended to odd-dimensional unit balls in
[46]. Few explicit computations of magnitude are known outside the realm of
compact domains in odd-dimensional Euclidean space, and even there the state-
of-the art [11, 12] can only provide asymptotic results in the semiclassical
limit and ensure existence of meromorphic extensions. In particular, nothing
was previously known about magnitude even in such a simple case as the unit
disk $X=B_{2}\subseteq\mathbb{R}^{2}$.

We provide a framework for a refined analysis and explicit computations for
solutions to Equation (1) when $X$ is a smooth, $n$-dimensional, compact
manifold with boundary, independent of the parity of $n$. The framework relies
on recent advances for pseudodifferential boundary problems and initiates
their semiclassical analysis. We work under certain regularity assumptions on
the distance function $\mathrm{d}$, firstly that its square is _regular at the
diagonal_ (see Definition 2.2) and secondly that it has _property (MR)_ (see
Definition 3.3 and 4.2). Our first assumption ensures that the distance
function behaves to leading term as a Euclidean distance, and is satisfied by
any geodesic distance function or a pullback thereof under an embedding. The
first assumption ensures that the diagonal behavior in Equation (1) is
governed by a pseudodifferential operator of order $-n-1$ which is elliptic
with parameter. Our second assumption – property (MR) – is a technical
condition to ensure that the off-diagonal behavior in Equation (1) is
negligible. Property (MR) is satisfied for subspace distances in manifolds
whose distance functions squared are smooth, but it in fact fails for higher
dimensional tori and real projective spaces.

The reader can note that there is an extended arXiv version of this paper [14]
containing more details and overview.

###  1.1. Main results

Let us summarize the main results of this paper. The results all circle around
the family of integral operators

$\mathcal{Z}_{X}(R)u(x):=\frac{1}{R}\int_{X}\mathrm{e}^{-R\mathrm{d}(x,y)}u(y)\mathrm{d}y,\quad
R\in\mathbb{C}\setminus\\{0\\}.$

Here $X$ is a compact manifold with boundary equipped with a distance function
$\mathrm{d}$ and a volume density $\mathrm{d}y$. We assume that
$\mathrm{d}^{2}$ is smooth in a small neighborhood of the diagonal $x=y$ and
there in local coordinates admits a Taylor expansion (for any $N>0$)

(5)
$\mathrm{d}(x,y)^{2}=H_{\mathrm{d}^{2},x}(v)+\sum_{j=3}^{\infty}C^{j}_{\mathrm{d}^{2}}(x;v)+O(|v|^{N+1}).$

Here $v=x-y$, and where $H_{\mathrm{d}^{2}}$ is a Riemannian metric on $X$ and
$C^{j}_{\mathrm{d}^{2}}$ in local coordinates is a symmetric $j$-form in $v$.
This condition can be summarized in the terminology that _$\mathrm{d}^{2}$ is
regular at the diagonal_, see Definition 2.2 and for more details on the
Taylor expansion, see Equation (9). We fix a function $\chi\in
C^{\infty}(X\times X)$ such that $\chi=1$ on a neighborhood of the diagonal
$x=y$ and $\mathrm{d}^{2}$ is smooth on the support of $\chi$. The
localization of $\mathcal{Z}$ to near the diagonal is the integral operator

$Q_{X}(R)u(x):=\frac{1}{R}\int_{X}\chi(x,y)\mathrm{e}^{-R\mathrm{d}(x,y)}u(y)\mathrm{d}y,\quad
R\in\mathbb{C}\setminus\\{0\\}.$

We remark that if $\mathrm{d}^{2}$ is smooth on all of $X\times X$, e.g. for a
domain or a submanifold in $\mathbb{R}^{n}$ with the induced metric, it holds
that $\mathcal{Z}_{X}-Q_{X}$ is smoothing with parameter on any sector
$\Gamma\subseteq\mathbb{C}_{+}$ with opening angle $<\pi/2$.

######  Theorem 1.1.

Let $X$ be a compact $n$-dimensional manifold with boundary and $\mathrm{d}$
such that $\mathrm{d}^{2}$ is regular at the diagonal (see Definition 2.2).
The family of integral operators $Q_{X}$ is an elliptic pseudodifferential
operator with parameter $R\in\mathbb{C}_{+}$ of order $-n-1$, and its
principal symbol is

$\overline{\sigma}_{-n-1}(Q_{X})(x,\xi,R)=n!\omega_{n}(R^{2}+g_{\mathrm{d}^{2}}(\xi,\xi))^{-(n+1)/2},$

where $g_{\mathrm{d}^{2}}$ is the dual metric to $H_{\mathrm{d}^{2}}$ from the
Taylor expansion (5). The properties of $Q_{X}$ can be summarized as follows:

  1. (1)

In each coordinate chart, the full symbol of $Q_{X}$ can be computed by an
iterative scheme as in Theorem 2.9.

  2. (2)

There exists an $R_{0}$ such that

$Q_{X}(R):\dot{H}^{-\frac{n+1}{2}}(X)\to\bar{H}^{\frac{n+1}{2}}(X),$

is invertible for $\mathrm{Re}(R)>R_{0}$ and $\mathrm{arg}(R)<\pi/(n+1)$. Here
$\dot{H}^{-\frac{n+1}{2}}(X)$, respectively $\bar{H}^{\frac{n+1}{2}}(X)$,
denote the Sobolev spaces of supported, respectively extendable distributions
in $X$ (see Section 4).

  3. (3)

If $\partial X=\emptyset$, then $Q_{X}(R)^{-1}$ is an elliptic
pseudodifferential operator of order $n+1$ whenever it exists. The full symbol
of $Q_{X}^{-1}$ can be computed by an iterative scheme as in Theorem 2.20.

Moreover, if $\mathrm{d}^{2}$ is smooth then all the properties above hold
also for $\mathcal{Z}_{X}$.

Theorem 1.1 is found in the bulk of the text as follows. The first statement
and item (1) is found in Theorem 2.9. Item (2) is proven in Theorem 4.1, see
also Corollary 2.21 of Theorem 2.22 for the simpler case of $\partial
X=\emptyset$. Item (3) follows from Theorem 2.20 and Corollary 2.21.

The operator $Q_{X}$ is generally more well behaved than $\mathcal{Z}_{X}$;
the off-diagonal singularities of $\mathrm{d}$ can create problems in
considering $\mathcal{Z}_{X}$ as a map between Sobolev spaces. For examples of
such phenomena, see Subsection 3.4. We impose one of two conditions on
$\mathrm{d}$; property (MR) and property (SMR) respectively to ensure that
$Q_{X}$ and $\mathcal{Z}_{X}$ share common functional analytic features as
operators between Sobolev space. The precise definition of property (MR) and
property (SMR) may be found in Definition 3.3 (for $\partial X=\emptyset$) and
Definition 4.2 (for $\partial X\neq\emptyset$). We note that property (MR) and
property (SMR) hold on any sector $\Gamma\subseteq\mathbb{C}\setminus\\{0\\}$
as soon as $\mathrm{d}^{2}$ is smooth on all of $X\times X$, e.g. for a domain
in $\mathbb{R}^{n}$ or more generally for a compact submanifold with boundary
in a manifold with $\mathrm{d}^{2}$ smooth.

######  Theorem 1.2.

Let $X$ be a compact $n$-dimensional manifold with boundary and let
$\mathrm{d}$ be a distance function such that $\mathrm{d}^{2}$ is regular at
the diagonal (see Definition 2.2). The family of operators

$Q_{X}(R):\dot{H}^{-\frac{n+1}{2}}(X)\to\bar{H}^{\frac{n+1}{2}}(X),\;R\in\mathbb{C}\setminus\\{0\\},$

is a holomorphic family of Fredholm operators that are invertible on a sector.
The inverse
$Q_{X}(R)^{-1}:\bar{H}^{\frac{n+1}{2}}(X)\to\dot{H}^{-\frac{n+1}{2}}(X)$,
$R\in\mathbb{C}\setminus\\{0\\}$, is a meromorphic family of Fredholm
operators.

If $\mathrm{d}$ satisfies property (SMR) on a sector $\Gamma$, then also

$\mathcal{Z}_{X}(R):\dot{H}^{-\frac{n+1}{2}}(X)\to\bar{H}^{\frac{n+1}{2}}(X),\;R\in\Gamma,$

is a holomorphic family of Fredholm operators whose inverse family
$\mathcal{Z}_{X}(R)^{-1}:\bar{H}^{\frac{n+1}{2}}(X)\to\dot{H}^{-\frac{n+1}{2}}(X)$,
$R\in\Gamma$, is a meromorphic family of Fredholm operators.

For the purposes described above, we are interested in precise asymptotic
information about solutions to $\mathcal{Z}_{X}(R)u=f$. We describe the
inverse operator $\mathcal{Z}_{X}^{-1}$ via Wiener-Hopf factorizations.

######  Theorem 1.3.

Let $X$ be a compact $n$-dimensional manifold with boundary and $\mathrm{d}$ a
distance function whose square is regular at the diagonal. For some $R_{0}\geq
0$ and any $R\in\Gamma_{\pi/(n+1)}(R_{0})$, we can write

$Q_{X}^{-1}=\chi_{1}A\chi_{1}^{\prime}+\chi_{2}(\varphi^{-1})^{*}W_{+}W_{-}\varphi^{*}\chi_{2}^{\prime}+S,$

where $A$ is a pseudodifferential parametrix of $Q_{X}$,
$\chi_{1},\chi_{1}^{\prime}\in C^{\infty}_{c}(X^{\circ})$, and
$\chi_{2},\chi_{2}^{\prime}\in C^{\infty}(X)$ are functions supported in a
collar neighborhood $U_{0}$ of $\partial X$ in $X$ such that

$\chi_{1}+\chi_{2}=1\quad\mbox{and}\quad\chi_{j}^{\prime}|_{\mathrm{supp}(\chi_{j})}=1,\;j=1,2,$

$\varphi:\partial X\times[0,1)\to U_{0}$ is a collar identification, and the
operators $S$, $W_{-}$ and $W_{+}$ satisfying the following as $R\to\infty$:

  1. (1)

$S:\overline{H}^{\mu}(X)\to\dot{H}^{-\mu}(X)$ is a continuous operator with
$\|S\|_{\overline{H}^{\mu}(X)\to\dot{H}^{-\mu}(X)}=O(R^{-\infty})$.

  2. (2)

$W_{+}:L^{2}(\partial X\times[0,\infty))\to\dot{H}^{-\mu}(\partial
X\times[0,\infty))$ is the properly supported pseudodifferential operator of
mixed-regularity $(\mu,0)$ from Definition 5.18 which is invertible for large
$R>0$ and in local coordinates has an asymptotic expansion modulo
$S^{\mu,-\infty}$ as in Lemma 5.21 and preserves support in $\partial
X\times[0,\infty)\subseteq\partial X\times\mathbb{R}$. Moreover, for
$\chi,\chi^{\prime}\in\mathbb{C}+C^{\infty}_{c}(\partial X\times[0,\infty))$
with $\chi\chi^{\prime}=0$, it holds that $\|\chi
W_{+}\chi^{\prime}\|_{L^{2}(\partial X\times[0,\infty))\to H^{-\mu}(\partial
X\times\mathbb{R})}=O(R^{-\infty})$.

  3. (3)

$W_{-}:\overline{H}^{\mu}(\partial X\times[0,\infty))\to L^{2}(\partial
X\times[0,\infty))$ is the properly supported pseudodifferential operator of
mixed-regularity $(\mu,0)$ from Definition 5.18 which is invertible for large
$R>0$ and in local coordinates has an asymptotic expansion modulo
$S^{\mu,-\infty}$ as in Lemma 5.21 and preserves support in $\partial
X\times(-\infty,0]\subseteq\partial X\times\mathbb{R}$. Moreover, for
$\chi,\chi^{\prime}\in\mathbb{C}+C^{\infty}_{c}(\partial X\times\mathbb{R})$
with $\chi\chi^{\prime}=0$, it holds that $\|\chi
W_{-}\chi^{\prime}\|_{H^{\mu}(\partial X\times\mathbb{R})\to L^{2}(\partial
X\times\mathbb{R})}=O(R^{-\infty})$.

Theorem 1.3 is stated as Theorem 5.22 in the body of the text. A key feature
of the construction in Theorem 1.3 is that it provides us with a method to
compute the symbolic structure of the inverse $Q_{X}^{-1}$.

######  Theorem 1.4.

Let $X$ be a compact $n$-dimensional manifold with boundary and $\mathrm{d}$ a
distance function whose square is regular at the diagonal. In the sector
$\mathrm{Re}(R)>R_{0}$ and $\mathrm{arg}(R)<\pi/(n+1)$, we have a complete
asymptotic expansion

$\langle
Q(R)^{-1}1_{X},1_{X}\rangle\sim\sum_{k=0}^{\infty}c_{k}(X,\mathrm{d})R^{n-k}+O(\mathrm{Re}(R)^{-\infty}),\quad\mbox{as
$\mathrm{Re}(R)\to\infty$ in $\Gamma$},$

where the coefficients $c_{k}(X,\mathrm{d})$ are given as

$c_{k}(X,\mathrm{d})=\int_{X}a_{k,0}(x,1)\mathrm{d}x+\int_{\partial
X}B_{\mathrm{d}^{2},k}(x)\mathrm{d}x^{\prime},$

where

  1. (1)

$a_{k,0}(\cdot,1)\in C^{\infty}(X)$ is an invariant polynomial in the entries
of the Taylor expansion (5) as described in Theorem 2.26 and can be computed
inductively using Lemma 2.24, with $a_{k,0}=0$ if $k$ is odd; and

  2. (2)

$B_{\mathrm{d}^{2},k}\in C^{\infty}(\partial X)$ is an invariant polynomial in
the entries of the Taylor coefficients of $\mathrm{d}^{2}$ at the diagonal in
$X$ near $\partial X$ as described in Proposition 6.9 and can be inductively
computed using Lemma 5.19.

In particular, we have that

$\displaystyle c_{0}(X,\mathrm{d})=$
$\displaystyle\frac{\mathrm{vol}(X)}{n!\omega_{n}},\quad
c_{1}(X,\mathrm{d})=\frac{(n+1)\mathrm{vol}(\partial X)}{2n!\omega_{n}},$
$\displaystyle c_{2}(X,\mathrm{d})=$ $\displaystyle\frac{n+1}{6\cdot
n!\omega_{n}}\int_{X}s_{\mathrm{d}^{2}}\mathrm{d}x+\frac{(n-1)(n+1)^{2}}{8\cdot
n!\omega_{n}}\int_{\partial X}H_{\mathrm{d}^{2}}\mathrm{d}x^{\prime}.$

where the scalar curvature $s_{\mathrm{d}^{2}}$ is defined as in Theorem 6.1
and the mean curvature $H_{\mathrm{d}^{2}}$ of the distance function is
defined as in Theorem 6.13.

Theorem 1.4 is stated as Theorem 6.13 in the body of the text.

######  Corollary 1.5.

Let $X$ be a compact $n$-dimensional manifold with boundary and $\mathrm{d}$ a
distance function whose square is regular at the diagonal. Write
$\mathcal{E}(R;X,\mathrm{d})$ for the ground state energy.

  1. (1)

If $\mathrm{d}$ has property (MR) on a sector $\Gamma$, the ground state
energy function $\mathcal{E}(R;X,\mathrm{d})$ is a well defined meromorphic
function of $R\in\Gamma$.

  2. (2)

If $\mathrm{d}$ has property (SMR) on a sector $\Gamma$, the ground state
energy function $\mathcal{E}(R;X,\mathrm{d})$ has the complete semiclassical
asymptotic expansion

$\mathcal{E}(R;X,\mathrm{d})\sim\sum_{k=0}^{\infty}\varepsilon_{k}(X,\mathrm{d})R^{-n-k}+O(\mathrm{Re}(R)^{-\infty}),\quad\mbox{as
$\mathrm{Re}(R)\to\infty$ in $\Gamma$},$

where

$\displaystyle\varepsilon_{0}(X,\mathrm{d})=$
$\displaystyle\frac{n!\omega_{n}}{\mathrm{vol}_{n}(X)},\quad\varepsilon_{1}(X,\mathrm{d})=-\frac{(n+1)\mathrm{vol}_{n-1}(X)}{2\mathrm{vol}_{n}(X)}$
$\displaystyle\varepsilon_{2}(X,\mathrm{d})=$
$\displaystyle\frac{(n+1)^{2}\mathrm{vol}_{n-1}(X)^{2}}{4\mathrm{vol}_{n}(X)^{2}}-\frac{n!\omega_{n}c_{2}(X,\mathrm{d})}{\mathrm{vol}_{n}(X)},$

and more generally

$\varepsilon_{k}(X,\mathrm{d})=p_{k,n}\left(\frac{c_{1}(X,\mathrm{d})}{\mathrm{vol}_{n}(X)},\ldots,\frac{c_{k}(X,\mathrm{d})}{\mathrm{vol}_{n}(X)}\right),$

for a universal polynomial $p_{k,n}$ of total degree $k$ (where each
$c_{j}(X,\mathrm{d})$ is declared to be of degree $j$).

###  1.2. Notational conventions

To avoid confusion, we will use the terms Riemannian metrics and distance
function to separate the notions of metrics that appear in Riemannian geometry
and metric geometry, respectively.

The Fourier transform on $\mathbb{R}^{n}$ is denoted by $\mathcal{F}$, where
we use the convention

$\mathcal{F}f(\xi)=\int_{\mathbb{R}^{n}}e^{-ix\cdot\xi}f(x)\mathrm{d}x$

for $f\in\mathcal{S}(\mathbb{R}^{n})$. We further write

$D_{x}=-i\frac{\partial}{\partial x}.$

For $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}$, we write
$|\alpha|=\sum_{j}\alpha_{j}$, $D^{\alpha}_{x}=D_{x_{1}}^{\alpha_{1}}\cdots
D_{x_{n}}^{\alpha_{n}}$ and $x^{\alpha}=x_{1}^{\alpha_{1}}\cdots
x_{n}^{\alpha_{n}}$. In this convention, for
$f\in\mathcal{S}(\mathbb{R}^{n})$,

$\mathcal{F}(D_{x}^{\alpha}f)(\xi)=\xi^{\alpha}\mathcal{F}f(\xi)\quad\mbox{and}\quad\mathcal{F}(x^{\alpha}f)(\xi)=(-D_{\xi})^{\alpha}\mathcal{F}f(\xi),$

and the product of pseudodifferential symbols is up to smoothing operators
defined from a symbol of the form

$p\\#q\sim\sum_{\alpha}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}pD_{x}^{\alpha}q=\sum_{\alpha}\frac{1}{\alpha!}D_{\xi}^{\alpha}p\partial_{x}^{\alpha}q,$

where $\alpha!=\prod_{j=1}^{n}\alpha_{j}!$.

We write $M$ for a manifold and $X$ for a compact manifold with boundary, or
occasionally a general compact metric space. We let $n$ denote the dimension
of $M$ or $X$ and use the notation

$\textstyle{\mu:=\frac{n+1}{2}}.$

We write $\mathrm{Diag}_{M}:=\\{(x,x):x\in M\\}$ for the diagonal in $M\times
M$. If $(X,\mathrm{d})$ is a compact metric space such that the matrix
$(\mathrm{e}^{-R\mathrm{d}(x,y)})_{x,y\in F}$ is positive definite for any
finite subset $F\subseteq X$, we say that $(X,\mathrm{d})$ is positive
definite. If $(X,R\mathrm{d})$ is positive definite for all $R>0$, we say that
$(X,\mathrm{d})$ is stably positive definite. This terminology follows [33].

For a manifold $M$, we write $C^{\infty}_{c}(M)$ for the Fréchet space of
compactly supported smooth functions and $\mathcal{D}^{\prime}(M)$ for its
topological dual – the distributions on $M$. A domain $X\subseteq M$ is a
subset which coincides with the closure of its interior points. If $X$ is a
compact manifold with boundary, it can always be embedded as a domain in a
manifold $M$ and we write $C^{\infty}(X)\subseteq C(X)$ for the restrictions
to $X$ of elements in $C^{\infty}(M)$.

A sector $\Gamma\subseteq\mathbb{C}$ is a conical subset, i.e.,
$\lambda\Gamma\subset\Gamma$ for all $\lambda>1$. Standard examples we use
throughout the paper are
$\mathbb{C}_{+}=\\{z\in\mathbb{C}:\mathrm{Re}(z)>0\\}$ and

$\Gamma_{\alpha}(R_{0}):=\\{z\in\mathbb{C}:|\mathrm{Arg}(z)|<\alpha,\mathrm{Re}(z)>R_{0}\\}.$

If we make a claim concerning $R\to+\infty$, it is implicitly assumed to be a
limit along the real line. We also note that for sectors
$\Gamma\subseteq\mathbb{C}_{+}$ of opening angle $\alpha<\pi/2$, there is a
$C_{\alpha}>0$ with $C_{\alpha}^{-1}|R|\leq\mathrm{Re}(R)\leq C_{\alpha}|R|$.

For two Banach spaces $V_{1}$ and $V_{2}$, we write $\mathbb{B}(V_{1},V_{2})$
for the space of bounded operators $V_{1}\to V_{2}$ and
$\mathbb{K}(V_{1},V_{2})$ for the space of compact operators $V_{1}\to V_{2}$.
Both form Banach spaces in the norm topology.

We write $\mathbb{N}=\\{0,1,2,3,\ldots\\}$ for the set of natural numbers. The
closed positive, respectively negative half-spaces are denoted by
$\mathbb{R}^{n}_{\pm}=\\{x\in\mathbb{R}^{n}:\pm x_{n}\geq 0\\}$.

###  1.3. Acknowledgments

We are most grateful to the referees for their careful reading that helped us
improve the paper. We thank Tony Carbery, Daniel Grieser, Gerd Grubb, Tom
Leinster, Rafe Mazzeo, Mark Meckes, Niels Martin Møller, Grigori Rozenblum,
Jan-Philip Solovej and Simon Willerton for fruitful and encouraging
discussions.

MG was supported by the Swedish Research Council Grant VR 2018-0350, NL by The
Maxwell Institute Graduate School in Analysis and its Applications, a Centre
for Doctoral Training funded by the UK Engineering and Physical Sciences
Research Council (Grant EP/L016508/01), the Scottish Funding Council, Heriot-
Watt University and the University of Edinburgh.

##  2\. The symbolic structure near the diagonal

To better understand the operator $\mathcal{Z}$ we first consider the case of
a manifold $M$. This analysis describes compact manifolds (see Subsection 2.4
and Section 3) and we carry it over to the interior of a compact manifold with
boundary below in Section 4 and 5. We formulate our results in terms of the
operator

(6)
$\mathcal{Z}(R)f(x):=\frac{1}{R}\int_{M}\mathrm{e}^{-R\mathrm{d}(x,y)}f(y)\mathrm{d}y,$

whose dependence on $R\neq 0$ is holomorphic under suitable assumptions
studied in Section 3 below. Here we are implicitly using a volume density on
$M$, and the operator depends on this choice. We shall later fix a certain
choice adapted to the distance function. We shall specify the domain and
codomain of this operator more precisely later on. For now, we can consider
$\mathcal{Z}$ an operator $C^{\infty}_{c}(M)\to\mathcal{D}^{\prime}(M)$ by
setting

$\langle\mathcal{Z}\varphi,\psi\rangle=\frac{1}{R}\int_{M\times
M}\mathrm{e}^{-R\mathrm{d}(x,y)}\varphi(y)\psi(x)\mathrm{d}x\mathrm{d}y,\quad\mbox{for}\quad\varphi,\psi\in
C^{\infty}_{c}(M).$

###  2.1. On a class of pseudodifferential operators with parameter

We pick a function $\chi\in C^{\infty}(M\times M)$ such that $\chi=1$ near
$\mathrm{Diag}_{M}$ and is supported in a small neighborhood of
$\mathrm{Diag}_{M}$. The precise choice of $\chi$ will not play an important
role, but we shall later specify conditions on its support. The operator
$\mathcal{Z}$ decomposes as

(7) $\mathcal{Z}=Q+L,\quad\mbox{where}\quad
Q(R)f(x):=\frac{1}{R}\int_{M}\chi(x,y)\mathrm{e}^{-R\mathrm{d}(x,y)}f(y)\mathrm{d}y.$

We call the operator $Q$ the _localization of $\mathcal{Z}$ near the
diagonal_. In this section we focus our attention to $Q$. Distance functions
might be non-smooth away from the diagonal despite being quite regular at the
diagonal and this off-diagonal behavior of the distance function dictates
whether or not $L$ is negligible. The remainder $L$ will be studied further in
Section 3.

This subsection is devoted to proving that the localized part $Q$ of
$\mathcal{Z}$ is a parameter-dependent pseudodifferential operator, see
Appendix A. We will treat a slightly more general class of operators than $Q$.
Consider a family of operators that take the form

(8) $Q_{G,\chi}(R)f(x):=\frac{1}{|R|_{\rm
Mc}}\int_{M}\chi(x,y)\mathrm{e}^{-|R|_{\rm
Mc}\sqrt{G(x,y)}}f(y)\mathrm{d}V(y).$

Here we have written

$|R|_{\rm Mc}:=\begin{cases}R,\;&\mathrm{Re}(R)>0,\\\
-R,\;&\mathrm{Re}(R)<0,\end{cases}$

for the McIntosh modulus which extends the absolute value to a holomorphic
function in $\mathbb{C}\setminus i\mathbb{R}$. We shall mainly be concerned
with the cases $R\in\mathbb{R}$ and $R\in\mathbb{C}_{+}$. The cut-off function
$\chi$ is as above with the additional constraint that $G$ is smooth on
$\mathrm{supp}(\chi)$. The function $G:M\times M\to[0,\infty)$ should be
_regular at the diagonal_ as made precise in Definition 2.2 below: to define
this notion we first introduce further terminology. Note that $T(M\times
M)=p_{1}^{*}TM\oplus p_{2}^{*}TM$ where $p_{j}:M\times M\to M$, $j=1,2$,
denotes the projection onto the $j$:th factor. Over the diagonal
$\mathrm{Diag}_{M}$, the map $Dp_{1}\oplus Dp_{2}:T(M\times
M)|_{\mathrm{Diag}_{M}}\to TM\oplus TM$ is an isomorphism. We define the
transversal tangent bundle to the diagonal to be

$T_{\rm tra}\mathrm{Diag}_{M}:=\ker(Dp_{1}+Dp_{2}:T(M\times
M)|_{\mathrm{Diag}_{M}}\to TM)\subseteq T(M\times M)|_{\mathrm{Diag}_{M}}.$

The restriction $Dp_{1}|:T_{\rm tra}\mathrm{Diag}_{M}\to TM$ is an
isomorphism.

######  Definition 2.1.

For a smooth function $G$ defined in a neighborhood of $\mathrm{Diag}_{M}$, we
define the transversal Hessian $H_{G}$ as the quadratic form on $T_{\rm
tra}\mathrm{Diag}_{M}$ obtained from restricting the Hessian of $G$ over the
diagonal to the transversal tangent bundle.

######  Definition 2.2.

A function $G:M\times M\to[0,\infty)$ is said to be _regular at the diagonal_
if there is a tubular neighborhood $U$ of the diagonal $\mathrm{Diag}_{M}$
such that $G|_{U}\in C^{\infty}(U)$ is a smooth function satisfying that

  * •

$G|_{U}$ and $\mathrm{d}G|_{U}$ vanish on $\mathrm{Diag}_{M}\subseteq U$;

  * •

$G(x)>0$ for $x\in U\setminus\mathrm{Diag}_{M}$; and

  * •

the transversal Hessian $H_{G}$ is positive definite in all points of
$\mathrm{Diag}_{M}$.

######  Remark 2.3.

The prototypical example of a function $G$ regular at the diagonal is
$G=\mathrm{d}^{2}$ for suitable distance functions $\mathrm{d}$. The function
$\mathrm{d}^{2}$ is regular at the diagonal for the Euclidean distance, or
when $\mathrm{d}$ is the geodesic distance on a Riemannian manifold (see
Example 2.17 below) or more generally a distance function induced from pulling
back a geodesic distance along an embedding of $M$ (see Example 2.16 below for
an example).

To show that $Q_{G,\chi}(R)$ is an elliptic pseudodifferential operator with
parameter, and to describe its full symbol, we shall use a slight detour. The
basic idea used in computing the full symbol of $Q_{G,\chi}(R)$ is to do an
inverse Fourier transform in $R$, and then Fourier transform all conormal
variables. When we Fourier transform in the $R$-variable the Schwartz kernel –
depending on $(x,y,R)$ – transforms to a conormal distribution on
$U\times\mathbb{R}$ (conormal to $\mathrm{Diag}_{M}\times\\{0\\}$) – depending
on $(x,y,\eta)$ – that we then Fourier transform in all the transversal
directions $(v,\eta)$ where $v=x-y$, thus producing the full symbol depending
on $(x,\xi,R)$. To compute the Fourier transform in the $R$-direction, we use
the following elementary lemma.

######  Proposition 2.4.

For a parameter $a\geq 0$, and
$F_{a}(R):=\mathrm{F.P.}\frac{\mathrm{e}^{-a|R|}}{|R|}$, we have that

$\mathcal{F}F_{a}(\eta)=-\log(\eta^{2}+a^{2})+\log(2)-2\gamma,$

where $\gamma$ is the Euler-Mascheroni constant.

###### Proof.

By taking a derivative in the parameter $a$ and using that the Fourier
transform of $R\mapsto\mathrm{e}^{-a|R|}$ is $2a(\eta^{2}+a^{2})^{-1}$, we see
that

$\frac{\partial}{\partial a}\mathcal{F}F_{a}(\eta)=-2a(\eta^{2}+a^{2})^{-1}.$

As such, $\mathcal{F}F_{a}(\eta)=-\log(\eta^{2}+a^{2})+c_{0}(\eta)$ for some
tempered distribution $c_{0}$. Setting $a=0$ and using Proposition A.1, we see
that $c_{0}$ is a constant. By Proposition A.1 we have that

$c_{0}=\beta_{0,1}=2\log(2)+\frac{1}{2}\psi(1/2)-\gamma=\log(2)-2\gamma.$

∎

######  Proposition 2.5.

Let $G:M\times M\to[0,\infty)$ be a function which is regular at the diagonal,
see Definition 2.2. For a neighborhood $U$ of $\mathrm{Diag}_{M}$ on which $G$
is smooth, define $\tilde{G}\in C^{\infty}(U\times\mathbb{R})$ by

$\tilde{G}(x,y,\eta):=\eta^{2}+G(x,y),$

and the conormal distribution $\log(\tilde{G})\in
I^{-n-1}(U\times\mathbb{R};\mathrm{Diag}_{M}\times\\{0\\})$ as in Proposition
A.2. Then there exists a canonical metric $g_{G}$ on $T^{*}M$ such that

$\sigma_{-n-1}(\log(\tilde{G}))(x,\xi,R)=-2\pi
n!\omega_{n}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2}.$

The canonical metric $g_{G}$ is dual to the transversal Hessian of $G$ under
the isomorphism $Dp_{1}|:T_{\rm tra}\mathrm{Diag}_{M}\to TM$.

In the following, unless specified otherwise we shall consider $M$ endowed
with the Riemannian metric $g_{G}$. This allows, in particular, to define a
Laplace operator on $M$.

###### Proof.

Since $G$ is regular at the diagonal, the function $\tilde{G}$ satisfies the
assumptions of Proposition A.2 and the result follows therefrom. ∎

To compute the full symbol of $Q_{G,\chi}$ we use a Taylor expansion in the
direction transversal to the diagonal. Consider a function $G:M\times
M\to[0,\infty)$ which is regular at the diagonal, see Definition 2.2. Consider
a coordinate chart $U_{0}\subseteq M$. The coordinates on $U_{0}$ induce
coordinates $(x,y)$ on $U_{0}\times U_{0}$ and we can identify a neighborhood
of $\mathrm{Diag}_{M}\cap(U_{0}\times U_{0})$ with a neighborhood of the zero
section in $T_{\rm tra}\mathrm{Diag}_{M}|_{U_{0}}$ via the map
$(x,y)\mapsto(x,x-y)$. If the coordinate chart $U_{0}$ on $M$ satisfies that
$G$ is smooth on $U_{0}\times U_{0}$, Taylor’s theorem implies that for any
$N\in\mathbb{N}$ we can on $U_{0}\times U_{0}$ write

(9) $G(x,y)=H_{G,x}(x-y)+\sum_{j=3}^{N}C_{G}^{(j)}(x;x-y)+r_{N}(x,x-y),$

for $|x-y|$ small enough, where $r_{N}$ is a smooth function with
$r_{N}(x,v)=O(|v|^{N+1})$ as $v\to 0$, $H_{G}$ is the transversal Hessian of
$G$, and $C_{G}^{(j)}:U_{0}\to\mathrm{Sym}^{j}(T_{\rm
tra}\mathrm{Diag}_{M}|_{U_{0}})$ takes values in the symmetric $j$-forms on
the transversal tangent bundle $T_{\rm tra}\mathrm{Diag}_{M}|_{U_{0}}$. A
short computation shows that $H_{G}$ indeed is a Riemannian metric on $M$
under the isomorphism $Dp_{1}|:T_{\rm tra}\mathrm{Diag}_{M}\to TM$. However,
each $C_{G}^{(j)}$ depends on the choice of coordinates, we nevertheless
suppress this dependence in the notation.

Since there is a canonical isomorphism $T_{\rm
tra}\mathrm{Diag}_{M}|_{U_{0}}\cong TM|_{U_{0}}$, the symmetric $j$-form
$C_{G}^{(j)}:U_{0}\to\mathrm{Sym}^{j}(T_{\rm tra}\mathrm{Diag}_{M}|_{U_{0}})$
appearing in the Taylor expansion (9) of $G$ defines a $j$:th order
differential operator

$C^{(j)}_{G}(x,-D_{\xi}):C^{\infty}(T^{*}M|_{U_{0}})\to
C^{\infty}(T^{*}M|_{U_{0}}),$

obtained by quantizing the coordinate functions, i.e.
$C^{(j)}_{G}(x,-D_{\xi})$ acts as multiplication operators by
$C^{(j)}_{G}(x,v)$ under the fiberwise inverse Fourier transform (in the
$v$-direction). For a $k\in\mathbb{N}_{+}$ and a multiindex
$\gamma\in\mathbb{N}^{k}_{\geq 3}$, we can define a differential operator
$C^{(\gamma)}_{G}(x,-D_{\xi})$ on $T^{*}M|_{U_{0}}$ by

$C^{(\gamma)}_{G}(x,-D_{\xi}):=\prod_{l=1}^{k}C^{(\gamma_{l})}_{G}(x,-D_{\xi}).$

Since each $C^{(\gamma_{l})}_{G}(x,-D_{\xi})$ acts as multiplication operators
under the inverse Fourier transform, the differential operators
$C^{(\gamma_{l})}_{G}(x,-D_{\xi})$, $l=1,\ldots,k$, commute. The order of
$C^{(\gamma)}_{G}(x,-D_{\xi})$ is $|\gamma|:=\sum_{l=1}^{k}\gamma_{l}$. For
$j\in\mathbb{N}$, define the finite set

$I_{j}:=\\{\gamma\in\cup_{k=1}^{\infty}\mathbb{N}^{k}_{\geq
3}:|\gamma|=j+2k\\}.$

For instance, we have that

$I_{1}=\\{3\\},\;I_{2}=\\{(3,3),4\\},\;\mbox{and}\;I_{3}=\\{(3,3,3),(4,3),(3,4),5\\}.$

The role of $I_{j}$ will become clear in Theorem 2.9 below describing the full
symbol of $Q_{G,\chi}$ from Equation (8) in a coordinate chart. For
$\gamma\in\mathbb{N}^{k}$, we set $\mathrm{rk}(\gamma):=k$. In other words,
$\gamma\in\cup_{k=1}^{\infty}\mathbb{N}^{k}_{\geq 3}$ belongs to $I_{j}$ if
and only if $j=|\gamma|-2\mathrm{rk}(\gamma)$. We remark that $|\gamma|\geq 3$
and $\mathrm{rk}(\gamma)>0$ is implicit for $\gamma\in I_{j}$ since
$I_{j}\subseteq\cup_{k=1}^{\infty}\mathbb{N}^{k}_{\geq 3}$. The number of
elements in $I_{j}\cap\mathbb{N}^{k}_{\geq 3}$ is the same as the number of
ways to write $j-k$ as a sum of $k$ natural numbers, and so

$\\#(I_{j}\cap\mathbb{N}^{k}_{\geq 3})=\begin{pmatrix}j-1\\\
k-1\end{pmatrix}.$

The following properties of $I_{j}$ follows.

######  Proposition 2.6.

Let $j>0$. The set $I_{j}\subseteq\cup_{k>0}\mathbb{N}^{k}_{\geq 3}$ satisfies
the following

  * •

$\max\\{|\gamma|:\gamma\in I_{j}\\}=3j$ and is attained at
$\gamma=\vec{3}\in\mathbb{N}^{j}$.

  * •

$\max\\{\gamma_{i}:\gamma\in I_{j}\\}=j+2$ and is attained at
$\gamma=j+2\in\mathbb{N}^{1}$.

  * •

$\\#I_{j}=2^{j-1}$.

For notational purposes, we introduce the following notation.

######  Definition 2.7.

For an integer $n\in\mathbb{N}\cup-2\mathbb{N}-1$, we introduce the notation

$\omega_{n}:=\frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2}+1\right)}.$

If $n>0$, then $\omega_{n}$ is the volume of the unit ball in $n$-dimensions.

To simplify the computations in the subsequent theorem, we note the following
relations for the $\Gamma$-function.

######  Proposition 2.8.

For natural numbers $n,k\in\mathbb{N}$ such that $n>2k+1$, we have that

$\Gamma\left(\frac{n+1}{2}-k\right)\Gamma\left(\frac{n-2k}{2}+1\right)=\sqrt{\pi}\frac{\Gamma(n-2k+1)}{2^{n-2k}}{=\sqrt{\pi}\frac{(n-2k)!}{2^{n-2k}}}.$

Moreover, we have the identities

$\displaystyle\frac{(-1)^{k+1}}{k!}\pi^{(n-1)/2}2^{n-2k}\Gamma\left(\frac{n+1}{2}-k\right)=$
$\displaystyle(-1)^{k+1}(n-2k)!\omega_{n-2k}\omega_{2k},\;$ for $2k<n$
$\displaystyle\frac{(-1)^{k+1}}{k!}\pi^{(n-1)/2}2^{n-2k}\Gamma\left(\frac{n+1}{2}-k\right)=$
$\displaystyle\frac{(-1)^{n/2+1}\omega_{2k}}{(2k-n)!\omega_{2k-n}}=$ for
$2k-n\in 2\mathbb{N}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{n/2+1}(2k-n+1)}{2(2\pi)^{2k-n}}\omega_{2k-n+1}\omega_{2k},$
$\displaystyle\frac{(-1)^{\frac{n+1}{2}}\pi^{(n-1)/2}}{2^{2k-n}\left(k-\frac{n+1}{2}\right)!k!}=$
$\displaystyle\frac{(-1)^{\frac{n+1}{2}}}{(2\pi)^{2k-n}}\omega_{2k}\omega_{2k-n-1},\;$
for $2k-n\in 2\mathbb{N}+1$

###### Proof.

The Legendre duplication formula
$\Gamma(\zeta)\Gamma(\zeta+1/2)=2^{1-2\zeta}\sqrt{\pi}\Gamma(2\zeta)$ applied
to $\zeta=n-2k+1$ implies the first stated identity, and the second one
follows from the identity $\Gamma(1/2-m)=(-4)^{m}m!/(2m)!$. Combining these
identities with the definition of $\omega_{n}$ produces the first and second
identities. The third identity follows from the definition of $\omega_{n}$. ∎

We now arrive at the main result of this subsection, describing the full
symbol of $Q_{G,\chi}$. The reader should keep in mind that we are primarily
interested in the function $G(x,y):=\mathrm{d}(x,y)^{2}$ for a distance
function $\mathrm{d}$ such that $\mathrm{d}^{2}$ is regular at the diagonal.
In this case $Q_{G,\chi}(R)=Q(R)$ is the localization of $\mathcal{Z}$ at the
diagonal for $\mathrm{Re}(R)>0$. We therefore formulate our results on the
sector $\mathbb{C}_{+}$, albeit for $Q_{G,\chi}$ they hold in the sector
$\mathbb{C}\setminus i\mathbb{R}$.

######  Theorem 2.9.

Let $M$ be an $n$-dimensional manifold and $G:M\times M\to[0,\infty)$ a
function which is regular at the diagonal. We denote the Riemannian metric on
$T^{*}M$ dual to the transversal Hessian $H_{G}$ by $g_{G}$, as in Proposition
2.5. Consider the operator

$Q_{G,\chi}(R)f:=\frac{1}{|R|_{\rm Mc}}\int_{M}\chi(x,y)\mathrm{e}^{-|R|_{\rm
Mc}\sqrt{G(x,y)}}f(y)\mathrm{d}y,$

where $\chi\in C^{\infty}(M\times M)$ a function with $\chi=1$ near
$\mathrm{Diag}_{M}$ and supported only where $G$ is smooth, and we use the
Riemannian volume density defined from $g_{G}$.

We have that $Q_{G,\chi}\in\Psi^{-n-1}_{\rm cl}(M;\mathbb{C}_{+})$ is a
classical elliptic pseudodifferential operator with parameter of order $-n-1$
with principal symbol

$\overline{\sigma}_{-n-1}(Q_{G,\chi})(x,\xi,R)=n!\omega_{n}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2}.$

In a coordinate chart $U_{0}$ on $M$, the full symbol $q$ of $Q_{G,\chi}$ has
a classical asymptotic expansion $q\sim\sum_{j=0}^{\infty}q_{j}$ computed from
the Taylor expansion (9) and each $q_{j}$ is the homogeneous symbol of degree
$-n-1-j$ which for $j>0$ and for $n$ odd is given by

$\displaystyle q_{j}(x,\xi,R)=$ $\displaystyle\sum_{\gamma\in
I_{j},\mathrm{rk}(\gamma)<(n+1)/2}\mathfrak{c}_{\mathrm{rk}(\gamma),n}C_{G}^{(\gamma)}(x,-D_{\xi})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)}-$
$\displaystyle-\sum_{\gamma\in
I_{j},\mathrm{rk}(\gamma)\geq(n+1)/2}\mathfrak{c}_{\mathrm{rk}(\gamma),n}C_{G}^{(\gamma)}(x,-D_{\xi})\left[(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)}\log(R^{2}+g_{G}(\xi,\xi))\right],$

and for $n$ even given as

$\displaystyle q_{j}(x,\xi,R)=$ $\displaystyle\sum_{\gamma\in
I_{j}}\mathfrak{c}_{\mathrm{rk}(\gamma),n}C_{G}^{(\gamma)}(x,-D_{\xi})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)}$

Here the coefficients are computed as

$\mathfrak{c}_{k,n}:=\begin{cases}(-1)^{k}(n-2k)!\omega_{n-2k}\omega_{2k},\;&\mbox{for
$2k<n$}\\\ \frac{(-1)^{1-n/2}\omega_{2k}}{(2k-n)!\omega_{2k-n}},\;&\mbox{for
$2k-n\in 2\mathbb{N}$}\\\
\frac{(-1)^{\frac{n+1}{2}}}{(2\pi)^{2k-n}}\omega_{2k}\omega_{2k-n-1},\;&\mbox{for
$2k-n\in 2\mathbb{N}+1$}\end{cases}$

######  Remark 2.10.

Exact expressions for $q_{1}$ and $q_{2}$ are given below in Proposition 2.13
and 2.14, respectively.

######  Remark 2.11.

From the expression for $q_{j}$ in Theorem 2.9, it is not clear that $q_{j}$
is homogeneous of degree $-n-1-j$ when $n$ is odd. We shall see in the proof
that this is in fact the case.

###### Proof of Theorem 2.9.

By Proposition 2.4, the Schwartz kernel
$\frac{1}{|R|}\chi(x,y)\mathrm{e}^{-|R|\sqrt{G(x,y)}}$ of $Q_{G,\chi}$ is the
Fourier transform in the $\eta$-direction of

$K(x,y,\eta)=-\frac{1}{2\pi}\chi(x,y)\log(\eta^{2}+G(x,y)).$

The extra $2\pi$ is coming from Fourier inversion in one dimension.

Let $U$ denote a neighborhood of the diagonal $\mathrm{Diag}_{M}$ on which $G$
is smooth. It follows from Proposition 2.5 (cf. Proposition A.2) that $K\in
I^{-n-1}(U\times\mathbb{R};\mathrm{Diag}_{M}\times\\{0\\})$ with principal
symbol

$\overline{\sigma}_{-n-1}(K)(x,\xi,R)=n!\omega_{n}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2}.$

Define $K_{0}(x,y,\eta):=\log(\eta^{2}+G(x,y))\in
I^{-n-1}(U\times\mathbb{R};\mathrm{Diag}_{M}\times\\{0\\})$. We compute in a
coordinate chart $U_{0}$ that $K_{0}\in
CI^{-n-1}(U\times\mathbb{R};\mathrm{Diag}_{M}\times\\{0\\})$ and using a
uniform asymptotic expansion we use Proposition A.5 to show that the Fourier
transform in the $\eta$-direction of $K$ is the Schwartz kernel of a
pseudodifferential operator with parameter.

In a coordinate chart $U_{0}$, we introduce the coordinates $(x,v)=(x,x-y)$ on
$U_{0}\times U_{0}$. Using Equation (9), we can write

$K_{0}(x,y,\eta)=-\log(\eta^{2}+H_{G}(v,v))-\log\left(1+\frac{\sum_{j=3}^{N}C_{G}^{(j)}(v,v)+r_{N}(x,v)}{\eta^{2}+H_{G}(v,v)}\right).$

For small $v$, we can Taylor expand

$K_{0}(x,y,\eta)=-\log(\eta^{2}+H_{G}(v,v))-\sum_{j=1}^{N}\sum_{\gamma\in
I_{j}}\frac{(-1)^{\mathrm{rk}(\gamma)}}{\mathrm{rk}(\gamma)}\frac{C^{(\gamma)}_{G}(v)}{(\eta^{2}+H_{G}(v,v))^{\mathrm{rk}(\gamma)}}+\tilde{r}_{N}(x,v,\eta).$

We note that, by the definition of $I_{j}$, each term in the second sum
$\sum_{\gamma\in
I_{j}}\frac{(-1)^{\mathrm{rk}(\gamma)}}{\mathrm{rk}(\gamma)}\frac{C^{(\gamma)}_{G}(v)}{(\eta^{2}+H_{G}(v,v))^{\mathrm{rk}(\gamma)}}$
is homogeneous of degree $j$. We also note that
$\tilde{r}_{N}(x,v,\eta)=O((|\eta|+|v|)^{N+1})$ and a short computation gives
that
$\partial^{\alpha}_{x}\partial^{\beta}_{v}\partial_{\eta}^{k}\tilde{r}_{N}=O((|\eta|+|v|)^{N+1-|\beta|-k})$
for any multiindices $\alpha$, $\beta$ and $k$. As such, we have a uniform
asymptotic expansion $K\sim\sum_{j=0}^{\infty}K_{j}$ (cf. Definition A.4)
where

(10)
$K_{j}(x,v,\eta)=\begin{cases}-\frac{1}{2\pi}\log(\eta^{2}+H_{G}(v,v)),\quad&j=0,\\\
\frac{1}{2\pi}\sum_{\gamma\in
I_{j}}\frac{(-1)^{\mathrm{rk}(\gamma)+1}}{\mathrm{rk}(\gamma)}\frac{C^{(\gamma)}_{G}(v)}{(\eta^{2}+H_{G}(v,v))^{\mathrm{rk}(\gamma)}},\quad&j>0.\end{cases}$

We conclude from Proposition A.5 that $Q_{G,\chi}\in\Psi^{-n-1}_{\rm
cl}(M;\mathbb{R})$ is a pseudodifferential operator with parameter.
Proposition A.5 implies that

$\overline{\sigma}_{-n-1}(Q_{G,\chi})(x,\xi,R)=\overline{\sigma}_{-n-1}(K)(x,\xi,R)=n!\omega_{n}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2}.$

This is invertible in $C^{\infty}(S(T^{*}M\oplus\mathbb{R}))$ so $Q_{G,\chi}$
is elliptic with parameter. It is readily seen that
$\overline{\sigma}_{-n-1}(Q_{G,\chi})(x,\xi,R)\neq 0$ also for
$R\in\mathbb{C}_{+}$, and the following symbol computation shows that
$Q_{G,\chi}\in\Psi^{-n-1}_{\rm cl}(M;\mathbb{C}_{+})$ is an elliptic
pseudodifferential operator with parameter.

Let us turn to the full symbol of $Q_{G,\chi}$. We will compute it from
Equation (10) and the Fourier transform computations of Section A of the
appendix. For $k>0$, denote the Fourier transform of
$(\eta^{2}+H_{G}(v,v))^{-k}$ in the $(v,\eta)$-direction by $F_{k}(x,\xi,R)$,
that is

$F_{k}(x,\xi,R):=\int_{T_{x}M\oplus\mathbb{R}}\frac{\mathrm{e}^{-i\xi.v-iR\eta}}{(\eta^{2}+H_{G}(v,v))^{k}}\mathrm{d}v\mathrm{d}\eta.$

Using homogeneity and rotational symmetries, $F_{k}$ can be computed as in
Proposition A.1 to be

$\frac{\pi^{(n+1)/2}2^{-2k+n+1}\Gamma\left(\frac{n+1}{2}-k\right)}{(k-1)!}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2+k},$

for $2k-n-1\notin 2\mathbb{N}$, and

(11)
$\frac{(-1)^{k-\frac{n+1}{2}}\pi^{(n+1)/2}}{2^{2k-n-1}\left(k-\frac{n+1}{2}\right)!(k-1)!}\left[(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2+k}\left(-\log(R^{2}+g_{G}(\xi,\xi))+\beta_{k-(n+1)/2,n+1}\right)\right],$

for $2k-n-1\in 2\mathbb{N}$. It follows that for $\gamma\in I_{j}$, the symbol
of the term $C^{(\gamma)}_{G}(v)(\eta^{2}+H_{G}(v,v))^{-\mathrm{rk}(\gamma)}$
is given by $C_{G}^{(\gamma)}(x,-D_{\xi})F_{\mathrm{rk}(\gamma)}(x,\xi,R)$
which is computed to be

$\frac{\pi^{(n+1)/2}2^{-2\mathrm{rk}(\gamma)+n+1}\Gamma\left(\frac{n+1}{2}-\mathrm{rk}(\gamma)\right)}{(\mathrm{rk}(\gamma)-1)!}C_{G}^{(\gamma)}(x,-D_{\xi})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)},$

for $2\mathrm{rk}(\gamma)-n-1\notin 2\mathbb{N}$, and

$\displaystyle\frac{(-1)^{\mathrm{rk}(\gamma)-\frac{n+1}{2}}\pi^{(n+1)/2}}{2^{2\mathrm{rk}(\gamma)-n-1}}$
$\displaystyle\left(\mathrm{rk}(\gamma)-\frac{n+1}{2}\right)!(\mathrm{rk}(\gamma)-1)!\cdot$
(12) $\displaystyle C_{G}^{(\gamma)}(x,-D_{\xi})$
$\displaystyle\left[(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)}\left(-\log(R^{2}+g_{G}(\xi,\xi))+\beta_{\mathrm{rk}(\gamma)-(n+1)/2,n+1}\right)\right],$

for $2\mathrm{rk}(\gamma)-n-1\in 2\mathbb{N}$. We conclude that the symbol
$q_{j}$ of $K_{j}$, for $j>0$, is (up to the term
$\beta_{\mathrm{rk}(\gamma),n+1}$) given by the formula in the statement of
the theorem for the pre-factors $\mathfrak{c}_{k,n}$:

$\displaystyle\mathfrak{c}_{k,n}=$
$\displaystyle\begin{cases}\frac{(-1)^{k+1}}{k!}\pi^{(n-1)/2}2^{n-2k}\Gamma\left(\frac{n+1}{2}-k\right),\;&\mbox{for
$2k-n-1\notin 2\mathbb{N}$,}\\\
\frac{(-1)^{\frac{n+1}{2}}\pi^{(n-1)/2}}{2^{2k-n}\left(k-\frac{n+1}{2}\right)!k!},\;&\mbox{for
$2k-n-1\in 2\mathbb{N}$.}\end{cases}$

Therefore $\mathfrak{c}_{k,n}$ takes the form prescribed in the theorem by
Proposition 2.8.

We finish the proof by showing that $\beta_{\mathrm{rk}(\gamma)-(n+1)/2,n+1}$
does not contribute in Equation (12) and that there is no logarithmic term
when expanding the $\xi$-derivatives in Equation (12). If
$2\mathrm{rk}(\gamma)-n-1\in 2\mathbb{N}$, then
$-(n+1)/2+\mathrm{rk}(\gamma)\in\mathbb{N}$ and so
$(R^{2}+g_{\mathrm{d}}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)}$ is a
polynomial of degree $2\mathrm{rk}(\gamma)-n-1\in 2\mathbb{N}$ in $\xi$. For
$\gamma\in I_{j}\subseteq\cup_{k}\mathbb{N}_{\geq 3}^{k}$, we have that
$j+2\mathrm{rk}(\gamma)=|\gamma|\geq 3\mathrm{rk}(\gamma)$. Therefore, if
$2\mathrm{rk}(\gamma)-n-1\in 2\mathbb{N}$ then
$C_{\mathrm{d}^{2}}^{(\gamma)}(x,D_{\xi})(R^{2}+g_{\mathrm{d}}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)}$
is a polynomial of degree
$2\mathrm{rk}(\gamma)-n-1-|\gamma|\leq-\mathrm{rk}(\gamma)-n-1<0$ and
therefore it must be identically zero. In other words, we have
$C_{G}^{(\gamma)}(x,-D_{\xi})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)}=0$
for $2\mathrm{rk}(\gamma)-n-1\in 2\mathbb{N}$. This equality proves that
$q_{j}$ can not contain a term with a logarithmic factor, and as such $q_{j}$
is homogeneous of degree $-n-1-j$ for all $j$, and

$\displaystyle C_{G}^{(\gamma)}$
$\displaystyle(x,-D_{\xi})\left[(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)}\left(-\log(R^{2}+g_{G}(\xi,\xi))+\beta_{\mathrm{rk}(\gamma)-(n+1)/2,n+1}\right)\right]=$
$\displaystyle=-C_{G}^{(\gamma)}(x,-D_{\xi})\left[(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)}\log(R^{2}+g_{G}(\xi,\xi))\right].$

∎

######  Remark 2.12.

The computation that
$q_{0}(x,\xi,R)=n!\omega_{n}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2}$ is compatible
with known symbol computations in $\mathbb{R}^{n}$ for $G(x,y)=|x-y|^{2}$ in
which case $q=q_{0}$ is the full symbol expansion when using Euclidean
coordinates. Indeed, this statement follows from the fact that
$\mathrm{e}^{-R|v|}$ is the Fourier transform of
$n!\omega_{n}(R^{2}+|\xi|^{2})^{-(n+1)/2}$, see for instance [2, Equation
(3)]. We remark that the coordinate dependent symbol computations of Theorem
2.9 will be used also in $\mathbb{R}^{n}$. The reason for using Theorem 2.9 in
$\mathbb{R}^{n}$ is that to describe the operator near the boundary of a
domain with the Wiener-Hopf factorization techniques of Section 5 below we
need to “straighten out the boundary”, i.e. choose coordinates in which the
boundary locally looks like a half-space. We make such computations more
precise in Subsection 2.2 below.

Let us give some further details in computing the symbols $q_{1}$ and $q_{2}$.
The precise information contained in $q_{1}$ and $q_{2}$ will be used later to
compute the first terms in the inverse of $Q$ and the asymptotics of its
expectation values in Section 6. Before entering into the symbol computations
of $q_{1}$ and $q_{2}$, let us introduce some notation. Since $g$ is a metric
on $T^{*}M$, it can be viewed as a symmetric tensor in $TM\otimes TM$ and for
a covector $\xi$, the contraction $\iota_{\xi}g_{G}$ takes values in $TM$. In
the coordinate chart, each $C^{j}_{G}$ takes values in the symmetric $j$-forms
on $TM|_{U_{0}}$. As such, expressions such as
$C^{3}_{G}(x,g_{G}\otimes\iota_{\xi}g_{G})$ or $C^{4}_{G}(x,g_{G}\otimes
g_{G})$, for instance, make sense. The reader should be aware that such
expressions are individually not coordinate invariant, the transformation
rules for these expressions can be deduced either from the Taylor expansion
(9) or from the transformation rules for pseudodifferential operators. For
computational purposes, we also note that

$\mathrm{d}_{\xi}g_{G}(\xi,\xi)=2\iota_{\xi}g_{G}.$

Here $\mathrm{d}_{\xi}$ denotes the fiberwise exterior differential and we are
implicitly using the canonical identification
$T^{*}(T^{*}M)=\pi^{*}TM\oplus\pi^{*}T^{*}M$ and that the $TM$-summand is
where $\mathrm{d}_{\xi}$ maps to. A useful tool in formulating the
computations is the Pochhammer $k$-symbol. For $x\in\mathbb{R}$,
$n\in\mathbb{N}$ and $k\in\mathbb{Z}$, we write

$(x)_{n,k}:=\underbrace{x(x+k)(x+2k)\cdots(x+(n-1)k)}_{n\;\mbox{\tiny
factors}},$

with the convention that $(x)_{0,k}=1$ for any $k$ and $(0)_{n,k}=1$ for any
$n$ and $k$.

######  Proposition 2.13.

Let $M$ and $Q_{G,\chi}$ be as in Theorem 2.9. In a coordinate chart on $M$,
the term $q_{1}$ of degree $-n-2$ appearing in the full symbol $q$ of
$Q_{G,\chi}$ is given by

$\displaystyle q_{1}(x,\xi,R)={-}i\bigg{(}$ $\displaystyle
6C^{3}_{G}(x,g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-2}-$
$\displaystyle-8C^{3}_{G}(x,\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-3}\bigg{)},$

if $n=1$, and if $n>1$ we have that

$\displaystyle q_{1}(x,\xi,R)=-i(n^{2}-1)\mathfrak{c}_{1,n}\bigg{(}$
$\displaystyle
3C^{3}_{G}(x,g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-1}-$
$\displaystyle-(n+3)C^{3}_{G}(x,\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-2}\bigg{)},$

where $\iota_{\xi}g_{G}$ denotes the contraction of the metric $g_{G}$ on
$T^{*}M$ along the covector $\xi$.

###### Proof.

We note that $I_{1}=\\{\gamma\in\cup_{k=1}^{\infty}\mathbb{N}^{k}_{\geq
3}:|\gamma|=1+2\mathrm{rk}(\gamma)\\}=\\{3\\}$. For $n=1$, we have
$\mathfrak{c}_{1,1}=-1/2$ and since we are in the critical case we compute as
follows

$\displaystyle q_{1}(x,\xi,R)=$
$\displaystyle-i\mathfrak{c}_{1,1}C^{3}(x,1)\partial_{\xi}^{3}\log(R^{2}+g_{G}(\xi,\xi))=iC^{3}(x,1)\partial_{\xi}^{2}\frac{\iota_{\xi}g_{G}}{R^{2}+g_{G}(\xi,\xi)}=$
$\displaystyle=$ $\displaystyle
iC^{3}_{G}(x,1)\bigg{(}-\frac{{6}g_{G}\iota_{\xi}g_{G}}{(R^{2}+g_{G}(\xi,\xi))^{2}}+\frac{8(\iota_{\xi}g_{G})^{3}}{(R^{2}+g_{G}(\xi,\xi))^{3}}\bigg{)},$

which proves the case $n=1$.

For $n>1$, we compute that

$\displaystyle q_{1}(x,\xi,R)=i(n+1)(n-1)\mathfrak{c}_{1,n}\bigg{(}-$
$\displaystyle
3C^{3}_{G}(x,g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-1}+$
$\displaystyle+(n+3)C^{3}_{G}(x,\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-2}\bigg{)}.$

∎

The next proposition follows from a similar computation.

######  Proposition 2.14.

Let $M$ and $Q_{G,\chi}$ be as in Theorem 2.9. In a coordinate chart on $M$,
the term $q_{2}$ of degree $-n-3$ appearing in the full symbol $q$ of
$Q_{G,\chi}$ is given as follows; for $n=1$ we have that

$\displaystyle q_{2}(x,\xi,R)=$
$\displaystyle\mathfrak{c}_{2,1}C^{3}(x,1)^{2}\partial_{\xi}^{6}\left((R^{2}+g_{G}(\xi,\xi))\log(R^{2}+g_{G}(\xi,\xi))\right)+\mathfrak{c}_{1,1}C^{4}(x,1)\partial_{\xi}^{4}\log(R^{2}+g_{G}(\xi,\xi))=$
$\displaystyle=$
$\displaystyle-\frac{C^{3}(x,1)^{2}}{16}\left(-\frac{2^{3}\cdot
15g_{G}^{3}}{(R^{2}+g_{G})^{2}}+\frac{2^{4}\cdot
90(\iota_{\xi}g_{G})^{2}g_{G}^{2}}{(R^{2}+g_{G})^{3}}-\frac{2^{4}\cdot
80(\iota_{\xi}g_{G})^{4}g_{G}}{(R^{2}+g_{G})^{4}}+\frac{2^{6}\cdot
24(\iota_{\xi}g_{G})^{6}}{(R^{2}+g_{G})^{5}}\right)$
$\displaystyle\quad\quad\quad-\frac{C^{4}(x,1)}{2}\left(-\frac{2^{2}\cdot
3g_{G}^{2}}{(R^{2}+g_{G})^{2}}+\frac{2^{3}\cdot
12(\iota_{\xi}g_{G})^{2}g_{G}}{(R^{2}+g_{G})^{3}}-\frac{2^{4}\cdot
6(\iota_{\xi}g_{G})^{4}}{(R^{2}+g_{G})^{4}}\right),$

and for $n=3$, we have that

$\displaystyle q_{2}(x,\xi,R)=$
$\displaystyle\mathfrak{c}_{2,3}C^{3}(x,\partial_{\xi})^{2}\left(\log(R^{2}+g_{G}(\xi,\xi))\right)+\mathfrak{c}_{1,3}C^{4}(x,\partial_{\xi})(R^{2}+g_{G}(\xi,\xi))^{-1}=$
$\displaystyle=$ $\displaystyle\mathfrak{c}_{2,3}240(C^{3}_{G}\otimes
C^{3}_{G})(g_{G}\otimes g_{G}\otimes g_{G})(R^{2}+g_{G}(\xi,\xi))^{-3}-$
$\displaystyle-\mathfrak{c}_{2,3}45\cdot
96C^{3}_{G}(x,g_{G}\otimes\iota_{\xi}g_{G})^{2}(R^{2}+g_{G}(\xi,\xi))^{-4}+$
$\displaystyle+\mathfrak{c}_{2,3}120\cdot
96C^{3}_{G}(x,\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})C^{3}_{G}(x,g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-5}-$
$\displaystyle-\mathfrak{c}_{2,3}80\cdot
96C^{3}_{G}(x,g_{G}\otimes\iota_{\xi}g_{G})^{2}(R^{2}+g_{G}(\xi,\xi))^{-6}+$
$\displaystyle+\mathfrak{c}_{1,3}24C^{4}(x,g_{G}\otimes
g_{G})(R^{2}+g_{G}(\xi,\xi))^{-3}-$
$\displaystyle{-\mathfrak{c}_{1,3}288C^{4}(x,g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-4}+}$
$\displaystyle{+\mathfrak{c}_{1,3}192C^{4}(x,\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-5}}$

and finally for $n\neq 1,3$, we have that

$\displaystyle q_{2}(x,\xi,R)=$
$\displaystyle-\mathfrak{c}_{2,n}C^{3}(x,\partial_{\xi})^{2}(R^{2}+g_{G}(\xi,\xi)))^{-(n-3)/2}+\mathfrak{c}_{1,n}C^{4}(x,\partial_{\xi})(R^{2}+g_{G}(\xi,\xi))^{-(n-1)/2}=$
$\displaystyle=$
$\displaystyle{-24\mathfrak{c}_{2,n}(n+3)_{4,-2}C^{3}(x,g_{G}\otimes\iota_{\xi}g_{G})^{2}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-2}+}$
$\displaystyle{+6\mathfrak{c}_{2,n}(n+5)_{5,-2}C^{3}(x,\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})C^{3}(x,g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-3}-}$
$\displaystyle{-\mathfrak{c}_{2,n}(n+7)_{6,-2}C^{3}(x,\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})^{2}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-4}{+}}$
$\displaystyle+3\mathfrak{c}_{2,n}(n+5)_{5,-2}(C^{3}\otimes
C^{3})(x,g_{G}\otimes g_{G}\otimes g_{G})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-1}+$
$\displaystyle{+3\mathfrak{c}_{1,n}(n^{2}-1)C^{4}(x,g_{G}\otimes
g_{G})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-1}-}$
$\displaystyle-6\mathfrak{c}_{1,n}(n+3)_{3,-2}C^{4}(x,g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-2}+$
$\displaystyle+\mathfrak{c}_{1,n}(n+5)_{4,-2}C^{4}(x,\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-3},$

where $\iota_{\xi}g_{G}$ denotes the contraction of the metric $g_{G}$ on
$T^{*}M$ along the covector $\xi$.

###  2.2. Examples and further structure in the symbol computations

In the preceding subsection we saw a detailed computation of the full symbol
of $Q_{\chi,G}$ in terms of the Taylor expansion. Let us consider a few
important special cases where further structures can be visible in the symbol
expansion, and proceed with a structural statement of for the entries in the
full symbol in the general case.

######  Example 2.15 (Symbol computations near a boundary in Euclidean space).

We consider the manifold $M=\mathbb{R}^{n}$ and the function
$G(x,y)=|x-y|^{2}$ which is regular at the diagonal. This example fits into
the bigger picture of the paper seeing that $G(x,y)=\mathrm{d}(x,y)^{2}$. It
will later in the paper be crucial to describe the symbol of $Q=Q_{G,\chi}$
near the boundary of a domain $X\subseteq\mathbb{R}^{n}$ with smooth boundary.
Fix a point $x_{0}\in\partial X$. Up to a rigid motion, we can assume that
$x_{0}=0$ and that the normal vector of $\partial X$ in $x_{0}$ is orthogonal
to the plane $x_{n}=0$, where we write $x=(x^{\prime},x_{n})$ for
$x^{\prime}\in\mathbb{R}^{n-1}$ and $x_{n}\in\mathbb{R}$. There is a
neighborhood $U_{0}=U_{00}\times(-\varepsilon,\varepsilon)$ of
$0\in\mathbb{R}^{n}$ and a smooth function $\varphi\in C^{\infty}(U_{00})$
such that $X\cap U_{0}=\\{x=(x^{\prime},x_{n})\in
U_{0}:\varphi(x^{\prime})\leq x_{n}\\}$. Since the normal vector of $\partial
X$ in $x_{0}$ is orthogonal to the plane $x_{n}=0$,
$\nabla_{x^{\prime}}\varphi(0)=0$. Near $x_{0}=0$, we use the coordinates

$(x^{\prime},x_{n})\mapsto(x^{\prime},x_{n}-\varphi(x^{\prime})).$

In these new coordinates, the domain $X$ looks like the half-space
$\\{(x^{\prime},x_{n}):x_{n}\geq 0\\}$ locally near $x_{0}=0$. We compute that
in these coordinates $G=G(x,y)$ can be written as

$\displaystyle|(x^{\prime},x_{n}-\varphi(x^{\prime}))-$
$\displaystyle(y^{\prime},y_{n}-\varphi(y^{\prime}))|^{2}=|x^{\prime}-y^{\prime}|^{2}+(x_{n}-y_{n}-(\varphi(x^{\prime})-\varphi(y^{\prime})))^{2}=$
$\displaystyle=$
$\displaystyle|x-y|^{2}-2(x_{n}-y_{n})(\varphi(x^{\prime})-\varphi(y^{\prime}))+(\varphi(x^{\prime})-\varphi(y^{\prime}))^{2}=$
$\displaystyle=$
$\displaystyle|v|^{2}+\sum_{j=2}^{N}\sum_{|\alpha^{\prime}|=j-1}\frac{-2\partial_{x^{\prime}}^{\alpha^{\prime}}\varphi(x^{\prime})}{\alpha^{\prime}!}v_{n}(v^{\prime})^{\alpha^{\prime}}+$
$\displaystyle+\sum_{j=2}^{N}\sum_{\begin{subarray}{c}|\alpha^{\prime}|+|\beta^{\prime}|=j,\\\
|\alpha^{\prime}|,|\beta^{\prime}|>0\end{subarray}}\frac{\partial_{x^{\prime}}^{\alpha^{\prime}}\varphi(x^{\prime})\partial_{x^{\prime}}^{\beta^{\prime}}\varphi(x^{\prime})}{\alpha^{\prime}!\beta^{\prime}!}(v^{\prime})^{\alpha^{\prime}+\beta^{\prime}}+O(|v|^{N+1}),$

where the sums over $\alpha^{\prime}$ and $\beta^{\prime}$ ranges over
$\alpha^{\prime},\beta^{\prime}\in\mathbb{N}^{n-1}$ and we have written
$v=x-y$, $v^{\prime}=x^{\prime}-y^{\prime}$ and $v_{n}=x_{n}-y_{n}$.
Introducing the notation

$\nabla^{j}\varphi(x^{\prime};v^{\prime}):=\sum_{|\alpha^{\prime}|=j}\frac{\partial_{x^{\prime}}^{\alpha^{\prime}}\varphi(x^{\prime})}{\alpha^{\prime}!}(v^{\prime})^{\alpha^{\prime}},$

we have in these coordinates that

$\displaystyle G(x,y)=$
$\displaystyle|v|^{2}-2v_{n}\nabla\varphi(x^{\prime})\cdot
v^{\prime}+(\nabla\varphi(x^{\prime})\cdot
v^{\prime})^{2}+\sum_{j=3}^{N}-2v_{n}\nabla^{j-1}\varphi(x^{\prime};v^{\prime})$
$\displaystyle+\sum_{j=3}^{N}\sum_{\begin{subarray}{c}k+l=j,\\\
k,l>0\end{subarray}}\nabla^{k}\varphi(x^{\prime};v^{\prime})\nabla^{l}\varphi(x^{\prime};v^{\prime})+O(|v|^{N+1})$

In particular, we can conclude that

$H_{G}(v)=|v|^{2}-2v_{n}\nabla\varphi(x^{\prime})\cdot
v^{\prime}+(\nabla\varphi(x^{\prime})\cdot v^{\prime})^{2},$

so $H_{G}$ is represented by the $n\times n$-matrix

$H_{G}=\begin{pmatrix}1_{n-1}+\nabla\varphi(x^{\prime})\nabla\varphi(x^{\prime})^{T}&-\nabla\varphi(x^{\prime})\\\
-\nabla\varphi(x^{\prime})^{T}&1\end{pmatrix}.$

Therefore, in the same basis we have that

$g_{G}=H_{G}^{-1}=\begin{pmatrix}1_{n-1}&\nabla\varphi(x^{\prime})\\\
\nabla\varphi(x^{\prime})^{T}&1+|\nabla\varphi(x^{\prime})|^{2}\end{pmatrix}.$

We note that $H_{G}|_{T\partial X}$ is the Riemannian metric on $\partial X$
induced from the Euclidean metric on $\mathbb{R}^{n}$ and the inclusion
$\partial X\hookrightarrow\mathbb{R}^{n}$. We also conclude that

(13) $C^{j}_{G}(x;v)=v_{n}C^{j,1}_{G}(x;v^{\prime})+C^{j,0}_{G}(x;v^{\prime})$

where

$C^{j,1}_{G}(x;v^{\prime}):=-2\nabla^{j-1}\varphi(x^{\prime};v^{\prime})\quad\mbox{and}\quad
C^{j,0}_{G}(x;v^{\prime}):=\sum_{k+l=j,k,l>0}\nabla^{k}\varphi(x^{\prime};v)\nabla^{l}\varphi(x^{\prime};v).$

Therefore, in these coordinates near the boundary of a domain in
$\mathbb{R}^{n}$, we have for $\gamma\in I_{j}$ that

$\displaystyle
C_{G}^{(\gamma)}(x,-D_{\xi})=(-1)^{|\gamma|}\prod_{l=1}^{\mathrm{rk}(\gamma)}\left[\sum_{|\alpha^{\prime}|=|\gamma_{l}|-1}\frac{-2\partial_{x^{\prime}}^{\alpha^{\prime}}\varphi(x^{\prime})}{\alpha^{\prime}!}D_{\xi^{\prime}}^{\alpha^{\prime}}D_{\xi_{n}}+\sum_{\begin{subarray}{c}|\alpha^{\prime}|+|\beta^{\prime}|=|\gamma_{l}|,\\\
|\alpha^{\prime}|,|\beta^{\prime}|>0\end{subarray}}\frac{\partial_{x^{\prime}}^{\alpha^{\prime}}\varphi(x^{\prime})\partial_{x^{\prime}}^{\beta^{\prime}}\varphi(x^{\prime})}{\alpha^{\prime}!\beta^{\prime}!}D_{\xi^{\prime}}^{\alpha^{\prime}+\beta^{\prime}}\right].$

This gives a method for computing the homogeneous symbol $q_{j}$ of degree
$-n-1-j$ for any $j$ following Theorem 2.9.

The principal symbol is computed as in Theorem 2.9. By Proposition 2.13 we
compute for $n>1$ that

$\displaystyle q_{1}(x,\xi,R)=$
$\displaystyle-i(n^{2}-1)\mathfrak{c}_{1,n}\bigg{(}6g_{G}(\nabla^{2}\varphi)g_{G}(\xi,\nabla\varphi-
e_{n})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-1}-$
$\displaystyle\qquad\qquad\qquad-2(n+3)\nabla^{2}\varphi(\iota_{\xi}g_{G},\iota_{\xi}g_{G})g_{G}(\xi,\nabla\varphi-
e_{n})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-2}\bigg{)}=$ $\displaystyle=$
$\displaystyle
i(n^{2}-1)\mathfrak{c}_{1,n}\bigg{(}6\xi_{n}g_{G}(\nabla^{2}\varphi)(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-1}-$
$\displaystyle\qquad\qquad\qquad-2(n+3)\xi_{n}\nabla^{2}\varphi((\xi^{\prime}+\xi_{n}\nabla\varphi)^{\otimes
2})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-2}\bigg{)},$

By Proposition 2.14 we compute for $n\neq 1,3$ that

$\displaystyle q_{2}(x,\xi,R)=$
$\displaystyle-24\mathfrak{c}_{2,n}(n+3)_{4,-2}4(g_{G}(\nabla^{2}\varphi))^{2}(g_{G}(\xi,\nabla\varphi-
e_{n}))^{2}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-2}+$
$\displaystyle+6\mathfrak{c}_{2,n}(n+5)_{5,-2}4g_{G}(\nabla^{2}\varphi)(g_{G}(\xi,\nabla\varphi-
e_{n}))^{2}\nabla^{2}\varphi(\iota_{\xi}g_{G},\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-3}-$
$\displaystyle-\mathfrak{c}_{2,n}(n+7)_{6,-2}4(\nabla^{2}\varphi(\iota_{\xi}g_{G},\iota_{\xi}g_{G}))^{2}(g_{G}(\xi,\nabla\varphi-
e_{n}))^{2}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-4}+$
$\displaystyle+3\mathfrak{c}_{2,n}(n+5)_{5,-2}\cdot
4(g_{G}(\nabla^{2}\varphi))^{2}g_{G}(\nabla\varphi-e_{n},\nabla\varphi-
e_{n})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-1}+$
$\displaystyle{+3\mathfrak{c}_{1,n}(n^{2}-1)\bigg{(}2(g_{G}\otimes
g_{G})(\nabla^{3}\varphi\otimes(\nabla\varphi-
e_{n}))+(g_{G}(\nabla^{2}\varphi))^{2}\bigg{)}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-1}-}$
$\displaystyle-6\mathfrak{c}_{1,n}(n+3)_{3,-2}\bigg{(}2(g_{G}\otimes\iota_{\xi}g_{G})(\nabla^{3}\varphi)g_{G}(\xi,\nabla\varphi-
e_{n})+$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+(g_{G}(\nabla^{2}\varphi))(\nabla^{2}\varphi(\iota_{\xi}g_{G},\iota_{\xi}g_{G}))\bigg{)}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-2}+$
$\displaystyle+\mathfrak{c}_{1,n}(n+5)_{4,-2}\bigg{(}2(\iota_{\xi}g_{G})^{\otimes
3}(\nabla^{3}\varphi)g_{G}(\xi,\nabla\varphi-e_{n})+$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+(\nabla^{2}\varphi(\iota_{\xi}g_{G},\iota_{\xi}g_{G}))^{2}\bigg{)}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-3}=$
$\displaystyle\qquad\qquad=$
$\displaystyle-48\mathfrak{c}_{2,n}(n+3)_{4,-2}(g_{G}(\nabla^{2}\varphi))^{2}\xi_{n}^{2}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-2}+$
$\displaystyle+24\mathfrak{c}_{2,n}(n+5)_{5,-2}g_{G}(\nabla^{2}\varphi)\xi_{n}^{2}\nabla^{2}\varphi((\xi^{\prime}+\xi_{n}\nabla\varphi)^{\otimes
2})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-3}-$
$\displaystyle-4\mathfrak{c}_{2,n}(n+7)_{6,-2}(\nabla^{2}\varphi((\xi^{\prime}+\xi_{n}\nabla\varphi)^{\otimes
2}))^{2}\xi_{n}^{2}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-4}+$
$\displaystyle+(12\mathfrak{c}_{2,n}(n+5)_{5,-2}+3\mathfrak{c}_{1,n}(n^{2}-1))(g_{G}(\nabla^{2}\varphi))^{2}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-1}+$
$\displaystyle-6\mathfrak{c}_{1,n}(n+3)_{3,-2}\bigg{(}-2\xi_{n}\nabla^{3}\varphi(1_{n-1}\otimes(\xi^{\prime}+\xi_{n}\nabla\varphi))+$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+(g_{G}(\nabla^{2}\varphi))(\nabla^{2}\varphi((\xi^{\prime}+\xi_{n}\nabla\varphi)^{\otimes
2})\bigg{)}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-2}+$
$\displaystyle+\mathfrak{c}_{1,n}(n+5)_{4,-2}\bigg{(}-2\xi_{n}(\nabla^{3}\varphi)((\xi^{\prime}+\xi_{n}\nabla\varphi)^{\otimes
3})+$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+(\nabla^{2}\varphi((\xi^{\prime}+\xi_{n}\nabla\varphi)^{\otimes
2})^{2}\bigg{)}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-3},$

and for $n=3$ that

$\displaystyle q_{2}(x,\xi,R)=$ $\displaystyle(\mathfrak{c}_{2,3}240\cdot
4+\mathfrak{c}_{1,3}24)(g_{G}(\nabla^{2}\varphi))^{2}(R^{2}+g_{G}(\xi,\xi))^{-3}-$
$\displaystyle-\mathfrak{c}_{2,3}45\cdot 96\cdot
4(g_{G}(\nabla^{2}\varphi))^{2}\xi_{n}^{2}(R^{2}+g_{G}(\xi,\xi))^{-4}+$
$\displaystyle+\mathfrak{c}_{2,3}120\cdot 96\cdot
4g_{G}(\nabla^{2}\varphi)\xi_{n}^{2}\nabla^{2}\varphi((\xi^{\prime}+\xi_{n}\nabla\varphi)^{\otimes
2})(R^{2}+g_{G}(\xi,\xi))^{-5}$ $\displaystyle-\mathfrak{c}_{2,3}80\cdot
96\cdot
4(g_{G}(\nabla^{2}\varphi))^{2}\xi_{n}^{2}(R^{2}+g_{G}(\xi,\xi))^{-6}+$
$\displaystyle-\mathfrak{c}_{1,3}288\bigg{(}-2\xi_{n}(\nabla^{3}\varphi)(1_{n-1}\otimes(\xi^{\prime}+\xi_{n}\nabla\varphi))+$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad(g_{G}(\nabla^{2}\varphi))(\nabla^{2}\varphi((\xi^{\prime}+\xi_{n}\nabla\varphi)^{\otimes
2}))\bigg{)}(R^{2}+g_{G}(\xi,\xi))^{-4}+$
$\displaystyle+\mathfrak{c}_{1,3}192\bigg{(}-2\xi_{n}(\nabla^{3}\varphi)((\xi^{\prime}+\xi_{n}\nabla\varphi)^{\otimes
3})+(\nabla^{2}\varphi((\xi^{\prime}+\xi_{n}\nabla\varphi)^{\otimes
2}))^{2}\bigg{)}(R^{2}+g_{G}(\xi,\xi))^{-5}.$

We note that $g_{G}(\nabla^{2}\varphi)(x_{0})$ is $(n-1)/2$ times the mean
curvature in $x_{0}$.

######  Example 2.16 (Symbol computations for a submanifold of Euclidean
space).

We consider a submanifold $M\subseteq\mathbb{R}^{N}$ and the function
$G(x,y)=|x-y|^{2}$ which is regular at the diagonal. This example fits into
the bigger picture of the paper seeing that $G(x,y)=\mathrm{d}(x,y)^{2}$ where
$\mathrm{d}$ is the distance function on $M$ making the inclusion
$M\subseteq\mathbb{R}^{N}$ isometric. To Taylor expand $G$ as in (9), we take
coordinates around a point $x_{0}\in M$ such that $M$ near $x_{0}$ is
parametrized by

$\begin{cases}x_{l}=x_{l},\;&l=1,\ldots,n\\\
x_{l}=\varphi_{l}(x_{1},\ldots,x_{n}),\;&l=n+1,\ldots,N\end{cases},$

for some functions $\varphi_{n+1},\ldots,\varphi_{N}$. Write
$x=(x_{1},\ldots,x_{n})$. In these coordinates,

$\displaystyle G(x,y)$
$\displaystyle=|x-y|^{2}+\sum_{l=n+1}^{N}|\varphi_{l}(x)-\varphi_{l}(y)|^{2}=$
$\displaystyle=|v|^{2}+\sum_{j=2}^{N_{0}}\sum_{l=n+1}^{N}\sum_{\begin{subarray}{c}|\alpha|+|\beta|=j,\\\
|\alpha|,|\beta|>0\end{subarray}}\frac{\partial_{x}^{\alpha}\varphi_{l}(x)\partial_{x}^{\beta}\varphi_{l}(x)}{\alpha!\beta!}v^{\alpha+\beta}+O(|v|^{N_{0}+1}),$

where $v=x-y$. Therefore, we conclude that

$H_{G}(v)=|v|^{2}+\sum_{l=n+1}^{N}(\nabla\varphi_{l}(x)\cdot v)^{2},$

and for $j>2$,

$C^{j}_{G}(x,v)=\sum_{l=n+1}^{N}\sum_{\begin{subarray}{c}|\alpha|+|\beta|=j,\\\
|\alpha|,|\beta|>0\end{subarray}}\frac{\partial_{x}^{\alpha}\varphi_{l}(x)\partial_{x}^{\beta}\varphi_{l}(x)}{\alpha!\beta!}v^{\alpha+\beta}=\sum_{l=n+1}^{N}\sum_{\begin{subarray}{c}i+k=j,\\\
i,k>0\end{subarray}}(\nabla^{i}\varphi_{l}\otimes\nabla^{k}\varphi_{l})(v).$

Computations similar to those in Example 2.15 can be carried out also for
submanifolds. To preserve the reader’s sanity, we spare the details.

######  Example 2.17 (Symbol computations for geodesic distances).

Consider a manifold $M$ equipped with a Riemannian metric $g_{M}$. To avoid
having to prescribe the distance between different components, we assume that
$M$ is connected. The geodesic distance $\mathrm{d}_{\rm geo}:M\times
M\to[0,\infty)$ is defined by

$\mathrm{d}_{\rm geo}(x,y):=\inf\\{L(c):\;\mbox{$c$ is a smooth path from $x$
to $y$}\\},$

where the length $L(c)$ of a path $c:[0,1]\to M$ is defined by

$L(c):=\int_{0}^{1}\sqrt{g_{M,c(t)}(\dot{c}(t),\dot{c}(t))}\mathrm{d}t.$

Here we write $\dot{c}:[0,1]\to TM$ for the derivative of the path, so
$\dot{c}(t)\in T_{c(t)}M$. For a suitable neighborhood $U\subseteq TM$ of the
zero section, the Riemannian metric defines an exponential map
$\mathrm{exp}:\boldsymbol{U}\to M$. More precisely, for a small enough $v\in
T_{x}M$, $\exp_{x}(v)=\exp(x,v)\in M$ is defined in local coordinates as
$\exp_{x}(v)=w_{x}(v;1)$ where $w_{x}(v;\cdot):[0,1]\to TM$ is the solution to
the second order ordinary differential equation

(14)
$\begin{cases}\ddot{w}_{x}(v,t)+\Gamma_{w_{x}(v;t)}(\dot{w}_{x}(v,t),\dot{w}_{x}(v,t))=0,\\\
w(v;0)=x,\\\ \dot{w}(v,0)=-v,\end{cases}$

where $\Gamma$ is the affine connection defined from $g$, which in local
coordinates is a vector valued symmetric bilinear form on the tangent bundle.
In these local coordinates, for $x$ and $y$ close enough we have that

$\mathrm{d}_{\rm geo}(x,y)=|\exp_{x}^{-1}(y)|_{g}^{2}.$

In particular, $\mathrm{d}_{\rm geo}^{2}$ is smooth in a neighborhood of the
diagonal.

Let us compute the Taylor expansion of $\mathrm{d}_{\rm geo}^{2}$ as in
Equation (9) and prove that $\mathrm{d}_{\rm geo}^{2}$ is regular at the
diagonal. We use a coordinate neighborhood as above, and write $v=x-y$. We are
looking for the Taylor expansion in $v$ of the coordinate function
$X_{x}(v)=\exp_{x}^{-1}(x-v)$. We Taylor expand

$w_{x}(v,t)=x-vt+\sum_{k=2}^{N}\frac{w^{(k)}_{x}(v;0)}{k!}t^{k}+O(|tv|^{N+1}).$

We note that $v\mapsto w^{(k)}_{x}(v;0)$ is a homogeneous polynomial of degree
$k$, we denote this by $W_{k}(x;v)$. It follows from Equation (14) that

(15) $\begin{cases}W_{1}(v)=-v,\\\ W_{2}(v)=-\Gamma(v,v),\\\
W_{3}(v)=-\mathrm{d}\Gamma(v,v,v)-2\Gamma(v,\Gamma(v,v)).\end{cases}$

The higher order terms $W_{4},W_{5},\ldots$ can be computed inductively from
Equation (14). Write the Taylor expansion of the unknown function $X_{x}$ in
$v$ as

$X_{x}(v)=\sum_{k=1}^{N}X^{(k)}(x;v)+O(|v|^{N+1}),$

where $X^{(k)}(x;\cdot)$ is a homogeneous polynomial of degree $k$ in $v$. The
identity $X_{x}(v)=\exp_{x}^{-1}(x-v)$ is equivalent to
$\exp_{x}(X_{x}(v))=x-v$ which implies that

(16)
$-\sum_{k=1}^{N}X^{(k)}(x;v)+\sum_{k=2}^{N}\frac{W_{k}(x,\sum_{l=1}^{N}X^{(l)}(x;v))}{k!}=-v+O(|v|^{N+1}).$

Considering the first order term, we see that $X^{(1)}(v)=v$. The higher order
terms can be inductively determined by considering each homogeneous term
separately:

(17)
$X^{(k)}(v)=\left[\sum_{j=2}^{k}\frac{W_{j}(x,\sum_{l=1}^{k-1}X^{(l)}(x;v))}{j!}\right]_{(k)},$

where $[\cdot]_{(k)}$ denotes the homogeneous term of degree $k$. Using
Equation (15) and (16), we compute the first terms to be

$\begin{cases}X^{(1)}(v)&=v,\\\
X^{(2)}(v)&=\frac{1}{2}W_{2}(x;v)=-\frac{1}{2}\Gamma(v,v),\\\
X^{(3)}(v)&=\frac{1}{6}W_{3}(x;v)+\frac{1}{2}\left[W_{2}\left(x;v-\frac{1}{2}\Gamma(v,v)\right)\right]_{(3)}=\\\
&=-\frac{1}{6}\mathrm{d}\Gamma(v,v,v)+\frac{1}{6}\Gamma(v,\Gamma(v,v)).\end{cases}$

We summarize these computations in a proposition.

######  Proposition 2.18.

Let $M$ be a manifold equipped with a Riemannian metric $g_{M}$ and let
$\mathrm{d}_{\rm geo}$ denote the geodesic distance. Then $\mathrm{d}_{\rm
geo}^{2}$ is regular at the diagonal and in a coordinate neighborhood the
Taylor expansion as in Equation (9) takes the form

$\mathrm{d}_{\rm geo}^{2}(x,y)=|v|^{2}_{g_{M}}+C^{3}_{\mathrm{d}_{\rm
geo}^{2}}(x;v)+C^{4}_{\mathrm{d}_{\rm geo}^{2}}(x;v)+O(|v|^{5}_{g_{m}}),$

where

$\displaystyle C^{3}_{\mathrm{d}_{\rm geo}^{2}}(x;v)=$ $\displaystyle-
g_{M}(v,\Gamma(v,v)),$ $\displaystyle C^{4}_{\mathrm{d}_{\rm geo}^{2}}(x;v)=$
$\displaystyle\frac{1}{4}|\Gamma(v,v)|^{2}_{g_{M}}+\frac{1}{3}\left(g(v,\mathrm{d}\Gamma(v,v,v))-g(v,\Gamma(v,\Gamma(v,v)))\right).$

The higher order terms $C^{j}_{\mathrm{d}_{\rm geo}^{2}}$ can be computed
inductively from Equation (17).

Let us return to the general case and describe the overall structure of the
terms appearing in the full symbol expansion of $Q_{G,\chi}$.

######  Lemma 2.19.

Let $M$ and $G$ be as in Theorem 2.9. The entries in the full symbol expansion
$q\sim\sum_{j=0}^{\infty}q_{j}$ of $Q_{G,\chi}\in\Psi^{-n-1}_{\rm
cl}(M;\mathbb{C}_{+})$ takes the form

$q_{j}(x,\xi,R)=\sum_{k=0}^{3j}P_{k,j}(x,\xi)(R^{2}+g_{G}(\xi,\xi))^{-\frac{n+1+j+k}{2}},$

where $P_{k,j}$ are of the form

  1. (1)

$P_{k,j}$ is a homogeneous polynomial of degree $k$ in $\xi$ and

$P_{k,j}\equiv 0\quad\mbox{if $j-k\notin 2\mathbb{Z}$}.$

  2. (2)

If $j-k\in 2\mathbb{Z}$, the polynomial $P_{k,j}$ takes the form

$P_{k,j}(x,\xi)=\sum_{\gamma\in
I_{j}}\rho_{\gamma,k,j}C^{(\gamma)}_{G}(x,\underbrace{\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}}_{k\;times}\otimes\underbrace{g_{G}\otimes\cdots\otimes
g_{G}}_{\frac{|\gamma|-k}{2}\;times}),$

for some coefficients
$\rho_{\gamma,k,j}\in\mathbb{Q}\pi^{n/2}+\mathbb{Q}\pi^{(n+1)/2}+\mathbb{Q}\pi^{(n-1)/2}$.

In the last item, we note that if $j-k\in 2\mathbb{Z}$, then $|\gamma|-k\in
2\mathbb{Z}$ for all $\gamma\in I_{j}$.

###### Proof.

It follows from the computations in Theorem 2.9 that $q_{j}$ can be written as
a finite sum

$q_{j}(x,\xi,R)=\sum_{k\geq
0}P_{k,j}(x,\xi)(R^{2}+g_{G}(\xi,\xi))^{-\frac{n+1+j+k}{2}},$

where $P_{k,j}$ is a polynomial in $\xi$ whose coefficients (as a polynomial
in $\xi$) are all polynomials in the Taylor coefficients of $G$ at the
diagonal. For the degrees to match, $P_{k,j}$ must be of degree $k$. Since the
powers $R^{2}+g_{G}(\xi,\xi)$ must differ from $-(n+1)/2$ by an integer,
$P_{k,j}=0$ unless $j-k\notin 2\mathbb{Z}$. Finally, the largest possible
degree $k$ for which $\xi^{\alpha}(R^{2}+g_{G}(\xi,\xi))^{-\frac{n+1+j+k}{2}}$
(with $|\alpha|=k$) can be a summand in $q_{j}$ is if the derivative
$C^{(\gamma)}_{G}(x,D_{\xi})$ acts only on
$(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)}$ (or
$(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)}\log(R^{2}+g_{G}(\xi,\xi))$
if $2\mathrm{rk}(\gamma)-n-1\in 2\mathbb{N}$) and in that case
$\alpha=\gamma$, so the maximal degree of $P_{k,j}$ is the size of the largest
index in $I_{j}$, i.e. $3j$. This proves item (1). To prove item (2), one
notices that by Theorem 2.9, all possible entries in $q_{j}$ consists of terms
of the form

$C^{(\gamma)}_{G}(x,\underbrace{\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}}_{k\;times}\otimes\underbrace{g_{G}\otimes\cdots\otimes
g_{G}}_{\frac{|\gamma|-k}{2}\;times})(R^{2}+g_{G}(\xi,\xi))^{-\frac{n+1+j+k}{2}},$

with a coefficient being a rational number times either $\pi^{n/2}$,
$\pi^{(n+1)/2}$ or $\pi^{(n-1)/2}$. ∎

###  2.3. The symbol structure of the parametrix

The operator $Q_{G,\chi}\in\Psi^{-n-1}_{\rm cl}(M;\mathbb{C}_{+})$ considered
in the previous subsection is elliptic with parameter by Theorem 2.9. In
particular, it admits a parametrix $A_{G,\chi}\in\Psi^{n+1}_{\rm
cl}(M;\mathbb{C}_{+})$, that is an operator with parameter so that
$A_{G,\chi}Q_{G,\chi}-1,Q_{G,\chi}A_{G,\chi}-1\in\Psi^{-\infty}_{\rm
cl}(M;\mathbb{C}_{+})$.

######  Theorem 2.20.

Let $M$ be an $n$-dimensional manifold and $G:M\times M\to[0,\infty)$ a
function which is regular at the diagonal. The full symbol $a$ in local
coordinates of the parametrix $A_{G,\chi}\in\Psi^{n+1}_{\rm
cl}(M;\mathbb{C}_{+})$ of $Q_{G,\chi}\in\Psi^{-n-1}_{\rm
cl}(M;\mathbb{C}_{+})$ has an asymptotic expansion
$a\sim\sum_{j=0}^{\infty}a_{j}$ where $a_{j}$ is constructed inductively from
the symbol expansion $q\sim\sum_{j}q_{j}$ of Theorem 2.9 by

$a_{j}=-a_{0}\sum_{k+l+|\alpha|=j,\,l<j}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}q_{k}D_{x}^{\alpha}a_{l}.$

Here $a_{0}(x,\xi,R)=\frac{1}{n!\omega_{n}}(R^{2}+g_{G}(\xi,\xi))^{(n+1)/2}$.
The next term is given by

$\displaystyle n=1:\quad a_{1}(x,\xi,R)=$
$\displaystyle-\frac{i}{2}(\partial_{\xi}g_{G})(\partial_{x}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-1}+\frac{3i}{2}C^{3}_{G}(x,g_{G}\otimes\iota_{\xi}g_{G})-$
$\displaystyle-2iC^{3}_{G}(x,\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-1}$
$\displaystyle n>1:\quad a_{1}(x,\xi,R)=$
$\displaystyle{-}\frac{(n+1)^{2}i}{n!\omega_{n}}g_{G}(\mathrm{d}_{x}g_{G}(\xi,\xi),\xi)(R^{2}+g_{G}(\xi,\xi))^{(n+1)/2-2}+$
$\displaystyle+\frac{3i\mathfrak{c}_{1,n}(n^{2}-1)}{(n!)^{2}\omega_{n}^{2}}C^{3}_{G}(x,g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{(n+1)/2-1}-$
$\displaystyle-\frac{i\mathfrak{c}_{1,n}(n+3)_{3,-2}}{(n!)^{2}\omega_{n}^{2}}C^{3}_{G}(x,\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{(n+1)/2-2}.$

The homogeneous terms $a_{j}$ each takes the form

$a_{j}(x,\xi,R)=\sum_{k=0}^{3j}\tilde{P}_{k,j}(x,\xi)(R^{2}+g_{G}(\xi,\xi))^{\frac{n+1-j-k}{2}},$

where $\tilde{P}_{k,j}$ are homogeneous polynomials of degree $k$ in $\xi$ and

$\tilde{P}_{k,j}\equiv 0\quad\mbox{if $j-k\notin 2\mathbb{Z}$}.$

The coefficients of $\tilde{P}_{k,j}$ as a polynomial in $\xi$ are all
polynomials in derivatives of the Taylor coefficients
$(C^{(\gamma)}_{G})_{\gamma\in\cup_{k\leq j}I_{k}}$ of $G$ at the diagonal
(from Equation (9)) of total degree $j$.

In the structural statement at the end of the proposition, the total degree
refers to the degree of the polynomial when counting an order $i$ derivative
of a Taylor coefficient $C^{(\gamma)}_{G}$ for $\gamma\in I_{k}$ to have
degree $k+i$.

###### Proof.

It follows from the parametrix construction for elliptic operators (see e.g.
[39, Chapter I.5]) that $a\sim\sum_{j=0}^{\infty}a_{j}$ where
$a_{0}=q_{0}^{-1}$ and
$a_{j}:=-a_{0}\sum_{k+l+|\alpha|=j,\,l<j}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}q_{k}D_{x}^{\alpha}a_{l}$
for $j>0$.

Let us prove the structural statement about $a_{j}$ by induction on $j$. It is
clear for $j=0$. Assume that the structural statement holds for $l<j+1$. We
have that

$\displaystyle q_{0}a_{j+1}=$
$\displaystyle-\sum_{k+l+|\alpha|=j+1,\,l<j+1}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}q_{k}D_{x}^{\alpha}a_{l}=$
$\displaystyle=$
$\displaystyle-\sum_{k+l+|\alpha|=j+1,\,l<j+1}\sum_{i_{1}=0}^{N(k)}\sum_{i_{2}=0}^{\tilde{N}(l)}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}\left[P_{i_{1},k}(x,\xi)(R^{2}+g_{G}(\xi,\xi))^{-\frac{n+1+k+i_{1}}{2}}\right]\cdot$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdot
D_{x}^{\alpha}\left[\tilde{P}_{i_{2},l}(x,\xi)(R^{2}+g_{G}(\xi,\xi))^{\frac{n+1-l-i_{2}}{2}}\right],$

which proves that $a_{j+1}$ has the claimed structure.

What remains is to compute $a_{1}$. We write

$\displaystyle a_{1}(x,\xi,R)=$ $\displaystyle-
a_{0}\sum_{|\alpha|=1}\partial^{\alpha}_{\xi}q_{0}D^{\alpha}_{x}a_{0}-a_{0}^{2}q_{1}=$
$\displaystyle=$
$\displaystyle-\frac{1}{n!\omega_{n}}(R^{2}+g_{G}(\xi,\xi))^{(n+1)/2}\sum_{|\alpha|=1}\partial^{\alpha}_{\xi}(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2}D^{\alpha}_{x}(R^{2}+g_{G}(\xi,\xi))^{(n+1)/2}$
$\displaystyle{+\frac{i(n^{2}-1)\mathfrak{c}_{1,n}}{(n!)^{2}\omega_{n}^{2}}(R^{2}+g_{G}(\xi,\xi))^{n+1}\bigg{(}3C^{3}_{G}(x,g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-1}-}$
$\displaystyle{\quad\quad\quad-(n+3)C^{3}_{G}(x,\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{-(n+1)/2-2}\bigg{)}=}$
$\displaystyle=$
$\displaystyle{-}\frac{(n+1)^{2}i}{n!\omega_{n}}g_{G}(\mathrm{d}_{x}g_{G}(\xi,\xi),\xi)(R^{2}+g_{G}(\xi,\xi))^{(n+1)/2-2}+$
$\displaystyle+\frac{3i\mathfrak{c}_{1,n}(n^{2}-1)}{(n!)^{2}\omega_{n}^{2}}C^{3}_{G}(x,g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{(n+1)/2-1}-$
$\displaystyle-\frac{i\mathfrak{c}_{1,n}(n+3)_{3,-2}}{(n!)^{2}\omega_{n}^{2}}C^{3}_{G}(x,\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G}\otimes\iota_{\xi}g_{G})(R^{2}+g_{G}(\xi,\xi))^{(n+1)/2-2}$

∎

We use the notation
$\Gamma_{\alpha}(R_{0}):=\\{z\in\mathbb{C}:|\mathrm{Arg}(z)|<\alpha,\mathrm{Re}(z)>R_{0}\\}$.

######  Corollary 2.21.

Let $M$ be a compact $n$-dimensional manifold and $G:M\times M\to[0,\infty)$ a
function which is regular at the diagonal. For some $R_{0}>0$, the operator
$Q_{G,\chi}(R)\in\Psi^{-n-1}_{\rm cl}(M)$ is invertible as an operator
$H^{-(n+1)/2}(M)\to H^{(n+1)/2}(M)$ for any $R\in\Gamma_{\pi/(n+1)}(R_{0})$
and

$Q_{G,\chi}^{-1}-A_{G,\chi}\in\Psi^{-\infty}_{\rm
cl}(M;\Gamma_{\pi/(n+1)}(R_{0})).$

In particular, $Q_{G,\chi}^{-1}$ is a pseudodifferential operator with
parameter in $\Gamma_{\pi/(n+1)}(R_{0})$ whose full symbol $a$ in local
coordinates has an asymptotic expansion $a\sim\sum_{j=0}^{\infty}a_{j}$ where
$a_{j}$ is as in Theorem 2.20.

Corollary 2.21 follows from Theorem 2.9 and 2.20 using standard techniques for
pseudodifferential operator with parameter. For the convenience of the reader,
we include its proof.

###### Proof.

Let $\Delta$ denote a positive Laplace operator on $M$ whose principal symbol
coincides with the metric $g_{G}$ dual to the transversal Hessian of $G$. It
follows from Theorem 2.9 that
$\overline{\sigma}_{-n-1}((R^{2}+\Delta)^{-\mu})=\overline{\sigma}_{-n-1}(Q)$.
Hence, the operator
$\mathfrak{r}(R):=1-(R^{2}+\Delta)^{\mu/2}Q(R^{2}+\Delta)^{\mu/2}$ is a
parameter-dependent pseudodifferential operator of order $-1$. We write

$Q_{G,\chi}=(R^{2}+\Delta)^{-\mu/2}(1-\mathfrak{r})(R^{2}+\Delta)^{-\mu/2}.$

The order of $\mathfrak{r}$ is $-1$, so by [39, Theorem 9.1] we have that
$\|\mathfrak{r}(R)\|_{L^{2}(M)\to L^{2}(M)}=O(\mathrm{Re}(R)^{-1})$ as
$\mathrm{Re}(R)\to\infty$. We conclude that there exists an $R_{0}>1$ such
that $\|\mathfrak{r}(R)\|_{L^{2}(M)\to L^{2}(M)}\leq\frac{1}{2}$ for
$R\in\Gamma_{\pi/(n+1)}(R_{0})$. Since $(R^{2}+\Delta)^{-\mu/2}:H^{s}(M)\to
H^{s+\mu}(M)$ is invertible for $R\in\Gamma_{\pi/(n+1)}(0)$ and any
$s\in\mathbb{R}$, we can for $R\in\Gamma_{\pi/(n+1)}(R_{0})$ invert
$Q_{G,\chi}$ as the absolutely convergent series of operators $H^{\mu}(M)\to
H^{-\mu}(M)$ given by

$Q_{G,\chi}^{-1}=\sum_{k=0}^{\infty}(R^{2}+\Delta)^{\mu/2}\mathfrak{r}(R)^{k}(R^{2}+\Delta)^{\mu/2}.$

It remains to prove that $Q_{G,\chi}^{-1}-A_{G,\chi}\in\Psi^{-\infty}_{\rm
cl}(M;\Gamma_{\pi/(n+1)}(R_{0}))$. By the construction above,
$Q_{G,\chi}^{-1}\in\Psi^{n+1}_{\rm cl}(M;\Gamma_{\pi/(n+1)}(R_{0}))$.
Therefore the classes $[Q_{G,\chi}^{-1}]$ and $[A_{G,\chi}]$ in the formal
symbol algebra $\Psi^{n+1}_{\rm cl}(M;\Gamma)/\Psi^{-\infty}_{\rm
cl}(M;\Gamma_{\pi/(n+1)}(R_{0}))$ are both inverses to

$[Q_{G,\chi}]\in\cup_{k\in\mathbb{Z}}\Psi^{k}_{\rm
cl}(M;\Gamma_{\pi/(n+1)}(R_{0}))/\Psi^{-\infty}_{\rm
cl}(M;\Gamma_{\pi/(n+1)}(R_{0})).$

By the uniqueness of inverses,
$[Q_{G,\chi}^{-1}]=[A_{G,\chi}]\in\Psi^{n+1}_{\rm
cl}(M;\Gamma_{\pi/(n+1)}(R_{0}))/\Psi^{-\infty}_{\rm
cl}(M;\Gamma_{\pi/(n+1)}(R_{0}))$. ∎

###  2.4. Analytic results for $Q$ on compact manifolds

The results of the previous subsections have analytic implications in the case
that $M$ is a compact manifold. We consider the scale of Hilbert spaces
$H^{s}_{R}(M):=(R^{2}+\Delta)^{-s/2}L^{2}(M)$ defined for
$R\in\mathbb{R}\setminus\\{0\\}$ and $s\in\mathbb{R}$ with the Hilbert space
structure making $(R^{2}+\Delta)^{-s/2}:L^{2}(M)\to H^{s}_{R}(M)$ unitary.
Here $\Delta$ could be any choice of Laplacian, but for the sake of simplicity
we fix the Laplacian associated with the Riemannian metric associated with a
function regular at the diagonal. By elliptic regularity,
$H^{s}(M)=H^{s}_{R}(M)$ as vector spaces with equivalent norms independently
of the choice of Laplacian, but the Hilbert space structure differs in a non-
uniform way as $R$ varies. At this stage, we shall start to concern ourselves
with extensions of $Q$ to the complex numbers, so we phrase our results in
terms of the operator

(18)
$Q(R)f(x):=\frac{1}{R}\int_{M}\chi(x,y)\mathrm{e}^{-R\mathrm{d}(x,y)}f(y)\mathrm{d}y,$

where $\mathrm{d}$ is a distance function whose square is regular at the
diagonal and $\chi$ a function being $1$ near the diagonal such that
$\mathrm{d}^{2}$ is smooth on the support of $\chi$. We note that
$Q(R)=Q_{\mathrm{d}^{2},\chi}(R)$ for $\mathrm{Re}(R)>0$.

######  Theorem 2.22.

Let $M$ be a compact $n$-dimensional manifold and $\mathrm{d}:M\times
M\to[0,\infty)$ a distance function whose square is regular at the diagonal.
Set $\mu:=(n+1)/2$. The operator

$Q(R):H^{-\mu}(M)\to H^{\mu}(M),$

defined from the expression (18) is a well defined Fredholm operator for all
$R\in\mathbb{C}\setminus\\{0\\}$ and there is an $R_{0}$ such that $Q(R)$ is
invertible for all $R\in\Gamma_{\pi/(n+1)}(R_{0})$. Moreover, the following
holds:

  1. a)

For each $R\in\mathbb{C}\setminus\\{0\\}$, $Q(R)\in\Psi^{-n-1}_{\rm cl}(M)$ is
an elliptic pseudodifferential operator and the family of operators

$(Q(R):H^{-\mu}(M)\to H^{\mu}(M))_{R\in\mathbb{C}\setminus\\{0\\}}$

depends holomorphically on $R\in\mathbb{C}\setminus\\{0\\}$. Moreover, we can
extend the holomorphic family $(Q(R)^{-1}:H^{\mu}(M)\to
H^{-\mu}(M))_{R\in\Gamma_{\pi/(n+1)}(R_{0})}$ meromorphically to
$R\in\mathbb{C}\setminus\\{0\\}$.

  2. b)

There is a $C>0$ such that

$C^{-1}\|f\|_{H^{-\mu}_{|R|}(M)}^{2}\leq\mathrm{Re}\langle
f,Q(R)f\rangle_{L^{2}}\leq C\|f\|_{H^{-\mu}_{|R|}(M)}^{2},$

for $R\in\Gamma_{\pi/(n+1)}(R_{0})$ and $f\in H^{-\mu}(M)$. In particular, for
$R\in\Gamma_{\pi/(n+1)}(R_{0})$, the norm
$\mathrm{Re}\langle\cdot,Q(R)\cdot\rangle$ is uniformly equivalent to the norm
on $H^{-\mu}_{|R|}(M)$.

###### Proof.

The first statements of the theorem follows from Corollary 2.21.

Part a) follows from the meromorphic Fredholm theorem (see [31, Proposition
1.1.8]) upon proving that $R\mapsto Q(R)\in\mathbb{B}(H^{-\mu}(M),H^{\mu}(M))$
is holomorphic with values in the set of Fredholm operators. We can write
$Q(R)$ as the integral operator with Schwartz kernel

(19)
$\frac{1}{R}\chi(x,y)\mathrm{e}^{-R\mathrm{d}(x,y)}=\frac{\chi(x,y)}{R}+\sum_{k=0}^{\infty}\frac{R^{k}}{(k+1)!}\chi(x,y)\mathrm{d}(x,y)^{k+1}.$

Using the fact that $\mathrm{d}^{2}$ is regular at the diagonal, an argument
as in the proof of Theorem 2.9 shows that $Q(R)$ is an elliptic
pseudodifferential operator of order $-n-1$. By Proposition A.1, the principal
symbol of $Q(R)$ (for fixed $R$) is given by

$\sigma_{-n-1}(Q(R))(x,\xi)=\pi^{(n-1)/2}2^{n}\Gamma\left(\frac{n+1}{2}\right)|\xi|_{g_{G}}^{-n-1}=n!\omega_{n}|\xi|^{-n-1}_{g_{\mathrm{d}^{2}}}.$

Therefore $R\mapsto Q(R)\in\mathbb{B}(H^{-\mu}(M),H^{\mu}(M))$ takes values in
the set of Fredholm operators. The expression (19), and again an argument as
in the proof of Theorem 2.9, shows that $R\mapsto
Q(R)\in\mathbb{B}(H^{-\mu}(M),H^{\mu}(M))$ depends holomorphically on
$R\in\mathbb{C}\setminus\\{0\\}$.

Part b) follows from the Gårding inequality (see Theorem A.3 on page A.3)
using that $\mathrm{Re}(Q)$ has positive principal symbol as a
pseudodifferential operator with parameter by Theorem 2.9 on page 2.9). ∎

###  2.5. Evaluation at $\xi=0$ of some symbols

For later purposes, we will be interested in knowing the value of the
homogeneous component of the full symbol of the parametrix of $Q$ at $\xi=0$,
constructed as in Theorem 2.20. The following lemma shows that the evaluations
of symbols of operators with parameter provides coordinate invariant
expressions, therefore containing invariants of a pseudodifferential operator
with parameter.

######  Lemma 2.23.

Assume that $M$ is a manifold and that $A\in\Psi^{m}_{\rm cl}(M;\Gamma)$ is a
properly supported pseudodifferential operator with parameter. Then there
exists a sequence $(a_{j,0})_{j\in\mathbb{N}}\subseteq
C^{\infty}(M\times\Gamma)$ such that

  1. i)

Each $a_{j,0}=a_{j,0}(x,R)$ is homogeneous of degree $m-j$ in $R$.

  2. ii)

In each local coordinate chart, $a_{j,0}(x,R)=a_{j}(x,0,R)$ where
$a\sim\sum_{j}a_{j}$ is a homogeneous expansion of the full symbol of $A$ in
that chart.

Moreover, for any $N\in\mathbb{N}$, we have that

(20)
$[A(R)1](x)=\sum_{j=0}^{N}a_{j}(x,R)+r_{N}(x,R)=\sum_{j=0}^{N}a_{j}(x,1)R^{m-j}+r_{N}(x,R),$

where $r_{N}\in C^{\infty}(M\times\Gamma)$ is a function such that for any
compact $K\subseteq M$ it holds that

$\sup_{x\in
K}|\partial_{x}^{\alpha}\partial_{R}^{k}r_{N}(x,R)|=O(\mathrm{Re}(R)^{m-N+|\alpha|+k}),\quad\mbox{as
$\mathrm{Re}(R)\to+\infty$}.$

###### Proof.

Choose a partition of unity $(\chi_{j})_{j}\subseteq C^{\infty}_{c}(M)$
subordinate to a locally finite covering by coordinate charts, and choose
$(\tilde{\chi}_{j})_{j}\subseteq C^{\infty}_{c}(M)$ such that
$\tilde{\chi}_{j}$ is supported in a coordinate chart and $\tilde{\chi}_{j}=1$
on $\mathrm{supp}(\chi_{j})$. We have that $\sum_{j}\chi_{j}A\tilde{\chi}_{j}$
converges in weak sense to a properly supported operator, and
$A-\sum_{j}\chi_{j}A\tilde{\chi}_{j}\in\Psi^{-\infty}_{\rm cl}(M;\Gamma)$.
Therefore, we can assume that $A$ is supported in a coordinate chart. In a
coordinate chart, and a homogeneous expansion $a\sim\sum_{j}a_{j}$ of the full
symbol of $A$, the method of stationary phase (see for instance [24, Chapter
VII.7]) implies that $[A(R)1](x)=\sum_{j=1}^{N}a_{j}(x,0,R)+r_{N}(x,R)$ as in
Equation (20). It is clear that that the function $A(R)1\in C^{\infty}(M)$ and
its asymptotic expansion is independent of choice of coordinates, and the
lemma follows. ∎

######  Lemma 2.24.

Let $M$ be an $n$-dimensional manifold, $G:M\times M\to[0,\infty)$ be a
function regular at the diagonal, and $(a_{j,0})_{j\in\mathbb{N}}\subseteq
C^{\infty}(M\times\mathbb{C}_{+})$ the sequence defined as in Lemma 2.23 from
$A_{G,\chi}\in\Psi^{n+1}_{\rm cl}(M;\mathbb{C}_{+})$ (cf. Theorem 2.20). The
functions $a_{j,0}$ are determined in local coordinates by the properties that

$a_{j,0}(x,R)=0,\quad\mbox{for all $j$ odd},$

and for even $j$, $a_{j,0}(x,R)$ is determined inductively from
$a_{0,0}(x,R)=\frac{1}{n!\omega_{n}}R^{n+1}$ and for $j>0$,

(21)
$a_{j,0}(x,R)=-\frac{1}{n!\omega_{n}}R^{n+1}\sum_{\begin{subarray}{c}k+2l+p=j\\\
2l<j,\,2|k+p\end{subarray}}i^{p}q_{k,p}(x,R).\nabla_{x}^{p}a_{j-2k,0}(x,R),$

where $q_{k,p}(x,R)$ denotes the $p$-linear form

$q_{k,p}(x,R).v:=\sum_{|\alpha|=p}\partial_{\xi}^{\alpha}q_{k}(x,\xi,R)v^{\alpha}|_{\xi=0}=\partial_{t}^{p}q_{k}(x,tv,R)|_{t=0}.$

###### Proof.

By Lemma 2.23, we can perform all computations in a coordinate chart. The
structural description of $a_{j}$ from Lemma 2.20 implies that for any
$(x,\xi,R)$ and $j\in\mathbb{N}$ it holds that

$a_{j}(x,-\xi,R)=(-1)^{j}a_{j}(x,\xi,R).$

We conclude that $a_{j}(x,0,R)=0$, and even that
$\nabla_{x}^{p}a_{j,0}(x,R)=0$ for any $p$, when $j$ is odd. To compute
$a_{j}$ for even $j$, we note that since
$a_{j}=-a_{0}\sum_{k+l+|\alpha|=j,\,l<j}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}q_{k}D_{x}^{\alpha}a_{l}$,
the formula (21) follows using that the only contributions are for even $l$,
and evenness of $j$ implies that $2|k+p$.

∎

######  Remark 2.25.

The formulas in Theorem 2.9 shows that for $n$ odd, and $j$ even,

$\displaystyle q_{j,0}(x,R)=$ $\displaystyle R^{-n-1-j}\sum_{\gamma\in
I_{j},\mathrm{rk}(\gamma)<(n+1)/2}\mathfrak{c}_{\mathrm{rk}(\gamma),n}(-1)^{\mathrm{rk}(\gamma)-1}\left(|\gamma|/2\right)!(n+1-2\mathrm{rk}(\gamma))_{|\gamma|/2,-2}C_{G}^{(\gamma)}(x,g_{G}^{\otimes|\gamma|/2})+$
$\displaystyle-R^{-n-1-j}\sum_{\gamma\in
I_{j},\mathrm{rk}(\gamma)\geq(n+1)/2}\mathfrak{c}_{\mathrm{rk}(\gamma),n}(-1)^{\mathrm{rk}(\gamma)-1}\left(|\gamma|/2\right)!(n+1-2\mathrm{rk}(\gamma))_{|\gamma|/2-1,-2}C_{G}^{(\gamma)}(x,g_{G}^{\otimes|\gamma|/2}),$

and for $n$ even, and $j$ even,

$\displaystyle q_{j,0}(x,R)=$ $\displaystyle R^{-n-1-j}\sum_{\gamma\in
I_{j}}\mathfrak{c}_{\mathrm{rk}(\gamma),n}(-1)^{\mathrm{rk}(\gamma)-1}\left(|\gamma|/2\right)!(n+1-2\mathrm{rk}(\gamma))_{|\gamma|/2,-2}C_{G}^{(\gamma)}(x,g_{G}^{\otimes|\gamma|/2}),$

and $\mathfrak{c}_{\mathrm{rk}(\gamma),n}$ is as in Theorem 2.9. Note that by
definition of the set $I_{j}$, $\gamma\in I_{j}$ satisfies that $|\gamma|$ is
even if and only if $j$ is even.

######  Theorem 2.26.

Let $M$ be an $n$-dimensional manifold, $G:M\times M\to[0,\infty)$ be a
function regular at the diagonal, and $(a_{j,0})_{j\in\mathbb{N}}\subseteq
C^{\infty}(M\times\mathbb{C}_{+})$ the restriction to $0$ of the full symbol
of $A_{G,\chi}$. Then for each $j>0$, $a_{j,0}$ is a polynomial in
$(C^{(\gamma)}_{G})_{\gamma\in\cup_{k\leq j}I_{k}}$ and its derivatives
contracted by the metric $g_{G}$ and its derivatives of total degree $j$ where
each $C^{(\gamma)}_{G}$, $\gamma\in I_{k}$, has degree $k$, the metric has
degree zero and $x$-derivatives increase the order by $1$.

In the special case $j=0$ we have

$a_{0,0}(x,R)=\frac{1}{n!\omega_{n}}R^{n+1},$

and for $j=2$ and $n=3$, we have that

$\displaystyle
a_{2,0}(x,R)=-\frac{24R^{2}}{(3!\omega_{3})^{2}}\bigg{(}10\mathfrak{c}_{1,3}C^{4}_{G}(x,g_{G}\otimes
g_{G})-\mathfrak{c}_{2,3}(C^{3}_{G}\otimes C^{3}_{G})(x,g_{G}\otimes
g_{G}\otimes g_{G})\bigg{)},$

while for $j=2$ and $n\neq 1,3$, we have that

(22) $\displaystyle
a_{2,0}(x,R)=-\frac{3R^{n-1}}{(n!\omega_{n})^{2}}\bigg{(}\mathfrak{c}_{1,n}(n^{2}-1)$
$\displaystyle C^{4}_{G}(x,g_{G}\otimes g_{G})-$
$\displaystyle-\mathfrak{c}_{2,n}(n+5)_{5,-2}(C^{3}_{G}\otimes
C^{3}_{G})(x,g_{G}\otimes g_{G}\otimes g_{G})\bigg{)},$

and for for $j=4$ we have the identity

$\displaystyle a_{4,0}(x,R)=-\frac{R^{2n+2}}{(n!\omega_{n})^{2}}$
$\displaystyle
q_{4,0}(x,R)-\frac{n+1}{R^{2}}g_{G}(\nabla^{2}_{x}a_{2,0}(x,R))-$ (23)
$\displaystyle-\frac{\mathfrak{c}_{1,n}(n^{2}-1)}{R^{2}n!\omega_{n}}(C^{3}_{G}\otimes\nabla_{x}a_{2,0})(g_{G}\otimes
g_{G})-\frac{R^{n+1}}{n!\omega_{n}}q_{2,0}(x,R)a_{2,0}(x,R).$

###### Proof.

The structural statement about $a_{j,0}(x,R)$ is readily deduced from Lemma
2.24 and induction by using the property that if $\gamma\in I_{j}$ and
$\gamma^{\prime}\in I_{j^{\prime}}$ then $(\gamma,\gamma^{\prime})\in
I_{j+j^{\prime}}$.

The equation for $a_{0,0}$ is immediate from Lemma 2.20. Using Lemma 2.24, we
have that $a_{2,0}(x,R)=-a_{0,0}^{2}q_{2,0}$ and the formula (22) follows from
that

$\displaystyle q_{2,0}(x,R)=$ $\displaystyle
3\mathfrak{c}_{1,n}(n^{2}-1)C^{4}_{G}(x,g_{G}\otimes
g_{G})R^{-n-3}+{3}\mathfrak{c}_{2,n}(n+5)_{5,-2}(C^{3}_{G}\otimes
C^{3}_{G})(x,g_{G}\otimes g_{G}\otimes g_{G})R^{-n-3},$

for $n\neq 1,3$ and a similar expression for $n=3$.

Let us compute $a_{4,0}(x,R)$. By Lemma 2.24 we have that

$\displaystyle a_{4,0}(x,R)=$
$\displaystyle-\frac{R^{n+1}}{n!\omega_{n}}\bigg{(}\sum_{k=0}^{4}i^{-k}q_{k,4-k}(x,R).\nabla_{x}^{4-k}\frac{R^{n+1}}{n!\omega_{n}}-\sum_{k=0}^{2}i^{-k}q_{k,2-k}(x,R).\nabla_{x}^{2-k}a_{2,0}(x,R)\bigg{)}=$
$\displaystyle=$
$\displaystyle\frac{R^{n+1}}{n!\omega_{n}}\bigg{(}q_{0,2}(x,R).\nabla_{x}^{2}a_{2,0}(x,R)-iq_{1,1}(x,R).\nabla_{x}a_{2,0}(x,R)-q_{2,0}(x,R)a_{2,0}(x,R)\bigg{)}-$
$\displaystyle\quad-\frac{R^{2n+2}}{(n!\omega_{n})^{2}}q_{4,0}(x,R),$

and the computation is complete upon using Proposition 2.13. ∎

######  Remark 2.27.

The full expression for $a_{4,0}$ can be computed from Equation (23). We omit
the full details, but let us note an expression for $q_{4,0}$. Since

$I_{4}=\\{6,(3,5),(4,4),(5,3),(3,3,4),(3,4,3),(4,3,3),(3,3,3,3)\\},$

we have that

$\displaystyle q_{4,0}(x,R)=$ $\displaystyle
R^{-n-5}\mathfrak{c}_{1,n}3!(n-1)_{3,-2}C_{G}^{6}(x,g_{G}^{\otimes
3})-R^{-n-5}\mathfrak{c}_{2,n}4!(n-3)_{4,-2}C_{G}^{(3,5)}(x,g_{G}^{\otimes
4})-$
$\displaystyle-R^{-n-5}\mathfrak{c}_{2,n}4!(n-3)_{4,-2}C_{G}^{(4,4)}(x,g_{G}^{\otimes
4})-R^{-n-5}\mathfrak{c}_{2,n}4!(n-3)_{4,-2}C_{G}^{(5,3)}(x,g_{G}^{\otimes
4})+$
$\displaystyle+R^{-n-5}\mathfrak{c}_{3,n}5!(n-5)_{5,-2}C_{G}^{(3,3,4)}(x,g_{G}^{\otimes
5})+R^{-n-5}\mathfrak{c}_{3,n}5!(n-5)_{5,-2}C_{G}^{(3,4,3)}(x,g_{G}^{\otimes
5})+$
$\displaystyle+R^{-n-5}\mathfrak{c}_{3,n}5!(n-5)_{5,-2}C_{G}^{(4,3,3)}(x,g_{G}^{\otimes
5})-R^{-n-5}\mathfrak{c}_{4,n}6!(n-7)_{6,-2}C_{G}^{(3,3,3,3)}(x,g_{G}^{\otimes
6}).$

######  Example 2.28 (Evaluations of symbols for domains in Euclidean space).

Let us return to the computations on Euclidean space from Example 2.15. We
consider $G(x,y)=|x-y|^{2}$ – the square of the Euclidean distance. Since
$q(x,\xi,R)=n!\omega_{n}(R^{2}+|\xi|^{2})^{-(n+1)/2}$ is a full symbol of
$Q_{G,\chi}$ in Euclidean coordinates,
$a(x,\xi,R)=\frac{1}{n!\omega_{n}}(R^{2}+|\xi|^{2})^{(n+1)/2}$ is a full
symbol of $A_{G,\chi}$. Therefore

(24) $a_{j,0}(x,R)=\begin{cases}\frac{1}{n!\omega_{n}}R^{n+1},\;&j=0,\\\
0,\;&j>0.\end{cases}$

By Lemma 2.24 this holds in any coordinate system on Euclidean space. We
remark that the bulk of computations carried out in Example 2.15 will mainly
be of interest when inverting $Q$ near the boundary of a domain, while the
computation (24) relates to interior terms.

######  Example 2.29 (Evaluations of symbols for submanifolds of Euclidean
space).

We return to submanifolds $M\subseteq\mathbb{R}^{N}$ and the function
$G(x,y)=|x-y|^{2}$ which is regular at the diagonal as in Example 2.16 above.
We take coordinates as in Example 2.16. By Theorem 2.26 we have that
$a_{0,0}(x,R)=\frac{1}{n!\omega_{n}}R^{n+1}$ and

$\displaystyle a_{2,0}(x,R)=$
$\displaystyle-\frac{3\mathfrak{c}_{1,n}(n^{2}-1)}{(n!\omega_{n})^{2}}\sum_{l=n+1}^{N}\sum_{\begin{subarray}{c}i+k=4,\\\
i,k>0\end{subarray}}(\nabla^{i}\varphi_{l}\otimes\nabla^{k}\varphi_{l})(g_{G}^{\otimes
2})R^{n-1}-$
$\displaystyle-\frac{3\mathfrak{c}_{2,n}(n+5)_{3,-2}}{(n!\omega_{n})^{2}}\sum_{l_{1},l_{2}=n+1}^{N}\sum_{\begin{subarray}{c}i_{1}+k_{1}=i_{2}+k_{2}=3,\\\
i_{1},i_{2},k_{1},k_{2}>0\end{subarray}}(\nabla^{i_{1}}\varphi_{l_{1}}\otimes\nabla^{k_{1}}\varphi_{l_{1}}\otimes\nabla^{i_{2}}\varphi_{l_{2}}\otimes\nabla^{k_{2}}\varphi_{l_{2}})(x,g_{G}^{\otimes
3})R^{n-1},$

where $g_{G}$ is the dual metric to the transversal Hessian:

$H_{G}(v)=|v|^{2}+\sum_{l=n+1}^{N}(\nabla\varphi_{l}(x)\cdot v)^{2}.$

######  Example 2.30 (Evaluations of symbols for geodesic distances).

Consider the geodesic distance on a Riemannian manifold $M$ as in Example
2.17. Let $g_{M}$ denote the Riemannian metric, and recall from Example 2.17
that $g_{M}$ coincides with the transversal Hessian of $\mathrm{d}_{\rm
geo}^{2}$ at the diagonal. We can compute $a_{2,0}$ in this case by means of
known Riemannian curvatures. We fix a point $x$ and choose coordinates so that
$\Gamma$ vanishes in that point (normal coordinates). In these coordinates,
$C^{3}_{\mathrm{d}^{2}_{\rm geo}}(x,v)=0$ and $C^{4}_{\mathrm{d}^{2}_{\rm
geo}}(x,\cdot)$ is a third of the Riemannian curvature in $x$ by Proposition
2.18. By Theorem 2.26 we conclude that for $n=3$, we have
$a_{2,0}(x,R)=-\frac{80R^{2}}{(3!\omega_{3})^{2}}s_{g}(x)$, while for $n\neq
1,3$, we have that

$a_{2,0}(x,R)=-\frac{R^{n-1}}{(n!\omega_{n})^{2}}\mathfrak{c}_{1,n}(n^{2}-1)s_{g}(x),$

where $s_{g}$ denotes the scalar curvature.

##  3\. Global behavior of $\mathcal{Z}$ on compact manifolds

In the previous section we studied the localization $Q$ of $\mathcal{Z}$ near
the diagonal. Here $\mathcal{Z}$ is the operator with parameter $R$ defined
from a distance function as in Equation (6). We now turn to study
$\mathcal{Z}$ by considering distance functions for which $L:=\mathcal{Z}-Q$
in a certain sense is negligible so that $Q$ dominates.

###  3.1. Controlling the off-diagonal part of $\mathcal{Z}$

In this subsection we shall study the remainder $L=\mathcal{Z}-Q$. We note
that

$L(R)f(x)=\frac{1}{R}\int_{M}(1-\chi(x,y))\mathrm{e}^{-R\mathrm{d}(x,y)}f(y)\mathrm{d}y.$

We start making two initial observations concerning the remainder term $L$.
The first observation concerns the behavior of the remainder term in $L^{2}$.

######  Proposition 3.1.

Let $M$ be a compact manifold and $\mathrm{d}:M\times M\to[0,\infty)$ a
distance function on $M$. Then $\mathbb{C}\setminus\\{0\\}\ni R\mapsto
L(R)\in\mathcal{L}^{2}(L^{2}(M))$ defines a holomorphic Hilbert-Schmidt valued
function. Moreover, $L$ satisfies that

$\|L(R)\|_{\mathcal{L}^{2}(L^{2}(M))}=O(\mathrm{Re}(R)^{-\infty})\quad\mbox{as
$\mathrm{Re}(R)\to+\infty$.}$

By the standard norm estimate $\|K\|_{\mathbb{B}}\leq\|K\|_{\mathcal{L}^{2}}$,
the statement in the proposition also holds in the operator norm. We remark
that by Theorem 2.9 and [39, Theorem 9.1], the localization to the diagonal
satisfies
$\|Q(R)\|_{\mathcal{L}^{p}(L^{2}(M))}=O(\mathrm{Re}(R)^{\frac{n}{p}-n-1})$ as
$\mathrm{Re}(R)\to+\infty$, for any $p>1+1/n$ (the bound is not uniform in
$p$).

###### Proof.

The function $\mathbb{C}\setminus\\{0\\}\ni
R\mapsto\frac{1}{R}(1-\chi(\cdot,\cdot))\mathrm{e}^{-R\mathrm{d}(\cdot,\cdot)}\in
C(M\times M)$ is clearly holomorphic in the norm on $C(M\times M)$. Since
$\chi=1$ in a neighborhood of the diagonal, compactness of $M$ implies that
$\|\frac{1}{R}(1-\chi(\cdot,\cdot))\mathrm{e}^{-R\mathrm{d}(\cdot,\cdot)}\|_{C(M\times
M)}=O(\mathrm{Re}(R)^{-\infty})$ as $\mathrm{Re}(R)\to+\infty$. Since $M$ is
compact, an integral operator $K$ with kernel $k\in C(M\times M)$ satisfies
the estimate for the Hilbert-Schmidt norm
$\|K\|_{\mathcal{L}^{2}(L^{2}(M))}\leq\mathrm{vol}(M)\|k\|_{C(M\times M)}$.
The proposition follows. ∎

The second observation concerns how an a priori estimate with respect to the
off-diagonal remainder $L$ affects distributional solutions to the magnitude
equation $\mathcal{Z}u=1$.

######  Proposition 3.2.

Let $M$ be a compact manifold and $\mathrm{d}:M\times M\to[0,\infty)$ a
distance function on $M$ such that $\mathrm{d}^{2}$ is regular at the
diagonal. Let $N\in\mathbb{N}$ and $\Gamma$ be a sector. Assume that $f\in
C^{\infty}(M)$ and that $(u_{R})_{R\in\Gamma}\subseteq\mathcal{D}^{\prime}(M)$
is a family of solutions to

$\mathcal{Z}(R)u_{R}=f$

satisfying that $\langle L(R)u_{R},\psi\rangle=O(\mathrm{Re}(R)^{-N})$ as
$\mathrm{Re}(R)\to+\infty$ in $\Gamma$ for all $\psi\in C^{\infty}(M)$. Then
for large enough $\mathrm{Re}(R)$,

$u_{R}=Q(R)^{-1}f+v_{R},$

where $(v_{R})_{R\in\Gamma}\subseteq\mathcal{D}^{\prime}(M)$ is a family
satisfying that $\langle v_{R},\psi\rangle=O(\mathrm{Re}(R)^{-N+n+1})$ as
$\mathrm{Re}(R)\to+\infty$ in $\Gamma$ for all $\psi\in C^{\infty}(M)$. In
particular, for any $\psi\in C^{\infty}(M)$,

$\langle u_{R},\psi\rangle=\langle
Q(R)^{-1}f,\psi\rangle+O(\mathrm{Re}(R)^{-N+n+1}),\quad\mbox{as
$\mathrm{Re}(R)\to+\infty$ in $\Gamma$}.$

###### Proof.

The equation $\mathcal{Z}(R)u_{R}=f$ implies that
$u_{R}=Q(R)^{-1}f-Q(R)^{-1}L(R)u_{R}$. Consider the distribution
$v_{R}:=-Q(R)^{-1}L(R)u_{R}\equiv u_{R}-Q(R)^{-1}f$. For $\psi\in
C^{\infty}(M)$, we have that

$\langle v_{R},\psi\rangle=\langle
L(R)u_{R},Q(R)^{-1}\psi\rangle=O(R^{-N+n+1}),$

because of the assumption on $L(R)u_{R}=O(\mathrm{Re}(R)^{-N})$ in a weak
sense and the fact that $Q(R)^{-1}$ is a pseudodifferential operator with
parameter of order $n+1$ preserving $C^{\infty}(M)$ with uniform norm
estimates $\|Q(R)^{-1}f\|_{C^{k}}\leq
C_{k}(1+\mathrm{Re}(R))^{n+1}\|f\|_{C^{n+k+1}}$. ∎

Proposition 3.1 implies that $L$ is small as an operator on $L^{2}$, and
Proposition 3.2 implies that the resulting remainder term will not alter the
asymptotic properties of solutions given an a priori estimate. However, as $Q$
is of negative order and acts compactly on $L^{2}$, well-posedness of the
problem in $L^{2}$ is not assured. We shall circumvent this problem by
imposing a regularity assumption on the distance function that forces the
magnitude equation naturally into a Sobolev space framework. We discuss
examples satisfying this assumption, as well as counterexamples, below in the
Subsections 3.2 and 3.4, respectively.

######  Definition 3.3 (Property (MR) of distance functions).

Let $M$ be an $n$-dimensional compact manifold and $\mathrm{d}:M\times
M\to[0,\infty)$ a distance function on $M$ such that $\mathrm{d}^{2}$ is
regular at the diagonal. Set $\mu:=(n+1)/2$. For a sector
$[1,\infty)\subseteq\Gamma\subseteq\mathbb{C}$, we say that $\mathrm{d}$ has
_property (MR)_ on $\Gamma$ if for any $R\in\Gamma$, $L(R)$ extends to a
continuous mapping $H^{-\mu}(M)\to H^{\mu}(M)$ with

$\|L(R)\|_{H^{-\mu}(M)\to
H^{\mu}(M)}=O(\mathrm{Re}(R)^{-\infty}),\quad\mbox{as
$\mathrm{Re}(R)\to+\infty$ in $\Gamma$}.$

If $L(R):H^{-\mu}(M)\to H^{\mu}(M)$ is a compact operator for $R\in\Gamma$ and
$\Gamma\ni R\mapsto L(R)\in\mathbb{K}(H^{-\mu}(M),H^{\mu}(M))$ is holomorphic
in norm sense, we say that $\mathrm{d}$ has _property (SMR)_ on $\Gamma$.

The acronyms MR and SMR stand for magnitude regularity and strong magnitude
regularity, respectively. Assuming these properties, the operator
$\mathcal{Z}$ inherits relevant analytic and geometric properties from $Q$.
Property (MR) will be used to compute asymptotic solutions to the magnitude
equation $R\mathcal{Z}u=1$, while property (SMR) will be used for constructing
meromorphic extensions of $\mathcal{Z}^{-1}$. If $\Gamma$ is a sector on which
a distance function has property (MR), it is clear that $0\notin\Gamma$. We
consider such results for compact manifolds below in Subsection 3.3.

###  3.2. Examples of distance functions satisfying property (MR) and (SMR)

Let us give a method to produce distance functions with property (SMR):

######  Proposition 3.4.

Let $M$ be a compact manifold embedded into a Riemannian manifold $i:M\to W$
such that the square of its geodesic distance $\mathrm{d}_{{\rm geo},W}^{2}$
is smooth, for instance $W=\mathbb{R}^{N}$, $W=\mathbb{H}_{N,\mathbb{R}}$ or
$W=\mathbb{H}_{N,\mathbb{C}}$, for some $N\in\mathbb{N}$.

The distance function $\mathrm{d}:M\times M\to[0,\infty)$,
$\mathrm{d}(x,y):=\mathrm{d}_{{\rm geo},W}(i(x),i(y))$ satisfies

  1. i)

$\mathrm{d}^{2}$ is smooth on $M\times M$ and regular at the diagonal.

  2. ii)

$L\in\Psi^{-\infty}(M;\mathbb{C}_{+})$ and $L(R)\in\Psi^{-\infty}(M)$ for any
$R\in\mathbb{C}\setminus\\{0\\}$.

  3. iii)

$\mathrm{d}$ has property (SMR) on $\mathbb{C}\setminus\\{0\\}$.

###### Proof.

Since $\mathrm{d}_{{\rm geo},W}^{2}$ is smooth, it is clear that
$\mathrm{d}^{2}$ is smooth on $M\times M$. It follows from Proposition 2.18
that $\mathrm{d}^{2}$ is regular at the diagonal, and in fact
$g_{\mathrm{d}^{2}}$ is the pullback metric $i^{*}g_{W}$ from the Riemannian
metric $g_{W}$ on $W$. Therefore the singular support support of $\mathrm{d}$
is the diagonal, and the integral kernel of $L(R)$ is smooth so $L(R)$ extends
to a continuous mapping $L(R):H^{s}(M)\to H^{t}(M)$ for any
$s,t\in\mathbb{R}$. Moreover, for any vector fields $X_{1},\ldots,X_{m}$ on
$M\times M$ we can estimate

$\left|X_{1}\cdots
X_{m}\left((1-\chi)\mathrm{e}^{-R\mathrm{d}}\right)\right|\leq
C_{m}|R|^{m-1}\mathrm{e}^{-\varepsilon\mathrm{Re}(R)}.$

Again, we write
$\varepsilon:=\inf\\{\mathrm{d}(x,y):(x,y)\in\chi^{-1}(1)\\}>0$. In
particular, we readily can deduce that $L\in\Psi^{-\infty}(M;\mathbb{C}_{+})$,
and for any $s,t\in\mathbb{R}$,

$\|L(R)\|_{H^{s}(M)\to
H^{t}(M)}=O(\mathrm{Re}(R)^{-\infty}),\quad\mbox{as}\quad\mathrm{Re}(R)\to+\infty.$

Therefore $\mathrm{d}$ has property (SMR) on $\mathbb{C}\setminus\\{0\\}$. ∎

In light of Proposition 3.4, we note the following corollary of Theorem 2.22.

######  Corollary 3.5.

Let $M$ be a compact manifold equipped with a distance function
$\mathrm{d}:M\times M\to[0,\infty)$ such that $\mathrm{d}^{2}$ is smooth, e.g.
a subspace distance as in Proposition 3.4. Then
$\mathcal{Z}(R)\in\Psi^{-n-1}_{\rm cl}(M)$ is an elliptic pseudodifferential
operator for any $R\in\mathbb{C}\setminus\\{0\\}$. Furthermore,
$\mathcal{Z}\in\Psi^{-n-1}_{\rm cl}(M;\mathbb{C}_{+})$ is elliptic with
parameter and its full symbol coincides with that of $Q$ as given in Theorem
2.9.

Another example of distance functions with property (MR) arises on spheres.

######  Proposition 3.6.

Let $\mathrm{d}$ denote the geodesic distance on a sphere $S^{n}$ in its round
metric. The distance function $\mathrm{d}$ has property (MR) on
$\mathbb{C}\setminus\\{0\\}$ but fails to satisfy property (SMR) on any
sector.

###### Proof.

The square of the geodesic distance is regular at the diagonal by Proposition
2.18. We note that $\mathrm{d}$ is smooth on $\\{(x,y)\in S^{n}\times
S^{n}:x\neq\pm y\\}$. Consider the operator $Uf(x):=f(-x)$. The operator $U$
acts via pullback along the antipodal mapping $\varphi(x):=-x$ which acts
isometrically due to $O(n)$-invariance of the Riemannian metric on $S^{n}$.
The operator $U$ extends to a unitary on all Sobolev spaces $H^{s}(S^{n})$,
$s\in\mathbb{R}$. Since it holds that
$\mathrm{d}(x,y)=\pi-\mathrm{d}(x,\varphi(y))$ we can conclude that the
integral kernel of $L(R)U$ is given by
$\chi_{\varphi}\mathrm{e}^{-R\mathrm{d}}\mathrm{e}^{-\pi R}$, where
$\chi_{\varphi}(x,y):=1-\chi(x,\varphi(y))$. In particular, since
$\chi_{\varphi}$ satisfies that $\chi_{\varphi}=1$ on the diagonal and
$\chi_{\varphi}=0$ on a neighborhood of the off-diagonal singularities of
$\mathrm{d}$, the operator $L(R)U\mathrm{e}^{\pi R}$ is an elliptic
pseudodifferential operator with parameter of order $-n-1$ by the same
argument as in Theorem 2.9. It follows that property (SMR) fails on any
sector. Using that $U$ is unitary, $L(R)$ extends to a continuous operator
$H^{-\mu}(S^{n})\to H^{\mu}(S^{n})$ with $\|L(R)U\mathrm{e}^{\pi
R}\|_{H^{-\mu}(S^{n})\to H^{\mu}(S^{n})}=O(1)$ as $\mathrm{Re}(R)\to\infty$.
Since $U$ is unitary, we deduce that

$\|L(R)\|_{H^{-\mu}(S^{n})\to
H^{\mu}(S^{n})}=O(\mathrm{e}^{-\pi\mathrm{Re}(R)})=O(\mathrm{Re}(R)^{-\infty})\quad\mbox{as}\quad\mathrm{Re}(R)\to\infty.$

∎

###  3.3. Analytic results for the operator $\mathcal{Z}$

We are now ready to extend the results of Section 2 for $Q$ to results on the
operator $\mathcal{Z}$ for compact manifolds with a distance function
satisfying property (MR) as defined in Definition 3.3.

######  Theorem 3.7.

Let $M$ be a compact $n$-dimensional manifold and $\mathrm{d}:M\times
M\to[0,\infty)$ a distance function on $M$ with property (MR) on the sector
$\Gamma$. Set $\mu:=(n+1)/2$. Then there is an $R_{0}\geq 0$ such that

$\mathcal{Z}(R):H^{-\mu}(M)\to H^{\mu}(M),$

is invertible for all $R\in\Gamma\cap\Gamma_{\pi/(n+1)}(R_{0})$. Moreover, for
$R\in\Gamma\cap\Gamma_{\pi/(n+1)}(R_{0})$

$\mathcal{Z}^{-1}=Q^{-1}+\mathcal{R},$

where $Q^{-1}\in\Psi^{n+1}_{\rm cl}(M;\Gamma_{\pi/(n+1)}(R_{0}))$ is the
elliptic pseudodifferential operator with parameter constructed in Corollary
2.21 and $\mathcal{R}:H^{\mu}(M)\to H^{-\mu}(M)$ is a family of operators
parametrized by $R\in\Gamma\cap\Gamma_{\pi/(n+1)}(R_{0})$ such that

$\|\mathcal{R}\|_{H^{\mu}(M)\to
H^{-\mu}(M)}=O(\mathrm{Re}(R)^{-\infty}),\quad\mbox{as
$\mathrm{Re}(R)\to+\infty$ in $\Gamma\cap\Gamma_{\pi/(n+1)}(R_{0})$}.$

Moreover, there is a constant $C>0$ such that

(25) $C^{-1}\|f\|_{H^{-\mu}_{|R|}(M)}^{2}\leq\mathrm{Re}\langle
f,\mathcal{Z}(R)f\rangle_{L^{2}}\leq C\|f\|_{H^{-\mu}_{|R|}(M)}^{2},$

for $R\in\Gamma\cap\Gamma_{\pi/(n+1)}(R_{0})$ and $f\in H^{-\mu}(M)$. In
particular, for $R\in\Gamma\cap\Gamma_{\pi/(n+1)}(R_{0})$, $\mathcal{Z}(R)$ is
coercive in form sense on $L^{2}(M)$ for the $H^{-\mu}$-norm.

###### Proof.

The operator $Q$ is invertible by Corollary 2.21. We can therefore write

$Q^{-1}\mathcal{Z}=1+Q^{-1}L,$

as operators on $H^{-\mu}(M)$. Since $Q^{-1}$ is a pseudodifferential operator
with parameter, we have that

(26) $\displaystyle\|Q(R)^{-1}L(R)\|_{H^{-\frac{n+1}{2}}(M)\to
H^{-\frac{n+1}{2}}(M)}$ $\displaystyle=O(\mathrm{Re}(R)^{-\infty}),$
$\displaystyle\mbox{as $\mathrm{Re}(R)\to+\infty$ in
$\Gamma\cap\Gamma_{\pi/(n+1)}(R_{0})$}.$

Therefore, for $\mathrm{Re}(R)\gg 0$ the operator $(1+Q^{-1}L)^{-1}Q^{-1}$
exists and is a left inverse to $\mathcal{Z}$. By an analogous argument,

(27) $\displaystyle\|L(R)Q(R)^{-1}\|_{H^{\frac{n+1}{2}}(M)\to
H^{\frac{n+1}{2}}(M)}$ $\displaystyle=O(\mathrm{Re}(R)^{-\infty}),$

as $\mathrm{Re}(R)\to+\infty$ in $\Gamma\cap\Gamma_{\pi/(n+1)}(R_{0})$, and
$\mathcal{Z}$ has the right inverse $Q^{-1}(1+LQ^{-1})^{-1}$ for
$\mathrm{Re}(R)\gg 0$. Therefore
$(1+Q^{-1}L)^{-1}Q^{-1}=Q^{-1}(1+LQ^{-1})^{-1}$ and this operator is an
inverse to $\mathcal{Z}$. By the estimates (26) and (27), it follows that

$\mathcal{R}=Z^{-1}-Q^{-1}=Q^{-1}\left((1+LQ^{-1})^{-1}-1\right)=\sum_{k=0}^{\infty}(-1)^{k}Q^{-1}(LQ^{-1})^{k},$

as a norm convergent sum and has the required decay property as
$\mathrm{Re}(R)\to+\infty$.

The estimate (25) follows from the decay property of $\mathcal{R}$ and Theorem
2.22. ∎

The following result is immediate from Lemma 2.23 and Theorem 3.7.

######  Corollary 3.8.

Let $M$ be a compact $n$-dimensional manifold and $\mathrm{d}:M\times
M\to[0,\infty)$ a distance function on $M$ with property (MR) on the sector
$\Gamma$. Take the sequence of homogeneous functions
$(a_{j,0})_{j\in\mathbb{N}}\subseteq
C^{\infty}(M\times\Gamma\cap\mathbb{C}_{+})$ as in Lemma 2.24. Then, for any
$N\in\mathbb{N}$, we have that

$[\mathcal{Z}(R)^{-1}1](x)=\sum_{j=0}^{N}a_{j}(x,R)+r_{N}(x,R),$

where $r_{N}\in C(\Gamma\cap\mathbb{C}_{+},H^{-\mu}(M))$, for $\mu=(n+1)/2$,
is a function such that

$\|r_{N}(\cdot,R)\|_{H^{-\mu}(M)}=O(\mathrm{Re}(R)^{n+1-N}),\quad\mbox{as
$\mathrm{Re}(R)\to+\infty$ in $\Gamma$}.$

######  Remark 3.9.

We remark that the role that property (MR) plays in Corollary 3.8 is to ensure
existence of a distributional solution to $\mathcal{Z}(R)u_{R}=1$. For
_computing_ the integrated asymptotics $\langle
1,\mathcal{Z}(R)^{-1}1\rangle$, property (MR) is not necessary. Indeed, with
no assumptions of property (MR), but assuming that $\mathrm{d}:M\times
M\to[0,\infty)$ is a distance function on $M$ such that $\mathrm{d}^{2}$ is
regular at the diagonal and that
$(u_{R})_{R>R_{0}}\subseteq\mathcal{D}^{\prime}(M)$ is a family of solutions
to

$\mathcal{Z}(R)u_{R}=1$

satisfying that $\langle L(R)u_{R},\psi\rangle=O(\mathrm{Re}(R)^{-N})$ as
$\mathrm{Re}(R)\to+\infty$ for all $\psi\in C^{\infty}(M)$, we have that

$\langle
u_{R},1\rangle=\sum_{j=1}^{N}\int_{M}a_{j}(x,R)\mathrm{d}x+O(\mathrm{Re}(R)^{n+1-N}),$

for any $N$. This follows from Lemma 2.23 and Proposition 3.2.

One instance where solutions to $\mathcal{Z}(R)u_{R}=1$ exist, yet the
distance function need not satisfy property (MR), is the geodesic distance on
a compact symmetric space $M=G/H$. See Proposition 3.17 below for examples of
symmetric spaces failing to satisfy property (MR). The problem
$\mathcal{Z}(R)u_{R}=1$ was studied for compact symmetric spaces in [45]. For
a compact symmetric space $M=G/H$, we use the normalized $G$-invariant measure
induced by the Haar measure. By symmetry, the function

$u_{R}(x)=\frac{1}{\int_{G/H}\mathrm{e}^{-R\mathrm{d}(x,y)}\mathrm{d}y},$

is constant and therefore solves $\mathcal{Z}(R)u_{R}=1$. It is readily
verified that $L(R)u_{R}=O(R^{-\infty})$ in distributional sense. We conclude
that each $a_{j}(x,R)$ is constant and that

$u_{R}(x)=u_{R}(eH)=\sum_{j=0}^{N}a_{j}(eH,R)+O(\mathrm{Re}(R)^{n+1-N}),$

for any $N$. For examples of computations of $u_{R}$ for compact symmetric
spaces, see [45].

The next result poses an obstruction to property (MR) for distance functions
and should be viewed as complementary to Theorem 3.16. Recall the following
terminology from [33]: a compact metric space $(X,\mathrm{d})$ is said to be
positive definite if for any finite subset $F\subseteq X$, the matrix
$(\mathrm{e}^{-\mathrm{d}(x,y)})_{x,y\in F}$ is positive definite. If
$(X,R\mathrm{d})$ is positive definite for all $R>0$, we say that
$(X,\mathrm{d})$ is stably positive definite.

######  Corollary 3.10.

Assume that $\mathrm{d}$ is a distance function on a compact manifold $M$ with
property (MR) on $[1,\infty)$. Then there exists an $R_{0}\geq 0$ such that
$(M,R\mathrm{d})$ is positive definite for all $R>R_{0}$.

###### Proof.

Consider the quadratic form $\mathfrak{q}_{R}(u)=\langle
u,\mathcal{Z}(R)u\rangle_{L^{2}}$, $u\in H^{-\mu}(M)$. By Theorem 3.7,
$\mathfrak{q}_{R}$ is positive definite for $R>R_{0}$, for some $R_{0}\geq 0$.
In particular, for any subspace $V\subseteq H^{-\mu}(M)$ the restriction of
$\mathfrak{q}_{R}$ to $V$ is also positive definite for $R>R_{0}$. For a
finite subset $F\subseteq M$, consider the subspace $V_{F}\subseteq
H^{-(\mu}(M)$ spanned by $\\{\delta_{x}:x\in F\\}$. In the basis
$(\delta_{x})_{x\in F}$, the quadratic form $\mathfrak{q}_{R}|_{V}$ is
represented by the $|F|\times|F|$-matrix
$(\mathrm{e}^{-R\mathrm{d}(x,y)})_{x,y\in F}$ and so it is positive definite
for $R>R_{0}$. ∎

######  Remark 3.11.

It was proven in [33, Subsection 3.2] that if $M$ is a compact Riemannian
manifold with $\pi_{1}(M)\neq 0$, the geodesic distance is not stably positive
definite, i.e. there exists an $R>0$ and a finite subset $F\subseteq M$ such
that $(\mathrm{e}^{-R\mathrm{d}_{\rm geo}(x,y)})_{x,y\in F}$ fails to be
positive definite. The manifold $M=S^{1}$ is not simply connected, and
therefore fails to be stably positive definite. but nevertheless Proposition
3.6 implies that $M=S^{1}$ with its geodesic distance has property (MR).
Therefore, property (MR) does not imply stably positive definiteness of the
metric space but only an asymptotic version thereof.

######  Theorem 3.12.

Let $M$ be an $n$-dimensional compact manifold with a distance function
$\mathrm{d}$ having property (SMR) on $\Gamma$ and set $\mu=(n+1)/2$. Then the
operator

$\mathcal{Z}(R):H^{-\mu}(M)\to H^{\mu}(M),$

is a well defined Fredholm operator for all $R\in\Gamma$ invertible for
$R\in\Gamma\cap\Gamma_{\pi/(n+1)}(R_{0})$. Moreover, the operator
$\mathcal{Z}(R):H^{-\mu}(M)\to H^{\mu}(M)$ depends holomorphically on $R$ in
$\Gamma$ and $\mathcal{Z}(R)^{-1}:H^{\mu}(M)\to H^{-\mu}(M)$ depends
holomorphically on $R\in\Gamma\cap\Gamma_{\pi/(n+1)}(R_{0})$ and admits a
meromorphic extension to $\Gamma$.

###### Proof.

By Theorem 2.22, $Q$ is a holomorphic function on $\Gamma$ with values in the
Fredholm operators and by property (SMR), $L$ is a compact valued holomorphic
function on $\Gamma$. Therefore $\mathcal{Z}$ defines a holomorphic function
on $\Gamma$ with values in the Fredholm operators. Since $\mathcal{Z}$ is
invertible for a large enough $R$, see Theorem 3.7, the theorem follows from
the meromorphic Fredholm theorem, see [31, Proposition 1.1.8]. ∎

######  Remark 3.13.

To ensure holomorphicity of $\mathcal{Z}$ and $\mathcal{Z}^{-1}$ on sectors,
the full property (SMR) is not needed. Indeed, if $\mathrm{d}$ has property
(MR) on a sector $\Gamma$ and $\Gamma\ni R\mapsto
L(R)\in\mathbb{B}(H^{-\mu}(M),H^{\mu}(M))$ is additionally holomorphic (in
norm sense) then by Theorem 3.7, for some $R_{0}\geq 0$, the mapping

$\Gamma\cap\Gamma_{\pi/(n+1)}(R_{0})\ni
R\mapsto\mathcal{Z}(R)\in\mathbb{B}(H^{-\mu}(M),H^{\mu}(M))$

is a holomorphic family of invertible operators. These properties are
inherited by its inverse,

$\Gamma\cap\Gamma_{\pi/(n+1)}(R_{0})\ni
R\mapsto\mathcal{Z}(R)^{-1}\in\mathbb{B}(H^{\mu}(M),H^{-\mu}(M)).$

By the proof of Proposition 3.6, this discussion applies to $S^{n}$ showing
that for some $R_{0}$, $\mathcal{Z}(R)^{-1}$ is holomorphic for
$R\in\Gamma\cap\Gamma_{\pi/(n+1)}(R_{0})$ in this case.

The following result follows from Corollary 3.5 and Theorem 3.12.

######  Theorem 3.14.

Let $M$ be a compact manifold with a distance function $\mathrm{d}$ such that
$\mathrm{d}^{2}$ is smooth on $M\times M$ and regular at the diagonal (cf.
Proposition 3.4), then $\tilde{\mathcal{Z}}(R):H^{-\mu}(M)\to H^{\mu}(M)$
depends holomorphically on $R\in\mathbb{C}\setminus\\{0\\}$ and the operator
$\mathcal{Z}(R)^{-1}:H^{\mu}(M)\to H^{-\mu}(M)$ extends meromorphically to
$\mathbb{C}\setminus\\{0\\}$.

###  3.4. Examples of distance functions that fail to satisfy property (MR)

Property (MR) of a distance function, as defined in Definition 3.3, is a
notable restriction on the singular support and singularity structure of the
distance function. We remark here that by a singular point, we mean any point
in the singular support, i.e. one in which the function is not $C^{\infty}$.
To better understand how these singularities affect the operator theoretic
properties of $L$, the reader is encouraged to review the proof of Proposition
3.6 where a crucial feature used in the proof is that the geodesic distance on
spheres near an off-diagonal singularity has the same singular features as it
has near the antipode of the singularity. An important property used there can
be stated as having control of the dimension of the off-diagonal singular
support of the metric. We make an elementary observation that follows from the
smoothness of the function $(0,\infty)\ni t\mapsto\sqrt{t}\in(0,\infty)$.

######  Proposition 3.15.

Let $\mathrm{d}$ be a distance function on a manifold $M$. Then it holds that

$\mathrm{singsupp}(\mathrm{d})\setminus\mathrm{Diag}_{M}=\mathrm{singsupp}(\mathrm{d}^{2})\setminus\mathrm{Diag}_{M}.$

Let us give a sufficient condition for a distance function (regular at the
diagonal) _not_ to satisfy property (MR). We note that an additional
obstruction was provided above in Proposition 3.10.

######  Theorem 3.16.

Let $M$ be a compact manifold and $\mathrm{d}$ a distance function such that
$\mathrm{d}^{2}$ is regular at the diagonal. Assume that there exists a
submanifold $N\subseteq M\times M$ with
$N\subseteq\mathrm{singsupp}(\mathrm{d}^{2})\setminus\mathrm{Diag}_{M}$ such
that any point $z_{0}\in N$ admits a neighborhood $U_{0}$ in $M\times M$ and a
coordinate chart

$\varphi:\mathbb{R}^{\dim(N)}_{t}\times\mathbb{R}^{\dim(M)-\dim(N)}_{s}\to
U_{0},$

with $\varphi(0)=x_{0}$, $\varphi^{-1}(U_{0}\cap
N)=\mathbb{R}^{\dim(N)}_{t}\times\\{0\\}$ and
$(t,s)\mapsto\varphi^{*}\mathrm{d}(t,s)-|s|$ being smooth in a neighborhood of
$0$. Then $L(R)$ does not extend to a continuous operator

$H^{-\mu}(M)\to H^{s}(M),$

for any $s>(3\dim(M)+1)/2-\dim(N)$ where $\mu=(\dim(M)+1)/2$. In particular,
if $\dim(N)>\dim(M)$ then $\mathrm{d}$ does not have property (MR) (see
Definition 3.3) on any sector containing a half-ray $[R_{0},\infty)$.

###### Proof.

We pick a point $(x_{0},y_{0})\in N$ and neighborhoods $U_{1}$ and $U_{2}$ of
$x_{0}$ and $y_{0}$ respectively, such that there exists a coordinate chart
$\varphi$ as in item ii) above on $U_{0}=U_{1}\times U_{2}$. Pick functions
$\chi_{1}\in C^{\infty}_{c}(U_{1})$ and $\chi_{2}\in C^{\infty}_{c}(U_{2})$
such that $\chi_{1}=1$ near $x_{0}$, $\chi_{2}=1$ near $y_{0}$ and such that
$(t,s)\mapsto\varphi^{*}\mathrm{d}(t,s)-|s|$ is smooth on
$\varphi^{-1}(\mathrm{supp}(\chi_{1})\times\mathrm{supp}(\chi_{2}))$. Clearly,
it suffices to prove that for any $R>0$, the operator $\chi_{1}L(R)\chi_{2}$
does not extend to a continuous operator $H^{-\mu}(M)\to H^{s}(M)$ for any
$s>(3\dim(M)+1)/2-\dim(N)$.

Set $k=\dim(N)$ and $n=\dim(M)$. The Schwartz kernel of $\chi_{1}L(R)\chi_{2}$
is a distribution on $M\times M$ supported in $U_{0}=U_{1}\times U_{2}$ and
pulling this kernel back along $\varphi$, we arrive at the distribution

$K(t,s)=\chi_{0}(s,t)\mathrm{e}^{-R|s|},$

where
$\chi_{0}=\varphi^{*}[(\chi_{1}\otimes\chi_{2})(1-\chi)]\mathrm{e}^{-R\psi}\in
C^{\infty}_{c}(\mathbb{R}^{2n})$ for
$\psi(t,s)=\varphi^{*}\mathrm{d}(t,s)-|s|$. It follows from Proposition A.1
and combining a Taylor expansion with asymptotic completeness, that $K\in
CI^{k-2n-1}(\mathbb{R}^{2n},\mathbb{R}^{k})$. We conclude that
$\chi_{1}L(R)\chi_{2}$ is a Fourier integral operator of order $k-2n-1$ and
this operator is elliptic in a neighborhood of $(x_{0},y_{0})$. Therefore,
since $\chi_{1}L(R)\chi_{2}$ is elliptic near $(x_{0},y_{0})$ of order
$k-2n-1$ it does not extend to a continuous operator $H^{-(n+1)/2}(M)\to
H^{s}(M)$ for any $s>-(n+1)/2-(k-2n-1)=(3n+1)/2-k$. ∎

######  Proposition 3.17.

Let $n>1$. The geodesic distance on the $n$-dimensional torus
$M=\mathbb{T}^{n}$ or the real projective space $M=\mathbb{R}P^{n}$ fails to
satisfy property (MR) (see Definition 3.3) on any sector containing a half-ray
$[R_{0},\infty)$. In fact, $L(R)$ does not extend to a continuous map
$H^{-\mu}(M)\to H^{s}(M)$ for $s>-n/2+3/2$.

###### Proof.

The proofs for both cases follow the same lines and rely on Theorem 3.16. For
$\mathbb{R}P^{n}$ we give a more geometric argument, and for $\mathbb{T}^{n}$
we give a coordinate oriented argument.

We first prove the result for real projective space. The projective space
$\mathbb{R}P^{n}$ is the quotient of $S^{n}$ by the antipodal map
$x\mapsto-x$. This quotient map is a covering map, and it is locally isometric
with respect to the geodesic distance. For $x\in\mathbb{R}P^{n}$, the equator
$\tilde{E}(x)\subseteq S^{n}$ is defined by

$\tilde{E}(x)=\\{v\in S^{n}:v\cdot x=0\\}.$

The condition $v\cdot x=0$ being invariant under the antipodal map,
$\tilde{E}(x)$ only depends on $x\in\mathbb{R}P^{n}$ and not on a choice of
pre-image of $x$ in $S^{n}$. We let $E(x)\subseteq\mathbb{R}P^{n}$ denote the
image under the quotient map. The off-diagonal singular support of the
geodesic distance on $\mathbb{R}P^{n}$ is the set

$\\{(x,y)\in\mathbb{R}P^{n}\times\mathbb{R}P^{n}:y\in E(x)\\}.$

Indeed, any geodesic in $\mathbb{R}P^{n}$ from a point $x$ lifts uniquely up
to a geodesic on $S^{n}$ up until the point it crosses $E(x)$ where the
geodesic distance $\mathrm{d}_{\rm geo}(x,\cdot)$ has a kink. The projection
mapping $p_{1}:\mathrm{singsupp}(\mathrm{d}_{\rm
geo})\setminus\mathrm{Diag}_{\mathbb{R}P^{n}}\to\mathbb{R}P^{n}$ is a locally
trivial $\mathbb{R}P^{n-1}$-bundle on $\mathbb{R}P^{n}$. Therefore,
$N=\mathrm{singsupp}(\mathrm{d}_{\rm
geo})\setminus\mathrm{Diag}_{\mathbb{R}P^{n}}\to\mathbb{R}P^{n}$ is a manifold
of dimension $2n-1>n$ and suitable local trivializations of
$p_{1}:N\to\mathbb{R}P^{n}$ satisfy the assumptions of item ii) in Theorem
3.16. We conclude from Theorem 3.16 that $L(R)$ has no bounded extension to an
operator $H^{-\mu}(\mathbb{R}P^{n})\to H^{s}(\mathbb{R}P^{n})$ for any
$s>-n/2+3/2$.

We now prove the result for the $n$-dimensional torus $\mathbb{T}^{n}$. Write
$\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$. While $\mathbb{T}^{n}$ is a
symmetric space, i.e. $\mathbb{T}^{n}=G/H$ for $G=\mathbb{T}^{n}$ and $H=1$,
it is instructive to consider $\mathrm{d}(x,y)$ for $x=0$. We have that
$\mathrm{d}(0,y)=|y|$ where we represent $y$ by as an element of the
fundamental domain $[-1/2,1/2)^{n}$, and $\mathbb{T}^{n}\ni
y\mapsto\mathrm{d}(0,y)$ as a function on $\mathbb{T}^{n}$ is the
$\mathbb{Z}^{n}$-periodic extension of $[-1/2,1/2)^{n}\ni y\mapsto|y|$.
Therefore $y\mapsto\mathrm{d}(0,y)$ has kinks on the image of
$\partial\left([-1/2,1/2)^{n}\right)$ in $\mathbb{T}^{n}$. In particular, we
see that the off-diagonal singular support of $\mathrm{d}$ is the set

$\\{(x,y)\in\mathbb{T}^{n}\times\mathbb{T}^{n}:x-y\in\partial\left([-1/2,1/2)^{n}\right)+\mathbb{Z}^{n}\\}.$

Consider the submanifold
$N_{0}:=\\{1/2\\}\times(-1/4,1/4)^{n-1}\subseteq\mathbb{R}^{n}$ and define the
$2n-1$-dimensional submanifold

$N:=\\{(x,y)\in\mathbb{T}^{n}\times\mathbb{T}^{n}:x-y\in
N_{0}+\mathbb{Z}^{n}\\}\subseteq\mathrm{singsupp}(\mathrm{d}^{2})\setminus\mathrm{Diag}_{\mathbb{T}^{n}}.$

Consider a point

$x_{0}=(x,1/2,y^{\prime})\in N.$

On the open ball of radius $1/4$ centred at $x_{0}$, we introduce the
coordinates $u=x_{1}-y_{1}-1/2$ and $t=(t^{\prime},y)$ where
$t^{\prime}=x^{\prime}-y^{\prime}$ in terms of standard coordinates
$x=(x_{1},x^{\prime})$ and $y=(y_{1},y^{\prime})$. In these coordinates, we
have that

$\mathrm{d}(x,y)=\sqrt{\left(\frac{1}{2}-|u|\right)^{2}+|t^{\prime}|^{2}}=\sqrt{-|u|+|t^{\prime}|^{2}+\frac{1}{4}+|u|^{2}}.$