# The magnitude and spectral geometry

Heiko Gimperlein, Magnus Goffeng, Nikoletta Louca  Heiko Gimperlein, Nikoletta
Louca
Maxwell Institute for Mathematical Sciences and
Department of Mathematics, Heriot-Watt University
Edinburgh EH14 4AS
United Kingdom

Magnus Goffeng,
Centre for Mathematical Sciences
University of Lund
Box 118, SE-221 00 Lund
Sweden
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study the geometric significance of Leinster’s notion of magnitude for a
smooth manifold with boundary of arbitrary dimension, motivated by open
questions for the unit disk in $\mathbb{R}^{2}$. For a large class of distance
functions, including embedded submanifolds of Euclidean space and Riemannian
manifolds satisfying a technical condition, we show that the magnitude
function is well defined for $R\gg 0$ and admits a meromorphic continuation to
sectors in $\mathbb{C}$. In the semiclassical limit $R\to\infty$, the
magnitude function admits an asymptotic expansion, which determines the
volume, surface area and integrals of generalized curvatures. Lower-order
terms are computed by black box computer algebra. We initiate the study of
magnitude analogues to classical questions in spectral geometry and prove an
asymptotic variant of the Leinster-Willerton conjecture.

###### Key words and phrases:

51F99 (primary), 28A75, 58J40, 58J50 (secondary)

##  1\. Introduction

The notion of _magnitude_ of an enriched category, and specifically for a
compact metric space, was introduced by Leinster [20] to capture the
“essential size” of an object. The magnitude has been shown to generalize the
cardinality of a set and the Euler characteristic, and it is even closely
related to measures of the diversity of a biological system. See [21] for an
overview.

For a finite metric space $(X,\mathrm{d})$, we say that $w:X\to\mathbb{R}$ is
a weight function provided that $\sum_{y\in
X}\mathrm{e}^{-\mathrm{d}(x,y)}w(y)=1$ for all $x\in X$. Given a weight
function $w$, the magnitude of a finite metric space $X$ is defined as
$\mathrm{mag}(X):=\sum_{x\in X}w(x)$. The magnitude $\mathrm{mag}(X)$ is
independent of the choice of $w$. More generally, for a compact, positive
definite metric space $(X,\mathrm{d})$ as in [23], the magnitude is defined as

(1) $\mathrm{mag}(X):=\sup\\{\mathrm{mag}(\Xi):\Xi\subset X\ \
\text{finite}\\}\ .$

This article finds a geometric origin of magnitude for smooth, compact
manifolds with boundary $X$ with a suitable distance function $\mathrm{d}$. In
particular, it provides a framework for the analysis and explicit computations
of magnitude when $X\subset\mathbb{R}^{n}$ is a compact domain, a long-
standing problem when $n$ is even. The framework combines classical results in
semiclassical analysis with current results for pseudodifferential boundary
problems. As an application, we initiate the study of magnitude analogues to
classical questions in spectral geometry and prove an asymptotic variant of
the Leinster-Willerton conjecture.

Instead of the magnitude of an individual space $(X,\mathrm{d})$, it proves
fruitful to study the function
$\mathcal{M}_{X}(R):=\mathrm{mag}(X,R\cdot\mathrm{d})$ for $R>0$. For a
compact, convex subset $X\subset\mathbb{R}^{n}$, Leinster and Willerton [22]
conjectured a surprising relation between magnitude and classical objects in
convex geometry, the intrinsic volumes $V_{k}(X)$:

######  Leinster-Willerton conjecture.

Suppose $X\subset\mathbb{R}^{n}$ is compact and convex. Then

$\displaystyle\mathcal{M}_{X}(R)=$
$\displaystyle\frac{1}{n!\omega_{n}}\mathrm{vol}_{n}(X)\
R^{n}+\frac{1}{2(n-1)!\omega_{n-1}}\mathrm{vol}_{n-1}(\partial X)\
R^{n-1}+\cdots+1$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\frac{1}{k!\omega_{k}}V_{k}(X)\ R^{k}\ .$

Here, $\omega_{k}$ is the volume of the $k$-dimensional unit ball.

This formula would imply an inclusion-exclusion principle
$\mathcal{M}_{X}+\mathcal{M}_{Y}=\mathcal{M}_{X\cup Y}+\mathcal{M}_{X\cap Y}$
for magnitude, a fundamental property of the Euler characteristic and the
other intrinsic volumes. See [2, 22] for further motivation.

The Leinster-Willerton conjecture is easily verified for an interval
$X\subset\mathbb{R}$, where
$\mathcal{M}_{X}(R)=\frac{\mathrm{Length}(X)}{2}R+\chi(X)$. More generally, it
holds for convex bodies $X\subset\mathbb{R}^{n}$ with the $\ell_{1}$, instead
of the Euclidean norm [21]. In dimension $n=5$, the Leinster-Willerton
conjecture was disproved by Barceló and Carbery [2], who computed that
$\mathcal{M}_{B_{5}}$ is a rational function, not a polynomial. In odd
dimensions their work related magnitude to a differential boundary problem,
which two of the authors [10] used to clarify the geometric content of
$\mathcal{M}_{X}$ when $X\subset\mathbb{R}^{n}$ is a compact, smooth domain,
$n$ odd: The magnitude function $\mathcal{M}_{X}$ extends to a meromorphic
function and admits an asymptotic expansion

$\mathcal{M}_{X}(R)\sim\frac{1}{n!\omega_{n}}\sum_{j=0}^{\infty}c_{j}(X)\
R^{n-j}\ \ \text{ for $R\to+\infty,$}$

in the sense that for any $N\in\mathbb{N}$,
$\mathcal{M}_{X}(R)-\sum_{j=0}^{N}c_{j}(X)R^{n-j}=\mathcal{O}(R^{n-N-1})$ as
$R\to+\infty$. Furthermore, for $j=0,1,2$ there exist
$\gamma_{n,j}\in\mathbb{Q}$ independent of $X$ such that
$c_{j}(X)=\gamma_{n,j}V_{j}(X)$ for convex $X$. The fourth term, $c_{3}(X)$,
is by the work [11] proportional to the Willmore energy $\int_{\partial
X}H^{2}$, a geometric quantity not predicted by the Leinster-Willerton
conjecture.

In this article we discuss extensions of the results of [10] to arbitrary
dimensions and to geometric settings, using a new, unified approach. For the
unit disk $B_{2}\subset\mathbb{R}^{2}$ we address long-standing questions
about the form of $\mathcal{M}_{B_{2}}$, popularized by Carbery: We find

$\mathcal{M}_{B_{2}}(R)=\frac{1}{2}R^{2}+\frac{3}{2}R+\frac{9}{8}+O(R^{-1}),$

and additionally that $\mathcal{M}_{B_{2}}(R)$ is not a polynomial. More
generally, our approach leads to an algorithm to calculate higher $c_{j}(X)$,
which has been implemented in python. We here state the result in the simplest
setting of smooth, compact, planar domains.

######  Theorem 1.1.

Let $X\subset\mathbb{R}^{2}$ be a smooth, compact domain.

  1. a)

$\mathcal{M}_{X}$ admits a meromorphic continuation to
$\mathbb{C}\setminus\\{0\\}$.

  2. b)

There exists an asymptotic expansion

$\mathcal{M}_{X}(R)\ \sim\
\frac{1}{2\pi}\sum_{j=0}^{\infty}c_{j}(X)R^{2-j}\quad\mbox{as $R\to+\infty$}.$

  3. c)

The first three coefficients are given by

$\displaystyle c_{0}(X)=\textnormal{Area}(X),\
c_{1}(X)=\frac{3}{2}\textnormal{Perim}(\partial X),\
c_{2}(X)=\frac{9}{8}\int_{\partial X}H\,\mathrm{d}S\ ,$

where $H$ is the mean curvature and $\textnormal{Perim}(\partial
X)=\int_{\partial X}\mathrm{d}S$ denotes the perimeter of the boundary.

######  Remark 1.2.

The computer code described in Appendix C leads to
$c_{3}(X)={\gamma}\int_{\partial X}H^{2}\,\mathrm{d}S$,
$c_{4}(X)={\delta}\int_{\partial X}H^{3}\,\mathrm{d}S$, where
$0\neq\gamma,\delta\in\mathbb{Q}$ are constants independent of $X$. In
particular, for a smooth, compact domain $X\subset\mathbb{R}^{2}$, the
magnitude function $\mathcal{M}_{X}$ is not a polynomial. The code is
available at this link.

The fact that $c_{j}(X)\propto\int_{\partial X}H^{j-1}\mathrm{d}S$ for
$j=1,2,3$ in odd dimensions (see [11]), and for $j=1,2,3,4$ in dimension
$n=2$, makes it natural to ask if $c_{j}(X)\propto\int_{\partial
X}H^{j-1}\mathrm{d}S$ for any $n>1$, $j>0$ and any smooth domain
$X\subseteq\mathbb{R}^{n}$.

Theorem 1.1 is a special case of Theorem 6.6, stated below for Euclidean
domains. Subsection 6.2 discusses extensions of Theorem 1.1 to manifolds with
boundary, with geometric implications the content of Section 3.

The starting point for this work is a reformulation of magnitude by Willerton
[29] and Meckes [23, 24]. Willerton [29] extended the notion of a weight
vector to the integral operator ${\mathcal{Z}}_{X}(R)$ on the space $M(X)$ of
finite Borel measures on $X$:

$\mathcal{Z}_{X}(R):M(X)\to
C(X),\quad\mathcal{Z}_{X}(R)\mu(x):=\frac{1}{R}\int_{X}\mathrm{e}^{-R\mathrm{d}(x,y)}\mathrm{d}\mu(y)\
.$

A _weight measure_ $\mu_{R}$ is a solution to the equation

$R\mathcal{Z}_{X}(R)\mu_{R}=1\ .$

If $(X,R\cdot\mathrm{d})$ is positive definite and admits a weight measure,
Meckes [23] showed that $\mathcal{M}_{X}(R)=\mu_{R}(X)$. We shall show that
for a domain $X\subset\mathbb{R}^{n}$ the operator $\mathcal{Z}_{X}(R)$ is a
pseudodifferential operator and therefore extends to the Sobolev space
$\dot{H}^{-\frac{n+1}{2}}(X)$ of distributions supported in $X$. The equation
$R\mathcal{Z}_{X}(R)u_{R}=1$ on $X$ admits a unique distributional solution
$u_{R}\in\dot{H}^{-\frac{n+1}{2}}(X)$. By relating this approach to Meckes’s
work [24], we conclude that $\mathcal{M}_{X}(R)=\langle u_{R},1\rangle_{X}$.

This pseudodifferential framework allows to use methods from semiclassical
analysis to study $\mathcal{M}_{X}$. A key ingredient is the construction of
an approximate inverse to $\mathcal{Z}_{X}$, based on recent advances for
pseudodifferential boundary problems [14]. The approach extends to subsets of
manifolds with a distance function, subject to appropriate technical
assumptions.

###  1.1. Acknowledgements

We thank Gerd Grubb and Mark Meckes for useful comments on this article and
Tony Carbery, Gerd Grubb, Tom Leinster, Rafe Mazzeo, Mark Meckes, Niels Martin
Møller, Grigori Rozenblum, Jan-Philip Solovej and Simon Willerton for fruitful
discussions.

MG was supported by the Swedish Research Council Grant VR 2018-0350. NL was
supported 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\. Overview and main results

Let $X$ be a compact manifold with boundary equipped with a distance function
$\mathrm{d}$ and a volume density $\mathrm{d}y$. Meckes’s abstract capacity-
theoretic approach to the magnitude function $\mathcal{M}_{X}$ in [24] relies
on the family of integral operators

(2)
$\displaystyle\mathcal{Z}_{X}(R)u(x):=\frac{1}{R}\int_{X}\mathrm{e}^{-R\mathrm{d}(x,y)}u(y)\mathrm{d}y,\quad
R\in\mathbb{C}\setminus\\{0\\},$

to study
$\mathcal{M}_{X}(R)=R^{-1}\langle\mathcal{Z}_{X}(R)^{-1}1,1\rangle_{X}$.
Explicit calculations of magnitude have avoided the solution of this integral
equation, based on reformulations available for certain homogeneous manifolds
[29] and odd-dimensional Euclidean domains [2, 10, 11, 30].

In this article we study the magnitude function in the general setting of a
manifold $X$ with boundary endowed with a distance function $\mathrm{d}$, by
making Meckes’s approach explicit. We require that $\mathrm{d}^{2}(x,y)$ is a
smooth function in a small neighborhood of the diagonal $x=y$. The operator
$\mathcal{Z}_{X}(R)$ then turns out to be a parameter-dependent
pseudodifferential operator on $X$, up to an error term which is often
negligible. This allows to adapt methods from semiclassical analysis and
recent developments for pseudodifferential boundary value problems to study
the inverse $\mathcal{Z}_{X}(R)^{-1}$ and the magnitude function
$\mathcal{M}_{X}(R)=R^{-1}\langle\mathcal{Z}_{X}(R)^{-1}1,1\rangle_{X}$ in
terms of the geometry of $(X,\mathrm{d})$.

The following theorem illustrates our results for the geometric content of the
magnitude function. We here state them for the geodesic distance function of a
Riemannian metric. See Remark 4.5 or [12, Section 3] for a discussion of the
properties (MR) and (SMR).

######  Theorem 2.1.

Let $X$ be a compact $n$-dimensional Riemannian manifold with boundary
equipped with its geodesic distance $\mathrm{d}$. Let $g$ denote its
Riemannian metric and $\mathrm{d}x$ its associated density. Assume that the
geodesic distance has property $(MR)$. Then there exists $R_{0}>0$ such that
$(X,R\cdot\mathrm{d})$ is a positive definite metric space for all $R>R_{0}$.
Its magnitude function $\mathcal{M}_{X}$ admits an asymptotic expansion

$\mathcal{M}_{X}(R)\ \sim\
\frac{1}{n!\omega_{n}}\sum_{j=0}^{\infty}c_{j}(X)R^{n-j},$

as $R\to\infty$. The geometric content and structure of the coefficients
$c_{j}$ are as follows:

  * •

$c_{0}(X)=\mathrm{vol}_{n}(X,g)$

  * •

$c_{1}(X)=\frac{n+1}{2}\mathrm{vol}_{n-1}(\partial X,g)$

  * •

If $n\geq 2$,
$c_{2}(X)=\frac{n+1}{6}\int_{X}s\mathrm{d}x+\frac{(n+1)^{2}(n-1)}{8}\int_{\partial
X}H\mathrm{d}S$, where $s$ is the scalar curvature and $H$ the mean curvature
of the boundary.

  * •

For $j\geq 4$, the coefficient $c_{j}(X)=c_{j}(X^{\circ})+c_{j}(X,\partial X)$
where $c_{j}(X^{\circ})$ can be computed as an integral over $X$ of a
universal polynomial in covariant derivatives of the curvature of total degree
$\leq j$ and $c_{j}(X,\partial X)$ is an integral over $\partial X$ of a
universal polynomial in covariant derivatives of the curvature as well as the
second fundamental form of $\partial X$ of total degree $\leq j$. Moreover,
$c_{j}(X^{\circ})=0$ for odd $j$.

If the distance function has property (SMR) on a sector
$\Gamma\subseteq\mathbb{C}$, e.g. if $\mathrm{d}^{2}$ is smooth and
$\Gamma=\mathbb{C}\setminus\\{0\\}$, then $\mathcal{M}_{X}$ extends
meromorphically to $\Gamma$.

Theorem 2.1 is a special case of Theorem 4.6 and Theorem 6.5, which includes
extensions to non-Riemannian distance functions. Non-Riemannian distance
functions are of interest, for example, for submanifolds of $\mathbb{R}^{n}$
with the subspace distance [29]. For Euclidean domains and the original
setting of the Leinster-Willerton conjecture, additional information is
obtained in Theorem 6.6.

Geometric applications are discussed in Section 3, motivated by classical
questions in spectral geometry and the category-theoretic orgin of magnitude.
For instance, we conclude the following results:

  1. (1)

The magnitude recovers the Euler characteristic of surfaces, see Subsection
3.1.

  2. (2)

The integral form of $c_{j}$ implies an inclusion-exclusion principle for
smooth, compact domains $X,Y,X\cap Y$ in a manifold which has property $(MR)$
(see Subsection 3.2):

$\mathcal{M}_{X\cup
Y}(R)\sim\mathcal{M}_{X}(R)+\mathcal{M}_{Y}(R)-\mathcal{M}_{X\cap Y}(R).$

  3. (3)

Analogous to Kac’ famous question [19], one can “magnitude the shape of a
drum” for balls: if $X$ is a smooth, compact domain in $\mathbb{R}^{n}$, $B$
is a ball and $\mathcal{M}_{X}(R)=\mathcal{M}_{B}(R)+o(R^{n-1})$, then $X$ is
isometric to $B$. See Subsection 3.3.

  4. (4)

Analogous to a theorem by Berger [3], magnitude characterizes Euclidean
domains whose boundary has constant mean curvature when the dimension $n$ is
odd or $n=2$: if $X$ and $Y$ are smooth, compact, $n$-dimensional domains with
$\mathcal{M}_{X}(R)=\mathcal{M}_{Y}(R)+o(R^{n-2})$, then $\partial X$ has
constant mean curvature if and only if $\partial Y$ has constant mean
curvature. See Subsection 3.4.

Let us illustrate Theorem 2.1 and its extensions by asymptotic computations of
the magnitude function in classical Riemannian and non-Riemannian geometries.
Beyond the intrinsic interest [2, 10, 22, 29, 30], these computations shed
light on continuity properties of $X\mapsto\mathcal{M}_{X}$, as raised by the
Leinster-Willerton conjecture, and on the technical assumption (MR).

######  Example 2.2 (Cylinders).

Suppose that $M\subseteq\mathbb{R}^{N}$ is a compact $n-1$-dimensional
submanifold, $T>0$ a parameter and $X_{T}=M\times[0,T]\subset\mathbb{R}^{N+1}$
is the cylinder on $M$ of height $T$. The space $X_{T}$ is a compact,
$n$-dimensional manifold with boundary, depicted in Figure 1(a). Equipping
$X_{T}$ with the subspace distance, Proposition 3.9 relates the magnitude of
$X_{T}$ with the magnitude of the base $M$:

$\frac{n!\omega_{n}\mathcal{M}_{X_{T}}(R)}{(n-1)!\omega_{n-1}\mathcal{M}_{M}(R)}=TR+\frac{n+1}{2}+T\cdot
O(1),\quad\mbox{as $R\to\infty$},$

and in particular

$n!\omega_{n}\mathcal{M}_{X_{T}}(R)=\mathrm{vol}_{n-1}(M)\left(TR^{n}+\frac{n+1}{2}R^{n-1}\right)+T\cdot
O(R^{n-2}),\quad\mbox{as $R\to\infty$}.$

Lower-order terms can be expressed in terms of the geometry of $M$ alone.

Figure 1. (a) Cylinder, Example 2.2, (b) spherical shell, Example 2.3,
(c) spherical cap, Example 2.4, (d) ball in hyperbolic plane (hyperboloid
model), Example 2.5, (e) toroidal armband, Example 2.6.

######  Example 2.3 (Spherical shells).

For $\varepsilon\in[0,1]$ and $n=2,3$ consider the family of spherical shells

$X_{\varepsilon}:=\\{x\in\mathbb{R}^{n}:1-\varepsilon\leq|x|\leq
1+\varepsilon\\},$

depicted in Figure 1(b). The limit case $\varepsilon=0$ corresponds to the
unit circle or sphere, $\varepsilon=1$ to the ball of radius $2$.

In dimension $n=2$, Theorem 2.1 shows

$\mathcal{M}_{X_{\varepsilon}}(R)=2\varepsilon R^{2}+3R+O(R^{-1})$

when $\varepsilon\in(0,1)$, while
$\mathcal{M}_{X_{1}}(R)=2R^{2}+3R+\frac{9}{8}+O(R^{-1})$ and
$\mathcal{M}_{X_{0}}(R)=\frac{\pi R}{1-e^{-\pi R}}\sim\pi R$. In the limit
$\varepsilon\to 1$, one observes continuity for the coefficients of $R^{2}$
and of $R$, but the $R^{0}$ term in the expansion jumps. For $\varepsilon\to
0$, the coefficients of $R^{2}$ and of $R^{0}$ are continuous, but the
coefficient of $R^{1}$ jumps.

This may be contrasted with the $n=3$-dimensional case, where the full
magnitude function can be computed:

$\mathcal{M}_{X_{\varepsilon}}(R)=\begin{cases}\frac{2R^{2}+2}{1-\mathrm{e}^{-\pi
R}},&\varepsilon=0,\\\
\frac{2\varepsilon^{2}+6\varepsilon}{3!}R^{3}+(2\varepsilon^{2}+2)R^{2}+4\varepsilon
R+2+\quad&\\\
\quad+\frac{\mathrm{e}^{-R(1-\varepsilon)}(R^{2}(1-\varepsilon)^{2}+1)+2R^{3}(1-\varepsilon)^{3}-3R^{2}(1-\varepsilon)^{2}+2R(1-\varepsilon)-1}{\sinh(2R(1-\varepsilon))-2R(1-\varepsilon)},\;&\varepsilon\in(0,1)\\\
\frac{8}{3!}R^{3}+4R^{2}+4R+1,&\varepsilon=1.\end{cases}$

See [10, Example 36] for the very similar computations for a spherical shell
$(2B_{3})\setminus B_{3}^{\circ}$, and [29] for $X_{0}=S^{2}$. Note that

$\lim_{\varepsilon\to
1}\mathcal{M}_{X_{\varepsilon}}(R)=\mathcal{M}_{X_{1}}(R)\ ,$
$\lim_{\varepsilon\to
0}\mathcal{M}_{X_{\varepsilon}}(R)=2R^{2}+2=\mathcal{M}_{X_{0}}(R)+O(R^{-\infty}).$

Still, the $R^{0}$ term in the expansion here is the Euler characteristic of
$\chi(X_{\varepsilon})$, which jumps when the topology changes. For
$\varepsilon\to 0$ there is a jump of dimension, and the pointwise values
correspondingly have a jump.

Because the family $X_{\varepsilon}$ is continuous in the Hausdorff distance,
we conclude that the magnitude function is not continuous with respect to this
topology, unlike for convex domains. Nevertheless, the asymptotics takes the
same form as $R\to\infty$.

######  Example 2.4 (Sphere).

Let $\theta\in[0,\pi]$ and $X_{\theta}\subseteq S^{2}$ denote the smooth,
compact spherical cap depicted in Figure 1(c),

$X_{\theta}:=\\{(x,y,z)\in S^{2}\subseteq\mathbb{R}^{3}:z\geq\cos(\theta)\\}.$

By [12, Proposition 3.6], the compact manifold $S^{2}$ with its geodesic
distance has property (MR), a property inherited by $X_{\theta}$. We compute
that for $\theta\in(0,\pi)$,

$\mathcal{M}_{X_{\theta}}(R)=(1-\cos(\theta))R^{2}+\sin(\theta)R+1+\frac{1}{8}\cos(\theta)+O(R^{-1}),$

using that the sphere $S^{2}$ has scalar curvature $2$, the mean curvature of
$\partial X_{\theta}$ is $H=-\mathrm{cotan}(\theta)$ and
$\mathrm{vol}_{2}(X_{\theta})=2\pi(1-\cos(\theta))$. In the limit cases
$\theta\in\\{0,\pi\\}$, $\mathcal{M}_{X_{0}}(R)=\mathcal{M}_{\rm pt}(R)=1$ and
$\mathcal{M}_{X_{\pi}}(R)=\mathcal{M}_{S^{2}}(R)=\frac{2R^{2}+2}{1-\mathrm{e}^{-\pi
R}}\sim 2R^{2}+2$ (see Example 2.3). Therefore, the coefficients $c_{j}$ jump
in the limit cases analogous to Example 2.3.

######  Example 2.5 (Hyperbolic space).

Consider the real hyperbolic $n$-space $M=\mathbb{H}^{n}$ with its Riemannian
distance function $\mathrm{d}$. Then $\mathrm{d}^{2}$ is smooth, so that $M$
and any compact, smooth domain $X\subset M$ have property (MR).

In dimension $n=3$ let $X_{r}\subset M$ be a ball of radius $r>0$, as depicted
in Figure 1(d) using the hyperboloid model of $M$ (for $n=2$). Theorem 2.1
shows

$\mathcal{M}_{X_{r}}(R)=\frac{\sinh(2r)-2r}{8}R^{3}+\frac{\cosh(2r)-1}{2}R^{2}+(H_{0}(r)-1)(\cosh(2r)-1)R+O(1).$

Indeed, the volume of $X_{r}$ is $\pi(\sinh(2r)-2r)$, the area of $\partial
X_{r}$ is $2\pi(\cosh(2r)-1)$, $s=-6$ in real hyperbolic $3$-space and
$H=H_{0}(r)$ is the mean curvature of the boundary of the ball, which only
depends on $r$ by rotational invariance.

######  Example 2.6 (Torus).

For $k,l\in\mathbb{N}$ consider the manifold $M_{1}=S^{k}\times\mathbb{R}^{l}$
and the closely related
$M_{2}=S^{k}\times\left(\mathbb{R}/2\mathbb{Z}\right)^{l}$, equipped with
their natural geodesic distances. By an extension of the argument for $S^{k}$
in [12, Proposition 3.6], $M_{1}$ and any compact, smooth domain $X\subset
M_{1}$ have property (MR). Property (MR) is not satisfied for $M_{2}$ if
$k,l>0$, following [12, Proposition 3.17]. Nevertheless, a domain $X\subset
M_{2}$ which is small enough to be contained in the image of
$S^{k}\times[0,1)^{l}\to M_{2}$ is isometric to a domain in $M_{1}$ and
therefore has property (MR).

We may therefore use Theorem 2.1 to compute the magnitude function of the
armband domain depicted in Figure 1(e), $X_{\varepsilon}$ defined as the image
of $S^{1}\times[0,\varepsilon]\to M_{2}$, for $\varepsilon\in(0,1)$. We obtain

$\mathcal{M}_{X_{\varepsilon}}(R)=\varepsilon R^{2}+2R+O(R^{-1}),$

since $s=0$ and $\int_{\partial X_{\varepsilon}}H\mathrm{d}S=0$ by sign
symmetry in the mean curvature. For $\varepsilon\geq 1$ property (MR) is not
satisfied, and Theorem 2.1 does not apply. The magnitude function of
$\widetilde{X}_{\varepsilon}=S^{1}\times[0,\varepsilon]\subset M_{1}$ is given
by $\mathcal{M}_{\widetilde{X}_{\varepsilon}}(R)=\varepsilon
R^{2}+2R+O(R^{-1})$ for all $\varepsilon>0$.

Let us now review the ideas and techniques behind the main results, including
Theorem 2.1. As mentioned at the beginning of this section, at the heart of
this paper is the analysis of the integral operator $\mathcal{Z}_{X}(R)$ from
(2) using pseudodifferential methods, when $X$ is a manifold with boundary.

Section 4 recalls the abstract function space framework introduced by Meckes
[24] for the magnitude operator $\mathcal{Z}_{X}(R)$ and connects it to
concrete Sobolev spaces of distributions in $X$ and the analytic techniques
developed in [12]. In particular, the approach in this article and in [12]
therefore, indeed, computes the magnitude function. The key result is Theorem
4.6: For $\mu=\frac{n+1}{2}$, $\mathrm{Re}(R)$ sufficiently large and
$\mathrm{arg}(R)<\frac{\pi}{n+1}$, the operator $\mathcal{Z}_{X}(R)$ defines a
holomorphic family of isomorphisms between the Sobolev spaces
$\dot{H}^{-\mu}(X)$ and $\overline{H}^{\mu}(X)$ of supported, respectively
extensible distributions. Results on the meromorphic continuation of
$\mathcal{M}_{X}$ to the complex plane follow from the proof, see Corollary
4.8. The abstract framework persists even when the boundary $\partial X$ is
merely $C^{0}$ (i.e. locally the graph of a continuous function).

The idea in the proof is to replace $\mathcal{Z}_{X}$ with a localization

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

Here, 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 $\chi$. We explicitly compute in Theorem 5.3 that the operator
$Q_{X}$ is a lower order perturbation of a fractional Laplacian with parameter
$(R^{2}+\Delta)^{-\mu}$. Under the above stated conditions on $R$, $Q_{X}$ is
invertible as a sufficiently small perturbation of the invertible fractional
Laplacian.

The main results of this article follow from the explicit analysis of the
pseudodifferential operator $Q_{X}$. We assume that $\mathrm{d}^{2}(x,y)$ is a
divergence in the sense of [1, Section 1.2], i.e. $\mathrm{d}^{2}(x,y)$ is a
smooth function in a small neighborhood of the diagonal $x=y$, and in local
coordinates it has a Taylor expansion (for any $N>0$)

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

where $H_{\mathrm{d}^{2}}(x,\cdot)$ is a Riemannian metric on $X$. In local
coordinates $C^{j}$ is a symmetric $j$-form in $x-y$. The Taylor expansion (3)
translates into an expansion of the symbol of the pseudodifferential operator
$Q_{X}$, see Theorem 5.3, where terms for larger $j$ contribute to terms of
the symbol of lower order jointly in $\xi$ and $R$.

Section 5 discusses the special case when $X=M$ is a closed manifold. In this
case the inverse $Q_{X}(R)^{-1}$ is a pseudodifferential operator, whenever it
exists, and the full symbol of $Q_{X}(R)^{-1}$ can be explicitly computed by
the iterative scheme described in Proposition 5.7. The operator $Q_{X}$ is
generally better behaved than $\mathcal{Z}_{X}$. The off-diagonal
singularities of $\mathrm{d}$ may create problems when considering
$\mathcal{Z}_{X}$ as a map between Sobolev spaces. Even for a Riemannian
manifold these relate to difficult geometric questions about the structure of
the cut-locus, cf. the discussion in [12, Section 3]. Under suitable
assumptions on $(X,\mathrm{d})$, such as (MR) and (SMR), properties of $Q_{X}$
are inherited by $\mathcal{Z}_{X}$, see Corollary 5.4. The asymptotic
expansion of the magnitude function

$\mathcal{M}_{X}(R)=R^{-1}\langle\mathcal{Z}_{X}(R)^{-1}1,1\rangle_{X}\sim
R^{-1}\langle Q_{X}(R)^{-1}1,1\rangle_{X},$

then follows with expansion coefficients $c_{j}$ computed from the symbol of
$Q_{X}^{-1}$, and therefore from $C^{j}$. See Theorem 5.9 and 5.10.

The reader can find a discussion of the properties (MR) and (SMR) in Remark
4.5 and details in [12, Section 3]. They are satisfied, for example, if
$\mathrm{d}^{2}$ is smooth on all of $X\times X$, such as for a domain or a
submanifold in $\mathbb{R}^{n}$ with the induced metric.

The analysis in Section 5 for a closed manifold illustrates the general
approach taken in this article. For a manifold with boundary the inverse
$Q_{X}(R)^{-1}$ decomposes into the previously studied interior part and a new
boundary contribution, as described in Theorem 6.3. Section 6 discusses the
boundary contribution using methods for pseudodifferential boundary value
problems, here of negative order $-\frac{n+1}{2}$. While such problems have a
long history [6], the magnitude problem connects to recent developments for
boundary problems for the fractional Laplacians [14, 16]. The resulting
boundary contribution to the expansion coefficients $c_{j}$ involves the
coefficients $C^{j}$ and the geometry of $\partial X$ inside $X$. Sections 5
and 6 rely on and motivate the purely analytic results for $\mathcal{Z}_{X}$
in [12], which we here exploit for geometric applications.

The methods presented in this article are amenable to a black-box computer
implementation, discussed in Appendix A. Appendix B lists pseudocode for
Euclidean domains and for submanifolds of Euclidean space, which are the basis
for the computational results in this article.

Beyond these main results of the article, the analysis of the boundary
contributions in Subsection 7.1 sheds light on the boundary behavior of the
weight distribution recently considered for applications in data science. More
precisely, in Subsection 7.1, we use Lemma 5.6 to obtain a weak form of the
structural properties conjectured in [4]. In Subsection 7.2, we connect to
results of Meckes [25] and address the Taylor expansion of the magnitude
function at $R=0$.

##  3\. Geometry of magnitude function and magnitude operator

The structural properties and geometric formulas for the expansion
coefficients of the magnitude function as $R\to\infty$ shed light on the
geometric content of magnitude. We here use Theorem 2.1, and more generally
results from Subsection 5.2 and 6.2 below, as a tool to generate geometric
consequences on magnitude.

###  3.1. Relation to Euler characteristic

Magnitude was originally motivated as a generalization of the Euler
characteristic to finite (enriched) categories, and Leinster and Willerton
conjectured that the expansion coefficient $c_{n}$ equals the Euler
characteristic for a convex body in $\mathbb{R}^{n}$. Therefore, the relation
between these two quantities is of interest.

######  Proposition 3.1.

Let $M$ be a compact Riemannian surface with geodesic distance $\mathrm{d}$
and Riemannian metric $g$. Assume that $M$ admits a family
$(u_{R})_{R>R_{0}}\subseteq\mathcal{D}^{\prime}(M)$ of distributional
solutions to

$\int_{M}\mathrm{e}^{-R\mathrm{d}(x,y)}u_{R}(y)\mathrm{d}y=1,\quad R>R_{0}.$

Assume that for any $\chi,\chi^{\prime}\in C^{\infty}(M)$ with disjoint
supports it holds that

$\int_{M}\chi(x)\mathrm{e}^{-R\mathrm{d}(x,y)}u_{R}(y)\chi^{\prime}(y)\mathrm{d}y=O(R^{-3}).$

Then the magnitude function $\mathcal{M}_{M}(R)$ is defined for $R>R_{0}$ and
there is an asymptotic expansion

$\mathcal{M}_{M}(R)=\frac{\mathrm{vol}_{2}(M,g)}{2\pi}R^{2}+\frac{\chi(M)}{4}+O(R^{-1}),$

where $\chi(M)$ denotes the Euler characteristic of $M$.

We note that if $M$ is a homogeneous surface of compact type, then it
satisfies the assumptions of the proposition as we can construct
$(u_{R})_{R>R_{0}}$ from an averaging procedure. See more in [29]. For further
discussion concerning this assumption, see Remark 5.11.

###### Proof.

The assumptions on the existence of the distributional solution and Theorem
5.10, a variant of Theorem 2.1 stated in the introduction, implies that
$\mathcal{M}_{M}(R)=\int_{M}u_{R}(y)\mathrm{d}y$ is defined for $R>R_{0}$.
Theorem 5.10 in fact gives us
$\mathcal{M}_{M}(R)\sim\frac{1}{2\pi}\sum_{k=0}^{2}c_{k}(M)R^{2-k}+O(R^{-1})$
for $c_{2}(M)=\frac{1}{2}\int_{M}\mathrm{s}$, where $s$ is the scalar
curvature. The assertion then follows from the Gauss-Bonnet theorem for
surfaces, $\int_{M}\mathrm{s}=\pi\chi(M)$. ∎

A corresponding connection between the constant term in $\mathcal{M}_{X}$ and
the Euler characteristic fails for domains $X\subset\mathbb{R}^{3}$. With $H$
the mean curvature of $\partial X$, $c_{3}(X)=\delta\int_{\partial X}H^{2}$ is
a multiple of the Willmore energy by [11]. The Willmore energy, however, can
be arbitrarily large on surfaces of genus $0$ and is not determined by the
Euler characteristic.

###  3.2. Inclusion-exclusion principles

As first shown in [10] for smooth, compact domains in a Euclidean space of odd
dimension, one of the fundamental properties of the Euler characteristic still
holds in an asymptotic form: the inclusion-exclusion principle. The results in
this article imply an inclusion-exclusion principle for smooth, compact
domains in manifolds of any dimension, under the condition (MR) from Remark
5.5.

######  Proposition 3.2.

Let $M$ be a manifold with a distance function which satisfies $(MR)$. Let
$X,Y\subset M$ be smooth, compact domains such that $X\cup Y$ and $X\cap Y$
are smooth. Then

$\mathcal{M}_{X\cup
Y}(R)=\mathcal{M}_{X}(R)+\mathcal{M}_{Y}(R)-\mathcal{M}_{X\cap
Y}(R)+O(R^{-\infty}).$

###### Proof.

The assertion follows from the local formulas for the expansion coefficients
$c_{j}$ from Theorem 6.5 and the fact that property $(MR)$ is inherited by
smooth, compact domains. ∎

###  3.3. Can you “magnitude the shape of a drum”

In spectral geometry, analogues of the expansion coefficients $c_{j}$ have
proven fruitful to find relationships between geometry and the eigenvalues of
the Laplace-Beltrami operator, see [28] for a recent overview. The guiding
question by M. Kac, “Can you hear the shape of a drum?” [19] has an analogue
for the magnitude function: Does the magnitude function $\mathcal{M}_{X}$
determine a compact domain $X$ up to isometry?

The answer is positive for the magnitude function of a ball:

######  Proposition 3.3.

Let $B\subset\mathbb{R}^{n}$ a ball. If $X\subset\mathbb{R}^{n}$ is a smooth,
compact domain with $\mathcal{M}_{X}=\mathcal{M}_{B}+o(R^{n-1})$ for
$R\to\infty$, then $X$ is isometric to $B$.

###### Proof.

By assumption

$\displaystyle n!\omega_{n}\mathcal{M}_{X}(R)$
$\displaystyle=\mathrm{vol}_{n}(X)R^{n}+{\textstyle\frac{n+1}{2}}\textnormal{vol}_{n-1}(\partial
X)R^{n-1}+o(R^{n-1})$
$\displaystyle=\mathrm{vol}_{n}(B)R^{n}+{\textstyle\frac{n+1}{2}}\textnormal{vol}_{n-1}(\partial
B)R^{n-1}+o(R^{n-1})=n!\omega_{n}\mathcal{M}_{B}(R)+o(R^{n-1}).$

Therefore $\mathrm{vol}_{n}(X)=\mathrm{vol}_{n}(B)$ and
$\mathrm{vol}_{n-1}(\partial X)=\mathrm{vol}_{n-1}(\partial B)$. Recalling the
isoperimetric inequality,
$n\omega_{n}^{1/n}\mathrm{vol}_{n}(X)^{(n-1)/n}\leq\mathrm{vol}_{n-1}(\partial
X)$, with equality if and only if $X$ is isometric to a ball, it follows that
$X$ is isometric to $B$. ∎

######  Remark 3.4.

For nonconvex domains, however, a counterexample by Meckes [9] shows that the
magnitude function $\mathcal{M}_{X}$ does not determine a compact domain $X$
up to isometry. Let $n$ be odd. Consider balls
$B_{1},B_{2}\subset\mathbb{R}^{n}$ of the same diameter which are contained in
the interior of a large ball $B$. Then $\mathcal{M}_{B\setminus
B_{1}^{\circ}}=\mathcal{M}_{B\setminus B_{2}^{\circ}}$, but generically
$B\setminus B_{1}^{\circ}$ and $B\setminus B_{2}^{\circ}$ are not isometric.

###  3.4. Constant mean curvature

The proof of Proposition 3.3 indicates the opportunities for studying the
relationship between the geometry of a domain $X$ and its magnitude function
$\mathcal{M}_{X}$ using techniques from spectral geometry, based on the
expansion of $\mathcal{M}_{X}$ for $R\to\infty$. Using such techniques we
obtain an analogue for magnitude of a theorem by Berger [3] that for a closed
Riemannian surface having constant sectional curvature is determined by the
eigenvalues of the Laplace-Beltrami operator.

######  Proposition 3.5.

Let $X,Y\subset\mathbb{R}^{n}$ be smooth, compact domains and $n$ odd or
$n=2$, and assume that $\partial X$ has constant mean curvature $H(\partial
X)=H$. If $\mathcal{M}_{X}=\mathcal{M}_{Y}+o(R^{n-3})$ for $R\to\infty$, then
also $\partial Y$ has constant mean curvature $H(\partial Y)=H$.

This result generalizes Proposition 3.3.

###### Proof.

Note that the polynomial $p_{Y}(z)=\int_{\partial Y}(z-H(\partial Y))^{2}$ has
a real zero if and only if the mean curvature of $\partial Y$ is constant.
Since $c_{j}(X)\propto\int_{\partial X}H^{j-1}\mathrm{d}S$ for $j=1,2,3$ if
$n=2$ (see Theorem 1.1 and Remark 1.2) or $n$ is odd (see [11]), there exist
constants $\alpha_{n},\beta_{n},\gamma_{n}$ depending only on the dimension
$n$ such that

$p_{Y}(z)=\alpha_{n}c_{1}(Y)z^{2}+\beta_{n}c_{2}(Y)z+\gamma_{n}c_{3}(Y)\ .$

Because $c_{j}(Y)=c_{j}(X)$ for $j=1,2,3$, $p_{Y}(z)=\int_{\partial
X}(z-H(\partial X))^{2}$. With $H(\partial X)=H$ constant, $p_{Y}$ vanishes in
$H$, and therefore $H(\partial Y)$ is constant and equals $H$. ∎

###  3.5. Relation to residues of manifolds

For a compact metric space $(X,\mathrm{d})$, the geometric relevance of the
magnitude operator $\mathcal{Z}_{X}(R)$ is not restricted to the magnitude
function $\mathcal{M}_{X}(R)=R\langle\mathcal{Z}_{X}(R)^{-1}1,1\rangle$.
Closely related quantities of interest are the meromorphic energy function

$B_{X}(z):=\mathrm{F.P.}|_{s=z}\int_{X\times X}\mathrm{d}(x,y)^{s}\
\mathrm{d}x\ \mathrm{d}y\ ,$

and the residue

$R_{X}(z):=\mathrm{res}_{s=z}\int_{X\times X}\mathrm{d}(x,y)^{s}\ \mathrm{d}x\
\mathrm{d}y\ .$

The energy $B_{M}(z)$ was first introduced by Brylinski [5] for knots in
$\mathbb{R}^{3}$ and by Fuller and Venmuri [7] for closed submanifolds of
$\mathbb{R}^{n}$. $B_{M}(z)$ and $R_{M}(z)$ may be expressed in terms of
$\mathcal{Z}_{X}$ by means of the formula

$\mathrm{d}^{-s}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}R^{s-1}e^{-R\mathrm{d}}\
\mathrm{d}R,$

valid for $\mathrm{Re}(s)>0$. Then

$\displaystyle\int_{X\times X}\mathrm{d}(x,y)^{-s}\ \mathrm{d}x\ \mathrm{d}y$
$\displaystyle=\frac{1}{\Gamma(s)}\int_{0}^{\infty}R^{s-1}\int_{X\times
X}e^{-R\mathrm{d}(x,y)}\ \mathrm{d}x\ \mathrm{d}y\ \mathrm{d}R$
$\displaystyle=\frac{1}{\Gamma(s)}\int_{0}^{\infty}R^{s-1}\langle\mathcal{Z}_{X}(R)1,1\rangle\
\mathrm{d}R\ .$

To understand the relation between the residues of

$f(s)=\int_{X\times X}\mathrm{d}(x,y)^{-s}\ \mathrm{d}x\ \mathrm{d}y$

and the expectation value of the magnitude operator,

$e(t)=\langle\mathcal{Z}_{X}(t)1,1\rangle=\int_{X\times
X}e^{-t\mathrm{d}(x,y)}\mathrm{d}x\ \mathrm{d}y\ ,$

we divide the integral on the right hand side and set $\lambda=R^{-1}$:

$\displaystyle f(s)$
$\displaystyle=\frac{1}{\Gamma(s)}\int_{0}^{1}R^{s-1}e(R)\
\mathrm{d}R+\frac{1}{\Gamma(s)}\int_{1}^{\infty}R^{s-1}e(R)\ \mathrm{d}R$
$\displaystyle=\frac{1}{\Gamma(s)}\int_{0}^{1}R^{s-1}e(R)\
\mathrm{d}R+\frac{1}{\Gamma(s)}\int_{0}^{1}\lambda^{-(s+1)}e(\lambda^{-1})\
\mathrm{d}\lambda\ .$

The proof of the following proposition is analogous to [17, Proposition 5.1].

######  Proposition 3.6.

Assume that $e(t)$ and $e(t^{-1})$ are holomorphic in $V_{\theta_{0}}$ (for
some $\theta_{0}\in(0,\frac{\pi}{2})$),

$V_{\theta_{0}}=\\{t=re^{i\theta}:\ 2>r>0,|\theta|<\theta_{0}\\}\ ,$

and $e(t)=O(|t|^{a})$, $e(t^{-1})=O(|t|^{b})$ for $t\to 0$ in $V_{\delta}$,
any $\delta<\theta_{0}$, for some $a,b\in\mathbb{R}$. Consider the function

$f(s)=\int_{0}^{\infty}t^{s-1}e(t)\ \mathrm{d}t,$

holomorphic for $\mathrm{Re}(s)>-a$. Then the following properties are
equivalent:

  1. (a)

$e(t)$ and $e(t^{-1})$ have asymptotic expansions for $t\to 0$,

$\displaystyle e(t)$ $\displaystyle\sim\sum_{j=0}^{\infty}a_{j}t^{\beta_{j}},\
\ \mbox{where}\ \ \beta_{j}\to+\infty,$ $\displaystyle e(t^{-1})$
$\displaystyle\sim\sum_{j=0}^{\infty}A_{j}t^{\gamma_{j}},\ \ \mbox{where}\ \
\gamma_{j}\to+\infty,$

uniformly for $t\in V_{\delta}$, for each $\delta<\theta_{0}$.

  2. (b)

$f(s)$ is meromorphic on $\mathbb{C}$ with the singularity structure

$\Gamma(s)f(s)\sim\sum_{j=0}^{\infty}\frac{a_{j}}{\beta_{j}+s}+\sum_{j=0}^{\infty}\frac{A_{j}}{\gamma_{j}-s},\
$

and for each real $C_{1},C_{2}$ and each $\delta<\theta_{0}$,

$|f(s)|\leq C(C_{1},C_{2},\delta)e^{-\delta|\mathrm{Im}(s)|},\ \
|\mathrm{Im}(s)|\geq 1,\ C_{1}\leq|\mathrm{Re}(s)|\leq C_{2}\ .$

The assumptions on $e(t)=\langle\mathcal{Z}_{X}(t)1,1\rangle$ from Proposition
3.6 are trivially satisfied; indeed $e(t)=O(t^{-\infty})$ as $t\to\infty$ and
$e(t)$ is real analytic near $0$. There is therefore, by Proposition 3.6, a
close relation between the expectation value of the magnitude operator
$\langle\mathcal{Z}_{X}(t)1,1\rangle$ and the residue for an arbitrary metric
space $(X,\mathrm{d})$.

The geometric content discovered for residues of knots, hypersurfaces and
domains therefore translates into geometric content for the corresponding
magnitude operator:

######  Proposition 3.7.

[5, 7, 27] Let $M$ be a compact hypersurface in $\mathbb{R}^{m+1}$ with
principal curvatures $\kappa_{i}$. Define
${\|h\|}^{2}=\sum_{i}{\kappa_{i}}^{2}$ and $o_{k}$ the surface area of the
unit $k$-sphere.

  1. (1)

The first residue is given by $\displaystyle R_{M}(-m)=\displaystyle
o_{m-1}\,\mbox{\rm Vol}\,(M).$

  2. (2)

The second residue is given by

$R_{M}(-m-2)=\frac{o_{m-1}}{8m}\int_{M}\left(2{\|h\|}^{2}-{m^{2}|H|}^{2}\right)\
\mathrm{d}x.$

######  Proposition 3.8.

[27] Let $\Omega\subset\mathbb{R}^{n}$ be a compact domain.

$\displaystyle R_{\Omega}(-n)$ $\displaystyle=\displaystyle{o_{n-1}}\text{\rm
Vol}\,(\Omega),$ $\displaystyle R_{\Omega}(-n-1)$
$\displaystyle=-\frac{o_{n-2}}{n-1}\,\textrm{\rm Vol}(\partial\Omega),$
$\displaystyle R_{\Omega}(-n-3)$
$\displaystyle=\displaystyle\frac{o_{n-2}}{24(n^{2}-1)}\int_{\partial\Omega}\left(3n^{2}H^{2}-2\mathrm{s}\right)\
\mathrm{d}x,$

Also an inclusion-exclusion principle was derived for residues [26].

###  3.6. Computations for cylinders

In the special case of a cylinder of the form $X_{T}=M\times[0,T]$, for $T>0$,
the magnitude function of $X_{T}$ relates to geometric invariants of $M$. Let
us compute an instance of this.

######  Proposition 3.9.

Assume that $M\subseteq\mathbb{R}^{N}$ is an $n-1$-dimensional compact
submanifold. For $T>0$, consider the compact submanifold with boundary

$X_{T}:=M\times[0,T]\subseteq\mathbb{R}^{N+1}.$

Equipping $M$ and $X_{T}$ with the subspace distance, we have that

$\displaystyle\mathcal{M}_{X_{T}}(R)\sim\ $
$\displaystyle\frac{\mathrm{vol}_{n-1}(M)}{n!\omega_{n}}TR^{n}+\frac{(n+1)\mathrm{vol}_{n-1}(M)}{2\cdot
n!\omega_{n}}R^{n-1}+$ $\displaystyle+\frac{n+1}{6\cdot
n!\omega_{n}}\int_{M}s_{\mathrm{d}^{2},M}\mathrm{d}x\
TR^{n-2}+\sum_{k=4}^{\infty}\rho_{k}(M)TR^{n-k},$

where $\rho_{k}(M)=\int_{M}a_{k,0}(x,1)\mathrm{d}x$ is computed as an integral
over $M$ from the local densities $a_{k,0}$ on $X_{T}$ of Theorem 6.5 and
$\rho_{k}(M)=0$ if $k$ is odd.

###### Proof.

On $X_{T}$, the last coordinate in $\mathbb{R}^{N+1}$ provides a global
coordinate $x_{n}$ for the direction transversal to the boundary, and it is a
normal for the Riemannian metric associated with the distance function.
Therefore, there are no boundary contributions to the magnitude asymptotics
par $c_{1}(X_{T})$ which is proportional to the volume of the boundary. The
format for $c_{0}(X_{T})$ and $c_{2}(X_{T})$ follows from Theorem 6.5 since
all structures are constant in the $x_{n}$-direction. Therefore, the
description
$c_{k}(X_{T})=\int_{X_{T}}a_{k,0}(x,1)\mathrm{d}x=T\int_{M}a_{k,0}(x,1)\mathrm{d}x=T\rho_{k}(M)$,
$k>2$, follows from Theorem 6.5. ∎

###  3.7. Asymptotically polynomial behavior in dimension $3$

In dimension $3$, the magnitude function of the unit ball is a polynomial by
[2]. To the knowledge of the authors, for Euclidean domains, this can only
happen in dimensions $1$ and $3$. Based on the computational evidence from
Theorem 6.6 and Remark 1.2 for the statement that
$c_{j}(X)\propto\int_{\partial X}H^{j-1}\mathrm{d}S$, for $j>0$ and
$X\subseteq\mathbb{R}^{n}$ a domain, we make the following observation.

######  Proposition 3.10.

Let $N>3$. The following two statements are equivalent:

  1. (1)

There are coefficients $\gamma_{1},\gamma_{2},\ldots,\gamma_{N}$ such that for
any domain $X\subseteq\mathbb{R}^{3}$ with smooth boundary we have that

$c_{j}(X)=\gamma_{j}\int_{\partial X}H^{j-1}\mathrm{d}S,\quad j=1,\ldots,N.$

  2. (2)

There is a coefficient $\gamma_{3}$ such that for any domain
$X\subseteq\mathbb{R}^{3}$ with smooth boundary we have that

$\mathcal{M}_{X}(R)=\frac{\mathrm{vol}_{3}(X)}{8\pi}R^{3}+\frac{\mathrm{vol}_{2}(\partial
X)}{4\pi}R^{2}+\frac{1}{2\pi}\int_{\partial
X}H\mathrm{d}SR+\gamma_{3}\int_{\partial X}H^{2}\mathrm{d}S+O(R^{2-N}).$

###### Proof.

It is clear from Theorem 6.6 that 2) implies 1), with $\gamma_{j}=0$ for
$j=4,\ldots,N$. If 1) holds, then we have for any domain
$X\subseteq\mathbb{R}^{3}$ with smooth boundary that

$\mathcal{M}_{X}(R)=\frac{\mathrm{vol}_{3}(X)}{8\pi}R^{3}+\frac{1}{8\pi}\sum_{j=1}^{N}\gamma_{j}\int_{\partial
X}H^{j-1}\mathrm{d}SR^{3-j}+O(R^{2-N}).$

By [2, Theorem 2], $\mathcal{M}_{X}(R)$ is a polynomial for the unit ball. In
particular, we must have that $\gamma_{j}=0$ for $j=4,\ldots,N$. The precise
form of $\gamma_{1}$ and $\gamma_{2}$ can be found in [10] or Theorem 6.6
below. ∎

##  4\. Weight distributions and magnitude

The work of Meckes [24] relates the magnitude to a capacity-like definition
for a reproducing kernel Hilbert space defined from $(X,\mathrm{d})$. We here
recall this approach in a form which connects it to the analytic techniques
developed in [12]. This will show that the approach of [12] indeed computes
the magnitude function.

Let $(M,\mathrm{d})$ be a compact metric space. Consider the vector space
$\mathrm{FM}(M)$ of finitely supported complex measures on $M$. For
$R\in\mathbb{C}$, we define the form
$\langle\cdot,\cdot\rangle_{\mathcal{W}_{R}}$ on $\mathrm{FM}(M)$ by

(4) $\langle\mu,\nu\rangle_{\mathcal{W}_{R}}:=\int_{M\times
M}\mathrm{e}^{-R\mathrm{d}(x,y)}\mathrm{d}\bar{\mu}(x)\mathrm{d}\nu(y).$

For real $R$, $\langle\mu,\nu\rangle_{\mathcal{W}_{R}}$ is a sesquilinear form
on $\mathrm{FM}(M)$. The following result follows from the definition of a
positive definite metric space.

######  Proposition 4.1.

Let $R>0$. The form $\langle\cdot,\cdot\rangle_{\mathcal{W}_{R}}$ is positive
definite on $\mathrm{FM}(M)$ if and only if $(M,R\cdot\mathrm{d})$ is positive
definite.

For $R>0$ such that $(M,R\cdot\mathrm{d})$ is positive definite, we define
$\mathcal{W}_{R}(M)$ as the completion of $\mathrm{FM}(M)$ in the inner
product $\langle\cdot,\cdot\rangle_{\mathcal{W}_{R}}$. Note that for any
compact $X\subseteq M$ there is an isometric inclusion of Hilbert spaces

$\mathcal{W}_{R}(X)\subseteq\mathcal{W}_{R}(M).$

Let $C^{1/2}(M,\mathrm{d})$ denote the Banach space of functions on $M$ which
are Hölder continuous of exponent $1/2$.

######  Proposition 4.2 (Proposition 3.2 of [24]).

For $R>0$ such that $(M,R\cdot\mathrm{d})$ is positive definite, the operator

$\mathcal{Z}(R)\mu(x):=\frac{1}{R}\int_{M}\mathrm{e}^{-R\mathrm{d}(x,y)}\mathrm{d}\mu(x),\quad\mu\in\mathrm{FM}(M)$

extends to a continuous mapping

$\mathcal{Z}(R):\mathcal{W}_{R}(M)\to C^{1/2}(M,\mathrm{d}).$

Moreover, for any $\mu\in\mathrm{FM}(M)$, $\nu\in\mathcal{W}_{R}(M)$,

(5)
$\langle\mu,\nu\rangle_{\mathcal{W}_{R}}=R\int_{M}[\mathcal{Z}(R)\nu](x)\mathrm{d}\bar{\mu}(x).$

We define $\mathcal{H}_{R}(M):=\mathcal{Z}(R)\mathcal{W}_{R}\subseteq
C^{1/2}(M,\mathrm{d})$. The vector space $\mathcal{H}_{R}(M)$ becomes a
Hilbert space by declaring
$\mathcal{Z}(R):\mathcal{W}_{R}(M)\to\mathcal{H}_{R}(M)$ to be a unitary
isomorphism. Equation (5) shows that there is a canonical identification
$\mathcal{H}_{R}(M)=\mathcal{W}_{R}(M)^{*}$ given by a pairing

(6) $\langle
f,\mu\rangle_{L^{2},R}:=\langle\mathcal{Z}(R)^{-1}f,\mu\rangle_{\mathcal{W}_{R}}=\langle
f,\mathcal{Z}(R)\mu\rangle_{\mathcal{H}_{R}},\quad
f\in\mathcal{H}_{R}(M),\;u\in\mathcal{W}_{R}(M).$

We call this pairing the $L^{2}$-pairing because, when $M$ is a manifold, the
$L^{2}$-pairing is independent of $R$ and coincides with the ordinary
$L^{2}$-pairing. By duality, for any compact $X\subseteq M$ the inclusion
$\mathcal{W}_{R}(X)\subseteq\mathcal{W}_{R}(M)$ induces a restriction mapping

$\mathcal{H}_{R}(M)\to\mathcal{H}_{R}(X).$

Since $\mathcal{W}_{R}(X)\subseteq W_{R}(M)$ as Hilbert spaces, the
restriction mapping $\mathcal{H}_{R}(M)\to\mathcal{H}_{R}(X)$ is a co-
isometry. By construction, we have a commuting diagram

$\begin{CD}\mathcal{W}_{R}(X)@>{\mathcal{Z}(R)}>{}>\mathcal{H}_{R}(X)\\\
@V{}V{}V@A{}A{}A\\\
\mathcal{W}_{R}(M)@>{\mathcal{Z}(R)}>{}>\mathcal{H}_{R}(M)\end{CD}$

We note that since $\mathcal{H}_{R}\subseteq C^{1/2}(M,\mathrm{d})$ is a
continuous inclusion, for any compact $X\subseteq M$ and $h\in\mathcal{H}_{R}$
the restriction $h|_{X}\in C^{1/2}(X,\mathrm{d})$ is well defined.

######  Theorem 4.3 (Section 3 of [24]).

Let $X\subseteq M$ be a compact subset and $R>0$ such that
$(M,R\cdot\mathrm{d})$ is positive definite. If there is an element
$u_{R}\in\mathcal{W}_{R}(X)$ such that
$h_{R}:=R\mathcal{Z}(R)\mu_{R}\in\mathcal{H}_{R}(M)$ satisfies

$h_{R}|_{X}=1,$

then

$\mathrm{mag}(X,R\cdot\mathrm{d})=\|u_{R}\|_{\mathcal{W}_{R}}^{2}.$

Since $\mathcal{W}_{R}(X)\subseteq\mathcal{W}_{R}(M)$, we trivially have that

(7)
$\|u\|_{\mathcal{W}_{R}(M)}^{2}=\|u\|_{\mathcal{W}_{R}(X)}^{2},\quad\forall
u\in\mathcal{W}_{R}(X).$

We note the following corollary that will play an important role for
applications to manifolds (that possibly have boundary).

######  Corollary 4.4.

Assume $(M,\mathrm{d})$ is a compact metric space and that $R_{0}>0$ is such
that $(M,R\cdot\mathrm{d})$ is positive definite for all $R>R_{0}$. Let
$X\subseteq M$ be a compact subset and assume the following:

  1. (1)

There are Hilbert spaces $\mathcal{H}(M)$, $\mathcal{H}(X)$, $\mathcal{W}(M)$
and $\mathcal{W}(X)$ such that for all $R>R_{0}$,

$\mathcal{H}_{R}(M)=\mathcal{H}(M),\;\mathcal{H}_{R}(X)=\mathcal{H}(X),\;\mathcal{W}_{R}(M)=\mathcal{W}(M)\;\mbox{and}\;\mathcal{W}_{R}(X)=\mathcal{W}(X),$

as vector spaces with equivalent norms.

  2. (2)

The space $\mathcal{H}(X)\subseteq C^{1/2}(X,\mathrm{d})$ contains the
constant function (the inclusion is induced from item (1)).

  3. (3)

The $L^{2}$-pairings between $\mathcal{W}(X)$ and $\mathcal{H}(X)$, and
between $\mathcal{W}(M)$ and $\mathcal{H}(M)$ induced from item (1) above are
independent of $R$.

Writing $\mathcal{Z}_{X}(R)$ for the operator
$\mathcal{W}(X)\to\mathcal{H}(X)$ induced from $\mathcal{Z}(R)$, we then have
that $\mathcal{Z}_{X}(R)$ is invertible for $R>R_{0}$ and the magnitude
function is given by

(8) $\mathcal{M}_{X}(R)=R^{-1}\langle
1,\mathcal{Z}_{X}(R)^{-1}1\rangle_{L^{2}},\quad\mbox{for $R>R_{0}$}.$

Moreover, if there is a connected domain
$\mathbb{R}_{>R_{0}}\subseteq\Gamma\subseteq\mathbb{C}$ such that
$\mathbb{R}_{>R_{0}}\ni
R\mapsto\mathcal{Z}_{X}(R)\in\mathbb{B}(\mathcal{W}(X),\mathcal{H}(X))$
extends to a holomorphic Fredholm valued function
$\Gamma\to\mathrm{Fred}(\mathcal{W}(X),\mathcal{H}(X))$, then the magnitude
function $\mathcal{M}_{X}$ extends to a meromorphic function on $\Gamma$.

Item (3) listed in the assumptions of Corollary 4.4 is purely cosmetic and
ensures that the $L^{2}$-pairing in Equation (9) does not depend on $R$. For
context, the reader should note that when $M$ is a manifold and $X\subseteq M$
is a smooth domain we can take $\mathcal{W}$ and $\mathcal{H}$ to be certain
Sobolev spaces by Theorem 4.6 below.

###### Proof.

Since $(M,R\cdot\mathrm{d})$ is positive definite for $R>R_{0}$, the operator
$\mathcal{Z}_{X}(R)$ is well defined and invertible using item (1). By item
(2), the constant function $1$ on $X$ is an element of $\mathcal{H}(X)$ and we
can define $u_{R}:=R^{-1}\mathcal{Z}_{X}(R)^{-1}1\in\mathcal{W}(X)$. Note that
by item (1), $u_{R}\in\mathcal{W}_{R}(X)$ for all $R>R_{0}$. By Theorem 4.3
and Equation (6), we have that

$\mathcal{M}_{X}(R)=\|u_{R}\|_{\mathcal{W}_{R}(M)}^{2}=R^{-1}\langle
1,u_{R}\rangle_{L^{2},R}=R^{-1}\langle
1,\mathcal{Z}_{X}(R)^{-1}1\rangle_{L^{2}}$

In the last equality we used item (3) to remove the dependence of $R$ in the
pairing.

The statement concerning the meromorphic extension of the magnitude function
follows from the meromorphic Fredholm theorem which shows that
$\mathbb{R}_{>R_{0}}\ni
R\mapsto\mathcal{Z}_{X}(R)^{-1}\in\mathbb{B}(\mathcal{H}(X),\mathcal{W}(X))$
extends to a meromorphic function
$\Gamma\to\mathbb{B}(\mathcal{H}(X),\mathcal{W}(X))$ which extends the
function $\mathcal{M}_{X}(R)=R^{-1}\langle
1,\mathcal{Z}_{X}(R)^{-1}1\rangle_{L^{2}}$ to a meromorphic function of
$R\in\Gamma$. ∎

######  Remark 4.5.

In the work [12] the notions of distance functions having property (MR) and
(SMR) were introduced. We refer the reader to [12, Section 3] for full
details. The reader should keep in mind that (SMR)$\Rightarrow$ (MR) and that
both (MR) and (SMR) are inherited by smooth subdomains. Domains and subspace
distances in Euclidean space satisfy property (SMR). More generally, a compact
subdomain or a subspace $X$ of a manifold $M$ satisfies (SMR) if (SMR) holds
on $M$, e.g. if the restriction of $\mathrm{d}^{2}$ to $X\times X$ is smooth.
For the geodisic distance this occurs when the diameter of $X$ is smaller than
the injectivity radius of $M$. The sphere $S^{n}$ with its geodesic distance
has property (MR) but not (SMR) by [12, Proposition 3.6]. Tori and real
projective space with their geodesic distances fail to satisfy (MR) in
dimension $>1$ by [12, Proposition 3.17]. In the special cases we know that
(MR) fails, the magnitude asymptotics can nevertheless be computed by the same
formalism as in Theorem 2.1, because Theorem 5.10 below applies.

To study the operator $\mathcal{Z}_{X}$ and the magnitude for manifolds,
potentially with boundary, we introduce the following scales of Sobolev
spaces. Let $X$ be a compact domain in a manifold $M$. For $s\in\mathbb{R}$,
write

$\dot{H}^{s}(X):=\\{u\in H^{s}(M):\mathrm{supp}(u)\subseteq
X\\},\quad\mbox{and}\quad\overline{H}^{s}(X):=H^{s}(M)/\dot{H}^{s}(M\setminus
X).$

Here $H^{s}(M)$ is the Sobolev space of order $s$ on $M$. If $M$ is compact,
$H^{s}(M)$ is unambiguously defined. If $M$ is non-compact, we can either
define the Sobolev scale in terms of a Riemannian structure or by replacing
the Sobolev spaces by local Sobolev spaces: the definitions of
$\dot{H}^{s}(X)$, $\overline{H}^{s}(X)$ on the compact domain $X$ do not
depend on this choice. We note that for $s=0$,
$\dot{H}^{0}(X)=\overline{H}^{0}(X)=L^{2}(X)$. The $L^{2}$-pairing between
$\dot{H}^{s}(X)$ and $\overline{H}^{-s}(X)$ is a perfect pairing inducing an
isomorphism $\dot{H}^{s}(X)^{*}\cong\overline{H}^{-s}(X)$. We recall a
relevant theorem from [12].

######  Theorem 4.6.

Let $X$ be an $n$-dimensional compact manifold with $C^{0}$-boundary and a
distance function $\mathrm{d}$ satisfying property (MR). Set $\mu:=(n+1)/2$.
Then there exists an $R_{0}>0$ such that

$\mathcal{Z}_{X}(R):\dot{H}^{-\mu}(X)\to\overline{H}^{\mu}(X),$

is a well defined invertible operator for all $R\in\Gamma$ with
$\arg(R)<\pi/(n+1)$ and $\mathrm{Re}(R)>R_{0}$. Moreover, $R_{0}$ can be
chosen so that the following holds:

  1. a)

There is a $C>0$ such that

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

for $R\in\\{R\in\Gamma:\arg(R)<\pi/(n+1)\;\mbox{and}\;\mathrm{Re}(R)>R_{0}\\}$
and $f\in\dot{H}^{-\mu}(X)$.

  2. b)

For $R>R_{0}$, the sesquilinear form
$R^{-1}\langle\cdot,\cdot\rangle_{\mathcal{W}_{R}}$ is uniformly equivalent to
the inner product of $\dot{H}^{-\mu}_{R}(X)$. In particular, for $R>R_{0}$:

     * •

The metric space $(X,R\cdot\mathrm{d})$ is positive definite.

     * •

We have the equalities
$\mathcal{W}_{R}(X)=\dot{H}^{-\mu}_{R}(X)=\dot{H}^{-\mu}(X)$ and
$\mathcal{H}_{R}(X)=\overline{H}^{\mu}_{R}(X)=\overline{H}^{\mu}(X)$ as vector
spaces with equivalent norms.

Moreover, if $\mathrm{d}$ has property (SMR) on a sector $\Gamma$ the operator
$\mathcal{Z}_{X}(R):\dot{H}^{-\mu}(X)\to\overline{H}^{\mu}(X)$ depends
holomorphically on $R\in\Gamma$ and
$\mathcal{Z}_{X}(R)^{-1}:\overline{H}^{\mu}(X)\to\dot{H}^{-\mu}(X)$ depends
holomorphically on
$R\in\\{R\in\Gamma:\arg(R)<\pi/(n+1)\;\mbox{and}\;\mathrm{Re}(R)>R_{0}\\}$
with a meromorphic extension to $R\in\Gamma$.

######  Remark 4.7.

As mentioned above, and proven in [12, Proposition 3.4], a distance function
$\mathrm{d}$ obtained by pulling back the Euclidean distance function along an
embedding $X\hookrightarrow\mathbb{R}^{N}$, has property $(SMR)$ on
$\mathbb{C}\setminus\\{0\\}$. In particular, for such distance functions the
operator

$\mathcal{Z}_{X}(R)^{-1}:\overline{H}^{\mu}(X)\to\dot{H}^{-\mu}(X),$

has a meromorphic extension to $R\in\mathbb{C}\setminus\\{0\\}$.

For the full proof of Theorem 4.6 we refer to [12, Section 4], but we sketch
the idea here. The idea in the proof is to use that property $(SMR)$ allows us
to replace $\mathcal{Z}_{X}$ with a localization $Q_{X}$ to an operator whose
integral kernel is supported near the diagonal. The operator $Q_{X}$ is an
elliptic pseudodifferential operator with parameter $R$ of order $-n-1$, we
discuss $Q_{X}$ in further detail below in Theorem 5.3. A computation of the
principal symbol of $Q_{X}$ shows that $Q_{X}$ is a lower order perturbation
of a fractional Laplacian with parameter $(R^{2}+\Delta)^{-(n+1)/2}$ – the
Laplacian comes from a Riemannian metric defined from the Taylor expansion of
the distance function and the fractional power is defined relative to an
extension to an ambient manifold. Therefore item b) follows from the Gårding
inequality. Item a) follows from item b) using the meromorphic Fredholm
theorem. Item c) is a direct consequence of item b).

######  Corollary 4.8.

Let $X$ be an $n$-dimensional compact manifold with $C^{0}$-boundary and a
distance function $\mathrm{d}$ satisfying property (MR) on a sector $\Gamma$
with non-trivial intersection with $[0,\infty)$. Then the magnitude function
is given by

(9) $\mathcal{M}_{X}(R)=R^{-1}\langle
1,\mathcal{Z}_{X}(R)^{-1}1\rangle_{L^{2}},\quad\mbox{for $R>R_{0}$},$

where $\mathcal{Z}_{X}(R):\dot{H}^{-\mu}(X)\to\overline{H}^{\mu}(X)$. If
moreover $\mathrm{d}$ satisfies property (SMR) on $\Gamma$, the magnitude
function $\mathcal{M}_{X}$ extends meromorphically to $\Gamma$ and is
holomorphic in the sector
$\\{R\in\Gamma:\arg(R)<\pi/(n+1)\;\mbox{and}\;\mathrm{Re}(R)>R_{0}\\}$ for
some $R_{0}>0$.

In particular, if $X\subseteq\mathbb{R}^{N}$ is a compact submanifold with
$C^{0}$-boundary (e.g. a domain with $C^{0}$-boundary) with the subspace
distance, then $\mathcal{M}_{X}$ extends meromorphically to
$\mathbb{C}\setminus\\{0\\}$ and can be computed from (9).

##  5\. The operator $\mathcal{Z}$ on a closed manifold

To better understand the operator $\mathcal{Z}$ we first consider the case of
a compact manifold $M$. We here give an informal review of the technical
considerations in the paper by [12]. As discussed in the previous section,
there are complications arising from the fact that distance function might be
non-smooth away from the diagonal despite being quite regular at the diagonal.

###  5.1. Localizing to the diagonal

We decompose the operator $\mathcal{Z}$ in a part near the diagonal and an
off-diagonal remainder:

$\mathcal{Z}=Q+L,$

where

(10)
$Q(R)f:=\frac{1}{R}\int_{M}\chi(x,y)\mathrm{e}^{-R\mathrm{d}(x,y)}f(y)\mathrm{d}y,$

and $\chi\in C^{\infty}(M\times M)$ satisfies $\chi=1$ near the diagonal
$\mathrm{Diag}_{M}:=\\{(x,x):x\in M\\}\subseteq M\times M$. We first study the
operator $Q$ and return below to study the remainder term $L$. In order to
control $Q$, we impose the following condition on the distance function
$\mathrm{d}$.

######  Definition 5.1.

Let $\mathrm{d}$ be a distance function on a manifold $M$. We say that
$\mathrm{d}$ is _regular at the diagonal_ if its square
$G:=\mathrm{d}^{2}:M\times M\to[0,\infty)$ satisfies that there is a
neighborhood $U$ of the diagonal $\mathrm{Diag}_{M}\subseteq M\times M$ such
that:

  * •

$G$ restricts to a smooth function on $U$;

  * •

$\mathrm{d}G$ vanishes on $\mathrm{Diag}_{M}$; and,

  * •

for each $x\in\mathrm{Diag}_{M}$, the transversal Hessian $H_{G}$ of $G$ in
$x$ is positively definite.

If $\mathrm{d}$ is regular at the diagonal, we write $g_{\mathrm{d}}$ for the
Riemannian metric on $T^{*}M$ dual to the transversal Hessian $H_{G}$.

######  Remark 5.2.

Examples of distance functions regular at the diagonal include subspace
distances on submanifolds in Euclidean space (see [12, Example 2.17]) or
geodesic distance functions on Riemannian manifolds (see [12, Example 2.18]).
Then the Riemannian metric coincides with $g_{\mathrm{d}}$.

Consider a distance function $\mathrm{d}$ regular at the diagonal. It follows
from Taylor’s theorem that for any $N\in\mathbb{N}$ in local coordinates on a
neighborhood $U_{0}$ we can write

(11)
$\mathrm{d}(x,y)^{2}=H_{\mathrm{d}^{2}}(x,x-y)+\sum_{j=3}^{N}C^{(j)}(x,x-y)+r_{N}(x,x-y),$

where $r_{N}$ is a smooth function with $r_{N}(x,v)=O(|v|^{N+1})$ as $v\to 0$,
$H_{\mathrm{d}^{2}}$ is the transversal Hessian of $\mathrm{d}^{2}$, and
$C^{(j)}:U_{0}\to\mathrm{Sym}^{j}(T^{*}M|_{U_{0}})$ are the Taylor
coefficients forming a locally defined symmetric $j$-form on $TM|_{U_{0}}$.

In the symbol computation of $Q$, the Taylor coefficients are used as
differential operators in the cotangent variable. For a $k\in\mathbb{N}_{+}$
and a multiindex $\gamma\in\mathbb{N}^{k}_{\geq 3}$, we can define a
differential operator $C^{(\gamma)}(x,D_{\xi})$ on $T^{*}M|_{U_{0}}$ defined
by

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

Here $D_{\xi}=-i\frac{\partial}{\partial\xi}$. The order of
$C^{(\gamma)}(x,-D_{\xi})$ 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 $\gamma\in\mathbb{N}^{k}$, we set $\mathrm{rk}(\gamma):=k$. In other
words, $\gamma\in\cup_{k}\mathbb{N}^{k}_{\geq 3}$ 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
$I_{j}\subseteq\cup_{k=1}^{\infty}\mathbb{N}^{k}_{\geq 3}$.

######  Theorem 5.3.

Let $M$ be an $n$-dimensional manifold equipped with a distance function
$\mathrm{d}$ regular at the diagonal. Consider the operator $Q$ from Equation
(10). Then $Q\in\Psi^{-n-1}_{\rm cl}(M;\mathbb{C}_{+})$ is elliptic with
parameter $R\in\mathbb{C}_{+}:=\\{R:\mathrm{Re}(R)>0\\}$. The principal symbol
of $Q$ is given by

$\sigma_{-n-1}(Q)(x,\xi,R)=n!\omega_{n}(R^{2}+g_{\mathrm{d}}(\xi,\xi))^{-(n+1)/2},$

where $\omega_{n}$ denotes the volume of the $n$-dimensional unit ball.

In local coordinates, the full symbol of $Q$ admits a classical asymptotic
expansion $\sum_{j=0}^{\infty}q_{j}$ where the $-n-1-j$-homogeneous functions
$q_{j}\in C^{\infty}((T^{*}M\oplus\mathbb{C}_{+})\setminus(M\times\\{0\\}))$
are given by $q_{0}=\sigma_{-n-1}(Q)(x,\xi,R)$ and for $j>0$, $q_{j}$ is in
coordinates given by

$\displaystyle q_{j}(x,\xi,R)=$ $\displaystyle\begin{cases}&\sum_{\gamma\in
I_{j},\mathrm{rk}(\gamma)<(n+1)/2}\mathfrak{c}_{\mathrm{rk}(\gamma),n}C^{(\gamma)}(x,-D_{\xi})(R^{2}+g_{\mathrm{d}}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)}+\qquad\mbox{for
$n$ odd}\\\ &-\sum_{\gamma\in
I_{j},\mathrm{rk}(\gamma)\geq(n+1)/2}\mathfrak{c}_{\mathrm{rk}(\gamma),n}C^{(\gamma)}(x,-D_{\xi})\left[(R^{2}+g_{\mathrm{d}}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)}\log(R^{2}+g_{G}(\xi,\xi))\right],\\\
{}\\\ &\sum_{\gamma\in
I_{j}}\mathfrak{c}_{\mathrm{rk}(\gamma),n}C^{(\gamma)}(x,-D_{\xi})(R^{2}+g_{\mathrm{d}}(\xi,\xi))^{-(n+1)/2+\mathrm{rk}(\gamma)},\
\mbox{for $n$ even.}\end{cases}$

The coefficients $\mathfrak{c}_{k,n}$ are given by

$\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}$

###### Sketch of proof.

For the full proof, see [12, Theorem 2.9]. We follow the notation of [12]. The
ideas in the proof rely on elementary techniques of calculus, and we recall
the salient features. We extend $Q$ to a function of $R\in\mathbb{C}\setminus
i\mathbb{R}$ by declaring $Q$ to be even in $R$. By taking a Fourier transform
in the $R$-direction, the Schwarz kernel of $Q$ is transformed to the
distribution

$K(x,y,\eta)=\chi(x,y)K_{0}(x,y,\eta),\quad\mbox{where}\quad
K_{0}(x,y,\eta):=-\log(\eta^{2}+g_{0}(x,y)).$

The statement of the theorem is local in nature, so it suffices to compute
with $K_{0}$ in local coordinates. Standard considerations show that $K_{0}$
is a conormal distribution $K_{0}\in
CI^{-n-1}(Z;\mathrm{Diag}_{M}\times\\{0\\})$ where $Z=U\times\mathbb{R}$ is a
neighborhood of $\mathrm{Diag}_{M}\times\mathbb{R}\subseteq M\times
M\times\mathbb{R}$. The arguments in [12, Theorem 2.9] ensures that $Q$ indeed
is a pseudodifferential operator with parameter of order $-n-1$ with the
prescribed principal symbol.

The computationally delicate issue is that of finding the full symbol in local
coordinates. We do so by expanding $K_{0}$ near $x=y$ and inverse transform in
$(x-y,\eta)$ to the symbol depending on $(\xi,R)$. Using the Taylor expansion
(11), with $v=x-y$, we write

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

For small $v=x-y$, we can Taylor expand

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

Each term in the second sum $\sum_{\gamma\in
I_{j}}\frac{(-1)^{\mathrm{rk}(\gamma)+1}}{\mathrm{rk}(\gamma)}\frac{C^{(\gamma)}(x,v)}{(\eta^{2}+H_{\mathrm{d}}(x,v))^{\mathrm{rk}(\gamma)}}$
is homogeneous of degree $j$. The error term $\tilde{r}_{n}$ is controlled in
[12]. To compute the full symbol we now compute the inverse Fourier transform
in $(v,\eta)$. For $l>0$, denote the Fourier transform of
$(\eta^{2}+H_{\mathrm{d}}(v,v))^{-l}$ in the $(v,\eta)$-direction by
$F_{l}(x,\xi,R)$. Using homogeneity and rotational invariance, we have that

$\displaystyle F_{l}(x,\xi,R):=$
$\displaystyle\int_{T_{x}M\oplus\mathbb{R}}\frac{\mathrm{e}^{-i\xi.v-iR\eta}}{(\eta^{2}+H_{\mathrm{d}}(v,v))^{l}}\mathrm{d}v\mathrm{d}\eta=$
$\displaystyle=$
$\displaystyle\begin{cases}\kappa_{n,l}(R^{2}+g_{\mathrm{d}}(\xi,\xi))^{-\frac{n+1}{2}+l},\quad&2l-n-1\notin
2\mathbb{N},\\\
(R^{2}+g_{\mathrm{d}}(\xi,\xi))^{-\frac{n+1}{2}+l}(\kappa_{n,l}\log(R^{2}+g_{\mathrm{d}}(\xi,\xi))+\beta_{n,l}),\quad&2l-n-1\in
2\mathbb{N},\end{cases}$

for suitable constants $\kappa_{n,l},\beta_{n,l}\in\mathbb{R}$ that are
explicitly computed in [12]. It follows that for $\gamma\in I_{j}$, the
Fourier transform of the term

$\frac{C^{(\gamma)}(x,v)}{(\eta^{2}+H_{\mathrm{d}}(v,v))^{l}},$

in the $(v,\eta)$-direction is given by
$C^{(\gamma)}(x,-D_{\xi})F_{l}(x,\xi,R)$. A long computation putting all of
these terms together gives the full symbol computation. ∎

Theorem 5.3 has some rather direct consequences on distance functions that are
smooth off the diagonal. We note the following consequence.

######  Corollary 5.4.

Let $M$ be an $n$-dimensional compact manifold with a distance function
$\mathrm{d}$ regular at the diagonal and smooth off-diagonally, e.g. the
subspace distance on a submanifold of Euclidean space. Set $\mu:=(n+1)/2$.
Then $\mathcal{Z}$ is an elliptic pseudodifferential operator with parameter
$R$ and order $-n-1$, and $\mathcal{Z}-Q$ is smoothing with parameter $R>0$.
In particular, for a suitable $R_{0}>0$ and $R>R_{0}$ the operator

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

is invertible. Moreover,

$\|\mathcal{Z}(R)^{-1}-Q(R)^{-1}\|_{H^{\mu}(M)\to
H^{-\mu}(M)}=O(R^{-\infty}),$

and

(12) $\mathcal{M}_{M}(R)=R^{-1}\langle Q(R)^{-1}1,1\rangle+O(R^{-\infty}).$

###### Sketch of proof.

If $\mathrm{d}$ is smooth off-diagonally, then

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

is smoothing and exponentially decaying as $R\to+\infty$. We conclude that
$Z=Q+L$ is an elliptic pseudodifferential operator with parameter $R$ and
order $-n-1$ from Theorem 5.3. Invertibility of $\mathcal{Z}$ for large enough
$R$ follows from the ellipticity with parameter of $Q$. And indeed, since
$\mathcal{Z}-Q$ is smoothing with parameter, so is $\mathcal{Z}^{-1}-Q^{-1}$
and the norm estimate follows. Therefore,

$\langle\mathcal{Z}(R)^{-1}1,1\rangle=\langle
Q(R)^{-1}1,1\rangle+O(R^{-\infty}).$

The equality (12) follows as in Corollary 4.8. ∎

######  Remark 5.5.

The assumption in Corollary 5.4 that the distance function is smooth off-
diagonally can be weakened to the distance function having property $(MR)$.

###  5.2. Asymptotic expansions for compact manifolds

For a compact manifold, Corollary 5.4 provides means of computing the
asymptotics of the magnitude function. Starting from [12, Subsection 2.6] we
compute the asymptotics of $\langle Q(R)^{-1}1,1\rangle$ as $R\to\infty$ using
semiclassical analysis of the pseudodifferential operator with parameter
$Q(R)^{-1}$. We mention two instrumental results in this direction.

######  Lemma 5.6.

Consider a properly supported pseudodifferential operator with parameter
$A\in\Psi^{m}_{\rm cl}(M;\Gamma)$ on a manifold $M$. Setting

$a_{j,0}(x,R):=a_{j}(x,0,R),$

in each coordinate chart where $\sum_{j}a_{j}$ is a homogeneous expansion of
the full symbol of $A$ in that chart, produces a sequence
$(a_{j,0})_{j\in\mathbb{N}}\subseteq C^{\infty}(M\times\Gamma)$ of functions
such that

  1. (1)

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

  2. (2)

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

(13)
$[A(R)1](x)=\sum_{j=0}^{N}a_{j,0}(x,R)+r_{N}(x,R)=\sum_{j=0}^{N}a_{j}(x,1)R^{m-j}+r_{N}(x,R),$

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
$\mathrm{Re}(R)\to+\infty$}.$

The proof of this lemma can be found in [12, Lemma 2.24]. One way to see why
it is true, is to note that it is a local statement and can as such be reduced
to a statement for compactly supported pseudodifferential operators with
parameter in $\mathbb{R}^{n}$. After this reduction, it is a short computation
with the Fourier transform that

$a(x,D,R)1=a(x,0,R).$

######  Proposition 5.7.

Let $M$ be an $n$-dimensional compact manifold with a distance function
$\mathrm{d}$ regular at the diagonal and smooth off-diagonally, e.g. the
subspace distance on a submanifold of Euclidean space. Set $\mu:=(n+1)/2$.
Then for a suitable $R_{0}>0$ and $R>R_{0}$ the operator

$\mathcal{Z}(R)^{-1}:H^{\mu}(M)\to H^{-\mu}(M),$

is an elliptic pseudodifferential operator with parameter of order $n+1$. In
each coordinate chart, we can compute the full symbol of $\mathcal{Z}(R)^{-1}$
as an asymptotic sum $\sum_{k}a_{k}$ where $a_{0},a_{1},\ldots$ are computed
iteratively from

$a_{0}(x,\xi,R)=\frac{1}{n!\omega_{n}}(R^{2}+g_{\mathrm{d}}(\xi,\xi))^{(n+1)/2},$

and

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

The proof of this proposition and further details on the precise form of the
symbols can be found in [12, Theorem 2.21]. The construction is standard in
pseudodifferential calculus and produces a parametrix that by abstract
nonsense reproduces $\mathcal{Z}(R)^{-1}$ up to operators smoothing with
parameter. The crucial feature of Proposition 5.7 is that it produces a way of
explicitly computing an inverse to arbitrary level of precision – where the
precision manifests itself through Lemma 5.6 as to which order of $R$ we can
compute $\langle\mathcal{Z}(R)^{-1}1,1\rangle$.

######  Theorem 5.8.

Let $M$ be an $n$-dimensional compact manifold with a distance function
$\mathrm{d}$ whose square is regular at the diagonal. Let
$(a_{j,0})_{j\in\mathbb{N}}\subseteq C^{\infty}(M;\mathbb{C}_{+})$ denote the
sequence of homogeneous functions obtained from restriction to $\xi=0$ of the
full symbol of $Q_{M}^{-1}$, as in [12, Section 2.5]. It holds that

$\langle
1,Q_{M}(R)^{-1}1\rangle=\frac{1}{n!\omega_{n}}\sum_{k=0}^{\infty}c_{k}(M,\mathrm{d})R^{n+1-k}+O(\mathrm{Re}(R)^{-\infty}),\quad\mbox{as
$\mathrm{Re}(R)\to+\infty$},$

where

$c_{k}(M,\mathrm{d})=n!\omega_{n}\int_{M}a_{k,0}(x,1)\mathrm{d}x.$

Here $\mathrm{d}x$ is the Riemannian volume density defined from
$g_{\mathrm{d}^{2}}$. The functions $a_{k,0}(x,1)$ are explicitly constructed
inductively from the Taylor expansion (11) using Lemma 5.6 and Proposition
5.7. In particular,

$\displaystyle c_{k}(M,\mathrm{d})=\begin{cases}0,\;&\mbox{when $k$ is
odd,}\\\ \mathrm{vol}(M,g_{\mathrm{d}}),\;&\mbox{when $k=0$},\\\
\frac{n+1}{6}\int_{X}s_{\mathrm{d}^{2}}\mathrm{d}x,\;&\mbox{when
$k=2$},\end{cases}$

where $s_{\mathrm{d}^{2}}$ in local coordinates is computed as the polynomial
in the Taylor coefficients of $\mathrm{d}^{2}$ at the diagonal given as

$\displaystyle s_{\mathrm{d}^{2}}(x):=$ $\displaystyle 3C^{4}(x,g\otimes
g)-3\frac{\mathfrak{c}_{2,n}(n+5)(n^{2}-9)}{\mathfrak{c}_{1,n}}(C^{3}\otimes
C^{3})(x,g\otimes g\otimes g),\quad\mbox{if $n\neq 1,3$}$ $\displaystyle
s_{\mathrm{d}^{2}}(x):=$ $\displaystyle 3\bigg{(}10C^{4}_{G}(x,g_{G}\otimes
g_{G})-\frac{\mathfrak{c}_{2,3}}{\mathfrak{c}_{1,3}}(C^{3}_{G}\otimes
C^{3}_{G})(x,g_{G}\otimes g_{G}\otimes g_{G})\bigg{)},\quad\mbox{if $n=3$}$

Theorem 5.8 follows by combining Lemma 5.6 and Proposition 5.7. For more
details on the structure of the local densities $a_{k,0}$, see [12, Subsection
2.5 and Theorem 6.1]. The notation $s_{\mathrm{d}^{2}}$ in Theorem 5.8 is
justified by [12, Example 2.31], which shows that in the case of the geodesic
distance function on a Riemannian manifold $s_{\mathrm{d}^{2}}$ is the scalar
curvature. The following theorem is a direct consequence of Corollary 4.4,
Corollary 5.4 and Theorem 5.8.

######  Theorem 5.9.

Let $M$ be an $n$-dimensional compact manifold with a distance function
$\mathrm{d}$ with property (MR) on $\Gamma$, e.g. if $\mathrm{d}$ is regular
at the diagonal and smooth off-diagonally. It holds that

$\mathcal{M}_{M}(R)=\frac{1}{n!\omega_{n}}\sum_{k=0}^{\infty}c_{k}(M,\mathrm{d})R^{n-k}+O(\mathrm{Re}(R)^{-\infty}),\quad\mbox{as
$\mathrm{Re}(R)\to+\infty$ in $\Gamma$},$

where $c_{k}(M,\mathrm{d})$ is as in Theorem 5.8. The structure of the
coefficients $c_{j}$ is as follows:

  * •

$c_{0}(M,\mathrm{d})=\mathrm{vol}_{n}(M,g_{\mathrm{d}})$.

  * •

For $j\geq 1$, the coefficient $c_{j}(M,\mathrm{d})$ is an integral over $M$
of a polynomial (universally constructed from the Taylor expansion (11) of
$g_{0}:=\mathrm{d}^{2}$ at the diagonal) in the covariant derivatives of the
curvature of $M$ where the total degree is $\leq j$.

We also have the following corollary, which follows from considerations in
[12] even for manifolds which do not satisfy property (MR).

######  Theorem 5.10.

Let $M$ be a compact $n$-dimensional Riemannian manifold equipped with its
geodesic distance $\mathrm{d}$. Let $g$ denote its Riemannian metric and
$\mathrm{d}y$ the associated volume density. Assume that $M$ admits a family
$(u_{R})_{R>R_{0}}\subseteq\mathcal{D}^{\prime}(M)$ of distributional
solutions to

$\int_{M}\mathrm{e}^{-R\mathrm{d}(x,y)}u_{R}(y)\mathrm{d}y=1,\quad R>R_{0}.$

Assume that for any $\chi,\chi^{\prime}\in C^{\infty}(M)$ with disjoint
supports it holds that

$\int_{M}\chi(x)\mathrm{e}^{-R\mathrm{d}(x,y)}u_{R}(y)\chi^{\prime}(y)\mathrm{d}y=O(R^{-N}).$

Then the magnitude function $\mathcal{M}_{M}(R)$ is defined for $R>R_{0}$ and
there is an asymptotic expansion

$\displaystyle n!\omega_{n}\mathcal{M}_{M}(R)=$
$\displaystyle\mathrm{vol}_{n}(M,g)R^{n}+\frac{n+1}{6}\int_{M}s\mathrm{d}xR^{n-1}+$
$\displaystyle+\sum_{k=4}^{N-1}c_{k}(M)R^{n-k}+O(\mathrm{Re}(R)^{n-N}),\quad\mbox{as
$\mathrm{Re}(R)\to+\infty$},$

where $s$ denotes the scalar curvature of $M$, $c_{j}(M)=0$ if $j$ is odd and
for $j\geq 4$, the coefficient $c_{j}$ is an integral over $M$ of a universal
polynomial in the covariant derivatives of the curvature of $M$ where the
total degree is $\leq j$.

######  Remark 5.11.

We note that the assumption of Theorem 5.10 is automatically satisfied if the
distance function $\mathrm{d}$ satsfies property (MR). In this case we can
take $u_{R}=\mathcal{Z}(R)^{-1}1$, which is smooth by elliptic regularity,
using an argument analogous to that in Proposition 5.7. The authors of this
paper do not know of a Riemannian manifold not satisfying the assumption of
Theorem 5.10. Indeed, the examples in [12, Section 3] of Riemannian manifolds
which do not satisfy property (MR) do satisfy the assumption of Theorem 5.10
by the results of [29].

###### Proof of Theorem 5.10.

The assumptions on the existence of the distributional solution implies that
$\mathcal{M}_{M}(R)=\int_{M}u_{R}(y)\mathrm{d}y$ is defined for $R>R_{0}$. By
[12, Proposition 3.2], we have that

$\mathcal{M}_{M}(R)=\frac{1}{n!\omega_{n}}\sum_{k=0}^{N}c_{k}(M)R^{n-k}+O(R^{n-N}),$

where the coefficients are computed as in Theorem 5.8.

By well known invariant theory, $c_{j}(M)$ is an integral over $M$ of a
universal polynomial in the covariant derivatives of the curvature of $M$
where the total degree is $\leq j$. The only possible invariants in degree $0$
is the volume and in degree $2$ it is the integral of the scalar curvature.
The universality implies that $c_{0}(M)=\alpha_{0}\mathrm{vol}_{n}(M,g)$ and
$c_{2}(M)=\alpha_{2}\int_{M}s\mathrm{d}V$ for dimensional constants
$\alpha_{0}$ and $\alpha_{2}$. The values $\alpha_{0}=1$ and
$\alpha_{2}=\frac{n+1}{6}$ can be read of from works of Willerton [29, Theorem
11]. Since the Taylor expansion of the geodesic distance function near the
boundary only contains even degree terms, $c_{j}(M)=0$ if $j$ is odd. ∎

##  6\. The structure of the magnitude function for manifolds with boundary

Let $X$ be an $n$-dimensional manifold with boundary equipped with a distance
function. We assume that $X$ is a domain with smooth boundary in a manifold
$M$. Using a variation of Theorem 4.6, we know that

$\mathcal{Z}_{X}(R):\dot{H}^{-\mu}(X)\to\bar{H}^{\mu}(X),$

is an isomorphism for sufficiently large $R$, as soon as $\mathrm{d}$ has
property $(MR)$, see [12, Theorem 4.7]. This occurs for instance when
$\mathrm{d}$ is regular at the diagonal and smooth off-diagonally. By a
variation of Corollary 4.8, we know how to compute $\mathcal{M}_{X}$ from
$\mathcal{Z}_{X}(R)^{-1}$. The problem is to describe the inverse
$\mathcal{Z}_{X}(R)^{-1}$ in the presence of boundary. This issue was solved
in [12] using ideas of Wiener-Hopf factorization dating back to Eskin [6] and
Hörmander [18]. We first give a brief overview of the computational tools
entering into this construction, after which we compute the asymptotics of the
magnitude function.

###  6.1. Wiener-Hopf factorization and inverting $\mathcal{Z}_{X}$

In order to invert $\mathcal{Z}_{X}$ we use standard ideas of parametrix
constructions in pseudodifferential calculus: to invert $\mathcal{Z}_{X}$ we
invert it in the interior and near the boundary. By the arguments in the
preceding sections, it suffices to construct $Q_{X}^{-1}$ under property
$(MR)$. We shall see that the inverse takes the form

$Q_{X}^{-1}=\chi A\tilde{\chi}+\mathcal{W}_{X}+S,$

where $A$ is a pseudodifferential parametrix with parameter (constructed in
the same way as in Proposition 5.7) to the localized operator $Q$,
$\chi,\tilde{\chi}\in C^{\infty}_{c}(X^{\circ})$ are cut off functions,
$\mathcal{W}_{X}$ is constructed from an inversion procedure near the boundary
and finally $S$ is a remainder term whose contribution to the magnitude
asymptotics is negligible.

Let us describe how to construct the operator $\mathcal{W}_{X}$ near the
boundary. By localizing the problem, we can consider an associated model
operator

$Q^{\partial}:\dot{H}^{-\mu}(\partial
X\times[0,\infty))\to\bar{H}^{\mu}(\partial X\times[0,\infty)).$

The operator $Q^{\partial}$ is constructed as coinciding with $Q_{X}$ in a
collar neighborhood of the boundary and extended to $\partial
X\times[0,\infty)$ as a fractional Laplacian so that
$\sigma_{-n-1}(Q^{\partial})$ is of the form in Theorem 5.3 and $Q^{\partial}$
commutes with translations outside a compact subset. The problem will be to

  1. (1)

Up to suitable errors, factor $Q^{\partial}$ as operators

(14) $\displaystyle\dot{H}^{-\mu}(\partial
X\times[0,\infty))\xrightarrow{Q^{\partial}_{+}}$
$\displaystyle\dot{H}^{0}(\partial X\times[0,\infty))=$ $\displaystyle=$
$\displaystyle\bar{H}^{0}(\partial
X\times[0,\infty))\xrightarrow{Q^{\partial}_{-}}\bar{H}^{\mu}(\partial
X\times[0,\infty)),$

where $Q^{\partial}_{\pm}$ are operators of order $-\mu$. For this
factorization to make sense, we need that $Q^{\partial}_{+}$ comes from an
operator on $\partial X\times\mathbb{R}$ that preserves support in $\partial
X\times[0,\infty)$ and similarly that $Q^{\partial}_{-}$ comes from an
operator on $\partial X\times\mathbb{R}$ that preserve supports in $\partial
X\times(-\infty,0]$.

  2. (2)

For the factorization to produce a useful outcome, we need that
$Q^{\partial}_{\pm}$ have suitable ellipticity properties, and their
parametrices $W^{\partial}_{\pm}$ also need the same domain preservation
properties. If this is the case, we can invert $Q^{\partial}$ up to suitable
error terms as the composition

(15) $\displaystyle\bar{H}^{\mu}(\partial X\times[0,\infty))$
$\displaystyle\xrightarrow{W^{\partial}_{-}}\bar{H}^{0}(\partial
X\times[0,\infty))=$ $\displaystyle=$ $\displaystyle\dot{H}^{0}(\partial
X\times[0,\infty))\xrightarrow{W^{\partial}_{+}}\dot{H}^{-\mu}(\partial
X\times[0,\infty)).$

A well known, yet crucial, observation to solve these two problems is that
support preservation in a half-space is characterized by Paley-Wiener’s
theorem. We write coordinates on $\partial X\times[0,\infty)$ as
$(x^{\prime},x_{n})$ and cotangent directions as $(\xi^{\prime},\xi_{n})$. In
light of Paley-Wiener’s theorem, an operator $A=a(x,D)$ preserving supports in
$\partial X\times[0,\infty)$ is characterized by the symbol
$a=a(x,\xi^{\prime},\xi_{n})$ having a holomorphic extension in $\xi_{n}$ to
the lower half-plane.

The first item in the list above is by Paley-Wiener’s theorem solved by
Wiener-Hopf factorization of the full symbol of $Q^{\partial}$ into factors
holomorphically invertible in the upper and lower half-plane respectively. A
technical issue that arises is that the factorization takes place in a space
of _mixed-regularity symbols_ , for more details see [12, Section 5]. This
issue is visible when we factor the principal symbol as

$n!\omega_{n}(R^{2}+g_{\mathrm{d}}(\xi,\xi))^{-\mu}=q_{0,+}(x,\xi^{\prime},\xi_{n})q_{0,-}(x,\xi^{\prime},\xi_{n}),$

where

(16)
$\begin{cases}q_{+,0}(x,\xi^{\prime},\xi_{n})&=n!\omega_{n}(\xi_{n}-h_{+}(x,\xi^{\prime},R))^{-\mu}\\\
q_{-,0}(x,\xi^{\prime},\xi_{n})&=h_{0}(x)^{-\mu}(\xi_{n}-h_{+}(x,\xi^{\prime},R))^{-\mu}.\end{cases}$

Here $h_{0}$ is a smooth function and $h_{\pm}$ are smooth functions (for
$\xi^{\prime}\neq 0$) of degree $+1$ in $\xi^{\prime}$. The functions $h_{0}$
and $h_{\pm}$ come from the polynomial factorization

$R^{2}+g_{\mathrm{d}}(\xi,\xi)=h_{0}(x)(\xi_{n}-h_{+}(x,\xi^{\prime},R))(\xi_{n}-h_{-}(x,\xi^{\prime},R)),$

with $h_{+}(x,\xi^{\prime},R)$ being the complex root in the upper half-plane
to the second order polynomial equation
$R^{2}+g_{\mathrm{d}}(\xi^{\prime},\xi_{n})=0$ in $\xi_{n}$. Similarly,
$h_{-}(x,\xi^{\prime},R)$ is the complex root in the lower half-plane. The
functions $q_{0,\pm}$ are not ordinary Hörmander symbols since derivatives in
the $\xi^{\prime}$ direction generally only improve symbol decay in the
$\xi^{\prime}$-direction (and not in the $\xi_{n}$-direction). The symbols
appearing rather satisfy a mixed-regularity symbol estimate described in [12,
Section 5.1]. The theory of these mixed-regularity symbols seems known to
experts and can be found reviewed in [12, Section 5.1]. A crucial feature is
that the theory of mixed-regularity symbols fits well with parameter
dependence. In particular, operators of mixed-regularity $(m,-\infty)$ have
operator norm $O(R^{-\infty})$ as operators between Sobolev spaces $H^{s}\to
H^{s-m}$. Therefore, for the purpose of obtaining asymptotics this technical
issue has little impact on the output.

By an abuse of notation, we write $q\sim\sum_{j}q_{j}$ for the full symbol of
$Q^{\partial}$ in a coordinate chart so each $q_{j}$ is computed near
$\partial X\times\\{0\\}$ as in Theorem 5.3. We have the following result.

######  Theorem 6.1 (Wiener-Hopf factorization).

There exists mixed-regularity symbols $q_{+},q_{-}\in S^{-\mu,0}(\partial
X\times\mathbb{R};\mathbb{C}_{+})$ with parameter $R\in\mathbb{C}_{+}$ such
that

  1. (1)

We have that

$q\sim\sum_{\alpha}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}q_{-}D_{x}^{\alpha}q_{+},$

up to terms in $S^{2\mu,-\infty}(\partial X\times\mathbb{R};\mathbb{C}_{+})$.

  2. (2)

The mixed-regularity symbols $q_{+},q_{-}\in S^{-\mu,0}(\partial
X\times\mathbb{R};\mathbb{C}_{+})$ admits asymptotic expansions

$q_{\pm}\sim\sum_{j=0}^{\infty}q_{\pm,j},$

up to terms in $S^{-\mu,-\infty}(\partial X\times\mathbb{R};\mathbb{C}_{+})$,
where $q_{\pm,j}$ can be constructed inductively by a partial fraction
decomposition from $q\sim\sum_{j}q_{j}$, and $q_{\pm,0}$ is as in Equation
(16).

  3. (3)

The symbols $q_{\pm,j}$ extend holomorphically to the half-plane
$\pm\mathrm{Im}(\xi_{n})>0$ and for $j>0$ can be written as .

$q^{\partial}_{\pm,j}(x,\xi,R)=\sum_{k=-1}^{j-1}b_{\pm,j,k}(x,\xi^{\prime},R)(\xi_{n}-h_{\pm}(x,\xi^{\prime},R))^{-\mu-j+k}\in
S^{-\mu-1,-j+1},$

where $b_{\pm,j,k}$ is homogeneous of degree $-k$ in $(\xi^{\prime},R)$.

In particular, the operators $Q_{\pm}^{\partial}:=Op(q_{\pm})$ satisfies that
$Q^{\partial}_{+}$ preserve supports in $\partial X\times[0,\infty)$,
$Q^{\partial}_{-}$ preserve supports in $\partial X\times(-\infty,0]$ and the
two operators factorize $Q^{\partial}$ as in Equation (14) up to an operator
of mixed-regularity $(2\mu,-\infty)$, i.e.
$Q^{\partial}-Q_{-}^{\partial}Q_{+}^{\partial}\in\Psi^{2\mu,-\infty}(\partial
X\times\mathbb{R};\mathbb{C}_{+})$.

The idea of Wiener-Hopf factorization goes gack to Eskin. The reader can find
further details in the general case in Hörmander [18] or Grubb [15, Theorem
2.7]. Let us briefly indicate how the construction of the terms $q_{\pm,j}$
goes in the special case of interest to this work. The identity
$q\sim\sum_{\alpha}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}q_{-}D_{x}^{\alpha}q_{+}$
is equivalent to requiring

$\frac{q^{\partial}_{+,j}}{q^{\partial}_{+,0}}+\frac{q^{\partial}_{-,j}}{q^{\partial}_{-,0}}=\frac{q^{\partial}_{j}}{q^{\partial}_{0}}-\frac{1}{q^{\partial}_{0}}\sum_{\begin{subarray}{c}k+l+|\alpha|=j\\\
k,l<j\end{subarray}}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}q_{-,k}^{\partial}D^{\alpha}_{x}q_{+,l}^{\partial}$

Starting from (16) and the identity
$q\sim\sum_{\alpha}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}q_{-}D_{x}^{\alpha}q_{+}$
we can proceed to inductively determine $q_{\boldsymbol{,}j}$ from $q_{\pm,k}$
for $k<j$ by performing a partial fraction decomposition

(17)
$\frac{q^{\partial}_{j}}{q^{\partial}_{0}}-\frac{1}{q^{\partial}_{0}}\sum_{\begin{subarray}{c}k+l+|\alpha|=j\\\
k,l<j\end{subarray}}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}q_{-,k}^{\partial}D^{\alpha}_{x}q_{+,l}^{\partial}=\mathfrak{q}_{+,j}+\mathfrak{q}_{-,j}.$

Indeed, a careful analysis of the left hand side in (17) shows that it is a
rational function and all denominators are products involving $\xi_{n}-h_{+}$
and $\xi_{n}-h_{-}$. As such, a partial fraction decomposition gives that

$\mathfrak{q}_{\pm,j}(x,\xi,R)=\sum_{k=-1}^{j-1}\mathfrak{b}_{\pm,j,k}(x,\xi^{\prime},R)(\xi_{n}-h_{\pm}(x,\xi^{\prime},R))^{-j+k}\in
S^{-1,-j+1},$

where $\mathfrak{b}_{\pm,j,k}$ is homogeneous of degree $-k$ in
$(\xi^{\prime},R)$ and can be explicitly computed. We now define

$q^{\partial}_{\pm,j}:=q^{\partial}_{\pm,0}\mathfrak{q}_{\pm,j}.$

The reader should note that the Wiener-Hopf factorization relies on the
decomposition in (17); for more general symbols (such as in [18]) the
decomposition requires the Cauchy integral that leads to less explicit
factorizations. The reader can consult [12, Section 5] for an explicit
computation of $q_{\pm,1}$.

What is of interest for us is that the leading symbols in the operators
$Q_{\pm}^{\partial}$, namely $q_{\pm,0}$, are nowhere vanishing in their
domains of holomorphicity. We can therefore proceed in the usual way with a
parametrix construction.

######  Proposition 6.2.

The operators

$\dot{H}^{-\mu}(\partial
X\times[0,\infty))\xrightarrow{Q^{\partial}_{+}}\dot{H}^{0}(\partial
X\times[0,\infty)),\quad\mbox{and}\quad\bar{H}^{0}(\partial
X\times[0,\infty))\xrightarrow{Q^{\partial}_{-}}\bar{H}^{\mu}(\partial
X\times[0,\infty)),$

are invertible for large enough $R$. Up to a operators of mixed-regularity
$(-1,-\infty)$, the inverse of $Q^{\partial}_{\pm}$ can be computed as
$W_{\pm}^{\partial}:=Op(w_{\pm})$ where the mixed-regularity symbols
$w_{+},w_{-}\in S^{\mu,0}(\partial X\times\mathbb{R};\mathbb{C}_{+})$ admits
asymptotic expansions

$w_{\pm}\sim\sum_{j=0}^{\infty}w_{\pm,j},$

up to terms in $S^{\mu,-\infty}(\partial X\times\mathbb{R};\mathbb{C}_{+})$,
where $w_{\pm,j}$ can be constructed inductively from
$q_{\pm}\sim\sum_{j}q_{\pm,j}$ as

$w_{\pm,0}(x,\xi,R):=(q^{\partial}_{\pm,0})^{-1}=\begin{cases}\frac{1}{n!\omega_{n}}(\xi_{n}-h_{+}(x,\xi^{\prime},R))^{\mu},\;&\mbox{for
$+$},\\\ {}\\\
h_{0}(x)^{\mu}(\xi_{n}-h_{-}(x,\xi^{\prime},R))^{\mu},\;&\mbox{for
$-$},\end{cases}$

and

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

The symbols $w_{\pm,j}$ extend holomorphically to the half-plane
$\mp\mathrm{Im}(\xi_{n})>0$ and for $j>0$ can be written as

$w^{\partial}_{\pm,j}(x,\xi,R)=\sum_{k=-1}^{j-1}\mathfrak{w}_{\pm,j,k}(x,\xi^{\prime},R)(\xi_{n}-h_{\pm}(x,\xi^{\prime},R))^{\mu-j+k}\in
S^{\mu-1,-j+1},$

where $\mathfrak{w}_{\pm,j,k}$ is homogeneous of degree $-k$ in
$(\xi^{\prime},R)$.

We can now conclude that
$W^{\partial}:=W_{-}^{\partial}W_{+}^{\partial}:\bar{H}^{\mu}(\partial
X\times[0,\infty))\to\dot{H}^{-\mu}(\partial X\times[0,\infty))$ is well
defined and inverts $Q^{\partial}$ up to operators of mixed-regularity
$(0,-\infty)$ in the sense that

$1-W^{\partial}Q^{\partial},1-Q^{\partial}W^{\partial}\in\Psi^{0,-\infty}(\partial
X\times\mathbb{R};\mathbb{C}_{+}).$

Gluing the inverse of $Q^{\partial}$ together with the interior parametrix
produces the following description of the inverse of $Q_{X}$.

######  Theorem 6.3.

Let $X$ be a compact manifold with boundary and $\mathrm{d}$ a distance
function regular at the diagonal. The inverse of

$Q_{X}:\dot{H}^{-\mu}(X)\to\bar{H}^{\mu}(X),$

can be written as

$Q_{X}^{-1}=\chi_{1}A\tilde{\chi}_{1}+\chi_{2}W_{-}^{\partial}W_{+}^{\partial}\tilde{\chi}_{2}+S,$

where $A$ is a pseudodifferential parametrix to the localized operator $Q$ on
$X$, $\chi_{1},\tilde{\chi}_{1},1-\chi_{2},1-\tilde{\chi}_{2}\in
C^{\infty}_{c}(X^{\circ})$ are cut off functions, $W_{\pm}^{\partial}$ are
constructed as in Proposition 6.2 from parametrices of the Wiener-Hopf factors
$Q_{\pm}^{\partial}$ and $S\in\Psi^{-2\mu,-\infty}(X;\mathbb{C}_{+})$.

###  6.2. Computing the asymptotics in presence of boundary

The main consequence of Theorem 6.3 that we utilize for magnitude computations
is that we can write

(18) $\langle Q_{X}^{-1}1,1\rangle_{L^{2}(X)}=\langle
A1,\chi_{1}\rangle_{L^{2}(X)}+\langle
W_{-}1,(W_{+})^{*}\chi_{2}\rangle_{L^{2}(\partial
X\times[0,\infty))}+O(R^{-\infty}).$

The right hand side of the expression (18) is particularly tractable as both
$W_{-}$ and $(W_{+})^{*}$ preserve supports in $\partial X\times(-\infty,0]$.
Writing out the right hand side (18) in terms of the homogeneous expansion of
the symbol, we can partially integrate term by term in the asymptotic sum and
arrive at the following computation for the asymptotics of $\langle
Q_{X}^{-1}1,1\rangle_{L^{2}(X)}$.

######  Corollary 6.4.

Let $X$ be a compact manifold with boundary and $\mathrm{d}$ a distance
function regular at the diagonal. We have that

$\displaystyle\langle Q_{X}^{-1}1,1\rangle_{L^{2}(X)}=$
$\displaystyle\sum_{j}\left(\int_{X}a_{0,j}(x,1)\mathrm{d}x+\int_{\partial
X}B_{j,\mathrm{d}^{2}}(x)\mathrm{d}x\right)R^{n+1-j},$

where $a_{0,j}$ are the local densities computed from a parametrix of $Q$ as
in Lemma 5.6, and $(B_{\mathrm{d}^{2},j})_{j>0}\subseteq C^{\infty}(\partial
X)$ is the sequence of functions defined by

$B_{\mathrm{d}^{2},j}(x^{\prime}):=\sum_{\begin{subarray}{c}j=|\beta|+\gamma_{n}+k+l\\\
\gamma_{n}>0\end{subarray}}\frac{i^{|\beta|+|\gamma_{n}|}(-1)^{|\beta|+1}}{\beta^{\prime}!(\beta_{n}+\gamma_{n})!}\partial_{x}^{\beta}w_{-,k}(x^{\prime},0,0,1)\partial_{x_{n}}^{\gamma_{n}-1}\partial_{\xi}^{\beta+(0,\gamma_{n})}w_{+,l}(x^{\prime},0,0,1),$

for $j>0$ and $B_{0}:=0$.

From the local nature of the computations of asymptotics in Corollary 6.4, we
can conclude the following statement for the asymptotics of magnitude. The
details in the computations can be found in [12, Subsection 6.3] (see also
[12, Appendix F]).

######  Theorem 6.5.

Let $X$ be a compact $n$-dimensional manifold with boundary equipped with a
distance function $\mathrm{d}$ with property $(MR)$, e.g. if $\mathrm{d}$ is
regular at the diagonal and smooth off-diagonally. The magnitude function
$\mathcal{M}_{X}$ admits an asymptotic expansion

$\mathcal{M}_{X}(R)\ =\
\frac{1}{n!\omega_{n}}\sum_{j=0}^{\infty}c_{j}(X,\mathrm{d})R^{n-j}+O(R^{-\infty}),$

as $R\to\infty$ along the positive axis. Here $\omega_{n}$ denotes the volume
of the unit ball in $\mathbb{R}^{n}$. The structure of the coefficients
$c_{j}$ is as follows:

  * •

$c_{0}(X,\mathrm{d})=\mathrm{vol}_{n}(X,g_{\mathrm{d}})$ computed in the
Riemannian metric $g_{\mathrm{d}}$ induced from the transversal Hessian of the
distance function at the diagonal;

  * •

$c_{1}(X)=\mu\mathrm{vol}_{n-1}(\partial X,g_{\mathrm{d}})$ computed in the
Riemannian metric induced from $g_{\mathrm{d}}$;

  * •

$c_{2}(X)=\frac{n+1}{6}\int_{X}s_{\mathrm{d}}\mathrm{d}V+\frac{\mu^{2}(n-1)}{2}\int_{\partial
X}H_{\mathrm{d}}\mathrm{d}S$, where $s_{\mathrm{d}}$ is a scalar curvature
like function defined from $\mathrm{d}$ and $H_{\mathrm{d}}$ is a mean
curvature like term of the boundary.

  * •

For $j\geq 3$, we have that

$c_{j}(X,\mathrm{d})=n!\omega_{n}\int_{X}a_{j,0}(x,1)\mathrm{d}x+n!\omega_{n}\int_{\partial
X}B_{\mathrm{d}^{2},j}(x)\mathrm{d}x,$

where $a_{j,0}$ is a polynomial (universally constructed from the Taylor
expansion (11) of $\mathrm{d}^{2}$ at the diagonal) in the covariant
derivatives of the curvature of $X$ where the total degree is $\leq j$.

In the special case of smooth, compact domains $X\subseteq\mathbb{R}^{n}$, one
readily computes that $a_{j,0}=0$ for $j>0$ (compare to [12, Example 2.29]).
We can in particular deduce the following simplified asymptotics for smooth,
compact Euclidean domains. For the computation of $c_{3}(X)$ for odd
dimensions, see [11], and for $c_{3}(X)$ in dimension $n=2$ see Remark 1.2.

######  Theorem 6.6.

Let $X\subseteq\mathbb{R}^{n}$ be a compact domain with smooth boundary
equipped with Euclidean distance. Let $\mathcal{M}_{X}$ denote its magnitude
function. Then there are constants $c_{j}(X)$, $j=0,1,2,\ldots$, such that

$\mathcal{M}_{X}(R)\ \sim\
\frac{1}{n!\omega_{n}}\sum_{j=0}^{\infty}c_{j}(X)R^{n-j},$

as $R\to\infty$ along the positive axis. The structure of the coefficients
$c_{j}$ is as follows:

  * •

$c_{0}(X)=\mathrm{vol}_{n}(X)$

  * •

$c_{1}(X)=\mu\mathrm{vol}_{n-1}(\partial X)$

  * •

$c_{2}(X)=\frac{\mu^{2}(n-1)}{2}\int_{\partial X}H\mathrm{d}S$, where $H$ is
the mean curvature of the boundary.

  * •

If $n=2$ or $n$ is odd, $c_{3}(X)=\alpha_{3}(n)\int_{\partial
X}H^{2}\mathrm{d}S$ for a dimensional constant $\alpha_{3}(n)>0$.

  * •

For $j\geq 3$, we have that

$c_{j}(X,\mathrm{d})=n!\omega_{n}\int_{\partial
X}B_{\mathrm{d}^{2},j}(x)\mathrm{d}x,$

where a $B_{\mathrm{d}^{2},j}(x)$ is explicitly computable as in Corollary 6.4
and is given by a universal polynomial in covariant derivatives of the second
fundamental form of $\partial X$ of total degree $\leq j$.

##  7\. Some remarks about finer structures

###  7.1. Localization of solutions to $A1$ and boundary terms

Let us provide an immediate consequence of Lemma 5.6 that highlights how the
solution to the equation $R\mathcal{Z}(R)U=1$ inside a manifold can be
described explicitly by means of the symbol of the parametrix $A$. An
interesting situation to keep in mind is when $M=X^{\circ}$ is the interior of
a manifold with boundary.

######  Corollary 7.1.

Let $M$ be an $n$-dimensional manifold equipped with a distance function
$\mathrm{d}$ whose square is smooth and $\Gamma\subseteq\mathbb{C}_{+}$ a
sector with positive angle to $i\mathbb{R}$. Write
$(a_{j,0})_{j\in\mathbb{N}}\subseteq C^{\infty}(M\times\mathbb{C}_{+})$ for
the sequence defined as in Lemma 5.6 from $Q^{-1}\in\Psi^{n+1}_{\rm
cl}(M;\mathbb{C}_{+})$.

Assume that $(u_{R})_{R\in\Gamma}\subseteq\mathcal{D}^{\prime}(M)$ is a family
of weak solutions to

$\int_{M}\mathrm{e}^{-R\mathrm{d}(x,y)}u_{R}(y)\mathrm{d}y=1,\quad x\in M.$

Then $(u_{R})_{R\in\Gamma}\subseteq C^{\infty}(M)$ and for any
$N\in\mathbb{N}$ and any compact $K\subseteq M$, there is a $C>0$ such that

$\left|u_{R}(x)-\sum_{j=0}^{n+N}a_{j,0}(x,1)R^{n-j}\right|\leq
C(1+|R|)^{-N},\quad\mbox{for all $x\in K$}.$

######  Remark 7.2.

In the special case that $M=X^{\circ}$ is the interior of a compact domain in
$\mathbb{R}^{n}$ and $\mathrm{d}$ is the Euclidean distance then for any
$N\in\mathbb{N}$ and any compact $K\subseteq M$, there is a $C>0$ such that

$\left|u_{R}(x)-\frac{R^{n}}{n!\omega_{n}}\right|\leq
C(1+|R|)^{-N},\quad\mbox{for all $x\in K$}.$

We note that for $n$ being odd, the results of [24] show that
$u_{R}=\frac{1}{n!\omega_{n}}(R^{2}-\Delta)^{n+1}h_{R}$ for a uniquely
determined $h_{R}\in H^{(n+1)/2}(\mathbb{R}^{n})$ solving
$(R^{2}-\Delta)^{n+1}h_{R}=0$ in $\mathbb{R}^{n}\setminus X$ and satisfying
$h_{R}=1$ in $X^{\circ}$. Therefore

$u_{R}(x)=\frac{R^{n}}{n!\omega_{n}}\quad\mbox{for all $x\in X^{\circ}$}.$

Since $u_{R}\in H^{-(n+1)/2}(\mathbb{R}^{n})$ is supported in $X$,
$u_{R}-\frac{R^{n}}{n!\omega_{n}}$ is supported on $\partial X$. If $n$ is
even, it is not clear if $u_{R}-\frac{R^{n}}{n!\omega_{n}}$ is supported on
$\partial X$. However, by our result $u_{R}-\frac{R^{n}}{n!\omega_{n}}$ tends
to $0$ away from $\partial X$ faster than any polynomial as $R\to\infty$. This
generalizes [4, Theorem 5] to even dimensions, as relevant for a boundary
detection method in data science applications.

###  7.2. Taylor expansion at $R=0$

In light of Corollary 4.8, it is of interest to study series expansions of
solutions to $R\mathcal{Z}(R)u_{R}=1$ near $R=0$. Indeed, meromorphicity
ensures that a Taylor expansion of the magnitude function at $R=0$ determines
the magnitude function completely. Recent results of Meckes [25] compute the
derivative of the magnitude function of a convex body in $\mathbb{R}^{n}$ at
$R=0$ in terms of an intrinsic volume. The following result provides a partial
description of the Taylor expansion at $R=0$ under a constraint on the
distance function similar to property (MR).

######  Theorem 7.3.

Let $X$ be a compact manifold with $C^{0}$-boundary with a distance function
$\mathrm{d}$ such that $\mathrm{d}^{2}$ is regular at the diagonal and that
$L(R)$ defines a holomorphic family of operators

$L(R):\dot{H}^{-\mu}(X)\to\overline{H}^{\mu}(X),$

for $R$ in a punctured neighborhood of $R=0$, for instance if $\mathrm{d}^{2}$
is smooth. Assume furthermore that the operator

$\mathcal{Z}_{1}:\dot{H}^{-\mu}(X)\to\overline{H}^{\mu}(X),\quad\mathcal{Z}_{1}u(x):=\int_{X}\mathrm{d}(x,y)u(y)\mathrm{d}y$

is invertible. Then there is a unique holomorphic family
$(u_{R})_{|R|\leq\delta^{-1}}\subseteq\dot{H}^{-\mu}(X)$, for some $\delta>0$,
which solves the equation

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

The holomorphic family $(u_{R})_{R}$ admits an expansion at $R=0$ as an
absolutely norm-convergent power series
$u_{R}=\sum_{k=0}^{\infty}u_{k}R^{k}\in\dot{H}^{-\mu}(X)$. The coefficients
$(u_{k})_{k}\subseteq\dot{H}^{-\mu}(X)$ are constructed from

$u_{0}:=\lambda_{1}g,\quad\mbox{where}\quad
g:=\mathcal{Z}_{1}^{-1}(1)\quad\mbox{and}\quad\lambda_{1}:=\frac{1}{\int_{X}g\mathrm{d}x},$

defining $u_{k+1}$ and an auxiliary parameter $\lambda_{k+2}$ from
$u_{0},\ldots,u_{k}$ and $\lambda_{k+1}$ as

$u_{k+1}=\sum_{l=0}^{k}(-1)^{k-l}\mathcal{Z}_{1}^{-1}\mathcal{Z}_{k+2-l}u_{l}+\lambda_{k+2}g,\quad\mbox{and}\quad\lambda_{k+2}=\frac{\lambda_{k+1}-\sum_{l=0}^{k}(-1)^{k-l}\int_{X}\mathcal{Z}_{1}^{-1}\mathcal{Z}_{k+2-l}u_{l}\mathrm{d}x}{\int_{M}\mathcal{Z}_{1}^{-1}(1)\mathrm{d}x}$

where
$\mathcal{Z}_{k}u(x):=\frac{1}{k!}\int_{X}\mathrm{d}(x,y)^{k}u(y)\mathrm{d}y$,
$k\geq 0$.

###### Proof.

We can decompose

$\mathcal{Z}_{k}=Q_{k}+L_{k},$

where
$Q_{k}(R)u(x)=\frac{1}{k!}\int_{X}\chi(x,y)\mathrm{d}(x,y)^{k}u(y)\mathrm{d}y$
is an elliptic pseudodifferential operator of order $-n-k$ for odd $k$ and
smoothing for even $k$ by a similar argument as in Theorem 5.3. We introduce
the notation $L_{0}=0$ and

$\mathcal{Z}_{0}u(x)=Q_{0}u(x):=\int_{X}u(y)\mathrm{d}y,$

for the projection onto the constant function, which is a smoothing operator.
Moreover, we can write

$R\mathcal{Z}(R)=\sum_{k=0}^{\infty}(-1)^{k}R^{k}\mathcal{Z}_{k},$

and

$RQ(R)=\sum_{k=0}^{\infty}(-1)^{k}R^{k}Q_{k},\quad\mbox{and}\quad
RL(R)=\sum_{k=0}^{\infty}(-1)^{k}R^{k}L_{k}.$

A short argument bounding the norm on $Q_{k}$ with Calderon-Vaillancourt’s
theorem shows that the expansion for $RQ(R)$ is an absolutely convergent
Taylor expansion on a neighborhood of $R=0$. Since we have assumed that $L(R)$
is a holomorphic function, the expansion for $RL(R)$ and $R\mathcal{Z}(R)$ are
absolutely convergent Taylor expansions in a neighborhood of $R=0$.

We search for an absolutely norm-convergent power series solving
$R\mathcal{Z}(R)u_{R}=1$. We make the formal ansatz
$u_{R}=\sum_{k=0}^{\infty}u_{k}R^{k}$ in which $R\mathcal{Z}(R)u_{R}=1$ is
equivalent to $L_{0}u_{0}=1$ and

(19) $\sum_{l=0}^{k}(-1)^{k-l}\mathcal{Z}_{k-l}u_{l}=0,\quad k>0.$

Note that the range of $\mathcal{Z}_{0}$ consists only of constants. We write
$\lambda_{k}:=\mathcal{Z}_{0}u_{k}$, note that $\lambda_{0}=1$. We set
$g:=\mathcal{Z}_{1}^{-1}(1)$. We can rewrite Equation (19), using that
$\mathcal{Z}_{1}$ is invertible, as

$u_{k-1}=\sum_{l=0}^{k-2}(-1)^{k-l}\mathcal{Z}_{1}^{-1}\mathcal{Z}_{k-l}u_{l}+\lambda_{k}g.$

Therefore, $u_{k}$ can be determined inductively as follows. We find $u_{0}$
as $u_{0}=\lambda_{1}g$, and then we determine $\lambda_{1}$ from
$\lambda_{1}\mathcal{Z}_{0}(g)=1$. The induction step is obtained as follows:
say we have determined $u_{0},\ldots,u_{k}$ and
$\lambda_{0},\ldots,\lambda_{k+1}$ we then define $u_{k+1}$ as

$u_{k+1}=\sum_{l=0}^{k}(-1)^{k-l}\mathcal{Z}_{1}^{-1}\mathcal{Z}_{k+2-l}u_{l}+\lambda_{k+2}g,$

where $\lambda_{k+2}$ is defined from
$\lambda_{k+2}\mathcal{Z}_{0}(g)=\lambda_{k+1}-\sum_{l=0}^{k}(-1)^{k-l}\mathcal{Z}_{0}\mathcal{Z}_{1}^{-1}\mathcal{Z}_{k+2-l}u_{l}$.The
fact that the Taylor expansion of $R\mathcal{Z}(R)$ converges implies that the
series $u_{R}=\sum_{k=0}^{\infty}u_{k}R^{k}$ is absolutely norm convergent in
a neighborhood of $R=0$. ∎

######  Corollary 7.4.

Under the assumptions of Theorem 7.3, $\mathcal{M}_{X}(R)$ is holomorphic at
$R=0$ where it admits the Taylor series

$\mathcal{M}_{X}(R)=1+\sum_{k=1}^{\infty}\lambda_{k}R^{k},$

for the sequence $(\lambda_{k})_{k>0}$ constructed in Theorem 7.3.

######  Remark 7.5.

In Theorem 7.3, the result relies on the right hand side of the equation
$R\mathcal{Z}(R)u=1$ being the constant function. Indeed, since
$R\mathcal{Z}(R)$ evaluates to the projection onto the constant functions at
$R=0$, there is no holomorphic family of solutions to $R\mathcal{Z}(R)u=f$
unless $f$ is constant.

######  Problem 1.

For a compact domain $X\subseteq\mathbb{R}^{n}$, does it hold true that the
operator

$\mathcal{Z}_{1}:\dot{H}^{-\mu}(X)\to\overline{H}^{\mu}(X),\quad\mathcal{Z}_{1}u(x):=\int_{X}|x-y|u(y)\mathrm{d}y,$

is invertible? Standard techniques with Gårding inequalities show that
$\mathcal{Z}_{1}$ is a Fredholm operator with vanishing index. Indeed, we can
write

$\langle\mathcal{Z}_{1}u,u\rangle_{L^{2}(X)}=\int_{\mathbb{R}^{n}}g(\xi)|\hat{u}(\xi)|^{2}\mathrm{d}\xi,\quad
u\in\dot{H}^{-\mu}(X),$

where $g$ is the Fourier transform of $x\mapsto|x|$. By [12, Proposition A.6],
and homogeneity arguments, we have that

$g(\xi)=-\pi^{\frac{n-1}{2}}2^{n}\Gamma\left(\frac{n+1}{2}\right)\mathrm{F.P.}|\xi|^{-n-1}.$

Therefore this problem could be susceptible to explicit quadratic form
estimates. Were the problem to have a positive solution for the unit ball
$X=B_{n}$ for odd $n$, [25, Theorem 4] implies that

(20) $\int_{X}\mathcal{Z}_{1}^{-1}1\mathrm{d}x=\frac{2}{V_{1}(X)},$

where $V_{1}(X)$ denotes the first intrinsic volume of $X=B_{n}$. The claim in
[25, Conjecture 5] is that the identity (20) holds for all compact convex
domains $X$.

##  Appendix A Algorithm for computing $c_{j}(X,\mathrm{d})$

We recall the expansion of $\mathcal{M}_{X}(R)$ from preceding chapters, given
by

$\mathcal{M}_{X}(R)=\sum_{k=0}^{\infty}c_{k}(X,\mathrm{d})R^{n+1-k}+O(R^{-\infty}),$

provided $(X,\mathrm{d})$ satisfies the relevant assumptions. The approach in
this article leads to an algorithmic procedure for computing the coefficients
$c_{k}$, which we summarize here and detail in special geometric situations.
Each stage of this procedure has been already presented and proved in [12],
and a pseudocode may be found in Appendix B.

For the convenience of the reader, we repeat some preliminary computations and
definitions:

$I_{j}:=\\{\gamma\in\cup_{k=1}^{\infty}\mathbb{N}^{k}_{\geq
3}:|\gamma|=j+2k\\},\,\mathrm{rk}(\gamma)=k\text{ for
}\gamma\in\mathbb{N}^{k}$
$\mathfrak{c}_{k,n}:=\begin{cases}(-1)^{k}(n-2k)!\omega_{n-2k}\omega_{2k},\;&\mbox{for
$2k-n-1<0$}\\\
\frac{(-1)^{1-n/2}\omega_{2k}}{(2k-n)!\omega_{2k-n}},\;&\mbox{for $2k-n-1\in
2\mathbb{N}+1$}\\\
\frac{(-1)^{\frac{n+1}{2}}}{(2\pi)^{2k-n}}\omega_{2k}\omega_{2k-n-1},\;&\mbox{for
$2k-n-1\in 2\mathbb{N}$}\end{cases},$
$q_{k,p}(x,R).v:=\sum_{|\alpha|=p}\partial_{\xi}^{\alpha}q_{k}(x,\xi,R)v^{\alpha}|_{\xi=0}=\partial_{t}^{p}q_{k}(x,tv,R)|_{t=0};$

By $C^{(\gamma)}(x,-D_{\xi})$, $\gamma\in I_{j}$, we denote the $|\gamma|$-th
order differential operator arising from the Taylor expansion of
$\mathrm{d}^{2}:M\times M\to[0,\infty]$ (see (11)) and write $g_{\mathrm{d}}$
for the dual Riemannian metric to $H_{\mathrm{d}^{2}}$. Using these
definitions, we calculate $c_{k}$ in the following steps.

  1. (1)

Calculate terms $q_{j}$ in the expansion of $q$ for $j\in[0,k]$
$q$ denotes the symbol of the operator $Q_{G,\chi}$ and it has an asymptotic
expansion $q\sim\sum_{j}q_{j}$, see Theorem 5.3. This is the operator we use
to establish results corresponding to the interior of $X$.

The first term is given by

$q_{0}(x,\xi,R)=(R^{2}+g_{\mathrm{d}}(\xi,\xi))^{-(n+1)/2},$

and for $j>0$ and $n$ odd we have that

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

and for $n$ even

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

When $M\subseteq\mathbb{R}^{N}$ is an $n$-dimensional submanifold of
$\mathbb{R}^{N}$, we determine $C^{(\gamma)}$ in the following way: take
coordinates around some $x_{0}\in M$ 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 $\varphi_{N+1},\ldots,\varphi_{n}$. Writing
$x=(x_{1},\ldots,x_{N})$, we have that

$H_{\mathrm{d}^{2}}(v)=|v|^{2}+\sum_{l=n+1}^{N}\left(\nabla\varphi_{l}(x)\cdot
v\right)^{2}$

and

$C^{j}(x,v)=\sum_{l=n+1}^{N}\sum_{\begin{subarray}{c}|\alpha|+|\beta|=j,\\\
|\alpha|,|\beta|>0\end{subarray}}\frac{\partial_{x}^{\alpha}\varphi_{l}(x)\partial_{x}^{\beta}\varphi_{l}(x)}{\alpha!\beta!}v^{\alpha+\beta},\quad
j>2.$

This is explained in [12, Example 2.17].

In the special case that $X\subset M=\mathbb{R}^{n}$ is a domain and
$\mathrm{d}(x,y)=|x-y|$ the Euclidean distance, $q_{j}=0$ for $j>0$ and
consequently $q=q_{0}=(R^{2}+|\xi|^{2})^{-(n+1)/2}$.

  2. (2)

Calculate terms $a_{j}$ in the expansion of $a$ at $\xi=0$ for $j\in[0,k]$

By $A$ we denote the parametrix to $Q$ in the interior of $X$, and its symbol
is denoted by $a$. It has an asymptotic expansion $a\sim\sum_{j}a_{j}$, and at
$\xi=0$, we write $a_{j,0}(x,R):=a_{j}(x,0,R)$. As described in [12, Lemma
2.25] the first term is given by

$a_{0,0}(x,R)=\frac{R^{(n+1)/2}}{n!\omega_{n}},$

and for $j>0$ the terms are determined inductively by

$\displaystyle a_{j,0}(x,R)=\begin{cases}0&\mbox{$j$ odd},\\\
-\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)&\mbox{$j$
even}.\end{cases}$

In the case that $X$ is a domain in $\mathbb{R}^{n}$, $a_{j}=0$ for $j>0$
since $q=q_{0}$, thus implying in a similar manner that $a=a_{0}$.

In the case that $M$ is an $n$-dimensional submanifold of $\mathbb{R}^{N}$, we
use the same parametrisation by $\varphi_{n+1},\ldots,\varphi_{N}$ as in the
previous step, see [12, Example 2.30].

  3. (3)

Calculate the terms $q^{\partial}_{j}$ in the expansion of $q^{\partial}$ near
the boundary for $j\in[0,k-1]$
By $Q^{\partial}$ we denote the localization of $Q$ near the boundary of $X$,
and we denote its symbol by $q^{\partial}$, see Subsection 6.1 above or [12,
Proposition 5.11]. It has an asymptotic expansion
$q^{\partial}\sim\sum_{j}q_{j}^{\partial}$ where $q^{\partial}_{j}=q_{j}$
using the coordinates induced by localizing near the boundary.

When $X=M$ is a compact submanifold of $\mathbb{R}^{N}$ it is immediate that
there are no contributions from this operator as the boundary is empty.

For smooth domains, we construct $q_{j}^{\partial}$ via a choice of
coordinates as in the pseudocode of Example B.1 in Appendix B. In the presence
of a boundary when $X\subset M=\mathbb{R}^{n}$, write
$x=(x^{\prime},x_{n})\in\mathbb{R}^{n-1}\times\mathbb{R}$ (resp.
$\xi=(\xi^{\prime},\xi_{n})$) and determine $Q^{\partial}$ in the following
way: fix a point $x_{0}$ on the boundary where the normal vector is orthogonal
to the plane $x_{n}=0$. Without loss of generality, this point can be
$x_{0}=0$. Pick a smooth $\varphi$ on the boundary such that
$\varphi(x^{\prime})<x_{n}$ in a neighborhood of $x_{0}$ that belongs to $X$.
Make a change of coordinates to
$(x^{\prime},x_{n})\mapsto(x^{\prime},x_{n}-\varphi(x^{\prime}))$. This
procedure is described in more detail in [12, Example 2.16].

In these new coordinates,

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

and

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

  4. (4)

Factorize $Q^{\partial}$ near the boundary into terms supported in the upper
$(+)$ and lower $(-)$ half planes

The next thing we have to do is factorize $Q^{\partial}$ as $Q_{-}Q_{+}$,
where $Q_{\pm}$ is supported in the upper resp. lower half plane. We denote
the symbols of $Q_{\pm}$ by $q_{\pm}$ and they have an asymptotic expansion
$q_{\pm}\sim\sum_{j}q_{\pm,j}^{\partial}$, see Theorem 6.1 above or [12,
Subsection 5.2]. Start by defining

$\displaystyle q^{\partial}_{+,0}:=$ $\displaystyle
n!\omega_{n}h_{0}^{-\mu}(\xi_{n}-h_{+})^{-\mu},$ $\displaystyle
q^{\partial}_{-,0}:=$ $\displaystyle(\xi_{n}-h_{-})^{-\mu},$

and construct $q^{\partial}_{\pm,j}$ inductively by

(21)
$\displaystyle\frac{q^{\partial}_{j}}{q^{\partial}_{0}}-\frac{1}{q^{\partial}_{0}}\sum_{\begin{subarray}{c}k+l+|\alpha|=j\\\
k,l<j\end{subarray}}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}q_{+,k}^{\partial}D^{\alpha}_{x}q_{-,l}^{\partial}=\mathfrak{q}_{+,j}+\mathfrak{q}_{-,j},$
$q^{\partial}_{\pm,j}:=q^{\partial}_{\pm,0}\mathfrak{q}_{\pm,j}.$

Here, the terms $h_{0}(x)$ is the leading $\xi_{n}$-term in the metric and
$h_{\pm}(x,\xi^{\prime},R)$ are the zeros in the $\pm$-plane, that all in all
are determined from the equation

$R^{2}+g_{\mathrm{d}}(\xi,\xi)=h_{0}(\xi_{n}-h_{+})(\xi_{n}-h_{-}),$

Split the LHS of (21) using a partial fraction decomposition into terms that
have denominators that are powers of $(\xi_{n}-h_{+})$ resp.
$(\xi_{n}-h_{-})$. The former belong to $\mathfrak{q}_{+,j}$ and the latter to
$\mathfrak{q}_{-,j}$. We note here that [12, Appendix E] contains general
results for these decompositions.

  5. (5)

Calculate the terms $w_{\pm,j}$ in the expansion of $w_{\pm}$ near the
boundary for $j\in[0,k-1]$
By $W_{\pm}$ we denote the inverse of $Q^{\partial}_{\pm}$, see Proposition
6.2 above or [12, Subsection 5.2]. We denote its symbol by $w_{\pm}$ and it
has an asymptotic expansion (see Proposition 6.2)
$w_{\pm}\sim\sum_{j=0}^{\infty}w_{\pm,j}$ where $w_{\pm,j}$ is constructed
inductively by

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

with

$w_{\pm,0}(x,\xi,R):=(q^{\partial}_{\pm,0})^{-1}=\begin{cases}\frac{1}{n!\omega_{n}}h_{0}(x)^{\mu}(\xi_{n}-h_{+}(x,\xi^{\prime},R))^{\mu}\;&\mbox{for
$+$},\\\ {}\\\ (\xi_{n}-h_{-}(x,\xi^{\prime},R))^{\mu}\;&\mbox{for
$-$}.\end{cases}$

  6. (6)

Compute the coefficient of $R^{n+1-k}$
We label this coefficient by $c_{k}$. It is calculated by

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

where

$\displaystyle B_{\mathrm{d}^{2},k}$ $\displaystyle(x^{\prime}):=$
$\displaystyle=\sum_{\begin{subarray}{c}k=|\beta|+\gamma_{n}+j+l\\\
\gamma_{n}>0\end{subarray}}\frac{i^{|\beta|+|\gamma_{n}|}(-1)^{|\beta|+1}}{\beta^{\prime}!(\beta_{n}+\gamma_{n})!}\partial_{x}^{\beta}w_{-,j}(x^{\prime},0,0,1)\partial_{x_{n}}^{\gamma_{n}-1}\partial_{\xi}^{\beta+(0,\gamma_{n})}w_{+,l}(x^{\prime},0,0,1).$

The details of this calculation are in [12, Subsection 6.2].

When $X$ has no boundary there are no boundary contributions and consequently
$c_{k}=\int_{X}a_{k}\;\mathrm{d}x$.

On the other hand, when $X$ is a domain in $\mathbb{R}^{n}$, if $k>0$ then
$c_{k}=\int_{\partial_{X}}B_{\mathrm{d}^{2},k}\mathrm{d}x^{\prime}$ as the
only contribution from the interior comes from $a_{0}$.

##  Appendix B Pseudocode in special examples

######  Example B.1 ($X\subset M=\mathbb{R}^{n}$ with boundary).

Input:

  * •

$n$: dimension of $M$

  * •

$(x,\xi,R)\in\mathbb{R}^{n}\times\mathbb{R}^{n}\times{\mathbb{R}}$

  * •

$k$: $n+1-k$ is the degree of term to be determined

  * •

$\varphi(x^{\prime})$: parametrise near $\partial X$

Definitions:

  * •

$D_{t}=-i\partial_{t}$

  * •

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

  * •

$\mathrm{rk}(\gamma)=$length$(\gamma)$

  * •

$\mathfrak{c}_{m,n}:=\begin{cases}(-1)^{m}(n-2m)!\omega_{n-2m}\omega_{2m},\;&\mbox{for
$2m-n-1<0$}\\\
\frac{(-1)^{1-n/2}\omega_{2m}}{(2m-n)!\omega_{2m-n}},\;&\mbox{for $2m-n-1\in
2\mathbb{N}+1$}\\\
\frac{(-1)^{\frac{n+1}{2}}}{(2\pi)^{2m-n}}\omega_{2m}\omega_{2m-n-1},\;&\mbox{for
$2m-n-1\in 2\mathbb{N}$}\end{cases},$

  * •

$a_{0}=\frac{1}{n!\omega_{n}}$, $a_{j}=0\,\forall j>0$

  * •

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

  * •

$g_{bdy}$: Riemannian metric near the boundary of $X$

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

  * •

$h_{0}:=1$

  * •

$h_{+}:=\frac{\xi^{\prime}\left(\nabla\varphi(x^{\prime})\right)}{h_{0}}+i\frac{\sqrt{R^{2}+\xi^{\prime
2}\left(1+\left|\nabla\varphi(x^{\prime})\right|^{2}\right)-\left(\xi^{\prime}\left(\nabla\varphi(x^{\prime})\right)\right)^{2}}}{\sqrt{h}_{0}}$

  * •

$h_{-}:=\frac{\xi^{\prime}\left(\nabla\varphi(x^{\prime})\right)}{h_{0}}-i\frac{\sqrt{R^{2}+\xi^{\prime
2}\left(1+\left|\nabla\varphi(x^{\prime})\right|^{2}\right)-\left(\xi^{\prime}\left(\nabla\varphi(x^{\prime})\right)\right)^{2}}}{\sqrt{h}_{0}}$

  * •

$q^{\partial}_{0}:=(R^{2}+g_{bdy})^{-(n+1)/2}$

  * •

$q^{\partial}_{+,0}:=n!\omega_{n}(\xi_{n}-h_{+})^{-(n+1)/2}$

  * •

$q^{\partial}_{+,0}:=h_{0}^{-(n+1)/2}(\xi_{n}-h_{-})^{-(n+1)/2}$

  * •

$w_{+,0}:=1/q^{\partial}_{+,0}$

  * •

$w_{-,0}:=1/q^{\partial}_{-,0}$

Steps to compute $c_{k}$:

  1. (1)

For $j\in[1,k-1]$ calculate $q_{j}^{\partial}$ by

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

for $n$ odd, and for $n$ even by

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

  2. (2)

For $j\in[1,k-1]$ calculate

(22)
$\displaystyle\frac{q^{\partial}_{j}}{q^{\partial}_{0}}-\frac{1}{q^{\partial}_{0}}\sum_{\begin{subarray}{c}k+l+|\alpha|=j\\\
k,l<j\end{subarray}}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}q_{+,k}^{\partial}D^{\alpha}_{x}q_{-,l}^{\partial}.$

  3. (3)

Using a partial fraction decomposition, partition the terms of (22) into ones
including factors of $(\xi_{n}-h_{+})$ and $(\xi_{n}-h_{-})$. Call the sum of
the former $\mathfrak{q}_{+,j}$ and the latter $\mathfrak{q}_{+,j}$.

  4. (4)

Set $q_{\pm,j}^{\partial}:=q_{\pm,0}^{\partial}\mathfrak{q}_{+,j}$

  5. (5)

For $j\in[1,k-1]$ calculate $w_{\pm,j}$ by

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

  6. (6)

Calculate $B_{\mathrm{d}^{2},k}$ by

$\displaystyle
B_{\mathrm{d}^{2},k}(x^{\prime}):=\sum_{\begin{subarray}{c}k=|\beta|+\gamma_{n}+j+l\\\
\gamma_{n}>0\end{subarray}}\frac{i^{|\beta|+|\gamma_{n}|}(-1)^{|\beta|+1}}{\beta^{\prime}!(\beta_{n}+\gamma_{n})!}\partial_{x}^{\beta}w_{-,j}(x^{\prime},0,0,1)\partial_{x_{n}}^{\gamma_{n}-1}\partial_{\xi}^{\beta+(0,\gamma_{n})}w_{+,l}(x^{\prime},0,0,1).$

For $k=0$ this is 0.

  7. (7)

Calculate $c_{k}$ by

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

######  Example B.2 ($X=M\subset\mathbb{R}^{N}$ without boundary).

Input:

  * •

$n$: dimension of $M$

  * •

$N$: dimension of Euclidean space

  * •

$(x,\xi,R)\in\mathbb{R}^{N}\times\mathbb{R}^{N}\times{\mathbb{R}}$

  * •

$k$: $n-k$ is the degree of term to be determined

  * •

$\varphi_{l}(x_{1},\ldots,x_{n})$ for $l\in[n+1,N]$: parametrize near a point
in $M$

Definitions:

  * •

$D_{t}=-i\partial_{t}$

  * •

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

  * •

$\mathrm{rk}(\gamma)=$length$(\gamma)$

  * •

$\mathfrak{c}_{m,n}:=\begin{cases}(-1)^{m}(n-2m)!\omega_{n-2m}\omega_{2m},\;&\mbox{for
$2m-n-1<0$}\\\
\frac{(-1)^{1-n/2}\omega_{2m}}{(2m-n)!\omega_{2m-n}},\;&\mbox{for $2m-n-1\in
2\mathbb{N}+1$}\\\
\frac{(-1)^{\frac{n+1}{2}}}{(2\pi)^{2m-n}}\omega_{2m}\omega_{2m-n-1},\;&\mbox{for
$2m-n-1\in 2\mathbb{N}$}\end{cases},$

  * •

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

  * •

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

Steps:

  1. (1)

For $j\in[1,k]$ calculate $q_{j}$ by

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

for $n$ odd, and for $n$ even by

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

  2. (2)

For $j\in[1,k]$ calculate $a_{j}$ by

$\displaystyle a_{j,0}(x,R)=\begin{cases}0&\mbox{$j$ odd},\\\
-\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)&\mbox{$j$
even}.\end{cases}$

  3. (3)

Calculate $c_{k}$ by

$c_{k}(M,\mathrm{d})=\int_{X}a_{k,0}(x,1)\mathrm{d}x.$

##  Appendix C Implementation of algorithm for 2-dimensional Euclidean domains

The algorithm from Appendix B was implemented in Python, for the case of
Euclidean domains. This was done with the use of the module sympy, a package
used in python for symbolic manipulation, and the resulting code for
$2$-dimensional domains is available in [13].

The algorithm computes the coefficient $c_{k}$ for arbitrary $k$, with two
limitations: The dimension $n$ is fixed, and the computational effort and
memory required to compute $c_{k}$ increase rapidly with $k$. The main
computational effort is the partial fraction decomposition, because of the
large number of terms involved.

We have used the code to compute $c_{k}$ up to $k=4$. We do not expect this
code to be used beyond $k=5$, because of the rapid increase in memory required
for subsequent $c_{k}$. As an illustration, the calculation of $q_{\pm,2}$
requires $6$ times more memory than $q_{\pm,1}$, and $q_{\pm,3}$ requires $40$
times more memory than $q_{\pm,2}$. Similarly, for $c_{3}$ the total number of
terms requiring a partial fraction decomposition is approximately $1000$,
whereas for $c_{4}$ this number increases to approximately $110000$.

## References

  * [1] S. Amari, _Information Geometry and Its Applications_ , Applied Mathematical Sciences 374, Springer Japan, Tokyo, 2016.
  * [2] J. A. Barceló, A. Carbery, _On the magnitudes of compact sets in Euclidean spaces_ , Amer. J. Math. 140 (2018), pp. 449-494.
  * [3] M. Berger, _Le spectre des variétés riemanniennes_ , Rev. Roumaine Math. Pures Appl. 13 (1968), pp. 915-931.
  * [4] E. Bunch, J. Kline, D. Dickinson, S. Bhat, G. Fung, _Weighting vectors for machine learning: numerical harmonic analysis applied to boundary detection_ , arXiv:2106.00827.
  * [5] J.-L. Brylinski, The beta function of a knot. Internat. J. Math. 10 (1999), pp. 415-423.
  * [6] G. I. Eskin, _Boundary value problems for elliptic pseudodifferential equations_ , Translations of Mathematical Monographs, 52. American Mathematical Society, Providence, R.I., 1981.
  * [7] E. J. Fuller, M. K. Vemuri. The Brylinski Beta Function of a Surface, Geometriae Dedicata 179 (2015), pp. 153-160.
  * [8] H. Gimperlein, M. Goffeng, _On the Magnitude Function of Domains in Euclidean Space, I_ , blog post at _The n-Category Cafe_ (2018), https://golem.ph.utexas.edu/category/2018/01/on_the_magnitude_function_of_d.html.
  * [9] H. Gimperlein, M. Goffeng, _On the Magnitude Function of Domains in Euclidean Space, IV_ , blog post at _The n-Category Cafe_ (2019), https://golem.ph.utexas.edu/category/2019/01/on_the_magnitude_function_of_d_3.html.
  * [10] H. Gimperlein, M. Goffeng, _On the magnitude function of domains in Euclidean space_ , Amer. J. Math. 143 (2021), pp. 939-967.
  * [11] H. Gimperlein, M. Goffeng, _The Willmore energy and the magnitude of Euclidean domains_ , arXiv:2109.10097
  * [12] H. Gimperlein, M. Goffeng, N. Louca, _Semiclassical analysis of a nonlocal boundary value problem related to magnitude_ , preprint.
  * [13] H. Gimperlein, M. Goffeng, N. Louca, _The magnitude and spectral geometry_ , ancillary file to arxiv preprint.
  * [14] G. Grubb, _Fractional Laplacians on domains, a development of Hörmander’s theory of $\mu$-transmission pseudodifferential operators_, Adv. Math. 268 (2015), pp. 478-528.
  * [15] G. Grubb, _Integration by parts and Pohozaev identities for space-dependent fractional-order operators_ , J. Differential Equations 261 (2016), pp. 1835–1879.
  * [16] G. Grubb, _Green’s formula and a Dirichlet-to-Neumann operator for fractional-order pseudodifferential operators_ , Comm. Partial Differential Equations 43 (2018), pp. 750-789.
  * [17] G. Grubb, R. T. Seeley, _Zeta and eta functions for Atiyah-Patodi-Singer operators_ , J. Geom. Anal. 6 (1996), pp. 31-77.
  * [18] L. Hörmander, _Seminar notes on pseudo-differential operators and boundary problems_ , Lectures at IAS Princeton 1965-66, available from Lund University, https://lup.lub.lu.se/record/7c9d8485-d7d6-4d47-80a5-a5533685c4ca.
  * [19] M. Kac, _Can one hear the shape of a drum?_ , The American Mathematical Monthly 73 (1966), pp. 1-23.
  * [20] T. Leinster, _The magnitude of metric spaces_ , Doc. Math. 18 (2013), pp. 857-905.
  * [21] T. Leinster, M. Meckes, _The magnitude of a metric space: from category theory to geometric measure theory_ , Measure theory in non-smooth spaces, 156–193, Partial Differ. Equ. Meas. Theory, De Gruyter Open, Warsaw, 2017.
  * [22] T. Leinster, S. Willerton, _On the asymptotic magnitude of subsets of Euclidean space_ , Geom. Dedicata 164(1) (2013), pp. 287-310.
  * [23] M. Meckes, _Positive definite metric spaces_ , Positivity 17 (2013), pp. 733-757.
  * [24] M. Meckes, _Magnitude, diversity, capacities and dimensions of metric spaces_ , Potential Anal. 42 (2015), pp. 549-572.
  * [25] M. Meckes, _On the magnitude and intrinsic volumes of a convex body in Euclidean space_ , Mathematika 66 (2020), pp. 343-355.
  * [26] J. O’Hara, _Residues of manifolds_ , arxiv:2012.01713.
  * [27] J. O’Hara, G. Solanes, Regularized Riesz energies of submanifolds, Math. Nachr. 291 (2018), pp. 1356-1373.
  * [28] C. Sutton, _Spectral geometry in the presence of symmetry_ , Notices of the Amer. Math. Soc. 68 (2021), pp. 869-883.
  * [29] S. Willerton, _On the magnitude of spheres, surfaces and other homogeneous spaces_ , Geom. Dedicata 168 (2014), pp. 291-310.
  * [30] S. Willerton, _The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials_ , Discrete Anal. 2020, Paper No. 5, 42 pp.