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arXiv:2201.11357v1 [math.AP] 27 Jan 2022
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE
PROBLEM RELATED TO MAGNITUDE
HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Abstract. We study a Dirichlet boundary problem related to the fractional Laplacian in a
manifold. Its variational formulation arises in the study of magnitude, an invariant of compact
metric spaces given by the reciprocal of the ground state energy. Using recent techniques
developed for pseudodifferential boundary problems we discuss the structure of the solution
operator and resulting properties of the magnitude. In a semiclassical limit we obtain an
asymptotic expansion of the magnitude in terms of curvature invariants of the manifold and
the boundary, similar to the invariants arising in short-time expansions for heat kernels.
Contents
1. Introduction 1
2. The symbolic structure near the diagonal 7
3. Global behavior of Zon compact manifolds 26
4. The operator Zon Sobolev spaces for a manifold with boundary 33
5. Structure of the inverse operator in the presence of a boundary 37
6. Conditional expectations of Q−1
Xand Z−1
X47
Appendix A. Overview of conormal distributions 54
Appendix B. Parameter dependent pseudodifferential operators 60
Appendix C. Inverting Qin an extended Boutet de Monvel calculus 65
Appendix D. The meromorphic Fredholm theorem 68
Appendix E. Partial fraction decompositions of symbols 68
Appendix F. Evaluation of some boundary symbols at zero 70
References 75
1. Introduction
The analysis of boundary problems for nonlocal operators has attracted much interest in
recent years, including Dirichlet and Neumann problems for fractional Laplacians in a Euclidean
domain. In this article we motivate and initiate the semiclassical analysis of related boundary
problems, with applications to the Leinster-Willerton conjecture for the magnitude invariant of
compact metric spaces.
To be specific, we consider the integral equation with parameter R > 0
(1) ZX
e−Rd(x,y)u(y)dy=f(x).
Here (X, d) is a compact metric space, and we focus on when Xis a manifold with boundary
and d is a distance function satisfying additional regularity assumptions. Already when X⊆R2
is the unit disc, close to nothing was known for the solutions to (1). We shall prove in this
paper that for Xa compact n-dimensional manifold with boundary, Equation (1) is well posed
for fin the Sobolev space H(n+1)/2(X) and admits a unique solution uR∈˙
H−(n+1)/2(X) for
sufficiently large R≫0 (for notation, see page 33). We relate the integral equation (1) to a
pseudodifferential boundary value problem which is elliptic with parameter.
Key words and phrases. 51F99 (primary), 28A75, 58J40, 58J50 (secondary).
1
2 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Our main results concern structural properties of solutions to Equation (1) such as asymptotic
behavior as R→ ∞ and meromorphic extensions in the parameter Rto sectors Γ ⊆C. The
methods for pseudodifferential boundary value problems that we use date far back, see the work
of Gregory Eskin [9] and Lars H¨ormander [21], but have in recent years seen much development
in work of Gerd Grubb [14, 15, 16, 17, 18].
The solution to (1) for the right hand side f= 1 enters in the so called magnitude function
of (X, d), studied extensively in for instance [2, 11, 25, 26, 28, 32, 33, 43, 44]. The empirical
properties of the solution to (1) have recently found applications in data science, leading to
precise conjectures for its structural properties [6, 7].
The case f= 1 can also be considered as a minimizing problem. Equation (1) relates to the
ground state energy
E(R;X, d) := inf ZXZX
e−Rd(x,y)dµ(x)dµ(y) : µa signed Borel measure with µ(X) = 1=
(2)
= inf
X
x,y∈X
e−Rd(x,y)c(x)c(y) : c:X→Rhas finite support and Px∈Xc(x) = 1
.
More precisely, if Ris such that (X, Rd) is positive definite (i.e. the matrix (e−Rd(x,y))x,y∈Fis
positive definite for any finite F⊆X), then by [33] a solution uRto Equation (1) with f= 1
satisfies ZX
uR(x)dx=1
E(R;X, d) .
Let us digest on the problem of finding E(R;X, d) and studying its semiclassical limit, as
it has been broadly studied in various mathematical communities. The ground state energy
E(R;X, d) is that of a signed distribution of finitely many particles on Xwhere a particle in x
interacts with a particle in yunder the potential e−Rd(x,y). As such, the scaling parameter R > 0
should be thought of as an order parameter with R→ ∞ corresponding to a semiclassical limit.
The non-locality of Equation (1) and the ground state energy E(R;X, d) makes the problem of
explicit computation an impossibility, however in the semiclassical limit the problem localizes
and is asymptotically described in terms of geometric invariants. Related problems concerning
ground state energies with nonlocal interaction potentials arise in the mean field description of
interacting particle systems, such as [10]. Specifically for log gases the dependence of the ground
state energy on the geometry has been of interest [46]. In complex geometry, Berman has
studied similar minimization problems from which geometric structures emerged [3, 4]. Related
operators also appear in image processing [1].
The integral equation (1) is, as mentioned above, related to magnitude – an invariant that
has been extensively studied since it was introduced by Tom Leinster [25]. We presume no
prerequisites from the reader on magnitude in this paper, but for the convenience of the reader
we summarize the implications to magnitude here and expand on this relation in the follow up
paper [12]. From its category-theoretic origin, magnitude has found unexpected applications
from algebraic topology [13, 27, 40] and applied category theory [8, 35] to data science [6, 7] and
mathematical biology [24].
For a metric space (X, d) this invariant leads to a function MX: (0,∞)→R∪ {∞}. When
the metric space (X, Rd) is positive definite, and in particular for compact sets X⊂Rn[32],
MX(R) is defined as MX(R) = RXuR(x)dx, where uRsatisfies the magnitude equation
(3) ZX
e−Rd(x,y)u(y)dy= 1.
The work [33] provides an abstract Hilbert space framework in which to pose this equation.
In the case of compact sets X⊂Rn, Meckes [33] gives an interpretation of the magnitude in
potential theoretic terms, as a generalized capacity:
(4) MX(R) = 1
Rn!ωn
inf nk(R2+ ∆)(n+1)/4hRk2
L2(Rn):hR∈H(n+1)/2(Rn), hR= 1 on Xo.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 3
Here ωndenotes the volume of the unit ball in Rn. The minimizer of (4) is attained at a function
hR∈H(n+1)/2(Rn) solving the non-local exterior problem
(5) ((R2+ ∆)(n+1)/2hR= 0,in Rn\X,
hR= 1,in X.
If nis odd, this was studied as a boundary value problem for the five-dimensional unit ball in
[2], and extended to odd-dimensional unit balls in [45]. Few explicit computations of magnitude
are known outside the realm of compact domains in odd-dimensional Euclidean space, and even
there the state-of-the art [11] can only provide asymptotic results in the semiclassical limit and
ensure existence of meromorphic extensions. In particular, nothing was previously known about
magnitude even in such a simple case as the unit disk X=B2⊆R2.
We provide a framework for a refined analysis and explicit computations for solutions to
Equation (1) when Xis a smooth, n-dimensional, compact manifold with boundary, independent
of the parity of n. The framework relies on recent advances for pseudodifferential boundary
problems and initiates their semiclassical analysis. We work under certain regularity assumptions
on the distance function d, firstly that its square is regular at the diagonal (see Definition 2.2)
and secondly that it has property (MR) (see Definition 3.3 and 4.2). Our first assumption ensures
that the distance function behaves to leading term as a Euclidean distance, and is satisfied by
any geodesic distance function or a pullback thereof under an embedding. The first assumption
ensures that the diagonal behavior in Equation (1) is governed by a pseudodifferential operator
of order −n−1 which is elliptic with parameter. Our second assumption – property (MR) –
is a technical condition to ensure that the off-diagonal behavior in Equation (1) is negligible.
Property (MR) is satisfied for subspace distances in manifolds whose distance functions squared
are smooth, but it in fact fails for higher dimensional tori and real projective spaces.
1.1. Main results. Let us summarize the main results of this paper. The results all circle
around the family of integral operators
ZX(R)u(x) := 1
RZX
e−Rd(x,y)u(y)dy, R ∈C\ {0}.
Here Xis a compact manifold with boundary equipped with a distance function d and a volume
density dy. We assume that d2is smooth in a small neighborhood of the diagonal x=yand
there in local coordinates admits a Taylor expansion (for any N > 0)
(6) d(x, y)2=Hd2,x (v) +
∞
X
j=3
Cj
d2(x;v) + O(|v|N+1).
Here v=x−y, and where Hd2is a Riemannian metric on Xand Cj
d2in local coordinates is a
symmetric j-form in v. This condition can be summarized in the terminology that d2is regular
at the diagonal, see Definition 2.2 and for more details on the Taylor expansion, see Equation
(10). We fix a function χ∈C∞(X×X) such that χ= 1 on a neighborhood of the diagonal
x=yand d2is smooth on the support of χ. The localization of Zto near the diagonal is the
integral operator
QX(R)u(x) := 1
RZX
χ(x, y)e−Rd(x,y)u(y)dy, R ∈C\ {0}.
We remark that if d2is smooth on all of X×X, e.g. for a domain or a submanifold in Rnwith
the induced metric, it holds that ZX−QXis smoothing with parameter on any sector Γ ⊆C+
with opening angle < π/2.
Theorem 1.1. Let Xbe a compact n-dimensional manifold with boundary and dsuch that d2
is regular at the diagonal (see Definition 2.2). The family of integral operators QXis an elliptic
pseudodifferential operator with parameter R∈C+of order −n−1, and its principal symbol is
σ−n−1(QX)(x, ξ, R) = n!ωn(R2+gd2(ξ, ξ))−(n+1)/2,
where gd2is the dual metric to Hd2from the Taylor expansion (6). The properties of QXcan
be summarized as follows:
4 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
(1) In each coordinate chart, the full symbol of QXcan be computed by an iterative scheme
as in Theorem 2.9.
(2) There exists an R0such that
QX(R) : ˙
H−n+1
2(X)→¯
Hn+1
2(X),
is invertible for Re(R)> R0and arg(R)< π/(n+ 1). Here ˙
H−n+1
2(X), respectively
¯
Hn+1
2(X), denote the Sobolev spaces of supported, respectively extendable distributions
in X(see Section 4).
(3) If ∂X =∅, then QX(R)−1is an elliptic pseudodifferential operator of order n+ 1 when-
ever it exists. The full symbol of Q−1
Xcan be computed by an iterative scheme as in
Theorem 2.21.
(4) If ∂X 6=∅,QXis elliptic with parameter of type −(n+ 1)/2and factorization index −∞
in an extended Boutet de Monvel calculus. In particular, there is a classical parameter
dependent parametrix in an extended Boutet de Monvel calculus of order n+ 1, type
(n+ 1)/2and factorization index −∞.
Moreover, if d2is smooth then all the properties above hold also for ZX.
Theorem 1.1 is found in the bulk of the text as follows. The first statement and item (1) is
found in Theorem 2.9. Item (2) is proven in Theorem 4.1, see also Corollary 2.22 of Theorem
2.23 for the simpler case of ∂X =∅. Item (3) follows from Theorem 2.21 and Corollary 2.22.
Item (4) is contained in Theorem 4.9.
The operator QXis generally more well behaved than ZX; the off-diagonal singularities of
d can create problems in considering ZXas a map between Sobolev spaces. For examples of
such phenomena, see Subsection 3.4. We impose one of two conditions on d; property (MR)
and property (SMR) respectively to ensure that QXand ZXshare common functional analytic
features as operators between Sobolev space. The reader can find the precise definition of
property (MR) and property (SMR) in Definition 3.3 (for ∂ X =∅) and Definition 4.2 (for
∂X 6=∅). We note that property (MR) and property (SMR) hold on any sector Γ ⊆C\ {0}as
soon as d2is smooth on all of X×X, e.g. for a domain in Rnor more generally for a compact
submanifold with boundary in a manifold with d2smooth.
Theorem 1.2. Let Xbe a compact n-dimensional manifold with boundary and let dbe a distance
function such that d2is regular at the diagonal (see Definition 2.2). The family of operators
QX(R) : ˙
H−n+1
2(X)→¯
Hn+1
2(X), R ∈C\ {0},
is a holomorphic family of Fredholm operators that are invertible on a sector. The inverse
QX(R)−1:¯
Hn+1
2(X)→˙
H−n+1
2(X), R ∈C\ {0},
is a meromorphic family of Fredholm operators.
If dsatisfies property (SMR) on a sector Γ, then also
ZX(R) : ˙
H−n+1
2(X)→¯
Hn+1
2(X), R ∈Γ,
is a holomorphic family of Fredholm operators whose inverse family
ZX(R)−1:¯
Hn+1
2(X)→˙
H−n+1
2(X), R ∈Γ,
is a meromorphic family of Fredholm operators.
For the purposes described above, we are interested in precise asymptotic information about
solutions to ZX(R)u=f. To this end, we were not able to utilize the extended Boutet de
Monvel calculus appearing in Theorem 1.1, item (4), but rather we describe the inverse operator
Z−1
Xvia Wiener-Hopf factorizations.
Theorem 1.3. Let Xbe a compact n-dimensional manifold with boundary and da distance
function whose square is regular at the diagonal. For some R0≥0and any R∈Γπ/(n+1)(R0),
we can write
Q−1
X=χ1Aχ′
1+χ2(ϕ−1)∗W+W−ϕ∗χ′
2+S,
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 5
where χ1, χ′
1∈C∞
c(X◦), and χ2, χ′
2∈C∞(X)are functions supported in a collar neighborhood
U0of ∂X in Xsuch that
χ1+χ2= 1 and χ′
j|supp(χj)= 1, j = 1,2,
ϕ:∂X ×[0,1) →U0is a collar identification, and the operators S,W−and W+satisfy the
following:
(1) S:Hµ(X)→˙
H−µ(X)is a continuous operator with
kSkHµ(X)→˙
H−µ(X)=O(R−∞),as R→ ∞.
(2) W+:L2(∂X ×[0,∞)) →˙
H−µ(∂X ×[0,∞)) is the properly supported pseudodifferential
operator of mixed-regularity (µ, 0) from Definitions 5.18 which is invertible for large
R > 0and in local coordinates has an asymptotic expansion modulo Sµ,−∞ as in Lemma
5.21 and preserves support in ∂X×[0,∞)⊆∂X ×R. Moreover, for χ, χ′∈C+C∞
c(∂X ×
[0,∞)) with χχ′= 0, it holds that
kχW+χ′kL2(∂X ×[0,∞))→H−µ(∂X ×R)=O(R−∞),as R→ ∞.
(3) W−:Hµ(∂X ×[0,∞)) →L2(∂X ×[0,∞)) is the properly supported pseudodifferential
operator of mixed-regularity (µ, 0) from Definitions 5.18 which is invertible for large R >
0and in local coordinates has an asymptotic expansion modulo Sµ,−∞ as in Lemma 5.21
and preserves support in ∂X ×(−∞,0] ⊆∂ X ×R. Moreover, for χ, χ′∈C+C∞
c(∂X ×R)
with χχ′= 0, it holds that
kχW−χ′kHµ(∂X ×R)→L2(∂X ×R)=O(R−∞),as R→ ∞.
The reader can find Theorem 1.3 stated as Theorem 5.22 in the body of the text. A key
feature of the construction in Theorem 1.3 is that it provides us with a method to compute the
symbolic structure of the inverse Q−1
X.
Theorem 1.4. Let Xbe a compact n-dimensional manifold with boundary and da distance
function whose square is regular at the diagonal. In the sector Re(R)> R0and arg(R)<
π/(n+ 1), we can write
hQ(R)−11X,1Xi=
∞
X
k=0
ck(X, d)Rn−k+O(Re(R)−∞),as Re(R)→ ∞ in Γ,
where the coefficients ck(X, d) are given as
ck(X, d) = ZX
ak,0(x, 1)dx+Z∂X
Bd2,k(x)dx′,
where
(1) ak,0(·,1) ∈C∞(X)is an invariant polynomial in the entries of the Taylor expansion (6)
as described in Theorem 2.27 and can be computed inductively using Lemma 2.25, with
ak,0= 0 if kis odd; and
(2) Bd2,k ∈C∞(∂X )is an invariant polynomial in the entries of the Taylor coefficients of
d2at the diagonal in Xnear ∂X as described in Proposition 6.9 and can be inductively
computed using Lemma 5.19.
In particular, we have that
c0(X, d) = vol(X)
n!ωn
c1(X, d) = (n+ 1)vol(∂X )
2n!ωn
,
c2(X, d) = n+ 1
6·n!ωnZX
sd2dx+(n−1)(n+ 1)2
8·n!ωnZ∂X
Hd2dx′.
where the scalar curvature sd2is defined as in Theorem 6.1 and the mean curvature Hd2of the
distance function is defined as in Theorem 6.13.
The reader can find Theorem 1.4 stated as Theorem 6.13 in the body of the text.
6 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Corollary 1.5. Let Xbe a compact n-dimensional manifold with boundary and da distance
function whose square is regular at the diagonal. Write E(R;X, d) for the ground state energy
from Equation (2).
(1) If dhas property (MR) on a sector Γ, the ground state energy function E(R;X, d) is a
well defined meromorphic function of R∈Γ.
(2) If dhas property (SMR) on a sector Γ, the ground state energy function E(R;X, d) has
the semiclassical limit
E(R;X, d) =
∞
X
k=0
εk(X, d)R−n−k+O(Re(R)−∞),as Re(R)→ ∞ in Γ,
where
ε0(X, d) = n!ωn
voln(X),
ε1(X, d) = −(n+ 1)voln−1(X)
2voln(X)
ε2(X, d) = (n+ 1)2voln−1(X)2
4voln(X)2−n!ωnc2(X, d)
voln(X),
and more generally
εk(X, d) = pk,n c1(X, d)
voln(X),...,ck(X, d)
voln(X),
for a universal polynomial pk,n of total degree k(where each cj(X, d) is declared to be of
degree j).
1.2. Notational conventions. To avoid confusion, we will use the terms Riemannian metrics
and distance function to separate the notions of metrics that appear in Riemannian geometry
and metric geometry, respectively.
The Fourier transform on Rnis denoted by F. We use the convention that
Dx=−i∂
∂x .
For α= (α1,...,αn)∈Nn, we write |α|=Pjαj,Dα
x=Dα1
x1···Dαn
xnand xα=xα1
1···xαn
n. In
this convention, for a Schwartz function fon Rn,
F(Dα
xf)(ξ) = ξαFf(ξ) and F(xαf)(ξ) = (−Dξ)αFf(ξ),
and the product of pseudodifferential symbols is up to smoothing operators defined from a symbol
of the form
p#q∼X
α
1
α!∂α
ξpDα
xq=X
α
1
α!Dα
ξp∂α
xq.
We write Mfor a manifold and Xfor a compact manifold with boundary, or occasionally a
general compact metric space. We let ndenote the dimension of Mor Xand use the notation
µ:= n+ 1
2.
We write DiagM:= {(x, x) : x∈M}for the diagonal in M×M. If (X, d) is a compact metric
space such that the matrix (e−Rd(x,y))x,y∈Fis positive definite for any finite subset F⊆X, we
say that (X, d) is positive definite. If (X, Rd) is positive definite for all R > 0, we say that (X, d)
is stably positive definite. This terminology follows [32].
For a manifold M, we write C∞
c(M) for the Fr´echet space of compactly supported smooth
functions and D′(M) for its topological dual – the distributions on M. If Xis a compact
manifold with boundary, it can always be embedded as a domain in a manifold Mand we write
C∞(X)⊆C(X) for the restrictions to Xof elements in C∞(M).
A sector Γ ⊆Cis a conical subset, i.e., λΓ⊂Γ for all λ > 1. Standard examples we use
throughout the paper are C+={z∈C: Re(z)>0}and
Γα(R0) := {z∈C:|Arg(z)|< α, Re(z)> R0}.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 7
If we make a claim concerning R→+∞, it is implicitly assumed to be a limit along the real
line. We also note that for sectors Γ ⊆C+of opening angle α < π/2, there is a Cα>0 with
C−1
α|R| ≤ Re(R)≤Cα|R|.
For two Banach spaces V1and V2, we write B(V1, V2) for the space of bounded operators
V1→V2and K(V1, V2) for the space of compact operators V1→V2. Both form Banach spaces
in the norm topology.
We write N={0,1,2,3,...}for the set of natural numbers.
1.3. Acknowledgments. We thank Tony Carbery, Gerd Grubb, Tom Leinster, Rafe Mazzeo,
Mark Meckes, Niels Martin Møller, Grigori Rozenblum, Jan-Philip Solovej and Simon Willerton
for fruitful and encouraging discussions.
MG was supported by the Swedish Research Council Grant VR 2018-0350. NL was supported
by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doc-
toral Training funded by the UK Engineering and Physical Sciences Research Council (Grant
EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of
Edinburgh.
2. The symbolic structure near the diagonal
To better understand the operator Zwe first consider the case of a manifold M. This analysis
describes compact manifolds (see Subsection 2.4 and Section 3) and we carry it over to the interior
of a compact manifold with boundary below in Section 4 and 5. We formulate our results in
terms of the operator
(7) Z(R)f(x) := 1
RZM
e−Rd(x,y)f(y)dy,
whose dependence on R6= 0 is holomorphic under suitable assumptions studied in Section 3
below. Here we are implicitly using a volume density on M, and the operator depends on this
choice. We shall later fix a certain choice adapted to the distance function. We shall specify the
domain and codomain of this operator more precisely later on. For now, we can consider Zan
operator C∞
c(M)→ D′(M) by setting
hZϕ, ψi=1
RZM×M
e−Rd(x,y)ϕ(y)ψ(x)dxdy, for ϕ, ψ ∈C∞
c(M).
2.1. On a class of pseudodifferential operators with parameter. We pick a function
χ∈C∞(M×M) such that χ= 1 near DiagMand is supported in a small neighborhood of
DiagM. The precise choice of χwill not play an important role, but we shall later specify
conditions on its support. The operator Zdecomposes as
(8) Z=Q+L, where Q(R)f(x) := 1
RZM
χ(x, y)e−Rd(x,y)f(y)dy.
We call the operator Qthe localization of Znear the diagonal. In this section we focus our
attention to Q. Distance functions might be non-smooth away from the diagonal despite being
quite regular at the diagonal and this off-diagonal behavior of the distance function dictates
whether or not Lis negligible. The remainder Lwill be studied further in Section 3.
This subsection is devoted to proving that the localized part Qof Zis a parameter-dependent
pseudodifferential operator, for a brief overview thereof, see Appendix B. We will treat a slightly
more general class of operators than Q. Consider a family of operators that take the form
(9) QG,χ(R)f(x) := 1
|R|Mc ZM
χ(x, y)e−|R|Mc √G(x,y)f(y)dV(y).
Here we have written
|R|Mc := (R, Re(R)>0,
−R, Re(R)<0,
for the McIntosh modulus which extends the absolute value to a holomorphic function in C\iR.
We shall mainly be concerned with the cases R∈Rand R∈C+. The cut-off function χis as
above with the additional constraint that Gis smooth on supp(χ). The function G:M×M→
[0,∞) should be regular at the diagonal as made precise in Definition 2.2 below: to define this
8 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
notion we first introduce further terminology. Note that T(M×M) = p∗
1T M ⊕p∗
2T M where
pj:M×M→M,j= 1,2, denotes the pro jection onto the j:th factor. Over the diagonal
DiagM, the map Dp1⊕Dp2:T(M×M)|DiagM→T M ⊕T M is an isomorphism. We define the
transversal tangent bundle to the diagonal to be
TtraDiagM:= ker(Dp1+Dp2:T(M×M)|DiagM→T M )⊆T(M×M)|DiagM.
The restriction Dp1|:Ttra DiagM→T M is an isomorphism.
Definition 2.1. For a smooth function Gdefined in a neighborhood of DiagM, we define the
transversal Hessian HGas the quadratic form on Ttra DiagMobtained from restricting the Hessian
of Gover the diagonal to the transversal tangent bundle.
Definition 2.2. A function G:M×M→[0,∞) is said to be regular at the diagonal if there is
a tubular neighborhood Uof the diagonal DiagMsuch that G|U∈C∞(U) is a smooth function
satisfying that
•G|Uand dG|Uvanish on DiagM⊆U;
•G(x)>0 for x∈U\DiagM; and
•the transversal Hessian HGis positive definite in all points of DiagM.
Remark 2.3.The prototypical example of a function Gregular at the diagonal is G= d2
for suitable distance functions d. The function d2is regular at the diagonal for the Euclidean
distance, or when d is the geodesic distance on a Riemannian manifold (see Example 2.18 below)
or more generally a distance function induced from pulling back a geodesic distance along an
embedding of M(see Example 2.17 below for an example).
To show that QG,χ(R) is an elliptic pseudodifferential operator with parameter, and to de-
scribe its full symbol, we shall use a slight detour. The basic idea used in computing the full
symbol of QG,χ(R) is to do an inverse Fourier transform in R, and then Fourier transform all
conormal variables. When we Fourier transform in the R-variable the Schwartz kernel – depend-
ing on (x, y, R) – transforms to a conormal distribution on U×R(conormal to DiagM× {0})
– depending on (x, y , η) – that we then Fourier transform in all the transversal directions (v, η)
where v=x−y, thus producing the full symbol depending on (x, ξ, R). To compute the Fourier
transform in the R-direction, we use the following elementary lemma.
Proposition 2.4. For a parameter a≥0, and Fa(R) := F.P.e−a|R|
|R|, we have that
FFa(η) = −log(η2+a2) + log(2) −2γ,
where γis the Euler-Mascheroni constant.
Proof. By taking a derivative in the parameter aand using that the Fourier transform of R7→
e−a|R|is 2a(η2+a2)−1, we see that
∂
∂a FFa(η) = −2a(η2+a2)−1.
As such, FFa(η) = −log(η2+a2) + c0(η) for some tempered distribution c0. Setting a= 0 and
using Proposition A.8, we see that c0is a constant. By Proposition A.6 we have that
c0=β0,1= 2 log(2) + 1
2ψ(1/2) −γ= log(2) −2γ.
Proposition 2.5. Let G:M×M→[0,∞)be a function which is regular at the diagonal, see
Definition 2.2. For a neighborhood Uof DiagMon which Gis smooth, define ˜
G∈C∞(U×R)
by ˜
G(x, y, η) := η2+G(x, y),
and the conormal distribution log( ˜
G)∈I−n−1(U×R; DiagM× {0})as in Proposition A.10.
Then there exists a canonical metric gGon T∗Msuch that
σ−n−1(log( ˜
G))(x, ξ, R) = −2πn!ωn(R2+gG(ξ, ξ))−(n+1)/2.
The canonical metric gGis dual to the transversal Hessian of Gunder the isomorphism Dp1|:
TtraDiagM→T M .
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 9
In the following, unless specified otherwise we shall consider Mendowed with the Riemannian
metric gG. This allows, in particular, to define a Laplace operator on M.
Proof. Since Gis regular at the diagonal, the function ˜
Gsatisfies the assumptions of Proposition
A.10 and the result follows therefrom.
To compute the full symbol of QG,χ we use a Taylor expansion in the direction transversal
to the diagonal. Consider a function G:M×M→[0,∞) which is regular at the diagonal, see
Definition 2.2. Consider a coordinate chart U0⊆M. The coordinates on U0induce coordinates
(x, y) on U0×U0and we can identify a neighborhood of DiagM∩(U0×U0) with a neighborhood
of the zero section in Ttra DiagM|U0via the map (x, y)7→ (x, x −y). If the coordinate chart U0
on Msatisfies that Gis smooth on U0×U0, Taylor’s theorem implies that for any N∈Nwe
can on U0×U0write
(10) G(x, y) = HG,x (x−y) +
N
X
j=3
C(j)
G(x;x−y) + rN(x, x −y),
for |x−y|small enough, where rNis a smooth function with rN(x, v) = O(|v|N+1 ) as v→0, HGis
the transversal Hessian of G, and C(j)
G:U0→Symj(TtraDiagM|U0) takes values in the symmetric
j-forms on the transversal tangent bundle Ttra DiagM|U0. A short computation shows that HG
indeed is a Riemannian metric on Munder the isomorphism Dp1|:TtraDiagM→T M . However,
each C(j)
Gdepends on the choice of coordinates, we nevertheless suppress this dependence in the
notation.
Since there is a canonical isomorphism TtraDiagM|U0∼
=T M |U0, the symmetric j-form C(j)
G:
U0→Symj(TtraDiagM|U0) appearing in the Taylor expansion (10) of Gdefines a j:th order
differential operator
C(j)
G(x, −Dξ) : C∞(T∗M|U0)→C∞(T∗M|U0),
obtained by quantizing the coordinate functions, i.e. C(j)
G(x, −Dξ) acts as multiplication op-
erators by C(j)
G(x, v) under the fiberwise inverse Fourier transform (in the v-direction). For a
k∈N+and a multiindex γ∈Nk
≥3, we can define a differential operator C(γ)
G(x, −Dξ) on T∗M|U0
by
C(γ)
G(x, −Dξ) :=
k
Y
l=1
C(γl)
G(x, −Dξ).
Since each C(γl)
G(x, −Dξ) acts as multiplication operators under the inverse Fourier transform,
the differential operators C(γl)
G(x, −Dξ), l= 1,...,k, commute. The order of C(γ)
G(x, −Dξ) is
|γ|:= Pk
l=1 γl. For j∈N, define the finite set
Ij:= {γ∈ ∪∞
k=1Nk
≥3:|γ|=j+ 2k}.
For instance, we have that
I1={3}, I2={(3,3),4},and I3={(3,3,3),(4,3),(3,4),5}.
The role of Ijwill become clear in Theorem 2.9 below describing the full symbol of QG,χ from
Equation (9) in a coordinate chart. For γ∈Nk, we set rk(γ) := k. In other words, γ∈ ∪∞
k=1Nk
≥3
belongs to Ijif and only if j=|γ| − 2rk(γ). We remark that |γ| ≥ 3 and rk(γ)>0 is implicit
for γ∈Ijsince Ij⊆ ∪∞
k=1Nk
≥3. The number of elements in Ij∩Nk
≥3is the same as the number
of ways to write j−kas a sum of knatural numbers, and so
#(Ij∩Nk
≥3) = j−1
k−1.
The following properties of Ijfollows.
Proposition 2.6. Let j > 0. The set Ij⊆ ∪k>0Nk
≥3satisfies the following
•max{|γ|:γ∈Ij}= 3jand is attained at γ=~
3∈Nj.
•max{γi:γ∈Ij}=j+ 2 and is attained at γ=j+ 2 ∈N1.
•#Ij= 2j−1.
10 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
For notational purposes, we introduce the following notation.
Definition 2.7. For an integer n∈N∪ −2N−1, we introduce the notation
ωn:= πn/2
Γn
2+ 1.
If n > 0, then ωnis the volume of the unit ball in n-dimensions.
To simplify the computations in the subsequent theorem, we note the following relations for
the Γ-function.
Proposition 2.8. For natural numbers n, k ∈Nsuch that n > 2k+ 1, we have that
Γn+ 1
2−kΓn−2k
2+ 1=√πΓ(n−2k+ 1)
2n−2k=√π(n−2k)!
2n−2k.
Moreover, we have the identities
(−1)k+1
k!π(n−1)/22n−2kΓn+ 1
2−k=(−1)k+1(n−2k)!ωn−2kω2k,for 2k < n
(−1)k+1
k!π(n−1)/22n−2kΓn+ 1
2−k=(−1)n/2+1ω2k
(2k−n)!ω2k−n
=for 2k−n∈2N
=(−1)n/2+1(2k−n+ 1)
2(2π)2k−nω2k−n+1ω2k,
(−1)n+1
2π(n−1)/2
22k−nk−n+1
2!k!=(−1)n+1
2
(2π)2k−nω2kω2k−n−1,for 2k−n∈2N+ 1
Proof. The Legendre duplication formula Γ(ζ)Γ(ζ+1/2) = 21−2ζ√πΓ(2ζ) applied to ζ=n−2k+
1 implies the first stated identity, and the second one follows from the identity Γ(1/2−m) =
(−4)mm!/(2m)!. Combining these identities with the definition of ωnproduces the first and
second identities. The third identity follows from the definition of ωn.
We now arrive at the main result of this subsection, describing the full symbol of QG,χ . The
reader should keep in mind that we are primarily interested in the function G(x, y) := d(x, y)2
for a distance function d such that d2is regular at the diagonal. In this case QG,χ (R) = Q(R)
is the localization of Zat the diagonal for Re(R)>0. We therefore formulate our results on the
sector C+, albeit for QG,χ they hold in the sector C\iR.
Theorem 2.9. Let Mbe an n-dimensional manifold and G:M×M→[0,∞)a function which
is regular at the diagonal. We denote the Riemannian metric on T∗Mdual to the transversal
Hessian HGby gG, as in Proposition 2.5. Consider the operator
QG,χ(R)f:= 1
|R|Mc ZM
χ(x, y)e−|R|Mc √G(x,y)f(y)dy,
where χ∈C∞(M×M)a function with χ= 1 near DiagMand supported only where Gis
smooth, and we use the Riemannian volume density defined from gG.
We have that QG,χ ∈Ψ−n−1
cl (M;C+)is a classical elliptic pseudodifferential operator with
parameter of order −n−1with principal symbol
σ−n−1(QG,χ)(x, ξ, R) = n!ωn(R2+gG(ξ, ξ ))−(n+1)/2.
In a coordinate chart U0on M, the full symbol qof QG,χ has a classical asymptotic expansion
q∼P∞
j=0 qjcomputed from the Taylor expansion (10) and each qjis the homogeneous symbol
of degree −n−1−jwhich for j > 0and for nodd is given by
qj(x, ξ, R) = X
γ∈Ij,rk(γ)<(n+1)/2
crk(γ),nC(γ)
G(x, −Dξ)(R2+gG(ξ, ξ ))−(n+1)/2+rk(γ)−
−X
γ∈Ij,rk(γ)≥(n+1)/2
crk(γ),nC(γ)
G(x, −Dξ)h(R2+gG(ξ, ξ ))−(n+1)/2+rk(γ)log(R2+gG(ξ, ξ))i,
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 11
and for neven given as
qj(x, ξ, R) = X
γ∈Ij
crk(γ),nC(γ)
G(x, −Dξ)(R2+gG(ξ, ξ))−(n+1)/2+rk(γ)
Here the coefficients are computed as
ck,n :=
(−1)k(n−2k)!ωn−2kω2k,for 2k < n
(−1)1−n/2ω2k
(2k−n)!ω2k−n,for 2k−n∈2N
(−1) n+1
2
(2π)2k−nω2kω2k−n−1,for 2k−n∈2N+ 1
Remark 2.10.Exact expressions for q1and q2are given below in Proposition 2.14 and 2.15,
respectively.
Remark 2.11.From the expression for qjin Theorem 2.9, it is not clear that qjis homogeneous
of degree −n−1−jwhen nis odd. We shall see in the proof that this is in fact the case.
Proof of Theorem 2.9. By Proposition 2.4, the Schwartz kernel 1
|R|χ(x, y)e−|R|√G(x,y)of QG,χ
is the Fourier transform in the η-direction of
K(x, y, η) = −1
2πχ(x, y) log(η2+G(x, y)).
The extra 2πis coming from Fourier inversion in one dimension.
Let Udenote a neighborhood of the diagonal DiagMon which Gis smooth. It follows from
Proposition 2.5 (cf. Proposition A.10) that K∈I−n−1(U×R; DiagM× {0}) with principal
symbol
σ−n−1(K)(x, ξ, R) = n!ωn(R2+gG(ξ , ξ))−(n+1)/2.
Define K0(x, y, η) := log(η2+G(x, y)) ∈I−n−1(U×R; DiagM×{0}). We compute in a coordinate
chart U0that K0∈CI −n−1(U×R; DiagM×{0}) and using a uniform asymptotic expansion we
use Proposition B.14 to show that the Fourier transform in the η-direction of Kis the Schwartz
kernel of a pseudodifferential operator with parameter.
In a coordinate chart U0, we introduce the coordinates (x, v) = (x, x −y) on U0×U0. Using
Equation (10), we can write
K0(x, y, η) = −log(η2+HG(v, v)) −log 1 + PN
j=3 C(j)
G(v, v) + rN(x, v)
η2+HG(v, v)!.
For small v, we can Taylor expand
K0(x, y, η) = −log(η2+HG(v, v)) −
N
X
j=1 X
γ∈Ij
(−1)rk(γ)
rk(γ)
C(γ)
G(v)
(η2+HG(v, v))rk(γ)+ ˜rN(x, v, η).
We note that, by the definition of Ij, each term in the second sum Pγ∈Ij
(−1)rk(γ)
rk(γ)
C(γ)
G(v)
(η2+HG(v,v))rk(γ)
is homogeneous of degree j. We also note that ˜rN(x, v, η) = O((|η|+|v|)N+1) and a short
computation gives that ∂α
x∂β
v∂k
η˜rN=O((|η|+|v|)N+1−|β|−k) for any multiindices α,βand k.
As such, we have a uniform asymptotic expansion K∼P∞
j=0 Kj(cf. Definition B.13) where
(11) Kj(x, v, η) = (−1
2πlog(η2+HG(v, v)), j = 0,
1
2πPγ∈Ij
(−1)rk(γ)+1
rk(γ)
C(γ)
G(v)
(η2+HG(v,v))rk(γ), j > 0.
We conclude from Proposition B.14 that QG,χ ∈Ψ−n−1
cl (M;R) is a pseudodifferential operator
with parameter. Proposition B.14 implies that
σ−n−1(QG,χ)(x, ξ, R) = σ−n−1(K)(x, ξ, R) = n!ωn(R2+gG(ξ , ξ))−(n+1)/2.
This is invertible in C∞(S(T∗M⊕R)) so QG,χ is elliptic with parameter. It is readily seen that
σ−n−1(QG,χ)(x, ξ, R)6= 0 also for R∈C+, and the following symbol computation shows that
QG,χ ∈Ψ−n−1
cl (M;C+) is an elliptic pseudodifferential operator with parameter.
12 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Let us turn to the full symbol of QG,χ. We will compute it from Equation (11) and the
Fourier transform computations of Subsection A.1 of the appendix. For k > 0, denote the
Fourier transform of (η2+HG(v, v))−kin the (v, η)-direction by Fk(x, ξ, R), that is
Fk(x, ξ, R) := ZTxM⊕R
e−iξ.v−iRη
(η2+HG(v, v))kdvdη.
Using homogeneity and rotational symmetries, Fkcan be computed as in Proposition A.7 to be
π(n+1)/22−2k+n+1Γn+1
2−k
(k−1)! (R2+gG(ξ, ξ))−(n+1)/2+k,
for 2k−n−1/∈2N, and
(12)
(−1)k−n+1
2π(n+1)/2
22k−n−1k−n+1
2!(k−1)! h(R2+gG(ξ, ξ))−(n+1)/2+k−log(R2+gG(ξ, ξ )) + βk−(n+1)/2,n+1i,
for 2k−n−1∈2N. It follows that for γ∈Ij, the symbol of the term
C(γ)
G(v)
(η2+HG(v, v))rk(γ),
is given by C(γ)
G(x, −Dξ)Frk(γ)(x, ξ, R) which is computed to be
π(n+1)/22−2rk(γ)+n+1Γn+1
2−rk(γ)
(rk(γ)−1)! C(γ)
G(x, −Dξ)(R2+gG(ξ, ξ))−(n+1)/2+rk(γ),
for 2rk(γ)−n−1/∈2N, and
(−1)rk(γ)−n+1
2π(n+1)/2
22rk(γ)−n−1rk(γ)−n+ 1
2!(rk(γ)−1)!·
C(γ)
G(x, −Dξ)h(R2+gG(ξ, ξ))−(n+1)/2+rk(γ)−log(R2+gG(ξ, ξ )) + βrk(γ)−(n+1)/2,n+1i,(13)
for 2rk(γ)−n−1∈2N. We conclude that the symbol qjof Kj, for j > 0, is (up to the term
βrk(γ),n+1) given by the formula in the statement of the theorem for the pre-factors ck,n:
ck,n =
(−1)k+1
k!π(n−1)/22n−2kΓn+1
2−k,for 2k−n−1/∈2N,
(−1) n+1
2π(n−1)/2
22k−n(k−n+1
2)!k!,for 2k−n−1∈2N.
Therefore ck,n takes the form prescribed in the theorem by Proposition 2.8.
We finish the proof by showing that βrk(γ)−(n+1)/2,n+1 does not contribute in Equation (13)
and that there is no logarithmic term when expanding the ξ-derivatives in Equation (13). If
2rk(γ)−n−1∈2N, then −(n+ 1)/2 + rk(γ)∈Nand so (R2+gd(ξ, ξ ))−(n+1)/2+rk(γ)is a
polynomial of degree 2rk(γ)−n−1∈2Nin ξ. For γ∈Ij⊆ ∪kNk
≥3, we have that
j+ 2rk(γ) = |γ| ≥ 3rk(γ).
Therefore, if 2rk(γ)−n−1∈2Nthen C(γ)
d2(x, Dξ)(R2+gd(ξ, ξ))−(n+1)/2+rk(γ)is a polynomial
of degree 2rk(γ)−n−1−|γ| ≤ −rk(γ)−n−1<0 and therefore it must be identically zero. In
other words, we have the equality
(14) C(γ)
G(x, −Dξ)(R2+gG(ξ, ξ))−(n+1)/2+rk(γ)= 0,
for 2rk(γ)−n−1∈2N. This equality proves that qjcan not contain a term with a logarithmic
factor, and as such qjis homogeneous of degree −n−1−jfor all j, and
C(γ)
G(x, −Dξ)h(R2+gG(ξ, ξ))−(n+1)/2+rk(γ)−log(R2+gG(ξ, ξ )) + βrk(γ)−(n+1)/2,n+1i=
=−C(γ)
G(x, −Dξ)h(R2+gG(ξ, ξ))−(n+1)/2+rk(γ)log(R2+gG(ξ, ξ ))i.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 13
We note the following corollary that will play a role later in the paper when we consider
manifolds with boundary, and study the inverse of Qin the Boutet de Monvel calculus. See
more in Theorem 4.9 and Appendix C. Recall that a complete classical parameter dependent
symbol p∼Plplof order mon a compact manifold Mis said to satisfy the µ-transmission
condition at a hypersurface Y⊆Mif in local coordinates x= (x′, xn), with Ybeing defined by
xn= 0, satisfies that
Dβ
xDα
ξDj
Rpl((x′,0),(0, ξn),0) = eπi(m−l−2µ−|α|−j)Dβ
xDα
ξDj
Rpl((x′,0),(0,−ξn),0).
One sometimes in short says that phas type µ. A classical pseudodifferential operator with
parameter is said to be of type µif its symbol is of type µ. See more in for instance [15,
Proposition 1]. Item (b) of Theorem 3.7 implies the following.
Corollary 2.12. Let Mbe a manifold and G:M×M→[0,∞)a function which is regular at
the diagonal. Set µ:= −(n+ 1)/2. The pseudodifferential operator with parameter QG,χ satisfies
the µ-transmission condition along any hypersurface in M. In fact, the asymptotic expansion
P∞
l=0 qlof the full symbol of Qfrom Theorem 3.7 satisfies that
Dβ
xDα
ξDj
Rql(x, ξ, R) = (−1)|α|+j+lDβ
xDα
ξDj
Rql(x, −ξ, −R),
for any multiindices α, β ∈Nn,x∈Mand (ξ, R)6= 0.
Remark 2.13.The computation that
q0(x, ξ, R) = n!ωn(R2+gG(ξ, ξ))−(n+1)/2,
is compatible with known symbol computations in Rnfor G(x, y ) = |x−y|2in which case q=q0
is the full symbol expansion when using Euclidean coordinates. Indeed, this statement follows
from the fact that e−R|v|is the Fourier transform of n!ωn(R2+|ξ|2)−(n+1)/2, see for instance [2,
Equation (3)]. We remark that the coordinate dependent symbol computations of Theorem 2.9
will be used also in Rn. The reason for using Theorem 2.9 in Rnis that to describe the operator
near the boundary of a domain with the Wiener-Hopf factorization techniques of Section 5 below
we need to “straighten out the boundary”, i.e. choose coordinates in which the boundary locally
looks like a half-space. We make such computations more precise in Subsection 2.2 below.
Let us give some further details in computing the symbols q1and q2. The precise information
contained in q1and q2will be used later to compute the first terms in the inverse of Qand the
asymptotics of its expectation values in Section 6. Before entering into the symbol computations
of q1and q2, let us introduce some notation. Since gis a metric on T∗M, it can be viewed as
a symmetric tensor in T M ⊗T M and for a covector ξ, the contraction ιξgGtakes values in
T M . In the coordinate chart, each Cj
Gtakes values in the symmetric j-forms on T M |U0. As
such, expressions such as C3
G(x, gG⊗ιξgG) or C4
G(x, gG⊗gG), for instance, make sense. The
reader should be aware that such expressions are individually not coordinate invariant, the
transformation rules for these expressions can be deduced either from the Taylor expansion (10)
or from the transformation rules for pseudodifferential operators. For computational purposes,
we also note that
dξgG(ξ, ξ) = 2ιξgG.
Here dξdenotes the fiberwise exterior differential and we are implicitly using the canonical
identification T∗(T∗M) = π∗T M ⊕π∗T∗Mand that the T M -summand is where dξmaps to.
A useful tool in formulating the computations is the Pochhammer k-symbol. For x∈R,n∈N
and k∈Z, we write
(x)n,k := x(x+k)(x+ 2k)···(x+ (n−1)k)
|{z }
nfactors
,
with the convention that (x)0,k = 1 for any kand (0)n,k = 1 for any nand k.
A combinatorial argument shows that
C3
G(x, −Dξ)(R2+gG(ξ, ξ ))−ν=−3·22i(ν)2,1C3
G(x, gG⊗ιξgG)(R2+gG(ξ, ξ ))−ν−2+
+ 23i(ν)3,1C3
G(x, ιξgG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ ))−ν−3.
14 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
C4
G(x, −Dξ)(R2+gG(ξ, ξ ))−ν=3 ·22(ν)2,1C4
G(x, gG⊗gG)(R2+gG(ξ, ξ ))−ν−2−
−6·23(ν)3,1C4
G(x, gG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ ))−ν−3+
+ 24(ν)4,1C4
G(x, ιξg⊗ιξgG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ ))−ν−4.
C3
G(x, −Dξ)2(R2+gG(ξ, ξ))−ν=−24 ·24(ν)4,1C3
G(x, gG⊗ιξgG)2(R2+gG(ξ, ξ))−ν−4+
+ 6 ·25(ν)5,1C3
G(x, ιξgG⊗ιξgG⊗ιξgG)C3
G(x, gG⊗ιξgG)(R2+gG(ξ, ξ))−ν−5−
−26(ν)6,1C3
G(x, ιξgG⊗ιξgG⊗ιξgG)2(R2+gG(ξ, ξ))−ν−6+
+ 3 ·23(ν)3,1(C3
G⊗C3
G)(x, gG⊗gG⊗gG)(R2+gG(ξ, ξ))−ν−3.
Proposition 2.14. Let Mand QG,χ be as in Theorem 2.9. In a coordinate chart on M, the
term q1of degree −n−2appearing in the full symbol qof QG,χ is given by
q1(x, ξ, R) = −i6C3
G(x, gG⊗ιξgG)(R2+gG(ξ, ξ))−2−
−8C3
G(x, ιξgG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ))−3,
if n= 1, and if n > 1we have that
q1(x, ξ, R) = −i(n2−1)c1,n 3C3
G(x, gG⊗ιξgG)(R2+gG(ξ, ξ))−(n+1)/2−1−
−(n+ 3)C3
G(x, ιξgG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ))−(n+1)/2−2,
where ιξgGdenotes the contraction of the metric gGon T∗Malong the covector ξ.
Proof. We note that I1={γ∈ ∪∞
k=1Nk
≥3:|γ|= 1 + 2rk(γ)}={3}. For n= 1, we have
c1,1=−1/2 and since we are in the critical case we compute as follows
q1(x, ξ, R) = −ic1,1C3(x, 1)∂3
ξlog(R2+gG(ξ, ξ )) = iC3(x, 1)∂2
ξ
ιξgG
R2+gG(ξ, ξ)=
=iC3
G(x, 1)−6gGιξgG
(R2+gG(ξ, ξ))2+8(ιξgG)3
(R2+gG(ξ, ξ))3,
which proves the case n= 1.
For n > 1, we compute using the identities above that
q1(x, ξ, R) = i(n+ 1)(n−1)c1,n −3C3
G(x, gG⊗ιξgG)(R2+gG(ξ, ξ))−(n+1)/2−1+
+ (n+ 3)C3
G(x, ιξgG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ))−(n+1)/2−2.
The next proposition follows from a similar computation.
Proposition 2.15. Let Mand QG,χ be as in Theorem 2.9. In a coordinate chart on M, the
term q2of degree −n−3appearing in the full symbol qof QG,χ is given as follows; for n= 1
we have that
q2(x, ξ, R) =c2,1C3(x, 1)2∂6
ξ(R2+gG(ξ, ξ)) log(R2+gG(ξ, ξ))+c1,1C4(x, 1)∂4
ξlog(R2+gG(ξ, ξ )) =
=−C3(x, 1)2
16 −23·15g3
G
(R2+gG)2+24·90(ιξgG)2g2
G
(R2+gG)3−24·80(ιξgG)4gG
(R2+gG)4+26·24(ιξgG)6
(R2+gG)5
−C4(x, 1)
2−22·3g2
G
(R2+gG)2+23·12(ιξgG)2gG
(R2+gG)3−24·6(ιξgG)4
(R2+gG)4,
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 15
and for n= 3, we have that
q2(x, ξ, R) =c2,3C3(x, ∂ξ)2log(R2+gG(ξ, ξ))+c1,3C4(x, ∂ξ)(R2+gG(ξ, ξ))−1=
=c2,3240(C3
G⊗C3
G)(gG⊗gG⊗gG)(R2+gG(ξ, ξ))−3−
−c2,345 ·96C3
G(x, gG⊗ιξgG)2(R2+gG(ξ, ξ))−4+
+c2,3120 ·96C3
G(x, ιξgG⊗ιξgG⊗ιξgG)C3
G(x, gG⊗ιξgG)(R2+gG(ξ, ξ))−5−
−c2,380 ·96C3
G(x, gG⊗ιξgG)2(R2+gG(ξ, ξ))−6+
+c1,324C4(x, gG⊗gG)(R2+gG(ξ, ξ))−3−
−c1,3288C4(x, gG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ))−4+
+c1,3192C4(x, ιξgG⊗ιξgG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ))−5
and finally for n6= 1,3, we have that
q2(x, ξ, R) = −c2,n C3(x, ∂ξ)2(R2+gG(ξ, ξ)))−(n−3)/2+c1,nC4(x, ∂ξ)(R2+gG(ξ, ξ))−(n−1)/2=
=−24c2,n(n+ 3)4,−2C3(x, gG⊗ιξgG)2(R2+gG(ξ, ξ))−(n+1)/2−2+
+6c2,n(n+ 5)5,−2C3(x, ιξgG⊗ιξgG⊗ιξgG)C3(x, gG⊗ιξgG)(R2+gG(ξ, ξ ))−(n+1)/2−3−
−c2,n(n+ 7)6,−2C3(x, ιξgG⊗ιξgG⊗ιξgG)2(R2+gG(ξ, ξ ))−(n+1)/2−4+
+ 3c2,n(n+ 5)5,−2(C3⊗C3)(x, gG⊗gG⊗gG)(R2+gG(ξ, ξ))−(n+1)/2−1+
+3c1,n(n2−1)C4(x, gG⊗gG)(R2+gG(ξ, ξ))−(n+1)/2−1−
−6c1,n(n+ 3)3,−2C4(x, gG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ ))−(n+1)/2−2+
+c1,n(n+ 5)4,−2C4(x, ιξgG⊗ιξgG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ ))−(n+1)/2−3,
where ιξgGdenotes the contraction of the metric gGon T∗Malong the covector ξ.
2.2. Examples and further structure in the symbol computations. In the preceding
subsection we saw a detailed computation of the full symbol of Qχ,G in terms of the Taylor
expansion. Let us consider a few important special cases where further structures can be visible
in the symbol expansion, and proceed with a structural statement of for the entries in the full
symbol in the general case.
Example 2.16 (Symbol computations near a boundary in Euclidean space).We consider the
manifold M=Rnand the function G(x, y) = |x−y|2which is regular at the diagonal. This
example fits into the bigger picture of the paper seeing that G(x, y) = d(x, y)2. It will later
in the paper be crucial to describe the symbol of Q=QG,χ near the boundary of a domain
X⊆Rnwith smooth boundary. Fix a point x0∈∂X . Up to a rigid motion, we can assume that
x0= 0 and that the normal vector of ∂X in x0is orthogonal to the plane xn= 0, where we write
x= (x′, xn) for x′∈Rn−1and xn∈R. There is a neighborhood U0=U00 ×(−ε, ε) of 0 ∈Rn
and a smooth function ϕ∈C∞(U00 ) such that X∩U0={x= (x′, xn)∈U0:ϕ(x′)< xn}. Since
the normal vector of ∂X in x0is orthogonal to the plane xn= 0, ∇x′ϕ(0) = 0. Near x0= 0, we
use the coordinates
(x′, xn)7→ (x′, xn−ϕ(x′)).
In these new coordinates, the domain Xlooks like the half-space {(x′, xn) : xn>0}locally near
x0= 0. We compute that in these coordinates G=G(x, y) can be written as
|(x′, xn−ϕ(x′))−(y′, yn−ϕ(y′))|2=|x′−y′|2+ (xn−yn−(ϕ(x′)−ϕ(y′)))2=
=|x−y|2−2(xn−yn)(ϕ(x′)−ϕ(y′)) + (ϕ(x′)−ϕ(y′))2=
=|v|2+
N
X
j=2 X
|α′|=j−1
−2∂α′
x′ϕ(x′)
α′!vn(v′)α′+
+
N
X
j=2 X
|α′|+|β′|=j,
|α′|,|β′|>0
∂α′
x′ϕ(x′)∂β′
x′ϕ(x′)
α′!β′!(v′)α′+β′+O(|v|N+1),
16 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
where the sums over α′and β′ranges over α′, β′∈Nn−1and we have written v=x−y,
v′=x′−y′and vn=xn−yn. Introducing the notation
∇jϕ(x′;v′) := X
|α′|=j
∂α′
x′ϕ(x′)
α′!(v′)α′,
we have in these coordinates that
G(x, y) =|v|2−2vn∇ϕ(x′)·v′+ (∇ϕ(x′)·v′)2+
N
X
j=3 −2vn∇j−1ϕ(x′;v′)
+
N
X
j=3 X
k+l=j,
k,l>0
∇kϕ(x′;v′)∇lϕ(x′;v′) + O(|v|N+1)
In particular, we can conclude that
HG(v) = |v|2−2vn∇ϕ(x′)·v′+ (∇ϕ(x′)·v′)2,
so HGis represented by the n×n-matrix
HG=1n−1+∇ϕ(x′)∇ϕ(x′)T−∇ϕ(x′)
−∇ϕ(x′)T1.
Therefore, in the same basis we have that
gG=H−1
G=1n−1∇ϕ(x′)
∇ϕ(x′)T1 + |∇ϕ(x′)|2.
We note that HG|T ∂X is the Riemannian metric on ∂X induced from the Euclidean metric on
Rnand the inclusion ∂X ֒→Rn. We also conclude that
(15) Cj
G(x;v) = vnCj,1
G(x;v′) + Cj,0
G(x;v′)
where
Cj,1
G(x;v′) := −2∇j−1ϕ(x′;v′) and Cj,0
G(x;v′) := X
k+l=j,k,l>0∇kϕ(x′;v)∇lϕ(x′;v).
Therefore, in these coordinates near the boundary of a domain in Rn, we have for γ∈Ijthat
C(γ)
G(x, −Dξ) = (−1)|γ|
rk(γ)
Y
l=1
X
|α′|=|γl|−1
−2∂α′
x′ϕ(x′)
α′!Dα′
ξ′Dξn+X
|α′|+|β′|=|γl|,
|α′|,|β′|>0
∂α′
x′ϕ(x′)∂β′
x′ϕ(x′)
α′!β′!Dα′+β′
ξ′
.
This gives a method for computing the homogeneous symbol qjof degree −n−1−jfor any j
following Theorem 2.9.
The principal symbol is computed as in Theorem 2.9. By Proposition 2.14 we compute for
n > 1 that
q1(x, ξ, R) = −i(n2−1)c1,n6gG(∇2ϕ)gG(ξ, ∇ϕ−en)(R2+gG(ξ, ξ ))−(n+1)/2−1−
−2(n+ 3)∇2ϕ(ιξgG, ιξgG)gG(ξ, ∇ϕ−en)(R2+gG(ξ, ξ ))−(n+1)/2−2=
=i(n2−1)c1,n6ξngG(∇2ϕ)(R2+gG(ξ, ξ))−(n+1)/2−1−
−2(n+ 3)ξn∇2ϕ((ξ′+ξn∇ϕ)⊗2)(R2+gG(ξ, ξ))−(n+1)/2−2,
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 17
By Proposition 2.15 we compute for n6= 1,3 that
q2(x, ξ, R) = −24c2,n (n+ 3)4,−24(gG(∇2ϕ))2(gG(ξ, ∇ϕ−en))2(R2+gG(ξ , ξ))−(n+1)/2−2+
+ 6c2,n(n+ 5)5,−24gG(∇2ϕ)(gG(ξ , ∇ϕ−en))2∇2ϕ(ιξgG, ιξgG)(R2+gG(ξ, ξ))−(n+1)/2−3−
−c2,n(n+ 7)6,−24(∇2ϕ(ιξgG, ιξgG))2(gG(ξ , ∇ϕ−en))2(R2+gG(ξ, ξ))−(n+1)/2−4+
+ 3c2,n(n+ 5)5,−2·4(gG(∇2ϕ))2gG(∇ϕ−en,∇ϕ−en)(R2+gG(ξ , ξ))−(n+1)/2−1+
+3c1,n(n2−1)2(gG⊗gG)(∇3ϕ⊗(∇ϕ−en)) + (gG(∇2ϕ))2(R2+gG(ξ , ξ))−(n+1)/2−1−
−6c1,n(n+ 3)3,−22(gG⊗ιξgG)(∇3ϕ)gG(ξ , ∇ϕ−en)+
+ (gG(∇2ϕ))(∇2ϕ(ιξgG, ιξgG))(R2+gG(ξ, ξ))−(n+1)/2−2+
+c1,n(n+ 5)4,−22(ιξgG)⊗3(∇3ϕ)gG(ξ, ∇ϕ−en)+
+ (∇2ϕ(ιξgG, ιξgG))2(R2+gG(ξ, ξ ))−(n+1)/2−3=
=−48c2,n(n+ 3)4,−2(gG(∇2ϕ))2ξ2
n(R2+gG(ξ, ξ ))−(n+1)/2−2+
+ 24c2,n(n+ 5)5,−2gG(∇2ϕ)ξ2
n∇2ϕ((ξ′+ξn∇ϕ)⊗2)(R2+gG(ξ, ξ ))−(n+1)/2−3−
−4c2,n(n+ 7)6,−2(∇2ϕ((ξ′+ξn∇ϕ)⊗2))2ξ2
n(R2+gG(ξ, ξ ))−(n+1)/2−4+
+ (12c2,n(n+ 5)5,−2+ 3c1,n (n2−1))(gG(∇2ϕ))2(R2+gG(ξ, ξ ))−(n+1)/2−1+
−6c1,n(n+ 3)3,−2−2ξn∇3ϕ(1n−1⊗(ξ′+ξn∇ϕ))+
+ (gG(∇2ϕ))(∇2ϕ((ξ′+ξn∇ϕ)⊗2)(R2+gG(ξ, ξ ))−(n+1)/2−2+
+c1,n(n+ 5)4,−2−2ξn(∇3ϕ)((ξ′+ξn∇ϕ)⊗3)+
+ (∇2ϕ((ξ′+ξn∇ϕ)⊗2)2(R2+gG(ξ, ξ ))−(n+1)/2−3,
and for n= 3 that
q2(x, ξ, R) =(c2,3240 ·4 + c1,324)(gG(∇2ϕ))2(R2+gG(ξ, ξ))−3−
−c2,345 ·96 ·4(gG(∇2ϕ))2ξ2
n(R2+gG(ξ, ξ ))−4+
+c2,3120 ·96 ·4gG(∇2ϕ)ξ2
n∇2ϕ((ξ′+ξn∇ϕ)⊗2)(R2+gG(ξ, ξ ))−5
−c2,380 ·96 ·4(gG(∇2ϕ))2ξ2
n(R2+gG(ξ, ξ ))−6+
−c1,3288−2ξn(∇3ϕ)(1n−1⊗(ξ′+ξn∇ϕ))+
(gG(∇2ϕ))(∇2ϕ((ξ′+ξn∇ϕ)⊗2))(R2+gG(ξ, ξ ))−4+
+c1,3192−2ξn(∇3ϕ)((ξ′+ξn∇ϕ)⊗3) + (∇2ϕ((ξ′+ξn∇ϕ)⊗2))2(R2+gG(ξ, ξ))−5.
We note that gG(∇2ϕ)(x0) is (n−1)/2 times the mean curvature in x0.
Example 2.17 (Symbol computations for a submanifold of Euclidean space).We consider a
submanifold M⊆RNand the function G(x, y) = |x−y|2which is regular at the diagonal. This
example fits into the bigger picture of the paper seeing that G(x, y) = d(x, y )2where d is the
distance function on Mmaking the inclusion M⊆RNisometric. To Taylor expand Gas in
(10), we take coordinates around a point x0∈Msuch that Mnear x0is parametrized by
(xl=xl, l = 1,...,n
xl=ϕl(x1,...,xn), l =n+ 1,...,N ,
18 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
for some functions ϕn+1,...,ϕN. Write x= (x1,...,xn). In these coordinates,
G(x, y) = |x−y|2+
N
X
l=n+1 |ϕl(x)−ϕl(y)|2=
=|v|2+
N0
X
j=2
N
X
l=n+1 X
|α|+|β|=j,
|α|,|β|>0
∂α
xϕl(x)∂β
xϕl(x)
α!β!vα+β+O(|v|N0+1),
where v=x−y. Therefore, we conclude that
HG(v) = |v|2+
N
X
l=n+1
(∇ϕl(x)·v)2,
and for j > 2,
Cj
G(x, v) =
N
X
l=n+1 X
|α|+|β|=j,
|α|,|β|>0
∂α
xϕl(x)∂β
xϕl(x)
α!β!vα+β=
N
X
l=n+1 X
i+k=j,
i,k>0
(∇iϕl⊗ ∇kϕl)(v).
Computations similar to those in Example 2.16 can be carried out also for submanifolds. To
preserve the reader’s sanity, we spare the details.
Example 2.18 (Symbol computations for geodesic distances).Consider a manifold Mequipped
with a Riemannian metric gM. To avoid having to prescribe the distance between different
components, we assume that Mis connected. The geodesic distance dgeo :M×M→[0,∞) is
defined by
dgeo(x, y) := inf{L(c) : cis a smooth path from xto y},
where the length L(c) of a path c: [0,1] →Mis defined by
L(c) := Z1
0qgM,c(t)( ˙c(t),˙c(t))dt.
Here we write ˙c: [0,1] →T M for the derivative of the path, so ˙c(t)∈Tc(t)M. For a suitable
neighborhood U⊆T M of the zero section, the Riemannian metric defines an exponential map
exp : U
U
U→M. More precisely, for a small enough v∈TxM, expx(v) = exp(x, v)∈Mis defined
in local coordinates as expx(v) = wx(v; 1) where wx(v;·) : [0,1] →T M is the solution to the
second order ordinary differential equation
(16)
¨wx(v, t) + Γwx(v;t)( ˙wx(v, t),˙wx(v, t)) = 0,
w(v; 0) = x,
˙w(v, 0) = −v,
where Γ is the affine connection defined from g, which in local coordinates is a vector valued
symmetric bilinear form on the tangent bundle. In these local coordinates, for xand yclose
enough we have that
dgeo(x, y) = |exp−1
x(y)|2
g.
In particular, d2
geo is smooth in a neighborhood of the diagonal.
Let us compute the Taylor expansion of d2
geo as in Equation (10) and prove that d2
geo is
regular at the diagonal. We use a coordinate neighborhood as above, and write v=x−y. We
are looking for the Taylor expansion in vof the coordinate function Xx(v) = exp−1
x(x−v). We
Taylor expand
wx(v, t) = x−vt +
N
X
k=2
w(k)
x(v; 0)
k!tk+O(|tv|N+1 ).
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 19
We note that v7→ w(k)
x(v; 0) is a homogeneous polynomial of degree k, we denote this by Wk(x;v).
It follows from Equation (16) that
(17)
W1(v) = −v,
W2(v) = −Γ(v, v),
W3(v) = −dΓ(v, v, v )−2Γ(v, Γ(v, v)).
The higher order terms W4, W5,... can be computed inductively from Equation (16). Write the
Taylor expansion of the unknown function Xxin vas
Xx(v) =
N
X
k=1
X(k)(x;v) + O(|v|N+1),
where X(k)(x;·) is a homogeneous polynomial of degree kin v. The identity Xx(v) = exp−1
x(x−v)
is equivalent to expx(Xx(v)) = x−vwhich implies that
(18) −
N
X
k=1
X(k)(x;v) +
N
X
k=2
Wk(x, PN
l=1 X(l)(x;v))
k!=−v+O(|v|N+1).
Considering the first order term, we see that X(1)(v) = v. The higher order terms can be
inductively determined by considering each homogeneous term separately:
(19) X(k)(v) =
k
X
j=2
Wj(x, Pk−1
l=1 X(l)(x;v))
j!
(k)
,
where [·](k)denotes the homogeneous term of degree k. Using Equation (17) and (18), we
compute the first terms to be
X(1)(v) = v,
X(2)(v) = 1
2W2(x;v) = −1
2Γ(v, v),
X(3)(v) = 1
6W3(x;v) + 1
2W2x;v−1
2Γ(v, v)(3) =
=−1
6dΓ(v, v, v ) + 1
6Γ(v, Γ(v, v )).
We summarize these computations in a proposition.
Proposition 2.19. Let Mbe a manifold equipped with a Riemannian metric gMand let dgeo de-
note the geodesic distance. Then d2
geo is regular at the diagonal and in a coordinate neighborhood
the Taylor expansion as in Equation (10) takes the form
d2
geo(x, y) = |v|2
gM+C3
d2
geo (x;v) + C4
d2
geo (x;v) + O(|v|5
gm),
where
C3
d2
geo (x;v) = −gM(v, Γ(v, v)),
C4
d2
geo (x;v) = 1
4|Γ(v, v)|2
gM+1
3(g(v, dΓ(v, v , v)) −g(v, Γ(v, Γ(v, v)))) .
The higher order terms Cj
d2
geo can be computed inductively from Equation (19).
Let us return to the general case and describe the overall structure of the terms appearing in
the full symbol expansion of QG,χ .
Lemma 2.20. Let Mand Gbe as in Theorem 2.9. The entries in the full symbol expansion
q∼P∞
j=0 qjof QG,χ ∈Ψ−n−1
cl (M;C+)takes the form
qj(x, ξ, R) =
3j
X
k=0
Pk,j (x, ξ)(R2+gG(ξ, ξ))−n+1+j+k
2,
where Pk,j are of the form
(1) Pk,j is a homogeneous polynomial of degree kin ξand
Pk,j ≡0if j−k /∈2Z.
20 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
(2) If j−k∈2Z, the polynomial Pk,j takes the form
Pk,j (x, ξ) = X
γ∈Ij
ργ,k,j C(γ)
G(x, ιξgG⊗ιξgG
|{z }
k times
⊗gG⊗ · ·· ⊗ gG
|{z }
|γ|−k
2times
),
for some coefficients ργ,k,j ∈Qπn/2+Qπ(n+1)/2+Qπ(n−1)/2.
In the last item, we note that if j−k∈2Z, then |γ| − k∈2Zfor all γ∈Ij.
Proof. It follows from the computations in Theorem 2.9 that qjcan be written as a finite sum
qj(x, ξ, R) = X
k≥0
Pk,j (x, ξ)(R2+gG(ξ, ξ))−n+1+j+k
2,
where Pk,j is a polynomial in ξwhose coefficients (as a polynomial in ξ) are all polynomials in the
Taylor coefficients of Gat the diagonal. For the degrees to match, Pk,j must be of degree k. Since
the powers R2+gG(ξ, ξ ) must differ from −(n+ 1)/2 by an integer, Pk,j = 0 unless j−k /∈2Z.
Finally, the largest possible degree kfor which ξα(R2+gG(ξ, ξ))−n+1+j+k
2(with |α|=k) can be
a summand in qjis if the derivative C(γ)
G(x, Dξ) acts only on (R2+gG(ξ, ξ ))−(n+1)/2+rk(γ)(or
(R2+gG(ξ, ξ))−(n+1)/2+rk(γ)log(R2+gG(ξ, ξ)) if 2rk(γ)−n−1∈2N) and in that case α=γ,
so the maximal degree of Pk,j is the size of the largest index in Ij, i.e. 3j. This proves item (1).
To prove item (2), one notices that by Theorem 2.9, all possible entries in qjconsists of terms
of the form
C(γ)
G(x, ιξgG⊗ιξgG
|{z }
k times
⊗gG⊗ · ·· ⊗ gG
|{z }
|γ|−k
2times
)(R2+gG(ξ, ξ))−n+1+j+k
2,
with a coefficient being a rational number times either πn/2,π(n+1)/2or π(n−1)/2.
2.3. The symbol structure of the parametrix. The operator QG,χ ∈Ψ−n−1
cl (M;C+) con-
sidered in the previous subsection is elliptic with parameter by Theorem 2.9. In particular,
it admits a parametrix AG,χ ∈Ψn+1
cl (M;C+), that is an operator with parameter so that
AG,χQG,χ −1, QG,χAG,χ −1∈Ψ−∞
cl (M;C+).
Theorem 2.21. Let Mbe an n-dimensional manifold and G:M×M→[0,∞)a function
which is regular at the diagonal. The full symbol ain local coordinates of the parametrix AG,χ ∈
Ψn+1
cl (M;C+)of QG,χ ∈Ψ−n−1
cl (M;C+)has an asymptotic expansion a∼P∞
j=0 ajwhere ajis
constructed inductively from the symbol expansion q∼Pjqjof Theorem 2.9 by
aj=−a0X
k+l+|α|=j, l<j
1
α!∂α
ξqkDα
xal.
For n= 1 the first two terms are given by
a0(x, ξ, R) = 1
2(R2+gG(ξ, ξ))
a1(x, ξ, R) = −i
2(∂ξgG)(∂xgG)(R2+gG(ξ, ξ))−1+3i
2C3
G(x, gG⊗ιξgG)−
−2iC3
G(x, ιξgG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ))−1
and for n > 1
a0(x, ξ, R) =(q0(x, ξ, R))−1=1
n!ωn
(R2+gG(ξ, ξ))(n+1)/2,
a1(x, ξ, R) =−(n+ 1)2i
n!ωn
gG(dxgG(ξ, ξ), ξ)(R2+gG(ξ, ξ))(n+1)/2−2+
+3ic1,n(n2−1)
(n!)2ω2
n
C3
G(x, gG⊗ιξgG)(R2+gG(ξ, ξ))(n+1)/2−1−
−ic1,n(n+ 3)3,−2
(n!)2ω2
n
C3
G(x, ιξgG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ))(n+1)/2−2.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 21
The homogeneous terms ajeach takes the form
aj(x, ξ, R) =
3j
X
k=0
˜
Pk,j (x, ξ)(R2+gG(ξ, ξ)) n+1−j−k
2,
where ˜
Pk,j are homogeneous polynomials of degree kin ξand
˜
Pk,j ≡0if j−k /∈2Z.
The coefficients of ˜
Pk,j as a polynomial in ξare all polynomials in derivatives of the Taylor
coefficients (C(γ)
G)γ∈∪k≤jIkof Gat the diagonal (from Equation (10)) of total degree j.
In the structural statement at the end of the proposition, the total degree refers to the degree
of the polynomial when counting an order iderivative of a Taylor coefficient C(γ)
Gfor γ∈Ikto
have degree k+i.
Proof. It follows from the parametrix construction for elliptic operators (see e.g. [38, Chapter
I.5]) that a∼P∞
j=0 ajwhere a0=q−1
0and aj:= −a0Pk+l+|α|=j, l<j 1
α!∂α
ξqkDα
xalfor j > 0.
Let us prove the structural statement about ajby induction on j. It is clear for j= 0. Assume
that the structural statement holds for l < j + 1. We have that
aj+1 =−a0X
k+l+|α|=j+1, l<j+1
1
α!∂α
ξqkDα
xal=
=−1
n!ωn
(R2+gG(ξ, ξ ))(n+1)/2X
k+l+|α|=j+1, l<j+1
N(k)
X
i1=0
˜
N(l)
X
i2=0
1
α!∂α
ξhPi1,k(x, ξ )(R2+gG(ξ, ξ))−n+1+k+i1
2i·
·Dα
xh˜
Pi2,l(x, ξ )(R2+gG(ξ, ξ)) n+1−l−i2
2i,
which proves that aj+1 has the claimed structure.
What remains is to compute a1. We write
a1(x, ξ, R) = −a0X
|α|=1
∂α
ξq0Dα
xa0−a2
0q1=
=−1
n!ωn
(R2+gG(ξ, ξ ))(n+1)/2X
|α|=1
∂α
ξ(R2+gG(ξ, ξ ))−(n+1)/2Dα
x(R2+gG(ξ, ξ ))(n+1)/2
+i(n2−1)c1,n
(n!)2ω2
n
(R2+gG(ξ, ξ ))n+13C3
G(x, gG⊗ιξgG)(R2+gG(ξ, ξ ))−(n+1)/2−1−
−(n+ 3)C3
G(x, ιξgG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ ))−(n+1)/2−2=
=−(n+ 1)2i
n!ωn
gG(dxgG(ξ, ξ ), ξ)(R2+gG(ξ, ξ))(n+1)/2−2+
+3ic1,n(n2−1)
(n!)2ω2
n
C3
G(x, gG⊗ιξgG)(R2+gG(ξ, ξ ))(n+1)/2−1−
−ic1,n(n+ 3)3,−2
(n!)2ω2
n
C3
G(x, ιξgG⊗ιξgG⊗ιξgG)(R2+gG(ξ, ξ ))(n+1)/2−2
We use the notation Γα(R0) := {z∈C:|Arg(z)|< α, Re(z)> R0}.
Corollary 2.22. Let Mbe a compact n-dimensional manifold and G:M×M→[0,∞)a
function which is regular at the diagonal. For some R0>0, the operator QG,χ(R)∈Ψ−n−1
cl (M)
is invertible as an operator H−(n+1)/2(M)→H(n+1)/2(M)for any R∈Γπ /(n+1)(R0)and
Q−1
G,χ −AG,χ ∈Ψ−∞
cl (M; Γπ/(n+1)(R0)).
In particular, Q−1
G,χ is a pseudodifferential with parameter in Γπ/(n+1)(R0)whose full symbol a
in local coordinates has an asymptotic expansion a∼P∞
j=0 ajwhere ajis as in Theorem 2.21.
22 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Corollary 2.22 follows from Theorem 2.9 and 2.21 using standard techniques for pseudodiffer-
ential with parameter. For the convenience of the reader, we include its proof.
Proof. Let ∆ denote a positive Laplace operator on Mwhose principal symbol coincides with the
metric gGdual to the transversal Hessian of G. It follows from Theorem 2.9 that σ−n−1((R2+
∆)−µ) = σ−n−1(Q). Hence, the operator r(R) := 1 −(R2+ ∆)µ/2Q(R2+ ∆)µ/2is a parameter-
dependent pseudodifferential operator of order −1. We write
QG,χ = (R2+ ∆)−µ/2(1 −r)(R2+ ∆)−µ/2.
The order of ris −1, so by Theorem B.11 (see page 63) we have that kr(R)kL2(M)→L2(M)=
O(Re(R)−1) as Re(R)→ ∞. We conclude that there exists an R0>1 such that kr(R)kL2(M)→L2(M)≤
1
2for R∈Γπ/(n+1)(R0). Since (R2+ ∆)−µ/2:Hs(M)→Hs+µ(M) is invertible for R∈
Γπ/(n+1)(0) and any s∈R, we can for R∈Γπ/(n+1) (R0) invert QG,χ as the absolutely conver-
gent series of operators Hµ(M)→H−µ(M) given by
Q−1
G,χ =
∞
X
k=0
(R2+ ∆)µ/2r(R)k(R2+ ∆)µ/2.
It remains to prove that Q−1
G,χ −AG,χ ∈Ψ−∞
cl (M; Γπ/(n+1)(R0)). By the construction above,
Q−1
G,χ ∈Ψn+1
cl (M; Γπ/(n+1)(R0)). Therefore the classes [Q−1
G,χ] and [AG,χ ] in the formal symbol
algebra Ψn+1
cl (M; Γ)/Ψ−∞
cl (M; Γπ/(n+1)(R0)) are both inverses to
[QG,χ]∈ ∪k∈ZΨk
cl(M; Γπ/(n+1) (R0))/Ψ−∞
cl (M; Γπ/(n+1)(R0)).
By the uniqueness of inverses, [Q−1
G,χ] = [AG,χ ]∈Ψn+1
cl (M; Γπ/(n+1)(R0))/Ψ−∞
cl (M; Γπ/(n+1)(R0)).
2.4. Analytic results for Qon compact manifolds. The results of the previous subsections
have analytic implications in the case that Mis a compact manifold. We consider the scale of
Hilbert spaces Hs
R(M) := (R2+∆)−s/2L2(M) defined for R∈R\{0}and s∈Rwith the Hilbert
space structure making (R2+ ∆)−s/2:L2(M)→Hs
R(M) unitary. Here ∆ could be any choice
of Laplacian, but for the sake of simplicity we fix the Laplacian associated with the Riemannian
metric associated with a function regular at the diagonal. By elliptic regularity, Hs(M) =
Hs
R(M) as vector spaces with equivalent norms independently of the choice of Laplacian, but
the Hilbert space structure differs in a non-uniform way as Rvaries. At this stage, we shall start
to concern ourselves with extensions of Qto the complex numbers, so we phrase our results in
terms of the operator
(20) Q(R)f(x) := 1
RZM
χ(x, y)e−Rd(x,y)f(y)dy,
where d is a distance function whose square is regular at the diagonal and χa function being 1
near the diagonal such that d2is smooth on the support of χ. We note that Q(R) = Qd2,χ(R)
for Re(R)>0.
Theorem 2.23. Let Mbe a compact n-dimensional manifold and d : M×M→[0,∞)a
distance function whose square is regular at the diagonal. Set µ:= (n+ 1)/2. The operator
Q(R) : H−µ(M)→Hµ(M),
defined from the expression (20) is a well defined Fredholm operator for all R∈C\{0}and there
is an R0such that Q(R)is invertible for all R∈Γπ/(n+1)(R0). Moreover, the following holds:
a) For each R∈C\ {0},Q(R)∈Ψ−n−1
cl (M)is an elliptic pseudodifferential operator and
the family of operators
(Q(R) : H−µ(M)→Hµ(M))R∈C\{0}
depends holomorphically on R∈C\ {0}. Moreover, the holomorphic family
(Q(R)−1:Hµ(M)→H−µ(M))R∈Γπ/(n+1)(R0),
extends meromorphically to R∈C\ {0}.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 23
b) There is a C > 0such that
C−1kfk2
H−µ
|R|(M)≤Rehf, Q(R)fiL2≤Ckfk2
H−µ
|R|(M),
for R∈Γπ/(n+1)(R0)and f∈H−µ(M). In particular, for R∈Γπ/(n+1)(R0),Q(R)
is coercive in form sense on L2(M)and the sesquilinear form Reh·, Q(R)·i is uniformly
equivalent to the inner product of H−µ
|R|(M).
Proof. The first statements of the theorem follows from Corollary 2.22.
Part a) follows from the meromorphic Fredholm theorem (see Appendix D) upon proving that
R7→ Q(R)∈B(H−µ(M), Hµ(M)) is holomorphic with values in the set of Fredholm operators.
We can write Q(R) as the integral operator with Schwartz kernel
(21) 1
Rχ(x, y)e−Rd(x,y)=χ(x, y )
R+
∞
X
k=0
Rk
(k+ 1)!χ(x, y)d(x, y )k+1.
Using the fact that d2is regular at the diagonal, an argument as in the proof of Theorem 2.9
shows that Q(R) is an elliptic pseudodifferential operator of order −n−1. By Proposition A.7,
the principal symbol of Q(R) (for fixed R) is given by
σ−n−1(Q(R))(x, ξ) = π(n−1)/22nΓn+ 1
2|ξ|−n−1
gG=n!ωn|ξ|−n−1
gd2.
Therefore R7→ Q(R)∈B(H−µ(M), Hµ(M)) takes values in the set of Fredholm operators. The
expression (21), and again an argument as in the proof of Theorem 2.9, shows that R7→ Q(R)∈
B(H−µ(M), Hµ(M)) depends holomorphically on R∈C\ {0}.
Part b) follows from the G˚arding inequality (see Corollary B.12 on page 64) using that Re(Q)
has positive principal symbol as a pseudodifferential operator with parameter by Theorem 2.9
on page 10).
2.5. Evaluation at ξ= 0 of some symbols. For later purposes, we will be interested in
knowing the value of the homogeneous component of the full symbol of the parametrix of Q
at ξ= 0, constructed as in Theorem 2.21. The following lemma shows that the evaluations
of symbols of operators with parameter provides coordinate invariant expressions, therefore
containing invariants of a pseudodifferential operator with parameter.
Lemma 2.24. Assume that Mis a manifold and that A∈Ψm
cl (M; Γ) is a properly supported
pseudodifferential operator with parameter. Then there exists a sequence (aj,0)j∈N⊆C∞(M×Γ)
such that
i) Each aj,0=aj,0(x, R)is homogeneous of degree m−jin R.
ii) In each local coordinate chart,
aj,0(x, R) = aj(x, 0, R),
where a∼Pjajis a homogeneous expansion of the full symbol of Ain that chart.
Moreover, for any N∈N, we have that
(22) [A(R)1](x) =
N
X
j=0
aj(x, R) + rN(x, R) =
N
X
j=0
aj(x, 1)Rm−j+rN(x, R),
where rN∈C∞(M×Γ) is a function such that for any compact K⊆Mit holds that
sup
x∈K|∂α
x∂k
RrN(x, R)|=O(Re(R)m−N+|α|+k),as Re(R)→+∞.
Proof. Choose a partition of unity (χj)j⊆C∞
c(M) subordinate to a locally finite covering
by coordinate charts, and choose (˜χj)j⊆C∞
c(M) such that ˜χjis supported in a coordinate
chart and ˜χj= 1 on supp(χj). We have that PjχjA˜χjconverges in weak sense to a properly
supported operator, and A−PjχjA˜χj∈Ψ−∞
cl (M; Γ). Therefore, we can assume that Ais
supported in a coordinate chart. In a coordinate chart, and a homogeneous expansion a∼Pjaj
of the full symbol of A, the method of stationary phase (see for instance [22, Chapter VII.7])
implies that [A(R)1](x) = PN
j=1 aj(x, 0, R) + rN(x, R) as in Equation (22). It is clear that
24 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
that the function A(R)1 ∈C∞(M) and its asymptotic expansion is independent of choice of
coordinates, and the lemma follows.
Lemma 2.25. Let Mbe an n-dimensional manifold, G:M×M→[0,∞)be a function
regular at the diagonal, and (aj,0)j∈N⊆C∞(M×C+)the sequence defined as in Lemma 2.24
from AG,χ ∈Ψn+1
cl (M;C+)(cf. Theorem 2.21). The functions aj,0are determined in local
coordinates by the properties that
aj,0(x, R) = 0,for all jodd,
and for even j,aj,0(x, R)is determined inductively from a0,0(x, R) = 1
n!ωnRn+1 and for j > 0,
(23) aj,0(x, R) = −1
n!ωn
Rn+1 X
k+2l+p=j
2l<j, 2|k+p
ipqk,p(x, R).∇p
xaj−2k,0(x, R),
where qk,p(x, R)denotes the p-linear form
qk,p(x, R).v := X
|α|=p
∂α
ξqk(x, ξ, R)vα|ξ=0 =∂p
tqk(x, tv, R)|t=0 .
Proof. By Lemma 2.24, we can perform all computations in a coordinate chart. The structural
description of ajfrom Lemma 2.21 implies that for any (x, ξ, R) and j∈Nit holds that
aj(x, −ξ, R) = (−1)jaj(x, ξ , R).
We conclude that aj(x, 0, R) = 0, and even that ∇p
xaj,0(x, R) = 0 for any p, when jis odd. To
compute ajfor even j, we note that since aj=−a0Pk+l+|α|=j, l<j 1
α!∂α
ξqkDα
xal, the formula (23)
follows using that the only contributions are for even l, and evenness of jimplies that 2|k+p.
Remark 2.26.The formulas in Theorem 2.9 shows that for nodd, and jeven,
qj,0(x, R) =R−n−1−jX
γ∈Ij,rk(γ)<(n+1)/2
crk(γ),n(−1)rk(γ)−1(|γ|/2)!(n+ 1 −2rk(γ))|γ|/2,−2C(γ)
G(x, g⊗|γ|/2
G)+
−R−n−1−jX
γ∈Ij,rk(γ)≥(n+1)/2
crk(γ),n(−1)rk(γ)−1(|γ|/2)!(n+ 1 −2rk(γ))|γ|/2−1,−2C(γ)
G(x, g⊗|γ|/2
G),
and for neven, and jeven,
qj,0(x, R) =R−n−1−jX
γ∈Ij
crk(γ),n(−1)rk(γ)−1(|γ|/2)!(n+ 1 −2rk(γ))|γ|/2,−2C(γ)
G(x, g⊗|γ|/2
G),
and crk(γ),n is as in Theorem 2.9. Note that by definition of the set Ij,γ∈Ijsatisfies that |γ|
is even if and only if jis even.
Theorem 2.27. Let Mbe an n-dimensional manifold, G:M×M→[0,∞)be a function
regular at the diagonal, and (aj,0)j∈N⊆C∞(M×C+)the restriction to 0of the full symbol of
AG,χ. Then for each j > 0,aj,0is a polynomial in (C(γ)
G)γ∈∪k≤jIkand its derivatives contracted
by the metric gGand its derivatives of total degree jwhere each C(γ)
G,γ∈Ik, has degree k, the
metric has degree zero and x-derivatives increase the order by 1.
In the special case j= 0 we have
a0,0(x, R) = 1
n!ωn
Rn+1,
and for j= 2 and n= 3, we have that
a2,0(x, R) = −24R2
(3!ω3)210c1,3C4
G(x, gG⊗gG)−c2,3(C3
G⊗C3
G)(x, gG⊗gG⊗gG),
while for j= 2 and n6= 1,3, we have that
a2,0(x, R) = −3Rn−1
(n!ωn)2c1,n(n2−1)C4
G(x, gG⊗gG)−(24)
−c2,n(n+ 5)5,−2(C3
G⊗C3
G)(x, gG⊗gG⊗gG),
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 25
and for for j= 4 we have the identity
a4,0(x, R) = −R2n+2
(n!ωn)2q4,0(x, R)−n+ 1
R2gG(∇2
xa2,0(x, R))−
−c1,n(n2−1)
R2n!ωn
(C3
G⊗ ∇xa2,0)(gG⊗gG)−Rn+1
n!ωn
q2,0(x, R)a2,0(x, R).(25)
Proof. The structural statement about aj,0(x, R) is readily deduced from Lemma 2.25 and in-
duction by using the property that if γ∈Ijand γ′∈Ij′then (γ, γ′)∈Ij+j′.
The equation for a0,0is immediate from Lemma 2.21. Using Lemma 2.25, we have that
a2,0(x, R) = −a2
0,0q2,0and the formula (24) follows from that
q2,0(x, R) =3c1,n(n2−1)C4
G(x, gG⊗gG)R−n−3+ 3c2,n(n+ 5)5,−2(C3
G⊗C3
G)(x, gG⊗gG⊗gG)R−n−3,
for n6= 1,3 and a similar expression for n= 3.
Let us compute a4,0(x, R). By Lemma 2.25 we have that
a4,0(x, R) = −Rn+1
n!ωn4
X
k=0
i−kqk,4−k(x, R).∇4−k
x
Rn+1
n!ωn−
2
X
k=0
i−kqk,2−k(x, R).∇2−k
xa2,0(x, R)=
=Rn+1
n!ωnq0,2(x, R).∇2
xa2,0(x, R)−iq1,1(x, R).∇xa2,0(x, R)−q2,0(x, R)a2,0(x, R)−
−R2n+2
(n!ωn)2q4,0(x, R),
and the computation is complete upon using Proposition 2.14.
Remark 2.28.The full expression for a4,0can be computed from Equation (25). We omit the
full details, but let us note an expression for q4,0. Since
I4={6,(3,5),(4,4),(5,3),(3,3,4),(3,4,3),(4,3,3),(3,3,3,3)},
we have that
q4,0(x, R) =R−n−5c1,n3!(n−1)3,−2C6
G(x, g⊗3
G)−R−n−5c2,n4!(n−3)4,−2C(3,5)
G(x, g⊗4
G)−
−R−n−5c2,n4!(n−3)4,−2C(4,4)
G(x, g⊗4
G)−R−n−5c2,n4!(n−3)4,−2C(5,3)
G(x, g⊗4
G)+
+R−n−5c3,n5!(n−5)5,−2C(3,3,4)
G(x, g⊗5
G) + R−n−5c3,n5!(n−5)5,−2C(3,4,3)
G(x, g⊗5
G)+
+R−n−5c3,n5!(n−5)5,−2C(4,3,3)
G(x, g⊗5
G)−R−n−5c4,n6!(n−7)6,−2C(3,3,3,3)
G(x, g⊗6
G).
Example 2.29 (Evaluations of symbols for domains in Euclidean space).Let us return to the
computations on Euclidean space from Example 2.16. We consider G(x, y) = |x−y|2– the
square of the Euclidean distance. Since q(x, ξ , R) = n!ωn(R2+|ξ|2)−(n+1)/2is a full symbol
of QG,χ in Euclidean coordinates, a(x, ξ, R) = 1
n!ωn(R2+|ξ|2)(n+1)/2is a full symbol of AG,χ .
Therefore
(26) aj,0(x, R) = (1
n!ωnRn+1, j = 0,
0, j > 0.
By Lemma 2.25 this holds in any coordinate system on Euclidean space. We remark that the
bulk of computations carried out in Example 2.16 will mainly be of interest when inverting Q
near the boundary of a domain, while the computation (26) relates to interior terms.
Example 2.30 (Evaluations of symbols for submanifolds of Euclidean space).We return to sub-
manifolds M⊆RNand the function G(x, y) = |x−y|2which is regular at the diagonal as in
Example 2.17 above. We take coordinates as in Example 2.17. By Theorem 2.27 we have that
26 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
a0,0(x, R) = 1
n!ωnRn+1 and
a2,0(x, R) = −3c1,n (n2−1)
(n!ωn)2
N
X
l=n+1 X
i+k=4,
i,k>0
(∇iϕl⊗ ∇kϕl)(g⊗2
G)Rn−1−
−3c2,n(n+ 5)3,−2
(n!ωn)2
N
X
l1,l2=n+1 X
i1+k1=i2+k2=3,
i1,i2,k1,k2>0
(∇i1ϕl1⊗ ∇k1ϕl1⊗ ∇i2ϕl2⊗ ∇k2ϕl2)(x, g⊗3
G)Rn−1,
where gGis the dual metric to the transversal Hessian:
HG(v) = |v|2+
N
X
l=n+1
(∇ϕl(x)·v)2.
Example 2.31 (Evaluations of symbols for geodesic distances).Consider the geodesic distance on
a Riemannian manifold Mas in Example 2.18. Let gMdenote the Riemannian metric, and recall
from Example 2.18 that gMcoincides with the transversal Hessian of d2
geo at the diagonal. We
can compute a2,0in this case by means of known Riemannian curvatures. We fix a point xand
choose coordinates so that Γ vanishes in that point (normal coordinates). In these coordinates,
C3
d2
geo (x, v) = 0 and C4
d2
geo (x, ·) is a third of the Riemannian curvature in xby Proposition 2.19.
By Theorem 2.27 we conclude that for n= 3, we have that
a2,0(x, R) = −80R2
(3!ω3)2sg(x),
while for n6= 1,3, we have that
a2,0(x, R) = −Rn−1
(n!ωn)2c1,n(n2−1)sg(x),
where sgdenotes the scalar curvature.
3. Global behavior of Zon compact manifolds
In the previous section we studied the localization Qof Znear the diagonal. Here Zis the
operator with parameter Rdefined from a distance function as in Equation (7). We now turn to
study Zby considering distance functions for which L:= Z − Qin a certain sense is negligible
so that Qdominates.
3.1. Controlling the off-diagonal part of Z.In this subsection we shall study the remainder
L=Z − Q. We note that
L(R)f(x) = 1
RZM
(1 −χ(x, y))e−Rd(x,y)f(y)dy.
We start making two initial observations concerning the remainder term L. The first observation
concerns the behavior of the remainder term in L2.
Proposition 3.1. Let Mbe a compact manifold and d : M×M→[0,∞)a distance function
on M. Then
C\ {0} ∋ R7→ L(R)∈ L2(L2(M)),
defines a holomorphic Hilbert-Schmidt valued function. Moreover, Lsatisfies that
kL(R)kL2(L2(M)) =O(Re(R)−∞)as Re(R)→+∞.
By the standard norm estimate kKkB≤ kKkL2, the statement in the proposition also holds
in the operator norm. We remark that by Theorem 2.9 and B.11, the localization to the diagonal
satisfies
kQ(R)kLp(L2(M)) =O(Re(R)n
p−n−1) as Re(R)→+∞,
for any p > 1 + 1/n (the bound is not uniform in p).
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 27
Proof. The function C\ {0} ∋ R7→ 1
R(1 −χ(·,·))e−Rd(·,·)∈C(M×M) is clearly holomorphic
in the norm on C(M×M). Since χ= 1 in a neighborhood of the diagonal, compactness of
Mimplies that k1
R(1 −χ(·,·))e−Rd(·,·)kC(M×M)=O(Re(R)−∞) as Re(R)→+∞. Since M
is compact, an integral operator Kwith kernel k∈C(M×M) satisfies the estimate for the
Hilbert-Schmidt norm kKkL2(L2(M)) ≤vol(M)kkkC(M×M). The proposition follows.
The second observation concerns how an a priori estimate with respect to the off-diagonal
remainder Laffects distributional solutions to the magnitude equation Zu= 1.
Proposition 3.2. Let Mbe a compact manifold and d : M×M→[0,∞)a distance function
on Msuch that d2is regular at the diagonal. Let N∈Nand Γbe a sector. Assume that
f∈C∞(M)and that (uR)R∈Γ⊆ D′(M)is a family of solutions to
Z(R)uR=f
satisfying that hL(R)uR, ψi=O(Re(R)−N)as Re(R)→+∞in Γfor all ψ∈C∞(M). Then
for large enough Re(R),
uR=Q(R)−1f+vR,
where (vR)R∈Γ⊆ D′(M)is a family satisfying that hvR, ψi=O(Re(R)−N+n+1)as Re(R)→+∞
in Γfor all ψ∈C∞(M). In particular, for any ψ∈C∞(M),
huR, ψi=hQ(R)−1f, ψi+O(Re(R)−N+n+1),as Re(R)→+∞in Γ.
Proof. The equation Z(R)uR=fimplies that
uR=Q(R)−1f−Q(R)−1L(R)uR.
Consider the distribution vR:= −Q(R)−1L(R)uR≡uR−Q(R)−1f. For ψ∈C∞(M), we have
that
hvR, ψi=hL(R)uR, Q(R)−1ψi=O(R−N+n+1),
because of the assumption on L(R)uR=O(Re(R)−N) in a weak sense and the fact that Q(R)−1
is a pseudodifferential operator with parameter of order n+ 1 preserving C∞(M) with uniform
norm estimates kQ(R)−1fkCk≤Ck(1 + Re(R))n+1kfkCn+k+1.
Proposition 3.1 implies that Lis small as an operator on L2, and Proposition 3.2 implies
that the resulting remainder term will not alter the asymptotic properties of solutions given an
a priori estimate. However, as Qis of negative order and acts compactly on L2, well-posedness
of the problem in L2is not assured. We shall circumvent this problem by imposing a regularity
assumption on the distance function that forces the magnitude equation naturally into a Sobolev
space framework. We discuss examples satisfying this assumption, as well as counterexamples,
below in the Subsections 3.2 and 3.4, respectively.
Definition 3.3 (Property (MR) of distance functions).Let Mbe an n-dimensional compact
manifold and d : M×M→[0,∞) a distance function on Msuch that d2is regular at the
diagonal. Set µ:= (n+ 1)/2. For a sector [1,∞)⊆Γ⊆C, we say that d has property (MR) on
Γ if for any R∈Γ, L(R) extends to a continuous mapping H−µ(M)→Hµ(M) with
kL(R)kH−µ(M)→Hµ(M)=O(Re(R)−∞),as Re(R)→+∞in Γ.
If L(R) : H−µ(M)→Hµ(M) is a compact operator for R∈Γ and
Γ∋R7→ L(R)∈K(H−µ(M), Hµ(M)),
is holomorphic in norm sense, we say that d has property (SMR) on Γ.
The acronyms MR and SMR stand for magnitude regularity and strong magnitude regularity,
respectively. Assuming these properties, the operator Zinherits relevant analytic and geometric
properties from Q. Property (MR) will be used to compute asymptotic solutions to the mag-
nitude equation RZu= 1, while property (SMR) will be used for constructing meromorphic
extensions of Z−1. If Γ is a sector on which a distance function has property (MR), it is clear
that 0 /∈Γ. We consider such results for compact manifolds below in Subsection 3.3.
28 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
3.2. Examples of distance functions satisfying property (MR) and (SMR). Let us give
a method to produce distance functions with property (SMR):
Proposition 3.4. Let Mbe a compact manifold embedded into a Riemannian manifold i:
M→Wsuch that the square of its geodesic distance d2
geo,W is smooth, for instance W=RN,
W=HN,Ror W=HN,C, for some N∈N.
The distance function d : M×M→[0,∞),d(x, y) := dgeo,W (i(x), i(y)) satisfies
i) d2is smooth on M×Mand regular at the diagonal.
ii) L∈Ψ−∞(M;C+)and L(R)∈Ψ−∞ (M)for any R∈C\ {0}.
iii) d has property (SMR) on C\ {0}.
Proof. Since d2
geo,W is smooth, it is clear that d2is smooth on M×M. It follows from Proposition
2.19 that d2is regular at the diagonal, and in fact gd2is the pullback metric i∗gWfrom the
Riemannian metric gWon W. Therefore the singular support support of d is the diagonal, and
the integral kernel of L(R) is smooth so L(R) extends to a continuous mapping L(R) : Hs(M)→
Ht(M) for any s, t ∈R. Moreover, for any vector fields X1,...,Xmon M×Mwe can estimate
X1···Xm(1 −χ)e−Rd≤Cm|R|m−1e−εRe(R).
Again, we write ε:= inf{d(x, y) : (x, y)∈χ−1(1)}>0. In particular, we readily can deduce
that L∈Ψ−∞(M;C+), and for any s, t ∈R,
kL(R)kHs(M)→Ht(M)=O(Re(R)−∞),as Re(R)→+∞.
Therefore d has property (SMR) on C\ {0}.
In light of Proposition 3.4, we note the following corollary of Theorem 2.23.
Corollary 3.5. Let Mbe a compact manifold equipped with a distance function d : M×M→
[0,∞)such that d2is smooth, e.g. a subspace distance as in Proposition 3.4. Then Z(R)∈
Ψ−n−1
cl (M)is an elliptic pseudodifferential operator for any R∈C\ {0}. Furthermore, Z ∈
Ψ−n−1
cl (M;C+)is elliptic with parameter and its full symbol coincides with that of Qas given in
Theorem 2.9.
Another example of distance functions with property (MR) arises on spheres.
Proposition 3.6. Let ddenote the geodesic distance on a sphere Snin its round metric. The
distance function dhas property (MR) on C\ {0}but fails to satisfy property (SMR) on any
sector.
Proof. The square of the geodesic distance is regular at the diagonal by Proposition 2.19. We note
that d is smooth on {(x, y)∈Sn×Sn:x6=±y}. Consider the operator U f(x) := f(−x). The
operator Uacts via pullback along the antipodal mapping ϕ(x) := −xwhich acts isometrically
due to O(n)-invariance of the Riemannian metric on Sn. The operator Uextends to a unitary on
all Sobolev spaces Hs(Sn), s∈R. Since it holds that d(x, y) = π−d(x, ϕ(y)) we can conclude
that the integral kernel of L(R)Uis given by χϕe−Rde−πR , where χϕ(x, y) := 1 −χ(x, ϕ(y)).
In particular, since χϕsatisfies that χϕ= 1 on the diagonal and χϕ= 0 on a neighborhood
of the off-diagonal singularities of d, the operator L(R)UeπR is an elliptic pseudodifferential
operator with parameter of order −n−1 by the same argument as in Theorem 2.9. It follows
that property (SMR) fails on any sector. Using that Uis unitary, L(R) extends to a continuous
operator H−µ(Sn)→Hµ(Sn) with
kL(R)UeπRkH−µ(Sn)→Hµ(Sn)=O(1),
as Re(R)→ ∞. Since Uis unitary, we deduce that
kL(R)kH−µ(Sn)→Hµ(Sn)=O(e−πRe(R)) = O(Re(R)−∞) as Re(R)→ ∞.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 29
3.3. Analytic results for the operator Z.We are now ready to extend the results of Section
2 for Qto results on the operator Zfor compact manifolds with a distance function satisfying
property (MR) as defined in Definition 3.3.
Theorem 3.7. Let Mbe a compact n-dimensional manifold and d : M×M→[0,∞)a distance
function on Mwith property (MR) on the sector Γ. Set µ:= (n+ 1)/2. Then there is an R0≥0
such that
Z(R) : H−µ(M)→Hµ(M),
is invertible for all R∈Γ∩Γπ/(n+1) (R0). Moreover, for R∈Γ∩Γπ/(n+1)(R0)
Z−1=Q−1+R,
where Q−1∈Ψn+1
cl (M; Γπ/(n+1)(R0)) is the elliptic pseudodifferential operator with parameter
constructed in Corollary 2.22 and R:Hµ(M)→H−µ(M)is a family of operators parametrized
by R∈Γ∩Γπ/(n+1)(R0)such that
kRkHµ(M)→H−µ(M)=O(Re(R)−∞),as Re(R)→+∞in Γ∩Γπ/(n+1) (R0).
Moreover, there is a constant C > 0such that
(27) C−1kfk2
H−µ
|R|(M)≤Rehf, Z(R)fiL2≤Ckfk2
H−µ
|R|(M),
for R∈Γ∩Γπ/(n+1)(R0)and f∈H−µ(M). In particular, for R∈Γ∩Γπ/(n+1)(R0),Z(R)is
coercive in form sense on L2(M)for the H−µ-norm.
Proof. The operator Qis invertible by Corollary 2.22. We can therefore write
Q−1Z= 1 + Q−1L,
as operators on H−µ(M). Since Q−1is a pseudodifferential operator with parameter, we have
that
kQ(R)−1L(R)kH−n+1
2(M)→H−n+1
2(M)=O(Re(R)−∞),(28)
as Re(R)→+∞in Γ ∩Γπ/(n+1) (R0).
Therefore, for Re(R)≫0 the operator (1 + Q−1L)−1Q−1exists and is a left inverse to Z. By
an analogous argument,
kL(R)Q(R)−1kHn+1
2(M)→Hn+1
2(M)=O(Re(R)−∞),(29)
as Re(R)→+∞in Γ ∩Γπ/(n+1) (R0),
and Zhas the right inverse Q−1(1 + LQ−1)−1for Re(R)≫0. Therefore (1 + Q−1L)−1Q−1=
Q−1(1+ LQ−1)−1and this operator is an inverse to Z. By the estimates (28) and (29), it follows
that
R=Z−1−Q−1=Q−1(1 + LQ−1)−1−1=
∞
X
k=0
(−1)kQ−1(LQ−1)k,
as a norm convergent sum and has the required decay property as Re(R)→+∞.
The estimate (27) follows from the decay property of Rand Theorem 2.23.
The following result is immediate from Lemma 2.24 and Theorem 3.7.
Corollary 3.8. Let Mbe a compact n-dimensional manifold and d : M×M→[0,∞)a distance
function on Mwith property (MR) on the sector Γ. Take the sequence of homogeneous functions
(aj,0)j∈N⊆C∞(M×Γ∩C+)as in Lemma 2.25. Then, for any N∈N, we have that
[Z(R)−11](x) =
N
X
j=0
aj(x, R) + rN(x, R),
where rN∈C(Γ ∩C+, H−µ(M)), for µ= (n+ 1)/2, is a function such that
krN(·, R)kH−µ(M)=O(Re(R)n+1−N),as Re(R)→+∞in Γ.
30 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Remark 3.9.We remark that the role that property (MR) plays in Corollary 3.8 is to ensure
existence of a distributional solution to Z(R)uR= 1. For computing the integrated asymptotics
h1,Z(R)−11i, property (MR) is not necessary. Indeed, with no assumptions of property (MR),
but assuming that d : M×M→[0,∞) is a distance function on Msuch that d2is regular at
the diagonal and that (uR)R>R0⊆ D′(M) is a family of solutions to
Z(R)uR= 1
satisfying that hL(R)uR, ψi=O(Re(R)−N) as Re(R)→+∞for all ψ∈C∞(M), we have that
huR,1i=
N
X
j=1 ZM
aj(x, R)dx+O(Re(R)n+1−N),
for any N. This follows from Lemma 2.24 and Proposition 3.2.
One instance where solutions to Z(R)uR= 1 exist, yet the distance function need not sat-
isfy property (MR), is the geodesic distance on a compact symmetric space M=G/H . See
Proposition 3.17 below for examples of symmetric spaces failing to satisfy property (MR). The
problem Z(R)uR= 1 was studied for compact symmetric spaces in [44]. For a compact symmet-
ric space M=G/H, we use the normalized G-invariant measure induced by the Haar measure.
By symmetry, the function
uR(x) = 1
RG/H e−Rd(x,y)dy,
is constant and therefore solves Z(R)uR= 1. It is readily verified that L(R)uR=O(R−∞ ) in
distributional sense. We conclude that each aj(x, R) is constant and that
uR(x) = uR(eH) =
N
X
j=0
aj(eH, R) + O(Re(R)n+1−N),
for any N. For examples of computations of uRfor compact symmetric spaces, see [44].
The next result poses an obstruction to property (MR) for distance functions and should
be viewed as complementary to Theorem 3.16. Recall the following terminology from [32]: a
compact metric space (X, d) is said to be positive definite if for any finite subset F⊆X, the
matrix (e−d(x,y))x,y∈Fis positive definite. If (X, Rd) is positive definite for all R > 0, we say
that (X, d) is stably positive definite.
Corollary 3.10. Assume that dis a distance function on a compact manifold Mwith property
(MR) on [1,∞). Then there exists an R0≥0such that (M, Rd) is positive definite for all
R > R0.
Proof. Consider the quadratic form qR(u) = hu, Z(R)uiL2,u∈H−µ(M). By Theorem 3.7, qRis
positive definite for R > R0, for some R0≥0. In particular, for any subspace V⊆H−µ(M) the
restriction of qRto Vis also positive definite for R > R0. For a finite subset F⊆M, consider
the subspace VF⊆H−(µ(M) spanned by {δx:x∈F}. In the basis (δx)x∈F, the quadratic
form qR|Vis represented by the |F| × |F|-matrix (e−Rd(x,y))x,y ∈Fand so it is positive definite
for R > R0.
Remark 3.11.It was proven in [32, Subsection 3.2] that if Mis a compact Riemannian manifold
with π1(M)6= 0, the geodesic distance is not stably positive definite, i.e. there exists an R > 0
and a finite subset F⊆Msuch that (e−Rdgeo (x,y))x,y∈Ffails to be positive definite. The
manifold M=S1is not simply connected, and therefore fails to be stably positive definite. but
nevertheless Proposition 3.6 implies that M=S1with its geodesic distance has property (MR).
Therefore, property (MR) does not imply stably positive definiteness of the metric space but
only an asymptotic version thereof.
Theorem 3.12. Let Mbe an n-dimensional compact manifold with a distance function dhaving
property (SMR) on Γand set µ= (n+ 1)/2. Then the operator
Z(R) : H−µ(M)→Hµ(M),
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 31
is a well defined Fredholm operator for all R∈Γinvertible for R∈Γ∩Γπ/(n+1) (R0). Moreover,
the operator Z(R) : H−µ(M)→Hµ(M)depends holomorphically on Rin Γand Z(R)−1:
Hµ(M)→H−µ(M)depends holomorphically on R∈Γ∩Γπ/(n+1) (R0)and admits a meromor-
phic extension to Γ.
Proof. By Theorem 2.23, Qis a holomorphic function on Γ with values in the Fredholm operators
and by property (SMR), Lis a compact valued holomorphic function on Γ. Therefore Zdefines
a holomorphic function on Γ with values in the Fredholm operators. Since Zis invertible for a
large enough R, see Theorem 3.7, the theorem follows from the meromorphic Fredholm theorem,
see Appendix D.
Remark 3.13.To ensure holomorphicity of Zand Z−1on sectors, the full property (SMR) is not
needed. Indeed, if d has property (MR) on a sector Γ and Γ ∋R7→ L(R)∈B(H−µ(M), H µ(M))
is additionally holomorphic (in norm sense) then by Theorem 3.7, for some R0≥0, the mapping
Γ∩Γπ/(n+1)(R0)∋R7→ Z(R)∈B(H−µ(M), Hµ(M))
is a holomorphic family of invertible operators. These properties are inherited by its inverse,
Γ∩Γπ/(n+1)(R0)∋R7→ Z(R)−1∈B(Hµ(M), H−µ(M)).
By the proof of Proposition 3.6, this discussion applies to Snshowing that for some R0,Z(R)−1
is holomorphic for R∈Γ∩Γπ /(n+1)(R0) in this case.
The following result follows from Corollary 3.5 and Theorem 3.12.
Theorem 3.14. Let Mbe a compact manifold with a distance function dsuch that d2is smooth
on M×Mand regular at the diagonal (cf. Proposition 3.4), then ˜
Z(R) : H−µ(M)→Hµ(M)
depends holomorphically on R∈C\{0}and the operator Z(R)−1:Hµ(M)→H−µ(M)extends
meromorphically to C\ {0}.
3.4. Examples of distance functions that fail to satisfy property (MR). Property (MR)
of a distance function, as defined in Definition 3.3, is a notable restriction on the singular
support and singularity structure of the distance function. We remark here that by a singular
point, we mean any point in the singular support, i.e. one in which the function is not C∞.
To better understand how these singularities affect the operator theoretic properties of L, the
reader is encouraged to review the proof of Proposition 3.6 where a crucial feature used in the
proof is that the geodesic distance on spheres near an off-diagonal singularity has the same
singular features as it has near the antipode of the singularity. An important property used
there can be stated as having control of the dimension of the off-diagonal singular support of the
metric. We make an elementary observation that follows from the smoothness of the function
(0,∞)∋t7→ √t∈(0,∞).
Proposition 3.15. Let dbe a distance function on a manifold M. Then it holds that
singsupp(d) \DiagM= singsupp(d2)\DiagM.
Let us give a sufficient condition for a distance function (regular at the diagonal) not to satisfy
property (MR). We note that an additional obstruction was provided above in Proposition 3.10.
Theorem 3.16. Let Mbe a compact manifold and da distance function such that d2is regular at
the diagonal. Assume that there exists a submanifold N⊆M×Mwith N⊆singsupp(d2)\DiagM
such that any point z0∈Nadmits a neighborhood U0in M×Mand a coordinate chart
ϕ:Rdim(N)
t×Rdim(M)−dim(N)
s→U0,
with ϕ(0) = x0,ϕ−1(U0∩N) = Rdim(N)
t× {0}and (t, s)7→ ϕ∗d(t, s)− |s|being smooth in a
neighborhood of 0. Then L(R)does not extend to a continuous operator
H−µ(M)→Hs(M),
for any s > (3 dim(M) + 1)/2−dim(N)where µ= (dim(M) + 1)/2. In particular, if dim(N)>
dim(M)then ddoes not have property (MR) (see Definition 3.3) on any sector containing a
half-ray [R0,∞).
32 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Proof. We pick a point (x0, y0)∈Nand neighborhoods U1and U2of x0and y0respectively,
such that there exists a coordinate chart ϕas in item ii) above on U0=U1×U2. Pick functions
χ1∈C∞
c(U1) and χ2∈C∞
c(U2) such that χ1= 1 near x0,χ2= 1 near y0and such that
(t, s)7→ ϕ∗d(t, s)−|s|is smooth on ϕ−1(supp(χ1)×supp(χ2)). Clearly, it suffices to prove that for
any R > 0, the operator χ1L(R)χ2does not extend to a continuous operator H−µ(M)→Hs(M)
for any s > (3 dim(M) + 1)/2−dim(N).
Set k= dim(N) and n= dim(M). The Schwartz kernel of χ1L(R)χ2is a distribution on
M×Msupported in U0=U1×U2and pulling this kernel back along ϕ, we arrive at the
distribution
K(t, s) = χ0(s, t)e−R|s|,
where χ0=ϕ∗[(χ1⊗χ2)(1 −χ)]e−Rψ ∈C∞
c(R2n) for ψ(t, s) = ϕ∗d(t, s)− |s|. It follows from
Proposition A.7 and combining a Taylor expansion with asymptotic completeness, that K∈
CI k−2n−1(R2n,Rk). We conclude that χ1L(R)χ2is a Fourier integral operator of order k−2n−1
and this operator is elliptic in a neighborhood of (x0, y0). Therefore, since χ1L(R)χ2is elliptic
near (x0, y0) of order k−2n−1 it does not extend to a continuous operator H−(n+1)/2(M)→
Hs(M) for any
s > −(n+ 1)/2−(k−2n−1) = (3n+ 1)/2−k.
Proposition 3.17. Let n > 1. The geodesic distance on the n-dimensional torus M=Tn
or the real projective space M=RPnfails to satisfy property (MR) (see Definition 3.3) on
any sector containing a half-ray [R0,∞). In fact, L(R)does not extend to a continuous map
H−µ(M)→Hs(M)for s > −n/2 + 3/2.
Proof. The proofs for both cases follow the same lines and rely on Theorem 3.16. For RPnwe
give a more geometric argument, and for Tnwe give a coordinate oriented argument.
We first prove the result for real pro jective space. The projective space RPnis the quotient
of Snby the antipodal map x7→ −x. This quotient map is a covering map, and it is locally
isometric with respect to the geodesic distance. For x∈RPn, the equator ˜
E(x)⊆Snis defined
by
˜
E(x) = {v∈Sn:v·x= 0}.
The condition v·x= 0 being invariant under the antipodal map, ˜
E(x) only depends on x∈RPn
and not on a choice of pre-image of xin Sn. We let E(x)⊆RPndenote the image under the
quotient map. The off-diagonal singular support of the geodesic distance on RPnis the set
{(x, y)∈RPn×RPn:y∈E(x)}.
Indeed, any geodesic in RPnfrom a point xlifts uniquely up to a geodesic on Snup until the
point it crosses E(x) where the geodesic distance dgeo (x, ·) has a kink. The projection mapping
p1: singsupp(dgeo)\DiagRPn→RPnis a locally trivial RPn−1-bundle on RPn. Therefore,
N= singsupp(dgeo)\DiagRPn→RPnis a manifold of dimension 2n−1> n and suitable local
trivializations of p1:N→RPnsatisfy the assumptions of item ii) in Theorem 3.16. We conclude
from Theorem 3.16 that L(R) has no bounded extension to an operator H−µ(RPn)→Hs(RPn)
for any s > −n/2 + 3/2.
We now prove the result for the n-dimensional torus Tn. Write Tn=Rn/Zn. While Tnis a
symmetric space, i.e. Tn=G/H for G=Tnand H= 1, it is instructive to consider d(x, y ) for
x= 0. We have that d(0, y) = |y|where we represent yby as an element of the fundamental
domain [−1/2,1/2)n, and Tn∋y7→ d(0, y) as a function on Tnis the Zn-periodic extension of
[−1/2,1/2)n∋y7→ |y|. Therefore y7→ d(0, y) has kinks on the image of ∂([−1/2,1/2)n) in Tn.
In particular, we see that the off-diagonal singular support of d is the set
{(x, y)∈Tn×Tn:x−y∈∂([−1/2,1/2)n) + Zn}.
Consider the submanifold N0:= {1/2}×(−1/4,1/4)n−1⊆Rnand define the 2n−1-dimensional
submanifold
N:= {(x, y)∈Tn×Tn:x−y∈N0+Zn} ⊆ singsupp(d2)\DiagTn.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 33
Consider a point
x0= (x, 1/2, y′)∈N.
On the open ball of radius 1/4 centred at x0, we introduce the coordinates u=x1−y1−1/2
and t= (t′, y) where t′=x′−y′in terms of standard coordinates x= (x1, x′) and y= (y1, y′).
In these coordinates, we have that
d(x, y) = s1
2− |u|2
+|t′|2=r−|u|+|t′|2+1
4+|u|2.
By shrinking the neighborhood of x0, we can Taylor expand
r−|u|+|t′|2+1
4+|u|2=r|t′|2+1
4+|u|2s1−|u|
|t′|2+1
4+|u|2=
∞
X
k=0
α
α
αk|u|k
(|t′|2+1
4+|u|2)k−1=
=|u|
∞
X
k=0
α
α
α2k+1 |u|2k
(|t′|2+1
4+|u|2)2k
|{z }
g(u,t′)
+
∞
X
k=0
α
α
α2k|u|2k
(|t′|2+1
4+|u|2)2k−1
|{z }
˜g(u,t′)
.
The functions gand ˜gare smooth near 0. Since g(0,0) 6= 0, we can define the new coordinate
s:= u˜g(u, t′). We conclude that in these coordinates d(x, y)− |s|is smooth. We conclude from
Theorem 3.16 that L(R) has no bounded extension to an operator H−µ(Tn)→Hs(Tn) for any
s > −n/2 + 3/2.
Remark 3.18.The analytical issues arising from the remainder term Lreflect fundamental prob-
lems in Riemannian geometry. The singular support of the geodesic distance is akin to the
conjugate locus of the Riemannian metric, which in general is hard to describe, see [5, 42] and
for related technical issues arising in the X-ray transform on a Riemannian manifold see [20].
Furthermore, Theorem 3.16 shows that even when the singular support is a tractable set, i.e.
when it looks like a submanifold near some point, dimensional obstructions to property (MR) ap-
pear. This gives rise to the analytic problem that the operator Lin the decomposition Z=Q+L
is in general of order higher than −n−1 in the Sobolev order, while it is infinitely decaying in
the parameter R. In this case Lis of higher order than Q, which is elliptic with parameter of
order −n−1 and which determines the analytic and geometric properties of Zin this article.
4. The operator Zon Sobolev spaces for a manifold with boundary
We now turn to compact manifolds with boundary. For simplicity, we tacitly assume that
Xis a compact domain in a manifold Mand, for the purposes of this section, it suffices to
assume that Xhas a C0-boundary. Recall that a domain is said to have C0-boundary if its
boundary can be realized as the graph of a continuous function. We call such spaces Xa compact
manifold with C0-boundary. We may then study the operators Zand Qin Mand deduce
results in Xby restriction to distributions supported in X. In this section we study analytic
properties and meromorphic extensions. Asymptotic properties are studied in the following
section under additional regularity assumptions on the boundary. For notational clarity, we
indicate the manifold on which an operator is defined by a subscript, e.g. ZXand ZMfor the
corresponding operator on Xand M, respectively.
We shall make use of the following scales of Sobolev spaces. For s∈Rand R > 0, write
˙
Hs
R(X) := {u∈Hs
R(M) : supp(u)⊆X},and Hs
R(X) := Hs
R(M)/˙
Hs
R(M\X).
An approximation argument shows that C∞
c(X◦)⊆˙
Hs
R(X) is dense for any s∈R, see [31, The-
orem 3.29]. We equip these Sobolev spaces with Hilbert space structure induced from Hs
R(M),
i.e. ˙
Hs
R(X)⊆Hs
R(M) as a subspace and Hs
R(X) as a quotient. We note that for s= 0,
˙
H0
R(X) = H0
R(X) = L2(X). The L2-pairing between ˙
Hs
R(X) and H−s
R(X) is a perfect pairing
and induces an isomorphism ˙
Hs
R(X)∗∼
=H−s
R(X) (uniformly in R). For R= 1, we omit Rfrom
the notation. We remark that any pseudodifferential operator with parameter A∈Ψm
cl (M; Γ)
induces a continuous operator
AX:˙
Hs(X)→Hs−m(X),
34 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
defined by the composition
˙
Hs(X)֒→Hs
c(M)A
−→ Hs−m
loc (M)→Hs−m(X).
Here the last map is the quotient map, Hs
c(M) denotes the space of compactly supported dis-
tributions that are s-Sobolev regular and Hs−m
loc (M) denotes the space of distributions that are
locally s−m-Sobolev regular.
Let us make two remarks regarding the operator AX:˙
Hs(X)→Hs−m(X). Firstly, for
any s∈Rdensity ensures that AX:˙
Hs(X)→Hs−m(X) is determined by continuity and the
restriction AX:C∞
c(X◦)→C∞(X). Secondly, if s=−m/2 and Ais formally self-adjoint, then
AXis determined from the polarization identity by the continuous quadratic form
qAX(u) := hu, AuiL2, u ∈˙
H−m/2(X),
defined from Aand the perfect L2-pairing ˙
Hm/2(X)×H−m/2(X)→C.
The analogue of Theorem 2.23 for QXis the following theorem.
Theorem 4.1. Let Xbe a compact n-dimensional manifold with C0-boundary and da distance
function on Xsuch that d2is regular at the diagonal. Set µ= (n+ 1)/2. Then the family of
operators
QX:= QM|X:˙
H−µ(X)→Hµ(X),
is a well defined family of Fredholm operators for all R∈C\ {0}. For some R0≥0,QX(R)is
invertible for all R∈Γπ/(n+1) (R0). Moreover, the following holds:
a) The family of operators
(QX(R) : ˙
H−µ(X)→Hµ(X))R∈C\{0},
depends holomorphically on R∈C\ {0}. Moreover, the holomorphic family
(QX(R)−1:Hµ(X)→˙
H−µ(X))R∈Γπ/(n+1)(R0),
extends meromorphically to R∈C\ {0}.
b) There are C, R0>0such that
C−1kfk2
˙
H−µ
|R|(X)≤Rehf, QX(R)fiL2≤Ckfk2
˙
H−µ
|R|(X),
for R∈Γπ/(n+1)(R0)and f∈˙
H−µ(X). In particular, for R∈Γπ/(n+1) (R0), the opera-
tor QX(R)is coercive on L2in the form sense and the sesquilinear form h·, QX(R)·iL2
is uniformly equivalent to the inner product of ˙
H−µ
R(X).
Proof. We first prove part b). This is a direct consequence of adapting Theorem 2.23, part b),
to compactly supported distributions that are −µ-Sobolev regular and using that ˙
Hs
|R|(X)⊆
Hs
|R|(M) is an isometric inclusion.
To prove part a), we note that for R∈C\ {0},QM(R) is a lower-order perturbation of
QM(R0) for any R0≫0. Therefore, the Rellich lemma implies that the quadratic form
qQ,R(u) := hu, Q(R)uiL2≡ hu, QX(R)uiL2, u ∈H−µ(X),
is a compact perturbation of qQ,R0. Therefore, the difference
QX(R)−QX(R0) : ˙
H−µ(X)→Hµ(X),
is a compact operator. Since part b) implies that QX(R0) is invertible for a large enough R0≫0,
we conclude that (QX(R) : ˙
H−µ(X)→Hµ(X))R∈C\{0}is a Fredholm family.
It remains to prove the assertion for the inverse of QX(R) The family of operators (QX(R) :
˙
H−µ(X)→Hµ(X))R∈C\{0}is obtained from the holomorphic family of operators (QM(R) :
H−µ
c(M)→Hµ
loc(M))R∈C\{0}(see Theorem 2.23, part a) via inclusions and pro jections, so it
is also holomorphic. As such, (QX(R)−1:Hµ(X)→˙
H−µ(X))R∈Γπ/(n+1)(R0)extends meromor-
phically to C\ {0}by the meromorphic Fredholm theorem (see Appendix D).
Similarly to the ideas in Section 3, we shall transfer the results of Theorem 4.1 to the operator
ZXusing property (MR). For a domain, let us make the notion of property (MR) more precise.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 35
Definition 4.2. Let Xbe a compact manifold with C0-boundary and d a distance function
on Xsuch that d2is regular at the diagonal. For a sector [1,∞)⊆Γ⊆C, we say that d has
property (MR) on Γ if (X, d) is isometrically embedded as a domain with smooth boundary in a
manifold Mequipped with a distance function dM, such that d2
Mis regular at the diagonal and
and for any R∈Γ, L(R) extends to a continuous mapping H−µ
c(M)→Hµ
loc(M) with
kL(R)k˙
H−µ(K)→¯
Hµ(K′)=O(Re(R)−∞),as Re(R)→+∞in Γ,
for any compact subsets K, K ′⊆M.
If, for any compact subsets K, K ′⊆M, the operator L(R) : ˙
H−µ(K)→Hµ(K′) is compact
for R∈Γ and
Γ∋R7→ L(R)∈K(˙
H−µ(K), Hµ(K)),
is holomorphic in norm sense, we say that d has property (SMR) on Γ.
In the absence of a boundary, the definition of property (MR) for a manifold with bound-
ary (Definition 4.2) is readily seen to be equivalent to property (MR) for a compact manifold
(Definition 3.3). The reader is encouraged to think of the definition of property (MR) for a
manifold with boundary as the distance function having “property (MR) on a neighborhood of
the manifold with boundary”. Let us consider two examples of distance functions with property
(SMR).
Example 4.3 (Domains in Riemannian manifolds with small diameter).Assume that X⊆Mis
a compact domain with C0-boundary in a Riemannian manifold with geodesic distance dgeo,M .
If the diameter of Xis strictly smaller than the injectivity radius of M, d2
geo,M is smooth on a
neighborhood of X. The same argument as in Proposition 3.4 shows that the distance function
dgeo := dgeo,M |Xon Xhas property (SMR) on C\ {0}. In fact, in this case, L∈Ψ−∞(X;C+)
and L(R)∈Ψ−∞(X) for any R∈C\ {0}.
Example 4.4 (Submanifolds with boundary).Assume that Xis a compact manifold with C0-
boundary embedded in a manifold i:X→Wand dWis a distance function on Wsuch that
the square d2
Wis smooth on W×Wand regular at the diagonal. This arises for instance for
W=RN,W=HN,Ror W=HN ,C, for some N∈N, with their geodesic distance.
The subspace distance function d : X×X→[0,∞), d(x, y ) := dW(i(x), i(y)) will then satisfy
i) d2is smooth on X×Xand regular at the diagonal.
ii) L∈Ψ−∞(X;C+) and L(R)∈Ψ−∞ (X) for any R∈C\ {0}.
iii) d has property (SMR) on C\ {0}.
This follows by the same arguments as in Proposition 3.4 by choosing a submanifold M⊆Win
which i(X) is a compact domain (with C0-boundary).
For a distance function with property (MR), we tacitly assume that the manifold with C0-
boundary is embedded into the manifold Mimplementing property (MR). The next result is
proven exactly as Theorem 3.7 but using Theorem 4.1 instead of Corollary 2.22.
Theorem 4.5. Let Xbe a compact n-dimensional manifold with C0-boundary and da distance
function on Xwith property (MR) on the sector Γ. Then there is an R0≥0such that
ZX(R) : ˙
H−µ(X)→Hµ(X),
is invertible for all R∈Γ∩Γπ/(n+1) (R0). Moreover,
Z−1
X=Q−1
X+RX,
where Q−1
Xis the inverse of QX(existing by Theorem 4.1) and R:Hµ(X)→˙
H−µ(X)is a
family of operators such that
kRkHµ(X)→˙
H−µ(X)=O(Re(R)−∞),as Re(R)→+∞in Γ∩Γπ/(n+1)(R0).
Moreover, there is a C > 0such that
(30) C−1kfk2
˙
H−µ
|R|(X)≤Rehf, ZX(R)fiL2≤Ckfk2
˙
H−µ
|R|(X),
for R∈Γ∩Γπ/(n+1)(R0)and f∈˙
H−µ(X). In particular, for R∈Γ∩Γπ/(n+1)(R0),ReZX(R)
is positive in form sense on L2(X)for the H−µ-norm.
36 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
The next result poses an obstruction to property (MR) for distance functions on manifolds
with boundary. It is proven in the same way as Corollary 3.10 but using Theorem 4.5 instead
of Theorem 3.7.
Corollary 4.6. Assume that dis a distance function on a manifold with C0-boundary Xwith
property (MR) on [1,∞). Then there exists an R0≥0such that for all finite subsets F⊆Xthe
|F| × |F|-matrix (e−Rd(x,y))x,y∈Fis positive definite for all R > R0.
Using Theorem 4.1 instead of Theorem 3.7, the next result is proven ad verbatim as Theorem
3.12.
Theorem 4.7. Let Xbe an n-dimensional compact manifold with C0-boundary and assume that
distance function dhas property (SMR) on Γ. There is an R0≥0such that the operator
ZX(R) : ˙
H−µ(X)→Hµ(X),
is a well defined Fredholm operator for all R∈Γand invertible for R∈Γ∩Γπ/(n+1) (R0).
Moreover, the operator ZX(R) : ˙
H−µ(X)→Hµ(X)depends holomorphically on Rin Γand
ZX(R)−1:Hµ(X)→˙
H−µ(X)depends meromorphically on R∈Γ.
Similarly to Corollary 3.5, we deduce the following special case of Theorem 4.7.
Corollary 4.8. Let Xbe an n-dimensional compact manifold with C0-boundary and assume
that dis a distance function satisfying that d2is smooth on X×Xand regular at the diagonal
(e.g. as in Example 4.3 or 4.4). Then the family of operators
(ZX(R) : ˙
H−µ(X)→Hµ(X))R∈C\{0},
depends holomorphically on R∈C\ {0}and for some R0the family
(ZX(R)−1:Hµ(X)→˙
H−µ(X))R∈Γπ/(n+1)(R0),
is holomorphic and extends meromorphically to R∈C\ {0}.
The above results in this subsection provide a functional analytic description of Zand its
dependence on R. The following result gives a slightly more precise description of Zin terms
of an extended Boutet de Monvel calculus; for its proof we refer to Appendix C. This result
requires the boundary to be smooth. Let us remark that while this result describes the inverse
of Zin terms of a well studied pseudodifferential calculus, we merely include it as an observation
as it is not used for computations below. The next section is devoted to an even more direct
asymptotic construction of the inverse of Zthat is put to use in the subsequent section.
Theorem 4.9. Let Xbe an n-dimensional compact manifold with boundary and consider a
distance function dsuch that d2is regular at the diagonal. Write µ:= (n+ 1)/2. The operator
QX:˙
H−µ(X)→Hµ(X),
is an elliptic pseudodifferential operator with parameter Rof order −n−1, type −(n+ 1)/2and
factorization index −∞ in the Boutet de Monvel calculus. In particular, QXadmits a classical
parameter dependent parametrix in an extended Boutet de Monvel calculus (made precise in
Appendix C).
If moreover dhas property (MR) on a sector Γ, then for R∈Γ∩Γπ/(n+1)(R0),Z−1differs
from a classical parameter dependent operator in an extended Boutet de Monvel calculus by an
operator R:Hµ(X)→˙
H−µ(X)of infinitely low order in Rin the sense that
kRkHµ(X)→˙
H−µ(X)=O(Re(R)−∞),as Re(R)→+∞in Γ∩Γπ/(n+1)(R0).
Remark 4.10.If d is the distance function pulled back along an embedding X ֒→W, where
the distance function on Wis smooth off-diagonally, then also Zis an elliptic pseudodifferential
operator with parameter Rof order −n−1, type −(n+ 1)/2 and factorization index −∞ in
the Boutet de Monvel calculus and admits a classical parameter dependent parametrix in an
extended Boutet de Monvel calculus. In particular, Z−1
Xis a classical parameter dependent
operator in an extended Boutet de Monvel calculus if Xis a domain in Euclidean space or
real/complex hyperbolic space.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 37
5. Structure of the inverse operator in the presence of a boundary
Consider a compact manifold with boundary Xequipped with a distance function d whose
square is regular at the diagonal. As above, we set µ:= (n+ 1)/2 where n:= dim(X). If d
has property (MR), Theorem 4.5 ensures that computations for QXrelate to computations for
ZXup to a term of infinitely low order in R(as Re(R)→ ∞), so we focus on the operator
QX. As proved in Theorem 4.1 above, the localized operator QX:˙
H−µ(X)→Hµ(X) is an
isomorphism for large enough Rin the sector Γπ/(n+1) . We shall now describe the inverse of
QXin more precise terms under the assumption that the boundary is smooth. The inverse
Q−1
X:Hµ(X)→˙
H−µ(X) will be computed as a sum of
•a pseudodifferential operator (with parameter) in the interior;
•a composition of two mixed-regularity pseudodifferential operator near the boundary,
where the two factors are obtained from inverting a Wiener-Hopf factorization of the
magnitude operator at the boundary; as well as
•an error term that acts as order 2µ, mapping Hµ(X)→˙
H−µ+1(X), whose norm is
O(|R|−∞) as R→ ∞.
Asymptotically, only the two first terms play a role, and in the next section we compute the
asymptotics of conditional expectations from the symbols of these first two terms. To describe
the decomposition, we shall need further terminology.
5.1. Mixed-regularity symbols and Sobolev spaces.
Definition 5.1. Let s, t ∈Rand n∈N>0. Write coordinates in Rnas ξ= (ξ′, ξn)∈Rn−1×R.
We define the Sobolev space of mixed-regularity (s, t) as
Hs,t(Rn) := f∈ S′(Rn) : ZRnhξishξ′it|ˆ
f(ξ)|2dξ < ∞.
If Ω ⊆Rnis a domain (so Ω = Ω◦), we define
˙
Hs,t(Ω) := {f∈Hs,t (Rn) : supp(f)⊆Ω},and Hs,t (Ω) := Hs,t(Rn)/H s,t(Ωc).
We also define ˙
Hs,t
c(Ω) and Hs,t
c(Ω) as the elements with compact support. The local Sobolev
spaces of mixed-regularity are defined as
˙
Hs,t
loc(Ω) := {f∈ S′(Rn) : χf ∈˙
Hs,t(Ω)∀χ∈C∞
c(Rn)},
Hs,t
loc(Ω) := Hs,t
loc(Rn)/H s,t
loc(Ωc).
We note that ˙
Hs,t(Ω) ⊆Hs,t (Rn) is a closed subspace. We call the quotient mapping
˙
Hs,t(Rn)→Hs,t (Ω) the restriction mapping. For large enough s > 0, we can identify ˙
Hs,t(Ω)
with a subspace of Hs,t (Ω) (but not for small s < 0). A standard computations with Fourier
transforms shows that the the identity operator induces continuous mappings Hs,t(Ω) →Hs′,t′
(Ω)
and ˙
Hs,t(Ω) →˙
Hs′,t′(Ω) if and only if s≥s′and t≥t′(which is locally compact if and only if
s > s′and t > t′).
Definition 5.2. Let u, m ∈Rand n∈N>0. Let Γ ⊆Cbe a sector and U⊆Rnan open subset.
Write coordinates in Rnas ξ= (ξ′, ξn)∈Rn−1×R. We say that a∈C∞(U×Rn×Γ) is a
symbol with parameter of mixed-regularity (u, m) if for any compact K⊆U,k∈N,α∈Nn
and β∈Nnthere is a constant C > 0 such that
sup
x∈K∂k
ξn∂β
(ξ′,R)∂α
xa(x, ξ′, ξn, R)≤Ch(ξ, R)iu−kh(ξ′, R)im−|β|,
for all (ξ, R)∈Rn+1. We let Su,m(U; Γ) denote the space of symbols with parameter of mixed-
regularity (u, m). We set Su,−∞(U; Γ) := ∩m∈RSu,m(U; Γ).
For a∈Su,m(U; Γ) we define
Op(a) : C∞
c(U)→C∞(U), Op(a)f(x) := 1
(2π)nZRn
a(x, ξ, R)ˆ
f(ξ)dξ.
Let Ψu,m(U; Γ) denote the linear space of operators C∞
c(U)→C∞(U) spanned by {Op(a) : a∈
Su,m(U; Γ)}and smoothing operators with parameter. We set Ψu,−∞(U; Γ) := ∩m∈RΨu,m(U; Γ).
38 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Example 5.3.Assume that 0 /∈Γ. Suppose that a(x, ξ, R) = b(x, ξ ′, R)(ξn−h(x, ξ′, R))uwhere
bis a homogeneous symbol with parameter of order mand his a homogeneous symbol with
parameter of order 1. A short computation shows that ∂k
ξn∂β
(ξ′,R)∂α
xais a sum of terms of the
form ˜
b(x, ξ′, R)(ξn−h(x, ξ′, R))u−k−lwhere ˜
bis homogeneous of order m− |β|+l. Therefore
a∈Su,m(U; Γ), and in fact a∈Su0,m+u−u0(U; Γ) for any u0≤u.
We note that since differentiation in the (ξ′, R)-direction improves the order of decay, we have
for a∈Su,m(U; Γ) and f∈C∞
c(U) that
Op(a)f(x) := 1
(2π)nZRn×Rn−1
a(x, ξ, R)Fyn→ξnf(y′, ξn)eiξnxn+iξ′(x′−y)′dy′dξ,
as an oscillatory integral. Moreover, we can conclude the following result from standard tech-
niques of oscillatory integrals (cf. [38, Chapter I.1]).
Proposition 5.4. Assume that A∈Ψu,m (U; Γ). Then for any χ, χ′∈C∞(U)with χχ′= 0
it holds that χAχ′∈Ψu,−∞(U; Γ). In particular, if χ, χ′∈C∞
c(U)satisfies χχ′= 0 then
kχAχ′kHs→Hs′=O(|R|−∞)as R→ ∞ for all s≥s′+u.
The last statement of the proposition follows from the next theorem.
Theorem 5.5. Let u, m, s, t, s′, t′∈Rand n∈N>0. Let A∈Ψu,m(Rn; Γ). Then Aextends to
a continuous operator
A:Hs,t
c(Rn)→Hs′,t′
loc (Rn),
if s≥s′+uand t≥t′+m. In this case, we have for any χ, χ′∈C∞
c(Rn)that there exists a
C=C(s, s′, A, χ, χ′)>0
kχAχ′kHs,t(Rn)→Hs′,t′(Rn)≤C(1 + |R|)t′−t+m.
Proof. The first part follows from the Calder´on-Vaillancourt theorem. The second part follows
from noting that Calder´on-Vaillancourt’s theorem proves the case t′+m=tand for A∈
Ψu,m(Rn; Γ) compactly supported, then (R2+∆′)(t−t′−m)/2A∈Ψu,t−t′(Rn; Γ) (where ∆′denotes
the Laplacian in the x′-direction).
Theorem 5.6. Let u, m, s, t, s′, t′∈Rand n∈N>0. Let a∈Su,m(Rn; Γ) be compactly supported
in the x-direction and assume that aextends to a holomorphic function in Im(ξn)<0. Set
A:= Op(a)Then Arestricts to and induces, respectively, continuous operators
˙
A:˙
Hs,t(Rn
+)→˙
Hs′,t′(Rn
+)and A:Hs,t(Rn
−)→Hs′,t′
(Rn
−),
if s≥s′+uand t≥t′+m.
Proof. Note that Ais well defined as the operator induced from A:Hs,t(Rn)→Hs′,t′(Rn) since
˙
Apreserves supports in Rn
+. The result follows from the Paley-Wiener theorem.
Remark 5.7.Under the assumptions of Theorem 5.6, Theorem 5.5 implies that for any sand s′
with s≥s′+uthere is a C > 0 such that for t≥t′+m
kAk˙
Hs,t(Rn
+)→˙
Hs′,t′(Rn
+)≤C(1 + |R|)t′−t+mand kAkHs,t(Rn
−)→Hs′,t′(Rn
−)≤C(1 + |R|)t′−t+m.
Definition 5.8. Let u∈R. Consider a sequence (aj)j∈Nof mixed-regularity symbols with
aj∈Su,mj(U; Γ) for a sequence mj→ −∞. Set m:= maxjmj. If a∈Su,m(U; Γ) satisfies that
for any N, there is an Msuch that a−PM
j=0 aj∈Su,−N(U; Γ), we write
a∼
∞
X
j=0
aj,
and call athe asymptotic sum of (aj)j∈N.
Proposition 5.9. Let u∈R. For any sequence (aj)j∈Nof mixed symbols with aj∈Su,mj(U; Γ)
for a sequence mj→ −∞, the asymptotic sum a∼P∞
j=0 ajexists in Su,m(U; Γ), where m:=
maxjmj. The asymptotic sum is uniquely determined modulo Su,−∞(U; Γ).
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 39
Proof. The proof of this proposition is carried out ad verbatim as in [23, Proposition 18.1.3]
upon replacing the ξin [23] with (ξ′, R).
Again, using that differentiation in the (ξ , R)-direction improves the order of decay, we can
conclude several results from the standard situation (cf. [38, Chapter I]). For instance, the ana-
logue of [38, Theorem 3.1, Chapter I.3] extends modulo Ψu,−∞ to Ψu,m which implies asymptotic
expansions of products and adjoints.
Proposition 5.10. Let A∈Ψu,m(U; Γ) and B∈Ψu′,m′(U; Γ) be mixed-regularity pseudodiffer-
ential operators out of which at least one is properly supported. Then AB ∈Ψu+u′,m+m′(U; Γ) is
a mixed-regularity pseudodifferential operator. Moreover, if a∈Su,m(U; Γ) and b∈Su′,m′(U; Γ)
are symbols with A−Op(a)and B−Op(b)being smoothing with parameter, then AB −Op(c)is
smoothing with parameter where c∈Su+u′,m+m′(U; Γ) is uniquely determined modulo Su+u′,−∞(U; Γ)
as the asymptotic sum
c∼X
α∈Nn
1
α!∂α
ξaDα
xb.
5.2. Wiener-Hopf factorization of QXnear the boundary. Using the machinery of the
previous subsection, we shall now factorize the operator QXnear the boundary into factors
that extend holomorphically into the upper respectively lower half-plane, and use Theorem 5.6
to (near the boundary) invert these individual factors as operators ˙
H−µ(X)→L2(X) and
L2(X)→Hµ(X), respectively. The reader should recall the structure of the full symbol of Q
from Theorem 2.9. We shorten the notation and write Cjfor Cj
d2, where Cj
d2are the Taylor
coefficients of d2as in Equation (10).
As above, we consider a compact manifold with boundary Xa distance function d on Xwhose
square is regular at the diagonal (cf. Definition 2.2). We tacitly fix a manifold Mcontaining X
as a smooth compact domain to which d extends as a distance function whose square is regular
at the diagonal. In particular, we can Taylor expand d2near any point in the diagonal as in
Equation (10) and its Taylor coefficients enter the full symbol of Qas in Theorem 2.9. To study
the behavior at the boundary, we first reduce to the model case that X=∂X ×[0,∞), as a
domain in ∂X ×R. We remark that ∂X ×[0,∞) is not compact, but we shall later on only use
the constructed operators in a form localized to near the compact boundary.
Proposition 5.11. Let Xbe a compact manifold with boundary with a distance function dwhose
square is regular at the diagonal, embedded into a manifold Mas in the preceding paragraph.
Consider the compact manifold Y=∂X and choose a tubular neighborhood U⊆Mof ∂X and
a diffeomorphism ϕ:U→Y×(−1,1), with ∂X =ϕ−1(Y× {0}). Then there exists a classical
elliptic pseudodifferential operator with parameter Q∂∈Ψ−n−1
cl (Y×R;C+)and a number ε > 0
such that
•Q∂is translation invariant outside a compact subset in the sense that there exists a
t0>0such that if f∈C∞
c(Y×R)is supported in {(y, t) : ±t > t0}then [Q∂f](· ∓ s) =
Q∂[f(· ∓ s)] for all s > 0.
•For all f∈C∞
c(Y×(−ε, ε)) it holds that Q(ϕ∗f)is supported in Uand
Q(ϕ∗f) = ϕ∗(Q∂f).
•the principal symbol of Q∂is given by
σ−n−1(x, ξ, R) = c(R2+g(ξ, ξ))−µ,
where gis a Riemannian metric on Y×Rwhich is translation invariant in the R-direction
outside a compact and coincides with ϕ∗gd2on Y×(−1,1).
Proof. Construct gfrom interpolating between gd2near 0 and something at infinity. Construct
Q∂from interpolation along the real line between Qnear 0 and (R2+ ∆g)−µat infinity. The
second part follows from that Qby construction has small propagation.
Henceforth, we shall fix a choice of Q∂and gas in Proposition 5.11.
40 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Remark 5.12.To fix a choice of a diffeomorphism ϕ:U→Y×(−1,1) is (up to a self-
diffeomorphism of Y) equivalent to choosing a vector field defined near Y=∂X which is
transversal to the boundary. This choice of vector field, or equivalently, the last entry of
ϕ:U→Y×(−1,1), gives rise to a coordinate that we denote by xn:U→(−1,1). We
remark that it is always possible to use the transversal vector field to be the metric normal to
the boundary, in which case we have that on ∂X:
gd2= dx2
n+g∂X,d2,
where g∂X,d2is the induced Riemannian metric on ∂X. For computational purposes, it becomes
clumsy to restrict to the the case when the transversal vector field is orthogonal to the boundary
but for some considerations it simplifies the formulas.
Following the notation of Subsection 5.1, we write xnfor this transversal coordinate and x′
for coordinates on ∂ X . Similarly, ξndenotes the cotangent variable in the transversal direction
and ξ′denotes the cotangent variables along ∂X .
We let q∂∈S−2µ
cl (Y×R; Γ) denote the full symbol of Q∂. We note that q∂∼P∞
j=0 q∂
jin
S−2µ(Y×R;C) where each q∂
j∈S−2µ−j(Y×R;C) is a homogeneous symbol in (ξ, R) of order
−2µ−j=−n−1−jand near t= 0, we have in any local coordinates on Ythat q∂
j=qjwhere
qjis computed as in Theorem 2.9 using the coordinates induced from Yand ϕ.
Proposition 5.13. There are unique homogeneous degree 1symbols
h±=h±(x, ξ′, R)∈S1(T∗Y×R, Y ×R;C),
that determine the complex solutions to the equation R2+gx((ξ′, ξn),(ξ′, ξn)) = 0 for fixed
(x, ξ′, R)with ±Im(ξn)>0. Furthermore, there is a unique function h0∈C∞(Y×R,R>0)such
that
R2+g(ξ, ξ) = h0(x)(ξn−h+(x, ξ′, R))(ξn−h−(x, ξ′, R)).
Moreover, we have that
h±(x, ξ′, R) = −ξ′(b(x))
h0(x)±ipR2+gY(ξ′, ξ′)−(ξ′(b))2
ph0(x),
for suitable band gYdetermined from the metric.
Proof. It is not hard to see that h0and h±are well defined and unique, but let us construct
them explicitly. We can decompose
(31) g=h0b
bTgY,
where gYis a metric on Yon each slice, and b∈C∞(Y×R, T Y ). Since gis translation invariant
outside a compact, h0,band gYare translation invariant outside a compact. We have that
R2+g(ξ, ξ) = R2+h0ξ2
n+ 2ξ′(b)ξn+gY(ξ′, ξ′).
We see that h0in the above definition is the h0in Equation (31) and that the complex roots are
as prescribed. The proposition follows.
Theorem 5.14. Let q∂∈S−2µ
cl (Y×R; Γ) be as above. Then there exists q∂
±∈S−µ,0(Y×R; Γ)
that are translation invariant outside a compact such that
(1) q∂
±∈S−µ,0(Y×R; Γ) admits asymptotic expansions (in S−µ,0in the sense of Definition
5.8 on page 38)
q∂
±∼
∞
X
j=0
q∂
±,j ,
where q∂
±,0∈S−µ,0and for j > 0,
q∂
±,j (x, ξ, R) =
j−1
X
k=−1
b±,j,k(x, ξ ′, R)(ξn−h±(x, ξ′, R))−µ−j+k∈S−µ−1,−j+1 ,
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 41
where b±,j,k is homogeneous of degree −kin (ξ′, R)and can be computed by an iter-
ative scheme of partial fraction decompositions as a homogeneous rational function in
derivatives of h0,h+and h−. The first terms are given by
q∂
+,0=n!ωn(ξn−h+)−µ,(32)
q∂
−,0=h−µ
0(ξn−h−)−µ,
and q±,1are computed in Proposition 5.16 below.
(2) The mixed-regularity symbols q∂
±∈S−µ,0(Y×R; Γ) and q∂
±,j ∈S−µ−1,−j+1 admit holo-
morphic extensions to ∓Im(ξn)>0.
(3) It holds that
q∂=X
α
1
α!∂α
ξq∂
−Dα
xq∂
+mod S−2µ,−∞.
Proof. Let us first massage the statements of the theorem. We want to construct q∂
±∼P∞
j=0 q∂
±,j ∈
S−µ,0admitting holomorphic extensions to ∓Im(ξn)>0 and satisfying q∂=Pα1
α!∂α
ξq∂
−Dα
xq∂
+
mod S−2µ,−∞. We remark that to ensure item (2), i.e. the holomorphic extension of q∂
±, it
suffices to construct each q∂
±,j so that it admits a holomorphic extension to ∓Im(ξn)>0. We
also note that the requirement on the composition is equivalent to
(33) q∂
j=X
k+l+|α|=j
1
α!∂α
ξq∂
−,kDα
xq∂
+,l.
We take the formula (32) as a definition, and note that q∂
±,0satisfy the structural statement in
item (1), extends holomorphically to ∓Im(ξn)>0 and q∂
0=q∂
−,0q∂
+,0. Using an idea described
in [21], Equation (33) is for j > 0 equivalent to
q∂
+,j
q∂
+,0
+q∂
−,j
q∂
−,0
=q∂
j
q∂
0−1
q∂
0X
k+l+|α|=j
k,l<j
1
α!∂α
ξq∂
−,kDα
xq∂
+,l
We proceed by induction. Assume that we have constructed q±,k for k < j satisfying the
statements of items (1), (2), and (3) in the relevant degrees. Using Lemma 2.20 and item (1) for
q±,k for k < j, we can use Lemma E.1 to uniquely partial fraction decompose
q∂
j
q∂
0−1
q∂
0X
k+l+|α|=j
k,l<j
1
α!∂α
ξq∂
−,kDα
xq∂
+,l =q+,j +q−,j ,
where q±,j by Proposition 2.6 and Theorem 2.9 takes the form
q±,j(x, ξ, R) =
j−1
X
k=−1
b±,j,k(x, ξ ′, R)(ξn−h±(x, ξ′, R))−j+k∈S−1,−j+1 ,
where b±,j,k is homogeneous of degree −kin (ξ′, R) and can be explicitly computed from the
results of Appendix E and Theorem 2.9. We now define
q∂
±,j := q∂
±,0q±,j,
and note that it by construction satisfies items (1), (2), and (3) in the relevant degrees.
Remark 5.15.As noted in Proposition 5.13, the symbols h+and h−are directly determined from
the metric and the choice of transversal to the boundary. Moreover, as the proof of Theorem
5.14 shows, each of the symbols b±,j,k =b±,j,k(x, ξ′, R) depends
(1) polynomially on (C(γ)
G)γ∈∪k≤jIkand its derivatives contracted by gGand ιngGand its
derivatives
(2) polynomially on h+,h−and rationally on h+−h−and h−1/2
0,
The total degree is jwhere each C(γ)
G,γ∈Ik, has degree k, the metric and h0have degree zero,
h+and h−have degree 1 and x-derivatives increase the order by 1.
42 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Proposition 5.16. For n > 1, the terms q±,1appearing in the expansion of q±in Theorem
5.14 are given by:
q∂
+,1=n!ωna0,+(x, ξ′, R)(ξn−h+)−µ−1+n!ωna1,+(x, ξ′, R)(ξn−h+)−µ−2,
q∂
−,1=a0,−(x, ξ′, R)h−µ
0(ξn−h−)−µ−1+a1,−(x, ξ′, R)h−µ
0(ξn−h−)−µ−2,
where the homogeneous symbols a+,0(of degree 0) and a+,1(of degree 1) are explicitly given in
Equation (34) and (35) below, and the homogeneous symbols a−,0(of degree 0) and a−,1(of
degree 1) are explicitly given in Equation (36) and (37) below.
Proof. Let us first make some preliminary computations with the decomposition from Lemma
E.1 using the explicit forms in Corollary E.4. We let ιndenote contraction by the unit normal.
We can write
C3(x, g ⊗ιξg) = C3(x, g ⊗ιng)ξn+C3(x, g ⊗ιξ′g).
Using Corollary E.4 we decompose
C3(x, g ⊗ιξg)(R2+gd(ξ, ξ))−1=h−1
0C3(x, g ⊗ιng)
h+−h−h+(ξn−h+)−1−h−(ξn−h−)−1
+C3(x, g ⊗ιξ′g)h−1
0
h+−h−(ξn−h+)−1−(ξn−h−)−1.
We can write
C3(x, ιξg⊗ιξg⊗ιξg) =C3(x, ιng⊗ιng⊗ιng)ξ3
n+ 3C3(x, ιξ′g⊗ιng⊗ιng)ξ2
n+
+ 3C3(x, ιξ′g⊗ιξ′g⊗ιng)ξn+C3(x, ιξ′g⊗ιξ′g⊗ιξ′g).
Using Corollary E.4 we decompose
C3(x, ιξg⊗ιξg⊗ιξg)(R2+gd(ξ, ξ))−2=
C3(x, ιng⊗ιng⊗ιng)h−2
0
(h+−h−)2h3
+(ξn−h+)−2+h3
−(ξn−h−)−2+
+h2
+(h+−3h−)
h+−h−
(ξn−h+)−1−h2
−(h−−3h+)
h+−h−
(ξn−h−)−1
+3h−2
0C3(x, ιξ′g⊗ιng⊗ιng)
(h+−h−)2h2
+(ξn−h+)−2+h2
−(ξn−h−)−2−
−2h+h−
h+−h−(ξn−h+)−1−(ξn−h−)−1
+3h−2
0C3(x, ιξ′g⊗ιξ′g⊗ιng)
(h+−h−)2h+(ξn−h+)−2+h−(ξn−h−)−2−
−h++h−
h+−h−(ξn−h+)−1−(ξn−h−)−1
+h−2
0C3(x, ιξ′g⊗ιξ′g⊗ιξ′g)
(h+−h−)2(ξn−h+)−2+ (ξn−h−)−2−
−2
h+−h−(ξn−h+)−1−(ξn−h−)−1.
Using Proposition 2.14, we compute that
q∂
1
q∂
0−1
q∂
0X
|α|=1
1
α!∂α
ξq∂
−,0Dα
xq∂
+,0=
=−3ic1,n(n2−1)
n!ωn
C3(x, g ⊗ιξg)(R2+gd(ξ, ξ))−1+
+ic1,n(n+ 3)3,−2
n!ωn
C3(x, ιξg⊗ιξg⊗ιξg)(R2+gd(ξ, ξ))−2+
+i(n+ 1)2h0
4(∇ξ′h−· ∇x′h+−∂xnh+)(R2+gd(ξ, ξ))−1.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 43
Decomposing as in the paragraphs above, we obtain
q+,1=n!ωna0,+(x, ξ′, R)(ξn−h+)−µ−1+n!ωna1,+(x, ξ′, R)(ξn−h+)−µ−2,
where
a0,+(x, ξ′, R) = −3ic1,n(n2−1)
n!ωnC3(x, g ⊗ιng)h++C3(x, g ⊗ιξ′g)h−1
0
h+−h−
+
(34)
+ic1,n(n+ 3)3,−2
n!ωn
h−2
0
(h+−h−)3
C3(x, ιng⊗ιng⊗ιng)h2
+(h+−3h−)−6h+h−C3(x, ιng⊗ιng⊗ιξ′g)−
−3(h++h−)C3(x, ιng⊗ιξ′g⊗ιξ′g)−2C3(x, ιξ′g⊗ιξ′g⊗ιξ′g)−
+i(n+ 1)2
4
(∇ξ′h−· ∇x′h+−∂xnh+)
h+−h−
,
and
a1,+(x, ξ′, R) = ic1,n(n+ 3)3,−2
n!ωn
h−2
0
(h+−h−)2
(35)
C3(x, ιng⊗ιng⊗ιng)h3
++ 3h2
+C3(x, ιξ′g⊗ιng⊗ιng)+
+ 3h+C3(x, ιξ′g⊗ιξ′g⊗ιng) + C3(x, ιng⊗ιξ′g⊗ιξ′g).
We also obtain
q−,1=a0,−(x, ξ′, R)h−µ
0(ξn−h−)−µ−1+a1,−(x, ξ′, R)h−µ
0(ξn−h−)−µ−2,
where
a0,−(x, ξ′, R) = 3ic1,n(n2−1)
n!ωnC3(x, g ⊗ιng)h−+C3(x, g ⊗ιξ′g)h−1
0
h+−h−−
(36)
−ic1,n(n+ 3)3,−2
n!ωn
h−2
0
(h+−h−)3
C3(x, ιng⊗ιng⊗ιng)h2
−(h−−3h+)−6h+h−C3(x, ιng⊗ιng⊗ιξ′g)−
−3(h++h−)C3(x, ιng⊗ιξ′g⊗ιξ′g)−2C3(x, ιξ′g⊗ιξ′g⊗ιξ′g)+
−i(n+ 1)2
4
(∇ξ′h−· ∇x′h+−∂xnh+)
h+−h−
,
and
a1,−(x, ξ′, R) = ic1,n(n+ 3)3,−2
n!ωn
h−2
0
(h+−h−)2
(37)
C3(x, ιng⊗ιng⊗ιng)h3
−+ 3h2
−C3(x, ιξ′g⊗ιng⊗ιng)+
+ 3h−C3(x, ιξ′g⊗ιξ′g⊗ιng) + C3(x, ιng⊗ιξ′g⊗ιξ′g).
The stated formulas follow.
Lemma 5.17. The operators Q∂
±:= Op(q∂
±)∈Ψ−µ,0(Y×R; Γ) satisfy that
(1) Q−Q−Q+∈Ψ−2µ,−∞(Y×R; Γ);
44 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
(2) Q+restricts to a well defined operator ˙
H−µ(Y×[0,∞)→L2(Y×[0,∞)) which is
invertible for large R;
(3) Q−restricts to a well defined operator L2(Y×[0,∞)) →Hµ(Y×[0,∞)) which is
invertible for large R.
Proof. Part (1) follows from Proposition 5.10 and Theorem 5.14. Parts (2) and (3) follow from
Theorem 5.5 and Theorem 5.6.
Definition 5.18. Define w±,j ∈Sµ,−jinductively by
w±,0(x, ξ, R) := (q∂
±,0)−1=
1
n!ωn(ξn−h+(x, ξ′, R))µ,for +,
h0(x)µ(ξn−h−(x, ξ′, R))µ,for −,
and then
w±,j := −w±,0X
k+l+|α|=j, l<j
1
α!∂α
ξq∂
±,kDα
xw±,l.
We also define w±:= Pjw±,j ∈Sµ,0(Y×R; Γ) and W±∈Ψµ,0(Y×R; Γ) is defined as
a properly supported modification of Op(w±) with the same full symbol which is translation
invariant outside a compact subset.
Lemma 5.19. Let w±∈Sµ,0(Y×R; Γ) be as above. Then
(1) The asymptotic expansion (in Sµ,0in the sense of Definition 5.8 on page 38) of w±∈
Sµ,0(Y×R; Γ)
w±∼
∞
X
j=0
w±,j ,
can for j > 0be expanded in a finite sum
w±,j (x, ξ, R) =
j−1
X
k=0
w±,j,k(x, ξ ′, R)(ξn−h±(x, ξ′, R))µ−j+k∈Sµ−1,−j+1,
where w±,j,k is homogeneous of degree −kin (ξ′, R)and can be computed by an iterative
scheme as a rational function of derivatives of h0,h+and h−.
(2) The mixed-regularity symbols w±∈Sµ,0(Y×R; Γ) and w±,j ∈Sµ−1,−j+1 admit holo-
morphic extensions to ∓Im(ξn)>0.
(3) The symbols w±,j,k =w±,j,k(x, ξ′, R)depends
(a) polynomially on (C(γ)
G)γ∈∪k≤jIkand its derivatives contracted by gGand ιngGand
its derivatives
(b) polynomially on h+,h−and rationally on h+−h−and h−1/2
0,
The total degree is jwhere each C(γ)
G,γ∈Ik, has degree k, the metric and h0have
degree zero, h+and h−have degree 1and x-derivatives increase the order by 1.
Proof. Items (1) and (2) follow from a short induction argument with the construction (in
Definition 5.18) and Theorem 5.14.
We now compute w±,1. By definition, we have that
w±,1=−w2
±,0q∂
±,1−w±,0X
|α|=1
∂α
ξq∂
±,0Dα
xw±,0.
A short algebraic manipulation with the computation of q±,1from Proposition 5.16 gives the
following formulas.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 45
Proposition 5.20. For n > 1, the terms w±,1appearing in the expansion of w±in Lemma 5.19
are given by:
w+,1(x, ξ′, ξn, R) = −1
n!ωn
a0,+(x, ξ′, R)(ξn−h+)µ−1−1
n!ωn
a1,+(x, ξ′, R)(ξn−h+)µ−2−
−i(n+ 1)2
4·n!ωn
(∂xnh+− ∇ξ′h+· ∇x′h+)(ξn−h+)µ−2,
w−,1(x, ξ′, ξn, R) = −a0,−(x, ξ′, R)hµ
0(ξn−h−)µ−1−a1,−(x, ξ′, R)hµ
0(ξn−h−)µ−2−
−i(n+ 1)2
4(∂xnh−− ∇ξ′h−· ∇x′h−)hµ
0(ξn−h−)µ−2−
−i(n+ 1)2
4(∂xnh0− ∇ξ′h−· ∇x′h0)hµ−1
0(ξn−h−)µ−1,
where the homogeneous symbols a+,0(of degree 0) and a+,1(of degree 1) were explicitly given
in Equation (34) and (35) above, and the homogeneous symbols a−,0(of degree 0) and a−,1(of
degree 1) were explicitly given in Equation (36) and (37) above.
Lemma 5.21. The operators W±∈Ψµ,0(Y×R; Γ) satisfy that
(1) 1 −W±Q∂
±,1−Q∂
±W±∈Ψ0,−∞(Y×R; Γ);
(2) W−preserves supports in Y×(−∞,0] and restricts to a well defined operator Hµ(Y×
[0,∞)) →L2(Y×[0,∞)which is invertible for large R;
(3) W+preserves supports in Y×[0,∞)and restricts to a well defined operator L2(Y×
[0,∞)) →˙
H−µ(Y×[0,∞)which is invertible for large R.
In particular, the operators
S0:= 1 −W+W−Q∂:˙
H−µ(Y×[0,∞)) →˙
H−µ(Y×[0,∞)),and
S1:= 1 −Q∂W+W−:Hµ(Y×[0,∞)) →Hµ(Y×[0,∞)),
are normbounded by O(R−∞).
Proof. Part (1) follows from Proposition 5.10 and Lemma 5.19. Parts (2) and (3) follow from
Theorem 5.5 and Theorem 5.6.
5.3. Decomposition of the inverse magnitude operator.
Theorem 5.22. Let Xbe an n-dimensional compact manifold with boundary and da distance
function whose square is regular at the diagonal. Set µ:= (n+1)/2. Let QX:˙
H−µ(X)→Hµ(X)
denote the restriction of Qχ,d2to Xand let A∈Ψn+1
cl (X;C+)denote a parametrix of Qχ,d2.
For some R0≥0and any R∈Γπ/(n+1) (R0), we can write
Q−1
X=χ1Aχ′
1+χ2(ϕ−1)∗W+W−ϕ∗χ′
2+S,
where χ1, χ′
1∈C∞
c(X◦), and χ2, χ′
2∈C∞(X)are functions supported in a collar neighborhood
U0of ∂X in Xsuch that
χ1+χ2= 1 and χ′
j|supp(χj)= 1, j = 1,2,
ϕ:∂X ×[0,1) →U0is a collar identification, and the operators S,W−and W+satisfy the
following:
(1) S:Hµ(X)→˙
H−µ(X)is a continuous operator with
kSkHµ(X)→˙
H−µ(X)=O(R−∞),as R→ ∞.
(2) W+:L2(∂X ×[0,∞)) →˙
H−µ(∂X ×[0,∞)) is the properly supported pseudodifferential
operator of mixed-regularity (µ, 0) from Definitions 5.18 which is invertible for large
R > 0and in local coordinates has an asymptotic expansion modulo Sµ,−∞ as in Lemma
5.21 and preserves support in ∂X×[0,∞)⊆∂X ×R. Moreover, for χ, χ′∈C+C∞
c(∂X ×
[0,∞)) with χχ′= 0, it holds that
kχW+χ′kL2(∂X ×[0,∞))→H−µ(∂X ×R)=O(R−∞),as R→ ∞.
46 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
(3) W−:Hµ(∂X ×[0,∞)) →L2(∂X ×[0,∞)) is the properly supported pseudodifferential
operator of mixed-regularity (µ, 0) from Definitions 5.18 which is invertible for large R >
0and in local coordinates has an asymptotic expansion modulo Sµ,−∞ as in Lemma 5.21
and preserves support in ∂X ×(−∞,0] ⊆∂ X ×R. Moreover, for χ, χ′∈C+C∞
c(∂X ×R)
with χχ′= 0, it holds that
kχW−χ′kHµ(∂X ×R)→L2(∂X ×R)=O(R−∞),as R→ ∞.
Proof. The properties of W+and W−listed in items (2) and (3) follow from the results of
Subsection 5.1. We note that it follows from the previous subsection that
χ1Q−1
Mχ′
1+χ2(ϕ−1)∗W+W−ϕ∗χ′
2:Hµ(X)→˙
H−µ(X),
is a well defined continuous operator. Pick a χ3supported close to ∂X with χ3= 1 on supp(χ′
2).
We compute that
(χ1Q−1
Mχ′
1+χ2(ϕ−1)∗W+W−ϕ∗χ′
2)QX=
=χ1+χ1Q−1
M(χ′
1−1)QX+χ2(ϕ−1)∗W+W−ϕ∗QXχ3+
+χ2(ϕ−1)∗W+W−ϕ∗(χ′
2−1)QXχ3+χ2(ϕ−1)∗W+W−ϕ∗χ′
2QX(1 −χ3) =
=χ1+χ1Q−1
M(χ′
1−1)QX+χ2(ϕ−1)∗W+W−Q∂ϕ∗χ3+
+χ2(ϕ−1)∗W+W−ϕ∗(χ′
2−1)QXχ3+χ2(ϕ−1)∗W+W−ϕ∗χ′
2QX(1 −χ3) =
=χ1+χ1Q−1
M(χ′
1−1)QX+χ2+ (ϕ−1)∗S0ϕ∗χ3+
+χ2(ϕ−1)∗W+W−ϕ∗(χ′
2−1)QXχ3+χ2(ϕ−1)∗W+W−ϕ∗χ′
2QX(1 −χ3) =
=1 + χ1Q−1
M(χ′
1−1)QX
|{z }
=:S2
+(ϕ−1)∗S0ϕ∗χ3+
+χ2(ϕ−1)∗W+W−ϕ∗(χ′
2−1)QXχ3
|{z }
=:S3
+χ2(ϕ−1)∗W+W−ϕ∗χ′
2QX(1 −χ3)
|{z }
=:S4
.
Since χ1(χ′
1−1) = 0, S2is a smoothing operator with parameter. Similarly, since χ′
2(1−χ3) = 0,
S4is a smoothing operator with parameter. Using Proposition 5.4 and Lemma 5.21, respectively,
we conclude that S3:˙
H−µ(X)→˙
H−µ(X) and (ϕ−1)∗S0ϕ∗χ3:˙
H−µ(X)→˙
H−µ(X) are
continuous with norms bounded by O(R−∞) as R→ ∞. In particular,
S5:= (χ1Q−1
Mχ′
1+χ2(ϕ−1)∗W+W−ϕ∗χ′
2)QX−1,
satisfies that S5:˙
H−µ(X)→˙
H−µ(X) is continuous and kS5k˙
H−µ(X)→˙
H−µ(X)=O(R−∞) as
R→ ∞. We conclude that (1 + S5)−1exists for large Rand k1−(1 + S5)−1k˙
H−µ(X)→˙
H−µ(X)=
O(R−∞) as R→ ∞. We therefore have that
Q−1
X= (1 + S5)−1(χ1Q−1
Mχ′
1+χ2(ϕ−1)∗W+W−ϕ∗χ′
2) = χ1Q−1
Mχ′
1+χ2(ϕ−1)∗W+W−ϕ∗χ′
2+S,
where
S:= (1 −(1 + S5)−1)(χ1Q−1
Mχ′
1+χ2(ϕ−1)∗W+W−ϕ∗χ′
2).
Since k1−(1 + S5)−1k˙
H−µ(X)→˙
H−µ(X)=O(R−∞) as R→ ∞, the same holds for Sand the
proof is complete.
By combining Theorem 4.5 with Theorem 5.22, we arrive at the following corollary.
Corollary 5.23. Let Xbe an n-dimensional compact manifold with boundary and da distance
function with property (MR) on Γ. For some R0≥0and any R∈Γπ/(n+1)(R0)∩Γ, we can
write
Z−1
X=χ1Aχ′
1+χ2(ϕ−1)∗W+W−ϕ∗χ′
2+˜
S,
where A,W+,W−and χ1, χ1, χ2, χ′
2∈C∞(X)are as in Theorem 5.22 and ˜
S:Hµ(X)→
˙
H−µ(X)is a continuous operator with
k˜
SkHµ(X)→˙
H−µ(X)=O(Re(R)−∞),as Re(R)→ ∞ in Γ.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 47
6. Conditional expectations of Q−1
Xand Z−1
X
A large motivation for this paper is the relation of the operator ZXwith magnitude. For
that purpose, we shall be interested in computing conditional expectations of Q−1
Xand Z−1
X
against the constant function 1. In an accompanying paper [12], we prove that this conditional
expectation of (RZX(R))−1coincides with the magnitude function. The section is divided into
three subsections: firstly, we study the case of no boundary, secondly we proceed to compute
the asymptotic expansion of the conditional expectation of Q−1
Xand finally we produce explicit
formulas for the asymptotic expansion and consider examples. As in the previous sections, we
perform computations for Qthat later translates into results for Zunder assumptions of property
(MR).
6.1. Asymptotic expansions for compact manifolds. Let us consider the case that X=M
is a compact manifold. Starting from Lemma 2.24 we compute the asymptotics of hQ(R)−11,1i
as R→ ∞ for a pseudodifferential operator with parameter R. Let vold(M) denote the volume
of Min the Riemannian metric defined from the transversal Hessian of d2at the diagonal.
Theorem 6.1. Let Mbe an n-dimensional compact manifold with a distance function dwhose
square is regular at the diagonal. Let (aj,0)j∈N⊆C∞(M;C+)denote the sequence of homoge-
neous functions obtained from restriction to ξ= 0 of the full symbol of Q−1
M, as in Lemma 2.24.
It holds that
h1, QM(R)−11i ∼
∞
X
k=0
ck(M, d)Rn+1−k+O(Re(R)−∞ ),as Re(R)→+∞,
where
ck(M, d) = ZM
ak,0(x, 1)dx.
Here dxis the Riemannian volume density defined from gd2. The functions ak,0(x, 1) depend on
the Taylor expansion (10) as described in Theorem 2.27 and can be computed inductively using
Lemma 2.25. In particular,
ck(M, d) =
0,when kis odd,
vold(M)
n!ωn,when k= 0,
n+1
6·n!ωnRXsd2dx, when k= 2,
where sd2in local coordinates is computed as the polynomial in the Taylor coefficients of d2at
the diagonal given as
sd2(x) :=3C4(x, g ⊗g)−3c2,n (n+ 5)(n2−9)
c1,n
(C3⊗C3)(x, g ⊗g⊗g),if n6= 1,3
sd2(x) :=310C4
G(x, gG⊗gG)−c2,3
c1,3
(C3
G⊗C3
G)(x, gG⊗gG⊗gG),if n= 3
Proof. The asymptotic expansion
hQ−1
M1,1i=
∞
X
k=0
ck(M, d)Rn+1−k+O(R−∞ ),
where ck(M, d) = RMak,0(x, 1)dxfollows directly from Lemma 2.24 and the fact that Q−1
Mis of
order n+ 1. It follows from Lemma 2.25 that ak(x, 0,1) = 0 for odd k. It follows from Theorem
2.27 that c0and c2take the prescribed form.
The justification for the notation sd2in Theorem 6.1 comes from Example 2.31 which shows
that for the geodesic distance on a Riemannian manifold, sd2is the scalar curvature. We further
conclude the following corollary.
48 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Corollary 6.2. Let Mbe a compact Riemannian manifold equipped with its geodesic distance.
Then
h1, QM(R)−11i=vold(M)
n!ωn
Rn+1 +n+ 1
6·n!ωnZM
sdxRn−1+O(Re(R)n−3),as Re(R)→+∞,
where sdenotes the scalar curvature of M.
Combining Theorem 3.7 with Theorem 6.1 we arrive at the following corollary.
Corollary 6.3. Let Mbe an n-dimensional compact manifold with a distance function dwith
property (MR) on Γ. It holds that
h1,ZM(R)−11i ∼
∞
X
k=0
ck(M, d)Rn+1−k+O(Re(R)−∞ ),as Re(R)→+∞in Γ,
where ck(M, d) is as in Theorem 6.1.
6.2. A lengthy exercise in integration by parts. To study the asymptotic expansions of
h1, Q−1
X1iin the presence of a boundary, we need a series of smaller lemmas. The reader should
recall the notation from Theorem 5.22.
Lemma 6.4. Let Xbe an n-dimensional compact manifold with boundary and da distance
function whose square is regular at the diagonal. It holds that
hQ−1
X1,1iL2(X)=hA1, χ1iL2(X)+hW−1,(W+)∗(χ2◦ϕ)iL2(∂X×[0,∞)) +O(R−∞ ).
Proof. We first note that W−preserves supports in ∂ X ×(−∞,0] ⊆∂X ×Rby Lemma 5.21 and
(W+)∗preserves supports in ∂ X ×(−∞,0] ⊆∂X ×Rsince W+preserves supports in ∂ X ×[0,∞)
by Lemma 5.21. Therefore, viewing 1 as an element of Hµ
loc(∂ X ×[0,∞)) and χ2◦ϕas an element
of Hµ
c(∂X ×[0,∞)), the images W−1∈L2
loc(∂ X ×[0,∞)) and (W+)∗(χ2◦ϕ)∈L2
c(∂X ×[0,∞))
are well defined and hW−1,(W+)∗(χ2◦ϕ)iL2(∂X ×[0,∞)) is well defined. By the same token,
hA1, χ1iL2(X)is defined as the inner product of χ1∈L2(X) with the restriction of A1M∈L2(M)
to X.
Since χ′
j|supp(χj)= 1, for j= 1,2, it follows that
hA1, χ1iL2(X)=hAχ′
1, χ1iL2(X)+O(R−∞) and
hW−1,(W+)∗(χ2◦ϕ)iL2(∂X×[0,∞)) =hW−(χ′
2◦ϕ),(W+)∗(χ2◦ϕ)iL2(∂X×[0,∞)) +O(R−∞ ).
The last equality follows from Proposition 5.4. Therefore, Theorem 5.22 reduces the statement
of the theorem to the property that hS1,1iL2(X)=O(R−∞). This is clear from the property of
Sthat kSkHµ→˙
H−µ=O(R−∞).
Lemma 6.5. Let Xbe an n-dimensional compact manifold with boundary and da distance
function whose square is regular at the diagonal. Let (aj,0)j∈N⊆C∞(M;C+)denote the sequence
of homogeneous functions obtained from restriction to ξ= 0 of the full symbol of A, as in Lemma
2.24. It holds that
hA1, χ1i=
∞
X
k=0
ck,χ1(X, d)Rn+1−k+O(R−∞),
where
ck,χ1(M, d) = ZX
χ1(x)ak,0(x, 1)dx.
Here dxis the Riemannian volume density defined from gd2.
Proof. The lemma follows immediately from Lemma 2.24 since χ1has compact support in X◦.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 49
Lemma 6.6. Let a=a(x′, ξ)∈C∞(Rn−1×Rn)be a polynomially bounded smooth function
with compact support in x′, and χ∈ S(Rn)a real even Schwartz function. Then as R→+∞,
1
(2π)nZRn−1ZRn
a(x′, ξ)Rnˆχ(−Rξ)e−Rix′ξ′dξdx=
=X
α∈Nn
Dα
xχ(0)
α!ZRn−1
Dα
ξ=0 a(x′, ξ)e−ix′ξ′dxR−|α|+O(R−∞).
In particular, if χis locally constant near 0, then
1
(2π)nZRn−1ZRn
a(x′, ξ)Rnˆχ(−Rξ)e−Rix′ξ′dξdx=χ(0) ZRn−1
a(x′,0)dx+O(R−∞).
Proof. Consider the distribution uR(ξ) := Rnˆχ(−Rξ). For any test function ϕ∈ S(Rn), we
compute that
(uR, ϕ) = ZRn
ˆχ(−ξ)ϕ(ξ/R)dξ=X
α∈Nn
Dα
xϕ(0)
α!ZRn
ˆχ(−ξ)ξαdξR−|α|+O(R−∞ ) =
= (2π)nX
α∈Nn
Dα
xχ(0)
α!R−|α|(δα, ϕ) + O(R−∞)
We conclude that in S′(Rn), we have an asymptotic expansion
uR= (2π)nX
α∈Nn
Dα
xχ(0)
α!R−|α|δα+O(R−∞).
Using standard methods for oscillatory integrals, we see that the same expansion holds also for
in the weak topology against polynomially bounded smooth functions.
We compute that
1
(2π)nZRn−1ZRn
a(x′, ξ)Rnˆχ(−Rξ)e−Rix′ξ′dξdx=1
(2π)nZRn−1ZRn
a(x′, ξ)uR(ξ)e−Rix′ξ′dξdx
=X
α∈Nn
Dα
xχ(0)
α!R−|α|ZRn−1
(δα
ξ, a(x′, ξ)e−ix′ξ′)dx+O(R−∞ ) =
=X
α∈Nn
Dα
xχ(0)
α!ZRn−1
Dα
ξ=0 a(x′, ξ)e−ix′ξ′dxR−|α|+O(R−∞ ).
Lemma 6.7. Let Xbe an n-dimensional compact manifold with boundary and da distance
function whose square is regular at the diagonal. We denote the symbols of W±by w±(as in
Definition 5.18 and Lemma 5.18). Then it holds that
hW−1, W ∗
+χ2i=
∞
X
k=0
ck,χ2(X, d)Rn+1−k+O(R−∞),
where
ck,χ2(X, d) = ZX
χ2(x)ak,0(x, 1)dx+
+X
k=|β|+γn+j+l
γn>0
i|β|+|γn|(−1)|β|+1
β′!(βn+γn)! Z∂X
∂β
xw−,j (x′,0,0,1)∂γn−1
xn∂β+(0,γn)
ξw+,l(x′,0,0,1)dx′.
Here dxis the Riemannian volume density on Xdefined from gd2and dx′the induced Riemann-
ian volume density on ∂X .
Proof. The computation can be reduced to one in local coordinates, so we can assume that w+
and w−are symbols of mixed-regularity (−µ, 0) in Rn, and up to O(R−∞ ) we can treat W+and
50 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
W−as compactly based. As such, we replace Mby Rnand Xby Rn
+in all computations. Let
w∗
+denote the symbol of W∗
+. By the same arguments as in [38, Chapter I.3], we have that
(38) w∗
+(x, ξ, R)∼X
α
1
α!∂α
ξDα
xw+(x, ξ, R),
in the sense of Definition 5.8. We note that Equation (38) only identifies w∗
+up to Sµ,−∞ but
this suffices as symbols from Sµ,−∞ will only contribute to the conditional expectation with
O(R−∞).
Using that W−and W∗
+preserves supports in Rn
−, we can consider χ2as an element of
C∞
c(Rn), and write
hW−1, W ∗
+χ2iL2(Rn
+)=ZRn
+
[W−1](x)[W∗
+χ2](x)dx,
where
W∗
+χ2(x) := 1
(2π)nZRn
w∗
+(x, ξ, R) ˆχ2(ξ)dξ,
is computed from the action of (Q−1
+)∗on χ∈C∞
c(Rn). We compute that
hW−1, W ∗
+χ2iL2(Rn
+)=1
(2π)nZRn
+ZRn
w−(x, 0, R)w∗
+(x, ξ, R) ˆχ2(ξ)eixξdξdx=
=1
(2π)nZRn
+ZRn
w−(x, 0, R)w∗
+(x, ξ, R) ˆχ2(−ξ)e−ixξdξdx=
=X
α
(−1)|α|
(2π)nα!ZRn
+ZRn
w−(x, 0, R)∂α
ξDα
xw+(x, ξ, R) ˆχ2(−ξ)e−ixξdξdx+O(R−∞ ) =
=X
α
(−1)αn
(2π)nα!ZRn
+ZRn
Dα′
x′(w−(x, 0, R)e−ixξ )Dαn
xn∂α
ξw+(x, ξ, R) ˆχ2(−ξ)dξdx+O(R−∞) =
=X
αX
γ′+β′=α′
(−1)αn
(2π)nβ′!γ′!αn!ZRn
+ZRn
Dβ′
x′w−(x, 0, R)Dαn
xn∂α
ξw+(x, ξ, R)(−iξ′)γ′ˆχ2(−ξ)e−ixξ dξdx+O(R−∞ ) =
=X
γ,β
1
(2π)nβ!γ!ZRn
+ZRn
Dβ
xw−(x, 0, R)∂γ+β
ξw+(x, ξ, R)(−iξ)γˆχ2(−ξ)e−ixξ dξdx+
+X
γ,β,
γn>0
βn
X
k=0
bγ,β,k ZRn−1ZRn
Dβ−(0,k)
xw−(x′,0,0, R)Dγn−1
xn∂β+γ
ξw+(x′,0, ξ, R)(iξ)(γ′,k)ˆχ2(−ξ)e−ix′ξ′dξdx+
+O(R−∞).
where
bγ,β ,k =i(−1)|γn|+1βn!
(2π)nβ′!γ′!(βn+γn)!k!(βn−k)!
By the composition formula for pseudodifferential operators (see [38, Chapter I.3]), we have
that
X
γ,β
1
(2π)nβ!γ!ZRn
+ZRn
Dβ
xw−(x, 0, R)∂γ+β
ξw+(x, ξ, R)(−iξ)γˆχ2(−ξ)e−ixξdξdx=
=ZRn
+
χ2(x)a(x, 0, R)dx+O(R−∞) = hQ−1
M1, χ2iL2(Rn
+)+O(R−∞).
We can therefore continue our calculation
hW−1, W ∗
+χ2iL2(Rn
+)=hQ−1
M1, χ2iL2(Rn
+)
+X
γ,β,
γn>0
βn
X
k=0
bγ,β,k ZRn−1ZRn
Dβ−(0,k)
xw−(x′,0,0, R)Dγn−1
xn∂β+γ
ξw+(x′,0, ξ, R)(iξ)(γ′,k)ˆχ2(−ξ)e−ixξ dξdx+
+O(R−∞)
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 51
To obtain an asymptotic expansion, we expand w±in its defining homogeneous expansion
w±∼Pw±,l from Definition 5.18 (see also Lemma 5.21). We see that
X
γ,β,
γn>0
βn
X
k=0
bγ,β,k ZRn−1ZRn
Dβ−(0,k)
xw−(x′,0,0, R)Dγn−1
xn∂β+γ
ξw+(x′,0, ξ, R)(iξ)(γ′,k)ˆχ2(−ξ)e−ixξ dξdx=
=
∞
X
i=0 X
i=|β|+|γ|+j+l,
γn>0
βn
X
k=0
bγ,β,k ZRn−1ZRn
Dβ−(0,k)
xw−,j (x′,0,0, R)Dγn−1
xn∂β+γ
ξw+,l(x′,0, ξ , R)(iξ)(γ′,k)ˆχ2(−ξ)e−ix′ξ′dξdx
Let us consider each of the terms
X
i=|β|+|γ|+j+l,
γn>0
βn
X
k=0
bγ,β,k ZRn−1ZRn
Dβ−(0,k)
xw−,j (x′,0,0, R)Dγn−1
xn∂β+γ
ξw+,l(x′,0, ξ , R)(iξ)(γ′,k)ˆχ2(−ξ)e−ix′ξ′dξdx=
=R2µ−iX
i=|β|+|γ|+j+l,
γn>0
βn
X
k=0
bγ,β,k ZRn−1ZRn
Dβ−(0,k)
xw−,j (x′,0,0,1)Dγn−1
xn∂β+γ
ξw+,l(x′,0,ξ
R,1)(iξ)(γ′,k)ˆχ2(−ξ)e−ix′ξ′dξdx.
We can compute each of the terms using Lemma 6.6 which implies that
1
(2π)nZRn−1ZRn
Dβ−(0,k)
xw−,j (x′,0,0,1)Dγn−1
xn∂β+γ
ξw+,l(x′,0, ξ, 1)(iξ)(γ′,k)Rnˆχ2(−Rξ)e−Rix′ξ′dξdx=
=(O(R−∞),if (γ′, k)6= 0,
RRn−1Dβ
xw−,j (x′,0,0,1)Dγn−1
xn∂β+(0,γn)
ξw+,l(x′,0,0,1)dx+O(R−∞ ),if (γ′, k) = 0
We conclude that
hQ−1
−1,(Q−1
+)∗χ2iL2(Rn
+)− hQ−1
M1, χ2iL2(Rn
+)=
=R2µ
∞
X
i=0 X
i=|β|+γn+j+l,
γn>0
i(−1)|γn|+1R−i
β′!(βn+γn)! ZRn−1
Dβ
xw−,j (x′,0,0,1)Dγn−1
xn∂β+(0,γn)
ξw+,l(x′,0,0,1)dx+O(R−∞ ).
After using D=−i∂, the boundary contributions have been computed.
The lemma now follows from Lemma 2.24 giving the asymptotic expansion
hQ−1
M1, χ2iL2(Rn
+)=X
k
Rn+1−kZRn
+
χ2(x)ak(x, 0,1)dx.
6.3. Asymptotic expansions for compact manifolds with boundary. We now study as-
ymptotic expansions of h1, Q−1
X1ifor a compact manifold with boundary, and give a procedure to
compute the coefficients. An important difference to the case of empty boundary is the boundary
contributions: we identify the from Lemma 6.7 as follows.
Definition 6.8. If Xis an n-dimensional compact manifold with boundary and d a distance
function whose square is regular at the diagonal, then we define the sequence of functions
(Bd2,k)k>0⊆C∞(∂X ) by
Bd2,k(x′) := X
k=|β|+γn+j+l
γn>0
i|β|+|γn|(−1)|β|+1
β′!(βn+γn)! ∂β
xw−,j (x′,0,0,1)∂γn−1
xn∂β+(0,γn)
ξw+,l(x′,0,0,1).
For notational simplicity, we set B0:= 0.
Proposition 6.9. Let Xbe an n-dimensional manifold with boundary, da distance function
whose square is regular at the diagonal, and (Bd2,k )k>0⊆C∞(∂X)as in Definition 6.8. Then
for each k > 0,Bd2,k is a polynomial in (C(γ)
G)γ∈∪k≤jIkand its derivatives contracted by the
metric gG, its contraction along the normal to the boundary, and its derivatives of total degree j
where each C(γ)
G,γ∈Ik, has degree k, the metric has degree zero and x-derivatives increase the
order by 1.
52 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Computing from the results of Appendix F, we can describe the special cases k= 1 and k= 2.
Proposition 6.10. Let (X, d) be as in Proposition 6.9. Then
Bd2,1(x′) = (n+ 1)
2·n!ωnph0(x′).
In particular, if xnis the transversal coordinate defined from the unit normal to ∂X (in gd2),
then
Bd2,1(x′) = (n+ 1)
2·n!ωn
.
Proof. By definition, we have that
Bd2,1(x′) = −iw−,0(x′,0,0,1)∂ξnw+,0(x′,0,0,1) =
=−i
n!ωn
h0(x′)µ(−h−(x′,0,1)))µ∂ξn[(ξn−h+(x, 0,1))µ]|ξn=0 =
=−i(n+ 1)
2·n!ωn
h0(x′)µ(−h−(x′,0,1)))µ(−h+(x, 0,1))µ−1=
=−i(n+ 1)
2·n!ωn
h0(x′)(−h−(x′,0,1))) = (n+ 1)
2·n!ωnph0(x).
Here we have used the identities of Lemma F.1. If xnis the transversal coordinate defined from
the unit normal to ∂X, then h0= 1.
Proposition 6.11. Let (X, d) be as in Proposition 6.9 and assume that xnis the transversal
coordinate defined from the unit normal ∂nto ∂X (in gd2). Then there are universal polynomials
α1and α2with rational coefficients such that
Bd2,2(x′) = α1(n)
n!ωn
C3(x′, ∂n⊗∂n⊗∂n) + α2(n)
n!ωn
C3(x′, g ⊗∂n).
Proof. By definition, we have that
Bd2,2(x) =1
2w−,0(x′,0,0,1)∂xn∂2
ξnw+,0(x′,0,0,1) −1
2∂xnw−,0(x′,0,0,1)∂2
ξnw+,0(x′,0,0,1)−
−iw−,1(x′,0,0,1)∂ξnw+,0(x′,0,0,1) −iw−,0(x′,0,0,1)∂ξnw+,1(x′,0,0,1)+
+∇x′w−,0(x′,0,0,1) · ∇ξ′∂ξnw+,0(x′,0,0,1).
These expressions were computed in Section F of the appendix. If xnis the transversal coordinate
defined from the unit normal to ∂X, then h0= 1 and b= 0. The lemmas of Section F, for h0= 1
and b= 0, shows that Bd2,2(x′) is in the linear span of C3(x′, ∂n⊗∂n⊗∂n) and C3(x′, g ⊗∂n),
and carefully inspecting the computations imply the existence of the universal polynomials α1
and α2.
Proposition 6.12. Let X⊆Rnbe a domain with smooth boundary equipped with the Euclidean
distance. Then for a universal polynomial β, it holds that
Bd2,2=β(n)
n!ωn
H,
where Hdenotes the mean curvature of the boundary.
Proof. We compute that
c1,n(n2−1)
2(n!ωn)2=−(n−2)!ωn−2ω2(n2−1)
2(n!ωn)2=−(n−1)!
(n−1)!
2πωn−2
nωn
n+ 1
4·n!ωn
=−n+ 1
4·n!ωn
.
Fix a point x0∈∂X and choose coordinates as in Example 2.16; in other words we write ∂X as
the graph of a function ϕ=ϕ(x′) with ∇ϕ(x0) = 0. The computations in Example 2.16 show
that h0(x′) = 1 + |∇ϕ(x′)|2is xn-independent and that b(x0) = ∇ϕ(x0) = 0. Moreover, C3(x, v)
is a first order order polynomial in vn, so C3(x, ιng⊗ιng⊗ιng) = 0. A short computation using
Equation (15) gives us that
C3(x0, g ⊗ιng) = −2gx0(∇2ϕ(x0)) = −(n−1)H(x0),
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 53
where Hdenotes the mean curvature. We conclude that
Bd2,2(x0) = n2−1
4·n!ωn
H(x0).
Combining Lemmas 6.4, 6.5 and 6.7 we arrive at the following theorem:
Theorem 6.13. Let Xbe an n-dimensional compact manifold with boundary and da distance
function whose square is regular at the diagonal. Denote the Riemannian volume density on X
defined from gd2by dxand the induced Riemannian volume density on ∂X by dx′. It holds that
(39) h1, Q−1
X1iL2(X)=
∞
X
k=0
ck(X, d)Rn+1−k+O(R−∞),as R→+∞,
where the coefficients ck(X, d) are given as
ck(X, d) = ZX
ak,0(x, 1)dx+Z∂X
Bd2,k(x)dx′,
where
(1) ak,0(·,1) ∈C∞(X)is an invariant polynomial in the entries of the Taylor expansion
(10) as described in Theorem 2.27 and can be computed inductively using Lemma 2.25,
with ak,0= 0 if kis odd; and
(2) Bd2,k ∈C∞(∂X )is an invariant polynomial in the entries of the Taylor coefficients of
d2at the diagonal in Xnear ∂X as described in Proposition 6.9 and can be inductively
computed using Lemma 5.19.
In particular, we have that
c0(X, d) = vol(X)
n!ωn
(40)
c1(X, d) = (n+ 1)vol(∂X )
2n!ωn
,(41)
c2(X, d) = n+ 1
6·n!ωnZX
sd2dx+(n−1)(n+ 1)2
8·n!ωnZ∂X
Hd2dx′.(42)
where the scalar curvature sd2is defined as in Theorem 6.1 and the mean curvature Hd2is an
explicit function in the linear span of C3(x′, ∂n⊗∂n⊗∂n)and C3(x′, g ⊗∂n)that coincides with
the usual mean curvature for domains in Rn.
Our notation Hd2in Theorem 6.13 is justified by Proposition 6.12 showing that Hd2=His
the mean curvature if Xis a domain in Euclidean space.
Proof. The expression in Equation (39) follows from Lemma 6.4 by adding together the compu-
tation of Lemma 6.5 with that in Lemma 6.7.
The computation (40) follows from the fact that a0,0(x, 1) = 1
n!ωn(see Theorem 2.27). The
computation (41) follows from Proposition 6.10. The computation (42) is a consequence of The-
orem 2.27 (computing the interior contribution) and Proposition 6.11 (computing the boundary
contribution).
Combining Theorem 4.5 with Theorem 6.13 we arrive at the following corollary.
Corollary 6.14. Let Xbe an n-dimensional compact manifold with boundary and da distance
function with property (MR) on [R0,∞), for some R0≥0. It holds that
h1,Z−1
X1iL2(X)=
∞
X
k=0
ck(X, d)Rn+1−k+O(R−∞),as R→+∞,
where the coefficients ck(X, d) are as in Theorem 6.13.
54 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Corollary 6.15. If X⊆Rnis a compact domain with smooth boundary, then
c0(X, d) = vol(X)
n!ωn
c1(X, d) = µvol(∂X)
n!ωn
,
c2(X, d) = µ2(n−1)
2·n!ωnZ∂X
HdS.
The computation of Corollary 6.15 is compatible with the computations of [11] for µ∈N. We
note that the precise proportionality constant in c2follows from the computation from [11] for
µ∈Nsince the pre-factor by Proposition 6.12 is determined as a universal polynomial in n.
Appendix A. Overview of conormal distributions
Pseudodifferential operators with parameters will play an important role in our study of
the operators Qand Z, both to prove meromorphic extensions and to compute asymptotic
expansions. We will use an approach to parameter dependence described in terms of conormal
distributions to which the operator Qis susceptible.
First we recall the basics of conormal distributions. We follow the approach of [23, Chapter
18.2]. For a tempered distribution u∈ S′(RN) we write Fuor sometimes ˆufor its Fourier
transform. We define the following subsets of RNas B1:= B(0,1) and Bj:= B(0,2j)\B(0,2j−1)
for j > 1. For a function u∈ S′(RN) we write u=P∞
j=1 ujin a distributional sense where uj
is determined by ˆuj= ˆuχBj. Following the notation of [23, Appendix B], for s∈R, we set
∞
Hs(RN) := {u∈ S′(RN) : sup
j∈N+
2sj kujkL2(RN)<∞}.
Written in terms of the standard notation for Besov spaces, ∞
Hs(RN) = Bs
∞,2(RN). By [23,
Theorem 18.2.9], the notion of belonging to ∞
Hs(RN) is coordinate invariant for compactly
supported distributions.
If Zis a smooth N-dimensional manifold, and u∈ D′(Z) satisfies that for each coordinate
chart κ:U→U′⊆RNand χ∈C∞
c(U), (κ−1)∗(χu)∈∞
Hs(RN) we say that uis locally ∞
Hs.
We define
∞
Hloc
s(Z) := {u∈ D′(Z) : uis locally ∞
Hs}.
Due to coordinate invariance on compacts of the space ∞
Hs(RN), it holds that
∞
Hloc
s(RN) = {u∈ D′(Rn) : χu ∈∞
Hs(RN)∀χ∈C∞
c(RN).
Due to the locality of the definition, if (κα)α∈Iis a locally finite atlas on a smooth manifold Z
and (χα)α∈Ia subordinate smooth partition of unity, u∈∞
Hloc
s(Z) if and only if (κ−1
α)∗(χαu)∈
∞
Hs(RN) for all α∈I.
Definition A.1 (Definition 18.2.6 of [23]).Let Zbe a smooth N-dimensional manifold and
Y⊆Za smooth submanifold which is closed in the topology of Z. We let Im(Z;Y) denote the
space of conormal distributions of order m. We say that u∈ D′(Z) is a distribution conormal
to Yof order mif for each collection of first order differential operator L1,...,Lp(with C∞-
coefficients) that are tangential to Y, it holds that
L1···Lpu∈∞
Hloc
−m−(N−k)/2(Z).
We let Im(Z, Y ) denote the space of distributions conormal to Yof order m.
We here define the order of conormal distributions in such a way that the order of the conormal
distribution matches the order of the operator it will define. Our convention differs from that in
[23] but is consistent with that in [39].
The reader should note that for u∈Im(Z, Y ), it holds that u|Z\Y∈C∞(Z\Y). Moreover,
if Lis an arbitrary differential operator (with C∞-coefficients) of order pthen Lu ∈Im+p(Z, Y )
whenever u∈Im(Z, Y ). However, if u∈Im(Z, Y ) then for any collection of first order differential
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 55
operator L1,...,Lp(with C∞-coefficients) that are tangential to Y, then L1···Lpu∈Im(Z, Y ).
By elliptic regularity, we have that
I−∞(Z, Y ) := ∩m∈RIm(Z, Y ) = C∞(Z).
Conormality of a distribution to Yencompasses being a smooth function away from Yand a
prescribed type of singularity as one approaches Y. The singularity at Yis characterized by a
symbol in the direction normal to Y.
Definition A.2. Let E→Ybe a real vector bundle over a k-dimensional smooth manifold.
We say that a∈C∞(E) is a symbol of order m∈Rif for each vector bundle trivialization
E|U→U×Rpover an open set U⊆Yevery compact K⊆Uand multiindices β∈Nk,γ∈Np
there is a constant C > 0 such that
|∂β
x∂γ
ξa(x, ξ)| ≤ C(1 + |ξ|2)(m−|γ|)/2,for (x, ξ )∈K×Rp.
Here we have identified awith a function on U×Rpusing the trivialization E|U→U×Rp.
We let Sm(E) denote the space of symbols of order mon E→Y. If Y=Rkand E=Y×Rp,
we write Sm(Rk×Rp) instead of Sm(E).
Let Z=RNand Y=Rkembedded into the first factor and write the coordinates on RNas
x= (y, z) where y∈Rkand z∈RN−k. It was proven in [23, Theorem 18.2.8] that the map
a7→ u, where
(43) u(x) := ZRN−k
a(y, z, ξ)eiz·ξdξ,
defines a surjection Sm(RN×RN−k)→Im(Z, Y ). Moreover, it was also proven in [23, Lemma
18.2.1] that the map ˜a7→ u, where
(44) u(x) := ZRN−k
˜a(y, ξ)eiz·ξdξ,
defines an isomorphism Sm(Rk×RN−k)→Im(Z, Y ). The relationship between the aappearing
in Equation (43) and the ˜aappearing in Equation (44) is stated in [23, Lemma 18.2.1] to be
˜a(y, ξ)∼X
α∈NN−k
1
α!∂α
zDα
ξa(y, z, ξ)|z=0 .
The right hand side is an asymptotic sum, for more details see [23], and is only well defined up
to smooth functions. Here ∼denotes equality up to smooth errors.
In the case of a general manifold, we need some further set up. Consider a smooth manifold
Zand a smooth submanifold Y⊆Zclosed in the topology of Z. We define the conormal bundle
N∗Y→Yas the kernel of the restriction mapping T∗Z|Y→T∗Yand the normal bundle
NY →Yas the vector bundle N Y := T Z|Y/T Y . By construction, N∗Y= (NY )∗as vector
bundles on Y. We equip the normal bundle with a fixed fiberwise volume density. After making
a choice of metric, one can find a tubular neighborhood UYof Y⊆Zand construct a fiber
preserving diffeomorphism ψ:UY→N Y mapping Yonto the zero section of N(Y). Here we
consider the tubular neighborhood to be a ball bundle over Y. We pick a χY∈C∞(UY) with
χY= 1 in a neighborhood of Ysuch that the pro jection mapping supp (χY)ψ
−→ N Y →Yis
proper. We define the mapping
q:Sm(N∗Y)→ D′(Z),
as the composition of the fiberwise Fourier transform Sm(N∗Y)→ D′(NY ) (with respect to the
volume density), pullback along ψand multiplication by χY. We can also define the mapping
σ:Im(Z, Y )→ D′(N∗Y),
as the composition of multiplication by χY, pullback along ψand Fourier transform in the fiber
direction (with respect to the volume density).
Theorem A.3. The maps qand σgive well defined maps
q:Sm(N∗Y)→Im(Z, Y )and σ:Im(Z, Y )→Sm(N∗Y),
56 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
that induce isomorphisms
q:Sm(N∗Y)/S−∞ (N∗Y)→Im(Z, Y )/C∞(Z)and
σ:Im(Z, Y )/C∞(Z)→Sm(N∗Y)/S−∞ (N∗Y).
The qoutient mappings
qm:Sm(N∗Y)/Sm−1(N∗Y)→Im(Z, Y )/Im−1(Z, Y )and
σm:Im(Z, Y )/Im−1(Z, Y )→Sm(N∗Y)/S m−1(N∗Y),
are mutual inverses and independent of the choice of ψand χY.
Proof. The details of the proof can be found in [23, Chapter XVIII.2]. The reader can also
consult [39, Chapter 4], see Theorem 4.3.2 and Theorem 4.3.16.
To digest this theorem, let us consider what this means in practice. Fix a tubular neighbor-
hood UYof Yinside Zand a bundle isomorphism ψ:UY→N Y . It says that u∈Im(Z, Y ) if
and only if uis C∞outside Yand we for any χ∈C∞
c(Z) supported in a small enough open set
U⊆UYcan write
χu(y, z ) = ZRN−k
a(y, z, ξ)eiz·ξdξ+r(x),
in local coordinates on Ufor a symbol a∈Sm(N∗Y) and an r∈C∞(U). We note that the
order of uis determined by the order of the symbol a.
An important subclass of conormal distributions that we will later make use of are the classical
ones. Classicality simplifies computations and are by definition given by asymptotic sums,
making them highly suitable for studying asymptotics of the magnitude with. Let us define
classicality in the context of both symbols and conormal distributions. For a vector bundle
E→Yand t > 0, we let λtdenote the dilation if functions and distributions in the fiber
direction. We say that a function or distribution uis homogeneous of degree m∈Cif λtu=tmu
for all t > 0.
Definition A.4. Consider a smooth N-dimensional manifold Zand a smooth k-dimensional
submanifold Y⊆Zclosed in the topology of Z. Fix a tubular neighborhood UYof Yinside Z,
a bundle isomorphism ψ:UY→NY and a . Let m∈C.
(1) We say that a∈C∞(N∗Y) is a classical symbol of order mif there is a collection
(aj)j∈N⊆C∞(N∗Y\Y), with ajbeing homogeneous of degree m−j, and a smooth
function χ∈C∞(N∗Y) which is 1 outside a fiberwise precompact neighborhood of Y,
such that
a∼
∞
X
j=0
χaj.
We let CSm(N∗Y) denote the space of classical symbols of order m.
(2) For m /∈Z, we say that u∈ D′(Z) is a classical conormal distribution of order mif u|Z\Y
is smooth and there is a collection (uj)j∈N∈ D′(N Y ), with ujbeing homogeneous of
degree j−m−N+k, and a smooth function χ∈C∞(Z) which is 1 in a neighborhood
of Ysuch that for each lthere is an Lwith
χu −
L
X
j=0
ψ∗uj∈Cl(Z).
We let CIm(Z, Y ) denote the space of classical conormal distributions of order m.
(3) For m∈Z, we say that u∈ D′(Z) is a classical conormal distribution of order mif u|Z\Y
is smooth and there is a collection (uj)j∈N∈ D′(N Y ), with ujbeing homogeneous of
degree j−m−N+k, a collection (pj)j∈N∈C∞(NY ) of fiber wise homogeneous
polynomials, with each pjbeing of degree j−m−N+k, and a smooth function χ∈
C∞(Z) which is 1 in a neighborhood of Ysuch that for each lthere is an Lwith
χu −
L
X
j=0
ψ∗(uj+pjlog | · |)∈Cl(Z).
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 57
We let CIm(Z, Y ) denote the space of classical conormal distributions of order m.
Let us make some remarks on the case m∈Z. Here log | · | ∈ C∞(NY \Y) denotes the
fiberwise logarithm in some choice of Riemannian metric. The reader should note that pj= 0
if j < m +N−k. The appearance of logarithms is due to the fact that despite symbols of
the form |ξ|−mbeing classical and homogeneous outside ξ6= 0, their Fourier transforms are not
homogeneous if m∈ −N+k−Nand contains logarithms. We shall explore this further in our
main example below where the Fourier transform of |ξ|k−Nis precisely the logarithm. This is of
relevance to us since the conormal distribution defined from the operator Qis a logarithm (see
Proposition 2.5).
Proposition A.5. It holds that CSm(N∗Y)⊆SRe(m)(N∗Y)and CIm(Z, Y )⊆IRe(m)(Z, Y ).
Moreover, the maps from Theorem A.3 induce mappings
q:CS m(N∗Y)→CIm(Z, Y )and σ:C I m(Z, Y )→CS m(N∗Y),
that induce isomorphisms
q:CS m(N∗Y)/S−∞(N∗Y)→C I m(Z, Y )/C∞(Z)and
σ:CI m(Z, Y )/C ∞(Z)→CSm(N∗Y)/S−∞(N∗Y),
qm:CS m(N∗Y)/CSm−1(N∗Y)→CIm(Z, Y )/CIm−1(Z, Y )and
σm:CI m(Z, Y )/CIm−1(Z, Y )→C Sm(N∗Y)/CSm−1(N∗Y)
The last two mappings are independent of the choices of ψand χYand are mutual inverses.
The reader should note that if we pick a Riemannian metric on N∗Y, the restriction map-
ping to the associated sphere bundle S N ∗Y⊆N∗Yof the leading term induces an isomor-
phism CS m(N∗Y)/CSm−1(N∗Y)∼
=C∞(SN ∗Y). We can as such view σmas an isomorphism
CI m(Z, Y )/C I m−1(Z, Y )→C∞(SN ∗Y).
We shall not give the details of the proof of Theorem A.5, but merely note that it is a trivial
consequence of Theorem A.3 in conjunction with homogeneity properties of the Fourier trans-
form. We shall give the details for the case of interest to us below in Subsection A.1.
Conormal distributions often arise from the Schwartz kernels of pseudodifferential operators.
To study the operator Q, we will need to apply conormal distributions to pseudodifferential
operators with parameters. But let us nevertheless provide the reader with some context first
by reviewing pseudodifferential operators from the point of view of conormal distributions.
Consider a manifold Mand set Z:= M×M. We let Y:= DiagM⊆Zdenote the diagonal.
Consider a continuous operator A:C∞
c(M)→C∞(M) with Schwartz kernel KA∈ D′(M×
M). In terms of conormal distributions, Ais called a pseudodifferential operator of order m∈
Rif KA∈Im(M×M, DiagM). The standard definition of a pseudodifferential operator is
equivalent by Theorem A.3. In the case at hand Y∼
=Mvia the projection map and N∗Y=
T∗M. As such, the symbol mapping induces isomorphisms Im(M×M, DiagM)/C∞(M×M)∼
=
Sm(T∗M)/S−∞ (T∗M) and analogous mappings in the classical setting.
A.1. Examples of conormal distributions. In this subsection we consider two examples
coming from powers and logarithms of distance functions in Euclidean space. Both examples
will play a pivotal role in our study of the operator Q. In lack of finding a good reference we
have included a large amount of details.
We consider Z=RNand Y=Rkviewed as a submanifold of Z. We write the Euclidean
coordinates as x= (y, z) for y∈Y=Rkand z∈RN−k. For t∈R+, define the automorphism
λtof C∞
c(Z) by λtϕ(y, z) := ϕ(y, tz). We define λton D′(Z) by
hλtu, ϕi:= t−khu, λ1/tϕi.
Take an α∈Cand define uα,0∈C∞(Z\Y) by uα,0(y, z) := |z|−α. If α /∈k−N+N, [22,
Theorem 3.2.3] guarantees that uα,0has a unique extension to a distribution uα∈ D′(Z) such
that λtuα=t−αuαfor all t > 0. A short computation shows that uαdepends holomorphically
on α∈C\(k−N+N). If f=f(w) is a function depending holomorphically on win a punctured
58 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
neighborhood of α, we write F.P.w=αffor the constant coefficient in the Laurent expansion of
faround w=α. For α∈C, we use the notation F.P.|z|−αto denote the distribution
hF.P.|z|−α, ϕi:= F.P.w=αhuw, ϕi.
It follows from the construction that if α∈C\(k−N+N), then F.P.|z|−αis homogeneous of
degree −α, i.e. λt.F.P.|z|−α=t−αF.P.|z|−αfor all t > 0. By [22, Theorem 3.2.3], F.P.|z|−α=uα
for α∈C\(k−N+N). The computation [22, Equation (3.2.24), page 76] shows that if
α∈k−N+Nwe have for all t > 0 that
λtF.P.|z|−α−t−αF.P.|z|−α= log(t)X
β∈Nk:|β|=α+N−k
cβ∂β
zδY,
for some coefficients cβand δY∈ D′(Z) is the distribution defined by
hδY, ϕi=ZRk
ϕ(y, 0)dy.
Proposition A.6. Let α∈C. Consider the distribution F.P.|z|−αon RN−kas constructed in
the preceding paragraph (for a fixed y). Then F.P.|z|−αis a tempered distribution on RN−kand
for ξ6= 0, we have that
FF.P.|z|α=
π(N−k)/22α+N−kΓ(α+N−k
2)
Γ(−α
2)|ξ|−N+k−α, α ∈C\(−N+k−2N),
π(N−k)/2(−1)l
22ll!Γ(N−k
2+l)|ξ|2l(−log |ξ|2+βl,N−k), α =−N+k−2l, l ∈N,
where
βl,N−k:= 2 log(2) + 1
2ψ((N−k)/2 + l)−Hl−γ,
and ψ(z) := Γ′(z)
Γ(z),H0= 0 and Hl:= Pl
j=1 1
jfor l > 0, and γis the Euler-Mascheroni constant.
We note that in the case α∈2N, the right hand side of the equality has a zero which is
compatible with that FF.P.|z|αis a sum of distributions supported at ξ= 0. A closely related
computation of hFF.P.|z|α, ϕifor test functions ϕwith vanishing moments may be found in [36],
Lemma 25.2. It implies the above result except for the value of βl,N−k.
Proof. The distribution F.P.|z|αis polynomially bounded and therefore a tempered distribution.
To compute its Fourier transform, we first assume that α /∈ −N+k−2N. In this case, F.P.|z|αis
homogeneous of degree α, so its Fourier transform must be homogeneous of degree −N+k−α.
Moreover, F.P.|z|αis O(N−k)-invariant, so its Fourier transform must be O(N−k)-invariant.
The space of O(N−k)-invariant tempered distributions of degree α /∈ −N+k−2Nis one-
dimensional and therefore there is a constant cαsuch that FF.P.|z|α=cαF.P.|ξ|−N+k−αif
α /∈(−N+k−2N)∪2Nand FF.P.|z|αis supported at ξ= 0 if α∈2N. To verify that
cα=π(N−k)/22α+N−kΓ(α+N−k
2)
Γ(−α
2)one restricts to αin the range (−N+k, −(N−k)/2) and
integrates against the standard Gaussian (see [37, Exemple 5, Chapitre VII.7] but beware of the
non-standard convention with 2πin the Fourier transform of [37]).
The case that α∈ −N+k−2Nfollows from that the Fourier transform is continuous and
therefore commutes with taking finite part values. Let us compute the details. Near αl=
−N+k−2lwe have that
cα=2al
α−αl
+bl+O(α−αl),
where
al=π(N−k)/2(−1)l
22ll!Γ N−k
2+land bl=π(N−k)/2(−1)l
22ll!Γ N−k
2+lβl,N−k.
The expressions for aland blfollows from the fact that the Γ-function has no zeroes and its
poles are situation in ζ∈ −Nwhere
Resζ=−kΓ(ζ) = (−1)k
k!and F.P.ζ=−kΓ(ζ) = (−1)k+1
k!(γ+Hk).
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 59
Taking the finite part of FF.P.|z|α=cαF.P.|ξ|−N+k−αat α=αl, gives us for ξ6= 0 that
FF.P.|z|−N+k−2l=bl|ξ|2l−2al|ξ|2llog |ξ|=bl|ξ|2l−al|ξ|2llog |ξ|2,
which produces the expression in the statement of the proposition.
Write CS m,k for the class of log-classical symbols and CIm,k for the corresponding class of
log-classical conormal distributions, see more in [30].
Proposition A.7. Let Z=RN,Y=Rkand α∈C\{0}. The distribution F.P.|z|−αconstructed
in the preceding paragraph satisfies that F.P.|z|−α∈CI α−N+k ,j (Z, Y ), where j∈ {0,1}and
j= 1 if and only if α∈N−k+ 2N, and
σα−N+kF.P.|z|−α(y, ξ) =
π(N−k)/22−α+N−kΓ(−α+N−k
2)
Γ(α
2)|ξ|−N+k+α, α ∈C\(N−k+ 2N),
π(N−k)/2(−1)l
22ll!Γ(N−k
2+l)|ξ|2l(−log |ξ|2+βl,N−k), α =N−k+ 2l, l ∈N.
Proof. For notational simplicity, set u:= F.P.|z|−α. Pick a compactly supported χ0∈C∞
c(RN−k)
with χ= 1 near z= 0. Write u=u1+u2where ˆu1=χˆuand u2=u−u1. The distribution u1is
constant in the y-direction. By Proposition A.6, u1is the Fourier transform in the z-direction of
the compactly supported distribution cαχF.P.|ξ|−N+k+α, so u1is smooth. It therefore suffices
to prove that u2∈CI α−N+k(Z, Y ). Consider the classical symbol a(ξ) := (1 −χ(ξ))ˆu(ξ). By
viewing aas a constant function of x= (y, z )∈Rn, we have that a∈CS−N+k+α,j (RN×RN−k)
by the computation of Proposition A.6. Moreover,
u2(y, z) = ZRN−k
a(y, z, ξ)eiz·ξdξ,
so u2∈CI α−N+k(Z, Y ) by Proposition A.5.
For Z=RNand Y=Rk, with coordinates x= (y, z ) for y∈Y=Rkand z∈RN−k,
we define the function u(y, z) := log |z|2. This function is locally integrable and defines a
distribution. Let us turn to studying its singularity at Y. First we need an elementary Fourier
transform computation that follows from Proposition A.6.
Proposition A.8. The function ˜u(z) := log |z|2on RN−kis a tempered distribution and for
ξ6= 0,
F˜u=−2π(N−k−1)!ωN−k−1|ξ|k−N,
where ωN−k−1denotes the volume of the unit ball in dimension N−k−1.
Proof. Since ˜uis locally integrable and polynomially bounded, it is a tempered distribution. To
compute its Fourier transform, we note that λt˜u−˜u= log(t). In other words, ˜uis homogeneous
of degree 0 up to a constant function. In particular, F˜umust therefore be homogeneous of degree
−n+kup to a multiple of δ0. Moreover, ˜uis O(N−k)-invariant, so F˜umust be O(N−k)-
invariant. The space of O(N−k)-invariant elements of S′(RN−k)/Cδ0that are homogeneous
of degree −N+kis one dimensional and is spanned by the equivalence class defined from
F.P.1
|ξ|N−k. It is clear that F˜uis not compactly supported, so its class in S′(RN−k)/Cδ0is
non-zero. It follows from the computation in Proposition A.7 that for ξ6= 0,
F˜u=−2N−kπ(N−k)/2ΓN−k
2|ξ|k−N.
A short computation, as in Proposition 2.8, gives that
ΓN−k
2ΓN−k−1
2+ 1=√π(N−k−1)!
2N−k−1.
Therefore we have that
2N−kπ(N−k)/2ΓN−k
2= 2π(N−k−1)! π(N−k−1)/2
ΓN−k−1
2+ 1= 2π(N−k−1)!ωN−k−1.
60 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Proposition A.9. Let Z=RNand Y=Rk. The distribution log |z|satisfies that log |z| ∈
CI −N+k(Z, Y )and
σ−N+k(log |z|) (y, ξ) = −2π(N−k−1)!ωN−k−1|ξ|k−N, ξ 6= 0.
The proof is analogous to that of Proposition A.7 and is therefore omitted.
For g∈C∞(Z) and a submanifold Y⊆Zwith a prescribed tubular neighborhood U∼
=NY ,
we define the transversal Hessian of gin y∈Yas the symmetric bilinear form on (N Y )ydefined
from the Hessian at the zero section of grestricted to (NY )yalong the tubular neighborhood.
Proposition A.10. Assume that Y⊆Zis an k-dimensional smooth submanifold of an N-
dimensional smooth manifold. Let ˜
G∈C∞(Z)be a smooth function such that
•˜
Gand d˜
Gvanishes on Y;
•˜
G(x)>0for x /∈Y; and
•for each y∈Y, the transversal Hessian H˜
Gof ˜
G(defined as in Definition 2.1) is a
positive definite quadratic form on the transversal tangent bundle of Y⊆Z.
Then log ˜
G∈CI −N+k(Z;Y)and its principal symbol σ−N+klog ˜
G∈C∞(N∗Y\Y)is given
by
σ−N+klog ˜
G(y, ξ) = −2π(N−k−1)!ωN−k−1|g˜
G(ξ, ξ)|−(N−k)/2, ξ 6= 0,
where g˜
Gis the metric dual to the transversal Hessian H˜
G.
Proof. The statement is local, so we can assume that there is an open set U⊆Rncontaining
0 such that Z=Uand Y=U∩Rk. Since dgvanishes on T Y , we can consider its restriction
to Yto be a section dg|Y:Y→N∗Y. Under the assumption that the transversal Hessian of
gis non-degenerate in all points of Y, we can assume that Uis taken small enough to be able
to choose coordinates (˜y, ˜z) on Usuch that g(˜y, ˜z) = |˜z|2for ˜z6= 0. For notational simplicity,
we assume that g(y, z ) = |z|2and that Z=U=RNand Y=Rk. The proposition now follows
from Proposition A.9.
Appendix B. Parameter dependent pseudodifferential operators
The reason for the above adventure through the seven circles of conormal distributions is
that it simplifies our description of the magnitude operator as a pseudodifferential operator with
parameter. We shall make heavy use of parameter dependent pseudodifferential operators, so let
us recall their definition and describe their kernel structure. In this subsection we only consider
classical symbols and operators.
Definition B.1. Let Γ ⊆Cbe a conical subset, m∈Cand n, p ∈N. A function a=a(x, ξ, R)∈
C∞(Rp×Rn×Γ) is said to be a classical symbol with parameter of order mif there exists a
sequence (aj)j∈N⊆C∞(Rp×((Rn×Γ) \{0})) of functions homogeneous of degree m−jin the
sense that
aj(x, tξ, tR) = tm−jaj(x, ξ , R),∀t > 0,
and a χ∈C∞
c(Rn×Γ) such that
a∼X
j
(1 −χ)aj.
We write CSm(Rp×Rn; Γ) for the space of classical symbols with parameter of order m.
For an n-dimensional manifold M, we define CSm(M; Γ) ⊆C∞(T∗M⊕Γ) as the space of
functions a∈C∞(T∗M⊕Γ) such that in each local chart φ:U⊆M→Rnand χ∈C∞
c(U),
(Dφ)∗(χa)∈CSm(Rn×Rn; Γ).
Remark B.2.Let us clarify what we mean by a∼Pj(1 −χ)aj. We write a∼Pj(1 −χ)aj
when for any α∈Np,β∈Nn,k, N ∈Nand compact K⊆Rpthere is a constant C > 0 and an
N0∈Nsuch that
∂α
x∂β
ξ∂k
R
a−
N0
X
j=0
(1 −χ)aj
(x, ξ, R)≤C(1 + |ξ|+|R|)−N, x ∈K, ξ ∈Rn, R ∈Γ.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 61
If this condition is satisfied, one can in fact take N0to be any natural number ≥N− |β| −
k+ Re(m). Conversely, if (aj)j∈N⊆C∞(Rp×((Rn×Γ) \ {0})) is a sequence of functions
homogeneous of degree m−jthen for any χ∈C∞
c(Rp×Γ) which equals 1 in a neighborhood
of 0 ∈Rn×Γ there exists an a∈CSm(Rp×Rn; Γ) with a∼Pj(1 −χ)aj.
Remark B.3.We use the notation S−∞(Rp×Rn; Γ) for the space of smoothing symbols, i.e. the
space of a∈C∞(Rp×Rn×Γ) satisfying that for any α∈Np,β∈Nn,k, N ∈Nand compact
K⊆Rpthere is a constant C > 0 such that
∂α
x∂β
ξ∂k
Ra≤C(1 + |ξ|+|R|)−N.
Note that S−∞(Rp×Rn; Γ) = ∩m∈RC Sm(Rp×Rn; Γ). For a, b ∈C∞(Rp×Rn×Γ) we write
a∼bif a−bis a smoothing symbol.
Remark B.4.For an n-dimensional manifold M, we shall implicitly use a Riemannian metric
on Mand write S(T∗M⊕Γ) →Mto denote fiber bundle of elements (ξ , R)∈T∗M⊕Γ
such that |ξ|2+|R|2= 1. An element a∈CS m(M; Γ) is determined modulo lower order
terms (i.e. CSm−1(M; Γ)) by its principal symbol. In other words, if a∼Pj(1 −χ)ajthen
a−a0∈CS m−1(M; Γ). Since a0is homogeneous of degree m, it is determined by its restriction
to S(T∗M⊕Γ). We deduce that a7→ a0|S(T∗M⊕Γ) defines an isomorphism
CS m(M; Γ)/CSm−1(M; Γ) ∼
=C∞(S(T∗M⊕Γ)).
We will mainly be interested in the case p= 2nor p=nwhere nis the dimension of
the underlying manifold. If Γ = Rour definition coincides with that of classical symbols on
Rp×Rn+1.
Let us turn to the quantization of symbols to operators. We can consider an element K∈
S−∞(R2n× {0}; Γ) ⊆C∞(R2n×Γ) as a family of operators K=K(R) : C∞
c(Rn)→C∞(Rn)
defined by
K(R)f(x) := ZRn
K(x, y, R)f(y)dy.
Such a family of operators is called a smoothing operator with parameter. We write Ψ−∞(Rn; Γ)
for the space of smoothing operators with parameters. If a=a(x, y, ξ, R)∈C Sm(R2n×Rn; Γ)
we can associate a family of operators Op(a) = Op(a)(R) : C∞
c(Rn)→C∞(Rn) defined by
Op(a)(R)f(x) := ZRnZRn
a(x, y, ξ, R)f(y)eiξ·(x−y)dydξ.
We note that in this order of integration, both integrals are well defined for f∈C∞
c(Rn). Al-
ternatively, we can consider this as an oscillatory integral in which the order of integration is
irrelevant. A family of operators A=A(R) : C∞
c(Rn)→C∞(Rn) is called a classical pseudodif-
ferential operator of order mwith parameter if there is a classical symbol a∈CSm(R2n×Rn; Γ)
and a smoothing operator K∈Ψ−∞ (Rn; Γ) such that
A=Op(a) + K.
We write Ψm
cl (Rn; Γ) for the space of classical pseudodifferential operators of order mwith
parameter. For two operators A1, A2:C∞
c(Rn)→C∞(Rn) we write A1∼A2if A1−A2is
smoothing with parameter. If A∈Ψm
cl (Rn; Γ) satisfies A=χAχ′for some χ, χ′∈C∞
r(Rn), we
say that Ais compactly supported.
Proposition B.5. Let m∈Cand consider a conical subset Γ⊆C. Then the following holds:
(1) Assume that Γ = Rand let A∈Ψm
cl (Rn;R)be a classical pseudodifferential operators
of order mwith parameter. There is an A′∈Ψm
cl (Rn;R)with A∼A′such that the
family of Schwartz kernels KA′(R)∈ D′(Rn×Rn)satisfies that FR→ηKA′∈CIm(Rn×
Rn×R;Rn× {0}), where FR→ηKA′denotes the Fourier transform in the R-direction
and we identify Rn× {0} ⊆ Rn×Rn×Ras a submanifold via the diagonal inclusion
Rn→Rn×Rn.
62 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
(2) Let A=Op(a) + K∈Ψm
cl (Rn; Γ) for an a∈CSm(R2n×Rn; Γ). For
˜a(x, ξ, R) := X
α∈Nn
1
α!∂α
yDα
ξa(x, y, ξ, R)|x=y∈CSm(Rn×Rn; Γ),
we have that Op(a)∼Op(˜a). In particular, we can write A=Op(˜a) + ˜
Kfor an
˜a∈CS m(Rn×Rn; Γ) and a smoothing operator with parameter ˜
K.
(3) An element Aas in (2) is uniquely determined up to smoothing operators by ˜amodulo
smoothing symbols. In particular, the mapping
Ψm
cl (Rn; Γ)/Ψ−∞(Rn; Γ) →CSm(Rn; Γ)/S −∞ (Rn; Γ), A 7→ ˜a,
is a well defined isomorphism.
(4) If A=Op(˜a)∈Ψm
cl (Rn; Γ) and B=Op(˜
b)∈Ψm′
cl (Rn; Γ) for ˜a∈CSm(Rn; Γ) and
˜
b∈CS m′(Rn; Γ) then AB ∈Ψm+m′
cl (Rn; Γ) and AB ∼Op(a#b)where
a#b(x, ξ)∼X
α
1
α!Dα
ξa(x, ξ, R)∂α
xb(x, ξ, R)∈CSm+m′(Rn×Rn; Γ).
Proof. For (1) we take A′to be the classical pseudodifferential operator defined from that KA′=
χKAwhere χ∈C∞(Rn×Rn) is defined from χ(x, y) = χ0(|x−y|2) for some χ0∈C∞
c(R) with
χ0= 1 in a neighborhood of 0. It is clear from standard arguments with oscillatory integrals
that KA′−KA∈Ψ−∞(Rn;R). Since KA′is properly supported, we can write A′=Op(a) for
an a∈CS m(R2n×Rn;R). It follows from the arguments around Equation (43) that
[FR→ηKA′](x, y, η) = ZRn+1
a(x, y, ξ, R)ei(x−y)ξ−iηRdRdξ,
defines an element of CIm(Rn×Rn×R;Rn× {0}).
Items (2), (3) and (4) follow as in the case of no parameters, see [38, Theorem 3.1], [39,
Theorem 4.3.2] (with ρ= 1 and δ= 0), and [38, Theorem 3.4], respectively.
Remark B.6.With regards to item (1) in Proposition B.5, we refer to a∈C Sm(R2n×Rn; Γ) as a
“two-variable symbol” and ˜a∈CSm(Rn×Rn; Γ) as a “one-variable symbol”. In the applications
we are concerned with, the two-variable symbol is easily obtained. Due to item (3), i.e. that the
two variable symbol is not determined by the operator, and item (4) we shall need to use the
transition from two-variable symbols to one-variable symbols. The isomorphism in item (3) of
Proposition B.5 is called the full symbol isomorphism.
For the remainder of this subsection let Mdenote an n-dimensional smooth manifold. For
simplicity we assume that Mis compact. For m∈Cand a conical subset Γ ⊆C, an family of
operators A=A(R) : C∞(M)→C∞(M) is a classical pseudodifferential operators of order m
with parameter if for any coordinate chart φ:U⊆M→U′⊆Rn, and any χ, χ′∈C∞
c(U′) the
composition of operators defined on f∈C∞
c(Rn) as
f7→ φ∗(χf)7→ Aφ∗(χf )7→ (φ−1)∗(χ′Aφ∗(χf )) ∈C∞(Rn),
is a classical pseudodifferential operators of order mwith parameter on Rn. We write Ψm
cl (M; Γ)
for the space of classical pseudodifferential operators of order mwith parameter on M. The space
of smoothing operators Ψ−∞ (M; Γ) is by definition isomorphic to the space S(Γ; C∞(M×M))
consisting of Schwartz functions from Γ into the Fr´echet space of smoothing operators C∞(M×
M).
Since the full symbol of a pseudodifferential operator with parameter is uniquely determined
modulo smoothing symbols in local coordinates, we obtain a full symbol isomorphism
Ψm
cl (M; Γ)/Ψ−∞(M; Γ) ∼
=CS m(M; Γ)/S−∞(M; Γ).
The next theorem contains well known statements that summarize the properties of pseu-
dodifferential operators with parameter we shall make use of. First, let us introduce some
notation. The 2n+ 1-dimensional manifold M×M×Rcarries an R-action by translation in
the third factor. We shall assume that Z⊆M×M×Ris an R-invariant open neighborhood
of DiagM× {0} ⊆ M×M×R, i.e. that there is an open neighborhood U⊆M×Mof
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 63
DiagMsuch that Z=U×R. From Proposition B.5 and the coordinate transformation laws of
pseudodifferential operators we deduce the following theorem.
Theorem B.7. Let m∈Cand let Γ⊆Cbe a sector.
(1) Assume that Γ = R. Item (1) of Proposition B.5 and the full symbol mapping, respec-
tively, induce isomorphisms
CI m(Z; DiagM× {0})/C∞(Z)∼
=Ψm
cl (M; Γ)/Ψ−∞(M; Γ) ∼
=
∼
=CS m(M; Γ)/S−∞(M; Γ),
and
CI m(Z; DiagM× {0})/CI m−1(Z; DiagM× {0})∼
=
∼
=CS m(M; Γ)/CSm−1(M; Γ) ∼
=C∞(S(T∗M⊕Γ)).
(2) The principal mapping that maps a pseudodifferential operator with parameter to its
leading homogeneous term defines an isomorphism
σm: Ψm
cl (M; Γ)/Ψm−1
cl (M; Γ) →C Sm(M; Γ)/CSm−1(M; Γ) ∼
=C∞(S(T∗M⊕Γ)).
(3) If A∈Ψm
cl (M; Γ) and B∈Ψm′
cl (M; Γ), then AB ∈Ψm+m′
cl (M; Γ). The full symbol
a#bof AB is computed in local coordinates from the full symbols aand bof Aand B,
respectively, and the formula
a#b(x, ξ)∼X
α
1
α!Dα
ξa(x, ξ, R)∂α
xb(x, ξ, R)∈CSm+m′(M; Γ).
Definition B.8. Let m∈C,Mbe a compact manifold and A∈Ψm
cl (M; Γ). We say that Ais
elliptic if σm(A)∈C∞(S(T∗M⊕Γ)) is an invertible element.
Remark B.9.The reader should beware that in item (3) of Theorem B.7, the full symbol depends
on the choice of coordinates.
The following proposition follows from item (2) of Theorem B.7 and asymptotic completeness
of the space of pseudodifferential operators with parameter.
Proposition B.10. Let A∈Ψm
cl (M; Γ) be an operator with full symbol a∈CSm(M; Γ). The
following are equivalent:
•Ais elliptic
•There exists b∈CS−m(M; Γ) with ab = 1 outside a compact subset of T∗M⊕Γ.
•There exists B∈Ψ−m
cl (M; Γ) such that AB −1, BA −1∈Ψ−∞ (M; Γ).
We recall that we have implicitly used a Riemannian metric on M. Using this metric, we
also define the scale of Hilbert spaces Hs(M) := (1 + ∆)−s/2L2(M) for s∈R, where ∆ is the
Laplace operator defined from the Riemannian metric. We also consider the scale Hs
R(M) :=
(R2+ ∆)−s/2L2(M) defined for R∈R\ {0}and s∈R. We note that elliptic regularity implies
that Hs(M) = Hs
R(M) as vector spaces with equivalent norms, but their Hilbert space structure
differs in a non-uniform way as Rvaries. The difference in Hilbert space structure is best seen
from the mapping properties of pseudodifferential operators with parameters, which is made
precise in the following theorem.
Theorem B.11. Let Γ⊆Γα(0) ∪ −Γα(0) be a bisector with opening angle < α ∈[0, π/2), and
A∈Ψm
cl (M; Γ). For any s, t ∈Rwith t≤s−Re(m)there is a constant Cs,t >0such that for
R∈Γ\ {0}kAfkHt
|R|≤Cs,t(1 + |R|)t−s+Re(m)kfkHs
|R|,∀f∈Hs(M).
Moreover, A:Hs
|R|(M)→Ht
|R|(M)is compact if t < s −Re(m). In particular, if Ais elliptic
then there are for any s∈Rconstants Cs, R0>0such that for |R|> R0
1
CskfkHs
|R|≤ kAfkHs−Re(m)
|R|≤CskfkHs
|R|,∀f∈Hs(M).
64 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Proof. It is clear that A:Hs
|R|(M)→Ht
|R|(M) is compact if t < s−Re(M) by the corresponding
statement without parameters. We start by showing the norm estimate kAf kHt
|R|≤Cs,t(1 +
|R|)t−s+Re(m)kfkHs
|R|. Since Γ does not contain the imaginary axis, the operator (R2+ ∆)m/2∈
Ψm
cl (M; Γ) is an invertible elliptic element. Assume that α∈[π/4, π /2). Since there is a positive
angle between Γ and the imaginary axis we have a lower quadratic form estimate
(R2+ ∆)∗(R2+ ∆) ≥(1 −cos2(2α))|R|4,for R∈Γ.
It therefore follows by interpolation that
(45) k(R2+ ∆)m/2fkHt
|R|≤Cs,t(1 + |R|)t−s+Re(m)kfkHs
|R|, f ∈Hs(M).
By combining Equation (45) with item (4) of Theorem B.7 we can reduce to t=s=m= 0.
To prove the theorem in the case t=s=m= 0, we need to show that point evaluation in
Rdefines a continuous representation Ψ0
cl(M; Γ) → B(L2(M)) with uniform seminorm bounds.
This statement follows from the fact that point evaluations in Rdefines a continuous mapping
Ψ0
cl(M; Γ) →Ψ0
cl(M) in the respective Fr´echet topologies with uniform seminorm bounds and the
Calder´on-Vaillancourt theorem implies that the action of zero order pseudodifferential operators
on L2(M) defines a continuous representation Ψ0
cl(M)→ B(L2(M)).
Fix an s∈R. We want to show that for some R0and Cs, we have the estimates 1
CskfkHs
|R|≤
kAfkHs−Re(m)
|R|≤CskfkHs
|R|for |R|> R0. By the preceding paragraph, the upper estimate
holds. To show the lower estimate we note that if A∈Ψm
cl (M; Γ) is elliptic, then there is a
B∈Ψ−m
cl (M; Γ) such that AB −1, BA −1∈Ψ−∞ (M; Γ). Write Asfor the continuous operator
Hs
|R|(M)→Hs−Re(m)
|R|(M) defined from Aand Bsfor the continuous operator Hs−Re(m)
|R|(M)→
Hs
|R|(M) defined from B. From the preceding paragraph, we have that k1−AsBsk,k1−BsAsk=
O(R−∞) so there is an R0such that Bs(R)−As(R)−1is a smoothing operator for |R|> R0. We
conclude from the preceding paragraph that there is a C′such that kBsgkHs
|R|≤C′kgkHs−Re(m)
|R|
for all g∈Hs−Re(m)
|R|(M). This shows that 1
C′kfkHs
|R|≤ kAfkHs−Re(m)
|R|
for all f∈Hs
|R|(M) and
|R|> R0, and we conclude the lower estimate.
Corollary B.12 (G˚arding inequality).Let Γ⊆Γα(0) ∪−Γα(0) be a bisector with opening angle
< α ∈[0, π/2), and A∈Ψm
cl (M; Γ) a formally self-adjoint operator with strictly positive principal
symbol, i.e. for some ε > 0,σ(A)(x, ξ , R)≥εfor all |ξ|2+|R|2= 1 and x∈M.
Then for large R, the quadratic form f7→ hAf, f iL2is continuous, positive and coercive on
HRe(m)/2
|R|. To be precise, there is an R0>0and a C > 0such that
1
CkfkHRe(m)/2
|R|≤ hAf, f iL2≤CkfkHRe(m)
|R|
,∀f∈Hs(M),|R|> R0.
Proof. It follows in the same way as in [38, Proposition 6.1], that there is a B∈Ψm/2
cl (M; Γ)
such that r:= A−B∗B∈Ψ−∞(M; Γ). It is clear that Bis elliptic and that Ris self-adjoint,
so the previous theorem implies that for some C0>0,
1
CkfkHRe(m)/2
|R|− hrf, f iL2≤ hAf, f iL2≤CkfkHRe(m)
|R|
+hrf, f iL2,∀f∈Hs(M).
Since r∈Ψ−∞(M; Γ), we have that kr(R)kL2→L2=O(|R|−∞) and the corollary follows.
One of the reasons for introducing conormal distributions above was that it will give us a
direct way of verifying that the magnitude operator is an elliptic pseudodifferential operator
with parameter. We will next present the relevant technical result needed for such an endeavor.
For a compact smooth n-dimensional manifold, identified with its diagonal in M×M, we let
Z=U×R⊆M×M×Rdenote an R-invariant tubular neighborhood of the diagonal of M. After
fixing some metric on Mthat we for notational simplicity assume to have injectivity radius >1,
we can assume that we have exponential coordinates on Zidentifying Zwith the ball bundle
BM ⊕R={(x, v, η)∈T M ⊕R:|v|<1}.
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 65
This is the ball bundle in the normal bundle of DiagM× {0} ⊆ Zunder the identification of
Mwith its diagonal. We denote the coordinate in the R-direction by ηsince for conormal
distributions it plays the role of a dual variable (cf. item (1) of Proposition B.5). In the
situation at hand, the conormal bundle of DiagM× {0} ⊆ Zis under the identification of M
with its diagonal given by T∗M⊕R→M. We denote coordinates in the fiber of the conormal
bundle by (ξ, R). We note that under these identifications, we have a canonical isomorphism
(46) CSm(N∗(DiagM× {0} ⊆ Z)) ∼
=CS m(M;R).
Compare this to Theorem B.7. A problem we need to address is that the correspondence be-
tween operators and symbols is only well behaved modulo smoothing terms, and the notions
of smoothing symbols and operators in the setting of conormal distributions differ from that in
the setting of the parameter dependent calculus. To address the issue at the level of operators,
instead of equivalence classes, we introduce a uniform notion of asymptotic expansions.
Assume that K∈CIm(Z;M× {0}) in exponential coordinates has a classical asymptotic
expansion of the form K∼P∞
j=0 χKjas in Definition A.4. Here χ∈C∞
c(U) is a function
with χ= 1 near the diagonal. In the above choice of coordinates, Kjis a smooth function on
T M ⊕R\(M× {0}). If m /∈Z,Kjis homogeneous of degree −m−j−n−1. If m∈Z,
Kj=uj+pjlog(|v|2+η2) where ujis homogeneous of degree −m−j−n−1 and pjis a
homogeneous polynomial in (v, η) of degree −m−j−n−1 (in particular pj= 0 if j < −m−n−1).
Definition B.13. We shall say that K∈CIm(Z;M× {0}) has a uniform asymptotic ex-
pansion if for any α∈Np,β∈Nn,k, N ∈Nthere is a constant C > 0 and an N0∈Nsuch
that for R6= 0 ∂α
x∂β
v∂k
η
K−
N0
X
j=0
χKj
≤C(1 + |v|+|η|)−N.
For x∈M, we write expx:TxM→Mfor the exponential map. Note that under our
assumption on the injectivity radius, expx:BxM→Mis a diffeomorphism onto its range.
Proposition B.14. Let Mbe a smooth n-dimensional manifold and Zis as in the preceding
paragraphs. Assume that K∈CIm(Z;M× {0})admits a uniform asymptotic expansion. Then
there is a pseudodifferential operator with parameter A∈Ψm(M;R)such that for any R, the
Schwartz kernel of A(R)is given by Fη→RK. In other words, for f∈C∞(M),A(R)fis defined
as the oscillatory integral
A(R)f(x) := ZBxM⊕R
K(x, v, η)f(expx(v))eiηRdvdη.
In particular, the full symbol of Ain CSm(M;R)/S−∞ (M;R)coincides with the full symbol of
Kin CS m(N∗(DiagM× {0} ⊆ Z))/S−∞ (N∗(DiagM× {0} ⊆ Z)) under the isomorphism (46).
Proof. If the uniform asymptotic expansion of Konly contains one term, i.e. K=K0, the
statement of the proposition holds by homogeneity properties of the Fourier transform. As such,
the proposition is in fact a statement concerning the asymptotic completeness of the space of
pseudodifferential operators with parameters. This is proven just as in the usual setting (see
[23, Proposition 18.1.3]).
Appendix C. Inverting Qin an extended Boutet de Monvel calculus
In this Appendix we prove Theorem 4.9. More generally, we discuss the magnitude operator
ZX(R) in the context of general techniques which have been developed for elliptic pseudodiffer-
ential boundary problems. While ZX(R) is not always a pseudodifferential operator, for distance
functions having property (MR) Theorem 4.5 shows that ZX(R) closely relates to the pseudodif-
ferential operator QX(R). While there is an extensive theory for boundary problems for elliptic
differential operators, developed over many decades, boundary problems for pseudodifferential
operators pose severe difficulties since the operators are nonlocal. Much of the recent study was
motivated by fractional powers of the Laplacian. In particular, for a larger class of operators
pseudodifferential methods were developed starting with [19], under the assumption that the op-
erator satisfies a µ-transmission condition at the boundary. They allowed to show, for example,
66 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
regularity results for solutions of elliptic Dirichlet problems and integration by parts formulas.
A basic observation is that such problems could be reduced to problems in the Boutet de Monvel
calculus, originally developed for differential boundary problems [19]. While the results in this
Appendix are not directly used in this article, they complement the specific analysis of Z(R)
by results and challenges for a large class of related problems. We refer to [15, 19] for further
context and notation.
As the techniques are local, it is sufficient to prove Theorem 4.9 for X⊂⊂ Rn. Consider the
Dirichlet problem
(47) r+PRuR=fin X
uR= 0 in Xc.
Generalizing QX(R), here PRis a parameter dependent, classical pseudodifferential operator on
Xwhich is elliptic of order −m=−n−1, of type −m/2 and has factorization index −m/2. We
make the further assumption that it is of infinite regularity.
We reduce (47) to a problem of type and order 0 in the Boutet de Monvel calculus, where a
parametrix can be constructed using classical techniques. The reduction relies on the following
operators to raise the order of PR, while preserving support in X, see [19, Section 2.5]:
Definition C.1. For l∈1
2Z, define λl
±∈Ψl
cl(Rn; Γ)
λl
−(ξ, R) = hξ′, Riψξn
bhξ′, Ri−iξnl
, λl
+(ξ, R) = λl
−(ξ, R).
Here ψ∈ S(R) with supp F−1ψ⊂R−and ψ(0) = 1, and b≥2 sup |∂tψ(t)|. We set Λl
±,R :=
Op λl
±(ξ, R).
The symbols λl
±are uniformly parameter-elliptic on Rn×Rn+1
+of order l, and they are of
factorization index and type l, respectively 0.
Composition of (47) on the left with r+Λm
2
−,Re+gives
r+Λm
2
−,Re+r+PRuR=r+Λm
2
−,RPRuR=r+Λm
2
−,Re+f.
Letting vR:= r+Λ−m
2
+,RuR,equivalently uR= Λ m
2
+,Re+vR,and denoting AR:= Λ m
2
−,RPRΛm
2
+,R gives
the following problem for vR:
(48) r+ARe+vR=r+Λm
2
−,Re+fin X
vr= 0 in M\X .
We next show that (48) is a problem in the Boutet de Monvel calculus, based on arguments
in [19, Section 2.5]:
Proposition C.2. The operator ARis an elliptic parameter-dependent pseudodifferential oper-
ator of order, type, and factorization index 0.
Using a parametrix constructed in this calculus, one recovers the solution to the original prob-
lem (47) by the transformation uR= Λ m
2
+,Re+vR. In particular, this implies the assertions about
QX(R) in Theorem 4.9. Unfortunately, for the purposes of this paper the general properties
of of the parametrix after c omposition with Λ m
2
+,R are not currently understood sufficiently to
directly yield results for the magnitude problem.
The second part of Theorem 4.9, about ZX(R), then follows from the decomposition ZX(R)−1=
QX(R)−1+RX(R) in Theorem 4.5.
Proof. That it is a pseudodifferential operator follows from the fact that it is a composition of
honest pseudodifferential operators. It is immediate that it is of order m
2−m+m
2= 0 and
factorization index −m
2+m
2= 0.
All that is left to show is that the composition λm/2
−,R #σ#λm/2
+,R is of type 0, where by σwe
denote the symbol of AR.
By our assumption on PRand due to the nature of the Λ operators, we see that we can regard
e
A(x, (ξ′, R), ξn) = AR(x, ξ, R) as a classical elliptic ψdo, with the parameter Rabsorbed in the
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 67
ξ′variable. For easier notation, we still write the new variable as ξ′, and use vand wto denote
the symbols λm
2
−,R and λm
2
+,R respectively.
We first show that the composition σ#wis of type 0.
Since σis of type −m
2, we have that
(49) Dβ
xDα
ξσ−m−j(x, ξ′,−N) = (−1)(−m−2(−m
2)−j−|α|)Dβ
xDα
ξσ−m−j(x, ξ′, N )
for all multiindices β, α, where σjare the terms in the polyhomogeneous expansion. Similarly,
since wis of type m
2it satisfies
(50) Dβ
xDα
ξwjm
2−j(x, ξ′,−N) = (−1)(m
2−2(m
2)−j−|α|)Dβ
xDα
ξwm
2−j(x, ξ′, N )
for all multiindices β, α, where wjare the terms in the polyhomogeneous expansion.
Similarly to above, denoting the terms in the polyhomogeneous expansion of σ#wby [σ#w]l,
we have that
[σ#w]−m
2−l="X
α
1
α!(Dα
ξσ−m−j)(Dα
xwm
2−k)#−m
2−l
=X
α:|α|=l−k−j
1
α!(Dα
ξσ−m−j)(Dα
xwm
2−k)
and thus
Dβ
xDγ
ξ[σ#w]−m
2−l=X
α:|α|=l−k−j
1
α!Dβ
xDγ
ξ(Dα
ξσ−m−j)(Dα
xwm
2−k)
=X
α:|α|=l−k−j
1
α!Dβ
x
X
ν:ν≤γγ
ν(Dα+ν
ξσ−m−j)(Dγ−ν
ξDα
xwm
2−k)
=X
α:|α|=l−k−j
1
α!X
ν:ν≤γ
ν′:ν′≤ββ
ν′γ
ν(Dβ−ν′
xDα+ν
ξσ−n−1−j)(Dα+ν′
xDγ−ν
ξwm
2−k).
By (49) and (50), we have that
Dβ−ν′
xDα+ν
ξσ−m−j(x, ξ′,−N) = (−1)(−j−|α+ν|)Dβ−ν′
xDα+ν
ξσ−m−j(x, ξ′, N )
and
Dα+ν′
xDγ−ν
ξwm
2−k(x, ξ′,−N) = (−1)(−m
2−k−|γ−m|)wm
2−k(x, ξ′, N ).
We thus have that
Dβ
xDγ
ξ[σ#w]−m
2−l(x, ξ′,−N)
=X
α:|α|=l−k−j
1
α!X
ν:ν≤γ
ν′:ν′≤ββ
ν′γ
ν(−1)(−m
2−(j+k)−(|α+γ|))
(Dβ−ν′
xDα+ν
ξσ−m−j(x, ξ′, N ))(Dα+ν′
xDγ−ν
ξwm
2−k(x, ξ′, N ))
= (−1)(−m
2−l−γ)X
α:|α|=l−k−j
1
α!Dβ
xDγ
ξDα
ξσ−m−j(x, ξ′, N )Dα
xwm
2−k(x, ξ′, N ))
= (−1)(−m
2−l−γ)Dβ
xDγ
ξ[σ#w]−m
2−l(x, ξ′, N )
and we see that this symbol is of type 0.
We now want to show that v#σ#wis of type 0. We let s:= σ#wand thus all we want to
show is that v#sis of type 0, and the calculation is analogous to the one above. Both vand s
are of type 0 and in addition the composition is of order 0.
68 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
As above,
[v#s]−l="X
α
1
α!Dα
ξv−j(Dα
xs−k)#−l
=X
α:|α|=l−k−j
1
α!Dα
ξv−j(Dα
xs−k)
and similarly
Dβ
xDγ
ξ[v#s]−l=X
α:|α|=l−k−j
1
α!X
ν:ν≤γ
ν′:ν′≤ββ
ν′γ
ν(Dβ−ν′
xDα+ν
ξv−j)(Dα+ν′
xDγ−ν
ξs−k).
Since vis such that
Dβ
xDα
ξv−j(x, ξ′,−N) = (−1)(−j−|α|)Dβ
xDα
ξv−j(x, ξ′, N )
and the same for sthe calculation follows from above.
Appendix D. The meromorphic Fredholm theorem
The meromorphic Fredholm theorem describes the inverse of a holomorphic family of Fredholm
operators. The version we consider can be found in [29, Proposition 1.1.8]. For two Hilbert spaces
H1and H2let Fred(H1,H2)⊆B(H1,H2) denote the set of Fredholm operators, which is open
in the operator norm.
Theorem D.1. Let D⊆Cbe a connected domain and T:D→Fred(H1,H2)a holomorphic
function. Assume that T(λ)is invertible for at least one λ∈D. Then the set
Z:= {z∈D: 0 ∈Spec(T(z))}
is a discrete subset of D, and the function λ7→ T(λ)−1exists as a meromorphic function
D→Fred(H1,H2). For λnear any z∈Z,λ7→ T(λ)−1has a pointwise norm-convergent
Laurent expansion
(51) T(λ)−1=
∞
X
k=−N
Tk(λ−z)k,
where Tkare finite rank operators whenever k < 0.
Appendix E. Partial fraction decompositions of symbols
We note the following structural result from basic calculus:
Lemma E.1. For all l, m ∈Nwith m < 2l, there exists homogeneous rational functions
(with rational coefficients) of (h+, h−)denoted by βl,m,0,±, βl,m,1,±,...,βl,m,l−1,±, where each
βl,m,j,±=bl,m,j,±(h+, h−)has homogeneous degree m−j−lin (h+, h−), such that
ξm
n(ξn−h+)−l(ξn−h−)−l=
l−1
X
j=0
βl,m,j,+(h+, h−)(ξn−h+)j−l+
l−1
X
j=0
βl,m,j,−(h+, h−)(ξn−h−)j−l.
For explicit computations of the first few terms, we require the precise form of βl,m,j,±:
Proposition E.2. Let h+, h−∈Cbe two distinct complex numbers and l∈N>0. Then for
ξn∈C\ {h+, h−}, we have that
(ξn−h+)−l(ξn−h−)−l
=1
(h+−h−)l
l−1
X
j=0
(−1)j
(h+−h−)jl+j−1
j(ξn−h+)−l+j+ (−1)l−j(ξn−h−)−l+j
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 69
Proof. We show the assertion by induction in l. The case l=1 is easily checked:
(ξn−h+)−1(ξn−h−)−1=1
h+−h−(ξn−h+)−1−(ξn−h−)−1.
Assume now that the assertion holds for l. To show it for l+ 1, we take derivatives ∂2
∂h+∂ h−on
both sides:
∂2
∂h+∂h−
(ξn−h+)−l(ξn−h−)−l=l2(ξn−h+)−l−1(ξn−h−)−l−1,
respectively,
∂2
∂h+∂h−
(ξn−h+)−l(ξn−h−)−l
=
l−1
X
j=0
(−1)jl+j−1
j∂2
∂h+∂h−1
(h+−h−)l+j(ξn−h+)−l+j+ (−1)l−j(ξn−h−)−l+j.
Consider
∂2
∂h+∂h−1
(h+−h−)l+j(ξn−h+)−l+j+ (−1)l−j(ξn−h−)−l+j
=∂h+∂h−
1
(h+−h−)l+j(ξn−h+)−l+j+ (−1)l−j(ξn−h−)−l+j
+∂h+
1
(h+−h−)l+j(−1)l−j∂h−(ξn−h−)−l+j
+∂h−
1
(h+−h−)l+j∂h+(ξn−h+)−l+j.
Using that
∂h±(ξn−h±)−l=l(ξn−h±)−l−1,
∂h+(h+−h−)−l=−l(h+−h−)−l−1, ∂h−(h+−h−)−l=l(h+−h−)−l−1,
as well as
∂h+∂h−(h+−h−)−l=−l(l+ 1)(h+−h−)−l−2,
we conclude
∂2
∂h+∂h−
(ξn−h+)−l(ξn−h−)−l
=
l−1
X
j=0
(−1)jl+j−1
j−(l+j)(l+j+ 1)
(h+−h−)l+j+2 (ξn−h+)−l+j+ (−1)l−j(ξn−h−)−l+j
+(l−j)(l+j)
(h+−h−)l+j+1 (ξn−h+)−l+j−1+ (−1)l−j+1 (l−j)(l+j)
(h+−h−)l+j+1 (ξn−h−)−l+j−1.
Note that the coefficient of (ξn−h+)−l+k= (ξn−h+)−(l+1)+(k+1) is given by
(−1)kl+k−1
k(−(l+k)(l+k+ 1)) + (−1)k+1l+k
k+ 1(l−k−1)(l+k+ 1)
= (−1)k+1 l+k−1
k(l+k) + l+k
k+ 1(l−k−1)(l+k+ 1)
= (−1)k+1 (l+k−1)!
k!(l−1)! (l+k) + (l+k)!
(k+ 1)!(l−1)!(l−k−1)(l+k+ 1)
= (−1)k+1l2l+k+ 1
k+ 1 .
70 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
A similar computation applies to the coefficient of (ξn−h−)−l+k= (ξn−h−)−(l+1)+(k+1), so
that
l2(ξn−h+)−l−1(ξn−h−)−l−1=∂2
∂h+∂h−
(ξn−h+)−l(ξn−h−)−l
=1
(h+−h−)l+1
l
X
j=0
(−1)jl2
(h+−h−)jl+j
j(ξn−h+)−l−1+j+ (−1)l+1−j(ξn−h−)−l−1+j.
The asserted formula follows for exponent l+ 1 after dividing by l2.
More generally, by differentiating with respect to h±we obtain:
Lemma E.3. For all l, m ∈Nand two distinct complex numbers h+, h−∈C, consider the
rational function:
Km,l(ξn) := (ξn−h+)−m(ξn−h−)−l.
This rational function can be decomposed as
Km,l(ξn)
=
m−1
X
j=0
(−1)j
(h+−h−)l+jl+j−1
j(ξn−h+)−m+j+
l−1
X
j=0
(−1)m
(h+−h−)m+jm+j−1
j(ξn−h−)−l+j.
These following formulas allow us to obtain explicit partial fraction decompositions for the
terms relevant to the factorization of ZR.
Corollary E.4. For all l∈Nand two distinct complex numbers h+, h−∈C,
ξn(ξn−h+)−l(ξn−h−)−l=Kl−1,l(ξn) + h+Kl,l(ξn),
ξ2
n(ξn−h+)−l(ξn−h−)−l=Kl−1,l−1(ξn) + h−Kl−1,l(ξn) + h+Kl,l−1(ξn) + h+h−Kl,l(ξn),
ξ3
n(ξn−h+)−l(ξn−h−)−l=Kl−2,l−1(ξn) + (2h++h−)Kl−1,l−1(ξn)+
+ (h2
−+h2
++h+h−)Kl−1,l(ξn) + h2
+h−Kl,l(ξn)
In particular, we have the formulas
(h+−h−)ξn(ξn−h+)−1(ξn−h−)−1=h+(ξn−h+)−1−h−(ξn−h−)−1,
(h+−h−)2ξn(ξn−h+)−2(ξn−h−)−2=h+(ξn−h+)−2+h−(ξn−h−)−2−
−h++h−
h+−h−
((ξn−h+)−1−(ξn−h−)−1),
(h+−h−)2ξ2
n(ξn−h+)−2(ξn−h−)−2=h2
+(ξn−h+)−2+h2
−(ξn−h−)−2−
−2h+h−
h+−h−
((ξn−h+)−1−(ξn−h−)−1),
(h+−h−)2ξ3
n(ξn−h+)−2(ξn−h−)−2=h3
+(ξn−h+)−2+h3
−(ξn−h−)−2+
+h2
+(h+−3h−)
h+−h−
(ξn−h+)−1−h2
−(h−−3h+)
h+−h−
(ξn−h−)−1.
Appendix F. Evaluation of some boundary symbols at zero
For the purpose of computations in Subsection 6.3, we are interested in evaluating some
symbols at ξ= 0 and R= 1. We will use the notations from the Sections 5 and 6 freely. We
tacitly assume that n > 1 to avoid limit cases.
Recall from Proposition 5.13 that
R2+g(ξ, ξ) = h0(ξn−h+)(ξn−h−),
where h±=h±(x, ξ′, R)∈S1(T∗Y×R, Y ×R;C) are of the form
h±(x, ξ′, R) = −ξ′(b(x))
h0(x)±ipR2+gY(ξ′, ξ′)−(ξ′(b))2
ph0(x).
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 71
Here we use the splitting of the metric
g=h0b
bTgY,
For xn= 0, i.e. on ∂X, we write x′instead of (x′,0). We conclude the following lemma from
elementary computations.
Lemma F.1. The following identities hold on ∂X :
(52) h0(x′)ν(−h+(x′,0,1))ν(−h−(x′,0,1))ν= 1,for all ν∈R.
(53) h±(x′,0,1) = ±ih0(x′)−1/2.
(54) ∂xnh±(x′,0,1) = ∓i
2
∂xnh0(x′)
h0(x′)3/2.
(55) ∇x′h±(x′,0,1) = ∓i
2∇x′h0(x′)
h0(x′)3/2.
(56) ∇ξ′h±(x′,0,1) = −b(x′)
h0(x′).
From Lemma F.1 we deduce the following series of lemmas. We use the notation µ=n+1
2.
Lemma F.2. The following identity holds on ∂X :
w−,0(x′,0,0,1)∂xn∂2
ξnw+,0(x′,0,0,1) = µ(µ−1)(µ−2)
2n!ωn
∂xnh0(x′).
Proof. We compute that
w−,0(x′,0,0,1)∂xn∂2
ξnw+,0(x′,0,0,1) =
=−µ(µ−1)(µ−2)
n!ωn
∂xnh+(x′,0,1)h0(x)µ(−h−(x′,0,1))µ(−h−(x′,0,1))µ−3=
=−µ(µ−1)(µ−2)
n!ωn
∂xnh+(x′,0,1)h0(x′)3(−h−(x′,0,1))3=
=−µ(µ−1)(µ−2)
n!ωn−i
2
∂xnh0(x′)
h0(x′)3/2h0(x′)3(ih0(x′)−1/2)3=
=µ(µ−1)(µ−2)
2n!ωn
∂xnh0(x′)
Lemma F.3. The following identity holds on ∂X :
∂xnw−,0(x′,0,0,1)∂2
ξnw+,0(x′,0,0,1) = −µ2(µ−1)
2·n!ωn
∂xnh0(x′).
Proof. We compute that
∂xnw−,0(x′,0,0,1)∂2
ξnw+,0(x′,0,0,1) =
=µ2(µ−1)
n!ωn∂xnh0(x′)h0(x′)µ−1(−h−(x′,0,1))µ−
−∂xnh−(x′,0,1)h0(x′)µ(−h−(x′,0,1))µ−1(−h+(x′,0,1))µ−2=
=µ2(µ−1)
n!ωn∂xnh0(x′)h−(x′,0,1)
h+(x′,0,1) +∂xnh−(x′,0,1) h0(x′)
h+(x′,0,1)=
=µ2(µ−1)
n!ωn−∂xnh0(x′) + i
2
∂xnh0(x′)
h0(x′)3/2
h0(x′)
ih0(x′)−1/2=−µ2(µ−1)
2·n!ωn
∂xnh0(x′)
72 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Lemma F.4. The following identity holds on ∂X :
w−,1(x′,0,0,1)∂ξnw+,0(x′,0,0,1) =
=iµc1,n(n2−1)
(n!ωn)23
2C3(x′, g ⊗ιng) + 17(n+ 3)
4h0(x)C3(x′, ιng⊗ιng⊗ιng)−
−7iµ3
4·n!ωn∂xnh0(x′) + b(x′)
h0(x′)· ∇x′h0(x′)
Proof. Recall the computation of w−,1from Proposition 5.20, and the homogeneous symbols
a−,0(of degree 0) and a−,1(of degree 1) computed in Proposition 5.16 and explicitly given in
Equation (36) and (37), respectively. We compute that
w−,1(x′,0,0,1)∂ξnw+,0(x′,0,0,1) =
=µ
n!ωn−a−,0(x′,0,1)h0(x)µ(−h−(x′,0,1))µ−1−a−,1(x′,0,1)h0(x)µ(−h−(x′,0,1))µ−2−
−iµ2(∂xnh−(x′,0,1) − ∇ξ′h−(x′,0,1) · ∇x′h−(x′,0,1))h0(x)µ(−h−(x′,0,1))µ−2−
−iµ2(∂xnh0(x′)− ∇ξ′h−(x′,0,1) · ∇x′h0(x′))h0(x′)µ−1(−h−(x′,0,1))µ−1·
·(−h+(x′,0,1))µ−1=
=µ
n!ωn−a−,0(x′,0,1)h0(x)−a−,1(x′,0,1)h0(x)(−h+(x′,0,1))−
−iµ2(∂xnh−(x′,0,1) − ∇ξ′h−(x′,0,1) · ∇x′h−(x′,0,1))h0(x)2(−h+(x′,0,1))−
−iµ2(∂xnh0(x′)− ∇ξ′h−(x′,0,1) · ∇x′h0(x′))=
=µ
n!ωn−a−,0(x′,0,1)h0(x) + ia−,1(x′,0,1)h0(x)3/2−
−iµ2i
2
∂xnh0(x′)
h0(x′)3/2+i
2
b(x′)
h0(x′)·∇x′h0(x′)
h0(x)3/2h0(x)2(−ih0(x′)−1/2)−
−iµ2∂xnh0(x′) + b(x′)
h0(x′)· ∇x′h0(x′)=
=µ
n!ωn−a−,0(x′,0,1)h0(x) + ia−,1(x′,0,1)h0(x)3/2−
−3i
2µ2∂xnh0(x′) + b(x′)
h0(x′)· ∇x′h0(x′).
(57)
Computing with Equation (36) and (37) gives us
a0,−(x, 0, R) =3ic1,n(n2−1)
n!ωn
C3(x′, g ⊗ιng)h0(x′)−1h−(x′,0,1)
h+(x′,0,1) −h−(x′,0,1)−
−ic1,n(n+ 3)3,−2
n!ωn
h0(x′)−2h−(x′,0,1)2(h−(x′,0,1) −3h+(x′,0,1))
(h+(x′,0,1) −h−(x′,0,1))3C3(x′, ιng⊗ιng⊗ιng)+
−i(n+ 1)2
4
(∇ξ′h−(x′,0,1) · ∇x′h+(x′,0,1) −∂xnh+(x′,0,1))
h+(x′,0,1) −h−(x′,0,1) =
=−ic1,n(n2−1)
n!ωn3
2h0(x′)C3(x′, g ⊗ιng) + 4(n+ 3)
h0(x)2C3(x′, ιng⊗ιng⊗ιng)+
+iµ2
4h0(x′)∂xnh0(x′) + b(x′)
h0(x′)· ∇x′h0(x′),
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 73
and
a1,−(x, ξ′, R) = ic1,n(n+ 3)3,−2
n!ωn
h−(x′,0,1)3
h0(x)2(h+(x′,0,1) −h−(x′,0,1))2C3(x′, ιng⊗ιng⊗ιng) =
=c1,n(n2−1)(n+ 3)
4·n!ωnh0(x′)5/2C3(x′, ιng⊗ιng⊗ιng)
Continuing from Equation (57), we have that
w−,1(x′,0,0,1)∂ξnw+,0(x′,0,0,1) =
=µ
n!ωnic1,n(n2−1)
n!ωn3
2C3(x′, g ⊗ιng) + 4(n+ 3)
h0(x)C3(x′, ιng⊗ιng⊗ιng)−
−iµ2
4∂xnh0(x′) + b(x′)
h0(x′)· ∇x′h0(x′)+ic1,n(n2−1)(n+ 3)
4·n!ωnh0(x′)C3(x′, ιng⊗ιng⊗ιng)−
−3i
2µ2∂xnh0(x′) + b(x′)
h0(x′)· ∇x′h0(x′)=
=iµc1,n(n2−1)
(n!ωn)23
2C3(x′, g ⊗ιng) + 17(n+ 3)
4h0(x)C3(x′, ιng⊗ιng⊗ιng)−
−7iµ3
4·n!ωn∂xnh0(x′) + b(x′)
h0(x′)· ∇x′h0(x′).
Lemma F.5. The following identity holds on ∂X :
w−,0(x′,0,0,1)∂ξnw+,1(x′,0,0,1) =
=ic1,n(n2−1)
(n!ωn)2−3(µ−1)
2C3(x′, g ⊗ιng) + µ(n+ 3)
4h0(x′)C3(x, ιng⊗ιng⊗ιng)+
+iµ2(3µ−5)
4·n!ωn∂xnh0(x′) + b(x′)
h0(x′)· ∇x′h0(x′).
Proof. Recall the computation of w+,1from Proposition 5.20, and the homogeneous symbols
a+,0(of degree 0) and a+,1(of degree 1) computed in Proposition 5.16 and explicitly given in
Equation (34) and (35), respectively. Note that a+,0,a+,1,h0and h±are independent of ξn.
We compute that
w−,0(x′,0,0,1)∂ξnw+,1(x′,0,0,1) =
=−1
n!ωn
h0(x′)µ(−h−(x′,0,1))µ·
·(µ−1)a+,0(x′,0,1)(−h+(x′,0,1))µ−2+ (µ−2)a+,1(x′,0,1)(−h+(x′,0,1))µ−3+
+iµ2(µ−2)(∂xnh+(x′,0,1) − ∇ξ′h+(x′,0,1) · ∇x′h+(x′,0,1))(−h+(x′,0,1))µ−3=
=−1
n!ωn(µ−1)a+,0(x′,0,1)h0(x′)2(−h−(x′,0,1))2+
+ (µ−2)a+,1(x′,0,1)h0(x′)3(−h−(x′,0,1))3+
+iµ2(µ−2)(∂xnh+(x′,0,1) − ∇ξ′h+(x′,0,1) · ∇x′h+(x′,0,1))h0(x′)3(−h−(x′,0,1))3=
=1
n!ωn(µ−1)a+,0(x′,0,1)h0(x′) + i(µ−2)a+,1(x′,0,1)h0(x′)3/2+
+iµ2(µ−2)
2∂xnh0(x′) + b(x′)
h0(x′)· ∇x′h0(x′).
(58)
74 HEIKO GIMPERLEIN, MAGNUS GOFFENG, NIKOLETTA LOUCA
Computing with Equation (34) and (35) gives us
a0,+(x′,0,1) = −3ic1,n(n2−1)
n!ωn
C3(x′, g ⊗ιng)h+(x′,0,1)
h0(x′)(h+(x′,0,1) −h−(x′,0,1))+
+ic1,n(n+ 3)3,−2
n!ωn
h+(x′,0,1)2(h+(x′,0,1) −3h−(x′,0,1))
h0(x)2(h+(x′,0,1) −h−(x′,0,1))3C3(x, ιng⊗ιng⊗ιng)+
+iµ2∇ξ′h−(x′,0,1) · ∇x′h+(x′,0,1) −∂xnh+(x′,0,1)
h+(x′,0,1) −h−(x′,0,1) =
=ic1,n(n2−1)
n!ωn−3C3(x′, g ⊗ιng)ih0(x)−1/2
h0(x′)2ih0(x)−1/2+
+ (n+ 3)(ih0(x′)−1/2)24ih0(x)−1/2
h0(x)2(2ih0(x′)−1/2)3C3(x, ιng⊗ιng⊗ιng)+
+iµ2
b(x′)
h0(x′)·i∇x′h0(x′)
2h0(x)3/2+i
2
∂xnh0(x′)
h0(x′)3/2
2ih0(x′)−1/2=
=ic1,n(n2−1)
n!ωn−3
2h0(x′)C3(x′, g ⊗ιng) + n+ 3
2h0(x′)2C3(x, ιng⊗ιng⊗ιng)+
+iµ2
4h0(x′)∂xnh0(x′) + b(x′)
h0(x′)· ∇x′h0(x′)
and
a1,+(x′,0,1) =ic1,n(n+ 3)3,−2
n!ωn
h+(x′,0,1)3
h0(x)2(h+(x′,0,1) −h−(x′,0,1))2C3(x, ιng⊗ιng⊗ιng) =
=−c1,n(n2−1)(n+ 3)
4·n!ωnh0(x′)5/2C3(x, ιng⊗ιng⊗ιng).
Continuing from Equation (58), we have that
w−,0(x′,0,0,1)∂ξnw+,1(x′,0,0,1) =
=1
n!ωn(µ−1)ic1,n (n2−1)
n!ωn−3
2C3(x′, g ⊗ιng) + n+ 3
2h0(x′)C3(x, ιng⊗ιng⊗ιng)+
+iµ2(µ−1)
4∂xnh0(x′) + b(x′)
h0(x′)· ∇x′h0(x′)+
−ic1,n(n2−1)(n+ 3)(µ−2)
4·n!ωnh0(x′)C3(x, ιng⊗ιng⊗ιng)+
+iµ2(µ−2)
2∂xnh0(x′) + b(x′)
h0(x′)· ∇x′h0(x′)=
=ic1,n(n2−1)
(n!ωn)2−3(µ−1)
2C3(x′, g ⊗ιng) + µ(n+ 3)
4h0(x′)C3(x, ιng⊗ιng⊗ιng)+
+iµ2(3µ−5)
4·n!ωn∂xnh0(x′) + b(x′)
h0(x′)· ∇x′h0(x′).
Lemma F.6. The following identity holds on ∂X :
∇x′w−,0(x′,0,0,1) · ∇ξ′∂ξnw+,0(x′,0,0,1) = −µ2(µ−1)
2·n!ωn
b(x′)
h0(x′)· ∇x′h0(x′).
SEMICLASSICAL ANALYSIS OF A NONLOCAL BOUNDARY VALUE PROBLEM 75
Proof. We compute that
∇x′w−,0(x′,0,0,1) · ∇ξ′∂ξnw+,0(x′,0,0,1) =
=−µ2(µ−1)
n!ωn∇x′h0(x′)h0(x′)µ−1(−h−(x′,0,1))µ− ∇x′h−(x′,0,1)h0(x′)µ(−h−(x′,0,1))µ−1·
· ∇ξ′h+(x′,0,1)(−h+(x′,0,1))µ−2=
=−µ2(µ−1)
n!ωn∇x′h0(x′)· ∇ξ′h+(x′,0,1)h−(x′,0,1)
h+(x′,0,1)+
+∇x′h−(x′,0,1) · ∇ξ′h+(x′,0,1) h0(x′)
h+(x′,0,1)=
=−µ2(µ−1)
n!ωnb(x′)
h0(x′)· ∇x′h0(x′)−b(x′)
2h0(x′)· ∇x′h0(x′)=−µ2(µ−1)
2·n!ωn
b(x′)
h0(x′)· ∇x′h0(x′)
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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:h.gimperlein@hw.ac.uk, nl24@hw.ac.uk, magnus.goffeng@math.lth.se
