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|  Risheng Liu received his B.Sc. (2007) and Ph.D. (2012) from Dalian
University of Technology, China. From 2010 to 2012, he was doing research as
joint Ph.D. in robotics institute at Carnegie Mellon University. From 2016 to
2018, He was doing research as Hong Kong Scholar at the Hong Kong Polytechnic
University. He is currently a full professor with the Digital Media Department
at International School of Information Science & Engineering, Dalian
University of Technology. He was awarded the “Outstanding Youth Science
Foundation” of the National Natural Science Foundation of China. His research
interests include optimization, computer vision and multimedia.
---|---
|  Jiaxin Gao received the B.S. degree in Applied Mathematics from Dalian
University of Technology, China, in 2018. She is currently pursuing the PhD
degree in software engineering at Dalian University of Technology, Dalian,
China. She is with the Key Laboratory for Ubiquitous Network and Service
Software of Liaoning Province, Dalian University of Technology, Dalian, China.
Her research interests include computer vision, machine learning and
optimization.
---|---
|  Jin Zhang received the B.A. degree in Journalism from the Dalian University
of Technology in 2007. He pursued a degree in mathematics and received the
M.S. degree in Operational Research and Cybernetics from the Dalian University
of Technology, China, in 2010, and the PhD degree in Applied Mathematics from
University of Victoria, Canada, in 2015. After working in Hong Kong Baptist
University for 3 years, he joined Southern University of Science and
Technology as a tenure-track assistant professor in the Department of
Mathematics. His broad research area is comprised of optimization, variational
analysis and their applications in economics, engineering and data science.
---|---
|  Deyu Meng (Member, IEEE) received the B.Sc. degree in information science,
the M.Sc. degree in applied mathematics, and the Ph.D. degree in computer
science from Xi’an Jiaotong University, Xi’an, China, in 2001, 2004, and 2008,
respectively. He is currently a Professor with the School of Mathematics and
Statistics, Xi’an Jiaotong University, and adjunct Professor with the Faculty
of Information Technology, The Macau University of Science and Technology,
Macao. From 2012 to 2014, he took his two-year sabbatical leave at Carnegie
Mellon University, Pittsburgh, PA, USA. His current research interests include
model-based deep learning, variational networks, and meta learning.
---|---
|  Zhouchen Lin (M’00-SM’08-F’18) received the PhD degree from Peking
University in 2000. He is currently a professor with the Key Laboratory of
Machine Perception, School of EECS, Peking University. His research interests
include computer vision, image processing, machine learning, pattern
recognition, and numerical optimization. He has been an area chair of CVPR,
ICCV, NIPS/NeurIPS, AAAI, IJCAI, ICLR and ICML many times, and is a Program
Co-Chair of ICPR 2022. He was an associate editor of the IEEE Transactions on
Pattern Analysis and Machine Intelligence and currently is an associate editor
of the International Journal of Computer Vision. He is a Fellow of IAPR and
IEEE.
---|---


Lennart Ronge Hadamard expansions for powers of causal Green’s operators and
“resolvents”

###### Abstract

We derive an asymptotic expansion analogous to the Hadamard expansion for
powers of advanced/retarded Green’s operators associated to a normally
hyperbolic operator $P$, as well as expansions for advanced/retarded Green’s
operators associated to $P-z$ for $z\in{\mathbb{C}}$. These expansions involve
the same Hadamard coefficients as the original Hadamard expansion.

##  1 Introduction

The Hadamard expansion is a formal expansion in differentiability orders of
Green’s operators associated to a wave operator. It takes the form

$\mathcal{K}(G)\sim\sum\limits_{k=0}^{\infty}V_{k}R(2k+2),$

where $\mathcal{K}(G)$ denotes the Schwartz kernel of a Green’s operator, the
$V_{k}$ are sections in two variables known as the Hadamard coefficients and
$R$ denotes a Lorentzian analogue to powers of the distance function. There
are Hadamard expansions both for advanced/retarded and for Feynman
Propagators, all involving the same Hadamard coefficients and different
distributions $R$. A construction for both types of Hadamard expansions,
showing both their similarities and differences, can be found e.g. in the
appendix of [BS20].

The Hadamard expansions give a canonical way to split off a singular part from
a Green’s operator. This is used e.g. in the renormalization of the Stress-
Energy Tensor on curved spacetimes (see e.g. [DF08]) and the Lorentzian
version of the local APS index theorem ([BS20]). The Hadamard coefficients are
also interesting in themselves, as they are the Lorentzian analogue for
Riemannian heat kernel coefficients and encode geometric properties like the
scalar curvature.

In [DW20], Dang and Wrochna develop Hadamard expansions for Feynman resolvents
and their powers and use these to investigate a spectral action on Lorentzian
scattering spaces. In [DW22], they use these expansions to investigate a
generalized Wodzicki residue.

Motivated by their work, we seek to develop similar expansions for powers of
the advanced/retarded Green’s operators. While the expansions we obtain are
similar to those for the Feynman propagators in [DW20], the methods used to
derive them are quite different.

In Section 2, we introduce the setting and background material. In order to
state the main results below and to have everything gathered in one place, we
give a summary of the core definitions here:

######  Definition/Notation.

$P$ is a normally hyperbolic operator, acting on sections of a vector bundle
$E$ over a globally hyperbolic manifold $M$. $U\subseteq M$ is an open,
geodesically convex, globally hyperbolic and causally compatible subset.
$\mathcal{K}(T)$ denotes the Schwartz kernel of an operator $T$. $G^{\pm}$
denotes the advanced/retarded Green’s operator for $P$ (2.4). $R_{\pm}$
denotes the Riesz distributions on $U$ (2.9). $V^{k}$ denotes the Hadamard
coefficients (2.13) associated to $P$ on $U$. $\sim$ denotes an asymptotic
expansion in differentiability orders (2.14).

In Section 3, we develop an expansion for powers of $G^{\pm}$ (see Theorem
3.3) – for $m\in\mathbb{Z}$, we have

$\mathcal{K}(G^{\pm
m})\sim\sum\limits_{k=0}^{\infty}\binom{m+k-1}{k}V^{k}R_{\pm}(2k+2m).$

In Section 4, we develop asymptotic expansions for $G^{\pm}_{P-z}$, the
advanced/retarded Green’s operator for $P-z$, with $z\in{\mathbb{C}}$. We
first show the formula for the Hadamard-coefficients of $P-z$,

$V^{k}(z)=\sum\limits_{m=0}^{k}\binom{k}{m}z^{m}V^{k-m},$

which immediately gives a double-sum expansion in which the $z$-dependence is
explicit (see Corollary 4.4):

$\mathcal{K}(G^{\pm}_{P-z})\sim\sum\limits_{k,m=0}^{\infty}\binom{k+m}{k}z^{m}V^{k}R_{\pm}(2k+2m+2).$

We then go on to define $z$-dependent analogues $R_{\pm}(z,2k)$ of the even
Riesz distributions, with which we can write the expansion for $G^{\pm}_{P-z}$
in the same form as that for $G^{\pm}$ only with the $z$-dependent Riesz
distributions instead of the usual ones (see Theorem 4.8):

$\mathcal{K}(G^{\pm}_{P-z})\sim\sum\limits_{k=0}^{\infty}V^{k}R_{\pm}(z,2k+2),$

or more generally

$\mathcal{K}(G^{\pm}_{P-z}{}^{m})\sim\sum\limits_{k=0}^{\infty}\binom{k+m-1}{k}V^{k}R_{\pm}(z,2k+2m).$

The results of this paper are from the author’s Ph. D. Thesis ([Ron23]), where
they are somewhat separate from the other content. The aim of this paper is to
provide the reader with an independent, shorter and more easily readable
exposition of these results. For this reason, the setup and proofs that do not
immediately concern the main results are kept more brief here, a more detailed
description can be found in the original thesis.

### Acknowledgements

I would like to thank Matthias Lesch and Koen van den Dungen for their advice
and support during the writing of both the thesis that this paper is based on
and the paper itself.

##  2 Preliminaries

###  2.1 Setting

Throughout this paper, we will assume that $M$ is a globally hyperbolic
Lorentzian manifold with metric $g$ and $P$ is a normally hyperbolic operator,
acting on sections of a vector bundle $E$ over $M$. Normally hyperbolic means
$P$ is a second order differential operator whose principal symbol is given by
the minus the Lorentzian metric. Global hyperbolicity of $M$ is just needed to
guarantee the existence and uniqueness of Green’s operators.

######  Remark 2.1.

As $M$ and $P$ are arbitrary, theorems proved and definitions made for $M$ or
$P$ will hold for arbitrary globally hyperbolic manifolds and arbitrary
normally hyperbolic operators. This will be exploited occasionally. Note that
the restriction of a normally hyperbolic operator to an open subset is again
normally hyperbolic.

###  2.2 Notation

We will often encounter definitions and statements that work in both
directions of time symetrically, usually distinguished by an index $+$ or $-$.
In order to avoid writing everything twice, we use the symbol $\pm$ to denote
both cases at once. The use of $\pm$ or $\mp$ in a definition or theorem
indicates that this should hold both with the upper sign chosen everywhere and
with the lower sign chosen everywhere in that definition or theorem. In some
cases we also need to change the wording depending on the choice of time
alignment. In that case we will use a ‘/’ to indicate that the first word is
to be used with the upper sign and the second word is to be used with the
lower sign.

For example, the statement ‘If $A$ is past/future compact, $J_{\pm}(A)\cap
J_{\mp}(x)$ is compact.’ means ‘If $A$ is past compact, $J_{+}(A)\cap
J_{-}(x)$ is compact and if $A$ is future compact, $J_{-}(A)\cap J_{+}(x)$ is
compact.’.

We write $J_{\pm}(x)$ for the causal future/past of a point or set $x$. We
write $\mathcal{K}(T)$ for the Schwartz kernel of an operator $T$. An operator
defined on functions on $M$ that is applied to a function on $M\times M$ is
taken to act in the first component, i.e.

$(Tf)(y,x):=(T(f(\cdot,x)))(y).$

More generally, an operator defined on distributions in $E$ is taken to act on
distributions in $E\boxtimes F$ (for any vector bundle $F$) in the first
component, i.e. for pure tensor products

$T(f\otimes g):=(Tf)\otimes g.$

This is defined so that for operators $T,S$ acting on sections of $E$, we have

$\mathcal{K}(TS)=T\mathcal{K}(S).$

###  2.3 Spaces of sections

In order to be able to define powers of Green’s operators, we need to define
them on the right domains. This requires us to talk about sections with past
or future compact support.

######  Definition 2.2.

A subset of $M$ is called past/future compact if and only if its intersection
with the past/future of every point in $M$ is compact.

We shall need the following spaces of sections and distributions:

######  Definition 2.3.

We denote by $\Gamma(E)$ the space of smooth sections and by $\Gamma^{k}(E)$
the space of k-times continuously differentiable sections of $E$. We denote by
$\mathcal{D}^{\prime}(E)$ the space of distributions in $E$. We use subscripts
to indicate restrictions on the support. We write these for $\Gamma$, but also
use the analogous notation with $\Gamma^{k}$ and $\mathcal{D}^{\prime}$.

  * •

For $A\subseteq M$, $\Gamma_{A}(E)$ denotes sections supported in $A$.

  * •

$\Gamma_{c}(E)$ denotes compactly supported sections.

  * •

$\Gamma_{\pm}(E)$ denotes sections with past/future compact support.

All these spaces carry canonical topologies such that all inclusions are
continuous and $\Gamma_{c}(E)$ is dense in all spaces. For example,
$\Gamma_{A}(E)$ carries the topology induced by all $C^{k}$-norms on compact
subsets and $\Gamma_{\pm}(E)$ carries the limit topology of all
$\Gamma_{A}(E)$ with $A$ past/future compact. See [Bär14, section 2] for a
more detailed description of these spaces.

###  2.4 Green’s operators

With this setup, the advanced and retarded Green’s operators can be defined as
inverses of $P$ on the appropriate spaces.

######  Definition/Proposition 2.4.

$P$ is invertible as a map of $\mathcal{D}^{\prime}_{\pm}(E)$ to itself. The
Green’s operator $G^{\pm}_{P}$ is defined to be the inverse of $P$ on
$\mathcal{D}^{\prime}_{\pm}(E)$. This is continuous and restricts to a
continuous map of $\Gamma_{\pm}(E)$ to itself. We omit the subscript $P$, if
it is clear what operator we are talking about. For any
$f\in\mathcal{D}^{\prime}_{\pm}(E)$, we have
$\operatorname{supp}(G^{\pm}f)\subseteq J_{\pm}(\operatorname{supp}(f))$.

This follows from [Bär14, Theorem 3.8, Corollary 3.11 and Lemma 4.1] (that $P$
is Green’s hyperbolic and thus the previous results apply, holds e.g. by
[BGP07, Corollary 3.4.3]). It is more customary to define Green’s operators as
maps $\Gamma_{c}(E)\rightarrow\Gamma(E)$. This is equivalent to the definition
used here via restriction/continuous extension. The definition we use has the
advantage that (integer) powers of Green’s operators are immediately well
defined, as the Green’s operators are automorphisms.

###  2.5 Suitable domains

Most considerations in this paper will not take place on all of $M$, but on
suitable subsets of $M$:

######  Definition 2.5.

A subset $U$ of $M$ shall be called GE, if it is open, geodesically convex,
globally hyperbolic, and causally compatible.

######  Fixed Notation 2.6.

From now on, fix an arbitrary GE set $U$ in $M$.

Openness is required to restrict distributions to $U$. Geodesic convexity will
be required to define Hadamard coefficients and Riesz distributions. Global
hyperbolicity ensures that $U$ has unique Green’s operators, while causal
compatibility ensures that those agree with the restriction of the Green’s
operators on $M$. Indeed, by [BGP07, Proposition 3.5.1.] (and continuity), we
have for any $f\in\mathcal{D}^{\prime}_{\pm}(E)$:

$G^{\pm}_{P|_{U}}(f|_{U})=(G^{\pm}_{P}f)|_{U}.$

In the following, we will not distinguish between $G^{\pm}_{P}$ and
$G^{\pm}_{P|_{U}}$ notationally, as the previous equality guarantees that this
does not lead to ambiguity. Moreover, we will often omit the restriction of
kernels to $U\times U$ from the notation.

######  Remark 2.7.

Again, any theorem and definition made in reference to $U$ works for arbitrary
GE subsets of globally hyperbolic manifolds. This includes Lorentzian vector
spaces as GE subsets of themselves.

The notion of GE sets is also sufficiently general to investigate local
properties around any point: [Min14, Corollary 2 and Remark 14] imply that
every point in $M$ has a neighborhood basis of GE-sets.

###  2.6 Riesz distributions and Hadamard coefficients

In this section, we introduce Riesz distributions and Hadamard coefficients,
which will play a central role in this paper. We will follow [BGP07] here,
they are also described in [Gün88] and [Fri75]. As all constructions rely
heavily on the exponential map, the objects will only be defined on convex
subsets of a Lorentzian manifold. We will define them for $U$ and omit the
index $U$ after the definition, unless we explicitly want to use them for a
different set.

The Riesz distributions on $U$ are roughly speaking a Lorentzian analogue to
‘powers of the distance function’. They are described in detail in [BGP07,
sections 1.2 and 1.4]. Here, they will play a role comparable to that of
powers of $x$ in a Taylor series.

######  Definition 2.8.

For $x,y\in U$, define

$\Gamma^{U}(y,x):=-g(exp_{x}^{-1}(y),exp_{x}^{-1}(y)).$

This is the Lorentzian analogue of the Riemannian distance squared.
$\Gamma(y,x)$ is positive if and only if $y$ is in the future or past of $x$.
We now define the Riesz distributions (see [BGP07, Definitions 1.2.1 and
1.4.1, Lemma 1.2.2 and Proposition 1.4.2] for the definition and the following
remarks)

######  Definition/Proposition 2.9.

For $\alpha\in{\mathbb{C}}$ with real part $\Re(\alpha)>d$ and $x\in U$,
define Riesz distributions $R^{U}_{\pm}(\alpha)(\cdot,x)$ as the distribution
on $U$ given by the function

$R^{U}_{\pm}(\alpha)(y,x)=\begin{cases}c_{\alpha}\Gamma(y,x)^{\frac{\alpha-d}{2}}&\text{,
if }x\in J_{\pm}(0)\\\ 0&\text{, if }x\notin J_{\pm}(0)\\\ \end{cases}$

with

$c_{\alpha}:=\frac{2^{1-\alpha}\pi^{\frac{2-d}{2}}}{\Gamma(\frac{a}{2})\Gamma(\frac{\alpha-d+2}{2})}.$

The map $\alpha\mapsto R^{U}_{\pm}(\alpha)(\cdot,x)$ is holomorphic as a map
into $\mathcal{D}^{\prime}(U)$ and extends uniquely to a holomorphic map on
all of ${\mathbb{C}}$. For arbitrary $\alpha\in{\mathbb{C}}$, define
$R^{U}_{\pm}(\alpha)(\cdot,x)$ to be the value of this holomorphic extension.
This defines Riesz distributions $R^{U}_{\pm}(\alpha)$ on $U\times U$ as
suggested by the notation:

$R^{U}_{\pm}(\alpha)[\psi]:=\int\limits_{U}R^{U}_{\pm}(\alpha)(\cdot,x)[\psi(\cdot,x)]dx$

for any test function $\psi\in C_{c}^{\infty}(U\times U)$.

The $\Gamma$ in the definition of $c_{\alpha}$ refers to the gamma function,
not the $\Gamma$ defined above. The prefactor is chosen such that

$\square
R^{{\mathbb{R}}^{d}}_{\pm}(\alpha+2)=R^{{\mathbb{R}}^{d}}_{\pm}(\alpha).$

The precise form of $c_{\alpha}$ will not be relevant in the following and
there will be no further occurences of gamma functions. We have

$R_{\pm}(0)=\delta,$

i.e. the 0-th Riesz distribution is the Dirac distribution on the diagonal.
The Riesz distributions on $U$ can be obtained from those on Minkowski space
via the exponential map:

$R^{U}_{\pm}(\alpha)(\cdot,x)=(exp_{x})^{-1*}R^{T_{x}M}_{\pm}(\alpha)(\cdot,0).$

They are supported in the causal past/future of the basepoint:

$\operatorname{supp}(R^{U}_{\pm}(\alpha)(\cdot,x))\subseteq J_{\pm}(x).$

As $\Gamma$ vanishes at the boundary of the causal past and future, looking at
the formula for $\Re(\alpha)>d$ shows that $R_{\pm}(\alpha)$ is $C^{n}$ if
$\Re(\alpha)>d+2n$.

On Minkowski space, the Riesz distributions $R^{{\mathbb{R}}^{d}}_{\pm}(2k)$
for $k\in{\mathbb{N}}$ can be thought of as fundamental solutions to
$\square^{k}$:

######  Proposition 2.10.

On a Lorentzian vector space $V$, we have

$R_{\pm}^{V}(2m)=G^{\pm m}_{\square}\delta,$

where $\delta$ denotes the dirac distribution on the diagonal and $\square$
the d’Alembertian on $V$.

###### Proof.

We have

$R_{\pm}^{V}(0)=\delta$

and

$\square_{V}R_{\pm}^{V}(2m+2)=R_{\pm}^{V}(2m).$

Thus

$R_{\pm}^{V}(2m+2)=G^{\pm}_{\square_{V}}R_{\pm}^{V}(2m).$

Iterating this, we obtain

$R_{\pm}^{V}(2m)=G^{\pm m}_{\square_{V}}R_{\pm}^{V}(0)=G^{\pm
m}_{\square_{V}}\delta.\qed$

We now turn to the second class of objects we want to define in this section:
the Hadamard coefficients. In order to define them, we introduce the following
differential operator.

######  Definition 2.11.

Define for $V\in\Gamma(E\boxtimes E^{*})$:

$\rho^{U}V(y,x):=\nabla_{\operatorname{grad}\Gamma(y,x)}V(y,x)-\left(\frac{1}{2}\square\Gamma(y,x)-d\right)V(y,x),$

where $\nabla$ is the connection induced by $P$.

The reader may immediately forget the definition of $\rho$ and only remember
the following property:

######  Proposition 2.12.

We have for $\alpha\in{\mathbb{C}}\backslash\\{0\\}$, $x\in U$ and
$V\in\Gamma(E\boxtimes E^{*}|_{U\times U})$

$P(VR_{\pm}(\alpha+2))=(PV)R_{\pm}(\alpha+2)-\frac{1}{\alpha}((\rho-\alpha)V)R_{\pm}(\alpha)$

###### Proof.

By [BGP07, Proposition 1.4.2], the Riesz distributions satisfy

$2\alpha\operatorname{grad}(R_{\pm}(\alpha+2))=(\operatorname{grad}\Gamma)R_{\pm}(\alpha)$

and

$\alpha\square
R_{\pm}(\alpha+2)=\left(\frac{1}{2}\square\Gamma-d+\alpha\right)R_{\pm}(\alpha).$

Using this and the Leibniz rule for normally hyperbolic operators, we
calculate for $\Re(\alpha)>d+4$ (so $R_{\pm}(\alpha,x)$ is $C^{2}$ and we
don’t have to worry about distributions)

$\displaystyle\alpha P(VR_{\pm}(\alpha+2,x))$ $\displaystyle=$
$\displaystyle\alpha\left((PV)R_{\pm}(\alpha+2,x)-2\nabla_{\operatorname{grad}R_{\pm}(\alpha+2,x)}V+V\square
R_{\pm}(\alpha+2,x)\right)$ $\displaystyle=$
$\displaystyle\alpha(PV)R_{\pm}(\alpha+2,x)-(\nabla_{\operatorname{grad}\Gamma}V)R_{\pm}(\alpha,x)+V\left(\frac{1}{2}\square\Gamma_{x}-d+\alpha\right)R_{\pm}(\alpha,x)$
$\displaystyle=$
$\displaystyle\alpha(PV)R_{\pm}(\alpha+2,x)-((\rho_{x}^{U}-\alpha)V)R_{\pm}(\alpha,x).$

As both sides are holomorphic, the equation holds for arbitrary $\alpha$. ∎

Pretty much exactly this calculation is done in [BGP07, equation 2.1] to
motivate the definition of the Hadamard coefficients, which we will give now:

######  Definition/Proposition 2.13.

There is a unique family of smooth sections

$V^{k,U}\in\Gamma(E\boxtimes E^{*}|_{U\times U})$

(indexed by $k\in{\mathbb{N}}$) that satisfies the transport equations

$(\rho-2k)V^{k,U}=2kPV^{k-1,U}$

(for $k=0$, the right hand side is set to $0$) subject to the initial
condition

$V^{0,U}(x,x)=1.$

We call these the Hadamard coefficients.

The motivation for defining the Hadamard coefficients is that we want to have
the formal equality

$P\sum\limits_{k=0}^{\infty}V^{k}R_{\pm}(2k+2)=\delta,$

which (together with support conditions) means that the infinite sum, if it
existed, would be the Schwartz kernel of $G^{\pm}$. In general, the sum does
not exist and we only get a form of asymptotic expansion.

###  2.7 Asymptotic expansions in differentiability orders

The Hadamard expansion, as well as the expansions developed in this paper, are
not exact equalities. Rather, the quantity on the left hand side can be
approximated up to arbitrary degree of differentiability by partial sums of
the right hand side:

######  Definition 2.14.

For distributions $F$ and $f_{k}$, we write

$F\sim\sum\limits_{k=0}^{\infty}f_{k}$

if and only if for any $m\in{\mathbb{N}}$ there is $N_{0}\in{\mathbb{N}}$ such
that for all $N>N_{0}$, the difference

$F-\sum\limits_{n=0}^{N}f_{n}$

is $m$ times continuously differentiable.

If we have multiple sums on the right hand side, we define this analogously
(as one would do with limits).

######  Remark 2.15.

As both Green’s kernels and Riesz distributions vanish outside the causal
past/future of the second argument, a $C^{k}$-remainder in the following
expansions also vanishes of order $k$ near the boundary of the past/future.

##  3 Powers of Green’s operators

We first derive an expansion for powers of the advanced/retarded Green’s
operators. Before we can start the main proof, we need to do some technical
work to guarantee that the Green’s operators do not decrease differentiability
orders too much (locally). This will be necessary to control the remainder
terms of our asymptotic expansion. The result is basically a consequence of
mapping smooth sections to smooth sections continuously.

######  Proposition 3.1.

Let $A\subset M$ be past/future compact compact and $W\subset M$ be open and
relatively compact. Let $r_{W}$ denote restriction to $W$. Then for every
$n\in{\mathbb{N}}$ there is $m\in{\mathbb{N}}$ such that $r_{W}\circ G^{\pm}$
maps $\Gamma^{m}_{A}(E)$ to $\Gamma^{n}(E|_{W})$ continuously.

###### Proof.

We know that

$G^{\pm}\colon\Gamma_{\pm}(E)\rightarrow\Gamma_{\pm}(E)$

is continuous. Since the topologies of $\Gamma_{A}(M)$ and
$\Gamma_{J_{\pm}(A)}(M)$ coincide with the subspace topology from
$\Gamma_{\pm}$ (as the latter carries the limit topology), we know that the
restriction

$G^{\pm}\colon\Gamma_{A}(E)\rightarrow\Gamma_{J_{\pm}(A)}(E)$

is also continuous. $C^{k}$\- norms on ${\overline{W}}$ are seminorms on the
target space. Thus by the seminorm-characterization of continuity, we know
that for any $n\in{\mathbb{N}}$, there is $c\in{\mathbb{R}}$,
$m\in{\mathbb{N}}$ and $K\subseteq M$ such that for all $\phi\in\Gamma_{A}(E)$

$\|G^{\pm}\phi\|_{C^{n}(\overline{W})}\leq c\|\phi\|_{C^{m}(K)}.$

The right hand side is also a seminorm of $\Gamma^{m}_{A}(E)$. Thus
$r_{W}\circ G^{\pm}|_{\Gamma_{A}(E)}$ extends continuously to a map
$\Gamma^{m}_{A}(E)\rightarrow\Gamma^{n}(E|_{W})$. As $C^{k}$\- convergence
implies distributional convergence, this continuous extension must coincide
with $r_{W}\circ G^{\pm}|_{\Gamma^{m}_{A}(E)}$, as the latter is
distributionally continuous. ∎

As Hadamard coefficients and Green’s kernels are sections in $E\boxtimes
E^{*}$ and we want differentiability in both coordinates, we need to extend
the above to the case where $G^{\pm}$ acts in the first component on sections
of a box product.

######  Proposition 3.2.

Let $A\subset M$ be past/future compact and $W\subset M$ be open and
relatively compact. Let $O$ be some manifold and $F$ some vector bundle over
$O$. Then for each $n\in{\mathbb{N}}$, there is $m\in{\mathbb{N}}$ such that
for any $f\in\Gamma^{m}(E\boxtimes F)$ supported in $A\times O$, we have
$G^{\pm}f|_{W\times O}\in\Gamma^{n}((E\boxtimes F)|_{W\times O})$, where
$G^{\pm}f(y,x):=G^{\pm}(f(\cdot,x))(y)$.

###### Proof.

As differentiability can be checked locally, i.e. in charts on which
$E\boxtimes F$ is trivial, it suffices that the partial derivatives up to
order $m$ of the component functions in these charts exist. Choose $m$ such
that $G^{\pm}$ maps $\Gamma^{m-n}_{A}(E)$ to $\Gamma^{n}(E)$ continuously. As
$\partial^{I}\circ G^{\pm}$ is continuous as a map into $C^{n-|I|}$-sections
for any multiindex with $|I|<n$, the differential quotients for partial
derivatives in the second component can be pulled inside the operator, where
the limits are well-defined and give functions of one less differentiability
order. Iterating this, starting with a $C^{m}$-section and moving all partial
derivatives in the second coordinates inside, the argument stays at least
$C^{m-n}$, so we obtain that all partial derivatives up to order $n$ of
$G^{\pm}f$ are well defined and continuous. ∎

We can now prove the main theorem of this section:

######  Theorem 3.3.

Let $U\subseteq M$ be a GE set and let $m\in\mathbb{Z}$. Then we have the
asymptotic expansion

$\mathcal{K}(G^{\pm m})|_{U\times
U}\sim\sum\limits_{k=0}^{\infty}\binom{m+k-1}{k}V^{k,U}R^{U}_{\pm}(2k+2m).$

For $m<0$, the right hand side is a finite sum, as all summands with $k+m>0$
vanish, and we have equality.

######  Remark 3.4.

Note that the special case $m=1$ reproduces the original Hadamard expansion.

######  Remark 3.5.

In case $m<0$, this is actually a formula for the kernel of powers of $P$. All
even Riesz distributions of order $0$ or less are the same in the advanced and
retarded case and supported only at the diagonal, so this expansion is not as
surprising as it might seem on first glance. From the case $m=-1$, we obtain
that

$\mathcal{K}(P)=V^{0}R_{\pm}(-2)-V^{1}\delta.$

In particular, $P$ is uniquely determined by the metric $g$ (which determines
the Riesz-distributions) and the first two Hadamard coefficients $V^{0}$ and
$V^{1}$.

###### Proof.

We proceed by two-way induction on $m$, showing that if the statement holds
for $m\in\mathbb{Z}$, it also holds for $m+1$ if $m\geq 0$ and for $m-1$, if
$m\leq 0$. The case $m=0$ follows from $R_{\pm}(0)=\delta$, as all summands
excapt the first vanish.

We make some preliminary calculations. For $k+m\neq 0$ we can use the
characterization of $\rho$ (2.12) and the transport equations (see 2.13) to
calculate

$\displaystyle\binom{k+m}{k}P(V^{k}R_{\pm}(2k+2m+2))$
$\displaystyle=\binom{k+m}{k}\Big{(}\frac{-1}{2k+2m}(\rho-2k-2m)V^{k}R_{\pm}(2k+2m)+(PV^{k})R_{\pm}(2k+2m+2)\Big{)}$
$\displaystyle=\binom{k+m}{k}\Big{(}\frac{-1}{2k+2m}((2kPV^{k-1}-2mV^{k})R_{\pm}(2k+2m))$
$\displaystyle+(PV^{k})R_{\pm}(2k+2m+2)\Big{)}$
$\displaystyle=\binom{k+m}{k}\frac{2m}{2k+2m}V^{k}R_{\pm}(2k+2m)-\binom{k+m}{k}\frac{2k}{2m+2k}(PV^{k-1})R_{\pm}(2k+2m)$
$\displaystyle+\binom{k+m}{k}(PV^{k})R_{\pm}(2k+2m+2)$
$\displaystyle=\binom{k+m-1}{k}V^{k}R_{\pm}(2k+2m)-\binom{k+m-1}{k-1}(PV^{k-1})R_{\pm}(2k+2m)$
$\displaystyle+\binom{k+m}{k}(PV^{k})R_{\pm}(2k+2m+2))$

In the special case $k=m=0$, we obtain:

$\displaystyle P(V^{0}R_{\pm}(2))$
$\displaystyle=\lim\limits_{\alpha\rightarrow 0}P(V^{0}R_{\pm}(\alpha+2))$
$\displaystyle=\lim\limits_{\alpha\rightarrow
0}P(V^{0})R_{\pm}(\alpha+2)-(\frac{1}{\alpha}\rho V^{0}-V^{0})R_{\pm}(\alpha)$
$\displaystyle=\lim\limits_{\alpha\rightarrow
0}P(V^{0})R_{\pm}(\alpha+2)+V^{0}R_{\pm}(\alpha)$
$\displaystyle=P(V^{0})R_{\pm}(2)+V^{0}R_{\pm}(0)$
$\displaystyle=P(V^{0})R_{\pm}(2)+V^{0}\delta$
$\displaystyle=\delta+P(V^{0})R_{\pm}(2),$

where we used that $V_{0}$ is $1$ on the diagonal. Either way, we obtain for
$k=0$:

$P(V^{0}R_{\pm}(2m+2))=V^{0}R_{\pm}(2m)+(PV^{0})R_{\pm}(2m+2)$

Putting everything together, for any $N\in{\mathbb{N}}$ in case $m\geq 0$ and
for any $N<-m$ if $m<0$, we have

$\displaystyle P\sum\limits_{k=0}^{N}\binom{m+k}{k}V^{k}R_{\pm}(2k+2m+2)$
$\displaystyle=V^{0}R_{\pm}(2m)+(PV^{0})R_{\pm}(2m+2)+\sum\limits_{k=1}^{N}\binom{k+m-1}{k}V^{k}R_{\pm}(2k+2m)$
$\displaystyle-\binom{k+m-1}{k-1}(PV^{k-1})R_{\pm}(2k+2m)+\binom{k+m}{k}(PV^{k})R_{\pm}(2k+2m+2)$
$\displaystyle=\sum\limits_{k=0}^{N}\binom{k+m-1}{k}V^{k}R_{\pm}(2k+2m)-\sum\limits_{k=0}^{N-1}\binom{k+m}{k}(PV^{k})R_{\pm}(2k+2m+2)$
$\displaystyle+\sum\limits_{k=0}^{N}\binom{k+m}{k}(PV^{k})R_{\pm}(2k+2m+2)$
$\displaystyle=\sum\limits_{k=0}^{N}\binom{k+m-1}{k}V^{k}R_{\pm}(2k+2m)+\binom{N+m}{N}(PV^{N})R_{\pm}(2N+2m+2)$
$\displaystyle=:\sum\limits_{k=0}^{N}\binom{k+m-1}{k}V^{k}R_{\pm}(2k+2m)+E_{N}$

where

$E_{N}:=\binom{N+m}{N}(PV^{N})R_{\pm}(2N+2m+2)$

is $C^{n}$ for $N\geq\frac{d}{2}+n-m$.

Case 1: $m+1\Rightarrow m$, $m<0$
Suppose for induction that

$\mathcal{K}(G^{\pm
m+1})=\sum\limits_{k=0}^{-m-1}\binom{m+k}{k}V^{k}R_{\pm}(2k+2m+2)$

(all further summands are 0). Let $N=-m-1$. We use the fact (see [BGP07, below
Proposition 2.3.1]) that on the diagonal $\Delta=\\{(x,x)|x\in U\\}$, we have

$PV^{N}|_{\Delta}=-V^{N+1}|_{\Delta}$

and the binomial identity

$\binom{-a}{b}=(-1)^{b}\binom{a+b-1}{b}$

for integer $b$, which implies in particular

$\binom{-1}{b}=(-1)^{b}.$

With these, we obtain

$\displaystyle E_{N}$ $\displaystyle=\binom{-1}{N}(PV^{N})R_{\pm}(0)$
$\displaystyle=(-1)^{N}(PV^{N})\delta$ $\displaystyle=(-1)^{N+1}V^{N+1}\delta$
$\displaystyle=\binom{-1}{N+1}V^{N+1}R_{\pm}(0)$
$\displaystyle=\binom{-1}{-m}V^{-m}R_{\pm}(0).$

We obtain from the inductive hypothesis and our previous calculations:

$\displaystyle\mathcal{K}(G^{\pm m})$ $\displaystyle=P\mathcal{K}(G^{\pm
m+1})$
$\displaystyle=P\sum\limits_{k=0}^{N}\binom{m+k}{k}V^{k}R_{\pm}(2k+2m+2)$
$\displaystyle=\sum\limits_{k=0}^{N}\binom{k+m-1}{k}V^{k}R_{\pm}(2k+2m)+E_{N}$
$\displaystyle=\sum\limits_{k=0}^{-m-1}\binom{k+m-1}{k}V^{k}R_{\pm}(2k+2m)+\binom{-1}{-m}V^{-m}R_{\pm}(0)$
$\displaystyle=\sum\limits_{k=0}^{-m}\binom{k+m-1}{k}V^{k}R_{\pm}(2k+2m)$

Case 2: $m\Rightarrow m+1$, $m\geq 0$
Fix $n\in{\mathbb{N}}$ and assume the theorem holds for some $m\geq 0$. We
first show that the expansion holds on relatively compact GE subsets of $U$.
Let $W,O\subseteq U$ be relatively compact (in $U$) and GE. Proposition 3.2
implies that there is $n^{\prime}\in{\mathbb{N}}$ such that $G^{\pm}$ maps
$C^{n^{\prime}}$-sections supported in $J^{U}_{\pm}(\overline{O})\times O$ to
sections that are $C^{n}$ on $W\times O$. By the inductive hypothesis, we may
choose $N\geq\frac{d}{2}+n^{\prime}$ such that

$F_{N}:=G^{\pm m}-\sum\limits_{k=0}^{N}\binom{m+k-1}{k}V^{k}R_{\pm}(2k+2m)$

is a $C^{n^{\prime}}$-section. We then have on $U\times U$

$\displaystyle\mathcal{K}(G^{\pm m+1})$
$\displaystyle=G^{\pm}\mathcal{K}(G^{\pm m})$
$\displaystyle=G^{\pm}\left(\sum\limits_{k=0}^{N}\binom{m+k-1}{k}V^{k}R_{\pm}(2k+2m)+F_{N}\right)$
$\displaystyle=G^{\pm}\left(P\sum\limits_{k=0}^{N}\binom{m+k}{k}V^{k}R_{\pm}(2k+2m+2)-E_{N}+F_{N}\right)$
$\displaystyle=\sum\limits_{k=0}^{N}\binom{m+k}{k}V^{k}R_{\pm}(2k+2m+2)+G^{\pm}(F_{N}-E_{N})$
$\displaystyle=:\sum\limits_{k=0}^{N}\binom{m+k}{k}V^{k}R_{\pm}(2k+2m+2)+\tilde{F}_{N}.$

with

$\tilde{F}_{N}:=\mathcal{K}(G^{\pm
m+1})-\sum\limits_{k=0}^{N}\binom{m+k}{k}V^{k}R_{\pm}(2k+2m+2)=G^{\pm}(F_{N}-E_{N}).$

By our choice of ${n^{\prime}}$ and $N$, $\tilde{F}_{N}$ is $C^{n}$ on
$W\times O$, as both $E_{N}|_{U\times O}$ and $F_{N}|_{U\times O}$ are
$C^{n^{\prime}}$ and supported in $J_{\pm}(O)\times O$. Note that the only
choices we made after choosing $W$ and $O$ were that of ${n^{\prime}}$ and
$N$. We thus want to make the expansion independent of the choice of $N$.

Let $N^{\prime}\geq\frac{d}{2}+n$ be arbitrary. For $k\geq N^{\prime}$, all
Riesz distributions $R_{\pm}(2k+2m+2)$ are given by $C^{n}$-functions. We see
that

$\displaystyle\tilde{F}_{N^{\prime}}$ $\displaystyle=\mathcal{K}(G^{\pm
m+1})-\sum\limits_{k=0}^{N^{\prime}}\binom{m+k}{k}V^{k}R_{\pm}(2k+2m+2)$
$\displaystyle=\sum\limits_{k=N^{\prime}+1}^{N}\binom{m+k}{k}V^{k}R_{\pm}(2k+2m+2)+\tilde{F}_{N}$

is $C^{n}$ on $W\times O$. As this is independent of $W$ and $O$, these can
replaced by arbitrary relatively compact GE sets. All points in $U$ have
relatively compact GE neighborhoods (choose any relatively compact
neighborhood and then choose a GE neighborhood contained in it). Thus any
$(y,x)\in U\times U$ has a neighborhood on which $\tilde{F}_{N^{\prime}}$ is
$C^{n}$, which means it is $C^{n}$ on all of $U\times U$. As $n$ was
arbitrary, we have the desired asymptotic expansion for $m+1$ and have thus
concluded our induction. ∎

##  4 Green’s “resolvent”

We now consider the Green’s operators for the operator $P-z$ for
$z\in{\mathbb{C}}$, which is still normally hyperbolic. If we view
$G^{\pm}_{P}$ as something like an inverse for $P$, then $G^{\pm}_{P-z}$ is
something like a resolvent. We seek to derive an expansion similar to the one
above, where only the Hadamard coefficients of $P$ rather than $P-z$ appear
and the $z$-dependence is instead shifted to a generalzed version of Riesz
distributions.

We want to exploit the asymptotic expansion that we already have in order to
obtain the one we would like to get. For this, we express the Hadamard
coefficients for $P-z$ in terms of those for $P$.

######  Definition 4.1.

For $z\in{\mathbb{C}}$, let $V^{k}(z)$ denote the Hadamard coefficients for
$P-z$.

######  Proposition 4.2.

We have

$V^{k}(z)=\sum\limits_{m=0}^{k}\binom{k}{m}z^{m}V^{k-m}.$

######  Remark 4.3.

The corresponding formula for the heat coefficients of a Laplace operator,

$a_{k}(\Delta-z)=\sum\limits_{m=0}^{k}\frac{1}{m!}z^{m}a_{k-m}(\Delta),$

can be shown analogously. Alternatively, that formula can also be deduced by
Taylor expanding $e^{tz}$ and multiplying out the expansions in

$e^{-t(\Delta-z)}=e^{tz}e^{-t\Delta}.$

###### Proof.

Let

$V_{k}(z):=\sum\limits_{m=0}^{k}\binom{k}{m}z^{m}V^{k-m}$

We need to show that $V_{0}(z)(x,x)=1$ and the $V_{k}(z)$ satisfy the
transport equations. The former holds, as $V_{0}(z)=V^{0}$. For the latter, we
calculate using the transport equation for $P$:

$\displaystyle(\rho-2k)V_{k}(z)$
$\displaystyle=\sum\limits_{m=0}^{k}\binom{k}{m}z^{m}(\rho-2k)V^{k-m}$
$\displaystyle=\sum\limits_{m=0}^{k}\binom{k}{m}z^{m}((\rho-2(k-m))V^{k-m}-2mV^{k-m})$
$\displaystyle=\sum\limits_{m=0}^{k}\binom{k}{m}z^{m}2(k-m)PV^{k-m-1}-\sum\limits_{m=0}^{k}\binom{k}{m}z^{m}2mV^{k-m}$
$\displaystyle=\sum\limits_{m=0}^{k}\binom{k-1}{m}z^{m}2kPV^{k-m-1}-\sum\limits_{m=1}^{k}\binom{k-1}{m-1}z^{m}2kV^{k-m}$
$\displaystyle=\sum\limits_{m=0}^{k-1}\binom{k-1}{m}z^{m}2kPV^{k-m-1}-\sum\limits_{m=0}^{k-1}\binom{k-1}{m}z^{m+1}2kV^{k-m-1}$
$\displaystyle=2k(P-z)\sum\limits_{m=0}^{k-1}\binom{k-1}{m}z^{m}V^{k-m-1}$
$\displaystyle=2k(P-z)V_{k-1}(z).$

The $V_{k}(z)$ thus satisfy the transport equations for $P-z$, so they are the
desired Hadamard coefficients. ∎

Inserting this into the Hadamard expansion, we obtain our first expansion for
the Green’s kernel of $P-z$, an asymptotic expansion in a double sum, where
the $z$ dependence is explicit:

######  Corollary 4.4.

For every GE set $U\subseteq M$, we have the asymptotic expansion

$\mathcal{K}(G^{\pm}_{P-z})|_{U\times
U}\sim\sum\limits_{k,m=0}^{\infty}\binom{k+m}{k}z^{m}V^{k,U}R^{U}_{\pm}(2k+2m+2).$

###### Proof.

Taking the standard Hadamard expansion to $P-z$, inserting our formula for the
$z$-dependent Hadamard coefficients (Proposition 4.2) and re-indexing the
summands, we obtain

$\displaystyle\mathcal{K}(G^{\pm}_{P-z})$
$\displaystyle\sim\sum\limits_{n=0}^{\infty}V^{n}(z)R_{\pm}(2n+2)$
$\displaystyle=\sum\limits_{n=0}^{\infty}\
\sum\limits_{k+m=n}\binom{n}{k}z^{m}V^{k}R_{\pm}(2n+2)$
$\displaystyle=\sum\limits_{k,m=0}^{\infty}\binom{k+m}{k}z^{m}V^{k}R_{\pm}(2k+2m+2)\qed$

This gives us an explicit expression for the $z$-dependence. We now seek to
shift the $z$-dependence into the Riesz distributions to obtain something
closer to the classical Hadamard expansion. The even Riesz distributions are
obtained from shifting fundamental solutions of the d’Alembert operator on
Minkowski space to the manifold

$R^{U}_{\pm}(2m)(\cdot,x)=(exp_{x})^{-1*}(G^{\pm m}_{\square}\delta_{0})$

(see Proposition 2.10 and the preceding remarks). A reasonable $z$-dependent
generalization is to do the same for $\square-z$ instead:

######  Definition 4.5.

For $z\in{\mathbb{C}}$, $x\in M$ and $m\in{\mathbb{N}}$, we define resolvent
Riesz distributions as

$R_{\pm}(z,2m)(\cdot,x):=(exp_{x}^{-1})^{*}(G^{\pm m}_{\square-z}\delta_{0}),$

and the versions on $U\times U$

$R_{\pm}(z,2m)[\psi]=\int\limits_{U}R_{\pm}(z,2m)(\cdot,x)[\psi(\cdot,x)]dx.$

Here $\square$ is the d’Alembertian on $T_{x}M$.

To construct our final expansion, we need an expansion for the resolvent Riesz
distributions in terms of the standard ones. We start by doing this in
Minkowski space:

######  Lemma 4.6.

On any Lorentzian vector space $V$, we have the asymptotic expansion

$G^{\pm
k}_{\square-z}\delta_{0}\sim\sum\limits_{m=0}^{\infty}\binom{k+m-1}{m}z^{m}R^{V}_{\pm}(2m+2k)(\cdot,0).$

###### Proof.

Theorem 3.3 applied to $\square-z$ gives us the asymptotic expansion

$G^{\pm k}_{\square-z}\delta_{0}=\mathcal{K}(G^{\pm
k}_{\square-z})(\cdot,0)\sim\sum\limits_{m=0}^{\infty}\binom{k+m-1}{m}V^{m}(z)(\cdot,0)R^{V}_{\pm}(2m+2k)(\cdot,0),$

where $V^{m}(z)$ denotes the Hadamard coefficients for $\square-z$. The
Hadamard coefficients of $\square$ are given by

$V^{k}=\begin{cases}{}^{1,\ \ k=0}_{0,\ \ k\neq 0}\end{cases},$

as this makes all transport equations become $0=0$. Thus, by Proposition 4.2,
the Hadamard coefficients for $\square-z$ are given by

$V^{k}(z)=z^{m}.$

Inserting this, we obtain

$G^{\pm
k}_{\square-z}\delta_{0}\sim\sum\limits_{m=0}^{\infty}\binom{k+m-1}{m}z^{m}R^{V}_{\pm}(2m+2k)(\cdot,0).\qed$

This carries over from Minkowski space to $M$ in a straightforward way:

######  Lemma 4.7.

We have the asymptotic expansion

$R_{\pm}(z,2k)\sim\sum\limits_{m=0}^{\infty}\binom{k+m-1}{m}z^{m}R^{U}_{\pm}(2m+2k).$

###### Proof.

As $U$ is convex, it is contractible and thus $TU$ is trivial. Let $\phi\colon
U\times{\mathbb{R}}^{d}\rightarrow TU$ be an isometric trivialization that
preserves time-orientation and denote by $\phi_{x}$ its restriction to the
fibre over $x$, identifying $\\{x\\}\times{\mathbb{R}}^{d}$ with
${\mathbb{R}}^{d}$. As $\phi_{x}$ is a time-orientation preserving isometry,
it preserves all objects defined only in terms of the metric and time-
orientation, i.e.

$\phi_{x}^{-1*}(R^{{\mathbb{R}}^{d}}_{\pm}(2m+2k))=R^{T_{x}M}_{\pm}(2m+2k)$

and

$\phi_{x}^{-1*}(G^{\pm k}_{\square_{{\mathbb{R}}^{d}}-z}\delta_{0})=G^{\pm
k}_{\square_{T_{x}M}-z}\delta_{0}.$

Let

$\operatorname{Exp}\colon\operatorname{Dom}(\operatorname{Exp})\subseteq
TU\rightarrow U\times U$

denote the exponential map with variable basepoint, i.e.
$\operatorname{Exp}(v)=(x,\exp_{x}(v))$ for $v\in T_{x}U$.

Let $n\in{\mathbb{N}}$ be arbitrary. For $N$ large enough such that

$G^{\pm
k}_{\square_{{\mathbb{R}}^{d}}-z}\delta_{0}=\sum\limits_{m=0}^{N}\binom{k+m-1}{m}z^{m}R^{{\mathbb{R}}^{d}}_{\pm}(2m+2k)(\cdot,0)+F$

with $F\in C^{n}({\mathbb{R}}^{d})$, we have

$\displaystyle R_{\pm}(z,2k)(\cdot,x)$ $\displaystyle=(\exp_{x})^{-1*}(G^{\pm
k}_{\square_{T_{x}M}-z}\delta_{0})$
$\displaystyle=(\exp_{x}\circ\phi_{x})^{-1*}(G^{\pm
k}_{\square_{{\mathbb{R}}^{d}}-z}\delta_{0})$
$\displaystyle=(\exp_{x}\circ\phi_{x})^{-1*}\left(\sum\limits_{m=0}^{N}\binom{k+m-1}{m}z^{m}R^{{\mathbb{R}}^{d}}_{\pm}(2m+2k)(\cdot,0)+F\right)$
$\displaystyle=\sum\limits_{m=0}^{N}\binom{k+m-1}{m}z^{m}(exp_{x})^{-1*}R^{T_{x}M}_{\pm}(2m+2k)(\cdot,0)+F\circ\phi_{x}^{-1}\circ
exp_{x}^{-1}$
$\displaystyle=\sum\limits_{m=0}^{N}\binom{k+m-1}{m}z^{m}R^{U}_{\pm}(2m+2k)(\cdot,x)+F\circ\phi^{-1}\circ\operatorname{Exp}^{-1}(\cdot,x).$

as $F\circ\phi^{-1}\circ\operatorname{Exp}^{-1}$ is $C^{n}$ (also as a
function of $x$) and $n$ was arbitrary, we have the desired expansion. ∎

We now have everything in place to prove the main result of this section: If,
in the asymptotic expansions for the Green’s operators of $P$, we replace the
standard Riesz distributions with the resolvent Riesz distributions for
$z\in{\mathbb{C}}$, then we obtain asymptotic expansions for the Green’s
operators of $P-z$.

######  Theorem 4.8.

If $U\subseteq M$ is GE, we have the asymptotic expansion

$\mathcal{K}(G^{\pm}_{P-z})|_{U\times
U}\sim\sum\limits_{k=0}^{\infty}V^{k,U}R^{U}_{\pm}(z,2k+2).$

More generally,

$\mathcal{K}(G^{\pm}_{P-z}{}^{m})|_{U\times
U}\sim\sum\limits_{k=0}^{\infty}\binom{k+m-1}{k}V^{k,U}R^{U}_{\pm}(z,2k+2m).$

###### Proof.

Inserting our formula for the $z$-dependent Hadamard coefficients (Proposition
4.2) into the expansion for powers in Theorem 3.3, applied with $P-z$ instead
of $P$, we obtain

$\displaystyle\mathcal{K}(G^{\pm}_{P-z}{}^{m})$
$\displaystyle\sim\sum\limits_{k=0}^{\infty}\binom{k+m-1}{k}V^{k}(z)R_{\pm}(2k+2m)$
$\displaystyle=\sum\limits_{k=0}^{\infty}\binom{k+m-1}{k}\sum\limits_{l+n=k}\binom{k}{n}z^{n}V^{l}R_{\pm}(2k+2m)$
$\displaystyle=\sum\limits_{l=0}^{\infty}\sum\limits_{n=0}^{\infty}\binom{n+l+m-1}{n+l}\binom{n+l}{n}z^{n}V^{l}R_{\pm}(2(n+m+l))$
$\displaystyle=\sum\limits_{l=0}^{\infty}\sum\limits_{n=0}^{\infty}\binom{n+l+m-1}{n}\binom{l+m-1}{l}z^{n}V^{l}R_{\pm}(2(n+m+l))$
$\displaystyle=\sum\limits_{l=0}^{\infty}\binom{l+m-1}{l}V^{l}\sum\limits_{n=0}^{\infty}\binom{n+l+m-1}{n}z^{n}R_{\pm}(2(n+m+l))$
$\displaystyle\sim\sum\limits_{l=0}^{\infty}\binom{l+m-1}{l}V^{l}R_{\pm}(z,2m+2l).$

Here we used the identity

$\binom{a}{b}\binom{b}{c}=\binom{a}{c}\binom{a-c}{b-c}$

and Lemma 4.7.

This proves the second claim, the first claim follows as the special case
$m=1$. ∎

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