# The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity

Sanchez Sanchez, Yafet  Schrohe, Elmar

###### Abstract

In this paper we estimate the Sobolev wavefront set for the causal propagator
$K_{G}$ of the Klein-Gordon equation in an ultrastatic spacetime when the
regularity of the metric is $C^{\tau}$ with $\tau>2$ and $C^{1,1}$. Our main
tools are a propagation of singularities result for non-smooth
pseudodifferential operators and eigenvalue asymptotics for elliptic operators
of low regularity.

————————————————————————

##  1 Introduction

The quantisation of the scalar field forms part of the basis for the subject
of Quantum Field Theory in Curved Spacetime and in particular for the most
spectacular prediction of that subject: Black Hole evaporation. While the main
mathematical framework was laid down already in the period 1980-2016 [20, 27,
36, 18, 21, 13, 24, 42, 40, 19, 28], it is now possible to approach certain
mathematical questions related to quantum fields propagating in spacetimes of
low regularity. This is motivated by the deep foundational work on causality
theory [11, 7, 34] and to advances in our understanding of non-linear
hyperbolic equations [9, 12, 29, 32] which were needed as a first step towards
a full understanding of Einstein’s equations.

The quantisation proceeds in two steps. First, one constructs an algebra of
observables, then one represents this algebra on a Hilbert space of suitable
states.

A common candidate for physical quantum states, $\omega$, are quasifree states
that satisfy the microlocal spectrum condition.

To state it, it is useful to introduce the sets

$\displaystyle C=\big{\\{}(p,\eta,q,\tilde{\eta})\in T^{*}(M\times
M)\backslash
0;g^{ab}(p)\eta_{a}{\eta}_{b}=g^{ab}(q)\tilde{\eta}_{a}\tilde{\eta}_{b}=0,(p,\eta)\sim(q,\tilde{\eta})\big{\\}}$
(1.1) $\displaystyle C^{+}=\left\\{(p,\eta,q,\tilde{\eta})\in C;\eta^{0}\geq
0,\tilde{\eta}^{0}\geq 0\right\\},$

where $(p,\eta)\sim(q,\tilde{\eta})$ means that $\eta,\tilde{\eta}$ are
cotangent to the null geodesic $\gamma$ at $p$ resp. $q$ and parallel
transports of each other along $\gamma$.

Using the above sets one can define the microlocal spectrum condition as
follows:

######  Definition 1.1.

A quasifree state $\omega_{H}$ on the algebra of observables satisfies the
microlocal spectrum condition if its two point function $\omega_{2H}$ is a
distribution in $\mathcal{D}^{\prime}(M\times M)$ and satisfies the following
wavefront set condition

$WF^{\prime}(\omega_{2H})=C^{+},$

where $WF^{\prime}(\omega_{2H}):=\\{(x_{1},\eta;x_{2},-\tilde{\eta})\in
T^{*}(M\times M);(x_{1},\eta;x_{2},\tilde{\eta})\in WF(\omega_{2H})\\}.$

These states are called Hadamard states [36, 18]. They include ground states
and KMS states.

A larger class of states characterised in terms of their Sobolev wavefront set
are called adiabatic states of order $N$. These states are the natural
generalisation of Hadamard states suitable for spacetimes of limited
regularity. Moreover, adiabatic states of different order produce different
physically measurable effects [26].

######  Definition 1.2.

A quasifree state $\omega_{N}$ on the algebra of observables is called an
adiabatic state of order $N\in\mathbb{R}$ if its two-point function
$\omega_{2N}$ is a bidistribution that satisfies the following
$H^{s}$-wavefront set condition for all $s\leq N+\frac{3}{2}$

$WF^{\prime s}(\omega_{2N})\subset C^{+},$

As a first result for the analysis of quasifree states in spacetimes of finite
regularity we analyse the wave front set of the causal propagator. This
propagator is a classical propagator in contrast with the two-point function
above described (see [15] for details on this convention). The analysis of the
classical propagators of the wave equation and its parametrices in low
regularity spacetimes introduces several technical challenges. Here we mention
some previous results. For example, Szeftel has constructed a parametrix which
requires only control over the $L^{2}$ curvature of the metric in order to
prove the $L^{2}$-curvature conjecture related to Einstein’s Field equations
[43, 29]. Moreover, Tataru [44] has constructed parametrices of the wave
equations in low regularity for metrics with $C^{1,1}$ coefficients as a
preliminary step to show suitable Strichartz estimates and analyse non-linear
PDE’s using phase space transforms. In addition, his results allowed even
lower regularity at the expense of showing weaker results. Finally, we mention
Smith’s construction of parametrices for the $C^{1,1}$ case using wave packets
(see [47] for a parametrix construction using Gaussians).

A main tool used in this paper are pseudodifferential operators with non-
smooth symbols. This topic has been a active area of research since the
introduction of the paradifferential calculus by Bony [8] (see also [6, 45,
33]). Fractional Laplacians and associated pseudodifferential problems have
been studied by Grubb and Abels in the non-smooth setting [1]. The main
contribution of our paper is establishing the singular structure of the causal
propagator when the regularity of the spacetime metric is $C^{2+\epsilon}$ for
$\epsilon>0$ (Theorem 6.3) and $C^{1,1}$ (Theorem 7.1). From the physical
point of view, $C^{1,1}$ metrics allow to describe spacetimes where the
curvature is finite but there are discontinuities in the energy-momentum
tensor e.g. certain interfaces or astrophysical models of stars. From the
mathematical point of view $C^{1,1}$ is the minimal condition where one has
existence and uniqueness of the geodesics and where the standard results from
smooth causality theory still holds [30].

###  1.1 The Smooth Setting

In this section we briefly review the results in the smooth setting.

Consider a pair $(M,g)$, where $M$ is a smooth manifold and $g$ is a smooth
Lorentzian metric. The Klein-Gordon operator on $(M,g)$ is given by

$g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi+m^{2}\phi=0$ (1.2)

where $g^{\mu\nu}$ is an inverse metric tensor, $\nabla_{\mu}$ is the
covariant derivative and $m$ is a positive real number.

The starting point is the notion of advanced and retarded Green operators in
this situation.

######  Definition 1.3.

Let $M$ be a time-oriented connected Lorentzian manifold and let $P$ be the
Klein-Gordon operator. An _advanced Green operator_ $G^{+}$ is a linear map
$G^{+}:{\mathcal{D}}(M)\to C^{\infty}(M)$ such that

  1. 1.

$P\circ G^{+}={\rm id}_{{\mathcal{D}}(M)}$

  2. 2.

$G^{+}\circ P|_{{\mathcal{D}}(M)}={\rm id}_{{\mathcal{D}}(M)}$

  3. 3.

${\operatorname{supp}}(G^{+}\phi)\subset J^{+}({\operatorname{supp}}(\phi))$
for all $\phi\in{\mathcal{D}}(M)$.

A _retarded Green operator_ $G^{-}$ satisfies $(1)$ and $(2)$, but $(3)$ is
replaced by the condition ${\operatorname{supp}}(G^{-}\phi)\subset
J^{-}({\operatorname{supp}}(\phi))$ for all $\phi\in{\mathcal{D}}(M)$.

In [5, Corollary 3.4.3] it is shown that these exist and are unique on a
globally hyperbolic manifold.

The advanced and retarded Green operators are then used to define the causal
propagator

$G:=G^{+}-G^{-}$

which maps ${{\mathcal{D}}(M)}$ to the space of spatially compact maps
$C^{\infty}_{{\rm sc}}(M)$, i.e. the smooth maps $\phi$ such that there exists
a compact subset $K\subset M$ with $\operatorname{supp}(\phi)\subset J(K)$. If
$M$ is globally hyperbolic then one has the following _exact sequence_ [5,
Theorem 3.4.7]:

${0}$${{\mathcal{D}}(M)}$${{\mathcal{D}}(M)}$${C^{\infty}_{{\rm
sc}}(M)}$${C^{\infty}_{{\rm
sc}}(M),}$$\scriptstyle{P}$$\scriptstyle{G}$$\scriptstyle{P}$

Since $G$ is a continuous linear operator, the Schwartz Kernel Theorem implies
that there exists one and only one distribution ${\displaystyle
K_{G}\in{\mathcal{D}}^{\prime}(X\times X)}$ such that

$K_{G}(u\otimes v)=\langle G(v),u\rangle,\quad u,v\in\mathcal{D}(M).$ (1.3)

It follows from Duistermaat and Hörmander’s characterisation using Fourier
Integral Operators that the kernel $K_{G}$ satisfies

$WF^{\prime}(K_{G})=C.$ (1.4)

More explicitly, they showed that $K_{G}\in I^{-\frac{3}{2}}(M\times
M,C^{\prime})$ where $I^{\mu}(X,\Lambda)$ denotes the space of Lagrangian
distributions of order $\mu$ over the manifold $X$ associated to the
Lagrangian submanifold $\Lambda$. In this case
$\Lambda=C^{\prime}=\\{(x_{1},\eta;x_{2},-\tilde{\eta});(x_{1},\eta;x_{2},\tilde{\eta})\in
C\\}$, [17, Theorem 6.5.3].

##  2 The Non-smooth Setting

In this section we will consider the case, where $g$ is a non-smooth metric.
We specify the precise regularity in each section.

The definition of the Green operators in the non-smooth setting will require
us to choose suitable spaces of functions as domain and range. We let

$\displaystyle V_{0}=$ $\displaystyle\\{\phi\in
H_{\text{comp}}^{2}(M);P\phi\in H^{1}_{\text{comp}}(M)\\}$ $\displaystyle
U_{0}=$ $\displaystyle H^{1}_{\text{comp}}(M)$ (2.1) $\displaystyle V_{sc}=$
$\displaystyle\\{\phi\in H_{\text{loc}}^{2}(M);P\phi\in H^{1}_{\text{loc}}(M)$
$\displaystyle\text{ and }\text{supp}(\phi)\subset J(K),\text{ where }K\text{
is a compact subset of }M\\}$

######  Definition 2.1.

An _advanced Green operator_ for $P$ is a linear map

$G^{+}:H_{\text{comp}}^{1}(M)\rightarrow H_{\text{loc}}^{2}(M)$

satisfying the properties

  1. 1.

$PG^{+}=\text{id}_{H_{\text{comp}}^{1}(M)}$,

  2. 2.

$G^{+}P|_{V_{0}}=\text{id}_{V_{0}}$,

  3. 3.

$\operatorname{supp}(G^{+}(f))\subset J^{+}(\operatorname{supp}(f))$ for all
$f\in H_{\text{comp}}^{1}(M)$,

A _retarded Green operator_ $G^{-}$ is defined similarly.

It was shown in [25, Theorem 5.8] that these operators exist and are unique on
Lorentzian manifolds that satisfy the condition of generalised hyperbolicity.
This condition is satisfied in particular for $C^{1,1}$ globally hyperbolic
spacetimes. Moreover, one obtains a short exact sequence for the low-
regularity causal propagators similar to that in the smooth case

${0}$${V_{0}}$${U_{0}}$${V_{\text{sc}}}$${H^{1}_{\text{loc}}(M).}$$\scriptstyle{P}$$\scriptstyle{G}$$\scriptstyle{P}$

###  2.1 The Ultrastatic Case

The ultrastatic case will be the main geometric setting of the paper. In this
case $M=\mathbb{R}\times\Sigma$ where $\Sigma$ is $3$-dimensional compact
manifold and the Lorentzian metric $g$ is of the form

$ds^{2}=dt^{2}-h_{ij}(x)dx^{i}dx^{j}$

where $h_{ij}(x)$ are the components of a time independent Riemannian metric
of Hölder regularity $C^{\tau}$ (when $\tau\in\mathbb{N}$ we will consider the
Zygmund spaces $C_{*}^{\tau}$, introduced in Definition 3.1).

We write local coordinates on $\mathbb{R}\times\Sigma$ in the form

$\displaystyle\tilde{x}=(t,x),\tilde{y}=(s,y)$ (2.2)

and the associated covariables as

$\displaystyle\tilde{\xi}=(\xi^{0},\xi),\tilde{\eta}=(\eta^{0},\eta).$ (2.3)

On the product $(\mathbb{R}\times\Sigma)\times(\mathbb{R}\times\Sigma)$ we use

$\displaystyle{\bf{x}}=(\tilde{x},\tilde{y}),{\boldsymbol{\xi}}=(\tilde{\xi},\tilde{\eta}).$
(2.4)

In these coordinates the Klein-Gordon operator $P$ on $M$ is

$P\phi=\partial_{tt}\phi-\Delta_{h}\phi+m^{2}\phi$ (2.5)

with
$\Delta_{h}\phi=\frac{1}{\sqrt{h}}\partial_{x^{i}}(h^{ij}\sqrt{h}\partial_{x^{j}}\phi)$
and $m>0$.

The causal propagator $G$ is given by
$\displaystyle{-\frac{\sin(A^{\frac{1}{2}}(t-s))}{A^{\frac{1}{2}}}}$ where
$A:=-\Delta_{h}\phi+m^{2}$ is self-adjoint on $L^{2}(\Sigma)$(see appendix
8.2). Moreover, the spectrum of $A$ is a discrete set of positive eigenvalues
which we denote them by $\\{\lambda_{j}^{2};j=1,2,\ldots\\}$, listed according
to their (finite) multiplicity. The associated set
$\\{\phi_{j}\\}_{j\in\mathbb{N}}$ of normalized real eigenfunctions is an
orthonormal basis of $L^{2}(\Sigma)$[31, Theorem 5.8]. For
$u,v\in{\mathcal{D}}(M)$ we have $G(v)\in\mathcal{D}^{\prime}(M)$ given by

$\displaystyle\langle G(v),u\rangle:=$ (2.6)
$\displaystyle=\int_{\Sigma}\int_{-\infty}^{\infty}\left(-\frac{\sin(A^{\frac{1}{2}}(t-s))}{A^{\frac{1}{2}}}v\right)(t,x)u(t,x)\sqrt{h(x)}dxdt$
$\displaystyle=-\int_{M}\left(\int_{-\infty}^{\infty}\sum_{j}\lambda_{j}^{-1}\sin(\lambda_{j}(t-s))\phi_{j}(x)\int_{\Sigma}\phi_{j}(y)v(s,y)\sqrt{h(y)}dyds\right)u(t,x)\sqrt{h(x)}dxdt,$

and the kernel of the causal propagator acting on the densities
$u\sqrt{h(x)}\otimes v\sqrt{h(y)}$ is given by

$\displaystyle K_{G}(u\otimes v)=$ (2.7) $\displaystyle=-\int_{M\times
M}\sum_{j}\lambda_{j}^{-1}\sin(\lambda_{j}(t-s))\phi_{j}(x)\phi_{j}(y)\sqrt{h(y)}\sqrt{h(x)}u(t,x)v(s,y)dsdydtdx,$

which gives the singular integral kernel representation

$K_{G}(t,x;s,y)=-\sum_{j}\lambda_{j}^{-1}\sin(\lambda_{j}(t-s))\phi_{j}(x)\phi_{j}(y).$
(2.8)

##  3 Pseudodifferential Operators with Non-smooth Symbols

###  3.1 Symbol Classes

Let $\\{\psi_{j};j=0,1,\ldots\\}$ be a Littlewood-Paley partition of unity on
$\mathbb{R}^{n}$, i.e., a partition of unity $1=\sum_{j=0}^{\infty}\psi_{j}$,
where $\psi_{0}\equiv 1$ for $|\xi|\leq 1$ and $\psi_{0}\equiv 0$ for
$|\xi|\geq 2$ and $\psi_{j}(\xi)=\psi_{0}(2^{j}\xi)-\psi_{0}(2^{1-j}\xi)$. The
support of $\psi_{j}$, $j\geq 1$, then lies in an annulus around the origin of
interior radius $2^{j}$ and exterior radius $2^{1+j}$.

######  Definition 3.1.

(a) For $\tau\in(0,\infty)$, the Hölder space $C^{\tau}(\mathbb{R}^{n})$ is
the set of all functions $f$ with

$\|f\|_{C^{\tau}}:=\displaystyle\sum_{|\alpha|\leq[\tau]}\|\partial^{\alpha}_{x}f\|_{L^{\infty}(\mathbb{R}^{n})}+\displaystyle\sum_{|\alpha|=[\tau]}\sup_{x\neq
y}\frac{|\partial^{\alpha}_{x}f(x)-\partial^{\alpha}_{x}f(y)|}{|x-y|^{\tau-[\tau]}}<\infty.$
(3.1)

(b) For $\tau\in\mathbb{R}$ the Zygmund space $C^{\tau}_{*}(\mathbb{R}^{n})$
consists of all functions $f$ with

$\|f\|_{C^{\tau}_{*}}=\sup_{j}2^{j\tau}\|\psi_{j}(D)f\|_{L^{\infty}}<\infty$
(3.2)

Here $\psi_{j}(D)$ is the Fourier multiplier with symbol $\psi_{j}$, i.e.,
$\psi_{j}(D)u=\mathcal{F}^{-1}\psi_{j}\mathcal{F}u$, where
$(\mathcal{F}u)(\xi)=(2\pi)^{-n/2}\int e^{-ix\xi}u(x)\,d^{n}x$ is the Fourier
transform.

We have the following relations $C^{\tau}=C^{\tau}_{*}$ if
$\tau\notin\mathbb{Z}$ and $C^{\tau}\subset C^{\tau}_{*}$ if
$\tau\in\mathbb{Z}$.

We next introduce symbol classes of finite Hölder or Zygmund regularity,
following Taylor [45]. We use the notation
$\langle\xi\rangle:=(1+|\xi|^{2})^{\frac{1}{2}}$, $\xi\in\mathbb{R}^{n}$.

######  Definition 3.2.

(a) Let $0\leq\delta<1$ A symbol $p(x,\xi)$ belongs to
$C^{\tau}_{*}S^{m}_{1,\delta}$ if

$\|D^{\alpha}_{\xi}p(x,\xi)\|_{C_{*}^{\tau}}\leq
C_{\alpha}\langle\xi\rangle^{m-|\alpha|+\tau\delta}\text{ and
}|D^{\alpha}_{\xi}p(x,\xi)|\leq C_{\alpha}\langle\xi\rangle^{m-|\alpha|}.$

(b) We obtain the symbol class $C^{\tau}S^{m}_{1,\delta}$ for $\tau>0$ by
requiring that

$\|D^{\alpha}_{\xi}p(x,\xi)\|_{C^{j}}\leq
C_{\alpha}\langle\xi\rangle^{m-|\alpha|+j\delta},\quad 0\leq j\leq\tau,\text{
and }|D^{\alpha}_{\xi}p(x,\xi)|\leq C_{\alpha}\langle\xi\rangle^{m-|\alpha|}.$

(c) A symbol $p(x,\xi)$ is in $C^{\tau}S_{cl}^{m}$ provided $p(x,\xi)\in
C^{\tau}S^{m}_{1,0}$ and $p(x,\xi)$ has a classical expansion

$p(x,\xi)\sim\sum_{j\geq 0}p_{m-j}(x,\xi)$

in terms $p_{m-j}$ homogeneous of degree $m-j$ in $\xi$ for $|\xi|\geq 1$, in
the sense that the difference between $p(x,\xi)$ and the sum over $0\leq j<N$
belongs to $C^{\tau}S^{m-N}_{1,0}$.

######  Remark 3.3.

The Klein-Gordon operator $P$ given by Eq.(2.5) has the symbol
$P(\tilde{x},\tilde{\xi})=(-\xi_{0}^{2}+h^{ij}\xi_{i}\xi_{j})+i\frac{1}{\sqrt{h}}\partial_{x^{i}}(h^{ij}\sqrt{h})\xi_{j}+m^{2}$.
For a metric of regularity $C^{\tau}$, the symbol $P(\tilde{x},\tilde{\xi})$
belongs to $C^{\tau-1}S^{2}_{cl}$.

The pseudodifferential operator $p(x,D_{x})$ with the symbol $p(x,\xi)\in
C^{\tau}S^{m}_{1,\delta}$ is given by

$p(x,D_{x})u=(2\pi)^{-n/2}\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}p(x,\xi)({\cal{F}}{u})(\xi)d^{n}\xi,\quad
u\in\mathcal{S}(\mathbb{R}^{n}).$ (3.3)

It extends to continuous maps

$p(x,D_{x}):H^{s+m}(\mathbb{R}^{n})\rightarrow
H^{s}(\mathbb{R}^{n}),\quad-\tau(1-\delta)<s<\tau.$ (3.4)

###  3.2 Symbol Smoothing

Given $p(x,\xi)\in C^{\tau}S_{1,\delta}^{m}$ and $\gamma\in(\delta,1)$ let

$p^{\\#}(x,\xi)=\displaystyle\sum_{j=0}^{\infty}J_{\epsilon_{j}}p(x,\xi)\psi_{j}(\xi).$
(3.5)

Here $J_{\epsilon}$ is the smoothing operator given by
$(J_{\epsilon}f)(x)=(\phi(\epsilon D)f)(x)$ with $\phi\in
C^{\infty}_{0}(\mathbb{R}^{n})$, $\phi(\xi)=1$ for $|\xi|\leq 1$, and we take
$\epsilon_{j}=2^{-j\gamma}$.

Letting $p^{b}(x,\xi)=p(x,\xi)-p^{\\#}(x,\xi)$ we obtain the decomposition

$p(x,\xi)=p^{\\#}(x,\xi)+p^{b}(x,\xi),$ (3.6)

where $p^{\\#}(x,\xi)\in S^{m}_{1,\gamma}$ and $p^{b}(x,\xi)\in
C^{\tau}S^{m-s(\gamma-\delta)}_{1,\gamma}$.

If $p\in C^{\tau}S^{m}_{1,0}$, then we additionally have $p^{b}\in
C^{\tau-t}S^{m-t\delta}_{1,0}$ with $\tau-t>0$ by [45, Proposition 1.3.B].
Furthermore, we have better estimates, see [45, Proposition 1.3.D]:

$\displaystyle
D_{x}^{\beta}p^{\\#}(x,\xi)\in\begin{cases}S^{m}_{1,\delta},\quad&|\beta|\leq\tau\text{
and}\\\ S^{m+\delta(|\beta|-r)}_{1,\delta},\quad&|\beta|>\tau\end{cases}$
(3.7)

###  3.3 Microlocal Sobolev Regularity and Propagation of Singularities for
Bisolution of the Klein-Gordon Operator

Let $p\in C^{\tau}S^{m}_{\rho,\delta}$, $\tau>0$, with $\delta<\rho$. Suppose
that there is a conic neighborhood $\Gamma$ of $(x_{0},\xi_{0})$ and constants
$c,C>0$ such that $|p(x,\xi)|\geq c|\xi|^{m}$ for $(x,\xi)\in\Gamma$,
$|\xi|\geq C$. Then $(x_{0},\xi_{0})$ is called non-characteristic. If $p$ has
a principal homogeneous symbol $p_{m}$, the condition is equivalent to
$p_{m}(x_{0},\xi_{0})\neq 0$. The complement of the set of non-characteristic
points is the set of characteristic points denoted by
${\operatorname{Char}}(p)$.

A distribution $u$ is microlocally in $H^{s}$ at $(x_{0},\xi_{0})\in
T^{*}M\backslash 0$ if there exists a conic neighbourhood ${\Gamma}$ of
$\xi_{0}$ and a smooth function $\varphi\in C_{0}^{\infty}(M)$ with
$\varphi(x_{0})\neq 0$ such that

$\int_{{\Gamma}}\langle\xi\rangle^{2s}|{\cal{F}}(\varphi
u)(\xi)|^{2}d^{n}\xi<\infty.$

Otherwise we say that $(x_{0},\xi_{0})$ lies in the $s$-wave front set
$WF^{s}(u)$.

If $u$ is microlocally in $H^{s}$ in an open conic subset ${\Gamma}\subset
T^{*}M\backslash 0$ we write $u\in H^{s}_{mcl}({\Gamma})$.

In the sequel we shall apply the Klein-Gordon operator also to functions and
distributions on $M\times M$. Using the coordinates in Eqs.(2.2),(2.3) and
(2.4), we distinguish the cases, where $P$ acts on the first set of variables
$(t,x)$ or on the second set $(s,y)$, and write $P_{(t,x)}$ and $P_{(s,y)}$,
respectively. The associated symbols $P_{(t,x)}({\bf{x}},{\boldsymbol{\xi}})$
and $P_{(s,y)}({\bf{x}},{\boldsymbol{\xi}})$ formally depend on the full set
of (co-)variables $({\bf{x}},{\boldsymbol{\xi}})$, however, only the
(co-)variables associated with either $(t,x)$ or $(s,y)$ show up:

$\displaystyle P_{(t,x)}({\bf{x}},{\boldsymbol{\xi}})$ $\displaystyle=$
$\displaystyle
P_{(t,x)}(\tilde{x},\tilde{\xi},\tilde{y},\tilde{\eta})=({-\xi^{0}}^{2}+h^{ij}(x)\xi_{i}\xi_{j})+{i\frac{1}{\sqrt{h}}\partial_{x^{i}}(h^{ij}\sqrt{h}(x))\xi_{j}}+{m^{2}}$
$\displaystyle P_{(s,y)}({\bf{x}},{\boldsymbol{\xi}})$ $\displaystyle=$
$\displaystyle
P_{(s,y)}(\tilde{x},\tilde{\xi},\tilde{y},\tilde{\eta})=(-{\eta^{0}}^{2}+h^{ij}(x)\eta_{i}\eta_{j})+{i\frac{1}{\sqrt{h}}\partial_{y^{i}}(h^{ij}\sqrt{h}(y))\eta_{j}}+{m^{2}}.$

In particular,

$\displaystyle\operatorname{Char}(P_{(t,x)})$ $\displaystyle=$
$\displaystyle\operatorname{Char}(P)\times
T^{*}M\cup\\{({\bf{x}},{\boldsymbol{\xi}})\in T^{*}(M\times
M)\backslash\\{0\\},\tilde{\xi}=0\\}$ (3.8)
$\displaystyle\operatorname{Char}(P_{(s,y)})$ $\displaystyle=$ $\displaystyle
T^{*}M\times\operatorname{Char}(P)\cup\\{({\bf{x}},{\boldsymbol{\xi}})\in
T^{*}(M\times M)\backslash\\{0\\},\tilde{\eta}=0\\}.$

######  Theorem 3.4.

Let the metric $g$ be of class $C^{\tau}$, $\tau>1$, $0\leq\sigma<\tau-1$ and
$v\in H_{loc}^{2+\sigma-\tau+\epsilon}(M\times M)$ for some ${\epsilon}>0$
with $P_{(t,x)}({\bf{x}},D_{\bf{x}})v=0$. Then

$WF^{\sigma+2}(v)\subset\operatorname{Char}(P_{(t,x)})$

###### Proof.

Since all the estimates are local, we may assume we are working in a single
coordinate patch. Let

$P_{(t,x)}({\bf{x}},{\boldsymbol{\xi}})=P_{(t,x)}(\tilde{x},\tilde{\xi},\tilde{y},\tilde{\eta})=\underbrace{({-\xi^{0}}^{2}+h^{ij}(x)\xi_{i}\xi_{j})}_{p_{2}({\bf{x}},{\boldsymbol{\xi}})}+\underbrace{i\frac{1}{\sqrt{h}}\partial_{x^{i}}(h^{ij}\sqrt{h}(x))\xi_{j}}_{p_{1}({\bf{x}},{\boldsymbol{\xi}})}+\underbrace{m^{2}}_{p_{0}({\bf{x}},{\boldsymbol{\xi}})}.$
(3.9)

The symbol smoothing (Eq.(3.6)) on $p_{2},p_{1}$ gives a decomposition

$\displaystyle p_{2}({\bf{x}},{\boldsymbol{\xi}})$
$\displaystyle=p_{2}^{\\#}({\bf{x}},{\boldsymbol{\xi}})+p_{2}^{b}({\bf{x}},{\boldsymbol{\xi}})$
$\displaystyle p_{1}({\bf{x}},{\boldsymbol{\xi}})$
$\displaystyle=p_{1}^{\\#}({\bf{x}},{\boldsymbol{\xi}})+p_{1}^{b}({\bf{x}},{\boldsymbol{\xi}})$
$\displaystyle P({\bf{x}},{\boldsymbol{\xi}})$
$\displaystyle=(p_{2}^{\\#}({\bf{x}},{\boldsymbol{\xi}})+p_{1}^{\\#}({\bf{x}},{\boldsymbol{\xi}}))+p_{2}^{b}({\bf{x}},{\boldsymbol{\xi}})+p_{1}^{b}({\bf{x}},{\boldsymbol{\xi}})+p_{0}({\bf{x}},{\boldsymbol{\xi}})$
$\displaystyle=q^{\\#}({\bf{x}},{\boldsymbol{\xi}})+p_{2}^{b}({\bf{x}},{\boldsymbol{\xi}})+p_{1}^{b}({\bf{x}},{\boldsymbol{\xi}})$

where

$q^{\\#}({\bf{x}},{\boldsymbol{\xi}})=(p_{2}^{\\#}({\bf{x}},{\boldsymbol{\xi}})+p_{1}^{\\#}({\bf{x}},{\boldsymbol{\xi}})+p_{0}({\bf{x}},{\boldsymbol{\xi}}))\in
S^{2}_{1,\delta},$ (3.10) $p_{2}^{b}({\bf{x}},{\boldsymbol{\xi}})\in
C^{\tau}S^{2-\tau\delta}_{1,\delta}\quad
p_{1}^{b}({\bf{x}},{\boldsymbol{\xi}})\in
C^{\tau-1}S^{1-(\tau-1)\delta}_{1,\delta}.$ (3.11)

Taking $0\leq\delta<1$ so close to $1$ that $2-\tau\delta<2-\tau+\epsilon$ we
have $v\in H_{loc}^{2+\sigma-\tau\delta}(M\times M)$ (notice this implies
$v\in H_{loc}^{1+\sigma-(\tau-1)\delta}(M\times M)$), we have

$q^{\\#}({\bf{x}},D_{{\bf{x}}})v=-(p_{2}^{b}({\bf{x}},D_{{\bf{x}}})+p_{1}^{b}({\bf{x}},D_{{\bf{x}}}))v=f,$
(3.12)

where $f\in H_{loc}^{\sigma}(M\times M)$, since
$p_{2}^{b}({\bf{x}},D_{{\bf{x}}})v\in H_{loc}^{\sigma}(M\times M)$ and
$p_{1}^{b}({\bf{x}},D_{{\bf{x}}})v\in H_{loc}^{\sigma+1-\delta}(M\times M)$.

Now if
$(\tilde{x}_{0},\tilde{\xi}_{0},\tilde{y}_{0},\tilde{\eta}_{0})\notin\operatorname{Char}(P_{(t,x)})$
there are $C,c>0$ such that

$|P({\bf{x}},{\boldsymbol{\xi}})|\geq c|{\boldsymbol{\xi}}|^{2}\text{ for
}|{\boldsymbol{\xi}}|\geq C$

in a conical neighbourhood $\Gamma$ that contains
$(\tilde{x}_{0},\tilde{\xi}_{0},\tilde{y}_{0},\tilde{\eta}_{0})$.

Since $p_{2}^{b}({\bf{x}},{\boldsymbol{\xi}})\in
C^{\tau}S^{2-\tau\delta}_{1,\delta}$ and
$p_{1}^{b}({\bf{x}},{\boldsymbol{\xi}})\in
C^{\tau-1}S^{1-(\tau-1)\delta}_{1,\delta}$ there exists a $\tilde{C}>0$ such
that

$\displaystyle|q^{\\#}({\bf{x}},{\boldsymbol{\xi}})|$ $\displaystyle\geq
C(1+|{\boldsymbol{\xi}}|^{2})-(1+|{\boldsymbol{\xi}}|^{2})^{\frac{2-\tau\delta}{2}}-(1+|{\boldsymbol{\xi}}|^{2})^{\frac{1-(\tau-1)\delta}{2}}$
$\displaystyle\geq\tilde{C}(1+|{\boldsymbol{\xi}}|^{2})\text{ for large
}|{\boldsymbol{\xi}}|.$

Therefore
$(\tilde{x}_{0},\tilde{\xi}_{0},\tilde{y}_{0},\tilde{\eta}_{0})\notin\operatorname{Char}(q^{\\#})$.

Since $q^{\\#}\in S^{2}_{1,\delta}$ and
$(\tilde{x}_{0},\tilde{\xi}_{0},\tilde{y}_{0},\tilde{\eta}_{0})\notin\operatorname{Char}(q^{\\#})$
there is a microlocal parametrix with symbol $\tilde{q}\in S^{-2}_{1,\delta}$
such that

$\displaystyle v+r({\bf{x}},D_{{\bf{x}}})v$
$\displaystyle=\tilde{q}({\bf{x}},D_{{\bf{x}}})q^{\\#}({\bf{x}},D_{{\bf{x}}})v=\tilde{q}({\bf{x}},D_{{\bf{x}}})f,$

where $(\tilde{x}_{0},\tilde{\xi}_{0},\tilde{y}_{0},\tilde{\eta}_{0})\notin
WF(r({\bf{x}},D_{{\bf{x}}})v)$ and $\tilde{q}({\bf{x}},D_{{\bf{x}}})f\in
H_{loc}^{\sigma+2}(M\times M)$ which shows that
$(\tilde{x}_{0},\tilde{\xi}_{0},\tilde{y}_{0},\tilde{\eta}_{0})\notin
WF^{\sigma+2}(\tilde{q}({\bf{x}},D_{{\bf{x}}})f)$. Since

$\displaystyle WF^{\sigma+2}(v)\subset
WF^{\sigma+2}(\tilde{q}({\bf{x}},D_{{\bf{x}}})f)\cup
WF^{\sigma+2}(r({\bf{x}},D_{{\bf{x}}})v),$ (3.13)

we see that
$(\tilde{x}_{0},\tilde{\xi}_{0},\tilde{y}_{0},\tilde{\eta}_{0})\notin
WF^{\sigma+2}(v)$, and the proof is complete. ∎

######  Remark 3.5.

Applying the symbol smoothing directly to $P_{(t,x)}\in C^{\tau-1}S^{2}_{1,0}$
would leave us with $P_{(t,x)}^{b}\in
C^{\tau-1}S^{2-{(\tau-1)}\delta}_{1,\delta}$. The advantage of the
decomposition in Theorem 3.4 is that we obtain the remainder
$p_{2}^{b}+p_{1}^{b}$ for $p^{b}_{2}\in C^{\tau}S^{2-\tau\delta}_{1,\delta}$
and $p^{b}_{1}\in C^{\tau-1}S^{1-(\tau-1)\delta}_{1,\delta}$. Therefore, given
$u\in H^{2+s-\tau\delta}$ we have $p^{b}_{2}u\in H^{s},p^{b}_{1}u\in
H^{s+1-\delta}$ and hence $p_{2}^{b}u+p_{1}^{b}u\in H^{s}$ for
$-{(1-\delta)}(\tau-1)<s<\tau-1$.

The theorem, below, will be crucial for our main result. Proofs can be found
in [45, Proposition 6.1.D] or [46, Proposition 11.4]. In [46, p.215], Taylor
points out that Zygmund regularity $C_{*}^{2}$ for the metric or the
boundedness of the Ricci tensor is sufficient for the result.

######  Theorem 3.6.

Let $u\in\mathcal{D}^{\prime}(M\times M)$ solve $P_{(t,x)}u=f$. Let $\gamma$
be a integral curve of the Hamiltonian vector field $H_{p_{2}}$ with $p_{2}$
as in Eq.(3.9). If for some $s\in\mathbb{R}$, $f\in H_{mcl}^{s}({\Gamma})$ and
$P_{(t,x)}^{b}u\in H_{mcl}^{s}({\Gamma)}$ where $\gamma\subset{\Gamma}$ with
${\Gamma}$ a conical neighbourhood and $u\in H_{mcl}^{s+1}(\gamma(0))$ then
$u\in H_{mcl}^{s+1}({\gamma})$.

######  Remark 3.7.

If $u\in H_{loc}^{2+s-\tau\delta}$, then $P_{(t,x)}^{b}u\in
H_{loc}^{s}(M\times M)$, see Remark (3.5). Moreover, using the divergence
structure of the operator if $u\in H_{loc}^{1+s-\tau\delta}(M\times M),f\in
H_{loc}^{s-1}(M\times M),u\in H_{mcl}^{s}({\gamma}(0))$, then $u\in
H_{mcl}^{s}({\gamma})$ for $-2(1-\delta)<s\leq 2$; see [46, p.210] for
details.

##  4 An Equivalent Sobolev Norm

The main result of this section is the following proposition and Corollary
4.4.

######  Proposition 4.1.

Let $\\{\phi_{j}\otimes\phi_{k};j,k=1,2,\ldots\\}$ be an orthonormal basis of
$L^{2}(\Sigma)\otimes L^{2}(\Sigma)$ associated to the eigenfunctions
$\\{\phi_{j}\\}$ of the operator $mI-\Delta_{h}$. Writing $u\in
L^{2}((\mathbb{R}\times\Sigma)\times(\mathbb{R}\times\Sigma))\cong
L^{2}(\mathbb{R}^{2}\times\Sigma^{2})\cong
L^{2}(\mathbb{R}^{2})\otimes_{H}L^{2}(\Sigma)\otimes_{H}L^{2}(\Sigma)$ in the
form

$\displaystyle{u(t,s,x,y)=\sum_{j,k}u_{jk}(t,s)\phi_{j}(x)\phi_{k}(y)}\quad\text{with
}u_{jk}=\langle u,\phi_{j}\otimes\phi_{k}\rangle\in L^{2}(\mathbb{R}^{2}),$
(4.1)

we obtain the following alternative description of the Sobolev spaces: For
$0\leq s\leq 2$

$\displaystyle H^{s}(\mathbb{R}^{2}\times\Sigma^{2})$
$\displaystyle=\\{u\in\mathcal{S}^{\prime}(\mathbb{R}^{2}\times\Sigma^{2});\sum_{j,k}\int_{\mathbb{R}^{2}}(|\xi_{0}|^{2}+|\eta_{0}|^{2}+\lambda_{j}^{2}+\lambda_{k}^{2})^{s}|(\mathcal{F}{u}_{jk)}(\xi_{0},\eta_{0})|^{2}d\xi_{0}d\eta_{0}<\infty\\}.$

Here $\mathcal{S}^{\prime}(\mathbb{R}^{2}\times\Sigma^{2})$ is the dual space
to
$\mathcal{S}(\mathbb{R}^{2}\times\Sigma^{2}):=\mathcal{S}(\mathbb{R}^{2})\hat{\otimes}_{\pi}C^{\infty}(\Sigma^{2})$.
First we show the result in the particular case $s=2$:

######  Lemma 4.2.

$\displaystyle H^{2}(\mathbb{R}^{2}\times\Sigma^{2})$
$\displaystyle=\\{u\in\mathcal{S}^{\prime}(\mathbb{R}^{2}\times\Sigma^{2});\sum_{j,k}\int_{\mathbb{R}^{2}}(|\xi_{0}|^{2}+|\eta_{0}|^{2}+\lambda_{j}^{2}+\lambda_{k}^{2})^{2}|(\mathcal{F}{u}_{jk})(\xi_{0},\eta_{0})|^{2}d\xi_{0}d\eta_{0}<\infty\\}$

###### Proof.

By definition (see Appendix 8.1).

$H^{2}(\mathbb{R}^{2}\times\Sigma^{2})=\\{u\in\mathcal{S}^{\prime}(\mathbb{R}^{2}\times\Sigma^{2});(I-\partial_{tt}-\partial_{ss}-\Delta_{h{{}_{x}}}-\Delta_{h{{}_{y}}})u\in
L^{2}(\mathbb{R}^{2}\times\Sigma^{2})\\},$ (4.2)

where we have equipped $\Sigma\times\Sigma$ with the product metric
$\tilde{h}$ induced by the metric $h$ on each of the components so that
$\Delta_{\tilde{h}}=\partial_{tt}+\partial_{ss}+\Delta_{h_{x}}+\Delta_{h_{y}}$,
where $\Delta_{h_{x}}$ and $\Delta_{h_{y}}$ are the Laplacians for the metric
$h$ on the first and second component of $\Sigma\times\Sigma$, respectively.
Since $m>0$ we may (at the expense of obtaining an equivalent norm) replace
$I$ in Eq.(4.2) by $2mI$. Writing $u\in
H^{2}(\mathbb{R}^{2}\times\Sigma^{2})\subset
L^{2}(\mathbb{R}^{2}\times\Sigma^{2})$in the form (4.1) and using the
orthonormality of the set $\\{\phi_{l}\\}_{l}$ in $L^{2}{(\Sigma)}$ we obtain

$\displaystyle\|u\|^{2}_{H^{2}(\mathbb{R}^{2}\times\Sigma^{2})}$
$\displaystyle:=\int_{\mathbb{R}^{2}}\int_{\Sigma\times\Sigma}|(2mI-\partial_{tt}-\partial_{ss}-\Delta_{h{{}_{x}}}-\Delta_{h{{}_{y}}})u|^{2}\sqrt{h(x)}\sqrt{h(y)}dtdsdxdy$
$\displaystyle=\int_{\mathbb{R}^{2}}\sum_{j,k}|-(\partial_{tt}+\partial_{ss})u_{jk}(t,s)+\lambda^{2}_{j}u_{jk}(t,s)+\lambda_{k}^{2}u_{jk}(t,s)|^{2}dtds.$

Applying the Fourier transform in $(s,t)$, Plancherel’s theorem shows that the
last expression equals

$\displaystyle=\sum_{j,k}\int_{\mathbb{R}^{2}}\left(\xi_{0}^{2}+\eta_{0}^{2}+\lambda^{2}_{j}+\lambda_{k}^{2}\right)^{2}|({\cal{F}}u_{jk})(\xi_{0},\eta_{0})|^{2}d\xi_{0}d\eta_{0}$

which proves the result. ∎

Before proving the main proposition we state the following result found in
Amann [2, I.(2.9.8)].

######  Theorem 4.3.

Let $A$ be a non-negative self-adjoint operator. Then we have the following
relation for the domains of the powers of $A$:

$\mathscr{D}(A^{(1-\theta)\alpha+\theta\beta})=[\mathscr{D}(A^{\alpha}),\mathscr{D}(A^{\beta})]_{\theta}$

for $0\leq{\rm Re}\,\alpha<{\rm Re}\,\beta$ and $0<\theta<1$. Here
$[\cdot,\cdot]_{\theta}$ denotes complex interpolation.

Proof of Proposition 4.1. Since
$(\mathbb{R}\times\Sigma)\times(\mathbb{R}\times\Sigma)$ is a complete
manifold, the operator $2mI-\Delta_{\tilde{h}}$ is positive and self-adjoint
(see Appendix 8.2). Using Theorem 4.3 we obtain for $0<\theta<1$

$\displaystyle
H^{2\theta}(\mathbb{R}^{2}\times\Sigma^{2})=\mathscr{D}((2mI-\Delta_{\tilde{h}})^{\theta})$
$\displaystyle=\\{u\in\mathcal{S}^{\prime}(\mathbb{R}^{2}\times\Sigma^{2});(2mI-\Delta_{\tilde{h}})^{\theta}u\in
L^{2}(\mathbb{R}^{2}\times\Sigma^{2})\\}.$

Since
$2mI-\Delta_{\tilde{h}}=2mI-\partial_{tt}-\partial_{ss}-\Delta_{h}{{}_{x}}-\Delta_{h}{{}_{y}}$
can be written as a multiplication operator in the form

$(2mI-\Delta_{\tilde{h}})u=\sum_{j,k}\phi_{j}(x)\phi_{k}(y)\int_{\mathbb{R}^{2}}e^{i(\xi_{0}t+\eta_{0}s)}\left(\xi_{0}^{2}+\eta_{0}^{2}+\lambda^{2}_{j}+\lambda_{k}^{2}\right)(\mathcal{F}u_{jk})(\xi_{0},\eta_{0})d\xi_{0}d\eta_{0},$

we infer from the orthonormality of the $\phi_{j}$ that
$(2mI-\Delta_{\tilde{h}})^{\theta}u$ in
$L^{2}(\mathbb{R}^{2}\times\Sigma^{2})$, if and only if

$\displaystyle\Big{\|}\sum_{j,k}\phi_{j}(x)\phi_{k}(y)\int_{\mathbb{R}^{2}}e^{i(\xi_{0}t+\eta_{0}s)}\left(\xi_{0}^{2}+\eta_{0}^{2}+\lambda^{2}_{j}+\lambda_{k}^{2}\right)^{\theta}({\cal{F}}u_{jk})(\xi_{0},\eta_{0})d\xi_{0}d\eta_{0}\Big{\|}^{2}_{L^{2}(\mathbb{R}^{2}\times\Sigma^{2})}$
$\displaystyle=$
$\displaystyle\sum_{j,k}\int_{\mathbb{R}^{2}}\left(\xi_{0}^{2}+\eta_{0}^{2}+\lambda^{2}_{j}+\lambda_{k}^{2}\right)^{2\theta}|({\mathcal{F}}u_{jk})(\xi_{0},\eta_{0})|^{2}d\xi_{0}d\eta_{0}<\infty.$

This establishes the required equivalence. $\Box$

######  Corollary 4.4.

For $-1\leq\theta\leq 0$ we obtain by $L^{2}$-duality that

$\displaystyle
H^{2\theta}(\mathbb{R}^{2}\times\Sigma^{2})=(H^{-2\theta}(\mathbb{R}^{2}\times\Sigma^{2}))^{\prime}$
$\displaystyle=\\{u\in\mathcal{S}^{\prime}(\mathbb{R}^{2}\times\Sigma^{2});\sum_{j,k}\int_{\mathbb{R}^{2}}(|\xi_{0}|^{2}+|\eta_{0}|^{2}+\lambda_{j}^{2}+\lambda_{k}^{2})^{2\theta}|\mathcal{F}u_{jk}(\xi_{0},\eta_{0})|^{2}d\xi_{0}d\eta_{0}<\infty\\},$

with $u_{j,k}=\langle
u,\phi_{j}\otimes\phi_{k}\rangle\in\mathcal{S}^{\prime}(\mathbb{R}^{2})$.

##  5 Support and Global Regularity of $K_{G}$

The following two lemmas contain the main results of this section. The first
lemma shows that only causally connected points belong to the support of
$K_{G}$. The second lemma establishes that $K_{G}\in
H_{loc}^{-\frac{1}{2}-\epsilon}(M\times M)$.

######  Lemma 5.1.

Let $(\tilde{x},\tilde{y})\in M\times M$ be such that $\tilde{x}$ and
$\tilde{y}$ are not causally related, i.e. $\tilde{x}\notin J(\tilde{y})$.
Then $(\tilde{x},\tilde{y})\notin\operatorname{supp}(K_{G})$.

###### Proof.

Since the support of $K_{G}$ is the complement of the largest open set where
$K_{G}$ vanishes, it is enough to show that there are open neighbourhoods $U$
of $\tilde{x}$ and $V$ of $\tilde{y}$ such that $K_{G}$ vanishes in $W=V\times
U$.

We construct the sets $U$ and $V$ as follows: Let
$\tilde{x}\in\\{t\\}\times\Sigma$ and $\tilde{y}\in\\{s\\}\times\Sigma$.
Without loss of generality we assume $t\leq s$. Since $M$ is globally
hyperbolic, $J(\tilde{y})\cap(\\{t\\}\times\Sigma)$ is compact. Therefore
there exists a compact neighbourhood $\tilde{V}\subset\\{t\\}\times\Sigma$
such that $\tilde{x}\in\tilde{V}$ and
$\tilde{V}\cap(J(\tilde{y})\cap(\\{t\\}\times\Sigma))=\emptyset$.

Similarly we can construct a compact neighbourhood $\tilde{U}$ such that
$\tilde{y}\in\tilde{U}$ and
$\tilde{U}\cap(J(\tilde{V})\cap(\\{s\\}\times\Sigma))=\emptyset$. Now we
consider the total domain of dependence of both sets i.e. $D(\tilde{U})$ and
$D(\tilde{V})$111Given a subset $S$ of $M$, the domain of dependence of $S$ is
the set of all points $p$ in $M$ such that every inextendible causal curve
through $p$ intersects $S$.. Notice that
$J(D(\tilde{V}))\cap\tilde{U}=\emptyset$ and $J(D(\tilde{U}))\cap
D(\tilde{V})=\emptyset$. Otherwise, we could construct a causal curve between
$\tilde{U}$ and $\tilde{V}$. We define $V:=\text{Int}D(\tilde{V})$ and
$U:=\text{Int}D(\tilde{U})$.

Now we show that $K_{G}$ vanishes in $W=V\times U$: Choose smooth functions
$\psi$ and $\phi$ with $\operatorname{supp}(\psi)\subset V$ and
$\operatorname{supp}(\phi)\subset U$. Then

$\displaystyle K_{G}(\psi\otimes\phi)$ $\displaystyle=\langle
G(\psi),\phi\rangle=\int_{M}G(\psi)\phi\sqrt{g}dx$
$\displaystyle=\int_{J(\operatorname{supp}(\psi))\cap\operatorname{supp}(\phi)}G(\psi)\phi\sqrt{g}dx$
$\displaystyle=\int_{J(V)\cap U}G(\psi)\phi\sqrt{g}dx=0.$

∎

######  Lemma 5.2.

$K_{G}\in H_{loc}^{-\frac{1}{2}-\epsilon}(M\times M)$ for every $\epsilon>0$.

###### Proof.

Let $\psi\in\mathcal{D}(M\times M)$. We will show $\psi K_{G}\in
H^{-\frac{1}{2}-\epsilon}(M\times M)$. Without loss of generality we take
$\psi(t,s,x,y)=\psi_{1}(t)\psi_{2}(s)$ with
$\psi_{1},\psi_{2}\in\mathcal{D}(\mathbb{R})$. According to Corollary 4.4

$\displaystyle\|\psi K_{G}\|^{2}_{H^{-\frac{1}{2}-\epsilon}(M\times M)}$
$\displaystyle=\sum_{j=k}\int_{\mathbb{R}^{2}}(|\xi_{0}|^{2}+|\eta_{0}|^{2}+\lambda_{j}^{2}+\lambda_{k}^{2})^{-\frac{1}{2}-\epsilon)}\Big{|}{\mathcal{F}_{(t,s)\to(\xi_{0},\eta_{0})}}\Big{(}\frac{\psi(t,s)}{\lambda_{j}}\sin{\lambda_{j}(t-s)}\Big{)}(\xi_{0},\eta_{0})\Big{|}^{2}d\xi_{0}d\eta_{0}.$

We have by direct computation that

$\displaystyle\Big{|}{\mathcal{F}_{(t,s)\to(\xi_{0},\eta_{0})}}\Big{(}\frac{\psi(t,s)}{\lambda_{j}}\sin{\lambda_{j}(t-s)}\Big{)}(\xi_{0},\eta_{0})\Big{|}^{2}$
$\displaystyle=\frac{1}{\lambda_{j}^{2}}\left|\frac{\psi_{1}(\xi_{0}-\lambda_{j})\psi_{2}(\eta_{0}+\lambda_{j})-\psi_{1}(\xi_{0}+\lambda_{j})\psi_{2}(\eta_{0}-\lambda_{j})}{2i}\right|^{2}$
$\displaystyle\leq\frac{1}{\lambda_{j}^{2}}\left(\frac{C}{\langle\xi_{0}-\lambda_{j}\rangle^{N}\langle\eta_{0}+\lambda_{j}\rangle^{N}}+\frac{C}{\langle\xi_{0}+\lambda_{j}\rangle^{N}\langle\eta_{0}-\lambda_{j}\rangle^{N}}\right)^{2}$
(5.1)

for all $N\in\mathbb{N}$ with a suitable constant $C$, since
$\psi_{1},\psi_{2}\in\mathcal{D}(M)$. This gives (with constants possibly
changing from line to line)

$\displaystyle\|\psi K_{G}\|_{H^{-\frac{1}{2}-\epsilon}(M\times M)}$
$\displaystyle=\sum_{j=k}\int_{\mathbb{R}^{2}}(|\xi_{0}|^{2}+|\eta_{0}|^{2}+\lambda_{j}^{2}+\lambda_{k}^{2})^{-\frac{1}{2}-\epsilon}\Big{|}{\mathcal{F}_{(t,s)\to(\xi_{0},\eta_{0})}}\Big{(}\frac{\psi(t,s)}{\lambda_{j}}\sin{\lambda_{j}(t-s)}\Big{)}(\xi_{0},\eta_{0})\Big{|}^{2}d\xi_{0}d\eta_{0}$
$\displaystyle\leq\sum_{j=k}\int_{\mathbb{R}^{2}}(\lambda_{j}^{2}+\lambda_{k}^{2})^{-\frac{1}{2}-\epsilon}\frac{1}{\lambda_{j}^{2}}\left(\frac{C}{(\langle\xi_{0}-\lambda_{j}\rangle\langle\eta_{0}+\lambda_{j}\rangle)^{N}}+\frac{C}{(\langle\xi_{0}+\lambda_{j}\rangle\langle\eta_{0}-\lambda_{j}\rangle)^{N}}\right)^{2}d\xi_{0}d\eta_{0}$
$\displaystyle\leq
C\sum_{j=k}\frac{(\lambda_{j}^{2}+\lambda_{k}^{2})^{-\frac{1}{2}-\epsilon}}{\lambda_{j}^{2}}$
$\displaystyle\leq C\sum_{j}\frac{1}{\lambda_{j}^{3+2\epsilon}}.$

Using Weyl’s law for non-smooth metrics [48, Theorem 1.1] we have the estimate
$l^{\frac{2}{3}}\leq C\lambda_{l}^{2}$ for a suitable constant $C$ which gives

$\displaystyle\|\psi{K}_{G}\|^{2}_{H^{-\frac{1}{2}-\epsilon}(M\times M)}$
$\displaystyle\leq\sum_{l}\frac{C}{\lambda_{l}^{3+2\epsilon}}\leq\sum_{l}\frac{C^{\prime}}{l^{1+\epsilon}}<\infty$

for a suitable constant $C^{\prime}$. ∎

It will be useful to consider the following bidistribution

######  Corollary 5.3.

Let $K_{A}\in\mathcal{D}^{\prime}(M\times M)$ be the bidistribution given by

$K_{A}(u\otimes v):=-\int_{M\times
M}\sum_{j}\lambda_{l}^{-2}\cos(\lambda_{l}(t-s))\phi_{l}(x)\phi_{l}(y)\sqrt{h(y)}\sqrt{h(x)}u(t,x)v(s,y)dsdydtdx$

Then,

$\displaystyle K_{A}\in H_{loc}^{\frac{1}{2}-\epsilon}(M\times M)\text{ for
every }\epsilon>0.$ (5.2)

###### Proof.

Chose $\psi$ as in the proof of Lemma 5.2. Direct computation shows that

$\displaystyle\Big{|}{\mathcal{F}_{(t,s)\to(\xi_{0},\eta_{0})}}\Big{(}\frac{\psi(t,s)}{\lambda_{j}}\cos({\lambda_{j}^{2}(t-s)})\Big{)}(\xi_{0},\eta_{0})\Big{|}^{2}$
$\displaystyle\leq\frac{1}{\lambda_{j}^{4}}\left(\frac{C}{\langle\xi_{0}-\lambda_{j}\rangle^{N}\langle\eta_{0}+\lambda_{j}\rangle^{N}}+\frac{C}{\langle\xi_{0}+\lambda_{j}\rangle^{N}\langle\eta_{0}-\lambda_{j}\rangle^{N}}\right)^{2}$

for all $N\in\mathbb{N}$ and a suitable constant $C$. This gives for
$\psi\in\mathcal{D}(M\times M)$

$\displaystyle\|\psi K_{A}\|_{H^{s}(M\times M)}$
$\displaystyle=\sum_{j=k}\int_{\mathbb{R}^{2}}(|\xi_{0}|^{2}+|\eta_{0}|^{2}+\lambda_{j}^{2}+\lambda_{k}^{2})^{s}\Big{|}{\mathcal{F}_{(t,s)\to(\xi_{0},\eta_{0})}}\Big{(}\frac{\psi(t,s)}{\lambda_{j}^{2}}\cos({\lambda_{j}(t-s)})\Big{)}(\xi_{0},\eta_{0})\Big{|}^{2}d\xi_{0}d\eta_{0}$
$\displaystyle\leq
C\sum_{j=k}\frac{(\lambda_{j}^{2}+\lambda_{k}^{2})^{s}}{\lambda_{j}^{4}}\leq
C^{\prime}\sum_{j}{\lambda_{j}^{2s-4}}$

for a suitable constant $C^{\prime}$.

According to Weyl’s law for non-smooth metrics [48, Theorem 1.1] we have the
estimate $l^{\frac{2}{3}}\leq C\lambda_{l}^{2}$ for a suitable constant $C$.
This gives for $s=\frac{1}{2}-\epsilon$

$\displaystyle\|\psi K_{A}\|^{2}_{H^{s}(M\times M)}$
$\displaystyle\leq\sum_{l}\frac{C^{\prime}}{\lambda_{l}^{3+2\epsilon}}\leq\sum_{l}\frac{C^{\prime\prime}}{l^{1+\epsilon}}<\infty.$
(5.3)

for a suitable constant $C^{\prime\prime}$. ∎

######  Remark 5.4.

Notice that $\partial_{t}K_{A}=K_{G}$.

##  6 The Main Theorems

In this section we prove the main results of the paper which are Theorem 6.3
and Theorem 6.5. First we show some helpful lemmas.

First, we establish the microlocal regularity of $K_{G}$ outside the set
$\operatorname{Char}(P)\times\operatorname{Char}(P)$.

######  Lemma 6.1.

For $\tau>2$ and any $\tilde{\epsilon}>0$

$WF^{-\frac{1}{2}-\tilde{\epsilon}+\tau}(K_{G})\subset\operatorname{Char}(P)\times\operatorname{Char}(P).$
(6.1)

###### Proof.

Since $K_{G}$ satisfies $(\partial_{t}+\partial_{s})K_{G}=0$ we conclude that
for all $l\in\mathbb{R}$

$WF^{l}(K_{G})\subset
WF(K_{G})\subset\operatorname{Char}(\partial_{t}+\partial_{s})=\\{(\tilde{x},\xi^{0},\xi,\tilde{y},\eta^{0},\eta)\in
T^{*}(M\times M)\backslash\\{0\\};\xi^{0}+\eta^{0}=0\\},$ (6.2)

where the second inclusion follows from the standard theory of
pseudodifferential operators.

Now we have
$P_{(t,x)}({\bf{x}},D_{\bf{x}})K_{A}=P_{(s,y)}({\bf{x}},D_{\bf{x}})K_{A}=0$.
Choose $\epsilon<\tilde{\epsilon}/2$. Since $K_{A}\in
H_{loc}^{\frac{1}{2}-\frac{\epsilon}{2}}(M\times
M)=H_{loc}^{(\frac{1}{2}-\epsilon)+\frac{\epsilon}{2}}(M\times M)$, an
application of Theorem 3.4 with $\sigma=-\frac{3}{2}+\tau-{\epsilon}<\tau-1$
shows that

$\displaystyle WF^{\frac{1}{2}+\tau-\tilde{\epsilon}}(K_{A})\subset$
$\displaystyle\operatorname{Char}(P_{(t,x)})\cap\operatorname{Char}(P_{(s,y)});$

here we assume without loss of generality that $\epsilon$ is so small that
$-\frac{3}{2}+\tau-{\epsilon}\geq 0$. Eq.(3.8) implies that

$\displaystyle WF^{\frac{1}{2}-\tilde{\epsilon}+\tau}(K_{A})\subset$
$\displaystyle(\operatorname{Char}(P)\times\operatorname{Char}(P))$
$\displaystyle\cup(\\{({\bf{x}},{\boldsymbol{\xi}})\in T^{*}(M\times
M)\backslash\\{0\\},\tilde{\xi}=0,(\tilde{y},\tilde{\eta})\in\operatorname{Char}(P)\\}$
$\displaystyle\cup\\{({\bf{x}},{\boldsymbol{\xi}})\in T^{*}(M\times
M)\backslash\\{0\\},(\tilde{x},\tilde{\xi})\in\operatorname{Char}(P),\tilde{\eta}=0\\}.$

If $\tilde{\eta}=0$, then $\eta_{0}=0$, and $\xi_{0}=0$ by Eq. (6.2). Since
$\operatorname{Char}P=\\{(\tilde{x},\tilde{\xi});(\xi_{0})^{2}={\sum_{i=1}^{3}}h^{ij}(x)\xi_{i}\xi_{j}\\}$
we then have $\tilde{\xi}=0$. Together with the corresponding argument for the
case $\tilde{\xi}=0$ this shows that

$WF^{\frac{1}{2}-\tilde{\epsilon}+\tau}(K_{A})\subset\operatorname{Char}(P)\times\operatorname{Char}(P);$
(6.3)

otherwise $0\in T^{*}(M\times M)$ will be in
$WF^{\frac{1}{2}-\tilde{\epsilon}+\tau}(K_{A})$.

Since $WF^{-\frac{1}{2}-\tilde{\epsilon}+\tau}(\partial_{t}K_{A})\subset
WF^{\frac{1}{2}-\tilde{\epsilon}+\tau}(K_{A})$ by [26, Proposition B.3], we
have

$WF^{-\frac{1}{2}-\tilde{\epsilon}+\tau}(K_{G})=WF^{-\frac{1}{2}-\tilde{\epsilon}+\tau}(\partial_{t}K_{A})\subset
WF^{\frac{1}{2}-\tilde{\epsilon}+\tau}(K_{A})\subset(\operatorname{Char}(P)\times\operatorname{Char}(P)).$

∎

Now we establish that points above the diagonal are of a specific form.

######  Lemma 6.2.

If $(\tilde{x},\tilde{\xi},\tilde{x},\tilde{\eta})\in
WF^{-\frac{3}{2}-\tilde{\epsilon}+\tau}(K_{G})$ for some $\tilde{\epsilon}>0$,
then $\tilde{\eta}=-\tilde{\xi}$.

###### Proof.

Note that
$(\tilde{x},\tilde{x})\in\text{sing}\operatorname{supp}(\partial_{t}K_{G})$
and therefore $(\tilde{x},\tilde{x})\in\text{sing}\operatorname{supp}(K_{G})$.
Suppose $\tilde{\eta}$ and $\tilde{\xi}$ are linearly independent, i.e.,
$\tilde{\eta}\neq\lambda\tilde{\xi}$ for $\lambda\in\mathbb{R}$. By Eq.(6.3)
$(\tilde{x},\tilde{x},\tilde{\xi},\tilde{\eta})\in\operatorname{Char}(P)\times\operatorname{Char}(P)$.
Now we choose a Cauchy hypersurface $\Sigma_{t_{0}}=\\{t_{0}\\}\times\Sigma$
such that the null geodesic with initial data $(\tilde{x},\tilde{\xi})$ and
the null geodesic with initial data $(\tilde{x},\tilde{\eta})$ intersect it.
These points of intersections are unique by global hyperbolicity. Moreover,
using the condition $\tilde{\eta}\neq\lambda\tilde{\xi}$, we can choose
$\Sigma_{t_{0}}$ such that these points are distinct. We denote these points
by $(t_{0},x_{0}),(t_{0},y_{0})$. Notice that these points are not causally
related.

Notice that $K_{A}$ satisfies $P_{(t,x)}K_{A}=0$, $P^{b}_{(t,x)}K_{A}\in
H^{-\frac{3}{2}-\epsilon+\tau}$ by taking into account Remark 3.5 for a choice
of $\delta$ close to $1$. This allows us to choose
$s=-\frac{3}{2}-\epsilon+\tau$, which is in the range $0<s<\tau-1$ in Theorem
3.6. This propagation of singularities result applied to the distribution
$K_{A}$ and the operator $P_{(t,x)}$ guarantees that if
$(x,x,\tilde{\xi},\tilde{\eta})\in
WF^{-\frac{3}{2}-\epsilon+\tau}(K_{G})\subset
WF^{-\frac{1}{2}-\epsilon+\tau}(K_{A})$ then the full null bicharacteristic is
contained in the wavefront set i.e.
$(\gamma(\tilde{x},\tilde{\xi}),(\tilde{x},\tilde{\eta}))\in{WF^{-\frac{1}{2}-\epsilon+\tau}(K_{A})}$,
where $\gamma(\tilde{x},\tilde{\xi})$ is the null bicharacteristic with
initial data $(\tilde{x},\tilde{\xi})$. Similarly, using the operator
$P_{(s,y)}$, we obtain
$(\gamma(\tilde{x},\tilde{\xi}),\gamma(\tilde{x},\tilde{\eta}))\in{WF^{-\frac{1}{2}-\epsilon+\tau}(K_{A})}$
where $\gamma(\tilde{x},\tilde{\eta})$ is the null bicharacteristic with
initial data $(\tilde{x},\tilde{\eta})$.

Now we show that this set of point is contained in
${WF^{-\frac{1}{2}-\epsilon+\tau}(K_{G})}$.

By Theorem 6.1.1’ from [17] we have

${WF^{-\frac{1}{2}-\epsilon+\tau}(K_{A})\backslash
WF^{-\frac{1}{2}-\epsilon+\tau}(K_{G})=WF^{-\frac{1}{2}-\epsilon+\tau}(K_{A})\backslash
WF^{-\frac{1}{2}-\epsilon+\tau}(\partial_{t}K_{A})}\subset\operatorname{Char}(\partial_{t})$
(6.4)

However, using Eq.(6.3) and that $(\partial_{t}+\partial_{s})K_{A}=0$, we have
the inclusion

$\displaystyle{WF^{-\frac{1}{2}-\epsilon+\tau}(K_{A})\subset
WF^{\frac{1}{2}-\epsilon+\tau}(K_{A})}\subset(\operatorname{Char}(P)\times\operatorname{Char}(P))\cap\operatorname{Char}(\partial_{t}+\partial_{s})$
(6.5)

Since
$(\operatorname{Char}(P)\times\operatorname{Char}(P))\cap\operatorname{Char}(\partial_{t}+\partial_{s})\cap\operatorname{Char}(\partial_{t})=\emptyset$,
taking the intersection between Eq.(6.4) and Eq.(6.5), we obtain that the left
hand side of Eq.(6.4) must be empty. Therefore,

${WF^{-\frac{1}{2}-\epsilon+\tau}(K_{A})\subset
WF^{-\frac{1}{2}-\epsilon+\tau}(K_{G}).}$ (6.6)

Hence,
$(\gamma(\tilde{x},\tilde{\xi}),\gamma(\tilde{x},\tilde{\eta}))\in{WF^{-\frac{1}{2}-\epsilon+\tau}}(K_{G})$.

In particular
$(t_{0},x_{0},t_{0},y_{0})\in\Pi({WF^{-\frac{1}{2}-\epsilon+\tau}}(K_{G}))$,
where $\Pi$ is the projection from $T^{*}(M\times M)$ to $M\times M$. However,
this is a contradiction to Proposition 5.1, since
$(t_{0},x_{0},t_{0},y_{0})\notin\operatorname{supp}(K_{G})$. Therefore,
$\tilde{\eta}=\lambda\tilde{\xi}$. Using Eq.(6.2) we have $\xi^{0}=-\eta^{0}$
which gives $\lambda=-1$ i.e. $\tilde{\eta}=-\tilde{\xi}$. ∎

In the above proof, we have used the distribution $K_{A}$, because a direct
application of Theorem 3.4 for $K_{G}$ is not possible, since for $\delta$
close to $1$ the above $\sigma$ cannot take the value $-\frac{1}{2}$.

Now we state one of the main results

######  Theorem 6.3.

Let $(M,g)$ be a $C^{\tau}$ ultrastatic spacetime with $\tau>2$ and $K_{G}$
the causal propagator. Then
$WF^{\prime-\frac{3}{2}-{\epsilon}+\tau}(K_{G})\subset C$ for every
${\epsilon}>0$ and $C$ as in Eq.(1.1).

###### Proof.

Let
$({\bf{x}},{\boldsymbol{\xi}})=(\tilde{x},\tilde{\xi},\tilde{y},-\tilde{\eta})\in
WF^{-\frac{3}{2}-{\epsilon}+\tau}(K_{G}){\subset
WF^{-\frac{1}{2}-\epsilon+\tau}(K_{A})}$, where the inclusion follows from
[26, Proposition B.3] since $K_{G}=\partial_{t}K_{A}$. The propagation of
singularities result (Theorem 3.6) implies that
$(\gamma(\tilde{x},\tilde{\xi}),\gamma(\tilde{y},-\tilde{\eta}))\in{WF^{-\frac{1}{2}-\epsilon+\tau}}(K_{A})$
where $\gamma(\tilde{x},\tilde{\xi})$ is the null bicharacteristic with
initial data $(\tilde{x},\tilde{\xi})$ and $\gamma(\tilde{y},-\tilde{\eta})$
is the null bicharacteristic with initial data $(\tilde{y},-\tilde{\eta})$.
Hence, by Eq.(6.6), we have
$(\gamma(\tilde{x},\tilde{\xi}),\gamma(\tilde{y},-\tilde{\eta}))\in{WF^{-\frac{1}{2}-\epsilon+\tau}}(K_{G})$.
Now we choose a Cauchy surface $\Sigma_{t_{1}}=\\{t_{1}\\}\times\Sigma$ and
notice that
$(t_{1},x_{1},{\xi}_{1},t_{1},x_{1},{\tilde{\eta}}_{1})=(\gamma(\tilde{x},\tilde{\xi}),\gamma(\tilde{y},\tilde{\eta}))\cap(\Sigma^{2})$
must satisfy Lemma 6.2 and therefore is of the form
$(t_{1},x_{1},{\tilde{\xi}}_{1},t_{1},x_{1},-{\tilde{\xi}}_{1}).$ Otherwise as
in the proof of this lemma, we could find
$(t_{1},x_{0},t_{1},y_{0})\in\Pi({WF^{-\frac{1}{2}-\epsilon+\tau}}(K_{G}))$
and $(t_{1},x_{0},t_{1},y_{0})\notin\operatorname{supp}(K_{G})$.

Furthermore, we define the null geodesic $\gamma:(-\infty,\infty)\rightarrow
M$ as follows

$\gamma(t)=\begin{cases}\Pi\gamma(\tilde{x},\tilde{\xi})(t)&t=(-\infty,t_{1})\\\
\Pi\gamma(\tilde{y},-\tilde{\eta})(-t)&t=(-t_{1},-\infty)\end{cases}$ (6.7)

where without loss of generality we assumed for $a<t_{1}<b$,
$\gamma(a)=\Pi\gamma(\tilde{x},\tilde{\xi})(a)=\tilde{x}$ and
$\gamma(b)=\Pi\gamma(\tilde{y},-\tilde{\eta})(b)=\tilde{y}$. Moreover
$g(\cdot,{\dot{\gamma}})|_{T_{\tilde{x}}M}=\tilde{\xi},g(\cdot,{\dot{\gamma}})|_{T_{\tilde{y}}M}=\tilde{\eta}$.

Therefore, there is a null geodesic between $\tilde{x}$ and $\tilde{y}$ with
cotangent vectors $\tilde{\xi}$ at $\tilde{x}$ and $\tilde{\eta}$ at
$\tilde{x}$ i.e. $({\bf{x}},{\boldsymbol{\xi}})\in
C^{\prime}:=\\{(\tilde{x},\tilde{\xi},\tilde{y},-\tilde{\eta});(\tilde{x},\xi;\tilde{y},\tilde{\eta})\in
C\\}$.

This shows

${WF^{-\frac{3}{2}-\epsilon+\tau}}(K_{G})\subset C^{\prime}$ (6.8)

Using the definition of
$WF^{l^{\prime}}(u):=\\{(\tilde{x},\tilde{\eta};\tilde{y},-\tilde{\eta})\in
T^{*}(M\times M);(\tilde{x},\tilde{\xi};\tilde{y},\tilde{\eta})\in
WF^{l}(u)\\}$ gives the result.

∎

For the analysis of adiabatic states it is enough to work with the inclusion
shown above. However, in the smooth case we have an equality of sets. In
Theorem 6.5, we show that this equality holds under stronger regularity
assumptions on the metric.

Before we show the following lemma

######  Lemma 6.4.

Let $(\tilde{x},\tilde{\xi})\in\operatorname{Char}(P)$ with $P$ as in Eq.
(2.5). Then $(\tilde{x},\tilde{x},\tilde{\xi},-\tilde{\xi})\in
WF^{\frac{3}{2}+{\epsilon}}(K_{G})$ for all ${\epsilon}>0.$

###### Proof.

Since $WF^{s_{1}}\subset WF^{s_{2}}$ for $s_{1}\leq s_{2}$, it is enough to
show the result for small $\epsilon$. Let $Q:=\mathbb{R}\times\Sigma^{2}$. We
define the embedding $f:Q\rightarrow M\times M$ by $f(s,x,y)=(s,x,s,y)$. The
set of normals of the map $f$ is

$\displaystyle N_{f}$ $\displaystyle=\\{(f(s,x,y),\tilde{\xi},\tilde{\eta})\in
T^{*}(M\times M);^{t}f^{\prime}(s,x,y)(\tilde{\xi},\tilde{\eta})=0\\}$
$\displaystyle=\\{(s,x,s,y,\xi^{0},0,-\xi^{0},0)\in T^{*}(M\times M)\\},$

where ${}^{t}f^{\prime}$ is the transpose of the differential of $f$. In
particular,
$N_{f}\cap(\operatorname{Char}P\times\operatorname{Char}P)=\emptyset$. By
Lemma 6.1

$WF^{\frac{1}{2}+\epsilon}(\partial_{t}K_{G})\cap N_{f}\subset
WF^{\frac{3}{2}+{\epsilon}}(K_{G})\cap N_{f}=\emptyset$

for suitably small $\epsilon>0$. Therefore Proposition B.7 from [26] implies
that the restriction of $\partial_{t}K_{G}$ to ${Q}$ is defined and satisfies

$\displaystyle WF^{\epsilon}(\partial_{t}K_{G}|_{Q})$ $\displaystyle\subset
f^{*}(WF^{{\frac{1}{2}+\epsilon}}(\partial_{t}K_{G}))$ (6.9)
$\displaystyle=\\{(s,x,y,^{t}f^{\prime}(\tilde{\xi},\tilde{\eta}))\in
T^{*}Q;(f(s,x,y),\tilde{\xi},\tilde{\eta})\in{WF^{\frac{1}{2}+\epsilon}}(\partial_{t}K_{G})\\}$

As a distribution, $\partial_{t}{K_{G}}|_{Q}$ is given by

$\partial_{t}K_{G}|_{Q}(s,x,y)=-\sum_{j}\phi_{j}(x)\phi_{j}(y),$

i.e., it acts on the density
$\psi_{1}(s)\psi_{2}(x)\psi_{3}(y)\sqrt{h(x)}\sqrt{h(y)}$, where
$\psi_{1}(s)\psi_{2}(x)\psi_{3}(y)$ are smooth functions of compact support,
by

$\langle\partial_{t}K_{G}|_{Q},\psi_{1}\psi_{2}\psi_{3}\rangle=-\int_{-\infty}^{\infty}\psi_{1}(p)dp\int_{\Sigma}\psi_{2}(w)\psi_{3}(w)\sqrt{h(w)}dw.$
(6.10)

Therefore its Fourier transform is given by

$({\cal{F}}(\partial_{t}K_{G}|_{Q}))({\chi},\xi,\eta)=\delta_{0}(\chi)\otimes\int_{\Sigma}e^{-iw(\xi+\eta)}\sqrt{h(w)}dw.$
(6.11)

Taking $\chi=0$ and $-\xi=\eta$ we see that
$({\cal{F}}(\partial_{t}K_{G}|_{Q}))(0,\xi,-\xi)$ does not decay and therefore
$(s,x,x,0,\xi,-\xi)\in WF^{\epsilon}(\partial_{t}K_{G}|_{Q})$.

Using Eq. (6.9) we find that $(s,x,s,x,\xi^{0},\xi,-\xi^{0},-\xi)\in
WF^{\frac{1}{2}+\epsilon}(\partial_{t}K_{G})$ for each $\xi\in T^{*}\Sigma$.

According to Proposition B.3 from [26]

$WF^{\frac{1}{2}+\epsilon}(\partial_{t}K_{G})\subset
WF^{\frac{3}{2}+\epsilon}(K_{G}).$ (6.12)

Since the wavefront set is contained in
$\operatorname{Char}(P)\times\operatorname{Char}(P)$ by Lemma 6.1 we obtain
that
$(s,x,s,x,\xi^{0},\xi,-\xi^{0},-\xi)\in\operatorname{Char}(P)\times\operatorname{Char}(P)$.
This gives the desired result. ∎

Now we show the equality of sets as in the smooth case.

######  Theorem 6.5.

Let $(M,g)$ be a $C^{\tau}$ ultrastatic spacetime with $\tau>3$ and $K_{G}$
the causal propagator. Then $C\subset
WF^{\prime-\frac{3}{2}+\tilde{\epsilon}}(K_{G})$ for all $\tilde{\epsilon}>0$
and $C$ as in Eq.(1.1).

###### Proof.

Under the additional regularity assumption we have
$P^{b}_{(t,x)}K_{A},P^{b}_{(s,y)}K_{A}\in H^{\frac{3}{2}+\tilde{\epsilon}}$
and therefore for $(\tilde{x},\tilde{\xi},\tilde{y},\tilde{\eta})\in
WF^{\frac{3}{2}+\tilde{\epsilon}}(K_{G})\subset
WF^{\frac{5}{2}+\tilde{\epsilon}}(K_{A})$ we can choose
$s=\frac{3}{2}+\tilde{\epsilon}$ in Theorem 3.6.

Now if
$({\bf{x}},{\boldsymbol{\xi}})=(\tilde{x},\tilde{\xi},\tilde{y},-\tilde{\eta})\in
C^{\prime}$ then there is a null geodesic $\gamma$ such that
$\gamma(t_{1})=\tilde{x},\gamma(t_{2})=\tilde{y}$ and
$g(\cdot,{\dot{\gamma}})|_{T_{\tilde{x}}M}=\tilde{\xi},g(\cdot,{\dot{\gamma}})|_{T_{\tilde{y}}M}=\tilde{\eta}$.
Now,$(\tilde{x},\tilde{\xi},\tilde{x},-\tilde{\xi})\in C^{\prime}$ and by
Lemma 6.4 $(\tilde{x},\tilde{\xi},\tilde{x},-\tilde{\xi})\in
WF^{-\frac{3}{2}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+\tau}-\tilde{\epsilon}}(K_{G})$.
Applying the Theorem 3.6 to $P_{(t,x)}K_{A},P_{(s,y)}K_{A}$ with the $s$ above
described we have
$(\gamma(\tilde{x},\tilde{\xi}),\gamma(\tilde{x},-\tilde{\xi}))\in
WF^{\frac{3}{2}+\tilde{\epsilon}}(K_{G})$. In particular this implies
$(\tilde{x},\tilde{\xi},\tilde{y},-\tilde{\eta})=({\bf{x}},{\boldsymbol{\xi}})\in
WF^{\frac{3}{2}+\tilde{\epsilon}}(K_{G})$. ∎

######  Remark 6.6.

The combination of Theorem 6.3 with Theorem 6.5 gives

$WF^{\prime-\frac{3}{2}+\tau-\tilde{\epsilon}}(K_{G})=C$

for $\tau>3$ and $\tilde{\epsilon}<\tau-3$.

##  7 The $C^{1,1}$ Case

The following theorem state the result for the case of $C^{1,1}$ regularity.

######  Theorem 7.1.

Let $(M,g)$ be a $C^{1,1}$ ultrastatic spacetime and $K_{G}$ the causal
propagator. Then $WF^{\prime\frac{1}{2}-\tilde{\epsilon}}(K_{G})\subset C$ for
all $\tilde{\epsilon}>0$.

Proof of Theorem 7.1. In order to show the theorem we will state how different
results of the paper change under this regularity.

From the comment above Theorem 3.6 we know that Theorem 3.6 still holds.

Also, notice that a $C^{1,1}$ metric guarantees the existence and uniqueness
of the Hamiltonian flow which is critical for the proof. Theorem 3.4 holds
even for $\tau>1$.

Lemma 5.1 requires no modification, since the results on global hyperbolicity
still hold for this regularity [39, Corollary 3.4]. The hypothesis in [48,
Theorem 1.1] is the requirement that the coefficients of the principal part
have one derivative that is Lipschitz which is clearly satisfied in the
$C^{1,1}$ case. Hence Lemma 5.2 holds.

For Lemma 6.2 and Theorem 6.3 the only thing to notice is that in this case
$P_{(t,x)}^{b}K_{A},P_{(s,y)}^{b}K_{A}\in H^{\frac{1}{2}-\tilde{\epsilon}}$
and therefore we can apply Theorem 3.6 for $s=\frac{1}{2}-\tilde{\epsilon}$. ∎

##  8 Appendix

###  8.1 Sobolev Spaces

$H^{s}(\mathbb{R}^{n}),\,s\in\mathbb{R}$, is the set of all tempered
distributions $u$ on $\mathbb{R}^{n}$ whose Fourier transforms ${\cal{F}}{u}$
are regular distributions satisfying

$\|u\|^{2}_{H^{s}(\mathbb{R}^{n})}:=\int\\!\langle\xi\rangle^{2s}|{\cal{F}}{u}(\xi)|^{2}\,d^{n}\xi<\infty.$

Let $(M,g)$ be a (possibly) non-compact Riemannian manifold which is
geodesically complete. The Laplace-Beltrami operator $-\Delta_{g}$ is
essentially selfadjoint if the regularity of the metric is $C^{\tau}$ for
$\tau\geq 2$, [41, 15]. For lower regularity see Appendix 8.2.

By $H^{s}(M)$ we denote the completion of $C_{0}^{\infty}(M)$ with respect to
the norm

$\|u\|_{H^{s}(M)}:=\|(I-\Delta_{g})^{s/2}u\|_{L^{2}(M)}.$

If $M$ is compact, $H^{s}(M)$ is independent of the metric.

For an open subset $U$ of $M$ we define the local Sobolev spaces:

$\displaystyle H^{s}_{loc}(U):=\\{u\in{\cal D}^{\prime}(U);\varphi u\in
H^{s}(M)\mbox{ for all }\varphi\in C^{\infty}_{0}(U)\\}.$

For $s\in 2\mathbb{N}_{0}\leq\tau$, the space $H^{2k}(M)$ is equivalent to the
usual Sobolev space defined in terms of local charts and a partition of unity.
We denote this for the moment by $\tilde{H}^{2k}(M)$. We show the equality of
both spaces for $k=1$ when the metric $g$ is of regularity $C^{1,1}$ and
$M=\mathbb{R}^{2}\times\Sigma^{2}$ with $\Sigma$ an $n$-dimensional compact
manifold.

######  Lemma 8.1.

$\tilde{H}^{2}(\mathbb{R}^{2}\times\Sigma^{2})={H}^{2}(\mathbb{R}^{2}\times\Sigma^{2})$.

###### Proof.

Let $u\in{\tilde{H}}^{2}(\mathbb{R}^{2}\times\Sigma^{2})$. In a coordinate
chart we have $\Delta
u=h^{ij}\partial_{ij}u+\frac{1}{\sqrt{h}}\partial_{j}u\partial_{i}(\sqrt{h}h^{ij})$.
From standard estimates we have $(I-\Delta)u\in
L^{2}(\mathbb{R}^{2}\times\Sigma^{2})$. Taking a sequence $u_{n}\rightarrow u$
in $\tilde{H}^{2}(\mathbb{R}^{2}\times\Sigma^{2})$ with $u_{n}\in
C_{c}^{\infty}(\mathbb{R}^{2}\times\Sigma^{2})$ gives the result.

Conversely, let $\\{u_{n}\\}\subset
C_{c}^{\infty}(\mathbb{R}^{2}\times\Sigma^{2})$ be a Cauchy sequence in
$H^{2}(\mathbb{R}^{2}\times\Sigma^{2})$ converging to $u$. By the inequality
$((I-\Delta)(u_{n}-u_{m}),(I-\Delta)(u_{n}-u_{m}))_{L^{2}}\geq((u_{n}-u_{m}),(u_{n}-u_{m}))_{L^{2}}$
we can conclude that $\\{u_{n}\\}$ is a Cauchy sequence in
$L^{2}(\mathbb{R}^{2}\times\Sigma^{2})$ and since
$H^{2}(\mathbb{R}^{2}\times\Sigma^{2})\subset
L^{2}(\mathbb{R}^{2}\times\Sigma^{2})$. This limit is $u$.

By [3, Theorem 1.1 ] with suitable modification for the case without boundary
(see [23, Section 5]) there is a continuous inverse
$(I-\Delta)^{-1}:L^{2}(\mathbb{R}^{2}\times\Sigma^{2})\rightarrow\tilde{H}^{2}(\mathbb{R}^{2}\times\Sigma^{2})$.
Here we have used the equivalence of Sobolev spaces in the smooth setting.
Since $(I-\Delta)u_{n}$ is a convergent sequence in
$L^{2}(\mathbb{R}^{2}\times\Sigma^{2})$, we have that $\\{u_{n}\\}$ is a
convergent sequence in $\tilde{H}^{2}(\mathbb{R}^{2}\times\Sigma^{2})$ which
must coincide with $u$. ∎

######  Remark 8.2.

Using the equality
$\tilde{H}^{2}(\mathbb{R}^{2}\times\Sigma^{2})={H}^{2}(\mathbb{R}^{2}\times\Sigma^{2})$
we have the equivalent characterisation

$H^{2}(\mathbb{R}^{2}\times\Sigma^{2})=\\{u\in\mathcal{S}^{\prime}(\mathbb{R}^{2}\times\Sigma^{2});(I-\partial_{tt}-\partial_{ss}-\Delta_{h{{}_{x}}}-\Delta_{h{{}_{y}}})u\in
L^{2}(\mathbb{R}^{2}\times\Sigma^{2})\\},$ (8.1)

###  8.2 Essential Self-adjointness of the Laplace-Beltrami Operator

######  Theorem 8.3.

Let $(\Sigma,h)$ be a smooth compact $n$-dimenisonal manifold equipped with a
Riemannian metric of regularity $C^{0}(\Sigma)\cap W^{1,n}(\Sigma)$. Then the
Laplace-Beltrami operator $\Delta_{h}$ is essentially self-adjoint.

We follow Strichartz’s article [41] that uses the following criterion [37,
Theorem X.1].

######  Lemma 8.4.

Let $A$ be any closed negative-definite symmetric, densely defined operator on
a Hilbert space $H$. Then $A=A^{*}$ if and only if there are no eigenvectors
with positive eigenvalue in the domain of $A^{*}$.

Now we will state the following helpful result

######  Proposition 8.5.

Let $u$ be an $L^{2}(\Sigma)$ function that satisfies $\Delta u=\lambda u$ for
some $\lambda>0$. Then $u$ is identically zero.

_Proof._ Let $u$ be a weak solution which by elliptic regularity satisfies
$u\in H^{2}({\Sigma})$ [22, Theorem 9.15]. Moreover, this regularity of $u$
combined with the regularity of $h$ is enough to apply the divergence theorem
[10, Proposition A.1].

Hence,

$\displaystyle\lambda\left(u,u\right)_{L^{2}(\Sigma)}=$ $\displaystyle(\Delta
u,u)_{L^{2}(\Sigma)}=-\left(du,du\right)_{L^{2}(\Sigma)}$ (8.2)

Now $\lambda\left(u,u\right)_{L^{2}(\Sigma)}\geq 0$ so we have $u=0$. ∎

_Proof of Lemma 8.3._ By direct computation $\Delta_{h}$ is negative-definite
and symmetric. That it is densely defined follows from the density of
$C^{\infty}(M)$ in $L^{2}(M)$ for continuous metrics (see [4, Proposition 7]
for even rougher cases). The application of Theorem 8.4 taking into account
Proposition 8.5 gives the result. ∎

For the non-compact case, one could follow the construction in Strichart’z
article. However, suitable modification are required under the regularity of
Theorem 8.3. For example, one would have to use an integral distance as in
[14] and [14, Theorem 5.11] to show the desired properties of the
approximations to unity. Then, one would need to verify that the elliptic
regularity results hold in that situaton as well. An alternative approach can
be found in [16, Lemma A.1].

Since we are only interested in the case of $M=\mathbb{R}^{2}\times\Sigma^{2}$
and the operator
$2mI-\partial_{tt}-\partial_{ss}-\Delta_{h_{x}}-\Delta_{h_{x}}$ under
$C^{1,1}$ regularity assumptions, we will proceed in a different manner.

######  Lemma 8.6.

The operator $2mI-\partial_{tt}-\partial_{ss}-\Delta_{h_{x}}-\Delta_{h_{x}}$
where $h_{x},h_{y}$ are Riemannian metrics of regularity $C^{1,1}$ is
essentially self-adjoint with domain $H^{2}(\mathbb{R}^{2}\times\Sigma^{2})$.

###### Proof.

By [38, Lemma 2.1] we obtain that
$-\partial_{tt}-\partial_{ss}-\Delta_{h_{x}}-\Delta_{h_{x}}$ is essentially
self-adjoint in in $L^{2}(\mathbb{R}^{2}\times\Sigma^{2})$ with domain
${\mathcal{D}}(-\partial_{tt})\otimes{\mathcal{D}}(-\partial_{ss})\otimes{\mathcal{D}}(-\Delta_{h_{x}}-\Delta_{h_{x}})=C^{\infty}_{c}(\mathbb{R})\otimes
C^{\infty}_{c}(\mathbb{R})\otimes C^{\infty}(\Sigma^{2})$. Since
$C^{\infty}_{c}(\mathbb{R})\otimes C^{\infty}_{c}(\mathbb{R})\otimes
C^{\infty}(\Sigma^{2})$ is dense in $H^{2}(\mathbb{R}^{2}\times\Sigma^{2})$
which carries the graph norm of
$-\partial_{tt}-\partial_{ss}-\Delta_{h_{x}}-\Delta_{h_{x}}$, we obtain that
the closure of the domain is $H^{2}(\mathbb{R}^{2}\times\Sigma^{2})$. Now
$-\partial_{tt}-\partial_{ss}-\Delta_{h_{x}}-\Delta_{h_{x}}$ and $2mI$ commute
and are self-adjoint. By [35, Lemma 4.16.1]
$2mI-\partial_{tt}-\partial_{ss}-\Delta_{h_{x}}-\Delta_{h_{x}}$ is self-
adjoint with domain $H^{2}(\mathbb{R}^{2}\times\Sigma^{2})$. ∎

Acknowledgements

We are grateful to Chris Fewster, Bernard Kay and James Vickers for helpful
discussions.

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Y. Sanchez Sanchez , Institut für Analysis, Leibniz Universität Hannover,
Welfengarten 1, 30167 Hannover, Germany

E-mail address<EMAIL_ADDRESS>
E. Schrohe, Institut für Analysis, Leibniz Universität Hannover, Welfengarten
1, 30167 Hannover, Germany

E-mail address<EMAIL_ADDRESS>


ab

# An Introduction to Map-making For CMB Experiments

Simon Biquard  Université Paris Cité, CNRS, Astroparticule et Cosmologie
F-75013 Paris, France

###### Abstract

The cosmic microwave background (CMB) anisotropies are a powerful probe of the
early universe, and have largely contributed to establishing the current
standard cosmological model. To extract the information encoded in those tiny
variations, one must first compress the raw, time-domain data collected by a
telescope into maps of the sky at the observed frequencies, in a procedure
known as map-making. I provide a general introduction to this problem, and
highlight a few specificities of the MAPPRAISER implementation.

##  1 Formalism

###  1.1 Data model

The time-ordered data (TOD) collected by the detectors is assumed to have been
calibrated and checked for quality during a pre-processing stage. It can then
be used to estimate maps of the sky signal observed at each frequency. This
_map-making_ operation represents a significant compression of the data
volume, from $\mathcal{O}(10^{12-14})$ time samples for modern CMB experiments
to typically $\mathcal{O}(10^{5-8})$ sky pixels depending on the particular
instrument and sky coverage. Ideally, this compression should not translate
into the loss of any cosmological information, which is why accurate modeling
of noise and systematics in the data is required.

The usual data model for the map-making problem links the vector of true sky
signal amplitudes $\mathbf{s}$ and the data vector $\mathbf{d}$ (a
concatenation of the TOD from all detectors) through the application of a
_pointing operator_ $\mathbf{P}$ (a tall and skinny matrix which encodes the
scanning pattern and orientation of the telescope):

$\mathbf{d}=\mathbf{P}\mathbf{s}+\mathbf{n}$ (1)

The time-domain vector $\mathbf{n}$ represents a stochastic contribution
(noise) that vanishes on average. A given row of the matrix $\mathbf{P}$
corresponds to the measurement of a pixel $p$ performed by one detector at a
given time $t$. Consequently, each row contains only a small number of non-
zero elements, which are determined by the telescope pointing information as
well as our knowledge of the instrument. In most applications, this number is
either 1 (for total intensity measurements) or 3 (for polarization-sensitive
measurements), which makes the pointing matrix typically very sparse and
computationally manageable. Below are two data model examples for a
polarization sensitive bolometer at an angle $\varphi$ with respect to the sky
coordinates:

$\displaystyle d_{t}$
$\displaystyle=I_{p_{t}}+\cos(2\varphi_{t})Q_{p_{t}}+\sin(2\varphi_{t})U_{p_{t}}+n_{t}$
(2) $\displaystyle d_{t}$
$\displaystyle=I_{p_{t}}+\cos(2\varphi_{t}+4\psi_{t})Q_{p_{t}}+\sin(2\varphi_{t}+4\psi_{t})U_{p_{t}}+n_{t}$
(3)

In the second case, an ideal half-wave plate (HWP) oriented with an angle
$\psi$ with respect to the instrument is used to modulate the incoming
polarization of the photons.

###  1.2 Sky signal estimators

Formally, map-making is then simply a _linear operation_ mapping the
measurements $\mathbf{d}$ to an estimate $\mathbf{m}$ of the sky amplitudes
with some operator $\mathbf{L}$:

$\mathbf{m}=\mathbf{L}\mathbf{d}$ (4)

Arguably the simplest possible choice is the _binning_ method,
$\mathbf{L}=\pab{\mathbf{P}^{\mathsf{T}}\mathbf{P}}^{-1}\mathbf{P}^{\mathsf{T}}$,
which corresponds to basic averaging of all measurements falling in each
pixel. This is an unbiased estimator which is also easy to compute, but is
generally very sub-optimal (noisy), because the correlated time-domain noise
is projected onto the sky without any prior weighting.

More generally, we see that requiring the estimator to be unbiased in general
constrains the map-making operator to be a left inverse of the pointing
matrix:

$\aab{\mathbf{m}}=\mathbf{L}\aab{\mathbf{d}}=\mathbf{L}\mathbf{P}\mathbf{s}+\underbrace{\mathbf{L}\aab{\mathbf{n}}}_{\text{vanishes}}=\mathbf{s}\quad\Longleftrightarrow\quad\mathbf{L}\mathbf{P}=\mathbf{I}$
(5)

A family of unbiased estimators is therefore obtained by taking

$\mathbf{L}=\pab{\mathbf{P}^{\mathsf{T}}\mathbf{W}\mathbf{P}}^{-1}\mathbf{P}^{\mathsf{T}}\mathbf{W}$
(6)

where the positive definite _weight matrix_ $\mathbf{W}$ determines the
resulting noise properties of the maps. A possible choice is
$\mathbf{W}=\mathbf{C}^{-1}_{\mathbf{n}}$, the inverse of the _noise
covariance matrix_. This corresponds to the generalized least squares (GLS)
solution of (1) and yields the best (i.e. lowest variance) linear unbiased
estimator of $\mathbf{s}$. This choice can be interpreted as _maximum
likelihood_ estimation with the prior that the statistics of the noise vector
are Gaussian.

Another possibility is to generalize the weight matrix using a set of temporal
modes, or _templates_ , which are to be de-projected from the data. If those
time-domain vectors are orthogonal and span the columns of a matrix
$\mathbf{T}$, then the _filtering and weighting_ operator
$\mathbf{F}_{\mathbf{T}}=\mathbf{W}\pab{\mathbf{I}-\mathbf{T}(\mathbf{T}^{\mathsf{T}}\mathbf{W}\mathbf{T})^{-1}\mathbf{T}^{\mathsf{T}}\mathbf{W}}$
effectively filters all those modes from the data and weights the remaining
ones by $\mathbf{W}$. That construction allows for unbiased time-domain
filtering [3].

Note, for completeness, that one may also compute a biased estimate of the
sky, for example of the form
$\mathbf{\tilde{m}}=\pab{\mathbf{P}^{\mathsf{T}}\mathbf{\Lambda}\mathbf{P}}^{-1}\mathbf{P}^{\mathsf{T}}\mathbf{F}\mathbf{d}$,
where $\mathbf{F}$ is a filtering operator and $\mathbf{\Lambda}$ a different
weight matrix (typically diagonal), which makes the computation much cheaper.
This approach is called _filter-and-bin_ map-making and has been successfully
used in many experiments, such as BICEP/Keck [1]. The bias must be corrected
for at a later stage of the analysis, typically by using an estimated transfer
function between the estimate and the true sky.

###  1.3 Computational aspects

Because of the size of the data involved, the map-making problem is typically
tackled using _iterative methods_. Indeed, direct methods that compute some
decomposition of the system matrix generally scale as the cube of the system
size, which makes them prohibitively expensive for large datasets. In other
words, the $\pab{\mathbf{P}^{\mathsf{T}}\mathbf{W}\mathbf{P}}^{-1}$ kernel of
(6), which can be a dense matrix, is never explicitly computed, and instead
the estimate is obtained by solving for $\mathbf{m}$ the map-making equation:

$\pab{\mathbf{P}^{\mathsf{T}}\mathbf{W}\mathbf{P}}\mathbf{m}=\mathbf{P}^{\mathsf{T}}\mathbf{W}\mathbf{d}$
(7)

in a way that only requires matrix-vector operations. The multiplication of
the system matrix $\mathbf{A}=\mathbf{P}^{\mathsf{T}}\mathbf{W}\mathbf{P}$
with a vector $\mathbf{x}$ is done from right to left in a series of three
products that can be performed efficiently.

Because the system matrix is symmetric and positive definite, conjugate
gradient techniques are appropriate for this task and have been widely used in
the map-making context. To speed up the convergence of the solver, a
_preconditioner_ is often employed: its role is to reduce the condition number
of the system. This approach is called the Preconditioned Conjugate Gradient
(PCG) method. A preconditioner $\mathbf{M}$ should approximate the inverse of
$\mathbf{A}$, such that solving the preconditioned system with matrix
$\mathbf{M}\mathbf{A}$ is faster than the original one. For the GLS solution
with $\mathbf{W}=\mathbf{C}^{-1}_{\mathbf{n}}$, a possible choice of
preconditioner which can be easily computed and stored in memory is the block-
diagonal matrix
$\mathbf{M}_{bd}=\pab{\mathbf{P}^{\mathsf{T}}\operatorname{diag}(\mathbf{C}^{-1}_{\mathbf{n}})\mathbf{P}}^{-1}$.

##  2 The MAPPRAISER library

In this section, I summarize the main features of the MAPPRAISER library, but
point the reader to the release paper [2] and references therein for a more
detailed and thorough description.

###  2.1 Overview

This software is built around two libraries written in C, available on GitHub.
The first one provides low-level operations such as sparse linear algebra for
pointing and template operators and custom communication schemes for
distributed data reduction. The second one builds on the first one and
implements different numerical techniques and map-making methods to compute
unbiased estimates of the sky signal. Both libraries use the MPI programming
model in order to operate on distributed computing systems and make use of a
large number of compute nodes. Finally, a Python wrapper allows the map-making
code to be used with the TOAST software package, a simulation and analysis
tool for time-ordered data.

The MAPPRAISER library implements two specific, optimized versions of the
operator $\mathbf{L}$ introduced in (6).

  1. 1.

the GLS solution $\mathbf{W}=\mathbf{C}^{-1}_{\mathbf{n}}$, assuming that the
noise is close to piece-wise stationary and uncorrelated between detectors:
the noise covariance matrix has a block-diagonal Toeplitz structure, which
allows for efficient matrix-vector operations using fast Fourier transforms.

  2. 2.

the _templates_ method, using the aforementioned filtering and weighting
operator
$\mathbf{F}_{\mathbf{T}}=\mathbf{W}\pab{\mathbf{I}-\mathbf{T}(\mathbf{T}^{\mathsf{T}}\mathbf{W}\mathbf{T})^{-1}\mathbf{T}^{\mathsf{T}}\mathbf{W}}$,
allowing for arbitrary time-domain templates but diagonal (uncorrelated) noise
weights.

###  2.2 Preconditioners

One of the flagship features of the MAPPRAISER library is the implementation
of two preconditioning options for the PCG solver. The first option is the
standard Block-Jacobi preconditioner $\mathbf{M}_{bd}$ introduced in
subsection 1.3, computed using the diagonal of the inverse noise covariance
matrix. It is block-diagonal, with one square block per sky pixel, whose size
is equal to the number of non-zero elements in a row of the pointing matrix.

The second option is the _two-level preconditioner_. The first level uses the
deflation technique and aims at removing a subspace containing the smallest
eigenvalues of the system matrix. The second level adds a correction to the
deflated matrix which shifts the eigenvalues of the subspace towards unity,
effectively reducing the condition number of the system matrix. The
preconditioner is formally defined as:

$\displaystyle\mathbf{M}_{2lvl}$
$\displaystyle=\overbrace{\mathbf{M}_{bd}\pab{\mathbf{I}-\mathbf{A}\mathbf{Q}}}^{\text{1st
level}}+\overbrace{\mathbf{Q}}^{\text{2nd level}}$ (8)
$\displaystyle\mathbf{Q}$
$\displaystyle=\mathbf{Z}\pab{\mathbf{Z}^{\mathsf{T}}\mathbf{A}\mathbf{Z}}^{-1}\mathbf{Z}^{\mathsf{T}}$
(9)

where $\mathbf{Z}$ is the deflation subspace matrix (its pixel-domain columns
span a subspace to be suppressed). The key element here is the construction of
$\mathbf{Z}$. In our tests, the most effective way of doing that has proven to
be the so-called _a posteriori_ construction, where $\mathbf{Z}$ is built by
estimating the relevant eigenpairs of the system matrix $\mathbf{A}$ with the
Lanczos algorithm. The performance of this preconditioner is illustrated in
Figure 1.

Figure 1: Relative reduction of residuals as a function of the iteration
number for the PCG solver (convergence is defined by reaching a $10^{-6}$
reduction factor). The Block-Jacobi preconditioner $\mathbf{M}_{bd}$ (in blue)
is compared to the a posteriori two-level preconditioner $\mathbf{M}_{2lvl}$
(other colors) for different dimensions of the deﬂation subspace. When
$\dim{\mathbf{Z}}=2048$, convergence is reached in one iteration. Figure taken
from the MAPPRAISER release paper.

##  3 Conclusion and future perspectives

The two-level preconditioning approach effectively solves the map-making
problem from a purely _computational_ point of view. Indeed, the
precomputation cost associated to a large deflation subspace can be
compensated by reusing it for solving many similar systems in a row (e.g. in
Monte Carlo simulations). While I expect this technique to be useful in the
context of increasing data volumes, it remains to be seen whether other
aspects of the problem such as I/O will not become limiting factors.

SB acknowledges funding from the European Research Council (ERC;
SCIPOL111https://scipol.in2p3.fr project) under the European Union’s Horizon
2020 research and innovation program (Grant agreement No. 101044073).

## References

## References

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  * [2] H. El Bouhargani et al, Astronomy and Computing 39, 100576 (2022).
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