# Microlocal Methods for The Elastic Travel Time Tomography Problem for
Transversely Isotropic Media

Yuzhou Zou  Mathematics Department
University of California, Santa Cruz
1156 High St, Santa Cruz, CA 95064, U.S.A<EMAIL_ADDRESS>
###### Abstract.

We study the travel time tomography problem for transversely isotropic media
using the artificial boundary method used in works in X-ray tomography and
boundary rigidity problems. The analysis involved requires a modification of
the scattering pseudodifferential calculus in order to provide parametrices
for operators which are degenerately elliptic; such a calculus is constructed
in this paper and used to solve the tomography problem of interest.

###### Contents

  1. 1 Introduction
  2. 2 Geometry of the blow-ups
  3. 3 The $bb$-symbol class
  4. 4 Operators quantized by the $bb$-symbol class
    1. 4.1 Oscillatory integral distributions and left reduction
    2. 4.2 Adjoints
    3. 4.3 Quantization: composition
  5. 5 Ellipticity of Operators from Elastic Inverse Problem
    1. 5.1 The localized operators
    2. 5.2 Scattering symbol
    3. 5.3 Symbol computation
  6. 6 The Local Recovery Problem

##  1\. Introduction

In this paper we consider the travel time tomography problem for transversely
isotropic media, using methods from microlocal analysis. The goal is to adapt
the arguments used in works in X-ray tomography [11, 7] and boundary rigidity
[8, 10] to this problem; such works made use of an “artificial boundary”
argument as well as extensive use of the scattering pseudodifferential
calculus. The works [9, 2] provide examples where such arguments were applied
to travel time tomography problems in elasticity. In particular, in the latter
work, the authors obtained recovery for some of the parameters involved, as
well as a description of the obstruction of the analysis to extend the
argument in recovering the remaining parameters: namely that some of the
operators involved were not elliptic considered as scattering
pseudodifferential operators, and hence were not known to have parametrices,
i.e. approximate inverses in a suitable sense. The main goal for this work is
to construct an operator calculus which provides parametrices to the operators
considered in [2], thus allowing the analogous arguments to be made in this
context.

The setting for this work is a combination of the settings in the previous
works, as well as the author’s previous work [13] which did not use the
artificial boundary argument of the above works. The essential aspects of this
setting are described below.

In elasticity, the linear elastic wave equation gives a linearized model for
the motion of elastic objects. The equation is given by

$u_{tt}=Eu,\quad(Eu)_{i}=\sum_{jkl}{\frac{1}{\rho(x)}\partial_{j}{c_{ijkl}(x)\partial_{l}u_{k}}},$

where $u:\mathbb{R}_{t}\times\mathbb{R}^{3}\to\mathbb{R}^{3}$ represents the
displacement of a material from a rest frame, $\rho(x)>0$ is the density of
the material, and $c_{ijkl}(x)$ are components of a tensor known as the
elasticity tensor. The principal symbol of the operator $u\mapsto u_{tt}-Eu$
is given by $-\tau^{2}\text{Id}+\sigma(-E)(x,\xi)$, where the matrix
$\sigma(-E)(x,\xi)$ is always symmetric and positive definite for all $x$ and
all $\xi\neq 0$, and hence in describing the microlocal wavefront set
$WF(u)\subset T^{*}(\mathbb{R}_{t}\times\mathbb{R}^{3}_{x})\backslash o$ of a
solution $u$, we have

$\displaystyle(t,x,\tau,\xi)\in WF(u)$
$\displaystyle\implies-\tau^{2}\text{Id}+\sigma(-E)(x,\xi)\text{ is not
invertible}$ $\displaystyle\iff\tau^{2}=G(x,\xi)\text{ where }G(x,\xi)\text{
is an eigenvalue of }\sigma(-E)(x,\xi).$

Under the further assumption that the multiplicity of the eigenvalues of
$\sigma(-E)$ remain constant in $(x,\xi)$, so that the eigenvalues can be
written as $G_{j}(x,\xi)$ for smooth functions $G_{j}$ on
$T^{*}\mathbb{R}^{3}$, it follows that $WF(u)\cap\\{\tau^{2}=G_{j}\\}$ is
invariant under the Hamilton flow of $\tau^{2}-G_{j}$. We are thus interested
in the dynamics of the so-called “elastic waves”, i.e. the Hamiltonian
dynamics of the eigenvalues $G_{j}$, the so-called “squared wave speeds” of
the material.

In the situation of transversely isotropic elasticity, the elasticity tensor
can be described in terms of an axis of isotropy, which we can represent
naturally as a line subbundle $\Sigma$ of the cotangent bundle
$T^{*}\mathbb{R}^{3}$, along with five material parameters. Following the
conventions of previous works, we denote these parameters as $a_{11}$,
$a_{33}$, $a_{55}$, $a_{66}$, and $E^{2}$. There are three kinds of waves,
denoted $qP$, $qSH$, and $qSV$, along with their squared wave speeds $G_{qP}$,
$G_{qSH}$, and $G_{qSV}$, and with respect to the parameters above, the
squared wave speeds can be written explicitly as

$G_{qSH}=a_{66}|\xi^{\prime}|^{2}+a_{55}\xi_{3}^{2}$

where $|\xi^{\prime}|^{2}=\xi_{1}^{2}+\xi_{2}^{2}$, and

$\displaystyle G_{\pm}$
$\displaystyle=(a_{11}+a_{55})|\xi^{\prime}|^{2}+(a_{33}+a_{55})\xi_{3}^{2}$
$\displaystyle\pm\sqrt{((a_{11}-a_{55})|\xi^{\prime}|^{2}+(a_{33}-a_{55})\xi_{3}^{2})^{2}-4E^{2}|\xi^{\prime}|^{2}\xi_{3}^{2}}$

where $+$ refers to the $qP$ wave speed and $-$ refers to the $qSV$ wave
speed. Here, we fix a point $x$ and choose coordinates
$(x_{1},x_{2},x_{3},\xi_{1},\xi_{2},\xi_{3})$ such that $dx_{3}$ aligns with
the axis of isotropy (i.e. $\text{span }(dx_{3})_{x}=\Sigma_{x}$), and
$(\xi_{1},\xi_{2},\xi_{3})$ is chosen with respect to $(x_{1},x_{2},x_{3})$ so
that overall the coordinates form canonical coordinates on
$T^{*}\mathbb{R}^{3}$, i.e. $(\xi_{1},\xi_{2},\xi_{3})$ corresponds to the
covector $\sum_{i=1}^{3}{\xi_{i}(dx_{i})_{x}}$.

The inverse problem we wish to investigate is the travel time tomography
problem. Consider a domain $\Omega\subset\mathbb{R}^{3}$, and suppose for each
pair of points on the boundary $\partial\Omega$, for each squared wave speed
$G$ there exists a unique Hamiltonian trajectory of $G$ (up to scaling) whose
projection to the physical space $\mathbb{R}^{3}_{x}$ connects the pair of
points. (Later on, it will suffice for this condition to hold for all
sufficiently close pair of points on the boundary). Suppose, for each pair of
points, we know the travel time of the trajectory associated to each wave
speed. Does that information suffice to determine the elasticity tensor of the
material?

We can formulate the main results of this paper which aim to answer the above
question. We assume that $\partial\Omega$ is convex with respect to all
Hamiltonian dynamics considered.

######  Theorem 1.1 (Local recovery of one parameter).

Let $p\in\partial\Omega$. Suppose that $\\{\tilde{a}_{\nu}\\}$ is another
collection of elastic parameters, such that $\tilde{a}_{\nu}=a_{\nu}$ for all
parameters except one, and that there exists a neighborhood $U$ of $p$ such
that the two collections of elastic parameters produces the same travel time
data between points on $U\cap\partial\Omega$. Then there is a (possibly
smaller) neighborhood $U^{\prime}$ such that $\tilde{a}_{\nu}=a_{\nu}$ on
$U^{\prime}$.

In other words, assuming just the convexity of $\partial\Omega$, we can
uniquely determine the elastic parameters in a neighborhood of
$\partial\Omega$ using the travel time data on $\partial\Omega$.

We will particularly be interested in the situation where our domain of
interest admits a _convex foliation_ , defined below:

######  Definition 1.1.

A manifold $\Omega$ with boundary $\partial\Omega$ admits a _convex foliation_
(with respect to a certain system of dynamics involved, such as geodesic
dynamics or more generally Hamiltonian dynamics associated to some
Hamiltonian) if there is a function $\rho:\overline{\Omega}\to[0,T]$ for some
$T>0$ such that $\rho^{-1}(0)=\partial\Omega$, $\rho^{-1}(t)$ is strictly
convex for $0\leq t<T$, and
$\overline{\Omega}\backslash\cup_{t\in[0,T)}{\rho^{-1}(t)}$ has measure zero.
The level sets $\rho^{-1}(t)$ for $0\leq t<T$ are called the _leaves_ of the
convex foliation.

By “strictly convex,” we mean that if $(X(s),\Xi(s))$ is a trajectory of the
relevant Hamiltonian dynamics (with $\frac{d}{ds}(X(s),\Xi(s))\neq 0$
everywhere), with $\rho(X(s_{0}))=t$ and
$\frac{d}{ds}|_{s=s_{0}}(\rho(X(s)))=0$ for some $0\leq t<T$, then
$\frac{d^{2}}{ds^{2}}|_{s=s_{0}}(-\rho(X(s)))>0$.

As mentioned above, the idea of a convex foliation was used in [11, 8, 10, 9,
2], etc., to upgrade local recovery results to global recovery results for
domains admitting a convex foliation. See [6] for a more detailed description
on how to interpret the convex foliation condition, as well as cases when
convex foliations must exist.

In the case where we have a convex foliation, we can extend our local recovery
result to a global recovery result.

######  Theorem 1.2 (Global recovery of one parameter).

Suppose $\Omega$ admits a convex foliation. Suppose that
$\\{\tilde{a}_{\nu}\\}$ is another collection of elastic parameters, such that
$\tilde{a}_{\nu}=a_{\nu}$ for all parameters except one, and the two
collections of elastic parameters produces the same travel time data between
every pair of points on $\partial\Omega$. Then we must have
$\tilde{a}_{\nu}=a_{\nu}$ everywhere in $\Omega$.

We expect the same recovery result to hold for the recovery of multiple
parameters (cf. the results in [2] and [13]); the computations are a bit more
complicated to write down, but the theory required should be the same as in
the one parameter recovery problem.

The main strategy is to take advantage of the Stefanov-Uhlmann
pseudolinearization argument, which offers a method to construct an operator
$I$ such that $I[\\{r_{\nu}\\}]=0$, where $r_{\nu}$ are the differences of the
parameters. The task is then to show the invertibility of the constructed
operator so as to show that $r_{\nu}=0$.

Explicitly, using the notation from [13], it was shown that if two sets of
parameters gave the same travel time data, then the difference
$r_{\nu}=\tilde{a}_{\nu}-a_{\nu}$ of the parameters must satisfy
$0=\sum{I^{\nu}(\nabla r_{\nu})}+\sum{\tilde{I}^{\nu}(r_{\nu})}$, where

(1.1)
$I^{\nu}[f_{1},f_{2},f_{3}](x,\xi)=-\int_{\mathbb{R}}{E^{\nu}(X(t),\Xi(t))\frac{\partial\tilde{\Xi}}{\partial\xi}(\tau(X(t),\Xi(t)),(X(t),\Xi(t)))\cdot\begin{pmatrix}f_{1}\\\
f_{2}\\\ f_{3}\end{pmatrix}(X(t))\,dt}$

and

(1.2)
$\tilde{I}^{\nu}[f](x,\xi)=\int_{\mathbb{R}}{\left(-\partial_{x}E^{\nu}\frac{\partial\tilde{\Xi}}{\partial\xi}(\tau(\cdot),\cdot)+\partial_{\xi}E^{\nu}\frac{\partial\tilde{\Xi}}{\partial
x}(\tau(\cdot),\cdot)\right)(X(t),\Xi(t))f(X(t))\,dt}.$

Here, $(x,\xi)$ are canonical coordinates on $T^{*}\mathbb{R}^{3}$, and
$(X(t),\Xi(t))=(X(t,x,\xi),\Xi(t,x,\xi))$ denotes the Hamilton flow with
respect to the wave speed $G(x,\xi)$. The function $E^{\nu}$ is defined in
[13] in terms of the two collection of parameters $\\{a_{\nu}\\}$ and
$\\{\tilde{a}_{\nu}\\}$; when $r_{\nu}$ is small one can think of $E^{\nu}$ as
approximately equal to $\frac{\partial G}{\partial\nu}$, i.e. the derivative
of the squared wave speed with respect to the parameter $\nu$ viewing $G$ as a
function of the parameters.

In [2], the authors obtained local recovery for the parameter $a_{11}$, which
combined with the existence of a convex foliation upgrades to a global
recovery result. The argument for the local recovery is a modification of the
local invertibility of the X-ray transform first studied in [11], where the
authors studied the invertibility of the geodesic X-ray transform (i.e.
showing that $u\equiv 0$ if $I[u]\equiv 0$, where $I$ is the geodesic X-ray
transform which takes a geodesic and a function and integrates the function on
the geodesic). The argument there follows as thus: assuming the convexity
assumptions, the authors showed that there exists a smooth function
$\tilde{x}$ such that $\tilde{x}(p)=0$, $d\tilde{x}\neq 0$ near $p$,
$\tilde{x}<0$ for $x\in\overline{\Omega}\backslash p$ near $p$, $\tilde{x}$ is
strictly convex with respect to all relevant dynamics, and for sufficiently
small $c>0$ we have
$\overline{\Omega_{c}}:=\\{\tilde{x}>-c\\}\cap\overline{\Omega}$ is compact.
(Indeed, if $\rho$ is a boundary defining function of $\partial{\Omega}$ near
$p$, with $\rho>0$ in the interior, so that $-\rho$ is strictly convex with
respect to all dynamics considered near $p$, then we can take $\tilde{x}$ to
be a slightly less convex modification of $-\rho$.) Near $p$, we view
$\overline{\Omega}$ as a closed subset of a manifold without boundary. We can
then fix $c>0$ sufficiently small, and let $x=\tilde{x}+c$, so that
$X=\\{x\geq 0\\}$ corresponds to $\\{\tilde{x}\geq-c\\}$. Note then that
$\\{x=0\\}$, equivalently $\\{\tilde{x}=-c\\}$, serves as the boundary for the
newly constructed space $X$; this boundary is what we refer to as an
“artificial boundary” (which itself depends on $c$). The situation is
illustrated in Figure 1.

Figure 1. Sketches of the domains. In the left figure, the blue curves are the
level sets of $\rho$, a boundary-defining function near $p$. The thick blue
curve denotes $\partial\Omega=\\{\rho=0\\}$. The red curves are the level sets
of $\tilde{x}$, a “less convex” modification of $-\rho$ (but still convex with
respect to the relevant dynamics). On the right, the thick red curve denotes
$\\{\tilde{x}+c=0\\}=\\{x_{c}=0\\}$ for some choice of $c>0$; this set denotes
the “artificial boundary,” with the original boundary (the thick blue curve)
constraining the support of the difference of parameters (but otherwise not
playing the role of a “boundary” in an analytic sense).

We can then choose two more smooth functions $y_{1},y_{2}$ so that
$(x,y_{1},y_{2})$ give coordinates in a neighborhood of
$\overline{\Omega_{c}}$. We then let

$Lv(z)=x^{-2}(z)\int_{S_{z}X}{\chi(\lambda(\nu)/x(z))v(\gamma_{\nu,z})\,d\nu}$

where $S_{z}X$ is the sphere bundle of $X$, and for a unit tangent vector
$\nu$ we let $\lambda(\nu)$ be the coefficient $\lambda$ appearing in writing
$\nu=\lambda\partial_{x}+\omega\cdot\partial_{y}$, and $\gamma_{\nu,z}$ is a
trajectory passing through $z$ with tangent vector $\nu$, and $\chi\in
C_{c}^{\infty}(\mathbb{R})$. Note that by convexity assumptions we can arrange
$\chi$ to be supported sufficiently close to $0$ such that trajectories $\nu$
with $\lambda/x\in\text{supp }\chi$ remain in $X$ for all times. Furthermore,
note the definition of $L$ above depends on the choice of $c>0$. The authors
then showed that for $\digamma>0$ and

$N_{\digamma}:=e^{-\digamma/x}LIe^{\digamma/x}$

we have that $N_{\digamma}$ is a _scattering pseudodifferential operator_ on
$X$, and furthermore that it is elliptic. (See [5] and [12] for an
introduction to the scattering pseudodifferential calculus and its
properties.) In particular, there exists a parametrix $G$ such that
$GN_{\digamma}=\text{Id}+E$, where $E\in\Psi^{-\infty,-\infty}$ is a “very
nice error.” The idea is that if $I[u]=0$, and we write
$u=e^{\digamma/x}\tilde{u}$, then $0=N_{\digamma}[\tilde{u}]$, and hence
applying a parametrix yields $\tilde{u}=E[\tilde{u}]$; if $E$ is sufficiently
nice (e.g. small enough in operator norm) then this allows us to conclude
$\tilde{u}=0$.

To formalize this, we now fix $\digamma>0$, but we allow $c$ to vary, and in
particular we consider what happens as $c\to 0$. We now write $N_{c}$ to
denote the operator $N_{\digamma}$ constructed for the given value of $c$, and
we let $E_{c}$ be the corresponding error. Then $N_{c}$ is a continuous
(w.r.t. the $\Psi^{-1,0}$ topology) family of scattering $\Psi$DOs, and
$E_{c}$ is a family of scattering $\Psi$DO continuous w.r.t. any
$\Psi^{-N,-N}$ topology. We also let
$\tilde{u}_{c}=e^{-\digamma/(\tilde{x}+c)}u$, where we interpret
$\tilde{u}_{c}$ as an element of
$L^{2}(\\{\tilde{x}>-c\\},\frac{d\tilde{x}\,dy}{(\tilde{x}+c)^{n+1}})$; then
$\tilde{u}_{c}$ is continuous in $c$ after identifying $\\{\tilde{x}>-c\\}$
with $[0,\infty)\times\mathbb{R}^{2}$ via the map
$(\tilde{x},y_{1},y_{2})\mapsto(\tilde{x}+c,y_{1},y_{2})$. We then have
$\tilde{u}_{c}=E_{c}[\tilde{u}_{c}]$. We now let $\phi_{c}$ satisfy
$0\leq\phi_{c}\leq 1$, $\phi_{c}\equiv 1$ on $\\{x<c\\}$, and $\text{supp
}\phi_{c}\subset\\{x<2c\\}$. Then
$\epsilon(c):=\|\phi_{c}E_{c}\phi_{c}\|_{L^{2}\to L^{2}}\to 0$, and since
$\phi_{c}\tilde{u}_{c}=\tilde{u}_{c}$, we have

$0=\tilde{u}_{c}+\phi_{c}E_{c}\phi_{c}\tilde{u}_{c}\implies\|\tilde{u}_{c}\|_{L^{2}}\leq\epsilon(c)\|\tilde{u}_{c}\|_{L^{2}},$

and hence $\tilde{u}_{c}\equiv 0$ for all sufficiently small $c>0$.

For the elastic tomography problem (and also boundary rigidity) problem, we
perform a similar argument, except that there are two operators of interest
$0=N[\nabla u]+\tilde{N}[u]$ (where $\nabla u=(\partial_{x}u,\partial_{y}u)$
are the “usual” derivatives with respect to the coordinates $(x,y)$, i.e. not
the scattering derivatives which have additional factors of $x^{2}$ and $x$).
Similar analysis would then show that $N$ admits an elliptic parametrix, with
the remaining term controlled via a Poincaré inequality argument.

In [2], the authors argued that one can recover $a_{11}$ from the travel time
data, assuming the other parameters are known, in large part because the
associated operator is elliptic. For the other parameters $a_{33}$ and
$E^{2}$, the associated operators were not elliptic.

In [13], an analogous problem was studied globally (i.e. without the
“artificial boundary” method of reducing to a local result). In that paper,
the analogous operators for $a_{33}$ and $E^{2}$ were not elliptic but did
resemble parabolic operators in an appropriate sense, and hence they admit a
parametrix in a calculus first studied by Boutet de Monvel [1].

We thus follow the geometric setup of [11, 2], etc., i.e. we will use the
“artificial boundary” method to reduce the problem to a local recovery
argument. Thus, we fix a point $p$, let $\tilde{x}$ satisfy the properties
described above, and work in the regime $c\to 0$. We analyze the operators
involved for fixed but sufficiently small $c>0$ (in particular we may take it
small enough to satisfy dynamical assumptions). We show that for fixed $c$ the
operators involved admit parametrices. Then in the inversion argument we take
$c\to 0$ to obtain a local recovery argument. Once the local recovery argument
is established, the process of obtaining global recovery is exactly the same
as in the previous papers.

The particular operators of interest are $N=L\circ I$, where we take our
formal adjoint $L$ to have the form

(1.3)
$Lv(x,y)=x^{-2}\int_{\mathbb{S}^{2}}{\chi(x^{\epsilon}\hat{\lambda})\exp\left(-\frac{\digamma^{2}\hat{\lambda}^{2}}{2\alpha}\right)v(\gamma_{\lambda,\omega})\,d\mathbb{S}^{2}(\lambda,\omega)}.$

where $0<\epsilon<1/2$ and $I=I^{\nu}$ as defined in (1.1) for some choice of
material parameter $\nu$. Here, $\gamma_{\lambda,\omega}$ is the trajectory
(passing through $(x,y)$) with tangent vector
$\lambda\partial_{x}+\omega\cdot\partial_{y}$, $\hat{\lambda}=\lambda/x$, and
$\alpha$ satisfies

$\gamma_{\lambda,\omega}-x=\lambda t+\alpha t^{2}+O(t^{3}).$

Note that if we let $\hat{t}=t/x$, we can rewrite the above equation as

$\frac{\gamma_{\lambda,\omega}-x}{x^{2}}=\hat{\lambda}\hat{t}+\alpha\hat{t}^{2}+O(x\hat{t}^{3}).$

We note that our choice of formal adjoint is slightly different than those in
previous works: namely, the previous works used a compactly supported cutoff
which is sufficiently close to a particular Gaussian; here we use a cutoff
which at the boundary is _equal_ to the Gaussian of reference, in order to
simplify computations at the boundary. As before, let
$N_{\digamma}=e^{-\digamma/x}Ne^{\digamma/x}$ for some fixed $\digamma>0$.

To show that $N_{\digamma}$ admits a parametrix, we will need to study the
inversion of scattering pseudodifferential operators whose symbols are not
quite elliptic, but degenerate in a certain way. We study the geometry
associated to the symbols’ degeneracy, as follows:

Let $M=(0,\infty)_{x}\times\mathbb{R}_{y}^{2}$, and consider the _scattering
cotangent bundle_ ${}^{sc}T^{*}M$. We can view this as the dual bundle of the
scattering tangent bundle, whose sections form the space of scattering vector
fields
$\mathcal{V}_{sc}=x\mathcal{V}_{b}=\text{span}_{C^{\infty}(M)}(x^{2}\partial_{x},\partial_{y_{1}},x\partial_{y_{2}})$;
alternatively we can view the bundle as spanned by the sections
$\frac{dx}{x^{2}}$ and $\frac{dy_{i}}{x}$, $i=1,2$; see [5] and [12] for more
details. Let $\mathscr{X}=\overline{{}^{sc}T^{*}M}$ be the compactification of
${}^{sc}T^{*}M$; this is the natural space on which symbols of scattering
pseudodifferential operators live. Note that $\mathscr{X}$ is a manifold with
corners, as it has two boundary hypersurfaces, namely a “fiber infinity”
obtained by radially compactifying each fiber, and a “base infinity111The term
“base infinity” corresponds to the identification of $M$ as a subset of the
upper half-sphere $\mathbb{S}_{+}^{3}$ and subsequently identifying its
interior with $\mathbb{R}^{3}$ via radial compactification; such an
identification sends a neighborhood of $\\{x=0\\}$ to a “neighborhood of
infinity” in $\mathbb{R}^{3}$.” corresponding to fibers over the boundary
$\\{x=0\\}$. Let $S\subset\mathscr{X}$ be the fiberwise span of the covector
field $\overline{\xi}$, and let $\Sigma$ be the intersection of $S$ with fiber
infinity. If we choose (local) coordinates $(x,y_{1},y_{2})$ such that $x$ is
a coordinate associated to the convex foliation and $y_{1}$ is a coordinate
associated to foliation generated by the one-form $\overline{\xi}$, and let
$(x,y,\xi,\eta)$ be the corresponding coordinates on the scattering cotangent
bundle (so that $(x,y,\xi,\eta)$ is associated to the element
$\left(\xi\frac{dx}{x^{2}}+\eta\cdot\frac{dy}{x}\right)|_{x,y}$), then

$S=\text{span }\frac{dy_{1}}{x}=\\{\xi=0,\eta_{2}=0\\}$

and

$\Sigma=\\{\xi=0,\eta_{2}=0,\rho=0\\}\quad\text{where}\quad\rho=\langle(\xi,\eta)\rangle^{-1}.$

If we let $\tau=(\rho\xi,\rho\eta_{2})$, then $(x,y,\rho,\tau)$ give local
coordinates on $\mathscr{X}$ near $\Sigma$, in which case we can write
$S=\\{\tau=0\\}$ and $\Sigma=\\{\rho=0,\tau=0\\}$.

In [2], it was shown that $N_{\digamma}\in\Psi^{-1}_{sc}$, and that the
principal symbol $\sigma(N_{\digamma})$ is elliptic at fiber infinity except
on $\Sigma$, and that it is elliptic at all finite points as well. In [13], it
was shown that $\sigma(N_{\digamma})$ has a subprincipal term which is purely
imaginary and is non-degenerate away from the boundary (i.e. in the region
$\\{x>0\\}$), but that it degenerates as a power of $x$ relative to the
principal symbol as $x\to 0$. This suggests that near fiber infinity (i.e.
near $\rho=0$) we can write

(1.4) $\sigma(N_{\digamma})=a_{-1}\rho+ixa_{-2}\rho^{2}+a_{-3}\rho^{3}$

where $a_{-1}$ and $a_{-2}$ are homogeneous of degree $0$ in the fiber
coordinates, $a_{-1}$ is positive except on $\Sigma$ where it degenerates
quadratically, $a_{-2}$ is nonzero near $\Sigma$, and $a_{-3}\in S^{0,0}$ is
positive near $\Sigma\cap\\{x=0\\}$. Note that we can write

$a_{-1}(x,y,\tau)=\langle q(x,y,\tau)\tau,\tau\rangle$

for some matrix $q$ which is positive definite near $\tau=0$.

Note that away from $x=0$, the symbol is non-elliptic on $\Sigma$ but has a
non-degenerate subprincipal part which is purely imaginary (relative to the
real principal symbol), and hence its inversion theory can be covered by a
calculus first studied by Boutet de Monvel [1]; see [13] for more details. One
way to understand this symbol calculus is to start with the fiber-compactified
cotangent bundle, blow up $\Sigma$ parabolically (thus creating an additonal
face), and consider symbols which are conormal to this blown-up space.
Explicitly, we let

$d_{\Sigma}=(|\tau|^{2}+\rho)^{1/2},$

and we blow up $\Sigma$ parabolically, so that $|\tau|^{2}$ and $\rho$ have
the same “order” in the blowup, to obtain
$\mathscr{X}_{1}=[\mathscr{X}:\Sigma]$, with $d_{\Sigma}$ the boundary
defining function of the new front face. Let
$\beta_{1}:\mathscr{X}_{1}\to\mathscr{X}$ be the blow-down map.

Figure 2. The first blow-up.

If we let $\tilde{\tau}=\tau/d_{\Sigma}$ (note that this is a smooth function
on the blow-up), then $\rho=d_{\Sigma}^{2}(1-|\tilde{\tau}|^{2})$, and (1.4)
becomes

(1.5) $a=\rho d_{\Sigma}^{2}\left(\langle
q(x,y,\tau)\tilde{\tau},\tilde{\tau}\rangle+ixa_{-2}(1-|\tilde{\tau}|^{2})+a_{-3}d_{\Sigma}^{2}(1-|\tilde{\tau}|^{2})^{2}\right).$

Note that, away from $x=0$, and near the new front face $\\{d_{\Sigma}=0\\}$,
the postfactor after $\rho d_{\Sigma}^{2}$ is uniformly nonzero. Indeed, it
would suffice to show the sum of the first two terms is uniformly nonzero, but
this follows since the first term is purely real, while the second term is
purely imaginary, and neither term vanishes simultaneously. It follows that
the symbol is now elliptic with respect to the blown-up space, and one can
apply the calculus in [1].

Near $x=0$, the subprincipal symbol degenerates, and thus the postfactor in
(1.5) vanishes at

$\Gamma:=\\{x=0,d_{\Sigma}=0,\tilde{\tau}=0\\}.$

Notice that $\Gamma$ is the lift of $S$ intersected with $x=0$ and
$d_{\Sigma}=0$. In particular, it is a sum of terms which vanish as
$O(|\tilde{\tau}|^{2})$, $O(x)$, and $O(d_{\Sigma}^{2})$. As such, we perform
a second blow-up by defining

$d_{\Gamma}=(x+d_{\Sigma}^{2}+|\tilde{\tau}|^{2})^{1/2}$

and letting $\mathscr{X}_{2}=[\mathscr{X}_{1}:\Gamma]$ be the corresponding
blow-up where $x^{1/2}$, $d_{\Sigma}$, and $\tilde{\tau}$ have the same degree
of homogeneity, with $d_{\Gamma}$ the new boundary-defining function. Let
$\beta_{2}:\mathscr{X}_{2}\to\mathscr{X}_{1}$ be the corresponding blow-down
map, and let $\beta:=\beta_{1}\circ\beta_{2}:\mathscr{X}_{2}\to\mathscr{X}$ be
the overall blow-down map from $\mathscr{X}_{2}$ to $\mathscr{X}$.

Figure 3. The second blow-up.

Having defined these blow-ups, we can consider a symbol calculus of symbols
conormal to the boundary of $\mathscr{X}_{2}$, and consider an operator
algebra consisting of quantizations of such symbols. This class of operators
turns out to be the right class of operators in which one can find a
parametrix for $N_{\digamma}$, allowing one to apply the arguments above to
prove Theorem 1.1, and hence Theorem 1.2.

In Section 2, we study the geometry of the blow-ups, and in particular for
each space we study the space of smooth vector fields tangent to its boundary.
In Section 3, we study properties of the calculus of symbols conormal to
$\partial\mathscr{X}_{2}$. In Section 4, we study the corresponding operator
algebra, proving that the symbol calculus and operator algebra shares many
desirable properties present in the standard scattering calculus. In Section
5, we analyze the symbol of $N_{\digamma}$ to show that it is elliptic
considered as an element of the operator algebra defined in Section 4; in
particular we show that $N_{\digamma}$ is still a scattering
pseudodifferential operator even though we adjusted our formal adjoint from a
compactly supported weight to a Gaussian weight. Finally, we apply the
machinery constructed to solve the inverse problem in Section 6.

######  Acknowledgments.

The author would like to thank András Vasy for helpful comments in this work.
The author gratefully acknowledges partial support from the National Science
Foundation under grant number DMS-1664683. Much of this article was written as
part of the author’s Ph.D. thesis at Stanford University.

##  2\. Geometry of the blow-ups

Recall that, given a smooth map $f:X_{1}\to X_{2}$ between two manifolds
$X_{1}$ and $X_{2}$, and vector fields $V_{i}$ on $X_{i}$, that $V_{1}$ and
$V_{2}$ are _$f$ -related_ if $f_{*}\circ V_{1}=V_{2}\circ f$, i.e. for all
$p\in X_{1}$ we have

$Df_{p}(V_{1})_{p}=(V_{2})_{f(p)}.$

If $f$ is surjective and $V_{1}$ and $V_{2}$ are $f$-related, we will say that
$V_{2}$ _lifts_ via $f$ to $V_{1}$, and we will write $V_{2}=f_{*}V_{1}$. We
will want to study under what conditions a vector field on $\mathscr{X}$ or
$\mathscr{X}_{1}$ lifts, via $\beta_{1}$ or $\beta_{2}$ or $\beta$, to a
smooth vector field on $\mathscr{X}_{1}$ or $\mathscr{X}_{2}$. Note that for
$f=\beta_{1}$, $\beta_{2}$, or $\beta$, that $f$ is a diffeomorphism of the
interiors of its domain and range, and hence if a continuous vector field
lifts to a continuous vector field, then such a lift is determined on a dense
subset and hence must be unique; as such the base vector field can in some
sense be identified with its lift.

To analyze vector fields tangent to the boundary of some manifold (for
example, obtained through a blow-up), we use the following observation,
formalized as a lemma:

######  Lemma 2.1.

Suppose $X$ is a manifold with corners, and let $X^{\circ}$ denote its
interior. Suppose $\\{y_{1},\dots,y_{N}\\}$ is a collection of functions in
$C^{\infty}(X)$ such that, for any $p\in\partial X$, there is a neighborhood
$U\ni p$ and a subset of functions $\\{y_{i_{1}},\dots,y_{i_{n}}\\}$ such that
$(U,(y_{i_{1}},\dots,y_{i_{n}}))$ forms a coordinate chart of $X$ near $p$.
Suppose $V$ is a $C^{\infty}(X^{\circ})$ vector field on $X^{\circ}$ such that
for every $1\leq i\leq N$, we have that $V(y_{i})$, initially defined as a
function on $X^{\circ}$, extends smoothly to a $C^{\infty}(X)$ function on all
of $X$. Then $V$ extends to a $C^{\infty}(X)$ vector field on all of $X$.

###### Proof.

It suffices to show, for every $p\in\partial X$, that $V$ is a $C^{\infty}(U)$
vector field for some neighborhood $U$ of $p$. Choosing
$(U,(y_{i_{1}},\dots,y_{i_{n}}))$ to be a coordinate chart, as is possible
from the hypothesis, and letting
$\\{\partial_{y_{i_{1}}},\dots,\partial_{y_{i_{n}}}\\}$ be the corresponding
partial derivative vector fields on $U$, we have that

$V=\sum_{j=1}^{n}{V(y_{i_{j}})\partial_{y_{i_{j}}}}\quad\text{in }U.$

Since $V(y_{i_{j}})\in C^{\infty}(X)$ for each $j$, it follows that the
coefficients of $V$ are in $C^{\infty}(U)$, i.e. $V$ is a $C^{\infty}(U)$
vector field, as desired. ∎

We recall that $\mathscr{X}=\overline{{}^{sc}T^{*}M}$ with
$M=(0,\infty)_{x}\times\mathbb{R}^{n-1}_{y}$ (in practice we take $n=3$, but
we can work with a general $n\geq 3$ as well), $\mathscr{X}_{1}$ is obtained
from $\mathscr{X}$ by blowing up $\Sigma=\\{\rho=0,\tau=0\\}$ parabolically,
and $\mathscr{X}_{2}$ is obtained from $\mathscr{X}_{1}$ by blowing up
$\Gamma=\\{x=0,d_{\Sigma}=0,\tilde{\tau}=0\\}$ in a way such that $x^{1/2}$,
$d_{\Sigma}$, and $\tilde{\tau}$ have the same degree of homogeneity.

We first study $\mathscr{X}_{1}$, and in particular the relationship between
vector fields on $\mathscr{X}_{1}$ tangent to its boundary with the
corresponding vector fields on $\mathscr{X}$. Note that on $\mathscr{X}_{1}$
we have that $d_{\Sigma}$, $\tilde{\tau}_{i}=\tau_{i}/d_{\Sigma}$, and
$\tilde{\rho}=\rho/d_{\Sigma}^{2}$ are smooth, with
$|\tilde{\tau}|^{2}+\tilde{\rho}=1$. Moreover, for any point in
$\partial\mathscr{X}_{1}$ there exists a neighborhood where some subset of the
following functions form local coordinates in the neighborhood:

$\\{x,y_{1},\dots,y_{n},d_{\Sigma},\tilde{\tau}_{1},\dots,\tilde{\tau}_{n-1},\tilde{\rho}\\}.$

Indeed, from the equation $|\tilde{\tau}|^{2}+\tilde{\rho}=1$ we can cover a
neighborhood of $\partial\mathscr{X}_{1}$ by the union

$\\{|\tilde{\tau}|^{2}<2/3\\}\cup\bigcup_{i=1}^{n-1}{\\{\tilde{\rho}<1/2,|\tilde{\tau}_{i}|>\epsilon\\}}$

for a sufficiently small but fixed $\epsilon>0$ (we can in fact take any
$\epsilon<\frac{1}{2(n-1)}$). In the first region, we have that
$(x,y,d_{\Sigma},\tilde{\tau})$ form local coordinates, with $x$ and
$d_{\Sigma}$ providing boundary-defining functions for the two faces of
$\partial\mathscr{X}_{1}$ in this region, while in the region
$\\{\tilde{\rho}<1/2,|\tilde{\tau}_{i}|>\epsilon\\}$ local coordinates are
given by
$(x,y,d_{\Sigma},\tilde{\rho},\tilde{\tau}_{1},\dots,\widehat{\tilde{\tau}_{i}},\dots,\tilde{\tau}_{n-1})$,
where $\widehat{\tilde{\tau}_{i}}$ denotes excluding $\tilde{\tau}_{i}$ from
the list; moreover in this case $x$, $d_{\Sigma}$, and $\tilde{\rho}$ provide
boundary defining functions of the three faces in this region.

We also recall that $S=\\{\tau=0\\}\subset\mathscr{X}$. Let $S_{1}$ be the
lift of $S$ to $\mathscr{X}_{1}$; explicitly this subset is
$\\{\tilde{\tau}=0\\}$.

We now have:

######  Lemma 2.2.

Let $V$ be a $C^{\infty}(\mathscr{X})$-vector field on $\mathscr{X}$ which is
tangent to $\partial\mathscr{X}$. Then

  1. (1)

$d_{\Sigma}V$ lifts to a $C^{\infty}(\mathscr{X}_{1})$-vector field on
$\mathscr{X}_{1}$ which is tangent to $\partial\mathscr{X}_{1}$.

  2. (2)

If in addition we have that $V$ is tangent to $S$, then $V$ itself lifts to a
$C^{\infty}(\mathscr{X}_{1})$-vector field on $\mathscr{X}_{1}$ tangent to
$\partial\mathscr{X}_{1}$, and furthermore it is tangent to $S_{1}$.

###### Proof.

In light of Lemma 2.1, it suffices to compute $d_{\Sigma}V$ (or $V$ itself if
it is tangent to $S$) applied to the functions

(2.1) $x,y_{i},d_{\Sigma},\tilde{\tau}_{i},\text{ and }\tilde{\rho}\text{ for
}1\leq i\leq n-1.$

Note that $V$ can be written as a $C^{\infty}(\mathscr{X})$ combination of the
vector fields

(2.2) $x\partial_{x},\partial_{y_{j}},\rho\partial_{\rho},\text{ and
}\partial_{\tau_{j}}\text{ for }1\leq j\leq n-1,$

while if it is furthermore tangent to $S$, then it can be written as a
combination of

(2.3) $x\partial_{x},\partial_{y_{j}},\rho\partial_{\rho}\text{ and
}\tau_{i}\partial_{\tau_{j}}\text{ for }1\leq i,j\leq n-1.$

We now note that $x\partial_{x}(x)=x$, $\partial_{y_{i}}(y_{i})=1$, and all of
the other preceding vector fields applied to $x$ (resp. $y_{i}$) yield $0$; in
addition, $x\partial_{x}$ and $\partial_{y_{i}}$ applied to the remaining
functions in (2.1) also yield zero. It thus remains to compute the vector
fields $\rho\partial_{\rho}$ and $d_{\Sigma}\partial_{\tau_{j}}$ against the
functions $d_{\Sigma}$, $\tilde{\tau}_{i}$, and $\tilde{\rho}$. We have

$\rho\partial_{\rho}(d_{\Sigma})=\frac{\rho}{2d_{\Sigma}}=\frac{1}{2}d_{\Sigma}\tilde{\rho},\quad
d_{\Sigma}\partial_{\tau_{j}}(d_{\Sigma})=\tau_{j}=d_{\Sigma}\tilde{\tau}_{j}$

and

$\rho\partial_{\rho}\left(\frac{\rho}{d_{\Sigma}^{2}}\right)=\frac{\rho}{d_{\Sigma}^{2}}-2\frac{\rho}{d_{\Sigma}^{3}}\frac{\rho}{2d_{\Sigma}}=\tilde{\rho}-\tilde{\rho}^{2},\quad
d_{\Sigma}\partial_{\tau_{j}}\left(\frac{\rho}{d_{\Sigma}^{2}}\right)=-2\frac{\rho}{d_{\Sigma}^{3}}\tau_{j}=-2\tilde{\rho}\tilde{\tau_{j}}$

and

$\rho\partial_{\rho}\left(\frac{\tau_{i}}{d_{\Sigma}}\right)=-\frac{\tau_{i}}{d_{\Sigma}^{2}}\frac{\rho}{2d_{\Sigma}}=-\frac{1}{2}\tilde{\rho}\tilde{\tau}_{i},\quad
d_{\Sigma}\partial_{\tau_{j}}\left(\frac{\tau_{i}}{d_{\Sigma}}\right)=\delta_{ij}-\frac{\tau_{i}}{d_{\Sigma}^{2}}\tau_{j}=\delta_{ij}-\tilde{\tau}_{i}\tilde{\tau}_{j}.$

All of these functions are in $C^{\infty}(\mathscr{X}_{1})$, and moreover
$\rho\partial_{\rho}$ and $d_{\Sigma}\partial_{\tau_{j}}$ applied to
$d_{\Sigma}$ and $\tilde{\rho}$ result in $C^{\infty}(\mathscr{X}_{1})$
multiples of the respective boundary-defining functions, i.e. they are tangent
to $\partial\mathscr{X}_{1}$. Moreover, since
$\tau_{i}\partial_{\tau_{j}}=\tilde{\tau}_{i}d_{\Sigma}\partial_{\tau_{j}}$,
it follows that if $V$ is tangent to $S$, then $V$ itself applied to the
functions in (2.1) are in $C^{\infty}(\mathscr{X}_{1})$, with $V$ applied to
the boundary-defining functions of $\partial\mathscr{X}_{1}$ resulting in
multiples of the respective boundary-defining functions, and since
$x\partial_{x}(\tilde{\tau}_{k})=\partial_{y_{j}}(\tilde{\tau}_{k})=0$ and

$\rho\partial_{\rho}(\tilde{\tau}_{k})=-\frac{1}{2}\tilde{\rho}\tilde{\tau}_{k},\quad\tau_{i}\partial_{\tau_{j}}(\tilde{\tau}_{k})=\tilde{\tau}_{i}d_{\Sigma}\partial_{\tau_{j}}(\tilde{\tau}_{k})=\tilde{\tau}_{i}(\delta_{jk}-\tilde{\tau}_{j}\tilde{\tau}_{k}),$

it follows that $V(\tilde{\tau}_{k})$ vanishes at $\tilde{\tau}=0$. Hence, its
$C^{\infty}(\mathscr{X}_{1})$-lift is tangent to $S_{1}=\\{\tilde{\tau}=0\\}$.
∎

We have a partial converse to the lifting result. Let $\mathcal{V}_{1}$ denote
the space of $C^{\infty}(\mathscr{X}_{1})$ vector fields on $\mathscr{X}_{1}$
which are tangent to $\partial\mathscr{X}_{1}$, and let
$\mathcal{V}_{1,\beta_{1}}$ denote the vector fields $V\in\mathcal{V}_{1}$
which are lifts of $C^{\infty}(\mathscr{X})$ vector fields on $\mathscr{X}$.
Note that if $V\in\mathcal{V}_{1,\beta_{1}}$, then $(\beta_{1})_{*}V$ is
automatically tangent to $\partial\mathscr{X}$, since the boundary-defining
functions of $\partial\mathscr{X}$ are products of boundary-defining functions
of $\partial\mathscr{X}_{1}$.

######  Lemma 2.3.

Any $V\in\mathcal{V}_{1}$ can be written as a finite sum

$V=\left(\sum_{i}{a_{i}V_{i}}\right)+\left(\sum_{j}{b_{j}d_{\Sigma}W_{j}}\right)$

where $a_{i},b_{j}\in C^{\infty}(\mathscr{X}_{1})$,
$V_{i}\in\mathcal{V}_{1,\beta_{1}}$ with $(\beta_{1})_{*}V_{i}$ tangent to
$S$, and $W_{j}$ are lifts222More accurately, $d_{\Sigma}W_{j}$ are lifts in
the interior of vector fields which are the product of $d_{\Sigma}$ and a
vector field which extends smoothly to a $C^{\infty}(\mathscr{X})$ vector
field on $\mathscr{X}$. of $C^{\infty}(\mathscr{X})$ vector fields on
$\mathscr{X}$ which are tangent to $\partial\mathscr{X}$ (but not necessarily
to $S$). Moreover, if $V$ is tangent to $S_{1}$, then only the first sum is
needed.

###### Proof.

Let $V\in\mathcal{V}_{1}$. We can write

$(\beta_{1})_{*}V=(V\rho)\partial_{\rho}+(V\tau)\cdot\partial_{\tau}+(Vx)\partial
x+(Vy)\cdot\partial_{y}$

where, to make sense of the RHS as a vector field on $\mathscr{X}$, we
interpret $V\rho$ as $(\beta_{1})^{*}V\rho$, etc.. Note that all coefficients
are in $C^{\infty}(\mathscr{X}_{1})$; moreover as $\partial_{y}$ is tangent to
$\partial\mathscr{X}_{1}$ (and is a $C^{\infty}(\mathscr{X})$ vector field on
$\mathscr{X}$), it follows that $(Vy)\cdot\partial_{y}\in
C^{\infty}(\mathscr{X}_{1})\otimes\mathcal{V}_{1,\beta_{1}}$. Furthermore,
since $x$ is still a boundary-defining function for one of the faces in
$\partial\mathscr{X}_{1}$, it follows (because $V$ is tangent to
$\partial\mathscr{X}_{1}$) that $Vx=xa$ where $a\in
C^{\infty}(\mathscr{X}_{1})$, and $x\partial_{x}$ is tangent to
$\mathscr{X}_{1}$, so $(Vx)\partial x=a\cdot x\partial_{x}\in
C^{\infty}(\mathscr{X}_{1})\otimes\mathcal{V}_{1,\beta_{1}}$. The same ends up
being true for $\rho$, since $\\{\rho=0\\}$ is contained in the boundary of
$\partial\mathscr{X}$ (more specifically, since
$\rho=\tilde{\rho}d_{\Sigma}^{2}$ where $\tilde{\rho}\in
C^{\infty}(\mathscr{X}_{1})$, it follows that

$V\rho=(V\tilde{\rho})d_{\Sigma}^{2}+2\tilde{\rho}(Vd_{\Sigma})d_{\Sigma};$

noting that $\tilde{\rho}$ and $d_{\Sigma}$ are boundary-defining functions of
two of the faces in $\partial\mathscr{X}$, it follows that $V\tilde{\rho}$ and
$Vd_{\Sigma}$ are $C^{\infty}(\mathscr{X}_{1})$ multiples of $\tilde{\rho}$
and $d_{\Sigma}$, respectively, so that overall $V\rho$ is a
$C^{\infty}(\mathscr{X}_{1})$ multiple of $\tilde{\rho}d_{\Sigma}^{2}=\rho$).
Hence, $(V\rho)\partial_{\rho}$ is a $C^{\infty}(\mathscr{X}_{1})$ multiple of
$\rho\partial_{\rho}$.

It remains to analyze the term $(V\tau)\cdot\partial_{\tau}$. Since

$\tau=d_{\Sigma}\tilde{\tau}\implies
V\tau=(Vd_{\Sigma})\tilde{\tau}+d_{\Sigma}(V\tilde{\tau}),$

it follows that $(V\tau)\cdot\partial_{\tau}$ is a sum of the terms
$(Vd_{\Sigma})\tilde{\tau}\cdot\partial_{\tau}$, which is a
$C^{\infty}(\mathscr{X}_{1})$ multiple of
$d_{\Sigma}\tilde{\tau}\cdot\partial_{\tau}=\tau\cdot\partial_{\tau}$ since
$Vd_{\Sigma}$ is a smooth multiple of $d_{\Sigma}$, as well as
$d_{\Sigma}(V\tilde{\tau})\cdot\partial_{\tau}$, which is a sum of
$C^{\infty}(\mathscr{X}_{1})$ multiples of $d_{\Sigma}\partial_{\tau_{i}}$
(i.e. of the form $b_{j}d_{\Sigma}W_{j}$ in the lemma statement).

Finally, we note that if $V$ is tangent to $S_{1}$, then
$V\tilde{\tau}_{i}=\sum{a_{j}\tilde{\tau}_{j}}$ for some $a_{j}\in
C^{\infty}(\mathscr{X}_{1})$, in which case

$d_{\Sigma}(V\tilde{\tau}_{i})\partial_{\tau_{i}}=\left(\sum_{j}{a_{j}d_{\Sigma}\tilde{\tau}_{j}}\right)\partial_{\tau_{i}}=\sum_{j}\left(a_{j}\tau_{j}\partial_{\tau_{i}}\right)$

and hence is a sum of $C^{\infty}(\mathscr{X}_{1})$ multiples of
$\tau_{j}\partial_{\tau_{i}}$. ∎

For the second blow-up $\beta_{2}:\mathscr{X}_{2}\to\mathscr{X}_{1}$, we have
similar results. We note that on $\mathscr{X}_{2}$, the functions
$d_{\Gamma}$, $\frac{d_{\Sigma}}{d_{\Gamma}}$,
$\frac{\tilde{\tau}_{i}}{d_{\Gamma}}$, and $\frac{x}{d_{\Gamma}^{2}}$ are in
$C^{\infty}(\mathscr{X}_{2})$. Furthermore, the second blow-down $\beta_{2}$
is a diffeomorphism away from the second front face, i.e. away from
$\Gamma=\\{x=0,d_{\Sigma}=0,d_{\Gamma}=0\\}$ in $\mathscr{X}_{1}$, it suffices
to study the geometry near the second front face in $\mathscr{X}_{2}$ (i.e.
near $\Gamma$ in $\mathscr{X}_{1}$), as the geometry is unchanged away from
there. As such, we note that any point in $\partial\mathscr{X}_{2}$ near the
second front face admits a neighborhood where a subset of the following
functions form local coordinates:

$\left\\{\frac{x}{d_{\Gamma}^{2}},y_{1},\dots,y_{n-1},d_{\Gamma},\frac{d_{\Sigma}}{d_{\Gamma}},\frac{\tilde{\tau}_{1}}{d_{\Gamma}},\dots,\frac{\tilde{\tau}_{n-1}}{d_{\Gamma}}\right\\}.$

Moreover, the boundary-defining functions for $\partial\mathscr{X}_{2}$ near
the second front face are given by $d_{\Gamma}$, $\frac{x}{d_{\Gamma}^{2}}$,
and $\frac{d_{\Sigma}}{d_{\Gamma}}$. The reasoning is similar to the reasoning
for the case of $\mathscr{X}_{1}$.

######  Lemma 2.4.

Let $V\in\mathcal{V}_{1}$. Then

  1. (1)

$d_{\Gamma}V$ lifts to a smooth vector field on $\mathscr{X}_{2}$ which is
tangent to $\partial\mathscr{X}_{2}$.

  2. (2)

If in addition we have that $V$ is tangent to $S_{1}$, then $V$ itself lifts
to a smooth vector field on $\mathscr{X}_{2}$ tangent to
$\partial\mathscr{X}_{2}$.

###### Proof.

Since the second blow-down $\beta_{2}$ is a diffeomorphism away from $\Gamma$
in $\mathscr{X}_{1}$, it suffices to analyze $V$ in a neighborhood of
$\Gamma$. In particular, $(x,y,d_{\Sigma},\tilde{\tau})$ forms local
coordinates of $\mathscr{X}_{1}$ near $\Gamma$, so vector fields in
$\mathcal{V}_{1}$ can be written as $C^{\infty}(\mathscr{X}_{1})$ combinations
of the vector fields $x\partial_{x}$,
$\partial_{y_{j}},d_{\Sigma}\partial_{d_{\Sigma}}$, and
$\partial_{\tilde{\tau}_{j}}$ for $1\leq j\leq n-1$ near $\Gamma$;
furthermore, vector fields which are also tangent to $S_{1}$ can be written as
a combination of $x\partial_{x}$, $\partial_{y_{j}}$,
$d_{\Sigma}\partial_{d_{\Sigma}}$, and
$\tilde{\tau}_{i}\partial_{\tilde{\tau}_{j}}$ for $1\leq i,j\leq n-1$.

We test vector fields like $d_{\Sigma}\partial_{d_{\Sigma}}$,
$d_{\Gamma}\partial_{\tilde{\tau}_{j}}$, and $x\partial_{x}$ against the
functions

$d_{\Gamma},\frac{d_{\Sigma}}{d_{\Gamma}},\frac{\tilde{\tau}_{i}}{d_{\Gamma}},\frac{x}{d_{\Gamma}^{2}},\quad
1\leq i\leq n-1.$

(Note that we can ignore the vector fields $\partial_{y_{j}}$ and the
coordinates functions $y_{i}$ for similar reasons as before.)

Thus, we first apply the vector fields to $d_{\Gamma}$, to get

$\displaystyle d_{\Sigma}\partial_{d_{\Sigma}}(d_{\Gamma})$
$\displaystyle=d_{\Sigma}\frac{d_{\Sigma}}{d_{\Gamma}}=d_{\Gamma}\left(\frac{d_{\Sigma}}{d_{\Gamma}}\right)^{2},$
$\displaystyle d_{\Gamma}\partial_{\tilde{\tau}_{j}}(d_{\Gamma})$
$\displaystyle=d_{\Gamma}\frac{\tilde{\tau}_{j}}{d_{\Gamma}},$ $\displaystyle
x\partial_{x}(d_{\Gamma})$
$\displaystyle=\frac{x}{2d_{\Gamma}}=d_{\Gamma}\left(\frac{x}{2d_{\Gamma}^{2}}\right).$

Note that all of the functions on the right-hand sides are smooth on
$\mathscr{X}_{2}$ and vanish at $\\{d_{\Gamma}=0\\}$.

We now check

$\displaystyle
d_{\Sigma}\partial_{d_{\Sigma}}\left(\frac{d_{\Sigma}}{d_{\Gamma}}\right)$
$\displaystyle=d_{\Sigma}\left(\frac{1}{d_{\Gamma}}-\frac{d_{\Sigma}}{d_{\Gamma}^{2}}\frac{d_{\Sigma}}{d_{\Gamma}}\right)=\frac{d_{\Sigma}}{d_{\Gamma}}\left(1-\left(\frac{d_{\Sigma}}{d_{\Gamma}}\right)^{2}\right),$
$\displaystyle
d_{\Gamma}\partial_{\tilde{\tau}_{j}}\left(\frac{d_{\Sigma}}{d_{\Gamma}}\right)$
$\displaystyle=-d_{\Gamma}\frac{d_{\Sigma}}{d_{\Gamma}^{2}}\frac{\tilde{\tau}_{j}}{d_{\Gamma}}=-\frac{d_{\Sigma}}{d_{\Gamma}}\left(\frac{\tilde{\tau}_{j}}{d_{\Gamma}}\right),$
$\displaystyle x\partial_{x}\left(\frac{d_{\Sigma}}{d_{\Gamma}}\right)$
$\displaystyle=-x\frac{d_{\Sigma}}{d_{\Gamma}^{2}}\frac{1}{2d_{\Gamma}}=-\frac{d_{\Sigma}}{d_{\Gamma}}\left(\frac{x}{2d_{\Gamma}^{2}}\right).$

These functions are smooth on $\mathscr{X}_{2}$ and vanish at
$\\{\frac{d_{\Sigma}}{d_{\Gamma}}=0\\}$.

We now check

$\displaystyle
d_{\Sigma}\partial_{d_{\Sigma}}\left(\frac{x}{d_{\Gamma}^{2}}\right)$
$\displaystyle=-2d_{\Sigma}\frac{x}{d_{\Gamma}^{3}}\frac{d_{\Sigma}}{d_{\Gamma}}=-2\frac{x}{d_{\Gamma}^{2}}\left(\frac{d_{\Sigma}}{d_{\Gamma}}\right)^{2},$
$\displaystyle
d_{\Gamma}\partial_{\tilde{\tau}_{j}}\left(\frac{x}{d_{\Gamma}^{2}}\right)$
$\displaystyle=-2d_{\Gamma}\frac{x}{d_{\Gamma}^{3}}\frac{\tilde{\tau}_{j}}{d_{\Gamma}}=-2\frac{x}{d_{\Gamma}^{2}}\left(\frac{\tilde{\tau}_{j}}{d_{\Gamma}}\right),$
$\displaystyle x\partial_{x}\left(\frac{x}{d_{\Gamma}^{2}}\right)$
$\displaystyle=\frac{x}{d_{\Gamma}^{2}}-2x\frac{x}{d_{\Gamma}^{3}}\frac{1}{2d_{\Gamma}}=\frac{x}{d_{\Gamma}^{2}}\left(1-\frac{x}{d_{\Gamma}^{2}}\right).$

These functions are smooth on $\mathscr{X}_{2}$ and vanish at
$\\{\frac{x}{d_{\Gamma}^{2}}=0\\}$.

Finally, we check

$\displaystyle
d_{\Sigma}\partial_{d_{\Sigma}}\left(\frac{\tilde{\tau}_{i}}{d_{\Gamma}}\right)$
$\displaystyle=-d_{\Sigma}\frac{\tilde{\tau}_{i}}{d_{\Gamma}^{2}}\frac{d_{\Sigma}}{d_{\Gamma}}=-\left(\frac{d_{\Sigma}}{d_{\Gamma}}\right)^{2}\frac{\tilde{\tau}_{i}}{d_{\Gamma}},$
$\displaystyle
d_{\Gamma}\partial_{\tilde{\tau}_{j}}\left(\frac{\tilde{\tau}_{i}}{d_{\Gamma}}\right)$
$\displaystyle=-d_{\Gamma}\frac{\tilde{\tau}_{i}}{d_{\Gamma}^{2}}\frac{\tilde{\tau}_{j}}{d_{\Gamma}}+d_{\Gamma}\frac{\delta_{ij}}{d_{\Gamma}}=-\frac{\tilde{\tau}_{j}}{d_{\Gamma}}\frac{\tilde{\tau}_{i}}{d_{\Gamma}}+\delta_{ij},$
$\displaystyle x\partial_{x}\left(\frac{\tilde{\tau}_{i}}{d_{\Gamma}}\right)$
$\displaystyle=-x\frac{\tilde{\tau}_{i}}{d_{\Gamma}^{2}}\frac{1}{2d_{\Gamma}}=-\frac{\tilde{\tau}_{i}}{d_{\Gamma}}\left(\frac{x}{2d_{\Gamma}^{2}}\right).$

These functions are smooth on $\mathscr{X}_{2}$. This implies that
$d_{\Gamma}V$ lifts to a $C^{\infty}(\mathscr{X}_{2})$ vector field on
$\mathscr{X}_{2}$.

Furthermore, since
$\tilde{\tau}_{i}\partial_{\tilde{\tau}_{j}}=\frac{\tilde{\tau}_{i}}{d_{\Gamma}}\cdot
d_{\Gamma}\partial_{\tilde{\tau}_{j}}$, it follows that if $V$ is tangent to
$S_{1}$, then $V$ itself lifts to a $C^{\infty}(\mathscr{X}_{2})$ vector field
on $\mathscr{X}_{2}$. ∎

######  Remark 2.5.

If $V$ is tangent to $S_{1}$, then similar arguments as before show that the
lift of $V$ will be tangent to
$\left\\{\frac{\tilde{\tau}}{d_{\Gamma}}=0\right\\}$. However, this fact will
not be used, so we do not emphasize this additional fact.

As a corollary of the above two lemmas, we have the following result regarding
the overall blow-down $\beta:\mathscr{X}_{2}\to\mathscr{X}$:

######  Corollary 2.6.

Suppose $V$ is a vector field on $\mathscr{X}$ which is tangent to
$\partial\mathscr{X}$. Then $d_{\Gamma}d_{\Sigma}V$ lifts to a smooth vector
field on $\mathscr{\mathscr{X}}_{2}$ which is tangent to
$\partial\mathscr{\mathscr{X}}_{2}$. Furthermore, if $V$ is tangent to $S$,
then $V$ itself lifts to a smooth vector field on $\mathscr{\mathscr{X}}_{2}$
which is tangent to $\partial\mathscr{\mathscr{X}}_{2}$.

Examples of vector fields on $\mathscr{X}$ tangent to $\partial\mathscr{X}$
include vector fields which are homogeneous of degree $0$ in the fiber
variables $(\xi,\eta)$–such vector fields are commonly used in testing
symbolic regularity.

The above corollary shows that certain vector fields tangent to
$\partial\mathscr{X}$ (namely those also tangent to $S$) can be lifted to a
smooth vector field on $\mathscr{X}_{2}$ tangent to $\partial\mathscr{X}_{2}$,
and all vector fields tangent to $\partial\mathscr{X}$ can be modified to lift
to a smooth vector field on $\mathscr{X}_{2}$ tangent to
$\partial\mathscr{X}_{2}$ by multiplying by $d_{\Gamma}d_{\Sigma}$.

Let $\mathcal{V}_{2}$ denote the space of $C^{\infty}(\mathscr{X}_{2})$ vector
fields on $\mathscr{X}_{2}$ which are tangent to $\partial\mathscr{X}_{2}$,
and let $\mathcal{V}_{2,\beta_{2}}$ denote the vector fields
$V\in\mathcal{V}_{2}$ which are $\beta_{2}$-lifts of
$C^{\infty}(\mathscr{X}_{1})$ vector fields on $\mathscr{X}_{1}$. Similar
calculations give the corresponding result to Lemma 2.3 considering blowing
down $\beta_{2}:\mathscr{X}_{2}\to\mathscr{X}_{1}$:

######  Lemma 2.7.

Any $V\in\mathcal{V}_{2}$ can be written as a finite sum

$V=\left(\sum_{i}{a_{i}V_{i}}\right)+\left(\sum_{j}{b_{j}d_{\Gamma}W_{j}}\right)$

where $a_{i},b_{j}\in C^{\infty}(\mathscr{X}_{2})$,
$V_{i}\in\mathcal{V}_{2,\beta_{2}}$ with $(\beta_{2})_{*}V$ tangent to
$S_{1}$, and $W_{j}$ are lifts of $C^{\infty}(\mathscr{X}_{1})$ vector fields
on $\mathscr{X}_{1}$ which are tangent to $\partial\mathscr{X}_{1}$ (but not
necessarily to $S_{1}$).

###### Proof.

We proceed similarly to the Lemma 2.3. Since the blowdown $\beta_{2}$ is a
diffeomorphism away from the second front face (corresponding to
$\Gamma=\partial S_{1}$ in $\mathscr{X}_{1}$), it suffices to analyze
$(\beta_{2})_{*}V$ for $V\in\mathcal{V}_{2}$ in a neighborhood of $S_{1}$.
Valid coordinates in that neighborhood are given by $d_{\Sigma}$,
$\tilde{\tau}_{i}$, $x$, and $y$. For $V\in\mathcal{V}_{2}$, we have

$(\beta_{2})_{*}V=(Vd_{\Sigma})\partial_{d_{\Sigma}}+(V\tilde{\tau})\cdot\partial_{\tilde{\tau}}+(Vx)\partial_{x}+(Vy)\cdot\partial_{y}.$

Since $\partial_{y}$ is still tangent to $\partial\mathscr{X}_{1}$, it follows
that $(Vy)\cdot\partial_{y}$ is of the form $a_{i}V_{i}$ above. Furthermore,
since $\\{d_{\Sigma}=0\\}$ and $\\{x=0\\}$ are still contained in
$\partial\mathscr{X}_{2}$, it follows similar from arguments above that
$Vd_{\Sigma}$ and $Vx$ are $C^{\infty}(\mathscr{X}_{2})$ multiples of
$d_{\Sigma}$ and $x$, respectively, so
$(Vd_{\Sigma})\partial_{d_{\Sigma}}+(Vx)\partial_{x}$ is of the form
$\sum{a_{i}V_{i}}$ above. Finally, we have

$(V\tilde{\tau})\cdot\partial_{\tilde{\tau}}=d_{\Gamma}V\left(\frac{\tilde{\tau}}{d_{\Gamma}}\right)\cdot\partial_{\tilde{\tau}}+(Vd_{\Gamma})\frac{\tilde{\tau}}{d_{\Gamma}}\cdot\partial_{\tilde{\tau}};$

the first term is of the form $b_{j}d_{\Gamma}W_{j}$ while the second term is
of the form $a_{i}V_{i}$, since $Vd_{\Gamma}$ is a
$C^{\infty}(\mathscr{X}_{2})$ multiple of $d_{\Gamma}$, i.e.
$\frac{Vd_{\Gamma}}{d_{\Gamma}}\in C^{\infty}(\mathscr{X}_{2})$. ∎

Let $\mathcal{V}_{2,\beta}$ denote vector fields in $\mathcal{V}_{2}$ which
are $\beta$-lifts of vector fields in $\mathcal{V}$. Combining these two
lemmas thus gives:

######  Corollary 2.8.

Any $V\in\mathcal{V}_{2}$ can be written as a finite sum

$V=\left(\sum_{i}{a_{i}V_{i}}\right)+\left(\sum_{k}{b_{j}d_{\Gamma}d_{\Sigma}W_{j}}\right)$

where $a_{i},b_{j}\in C^{\infty}(\mathscr{X}_{2})$,
$V_{i}\in\mathcal{V}_{2,\beta}$ with $\beta_{*}V_{i}$ tangent to $S$, and
$W_{j}$ are lifts of $C^{\infty}(\mathscr{X})$ vector fields which are tangent
to $\partial\mathscr{X}$ (but not necessarily to $S$).

###### Proof.

From Lemma 2.7, we have that $V\in\mathcal{V}_{2}$ can be written as
$V=\left(\sum_{i}{a_{i}\tilde{V}_{i}}\right)+\left(\sum_{j}{b_{j}d_{\Gamma}\tilde{W}_{j}}\right)$,
with $\tilde{V}_{i}$ a $\beta_{2}$-lift of a vector field in $\mathcal{V}_{1}$
which is also tangent to $S_{1}$, and $d_{\Gamma}\tilde{W}_{j}$ a
$\beta_{2}$-lift of $d_{\Gamma}$ times a vector field in $\mathcal{V}_{1}$.
From Lemma 2.3, we know that $\beta_{2}^{*}\tilde{V}_{i}$ is itself a
$\beta_{1}$-lift of a vector field in $\mathcal{V}$ tangent to $S$, while
$\beta_{2}^{*}\tilde{W}_{j}=\sum{\tilde{a}_{k}V^{\prime}_{j,k}}+\sum{\tilde{b}_{l}d_{\Sigma}W^{\prime}_{j,l}}$,
with $V^{\prime}_{j,k}\in\mathcal{V}_{1,\beta_{1}}$ and
$d_{\Sigma}W^{\prime}_{j,l}$ a $\beta_{1}$-lift of $d_{\Sigma}$ times a vector
field in $\mathcal{V}$ not necessarily tangent to $S$. It follows that

$V=\left(\sum_{i}{a_{i}\tilde{V}_{i}}+\sum_{j,k}{b_{j}\tilde{a}_{k}d_{\Gamma}V^{\prime}_{j,k}}\right)+\sum_{j,l}{b_{j}\tilde{b}_{l}d_{\Gamma}d_{\Sigma}W^{\prime}_{j,l}}.$

The terms in the first parentheses are all in
$C^{\infty}(\mathscr{X}_{2})\otimes\mathcal{V}_{2,\beta}$, giving the form
claimed in the Corollary. ∎

######  Corollary 2.9.

We have that $\mathcal{V}_{2}$ is generated over $C^{\infty}(\mathscr{X}_{2})$
by the vector fields

$x\partial_{x},\partial_{y_{j}},\rho\partial_{\rho},d_{\Gamma}d_{\Sigma}\partial_{\tau_{j}},\quad
1\leq j\leq n-1.$

Note as well that we are usually interested in the behavior of functions on
$\mathscr{X}_{2}$ near the boundary $\partial\mathscr{X}_{2}$, and hence we
can take the boundary-defining functions $\rho$, $x$, $d_{\Sigma}$, and
$d_{\Gamma}$ to be bounded from above away from their front faces (and in
particular away from the boundary). This is a convention we will adopt from
now on.

##  3\. The $bb$-symbol class

Much like scattering symbols are, up to factors of $\rho$ and $x$, functions
in the interior of $\mathscr{X}$ which extend to distributions conormal to the
boundary $\partial\mathscr{X}$, we can consider a symbol class based on
distributions conormal to $\partial\mathscr{X}_{2}$. We recall that the
interior of $\mathscr{X}$ is identified with ${}^{sc}T^{*}M$ with
$M=(0,\infty)\times\mathbb{R}^{n-1}$, and we will identify the interior of
$\partial\mathscr{X}_{2}$ with ${}^{sc}T^{*}M$ as well via the blow-down
$\beta$.

######  Definition 3.1.

Define $S^{0,0,0,0}(M)$ to consist of functions
$a:{{}^{sc}T^{*}M}\to\mathbb{C}$ which are smooth on the interior
$\mathscr{X}_{2}^{\circ}={{}^{sc}T^{*}M}$ and are $L^{\infty}$-conormal to the
boundary $\partial\mathscr{\mathscr{X}}_{2}$, i.e. satisfy the estimates

$|V^{\alpha}a|\leq C_{\alpha}$

on any subset of ${}^{sc}T^{*}M$ which is pre-compact333That is, the estimate
should hold with a uniform constant on any set whose projection to the base is
pre-compact; in particular on sets of the form $\\{x\leq C\\}$ when using the
$(x,y)$ spatial variables. Notably the uniform estimate should hold on the
entire fiber, including at fiber infinity. In most cases we will be concerned
with distributions which are compactly supported, in which case we may
consider without loss of generality symbols whose spatial support is compact,
in which case the estimates are uniform and not just locally uniform. (when
considering ${}^{sc}T^{*}M$ as a subset of$\overline{{}^{sc}T^{*}M}$) whenever
$V^{\alpha}$ is a product of vector fields in $\mathcal{V}_{2}$, i.e. tangent
to $\partial\mathscr{X}_{2}$. Define

$S^{m,l,k,j}(M):=\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}S^{0,0,0,0}(M).$

We will often write $S^{m,l,k,j}$ in place of $S^{m,l,k,j}(M)$ if the ambient
space is clear or not important.

We will call the collection of symbols belonging in some $S^{m,l,k,j}$ the
class of $bb$-scattering symbols (or $bb$-symbols for short). The notation can
be interpreted two ways: first, this is the symbol class associated to making
two blowups to the radially compactified cotangent bundle (hence the two
$b$’s), and secondly this is in some sense a combination of the _B_ outet de
Monvel calculus studied in [1] in the presence of an additional _b_ oundary.

######  Example 3.1.

  1. (1)

The space $C^{\infty}(\mathscr{X}_{2})$ is contained in $S^{0,0,0,0}$, since
functions in $C^{\infty}(\mathscr{X}_{2})$ satisfy estimates upon application
of _any_ vector fields with $C^{\infty}(\mathscr{X}_{2})$ coefficients, and
not just those which are tangent to $\partial\mathscr{X}_{2}$.

  2. (2)

The boundary-defining functions $\rho$ and $x$ belong to $S^{-1,0,0,0}$ and
$S^{0,-1,0,0}$, respectively. On the other hand, we can also write

$\rho=d_{\Sigma}^{2}\tilde{\rho}=d_{\Sigma}d_{\Gamma}\frac{d_{\Sigma}}{d_{\Gamma}}\tilde{\rho}=d_{\Gamma}^{2}\left(\frac{d_{\Sigma}}{d_{\Gamma}}\right)^{2}\tilde{\rho}\quad\text{where
}\tilde{\rho}=\frac{\rho}{d_{\Sigma}^{2}},$

and since $\tilde{\rho}$ and $\frac{d_{\Sigma}}{d_{\Gamma}}$ both belong to
$C^{\infty}(\mathscr{X}_{2})\subset S^{0,0,0,0}$, we can also write

$\rho\in S^{0,0,-2,0}\cap S^{0,0,-1,-1}\cap S^{0,0,0,-2}.$

Similarly, we have

$x=d_{\Gamma}^{2}\tilde{x}\quad\text{where
}\tilde{x}=\frac{x}{d_{\Gamma}^{2}},$

so $x\in S^{0,0,0,-2}$. Later on we will see that $S^{-1,0,0,0}\subset
S^{0,0,-2,0}\subset S^{0,0,-1,-1}\subset S^{0,0,0,-2}$ and
$S^{0,-1,0,0}\subset S^{0,0,0,-2}$, so these observations are not
contradictory.

  3. (3)

For any $1\leq i\leq n-1$, we have

$\tau_{i}=d_{\Sigma}\tilde{\tau}_{i}=d_{\Sigma}d_{\Gamma}\frac{\tilde{\tau}_{i}}{d_{\Gamma}}.$

Hence, we have

(3.1) $\tau_{i}\in S^{0,0,-1,-1}.$

The expected algebra property holds:

######  Proposition 3.1.

Let $a\in S^{m,l,k,j}$ and $b\in
S^{m^{\prime},l^{\prime},k^{\prime},j^{\prime}}$. Then we have $ab\in
S^{m+m^{\prime},l+l^{\prime},k+k^{\prime},j+j^{\prime}}$.

###### Proof.

It suffices to consider the case
$(m,l,k,j)=(m^{\prime},l^{\prime},k^{\prime},j^{\prime})=(0,0,0,0)$. Noting
that $V(ab)=(Va)b+a(Vb)$, it follows by induction that for a product
$V^{\alpha}$ of vector fields in $\mathcal{V}_{2}$, we have that
$V^{\alpha}(ab)$ is a sum of products of the form $(V^{\beta}a)(V^{\gamma}b)$
for some products $V^{\beta}$, $V^{\gamma}$ of vector fields in
$\mathcal{V}_{2}$. All of these terms satisfy the uniform estimates, so
$V^{\alpha}(ab)$ does as well. ∎

######  Proposition 3.2.

Suppose that $a\in S^{m,l,k,j}$ and $V^{\alpha}$ is a product of vector fields
in $\mathcal{V}_{2}$. Then $V^{\alpha}a\in S^{m,l,k,j}$ as well.

###### Proof.

If $(m,l,k,j)=(0,0,0,0)$, then this follows easily from the definition.
Otherwise, take $a=\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}\tilde{a}$,
with $\tilde{a}\in S^{0,0,0,0}$. It suffices to prove the statement for
$|\alpha|=1$ as the general case then follows by induction. Since $\rho$, $x$,
$d_{\Sigma}$, and $d_{\Gamma}$ are all products of boundary-defining functions
of $\mathcal{X}_{2}$, it follows that for $V\in\mathcal{V}_{2}$ we have

$V(\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j})\in\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}C^{\infty}(\mathscr{X}_{2})\subset
S^{m,l,k,j}.$

Hence

$Va=V(\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j})\tilde{a}+\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}V\tilde{a}\in
S^{m,l,k,j}S^{0,0,0,0}+\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}S^{0,0,0,0}=S^{m,l,k,j},$

as desired. ∎

We can thus reformulate the definition of $S^{m,l,k,j}$ as follows:

######  Corollary 3.3.

A function $a$ is in $S^{m,l,k,j}$ if and only if it satisfies estimates of
the form

$|V^{\alpha}a|\leq C_{\alpha}\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}$

on any pre-compact subset of ${}^{sc}T^{*}M$ whenever $V^{\alpha}$ is a
product of vector fields in $\mathcal{V}_{2}$.

###### Proof.

From the above proposition, if $a\in S^{m,l,k,j}$, then $V^{\alpha}a\in
S^{m,l,k,j}$, and hence it equals
$\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}$ times a function in
$S^{0,0,0,0}$, which satisfies uniform estimates, so overall $V^{\alpha}a$ wil
satisfy the desired estimate above. Conversely, if $a$ satisfies the above
estimates, we’d like to show that $\rho^{m}x^{l}d_{\Sigma}^{k}d_{\Gamma}^{j}a$
is in $S^{0,0,0,0}$. Similar arguments from the above paragraphs show that
$V^{\alpha}(\rho^{m}x^{l}d_{\Sigma}^{k}d_{\Gamma}^{j}a)$ has the form

$V^{\alpha}(\rho^{m}x^{l}d_{\Sigma}^{k}d_{\Gamma}^{j}a)=\sum_{\beta\leq\alpha}{\rho^{m}x^{l}d_{\Sigma}^{k}d_{\Gamma}^{j}(V^{\beta}a)b_{\beta}}$

for some functions $b_{\beta}\in C^{\infty}(\mathscr{X}_{2})$. Then by
hypothesis we have that
$\rho^{m}x^{l}d_{\Sigma}^{k}d_{\Gamma}^{j}(V^{\beta}a)$ satisfies zeroth-order
uniform estimates (i.e. the symbol estimates for $S^{0,0,0,0}$ symbols), while
the $b_{\beta}$ also satisfy zeroth-order estimates, and hence overall
$V^{\alpha}(\rho^{m}x^{l}d_{\Sigma}^{k}d_{\Gamma}^{j}a)$ satisfies zeroth-
order estimates, i.e. $\rho^{m}x^{l}d_{\Sigma}^{k}d_{\Gamma}^{j}a\in
S^{0,0,0,0}$, as desired. ∎

Such symbols also satisfy the following estimate:

######  Proposition 3.4.

Suppose $W^{\alpha}$ and $V^{\beta}$ are products vector fields on
$\mathscr{X}$ which are homogeneous of degree $0$, with the latter a product
of vector fields also tangent to $S$. Then for $a\in S^{m,l,k,j}$, we have
$W^{\alpha}V^{\beta}a\in S^{m,l,k+|\alpha|,j+|\alpha|}$, and in particular

$|W^{\alpha}V^{\beta}a|\leq
C_{\alpha,\beta}\rho^{-m}x^{-l}d_{\Sigma}^{-k-|\alpha|}d_{\Gamma}^{-j-|\alpha|}.$

This follows almost immediately from Corollary 2.6, noting that if
$V,W\in\mathcal{V}$ with $V$ tangent to $S$, then $V$ and
$d_{\Sigma}d_{\Gamma}W$ are444More accurately, they lift to vector fields in
$\mathcal{V}_{2}$ on $\mathscr{X}_{2}$. also tangent to
$\partial\mathscr{X}_{2}$.

We have the following inclusions:

######  Proposition 3.5.

  1. (1)

If $m\leq m^{\prime}$, $l\leq l^{\prime}$, $k\leq k^{\prime}$, and $j\leq
j^{\prime}$, then $S^{m,l,k,j}\subset
S^{m^{\prime},l^{\prime},k^{\prime},j^{\prime}}$.

  2. (2)

If $k\geq k^{\prime}$, then $S^{m,l,k,j}\subset
S^{m+(k-k^{\prime})/2,l,k^{\prime},j}$.

  3. (3)

If $j\geq j^{\prime}$, then $S^{m,l,k,j}\subset
S^{m,l+(j-j^{\prime})/2,k,j^{\prime}}\cap
S^{m,l,k+(j-j^{\prime}),j^{\prime}}$.

  4. (4)

We have

$S^{m,l,k,j}\subset
S^{m+(k-k^{\prime}+s(j-j^{\prime})_{+})_{+}/2,l+t(j-j^{\prime})_{+}/2,k^{\prime},j^{\prime}}$

for any $m,l,k,j,k^{\prime},j^{\prime}$ and any $s,t$ with $s+t=1$, where
$x_{+}=\max(0,x)$. In particular, we have $S^{m,l,k,j}\subset
S^{m+(k+sj_{+})_{+}/2,l+tj_{+}/2,0,0}$ for all $s+t=1$, and
$S^{m-1,l-1,k-1,j-1}\subset S^{m-1/2,l-1/2,k,j}$.

Part 2 roughly says that we can arbitrarily improve our $k$ order (i.e. order
with respect to $d_{\Sigma}$) at the cost of $1/2$ order in $\rho$, while
parts 3 and 4 says that we can improve our $j$ order (with respect to
$d_{\Gamma}$) at the cost of $1/2$ order in either $x$ or $\rho$.

###### Proof.

  1. (1)

We recall that we may assume without loss of generality that $\rho$, $x$,
$d_{\Sigma}$, and $d_{\Gamma}$ are bounded away from their associated front
faces. Hence, we have

$\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}=(\rho^{m^{\prime}-m}x^{l^{\prime}-l}d_{\Sigma}^{k^{\prime}-k}d_{\Gamma}^{j^{\prime}-j})\rho^{-m^{\prime}}x^{-l^{\prime}}d_{\Sigma}^{-k^{\prime}}d_{\Gamma}^{-j^{\prime}}\leq
C\rho^{-m^{\prime}}x^{-l^{\prime}}d_{\Sigma}^{-k^{\prime}}d_{\Gamma}^{-j^{\prime}}$

since $m^{\prime}-m,l^{\prime}-l,k^{\prime}-k,j^{\prime}-j\geq 0$, and hence
$a\in S^{m,l,k,j}$ implies

$\displaystyle|V^{\alpha}a|\leq
C_{\alpha}\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}\leq
C_{\alpha}C\rho^{-m^{\prime}}x^{-l^{\prime}}d_{\Sigma}^{-k^{\prime}}d_{\Gamma}^{-j^{\prime}},$

i.e. $a\in S^{m^{\prime},l^{\prime},k^{\prime},j^{\prime}}$.

  2. (2)

We have $d_{\Sigma}\geq\rho^{1/2}$, so for $s\leq 0$ we have
$d_{\Sigma}^{s}\leq\rho^{s/2}$. Thus for $k^{\prime}-k\leq 0$ we have
$d_{\Sigma}^{k^{\prime}-k}\leq\rho^{-(k-k^{\prime})/2}$, and hence

$\rho^{-m}d_{\Sigma}^{-k}=\rho^{-m}d_{\Sigma}^{k^{\prime}-k}d_{\Sigma}^{-k^{\prime}}\leq\rho^{-(m+(k-k^{\prime})/2)}d_{\Sigma}^{-k^{\prime}}.$

  3. (3)

Similar reasoning as above, noting that $d_{\Gamma}\geq x^{1/2}$ and
$d_{\Gamma}\geq d_{\Sigma}$.

  4. (4)

We have

$\displaystyle S^{m,l,k,j}$ $\displaystyle\subset
S^{m,l+t(j-j^{\prime})_{+}/2,k,j^{\prime}+s(j-j^{\prime})}$
$\displaystyle\subset
S^{m,l+t(j-j^{\prime})_{+}/2,k+s(j-j^{\prime})_{+},j^{\prime}}$
$\displaystyle\subset
S^{m+(k-k^{\prime}+s(j-j^{\prime})_{+})_{+}/2,l+t(j-j^{\prime})_{+}/2,k^{\prime},j^{\prime}},$

using parts 3, 3, and 2, respectively.

∎

We also have the following ellipticity result:

######  Proposition 3.6.

Suppose $a\in S^{m,l,k,j}$ satisfies an “ellipticity” estimate

$|a|\geq c\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}$

near $\partial\mathscr{X}_{2}$. Let $r\in\mathbb{Z}$, and $b\in
C^{\infty}(^{sc}T^{*}M)$ agrees with $a^{r}$ near $\partial\mathscr{X}_{2}$.
Then $b\in S^{rm,rl,rk,rj}$. If $a$ takes values in $\mathbb{R}_{+}$ in a
neighborhood of $\partial\mathscr{X}_{2}$, then the same conclusion holds for
any $r\in\mathbb{R}$.

In particular, if $a\in S^{m,l,k,j}$ satisfies the ellipticity estimate, then
any smooth function equaling $a^{-1}$ near $\partial\mathscr{X}_{2}$ is a
symbol in $S^{-m,-l,-k,-j}$.

The properties follow from standard arguments in microlocal analysis, but we
repeat them here for convenience:

###### Proof.

From the generalized power rule

$V\left(a^{\beta}\right)=\beta a^{\beta-1}Va,$

it follows by induction that

$V^{\alpha}\left(a^{r}\right)\in\sum_{i=1}^{|\alpha|}a^{r-i}\prod_{i^{\prime}=1}^{i}S^{m,l,k,j},$

i.e. it is a sum of terms of the form $a^{\beta-i}$ times products of $i$
symbols in $S^{m,l,k,j}$. It follows, from the lower bound on $a$, that

$\left|V^{\alpha}\left(a^{r}\right)\right|\leq
C\sum_{i=1}^{|\alpha|}\frac{(\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j})^{i}}{(\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j})^{i-r}}=C^{\prime}\rho^{-rm}x^{-rl}d_{\Sigma}^{-rk}d_{\Gamma}^{-rj},$

as desired. ∎

Note that the above proposition is an “ellipticity” statement showing that
symbols which are “elliptic” in this symbol class admit an inverse which also
belongs to this symbol class. To make use of this ellipticity result, we give
an example of a class of operators which turn out to be elliptic in this new
symbol class. Our operators of interest in the local inverse problem turn out
to have symbols of this form, making the following lemma quite applicable:

######  Lemma 3.7.

Suppose $a$ is a scattering symbol satisfying the asymptotic expansion (in the
fiber variables)

$a\sim\sum_{j\geq 1}{a_{-j}(x,y,\tau)\rho^{j}}$

where $a_{-j}\in S^{0,0}$, and the following properties hold regarding
$a_{-1}$, $a_{-2}$, and $a_{-3}$:

  * •

$a_{-1}$ is nonnegative, and it vanishes only on $\Sigma$, where it vanishes
non-degenerately quadratically.

  * •

$a_{-2}=ix\tilde{a}$, where $\tilde{a}\in S^{0,0}$ is real-valued and non-
vanishing on $\Sigma$.

  * •

$a_{-3}$ is positive on $\Sigma\cap\\{x=0\\}$.

Then $a$ is a symbol in $S^{-1,0,-2,-2}$ which is elliptic near fiber
infinity, i.e. it satisfies the lower bound $|a|\geq c\rho
d_{\Sigma}^{2}d_{\Gamma}^{2}$ in a neighborhood of $\rho=0$.

Note that the hypotheses of this lemma later on end up being the results of
Lemma 5.3.

###### Proof.

The assumptions give uniform bounds $a_{-1}\geq C|\tau|^{2}$ everywhere, as
well as bounds $|a_{2}|\geq Cx$ near $\Sigma$, and $a_{-3}>C$ in a
neighborhood of $\Sigma\cap\\{x=0\\}$. We first analyze

$a^{0}:=a_{-1}(x,y,\tau)\rho+a_{-2}(x,y,0)\rho^{2}+a_{-3}(0,y,0)\rho^{3}.$

Since $a_{-1}$ and $a_{-3}$ are real (in fact nonnegative), while $a_{-2}$ is
purely imaginary, we have

$\displaystyle|a^{0}|\geq C(|a_{-1}\rho+a_{-3}\rho^{3}|+|a_{-2}\rho^{2}|)$
$\displaystyle\geq C\rho(|\tau|^{2}+\rho^{2}+x\rho)$ $\displaystyle=C\rho
d_{\Sigma}^{2}\left(|\tilde{\tau}|^{2}+d_{\Sigma}^{2}\left(\frac{\rho}{d_{\Sigma}^{2}}\right)^{2}+x\frac{\rho}{d_{\Sigma}^{2}}\right)$
$\displaystyle=C\rho
d_{\Sigma}^{2}d_{\Gamma}^{2}\left(\frac{|\tilde{\tau}|^{2}}{d_{\Gamma}^{2}}+\frac{d_{\Sigma}^{2}}{d_{\Gamma}^{2}}\left(\frac{\rho}{d_{\Sigma}^{2}}\right)^{2}+\frac{x}{d_{\Gamma}^{2}}\frac{\rho}{d_{\Sigma}^{2}}\right).$

On the region $\frac{\rho}{d_{\Sigma}^{2}}\geq 1/2$, the factor in the
parentheses on the RHS is at least

$\frac{|\tilde{\tau}|^{2}}{d_{\Gamma}^{2}}+\frac{d_{\Sigma}^{2}}{d_{\Gamma}^{2}}\left(\frac{1}{2}\right)^{2}+\frac{x}{d_{\Gamma}^{2}}\cdot\frac{1}{2}\geq\frac{1}{4}\left(\frac{|\tilde{\tau}|^{2}}{d_{\Gamma}^{2}}+\frac{d_{\Sigma}^{2}}{d_{\Gamma}^{2}}+\frac{x}{d_{\Gamma}^{2}}\right)=\frac{1}{4}.$

Otherwise, since $\frac{\rho}{d_{\Sigma}^{2}}+|\tilde{\tau}|^{2}=1$, we have
$|\tilde{\tau}|^{2}\geq 1/2$, and we can always w.l.o.g. bound $d_{\Gamma}$
from above, so the term $|\tilde{\tau}|^{2}/d_{\Gamma}^{2}$ would be bounded
from below. In any case, we have the desired ellipticity bound.

We now analyze how $a$ differs from $a^{0}$. Note that

$a_{-2}(x,y,\tau)-a_{-2}(x,y,0)=i\tau\cdot\tilde{a}_{-2}(x,y,\tau)$

where $\tilde{a}_{-2}$ is a real-(vector-)valued symbol in $S^{0,0}$, and

$a_{-3}(x,y,\tau)-a_{-3}(0,y,0)=O(x)+O(\tau)$

and

$a-\sum_{j=1}^{3}{a_{-j}\rho^{j}}=O(\rho^{4}).$

Here, the “big-O” notation should be interpreted as saying that $O(f)$ means
$f$ times a symbol in $S^{0,0}$, or in the case of $O(\tau)$, meaning a
$S^{0,0}$ combination of the functions $\tau_{i}$. Thus overall we have

$a-a^{0}=i\rho^{2}\tau\cdot\tilde{a}_{-2}+O(x\rho^{3})+O(\tau\rho^{3})+O(\rho^{4}).$

We first show $a^{0}+i\rho^{2}\tau\cdot\tilde{a}_{-2}$ still satisfies the
ellipticity bound; note that the additional term is still in $S^{-1,0,-2,-2}$
since $\rho^{2}\tau_{i}\in S^{-2,0,-1,-1}\subset S^{-1,0,-2,-2}$). Note that
the additional term $i\rho^{2}\tau\cdot\tilde{a}_{-2}$ is purely imaginary,
which helps in that it doesn’t “compete” with the real-valued terms in
$\tilde{a}$. We can argue as follows:

  * •

Suppose we are in the region where $|\tau\cdot\tilde{a}_{-2}|\leq|a_{-2}|/2$.
Then the additional term $i\rho^{2}\tau\cdot\tilde{a}_{-2}$ can be absorbed
into the $a_{-2}\rho^{2}$ term.

  * •

Otherwise, we have $|\tau\cdot\tilde{a}_{-2}|\geq|a_{-2}|/2$. From the bound
$|a_{-2}|\geq Cx$, in this region we necessarily have $|\tau|\geq Cx/2$ for
$c=C/2$. This implies $x\leq 2C^{-1}d_{\Sigma}|\tilde{\tau}|\leq
C^{-1}(d_{\Sigma}^{2}+|\tilde{\tau}|^{2})$, and thus

$d_{\Gamma}^{2}=x+d_{\Sigma}^{2}+|\tilde{\tau}|^{2}\leq(1+C^{-1})(d_{\Sigma}^{2}+|\tilde{\tau}|^{2}),$

and hence $d_{\Sigma}^{2}+|\tilde{\tau}|^{2}\geq(1+C^{-1})^{-1}d_{\Gamma}^{2}$
in this region. Then

$\displaystyle|a^{0}+i\rho^{2}\tau\cdot\tilde{a}_{-2}|$
$\displaystyle\geq|\text{Re}(\tilde{a}+i\rho^{2}\tau\cdot\tilde{a}_{-2})|$
$\displaystyle=a_{-1}\rho+a_{-3}\rho^{3}$ $\displaystyle\geq
C^{\prime}\rho(|\tau|^{2}+\rho^{2})$ $\displaystyle=C^{\prime}\rho
d_{\Sigma}^{2}(|\tilde{\tau}|^{2}+d_{\Sigma}^{2}(1-|\tilde{\tau}|^{2})^{2}).$

The term in the parentheses is bounded from below by $d_{\Gamma}^{2}$: indeed,
in the region where $|\tilde{\tau}|\geq 1/2$ we can take $d_{\Gamma}$ to be
bounded from above, while in the region where $|\tilde{\tau}|\leq 1/2$ we can
use the above estimate to see that in this region we have

$|\tilde{\tau}|^{2}+d_{\Sigma}^{2}(1-|\tilde{\tau}|^{2})^{2}\geq|\tilde{\tau}|^{2}+\frac{9}{16}d_{\Sigma}^{2}\geq
cd_{\Gamma}^{2}$

for $c=\frac{9}{16(1+C^{-1})}$.

Thus, $a^{0}+i\rho^{2}\tau\cdot\tilde{a}_{-2}$ satisfies elliptic estimates,
regardless of whatever $\tilde{a}_{-2}$ happens to be555It turns out to vanish
at $\\{x=0,\tau=0\\}$ which would have made computations easier, but this fact
is not needed for this argument.

We now show the remaining terms can be absorbed into the elliptic estimate, at
least near fiber infinity $\rho=0$. Indeed, note that $x\leq d_{\Gamma}^{2}$,
$|\tau_{j}|=d_{\Sigma}|\tilde{\tau}_{j}|\leq d_{\Sigma}d_{\Gamma}\leq
d_{\Gamma}^{2}$, and $\rho\leq d_{\Sigma}^{2}$, and hence

$x\rho^{3}\leq\rho^{2}d_{\Sigma}^{2}d_{\Gamma}^{2},\quad\tau_{j}\rho^{3}\leq\rho^{2}d_{\Sigma}^{2}d_{\Gamma}^{2},\quad\rho^{4}\leq\rho^{2}d_{\Sigma}^{4}\leq\rho^{2}d_{\Sigma}^{2}d_{\Gamma}^{2}.$

Hence the remaining terms are $\rho O(\rho d_{\Sigma}^{2}d_{\Gamma}^{2})$, and
hence the lower bound on the main terms allow the remaining terms to be
absorbed into the elliptic estimate, at least near $\rho=0$. This shows that
$a$ satisfies the elliptic estimate near fiber infinity. ∎

We note that this lemma does not establish ellipticity behavior at finite
points; such estimates would need to be proven separately.

We now establish a criterion for a class of symbols to belong to the
$bb$-symbol class that will be useful for proving certain inclusions:

######  Lemma 3.8.

Suppose $S$ is a subspace of functions which are smooth on the interior
${{}^{sc}T^{*}M}$ which satisfy the following properties:

  1. (1)

Every function $a\in S$ satisfies a bound $|a|\leq
C_{a}\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}$ for some constant $C_{a}$.

  2. (2)

$S$ is closed under the action of $\mathcal{V}_{2}$, that is we have $Va\in S$
whenever $a\in S$ and $V\in\mathcal{V}_{2}$.

Then $S\subset S^{m,l,k,j}$.

###### Proof.

For any product $V^{\alpha}$ of vector fields in $\mathcal{V}_{2}$, we have
that $V^{\alpha}a\in S$ for any $a\in S$ by repeated applications of the
second property in the hypothesis. The first property in the hypothesis then
gives $|V^{\alpha}a|\leq
C_{\alpha}\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}$, as desired. ∎

######  Corollary 3.9.

Suppose $S$ is a subspace of functions which are smooth on the interior
${{}^{sc}T^{*}M}$ which satisfy the following properties:

  1. (1)

Every function $a\in S$ satisfies a bound $|a|\leq
C_{a}\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}$ for some constant $C_{a}$.

  2. (2)

For any $a\in S$ and any $V\in\mathcal{V}_{2}$, we have $Va\in
S^{0,0,0,0}\otimes S$.

Then $S\subset S^{m,l,k,j}$.

###### Proof.

Let $\tilde{S}=S^{0,0,0,0}\otimes S$. Then $\tilde{S}$ satisfies the
hypothesis in Lemma 3.8, so $\tilde{S}\subset S^{m,l,k,j}$. Since
$S\subset\tilde{S}$ as well since $1\in S^{0,0,0,0}$, it follows that
$S\subset S^{m,l,k,j}$, as desired. ∎

Having introduced the class of symbols we wish to consider, we now consider a
slightly modified version of the class of vector fields used to test the
regularity of these symbols. We will consider
$\mathcal{V}_{2,c}:=S^{0,0,0,0}\otimes\mathcal{V}_{2}$, the space of vector
fields whose coefficients are in $S^{0,0,0,0}$ (i.e. functions which are not
necessarily smooth up to $\partial\mathscr{X}_{2}$, but merely conormal to
$\partial\mathscr{X}_{2}$).

######  Lemma 3.10.

For any product $V^{\alpha}$ of vector fields in $\mathcal{V}_{2,c}$, we have
$V^{\alpha}a\in S^{m,l,k,j}$ for $a\in S^{m,l,k,j}$.

###### Proof.

It suffices by induction to show this when $|\alpha|=1$. In that case, we can
write $V=\sum{b_{i}V_{i}}$ with $b_{i}\in S^{0,0,0,0}$ and
$V_{i}\in\mathcal{V}_{2}$, in which case $b_{i}V_{i}a\in S^{m,l,k,j}$ as
$V_{i}a\in S^{m,l,k,j}$. ∎

We have a version of Corollary 2.9:

######  Lemma 3.11.

$\mathcal{V}_{2,c}$ is generated over $S^{0,0,0,0}$ by the vector fields

(3.2)
$x\partial_{x},\partial_{y_{j}},\rho\partial_{\rho},\tau_{i}\partial_{\tau_{j}},\rho\partial_{\tau_{j}},x^{1/2}\rho^{1/2}\partial_{\tau_{j}},1\leq
i,j\leq n-1.$

###### Proof.

We first show that $\rho\partial_{\tau_{j}}$ and
$x^{1/2}\rho^{1/2}\partial_{\tau_{j}}$ do belong to $\mathcal{V}_{2,c}$.
Indeed, since $\rho\in S^{-1,0,0,0}\subset S^{0,0,-1,-1}$ and
$x^{1/2}\rho^{1/2}\in S^{-1/2,-1/2,0,0}\subset S^{0,0,-1,-1}$, it follows that
both $\rho$ and $x^{1/2}\rho^{1/2}$ are $S^{0,0,0,0}$ multiples of
$d_{\Sigma}d_{\Gamma}$, and hence these vector fields are a $S^{0,0,0,0}$
multiple of $d_{\Sigma}d_{\Gamma}\partial_{\tau_{j}}$, which are in
$\mathcal{V}_{2}$.

From Corollary 2.9, it now suffices to show that
$d_{\Gamma}d_{\Sigma}\partial_{\tau_{j}}$ can be obtained as a $S^{0,0,0,0}$
combination of the vector fields $\rho\partial_{\tau_{j}}$,
$x^{1/2}\rho^{1/2}\partial_{\tau_{j}}$, and $\tau_{i}\partial_{\tau_{j}}$; in
particular it suffices to show that $d_{\Sigma}d_{\Gamma}$ can be written as a
$S^{0,0,0,0}$ combination of $\rho$, $\rho^{1/2}x^{1/2}$, and $\tau_{i}$. Note

$d_{\Sigma}-\rho^{1/2}=\frac{d_{\Sigma}^{2}-\rho}{d_{\Sigma}+\rho^{1/2}}=\frac{|\tau|^{2}}{d_{\Sigma}+\rho^{1/2}}=\sum_{i=1}^{n-1}{\frac{\tau_{i}}{d_{\Sigma}+\rho^{1/2}}\tau_{i}}.$

We note that $\rho^{1/2}\in S^{-1/2,0,0,0}\subset S^{0,0,-1,0}$, and hence
$d_{\Sigma}+\rho^{1/2}\in S^{0,0,-1,0}$, and is elliptic as an element of
$S^{0,0,-1,0}$ since $|d_{\Sigma}+\rho^{1/2}|\geq d_{\Sigma}$; hence we have
$\frac{1}{d_{\Sigma}+\rho^{1/2}}\in S^{0,0,1,0}$. Moreover, from (3.1), we
have $\tau_{i}\in S^{0,0,-1,-1}$. Thus, we have

$d_{\Sigma}=\rho^{1/2}+\sum_{i=1}^{n-1}{a_{i}\tau_{i}},\quad
a_{i}=\frac{\tau_{i}}{d_{\Sigma}+\rho^{1/2}}\in S^{0,0,0,-1}.$

Similarly, we have

$\displaystyle d_{\Gamma}=x^{1/2}+\frac{d_{\Gamma}^{2}-x}{d_{\Gamma}+x^{1/2}}$
$\displaystyle=x^{1/2}+\frac{d_{\Sigma}^{2}+|\tilde{\tau}|^{2}}{d_{\Gamma}+x^{1/2}}$
$\displaystyle=x^{1/2}+\frac{\rho+(d_{\Sigma}^{2}+1)|\tilde{\tau}|^{2}}{d_{\Gamma}+x^{1/2}}$
$\displaystyle=x^{1/2}+\frac{\rho^{1/2}}{d_{\Gamma}+x^{1/2}}\rho^{1/2}+\sum_{i=1}^{n-1}{\frac{(1+d_{\Sigma}^{2})\tilde{\tau}_{i}}{d_{\Gamma}+x^{1/2}}\tilde{\tau}_{i}}$
$\displaystyle=x^{1/2}+\frac{\rho^{1/2}}{d_{\Gamma}+x^{1/2}}\rho^{1/2}+\sum_{i=1}^{n-1}{(1+d_{\Sigma}^{2})\frac{d_{\Gamma}}{d_{\Gamma}+x^{1/2}}\frac{\tilde{\tau}_{i}}{d_{\Gamma}}\frac{1}{d_{\Sigma}}\tau_{i}}.$

As before, we have that $d_{\Gamma}+x^{1/2}$ is elliptic as an element of
$S^{0,0,0,-1}$, and hence $\frac{1}{d_{\Gamma}+x^{1/2}}\in S^{0,0,0,1}$.
Hence,

$\rho^{1/2}\in S^{0,0,-1,0}\implies\frac{\rho^{1/2}}{d_{\Gamma}+x^{1/2}}\in
S^{0,0,-1,1}\subset S^{0,0,0,0}$

and

$(1+d_{\Sigma}^{2})\frac{d_{\Gamma}}{d_{\Gamma}+x^{1/2}}\frac{\tilde{\tau}_{i}}{d_{\Gamma}}\frac{1}{d_{\Sigma}}\in
S^{0,0,1,0}.$

Hence, we have

$d_{\Gamma}=x^{1/2}+b\rho^{1/2}+\sum_{i=1}^{n-1}c_{i}\tau_{i},\quad
b=\frac{\rho^{1/2}}{d_{\Gamma}+x^{1/2}}\in
S^{0,0,0,0},c_{i}=(1+d_{\Sigma}^{2})\frac{d_{\Gamma}}{d_{\Gamma}+x^{1/2}}\frac{\tilde{\tau}_{i}}{d_{\Gamma}}\frac{1}{d_{\Sigma}}\in
S^{0,0,1,0}.$

Thus we have

$\displaystyle d_{\Gamma}d_{\Sigma}$
$\displaystyle=d_{\Gamma}\rho^{1/2}+\sum_{i=1}^{n-1}{d_{\Gamma}a_{i}\tau_{i}}$
$\displaystyle=\left(x^{1/2}+b\rho^{1/2}+\sum_{i=1}^{n-1}c_{i}\tau_{i}\right)\rho^{1/2}+\sum_{i=1}^{n-1}{d_{\Gamma}a_{i}\tau_{i}}$
$\displaystyle=x^{1/2}\rho^{1/2}+b\rho+\sum_{i=1}^{n-1}{(c_{i}\rho^{1/2}+d_{\Gamma}a_{i})\tau_{i}}.$

Noting that $a_{i}\in S^{0,0,0,-1}\implies d_{\Gamma}a_{i}\in
S^{0,0,0,-2}\subset S^{0,0,0,0}$ and $\rho^{1/2}\in S^{0,0,-1,0},c_{i}\in
S^{0,0,1,0}\implies\rho^{1/2}c_{i}\in S^{0,0,0,0}$, we see that
$d_{\Gamma}d_{\Sigma}$ can indeed be written as a $S^{0,0,0,0}$ combination of
$x^{1/2}\rho^{1/2}$, $\rho$, and $\tau_{i}$. ∎

######  Corollary 3.12.

A function $a:{{}^{sc}T^{*}M}\to\mathbb{C}$ which is smooth on the interior
belongs to $S^{m,l,k,j}$ if and only if

(3.3) $|W^{\alpha}V^{\beta}a|\leq
C_{\alpha,\beta}\min(\rho^{-m-|\alpha|}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j},\rho^{-m-|\alpha|/2}x^{-l-|\alpha|/2}d_{\Sigma}^{-k}d_{\Gamma}^{-j})$

whenever $W^{\alpha}$ and $V^{\beta}$ are products vector fields on
$\mathscr{X}$ which are homogeneous of degree $0$, with the latter a product
of vector fields also tangent to $S$.

We compare this result to the definition of the Boutet de Monvel calculus
developed in [1], where the symbolic estimates are stated in the above manner
of losing decay upon applying vector fields which are not tangent to $S$, as
opposed to symbolic estimates formulated in terms of considering vector fields
tangent to the boundary of a blow-up.

We now compare the $bb$-class of symbols to the “Hörmander classes” of
symbols:

######  Definition 3.2.

The symbol class $S_{\delta}^{m,l}$ where $\delta\in[0,1]$ consists of
functions $a:{{}^{sc}T^{*}M}\to\mathbb{C}$ which are smooth on the interior
and satisfy the estimates

$|V^{\alpha}a|\leq C_{\alpha}\rho^{-m-\delta|\alpha|}x^{-l-\delta|\alpha|}$

whenever $V^{\alpha}$ is a product of vector fields tangent to
$\partial\mathscr{X}$.

Then we have the following relation between the Hörmander classes of symbols
and the $bb$-class of symbols, using Corollary 3.12:

######  Proposition 3.13.

We have $S_{0}^{m,l}\subset S^{m,l,0,0}\subset S_{1/2}^{m,l}$.

###### Proof.

For the first inclusion666For the case $m=l=0$, the statement is just that
distributions conormal to $\partial\mathscr{X}$ are also distributions
conormal to $\partial\mathscr{X}_{2}$ by identifying the interiors of
$\mathscr{X}$ and $\mathscr{X}_{2}$. This is a standard fact using that the
blow-down map $\beta:\mathscr{X}_{2}\to\mathscr{X}$ is a $b$-map, cf. Chapter
6 in [4], we note that any $a\in S_{0}^{m,l}$ certainly satisfies the estimate
in (3.3), as it satisfies the stronger estimate $|W^{\alpha}V^{\beta}a|\leq
C\rho^{-m}x^{-l}$ that does not depend on whether the vector fields being
applied are tangent to $S$ or not. Hence, by Corollary 3.12, we have that
$a\in S^{m,l,0,0}$.

For the second inclusion, we use Proposition 3.4 and Part 4 of Proposition 3.5
(with $s=0$ and $t=1$) to note that if $V^{\alpha}$ is a product of vector
fields tangent to $\partial\mathscr{X}$, we have

$V^{\alpha}a\in S^{m,l,|\alpha|,|\alpha|}\subset
S^{m+|\alpha|/2,l+|\alpha|/2,0,0}$

and hence $|V^{\alpha}a|\leq
C_{\alpha}\rho^{-(m+|\alpha|/2)}x^{-(l+|\alpha|)}$. This shows $a\in
S_{1/2}^{m,l}$. ∎

For scattering symbols, there is a well-defined notion of principal symbol
identifying elements of $S^{m,l}$ with its equivalence class in
$S^{m,l}/S^{m-1,l-1}$. It turns out for the $bb$-class of symbols that the
correct “lower-order” space for symbols $S^{m,l,k,j}$ is
$S^{m-1,l-1,k+1,j+1}$. Since

$S^{m-1,l-1,k+1,j+1}\subset S^{m-1/2,l-1/2,k,j},$

we see that elements in $S^{m-1,l-1,k+1,j+1}$ are at least $1/2$ order better
in both $x$ and $\rho$ as elements in $S^{m,l,k,j}$, so it is appropriate to
say that those elements are of lower-order. We formalize the subsequent notion
of principal symbol as follows:

######  Definition 3.3.

For $a\in S^{m,l,k,j}$, its _principal symbol_ $\sigma(a)$ is the equivalence
class of $a$ in $S^{m,l,k,j}/S^{m-1,l-1,k+1,j+1}$.

In addition, we want to characterize the class of “residual” symbols which
belong in spaces of “arbitrary lower order,” i.e. of spaces of the form
$S^{m-i,l-i,k+i,j+i}$ for all $i\geq 0$. It turns out that this class
coincides with the space of residual scattering symbols:

######  Proposition 3.14.

Let

$S^{-\infty,-\infty}=\cap_{m,l}{S^{m,l}}$

be the class of residual scattering symbols. Then, for any fixed $(m,l,k,j)$,
we have

$\cap_{i\geq 0}{S^{m-i,l-i,k+i,j+i}}=S^{-\infty,-\infty}.$

###### Proof.

Since

$S^{-N,-N}\subset S^{-N,-N,0,0}\subset S^{-N+(-k)_{+}/2,-N+(-j)_{+}/2,k,j}$

for any $N>0$, it follows that we have $S^{-N,-N}\subset S^{m,l,k,j}$ for $N$
sufficiently large. Any element in $S^{-\infty,-\infty}$ belongs to
$S^{-N,-N}$ for any $N>0$, so it follows that any element in
$S^{-\infty,-\infty}$ belongs in $S^{m,l,k,j}$. The same logic applies to
$S^{m-i,l-i,k+i,j+i}$ for any $i\geq 0$, so we have the inclusion
$S^{-\infty,-\infty}\subset S^{m-i,l-i,k+i,j+i}$.

Conversely, if $a\in\cap_{i\geq 0}{S^{m-i,l-i,k+i,j+i}}$, we’d like to show
that $a\in S^{-N,-N}$ for any fixed $N$. We thus need to show that for any
product $V^{\alpha}$ of vector fields tangent to $\partial\mathscr{X}$, we
have

$|V^{\alpha}a|\leq C\rho^{N}x^{N}.$

Using Propositions 3.13 and 3.5, we have

$S^{m-i,l-i,k+i,j+i}\subset S^{m-i+(k+i)_{+},l-i+(j+i)_{+},0,0}\subset
S_{1/2}^{m-i+(k+i)_{+},l-i+(j+i)_{+}},$

so it follows that by taking $i$ sufficiently large we have

$a\in S_{1/2}^{-(N+|\alpha|/2),-(N+|\alpha|/2)}.$

But then this would give

$|V^{\alpha}a|\leq
C\rho^{(N+|\alpha|/2)-|\alpha|/2}x^{(N+|\alpha|/2)-|\alpha|/2}=C\rho^{N}x^{N},$

as desired. ∎

To establish the theory of the quantization of these symbols, it will be
necessary to consider symbols depending on more variables. We give the formal
definition here:

######  Definition 3.4.

Define $S^{0,0,0,0}_{full}$ to consist of functions
$a(x,y,X,Y,\xi,\eta):(^{sc}T^{*}M)_{x,y,\xi,\eta}\times\mathbb{R}^{n}_{X,Y}\to\mathbb{C}$
which satisfy the estimates $|V^{\alpha}a|\leq C$ whenever $V^{\alpha}$ is a
product of the following vector fields:

  * •

Vector fields with no component in the $(X,Y)$ direction which are also
tangent to $\partial\mathscr{X}_{2}\times\mathbb{R}^{n}_{X,Y}$,

  * •

The vector fields $\frac{1}{x}\partial_{X}$ and $\frac{1}{x}\partial_{Y_{j}}$.

Define
$S^{m,l,k,j}_{full}:=\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}S^{0,0,0,0}_{full}$.

It follows via the arguments above that we can rephrase the estimates as
requiring

$|V^{\alpha}D^{\beta}_{X,Y}a|\leq
C_{\alpha,\beta}\rho^{-m}x^{-l+|\beta|}d_{\Sigma}^{-k}d_{\Gamma}^{-j}$

whenever $V^{\alpha}$ is a product of vector fields on $\mathscr{X}$ which are
tangent to $\partial\mathscr{X}_{2}$ (interpreted as vector fields on
${}^{sc}T^{*}M\times\mathbb{R}^{n}_{X,Y}$ with no $(X,Y)$ components).

##  4\. Operators quantized by the $bb$-symbol class

###  4.1. Oscillatory integral distributions and left reduction

We want to consider operators whose Schwartz kernels are of the form

(4.1) $(2\pi)^{-n}\left(\int_{\mathbb{R}^{n}}{e^{i(\xi X+\eta\cdot
Y)}a(x,y,\xi,\eta)\,d\xi\,d\eta}\right)\,dX\,dY$

with $a\in S^{m,l,k,j}$ where $X=\frac{x^{\prime}-x}{x^{2}}$ and
$Y=\frac{y^{\prime}-y}{x}$, and we interpret $(x,y,\xi,\eta)$ as the element
$\xi\frac{dx}{x^{2}}+\eta\cdot\frac{dy}{x}\in^{sc}T^{*}M$. (The above
expression for the right density should be interpreted as saying that the
corresponding operator is given by

$Au(x,y)=(2\pi)^{-n}\int_{\mathbb{R}^{n}_{\xi,\eta}\times\mathbb{R}^{n}_{X,Y}}{e^{i(\xi
X+\eta\cdot Y)}a(x,y,\xi,\eta)u(x+x^{2}X,y+xY)\,d\xi\,d\eta\,dX\,dY};$

note that $x^{\prime}=x+x^{2}X$ and $y^{\prime}=y+xY$.)

To study the properties of such operators, we first consider the oscillatory
integral behavior of the Schwartz kernel viewed as a distribution in the
variables $X$ and $Y$, before assigning the interpretation of $X$ and $Y$ as
coordinates on the scattering double space. As such, consider oscillatory
integral distributions of the form

(4.2) $K_{a}(x,y,X,Y)=(2\pi)^{-n}\int_{\mathbb{R}^{n}}{e^{i(\xi X+\eta\cdot
Y)}a(x,y,\xi,\eta)\,d\xi\,d\eta},\quad a\in S^{m,l,k,j}.$

Here, we consider $X$ and $Y$ just as variables; later we will include the
interpretation $X=\frac{x^{\prime}-x}{x^{2}}$, and $Y=\frac{y^{\prime}-y}{x}$.
We view this as a family of distributions on $\mathbb{R}^{n}_{X,Y}$
parametrized by $(x,y)$. We can thus write

$K_{a}(x,y,X,Y)=\mathcal{F}^{-1}_{(\xi,\eta)\to(X,Y)}(a(x,y,\cdot,\cdot))$

where $\mathcal{F}^{-1}_{(\xi,\eta)\to(X,Y)}$ is the inverse Fourier transform
in the $(\xi,\eta)$ variables.

It will be convenient to consider more general oscillatory integral
distributions of the form

(4.3) $K_{\tilde{a}}(x,y,X,Y)=(2\pi)^{-n}\int_{\mathbb{R}^{n}}{e^{i(\xi
X+\eta\cdot Y)}\tilde{a}(x,y,X,Y\xi,\eta)\,d\xi\,d\eta},\quad\tilde{a}\in
S^{m,l,k,j}_{full}.$

We show that this more general class in fact coincides with our first
definition. Formally, we write:

######  Proposition 4.1.

Let $\tilde{a}\in S^{m,l,k,j}_{full}$ and let $K_{\tilde{a}}$ be defined as in
(4.3). Then there is a symbol $a\in S^{m,l,k,j}$ such that
$K_{a}=K_{\tilde{a}}$, where $K_{a}$ is defined as in (4.2). Moreover, $a$
satisfies an asymptotic series formula

$a(x,y,\xi,\eta)\sim\sum_{\alpha}{\frac{i^{|\alpha|}}{\alpha!}\partial^{\alpha}_{(\xi,\eta)}\partial^{\alpha}_{(X,Y)}\tilde{a}(x,y,0,0,\xi,\eta)},$

in that
$a-\sum_{|\alpha|<N}{\frac{i^{|\alpha|}}{\alpha!}\partial_{(\xi,\eta)}^{\alpha}\partial_{(X,Y)}^{\alpha}\tilde{a}|_{(X,Y)=(0,0)}}\in
S^{m-N,l-N,k+N,j+N}$ for every $N$. In particular,
$\sigma(a)=\sigma(\tilde{a}|_{(X,Y)=(0,0)})$.

Note that $a$ can be recovered from $K_{a}$ by taking the Fourier transform
$a(x,y,\xi,\eta)=\mathcal{F}_{(X,Y)\to(\xi,\eta)}(K_{a}(x,y,\cdot,\cdot))$.
Hence, to prove Proposition 4.1, it suffices to show that
${\mathcal{F}_{(X,Y)\to(\xi,\eta)}(K_{\tilde{a}}(x,y,\cdot,\cdot))}$ belongs
in $S^{m,l,k,j}$. The proposition follows from the following two lemmas:

######  Lemma 4.2.

Let $\tilde{a}\in S^{m,l,k,j}_{full}$ and $K_{\tilde{a}}$ be defined as in
(4.3), and let
$a(x,y,\xi,\eta)=\mathcal{F}_{(X,Y)\to(\xi,\eta)}(K_{\tilde{a}}(x,y,\cdot,\cdot))$.
Then for every $N>0$ there exists $a_{N}\in S^{m,l,k,j}$ and $\tilde{a}_{N}\in
S^{-N,-N,0,0}_{full}$ such that

$K_{\tilde{a}}=K_{a_{N}}+K_{\tilde{a}_{N}}.$

Moreover, the $a_{N}$ can be chosen to satisfy

$a_{N}=\sum_{|\alpha|\leq
N^{\prime}}{\frac{i^{|\alpha|}}{\alpha!}\partial^{\alpha}_{(\xi,\eta)}\partial^{\alpha}_{(X,Y)}\tilde{a}(x,y,0,0,\xi,\eta)}$

for sufficiently large $N^{\prime}$.

######  Lemma 4.3.

Suppose $\tilde{a}\in S^{-N,-N,0,0}_{full}$, and let
$a=\mathcal{F}_{(X,Y)\to(\xi,\eta)}{K_{\tilde{a}}(x,y,\cdot,\cdot)}$. Then $a$
satisfies the estimate

$|V^{\beta}a|\leq C\rho^{N-(|\beta|/2+n+1)}x^{N}$

whenever $V^{\beta}$ is a product of vector fields tangent to
$\partial\mathscr{X}_{2}$.

Lemma 4.3 is essentially a weak version of Proposition 4.1, since the
requirements for symbolicity is that it satisfies the estimate there for an
arbitrary number of vector fields. However, the main content of the
Proposition is in Lemma 4.2, which essentially gives a finite number of terms
in an asymptotic expansion with a very weak remainder; the purpose of Lemma
4.3 is to give an estimate on this remainder.

###### Proof of Lemma 4.2.

The proof is the same as the classical case: if we write $Z=(X,Y)$ and
suppress all other variables, then we use Taylor’s theorem with remainder to
write

$\tilde{a}(Z)=\tilde{a}(0)+\sum_{|\alpha|\leq
N^{\prime}}{Z^{\alpha}\frac{\partial_{Z}^{\alpha}\tilde{a}(0)}{\alpha!}}+\sum_{|\beta|=N^{\prime}+1}{Z^{\beta}R_{\beta}\tilde{a}(Z)}$

where

$R_{\beta}\tilde{a}(Z)=\frac{|\beta|}{\beta!}\int_{0}^{1}{(1-t)^{|\beta|-1}\partial_{Z}^{\beta}\tilde{a}(tX)\,dt}.$

Note that if $\tilde{a}\in S^{m,l,k,j}_{full}$, then $R_{\beta}\tilde{a}\in
S^{m,l-|\beta|,k,j}_{full}$ for any $\beta$. It follows that

$\displaystyle\int{e^{i(\xi X+\eta\cdot Y)}\tilde{a}(Z)\,d\xi\,d\eta}$
$\displaystyle=\sum_{|\alpha|\leq
N^{\prime}}{\frac{1}{\alpha!}\int{Z^{\alpha}e^{i(\xi,\eta)\cdot
Z}\partial_{Z}^{\alpha}\tilde{a}(0)\,d\xi\,d\eta}}+\sum_{|\beta|=N^{\prime}+1}{\int{Z^{\beta}e^{i(\xi,\eta)\cdot
Z}R_{\beta}\tilde{a}(Z)\,d\xi\,d\eta}}$
$\displaystyle=\int{e^{i(\xi,\eta)\cdot Z}\left(\sum_{|\alpha|\leq
N^{\prime}}{\frac{i^{|\alpha|}}{\alpha!}\partial^{\alpha}_{(\xi,\eta)}\partial^{\alpha}_{Z}\tilde{a}(0)}+\sum_{|\beta|=N^{\prime}+1}{i^{|\beta|}\partial^{\beta}_{(\xi,\eta)}R_{\beta}(Z)}\right)\,d\xi\,d\eta}.$

Notice that $\xi$ and $\eta$ derivatives are $\rho$ times a vector field
tangent to $\partial\mathscr{X}$, so
$\partial^{\alpha}_{(\xi,\eta)}\partial^{\alpha}_{Z}\tilde{a}(0)\in
S^{m-|\alpha|,l-|\alpha|,k+|\alpha|,j+|\alpha|}$, and
$\partial^{\beta}_{(\xi,\eta)}R_{\beta}(Z)\in
S^{m-|\beta|,l-|\beta|,k+|\beta|,j+|\beta|}_{full}=S^{m-(N^{\prime}+1),l-(N^{\prime}+1),k+(N^{\prime}+1),j+(N^{\prime}+1)}_{full}$
when $|\beta|=N^{\prime}+1$. We note that, for a fixed $N>0$, we have

$S^{m-(N^{\prime}+1),l-(N^{\prime}+1),k+(N^{\prime}+1),j+(N^{\prime}+1)}_{full}\subset
S^{-N,-N,0,0}_{full}$

for sufficiently large $N^{\prime}$. Thus, for $N^{\prime}$ sufficiently
large, if we let

$a_{N}=\sum_{|\alpha|\leq
N^{\prime}}{\frac{i^{|\alpha|}}{\alpha!}\partial^{\alpha}_{(\xi,\eta)}\partial^{\alpha}_{Z}\tilde{a}(0)}\quad\text{and}\quad\tilde{a}_{N}=\sum_{|\beta|=N^{\prime}+1}{i^{|\beta|}\partial^{\beta}_{(\xi,\eta)}R_{\beta}(Z)}$

then $a_{N}\in S^{m,l,k,j}$, $\tilde{a}_{N}\in S^{-N,-N,0,0}_{full}$, and
$K_{\tilde{a}}=K_{a_{N}}+K_{\tilde{a}_{N}}$, as desired.∎

###### Proof of Lemma 4.3.

We recall from Lemma 3.11 that $\mathcal{V}_{2,c}$ is generated over
$S^{0,0,0,0}$ by the vector fields

$x\partial_{x},\partial_{y_{j}},\rho\partial_{\rho},\tau_{i}\partial_{\tau_{j}},\rho\partial_{\tau_{j}},x^{1/2}\rho^{1/2}\partial_{\tau_{j}}.$

Rewriting these vector fields in the $(\xi,\eta)$ coordinates, we see that
$\mathcal{V}_{2,c}$ is generated over $S^{0,0,0,0}$ by the vector fields

(4.4)
$x\partial_{x},\partial_{y_{j}},\partial_{\xi},\partial_{\eta_{j}},\xi\partial_{\xi},\xi\partial_{\eta_{j}},\eta_{i}\partial_{\xi},\eta_{i}\partial_{\eta_{j}},\eta_{1}\partial_{\eta_{1}},x^{1/2}\langle(\xi,\eta)\rangle^{1/2}\partial_{\xi},x^{1/2}\langle(\xi,\eta)\rangle^{1/2}\partial_{\eta_{i}}.$

It suffices to prove the lemma when $V^{\beta}$ is a product of the above
vector fields. We have the Fourier transform commutation relations

$\xi\mathcal{F}_{(X,Y)\to(\xi,\eta)}K=\mathcal{F}_{(X,Y)\to(\xi,\eta)}{D_{X}K},\eta_{i}\mathcal{F}_{(X,Y)\to(\xi,\eta)}K=\mathcal{F}_{(X,Y)\to(\xi,\eta)}{D_{Y_{i}}K}$

and

$D_{\xi}\mathcal{F}_{(X,Y)\to(\xi,\eta)}K=\mathcal{F}_{(X,Y)\to(\xi,\eta)}{(-XK)},D_{\eta_{i}}\mathcal{F}_{(X,Y)\to(\xi,\eta)}K=\mathcal{F}_{(X,Y)\to(\xi,\eta)}{(-Y_{i}K)}.$

We have similar relations in applying differentiation/multiplication in
$(X,Y)$ to (4.3); the only difference is that the symbol also depends on
$(X,Y)$, so applying a derivative in $(X,Y)$ results in another term.
Nonetheless, we have

$XK_{\tilde{a}}=K_{-D_{\xi}\tilde{a}},Y_{i}K_{\tilde{a}}=K_{-D_{\eta_{i}}\tilde{a}}$

and

(4.5)
$D_{X}K_{\tilde{a}}=K_{\xi\tilde{a}+D_{X}\tilde{a}},D_{Y_{i}}K_{\tilde{a}}=K_{\eta_{i}\tilde{a}+D_{Y_{i}}\tilde{a}}.$

Thus, we have

$\displaystyle\eta_{i}\partial_{\eta_{j}}(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}})$
$\displaystyle=\eta_{i}\mathcal{F}_{(X,Y)\to(\xi,\eta)}(-Y_{j}K_{\tilde{a}})$
$\displaystyle=\eta_{i}\mathcal{F}_{(X,Y)\to(\xi,\eta)}{K_{\partial_{\eta_{j}}\tilde{a}}}$
$\displaystyle=\mathcal{F}_{(X,Y)\to(\xi,\eta)}{D_{Y_{i}}K_{\partial_{\eta_{j}}\tilde{a}}}$
$\displaystyle=\mathcal{F}_{(X,Y)\to(\xi,\eta)}{K_{\eta_{i}\partial_{\eta_{j}}\tilde{a}+D_{Y_{i}}\partial_{\eta_{j}}\tilde{a}}},$

i.e. for $V=\eta_{i}\partial_{\eta_{j}}$ we have

$V(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}})=\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{V\tilde{a}+\tilde{a}^{\prime}}$

where $\tilde{a}^{\prime}\in S^{-N,-N-1,0,0}_{full}$. It follows that
$\tilde{a}^{\prime\prime}=V\tilde{a}+\tilde{a}^{\prime}\in
S^{-N,-N,0,0}_{full}$, and hence
$V(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}})=\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}^{\prime\prime}}$
with $\tilde{a}^{\prime\prime}\in S^{-N,-N,0,0}_{full}$. More generally, the
same logic applies if $V$ is one of the following vector fields:

(4.6)
$x\partial_{x},\partial_{y_{j}},\partial_{\xi},\partial_{\eta_{j}},\xi\partial_{\xi},\xi\partial_{\eta_{j}},\eta_{i}\partial_{\xi},\eta_{i}\partial_{\eta_{j}},\eta_{1}\partial_{\eta_{1}}.$

It follows that if $V^{\beta}$ is a product of the above vector fields (i.e. a
product of any of the vector fields in (4.4) except
$x^{1/2}\langle(\xi,\eta)\rangle^{1/2}\partial_{\xi}$ and
$x^{1/2}\langle(\xi,\eta)\rangle^{1/2}\partial_{\eta_{i}}$), then

$V^{\beta}(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}})=(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}^{\prime}})\text{
for some }\tilde{a}^{\prime}\in S^{-N,-N,0,0}_{full}$

In addition, we have

$\eta_{j}(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}})=\mathcal{F}_{(X,Y)\to(\xi,\eta)}{K_{\eta_{j}\tilde{a}+D_{Y_{j}}\tilde{a}}}$

with $\eta_{j}\tilde{a}+D_{Y_{j}}\tilde{a}\in
S^{-N+1,-N,0,0}_{full}+S^{-N,-N-1,0,0}_{full}\in S^{-N+1,-N,0,0}_{full}$. Thus
in general we have

$(\xi,\eta)^{\alpha}V^{\beta}(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}})=(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}^{\prime}})\text{
for some }\tilde{a}^{\prime}\in S^{-N+|\alpha|,-N,0,0}_{full}.$

If $\tilde{a}\in S^{m,l,0,0}_{full}$ with $m<-n$, then the integrand in the
Fourier transform is absolutely integrable, and hence we can crudely bound
$|(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}^{\prime}})|\leq Cx^{-l}$, and
hence

$|(\xi,\eta)^{\alpha}V^{\beta}(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}})|\leq
Cx^{-l}\text{ as long as }-N+|\alpha|<-n,i.e.|\alpha|<N-n.$

Since this holds for all $\alpha$ with $|\alpha|<N-n$, then we have

(4.7) $|V^{\beta}(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}})|\leq
C\langle(\xi,\eta)\rangle^{-(N-n-1)}x^{N}.$

Finally, for $V=x^{1/2}\langle(\xi,\eta)\rangle^{1/2}\partial_{\xi}$ and
$x^{1/2}\langle(\xi,\eta)\rangle^{1/2}\partial_{\eta_{i}}$, we crudely rewrite
these as $(x^{1/2}\langle(\xi,\eta)\rangle^{1/2})(\partial_{\xi})$ and
$(x^{1/2}\langle(\xi,\eta)\rangle^{1/2})(\partial_{\eta_{i}})$. Hence, if we
consider a product $V^{\beta}(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}})$
where $V^{\beta}$ may contain products of
$x^{1/2}\langle(\xi,\eta)\rangle^{1/2}\partial_{\xi}$ or
$x^{1/2}\langle(\xi,\eta)\rangle^{1/2}\partial_{\eta_{i}}$, then if there are
$k$ of such products, we can rewrite
$V^{\beta}(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}})$ as a
$C^{\infty}(\mathscr{X}_{2})$-combination of terms of the form
$x^{j/2}\langle(\xi,\eta)\rangle^{j/2}\tilde{V}^{\beta}(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}})$
for $0\leq j\leq k$, where now $\tilde{V}^{\beta}$ is a product of vector
fields of the form (4.6) (by moving factors
$x^{1/2}\langle(\xi,\eta)\rangle^{1/2}$ all the way to the left since
commutators behave nicely). For each term we can bound

$\displaystyle|x^{j/2}\langle(\xi,\eta)\rangle^{j/2}\tilde{V}^{\beta}(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}})|$
$\displaystyle\leq
C\langle(\xi,\eta)\rangle^{j/2}|\tilde{V}^{\beta}(\mathcal{F}_{(X,Y)\to(\xi,\eta)}K_{\tilde{a}})|$
$\displaystyle\leq C\langle(\xi,\eta)\rangle^{|\beta|/2-(N-n-1)}x^{N}$

using (4.7) (note that $j\leq k\leq|\beta|$). Hence, overall we have

$|V^{\beta}a|\leq C\langle(\xi,\eta)\rangle^{|\beta|/2+n+1-N}x^{N}\leq
C\rho^{N-(|\beta|/2+n+1)}x^{N},$

as claimed. ∎

###### Proof of Proposition 4.1 from Lemma 4.2 and 4.3.

For $\tilde{a}\in S^{m,l,k,j}_{full}$, let
$a=\mathcal{F}_{(X,Y)\to(\xi,\eta)}(K_{\tilde{a}})$. We want to show that
$a\in S^{m,l,k,j}$ and
$a-\sum_{|\alpha|<N}{\frac{i^{|\alpha|}}{\alpha!}\partial_{(\xi,\eta)}^{\alpha}\tilde{a}|_{(X,Y)=(0,0)}}\in
S^{m-N,l-N,k+N,j+N}$ for any $N$. The former involves showing that, for any
product $V^{\beta}$ of vector fields in $\mathcal{V}_{2}$, we have
$|V^{\beta}a|\leq C\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}$.

By Lemma 4.2, for any $N>0$ we can write
$K_{\tilde{a}}=K_{a_{N}}+K_{\tilde{a}_{N}}$ where

$a_{N}=\sum_{|\alpha|\leq
N^{\prime}}{\frac{i^{|\alpha|}}{\alpha!}\partial^{\alpha}_{(\xi,\eta)}\partial^{\alpha}_{Z}\tilde{a}(0)}\in
S^{m,l,k,j}$

for $N^{\prime}$ sufficiently large and $\tilde{a}_{N}\in S^{-N,-N,0,0}$;
we’ll consider $N$ very large. In particular, we’ll take $N$ large enough such
that

$\rho^{N-(|\beta|/2+n+1)}x^{N}\leq
C\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}.$

Then we have

$\displaystyle a=\mathcal{F}_{(X,Y)\to(\xi,\eta)}(K_{\tilde{a}})$
$\displaystyle=\mathcal{F}_{(X,Y)\to(\xi,\eta)}(K_{a_{N}})+\mathcal{F}_{(X,Y)\to(\xi,\eta)}(K_{\tilde{a}_{N}})$
$\displaystyle=a_{N}+\mathcal{F}_{(X,Y)\to(\xi,\eta)}(K_{\tilde{a}_{N}}).$

Hence, we have

$|V^{\beta}a|\leq|V^{\beta}a_{N}|+|V^{\beta}\left(\mathcal{F}_{(X,Y)\to(\xi,\eta)}(K_{\tilde{a}_{N}})\right)|.$

The term $|V^{\beta}a_{N}|$ is bounded by
$C\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}$ since $a_{N}\in S^{m,l,k,j}$.
By Lemma 4.3, the second term
$|V^{\beta}\left(\mathcal{F}_{(X,Y)\to(\xi,\eta)}(K_{\tilde{a}_{N}})\right)|$
is bounded by $C\rho^{N-(|\beta|/2+n+1)}x^{N}$, which in turn is bounded by
$C\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}$ since we assume $N$ is
sufficiently large. Thus overall we have the bound $|V^{\beta}a|\leq
C\rho^{-m}x^{-l}d_{\Sigma}^{-k}d_{\Gamma}^{-j}$, as desired. This shows that
$a\in S^{m,l,k,j}$.

Moreover, the same argument shows the asymptotic series statement: indeed, for
a fixed $N_{0}$, for sufficiently large $N$ we have $S^{-N,-N,0,0}\subset
S^{m-N_{0},l-N_{0},k+N_{0},j+N_{0}}$, in which case

$a=a_{N}+\mathcal{F}_{(X,Y)\to(\xi,\eta)}(K_{\tilde{a}_{N}})$

where $\tilde{a}_{N}\in S_{full}^{-N,-N,0,0}$, and hence from the above result
we now know that $\mathcal{F}_{(X,Y)\to(\xi,\eta)}(K_{\tilde{a}_{N}})\in
S^{-N,-N,0,0}\subset S^{m-N_{0},l-N_{0},k+N_{0},j+N_{0}}$. Since we can
arrange for

$a_{N}=\sum_{|\alpha|\leq
N^{\prime}}{\frac{i^{|\alpha|}}{\alpha!}\partial^{\alpha}_{(\xi,\eta)}\partial^{\alpha}_{Z}\tilde{a}(0)}$

for $N^{\prime}$ sufficiently large (in particular larger than $N_{0}$), it
follows that
${a-\sum_{|\alpha|<N}{\frac{i^{|\alpha|}}{\alpha!}\partial_{(\xi,\eta)}^{\alpha}\partial_{(X,Y)}^{\alpha}\tilde{a}|_{(X,Y)=(0,0)}}}\in
S^{m-N_{0},l-N_{0},k+N_{0},j+N_{0}}$, as desired. ∎

As an immediate corollary, we note that if $\chi\in
C_{c}^{\infty}(\mathbb{R})$ with $\chi\equiv 1$ in a neighborhood of the
origin, then the distributions
$\chi(xX,xY)K_{a}=K_{a(x,y,\xi,\eta)\chi(xX,xY)}$ equals $K_{a^{\prime}}$ for
some $a^{\prime}\in S^{m,l,k,j}$; moreover $a-a^{\prime}\in
S^{-\infty,-\infty}$. So philosophically we can multiply by a cutoff of the
form $\chi(xX,xY)$ without “changing the symbol $a$ by too much”. This is
convenient once we introduce the interpretation of $X$ and $Y$ as the
variables on the scattering double space, i.e. $X=\frac{x^{\prime}-x}{x^{2}}$
and $Y=\frac{y^{\prime}-y}{x}$. In that case, we need the cutoff $\chi(xX,xY)$
to guarantee that the distribution $K_{a}$ vanishes (to infinite order) at the
left and right faces ($\\{x^{\prime}>0,x=0\\}$ and $\\{x^{\prime}=0,x>0\\}$)
and the $b$-front face away from the scattering front face (i.e.
$\\{x+x^{\prime}=0,\frac{x^{\prime}-x}{x+x^{\prime}}\neq 0\\}$).

We now explore some basic properties of distributions of the form (4.2).

######  Lemma 4.4.

Let $K=K_{a}$ as in (4.2) for some $a\in S^{m,l,k,j}$. Then:

  * •

$K$ is smooth on $\\{x>0,|(X,Y)|\neq 0\\}$.

  * •

If we let $X_{b}=xX$ and $Y_{b}=xY$, then in the limit where $x\to 0$ but
$(X_{b},Y_{b})$ is a fixed nonzero value, we have that $K$ is smooth and
vanishes to infinite order as $x\to 0$.

Once we assign the interpretation of $X=\frac{x^{\prime}-x}{x^{2}}$ and
$Y=\frac{y^{\prime}-y}{x}$ on the double space (so that
$X_{b}=\frac{x^{\prime}-x}{x}$ and $Y_{b}=y^{\prime}-y$), then the last
condition can be rephrased as saying that $K$ vanishes to infinite order on
the b-face of the scattering double space (see e.g. [3]).

###### Proof.

  * •

On $\\{x>0,|(X,Y)|\neq 0\\}$, at least one of $xX$ or $xY_{j}$ ($1\leq j\leq
n-1$) is nonzero; in fact any point admits a neighborhood on which one of
these functions is bounded from below. If $xX$ is bounded from below in this
neighborhood, we have

(4.8) $(xX)^{N}K_{a}=K_{x^{N}(-D_{\xi})^{N}a},\quad\text{with
}x^{N}(-D_{\xi})^{N}a\in S^{m-N,l-N,k+N,j+N}.$

Taking $N$ sufficiently large so that $S^{m-N,l-N,k+N,j+N}\subset
S^{-n-1,0,0,0}$, we see that $(xX)^{N}K_{a}$ is continuous, since the integral
defining the inverse Fourier transform is integrable. A similar argument
applies for the remaining coordinate functions777Note that for $Y_{1}$ the
situation is in fact even better since $x^{N}(-D_{\eta_{1}})^{N}a\in
S^{m-N,l-N,k,j}$. to show that $K_{a}$ is continuous in
$\\{x>0,(X,Y)\neq(0,0)\\}$.

Furthermore, from (4.5), we see that

$D_{X,Y}^{\alpha}K_{a}=K_{\tilde{a}},\quad\tilde{a}\in S^{m+|\alpha|,l,k,j}.$

From the above result, we have that $D_{X,Y}^{\alpha}$ is also continuous in
$\\{x>0,|(X,Y)|\neq 0\\}$. This shows that $K_{a}$ is smooth in the region
$\\{x>0,|(X,Y)|\neq 0\\}$.

  * •

Suppose we are in the regime where $x\to 0$ but $X_{b}$ is bounded away from
zero. Then (4.8) gives

$X_{b}^{N}K_{a}=K_{\tilde{a}},\quad\tilde{a}\in S^{m-N,l-N,k+N,j+N}.$

For any fixed $N_{0}$, we can take $N$ sufficiently large so that
$S^{m-N,l-N,k+N,j+N}\subset S^{-n-1,-N_{0},0,0}$, in which case we have

$|X_{b}^{N}K_{a}|\leq Cx^{N_{0}}.$

Thus, in the regime where $X_{b}$ is bounded away from zero, we have that
$|K_{a}|\leq Cx^{N_{0}}$ for some constant $C$ for all $N_{0}$, i.e. it
vanishes to infinite order in $x$. A similar argument holds if $Y_{b}$ is
bounded away from zero.

∎

As such, we have that $K_{a}$ vanishes to infinite order on the b-face for any
$a\in S^{m,l,k,j}$. However, there is no guarantee that it vanishes on the
left and right faces $\\{x^{\prime}=0\\}\cap\\{x>0\\}$ and
$\\{x=0\\}\cap\\{x^{\prime}>0\\}$, which the Schwartz kernels of (ordinary)
scattering $\Psi$DO must satisfy (cf. [5]). Hence, in defining our operator
class, we build the vanishing on the left and right faces into the definition,
as follows:

######  Definition 4.1.

The class of $bb$-pseudodifferential operators $\Psi^{m,l,k,j}$ is defined as
the operators whose Schwartz kernels $K(x,y,X,Y)$ satisfy the following
properties:

  * •

$K=K_{a}$ as defined in (4.2) for some $a\in S^{m,l,k,j}$, and

  * •

$K$ vanishes to infinite order at the left and right faces
$\\{x^{\prime}=0\\}\cap\\{x>0\\}$ and $\\{x=0\\}\cap\\{x^{\prime}>0\\}$.

A consequence of Proposition 4.1 is the following: if we start with _any_
symbol $a\in S^{m,l,k,j}$, then the oscillatory integral $K_{a}$ defined in
(4.2) may not satisfy the property that it vanishes to infinite order at the
left and right faces $\\{x^{\prime}=0\\}\cap\\{x>0\\}$ and
$\\{x=0\\}\cap\\{x^{\prime}>0\\}$ when we introduce $x^{\prime}=x+x^{2}X$ and
$y^{\prime}=y+xY$; however, by multiplying $K_{a}$ by a cutoff $\chi(xX,xY)$
where $\chi$ is identically $1$ near $0$ but supported in the ball of radius
$1$, then the oscillatory distribution _will_ vanish to infinite order at the
left and right faces; moreover, that oscillatory distribution will still equal
$K_{\tilde{a}}$ for some $\tilde{a}\in S^{m,l,k,j}$, and we can be sure that
$a$ and $\tilde{a}$ differ by a residual difference in $S^{-\infty,-\infty}$.
In other words, we can quantize any symbol in $S^{m,l,k,j}$, so long as we
first modify it by some residual symbol in $S^{-\infty,-\infty}$. We summarize
the results as follows:

######  Proposition 4.5.

Let $a\in S^{m,l,k,j}$. Then there exists $\tilde{a}\in S^{m,l,k,j}$ with
$\tilde{a}-a\in S^{-\infty,-\infty}$ such that $K_{\tilde{a}}$ vanishes to
infinite order at the left and right faces $\\{x^{\prime}=0\\}\cap\\{x>0\\}$
and $\\{x=0\\}\cap\\{x^{\prime}>0\\}$, i.e. that
$K_{\tilde{a}}\in\Psi^{m,l,k,j}$. Moreover, if $\tilde{a}^{\prime}\in
S^{m,l,k,j}$ satisfies the same properties as $\tilde{a}$ in the previous
sentence, then
$K_{\tilde{a}^{\prime}}-K_{\tilde{a}}\in\Psi^{-\infty,-\infty}$, i.e. the two
operators differ by a (scattering) residual operator.

As such, we can define:

######  Definition 4.2.

Let $a\in S^{m,l,k,j}$. We will say that $A\in\Psi^{m,l,k,j}$ is _a
quantization_ of $a$ if $A=K_{\tilde{a}}$, where $\tilde{a}-a\in
S^{-\infty,-\infty}$.

The above proposition then says that for any $a\in S^{m,l,k,j}$, there exists
a quantization of $a$, and such a quantization is uniquely determined up to a
residual operator in $\Psi^{-\infty,-\infty}$.

###  4.2. Adjoints

We would like these operators to satisfy the usual boundedness condition:
namely, that order $0$ symbols are bounded on $L^{2}$, so that all other
symbols are bounded between appropriately defined Sobolev spaces (which can be
defined in terms of this calculus). The volume form we take will be that
associated to a “scattering metric”; explicitly in local coordinates such a
volume can be given by $d\text{Vol}_{g_{sc}}=\frac{dx\,dy}{x^{(n+1)}}$. To do
so, we consider adjoints with respect to the above volume form, and we show
these can be expressed as quantizations of another symbol, with “principal
symbol” equal to $\overline{a}$. This, combined with the composition result
obtained in the next section, will give $L^{2}$-boundedness, via the usual
arguments in microlocal analysis.

For convenience, we will consider quantizations of full symbols supported in
$\\{(xX,xY)\\}<1/2$; this is permissible in light of the above discussion.

######  Proposition 4.6.

Let

$Au(x,y)=(2\pi)^{-n}\int_{\mathbb{R}^{n}_{X,Y}\times\mathbb{R}^{n}_{\xi,\eta}}{e^{i(\xi
X+\eta\cdot Y)}a(x,y,X,Y,\xi,\eta)u(x+x^{2}X,y+xY)\,dX\,dY\,d\xi\,d\eta}$

with $a\in S^{m,l,k,j}_{full}$ supported in $|(xX,xY)|<1/2$. Then the adjoint
with respect to the $L^{2}$ inner product associated with the measure
$\frac{dx}{x^{n+1}}dy$ on $[0,\infty)_{x}\times\mathbb{R}^{n-1}_{y}$ is
another operator quantized by a full symbol whose principal symbol is
$(x,y,\xi,\eta)\mapsto\overline{a}(x,y,0,0,\xi,\eta)$.

###### Proof.

Since we can write the “$(X,Y)$-Schwartz kernel” $K_{A}(x,y,X,Y)$ (satisfying
$Au(x,y)=\int{K_{A}(x,y,X,Y)u(x+x^{2}X,y+xY)\,dX\,dY}$) as

$K_{A}(x,y,X,Y)=(2\pi)^{-n}\int_{\mathbb{R}^{n}}{e^{i(\xi X+\eta\cdot
Y)}a(x,y,X,Y,\xi,\eta)\,d\xi\,d\eta},$

it will be helpful to obtain an expression for the “$(X,Y)$-Schwartz kernel”
$K_{A*}$ of the adjoint $A^{*}$ in terms of $K_{A}$. We note that

$\displaystyle\langle Au,v\rangle$
$\displaystyle=\int{K_{A}(x,y,X,Y)u(x+x^{2}X,y+xY)\overline{v(x,y)}\,dX\,dY\,\frac{dx}{x^{n+1}}\,dy}$
$\displaystyle=\int{K_{A}\left(x,y,\frac{x^{\prime}-x}{x^{2}},\frac{y^{\prime}-y}{x}\right)u(x^{\prime},y^{\prime})\overline{v(x,y)}\,dx\,dy\,dx^{\prime}\,dy^{\prime}}$
$\displaystyle=\int{\overline{K_{A^{*}}}\left(x^{\prime},y^{\prime},\frac{x-x^{\prime}}{(x^{\prime})^{2}},\frac{y-y^{\prime}}{(y^{\prime})^{2}}\right)\overline{v(x,y)}u(x^{\prime},y^{\prime})\,dx^{\prime}\,dy^{\prime}\,dx\,dy}$
$\displaystyle=\overline{\langle A^{*}v,u\rangle}=\langle u,A^{*}v\rangle,$

as long as

$\overline{K_{A^{*}}}\left(x^{\prime},y^{\prime},\frac{x-x^{\prime}}{(x^{\prime})^{2}},\frac{y-y^{\prime}}{(y^{\prime})^{2}}\right)=K_{A}\left(x,y,\frac{x^{\prime}-x}{x^{2}},\frac{y^{\prime}-y}{x}\right).$

Changing the roles of $(x,y)$ and $(x^{\prime},y^{\prime})$, and noting that

$\frac{x-x^{\prime}}{(x^{\prime})^{2}}=-\frac{(x^{\prime}-x)/x^{2}}{(x^{\prime}/x)^{2}}=-\frac{X}{(1+xX)^{2}}$

and

$\frac{y-y^{\prime}}{x^{\prime}}=-\frac{(y^{\prime}-y)/x}{x^{\prime}/x}=-\frac{Y}{1+xX},$

it follows that

$\displaystyle K_{A^{*}}(x,y,X,Y)$
$\displaystyle=\overline{K_{A}}\left(x+x^{2}X,y+xY,-\frac{X}{(1+xX)^{2}},-\frac{Y}{1+xX}\right)$
$\displaystyle=(2\pi)^{-n}\overline{\int_{\mathbb{R}^{n}}{e^{i(\xi\cdot\frac{-X}{(1+xX)^{2}}+\eta\cdot\frac{-Y}{1+xX})}a\left(x+x^{2}X,y+xY,-\frac{X}{(1+xX)^{2}},-\frac{Y}{1+xX},\xi,\eta\right)\,d\xi\,d\eta}}$
$\displaystyle=(2\pi)^{-n}\int_{\mathbb{R}^{n}}{e^{i(\xi X+\eta\cdot
Y)}\tilde{a}(x,y,X,Y,\xi,\eta),d\xi\,d\eta},$

where

$\displaystyle\tilde{a}(x,y,X,Y,\xi,\eta)$
$\displaystyle=\overline{a}\left(x+x^{2}X,y+xY,-\frac{X}{(1+xX)^{2}},-\frac{Y}{1+xX},(1+xX)^{2}\xi,(1+xX)\eta\right)(1+xX)^{n+1}.$

Note that in going from the second-to-last line to the last line we rescaled
$\xi$ and $\eta$ to $(1+xX)^{2}\xi$ and $(1+xX)\eta$, respectively.

We can now check that $\tilde{a}\in S^{m,l,k,j}_{full}$. The idea is that
$1+xX$ is bounded from above and below in the support of $a$; moreover we can
compute quantities like

$\displaystyle x\partial_{x}\tilde{a}=$
$\displaystyle\left[(x+2x^{2}X)\partial_{x}+xY\cdot\partial_{y}+\frac{2(xX)^{2}}{(1+xX)^{3}}\frac{1}{x}\partial_{X}+\frac{(xX)(xY)}{(1+xX)^{2}}\frac{1}{x}\partial_{Y}+2xX(1+xX)\xi\partial_{\xi}+xX\eta\cdot\partial_{\eta}\right]\overline{a}$

and

$\displaystyle\frac{1}{x}\partial_{X}\tilde{a}=$
$\displaystyle\left[x\partial_{x}+\left(-\frac{1}{(1+xX)^{2}}+\frac{2xX}{(1+xX)^{3}}\right)\frac{1}{x}\partial_{X}-\frac{xY}{(1+xX)^{2}}\frac{1}{x}\partial_{Y}+2(1+xX)\xi\partial_{\xi}+\eta\cdot\partial_{\eta}\right]\overline{a}.$

to show that $\tilde{a}$ indeed satisfies the desired estimates. We note that
the principal symbol of $\tilde{a}$ is given by

$\tilde{a}(x,y,0,0,\xi,\eta)=\overline{a}(x,y,0,0,\xi,\eta),$

so that in particular if $a(x,y,X,Y,\xi,\eta)$ is given by a left-reduced
symbol $a(x,y,\xi,\eta)$ times a cutoff in $(xX,xY)$, then the principal
symbol of the adjoint is given by $\overline{a}(x,y,\xi,\eta)$, as expected. ∎

###  4.3. Quantization: composition

We now investigate composition. We assume all quantities are supported in a
fixed (foliated) chart.

Thus, let $(x,y)$, $(x^{\prime},y^{\prime})$, and
$(x^{\prime\prime},y^{\prime\prime})$ denote the necessary coordinates. Note
if $X=\frac{x^{\prime}-x}{x^{2}}$ and $Y=\frac{y^{\prime}-y}{x}$, then
$x^{\prime}=x+x^{2}X$ and $y^{\prime}=y+xY$. We note that if
$X^{\prime}=\frac{x^{\prime\prime}-x^{\prime}}{(x^{\prime})^{2}}$ and
$Y^{\prime}=\frac{y^{\prime\prime}-y^{\prime}}{x^{\prime}}$, then we can write

$X^{\prime}=\frac{x^{\prime\prime}-x-x^{2}X}{x^{2}(1+xX)^{2}}=\frac{1}{(1+xX)^{2}}\tilde{X}-\frac{X}{(1+xX)^{2}}$

where $\tilde{X}=\frac{x^{\prime\prime}-x}{x^{2}}$. Similarly

$Y^{\prime}=\frac{y^{\prime\prime}-y-xY}{x(1+xX)}=\frac{1}{1+xX}\tilde{Y}-\frac{Y}{1+xX},\quad\tilde{Y}=\frac{y^{\prime\prime}-y}{x^{2}}.$

Thus, suppose $A$ and $B$ are quantized by $a$ and $b$. Let $\tilde{A}$ be the
operator whose Schwartz kernel is that of $A$ times $\chi(xX,xY)$ where $\chi$
is identically $1$ in a neighborhood of $0$ and is supported in the ball of
radius $1/2$; note then that $(A-\tilde{A})B\in\Psi^{-\infty,-\infty}$. We
have

$\displaystyle\tilde{A}Bu(x,y)$ $\displaystyle=(2\pi)^{-n}\int{e^{i(\xi
X+\eta\cdot
Y)}a(x,y,\xi,\eta)\chi(xX,xY)Bu(x+x^{2}X,y+xY)\,dX\,dY\,d\xi\,d\eta}$
$\displaystyle=(2\pi)^{-2n}\int{e^{i(\xi X+\eta\cdot
Y+\xi^{\prime}X^{\prime}+\eta^{\prime}\cdot
Y^{\prime})}a(x,y,\xi,\eta)b(x+x^{2}X,y+xY,\xi^{\prime},\eta^{\prime})\chi(xX,xY)}$
$\displaystyle\cdot
u(x^{\prime\prime},y^{\prime\prime})\,dX^{\prime}\,dY^{\prime}\,d\xi^{\prime}\,d\eta^{\prime}\,dX\,dY\,d\xi\,d\eta$

where
$(x^{\prime\prime},y^{\prime\prime})=(x^{\prime}+(x^{\prime})^{2}X^{\prime},y^{\prime}+x^{\prime}Y^{\prime})$.
Changing coordinates from $(X^{\prime},Y^{\prime})$ to
$(X^{\prime\prime},Y^{\prime\prime})$ (in which case
$dX^{\prime}\,dY^{\prime}$ transforms as
$\frac{1}{(1+xX)^{n+1}}\,d\tilde{X}\,d\tilde{Y}$, the integral becomes

$\displaystyle(2\pi)^{-2n}\int$ $\displaystyle e^{i\left(\xi X+\eta\cdot
Y+\frac{\xi^{\prime}}{(1+xX)^{2}}(\tilde{X}-X)+\frac{\eta^{\prime}}{1+xX}\cdot(\tilde{Y}-Y)\right)}a(x,y,\xi,\eta)b(x+x^{2}X,y+xY,\xi^{\prime},\eta^{\prime})\chi(xX,xY)$
$\displaystyle\cdot
u(x+x^{2}\tilde{X},y+x\tilde{Y})\frac{d\tilde{X}\,d\tilde{Y}}{(1+xX)^{n+1}}d\xi^{\prime}\,d\eta^{\prime}\,dX\,dY\,d\xi\,d\eta.$

If we now let $\tilde{\xi}=\frac{\xi^{\prime}}{(1+xX)^{2}}$ and
$\tilde{\eta}=\frac{\eta^{\prime}}{1+xX}$, then the integral becomes

$\displaystyle(2\pi)^{-2n}\int$ $\displaystyle e^{i\left(\xi X+\eta\cdot
Y+\tilde{\xi}(\tilde{X}-X)+\tilde{\eta}\cdot(\tilde{Y}-Y)\right)}a(x,y,\xi,\eta)b(x+x^{2}X,y+xY,(1+xX)^{2}\tilde{\xi},(1+xX)\tilde{\eta})$
$\displaystyle\cdot\chi(xX,xY)u(x+x^{2}\tilde{X},y+x\tilde{Y})\,d\tilde{X}\,d\tilde{Y}\,d\tilde{\xi}\,d\tilde{\eta}\,dX\,dY\,d\xi\,d\eta.$

Replacing $\xi$ and $\eta$ by $\xi+\tilde{\xi}$ and $\eta+\tilde{\eta}$, it
follows that the integral becomes

$(2\pi)^{-n}\int{e^{i(\tilde{\xi}\tilde{X}+\tilde{\eta}\cdot\tilde{Y})}c(x,y,\tilde{\xi},\tilde{\eta})u(x+x^{2}\tilde{X},y+x\tilde{Y})d\tilde{X}\,d\tilde{Y}\,d\tilde{\xi}\,d\tilde{\eta}}$

where

$\displaystyle c(x,y,\tilde{\xi},\tilde{\eta})=$ $\displaystyle(2\pi)^{-n}\int
e^{i(\xi X+\eta\cdot Y)}a(x,y,\xi+\tilde{\xi},\eta+\tilde{\eta})$
$\displaystyle\cdot
b(x+x^{2}X,y+xY,(1+xX)^{2}\tilde{\xi},(1+xX)\tilde{\eta})\chi(xX,xY)\,dX\,dY\,d\xi\,d\eta.$

We now note that

$\tilde{b}(x,y,X,Y,\xi,\eta)=\chi(xX,xY)b(x+x^{2}X,y+xY,(1+xX)^{2}\xi,(1+xX)\eta)$

is a full symbol, with $\tilde{b}(x,y,0,0,\xi,\eta)=b(x,y,\xi,\eta)$: the main
calculations to note are

$\displaystyle x\partial_{x}\tilde{b}=$
$\displaystyle(xX\partial_{x}\chi(xX)+xY\cdot\partial_{y}\chi(xY))b+\chi(xX)\cdot$
$\displaystyle\left((1+2xX)x\partial_{x}b+xY\cdot\partial_{y}b+2xX(1+xX)\xi\partial_{\xi}b+xX\eta\cdot\partial_{\eta}b\right)$

where $b$ and its derivatives are evaluated at
$(x+x^{2}X,y+xY,(1+xX)^{2}\xi,(1+xX)\eta)$, and

$\frac{1}{x}\partial_{X}\tilde{b}=\partial_{x}\chi(xX,xY)b+\chi(xX,xY)(x\partial_{x}b+2(1+xX)\xi\cdot\partial_{\xi}b+\eta\cdot\partial_{\eta}b).$

As such, for $a\in S^{m,l,k,j}$ and $b\in
S^{m^{\prime},l^{\prime},k^{\prime},j^{\prime}}_{full}$, define

(4.9)
$(a\\#b)(x,y,\xi,\eta):=(2\pi)^{-n}\int_{\mathbb{R}^{2n}}{e^{i(\xi^{\prime}\cdot
X+\eta^{\prime}\cdot
Y)}a(x,y,\xi^{\prime}+\xi,\eta^{\prime}+\eta)b(x,y,X,Y,\xi,\eta)\,dX\,dY\,d\xi^{\prime}\,d\eta^{\prime}}.$

Note that by replacing $\xi^{\prime}$ and $\eta^{\prime}$ by
$\xi^{\prime}-\xi$ and $\eta^{\prime}-\eta$, we can also rewrite the above
oscillatory integral as

$(a\\#b)=(2\pi)^{-n}\int_{\mathbb{R}^{2n}}{e^{i[(\xi^{\prime}-\xi)\cdot
X+(\eta^{\prime}-\eta)\cdot
Y]}a(x,y,\xi^{\prime},\eta^{\prime})b(x,y,X,Y,\xi,\eta)\,dX\,dY\,d\xi^{\prime}\,d\eta^{\prime}}.$

The main result is:

######  Theorem 4.7.

For $a\in S^{m,l,k,j}$ and $b\in
S^{m^{\prime},l^{\prime},k^{\prime},j^{\prime}}_{full}$ we have $a\\#b\in
S^{M,L,K,J}$, where $M=m+m^{\prime}$, $L=l+l^{\prime}$, $K=k+k^{\prime}$, and
$J=j+j^{\prime}$. Moreover,

$(a\\#b)(x,y,\xi,\eta)-a(x,y,\xi,\eta)b(x,y,0,0,\xi,\eta)\in
S^{M-1,L-1,K+1,J+1},$

i.e. the principal symbol of $a\\#b$ equals888In the sense that they have the
same equivalence class under the principal symbol mapping
$a(x,y,\xi,\eta)b(x,y,0,0,\xi,\eta)$.

For notational convenience, we will write
$a(x,y,\xi,\eta)b(x,y,0,0,\xi,\eta)$, the expected principal symbol, as
$ab|_{X=0,Y=0}$.

This will follow from the following lemmas:

######  Lemma 4.8 (Asymptotic series).

For any $N$, we have

(4.10)
$a\\#b-\sum_{|\alpha|<N}{\frac{i^{|\alpha|}}{|\alpha|!}\partial^{\alpha}_{\xi,\eta}a\partial^{\alpha}_{X,Y}b|_{X=0,Y=0}}=\sum_{i=1}^{N^{\prime}}{a_{i}\\#b_{i}}$

for some choices of $a_{i}\in S^{m-N,l-N,k+N,j+N}$ and $b_{i}\in
S^{m^{\prime},l^{\prime},k^{\prime},j^{\prime}}_{full}$.

######  Lemma 4.9 ($0$th order estimate).

For $a\in S^{m,l,k,j}$ and $b\in
S^{m^{\prime},l^{\prime},k^{\prime},j^{\prime}}_{full}$, we have
$|(a\\#b)|\leq C\rho^{-M}x^{-L}d_{\Sigma}^{-K}d_{\Gamma}^{-J}$.

(The proof in fact uses the asymptotic series expansion in Lemma 4.8.)

######  Lemma 4.10 (Vector fields).

Let $V$ be any vector field tangent to $\partial\mathscr{X}_{2}$. Then

$V(a\\#b)=\sum_{i=1}^{N}{a_{i}\\#b_{i}}$

for some choices of $a_{i}\in S^{m_{i},l_{i},k_{i},j_{i}}$ and $b_{i}\in
S^{m^{\prime}_{i},l^{\prime}_{i},k^{\prime}_{i},j^{\prime}_{i}}_{full}$
satisfying $m_{i}+m^{\prime}_{i}=M$, $l_{i}+l^{\prime}_{i}=L$,
$k_{i}+k^{\prime}_{i}=K$, and $j_{i}+j^{\prime}_{i}=J$.

###### Proof of Theorem 4.7.

Note that directly from Lemma 4.9 we have that $a\\#b$ satisfies the zeroth
order symbol estimate $|(a\\#b)|\leq
C\rho^{-M}x^{-L}d_{\Sigma}^{-K}d_{\Gamma}^{-J}$, so it suffices to show the
same estimate holds after applying a product of vector fields $V_{1}\dots
V_{N}$ which are tangent to $\partial\mathscr{X}_{2}$. But by applying Lemma
4.10 repeatedly, we see that

$V_{1}\dots V_{N}(a\\#b)=\sum_{i=1}^{N^{\prime}}{a_{i}\\#b_{i}}$

where $a_{i}\in S^{m_{i},l_{i},k_{i},j_{i}}$ and $b_{i}\in
S^{m^{\prime}_{i},l^{\prime}_{i},k^{\prime}_{i},j^{\prime}_{i}}_{full}$ with
$m_{i}+m^{\prime}_{i}=M$, $l_{i}+l^{\prime}_{i}=L$, $k_{i}+k^{\prime}_{i}=K$,
and $j_{i}+j^{\prime}_{i}=J$. Then, applying the zeroth order estimate in
Lemma 4.9 gives

$|V_{1}\dots V_{N}(a\\#b)|\leq
C\rho^{-M}x^{-L}d_{\Sigma}^{-K}d_{\Gamma}^{-J},$

as desired. Hence, we have shown that $a\\#b\in S^{M,L,K,J}$ for $a\in
S^{m,l,k,j}$ and $b\in
S^{m^{\prime},l^{\prime},k^{\prime},j^{\prime}}_{full}$.

To show the principal symbol statement, we observe that applying Lemma 4.8
with $N=1$ gives $a\\#b-ab|_{X=0,Y=0}=\sum_{i=1}^{N^{\prime}}{a_{i}\\#b_{i}}$
with $a_{i}\in S^{m-1,l-1,k+1,j+1}$ and $b_{i}\in
S^{m^{\prime},l^{\prime},k^{\prime},j^{\prime}}_{full}$. Applying the
conclusion of the above paragraph gives $a\\#b-ab|_{X=0,Y=0}\in
S^{M-1,L-1,K+1,J+1}$, as desired. ∎

###### Proof of Lemma 4.8.

The terms in the asymptotic series can be derived by applying the method of
stationary phase; however in this case we can make a more direct integration
by parts argument similar to the left reduction argument: indeed, letting
$Z=(X,Y)$ and suppressing the variables $x$, $y$, $\xi$, and $\eta$, we have

$b(Z)=\sum_{0\leq|\alpha|<N}{Z^{\alpha}\frac{\partial_{Z}^{\alpha}b(0)}{\alpha!}}+\sum_{|\beta|=N}{Z^{\beta}R_{\beta}b(Z)}$

where

$R_{\beta}b(Z)=\frac{|\beta|}{\beta!}\int_{0}^{1}{(1-t)^{|\beta|-1}\partial_{Z}^{\beta}b(tX)\,dt}.$

Note that $\partial_{Z}^{\alpha}b(0)\in
S^{m^{\prime},l^{\prime}-|\alpha|,k^{\prime},j^{\prime}}$ and $R_{\beta}b\in
S^{m^{\prime},l^{\prime}-|\beta|,k^{\prime},j^{\prime}}_{full}$. It follows
that

$\displaystyle a\\#b$ $\displaystyle=\sum_{0\leq|\alpha|\leq
n}{(2\pi)^{-n}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}{e^{i[(\xi^{\prime}-\xi)X+(\eta^{\prime}-\eta)\cdot
Y]}a(\xi^{\prime},\eta^{\prime})(X,Y)^{\alpha}\frac{(\partial_{X,Y}^{\alpha}b)|_{(X,Y)=(0,0)}}{\alpha!}\,d\xi^{\prime}\,d\eta^{\prime}\,dX\,dY}}$
$\displaystyle+\sum_{|\beta|=N}{(2\pi)^{-n}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}{e^{i[(\xi^{\prime}-\xi)X+(\eta^{\prime}-\eta)\cdot
Y]}a(\xi^{\prime},\eta^{\prime})(X,Y)^{\beta}R_{\beta}(X,Y)\,d\xi^{\prime}\,d\eta^{\prime}\,dX\,dY}}.$

To evaluate the terms in the first sum, note that
$(\partial_{X,Y}^{\alpha}b)|_{(X,Y)=(0,0)}$ is independent of the variables of
integration, so we can write the term as

$\displaystyle(2\pi)^{-n}\frac{(\partial_{X,Y}^{\alpha}b)|_{(X,Y)=(0,0)}}{\alpha!}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}{e^{i[(\xi^{\prime}-\xi)X+(\eta^{\prime}-\eta)\cdot
Y]}a(\xi^{\prime},\eta^{\prime})(X,Y)^{\alpha}\,d\xi^{\prime}\,d\eta^{\prime}\,dX\,dY}$
$\displaystyle=(2\pi)^{-n}\frac{i^{|\alpha|}(\partial_{X,Y}^{\alpha}b)|_{(X,Y)=(0,0)}}{\alpha!}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}{(-\partial_{\xi,\alpha})^{\alpha}\left(e^{i[(\xi^{\prime}-\xi)X+(\eta^{\prime}-\eta)\cdot
Y]}\right)a(\xi^{\prime},\eta^{\prime})\,d\xi^{\prime}\,d\eta^{\prime}\,dX\,dY}$
$\displaystyle=(2\pi)^{-n}\frac{i^{|\alpha|}(\partial_{X,Y}^{\alpha}b)|_{(X,Y)=(0,0)}}{\alpha!}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}{e^{i[(\xi^{\prime}-\xi)X+(\eta^{\prime}-\eta)\cdot
Y]}\partial_{\xi,\eta}^{\alpha}a(\xi^{\prime},\eta^{\prime})\,d\xi^{\prime}\,d\eta^{\prime}\,dX\,dY}$
$\displaystyle=\frac{i^{|\alpha|}(\partial_{X,Y}^{\alpha}b)|_{(X,Y)=(0,0)}}{\alpha!}\int_{\mathbb{R}^{n}}{\partial_{\xi,\eta}^{\alpha}a(\xi^{\prime},\eta^{\prime})\delta(\xi^{\prime}-\xi,\eta^{\prime}-\eta)\,d\xi^{\prime}\,d\eta^{\prime}}$
$\displaystyle=\frac{i^{|\alpha|}}{\alpha!}\partial_{\xi,\eta}^{\alpha}a(\xi,\eta)\partial_{X,Y}^{\alpha}b|_{(X,Y)=(0,0)}.$

For each term in the second sum, we apply the same integration by parts
argument to rewrite the term as

$(2\pi)^{-n}i^{|\beta|}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}{e^{i[(\xi^{\prime}-\xi)X+(\eta^{\prime}-\eta)\cdot
Y]}\partial_{\xi,\eta}^{\beta}a(\xi^{\prime},\eta^{\prime})R_{\beta}(X,Y)\,d\xi^{\prime}\,d\eta^{\prime}\,dX\,dY}.$

By definition, this is
$(i^{|\beta|}\partial_{\xi,\eta}^{\beta}a)\\#(R_{\beta}b)$. Noting that
$\partial_{\xi,\eta}^{\beta}a\in
S^{m-|\beta|,l,k+|\beta|,j+|\beta|}=S^{m-N,l,k+N,j+N}$ and $R_{\beta}b\in
S^{m^{\prime},l^{\prime}-N,k^{\prime},j^{\prime}}$, the desired statement
follows by noting that we can multiply $\partial_{\xi,\eta}^{\beta}a$ and
divide $R_{\beta}b$ by $x^{N}$ without changing the overall integral. ∎

###### Proof of Lemma 4.9.

We first prove the much weaker statement: for $m_{0}\in\mathbb{N}$, we have

$m<-n-m_{0},a\in S^{m,l,0,0},b\in
S^{m^{\prime},l^{\prime},k^{\prime},j^{\prime}}\implies|a\\#b|\leq
C\rho^{m_{0}-m^{\prime}}x^{-(l+l^{\prime})}d_{\Sigma}^{-k^{\prime}}d_{\Gamma}^{-j^{\prime}}.$

By multiplying $a$ by a power of $x$ and $b$ by powers of $\rho$, $x$,
$d_{\Sigma}$, and $d_{\Gamma}$, we may assume that
$l=m^{\prime}=l^{\prime}=k^{\prime}=j^{\prime}=0$. We can write the integral
as

$(a\\#b)=\int_{\mathbb{R}^{n}}{e^{-i(\xi X+\eta\cdot
Y)}\mathcal{F}_{(\xi^{\prime},\eta^{\prime})\to(X,Y)}^{-1}a(x,y,X,Y)b(x,y,X,Y,\xi,\eta)\,dX\,dY}.$

We note that $\mathcal{F}^{-1}a(x,y,X,Y)$ is Schwartz in $(X,Y)$ away from the
origin $(X,Y)=(0,0)$, and moreover it satisfies uniform Schwartz estimates in
$(x,y)$, since $a$ is smooth uniformly in $(x,y)$. Near the origin, we have
that $\mathcal{F}^{-1}a(x,y,X,Y)$ is in $C^{m_{0}}$, with uniformly bounded
derivatives up to order $m_{0}$: indeed, note that

$D_{X,Y}^{\alpha}\mathcal{F}^{-1}a(x,y,X,Y)=\mathcal{F}^{-1}\left((i(\xi,\eta))^{\alpha}a\right)(x,y,X,Y),$

and for $|\alpha|\leq m_{0}$ we have $|(i(\xi,\eta))^{\alpha}a|\leq
C\langle(\xi,\eta)\rangle^{|\alpha|+m}$ with $|\alpha|+m\leq m_{0}+m<-n$, and
hence the integral defining its inverse Fourier transform is uniformly
integrable, i.e. $\mathcal{F}^{-1}\left((i(\xi,\eta))^{\alpha}a\right)$ is
uniformly bounded. As such, we note that for $|\alpha|\leq m_{0}$ we have

$\displaystyle(\xi,\eta)^{\alpha}(a\\#b)$
$\displaystyle=\int_{\mathbb{R}^{n}}{(-D_{X,Y})^{\alpha}\left(e^{-i(\xi
X+\eta\cdot
Y)}\right)\mathcal{F}_{(\xi^{\prime},\eta^{\prime})\to(X,Y)}^{-1}a(x,y,X,Y)b(x,y,X,Y,\xi,\eta)\,dX\,dY}$
$\displaystyle=\int_{\mathbb{R}^{n}}{e^{-i(\xi X+\eta\cdot
Y)}D_{X,Y}^{\alpha}\left(\mathcal{F}_{(\xi^{\prime},\eta^{\prime})\to(X,Y)}^{-1}a(x,y,X,Y)b(x,y,X,Y,\xi,\eta)\right)\,dX\,dY}$

Noting that
$D_{X,Y}^{\alpha}\left(\mathcal{F}_{(\xi^{\prime},\eta^{\prime})\to(X,Y)}^{-1}a(x,y,X,Y)b(x,y,X,Y,\xi,\eta)\right)$
is uniformly bounded near $(X,Y)=(0,0)$ since $\mathcal{F}^{-1}a$ is in
$C^{m_{0}}$ with $|\alpha|\leq m_{0}$, and the term is also uniformly Schwartz
away from $(X,Y)=(0,0)$, it follows that $(\xi,\eta)^{\alpha}(a\\#b)$ is
uniformly bounded for all $|\alpha|\leq m_{0}$. This implies that
$\langle(\xi,\eta)\rangle^{m_{0}}(a\\#b)$ is uniformly bounded, i.e.
$|(a\\#b)|\leq C\rho^{m_{0}}$, as desired.

To show the lemma in full generality, we use the asymptotic expansion (4.10)
to write, for any $N$,

$a\\#b=\sum_{|\alpha|<N}{\frac{i^{|\alpha|}}{|\alpha|!}\partial^{\alpha}_{\xi,\eta}a\partial^{\alpha}_{X,Y}b|_{X=0,Y=0}}+\sum_{i=1}^{N^{\prime}}{a_{i}\\#b_{i}},$

where $a_{i}\in S^{m-N,l-N,k+N,j+N}$ and $b_{i}\in
S^{m^{\prime},l^{\prime},k^{\prime},j^{\prime}}_{full}$. Each term in the
first sum is in $S^{M-|\alpha|,L-|\alpha|,K+|\alpha|,J+|\alpha|}\subset
S^{M-|\alpha|/2,L-|\alpha|/2,K,J}$ and hence bounded by
$C\rho^{-M}x^{-L}d_{\Sigma}^{-K}d_{\Gamma}^{-J}$. For the terms
$a_{i}\\#b_{i}$ in the remainder term, we note that if $N$ is sufficiently
large so that $k+N$ and $j+N$ are both positive, then we have

$a_{i}\in S^{m-N,l-N,k+N,j+N}\subset S^{m+k/2-N/2,l+j/2-N/2,0,0}.$

If we take $m_{0}$ sufficiently large so that $m_{0}+m\geq(k_{-})/2$, and we
take $N$ sufficiently large so that $m+k/2-N/2<-n-m_{0}$, then by the weaker
estimate proven above, we have

$\displaystyle\left|\sum_{i=1}^{N^{\prime}}{a_{i}\\#b_{i}}\right|$
$\displaystyle\leq
C\rho^{m_{0}-m^{\prime}}x^{N/2-j/2-(l+l^{\prime})}d_{\Sigma}^{-k^{\prime}}d_{\Gamma}^{-j^{\prime}}$
$\displaystyle=C\rho^{m_{0}+m}x^{N/2-j/2}\left(\rho^{-(m+m^{\prime})}x^{-(l+l^{\prime})}d_{\Sigma}^{-k^{\prime}}d_{\Gamma}^{-j^{\prime}}\right).$

If we also require $N$ to be sufficiently large so that
$N/2-j/2\geq(j_{-})/2$, then we have

$\rho^{m_{0}+m}x^{N/2-j/2}\leq\rho^{(k_{-})/2}x^{(j_{-})/2}\leq
Cd_{\Sigma}^{-k}d_{\Gamma}^{-j}.$

Thus, if $N$ is sufficiently large, we have that the remainder term is also
bounded by $C\rho^{-M}x^{-L}d_{\Sigma}^{-K}d_{\Gamma}^{-J}$, thus showing the
estimate in general. ∎

###### Proof of Lemma 4.10.

Since the collection of vector fields tangent to $\partial\mathscr{X}_{2}$ are
generated over $S^{0,0,0,0}$ by the vector fields

$x\partial_{x},\partial_{y},\xi\partial_{\xi},\eta_{i}\partial_{\xi},\xi\partial_{\eta_{j}},\eta_{i}\partial_{\eta_{j}},\eta_{j}\partial_{\eta_{1}},\partial_{\xi},\partial_{\eta_{j}},x^{1/2}\rho^{-1/2}\partial_{\xi},x^{1/2}\rho^{-1/2}\partial_{\eta_{i}},\quad
i=2,\dots n-1,j=1,\dots,n-1,$

and that furthermore $c(a\\#b)=a\\#(cb)$ when $c\in S^{0,0,0,0}$, it suffices
to show the lemma for the above vector fields.

For the horizontal vector fields $V=x\partial_{x},\partial_{y}$, from (4.9) we
have

$V(a\\#b)=(Va)\\#b+a\\#(Vb).$

Thus the lemma holds for these vector fields since symbols of a given order
are preserved under $V$.

For the vertical vector fields, we rewrite (4.9) as

(4.11)