# Microlocal analysis of $d$-plane transform on the Euclidean space

Hiroyuki Chihara  College of Education, University of the Ryukyus, Nishihara,
Okinawa 903-0213, Japan<EMAIL_ADDRESS>
###### Abstract.

We study the basic properties of $d$-plane transform on the Euclidean space as
a Fourier integral operator, and its application to the microlocal analysis of
streaking artifacts in its filtered back-projection. The $d$-plane transform
is defined by integrals of functions on the $n$-dimensional Euclidean space
over all the $d$-dimensional planes, where $0<d<n$. This maps functions on the
Euclidean space to those on the affine Grassmannian $G(d,n)$. This is said to
be X-ray transform if $d=1$ and Radon transform if $d=n-1$. When $n=2$ the
X-ray transform is thought to be measurements of CT scanners. In this paper we
obtain concrete expression of the canonical relation of the $d$-plane
transform and quantitative properties of the filtered back-projection of the
product of the images of the $d$-plane transform. The latter one is related to
the metal streaking artifacts of CT images, and some generalization of recent
results of Park-Choi-Seo (2017) and Palacios-Uhlmann-Wang (2018) for the X-ray
transform on the plane.

###### Key words and phrases:

Radon transform, $d$-plane transform, Fourier integral operators, wave front
set

###### 2020 Mathematics Subject Classification:

Primary 58J40, Secondary 44A12, 53D22

Supported by the JSPS Grant-in-Aid for Scientific Research #19K03569.

##  1\. Introduction

Let $n$ and $d$ be positive integers with $n\geqq 2$ and
$1\leqq{d}\leqq{n-1}$. We study the microlocal analysis of $d$-plane transform
of functions on the $n$-dimensional Euclidean space, and their applications
related to metal streaking artifacts of CT images. Following Helgason’s
celebrated textbook [8] we begin with the definition of the $d$-plane
transform and well-known facts on it. Let $G_{d,n}$ be a Grassmannian, that
is, the set of all $d$-dimensional vector subspaces of $\mathbb{R}^{n}$. For
$\sigma\in G_{d,n}$ we denote by $\sigma^{\perp}$ the orthogonal complement of
$\sigma$ in $\mathbb{R}^{n}$. Set

$G(d,n):=\\{x^{\prime\prime}+\sigma:\sigma\in
G_{d,n},x^{\prime\prime}\in\sigma^{\perp}\\},$

which is the set of all $d$-dimensional planes of $\mathbb{R}^{n}$ and is said
to be an affine Grassmannian. We sometimes denote $x^{\prime\prime}+\sigma\in
G(d,n)$ by $(\sigma,x^{\prime\prime})$. Here we introduce the $d$-plane
transform $\mathcal{R}_{d}$. For a function
$f(x)=\mathcal{O}(\langle{x}\rangle^{-d-\varepsilon})$ of
$x=(x_{1},\dotsc,x_{n})=x^{\prime}+x^{\prime\prime}\in\sigma\oplus\sigma^{\perp}=\mathbb{R}^{n}$
with $\sigma\in G_{d,n}$, we define $\mathcal{R}_{d}f$ by

$\mathcal{R}_{d}f(\sigma,x^{\prime\prime}):=\int_{\sigma}f(x^{\prime}+x^{\prime\prime})dx^{\prime},$

where $\langle{x}\rangle=\sqrt{1+\lvert{x}\rvert^{2}}$,
$\lvert{x}\rvert^{2}=x{\cdot}x$, $x{\cdot}y$ is the standard inner product of
$x,y\in\mathbb{R}^{n}$, and $dx^{\prime}$ is the $d$-dimensional Lebesgue
measure on $\sigma$. $\mathcal{R}_{1}f$ is said to be the X-ray transform of
$f$, and $\mathcal{R}_{n-1}f$ is said to be the Radon transform of $f$.

Next we recall inversion formula of $\mathcal{R}_{d}$. Roughly speaking, the
formal adjoint of $\mathcal{R}_{d}$ is given as integrals of functions over
the set of all $d$-planes passing through arbitrary fixed point
$x\in\mathbb{R}^{n}$. More precisely

$\displaystyle\mathcal{R}_{d}^{\ast}\varphi(x)$
$\displaystyle:=\frac{1}{C(d,n)}\int_{\\{\Xi\in
G(d,n):x\in\Xi\\}}\varphi(\Xi)d\mu(\Xi)$
$\displaystyle=\frac{1}{C(d,n)}\int_{O(n)}\varphi(x+k\cdot\sigma)dk,$

where $\varphi\in C\bigl{(}G(d,n)\bigr{)}$, $C\bigl{(}G(d,n)\bigr{)}$ is the
set of all continuous functions on $G(d,n)$,
$C(d,n)=(4\pi)^{d/2}\Gamma(n/2)/\Gamma\bigl{(}(n-1)/2\bigr{)}$,
$\Gamma(\cdot)$ is the gamma function, $O(n)$ is the orthogonal group, $d\mu$
and $dk$ are the normalized measures which are invariant under rotations, and
$\sigma\in G_{d,n}$ is arbitrary. The inversion formula of $\mathcal{R}_{d}$
is given as follows.

######  Proposition 1.1 ([8, Theorem 6.2]).

For $f(x)=\mathcal{O}(\langle{x}\rangle^{-d-\varepsilon})$

$f=(-\Delta_{x})^{d/2}\mathcal{R}_{d}^{\ast}\mathcal{R}_{d}f=\mathcal{R}_{d}^{\ast}(-\Delta_{x^{\prime\prime}})^{d/2}\mathcal{R}_{d}f,$

where $-\Delta_{x}=-\partial_{x_{1}}^{2}-\dotsb-\partial_{x_{n}}^{2}$ and
$-\Delta_{x^{\prime\prime}}$ is the Laplacian on $\sigma^{\perp}$.

Operators $\mathcal{R}_{d}^{\ast}$ and
$(-\Delta_{x})^{d/2}\mathcal{R}_{d}^{\ast}=\mathcal{R}_{d}^{\ast}(-\Delta_{x^{\prime\prime}})^{d/2}$
are said to be the unfiltered back-projection operator and the filtered back-
projection operator respectively.

Here we recall the mapping properties of $\mathcal{R}_{d}$. To state them we
introduce some function spaces. Let $X$ be a smooth manifold. We denote by
$C^{\infty}(X)$ the set of all smooth functions on $X$. The set of all
compactly supported smooth functions on $X$ is denoted by $C^{\infty}_{0}(X)$.
$\mathcal{D}(X)$ is the locally convex space defined by adding the inductive
limit topology to $C^{\infty}_{0}(X)$. We denote by $\mathcal{D}^{\prime}(X)$
the space of Schwartz distributions on $X$, which is the topological dual of
$\mathcal{D}(X)$. We denote by $\mathcal{D}_{H}\bigl{(}G(d,n)\bigr{)}$ the set
of all $\varphi\in\mathcal{D}\bigl{(}G(d,n)\bigr{)}$ satisfying the following
moment conditions: for any $k=0,1,2,\dotsc$ there exists a homogeneous
polynomial $P_{k}$ on $\mathbb{R}^{n}$ of degree $k$ such that for any
$\sigma\in G_{d,n}$

$P_{k}(\xi^{\prime\prime})=\int_{\sigma^{\perp}}(\xi^{\prime\prime}\cdot
x^{\prime\prime})^{k}\varphi(\sigma,x^{\prime\prime})dx^{\prime\prime},\quad\xi^{\prime\prime}\in\sigma^{\perp}.$

The range of $\mathcal{R}_{d}$ of $\mathcal{D}(\mathbb{R}^{n})$ is
characterized as follows.

######  Proposition 1.2 ([8, Theorem 6.3]).

$\mathcal{R}_{d}$ is a linear bijection of $\mathcal{D}(\mathbb{R}^{n})$ onto
$\mathcal{D}_{H}\bigl{(}G(d,n)\bigr{)}$.

It is well-known that $\mathcal{R}_{d}$ is an elliptic Fourier integral
operator. See [6]. Moreover the canonical relation of $\mathcal{R}_{n-1}$ is
well-known and applied to the microlocal analysis of CT images for $n=2$. See
[1], [13], [15], [16], [17] and [18]. As far as the author knows the canonical
relation of $\mathcal{R}_{d}$ for $d{\neq}n-1$ was rarely considered so far.
It is important to obtain a comprehensive and useful expression of them.
Indeed the canonical relation of the X-ray transform $\mathcal{R}_{1}$ on
$\mathbb{R}^{3}$ is expected to play a crucial role in the microlocal analysis
of the CT scanners of the fourth generation or later ones. See, e.g., [3] for
the history of actual CT scanners including the fourth generation until 2002.

Following Epstein’s textbook [3] on mathematical theory of medical imaging and
a recent paper [16] of Park, Choi and Seo, we review the outline of computed
tomography (CT). In what follows we assume for a while that $n=2$ and
$d=1(=n-1)$. Throughout the present paper the X-ray beam is supposed to have
no width. In medical imaging $f(x)$ and $\mathcal{R}_{1}f$ are thought to be
the distribution of attenuation coefficients of the section of a human body
and the measurements $P(\sigma,x^{\prime\prime})$ of a CT scanner for this
section. If a human body consists only of normal tissue, then it follows from
the Beer-Lambert law that $P=\mathcal{R}_{1}f$ holds, and $f$ is reconstructed
by the filtered back-projection. However there are some factors causing
artifacts in actual CT images: beam width, partial volume effect, beam
hardening, noise in measurements, numerical errors and etc. We focus on metal
streaking artifacts caused by the beam hardening. If a human body contains
metal such as implants, dental fillings, stents, metal bones and etc,
streaking artifacts occur on lines tangent to two or more boundaries of metal
region in its CT images. This is basically due to the fluctuation of the
energy level $E\in[0,\infty)$ of X-ray beams for normal tissue and metal
regions. We explain this more concretely. In what follows we denote the
distribution of attenuation coefficients by $f_{E}(x)$ with $E\in[0,\infty)$.
The measurements of CT scanners are described by a spectral function
$\rho(E)$, which is a probability density function on $[0,\infty)$:

$P=-\log\left\\{\int_{0}^{\infty}\rho(E)\exp(-\mathcal{R}_{1}f_{E})dE\right\\}.$

If a human body consists only of normal tissue, we may assume that the X-ray
beams are monochromatic with a fixed energy $E_{0}>0$, that is, $f_{E}$ is
independent of $E$ and $f_{E}=f_{E_{0}}$. In this case the measurements become

$P=-\log\left\\{\exp(-\mathcal{R}_{1}f_{E_{0}})\cdot\int_{0}^{\infty}\rho(E)dE\right\\}=\mathcal{R}_{1}f_{E_{0}}.$

The beam hardening gives nonlinear effects to the measurements of CT scanners
and cause the metal streaking artifacts in CT images.

Recently the metal streaking artifacts in CT images were studied by using
microlocal analysis. To explain this we introduce some setting and notation.
Let $D$ be the metal region in $\mathbb{R}^{2}$. We remark that the singular
support of the characteristic function of $D$ is its boundary $\partial{D}$
and the singular directions are the normal directions on $\partial{D}$.
Consider the case that the spectral function is a normalized characteristic
function of an interval $[E_{0}-\varepsilon,E_{0}+\varepsilon]$, that is,

$\rho(E)=\frac{1}{2\varepsilon}\chi_{[E_{0}-\varepsilon,E_{0}+\varepsilon]}(E).$

Suppose that $f_{E}(x)$ is a sum of the normal tissue part $f_{E_{0}}(x)$ and
the metal part of the form

$f_{E}(x)=f_{E_{0}}(x)+\alpha(E-E_{0})\chi_{D}(x)$

with some small parameter $\alpha>0$. If $\alpha\varepsilon>0$ is small
enough, the measurements become

$\displaystyle P$
$\displaystyle=\mathcal{R}_{1}f_{E_{0}}-\log\left\\{\frac{\sinh(\alpha\varepsilon\mathcal{R}_{1}\chi_{D})}{\alpha\varepsilon\mathcal{R}_{1}\chi_{D}}\right\\}$
$\displaystyle=\mathcal{R}_{1}f_{E_{0}}+\sum_{l=1}A_{l}(\alpha\varepsilon\mathcal{R}_{1}\chi_{D})^{2l}$

with some sequence of real numbers $A_{1},A_{2},A_{3},\dotsc$. In their
pioneering work [16] Park, Choi and Seo first studied the microlocal analysis
of this phenomenon. Roughly speaking they proved that if there are two points
on $\partial{D}$ which are connected by a common tangential line, then the
streaking artifacts appear on the tangential line in the filtered back-
projection of $P$. Note that in this case the singular directions of the two
points are the same. They also described the streaking artifacts as the wave
front set of the filtered back-projection of $P$. Their results show the
qualitative properties of the metal streaking artifacts. Moreover in [15]
Palacios, Uhlmann and Wang developed the results of [16]. By using the notion
of paired Lagrangian distributions introduced in [4], they proved that the
metal streaking artifacts are conormal distributions supported on all the
common tangential lines. These results are the quantitative properties of the
metal streaking artifacts.

Here we show Figure 1 illustrating a characteristic function $f$ of two disks
on the plane, its X-ray transform $\mathcal{R}_{1}f$, and the filtered back-
projections of $\mathcal{R}_{1}f$ and $(\mathcal{R}_{1}f)^{2}$. We see the
streaking artifacts on the four lines tangent to both two disks in the
filtered back-projection of $(\mathcal{R}_{1}f)^{2}$.

Figure 1.

In the present paper we shall obtain the canonical relations for all the
$d$-plane transforms on $\mathbb{R}^{n}$ for general $n=2,3,4,\dotsc$ and
$0<d<n$. Our expression of them is useful to understand the relationship
between some conormal bundles in the cotangent bundles of $\mathbb{R}^{n}$ and
$G(d,n)$. Based on the canonical relation we shall also study the metal
streaking artifacts related to them. The organization of the present paper is
as follows. In Section 2 we quickly review microlocal analysis used in the
present paper. In Section 3 we obtain a concrete expression of the cotangent
bundle of $G(d,n)$, and compute the canonical relation of $\mathcal{R}_{d}$.
In Section 4 we study the geometry of the conormal bundle of $\partial{D}$ and
its canonical transformation associated to $\mathcal{R}_{d}$ under the
assumption that $D$ is a disjoint union of finite number of strictly convex
bounded domains with smooth boundaries. In Section 5, following [4] we
introduce paired Lagrangian distributions and study the product
$\mathcal{R}_{d}\chi_{D}\cdot\mathcal{R}_{d}\chi_{D}$. Finally in Section 6 we
prove that the metal streaking artifacts are conormal distributions whose
singular supports are $(n-1)$-dimensional or $(n-2)$-dimensional planes which
are tangent to two or more boundaries of $D$. In case of $d=n-1$ the planes of
singular supports of the conormal distributions are only $(n-1)$-dimensional
planes. On one hand, in case that the singular support of the conormal
distribution is an $(n-1)$-dimensional plane, the directions of the
singularities of the tangent points are in the same normal direction and the
singularities spread in the whole hyperplane. On the other hand, in case that
the singular support of the conormal distribution is an $(n-2)$-dimensional
plane, the propagation of singularities is somewhat curious and natural as
follows. In this case there are two normal directions of the singularities at
the tangent points. All the two-dimensional singularities spanned by those at
tangent points do not necessarily propagate, but the singularities of only two
normal directions at the tangent points spread in the whole
$(n-2)$-dimensional plane. Our method of analysis of the streaking artifacts
is based on the idea of [15], and the results are some generalization of
recent results of [16] and [15] for the X-ray transform on the two-dimensional
plane. The case that $D$ is a disjoint union of finite numbers of convex
polygons in $\mathbb{R}^{2}$ was also studied in [16] and [15]. It is possible
to study the case that $D$ is a disjoint union of finite numbers of convex
polyhedra in $\mathbb{R}^{3}$. However we omit this since we need to take many
kinds of relations such as vertex-vertex, vertex-edge, vertex-polygon, edge-
edge, edge-polygon and polygon-polygon into account.

Throughout the present paper we make use of microlocal analysis freely. See
Hörmander’s four volumes of celebrated textbooks on microlocal analysis [9],
[10], [11] and [12], and textbooks on Fourier integral operators [2] and [5].

##  2\. Quick review of microlocal analysis

We shall briefly review some basic notions of microlocal analysis used later.
We begin with the definition of the Fourier transform in $\mathbb{R}^{N}$ with
$N=1,2,3,\dotsc$. For a rapidly decreasing function $f(x)$ on
$\mathbb{R}^{N}$, its Fourier transform is defined by

$\hat{f}(\xi)=\mathcal{F}f(\xi):=\int_{\mathbb{R}^{N}}e^{-ix\cdot\xi}f(x)dx,\quad\xi\in\mathbb{R}^{N},$

and the inverse Fourier transform of a rapidly decreasing function $g(\xi)$ on
$\mathbb{R}^{N}$ is defined by

$\mathcal{F}^{\ast}{g}(x):=(2\pi)^{-N}\mathcal{F}g(-x)=\frac{1}{(2\pi)^{N}}\int_{\mathbb{R}^{N}}e^{ix\cdot\xi}g(\xi)d\xi,\quad
x\in\mathbb{R}^{N},$

Here we introduce wave front sets of distributions.

######  Definition 2.1 (Wave front set).

Let $X$ be a smooth manifold. $T^{\ast}{X}$ denotes the cotangent bundle of
$X$. For $u\in\mathscr{D}^{\prime}(X)$ and $(x,\xi)\in T^{\ast}{X}\setminus
0$, we say that $(x,\xi)\not\in\operatorname{WF}(u)$ if there exists
$\phi(y)\in C^{\infty}_{0}(X)$ with $\phi(x)\neq 0$ and a conic neighborhood
of $V$ at $\eta=\xi$ such that for any $M>0$

$\widehat{\phi{u}}(\eta)=\mathcal{O}(\langle\eta\rangle^{-M})\quad\text{for}\quad\eta\in{V}.$

We denote this by
$\widehat{\phi{u}}(\xi)=\mathcal{O}(\langle\xi\rangle^{-\infty})$.
$\operatorname{WF}(u)$ is said to be the wave front set of $u$.

Equivalently, if we set

$\Sigma_{x}(u):=\bigcap_{\begin{subarray}{c}\phi(y)\in C^{\infty}_{0}(X)\\\
\phi(x)\neq 0\end{subarray}}\\{\eta{\in}T_{x}^{\ast}{X}\setminus
0:\widehat{\phi{u}}(\eta)\neq\mathcal{O}(\langle\eta\rangle^{-\infty})\\},$

then we have

$\operatorname{WF}(u)=\\{(x,\xi){\in}T^{\ast}{X}\setminus
0:\xi\in\Sigma_{x}(u)\\}.$

Consider a characteristic function $g(x,y)$ of a half plane in
$\mathbb{R}^{2}$: $g(x,y)=1$ if $x>0$ and $g(x,y)=0$ otherwise. For
$\phi(x,y)\in C^{\infty}_{0}(\mathbb{R}^{2})$ we deduce that

  * •

If $\phi(0,y)=0$, then $\widehat{\phi{g}}(\xi,\eta)=\hat{\phi}(\xi,\eta)$ or
$\widehat{\phi{g}}(\xi,\eta)=0$. We have
$\widehat{\phi{g}}(\xi,\eta)=\mathcal{O}(\langle\xi;\eta\rangle^{-\infty})$,
where
$\langle\xi;\eta\rangle=\sqrt{1+\lvert\xi\rvert^{2}+\lvert\eta\rvert^{2}}$.

  * •

If $\phi(0,y)\neq 0$, then

$\widehat{\phi{g}}(\xi,\eta)=\text{constant}\cdot\operatorname{VP}\int_{\mathbb{R}}\frac{\hat{\phi}(\zeta,\eta)}{\xi-\zeta}d\zeta=\mathcal{O}(\langle\eta\rangle^{-\infty}).$

Hence we have $\operatorname{WF}(g)=\\{(0,y,\xi,0)\ |\
y,\xi\in\mathbb{R},\xi\neq 0\\}$. $g(x,y)$ is a typical example of conormal
distributions. Using this notion we can describe and compute microlocal
singularities quantitatively. Conormal distributions and their generalization
called Lagrangian distributions play crucial role in the present paper.
Following Hörmander’s textbook [11, Section 18.2] we introduce basic facts on
conormal distributions. To state the definition of conormal distributions we
introduce function spaces on $\mathbb{R}^{N}$. Split $\mathbb{R}^{N}$ into a
disjoint family of a disk $X_{0}$ and annuli $X_{1},X_{2},X_{3},\dotsc$
defined by

$X_{0}:=\\{\lvert\xi\rvert<1\\},\qquad
X_{j}:=\\{2^{j-1}\leqq\lvert\xi\rvert<2^{j}\\},\quad(j=1,2,3,\dotsc).$

Let $s$ be a real number. We say that a tempered distribution $u$ on
$\mathbb{R}^{N}$ belongs to ${}^{\infty}{H}_{(s)}(\mathbb{R}^{N})$ if

$\lVert{u}\rVert_{{}^{\infty}H_{(s)}(\mathbb{R}^{N})}:=\sup_{j=0,1,2,\dotsc}\left(\int_{X_{j}}\langle\xi\rangle^{2s}\lvert\hat{u}(\xi)\rvert^{2}d\xi\right)^{1/2}<\infty.$

Using smooth cut-off functions, we can define the local space
${}^{\infty}{H}_{(s)}^{\text{loc}}(X)$ of distributions on an $N$-dimensional
smooth manifold $X$. Here we define conormal distributions.

######  Definition 2.2 (Conormal distribution).

Let $X$ be an $N$-dimensional smooth manifold, and let $Y$ be a closed
submanifold of X. $u\in\mathscr{D}^{\prime}(X)$ is said to be conormal with
respect to $Y$ of degree $m$ if

$L_{1}{\dotsb}L_{M}u\in{{}^{\infty}}H^{\text{loc}}_{(-m-N/4)}(X)$

for all $M=0,1,2,\dotsc$ and all vector fields $L_{1},\dotsc,L_{M}$ tangential
to $Y$. Denote by $I^{m}(X,N^{\ast}{Y})$ or $I^{m}(N^{\ast}{Y})$, the set of
all distributions on $X$ conormal with respect to $Y$ of degree $m$. Here
$N^{\ast}{Y}$ is the conormal bundle of $Y$, which is a conic Lagrangian
submanifold of $T^{\ast}{X}$.

Roughly speaking, $I^{m}(X,N^{\ast}{Y})$ is the set of all distributions $u$
on $X$ such that $\operatorname{singsupp}u\subset Y$ and $u$ has some upper
bound of the order of singularities whose directions are perpendicular to $Y$.
Conormal distributions can be characterized by oscillatory integrals which are
similar to the distribution kernels of pseudodifferential operators. Here we
introduce symbol classes.

######  Definition 2.3.

For $m\in\mathbb{R}$ we denote by $S^{m}(\mathbb{R}^{N}\times\mathbb{R}^{k})$
the set of all smooth functions $a(x,\xi^{\prime})$ on
$\mathbb{R}^{N}\times\mathbb{R}^{k}$ such that for any compact set $K$ in
$\mathbb{R}^{N}$ and for any multi-indices $\alpha$ and $\beta$

$\lvert\partial_{x}^{\beta}\partial_{\xi^{\prime}}^{\alpha}a(x,\xi^{\prime})\rvert\leqq
C_{K,\alpha,\beta}\langle\xi^{\prime}\rangle^{m-\lvert\alpha\rvert},\quad(x,\xi^{\prime})\in
K\times\mathbb{R}^{k}$

with some positive constant $C_{K,\alpha,\beta}$.

Here we state the characterization of conormal distributions.

######  Proposition 2.4 ([11, Theorem 18.2.8]).

Let $X$ be an $N$-dimensional smooth manifold, and let $Y$ be a closed
submanifold of $X$ with $\operatorname{codim}Y=k$. Fix arbitrary $y\in Y$ and
choose local coordinates
$x=(x^{\prime},x^{\prime\prime})\in\mathbb{R}^{k}\times\mathbb{R}^{N-k}$ such
that $x(y)=0$ and $Y$ is represented as $\\{x^{\prime}=0\\}$ locally. For
$u\in I^{m+k/2-N/4}(N^{\ast}{Y})$ there exists an amplitude
$a(x^{\prime\prime},\xi^{\prime})\in
S^{m}(\mathbb{R}^{N-k}\times\mathbb{R}^{k})$ such that

$u(x)=\int_{\mathbb{R}^{k}}e^{ix^{\prime}\cdot\xi^{\prime}}a(x^{\prime\prime},\xi^{\prime})d\xi^{\prime}\quad\text{near}\quad
x=0.$ (2.1)

Conversely every $u$ of this form is in $I^{m+k/2-N/4}(N^{\ast}{Y})$.

Here we remark that if there exists an amplitude $b(x,\xi^{\prime})\in
S^{m}(\mathbb{R}^{N}\times\mathbb{R}^{k})$ such that

$u(x)=\int_{\mathbb{R}^{k}}e^{ix^{\prime}\cdot\xi^{\prime}}b(x,\xi^{\prime})d\xi^{\prime}\quad\text{near}\quad
x=0.,$ (2.2)

then this can be reduced to (2.1). Indeed, if (2.2) holds, then (2.1) holds
with an amplitude given by

$a(x^{\prime\prime},\xi^{\prime}):=e^{-iD_{x^{\prime}}\cdot
D_{\xi^{\prime}}}b(x,\xi^{\prime})|_{x^{\prime}=0}\sim\sum_{j=0}^{\infty}\frac{(-iD_{x^{\prime}}\cdot
D_{\xi^{\prime}})^{j}b(x,\xi^{\prime})|_{x^{\prime}=0}}{j!},$

where $D_{x^{\prime}}=-i\partial_{x^{\prime}}$ and $\sim$ means the asymptotic
expansion of symbols. See [11, Lemma 18.2.1].

We apply Proposition 2.4 to two elementary examples. The characteristic
function $g(x,y)$ of the half plane in $\mathbb{R}^{2}$ belongs to

$I^{-1}\bigl{(}N^{\ast}\\{(0,y):y\in\mathbb{R}\\}\bigr{)}=I^{-1}\bigl{(}\mathbb{R}^{2},\\{(0,y,\xi,0)\\}:y,\xi\in\mathbb{R}\bigr{)}.$

Consider a pseudodifferential operator

$\operatorname{Op}(a)u(x)=\frac{1}{(2\pi)^{N}}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}e^{i(x-y)\cdot\xi}a(x,\xi)u(y)dyd\xi,\quad
u\in\mathcal{D}(\mathbb{R}^{N})$

with $a(x,\xi)\in S^{m}(\mathbb{R}^{N}\times\mathbb{R}^{N})$. Set
$\Delta=\\{(x,x):x\in\mathbb{R}^{N}\\}$. This is the diagonal part of
$\mathbb{R}^{N}\times\mathbb{R}^{N}$. The distribution kernel of
$\operatorname{Op}(a)$ is given by an oscillatory integral

$K(x,y)=\frac{1}{(2\pi)^{N}}\int_{\mathbb{R}^{N}}e^{i(x-y)\cdot\xi}a(x,\xi)d\xi$

and belongs to

$I^{m}(N^{\ast}\Delta)=I^{m}\bigl{(}\mathbb{R}^{N}\times\mathbb{R}^{N},\\{(x,x,\xi,-\xi):x,\xi\in\mathbb{R}^{N}\\}\bigr{)}.$

Conormal distributions are typical examples of Lagrangian distributions, and
pseudodifferential operators are typical examples of Fourier integral
operators. The distribution kernels of Fourier integral operators are
Lagrangian distributions. We need more general notions like them later since
the $d$-plane transform is an elliptic Fourier integral operator. In what
follows we shall quickly introduce Lagrangian distributions, and Fourier
integral operators and their canonical relations. We begin with phase
functions which are general form of $(x-y)\cdot\xi$ in $e^{i(x-y)\cdot\xi}$
for pseudodifferential operators.

######  Definition 2.5 (non-degenerate phase function).

Let $X$ be an $N$-dimensional smooth manifold. We say that a real-valued
function $\varphi(x,\theta)\in
C^{\infty}\bigl{(}X{\times}(\mathbb{R}^{k}\setminus\\{0\\})\bigr{)}$ is a non-
degenerate phase function if

  * •

$\varphi(x,t\theta)=t\varphi(x,\theta)$ for $(x,\theta)\in
X{\times}(\mathbb{R}^{k}\setminus\\{0\\})$ and $t>0$.

  * •

$d\varphi(x,\theta)\neq 0$.

  * •

If $\varphi^{\prime}_{\theta}(x,\theta)=0$, then
$\operatorname{rank}[d\varphi^{\prime}_{\theta}(x,\theta)]\equiv k$.

In this case
$C_{\varphi}=\\{(x,\theta):\varphi^{\prime}_{\theta}(x,\theta)=0\\}$ is an
$N$-dimensional submanifold of $X\times(\mathbb{R}^{k}\setminus\\{0\\})$. Set
$\Lambda_{\varphi}:=\bigl{\\{}\bigl{(}x,\varphi^{\prime}_{x}(x,\theta)\bigr{)}:\varphi^{\prime}_{\theta}(x,\theta)=0\bigr{\\}}$.
Then

$C_{\varphi}\ni(x,\theta)\mapsto\bigl{(}x,\varphi^{\prime}_{x}(x,\theta)\bigr{)}\in\Lambda_{\varphi}$

is a diffeomorphism and $\Lambda_{\varphi}$ becomes a conic Lagrangian
submanifold of $T^{\ast}{X}\setminus 0$, that is, a conic submanifold with
$d\xi{\wedge}dx\equiv 0$ on $\Lambda_{\varphi}$. Here we introduce Lagrangian
distributions.

######  Definition 2.6 (Lagrangian distribution).

Let $\Lambda$ be a conic Lagrangian submanifold of $T^{\ast}{X}\setminus 0$.
We denote by $I^{m}(X,\Lambda)=I^{m}(\Lambda)$ the set of all
$u\in\mathscr{D}^{\prime}(X)$ satisfying

  * •

$\operatorname{WF}(u)\subset\Lambda$.

  * •

For any $(x_{0},\xi_{0})\in\Lambda$ there exist a non-degenerate phase
function $\varphi(x,\theta)$ and an amplitude $a(x,\theta)\in
S^{m+N/4-k/2}(X\times\mathbb{R}^{k})$ with some $k=1,2,3,\dotsc$ such that
$\Lambda=\Lambda_{\varphi}$ near $(x_{0},\xi_{0})$ and

$u(x)=\int e^{i\varphi(x,\theta)}a(x,\theta)d\theta\quad\text{microlocally
near}\quad(x_{0},\xi_{0}).$

Finally in the present section we introduce Fourier integral operators and
their canonical relations. Let $X$ and $Y$ be manifolds, and let $\Lambda$ be
a conic Lagrangian submanifold of $T^{\ast}(X{\times}Y)\setminus 0$. For
$K(x,y)\in I^{m}(X{\times}Y,\Lambda)$,

$Au(x):=\int_{Y}K(x,y)u(y)dy,\quad u\in\mathscr{D}(Y)$

defines a linear operator of $\mathscr{D}(Y)$ to $\mathscr{D}^{\prime}(X)$.
$A$ is said to be a Fourier integral operator. Indeed $A$ is locally given by

$Au(x):=\iint e^{i\varphi(x,y,\theta)}a(x,y,\theta)u(y)dyd\theta$

with a non-degenerate phase function $\varphi(x,y,\theta)$ and an amplitude
$a(x,y,\theta)$. If the principal part of $a$ does not vanish, then $A$ is
called an elliptic Fourier integral operator.

$\Lambda^{\prime}:=\\{(x,y;\xi,\eta):(x,y;\xi,-\eta)\in\Lambda\\}$ is said to
be the canonical relation of $A$. We have
$\operatorname{WF}(Au)\subset\Lambda^{\prime}\circ\operatorname{WF}(u)$ and if
$A$ is elliptic then
$\operatorname{WF}(Au)=\Lambda^{\prime}\circ\operatorname{WF}(u)$ holds. Here
$\circ$ denotes the composition defined as follows. For $W\subset
T^{\ast}{X}\times T^{\ast}{Y}$ and $Z\subset T^{\ast}{Y}$ we set

$W{\circ}Z:=\\{(x,\xi)\in T^{\ast}{X}:(x,y;\xi,\eta)\in W\ \text{with some}\
(y,\eta)\in Z\\}.$

The canonical relation describes the correspondence between the microlocal
singularities of $u$ and those of $Au$. In other words this correspondence is
a mapping

$T^{\ast}{Y}\ni\bigl{(}y,-\varphi^{\prime}_{y}(x,y,\theta)\bigr{)}\mapsto\bigl{(}x,\varphi^{\prime}_{x}(x,y,\theta)\bigr{)}\in
T^{\ast}{X}.$ (2.3)

This is called a canonical transformation.

Formally the adjoint of $A$ is given by

$A^{\ast}{v(y)}=\iint
e^{-i\varphi(x,y,\theta)}\overline{a(x,y,\theta)}v(x)dxd\theta.$

Hence if $\Lambda^{\prime}$ is the canonical relation of $A$, then the phase
function is $-\varphi$ and the canonical relation $(\Lambda^{\prime})^{\ast}$
of $A^{\ast}$ is given by

$(\Lambda^{\prime})^{\ast}=\\{(y,x;-\varphi_{y}^{\prime},\varphi_{x}^{\prime}):\varphi^{\prime}_{\theta}=0\\}=\\{(y,x;\eta,\xi):(x,y;\xi,\eta)\in\Lambda^{\prime}\\}.$

##  3\. Canonical relations of the $d$-plane transform

In this section we present the basic properties of the $d$-plane transform as
a Fourier integral operator. More precisely we show that the distribution
kernel of the $d$-plane transform is a Lagrangian distribution on
$G(d,n)\times\mathbb{R}^{n}$ with some order and some conic Lagrangian
submanifold of $T^{\ast}\bigl{(}G(d,n)\times\mathbb{R}^{n}\bigr{)}\setminus
0$. So we begin with the expression of $T^{\ast}{G(d,n)}$. We should
understand the structure of the mapping (2.3) for $Y=\mathbb{R}^{n}$ and
$X=G(d,n)$ concretely and clearly so that we can make full use of microlocal
analysis. This depends on the expression of $T^{\ast}{G(d,n)}$. Here we recall
that $\operatorname{dim}G_{d,n}=d(n-d)$ and
$N(d,n):=\operatorname{dim}G(d,n)=(d+1)(n-d)$.

######  Lemma 3.1 (The expression of $T^{\ast}{G(d,n)}$).

$T^{\ast}{G(d,n)}=\\{(\sigma,x^{\prime\prime};\eta_{1},\dotsc,\eta_{d},\xi):\sigma{\in}G_{d,n},\
x^{\prime\prime},\eta_{1},\dotsc,\eta_{d},\xi\in\sigma^{\perp}\\}.$

###### Proof.

Fix arbitrary $(\sigma,x^{\prime\prime})\in G(d,n)$. It is easy to see that
the set of cotangent vectors corresponding to $x^{\prime\prime}$ is
$\xi\in\sigma^{\perp}$. There exists an orthonormal system
$\\{\omega_{1},\dotsc,\omega_{d}\\}$ of $\mathbb{R}^{n}$ such that
$\sigma=\langle\omega_{1},\dotsc,\omega_{d}\rangle$. We remark that the
covariant derivative of a function on $\sigma$ with respect to $\omega_{j}$ is
a vector in $\omega_{j}^{\perp}$. Recall that the set of cotangent vectors
corresponding to $\sigma$ is a $d(n-d)$-dimensional real vector space. If we
replace $\omega_{j}$ by $c\omega_{j}$ with $c\neq 0$, then

$\langle\omega_{1},\dotsc,\omega_{j-1},c\omega_{j},\omega_{j+1},\dotsc,\omega_{d}\rangle=\langle\omega_{1},\dotsc,\omega_{d}\rangle,\quad
j=1,\dotsc,d.$

This means that any cotangent vector corresponding $\sigma$ must belong to
$\omega_{1}^{\perp}\cap\dotsb\cap\omega_{d}^{\perp}=\sigma^{\perp}$. Hence the
vector space of all cotangent vectors corresponding $\sigma$ can be seen as
the $d$-product $\sigma^{\perp}\times\dotsb\times\sigma^{\perp}$. ∎

We denote by $\pi_{\sigma}$ the orthogonal projection of $\mathbb{R}^{n}$ onto
$\sigma\in G_{d,n}$. If $\sigma$ is spanned by an orthonormal system
$\\{\omega_{1},\dotsc,\omega_{d}\\}$ of $\mathbb{R}^{n}$, then $\pi_{\sigma}$
is given by

$\pi_{\sigma}=\sum_{j=1}^{d}\omega\omega^{T}.$

We denote this case by
$\sigma=\langle\omega_{1},\dotsc,\omega_{d}\rangle_{\operatorname{ON}}$. This
notation depends on the choice of an orthonormal basis of $\sigma$, and we
need to take care of it below. We have the quantitative properties of
$\mathcal{R}_{d}$ as follows.

######  Theorem 3.2.

$\mathcal{R}_{d}$ is an elliptic Fourier integral operator whose distribution
kernel belongs to

$I^{0-(N(d,n)+n)/4+(n-d)/2}\bigl{(}G(d,n)\times\mathbb{R}^{n},\Lambda_{\phi}\bigr{)},$
$0-\frac{N(d,n)+n}{4}+\frac{n-d}{2}=-\frac{d(n-d+1)}{4},$
$\displaystyle\Lambda_{\phi}^{\prime}$
$\displaystyle=\bigl{\\{}\bigl{(}\sigma,y-\pi_{\sigma}{y},y;\eta(y\cdot\omega_{1},\dotsc,y\cdot\omega_{d},1,1)\bigr{)}:$
$\displaystyle\qquad\qquad\sigma=\langle\omega_{1},\dotsc,\omega_{d}\rangle_{\operatorname{ON}}{\in}G_{d,n},y\in\mathbb{R}^{n},\eta\in\sigma^{\perp}\setminus\\{0\\}\bigr{\\}}$
$\displaystyle=\bigl{\\{}\bigl{(}\sigma,y-{\pi_{\sigma}}y,y;\eta(y\cdot\omega_{1},\dotsc,y\cdot\omega_{d},1,1)\bigr{)}:$
$\displaystyle\qquad\qquad(y,\eta)\in T^{\ast}\mathbb{R}^{n}\setminus
0,\sigma=\langle\omega_{1},\dotsc,\omega_{d}\rangle_{\operatorname{ON}}\in
G_{d,n},\sigma\subset\eta^{\perp}\bigr{\\}}$
$\displaystyle=\bigl{\\{}\bigl{(}\sigma,x^{\prime\prime},x^{\prime\prime}+t_{1}\omega_{1}+\dotsb+t_{d}\omega_{d};\xi(t_{1},\dotsc,t_{d},1,1)\bigr{)}$
$\displaystyle\qquad\qquad:(\sigma,x^{\prime\prime}){\in}G(d,n),\sigma=\langle\omega_{1},\dotsc,\omega_{d}\rangle_{\operatorname{ON}}\in
G_{d,n},t_{1},\dotsc,t_{d}\in\mathbb{R},\xi\in\sigma^{\perp}\setminus\\{0\\}\bigr{\\}}.$

###### Proof.

We begin with the generalized Fourier slice theorem. For any
$\sigma{\in}G_{d,n}$ and $\xi\in\sigma^{\perp}$, we have
$x^{\prime}\cdot\xi=0$ for $x^{\prime}\in\sigma$ and

$\displaystyle\int_{\sigma^{\perp}}e^{-ix^{\prime\prime}\cdot\xi}\mathcal{R}_{d}f(\sigma,x^{\prime\prime})dx^{\prime\prime}$
$\displaystyle=\int_{\sigma^{\perp}}\int_{\sigma}e^{-ix^{\prime\prime}\cdot\xi}f(x^{\prime}+x^{\prime\prime})dx^{\prime}dx^{\prime\prime}$
$\displaystyle=\int_{\sigma^{\perp}}\int_{\sigma}e^{-i(x^{\prime}+x^{\prime\prime})\cdot\xi}f(x^{\prime}+x^{\prime\prime})dx^{\prime}dx^{\prime\prime}$
$\displaystyle=\int_{\mathbb{R}^{n}}e^{-ix\cdot\xi}f(x)dx=\hat{f}(\xi),\quad\xi\in\sigma^{\perp}.$

Then for any $(\sigma,x^{\prime\prime}){\in}G(d,n)$, the Fourier inversion
formula on $\sigma^{\perp}$ implies that

$\displaystyle\mathcal{R}_{d}f(\sigma,x^{\prime\prime})$
$\displaystyle=\frac{1}{(2\pi)^{n-d}}\int_{\sigma^{\perp}}e^{ix^{\prime\prime}\cdot\xi}\hat{f}(\xi)d\xi$
$\displaystyle=\frac{1}{(2\pi)^{n-d}}\int_{\sigma^{\perp}}\int_{\mathbb{R}^{n}}e^{i(x^{\prime\prime}-y)\cdot\xi}f(y)dyd\xi$
$\displaystyle=\frac{1}{(2\pi)^{n-d}}\int_{\sigma^{\perp}}\int_{\mathbb{R}^{n}}e^{i\phi(\sigma,x^{\prime\prime},y,\xi)}f(y)dyd\xi,$
$\displaystyle\phi(\sigma,x^{\prime\prime},y,\xi)$
$\displaystyle=(x^{\prime\prime}-y)\cdot\xi,\quad(\sigma,x^{\prime\prime}){\in}G(d,n),y\in\mathbb{R}^{n},\xi\in\sigma^{\perp}.$

It suffices to show that $\phi$ is a non-degenerate phase function, and to
obtain the Lagrangian submanifold $\Lambda_{\phi}$. If this is true, then it
follows that the amplitude is the function which is identically equal to
$1/(2\pi)^{n-d}$. This symbol belongs to
$S^{0}\Bigl{(}\bigl{(}G(d,n)\times\mathbb{R}^{n}\bigr{)}\times\mathbb{R}^{n-d}\Bigr{)}$
and the order of the Lagrangian distribution is

$0+\frac{N(d,n)+n}{4}-\frac{n-d}{2}.$

We find the critical points define by $\phi_{\xi}^{\prime}=0$ for
$\xi\in\sigma^{\perp}$. Set

$\psi_{1}(\sigma,x^{\prime\prime},y,\Xi):=\phi(\sigma,x^{\prime\prime},y,\Xi-\pi_{\sigma}\Xi)=(x^{\prime\prime}-y)\cdot(\Xi-\pi_{\sigma}\Xi)=(x^{\prime\prime}-y+\pi_{\sigma}{y})\cdot\Xi$

for $\Xi\in\mathbb{R}^{n}$. Then $\Xi$ moves essentially in $\sigma^{\perp}$
and not in the whole space $\mathbb{R}^{n}$ for $\psi_{1}$. Then we have

$\phi_{\xi}^{\prime}=\nabla_{\Xi}\psi_{1}|_{\Xi=\xi}=x^{\prime\prime}-(y-\pi_{\sigma}{y}).$

Hence the critical points of $\phi$ are characterized by
$x^{\prime\prime}=y-\pi_{\sigma}{y}$. Let $E$ be the $n{\times}n$ identity
matrix. We check the non-degeneracy of the critical points. We have

$\phi^{\prime\prime}_{\xi
y}=-E+\pi_{\sigma},\quad\operatorname{rank}\phi^{\prime\prime}_{\xi y}=n-d.$

Then we have
$\operatorname{rank}(d\phi^{\prime}_{\xi})|_{x=y-\pi_{\sigma}{y}}=n-d$ since
$\xi$ moves in the $(n-d)$-dimensional vector subspace $\sigma^{\perp}$.
Finally we obtain $\phi^{\prime}_{\sigma,x^{\prime\prime},y,\xi}\neq 0$ for
$\xi\in\sigma^{\perp}\setminus\\{0\\}$ since $\phi^{\prime}_{y}=-\xi$. Hence
we have proved that $\phi(\sigma,x^{\prime\prime},y,\xi)$ is a non-degenerate
phase function with critical points given by
$x^{\prime\prime}=y-\pi_{\sigma}{y}$.

Finally we compute to find the Lagrangian submanifold

$\Lambda_{\phi}=\\{(\sigma,x^{\prime\prime},y;\phi^{\prime}_{\sigma},\phi^{\prime}_{x^{\prime\prime}},\phi^{\prime}_{y}):(\sigma,x^{\prime\prime})\in
G(d,n),y\in\mathbb{R}^{n},\xi\in\sigma^{\perp},x^{\prime\prime}=y-\pi_{\sigma}{y}\\}$

of $T^{\ast}\bigl{(}G(d,n)\times\mathbb{R}^{n}\bigr{)}\setminus 0$. Recall
that $\phi^{\prime}_{y}=-\xi$. Set

$\psi_{2}(\sigma,x,y,\xi):=\phi(\sigma,x-\pi_{\sigma}{x},y,\xi)=(x-y)\cdot\xi$

for $(\sigma,x,y,\xi)\in
G_{d,n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\sigma^{\perp}$. Then we
have
$\phi_{x^{\prime\prime}}^{\prime}=\nabla_{x}\psi_{2}|_{x=x^{\prime\prime}}=\xi$.
Next we compute $\phi^{\prime}_{\sigma}$. Suppose that $\sigma$ is spanned by
an orthonormal system $\\{\omega_{1},\dotsc,\omega_{d}\\}$ in
$\mathbb{R}^{n}$. We compute $\phi^{\prime}_{\omega_{j}}$. Consider a small
perturbation of $\sigma$ of the form

$\tilde{\sigma}=\langle\mu_{1},\dotsc.\mu_{d}\rangle=\sigma+o(1),\quad\mu_{1},\dotsc.\mu_{d}\in\mathbb{R}^{n}.$

Set

$\psi_{3}(\tilde{\sigma},x^{\prime\prime},y,\xi):=\left(x-y+\sum_{j=1}^{d}(y\cdot\mu_{j})\mu_{j}\right)\cdot\xi=(x^{\prime\prime}-y)\cdot\xi+\sum_{j=1}^{d}(y\cdot\mu_{j})(\xi\cdot\mu_{j}).$

Then we have

$\phi^{\prime}_{\omega_{j}}=\bigl{(}\nabla_{\mu_{j}}\psi_{3}-(\mu_{j}\cdot\nabla_{\mu_{j}}\psi_{3})\mu_{j}\bigr{)}|_{\mu_{j}=\omega_{j}}=(\xi\cdot\omega_{j})y+(y\cdot\omega_{j})\xi=(y\cdot\omega_{j})\xi$

since $\xi\in\langle\omega_{1},\dotsc,\omega_{d}\rangle^{\perp}$. Hence we
obtain

$\phi^{\prime}_{\sigma,x^{\prime\prime},y}=\xi(y\cdot\omega_{1},\dotsc,y\cdot\omega_{d},1,-1)$

for $x^{\prime\prime}\in\sigma^{\perp}$, $y\in\mathbb{R}^{n}$ and
$\xi\in\sigma^{\perp}\setminus\\{0\\}$. Summing up the above computation, we
deduce that

$\displaystyle\Lambda_{\phi}$
$\displaystyle=\bigl{\\{}\bigl{(}\sigma,y-\pi_{\sigma}{y},y;\eta(y\cdot\omega_{1},\dotsc,y\cdot\omega_{d},1,-1)\bigr{)}:$
$\displaystyle\qquad\qquad\sigma=\langle\omega_{1},\dotsc,\omega_{d}\rangle_{\operatorname{ON}}{\in}G_{d,n},y\in\mathbb{R}^{n},\eta\in\sigma^{\perp}\setminus\\{0\\}\bigr{\\}}$
$\displaystyle=\bigl{\\{}\bigl{(}\sigma,y-{\pi_{\sigma}}y,y;\eta(y\cdot\omega_{1},\dotsc,y\cdot\omega_{d},1,-1)\bigr{)}:$
$\displaystyle\qquad\qquad(y,\eta)\in T^{\ast}\mathbb{R}^{n}\setminus
0,\sigma=\langle\omega_{1},\dotsc,\omega_{d}\rangle_{\operatorname{ON}}\in
G_{d,n},\sigma\subset\eta^{\perp}\bigr{\\}}$
$\displaystyle=\bigl{\\{}\bigl{(}\sigma,x^{\prime\prime},x^{\prime\prime}+t_{1}\omega_{1}+\dotsb+t_{d}\omega_{d};\xi(t_{1},\dotsc,t_{d},1,-1)\bigr{)}$
$\displaystyle\qquad\qquad:(\sigma,x^{\prime\prime}){\in}G(d,n),\sigma=\langle\omega_{1},\dotsc,\omega_{d}\rangle_{\operatorname{ON}}\in
G_{d,n},t_{1},\dotsc,t_{d}\in\mathbb{R},\xi\in\sigma^{\perp}\setminus\\{0\\}\bigr{\\}}.$

This completes the proof. ∎

##  4\. Geometry of metal regions

In Section 1 we review the known results on the artifacts caused by metal
region $D$ in $\mathbb{R}^{2}$. In the present section we study the metal
streaking artifacts in general space dimension $n$. Let $E\in[0,\infty)$ be
the energy level of X-rays and let $E_{0}>0$ be the standard level for human
tissue. Suppose that the distribution of attenuation coefficients on
$\mathbb{R}^{n}$ depends on $E$ of the form

$f_{E}(x)=f_{E_{0}}(x)+\alpha(E-E_{0})\chi_{D}(x),\quad x\in\mathbb{R}^{n},$

where $\alpha>0$ is a constant. The spectral function is supposed to be a
probability density of uniformly distributed on a closed interval
$[E_{0}-\varepsilon,E_{0}+\varepsilon]\subset[0,\infty)$ of the form

$\rho(E)=\frac{1}{2\varepsilon}\chi_{[E_{0}-\varepsilon,E_{0}+\varepsilon]}(E),\quad
E\in[0,\infty).$

Then the measurement is not $\mathcal{R}_{d}f_{E}(\sigma,x^{\prime\prime})$
but

$\displaystyle P_{d}(\sigma,x^{\prime\prime})$
$\displaystyle=-\log\left\\{\frac{1}{2\varepsilon}\int_{E_{0}-\varepsilon}^{E_{0}+\varepsilon}\rho(E)\exp\bigl{(}-\mathcal{R}_{d}f_{E}(\sigma,x^{\prime\prime})\bigr{)}dE\right\\}$
$\displaystyle=\mathcal{R}_{d}f_{E_{0}}(\sigma,x^{\prime\prime})+P_{d,\text{MA}}(\sigma,x^{\prime\prime}),$
$\displaystyle P_{d,\text{MA}}(\sigma,x^{\prime\prime})$
$\displaystyle=\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\left\\{\sum_{l=1}^{\infty}\frac{\bigl{(}\alpha\varepsilon\mathcal{R}_{d}\chi_{D}(\sigma,x^{\prime\prime})\bigr{)}^{2l}}{(2l+1)!}\right\\}^{k}$
$\displaystyle=\sum_{l=1}^{\infty}A_{l}\bigl{(}\alpha\varepsilon\mathcal{R}_{d}\chi_{D}(\sigma,x^{\prime\prime})\bigr{)}^{2l},$

where $\\{A_{l}\\}_{l=1}^{\infty}$ is a sequence of real numbers. The CT image
is denoted by $f_{\text{CT}}$, that is,

$\displaystyle f_{\text{CT}}$
$\displaystyle:=\mathcal{R}_{d}^{\ast}(-\Delta_{x^{\prime\prime}})^{d/2}P_{d}=f_{E_{0}}+f_{\text{MA}},$
$\displaystyle f_{\text{MA}}$
$\displaystyle=\mathcal{R}_{d}^{\ast}(-\Delta_{x^{\prime\prime}})^{d/2}P_{d,\text{MA}}=\sum_{l=1}^{\infty}A_{l}(\alpha\varepsilon)^{2l}\mathcal{R}_{d}^{\ast}(-\Delta_{x^{\prime\prime}})^{d/2}\bigl{[}(\mathcal{R}_{d}\chi_{D})^{2l}\bigr{]}$

The metal streaking artifacts are nonlinear effects and we need to study the
microlocal singularities of

$(\mathcal{R}_{d}\chi_{D})^{2l},\quad\mathcal{R}_{d}^{\ast}(-\Delta_{x^{\prime\prime}})^{d/2}\bigl{[}(\mathcal{R}_{d}\chi_{D})^{2l}\bigr{]}\quad(l=1,2,3,\dotsc).$

Throughout the present paper we assume the following:

  * (A)

$D$ is a disjoint union of bounded strictly convex smooth domains
$D_{1},\dotsc,D_{J}\subset\mathbb{R}^{n}$ with $J=1,2,3,\dotsc$.

Set $\Sigma_{j}:=\partial{D}_{j}$ and
$\Sigma:=\displaystyle\bigcup_{j=1}^{J}\Sigma_{j}$. Then $\Sigma=\partial{D}$.
We denote by $\pi_{G(d,n)}$ the projection of $T^{\ast}{G(d,n)}$ onto
$G(d,n)$. We prepare some lemmas related with basic properties of
$\Lambda_{\phi}^{\prime}{\circ}N^{\ast}\Sigma_{j}$. The first lemma says that
$\Lambda_{\phi}^{\prime}{\circ}N^{\ast}\Sigma_{j}$ is also a conormal bundle
of a hypersurface of $G(d,n)$.

######  Lemma 4.1.

Set $S_{j}:=\pi_{G(d,n)}(\Lambda_{\phi}^{\prime}{\circ}N^{\ast}\Sigma_{j})$.
Then $\operatorname{codim}S_{j}=1$ and
$N^{\ast}{S_{j}}=\Lambda_{\phi}^{\prime}{\circ}N^{\ast}\Sigma_{j}$.

###### Proof.

Recall that $N^{\ast}\Sigma_{j}$ and $\Lambda_{\phi}^{\prime}$ are concretely
given by

$N^{\ast}\Sigma_{j}=\\{(y,\eta)\in
T^{\ast}\mathbb{R}^{n}:y\in\Sigma_{j},\eta\in N^{\ast}_{y}\Sigma_{j}\\},$

and

$\displaystyle\Lambda_{\phi}^{\prime}$
$\displaystyle=\bigl{\\{}\bigl{(}\sigma,y-{\pi_{\sigma}}y,y;\eta(y\cdot\omega_{1},\dotsc,y\cdot\omega_{d},1,1)\bigr{)}:$
$\displaystyle\qquad\qquad(y,\eta)\in T^{\ast}\mathbb{R}^{n}\setminus
0,\sigma=\langle\omega_{1},\dotsc,\omega_{d}\rangle_{\operatorname{ON}}\in
G_{d,n},\sigma\subset\eta^{\perp}\bigr{\\}}$

respectively. Then we have

$\displaystyle\Lambda_{\phi}^{\prime}{\circ}N^{\ast}\Sigma_{j}$
$\displaystyle=\bigl{\\{}\bigl{(}\sigma,y-{\pi_{\sigma}}y;\eta(y\cdot\omega_{1},\dotsc,y\cdot\omega_{d},1)\bigr{)}:$
$\displaystyle\qquad\qquad(y,\eta){\in}N^{\ast}\Sigma_{j}\setminus
0,\sigma=\langle\omega_{1},\dotsc,\omega_{d}\rangle_{\operatorname{ON}}\in
G_{d,n},\sigma\subset\eta^{\perp}\bigr{\\}}$

and therefore

$S_{j}=\pi_{G(d,n)}(\Lambda_{\phi}^{\prime}{\circ}N^{\ast}\Sigma_{j})=\\{(\sigma,y-{\pi_{\sigma}}y):y\in\Sigma_{j},\sigma\in
G_{d,n},\sigma\subset T_{y}\Sigma_{j}\\}.$

Then we deduce that

$\displaystyle\operatorname{dim}S_{j}$
$\displaystyle=\operatorname{dim}\Sigma_{j}+\operatorname{dim}G_{d,n-1}=(n-1)+d(n-1-d)$
$\displaystyle=(d+1)(n-d)-1=N(d,n)-1.$

This shows that $S_{j}$ is a hypersurface in $G(d,n)$ and
$\operatorname{codim}S_{j}=1$. The canonical symplectic form of
$T^{\ast}\mathbb{R}^{n}$ vanishes on $N^{\ast}\Sigma_{j}$, and the canonical
transform preserves the canonical symplectic form. Hence the canonical
symplectic form of $T^{\ast}{G(d,n)}$ also vanishes on
$\Lambda_{\phi}^{\prime}{\circ}N^{\ast}\Sigma_{j}$. Combining these facts we
deduce that $N^{\ast}{S_{j}}=\Lambda_{\phi}^{\prime}{\circ}N^{\ast}\Sigma_{j}$
since $N^{\ast}{S_{j}}$ and $\Lambda_{\phi}^{\prime}{\circ}N^{\ast}\Sigma_{j}$
are conic Lagrangian submanifolds with the same base space $S_{j}$. ∎

Next we discuss the intersection $S_{jk}:=S_{j}{\cap}S_{k}$ for $j{\neq}k$.
The structure of the intersection of Lagrangian submanifolds determines some
properties of the product of Lagrangian distributions. So the intersection
calculus of $S_{jk}$ for $j{\neq}k$ plays a crucial role in the microlocal
analysis of nonlinear effects of $\mathcal{R}_{d}\chi_{D}$. In particular we
need to know $\operatorname{codim}S_{jk}$ and the relationship between
$N^{\ast}{S_{j}}$, $N^{\ast}{S_{k}}$ and $N^{\ast}{S_{jk}}$. We begin with
some basic properties of the intersection $S_{jk}$ for $j{\neq}k$. We first
remark that $S_{jk}\neq\emptyset$ holds if and only if there exist
$(\sigma,x^{\prime\prime})\in S_{jk}$, $y_{j}\in\Sigma_{j}$ and
$y_{k}\in\Sigma_{k}$ such that $y_{j}{\neq}y_{k}$ since
$\Sigma_{j}\cap\Sigma_{k}=\emptyset$, and

$x^{\prime\prime}=y_{j}-\pi_{\sigma}{y_{j}}=y_{k}-\pi_{\sigma}{y_{k}},\quad\sigma\subset
T_{y_{j}}\Sigma_{j}{\cap}T_{y_{k}}\Sigma_{k}.$

If $N^{\ast}_{y_{j}}\Sigma=N^{\ast}_{y_{k}}\Sigma$, then
$T_{y_{j}}\Sigma_{j}=T_{y_{k}}\Sigma_{k}$, otherwise

$T_{y_{j}}\Sigma_{j}{\cap}T_{y_{k}}\Sigma_{k}=\bigl{(}N^{\ast}_{y_{j}}\Sigma_{j}\bigr{)}^{\perp}\cap\bigl{(}N^{\ast}_{y_{k}}\Sigma_{k}\bigr{)}^{\perp}$

is an $(n-2)$-dimensional vector subspace. See Figures 2 and 3 below.

Figure 2. This illustrates the case of
$N^{\ast}_{y_{j}}\Sigma=N^{\ast}_{y_{k}}\Sigma$ with $\eta_{j},\eta_{k}\in
N^{\ast}\Sigma_{j}\setminus\\{0\\}=N^{\ast}\Sigma_{k}\setminus\\{0\\}$.

Figure 3. This illustrates the case of
$N^{\ast}_{y_{j}}\Sigma{\neq}N^{\ast}_{y_{k}}\Sigma$ with $\eta_{j}\in
N^{\ast}\Sigma_{j}\setminus\\{0\\}$ and $\eta_{k}\in
N^{\ast}\Sigma_{k}\setminus\\{0\\}$.

We split ${S_{jk}}$ into two parts:

$\displaystyle S_{jk}^{(1)}$ $\displaystyle=\\{(\sigma,x^{\prime\prime})\in
S_{jk}:N^{\ast}_{y_{j}}\Sigma=N^{\ast}_{y_{k}}\Sigma\\},$ $\displaystyle
S_{jk}^{(2)}$ $\displaystyle=\\{(\sigma,x^{\prime\prime})\in
S_{jk}:N^{\ast}_{y_{j}}\Sigma{\neq}N^{\ast}_{y_{k}}\Sigma\\}.$

Clearly we have a disjoint union $S_{jk}=S_{jk}^{(1)}{\cup}S_{jk}^{(2)}$, and
in particular $S_{jk}=S_{jk}^{(1)}$ and $S_{jk}^{(2)}=\emptyset$ if $d=n-1$.
Here we introduce some basic notions of intersection calculus, those are,
transversal intersections and clean intersections of two submanifolds. See
[14] and [7] for the detail.

######  Definition 4.2.

Let $X$ be a smooth manifold, and let $Y$ and $Z$ be submanifolds of $X$.

  * •

We say that $Y$ and $Z$ intersect transversally if
$N^{\ast}_{x}Y{\cap}N^{\ast}_{x}Z=\\{0\\}$ for all $x\in Y{\cap}Z$. Note that
this condition is equivalent to that $T_{x}Y{\cup}T_{x}Z=T_{x}X$ for all $x\in
Y{\cap}Z$.

  * •

We say that $Y$ and $Z$ intersect cleanly if $Y{\cap}Z$ is smooth and
$T_{x}Y{\cap}T_{x}Z=T_{x}(Y{\cap}Z)$ for all $x\in Y{\cap}Z$.

It is well-known that if $Y$ and $Z$ intersect transversally, then

$\operatorname{codim}(Y{\cap}Z)=\operatorname{codim}(Y)+\operatorname{codim}(Z),$

and $Y$ and $Z$ intersect cleanly. Here we state the properties of the
intersection $S_{jk}$.

######  Lemma 4.3.

Suppose that $j{\neq}k$ and $S_{jk}\neq\emptyset$. Then $S_{j}$ and $S_{k}$
intersect transversally.

###### Proof.

Fix arbitrary $(\sigma,x^{\prime\prime})\in S_{jk}$. There exist
$y_{j}\in\Sigma_{j}$ and $y_{k}\in\Sigma_{k}$ such that $\sigma\subset
T_{y_{j}}\Sigma_{j}{\cap}T_{y_{k}}\Sigma_{k}$. Hence we can choose an
orthonormal system $\\{\omega_{0},\omega_{1}\\}\subset\sigma$ and
$t_{0},t_{j},t_{k}\in\mathbb{R}$ such that

$y_{j}=x^{\prime\prime}+t_{0}\omega_{0}+t_{j}\omega_{1},\quad
y_{k}=x^{\prime\prime}+t_{0}\omega_{0}+t_{k}\omega_{1}.$

More precisely $\omega_{1}$ is a vector of the direction of $y_{k}-y_{j}$ and
we can set $\omega_{1}:=(y_{k}-y_{j})/\lvert{y_{k}-y_{j}\rvert}$. $\omega_{0}$
is a perpendicular direction from $x^{\prime\prime}$ to the line connecting
$y_{j}$ and $y_{k}$. In case $d=1$ we think that $t_{0}=0$. We have
$t_{j}{\neq}t_{k}$ since $\Sigma_{j}\cap\Sigma_{k}=\emptyset$. Recall Lemma
3.1. In the same way as the proof of Lemma 4.1, we have

$\displaystyle N^{\ast}_{(\sigma,x^{\prime\prime})}{S_{j}}$
$\displaystyle=\\{\eta_{j}(t_{0},t_{j},0,\dotsc,0,1):\eta_{j}\in
N^{\ast}\Sigma_{j}\\},$ $\displaystyle
N^{\ast}_{(\sigma,x^{\prime\prime})}{S_{k}}$
$\displaystyle=\\{\eta_{k}(t_{0},t_{k},0,\dotsc,0,1):\eta_{k}\in
N^{\ast}\Sigma_{k}\\}.$

If $\eta_{j}\neq 0$ and $\eta_{k}\neq 0$, then vectors
$\eta_{j}(t_{0},t_{j},0,\dotsc,0,1)$ and $\eta_{k}(t_{0},t_{k},0,\dotsc,0,1)$
are linearly independent in $T^{\ast}_{(\sigma,x^{\prime\prime})}G(d,n)$ since
$t_{j}{\neq}t_{k}$, and therefore
$N^{\ast}_{(\sigma,x^{\prime\prime})}{S_{j}}{\cap}N^{\ast}_{(\sigma,x^{\prime\prime})}{S_{k}}=\\{0\\}$.
This completes the proof. ∎

Lemma 4.3 implies that $\operatorname{codim}(S_{jk})=2$. Next we obtain the
concrete expression of $N^{\ast}{S_{jk}}$ which is required for computing the
composition $(\Lambda_{\phi}^{\prime}){\circ}N^{\ast}{S_{jk}}$.

######  Lemma 4.4.

$\displaystyle N^{\ast}{S_{jk}^{(1)}}$
$\displaystyle=\bigl{\\{}\bigl{(}\sigma,x^{\prime\prime};\eta(t_{0},t,0,\dotsc,0,1)\bigr{)}:N^{\ast}_{y_{j}}\Sigma_{j}=N^{\ast}_{y_{k}}\Sigma_{k},$
$\displaystyle\qquad\qquad\sigma\subset
T_{y_{j}}\Sigma_{j},x^{\prime\prime}=y_{j}-\pi_{\sigma}{y_{j}}=y_{k}-\pi_{\sigma}{y_{k}},$
$\displaystyle\qquad\qquad t\in\mathbb{R},\eta\in N^{\ast}_{y_{j}}\Sigma_{j}\
\text{with some}\
t_{0}\in\mathbb{R},y_{j}\in\Sigma_{j},y_{k}\in\Sigma_{k}\bigr{\\}},$
$\displaystyle N^{\ast}{S_{jk}^{(2)}}$
$\displaystyle=\bigl{\\{}\bigl{(}\sigma,x^{\prime\prime};\Xi\bigr{)}:N^{\ast}_{y_{j}}\Sigma_{j}\neq
N^{\ast}_{y_{k}}\Sigma_{k}$ $\displaystyle\qquad\qquad\sigma\subset
T_{y_{j}}\Sigma_{j}{\cap}T_{y_{k}}\Sigma_{k},x^{\prime\prime}=y_{j}-\pi_{\sigma}{y_{j}}=y_{k}-\pi_{\sigma}{y_{k}},$
$\displaystyle\qquad\qquad\Xi\in
N^{\ast}_{(\sigma,x^{\prime\prime})}\Sigma_{j}\oplus
N^{\ast}_{(\sigma,x^{\prime\prime})}\Sigma_{k}\ \text{with some}\
y_{j}\in\Sigma_{j},y_{k}\in\Sigma_{k}\bigr{\\}}.$

###### Proof.

The statement on $N^{\ast}{S_{jk}^{(2)}}$ is obvious. We shall obtain
$N^{\ast}{S_{jk}^{(1)}}$. Fix arbitrary $(\sigma,x^{\prime\prime})\in
S_{jk}^{(1)}$. We use the same notation in the proof of Lemma 4.3. In view of
Lemma 4.3 and its proof, we deduce that

$N^{\ast}_{(\sigma,x^{\prime\prime})}S_{jk}^{(1)}=\langle\eta(t_{0},t_{j},0,\dotsc,0,1),\eta(t_{0},t_{k},0,\dotsc,0,1)\rangle$

with some $\eta\in N^{\ast}_{y_{j}}\Sigma_{j}\setminus\\{0\\}$. Recall that
$t_{j}{\neq}t_{k}$. For any $t\in\mathbb{R}$, set $\alpha:=t_{k}-t$ and
$\beta:=t-t_{j}$. Then $\alpha+\beta=t_{k}-t_{j}\neq 0$,
$t=(\alpha{t_{j}}+\beta{t_{k}})/(\alpha+\beta)$ and

$\displaystyle\eta(t_{0},t,0,\dotsc,0,1)$
$\displaystyle=\eta\left(t_{0},\frac{\alpha{t_{j}}+\beta{t_{k}}}{\alpha+\beta},0,\dotsc,0,1\right)$
$\displaystyle=\frac{\alpha}{\alpha+\beta}\eta(t_{0},t_{j},0,\dotsc,0,1)+\frac{\beta}{\alpha+\beta}\eta(t_{0},t_{k},0,\dotsc,0,1).$

Hence we get $\\{\eta(t_{0},t,0,\dotsc,0,1):t\in\mathbb{R},\eta\in
N^{\ast}_{y_{j}}\Sigma_{j}\\}\subset
N^{\ast}_{(\sigma,x^{\prime\prime})}S_{jk}$. The converse inclusion relation
is obvious and we obtain $\\{\eta(t_{0},t,0,\dotsc,0,1):t\in\mathbb{R},\eta\in
N^{\ast}_{y_{j}}\Sigma_{j}\\}=N^{\ast}_{(\sigma,x^{\prime\prime})}S_{jk}$.
This completes the proof. ∎

Here we set

$S:=\bigcup_{j=1}^{J}S_{j},\quad SS:=\bigcup_{1\leqq j<k\leqq J}S_{jk},\quad
SS^{(i)}:=\bigcup_{1\leqq j<k\leqq J}S_{jk}^{(i)}\quad(i=1,2)$

for the sake of convenience. In order to describe the canonical transformation
of $N^{\ast}{S_{j}}$, $N^{\ast}{S_{jk}}$, $N^{\ast}{S_{jk}^{(1)}}$ and
$N^{\ast}{S_{jk}^{(2)}}$, we here introduce notation of the union of conormal
bundles of common hyperplanes or $(n-2)$-dimensional planes in
$\mathbb{R}^{n}$. For $j{\neq}k$ set

$\displaystyle N^{\ast}\mathcal{L}_{jk}^{(1)}$
$\displaystyle:=\bigcup\\{(y,\eta)\in N^{\ast}\Pi_{jk}:$
$\displaystyle\qquad\qquad\text{$\Pi_{jk}$ is a hyperplane tangent to both
$\Sigma_{j}$ and $\Sigma_{k}$.}\\},$ $\displaystyle
N^{\ast}\mathcal{L}_{jk}^{(2)}$ $\displaystyle:=\bigcup\\{(y,\eta)\in
N^{\ast}\Pi_{jk}:$ $\displaystyle\qquad\qquad\text{$\Pi_{jk}$ is an
$(n-2)$-dimensional plane tangent to both $\Sigma_{j}$ and $\Sigma_{k}$.}\\},$
$\displaystyle\widetilde{N^{\ast}\mathcal{L}_{jk}^{(2)}}$
$\displaystyle:=\bigcup\\{(y,\eta)\in N^{\ast}\mathcal{L}_{jk}^{(2)}:$ There
exists $y_{j}\in\Sigma_{j}$ such that $\eta\in N^{\ast}_{y_{j}}\Sigma_{j}$
$\displaystyle\qquad\qquad\text{or there exists $y_{k}\in\Sigma_{k}$ such that
$\eta\in N^{\ast}_{y_{k}}\Sigma_{k}$.}\\},$ $\displaystyle
N^{\ast}\mathcal{L}_{jk}$
$\displaystyle:=N^{\ast}\mathcal{L}_{jk}^{(1)}{\cup}N^{\ast}\mathcal{L}_{jk}^{(2)},\quad\widetilde{N^{\ast}\mathcal{L}^{(2)}}:=\bigcup_{1\leqq
j<k\leqq J}\widetilde{N^{\ast}\mathcal{L}_{jk}^{(2)}},$ $\displaystyle
N^{\ast}\mathcal{L}^{(1)}$ $\displaystyle:=\bigcup_{1\leqq j<k\leqq
J}N^{\ast}\mathcal{L}_{jk}^{(1)},\quad N^{\ast}\mathcal{L}:=\bigcup_{1\leqq
j<k\leqq J}N^{\ast}\mathcal{L}_{jk}.$

We have the following.

######  Lemma 4.5.

We have

$(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S_{j}}=N^{\ast}\Sigma_{j},\quad(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S}=N^{\ast}\Sigma,$
$(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S_{jk}^{(1)}}=N^{\ast}\mathcal{L}_{jk}^{(1)},\quad(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{SS}^{(1)}=N^{\ast}\mathcal{L}^{(i)},$
$(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S_{jk}^{(2)}}=\widetilde{N^{\ast}\mathcal{L}_{jk}^{(2)}},\quad(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{SS}^{(2)}=\widetilde{N^{\ast}\mathcal{L}^{(2)}},$
$(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S_{jk}}\subset
N^{\ast}\mathcal{L}_{jk},\quad(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{SS}\subset
N^{\ast}\mathcal{L}.$

###### Proof.

We have
$(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S_{j}}=(\Lambda_{\phi}^{\prime})^{\ast}\circ\Lambda_{\phi}^{\prime}{\circ}N^{\ast}\Sigma_{j}=N^{\ast}\Sigma_{j}$
immediately. It suffices to prove that

$(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S_{jk}^{(1)}}=N^{\ast}\mathcal{L}_{jk}^{(1)},\quad(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S_{jk}^{(2)}}=\widetilde{N^{\ast}\mathcal{L}_{jk}^{(2)}}.$

Recall

$\displaystyle(\Lambda_{\phi}^{\prime})^{\ast}$
$\displaystyle=\bigl{\\{}\bigl{(}x^{\prime\prime}+t_{1}\omega_{1}+\dotsb+t_{d}\omega_{d},\sigma,x^{\prime\prime};\xi(1,t_{1},\dotsc,t_{d},1)\bigr{)}:$
$\displaystyle\qquad\qquad(\sigma,x^{\prime\prime}){\in}G(d,n),\sigma=\langle\omega_{1},\dotsc,\omega_{d}\rangle_{\operatorname{ON}}\in
G_{d,n},t_{1},\dotsc,t_{d}\in\mathbb{R},\xi\in\sigma^{\perp}\setminus\\{0\\}\bigr{\\}}.$

We remark that only the points in $T^{\ast}{G(d,n)}$ of the form

$\bigl{(}\sigma,x^{\prime\prime};\xi(t_{1},\dotsc,t_{d},1)\bigr{)},\quad\xi\in\sigma^{\perp},t_{1},\dotsc,t_{d}\in\mathbb{R}$

are accepted by $(\Lambda_{\phi}^{\prime})^{\ast}$ for the composition. Hence
for $j{\neq}k$ all the elements in $N^{\ast}{S_{jk}^{(1)}}$ are accepted.
However the elements in $N^{\ast}{S_{jk}^{(2)}}$ of the form

$\bigl{(}\sigma,x^{\prime\prime};\xi(t_{1},\dotsc,t_{d},1)\bigr{)},\quad\xi\in
N_{y_{j}}^{\ast}\Sigma_{j}{\cup}N^{\ast}\Sigma_{k},t_{1},\dotsc,t_{d}\in\mathbb{R}$

are accepted and not for $\xi\in
N_{y_{j}}^{\ast}\Sigma_{j}{\oplus}N^{\ast}_{y_{k}}\Sigma_{k}{\setminus}N_{y_{j}}^{\ast}\Sigma_{j}{\cup}N^{\ast}_{y_{k}}\Sigma_{k}$.

By using Lemma 4.4, We have

$\displaystyle(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S_{jk}}^{(1)}$
$\displaystyle=\bigl{\\{}\bigl{(}x^{\prime\prime}+t_{1}\omega_{1}+\dotsb+t_{d}\omega_{d},;\xi\bigr{)}:$
$\displaystyle\qquad\qquad(\sigma,x^{\prime\prime}){\in}G(d,n),\sigma=\langle\omega_{1},\dotsc,\omega_{d}\rangle_{\operatorname{ON}}\in
G_{d,n},$ $\displaystyle\qquad\qquad
t_{1},\dotsc,t_{d}\in\mathbb{R},\xi\in\sigma^{\perp}\setminus\\{0\\},$
$\displaystyle\qquad\qquad
x^{\prime\prime}=y_{j}-\pi_{\sigma}{y_{j}}=y_{k}-\pi_{\sigma}{y_{k}},\sigma\subset
T_{y_{j}}\Sigma_{j}=T_{y_{k}}\Sigma_{k}$ $\displaystyle\qquad\qquad\text{with
some $y_{j}\in\Sigma_{j}$ and $y_{k}\in\Sigma_{k}$}\\}$
$\displaystyle=\\{(x^{\prime\prime}+x^{\prime};\xi):(\sigma,x^{\prime\prime}){\in}G(d,n),x^{\prime\prime}=y_{j}-\pi_{\sigma}{y_{j}}=y_{k}-\pi_{\sigma}{y_{k}},$
$\displaystyle\qquad\qquad\sigma\subset
T_{y_{j}}\Sigma_{j}=T_{y_{k}}\Sigma_{k},x^{\prime}\in\sigma,$
$\displaystyle\qquad\qquad\xi\in N^{\ast}_{y_{j}}\Sigma_{j}\ \text{with some
$y_{j}\in\Sigma_{j}$ and $y_{k}\in\Sigma_{k}$}\\}$
$\displaystyle=\\{(y;\eta)\in N^{\ast}\Pi_{jk}:\text{$\Pi_{jk}$ is a
hyperplane tangent to both $\Sigma_{j}$ and $\Sigma_{k}$.}\\}$
$\displaystyle=N^{\ast}\mathcal{L}_{jk}^{(1)},$
$\displaystyle(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S_{jk}}^{(2)}$
$\displaystyle=\bigl{\\{}\bigl{(}x^{\prime\prime}+t_{1}\omega_{1}+\dotsb+t_{d}\omega_{d},;\xi\bigr{)}:$
$\displaystyle\qquad\qquad(\sigma,x^{\prime\prime}){\in}G(d,n),\sigma=\langle\omega_{1},\dotsc,\omega_{d}\rangle_{\operatorname{ON}}\in
G_{d,n},$ $\displaystyle\qquad\qquad t_{1},\dotsc,t_{d}\in\mathbb{R},\xi\in
N_{y_{j}}^{\ast}\Sigma_{j}{\cup}N^{\ast}_{y_{k}}\Sigma_{k},$
$\displaystyle\qquad\qquad
x^{\prime\prime}=y_{j}-\pi_{\sigma}{y_{j}}=y_{k}-\pi_{\sigma}{y_{k}},\sigma\subset
T_{y_{j}}\Sigma_{j}{\cap}T_{y_{k}}\Sigma_{k}$
$\displaystyle\qquad\qquad\text{with some $y_{j}\in\Sigma_{j}$ and
$y_{k}\in\Sigma_{k}$}\\}$
$\displaystyle=\\{(x^{\prime\prime}+x^{\prime};\xi):(\sigma,x^{\prime\prime}){\in}G(d,n),x^{\prime\prime}=y_{j}-\pi_{\sigma}{y_{j}}=y_{k}-\pi_{\sigma}{y_{k}},$
$\displaystyle\qquad\qquad\sigma\subset
T_{y_{j}}\Sigma_{j}{\cap}T_{y_{k}}\Sigma_{k},x^{\prime}\in\sigma,\xi\in
N^{\ast}_{y_{j}}\Sigma_{j}{\cup}N^{\ast}_{y_{k}}\Sigma_{k}$
$\displaystyle\qquad\qquad\text{with some $y_{j}\in\Sigma_{j}$ and
$y_{k}\in\Sigma_{k}$}\\}$ $\displaystyle=\\{(y;\eta)\in N^{\ast}\Pi_{jk}:$
$\Pi_{jk}$ is an $(n-2)$-dimensional plane tangent to both $\Sigma_{j}$ and
$\Sigma_{k}$. There exists $y_{j}\in\Sigma_{j}$ such that $\eta\in
N^{\ast}_{y_{j}}\Sigma_{j}$ $\displaystyle\qquad\qquad\text{or there exists
$y_{k}\in\Sigma_{k}$ such that $\eta\in N^{\ast}_{y_{k}}\Sigma_{k}$.}\\},$
$\displaystyle=\widetilde{N^{\ast}\mathcal{L}_{jk}^{(2)}}.$

This completes the proof. ∎

##  5\. Paired Lagrangian distributions and their products

In the present section we study basic facts used for our quantitative analysis
of the metal streaking artifacts. We begin with identifying the classes of
conormal distributions to which $\chi_{D}$ and $\mathcal{R}_{d}\chi_{D}$
belong respectively.

######  Lemma 5.1.

$\displaystyle\chi_{D_{j}}$ $\displaystyle\in
I^{-1/2-n/2}(N^{\ast}\Sigma_{j}),\quad\chi_{D}\in
I^{-1/2-n/2}(N^{\ast}\Sigma),$ $\displaystyle\mathcal{R}_{d}\chi_{D_{j}}$
$\displaystyle\in
I^{-(d+2)/2+1/2-N(d,n)/4}(N^{\ast}{S}_{j})=I^{-(d+1)(n-d+2)/4}(N^{\ast}{S_{j}}),$
$\displaystyle\mathcal{R}_{d}\chi_{D}$ $\displaystyle\in
I^{-(d+2)/2+1/2-N(d,n)/4}(N^{\ast}{S}).$

Moreover the principal symbols of these conormal distributions do not vanish
everywhere on the associating conormal bundles respectively.

###### Proof.

We study only $\chi_{D_{j}}$ and $\mathcal{R}_{d}\chi_{D_{j}}$. It is easy to
see that

$\chi_{D_{j}}\in
I^{-1+1/2-n/4}(N^{\ast}\Sigma_{j})=I^{-1/2-n/4}(N^{\ast}\Sigma_{j}).$

Indeed if we choose appropriate local coordinates in $x\in\mathbb{R}^{n}$,
then $\chi_{D_{j}}$ can be expressed as a characteristics function of a half
space $\\{(0,x_{2},\dotsc,x_{n}):x_{2},\dotsc,x_{n}\in\mathbb{R}\\}$ and

$\chi_{D_{j}}(x)=\int_{\mathbb{R}}e^{ix_{1}\xi_{1}}a(x_{2},\dotsc,x_{n},\xi_{1})d\xi_{1}$

locally near $(x_{1},x_{2},\dotsc,x_{n})=(0,0,\dotsc,0)$ with some amplitude

$a(x_{2},\dotsc,x_{n},\xi_{1})\sim\frac{1}{\xi_{1}}\in
S^{-1}\bigl{(}\mathbb{R}^{n-1}\times(\mathbb{R}\setminus\\{0\\})\bigr{)}.$

It is also easy to see that the principal symbol of
$a(x_{2},\dotsc,x_{n},\xi_{1})$ does not vanish. Combining this, Theorem 3.2
and Lemmas 4.1, we deduce that

$\mathcal{R}_{d}\chi_{D_{j}}\in
I^{-(N(d,n)+n)/4+(n-d)/2-1/2-n/4}(\Lambda_{\phi}^{\prime}{\circ}N^{\ast}\Sigma_{j})=I^{-(d+2)/2+1/2-N(d,n)/4}(N^{\ast}{S_{j}}),$

and the principal symbol of $\mathcal{R}_{d}\chi_{D_{j}}$ does not vanish
since the principal symbol of $\chi_{D_{j}}$ does not vanish and
$\mathcal{R}_{d}$ is an elliptic Fourier integral operator. This completes the
proof. ∎

In the present section we concentrate on the analysis of

$(\mathcal{R}_{d}\chi_{D})^{2}=\sum_{j=1}^{J}(\mathcal{R}_{d}\chi_{D_{j}})^{2}+2\sum_{1\leqq
j<k\leqq J}\mathcal{R}_{d}\chi_{D_{j}}\cdot\mathcal{R}_{d}\chi_{D_{k}}.$

So we need to study $(\mathcal{R}_{d}\chi_{D_{j}})^{2}$ and
$\mathcal{R}_{d}\chi_{D_{j}}\cdot\mathcal{R}_{d}\chi_{D_{k}}$ with $j{\neq}k$.
We begin with the analysis of $(\mathcal{R}_{d}\chi_{D_{j}})^{2}$. This will
be done by symbolic calculus of conormal distributions. In general we have the
following.

######  Lemma 5.2.

If $u,v\in I^{-(d+1)(n-d+2)/4}(N^{\ast}{S_{j}})$, then $uv\in
I^{-(d+1)(n-d+2)/4}(N^{\ast}{S_{j}})$.

###### Proof.

The outline of the proof of Lemma 5.2 is similar to that of [15, Lemma 4.1].
However we introduce some cut-off functions for the frequency space and obtain
fine evaluation. Suppose that $u,v\in I^{-(d+1)(n-d+2)/4}(N^{\ast}{S_{j}})$.
Fix arbitrary $x_{0}\in S_{j}$ and consider the product $uv$ near $x_{0}$.
Note that the codimension of $S_{j}$ is one and the order of the amplitudes of
$u$ and $v$ is

$-\frac{(d+1)(n-d+2)}{4}-\frac{1}{2}+\frac{N(d,n)}{4}=-\frac{d}{2}-1.$

Then there exist amplitudes $a(x,\xi_{1}),b(x,\xi_{1})\in
S^{-d/2-1}(\mathbb{R}^{N(d,n)}\times\mathbb{R})$ which are compactly supported
in $x$ near $x_{0}$ such that $u(x)$ and $v(x)$ are given by

$u(x)=\int_{\mathbb{R}}e^{ix_{1}\xi_{1}}a(x,\xi_{1})d\xi_{1},\quad
v(x)=\int_{\mathbb{R}}e^{ix_{1}\xi_{1}}b(x,\xi_{1})d\xi_{1}$

near $x=x_{0}$ respectively. Hence we have the explicit formula of the product
$uv$ as

$u(x)v(x)=\int_{\mathbb{R}}\int_{\mathbb{R}}e^{ix_{1}(\xi_{1}+\eta_{1})}a(x,\xi_{1})b(x,\eta_{1})d\xi_{1}d\eta_{1}=\int_{\mathbb{R}}e^{ix_{1}\xi_{1}}c_{1}(x,\xi_{1})d\xi_{1},$
$c_{1}(x,\xi_{1})=\int_{\mathbb{R}}a(x,\xi_{1}-\eta_{1})b(x,\eta_{1})d\eta_{1}$

near $x=x_{0}$. It is easy to see that $c_{1}(x,\xi_{1})$ is compactly
supported in $x$ near $x_{0}$. It suffices to show that $c_{1}(x,\xi_{1})\in
S^{-d/2-1}(\mathbb{R}^{N(d,n)}\times\mathbb{R})$. For this purpose we split
$\eta_{1}$-space into two parts by using cut-off functions. Pick up
$\psi(t)\in C^{\infty}_{0}(\mathbb{R})$ such that

$0\leqq\psi(t)\leqq 1,\quad\psi(t)\equiv\begin{cases}1&\ (\lvert{t}\rvert\leqq
1/2),\\\ 0&\ (\lvert{t}\rvert\geqq
3/4),\end{cases}\quad\psi(t)>0\quad(\lvert{t}\rvert<3/4).$

Set
$\Psi_{1}(\xi_{1},\eta_{1}):=\phi(\langle\xi_{1}-\eta_{1}\rangle/\langle\xi_{1}\rangle)$.
Then the properties of $\Psi_{1}(\xi_{1},\eta_{1})$ are the following.

  * (i)

$\Psi_{1}(\xi_{1},\eta_{1})>0$ is equivalent to
$\langle\xi_{1}-\eta_{1}\rangle\leqq 3\langle\xi_{1}\rangle/4$, which implies
$C_{1}^{-1}\langle\xi_{1}\rangle\leqq\langle\eta_{1}\rangle\leqq
C_{1}\langle\xi_{1}\rangle$ with some $C_{1}>1$. Indeed
$\langle\xi_{1}-\eta_{1}\rangle\leqq 3\langle\xi_{1}\rangle/4$ implies that
$\lvert\xi_{1}-\eta_{1}\rvert\leqq 3\lvert\xi_{1}\rvert/4$ since

$1+\lvert\xi_{1}-\eta_{1}\rvert^{2}\leqq\frac{9}{16}+\frac{9\lvert\xi_{1}\rvert^{2}}{16}<1+\frac{9\lvert\xi_{1}\rvert^{2}}{16}.$

Applying this to
$\pm(\lvert\xi_{1}\rvert-\lvert\eta_{1}\rvert)\leqq\lvert\xi_{1}-\eta_{1}\rvert$,
we have $\lvert\xi_{1}\rvert/4\leqq\lvert\eta_{1}\rvert\leqq
7\lvert\xi_{1}\rvert/4$.

  * (ii)

$1-\Psi_{1}(\xi_{1},\eta_{1})>0$ is equivalent to
$\langle\xi_{1}-\eta_{1}\rangle>\langle\xi_{1}\rangle/2$.

  * (iii)

The results of (i) and (ii) implies that
$\operatorname{supp}\psi^{\prime}(\langle\xi_{1}-\eta_{1}\rangle/\langle\xi_{1}\rangle)\subset
E_{1}$, where

$E_{1}=\\{(\xi_{1},\eta_{1})\in\mathbb{R}^{2}:\langle\xi_{1}\rangle/2\leqq\langle\xi_{1}-\eta_{1}\rangle\leqq
3\langle\xi_{1}\rangle/4\\}.$

In particular for $(\xi_{1},\eta_{1})\in E_{1}$

$C_{1}^{-1}\langle\eta_{1}\rangle/2\leqq\langle\xi_{1}\rangle/2\leqq\langle\xi_{1}-\eta_{1}\rangle\leqq
3\langle\xi\rangle/4\leqq 3C_{1}\langle\eta_{1}\rangle/4.$ (5.1)

  * (iv)

Let $\alpha$ and $\beta$ be non-negative integers with $\alpha+\beta>0$. By
using the chain rule of differentiation and (5.1), we have

$\displaystyle\partial_{\xi_{1}}^{\alpha}\partial_{\eta_{1}}^{\beta}\Psi_{1}(\xi_{1},\eta_{1})$
$\displaystyle=\sum_{l=1}^{\alpha+\beta}\sum_{\begin{subarray}{c}\alpha_{1}+\dotsb+\alpha_{l}=\alpha\\\
\beta_{1}+\dotsb+\beta_{l}=\beta\end{subarray}}C^{\alpha_{1},\dotsc,\alpha_{l}}_{\beta_{1},\dotsc,\beta_{l}}\cdot\psi^{(l)}\left(\frac{\langle\xi_{1}-\eta_{1}\rangle}{\langle\xi_{1}\rangle}\right)$
$\displaystyle\times\prod_{m=1}^{l}\partial_{\xi_{1}}^{\alpha_{m}}\partial_{\eta_{1}}^{\beta_{m}}\left(\frac{\langle\xi_{1}-\eta_{1}\rangle}{\langle\xi_{1}\rangle}\right)$
$\displaystyle=\mathcal{O}(\langle\xi\rangle^{-(\alpha+\beta)}),$

where $C^{\alpha_{1},\dotsc,\alpha_{l}}_{\beta_{1},\dotsc,\beta_{l}}$, with
$\alpha_{1}+\dotsb+\alpha_{l}=\alpha$ and $\beta_{1}+\dotsb+\beta_{l}=\beta$
are some constants. Recall that $0\leqq\Psi_{1}(\xi_{1},\eta_{1})\leqq 1$.
Then we deduce that
$\partial_{\xi_{1}}^{\alpha}\partial_{\eta_{1}}^{\beta}\Psi_{1}(\xi_{1},\eta_{1})=\mathcal{O}(\langle\xi\rangle^{-(\alpha+\beta)})$
for all non-negative integers $\alpha$ and $\beta$.

We make use of the Leibniz formula,
$\partial_{\xi_{1}}^{\alpha}a(x,\xi_{1}-\eta_{1})=(-1)^{\alpha}\partial_{\eta_{1}}^{\alpha}a(x,\xi_{1}-\eta_{1})$,
integration by parts and the properties of $\Psi_{1}(\xi_{1},\eta_{1})$ in the
order, and confirm that
$\partial_{x}^{\gamma}\partial_{\xi_{1}}^{\alpha}c_{1}(x,\xi_{1})=\mathcal{O}(\langle\xi_{1}\rangle^{-d/2-1-\alpha})$.
We deduce that

$\displaystyle\partial_{x}^{\gamma}\partial_{\xi_{1}}^{\alpha}c_{1}(x,\xi_{1})$
$\displaystyle=\sum_{\gamma_{1}+\gamma_{2}=\gamma}\frac{\gamma!}{\gamma_{1}!\gamma_{2}!}\int_{\mathbb{R}}\partial_{x}^{\gamma_{1}}\partial_{\xi_{1}}^{\alpha}a(x,\xi_{1}-\eta_{1})\partial_{x}^{\gamma_{2}}b(x,\eta_{1})d\eta_{1}$
$\displaystyle=F_{1}+G_{1},$ (5.2) $\displaystyle F_{1}$
$\displaystyle=\sum_{\gamma_{1}+\gamma_{2}=\gamma}\frac{\gamma!}{\gamma_{1}!\gamma_{2}!}\int_{\mathbb{R}}\bigl{(}1-\Psi_{1}(\xi_{1},\eta_{1})\bigr{)}\partial_{x}^{\gamma_{1}}\partial_{\xi_{1}}^{\alpha}a(x,\xi_{1}-\eta_{1})\partial_{x}^{\gamma_{2}}b(x,\eta_{1})d\eta_{1},$
$\displaystyle G_{1}$
$\displaystyle=\sum_{\gamma_{1}+\gamma_{2}=\gamma}\frac{\gamma!}{\gamma_{1}!\gamma_{2}!}\int_{\mathbb{R}}\Psi_{1}(\xi_{1},\eta_{1})\partial_{x}^{\gamma_{1}}(-\partial_{\eta_{1}})^{\alpha}a(x,\xi_{1}-\eta_{1})\partial_{x}^{\gamma_{2}}b(x,\eta_{1})d\eta_{1}.$

The region of the integration $F_{1}$ satisfies
$\langle\xi_{1}-\eta_{1}\rangle>\langle\xi_{1}\rangle/2$. We deduce that

$\displaystyle F_{1}$
$\displaystyle=\int_{\langle\xi_{1}-\eta_{1}\rangle>\langle\xi_{1}\rangle/2}\mathcal{O}(\langle\xi_{1}-\eta_{1}\rangle^{-d/2-1-\alpha}\langle\eta_{1}\rangle^{-d/2-1})d\eta_{1}$
$\displaystyle=\int_{\langle\xi_{1}-\eta_{1}\rangle>\langle\xi_{1}\rangle/2}\mathcal{O}(\langle\xi_{1}\rangle^{-d/2-1-\alpha}\langle\eta_{1}\rangle^{-d/2-1})d\eta_{1}$
$\displaystyle=\mathcal{O}(\langle\xi_{1}\rangle^{-d/2-1-\alpha})$ (5.3)

since $\langle\eta_{1}\rangle^{-d/2-1}$ is integrable in $\eta_{1}$ on
$\mathbb{R}$.

The region of the integration $G_{1}$ satisfies
$C_{1}^{-1}\langle\xi_{1}\rangle\leqq\langle\eta_{1}\rangle\leqq
C_{1}\langle\xi_{1}\rangle$. We deduce that

$\displaystyle G_{1}$
$\displaystyle=\sum_{\alpha_{1}+\alpha_{2}=\alpha}\frac{\alpha!}{\alpha_{1}!\alpha_{2}!}\sum_{\gamma_{1}+\gamma_{2}=\gamma}\frac{\gamma!}{\gamma_{1}!\gamma_{2}!}$
$\displaystyle\times\int_{\mathbb{R}}\partial_{\eta_{1}}^{\alpha_{1}}\Psi_{1}(\xi_{1},\eta_{1})\cdot\partial_{x}^{\gamma_{1}}a(x,\xi_{1}-\eta_{1})\cdot\partial_{x}^{\gamma_{2}}\partial_{\eta_{1}}^{\alpha_{2}}b(x,\eta_{1})d\eta_{1}$
$\displaystyle=\sum_{\alpha_{1}+\alpha_{2}=\alpha}\int_{C_{1}^{-1}\langle\xi_{1}\rangle\leqq\langle\eta_{1}\rangle\leqq
C_{1}\langle\xi_{1}\rangle}$
$\displaystyle\qquad\qquad\qquad\times\mathcal{O}(\langle{\xi_{1}}\rangle^{-\alpha_{1}}\langle\xi_{1}-\eta_{1}\rangle^{-d/2-1}\langle\eta_{1}\rangle^{-d/2-1-\alpha_{2}})d\eta_{1}$
$\displaystyle=\int_{\mathbb{R}}\mathcal{O}(\langle\xi_{1}-\eta_{1}\rangle^{-d/2-1}\langle\xi_{1}\rangle^{-d/2-1-\alpha})d\eta_{1}$
$\displaystyle=\mathcal{O}(\langle\xi_{1}\rangle^{-d/2-1-\alpha})$ (5.4)

since $\langle\xi_{1}-\eta_{1}\rangle^{-d/2-1}$ is integrable in $\eta_{1}$ on
$\mathbb{R}$.

Substitute (5.3) and (5.4) into (5.2). We obtain
$\partial_{x}^{\gamma}\partial_{\xi_{1}}^{\alpha}c_{1}(x,\xi_{1})=\mathcal{O}(\langle\xi_{1}\rangle^{-d/2-1-\alpha})$.
This completes the proof. ∎

Next we study the microlocal singularity of
$\mathcal{R}_{d}\chi_{D_{j}}\cdot\mathcal{R}_{d}\chi_{D_{k}}$ on $S_{jk}$. For
this purpose we here introduce some classes of distributions. They are called
paired Lagrangian distributions. See [14], [7] and [4].

######  Definition 5.3.

Let $X$ be a smooth manifold, and let $\mu,\nu\in\mathbb{R}$. Suppose that
$\Lambda_{0}$ and $\Lambda_{1}$ are cleanly intersecting conic Lagrangian
submanifold of $T^{\ast}{X}\setminus 0$. We define a class of distributions
$I^{\mu,\nu}(\Lambda_{0},\Lambda_{1})$ on $X$ which is said to be a paired
Lagrangian distribution associated to the pair $(\Lambda_{0},\Lambda)$ of
order $\mu,\nu$ as follows. We say that $u\in
I^{\mu,\nu}(\Lambda_{0},\Lambda_{1})$ if $u\in\mathcal{D}^{\prime}(X)$,
$\operatorname{WF}(u)\subset\Lambda_{0}\cup\Lambda_{1}$ and microlocally away
from $\Lambda_{0}\cap\Lambda_{1}$

$I^{\mu,\nu}(\Lambda_{0},\Lambda_{1})\subset
I^{\mu+\nu}(\Lambda_{0}\setminus\Lambda_{1}),\quad
I^{\mu,\nu}(\Lambda_{0},\Lambda_{1})\subset I^{\mu}(\Lambda_{1}).$

It is known that paired Lagrangian distributions associated to a pair of two
conormal bundles can be characterized as oscillatory integrals as follows. Let
$X$ be an $N$-dimensional smooth manifold, and let $Y$ and $Z$ be
transversally intersecting submanifolds of $X$. It follows that $N^{\ast}{Y}$
and $N^{\ast}(Y{\cap}Z)$ intersect cleanly. Set $\operatorname{codim}{Y}=k$
and $\operatorname{codim}(Y{\cap}Z)=k+l$. Fix arbitrary $p_{0}\in X$, and
choose local coordinates
$x=(x_{1},\dotsc,x_{N})=(x^{\prime},x^{\prime\prime},x^{\prime\prime\prime})\in\mathbb{R}^{k}\times\mathbb{R}^{l}\times\mathbb{R}^{N-k-l}$
such that $x(p_{0})=0$ and

$\displaystyle Y$
$\displaystyle=\\{x=(x^{\prime},x^{\prime\prime},x^{\prime\prime\prime}):x^{\prime}=(x_{1},\dotsc,x_{k})=0\\},$
$\displaystyle Z$
$\displaystyle=\\{x=(x^{\prime},x^{\prime\prime},x^{\prime\prime\prime}):x^{\prime}=(x_{1},\dotsc,x_{k})=0,x^{\prime\prime}=(x_{k+1},\dotsc,x_{k+l})=0\\}$

near $p_{0}$. It is known that $u\in
I^{\mu,\nu}(N^{\ast}(Y{\cap}Z),N^{\ast}{Y})$ if and only if there exists a
symbol $a(x,\xi^{\prime},\xi^{\prime\prime})\in
S^{\mu-k/2+N/4,\nu-l/2}\bigl{(}\mathbb{R}^{N}\times(\mathbb{R}^{k}\setminus\\{0\\})\times\mathbb{R}^{l}\bigr{)}$
such that

$u(x)=\iint_{\mathbb{R}^{k}\times\mathbb{R}^{l}}e^{i(x^{\prime}\cdot\xi^{\prime}+x^{\prime\prime}\cdot\xi^{\prime\prime})}a(x,\xi^{\prime},\xi^{\prime\prime})d\xi^{\prime}d\xi^{\prime\prime}$

near $x=0$. Here
$S^{m,m^{\prime}}\bigl{(}\mathbb{R}^{N}\times(\mathbb{R}^{k}\setminus\\{0\\})\times\mathbb{R}^{l}\bigr{)}$
is the set of all smooth functions $a(x,\xi,\eta)$ on
$\mathbb{R}^{N}\times(\mathbb{R}^{k}\setminus\\{0\\})\times\mathbb{R}^{l}$
with the following conditions: for any compact set $K$ in $\mathbb{R}^{N}$ and
for any multi-indices $\alpha$, $\beta$, and $\gamma$ there exists a constant
$C(K,\alpha,\beta,\gamma)>0$ such that

$\lvert\partial_{x}^{\gamma}\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}a(x,\xi,\eta)\rvert\leqq
C(K,\alpha,\beta,\gamma)\langle\xi;\eta\rangle^{m-\lvert\alpha\rvert}\langle\eta\rangle^{m^{\prime}-\lvert\beta\rvert}$

for $(x,\xi,\eta)\in
K\times(\mathbb{R}^{k}\setminus\\{0\\})\times\mathbb{R}^{l}$.

We compute the microlocal singularity of
$\mathcal{R}_{d}\chi_{D_{j}}\cdot\mathcal{R}_{d}\chi_{D_{k}}$ near $S_{jk}$.
For this purpose we make full use of the following strong results due to
Greenleaf and Uhlmann in [4].

######  Lemma 5.4 ([4, Lemma 1.1]).

Let $X$ be an $N$-dimensional smooth manifold, and let $Y$ and $Z$ be
transversally intersecting smooth submanifolds of $X$ with
$\operatorname{codim}Y=k_{1}$ and $\operatorname{codim}Z=k_{2}$ respectively.
Set $l_{1}=\operatorname{codim}(Y{\cap}Z)-k_{1}$ and
$l_{2}=\operatorname{codim}(Y{\cap}Z)-k_{2}$. Let $m_{1}$ and $m_{2}$ be real
numbers. Then we have

$\displaystyle I^{m_{1}+k_{1}/2-N/4}(N^{\ast}{Y})\cdot
I^{m_{2}+k_{2}/2-N/4}(N^{\ast}{Z})$ $\displaystyle\qquad\subset
I^{m_{1}+k_{1}/2-N/4,m_{2}+l_{1}/2}(N^{\ast}(Y{\cap}Z),N^{\ast}{Y})$
$\displaystyle\qquad+I^{m_{2}+k_{2}/2-N/4,m_{1}+l_{2}/2}(N^{\ast}(Y{\cap}Z),N^{\ast}{Z}).$

Applying Lemma 5.4 to
$\mathcal{R}_{d}\chi_{D_{j}}\cdot\mathcal{R}_{d}\chi_{D_{k}}$, we have the
following results.

######  Theorem 5.5.

  * (I)

$\mathcal{R}_{d}\chi_{D_{j}}\cdot\mathcal{R}_{d}\chi_{D_{k}}$ belongs to

$\displaystyle\mathcal{A}_{jk}:=$ $\displaystyle
I^{-(d+2)/2+1/2-N(d,n)/4,-(d+2)/2+1/2}(N^{\ast}{S_{jk}},N^{\ast}{S_{j}})$
$\displaystyle+$ $\displaystyle
I^{-(d+2)/2+1/2-N(d,n)/4,-(d+2)/2+1/2}(N^{\ast}{S_{jk}},N^{\ast}{S_{k}}).$

Moreover the principal symbol of the product is non-vanishing on
$N^{\ast}{S_{jk}}$.

  * (II)

$\mathcal{R}_{d}^{\ast}(-\Delta_{x^{\prime\prime}})^{d/2}\bigl{\\{}\mathcal{R}_{d}\chi_{D_{j}}\cdot\mathcal{R}_{d}\chi_{D_{k}}\bigr{\\}}$
belongs to

$\displaystyle\mathcal{X}_{jk}:=$ $\displaystyle
I^{-1+1/2-n/4,-(d+2)/2+1/2}(N^{\ast}\mathcal{L}_{jk},N^{\ast}\Sigma_{j})$
$\displaystyle+$ $\displaystyle
I^{-1+1/2-n/4,-(d+2)/2+1/2}(N^{\ast}\mathcal{L}_{jk},N^{\ast}\Sigma_{k}).$

In particular,
$\mathcal{R}_{d}^{\ast}(-\Delta_{x^{\prime\prime}})^{d/2}\bigl{\\{}\mathcal{R}_{d}\chi_{D_{j}}\cdot\mathcal{R}_{d}\chi_{D_{k}}\bigr{\\}}\in
I^{-(d+2)/2-n/4}(N^{\ast}\mathcal{L}_{jk})$ microlocally away from
$N^{\ast}\Sigma_{j}{\cup}N^{\ast}\Sigma_{k}$. Moreover the principal symbol is
non-vanishing on $N^{\ast}\mathcal{L}_{jk}$.

###### Proof.

Recall Lemma 4.1. To prove (I), we apply Lemma 5.4 to
$\mathcal{R}_{d}\chi_{D_{j}}\cdot\mathcal{R}_{d}\chi_{D_{k}}$ with

$N=N(d,n),\quad m_{1}=m_{2}=-\frac{d+2}{2},\quad k_{1}=k_{2}=1,\quad
l_{1}=l_{2}=1.$

Then we deduce that

$\displaystyle\mathcal{R}_{d}\chi_{D_{j}}\cdot\mathcal{R}_{d}\chi_{D_{k}}$
$\displaystyle\qquad\in
I^{-(d+2)/2+1/2-N(d,n)/4,-(d+2)/2+1/2}\bigl{(}N^{\ast}{S_{jk}},N^{\ast}{S_{j}}\bigr{)}$
$\displaystyle\qquad+I^{-(d+2)/2+1/2-N(d,n)/4,-(d+2)/2+1/2}\bigl{(}N^{\ast}{S_{jk}},N^{\ast}{S_{k}}\bigr{)}$
$\displaystyle\qquad=\mathcal{A}_{jk}.$

To prove that the principal symbol of
$\mathcal{R}_{d}\chi_{D_{j}}\cdot\mathcal{R}_{d}\chi_{D_{k}}$ does not vanish
everywhere on $N^{\ast}{S_{jk}}$, we need to go back to the proof of Lemma
5.4. Here we describe its outline. Recall that $S_{j}$ and $S_{k}$ intersect
transversally. Fix arbitrary $p_{0}\in S_{jk}$. We can choose local
coordinates $(x,y,z)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{N(d,n)-2}$
near $p_{0}$ with $\bigl{(}x(p_{0}),y(p_{0}),z(p_{0})\bigr{)}=0$ such that
$S_{j}=\\{x=0\\}$ and $S_{k}=\\{y=0\\}$ near $p_{0}$. Moreover there exist
symbols $a(y,z,\xi)$ and $b(x,z,\eta)$ such that the principal part of
$a(y,z,\xi)$ is non-vanishing on
$N^{\ast}_{p_{0}}{S_{j}}=\\{(0,y,z;\xi,0,0,)\\}$, the principal part of
$b(x,z,\eta)$ is non-vanishing on
$N^{\ast}_{p_{0}}{S_{k}}=\\{(x,0,z,\xi,0,0)\\}$, and

$\displaystyle\mathcal{R}_{d}\chi_{D_{j}}(x,y,z)$
$\displaystyle=\int_{\mathbb{R}}e^{ix\xi}a(y,z,\xi)d\xi,$
$\displaystyle\mathcal{R}_{d}\chi_{D_{k}}(x,y,z)$
$\displaystyle=\int_{\mathbb{R}}e^{iy\eta}b(z,z,\eta)d\eta$

near $p_{0}$. Hence

$\mathcal{R}_{d}\chi_{D_{j}}(x,y,z)\cdot\mathcal{R}_{d}\chi_{D_{k}}(x,y,z)=\iint_{\mathbb{R}\times\mathbb{R}}e^{i(x\xi+y\eta)}a(y,z,\xi)b(x,z,\eta)d\xi
d\eta$

near $p_{0}$. This shows that the symbol of
$\mathcal{R}_{d}\chi_{D_{j}}(x,y,z)\cdot\mathcal{R}_{d}\chi_{D_{k}}(x,y,z)$ is
$a(y,z,\xi)b(x,z,\eta)$ whose principal part is non-vanishing on
$N^{\ast}_{p_{0}}{S_{jk}}=\\{(0,0,z;\xi,\eta,0)\\}$ near $p_{0}$.

Next we consider (II).
$\mathcal{R}_{d}^{\ast}(-\Delta_{x^{\prime\prime}})^{d/2}$ is an elliptic
Fourier integral operator of order

$\frac{d}{2}+\frac{N(d,n)}{4}-\frac{n}{4},$

and its canonical relation is the same as that of $\mathcal{R}_{d}^{\ast}$,
that is, $(\Lambda_{\phi}^{\prime})^{\ast}$ since
$(-\Delta_{x^{\prime\prime}})^{d/2}$ is an elliptic pseudodifferential
operator. Hence we deduce that

$\displaystyle\mathcal{R}_{d}^{\ast}(-\Delta_{x^{\prime\prime}})^{d/2}\bigl{\\{}\mathcal{R}_{d}\chi_{D_{j}}(x,y,z)\cdot\mathcal{R}_{d}\chi_{D_{k}}\bigr{\\}}$
$\displaystyle\qquad\in
I^{-1+1/2-n/4,-(d+2)/2+1/2}\bigl{(}(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S_{jk}},(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S_{j}}\bigr{)}$
$\displaystyle\qquad+I^{-1+1/2-n/4,-(d+2)/2+1/2}\bigl{(}(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S_{jk}},(\Lambda_{\phi}^{\prime})^{\ast}{\circ}N^{\ast}{S_{k}}\bigr{)}$
$\displaystyle\qquad\subset
I^{-1+1/2-n/4,-(d+2)/2+1/2}\bigl{(}N^{\ast}\mathcal{L}_{jk},N^{\ast}\Sigma_{j}\bigr{)}$
$\displaystyle\qquad+I^{-1+1/2-n/4,-(d+2)/2+1/2}\bigl{(}N^{\ast}\mathcal{L}_{jk},N^{\ast}\Sigma_{k}\bigr{)}$
$\displaystyle\qquad=\mathcal{X}_{jk}$

The principal part of the symbol of
$\mathcal{R}_{d}^{\ast}(-\Delta_{x^{\prime\prime}})^{d/2}\bigl{\\{}\mathcal{R}_{d}\chi_{D_{j}}(x,y,z)\cdot\mathcal{R}_{d}\chi_{D_{k}}(x,y,z)\bigr{\\}}$
is non-vanishing on $N^{\ast}\mathcal{L}_{jk}$ since the principal symbol of
$\mathcal{R}_{d}\chi_{D_{j}}(x,y,z)\cdot\mathcal{R}_{d}\chi_{D_{k}}(x,y,z)$ is
non-vanishing on $N^{\ast}{S_{jk}}$ and
$\mathcal{R}_{d}^{\ast}(-\Delta_{x^{\prime\prime}})^{d/2}$ is an elliptic
Fourier integral operator. This completes the proof. ∎

##  6\. Generalized beam hardening effects

Finally in the present section we prove our main theorem on what the metal
streaking artifacts are. Recall that the CT image of the metal region is given
by

$f_{\text{MA}}=\sum_{l=1}^{\infty}A_{l}(\alpha\varepsilon)^{2l}\mathcal{R}_{d}^{\ast}(-\Delta_{x^{\prime\prime}})^{d/2}\bigl{[}(\mathcal{R}_{d}\chi_{D})^{2l}\bigr{]},$

where $\\{A_{l}\\}_{l=1}^{\infty}$ is a sequence of real numbers, and $\alpha$
and $\varepsilon$ are small positive constants. Our main results are the
following.

######  Theorem 6.1.

Suppose (A) on the metal region $D$. Then away from $N^{\ast}\Sigma$, the
metal streaking artifacts $f_{\text{MA}}$ belongs to
$I^{-(d/2+1+n/4)}(N^{\ast}\mathcal{L})$ and $\operatorname{WF}(f_{\text{{\rm
MA}}})\subset
N^{\ast}\mathcal{L}^{(1)}\cup\widetilde{N^{\ast}\mathcal{L}^{(2)}}$.

Roughly speaking, Theorem 6.1 asserts that $f_{\text{MA}}$ is a conormal
distribution whose singular support consists of all the hyperplanes tangent to
two of $\Sigma_{1},\dotsc,\Sigma_{J}$ and all the $(n-2)$-dimensional planes
tangent to two of $\Sigma_{1},\dotsc,\Sigma_{J}$ if $d\leqq n-2$.

Basically our strategy of the proof of Theorem 6.1 is the same as that of [15,
Theorem 4.7] due to Palacios, Uhlmann and Wang. Firstly we state our strategy
of the proof of Theorem 6.1 shortly. Set

$\mathcal{A}:=\sum_{1\leqq j<k\leqq
J}\mathcal{A}_{jk},\quad\mathcal{X}:=\sum_{1\leqq j<k\leqq
J}\mathcal{X}_{jk},$

In terms of this notation Theorem 6.1 asserts that
$f_{\text{MA}}\in\mathcal{X}$. Lemmas 5.2 and 5.5 essentially show that
$(\mathcal{R}_{d}\chi_{D})^{2}\in\mathcal{A}$,
$\mathcal{R}_{d}(\mathcal{A})\subset\mathcal{X}$, and that away from
$N^{\ast}\Sigma$,

$u\in I^{-(d/2+1+n/4)}(N^{\ast}\mathcal{L}),\quad\operatorname{WF}(u)\subset
N^{\ast}\mathcal{L}^{(1)}\cup\widetilde{N^{\ast}\mathcal{L}^{(2)}}$

for $u\in\mathcal{A}$. If $\mathcal{A}$ is an algebra, then Theorem 6.1 holds.
So we shall prove that $\mathcal{A}$ is an algebra.

######  Lemma 6.2.

Suppose that $j{\neq}k$. Then $\mathcal{A}_{jk}$ is an algebra, and so is
$\mathcal{A}$. More concretely we have the following.

  * (I)

If

$u,v\in
I^{-(d+2)/2+1/2-N(d,n)/4,-(d+2)/2+1/2}\bigl{(}N^{\ast}{S_{jk}},N^{\ast}{S_{j}}\bigr{)},$

then

$uv\in
I^{-(d+2)/2+1/2-N(d,n)/4,-(d+2)/2+1/2}\bigl{(}N^{\ast}{S_{jk}},N^{\ast}{S_{j}}\bigr{)}.$

  * (II)

If

$u\in
I^{-(d+2)/2+1/2-N(d,n)/4,-(d+2)/2+1/2}\bigl{(}N^{\ast}{S_{jk}},N^{\ast}{S_{j}}\bigr{)},$
$v\in
I^{-(d+2)/2+1/2-N(d,n)/4,-(d+2)/2+1/2}\bigl{(}N^{\ast}{S_{jk}},N^{\ast}{S_{k}}\bigr{)},$

then $uv\in\mathcal{A}_{jk}$.

In what follows, let $\psi(t)$ be the same as in the proof of Lemma 5.2.
Firstly we prove (I) of Lemma 6.2.

###### Proof of Lemma 6.2 (I).

Suppose that

$u,v\in
I^{-(d+2)/2+1/2-N(d,n)/4,-(d+2)/2+1/2}\bigl{(}N^{\ast}{S_{jk}},N^{\ast}{S_{j}}\bigr{)}.$

Fix arbitrary $p_{0}\in S_{jk}$, and choose appropriate local coordinates
$(x,y,z)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{N(d,n)-2}$ such that
$\bigl{(}x(p_{0}),y(p_{0}),z(p_{0})\bigr{)}=0$, and

$\displaystyle N^{\ast}{S}_{j}$
$\displaystyle=\\{(0,y,z;\xi,0,0):\xi,y\in\mathbb{R},z\in\mathbb{R}^{N(d,n)-2}\\},$
$\displaystyle N^{\ast}{S}_{k}$
$\displaystyle=\\{(x,0,z;0,\eta,0):x,\eta\in\mathbb{R},z\in\mathbb{R}^{N(d,n)-2}\\},$
$\displaystyle N^{\ast}{S}_{jk}$
$\displaystyle=\\{(0,0,z;\xi,\eta,0):\xi,\eta\in\mathbb{R},z\in\mathbb{R}^{N(d,n)-2}\\}$

near $(x,y,z)=(0,0,0)$. Then there exist symbols

$a(z,\xi,\eta),b(z,\xi,\eta)\in
S^{-(d+2)/2,-(d+2)/2}\bigl{(}\mathbb{R}^{N(d,n)-2}\times(\mathbb{R}\setminus\\{0\\})\times\mathbb{R}\bigr{)}$

such that $a(z,\xi,\eta)$ and $b(z,\xi,\eta)$ are compactly supported in $z$
near $z=0$, and $u$ and $v$ are represented by oscillatory integrals

$\displaystyle u(x,y,z)$
$\displaystyle=\iint_{\mathbb{R}\times\mathbb{R}}e^{i(x\xi+y\eta)}a(z,\xi,\eta)d\xi
d\eta,$ $\displaystyle v(x,y,z)$
$\displaystyle=\iint_{\mathbb{R}\times\mathbb{R}}e^{i(x\xi+y\eta)}b(z,\xi,\eta)d\xi
d\eta$

near $(x,y,z)=(0,0,0)$. Then we have

$\displaystyle(uv)(x,y,z)$
$\displaystyle=\iint_{\mathbb{R}\times\mathbb{R}}e^{i(x\xi+y\eta)}c_{2}(z,\xi,\eta)d\xi
d\eta,$ $\displaystyle c_{2}(z,\xi,\eta)$
$\displaystyle=\iint_{\mathbb{R}\times\mathbb{R}}a(z,\xi-\zeta,\eta-\tau)b(z,\zeta,\tau)d\zeta
d\tau$

near $(x,y,z)=(0,0,0)$. Note that $c_{2}(z,\xi,\eta)$ is compactly supported
in $z$ near $z=0$. It suffices to show that

$c_{2}(z,\xi,\eta)\in
S^{-(d+2)/2,-(d+2)/2}\bigl{(}\mathbb{R}^{N(d,n)-2}\times(\mathbb{R}\setminus\\{0\\})\times\mathbb{R}\bigr{)}.$

For this purpose we introduce two cut-off functions defined by $\psi(t)$:

$\displaystyle\Psi_{2}(\xi,\eta,\zeta,\tau)$
$\displaystyle=\psi\left(\frac{\langle\xi-\zeta;\eta-\tau\rangle}{\langle\xi;\eta\rangle}\right),$
$\displaystyle\Psi_{3}(\eta,\tau)$
$\displaystyle=\psi\left(\frac{\langle\eta-\tau\rangle}{\langle\eta\rangle}\right).$

In the same way as $\Psi_{1}$ in the Proof of Lemma 5.2, we can deduce the
properties of $\Psi_{2}$ and $\Psi_{3}$ as follows.

  * (i)

If $\Psi_{2}(\xi,\eta,\zeta,\tau)>0$, then
$C_{1}^{-1}\langle\xi;\eta\rangle\leqq\langle\zeta;\tau\rangle\leqq{C_{1}}\langle\xi;\eta\rangle$
and

$\partial_{\zeta}^{\alpha}\partial_{\tau}^{\beta}\Psi_{2}(\xi,\eta,\zeta,\tau)=\mathcal{O}(\langle\xi;\eta\rangle^{-(\alpha+\beta)})$

for all non-negative integers $\alpha$ and $\beta$.

  * (ii)

$1-\Psi_{2}(\xi,\eta,\zeta,\tau)>0$ is equivalent to
$\langle\xi-\zeta;\eta-\tau\rangle>\langle\xi;\eta\rangle/2$.

  * (iii)

If $\Psi_{3}(\eta,\tau)>0$, then
$C_{1}^{-1}\langle\eta\rangle\leqq\langle\tau\rangle\leqq{C_{1}}\langle\eta\rangle$
and
$\partial_{\tau}^{\beta}\Psi_{3}(\eta,\tau)=\mathcal{O}(\langle\eta\rangle^{-\beta})$
for any non-negative integer $\beta$.

  * (iv)

$1-\Psi_{3}(\eta,\tau)>0$ is equivalent to
$\langle\eta-\tau\rangle>\langle\eta\rangle/2$.

We make use of

$1=\Psi_{2}\Psi_{3}+\Psi_{2}(1-\Psi_{3})+(1-\Psi_{2})\Psi_{3}+(1-\Psi_{2})(1-\Psi_{3}),$
$\frac{\partial}{\partial\xi}(\xi-\zeta)=-\frac{\partial}{\partial\zeta}(\xi-\zeta),\quad\frac{\partial}{\partial\eta}(\eta-\tau)=-\frac{\partial}{\partial\tau}(\eta-\tau),$

and integration by parts. We split
$\partial_{z}^{\gamma}\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}c_{2}(z,\xi,\eta)$
into four parts:

$\displaystyle\partial_{z}^{\gamma}\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}c_{2}(z,\xi,\eta)$
$\displaystyle=\sum_{\gamma_{1}+\gamma_{2}=\gamma}\frac{\gamma!}{\gamma_{1}!\gamma_{2}!}\iint_{\mathbb{R}\times\mathbb{R}}\partial_{z}^{\gamma_{1}}\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}a(z,\xi-\zeta,\eta-\tau)$
$\displaystyle\qquad\qquad\qquad\qquad\times\partial_{z}^{\gamma_{2}}b(z,\zeta,\eta)d\zeta
d\tau$ $\displaystyle=F_{2}+G_{2}+H_{2}+I_{2},$ (6.1) $\displaystyle F_{2}$
$\displaystyle=\sum_{\gamma_{1}+\gamma_{2}=\gamma}\frac{\gamma!}{\gamma_{1}!\gamma_{2}!}\iint_{\mathbb{R}\times\mathbb{R}}\partial_{z}^{\gamma_{1}}(-\partial_{\zeta})^{\alpha}(-\partial_{\tau})^{\beta}a(z,\xi-\zeta,\eta-\tau)$
$\displaystyle\qquad\qquad\qquad\qquad\times\Psi_{2}\Psi_{3}\partial_{z}^{\gamma_{2}}b(z,\zeta,\eta)d\zeta
d\tau,$ $\displaystyle G_{2}$
$\displaystyle=\sum_{\gamma_{1}+\gamma_{2}=\gamma}\frac{\gamma!}{\gamma_{1}!\gamma_{2}!}\iint_{\mathbb{R}\times\mathbb{R}}\partial_{z}^{\gamma_{1}}(-\partial_{\zeta})^{\alpha}\partial_{\eta}^{\beta}a(z,\xi-\zeta,\eta-\tau)$
$\displaystyle\qquad\qquad\qquad\qquad\times\Psi_{2}(1-\Psi_{3})\partial_{z}^{\gamma_{2}}b(z,\zeta,\eta)d\zeta
d\tau,$ $\displaystyle H_{2}$
$\displaystyle=\sum_{\gamma_{1}+\gamma_{2}=\gamma}\frac{\gamma!}{\gamma_{1}!\gamma_{2}!}\iint_{\mathbb{R}\times\mathbb{R}}\partial_{z}^{\gamma_{1}}\partial_{\xi}^{\alpha}(-\partial_{\tau})^{\beta}a(z,\xi-\zeta,\eta-\tau)$
$\displaystyle\qquad\qquad\qquad\qquad\times(1-\Psi_{2})\Psi_{3}\partial_{z}^{\gamma_{2}}b(z,\zeta,\eta)d\zeta
d\tau,$ $\displaystyle I_{2}$
$\displaystyle=\sum_{\gamma_{1}+\gamma_{2}=\gamma}\frac{\gamma!}{\gamma_{1}!\gamma_{2}!}\iint_{\mathbb{R}\times\mathbb{R}}\partial_{z}^{\gamma_{1}}\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}a(z,\xi-\zeta,\eta-\tau)$
$\displaystyle\qquad\qquad\qquad\qquad\times(1-\Psi_{2})(1-\Psi_{3})\partial_{z}^{\gamma_{2}}b(z,\zeta,\eta)d\zeta
d\tau.$

The region of the integration $F_{2}$ satisfies

$C_{1}^{-1}\langle\xi;\eta\rangle\leqq\langle\zeta;\tau\rangle\leqq{C_{1}}\langle\xi;\eta\rangle,\quad
C_{1}^{-1}\langle\eta\rangle\leqq\langle\tau\rangle\leqq{C_{1}}\langle\eta\rangle.$

We deduce that

$\displaystyle F_{2}$
$\displaystyle=\sum_{\alpha_{2}+\alpha_{1}=\alpha}\frac{\alpha!}{\alpha_{2}!\alpha_{1}!}\sum_{\beta_{2}+\beta_{3}+\beta_{1}=\beta}\frac{\beta!}{\beta_{2}!\beta_{3}!\beta_{1}!}\sum_{\gamma_{1}+\gamma_{2}=\gamma}\frac{\gamma!}{\gamma_{1}!\gamma_{2}!}$
$\displaystyle\times\iint_{\begin{subarray}{c}C_{1}^{-1}\langle\xi;\eta\rangle\leqq\langle\zeta;\tau\rangle\leqq{C_{1}}\langle\xi;\eta\rangle\\\
C_{1}^{-1}\langle\eta\rangle\leqq\langle\tau\rangle\leqq{C_{1}}\langle\eta\rangle\end{subarray}}\partial_{z}^{\gamma_{1}}a(z,\xi-\zeta,\eta-\tau)$
$\displaystyle\times\partial_{\zeta}^{\alpha_{2}}\partial_{\tau}^{\beta_{2}}\Psi_{2}\cdot\partial_{\tau}^{\beta_{3}}\Psi_{3}\cdot\partial_{z}^{\gamma_{2}}\partial_{\zeta}^{\alpha_{1}}\partial_{\tau}^{\beta_{1}}b(z,\zeta,\eta)d\zeta
d\tau$
$\displaystyle=\sum_{\alpha_{2}+\alpha_{1}=\alpha}\sum_{\beta_{2}+\beta_{3}+\beta_{1}=\beta}\iint_{\begin{subarray}{c}C_{1}^{-1}\langle\xi;\eta\rangle\leqq\langle\zeta;\tau\rangle\leqq{C_{1}}\langle\xi;\eta\rangle\\\
C_{1}^{-1}\langle\eta\rangle\leqq\langle\tau\rangle\leqq{C_{1}}\langle\eta\rangle\end{subarray}}$
$\displaystyle\times\mathcal{O}(\langle\xi-\zeta;\eta-\tau\rangle^{-(d+2)/2}\langle\eta-\tau\rangle^{-(d+2)/2}\langle\xi;\eta\rangle^{-\alpha_{2}-\beta_{2}}$
$\displaystyle\times\langle\eta\rangle^{-\beta_{3}}\langle\zeta;\tau\rangle^{-(d+2)/2-\alpha_{1}}\langle\tau\rangle^{-(d+2)/2-\beta_{1}})d\zeta
d\tau$
$\displaystyle=\iint_{\mathbb{R}\times\mathbb{R}}\mathcal{O}(\langle\xi-\zeta\rangle^{-(d+2)/2}\langle\eta-\tau\rangle^{-(d+2)/2}\langle\xi;\eta\rangle^{-(d+2)/2-\alpha}\langle\eta\rangle^{-(d+2)/2-\beta})d\zeta
d\tau$
$\displaystyle=\mathcal{O}(\langle\xi;\eta\rangle^{-(d+2)/2-\alpha}\langle\eta\rangle^{-(d+2)/2-\beta}).$
(6.2)

since $(d+2)/2=1+d/2>1$ guarantees that $\langle\xi-\zeta\rangle^{-(d+2)/2}$
is integrable in $\zeta$ on $\mathbb{R}$, and
$\langle\eta-\tau\rangle^{-(d+2)/2}$ is integrable in $\tau$ on $\mathbb{R}$.

The region of the integration $G_{2}$ satisfies

$C_{1}^{-1}\langle\xi;\eta\rangle\leqq\langle\zeta;\tau\rangle\leqq{C_{1}}\langle\xi;\eta\rangle,\quad\langle\eta\rangle/2\leqq\langle\eta-\tau\rangle.$

We deduce that

$\displaystyle G_{2}$
$\displaystyle=\sum_{\alpha_{2}+\alpha_{1}=\alpha}\frac{\alpha!}{\alpha_{2}!\alpha_{1}!}\sum_{\gamma_{1}+\gamma_{2}=\gamma}\frac{\gamma!}{\gamma_{1}!\gamma_{2}!}$
$\displaystyle\times\iint_{\begin{subarray}{c}C_{1}^{-1}\langle\xi;\eta\rangle\leqq\langle\zeta;\tau\rangle\leqq{C_{1}}\langle\xi;\eta\rangle\\\
\langle\eta\rangle/2\leqq\langle\eta-\tau\rangle\end{subarray}}\partial_{z}^{\gamma_{1}}\partial_{\eta}^{\beta}a(z,\xi-\zeta,\eta-\tau)$
$\displaystyle\times\partial_{\zeta}^{\alpha_{2}}\Psi_{2}\cdot(1-\Psi_{3})\cdot\partial_{\zeta}^{\alpha_{1}}\partial_{z}^{\gamma_{2}}b(z,\zeta,\eta)d\zeta
d\tau$
$\displaystyle=\sum_{\alpha_{2}+\alpha_{1}=\alpha}\iint_{\begin{subarray}{c}C_{1}^{-1}\langle\xi;\eta\rangle\leqq\langle\zeta;\tau\rangle\leqq{C_{1}}\langle\xi;\eta\rangle\\\
\langle\eta\rangle/2\leqq\langle\eta-\tau\rangle\end{subarray}}$
$\displaystyle\times\mathcal{O}(\langle\xi-\zeta;\eta-\tau\rangle^{-(d+2)/2}\langle\eta-\tau\rangle^{-(d+2)/2-\beta}$
$\displaystyle\times\langle\xi;\eta\rangle^{-\alpha_{2}}\langle\zeta;\tau\rangle^{-(d+2)/2-\alpha_{1}}\langle\tau\rangle^{-(d+2)/2})d\zeta
d\tau$
$\displaystyle=\iint_{\mathbb{R}\times\mathbb{R}}\mathcal{O}(\langle\xi-\zeta\rangle^{-(d+2)/2}\langle\eta\rangle^{-(d+2)/2-\beta}\langle\xi;\eta\rangle^{-(d+2)/2-\alpha}\langle\tau\rangle^{-(d+2)/2})d\zeta
d\tau$
$\displaystyle=\mathcal{O}(\langle\xi;\eta\rangle^{-(d+2)/2-\alpha}\langle\eta\rangle^{-(d+2)/2-\beta})$
(6.3)

since $(d+2)/2=1+d/2>1$ guarantees that $\langle\xi-\zeta\rangle^{-(d+2)/2}$
is integrable in $\zeta$ on $\mathbb{R}$, and $\langle\tau\rangle^{-(d+2)/2}$
is integrable in $\tau$ on $\mathbb{R}$.

The region of the integration $H_{2}$ satisfies

$\langle\xi;\eta\rangle/2\leqq\langle\xi-\zeta;\eta-\tau\rangle,\quad
C_{1}^{-1}\langle\eta\rangle\leqq\langle\tau\rangle\leqq{C_{1}}\langle\eta\rangle.$

We deduce that

$\displaystyle H_{2}$
$\displaystyle=\sum_{\beta_{2}+\beta_{3}+\beta_{1}=\beta}\frac{\beta!}{\beta_{2}!\beta_{3}!\beta_{1}!}\sum_{\gamma_{1}+\gamma_{2}=\gamma}\frac{\gamma!}{\gamma_{1}!\gamma_{2}!}$
$\displaystyle\times\iint_{\begin{subarray}{c}\langle\xi;\eta\rangle/2\leqq\langle\xi-\zeta;\eta-\tau\rangle\\\
C_{1}^{-1}\langle\eta\rangle\leqq\langle\tau\rangle\leqq{C_{1}}\langle\eta\rangle\end{subarray}}\partial_{z}^{\gamma_{1}}\partial_{\xi}^{\alpha}a(z,\xi-\zeta,\eta-\tau)$
$\displaystyle\times\partial_{\tau}^{\beta_{2}}(1-\Psi_{2})\cdot\partial_{\tau}^{\beta_{3}}\Psi_{3}\cdot\partial_{z}^{\gamma_{2}}\partial_{\tau}^{\beta_{1}}b(z,\zeta,\eta)d\zeta
d\tau$
$\displaystyle=\sum_{\beta_{2}+\beta_{3}+\beta_{1}=\beta}\iint_{\begin{subarray}{c}\langle\xi;\eta\rangle/2\leqq\langle\xi-\zeta;\eta-\tau\rangle\\\
C_{1}^{-1}\langle\eta\rangle\leqq\langle\tau\rangle\leqq{C_{1}}\langle\eta\rangle\end{subarray}}$
$\displaystyle\times\mathcal{O}(\langle\xi-\zeta;\eta-\tau\rangle^{-(d+2)/2-\alpha}\langle\eta-\tau\rangle^{-(d+2)/2}$
$\displaystyle\times\langle\xi;\eta\rangle^{-\beta_{2}}\langle\eta\rangle^{-\beta_{3}}\langle\zeta;\tau\rangle^{-(d+2)/2}\langle\tau\rangle^{-(d+2)/2-\beta_{1}})d\zeta
d\tau$
$\displaystyle=\iint_{\mathbb{R}\times\mathbb{R}}\mathcal{O}(\langle\xi;\eta\rangle^{-(d+2)/2-\alpha}\langle\eta-\tau\rangle^{-(d+2)/2}\langle\zeta\rangle^{-(d+2)/2}\langle\eta\rangle^{-(d+2)/2-\beta})d\zeta
d\tau$
$\displaystyle=\mathcal{O}(\langle\xi;\eta\rangle^{-(d+2)/2-\alpha}\langle\eta\rangle^{-(d+2)/2-\beta})$
(6.4)

since $(d+2)/2=1+d/2>1$ guarantees that $\langle\zeta\rangle^{-(d+2)/2}$ is
integrable in $\zeta$ on $\mathbb{R}$, and
$\langle\eta-\tau\rangle^{-(d+2)/2}$ is integrable in $\tau$ on $\mathbb{R}$.

The region of the integration $I_{2}$ satisfies

$\langle\xi;\eta\rangle/2\leqq\langle\xi-\zeta;\eta-\tau\rangle,\quad\langle\eta\rangle/2\leqq\langle\eta-\tau\rangle.$

We deduce that

$\displaystyle I_{2}$
$\displaystyle=\sum_{\gamma_{1}+\gamma_{2}=\gamma}\frac{\gamma!}{\gamma_{1}!\gamma_{2}!}\iint_{\begin{subarray}{c}\langle\xi;\eta\rangle/2\leqq\langle\xi-\zeta;\eta-\tau\rangle\\\
\langle\eta\rangle/2\leqq\langle\eta-\tau\rangle\end{subarray}}\partial_{z}^{\gamma_{1}}\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}a(z,\xi-\zeta,\eta-\tau)$
$\displaystyle\times(1-\Psi_{2})\cdot(1-\Psi_{3})\cdot\partial_{z}^{\gamma_{2}}b(z,\zeta,\eta)d\zeta
d\tau$
$\displaystyle=\iint_{\begin{subarray}{c}\langle\xi;\eta\rangle/2\leqq\langle\xi-\zeta;\eta-\tau\rangle\\\
\langle\eta\rangle/2\leqq\langle\eta-\tau\rangle\end{subarray}}\mathcal{O}(\langle\xi-\zeta;\eta-\tau\rangle^{-(d+2)/2-\alpha}\langle\eta-\tau\rangle^{-(d+2)/2-\beta}$
$\displaystyle\times\langle\zeta;\tau\rangle^{-(d+2)/2}\langle\tau\rangle^{-(d+2)/2})d\zeta
d\tau$
$\displaystyle=\iint_{\mathbb{R}\times\mathbb{R}}\mathcal{O}(\langle\xi;\eta\rangle^{-(d+2)/2-\alpha}\langle\eta\rangle^{-(d+2)/2-\beta}\langle\zeta\rangle^{-(d+2)/2}\langle\tau\rangle^{-(d+2)/2})d\zeta
d\tau$
$\displaystyle=\mathcal{O}(\langle\xi;\eta\rangle^{-(d+2)/2-\alpha}\langle\eta\rangle^{-(d+2)/2-\beta})$
(6.5)

since $(d+2)/2=1+d/2>1$ guarantees that $\langle\zeta\rangle^{-(d+2)/2}$ is
integrable in $\zeta$ on $\mathbb{R}$, and $\langle\tau\rangle^{-(d+2)/2}$ is
integrable in $\tau$ on $\mathbb{R}$.

Substitute (6.2), (6.3), (6.4) and (6.5) into (6.1). We have

$\partial_{z}^{\gamma}\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}c_{2}(z,\xi,\eta)=\mathcal{O}(\langle\xi;\eta\rangle^{-(d+2/2)-\alpha}\langle\eta\rangle^{-(d+2)/2-\beta}).$

This completes the proof. ∎

Finally we prove (II) of Lemma 6.2.

###### Proof of Lemma 6.2 (II).

Suppose that

$u\in
I^{-(d+2)/2+1/2-N(d,n)/4,-(d+2)/2+1/2}\bigl{(}N^{\ast}{S_{jk}},N^{\ast}{S_{j}}\bigr{)},$
$v\in
I^{-(d+2)/2+1/2-N(d,n)/4,-(d+2)/2+1/2}\bigl{(}N^{\ast}{S_{jk}},N^{\ast}{S_{k}}\bigr{)}.$

In the same way as the proof of (I), fix arbitrary $p_{0}\in S_{jk}$, and
choose appropriate local coordinates
$(x,y,z)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{N(d,n)-2}$ such that
$\bigl{(}x(p_{0}),y(p_{0}),z(p_{0})\bigr{)}=0$, and

$\displaystyle N^{\ast}{S}_{j}$
$\displaystyle=\\{(0,y,z;\xi,0,0):\xi,y\in\mathbb{R},z\in\mathbb{R}^{N(d,n)-2}\\},$
$\displaystyle N^{\ast}{S}_{k}$
$\displaystyle=\\{(x,0,z;0,\eta,0):x,\eta\in\mathbb{R},z\in\mathbb{R}^{N(d,n)-2}\\},$
$\displaystyle N^{\ast}{S}_{jk}$
$\displaystyle=\\{(0,0,z;\xi,\eta,0):\xi,\eta\in\mathbb{R},z\in\mathbb{R}^{N(d,n)-2}\\}$

near $(x,y,z)=(0,0,0)$. Then there exist symbols

$a(z,\xi,\eta),b(z,\eta,\xi)\in
S^{-(d+2)/2,-(d+2)/2}\bigl{(}\mathbb{R}^{N(d,n)-2}\times(\mathbb{R}\setminus\\{0\\})\times\mathbb{R}\bigr{)}$

such that $a(z,\xi,\eta)$ and $b(z,\eta,\xi)$ are compactly supported in $z$
near $z=0$, and $u$ and $v$ are represented by oscillatory integrals

$\displaystyle u(x,y,z)$
$\displaystyle=\iint_{\mathbb{R}\times\mathbb{R}}e^{i(x\xi+y\eta)}a(z,\xi,\eta)d\xi
d\eta,$ $\displaystyle v(x,y,z)$
$\displaystyle=\iint_{\mathbb{R}\times\mathbb{R}}e^{i(x\xi+y\eta)}b(z,\eta,\xi)d\xi
d\eta$

near $(x,y,z)=(0,0,0)$. We need to take care of the order of the variables of
$b(z,\eta,\xi)$. Then we have

$\displaystyle(uv)(x,y,z)$
$\displaystyle=\iint_{\mathbb{R}\times\mathbb{R}}e^{i(x\xi+y\eta)}c_{3}(z,\xi,\eta)d\xi
d\eta,$ $\displaystyle c_{3}(z,\xi,\eta)$
$\displaystyle=\iint_{\mathbb{R}\times\mathbb{R}}a(z,\xi-\zeta,\eta-\tau)b(z,\tau,\zeta)d\zeta
d\tau$

near $(x,y,z)=(0,0,0)$. Note that $c_{3}(z,\xi,\eta)$ is compactly supported
in $z$ near $z=0$. It suffices to show that

$\displaystyle c_{3}(z,\xi,\eta)$ $\displaystyle\in
S^{-(d+2)/2,-(d+2)/2}\bigl{(}\mathbb{R}_{z}^{N(d,n)-2}\times(\mathbb{R_{\xi}}\setminus\\{0\\})\times\mathbb{R_{\eta}}\bigr{)}$
$\displaystyle+S^{-(d+2)/2,-(d+2)/2}\bigl{(}\mathbb{R}_{z}^{N(d,n)-2}\times(\mathbb{R}_{\eta}\setminus\\{0\\})\times\mathbb{R}_{\xi}\bigr{)}.$

In the same way as the proof of (I), we can obtain

$\partial_{z}^{\gamma}\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}c_{3}(z,\xi,\eta)=\mathcal{O}(\langle\xi\rangle^{-(d+2)/2-\alpha}\langle\eta\rangle^{-(d+2)/2-\beta}).$
(6.6)

To complete the proof we split $c_{3}(z,\xi,\eta)$ into two parts smoothly.
For this purpose we introduce two cut-off functions defined by $\psi(t)$. Set
$\Psi_{4}(\xi,\eta):=\psi(\langle\xi\rangle/\langle\eta\rangle)$. We obtain
the following properties.

  * (i)

If $\Psi_{4}(\xi,\eta)>0$, then $\langle\xi;\eta\rangle\leqq
C_{2}\langle\eta\rangle$ with some $C_{2}>1$ since $\langle\xi\rangle\leqq
3\langle\eta\rangle/4$.

  * (ii)

If $1-\Psi_{4}(\xi,\eta)>0$, then $\langle\xi;\eta\rangle\leqq
C_{2}\langle\xi\rangle$ with some $C_{2}>1$ since
$\langle\xi\rangle\geqq\langle\eta\rangle/2$.

  * (iii)

We have that

$\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}\Psi_{4}(\xi,\eta),\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}\bigl{(}1-\Psi_{4}(\xi,\eta)\bigr{)}=\mathcal{O}(\langle\xi;\eta\rangle^{-\alpha-\beta})$

for all the non-negative integers $\alpha$ and $\beta$ since
$0\leqq\Psi_{4}(\xi,\eta)\leqq 1$ for all $(\xi,\eta)\in\mathbb{R}^{2}$.
Moreover $\langle\xi;\eta\rangle\leqq C_{2}\langle\eta\rangle$ and
$\langle\xi;\eta\rangle\leqq C_{2}\langle\xi\rangle$ hold in
$\operatorname{supp}\psi^{\prime}(\langle\xi\rangle/\langle\eta\rangle)$.

Set $c_{3,1}(z,\xi,\eta):=\bigl{(}1-\Psi_{4}(\xi,\eta)\bigr{)}c(z,\xi,\eta)$
and $c_{3,2}(z,\xi,\eta):=\Psi_{4}(\xi,\eta)c(z,\xi,\eta)$. Combining (6.6)
and the properties (i), (ii) and (iii) above, we shall show

$\displaystyle c_{3,1}(z,\xi,\eta)$ $\displaystyle\in
S^{-(d+2)/2,-(d+2)/2}\bigl{(}\mathbb{R}_{z}^{N(d,n)-2}\times(\mathbb{R_{\xi}}\setminus\\{0\\})\times\mathbb{R_{\eta}}\bigr{)}$
$\displaystyle c_{3,2}(z,\xi,\eta)$ $\displaystyle\in
S^{-(d+2)/2,-(d+2)/2}\bigl{(}\mathbb{R}_{z}^{N(d,n)-2}\times(\mathbb{R}_{\eta}\setminus\\{0\\})\times\mathbb{R}_{\xi}\bigr{)}.$

Since $\operatorname{supp}c_{3,1}(z,\cdot,\cdot)$ is contained in
$\\{\langle\xi;\eta\rangle\leqq C_{2}\langle\xi\rangle\\}$, we deduce that

$\displaystyle\partial_{z}\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}c_{3,1}(z,\xi,\eta)$
$\displaystyle=\sum_{\alpha_{1}+\alpha_{2}=\alpha}\sum_{\beta_{1}+\beta_{2}=\beta}\frac{\alpha!}{\alpha_{1}!\alpha_{2}!}\frac{\beta!}{\beta_{1}!\beta_{2}!}$
$\displaystyle\times\partial_{\xi}^{\alpha_{1}}\partial_{\eta}^{\beta_{1}}\bigl{(}1-\Psi_{4}(\xi,\eta)\bigr{)}\cdot\partial_{z}\partial_{\xi}^{\alpha_{2}}\partial_{\eta}^{\beta_{2}}c_{3}(z,\xi,\eta)$
$\displaystyle=\sum_{\alpha_{1}+\alpha_{2}=\alpha}\sum_{\beta_{1}+\beta_{2}=\beta}$
$\displaystyle\times\mathcal{O}(\langle\xi;\eta\rangle^{-\alpha_{1}-\beta_{1}}\langle\xi\rangle^{-(d+2)/2-\alpha_{2}}\langle\eta\rangle^{-(d+2)/2-\beta_{2}})$
$\displaystyle=\mathcal{O}(\langle\xi;\eta\rangle^{-(d+2)/2-\alpha}\langle\eta\rangle^{-(d+2)/2-\beta}).$

Since $\operatorname{supp}c_{3,2}(z,\cdot,\cdot)$ is contained in
$\\{\langle\xi;\eta\rangle\leqq C_{2}\langle\eta\rangle\\}$, we deduce that

$\displaystyle\partial_{z}\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}c_{3,2}(z,\xi,\eta)$
$\displaystyle=\sum_{\alpha_{1}+\alpha_{2}=\alpha}\sum_{\beta_{1}+\beta_{2}=\beta}\frac{\alpha!}{\alpha_{1}!\alpha_{2}!}\frac{\beta!}{\beta_{1}!\beta_{2}!}$
$\displaystyle\times\partial_{\xi}^{\alpha_{1}}\partial_{\eta}^{\beta_{1}}\Psi_{4}(\xi,\eta)\cdot\partial_{z}\partial_{\xi}^{\alpha_{2}}\partial_{\eta}^{\beta_{2}}c_{3}(z,\xi,\eta)$
$\displaystyle=\sum_{\alpha_{1}+\alpha_{2}=\alpha}\sum_{\beta_{1}+\beta_{2}=\beta}$
$\displaystyle\times\mathcal{O}(\langle\xi;\eta\rangle^{-\alpha_{1}-\beta_{1}}\langle\xi\rangle^{-(d+2)/2-\alpha_{2}}\langle\eta\rangle^{-(d+2)/2-\beta_{2}})$
$\displaystyle=\mathcal{O}(\langle\xi;\eta\rangle^{-(d+2)/2-\beta}\langle\xi\rangle^{-(d+2)/2-\alpha}).$

This completes the proof. ∎

## References

  * [1] H. Chihara, Bargmann transform on the space of hyperplanes, submitted, arXiv:2007.12365.
  * [2] J. J. Duistermaat, “Fourier Integral Operators”, Birkhäuser, Boston, 1995.
  * [3] C. L. Epstein, ”Introduction to the Mathematics of Medical Imaging, Second Edition”, SIAM, Philadelphia, 2008.
  * [4] A. Greenleaf and G. Uhlmann, Recovering singularities of a potential from singularities of scattering data, Comm. Math. Phys., 157 (1993), pp.549–572.
  * [5] A. Grigis and J. Sjöstrand, “Microlocal Analysis for Differential Operators: An Introduction”, Cambridge University Press, Cambridge, 1994.
  * [6] V. Guillemin and S. Sternberg, “Geometric Asymptotics”, Mathematical Surveys, 14, American Mathematical Society, Providence, RI, 1977.
  * [7] V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), pp.251–-267.
  * [8] S. Helgason, “Integral Geometry and Radon Transforms”, Springer-Verlag, New York, NY, 2011.
  * [9] L. Hörmander, “The Analysis of Linear Partial Differential Operators I”, Springer-Verlag, Berlin, 1983.
  * [10] L. Hörmander, “The Analysis of Linear Partial Differential Operators II”, Springer-Verlag, Berlin, 1983.
  * [11] L. Hörmander, “The Analysis of Linear Partial Differential Operators III”, Springer-Verlag, Berlin, 1985.
  * [12] L. Hörmander, “The Analysis of Linear Partial Differential Operators IV”, Springer-Verlag, Berlin, 1985.
  * [13] V. P. Krishnan and E. T. Quinto, Microlocal analysis in tomography, Handbook of mathematical methods in imaging, Vol. 1, 2, 3, pp.847–902, Springer-Verlag, New York, 2015.
  * [14] R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math., 32 (1979), pp.483–-519.
  * [15] B. Palacios, G. Uhlmann and Y. Wang, Quantitative analysis of metal artifacts in X-ray tomography, SIAM J. Math. Anal., 50 (2018), pp.4914–4936.
  * [16] H. S. Park, J. K. Choi and J. K. Seo, Characterization of metal artifacts in X-ray computed tomography, Comm. Pure Appl. Math., 70 (2017), pp.2191–-2217.
  * [17] E. T. Quinto, Singularities of the X-Ray transform and limited data tomography in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$, SIAM Journal on Mathematical Analysis, 24 (1993), pp.1215–1225.
  * [18] E. T. Quinto, An introduction to X-ray tomography and Radon transforms, ”The Radon Transform, Inverse Problems, and Tomography”, Proceedings of Symposia in Applied Mathematics, 63, pp.1–23, American Mathematical Society, Providence, RI, 2006.