# Unique continuation for the momentum ray transform

Joonas Ilmavirta, Pu-Zhao Kow and Suman Kumar Sahoo  Department of Mathematics
and Statistics, University of Jyväskylä, Finland.
E-mail<EMAIL_ADDRESS>Department of Mathematics and Statistics,
University of Jyväskylä, Finland.
E-mail<EMAIL_ADDRESS>Department of Mathematics and Statistics,
University of Jyväskylä, Finland.
E-mail<EMAIL_ADDRESS>
###### Abstract.

The present article focuses on a unique continuation result for certain
weighted ray transforms, utilizing the unique continuation property (UCP) of
the fractional Laplace operator. Specifically, we demonstrate a conservative
property for momentum ray transforms acting on tensors, as well as the
antilocality property for both weighted ray and cone transforms acting on
functions.

###### Key words and phrases:

ray transform, tensor tomography, antilocality property, unique continuation
property (UCP), fractional Laplacian, fractional elliptic operator, Saint
Venant operator

###### 2020 Mathematics Subject Classification:

Primary 46F12, 45Q05, 35R11

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Momentum ray transform on Schwartz space
    2. 2.2 Momentum ray transform on compactly supported tensor field distributions
    3. 2.3 Normal operator of momentum ray transforms
    4. 2.4 Some differential operators
    5. 2.5 Generalized Saint-Venant operator
  3. 3 A generalization of the curl-curl identity
  4. 4 A solenoidal decomposition theorem
  5. 5 Main results
    1. 5.1 Unique continuation property for momentum ray transform on tensors
    2. 5.2 Measurable unique continuation property of momentum ray transform
    3. 5.3 Generalization of momentum ray transform: Fractional momentum ray transform
    4. 5.4 Antilocality property of the weighted ray transform
    5. 5.5 Antilocality property of the cone transform
  6. A Negative power of elliptic operator

##  1\. Introduction

To what extent is a tensor field determined by its line integrals? Depending
on the weighting and other choices in the line integrals, there may or may not
be a gauge freedom or other non-uniqueness. We will focus on the so-called
momentum ray transform, and study its uniqueness with the help of the unique
continuation principle or property (UCP).

The space of $m$-tensor fields in $\mathbb{R}^{n}$ is denoted by $T^{m}\equiv
T^{m}(\mathbb{R}^{n})$, while its subspace of covariant symmetric $m$-tensor
fields on $\mathbb{R}^{n}$ is denoted by $S^{m}\equiv S^{m}(\mathbb{R}^{n})$.
In Cartesian coordinates, an element $f$ can be written as

$f(x)=f_{i_{1}\dots i_{m}}(x)\,\mathrm{d}x^{i_{1}}\cdots\mathrm{d}x^{i_{m}},$

where $f_{i_{1}\dots i_{m}}(x)$ is symmetric in all indices
$i_{1},\dots,i_{m}\in\\{1,\dots,n\\}$. For repeated indices, Einstein
summation convention will be assumed throughout this article.

Let $\mathcal{S}(S^{m})=\mathcal{S}(\mathbb{R}^{n};S^{m}(\mathbb{R}^{n}))$ be
the Schwartz class of symmetric $m$ tensor fields in $\mathbb{R}^{n}$. We
denote $\mathcal{S}(S^{0})=\mathcal{S}(\mathbb{R}^{n})$. The integral
transforms in the focus of our study can be defined on these spaces as
follows:

  1. (1)

The ray transform $If$ of $f\in\mathcal{S}(S^{m})$ is defined by

$\displaystyle If(x,\xi):=\int\limits_{-\infty}^{\infty}f_{i_{1}\cdots
i_{m}}(x+t\xi)\,\xi_{i_{1}}\cdots\xi_{i_{m}}\mathrm{d}t.$

  2. (2)

The $k^{\rm th}$ momentum ray transform (MRT) $I^{k}f$ of
$f\in\mathcal{S}(S^{m})$ is defined by

$\displaystyle
I^{k}f(x,\xi):=\int\limits_{-\infty}^{\infty}t^{k}\,f_{i_{1}\cdots
i_{m}}(x+t\xi)\,\xi_{i_{1}}\cdots\xi_{i_{m}}\mathrm{d}t\quad\mbox{for all
integers $k\geq 0$}.$

  3. (3)

For $s\in(0,\frac{n}{2})$, the fractional momentum ray transform $I^{2s-1}f$
of $f\in\mathcal{S}(\mathbb{R}^{n})$ is defined by

$\big{(}\mathcal{X}_{s}f\big{)}(x,\xi):=\big{(}I^{2s-1}f\big{)}(x,\xi)\equiv\int\limits_{0}^{\infty}t^{2s-1}f(x+t\xi)\,\mathrm{d}t.$
(1.1)

In the case $k=0$ the momentum ray transform reduces to the classical ray
transform: $I^{0}=I$. This transform has been studied extensively due to its
wide range of potential applications in various scientific fields; see
[PSU23]. MRTs were first introduced by Sharafutdinov [Sha94] and studied
further in the works [KMSS19, KMSS20, MS21]. For functions, $I^{k}f$ appears
to give the inversion formula of cone transform and conical Radon transform
whereas the latter transform has promising application in Compton cameras; see
[KT17]. For tensors, MRTs appear to solve Calderón type inverse problems for
polyharmonic operators; see [BKS21, SS23]. Another motivation to study MRTs
are its connection with the exponential ray transform defined as
$(I_{\exp}^{\alpha}f)(x,\xi):=\int_{\mathbb{R}}e^{\alpha t}f_{i_{1}\cdots
i_{m}}(x+t\xi)\,\xi_{i_{1}}\cdots\xi_{i_{m}}\mathrm{d}t$, for some real number
$\alpha$. Formally (and truly whenever the series is convergent in a suitable
sense) the exponential ray transform is related to the momentum ray transforms
by

$I_{\exp}^{\alpha}f=\sum_{k=0}^{\infty}\frac{\alpha^{k}}{k!}I^{k}f.$

We now state the main results (not all) of our article. We refer the reader
Section 2 for the definition of $R^{k}f$.

######  Theorem I.

Let $m\in\mathbb{N}$ and an integer $0\leq k\leq m$. Suppose that

$\begin{cases}f\in L^{1}(\mathbb{R}^{n};S^{m})\cap L_{\rm
loc}^{2}(\mathbb{R}^{n};S^{m})&\text{when $0\leq k\leq n-1$,}\\\ f\in
L^{2}(\mathbb{R}^{n};S^{m})\text{ with compact support}&\text{when $n\leq
k\leq m$.}\end{cases}$ (1.2)

If there exists an nonempty open set $U$ in $\mathbb{R}^{n}$ such that

$R^{k}f|_{U}=0\quad\text{for some $0\leq k\leq m$. }$

Suppose that there exists $x_{0}\in U$ such that for each $0\leq p\leq k$ the
following assumption holds:

$N^{p}f$ vanishes at $x_{0}$ of infinite derivative order (1.3)

then $R^{k}f\equiv 0$ in $\mathbb{R}^{n}$. If we additionally assuming that
$f$ has compact support, then $f$ is a generalized potential field, that is,
$f=\mathrm{d}^{k+1}v$ for some
$v\in\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m-k-1})$.

######  Theorem II.

Suppose $U\subseteq\mathbb{R}^{n}$ be any non empty open set and $n\geq 2$.
Suppose that $N^{0}f\in H^{\frac{1}{2}}(\mathbb{R}^{n})$. If $f|_{U}=0$ and
there exists a positive measure set $E$ in $U$ such that
$\left(N^{0}f\right)|_{E}=0$, then $f\equiv 0$ in $\mathbb{R}^{n}$.

######  Theorem III.

Let $n\geq 2$ be an integer, $0<s\leq\frac{n}{4}$ with $s\neq\mathbb{Z}$ and
let $f\in L^{\frac{2n}{n+4s}}(\mathbb{R}^{n})$. If there exists a non-empty
open set $U$ in $\mathbb{R}^{n}$ such that

$f=\mathcal{A}_{s}f=0\quad\text{in $U$}\quad\implies
f=0\quad\mbox{in}\quad\mathbb{R}^{n},$

where
$(\mathcal{A}_{s}f)(x)=c(n,-s)\int_{\mathbb{S}^{n-1}}(\mathcal{X}_{s}f)(x,\xi)\,\mathrm{d}S(\xi)$
with $c(n,s):=\frac{2^{2s}\Gamma(\frac{n+2s}{2})}{\pi^{n/2}|\Gamma(-s)|}$.

The following table contains the list of some existing results in this
direction:

Uniqueness results
---
Transforms |  Full data |  UCP |  Support theorem
Ray transform |  [Hel99, Sha94] |  [IM20, IM21, AKS22] |  [Kri07]
$d$ plane transform |  [Hel99] |  [CMR21] |  [Hel99]
MRT |  [KMSS19] |  [AKS22], current article |  [AM19]
Cone transform |  [KT17] |  current article |  unknown

The ray transform with full data is a classical concept, and we have only
given text book references rather than original articles for it.

The unique continuation result for ray transform can be considered as a
partial data uniqueness result, in which one considers the following question:
If a function or a symmetric tensor field $f$ satisfies $If=0$ and $f=0$ in
some open set $U$, then can one conclude that $f=0$ (for functions) or $f$ is
a potential field (for symmetric tensor fields) in the whole space?

This question is related to the _interior tomography problem_ or the _region
of interest (ROI) tomography_ problem. The goal of ROI tomography is to
reconstruct the function within an open set (the region of interest) from the
line integrals over all lines through that set. It is well known that from
this information one cannot recover the whole function in the region of
interest; see [KQR15, Example 2.1] However, it is possible to recover the
singularities of the function in the ROI because of the fact that ray
transform $If$ is an elliptic Fourier integral operator (FIO) and its normal
operator $I^{*}If$ is an elliptic pseudo-differential operator. Therefore by
pseudo-local property, we have $\mbox{WF}(f)=\mbox{WF}(I^{*}If)$, see [Qui06].
This fact serves as a motivation for Lambda tomography [FRS92], which is a
local reconstruction method that uses ROI data to reconstruct singularities.

However, the problem we pose is different: We only use the rays that meet the
set $V$, but our goal is to reconstruct the scalar or tensor field outside
$V$, not inside it. We assume that something about the field is known in $V$.
The simplest choice is $f|_{V}=0$, but this can be relaxed to the gauge
invariant assumption $Rf|_{V}=0$, where $R$ is the Saint Venant operator. The
assumption that the integrals over all lines vanish can be relaxed to
$(I^{k})^{*}I^{k}f$ vanishing in a subset of positive measure of $V$ or
vanishing to infinite order at a point of $V$. To make sense of
$(I^{k})^{*}I^{k}f$ vanishing to infinite order, we need it to be smooth, and
that follows from ellipticity of the normal operator $(I^{k})^{*}I^{k}$ as a
pseudo-differential operator.

The following are equivalent for $f\in\mathcal{S}(S^{m})$ and $0\leq k\leq m$:

  1. (1)

$I^{k}f=0$ for all lines

  2. (2)

$R^{k}f=0$ (the $k^{\rm th}$ power of the Saint-Venant operator)

  3. (3)

$f$ is the $(k+1)^{\rm st}$ symmetrized covariant derivative of some
$v\in\mathcal{S}(S^{m-k-1})$ (with the convention that tensor fields of order
$-1$ are identically zero)

To prove a similar conclusion for our partial data setting, we aim to pass
from partial data on $I^{k}f$ to the global conclusion $R^{k}f=0$, from which
the conclusion follows. The advantage of working with $R^{k}f$ is that it is a
differential operator which is local in nature — unlike the integral operator
$I^{k}$.

By connecting the normal operator of ray transform with fractional Laplacian,
we see that the UCP for ray transforms is a consequence of UCP (more
precisely, the antilocality property) of fractional Laplacian. UCP (Unique
Continuation Property) is a useful tool for studying inverse problems, where
one aims to recover an unknown function from measured data. UCP ensures that
if a solution to a PDE vanishes in a suitable sense, then it must be
identically zero. UCP has been extensively studied for local operators (e.g.,
Laplace and wave operators) [Isa07] and more recently for nonlocal operators
(e.g., fractional Laplace equation, and fractional wave equation of
peridynamic type) [GSU20, Gho22]. Interestingly, the proof of UCP for some
nonlocal elliptic operators requires the use of UCP for local elliptic
equations, see [GFR19].

The rest of the article is organized as follows. In Section 2, we provide a
recap of some preliminary results, including definitions and notation that we
will use in the upcoming sections. In Section 3, we study more properties of
generalized Saint Venant operator, and a new decomposition theorem of
symmetric tensor fields is presented in Section 4. Finally, we recall our main
results and prove them in Section 5. In Appendix A, we provide the domain of a
negative fractional power of a general elliptic operator (see Definition A.3)
analogously to the fractional Laplace operator.

## Acknowledgements

J. I. was supported by the Academy of Finland (grants 332890 and 351665).
P.-Z. K and S. K. S were partly supported by the Academy of Finland (Centre of
Excellence in Inverse Modelling and Imaging, grant 312121) and by the European
Research Council under Horizon 2020 (ERC CoG 770924).

##  2\. Preliminaries

In order to make the paper self-contained, in this section, we introduce some
operators – especially the momentum ray transforms and Saint-Venant operators
– as well as some of their known properties. We are not going to exhaust all
the details here, see e.g. [AKS22, KMSS19, Sha94] and references therein for
more details. Readers who are already familiar with the subject may proceed to
the next section.

###  2.1. Momentum ray transform on Schwartz space

For each vectors $\xi^{(1)},\cdots,\xi^{(m)}\in\mathbb{R}^{n}$, their _tensor
product_ (or _juxtaposition_) $\xi^{(1)}\otimes\cdots\otimes\xi^{(m)}$ is
defined by

$(\xi^{(1)}\otimes\cdots\otimes\xi^{(m)})_{i_{1}\cdots
i_{m}}:=\xi_{i_{1}}^{(1)}\cdots\xi_{i_{m}}^{(m)}.$

If $\xi^{(1)}=\cdots=\xi^{(m)}=\xi$, we simply denote

$\xi^{\otimes m}:=\xi^{(1)}\otimes\cdots\otimes\xi^{(m)}\in
S^{m}\quad\text{and}\quad\xi_{i_{1}\cdots i_{m}}^{\otimes m}:=(\xi^{\otimes
m})_{i_{1}\cdots i_{m}}.$

We also denote $\langle\cdot,\cdot\rangle:S^{m}\times
S^{m}\rightarrow\mathbb{R}$ by

$\langle f,g\rangle:=f_{i_{1}\cdots i_{m}}g_{i_{1}\cdots i_{m}}.$

Let ${\mathcal{S}}(\mathbb{R}^{n};S^{m})$ be the Schwartz class of symmetric
$m$ tensor fields in $\mathbb{R}^{n}$. Given any non-negative integer $k$, we
define the mapping $J^{k}:{\mathcal{S}}(\mathbb{R}^{n};S^{m})\rightarrow
C^{\infty}(\mathbb{R}^{n}\times(\mathbb{R}^{n}\setminus\\{0\\}))$ by

$(J^{k}f)(x,\xi):=\int\limits_{-\infty}^{\infty}t^{k}\,\langle
f(x+t\xi),\xi^{\otimes
m}\rangle\,\mathrm{d}t\equiv\int\limits_{-\infty}^{\infty}t^{k}\,f_{i_{1}\dots
i_{m}}(x+t\xi)\,\xi_{i_{1}}\cdots\xi_{i_{m}}\,\mathrm{d}t$ (2.1)

for all $(x,\xi)\in\mathbb{R}^{n}\times(\mathbb{R}^{n}\setminus\\{0\\})$.
Denote the tangent bundle of the unit sphere by

$T\mathbb{S}^{n-1}:=\begin{Bmatrix}\begin{array}[]{l|l}(x,\xi)\in\mathbb{R}^{n}\times\mathbb{S}^{n-1}&\langle
x,\xi\rangle=0\end{array}\end{Bmatrix}.$

Since each point $(x,\xi)\in T{\mathbb{S}}^{n-1}$ determines a unique line
$x+t\xi$ with $t\in\mathbb{R}$, then it make sense to denote $I^{k}\equiv
J^{k}|_{T\mathbb{S}^{n-1}}$ be the restriction of $J^{k}$ on
$T\mathbb{S}^{n-1}$. Let $\mathcal{S}(T\mathbb{S}^{n-1})$ be the space of
smooth functions $\varphi(x,\xi)$ on $T\mathbb{S}^{n-1}$ such that all their
derivatives decrease rapidly in the first argument [Sha94, Section 2.1]. It is
easily see that

$I^{k}\equiv
J^{k}|_{T\mathbb{S}^{n-1}}:\mathcal{S}(\mathbb{R}^{n};S^{m})\rightarrow\mathcal{S}(T\mathbb{S}^{n-1})\quad\mbox{is
bounded}.$ (2.2)

######  Definition 2.1.

For each $m\in\mathbb{Z}_{\geq 0}$ and integer $0\leq k\leq m$, we call (2.2)
the _$k^{\rm th}$ momentum ray transform_ of a symmetric $m$-tensor field.

######  Remark 2.2 (An equivalent relation).

From [KMSS19, (2.6)] we have

$(J^{q}f)(x,\xi)=|\xi|^{m-2q-1}\sum_{\ell=0}^{q}(-1)^{q-\ell}\binom{q}{\ell}|\xi|^{\ell}\langle\xi,x\rangle^{q-\ell}(I^{\ell}f)\underbrace{\left(x-\frac{\langle
x,\xi\rangle}{|\xi|^{2}}\xi,\frac{\xi}{|\xi|}\right)}_{\in\,T\mathbb{S}^{n-1}}$
(2.3)

which implies, for each integer $0\leq k\leq m$, that

$\displaystyle(I^{0}f,I^{1}f,\cdots,I^{k}f)=(0,0,\cdots,0)\iff(J^{0}f,J^{1}f,\cdots,J^{k}f)=(0,0,\cdots,0).$

###  2.2. Momentum ray transform on compactly supported tensor field
distributions

It is well-known that the (classical) ray transform $I^{0}$ is well-defined on
compactly supported tensor field distributions [Sha94]. Similar extension for
momentum ray transforms was considered in [BKS21] in order to study the
polyharmonic operator inverse problem. Similar ideas also work for momentum
ray transforms [AKS22, Section 2.3], which will be presented here in order to
make the paper self-contained. Let
$\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m})$ be the compactly supported
symmetric $m$ tensor distribution fields in $\mathbb{R}^{n}$, and we define
the bounded linear operator
$I^{k}:\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m})\to\mathcal{E}^{\prime}(T\mathbb{S}^{n-1})$
by

$(I^{k}f,g)_{T\mathbb{S}^{n-1}}=\left(f,(I^{k})^{*}g\right)_{\mathbb{R}^{n};S^{m}}\quad\text{for
all $f\in\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m})$ and $g\in
C^{\infty}(T\mathbb{S}^{n-1})$,}$ (2.4)

where the distributional adjoint
$(I^{k})^{*}:C^{\infty}(T\mathbb{S}^{n-1})\rightarrow
C^{\infty}(\mathbb{R}^{n};S^{m})$ is defined by

$\left((I^{k})^{*}g\right)_{i_{1}\cdots
i_{m}}(x):=\int\limits_{\mathbb{S}^{n-1}}\langle
x,\xi\rangle^{k}\xi_{i_{1}}\dots\xi_{i_{m}}\,g(x-\langle
x,\xi\rangle\xi,\xi)\,\mathrm{d}S_{\xi}\quad\text{for all
$x\in\mathbb{R}^{n}$.}$ (2.5)

Similarly, we define
$J^{k}:\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m})\rightarrow\mathcal{D}^{\prime}(\mathbb{R}^{n}\times\mathbb{S}^{n-1})$
by

$(J^{k}f,g)_{\mathbb{R}^{n}\times\mathbb{S}^{n-1}}=\left(f,(J^{k})^{*}g\right)_{\mathbb{R}^{n};S^{m}}\quad\text{for
all $f\in\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m})$ and $g\in
C_{c}^{\infty}(\mathbb{R}^{n}\times\mathbb{S}^{n-1})$,}$

where the distributional adjoint
$(J^{k})^{*}:C_{c}^{\infty}(\mathbb{R}^{n}\times\mathbb{S}^{n-1})\rightarrow
C^{\infty}(\mathbb{R}^{n};S^{m})$ is defined by

$\left((J^{k})^{*}g\right)_{i_{1}\cdots
i_{m}}(x):=\int\limits_{\mathbb{S}^{n-1}}\int\limits_{\mathbb{R}}t^{k}g(x-t\xi,\xi)\xi_{i_{1}}\cdots\xi_{i_{m}}\,\mathrm{d}t\,\mathrm{d}S_{\xi}\quad\text{for
all $x\in\mathbb{R}^{n}$.}$

###  2.3. Normal operator of momentum ray transforms

Using (2.2), for each integer $0\leq k\leq m$, let us denote
$N^{k}=(I^{k})^{*}I^{k}:\mathcal{S}(\mathbb{R}^{n};S^{m})\to
C^{\infty}(\mathbb{R}^{n};S^{m})$ the _normal operator_ of the $k^{\rm th}$
momentum ray transform in Definition 2.1. For each
$f\in\mathcal{S}(\mathbb{R}^{n};S^{m})$, we have

$\displaystyle(N^{k}f)_{i_{1}\cdots i_{m}}(x)$ (2.6)
$\displaystyle\quad=2\sum\limits_{\ell=0}^{k}\binom{k}{\ell}(-1)^{\ell}x_{p_{1}\cdots
p_{2k-\ell}}^{\otimes(2k-\ell)}\left(f_{j_{1}\dots j_{m}}*\Xi_{p_{1}\cdots
p_{2k-\ell}i_{1}\cdots i_{m}j_{1}\dots j_{m}}\right)(x),$

for all $x\in\mathbb{R}^{n}$, where

$\Xi_{p_{1}\cdots p_{2k-\ell}i_{1}\cdots i_{m}j_{1}\dots
j_{m}}(z):=\frac{z_{p_{1}\cdots p_{2k-\ell}i_{1}\cdots i_{m}j_{1}\dots
j_{m}}^{\otimes(2m+2k-\ell)}}{|z|^{2m+2k-2\ell+n-1}}\quad\text{for all
$z\in\mathbb{R}^{n}$,}$

see [AKS22, (2.13)], therefore the normal operators extend to the mapping

$(N^{0},\cdots,N^{m}):\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m})\rightarrow(\mathcal{S}^{\prime}(\mathbb{R}^{n};S^{m}))^{m+1}$
(2.7)

as the convolution of a compactly supported distribution and a tempered
distribution. Since

$|\Xi_{p_{1}\cdots p_{2k-\ell}i_{1}\cdots i_{m}j_{1}\dots
j_{m}}(z)|\leq\frac{1}{|z|^{n-1-\ell}}\quad\text{for all
$\ell\in\mathbb{Z}_{\geq 0}$,}$

then the convolution with $\Xi_{p_{1}\cdots p_{2k-\ell}i_{1}\cdots
i_{m}j_{1}\dots j_{m}}(z)$ is the mapping from $L^{p}(\mathbb{R}^{n})$ to
$L^{q}(\mathbb{R}^{n})$ provided $\frac{1}{q}=\frac{1}{p}-\frac{1+\ell}{n}$
when $0\leq\ell\leq n-2$, see (5.9). When $\ell=n-1$, it is trivial to see
that the convolution with $\Xi_{p_{1}\cdots p_{2k-\ell}i_{1}\cdots
i_{m}j_{1}\dots j_{m}}(z)$ is the mapping from $L^{1}(\mathbb{R}^{n})$ to
itself. Therefore the normal operators also extend to the mapping

$N^{k}:L^{1}(\mathbb{R}^{n};S^{m})\rightarrow\mathcal{D}^{\prime}(\mathbb{R}^{n};S^{m})\quad\text{for
all integer $0\leq k\leq n-1$.}$ (2.8)

When $m=k=0$, (2.6) becomes

$\left(N^{0}|_{S^{0}}f\right)=2f*\frac{1}{|\cdot|^{n-1}}\quad\text{for all
$f\in\mathcal{E}^{\prime}(\mathbb{R}^{n})\cup\mathcal{S}(\mathbb{R}^{n})$,}$
(2.9)

and for each integer $\ell\geq 0$ we know that

$N^{0}|_{S^{0}}:W^{\ell,p}(\mathbb{R}^{n})\rightarrow
W^{\ell,q}(\mathbb{R}^{n})\text{ is bounded provided
}\frac{1}{q}=\frac{1}{p}-\frac{1}{n},$ (2.10)

see (5.10).

###  2.4. Some differential operators

We now recall certain differential operators from [Sha94, Section 2.1]. The
_symmetrization with respect to a part of indices_ is formally given by

$\sigma(i_{1},\cdots,i_{p})u_{i_{1},\cdots,i_{m}}=\frac{1}{p!}\sum_{\pi\in\Pi_{p}}u_{i_{\pi(1)}\cdots
i_{\pi(p)}i_{p+1}\cdots i_{m}},$

where $\Pi_{p}$ is the set of $p$-permutations. Let
$\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m})$ be the symmetric $m$ tensor
distribution fields in $\mathbb{R}^{n}$. The _inner derivative_ or
_symmetrized derivative_ is denoted as
$\mathrm{d}:\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m})\to\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m+1})$
given by

$(\mathrm{d}f)_{i_{1}\cdots i_{m+1}}=\sigma(i_{1}\cdots i_{m+1})\frac{\partial
f_{i_{1}\cdots i_{m}}}{\partial
x_{i_{m+1}}}\equiv\frac{1}{(m+1)!}\sum_{\pi\in\Pi_{m+1}}\frac{\partial
f_{i_{\pi(1)}\cdots i_{\pi(m)}}}{\partial x_{i_{\pi(m+1)}}}$ (2.11)

The _divergence_
$\delta:\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m})\to\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m-1})$
is defined by

$(\delta f)_{i_{1}\dots i_{m-1}}=\frac{\partial f_{i_{1}\dots i_{m}}}{\partial
x^{i_{m}}}.$ (2.12)

The operators $\mathrm{d}$ and $-\delta$ are formally dual to each other with
respect to $L^{2}$ inner product in the sense of

$(\mathrm{d}u,v)_{\mathbb{R}^{n};S^{m+1}}=-(u,\delta
v)_{\mathbb{R}^{n};S^{m}}$

for all $u\in C_{c}^{\infty}(\mathbb{R}^{n};S^{m})$ and $v\in
C_{c}^{\infty}(\mathbb{R}^{n};S^{m+1})$, see [Sha94, (2.1.8)]. It is also
interesting to mention that $\delta^{k+1}N^{k}f=0$ and for each integer $0\leq
r\leq k$ that

$\displaystyle\left(\delta^{r}N^{k}f\right)(x)$
$\displaystyle=\frac{k!}{(k-r)!}\int\limits_{\mathbb{S}^{n-1}}\langle
x,\xi\rangle^{k-r}\xi^{\otimes(m-r)}\left(I^{k}f\right)(x-\langle
x,\xi\rangle\xi,\xi)\,\mathrm{d}S_{\xi}$ (2.13)
$\displaystyle=\frac{k!}{(k-r)!}\sum\limits_{\ell=0}^{k}\binom{k}{\ell}\int\limits_{\mathbb{S}^{n-1}}\langle
x,\xi\rangle^{2k-r-\ell}\xi^{\otimes(m-r)}\left(J^{\ell}f\right)(x,\xi)\,\mathrm{d}S_{\xi},$

for all
$f\in\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m})\cup\mathcal{S}(\mathbb{R}^{n};S^{m})$,
see [AKS22, (2.14)]. Given any $m\in\mathbb{Z}_{\geq 0}$ and an integer $0\leq
k\leq m$, it is well-known that the momentum ray transforms (in Definition
2.1) satisfy the following property:

$I^{k}|_{S^{m}}(\mathrm{d}v)=-kI^{k-1}|_{S^{m-1}}(v)\quad\text{for all
$v\in\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m-1})\cup\mathcal{S}(\mathbb{R}^{n};S^{m-1})$,}$
(2.14)

where $\mathrm{d}$ is given in (2.11), see e.g. [KMSS19, Section 2]. By
utilizing (2.14), we can easily show that the similar scaling property also
holds for their corresponding normal operators.

######  Lemma 2.3.

Given any $m\in\mathbb{Z}_{\geq 0}$ and integer $0\leq k\leq m$, there holds

$\delta N^{k}|_{S^{m}}(\mathrm{d}v)=-k^{2}N^{k-1}|_{S^{m-1}}(v)\quad\text{for
all
$v\in\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m-1})\cup\mathcal{S}(\mathbb{R}^{n};S^{m-1})$.}$

###### Proof.

Choosing $f=\mathrm{d}v$ and $r=1$ in (2.13), using (2.14) one obtain

$\displaystyle\left(\delta N^{k}|_{S^{m}}\mathrm{d}v\right)(x)$
$\displaystyle=k\int\limits_{\mathbb{S}^{n-1}}\langle
x,\xi\rangle^{k-1}\xi^{\otimes(m-1)}\left(I^{k}|_{S^{m}}(\mathrm{d}v)\right)(x-\langle
x,\xi\rangle\xi,\xi)\,\mathrm{d}S_{\xi}$
$\displaystyle=-k^{2}\int\limits_{\mathbb{S}^{n-1}}\langle
x,\xi\rangle^{k-1}\xi^{\otimes(m-1)}\left(I^{k-1}|_{S^{m-1}}(v)\right)(x-\langle
x,\xi\rangle\xi,\xi)\,\mathrm{d}S_{\xi}$
$\displaystyle=-k^{2}\left(N^{k-1}|_{S^{m-1}}(v)\right)(x),$

which immediately implies our lemma. ∎

###  2.5. Generalized Saint-Venant operator

We now denote $S^{m_{1}}\times S^{m_{2}}$ the set of $(m_{1}+m_{2})$-tensors
symmetric with respect to the group first $m_{1}$ indices and last $m_{2}$
indices. Accordingly, we introduce the generlized Saint-Venent operator as in
[AKS22, (2.8)], which is a generalization of curl on $\mathbb{R}^{n}$, see
Remark 2.6.

######  Definition 2.4.

For $m\in\mathbb{Z}_{\geq 0}$ and integer $0\leq k\leq m$, the $k^{\rm th}$
_generalized Saint-Venant operator_
$W^{k}:\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m})\to\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m-k}\otimes
S^{m})$ is defined as

$\displaystyle(W^{k}f)_{p_{1}\cdots p_{m-k}q_{1}\cdots q_{m-k}}^{i_{1}\cdots
i_{k}}$ (2.15)
$\displaystyle\quad\coloneqq\sigma(p_{1},\cdots,p_{m-k})\sigma(q_{1},\cdots,q_{m-k},i_{1}\dots
i_{k})\times$
$\displaystyle\quad\qquad\times\sum_{\ell=0}^{m-k}(-1)^{\ell}\binom{m-k}{\ell}\frac{\partial^{m-k}f^{i_{1}\cdots
i_{k}}_{p_{1}\dots p_{m-k-\ell}q_{1}\cdots q_{\ell}}}{\partial
x^{p_{m-k-\ell+1}}\cdots\partial x^{p_{m-k}}\partial
x^{q_{\ell+1}}\cdots\partial x^{q_{m-k}}}.$

When $k=0$, the $0^{\rm th}$ generalized Saint-Venant operator $W^{0}$ reduces
to the (classical) Saint-Venent operator on $S^{m}$ [Sha94, (2.8)]. When
$k=m$, we have $W^{m}=\mathbb{I}$, where $\mathbb{I}$ is the identity
operator.

######  Remark 2.5 (An equivalent definition).

For each $m\in\mathbb{Z}_{\geq 0}$ and integer $0\leq k\leq m$, we consider
the operator $R^{k}\equiv
R^{k}|_{S^{m}}:\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m})\rightarrow\mathscr{D}^{\prime}(\mathbb{R}^{n};T^{2m})$
given by

$(R^{k}f)_{p_{1}q_{1}\dots p_{m-k}q_{m-k}}^{i_{1}\dots
i_{k}}:=\alpha(p_{1}q_{1})\dots\alpha(p_{m-k}q_{m-k})\frac{\partial^{m-k}f^{i_{1}\dots
i_{k}}_{p_{1}\dots p_{m-k}}}{\partial x^{q_{1}}\dots\partial x^{q_{m-k}}}$
(2.16)

where the _alternation of two indices_ is defined as:

$\alpha(i_{1}i_{2})u_{i_{1}i_{2}j_{1}\cdots
j_{p}}:=\frac{1}{2}(u_{i_{1}i_{2}j_{1}\cdots j_{p}}-u_{i_{2}i_{1}j_{1}\cdots
j_{p}}).$

In particular, one has

$\left(R^{k}|_{S^{m}}f\right)_{p_{1}q_{1}\dots p_{m-k}q_{m-k}}^{i_{1}\dots
i_{k}}=\left(R^{0}|_{S^{m-k}}f^{i_{1}\cdots i_{k}}\right)_{p_{1}q_{1}\cdots
p_{m-k}q_{m-k}},$ (2.17)

see [AKS22, (2.6)]. For each $m\in\mathbb{Z}_{\geq 0}$ and integer $0\leq
k\leq m$, one can show that111We again remind readers about the typos in
[Sha94, (2.4.6)(2.4.7)], see [AKS22, (2.6)(2.7)] for corrected statement.

$\displaystyle(W^{k}f)_{p_{1}\cdots p_{m-k}q_{1}\cdots q_{m-k}}^{i_{1}\cdots
i_{k}}$ $\displaystyle\quad=2^{m-k}\sigma(q_{1}\cdots q_{m-k}i_{1}\cdots
i_{k})\sigma(p_{1}\cdots p_{m-k})(R^{k}f)_{p_{1}q_{1}\cdots
p_{m-k}q_{m-k}}^{i_{1}\cdots i_{k}},$

and

$\displaystyle(R^{k}f)_{p_{1}q_{1}\cdots p_{m-k}q_{m-k}}^{i_{1}\cdots i_{k}}$
$\displaystyle\quad=\frac{1}{m-k+1}\binom{m}{k}\alpha(p_{1}q_{1})\cdots\alpha(p_{m-k}q_{m-k})(W^{k}f)_{p_{1}\cdots
p_{m-k}q_{1}\cdots q_{m-k}}^{i_{1}\cdots i_{k}},$

see [AKS22, Lemma 4.1]. Hence, for each open set $U\subset\mathbb{R}^{n}$ and
$f\in\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m})$ there holds

$W^{k}f=0\text{ in $U$}\iff R^{k}f=0\text{ in $U$.}$ (2.18)

Based on the above observation, we can slightly abuse the terminology by also
referring (2.16) the $k^{\rm th}$ generalized Saint-Venant operator. In view
of the reduction formula (2.17), it is more convenient to work with $R^{k}$
rather than $W^{k}$.

######  Remark 2.6.

By identifying
$\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{1})\cong\left((\mathscr{D}^{\prime}(\mathbb{R}^{n}))\right)^{n}$,
when $m=1$ and $k=0$, we see that

$\left(R^{0}|_{S^{1}}f\right)_{p,q}=\alpha(p,q)\frac{\partial f_{p}}{\partial
x^{q}}=\frac{1}{2}\left(\frac{\partial f_{p}}{\partial x^{q}}-\frac{\partial
f_{q}}{\partial x^{p}}\right)\equiv\frac{1}{\sqrt{2}}\left({\rm
curl}\,(f)\right)_{p,q}$ (2.19)

for all
$f\in\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{1})\cong\left(\mathscr{D}^{\prime}(\mathbb{R}^{n})\right)^{n}$,
see also Remark 3.2 for more details about (2.19).

For $u\in S^{k}$, we denote by $\mathfrak{i}_{u}:S^{m}\rightarrow S^{m+k}$ the
operator of symmetric multiplication by $u$ and by
$\mathfrak{j}_{u}:S^{m+k}\rightarrow S^{m}$ the corresponding dual operator,
and defined as

$\displaystyle\left(\mathfrak{i}_{u}v\right)_{i_{1}\cdots i_{m+k}}$
$\displaystyle:=\sigma(i_{1}\cdots i_{m+k})u_{i_{1}\cdots
i_{k}}v_{i_{k+1}\cdots i_{k+m}}\quad\text{for all $v\in S^{m}$},$
$\displaystyle\left(\mathfrak{j}_{u}w\right)_{i_{1}\cdots i_{m+k}}$
$\displaystyle:=w_{i_{1}\cdots i_{m+k}}u^{i_{m+1}\cdots i_{m+k}}\quad\text{for
all $w\in S^{m+k}$,}$

see [Sha94, (2.1.5)]. Let $\mathfrak{e}^{k}\in S^{k}$ be the Euclidean metric
tensor, given by

$\mathfrak{e}_{i_{1}\cdots i_{k}}^{k}=\begin{cases}1&\text{if
$i_{1}=\cdots=i_{k}$,}\\\ 0&\text{otherwise,}\end{cases}$

and we write
$\mathfrak{i}_{(k)}:=\mathfrak{i}_{\mathfrak{e}^{k}},\quad\mathfrak{j}_{(k)}:=\mathfrak{j}_{\mathfrak{e}^{k}}$.
We end this section by recalling [AKS22, Proposition 4.4], which gives a
connection between normal operator of momentum ray transform and generalized
Saint-Venant operator.

######  Proposition 2.7.

[AKS22, Proposition 4.4] Given $m\in\mathbb{Z}_{\geq 0}$ and integer $0\leq
k\leq m$, we consider
$f\in\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m})\cup\mathcal{S}(\mathbb{R}^{n};S^{m})$.
If $N^{0}|_{S^{0}}$ be the operator given in (2.9), then there holds

$\begin{split}&m!N^{0}|_{S^{0}}\left((R^{0}|_{S^{m-k}}f^{i_{1}\cdots
i_{k}})_{p_{1}q_{1}\cdots p_{m-k}q_{m-k}}\right)\\\ &\quad=\sigma(i_{1}\dots
i_{k})\sum\limits_{r=0}^{k}(-1)^{r}\binom{k}{r}\frac{\partial^{r}}{\partial
x^{i_{1}}\cdots\partial
x^{i_{r}}}(R^{k}|_{S^{m-r}}\left(G_{m-r})\right)_{p_{1}q_{1}\cdots
p_{m-k}q_{m-k}}^{i_{r+1}\cdots i_{k}},\end{split}$ (2.20)

where $G_{m-r}$ is a symmetric $(m-r)$-tensor given by

$\displaystyle
G_{m-r}=\sum\limits_{\ell=0}^{\lfloor\frac{m-r}{2}\rfloor}c_{\ell,m-r}\mathfrak{i}_{(2)}^{\ell}\mathfrak{j}_{(2)}^{\ell}\sum\limits_{p=0}^{r}(-1)^{r-p}\frac{1}{p!}\binom{r}{p}\mathfrak{j}_{x^{\otimes(r-p)}}\delta^{p}N^{p}f,$

with coefficients

$c_{\ell,s}=\left(\prod_{p=0}^{s-\ell-1}(n-1+2p)\right)\frac{(-1)^{\ell}s!}{2^{\ell}\ell!(s-2\ell)!},$

where $N^{p}$ is the normal operator of the $k^{\rm th}$ in momentum ray
transform given in (2.6) and for each $\alpha\in\mathbb{R}$ the “floor”
$\lfloor\alpha\rfloor$ denotes the largest integer with $\leq\alpha$.

######  Remark 2.8.

In particular when $m=1$ and $k=0$, (LABEL:key_equation_mrt) reads
$N^{0}|_{S^{0}}\left(\left(R^{0}|_{S^{1}}f\right)_{ij}\right)=(n-1)\left(R^{0}|_{S^{1}}\left(N^{0}|_{S^{1}}f\right)\right)_{ij}\,\,\,\text{
$\forall\,\,f\in\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{1})\cup\mathcal{S}(\mathbb{R}^{n};S^{1})$.}$
(2.21a) Plugging (2.19) into (2.21a), we conclude
$N^{0}|_{S^{0}}\left(\left({\rm curl}\,(f)\right)_{ij}\right)=(n-1)\left({\rm
curl}\,\left(N^{0}|_{S^{1}}f\right)\right)_{ij}\quad\text{
$\forall\,\,f\in\left(\mathcal{E}^{\prime}(\mathbb{R}^{n})\right)^{n}\cup\left(\mathcal{S}(\mathbb{R}^{n})\right)^{n}$.}$
(2.21b)

##  3\. A generalization of the curl-curl identity

We shall prove the following useful property of generalized Saint-Venant
operator (Definition 2.4), which is new based on our knowledge. As an
application, we also prove a smoothing property of generalized Saint-Venant
operator in Lemma 3.3.

######  Proposition 3.1.

Given any integer $m\geq 0$, the identity

$\displaystyle\frac{\partial^{\ell}}{\partial x^{j_{1}}\cdots\partial
x^{j_{\ell}}}\left(R^{0}|_{S^{m}}f\right)_{i_{1}j_{1}\cdots
i_{\ell}j_{\ell}i_{\ell+1}j_{\ell+1}\cdots i_{m}j_{m}}$ (3.1)
$\displaystyle\quad=\frac{1}{2^{\ell}}\sigma(i_{1}\cdots
i_{\ell})\sum\limits_{p=0}^{\ell}\,\binom{\ell}{p}(-1)^{p}\frac{\partial^{p}}{\partial
x^{i_{1}}\cdots\partial
x^{i_{p}}}\Delta^{\ell-p}\left(R^{0}|_{S^{m-\ell}}\left((\delta^{p}f)^{i_{p+1}\cdots
i_{\ell}}\right)\right)_{i_{\ell+1}j_{\ell+1}\cdots i_{m}j_{m}}$

holds true for all $f\in\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m})$ and
integer $0\leq\ell\leq m$.

######  Remark 3.2.

When $m=\ell=1$, (3.7) reduces to the following well-known curl-curl identity

$\Delta f^{i}-\frac{\partial}{\partial x^{i}}(\delta
f)=2\frac{\partial}{\partial
x^{j}}(R^{0}|_{S^{1}}f)_{i,j}=\frac{\partial}{\partial
x^{j}}\left(\frac{\partial f_{i}}{\partial x^{j}}-\frac{\partial
f_{j}}{\partial x^{i}}\right)=-\left({\rm curl}^{\intercal}\,{\rm
curl}\,(f)\right)_{i},$ (3.2)

where $\left({\rm
curl}^{\intercal}\,(g)\right)_{i}:=\frac{1}{\sqrt{2}}\frac{\partial}{\partial
x^{j}}(g_{i,j}-g_{j,i})$ is the _formal transpose of curl_. In 3-dimensional
case, one even can reduce (3.2) to

$\Delta f-\nabla(\nabla\cdot f)=-\nabla\times(\nabla\times f)\quad\text{for
all $f\in\left(\mathscr{D}^{\prime}(\mathbb{R}^{3})\right)^{3}$,}$ (3.3)

where
$\nabla\times\cdot:\mathscr{D}^{\prime}(\mathbb{R}^{3})\rightarrow\mathscr{D}^{\prime}(\mathbb{R}^{3})$
is the usual curl on $\mathbb{R}^{3}$.

###### Proof of Proposition 3.1.

The identity (3.1) is trivial when $\ell=0$, we only need to prove (3.1) for
$\ell\geq 1$.

Step 1: Basic case. We first prove (3.1) when $\ell=1$. Given any integer
$m\geq 1$, by applying the divergence operator $\delta$ on the classical
Saint-Venant operator
$R^{0}:\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m})\rightarrow\mathscr{D}^{\prime}(\mathbb{R}^{n};T^{2m})$,
we obtain

$\displaystyle 2\frac{\partial}{\partial
x^{j_{1}}}\left(R^{0}|_{S^{m}}f\right)_{i_{1}j_{1}i_{2}j_{2}\cdots
i_{m}j_{m}}$ (3.4) $\displaystyle\quad=2\frac{\partial}{\partial
x^{j_{1}}}\left(\alpha(i_{1}j_{1})\cdots\alpha(i_{m}j_{m})\frac{\partial^{m}}{\partial
x^{j_{1}}\cdots\partial x^{j_{m}}}f_{i_{1}i_{2}\cdots i_{m}}\right)$
$\displaystyle\quad=\Delta\alpha(i_{2}j_{2})\cdots\alpha(i_{m}j_{m})\frac{\partial^{m-1}}{\partial
x^{j_{2}}\cdots\partial x^{j_{m}}}f_{i_{2}\cdots i_{m}}^{i_{1}}$
$\displaystyle\quad\qquad-\frac{\partial}{\partial
x^{i_{1}}}\alpha(i_{2}j_{2})\cdots\alpha(i_{m}j_{m})\frac{\partial^{m-1}}{\partial
x^{j_{2}}\cdots\partial x^{j_{m}}}(\delta f)_{i_{2}\cdots i_{m}}$
$\displaystyle\quad=\left(\Delta
R^{0}|_{S^{m-1}}f^{i_{1}}-\frac{\partial}{\partial
x^{i_{1}}}R^{0}|_{S^{m-1}}(\delta f)\right)_{i_{2}j_{2}\cdots i_{m}j_{m}}$

for all $f\in\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m})$.

Step 2: Induction on $\ell$. Assume that there exists an integer $\ell\geq 1$
such that (3.1) holds true for all $m\geq\ell$. Taking the derivative
$\frac{\partial}{\partial x^{j_{\ell+1}}}$ on the induction hypothesis gives

$\displaystyle\qquad 2^{\ell}\frac{\partial^{\ell+1}}{\partial
x^{j_{1}}\cdots\partial
x^{j_{\ell+1}}}\left(R^{0}|_{S^{m}}f\right)_{i_{1}j_{1}\cdots
i_{\ell+1}j_{\ell+1}i_{\ell+2}j_{\ell+2}\cdots i_{m}j_{m}}$ (3.5)
$\displaystyle=\sigma(i_{1}\cdots
i_{\ell+1})2^{\ell}\frac{\partial^{\ell+1}}{\partial x^{j_{1}}\cdots\partial
x^{j_{\ell+1}}}\left(R^{0}|_{S^{m}}f\right)_{i_{1}j_{1}\cdots
i_{\ell+1}j_{\ell+1}i_{\ell+2}j_{\ell+2}\cdots i_{m}j_{m}}$
$\displaystyle=\sigma(i_{1}\cdots
i_{\ell+1})\sum\limits_{p=0}^{\ell}\,\binom{\ell}{p}(-1)^{p}\frac{\partial^{p}}{\partial
x^{i_{1}}\cdots\partial x^{i_{p}}}\Delta^{\ell-p}\frac{\partial}{\partial
x^{j_{\ell+1}}}\left(R^{0}|_{S^{m-\ell}}\left((\delta^{p}f)^{i_{p+1}\cdots
i_{\ell}}\right)\right)_{i_{\ell+1}j_{\ell+1}\cdots i_{m}j_{m}},$

here we utilize the fact that $\sigma(i_{1}\cdots
i_{\ell+1})\sigma(i_{1}\cdots i_{\ell})=\sigma(i_{1}\cdots i_{\ell+1})$.
Similar to (3.4), we have

$\displaystyle(\delta^{p+1}f)_{i_{\ell+2}\cdots i_{m}}^{i_{p+1}\cdots
i_{\ell}}=(\delta^{p+1}f)_{i_{\ell+2}\cdots i_{m}}^{i_{p+2}\cdots
i_{\ell+1}}\quad\text{(because $\delta^{p+1}f\in
C^{\infty}(\mathbb{R}^{n};S^{m-(p+1)})$),}$
$\displaystyle\frac{\partial}{\partial
x^{i_{\ell+1}}}R^{0}|_{S^{m-(\ell+1)}}\left(\delta^{p+1}f\right)^{i_{p+2}\cdots
i_{\ell+1}}=\frac{\partial}{\partial
x^{i_{p+1}}}R^{0}|_{S^{m-(\ell+1)}}\left(\delta^{p+1}f\right)^{i_{p+2}\cdots
i_{\ell+1}}.$

This implies

$\displaystyle 2\frac{\partial}{\partial
x^{j_{\ell+1}}}\left(R^{0}|_{S^{m-\ell}}\left((\delta^{p}f)^{i_{p+1}\cdots
i_{\ell}}\right)\right)_{i_{\ell+1}j_{\ell+1}\cdots i_{m}j_{m}}$ (3.6)
$\displaystyle\quad=2\frac{\partial}{\partial
x^{j_{\ell+1}}}\alpha(i_{\ell+1}j_{\ell+1})\cdots\alpha(i_{m}j_{m})\frac{\partial^{m-\ell}}{\partial
x^{j_{\ell+1}}\cdots\partial
x^{j_{m}}}\left(\delta^{p}f\right)_{i_{\ell+1}\cdots i_{m}}^{i_{p+1}\cdots
i_{\ell}}$
$\displaystyle\quad=\Delta\alpha(i_{\ell+2}j_{\ell+2})\cdots\alpha(i_{m}j_{m})\frac{\partial^{m-(\ell+1)}}{\partial
x^{j_{\ell+2}}\cdots\partial x^{j_{m}}}(\delta^{p}f)_{i_{\ell+2}\cdots
i_{m}}^{i_{p+1}\cdots i_{\ell}i_{\ell+1}}$
$\displaystyle\quad\qquad-\frac{\partial}{\partial
x^{i_{\ell+1}}}\alpha(i_{\ell+2}j_{\ell+2})\cdots\alpha(i_{m}j_{m})\frac{\partial^{m-(\ell+1)}}{\partial
x^{j_{\ell+2}}\cdots\partial x^{j_{m}}}(\delta^{p+1}f)_{i_{\ell+2}\cdots
i_{m}}^{i_{p+2}\cdots i_{\ell+1}}$ $\displaystyle\quad=\left(\Delta
R^{0}|_{S^{m-(\ell+1)}}\left(\delta^{p}f\right)^{i_{p+1}\cdots
i_{\ell+1}}-\frac{\partial}{\partial
x^{i_{p+1}}}R^{0}|_{S^{m-(\ell+1)}}\left(\delta^{p+1}f\right)^{i_{p+2}\cdots
i_{\ell+1}}\right)_{i_{\ell+2}j_{\ell+2}\cdots i_{m}j_{m}}.$

We now combine (3.5) and (3.6) to obtain

$\displaystyle\qquad 2^{\ell+1}\frac{\partial^{\ell+1}}{\partial
x^{j_{1}}\cdots\partial
x^{j_{\ell+1}}}\left(R^{0}|_{S^{m}}f\right)_{i_{1}j_{1}\cdots
i_{\ell+1}j_{\ell+1}i_{\ell+2}j_{\ell+2}\cdots i_{m}j_{m}}$
$\displaystyle=\sigma(i_{1}\cdots
i_{\ell+1})\sum_{p=0}^{\ell}\binom{\ell}{p}(-1)^{p}\frac{\partial^{p}}{\partial
x^{i_{1}}\cdots\partial
x^{i_{p}}}\Delta^{(\ell+1)-p}\left(R^{0}|_{S^{m-(\ell+1)}}\left(\delta^{p}f\right)^{i_{p+1}\cdots
i_{\ell+1}}\right)_{i_{\ell+2}j_{\ell+2}\cdots i_{m}j_{m}}$
$\displaystyle\quad-\sigma(i_{1}\cdots
i_{\ell+1})\sum_{p=0}^{\ell}\binom{\ell}{p}(-1)^{p}\frac{\partial^{p+1}}{\partial
x^{i_{1}}\cdots\partial
x^{i_{p+1}}}\Delta^{\ell-p}\left(R^{0}|_{S^{m-(\ell+1)}}\left(\delta^{p+1}f\right)^{i_{p+2}\cdots
i_{\ell+1}}\right)_{i_{\ell+2}j_{\ell+2}\cdots i_{m}j_{m}}$
$\displaystyle=\sigma(i_{1}\cdots
i_{\ell+1})\sum_{p=0}^{\ell}\binom{\ell}{p}(-1)^{p}\frac{\partial^{p}}{\partial
x^{i_{1}}\cdots\partial
x^{i_{p}}}\Delta^{(\ell+1)-p}\left(R^{0}|_{S^{m-(\ell+1)}}\left(\delta^{p}f\right)^{i_{p+1}\cdots
i_{\ell+1}}\right)_{i_{\ell+2}j_{\ell+2}\cdots i_{m}j_{m}}$
$\displaystyle\quad+\sigma(i_{1}\cdots
i_{\ell+1})\sum_{p=1}^{\ell+1}\binom{\ell}{p-1}(-1)^{p}\frac{\partial^{p}}{\partial
x^{i_{1}}\cdots\partial
x^{i_{p}}}\Delta^{(\ell+1)-p}\left(R^{0}|_{S^{m-(\ell+1)}}\left(\delta^{p}f\right)^{i_{p+1}\cdots
i_{\ell+1}}\right)_{i_{\ell+2}j_{\ell+2}\cdots i_{m}j_{m}}$
$\displaystyle=\sigma(i_{1}\cdots
i_{\ell+1})\sum_{p=0}^{\ell+1}\binom{\ell+1}{p}(-1)^{p}\frac{\partial^{p}}{\partial
x^{i_{1}}\cdots\partial
x^{i_{p}}}\Delta^{(\ell+1)-p}\left(R^{0}|_{S^{m-(\ell+1)}}\left(\delta^{p}f\right)^{i_{p+1}\cdots
i_{\ell+1}}\right)_{i_{\ell+2}j_{\ell+2}\cdots i_{m}j_{m}}.$

This finishes the induction step and completes the proof. ∎

When $\ell=m$, (3.1) in Proposition 3.1 gives

$\displaystyle\frac{\partial^{m}}{\partial x^{j_{1}}\cdots\partial
x^{j_{m}}}\left(R^{0}|_{S^{m}}f\right)_{i_{1}j_{1}\cdots i_{m}j_{m}}$ (3.7)
$\displaystyle\quad=\frac{1}{2^{m}}\sigma(i_{1}\cdots
i_{m})\sum\limits_{p=0}^{m}\,\binom{m}{p}(-1)^{p}\frac{\partial^{p}}{\partial
x^{i_{1}}\cdots\partial
x^{i_{p}}}\Delta^{m-p}\left((\delta^{p}f)^{i_{p+1}\cdots i_{m}}\right).$

As an application of (3.7), we now able to prove the following smoothing
property of generalized Saint-Venant operator (Definition 2.4), which is also
true by replacing $R^{k}$ with $W^{k}$ due to (2.18).

######  Lemma 3.3.

Given $m\in\mathbb{Z}_{\geq 0}$, let $U$ be an open set in $\mathbb{R}^{n}$
and $g\in\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m})$. If there exists an
integer $0\leq\mathsf{k}\leq m$ such that

$R^{\mathsf{k}}g=0$ and $\delta^{\mathsf{k}+1}g=0$ in $U$, (3.8)

then $g\in C^{\infty}(U;S^{m})$. In addition when
$g\in\mathscr{E}^{\prime}(\mathbb{R}^{n};S^{m})$ and (3.8) holds for
$U=\mathbb{R}^{n}$, then we conclude that $g\equiv 0$ in $\mathbb{R}^{n}$.

###### Proof.

From (2.16), we have

$\displaystyle(R^{k-1}|_{S^{m}}f)_{p_{1}q_{1}\dots
p_{m-k+1}q_{m-k+1}}^{i_{1}\dots i_{k-1}}$ (3.9)
$\displaystyle\quad=\alpha(p_{m-k+1}q_{m-k+1})\frac{\partial}{\partial
x^{q_{m-k+1}}}\left(\alpha(p_{1}q_{1})\dots\alpha(p_{m-k}q_{m-k})\frac{\partial^{m-k}f_{p_{1}\dots
p_{m-k+1}}^{i_{1}\dots i_{k-1}}}{\partial x^{q_{1}}\dots\partial
x^{q_{m-k}}}\right)$
$\displaystyle\quad=\alpha(p_{m-k+1}q_{m-k+1})\frac{\partial}{\partial
x^{q_{m-k+1}}}\left(\alpha(p_{1}q_{1})\dots\alpha(p_{m-k}q_{m-k})\frac{\partial^{m-k}f_{p_{1}\dots
p_{m-k}}^{i_{1}\dots i_{k-1}p_{m-k+1}}}{\partial x^{q_{1}}\dots\partial
x^{q_{m-k}}}\right)$
$\displaystyle\quad=\alpha(p_{m-k+1}q_{m-k+1})\frac{\partial}{\partial
x^{q_{m-k+1}}}(R^{k}|_{S^{m}}f)_{p_{1}q_{1}\cdots p_{m-k}q_{m-k}}^{i_{1}\cdots
i_{k}p_{m-k+1}}$

for all $1\leq k\leq m$ and $f\in\mathscr{D}^{\prime}(\mathbb{R}^{n};S^{m})$.
Therefore the first assumption in (3.8) and (2.17) implies

$\left(R^{0}|_{S^{m-k}}g^{i_{1}\cdots i_{k}}\right)_{p_{1}q_{1}\cdots
p_{m-k}q_{m-k}}=0\quad\text{in $U$}\quad\text{for all $0\leq
k\leq\mathsf{k}$,}$

therefore we have

$\left(R^{0}|_{S^{m-k}}(\delta^{k}g)\right)_{p_{1}q_{1}\cdots
p_{m-k}q_{m-k}}=0\quad\text{in $U$}\quad\text{for all $0\leq
k\leq\mathsf{k}$.}$ (3.10)

Choosing $f=\delta^{k}g$ in (3.7), from (3.10) and second assumption in (3.8),
we see that

$\displaystyle 0$ $\displaystyle=2^{m}\frac{\partial^{m}}{\partial
x^{j_{1}}\cdots\partial
x^{j_{m}}}\left(R^{0}|_{S^{m-k}}(\delta^{k}g)\right)_{i_{1}j_{1}\cdots
i_{m}j_{m}}$ (3.11) $\displaystyle=\sigma(i_{1}\cdots
i_{m})\sum_{p=0}^{\mathsf{k}-k}\binom{m}{p}(-1)^{p}\frac{\partial^{p}}{\partial
x^{i_{1}}\cdots\partial
x^{i_{p}}}\Delta^{m-p}\left((\delta^{p+k}g)^{i_{p+1}\cdots
i_{m}}\right)\quad\text{in $U$.}$ By choosing $k=\mathsf{k}$ in (3.11), one
see that $\Delta^{m}\left((\delta^{\mathsf{k}}g)^{i_{1}\cdots i_{m}}\right)=0$
in $U$. Therefore, by local elliptic regularity, one know that
$\delta^{\mathsf{k}}g\in C^{\infty}(U;S^{m-\mathsf{k}}).$ (3.12a) In addition,
when $g\in\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m})$ and $U=\mathbb{R}^{n}$,
we know that $\delta^{\mathsf{k}}g\equiv 0\quad\text{in $\mathbb{R}^{n}$}.$
(3.12b) On the other hand, for each $0\leq k<\mathsf{k}$, we can write (3.11)
as $\displaystyle\Delta^{m}\left((\delta^{k}g)\right)^{i_{1}\cdots
i_{m}}=\sigma(i_{1}\cdots
i_{m})\Delta^{m}\left((\delta^{k}g)\right)^{i_{1}\cdots i_{m}}$
$\displaystyle\quad=-\sigma(i_{1}\cdots
i_{m})\sum_{p=1}^{\mathsf{k}-k}\binom{m}{p}(-1)^{p}\frac{\partial^{p}}{\partial
x^{i_{1}}\cdots\partial
x^{i_{p}}}\Delta^{m-p}\left((\delta^{p+k}g)^{i_{p+1}\cdots i_{m}}\right)$
$\displaystyle\quad=\sigma(i_{1}\cdots
i_{m})\sum_{p=0}^{\mathsf{k}-k-1}\binom{m}{p+1}(-1)^{p}\frac{\partial^{p+1}}{\partial
x^{i_{1}}\cdots\partial
x^{i_{p+1}}}\Delta^{m-p-1}\left((\delta^{p+k+1}g)^{i_{p+2}\cdots
i_{m}}\right)$ in $U$. Therefore, for each $0\leq k<\mathsf{k}$ we have the
implication $\delta^{k+1}g\in
C^{\infty}(U;S^{m-k-1})\quad\text{implies}\quad\delta^{k}g\in
C^{\infty}(U;S^{m-k}).$ (3.13a) In addition, when
$g\in\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m})$ and $U=\mathbb{R}^{n}$, for
each $0\leq k<\mathsf{k}$ we have the implication $\delta^{k+1}g\equiv 0\text{
in $\mathbb{R}^{n}$}\quad\text{implies}\quad\delta^{k}g\equiv 0\text{ in
$\mathbb{R}^{n}$.}$ (3.13b)

Combining (3.12a) and (3.13a) for the general case, and combining (3.12b) and
(3.13b) for the case when $g\in\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m})$ and
$U=\mathbb{R}^{n}$, we conclude our result. ∎

##  4\. A solenoidal decomposition theorem

The main theme of this section is to prove a generalized solenoidal potential
decomposition theorem, which is also new according to our best knowledge.

######  Proposition 4.1.

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^{n}$. For each $f\in
H^{\alpha}(\Omega;S^{m})$ with $\alpha\in\mathbb{Z}_{\geq 0}$, there exists a
unique decomposition

$f=\tilde{f}+\mathrm{d}^{k}v\quad\text{in $\Omega$}$ (4.1)

with $\tilde{f}\in H^{\alpha}(\Omega;S^{m})$ and $v\in
H^{\alpha+k}(\Omega;S^{m-k})\cap H_{0}^{k}(\Omega;S^{m-k})$.

We shall borrow some ideas from [SS23, Lemma 7.1] (see also [DS11, Theorem
1.5] and [Sha94, Theorem 3.3.2]) to proof Proposition 4.1. Formally acting
$\delta^{k}$ on (4.1), one reach the equation
$\delta^{k}f=\delta^{k}\mathrm{d}^{k}v$. This suggests us to prove the
following lemma.

######  Lemma 4.2.

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^{n}\,(n\geq 2)$, and
given $k\in\mathbb{N}$ and $m\in\mathbb{N}$. Given any $h\in
H^{\ell}(\Omega;S^{m})$ with integer $\ell\geq-k$, there exists a unique $w\in
H^{2k+\ell}(\Omega;S^{m})\cap H_{0}^{k}(\Omega;S^{m})$ such that

$(-1)^{k}\delta^{k}\mathrm{d}^{k}w=h\quad\text{in $\Omega$.}$ (4.2)

###### Proof.

Since the principal symbol of $\delta^{k}$ (resp. $\mathrm{d}^{k}$) is
$\mathbf{i}^{k}\mathfrak{j}_{\xi^{\otimes k}}$ (resp.
$\mathbf{i}^{k}\mathfrak{i}_{\xi^{\otimes k}}$), then the principal symbol of
$(-1)^{k}\delta^{k}\mathrm{d}^{k}$ is $\mathfrak{j}_{\xi^{\otimes
k}}\mathfrak{i}_{\xi^{\otimes k}}$. By using [Sha94, Lemma 3.3.3], one have

$\mathfrak{j}_{\xi}\mathfrak{i}_{\xi}=\frac{1}{m+1}|\xi|^{2}\mathbb{I}+\frac{m}{m+1}\mathfrak{i}_{\xi}\mathfrak{j}_{\xi}>0\quad\text{in
$S^{m}$},$

where $\mathbb{I}$ is the identity operator. Since
$\mathfrak{j}_{\xi^{\otimes(\ell+1)}}\mathfrak{i}_{\xi^{\otimes(\ell+1)}}=\mathfrak{j}_{\xi}(\mathfrak{j}_{\xi^{\otimes\ell}}\mathfrak{i}_{\xi^{\otimes\ell}})\mathfrak{i}_{\xi}$
for all $\ell\in\mathbb{N}$, then we know that $\mathfrak{j}_{\xi^{\otimes
k}}\mathfrak{i}_{\xi^{\otimes k}}>0$ in $S^{m}$. Therefore from [Tay11,
Exercises 5.11.3 and 5.11.4], we know that the mapping

$(-1)^{k}\delta^{k}\mathrm{d}^{k}:H^{2k+\ell}(\Omega)\cap
H_{0}^{k}(\Omega)\rightarrow H^{\ell}(\Omega)\text{ is Fredholm of index
zero.}$

In view of Fredholm theory, it remains to show the solution of (4.2) is
unique. In particular, if $w\in H_{0}^{k}(\Omega;S^{m})$ satisfies
$\delta^{k}\mathrm{d}^{k}w=0$, integration by parts yields

$\langle\delta^{k}\mathrm{d}^{k}w,w\rangle=\langle\mathrm{d}^{k}w,\mathrm{d}^{k}w\rangle=0,$

which implies $\mathrm{d}^{k}w=0$. Finally, repeating using [DS11, Theorem
1.3] as in the proof of [SS23, Lemma 7.1], we conclude our lemma. ∎

We are now ready to prove the main result of this section.

###### Proof of Proposition 4.1.

By choosing $h=(-1)^{k}\delta^{k}f\in H^{\alpha-k}(\Omega;S^{m-k})$ in Lemma
4.2, there exists a unique $v\in H^{\alpha+k}(\Omega;S^{m-k})\cap
H_{0}^{k}(\Omega;S^{m-k})$ such that

$\delta^{k}\mathrm{d}^{k}v=\delta^{k}f.$

Clearly, if we define $\tilde{f}:=f-\mathrm{d}^{k}v$, then we have
$\delta^{k}\tilde{f}=0$. The uniqueness of the decomposition also easily
follows from the uniqueness of solution in Lemma 4.2. ∎

##  5\. Main results

###  5.1. Unique continuation property for momentum ray transform on tensors

We say that a tensor $g\in C^{\infty}(\mathbb{R}^{n};S^{m})$ _vanishes at
$x_{0}$ of infinite derivative order_ if

$\left.\frac{\partial^{\alpha}}{\partial x^{\alpha}}g_{i_{1}\cdots
i_{m}}\right|_{x=x_{0}}=0\quad\text{for all multi-index $\alpha$.}$

In this section, we generalize [AKS22, Theorem 2.4] in the following theorem.
See I

In particular the assumption (1.3) makes sense by the following lemma.

######  Lemma 5.1.

Let $m\in\mathbb{N}$ and an integer $0\leq k\leq m$. Suppose that $f$
satisfies (1.2). If there exists an open set $U$ in $\mathbb{R}^{n}$ such that
$R^{k}f=0$ in $U$, then $N^{p}f$ is smooth in $U$ for all $0\leq p\leq k$.

###### Proof.

Fix $x_{0}\in U$, and let $B$ be a ball in $U$ containing $x_{0}$. Using
Proposition 4.1 (with $\alpha=0$), one can decompose $f\in L^{2}(B)$ in $B$ as

$f=\tilde{f}+\mathrm{d}^{k+1}v\quad\text{in $B$}$

where $\tilde{f}\in L^{2}(B;S^{m})$ satisfying $\delta^{k+1}\tilde{f}=0$ in
$B$ and $v\in H_{0}^{k+1}(B;S^{m-k-1})$. Note that the zero extension
$v\chi_{B}$ of $v$ is in $H^{k+1}(\mathbb{R}^{n};S^{m-k-1})$ and satisfies

$f=\tilde{f}+\mathrm{d}^{k+1}(v\chi_{B})\quad\text{in $\mathbb{R}^{n}$},$
(5.1)

by extending $\tilde{f}=f$ outside $B$. By using [Sha94, Theorem 2.17.2] (with
$F=\mathrm{d}^{k+1}(v\chi_{B})\in\mathcal{E}^{\prime}(\mathbb{R}^{n};S^{m})$)
and the equivalence in Remark 2.5, we have
$R^{k}(\mathrm{d}^{k+1}(v\chi_{B}))=0$. Therefore, acting $R^{k}$ on (5.1) one
see that

$R^{k}f=R^{k}\tilde{f}\quad\text{in $\mathbb{R}^{n}$,}$ (5.2)

therefore we conclude that $R^{k}\tilde{f}|_{B}=0$. Since
$\delta^{k+1}\tilde{f}=0$ in $B$, Lemma 3.3 guarantees that $\tilde{f}$ is
smooth in $B$. Since (1.2), we know that $N^{p}f$ is well-defined by (2.8)
(when $0\leq k\leq n-2$) and (2.7) (when $n-1\leq k\leq m$). By using (2.14),
for each $0\leq p\leq k$, acting $N^{p}$ on (5.1) one see that

$N^{p}f=N^{p}\tilde{f}\quad\text{in $\mathbb{R}^{n}$,}$ (5.3)

which implies that $N^{p}f$ is smooth in $B$, and hence smooth near $x_{0}$.
By arbitrariness of $x_{0}\in U$, we conclude our lemma. ∎

With Proposition 2.7 and a unique continuation principal of ray transform of
scalar functions in [IM20, Theorem 1.1] at hand, we now able to prove Theorem
I.

###### Proof of Theorem I.

Since $N^{p}f$ vanishes at $x_{0}$ of infinite derivative order, by using
Proposition 2.7 we know that
$N^{0}|_{S^{0}}\left((R^{0}|_{S^{m-k}}f^{i_{1}\dots i_{k}})_{p_{1}q_{1}\dots
p_{m-k}q_{m-k}}\right)$ vanishes at $x_{0}$ of infinite derivative order as
well. Since $R^{0}|_{S^{m-k}}(f^{i_{1}\cdots i_{k}})=R^{k}|_{S^{m}}f=0$ in
$U$, using [IM20, Theorem 1.1] and the equivalence (2.18) we reach
$R^{k}f=W^{k}f=0$ in $\mathbb{R}^{n}$. Then our result follows from [Sha94,
Theorem 2.17.2]. ∎

######  Remark 5.2.

Following the ideas of [IM22], it is possible to improve the Theorem I by
replacing the assumption $R^{k}f=0$ in $U$ with $P(D)R^{k}f=0$ in $U$, where
$P(D)$ is a constant coefficient differential operator.

###  5.2. Measurable unique continuation property of momentum ray transform

We first recall the following unique continuation property from positive
measure set (MUCP) for fractional Laplace operator from [GRSU20] (or [GFR19]).

######  Lemma 5.3.

[GRSU20, Proposition 5.1]. Let $n\geq 1$ and $\Omega$ be an open set in
$\mathbb{R}^{n}$. Let $q\in L^{\infty}(\Omega)$ and assume that $u\in
H^{s}(\mathbb{R}^{n}))$ with $s\in[\frac{1}{4},1)$ satisfies

$\displaystyle((-\Delta)^{s}+q)u=0\quad\mbox{in}\quad\Omega.$

If there exists a measurable set $E\subset\Omega$ with positive measure such
that $u=0$ in $E$, then $u\equiv 0$ in $\mathbb{R}^{n}$.

######  Remark 5.4.

See also [GFR19, Theorem 4] for more general results. We also refer to Lemma
5.10 in the next section for antilocality property for fractional elliptic
operators, which implies the unique continuation property from nonempty open
sets.

By utilizing the above lemma, we can prove some MUCP results for ray
transforms. See II

###### Proof of Theorem II.

It is well-known that (see e.g. [Ilm21, Exercise 12.10]) there exists a
constant $c_{n}\neq 0$ such that

$(-\Delta)^{\frac{1}{2}}N^{0}f=c_{n}f\quad\text{in $\mathbb{R}^{n}$.}$ (5.4)

Now $f|_{U}=0$ entails that
$\left((-\Delta)^{1/2}\left(N^{0}f\right)\right)|_{U}=0$. Since
$\left.\left(N^{0}f\right)\right|_{E}=0$, by applying Lemma 5.3 (with
$s=\frac{1}{2}$, $u=N^{0}f$ and $q=0$), this implies that $N^{0}f=0$ in
$\mathbb{R}^{n}$, and our result follows from (5.4). ∎

By using (2.21b), we also can obtain an analogue for vector fields.

######  Theorem 5.5.

Suppose $U\subseteq\mathbb{R}^{n}$ be any non empty open set and $n\geq 2$.
Suppose that ${\rm curl}\,(N^{0}|_{S^{1}}f)\in
H^{\frac{1}{2}}(\mathbb{R}^{n})$ (see Section 2.2). If $\left({\rm
curl}\,(f)\right)|_{U}=0$ and there exists a positive measure set $E$ in $U$
such that $\left.{\rm curl}\,(N^{0}|_{S^{1}}f)\right|_{E}=0$, then ${\rm
curl}\,(f)\equiv 0$ in $\mathbb{R}^{n}$.

######  Remark 5.6.

We now consider the case when $n=2,3$. Let $U$ be an open set in
$\mathbb{R}^{n}$ which is star-shaped with respect to some $x_{0}\in U$. If
$f\in(C^{1}(U))^{n}$ and ${\rm curl}\,(f)=0$ in $U$, then it is well-known
that there exists a potential $p\in C^{2}(U)$ such that $f=\nabla p$ in $U$,
which is an consequence of Poincaré lemma for de Rham cohomology groups. For
any dimension $n\geq 2$, one can use [Sha94, Theorem 2.17.2] to conclude the
same result.

###### Proof of Theorem 5.5.

Combining (2.21b) and (5.4), one see that there exists a constant
$c_{n}^{\prime}\neq 0$ such that

$(-\Delta)^{\frac{1}{2}}\left({\rm
curl}\,(N^{0}|_{S^{1}}f)\right)=c_{n}^{\prime}{\rm curl}\,(f)\quad\text{in
$\mathbb{R}^{n}$.}$ (5.5)

Now $\left.{\rm curl}\,(f)\right|_{U}=0$ entails that
$\left.(-\Delta)^{\frac{1}{2}}\left({\rm
curl}\,(N^{0}|_{S^{1}}f)\right)\right|_{U}=0$. Since $\left.{\rm
curl}\,(N^{0}|_{S^{1}}f)\right|_{E}=0$, by applying Lemma 5.3 (with
$s=\frac{1}{2}$, $u={\rm curl}\,(N^{0}|_{S^{1}}f)$ and $q=0$), this implies
that ${\rm curl}\,(N^{0}|_{S^{1}}f)=0$ in $\mathbb{R}^{n}$. Combining this
with (5.5), we conclude our theorem. ∎

###  5.3. Generalization of momentum ray transform: Fractional momentum ray
transform

The main theme of this section is to generalize the momentum ray transform to
a more general weighted ray transform, and explain the connection with
fractional elliptic operators.

Let $n\geq 2$ be an integer. For each real number $0<s<\frac{n}{2}$, we
consider the fractional momentum ray transform
$\mathcal{X}_{s}:\mathcal{S}(\mathbb{R}^{n})\rightarrow
C^{\infty}(\mathbb{R}^{n}\times\mathbb{S}^{n-1})$ defined by

$\big{(}\mathcal{X}_{s}f\big{)}(x,\xi):=\big{(}I^{2s-1}f\big{)}(x,\xi)\equiv\int\limits_{0}^{\infty}\tau^{2s-1}f(x+\tau\xi)\,\mathrm{d}\tau\quad\text{for
all }f\in\mathcal{S}(\mathbb{R}^{n})$ (5.6)

and for all $(x,\xi)\in\mathbb{R}^{n}\times\mathbb{S}^{n-1}$. Rather than the
normal operator $N^{2s-1}$, here we alternative consider the average
$\mathcal{A}_{s}:\mathcal{S}(\mathbb{R}^{n})\rightarrow
C^{\infty}(\mathbb{R}^{n})$ over the sphere $\mathbb{S}^{n-1}$ is defined by

$\big{(}\mathcal{A}_{s}f\big{)}(x):=c(n,-s)\int\limits_{\mathbb{S}^{n-1}}\big{(}\mathcal{X}_{s}f\big{)}(x,\xi)\,\mathrm{d}S_{\xi},$

where $c(n,s):=\frac{2^{2s}\Gamma(\frac{n+2s}{2})}{\pi^{n/2}|\Gamma(-s)|}$. We
compute that

$\displaystyle(\mathcal{A}_{s}f)(x)$
$\displaystyle=c(n,-s)\int\limits_{\mathbb{S}^{n-1}}\int\limits_{0}^{\infty}\tau^{2s-1}f(x+\tau\xi)\,\mathrm{d}\tau\,\mathrm{d}S_{\xi}$
$\displaystyle=c(n,-s)\int\limits_{\mathbb{S}^{n-1}}\int\limits_{0}^{\infty}\tau^{2s-n}f(x+\tau\xi)\,\big{(}\tau^{n-1}\mathrm{d}\tau\,\mathrm{d}S_{\xi}\big{)}$
$\displaystyle=c(n,-s)\int\limits_{\mathbb{R}^{n}}|y|^{2s-n}f(x+y)\,\mathrm{d}y,$

In other words, we observe that
$\mathcal{A}_{s}:\mathcal{S}(\mathbb{R}^{n})\rightarrow
C^{\infty}(\mathbb{R}^{n})$ is simply the Riesz potential, and in particular
from [Sti19, Theorem 5] for each $f\in\mathcal{S}(\mathbb{R}^{n})$ we have

$(-\Delta)^{-s}f(x)=\frac{1}{\Gamma(s)}\int\limits_{0}^{\infty}e^{t\Delta}f(x)\,\frac{\mathrm{d}t}{t^{1-s}}=(\mathcal{A}_{s}f)(x)\quad\text{for
all $x\in\mathbb{R}^{n}$,}$ (5.7)

where $(-\Delta)^{-s}$ is the negative power of the Laplacian and
$\\{e^{t\Delta}\\}_{t\geq 0}$ is the classical heat diffusion semigroup, see
also [Kwa17, Definition 2.3]. This corresponds to the numerical identity

$\lambda^{-s}=\frac{1}{\Gamma(s)}\int\limits_{0}^{\infty}e^{-t\lambda}\,\frac{\mathrm{d}t}{t^{1-s}}\quad\text{for
all $\lambda>0$,}$ (5.8)

see e.g. [Kwa17, Section 2.7]. For each $p\in[1,\frac{n}{2s})$, by utilizing
weak Young’s inequality, one can show that

$\mathcal{A}_{s}:L^{p}(\mathbb{R}^{n})\rightarrow L^{q}(\mathbb{R}^{n})\text{
is bounded provided }\frac{1}{q}=\frac{1}{p}-\frac{2s}{n},$ (5.9)

see also e.g. [Kwa17, Section 2.7]. Using convolution theorem we observe that
$\partial^{\alpha}(\mathcal{A}_{s}f)=\mathcal{A}_{s}(\partial^{\alpha}f)$,
then in particular

$\mathcal{A}_{s}:W^{m,p}(\mathbb{R}^{n})\rightarrow
W^{m,q}(\mathbb{R}^{n})\text{ is bounded provided
}\frac{1}{q}=\frac{1}{p}-\frac{2s}{n},$ (5.10)

for all non-negative integer $m$. We are interested in the following
particular case:

######  Lemma 5.7.

Let $n\geq 2$ be an integer and let $0<s\leq\frac{n}{4}$, then for each non-
negative integer $m$ we know that

$\mathcal{A}_{s}:W^{m,\frac{2n}{n+4s}}(\mathbb{R}^{n})\rightarrow
W^{m,2}(\mathbb{R}^{n})\equiv H^{m}(\mathbb{R}^{n})\text{ is bounded,}$

and $(-\Delta)^{s}\mathcal{A}_{s}f=f$ for all $f\in
W^{m,\frac{2n}{n+4s}}(\mathbb{R}^{n})$. Here, $(-\Delta)^{s}$ is the usual
Fourier fractional Laplacian222See also e.g. [Sti19, Definition 1] for
definition of fractional Laplacian in some suitable distribution sense.

######  Remark 5.8 (Critical case).

When $s=\frac{n}{2}$, we define the weighted ray transform
$\mathcal{X}_{\frac{n}{2}}:\mathcal{S}(\mathbb{R}^{n})\rightarrow
C^{\infty}(\mathbb{R}^{n}\times\mathbb{S}^{n-1})$ by

$\big{(}\mathcal{X}_{\frac{n}{2}}f\big{)}(x,\xi):=\int_{0}^{\infty}(-2\log\tau-\gamma_{\rm
EM})f(x+\tau\xi)\,\mathrm{d}\tau\quad\text{for all
$(x,\xi)\in\mathbb{R}^{n}\times\mathbb{S}^{n-1}$,}$ (5.11)

where the Euler-Mascheroni constant $\gamma_{\rm EM}$ is given by $\gamma_{\rm
EM}:=-\int_{0}^{\infty}e^{-t}\log t\,\mathrm{d}t\approx 0.577215$. The the
average operator
$\mathcal{A}_{\frac{n}{2}}:\mathcal{S}(\mathbb{R}^{n})\rightarrow
C^{\infty}(\mathbb{R}^{n})$ is then given by

$\big{(}\mathcal{A}_{\frac{n}{2}}f\big{)}(x):=\frac{1}{\Gamma(\frac{n}{2})(4\pi)^{\frac{n}{2}}}\int\limits_{\mathbb{S}^{n-1}}\big{(}\mathcal{X}_{\frac{n}{2}}f\big{)}(x,\xi)\,\mathrm{d}S_{\xi}.$

We compute that

$\displaystyle(\mathcal{A}_{\frac{n}{2}}f)(x)$
$\displaystyle=\frac{1}{\Gamma(\frac{n}{2})(4\pi)^{\frac{n}{2}}}\int\limits_{\mathbb{S}^{n-1}}\int\limits_{0}^{\infty}(-2\log\tau-\gamma_{\rm
EM})f(x+\tau\xi)\,\mathrm{d}\tau\,\mathrm{d}S_{\xi}$
$\displaystyle=\frac{1}{\Gamma(\frac{n}{2})(4\pi)^{\frac{n}{2}}}\int\limits_{\mathbb{S}^{n-1}}\int\limits_{0}^{\infty}(-2\log\tau-\gamma_{\rm
EM})f(x+\tau\xi)\,\big{(}\tau^{n-1}\mathrm{d}\tau\,\mathrm{d}S_{\xi}\big{)}$
$\displaystyle=\frac{1}{\Gamma(\frac{n}{2})(4\pi)^{\frac{n}{2}}}\int\limits_{\mathbb{R}^{n}}(-2\log|y|-\gamma_{\rm
EM})f(x+y)\,\mathrm{d}y.$

Since $|\xi|^{-n}$ is not a tempered distribution, then it is interesting to
mention that [Sti19, Theorem 5] showed that

$((-\Delta)^{-\frac{n}{2}}f)(x)=(\mathcal{A}_{\frac{n}{2}}f)(x)\quad\text{for
all $x\in\mathbb{R}^{n}$}$

for all $f\in\mathcal{S}(\mathbb{R}^{n})$ with
$\int\limits_{\mathbb{R}^{n}}f(x)\,\mathrm{d}x=0$.

For $0<s<\frac{n}{2}$, we now express the weighted ray transform (5.6) in
terms of (Gauss-Weierstrass) heat kernel

$k_{t}(x,y):=\frac{1}{(4\pi t)^{\frac{n}{2}}}e^{-\frac{|x-y|^{2}}{4t}}.$

It is know that

$(e^{t\Delta}f)(x)=\int\limits_{\mathbb{R}^{n}}k_{t}(x,y)f(y)\,\mathrm{d}y\quad\text{for
all $x\in\mathbb{R}^{n}$}$

and $(e^{t\Delta}f)^{\wedge}(x)=e^{-t|\xi|^{2}}\hat{f}(\xi)$ for all
$f\in\mathcal{S}(\mathbb{R}^{n})$. In addition, from [Kwa17, Section 2.7] we
have

$\frac{1}{\Gamma(s)}\int\limits_{0}^{\infty}k_{t}(x,y)\,\frac{\mathrm{d}t}{t^{1-s}}=c(n,-s)|x-y|^{2s-n}.$

Accordingly, we observe that

$|x-y|^{2s-1}=\frac{1}{\Gamma(s)c(n,-s)}|x-y|^{n-1}\int\limits_{0}^{\infty}k_{t}(x,y)\,\frac{\mathrm{d}t}{t^{1-s}}\quad\mbox{for
all $x\neq y\in\mathbb{R}^{n}$.}$

Therefore the X-ray transform (5.6) and the corresponding averaging operator
can be rephrase as

$\displaystyle(\mathcal{X}_{s}f)(x,\xi)$
$\displaystyle=\frac{1}{\Gamma(s)c(n,-s)}\int\limits_{0}^{\infty}\tau^{n-1}\bigg{(}\int\limits_{0}^{\infty}k_{t}(x,x+\tau\xi)\,\frac{\mathrm{d}t}{t^{1-s}}\bigg{)}f(x+\tau\xi)\,\mathrm{d}\tau$
(5.12) $\displaystyle\left((-\Delta)^{-s}f\right)(x)$
$\displaystyle\equiv(\mathcal{A}_{s}f)(x)=\frac{1}{\Gamma(s)}\int_{\mathbb{R}^{n}}\bigg{(}\int\limits_{0}^{\infty}k_{t}(x,y)\,\frac{\mathrm{d}t}{t^{1-s}}\bigg{)}f(y)\,\mathrm{d}y$

for all $f\in\mathcal{S}(\mathbb{R}^{n})$. Based on the above observation, it
is possible to definine negative power of elliptic operators with some
suitable domain, see Appendix A for more details.

######  Remark 5.9.

We have defined fractional MRT in (5.6) by taking integration over half lines.
However one can define fractional MRT by taking integration over whole lines
in the following way:

$\displaystyle
J^{s}f(x,\xi):=\int_{\mathbb{R}}|t|^{s}\,f(x+t\xi)\,\mathrm{d}t\quad\mbox{for
$f\in\mathcal{S}(\mathbb{R}^{n})$ and $-1<s<1$}.$

Note that, the weight function we have considered here is not smooth at the
origin. But after taking the average over $\mathbb{S}^{n-1}$ we see that
$A_{s}=\int_{\mathbb{S}^{n-1}}J^{s}(x,\xi)\,\mathrm{d}S(\xi)$ is same as
$\mathcal{A}_{2s-1}$ upto a constant. Using this we can study the UCP of
$A_{s}$ as well as $J^{s}$.

###  5.4. Antilocality property of the weighted ray transform

Using [GFR19, Proposition 1.9] (which is valid for some class of general
elliptic operators) and the smoothing argument as in the proof of [GSU20,
Theorem 1.2], we have the following lemma.

######  Lemma 5.10.

Let $n\geq 1$ be an integer, $s>0$ with $s\notin\mathbb{Z}$. Let $u\in
H^{r}(\mathbb{R}^{n})$ for some $r\in\mathbb{R}$. If

$u=(-\Delta)^{s}u=0\quad\text{in some open set in $\mathbb{R}^{n}$,}$

then $u\equiv 0$ in $\mathbb{R}^{n}$.

Based on Lemma 5.7, we can obtain the antilocality property of the weighted
ray transform $\mathcal{A}_{s}$ using the antilocality property of the
fractional Laplacian. See III

######  Remark 5.11.

If $\big{(}\mathcal{X}_{s}f\big{)}(x,\xi)=0$ for all $x\in U$ and
$\xi\in\mathbb{S}^{n-1}$, then $\mathcal{A}_{s}f=0$ in $U$.

###### Proof of Theorem III.

Using Lemma 5.7, we know that

$g:=\mathcal{A}_{s}f\in L^{2}(\mathbb{R}^{n}),\quad g=(-\Delta)^{s}g=0\text{
in $U$.}$

From Lemma 5.10 we know that $g\equiv 0$ in $\mathbb{R}^{n}$. Consequently,
again using Lemma 5.7 we conclude $f\equiv(-\Delta)^{s}g\equiv 0$ in
$\mathbb{R}^{n}$. ∎

We also have the following interesting observation.

######  Lemma 5.12 (A support theorem for $\mathcal{A}_{s}$).

Let $n\geq 2$ be an integer, $0<s<1$ and let $f\in W^{1,p}(\mathbb{R}^{n})$
for some $\max\\{1,\frac{2n}{n+4s}\\}\leq p<\frac{n}{2s}$. If there exists a
nonempty bounded Lipschitz domain $\Omega$ such that

$\mathcal{A}_{s}f=0\text{ in $\mathbb{R}^{n}\setminus\overline{\Omega}$},\quad
f=0\text{ in $\Omega$,}$

then $f\equiv 0$ in $\mathbb{R}^{n}$.

###### Proof.

Using (5.10), we know that $g:=\mathcal{A}_{s}f\in W^{1,p}(\mathbb{R}^{n})$
for some $p\geq 2$. Since $g$ has compact support and $0<s<1$, then $g\in
H^{1}(\mathbb{R}^{n})\subset H^{s}(\mathbb{R}^{n})$. From (5.7) we have

$(-\Delta)^{s}g=0\text{ in $\Omega$},\quad g=0\text{ in
$\mathbb{R}^{n}\setminus\overline{\Omega}$.}$

Since 0 is not an eigenvalue of $(-\Delta)^{s}$ (a consequence of [LL19,
Proposition 3.3] or [RO16, Corollary 5.2]), then we know that $g\equiv 0$ in
$\mathbb{R}^{n}$. Consequently, again using Lemma 5.7 we conclude
$f\equiv(-\Delta)^{s}g\equiv 0$ in $\mathbb{R}^{n}$. ∎

###  5.5. Antilocality property of the cone transform

This weighted ray transform is related to the cone transform. Cone transform
appears in different imaging approaches, most notably in the modeling of data
provided by the so-called Compton camera, which has unique applications in
domains such as medical and industrial imaging, homeland security, and gamma
ray astronomy, see [KT17]. Let $\mathfrak{C}(u,\beta,\psi)$ be the cone with
vertex $u\in\mathbb{R}^{n}$, central axis $\beta\in\mathbb{S}^{n-1}$ and
opening angle $\psi\in(0,\pi)$, that is,

$\mathfrak{C}(u,\beta,\psi):=\begin{Bmatrix}\begin{array}[]{l|l}x\in\mathbb{R}^{n}&(x-u)\cdot\beta=|x-u|\cos\psi\end{array}\end{Bmatrix}.$

Following [KT17], we next introduce the weighted cone transform.

######  Definition 5.13.

For each $-1<k<n-1$ and $f\in\mathcal{S}(\mathbb{R}^{n})$, the $k$-th weighted
cone transform is defined as:

$\displaystyle(\mathcal{C}^{k}f)(u,\beta,\psi)=\int\limits_{\mathfrak{C}(u,\beta,\psi)}f(x)\,|x-u|^{k-n+2}\,\mathrm{d}S_{x},$

where $\mathrm{d}S$ is the surface measure on the cone
$\mathfrak{C}(u,\beta,\psi)$.

It is worth-mentioning that the cone transform is related to the momentum ray
transform $I^{k}\equiv\mathcal{X}_{\frac{k+1}{2}}$, precisely,

$\int\limits_{0}^{\pi}(\mathcal{C}^{k}f)(u,\beta,\psi)h(\cos(\psi))\,\mathrm{d}\psi=\int\limits_{\mathbb{S}^{n-1}}(I^{k}f)(u,\sigma)h(\sigma\cdot\beta)\,\mathrm{d}\sigma\quad\text{for
all $f\in\mathcal{S}(\mathbb{R}^{n})$}$ (5.13)

for all distribution $h\in\mathcal{D}^{\prime}(\mathbb{R}^{1})$ which are
regular near $t=\pm 1$, see [KT17, (19)]. By writing
$s=\frac{k+1}{2}\in(0,\frac{n}{2})$ and choosing $h\equiv c(n,-s)$ in (5.13)
yields

$c(n,-s)\int\limits_{0}^{\pi}(\mathcal{C}^{2s-1}f)(u,\beta,\psi)\,\mathrm{d}\psi=c(n,-s)\int\limits_{\mathbb{S}^{n-1}}(\mathcal{X}_{s}f)(u,\sigma)\,\mathrm{d}\sigma=(\mathcal{A}_{s}f)(u)\quad\text{for
all $u\in\mathbb{R}^{n}$}$ (5.14)

for all $f\in\mathcal{S}(\mathbb{R}^{n})$. Therefore, as a corollary of
Theorem III, we immediately obtain the antilocality property for cone
transform.

######  Corollary 5.14.

Let $n\geq 2$ be an integer, $0<s\leq\frac{n}{4}$ with $s\neq\mathbb{Z}$ and
let $f\in L^{\frac{2n}{n+4s}}(\mathbb{R}^{n})$. Suppose that there exists
$\beta\in\mathbb{S}^{n-1}$ and a non-empty open set $U$ in $\mathbb{R}^{n}$
such that $f=0$ in $U$ and

$(\mathcal{C}^{2s-1}f)(u,\beta,\psi)=0\quad\text{for all $(u,\psi)\in
U\times(0,\pi)$,}$

then $f\equiv 0$ in $\mathbb{R}^{n}$.

##  Appendix A Negative power of elliptic operator

The main theme of this appendix is to explain the connection between some
weighted x-ray transform with negative power of some class of elliptic
operators on some natural domain. Let
$A(x)=(a_{ij}(x))_{i,j=1}^{n}\in(C^{\infty}(\mathbb{R}^{n}))^{n\times n}$ such
that $a_{ij}=a_{ji}$ for all $i,j=1,\cdots,n$. Suppose that $A$ satisfies the
following ellipticity condition: there exists a constant $0<c<1$ such that

$c|\xi|^{2}\leq\xi\cdot A(x)\xi\leq c^{-1}|\xi|^{2}\quad\text{for all
}x\in\mathbb{R}^{n}.$ (A.1)

It is known that the operator $-\nabla\cdot A\nabla$ with domain
$H^{2}(\mathbb{R}^{n})$ is the maximal extension such that it is self-
adjoint333The operator $-\nabla\cdot A\nabla$ is not self-adjoint on the
domain $C_{c}^{\infty}(\mathbb{R}^{n})$. and densly defined in
$L^{2}(\mathbb{R}^{n})$, see [Gri09, Theorem 4.6]. Using the spectral theorem
for self-adjoint operator in a real Hilbert space as in [Gri09, Appendix
A.5.4], there exists a unique _spectral resolution_ (also known as _resolution
of identity_) $\\{E_{\lambda}\\}$ in $L^{2}(\mathbb{R}^{n})$ corresponding to
$-\nabla\cdot A\nabla$ such that

$\left(-\nabla\cdot A\nabla
f,g\right)_{L^{2}(\mathbb{R}^{n})}=\int_{0}^{\infty}\lambda\,\mathrm{d}(E_{\lambda}f,g)_{L^{2}(\mathbb{R}^{n})}$

for all $f\in H^{2}(\mathbb{R}^{n})$ and $g\in L^{2}(\mathbb{R}^{n})$. The
detailed properties of spectral resolution can be found in [Gri09, Appendix
A.5.3], here we only state some of them. Let $\lambda\mapsto\varphi(\lambda)$
be a Borel function on $\mathbb{R}$. By utilizing Riesz representation theorem
(see also [Gri09, (A.38)]), one can define the self-adjoint operator
$\varphi(-\nabla\cdot A\nabla)$ by

$\left(\varphi(-\nabla\cdot
A\nabla)f,g\right)_{L^{2}(\mathbb{R}^{n})}=\int\limits_{0}^{\infty}\varphi(\lambda)\,\mathrm{d}(E_{\lambda}f,g)_{L^{2}(\mathbb{R}^{n})}\equiv\bigg{(}\int\limits_{0}^{\infty}\varphi(\lambda)\,\mathrm{d}E_{\lambda}f,g\bigg{)}_{L^{2}(\mathbb{R}^{n})}$
(A.2)

for all $g\in L^{2}(\mathbb{R}^{n})$ and

$f\in{\rm dom}\,\left(\varphi(-\nabla\cdot
A\nabla)\right):=\begin{Bmatrix}\begin{array}[]{l|l}f\in
L^{2}(\mathbb{R}^{n})&\displaystyle{\int\limits_{0}^{\infty}}|\varphi(\lambda)|^{2}\,\mathrm{d}\|E_{\lambda}f\|_{L^{2}(\mathbb{R}^{n})}^{2}<\infty\end{array}\end{Bmatrix}.$
(A.3)

One may define $(-\nabla\cdot A\nabla)^{-s}$ using the mapping
$\varphi(\lambda)=\lambda^{-s}$. But however in general $\nabla\cdot
A\nabla:H^{2}(\mathbb{R}^{n})\rightarrow L^{2}(\mathbb{R}^{n})$ is not
injective, therefore the domain ${\rm dom}\,((-\nabla\cdot A\nabla)^{-s})$ is
somehow artificial. The main theme of this appendix is to define
$(-\nabla\cdot A\nabla)^{-s}$ on a suitable domain.

By using [Gri09, (A.32)], we know that ${\rm dom}\,(e^{t\nabla\cdot
A\nabla})=L^{2}(\mathbb{R}^{n})$ for all $t>0$. Based on this we can define
the action of the heat semigroup and of the fractional powers (of order $s>0$)
of $-\nabla\cdot A\nabla$ as

$\displaystyle e^{t\nabla\cdot A\nabla}f$
$\displaystyle:=\int\limits_{0}^{\infty}e^{-t\lambda}\,\mathrm{d}E_{\lambda}f\quad\text{for
all $f\in L^{2}(\mathbb{R}^{n})$ and $t>0$,}$ $\displaystyle(-\nabla\cdot
A\nabla)^{s}f$
$\displaystyle:=\int\limits_{0}^{\infty}\lambda^{s}\,\mathrm{d}E_{\lambda}f\quad\text{for
all $f\in{\rm dom}\,\left((-\nabla\cdot A\nabla)^{s}\right)$.}$

Using [Gri09, Theorems 7.6, 7.7 and 7.13], we know that the bounded operator
$e^{t\nabla\cdot A\nabla}$ admits a _unique_ symmetric (heat) kernel
$k_{t}^{A}(x,y)$: For each $0<t<\infty$ and $f\in L^{2}(\mathbb{R}^{n})$, we
have

$\big{(}e^{t\nabla\cdot
A\nabla}f\big{)}(x)=\int\limits_{\mathbb{R}^{n}}k_{t}^{A}(x,y)f(y)\,\mathrm{d}y\quad\mbox{for
all}\quad x\in\mathbb{R}^{n},$ (A.4)

see also [QX21, Theorem 2]. In addition, there exist positive constants
$b_{1},b_{2},c_{1},c_{2}$ such that

$c_{1}t^{-\frac{n}{2}}\exp\bigg{(}-b_{1}\frac{|x-y|^{2}}{t}\bigg{)}\leq
k_{t}^{A}(x,y)=k_{t}^{A}(y,x)\leq
c_{2}t^{-\frac{n}{2}}\exp\bigg{(}-b_{2}\frac{|x-y|^{2}}{t}\bigg{)},$ (A.5)

for all $t>0$ and $x,y\in\mathbb{R}^{n}$, see [Dav89, Chapter 3]. We refer to
[GT12] for the proof of two-sided estimates for heat kernels $k_{t}^{M}$ on
abstract metric measure spaces $M$, extending those already known in
Riemannian manifolds and in various types of fractals. Moreover, the heat
kernel is also _conservative_ (or _stochastically complete_)444These
terminologies are found in [GHL14, GT12]. We also refer to [Dav89, Theorem
1.4.4] for an abstract result., that is,

$\int\limits_{\mathbb{R}^{n}}k_{t}^{A}(x,y)\,\mathrm{d}y=1\quad\text{for all
$t>0$ and $x\in\mathbb{R}^{n}$.}$

In view of (5.12), now it is natural to consider the following definition.

######  Definition A.1 (Weighted x-ray transform).

Let $n\geq 2$ be an integer, and let $0<s<\frac{n}{2}$. For each
$f\in\mathcal{S}(\mathbb{R}^{n})$, we define

$\big{(}\mathcal{X}_{s}^{A}f\big{)}(x,\xi):=\int\limits_{0}^{\infty}w(x,x+\tau\xi)f(x+\tau\xi)\,\mathrm{d}\tau,$

where the weight is given by

$w(x,y):=\frac{1}{\Gamma(s)c(n,-s)}|y-x|^{n-1}\int\limits_{0}^{\infty}k_{t}^{A}(x,y)\,\frac{\mathrm{d}t}{t^{1-s}}\quad\mbox{for
all $x\neq y\in\mathbb{R}^{n}$,}$

and the average over the sphere $\mathbb{S}^{n-1}$ is defined by

$\big{(}\mathcal{A}_{s}^{A}f\big{)}(x):=c(n,-s)\int\limits_{\mathbb{S}^{n-1}}\big{(}\mathcal{X}_{s}^{A}f\big{)}(x,\xi)\,\mathrm{d}S_{\xi}\equiv\frac{1}{\Gamma(s)}\int_{\mathbb{R}^{n}}\bigg{(}\int\limits_{0}^{\infty}k_{t}^{A}(x,y)\,\frac{\mathrm{d}t}{t^{1-s}}\bigg{)}f(y)\,\mathrm{d}y.$
(A.6)

From (5.9) and (A.5), we can easily conclude the following lemma.

######  Lemma A.2.

Let $n\geq 2$ be an integer and let $1\leq p<\frac{n}{2s}$ provided
$0<s<\frac{n}{2}$. Let
$A(x)=(a_{ij}(x))_{i,j=1}^{n}\in(C^{\infty}(\mathbb{R}^{n}))^{n\times n}$ such
that $a_{ij}=a_{ji}$ for all $i,j=1,\cdots,n$ and satisfies the ellipticity
condition (A.1). Then

$\mathcal{A}_{s}^{A}:L^{p}(\mathbb{R}^{n})\rightarrow
L^{q}(\mathbb{R}^{n})\text{ is bounded,}$

where $\frac{1}{q}=\frac{1}{p}-\frac{2s}{n}$. In particular when
$0<s<\frac{n}{4}$, the operators

$\displaystyle\mathcal{A}_{s}^{A}:L^{2}(\mathbb{R}^{n})\rightarrow
L^{\frac{2n}{n-4s}}(\mathbb{R}^{n})$
$\displaystyle\mathcal{A}_{s}^{A}:L^{\frac{2n}{n+4s}}(\mathbb{R}^{n})\rightarrow
L^{2}(\mathbb{R}^{n})$

are bounded.

Using the lemma above, we know that

$(\mathcal{A}_{s}^{A}|f|)(x)=\frac{1}{\Gamma(s)}\int\limits_{\mathbb{R}^{n}}\int\limits_{0}^{\infty}\left|k_{t}^{A}(x,y)\frac{1}{t^{1-s}}f(y)\right|\,\mathrm{d}t\,\mathrm{d}y<\infty\quad\text{for
a.e. $x\in\mathbb{R}^{n}$,}$

whenever $f\in L^{p}(\mathbb{R}^{n})$ for some $p\in[1,\frac{n}{2s})$.
Therefore we can apply Fubini’s theorem on (A.6) yields

$\big{(}\mathcal{A}_{s}^{A}f\big{)}(x)=\frac{1}{\Gamma(s)}\int\limits_{0}^{\infty}\bigg{(}\int\limits_{\mathbb{R}^{n}}k_{t}^{A}(x,y)f(y)\mathrm{d}y\bigg{)}\,\frac{\mathrm{d}t}{t^{1-s}}\equiv\frac{1}{\Gamma(s)}\int\limits_{0}^{\infty}(e^{t\nabla\cdot
A\nabla}f)(x)\,\frac{\mathrm{d}t}{t^{1-s}}$ (A.7)

for all $f\in L^{p}(\mathbb{R}^{n})$ for some $p\in[1,\frac{n}{2s})$. In view
of (5.7), then we can define $(-\nabla\cdot A\nabla)^{-s}$ with a suitable
domain rather than the artificial domain ${\rm dom}\,((-\nabla\cdot
A\nabla)^{-s})$ given in (A.3).

######  Definition A.3.

Let $n\geq 2$ be an integer and let $1\leq p<\frac{n}{2s}$ provided
$0<s<\frac{n}{2}$. Let
$A(x)=(a_{ij}(x))_{i,j=1}^{n}\in(C^{\infty}(\mathbb{R}^{n}))^{n\times n}$ such
that $a_{ij}=a_{ji}$ for all $i,j=1,\cdots,n$ and satisfies the ellipticity
condition (A.1). Then the bounded linear operator $(-\nabla\cdot
A\nabla)^{-s}:L^{p}(\mathbb{R}^{n})\rightarrow L^{q}(\mathbb{R}^{n})$ with
$\frac{1}{q}=\frac{1}{p}-\frac{2s}{n}$ is defined by

$(-\nabla\cdot A\nabla)^{-s}f:=\mathcal{A}_{s}^{A}f$

for all $f\in L^{p}(\mathbb{R}^{n})$.

######  Remark A.4.

The operator $(-\nabla\cdot A\nabla)^{-s}$ for $s\in\mathbb{C}$ with
$\Re(s)>0$ also can be defined using Balakrishnan operator with domain ${\rm
range}\,(A^{m})$ where $m$ is the minimum integer such that $m>\Re(s)$, see
[MCSA01, Definition 7.2.1].

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11institutetext:  Jožef Stefan Institute, Parallel and Distributed Systems
Laboratory, Ljubljana, Slovenia 11email:
<EMAIL_ADDRESS>22institutetext:
Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
33institutetext: Jožef Stefan International Postgraduate School, Ljubljana,
Slovenia

# Oscillatory behaviour of the RBF-FD approximation accuracy under increasing
stencil size

Andrej Kolar-Požun 1122 0000-0001-9694-3828 Mitja Jančič 1133
0000-0003-1850-412X Miha Rot 1133 0000-0002-8869-4507 Gregor Kosec 11
0000-0002-6381-9078

###### Abstract

When solving partial differential equations on scattered nodes using the
Radial Basis Function generated Finite Difference (RBF-FD) method, one of the
parameters that must be chosen is the stencil size. Focusing on Polyharmonic
Spline RBFs with monomial augmentation, we observe that it affects the
approximation accuracy in a particularly interesting way - the solution error
oscillates under increasing stencil size. We find that we can connect this
behaviour with the spatial dependence of the signed approximation error. Based
on this observation we are then able to introduce a numerical quantity that
indicates whether a given stencil size is locally optimal.

###### Keywords:

Meshless Stencil RBF-FD PHS

##  1 Introduction

Radial Basis Function generated Finite Differences (RBF-FD) is a method for
solving Partial Differential Equations (PDEs) on scattered nodes that has
recently been increasing in popularity. It uses Radial Basis Functions (RBFs)
to locally approximate a linear differential operator in a chosen
neighbourhood, generalising the well known finite difference methods. This
neighbourhood used for the approximation is referred to as the stencil of a
given point and is commonly chosen to simply consist of its $n$ closest
neighbours.

Among the different possible choices of a RBF used, the Polyharmonic Splines
(PHS) with appropriate polynomial augmentation stand out due to the fact that
they possess no shape parameter, eliminating all the hassle that comes with
having to find its optimal value. PHS RBF-FD has been studied extensively and
proved to work well in several different contexts [6, 7, 8, 1, 2]. Unlike in
the case of RBFs with a shape parameter, where the approximation order is
determined by the stencil size [3], in PHS RBF-FD it is determined by the
degree of the monomials included in the augmentation [1]. Despite that, the
choice of an appropriate stencil size can have a substantional impact on the
accuracy. More precisely, the accuracy of the method displays an oscillatory
behaviour under increasing stencil size.

In the remainder of the paper, we present this observation and our findings.
Ideally we would like to be able to predict the stencil sizes that correspond
to the accuracy minima or at least provide some indicator on whether a given
stencil size is near the minimum.

The following section describes our problem setup along with the numerical
solution procedure and in section 3 our results are discussed.

##  2 Problem setup

Our analyses are performed on the case of the Poisson equation

$\nabla^{2}u(\textbf{x})=f(\textbf{x}),$ (1)

where the domain is a disc $\Omega=\\{\textbf{x}\in\mathbb{R}^{2}:\left\lVert
x-(0.5,0.5)\right\rVert\leq 0.5\\}$. We choose the function $f(\textbf{x})$
such, that the problem given by Equation (1) has a known analytic solution.
Concretely, we choose

$\displaystyle u(x,y)=$ $\displaystyle\sin(\pi x)\sin(\pi y),$ (2)
$\displaystyle f(x,y)=$ $\displaystyle-2\pi^{2}\sin(\pi x)\sin(\pi y)$ (3)

with the Dirichlet boundary conditions given by a restriction of
$u(\textbf{x})$ to the boundary $\partial\Omega$.

We discretise the domain with the discretisation distance $h=0.01$, first
discretising the boundary and then the interior using the algorithm proposed
in [4]. The Laplacian is then discretised using the RBF-FD algorithm as
described in [5], where we choose the radial cubics as our PHS
($\phi(r)=r^{3}$) augmented with monomials up to degree $m=3$, inclusive. This
requires us to associate to each discretisation point $\textbf{x}_{i}$ its
stencil, which we take to consist of its $n$ nearest neighbours. We can now
convert the PDE (1) into a sparse linear system, which we then solve to obtain
an approximate solution $\hat{u}(\textbf{x})$. The source code is readily
available in our git
repository111https://gitlab.com/e62Lab/public/2023_cp_iccs_stencil_size_effect.

The chosen analytical solution $u(\textbf{x})$ is displayed in Figure 1, which
additionally serves as a visual representation of how the domain is
discretised.

Figure 1: The analytical solution to the considered Poisson problem.

Having both the analytical and approximate solutions, we will be interested in
the approximation error. It will turn out to be useful to consider the signed
pointwise errors of both the solution and the Laplacian approximation:

$\displaystyle e_{\mathrm{poiss}}^{\pm}(\textbf{x}_{i})=\hat{u}_{i}-u_{i},$
(4) $\displaystyle
e_{\mathrm{lap}}^{\pm}(\textbf{x}_{i})=\tilde{\nabla}^{2}u_{i}-f_{i},$ (5)

where $\tilde{\nabla}^{2}$ is the discrete approximation of the Laplacian and
we have introduced the notation $u_{i}=u(\textbf{x}_{i})$. The "poiss" and
"lap" subscripts may be omitted in the text when referring to both errors at
once.

As a quantitative measure of the approximation quality, we will also look at
the average/max absolute value error:

$\displaystyle
e_{\mathrm{poiss}}^{\mathrm{max}}=\max_{\textbf{x}_{i}\in\mathring{\Omega}}|e_{\mathrm{poiss}}^{\pm}(\textbf{x}_{i})|,$
(6) $\displaystyle
e_{\mathrm{poiss}}^{\mathrm{avg}}=\frac{1}{N_{\mathrm{int}}}\sum_{\textbf{x}_{i}\in\mathring{\Omega}}|e_{\mathrm{poiss}}^{\pm}(\textbf{x}_{i})|$
(7)

and analogously for $e_{\mathrm{lap}}^{\mathrm{max}}$ and
$e_{\mathrm{lap}}^{\mathrm{avg}}$. $N_{\mathrm{int}}$ is the number of
discretisation points inside the domain interior $\mathring{\Omega}$.

In the next section we will calculate the approximation error for various
stencil sizes $n$ and further investigate its (non-trivial) behaviour.

It is worth noting that the setup considered is almost as simple as it can be.
The fact that we have decided not to consider a more complicated problem is
intentional - there is no need to complicate the analysis by considering a
more complex problem if the investigated phenomena already appears in a
simpler one. This reinforces the idea that such behaviour arises from the
properties of the methods used and not from the complexity of the problem
itself.

##  3 Results

In Figure 2 we see that $e_{\mathrm{poiss}}^{\mathrm{max}}(n)$ oscillates with
several local minima (at stencil sizes $n=28,46$) and maxima (stencil sizes
$n=17,46$). The dependence $e_{\mathrm{poiss}}^{\mathrm{max}}(n)$ seems to
mostly resemble a smooth function. This is even more evident in
$e_{\mathrm{poiss}}^{\mathrm{avg}}(n)$. The errors of the Laplacian are also
plotted and we can observe that $e_{\mathrm{lap}}^{\mathrm{avg}}(n)$ has local
minima and maxima at same stencil sizes. Such regularity implies that the
existence of the minima in the error is not merely a coincidence, but a
consequence of a certain mechanism that could be explained. Further
understanding of this mechanism would be beneficial, as it could potentially
allow us to predict the location of these local minima a priori. Considering
that the error difference between the neighbouring local maxima and minima can
be over an order of magnitude apart this could greatly increase the accuracy
of the method without having to increase the order of the augmentation or the
discretisation density. Note that the behaviour of
$e_{\mathrm{lap}}^{\mathrm{max}}$ stands out as it is much more irregular.
This implies that in order to explain the observed oscillations, we have to
consider the collective behaviour of multiple points. This will be confirmed
later on, when we consider the error’s spatial dependence.

An immediate idea is that the choice of a sparse solver employed at the end of
the solution procedure is responsible for the observed behaviour. We have
eliminated this possibility by repeating the analysis with both the SparseLU
and BiCGSTAB solvers, where no difference has been observed. The next idea we
explore is the possibility of the discretisation being too coarse.

Figure 2: Dependence of the approximation errors on the stencil size $n$.

Figure 3 shows that under discretisation refinement
$e_{\mathrm{poiss}}^{\mathrm{max}}(n)$ maintains the same shape and is just
shifted vertically towards a lower error. The latter shift is expected, as we
are dealing with a convergent method, for which the solution error behaves as
$e\propto h^{p}$ as $h\to 0$, where $p$ is the order of the method. We also
show $e_{\mathrm{poiss}}^{\mathrm{max}}(h)$ in a log-log scale for some
different stencil sizes. It can be seen that the slopes and therefore the
orders $p$ generally do not change with the stencil size and that the observed
oscillations mainly affect the proportionality constant in $e\propto h^{p}$.
The stencil dependence of the error proportionality constant has already been
observed in similar methods [3, 9].

Figure 3: Behaviour of the approximation errors under a refinement of the
discretisation.

Next we check if boundary stencils are responsible for the observed behaviour
as it is known that they can be problematic due to their one-sideness [2]. In
Figure 4 we have split our domain into two regions - the nodes near the
boundary
$\\{\textbf{x}_{i}\in\Omega:\left\lVert\textbf{x}_{i}-(0.5,0.5)\right\rVert>0.4\\}$
are coloured red, while the nodes far from the boundary
$\\{\textbf{x}_{i}\in\Omega:\left\lVert\textbf{x}_{i}-(0.5,0.5)\right\rVert\leq
0.4\\}$ are black. We can see that the dependence of
$e_{\mathrm{poiss}}^{\mathrm{max}}(n)$ marginally changes if we keep the
stencil size near the boundary fixed at $n=28$ (corresponding to one of the
previously mentioned minima), while only changing the stencil sizes of the
remaining nodes. This shows that the observed phenomena is not a consequence
of the particularly problematic boundary stencils.

Figure 4: The seperation of the domain into two regions is seen on the left,
where the green circles show the radii of the biggest stencils considered
($n=69$). The right graph shows the error dependence when either of the
regions is at a fixed stencil size $n=28$. The previous result with no fixed
stencil size regions is also shown.

Figure 5 provides some more insight into the mechanism behind the oscillating
error. Here we have plotted the spatial dependence of the signed error
$e_{\mathrm{poiss}}^{\pm}$ for those stencils that correspond to the marked
local extrema. We can observe that in the maxima, the error has the same sign
throughout the whole domain. On the other hand, near the values of $n$ that
correspond to the local minima there are parts of the domain that have
differing error signs. Concretely, the sign of $e_{\mathrm{poiss}}^{\pm}$ is
negative for stencil sizes between $17$ and $27$ inclusive. In the minima at
$n=28$ both error signs are present, while for bigger stencil sizes (between
$29$ and $45$ inclusive) the error again has constant sign only this time
positive. The story turns out to repeat on the next minimum.

Figure 5: Spatial dependence of $e_{\mathrm{poiss}}^{\pm}$ in some local
extrema. The colour scale is the same for all drawn plots.

This connection between the sign of $e_{\mathrm{poiss}}^{\pm}$ and the minima
in $e_{\mathrm{poiss}}^{\mathrm{max}}(n)$ motivates us to define a new
quantity:

$\delta
N_{\mathrm{poiss}}^{\pm}=\frac{1}{N_{\mathrm{int}}}\left(|\\{\textbf{x}_{i}\in\mathring{\Omega}:e_{\mathrm{poiss}}^{\pm}(\textbf{x}_{i})>0\\}|-|\\{\textbf{x}_{i}\in\mathring{\Omega}:e_{\mathrm{poiss}}^{\pm}(\textbf{x}_{i})<0\\}|\right)$
(8)

and analogously for $\delta N_{\mathrm{lap}}^{\pm}$. Simply put, the quantity
$\delta N_{\mathrm{poiss}}^{\pm}$ is proportional to the difference between
the number of nodes with positively and negatively signed error. Assigning
values of $\pm 1$ to positive/negative errors respectively, this quantity can
be roughly interpreted as the average sign of the error. It should hold that
$|\delta N_{\mathrm{poiss}}^{\pm}|$ is approximately equal to $1$ near the
maxima and lowers in magnitude as we approach the minima. Figure 6 confirms
this intuition - $\delta N_{\mathrm{poiss}}^{\pm}(n)$ changes its values
between $\pm 1$ very abruptly only near the $n$ that correspond to the minima
of $e_{\mathrm{poiss}}^{\mathrm{max}}(n)$. A similar conclusion can be made
for $\delta N_{\mathrm{lap}}^{\pm}$, which acts as a sort of "smoothed out"
version of $\delta N_{\mathrm{poiss}}^{\pm}(n)$.

At a first glance, $N_{\mathrm{lap}}^{\pm}$ seems like a good candidate for an
error indicator - it has a well-behaved dependence on $n$, approaches $\pm 1$
as we get closer to the error maxima and has a root near the error minima. The
major downside that completely eliminates its applicability in the current
state is the fact that we need access to the analytical solution to be able to
compute it.

Figure 6: The quantities $\delta N^{\pm}(n)$ along with the spatial profiles
of the signs of $e^{\pm}$ for some chosen stencil sizes. For convenience,
$\delta N^{\pm}=0$ is marked with an orange line.

##  4 Conclusions

Our study started with a simple observation - when solving a Poisson problem
with PHS RBF-FD, the approximation accuracy depends on the stencil size $n$ in
a non-trivial manner. In particular, there exist certain stencil sizes where
the method is especially accurate. A priori knowledge of these stencil sizes
could decrease the solution error without any additional effort and is
therefore strongly desirable. We have made a small step towards understanding
this phenomena, eliminating various common numerical issues as the cause.
Looking at the spatial dependence of the signed solution error, we have
noticed that in the stencil sizes corresponding to the local error minima, the
signed solution error is not strictly positive or negative. This is unlike the
generic stencil sizes, where the error has the same sign throughout the
domain. Motivated by this observation, we have introduced a quantity that is
roughly the average sign of the pointwise Laplace operator errors and appears
to have a root only near the stencils corresponding to the local error minima.

The research presented is a step towards defining a more practically useful
indicator, which would reveal the most accurate stencil sizes even without
having access to the analytical solution and is a part of our ongoing
research. Additional future work includes more rigorous theoretical
explanations for the observations presented. Further experimental
investigations should also be made, particularly to what extent our
observations carry over to different problem setups - other differential
equations and domain shapes.

###  4.0.1 Acknowledgements

The authors would like to acknowledge the financial support of Slovenian
Research Agency (ARRS) in the framework of the research core funding No.
P2-0095 and the Young Researcher programs PR-10468 and PR-12347.

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