# Ray transform on Sobolev spaces of symmetric tensor fields, II: Range
characterization

Venkateswaran P. Krishnan  Centre for Applicable Mathematics, Tata Institute
of Fundamental Research, India<EMAIL_ADDRESS>and  Vladimir A.
Sharafutdinov  Sobolev Institute of Mathematics, 4 Koptyug Av., 630090,
Novosibirsk, Russia<EMAIL_ADDRESS>
###### Abstract.

The ray transform $I$ integrates symmetric $m$-tensor field in
$\mathbb{R}^{n}$ over lines. This transform in Sobolev spaces was studied in
our earlier work where higher order Reshetnyak formulas (isometry relations)
were established. The main focus of the current work is the range
characterization. In dimensions $n\geq 3$, the range characterization of the
ray transform in Schwartz spaces is well-known; the main ingredient of the
characterization is a system of linear differential equations of order
$2(m+1)$ which are called John equations. Using the higher order Reshetnyak
formulas, the range of the ray transform on Sobolev spaces is characterized in
dimensions $n\geq 3$ in this paper.

##  1\. Introduction

The ray transform integrates functions or more generally symmetric tensor
fields over lines and the Radon transform integrates functions over
hyperplanes in $\mathbb{R}^{n}$. The ray transform of functions is the main
mathematical tool of computer tomography. The ray transform of vector fields
and that of symmetric second rank tensor fields arise in Doppler tomography
and travel time tomography, respectively. The Radon transform originally arose
while trying to decompose a solution of the wave equation in terms of plane
waves. In 2-dimensions, the ray transform of functions coincides, up to
notation, with the Radon transform. The Radon transform is extensively studied
in [3, 10] and a systematic treatment of the ray transform of symmetric tensor
fields in Euclidean and Riemannian manifold settings is performed in [12]. The
ray and Radon transforms will be denoted by $I$ and $R$, respectively, in this
paper.

Three questions naturally arise in the study of the Radon transform and ray
transform of symmetric tensor fields (or any other transform with tomographic
applications) (i) inversion formulas (ii) stability and (iii) range
characterization. We do not discuss inversion formulas in this paper. Instead,
we refer the interested reader to the aforementioned books [3, 10, 12] where
explicit inversion formulas are derived for such transforms. The second
question is that of stability. As we will see, this is related to the third
question of range characterization; the main topic of this paper. For this
reason, we begin with a discussion of stability for the Radon transform. To
make the discussion accessible to a broader audience, we have chosen to be
informal in the Introduction. Our current work is a companion paper to our
prior work on this topic [8], and we largely follow the notation from that
paper. To avoid repetition, we have chosen to give only a condensed version of
the relevant material to make the discussion self-contained. We recommend the
interested readers to read this in conjunction with our earlier work as and
when required.

The Radon transform $R$ integrates a function along hyperplanes. The set of
hyperplanes can be parameterized by points of
$\mathbb{S}^{n-1}\times\mathbb{R}$. Then $R$ is defined by

$Rf(\xi,p)=\int\limits_{\langle\xi,x\rangle=p}f(x)\,dx\quad\big{(}(\xi,p)\in\mathbb{S}^{n-1}\times\mathbb{R}\big{)},$

where $\langle\cdot\>,\cdot\rangle$ is the standard dot-product in
$\mathbb{R}^{n}$ and $dx$ is the $(n-1)$-dimensional Lebesgue measure on the
hyperplane $\\{x\mid\langle\xi,x\rangle=p\\}$. Some condition on $f$ should be
imposed for the integral above to converge. The Reshetnyak formulas for the
Radon transform is a family of isometry relations (the best stability
estimates possible) involving certain weighted Sobolev spaces on
$\mathbb{R}^{n}$ and on $\mathbb{S}^{n-1}\times\mathbb{R}$. For the precise
definitions of the weighted Sobolev spaces appearing below, we refer the
reader to [3, 14, 13]. Section 2.1 in [8] also provides a good summary of
these works. We have the following _family of higher order Reshetnyak
formulas_ for the Radon transform [13]:

$\lVert f\rVert_{H_{t}^{(r,s)}(\mathbb{R}^{n})}=\lVert
Rf\rVert_{H^{r,s+(n-1)/2}_{t+(n-1)/2}\left(\mathbb{S}^{n-1}\times\mathbb{R}\right)},$

for arbitrary reals $s,r$ and for $t>-n/2$. We discuss a few special cases of
this formula. The case when $r=s=t=0$ is the original isometry relation
between $f$ and its Radon transform, due to Reshetnyak in the 1960s. The case
when $r=0$, $s$ arbitrary and $t>-n/2$ was given by the second author in [14].

The Reshetnyak formulas are closely related to the range characterization of
the Radon transform. The classical Gel’fand-Helgason-Ludwig (GHL) range
characterization theorem for the Radon transform on the Schwartz space is the
following (below $\mathcal{S}(\mathbb{S}^{n-1}\times\mathbb{R})$ denotes the
Schwartz space of functions on $\mathbb{S}^{n-1}\times\mathbb{R}$):

A function $g\in\mathcal{S}(\mathbb{S}^{n-1}\times\mathbb{R})$ is the Radon
transform of a function $f\in\mathcal{S}(\mathbb{R}^{n})$ if and only if the
following two conditions are satisfied:

(1) $g(\omega,p)=g(-\omega,-p)$,

(2) for any integer $k\geq 0$, $\int_{\mathbb{R}}p^{k}g(\omega,p)\,dp$ is a
homogeneous polynomial of degree at most $k$ in $\omega$.

A natural question is the range characterization of the Radon transform in
weighted Sobolev spaces $H_{t}^{(r,s)}(\mathbb{R}^{n})$. The following result
is obtained in [13]: For arbitrary reals $r,s$ and $t>-n/2$, the Radon
transform
$R:\mathcal{S}(\mathbb{R}^{n})\to\mathcal{S}(\mathbb{S}^{n-1}\times\mathbb{R})$
uniquely extends to a bijective isometry of Hilbert spaces

$R:H^{(r,s)}_{t}(\mathbb{R}^{n})\to
H^{(r,s+(n-1)/2)}_{t+(n-1)/2,e}(\mathbb{S}^{n-1}\times\mathbb{R}).$

Hereafter, the index $e$ stands for “even”, i.e.,
$H^{(r,s}_{t,e}(\mathbb{S}^{n-1}\times\mathbb{R})$ is the subspace of
$H^{(r,s}_{t}(\mathbb{S}^{n-1}\times\mathbb{R})$ consisting of functions
satisfying $g(\omega,p)=g(-\omega,-p)$. We make one crucial remark here. The
second part of the GHL condition disappears in the range characterization when
one extends the Radon transform from the Schwartz space to the weighted
Sobolev spaces. The special case of this result, when $r=s=t=0$ can be found
in [3], and for the case when $r=0$, $s$ arbitrary and $t>-n/2$ in [14]. To
show the bijectivity of the Radon transform on these spaces, one proceeds as
follows. The Reshetnyak formula shows that, being an isometry, the range of
the operator $R|_{H^{(r,s)}_{t}(\mathbb{R}^{n})}$ is a closed subspace of
$H^{(r,s+(n-1)/2)}_{t+(n-1)/2,e}(\mathbb{S}^{n-1}\times\mathbb{R})$, and the
proof is completed by showing that the orthogonal complement of the latter
subspace (with respect to the
$H^{(r,s+(n-1)/2)}_{t+(n-1)/2}(\mathbb{S}^{n-1}\times\mathbb{R})$ scalar
product) is equal to zero.

We next discuss the existence of Reshetnyak formulas for the ray transform.
Let $S^{m}\mathbb{R}^{n}$ be the ${n+m-1}\choose m$-dimensional complex vector
space of rank $m$ symmetric tensor fields on ${\mathbb{R}}^{n}$. The family of
oriented straight lines in ${\mathbb{R}}^{n}$ is parameterized by points of
the manifold

$T{\mathbb{S}}^{n-1}=\\{(x,\xi)\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\mid|\xi|=1,\langle
x,\xi\rangle=0\\}\subset{\mathbb{R}}^{n}\times{\mathbb{R}}^{n},$

that is, the tangent bundle of the unit sphere ${\mathbb{S}}^{n-1}$. A point
$(x,\xi)\in T{\mathbb{S}}^{n-1}$ determines the line $\\{x+t\xi\mid
t\in{\mathbb{R}}\\}$. Let $\mathcal{S}(\mathbb{R}^{n};S^{m}\mathbb{R}^{n})$
denote the Schwartz space of $S^{m}\mathbb{R}^{n}$-valued functions on
$\mathbb{R}^{n}$ and $\mathcal{S}(T\mathbb{S}^{n-1})$ denote the Schwartz
space of functions on $T\mathbb{S}^{n-1}$. The ray transform $I$ is the linear
continuous operator

$I:\mathcal{S}(\mathbb{R}^{n};S^{m}\mathbb{R}^{n})\to\mathcal{S}(T\mathbb{S}^{n-1})$
(1.1)

that is defined, for $f=(f_{i_{1}\dots
i_{m}})\in\mathcal{S}(\mathbb{R}^{n};S^{m}\mathbb{R}^{n})$, by

$If(x,\xi)=\int\limits_{-\infty}^{\infty}f_{i_{1}\dots
i_{m}}(x+t\xi)\,\xi^{i_{1}}\dots\xi^{i_{m}}\,\mathrm{d}t=\int\limits_{-\infty}^{\infty}\langle
f(x+t\xi),\xi^{m}\rangle\,dt\quad\big{(}(x,\xi)\in
T{\mathbb{S}}^{n-1}\big{)}.$ (1.2)

Here and henceforth, we use the Einstein summation rule: the summation from 1
to $n$ is assumed over every index repeated in lower and upper positions in a
monomial. We use either lower or upper indices for denoting coordinates of
vectors and tensors. Since we work in Cartesian coordinates only, there is no
difference between covariant and contravariant tensors. For $m>0$, the ray
transform $I$ has an infinite dimensional kernel consisting of so-called
potential tensor fields. Therefore it is natural to restrict the ray transform
to the orthogonal complement of potential tensor fields, called solenoidal
tensor fields. Analogous to the higher order Reshetnyak formulas for the Radon
transform, the following family of higher order Reshetnyak formulas for the
ray transform of symmetric tensor fields was derived recently in our work [8]:
For an integer $r\geq 0$, real $s$ and $t>-n/2$, the $r^{\mathrm{th}}$ order
Reshetnyak formula for the ray transform restricted to solenoidal tensor
fields is

$\lVert f\rVert_{H^{(r,s)}_{t}(\mathbb{R}^{n};S^{m}\mathbb{R}^{n})}=\lVert
If\rVert_{H^{\left(r,s+1/2\right)}_{t+1/2}(T\mathbb{S}^{n-1})}.$ (1.3)

Definitions of the spaces in the formula are given in the next section. Two
special cases of this Reshetnyak formula were derived by the second author.
The first was for the case $r=0$ in [14], and a first order Reshetnyak formula
corresponding to $r=1$ (with a slightly different definition for the weighted
Sobolev spaces) for the case of the ray transform of functions was given in
[13].

Now we come to the question of range characterization for the ray transform.
John [5] studied the range characterization of the ray transform of Schwartz
class functions in $\mathbb{R}^{3}$ and showed that a function is in the range
of this transform if and only if it solves a certain second order linear
differential equation, the so-called John equation. Helgason [2] generalized
this result to an arbitrary $n\geq 3$ dimension. A system of linear second
order differential equations appears in the case of $n\geq 4$, they are still
called John equations. The second author proved a range characterization
result for the ray transform of symmetric tensor fields in Schwartz space in
dimensions $n\geq 3$ [12, Theorem 2.10.1].

For the $m$-tensor case, a big system of linear differential equations of
order $2(m+1)$ appears, see equations (2.5) below. Observe that the left-hand
side of every equation (2.5) is actually a product of $m$ John operators
defined by (2.6). Equations (2.5) are not independent, there are many linear
relations between them since John operators commute with each other. One would
like to distinguish a minimal subsystem of (2.5) which is still equivalent to
the whole system. We do not try to minimize the system (2.5) in the present
paper. Nevertheless, let us mention that, in the 3-dimensional case, the
system (2.5) is equivalent to one linear differential equation of order
$2(m+1)$ if an appropriate parametrization of the family of lines in
${\mathbb{R}}^{3}$ has been chosen [9]. In an arbitrary dimension,
minimization of the system (2.5) is discussed in [1].

Integral GHL conditions for the Radon transform are very different of John
differential equations for the ray transform. Nevertheless, they are related
as is shown by Denisiuk [1] by restricting the ray transform to 2D planes.

Unlike GHL conditions, the John equations persist in the passage from the
Schwartz space to weighted Sobolev spaces. In the scalar case, this fact is
proved in [15]. The main goal of the present article is the proof of the
latter fact for the ray transform of symmetric tensor fields. Namely, being
understood in the distribution sense, the John equations characterize the
range of the ray transform on the spaces
$H^{(r,s)}_{t}(\mathbb{R}^{n};S^{m}\mathbb{R}^{n})$. Unlike the case of the
Schwartz space, Reshetnyak formulas play an important role in the
characterization.

In our opinion, the difference between GHL conditions for the Radon transform
and John equations for the ray transform is caused by the following. The ray
transform is over-determined in dimensions $n\geq 3$: we try to recover a
tensor field depending on $n$ variables from a $(2n-2)$-dimensional set of
information. On the other hand, in the case of the Radon transform, the
problem of recovering $f$ from $Rf$ is formally determined.

Finally, we shortly discuss the ray transform in the 2-dimensional case. As we
have mentioned, the ray transform of functions coincides with the Radon
transform in 2-dimensions. For symmetric $m$-tensor fields, the range
characterization of the 2-dimensional ray transform in the Scwartz space was
obtained by Pantjukhina [11]; some version of integral GHL conditions
participates here. The range characterization of the 2D ray transform in
weighted Sobolev spaces of symmetric $m$-tensor fields is not studied so far
for $m>0$. The following interesting question remains open: What kind of
conditions, either John’s differential equations or GHL integral conditions,
should be used in the latter problem?

##  2\. Preliminaries and statement of the main result

In this section, we state all preliminary definitions and results required for
a precise statement of the range characterization.

###  2.1. Ray transform of symmetric tensor fields

Recall from the previous section that
$\mathcal{S}(\mathbb{R}^{n};S^{m}\mathbb{R}^{n})$ is the Schwartz space of
symmetric $m$-tensor valued functions on $\mathbb{R}^{n}$ (each component of
the tensor field is a Schwartz class function) and we also introduced the
Schwartz space ${\mathcal{S}}(T{\mathbb{S}}^{n-1})$ that is defined as
follows. Given a function $\varphi\in C^{\infty}(T{\mathbb{S}}^{n-1})$, we
extend it to some neighborhood of $T{\mathbb{S}}^{n-1}$ in
${\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$ so that (the extension is again
denoted by $\varphi$)

$\varphi(x,r\xi)=\varphi(x,\xi)\
(r>0),\quad\varphi(x+r\xi,\xi)=\varphi(x,\xi)\ (r\in{\mathbb{R}}).$

We say that a function $\varphi\in C^{\infty}(T{\mathbb{S}}^{n-1})$ belongs to
${\mathcal{S}}(T{\mathbb{S}}^{n-1})$ if the seminorm

$\|\varphi\|_{k,\alpha,\beta}=\sup\limits_{(x,\xi)\in
T{\mathbb{S}}^{n-1}}\left|(1+|x|)^{k}\partial^{\alpha}_{x}\partial^{\beta}_{\xi}\varphi(x,\xi)\right|$

is finite for every $k\in{\mathbb{N}}$ and for all multi-indices $\alpha$ and
$\beta$. The family of these seminorms defines the topology on
${\mathcal{S}}(T{\mathbb{S}}^{n-1})$.

Recall that the ray transform (1.1) is defined by (1.2). In the case of an
even $m$, the ray transform is the linear continuous operator

$I:{\mathcal{S}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})\rightarrow{\mathcal{S}}_{e}(TS^{n-1}),$

and in the case of an odd $m$,

$I:{\mathcal{S}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})\rightarrow{\mathcal{S}}_{o}(TS^{n-1}),$

where ${\mathcal{S}}_{e}(TS^{n-1})\ ({\mathcal{S}}_{o}(TS^{n-1}))$ is the
subspace of ${\mathcal{S}}_{e}(TS^{n-1})$ consisting of functions satisfying
$\varphi(x,-\xi)=\varphi(x,\xi)$ (satisfying
$\varphi(x,-\xi)=-\varphi(x,\xi)$). To unite these formulas, let us introduce
the parity of $m$

$\pi(m)=\left\\{\begin{array}[]{l}e\ \mbox{if}\ m\ \mbox{is even},\\\ o\
\mbox{if}\ m\ \mbox{is odd}.\end{array}\right.$

Then the ray transform can be initially considered as a linear continuous
operator

$I:{\mathcal{S}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})\rightarrow{\mathcal{S}}_{\pi(m)}(TS^{n-1}).$
(2.1)

###  2.2. Fourier slice theorem

The Fourier transform of symmetric tensor fields

$F:\mathcal{S}(\mathbb{R}^{n};S^{m}\mathbb{R}^{n})\to\mathcal{S}(\mathbb{R}^{n};S^{m}\mathbb{R}^{n}),\quad
f\mapsto\widehat{f}$

is defined component wise (hereafter i is the imaginary unit):

$\widehat{f}_{i_{1}\cdots
i_{m}}(y)=\frac{1}{(2\pi)^{n/2}}\int\limits_{\mathbb{R}^{n}}e^{-\textsl{i}\langle
y,x\rangle}f_{i_{1}\cdots i_{m}}(x)\,dx.$

The Fourier transform
$F:\mathcal{S}(T\mathbb{S}^{n-1})\to\mathcal{S}(T\mathbb{S}^{n-1}),\
\varphi\mapsto\widehat{\varphi}$ is defined as the $(n-1)$-dimensional Fourier
transform over the subspace $\xi^{\perp}$:

$\widehat{\varphi}(y,\xi)=\frac{1}{(2\pi)^{(n-1)/2}}\int\limits_{\xi^{\perp}}e^{-\textsl{i}\langle
y,x\rangle}\varphi(x,\xi)\,dx\quad\big{(}(y,\xi)\in
T{\mathbb{S}}^{n-1}\big{)}.$

The Fourier slice theorem [12, formula (2.1.5)] states:

$\widehat{If}(y,\xi)=\sqrt{2\pi}\langle\widehat{f}(y),\xi^{m}\rangle\mbox{ for
}(y,\xi)\in T\mathbb{S}^{n-1}.$ (2.2)

###  2.3. Range characterization of the ray transform on the Schwartz space

We first observe that the right-hand side of (1.2) makes sense for all
$(x,\xi)\in\mathbb{R}^{n}\times\mathbb{R}^{n}\setminus\\{0\\}$, and we use
notation $J$ to denote the extended operator. Strictly speaking, we define the
continuous linear operator

$J:\mathcal{S}(\mathbb{R}^{n};S^{m}\mathbb{R}^{n})\to
C^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n}\setminus\\{0\\})$

by

$Jf(x,\xi)=\int\limits_{-\infty}^{\infty}f_{i_{1}\dots
i_{m}}(x+t\xi)\,\xi^{i_{1}}\dots\xi^{i_{m}}\,dt\quad(x\in{\mathbb{R}}^{n},0\neq\xi\in{\mathbb{R}}^{n}).$
(2.3)

This extension allows us to take partial derivatives of $Jf$ with respect to
variables $x$ and $\xi$ freely. One can go back and forth between the original
definition and the extended one by

$Jf|_{T\mathbb{S}^{n-1}}=If\quad\mbox{and}\quad
Jf(x,\xi)=|\xi|^{m-1}If\left(x-\frac{\langle
x,\xi\rangle\xi}{|\xi|^{2}},\frac{\xi}{|\xi|}\right).$

Given a function $\varphi\in T\mathbb{S}^{n-1}$, let us define an extension
$\psi$ of this function to
$\mathbb{R}^{n}\times\mathbb{R}^{n}\setminus\\{0\\}$ as follows:

$\psi(x,\xi)=|\xi|^{m-1}\varphi\left(x-\frac{\langle
x,\xi\rangle\xi}{|\xi|^{2}},\frac{\xi}{|\xi|}\right).$ (2.4)

The range characterization theorem for the ray transform of symmetric tensor
fields in dimension $n\geq 3$ is as follows:

######  Theorem 2.1.

A function $\varphi\in{\mathcal{S}}(T{\mathbb{S}}^{n-1})\ (n\geq 3)$ belongs
to the range of the operator (1.1) if and only if the following two conditions
hold:

  1. (1)

$\varphi(x,-\xi)=(-1)^{m}\varphi(x,\xi)$;

  2. (2)

the function $\psi\in
C^{\infty}\big{(}{\mathbb{R}}^{n}\times({\mathbb{R}}^{n}\setminus\\{0\\})\big{)}$,
defined by (2.4), satisfies the equations

$\Big{(}\frac{\partial^{2}}{\partial
x^{i_{1}}\partial\xi^{j_{1}}}-\frac{\partial^{2}}{\partial
x^{j_{1}}\partial\xi^{i_{1}}}\Big{)}\dots\Big{(}\frac{\partial^{2}}{\partial
x^{i_{m+1}}\partial\xi^{j_{m+1}}}-\frac{\partial^{2}}{\partial
x^{j_{m+1}}\partial\xi^{i_{m+1}}}\Big{)}\psi=0$ (2.5)

for all indices $1\leq i_{1},j_{1},\dots,i_{m+1},j_{m+1}\leq n$.

We call (2.5) the John equations and the differential operators

$J_{ij}=\frac{\partial^{2}}{\partial
x^{i}\partial\xi^{j}}-\frac{\partial^{2}}{\partial
x^{j}\partial\xi^{i}}:C^{\infty}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})\rightarrow
C^{\infty}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n}),$ (2.6)

the John operators. (We hope the reader is not confused by similarity between
the notation $J$ for the operator (2.3) and $J_{ij}$ for John operators; both
the notations are standard ones.) In the case of $(m,n)=(0,3)$, Theorem 2.1 is
equivalent to John’s result [5]. In the case of $m=0$, Theorem 2.1 was proved
by Helgason [2]. For the proof in the general case see [12, Theorem 2.10.1].

Our goal, as already mentioned, is to generalize the above result to weighted
Sobolev spaces. The spaces are discussed in next few paragraphs.

###  2.4. The vector fields $X_{i}$ and $\Xi_{i}$ and intrinsic John operators

The following first order differential operators on $T\mathbb{S}^{n-1}$ were
introduced in [6, 7]. Consider $\mathbb{R}^{n}\times\mathbb{R}^{n}$ with
variables $(x,\xi)$ and introduce the following vector fields:

$\displaystyle\widetilde{X}_{i}$ $\displaystyle=\frac{\partial}{\partial
x_{i}}-\xi_{i}\xi^{p}\frac{\partial}{\partial x^{p}},$
$\displaystyle\widetilde{\Xi}_{i}$
$\displaystyle=\frac{\partial}{\partial\xi_{i}}-x_{i}\xi^{p}\frac{\partial}{\partial
x^{p}}-\xi_{i}\xi^{p}\frac{\partial}{\partial\xi^{p}}.$

These vector fields are tangent to $T\mathbb{S}^{n-1}$ at every point
$(x,\xi)\in T\mathbb{S}^{n-1}$, as was shown in [6], and therefore can be
viewed as vector fields on $T\mathbb{S}^{n-1}$. Let $X_{i}$ and $\Xi_{i}$ be
the restrictions of these vector fields to $T\mathbb{S}^{n-1}$.

The operators $X_{i}$ and $\Xi_{i}$ are related to the Fourier transform by
the equalities [6]:

$\widehat{X_{i}\varphi}=\textsl{i}\,y_{i}\,\widehat{\varphi},\quad\widehat{\Xi_{i}\varphi}=\Xi_{i}\widehat{\varphi},\quad$

which hold for every function $\varphi\in{\mathcal{S}}(T{\mathbb{S}}^{n-1})$
and for every $1\leq i\leq n$.

We use the vector fields $X_{i}$ and $\Xi_{i}$ to introduce the second order
differential operators

${\mathcal{J}}_{ij}:C^{\infty}(T{\mathbb{S}}^{n-1})\rightarrow
C^{\infty}(T{\mathbb{S}}^{n-1})\quad(1\leq i,j\leq n)$

by ${\mathcal{J}}_{ij}=X_{i}\Xi_{j}-X_{j}\Xi_{i}$. The operator
$\mathcal{J}_{ij}$ are called intrinsic John operators.

###  2.5. The spaces $H^{(r,s)}_{t}(T\mathbb{S}^{n-1})$

We define the second order differential operator $\Delta_{\xi}$ on
$T\mathbb{S}^{n-1}$ by $\Delta_{\xi}=-\sum\limits_{i=1}^{n}\Xi_{i}^{2}$. This
operator is used for defining the weighted Sobolev spaces on
$T\mathbb{S}^{n-1}$. The following scalar product was introduced in [14]. For
$\varphi_{j}\in{\mathcal{S}}(T{\mathbb{S}}^{n-1})\ (j=1,2)$,

$(\varphi_{1},\varphi_{2})_{H^{s}_{t}(T{\mathbb{S}}^{n-1})}=\frac{\Gamma\big{(}\frac{n-1}{2}\big{)}}{4\pi^{(n+1)/2}}\int\limits_{{\mathbb{S}}^{n-1}}\int\limits_{\xi^{\bot}}|y|^{2t}(1+|y|^{2})^{s-t}\widehat{\varphi_{1}}(y,\xi)\,\overline{\widehat{\varphi_{2}}(y,\xi)}\,dyd\xi.$
(2.7)

We showed in [8] that (1.1) is a positive semi-definite operator with respect
to the scalar product (2.7) for any real $s$ and $t>-(n-1)/2$.

######  Definition 2.2.

[8, Def 3.4] For an integer $r\geq 0$, real $s$ and $t>-(n-1)/2$, introduce
the norm on $\mathcal{S}(T\mathbb{S}^{n-1})$

$\|\varphi\|^{2}_{H^{(r,s)}_{t}(T\mathbb{S}^{n-1})}=\big{(}(\mathrm{Id}+\Delta_{\xi})^{r}\varphi,\varphi\big{)}_{H^{s}_{t}(T{\mathbb{S}}^{n-1})}=\sum\limits_{l=0}^{r}{r\choose
l}\big{(}\Delta_{\xi}^{l}\varphi,\varphi\big{)}_{H^{s}_{t}(T{\mathbb{S}}^{n-1})}$
(2.8)

and define the Hilbert space $H^{(r,s)}_{t}(T{\mathbb{S}}^{n-1})$ as the
completion of ${\mathcal{S}}(T{\mathbb{S}}^{n-1})$ with respect to the norm
(2.8).

The Hilbert spaces $H^{(r,s)}_{t}(\mathbb{R}^{n};S^{m}\mathbb{R}^{n})$ are
defined in such a way that there is the equality in (1.3); see [8, Def 5.2].
We do not oresent the precise definition of
$H^{(r,s)}_{t}(\mathbb{R}^{n};S^{m}\mathbb{R}^{n})$ here since it does not
participate in calculations related to the range characterization result
below.

Recall that
${\mathcal{S}}_{\mathrm{s}ol}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$ is the
subspace of ${\mathcal{S}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$ consisting
of solenoidal tensor fields satisfying

$\sum\limits_{p=1}^{n}\frac{\partial f_{pi_{2}\dots i_{m}}}{\partial
x^{p}}=0.$ (2.9)

By [8, Theorem 1.1], for all $n\geq 2$ and $m\geq 0$, the ray transform

$I:{\mathcal{S}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})\rightarrow{\mathcal{S}}_{\pi(m)}(T{\mathbb{S}}^{n-1})$

extends to the isometric embedding of Hilbert spaces

$I:H^{(r,s)}_{t,\mathrm{sol}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})\rightarrow
H^{(r,s+1/2)}_{t+1/2,\pi(m)}(T{\mathbb{S}}^{n-1})$ (2.10)

for every integer $r\geq 0$, every real $s$ and every $t>-n/2$. See (2.1) for
the additional index $\pi(m)$ on the right-hand side of (2.10). Therefore the
range of the operator (2.10) is a closed subspace of
$H^{(r,s+1/2)}_{t+1/2,\pi(m)}(T{\mathbb{S}}^{n-1})$.

Let ${\mathcal{S}}^{\prime}(T{\mathbb{S}}^{n-1})$ be the space of tempered
distributions on $T{\mathbb{S}}^{n-1}$. By $\langle\varphi|\psi\rangle$ we
denote the value of a distribution
$\varphi\in{\mathcal{S}}^{\prime}(T{\mathbb{S}}^{n-1})$ on a test function
$\psi\in{\mathcal{S}}(T{\mathbb{S}}^{n-1})$. The embedding
${\mathcal{S}}(T{\mathbb{S}}^{n-1})\subset{\mathcal{S}}^{\prime}(T{\mathbb{S}}^{n-1})$
is defined by

$\langle\varphi|\psi\rangle=\int\limits_{{\mathbb{S}}^{n-1}}\int\limits_{\xi^{\bot}}\varphi(x,\xi)\psi(x,\xi)\,dxd\xi\quad\mbox{for}\quad\varphi,\psi\in{\mathcal{S}}(T{\mathbb{S}}^{n-1}).$
(2.11)

######  Proposition 2.3.

For an integer $r\geq 0$, real $s$ and $t\in\big{(}-(n-1)/2,(n-1)/2\big{)}$,
the space $H^{(r,s)}_{t}(T{\mathbb{S}}^{n-1})$ consists of tempered
distributions. More precisely, the identity map of
${\mathcal{S}}(T{\mathbb{S}}^{n-1})$ extends to a continuous embedding
$H^{(r,s)}_{t}(T{\mathbb{S}}^{n-1})\subset{\mathcal{S}}^{\prime}(T{\mathbb{S}}^{n-1})$.

###### Proof.

There exists the continuous embedding
$H^{(r,s)}_{t}(T{\mathbb{S}}^{n-1})\subset H^{s}_{t}(T{\mathbb{S}}^{n-1})$.
Therefore it suffices to prove Proposition 2.3 in the case of $r=0$, i.e., to
prove that the identical map of ${\mathcal{S}}(T{\mathbb{S}}^{n-1})$ extends
to a continuous embedding
$H^{s}_{t}(T{\mathbb{S}}^{n-1})\subset{\mathcal{S}}^{\prime}(T{\mathbb{S}}^{n-1})$.
To this end we will demonstrate the validity of the estimate

$|\langle\varphi|\psi\rangle|\leq
C_{s,t}\|\varphi\|_{H^{s}_{t}(T{\mathbb{S}}^{n-1})}\|\psi\|_{H^{-s}_{-t}(T{\mathbb{S}}^{n-1})}\quad\mbox{for}\quad\varphi,\psi\in{\mathcal{S}}(T{\mathbb{S}}^{n-1}).$

With the help of the Plancherel formula, (2.11) can be written as

$\langle\varphi|\psi\rangle=\int\limits_{{\mathbb{S}}^{n-1}}\int\limits_{\xi^{\bot}}\widehat{\varphi}(y,\xi)\widehat{\psi}(y,\xi)\,dyd\xi.$

Writing this in the form

$\langle\varphi|\psi\rangle=\int\limits_{{\mathbb{S}}^{n-1}}\int\limits_{\xi^{\bot}}\Big{(}|y|^{t}(1+|y|^{2})^{(s-t)/2}\widehat{\varphi}(y,\xi)\Big{)}\Big{(}|y|^{-t}(1+|y|^{2})^{-(s-t)/2}\widehat{\psi}(y,\xi)\Big{)}\,dyd\xi$

and applying the Cauchy – Bunyakovski inequality, we obtain

$\displaystyle|\langle\varphi|\psi\rangle|$
$\displaystyle\leq\Big{(}\int\limits_{{\mathbb{S}}^{n-1}}\int\limits_{\xi^{\bot}}|y|^{2t}(1+|y|^{2})^{s-t}|\widehat{\varphi}(y,\xi)|^{2}\,dyd\xi\Big{)}^{1/2}\times$
$\displaystyle\times\Big{(}\int\limits_{{\mathbb{S}}^{n-1}}\int\limits_{\xi^{\bot}}|y|^{-2t}(1+|y|^{2})^{t-s}|\widehat{\psi}(y,\xi)|^{2}\,dyd\xi\Big{)}^{1/2}=C_{s,t}\|\varphi\|_{H^{s}_{t}(T{\mathbb{S}}^{n-1})}\|\psi\|_{H^{-s}_{-t}(T{\mathbb{S}}^{n-1})}.$

∎

By Proposition 2.3, we have the linear continuous operator

$\displaystyle{\mathcal{J}}_{i_{1}j_{1}}\Big{(}{\mathcal{J}}_{i_{2}j_{2}}-(\xi_{i_{2}}X_{j_{2}}-\xi_{j_{2}}X_{i_{2}})\Big{)}\Big{(}{\mathcal{J}}_{i_{3}j_{3}}-2(\xi_{i_{3}}X_{j_{3}}-\xi_{j_{3}}X_{i_{3}})\Big{)}\dots$
(2.12)
$\displaystyle\dots\Big{(}{\mathcal{J}}_{i_{m+1}j_{m+1}}-m(\xi_{i_{m+1}}X_{j_{m+1}}-\xi_{j_{m+1}}X_{i_{m+1}})\Big{)}:H^{(r,s+1/2)}_{t+1/2,\pi(m)}(T{\mathbb{S}}^{n-1})\rightarrow{\mathcal{S}}^{\prime}(T{\mathbb{S}}^{n-1})$

under the only restriction $t\in\big{(}-n/2,(n-2)/2\big{)}$.

We are now ready to state the main result of this paper.

######  Theorem 2.4 (Main theorem).

Let $n\geq 3$ and $t\in\big{(}-(n-1)/2,(n-2)/2\big{)}$. For every integers
$m\geq 0,r\geq 0$ and every real $s$, the range of the operator (2.10)
coincides with the intersection of kernels of differential operators (2.12)
for all $1\leq i_{1},j_{1},\dots,i_{m+1},j_{m+1}\leq n$. In other words, a
“function” $\varphi\in H^{(r,s+1/2)}_{t+1/2,\pi(m)}(T{\mathbb{S}}^{n-1})$
belongs to the range of the operator (2.10) if and only if it satisfies the
equations

$\displaystyle{\mathcal{J}}_{i_{1}j_{1}}\Big{(}{\mathcal{J}}_{i_{2}j_{2}}-(\xi_{i_{2}}X_{j_{2}}-\xi_{j_{2}}X_{i_{2}})\Big{)}\Big{(}{\mathcal{J}}_{i_{3}j_{3}}-2(\xi_{i_{3}}X_{j_{3}}-\xi_{j_{3}}X_{i_{3}})\Big{)}\dots$
(2.13)
$\displaystyle\dots\Big{(}{\mathcal{J}}_{i_{m+1}j_{m+1}}-m(\xi_{i_{m+1}}X_{j_{m+1}}-\xi_{j_{m+1}}X_{i_{m+1}})\Big{)}\varphi=0$

for all $1\leq i_{1},j_{1},\dots,i_{m+1},j_{m+1}\leq n$. Equations (2.13) are
understood in the distribution sense.

##  3\. Range characterization for ray transform of Schwartz class tensor
fields using intrinsic John operators

We start with the following lemma.

######  Lemma 3.1.

Assume a function $\psi\in
C^{\infty}\big{(}{\mathbb{R}}^{n}\times({\mathbb{R}}^{n}\setminus\\{0\\})\big{)}$
to be positively homogeneous of degree $\lambda$ in $\xi$ and to satisfy

$\psi(x+t\xi,\xi)=\psi(x,\xi)\quad(t\in{\mathbb{R}}).$ (3.1)

Let $\varphi\in C^{\infty}(T{\mathbb{S}}^{n-1})$ be the restriction of $\psi$
to $T{\mathbb{S}}^{n-1}$. Then the equality

$(J_{ij}\psi)|_{T{\mathbb{S}}^{n-1}}=\big{(}{\mathcal{J}}_{ij}-(\lambda+1)(\xi_{i}X_{j}-\xi_{j}X_{i})\big{)}\varphi$
(3.2)

holds for any $1\leq i,j\leq n$.

###### Proof.

(Compare with [15, Lemma 2.4]) By the very definition of the operators $X_{i}$
and $\Xi_{i}$, the equality

${\mathcal{J}}_{ij}\varphi=(X_{i}\Xi_{j}-X_{j}\Xi_{i})\varphi=({\tilde{X}}_{i}{\tilde{\Xi}}_{j}-{\tilde{X}}_{j}{\tilde{\Xi}}_{i})\psi$
(3.3)

holds on $T{\mathbb{S}}^{n-1}$.

By the definition of the operators ${\tilde{X}}_{i}$ and ${\tilde{\Xi}}_{i}$,

$\displaystyle{\tilde{X}}_{i}{\tilde{\Xi}}_{j}-{\tilde{X}}_{j}{\tilde{\Xi}}_{i}$
$\displaystyle=\Big{(}\frac{\partial}{\partial
x^{i}}-\xi_{i}\xi^{p}\frac{\partial}{\partial
x^{p}}\Big{)}\Big{(}\frac{\partial}{\partial\xi^{j}}-x_{j}\xi^{q}\frac{\partial}{\partial
x^{q}}-\xi_{j}\xi^{q}\frac{\partial}{\partial\xi^{q}}\Big{)}$
$\displaystyle-\Big{(}\frac{\partial}{\partial
x^{j}}-\xi_{j}\xi^{p}\frac{\partial}{\partial
x^{p}}\Big{)}\Big{(}\frac{\partial}{\partial\xi^{i}}-x_{i}\xi^{q}\frac{\partial}{\partial
x^{q}}-\xi_{i}\xi^{p}\frac{\partial}{\partial\xi^{q}}\Big{)}.$

After opening parentheses, this becomes

$\displaystyle{\tilde{X}}_{i}{\tilde{\Xi}}_{j}-{\tilde{X}}_{j}{\tilde{\Xi}}_{i}$
$\displaystyle=\Big{(}\frac{\partial^{2}}{\partial
x^{i}\partial\xi^{j}}-\frac{\partial^{2}}{\partial
x^{j}\partial\xi^{i}}\Big{)}+\Big{(}x_{i}\xi^{p}\frac{\partial^{2}}{\partial
x^{j}\partial x^{p}}-x_{j}\xi^{p}\frac{\partial^{2}}{\partial x^{i}\partial
x^{p}}\Big{)}$ (3.4)
$\displaystyle+\Big{(}\xi_{i}\xi^{p}\frac{\partial^{2}}{\partial
x^{j}\partial\xi^{p}}-\xi_{j}\xi^{p}\frac{\partial^{2}}{\partial
x^{i}\partial\xi^{p}}\Big{)}-\Big{(}\xi_{i}\xi^{p}\frac{\partial^{2}}{\partial
x^{p}\partial\xi^{j}}-\xi_{j}\xi^{p}\frac{\partial^{2}}{\partial
x^{p}\partial\xi^{i}}\Big{)}$
$\displaystyle-(x_{i}\xi_{j}-x_{j}\xi_{i})\xi^{p}\xi^{q}\frac{\partial^{2}}{\partial
x^{p}\partial x^{q}}.$

We transform the fourth term on the right-hand side as follows:

$\displaystyle\xi_{i}\xi^{p}\frac{\partial^{2}}{\partial
x^{p}\partial\xi^{j}}-\xi_{j}\xi^{p}\frac{\partial^{2}}{\partial
x^{p}\partial\xi^{i}}$
$\displaystyle=\xi_{i}\frac{\partial}{\partial\xi^{j}}\Big{(}\xi^{p}\frac{\partial}{\partial
x^{p}}\Big{)}-\xi_{i}\frac{\partial}{\partial
x^{j}}-\xi_{j}\frac{\partial}{\partial\xi^{i}}\Big{(}\xi^{p}\frac{\partial}{\partial
x^{p}}\Big{)}+\xi_{j}\frac{\partial}{\partial x^{i}}$
$\displaystyle=\Big{(}\xi_{i}\frac{\partial}{\partial\xi^{j}}-\xi_{j}\frac{\partial}{\partial\xi^{i}}\Big{)}\xi^{p}\frac{\partial}{\partial
x^{p}}-\Big{(}\xi_{i}\frac{\partial}{\partial
x^{j}}-\xi_{j}\frac{\partial}{\partial x^{i}}\Big{)}.$

Substitute this expression into (3.4)

$\displaystyle{\tilde{X}}_{i}{\tilde{\Xi}}_{j}-{\tilde{X}}_{j}{\tilde{\Xi}}_{i}$
$\displaystyle=\Big{(}\frac{\partial^{2}}{\partial
x^{i}\partial\xi^{j}}-\frac{\partial^{2}}{\partial
x^{j}\partial\xi^{i}}\Big{)}+\Big{(}x_{i}\xi^{p}\frac{\partial^{2}}{\partial
x^{j}\partial x^{p}}-x_{j}\xi^{p}\frac{\partial^{2}}{\partial x^{i}\partial
x^{p}}\Big{)}$
$\displaystyle+\Big{(}\xi_{i}\xi^{p}\frac{\partial^{2}}{\partial
x^{j}\partial\xi^{p}}-\xi_{j}\xi^{p}\frac{\partial^{2}}{\partial
x^{i}\partial\xi^{p}}\Big{)}$
$\displaystyle-\Big{(}\xi_{i}\frac{\partial}{\partial\xi^{j}}-\xi_{j}\frac{\partial}{\partial\xi^{i}}\Big{)}\xi^{p}\frac{\partial}{\partial
x^{p}}+\Big{(}\xi_{i}\frac{\partial}{\partial
x^{j}}-\xi_{j}\frac{\partial}{\partial x^{i}}\Big{)}$
$\displaystyle-(x_{i}\xi_{j}-x_{j}\xi_{i})\xi^{p}\xi^{q}\frac{\partial^{2}}{\partial
x^{p}\partial x^{q}}.$

On using the operators $J_{ij},\
\langle\xi,\partial_{x}\rangle=\xi^{p}\frac{\partial}{\partial x^{p}}$ and
$\langle\xi,\partial_{\xi}\rangle=\xi^{p}\frac{\partial}{\partial\xi^{p}}$, we
write this in the form

$\displaystyle{\tilde{X}}_{i}{\tilde{\Xi}}_{j}-{\tilde{X}}_{j}{\tilde{\Xi}}_{i}$
$\displaystyle=J_{ij}+\Big{(}x_{i}\frac{\partial}{\partial
x^{j}}-x_{j}\frac{\partial}{\partial
x^{i}}\Big{)}\langle\xi,\partial_{x}\rangle+\Big{(}\xi_{i}\frac{\partial}{\partial
x^{j}}-\xi_{j}\frac{\partial}{\partial
x^{i}}\Big{)}\langle\xi,\partial_{\xi}\rangle$ (3.5)
$\displaystyle-\Big{(}\xi_{i}\frac{\partial}{\partial\xi^{j}}-\xi_{j}\frac{\partial}{\partial\xi^{i}}\Big{)}\langle\xi,\partial_{x}\rangle+\Big{(}\xi_{i}\frac{\partial}{\partial
x^{j}}-\xi_{j}\frac{\partial}{\partial
x^{i}}\Big{)}-(x_{i}\xi_{j}-x_{j}\xi_{i})\langle\xi,\partial_{x}\rangle^{2}.$

Since $\psi$ is positively homogeneous of degree $\lambda$ in $\xi$ and
satisfies (3.1), the equations

$\langle\xi,\partial_{x}\rangle\psi=0,\quad\langle\xi,\partial_{\xi}\rangle\psi=\lambda\psi$

hold. Therefore the equality (3.5) for the function $\psi$ looks as follows:

$({\tilde{X}}_{i}{\tilde{\Xi}}_{j}-{\tilde{X}}_{j}{\tilde{\Xi}}_{i})\psi=J_{ij}\psi+(\lambda+1)\Big{(}\xi_{i}\frac{\partial}{\partial
x^{j}}-\xi_{j}\frac{\partial}{\partial x^{i}}\Big{)}\psi.$ (3.6)

The relation

$\xi_{i}\frac{\partial}{\partial x^{j}}-\xi_{j}\frac{\partial}{\partial
x^{i}}=\xi_{i}X_{j}-\xi_{j}X_{i}$

holds on $T{\mathbb{S}}^{n-1}$ as is seen from the definition of $X_{i}$.
Therefore (3.6) implies that

$({\tilde{X}}_{i}{\tilde{\Xi}}_{j}-{\tilde{X}}_{j}{\tilde{\Xi}}_{i})\psi=J_{ij}\psi+(\lambda+1)(\xi_{i}X_{j}-\xi_{j}X_{i})\varphi$

on $T{\mathbb{S}}^{n-1}$. Together with (3.3), this gives (3.2). ∎

######  Lemma 3.2.

Assume a function $\psi\in
C^{\infty}\big{(}{\mathbb{R}}^{n}\times({\mathbb{R}}^{n}\setminus\\{0\\})\big{)}$
to be positively homogeneous of degree $\lambda$ in $\xi$ and to satisfy
$\psi(x+t\xi,\xi)=\psi(x,\xi)$ for $t\in{\mathbb{R}}$. Let $\varphi\in
C^{\infty}(T{\mathbb{S}}^{n-1})$ be the restriction of $\psi$ to
$T{\mathbb{S}}^{n-1}$. Then, for every $k=1,2,\dots$, the equality

$\displaystyle(J_{i_{1}j_{1}}\dots J_{i_{k}j_{k}}\psi)|_{T{\mathbb{S}}^{n-1}}$
(3.7)
$\displaystyle=\Big{(}{\mathcal{J}}_{i_{1}j_{1}}-c_{1}(\lambda,k)\big{(}\xi_{i_{1}}X_{j_{1}}-\xi_{j_{1}}X_{i_{1}}\big{)}\Big{)}\Big{(}{\mathcal{J}}_{i_{2}j_{2}}-c_{2}(\lambda,k)\big{(}\xi_{i_{2}}X_{j_{2}}-\xi_{j_{2}}X_{i_{2}}\big{)}\Big{)}$
$\displaystyle\dots\Big{(}{\mathcal{J}}_{i_{k}j_{k}}-c_{k}(\lambda,k)\big{(}\xi_{i_{k}}X_{j_{k}}-\xi_{j_{k}}X_{i_{k}}\big{)}\Big{)}\varphi$

holds for any indices $1\leq i_{1},j_{1},\dots,i_{k},j_{k}\leq n$, where

$c_{l}(\lambda,k)=\lambda-k+l+1.$ (3.8)

###### Proof.

The proof is going by induction in $k$.

By Lemma 3.1,

$(J_{i_{k}j_{k}}\psi)|_{T{\mathbb{S}}^{n-1}}=\Big{(}{\mathcal{J}}_{i_{k}j_{k}}-(\lambda+1)\big{(}\xi_{i_{k}}X_{j_{k}}-\xi_{j_{k}}X_{i_{k}}\big{)}\Big{)}\varphi.$
(3.9)

This coincides with (3.7) for $k=1$ with

$c_{1}(\lambda,1)=\lambda+1.$ (3.10)

Fix indices $i_{k}$ and $j_{k}$ and set $\psi^{\prime}=J_{i_{k}j_{k}}\psi$.
Formula (3.9) can be rewritten as

$\psi^{\prime}|_{T{\mathbb{S}}^{n-1}}=\Big{(}{\mathcal{J}}_{i_{k}j_{k}}-c_{1}(\lambda,1)\big{(}\xi_{i_{k}}X_{j_{k}}-\xi_{j_{k}}X_{i_{k}}\big{)}\Big{)}\varphi.$
(3.11)

The function $\psi^{\prime}$ is homogeneous of degree $\lambda-1$ in $\xi$.
Besides this, it satisfies $\langle\xi,\partial_{x}\rangle\psi^{\prime}=0$
since the operators $\langle\xi,\partial_{x}\rangle$ and $J_{kl}$ commute as
easily follows from definitions of these operators. By the induction
hypothesis,

$\displaystyle(J_{i_{1}j_{1}}\dots J_{i_{k-1}j_{k-1}}\psi^{\prime})$
$\displaystyle|_{T{\mathbb{S}}^{n-1}}=\Big{(}{\mathcal{J}}_{i_{1}j_{1}}-c_{1}(\lambda-1,k-1)\big{(}\xi_{i_{1}}X_{j_{1}}-\xi_{j_{1}}X_{i_{1}}\big{)}\Big{)}\dots$
$\displaystyle\dots\Big{(}{\mathcal{J}}_{i_{k-1}j_{k-1}}-c_{k}(\lambda-1,k-1)\big{(}\xi_{i_{k-1}}X_{j_{k-1}}-\xi_{j_{k-1}}X_{i_{k-1}}\big{)}\Big{)}\big{(}\psi^{\prime}|_{T{\mathbb{S}}^{n-1}}\big{)}.$

Substituting the expression (3.11) into the right-hand side of the last
formula and the value $\psi^{\prime}=J_{i_{k}j_{k}}\psi$ into the left-hand
side, we obtain (3.7) with the coefficients

$c_{1}(\lambda,k)=c_{1}(\lambda\\!-\\!1,k\\!-\\!1),\dots,c_{k-1}(\lambda,k)=c_{k-1}(\lambda\\!-\\!1,k\\!-\\!1);\
c_{k}(\lambda,k)=c_{1}(\lambda,1).$

Together with (3.10), last equalities imply (3.8) ∎

Let us distinguish the most important case when $k=m+1$ and $\lambda=m-1$.

######  Corollary 3.3.

For an integer $m\geq 0$, assume a function $\psi\in
C^{\infty}\big{(}{\mathbb{R}}^{n}\times({\mathbb{R}}^{n}\setminus\\{0\\})\big{)}$
to be positively homogeneous of degree $m-1$ in $\xi$ and to satisfy
$\psi(x+t\xi,\xi)=\psi(x,\xi)$ for $t\in{\mathbb{R}}$. Let $\varphi\in
C^{\infty}(T{\mathbb{S}}^{n-1})$ be the restriction of $\psi$ to
$T{\mathbb{S}}^{n-1}$. Then the equality

$\displaystyle(J_{i_{1}j_{1}}\dots
J_{i_{m+1}j_{m+1}}\psi)|_{T{\mathbb{S}}^{n-1}}$
$\displaystyle={\mathcal{J}}_{i_{1}j_{1}}\Big{(}{\mathcal{J}}_{i_{2}j_{2}}-\big{(}\xi_{i_{2}}X_{j_{2}}-\xi_{j_{2}}X_{i_{2}}\big{)}\Big{)}\Big{(}{\mathcal{J}}_{i_{3}j_{3}}-2\big{(}\xi_{i_{3}}X_{j_{3}}-\xi_{j_{3}}X_{i_{3}}\big{)}\Big{)}$
(3.12)
$\displaystyle\dots\Big{(}{\mathcal{J}}_{i_{m+1}j_{m+1}}-m\big{(}\xi_{i_{m+1}}X_{j_{m+1}}-\xi_{j_{m+1}}X_{i_{m+1}}\big{)}\Big{)}\varphi$

holds for any indices $1\leq i_{1},j_{1},\dots,i_{m+1},j_{m+1}\leq n$.

######  Theorem 3.4 (Intrinsic form of Theorem 2.1).

Let $n\geq 3$. A function
$\varphi\in{\mathcal{S}}_{\pi(m)}(T{\mathbb{S}}^{n-1})$ belongs to the range
of the operator (1.1) if and only if it satisfies

$\displaystyle{\mathcal{J}}_{i_{1}j_{1}}\Big{(}{\mathcal{J}}_{i_{2}j_{2}}-\big{(}\xi_{i_{2}}X_{j_{2}}-\xi_{j_{2}}X_{i_{2}}\big{)}\Big{)}\Big{(}{\mathcal{J}}_{i_{3}j_{3}}-2\big{(}\xi_{i_{3}}X_{j_{3}}-\xi_{j_{3}}X_{i_{3}}\big{)}\Big{)}$
(3.13)
$\displaystyle\dots\Big{(}{\mathcal{J}}_{i_{m+1}j_{m+1}}-m\big{(}\xi_{i_{m+1}}X_{j_{m+1}}-\xi_{j_{m+1}}X_{i_{m+1}}\big{)}\Big{)}\varphi=0$

for all indices $1\leq i_{1},j_{1},\dots,i_{m+1},j_{m+1}\leq n$.

###### Proof.

Necessity. Let a function
$\varphi\in{\mathcal{S}}_{\pi(m)}(T{\mathbb{S}}^{n-1})$ belong to the range of
the operator (1.1). Define the function $\psi\in
C^{\infty}\big{(}{\mathbb{R}}^{n}\times({\mathbb{R}}^{n}\setminus\\{0\\})\big{)}$
by (2.4). Then $\psi|_{T{\mathbb{S}}^{n-1}}=\varphi$. By Theorem 2.1, the
function $\psi$ satisfies the John equations (2.5) for all indices $1\leq
i_{1},j_{1},\dots,i_{m+1},j_{m+1}\leq n$. The left-hand side of equation
(3.12) is equal to zero. Equating the right-hand side of (3.12) to zero, we
arrive to (3.13).

Sufficiency. Let a function
$\varphi\in{\mathcal{S}}_{\pi(m)}(T{\mathbb{S}}^{n-1})$ satisfy (3.13). We
again define $\psi\in
C^{\infty}\big{(}{\mathbb{R}}^{n}\times({\mathbb{R}}^{n}\setminus\\{0\\})\big{)}$
by (2.3). Then $\psi|_{T{\mathbb{S}}^{n-1}}=\varphi$ and Corollary 3.3 applies
to the functions $\psi$ and $\varphi$. By (3.13), the right-hand side of
(3.12) is equal to zero. Equating the left-hand side of (3.12) to zero, we
obtain $(J_{i_{1}j_{1}}\dots
J_{i_{k-1}j_{k-1}}\psi)|_{T{\mathbb{S}}^{n-1}}=0$. Since $\psi$ satisfies

$\psi(x+t\xi,\xi)=\psi(x,\xi)\
(t\in\mathbb{R}),\quad\psi(x,t\xi)=\mbox{sgn}(t)\psi(x,\xi)\ (0\neq
t\in\mathbb{R}),$

the latter equation implies $J_{i_{1}j_{1}}\dots J_{i_{k-1}j_{k-1}}\psi=0$.
Applying Theorem 2.1, we infer that $\varphi$ belongs to the range of the
operator (1.1). ∎

Recall that
${\mathcal{S}}_{\mathrm{s}ol}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$ is the
space of smooth solenoidal fast decaying tensor fields defined by (2.9).

######  Lemma 3.5 (Main lemma).

Let $n\geq 3$ and $t\in\big{(}-(n-1)/2,(n-2)/2\big{)}$. For every integer
$m\geq 0$ and every real $s$, the following statement is valid.

Let a “function” $\varphi\in H^{s+1/2}_{t+1/2,\pi(m)}(T{\mathbb{S}}^{n-1})$
satisfy equations (2.13) in the distribution sense for all $1\leq
i_{1},j_{1},\dots,i_{m+1},j_{m+1}\leq n$. Assume also that

$(\varphi,If)_{H^{s+1/2}_{t+1/2}(T{\mathbb{S}}^{n-1})}=0\quad\mbox{\rm for
every tensor field}\quad
f\in{\mathcal{S}}_{sol}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n}).$ (3.14)

Then $\varphi=0$.

We emphasize that the integer $r$ does not participate in Lemma 3.5 (more
precisely, $r=0$ in the lemma). It is the main advantage of Lemma 3.5 as
compared with Theorem 2.4. Even in the case of $r=0$, Lemma 3.5 has one more
advantage: we do not need to recall the definition of the norm
$\|\cdot\|_{H^{(0,s)}_{t,\mathrm{sol}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})}$.

The proof of Lemma 3.5 will be presented in the next section. Now, we present
the proof of Theorem 2.4 assuming Lemma 3.5 to be valid.

###### Proof of Theorem 2.4.

We start with the easy proof of the “only if” statement of Theorem 2.4. Given
$f\in H^{(r,s)}_{t,\mathrm{sol}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$, set
$\varphi=If\in H^{(r,s+1/2)}_{t+1/2,\pi(m)}(T{\mathbb{S}}^{n-1})$. Since
${\mathcal{S}}_{\mathrm{sol}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$ is a
dense subspace of
$H^{(r,s)}_{t,\mathrm{sol}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$, we can
choose a sequence of tensor fields
$f_{k}\in{\mathcal{S}}_{sol}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})\
(k=1,2,\dots)$ such that

$f_{k}\to f\quad\mbox{in}\quad
H^{(r,s)}_{t,\mathrm{sol}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})\quad\mbox{as}\quad
k\to\infty.$ (3.15)

Set $\varphi_{k}=If_{k}$. By Theorem 3.4, every function
$\varphi_{k}\in{\mathcal{S}}_{\pi(m)}(T{\mathbb{S}}^{n-1})$ satisfies the
equations

$\displaystyle{\mathcal{J}}_{i_{1}j_{1}}\Big{(}{\mathcal{J}}_{i_{2}j_{2}}-(\xi_{i_{2}}X_{j_{2}}-\xi_{j_{2}}X_{i_{2}})\Big{)}\Big{(}{\mathcal{J}}_{i_{3}j_{3}}-2(\xi_{i_{3}}X_{j_{3}}-\xi_{j_{3}}X_{i_{3}})\Big{)}\dots$
(3.16)
$\displaystyle\dots\Big{(}{\mathcal{J}}_{i_{m+1}j_{m+1}}-m(\xi_{i_{m+1}}X_{j_{m+1}}-\xi_{j_{m+1}}X_{i_{m+1}})\Big{)}\varphi_{k}=0$

for all $1\leq i_{1},j_{1},\dots,i_{m+1},j_{m+1}\leq n$. Of course (3.15)
implies that

$\varphi_{k}\to\varphi\quad\mbox{in}\quad{\mathcal{S}}^{\prime}(T{\mathbb{S}}^{n-1})\quad\mbox{as}\quad
k\to\infty.$

Passing to limit as $k\to\infty$ in (3.16), we arrive to (2.13).

Now, we prove the “if” statement of Theorem 2.4. The following diagram is
helpful for better understanding our arguments

$\displaystyle
H^{(r,s)}_{t,\mathrm{sol}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})\quad{\stackrel{{\scriptstyle
I}}{{\longrightarrow}}}\quad
H^{(r,s+1/2)}_{t+1/2,\pi(m)}(T{\mathbb{S}}^{n-1})$ (3.17) $\displaystyle\qquad
i\downarrow\qquad\qquad\qquad\qquad\qquad\qquad\downarrow j$ $\displaystyle
H^{s}_{t,\mathrm{sol}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})\quad{\stackrel{{\scriptstyle
I}}{{\longrightarrow}}}\quad H^{s+1/2}_{t+1/2,\pi(m)}(T{\mathbb{S}}^{n-1})$

Here horizontal arrows (the ray transform) are isometric embeddings of Hilbert
spaces while vertical arrows are continuous embeddings. The diagram is
commutative, i.e., $iI=Ij$.

Let $({\rm{Ker}}\,J)^{(r,s+1/2)}_{t+1/2}$ be the closed subspace of
$H^{(r,s+1/2)}_{t+1/2,\pi(m)}(T{\mathbb{S}}^{n-1})$ consisting of “functions”
$\varphi$ satisfying equations (3.16) for all $1\leq
i_{1},j_{1},\dots,i_{m+1},j_{m+1}\leq n$. We denote the range of the operator
(2.10) by $({\rm{Ran}}\,I)^{(r,s+1/2)}_{t+1/2}$. It is also a closed subspace
of $H^{(r,s+1/2)}_{t+1/2,\pi(m)}(T{\mathbb{S}}^{n-1})$ by the Reshetnyak
formula. By the “only if” statement of Theorem 2.4,

$({\rm{Ran}}\,I)^{(r,s+1/2)}_{t+1/2}\subset({\rm{Ker}}\,J)^{(r,s+1/2)}_{t+1/2}.$
(3.18)

We have to prove that actually there is the equality in (3.18). To this end we
represent $({\rm{Ker}}\,J)^{(r,s+1/2)}_{t+1/2}$ as

$({\rm{Ker}}\,J)^{(r,s+1/2)}_{t+1/2}=({\rm{Ran}}\,I)^{(r,s+1/2)}_{t+1/2}\oplus\Big{(}({\rm{Ran}}\,I)^{(r,s+1/2)}_{t+1/2}\big{)}^{\bot},$
(3.19)

where $\Big{(}({\rm{Ran}}\,I)^{(r,s+1/2)}_{t+1/2}\big{)}^{\bot}$ is the
orthogonal complement of $({\rm{Ran}}\,I)^{(r,s+1/2)}_{t+1/2}$ in
$({\rm{Ker}}\,J)^{(r,s+1/2)}_{t+1/2}$ with respect to the scalar product
$(\cdot,\cdot)_{H^{s+1/2}_{t+1/2}(T{\mathbb{S}}^{n-1})}$. The embedding (3.18)
is the equality if and only if the second summand on the right-hand side of
(3.19) is equal to zero.

Let $\varphi\in\Big{(}({\rm{Ran}}\,I)^{(r,s+1/2)}_{t+1/2}\big{)}^{\bot}$. Then
the “function” $\varphi$ satisfies equations (2.13) for all $1\leq
i_{1},j_{1},\dots,i_{m+1},j_{m+1}\leq n$ and the statement (3.14) holds for
$\varphi$.

Recall that $j$ denotes the right vertical arrow in the diagram (3.17). The
embedding $j$ obviously preserves the scalar product
$(\cdot,\cdot)_{H^{s+1/2}_{t+1/2}(T{\mathbb{S}}^{n-1})}$. Therefore (3.14)
implies

$(j\varphi,If)_{H^{s+1/2}_{t+1/2}(T{\mathbb{S}}^{n-1})}=0\quad\mbox{\rm for
every tensor field}\quad
f\in{\mathcal{S}}_{sol}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n}).$

Besides this, the “function” $j\varphi$ satisfies equations (2.13) for all
$1\leq i_{1},j_{1},\dots,i_{m+1},j_{m+1}\leq n$ since these equations are
understood in the distribution sense. Thus, the “function” $j\varphi\in
H^{s+1/2}_{t+1/2,\pi(m)}(T{\mathbb{S}}^{n-1})$ satisfies all hypotheses of
Lemma 3.5. By the lemma, $j\varphi=0$. Since $j$ is an injective operator,
this implies $\varphi=0$. ∎

##  4\. Proof of Lemma 3.5

Recall that the weighted $L^{2}$-space $L^{2,s}_{t}({\mathbb{R}}^{n})$ was
defined in [15, Section 3] for $s\in\mathbb{R}$ and $t>-n/2$ as the completion
of ${\mathcal{S}}({\mathbb{R}}^{n})$ with respect to the norm

$\|f\|^{2}_{L^{2,s}_{t}({\mathbb{R}}^{n})}=\int\limits_{{\mathbb{R}}^{n}}|y|^{2t}(1+|y|^{2})^{s-t}|f(y)|^{2}\,dy.$

The Hilbert space $L^{2,s}_{t,e}(T{\mathbb{S}}^{n-1})$ of even “functions” was
defined in [15, Section 3] for $s\in\mathbb{R}$ and $t>-(n-1)/2$ as the
completion of ${\mathcal{S}}_{e}(T{\mathbb{S}}^{n-1})$ (the index $e$ stands
for “even”) with respect to the norm

$\|\varphi\|^{2}_{L^{2,s}_{t}(T{\mathbb{S}}^{n-1})}=\frac{\Gamma\big{(}\frac{n-1}{2}\big{)}}{4\pi^{(n+1)/2}}\int\limits_{{\mathbb{S}}^{n-1}}\int\limits_{\xi^{\bot}}|y|^{2t}(1+|y|^{2})^{s-t}|\varphi(y,\xi)|^{2}\,dyd\xi.$
(4.1)

By [15, Lemma 3.1], the Fourier transform
$F:{\mathcal{S}}_{e}(T{\mathbb{S}}^{n-1})\rightarrow{\mathcal{S}}_{e}(T{\mathbb{S}}^{n-1})$
extends to the bijective isometry of Hilbert spaces

$F:H^{s}_{t,e}(T{\mathbb{S}}^{n-1})\rightarrow
L^{2,s}_{t,e}(T{\mathbb{S}}^{n-1}).$

Quite similarly, the weighted $L^{2}$-space
$L^{2,s}_{t,o}(T{\mathbb{S}}^{n-1})$ of odd “functions” is defined for
$s\in\mathbb{R}$ and $t>-(n-1)/2$ as the completion of
${\mathcal{S}}_{o}(T{\mathbb{S}}^{n-1})$ (the index $o$ stands for “odd”) with
respect to the same norm (4.1). The Fourier transform
$F:{\mathcal{S}}_{o}(T{\mathbb{S}}^{n-1})\rightarrow{\mathcal{S}}_{o}(T{\mathbb{S}}^{n-1})$
extends to the bijective isometry of Hilbert spaces

$F:H^{s}_{t,o}(T{\mathbb{S}}^{n-1})\rightarrow
L^{2,s}_{t,o}(T{\mathbb{S}}^{n-1}).$

We will first prove the following statement.

######  Lemma 4.1.

Let $m\geq 0$ and $n\geq 3$ be integers, $s\in\mathbb{R}$ and
$t\in\big{(}-(n-1)/2,(n-1)/2\big{)}$. Given a “function” $\varphi\in
H^{s}_{t,\pi(m)}(T{\mathbb{S}}^{n-1})$ satisfying equations (2.13) in the
distribution sense for all $1\leq i_{1},j_{1},\dots,i_{m+1},j_{m+1}\leq n$,
let $\widehat{\varphi}\in L^{2,s}_{t,\pi(m)}(T{\mathbb{S}}^{n-1})$ be the
Fourier transform of $\varphi$. There exists a unique symmetric tensor field
$\widehat{\Phi}=(\widehat{\Phi}_{i_{1}\dots i_{m}})\in
L^{2,s}_{t}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$ satisfying

$y^{p}\,\widehat{\Phi}_{pi_{2}\dots i_{m}}(y)=0$ (4.2)

and such that

$\widehat{\varphi}(y,\xi)=\widehat{\Phi}_{i_{1}\dots
i_{m}}(y)\xi^{i_{1}}\dots\xi^{i_{m}}\quad\big{(}(y,\xi)\in
T{\mathbb{S}}^{n-1}\big{)}.$ (4.3)

Before we begin the proof of this lemma, we define a suitable smooth
hypersurface $V_{n}$ in
$({\mathbb{R}}^{n}\setminus\\{0\\})\times({\mathbb{R}}^{n}\setminus\\{0\\})$
in which the operators appearing in (2.13) (in the Fourier variables)
simplify.

Let

${\mathbb{S}}^{n-1}_{0}=\\{(0,\xi)\mid\xi\in{\mathbb{S}}^{n-1}\\}\subset
T{\mathbb{S}}^{n-1}$

and

$V_{n}=\\{(x,\xi)\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\mid x\neq 0,\xi\neq
0,\langle
x,\xi\rangle=0\\}\subset({\mathbb{R}}^{n}\setminus\\{0\\})\times({\mathbb{R}}^{n}\setminus\\{0\\}).$

$V_{n}$ is a hypersurface in
$({\mathbb{R}}^{n}\setminus\\{0\\})\times({\mathbb{R}}^{n}\setminus\\{0\\})$.
The manifold $V_{n}$ is diffeomorphic to
$(T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0})\times(0,\infty)$, the
diffeomorphism is defined by

${(T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0})\times(0,\infty)}\longrightarrow
V_{n},\quad(x,\xi;\rho)\mapsto(x,\rho\xi).$

On the other hand, $T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0}$ is a
hypersurface in $V_{n}$ and there is the projection

$p:V_{n}\rightarrow T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0},\quad
p(x,\xi)=(x,\xi/|\xi|).$

For the manifold $V_{n}$, we repeat arguments of [6, Section 4] with small
modifications. Namely, we define vector fields
$\tilde{X^{\prime}_{i}},\tilde{\Xi^{\prime}_{i}}\ (1\leq i\leq n)$ on
${\mathbb{R}}^{n}\times({\mathbb{R}}^{n}\setminus\\{0\\})=\\{(x,\xi)\mid\xi\neq
0\\}$ by

$\displaystyle\tilde{X^{\prime}_{i}}$ $\displaystyle=\frac{\partial}{\partial
x^{i}}-\frac{1}{|\xi|^{2}}\,\xi_{i}\xi^{p}\frac{\partial}{\partial
x^{p}}\quad(1\leq i\leq n),$ (4.4) $\displaystyle\tilde{\Xi^{\prime}_{i}}$
$\displaystyle=\frac{\partial}{\partial\xi^{i}}-\frac{1}{|\xi|^{2}}\,x_{i}\xi^{p}\frac{\partial}{\partial
x^{p}}-\frac{1}{|\xi|^{2}}\,\xi_{i}\xi^{p}\frac{\partial}{\partial\xi^{p}}\quad(1\leq
i\leq n).$

As compared with formulas (4.3) of [6], there is the additional factor
$1/|\xi|^{2}$ at some terms on right-hand sides of (4.4). The factor is added
to make $\tilde{X^{\prime}_{i}}$ and $\tilde{\Xi^{\prime}_{i}}$ homogeneous in
$\xi$. At points of $T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0}$, the
vector fields $\tilde{X^{\prime}_{i}}$ and $\tilde{\Xi^{\prime}_{i}}$ coincide
with $\tilde{X}_{i}$ and $\tilde{\Xi}_{i}$ respectively.

At every point $(x,\xi)\in V_{n}$, vectors ${\tilde{X}^{\prime}}_{i}(x,\xi)$
and ${\tilde{\Xi}^{\prime}}_{i}(x,\xi)\ (1\leq i\leq n)$ are tangent to
$V_{n}$. This follows from the obvious equalities

$\tilde{X^{\prime}_{i}}\langle
x,\xi\rangle=0,\quad\tilde{\Xi^{\prime}_{i}}\langle x,\xi\rangle=0\quad(1\leq
i\leq n)$

which hold on $V_{n}$. Let $X_{i}$ and $\Xi_{i}$ be the restrictions of vector
fields ${\tilde{X}^{\prime}}_{i}$ and ${\tilde{\Xi}^{\prime}}_{i}$ to the
manifold $V_{n}$ respectively. Thus, $X_{i}$ and $\Xi_{i}$ are smooth vector
fields on $V_{n}$ and can be considered as first order differential operators

$X_{i},\Xi_{i}:C^{\infty}(V_{n})\rightarrow C^{\infty}(V_{n}).$

The restrictions of the vector fields $X_{i}$ and $\Xi_{i}$ to the submanifold
$T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0}\subset V_{n}$ coincide
with the vector fields on $T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0}$
which were denoted by the same notations $X_{i}$ and $\Xi_{i}$ before. The
coincidence of notations should not imply any misunderstanding since the new
vector fields $X_{i}$ and $\Xi_{i}$ are natural extensions of old $X_{i}$ and
$\Xi_{i}$ from the manifold
$T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0}$ to $V_{n}$.

The main advantage of the manifold $V_{n}$, as compared with
$T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0}$, is the following fact.
The vector fields
$\langle\xi,\partial_{\xi}\rangle=\xi^{p}\frac{\partial}{\partial\xi^{p}}$ and
$x_{i}\frac{\partial}{\partial\xi^{j}}-x_{j}\frac{\partial}{\partial\xi^{i}}\
(1\leq i,j\leq n)$ are tangent to $V_{n}$ at every point $(x,\xi)\in V_{n}$.
This follows from the equalities

$\langle\xi,\partial_{\xi}\rangle(\langle
y,\xi\rangle)=0,\quad\Big{(}x_{i}\frac{\partial}{\partial\xi^{j}}-x_{j}\frac{\partial}{\partial\xi^{i}}\Big{)}\langle
y,\xi\rangle=0$

which hold at points $(x,\xi)\in V_{n}$. Thus, we have the well defined first
order differential operators

$\langle\xi,\partial_{\xi}\rangle:C^{\infty}(V_{n})\rightarrow
C^{\infty}(V_{n}),\quad
x_{i}\frac{\partial}{\partial\xi^{j}}-x_{j}\frac{\partial}{\partial\xi^{i}}:C^{\infty}(V_{n})\rightarrow
C^{\infty}(V_{n})\quad(1\leq i,j\leq n).$ (4.5)

Let ${\mathcal{D}}^{\prime}(V_{n})$ be the topological vector space of
distributions on the manifold $V_{n}$. Since differentiation and
multiplication of a distribution by a smooth function are well defined,

$\langle\xi,\partial_{\xi}\rangle:{\mathcal{D}}^{\prime}(V_{n})\rightarrow{\mathcal{D}}^{\prime}(V_{n}),\quad
y_{i}\frac{\partial}{\partial\xi^{j}}-y_{j}\frac{\partial}{\partial\xi^{i}}:{\mathcal{D}}^{\prime}(V_{n})\rightarrow{\mathcal{D}}^{\prime}(V_{n})\quad(1\leq
i,j\leq n)$

are well defined differential operators. We note that the latter formula is
written in the variables $(y,\xi)$ that are Fourier dual of $(x,\xi)$.

The operators $x_{i}\Xi_{j}-x_{j}\Xi_{i}:C^{\infty}(V_{n})\rightarrow
C^{\infty}(V_{n})$ can be expressed through operators (4.5):

$x_{i}\Xi_{j}-x_{j}\Xi_{i}=x_{i}\frac{\partial}{\partial\xi^{j}}-x_{j}\frac{\partial}{\partial\xi^{i}}-\frac{1}{|\xi|^{2}}\,(x_{i}\xi_{j}-x_{j}\xi_{i})\langle\xi,\partial_{\xi}\rangle\quad(1\leq
i,j\leq n).$ (4.6)

This easily follows from the definition (4.4).

######  Lemma 4.2.

Let a function $\varphi\in
C^{\infty}(T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0})$ satisfy the
equations

$\displaystyle\Big{(}x_{i_{m+1}}\Xi_{j_{m+1}}\\!-\\!x_{j_{m+1}}\Xi_{i_{m+1}}\Big{)}\Big{(}x_{i_{m}}\Xi_{j_{m}}\\!-\\!x_{j_{m}}\Xi_{i_{m}}+(x_{i_{m}}\xi_{j_{m}}\\!-\\!x_{j_{m}}\xi_{i_{m}})\Big{)}$
(4.7)
$\displaystyle\Big{(}x_{i_{m\\!-\\!1}}\Xi_{j_{m\\!-\\!1}}\\!-\\!x_{j_{m\\!-\\!1}}\Xi_{i_{m\\!-\\!1}}\\!+\\!2(x_{i_{m\\!-\\!1}}\xi_{j_{m\\!-\\!1}}\\!-\\!x_{j_{m\\!-\\!1}}\xi_{i_{m\\!-\\!1}})\Big{)}\dots$
$\displaystyle\dots\Big{(}x_{i_{1}}\Xi_{j_{1}}\\!-\\!x_{j_{1}}\Xi_{i_{1}}\\!+\\!m(x_{i_{1}}\xi_{j_{1}}\\!-\\!x_{j_{1}}\xi_{i_{1}})\Big{)}\varphi=0$

for all indices $1\leq i_{1},j_{1},\dots i_{m+1},j_{m+1}\leq n$. Define the
function $\psi\in C^{\infty}(V_{n})$ by

$\psi(x,\xi)=|\xi|^{m}\varphi\Big{(}x,\frac{\xi}{|\xi|}\Big{)}.$ (4.8)

This function satisfies the equations

$\Big{(}x_{i_{1}}\frac{\partial}{\partial\xi^{j_{1}}}-x_{j_{1}}\frac{\partial}{\partial\xi^{i_{1}}}\Big{)}\dots\Big{(}x_{i_{m+1}}\frac{\partial}{\partial\xi^{j_{m+1}}}-x_{j_{m+1}}\frac{\partial}{\partial\xi^{i_{m+1}}}\Big{)}\psi=0$
(4.9)

also for all indices.

###### Proof.

Being defined by (4.8), the function $\psi\in C^{\infty}(V_{n})$ is positively
homogeneous of degree $m$ in $\xi$ and satisfies
$\psi|_{T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0}}=\varphi$.

Let us fix indices $i_{1},j_{1},\dots,i_{m+1},j_{m+1}$ and define the
functions $\psi_{1},\psi_{2},\dots,\psi_{m+1}\in C^{\infty}(V_{n})$ by the
recurrent relations

$\psi_{1}=\Big{(}x_{i_{1}}\Xi_{j_{1}}-x_{j_{1}}\Xi_{i_{1}}+\frac{m}{|\xi|^{2}}(x_{i_{1}}\xi_{j_{1}}-x_{j_{1}}\xi_{i_{1}})\Big{)}\psi,$
(4.10)
$\psi_{k}=\Big{(}x_{i_{k}}\Xi_{j_{k}}-x_{j_{k}}\Xi_{i_{k}}+\frac{m-k+1}{|\xi|^{2}}(x_{i_{k}}\xi_{j_{k}}-x_{j_{k}}\xi_{i_{k}})\Big{)}\psi_{k-1}\quad(k=2,\dots,m+1),$
(4.11)

where $\Xi_{i}$ are now understood as differential operators on the manifold
$V_{n}$. Observe that every $\psi_{k}(x,\xi)$ is positively homogeneous of
degree $m-k$ in $\xi$. Combining (4.10)–(4.11), we see that

$\displaystyle\psi_{m+1}=$
$\displaystyle\Big{(}x_{i_{m+1}}\Xi_{j_{m+1}}\\!-\\!x_{j_{m+1}}\Xi_{i_{m+1}}\Big{)}\Big{(}x_{i_{m}}\Xi_{j_{m}}\\!-\\!x_{j_{m}}\Xi_{i_{m}}+\frac{1}{|\xi|^{2}}(x_{i_{m}}\xi_{j_{m}}\\!-\\!x_{j_{m}}\xi_{i_{m}})\Big{)}$
$\displaystyle\Big{(}x_{i_{m\\!-\\!1}}\Xi_{j_{m\\!-\\!1}}\\!-\\!x_{j_{m\\!-\\!1}}\Xi_{i_{m\\!-\\!1}}\\!+\\!\frac{2}{|\xi|^{2}}(x_{i_{m\\!-\\!1}}\xi_{j_{m\\!-\\!1}}\\!-\\!x_{j_{m\\!-\\!1}}\xi_{i_{m\\!-\\!1}})\Big{)}\\!\dots$
$\displaystyle\dots\Big{(}x_{i_{1}}\Xi_{j_{1}}\\!-\\!x_{j_{1}}\Xi_{i_{1}}\\!+\\!\frac{m}{|\xi|^{2}}(x_{i_{1}}\xi_{j_{1}}\\!-\\!x_{j_{1}}\xi_{i_{1}})\Big{)}\psi.$

Equation (4.7) is now written as

$\psi_{m+1}|_{T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0}}=0.$ (4.12)

Since the function $\psi_{m+1}\in C^{\infty}(V_{n})$ is positively homogeneous
of degree $-1$ in $\xi$ and satisfies (4.12), it must be identically equal to
zero, i.e.,

$\psi_{m+1}=0.$ (4.13)

Let us simplify the formula (4.10). By (4.6),

$x_{i_{1}}\Xi_{j_{1}}-x_{j_{1}}\Xi_{i_{1}}=x_{i_{1}}\frac{\partial}{\partial\xi^{j_{1}}}-x_{j_{1}}\frac{\partial}{\partial\xi^{i_{1}}}-\frac{1}{|\xi|^{2}}\,(x_{i_{1}}\xi_{j_{1}}-x_{j_{1}}\xi_{i_{1}})\langle\xi,\partial_{\xi}\rangle.$

Substituting this expression into (4.10), we obtain

$\psi_{1}=\Big{(}x_{i_{1}}\frac{\partial}{\partial\xi^{j_{1}}}-x_{j_{1}}\frac{\partial}{\partial\xi^{i_{1}}}+\frac{1}{|\xi|^{2}}(x_{i_{1}}\xi_{j_{1}}-x_{j_{1}}\xi_{i_{1}})(m-\langle\xi,\partial_{\xi}\rangle)\Big{)}\psi.$
(4.14)

Since the function $\psi(x,\xi)$ is a positively homogeneous of degree $m$ in
$\xi$, it satisfies

$(m-\langle\xi,\partial_{\xi}\rangle)\psi=0.$

Formula (4.14) is simplified to the following one:

$\psi_{1}=\Big{(}x_{i_{1}}\frac{\partial}{\partial\xi^{j_{1}}}-x_{j_{1}}\frac{\partial}{\partial\xi^{i_{1}}}\Big{)}\psi.$
(4.15)

In the same way we simplify the formula (4.11) to the following one:

$\psi_{k}=\Big{(}x_{i_{k}}\frac{\partial}{\partial\xi^{j_{k}}}-x_{j_{k}}\frac{\partial}{\partial\xi^{i_{k}}}\Big{)}\psi_{k-1}\quad(k=2,\dots,m+1).$
(4.16)

Combining (4.15)–(4.16), we see that

$\psi_{m+1}=\Big{(}x_{i_{m+1}}\frac{\partial}{\partial\xi^{j_{m+1}}}-x_{j_{m+1}}\frac{\partial}{\partial\xi^{i_{m+1}}}\Big{)}\dots\Big{(}x_{i_{1}}\frac{\partial}{\partial\xi^{j_{1}}}-x_{j_{1}}\frac{\partial}{\partial\xi^{i_{1}}}\Big{)}\psi.$

Therefore the equation (4.13) is now written as

$\Big{(}x_{i_{m+1}}\frac{\partial}{\partial\xi^{j_{m+1}}}-x_{j_{m+1}}\frac{\partial}{\partial\xi^{i_{m+1}}}\Big{)}\dots\Big{(}x_{i_{1}}\frac{\partial}{\partial\xi^{j_{1}}}-x_{j_{1}}\frac{\partial}{\partial\xi^{i_{1}}}\Big{)}\psi=0.$
(4.17)

This equation holds for any indices. Observe that the order of factors on the
left-hand side of (4.17) does not matter since the operators
$x_{i}\frac{\partial}{\partial\xi^{j}}-x_{j}\frac{\partial}{\partial\xi^{i}}$
and
$x_{k}\frac{\partial}{\partial\xi^{l}}-x_{l}\frac{\partial}{\partial\xi^{k}}$
commute. We have thus proved (4.9). ∎

###### Proof of Lemma 4.1.

(Compare with the proof of [15, Lemma 4.10].) We restrict the function
$\widehat{\varphi}\in L^{2,s}_{t,\pi(m)}(T{\mathbb{S}}^{n-1})$ to the open set
$T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0}$ and consider the
restriction as a distribution, i.e.,
$\widehat{\varphi}\in{\mathcal{D}}^{\prime}(T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0})$,
see Proposition 2.3.

Given
$\widehat{\varphi}\in{\mathcal{D}}^{\prime}(T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0})$,
similar to what was done in Lemma 4.2, we define
$\widehat{\psi}\in{\mathcal{D}}^{\prime}(V_{n})$ by

$\widehat{\psi}(y,\xi)=|\xi|^{m}\widehat{\varphi}\Big{(}y,\frac{\xi}{|\xi|}\Big{)}.$
(4.18)

The distribution $\widehat{\psi}(y,\xi)$ is positively homogeneous of degree
$m$ in $\xi$. Hereafter we use some basic facts of theory of homogeneous
distributions. Theory of homogeneous distributions on ${\mathbb{R}}^{n}$ is
presented in [4, Section 3.2]. In our case, $\widehat{\psi}(y,\xi)$ is a
homogeneous distribution of $\xi$ depending on the additional argument $y$. We
use only facts of the theory which can be generalized to distributions with a
parameter.

By Lemma 4.2, the equations (2.13) are written in terms of $\widehat{\psi}$ as
follows:

$\Big{(}y_{i_{1}}\frac{\partial}{\partial\xi^{j_{1}}}-y_{j_{1}}\frac{\partial}{\partial\xi^{i_{1}}}\Big{)}\dots\Big{(}y_{i_{m+1}}\frac{\partial}{\partial\xi^{j_{m+1}}}-y_{j_{m+1}}\frac{\partial}{\partial\xi^{i_{m+1}}}\Big{)}\widehat{\psi}=0\quad(1\leq
i_{1},j_{1},\cdots,i_{m+1},j_{m+1}\leq n).$ (4.19)

Lemma 4.2 is formulated for smooth functions. Nevertheless, it is true for
distributions as well. The proof is the same with minor modifications.

The projection

$q:V_{n}\to{\mathbb{R}}^{n}\setminus\\{0\\},\quad q(y,\xi)=y$ (4.20)

is a smooth fiber bundle with fibers $q^{-1}(y)=y^{\bot}\setminus\\{0\\}$
diffeomorphic to ${\mathbb{R}}^{n-1}\setminus\\{0\\}$. It determines the
linear continuous operator

$q^{*}:{\mathcal{D}}^{\prime}({\mathbb{R}}^{n}\setminus\\{0\\})\to{\mathcal{D}}^{\prime}(V_{n}),\quad
q^{*}f=f\circ q.$

See [4, Section 6.1] for the definition of the composition of a distribution
with a smooth map.

Let us consider first order differential operators

$A_{ij}=y_{i}\frac{\partial}{\partial\xi^{j}}-y_{j}\frac{\partial}{\partial\xi^{i}}:C^{\infty}(V_{n})\to
C^{\infty}(V_{n})\quad(1\leq i<j\leq n)$

as vector fields on the manifold $V_{n}$. These vector fields are tangent to
fibers of the bundle (4.20). At every point $(y,\xi)\in V_{n}$, the linear
hull of the vectors $A_{ij}(y,\xi)$ coincided with the tangent space
$T_{(y,\xi)}\big{(}q^{-1}(y)\big{)}$ of the fiber $q^{-1}(y)$.

For an arbitrary point $(y_{0},\xi_{0})\in V_{n}$, we can choose a
neighborhood $U\subset V_{n}$ of $(y_{0},\xi_{0})$ and local coordinates
$(y^{1},\dots,y^{n-1},\xi^{1},\dots,\xi^{n-1})$ defined in $U$ so that the
restriction of the projection (4.20) to $U$ is defined in these coordinates by
$q(y^{1},\dots,y^{n-1},\xi^{1},\dots,\xi^{n-1})=(y^{1},\dots,y^{n-1})$.
Moreover, we can assume without lost of generality that the coordinate map is
a diffeomorphism of $U$ onto
${\mathbb{R}}^{n-1}\times({\mathbb{R}}^{n-1}\setminus\\{0\\})$. Equations
(4.19) are written in these coordinates as

$\frac{\partial^{m+1}\widehat{\psi}}{\partial\xi^{\alpha_{1}}\dots\partial\xi^{\alpha_{m+1}}}=0\quad(1\leq\alpha_{1},\dots,\alpha_{m+1}\leq
n-1).$ (4.21)

The summation over repeated Greek indices is performed from 1 to $n-1$. Let us
use the following easy statement

If a distribution
$\widehat{\psi}\in{\mathcal{D}}^{\prime}\big{(}{\mathbb{R}}^{n-1}\times({\mathbb{R}}^{n-1}\setminus\\{0\\})\big{)}\
(n\geq 3)$ on
${\mathbb{R}}^{n-1}\times({\mathbb{R}}^{n-1}\setminus\\{0\\})=\\{(y^{1},\dots,y^{n-1},\xi^{1},\dots,\xi^{n-1})\\}$
satisfies equations (4.21), then it is a homogeneous polynomial of degree $m$
in $\xi$, i.e., there exist distributions
$\widehat{\Phi}_{\alpha_{1}\dots\alpha_{m}}\in{\mathcal{D}}^{\prime}({\mathbb{R}}^{n-1})$
such that

$\widehat{\psi}=(\tilde{q}\,{}^{*}\widehat{\Phi}_{\alpha_{1}\dots\alpha_{m}})\xi^{\alpha_{1}}\dots\xi^{\alpha_{m}},$
(4.22)

where
$\tilde{q}:{\mathbb{R}}^{n-1}\times({\mathbb{R}}^{n-1}\setminus\\{0\\})\rightarrow{\mathbb{R}}^{n-1}$
is defined by $\tilde{q}(y,\xi)=y$.

Applying this statement, we obtain the representation (4.22) for
$\widehat{\psi}$ in some local coordinates. Returning to Cartesian coordinates
in ${\mathbb{R}}^{n}$, we write the representation in the form

$\widehat{\psi}=(q^{*}\widehat{\Phi}_{i_{1}\dots
i_{m}})\xi^{i_{1}}\dots\xi^{i_{m}}$ (4.23)

with $\widehat{\Phi}_{i_{1}\dots
i_{m}}\in{\mathcal{D}}^{\prime}({\mathbb{R}}^{n}\setminus\\{0\\})$. The
summation over repeated Latin indices in (4.23) is performed from 1 to $n$. We
will also write (4.23) in the traditional form

$\widehat{\psi}(y,\xi)=\widehat{\Phi}_{i_{1}\dots
i_{m}}(y)\xi^{i_{1}}\dots\xi^{i_{m}}\quad(y\in{\mathbb{R}}^{n}\setminus\\{0\\},\xi\in
y^{\bot}\setminus\\{0\\})$ (4.24)

although it is understood in the distribution sense. Of course we can assume
$\widehat{\Phi}_{i_{1}\dots i_{m}}$ to be symmetric in all indices, i.e.,
$\widehat{\Phi}=(\widehat{\Phi}_{i_{1}\dots
i_{m}})\in{\mathcal{D}}^{\prime}({\mathbb{R}}^{n}\setminus\\{0\\};S^{m}{\mathbb{R}}^{n})$.

The tensor field $\widehat{\Phi}$ is not uniquely determined by (4.24) since
the equation (4.24) holds for $\xi\in y^{\bot}$ only (in the distribution
sense: the equation (4.23) holds on $V_{n}$). Nevertheless, we can state:
there exists a unique tensor field $\widehat{\Phi}=(\widehat{\Phi}_{i_{1}\dots
i_{m}})\in{\mathcal{D}}^{\prime}({\mathbb{R}}^{n}\setminus\\{0\\};S^{m}{\mathbb{R}}^{n})$
satisfying (4.24) and (4.2), where (4.2) is also understood in the
distribution sense. Indeed, let
$\widehat{\Phi^{\prime}}\in{\mathcal{D}}^{\prime}({\mathbb{R}}^{n}\setminus\\{0\\};S^{m}{\mathbb{R}}^{n})$
be an arbitrary tensor field satisfying (4.24). Define
$\widehat{\Phi}=(\widehat{\Phi}_{i_{1}\dots
i_{m}})\in{\mathcal{D}}^{\prime}({\mathbb{R}}^{n}\setminus\\{0\\};S^{m}{\mathbb{R}}^{n})$
by

$\widehat{\Phi}_{i_{1}\dots
i_{m}}(y)=\Big{(}\delta_{i_{1}}^{j_{1}}-\frac{y_{i_{1}}y^{j_{1}}}{|y|^{2}}\Big{)}\dots\Big{(}\delta_{i_{m}}^{j_{m}}-\frac{y_{i_{m}}y^{j_{m}}}{|y|^{2}}\Big{)}\widehat{\Phi^{\prime}}_{j_{1}\dots
j_{m}}(y)\quad(0\neq y\in{\mathbb{R}}^{n}),$

where $\delta_{i}^{j}$ is the Kronecker tensor. The latter formula can be also
understood in the distribution sense. The tensor field $\widehat{\Phi}$
satisfies (4.24) and (4.2) and is uniquely determined by these relations. In
terms of [12, Lemma 2.6.1], $\widehat{\Phi}$ is the tangential component of
$\widehat{\Phi^{\prime}}$.

The projection (4.20) is the composition $q=\pi\circ p$, where the smooth maps

$V_{n}\ {\stackrel{{\scriptstyle p}}{{\longrightarrow}}}\
T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0}\
{\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}}\
{\mathbb{R}}^{n}\setminus\\{0\\}$

are defined by $p(y,\xi)=(y,\xi/|\xi|)$ and $\pi(y,\xi)=y$. Actually $p$ and
$\pi$ are also smooth fiber bundles. The formula (4.23) can be written as

$\widehat{\psi}=|\xi|^{m}(p^{*}\pi^{*}\widehat{\Phi}_{i_{1}\dots
i_{m}})\frac{\xi^{i_{1}}}{|\xi|}\dots\frac{\xi^{i_{m}}}{|\xi|}.$

Since $\frac{\xi^{i}}{|\xi|}=\xi^{i}\circ p$, the formula can be also written
as

$\widehat{\psi}=|\xi|^{m}p^{*}\big{(}(\pi^{*}\widehat{\Phi}_{i_{1}\dots
i_{m}})\xi^{i_{1}}\dots\xi^{i_{m}}\big{)}.$ (4.25)

By the definition (4.18),

$\widehat{\psi}=|\xi|^{m}p^{*}\widehat{\varphi}.$

This gives together with (4.25)

$p^{*}\widehat{\varphi}=p^{*}\big{(}(\pi^{*}\widehat{\Phi}_{i_{1}\dots
i_{m}})\xi^{i_{1}}\dots\xi^{i_{m}}\big{)}.$

Since $p^{*}$ is an injective operator, this implies

$\widehat{\varphi}=(\pi^{*}\widehat{\Phi}_{i_{1}\dots
i_{m}})\xi^{i_{1}}\dots\xi^{i_{m}}$

or in the traditional form

$\widehat{\varphi}(y,\xi)=\widehat{\Phi}_{i_{1}\dots
i_{m}}(y)\xi^{i_{1}}\dots\xi^{i_{m}}\quad\big{(}(y,\xi)\in
T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0}\big{)}.$ (4.26)

We have thus proved that the statement (4.3) of Lemma 4.1 holds on
$T{\mathbb{S}}^{n-1}\setminus{\mathbb{S}}^{n-1}_{0}$. By the hypothesis of the
lemma, $\widehat{\varphi}\in L^{2,s}_{t,\pi(m)}(T{\mathbb{S}}^{n-1})$.
Together with (4.26), this easily implies that $\widehat{\Phi}\in
L^{2,s}_{t}({\mathbb{R}}^{n}\setminus\\{0\\};S^{m}{\mathbb{R}}^{n})$ and the
formula (4.3) holds in the conventional sense at almost all points $(y,\xi)\in
T{\mathbb{S}}^{n-1}$. ∎

###### Proof of Lemma 3.5.

By the definition of the $H^{s+1/2}_{t+1/2}(T{\mathbb{S}}^{n-1})$-norm, the
equation (3.14) is written as follows:

$\int\limits_{{\mathbb{S}}^{n-1}}\int\limits_{\xi^{\bot}}|y|^{2t+1}(1+|y|^{2})^{s-t}\widehat{\varphi}(y,\xi)\overline{\widehat{If}(y,\xi)}\,dyd\xi=0\quad\big{(}f\in{\mathcal{S}}_{sol}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})\big{)}.$
(4.27)

Applying Lemma 4.1 to the function $\widehat{\varphi}\in
L^{2,s+1/2}_{t+1/2.\pi(m)}(T{\mathbb{S}}^{n-1})$, we can state the existence
of a tensor field $\widehat{\Phi}\in
L^{2,s+1/2}_{t+1/2}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$ satisfying (4.2)
and (4.3). With the help of (4.3), the equation (4.27) becomes

$\int\limits_{{\mathbb{S}}^{n-1}}\int\limits_{\xi^{\bot}}|y|^{2t+1}(1+|y|^{2})^{s-t}{\widehat{\Phi}}{}^{i_{1}\dots
i_{m}}(y)\overline{\widehat{If}(y,\xi)}\,\xi_{i_{1}}\dots\xi_{i_{m}}\,dyd\xi=0.$
(4.28)

By (2.2),

$\widehat{If}(y,\xi)=(2\pi)^{1/2}\widehat{f}_{i_{1}\dots
i_{m}}(y)\xi^{i_{1}}\dots\xi^{i_{m}}\quad\big{(}(y,\xi)\in
T{\mathbb{S}}^{n-1}\big{)}.$

Substituting this expression into (4.28), we obtain

$\int\limits_{{\mathbb{S}}^{n-1}}\int\limits_{\xi^{\bot}}|y|^{2t+1}(1+|y|^{2})^{s-t}{\widehat{\Phi}}{}^{i_{1}\dots
i_{m}}(y)\overline{\widehat{f}^{j_{1}\dots
j_{m}}(y)}\,\xi_{i_{1}}\dots\xi_{i_{m}}\xi_{j_{1}}\dots\xi_{j_{m}}\,dyd\xi=0.$

Changing the order of integrations with the help of [12, Lemma 2.15.3], we
write this in the form

$\int\limits_{{\mathbb{R}}^{n}}|y|^{2t}(1+|y|^{2})^{s-t}{\widehat{\Phi}}{}^{i_{1}\dots
i_{m}}(y)\overline{\widehat{f}^{j_{1}\dots
j_{m}}(y)}\bigg{[}\int\limits_{{\mathbb{S}}^{n-1}\cap\xi^{\bot}}\xi_{i_{1}}\dots\xi_{i_{m}}\xi_{j_{1}}\dots\xi_{j_{m}}\,d^{n-2}\xi\bigg{]}\,dy=0.$
(4.29)

By [12, Lemma 2.15.4],

$\int\limits_{{\mathbb{S}}^{n-1}\cap\xi^{\bot}}\xi_{i_{1}}\dots\xi_{i_{m}}\xi_{j_{1}}\dots\xi_{j_{m}}\,d^{n-2}\xi=c_{m,n}\varepsilon^{m}_{i_{1}\dots
i_{m}j_{1}\dots j_{m}}(y)$

with some positive constant $c_{m,n}$, where

$\varepsilon_{ij}(y)=\delta_{ij}-y_{i}y_{j}/|y|^{2}$ (4.30)

and $\varepsilon^{m}$ is the $m^{\mathrm{th}}$ symmetric power of
$\varepsilon$. The formula (4.29) becomes

$\int\limits_{{\mathbb{R}}^{n}}|y|^{2t}(1+|y|^{2})^{s-t}\varepsilon^{m}_{i_{1}\dots
i_{m}j_{1}\dots j_{m}}(y){\widehat{\Phi}}{}^{i_{1}\dots
i_{m}}(y)\overline{\widehat{f}^{j_{1}\dots j_{m}}(y)}\,dy=0.$ (4.31)

This equality holds for every
$f\in{\mathcal{S}}_{\mathrm{sol}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$.

We write the equation (4.31) in the form

$\int\limits_{{\mathbb{R}}^{n}}\varepsilon^{m}_{i_{1}\dots i_{m}j_{1}\dots
j_{m}}(y)\Big{(}|y|^{t+1/2}(1+|y|^{2})^{(s-t)/2}{\widehat{\Phi}}{}^{i_{1}\dots
i_{m}}(y)\Big{)}\Big{(}|y|^{t-1/2}(1+|y|^{2})^{(s-t)/2}\overline{\widehat{f}^{j_{1}\dots
j_{m}}(y)}\Big{)}dy=0.$ (4.32)

Since the tensor field $\widehat{\Phi}$ belongs to
$L^{2,s+1/2}_{t+1/2}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$ and satisfies
(4.2), the tensor field
$\tilde{\Phi}=|y|^{t+1/2}(1+|y|^{2})^{(s-t)/2}{\widehat{\Phi}}$ belongs to
$L^{2}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$ and satisfies

$y_{p}\tilde{\Phi}{}^{pj_{2}\dots j_{m}}(y)=0.$ (4.33)

Recall that $2t-1>-n$ by the hypothesis $t\in\big{(}-(n-1)/2,(n-2)/2\big{)}$
of Lemma 3.5. Since
$f\in{\mathcal{S}}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$ is a solenoidal
tensor field, the tensor field $g=|y|^{t-1/2}(1+|y|^{2})^{(s-t)/2}\widehat{f}$
belongs to $L^{2}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$ and satisfies

$y_{p}g^{pj_{2}\dots j_{m}}(y)=0.$ (4.34)

The equation (4.32) is written in terms of $\tilde{\Phi}$ and $g$ as follows:

$\int\limits_{{\mathbb{R}}^{n}}\varepsilon^{m}_{i_{1}\dots i_{m}j_{1}\dots
j_{m}}(y){\tilde{\Phi}}{}^{i_{1}\dots i_{m}}(y)\overline{g^{j_{1}\dots
j_{m}}(y)}\,dy=0.$ (4.35)

In terms of [8, Section 4], (4.33) and (4.34) mean that $\tilde{\Phi}$ and $g$
are tangential tensor fields. Let
$L^{2}_{\top}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$ be the space of all
such tensor fields.

Since the tensor fields $\tilde{\Phi}$ and $g$ belong to
$L^{2}_{\top}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$, the second summand on
the right-hand side of (4.30) gives no contribution to the integral (4.35),
i.e. $\varepsilon$ can be replaced with the Kronecker tensor $\delta$ in
(4.35). We thus arrive at the equation

$\int\limits_{{\mathbb{R}}^{n}}\delta^{m}_{i_{1}\dots i_{m}j_{1}\dots
j_{m}}(y){\tilde{\Phi}}{}^{i_{1}\dots i_{m}}(y)\overline{g^{j_{1}\dots
j_{m}}(y)}\,dy=0.$ (4.36)

Now, we apply some arguments of [12, Section 2.15] to equation (4.36). Indeed,
the integrand in (4.35) is of the same structure as that in [12, formula
2.15.11]. The latter formula was transformed in [12, Section 2.15] to a form
similar to (4.36). We need only the statement following from further arguments
presented in [12, Section 2.15]: the integral (4.36) defines a positive scalar
product on the space $L^{2}_{\top}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$.

Being valid for every $g\in
L^{2}_{\top}({\mathbb{R}}^{n};S^{m}{\mathbb{R}}^{n})$, the equation (4.36)
implies $\tilde{\Phi}=0$. Together with the equality
$\tilde{\Phi}=|y|^{t+1/2}(1+|y|^{2})^{(s-t)/2}{\widehat{\Phi}}$, this gives
${\widehat{\Phi}}=0$. With the help of (4.26), the latter equality implies
$\widehat{\varphi}=0$ and $\varphi=0$. ∎

Remark. The hypothesis $n\geq 3$ of Lemma 3.5 was used in our arguments as
follows. It is important that ${\mathbb{R}}^{n-1}\setminus\\{0\\}$ is a
connected space for the validity of (4.22). In the case of $n=2$, the right-
hand side of (4.22) can be represented by two different polynomials on two
connected components of ${\mathbb{R}}^{1}\setminus\\{0\\}$.

## Acknowledgments

The first author would like to thank the Isaac Newton Institute for
Mathematical Sciences, Cambridge, UK, for support and hospitality during the
semester programme, _Rich and Nonlinear Tomography - a multidisciplinary
approach_ in 2023, supported by EPSRC Grant Number EP/R014604/1. The work of
the second author was performed according to the Russian Government research
assignment for IM SB RAS, project FWNF-2022-0006.

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