# Rotational smoothing

Pedro Caro  BCAM - Basque Center of Applied Mathematics, Bilbao, Spain.
<EMAIL_ADDRESS>,  Cristóbal J. Meroño  Universidad Politécnica de Madrid,
ETSI Caminos, Departmento de Matemática e Informática, Madrid, Spain.
<EMAIL_ADDRESS>and  Ioannis Parissis  Departamento de Matemáticas,
Universidad del Pais Vasco, Aptdo. 644, 48080 Bilbao, Spain and Ikerbasque,
Basque Foundation for Science, Bilbao, Spain<EMAIL_ADDRESS>
###### Abstract.

Rotational smoothing is a phenomenon consisting in a gain of regularity by
means of averaging over rotations. This phenomenon is present in operators
that regularize only in certain directions, in contrast to operators
regularizing in all directions. The gain of regularity is the result of
rotating the directions where the corresponding operator performs the
smoothing effect. In this paper we carry out a systematic study of the
rotational smoothing for a class of operators that includes $k$-vector-space
Riesz potentials in $\mathbb{R}^{n}$ with $k<n$, and the convolution with
fundamental solutions of elliptic constant-coefficient differential operators
acting on $k$-dimensional linear subspaces. Examples of the latter type of
operators are the planar Cauchy transform in $\mathbb{R}^{n}$, or a solution
operator for the transport equation in $\mathbb{R}^{n}$. The analysis of
rotational smoothing is motivated by the resolution of some inverse problems
under low-regularity assumptions.

###### Key words and phrases:

Rotational smoothing, multipliers with singular symbols, Riesz potentials,
Cauchy transform, special orthogonal group.

##  1\. Introduction

In this paper we introduce the concept of rotational smoothing to refer to a
phenomenon consisting in the gain of regularity by means of averaging over
rotations. In order to illustrate this phenomenon, let us consider an elliptic
homogeneous polynomial of order $m\in\mathbb{N}$ with $k\in\mathbb{N}$
variables and complex coefficients

(1) $p(\xi)=\sum_{|\alpha|=m}c_{\alpha}\xi^{\alpha}.$

Here $c_{\alpha}\in\mathbb{C}$, $\xi=(\xi_{1},\dots,\xi_{k})$,
$\alpha=(\alpha_{1},\dots,\alpha_{k})\in\mathbb{N}_{0}^{k}$ is a multi-index
so that $|\alpha|=\alpha_{1}+\dots+\alpha_{k}$ and
$\xi^{\alpha}=\xi_{1}^{\alpha_{1}}\dots\xi_{k}^{\alpha_{k}}$. By calling this
homogeneous polynomial elliptic, we mean that if $p(\xi)=0$ for
$\xi\in\mathbb{R}^{k}$ then $\xi=0$. It is well known that the differential
operator

(2)
$P={p(-\mathrm{i}\partial)}=\sum_{|\alpha|=m}(-\mathrm{i})^{m}c_{\alpha}\partial^{\alpha}$

associated to (1) —by analogy
$\partial^{\alpha}=\partial_{x_{1}}^{\alpha_{1}}\dots\partial_{x_{k}}^{\alpha_{k}}$—
admits a fundamental solution $E$ in $\mathbb{R}^{k}$ so that $u=E\star f$ is
a solution of the equation $Pu=f$ in $\mathbb{R}^{k}$ and

(3) $\|u\|_{\dot{H}^{m}(\mathbb{R}^{k})}\lesssim\|f\|_{L^{2}(\mathbb{R}^{k})}$

for all $f\in L^{2}(\mathbb{R}^{k})$.111Throughout the paper we write
$a\lesssim b$ or equivalently $b\gtrsim a$, when $a$ and $b$ are positive
constants and there exists $C>0$ so that $a\leq Cb$. We refer to $C$ as the
implicit constant. Additionally, if $a\lesssim b$ and $b\lesssim a$, we write
$a\eqsim b$. Here and throughout the article the semi-norm of the homogeneous
Sobolev space $\dot{H}^{s}(\mathbb{R}^{n})$ with $s\in\mathbb{R}$ and
$n\in\mathbb{N}$ is given by

$\|\phi\|_{\dot{H}^{s}(\mathbb{R}^{n})}=\Big{(}\int_{\mathbb{R}^{n}}|\xi|^{2s}|\widehat{\phi}(\xi)|^{2}\,\mathrm{d}\xi\Big{)}^{1/2},$

where
$\widehat{\phi}(\xi)=(2\pi)^{-n/2}\int_{\mathbb{R}^{n}}e^{-\mathrm{i}\xi\cdot
x}\phi(x)\,\mathrm{d}x$ denotes the Fourier transform of
$\phi\in\mathcal{S}(\mathbb{R}^{n})$ in the Schwartz class. Note that
inequality (3) entails a gain of $m$-derivatives for $u$ with respect to the
regularity of $f$.

Suppose now that $f$ depends on $n$ variables $x=(x_{1},\dots,x_{n})$ and $P$
still denotes the differential operator (2) of $k$ variables
$x^{\prime}=(x_{1},\dots,x_{k})$ with $n>k$. If $u$ denotes the solution of
$Pu=f$ in $\mathbb{R}^{n}$ given by $u=\tilde{E}\star f$ with
$\tilde{E}=E\otimes\delta_{0}$ and $\delta_{0}$ denoting the Dirac
distribution in $\mathbb{R}^{n-k}$ supported at the origin, then we can only
expect that

$\Big{(}\int_{\mathbb{R}^{n-k}}\|u(\centerdot,x^{\prime\prime})\|_{\dot{H}^{m}(\mathbb{R}^{k})}^{2}\mathrm{d}x^{\prime\prime}\Big{)}^{1/2}\lesssim\|f\|_{L^{2}(\mathbb{R}^{n})}.$

This means that the only possible regularity gain happens on the first $k$
variables and definitely not on the latter $n-k$ ones. This is due to the fact
that the convolution with $\tilde{E}$ in $x=(x^{\prime},x^{\prime\prime})$ has
the component $E$ which regularizes in $x^{\prime}$ and the component
$\delta_{0}$ which acts as the identity on the $x^{\prime\prime}$.

After this discussion the question of rotational smoothing can be set up as
follows. Consider the family of polynomials of $n$ variables
$\\{\tilde{p}_{Q}(\xi):Q\in\mathrm{SO}(n)\\}$ given by

(4) $\tilde{p}_{Q}(\xi)=p(Qe_{1}\cdot\xi,\dots,Qe_{k}\cdot\xi)$

with $p$ the polynomial function in (1). Here and throughout the paper
$\\{e_{1},\ldots,e_{n}\\}$ denotes the standard basis of $\mathbb{R}^{n}$ and

$\mathrm{SO}(n)=\\{Q\in\mathrm{GL}(n):\,Q^{\mathsf{T}}Q=\mathrm{id},\,\det
Q=1\\},$

where $\mathrm{GL}(n)$ is the general linear group of $n\times n$ real
matrices, $Q^{\mathsf{T}}$ is the transpose of $Q$ and $\mathrm{id}$ stands
for the identity matrix. Note that $\tilde{p}_{Q}(\xi)$ is not elliptic any
more since it vanishes on linear subspaces of codimension $k$. We continue by
writing $\tilde{P}(Q)$ for the differential operator associated to the
polynomial $\tilde{p}_{Q}(\xi)$, and let $u(\centerdot,Q)$ be the solution of
the equation $\tilde{P}(Q)u(\centerdot,Q)=f$ in $\mathbb{R}^{n}$ given by

$u(x,Q)=([Q^{\mathsf{T}}]^{\ast}\tilde{E}\star
f)(x),\qquad(x,Q)\in\mathbb{R}^{n}\times\mathrm{SO}(n),$

where $[Q^{\mathsf{T}}]^{\ast}\tilde{E}$ denotes the pull-back of $\tilde{E}$
by the map $x\in\mathbb{R}^{n}\mapsto Q^{\mathsf{T}}x\in\mathbb{R}^{n}$:
namely the pull-back acts as
$[Q^{\mathsf{T}}]^{\ast}\phi(x)=\phi(Q^{\mathsf{T}}x)$ for all
$\phi\in\mathcal{S}(\mathbb{R}^{n})$ and $x\in\mathbb{R}^{n}$. Then, the
rotational smoothing phenomenon consists of a gain of $m$-derivatives in all
the variables when performing an average over rotations $Q\in\mathrm{SO}(n)$.
To put in quantitative terms, we ask whether it is possible to have an
inequality of the following type

$\bigg{(}\int_{\mathrm{SO}(n)}\|u(\centerdot,Q)\|_{\dot{H}^{m}(\mathbb{R}^{n})}^{2}\mathrm{d}\mu(Q)\bigg{)}^{1/2}\lesssim\|f\|_{L^{2}(\mathbb{R}^{n})}$

for all $f\in L^{2}(\mathbb{R}^{n})$, where $\mu$ stands for the normalized
Haar measure on the special orthogonal group $\mathrm{SO}(n)$.

###  1.1. Main results

In this article, we carry out a systematic study of the phenomenon of
rotational smoothing for a general class of operators. These operators are
defined by considering, for each $\alpha\in\mathbb{R}_{+}=(0,\infty)$, a
continuous function $q_{\alpha}:\mathbb{R}^{k}\to\mathbb{C}$ satisfying the
following properties:

  1. (a)

If $q_{\alpha}(\eta)=0$ then $\eta=0$.

  2. (b)

The equality $q_{\alpha}(\lambda\eta)=\lambda^{\alpha}q_{\alpha}(\eta)$ holds
whenever $\lambda\in\mathbb{R}_{+}$ and
$\eta\in\mathbb{R}^{k}\setminus\\{0\\}$.

Then, define the symbols

$p_{\alpha}(\xi,Q)=q_{\alpha}(Qe_{1}\cdot\xi,\dots,Qe_{k}\cdot\xi),\qquad(\xi,Q)\in\mathbb{R}^{n}\times\mathrm{SO}(n).$

Let $P_{\alpha}(Q)$ be the operator defined by

$[P_{\alpha}(Q)f](x)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^{n}}e^{\mathrm{i}x\cdot\xi}\,p_{\alpha}(\xi,Q)\,\widehat{f}(\xi)\mathrm{d}\xi,\quad(x,Q)\in\mathbb{R}^{n}\times\mathrm{SO}(n),$

for $f\in\mathcal{S}(\mathbb{R}^{n})$. We ask whether we can provide a
solution operator $S_{\alpha}$ so that if $u(x,Q)=S_{\alpha}f(x,Q)$, then
$P_{\alpha}(Q)u(\centerdot,Q)=f$ in $\mathbb{R}^{n}$ for all
$Q\in\mathrm{SO}(n)$ and

(5)
$\bigg{(}\int_{\mathrm{SO}(n)}\|S_{\alpha}f(\centerdot,Q)\|_{\dot{H}^{\alpha}(\mathbb{R}^{n})}^{2}\mathrm{d}\mu(Q)\bigg{)}^{1/2}\lesssim\|f\|_{L^{2}(\mathbb{R}^{n})}$

for all $f\in L^{2}(\mathbb{R}^{n})$. Note again that these operators are not
elliptic since their symbols vanish on linear subspaces of codimension $k$.

In order to clarify how this framework generalizes the previous discussion
about differential operators, it may be convenient to note that if we choose
$q_{m}=p$ with $p$ the polynomial function in (1), then $p_{m}(\xi,Q)$ is the
polynomial $\tilde{p}_{Q}(\xi)$ in (4) and $P_{m}(Q)$ is its corresponding
differential operator $\tilde{P}(Q)$. The generalization described above
allows us to also introduce non-local operators in our analysis, such as the
fractional Laplacian acting on $k$-dimensional vector subspaces of
$\mathbb{R}^{n}$. Note that if $q_{\alpha}(\eta)=|\eta|^{\alpha}$ for all
$\eta\in\mathbb{R}^{k}$, then
$P_{\alpha}(\mathrm{id})=(-\Delta_{x^{\prime}})^{\alpha/2}$, where
$\Delta_{x^{\prime}}$ denotes
$\partial_{x_{1}}^{2}+\dots+\partial_{x_{k}}^{2}$.

Before stating the result quantifying the rotational smoothing phenomenon, we
need to complement the conditions (a) and (b) with an additional cancellation
assumption, whenever $\alpha-k\in\mathbb{N}_{0}$:

  1. (c)

If $\alpha-k\in\mathbb{N}_{0}$, then $q_{\alpha}$ satisfies that

$\int_{\SS^{k-1}}\frac{\theta^{\beta}}{q_{\alpha}(\theta)}\,\mathrm{d}S(\theta)=0,\qquad\forall\beta\in\mathbb{N}_{0}^{k},\quad|\beta|=\alpha-k,$

where the volume form $\mathrm{d}S$ denotes the usual spherical measure on
$\SS^{k-1}$. In the case $k=1$ we have that $\SS^{0}=\\{-1,1\\}$, and (c) is
interpreted as

$\int_{\SS^{k-1}}\frac{\theta^{\beta}}{q_{\alpha}(\theta)}\,\mathrm{d}S(\theta)=\frac{(-1)^{\alpha-1}}{q_{\alpha}(-1)}+\frac{1}{q_{\alpha}(1)}=0.$

This discussion presents the bare minimum framework and motivation to state
our main result. From now on we set

$\lceil\alpha-k\rceil=\min\\{l\in\mathbb{N}_{0}:\alpha-k<l\\}.$

######  Theorem 1.

Assume $n\in\mathbb{N}$ with $n\geq 2$. Let $k\in\\{1,\dots,n-1\\}$ and
$\alpha>0$ be given, and consider $q_{\alpha}\in C(\mathbb{R}^{k})$ satisfying
(a), (b) and (c). There exists a solution operator $S_{\alpha}$ and a constant
$C>0$ depending on $n$, $k$, $\alpha$ and $\delta$ such that, for all
$f\in\mathcal{S}(\mathbb{R}^{n})$ we have that

$\|S_{\alpha}f\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}=C\|f\|_{L^{2}(\mathbb{R}^{n})}$

with $\delta=\lceil\alpha-k\rceil$ if $\alpha-k/2<\lceil\alpha-k\rceil$, and
$\delta>\alpha-k/2$ if $\alpha-k/2\geq\lceil\alpha-k\rceil$.

Note that in the range $0<\alpha<k/2$, we can choose $\delta=0$ so that the
identity in 1 implies the inequality (5). Unfortunately, whenever $\alpha\geq
k/2$ we have to replace the $L^{2}$-average on the rotations by a weaker
quantity captured by the $H^{-\delta}$-norm on $\mathrm{SO}(n)$. Nevertheless,
this result is satisfactory in terms of the regularity gain because the
$-\delta$ derivatives of the Sobolev space in $\mathrm{SO}(n)$ act directly on
the operator and not on the functions of its domain. Thus, we still obtain a
gain of $\alpha$ derivatives in $\mathbb{R}^{n}$.

At this point, the statement of the result is vague in the sense that it is
not clear if the solution operator $S_{\alpha}$ can be explicitly defined.
Additionally, the restrictions on $\delta$ imposed by the relations between
$\alpha$ and $k$ may seem mysterious. For this reason, we present in Section 2
a more precise version of 1 where the definition of $S_{\alpha}$ and the
constant $C$ are explicit. The discussion in Section 2 also clarifies the
restrictions on $\delta$. For the case $0<\alpha<k$, the definition of
$S_{\alpha}$ is very simple and can be found in (11).

In order to analyse the need of replacing the $L^{2}$-average on the rotations
by the $H^{-\delta}$-norm on $\mathrm{SO}(n)$, we choose
$q_{\alpha}(\eta)=|\eta|^{\alpha}$ for all $\eta\in\mathbb{R}^{k}$ with
$0<\alpha<k$ and define the solution map, denoted in this case by
$I_{\alpha}$, as

$I_{\alpha}f(x,Q)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^{n}}e^{\mathrm{i}x\cdot\xi}\,\frac{\widehat{f}(\xi)}{|(Qe_{1}\cdot\xi)^{2}+\dots+(Qe_{k}\cdot\xi)^{2}|^{\alpha/2}}\,\mathrm{d}\xi,\qquad(x,Q)\in\mathbb{R}^{n}\times\mathrm{SO}(n),$

for $f\in\mathcal{S}(\mathbb{R}^{n})$. The solution operator $I_{\alpha}$ is
the $k$-vector-space Riesz potential, and its definition is possible because
the function

$\xi\in\mathbb{R}^{n}\setminus\Sigma_{Q}\longmapsto|(Qe_{1}\cdot\xi)^{2}+\dots+(Qe_{k}\cdot\xi)^{2}|^{-\alpha/2},$

with
$\Sigma_{Q}=\\{\xi\in\mathbb{R}^{n}:Qe_{1}\cdot\xi=\dots=Qe_{k}\cdot\xi=0\\}$,
can be extended to a locally-integrable function in $\mathbb{R}^{n}$. However,
if $k/2\leq\alpha<k$ and $f\in\mathcal{S}(\mathbb{R}^{n})$ satisfies that
$\widehat{f}(\xi)\neq 0$ whenever $\xi\in\Sigma_{Q}$ then

$\|I_{\alpha}f(\centerdot,Q)\|_{\dot{H}^{\alpha}(\mathbb{R}^{n})}^{2}=\int_{\mathbb{R}^{n}}\frac{|\xi|^{2\alpha}}{|(Qe_{1}\cdot\xi)^{2}+\dots+(Qe_{k}\cdot\xi)^{2}|^{\alpha}}|\widehat{f}(\xi)|^{2}\,\mathrm{d}\xi=\infty,\qquad\forall\,Q\in\mathrm{SO}(n).$

This follows from the fact that the function

$\xi\in\mathbb{R}^{n}\setminus\Sigma_{Q}\mapsto|\xi|^{\alpha}/|(Qe_{1}\cdot\xi)^{2}+\dots+(Qe_{k}\cdot\xi)^{2}|^{\alpha/2}$

cannot be extended to a locally-square-integrable function in
$\mathbb{R}^{n}$. We remark that the first identity above is a consequence of
Plancherel’s theorem. Thus, an $L^{2}$-average version of rotational smoothing
in the range $k/2\leq\alpha<k$ cannot be global. A possible way around this is
to postulate a local version of the rotational smoothing estimate where the
left-hand side of the identity in 1 would be replaced, in the case of the
$k$-vector-space Riesz potential, by

$\bigg{(}\int_{\mathrm{SO}(n)}\|(-\Delta)^{\alpha/2}I_{\alpha}f(\centerdot,Q)\|^{2}_{L^{2}(B_{R})}\,\mathrm{d}\mu(Q)\bigg{)}^{1/2}.$

Here $(-\Delta)^{\alpha/2}$ denotes the fractional Laplacian in
$\mathbb{R}^{n}$ and $B_{R}=\\{x\in\mathbb{R}^{n}:|x|<R\\}$. However, there is
no hope even for an inequality of this type without loss of derivatives. In
fact, we will see in the following result that, when localizing in balls and
then averaging in $\mathrm{SO}(n)$, the $k$-vector-space Riesz potential
exhibits a loss of derivatives in the range $k/2\leq\alpha<k$.

######  Theorem 2.

Assume $n\in\mathbb{N}$ with $n\geq 2$. Let $k\in\\{1,\dots,n-1\\}$ and
$k/2\leq\alpha<k$ be given. For every $R>0$ there exists a sequence of
functions $\\{f_{N}:N\in\mathbb{N}\\}$ such that
$\|f_{N}\|_{\dot{H}^{s}(\mathbb{R}^{n})}\eqsim N^{s}$ with $s\in\mathbb{R}$
and

$\int_{\mathrm{SO}(n)}\|(-\Delta)^{\alpha/2}I_{\alpha}f_{N}(\centerdot,Q)\|^{2}_{L^{2}(B_{R})}\,\mathrm{d}\mu(Q)\gtrsim\mathfrak{g}(RN)$

for all $N>\pi/(2R)$. The growth function $\mathfrak{g}$ is given by
$\mathfrak{g}(t)=\log t$ if $\alpha=k/2$ and $\mathfrak{g}(t)=t^{2\alpha-k}$
if $\alpha>k/2$. The implicit constants are independent of $R$ and $N$.

This theorem ensures a loss of at least $s$ derivatives with $s=\alpha-k/2$ if
$\alpha>k/2$ and $s>0$ if $\alpha=k/2$, compared to the maximum expected gain
of $\alpha$ derivatives. The proof of 2 is given in Section 5. Putting
together Theorems 1 and 2 we see that the $s$ uncontrolled derivatives of 2
are regularized by $-\delta$ derivatives in $\mathrm{SO}(n)$ of 1 —note that
$\delta\geq s$ in the range $k>\alpha\geq k/2$.

The homogeneity assumption (b) for the function $q_{\alpha}$ plays a key role
in the proof of 1. This property is immediately transferred to the
distribution in $\mathbb{R}^{k}\setminus\\{0\\}$ determined by the function
$\eta\in\mathbb{R}^{k}\setminus\\{0\\}\mapsto 1/q_{\alpha}(\eta)$. To
construct the solution operator $S_{\alpha}$, we need to extend this
distribution to belong to $\mathcal{S}^{\prime}(\mathbb{R}^{k})$. However, in
order for the extension to preserve the homogeneity in the case
$\alpha-k\in\mathbb{N}_{0}$, we need to impose the cancellation condition (c).
This is analysed in Section 3. Furthermore, as derived from the following
result, this cancellation property is needed not only to preserve the
homogeneity but also to guarantee that the rotational smoothing phenomenon
yields the maximum gain of $\alpha$ derivatives.

######  Theorem 3.

Assume $n\in\mathbb{N}$ with $n\geq 2$. Let $k\in\\{1,\dots,n-1\\}$ and
$\alpha>0$ be such that $\alpha-k\in\mathbb{N}_{0}$. Consider $q_{\alpha}\in
C(\mathbb{R}^{k})$ satisfying (a), (b), but assume that (c) does not hold.
Then, there exists a sequence of functions
$\\{f_{N}\in\mathcal{S}(\mathbb{R}^{n}):N\in\mathbb{N}\\}$ satisfying that
$\|f_{N}\|_{L^{2}(\mathbb{R}^{n})}=1$ and a number $N_{0}\in\mathbb{N}$ such
that

$\|S_{\alpha}f_{N}\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}\gtrsim\log
N,$

for all $N\geq N_{0}$.

This theorem ensures a logarithmic loss of derivatives with respect to the
maximum expected gain of $\alpha$ derivatives. The proof of 3 is given in
Section 5.

###  1.2. Rotational smoothing and inverse problems

The study of rotational smoothing is motivated by certain inverse boundary
value problems in a non-regular setting. In order to give the relevant
examples that drive the research reported in this paper, we consider the
inverse problem consisting in determining the advection term in the convection
equation from non-invasive measurements. One can consider two different
physical situations when formulating this problem.

In a first approach, we can assume the advection to be described by a static
vector field $A$ and data to be obtained from boundary measurements of
solutions of the steady-state convection equation $\Delta u+A\cdot\nabla u=0$
in a bounded open subset $D\subset\mathbb{R}^{n}$ with $n\geq 3$. Usually,
boundary data is encoded in the Dirichlet-to-Neumann map defined by

$\Lambda_{A}:f\mapsto\nu\cdot\nabla u|_{\partial D}$

where $\partial D$ denotes the boundary of $D$, $\nu$ is the outward unit
normal vector along $\partial D$, and $u$ is the solution of the convection
equation in $D$ such that $u|_{\partial D}=f$ —for more details on the
formulation of this inverse boundary value problem see [3, 8, 7]. The only
currently available procedure for determining $A$ from $\Lambda_{A}$ proceeds
by constructing some special solutions for the differential operator
$\Delta+A\cdot\nabla$. These solutions are called _complex geometrical optics_
and they are expressed as follows

$u_{\zeta}(x)=e^{\zeta\cdot x}(v_{\zeta}(x)+w_{\zeta}(x)),$

where $\zeta\in\mathbb{C}^{n}$ is such that $\zeta\cdot\zeta=0$; equivalently
$|\Re\zeta|=|\Im\zeta|$ and $\Re\zeta\cdot\Im\zeta=0$ with $\Re\zeta$ and
$\Im\zeta$ denoting the real and imaginary parts of $\zeta$ respectively. The
_amplitude_ $v_{\zeta}$ and the _correction term_ $w_{\zeta}$ are constructed
as solutions of the equations

(6) $2\zeta\cdot\nabla v_{\zeta}+\zeta\cdot Av_{\zeta}=0,$

and

(7) $(\Delta+2\zeta\cdot\nabla+A\cdot\nabla+\zeta\cdot
A)w_{\zeta}=-(\Delta+A\cdot\nabla)v_{\zeta}.$

In order to solve (6) we choose $\zeta=\tau Q(e_{1}+ie_{2})$ with
$Q\in\mathrm{SO}(n)$ and the amplitude in the form
$v_{\zeta}=e^{\psi_{\zeta}}$, with $\psi_{\zeta}$ solving

(8) $Q(e_{1}+ie_{2})\cdot\nabla\psi_{\zeta}=-\frac{1}{2}Q(e_{1}+ie_{2})\cdot
A.$

As we see from (7), the regularity of $v_{\zeta}$ is important in order to
ensure that the right-hand side $-(\Delta+A\cdot\nabla)v_{\zeta}$ is in an
appropriate space so that we can solve for $w_{\zeta}$. In the case that the
components of $A$ are just in $L^{\infty}(D)$, the a priori regularity of
solutions of the equation (8) is not sufficient. For this reason, one needs to
look for a hidden smoothing phenomenon, namely rotational smoothing. Note that
solutions of equation (8) fit in the framework of (11), with $\alpha=1$, $k=2$
and $q_{1}(\eta)=i\eta_{1}-\eta_{2}$ for all $\eta\in\mathbb{R}^{2}$, so that

$p_{1}(\xi,Q)=q_{1}(Qe_{1}\cdot\xi,Qe_{2}\cdot\xi)=\xi\cdot Q(ie_{1}-e_{2}).$

Indeed the solutions of (8) are given by the Cauchy transform along planes in
$\mathbb{R}^{n}$ with $n\geq 3$: for $f\in\mathcal{S}(\mathbb{R}^{n})$ and
$(x,Q)\in\mathbb{R}^{n}\times\mathrm{SO}(n)$,

$\mathcal{C}f(x,Q)=\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}\frac{\hat{f}(\xi)}{\xi\cdot
Q(ie_{1}-e_{2})}\mathrm{d}\xi.$

The second physical situation where one can consider this inverse problem
corresponds to allowing advection terms modelled by time-dependent vector
fields. Under these conditions, one has to consider the equation
$\partial_{t}u-\Delta u+A\cdot\nabla u=0$ in $D$ throughout the time interval
$(0,T)$. For a full description of the problem, see [2]. For the resolution of
this inverse problem, one constructs geometric optics

$u_{\kappa}(t,x)=e^{|\kappa|^{2}t+\kappa\cdot
x}(v_{\kappa}(t,x)+w_{\kappa}(t,x)),$

where $\kappa\in\mathbb{R}^{n}$. The _amplitude_ $v_{\kappa}$ and the
_correction term_ $w_{\kappa}$ are constructed as solutions of the equations

(9) $2\kappa\cdot\nabla v_{\kappa}-\kappa\cdot Av_{\kappa}=0$

and

$(\partial_{t}-\Delta-2\kappa\cdot\nabla+A\cdot\nabla+\kappa\cdot
A)w_{\kappa}=(-\partial_{t}+\Delta-A\cdot\nabla)v_{\kappa}.$

In order to solve (9) we choose $\kappa=\tau Qe_{1}$ with
$Q\in\mathrm{SO}(n)$, and the amplitude in the form
$v_{\kappa}=e^{\psi_{\kappa}}$ with $\psi_{\kappa}$ solving

(10) $Qe_{1}\cdot\nabla\psi_{\kappa}=\frac{1}{2}Qe_{1}\cdot A.$

In the case that the components of $A$ are just in $L^{\infty}((0,T)\times
D)$, one needs again to look for a rotational smoothing phenomenon. Note that
equation (10) fits the framework of Theorem 1 with $\alpha=k=1$ and
$q_{\alpha}(\eta)=i\eta$ for all $\eta\in\mathbb{R}$. Note that $\alpha-k=0$
and it also satisfies the cancellation property
$1/q_{\alpha}(-1)+1/q_{\alpha}(1)=0$.

We expect that the systematic analysis of rotational smoothing carried out in
this article can be applied to the resolution of the previous and possibly
other inverse problems in non-regular settings.

###  1.3. Outline

The paper contains five more sections and an appendix. In Section 2 we provide
a more detailed formulation of 1. Additionally, we give a straightforward
proof of the rotational smoothing phenomenon for the range $0<\alpha<k/2$.
Section 3 contains the construction of the extension of the distribution
determined by the function $\eta\in\mathbb{R}^{k}\setminus\\{0\\}\mapsto
1/q_{\alpha}(\eta)$. In Section 4 we prove 1 assuming some technical points
that will be postponed. Section 5 is devoted to prove Theorems 2 and 3. In
Section 6 we provide the necessary analytical framework to cover the gaps
assumed along the proof of 1. Lastly, in Appendix A we collect and prove some
key properties about $\mathrm{SO}(n)$.

######  Acknowledgements.

P.C. is supported by the grants BERC 2018-2021, Severo Ochoa SEV-2017-0718 and
PGC2018-094528-B-I00, and the Ikerbasque foundation. He wants to thank D.
Faraco and K. Rogers for some fruitful discussions on the topic some years
ago. C.J.M. is supported by MTM2017-85934-C3-3-P. I.P. is supported by
Ikerbasque, PGC2018-094528-B-I00 (AEI/FEDER, UE) and the Basque Government
grant T1247-19.

##  2\. More precise statements of rotational smoothing

In this section we restate 1 providing an explicit definition of the solution
operator $S_{\alpha}$ and computing the constant $C$. The statement is
reformulated in two theorems, one corresponding the to the range $0<\alpha<k$
and the other for $\alpha\geq k$.

Let $q_{\alpha}$ satisfy (a) and (b) with $0<\alpha<k$ and write
$p_{\alpha}(\xi,Q)=q_{\alpha}(Qe_{1}\cdot\xi,\dots,Qe_{k}\cdot\xi)$ for all
$(\xi,Q)\in\mathbb{R}^{n}\times\mathrm{SO}(n)$. We define the solution map
$S_{\alpha}$ for $f\in\mathcal{S}(\mathbb{R}^{n})$ by

(11)
$S_{\alpha}f(x,Q)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^{n}}e^{\mathrm{i}x\cdot\xi}\,\frac{\widehat{f}(\xi)}{p_{\alpha}(\xi,Q)}\,\mathrm{d}\xi,\qquad(x,Q)\in\mathbb{R}^{n}\times\mathrm{SO}(n).$

This definition is possible because the function
$\xi\in\mathbb{R}^{n}\setminus\Sigma_{Q}\mapsto 1/p_{\alpha}(\xi,Q)$ with
$\Sigma_{Q}=\\{\xi\in\mathbb{R}^{n}:Qe_{1}\cdot\xi=\dots=Qe_{k}\cdot\xi=0\\}$
can be extended to a locally-integrable function in $\mathbb{R}^{n}$ with
temperate growth. Let $r_{\alpha}$ denote the restriction to $\SS^{n-1}$ of
the function

(12)
$(\xi^{\prime},\xi^{\prime\prime})\in\mathbb{R}^{k}\times\mathbb{R}^{n-k}\longmapsto\left\\{\begin{aligned}
&1/q_{\alpha}(\xi^{\prime})&&\textnormal{if }\xi^{\prime}\neq 0,\\\
&0&&\textnormal{if }\xi^{\prime}=0.\end{aligned}\right.$

Thus, the rotational smoothing phenomenon can be formulated as follows.

######  Theorem 2.1.

Assume $n\in\mathbb{N}$ with $n\geq 2$. Let $k\in\\{1,\dots,n-1\\}$ and
$\alpha>0$ be given. Consider $q_{\alpha}\in C(\mathbb{R}^{k})$ satisfying (a)
and (b) with $\alpha<k$. Then, for all $f\in\mathcal{S}(\mathbb{R}^{n})$ we
have that

$\|S_{\alpha}f\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}=S(\SS^{n-1})^{-1/2}\|r_{\alpha}\|_{H^{-\delta}(\SS^{n-1})}\|f\|_{L^{2}(\mathbb{R}^{n})}$

with $\delta=0$ if $\alpha<k/2$ and $\delta>\alpha-k/2$ if $k/2\leq\alpha<k$.
Here $S(\SS^{n-1})$ stands for the measure of the unit sphere.

Theorem 2.1 be proved in Section 4.2. This result is a more explicit
formulation of 1 for the case $0<\alpha<k$. From this formulation we see that
the restriction on $\delta$ comes from the need of having $r_{\alpha}\in
H^{-\delta}(\SS^{n-1})$. Notice also that this identity shows explicitly that
the $-\delta$ derivatives affect only $r_{\alpha}$, and hence just the
operator, as we mentioned in the introduction.

Before proceeding with the task of providing a more explicit formulation of 1
for the range $\alpha\geq k$, we will include here a straightforward proof of
Theorem 2.1 in the special case where $\alpha<k/2$. That is, we prove that if
$\alpha<k/2$ then

(13)
$\|S_{\alpha}f\|_{L^{2}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}=S(\SS^{n-1})^{-1/2}\|r_{\alpha}\|_{L^{2}(\SS^{n-1})}\|f\|_{L^{2}(\mathbb{R}^{n})}$

for all $f\in\mathcal{S}(\mathbb{R}^{n})$. Despite the fact that the proof of
the case $\alpha<k/2$ will be contained in the argument presented in the
subsequent sections, the reader can use this more immediate proof as a roadmap
for the general case.

###### Proof of the identity (13)..

Using Plancherel’s identity, together with the identities (11), and switching
the order of integration we have that

$\int_{\mathrm{SO}(n)}\|S_{\alpha}f(\centerdot,Q)\|^{2}_{\dot{H}^{\alpha}(\mathbb{R}^{n})}\,\mathrm{d}\mu(Q)=\int_{\mathbb{R}^{n}}\|m_{\alpha}(\xi,\centerdot)\|^{2}_{L^{2}(\mathrm{SO}(n))}|\widehat{f}(\xi)|^{2}\,\mathrm{d}\xi,$

where

(14)
$m_{\alpha}(\xi,Q)=\frac{|\xi|^{\alpha}}{p_{\alpha}(\xi,Q)}\qquad(\xi,Q)\in\mathbb{R}^{n}\times\mathrm{SO}(n).$

Let $M\in\mathrm{SO}(n)$ be such that $\xi=|\xi|Me_{1}$; then

$m_{\alpha}(\xi,Q)=\frac{|\xi|^{\alpha}}{q_{\alpha}(|\xi|Qe_{1}\cdot
Me_{1},\dots,|\xi|Qe_{k}\cdot Me_{1})}=r_{\alpha}(Q^{\mathsf{T}}Me_{1}).$

In the last identity we have used the homogeneity of $q_{\alpha}$ given in (b)
and the fact that at this point $r_{\alpha}$ denotes the restriction of (12)
to $\SS^{n-1}$. Therefore,

$\begin{split}\|S_{\alpha}f\|_{L^{2}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}=\int_{\mathbb{R}^{n}}\bigg{(}\int_{\mathrm{SO}(n)}|r_{\alpha}(Q^{\mathsf{T}}Me_{1})|^{2}\mathrm{d}\mu(Q)\bigg{)}|\widehat{f}(\xi)|^{2}\,\mathrm{d}\xi.\end{split}$

Since $Q^{\mathsf{T}}M=(M^{\mathsf{T}}Q)^{\mathsf{T}}$ and the Haar measure is
left-invariant and invariant under transposition, see Proposition A.2, we have
again by Plancherel’s identity that

$\|S_{\alpha}f\|_{L^{2}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}=\bigg{(}\int_{\mathrm{SO}(n)}|r_{\alpha}(Qe_{1})|^{2}\mathrm{d}\mu(Q)\bigg{)}^{1/2}\|f\|_{L^{2}(\mathbb{R}^{n})}.$

Finally, by a very well-known relation between the Haar measure $\mu$ and the
hypersurface measure on $\SS^{n-1}$, see Proposition A.3, we have that

(15)
$\bigg{(}\int_{\mathrm{SO}(n)}|r_{\alpha}(Qe_{1})|^{2}\mathrm{d}\mu(Q)\bigg{)}^{1/2}=S(\SS^{n-1})^{-1/2}\|r_{\alpha}\|_{L^{2}(\SS^{n-1})}.$

This proves the identity (13) for $\alpha<k/2$. The fact that $r_{\alpha}\in
L^{2}(\SS^{n-1})$ when $\alpha<k/2$ is derived from the discussion in Section
4.4. ∎

We continue with the task of providing a more explicit formulation of 1 for
the range $\alpha\geq k$. Let $q_{\alpha}$ satisfy (a), (b) and (c). We pay
attention only to the case $\alpha\geq k$, but note that the more general
formulation presented in the following lines coincides with the previous one
for the case that $\alpha<k$. In Section 3 we will see that there is an
explicitly defined distribution
$q_{\alpha}^{-1}\in\mathcal{S}^{\prime}(\mathbb{R}^{k})$ that extends the
distribution

(16)
$\phi\in\mathcal{D}(\mathbb{R}^{k}\setminus\\{0\\})\longmapsto\int_{\mathbb{R}^{k}\setminus\\{0\\}}\frac{\phi(\eta)}{q_{\alpha}(\eta)}\,\mathrm{d}\eta.$

With the tempered distribution $q_{\alpha}^{-1}$ at hand, we first introduce
$q_{\alpha}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}$ in $\mathbb{R}^{n}$ as
the tensor product of $q_{\alpha}^{-1}\in\mathcal{S}^{\prime}(\mathbb{R}^{k})$
and the distribution corresponding to the function
$\mathbf{1}_{\mathbb{R}^{n-k}}(\kappa)=1$ for all $\kappa\in\mathbb{R}^{n-k}$.
Note that this latter tempered distribution in $\mathbb{R}^{n}$ generalizes
(12) for $\alpha\geq k$. In the same spirit as for the case $\alpha<k$, the
distribution $q_{\alpha}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}$ will
determine another distribution $r_{\alpha}$ in $\SS^{n-1}$ such that
$r_{\alpha}\in H^{-\delta}(\SS^{n-1})$ with $\delta=\lceil\alpha-k\rceil$ if
$\alpha-k/2<\lceil\alpha-k\rceil$ and $\delta>\alpha-k/2$ if
$\alpha-k/2\geq\lceil\alpha-k\rceil$. For convenience recall here that

$\lceil\alpha-k\rceil=\min\\{l\in\mathbb{N}_{0}:\alpha-k<l\\}.$

Additionally, recall that in the range $\alpha<k$ the function $r_{\alpha}$
was chosen precisely as the restriction of (12) to $\SS^{n-1}$. By analogy, it
is convenient to visualize $r_{\alpha}$ as a formal restriction of
$q_{\alpha}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}$ to $\SS^{n-1}$. Then, we
consider the family of tempered distributions
$\\{p_{\alpha}^{-1}(\centerdot,Q)\in\mathcal{S}^{\prime}(\mathbb{R}^{n}):\,Q\in\mathrm{SO}(n)\\}$
defined as

$p_{\alpha}^{-1}(\centerdot,Q)=[Q^{\mathsf{T}}]^{\ast}(q_{\alpha}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}),$

and the solution operator defined for $f\in\mathcal{S}(\mathbb{R}^{n})$ as

(17) $S_{\alpha}f(x,Q)=\frac{1}{(2\pi)^{n/2}}\langle
p_{\alpha}^{-1}(\centerdot,Q),\widehat{f(\centerdot+x)}\rangle,\quad(x,Q)\in\mathbb{R}^{n}\times\mathrm{SO}(n).$

Here the duality pairing represents the action of the tempered distribution
$p_{\alpha}^{-1}(\centerdot,Q)$ on the Fourier transform of the function
$y\in\mathbb{R}^{n}\mapsto f(y+x)$.

######  Theorem 2.2.

Assume $n\in\mathbb{N}$ with $n\geq 2$. Let $k\in\\{1,\dots,n-1\\}$ and
$\alpha\geq k$ be given, and consider $q_{\alpha}\in C(\mathbb{R}^{k})$
satisfying (a), (b) and (c). Define the solution operator $S_{\alpha}$ as in
(17). Then, there exists a distribution $r_{\alpha}\in H^{-\delta}(\SS^{n-1})$
determined by $q_{\alpha}$ such that, for all
$f\in\mathcal{S}(\mathbb{R}^{n})$ we have that

$\|S_{\alpha}f\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}=S(\SS^{n-1})^{-1/2}\|r_{\alpha}\|_{H^{-\delta}(\SS^{n-1})}\|f\|_{L^{2}(\mathbb{R}^{n})}$

with $\delta=\lceil\alpha-k\rceil$ if $\alpha-k/2<\lceil\alpha-k\rceil$ and
$\delta>\alpha-k/2$ if $\alpha-k/2\geq\lceil\alpha-k\rceil$. Here
$S(\SS^{n-1})$ stands for the measure of the unit sphere.

We give an explicit formula for $r_{\alpha}$ in Proposition 4.4 for a specific
choice of local coordinates on the sphere $\SS^{n-1}$. Theorem 2.2 be proved
in Section 4.2.

##  3\. The tempered distribution $q_{\alpha}^{-1}$

Here we discuss the extension of the distributions

(18)
$\phi\in\mathcal{D}(\mathbb{R}^{k}\setminus\\{0\\})\longmapsto\int_{\mathbb{R}^{k}\setminus\\{0\\}}\frac{\phi(\eta)}{q_{\alpha}(\eta)}\,\mathrm{d}\eta$

that we use to define the solution operator $S_{\alpha}$ of Theorem 2.2 as
explained in Section 2. Recall that
$\mathcal{D}(\mathbb{R}^{k}\setminus\\{0\\})$ denotes the space of smooth
functions in $\mathbb{R}^{k}\setminus\\{0\\}$ with compact support. As
mentioned earlier, the extension of (18) will be denoted by $q_{\alpha}^{-1}$,
it will belong to $\mathcal{S}^{\prime}(\mathbb{R}^{k})$ and it will have the
same homogeneity property as (18). More precisely, we will construct the
distribution $q_{\alpha}^{-1}\in\mathcal{S}^{\prime}(\mathbb{R}^{k})$
satisfying:

  1. (i)

For every $\phi\in\mathcal{S}(\mathbb{R}^{k})$ such that
$\operatorname*{supp}\phi\subset\mathbb{R}^{k}\setminus\\{0\\}$ we have that

$\langle
q_{\alpha}^{-1},\phi\rangle=\int_{\mathbb{R}^{k}\setminus\\{0\\}}\frac{\phi(\eta)}{q_{\alpha}(\eta)}\,\mathrm{d}\eta.$

  2. (ii)

For every $\phi\in\mathcal{S}(\mathbb{R}^{k})$ and $\lambda>0$ we have that

$\langle q_{\alpha}^{-1},\phi\rangle=\lambda^{-\alpha}\langle
q_{\alpha}^{-1},\phi_{\lambda}\rangle,$

where $\phi_{\lambda}(\eta)=\lambda^{k}\phi(\lambda\eta)$ for
$\eta\in\mathbb{R}^{k}$.

We establish the extension of the distribution in (18) in two different
lemmas. First when $\alpha-k\notin\mathbb{N}_{0}$ and then for
$\alpha-k\in\mathbb{N}_{0}$. As we will see below, separating these cases is
convenient since for $\alpha-k\notin\mathbb{N}_{0}$ there is a unique
extension satisfying (ii). On the other hand, for $\alpha-k\in\mathbb{N}_{0}$
there are no extensions satisfying (ii) unless the cancellation condition (c)
holds. Also, in the latter case the homogeneity does not uniquely determine
the extension. In order to state the extension lemmas we recall once again
that

$\lceil\alpha-k\rceil=\min\\{l\in\mathbb{N}_{0}:\alpha-k<l\\},$

and introduce

$\lfloor\alpha-k\rfloor=\max\\{l\in\mathbb{N}_{0}:l\leq\alpha-k\\}.$

######  Lemma 3.1.

Let $k\in\mathbb{N}$ and $\alpha>0$ be given, and consider $q_{\alpha}\in
C(\mathbb{R}^{k})$ satisfying (a) and (b). If $\alpha-k\notin\mathbb{N}_{0}$,
then the linear functional

$\phi\in\mathcal{S}(\mathbb{R}^{k})\longmapsto\begin{cases}\displaystyle\int_{\mathbb{R}^{k}}\frac{\phi(\eta)}{q_{\alpha}(\eta)}\,\mathrm{d}\eta,&\alpha-k<0,\vspace{1.2em}\\\
\displaystyle\frac{\Gamma(\alpha-k-\lfloor\alpha-k\rfloor)}{\Gamma(\alpha-k+1)}\int_{\mathbb{R}^{k}}\frac{(\eta\cdot\nabla)^{\lceil\alpha-k\rceil}\phi(\eta)}{q_{\alpha}(\eta)}\,\mathrm{d}\eta,&\alpha-k>0,\end{cases}$

denoted by $q_{\alpha}^{-1}$, belongs to
$\mathcal{S}^{\prime}(\mathbb{R}^{k})$ and it is the only tempered
distribution in $\mathbb{R}^{k}$ satisfying the properties (i) and (ii). Here
$\Gamma$ denotes the Gamma function.

The extension of the distribution in (18) in the case
$\alpha-k\in\mathbb{N}_{0}$ is the content of the following lemma.

######  Lemma 3.2.

Let $k\in\mathbb{N}$ and $\alpha>0$ be given and consider $q_{\alpha}\in
C(\mathbb{R}^{k})$ satisfying (a) and (b). If $\alpha-k\in\mathbb{N}_{0}$,
then the linear functional

$\phi\in\mathcal{S}(\mathbb{R}^{k})\longmapsto\frac{-1}{(\alpha-k)!}\int_{\mathbb{R}^{k}}\frac{\log|\eta|}{q_{\alpha}(\eta)}(\eta\cdot\nabla)^{\alpha-k+1}\phi(\eta)\,\mathrm{d}\eta,$

denoted by $q_{\alpha}^{-1}$, belongs to
$\mathcal{S}^{\prime}(\mathbb{R}^{k})$ and satisfies (i). Moreover,
$q_{\alpha}^{-1}$ satisfies the homogeneity property (ii) if and only if
$q_{\alpha}$ satisfies (c).

Before going into the proof of each one of the two lemmas above we make some
comments that are relevant for any $k\in\mathbb{N}$ and $\alpha>0$.

Note that for $\phi\in\mathcal{D}(\mathbb{R}^{k}\setminus\\{0\\})$ we can use
spherical coordinates in order to write

(19)
$\int_{\mathbb{R}^{k}\setminus\\{0\\}}\frac{\phi(\eta)}{q_{\alpha}(\eta)}\,\mathrm{d}\eta=\int_{\SS^{k-1}}\frac{1}{q_{\alpha}(\theta)}\bigg{(}\int_{0}^{\infty}\rho^{k-\alpha-1}\phi(\rho\theta)\,\mathrm{d}\rho\bigg{)}\,\mathrm{d}S(\theta),$

where the volume form $\mathrm{d}S$ denotes the usual spherical measure on
$\SS^{k-1}$. With this identity in hand we see that if we aim at extending
(18) it would be convenient to first extend the distribution in
$\mathbb{R}\setminus\\{0\\}$, given by

(20)
$\psi\in\mathcal{D}(\mathbb{R}\setminus\\{0\\})\longmapsto\int_{0}^{\infty}\rho^{k-\alpha-1}\psi(\rho)\,\mathrm{d}\rho.$

In order to carry out this extension, consider for $a\in\mathbb{R}$ the
function ${\rm x}_{+}^{a}:\mathbb{R}\setminus\\{0\\}\rightarrow[0,\infty)$
given by the formula

(21) $\mathrm{x}_{+}^{a}(x)=\begin{cases}x^{a},&x>0,\vspace{.9em}\\\
0,&x<0.\end{cases}$

If $-1<a$ then we can extend the function ${\rm x}_{+}^{a}$ to a function
defined in the whole set $\mathbb{R}$ by setting for example ${\rm
x}_{+}^{a}(0)=0$; this extension is locally integrable and consequently
defines a tempered distribution in $\mathbb{R}$, still denoted by ${\rm
x}_{+}^{a}$:

(22) $\langle{\rm
x}_{+}^{a},\psi\rangle=\int_{0}^{\infty}x^{a}\psi(x)\,\mathrm{d}x,\qquad\psi\in\mathcal{S}(\mathbb{R}),\,a>-1.$

Our next step is to construct a distribution on $\mathbb{R}$ which extends the
function ${\rm x}_{+}^{a}$ when $a\leq-1$, defined by (21) in
$\mathbb{R}\setminus\\{0\\}$. That is, we want to extend the functional

(23)
$\psi\in\mathcal{D}(\mathbb{R}\setminus\\{0\\})\longmapsto\int_{0}^{\infty}x^{a}\psi(x)\,\mathrm{d}x,$

for $a\leq-1$. We first deal with the case $a=-1$. If
$\psi\in\mathcal{D}(\mathbb{R}\setminus\\{0\\})$ we have that

$\int_{0}^{\infty}x^{-1}\psi(x)\,\mathrm{d}x=-\int_{0}^{\infty}\log(x)\,\psi^{\prime}(x)\,\mathrm{d}x,$

where $\psi^{\prime}$ denotes the first-order derivative of $\psi$. Note that
the right-hand side of the previous inequality is finite for every
$\psi\in\mathcal{S}(\mathbb{R})$. Thus, we define the desired extension as

(24) $\langle{\rm
x}_{+}^{-1},\psi\rangle=-\int_{0}^{\infty}\log(x)\,\psi^{\prime}(x)\,\mathrm{d}x,\qquad\forall\,\psi\in\mathcal{S}(\mathbb{R}).$

In order to perform the extension in the range $a<-1$, we first note that for
$l\in\mathbb{N}$ and $b\notin\\{0,\dots,l-1\\}$ the function in (21) satisfies
the identity

$\frac{d^{l}}{dx^{l}}{\rm x}_{+}^{b}=b\dots(b-(l-1)){\rm x}_{+}^{b-l},$

or equivalently, if $a\notin\\{-1,\dots,-l\\}$

${\rm x}_{+}^{a}=\frac{1}{(a+l)\dots(a+1)}\frac{d^{l}}{dx^{l}}{\rm
x}_{+}^{a+l}.$

This means that, if $l\in\mathbb{N}$, $a\notin\\{-1,\dots,-l\\}$ and
$\psi\in\mathcal{D}(\mathbb{R}\setminus\\{0\\})$ then

$\int_{0}^{\infty}x^{a}\psi(x)\,\mathrm{d}x=\frac{(-1)^{l}}{(a+l)\dots(a+1)}\int_{0}^{\infty}x^{a+l}\,\psi^{(l)}(x)\,\mathrm{d}x.$

Here $\psi^{(l)}$ denotes the $l^{\rm th}$-order derivative of $\psi$.
Therefore, whenever $-l-1\leq a<-l$ with $l\in\mathbb{N}$, we can define the
tempered distribution ${\rm x}_{+}^{a}$ by

(25) $\langle{\rm
x}_{+}^{a},\psi\rangle=\frac{(-1)^{l}}{(a+l)\dots(a+1)}\langle{\rm
x}_{+}^{a+l},\psi^{(l)}\rangle,\qquad\psi\in\mathcal{S}(\mathbb{R}).$

The procedure described above defines for all $a\in\mathbb{R}$ a tempered
distribution ${\rm x}_{+}^{a}$ on $\mathbb{R}$ that extends the distribution
given in (23) initially defined on $\mathbb{R}\setminus\\{0\\}$. We now use
this distribution ${\rm x}_{+}^{a}$ in order to define a linear functional
that eventually will represent the tempered distribution $q_{\alpha}^{-1}$ in
the whole of $\mathbb{R}^{k}$. This linear functional is

(26)
$\phi\in\mathcal{S}(\mathbb{R}^{k})\longmapsto\int_{\SS^{k-1}}\frac{1}{q_{\alpha}(\theta)}\langle{\rm
x}_{+}^{k-\alpha-1},\psi_{\theta}\rangle\,\mathrm{d}S(\theta)$

where $\psi_{\theta}(x)=\phi(x\theta)$ for $x\in\mathbb{R}$. Note that this
functional is indeed a tempered distribution on $\mathbb{R}^{k}$.

Using (19) and the fact that ${\rm x}_{+}^{k-\alpha-1}$ extends the functional
(20) we conclude that

$\int_{\SS^{k-1}}\frac{1}{q_{\alpha}(\theta)}\langle{\rm
x}_{+}^{k-\alpha-1},\psi_{\theta}\rangle\,\mathrm{d}S(\theta)=\int_{\mathbb{R}^{k}\setminus\\{0\\}}\frac{\phi(\eta)}{q_{\alpha}(\eta)}\,\mathrm{d}\eta$

for all $\phi\in\mathcal{S}(\mathbb{R}^{k})$ such that
$\operatorname*{supp}\phi\subset\mathbb{R}^{k}\setminus\\{0\\}$ with
$\psi_{\theta}(x)=\phi(x\theta)$ for $x\in\mathbb{R}$. This shows that the
functional (26) satisfies the property (i).

In the proofs of Lemma 3.1 and Lemma 3.2 we show that the tempered
distribution (26) coincides with the linear functionals defined in the
corresponding statements, and satisfies the homogeneity property (ii).
Moreover, when $\alpha-k\notin\mathbb{N}_{0}$ we check that this is the unique
extension of the distribution (18).

###### Proof of Lemma 3.1.

The fact that

$\int_{\SS^{k-1}}\frac{1}{q_{\alpha}(\theta)}\langle{\rm
x}_{+}^{k-\alpha-1},\psi_{\theta}\rangle\,\mathrm{d}S(\theta)=\int_{\mathbb{R}^{k}}\frac{\phi(\eta)}{q_{\alpha}(\eta)}$

whenever $\alpha-k<0$ follows immediately from the definition (26) for
$q_{\alpha}^{-1}$ and the identity (22). One only needs to undo the spherical
change of coordinates. The equality

$\int_{\SS^{k-1}}\frac{1}{q_{\alpha}(\theta)}\langle{\rm
x}_{+}^{k-\alpha-1},\psi_{\theta}\rangle\,\mathrm{d}S(\theta)=\frac{\Gamma(\alpha-k-\lfloor\alpha-k\rfloor)}{\Gamma(\alpha-k+1)}\int_{\mathbb{R}^{k}}\frac{(\eta\cdot\nabla)^{\lceil\alpha-k\rceil}\phi(\eta)}{q_{\alpha}(\eta)}$

for $\alpha-k>0$ follows from the identities (25) and (22). In this case one
needs to use the properties of the Gamma function in order to calculate the
precise constant, and note that

(27)
$\psi_{\theta}^{(l)}(x)=(\theta\cdot\nabla)^{l}\phi(x\theta),\quad\forall\,x\in\mathbb{R},\enspace
l\in\mathbb{N}.$

The homogeneity property (ii) follows by performing the change of variables
$\eta=\lambda\zeta$ in the expressions of $q_{\alpha}^{-1}$ written in the
statement of the lemma.

Finally, we discuss the issue of uniqueness. The difference between
$q_{\alpha}^{-1}$ and any other distribution satisfying (i) and (ii) is a
distribution supported at $\\{0\\}$. This means that it has to be a finite
linear combination of derivatives of $\delta_{0}$, the Dirac distribution in
$\mathbb{R}^{k}$ supported at $\\{0\\}$. That is, the difference can be
written as $\sum_{|\beta|\leq l}c_{\beta}\partial^{\beta}\delta_{0}$ with
$c_{\beta}\in\mathbb{C}$, $\beta\in\mathbb{N}_{0}^{k}$ and
$l\in\mathbb{N}_{0}$. However, this difference has to satisfy (ii), namely it
is homogeneous of order $-\alpha$. This can only happen if $c_{\beta}=0$ for
all $\beta\in\mathbb{N}_{0}^{k}$ such that $|\beta|\leq l$. Indeed,

$\langle\partial^{\beta}\delta_{0},\phi_{\lambda}\rangle=\lambda^{k+|\beta|}\langle\partial^{\beta}\delta_{0},\phi\rangle,$

which means that $\partial^{\beta}\delta_{0}$ is homogeneous of order
$-k-|\beta|$. Since $\alpha-k\notin\mathbb{N}_{0}$, $\alpha\neq k+|\beta|$ for
any $\beta\in\mathbb{N}_{0}^{k}$. This completes the proof of Lemma 3.1. ∎

###### Proof of Lemma 3.2.

The equality

$\int_{\SS^{k-1}}\frac{1}{q_{\alpha}(\theta)}\langle{\rm
x}_{+}^{k-\alpha-1},\psi_{\theta}\rangle\,\mathrm{d}S(\theta)=\frac{-1}{(\alpha-k)!}\int_{\mathbb{R}^{k}}\frac{\log|\eta|}{q_{\alpha}(\eta)}(\eta\cdot\nabla)^{\alpha-k+1}\phi(\eta)\,\mathrm{d}\eta$

for $\alpha-k\in\mathbb{N}_{0}$ is a consequence of the identity (24) when
$\alpha=k$, and it follows (25) and (24) when $\alpha-k\in\mathbb{N}$. Again
one needs to use the relation (27) to verify this.

Let us now address the homogeneity of $q_{\alpha}^{-1}$. In this case
$\alpha-k\in\mathbb{N}_{0}$ and we will have that the homogeneity holds if and
only if we assume condition (c).

After the change of variables $\eta=\lambda\zeta$ we have that

(28)
$\begin{split}&\int_{\mathbb{R}^{k}}\frac{\log|\eta|}{q_{\alpha}(\eta)}(\eta\cdot\nabla)^{\alpha-k+1}\phi(\eta)\,\mathrm{d}\eta\\\
&\qquad\qquad=\lambda^{-\alpha}\int_{\mathbb{R}^{k}}\frac{\log|\zeta|}{q_{\alpha}(\zeta)}(\zeta\cdot\nabla)^{\alpha-k+1}[\lambda^{k}\phi(\lambda\centerdot)](\zeta)\,\mathrm{d}\zeta\\\
&\qquad\qquad\quad+\lambda^{-\alpha}\log\lambda\int_{\mathbb{R}^{k}}\frac{1}{q_{\alpha}(\zeta)}(\zeta\cdot\nabla)^{\alpha-k+1}[\lambda^{k}\phi(\lambda\centerdot)](\zeta)\,\mathrm{d}\zeta.\end{split}$

Hence, we can see that $q_{\alpha}^{-1}$ satisfies (ii) if and only if the
last term on the right-hand side of the previous identity vanishes for all
$\phi\in\mathcal{S}(\mathbb{R}^{k})$. Let us compute this term explicitly.

Using spherical coordinates we have for any
$\phi\in\mathcal{S}(\mathbb{R}^{k})$ that

$\int_{\mathbb{R}^{k}}\frac{1}{q_{\alpha}(\zeta)}(\zeta\cdot\nabla)^{\alpha-k+1}\phi(\zeta)\,\mathrm{d}\zeta=\int_{\SS^{k-1}}\frac{1}{q_{\alpha}(\theta)}\int_{0}^{\infty}\psi_{\theta}^{(\alpha-k+1)}(\rho)\,\mathrm{d}\rho\,\mathrm{d}S(\theta)$

with $\psi_{\theta}(\rho)=\phi(\rho\theta)$ for $\rho>0$, and
$\psi_{\theta}^{(\alpha-k+1)}$ denoting the $(\alpha-k+1)^{\rm th}$-order
derivative of $\psi_{\theta}$. Since by (27)

$\int_{0}^{\infty}\psi_{\theta}^{(\alpha-k+1)}(\rho)\,\mathrm{d}\rho=-(\theta\cdot\nabla)^{\alpha-k}\phi(0),$

we have that

$\begin{split}\int_{\mathbb{R}^{k}}\frac{1}{q_{\alpha}(\zeta)}(\zeta\cdot\nabla)^{\alpha-k+1}\phi(\zeta)\,\mathrm{d}\zeta&=-\int_{\SS^{k-1}}\frac{(\theta\cdot\nabla)^{\alpha-k}\phi(0)}{q_{\alpha}(\theta)}\,\mathrm{d}S(\theta)\\\
&=-\sum_{|\beta|=\alpha-k}c_{\beta}\partial^{\beta}\phi(0)\int_{\SS^{k-1}}\frac{\theta^{\beta}}{q_{\alpha}(\theta)}\,\mathrm{d}S(\theta)\end{split}$

where the coefficients $c_{\beta}$ are strictly positive. Therefore, the
right-hand side vanishes for all $\phi\in\mathcal{S}(\mathbb{R}^{k})$ if and
only if (c) holds. Consequently, if the cancellation property (c) is verified
then

$\int_{\mathbb{R}^{k}}\frac{1}{q_{\alpha}(\zeta)}(\zeta\cdot\nabla)^{\alpha-k+1}\phi(\zeta)\,\mathrm{d}\zeta=0$

for all $\phi\in\mathcal{S}(\mathbb{R}^{k})$, and by (28), property (ii) is
satisfied as desired. The proof of the lemma is complete ∎

######  Remark 3.3.

Condition (c) in the statement of Lemma 3.2 is necessary: if $q_{\alpha}$ does
not satisfy (c) then there is no distribution in
$\mathcal{S}^{\prime}(\mathbb{R}^{k})$ such that (i) and (ii) are satisfied
simultaneously. To see this let $t_{\alpha}^{-1}$ be a tempered distribution
satisfying (i). As mentioned in the proof of Lemma 3.1 $t_{\alpha}^{-1}$
differs from $q_{\alpha}^{-1}$ by a linear combination of derivatives of a
Dirac delta supported at $\\{0\\}$. It is easy to verify that a linear
combination of derivatives of $\delta_{0}$ does not affect the second summand
in the right hand side of (28). Thus, if $t_{\alpha}^{-1}$ satisfies (i) and
(ii) then the second summand in the right hand side of (28) must vanish and
the same proof yields that $t_{\alpha}^{-1}$ must satisfy (c).

For future reference it is important to note that $q_{\alpha}^{-1}\in
H^{-\delta}_{\rm loc}(\mathbb{R}^{k})$ with $\delta=\lceil\alpha-k\rceil$ if
$\alpha-k/2<\lceil\alpha-k\rceil$ and $\delta>\alpha-k/2$ if
$\alpha-k/2\geq\lceil\alpha-k\rceil$. In order to prove this property we show
that $\chi q_{\alpha}^{-1}\in H^{-\delta}(\mathbb{R}^{k})$ for any
$\chi\in\mathcal{D}(\mathbb{R}^{n})$. By definition we have for all
$\phi\in\mathcal{S}(\mathbb{R}^{k})$ that

(29) $|\langle\chi
q_{\alpha}^{-1},\phi\rangle|\lesssim\Big{(}\int_{K}\Big{(}\log\frac{1+|\eta|}{|\eta|}\Big{)}^{p}\frac{|\eta|^{p\lceil\alpha-k\rceil}}{|q_{\alpha}(\eta)|^{p}}\,\mathrm{d}\eta\Big{)}^{1/p}\|\phi\|_{W^{\lceil\alpha-k\rceil,p^{\prime}}(\mathbb{R}^{k})};$

here $K=\operatorname*{supp}\chi$. Using spherical coordinates one can check
that the integral in $K$ is finite whenever
$p<k/[k-(\lceil\alpha-k\rceil-(\alpha-k))]$. Since
$\alpha-k/2<\lceil\alpha-k\rceil$ if and only if
$2<k/[k-(\lceil\alpha-k\rceil-(\alpha-k))]$, we can choose $p=2$ in (29) and
deduce by duality that $\chi q_{\alpha}^{-1}\in
H^{-\lceil\alpha-k\rceil}(\mathbb{R}^{k})$ whenever
$\alpha-k/2<\lceil\alpha-k\rceil$. If $\alpha-k/2\geq\lceil\alpha-k\rceil$, by
Sobolev’s embedding theorem we have that

$\begin{split}\|\phi\|_{W^{\lceil\alpha-k\rceil,p^{\prime}}(\mathbb{R}^{k})}&\eqsim\|(I-\Delta)^{\lceil\alpha-k\rceil/2}\phi\|_{L^{p^{\prime}}(\mathbb{R}^{k})}\\\
&\lesssim\|(I-\Delta)^{\lceil\alpha-k\rceil/2}\phi\|_{H^{s}(\mathbb{R}^{k})}=\|\phi\|_{H^{s+\lceil\alpha-k\rceil}(\mathbb{R}^{k})}\end{split}$

for $s-k/2=-k/p^{\prime}$. The upper bound on $p$ imposes the lower bound
$s+\lceil\alpha-k\rceil>\alpha-k/2$. Thus, if
$\alpha-k/2\geq\lceil\alpha-k\rceil$, again by duality we can ensure that
$\chi q_{\alpha}^{-1}\in H^{-\delta}(\mathbb{R}^{k})$ for all
$\delta>\alpha-k/2$.

We conclude this section by recording some useful remarks. Let
$\psi\in\mathcal{S}(\mathbb{R}^{k})$ be such that
$\int_{\mathbb{R}^{k}}\psi(\eta)\,\mathrm{d}\eta=1$ and for
$t\in\mathbb{R}_{+}$ we set $\psi_{t}(\eta)=t^{k}\psi(t\eta)$ for
$\eta\in\mathbb{R}^{k}$. We introduce the smooth function

(30) $q_{\alpha,t}^{-1}(\eta)=(q_{\alpha}^{-1}\star\psi_{t})(\eta)=\langle
q_{\alpha}^{-1},\psi_{t}(\eta-\centerdot)\rangle,\qquad\eta\in\mathbb{R}^{k}.$

First, one can verify that

(31) $\lim_{t\to 0}\|q_{\alpha,t}^{-1}\|_{L^{\infty}(\mathbb{R}^{k})}=0,$

This follows from (ii) which implies that

$q_{\alpha,t}^{-1}(\eta)=\langle
q_{\alpha}^{-1},\psi_{t}(\eta-\centerdot)\rangle=t^{\alpha}\langle
q_{\alpha}^{-1},\psi(t\eta-\centerdot)\rangle.$

Secondly, we have that for every compact subset $K$ of $\mathbb{R}^{k}$ such
that $0\notin K$

(32) $\lim_{t\to\infty}\sup_{\eta\in
K}\Big{|}\frac{1}{q_{\alpha}(\eta)}-q_{\alpha,t}^{-1}(\eta)\Big{|}=0.$

Thirdly, for every $\phi\in\mathcal{S}(\mathbb{R}^{k})$ we have that

(33)
$\lim_{t\to\infty}\int_{\mathbb{R}^{k}}q_{\alpha,t}^{-1}(\eta)\phi(\eta)\,\mathrm{d}\eta=\langle
q_{\alpha}^{-1},\phi\rangle.$

Lastly, for every $\chi\in\mathcal{D}(\mathbb{R}^{n})$ we have that

(34)
$\lim_{t\to\infty}\|\chi(q_{\alpha}^{-1}-q_{\alpha,t}^{-1})\|_{H^{-\delta}(\mathbb{R}^{k})}=0,$

with $\delta=\lceil\alpha-k\rceil$ if $\alpha-k/2<\lceil\alpha-k\rceil$ and
$\delta>\alpha-k/2$ if $\alpha-k/2\geq\lceil\alpha-k\rceil$.

##  4\. Rotational smoothing in the general case

This section is divided into four subsections. In Section 4.1 we provide some
preliminary facts about the functional spaces and an important property of
Sobolev spaces in $\mathrm{SO}(n)$, and in Sections 4.2 and 4.3 we prove
Theorems 2.1 and 2.2. Recall from our discussion in the Section 2 and the
statement of Theorem 2.2, the need of introducing a distribution $r_{\alpha}$
standing formally as the restriction to $\SS^{n-1}$ of
$q_{\alpha}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}$. In Section 4.4 we give
the explicit definition of $r_{\alpha}$.

###  4.1. Some preliminary notions

We now summarize the key properties of Sobolev spaces and $\mathrm{SO}(n)$
that one needs in order to prove Theorems 2.1 and 2.2. We provide the
statements without proofs in this section, leaving the more rigorous analysis
of the functional framework for Section 6.

Let $(M,g)$ stand for a compact boundaryless Riemannian manifold of dimension
$m\in\mathbb{N}$. We denote the canonical Laplace-Beltrami in $M$ by
$\Delta_{g}$. We denote by $\lambda_{j}$, $j\in\mathbb{N}_{0}$, the
eigenvalues of $-\Delta_{g}$, which are always non-negative, and denote by
$\Pi_{j}$ the projector onto the corresponding invariant subspace of
eigenfunctions associated to $\lambda_{j}$. This eigenvalue expansion yields a
natural way to define the fractional Bessel potentials of order
$s\in\mathbb{R}$ for every $u\in C^{\infty}(M)$

(35)
$(\mathrm{Id}-\Delta_{g})^{s/2}u=\sum_{j\in\mathbb{N}_{0}}(1+\lambda_{j})^{s/2}\Pi_{j}u.$

We can now define the fractional Sobolev spaces $H^{s}(M)$ for
$s\in\mathbb{R}$ in an intrinsic way as the Banach completion of
$C^{\infty}(M)$ with respect to the norm

$\|u\|_{H^{s}(M)}=\|(\mathrm{Id}-\Delta_{g})^{s/2}u\|_{L^{2}(M)}=\Big{(}\sum_{j\in\mathbb{N}_{0}}(1+\lambda_{j})^{s}\|\Pi_{j}u\|_{L^{2}(M)}^{2}\Big{)}^{1/2}.$

The next proposition yields an important relation between Sobolev spaces in
$\mathrm{SO}(n)$ and $\SS^{n-1}$ which plays an important role in the proof of
Theorems 2.1 and 2.2.

######  Proposition 4.1.

Fix any $v\in\SS^{n-1}$, and let $f$ be in $H^{s}(\SS^{n-1})$. Consider the
function $F_{v}:\mathrm{SO}(n)\to\SS^{n-1}$ given by
$F_{v}(Q)=Q^{\mathsf{T}}v$ for every rotation $Q\in\mathrm{SO}(n)$. Then, it
holds that

$\|f\circ
F_{v}\|_{H^{s}(\mathrm{SO}(n))}=S(\SS^{n-1})^{-1/2}\|f\|_{H^{s}(\SS^{n-1})}$

for all $s\in\mathbb{R}$.

This result is a generalization for $s\neq 0$ of the well known identity that
we have already stated in (15). The proof requires some preliminary notions
about $\mathrm{SO}(n)$ and is given in Appendix A. It essentially follows from
an identity relating the operators $\Delta_{\mathrm{SO}(n)}$ and
$\Delta_{\SS^{n-1}}$ that we prove in Lemma A.6.

The following proposition, which we prove in Section 6, will allow us to work
with Sobolev spaces on local charts.

######  Proposition 4.2.

Let $(M,g)$ be a compact boundaryless Riemannian manifold of dimension
$m\in\mathbb{N}$. Let $\\{(U_{1},\varphi_{1}),\dots,(U_{l},\varphi_{l})\\}$ be
a smooth atlas for $M$, and consider $\\{\chi_{1},\dots,\chi_{l}\\}$ a smooth
partition of unity subordinated to the atlas. Then, for $s\in\mathbb{R}$ we
have that

$\|u\|_{H^{s}(M)}\eqsim\sum_{j=1}^{l}\|(\chi_{j}u)\circ\varphi_{j}^{-1}\|_{H^{s}(\mathbb{R}^{m})}$

for all $u\in C^{\infty}(M)$. The implicit constants only depend on $m$, $s$,
the atlas and the partition of unity.

We now define the spaces $H^{s}(M;\dot{H}^{t}(\mathbb{R}^{n}))$ that appear in
the statement of Theorems 2.1 and 2.2.

######  Definition 4.3.

Let $\mathcal{A}(\mathbb{R}^{n}\times M)$ denote the set of $u\in
C^{\infty}(\mathbb{R}^{n}\times M)$ such that for every
$\alpha,\beta\in\mathbb{N}_{0}^{n}$ and every $k\in\mathbb{N}_{0}$ there
exists $C>0$ such that

$|x^{\beta}(\mathrm{Id}-\Delta_{g})^{k}\partial^{\alpha}u(x,p)|\leq
C\qquad\forall\,(x,p)\in\mathbb{R}^{n}\times M.$

A function $u\in C^{\infty}(\mathbb{R}^{n}\times M)$ belongs to
$\mathcal{A}(\mathbb{R}^{n}\times M)$ if and only if its Fourier transform,
denoted by $\widehat{u}$ and defined by

(36) $(\xi,p)\in\mathbb{R}^{n}\times
M\longmapsto\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^{n}}e^{-\mathrm{i}\xi\cdot
x}u(x,p)\,\mathrm{d}x\in\mathbb{C},$

belongs to $\mathcal{A}(\mathbb{R}^{n}\times M)$. In Section 6 we show that
the non-negative functional

(37) $u\in\mathcal{A}(\mathbb{R}^{n}\times
M)\longmapsto\Big{(}\int_{M}\|(\mathrm{Id}-\Delta_{g})^{s/2}u(\centerdot,p)\|_{\dot{H}^{t}(\mathbb{R}^{n})}^{2}\,\mathrm{d}\mu_{g}(p)\Big{)}^{1/2}$

is well defined for all $s\in\mathbb{R}$ and $t\in\mathbb{R}_{+}$. It defines
a norm on $\mathcal{A}(\mathbb{R}^{n}\times M)$ that will be denoted from now
on by $\|\centerdot\|_{H^{s}(M;\dot{H}^{t}(\mathbb{R}^{n}))}$. Then, the space
$H^{s}(M;\dot{H}^{t}(\mathbb{R}^{n}))$ is the Banach completion of
$\mathcal{A}(\mathbb{R}^{n}\times M)$ with respect to the previous norm. For
the convenience of the reader let us point out that
$L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))$ and
$H^{0}(M;\dot{H}^{t}(\mathbb{R}^{n}))$ denote the same space. In Lemma 6.6 we
also prove that

(38)
$\|u\|_{H^{s}(M;\dot{H}^{t}(\mathbb{R}^{n}))}=\Big{(}\int_{\mathbb{R}^{n}}|\xi|^{2t}\|\widehat{u}(\xi,\centerdot)\|_{H^{s}(M)}^{2}\,\mathrm{d}\xi\Big{)}^{1/2}$

for all $u\in\mathcal{A}(\mathbb{R}^{n}\times M)$, $s\in\mathbb{R}$ and
$t\in\mathbb{R}_{+}$. Finally, we show in Proposition 6.8 that every $u\in
H^{s}(M;\dot{H}^{t}(\mathbb{R}^{n}))$ with $s\in\mathbb{R}$ and
$t\in\mathbb{R}_{+}$ determines a unique distribution in $\mathbb{R}^{n}\times
M$.

With these ingredients we are now ready to prove Theorems 2.1 and 2.2. We
remind the interested reader that detailed statements and proofs of the
preliminary results above are contained in Section 6.

###  4.2. Proof of Theorems 2.1 and 2.2

This section contains the full scheme needed to prove the rotational smoothing
phenomenon.

Consider the sequence $\\{q_{\alpha,N}^{-1}:N\in\mathbb{N}\\}$ with
$q_{\alpha,N}^{-1}$ defined in (30), and define for $N\in\mathbb{N}$ the
symbol

$p_{\alpha,N}^{-1}(\xi,Q)=[q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](Q^{\mathsf{T}}\xi),\qquad(\xi,Q)\in\mathbb{R}^{n}\times\mathrm{SO}(n),$

and the operator

(39)
$S_{\alpha,N}f(x,Q)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^{n}}e^{\mathrm{i}x\cdot\xi}p_{\alpha,N}^{-1}(\xi,Q)\widehat{f}(\xi)\,\mathrm{d}\xi,\qquad(x,Q)\in\mathbb{R}^{n}\times\mathrm{SO}(n),$

with $f\in\mathcal{S}(\mathbb{R}^{n})$. One can check that
$S_{\alpha,N}f\in\mathcal{A}(\mathbb{R}^{n}\times\mathrm{SO}(n))$ for all
$f\in\mathcal{S}(\mathbb{R}^{n})$ and all $N\in\mathbb{N}$. By (38) we have
that for every $N\in\mathbb{N}$ and $f\in\mathcal{S}(\mathbb{R}^{n})$, the
following identity holds

(40) $\displaystyle\|S_{\alpha,N}f$
$\displaystyle\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}^{2}=\int_{\mathbb{R}^{n}}|\xi|^{2\alpha}\|p_{\alpha,N}^{-1}(\xi,\centerdot)\|_{H^{-\delta}(\mathrm{SO}(n))}^{2}|\widehat{f}(\xi)|^{2}\,\mathrm{d}\xi.$

For $v={\xi}/|\xi|$, we can write

$p_{\alpha,N}^{-1}(\xi,Q)=[q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](|\xi|Q^{\mathsf{T}}v),\qquad(\xi,Q)\in\mathbb{R}^{n}\times\mathrm{SO}(n),$

and by Proposition 4.1 we have that

(41)
$\begin{split}\|p_{\alpha,N}^{-1}(\xi,\centerdot)\|_{H^{-\delta}(\mathrm{SO}(n))}&=\|[q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](|\xi|F_{v})\|_{H^{-\delta}(\mathrm{SO}(n))}\\\
&=S(\SS^{n-1})^{-1/2}\|[q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](|\xi|\centerdot)\|_{H^{-\delta}(\SS^{n-1})}.\end{split}$

Our first step towards deriving Theorems 2.1 and 2.2 will be to show that, for
all $f\in\mathcal{S}(\mathbb{R}^{n})$, the sequence
$\\{S_{\alpha,N}f:N\in\mathbb{N}\\}$ is a Cauchy sequence with respect to the
norm
$\|\centerdot\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}$.
Start by considering an arbitrary $f\in\mathcal{S}(\mathbb{R}^{n})$ and noting
that for $L,N\in\mathbb{N}$ we have that

$\begin{split}S(\SS^{n-1})&\|S_{\alpha,N+L}f-S_{\alpha,N}f\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}^{2}\\\
&=\int_{\mathbb{R}^{n}}|\xi|^{2\alpha}\|[(q_{\alpha,N+L}^{-1}-q_{\alpha,N}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](|\xi|\centerdot)\|_{H^{-\delta}(\SS^{n-1})}^{2}|\widehat{f}(\xi)|^{2}\,\mathrm{d}\xi.\end{split}$

Recall that $\delta=\lceil\alpha-k\rceil$ if
$\alpha-k/2<\lceil\alpha-k\rceil$, and $\delta>\alpha-k/2$ if
$\alpha-k/2\geq\lceil\alpha-k\rceil$. We make the following claims:

  1. (I)

For every $\xi\in\mathbb{R}^{n}\setminus\\{0\\}$, we have that

$\lim_{N\to\infty}|\xi|^{\alpha}\|[(q_{\alpha,N+L}^{-1}-q_{\alpha,N}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](|\xi|\centerdot)\|_{H^{-\delta}(\SS^{n-1})}=0$

uniformly for all $L\in\mathbb{N}$.

  2. (II)

There exists a constant $C>0$ such that

$|\xi|^{\alpha}\|[q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](|\xi|\centerdot)\|_{H^{-\delta}(\SS^{n-1})}\leq
C$

for all $\xi\in\mathbb{R}^{n}\setminus\\{0\\}$ and all $N\in\mathbb{N}$.

  3. (III)

There is a distribution $r_{\alpha}\in H^{-\delta}(\SS^{n-1})$ such that

$\lim_{N\to\infty}\|\lambda^{\alpha}[q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\lambda\centerdot)-r_{\alpha}\|_{H^{-\delta}(\SS^{n-1})}=0,$

for every $\lambda>0$, where $r_{\alpha}$ is independent of $\lambda$.

We postpone the proofs of the claims (I), (II) and (III) until Section 4.3
below. Assuming (I) and (II), the dominated convergence theorem implies that

$\lim_{N\to\infty}\|S_{\alpha,N+L}f-S_{\alpha,N}f\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}=0$

uniformly in $L\in\mathbb{N}$, which means that
$\\{S_{\alpha,N}f:N\in\mathbb{N}\\}$ is Cauchy with respect to the norm
$\|\centerdot\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}$.

We proceed to proving the identity in the statement of the theorem. Let us
denote by $u_{\alpha}^{f}\in
H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))$ the equivalence
class of the Cauchy sequence $\\{S_{\alpha,N}f:N\in\mathbb{N}\\}$; we have by
(40) and (41) that

$\begin{split}S(\SS^{n-1})\|&u_{\alpha}^{f}\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}^{2}\\\
&=\lim_{N\to\infty}\int_{\mathbb{R}^{n}}|\xi|^{2\alpha}\|[q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](|\xi|\centerdot)\|_{H^{-\delta}(\SS^{n-1})}^{2}|\widehat{f}(\xi)|^{2}\,\mathrm{d}\xi.\end{split}$

On the one hand, as mentioned in Section 4.1, $u_{\alpha}^{f}$ determines a
distribution in $\mathbb{R}^{n}\times\mathrm{SO}(n)$. On the other hand, we
have that

$S_{\alpha}f(x,Q)-S_{\alpha,N}f(x,Q)=\frac{1}{(2\pi)^{n/2}}\langle(q_{\alpha}^{-1}-q_{\alpha,N}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}},\widehat{f(Q(\centerdot+x))}\rangle.$

By (33) we can ensure that for every compact $K\subset\mathbb{R}^{n}$ we have

$\lim_{N\to\infty}\|S_{\alpha}f-S_{\alpha,N}f\|_{L^{\infty}(K\times\mathrm{SO}(n))}=0.$

Hence $S_{\alpha}f\in C(\mathbb{R}^{n}\times\mathrm{SO}(n))$ can be identified
with a distribution in $\mathbb{R}^{n}\times\mathrm{SO}(n)$ and

$S_{\alpha,N}f\xrightarrow[N\to\infty]{}S_{\alpha}f\quad\textnormal{in}\quad\mathcal{D}^{\prime}(\mathbb{R}^{n}\times\mathrm{SO}(n)).$

This implies that $u_{\alpha}^{f}=S_{\alpha}f$ which yields that

(42) $\displaystyle S(\SS^{n-1})$
$\displaystyle\|S_{\alpha}f\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}^{2}$
$\displaystyle=\lim_{N\to\infty}\int_{\mathbb{R}^{n}}|\xi|^{2\alpha}\|[q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](|\xi|\centerdot)\|_{H^{-\delta}(\SS^{n-1})}^{2}|\widehat{f}(\xi)|^{2}\,\mathrm{d}\xi.$

Now (III) implies that

$\|r_{\alpha}\|_{H^{-\delta}(\SS^{n-1})}=\lim_{N\to\infty}|\xi|^{\alpha}\|[q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](|\xi|\centerdot)\|_{H^{-\delta}(\SS^{n-1})}.$

This identity together with (II) turn, by the dominated convergence theorem,
the identity (42) into

$\|S_{\alpha}f\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}^{2}=S(\SS^{n-1})^{-1}\|r_{\alpha}\|_{H^{-\delta}(\SS^{n-1})}^{2}\int_{\mathbb{R}^{n}}|\widehat{f}(\xi)|^{2}\,\mathrm{d}\xi.$

Consequently, Plancherel’s identity in $\mathbb{R}^{n}$ yields the identities
stated in Theorems 2.1 and 2.2. This completes their proofs, up to showing the
claims (I), (II) and (III).

###  4.3. Proofs of claims (I), (II) and (III)

Before addressing these claims, we make a simple observation about the set of
distributions
$\\{q_{\alpha,t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}:t\in\mathbb{R}_{+}\\}$
with $q_{\alpha,t}^{-1}$ defined in (30). This is, for
$\lambda\in\mathbb{R}_{+}$ and $\theta\in\SS^{n-1}$ we have that

(43)
$\lambda^{\alpha}[q_{\alpha,t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\lambda\theta)=[q_{\alpha,\lambda
t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\theta).$

As we see in the following identities this is a consequence of the homogeneity
property of $q_{\alpha}^{-1}$ (see Lemmas 3.1 and 3.2):

$\begin{split}\lambda^{\alpha}q_{\alpha,t}^{-1}(\lambda\eta)&=\lambda^{\alpha}\langle
q_{\alpha}^{-1},\psi_{t}(\lambda\eta-\centerdot)\rangle=\lambda^{\alpha}\lambda^{-\alpha}\langle
q_{\alpha}^{-1},\lambda^{k}\psi_{t}(\lambda\eta-\lambda\centerdot)\rangle\\\
&=\langle q_{\alpha}^{-1},\psi_{\lambda
t}(\eta-\centerdot)\rangle=q_{\alpha,\lambda t}^{-1}(\eta),\end{split}$

where $\psi_{t}(\eta)=t^{k}\psi(t\eta)$.

We start by proving (I). In order to deal with the norm of
$H^{-\delta}(\SS^{n-1})$ we will use Proposition 4.2. For this reason we
introduce an atlas $\\{(U_{1},\varphi_{1}),\dots,(U_{2n},\varphi_{2n})\\}$ for
$\SS^{n-1}$. Consider the open subset of $\mathbb{R}^{n-1}$ given by

(44) $V=\\{y\in\mathbb{R}^{n-1}:|y|^{2}<(n-1/2)/n\\},$

and let $\phi$ denote the function $\phi(y)=(1-|y|^{2})^{1/2}$ for all $y\in
V$. The set $U=\\{(y,\phi(y))\in\mathbb{R}^{n}:y\in V\\}$ is an open patch in
$\SS^{n-1}$, and

$\omega\in U\Rightarrow|e_{n}\cdot\omega|^{2}>1/(2n).$

From this patch, we obtain an atlas of $\SS^{n-1}$. To do so, consider the
reflections $R_{jn}$ and $P_{j}$ with $j\in\\{1,\dots,n\\}$ that satisfy:

  1. (i)

$R_{jn}e_{j}=e_{n}$, $R_{jn}e_{n}=e_{j}$ and $R_{jn}e_{l}=e_{l}$ for
$l\notin\\{j,n\\}$,

  2. (ii)

$P_{j}e_{j}=-e_{j}$ and $P_{j}e_{l}=e_{l}$ for $l\neq j$.

With this choice $R_{nn}=\mathrm{id}$. The atlas that we consider consists of
$2n$ charts denoted by $(U_{1},\varphi_{1}),\dots,(U_{2n},\varphi_{2n})$. The
odd charts are defined by

(45) $\begin{split}U_{2j-1}&=\\{R_{jn}\omega:\omega\in U\\},\\\
\varphi_{2j-1}(\theta)&=(e_{1}\cdot
R_{jn}^{\mathsf{T}}\theta,\dots,e_{n-1}\cdot
R_{jn}^{\mathsf{T}}\theta),\quad\theta\in U_{2j-1},\end{split}$

with $j\in\\{1,\dots,n\\}$. The even charts are given by

(46) $\begin{split}U_{2j}&=\\{P_{j}R_{jn}\omega:\omega\in U\\},\\\
\varphi_{2j}(\theta)&=(e_{1}\cdot
R_{jn}^{\mathsf{T}}P_{j}^{\mathsf{T}}\theta,\dots,e_{n-1}\cdot
R_{jn}^{\mathsf{T}}P_{j}^{\mathsf{T}}\theta),\quad\theta\in
U_{2j},\end{split}$

with $j\in\\{1,\dots,n\\}$. This choice ensures that $\varphi_{m}(U_{m})=V$
for every $m\in\\{1,\dots,2n\\}$.

These charts satisfy some simple properties that we will refer to in a later
analysis. For the odd charts we have that if $j\in\\{1,\dots,n-1\\}$ then

(47) $\zeta\in
V\Rightarrow\begin{cases}e_{l}\cdot\varphi_{2j-1}^{-1}(\zeta)=\zeta_{l},&\quad\forall
l\in\\{1,\dots,n-1\\}\setminus\\{j\\},\vspace{.7em}\\\
e_{n}\cdot\varphi_{2j-1}^{-1}(\zeta)=\zeta_{j}.&\end{cases}$

Additionally,

(48) $\zeta\in V\Rightarrow\begin{aligned}
&e_{l}\cdot\varphi_{2n-1}^{-1}(\zeta)=\zeta_{l}\qquad\forall
l\in\\{1,\dots,n-1\\}.\end{aligned}$

Moreover,

(49) $\theta\in U_{2j-1}\Rightarrow|e_{j}\cdot\theta|^{2}>1/(2n),$

or equivalently,

(50)
$\zeta\in{V}\Rightarrow|e_{j}\cdot\varphi_{2j-1}^{-1}(\zeta)|^{2}>1/(2n).$

For the even charts we similarly have that if $j\in\\{1,\dots,n-1\\}$ then

(51) $\zeta\in
V\Rightarrow\begin{cases}e_{l}\cdot\varphi_{2j}^{-1}(\zeta)=\zeta_{l},&\quad\forall
l\in\\{1,\dots,n-1\\}\setminus\\{j\\},\vspace{.7em}\\\
e_{n}\cdot\varphi_{2j}^{-1}(\zeta)=\zeta_{j}.&\end{cases}$

In addition,

(52) $\zeta\in V\Rightarrow\begin{aligned}
&e_{l}\cdot\varphi_{2n}^{-1}(\zeta)=\zeta_{l}\qquad\forall
l\in\\{1,\dots,n-1\\}.\end{aligned}$

Moreover,

(53) $\theta\in U_{2j}\Rightarrow|e_{j}\cdot\theta|^{2}>1/(2n),$

or equivalently,

(54) $\zeta\in{V}\Rightarrow|e_{j}\cdot\varphi_{2j}^{-1}(\zeta)|^{2}>1/(2n).$

After having introduced the atlas
$\\{(U_{1},\varphi_{1}),\dots,(U_{2n},\varphi_{2n})\\}$ for $\SS^{n-1}$, let
us focus on the task of proving the limit in (I). By (43) and Proposition 4.2
we have for all $\lambda,t\in\mathbb{R}_{+}$ and $s\geq 0$, that

$\begin{split}\lambda^{\alpha}\|[(q_{\alpha,t+s}^{-1}&-q_{\alpha,t}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\lambda\centerdot)\|_{H^{-\delta}(\SS^{n-1})}\\\
&=\|(q_{\alpha,\lambda(t+s)}^{-1}-q_{\alpha,\lambda
t}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}\|_{H^{-\delta}(\SS^{n-1})}\\\
&\eqsim\sum_{i=1}^{2n}\|[\chi_{i}(q_{\alpha,\lambda(t+s)}^{-1}-q_{\alpha,\lambda
t}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}]\circ\varphi_{i}^{-1}\|_{H^{-\delta}(\mathbb{R}^{n-1})}.\end{split}$

By (49) and (53), the function defined in (12) is bounded in
$U_{1}\cup\dots\cup U_{2k}$, so we bound the first $2k$ terms of the previous
sum as follows

$\begin{split}\sum_{i=1}^{2k}\|[\chi_{i}(q_{\alpha,\lambda(t+s)}^{-1}&-q_{\alpha,\lambda
t}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}]\circ\varphi_{i}^{-1}\|_{H^{-\delta}(\mathbb{R}^{n-1})}\\\
&\lesssim\sum_{i=1}^{2k}\|[\chi_{i}(q_{\alpha,\lambda(t+s)}^{-1}-q_{\alpha,\lambda
t}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}]\circ\varphi_{i}^{-1}\|_{L^{2}(\mathbb{R}^{n-1})}\\\
&\lesssim\sum_{i=1}^{2k}\|(q_{\alpha,\lambda(t+s)}^{-1}-q_{\alpha,\lambda
t}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}\|_{L^{\infty}(U_{i})}.\end{split}$

Now, by (32), (49) and (53) we have that

$\lim_{t\to\infty}\sum_{i=1}^{2k}\|(q_{\alpha,\lambda(t+s)}^{-1}-q_{\alpha,\lambda
t}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}\|_{L^{\infty}(U_{i})}=0$

uniformly for in $s\geq 0$. This corresponds to the first $2k$ charts. We turn
now our attention to the last $2(n-k)$ charts. From the identities (47), (48),
(51) and (52) we know that if $i\in\\{2k+1,\dots,2n\\}$ then

$(q_{\alpha,t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}})(\varphi_{i}^{-1}(\zeta))=(q_{\alpha,t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k-1}})(\zeta)\qquad\forall\,\zeta\in\varphi_{i}(U_{i}),$

and consequently that

$\begin{split}\sum_{i=2k+1}^{2n}\|[\chi_{i}(q_{\alpha,\lambda(t+s)}^{-1}&-q_{\alpha,\lambda
t}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}]\circ\varphi_{i}^{-1}\|_{H^{-\delta}(\mathbb{R}^{n-1})}\\\
=&\sum_{i=2k+1}^{2n}\|\tilde{\chi}_{i}(q_{\alpha,\lambda(t+s)}^{-1}-q_{\alpha,\lambda
t}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k-1}}\|_{H^{-\delta}(\mathbb{R}^{n-1})},\end{split}$

where $\tilde{\chi}_{i}(\zeta)=\chi_{i}\circ\varphi_{i}^{-1}(\zeta)$ for
$\zeta\in\mathbb{R}^{n-1}$. Additionally,

$\begin{split}\big{\|}\tilde{\chi}_{i}(q_{\alpha,\lambda(t+s)}^{-1}&-q_{\alpha,\lambda
t}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k-1}}\big{\|}_{H^{-\delta}(\mathbb{R}^{n-1})}\\\
&\leq\bigg{(}\int_{\mathbb{R}^{n-k-1}}\big{\|}\tilde{\chi}_{i}(\centerdot,\kappa)(q_{\alpha,\lambda(t+s)}^{-1}-q_{\alpha,\lambda
t}^{-1})\big{\|}_{H^{-\delta}(\mathbb{R}^{k})}^{2}\,\mathrm{d}\kappa\bigg{)}^{1/2}.\end{split}$

By (34), we know that

$\lim_{t\to\infty}\sum_{i=2k+1}^{2n}\bigg{(}\int_{\mathbb{R}^{n-k-1}}\big{\|}\tilde{\chi}_{i}(\centerdot,\kappa)(q_{\alpha,\lambda(t+s)}^{-1}-q_{\alpha,\lambda
t}^{-1})\big{\|}_{H^{-\delta}(\mathbb{R}^{k})}^{2}\,\mathrm{d}\kappa\bigg{)}^{1/2}=0$

uniformly in $s\geq 0$. This corresponds to the last $2(n-k)$ charts. Summing
up, we have proved that for every $\lambda>0$

(55)
$\lim_{t\to\infty}\lambda^{\alpha}\big{\|}[(q_{\alpha,t+s}^{-1}-q_{\alpha,t}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\lambda\centerdot)\big{\|}_{H^{-\delta}(\SS^{n-1})}=0$

uniformly in $s\geq 0$, which proves (I).

As a consequence of (55) in the case $\lambda=1$, we can derive that (III)
holds. Indeed, thanks to (55) we have that, for every sequence
$\\{a_{N}\in\mathbb{R}_{+}:N\in\mathbb{N}\\}$ such that
$\lim_{N\to\infty}a_{N}=\infty$, the sequence
$\\{[q_{\alpha,a_{N}}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\centerdot):N\in\mathbb{N}\\}$
is Cauchy in $H^{-\delta}(\SS^{n-1})$. Moreover, given two sequences
$\\{a_{N}\in\mathbb{R}_{+}:N\in\mathbb{N}\\}$ and
$\\{b_{N}\in\mathbb{R}_{+}:N\in\mathbb{N}\\}$ such that
$\lim_{N\to\infty}a_{N}=\lim_{N\to\infty}b_{N}=\infty$, we have that the
sequence $\\{c_{N}\in\mathbb{R}_{+}:N\in\mathbb{N}\\}$ with $c_{2L-1}=a_{L}$
and $c_{2L}=b_{L}$ for $L\in\mathbb{N}$ also satisfies that
$\lim_{N\to\infty}c_{N}=\infty$. Therefore, the sequence
$\\{[q_{\alpha,c_{N}}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\centerdot):N\in\mathbb{N}\\}$
is also Cauchy in $H^{-\delta}(\SS^{n-1})$. Hence, the Cauchy sequences
$\\{[q_{\alpha,a_{N}}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\centerdot):N\in\mathbb{N}\\}$
and
$\\{[q_{\alpha,b_{N}}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\centerdot):N\in\mathbb{N}\\}$
belong to the same equivalence class. Consequently, all these Cauchy sequences
in $H^{-\delta}(\SS^{n-1})$ belong to the same equivalence class. Let
$r_{\alpha}$ be the equivalence class determined by these Cauchy sequences.
Then, for every sequence $\\{a_{N}\in\mathbb{R}_{+}:N\in\mathbb{N}\\}$ such
that $\lim_{N\to\infty}a_{N}=\infty$, we have that

(56)
$\lim_{N\to\infty}\big{\|}[q_{\alpha,a_{N}}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\centerdot)-r_{\alpha}\big{\|}_{H^{-\delta}(\SS^{n-1})}=0.$

Eventually, by (43) we have for all $\lambda\in\mathbb{R}_{+}$ that

$\lim_{N\to\infty}\big{\|}[\lambda^{\alpha}[q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\lambda\centerdot)-r_{\alpha}\big{\|}_{H^{-\delta}(\SS^{n-1})}=\lim_{N\to\infty}\big{\|}[q_{\alpha,\lambda
N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\centerdot)-r_{\alpha}\big{\|}_{H^{-\delta}(\SS^{n-1})}=0.$

This proves (III).

Finally, we address (II). In order to show that it holds, note that the
function

$t\in(0,\infty)\longmapsto\|[q_{\alpha,t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\centerdot)\|_{H^{-\delta}(\SS^{n-1})}$

is continuous and by (31) it follows that

$\lim_{t\to
0}\|[q_{\alpha,t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\centerdot)\|_{H^{-\delta}(\SS^{n-1})}=0.$

This together with (56) imply that there is a constant $C$ that only depends
on $k$, $n$ and $\delta$ such that

$\sup_{t>0}\big{\|}[q_{\alpha,t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\centerdot)\big{\|}_{H^{-\delta}(\SS^{n-1})}\leq
C.$

Again the identity (43) implies that

$\lambda^{\alpha}\big{\|}[q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\lambda\centerdot)\big{\|}_{H^{-\delta}(\SS^{n-1})}\leq
C$

for all $\lambda\in\mathbb{R}_{+}$ and $N\in\mathbb{N}$. This concludes the
proof of (II).

###  4.4. The distribution $r_{\alpha}$ on $\SS^{n-1}$

On the one hand, $r_{\alpha}$ determines a unique distribution on $\SS^{n-1}$
(see Corollary 6.3). On the other hand, distributions on $\SS^{n-1}$ are a
family of distributions in open subsets of $\mathbb{R}^{n-1}$ associated to
local charts (see Section 6.1). Therefore, in order to describe the
distribution on $\SS^{n-1}$ determined by $r_{\alpha}$, we only have to
compute the distributional limit of

$\lim_{N\to\infty}[q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}]\circ\varphi_{m}^{-1}$

in $\varphi_{m}(U_{m})$, where
$\\{(U_{1},\varphi_{1}),\dots,(U_{2n},\varphi_{2n})\\}$ is the atlas
introduced in Section 4.3. For convenience, recall that $\varphi_{m}(U_{m})=V$
for all $m\in\\{1,\dots,2n\\}$, where $V$ was defined in (44).

######  Proposition 4.4.

Let $r_{\alpha,m}\in\mathcal{D}^{\prime}(V)$ with $m\in\\{1,\dots,2n\\}$ be
defined by

$\langle
r_{\alpha,m},\phi\rangle=\begin{cases}\displaystyle\int_{V}\frac{\phi(\zeta)}{q_{\alpha}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}(\varphi_{m}^{-1}(\zeta))}\,\mathrm{d}\zeta,&\quad
1\leq m\leq 2k,\vspace{.7em}\\\ \langle
q_{\alpha}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k-1}},\phi\rangle,&\quad
2k<m\leq 2n,\end{cases}$

for all $\phi\in\mathcal{D}(V)$, where
$q_{\alpha}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}(\xi)=q_{\alpha}(e_{1}\cdot\xi,\dots,e_{k}\cdot\xi)$
for all $\xi\in\mathbb{R}^{n}$. Then, the sequence $\\{r_{\alpha}^{N}\in
C^{\infty}(\SS^{n-1}):N\in\mathbb{N}\\}$ given by

$r_{\alpha}^{N}(\theta)=(q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}})(\theta),\qquad\theta\in\SS^{n-1},$

satisfies that

(57)
$\lim_{N\to\infty}\int_{V}r_{\alpha}^{N}(\varphi_{m}^{-1}(\zeta))\phi(\zeta)\,\mathrm{d}\zeta=\langle
r_{\alpha,m},\phi\rangle,$

for all $\phi\in\mathcal{D}(V)$ and $m\in\\{1,\dots,2n\\}$.

Note that the integral in $V$ for $1\leq m\leq 2k$ is finite for all
$\phi\in\mathcal{D}(V)$ since (50) and (54) imply
$|q_{\alpha}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}(\varphi_{m}^{-1}(\zeta))|>0$
for all $\zeta\in V$.

###### Proof.

By definition,

$\int_{V}r_{\alpha}^{N}(\varphi_{m}^{-1}(\zeta))\phi(\zeta)\,\mathrm{d}\zeta=\int_{V}(q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}})(\varphi_{m}^{-1}(\zeta))\phi(\zeta)\,\mathrm{d}\zeta.$

From (32), (49) and (53) we know that whenever $1\leq m\leq 2k$

$\lim_{N\to\infty}\sup_{\theta\in\overline{U_{m}}}\Big{|}\frac{1}{q_{\alpha}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}(\theta)}-(q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}})(\theta)\Big{|}=0,$

and consequently

$\lim_{N\to\infty}\int_{V}r_{\alpha}^{N}(\varphi_{m}^{-1}(\zeta))\phi(\zeta)\,\mathrm{d}\zeta=\int_{V}\frac{\phi(\zeta)}{q_{\alpha}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}(\varphi_{m}^{-1}(\zeta))}\,\mathrm{d}\zeta.$

This proves (57) whenever $1\leq m\leq 2k$. We focus now on the case $2k<m\leq
2n$. From the identities (47), (48), (51) and (52) we know that

$(q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}})(\varphi_{m}^{-1}(\zeta))=(q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k-1}})(\zeta)\quad\forall\,\zeta\in
V,$

and consequently that

$\begin{split}\int_{V}r_{\alpha}^{N}(\varphi_{m}^{-1}(\zeta))\phi(\zeta)\,\mathrm{d}\zeta&=\int_{V}(q_{\alpha,N}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k-1}})(\zeta)\phi(\zeta)\,\mathrm{d}\zeta\\\
&=\int_{\mathbb{R}^{k}}q_{\alpha,N}^{-1}(\eta)\Big{(}\int_{\mathbb{R}^{n-k-1}}\phi(\eta,\kappa)\,\mathrm{d}\kappa\Big{)}\,\mathrm{d}\eta.\end{split}$

By (33), we know that

$\begin{split}\lim_{n\to\infty}\int_{\mathbb{R}^{k}}q_{\alpha,N}^{-1}(\eta)&\Big{(}\int_{\mathbb{R}^{n-k-1}}\phi(\eta,\kappa)\,\mathrm{d}\kappa\Big{)}\,\mathrm{d}\eta=\langle
q_{\alpha}^{-1},\int_{\mathbb{R}^{n-k-1}}\phi(\centerdot,\kappa)\,\mathrm{d}\kappa\rangle\\\
&=\langle
q_{\alpha}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k-1}},\phi\rangle.\end{split}$

This means that

$\lim_{n\to\infty}\int_{V}r_{\alpha}^{N}(\varphi_{m}^{-1}(\zeta))\phi(\zeta)\,\mathrm{d}\zeta=\langle
q_{\alpha}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k-1}},\phi\rangle,$

which proves (57) when $2k<m\leq 2n$. This completes the proof of Proposition
4.4 for every $m\in\\{1,\dots,2n\\}$. ∎

##  5\. Proofs of Theorems 2 and 3

This section contains two subsections devoted to the proofs of Theorem 2 and
3, respectively.

###  5.1. Loss of derivatives when averaging in $L^{2}(\mathrm{SO}(n))$

In this section we prove 2, which asserts a loss of at least $s$ derivatives
for the $k$-plane Riesz potential of order $\alpha$ with $s=\alpha-k/2$ if
$\alpha>k/2$ and $s>0$ if $\alpha=k/2$. Before providing the explicit
expression of the sequence $\\{f_{N}:N\in\mathbb{N}\\}$ stated in that
theorem, note that by identities (11) and (14) with
$p_{\alpha}(\xi,Q)=(|\xi\cdot Qe_{1}|^{2}+\dots+|\xi\cdot
Qe_{k}|^{2})^{\alpha/2}$ we have

$\begin{split}\int_{\mathrm{SO}(n)}&\|(-\Delta)^{\alpha/2}I_{\alpha}f_{N}(\centerdot,Q)\|^{2}_{L^{2}(B_{R})}\,\mathrm{d}\mu(Q)\\\
&=\int_{\mathrm{SO}(n)}\int_{B_{R}}\Big{|}\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^{n}}e^{\mathrm{i}x\cdot\xi}m_{\alpha}(\xi,Q)\widehat{f_{N}}(\xi)\,\mathrm{d}\xi\Big{|}^{2}\,\mathrm{d}x\,\mathrm{d}\mu(Q)\\\
&=\int_{B_{R}}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}e^{\mathrm{i}x\cdot(\xi-\eta)}K_{\alpha}(\xi,\eta)\widehat{f_{N}}(\xi)\overline{\widehat{f_{N}}(\eta)}\,\mathrm{d}(\xi,\eta)\,\mathrm{d}x,\end{split}$

where

$K_{\alpha}(\xi,\eta)=\frac{1}{(2\pi)^{n}}\int_{\mathrm{SO}(n)}m_{\alpha}(\xi,Q)m_{\alpha}(\eta,Q)\,\mathrm{d}\mu(Q),\qquad(\xi,\eta)\in\mathbb{R}^{n}\times\mathbb{R}^{n}.$

Since $x$ is localized in $B_{R}$, we will localize $\xi-\eta$ in a ball of
radius of the order $1/R$ so that $e^{ix\cdot(\xi-\eta)}$ does not oscillate.
Additionally, we want $\widehat{f_{N}}$ to be suitably supported on the high
frequencies of the order $N$. Thus we choose

$\widehat{f_{N}}(\xi)=R^{n/2}\mathbf{1}_{B(Ne_{1};\pi/(8R))}(\xi),\qquad\xi\in\mathbb{R}^{n}.$

Here $\mathbf{1}_{B(Ne_{1};\pi/(8R))}(\xi)$ denotes the characteristic
function of the ball $B(Ne_{1};\pi/(8R))$ centred at $Ne_{1}$ and having
radius $\pi/(8R)$. The normalization factor $R^{n/2}$ easily implies the
estimate $\|f_{N}\|_{\dot{H}^{s}(\mathbb{R}^{n})}\eqsim N^{s}$ whenever
$N>\pi/(4R)$. With this choice for $f_{N}$ we have that

$\begin{split}\int_{\mathrm{SO}(n)}&\|(-\Delta)^{\alpha/2}I_{\alpha}f_{N}(\centerdot,Q)\|^{2}_{L^{2}(B_{R})}\,\mathrm{d}\mu(Q)\\\
&=R^{n}\int_{B_{R}}\int_{\Sigma_{N}^{R}}\cos(x\cdot(\xi-\eta))K_{\alpha}(\xi,\eta)\,\mathrm{d}(\xi,\eta)\,\mathrm{d}x,\end{split}$

where $\Sigma_{N}^{R}=B(Ne_{1};\pi/(8R))\times B(Ne_{1};\pi/(8R))$. Because of
the localization $x\in B_{R}$ and $(\xi,\eta)\in\Sigma_{N}^{R}$, we have that
$\cos(x\cdot(\xi-\eta))\geq 1/\sqrt{2}$ and consequently

$\int_{\mathrm{SO}(n)}\|(-\Delta)^{\alpha/2}I_{\alpha}f_{N}(\centerdot,Q)\|^{2}_{L^{2}(B_{R})}\,\mathrm{d}\mu(Q)\gtrsim
R^{2n}\int_{\Sigma_{N}^{R}}K_{\alpha}(\xi,\eta)\,\mathrm{d}(\xi,\eta).$

By the translations $\zeta=\xi-Ne_{1}$ and $\kappa=\eta-Ne_{1}$, the right-
hand side of the previous inequality is equal to

$R^{2n}\int_{\Sigma_{0}^{R}}K_{\alpha}(Ne_{1}+\zeta,Ne_{1}+\kappa)\,\mathrm{d}(\zeta,\kappa).$

The fact that $\zeta\in B(0;\pi/(8R))$ implies that

$m_{\alpha}(Ne_{1}+\zeta,Q)\gtrsim\frac{1}{(|e_{1}\cdot
Qe_{1}|^{2}+\dots+|e_{1}\cdot Qe_{k}|^{2}+(RN)^{-2})^{\alpha/2}}$

whenever $N>\pi/(2R)$. Thus,

$\begin{split}R^{2n}\int_{\Sigma_{0}^{R}}&K_{\alpha}(Ne_{1}+\zeta,Ne_{1}+\kappa)\,\mathrm{d}(\zeta,\kappa)\\\
&\gtrsim\int_{\mathrm{SO}(n)}\frac{1}{(|e_{1}\cdot
Qe_{1}|^{2}+\dots+|e_{1}\cdot
Qe_{k}|^{2}+(RN)^{-2})^{\alpha}}\,\mathrm{d}\mu(Q)\\\
&\eqsim\int_{\SS^{n-1}}\frac{1}{(|\theta\cdot e_{1}|^{2}+\dots+|\theta\cdot
e_{k}|^{2}+(RN)^{-2})^{\alpha}}\,\mathrm{d}S(\theta).\end{split}$

In the last step we have used the well known identity for the Haar measure
stated in Proposition A.3. Finally, in order to complete the proof of 2 we
show that the integral on the sphere has a lower bound of the order
$\mathfrak{g}(RN)$ whenever $N>\pi/(2R)$.

In the statement of the lemma below we remember that the growth function
$\mathfrak{g}$ is defined as $\mathfrak{g}(t)=\log t$ if $\alpha=k/2$ and
$\mathfrak{g}(t)=t^{2\alpha-k}$ if $\alpha>k/2$.

######  Lemma 5.1.

We have that

$\int_{\SS^{n-1}}\frac{1}{(|\theta\cdot e_{1}|^{2}+\dots+|\theta\cdot
e_{k}|^{2}+\varepsilon^{2})^{\alpha}}\,\mathrm{d}S(\theta)\gtrsim\mathfrak{g}(1/\varepsilon)$

whenever $\varepsilon<1$.

###### Proof.

To prove this lemma we could use two charts consisting on half of the sphere
each one. However, for simplicity we use again the same atlas introduced in
(44),(45) and (46). Let $\\{\chi_{1},\dots,\chi_{2n}\\}$ be a partition of
unity subordinated to this atlas such that $\chi_{j}(\varphi_{j}^{-1}(y))\geq
c>0$ whenever $|y|^{2}\leq(n-1)/n$. By (44) this condition can be satisfied
for all $y\in V$. We start by neglecting the charts where the integrand is
bounded

$\begin{split}\int_{\SS^{n-1}}&\frac{1}{(|\theta\cdot
e_{1}|^{2}+\dots+|\theta\cdot
e_{k}|^{2}+\varepsilon^{2})^{\alpha}}\,\mathrm{d}S(\theta)\\\
&\geq\sum_{j=2k+1}^{2n}\int_{\SS^{n-1}}\frac{\chi_{j}(\theta)}{(|\theta\cdot
e_{1}|^{2}+\dots+|\theta\cdot
e_{k}|^{2}+\varepsilon^{2})^{\alpha}}\,\mathrm{d}S(\theta).\end{split}$

As mentioned, we have neglected the first $2k$ charts because $|\theta\cdot
e_{j}|^{2}>1/(2n)$ for all $\theta\in U_{2j-1}\cup U_{2j}$ and
$j\in\\{1,\dots,k\\}$. Expressing the remaining integrals explicitly in
coordinates the right-hand side of the previous inequality is bounded below by

$2(n-k)c\int_{W}\frac{(1+|\nabla\phi(y)|^{2})^{1/2}}{(|y^{\prime}|^{2}+\varepsilon^{2})^{\alpha}}\,\mathrm{d}y\gtrsim\int_{W^{\prime}}\frac{1}{(|y^{\prime}|^{2}+\varepsilon^{2})^{\alpha}}\,\mathrm{d}y^{\prime}$

where $W=\\{y\in\mathbb{R}^{n-1}:|y|^{2}\leq(n-1)/n\\}\subset V$,
$y=(y^{\prime},y^{\prime\prime})\in\mathbb{R}^{k}\times\mathbb{R}^{n-1-k}$ and
$W^{\prime}=\\{y^{\prime}\in\mathbb{R}^{k}:|y^{\prime}|^{2}\leq(n-1)/(2n)\\}$.
Finally, changing to spherical coordinates in $\mathbb{R}^{k}$, we see that
right-hand side of the last inequality is of the order
$\mathfrak{g}(1/\varepsilon)$ when $\varepsilon\to 0$. ∎

###  5.2. The cancellation property (c) is necessary

In this section we prove 3, which implies the need of the cancellation
property (c) to ensure the maximum gain in the rotational smoothing.

Let $f\in\mathcal{S}(\mathbb{R}^{n})$ be any function with
$\|f\|_{L^{2}(\mathbb{R}^{n})}=1$ such that $\widehat{f}$ has compact support
in the set $\\{\xi\in\mathbb{R}^{n}:1<|\xi|<2\\}$. We then define $f_{N}$ so
that

$\widehat{f}_{N}(\xi)=N^{-n/2}\widehat{f}(\xi/N)$

holds. Thus, the support of $\widehat{f}_{N}$ is contained in the set
$\\{\xi\in\mathbb{R}^{n}:N<|\xi|<2N\\}$.

Consider the operator $S_{\alpha,t}$ defined in (39). Then by a
straightforward change of variables, and by the discussion in Section 4.2 one
has that

(58)
$\|S_{\alpha,t}f_{N}\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}^{2}=\int_{\mathbb{R}^{n}}|\xi|^{2\alpha}\|[q_{\alpha,t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](|\xi|\centerdot)\|_{H^{-\delta}(\SS^{n-1})}^{2}|\widehat{f}_{N}(\xi)|^{2}\,\mathrm{d}\xi\\\
=\int_{1<|\xi|<2}N^{2\alpha}|\xi|^{2\alpha}\|[q_{\alpha,t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](N|\xi|\centerdot)\|_{H^{-\delta}(\SS^{n-1})}^{2}|\widehat{f}(\xi)|^{2}\,\mathrm{d}\xi.$

From (28) one gets, for all $\phi\in\mathcal{S}(\mathbb{R}^{k})$, that

(59) $\langle q_{\alpha}^{-1},\phi\rangle=\lambda^{-\alpha}\langle
q_{\alpha}^{-1},\lambda^{k}\phi(\lambda\centerdot)\rangle+\lambda^{-\alpha}\log\lambda\langle
d_{\alpha},\lambda^{k}\phi(\lambda\centerdot)\rangle,$

where

$\langle
d_{\alpha},\phi\rangle=\int_{\mathbb{R}^{k}}\frac{1}{q_{\alpha}(\zeta)}(\zeta\cdot\nabla)^{\alpha-k+1}\phi(\zeta)\,\mathrm{d}\zeta.$

Consider $q_{\alpha,t}^{-1}$ as defined in (30), and let
$d_{\alpha,t}(\eta)=\langle d_{\alpha},\psi_{t}(\eta-\centerdot)\rangle$ with
$\psi_{t}(\eta)=t^{k}\psi(t\eta)$ $\eta\in\mathbb{R}^{k}$ and $t>0$. For all
$\eta\in\mathbb{R}^{k}$, $t>0$ and $\lambda>0$ we have by (59) that

$\begin{split}\lambda^{\alpha}q_{\alpha,t}^{-1}(\lambda\eta)&=\lambda^{\alpha}\langle
q_{\alpha}^{-1},\psi_{t}(\lambda\eta-\centerdot)\rangle=\langle
q_{\alpha}^{-1},\lambda^{k}\psi_{t}(\lambda\eta-\lambda\centerdot)\rangle+\log\lambda\langle
d_{\alpha},\lambda^{k}\psi_{t}(\lambda\eta-\lambda\centerdot)\rangle\\\
&=\langle q_{\alpha}^{-1},\psi_{\lambda
t}(\eta-\centerdot)\rangle+\log\lambda\langle d_{\alpha},\psi_{\lambda
t}(\eta-\centerdot)\rangle=q_{\alpha,\lambda
t}^{-1}(\eta)+\log\lambda\,d_{\alpha,\lambda t}(\eta).\end{split}$

Thus, for every $\lambda,t\in\mathbb{R}_{+}$ and $\theta\in\SS^{n-1}$ we have
that

(60)
$\lambda^{\alpha}[q_{\alpha,t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\lambda\theta)=[q_{\alpha,\lambda
t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\theta)+\log\lambda[d_{\alpha,\lambda
t}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](\theta).$

We now claim that

(61)
$\lim_{t\to\infty}\big{\|}(q_{\alpha,t+s}^{-1}-q_{\alpha,t}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}\big{\|}_{H^{-\delta}(\SS^{n-1})}=0,$

and

(62)
$\lim_{t\to\infty}\big{\|}(d_{\alpha,t+s}-d_{\alpha,t})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}\big{\|}_{H^{-\delta}(\SS^{n-1})}=0,$

uniformly for $s\geq 0$. The proof of this claim is, in both cases, exactly
the same as the proof of (55) with $\lambda=1$, so we omit it. As a
consequence of this, there exist some constants $A>0$ and $B>0$ independent of
$t$ and $\lambda$ such that

(63) $\|q_{\alpha,\lambda
t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}\|_{H^{-\delta}(\SS^{n-1})}\leq
A\quad\text{and}\quad\|d_{\alpha,\lambda
t}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}\|_{H^{-\delta}(\SS^{n-1})}\geq B$

for all $t>1$ and $\lambda\geq\lambda_{0}$ for a large enough $\lambda_{0}>0$.
Notice that one needs (c) to be false in order to get the second estimate,
since this guarantees that $d_{\alpha}$ and its restriction to the sphere
defined by (62) are non-vanishing (see the proof of Lemma 3.2).

From the previous discussion with $\lambda=|\xi|N$ we get that

$\displaystyle N^{2\alpha}|\xi|^{2\alpha}\|[q_{\alpha,t}^{-1}$
$\displaystyle\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](N|\xi|\centerdot)\|_{H^{-\delta}(\SS^{n-1})}^{2}$
$\displaystyle=\|q_{\alpha,N|\xi|t}^{-1}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}+\log(|\xi|N)\,d_{\alpha,N|\xi|t}\otimes\mathbf{1}_{\mathbb{R}^{n-k}}\|_{H^{-\delta}(\SS^{n-1})}^{2}$
$\displaystyle\geq\left(B\log(|\xi|N)-A\right)^{2}\geq\left(B\log
N-A\right)^{2}\gtrsim(\log N)^{2}$

for all $1<|\xi|<2$, $t>1$, and for all $N\geq N_{0}$ for a large enough
$N_{0}\in\mathbb{N}$ independent of $\xi$ and $t$. Combining this with (58)
yields

$\|S_{\alpha,t}f_{N}\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}\gtrsim\log
N,$

for all $t>1$ and $N\geq N_{0}$. Since the previous estimate is uniform in
$t$, to conclude the proof of the theorem it is enough to justify that we can
pass to the limit $t$ tends to $\infty$. We now do this. First observe that
(60),(61) and (62) imply that for any fixed $\lambda>0$

$\lambda^{\alpha}\lim_{t\to\infty}\big{\|}(q_{\alpha,t+s}^{-1}-q_{\alpha,t}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}(\lambda\centerdot)\big{\|}_{H^{-\delta}(\SS^{n-1})}=0$

uniformly in $s\geq 0$. Hence, if $M,L\in\mathbb{N}$,

$\begin{split}&\lim_{M\to
0}S(\SS^{n-1})\|S_{\alpha,M+L}f_{N}-S_{\alpha,M}f_{N}\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}^{2}\\\
&=\int_{N<|\xi|<2N}\lim_{M\to
0}|\xi|^{2\alpha}\|[(q_{\alpha,|\xi|(M+L)}^{-1}-q_{\alpha,|\xi|M}^{-1})\otimes\mathbf{1}_{\mathbb{R}^{n-k}}](|\xi|\centerdot)\|_{H^{-\delta}(\SS^{n-1})}^{2}|\widehat{f}_{N}(\xi)|^{2}\,\mathrm{d}\xi=0,\end{split}$

for $N\geq N_{0}$. Since $|\xi|\simeq N$ in the domain of the integral, here
the use of the dominated convergence theorem is allowed for $N$ large enough
by the first estimate of (63). Therefore
$\\{S_{\alpha,M}f_{N}:M\in\mathbb{N}\\}$ is a Cauchy sequence for each fixed
$N\geq N_{0}$ so, by the same uniqueness on limit arguments explained at
Section 4.2, we conclude that

$\|S_{\alpha}f_{N}\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}=\lim_{M\to\infty}\|S_{\alpha,M}f_{N}\|_{H^{-\delta}(\mathrm{SO}(n);\dot{H}^{\alpha}(\mathbb{R}^{n}))}\gtrsim\log
N,$

for all $N\geq N_{0}$. This concludes the proof of 3.

##  6\. The functional analytical framework

In this section we adopt a rather abstract point of view and establish the
analytical framework that we have used in order to state and prove Theorems
2.1 and 2.2. The section is divided in two, Sections 6.1 and 6.2. In Section
6.1 we recall the notion of distributions on smooth manifolds and prove a
technical lemma that will be very useful in Section 6.2. In Section 6.2 we
give an intrinsic description of fractional Sobolev spaces on compact
boundaryless Riemannian manifolds. Then, we introduce the mixed-norm Sobolev
spaces that we have used in Theorems 2.1 and 2.2 in order to quantify the
effect of rotational smoothing.

###  6.1. Distributions on a smooth manifolds

We begin by recalling the definition of a distribution on a smooth manifold
$M$. For convenience, if $U$ and $V$ are open subsets of $\mathbb{R}^{m}$ with
$m\in\mathbb{N}$, $U\subset V$ and $\phi\in\mathcal{D}(U)$, then we will still
write $\phi$ to denote the extension of this function to $V$, defined to be
zero in $V\setminus U$. In the same way, if $u\in\mathcal{D}^{\prime}(V)$ we
still write $u$ to refer to the distribution
$\phi\in\mathcal{D}(U)\mapsto\langle u,\phi\rangle$ that belongs to
$\mathcal{D}^{\prime}(U)$. Let $(M,\mathcal{A})$ be a smooth manifold where
$M$ is a topological manifold of dimension $m\in\mathbb{N}$ and $\mathcal{A}$
is a maximal smooth atlas. Assume that for every chart
$(U_{\alpha},\varphi_{\alpha})\in\mathcal{A}$ there is a distribution
$u_{\alpha}$ in $\varphi_{\alpha}(U_{\alpha})\subset\mathbb{R}^{m}$. If for
every $(U_{\alpha},\varphi_{\alpha})$ and $(U_{\beta},\varphi_{\beta})$ in
$\mathcal{A}$ with $U_{\alpha}\cap U_{\beta}\neq\emptyset$ we have that

(64) $\langle
u_{\alpha},\phi\rangle=\langle(\varphi_{\beta}\circ\varphi_{\alpha}^{-1})^{\ast}u_{\beta},\phi\rangle\qquad\forall\,\phi\in\mathcal{D}(\varphi_{\alpha}(U_{\alpha}\cap
U_{\beta})),$

then the family of all distributions $u_{\alpha}$ is said to be a distribution
$u\in\mathcal{D}^{\prime}(M)$. Given a distribution
$u\in\mathcal{D}^{\prime}(M)$ it is convenient to denote the corresponding
$u_{\alpha}$ by $u\circ\varphi_{\alpha}^{-1}$. According to [5, Theorem 6.3.4]
a distribution $u\in\mathcal{D}^{\prime}(M)$ is determined only by the family
of distributions $u_{\alpha}$ corresponding to the charts
$(U_{\alpha},\varphi_{\alpha})$ in an atlas $\mathcal{B}$ that is not
necessarily maximal. More precisely, assume that
$\mathcal{B}\subset\mathcal{A}$ is an atlas and that for every chart
$(U_{\alpha},\varphi_{\alpha})\in\mathcal{B}$ there is a distribution
$u_{\alpha}$ in $\varphi_{\alpha}(U_{\alpha})\subset\mathbb{R}^{m}$. If for
every $(U_{\alpha},\varphi_{\alpha})$ and $(U_{\beta},\varphi_{\beta})$ in
$\mathcal{B}$ with $U_{\alpha}\cap U_{\beta}\neq\emptyset$ the identity (64)
holds, then there exists a unique $u\in\mathcal{D}^{\prime}(M)$ such that
$u\circ\varphi_{\alpha}^{-1}=u_{\alpha}$ for all
$(U_{\alpha},\varphi_{\alpha})\in\mathcal{B}$.

In the remaining of Section 6.1 we will show that elements in the Banach
completion of a normed subspace of the smooth functions on a manifold
determine distributions in the manifold.

For convenience, we first recall briefly the definition of the Banach
completion of a normed space $(V,\|\centerdot\|_{V})$. Let $B$ denote the set
of all equivalence classes of Cauchy sequences in the normed space obtained by
the following relation: Given two Cauchy sequences
$\\{x_{N}:N\in\mathbb{N}\\}$ and $\\{y_{N}:N\in\mathbb{N}\\}$ in $V$ we say
that they are related if and only if

$\lim_{N\to\infty}\|x_{N}-y_{N}\|_{V}=0.$

The vector space structure of $V$ naturally induces a vector structure on $B$.
Furthermore, we can endow $B$ with a norm defined as follows: if $x\in B$
denotes the equivalence class of the Cauchy sequence
$\\{x_{N}:N\in\mathbb{N}\\}\subset V$, we define the norm

$\|x\|_{B}=\lim_{N\to\infty}\|x_{N}\|_{V}.$

Then, $(B,\|\centerdot\|_{B})$ is a Banach space and contains a dense subspace
isometric to $V$. The space $(B,\|\centerdot\|_{B})$ is called the Banach
completion of $(V,\|\centerdot\|_{V})$.

In order to address the last endeavour of Section 6.1, let $(M,\mathcal{A})$
be a smooth manifold of dimension $m\in\mathbb{N}$. Assume that there exists
an atlas $\\{(U_{1},\varphi_{1}),\dots,(U_{l},\varphi_{l})\\}$ with a finite
number $l\in\mathbb{N}$ of charts and choose $\\{\chi_{1},\dots,\chi_{l}\\}$ a
smooth partition of unity subordinated to this atlas. For convenience, we
agree from now on that whenever $u\in C^{\infty}(M)$, the extension by zero of
the functions $(\chi_{j}u)\circ\varphi_{j}^{-1}$ to
$\mathbb{R}^{m}\setminus\varphi_{j}(U_{j})$ with $j\in\\{1,\dots,l\\}$ will be
still denoted by $(\chi_{j}u)\circ\varphi_{j}^{-1}$. Let $\mathscr{V}(M)$
denote a vector subspace of $C^{\infty}(M)$ such that if $u\in\mathscr{V}(M)$,
then $(\chi_{j}u)\circ\varphi_{j}^{-1}\in\mathcal{S}(\mathbb{R}^{m})$ for all
$j\in\\{1,\dots,l\\}$. Given a norm $\|\centerdot\|$ on
$\mathcal{S}(\mathbb{R}^{m})$ such that convergent sequences with respect to
this norm also converge in $\mathcal{S}^{\prime}(\mathbb{R}^{m})$, we endow
$\mathscr{V}(M)$ with the norm defined by

$\|u\|_{\mathscr{V}(M)}=\sum_{j=1}^{l}\big{\|}(\chi_{j}u)\circ\varphi_{j}^{-1}\big{\|},\qquad
u\in\mathscr{V}(M).$

Let $(\mathscr{B}(M),\|\centerdot\|_{\mathscr{B}(M)})$ denote the Banach
completion of $(\mathscr{V}(M),\|\centerdot\|_{\mathscr{V}(M)})$. Under these
conditions we can see that the elements of $\mathscr{B}(M)$ determine
distributions in $M$.

######  Lemma 6.1.

Assume $M$ and $\mathscr{B}(M)$ to be as above. Then every element in
$\mathscr{B}(M)$ determines a unique distribution in $M$.

###### Proof.

Let $u\in\mathscr{B}(M)$ be given and consider a Cauchy sequence
$\\{u_{N}:N\in\mathbb{N}\\}\subset\mathscr{V}(M)$ in the equivalence class
denoted by $u$. We will show that $u$ determines a unique distribution in $M$.
For every $\phi\in\mathcal{D}(\varphi_{j}(U_{j}))$ with $j\in\\{1,\dots,l\\}$
the following limit exists

(65)
$\lim_{N\to\infty}\int_{\varphi_{j}(U_{j})}u_{N}\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x.$

Indeed,

$\int_{\varphi_{j}(U_{j})}u_{N}\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x=\sum_{i=1}^{l}\int_{\varphi_{j}(U_{j})}(\chi_{i}u_{N})\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x.$

On the one hand if $U_{i}\cap U_{j}=\emptyset$, then

$\int_{\varphi_{j}(U_{j})}(\chi_{i}u_{N})\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x=0.$

On the other hand, $U_{i}\cap U_{j}\neq\emptyset$, then

$\int_{\varphi_{j}(U_{j})}(\chi_{i}u_{N})\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x=\int_{\varphi_{j}(U_{i}\cap
U_{j})}(\chi_{i}u_{N})\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x.$

Changing variable according to $x=\varphi_{j}\circ\varphi_{i}^{-1}(y)$ we have
that the right-hand side equals

(66) $\int_{\varphi_{i}(U_{i}\cap
U_{j})}(\chi_{i}u_{N})\circ\varphi_{i}^{-1}(y)\phi\big{(}\varphi_{j}\circ\varphi_{i}^{-1}(y)\big{)}\big{|}\det\big{(}\nabla\varphi_{j}\circ\varphi_{i}^{-1}(y)\big{)}\big{|}\,\mathrm{d}y.$

Recall that we have assumed that convergent sequences with respect to the norm
$\|\centerdot\|$ also converge in $\mathcal{S}^{\prime}(\mathbb{R}^{m})$.
Then, since $\\{(\chi_{i}u_{N})\circ\varphi_{i}^{-1}:N\in\mathbb{N}\\}$ with
$i\in\\{1,\dots,l\\}$ so that $U_{i}\cap U_{j}\neq\emptyset$ are Cauchy
sequences with respect to $\|\centerdot\|$, we know that they converge in
$\mathcal{S}^{\prime}(\mathbb{R}^{m})$. Therefore the limits of the integrals
(66) with $i\in\\{1,\dots,l\\}$ so that $U_{i}\cap U_{j}\neq\emptyset$ exist
as $N\to\infty$. This implies that the limit in (65) also exists.

Furthermore, the limit (65) is independent of the Cauchy sequence chosen in
the equivalence class denoted by $u$. Namely, if $\\{v_{N}:N\in\mathbb{N}\\}$
is another Cauchy sequence in the equivalence class of $u$, then

$\lim_{N\to\infty}\int_{\varphi_{j}(U_{j})}u_{N}\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x=\lim_{N\to\infty}\int_{\varphi_{j}(U_{j})}v_{N}\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x.$

Indeed,

$\begin{split}&\bigg{|}\int_{\varphi_{j}(U_{j})}u_{N}\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x-\int_{\varphi_{j}(U_{j})}v_{N}\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x\bigg{|}\\\
&\qquad\leq\sum_{i=1}^{l}\bigg{|}\int_{\varphi_{j}(U_{j})}(\chi_{i}u_{N}-\chi_{i}v_{N})\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x\bigg{|}.\end{split}$

As we argued earlier, if $U_{i}\cap U_{j}=\emptyset$, then

$\int_{\varphi_{j}(U_{j})}(\chi_{i}u_{N}-\chi_{i}v_{N})\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x=0.$

Otherwise, if $U_{i}\cap U_{j}\neq\emptyset$, then

$\begin{split}\int_{\varphi_{j}(U_{j})}&(\chi_{i}u_{N}-\chi_{i}v_{N})\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x\\\
&=\int_{\varphi_{j}(U_{i}\cap
U_{j})}(\chi_{i}u_{N}-\chi_{i}v_{N})\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x.\end{split}$

Changing variable according to $x=\varphi_{j}\circ\varphi_{i}^{-1}(y)$ we have
that the right-hand side equals

(67) $\int_{\varphi_{i}(U_{i}\cap
U_{j})}(\chi_{i}u_{N}-\chi_{i}v_{N})\circ\varphi_{i}^{-1}(y)\psi(y)\,\mathrm{d}y$

with
$\psi(y)=\phi\big{(}\varphi_{j}\circ\varphi_{i}^{-1}(y)\big{)}\big{|}\det\big{(}\nabla\varphi_{j}\circ\varphi_{i}^{-1}(y)\big{)}\big{|}$.
Since
$\\{(\chi_{i}u_{N}-\chi_{i}v_{N})\circ\varphi_{i}^{-1}:N\in\mathbb{N}\\}$ with
$i\in\\{1,\dots,l\\}$ so that $U_{i}\cap U_{j}\neq\emptyset$ converges to zero
with respect to $\|\centerdot\|$, we know that it also does so in
$\mathcal{S}^{\prime}(\mathbb{R}^{m})$, and therefore the limits of the
integrals (67) with $i\in\\{1,\dots,l\\}$ so that $U_{i}\cap
U_{j}\neq\emptyset$ converge to zero.

Thus, given $u\in\mathscr{B}(M)$, the limit (65) is well defined. Then we
introduce $u^{j}\in\mathcal{D}^{\prime}(\varphi_{j}(U_{j}))$ for
$j\in\\{1,\dots,l\\}$ as

$\langle
u^{j},\phi\rangle=\lim_{N\to\infty}\int_{\varphi_{j}(U_{j})}u_{N}\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x,\qquad\forall\phi\in\mathcal{D}(\varphi_{j}(U_{j})).$

Note that if $(U_{j},\varphi_{j})$ and $(U_{i},\varphi_{i})$ satisfy
$U_{j}\cap U_{i}\neq\emptyset$ then

$\int_{\varphi_{j}(U_{j}\cap
U_{i})}u_{N}\circ\varphi_{j}^{-1}(x)\phi(x)\,\mathrm{d}x=\int_{\varphi_{j}(U_{j}\cap
U_{i})}(\varphi_{i}\circ\varphi_{j}^{-1})^{\ast}(u_{N}\circ\varphi_{i}^{-1})(x)\phi(x)\,\mathrm{d}x$

for all $\phi\in\mathcal{D}(\varphi_{j}(U_{j}\cap U_{i}))$. Then, by [5,
Theorem 6.1.2] we have

$\langle
u^{j},\phi\rangle=\langle(\varphi_{i}\circ\varphi_{j}^{-1})^{\ast}u^{i},\phi\rangle,\qquad\forall\,\phi\in\mathcal{D}(\varphi_{j}(U_{j}\cap
U_{i})).$

Consequently, [5, Theorem 6.3.4] states that $\\{u^{1},\dots,u^{l}\\}$
determines a unique distribution in $M$. This concludes the proof of this
lemma. ∎

###  6.2. A convenient description of Sobolev spaces

We start this section by describing an intrinsic notion of Sobolev spaces on a
compact boundaryless Riemannian manifold $(M,g)$. Then we introduce the mixed-
norm Sobolev spaces on $\mathbb{R}^{n}\times M$ used to capture the rotational
smoothing phenomenon.

####  6.2.1. Sobolev spaces on compact boundaryless Riemannian manifolds

Let $(M,g)$ as above and denote the Laplace–Beltrami by $\Delta_{g}$ as in
Section 4.1. By Rellich’s theorem, the mapping $L^{2}(M)\ni f\mapsto u\in
L^{2}(M)$, satisfying $-\Delta_{g}u=f$ in $M$, is compact. Additionally,
$-\Delta_{g}$ is self-adjoint and non-negative. By the spectral theorem there
exists a sequence $\\{\lambda_{j}:j\in\mathbb{N}_{0}\\}$ of non- negative real
eigenvalues and an orthonormal basis of $L^{2}(M)$ formed by eigenfunctions
$\\{\psi_{j}^{i}:j\in\mathbb{N}_{0},\,i\in\mathbb{N},\,i\leq d_{j}\\}$ such
that $\lambda_{0}=0$, $\lambda_{j}<\lambda_{j+1}$, and
$-\Delta_{g}\psi_{j}^{i}=\lambda_{j}\psi_{j}^{i}$ with $\psi_{j}^{i}\in
C^{\infty}(M)$ for $i\in\\{1,\dots,d_{j}\\}$ and all $j\in\mathbb{N}_{0}$. Set
$H_{j}=\operatorname*{span}\\{\psi_{j}^{i}:i\in\mathbb{N},\,i\leq d_{j}\\}$
and define, for $u\in L^{1}(M)$, the projector $\Pi_{j}$ on $H_{j}$ as

$\Pi_{j}u(p)=\sum_{i=1}^{d_{j}}\Big{(}\int_{M}\overline{\psi_{j}^{i}}u\,\mathrm{d}\mu_{g}\Big{)}\psi_{j}^{i}(p),\qquad
p\in M.$

Here $\mu_{g}$ denotes the Riemannian density. Note that for $u\in
C^{\infty}(M)$ we have that
$\Pi_{j}[(\mathrm{Id}-\Delta_{g})u]=(1+\lambda_{j})\Pi_{j}u$. Consequently,

(68)
$\Pi_{j}[(\mathrm{Id}-\Delta_{g})\circ\overset{k}{\cdots}\circ(\mathrm{Id}-\Delta_{g})u]=(1+\lambda_{j})^{k}\Pi_{j}u\qquad\forall\,u\in
C^{\infty}(M).$

The fact that $u\in C^{\infty}(M)$ implies
$(\mathrm{Id}-\Delta_{g})\circ\overset{k}{\cdots}\circ(\mathrm{Id}-\Delta_{g})u\in
L^{2}(M)$ for every $k\in\mathbb{N}_{0}$. This conclusion together with the
identity (68) ensures that

(69)
$(\mathrm{Id}-\Delta_{g})\circ\overset{k}{\cdots}\circ(\mathrm{Id}-\Delta_{g})u=\sum_{j\in\mathbb{N}_{0}}(1+\lambda_{j})^{k}\Pi_{j}u$

and for any $k\in\mathbb{N}_{0}$

$\sum_{j\in\mathbb{N}_{0}}(1+\lambda_{j})^{2k}\|\Pi_{j}u\|_{L^{2}(M)}^{2}<\infty.$

Thus, we know that
$\\{\sum_{j=0}^{N}(1+\lambda_{j})^{s/2}\Pi_{j}u:N\in\mathbb{N}_{0}\\}$ with
$u\in C^{\infty}(M)$ is a Cauchy sequence in $L^{2}(M)$ for every
$s\in\mathbb{R}$. The limit of this sequence defines the Bessel potential of
order $s$. This is the definition given in (35) for $u\in C^{\infty}(M)$ and
$s\in\mathbb{R}$. Note that for $s\in\mathbb{R}$ and $u\in C^{\infty}(M)$, we
have that

(70)
$\|(\mathrm{Id}-\Delta_{g})^{s/2}u\|_{L^{2}(M)}=\Big{(}\sum_{j\in\mathbb{N}_{0}}(1+\lambda_{j})^{s}\|\Pi_{j}u\|_{L^{2}(M)}^{2}\Big{)}^{1/2}.$

Furthermore, the non-negative functional

$\|\centerdot\|_{H^{s}(M)}:u\in
C^{\infty}(M)\longmapsto\|(\mathrm{Id}-\Delta_{g})^{s/2}u\|_{L^{2}(M)}$

defines a norm. Identity (70) gives a simple way of computing
$\|u\|_{H^{s}(M)}$ whenever $u$ belongs to $C^{\infty}(M)$. Note that the norm
for $s=0$ coincides with the norm of ${L^{2}(M)}$. With this norm we define
the Sobolev space of order $s$ in an intrinsic way.

######  Definition 6.2.

For $s\in\mathbb{R}$ define $H^{s}(M)$ as the Banach completion of the normed
space $(C^{\infty}(M),\|\centerdot\|_{H^{s}(M)})$.

By continuity, the Bessel potential of order $s$ can be defined for every
$u\in H^{s}(M)$ and

(71) $\|u\|_{H^{s}(M)}=\|(\mathrm{Id}-\Delta_{g})^{s/2}u\|_{L^{2}(M)},$

holds for all $u\in H^{s}(M)$.

The intrinsic definition of the Sobolev spaces given above has an equivalent
description in terms of local charts, as we have stated in Proposition 4.2.
This proposition is a consequence of the results in [9, 11]. Let us mention
the main idea of the proof. Consider first the case where $s=2k$ with
$k\in\mathbb{N}$. As a consequence of the identities (71) and (69) and the
fact that, in every chart $(U_{j},\varphi_{j})$, the operator

$(\mathrm{Id}-\Delta_{g})^{k}=(\mathrm{Id}-\Delta_{g})\circ\overset{k}{\cdots}\circ(\mathrm{Id}-\Delta_{g})$

is identified with a differential operator of order $2k$ in $\mathbb{R}^{m}$,
one sees that for every $k\in\mathbb{N}$

(72)
$\|u\|_{H^{2k}(M)}\lesssim\sum_{j=1}^{N}\|(\chi_{j}u)\circ\varphi_{j}^{-1}\|_{H^{2k}(\mathbb{R}^{m})}.$

The reverse inequality follows from the well-posedness and the regularity
theory for the equation $u-\Delta_{g}u=f$ in $M$. This is fully detailed in
the book [10]. This is a sketch of the proof in the cases where $s=2k$ with
$k\in\mathbb{N}$. For general $s\in\mathbb{R}$ one first needs to show that in
every chart $(U_{j},\varphi_{j})$, the operator
$(\mathrm{Id}-\Delta_{g})^{s/2}$ is identified with an elliptic
pseudodifferential operator of order $s$ with a principal symbol given by
$(\sum_{i,k=1}^{m}g^{ik}(x)\xi_{i}\xi_{k})^{s/2}$, for
$x\in\varphi_{j}(U_{j})$ and $|\xi|\gtrsim 1$ (see the details in [9, 11]).
Thus, the generalization of (72) to $s\in\mathbb{R}$ follows from the
boundedness of pseudodifferential operators of order $s$. The reverse
inequality is a consequence of the ellipticity and the possibility of
constructing the corresponding parametrix for $(\mathrm{Id}-\Delta_{g})^{s/2}$
in each chart. This finishes the sketch of the proof in the general case
$s\in\mathbb{R}$.

We finish the discussion of Sobolev spaces on compact boundaryless Riemannian
manifolds by showing that the elements of $H^{s}(M)$ are distributions in $M$.

######  Corollary 6.3.

Let $(M,g)$ be a compact boundaryless Riemannian manifold of dimension
$m\in\mathbb{N}$. Every element in $H^{s}(M)$ with $s\in\mathbb{R}$ determines
a unique distribution in $M$.

###### Proof.

This follows from Lemma 6.1 and Proposition 4.2. ∎

####  6.2.2. Mixed-norm Sobolev spaces

As previously agreed, $(M,g)$ stands for a compact boundaryless Riemannian
manifold of dimension $m\in\mathbb{N}$. Recall the definition of
$\mathcal{A}(\mathbb{R}^{n}\times M)$ given in Definition 4.3. As mentioned
then, a function $u\in C^{\infty}(\mathbb{R}^{n}\times M)$ belongs to
$\mathcal{A}(\mathbb{R}^{n}\times M)$ if and only if its Fourier transform,
denoted by $\widehat{u}$ or $\mathcal{F}u$ and defined by (36), belongs to
$\mathcal{A}(\mathbb{R}^{n}\times M)$. Consequently, for all
$t\in\mathbb{R}_{+}$ and all $k\in\mathbb{N}_{0}$ there exists a constant
$C>0$ such that

(73)
$\int_{\mathbb{R}^{n}}|\xi|^{2t}\|(\mathrm{Id}-\Delta_{g})^{k}\widehat{u}(\xi,\centerdot)\|_{L^{2}(M)}^{2}\,\mathrm{d}\xi\leq
C.$

Note that for $k=0$ we have

$\int_{M}\|u(\centerdot,p)\|_{\dot{H}^{t}(\mathbb{R}^{n})}^{2}\,\mathrm{d}\mu_{g}(p)=\int_{\mathbb{R}^{n}}|\xi|^{2t}\|\widehat{u}(\xi,\centerdot)\|_{L^{2}(M)}^{2}\,\mathrm{d}\xi.$

This implies that for all $t\in\mathbb{R}_{+}$ we have that the non-negative
functional

$u\in\mathcal{A}(\mathbb{R}^{n}\times
M)\longmapsto\bigg{(}\int_{M}\|u(\centerdot,p)\|_{\dot{H}^{t}(\mathbb{R}^{n})}^{2}\,\mathrm{d}\mu_{g}(p)\bigg{)}^{1/2}$

is well defined. Furthermore, it defines norm on
$\mathcal{A}(\mathbb{R}^{n}\times M)$ that will be denoted from now on by
$\|\centerdot\|_{L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))}$.

######  Definition 6.4.

For $t\in\mathbb{R}_{+}$, we define $L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))$ as
the Banach completion of the normed space $(\mathcal{A}(\mathbb{R}^{n}\times
M),\|\centerdot\|_{L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))})$.

######  Proposition 6.5.

For $t\in\mathbb{R}_{+}$, we have that every $u\in
L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))$ determines a unique distribution in
$\mathbb{R}^{n}\times M$.

###### Proof.

This statement is a straightforward conclusion of Lemma 6.1. This becomes
clear when noting that $\mathbb{R}^{n}\times M$ is a smooth manifold that can
be covered by an atlas $\\{(\mathbb{R}^{n}\times
U_{1},\mathrm{id}\otimes\varphi_{1}),\dots,(\mathbb{R}^{n}\times
U_{l},\mathrm{id}\otimes\varphi_{l})\\}$ with a finite number $l\in\mathbb{N}$
of charts. Here the charts $(U_{1},\varphi_{1}),\dots,(U_{l},\varphi_{l})$
form an atlas for the compact Riemannian manifold $M$. Then, we choose
$\\{\chi_{1},\dots,\chi_{l}\\}$ a smooth partition of unity subordinated to
the atlas for $M$ and set $\mathscr{V}(\mathbb{R}^{n}\times
M)=\mathcal{A}(\mathbb{R}^{n}\times M)$ with the norm

$\|u\|_{\mathscr{V}(\mathbb{R}^{n}\times
M)}=\sum_{j=1}^{l}\Big{(}\int_{\mathbb{R}^{m}}\|\chi_{j}\circ\varphi_{j}^{-1}(y)u(\centerdot,\varphi_{j}^{-1}(y))\|_{\dot{H}^{t}(\mathbb{R}^{n})}^{2}|g_{j}(y)|^{1/2}\mathrm{d}y\Big{)}^{1/2},$

where $|g_{j}(y)|$ denotes the determinant of the matrix representing the
metric $g$ in the chart $(U_{j},\varphi_{j})$. The only point left to conclude
this proposition from Lemma 6.1 is to observe that a sequence in
$\mathscr{V}(\mathbb{R}^{n}\times M)=\mathcal{A}(\mathbb{R}^{n}\times M)$ is
Cauchy with respect to the norm
$\|\centerdot\|_{\mathscr{V}(\mathbb{R}^{n}\times M)}$ if and only if is
Cauchy with respect to
$\|\centerdot\|_{L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))}$. ∎

From the discussion on Section 6.2, we know that
$(\mathrm{Id}-\Delta_{g})^{s}u(x,\centerdot)\in L^{2}(M)$ for all
$u\in\mathcal{A}(\mathbb{R}^{n}\times M)$ and $s\in\mathbb{R}$. Furthermore,
whenever $k\in\mathbb{N}_{0}$ with $k>s$ we have

$\begin{split}\|(\mathrm{Id}-\Delta_{g})^{s}u(x,\centerdot)\|_{L^{2}(M)}&\leq\|(\mathrm{Id}-\Delta_{g})^{k}u(x,\centerdot)\|_{L^{2}(M)}\\\
&\leq\mu_{g}(M)^{1/2}\|(\mathrm{Id}-\Delta_{g})^{k}u(x,\centerdot)\|_{L^{\infty}(M)}\end{split}$

for all $x\in\mathbb{R}^{n}$. The fact that
$u\in\mathcal{A}(\mathbb{R}^{n}\times M)$ implies that the function
$x\mapsto\|(\mathrm{Id}-\Delta_{g})^{k}u(x,\centerdot)\|_{L^{\infty}(M)}$
belongs to $L^{2}(\mathbb{R}^{n})$. Hence, $(\mathrm{Id}-\Delta_{g})^{s}u\in
L^{2}(\mathbb{R}^{n}\times M)$ for all $u\in\mathcal{A}(\mathbb{R}^{n}\times
M)$ and $s\in\mathbb{R}$. Below, we will see that in fact
$(\mathrm{Id}-\Delta_{g})^{s}u\in L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))$ for
all $t\in\mathbb{R}_{+}$, and we will provide a convenient way of computing
its norm. Start by noticing that from the identity (69) we have that

(74)
$\|(\mathrm{Id}-\Delta_{g})^{k}\widehat{u}(\xi,\centerdot)\|_{L^{2}(M)}^{2}=\sum_{j\in\mathbb{N}_{0}}(1+\lambda_{j})^{2k}\|\Pi_{j}\widehat{u}(\xi,\centerdot)\|_{L^{2}(M)}^{2}$

for all $\xi\in\mathbb{R}^{n}$ and all $k\in\mathbb{N}_{0}$. Then, by the
inequality (73), we have that for all $t\in\mathbb{R}_{+}$ and
$k\in\mathbb{N}_{0}$ there exists a constant $C>0$ such that

$\sum_{j\in\mathbb{N}_{0}}(1+\lambda_{j})^{2k}\int_{\mathbb{R}^{n}}|\xi|^{2t}\|\Pi_{j}\widehat{u}(\xi,\centerdot)\|_{L^{2}(M)}^{2}\,\mathrm{d}\xi\leq
C.$

In particular, from (70) we know that the non-negative functional

$u\in\mathcal{A}(\mathbb{R}^{n}\times
M)\longmapsto\Big{(}\int_{\mathbb{R}^{n}}|\xi|^{2t}\|\widehat{u}(\xi,\centerdot)\|_{H^{s}(M)}^{2}\,\mathrm{d}\xi\Big{)}^{1/2}$

is well-defined for all $s\in\mathbb{R}$ and $t\in\mathbb{R}_{+}$.

######  Lemma 6.6.

For all $u\in\mathcal{A}(\mathbb{R}^{n}\times M)$ we have that
$(\mathrm{Id}-\Delta_{g})^{s}u\in L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))$ for
all $s\in\mathbb{R}$ and $t\in\mathbb{R}_{+}$. Moreover,

$\|(\mathrm{Id}-\Delta_{g})^{s/2}u\|_{L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))}=\Big{(}\int_{\mathbb{R}^{n}}|\xi|^{2t}\|\widehat{u}(\xi,\centerdot)\|_{H^{s}(M)}^{2}\,\mathrm{d}\xi\Big{)}^{1/2}$

for all $u\in\mathcal{A}(\mathbb{R}^{n}\times M)$, $s\in\mathbb{R}$ and
$t\in\mathbb{R}_{+}$.

###### Proof.

Observe that $\sum_{j=0}^{N}(1+\lambda_{j})^{s/2}\Pi_{j}u$ belongs to
$\mathcal{A}(\mathbb{R}^{n}\times M)$ for an arbitrary
$u\in\mathcal{A}(\mathbb{R}^{n}\times M)$, $N\in\mathbb{N}_{0}$ and
$s\in\mathbb{R}$. Moreover,

(75)
$\Big{\\{}\sum_{j=0}^{N}(1+\lambda_{j})^{s/2}\Pi_{j}u:N\in\mathbb{N}\Big{\\}}$

is a Cauchy sequence in $L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))$ for every
$s\in\mathbb{R}$ and $t\in\mathbb{R}_{+}$. Indeed, since
$\mathcal{F}(\Pi_{j}u)=\Pi_{j}\widehat{u}$, we have for every
$L,N\in\mathbb{N}$ that

(76)
$\begin{split}\Big{\|}\sum_{j=N}^{N+L}(1+\lambda_{j})^{s/2}\Pi_{j}u\Big{\|}_{L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))}^{2}&=\int_{\mathbb{R}^{n}}|\xi|^{2t}\Big{\|}\sum_{j=N}^{N+L}(1+\lambda_{j})^{s/2}\Pi_{j}\widehat{u}(\xi,\centerdot)\Big{\|}_{L^{2}(M)}^{2}\,\mathrm{d}\xi\\\
&=\int_{\mathbb{R}^{n}}|\xi|^{2t}\sum_{j=N}^{N+L}(1+\lambda_{j})^{s}\|\Pi_{j}\widehat{u}(\xi,\centerdot)\|_{L^{2}(M)}^{2}\,\mathrm{d}\xi.\end{split}$

From (74) we have for $k\in\mathbb{N}_{0}$ such that $k>s/2$ that

$\sum_{j=N}^{N+L}(1+\lambda_{j})^{s}\|\Pi_{l}\widehat{u}(\xi,\centerdot)\|_{L^{2}(M)}^{2}\leq\|(\mathrm{Id}-\Delta_{g})^{k}\widehat{u}(\xi,\centerdot)\|_{L^{2}(M)}^{2}.$

By the fact that $\widehat{u}\in\mathcal{A}(\mathbb{R}^{n}\times M)$ we have
that the function
$\xi\mapsto|\xi|^{t}\|(\mathrm{Id}-\Delta_{g})^{k}\widehat{u}(\xi,\centerdot)\|_{L^{2}(M)}$
belongs to $L^{2}(\mathbb{R}^{n})$. Then, by the dominated convergence theorem
we have that

$\lim_{N\to\infty}\int_{\mathbb{R}^{n}}|\xi|^{2t}\sum_{j=N}^{N+L}(1+\lambda_{j})^{s}\|\Pi_{j}\widehat{u}(\xi,\centerdot)\|_{L^{2}(M)}^{2}\,\mathrm{d}\xi=0$

uniformly in $L\in\mathbb{N}_{0}$, which implies that

$\lim_{N\to\infty}\Big{\|}\sum_{j=N}^{N+L}(1+\lambda_{j})^{s/2}\Pi_{j}u\Big{\|}_{L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))}=0,$

uniformly in $L\in\mathbb{N}_{0}$. Consequently the sequence (75) is Cauchy in
$L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))$. Therefore, there is $v_{s}\in
L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))$ defined by the Cauchy sequence (75). By
a standard limiting argument applied to (76) with $N=0$ and $L$ tending to
infinity, we have that

$\|v_{s}\|_{L^{2}(M;\dot{H}^{t}(\mathbb{R}^{n}))}=\Big{(}\int_{\mathbb{R}^{n}}|\xi|^{2t}\|\widehat{u}(\xi,\centerdot)\|_{H^{s}(M)}^{2}\,\mathrm{d}\xi\Big{)}^{1/2}.$

It remains to prove that $v_{s}=(\mathrm{Id}-\Delta_{g})^{s/2}u$.

Right before this lemma we saw that $(\mathrm{Id}-\Delta_{g})^{s/2}u\in
L^{2}(\mathbb{R}^{n}\times M)$ since $u\in\mathcal{A}(\mathbb{R}^{n}\times
M)$. Furthermore, recall that for every $x\in\mathbb{R}^{n}$, we defined
$(\mathrm{Id}-\Delta_{g})^{s/2}u(x,\centerdot)$ as the equivalence class of

$\Big{\\{}\sum_{j=0}^{N}(1+\lambda_{j})^{s/2}\Pi_{j}u(x,\centerdot):N\in\mathbb{N}\Big{\\}}$

which is a Cauchy sequence in $L^{2}(M)$ for every $x\in\mathbb{R}^{n}$.
Hence, whenever $k\in\mathbb{N}_{0}$ with $k>s$ and $N\in\mathbb{N}_{0}$ we
have that

$\begin{split}\Big{\|}(\mathrm{Id}-\Delta_{g})^{s/2}u(x,\centerdot)&-\sum_{j=0}^{N}(1+\lambda_{j})^{s/2}\Pi_{j}u(x,\centerdot)\Big{\|}_{L^{2}(M)}\\\
&\leq\Big{(}\sum_{j=N+1}^{\infty}(1+\lambda_{j})^{k}\|\Pi_{j}u(x,\centerdot)\|_{L^{2}(M)}^{2}\Big{)}^{1/2}\\\
&\leq\mu_{g}(M)^{1/2}\|(\mathrm{Id}-\Delta_{g})^{k}u(x,\centerdot)\|_{L^{\infty}(M)}\end{split}$

for all $x\in\mathbb{R}^{n}$. Since $u\in\mathcal{A}(\mathbb{R}^{n}\times M)$,
the function
$x\mapsto\|(\mathrm{Id}-\Delta_{g})^{k}u(x,\centerdot)\|_{L^{\infty}(M)}$
belongs to $L^{2}(\mathbb{R}^{n})$. From here one can derive that

$\lim_{N\to\infty}\int_{\mathbb{R}^{n}}\Big{\|}(\mathrm{Id}-\Delta_{g})^{s/2}u(x,\centerdot)-\sum_{j=0}^{N}(1+\lambda_{j})^{s/2}\Pi_{j}u(x,\centerdot)\Big{\|}_{L^{2}(M)}\,\mathrm{d}x=0.$

At this point we are in the position to ensure that
$v_{s}=(\mathrm{Id}-\Delta_{g})^{s/2}u$. Indeed, on the one hand we know from
Proposition 6.5 that $v_{s}$ determines a unique distribution in
$\mathbb{R}^{n}\times M$ defined from the Cauchy sequence in (75). On the
other hand, this same Cauchy sequence converges to
$(\mathrm{Id}-\Delta_{g})^{s/2}u$ in $L^{2}(\mathbb{R}^{n}\times M)$. Since
$(\mathrm{Id}-\Delta_{g})^{s/2}u$ is in particular a distribution in
$\mathbb{R}^{n}\times M$, we can conclude that
$(\mathrm{Id}-\Delta_{g})^{s/2}u$ is the distribution determined by $v_{s}$.
This concludes the proof of this lemma. ∎

After Lemma 6.6 we know that the non-negative functional defined in (37) is
well defined for all $s\in\mathbb{R}$, $t\in\mathbb{R}_{+}$ and
$u\in\mathcal{\mathcal{missing}}{A}(\mathbb{R}^{n}\times M)$.

######  Definition 6.7.

For $s\in\mathbb{R}$ and $t\in\mathbb{R}_{+}$, we define
$H^{s}(M;\dot{H}^{t}(\mathbb{R}^{n}))$ as the Banach completion of the set
$\mathcal{A}(\mathbb{R}^{n}\times M)$ endowed with the norm
$\|\centerdot\|_{H^{s}(M;\dot{H}^{t}(\mathbb{R}^{n}))}$.

We now can prove the following proposition.

######  Proposition 6.8.

For $s\in\mathbb{R}$ and $t\in\mathbb{R}_{+}$ we have that every $u\in
H^{s}(M;\dot{H}^{t}(\mathbb{R}^{n}))$ determines a unique distribution in
$\mathbb{R}^{n}\times M$.

###### Proof.

Again this statement follows from Lemma 6.1. Recall that $\mathbb{R}^{n}\times
M$ is smooth manifold that can be covered by an atlas
$\\{(\mathbb{R}^{n}\times
U_{1},\mathrm{id}\otimes\varphi_{1}),\dots,(\mathbb{R}^{n}\times
U_{l},\mathrm{id}\otimes\varphi_{l})\\}$ with a finite number $l\in\mathbb{N}$
of charts, where $\\{(U_{1},\varphi_{1}),\dots,(U_{l},\varphi_{l})\\}$ was an
atlas for the compact Riemannian manifold $M$. Then, with the choice of a
smooth partition of unity $\\{\chi_{1},\dots,\chi_{l}\\}$ subordinated to the
atlas for $M$, we set $\mathscr{V}(\mathbb{R}^{n}\times
M)=\mathcal{A}(\mathbb{R}^{n}\times M)$ with the norm

$\|u\|_{\mathscr{V}(\mathbb{R}^{n}\times
M)}=\sum_{j=1}^{l}\bigg{(}\int_{\mathbb{R}^{n}}|\xi|^{2t}\|\chi_{j}\circ\varphi_{j}^{-1}(y)\widehat{u}(\xi,\varphi_{j}^{-1}(\centerdot))\|_{H^{s}(\mathbb{R}^{m})}^{2}\,\mathrm{d}\xi\bigg{)}^{1/2}.$

The only point left in order to conclude this proposition from Lemma 6.1 is to
check that a sequence in $\mathscr{V}(\mathbb{R}^{n}\times
M)=\mathcal{A}(\mathbb{R}^{n}\times M)$ is Cauchy with respect to the norm
$\|\centerdot\|_{\mathscr{V}(\mathbb{R}^{n}\times M)}$ if and only if is
Cauchy with respect to
$\|\centerdot\|_{H^{s}(M;\dot{H}^{t}(\mathbb{R}^{n}))}$. The fact that this is
the case follows from Proposition 4.2. ∎

##  Appendix A The special orthogonal group

In this appendix we collect some facts about the special orthogonal group
$\mathrm{SO}(n)$ in order to be able to prove Proposition 4.1. Some of the
properties stated here are well-known, other not so much. For the sake of
completeness, we have included the proofs of all the results collected.

Let $\mathrm{M}(n)$ denote the vector space of real $n\times n$ matrices. The
general linear group

$\mathrm{GL}(n)=\\{Q\in\mathrm{M}(n):\det\,Q\neq 0\\}$

is an open subset of $\mathrm{M}(n)$ and consequently a smooth manifold of
dimension $n^{2}$. Endowed with the matrix multiplication, it becomes a Lie
group. The standard coordinates on $\mathrm{GL}(n)$ are denoted by
$x^{\alpha}(Q)=e_{\alpha_{1}}\cdot Qe_{\alpha_{2}}$ with
$\alpha=(\alpha_{1},\alpha_{2})$ and $\alpha_{j}\in\\{1,\dots,n\\}$. Its
tangent space at a point $Q\in\mathrm{GL}(n)$ is defined as follows

$T_{Q}\mathrm{GL}(n)=\operatorname*{span}\\{\partial_{x^{\alpha}}|_{Q}:\alpha=(\alpha_{1},\alpha_{2})\in\\{1,\ldots,n\\}^{2}\\}.$

Every $X=\sum_{\alpha}X^{\alpha}\partial_{x^{\alpha}}$ in $T_{Q}GL(n)$ can be
identified with a matrix whose components are $X^{\alpha}$ with
$\alpha=(\alpha_{1},\alpha_{2})$, $\alpha_{j}\in\\{1,\dots,n\\}$ and
$\alpha_{1}$ enumerating the rows and $\alpha_{2}$ the columns.222The
summatory $\sum_{\alpha}$ corresponds to the summation of all indices
$\alpha=(\alpha_{1},\alpha_{2})$ with $\alpha_{j}\in\\{1,\dots,n\\}$. Thus, we
identify $T_{Q}\mathrm{GL}(n)$ with $\mathrm{M}(n)$, and we will use the same
notation for the corresponding elements of $T_{Q}\mathrm{GL}(n)$ and
$\mathrm{M}(n)$. The euclidean metric on $T_{Q}\mathrm{GL}(n)$ is defined by

$e(X,Y)=\operatorname*{tr}(X^{\mathsf{T}}Y),\qquad X,Y\in
T_{Q}\mathrm{GL}(n),$

where $\operatorname*{tr}(X^{\mathsf{T}}Y)$ denotes the trace of the matrix
$X^{\mathsf{T}}Y$. The orthogonal group

$\mathrm{O}(n)=\\{Q\in\mathrm{GL}(n):Q^{\mathsf{T}}Q=\mathrm{id}\\}$

is the level set $\Phi^{-1}(\mathrm{id})$ of the function
$\Phi:\mathrm{GL}(n)\rightarrow\mathrm{M}(n)$ defined by
$\Phi(Q)=Q^{\mathsf{T}}Q$. Thus, $\mathrm{O}(n)$ is a compact Lie subgroup of
$\mathrm{GL}(n)$ of dimension $n(n-1)/2$. The special orthogonal group
$\mathrm{SO}(n)$ is the connected component of $\mathrm{O}(n)$, consisting of
matrices with positive determinant. Thus, it is also a compact manifold of
dimension $n(n-1)/2$ and becomes a Lie group under matrix multiplication. The
tangent space of $\mathrm{SO}(n)$ at $Q$ is the vector space

$T_{Q}\mathrm{SO}(n)=\\{X\in T_{Q}\mathrm{GL}(n):X^{T}Q+Q^{T}X=0\\}.$

On each tangent space $T_{Q}\mathrm{SO}(n)$ we consider the metric

$g(X,Y)=\frac{1}{2}\operatorname*{tr}(X^{\mathsf{T}}Y),\qquad X,Y\in
T_{Q}\mathrm{SO}(n).$

Thus, $(\mathrm{SO}(n),g)$ becomes a compact Riemannian manifold. The metric
$g$ has some simple invariances that can be described by the following three
bijective mappings: For $P\in\mathrm{SO}(n)$, we define the left and right
translations, as $L_{P}(Q)=PQ$ and $R_{P}(Q)=QP$ for all $Q\in\mathrm{SO}(n)$.
We also define the transposition as $\mathsf{T}(Q)=Q^{\mathsf{T}}$ for all
$Q\in\mathrm{SO}(n)$. Note that if $Q\in\mathrm{SO}(n)$ and $X\in
T_{Q}\mathrm{SO}(n)$, then the corresponding push-forwards satisfy:

$\displaystyle(L_{P})_{\ast}X\in T_{PQ}\mathrm{SO}(n),$
$\displaystyle(L_{P})_{\ast}X=PX,$ $\displaystyle(R_{P})_{\ast}X\in
T_{QP}\mathrm{SO}(n),$ $\displaystyle(R_{P})_{\ast}X=XP,$
$\displaystyle\mathsf{T}_{\ast}X\in T_{Q^{\mathsf{T}}}\mathrm{SO}(n),$
$\displaystyle T_{\ast}X=X^{\mathsf{T}}.$

Thus, given $Q\in\mathrm{SO}(n)$ and $X,Y\in T_{Q}\mathrm{SO}(n)$ we have the
following

(77)
$\displaystyle[(L_{P})^{\ast}g]_{Q}(X,Y)={\frac{1}{2}}\operatorname*{tr}(X^{\mathsf{T}}P^{\mathsf{T}}PY)={\frac{1}{2}}\operatorname*{tr}(X^{\mathsf{T}}Y)=g_{Q}(X,Y),$
(78)
$\displaystyle[(R_{P})^{\ast}g]_{Q}(X,Y)={\frac{1}{2}}\operatorname*{tr}(P^{\mathsf{T}}X^{\mathsf{T}}YP)={\frac{1}{2}}\operatorname*{tr}(X^{\mathsf{T}}Y)=g_{Q}(X,Y),$
(79)
$\displaystyle[\mathsf{T}^{\ast}g]_{Q}(X,Y)={\frac{1}{2}}\operatorname*{tr}(XY^{\mathsf{T}})={\frac{1}{2}}\operatorname*{tr}(X^{\mathsf{T}}Y)=g_{Q}(X,Y),$

since $P\in\mathrm{SO}(n)$. This means that $g$ is left and right invariant,
that is bi-invariant, and transposition invariant.

In the following lines we will construct the normalized Haar measure $\mu$ in
$\mathrm{SO}(n)$ using the language of Riemannian geometry. Since we are
considering a very particular case, it is worth presenting a concrete
construction. Consider

(80) $E_{jk}=\partial_{x^{jk}}-\partial_{x^{kj}}$

with $j,k\in\\{1,\dots,n\\}$ for $j<k$.333The Greek letters are used for
multi-indices, while the Latin ones are reserved for indices. Thus,
$x^{\alpha}=x^{jk}$ if $\alpha=(j,k)$. Note that every $E_{\alpha}$ belongs to
$T_{\mathrm{id}}\mathrm{SO}(n)$ and
$\\{E_{\alpha}:1\leq\alpha_{1}<\alpha_{2}\leq n\\}$ is an orthonormal basis of
$T_{\mathrm{id}}\mathrm{SO}(n)$. Choose an orientation for
$T_{\mathrm{id}}\mathrm{SO}(n)$ so that the ordered basis

$\\{E_{1\,2},\dots,E_{1\,n},E_{2\,3},\dots,E_{2\,n},\dots,E_{n-1\,n}\\}$

is positively oriented. Consider the covector
$\varepsilon^{jk}=2^{-1}(dx^{jk}-dx^{kj})$ in the cotangent space
$T^{\ast}_{\mathrm{id}}\mathrm{SO}(n)$. The set
$\\{\varepsilon^{\alpha}:1\leq\alpha_{1}<\alpha_{2}\leq n\\}$ forms an
orthonormal basis of $T^{\ast}_{\mathrm{id}}\mathrm{SO}(n)$. With these
covectors we will define the Riemannian or Haar volume form $\nu$ at $Q$ as

$\nu_{Q}(X_{1},\dots,X_{n(n-1)/2})=\big{[}(L_{Q^{\mathsf{T}}})^{\ast}\bigwedge_{\alpha_{1}<\alpha_{2}}\varepsilon^{\alpha}\big{]}(X_{1},\dots,X_{n(n-1)/2})$

for $X_{1},\dots,X_{n(n-1)/2}\in T_{Q}\mathrm{SO}(n)$. Here
$(L_{Q^{\mathsf{T}}})^{\ast}\bigwedge_{\alpha_{1}<\alpha_{2}}\varepsilon^{\alpha}$
denotes the pull-back of the $n(n-1)/2$-covector

$\bigwedge_{\alpha_{1}<\alpha_{2}}\varepsilon^{\alpha}=\varepsilon^{1\,2}\wedge\cdots\wedge\varepsilon^{1\,n}\wedge\varepsilon^{2\,3}\wedge\cdots\wedge\varepsilon^{2\,n}\wedge\cdots\wedge\varepsilon^{n-1\,n}$

through the mapping $L_{Q^{\mathsf{T}}}$. The quantity
$\varpi_{n}=\int_{\mathrm{SO}(n)}\nu$ is positive and, since $\mathrm{SO}(n)$
is compact, it is also finite. Thus, we define the normalized Haar measure
$\mu$ as the representative of the functional

$f\in
C(\mathrm{SO}(n))\longmapsto\frac{1}{\varpi_{n}}\int_{\mathrm{SO}(n)}f\nu\in\mathbb{C},$

where $C(\mathrm{SO}(n))$ stands for the continuous functions on
$\mathrm{SO}(n)$. The measure $\mu$ has several useful invariances with
respect to the mappings $L_{P}$, $R_{P}$ and $\mathsf{T}$.

######  Lemma A.1.

We have that the Haar volume form is left and right invariant —bi-invariant—
and transposition invariant:

(81) $\displaystyle(L_{P})^{\ast}\nu=\nu\qquad\forall\,P\in\mathrm{SO}(n),$
(82) $\displaystyle(R_{P})^{\ast}\nu=\nu\qquad\forall\,P\in\mathrm{SO}(n),$
(83) $\displaystyle\mathsf{T}^{\ast}\nu=\nu.$

###### Proof.

These identities follow from the invariances of the metric on
$\mathrm{SO}(n)$. Recall that if $M$ and $N$ are oriented manifolds, $g$ is a
Riemannian metric on $N$, and $F:M\rightarrow N$ is a smooth immersion, then
the pull-back $F^{\ast}g$ is a Riemannian metric on $M$. Furthermore, by the
definition of Riemannian volume form we have that

$\mathrm{d}V_{F^{\ast}g}=F^{\ast}\mathrm{d}V_{g}.$

Since $L_{P}$, $R_{P}$ and $\mathsf{T}$ are smooth immersions, the volume
forms generated by $(L_{P})^{\ast}g$, $(R_{P})^{\ast}g$ and $T^{\ast}g$ are
$(L_{P})^{\ast}\nu$, $(R_{P})^{\ast}\nu$ and $T^{\ast}\nu$ respectively. Thus,
by the invariances (77), (78) and (79) for the metric $g$, we derive the
invariances (81), (82) and (83) for $\nu$. ∎

######  Proposition A.2.

If $f\in L^{1}(\mathrm{SO}(n))$, then

(84)
$\begin{split}\int_{\mathrm{SO}(n)}f(PQ)\,\mathrm{d}\mu(Q)&=\int_{\mathrm{SO}(n)}f(Q)\,\mathrm{d}\mu(Q)\quad\forall\,P\in\mathrm{SO}(n),\\\
\int_{\mathrm{SO}(n)}f(QP)\,\mathrm{d}\mu(Q)&=\int_{\mathrm{SO}(n)}f(Q)\,\mathrm{d}\mu(Q)\quad\forall\,P\in\mathrm{SO}(n),\\\
\int_{\mathrm{SO}(n)}f(Q^{\mathsf{T}})\,\mathrm{d}\mu(Q)&=\int_{\mathrm{SO}(n)}f(Q)\,\mathrm{d}\mu(Q).\end{split}$

###### Proof.

These identities are immediately seen to hold from Lemma A.1 and the
definition of $\mu$ for functions $f\in C^{\infty}(\mathrm{SO}(n))$. The
general case follows by density. ∎

The next invariance shows a relation between $\mu$ and the hypersurface
measure on $\SS^{n-1}$.

######  Proposition A.3.

Let $F:\mathrm{SO}(n)\to\SS^{n-1}$ be given either by $F(Q)=Qe_{1}$ or by
$F(Q)=Q^{\mathsf{T}}e_{1}$. If $f\in L^{1}(\SS^{n-1})$ then

$\int_{\mathrm{SO}(n)}f(F(Q))\,\mathrm{d}\mu(Q)=\frac{1}{S(\SS^{n-1})}\int_{\SS^{n-1}}f(\theta)\,\mathrm{d}S(\theta),$

where $S(\SS^{n-1})=\int_{\SS^{n-1}}\,\mathrm{d}S(\theta)$.

###### Proof.

By Proposition A.2 it is enough to verify the case of $F(Q)=Qe_{1}$. Consider
the smooth map $F(Q)=Qe_{1}$. Let $\mu_{F}$ denote the image measure of $\mu$
on $\SS^{n-1}$, that is, given a Borel $\Sigma\subset\SS^{n-1}$,
$\mu_{F}(\Sigma)=\mu(F^{-1}(\Sigma))$. The identity

$\int_{\mathrm{SO}(n)}f(Qe_{1})\,\mathrm{d}\mu(Q)=\int_{\SS^{n-1}}f(\theta)\,\mathrm{d}\mu_{F}(\theta)$

holds trivially for simple functions, and can be extended by density. It
remains to prove that $\mu_{F}(\Sigma)=S(\Sigma)/S(\SS^{n-1})$ for every Borel
$\Sigma\subset\SS^{n-1}$. From the identity (84) in Proposition A.2, one can
derive that if $\Sigma$ is a Borel set of $\SS^{n-1}$ then
$\mu_{F}(P\Sigma)=\mu_{F}(\Sigma)$ for every $P\in\mathrm{SO}(n)$. The same
holds for $S(P\Sigma)=S(\Sigma)$. This common invariance means that $\mu_{F}$
and $S$ are both uniformly distributed on $\SS^{n-1}$. Therefore, there exists
a $c>0$ such that $\mu_{F}(\Sigma)=cS(\Sigma)$ for all Borel sets $\Sigma$
(see for example [6, Theorem 3.4]). By testing the identity in $\SS^{n-1}$ one
gets $S(\SS^{n-1})=\mu_{F}(\SS^{n-1})/c=\mu(\mathrm{SO}(n))/c=1/c$ and this
completes the proof of this proposition. ∎

The next step is to transfer these invariances to Sobolev spaces. To do so we
first collect some invariances of the Laplace–Beltrami operator
$\Delta_{\mathrm{SO}(n)}$.

######  Lemma A.4.

Assume $f\in C^{\infty}(\mathrm{SO}(n))$ and for $P,Q\in\mathrm{SO}(n)$ set