# Global hypoellipticity of $G$-invariant operators on homogeneous vector
bundles

Duván Cardona  Duván Cardona: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium E-mail address
<EMAIL_ADDRESS>and  André Pedroso Kowacs  André Pedroso
Kowacs: Department of Mathematics Universidade Federal do Paraná (UFPR),
Brazil E-mail address<EMAIL_ADDRESS>
###### Abstract.

We establish necessary and sufficient conditions for the global
hypoellipticity of $G$-invariant operators on homogeneous vector bundles.
These criteria are established in terms of the corresponding matrix-valued
symbols as developed by Ruzhansky and Turunen and extended in [7] to
homogeneous vector-bundles.

###### Key words and phrases:

Global hypoellipticity, Homogeneous operators, Compact Lie groups, Homogeneous
vector bundles

###### 1991 Mathematics Subject Classification:

Primary 22E30, 43A77 ; Secondary 58J40

The authors are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and
Partial Differential Equations and by the Methusalem programme of the Ghent
University Special Research Fund (BOF) (Grant number 01M01021). Duván Cardona
has been supported by the FWO Fellowship grant No 1204824N of the Belgian
Research Foundation FWO. André Kowacs was supported in part by the Coordenao
de Aperfeioamento de Pessoal de Nível Superior - Brasil (CAPES)

##  1\. Introduction

###  1.1. Outline

The aim of this note is to apply the matrix-valued quantisation in [7] on
homogeneous vector bundles over compact homogeneous manifolds to the
characterisation of hypoellitic $G$-invariant operators, see Subsection 2.0.4
for details. The results below were presented by the second author at the
Ghent Analysis and PDE seminar and were announced in [5].

Since the notion of hypoellipticity is the main topic of this note we present
this definition as follows. Let $M$ be a compact manifold without boundary. By
following Hörmander [28, 29], a differential operator $D$ with
$C^{\infty}$-coefficients acting on the space of distributions
$\mathscr{D}^{\prime}(M)$ is called globally hypoelliptic if the equation

(1.1) $Du=f,$

only has solution $u\in C^{\infty}(M)$ when $f\in C^{\infty}(M).$ This
definition is easily extended when $D$ is a pseudo-differential operator
acting on smooth sections $\Gamma^{\infty}(E)$ of a vector-bundle $E$ over
$M.$ In the case where $M=G/K$ is a compact homogeneous manifold and $E$ is a
$G$-vector bundle on $M,$ this work aims to investigate necessary and
sufficient conditions for the global hypoellipticity of $G$-invariant pseudo-
differential operators acting on smooth sections of $E.$ In view of the
characterisation of hypoelliptic differential operators with constant
coefficients, see [30], where criteria can be obtained in terms of the
principal symbol of the operator, here we adopt a new perspective and our
criteria are developed in terms of the (global) matrix-valued symbols as
developed by Ruzhansky and Turunen [18]. The global hypoellipticity property
for pseudo-differential operators has been widely studied, see for instance
[22, 23, 24, 26, 27, 31, 32, 33]. The main results of this note are presented
in Section 3.

###  1.2. State-of-the-art

The hypoellipticity of $G$-invariant pseudo-differential operators $D$ is
considered in the setting of $G$-vector bundles. The research about the
hypoellipticity of a differential operator with a principal symbol having
real-valued coefficients and constant signs has been a field of intensive
research, where typically, the following two methods are applied. The first
approach is based on an a priori estimate of solutions in the scale of the
Sobolev spaces. Once one finds such an inequality, one can prove the
hypoellipticity of the operator with the aid of interpolation inequalities or
with the application of the theory of pseudo-differential operators. The
second approach is based on the study of fundamental solutions, which is a
very well-known difficult problem, in particular if the operator has variable
coefficients. For a general analysis about the hypoellicity of linear partial
differential operators we refer to Hörmander [30].

As it is well-known, the extensive class of elliptic operators provides
fundamental examples of hypoelliptic operators. This fact suggests that the
principal symbol of hypoelliptic operators has a kind of positivity. In 1967,
Hörmander [29] proved the hypoellipticity of second order differential
operators when the principal symbol does not change of sign when the dual
variables vary. The following is a natural question that arose after
Hörmander’s work [29]

“Is it necessary for hypoellipticity that the principal symbol does not change
sign when the space variables vary?

In 1971, Kannai [10] proved that

$L_{1}=\partial_{t}+t\partial_{x}^{2},$

acting in smooth functions on $\mathbb{R}^{2}$ is hypoelliptic, while

$L_{2}=\partial_{t}-t\partial_{x}^{2}$

is not, where the key remark is that $L_{1}$ and $L_{2}$ are typical examples
of operators with sign-changing principal symbols. This example illustrates
that the semi-definiteness of the principal symbol is not necessary for
hypoellipticity and that the type of changing sign is important. The sign of
the principal symbol of $L_{2}$ above changes from minus to plus as $t$
increases. This is a condition similar to the Nirenberg-Treves criterion for
local solvability of differential equations of principal type, see [16]. In
1976, Beals and Fefferman [3] extended Kannai’s result to higher dimensional
cases. Their result was obtained by obtaining a suitable a priori estimate
with weight and using the general theory of pseudo-differential operators due
to Beals [1, 2]. An important characteristic of this method is that the
hypoellipticity property follows from a single a priori estimate. However,
there are many restrictions on the weight. Therefore, the class of functions
that control the sign of the principal symbol is strictly limited. Beals and
Fefferman’s result was extended by many authors, see e.g. Kumano-go and
Taniguchi [13] and Lanconelli [14]. We also refer to the seminal 1987’s work
of Morimoto [15]. As we will show in this note, the hypoellipticity of
$G$-invariant pseudo-differential operators can be effectively analysed by
applying the matrix-valued quantisation formula (2.6). Indeed, this is a new
point of view in counterpart to the aforementioned results based on the notion
of principal symbol.

In the setting of a compact Lie group $G$, pseudo-differential operators can
be analysed in terms of the corresponding global matrix-valued symbols as
introduced by Ruzhansky and Turunen [18]. Indeed, a pseudo-differential
operator $A$ can be writen in terms of the group Fourier transform $\hat{}$ as

$Af(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}\left(\xi(x)\sigma_{A}(x,\xi)\widehat{f}(\xi)\right),$

where the sum runs over the elements of the unitary dual $\widehat{G}$ of $G,$
namely, the set formed by all equivalence classes of continuous irreducible
unitary representation of the group $[\xi],$ see Subsection 2 for details.
Ruzhansky-Turunen’s matrix-valued symbols can be extended also in the setting
of homogeneous vector bundles, see Section 2.

While in the standard theory of pseudo-differential operators on manifolds
many criteria for the properties of the operator depend on the principal
symbol, on compact Lie groups, properties of the operator involve the (full)
matrix-valued symbol. We refer the reader to [6] and the extensive list of
references therein that include matrix-valued symbol criteria for several
properties of pseudo-differential operators: boundedness of pseudo-
differential oeprators on $L^{p}$ spaces, on $L^{p}$-Sobolev spaces, on
Triebel-Lizorkin spaces, on Besov spaces, etc., as well as applications of the
matrix-valued symbols to the analysis of elliptic and parabolic equations,
Gårding type inequalities and global well-posedness of pseudo-differential
problems. We remark that as for the analysis of the hypoellipticity of
homogeneous differential operators on homogeneous vector-bundles, the main
motivation comes from the index-type theorems in this setting, see the seminal
work of Bott [4] for details.

###  1.3. Structure of the note

We organise this note as follows. In Section 2 we introduce the notation used
in this work. Finally in Section 3 state our main results, we present their
proofs of our results and we provide some examples.

##  2\. Preliminaries

In this section, we introduce the notations for the development of this work.
We first present the basics of representation theory on compact groups, which
can be found in more detail in [18, Chapter 7].

####  2.0.1. The group Fourier transform

Let $G$ be a compact Lie group. We denote by $\widehat{G}$ its unitary dual.
As every $[\xi]\in\widehat{G}$ is finite dimensional, we can always choose a
representative that is matrix-valued. As a conseuquence of the Peter-Weyl
Theorem, the collection of the coefficient functions of all such matrices is
an orthogonal basis for $L^{2}(G)$, where integration is taken with respect to
the Haar measure in $G$. Then, define the matrix-valued Fourier coefficients
of $f\in L^{2}(G)$ by

$\widehat{f}(\xi)\doteq\smallint_{G}f(x)\xi(x)^{*}dx,$

for $[\xi]\in\widehat{G}$, and write the Fourier inversion formula as

$f(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}\left(\xi(x)\widehat{f}(\xi)\right),\,\forall
x\in G,$

where $d_{\xi}\doteq\dim(\xi)$. As the coefficient functions of elements of
$\widehat{G}$ are smooth, we can extend this definition to the set of
distributions $\mathcal{D}^{\prime}(G)$ by

$\widehat{u}(\xi)=\langle u,\xi^{*}\rangle,$

for $u\in\mathcal{D}^{\prime}(G)$ and $[\xi]\in\widehat{G}$, where this
evaluation is to be understood coefficient-wise.
Now we summarize the quantisation of pseudo-differential operators by matrix-
valued symbols, following [18, Chapter 10].

####  2.0.2. The quantisation formula on compact Lie groups

To a continuous linear operator $A:C^{\infty}(G)\to C^{\infty}(G)$ we can
associate its global matrix-valued symbol $\sigma_{A}$ by

$\sigma_{A}(x,\xi)=\xi(x)^{*}A\xi(x),\,\forall(x,[\xi])\in
G\times\widehat{G},$

so that we can write the quantisation

$Af(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}\left(\xi(x)\sigma_{A}(x,\xi)\widehat{f}(\xi)\right)\,\forall
x\in G.$

In this setting, left-invariant continuous linear operators are Fourier
multipliers characterized by having a matrix-valued symbol which independent
on the variable $x\in G$, i.e.: $\sigma_{A}(x,\xi)=\sigma_{A}(\xi)$.

####  2.0.3. Vector Bundles

Let $E,X$ be topological spaces, and let $\mathbb{K}$ be a field. Following
[21, Chapter 1], we say that $E$ is a $\mathbb{K}$-vector bundle over $X$ if
there exists a continuous mapping $p:E\to X$ such that

  1. (1)

For every $x\in X$, $E_{x}\doteq p^{-1}(x)$ is a finite dimensional
$\mathbb{K}$-vector space.

  2. (2)

For each $x\in X$, there exists a neighbourhood $U$ of $x$ and a homeomorphism
$\psi:p^{-1}(U)\to U\times\mathbb{K}^{n}$ such that $\psi(v)=(p(v),f(v))$,
where $f:E_{p(v)}\to\mathbb{K}^{n}$ is a linear map between vector spaces.

Let $p:E\to X$ be a vector bundle. A continuous map $s:X\to E$ is called a
section of $E$ if for all $x\in X$, $p(s(x))=p(x)$, or equivalently, $s(x)\in
E_{x}$. We denote by $\Gamma(E)$ the set of all sections of $E$. If $X,E$ are
smooth manifolds, we also define $\Gamma^{\infty}(E)$ the set of all smooth
sections of $E$. If $X$ is orientable, the space $L^{q}(E)$, $1\leq q<\infty$,
is then defined as the completion of the set of all smooth sections
$s\in\Gamma^{\infty}(E)$ such that

(2.1)
$\|s\|_{L^{q}(E)}\doteq\left(\smallint_{X}\|s(x)\|_{E_{x}}^{q}dx\right)^{\frac{1}{q}}<\infty.$

We shall denote by $\mathcal{D}^{\prime}(E)$ the set of continuous linear
functionals over $\Gamma^{\infty}(E)$.

####  2.0.4. Homogeneous vector bundles

Next, we record some notions about the basic theory of homogeneous vector
bundles, following [21, Chapter 5]. Let $G$ be a compact Lie group and $K$ a
closed subgroup of $G$. Denote the quotient $M=G/K$ equipped with its natural
compact manifold topology. There exists a natural left action of $G$ on $M$
given by $g\cdot hK=ghK$, for every $g,h\in G$. We say that a
$\mathbb{C}$-vector bundle (or $\mathbb{R}$-vector bundle) $p:E\to M$ is a
homogeneous vector bundle over $M$ if $G$ acts on $E$ on the left and this
action satisfies:

  1. (1)

$g\cdot E_{x}=E_{gx}$, for all $x\in M$, $g\in G$.

  2. (2)

The previously induced mappings from $E_{x}$ to $E_{gx}$ are linear.

There is natural left action of $G$ on $\Gamma(E)$,
$G\times\Gamma(E)\to\Gamma(E)$ given by

$(g\cdot s)(x)=g\cdot s(g^{-1}x),$

for all $x\in X$, $g\in G$. Consider $p_{1}:E\to M$, $p_{2}:F\to M$, $M=G/K$
to be homogeneous vector bundles, where $K<G$ are compact Lie groups. Let
$E_{0}=p_{1}^{-1}(K)$, $F_{0}=p_{2}^{-1}(K)$ be the fibers at the identity
coset. For each $\tau\in\text{Hom}(K,\text{End}(E_{0}))$, there is a natural
right action of $K$ on $G\times E_{0}$ by $(g,v)=(gk,\tau^{-1}(k)v)$. We
denote by $G\times_{\tau}E_{0}$ the quotient $(G\times E_{0})/K$ under this
action. Similarly, we define $G\times_{\omega}F_{0}$. One can show that there
always exist $\tau\in\text{Hom}(K,\text{End}(E_{0}))$,
$\omega\in\text{Hom}(K,\text{End}(F_{0}))$ such that

$E\cong G\times_{\tau}E_{0},\qquad F\cong G\times_{\omega}F_{0},$

are isomorphic as homogeneous vector bundles.
Now define $C^{\infty}(G,E_{0})^{\tau}\subset C^{\infty}(G,E_{0})$ by

(2.2) $C^{\infty}(G,E_{0})^{\tau}=\\{f\in C^{\infty}(G,E_{0})|\forall g\in
G,\forall k\in K,f(gk)=\tau(k)^{-1}f(g)\\}.$

In a similar way we define $L^{q}(G,E_{0})^{\tau}$ for all $q\geq 1$ and for
the homogeneous vector bundle $p_{2}:F\to M$.

######  Remark 2.1.

As in [21, Chapter 5], consider the map $\chi_{\tau}:\Gamma^{\infty}(E)\to
C^{\infty}(G,E_{0})^{\tau}$, given by

(2.3) $\chi_{\tau}(s)(g)\doteq g^{-1}\cdot s(gK)\equiv(g^{-1}\cdot s)(e_{G}K)$

where $e_{g}\in G$ is the group identity. This mapping extends to a unitary
isomorphism between $L^{2}(E)$ and $L^{2}(G,E_{0})^{\tau}$, so that we may
identify $\Gamma^{\infty}(E)\cong C^{\infty}(G,E_{0})^{\tau}$.

Let $\tilde{D}:\Gamma(E)\to\Gamma(F)$ be a linear operator. We say that
$\tilde{D}$ is $G$-invariant (or homogeneous) if

$\tilde{D}(g\cdot s)=g\cdot(\tilde{D}s),$

for all $s\in\Gamma(E),\,g\in G$.
As mentioned in [21, Page 120], every continuous linear map
$\tilde{A}:\Gamma^{\infty}(E)\to\Gamma^{\infty}(F)$ induces a continuous
linear map $A:C^{\infty}(G,E_{0})^{\tau}\to C^{\infty}(F,F_{0})^{\omega}$ by

(2.4) $A\doteq\chi_{\omega}\circ\tilde{A}\circ\chi_{\tau}^{-1}.$

######  Remark 2.2.

In the case where $\tilde{D}:\Gamma^{\infty}(E)\to\Gamma^{\infty}(F)$ is a
$G$-invariant (homogeneous) continuous linear map, the mapping
$D:C^{\infty}(G,E_{0})^{\tau}\to C^{\infty}(F,F_{0})^{\omega}$ is a vector-
valued Fourier multiplier (as explained below).

####  2.0.5. The quantisation formula on homogeneous vector-bundles

Next we define the vector valued group Fourier transform, following [7], in
order to obtain a vector valued quantisation formula. Then, we use the
previous identifications to obtain a quantisation formula for homogeneous
operators. Choose orthonormal basis
$B_{E_{0}}=\\{e_{i,E_{0}}\\}_{i=1}^{d_{\tau}}$ and
$B_{F_{0}}=\\{e_{i,F_{0}}\\}_{i=1}^{d_{\omega}}$ for $E_{0}$ and $F_{0}$
respectively. Denoting by $e_{i,E_{0}}^{*}(v)\doteq\langle
v,e_{i,E_{0}}\rangle_{E_{0}}$, $e_{i,F_{0}}^{*}(w)\doteq\langle
w,e_{i,F_{0}}\rangle_{F_{0}}$, for $v\in E_{0}$, $w\in F_{0}$, the vector
valued $B_{E_{0}}$-Fourier transform of $f\in C^{\infty}(G,E_{0})$ is given by

$\widehat{f}(i,\xi)\doteq\smallint_{G}e_{i,E_{0}}^{*}(f(x))\xi(x)^{*}dx,\,\forall[\xi]\in\widehat{G},\,\forall
1\leq i\leq\dim(E_{0})=d_{\tau}.$

As the coefficient functions of elements of $\widehat{G}$ are smooth, we can
extend this definition to the set of distributions
$\mathcal{D}^{\prime}(G,E_{0})$ by

$\widehat{u}(i,\xi)=\langle u,\xi^{*}\otimes e_{i,E_{0}}\rangle$

for $u\in\mathcal{D}^{\prime}(G,E_{0})$ and $[\xi]\in\widehat{G}$, where this
evaluation is to be understood coefficient-wise. From the Peter-Weyl Theorem,
the inversion formula can be written as

$f(x)=\sum_{i=1}^{d_{\tau}}\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}\left(\xi(x)\widehat{f}(i,\xi)\right)e_{i,E_{0}},\,\forall
x\in G.$

Let $A:C^{\infty}(G,E_{0})\to C^{\infty}(G,F_{0})$ be a continuous linear
operator. For $1\leq i\leq d_{\tau}=\dim(E_{0})$, $1\leq r\leq
d_{\omega}=\dim(F_{0})$, define the matrix symbol

(2.5) $\sigma_{A}(i,r,x,\xi)\doteq\xi(x)^{*}[e^{*}_{r,F_{0}}[A(\xi_{uv}\otimes
e_{r,E_{0}})(x)]]_{u,v=1}^{d_{\xi}}$

where $x\in G$, $\xi\in\widehat{G}$. The matrix-valued
$(B_{E_{0}},B_{F_{0}})$-symbol of $A$ is then defined as the mapping
$\sigma_{A}:\\{1\leq i\leq d_{\tau}\\}\times\\{1\leq i\leq d_{\omega}\\}\times
G\times\widehat{G}\to\bigcup\left\\{\mathbb{C}^{d_{\xi}\times
d_{\xi}}|[\xi]\in\widehat{G}\right\\}$ given by (2.5). One can prove that the
quantisation formula can then be written as

(2.6) $\displaystyle
Af(x)=\sum_{r=1}^{d_{\omega}}\sum_{i=1}^{d_{\tau}}\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}\left(\xi(x)\sigma_{A}(i,r,x,\xi)\widehat{f}(i,\xi)\right)e_{i,F_{0}}.$

Now let $\tilde{A}:\Gamma^{\infty}(E)\to\Gamma^{\infty}(F)$ be a continuous
linear operator. Then by (2.4) we can define the martrix-valued
$(B_{E_{0}},B_{F_{0}})$-symbol of $\tilde{A}$ as
$\sigma_{\tilde{A}}=\sigma_{A}$. It follows from $\eqref{vectorquantisation}$
that in this case, the quantisation formula can be written as

$\displaystyle\tilde{A}s(gK)$
$\displaystyle=\chi_{\omega}^{-1}\left(\sum_{r=1}^{d_{\omega}}\sum_{i=1}^{d_{\tau}}\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}\left(\xi(x)\sigma_{\tilde{A}}(i,r,g,\xi)\widehat{\chi_{\tau}s}(i,\xi)\right)e_{i,F_{0}}\right)$
$\displaystyle\equiv\left(g,\sum_{r=1}^{d_{\omega}}\sum_{i=1}^{d_{\tau}}\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}\left(\xi(x)\sigma_{\tilde{A}}(i,r,g,\xi)\widehat{\chi_{\tau}s}(i,\xi)\right)e_{i,F_{0}}\right)\cdot
K,$

for every $gK\in M=G/K$, and every section $s\in\Gamma^{\infty}(E)$.

######  Remark 2.3.

Notice that these definitions all depend on the choice of basis $B_{E_{0}}$
and $B_{F_{0}}$ of $E_{0}$ and $F_{0}$, respectively. In fact, if
$B^{\prime}_{E_{0}}$$B^{\prime}_{F_{0}}$is another choice of respective basis,
then if we denote by $U_{E}\in End(E_{0})$, $U_{F}\in End(F_{0})$ the
corresponding unitary change of basis operators, then for every continuous
linear operator $A:C^{\infty}(G,E_{0})\to C^{\infty}(G,F_{0})$ we have

$\sigma_{A}(i,r,x,\xi)=\sigma_{U_{F}AU_{E}^{*}}(i,r,x,\xi),$

where on the left we consider the $(B_{E_{0}},B_{F_{0}})$-symbol of ${A}$ and
on the right we consider the $(B_{E_{0}}^{\prime},B_{F_{0}}^{\prime})$-symbol
of $U_{F}AU^{*}_{E}$.

##  3\. Main Results

In this section we present the contributions of this note. We record the
following fundamental notations/facts:

  * •

For every $s\in\mathbb{R},$ the Sobolev space
$H^{s}(G)\subset\mathcal{D}^{\prime}(G)$ is defined by

$H^{s}(G)=\\{u\in\mathcal{D}^{\prime}(G)|\|u\|_{H^{s}(G)}<+\infty\\}$

where

$\|u\|_{H^{s}(G)}=\left(\sum_{[\xi]\in\widehat{G}}d_{\xi}\langle\xi\rangle^{2s}\|\widehat{u}(\xi)\|_{HS}^{2}\right)^{\frac{1}{2}}.$

Here, $\langle\xi\rangle=\sqrt{1+\nu_{\xi}}$ is the common eigenvalue of
$(\text{Id}+\mathcal{L}_{G})^{\frac{1}{2}}$ corresponding to the coefficient
functions of $[\xi]\in\widehat{G}$, where $\mathcal{L}_{G}$ is the positive
Laplace-Beltrami operator on $G$. Also,
$\|A\|_{HS}=\sqrt{\operatorname{Tr}(A^{*}A)}$ for any square matrix $A$.

  * •

For any matrix $A$, define $\lambda_{\min}[A]\geq 0$ its smallest singular
value.

  * •

We shall also use the following well-known characterizations:

(3.1) $\mathcal{D}^{\prime}(G)=\bigcup_{s\in\mathbb{R}}H^{s}(G),\qquad
C^{\infty}(G)=\bigcap_{s\in\mathbb{R}}H^{s}(G).$

  * •

The following fact is a consequence of the Weyl eigenvalue counting formula
for the Laplacian, see [21, Chapter 5]:

$\sum_{[\xi]\in\widehat{G}}d_{\xi}^{2}\langle\xi\rangle^{-2t}<\infty\iff
t>\frac{n}{2}.$

  * •

The Sobolev space $H^{s}(E)$ (and similarly $H^{s}(F)$) for any
$s\in\mathbb{R}$, is defined by the norm

(3.2)
$\|u\|_{H^{s}(E)}^{2}=\sum_{i=1}^{d_{\tau}}\sum_{[\xi]\in\widehat{G}}d_{\xi}\langle\xi\rangle^{2s}\|\widehat{\chi_{\tau}u}(i,\xi)\|_{HS}^{2}.$

  * •

Distributions and smooth functions are characterised in terms of Sobolev
spaces as follows:

(3.3)
$\mathcal{D}^{\prime}(E)=\bigcup_{s\in\mathbb{R}}H^{s}(E),\qquad\Gamma^{\infty}(E)=\bigcap_{s\in\mathbb{R}}H^{s}(E).$

######  Remark 3.1.

It is possible that for all $u\in\mathcal{D}^{\prime}(E)$,
$\widehat{\chi_{\tau}u}(i,\xi)=0$ for all $1\leq i\leq d_{\tau}$, for some
$[\xi]\in\widehat{G}$. We denote the set of all such $[\xi]$ by
$\widehat{G}(E)$. Hence, (3.2) can be rewritten as

(3.4)
$\|u\|_{H^{s}(E)}^{2}=\sum_{i=1}^{d_{\tau}}\sum_{[\xi]\in\widehat{G}(E)}d_{\xi}\langle\xi\rangle^{2s}\|\widehat{\chi_{\tau}u}(i,\xi)\|_{HS}^{2}.$

######  Remark 3.2.

In the case $E=G\times_{\widehat{1}}\mathbb{C}$ is the trivial bundle,
$\Gamma^{\infty}(E)$ coincides with $C^{\infty}(M)$, and
$\chi_{\tau}:C^{\infty}(M)\to C^{\infty}(G)$ is just the projective lifting

$\chi_{\tau}f(g)\equiv\dot{f}(g)\doteq f(gK),\,\forall g\in G.$

Also in this case, the set $\widehat{G}(E)$ this coincides with the set of all
$[\xi]$ such that $\Pi_{M}\xi=0$, where $\Pi_{M}$ is the orthogonal projection
of $L^{2}(G)$ onto $L^{2}(M)$.

###  3.1. Statements

Our first result provides a necessary and sufficient condition on the symbol
of a left-invariant pseudo-differential operator (Fourier multiplier) on a
general compact Lie group for the operator to be globally hypoelliptic. This
extends Theorem 2.1 in [33] to the setting of compact Lie groups.

######  Theorem 3.1.

Let $G$ be a compact Lie group. If $D$ is a left invariant pseudo-differential
operator (Fourier multiplier) on $G$, then $D$ is globally hypoelliptic if and
only if there exists $C>0$, $k\in\mathbb{R}$ such that

(3.5) $\lambda_{\min}[\sigma_{D}(\xi)]\geq C\langle\xi\rangle^{k},$

for all but a finite number of $[\xi]\in\widehat{G}$.

Next, we extend the previous theorem to the setting of homogeneous operators
(Fourier multipliers) on homogeneous vector bundles.

######  Theorem 3.2.

Let $p:E\to M$, $p:F\to M$, $M=G/K$ be homogeneous vector bundles, where $K<G$
are compact Lie groups, $E=G\times_{\tau}E_{0}$, $F=G\times_{\omega}F_{0}$.
Let $D:\Gamma^{\infty}(E)\to\Gamma^{\infty}(F)$ be a continuous homogeneous
operator (continuous $G$-invariant operator) from the smooth sections of $E$
to the smooth sections of $F$. For every $[\xi]\in\widehat{G}(E)$ set

$m_{\xi}(D)=\min_{1\leq i\leq
d_{\tau}}\sum_{r=1}^{d_{\omega}}\lambda_{\min}[\sigma_{{D}}(i,r,\xi)]$

Then $D$ is globally hypoelliptic if and only if there exists $C>0$,
$k\in\mathbb{R}$ such that

(3.6) $m_{\xi}(D)\geq C\langle\xi\rangle^{k}$

for all but a finite number of $[\xi]\in\widehat{G}(E)$.

As a consequence, we obtain a similar characterization of globally
hypoelliptic Fourier multipliers on compact homogeneous spaces. Here we are
going to follow the following notation from Connolly [8]. Let $M=G/K$ be a
smooth homogeneous manifold, where $K<G$ are compact Lie groups. Set

(3.7) $\widehat{M}=\\{[\xi]\in\widehat{G}|\Pi_{M}(\xi)\not\equiv 0\\}$

where $\Pi_{M}:C^{\infty}(G)\to C^{\infty}(M)$ is the projection given by

$(\Pi_{M}f)(gK)=\smallint_{K}f(gk)d\mu_{G}(k).$

For each $[\xi]\in\widehat{M}$, let $d_{\xi}^{K}\in\mathbb{N}$ be the number
of non-zero rows in the matrix coefficients of $\Pi_{M}\xi$, which we can
assume all to be ordered first. Now we present the following criterion.

######  Corollary 3.1.

Let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a Fourier multiplier on $M$, and
$\tilde{D}$ its projective lifting on $G$. Then $D$ is globally hypoelliptic
if and only if there exists $C>0$, $k\in\mathbb{R}$ such that

(3.8) $\lambda_{\min}[[\sigma_{\tilde{D}}(\xi)]_{d_{\xi}\times
d_{\xi}^{K}}]\geq C\langle\xi\rangle^{k},$

where $[A]_{m\times n}$ denotes the upper left $m$ by $n$ block of the matrix
$A$, for all but a finite number of $[\xi]\in\widehat{M}$.

###  3.2. Proofs

Now, we are going to present the proof of the previous results. We start with
the following lemma:

######  Lemma 3.1.

Let $A\in\mathbb{C}^{m\times n}$ and $B\in\mathbb{C}^{n\times p}$, where
$m,n,p\in\operatorname{\mathbb{N}}$, be two compatible matrices of complex
entries. Then

$\|AB\|_{HS}\geq\lambda_{\min}[A]\|B\|_{HS}.$

###### Proof.

Since $A^{*}A$ is a normal square matrix, it follows from the spectral theorem
that we can write $A^{*}A=Q^{*}\Lambda Q$, where
$\Lambda\in\mathbb{C}^{n\times n}$ is a diagonal matrix whose entries are
given by the eigenvalues of $A^{*}A$, which correspond to the singular values
of $A$ squared, and $Q\in\mathbb{C}^{n\times n}$ is unitary. Then

$\displaystyle\|AB\|_{HS}^{2}$ $\displaystyle=\operatorname{Tr}(B^{*}A^{*}AB)$
$\displaystyle=\operatorname{Tr}(B^{*}Q^{*}\Lambda QB)$
$\displaystyle=\operatorname{Tr}((QB)^{*}\Lambda QB)$
$\displaystyle\geq\lambda_{\min}[A]^{2}\operatorname{Tr}((QB)^{*}QB)$
$\displaystyle=\lambda_{\min}[A]^{2}\operatorname{Tr}(B^{*}B)=\lambda_{\min}[A]^{2}\|B\|_{HS}^{2},$

which implies the claim. ∎

###### Proof of Theorem 3.1.

Suppose $D$ satisfies (3.5) for all $[\xi]\in V$, with $\widehat{G}\backslash
V$ finite. Then, if $u\in\mathcal{D}^{\prime}(G)$ is such that $Du\in
C^{\infty}(G)$, we have for $s\in\operatorname{\mathbb{R}}$ that

(3.9) $\displaystyle+\infty>\|Du\|_{H^{s}(G)}^{2}\geq\sum_{[\xi]\in
V}d_{\xi}\langle\xi\rangle^{2s}\|\widehat{Du}(\xi)\|_{HS}^{2}=\sum_{[\xi]\in
V}d_{\xi}\langle\xi\rangle^{2s}\|\sigma_{D}(\xi)\widehat{u}(\xi)\|_{HS}^{2}.$

Applying Lemma 3.1 to this inequality yields

$\displaystyle+\infty>\|Du\|_{H^{s}(G)}^{2}$ $\displaystyle\geq\sum_{[\xi]\in
V}d_{\xi}\langle\xi\rangle^{2s}\lambda_{\min}[\sigma_{D}(\xi)]^{2}\|\widehat{u}(\xi)\|_{HS}^{2}$
$\displaystyle\geq C^{2}\sum_{[\xi]\in
V}d_{\xi}\langle\xi\rangle^{2(s+k)}\|\widehat{u}(\xi)\|_{HS}^{2}.$

Since $\widehat{G}\backslash V$ is finite, we conclude that
$\|u\|_{H^{s+k}(G)}<+\infty$ also. Since this holds for any $s\in\mathbb{R}$,
we conclude, by (3.1), that $u\in C^{\infty}(G)$ and thus $D$ is globally
hypoelliptic. Suppose now that $D$ does not satisfy (3.5). Thus, for each
$k\in\mathbb{N}$, there exists distinct $\xi_{k}\in\widehat{G}$,
$v_{k}\in\mathbb{C}^{d_{\xi_{k}}}$, $\|v_{k}\|_{2}=1$, $\lambda_{k}\geq 0$
such that

$\sigma_{D}(\xi_{k})^{*}\sigma_{D}(\xi_{k})v_{k}=\lambda_{k}^{2}v_{k},$

and

$\lambda_{k}<2^{-k}\langle\xi_{k}\rangle^{-k},$

for every $k\in\operatorname{\mathbb{N}}$. Let $u\in\mathcal{D}^{\prime}(G)$
be defined by the Fourier coefficients

$\displaystyle\widehat{u}(\xi)=\begin{bmatrix}|&|&\dots&|\\\
(v_{k})^{T}&0&\dots&0\\\ |&|&\dots&|\end{bmatrix},$

if $\xi=\xi_{k}$, and $\widehat{u}(\xi)=0$ otherwise. Then
$u\in\mathcal{D}^{\prime}(G)\backslash C^{\infty}(G)$ clearly, as
$\|\widehat{u}(\xi_{k})\|_{HS}=1$, for all $k\in\operatorname{\mathbb{N}}$,
but on the other hand

$\displaystyle\|\widehat{Du(\xi_{k})}\|_{HS}=\|\sigma_{D}(\xi_{k})\widehat{u}(\xi_{k})\|_{HS}=\lambda_{k}\|\widehat{u}(\xi_{k})\|_{HS}<2^{-k}\langle\xi_{k}\rangle^{-k},$

for all $k\in\operatorname{\mathbb{N}}$ and $\|\widehat{Du}(\xi)\|_{HS}=0$ for
all other $[\xi]\in\widehat{G}$, therefore $Du\in C^{\infty}(G)$ and so $D$ is
not globally hypoelliptic. ∎

######  Remark 3.3.

The proof of Theorem 3.2 is similar to the one of Theorem 3.1, but there are
some additional facts to take in consideration for the vector-valued setting,
as well as the need to consider the restriction to the set $\widehat{G}(E)$.

###### Proof of Theorem 3.2.

Indeed, suppose $D$ satisfies the inequality over $V\subset\widehat{G}(E)$.
Then for $u\in\mathcal{D}^{\prime}(E)$ such that $Du\in\Gamma^{\infty}(E)$,
and for any $s\in\mathbb{R}$:

(3.10) $\displaystyle+\infty>\|Du\|^{2}_{H^{s}(E)}$
$\displaystyle\geq\sum_{i=1}^{d_{\tau}}\sum_{[\xi]\in\widehat{G}(E)}d_{\xi}\langle\xi\rangle^{2s}\sum_{r=1}^{d_{\omega}}\|\sigma_{D}(i,r,\xi)\widehat{\chi_{\tau}u}(i,\xi)\|_{HS}^{2}.$

But by Lemma 3.1:

$\displaystyle\sum_{r=1}^{d_{\omega}}\|\sigma_{D}(i,r,\xi)\widehat{\chi_{\tau}u}(i,\xi)\|_{HS}^{2}\geq\sum_{r=1}^{d_{\omega}}\lambda_{\min}[\sigma_{D}(i,r,\xi)]^{2}\|\widehat{\chi_{\tau}u}(i,\xi)\|_{HS}^{2}\geq\frac{1}{2}m_{\xi}(D)^{2}\|\widehat{\chi_{\tau}u}(i,\xi)\|_{HS}^{2},$

for every $1\leq i\leq d_{\tau}$. Hence, (3.10) implies that

$\displaystyle+\infty>\|Du\|^{2}_{H^{s}(E)}$
$\displaystyle\geq\frac{1}{2}\sum_{i=1}^{d_{\tau}}\sum_{[\xi]\in
V}d_{\xi}\langle\xi\rangle^{s}m_{\xi}(D)^{2}\|\widehat{\chi_{\tau}u}(i,\xi)\|_{HS}^{2}$
$\displaystyle\geq\frac{C^{2}}{2}\sum_{i=1}^{d_{\tau}}\sum_{[\xi]\in
V}\langle\xi\rangle^{2(s+k)}d_{\xi}\|\widehat{\chi_{\tau}u}(i,\xi)\|_{HS}^{2}.$

Since $\widehat{G}(E)\backslash V$ is finite, we conclude that
$\|u\|_{H^{s+k}(E)}<+\infty$ also. Since this holds for any $s\in\mathbb{R}$,
we conclude by (3.3) that $u\in\Gamma^{\infty}(E)$ and thus $D$ is globally
hypoelliptic. Suppose now that $D$ does not satisfy (3.6). Then, in
particular, for each $k\in\mathbb{N}$, there exists distinct
$\xi_{k}\in\widehat{G}(E)$, $1\leq i_{k}\leq d_{\tau},\,1\leq r_{k}\leq
d_{\omega}$ $v_{k}\in\mathbb{C}^{d_{\xi_{k}}}$, $\|v_{k}\|_{2}=1$,
$\lambda_{k}\geq 0$ such that

$[\sigma_{{D}}(i_{k},r_{k},\xi_{k})]^{*}[\sigma_{{D}}(i_{k},r_{k},\xi_{k})]v_{k}=\lambda_{k}^{2}v_{k},$

and

$\lambda_{k}<2^{-k}\langle\xi_{k}\rangle^{-k},$

for every $k\in\operatorname{\mathbb{N}}$. Let $u\in\mathcal{D}^{\prime}(E)$
be defined by the Fourier coefficients

$\displaystyle\widehat{\chi_{\tau}u}(i,\xi)=\begin{bmatrix}|&|&\dots&|\\\
(v_{k})^{T}&0&\dots&0\\\ |&|&\dots&|\end{bmatrix},$

if $i=i_{k}$, $\xi=\xi_{k}$, and $\widehat{\chi_{\tau}u}(i,\xi)=0$ otherwise.
Then $u\in\mathcal{D}^{\prime}(E)\backslash\Gamma^{\infty}(E)$ clearly, as
$\|\widehat{\chi_{\tau}u}(i_{k},\xi_{k})\|_{HS}=1$, for all $k\in\mathbb{N}$,
and $\widehat{\chi_{\tau}u}(i,\xi)=0$ for all $[\xi]\not\in\widehat{G}(E)$,
but on the other hand

$\displaystyle\|\widehat{\chi_{\tau}{Du}}(i_{k},\xi_{k})\|_{HS}=\left\|\sum_{r=1}^{d_{\omega}}\sigma_{{D}}(i_{k},r_{k},\xi_{k})\widehat{\chi_{\tau}u}(i_{k},\xi_{k})\right\|_{HS}=r\lambda_{k}\|\widehat{\chi_{\tau}u}(i_{k},\xi_{k})\|_{HS}<2^{-k}r\langle\xi_{k}\rangle^{-k},$

for all $k\in\operatorname{\mathbb{N}}$, and
$\|\widehat{\chi_{\tau}{Du}}(i,\xi)\|_{HS}$ for all other $1\leq i\leq
d_{\tau}$, $[\xi]\in\widehat{G}(E)$, therefore $Du\in\Gamma^{\infty}(E)$ and
so $D$ is not globally hypoelliptic. ∎

######  Remark 3.4.

As before, the proof of Corollary 3.1 is also similar to the proof of 3.1, but
now extra care needs to be taken to discard the frequencies which are not
relevant for the lifting of operators on the homogeneous space.

###### Proof of Corollary 3.1.

First, notice that smooth functions on $M$ may be seen as smooth sections on
the trivial bundle $p:E=G\times_{\widehat{1}}\mathbb{C}\to M$, and so by the
remarks given in Section 2, this result follows from the Theorem 3.2. More
precisely, note that the Sobolev norms for $u\in H^{s}(M)$ are given by:

$\|u\|_{H^{s}(M)}^{2}=\sum_{[\xi]\in\widehat{M}}{d_{\xi}}\langle\xi\rangle^{s}\|\widehat{\dot{u}}(\xi)\|^{2}_{HS}$

where $\dot{u}$ is the projective lifting of $u$ on $G$, for
$s\in\operatorname{\mathbb{R}}$ as $\widehat{G}(E)=\widehat{M}$. Also, for
$u\in\mathcal{D}^{\prime}(M)$, its symbol at $[\xi]\in\widehat{M}$ has
possibly only the first $d_{\xi}^{K}$ lines different from zero. Similarly,
the projective lifting of $D$, $\tilde{D}$, has possibly only the first
$d_{\xi}^{K}$ columns different from $0$. Hence, it follows from Lemma 3.1
that

$\displaystyle\|\widehat{\dot{Du}}(\xi)\|_{HS}^{2}=\|\sigma_{\tilde{D}}(\xi)\widehat{\dot{u}}(\xi)\|_{HS}^{2}$
$\displaystyle=\|[\sigma_{\tilde{D}}(\xi)\widehat{\dot{u}}(\xi)]_{d_{\xi}^{K}\times
d_{\xi}}\|_{HS}^{2}$
$\displaystyle\geq\lambda_{\min}[[\sigma_{\tilde{D}}(\xi)]_{d_{\xi}\times
d_{\xi}^{K}}]^{2}\|\widehat{\dot{u}}(\xi)_{d_{\xi}^{K}\times
d_{\xi}}\|_{HS}^{2}.$

Suppose $D$ satisfies (3.8) for all $[\xi]\in V$, $\widehat{M}\backslash V$
finite. Then, if $u\in\mathcal{D}^{\prime}(M)$ is such that $Du\in
C^{\infty}(M)$, we have for $s\in\operatorname{\mathbb{R}}$ that

$\displaystyle+\infty>\|Du\|_{H^{s}(M)}^{2}$
$\displaystyle=\sum_{[\xi]\in\widehat{M}}d_{\xi}\langle\xi\rangle^{2s}\|\widehat{\dot{Du}}(\xi)\|_{HS}^{2}\geq\sum_{[\xi]\in
V}d_{\xi}\langle\xi\rangle^{2s}\|\widehat{\dot{Du}}(\xi)\|_{HS}^{2}$
$\displaystyle\geq\sum_{[\xi]\in
V}d_{\xi}\langle\xi\rangle^{2s}\lambda_{\min}[[\sigma_{\tilde{D}}(\xi)]_{d_{\xi}\times
d_{\xi}^{K}}]^{2}\|\widehat{\dot{u}}(\xi)\|_{HS}^{2}$ $\displaystyle\geq
C^{2}\sum_{[\xi]\in
V}d_{\xi}\langle\xi\rangle^{2(s+k)}\|\widehat{\dot{u}}(\xi)\|_{HS}^{2}.$

Since $\widehat{M}\backslash V$ is finite, we conclude that
$\|u\|_{H^{s+k}(M)}<+\infty$ also. Since this holds for any $s\in\mathbb{R}$,
we conclude by (3.3) that $u\in C^{\infty}(M)$ and thus $D$ is globally
hypoelliptic. Suppose now that $D$ does not satisfy (3.8). Then, for each
$k\in\mathbb{N}$, there exist distinct $\xi_{k}\in\widehat{M}$,
$v_{k}\in\mathbb{C}^{d_{\xi_{k}}^{K}}$, $\|v_{k}\|_{2}=1$, $\lambda_{k}\geq 0$
such that

$[\sigma_{\tilde{D}}(\xi)]_{d_{\xi}^{K}\times
d_{\xi}}^{*}[\sigma_{\tilde{D}}(\xi)]_{d_{\xi}\times
d_{\xi}^{K}}v_{k}=\lambda_{k}^{2}v_{k},$

and

$\lambda_{k}<2^{-k}\langle\xi_{k}\rangle^{-k}.$

Let $u\in\mathcal{D}^{\prime}(M)$ be defined by the Fourier coefficients

$\displaystyle\widehat{\dot{u}}(\xi)=\begin{bmatrix}|&|&\dots&|\\\
(v_{k})^{T}&0&\dots&0\\\ |&|&\dots&|\\\
0_{d_{\xi}-d_{\xi}^{K}}&0_{d_{\xi}-d_{\xi}^{K}}&\dots&0_{d_{\xi}-d_{\xi}^{K}}\end{bmatrix},$

if $\xi=\xi_{k}$, and $\widehat{u}(\xi)=0$ otherwise. Then
$u\in\mathcal{D}^{\prime}(M)\backslash C^{\infty}(M)$ clearly, as
$\|\widehat{\dot{u}}(\xi_{k})\|_{HS}=1$, for all
$k\in\operatorname{\mathbb{N}}$, also, $\widehat{u}(\xi)=0$ for
$[\xi]\not\in\widehat{M}$ and only the first $d_{\xi}^{K}$ lines of
$\widehat{u}(\xi)$ are possibly different from zero. But on the other hand

$\displaystyle\|\widehat{\dot{Du}}(\xi_{k})\|_{HS}=\|\sigma_{\tilde{D}}(\xi_{k})\widehat{\dot{u}}(\xi_{k})\|_{HS}=\lambda_{k}\|\widehat{\dot{u}}(\xi_{k})\|_{HS}<2^{-k}\langle\xi_{k}\rangle^{-k},$

for all $k\in\operatorname{\mathbb{N}}$ and
$\|\widehat{\dot{Du}}(\xi)\|_{HS}=0$ for all other $[\xi]\in\widehat{M}$,
therefore $Du\in C^{\infty}(M)$ and so $D$ is not globally hypoelliptic. ∎

###  3.3. Remarks and examples

We illustrate our main results in the following examples:

######  Example 3.1.

Notice that if $D$ is a left-invariant pseudo-differential operator on a
compact Lie group $G$ with diagonal symbol, e.g. $D$ is given by $cX+d$, where
$X$ is a left-invariant vector field on $G$, $c,d\in\mathbb{C}$, the
inequality in (3.5) of Theorem 3.1 can be rewritten as

$|\sigma_{D}(\xi)_{jj}|\geq C\langle\xi\rangle^{k}$

for some $C>0$, $k\in\mathbb{R}$ and all but finitely many $1\leq j\leq
d_{\xi},\,[\xi]\in\widehat{G}$.

######  Example 3.2.

Let $c\in\mathbb{C}$ and consider $D=\partial_{0}+c$, a perturbation of the
neutral operator $\partial_{0}$ on the compact Lie group $SU(2)$. Under the
usual identification
$\widehat{SU(2)}\sim\frac{1}{2}\operatorname{\mathbb{N}}_{0}$, its symbol is
given by:

$\sigma_{\partial_{0}+c}(\ell)_{ij}=(c+i)\delta_{ij}$

where $-\ell\leq i,j\leq\ell,\,\ell-i,\ell-j\in\mathbb{Z}$, and so if
$c\in\frac{1}{2}\operatorname{\mathbb{Z}}$ we have that,

$\lambda_{\min}[\sigma_{\partial_{0}+c}]=0$

for infinitely many $\ell\in\frac{1}{2}\operatorname{\mathbb{N}}_{0}$ and so
it does not satisfy inequality (3.5). In this case, Theorem 3.1 implies that
$\partial_{0}+c$ is not globally hypoelliptic. On the other hand, if
$c\not\in\frac{1}{2}\operatorname{\mathbb{Z}}$, it is possible to show that

$|\sigma_{D}(\xi)_{jj}|\geq\epsilon$

for some $\epsilon>0$, therefore by the same theorem and Example 3.1 we
conclude that in this case $\partial_{0}+c$ is globally hypoelliptic.

######  Example 3.3.

Let $G$ be a compact Lie group and $\mathcal{L}$ a sub-Laplacian satisfying
the Hörmander condition with step $\kappa$. It is negative definite formally
self-adjoint, and so by choosing an appropriate basis for the representation
spaces of $\widehat{G}$, its symbol is diagonal and given by

$\sigma_{\mathcal{L}}(\xi)=\text{diag}\left(-\nu_{1}^{2},\dots,-\nu_{d_{\xi}}^{2}\right),$

for some $\nu_{j}(\xi)\geq 0$ and for all $[\xi]\in\widehat{G}$. It follows
from the functional calculus, that for any $s\in\mathbb{R}$ the symbol of the
operator $(\text{Id}-\mathcal{L})^{\frac{s}{2}}$ is given by

$\sigma_{(\text{Id}-\mathcal{L})^{\frac{s}{2}}}(\xi)=\text{diag}\left((1+\nu_{1}^{2})^{\frac{s}{2}},\dots,(1+\nu_{d_{\xi}}^{2})^{\frac{s}{2}}\right),$

for all $[\xi]\in\widehat{G}$. As proved in Proposition 3.1 of [25] there
exist $C_{1},C_{2}>0$ such that

$C_{1}\langle\xi\rangle\geq(1+\nu_{j}(\xi)^{2})^{\frac{1}{2}}\geq
C_{2}\langle\xi\rangle^{\frac{1}{\kappa}},$

for all $1\leq j\leq d_{\xi}$, $[\xi]\in\widehat{G}$. It follows from Theorem
3.1 and Example 3.1 that the operator $(\text{Id}-\mathcal{L})^{\frac{s}{2}}$
is globally hypoelliptic for any $s\in\mathbb{R}$.

######  Example 3.4.

Let $G$ be a compact Lie group, $\mathcal{L}$ a subelliptic operator such that

(3.11) $\|u\|_{H^{\frac{2}{r}}(G)}\leq
C\left(\|\mathcal{L}u\|_{L^{2}(G)}+\|u\|_{L^{2}(G)}\right).$

Theorem 3.1 provides an alternative proof of the fact that $\mathcal{L}$ is
globally hypoelliptic. Indeed, given $[\xi]\in\widehat{G}$, let
$v_{\xi}\in\mathbb{C}^{d_{\xi}}$ be a unit vector such that

(3.12)
$\sigma_{\mathcal{L}}(\xi)^{*}\sigma_{\mathcal{L}}(\xi)v_{\xi}=\lambda_{\min}[\sigma_{\mathcal{L}}(\xi)]^{2}v_{\xi}.$

Let $u\in\mathcal{D}^{\prime}(G)$ be defined by the Fourier coefficients

$\widehat{u}(\xi)=\begin{bmatrix}|&|&\dots&|\\\ (v_{\xi})^{T}&0&\dots&0\\\
|&|&\dots&|\end{bmatrix},$

and $\widehat{u}(\xi^{\prime})=0$ for all other
$[\xi^{\prime}]\in\widehat{G}$. Then by (3.11) and Plancherel’s Theorem

$\displaystyle d_{\xi}\langle\xi\rangle^{\frac{2}{r}}$ $\displaystyle\leq
C(d_{\xi}\|\sigma_{\mathcal{L}}(\xi)\widehat{u}(\xi)\|_{HS}^{2}+d_{\xi})$
$\displaystyle\leq
C(d_{\xi}\lambda_{\min}[\sigma_{\mathcal{L}}(\xi)]^{2}+d_{\xi})$

which implies

$\lambda_{\min}[\sigma_{\mathcal{L}}(\xi)]\geq\sqrt{\frac{1}{C}\langle\xi\rangle^{\frac{2}{r}}-1}.$

Since $[\xi]\in\widehat{G}$ is arbitrary, we conclude this holds for every
$[\xi]\in\widehat{G}$. Therefore, for all but finitely many
$[\xi]\in\widehat{G}$, we have the estimate

$\lambda_{\min}[\sigma_{\mathcal{L}}(\xi)]\gg\langle\xi\rangle^{\frac{1}{r}},$

and so by Theorem 3.1 we conclude $\mathcal{L}$ is globally hypoelliptic. Note
that a typical example of an operator $\mathcal{L}$ satisfying the properties
above is an arbitrary Hörmander sub-Laplacian [29]. In this case we refer to
Rothschild and Stein [17] for the validity of the subelliptic estimate (3.11).

######  Example 3.5.

Consider the compact homogeneous space
$M=\mathbb{S}^{2}=SU(2)/\mathbb{T}^{1}$, where $K=\mathbb{T}^{1}$ is the
(maximal torus) subgroup of $SU(2)$ given in Euler angles by

$\left\\{\begin{pmatrix}e^{i\psi/2}&ie^{-i\psi/2}\\\
ie^{i\psi/2}&e^{-i\psi/2}\end{pmatrix}:\psi\in[-2\pi,2\pi)\right\\}.$

In this case, note that the only $K$ invariant elements of $\widehat{SU(2)}$
are given by $\mathfrak{t}^{\ell}_{n0}$,
$\ell\in\operatorname{\mathbb{N}}_{0}$, and indeed
$\Pi_{\mathbb{S}^{2}}\mathfrak{t}^{\ell}\equiv 0$
(i.e.:$\Pi_{\mathbb{S}^{2}}\mathfrak{t}^{\ell}_{nm}\equiv 0,$ for all
$-\ell\leq m,n\leq\ell$) if
$\ell\in\frac{1}{2}\mathbb{N}_{0}\backslash\mathbb{N}_{0}$. Therefore, when
$M=\mathbb{S}^{2}$ in (3.7), $\widehat{\mathbb{S}^{2}}$ can be parametrized by
$\operatorname{\mathbb{N}}_{0}$ and the Fourier inversion formula can be
written (in the usual notation) as

$f(\phi,\theta)=\sum_{\ell\in\operatorname{\mathbb{N}}_{0}}(2\ell+1)\sum_{-\ell\leq
n\leq\ell}\mathfrak{t}^{\ell}_{n0}(\phi,\theta,0)\widehat{\dot{f}}(\ell)_{0n},$

where $\dot{f}(\phi,\theta,\psi)=f(\phi,\theta)$ for $f\in
C^{\infty}(\mathbb{S}^{2})$, $0\leq\phi<2\pi,\,0\leq\theta\leq\pi$ and the sum
ranges over $n-\ell\in\mathbb{Z}$.
Hence, by Corollary 3.1, a homogeneous operator $D$ on $\mathbb{S}^{2}$ is
globally hypoelliptic if and only if it satisfies

$\lambda_{\min}[\text{middle column of
}\sigma_{\tilde{D}}(\xi)]=\sqrt{\sum_{-\ell\leq
n\leq\ell}|\sigma_{\tilde{D}}(\ell)_{n0}|^{2}}\geq C(1+\ell)^{k},$

for some $C>0$, $k\in\mathbb{R}$ and all but finitely many
$\ell\in\operatorname{\mathbb{N}}_{0}$.

Taking for instance the Laplacian on the sphere $\Delta_{\mathbb{S}^{2}}$,
then its symbol, in this notation, is given by

$\sigma_{\tilde{\Delta_{{\mathbb{S}^{2}}}}}(\ell)=\text{diag}\big{(}0,\dots,0,\ell^{2}+\ell,0,\dots,0\big{)}$

so the inequality above can be re-written as

$\ell^{2}+\ell\geq C(1+\ell)^{k},$

which is indeed true for $C=k=1$,
$\forall\ell\in\operatorname{\mathbb{N}}_{0},\,\ell\neq 0$.

###  3.4. Acknowledgement

The authors would like to thank Alexandre Kirilov and Wagner A. Almeida de
Moraes for discussions.

### Conflict of interests statement.

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

### Data Availability Statements.

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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