# Vector Valued Gårding Inequality for Pseudo-differential Operators on
Compact Homogeneous Manifolds

André Kowacs and Michael Ruzhansky

###### Abstract.

We prove sufficient conditions in order to obtain a sharp Gårding inequality
for pseudo-differential operators acting on vector-valued functions on compact
Lie groups. As a consequence, we obtain a sharp Gårding inequality for compact
homogeneous vector bundles and compact homogeneous manifolds. The sharp
Gårding inequality is the strongest lower bound estimate known to hold for
systems on $\operatorname{\mathbb{R}}^{n}$, and the aim of this paper is to
extend this property to systems on compact Lie groups and compact homogeneous
manifolds. Our results extend previous works by Lax and Nirenberg [Comm. Pure
Appl. Math., Vol. 8, 129-209, (1966)], and by Ruzhansky and Turunen [J. Funct.
Anal., Vol. 267, 144-172, (2011)]. As an application, we establish existence
and uniqueness of solution to a class of systems of initial value problems of
pseudo-differential equations on compact Lie groups and compact homogeneous
manifolds.

###### Key words and phrases:

Gårding inequality, Compact Lie group, Pseudo-differential operator, Fourier
Analysis.

###### 1991 Mathematics Subject Classification:

Primary 22E30, 43A77. Secondary 58J40, 35S10

The authors are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and
Partial Differential Equations, by the Methusalem programme of the Ghent
University Special Research Fund (BOF) (Grant number 01M01021) and by the FWO
grant G011522N. Michael Ruzhansky was also supported by EPSRC grant
EP/V005529/1. André Kowacs was supported in part by the Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code
001”;

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.0.1 The group Fourier transform
    2. 2.0.2 The quantisation on compact Lie groups
    3. 2.1 The vector-valued quantisation on compact Lie groups
    4. 2.2 Even and odd functions on compact Lie groups
  3. 3 Main Results
    1. 3.1 Sharp Gårding inequality on compact Lie groups
    2. 3.2 Sharp Gårding inequality on compact homogeneous vector bundles
  4. 4 Application for vector-valued diffusion problems

##  1\. Introduction

The so called Gårding inequality was first proved by Gårding in his paper [6]
and can be stated as follows:
Let $P$ be an elliptic self-adjoint pseudo-differential operator of order
$m\in\operatorname{\mathbb{R}}$ on an open set
$U\subset\operatorname{\mathbb{R}}^{n}$. Then, for any compact set $Q\subset
U$ and $\gamma<m/2$, there exist constants $c_{\gamma,Q},\,C_{\gamma,Q}>0$
such that

$(Pu,u)_{L^{2}}\geq
c_{\gamma,Q}\|u\|^{2}_{H^{m/2}}-C_{\gamma,Q}\|u\|^{2}_{H^{\gamma}},$

for every $u\in C_{0}^{\infty}(Q)$. In order to apply it to different
problems, Hörmander [7] and Lax and Nirenberg [8] then adapted this result and
proved the so called sharp Gårding inequality for pseudo-differential
operators for symbols in the Kohn-Nirenberg class
$S^{m}(\operatorname{\mathbb{R}}^{2n})$. Their result states that if a symbol
$p\in S^{m}(\operatorname{\mathbb{R}}^{2n})$, for
$m\in\operatorname{\mathbb{R}}$ satisfies
$\operatorname{\text{Re}}p(x,\xi)\geq 0$, then the pseudo-differential
operator $p(x,D)$ associated to this symbol satisfies the estimate

$\operatorname{\text{Re}}(p(x,D)u,u)_{L^{2}(\operatorname{\mathbb{R}}^{n})}\geq-C\|u\|_{H^{\frac{m-1}{2}}}^{2},$

for some $C>0$ and every $u\in
H^{\frac{m-1}{2}}(\operatorname{\mathbb{R}}^{n})$, where the norm on the right
hand side of the inequality denotes the usual norm in the Sobolev space
$H^{\frac{m-1}{2}}(\operatorname{\mathbb{R}}^{n})$. Since then, this result
has been extended, by many authors, to different settings and symbol classes.
In particular, this type of inequality has been extended to the global symbol
classes on compact Lie groups for scalar valued functions. In this paper,
inspired by the works on Gårding inequality on compact Lie group [11], on
sharp Gårding inequality on compact Lie groups [10], on sharp Gårding
inequality for subelliptic operators on compact Lie groups [3] and on Gårding
inequality on graded Lie groups [2], we prove a type of sharp Gårding
inequality for pseudo-differential operators acting on vector-valued functions
on compact Lie groups. In order to do so, we also extend the notion of pseudo-
differential operators defined by amplitudes to the vector-valued compact Lie
group setting. As a consequence, we obtain a sharp Gårding inequality for
pseudo-differential operators on compact homogeneous vector bundles and
compact homogeneous manifolds. Finally, in the end we present an application
of these result proving existence and uniqueness of solution to a class of
systems of vector valued Cauchy problems of pseudo-differential equations. It
is worth mentioning that the symbol classes considered are the global symbol
classes defined in [9]. These classes of symbols coincide with the usual
Hörmander local symbol classes on manifolds in the range that the latter are
well defined, that is $0\leq\delta<\rho\leq 1$, $\rho\geq 1-\delta$. The
global symbols for compact Lie groups developed in [9] allow us to extend this
range to include the cases $\rho<1-\delta$, and thus are more general to deal
with. It is also relevant to mention that the sharp Gåarding inequality is the
strongest lower bound estimate known to hold for systems on
$\operatorname{\mathbb{R}}^{n}$ (vector-valued functions). The aim of this
paper is to extend this property for global quantisation of operators on
compact Lie groups.
This paper is organized as follows:
In Section 2 we recall some of the theory about pseudo-differential operators
on compact Lie groups. Next we also develop and prove some results concerning
amplitudes for pseudo-differential operators acting on vector-valued functions
on compact Lie groups. At the end of this section, we also recollect some
basic facts about parity of functions on Lie groups.
In Section 3 state and prove our mains results. We also prove a few lemmas
which aid in its proofs.
Finally, in Section 4 we present an application of our main result, in which
we prove existence and uniqueness of solution to a certain class of systems of
pseudo-differential equations.

##  2\. Preliminaries

####  2.0.1. The group Fourier transform

Let $G$ be a compact Lie group. Denote by $\text{Rep}(G)$ the set of all
continuous unitary representations of $G$, that is, the set of all continuous
$\xi\in\text{Hom}(G,\text{Aut}(\mathcal{H}_{\xi}))$, where $\mathcal{H}_{\xi}$
is some Hilbert space and $\xi(g)$ is unitary for every $g\in G$. It is well
known that for all $\xi\in\text{Rep}(G)$, the spaces $\mathcal{H}_{\xi}$ are
finite dimensional, and we denote their dimensions by $d_{\xi}$. Define the
equivalence relation $\xi\sim\eta$ if there exists a bijection
$A:\mathcal{H}_{\xi}\to H_{\eta}$ such that $\eta(g)A=A\xi(g)$, for all $g\in
G$. The set of all such equivalence classes, denoted by $\widehat{G}$, is
known as the unitary dual of $G$. As every class $[\xi]\in\widehat{G}$ is
finite dimensional, we can always choose a representative that is matrix-
valued. It follows from the Peter-Weyl Theorem that the collection of all
coefficient functions of elements of $\widehat{G}$ is an orthogonal basis for
$L^{2}(G)$, where integration is taken with respect to the Haar measure in
$G$. This motivates the definition of the matrix-valued Fourier coefficients
of $f\in L^{2}(G)$ by

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

for $\xi\in\text{Rep}(G)$. This gives the Fourier inversion formula as

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

for almost every $x$ in $G$, where in the sum we choose a single
representative for each class in $\widehat{G}$. As in the Euclidean case,
there is also an analogue of Plancherel’s Theorem, namely

$\|f\|_{L^{2}(G)}=\left(\int_{G}|f(x)|^{2}dx\right)^{\frac{1}{2}}=\left(\sum_{[\xi]\in\widehat{G}}d_{\xi}\|\widehat{f}(\xi)\|_{HS}^{2}\right)^{\frac{1}{2}}=\|\widehat{f}\|_{L^{2}(\widehat{G})},$

for any $f\in L^{2}(G)$, where $\|A\|_{HS}^{2}\doteq\operatorname{Tr}(A^{*}A)$
for any matrix $A$. When $G$ is also a Lie group, these definitions can be
extended to the set $\mathcal{D}^{\prime}(G)$ of distributions on $G$, through
the formula

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

for $u\in\mathcal{D}^{\prime}(G)$ and $\xi\in\text{Rep}(G)$, where evaluation
should be understood coefficient-wise.

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

Now assume that $G$ is also a Lie group. Given a continuous linear operator
$L:C^{\infty}(G)\to C^{\infty}(G)$, define its matrix-valued global symbol
$\sigma_{L}$ by

$\sigma_{L}(x,\xi)\doteq\xi(x)^{*}L\xi(x),$

for all $(x,[\xi])\in G\times\widehat{G}$. In [9] the authors proved the
quantisation formula

(1)
$Lf(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}\left[\xi(x)\sigma_{L}(x,\xi)\widehat{f}(\xi)\right]$

for all $x\in G$, $f\in C^{\infty}(G)$. In order to classify such symbols, a
difference operator of order $k\in\mathbb{N}$, $\Delta_{q}$ on $\widehat{G}$ ,
where $q\in C^{\infty}(G)$ vanishes of order $k$ at the group identity
$e=e_{G}$, is defined by

$\Delta_{q}\widehat{f}(\xi)\doteq\widehat{qf}(\xi),$

for every $[\xi]\in\widehat{G}$.
A family of difference operators $\\{\Delta_{q_{1}},\dots,\Delta_{q_{r}}\\}$
of order $1$ is said to be strongly admissible if

$\text{rank}\\{\nabla q_{j}(e)|1\leq j\leq r\\}=\dim(G)$

and

$\bigcap_{j=1}^{r}\\{x\in G|q_{j}(x)=0\\}=\\{e_{G}\\}.$

We fix such a family and define the multi-index notation for higher order
difference operators by

$q_{\alpha}\doteq q_{1}^{\alpha_{1}}\dots q_{k}^{\alpha_{k}},$
$\Delta^{\alpha}_{\xi}\doteq\Delta_{q_{1}}^{\alpha_{1}}\dots\Delta_{q_{r}}^{\alpha_{r}}\equiv\Delta_{q_{\alpha}},$

for a multi-index $\alpha$. We recall (see [3]) that we may choose such family
so that an analog of the Leibniz’s rule holds as follows:

######  Proposition 2.1 (Leibniz’s like formula for Difference Operators).

For any multi-index $\alpha$ there exist constants $C_{\lambda,\mu}\geq 0$
such that

$\Delta_{\xi}^{\alpha}[\widehat{f}(\xi)\widehat{g}(\xi)]=\sum_{|\mu|,|\lambda|\leq|\alpha|\leq|\lambda+\mu|}C_{\lambda,\mu}(\Delta_{\xi}^{\lambda}\widehat{f}(\xi))(\Delta_{\xi}^{\mu}\widehat{g}(\xi)),$

for any $f,g\in\mathcal{D}^{\prime}(G)$, and all $[\xi]\in\widehat{G}$.

Next, recall that the vector space of left-invariant vector fields of $G$,
denoted by $\mathfrak{g}$, may be identified with the tangent space at the
identity $e$, and each of its non-zero elements corresponds to a first-order
differential operator $X:C^{\infty}(G)\to C^{\infty}(G)$ given by

$Xf(x)\doteq\frac{d}{dt}\bigg{|}_{t=0}f(x\exp(tX)),$

for $x\in G$, where $\exp:\mathfrak{g}\to G$ denotes the standard Lie
exponential mapping. If $\\{X_{1},\dots,X_{n}\\}$ is a basis for
$\mathfrak{g}$, we will use the standard multi-index notation

$\partial^{\alpha}=X_{1}^{\alpha_{1}}\dots X_{n}^{\alpha_{n}},$

for a standard left-invariant differential operator of order $|\alpha|$.
Notice that, in general, these operators do not commute, hence to represent
all differential operators of a certain order, one would need to specify their
order of composition. As this particular order will not be relevant, we will
consider $\partial^{\alpha}$ as denoting an arbitrary differential operator of
order $|\alpha|$.
Consider now the positive Laplace-Beltrami operator of $G$, $\mathcal{L}_{G}$.
This is a second order bi-invariant differential operator and the coefficient
functions $\\{\xi_{ij}\\}_{i,j=1}^{d_{\xi}}$ of $[\xi]\in\widehat{G}$ are all
eigenfunctions correponding to the same respective eigenvalue
$\lambda_{[\xi]}\geq 0$. Defining the weight

$\langle\xi\rangle\doteq(1+\lambda_{[\xi]})^{1/2},$

corresponding to the eigenvalues of the first order differential operator
$(\text{Id}+\mathcal{L}_{G})^{\frac{1}{2}}$. We use these to describe the
symbol classes as follows.

######  Definition 2.2.

Let $G$ be a compact Lie group, $m\in\operatorname{\mathbb{R}}$,
$0\leq\rho,\delta\leq 1$. We shall say that a matrix-valued function

$\sigma:G\times\widehat{G}\to\bigcup_{[\xi]\in\widehat{G}}\mathbb{C}^{d_{\xi}\times
d_{\xi}},$

smooth in $x\in G$, is in the symbol class $\mathcal{S}_{\rho,\delta}^{m}(G)$
if

$\|\partial^{\beta}_{x}\Delta^{\alpha}_{\xi}\sigma(x,\xi)\|_{op}\leq
C_{\alpha,\beta}\langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|},$

for every multi-index $\alpha,\beta$ and for all $(x,[\xi])\in
G\times\widehat{G}$. For $\sigma_{L}\in\mathcal{S}_{\rho,\delta}^{m}(G)$ we
will write $L\in\Psi^{m}_{\rho,\delta}(G)\equiv\text{Op}(\sigma_{L})$ for the
continuous linear operator defined by formula (1).

The operators associated with such symbol classes are called pseudo-
differential operators, the real number $m$ is said to be its order.
As $G$ is in particular also a smooth manifold, one could have defined the
Hormander classes of pseudo-differential operators by local coordinate
systems, denoted $\Psi_{\rho,\delta}^{m}(G,\text{loc})$ for
$m\in\operatorname{\mathbb{R}}$, $0\leq\delta<\rho\leq 1$, $\rho\geq
1-\delta$. In fact, the first definition extends the second in the following
sense.

######  Theorem 2.3 (Equivalence of classes, [9],[5]).

Let $L:C^{\infty}(G)\to C^{\infty}(G)$ be a continuous linear operator and
$m\in R$, $0\leq\delta<\rho\leq 1$, $\rho\geq 1-\delta$. Then,
$L\in\Psi_{\rho,\delta}^{m}(G,\text{loc})$ if, and only if,
$\sigma_{L}\in\mathcal{S}_{\rho,\delta}^{m}$. Consequently,

$\Psi_{\rho,\delta}^{m}(G,\text{loc})=\Psi^{m}_{\rho,\delta}(G),\,0\leq\delta<\rho\leq
1,\,\rho\geq 1-\delta,\,m\in\operatorname{\mathbb{R}}.$

In this paper, we shall make use of the following lemma, whose proof can be
found in [5].

######  Lemma 2.4 (Taylor Expansion in Compact Lie Groups).

Let $G$ be a compact Lie group of dimension $d$, and let
$\\{\Delta_{q_{1}},\dots,\Delta_{q_{d}}\\}$ be a strongly admissible
collection of difference operators. Then there exists a basis of left-
invariant vector fields $\\{X_{1},\dots,X_{d}\\}$ such that

$X_{j}q_{k}(x^{-1})|_{x=e_{G}}=\delta_{jk},$

for all $1\leq j,k\leq d$. Moreover, by using a multi-index notation relative
to both this collection of difference operators, and this basis of left
invariant vector fields, any $f\in C^{\infty}(G)$, can be written as

$f(xy)=f(x)+\sum_{1\leq|\alpha|<N}\frac{1}{\alpha!}\partial^{\alpha}f(x)q_{\alpha}(y^{-1})+R_{x,N}^{f}(y),$

for any $x,y\in G$, $N\in\operatorname{\mathbb{N}}$, where the last term is
referred to as the Taylor remainder of order $N$ and satisfies

$|R_{x,N}^{f}(y)|\lesssim|y|^{N}\max_{|\alpha|\leq
N}\|\partial^{\alpha}f\|_{L^{\infty}(G)}.$

###  2.1. The vector-valued quantisation on compact Lie groups

Let $E_{0}$ be a $n$-dimensional $\mathbb{C}$-vector space, where
$n\in\operatorname{\mathbb{N}}$. Consider $B_{0}=\\{e_{1},\dots,e_{n}\\}$ an
orthonormal basis on $E_{0}$. We may identify
$E_{0}\cong\mathbb{C}^{n}\cong\mathbb{C}^{n\times 1}$ by identifying each
element in $B_{0}$ with the canonical basis in $\mathbb{C}^{n}$. Hence, given
a mapping $f:G\to E_{0}$, we may write it as

$f(x)=(f_{1}(x),\dots,f_{n}(x))^{T}\in\mathbb{C}^{n\times 1},$

where $f_{j}(x)=\langle f(x),e_{j}\rangle_{E_{0}}\in\mathbb{C}$, for each
$x\in G$. Following [4], for $f\in L^{1}(G,E_{0})$ we define the Fourier
coefficients of $f$ at $[\xi]\in\widehat{G}$ by

$\widehat{f}(\xi)=(\widehat{f}_{1}(\xi),\dots,\widehat{f}_{n}(\xi))^{T},$

where $[\xi]\in\widehat{G}$. If $A:C^{\infty}(G,E_{0})\to C^{\infty}(G,E_{0})$
is a continuous linear operator, then by defining its matrix-valued symbol

(2) $\sigma_{A}(i,r,x,\xi)=\xi(x)^{*}e_{r}^{*}[A(\xi\otimes e_{i})(x)],$

for $1\leq i,r\leq n$, $x\in G,[\xi]\in\widehat{G}$, it was proved in [4] that
the quantisation formula

(3)
$Af(x)=\sum_{i,r=1}^{n}\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}[\xi(x)\sigma_{A}(i,r,x,[\xi])\widehat{f}_{i}(\xi)]e_{r},$

holds true for every $x\in G$, $f\in C^{\infty}(G,E_{0})$. Equivalently, one
can also write it as

$Af(x)=\left(\sum_{i=1}^{n}\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}[\xi(x)\sigma_{A}(i,1,x,[\xi])\widehat{f}_{i}(\xi)],\dots,\sum_{i=1}^{n}\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}[\xi(x)\sigma_{A}(i,n,x,[\xi])\widehat{f}_{i}(\xi)]\right)^{T},$

or even,

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

where $\sigma_{A}(x,[\xi])\in(\mathbb{C}^{d_{\xi}\times d_{\xi}})^{n\times n}$
is given by:

$\sigma_{A}(x,[\xi])_{ir}=\sigma_{A}(i,r,x,[\xi]),$

for $1\leq i,r\leq n$, $(x,[\xi])\in G\times\widehat{G}$, and $\xi(x)^{\otimes
n}$ is the block diagonal $(\mathbb{C}^{d_{\xi}\times d_{\xi}})^{n\times
n}$-matrix given by

(4) $\xi(x)^{\otimes n}=\text{diag}(\xi(x),\dots,\xi(x)),$

for all $(x,[\xi])\in G\times\widehat{G}$. Finally, for a matrix vector $v$
the trace above should be understood component-wise, i.e.:
$(\operatorname{Tr}(v))_{j}=\operatorname{Tr}(v_{j})$, for $1\leq j\leq n$.
As before, we introduce the symbol classes with respect to these quantisation
formulas.

######  Definition 2.5.

Let $J_{n}\doteq\\{1,\dots,n\\}$, $0\leq\rho,\delta\leq 1$. A symbol
$\sigma\in\mathcal{S}_{\rho,\delta}^{m}(G\otimes\text{End}(E_{0}))$ is a
mapping from $J_{n}\times J_{n}\times G\times\text{Rep}(G)$, smooth in $x$,
such that, for any $\xi\in\text{Rep}(G)$ and $i,r\in
J_{n},\,\sigma(i,r,x,\xi)\in\mathbb{C}^{d_{\xi}\times d_{\xi}}$, and for any
admissible collection of difference operators $\Delta_{\xi}^{\alpha}$, it
satisfies

$\|\Delta_{\xi}^{\alpha}\partial_{x}^{\beta}\sigma(i,r,x,\xi)\|_{op}\leq
C_{\alpha\beta}\operatorname{\langle\xi\rangle}^{m-\rho|\alpha|+\delta|\beta|},$

for some $C_{\alpha,\beta}>0$, for all multi-indices $\alpha,\beta$, and all
$(i,r,x,[\xi])\in J_{n}\times J_{n}\times G\times\widehat{G}$.
For a symbol $\sigma$, its associated operator
$\text{Op}(\sigma):C^{\infty}(G,E_{0})\to\mathcal{D}^{\prime}(G,E_{0})$ is
defined by

(5)
$(\text{Op}(\sigma)u)_{r}(x)\doteq\sum_{[\eta]\in\widehat{G}}d_{\eta}\operatorname{Tr}\left[\eta(x)\sum_{i=1}^{n}\sigma(i,r,x,\eta)\widehat{u_{i}}(\eta)\right],$

for every $1\leq r\leq n$. We then say $\text{Op}(\sigma)$ is a vector-valued
pseudo-differential operator of order $m$, and write
$\text{Op}(\sigma)\in\Psi^{m}_{\rho,\delta}(G\otimes\text{End}(E_{0}))$.

The vector-valued Sobolev space $H^{s}(G,E_{0})$,
$s\in\operatorname{\mathbb{R}}$, is defined, as the completion of
$C^{\infty}(G,E_{0})$ under the Sobolev norm

$\|u\|_{H^{s}(G,E_{0})}^{2}=\sum_{[\xi]\in\widehat{G}}d_{\xi}\langle\xi\rangle^{2s}\sum_{i=1}^{d_{\tau}}\|\widehat{u}(i,\xi)\|_{HS}^{2}.$

The following theorem will be useful and its proof can be found in [4].

######  Theorem 2.6.

Let $A:C^{\infty}(G,E_{0})\to C^{\infty}(G,E_{0})$ be a continuous linear
operator with symbol
$a\in\mathcal{S}_{\rho,\delta}^{m}(G\otimes\text{End}(E_{0}))$,
$0\leq\delta<\rho\leq 1$. Then $A:H^{s}(G,E_{0})\to H^{s-m}(G,E_{0})$ extends
to a bounded operator for all $s\in\operatorname{\mathbb{R}}$.

Next, we generalize the concept of symbols to amplitudes, as follows.

######  Definition 2.7.

An amplitude $a\in\mathcal{A}_{\rho,\delta}^{m}(G\otimes\text{End}(E_{0}))$,
where $m\in\operatorname{\mathbb{R}}$, $0\leq\rho,\delta\leq 1$, is a mapping
from $J_{n}\times J_{n}\times G\times G\times\text{Rep}(G)$, smooth in $x$ and
$y$, such that, for any
$\xi\in\text{Rep}(G),\,a(i,r,x,y,\xi)\in\mathbb{C}^{d_{\xi}\times d_{\xi}}$,
and for any admissible collection of difference operators
$\Delta_{\xi}^{\alpha}$, it satisfies

$\|\Delta_{\xi}^{\alpha}\partial_{x}^{\beta}\partial_{y}^{\gamma}a(i,r,x,y,\xi)\|_{op}\leq
C_{\alpha\beta\gamma}\operatorname{\langle\xi\rangle}^{m-\rho|\alpha|+\delta|\beta+\gamma|},$

for all multi-indices $\alpha,\beta,\gamma$, $(i,r,x,y,[\xi])\in J_{n}\times
J_{n}\times G\times G\times\widehat{G}$.
For an amplitude $a$, the amplitude operator
$\text{Op}(a):C^{\infty}(G,E_{0})\to\mathcal{D}^{\prime}(G,E_{0})$ is defined
by

(6)
$(\text{Op(a)}u)_{r}(x)\doteq\sum_{[\eta]\in\widehat{G}}d_{\eta}\operatorname{Tr}\left[\eta(x)\int_{G}\sum_{i=1}^{n}a(i,r,x,y,\eta)u_{i}(y)\eta^{*}(y)\mathop{dy}\right]$

for every $1\leq r\leq n$. Notice that if $A$ is a pseudo-differential
operator $A:C^{\infty}(G,E_{0})\to C^{\infty}(G,E_{0})$ defined as before, and
if $a(i,r,x,y,\eta)=\sigma_{A}(i,r,x,\eta)$, then $\text{Op}(a)=A$.

######  Proposition 2.8.

Let $0\leq\delta<1$ and $0\leq\rho\leq 1$, and
$a\in\mathcal{A}_{\rho,\delta}^{m}(G\otimes\text{End}(E_{0}))$. Then
$\text{Op}(a)$ is a continuous linear operator from $C^{\infty}(G,E_{0})$ to
$C^{\infty}(G,E_{0})$.

###### Proof.

Recall that by definition,
$(1+\mathcal{L}_{G})\eta=\langle\eta\rangle^{2}\eta$. Consequently,
integrating by parts in the $dy$-integral in (6) with respect to
$\langle\eta\rangle^{-2}(1+\mathcal{L}_{G})$ arbitrarily many times, we obtain
that the $\eta$-series in (6) converges uniformly and absolutely, so that
$\text{Op}(a)\in C^{\infty}(G,E_{0})$ provided $u\in C^{\infty}(G,E_{0})$. The
continuity of $\text{Op}(a)$ follows similarly. ∎

######  Proposition 2.9.

Let $m\in\operatorname{\mathbb{R}}$, $0\leq\delta<\rho\leq 1$ and
$a\in\mathcal{A}_{\rho,\delta}^{m}(G\otimes\text{End}(E_{0}))$. Then
$A=\text{Op}(a)$ is a pseudo-differential operator with matrix symbol
$\sigma_{A}\in\mathcal{S}_{\rho,\delta}^{m}(G\otimes\text{End}(E_{0}))$.
Moreover, $\sigma_{A}$ has the asymptotic expansion

$\sigma_{A}(i,r,x,\xi)\sim\sum_{\alpha\geq
0}\frac{1}{\alpha!}\partial_{y}^{\alpha}\Delta_{\xi}^{\alpha}a(i,r,x,y,\xi)|_{y=x}$

for each $1\leq i,r,\leq n$, in the sense that

$\left(\sigma_{A}(i,r,x,\xi)-\sum_{0\leq|\alpha|<N}\frac{1}{\alpha!}\partial_{y}^{\alpha}\Delta_{\xi}^{\alpha}a(i,r,x,y,\xi)|_{y=x}\right)\in\mathcal{S}_{\rho,\delta}^{m-(\rho-\delta)N}(G\otimes\text{End}(E_{0})),$

for every $N\in\operatorname{\mathbb{N}}$ sufficiently big.

###### Proof.

Indeed, by Proposition 2.8, $A$ is a continuous linear operator acting on
$C^{\infty}(G,E_{0})$ and therefore admits symbol $\sigma_{A}$, the matrix
symbol of $A$, given by:

$\sigma_{A}(i,r,x,\xi)=\xi(x)^{*}e_{r}^{*}[A(\xi\otimes e_{i})(x)].$

Therefore

$\displaystyle\sigma_{A}(i,r,x,\xi)_{mn}$
$\displaystyle=\sum_{l=1}^{d_{\xi}}\xi(x^{-1})_{ml}(A(\xi_{ln}\otimes
e_{i}))_{r}(x)$
$\displaystyle=\sum_{l=1}^{d_{\xi}}\xi(x^{-1})_{ml}\int_{G}\sum_{[\eta]\in\widehat{G}}d_{\eta}\operatorname{Tr}[\eta(x)a(i,r,x,y,\eta)\xi(y)_{ln}\eta(y)^{*}]dy$
$\displaystyle=\int_{G}\sum_{l=1}^{d_{\xi}}\xi(x^{-1})_{ml}\xi(y)_{ln}\sum_{[\eta]\in\widehat{G}}d_{\eta}\operatorname{Tr}[\eta(y^{-1})\eta(x)a(i,r,x,y,\eta)]dy$
$\displaystyle=\int_{G}\xi(x^{-1}y)_{mn}\sum_{[\eta]\in\widehat{G}}d_{\eta}\operatorname{Tr}[\eta(y^{-1}x)a(i,r,x,y,\eta)]dy.$

Performing the change of variables $z=y^{-1}x$ (justified by the invariance of
the Haar measure), we obtain that

$\displaystyle\sigma_{A}(i,r,x,\xi)_{mn}$
$\displaystyle=\int_{G}\xi(z^{-1})_{mn}\sum_{[\eta]\in\widehat{G}}d_{\eta}\sum_{j,k=1}^{d_{\eta}}\eta(z)_{jk}\,a(i,r,x,xz^{-1},\eta)_{kj}dz$
$\displaystyle=\sum_{0\leq|\alpha|<N}\frac{1}{\alpha!}\partial_{y}^{\alpha}|_{y=x}\int_{G}\xi(z^{-1})_{mn}q_{\alpha}(z)\sum_{[\eta]\in\widehat{G}}d_{\eta}\sum_{j,k=1}^{d_{\eta}}\eta(z)_{jk}a(i,r,x,y,\eta)_{kj}dz$
$\displaystyle+\int_{G}\xi(z^{-1})_{mn}\sum_{[\eta]\in\widehat{G}}d_{\eta}\sum_{j,k=1}^{d_{\eta}}\eta(z)_{jk}R_{N}(i,r,z,\eta)_{kj}dz$
$\displaystyle=\sum_{0\leq|\alpha|<N}\frac{1}{\alpha!}\int_{G}\xi(z)_{mn}^{*}q_{\alpha}(z)\sum_{[\eta]\in\widehat{G}}d_{\eta}\operatorname{Tr}[\eta(z)\partial_{y}^{\alpha}|_{y=x}a(i,r,x,y,\eta)]dz$
$\displaystyle+\int_{G}\xi(z^{-1})_{mn}\sum_{[\eta]\in\widehat{G}}d_{\eta}\sum_{j,k=1}^{d_{\eta}}\eta(z)_{jk}R_{N}(i,r,z,\eta)_{kj}dz$
$\displaystyle=\sum_{0\leq|\alpha|<N}\frac{1}{\alpha!}\partial_{y}^{\alpha}\Delta_{\xi}^{\alpha}a(i,r,x,y,\xi)_{mn}|_{y=x}$
(7)
$\displaystyle+\sum_{[\eta]\in\widehat{G}}d_{\eta}\sum_{j,k=1}^{n}\int_{G}\xi(z^{-1})_{mn}\eta(z)_{jk}R_{N}(i,r,z,\eta)_{kj}dz,$

where $R_{N}(i,r,z,\eta)_{kj}$ is the remainder of the Taylor expansion in $z$
of order $N\in\operatorname{\mathbb{N}}$ of $a(i,r,x,z,\eta)_{kj}$ centered at
$x$. Therefore, to prove the claim we only need to analyze these last series
of integrals. For $(x,y)\in G\times G$, $1\leq i,r\leq n$, define
$k_{A,x,y}(i,r,\cdot)$ to be the distribution defined by

$a(i,r,x,y,\xi)=\widehat{k_{A,x,y}}(i,r,\xi),\,[\xi]\in\widehat{G}.$

Similarly, denote by $k_{\sigma,x}(i,r,\cdot)$ to be the right convolution
kernels of $A$, that is, the distributions which satisfy

$\sigma_{A}(i,r,x,\xi)=\widehat{k_{\sigma,x}}(i,r,\xi),\,[\xi]\in\widehat{G}.$

Since

$\displaystyle Af(x)_{r}$
$\displaystyle=\int_{G}\sum_{i=1}^{n}f_{i}(y)k_{A,x,y}(i,r,y^{-1}x)dy$
$\displaystyle=\int_{G}\sum_{i=1}^{n}f(xz^{-1})k_{A,x,xz^{-1}}(z)dz,$

and

$Af(x)_{r}=\int_{G}\sum_{i=1}^{n}f_{i}(y)k_{\sigma,x}(i,r,y^{-1}x)=\sum_{i=1}^{n}f_{i}(xz^{-1})k_{\sigma,x}(i,r,z),$

we have that

$k_{A,x,y}(i,r,y^{-1}x)=k_{\sigma,x}(i,r,y^{-1}x),\,k_{A,x,xz^{-1}}(i,r,z)=k_{\sigma,x}(i,r,z),$

for all $1\leq i\leq r,\leq n$, $x,y,z\in G$, in the sense of distributions on
$G$. Now, fix $1\leq i,r\leq n$, and notice that the last term in (7) is the
inverse Fourier transform of

$k_{\sigma,x}(i,r,z)-\sum_{|\alpha|<N}q_{\alpha}(z)\partial_{z}^{\alpha}k_{A,x,xz_{1}^{-1}}(i,r,z)|_{z_{1}=e}.$

Our goal now is to prove that the Fourier transform of the expression above is
in $\mathcal{S}_{\rho,\delta}^{m-(\rho-\delta)N}(G\otimes\text{End}(E_{0}))$
for every $N$ sufficiently big, which we do as follows. For any multi-indices
$\gamma,\beta$, let $M^{\prime}\in 2\operatorname{\mathbb{N}}_{0}$ such that

$M^{\prime}>\rho|\gamma|-\delta|\beta|-m+(\rho-\delta)N,$

and

(8) $\frac{M^{\prime}}{2}>-\rho|\gamma|+\delta|\beta|+m+2(\rho-\delta)N.$

Then

$\displaystyle\left\|\langle\xi\rangle^{\rho|\gamma|-\delta|\beta|-m+(\rho-\delta)N}\Delta^{\gamma}\partial_{x}^{\beta}\left[\sum_{[\eta]\in\widehat{G}}d_{\eta}\sum_{i=1}^{n}\sum_{j,k=1}^{n}\int_{G}\xi(z^{-1})_{mn}\eta(z)_{jk}R_{N}(i,r,z,\eta)_{kj}dz\right]\right\|_{op}$
$\displaystyle\leq\left\|\langle\xi\rangle^{M^{\prime}}\Delta^{\gamma}\partial_{x}^{\beta}\left[\sum_{[\eta]\in\widehat{G}}d_{\eta}\sum_{i=1}^{n}\sum_{j,k=1}^{n}\int_{G}\xi(z^{-1})_{mn}\eta(z)_{jk}R_{N}(i,r,z,\eta)_{kj}dz\right]\right\|_{op}$
$\displaystyle\leq\|(\text{Id}+\mathcal{L})^{\frac{M^{\prime}}{2}}[q_{\gamma}(z)\partial_{x}^{\beta}[k_{\sigma,x}(i,r,z)-\sum_{|\alpha|<N}q_{\alpha}(z)\partial_{z}^{\alpha}k_{A,x,xz_{1}^{-1}}(i,r,z)|_{z_{1}=e}]\|_{L^{1}(G)_{z}}$
(9)
$\displaystyle=\|(\text{Id}+\mathcal{L})^{\frac{M^{\prime}}{2}}[q_{\gamma}(z)\partial_{x}^{\beta}R_{x,N}^{k_{A,x,x}}(i,r,z)]\|_{L^{1}(G)_{z}},$

where

$k_{\sigma,x}(i,r,z)=k_{A,x,xz^{-1}}(i,r,z)=\sum_{|\alpha|<N}q_{\alpha}(z)\partial_{z_{1}}^{\alpha}k_{A,x,xz_{1}^{-1}}(i,r,z)|_{z_{1}=e}+R_{x,N}^{k_{A,x}}(i,r,z)$

for all $x,y,z\in G$ by the Taylor expansion theorem. Moreover, by [5], Lemma
7.4, the remainder satisfies the estimate

$|R_{x,N}(i,r,z)|\leq C|z|^{N}\max_{|\alpha|\leq
N}\|\partial^{\alpha}_{x}k_{A,x,x}(i,r,z)\|_{L^{\infty}(G)_{x}}.$

Now applying the definition of $\mathcal{L}$ and the Leibniz’s rule on (9),
gives us

$\displaystyle\|(\text{Id}+\mathcal{L})^{\frac{M^{\prime}}{2}}[q_{\gamma}(z)\partial_{x}^{\beta}$
$\displaystyle R_{x,N}^{k_{A,x,x}}(i,r,z)]\|_{L^{1}(G)_{z}}$
$\displaystyle\lesssim\sum_{1\leq i_{1}\leq\dots\leq i_{d}\leq
d,\,|\lambda|\leq M^{\prime}}\|X_{i_{1},z}^{\lambda_{1}}\dots
X_{i_{d},z}^{\lambda_{d}}[R_{x,N}^{q_{\gamma}(z)\partial_{x}^{\beta}k_{A,x,x}}(i,r,z)\|_{L^{1}(G)_{z}}.$

By estimates similar to the ones above, we have

$|X_{i_{1},z}^{\lambda_{1}}\dots
X_{i_{d},z}^{\lambda_{d}}[R_{x,N}^{q_{\gamma}(z)\partial_{x}^{\beta}k_{A,x,x}}(i,r,z)]|\lesssim|z|^{N-|\lambda|}\max_{|\alpha|\leq
N-|\lambda|}\|\partial_{z}^{\alpha+\lambda}(q_{\gamma}(z)\partial_{x}^{\beta}k_{A,x,x}(i,r,z))\|_{L^{\infty}(G)_{x}},$

and since
$\\{\partial_{z}^{\alpha+\lambda}(q_{\gamma}(\cdot)\partial_{x}^{\beta}k_{A,x,x}(i,r,\cdot))\\}_{i,r=1}^{n}$
are the right-convolution kernels of a pseudo-differential operator of order

$s^{\prime}=m+\delta|\beta|+\delta|\alpha|+\delta|\lambda|-\rho|\gamma|,$

by Proposition 6.7 of [5] we have that

$\displaystyle\|\partial_{z}^{\alpha+\lambda}(q_{\gamma}(z)\partial_{x}^{\beta}k_{A,x,x}(i,r,z))\|_{L^{\infty}(G)_{z}}\lesssim|z|^{-\frac{s^{\prime}+d}{\rho}},$

for $1\leq i,r\leq n$. Putting these previous estimates together, one obtains

(10)
$\|(\text{Id}+\mathcal{L})^{\frac{M^{\prime}}{2}}[q_{\gamma}(z)\partial_{x}^{\beta}R_{x,N}^{k_{A,x,x}}(i,r,z)]\|_{L^{1}(G)_{z}}\lesssim\int_{G}|z|^{N-|\lambda|}|z|^{-\frac{s^{\prime}+d}{\rho}},$

and the integral on the right-hand-side is finite if

$\displaystyle\rho(|\lambda|-N)+s^{\prime}+d<\rho d.$

But indeed, for $N$ big enough, we have $M^{\prime}\lesssim(\rho-\delta)N$. By
choosing $M^{\prime}$ so that $(\rho-\delta)N/2\lesssim M^{\prime}$ also, we
have that $(\rho-\delta)N\asymp M^{\prime}$. Hence, using that
$|\alpha|+|\lambda|\leq N$ and (8) yields

$\displaystyle\rho(|\lambda|-N)+s^{\prime}+d$
$\displaystyle=\rho|\lambda|-\rho
d+m+\delta|\beta|+\delta|\alpha|+\delta|\lambda|-\rho|\gamma|+d$
$\displaystyle<\rho|\lambda|-\rho
N+m+\delta|\beta|+\rho|\alpha|+\rho|\lambda|-\rho|\gamma|+n$
$\displaystyle=\rho|\lambda|-\rho(N-|\alpha|-|\lambda|)+m+\delta|\beta|-\rho|\gamma|+d$
$\displaystyle\leq\rho|\lambda|+m+\delta|\beta|-\rho|\gamma|+d$
$\displaystyle\leq\rho
M^{\prime}+\frac{M^{\prime}}{2}-2N(\rho-\delta)+d\leq\frac{3M^{\prime}}{2}-2N(\rho-\delta)+d$
$\displaystyle\asymp\frac{3(\rho-\delta)N}{2}-2N(\rho-\delta)+d=-\frac{(\rho-\delta)N}{2}+d.$

By choosing $N\geq N_{0}\geq 2d(\rho-1)/(\rho-\delta)$, we get that
$|\lambda|-N+s^{\prime}+d<\rho d$, which by (10) and the previous inequalities
proves that the remainder term in the asymptotic expansion is in
$\mathcal{S}_{\rho,\delta}^{m-(\rho-\delta)N}(G\otimes\text{End}(E_{0}))$, for
all such $N$. ∎

###  2.2. Even and odd functions on compact Lie groups

Recall that $f\in C^{\infty}(G)$ is called central if $f(xy)=f(yx)$, for every
$x,y\in G$. Following [10], we will say $f\in C^{\infty}(G)$ is even if it is
invariant under inversions, that is: $f(x^{-1})=f(x)$ for every $x\in G$.
Similarly, we will say that $f\in C^{\infty}(G)$ is odd if $f(x^{-1})=-f(x)$,
for every $x\in G$. The following results are then easy to verify:

######  Lemma 2.10.

Let $f\in C^{\infty}(G)$ be central and even. Then for any left-invariant
vector field $X$, the function $Xf$ is odd. Moreover, if $g\in C^{\infty}$ is
odd, then $fg$ is also odd.

######  Lemma 2.11.

Let $f\in C^{\infty}(G)$ be odd. Then $\int_{G}f(x)dx=0$.

##  3\. Main Results

###  3.1. Sharp Gårding inequality on compact Lie groups

Here we present the main result of this paper. Its proof will be then carried
out through several lemmas.

######  Theorem 3.1.

Let $G$ be a compact Lie group, $\dim(G)=d$, $E_{0}$ an finite a
$n$-dimensional $\mathbb{C}$-vector space. Let
$A=\text{Op}(\sigma_{A})\in\Psi^{m}_{\rho,\delta}(G\otimes\text{End}(E_{0}))$,
$0\leq\delta<\rho\leq 1$, be such that its matrix-valued symbol
$\sigma_{A}(x,\xi)$ is positive semi-definite for every $(x,[\xi])\in
G\times\widehat{G}$, in the sense that
$B^{*}\sigma_{A}(x,\xi)B\in\mathbb{C}^{d_{\xi}\times d_{\xi}}$ is positive
semi-definite for all $B\in(\mathbb{C}^{d_{\xi}\times d_{\xi}})^{n\times 1}$.
Equivalently, this means that for any
$B_{1},\dots,B_{n}\in\mathbb{C}^{d_{\xi}\times 1}$, the inequality

$\displaystyle\sum_{i,r=1}^{n}\overline{B_{i}}^{T}\sigma_{A}(i,r,x,\xi)B_{r}\geq
0,$

holds for all $(x,[\xi])\in G\times\widehat{G}$. Then there exists $C>0$ such
that, for every $u\in C^{\infty}(G,E_{0})$ we have

(11)
$\text{Re}(Au,u)_{L^{2}(G,E_{0})}\geq-C\|u\|^{2}_{H^{(m-(\rho-\delta))/2}(G,E_{0})}.$

######  Remark 3.2.

In the previous statement and throughout the rest of this paper, for a column-
vector of matrices $B\in(\mathbb{C}^{d_{\xi}\times d_{\xi}})^{n\times 1}$ we
denote by $B^{*}\in(\mathbb{C}^{d_{\xi}\times d_{\xi}})^{1\times n}$ the row-
vector of matrices given by

$B^{*}=\left(B_{1}^{*},\dots,B_{n}^{*}\right).$

First, notice that if $Q:H^{\frac{m-(\rho-\delta)}{2}}(G,E_{0})\to
H^{-\frac{m-(\rho-\delta)}{2}}(G,E_{0})$ is a bounded linear operator, we have

$\displaystyle\text{Re}(Qu,u)_{L^{2}(G,E_{0})}$
$\displaystyle\geq-|(Qu,u)_{L^{2}(G,E_{0})}|$
$\displaystyle\geq-\|Qu\|_{H^{-\frac{m-(\rho-\delta)}{2}}(G,E_{0})}\|u\|_{H^{\frac{m-(\rho-\delta)}{2}}(G,E_{0})}$
$\displaystyle\geq-\|Q\|_{op}\|u\|_{H^{\frac{m-(\rho-\delta)}{2}}(G,E_{0})}^{2}.$

Hence, Theorem 3.1 will follow once we show that $A$ can be written as
$A=P+Q$, where $P$ is positive and $Q$ is as above.
First, following the ideas in [10] and [3], we construct an auxiliary function
$w_{\xi}:G\to\mathbb{C}$, for each $\xi\in\text{Rep}(G)$.
We can assume $G$ is a closed subgroup of $\text{GL}(N,\mathbb{R})$ for some
$N\in\operatorname{\mathbb{N}}$. Then its Lie algebra
$\mathfrak{g}\subset\operatorname{\mathbb{R}}^{N\times N}$ is a
$d$-dimensional vector subspace such that $[A,B]\doteq AB-BA\in\mathfrak{g}$,
for every $A,B\in\mathfrak{g}$. Let $U\subset G$ and $V\subset\mathfrak{g}$ be
neighbourhoods of the identity $\text{Id}=e\in G$ and $0\in\mathfrak{g}$,
respectively, so that the matrix exponential mapping is a diffeomorphism
$\exp:V\to U$. Without loss of generality, we may assume that $V$ is the open
ball $V=B(0,r)=\\{z\in\operatorname{\mathbb{R}}^{d}|\,|z|<r\\}$, of radius
$r>0$. Let $\phi:[0,\infty)\to[0,\infty)$ be a smooth function such that the
mapping

$\displaystyle\mathfrak{g}$ $\displaystyle\to\operatorname{\mathbb{R}}$
$\displaystyle z$ $\displaystyle\mapsto\phi(|z|)$

is supported in $V$ and such that $\phi(s)=1$ for all sufficiently small
$s>0$. For every $\xi\in\text{Rep}(G)$ define

$\displaystyle w_{\xi}:G$ $\displaystyle\to\operatorname{\mathbb{R}}$
$\displaystyle x$
$\displaystyle\mapsto\phi(|\exp^{-1}(x)|\langle\xi\rangle^{\frac{\rho+\delta}{2}})\psi(\exp^{-1}(x))\langle\xi\rangle^{\frac{d(\rho+\delta)}{4}},$

where $\psi(y)\doteq C_{0}|\det D\exp(y)|^{-\frac{1}{2}}f(y)^{-\frac{1}{2}}$,
for every $y\in\mathfrak{g}\cong\operatorname{\mathbb{R}}^{d}$, $D\exp$ is the
Jacobi matrix of the mapping $\exp$, $f$ is the density with respect to the
Lebesgue measure of the Haar measure on $G$ pulled back to
$\mathfrak{g}\cong\operatorname{\mathbb{R}}^{d}$ by the exponential mapping,
and with
$C_{0}=(\int_{\operatorname{\mathbb{R}}^{d}}\phi(|z|)dz)^{-\frac{1}{2}}$.

The following lemma states the main properties of $w_{\xi}$ that will be used
in this paper, and its proof can be found in [3].

######  Lemma 3.3.

The functions $w_{\xi}$ defined above are smooth, for every
$\xi\in\text{Rep}(G)$. Moreover, they satisfy the following properties:

  1. (1)

$w_{\xi}(e)=C_{0}\langle\xi\rangle^{\frac{d(\rho+\delta)}{4}}$;

  2. (2)

$w_{\xi}$ is central and inversion invariant;

  3. (3)

$\text{dist}(x,e)\sim|\exp^{-1}(x)|\lesssim\operatorname{\langle\xi\rangle}^{-\frac{\rho+\delta}{2}}$
on the support of $w_{\xi}$;

  4. (4)

$\|w_{\xi}\|_{L^{2}(G)}=1$;

  5. (5)

$(x,[\xi])\mapsto
w_{\xi}(x)\text{Id}_{d_{\xi}}\in\mathcal{S}^{d(\rho+\delta)/4}_{\rho,(\rho+\delta)/2}(G)$,

for every $\xi\in\text{Rep}(G)$.

######  Lemma 3.4.

Let $\sigma_{A}$ be the matrix valued symbol of $A$ as in Theorem 3.1. Define
the amplitude

(12)
$p(i,r,x,y,\xi)\doteq\int_{G}w_{\xi}(xz^{-1})w_{\xi}(yz^{-1})\sigma_{A}(i,r,z,\xi)dz\in(\mathbb{C}^{d_{\xi}\times
d_{\xi}})^{n\times n},$

for every $1\leq i,r\leq n,\,x,y\in G$, $[\xi]\in\widehat{G}$, where $w_{\xi}$
is as in Lemma 3.3. Then
$p\in\mathcal{A}_{\rho,\frac{\rho+\delta}{2}}^{m}(G\otimes\text{End}(E_{0}))$
and the linear operator $P:C^{\infty}(G,E_{0})\to C^{\infty}(G,E_{0})$ given
by

$(Pu)_{r}(x)=\int_{G}\sum_{i=1}^{n}\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}(\xi(y^{-1}x)p(i,r,x,y,\xi)u_{i}(y))dy,\quad
1\leq r\leq n,$

for every $x\in G$ is a well defined pseudo-differential operator of order
$m$, and also a positive linear operator on $L^{2}(G,E_{0})$.

###### Proof.

First, we verify that
$p\in\mathcal{A}_{\rho,\frac{\rho+\delta}{2}}^{m}(G\otimes\text{End}(E_{0}))$.
Indeed, note that by the Leibniz’s rule, for any multi-indices
$\alpha,\beta,\gamma$, the matrix
$\partial_{x}^{\beta}\partial_{y}^{\gamma}\Delta_{\xi}^{\alpha}p(i,r,x,y,\xi)$
is given by a sum of terms of the form

$\int_{G}(\Delta_{\xi}^{\eta}\partial_{x}^{\beta}w_{\xi}(xz^{-1}))(\Delta_{\xi}^{\lambda}\partial_{y}^{\gamma}w_{\xi}(yz^{-1}))(\Delta_{\xi}^{\mu}\sigma_{A}(i,r,z,\xi))dz$

where $|\eta+\lambda+\mu|\geq|\alpha|$. Also, due to properties of $w_{\xi}$
from Lemma 3.3, we have

$\displaystyle\|(\Delta_{\xi}^{\eta}\partial_{x}^{\beta}w_{\xi}(xz^{-1}))(\Delta_{\xi}^{\lambda}\partial_{y}^{\gamma}w_{\xi}(yz^{-1}))\|_{op}$
$\displaystyle\lesssim\operatorname{\langle\xi\rangle}^{d\frac{(\rho+\delta)}{2}-\rho(|\eta|+|\lambda|)+\frac{(\rho+\delta)}{2}(|\beta|+|\gamma|)}.$

Therefore, since $\text{supp}(w_{\xi})$ is contained in a set of measure
proportional to
$\operatorname{\langle\xi\rangle}^{-d\frac{(\rho+\delta)}{2}}$, we can
estimate

$\displaystyle\left\|\int_{G}(\Delta_{\xi}^{\eta}\partial_{x}^{\beta}w_{\xi}(xz^{-1}))(\Delta_{\xi}^{\lambda}\partial_{y}^{\gamma}w_{\xi}(yz^{-1}))(\Delta_{\xi}^{\mu}\sigma_{A}(i,r,z,\xi))dz\right\|_{op}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\lesssim\operatorname{\langle\xi\rangle}^{d\frac{(\rho+\delta)}{2}-\rho(|\eta|+|\lambda|)+\frac{(\rho+\delta)}{2}(|\beta|+|\gamma|)-d\frac{(\rho+\delta)}{2}+m-\rho|\mu|}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\operatorname{\langle\xi\rangle}^{m-\rho|\alpha|+\frac{(\rho+\delta)}{2}(|\beta|+|\gamma|)}$

and so
$\|\partial_{x}^{\beta}\partial_{y}^{\gamma}\Delta_{\xi}^{\alpha}p(i,r,x,y,\xi)\|_{op}\lesssim\operatorname{\langle\xi\rangle}^{m-\rho|\alpha|+\frac{(\rho+\delta)}{2}(|\beta|+|\gamma|)}$
also, which proves the first claim. Next, notice that

$\displaystyle(Pu,u)_{L^{2}(G,E_{0})}$ $\displaystyle=\int_{G}\langle
Pu(x),u(x)\rangle_{E_{0}}dx$
$\displaystyle=\int_{G}\int_{G}\sum_{r,i=1}^{n}\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}[\xi(y)^{*}\xi(x)p(i,r,x,y,\xi)u_{i}(y)]\mathop{dy}\overline{u_{r}(x)}\mathop{dx}.$

Substituting $p(i,r,x,y,\xi)$ by its definition in the previous expression,
one obtains

(13)
$\displaystyle\int_{G}\sum_{[\xi]\in\widehat{G}}\sum_{r,i=1}^{n}\operatorname{Tr}[\xi(x)\int_{G}w_{\xi}(xz^{-1})w_{\xi}(yz^{-1})\sigma_{A}(i,r,z,\xi)\mathop{dz}u_{i}(y)\xi(y)^{*}]\mathop{dy}\overline{u_{r}(x)}\mathop{dx}.$

Let

$M(i,z,\xi)\doteq\int_{G}w_{\xi}(yz^{-1})\xi(yz^{-1})^{*}u_{i}(y)dy\in\mathbb{C}^{d_{\xi}\times
d_{\xi}}.$

Then (13) can be written as

$\displaystyle\int_{G}\sum_{i,r=1}^{n}\sum_{[\xi]\in\widehat{G}}d_{\xi}\operatorname{Tr}[M(i,z,\xi)^{*}\sigma_{A}(i,r,z,\xi)M(r,z,\xi)]dz,$

which is non-negative by our hypothesis since

$\displaystyle\operatorname{Tr}[M(i,z,\xi)^{*}\sigma_{A}(i,r,z,\xi)M(r,z,\xi)]$
$\displaystyle=\sum_{k=1}^{d_{\xi}}f_{k}^{*}M(i,z,\xi)^{*}\sigma_{A}(i,r,z,\xi)M(r,z,\xi)f_{k}\geq
0,$

as $M(r,z,\xi)f_{k}\in\mathbb{C}^{d_{\xi}\times 1}$, where
$\\{f_{k}\\}_{k=1}^{d_{\xi}}$ is any orthonormal basis of column vectors in
$\mathbb{C}^{d_{\xi}}$. ∎

######  Lemma 3.5.

Following the previous notation, we have that $(i,r,x,[\xi])\mapsto
p(i,r,x,x,\xi)-\sigma_{A}(x,\xi)$ is the symbol of a pseudo-differential
operator bounded from $H^{s}(G,E_{0})$ to
$H^{s-(m-(\rho-\delta))}(G,E_{0})$, for any $s\in\operatorname{\mathbb{R}}$.

###### Proof.

By Theorem 2.6, it is enough to show that

$(i,r,x,[\xi])\mapsto
p(i,r,x,x,\xi)-\sigma_{A}(i,r,x,\xi)\in\mathcal{S}^{m-(\rho-\delta)}_{\rho,\delta}(G\otimes\text{End}(E_{0})).$

Notice that, by Lemma 3.3,

$\displaystyle p(i,r,x,x,\xi)-\sigma_{A}(i,r,x,\xi)$
$\displaystyle=\int_{G}w_{\xi}(z)^{2}\sigma_{A}(i,r,xz^{-1},\xi)dz-\sigma_{A}(i,r,x,\xi)$
$\displaystyle=\int_{G}w_{\xi}(z)^{2}(\sigma_{A}(i,r,xz^{-1},\xi)-\sigma_{A}(i,r,x,\xi))dz.$

Using the Taylor expansion of $\sigma_{A}(i,r,xz^{-1},\xi)$ at $x$ we can
write

$\displaystyle\sigma_{A}(i,r,xz^{-1},\xi)=\sigma_{A}(i,r,x,\xi)+\sum_{|\gamma|=1}\partial_{x}^{\gamma}\sigma_{A}(i,r,x,\xi)q_{\gamma}(z)+R_{x}(i,r,z,\xi),$

where
$R_{x}\in\mathcal{S}_{\rho,\delta}^{m+2\delta}(G\otimes\text{End}(E_{0}))$ is
the Taylor remainder of order $2$ of $\sigma_{A}(\cdot,\cdot,x,\cdot)$. Notice
that we can choose the “polynomials” $q_{\gamma}$ so that they are odd for all
$|\gamma|=1$, and using that $w_{\xi}$ is even, we can conclude that

$\int_{G}w_{\xi}^{2}(z)q_{\gamma}(z)dz=0,$

for all $|\gamma|=1$. Hence, for multi-indices $\alpha,\beta$:

$\displaystyle\Delta^{\alpha}_{\xi}\partial_{x}^{\beta}(p(i,r,x,x,\xi)-\sigma_{A}(i,r,x,\xi))=\Delta^{\alpha}_{\xi}\int_{G}w_{\xi}(z)^{2}\partial_{x}^{\beta}R_{x}(i,r,z,\xi)\mathop{dz}.$

Applying the Leibniz rule, we can write the expression above as a sum of terms
of the form

$\displaystyle\int_{G}\partial_{x}^{\beta}\Delta_{\xi}^{\alpha_{1}}R_{x}(i,r,z,\xi)\Delta_{\xi}^{\alpha_{2}}w_{\xi}(z)\Delta_{\xi}^{\alpha_{3}}w_{\xi}(z)dz,$

where $|\alpha_{1}+\alpha_{2}+\alpha_{3}|\geq|\alpha|$. By Lemma 7.4 of [5],
the remainder satisfies

$\|\Delta_{\xi}^{\alpha_{1}}\partial_{x}^{\beta}R_{x}(i,r,z,\xi)\|_{op}\leq
C|z|^{2}\max_{|\gamma|\leq 2}\sup_{x\in
G}\|\Delta_{\xi}^{\alpha_{1}}\partial_{x}^{\gamma}\partial_{x}^{\beta}\sigma_{A}(i,r,x,\xi)\|_{op}\lesssim|z|^{2}\operatorname{\langle\xi\rangle}^{m-\rho|\alpha_{1}|+\delta(2+|\beta|)},$

for all $z\in G$, $|z|\lesssim\langle\xi\rangle^{-\frac{\rho+\delta}{2}}$ on
$\text{supp}(w_{\xi})$ by Lemma 3.3, and
$|\text{supp}(w_{\xi})|\lesssim\langle\xi\rangle^{-d\frac{(\rho+\delta)}{2}}$,
we conclude that

$\displaystyle\left\|\int_{G}\partial_{x}^{\beta}\Delta_{\xi}^{\alpha_{1}}R_{x}(i,r,z,\xi)\Delta_{\xi}^{\alpha_{2}}w_{\xi}(z)\Delta_{\xi}^{\alpha_{3}}w_{\xi}(z)dz\right\|_{op}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\lesssim\operatorname{\langle\xi\rangle}^{-\rho-\delta+m-\rho|\alpha_{1}|+\delta(2+|\beta|)+d\frac{(\rho+\delta)}{4}-\rho|\alpha_{2}|+d\frac{(\rho+\delta)}{4}-\rho|\alpha_{3}|-d\frac{(\rho+\delta)}{2}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\operatorname{\langle\xi\rangle}^{m-(\rho-\delta)-\rho|\alpha|+\delta|\beta|},$

which implies

$\|\Delta^{\alpha}_{\xi}\partial_{x}^{\beta}(p(i,r,x,x,\xi)-\sigma_{A}(i,r,x,\xi))\|_{op}\lesssim\operatorname{\langle\xi\rangle}^{m-(\rho-\delta)-\rho|\alpha|+\delta|\beta|}$

proving the claim. ∎

######  Lemma 3.6.

Let $\sigma_{P}$ be the symbol of the pseudo-differential operator $P$ defined
in Lemma 3.4. Then the pseudo-differential operator with symbol
$\\{\sigma_{P}(i,r,x,\xi)-p\\}_{i,r=1}^{n}$ is bounded from $H^{s}(G,E_{0})$
to $H^{s-(m-(\rho-\delta))}(G,E_{0})$, for any
$s\in\operatorname{\mathbb{R}}$.

###### Proof.

As in the proof of the previous lemma, it is enough to prove that

$(i,r,x,[\xi])\mapsto\sigma_{P}(i,r,x,\xi)-p(i,r,x,x,\xi)\in\mathcal{S}_{\rho,\frac{(\rho+\delta)}{2}}^{m-(\rho-\delta)}(G\otimes\text{End}(E_{0})).$

By Proposition 2.9, we have the asymptotic expansion

$\sigma_{P}(i,r,x,\xi)\sim\sum_{\alpha\geq
0}\frac{1}{\alpha!}\Delta_{\xi}^{\alpha}\partial_{y}^{\alpha}p(i,r,x,y,\xi)|_{y=x},$

with the properties specified in Proposition 2.9. For now, recalling the
formula (12), notice that after a change of variables $z^{-1}x\mapsto z$ one
obtains

$\Delta_{\xi}^{\alpha}\partial_{y}^{\alpha}p(i,r,x,y,\xi)|_{y=x}=\Delta_{\xi}^{\alpha}\int_{G}w_{\xi}(z)\partial_{z}^{\alpha}w_{\xi}(z)\sigma_{A}(i,r,z^{-1}x,\xi)\mathop{dz}.$

Let $N\in\operatorname{\mathbb{N}}$ to be chosen later. Define

$S_{N}(i,r,x,\xi)=\sigma_{P}(i,r,x,\xi)-p(i,r,x,x)-R_{N}(i,r,x,\xi),$

where
$R_{N}\in\mathcal{S}_{\rho,\delta}^{m-\frac{(\rho+\delta)}{2}(N+1)}(G\otimes\text{End}(E_{0}))$
is the remainder term given by the asymptotic expansion above. Then applying
the Taylor expansion to the formula above yields

$\displaystyle S_{N}(i,r,x,\xi)$ $\displaystyle=\sum_{1\leq|\alpha|\leq
N}\Delta_{\xi}^{\alpha}\int_{G}w_{\xi}(z)\partial_{z}^{\alpha}w_{\xi}(z)\sigma_{A}(i,r,x,\xi)\mathop{dz}$
$\displaystyle+\sum_{1\leq|\alpha|\leq
N}\Delta_{\xi}^{\alpha}\int_{G}w_{\xi}(z)\partial_{z}^{\alpha}w_{\xi}(z)R_{\sigma,1}(i,r,z^{-1}x,\xi)\mathop{dz}$
$\displaystyle\doteq I(i,r,x,\xi)+J(i,r,x,\xi),$

where $R_{\sigma,1}$ is the remainder in the Taylor expansion of order $1$ of
$\sigma_{A}$, centered at $x$. Therefore, if we can prove that $I,J$ and
$R_{N}$ all belong to the specified symbol class, we will have proved the
lemma. Let is examine each one at a time. First, we see that the first term in
the sums in $I$ and $J$ (of order 0) vanishes since

$\int_{G}w_{\xi}(z)\partial_{z}^{\alpha}w_{\xi}(z)\mathop{dz}=0,$

for $|\alpha|=1$ because the functions $w_{\xi}$ and
$\partial^{\alpha}w_{\xi}$ are even and odd, respectively, by Lemmas 2.10,
2.11 and 3.3. In particular, $I$ is given by

$\displaystyle\sum_{2\leq|\alpha|\leq
N}\Delta_{\xi}^{\alpha}\int_{G}w_{\xi}(z)\partial_{z}^{\alpha}w_{\xi}(z)$
$\displaystyle\mathop{dz}\sigma_{A}(i,r,x,\xi)$
$\displaystyle=\sum_{2\leq|\alpha|\leq
N}\sum_{\kappa,\lambda,\mu}C_{\kappa,\lambda,\mu}\int_{G}(\Delta_{\xi}^{\kappa}w_{\xi}(z))(\Delta_{\xi}^{\lambda}\partial_{z}^{\alpha}w_{\xi}(z))\mathop{dz}\Delta_{\xi}^{\mu}\sigma_{A}(i,r,x,\xi),$

for some constants $C_{\kappa,\lambda,\mu}$, where the sum is taken over
$|\kappa+\lambda+\mu|\geq|\alpha|$, where here we have used the “Leibniz’s”
rule for difference operators, that is, Lemma 2.1. Recalling that by Lemma 3.3
$(x,[\xi])\mapsto w_{\xi}(x)\text{Id}_{d_{\xi}}$ is in
$\mathcal{S}_{\rho,(\rho+\delta)/2}^{d(\rho+\delta)/4}(G)$, and using that
$\sigma_{A}\in\mathcal{S}_{\rho,\delta}^{m}(G\otimes\text{End}(E_{0}))$, we
get that

(14)
$\int_{G}|(\Delta_{\xi}^{\kappa}w_{\xi}(z))(\Delta_{\xi}^{\lambda}\partial_{z}^{\alpha}w_{\xi}(z))|\mathop{dz}\lesssim\operatorname{\langle\xi\rangle}^{d\frac{(\rho+\delta)}{2}-\rho(|\kappa|+|\lambda|)+|\alpha|\frac{(\rho+\delta)}{2}}\operatorname{\langle\xi\rangle}^{-d\frac{(\rho+\delta)}{2}}=\operatorname{\langle\xi\rangle}^{-\rho(|\kappa|+|\lambda|)+|\alpha|\frac{(\rho+\delta)}{2}},$

where we have taken into account that the support of $z\mapsto w_{\xi}(z)$ is
contained in a set of measure
$\sim\operatorname{\langle\xi\rangle}^{-d\frac{(\rho+\delta)}{2}}$, by Lemma
3.3, and that taking differences in $\xi$ does not increase the support in
$z$. Thus we get

$\displaystyle\|I(i,r,x,\xi)\|_{op}$
$\displaystyle\lesssim\sum_{2\leq|\alpha|\leq
N}\sum_{\kappa,\lambda,\mu}C_{\kappa,\lambda,\mu}\operatorname{\langle\xi\rangle}^{-\rho(|\kappa|+|\lambda|)+|\alpha|\frac{(\rho+\delta)}{2}}\operatorname{\langle\xi\rangle}^{m-\rho|\mu|}$
$\displaystyle\lesssim\sum_{2\leq|\alpha|\leq
N}\operatorname{\langle\xi\rangle}^{m-\rho|\alpha|+|\alpha|\frac{(\rho+\delta)}{2}}\lesssim\operatorname{\langle\xi\rangle}^{m-(\rho-\delta)},$

since

$\displaystyle-\rho|\alpha|+|\alpha|\frac{(\rho+\delta)}{2}=-|\alpha|\frac{(\rho-\delta)}{2}\leq-(\rho-\delta).$

Applying a similar argument to
$\Delta_{\xi}^{\gamma}\partial_{x}^{\beta}I(i,r,x,\xi)$ we obtain the
respective decay estimates, which allow us to conclude that
$I\in\mathcal{S}_{\rho,\frac{(\rho+\delta)}{2}}^{m-(\rho-\delta)}(G\otimes\text{End}(E_{0}))$,
as desired. We now consider the term $J$. In this case,

$\|J(i,r,x,\xi)\|_{op}\lesssim\sum_{1\leq|\alpha|\leq
N}\sum_{\kappa,\lambda,\mu}\int_{G}\|(\Delta_{\xi}^{\kappa}w_{\xi}(z))(\Delta_{\xi}^{\lambda}\partial_{z}^{\alpha}w_{\xi}(z))(\Delta_{\xi}^{\mu}R_{\sigma,1}(i,r,z^{-1}x,\xi)\|_{op}\mathop{dz}$

where again we have used the Leibniz’s rule for difference operators, and the
middle sum is over a finite set satisfying $|\kappa+\lambda+\mu|\geq|\alpha|$.
Using Lemma 3.3 and the estimates for the remainder in the Taylor expansion,
we get that

$\displaystyle\|J(i,r,x,\xi)\|_{op}$
$\displaystyle\lesssim\sum_{1\leq|\alpha|\leq
N}\sum_{\kappa,\lambda,\mu}\int_{\text{supp}(w_{\xi})}\operatorname{\langle\xi\rangle}^{d\frac{(\rho+\delta)}{2}-\rho(|\kappa|+|\lambda|)+|\alpha|\frac{(\rho+\delta)}{2})}|z|^{1}\mathop{dz}\times$
$\displaystyle\times\max_{|\gamma|\leq
1}\|\|\Delta_{\xi}^{\mu}\partial_{y}^{\gamma}\sigma_{A}(i,r,y,\xi)\|_{L^{\infty}(G)_{y}}\|_{op}$
$\displaystyle\lesssim\sum_{1\leq|\alpha|\leq
N}\operatorname{\langle\xi\rangle}^{d\frac{(\rho+\delta)}{2}-\rho(|\alpha|)+|\alpha|\frac{(\rho+\delta)}{2}-\frac{(\rho+\delta)}{2}-d\frac{(\rho+\delta)}{2}+m+\delta}$
$\displaystyle\lesssim\operatorname{\langle\xi\rangle}^{m-\frac{(\rho-\delta)}{2}-\frac{(\rho-\delta)}{2}}=\operatorname{\langle\xi\rangle}^{m-(\rho-\delta)}.$

Again, applying a similar argument to
$\Delta_{\xi}^{\gamma}\partial_{x}^{\beta}J(i,r,x,\xi)$ we obtain the
respective decay estimates, which allow us to conclude that
$J\in\mathcal{S}_{\rho,\frac{(\rho+\delta)}{2}}^{m-(\rho-\delta)}(G\otimes\text{End}(E_{0}))$,
as desired. It only remains necessary to study the remainder $R_{N}$. In this
case, by Proposition 2.9, we have that
$R_{N}\in\mathcal{S}_{\rho,\frac{\rho+\delta}{2}}^{m-\frac{(\rho+\delta)}{2}(N+1)}(G\otimes\text{End}(E_{0}))$.
By taking $N$ sufficiently large, we obtain that $R_{N}$ belongs to the
desired symbol class, which concludes the proof.∎

###### Proof of Theorem 3.1.

Let $Q=A-P$, with the operator $P$ as in Lemma 3.4. Let $u\in
C^{\infty}(G,E_{0})$. Then $A=P+Q$ and the positivity of $P$ implies

$\displaystyle\operatorname{\text{Re}}(Au,u)_{L^{2}(G,E_{0})}$
$\displaystyle=\operatorname{\text{Re}}(Pu,u)_{L^{2}(G,E_{0})}+\operatorname{\text{Re}}(Qu,u)_{L^{2}(G,E_{0})}$
$\displaystyle\geq\operatorname{\text{Re}}(Qu,u)_{L^{2}(G,E_{0})}.$

Let now $P_{0}=\text{Op}(\\{p(i,r,x,x,\xi)_{i,r=1}^{n}\\})$. Writing
$Q=(A-P_{0})+(P_{0}-P)$, we have

$\sigma_{A-P_{0}}(i,r,x,\xi)=\sigma_{A}(i,r,x,\xi)-p(i,r,x,x,\xi)$

and

$\sigma_{P_{0}-P}(i,r,x,\xi)=p(i,r,x,x,\xi)-\sigma_{P}(i,r,x,\xi).$

Consequently, both $A-P_{0}$ and $P_{0}-P$ are bounded from
$H^{\frac{m-(\rho-\delta)}{2}}(G,E_{0})$ to
$H^{-\frac{m-(\rho-\delta)}{2}}(G,E_{0})$ by Lemmas 3.5 and 3.6, respectively.
It follows that $Q$ is also bounded between these spaces so that

$\displaystyle|\operatorname{\text{Re}}(Qu,u)_{L^{2}(G,E_{0})}|$
$\displaystyle\leq\|Qu\|_{H^{-\frac{m-(\rho-\delta)}{2}}(G,E_{0})}\|u\|_{H^{\frac{m-(\rho-\delta)}{2}}(G,E_{0})}$
$\displaystyle\lesssim\|u\|_{H^{\frac{m-(\rho-\delta)}{2}}(G,E_{0})}^{2},$

completing the proof of Theorem 3.1. ∎

This result yields the sharp Gårding inequality for homogeneous vector bundles
over compact manifolds as a corollary. Before stating it precisely, we will
recall the quantisation on homogeneous vector bundles as introduced in the
setting of compact homogeneous manifolds, [4]. But firs, we recall the setting
of compact homogeneous manifolds, also following [13].

###  3.2. Sharp Gårding inequality on compact homogeneous vector bundles

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 $x\in X$, $p(s(x))=p(x)$. 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

(15)
$\|s\|_{L^{q}(E)}\doteq\left(\smallint_{X}\|s(x)\|_{E_{x}}^{q}dx\right)^{\frac{1}{q}}<\infty.$

Now, consider $G$ be a compact Lie group and $K$ a closed subgroup of $G$. Let
$M=G/K$ be 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 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 $p:E\to M$. Let $E_{0}=p^{-1}(K)$, be the
fiber at the identity coset. As proved in [1], there exists
$\tau\in\text{Hom}(K,\text{End}(E_{0}))$, and 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. In
[1], Bott shows that $G\times_{\tau}E_{0}$ admits a natural homogeneous vector
bundle structure, and that in fact there exists
$\tau\in\text{Hom}(K,\text{End}(E_{0}))$ such that

$E\cong G\times_{\tau}E_{0}$

as vector bundles. Now consider the vector subspace
$C^{\infty}(G,E_{0})^{\tau}\subset C^{\infty}(G,E_{0})$ given by

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

and likewise $L^{2}(G,E_{0})^{\tau}\subset L^{2}(G,E_{0})$ by

$L^{2}(G,E_{0})^{\tau}=\\{f\in L^{2}(G,E_{0})|f(gk)=\tau(k)^{-1}f(g)\text{ for
a. e. }g,k\in G\\}.$

It can be shown that the bijection $\chi_{\tau}:\Gamma^{\infty}(E)\to
C^{\infty}(G,E_{0})^{\tau}$, given by

$\chi_{\tau}(s)(g)\doteq g^{-1}\cdot s(gK)$

for every $g\in G$ extends to a surjective isometry from $L^{2}(E)$ into
$L^{2}(G,E_{0})^{\tau}$, so that we may identify $\Gamma^{\infty}(E)\cong
C^{\infty}(G,E_{0})^{\tau}$, $L^{2}(E)\cong L^{2}(G,E_{0})^{\tau}$. The
Sobolev space $H^{s}(E)$ for $s\in\operatorname{\mathbb{R}}$ is then defined
as the completion of the set of smooth sections under the norm

$\|u\|_{H^{s}(E)}\doteq\|\chi_{\tau}u\|_{H^{s}(G,E_{0})}.$

Now let $\tilde{A}:\Gamma^{\infty}(E)\to\Gamma^{\infty}(E)$ be a continuous
linear operator. Then it induces a continuous linear map
$A:C^{\infty}(G,E_{0})^{\tau}\to C^{\infty}(G,E_{0})^{\tau}$ by

$A=\chi_{\tau}\circ\tilde{A}\circ\chi_{\tau}^{-1}.$

If $A\in\Psi_{\rho,\delta}^{m}(G\otimes\text{End}(E_{0}))$, we say that
$\tilde{A}\in\Psi^{m}_{\rho,\delta}(E)$, and define its symbol by
$\sigma_{\tilde{A}}\doteq\sigma_{A}$, where $\sigma_{A}$ is the matrix-valued
symbol defined in (2). The quantisation formula (3) then implies

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

######  Corollary 3.7.

Let $p:E\to M=G/K$ be a homogeneous vector bundle over a compact homogeneous
manifold $M$, where $K<G$ are compact Lie groups, $E\cong
G\times_{\tau}E_{0}$. Let $\tilde{A}\in\Psi^{m}_{\rho,\delta}(E)$,
$0\leq\delta<\rho\leq 1$, be such that its matrix-valued symbol
$\sigma_{A}(x,\xi)$ is positive semi-definite for every $(x,[\xi])\in
G\times\widehat{G}$, in the sense of Theorem 3.1. Then there exists $C>0$ such
that

$\operatorname{\text{Re}}(\tilde{A}s,s)_{L^{2}(E)}\geq-C\|s\|^{2}_{H^{\frac{m-(\rho-\delta)}{2}}(E)},$

for every $s\in\Gamma^{\infty}(E)$.

###### Proof.

Let $A=\text{Op}(\sigma_{A})$ be the vector-valued operator associated with
$\tilde{A}$. For $s\in\Gamma^{\infty}(E)$, set $u=\chi_{\tau}s$. Theorem 3.1
implies that there exists $C>0$ such that

$\displaystyle\operatorname{\text{Re}}(Au,u)_{L^{2}(G,E_{0})}$
$\displaystyle\geq-C\|u\|^{2}_{H^{\frac{m-(\rho-\delta)}{2}}(G,E_{0})}$
$\displaystyle=-C\|s\|^{2}_{H^{\frac{m-(\rho-\delta)}{2}}(E)}.$

The result then follows from the fact that

$\operatorname{\text{Re}}(\tilde{A}s,s)_{L^{2}(E)}=\operatorname{\text{Re}}(\chi_{\tau}^{-1}A\chi_{\tau}\chi_{\tau}^{-1}u,\chi_{\tau}^{-1}u)_{L^{2}(E)}=\operatorname{\text{Re}}(A\chi_{\tau}\chi_{\tau}^{-1}u,u)_{L^{2}(G,E_{0})}=\operatorname{\text{Re}}(Au,u)_{L^{2}(G,E_{0})},$

where we have used that $\chi_{\tau}^{-1}:L^{2}(G,E_{0})^{\tau}\to L^{2}(E)$
is an isometry. ∎

Finally, note that if $E=G\times_{\widehat{1}}\mathbb{C}\cong M$ is the
trivial bundle, then we can identify $\Gamma^{\infty}(E)\cong C^{\infty}(M)$,
and $\chi_{\widehat{1}}:C^{\infty}(M)\to C^{\infty}(G)^{K}$ is just the
projective lifting

$\chi_{\widehat{1}}f(g)\equiv\dot{f}(g)\doteq f(gK),\,\forall g\in G.$

The pseudo-differential operator classes and Sobolev spaces of $M$ are then
defined likewise.

######  Corollary 3.8.

Let $M=G/K$ be a compact homogeneous manifold, where $K<G$ are compact Lie
groups. Let $\tilde{A}\in\Psi^{m}_{\rho,\delta}(M),\,0\leq\delta<\rho\leq 1$,
be such that its matrix valued symbol
$\sigma_{A}=\sigma_{A}(x,\xi)\in\mathcal{S}^{m}_{\rho,\delta}(G)$ is positive
semi-definite for every $(x,[\xi])\in G\times\widehat{G}$. Then there exists
$C>0$ such that

$\displaystyle\operatorname{\text{Re}}(\tilde{A}u,u)_{L^{2}(M)}$
$\displaystyle\geq-C\|u\|^{2}_{H^{\frac{m-(\rho-\delta)}{2}}(M)},$

for every $u\in C^{\infty}(M).$

##  4\. Application for vector-valued diffusion problems

In this section we will apply our results to study well-posedness of pseudo-
differential evolution Cauchy problems. Let $T>0$, we are interested in
proving existence and uniqueness for the following Cauchy problem

(16) $\displaystyle\begin{cases}\frac{\partial v}{\partial t}-K(t,x)v=f,\quad
v\in\mathcal{D}^{\prime}((0,T)\times G,E_{0}),\\\ v(0)=u_{0},\end{cases}$

where $u_{0}\in L^{2}(G)$, $f\in L^{2}([0,T],L^{2}(G,E_{0}))$,
$K(t)=K(t,x)\in\Psi^{m}_{\rho,\delta}(G\otimes\text{End}(E_{0}))$,
$0\leq\delta<\rho\leq 1$, $m\leq\rho-\delta$. For that, we will assume
appropriate conditions on $K(t)$ in order to apply our previous results. We
begin with the following proposition:

######  Proposition 4.1.

Let $K(t)=K(t,x)\in\Psi^{m}_{\rho,\delta}(G\otimes\text{End}(E_{0}))$, for
every $0\leq t\leq T$, and some $0\leq\delta<\rho\leq 1$, be a pseudo-
differential operator of order $m\leq\rho-\delta$, such that the matrix valued
symbol $\sigma_{-K}(t,x,\xi)$ is positive semi-definite in the sense of
Theorem 3.1, and depends continuously on $t$. Then if $v\in
C^{1}([0,T],L^{2}(G,E_{0}))\cap C([0,T],H^{\frac{m-(\rho-\delta)}{2}})$, then
there exist $C^{\prime},C^{\prime\prime}>0$ such that

(17)
$\|v(t)\|_{L^{2}(G,E_{0}))}^{2}\leq\left(C^{\prime}\|v(0)\|_{L^{2}(G,E_{0}))}^{2}+C^{\prime\prime}\int_{0}^{T}\|Q(t)v(t)\|_{L^{2}(G,E_{0})}^{2}dt\right),$

and

(18)
$\|v(t)\|_{L^{2}(G,E_{0}))}^{2}\leq\left(C^{\prime}\|v(T)\|_{L^{2}(G,E_{0}))}^{2}+C^{\prime\prime}\int_{0}^{T}\|Q(t)^{*}v(t)\|_{L^{2}(G,E_{0})}^{2}dt\right),$

for every $0\leq t\leq T$, where $Q(t)=\partial_{t}-K(t)$, and $Q(t)^{*}$
denotes the adjoint of $Q(t)$.

###### Proof.

First, define

$f(t)\doteq Q(t)v(t),$

where $Q(t)=\partial_{t}-K(t),\,0\leq t\leq T$. Then

$\displaystyle\frac{d}{dt}\|v(t)\|_{L^{2}(G,E_{0})}^{2}$
$\displaystyle=\left(\frac{dv(t)}{dt},v\right)_{L^{2}(G,E_{0})}+\left(v,\frac{dv(t)}{dt}\right)_{L^{2}(G,E_{0})}$
$\displaystyle=\left(f(t)+K(t)v(t),v\right)_{L^{2}(G,E_{0})}+\left(v,f(t)+K(t)v(t)\right)_{L^{2}(G,E_{0})}$
$\displaystyle=2\operatorname{\text{Re}}(K(t)v(t),v(t))+2\operatorname{\text{Re}}(f(t),v(t)).$

From the parallelogram law

$2\operatorname{\text{Re}}(f(t),v(t))\leq
2\|f(t)\|_{L^{2}(G,E_{0})}^{2}+2\|v(t)\|_{L^{2}(G,E_{0})}^{2}.$

From the sharp Garding inequality (11), and the continuity of the symbol on
$t\in[0,T]$:

$\operatorname{\text{Re}}(-Kv,v)_{L^{2}(G,E_{0})}^{2}\geq-C\|v\|^{2}_{H^{\frac{m-(\rho-\delta)}{2}}(G,E_{0})}\geq-C\|v\|^{2}_{L^{2}(G,E_{0})},$

where the last inequality comes from our restriction on $m$ and the continuous
inclusion $L^{2}(G,E_{0})\hookrightarrow H^{-\epsilon}(G,E_{0})$. Together,
these inequalities then imply

$\frac{d}{dt}\|v(t)\|_{L^{2}(G,E_{0})}^{2}\leq
C^{\prime}\|v(t)\|_{L^{2}(G,E_{0})}^{2}+C^{\prime\prime}\|f(t)\|_{{L^{2}(G,E_{0})}}^{2},$

for $C^{\prime}=(C+2),C^{\prime\prime}=2>0$. Applying Gronwall’s Lemma to this
inequality we obtain the desired estimate. The other estimate follows
analogously, applying the previous analysis to $v(T-\cdot)$ instead of
$v(\cdot)$, $f(T-\cdot)$ instead of $f(\cdot)$, and
$Q(t)^{*}=-\partial_{t}-K(t)^{*}$ instead of $Q(t)$ and using that
$\operatorname{\text{Re}}(K(T-t)^{*})=\operatorname{\text{Re}}(K(T-t))$. ∎

######  Theorem 4.2.

Let $u_{0}\in L^{2}(G,E_{0})$, $f\in L^{2}([0,T],L^{2}(G,E_{0}))$. Let also
$K(t)$ satisfy the conditions of Proposition 4.1. Then there exists a unique
$v\in C^{1}([0,T],L^{2}(G,E_{0}))\cap C([0,T],$
$H^{\frac{m-(\rho-\delta)}{2}}(G,E_{0}))$ which solves the Cauchy problem
(16). Moreover, $v$ satisfies the energy estimate

(19)
$\|v(t)\|_{L^{2}(G,E_{0}))}^{2}\leq\left(C^{\prime}\|v(0)\|_{L^{2}(G,E_{0}))}^{2}+C^{\prime\prime}\int_{0}^{T}\|f(t)\|_{L^{2}(G,E_{0})}^{2}dt\right),$

for every $0\leq t\leq T$.

###### Proof.

We have already proved the estimate (19) for $v\in X_{m}\doteq
C^{1}([0,T],L^{2}(G))\cap$$\\\
C([0,T],H^{\frac{m-(\rho-\delta)}{2}}(G,E_{0}))$. We will now prove that the
solution exists and belongs to this space. Let us again denote
$Q(t)\doteq\partial_{t}-K(t)$. Consider the space

$E\doteq\\{\phi\in C^{1}([0,T],L^{2}(G))\cap
C([0,T],H^{\frac{m-(\rho-\delta)}{2}}(G,E_{0}))|\phi(T)=0\\},$

and also $Q^{*}E\doteq\\{Q^{*}\phi|\phi\in E\\}$. Define the linear functional
$\beta\in(Q^{*}E)^{\prime}$ by

$\beta(Q^{*}\phi)\doteq\int_{0}^{T}(\phi(t),f(t))_{L^{2}(G,E_{0})}dt+(\phi(0),u_{0})_{L^{2}(G,E_{0})}.$

It follows from (18) that for every $\phi\in E$:

(20) $\|\phi(t)\|_{L^{2}(G,E_{0})}^{2}\leq
C^{\prime\prime}\int_{0}^{T}\|Q(t)^{*}\phi(t)\|_{L^{2}(G,E_{0})}^{2}dt.$

Hence

$\displaystyle|\beta(Q^{*}\phi)|$
$\displaystyle\leq\int_{0}^{T}\|f(t)\|_{L^{2}(G,E_{0})}\|\phi(t)\|_{L^{2}(G,E_{0})}dt+\|u_{0}\|_{L^{2}(G,E_{0})}\|\phi(0)\|_{L^{2}(G,E_{0})}$
$\displaystyle\leq\|f\|_{L^{2}([0,T],L^{2}(G,E_{0}))}\|\phi\|_{L^{2}([0,T],L^{2}(G,E_{0}))}+\|u_{0}\|_{L^{2}(G,E_{0})}\|\phi(0)\|_{L^{2}(G,E_{0})}$
$\displaystyle\lesssim_{T}\left(\|f\|_{L^{2}([0,T],L^{2}(G,E_{0}))}+\|u_{0}\|_{L^{2}(G,E_{0})}\right)\|Q^{*}\phi\|_{L^{2}([0,T],L^{2}(G,E_{0}))},$

where the last inequality is partly due to the estimate (20). This shows that
$\beta$ is a bounded functional on $\mathcal{I}\doteq Q^{*}E\cap
L^{2}([0,T],L^{2}(G,E_{0}))$, with the topology induced by the usual norm in
$L^{2}([0,T],L^{2}(G,E_{0}))$. The Hahn-Banach extension theorem then implies
that we may extend $\beta$ to a bounded functional $\tilde{\beta}$ defined on
all $L^{2}([0,T],L^{2}(G,E_{0}))$, and by the Riesz representation theorem for
Hilbert spaces, there exists $v\in L^{2}([0,T],L^{2}(G))$ such that

$\tilde{\beta}(\psi)=(\psi,v)_{L^{2}([0,T],L^{2}(G,E_{0}))},$

for every $\psi\in L^{2}([0,T],L^{2}(G,E_{0}))$. In particular, for
$\psi=Q^{*}\phi\in\mathcal{I}$, we have

$\beta(Q^{*}\phi)=\tilde{\beta}(Q^{*}\phi)=(Q^{*}\phi,v)_{L^{2}([0,T],L^{2}(G,E_{0}))}.$

Notice that we can view $C^{\infty}_{0}((0,T),C^{\infty}(G))$ as a subspace of
$E$, that is

$C^{\infty}_{0}((0,T),C^{\infty}(G))\subset\\{\phi\in
C^{1}([0,T],L^{2}(G))\cap
C([0,T],H^{\frac{m-(\rho-\delta)}{2}}(G,E_{0}))|\phi(T)=0\\},$

and so we have the identity

$\displaystyle(\phi,f)_{L^{2}([0,T],L^{2}(G,E_{0}))}$
$\displaystyle=\int_{0}^{T}(\phi(t),f(t))_{L^{2}(G,E_{0})}dt$
$\displaystyle=\int_{0}^{T}(\phi(t),f(t))_{L^{2}(G,E_{0})}dt+(\phi(0),u_{0})_{L^{2}(G,E_{0})}$
$\displaystyle=\beta(Q^{*}\phi)$
$\displaystyle=(Q^{*}\phi,v)_{L^{2}([0,T],L^{2}(G,E_{0}))},$

for every $\phi\in C^{\infty}_{0}((0,T),C^{\infty}(G))$, as $\phi(0)\equiv 0$.
This implies $v\in\text{Dom}(Q^{**})$, and from $Q^{**}=Q$ we obtain

$(\phi,Qv)=(Q^{*}\phi,v)=(\phi,f),\,\forall\phi\in
C^{\infty}_{0}((0,T),C^{\infty}(G)).$

This implies $Qv=f$. Moreover, we claim that $v(0)=u_{0}$. Indeed, note that
for $\phi\in E$, integration by parts yields

$\displaystyle\int_{0}^{T}\left({\phi(t)},\frac{dv(t)}{dt}\right)dt=-(\phi(0),v(0))-\int_{0}^{T}\left(\frac{d\phi(t)}{dt},v(t)\right)dt.$

On the other hand, it is also true that

$\displaystyle\int_{0}^{T}\left(\phi(t),\frac{dv(t)}{dt}\right)dt$
$\displaystyle=\int_{0}^{T}(\phi(t),f(t)+K(t)v(t))dt$
$\displaystyle=\int_{0}^{T}(\phi(t),f(t))dt+\int_{0}^{T}(K(t)^{*}\phi(t),v(t))dt.$

Combining these two, we get that

$\int_{0}^{T}(\phi(t),f(t))dt+\int_{0}^{T}(K(t)^{*}\phi(t),v(t))dt=-(\phi(0),v(0))-\int_{0}^{T}\left(\frac{d\phi(t)}{dt},v(t)\right)dt,$

so that

$\displaystyle\int_{0}^{T}(\phi(t),f(t))dt+(\phi(0),v(0))$
$\displaystyle=\int_{0}^{T}\left(-\frac{d\phi(t)}{dt}-K(t)^{*}\phi(t),v(t)\right)dt$
$\displaystyle=\int_{0}^{T}(Q(t)^{*}\phi(t),v(t))dt$
$\displaystyle=(Q^{*}\phi,v)=\beta(Q^{*}\phi)$
$\displaystyle=\int_{0}^{T}(\phi(t),f(t))dt+(\phi(0),u_{0}),$

and so $v(0)=u_{0}$. Finally, to prove the uniqueness, assume that $u\in
C^{1}([0,T],L^{2}(G,E_{0}))\cap
C([0,T],H^{\frac{m-(\rho-\delta)}{2}(G,E_{0})})$ is also a solution to (16).
Then the function $w\doteq v-u$ solves the Cauchy problem

$\begin{cases}\frac{dw(t)}{dt}-K(t,x)w=0,\\\ w(0)=0,\end{cases}$

and from Proposition 4.1, $\|w(t)\|_{L^{2}(G,E_{0})}=0$ for all $0\leq t\leq
T$. Hence we have that $v(t,x)=u(t,x)$ for all $t\in[0,T]$ and a.e. $x\in G$.
∎

We may apply the ideas of Corollaries 3.7 and 3.8 to obtain the following
corollary to Theorem 4.2.

######  Corollary 4.3.

Let $M=G/K$ be a compact homogeneous manifold. Let $u_{0}\in L^{2}(M)$, $f\in
L^{2}([0,T],L^{2}(M))$. Let
$\tilde{K}(t)=\tilde{K}(t,x)\in\Psi^{m}_{\rho,\delta}(M)$, for every $0\leq
t\leq T$, and some $0\leq\delta<\rho\leq 1$, be a pseudo-differential operator
of order $m\leq\rho-\delta$, such that the matrix valued symbol
$\sigma_{-K}(t,x,\xi)$ is positive semi-definite for all $t,x,\xi$, and
depends continuously on $t$. Then there exists a unique $v\in
C^{1}([0,T],L^{2}(M))\cap C([0,T],H^{\frac{m-(\rho-\delta)}{2}}(M))$ which
solves the Cauchy problem:

(21) $\displaystyle\begin{cases}\frac{\partial v}{\partial t}-K(t,x)v=f,\\\
v(0,x)=u_{0}(x).\end{cases}$

Moreover, $v$ satisfies the energy estimate

(22)
$\|v(t)\|_{L^{2}(M)}^{2}\leq\left(C^{\prime}\|v(0)\|_{L^{2}(M)}^{2}+C^{\prime\prime}\int_{0}^{T}\|f(t)\|_{L^{2}(M)}^{2}dt\right),$

for every $0\leq t\leq T$.

We exemplify an application of Theorem 4.2 to prove existence and uniqueness
to the following system of differential equations.

Denote by $\Xi=(I+\mathcal{L}_{G})^{1/2}\in\Psi^{1}_{1,0}(G)$ the Bessel
potential of order $1$. Consider the vector-valued Cauchy problem given by

(23) $\begin{cases}{\frac{d}{dt}}\begin{pmatrix}u_{1}(t)\\\
u_{2}(t)\end{pmatrix}-\begin{pmatrix}-\Xi&-\Xi\\\
-\Xi&-\Xi\end{pmatrix}\begin{pmatrix}u_{1}(t)\\\
u_{2}(t)\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix},\\\
\begin{pmatrix}u_{1}(0)\\\ u_{2}(0)\end{pmatrix}=\begin{pmatrix}u_{10}\\\
u_{20}\end{pmatrix}.\end{cases}$

Viewing the matrix $K(t)\equiv K=\begin{pmatrix}-\Xi&-\Xi\\\
-\Xi&-\Xi\end{pmatrix}$ as pseudo-differential operator in the class
$\Psi^{1}_{1,0}(G\otimes\text{End}(\mathbb{C}^{2}))$, its symbol is given by

$\sigma_{-K}(x,\xi)=\begin{pmatrix}\langle\xi\rangle
I_{d_{\xi}}&\langle\xi\rangle I_{d_{\xi}}\\\ \langle\xi\rangle
I_{d_{\xi}}&\langle\xi\rangle I_{d_{\xi}}\end{pmatrix},$

for every $(x,[\xi])\in G\times\widehat{G}$. It is clear that in this case
$\sigma_{-K}$ satisfies the previous hypothesis and so the last theorem holds
true for the Cauchy problem (23). With a simple computation one may find a
solution given by

(24) $\begin{cases}\widehat{u_{1}}(t,\xi)=\frac{1}{2}e^{-2\langle\xi\rangle
t}({\widehat{u_{10}}(\xi)+\widehat{u_{20}}(\xi)})-\frac{1}{2}({\widehat{u_{10}}(\xi)-\widehat{u_{20}}(\xi)})\\\
\widehat{u_{2}}(t,\xi)=\frac{1}{2}e^{-2\langle\xi\rangle
t}({\widehat{u_{10}}(\xi)+\widehat{u_{20}}(\xi)})+\frac{1}{2}({\widehat{u_{10}}(\xi)-\widehat{u_{20}}(\xi)}).\end{cases}$

Theorem 4.2 then implies that this is the unique solution to (23), and that it
satisfies the estimates

$\|u_{1}(t)\|_{L^{2}(G)}^{2}+\|u_{2}(t)\|_{L^{2}(G)}^{2}\leq
C^{\prime}(\|u_{10}\|_{L^{2}(G)}^{2}+\|u_{20}\|_{L^{2}(G)}^{2}),$

for some $C^{\prime}>0$. Note that indeed, by squaring (24), adding both
equations and applying Plancherel’s Identity, one can easily obtain that the
above inequality holds for $C^{\prime}=\frac{1}{2}.$

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