#  Subelliptic sharp Gårding inequality on compact Lie groups

Duván Cardona  Duván Cardona: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium E-mail address
<EMAIL_ADDRESS>,  Serena Federico  Serena Federico: Department
of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University,
Belgium E-mail address<EMAIL_ADDRESS>and  Michael Ruzhansky
Michael Ruzhansky: Department of Mathematics: Analysis, Logic and Discrete
Mathematics Ghent University, Belgium and School of Mathematical Sciences
Queen Mary University of London United Kingdom E-mail address
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this work we establish a subelliptic sharp Gårding inequality on compact
Lie groups for pseudo-differential operators with symbols belonging to global
subelliptic Hörmander classes. In order for the inequality to hold we require
the global matrix-valued symbol to satisfy the suitable classical
nonnegativity condition in our setting. Our result extends to
$\mathscr{S}^{m}_{\rho,\delta}(G)$-classes, $0\leq\delta<\rho$, the one in
[28] about the validity of the sharp Gårding inequality for the class
$\mathscr{S}^{m}_{1,0}(G)$. We remark that the result we prove here is already
new and sharp in the case of the torus.

###### Key words and phrases:

Sub-Laplacian, Compact Lie group, Pseudo-differential operator, Fourier
analysis, Gårding type inequalities

###### 2010 Mathematics Subject Classification:

Primary 22E30; 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). Michael
Ruzhansky is also supported by the EPSRC grant EP/R003025/2.
This project has received funding from the European Union’s Horizon 2020
research and the innovation programme under the Marie Skłodowska-Curie grant
agreement No 838661.

###### Contents

  1. 1 Introduction
    1. 1.1 Outline and historical remarks
    2. 1.2 The subelliptic sharp Gårding inequality
  2. 2 Sub-Laplacians and pseudo-differential operators on compact Lie groups
    1. 2.1 Pseudo-differential operators via localisations
    2. 2.2 The positive sub-Laplacian and pseudo-differential operators via global symbols
    3. 2.3 Odd and even functions on compact Lie groups
    4. 2.4 Subelliptic Hörmander classes on compact Lie groups
  3. 3 Proof of the main Theorem 1.1
  4. 4 Final remarks

##  1\. Introduction

###  1.1. Outline and historical remarks

In this work we establish the sharp Gårding inequality for pseudo-differential
operators with symbols in the global subelliptic Hörmander classes on compact
Lie groups [4]. As a byproduct we obtain the extension to the global Hörmander
classes $\mathscr{S}^{m}_{\rho,\delta}(G),$ for all $0\leq\delta<\rho\leq 1$,
of the sharp Gårding inequality proved in [28] for the Kohn-Nirenberg classes
$\mathscr{S}^{m}_{1,0}(G)$.

Before describing in detail the main result of this paper concerning operators
on compact Lie groups, let us briefly go back to the Euclidean case and
describe the celebrated microlocal result which inspired the analysis of the
problem we consider here.

Gårding's type inequalities have played a crucial role in the study of several
problems related with partial and pseudo-differential operators. These
inequalities are $L^{2}$-lower bounds which can be applied, in different
contexts, to obtain results about the existence and uniqueness of solutions of
differential and pseudo-differential equations. The starting point in the
investigation of these fundamental lower bounds was the celebrated work of
Gårding [12] in which he proved the so called Gårding inequality for elliptic
operators:

Let $P$ be an elliptic self-adjoint pseudo-differential operator of order $m$
on an open set $\Omega\subset\mathbb{R}^{n}$, then, for any $\mu<m/2$ and any
compact $K\subset\Omega$, there exist two positive constants $c_{\mu,K}$ and
$C_{\mu,K}$ such that111We denote by $H^{s}$ the standard Sobolev space of
order $s$ defined as the completion of $C^{\infty}_{0}(\mathbb{R}^{n})$ with
respect to the norm $\|u\|_{H^{s}}:=\|(1-\Delta)^{\frac{s}{2}}u\|_{L^{2}}$,
where $\Delta$ is the standard Laplacian on $\mathbb{R}^{n}$.

$(Pu,u)\geq
c_{\mu,K}\|u\|^{2}_{H^{\frac{m}{2}}}-C_{\mu,K}\|u\|_{H^{\mu}}^{2},\quad\forall
u\in C_{0}^{\infty}(K).$ (1.1)

Inequality (1.1) was used by Gårding to derive the existence of solutions of
the Dirichlet problem for elliptic operators as well as to study the
distribution of the eigenvalues. However, in order to deal with non-elliptic
problems, some refinements of the latter are needed. In particular, Hörmander
proved in [14] the following Sharp Gårding inequality for operators with
symbols having nonegative real part:

Let $P$ be a pseudo-differential operator of order $m$ defined on an open set
$\Omega\subset\mathbb{R}^{n}$, and let $p\in S^{m}_{1,0}(\Omega)$ be its
symbol. If $\mathsf{Re}(p(x,\xi))\geq 0$ for all $(x,\xi)\in
T^{*}\Omega\setminus 0$, then for any compact subset $K\subset\Omega$ there
exists a constant $C_{K}>0$ such that

$\mathsf{Re}(Pu,u)\geq-C_{K}\|u\|^{2}_{\frac{m-1}{2}},\quad\forall u\in
C_{0}^{\infty}(K).$ (1.2)

After (1.2) some generalizations were proved, that is, specifically, the
suitable version for operators with symbols in
$S^{m}_{\rho,\delta}(\mathbb{R}^{n})$, and the corresponding version for
systems (see, for instance, [33]).

Note that the previous inequalities are both based on a sign property of the
symbol, and that no geometric property of the characteristic set is taken into
account.

Further improvements of (1.2) have been established by means of a deeper
analysis of the geometry of the characteristic set and of the invariants
associated with the operators, like, for instance, the principal and the
subprincipal symbol. In this direction we have the Melin inequality proved in
[20] and the Hörmander inequality proved in [16], the latter improved by
Parenti and Parmeggiani in [22] (see also [23]). A somehow different approach
was adopted by Fefferman and Phong in [8] where they derived the sharpest
result only by requiring the nonnegativity of the total symbol of the
operator. For a survey about the fundamental lower bounds mentioned above we
refer to [26].

Let us stress that these refinements not only allow the study of nonelliptic
operators, but can also be used to obtain microlocal energy estimates leading
to results on propagation of singularities (see Hörmander [18]). Additionally,
the (sharp) Gårding inequality and its generalizations become a fundamental
tool to analyze the existence of solutions of a wide class of boundary value
problems (like the $\overline{\delta}$-Neumann problem), and to investigate
the global solvability of evolution problems and the local well-posedness of
the Cauchy problem for evolution equations.

Let us also mention that some results, both positive and negative, about the
validity of some of the fundamental lower bounds mentioned above are known for
systems as well, and we refer the interested reader to [24, 25] for an
overview of this topic. In the setting of compact Lie groups the validity of
such estimates is more delicate to analyze. Indeed, only some of the lower
bounds presented above have been proved so far. For instance, in the case of a
compact Lie group $G$, and more generally on manifolds, (1.1) remains valid
for operators with symbols in the usual $S^{m}_{\rho,\delta}(M)$ Hörmander
classes, where $0\leq\delta<\rho\leq 1,$ and $\rho\geq 1-\delta$.

We recall that on closed manifolds $(\rho,\delta)$-pseudo-differential
operators are well defined provided that $0\leq\delta<\rho\leq 1,$ $\rho\geq
1-\delta,$ and that, under these assumptions, the global classes
$\mathscr{S}^{m}_{\rho,\delta}(G)$ defined in [27] coincide with the usual
Hörmander classes on compact manifolds. However bear in mind that the
$\mathscr{S}^{m}_{\rho,\delta}(G)$-classes are defined for all
$0\leq\delta<\rho\leq 1$ without the restriction $\rho\geq 1-\delta$.

On compact Lie groups, (1.1) for global symbols in the classes
$\mathscr{S}^{m}_{\rho,\delta}(G),$ for $0\leq\delta<\rho\leq 1$ (the whole
range), was proved by the third author and J. Wirth in [30] and extended for
subelliptic classes of pseudo-differential operators by the first and third
author in [4, Page 96]. We refer the reader to [5, Page 27] for Gårding type
inequalities on smooth manifolds, with or without boundary, using global
symbol criteria.

As for the sharp Gärding inequality in the general manifold setting, since a
condition on the whole symbol (and not only on the principal symbol) is
needed, this makes the result far reaching in this generality (symbols are not
invariantly defined on manifolds, while the principal symbol is). However, by
using the description of Hörmander classes
$\Psi^{m}_{\rho,\delta}(G,\mathrm{loc}),$ $0\leq\delta<\rho\leq 1,$ $\rho\geq
1-\delta,$ in terms of global symbols defined on the phase space
$G\times\widehat{G}$ 222Here, $\widehat{G}$ is the unitary dual of $G$ which
consists of all equivalence classes $[\xi]$ of irreducible, unitary continuous
representations of $G,$ $\xi\in\textnormal{End}(G,H_{\xi})$ on a finite
dimensional vector space $H_{\xi}\cong\mathbb{C}^{d_{\xi}}.$ (see (2.8) for
details about these classes of operators), the third author and Turunen proved
in [28] the following sharp Gärding inequality on any compact Lie group:

Let $A\in\Psi^{m}_{1,0}(G,\mathrm{loc})$ be such that the almost positivity
condition $a(x,[\xi])\geq 0$ on the global (matrix-valued) symbol of $A$ holds
true, then

$\mathsf{Re}(Au,u)_{L^{2}(G)}\geq-C\|u\|_{H^{\frac{m-1}{2}}(G)},\quad\forall
u\in C^{\infty}(G).$ (1.3)

We remark again that, in contrast with (1.1), this result requires a condition
on the global symbol of $A$. This is a nontrivial difference since results
involving conditions on the principal symbol only can be easily extended to
manifolds, while results requiring conditions on the total symbol are, in
general, not (yet) available in the manifold setting.

In the present paper we will focus on the validity of what we shall call
subelliptic sharp Gårding inequality, that is, on the suitable formulation of
the sharp Gårding inequality on compact Lie groups for pseudo-differential
operators with symbols belonging to global subelliptic classes. The
aforementioned subelliptic classes, and the corresponding pseudo-differential
calculus, are developed in [4] by using the sub-Riemannian structure of an
arbitrary compact Lie group $G$ and the matrix-valued quantization developed
in [27].

Since any sub-Riemannian structure on $G$ is encoded in terms of a positive
sub-Laplacian $\mathcal{L}$ over a compact Lie group $G,$ the global
subelliptic Hörmander classes of symbols in [4] were denoted by
$S^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}),$ where
$m\in\mathbb{R}$ and $0\leq\delta<\rho\leq 1.$

###  1.2. The subelliptic sharp Gårding inequality

The statement of our main result, that is of the subelliptic sharp Gårding
inequality on a compact Lie group $G$, is given in Theorem 1.1 below. Here and
in the rest of the paper we shall denote by $H^{s,\,\mathcal{L}}(G)$, for
$s\in\mathbb{R}$, the subelliptic Sobolev space of order $s$ associated with a
fixed positive sub-Laplacian $\mathcal{L}$, that is, the space defined as the
completion of $C^{\infty}(G)$ with respect to the norm
$\|u\|_{H^{s,\,\mathcal{L}}(G)}:=\|(1+\mathcal{L})^{\frac{s}{2}}u\|_{L^{2}(G)}.$

######  Theorem 1.1 (Subelliptic sharp Gårding inequality).

Let $G$ be a compact Lie group and let
$\mathcal{L}=\mathcal{L}_{X}$333$\mathcal{L}_{X}:=-X_{1}^{2}-X_{2}^{2}-\cdots-
X_{k}^{2}.$ be the (positive) sub-Laplacian associated with a system
$X=\\{X_{i}\\}_{i=1}^{k}$ of left-invariant vector fields satisfying
Hörmander's condition of step $\kappa$ 444which means that the vector fields
$X_{1},\cdots,X_{k},$ together with their commutators of length at most
$\kappa$ span the Lie algebra $\mathfrak{g}$ of $G$ (under the identification
$\mathfrak{g}\cong T_{e}G$, with $e\in G$ being the neutral element). If
$\mathcal{L}_{X}$ is a positive Laplacian, we trivially have $\kappa=1.$. For
$0<\rho\leqslant 1$ and $0\leq\delta<(2\kappa-1)^{-1}\rho$, and for
$m\in\mathbb{R}$, let

$A\equiv a(x,D):C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$

be a continuous linear operator with global symbol
$a\in{S}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$. Then, if
$a(x,[\xi])\geq 0$ for all $(x,[\xi])\in G\times\widehat{G}$, there exists a
positive constant $C$ such that

$\mathsf{Re}(Au,u)\geq-C\|u\|_{H^{\frac{m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)}{2},\,\mathcal{L}}(G)}^{2},$
(1.4)

for all $u\in C^{\infty}(G)$, where $(\cdot,\cdot)$ stands for the
$L^{2}$-scalar product.

As a consequence of the previous result we obtain in Corollary 1.2 below the
elliptic sharp Gårding inequality for operators with symbols in the Hörmander
classes $\mathscr{S}^{m}_{\rho,\delta}(G)$ (corresponding to the case when
$\kappa=1$).

Corollary 1.2 extends the main result in [28] holding for operators with
symbols belonging to the particular class $\mathscr{S}^{m}_{1,0}(G)$.

######  Corollary 1.2 (Elliptic sharp Gårding Inequality).

For $0\leqslant\delta<\rho\leqslant 1,$ let $A\equiv
a(x,D):C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ be a continuous linear
operator with symbol $a\in\mathscr{S}^{m}_{\rho,\delta}(G)$, $m\in\mathbb{R}$.
Let us assume that $a(x,[\xi])\geq 0$ for every $(x,[\xi])\in
G\times\widehat{G}.$ Then, there exists $C>0$, such that

$\mathsf{Re}(Au,u)\geq-C\|u\|_{H^{\frac{m-(\rho-\delta)}{2}}(G)}^{2}$ (1.5)

for all $u\in C^{\infty}(G).$

Let us briefly discuss some immediate consequences of our main Theorem 1.1.

######  Remark 1.3.

Notice that when $\mathcal{L}_{X}=\Delta$ is the Laplacian on the group (that
is $X=\\{X_{i}\\}_{i=1}^{n}$ is a basis of the Lie algebra $\mathfrak{g}$, or,
equivalently, $\kappa=1$), then Theorem 1.1 provides the sharp Gårding
inequality for operators with nonnegative symbols in the classes
$\mathscr{S}^{m}_{\rho,\delta}(G)$ with $0\leq\delta<\rho\leq 1$ (see
Corollary 1.2). This, in particular, shows that our result extends the one in
[28] where the elliptic Sharp Gårding inequality (namely for the standard
global non subelliptic symbols defined in [27]) was proved only for
$(\rho,\delta)=(1,0)$. More remarkably, the result applies to all
$\mathscr{S}^{m}_{\rho,\delta}(G)$ classes of global symbols with
$0\leq\delta<\rho\leq 1$, and not only to those where $\rho\geq 1-\delta$ and
corresponding to the standard Hörmander classes.

######  Remark 1.4.

Observe that the elliptic Sharp Gårding inequality for
$\mathscr{S}^{m}_{\rho,\delta}(G)$-classes with $(\rho,\delta)$ in the whole
admissible range (see Corollary 1.2), is already a new result in the case of
the torus $G=\mathbb{T}^{n}$. As remarked above, the standard local theory
allows to consider a restricted range for $\rho$ and $\delta$, therefore our
result is much better that the one possibly obtainable trough the local
theory, since here the parameters $\rho$ and $\delta$ can be taken in the full
range $0\leq\delta<\rho\leq 1$, allowing also the case where $\rho<1-\delta.$

As regards the purely subelliptic seeting, the appearence of the parameter
$\kappa$ (which is related to the subelliptic order of the fixed sub-
Laplacian) in the subelliptic Sobolev norms in (1.4), is dictated by the
combination of the noncommutativity of the group and the noncommutativity
property of the symbols of sub-Laplacians. Indeed, while the Laplacian is a
central operator having matrix-valued global symbol commuting with any other
symbol, no sub-Laplacian has the same commutativity properties. Therefore, in
order not to restrict our analysis to very particular subclasses of symbols,
we combined elliptic with subelliptic strategies.

Note that, as observed by the third author and Fischer in [10], the same
problem arises in the nilpotent Lie group setting. In fact, due the intrinsic
noncommutativity and subellipticity of the setting, in [10] the sharp Gårding
inequality in the nilpotent setting is announced for very particular
operators, that is, roughly speaking, for those commuting with the fixed sub-
Laplacian.

We want to stress that no such commutativity condition is assumed in Theorem
1.1. However there is a price to pay to work in this general framework, price
that is given by a restriction on the classes to which the result applies,
namely those such that $0\leq\delta<(2\kappa-1)^{-1}\rho$. Note that the case
$(\rho,\delta)=(1,0)$ is covered by our result.

Let us mention that even imposing a commutativity condition in the same spirit
as in [10], the use of the global pseudo-differential subelliptic calculus
does not lead to the expected result (that is (1.4) without the appearence of
$\kappa$ in the Sobolev norm), and that the expeceted result for very special
classes can most probably be reached via the global functional calculus.

More remarks about the strenght of our result in the purely subelliptic
setting are given in Section 4. There we also show that our subelliptic result
does not follow from the elliptic one.

We now conclude this introduction by giving the plan of the paper.

  * •

In Section 2 we recall some basic facts about pseudo-differential operators on
compact Lie groups, we recall the subelliptic global symbol classes in [4]
that are the object of our analysis. At the end of the section an analysis of
amplitude subelliptic operators is consistently developed.

  * •

In Section 3 we focus on the proof of the main theorem about the subelliptic
sharp Gårding inequality.

  * •

Finally, Section 4 is devoted to some remarks about our result in the purely
subelliptic setting.

##  2\. Sub-Laplacians and pseudo-differential operators on compact Lie groups

###  2.1. Pseudo-differential operators via localisations

Pseudo-differential operators on compact manifolds, and consequently on
compact Lie groups, can be defined by using local coordinate charts (see
Hörmander [17] and also M. Taylor [32] for a good introductory background on
the subject).

Let us briefly introduce these classes starting with the definition in the
Euclidean setting. Let $U$ be an open subset of $\mathbb{R}^{n}.$ We say that
the ``symbol" $a\in C^{\infty}(U\times\mathbb{R}^{n},\mathbb{C})$ belongs to
the Hörmander class of order $m$ and of $(\rho,\delta)$-type,
$S^{m}_{\rho,\delta}(U\times\mathbb{R}^{n}),$ $0\leqslant\rho,\delta\leqslant
1,$ if for every compact subset $K\subset U$ and for all
$\alpha,\beta\in\mathbb{N}_{0}^{n}$, the symbol inequalities

$|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|\leqslant
C_{\alpha,\beta,K}(1+|\xi|)^{m-\rho|\alpha|+\delta|\beta|},$

hold true uniformly in $x\in K$ for all $\xi\in\mathbb{R}^{n}.$ Then, a
continuous linear operator $A:C^{\infty}_{0}(U)\rightarrow C^{\infty}(U)$ is a
pseudo-differential operator of order $m$ of $(\rho,\delta)$-type, if there
exists a symbol $a\in S^{m}_{\rho,\delta}(U\times\mathbb{R}^{n})$ such that

$Af(x)=\int\limits_{\mathbb{R}^{n}}e^{2\pi
ix\cdot\xi}a(x,\xi)(\mathscr{F}_{\mathbb{R}^{n}}{f})(\xi)d\xi,$

for all $f\in C^{\infty}_{0}(U),$ where

$(\mathscr{F}_{\mathbb{R}^{n}}{f})(\xi):=\int\limits_{U}e^{-i2\pi
x\cdot\xi}f(x)dx$

is the Euclidean Fourier transform of $f$ at $\xi\in\mathbb{R}^{n}.$

Once the definition of Hörmander classes on open subsets of $\mathbb{R}^{n}$
is established, it can be extended to smooth manifolds as follows. Given a
$C^{\infty}$-manifold $M,$ a linear continuous operator
$A:C^{\infty}_{0}(M)\rightarrow C^{\infty}(M)$ is a pseudo-differential
operator of order $m$ of $(\rho,\delta)$-type, with $\rho\geqslant 1-\delta,$
and $0\leq\delta<\rho\leq 1,$ if for every local coordinate patch
$\omega:M_{\omega}\subset M\rightarrow U_{\omega}\subset\mathbb{R}^{n},$ and
for every $\phi,\psi\in C^{\infty}_{0}(U_{\omega}),$ the operator

$Tu:=\psi(\omega^{-1})^{*}A\omega^{*}(\phi u),\,\,u\in
C^{\infty}(U_{\omega}),$

is a standard pseudo-differential operator with symbol $a_{T}\in
S^{m}_{\rho,\delta}(U_{\omega}\times\mathbb{R}^{n}).$ In this case we write
$A\in\Psi^{m}_{\rho,\delta}(M,\textnormal{loc}).$

###  2.2. The positive sub-Laplacian and pseudo-differential operators via
global symbols

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}\simeq
T_{e_{G}}G$, where $e_{G}$ is the neutral element of $G$, and let

$X=\\{X_{1},\cdots,X_{k}\\}\subset\mathfrak{g}$

be a system of $C^{\infty}$-vector fields. For all
$I=(i_{1},\cdots,i_{\omega})\in\\{1,2,\cdots,k\\}^{\omega}$ of length
$\omega\geqslant 1$, we denote by

$X_{I}:=[X_{i_{1}},[X_{i_{2}},\cdots[X_{i_{\omega-1}},X_{i_{\omega}}]\cdots]]$

a commutator of length $\omega$, where $X_{I}:=X_{i}$ when $\omega=1$ and
$I=(i)$. The system $X$ is said to satisfy Hörmander's condition of step (or
order) $\kappa$ if $\mathfrak{g}=\mathrm{span}\\{X_{I}:|I|\leq\kappa\\}$, that
is, in other words, the vector fields $X_{j}$, $j=1,\ldots,k$, together with
their commutator up to length $\kappa$, generate the whole Lie algebra
$\mathfrak{g}$.

Note that we are assuming that there is no subsystem
$Y=\\{Y_{1},\cdots,Y_{\ell}\\}\subset X$, $\ell<k$, of smooth vector fields
such that $\mathfrak{g}=\mathrm{span}\\{Y_{I}:|I|\leq\kappa\\}$. In this case
we call $X$ a system of Hörmander's vector fields.

Given a system $X=\\{X_{1},\cdots,X_{k}\\}$ of Hörmander's vector fields, then
the operator defined as

$\mathcal{L}\equiv\mathcal{L}_{X}:=-(X_{1}^{2}+\cdots+X_{k}^{2}),$

is a hypoelliptic operator by Hörmander theorem on sums of the squares of
vector fields (see Hörmander [15]). In particular the operator $\mathcal{L}$
is also subelliptic, and it is called the subelliptic Laplacian associated
with the system $X$, or simply sub-Laplacian. It is clear from the definition
that one can define different sub-Laplacians by using different systems of
Hörmander's vector fields (and that satisfy Hörmander condition of different
step).

We will not treat other aspects of the analysis of sub-Laplacians here, we
refer the interested reader to Agrachev et al. [1], Bismut [3], Domokos et al.
[7], and to the fundamental book of Montgomery [21]. For some applications of
Hörmander's vector fields we refer to the book of Bramanti [2].

Let us now introduce the Hausdorff dimension associated with the sub-Laplacian
$\mathcal{L}$. For all $x\in G,$ let $H_{x}^{\omega}G$ be the linear subspace
of the tangent space $T_{x}G$ generated by the $X_{i}$'s and by all the Lie
brackets

$[X_{j_{1}},X_{j_{2}}],[X_{j_{1}},[X_{j_{2}},X_{j_{3}}]],\cdots,[X_{j_{1}},[X_{j_{2}},[X_{j_{3}},\cdots,X_{j_{\omega}}]]],$

with $\omega\leqslant\kappa.$ Then clearly Hörmander condition can be stated
as $H_{x}^{\kappa}G=T_{x}G$ for all $x\in G$, where the following inclusions
hold

$H_{x}^{1}G\subset H_{x}^{2}G\subset H_{x}^{3}G\subset\cdots\subset
H_{x}^{\kappa-1}G\subset H_{x}^{\kappa}G=T_{x}G,\,\,x\in G.$

Note that the dimension of every $H_{x}^{\omega}G$ is constant in $x\in G$, so
we set $\dim H^{\omega}G:=\dim H_{x}^{\omega}G,$ for all $x\in G$, and have
that the Hausdorff dimension can be defined as (see e.g. [19, p. 6]),

$Q:=\dim(H^{1}G)+\sum_{i=1}^{\kappa-1}(i+1)(\dim H^{i+1}G-\dim H^{i}G).$ (2.1)

As already mentioned in the introduction, we will make use of the quantization
process developed by the third author and V. Turunen in [27]. We briefly
recall below how this global quantization is defined.

Let $A$ be a continuous linear operator from $C^{\infty}(G)$ into
$\mathscr{D}^{\prime}(G),$ and let $\widehat{G}$ be the algebraic unitary dual
of $G.$ Then, there exists a function

$a:G\times\widehat{G}\rightarrow\cup_{\ell\in\mathbb{N}}\mathbb{C}^{\ell\times\ell},$
(2.2)

that we call the symbol of $A,$ such that
$a(x,\xi):=a(x,[\xi])\in\mathbb{C}^{d_{\xi}\times d_{\xi}}$ for every
equivalence class $[\xi]\in\widehat{G},$ where
$\xi:G\rightarrow\textnormal{Hom}(H_{\xi}),$
$H_{\xi}\cong\mathbb{C}^{d_{\xi}},$ and such that

$Af(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}{\textnormal{Tr}}[\xi(x)a(x,\xi)\widehat{f}(\xi)],\,\,\forall
f\in C^{\infty}(G).$ (2.3)

Note that we have denoted by

$\widehat{f}(\xi)\equiv(\mathscr{F}f)(\xi):=\int\limits_{G}f(x)\xi(x)^{*}dx\in\mathbb{C}^{d_{\xi}\times
d_{\xi}},\,\,\,[\xi]\in\widehat{G},$

the Fourier transform of $f$ at $\xi\cong(\xi_{ij})_{i,j=1}^{d_{\xi}},$ where
the matrix representation of $\xi$ is induced by an orthonormal basis of the
representation space $H_{\xi}.$ The function $a$ in (2.2) satisfying (2.3) is
unique, and satisfies the identity

$a(x,\xi)=\xi(x)^{*}(A\xi)(x),\,\,A\xi:=(A\xi_{ij})_{i,j=1}^{d_{\xi}},\,\,\,[\xi]\in\widehat{G}.$

Note that the previous identity is well defined. Indeed, it is well known that
the functions $\xi_{ij}$, which are of $C^{\infty}$-class, are the
eigenfunctions of the positive Laplace operator $\mathcal{L}_{G}$, that is
$\mathcal{L}_{G}\xi_{ij}=\lambda_{[\xi]}\xi_{ij}$ for some non-negative real
number $\lambda_{[\xi]}\geq 0$ depending only of the equivalence class $[\xi]$
and not on the representation $\xi$.

In general, we refer to the function $a$ as the (global or full) symbol of the
operator $A,$ and we will use the notation $A=\textnormal{Op}(a)$ to indicate
that $a:=\sigma_{A}$ is the symbol associated with the operator $A.$

In order to classify symbols in the Hörmander classes, in [27] the authors
defined the notion of difference operators, which endows $\widehat{G}$ with a
difference structure. Following [31], a difference operator
$Q_{\xi}:\mathscr{D}^{\prime}(\widehat{G})\rightarrow\mathscr{D}^{\prime}(\widehat{G})$
of order $k$ is defined as

$Q_{\xi}\widehat{f}(\xi)=\widehat{qf}(\xi),\,[\xi]\in\widehat{G},$ (2.4)

for some function $q$ vanishing of order $k$ at the neutral element $e=e_{G}.$
We will denote by $\textnormal{diff}^{k}(\widehat{G})$ the class of all
difference operators of order $k.$ For a fixed smooth function $q,$ the
associated difference operator will be denoted by $\Delta_{q}\equiv Q_{\xi}.$
A system of difference operators (see e.g. [31])

$\Delta_{\xi}^{\alpha}:=\Delta_{q_{(1)}}^{\alpha_{1}}\cdots\Delta_{q_{(i)}}^{\alpha_{i}},\,\,\alpha=(\alpha_{j})_{1\leqslant
j\leqslant i},$

with $i\geq n$, is called admissible if

$\textnormal{rank}\\{\nabla q_{(j)}(e):1\leqslant j\leqslant
i\\}=\textnormal{dim}(G),\textnormal{ and
}\Delta_{q_{(j)}}\in\textnormal{diff}^{1}(\widehat{G}).$

An admissible collection is said to be strongly admissible if, additionally,

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

######  Remark 2.1.

Matrix components of unitary representations induce difference operators.
Indeed, if $\xi_{1},\xi_{2},\cdots,\xi_{k},$ are fixed irreducible and unitary
representation of $G$, which not necessarily belong to the same equivalence
class, then each coefficient of the matrix

$\xi_{\ell}(g)-I_{d_{\xi_{\ell}}}=[\xi_{\ell}(g)_{ij}-\delta_{ij}]_{i,j=1}^{d_{\xi_{\ell}}},\,\quad
g\in G,\,\,1\leq\ell\leq k,$ (2.5)

that is each function $q^{\ell}_{ij}(g):=\xi_{\ell}(g)_{ij}-\delta_{ij}$,
$g\in G,$ defines a difference operator

$\mathbb{D}_{\xi_{\ell},i,j}:=\mathscr{F}(\xi_{\ell}(g)_{ij}-\delta_{ij})\mathscr{F}^{-1}.$
(2.6)

We can fix $k\geq\mathrm{dim}(G)$ of these representations in such a way that
the corresponding family of difference operators is admissible, that is,

$\textnormal{rank}\\{\nabla q^{\ell}_{i,j}(e):1\leqslant\ell\leqslant
k\\}=\textnormal{dim}(G).$

To define higher order difference operators of this kind, let us fix a unitary
irreducible representation $\xi_{\ell}$. Since the representation is fixed we
omit the index $\ell$ of the representations $\xi_{\ell}$ in the notation that
will follow. Then, for any given multi-index
$\alpha\in\mathbb{N}_{0}^{d_{\xi_{\ell}}^{2}}$, with
$|\alpha|=\sum_{i,j=1}^{d_{\xi_{\ell}}}\alpha_{i,j}$, we write

$\mathbb{D}^{\alpha}:=\mathbb{D}_{1,1}^{\alpha_{11}}\cdots\mathbb{D}^{\alpha_{d_{\xi_{\ell}},d_{\xi_{\ell}}}}_{d_{\xi_{\ell}}d_{\xi_{\ell}}}$

for a difference operator of order $|\alpha|$.

The difference operators endow the unitary dual $\widehat{G}$ with a
difference structure. For difference operators of the previous form, the
following finite Leibniz-like formula holds true (see [29] for details). Note
that below we are still assuming that the representation $\xi_{\ell}$ is
fixed.

######  Proposition 2.2 (Leibniz rule for difference operators).

Let $G$ be a compact Lie group and let $\mathbb{D}^{\alpha},$
$\alpha\in\mathbb{N}^{d_{\xi_{\ell}}}_{0},$ be the family of difference
operators defined in (2.6). Then, the following Leibniz rule

$\displaystyle(\mathbb{D}^{\alpha}a_{1}a_{2})(x_{0},\xi)(x_{0},\xi))=\sum_{|\gamma|,|\varepsilon|\leqslant|\alpha|\leqslant|\gamma|+|\varepsilon|}C_{\varepsilon,\gamma}(\mathbb{D}^{\gamma}a_{1})(x_{0},\xi)(\mathbb{D}^{\varepsilon}a_{2})(x_{0},\xi),\quad
x_{0}\in G,$

holds for all $a_{1},a_{2}\in
C^{\infty}(G)\times\mathscr{S}^{\prime}(\widehat{G})$, where the summation is
taken over all $\varepsilon,\gamma$ such that
$|\varepsilon|,|\delta|\leq|\alpha|\leq|\gamma|+|\varepsilon|$.

Note that for different kind of difference operators, namely for those given
by compositions of difference operators of higher order associated with
different representations, a Leibniz-like formula still holds true by
iteration. For more details about difference operators and Leibniz-like
formulas for admissible collections see also Corollary 5.13 in [9].

We are now going to introduce the global Hörmander classes of symbols defined
in [27]. First let us recall that every left-invariant vector field
$Y\in\mathfrak{g}$ can be identified with the first order differential
operator $\partial_{Y}:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ given
by

$\partial_{Y}f(x)=(Y_{x}f)(x)=\frac{d}{dt}f(x\exp(tY))|_{t=0}.$

If $\\{X_{1},\cdots,X_{n}\\}$ is a basis of the Lie algebra $\mathfrak{g},$
then we will use the standard multi-index notation

$\partial_{X}^{\alpha}=X_{x}^{\alpha}=\partial_{X_{1}}^{\alpha_{1}}\cdots\partial_{X_{n}}^{\alpha_{n}},$

for a canonical left-invariant differential operator of order $|\alpha|.$

By using this property, together with the following notation for the so-called
elliptic weight

$\langle\xi\rangle:=(1+\lambda_{[\xi]})^{1/2},\,\,[\xi]\in\widehat{G},$

we can finally give the definition of global symbol classes.

######  Definition 2.3.

Let $G$ be a compact Lie group and let $0\leqslant\delta,\rho\leqslant 1.$ Let

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

be a matrix-valued function such that for any $[\xi]\in\widehat{G},$
$\sigma(\cdot,[\xi])$ is of $C^{\infty}$-class, and such that, for any given
$x\in G$ there is a distribution $k_{x}\in\mathscr{D}^{\prime}(G),$ smooth in
$x,$ satisfying $\sigma(x,\xi)=\widehat{k}_{x}(\xi),$ $[\xi]\in\widehat{G}$.
We say that $\sigma\in\mathscr{S}^{m}_{\rho,\delta}(G)$ if the following
symbol inequalities

$\|\partial_{X}^{\beta}\Delta_{\xi}^{\gamma}\sigma_{A}(x,\xi)\|_{\textnormal{op}}\leqslant
C_{\alpha,\beta}\langle\xi\rangle^{m-\rho|\gamma|+\delta|\beta|},$ (2.7)

are satisfied for all $\beta$ and $\gamma$ multi-indices and for all
$(x,[\xi])\in G\times\widehat{G}.$ For
$\sigma_{A}\in\mathscr{S}^{m}_{\rho,\delta}(G)$ we will write
$A\in\Psi^{m}_{\rho,\delta}(G)\equiv\textnormal{Op}(\mathscr{S}^{m}_{\rho,\delta}(G)).$

The global Hörmander classes on compact Lie groups can be used to describe the
Hörmander classes defined by local coordinate systems. We present the
corresponding statement as follows.

######  Theorem 2.4 (Equivalence of classes, [9, 27, 29]).

Let $A:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ be a continuous linear
operator and let $0\leq\delta<\rho\leq 1,$ with $\rho\geq 1-\delta.$ Then,
$A\in\Psi^{m}_{\rho,\delta}(G,\textnormal{loc}),$ if and only if
$\sigma_{A}\in\mathscr{S}^{m}_{\rho,\delta}(G).$ Consequently,

$\textnormal{Op}(\mathscr{S}^{m}_{\rho,\delta}(G))=\Psi^{m}_{\rho,\delta}(G,\textnormal{loc}),\,\,\,0\leqslant\delta<\rho\leqslant
1,\,\rho\geqslant 1-\delta.$ (2.8)

###  2.3. Odd and even functions on compact Lie groups

Since we will use some properties of odd and even functions on $G$ in this
paper, for completeness we recall these definitions below.

######  Definition 2.5.

On a group $G,$ a function $f:G\rightarrow\mathbb{C}$ is

  * •

even, if $f(x^{-1})=f(x),$ for every $x\in G$;

  * •

central, if $f(xy)=f(yx),$ for every $x,y\in G$;

  * •

odd, if $f(x^{-1})=-f(x),$ for every $x\in G$.

Next, we summarize the action of vector fields on even and odd functions (see
Proposition 3.11 of [28]).

######  Proposition 2.6.

Let $G$ be a Lie group and $f\in C^{\infty}(G).$ Let $X_{i},$ $1\leq i\leq s$,
be an arbitrary system of vector fields in $\mathfrak{g}.$ Then

$\partial_{X_{1}}\cdots\partial_{X_{s}}f(x^{-1})=\pm\partial_{X_{s}}\cdots\partial_{X_{1}}f(x),$
(2.9)

and

$\partial_{X_{1}}\cdots\partial_{X_{s}}f(x^{-1})=(-1)^{s+1}\partial_{X_{s}}\cdots\partial_{X_{1}}f(x)$

if $f$ is an odd central function.

###  2.4. Subelliptic Hörmander classes on compact Lie groups

In order to define the subelliptic Hörmander calculus, we will use a suitable
basis of the Lie algebra arising from Taylor expansions. We explain the choice
of this basis by means of the following lemma (see Lemma 7.4 in [9]).

######  Lemma 2.7.

Let $G$ be a compact Lie group of dimension $n.$ Let
$\mathfrak{D}=\\{\Delta_{q_{(j)}}\\}_{1\leqslant j\leqslant n}$ be a strongly
admissible admissible collection of difference operators, that is

$\textnormal{rank}\\{\nabla q_{(j)}(e):1\leqslant j\leqslant
n\\}=n,\,\,\,\bigcap_{j=1}^{n}\\{x\in G:q_{(j)}(x)=0\\}=\\{e_{G}\\}.$

Then there exists a basis
$X_{\mathfrak{D}}=\\{X_{1,\mathfrak{D}},\cdots,X_{n,\mathfrak{D}}\\}$ of
$\mathfrak{g}$ such that

$X_{j,\mathfrak{D}}q_{(k)}(\cdot^{-1})(e_{G})=\delta_{jk}.$

Moreover, by using the multi-index notation

$\partial_{X}^{(\beta)}=\partial_{X_{1,\mathfrak{D}}}^{\beta_{1}}\cdots\partial_{X_{n,\mathfrak{D}}}^{\beta_{n}},$

for any $\beta\in\mathbb{N}_{0}^{n},$ where

$\partial_{X_{i,\mathfrak{D}}}f(x)=\frac{d}{dt}f(x\exp(tX_{i,\mathfrak{D}}))|_{t=0},\,\,f\in
C^{\infty}(G),$

and denoting by

$R_{x,N}^{f}(y)=f(xy)-\sum_{|\alpha|<N}q_{(1)}^{\alpha_{1}}(y^{-1})\cdots
q_{(n)}^{\alpha_{n}}(y^{-1})\partial_{X}^{(\alpha)}f(x)$

the Taylor remainder, we have that

$|R_{x,N}^{f}(y)|\leqslant C|y|^{N}\max_{|\alpha|\leqslant
N}\|\partial_{X}^{(\alpha)}f\|_{L^{\infty}(G)},$

where the constant $C>0$ is dependent on $N,$ $G$ and $\mathfrak{D}$ (but not
on $f\in C^{\infty}(G)).$ In addition we have that
$\partial_{X}^{(\beta)}|_{x_{1}=x}R_{x_{1},N}^{f}=R_{x,N}^{\partial_{X}^{(\beta)}f}$,
and

$|\partial_{X}^{(\beta)}|_{y_{1}=y}R_{x,N}^{f}(y_{1})|\leqslant
C|y|^{N-|\beta|}\max_{|\alpha|\leqslant
N-|\beta|}\|\partial_{X}^{(\alpha+\beta)}f\|_{L^{\infty}(G)},$

provided that $|\beta|\leqslant N.$

Using the notation above, and denoting by
$\Delta_{\xi}^{\alpha}:=\Delta_{q_{(1)}}^{\alpha_{1}}\cdots\Delta_{q_{(n)}}^{\alpha_{n}},$
we can introduce the subelliptic Hörmander class of symbols of order
$m\in\mathbb{R}$ of type $(\rho,\delta)$.

######  Definition 2.8 (Subelliptic Hörmander classes).

Let $G$ be a compact Lie group and let $0\leqslant\delta,\rho\leqslant 1.$ Let
us consider a sub-Laplacian $\mathcal{L}=-(X_{1}^{2}+\cdots+X_{k}^{2})$ on
$G,$ where the system of vector fields $X=\\{X_{i}\\}_{i=1}^{k}$ satisfies the
Hörmander condition of step $\kappa$. We say that
$\sigma\in{S}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$ if

$p_{\alpha,\beta,\rho,\delta,m,\textnormal{left}}(\sigma)^{\prime}:=\sup_{(x,[\xi])\in
G\times\widehat{G}}\|\widehat{\mathcal{M}}(\xi)^{(\rho|\alpha|-\delta|\beta|-m)}\partial_{X}^{(\beta)}\Delta_{\xi}^{\alpha}\sigma(x,\xi)\|_{\textnormal{op}}<\infty,$
(2.10)
$p_{\alpha,\beta,\rho,\delta,m,\textnormal{right}}(\sigma)^{\prime}:=\sup_{(x,[\xi])\in
G\times\widehat{G}}\|(\partial_{X}^{(\beta)}\Delta_{\xi}^{\alpha}\sigma(x,\xi))\widehat{\mathcal{M}}(\xi)^{(\rho|\alpha|-\delta|\beta|-m)}\|_{\textnormal{op}}<\infty,$
(2.11)

holds true for all $\alpha,\beta\in\mathbb{N}^{n}_{0}$.

By following the usual nomenclature, we define:

$\textnormal{Op}({S}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})):=\\{A:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G):\sigma_{A}\equiv\widehat{A}(x,\xi)\in{S}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})\\},$

with

$Af(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{{Tr}}(\xi(x)\widehat{A}(x,\xi)\widehat{f}(\xi)),\,\,\,f\in
C^{\infty}(G),\,x\in G.$

We will use the notation $\widehat{\mathcal{M}}$ for the matrix-valued symbol
of the operator $\mathcal{M}:=(1+\mathcal{L})^{\frac{1}{2}},$ and, for every
$[\xi]\in\widehat{G}$ and $s\in\mathbb{R},$ we define

$\widehat{\mathcal{M}}(\xi)^{s}:=\textnormal{diag}[(1+\nu_{ii}(\xi)^{2})^{\frac{s}{2}}]_{1\leqslant
i\leqslant d_{\xi}},$

where
$\widehat{\mathcal{L}}(\xi)=:\textnormal{diag}[\nu_{ii}(\xi)^{2}]_{1\leqslant
i\leqslant d_{\xi}}$ is the symbol of the sub-Laplacian $\mathcal{L}$ at
$[\xi].$

######  Definition 2.9 (Subelliptic amplitudes).

A function $a:G\times
G\times\widehat{G}\rightarrow\cup_{[\xi]\in\widehat{G}}\mathbb{C}^{d_{\xi}\times
d_{\xi}}$ is an amplitude symbol if for every $[\xi]\in\widehat{G},$
$a(\cdot,\cdot,[\xi])$ is smooth. In addition, $a$ belongs to the subelliptic
amplitude class of order $m$ and type $(\rho,\delta)$,
$\mathcal{A}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times G\times\widehat{G})$ if

$\sup_{(x,y,[\xi])\in G\times
G\times\widehat{G}}\|\widehat{\mathcal{M}}(\xi)^{(\rho|\alpha|-\delta(|\beta|+|\gamma|)-m)}\partial_{X}^{(\beta)}\partial_{Y}^{(\gamma)}\Delta_{\xi}^{\alpha}a(x,y,\xi)\|_{\textnormal{op}}<\infty,$
(2.12)

and

$\sup_{(x,y,[\xi])\in G\times
G\times\widehat{G}}\|(\partial_{X}^{(\beta)}\partial_{Y}^{(\gamma)}\Delta_{\xi}^{\alpha}a(x,y,\xi))\widehat{\mathcal{M}}(\xi)^{(\rho|\alpha|-\delta(|\beta|+|\gamma|)-m)}\|_{\textnormal{op}}<\infty.$
(2.13)

The amplitude operator associated with an amplitude
$a\in\mathcal{A}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times G\times\widehat{G})$
is defined via

$Af(x)\equiv\textnormal{AOp}(a)f(x):=\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{{Tr}}\left(\xi(x)\int\limits_{G}a(x,y,\xi)\xi(y)^{*}f(y)dy\right),$

where $f\in C^{\infty}(G).$

The decay properties of subelliptic symbols are summarized in the following
lemma (see [4, Chapter 4]).

######  Lemma 2.10.

Let $G$ be a compact Lie group and let $0\leqslant\delta,\rho\leqslant 1.$ If
$a\in{S}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}),$ then for every
$\alpha,\beta\in\mathbb{N}_{0}^{n},$ there exists $C_{\alpha,\beta}>0$
satisfying the estimates

$\|\partial_{X}^{(\beta)}\Delta_{\xi}^{\alpha}a(x,\xi)\|_{\textnormal{op}}\leqslant
C_{\alpha,\beta}\sup_{1\leqslant i\leqslant
d_{\xi}}(1+\nu_{ii}(\xi)^{2})^{\frac{m-\rho|\alpha|+\delta|\beta|}{2}},$

uniformly in $(x,[\xi])\in G\times\widehat{G}.$

In the next theorem we describe some fundamental properties of the subelliptic
calculus [4].

######  Theorem 2.11.

Let $0\leqslant\delta<\rho\leqslant 1,$ and let
$\Psi^{m,\,\mathcal{L}}_{\rho,\delta}:=\textnormal{Op}({S}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})),$
for every $m\in\mathbb{R}.$ Then,

  * -

The mapping $A\mapsto
A^{*}:\Psi^{m,\,\mathcal{L}}_{\rho,\delta}\rightarrow\Psi^{m,\,\mathcal{L}}_{\rho,\delta}$
is a continuous linear mapping between Fréchet spaces and the symbol of
$A^{*},$ $\sigma_{A^{*}}(x,\xi)$ satisfies the asymptotic expansion,

$\widehat{A^{*}}(x,\xi)\sim\sum_{|\alpha|=0}^{\infty}\Delta_{\xi}^{\alpha}\partial_{X}^{(\alpha)}(\widehat{A}(x,\xi)^{*}).$

This means that, for every $N\in\mathbb{N},$ and for all $\ell\in\mathbb{N},$

$\footnotesize{\Delta_{\xi}^{\alpha_{\ell}}\partial_{X}^{(\beta)}\left(\widehat{A^{*}}(x,\xi)-\sum_{|\alpha|\leqslant
N}\Delta_{\xi}^{\alpha}\partial_{X}^{(\alpha)}(\widehat{A}(x,\xi)^{*})\right)\in{S}^{m-(\rho-\delta)(N+1)-\rho\ell+\delta|\beta|,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})},$

where $|\alpha_{\ell}|=\ell.$

  * -

The mapping $(A_{1},A_{2})\mapsto A_{1}\circ
A_{2}:\Psi^{m_{1},\,\mathcal{L}}_{\rho,\delta}\times\Psi^{m_{2},\,\mathcal{L}}_{\rho,\delta}\rightarrow\Psi^{m_{3},\,\mathcal{L}}_{\rho,\delta}$
is a continuous bilinear mapping between Fréchet spaces, and the symbol of
$A=A_{1}\circ A_{2}$ satisfies is given by the asymptotic formula

$\sigma_{A}(x,\xi)\sim\sum_{|\alpha|=0}^{\infty}(\Delta_{\xi}^{\alpha}\widehat{A}_{1}(x,\xi))(\partial_{X}^{(\alpha)}\widehat{A}_{2}(x,\xi)),$

which, in particular, means that, for every $N\in\mathbb{N},$ and for all
$\ell\in\mathbb{N},$

$\displaystyle\Delta_{\xi}^{\alpha_{\ell}}\partial_{X}^{(\beta)}\left(\sigma_{A}(x,\xi)-\sum_{|\alpha|\leqslant
N}(\Delta_{\xi}^{\alpha}\widehat{A}_{1}(x,\xi))(\partial_{X}^{(\alpha)}\widehat{A}_{2}(x,\xi))\right)$
$\displaystyle\hskip
56.9055pt\in{S}^{m_{1}+m_{2}-(\rho-\delta)(N+1)-\rho\ell+\delta|\beta|,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}),$

for all $\alpha_{\ell}\in\mathbb{N}_{0}^{n}$ with $|\alpha_{\ell}|=\ell.$

  * -

For $0\leqslant\delta<\rho\leqslant 1,$ (or for $0\leq\delta\leq\rho\leq 1,$
$\delta<1/\kappa$) let us consider a continuous linear operator
$A:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ with symbol
$\sigma\in{S}^{0,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$. Then $A$
extends to a bounded operator from $L^{2}(G)$ to $L^{2}(G).$

######  Remark 2.12.

Let us remark that for all $m>0$, we have

$\mathscr{S}^{\frac{m}{\kappa}}_{\rho,\frac{\delta}{\kappa}}(G)\subset
S^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}).$

To show this property we will make use of the estimate

$\langle\xi\rangle^{\frac{1}{\kappa}}\lesssim(1+\nu_{ii}(\xi)^{2})^{\frac{1}{2}}\lesssim\langle\xi\rangle$
(2.14)

proved in Proposition 3.1 of [13]. Indeed, for
$\sigma\in\mathscr{S}^{\frac{m}{\kappa}}_{\rho,\frac{\delta}{\kappa}}(G),$ we
have that

$\|\partial_{X}^{(\beta)}\Delta_{\xi}^{\alpha}\sigma(x,\xi)\|_{\textnormal{op}}\lesssim\langle\xi\rangle^{\frac{m}{\kappa}-\rho|\alpha|+\frac{\delta}{\kappa}|\beta|},$

and, consequently, we have

$\displaystyle\|\widehat{M}(\xi)^{-m+\rho|\alpha|-\delta|\beta|}\partial_{X}^{(\beta)}\Delta_{\xi}^{\alpha}\sigma(x,\xi)\|_{\textnormal{op}}\leq\sup_{1\leq
i\leq
d_{\xi}}\langle\nu_{ii}(\xi)\rangle^{-m+\rho|\alpha|-\delta|\beta|}\langle\xi\rangle^{\frac{m}{\kappa}-\rho|\alpha|+\frac{\delta}{\kappa}|\beta|}.$

From the previous inequality, for $-m+\rho|\alpha|+\delta|\beta|<0,$ we obtain

$\displaystyle\langle\nu_{ii}(\xi)\rangle^{-m+\rho|\alpha|-\delta|\beta|}\langle\xi\rangle^{\frac{m}{\kappa}-\rho|\alpha|+\frac{\delta}{\kappa}|\beta|}$
$\displaystyle\lesssim\langle\xi\rangle^{\frac{-m+\rho|\alpha|-\delta|\beta|}{\kappa}}\langle\xi\rangle^{\frac{m}{\kappa}-\rho|\alpha|+\frac{\delta}{\kappa}|\beta|}\lesssim
1,$

while for $-m-\rho|\alpha|+\delta|\beta|\geq 0,$ we have

$\displaystyle\langle\nu_{ii}(\xi)\rangle^{-m+\rho|\alpha|-\delta|\beta|}\langle\xi\rangle^{\frac{m}{\kappa}-\rho|\alpha|+\frac{\delta}{\kappa}|\beta|}$
$\displaystyle\lesssim\langle\xi\rangle^{-m+\rho|\alpha|-\delta|\beta|}\langle\xi\rangle^{\frac{m}{\kappa}-\rho|\alpha|+\frac{\delta}{\kappa}|\beta|}\lesssim
1,$

proving that $\sigma\in
S^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}).$

######  Remark 2.13.

The last assertion in Theorem 2.11 remains valid if we consider
$0\leq\delta\leq\rho\leq 1,$ $\delta<1/\kappa.$ This is the subelliptic
Calderón-Vaillancourt theorem proved in [4], which gives the boundedness of
pseudo-differential operators in the subelliptic calculus in subelliptic
Sobolev spaces (see Theorem 2.15).

######  Proposition 2.14.

Let $A:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ be a continuous linear
operator with symbol $a\in
S^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}),$ $0\leq\delta<\rho\leq
1.$ Then $A:H^{s,\,\mathcal{L}}(G)\rightarrow H^{s-m,\,\mathcal{L}}(G)$
extends to a bounded operator for all $s\in\mathbb{R}.$

###### Proof.

In view of the closed graph Theorem, we only need to show that there exists
$C>0$ such that

$\|Au\|_{H^{s-m,\,\mathcal{L}}(G)}=\|\mathcal{M}_{s-m}Au\|_{L^{2}(G)}\leq
C\|u\|_{H^{s,\,\mathcal{L}}(G)},\,\,u\in C^{\infty}(G).$ (2.15)

By replacing $u$ by $\mathcal{M}_{-s}u,$ we can see that (2.15) is equivalent
to the following estimate

$\|\mathcal{M}_{s-m}A\mathcal{M}_{-s}u\|_{L^{2}(G)}\leq
C\|u\|_{L^{2}(G)},\,\,u\in C^{\infty}(G),$ (2.16)

which, once again by the closed graph theorem, is equivalent to show that
$A_{s}:=\mathcal{M}_{s-m}A\mathcal{M}_{-s}$ admits a bounded extension from
$C^{\infty}(G)$ to $L^{2}(G).$ By the subelliptic calculus $A_{s}\in
S^{0,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}).$ This, finally, gives
the existence of a bounded extension of $A_{s}$ as a consequence of the
subelliptic Calderón-Vaillancourt Theorem. ∎

######  Proposition 2.15.

Let $0\leq\delta<\rho\leq 1,$ and let
$a\in\mathcal{A}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times G\times\widehat{G}).$
Then $A\equiv\textnormal{AOp}(a)$ is a subelliptic pseudo-differential
operator with symbol $\sigma\in
S^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G}),$ that is
$A\equiv\textnormal{Op}(\sigma),$ which obeys to the formula

$\sigma(x,\xi)\sim\sum_{\alpha\in\mathbb{N}_{0}^{n}}(\partial_{Y}^{(\alpha)}\Delta_{\xi}^{\alpha}a(x,y,\xi))|_{y=x},$
(2.17)

in the sense that, for all $N\in\mathbb{N}$, and for all $\ell\in\mathbb{N},$

$\displaystyle\Delta_{\xi}^{\alpha_{\ell}}\partial_{X}^{(\beta)}\left(\sigma(x,\xi)-\sum_{|\alpha|\leqslant
N}(\partial_{Y}^{(\alpha)}\Delta_{\xi}^{\alpha}a(x,y,\xi))|_{y=x}\right)\in{S}^{m-(\rho-\delta)(N+1)-\rho\ell+\delta|\beta|,\,\mathcal{L}}_{\rho,\delta},$
(2.18)

for every $\alpha_{\ell}\in\mathbb{N}_{0}^{n}$ with $|\alpha_{\ell}|=\ell.$

###### Proof.

The proof of the asymptotic expansion (2.17) is similar to the one for the
analogous statement for elliptic Hörmander classes. Indeed, as in [28, Page
2891], we have that, for any amplitude operator $A$,

$\sigma_{A}(x,\xi)=\int\limits_{G}\xi(z)^{*}\sum_{[\eta]\in\widehat{G}}d_{\eta}\textnormal{Tr}[\eta(z)a(x,xz^{-1},\eta)]dz.$

Hence, applying Taylor expansion (see Lemma 2.7), we have

$a(x,xz^{-1},\eta)\sim\sum_{\alpha\in\mathbb{N}_{0}^{n}}(\partial_{Y}^{(\alpha)}a(x,y,\xi))|_{y=x}q_{(\alpha)}(z),$

and, consequently, we get

$\displaystyle\sigma_{A}(x,\xi)$
$\displaystyle\sim\sum_{\alpha\in\mathbb{N}_{0}^{n}}\int\limits_{G}\xi(z)^{*}\sum_{[\eta]\in\widehat{G}}d_{\eta}\textnormal{Tr}[\eta(z)\partial_{Y}^{(\alpha)}a(x,y,\xi))|_{y=x}q_{(\alpha)}(z)]dz$
$\displaystyle=\sum_{\alpha\in\mathbb{N}_{0}^{n}}\int\limits_{G}\xi(z)^{*}\sum_{[\eta]\in\widehat{G}}d_{\eta}\textnormal{Tr}[\eta(z)\partial_{Y}^{(\alpha)}a(x,y,\xi))|_{y=x}q_{(\alpha)}(z)]dz$
$\displaystyle=\sum_{\alpha\in\mathbb{N}_{0}^{n}}\int\limits_{G}\xi(z)^{*}q_{(\alpha)}(z)\sum_{[\eta]\in\widehat{G}}d_{\eta}\textnormal{Tr}[\eta(z)\partial_{Y}^{(\alpha)}a(x,y,\xi))|_{y=x}]dz$
$\displaystyle=\sum_{\alpha\in\mathbb{N}_{0}^{n}}\int\limits_{G}\xi(z)^{*}q_{(\alpha)}(z)\sum_{[\eta]\in\widehat{G}}d_{\eta}\textnormal{Tr}[\eta(z)\partial_{Y}^{(\alpha)}a(x,y,\xi))|_{y=x}]dz$
$\displaystyle=\sum_{\alpha\in\mathbb{N}_{0}^{n}}(\partial_{Y}^{(\alpha)}\Delta_{\xi}^{\alpha}a(x,y,\xi))|_{y=x}.$

Now we analyze the remainder term of the asymptotic expansion above. Our goal
is to prove that

$\displaystyle\Delta_{\xi}^{\alpha_{\ell}}\partial_{X}^{(\beta)}\left(\sigma(x,\xi)-\sum_{|\alpha|\leqslant
N}(\partial_{Y}^{(\alpha)}\Delta_{\xi}^{\alpha}a(x,y,\xi))|_{y=x}\right)\in{S}^{m-(\rho-\delta)(N+1)-\rho\ell+\delta|\beta|,\,\mathcal{L}}_{\rho,\delta}$
(2.19)

for all $\alpha_{\ell}$ with $|\alpha_{\ell}|=\ell.$ Below we will only prove
the existence of $N_{0}\in\mathbb{N}$ such that (2.19) is true for any given
$N\geq N_{0}$. The case when $N<N_{0}$ follows from the previous one by
standard arguments.

For any given $(x,y)\in G\times G$ let $k_{A,x,y}$ be the distribution defined
via

$a(x,y,\xi)=\widehat{k}_{A,x,y}(\xi),\,\,[\xi]\in\widehat{G}.$

Denote by $k_{\sigma,x}$ the right-convolution kernel of $A,$ that is the
distribution that satisfies $\sigma(x,\xi)=\widehat{k}_{\sigma,x}(\xi)$ for
all $[\xi]\in\widehat{G}.$ Because of the identities

$Af(x)\equiv\textnormal{AOp}(a)f(x)=\int\limits_{G}f(y)k_{A,x,y}(y^{-1}x)dy=\int\limits_{G}f(xz^{-1})k_{A,x,xz^{-1}}(z)dz,$

and

$Af(x)\equiv\textnormal{Op}(\sigma)f(x)=\int\limits_{G}f(y)k_{\sigma,x}(y^{-1}x)dy=\int\limits_{G}f(xz^{-1})k_{\sigma,x}(z)dz,$

we have

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

for all $x,y,z\in G.$ Moreover, by using Taylor expansion at $z^{-1}=e$ (see
Lemma 2.7), we can write

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

where the remainder term satisfies the estimate

$|R_{x,N}^{k_{A,x,\cdot}}(z)|=|R_{x,N}^{k_{A,x,x}}(z)|\leqslant
C|z|^{N}\max_{|\alpha|\leqslant
N}\|\partial_{X}^{(\alpha)}k_{A,x,x}(z)\|_{L_{x}^{\infty}(G)}.$

Observe now that the inverse Fourier transform of the term on the left hand
side of (2.19) is given by

$q_{\alpha_{\ell}}(z)\partial_{X}^{(\beta)}\left(k_{\sigma,x}(z)-\sum_{|\alpha|<N}q_{\alpha}(z)\partial_{Z_{1}}^{(\alpha)}k_{A,x,xz_{1}^{-1}}(z)|_{z_{1}=e}\right)=q_{\alpha_{\ell}}(z)\partial_{X}^{(\beta)}R_{x,N}^{k_{A,x,x}}(z),$

and that, for any given $M^{\prime}\geq 0$, with $M^{\prime}\equiv 0\mod{2}$
and such that

$M^{\prime}>\rho\ell-\delta|\beta|-m+(\rho-\delta)N,$ (2.20)

and

$\frac{M^{\prime}}{2}>-\rho\ell+\delta|\beta|+m+2(\rho-\delta)N,$ (2.21)

we have

$\displaystyle\|\Delta_{\xi}^{\alpha_{\ell}}\partial_{X}^{(\beta)}\left(\sigma(x,\xi)-\sum_{|\alpha|<N}(\partial_{Y}^{(\alpha)}\Delta_{\xi}^{\alpha}a(x,y,\xi))|_{y=x}\right)\widehat{\mathcal{M}}(\xi)^{(\rho\ell-\delta|\beta|-(m-(\rho-\delta)N))}\|_{\textnormal{op}}$
$\displaystyle=\|\Delta_{\xi}^{\alpha_{\ell}}\partial_{X}^{(\beta)}\left(\sigma(x,\xi)-\sum_{|\alpha|<N}(\partial_{Y}^{(\alpha)}\Delta_{\xi}^{\alpha}a(x,y,\xi))|_{y=x}\right)$
$\displaystyle\hskip
142.26378pt\times\widehat{\mathcal{M}}(\xi)^{M^{\prime}}\widehat{\mathcal{M}}(\xi)^{(\rho\ell-\delta|\beta|-(m-(\rho-\delta)N))-M^{\prime}}\|_{\textnormal{op}}$
$\displaystyle\leq\|\Delta_{\xi}^{\alpha_{\ell}}\partial_{X}^{(\beta)}\left(\sigma(x,\xi)-\sum_{|\alpha|<N}(\partial_{Y}^{(\alpha)}\Delta_{\xi}^{\alpha}a(x,y,\xi))|_{y=x}\right)\widehat{\mathcal{M}}(\xi)^{M^{\prime}}\|_{\textnormal{op}}$
$\displaystyle\lesssim\|(1+\mathcal{L}_{z})^{\frac{M^{\prime}}{2}}[q_{\alpha_{\ell}}(z)\partial_{X}^{(\beta)}R_{x,N}^{k_{A,x,x}}(z)]\|_{L^{1}(G,dz)}.$

Now, the application of Leibniz rule gives

$\displaystyle\|(1+\mathcal{L}_{z})^{\frac{M^{\prime}}{2}}[q_{\alpha_{\ell}}(z)\partial_{X}^{(\beta)}R_{x,N}^{k_{A,x,x}}(z)]\|_{L^{1}(G,dz)}=\|(1+\mathcal{L}_{z})^{\frac{M^{\prime}}{2}}[R_{x,N+1}^{q_{\alpha_{\ell}}(z)\partial_{X}^{(\beta)}k_{A,x,x}}(z)]\|_{L^{1}(G,dz)}$
$\displaystyle\lesssim\sum_{1\leqslant i_{1}\leqslant
i_{2}\leqslant\cdots\leqslant i_{k}\leqslant k\,,|\gamma|\leqslant
M^{\prime}}\|X^{\gamma_{1}}_{i_{1},z}\cdots
X^{\gamma_{k}}_{i_{k},z}[R_{x,N}^{q_{\alpha_{\ell}}(z)\partial_{X}^{(\beta)}k_{A,x,x}}(z)]\|_{L^{1}(G,dz)}.$

By the estimates in Lemma 2.7 we have

$\displaystyle|X^{\gamma_{1}}_{i_{1},z}\cdots
X^{\gamma_{k}}_{i_{k},z}[R_{x,N+1}^{q_{\alpha_{\ell}}(z)\partial_{X}^{(\beta)}k_{A,x,x}}(z)]|\lesssim|z|^{N-|\gamma|}\max_{|\alpha|\leq
N-|\gamma|}\|\partial_{Z}^{(\alpha+\gamma)}({q_{\alpha_{\ell}}(z)\partial_{X}^{(\beta)}k_{A,x,x}}(z))\|_{L^{\infty}(G)},$

and, since
$\partial_{Z}^{(\gamma)}({q_{\alpha_{\ell}}(z)\partial_{X}^{(\beta)}k_{A,x,x}}(z))$
is the right-convolution kernel of a subelliptic operator of order

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

by Proposition 4.24 of [4] we have

$\max_{|\alpha|\leq
M^{\prime}}\|\partial_{Z}^{(\alpha)}({q_{\alpha_{\ell}}(z)\partial_{X}^{(\beta)}k_{A,x,x}}(z))\|_{L^{\infty}(G)}\lesssim_{a}|z|^{-\frac{s^{\prime}+Q}{\rho}}.$

Therefore, putting the previous estimates together, we obtain

$\displaystyle\|(1+\mathcal{L}_{z})^{\frac{M^{\prime}}{2}}[q_{\alpha_{\ell}}(z)\partial_{X}^{(\beta)}R_{x,N}^{k_{A,x,x}}(z)]\|_{L^{1}(G,dz)}\lesssim_{a}\int\limits_{G}|z|^{N-|\gamma|}|z|^{-\frac{s^{\prime}+Q}{\rho}}dz,$
(2.22)

where the integral on the right hand side is finite if

$\rho(|\gamma|-N)+s^{\prime}+Q<n{\rho}.$

Notice that, for $N$ big enough, we have $(\rho-\delta)N\lesssim M^{\prime}$.
However, by choosing $M^{\prime}$ in such a way that $M^{\prime}\lesssim
100(\rho-\delta)N,$ we have that $(\rho-\delta)N\asymp M^{\prime}.$ Hence, by
using the inequality $|\alpha|+|\gamma|\leq N$ together with (2.21), we have

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

Now, by choosing $N\gtrsim N_{0}\geq 2(n\rho-Q)/(\rho-\delta))$, we get that
$|\gamma|-N+s^{\prime}+Q<n\rho$, which, in particular, gives the validity of
(2.18) for $N$ large enough. Finally, the case $N\lesssim N_{0}$ follows from
standard arguments from the previous case. This concludes the proof. ∎

##  3\. Proof of the main Theorem 1.1

Notations. In this section the following notations will be adopted:

  * -

$q_{\gamma}(x):=q_{(1)}^{\gamma_{1}}(x)\ldots q_{(n)}^{\gamma_{n}}(x)$, for
any given $\gamma\in\mathbb{N}_{0}^{n}$;

  * -

$X=\\{X_{1},\cdots,X_{k}\\}$ will be a system of vector fields satisfying
Hörmander condition of step $\kappa;$

  * -

$\mathcal{L}:=\mathcal{L}_{X}$ will be the positive sub-Laplacian associated
with the system $X$;

  * -

$\mathcal{M}$ will be the operator defined as
$\mathcal{M}:=(I+\mathcal{L})^{1/2}$, while $\widehat{\mathcal{M}}(\xi)$ will
denote the corresponding matrix-valued symbol
$\widehat{\mathcal{M}}(\xi)=\mathrm{diag}[\langle\nu_{ii}(\xi)\rangle]_{1\leq
i\leq d_{\xi}},$ where
$\langle\nu_{ii}(\xi)\rangle:=(1+\nu_{ii}(\xi)^{2})^{1/2}$;

  * -

$\|\cdot\|_{m,\,\mathcal{L}}$ will denote the norm in the Sobolev space
$H^{m,\,\mathcal{L}}(G)$ associated with the subelliptic operator
$\mathcal{L}$ on $G$;

  * -

$\|\cdot\|_{op}$ will denote the $\ell^{2}\rightarrow\ell^{2}$ operator norm
of the linear finite dimensional mapping (matrix multiplication by)
$\sigma_{A}(x,\xi)$, that is,

$\|\sigma_{A}(x,\xi)\|_{op}=\sup\\{\|\sigma_{A}(x,\xi)v\|_{\ell^{2}}:v\in\mathbb{C}^{d_{\xi}},\|v\|_{\ell^{2}}=1\\}.$

######  Remark 3.1.

The proof of the main theorem relies on the construction of a positive pseudo-
differential operator
$P\in\textnormal{Op}(S^{m,\,\mathcal{L}}_{\rho,\frac{\rho+\delta}{2}}(G\times\widehat{G}))$
such that $A-P=Q,$ with $A$ as in the hypotheses of Theorem 1.1, and

$Q:H^{\frac{m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)}{2},\,\,\mathcal{L}}(G)\rightarrow
H^{\frac{-(m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta))}{2},\,\,\mathcal{L}}(G)$

being a bounded operator. In fact, if such a decomposition were true, then
this would immediately lead to the desired inequality by simply observing that

${\mathrm{Re}}(Au,u)=\mathrm{Re}(Pu,u)+\mathrm{Re}(Qu,u)\geq-C\|u\|_{\frac{m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)}{2},\,\,\mathcal{L}},$

for all $u\in C^{\infty}(G)$.

As in the proof of the Sharp Gårding inequality on compact Lie groups (see
[28]), the key point here is the construction of the so called weight function
$w_{\xi}$, which, essentially, corresponds to the construction of the operator
$P$ with the properties mentioned in Remark 3.1.

The aforementioned construction follows the lines of the Euclidean case.
However, the adaptation to the Lie-group setting is linked to the deep group
structure. Indeed, a crucial key point in our proof will be the choice of the
suitable power of $\langle\xi\rangle$ inside the expression of $w_{\xi}$,
choice that depends on the relation between the eigenvalues of the Laplacian
and those of the sublaplacian on the group.

We remark that the use of an elliptic weight $w_{\xi}$, namely depending on
$\langle\xi\rangle$, instead of a subelliptic one, that is depending on
$\widehat{\mathcal{M}}(\xi)$, is fundamental here. This, in particular, is due
to the fact that while the Laplacian is a central operator on the group and
its (matrix valued) symbol commutes with all the other symbols, the same
property does not hold for any sublaplacian on the group. The noncommutativity
property of the symbol of sub-Laplacians, toghether with the intrinsic
noncommutativity of the group, causes several technical issues not allowing
the use of a subelliptic weight.

Notice also that similar anomalies occur in the nilpotent Lie group setting,
where, again, sub-Laplacians (and more generally Rockland operators), do not
have symbols with the same commutaivity property as the symbol of the
Laplacian on compact Lie groups. This main difference, as remarked in [10], is
very likely the reason why the full sharp Gårding inequality, that is for any
operator with nonnegative symbol, has not been proved yet in the nilpotent Lie
group setting (see [10] where the result for specific operators has been
announced).

Taking into account the previous clarifications, we can now build the proof of
Theorem 1.1 starting from the construction of the weight function $w_{\xi}$.

Let us consider $G$ as a closed subgroup of
$GL(N,\mathbb{R})\subset\mathbb{R}^{N\times N}$ for some $N\in\mathbb{N}$, so
that its Lie algebra $\mathfrak{g}$ is an $n$-dimensional vector subspace such
that

$[A,B]:=AB-BA\in\mathfrak{g}$

for every $A,B\in\mathfrak{g}$. Let $e\in G$ be the neutral element, $U\subset
G$ a neighborhood of $e$, and $V\subset\mathfrak{g}$ a neighborhood of
$0\in\mathfrak{g}$ such that the matrix exponential mapping

$\exp:V\rightarrow U$

is a diffeomorphism. We define on $\mathfrak{g}$ the central norm $|\cdot|$
(that we shall use only in the definition of the function $w_{\xi}$ below) as
follows

$|X|:=\int\limits_{G}|uXu^{-1}|_{0}du,$ (3.1)

where the product under the integral is the product of matrices, and where
$|\cdot|_{0}$ stands for the Euclidean norm on $\mathfrak{g}$. Note that
$\exp^{-1}$ is central with respect to the norm (3.1), i.e.,
$|\exp^{-1}(xy)|=|\exp^{-1}(yx)|$, and that the norm (3.1) is invariant by the
adjoint representation.

We now assume that the neighborhood $V$ of $0_{\mathfrak{g}}$ is the open ball
$V=\mathbb{B}(0,r)=\\{Z\in\mathbb{R}^{n}:|Z|<r\\}$, with $r>0$, and consider a
real smooth function $\phi:[0,r)\rightarrow[0,\infty)$, radial on
$\mathfrak{g}$, supported on $V$, and such that $\phi(s)=\phi(|Z|)=1$ for
$s>0$ small. Then we define the function

$w_{\xi}(x):=\phi(|\exp^{-1}(x)|\langle\xi\rangle^{\frac{(\rho+\delta)}{2\kappa}})\psi(\exp^{-1}(x))\langle\xi\rangle^{\frac{n(\rho+\delta)}{4\kappa}}$
(3.2)

with

$\psi(Y)=C_{0}|\det D\exp(Y)|^{-1/2}f(Y)^{-1/2},\quad
C_{0}=(\int\limits_{\mathbb{R}^{n}}\phi(|Z|)^{2}dZ)^{-1/2},$

where, recall, $\rho+\delta<2$,
$\langle\xi\rangle=(1+\lambda_{\xi}^{2})^{\frac{1}{2}}$, i.e.
$\sigma_{I-\Delta}(\xi)=\langle\xi\rangle^{2}I_{d_{\xi}}$, $D\exp$ is the
Jacobian matrix of $\exp$, and $f(Y)$ is the density with respect to the
Lebesgue measure of the pull-back on $\mathfrak{g}$ of the Haar measure on $G$
by the exponential mapping.

Note that, given the geodesic distance $d$, we have
$d(x,e)\simeq|\exp^{-1}(x)|$ and $\mathrm{supp}\,{w_{\xi}}=\\{x\in
G:d(x,e)\leq r\langle\xi\rangle^{-\frac{(\rho+\delta)}{2\kappa}}\\}\subset
U_{e}$.

######  Remark 3.2.

Observe that from Remark 2.12 and the fact that
$\langle\xi\rangle^{\frac{n(\rho+\delta)}{4\kappa}}I_{d_{\xi}}\in\mathscr{S}^{\frac{n(\rho+\delta)}{4\kappa}}_{1,0}(G),$
we deduce that
$\langle\xi\rangle^{\frac{n(\rho+\delta)}{4\kappa}}I_{d_{\xi}}\in
S^{\frac{n(\rho+\delta)}{4},\,\mathcal{L}}_{1,0}(G\times\widehat{G})\subset
S^{\frac{n(\rho+\delta)}{4},\,\mathcal{L}}_{\rho,0}(G\times\widehat{G})$, for
all $\rho\in(0,1]$.

We are now ready to state a key lemma about the basic properties of the
function $w_{\xi}$.

######  Lemma 3.3.

Let $w_{\xi}$ be as above. Then
$w_{\xi}(e)=C_{0}\langle\xi\rangle^{\frac{n(\rho+\delta)}{4\kappa}}$,
$w_{\xi}$ is central and inversion invariant, and $d(x,e)\leq
r\langle\xi\rangle^{-\frac{(\rho+\delta)}{2\kappa}}$ on
$\mathrm{supp}\,(w_{\xi})$. Moreover, $\|w_{\xi}\|_{L^{2}(G)}=1$ for all
$\xi\in\mathrm{Rep}(G),$ and $(x,\xi)\mapsto
w_{\xi}(x)I_{d_{\xi}}\in\mathscr{S}^{\frac{n(\rho+\delta)}{4\kappa}}_{1,\frac{(\rho+\delta)}{2\kappa}}(G)\subset
S^{\frac{n(\rho+\delta)}{4},\,\mathcal{L}}_{\rho,\frac{(\rho+\delta)}{2}}(G\times\widehat{G})$,
for all $\rho\in(0,1]$ and $0\leq\delta<\rho$.

###### Proof.

Notice that, due to the properties of $\phi$, we immediately have that
$w_{\xi}(e)=C_{0}\langle\xi\rangle^{\frac{n(\rho+\delta)}{4\kappa}}$,
$w_{\xi}$ is central and inversion invariant, and $d(x,e)\leq
r\langle\xi\rangle^{-\frac{(\rho+\delta)}{2\kappa}}$ on
$\mathrm{supp}\,w_{\xi}$. As for the $\|w_{\xi}\|_{L^{2}(G)}$, we have

$\begin{split}\int\limits_{G}|w_{\xi}(x)|^{2}dx&=\langle\xi\rangle^{\frac{n(\rho+\delta)}{2\kappa}}\int\limits_{\mathbb{R}^{n}}\phi_{\xi}(|Y|\langle\xi\rangle^{\frac{(\rho+\delta)}{2\kappa}})^{2}|\psi(|Y|)|^{2}|detD\exp(Y)|f(Y)dY\\\
&=\int\limits_{\mathbb{R}^{n}}\phi_{\xi}(|Z|)^{2}|\psi(|Z|\langle\xi\rangle^{-\frac{(\rho+\delta)}{2\kappa}})|^{2}|detD\exp(Z\langle\xi\rangle^{-\frac{(\rho+\delta)}{2\kappa}})|f(Z\langle\xi\rangle^{-\frac{(\rho+\delta)}{2\kappa}})dZ\\\
&=C_{0}^{2}\int\limits_{\mathbb{R}^{n}}\phi_{\xi}(|Z|)^{2}dZ=1,\end{split}$

where in the second line we applied the change of variables
$Z=Y\langle\xi\rangle^{\frac{(\rho+\delta)}{2\kappa}}$, while in the third
line we simply used the expression of $\psi$.

We are now left with the proof of

$(x,\xi)\mapsto
w_{\xi}(x)I_{d_{\xi}}\in\mathscr{S}^{\frac{n(\rho+\delta)}{4\kappa}}_{\rho,\frac{(\rho+\delta)}{2\kappa}}(G)\subset
S^{\frac{n(\rho+\delta)}{4},\,\mathcal{L}}_{\rho,\frac{(\rho+\delta)}{2}}(G\times\widehat{G}).$
(3.3)

From the compactness of $G$, proving (3.3) is equivalent to showing that, for
every multi-index $\beta$ and for any fixed $x\in G$,
$(\partial^{(\beta)}_{X}w_{\xi})(x)I_{d_{\xi}}\in\mathscr{S}_{\rho,0}^{\frac{n(\rho+\delta)}{4\kappa}+\frac{(\rho+\delta)}{2\kappa}|\beta|}(G)$
(see Lemma 3.3 in [28]). First observe that

$\partial^{(\beta)}_{X}w_{\xi}(x)I_{d_{\xi}}=\sum_{\alpha;|\alpha|\leq|\beta|}C_{\alpha,\beta}\Big{[}\partial_{X}^{(\alpha)}\phi(|\exp^{-1}(x)|\langle\xi\rangle^{\frac{(\rho+\delta)}{2\kappa}})\Big{]}\partial_{X}^{(\beta-\alpha)}\psi(\exp^{-1}(x))\langle\xi\rangle^{\frac{n(\rho+\delta)}{4\kappa}}I_{d_{\xi}}$
$=\sum_{\alpha;|\alpha|\leq|\beta|}C_{\alpha,\beta}\phi_{\alpha}(|\exp^{-1}(x)|\langle\xi\rangle^{\frac{(\rho+\delta)}{2\kappa}})\langle\xi\rangle^{\frac{(\rho+\delta)|\alpha|}{2\kappa}}\chi_{\beta-\alpha}(\exp^{-1}(x))\langle\xi\rangle^{\frac{n(\rho+\delta)}{4\kappa}}I_{d_{\xi}},$

where $\phi_{\alpha},\chi_{\beta-\alpha}$ are suitable functions such that
$\phi_{\alpha}\in C_{0}^{\infty}(\mathbb{R})$ and is constant near the origin,
while $\chi_{\beta-\alpha}\in C_{0}^{\infty}(V)$. Since

$\langle\xi\rangle^{\frac{n(\rho+\delta)}{4\kappa}+\frac{(\rho+\delta)|\alpha|}{2\kappa}}I_{d_{\xi}}\in\mathscr{S}^{\frac{n(\rho+\delta)}{4\kappa}+\frac{(\rho+\delta)}{2\kappa}|\beta|}_{1,0}(G),$

then, for every (fixed) $x\in G$,

$\partial^{\beta}_{x}w_{\xi}(x)I_{d_{\xi}}\in\mathscr{S}_{1,0}^{\frac{n(\rho+\delta)}{4\kappa}+\frac{(\rho+\delta)}{2\kappa}|\beta|}(G)$

if
$\phi_{\alpha}(|\exp^{-1}(x)|\langle\xi\rangle^{\frac{(\rho+\delta)}{2\kappa}})\chi_{\beta-\alpha}(\exp^{-1}(x))\in\mathscr{S}^{0}_{1,0}(G)$
for all $\alpha$ and $\beta$ as above. Therefore, to complete the proof it is
enough to check that these last terms are standard global symbols of order 0.

Now, given $x\in G$, it is easy to see that

$\phi_{\alpha}(|\exp^{-1}(x)|\langle\xi\rangle^{\frac{(\rho+\delta)}{2\kappa}})\chi_{\beta-\alpha}(\exp^{-1}(x))\leq
C.$ (3.4)

In fact, if $x$ is such that $\exp^{-1}(x)=0$, then $\phi_{\alpha}$ is
constant and the inequality follows. If, instead, $\exp^{-1}(x)\neq 0$, then,
since $\phi_{\alpha}$ is compactly supported in $\xi$, we get that the symbol
in the left hand side of (3.4) is compactly supported, then smoothing, and the
inequality follows. This concludes the proof of (3.3) from which the result
follows. ∎

######  Remark 3.4.

Note that from Remark 2.12, for all $m>0$, we have
$\mathscr{S}^{\frac{m}{\kappa}}_{\rho,\frac{\delta}{\kappa}}(G)\subset
S^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$. This fact, together
with the property $(x,\xi)\mapsto
w_{\xi}(x)I_{d_{\xi}}\in\mathscr{S}^{\frac{n(\rho+\delta)}{4\kappa}}_{\rho,\frac{(\rho+\delta)}{2\kappa}}(G)$,
immediately gives $(x,\xi)\mapsto w_{\xi}(x)I_{d_{\xi}}\in
S^{\frac{n(\rho+\delta)}{4},\,\mathcal{L}}_{\rho,\frac{(\rho+\delta)}{2}}(G)$.
Note also that the symbol $w_{\xi}(x)I_{d_{\xi}}$ commutes with any
subelliptic symbol $b(x,\xi)$, for all $\xi\in\widehat{G}$.

######  Proposition 3.5.

Let $\sigma_{A}\in S^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$ and
let $p(x,y,\xi)$ be the amplitude

$p(x,y,\xi):=\int\limits_{G}w_{\xi}(xz^{-1}){w_{\xi}}(yz^{-1})\sigma_{A}(z,\xi)dz,$
(3.5)

where $w_{\xi}$ is as in (3.2). Then
$p\in\mathcal{A}^{m,\,\mathcal{L}}_{\rho,\frac{(\rho+\delta)}{2}}(G\times
G\times\widehat{G})$ and the amplitude operator $P=\textnormal{Op}(p)$ given
by

$Pu(x)=\int\limits_{G}\sum_{[\xi]\in\widehat{G}}d_{\xi}\mathrm{Tr}\left(\xi(y^{-1}x)p(x,y,\xi)\right)u(y)dy$

is positive.

###### Proof.

Recall that
$p\in\mathcal{A}^{m,\,\mathcal{L}}_{\rho,\frac{(\rho+\delta)}{2}}(G\times
G\times\widehat{G})$ if

$\sup_{(x,y,[\xi])\in G\times
G\times\widehat{G}}\|\widehat{\mathcal{M}}(\xi)^{-m+\rho|\alpha|-\frac{(\rho+\delta)}{2}(|\beta|+|\gamma|)}\partial_{X}^{(\beta)}\partial_{Y}^{(\gamma)}\Delta_{\xi}^{\alpha}p(x,y,\xi)\|_{\textnormal{op}}<\infty,$

and that, by Leibniz rule,
$\partial_{X}^{(\beta)}\partial_{Y}^{(\gamma)}\Delta_{\xi}^{\alpha}p(x,y,\xi)$
is a sum of terms of the form

$\int\limits_{G}(\Delta_{\xi}^{\eta}\partial_{X}^{(\beta)}w_{\xi}(xz^{-1}))(\Delta_{\xi}^{\lambda}\partial_{Y}^{(\gamma)}w_{\xi}(yz^{-1}))(\Delta_{\xi}^{\mu}\sigma_{A}(z,\xi))dz,$

where $|\eta+\lambda+\mu|\geq|\alpha|$. Moreover, due to the properties of
$w_{\xi}$, we have

$\|(\Delta_{\xi}^{\eta}\partial_{X}^{(\beta)}w_{\xi}(xz^{-1}))(\Delta_{\xi}^{\lambda}\partial_{Y}^{(\gamma)}w_{\xi}(yz^{-1}))\|_{op}$
$\leq
C\langle\xi\rangle^{\frac{n(\rho+\delta)}{4\kappa}-\rho|\eta|+\frac{(\rho+\delta)}{2}\frac{|\beta|}{\kappa}}\langle\xi\rangle^{\frac{n(\rho+\delta)}{4\kappa}-\rho|\lambda|+\frac{(\rho+\delta)}{2}\frac{|\gamma|}{\kappa}}$
$\leq
C\langle\xi\rangle^{\frac{n(\rho+\delta)}{2\kappa}}\sup_{i=1,...,d_{\xi}}\langle\nu_{ii}(\xi)\rangle^{-\rho(|\lambda|+|\eta|)+\frac{(\rho+\delta)}{2}(|\beta|+|\gamma|)},$

therefore, since $\mathrm{supp}\,w_{\xi}=\\{z;d(z,e)\leq
r\langle\xi\rangle^{-\frac{(\rho+\delta)}{2\kappa}}\\}$ is contained in a set
of measure $r\langle\xi\rangle^{-\frac{(\rho+\delta)n}{2\kappa}}$, we get

$\|\int\limits_{G}\widehat{\mathcal{M}}(\xi)^{-\rho|\mu|+\rho|\alpha|-\frac{(\rho+\delta)}{2}(|\beta|+|\gamma|)}(\Delta_{\xi}^{\eta}\partial_{X}^{(\beta)}w_{\xi}(xz^{-1}))(\Delta_{\xi}^{\lambda}\partial_{Y}^{(\gamma)}w_{\xi}(yz^{-1}))$
$\times\widehat{\mathcal{M}}(\xi)^{-m+\rho|\mu|}(\Delta_{\xi}^{\mu}\sigma_{A}(z,\xi))dz\|_{op}$
$\leq
C\langle\xi\rangle^{-\frac{n(\rho+\delta)}{2\kappa}}\|\Delta_{\xi}^{\eta}\partial_{X}^{(\beta)}w_{\xi}(xz^{-1})\Delta_{\xi}^{\lambda}\partial_{Y}^{(\gamma)}w_{\xi}(yz^{-1}))\widehat{\mathcal{M}}(\xi)^{-\rho|\mu|+\rho|\alpha|-\frac{(\rho+\delta)}{2}(|\beta|+|\gamma|)}\|_{op}$
$\times\|\widehat{\mathcal{M}}(\xi)^{-m+\rho|\mu|}(\Delta_{\xi}^{\mu}\sigma_{A}(z,\xi))\|_{op}$
$\leq
C\langle\xi\rangle^{-\frac{n(\rho+\delta)}{2\kappa}}\langle\xi\rangle^{\frac{n(\rho+\delta)}{2\kappa}}\|\langle\xi\rangle^{-\rho(|\lambda|+|\eta|)+\frac{(\rho+\delta)}{2\kappa}(|\beta|+|\gamma|)}\widehat{\mathcal{M}}(\xi)^{-\rho|\mu|+\rho|\alpha|-\frac{(\rho+\delta)}{2}(|\beta|+|\gamma|)}\|_{op}$

$\leq
C\sup_{i=1,...,d_{\xi}}\langle\nu_{ii}(\xi)\rangle^{\rho|\alpha|-\frac{(\rho+\delta)}{2}(|\beta|+|\gamma|)-\rho(|\eta|+|\lambda|+|\mu|)+\frac{(\rho+\delta)}{2}(|\beta|+|\gamma|)}\leq
C.$

Finally, since
$\|\widehat{\mathcal{M}}(\xi)^{-m+\rho|\alpha|-\frac{(\rho+\delta)}{2}(|\beta|+|\gamma|)}\partial_{X}^{(\beta)}\partial_{Y}^{(\gamma)}\Delta_{\xi}^{\alpha}p(x,y,\xi)\|_{\textnormal{op}}$
is estimated by a sum of terms of the previous form, the first result follows.

To see that $P$ is positive, on denoting

$M(z,\xi):=\int\limits_{G}w_{\xi}(yz^{-1})\xi(yz^{-1})\overline{u(y)}dy,$

we have

$(Pu,u)=\int\limits_{G}\int\limits_{G}\sum_{[\xi]\in\widehat{G}}d_{\xi}\mathrm{Tr}(\xi(x)p(x,y,\xi)u(y)\xi(y)^{*}dy)\overline{u(x)}dx$

$=\int\limits_{G}\int\limits_{G}\sum_{[\xi]\in\widehat{G}}d_{\xi}\mathrm{Tr}\left(\xi(x)\int\limits_{G}w_{\xi}(xz^{-1}){w_{\xi}}(yz^{-1})\sigma_{A}(z,\xi)dzu(y)\xi(y)^{*}dy\right)\overline{u(x)}dx$

$=\int\limits_{G}\sum_{[\xi]\in\widehat{G}}d_{\xi}\mathrm{Tr}\left(\left(\int\limits_{G}\xi(xz^{-1})w_{\xi}(xz^{-1})\overline{u(x)}dx\right)\xi(z)\sigma_{A}(z,\xi)\xi(z)^{*}\right.$
$\left.\times\left(\int\limits_{G}{w_{\xi}}(yz^{-1})\xi(yz^{-1})^{*}u(y)dy\right)\right)dz$

$=\int\limits_{G}\sum_{[\xi]\in\widehat{G}}d_{\xi}\mathrm{Tr}\left(M(z,\xi)\xi(z)\sigma_{A}(x,\xi)\xi(z)^{*}M(z,\xi)^{*}\right)dz,$

which is non-negative due to the non-negativity of $\sigma_{A}$ and the fact
that the positivity of matrices is invariant under unitary transformations.
This concludes the proof. ∎

######  Lemma 3.6.

Let $s\in\mathbb{R}$ and
$p\in\mathcal{A}^{m,\,\mathcal{L}}_{\rho,\frac{(\rho+\delta)}{2}}(G\times
G\times\widehat{G})$ as in (3.5). Let us assume that $0\leq\delta<\rho\leq 1$.
Then the operator with symbol $p(x,x,\xi)-\sigma_{A}(x,\xi)$ is bounded from
$H^{s,\,\mathcal{L}}(G)$ to
$H^{s-(m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)),\,\mathcal{L}}(G)$.

###### Proof.

By the properties of pseudo-differential operators with symbols in subelliptic
classes, in order to prove the theorem it suffices to show that

$p(x,x,\xi)-\sigma_{A}(x,\xi)\in
S^{m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta),\,\mathcal{L}}_{\rho,\max\\{\delta,\frac{(\rho+\delta)}{2}\\}}(G\times\widehat{G})=S^{m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta),\,\mathcal{L}}_{\rho,\frac{(\rho+\delta)}{2}}(G\times\widehat{G}).$
(3.6)

Note that, since $\|w_{\xi}\|_{L^{2}(G)}=1$, we have

$p(x,x,\xi)-\sigma_{A}(x,\xi)=\int\limits_{G}w_{\xi}(xz^{-1})^{2}\sigma_{A}(z,\xi)dz-\sigma_{A}(x,\xi)$
$=\int\limits_{G}w_{\xi}(z)^{2}(\sigma_{A}(z^{-1}x,\xi)-\sigma_{A}(x,\xi))dz,$

where, recall, $\mathrm{supp}\,w_{\xi}=\\{z\in G:d(z,e)\leq
r\langle\xi\rangle^{-\frac{(\rho+\delta)}{2\kappa}}\\}$. Then we consider the
Taylor expansion of order one of $\sigma_{A}(z^{-1}x,\xi)$ with respect to
$z^{-1}$ at $z^{-1}=e$, namely,

$\sigma_{A}(z^{-1}x,\xi)=\sigma_{A}(x,\xi)+\sum_{|\gamma|=1}\partial_{X}^{(\gamma)}\sigma_{A}(x,\xi)q_{\gamma}(z)+\sum_{|\gamma|=2}q_{\gamma}(z)\sigma_{A,\gamma}(z^{-1}x,\xi),$

where the last term in the right hand side represents the remainder of Taylor
expansion of order 1. Note also that we can choose the polynomials
$q_{\gamma}$ to be odd when $|\gamma|=1$, that is,
$q_{\gamma}(z)=-q_{\gamma}(z^{-1})$. By using the expansion above together
with the property $\int\limits_{G}|w_{\xi}(z)|^{2}q_{\gamma}(z)dz=0$ when
$|\gamma|=1$ (since $w_{\xi}$ is even while $q_{\gamma}$ is odd), we obtain

$p(x,x,\xi)-\sigma_{A}(x,\xi)=\int\limits_{G}w_{\xi}(z)^{2}\Big{(}\sigma_{A}(x,\xi)-\sigma_{A}(x,\xi)$
$+\sum_{|\gamma|=1}\partial_{X}^{(\gamma)}\sigma_{A}(x,\xi)q_{\gamma}(z)+\sum_{|\gamma|=2}\sigma_{A,\gamma}(z^{-1}x,\xi)q_{\gamma}(z)\Big{)}dz$
$=\sum_{|\gamma|=2}\int\limits_{G}\sigma_{A,\gamma}(z^{-1}x,\xi)w_{\xi}(z)^{2}q_{\gamma}(z)dz,$

where the symbols $\sigma_{A,\gamma}\in
S^{m+\delta|\gamma|,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$ come
from the remainder term of Taylor expansion. Now, by Leibniz rule and the left
invariance of $\partial^{(\beta)}_{X}$, we have that the quantity

$\partial^{(\beta)}_{X}\triangle_{\xi}^{\alpha}(p(x,x,\xi)-\sigma_{A}(x,\xi))=\partial^{(\beta)}_{X}\triangle_{\xi}^{\alpha}\sum_{|\gamma|=2}\int\limits_{G}\sigma_{A,\gamma}(z^{-1}x,\xi)w_{\xi}(z)^{2}q_{\gamma}(z)dz$

can be written as a sum of terms of the form

$\int\limits_{G}(\partial^{(\beta)}_{X}\triangle_{\xi}^{\alpha_{1}}\sigma_{A,\gamma}(z^{-1}x,\xi))(\triangle_{\xi}^{\alpha_{2}}w_{\xi}(z))(\triangle_{\xi}^{\alpha_{3}}w_{\xi}(z))q_{\gamma}(z)dz$
$=\int\limits_{G}(\partial^{(\beta)}_{Y}\triangle_{\xi}^{\alpha_{1}}\sigma_{A,\gamma}(y,\xi))|_{y=z^{-1}x}(\triangle_{\xi}^{\alpha_{2}}w_{\xi}(z))(\triangle_{\xi}^{\alpha_{3}}w_{\xi}(z))q_{\gamma}(z)dz,$
(3.7)

where $|\gamma|=2$ and $|\alpha_{1}+\alpha_{2}+\alpha_{3}|\geq|\alpha|$.

Since for $\gamma$ such that $|\gamma|=2$ we have $|q_{\gamma}(z)|\leq
C\langle\xi\rangle^{-\frac{(\rho+\delta)}{\kappa}}$ on the support of
$w_{\xi}$, and since
$\mathrm{meas}(\mathrm{supp}(w_{\xi}))\lesssim\langle\xi\rangle^{-\frac{n(\rho+\delta)}{2\kappa}}$,
in view of (2.14) each therm of the form (3.7) satisfies

$\|\widehat{\mathcal{M}}(\xi)^{-m+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)+\rho|\alpha|-\max\\{\delta,\frac{(\rho+\delta)}{2}\\}|\beta|}\int\limits_{G}(\partial^{(\beta)}_{Y}\triangle_{\xi}^{\alpha_{1}}\sigma_{A,\gamma}(y,\xi))|_{y=z^{-1}x}(\triangle_{\xi}^{\alpha_{2}}w_{\xi}(z))$
$\times(\triangle_{\xi}^{\alpha_{3}}w_{\xi}(z))q_{\gamma}(z)dz\|_{op}$ $\leq
C\sup_{x\in
G}\|\widehat{\mathcal{M}}(\xi)^{-m+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)+\rho|\alpha|-\frac{(\rho+\delta)}{2}|\beta|}\partial^{(\beta)}_{X}\triangle_{\xi}^{\alpha_{1}}\sigma_{A,\gamma}(x,\xi))\|_{op}$
$\sup_{z_{1}\in
G}\|\triangle_{\xi}^{\alpha_{2}}w_{\xi}(z_{1})\|_{op}\sup_{z_{2}\in
G}\|\triangle_{\xi}^{\alpha_{3}}w_{\xi}(z_{2})\|_{op}\int\limits_{\mathrm{supp}\,(w_{\xi})}|q_{\gamma}(z)|\,dz$
$\leq
C\sup_{i=1,...,d_{\xi}}\langle\nu(\xi)_{ii}\rangle^{-m+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)+\rho|\alpha|-\delta|\beta|+m+2\delta+|\beta|\delta-|\alpha_{1}|\rho}$
$\times\,\,\langle\xi\rangle^{\frac{n(\rho+\delta)}{2\kappa}-(|\alpha_{2}|+|\alpha_{3}|)\rho-\frac{n(\rho+\delta)}{2\kappa}-\frac{(\rho+\delta)}{\kappa}}$
$\leq
C\sup_{i=1,...,d_{\xi}}\langle\nu(\xi)_{ii}\rangle^{\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)+\rho|\alpha|-\delta|\beta|+2\delta+|\beta|\delta-(|\alpha_{1}|+|\alpha_{2}|+|\alpha_{3}|)\rho-\frac{(\rho+\delta)}{\kappa}}\leq
C.$

This, finally, shows (3.6) and concludes the proof. ∎

######  Lemma 3.7.

Let $s\in\mathbb{R}.$ Let us assume that $0<\rho\leqslant 1$ and
$0\leq\delta<(2\kappa-1)^{-1}\rho\leq\rho.$ Then the pseudo-differential
operator with symbol $\sigma_{P}(x,\xi)-p(x,x,\xi)$ is bounded from
$H^{s,\,\mathcal{L}}(G)$ to
$H^{s-(m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)),\,\mathcal{L}}(G)$.

###### Proof.

As in the proof of Lemma 3.7 we need to show that

$\sigma_{P}(x,\xi)-p(x,x,\xi)\in
S^{m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta),\,\mathcal{L}}_{\rho,\max\\{\delta,\frac{(\rho+\delta)}{2}\\}}=S^{m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta),\,\mathcal{L}}_{\rho,\frac{(\rho+\delta)}{2}}.$
(3.8)

To prove (3.8) we use the asymptotic expansion in (2.18), which implies that

$\sigma_{P}(x,\xi)-p(x,x,\xi)\sim\sum_{|\beta|\geq
1}\partial_{Y}^{(\beta)}\Delta^{\beta}_{\xi}p(x,y,\xi)|_{y=x}.$

The asymptotic formula above means that, for all $N\in\mathbb{N}_{0}$,

$\sigma_{P}(x,\xi)-p(x,x,\xi)=\sum_{1\leq|\beta|\leq
N}\Delta_{\xi}^{\beta}\int\limits_{G}w_{\xi}(z)(\partial_{Z}^{(\beta)}w_{\xi}(z))\sigma_{A}(z^{-1}x,\xi)dz+r_{N}(x,\xi),$

with $r_{N}(x,\xi)\in
S^{m-\frac{(\rho+\delta)}{2}(N+1),\,\mathcal{L}}_{\rho,\frac{(\rho+\delta)}{2}}(G\times\widehat{G})$.
Let $N\geq 1$ and define

$S_{N}(x,\xi):=\sigma_{P}(x,\xi)-p(x,x,\xi)-r_{N}(x,\xi).$

Now we expand $\sigma_{A}(z^{-1}x,\xi)$ by using Taylor expansion with respect
to $z^{-1}$ at $z^{-1}=e$ as in the proof of Lemma 3.6, and have

$S_{N}(x,\xi)=\sum_{1\leq|\beta|\leq
N}\Delta_{\xi}^{\beta}\int\limits_{G}w_{\xi}(z)(\partial_{Z}^{(\beta)}w_{\xi}(z))dz\,\sigma_{A}(x,\xi)$
$+\sum_{|\gamma|=1}\sum_{1\leq|\beta|\leq
N}\Delta_{\xi}^{\beta}\int\limits_{G}w_{\xi}(z)(\partial_{Z}^{(\beta)}w_{\xi}(z))q_{\gamma}(z)\,\sigma_{A,\gamma}(z^{-1}x,\xi)dz$
$=I(x,\xi)+J(x,\xi),$

where $\sigma_{A,\gamma}\in
S^{m+\delta|\gamma|,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$
($|\gamma|=1$) comes, as before, from the remainder term of Taylor expansion.
To have (3.8) we need $I,J$ and $r_{N}$ to belong to the subelliptic class in
(3.8). We then analyse the three terms separately starting from $I$.

Note that, when $|\beta|=1$, $w_{\xi}$ and $\partial^{(\beta)}_{X}w_{\xi}$ are
even and odd respectively (see Proposition 3.11 in [28]), so we have

$\int\limits_{G}w_{\xi}(z)(\partial_{Z}^{(\beta)}w_{\xi}(z))dz=0,\quad\text{for}\,\,|\beta|=1,$

and

$I(x,\xi)=\sum_{2\leq|\beta|\leq
N}\Delta_{\xi}^{\beta}\int\limits_{G}w_{\xi}(z)(\partial_{Z}^{(\beta)}w_{\xi}(z))dz\,\sigma_{A}(x,\xi).$

In particular $I(x,\xi)$ will be given by a sum of terms of the form

$\sum_{2\leq|\beta|\leq
N}\int\limits_{G}(\Delta_{\xi}^{\eta}w_{\xi}(z))(\Delta_{\xi}^{\lambda}\partial_{Z}^{(\beta)}w_{\xi}(z))dz\,\Delta_{\xi}^{\mu}\sigma_{A}(x,\xi),$
(3.9)

with $|\eta+\lambda+\mu|\geq|\beta|$. Due to the measure of the support of
$w_{\xi}$, each term of the form (3.9) satisfies

$\left|\int\limits_{G}(\Delta_{\xi}^{\eta}w_{\xi}(z))(\Delta_{\xi}^{\lambda}\partial_{Z}^{(\beta)}w_{\xi}(z))dz\right|\leq
C\langle\xi\rangle^{-\frac{(\rho+\delta)}{2\kappa}n}\langle\xi\rangle^{\frac{n(\rho+\delta)}{4\kappa}-\rho|\eta|}\langle\xi\rangle^{\frac{n(\rho+\delta)}{4\kappa}-\rho|\lambda|+\frac{(\rho+\delta)}{2\kappa}|\beta|}.$
(3.10)

Then for $I$ we get

$\|\widehat{\mathcal{M}}(\xi)^{-m+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)}I(x,\xi)\|_{op}\leq
C\sum_{2\leq|\beta|\leq
N}\,\,\sum_{\lambda,\eta,\mu;|\eta+\lambda+\mu|\geq|\beta|}\|\widehat{\mathcal{M}}(\xi)^{-m+\rho|\mu|}\Delta_{\xi}^{\mu}\sigma_{A}(x,\xi)\|_{op}$
$\times\langle\xi\rangle^{-\rho(|\lambda|+|\eta|)+\frac{(\rho+\delta)}{2\kappa}|\beta|}\|\widehat{\mathcal{M}}(\xi)^{-\rho|\mu|+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)}\|_{op}$
$\leq C\sum_{2\leq|\beta|\leq
N}\,\,\sum_{\lambda,\eta,\mu;|\eta+\lambda+\mu|\geq|\beta|}\sup_{i=1,...,d_{\xi}}\langle\nu_{ii}(\xi)\rangle^{-\rho(|\lambda|+|\eta|+|\mu|)+\frac{(\rho+\delta)}{2}|\beta|+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)}\lesssim
C,$

since

$-\rho(|\lambda|+|\eta|+|\mu|)+\frac{(\rho+\delta)}{2}|\beta|=-\rho(|\lambda|+|\eta|+|\mu|-|\beta|)+\frac{\delta-\rho}{2}|\beta|$
$<\frac{\delta-\rho}{2}|\beta|\leq\delta-\rho<0,$

and $\delta-\rho+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)\leq 0$ for $\delta$
such that $\rho-(2\kappa-1)\delta>0$. Finally, by using the same kind of
estimates as above in combination with Leibniz rule (see Remark 2.1), we can
also conclude the more general estimate

$\|\widehat{\mathcal{M}}(\xi)^{-m+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)+|\alpha|\rho+|\beta|\max\\{\delta,\frac{(\rho+\delta)}{2}\\}}\partial_{X}^{(\beta)}\triangle_{\xi}^{\alpha}I(x,\xi)\|_{op}\leq
C$

which, in particular, gives that $I(x,\xi)\in
S^{m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta),\,\mathcal{L}}_{\rho,\max\\{\delta,\frac{(\rho+\delta)}{2}\\}}=S^{m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta),\,\mathcal{L}}_{\rho,\frac{(\rho+\delta)}{2}}$.

We now consider the term $J(x,\xi)$. In this case we have

$\|\widehat{\mathcal{M}}(\xi)^{-m+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)}J(x,\xi)\|_{op}\leq
C\sum_{1\leq|\beta|\leq
N}\,\,\sum_{\lambda,\eta,\mu;|\eta+\lambda+\mu|\geq|\beta|}\sum_{\gamma,|\gamma|=1}\|\widehat{\mathcal{M}}(\xi)^{-m+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)}$
$\int\limits_{G}(\Delta_{\xi}^{\eta}w_{\xi}(z))(\Delta_{\xi}^{\lambda}\partial_{Z}^{(\beta)}w_{\xi}(z))q_{\gamma}(z)\,\Delta_{\xi}^{\mu}\sigma_{A,\gamma}(z^{-1}x,\xi)dz\|_{op},$

so, as for $I(x,\xi)$, we prove that each term in the sum is bounded. Arguing
as before (and using the fact that
$(\Delta_{\xi}^{\eta}w_{\xi}(z))(\Delta_{\xi}^{\lambda}\partial_{Z}^{(\beta)}w_{\xi}(z))$
commutes with $\Delta_{\xi}^{\mu}\sigma_{A,\gamma}(z^{-1}x,\xi)$) we get

$\|\widehat{\mathcal{M}}(\xi)^{-m+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)}\int\limits_{G}(\Delta_{\xi}^{\eta}w_{\xi}(z))(\Delta_{\xi}^{\lambda}\partial_{Z}^{(\beta)}w_{\xi}(z))q_{\gamma}(z)\Delta_{\xi}^{\mu}\sigma_{A,\gamma}(z^{-1}x,\xi)dz\|_{op}$
$\leq
C\sup_{i=1,...,d_{\xi}}\langle\nu_{ii}(\xi)\rangle^{-\frac{(\rho+\delta)}{2\kappa}-\rho(|\eta|+|\lambda|+|\mu|)+\frac{(\rho+\delta)}{2}|\beta|+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)}$
$\leq
C\sup_{i=1,...,d_{\xi}}\langle\nu_{ii}(\xi)\rangle^{-\frac{(\rho+\delta)}{2\kappa}-\rho(|\eta|+|\lambda|+|\mu|-|\beta|)+\frac{(\delta-\rho)}{2}|\beta|+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)}\leq
C.$

The more general estimate (for $|\alpha|,|\beta|\neq 0$)

$\|\widehat{\mathcal{M}}(\xi)^{-m+\frac{1}{\kappa}(\rho-(2\kappa-1)\delta)+|\alpha|\rho+|\beta|\max\\{\delta,\frac{(\rho+\delta)}{2}\\}}\partial_{X}^{(\beta)}\triangle_{\xi}^{\alpha}J(x,\xi)\|_{op}\leq
C$

giving that $J\in
S^{m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta),\,\mathcal{L}}_{\rho,\max\\{\delta,\frac{(\rho+\delta)}{2}\\}}(G\times\widehat{G})=S^{m-\frac{1}{\kappa}(\rho-(2\kappa-1)\delta),\,\mathcal{L}}_{\rho,\frac{(\rho+\delta)}{2}}(G\times\widehat{G})$
can be derived by following the same steps as in the proof of Lemma 3.6,
therefore we omit the proof.

We are now left with the study of $r_{N}$. In this case, by the properties of
the remainder in the asymptotic formula, we have that, for all
$N\in\mathbb{N}$, $r_{N}\in
S^{m-\frac{(\rho+\delta)}{2}(N+1),\,\mathcal{L}}_{\rho,\frac{(\rho+\delta)}{2}}(G\times\widehat{G})$,
for all $0\leq\delta<\rho\leq 1$ such that $\rho-(2\kappa-1)\delta>0$. This
shows (3.8) and concludes the proof. ∎

###### Proof of Theorem 1.1.

Let us consider the amplitude operator $P$ in Proposition 3.5. In view of
Remark 3.1, it suffices to prove that

$Q:H^{\frac{m-\frac{1}{\kappa}(\rho-(\kappa-1)\delta)}{2},\,\mathcal{L}}(G)\rightarrow
H^{\frac{-(m-\frac{1}{\kappa}(\rho-(\kappa-1)\delta))}{2},\,\mathcal{L}}(G)$

is a bounded operator. To show this property it is enough to observe that

$\sigma_{Q}(x,\xi):=(\sigma_{A}(x,\xi)-p(x,x,\xi))+(p(x,x,\xi)-\sigma_{P}(x,\xi)),$

therefore, by Lemma 3.6 and Lemma 3.7, the required boundedness of $Q$
follows. This completes the proof of the subelliptic sharp Gårding inequality.
∎

##  4\. Final remarks

Due to the statement of our main result some remarks are in order.

Considering the general version of the Euclidean sharp Gårding inequality for
$(\rho,\delta)$-classes, and its correspondent in the compact Lie group
setting given by Theorem 1.1 in the case $\kappa=1$, that is Corollary 1.2,
one would expect that the suitable version of the subelliptic sharp Gårding
inequality should be as follows:

Conjecture. Let $G$ be a compact Lie group and let
$\mathcal{L}=\mathcal{L}_{X}$ be the (positive) sub-Laplacian associated with
a system $X=\\{X_{i}\\}_{i=1}^{k}$ of left-invariant vector fields satisfying
Hörmander's condition of step $\kappa$. For $0\leq\delta<\rho\leq 1$ and
$m\in\mathbb{R}$, let $A\equiv
a(x,D):C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ be a continuous linear
operator with symbol
$a\in{S}^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$. Then, if
$a(x,[\xi])\geq 0$ for all $(x,[\xi])\in G\times\widehat{G}$, there exists a
positive constant $C$ such that

$\mathsf{Re}(Au,u)\geq-C\|u\|_{H^{\frac{m-(\rho-\delta)}{2},\,\mathcal{L}}(G)}^{2},$
(4.1)

for all $u\in C^{\infty}(G)$.

The subelliptic setting and the noncommutativity of the group have not allowed
us to prove the expected version (4.1) of the subelliptic sharp Gårding
inequality. However, it is worth to make some considerations to stress the
importance and nontriviality of our result.

Let us first recall that some inclusions between global subelliptic classes
and standard global classes can be established. Specifically, one has that

  * 1.

If $m>0$, $0\leq\delta<\frac{\rho}{\kappa}$, and $\rho\leq 1$, then
$S^{m,\mathcal{L}}_{\rho,\delta}\subset\mathscr{S}^{m}_{\frac{\rho}{\kappa},\delta}(G)$.

  * 2.

If $m\leq 0$, $0\leq\delta<\frac{\rho}{k}$, and $\rho\leq 1$, then
$S^{m,\mathcal{L}}_{\rho,\delta}\subset\mathscr{S}^{\frac{m}{\kappa}}_{\frac{\rho}{k},\delta}(G)$.

Three immediate consequences of the inclusions above and of Corollary 1.2 are
described below.

Consequence 1. Let $m>\frac{\rho}{\kappa}-\delta>0$,
$0\leq\delta<\frac{\rho}{\kappa}\leq\frac{1}{\kappa}$, and $a\in
S^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$. Then, if
$a(x,[\xi])\geq 0$ for all $(x,[\xi])\in G\times\widehat{G}$, there exists a
positive constant $C$ such that

$\mathsf{Re}(Au,u)\geq-C\|u\|_{H^{\frac{m-(\frac{\rho}{\kappa}-\delta)}{2}}(G)}^{2}\geq-C\|u\|_{H^{\frac{\kappa
m-(\frac{\rho}{\kappa}-\delta)}{2},\,\mathcal{L}}(G)}^{2},$ (4.2)

for all $u\in C^{\infty}(G)$.

Consequence 2. Let $0<m\leq\frac{\rho}{\kappa}-\delta$,
$0\leq\delta<\frac{\rho}{\kappa}\leq\frac{1}{\kappa}$, and $a\in
S^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$. Then, if
$a(x,[\xi])\geq 0$ for all $(x,[\xi])\in G\times\widehat{G}$, there exists a
positive constant $C$ such that

$\mathsf{Re}(Au,u)\geq-C\|u\|_{H^{\frac{m-(\frac{\rho}{\kappa}-\delta)}{2}}(G)}^{2}\geq-C\|u\|_{H^{\frac{m-(\frac{\rho}{\kappa}-\delta)}{2},\,\mathcal{L}}(G)}^{2},$
(4.3)

for all $u\in C^{\infty}(G)$.

Consequence 3. Let $m\leq 0$,
$0\leq\delta<\frac{\rho}{\kappa}\leq\frac{1}{\kappa}$, and $a\in
S^{m,\,\mathcal{L}}_{\rho,\delta}(G\times\widehat{G})$. Then, if
$a(x,[\xi])\geq 0$ for all $(x,[\xi])\in G\times\widehat{G}$, there exists a
positive constant $C$ such that

$\mathsf{Re}(Au,u)\geq-C\|u\|_{H^{\frac{\frac{m}{\kappa}-(\frac{\rho}{\kappa}-\delta)}{2}}(G)}^{2}\geq-C\|u\|_{H^{\frac{\frac{m}{\kappa}-(\frac{\rho}{\kappa}-\delta)}{2},\,\mathcal{L}}(G)}^{2},$
(4.4)

for all $u\in C^{\infty}(G)$.

The previous inequalities, namely (4.2), (4.3) and (4.4), show that Theorem
1.1 for subelliptic classes ($\kappa\neq 1$) does not follow from Theorem 1.1
applied to standard global classes ($\kappa=1$), that is, from our Corollary
1.2. In other words, the subelliptic sharp Gårding inequality is not a
consequence of the so called elliptic sharp Gårding inequality. Note also that
Theorem 1.1 applied to subelliptic classes gives results better than those in
(4.2), (4.3) and (4.4).

To conclude, let us say that it is natural to conjecture that (4.1) holds true
in the subelliptic setting. Of course one can find particular operators for
which (4.1) is satisfied. An immediate example is given by operators of the
form $A=a(x)\mathcal{L}$, where $a(x)$ is a nonnegative smooth function and
$\mathcal{L}$ is a positive sub-Laplacian. However, this is a very specific
operator for which the validity of (4.1) follows from simple and direct
computations. For the sake of completeness we will briefly prove this fact
below.

Note that we can write

$\displaystyle\mathsf{Re}(a(x)\mathcal{L}u,u)$
$\displaystyle=\mathsf{Re}\big{(}\big{[}M_{a},\mathcal{L}^{1/2}\big{]}\mathcal{L}^{1/2}u,u\big{)}+\mathsf{Re}(\mathcal{L}^{1/2}a\mathcal{L}^{1/2}u,u),$
$\displaystyle=\big{(}\big{[}M_{a},\mathcal{L}^{1/2}\big{]}\mathcal{L}^{1/2}u,u\big{)}+\|\sqrt{a}\mathcal{L}^{1/2}u\|^{2}_{L^{2}(G)},$

where $\big{[}M_{a},\mathcal{L}^{1/2}\big{]}$ stands for the commutator
between the multiplicative operator $M_{a}f:=af$ and $\mathcal{L}^{1/2}$. In
this particular case (it is not true in general in our non Euclidean setting),
we have that $\big{[}M_{a},\mathcal{L}^{1/2}\big{]}$ is of subelliptic order
$0$, namely its symbol belongs to the class
$S^{0,\mathcal{L}}_{1,0}(G\times\widehat{G})$. This allows us to estimate
$\mathsf{Re}\big{(}\big{[}M_{a},\mathcal{L}^{1/2}\big{]}\mathcal{L}^{1/2}u,u\big{)}=\mathsf{Re}\big{(}\mathcal{L}^{-1/4}\big{[}M_{a},\mathcal{L}^{1/2}\big{]}\mathcal{L}^{1/2}u,\mathcal{L}^{1/4}u\big{)}\leq
C\|u\|_{1/2,\mathcal{L}}^{2}$, for some $C>0$, and to conclude that

$\mathsf{Re}(a(x)\mathcal{L}u,u)\geq-C\|u\|_{1/2,\mathcal{L}}^{2},\quad\forall
u\in C^{\infty}(G),$

where the index $\frac{1}{2}=\frac{m-1}{2}$ ($m=2$) is ``optimal''.

Of course, by using the steps above, we have that (4.1) holds true for any
operator of the form $A=a(x)\mathcal{L}^{m}$ with $m\in\mathbb{R}$.

That said, it is important to underline once more that the noncommutative
structure plays a determinant role in the validity of fundamental a priori
estimates such as the one studied here, therefore it would not be surprising
if the expected optimal result in (4.1) can be attained only under very
particular conditions and/or with tools still to be developed.

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