#  $M$-Ellipticity of Fredholm Pseudo-Differential Operators on
$L^{p}(\mathbb{R}^{n})$ and Gårding’s Inequality

Aparajita Dasgupta  Aparajita Dasgupta: Department of Mathematics Indian
Institute of Technology, Delhi, Hauz Khas New Delhi-110016 India E-mail
address<EMAIL_ADDRESS>and  Lalit Mohan  Lalit Mohan: Department
of Mathematics Indian Institute of Technology, Delhi India E-mail address
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the $M$-ellipticity of Fredholm pseudo-differential
operators associated with weighted symbols on $L^{p}(\mathbb{R}^{n})$,
$1<p<\infty$. We also prove the Gårding’s inequality for $M$-elliptic
operators and the hybrid class of pseudo-differential operators, namely
SG$M$-elliptic operators.

###### Key words and phrases:

$M$-ellipticity, Fredholm Operators, Gårding’s Inequality

###### 1991 Mathematics Subject Classification:

Primary 47G30; Secondary 35S05

###### Contents

  1. 1 Introduction
  2. 2 $M$-Ellipticity of Fredholm Pseudo-Differential Operators
  3. 3 Gårding’s Inequality for $M$-elliptic operators

##  1\. Introduction

A general symbolic calculus and corresponding definition of global ellipticity
for partial differential operators and pseudo-differential operators on
noncompact manifolds represent a relevant issue of the modern Mathematical
Analysis. In the case of the Euclidean space $\mathbb{R}^{n}$, one can refer
to the pseudo-differential operators corresponding to the Hörmander symbol
class, $S^{m}_{\rho,\delta}$, $m\in\mathbb{R}$, $0\leq\rho,\delta\leq 1,$ [12,
19]. The other is given by the so-called SG classes. In [7], they are also
called pseudo-differential operators with exit behavior. SG pseudo-
differential operators and related topics can be found in [3, 5, 6, 13, 14]
and the references therein. In general to prove $L^{p}$ boundedness of pseudo-
differential operators associated with the Hörmander symbols, $S^{0}_{1,0},$
the key ingredient is the Mikjlin-Hörmander theorem about Fourier multipliers.
But unfortunately, $L^{p}$-boundedness theorem for $p\neq 2$ doesnot hold for
operators in $S^{0}_{\rho,0},$ for $0<\rho<1.$ Taylor in [15] introduced a
suitable symbol subclass, $M^{m}_{\rho,0},$ of $S^{m}_{\rho,0}$ giving a
continuous operator for every $1<p<\infty$ and $0<\rho\leq 1.$ Further to
this, Garello and Morando introduced the subclass $M^{m}_{\rho,\Lambda}$ of
$S^{m}_{\rho,\Lambda},$ which are just weighted version of the ones introduced
by Taylor in [8, 9, 10] and developed the symbolic calculus for the associated
pseudo-differential operators with many applications to study the regularity
of multi-quasi-elliptic operators. Many studies on various properties of
$M$-elliptic pseudo-differential operators can be found in [8, 18, 2, 4, 10]
and [17]. But till now not much has been done on the $M$-ellipticity of
Fredholm pseudo-differential operators. In this paper, we prove that Fredholm
pseudo differential operators with weighted symbol are $M$-elliptic pseudo-
differential operators on $\mathbb{R}^{n}$. Finally we prove the Gårding’s
Inequality for $M-$elliptic operators. Let us start with a few historical
notes and basic definitions.

A positive function $\Lambda\in C^{\infty}(\mathbb{R}^{n})$ is said to be a
weight function with polynomial growth if there exists positive constants
$\mu_{0}$, $\mu_{1}$, $C_{0}$ and $C_{1}$ be such that $\mu_{0}\leq\mu_{1}$
and $C_{0}\leq C_{1}$, for which we have following conditions:

$C_{0}(1+|\xi|)^{\mu_{0}}\leq\Lambda(\xi)\leq C_{1}(1+|\xi|)^{\mu_{1}},\hskip
8.5359pt\xi\in\mathbb{R}^{n}.$

Also, we assume that there exists a real constant $\mu$ such that
$\mu\geq\mu_{1}$ and for all multi-indices $\alpha$,$\gamma$
$\in\mathbb{Z}^{n}$ with $\gamma_{j}\in\\{0,1\\},j=1,2,...,n$, we can find a
positive constant say $C_{\alpha,\gamma}$ such that

$\displaystyle|\xi^{\gamma}(\partial^{\alpha+\gamma}\Lambda)(\xi)|\leq
C_{\alpha,\gamma}\Lambda(\xi)^{1-\frac{1}{\mu}|\alpha|},\hskip
8.5359pt\xi\in\mathbb{R}^{n}.$ (1.1)

Let us take $m\in\mathbb{R}$ and let $\rho\in(0,\frac{1}{\mu}]$. Then we say a
function $\sigma\in C^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ is in
$S_{\rho,\Lambda}^{m}$ class if for all multi-indices $\alpha$ and $\beta$ we
can find a positive constant $C_{\alpha,\beta}$ such that

$|(\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\sigma)(x,\xi)|\leq
C_{\alpha,\beta}\Lambda(\xi)^{m-\rho|\beta|},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

We call such $\sigma$ be a symbol of order $m$ and type $\rho$ with weight
$\Lambda$. Also, we say a symbol $\sigma\in M_{\rho,\Lambda}^{m}$ if for all
multi-indices $\gamma$ with $\gamma_{j}\in\\{0,1\\},j=1,2,...,n$, we have

$\xi^{\gamma}(\partial_{\xi}^{\gamma}\sigma)(x,\xi)\in S_{\rho,\Lambda}^{m}.$

Let $\sigma\in S_{\rho,\Lambda}^{m}$. Define pseudo-differential operator
$T_{\sigma}$ associate with symbol $\sigma$ by

$(T_{\sigma}\varphi)(x)=(2\pi)^{-\frac{n}{2}}\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}\sigma(x,\xi)\hat{\varphi}(\xi)\,d\xi,\hskip
8.5359ptx\in\mathbb{R}^{n},$

where $\varphi$ is in Schwartz space $\mathcal{S}$ and

$\hat{\varphi}(\xi)=(2\pi)^{-\frac{n}{2}}\int_{\mathbb{R}^{n}}e^{-ix\cdot\xi}\varphi(x)\,dx,\hskip
8.5359pt\xi\in\mathbb{R}^{n}.$

Now we give a brief introduction to the properties of $M$-elliptic pseudo-
differential operators. For this, we will start by defining a class of symbols
$S_{\rho,\Lambda}^{m}$, where $m$, $\in\mathbb{R}$ and
$\rho\in(0,\frac{1}{\mu}]$.
We say a symbol $\sigma\in C^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ is
in $S_{\rho,\Lambda}^{m}$ if for all multi-indices $\alpha$ we can find a
positive constant $C_{\alpha}$ such that

$|(\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\sigma)(x,\xi)|\leq
C_{\alpha}\Lambda(\xi)^{m-\rho|\alpha|},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

Also, we say a symbol $\sigma\in M_{\rho,\Lambda}^{m}$ if for all multi-
indices $\gamma$ with $\gamma_{j}\in\\{0,1\\},j=1,2,...,n$, we have

$\xi^{\gamma}(\partial_{\xi}^{\gamma}\sigma)(x,\xi)\in S_{\rho,\Lambda}^{m}.$

Let $\sigma\in S_{\rho,\Lambda}^{m}$. Define pseudo-differential operator
$T_{\sigma}$ associated with symbol $\sigma$ by

$\displaystyle(T_{\sigma}\varphi)(x)=(2\pi)^{-\frac{n}{2}}\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}\sigma(x,\xi)\hat{\varphi}(\xi)\,d\xi,\hskip
8.5359ptx\in\mathbb{R}^{n},$ (1.2)

where $\varphi$ is in Schwartz space $\mathcal{S}$ and

$\hat{\varphi}(\xi)=(2\pi)^{-\frac{n}{2}}\int_{\mathbb{R}^{n}}e^{-ix\cdot\xi}\varphi(x)\,dx,\hskip
8.5359pt\xi\in\mathbb{R}^{n}.$

Note that it can be easily shown that
$T_{\sigma}:\mathcal{S}\rightarrow\mathcal{S}$ is a continuous linear mapping.
The following results can be found in [6].

######  Theorem 1.1.

Let $\sigma\in M_{\rho,\Lambda}^{m}$ and $\tau\in M_{\rho,\Lambda}^{\mu}$.
Then $T_{\sigma}T_{\tau}=T_{\lambda}$, where $\lambda\in
M_{\rho,\Lambda}^{m+\mu}$ and

$\lambda\sim\sum_{\mu}\frac{(-i)^{|\mu|}}{\mu!}(\partial_{\xi}^{\mu}\sigma)(\partial_{x}^{\mu}\tau).$

Here the asymptotic expansion means that for every positive integer $M$, there
exists a positive integer $N$ such that

$\lambda-\sum_{|\mu|<N}\frac{(-i)^{|\mu|}}{\mu!}(\partial_{\xi}^{\mu}\sigma)(\partial_{x}^{\mu}\tau)\in
M_{\rho,\Lambda}^{m+\mu-\rho M}.$

######  Theorem 1.2.

Let $\sigma\in M_{\rho,\Lambda}^{m}$. Then the formal adjoint
$T_{\sigma}^{\ast}$ of $T_{\sigma}$ is the pseudo-differential operator
$T_{\tau}$, where $\tau\in M_{\rho,\Lambda}^{m}$ and

$\tau\sim\sum_{\mu}\frac{(-i)^{|\mu|}}{\mu!}\partial_{\xi}^{\mu}\partial_{x}^{\mu}\overline{\sigma}.$

Here the asymptotic expansion means that for every positive integer $M$, there
exists a positive integer $N$ such that

$\tau-\sum_{|\mu|<N}\frac{(-i)^{|\mu|}}{\mu!}\partial_{\xi}^{\mu}\partial_{x}^{\mu}\overline{\sigma}\in
M_{\rho,\Lambda}^{m-\rho M}.$

Now using formal adjoint, we can extend the definition of a pseudo-
differential operator from the Schwartz space $\mathcal{S}$ to the space
$\mathcal{S}^{\prime}$ of all tempered distributions. For this, let $\sigma\in
M_{\rho,\Lambda}^{m}$. Then for all $u$ in $\mathcal{S}^{\prime}$, we define
$T_{\sigma}u:\mathcal{S}\rightarrow\mathbb{C}$ such that

$\displaystyle(T_{\sigma}u)(\varphi)=u(\overline{T_{\sigma}^{\ast}\overline{\varphi}}).$
(1.3)

It is easy to check that $T_{\sigma}$ maps $\mathcal{S}^{\prime}$ into
$\mathcal{S}^{\prime}$ continuously. In fact, we have the following theorem.

######  Theorem 1.3.

Let $\sigma\in M_{\rho,\Lambda}^{0}$. Then
$T_{\sigma}:L^{p}(\mathbb{R}^{n})\rightarrow L^{p}(\mathbb{R}^{n})$ is a
bounded linear operator for $1<p<\infty$.

Proof of this theorem can be found in from Theorem 1.4 in [18].

Let $\sigma\in M_{\rho,\Lambda}^{m}$, where $m\in\mathbb{R}$. Then $\sigma$ is
said to be $M$-elliptic if there exists positive constants $C$ and $R$ such
that

$|\sigma(x,\xi)|\geq C\Lambda(\xi)^{m},\hskip 8.5359pt|\xi|\geq R.$

######  Theorem 1.4.

Let $\sigma\in M_{\rho,\Lambda}^{m}$ be $M$-elliptic. Then there exists a
symbol $\tau\in M_{\rho,\Lambda}^{-m}$ such that

$T_{\tau}T_{\sigma}=I+R$

and

$T_{\sigma}T_{\tau}=I+S,$

where $R$ and $S$ are pseudo-differential operators with symbol in
$\bigcap_{k\in\mathbb{R}}M_{\rho,\Lambda}^{k}$.

The pseudo-differential operator $T_{\tau}$ in Theorem 1.4 is known as
parametrix of the $M$-elliptic pseudo-differential operator $T_{\sigma}$.
To make this paper self contained we recall here the definition of the Sobolev
spaces.

For $m\in\mathbb{R}$ and $1\,<\,p\,<\,\infty$, we define the Sobolev space
$H_{\Lambda}^{m,p}$ by

$H_{\Lambda}^{m,p}=\\{u\in\mathcal{S}^{\prime}:J_{-m2}\,u\in\,L^{p}(\mathbb{R}^{n})\\}.$

Then $H_{\Lambda}^{m,p}$ is a Banach space in which the norm $\|\,\|_{m,p}$ is
given by

$\|u\|_{m,p,\Lambda}=\|J_{-m}\,u\|_{L^{p}(\mathbb{R}^{n})},\hskip
8.5359ptu\,\in\,H_{\Lambda}^{m,p}.$

Note that $H_{\Lambda}^{0,p}=L^{p}(\mathbb{R}^{n})$.

All the above results and definitions for SG-$M$ elliptic case can be found in
[1].

The main aim of this paper is to prove the ellipticity of fredholm pseudo-
differential operators associated with weighted symbols and to prove the
Gårding’s Inequality for $M$-elliptic operators. In Section 2, first, we prove
that Fredholm pseudo-differential operators are $M$-elliptic and then show
that ellipticity of fredholmn pseudo-differential operators for the hybrid
case. In Section 3, we prove the Gårding’s inequality for both $M$-elliptic
operators and SG $M$-elliptic operators.

##  2\. $M$-Ellipticity of Fredholm Pseudo-Differential Operators

In this section, our aim is to show that Fredholm pseudo-differential operator
is $M$-elliptic on $L^{p}(\mathbb{R}^{n})$. For this, we need some technical
preparations which are done in [5] and [19].

######  Definition 2.1.

Let $\lambda>0,\tau\geq 0$ and $x_{0},\xi_{0}\in\mathbb{R}^{n}$. For
$1<p<\infty$, we define the operator
$R_{\lambda,\tau}\left(x_{0},\xi_{0}\right):L^{p}\left(\mathbb{R}^{n}\right)\rightarrow
L^{p}\left(\mathbb{R}^{n}\right)$, by

$\left(R_{\lambda,\tau}\left(x_{0},\xi_{0}\right)u\right)(x)=\lambda^{\tau
n/p}\,e^{i\lambda
x\cdot\xi_{0}}\,u\left(\lambda^{\tau}\left(x-x_{0}\right)\right),\quad
x\in\mathbb{R}^{n},$

for all $u\in L^{p}\left(\mathbb{R}^{n}\right)$.

######  Proposition 2.2.

The operator
$R_{\lambda,\tau}\left(x_{0},\xi_{0}\right):L^{p}\left(\mathbb{R}^{n}\right)\rightarrow
L^{p}\left(\mathbb{R}^{n}\right)$ is a surjective isometry and the inverse is
given by

$\left(R_{\lambda,\tau}\left(x_{0},\xi_{0}\right)^{-1}u\right)(x)=\lambda^{-\tau
n/p}\,e^{-i\lambda(x_{0}+\lambda^{-\top}x)\cdot\xi_{0}}\,u\left(x_{0}+\lambda^{-\tau}x\right),\quad
x\in\mathbb{R}^{n},$

for all $u\in L^{p}\left(\mathbb{R}^{n}\right)$.

######  Proposition 2.3.

Let $1<p<\infty$ and $\tau>0$. Then, for all $u\in
L^{p}\left(\mathbb{R}^{n}\right)$ and $v\in
L^{p^{\prime}}\left(\mathbb{R}^{n}\right)$, where
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$,

$\left(R_{\lambda,\tau}\left(x_{0},\xi_{0}\right)u,v\right)\rightarrow 0$

as $\lambda\rightarrow\infty$.

The following theorem is one of the main theorems of this paper.

######  Theorem 2.4.

Let $\sigma\in M_{\rho,\Lambda}^{0}$ be such that
$T_{\sigma}:L^{p}\left(\mathbb{R}^{n}\right)\rightarrow
L^{p}\left(\mathbb{R}^{n}\right)$ is a Fredholm operator for $1<p<\infty.$
Then $\sigma$ is $M$-elliptic.

To prove this theorem, we need following lemma.

######  Lemma 2.5.

Let $\sigma\in M_{\rho,\Lambda}^{m}$, where $m\in(-\infty,\infty).$ Then

$\displaystyle
R_{\lambda,\tau}\left(x_{0},\xi_{0}\right)^{-1}T_{\sigma}R_{\lambda,\tau}\left(x_{0},\xi_{0}\right)=T_{\sigma_{\lambda}\tau},$
(2.1)

where

$\displaystyle\sigma_{\lambda,\tau}(x,\eta)=\sigma\left(x_{0}+\lambda^{-\tau}x,\lambda\xi_{0}+\lambda^{\top}\eta\right),\quad
x,\eta\in\mathbb{R}^{n}.$ (2.2)

Moreover, if $\sigma\in M_{\rho,\Lambda}^{0},\lambda\geq
1,0\leq\tau\leq\rho\mu_{0}/(1+\rho\mu_{0})$ and $\xi_{0}\neq 0$, then for all
multi-indices $\alpha$ and $\beta$, there exists a positive constant
$C_{\beta}$ such that

$\displaystyle\left|\left(\partial_{x}^{\alpha}\partial_{\eta}^{\beta}\sigma_{\lambda,\tau}\right)(x,\eta)\right|\leq
C_{\beta}p_{\alpha,\beta}(\sigma)\frac{\left(\Lambda(\eta)\right)^{\rho|\beta|}}{\left|\xi_{0}\right|^{\rho\mu_{0}|\beta|}}\lambda^{-\tau|\alpha|}\lambda^{-(\rho\mu_{0}-(1+\rho\mu_{0})\tau)|\beta|},\quad
x,\eta\in\mathbb{R}^{n}.$ (2.3)

Here $p_{\alpha,\beta}$ denotes the corresponding norm in
$M_{\rho,\Lambda}^{0}.$

Proof  Fix $x_{0},\xi_{0}\in\mathbb{R}^{n}$. Let $u\in\mathcal{S}$. Then for
all $x\in\mathbb{R}^{n}$, we can write
$\left(R_{\lambda,\tau}\left(x_{0},\xi_{0}\right)^{-1}T_{\sigma}R_{\lambda,\tau}\left(x_{0},\xi_{0}\right)u\right)(x)$
in the form

$=e^{-i\lambda\left(x_{0}+\lambda^{-\top}x\right)\cdot\xi_{0}}\,(2\pi)^{-n}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i\lambda\left(x_{0}+\lambda^{-\tau}x-y\right)\cdot\xi}\sigma\left(x_{0}+\lambda^{-\tau}x,\xi\right)e^{i\lambda
y\cdot\xi_{0}}u\left(\lambda^{\tau}\left(y-x_{0}\right)\right)dyd\xi.$

Then by substituting $z=\lambda^{\tau}(y-x_{0})$, the above expression takes
the form

$=\lambda^{-\tau
n}(2\pi)^{-n}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i\lambda^{-\tau}\left(\xi-\lambda\xi_{0}\right)\cdot(x-z)}\sigma\left(x_{0}+\lambda^{-\tau}x,\xi\right)u(z)dzd\xi.$

Now by substituting $\eta=\lambda^{-\tau}(\xi-\lambda\xi_{0})$, the above
expression takes the form

$=(2\pi)^{-n}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{i(x-z)\cdot\eta}\sigma\left(x_{0}+\lambda^{-\tau}x,\lambda\xi_{0}+\lambda^{\tau}\eta\right)u(z)dzd\eta.$

Thus, we get (2.1) and (2.2), as asserted. Now, using (2.2), the chain rule
and Peetre’s inequality,

$\displaystyle\left|\left(\partial_{x}^{\alpha}\partial_{\eta}^{\beta}\sigma_{\lambda,\tau}\right)(x,\eta)\right|$
$\displaystyle=\left|\left(\partial_{x}^{\alpha}\partial_{\eta}^{\beta}\sigma\right)\left(x_{0}+\lambda^{-\tau}x,\lambda\xi_{0}+\lambda^{\tau}\eta\right)\lambda^{-\tau|\alpha|}\lambda^{\tau|\beta|}\right|$
$\displaystyle\leq
p_{\alpha,\beta}(\sigma)\left(\Lambda(\lambda\xi_{0}+\lambda^{\tau}\eta)\right)^{-\rho|\beta|}\lambda^{-\tau|\alpha|}\lambda^{\tau|\beta|}$
$\displaystyle\leq
C_{\beta}p_{\alpha,\beta}(\sigma)\left\langle\lambda\xi_{0}+\lambda^{\tau}\eta\right\rangle^{-\rho\mu_{0}|\beta|}\lambda^{-\tau|\alpha|}\lambda^{\tau|\beta|}$
$\displaystyle\leq
C_{\beta}p_{\alpha,\beta}(\sigma)\left(\lambda\xi_{0}\right\rangle^{-\rho\mu_{0}|\beta|}\left\langle\lambda^{\tau}\eta\right\rangle^{\rho\mu_{0}|\beta|}\lambda^{-\tau|\alpha|}\lambda^{\tau|\beta|}$
$\displaystyle\leq
C_{\beta}p_{\alpha,\beta}(\sigma)\left|\xi_{0}\right|^{-\rho\mu_{0}|\beta|}\langle\eta\rangle^{\rho\mu_{0}|\beta|}\lambda^{-(\rho\mu_{0}-(1+\rho\mu_{0})\tau)|\beta|}\lambda^{-\tau|\alpha|}$
$\displaystyle\leq
C_{\beta}^{\prime}p_{\alpha,\beta}(\sigma)\left|\xi_{0}\right|^{-\rho\mu_{0}|\beta|}\left(\Lambda(\eta)\right)^{\rho|\beta|}\lambda^{-\left(\rho\mu_{0}-(1+\rho\mu_{0})\tau\right)|\beta|}\lambda^{-\tau|\alpha|}$

which completes proof of Equation (2.3). $\Box$
Proof of Theorem 2.4 Since $T_{\sigma}$ is a Fredholm operator, so by Theorem
20.5 in [19], we can find a non-zero bounded linear operator $S$ on
$L^{p}\left(\mathbb{R}^{n}\right)$ and a compact operator $K$ on
$L^{p}\left(\mathbb{R}^{n}\right)$ such that

$ST_{\sigma}=I+K.$

Let $M$ be the set of all points $\xi$ in $\mathbb{R}^{n}$ such that there
exists a point $x$ in $\mathbb{R}^{n}$ for which

$|\sigma(x,\xi)|\leq\frac{1}{2\|S\|}.$

Now, if $M$ is bounded, then there exists a positive number $R$ such that

$|\xi|<R,\quad\xi\in M.$

Thus, for each point $\xi\in\mathbb{R}^{n}$ with $|\xi|\geq R$, we get, for
all $x\in\mathbb{R}^{n}$,

$|\sigma(x,\xi)|\geq\frac{1}{2\|S\|},$

and this implies that that $\sigma$ is $M$-elliptic. So, suppose that $M$ is
not bounded. Then there exists a sequence
$\left\\{\left(x_{k},\xi_{k}\right)\right\\}$ in
$\mathbb{R}^{n}\times\mathbb{R}^{n}$ such that

$\left|\xi_{k}\right|\rightarrow\infty$

as $k\rightarrow\infty$ and

$\left|\sigma\left(x_{k},\xi_{k}\right)\right|\leq\frac{1}{2\|S\|},\quad
k=1,2,\ldots$

Thus, there exists a subsequence of
$\left\\{\left(x_{k},\xi_{k}\right)\right\\}$, again denoted by
$\left\\{\left(x_{k},\xi_{k}\right)\right\\}$, such that

$\sigma\left(x_{k},\xi_{k}\right)\rightarrow\sigma_{\infty}$

for some complex number $\sigma_{\infty}$ as $k\rightarrow\infty.$ Therefore

$\displaystyle\left|\sigma_{\infty}\right|\leq\frac{1}{2\|S\|}.$ (2.4)

For $k=1,2,\ldots$, let $\lambda_{k}=\left|\xi_{k}\right|$. Then, by Lemma
2.5, we have

$R_{\lambda_{k},\tau}\left(x_{k},\frac{\xi_{k}}{\left|\xi_{k}\right|}\right)^{-1}T_{\sigma}R_{\lambda_{h},\tau}\left(x_{k},\frac{\xi_{k}}{\left|\xi_{k}\right|}\right)=T_{\sigma_{\lambda_{k},\tau}},$

where

$\sigma_{\lambda_{k},\tau}(x,\eta)=\sigma\left(x_{k}+\lambda_{k}^{-\tau}x,\xi_{k}+\lambda_{k}^{\tau}\eta\right),\quad
x,\eta\in\mathbb{R}^{n}.$

Let $\alpha$ and $\beta$ be arbitrary multi-indices. Then, by Equation (2.3),
there exists a positive constant $C_{\beta}$ such that

$\displaystyle\left|\left(\partial_{x}^{\alpha}\partial_{\eta}^{\beta}\sigma_{\lambda_{k},\tau}\right)(x,\eta)\right|\leq
C_{\beta}p_{\alpha,\beta}(\sigma)\left(\Lambda(\eta)\right)^{\rho|\beta|}\lambda_{k}^{-\tau|\alpha|}\lambda_{k}^{-(\rho\mu_{0}-(1+\rho\mu_{0})\tau)|\beta|},\quad
x,\eta\in\mathbb{R}^{n}.$ (2.5)

For $k=1,2,\ldots$, we define $\sigma_{k}^{\infty}$ by

$\sigma_{k}^{\infty}=\sigma_{\lambda_{k},\tau}(0,0)=\sigma\left(x_{k},\xi_{k}\right).$

Then, by using Theorem 7.3 in [19] and the estimate (2.5), we get

$\displaystyle\hskip
13.6572pt\left|\sigma_{\lambda_{k},\tau}(x,\eta)-\sigma_{k}^{\infty}\right|$
(2.6)
$\displaystyle=\left|\sigma_{\lambda_{k},\tau}(x,\eta)-\sigma_{\lambda_{k},\tau}(0,0)\right|$
$\displaystyle=\left|\sum_{|\gamma+\mu|=1}x^{\gamma}\eta^{\mu}\int_{0}^{1}\left(\partial_{x}^{\gamma}\partial_{\eta}^{\mu}\sigma_{\lambda_{k},\tau}\right)(\theta
x,\theta\eta)d\theta\right|$
$\displaystyle\leq\sum_{|\gamma+\mu|=1}|x|^{|\gamma|}|\eta|^{|\mu|}\int_{0}^{1}C_{\mu}p_{\gamma,\mu}(\sigma)\left(\Lambda(\eta)\right)^{\rho|\mu|}\lambda_{k}^{-\tau|\gamma|}\lambda_{k}^{-(\rho\mu_{0}-(1+\rho\mu_{0})\tau)|\mu|}d\theta\rightarrow
0$

uniformly for $(x,\eta)$ on every compact subset $K$ of
$\mathbb{R}^{n}\times\mathbb{R}^{n}$ as $k\rightarrow\infty$. Let
$u\in\mathcal{S}$. Then

$\left(T_{\sigma_{\lambda_{k}},\tau}u\right)(x)-\sigma_{k}^{\infty}u(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^{n}}e^{ix\cdot\eta}\left(\sigma_{\lambda_{k},\tau}(x,\eta)-\sigma_{k}^{\infty}\right)\hat{u}(\eta)d\eta$

for all $x\in\mathbb{R}^{n}$. By (2.6), the assumption that $\sigma\in
M_{\rho,\Lambda}^{0}$ and Lebesgue’s dominated convergence theorem,

$\left(T_{\sigma_{\lambda_{k},\tau}}u\right)(x)\rightarrow\sigma_{k}^{\infty}u(x)$

for all $x\in\mathbb{R}^{n}$ as $k\rightarrow\infty$. Moreover, for all
$l\in\mathbb{N}$, using (2.5) and an integration by parts, we can find a
positive constant $C_{\mu}$ for each $\mu$ with $|\mu|\leq 2l$ such that

$\displaystyle\left|\langle
x)^{2l}\left(T_{\sigma_{\lambda_{k},\tau}}u\right)(x)\right|$ $\displaystyle=$
$\displaystyle\left|\langle
x\rangle^{2l}(2\pi)^{-n/2}\int_{\mathbb{R}^{n}}e^{ix\cdot\eta}\sigma_{\lambda_{k},\tau}(x,\eta)\hat{u}(\eta)d\eta\right|$
$\displaystyle=$
$\displaystyle\left|(2\pi)^{-n/2}\int_{\mathbb{R}^{n}}\left(\left(I-\Delta_{\xi}\right)^{l}e^{ix\cdot\xi}\right)\sigma_{\lambda_{k},\tau}(x,\xi)\hat{u}(\xi)d\xi\right|$
$\displaystyle\leq$
$\displaystyle(2\pi)^{-n/2}\int_{\mathbb{R}^{n}}\left|\sum_{|\mu|\leq
2l}\frac{1}{\mu!}\left(P^{\mu}(D)\hat{u}\right)(\xi)\right|C_{\mu}(\Lambda(\xi))^{|\mu|}d\xi$
$\displaystyle\leq$
$\displaystyle(2\pi)^{-n/2}\int_{\mathbb{R}^{n}}\sum_{|\mu|\leq
2l}\frac{C_{\mu}^{\prime}}{\mu!}|\hat{u}(\xi)|\langle\xi\rangle^{2l-|\mu|}\langle\xi\rangle^{\mu_{1}|\mu|}d\xi$

for all $x\in\mathbb{R}^{n}$, where $P(D)=(I-\Delta)^{l}$. So, there exists a
positive constant $C$ such that

$\left|\left(T_{\sigma_{\lambda_{k},\tau}}u\right)(x)\right|^{p}\leq C\langle
x\rangle^{-2lp},\quad x\in\mathbb{R}^{n}.$

Now, if $2lp>n$, then $\langle x\rangle^{-2lp}\in
L^{1}\left(\mathbb{R}^{n}\right)$. So, there exists a positive constant
$C_{1}$ such that

$\left|\left(T_{\sigma_{\lambda_{k},\tau}}-\sigma_{k}^{\infty}\right)u(x)\right|^{p}\leq
C_{1}\langle x\rangle^{-2lp},\quad x\in\mathbb{R}^{n}.$

Thus,

$T_{\sigma_{\lambda_{k},\tau}}u\rightarrow\sigma_{\infty}u$

in $L^{p}\left(\mathbb{R}^{n}\right)$ as $k\rightarrow\infty$. Let $u$ be a
nonzero function in $L^{p}\left(\mathbb{R}^{n}\right).$ Since
$R_{\lambda_{k},\tau}\left(x_{k},\frac{\xi_{k}}{\xi_{k}\mid}\right)$ is an
isometry, it follows that

$\displaystyle 0<\|u\|_{p}=$
$\displaystyle\left\|R_{\lambda_{k},\tau}\left(x_{k},\frac{\xi_{k}}{\left|\xi_{k}\right|}\right)u\right\|_{p}$
(2.7) $\displaystyle=$
$\displaystyle\left\|\left(ST_{\sigma}-K\right)R_{\lambda_{k},\tau}\left(x_{k},\frac{\xi_{k}}{\left|\xi_{k}\right|}\right)u\right\|_{p}$
$\displaystyle\leq$
$\displaystyle\left\|ST_{\sigma}R_{\lambda_{k},\tau}\left(x_{k},\frac{\xi_{k}}{\left|\xi_{k}\right|}\right)u\right\|{p}+\left\|{KR_{\lambda_{k},\tau}}\left(x_{k},\frac{\xi_{k}}{\left|\xi_{k}\right|}\right)u\right\|_{p}$
$\displaystyle\leq$
$\displaystyle\left\|R_{\lambda_{k},\tau}\left(x_{k},\frac{\xi_{k}}{\left|\xi_{k}\right|}\right)^{-1}T_{\sigma}R_{\lambda_{k},\tau}\left(x_{k},\frac{\xi_{k}}{\left|\xi_{k}\right|}\right)u\right\|_{p}\|S\|+\left\|KR_{\lambda_{k},\tau}\left(x_{k},\frac{\xi_{k}}{\left|\xi_{k}\right|}\right)u\right\|_{p}.$

Now, using the fact that $K$ is a compact operator and Proposition 2.3, it
follows that

$\left\|KR_{\lambda_{k},\tau}\left(x_{k},\frac{\xi_{k}}{\left|\xi_{k}\right|}\right)u\right\|_{p}\rightarrow
0$

as $k\rightarrow\infty$. Then, by Equation (2.7),

$\displaystyle\|u\|_{p}\leq\|S\|\left|\sigma_{\infty}\right|\|u\|_{p}.$ (2.8)

Thus, Equation (2.4) and Equation (2.8) give the contradiction that

$\frac{1}{\|S\|}\leq\left|\sigma_{\infty}\right|\leq\frac{1}{2\|S\|},$

which completes the proof of the theorem.

The preceding theorem can be generalized to the following theorem.

######  Theorem 2.6.

Let $\sigma\in M_{\rho,\Lambda}^{m},$ where $m\in(-\infty,\infty)$, and let
$T_{\sigma}:H_{\Lambda}^{s,p}\rightarrow H_{\Lambda}^{s-m,p}$ be a Fredholm
operator for some $s\in(-\infty,\infty).$ Then $T_{\sigma}$ is an $M$-elliptic
operator.

Proof  By Theorem 1.6 in [18], the operators $T_{\sigma}:H^{s,p}\rightarrow
H_{\Lambda}^{s-m,p},J_{-s}:H_{\Lambda}^{s,p}\rightarrow
L^{p}\left(\mathbb{R}^{n}\right),J_{s}:L^{p}\left(\mathbb{R}^{n}\right)\rightarrow
H_{\Lambda}^{s,p},J_{m-s}:H^{s-m,p}\rightarrow
L^{p}\left(\mathbb{R}^{n}\right)$ and
$J_{s-m}:L^{p}\left(\mathbb{R}^{n}\right)\rightarrow H^{s-m,p}$ are bounded
linear operators. Here $J_{m}$ is a pseudo-differential operator with symbol
$\sigma_{m}\in M_{\rho,\Lambda}^{-m}$, where

$\sigma_{m}(\xi)=(\Lambda(\xi))^{-m},\quad\xi\in\mathbb{R}^{n}.$

Let

$\displaystyle J_{m-s}T_{\sigma}J_{s}=T_{\tau}.$ (2.9)

Then

$T_{\tau}:L^{p}\left(\mathbb{R}^{n}\right)\rightarrow
L^{p}\left(\mathbb{R}^{n}\right),$

where $\tau\in M_{\rho,\Lambda}^{0}.$ Since $J_{s}$ is bijective, it follows
that $J_{s}$ is Fredholm and $M$-elliptic for all $s\in(-\infty,\infty).$ So,
by Theorem 2.4, $T_{\tau}$ is $M$-elliptic. By Equation (2.9), the fact that
$J_{s},s\in\mathbb{R}$, is bijective and product of two $M$-elliptic operators
is again an $M$-elliptic operator, we get $T_{\sigma}$ is $M$-elliptic. $\Box$
The following theorem is a simple consequence of Theorem 2.4 and Theorem 2.6,
which proves the $M$-ellipticity of the Fredholm SG pseudo-differential
operators on $L^{p}(\mathbb{R}^{n})$. Details about the Sobolev spaces,
$H_{\Lambda}^{s_{1},s_{2},p},$ can be found in [1].

######  Theorem 2.7.

Let $\sigma\in M_{\rho,\Lambda}^{m_{1},m_{2}}$, where
$m_{1},m_{2}\in(-\infty,\infty)$ and let

$T_{\sigma}:H_{\Lambda}^{s_{1},s_{2},p}\rightarrow
H_{\Lambda}^{s_{1}-m_{1},s_{2}-m_{2},p}$

is a Fredholm operator for some $s_{1},s_{2}\in(-\infty,\infty)$. Then
$T_{\sigma}$ is a SG $M$-elliptic operator.

##  3\. Gårding’s Inequality for $M$-elliptic operators

We begin with a definition.

######  Definition 3.1.

Let $\sigma\in M_{\rho,\Lambda}^{m}$, where $m\in\mathbb{R}$. Then $\sigma$ is
said to be strongly $M$\- elliptic if there exist positive constants $C$ and
$R$ for which

$\operatorname{Re}(\sigma(x,\xi))\geq C\,\Lambda(\xi)^{m},\hskip
8.5359pt|\xi|\geq R.$

######  Theorem 3.2.

(Gårding’s Inequality For M-elliptic Operators)
Let $\sigma\in M_{\rho,\Lambda}^{2m}$, where $m\in\mathbb{R}$, be strongly
$M$\- elliptic symbol. Then we can find positive constants $C^{\prime}$ and
$C_{s}$ for every real number $s\geq\frac{\rho}{2}$ such that

$\operatorname{Re}(T_{\sigma}\,\varphi,\varphi)\geq
C^{\prime}\,\|\varphi\|_{m,2,\Lambda}^{2}-C_{s}\,\|\varphi\|_{m-s,2,\Lambda}^{2},\hskip
8.5359pt\varphi\in\mathcal{S}.$

To prove this theorem, we need the following three lemmas.

######  Lemma 3.3.

Let $\sigma\in S_{\rho,\Lambda}^{0}$ and $F$ be a $C^{\infty}$\- function on
the complex plane $\mathbb{C}$. Then $F\circ\sigma\in S_{\rho,\Lambda}^{0}$.

Proof  We need to prove that for any two multi-indices $\alpha$ and $\beta$,
we can find a positive constant $C_{\alpha,\beta}$ such that

$\displaystyle|(\partial_{x}^{\alpha}\partial_{\xi}^{\beta}(F\circ\sigma))(x,\xi)|\leq
C_{\alpha,\beta}\,\Lambda(\xi)^{-\rho|\beta|},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$ (3.1)

Let $\alpha$ and $\beta$ be two multi-indices such that $|\alpha+\beta|=0$,
i.e., $|\alpha|=0$ and $|\beta|=0$. Since $\sigma\in S_{\rho,\Lambda}^{0}$, so
we can find a positive constant $C$ such that

$|\sigma(x,\xi)|\leq C,\hskip 8.5359ptx,\xi\in\mathbb{R}^{n}.$

Thus, $F\circ\sigma$ is a bounded and $C^{\infty}$ function on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$. So we can find another positive constant
$C_{0}$ such that

$|(F\circ\sigma)(x,\xi)|\leq C_{0},\hskip 8.5359ptx,\xi\in\mathbb{R}^{n}.$

Hence Equation (3.1) is true for all multi-indices $\alpha$ and $\beta$ with
$|\alpha+\beta|=0$. Now, suppose that Equation (3.1) is true for all
$C^{\infty}$ functions $F$ on $\mathbb{C}$, $\sigma\in S_{\rho,\Lambda}^{0}$
and multi-indices $\alpha$ and $\beta$ with $|\alpha+\beta|\leq k$. Let
$\alpha$ and $\beta$ be multi-indices with $|\alpha+\beta|=k+1$. We first
suppose that

$\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}=\partial_{x}^{\alpha}\,\partial_{\xi}^{\gamma}\,\partial_{\xi_{j}}$

for some multi-index $\gamma$ with $|\alpha+\gamma|=k$ and for some
$j=1,2,...,n$. Then, by chain rule,

$(\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}(F\circ\sigma))(x,\xi)=\partial_{x}^{\alpha}\,\partial_{\xi}^{\gamma}\\{(F_{1}\circ\sigma)\,\partial_{\xi_{j}}\sigma+(F_{2}\circ\sigma)\,\partial_{\xi_{j}}\sigma\\}(x,\xi)$

for all $x$ and $\xi$ in $\mathbb{R}^{n}$, where $F_{1}$ and $F_{2}$ are the
partial derivatives of $F$ with respect to the first and second variables
respectively. Now, by Leibniz’s formula and the induction hypothesis, we can
find two positive constants $C_{\delta_{0},\delta}$ and
$C_{\alpha,\delta_{0},\gamma,\delta,j}$ such that

$\displaystyle|(\partial_{x}^{\alpha}\,\partial_{\xi}^{\gamma}\\{(F_{1}\circ\sigma)\,\partial_{\xi_{j}}\sigma\\})(x,\xi)|$
$\displaystyle\leq\sum_{\begin{subarray}{c}\delta_{0}\leq\alpha\\\
\delta\leq\gamma\end{subarray}}{\alpha\choose\delta_{0}}\,{\gamma\choose\delta}\,|(\partial_{x}^{\delta_{0}}\,\partial_{\xi}^{\delta}(F_{1}\circ\sigma))(x,\xi)|\,|(\partial_{x}^{\alpha-\delta_{0}}\,\partial_{\xi}^{\gamma-\delta}\,(\partial_{\xi_{j}}\sigma))(x,\xi)|$
$\displaystyle\leq\sum_{\begin{subarray}{c}\delta_{0}\leq\alpha\\\
\delta\leq\gamma\end{subarray}}{\alpha\choose\delta_{0}}\,{\gamma\choose\delta}\,C_{\delta_{0},\delta}\,\Lambda(\xi)^{-\rho|\delta|}\,C_{\alpha,\delta_{0},\gamma,\delta,j}\,\Lambda(\xi)^{-\rho(|\gamma|-|\delta|+1)}$
$\displaystyle\leq C_{\alpha,\gamma,j}\,\Lambda(\xi)^{-\rho(|\gamma|+1)}$
$\displaystyle=C_{\alpha,\gamma,j}\,\Lambda(\xi)^{-\rho(|\beta|)},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n},$

where

$C_{\alpha,\gamma,j}=\sum_{\begin{subarray}{c}\delta_{0}\leq\alpha\\\
\delta\leq\gamma\end{subarray}}{\alpha\choose\delta_{0}}\,{\gamma\choose\delta}\,C_{\delta_{0},\delta}\,C_{\alpha,\delta_{0},\gamma,\delta,j}.$

Similarly, we can find another positive constant
$C_{\alpha,\gamma,j}^{\prime}$ such that

$|(\partial_{x}^{\alpha}\,\partial_{\xi}^{\gamma}\\{(F_{2}\circ\sigma)\,\partial_{\xi_{j}}\sigma\\})(x,\xi)|\leq
C_{\alpha,\gamma,j}^{\prime}\,\Lambda(\xi)^{-\rho(|\beta|)},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

Hence

$|(\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}(F\circ\sigma))(x,\xi)|\leq(C_{\alpha,\gamma,j}+C_{\alpha,\gamma,j}^{\prime})\,\Lambda(\xi)^{-\rho(|\beta|)},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

Now, we assume that

$\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}=\partial_{x}^{\gamma}\,\partial_{x_{j}}\,\partial_{\xi}^{\beta}$

for some multi-index $\gamma$ with $|\gamma+\beta|=k$ and for some
$j=1,2,...,n$. Then, as before, we can find a positive constant
$C_{\alpha,\gamma,j}^{\prime\prime}$ such that

$|(\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}(F\circ\sigma))(x,\xi)|\leq
C_{\alpha,\gamma,j}^{\prime\prime}\,\Lambda(\xi)^{-\rho(|\beta|)},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

So, by the principle of mathematical induction, Equation (3.1) follows. $\Box$

######  Lemma 3.4.

Let $\sigma\in M_{\rho,\Lambda}^{0}$ and $F$ be a $C^{\infty}$\- function on
the complex plane $\mathbb{C}$. Then $F\circ\sigma\in M_{\rho,\Lambda}^{0}$.

Proof  Let
$S=\\{\gamma\in\mathbb{Z}_{0}^{n}:\gamma_{j}\in\\{0,1\\},j=1,2,...,n;n\in\mathbb{N}\\}$,
where $\mathbb{Z}_{0}=\mathbb{Z}_{+}\,\cup\,\\{0\\}.$
Then, we need to show that, for all $\gamma\in S$, we have

$\xi^{\gamma}\,\partial_{\xi}^{\gamma}\,(F\circ\sigma)(x,\xi)\in
S_{\rho,\Lambda}^{0},$

i.e., for all multi-indices $\alpha,\beta$, we can find a positive constant
$C_{\alpha,\beta}$ such that

$\displaystyle|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}(\xi^{\gamma}\,\partial_{\xi}^{\gamma}\,(F\circ\sigma)(x,\xi))|\leq
C_{\alpha,\beta}\,(\Lambda(\xi))^{-\rho\,|\beta|},\hskip
8.5359pt\forall\gamma\in S,\hskip 8.5359ptx,\xi\in\mathbb{R}^{n}.$ (3.2)

Let $|\alpha+\beta|=0.$ This implies $|\alpha|=0$ and $|\beta|=0.$ Then we
need to show that there exists a positive constant $C_{00}$ such that

$\displaystyle|(\xi^{\gamma}\,\partial_{\xi}^{\gamma}\,(F\circ\sigma)(x,\xi))|\leq
C_{00},\hskip 8.5359pt\forall\gamma\in S,\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$ (3.3)

Let $|\gamma|=0$. Then by Lemma 3.3, we can find a positive constant $C_{00}$
such that

$|(F\circ\sigma)(x,\xi)|\leq C_{00},\hskip 8.5359ptx,\xi\in\mathbb{R}^{n}.$

Now, suppose that Equation (3.3) is true for all $C^{\infty}$ functions $F$ on
$\mathbb{C}$, $\sigma\in M_{\rho,\Lambda}^{0}$ and multi-indices $\gamma$ with
$|\gamma|\leq k$. Let $\gamma$ be a multi-index with $|\gamma|=k+1$. We
suppose that
$\partial_{\xi}^{\gamma}=\partial_{\xi}^{\gamma_{0}}\,\partial_{\xi_{j}}$, for
some $j=1,2,...,n$ and multi-index $\gamma_{0}$ with $|\gamma_{0}|=k$. Then by
chain rule,

${\xi}^{\gamma}\,\partial_{\xi}^{\gamma}(F\circ\sigma)(x,\xi)={\xi}^{\gamma}\,\partial_{\xi}^{\gamma_{0}}\\{(F_{1}\circ\sigma)\,\partial_{\xi_{j}}\sigma+(F_{2}\circ\sigma)\,\partial_{\xi_{j}}\sigma\\}(x,\xi)$

for all $x$ and $\xi$ in $\mathbb{R}^{n}$, where $F_{1}$ and $F_{2}$ are the
partial derivatives of $F$ with respect to the first and second variables
respectively. Now, by Leibniz’s formula,

$\displaystyle{\xi}^{\gamma}\,\partial_{\xi}^{\gamma_{0}}((F_{1}\circ\sigma)(x,\xi)\,.\,\partial_{\xi_{j}}(\sigma(x,\xi)))$
$\displaystyle={\xi}^{\gamma}\,\sum_{\delta_{0}\leq\gamma_{0}}{\delta_{0}\choose\gamma_{0}}\,\partial_{\xi}^{\delta_{0}}(F_{1}\circ\sigma)(x,\xi)\,.\,\partial_{\xi}^{\gamma_{0}-\delta_{0}}(\partial_{\xi_{j}}\sigma(x,\xi))$
$\displaystyle=\sum_{\delta_{0}\leq\gamma_{0}}{\delta_{0}\choose\gamma_{0}}\,({\xi}^{\delta_{0}}\,\partial_{\xi}^{\delta_{0}}(F_{1}\circ\sigma)(x,\xi))\,.\,({\xi}^{\gamma_{0}-\delta_{0}+1}\partial_{\xi}^{\gamma_{0}-\delta_{0}+1}(\sigma(x,\xi))).$

Thus by induction hypothesis and using the fact that $\sigma\in
M_{\rho,\Lambda}^{0}$, we can find positive constants $C_{\delta_{0}}$ and
$C_{\gamma_{0},\delta_{0}}$ such that

$\displaystyle|{\xi}^{\gamma}\,\partial_{\xi}^{\gamma_{0}}((F_{1}\circ\sigma)(x,\xi)\,.\,\partial_{\xi_{j}}(\sigma(x,\xi)))|$
$\displaystyle\leq\sum_{\delta_{0}\leq\gamma_{0}}{\delta_{0}\choose\gamma_{0}}\,|({\xi}^{\delta_{0}}\,\partial_{\xi}^{\delta_{0}}(F_{1}\circ\sigma)(x,\xi))|\,.\,|({\xi}^{\gamma_{0}-\delta_{0}+1}\partial_{\xi}^{\gamma_{0}-\delta_{0}+1}(\sigma(x,\xi)))|$
$\displaystyle\leq\sum_{\delta_{0}\leq\gamma_{0}}{\delta_{0}\choose\gamma_{0}}\,C_{\delta_{0}}\,.\,C_{\gamma_{0},\delta_{0}}.$

So, we have

$|{\xi}^{\gamma}\,\partial_{\xi}^{\gamma_{0}}((F_{1}\circ\sigma)(x,\xi)\,.\,\partial_{\xi_{j}}(\sigma(x,\xi)))|\leq
C_{00}^{\prime},\hskip 8.5359ptx,\xi\in\mathbb{R}^{n},$

where

$C_{00}^{\prime}=\sum_{\delta_{0}\leq\gamma_{0}}{\delta_{0}\choose\gamma_{0}}\,C_{\delta_{0}}\,.\,C_{\gamma_{0},\delta_{0}}.$

Similarly, we can find a positive constant $C_{00}^{\prime\prime}$ such that

$|{\xi}^{\gamma}\,\partial_{\xi}^{\gamma_{0}}((F_{2}\circ\sigma)(x,\xi)\,.\,\partial_{\xi_{j}}(\sigma(x,\xi)))|\leq
C_{00}^{\prime\prime},\hskip 8.5359ptx,\xi\in\mathbb{R}^{n}.$

Hence

$|{\xi}^{\gamma}\,\partial_{\xi}^{\gamma}(F\circ\sigma)(x,\xi)|\leq(C_{00}^{\prime}+C_{00}^{\prime\prime}),\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

Thus, by the principle of mathematical induction, Equation (3.3) follows.
Now, suppose that Equation (3.2) is true for all $C^{\infty}$ functions $F$ on
$\mathbb{C}$, $\sigma\in M_{\rho,\Lambda}^{0}$ and multi-indices $\alpha$ and
$\beta$ with $|\alpha+\beta|\leq k$. Let $\alpha$ and $\beta$ be multi-indices
with $|\alpha+\beta|=k+1$. Then we need to show that we can find a positive
constant $C_{\alpha,\beta}$ such that

$\displaystyle|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}(\xi^{\gamma}\,\partial_{\xi}^{\gamma}\,(F\circ\sigma)(x,\xi))|\leq
C_{\alpha,\beta}\,(\Lambda(\xi))^{-\rho\,|\beta|},\hskip
8.5359pt\forall\gamma\in S,\hskip 8.5359ptx,\xi\in\mathbb{R}^{n}.$ (3.4)

We first suppose that

$\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}=\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta_{0}}\,\partial_{\xi_{j}}$

for some multi-index $\beta_{0}$ with $|\alpha+\beta_{0}|=k$ and for some
$j=1,2,...,n$.
Let $|\gamma|=0$. Then by Lemma 3.3, we can find a positive constant
$C_{\alpha,\beta}$ such that

$|(\partial_{x}^{\alpha}\partial_{\xi}^{\beta}(F\circ\sigma))(x,\xi)|\leq
C_{\alpha,\beta}\,\Lambda(\xi)^{-\rho|\beta|},\hskip
8.5359pt\xi\in\mathbb{R}^{n}.$

Now, suppose that Equation (3.4) is true for all $C^{\infty}$ functions $F$ on
$\mathbb{C}$, $\sigma\in M_{\rho,\Lambda}^{0}$ and multi-indices $\gamma$ with
$|\gamma|\leq k$. Let $\gamma$ be a multi-index with $|\gamma|=k+1$. We
suppose that
$\partial_{\xi}^{\gamma}=\partial_{\xi}^{\gamma_{0}}\,\partial_{\xi_{i}}$, for
some $i=1,2,...,n$ and multi-index $\gamma_{0}$ with $|\gamma_{0}|=k$. Then,
by chain rule,

$\displaystyle\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}(\xi^{\gamma}\,\partial_{\xi}^{\gamma}\,(F\circ\sigma)(x,\xi))$
$\displaystyle=\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}(\xi^{\gamma}\,\partial_{\xi}^{\gamma_{0}}\,\partial_{\xi_{i}}(F\circ\sigma)(x,\xi)))$
$\displaystyle=\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}\,({\xi}^{\gamma}\,\partial_{\xi}^{\gamma_{0}}\\{(F_{1}\circ\sigma)\,\partial_{\xi_{i}}\sigma+(F_{2}\circ\sigma)\,\partial_{\xi_{i}}\sigma\\}(x,\xi)).$

Thus, by Leibniz’s formula,

$\displaystyle\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}\,({\xi}^{\gamma}\,\partial_{\xi}^{\gamma_{0}}((F_{1}\circ\sigma)\,.\,\partial_{\xi_{i}}\sigma)(x,\xi))$
$\displaystyle=\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}\,\left({\xi}^{\gamma}\,\left(\sum_{\delta_{0}\leq\gamma_{0}}{\delta_{0}\choose\gamma_{0}}\,\partial_{\xi}^{\delta_{0}}(F_{1}\circ\sigma)(x,\xi)\,.\,\partial_{\xi}^{\gamma_{0}-\delta_{0}}(\partial_{\xi_{i}}\sigma(x,\xi))\right)\right)$
$\displaystyle=\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta_{0}}\,\partial_{\xi_{j}}\,\left(\sum_{\delta_{0}\leq\gamma_{0}}{\delta_{0}\choose\gamma_{0}}\,(\xi^{\delta_{0}}\,\partial_{\xi}^{\delta_{0}}(F_{1}\circ\sigma)(x,\xi))\,.\,({\xi}^{\gamma_{0}-\delta_{0}+1}\partial_{\xi}^{\gamma_{0}-\delta_{0}+1}\sigma(x,\xi))\right)$
$\displaystyle=\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta_{0}}\,\left(\sum_{\delta_{0}\leq\gamma_{0}}{\delta_{0}\choose\gamma_{0}}\,\partial_{\xi_{j}}\left((\xi^{\delta_{0}}\,\partial_{\xi}^{\delta_{0}}(F_{1}\circ\sigma)(x,\xi))\,.\,({\xi}^{\gamma_{0}-\delta_{0}+1}\partial_{\xi}^{\gamma_{0}-\delta_{0}+1}\sigma(x,\xi))\right)\right)$
$\displaystyle=\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta_{0}}\,\left(\sum_{\delta_{0}\leq\gamma_{0}}{\delta_{0}\choose\gamma_{0}}\,\partial_{\xi_{j}}\left(\xi^{\delta_{0}}\,\partial_{\xi}^{\delta_{0}}(F_{1}\circ\sigma)(x,\xi)\right)\,.\,({\xi}^{\gamma_{0}-\delta_{0}+1}\partial_{\xi}^{\gamma_{0}-\delta_{0}+1}\sigma(x,\xi))\right)$
$\displaystyle\hskip
11.38092pt+\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta_{0}}\,\left(\sum_{\delta_{0}\leq\gamma_{0}}{\delta_{0}\choose\gamma_{0}}\,(\xi^{\delta_{0}}\,\partial_{\xi}^{\delta_{0}}(F_{1}\circ\sigma)(x,\xi))\,.\,\partial_{\xi_{j}}\left(({\xi}^{\gamma_{0}-\delta_{0}+1}\partial_{\xi}^{\gamma_{0}-\delta_{0}+1}\sigma(x,\xi))\right)\right).$

First, consider

$\displaystyle\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta_{0}}\,\left(\sum_{\delta_{0}\leq\gamma_{0}}{\delta_{0}\choose\gamma_{0}}\,\partial_{\xi_{j}}\left(\xi^{\delta_{0}}\,\partial_{\xi}^{\delta_{0}}(F_{1}\circ\sigma)(x,\xi)\right)\,.\,({\xi}^{\gamma_{0}-\delta_{0}+1}\partial_{\xi}^{\gamma_{0}-\delta_{0}+1}\sigma(x,\xi))\right)$
$\displaystyle=\sum_{\delta_{0}\leq\gamma_{0}}\sum_{p_{0}\leq\beta_{0}}\sum_{p\leq\alpha}{\delta_{0}\choose\gamma_{0}}{p_{0}\choose\beta_{0}}{p\choose\alpha}\,\partial_{x}^{p}\,\partial_{\xi}^{p_{0}+1}\left(\xi^{\delta_{0}}\,\partial_{\xi}^{\delta_{0}}(F_{1}\circ\sigma)(x,\xi)\right).\,\partial_{x}^{\alpha-p}\,\partial_{\xi}^{\beta_{0}-p_{0}}\left({\xi}^{\gamma_{0}-\delta_{0}+1}\partial_{\xi}^{\gamma_{0}-\delta_{0}+1}\sigma(x,\xi)\right).$

Thus by induction hypothesis and using the fact that $\sigma\in
M_{\rho,\Lambda}^{0}$, we can find positive constants $C_{p,p_{0}}$ and
$C_{p,p_{0},\alpha,\beta_{0}}$ such that

$\displaystyle\left|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta_{0}}\,\left(\sum_{\delta_{0}\leq\gamma_{0}}{\delta_{0}\choose\gamma_{0}}\,\partial_{\xi_{j}}\left(\xi^{\delta_{0}}\,\partial_{\xi}^{\delta_{0}}(F_{1}\circ\sigma)(x,\xi)\right)\,.\,({\xi}^{\gamma_{0}-\delta_{0}+1}\partial_{\xi}^{\gamma_{0}-\delta_{0}+1}\sigma(x,\xi))\right)\right|$
$\displaystyle\leq\sum_{\delta_{0}\leq\gamma_{0}}\sum_{p_{0}\leq\beta_{0}}\sum_{p\leq\alpha}{\delta_{0}\choose\gamma_{0}}{p_{0}\choose\beta_{0}}{p\choose\alpha}\,C_{p,p_{0}}\,(\Lambda(\xi))^{-\rho|p_{0}+1|}\,.\,C_{p,p_{0},\alpha,\beta_{0}}\,(\Lambda(\xi))^{-\rho|\beta_{0}-p_{0}|}$
$\displaystyle=C_{\alpha,\beta}^{\prime}\,(\Lambda(\xi))^{-\rho|\beta|},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n},$

where

$C_{\alpha,\beta}^{\prime}=\sum_{\delta_{0}\leq\gamma_{0}}\sum_{p_{0}\leq\beta_{0}}\sum_{p\leq\alpha}{\delta_{0}\choose\gamma_{0}}{p_{0}\choose\beta_{0}}{p\choose\alpha}\,C_{p,p_{0}}\,C_{p,p_{0},\alpha,\beta_{0}}.$

Similarly, we can find a positive constant $C_{\alpha,\beta}^{\prime\prime}$
such that

$\displaystyle\left|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta_{0}}\,\left(\sum_{\delta_{0}\leq\gamma_{0}}{\delta_{0}\choose\gamma_{0}}\,(\xi^{\delta_{0}}\,\partial_{\xi}^{\delta_{0}}(F_{1}\circ\sigma)(x,\xi))\,.\,\partial_{\xi_{j}}\left(({\xi}^{\gamma_{0}-\delta_{0}+1}\partial_{\xi}^{\gamma_{0}-\delta_{0}+1}\sigma(x,\xi))\right)\right)\right|$
$\displaystyle\leq
C_{\alpha,\beta}^{\prime\prime}\,(\Lambda(\xi))^{-\rho|\beta|},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

Hence

$|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}\,({\xi}^{\gamma}\,\partial_{\xi}^{\gamma_{0}}((F_{1}\circ\sigma)\,.\,\partial_{\xi_{i}}\sigma)(x,\xi))|\leq(C_{\alpha,\beta}^{\prime}+C_{\alpha,\beta}^{\prime\prime})\,(\Lambda(\xi))^{-\rho|\beta|},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

Similarly, we can find a positive constant
$C_{\alpha,\beta}^{\prime\prime\prime}$ such that

$|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}\,({\xi}^{\gamma}\,\partial_{\xi}^{\gamma_{0}}((F_{2}\circ\sigma)\,.\,\partial_{\xi_{i}}\sigma)(x,\xi))|\leq
C_{\alpha,\beta}^{\prime\prime\prime}\,(\Lambda(\xi))^{-\rho|\beta|},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

Hence

$|\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}(\xi^{\gamma}\,\partial_{\xi}^{\gamma}\,(F\circ\sigma)(x,\xi))|\leq(C_{\alpha,\beta}^{\prime}+C_{\alpha,\beta}^{\prime\prime}+C_{\alpha,\beta}^{\prime\prime\prime})\,(\Lambda(\xi))^{-\rho|\beta|},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

Now, we assume that

$\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}=\partial_{x}^{\gamma}\,\partial_{x_{j}}\,\partial_{\xi}^{\beta}$

for some multi-index $\gamma$ with $|\gamma+\beta|=k$ and for some
$j=1,2,...,n$. Then, as before, we can find a positive constant
$C_{\alpha,\beta}^{\prime\prime\prime\prime}$ such that

$|(\partial_{x}^{\alpha}\,\partial_{\xi}^{\beta}(F\circ\sigma))(x,\xi)|\leq
C_{\alpha,\beta}^{\prime\prime\prime\prime}\,\Lambda(\xi)^{-\rho(|\beta|)},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

So, by the principle of mathematical induction, Equation (3.4) follows.
This completes the proof of the Equation (3.2). $\Box$

######  Lemma 3.5.

Let $\sigma$ be a strongly $M$-elliptic symbol in $M_{\rho,\Lambda}^{2m}$,
where $m\in\mathbb{R}$. Then we can find two positive constants $\eta$ and
$\kappa$ such that

$\operatorname{Re}(\sigma(x,\xi))\geq\eta\,(\Lambda(\xi))^{2m}-\kappa\,(\Lambda(\xi))^{2m-\rho},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

Proof  By strong ellipticity, there exist positive constants $C$ and $R$ such
that

$\operatorname{Re}(\sigma(x,\xi))\geq C\,(\Lambda(\xi))^{2m},\hskip
8.5359pt|\xi|\geq R.$

Since $\sigma\in M_{\rho,\Lambda}^{2m}$, we can find a positive constant $K$
such that

$|\sigma(x,\xi)|\leq K\,(\Lambda(\xi))^{2m},\hskip 8.5359ptx,\xi\in R^{n}.$

Therefore, if $m\geq 0$, then

$\displaystyle|\operatorname{Re}(\sigma(x,\xi))|\leq
K\,(\Lambda(\xi))^{2m}\leq K_{2m}\,(1+|\xi|)^{2\mu_{1}.m}\leq
K_{2m}\,(1+R)^{2\mu_{1}.m},\hskip 8.5359pt|\xi|\leq R,$ (3.5)

where

$K_{2m}=K\,.\,C_{1}^{2m}.$

And if $m<0$, then

$\displaystyle|\operatorname{Re}(\sigma(x,\xi))|\leq
K\,(\Lambda(\xi))^{2m}\leq K,\hskip 8.5359pt|\xi|\leq R.$ (3.6)

By Equation (3.5) and Equation (3.6), for given $m\in\mathbb{R}$, we can find
a positive constant $M$ such that

$\operatorname{Re}(\sigma(x,\xi))\geq-M,\hskip 8.5359pt|\xi|\leq R.$

Since $\frac{\operatorname{Re}\sigma}{(\Lambda(.))^{2m-\rho}}$ is continuous
on the compact set $\\{\xi\in\mathbb{R}^{n}:|\xi|\leq R\\}$, so we can find a
positive constant $\kappa$ such that

$\frac{\operatorname{Re}\sigma}{(\Lambda(\xi))^{2m-\rho}}>-\kappa,\hskip
8.5359pt|\xi|\leq R.$

Therefore

$\operatorname{Re}(\sigma(x,\xi))+\kappa\,(\Lambda(\xi))^{2m-\rho}>0,\hskip
8.5359pt|\xi|\leq R.$

Since
$\frac{\operatorname{Re}\sigma+\kappa\,(\Lambda(.))^{2m-\rho}}{(\Lambda(.))^{2m}}$
is a positive and continuous function on the compact set
$\\{\xi\in\mathbb{R}^{n}:|\xi|\leq R\\}$, so we can find another positive
constant $\delta$ such that

$\frac{\operatorname{Re}\sigma+\kappa\,(\Lambda(\xi))^{2m-\rho}}{(\Lambda(\xi))^{2m}}\geq\delta,\hskip
8.5359pt|\xi|\leq R.$

So, the lemma is proved if we let $\eta=min(C,\delta)$. $\Box$

Proof of Theorem 3.2 Let $T_{\tau}=J_{m}\,T_{\sigma}\,J_{m}$, where
$J_{m}=T_{\sigma_{m}}$ and $\sigma_{m}(\xi)=(\Lambda(\xi))^{-m}$. Then, using
the asymptotic expansion for the product of two pseudo-differential operators
in Theorem 1.2 in [18],

$T_{\sigma}\,J_{m}=T_{\tau_{1}},$

where

$\displaystyle\tau_{1}-(\Lambda(.))^{-m}\sigma\in M_{\rho,\Lambda}^{m-\rho}.$
(3.7)

Similarly,

$T_{\tau}=J_{m}\,T_{\tau_{1}}$

and

$\displaystyle\tau-(\Lambda(.))^{-m}\tau_{1}\in M_{\rho,\Lambda}^{-\rho}.$
(3.8)

Multiplying Equation (3.7) by $(\Lambda(.))^{-m}$ and adding the result to
Equation (3.8), we get

$\tau-(\Lambda(.))^{-2m}\sigma\in M_{\rho,\Lambda}^{-\rho}.$

Therefore

$\tau=(\Lambda(.))^{-2m}\sigma+r,$

where $r\in M_{\rho,\Lambda}^{-\rho}$. So, by Lemma 3.5,

$\operatorname{Re}\tau=(\Lambda(.))^{-2m}\,\operatorname{Re}\sigma+\operatorname{Re}r\geq\eta-\kappa(\Lambda(.))^{-\rho}+\operatorname{Re}r\geq\eta-\kappa^{\prime}\,(\Lambda(.))^{-\rho},$

where $\kappa^{\prime}$ is another positive constant. Therefore $\tau$
satisfies the conclusion of Lemma 3.5 with $m=0$. Let us suppose for a moment
that Gårding’s inequality is valid for $m=0$. Then we can find a positive
constant $C^{\prime}$ and a positive constant $C_{s}$ for every real number
$s\geq\frac{\rho}{2}$ such that

$\displaystyle\operatorname{Re}\left(T_{\sigma}\varphi,\varphi\right)$
$\displaystyle=\operatorname{Re}\left(J_{-m}T_{\tau}J_{-m}\varphi,\varphi\right)$
$\displaystyle=\operatorname{Re}\left(T_{\tau}J_{-m}\varphi,J_{-m}\varphi\right)$
$\displaystyle\geq
C^{\prime}\,\|J_{-m}\varphi\|_{0,2,\Lambda}^{2}-C_{s}\,\|J_{-m}\varphi\|_{-s,2,\Lambda}^{2}$
$\displaystyle=C^{\prime}\,\|\varphi\|_{m,2,\Lambda}^{2}-C_{s}\,\|\varphi\|_{m-s,2,\Lambda}^{2}$

for all $\varphi$ in $\mathcal{S}$. Now we need only to prove Gårding’s
inequality for $m=0$. Let $\sigma\in M_{\rho,\Lambda}^{0}$. Then, by Lemma
3.5, we can find positive constants $\eta$ and $\kappa$ such that

$\operatorname{Re}\sigma+\kappa\,(\Lambda(.))^{-\rho}\geq\eta.$

Let $F$ be a $C^{\infty}$ function on $\mathbb{C}$ such that

$F(z)=\sqrt{\frac{\eta}{2}+z},\hskip 8.5359ptz\in[0,\infty).$

Let $\tau$ be the function defined on $\mathbb{R}^{n}\times\mathbb{R}^{n}$ by

$\tau(x,\xi)=F\left(2\left(\operatorname{Re}\sigma(x,\xi)+\kappa\,(\Lambda(\xi))^{-\rho}-\eta\right)\right),\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

Then, by Lemma 3.4, $\tau\in M_{\rho,\Lambda}^{0}$, and for all $x$ and $\xi$
in $\mathbb{R}^{n}$,

$\displaystyle\tau(x,\xi)$
$\displaystyle=\sqrt{\frac{\eta}{2}+2\operatorname{Re}\sigma(x,\xi)+2\kappa\,(\Lambda(\xi))^{-\rho}-2\eta}$
$\displaystyle=\sqrt{2\operatorname{Re}\sigma(x,\xi)+2\kappa(\Lambda(\xi))^{-\rho}-\frac{3}{2}\eta}.$

Using the asymptotic expansion for the formal adjoint of a pseudo-
differential operator in Theorem 1.3 in [18], we get

$T_{\tau}^{\ast}=T_{\tau^{\ast}},$

where $\tau^{\ast}\in M_{\rho,\Lambda}^{0}$ and $\tau-\tau^{\ast}\in
M_{\rho,\Lambda}^{-\rho}$. Again, by using Theorem 1.2 in [18], we have

$T_{\tau}^{\ast}T_{\tau}=T_{\lambda},$

where

$\lambda-\tau^{\ast}\tau\in M_{\rho,\Lambda}^{-\rho}.$

If we let $r_{1}$ and $r_{1}^{\prime}$ in $M_{\rho,\Lambda}^{-\rho}$ be such
that

$\tau^{\ast}=\tau+r_{1}$

and

$\lambda=\tau^{\ast}\tau+r_{1}^{\prime}.$

Then

$\lambda=\left(\tau+r_{1}\right)\tau+r_{1}^{\prime}=2\operatorname{Re}\sigma+2\kappa\,(\Lambda(.))^{-\rho}-\frac{3}{2}\eta+r_{2},$

where

$r_{2}=r_{1}\tau+r_{1}^{\prime}\in M_{\rho,\Lambda}^{-\rho}.$

So, if we let $r_{3}=2\kappa\,(\Lambda(.))^{-\rho}+r_{2}\in
M_{\rho,\Lambda}^{-\rho}$, then we get

$\lambda=2\operatorname{Re}\sigma-\frac{3}{2}\eta+r_{3}.$

But

$2\operatorname{Re}\sigma=\sigma+\bar{\sigma}=\sigma+\sigma^{\ast}+r_{4}$

for some $r_{4}$ in $M_{\rho,\Lambda}^{-\rho}$. Therefore

$\lambda=\sigma+\sigma^{\ast}-\frac{3}{2}\eta+r_{5}$

for some $r_{5}$ in $M_{\rho,\Lambda}^{-\rho}$. Thus,

$\sigma+\sigma^{\ast}=\lambda+\frac{3}{2}\eta-r_{5}.$

Since

$\left(T_{\lambda}\varphi,\varphi\right)=\left(T_{\tau}\varphi,T_{\tau}\varphi\right)\geq
0,\hskip 8.5359pt\varphi\in\mathcal{S},$

it follows that

$\displaystyle 2\operatorname{Re}\left(T_{\sigma}\,\varphi,\varphi\right)$
$\displaystyle=\left(T_{\sigma}\,\varphi,\varphi\right)+\left(T_{\sigma}^{\ast}\,\varphi,\varphi\right)=\left(T_{\sigma+\sigma^{\ast}}\,\varphi,\varphi\right)$
$\displaystyle=\left(T_{\lambda}\,\varphi,\varphi\right)+\frac{3}{2}\eta\,\|\varphi\|_{0,2,\Lambda}^{2}-\left(T_{r_{5}}\varphi,\varphi\right)$
$\displaystyle\geq\eta\,\|\varphi\|_{0,2,\Lambda}^{2}+\left\\{\frac{\eta}{2}\,\|\varphi\|_{0,2,\Lambda}^{2}-\left\|T_{r_{5}}\,\varphi\right\|_{\frac{\rho}{2},2,\Lambda}\,\|\varphi\|_{-\frac{\rho}{2},2,\Lambda}\right\\},\hskip
8.5359pt\varphi\in\mathcal{S}.$

Then, by Theorem 1.6 in [18] and using the fact that $r_{5}\in
M_{\rho,\Lambda}^{-\rho}$, we can find a positive constant $\nu$ such that

$2\operatorname{Re}\left(T_{\sigma}\varphi,\varphi\right)\geq\eta\,\|\varphi\|_{0,2,\Lambda}^{2}+\left\\{\frac{\eta}{2}\,\|\varphi\|_{0,2,\Lambda}^{2}-\nu\,\|\varphi\|_{-\frac{\rho}{2},2,\Lambda}^{2}\right\\},\hskip
8.5359pt\varphi\in\mathcal{S}.$

But

$\nu\|\varphi\|_{-\frac{\rho}{2},2,\Lambda}^{2}=\int_{\mathbb{R}^{n}}\nu\,(\Lambda(\xi))^{-\rho}\,|\hat{\varphi}(\xi)|^{2}d\xi=I+J,$

where

$I=\int_{\nu\,(\Lambda(\xi))^{-\rho}\leq\frac{\eta}{2}}\nu\,(\Lambda(\xi))^{-\rho}\,|\hat{\varphi}(\xi)|^{2}\,d\xi$

and

$J=\int_{\nu(\Lambda(\xi))^{-\rho}\geq\frac{\eta}{2}}\nu\,(\Lambda(\xi))^{-\rho}\,|\hat{\varphi}(\xi)|^{2}\,d\xi.$

Obviously,

$I\leq\frac{\eta}{2}\int_{\mathbb{R}^{n}}|\hat{\varphi}(\xi)|^{2}d\xi=\frac{\eta}{2}\,\|\varphi\|_{0,2,\Lambda}^{2}.$

To estimate $J$, we note that

$\nu\,(\Lambda(\xi))^{-\rho}\geq\frac{\eta}{2}\Rightarrow\Lambda(\xi)\leq\left(\frac{2\nu}{\eta}\right)^{\frac{1}{\rho}}.$

So, for $\nu\,(\Lambda(\xi))^{-\rho}\geq\frac{\eta}{2}$, we get, for every
real number $s\geq\frac{\rho}{2}$,

$\displaystyle\nu\,(\Lambda(\xi))^{-\rho}$
$\displaystyle=\nu\,(\Lambda(\xi))^{2s-\rho}\,(\Lambda(\xi))^{-2s}$
$\displaystyle\leq\nu\left(\frac{2\nu}{\eta}\right)^{\frac{2s-\rho}{\rho}}\,(\Lambda(\xi))^{-2s}.$

Thus for every real number $s\geq\frac{\rho}{2}$,

$J\leq\nu\left(\frac{2\nu}{\eta}\right)^{\frac{2s-\rho}{\rho}}\int_{\mathbb{R}^{n}}(\Lambda(\xi))^{-2s}\,|\hat{\varphi}(\xi)|^{2}\,d\xi=C_{s}^{\prime}\,\|\varphi\|_{-s,2,\Lambda}^{2}.$

where
$C_{s}^{\prime}=\nu\,\left(\frac{2\nu}{\eta}\right)^{\frac{2s-\rho}{\rho}}$.
Therefore

$2\operatorname{Re}(T_{\sigma}\,\varphi,\varphi)\geq\eta\,\|\varphi\|_{0,2,\Lambda}^{2}-C_{s}^{\prime}\,\|\varphi\|_{-s,2,\Lambda}^{2},\hskip
8.5359pt\varphi\in\mathcal{S},$

and this completes the proof of the theorem.

######  Definition 3.6.

Let $\sigma\in M_{\rho,\Lambda}^{m_{1},m_{2}}$, where
$m_{1},m_{2}\in\mathbb{R}$. Then $\sigma$ is said to be strongly SG
$M$-elliptic if there exist positive constants $C$ and $R$ such that

$\operatorname{Re}(\sigma(x,\xi))\geq
C\Lambda(x)^{m_{2}}\Lambda(\xi)^{m_{1}},\hskip 8.5359pt|x|^{2}+|\xi|^{2}\geq
R^{2}.$

######  Theorem 3.7.

(Gårding’s Inequality For SG M-elliptic Operators)
Let $\sigma\in M_{\rho,\Lambda}^{2m_{1},2m_{2}}$, where
$m_{1},m_{2}\in\mathbb{R}$, be strongly SG $M$-elliptic symbol. Then we can
find positive constants $C^{\prime}$ and $C_{s_{1},s_{2}}$ for every real
number $s_{1}\leq\frac{\rho}{2}$ , $s_{2}\geq\frac{\rho}{2}$ such that

$\operatorname{Re}(T_{\sigma}\,\varphi,\varphi)\geq
C^{\prime}\,\|\varphi\|_{m_{1},m_{2},2,\Lambda}^{2}-C_{s_{1},s_{2}}\,\|\varphi\|_{m_{1}-s_{1},m_{2}-s_{2},2,\Lambda}^{2},\hskip
8.5359pt\varphi\in\mathcal{S}.$

To prove this theorem, we need the following two lemmas.

######  Lemma 3.8.

Let $\sigma\in M_{\rho,\Lambda}^{0,0}$ and $F$ be a $C^{\infty}$\- function on
the complex plane $\mathbb{C}$. Then $F\circ\sigma\in M_{\rho,\Lambda}^{0,0}$.

Proof of the above lemma follows from the similar techniques as in Lemma 3.3
and Lemma 3.4.

######  Lemma 3.9.

Let $\sigma\in M_{\rho,\Lambda}^{2m_{1},2m_{2}}$, where
$m_{1},m_{2}\in\mathbb{R}$, be strongly SG $M$-elliptic symbol. Then we can
find two positive constants $\eta$ and $\kappa$ such that

$\operatorname{Re}(\sigma(x,\xi))\geq\eta\,(\Lambda(x))^{2m_{2}}\,(\Lambda(\xi))^{2m_{1}}-\kappa\,(\Lambda(x))^{2m_{2}-\rho}\,(\Lambda(\xi))^{2m_{1}-\rho},\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

Proof of the above lemma follows from the similar techniques as in Lemma 3.5.

Proof of Theorem 3.7 Let
$T_{\tau}=J_{m_{1},m_{2}}\,T_{\sigma}\,J_{m_{1},m_{2}}$, where
$J_{m_{1},m_{2}}=T_{\sigma_{m_{1},m_{2}}}$ and
$\sigma_{m_{1},m_{2}}(x,\xi)=(\Lambda(x))^{-m_{2}}\,(\Lambda(\xi))^{-m_{1}}$.
Then, by Theorem 1.1, we have

$T_{\sigma}\,J_{m_{1},m_{2}}=T_{\tau_{1}},$

where

$\displaystyle\tau_{1}-\sigma_{m_{1},m_{2}}\sigma\in
M_{\rho,\Lambda}^{m_{1}-\rho,m_{2}-\rho}.$ (3.9)

Similarly,

$T_{\tau}=J_{m_{1},m_{2}}\,T_{\tau_{1}}$

and

$\displaystyle\tau-\sigma_{m_{1},m_{2}}\tau_{1}\in
M_{\rho,\Lambda}^{-\rho,-\rho}.$ (3.10)

Multiplying Equation (3.9) by $\sigma_{m_{1},m_{2}}$ and adding the result to
Equation (3.10), we get

$\tau-\sigma_{m_{1},m_{2}}^{2}\sigma\in M_{\rho,\Lambda}^{-\rho,-\rho}.$

Therefore

$\tau=\sigma_{m_{1},m_{2}}^{2}\sigma+r,$

where $r\in M_{\rho,\Lambda}^{-\rho,-\rho}$. So, by Lemma 3.9,

$\operatorname{Re}\tau=\sigma_{m_{1},m_{2}}^{2}\,\operatorname{Re}\sigma+\operatorname{Re}r\geq\eta-\kappa\,(\Lambda(x))^{-\rho}\,(\Lambda(\xi))^{-\rho}+\operatorname{Re}r\geq\eta-\kappa^{\prime}\,(\Lambda(x))^{-\rho}\,(\Lambda(\xi))^{-\rho},$

where $\kappa^{\prime}$ is another positive constant. Therefore $\tau$
satisfies the conclusion of Lemma 3.9 with $m_{1}=0$ and $m_{2}=0$. Let us
suppose for a moment that Gårding’s inequality is valid for $m_{1}=0$ and
$m_{2}=0$. Then we can find a positive constant $C^{\prime}$ and a positive
constant $C_{s_{1},s_{2}}$ for every real number $s_{1}\leq\frac{\rho}{2}$ ,
$s_{2}\geq\frac{\rho}{2}$ such that

$\displaystyle\operatorname{Re}\left(T_{\sigma}\varphi,\varphi\right)$
$\displaystyle=\operatorname{Re}\left(J_{-m_{1},-m_{2}}T_{\tau}J_{-m_{1},-m_{2}}\,\varphi,\varphi\right)$
$\displaystyle=\operatorname{Re}\left(T_{\tau}J_{-m_{1},-m_{2}}\,\varphi,J_{-m_{1},-m_{2}}\,\varphi\right)$
$\displaystyle\geq
C^{\prime}\,\|J_{-m_{1},-m_{2}}\,\varphi\|_{0,0,2,\Lambda}^{2}-C_{s_{1},s_{2}}\,\|J_{-m_{1},-m_{2}}\,\varphi\|_{-s_{1},-s_{2},2,\Lambda}^{2}$
$\displaystyle=C^{\prime}\,\|\varphi\|_{m_{1},m_{2},2,\Lambda}^{2}-C_{s_{1},s_{2}}\,\|\varphi\|_{m_{1}-s_{1},m_{2}-s_{2},2,\Lambda}^{2}$

for all $\varphi$ in $\mathcal{S}$. Now we need only to prove Gårding’s
inequality for $m_{1}=0$ and $m_{2}=0$. Let $\sigma\in
M_{\rho,\Lambda}^{0,0}$. Then, by Lemma 3.9, we can find positive constants
$\eta$ and $\kappa$ such that

$\operatorname{Re}\sigma+\kappa\,(\Lambda(x))^{-\rho}\,(\Lambda(\xi))^{-\rho}\geq\eta.$

Let $F$ be a $C^{\infty}$ function on $\mathbb{C}$ such that

$F(z)=\sqrt{\frac{\eta}{2}+z},\hskip 8.5359ptz\in[0,\infty).$

Let $\tau$ be the function defined on $\mathbb{R}^{n}\times\mathbb{R}^{n}$ by

$\tau(x,\xi)=F\left(2\left(\operatorname{Re}\sigma(x,\xi)+\kappa\,(\Lambda(x))^{-\rho}\,(\Lambda(\xi))^{-\rho}-\eta\right)\right),\hskip
8.5359ptx,\xi\in\mathbb{R}^{n}.$

Then, by Lemma 3.8, $\tau\in M_{\rho,\Lambda}^{0,0}$, and for all $x$ and
$\xi$ in $\mathbb{R}^{n}$,

$\displaystyle\tau(x,\xi)$
$\displaystyle=\sqrt{\frac{\eta}{2}+2\operatorname{Re}\sigma(x,\xi)+2\kappa\,(\Lambda(x))^{-\rho}\,(\Lambda(\xi))^{-\rho}-2\eta}$
$\displaystyle=\sqrt{2\operatorname{Re}\sigma(x,\xi)+2\kappa\,(\Lambda(x))^{-\rho}\,(\Lambda(\xi))^{-\rho}-\frac{3}{2}\eta}.$

Hence, by Theorem 1.2, we get

$T_{\tau}^{\ast}=T_{\tau^{\ast}},$

where $\tau^{\ast}\in M_{\rho,\Lambda}^{0,0}$ and $\tau-\tau^{\ast}\in
M_{\rho,\Lambda}^{-\rho,-\rho}$. Again, by using Theorem 1.1, we have

$T_{\tau}^{\ast}T_{\tau}=T_{\lambda},$

where

$\lambda-\tau^{\ast}\tau\in M_{\rho,\Lambda}^{-\rho,-\rho}.$

If we let $r_{1}$ and $r_{1}^{\prime}$ in $M_{\rho,\Lambda}^{-\rho,-\rho}$ be
such that

$\tau^{\ast}=\tau+r_{1}$

and

$\lambda=\tau^{\ast}\tau+r_{1}^{\prime}.$

Then

$\lambda=\left(\tau+r_{1}\right)\tau+r_{1}^{\prime}=2\operatorname{Re}\sigma+2\kappa\,(\Lambda(x))^{-\rho}\,(\Lambda(\xi))^{-\rho}-\frac{3}{2}\eta+r_{2},$

where

$r_{2}=r_{1}\tau+r_{1}^{\prime}\in M_{\rho,\Lambda}^{-\rho,-\rho}.$

So, if we let
$r_{3}=2\kappa\,(\Lambda(x))^{-\rho}\,(\Lambda(\xi))^{-\rho}+r_{2}\in
M_{\rho,\Lambda}^{-\rho,-\rho}$, then we get

$\lambda=2\operatorname{Re}\sigma-\frac{3}{2}\eta+r_{3}.$

But

$2\operatorname{Re}\sigma=\sigma+\bar{\sigma}=\sigma+\sigma^{\ast}+r_{4}$

for some $r_{4}$ in $M_{\rho,\Lambda}^{-\rho,-\rho}$. Therefore

$\lambda=\sigma+\sigma^{\ast}-\frac{3}{2}\eta+r_{5}$

for some $r_{5}$ in $M_{\rho,\Lambda}^{-\rho,-\rho}$. Thus,

$\sigma+\sigma^{\ast}=\lambda+\frac{3}{2}\eta-r_{5}.$

Since

$\left(T_{\lambda}\varphi,\varphi\right)=\left(T_{\tau}\varphi,T_{\tau}\varphi\right)\geq
0,\hskip 8.5359pt\varphi\in\mathcal{S},$

it follows that

$\displaystyle 2\operatorname{Re}\left(T_{\sigma}\,\varphi,\varphi\right)$
$\displaystyle=\left(T_{\sigma}\,\varphi,\varphi\right)+\left(T_{\sigma}^{\ast}\,\varphi,\varphi\right)=\left(T_{\sigma+\sigma^{\ast}}\,\varphi,\varphi\right)$
$\displaystyle=\left(T_{\lambda}\,\varphi,\varphi\right)+\frac{3}{2}\eta\,\|\varphi\|_{0,0,2,\Lambda}^{2}-\left(T_{r_{5}}\varphi,\varphi\right)$
$\displaystyle\geq\eta\,\|\varphi\|_{0,0,2,\Lambda}^{2}+\left\\{\frac{\eta}{2}\,\|\varphi\|_{0,0,2,\Lambda}^{2}-\left\|T_{r_{5}}\,\varphi\right\|_{\frac{\rho}{2},\frac{\rho}{2},2,\Lambda}\,\|\varphi\|_{-\frac{\rho}{2},-\frac{\rho}{2},2,\Lambda}\right\\},\hskip
8.5359pt\varphi\in\mathcal{S}.$

Then, by Theorem LABEL:bdd and using the fact that $r_{5}\in
M_{\rho,\Lambda}^{-\rho,-\rho}$, we can find a positive constant $\nu$ such
that

$2\operatorname{Re}\left(T_{\sigma}\varphi,\varphi\right)\geq\eta\,\|\varphi\|_{0,0,2,\Lambda}^{2}+\left\\{\frac{\eta}{2}\,\|\varphi\|_{0,0,2,\Lambda}^{2}-\nu\,\|\varphi\|_{-\frac{\rho}{2},-\frac{\rho}{2},2,\Lambda}^{2}\right\\},\hskip
8.5359pt\varphi\in\mathcal{S}.$

Since $\frac{\rho}{2}>0$, so by Theorem LABEL:contain-thm, there exists a
positive constant $C_{0}$ such that

$|\varphi\|_{-\frac{\rho}{2},0,2,\Lambda}^{2}\leq
C_{0}\,\|\varphi\|_{0,0,2,\Lambda}^{2},\hskip 8.5359pt\varphi\in\mathcal{S}.$

Now

$\nu\|\varphi\|_{-\frac{\rho}{2},-\frac{\rho}{2},2,\Lambda}^{2}=\int_{\mathbb{R}^{n}}\nu\,(\Lambda(x))^{-\rho}\,|(T_{\sigma_{\frac{\rho}{2}}}\varphi)(x)|^{2}dx=I+J,$

where

$I=\int_{\nu\,(\Lambda(x))^{-\rho}\leq\frac{\eta}{2C_{0}}}\nu\,(\Lambda(x))^{-\rho}\,|(T_{\sigma_{\frac{\rho}{2}}}\varphi)(x)|^{2}dx$

and

$J=\int_{\nu(\Lambda(x))^{-\rho}\geq\frac{\eta}{2C_{0}}}\nu\,(\Lambda(x))^{-\rho}\,|(T_{\sigma_{\frac{\rho}{2}}}\varphi)(x)|^{2}dx.$

Obviously,

$I\leq\frac{\eta}{2C_{0}}\int_{\mathbb{R}^{n}}|(T_{\sigma_{\frac{\rho}{2}}}\varphi)(x)|^{2}dx=\frac{\eta}{2C_{0}}\,\|\varphi\|_{-\frac{\rho}{2},0,2,\Lambda}^{2}\leq\frac{\eta}{2}\,\|\varphi\|_{0,0,2,\Lambda}^{2}.$

To estimate $J$, we note that

$\nu\,(\Lambda(x))^{-\rho}\geq\frac{\eta}{2C_{0}}\Rightarrow\Lambda(x)\leq\left(\frac{2\nu
C_{0}}{\eta}\right)^{\frac{1}{\rho}}.$

So, for $\nu\,(\Lambda(x)^{-\rho}\geq\frac{\eta}{2C_{0}}$, we get, for every
real number $s_{2}\geq\frac{\rho}{2}$,

$\displaystyle\nu\,(\Lambda(x))^{-\rho}$
$\displaystyle=\nu\,(\Lambda(x))^{2s_{2}-\rho}\,(\Lambda(x))^{-2s_{2}}$
$\displaystyle\leq\nu\left(\frac{2\nu
C_{0}}{\eta}\right)^{\frac{2s_{2}-\rho}{\rho}}\,(\Lambda(x))^{-2s_{2}}.$

Thus for every real number $s_{2}\geq\frac{\rho}{2}$,

$J\leq\nu\left(\frac{2\nu
C_{0}}{\eta}\right)^{\frac{2s_{2}-\rho}{\rho}}\int_{\mathbb{R}^{n}}(\Lambda(x))^{-2s_{2}}\,|(T_{\sigma_{\frac{\rho}{2}}}\varphi)(x)|^{2}dx=C_{s_{2}}^{\prime}\,\|\varphi\|_{-\frac{\rho}{2},-s_{2},2,\Lambda}^{2},$

where $C_{s_{2}}^{\prime}=\nu\left(\frac{2\nu
C_{0}}{\eta}\right)^{\frac{2s_{2}-\rho}{\rho}}$. Therefore

$2\operatorname{Re}(T_{\sigma}\,\varphi,\varphi)\geq\eta\,\|\varphi\|_{0,0,2,\Lambda}^{2}-C_{s_{2}}^{\prime}\,\|\varphi\|_{-\frac{\rho}{2},-s_{2},2,\Lambda}^{2},\hskip
8.5359pt\varphi\in\mathcal{S}.$

Since $\frac{\rho}{2}\geq s_{1}$, so by Theorem LABEL:contain-thm, there
exists a positive constant $C_{s_{1}}^{\prime}$ such that

$|\varphi\|_{-\frac{\rho}{2},-s_{2},2,\Lambda}^{2}\leq
C_{s_{1}}^{\prime}\,\|\varphi\|_{-s_{1},-s_{2},2,\Lambda}^{2},\hskip
8.5359pt\varphi\in\mathcal{S}.$

Thus for every real number $s_{1}\leq\frac{\rho}{2}$ and
$s_{2}\geq\frac{\rho}{2}$, we have

$2\operatorname{Re}(T_{\sigma}\,\varphi,\varphi)\geq\eta\,\|\varphi\|_{0,0,2,\Lambda}^{2}-C_{s_{1},s_{2}}\,\|\varphi\|_{-\frac{\rho}{2},-s_{2},2,\Lambda}^{2},\hskip
8.5359pt\varphi\in\mathcal{S},$

where $C_{s_{1},s_{2}}=C_{s_{1}}^{\prime}\,C_{s_{2}}^{\prime}$ and this
completes the proof of the theorem.

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