# Degenerate Schrödinger equations with irregular potentials

Duván Cardona  Duván Cardona: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium E-mail address
<EMAIL_ADDRESS>,  Marianna Chatzakou  Marianna Chatzakou:
Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent
University, Belgium E-mail address<EMAIL_ADDRESS>,  Julio
Delgado  Julio Delgado: Departmento de Matematicas Universidad del Valle Cali-
Colombia 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>
###### Abstract.

In this work we investigate a class of degenerate Schrödinger equations
associated to degenerate elliptic operators with irregular potentials on
$\mathbb{R}^{n}$ by introducing a suitable Hörmander metric $g$ and a
$g$-weight $m$. We establish the well-posedness for the corresponding
degenerate Schrödinger and degenerate parabolic equations. When the
subelliticity is available on the degenerate elliptic operator we deduce
spectral properties for a class of degenerate Hamiltonians. We also study the
$L^{p}$ mapping properties for operators with symbols in the $S(m^{-\beta},g)$
classes in the spirit of classical Fefferman's $L^{p}$-bounds for the
$(\rho,\delta)$ calculus. Finally, within our $S(m,g)$-classes, sharp
$L^{p}$-estimates and Schatten properties for Schrödinger operators for
Hörmander sums of squares are also investigated.

###### Key words and phrases:

Degenerate harmonic oscillators, degenerate elliptic operators, Schrödinger
equation, $L^{p}$-bounds, Schatten-von Neumann ideals

###### 2020 Mathematics Subject Classification:

35L80, 47G30, 35L40, 35A27

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). J. Delgado is
also supported by Vice. Inv. Universidad del Valle Grant CI 71329, MathAmSud
and Minciencias-Colombia under the project MATHAMSUD21-MATH-03. Marianna
Chatzakou is also supported by the FWO Fellowship grant No 12B1223N. Michael
Ruzhansky is also supported by EPSRC grant EP/R003025/2.

###### Contents

  1. 1 Introduction
    1. 1.1 Outline
    2. 1.2 State of the art and contributions
  2. 2 Preliminaries: Weyl-Hörmander calculus
  3. 3 The Hörmander metric adapted to the degenerate Hamiltonian
  4. 4 Some applications to well-posedness for degenerate Schrödinger and degenerate parabolic equations
  5. 5 $L^{p}$-bounds, $H^{1}$-$L^{1}$ and $L^{\infty}$-$BMO$ estimates for degenerate operators
    1. 5.1 $L^{\infty}$-$BMO$ and $H^{1}$-$L^{1}$ boundedness
    2. 5.2 $L^{p}$-boundedness
  6. 6 Improved $L^{p}$ estimates and Schatten properties for Schrödinger operators for Hörmander sums of squares
  7. 7 Schatten-von Neumann classes of the Hamiltonians

##  1\. Introduction

###  1.1. Outline

In this work we study a class of degenerate Schrödinger equations
corresponding to a Hamiltonian: $\mathcal{H}_{V}=a_{2}(x,D)+V(x)$, where
$a_{2}(x,D)$ is a second order degenerate elliptic operator on
$\mathbb{R}^{n}$ and the potential $V$ is a real-valued measurable function of
quadratic order at $\infty$. Our analysis covers the case where the principal
symbol $a_{2}(x,\xi)$ of the operator $a_{2}(x,D)$ which we assume to be
positive belongs to the Kohn-Nirenberg class of second order, that is, it
satisfies estimates of the type

$|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a_{2}(x,\xi)|\leq
C_{\alpha,\beta}(1+|\xi|)^{2-|\alpha|},\,\,(x,\xi)\in T^{*}\mathbb{R}^{n}.$
(1.1)

Note that by the spectral theory of second-order self-adjoint operators, the
positivity condition $a_{2}(x,\xi)\geq 0$ also covers the case of differential
operators given by

$a_{2}(x,D)=-\sum a_{ij}(x)\frac{\partial^{2}}{\partial x_{i}\partial
x_{j}}f+\mbox{ lower order terms},\quad\quad f\in
C_{0}^{\infty}(\mathbb{R}^{n}),$ (1.2)

with the coefficients $a_{ij}\in C_{b}^{\infty}(\mathbb{R}^{n})$, belonging to
the space of real-valued smooth functions which are uniformly bounded on
$\mathbb{R}^{n}$, together with all their derivatives, and so that
$A(x)=(a_{ij}(x))$ is a positive semi-definite matrix for every
$x\in\mathbb{R}^{n}.$ Then, our setting also includes Hörmander sub-Laplacians

$a_{2}(x,D)=-\sum\limits_{j=1}^{k}X_{j}^{2}$

that are sums of squares of real vector fields. For a differential operator
$a_{2}(x,D)$ whose symbol is as in (1.1), we introduce a metric $g$ on the
phase-space defined by

$g_{X}(dx,d\xi):=\frac{(\langle\xi\rangle^{2}+|x|^{2})}{m(x,\xi)}\left(dx^{2}+\frac{d\xi^{2}}{\langle\xi\rangle^{2}+|x|^{2}}\right)\,,$
(1.3)

where

$m(x,\xi)=a(x,\xi)+\langle(x,\xi)\rangle,$ (1.4)

with

$a(x,\xi)=a_{2}(x,\xi)+|x|^{2},\,\,X=(x,\xi),\,\,\langle(x,\xi)\rangle=(1+|x|^{2}+|\xi|^{2})^{\frac{1}{2}}.$

The metric and the weight above are sharp with respect to the Hörmander
conditions: temperateness, slow continuity, and the uncertainty principle,
that defines the $S(m,g)$ class that contains the Schrödinger operators
$a_{2}(x,D)+|x|^{2}.$

After establishing that we are in a suitable $S(m,g)$ pseudodifferential
calculus, i.e, the metric $g$ is indeed a Hörmander metric and $m$ is a
$g$-weight. We consider $V$ to be a real-valued Borel function defined on
$\mathbb{R}^{n}$ such that for appropriate constants $C,C_{1},C_{2}>0$,
$|V(x)|\leq C|x|^{2},$ for a.e.$|x|\geq C_{1}$ and $V(x)\geq-C_{2}$, for a.e.
$x\in\mathbb{R}^{n}$. Then, by using the machinery of the Weyl-Hörmander
calculus, we:

  * •

obtain the well-posedness of the following Cauchy problem on the Sobolev space
$H(M,g)$:

$\left\\{\begin{array}[]{rl}i\partial_{t}u&=\mathcal{H}_{V}u,\\\
u(0)&=f.\end{array}\right.$ (1.5)

  * •

We analyse the well posedness for degenerate parabolic equations of the form

$\left\\{\begin{array}[]{rl}\partial_{t}u&=\mathcal{H}u,\\\
u(0)&=f,\end{array}\right.$ (1.6)

on $L^{2}(\mathbb{R}^{n}).$ The Hamiltonian $\mathcal{H}$ is considered to be
irregular.

  * •

We establish the $L^{p}$-boundedness theory for the
$S(m^{-\frac{n\varepsilon}{2}},g)$-classes with the metric $g$ and the weight
$m$ as in (1.3) and in (1.4), respectively. Moreover, we give the sharp index
$\varepsilon$ in the case where $a_{2}(x,D)$ is the principal symbol of a
Hörmander sub-Laplacian. Consequently, we obtain the $L^{p}$-boundedness of
negative powers for our degenerate Schrödinger type operators in that case.

  * •

We investigate the membership of the operators in the classes $S(m^{-\mu},g)$
to the Schatten-von Neumann classes $S_{r}(L^{2}(\mathbb{R}^{n})).$
Consequently, the information about the distribution of the eigenvalues for a
class of Schrödinger type operators is obtained, as well as the rate of growth
of their eigenvalues.

###  1.2. State of the art and contributions

The study of the interplay of a Hamiltonian between Schrödinger semigroups and
heat semigroups in order to analyse the behaviour of the eigenfunctions of the
Hamiltonian on $L^{p}$ spaces has been of an active interest in the last
decades since the seminal works of B. Simon and Carmona [76], [77], [75],
[17]. See also [1], [53], [57] for further developments on this matter. Thus,
the presentation here for the $L^{2}$ setting is somewhat in the same spirit.
We will also obtain spectral properties for Hamiltonians of the form
$\mathcal{H}_{V}+J_{\langle(x,\xi)\rangle}$, where $J_{\langle(x,\xi)\rangle}$
is the pseudodifferential operator with symbol $\langle(x,\xi)\rangle$, by
means of the analysis of Schatten-von Neumann classes and where we will
require some growth condition on the potential $V$ at $\infty$ to guarantee
the existence of a discrete spectrum, as it is well-known from the classical
elliptic theory of Schrödinger operators. This will be the prototypical case
if e.g. $V=|x|^{2}.$ We obtain the spectral properties for
$\mathcal{H}_{V}+J_{\langle(x,\xi)\rangle}$ by studying negative powers of
such operators in suitable Schatten-von Neumann classes.

In order to avoid some eventual misunderstanding regarding the terminology of
degeneracies in this work and related works on the Schrödinger equation we
recall some points. In the context of the study of the Schrödinger equation,
it is customary to use the term degenerate in a different sense. Indeed, in
quantum mechanics, an energy level $E$ is called degenerate if there are two
or more states corresponding to the energy level $E$ of the quantum system.
The number of different states corresponding to $E$ is known as the degree of
degeneracy of the level in that setting. The investigation of the
hypoellipticity and $L^{2}$ estimates for degenerate Schrödinger operators has
been considered by Y. Morimoto in [68] as an extension of the Theorem 4 in
Chap. II of the seminal work of C. Fefferman [44].

An important feature of our results up to this point is that we do not impose
any subellipticity condition on the operator $a_{2}(x,D)$, allowing to have
for instance sum of squares without the Hörmander condition, and in particular
infinitely degenerate elliptic operators. It is also clear that our metric
neither satisfies the so-called strong uncertainty principle ([6], [71],
[72]), nor the Beals condition ([2], [3]) to guarantee the $L^{p}$ boundedness
of operators with simbols in $S(1,g)$, making it somehow patological from
other perspectives.

Moreover, we also establish the well-posedness for degenerate parabolic
equations when

$a_{2}(x,D)=-\sum\limits_{j=1}^{k}X_{j}^{2}$

is a sum of squares of real vector fields. We also extend the above results on
the well-posedness for positive powers of the Hamiltonian. Further, by using
the so-called geodesic temperateness of the metric $g$ ([62], [9]) and
assuming that $a_{2}(x,D)$ is subelliptic, we prove that

$\mathcal{H}_{V}+C:H(m,g)\rightarrow L^{2}(\mathbb{R}^{n}),$ (1.7)

is an isomorphism for $C>0$ large enough.

In particular, one can consider the important case of $a_{2}(x,\xi)$ being the
symbol of a sum of squares

$L=-\sum\limits_{j=1}^{k}X_{j}^{2},$

where $X_{j}$ are real vector fields on $\mathbb{R}^{n}$. In the case of
$n=2$, we will give an improved version of the $L^{p}$ bounds due to the
specific knowledge of the Hörmander condition when
$a_{2}(x,\xi)=\xi_{1}^{2}+\tilde{x_{1}}^{2}\xi_{2}^{2}$, where $\tilde{x_{1}}$
is given by (1.11). The (rescaled) operator under investigation that is now
expressed as follows

$a(x,D):=-(\partial_{x_{1}}^{2}+\tilde{x}_{1}^{2}\partial_{x_{2}}^{2})+|x|^{2}\,,$
(1.8)

boils down to a particular case of the so-called degenerate Schrödinger
operator, also known as a degenerate harmonic oscillator. To justify the use
of term ``degenerate harmonic oscillator'' let us consider the case where
$\tilde{x}_{1}\equiv 1$; in this case the second order differential operator
on $\mathbb{R}^{2}$ given by

$a(x,D)=-(\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2})+|x|^{2}$

corresponds to the quantum harmonic oscillator on $\mathbb{R}^{2}$; see e.g.
the monograph [70]. In the degenerate situation described by (1.8) the
operators are parametrised by a suitable family of smooth functions,
$\tilde{x}_{i}\in C^{\infty}(\mathbb{R})$ that satisfy

$\displaystyle\tilde{x}_{1}:=\left\\{\begin{array}[]{ll}\textnormal{sgn}(x_{1})\,x_{1},&\quad|x_{1}|\leq
2\\\ c^{\prime}\,\textnormal{sgn}(x_{1}),&\quad|x_{1}|\geq
4.\end{array}\right.$ (1.11)

where $c^{\prime}\in\mathbb{R},$ with $c^{\prime}\neq 0$.

Our next aim is the mathematical investigation of the qualitative $L^{p}$
mapping properties for the negative powers of our degenerate Schrödinger
operators within the $S(m^{-t},g)$ classes. For other works on different
directions on the analysis of degenerate Schrödinger equations; see for
example, Doi [40, 41], Hara [46], Ichinose [55, 56], Hiroshi and Kajitani
[47], Mizohata [67] and Takeuchi [79] and the references therein.

The $L^{\infty}$-$BMO,$ the $H^{1}$-$L^{1}$ and the $L^{p}$-boundedness for
the pseudo-differential operators in the classes $S(m^{-t},g)$ will be studied
in Section 5, generalising to our context the estimates in Fefferman [42]. It
is important to point out that in the study of $L^{p}$ bounds for pseudo-
differential operators in $S(m,g)$ classes one has to take into account Beal's
condition. Indeed, R. Beals has characterised in [2] and [3] the Hörmander's
metrics $g$ ensuring the $L^{p}$ boundedness for operators in $OpS(1,g)$ and
$1<p<\infty$. The condition discovered by Beals for a split metric
$g_{x,\xi}(z,\theta)=g_{x,\xi}(z,0)+g_{x,\xi}(0,\theta)$ is

$g_{x,\xi}(0,\theta)\leq C.$

In the case of $S_{\rho,\delta}^{0}$ classes, the Beals condition is nothing
else but the well known restriction $\rho=1$. For our metric $g$ this
condition is equivalent to the inequality

$g_{\tilde{x},\tilde{\xi}}(0,\xi)=\frac{|\xi|^{2}}{a(x,\xi)+\langle
X\rangle}\leq C,$

which does not hold unless $a$ is elliptic. Since we are interested in the
degenerate case, we do not dispose of the $L^{p}$ boundedness for operators in
$OpS(1,g)$, however one can obtain $L^{p}$ boundedness for certain intervals
centered at $p=2$ and suitable negative powers of the weight $m(x,\xi)$
defined by (1.4). The latter is a well known situation in the case of
$S_{\rho,\delta}^{-t}$ classes when $\rho<1$ and has been established by
Fefferman ([42]). Other subsequent generalizations for $S(m,g)$-classes of the
Fefferman boundedness theorem [42] have been established by one of the authors
of this work in [20, 26]. Extensions of the Fefferman results to other non-
commutative structures can be found in [15] and [34]. $L^{p}$-bounds in the
context of the harmonic oscillator were investigated in [16].

Regarding the boundedness properties for the class
$\textnormal{Op}(S(m^{-t},g)),$ we found the following facts. For a non-
negative symbol $a_{2}=a_{2}(x,\xi)\in
S^{2}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ of a differential operator, we
have:

  * $\bullet$

If $\frac{3}{4}\leq\beta<1,$ then
$\textnormal{Op}(S(m^{-\frac{n}{2}\beta},g))\subset\mathscr{B}(L^{\infty}(\mathbb{R}^{n}),BMO(\mathbb{R}^{n})).$

  * $\bullet$

If $0\leq\beta<\frac{3}{4}$ and
$\sigma(x,D)\in\textnormal{Op}(S(m^{-\frac{n}{2}\beta},g))$, then
$\sigma(x,D):L^{p}(\mathbb{R}^{n})\rightarrow L^{p}(\mathbb{R}^{n})$ extends
to a bounded operator, provided that

$\left|\frac{1}{p}-\frac{1}{2}\right|\leq\frac{n}{2}\beta.$ (1.12)

This paper is organized as follows. In Section 2 we will present the necessary
background on the theory of pseudo-differential operators that we will use in
this paper, in particular some basics about the Weyl-Hörmander classes. In
Section 3, we prove, among other things, that the metric $g$ defined in (1.3)
is a Hörmander metric and the weight $m$ is indeed, a $g$-weight. In Section 4
we present some applications to the well-posedness for degenerate Schrödinger
operators and degenerate parabolic equations as was summarised above in the
$L^{2}$-setting. Sections 5 and 6 are devoted to the study of the $L^{p}$
boundedness for the classes $S(m^{-\beta},g)$ as we have described above. We
finish the paper with Section 7 where we deduce some consequences on spectral
properties for our degenerate Schrödinger operators.

##  2\. Preliminaries: Weyl-Hörmander calculus

In this section we summarise some preliminary topics about the Weyl
quantisation and the Weyl-Hörmander calculus. For this we will follow
Hörmander [52, Section 18.5] and Chapter 2 of [62].

The Weyl quantisation of
$a\in\mathscr{S}^{\prime}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ is the operator
$a(x,D):\mathscr{S}(\mathbb{R}^{n})\rightarrow\mathscr{S}^{\prime}(\mathbb{R}^{n})$
defined in the weak sense by the symmetric expression

$\displaystyle
a^{w}(x,D)u(x):=\int\limits_{\mathbb{R}^{n}}\int\limits_{\mathbb{R}^{n}}e^{2\pi
i\langle
x-y,\xi\rangle}a((x+y)/2,\xi)u(y)dyd\xi,\,\,u\in\mathscr{S}(\mathbb{R}^{n}).$

The Weyl quantisation of $a(\cdot,\cdot)$ is the special case of the
$\tau$-quantisation of $a(\cdot,\cdot)$ defined for $0\leq\tau\leq 1,$ by

$a^{\tau}(x,D)f(x):=\int\limits_{\mathbb{R}^{n}}\int\limits_{\mathbb{R}^{n}}e^{2\pi
i\langle x-y,\xi\rangle}a(\tau x+(1-\tau)y,\xi)u(y)dyd\xi,$

also defined in the weak sense. So, the Weyl quantisation of $a(\cdot,\cdot)$
is $(1/2)$-quantisation of $a(\cdot,\cdot),$ i.e.
$a^{\frac{1}{2}}(x,D)=a^{w}(x,D).$ The $1$-quantisation (Kohn-Nirenberg
quantisation) of $a(\cdot,\cdot),$ recovers the usual definition of the
pseudo-differential operator associated to the `symbol' $a(\cdot,\cdot):$

$a(x,D)u(x)\equiv a^{1}(x,D)u(x)=\int\limits_{\mathbb{R}^{n}}e^{2\pi i\langle
x,\xi\rangle}a(x,\xi)\widehat{u}(\xi)d\xi,$

where the equivalence of both representations is a consequence of the fact
that the Fourier transform is an isomorphism on the Schwartz space implying
that $\widehat{u}\in\mathscr{S}(\mathbb{R}^{n})$ where by abuse of notation we
have written $a(x,\xi)\widehat{u}:=a(x,\xi)\widehat{u}(\xi)$ which makes sense
if e.g. the distribution $a(\cdot,\cdot)$ agrees with a locally integrable
function. For $a,b\in\mathscr{S}(\mathbb{R}^{n}),$ and
$0\leq\tau,\tau^{\prime}\leq 1,$ we have the equality
$a^{\tau}(x,D)=b^{\tau^{\prime}}(x,D)$ if and only if

$a(x,\xi)=\int\limits_{\mathbb{R}^{n}}e^{-2\pi
i\langle\xi-\eta,z\rangle}b(x+(\tau^{\prime}-\tau)z,\eta)d\eta,$

which means that we always can pass from one quantisation to other one by
using the semigroup $(J_{t})_{t\in\mathbb{R}},$ defined by

$J_{t}a(x,\xi):=\int\limits_{\mathbb{R}^{n}}e^{-2\pi
i\langle\xi-\eta,z\rangle}b(x+tz,\eta)dyd\eta,$

which satisfies $J_{t+s}=J_{t}J_{s},$ $t,s\in\mathbb{R}.$ In particular, for
the Weyl quantisation and the Kohn-Nirenberg quantisation of $a$,
respectively, we have

$a(x,D)=(J_{1/2}a)^{w}(x,D),\quad\,a^{w}(x,D)=(J_{-1/2}a)(x,D).$

######  Remark 2.1 (The adjoint of the Weyl quantisation).

The following remarkable property: $a^{w}(x,D)^{*}=\overline{a}^{w}(x,D),$
implies that $a^{w}(x,D)^{*}={a}^{w}(x,D)$ for real valued symbols and it is
one of the reasons why Hermann Weyl introduced this quantisation for the
purposes of quantum mechanics (see Hörmander [52, page 151]). The other
important property is its symplectic invariance that we will not discuss here.

######  Remark 2.2 (Composition in the Weyl quantisation).

Let us write $X=(x,\xi),$ (or $Y=(y,\eta)$ etc.) for points on the phase space
$\mathbb{R}^{2n}=\mathbb{R}^{n}_{x}\times\mathbb{R}^{n}_{\xi},$ (resp.
$\mathbb{R}^{n}_{y}\times\mathbb{R}^{n}_{\eta}).$ The composition $a^{w}\circ
b^{w}(x,D)\equiv a^{w}(x,D)\circ b^{w}(x,D)$ in the Weyl quantisation is
related with the symplectic structure of
$\mathbb{R}^{n}\times\mathbb{R}^{n}=T^{*}\mathbb{R}^{n}$. Indeed, for $a,b\in
S(\mathbb{R}^{n})$, let us define

$(a\\#b)(X):=\frac{1}{\pi^{{2n}}}\int\limits_{\mathbb{R}^{2n}}\int\limits_{\mathbb{R}^{2n}}e^{-2i\sigma(X-Y_{1},X-Y_{2})}a(Y_{1})b(Y_{2})dY_{1}dY_{2},$

where $\sigma(X,Y):=\langle y,\xi\rangle-\langle x,\eta\rangle$ is the
symplectic form on $\mathbb{R}^{2n}.$ Then, in terms of the operation $\\#,$
we have $a^{w}(x,D)\circ b^{\omega}(x,D)=(a\\#b)^{\omega}(x,D)$.

In order to introduce the Weyl-Hörmander calculus we need the notion of a
Hörmander metric. One reason for this, is that if we want to study the
properties of some operator $L,$ we can to associate to $L$ this kind of
metric.

######  Definition 2.3 (Hörmander's metric).

For $X\in\mathbb{R}^{2n}$, let $g_{X}(\cdot)$ be a positive definite quadratic
form on $\mathbb{R}^{2n}$. We say that $g(\cdot)$ is a Hörmander metric if the
following three conditions are satisfied:

  1. i.

Continuity or slowness. There exist a constant $C>0$ such that

$g_{X}(X-Y)\leq
C^{-1}\Longrightarrow\left(\frac{g_{X}(T)}{g_{Y}(T)}\right)^{\pm 1}\leq C,$

for all $X,Y\in\mathbb{R}^{2n}$, $T\in\mathbb{R}^{2n}\setminus\\{0\\}$.

  2. ii.

Uncertainty principle. In terms of the symplectic form $\sigma(Y,Z):=\langle
z,\eta\rangle-\langle y,\zeta\rangle$, we define

$g_{X}^{\sigma}(T):=\sup_{W\neq 0}\frac{\sigma(T,W)^{2}}{g_{X}(W)}.$ (2.1)

We say that $g$ satisfies the uncertainty principle  if

$\lambda_{g}(X)=\inf_{T\neq
0}\left(\frac{g_{X}^{\sigma}(T)}{g_{X}(T)}\right)^{1/2}\geq 1,$

for all $X,T\in\mathbb{R}^{2n}$.

  3. iii.

Temperateness. We say that $g$ is temperate if there exist $\overline{C}>0$
and $J\in\mathbb{N}$ such that

$\left(\frac{g_{X}(T)}{g_{Y}(T)}\right)^{\pm
1}\leq\overline{C}(1+g_{Y}^{\sigma}(X-Y))^{J},$

for all $X,Y,T\in\mathbb{R}^{2n}$.

Let $g$ be a Hörmander metric. The uncertainty parameter or the Planck
function associated to $g$ is defined by

$h_{g}(X)^{2}=\sup_{T\neq 0}\frac{g_{X}(T)}{g_{X}^{\sigma}(T)}.$

Note that since $h_{g}(X)=(\lambda_{g}(X))^{-1}$, the uncertainty principle
can be translated into the condition

$h_{g}(X)\leq 1.$

######  Definition 2.4 (g-weight).

Let $M:\mathbb{R}^{2n}\rightarrow\mathbb{R}^{+}$ be a function. Then,

  * •

we say that $M$ is $g$-continuous if there exists $\tilde{C}>0$ such that

$g_{X}(X-Y)\leq\frac{1}{\tilde{C}}\Longrightarrow\left(\frac{M(X)}{M(Y)}\right)^{\pm
1}\leq\tilde{C}\,.$

for all $X,Y\in\mathbb{R}^{2n}$.

  * •

we say that $M$ is $g$-temperate if there exist $\tilde{C}>0$ and
$N\in\mathbb{N}$ such that

$\left(\frac{M(X)}{M(Y)}\right)^{\pm
1}\leq\tilde{C}(1+g_{Y}^{\sigma}(X-Y))^{N}\,,$

for all $X,Y\in\mathbb{R}^{2n}$.

We will say that $M$ is a $g$-weight if it is $g$-continuous and
$g$-temperate.

We can now define the classes of symbols adapted to a Hörmander metric $g$ and
a $g$-weight $M$.

######  Definition 2.5.

For a Hörmander metric $g$ and a $g$-weight $M$, the class $S(M,g)$ consists
of all smooth functions $\sigma\in C^{\infty}(\mathbb{R}^{2n})$ such that for
any $k\in\mathbb{N}$ there exists $C_{k}>0$, such that for all
$X,T_{1},...,T_{k}\in\mathbb{R}^{2n}$ we have

$|\sigma^{(k)}(X;T_{1}\otimes\cdots\otimes T_{k})|\leq
C_{k}M(X)\prod_{i=1}^{k}g_{X}^{1/2}(T_{i})\,.$ (2.2)

The notation $\sigma^{(k)}$ stands for the $k^{th}$ derivative of $\sigma$ and
$\sigma^{(k)}(X;T_{1},...,T_{k})$ denotes the $k^{th}$ derivative tensor of
$a$ at $X$ in the directions $T_{1},...,T_{k}$. For $\sigma\in S(M,g)$ we
denote by $\parallel\sigma\parallel_{k,S(M,g)}$ the minimum $C_{k}$ satisfying
the above inequality. The class $S(M,g)$ becomes a Fréchet space endowed with
the family of seminorms $\parallel\cdot\parallel_{k,S(M,g)}$.

The action of the Weyl quantization on the Sobolev spaces is determined by the
following theorem (cf. [8]).

######  Theorem 2.6.

Let $g$ be a Hörmander metric, and let $M_{1},M_{2}$ be two $g$-weights. Then
for any $a\in S(M,g)$ we have that

$a^{\tau}(x,D):H(M_{1},g)\longrightarrow H(M_{1}/M,g)\,,$

admits a bounded extension for all $\tau$. Since $H(1,g)=L^{2},$ in the
particular case $M_{1}=M,$ $a^{\tau}:H(M,g)\longrightarrow L^{2},$ admits a
bounded extension.

For the general aspects, about the $L^{p}$ and Hölder boundedness of pseudo-
differential operators we refer the reader to Beals [2, 3] and to more recent
works [20, 26] and [13].

##  3\. The Hörmander metric adapted to the degenerate Hamiltonian

Motivated by the study of the family of operators given by (1.8), in this
section, we will study the general metric (3.1) associated with the bigger
class of pseudo-differential operators with symbol as in (1.7). In particular,
in Lemma 3.7 we show that $a\in S(m,g)$ for $m,g$ as in the next theorem that
applies to the general situation on $\mathbb{R}^{n}$.

######  Theorem 3.1.

Let $a_{2}\in S^{2}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ be non-negative
$(\geq 0)$. We define the following Riemannian metric on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$,

$g_{X}(dx,d\xi):=m^{-1}(x,\xi)(\langle\xi\rangle^{2}+|x|^{2})dx^{2}+m^{-1}(x,\xi)d\xi^{2}\,,$
(3.1)

where

$m(x,\xi)=a(x,\xi)+\langle X\rangle,$ (3.2)

and the symbol $a$ is given by

$a(x,\xi)=a_{2}(x,\xi)+|x|^{2}\,.$ (3.3)

Then $g$ is a Hörmander metric.

Before giving the proof, the following inequality will be useful to prove the
continuity for the metric $g$ and it follows by a simple argument using
Taylor's formula of order $2$.

######  Proposition 3.2.

Let $f$ be a non-negative function in $C^{2}(\mathbb{R})$. Then

$(f^{\prime}(t))^{2}\leq\left\lVert
f^{\prime\prime}\right\rVert_{L^{\infty}}f(t),\mbox{ for all }t\in\mathbb{R}.$

Also the following lemma will be used to simplify the proof of continuity and
temperateness of the metric.

######  Lemma 3.3.

Let $g_{s}$ be the Riemannian metric on $\mathbb{R}^{n}\times\mathbb{R}^{n}$
defined by

$(g_{s})_{X}:=dx^{2}+\frac{d\xi^{2}}{\langle\xi\rangle^{2}+|x|^{2}}.$

Then $g_{s}$ is a Hörmander metric.

###### Proof.

Since $1\leq\langle\xi\rangle^{2}+|x|^{2},$ and
$g_{s}^{\sigma}(dx,d\xi)=(\langle\xi\rangle^{2}+|x|^{2})dx^{2}+d\xi^{2}$, the
uncertainty principle follows.

We note that

$(G_{s})_{X}(dx,d\xi):=\frac{dx^{2}}{1+|\xi|^{2}+|x|^{2}}+\frac{d\xi^{2}}{1+|\xi|^{2}+|x|^{2}}\leq
C(g_{s})_{X}.$

Since the weight $1+|\xi|^{2}+|x|^{2}$ is continuous with respect to the
Shubin metric $G_{s}$, it follows that it is also continuous with respect to
$g_{s}$. Hence $g_{s}$ is continuous.

To prove the temperateness of $g_{s}$, we will see that

$\frac{\langle\xi\rangle^{2}+|x|^{2}}{\langle\eta\rangle^{2}+|y|^{2}}\leq
C((g_{s})_{Y}^{\sigma}(X-Y)+1)^{N},$

for suitable $C,N$.

We observe that

$\frac{\langle\xi\rangle^{2}+|x|^{2}}{\langle\eta\rangle^{2}+|y|^{2}}=\frac{\langle\xi\rangle^{2}+|x|^{2}-\langle\eta\rangle^{2}-|y|^{2}}{\langle\eta\rangle^{2}+|y|^{2}}+1=\frac{\langle\xi\rangle^{2}-\langle\eta\rangle^{2}+|x|^{2}-|y|^{2}}{\langle\eta\rangle^{2}+|y|^{2}}+1.$

Now

$\frac{\langle\xi\rangle^{2}-\langle\eta\rangle^{2}}{\langle\eta\rangle^{2}+|y|^{2}}\leq
C\frac{(|\xi-\eta|^{2}+|\eta|^{2})}{\langle\eta\rangle^{2}+|y|^{2}}\leq
C(|\xi-\eta|^{2}+1)\leq C((g_{s})_{Y}^{\sigma}(X-Y)+1).$ (3.4)

The other term $\frac{|x|^{2}-|y|^{2}}{\langle\eta\rangle^{2}+|y|^{2}}$ can be
estimated similarly. Thus $g_{s}$ is temperate and this concludes the proof of
the lemma. ∎

###### Proof.

(Proof of Theoren 3.1.) The uncertainty principle follows from

$\langle\xi\rangle^{2}+|x|^{2}\leq(a_{2}(x,\xi)+|x|^{2}+\langle
X\rangle)^{2},$

which holds immediately.

To prove the continuity, we note that

$g_{X}=\frac{\langle\xi\rangle^{2}+|x|^{2}}{a_{2}+|x|^{2}+\langle
X\rangle}\left(dx^{2}+\frac{d\xi^{2}}{\langle\xi\rangle^{2}+|x|^{2}}\right)\geq
C^{\prime}(g_{s})_{X}\geq C(G_{s})_{X},$ (3.5)

since $\frac{\langle\xi\rangle^{2}+|x|^{2}}{m(X)}\geq C^{\prime}$, and where
$G_{s}$ is the Shubin metric as in the proof of Lemma 3.3. But
$\langle\xi\rangle^{2}+|x|^{2}$ is a continuous weight with respect to
$G_{s}$, so the same holds true with respect to $g$ as well.

The continuity of $g$ is now reduced to the study of the ratio
$\frac{m(X)}{m(Y)}$, which can be reduced to the analysis of
$\,\frac{m(X)-m(Y)}{m(Y)}$.

So we first note that

$m(X)-m(Y)=(a_{2}(X)-a_{2}(Y))+|x|^{2}-|y|^{2}+\langle X\rangle-\langle
Y\rangle.$

For the term $a_{2}(X)-a_{2}(Y)$, we use the Taylor's inequality of order $2$:

$|a_{2}(X)-a_{2}(Y)-da_{2}(Y)\cdot(X-Y)|\leq
C\sup_{t\in[0,1]}|d^{2}a_{2}(Y+t(X-Y))\cdot(X-Y)^{2}|\,.$

Since $a_{2}\in S^{2}(\mathbb{R}^{n}\times\mathbb{R}^{n})$, we have

$\sup_{t\in[0,1]}|d^{2}a_{2}(Y+t(X-Y))\cdot(X-Y)^{2}|\leq
C(\langle\eta\rangle^{2}|x-y|^{2}+|\xi-\eta|^{2}).$

On the other hand, by Proposition 3.2 applied to the partial derivatives of
$a_{2}$, we get

$\displaystyle\sup_{t\in[0,1]}|da_{2}(Y+t(X-Y))\cdot(X-Y)|\leq$ $\displaystyle
C(|\partial_{x}a_{2}(Y)\cdot(x-y)|+|\partial_{\xi}a_{2}(Y)\cdot(\xi-\eta)|.$
$\displaystyle\leq$ $\displaystyle
Ca_{2}^{\frac{1}{2}}(Y)\langle\eta\rangle|x-y|+Ca_{2}^{\frac{1}{2}}(Y)|\xi-\eta|.$

Therefore

$|a_{2}(X)-a_{2}(Y)|\leq
C(a_{2}^{\frac{1}{2}}(Y)\langle\eta\rangle|x-y|+a_{2}^{\frac{1}{2}}(Y)|\xi-\eta|+\langle\eta\rangle^{2}|x-y|^{2}+|\xi-\eta|^{2})\,.$

Now

$\frac{|a_{2}(X)-a_{2}(Y)|^{2}}{m^{2}(Y)}\leq$ $\displaystyle\leq$
$\displaystyle
C\left(\frac{a_{2}(Y)\langle\eta\rangle^{2}|x-y|^{2}}{(a+\langle
Y\rangle)^{2}}+\frac{a_{2}(Y)|\xi-\eta|^{2}}{(a+\langle
Y\rangle)^{2}}+\frac{\langle\eta\rangle^{4}|x-y|^{4}}{(a+\langle
Y\rangle)^{2}}+\frac{|\xi-\eta|^{4}}{(a+\langle Y\rangle)^{2}}\right)$
$\displaystyle\leq$ $\displaystyle
C\left(\frac{(\langle\eta\rangle^{2}+|y|^{2})|x-y|^{2}}{a+\langle
Y\rangle}+\frac{|\xi-\eta|^{2}}{a+\langle
Y\rangle}\right)+C\left(\frac{(\langle\eta\rangle^{2}+|y|^{2})^{2}|x-y|^{4}}{(a+\langle
Y\rangle)^{2}}+\frac{|\xi-\eta|^{4}}{(a+\langle Y\rangle)^{2}}\right)$
$\displaystyle\leq$ $\displaystyle C(g_{Y}(X-Y)+(g_{Y}(X-Y))^{2}).$

Hence

$\frac{|a_{2}(X)-a_{2}(Y)|}{m(Y)}\leq
C\left(g_{Y}(X-Y))^{\frac{1}{2}}+g_{Y}(X-Y)\right).$ (3.6)

For the term $\frac{\langle X\rangle-\langle Y\rangle}{m(Y)}$, we note that
$|\langle X\rangle-\langle Y\rangle|\leq C(||\xi|-|\eta||+||x|-|y||)$. We have

$\frac{(|\xi|-|\eta|)^{2}}{m^{2}(Y)}\leq\frac{|\xi-\eta|^{2}}{m^{2}(Y)}\leq\frac{|\xi-\eta|^{2}}{a+\langle
Y\rangle}\leq Cg_{Y}(X-Y).$

We also have

$\frac{(|x|-|y|)^{2}}{m^{2}(Y)}\leq\frac{|x-y|^{2}}{m^{2}(Y)}\leq\frac{(\langle\eta\rangle^{2}+|y|^{2})|x-y|^{2}}{a+\langle
Y\rangle}\leq Cg_{Y}(X-Y).$

Thus

$\frac{|\langle\xi\rangle-\langle\eta\rangle|}{m(Y)}\leq
C(g_{Y}(X-Y))^{\frac{1}{2}}.$ (3.7)

The term $\frac{||x|^{2}-|y|^{2}|}{m(Y)}$ can be analysed in a similar way,
obtaining

$\frac{||x|^{2}-|y|^{2}|}{m(Y)}\leq C\frac{|x-y|^{2}+|y|^{2}}{m(Y)}\leq
C\left(\frac{(\langle\eta\rangle^{2}+|y|^{2})|x-y|^{2}}{m(Y)}+1\right)=C(g_{Y}(X-Y)+1).$
(3.8)

By (3.6), (3.7) and (3.8), we get

$\frac{m(X)}{m(Y)}\leq C(g_{Y}(X-Y)+g_{Y}^{\frac{1}{2}}(X-Y))+1)\leq
C(g_{Y}(X-Y)+1).$ (3.9)

Therefore, the metric $g$ is continuous.

In order to prove the temperateness of $g$, we first observe that from the
proof of Lemma 3.3, by using the first inequality in (3.4), we obtain

$\displaystyle\frac{|\langle\xi\rangle^{2}-\langle\eta\rangle^{2}|}{\langle\eta\rangle^{2}+|y|^{2}}\leq$
$\displaystyle
C\left(\frac{|\xi-\eta|^{2}}{\langle\eta\rangle^{2}+|y|^{2}}+1\right)$
$\displaystyle\leq$ $\displaystyle C\left(\frac{a(Y)+\langle
Y\rangle}{\langle\eta\rangle^{2}+|y|^{2}}|\xi-\eta|^{2}+1\right)$
$\displaystyle\leq$ $\displaystyle C(g_{Y}^{\sigma}(X-Y)+1).$

In a similar way, for the term
$\frac{|x|^{2}-|y|^{2}}{\langle\eta\rangle^{2}+|y|^{2}}$ we have

$\displaystyle\frac{||x|^{2}-|y|^{2}|}{\langle\eta\rangle^{2}+|y|^{2}}\leq$
$\displaystyle
C\left(\frac{|x-y|^{2}}{\langle\eta\rangle^{2}+|y|^{2}}+1\right)$
$\displaystyle\leq$ $\displaystyle C\left((a(Y)+\langle
Y\rangle)|x-y|^{2}+1\right)$ $\displaystyle\leq$ $\displaystyle
C(g_{Y}^{\sigma}(X-Y)+1).$

On the other hand, for the ratio $\frac{m(X)}{m(Y)}=\frac{a(X)+\langle
X\rangle}{a(Y)+\langle Y\rangle}$, we note that by (3.9) and since $g\leq
g^{\sigma}$ , we get

$\frac{m(X)}{m(Y)}\leq C(g_{Y}(X-Y)+1)\leq C(g_{Y}^{\sigma}(X-Y)+1).$ (3.10)

Therefore $g$ is temperate and we conclude the proof. ∎

As a consequence of (3.10) we have that $m$ is indeed a $g$-weight.

######  Corollary 3.4.

The function $m$ given in (3.2) is a $g$-weight with respect to the metric $g$
defined by (3.1).

######  Lemma 3.5.

Let us consider the Hörmander metric $g$ defined by (3.1) and the $g$-weight
in (3.2). Then, a smooth symbol $\sigma:=\sigma(x,\xi)\in S(m,g)$ if and only
if

$|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}\sigma(x,\xi)|\leq
C_{\alpha\beta}m^{1-\frac{|\alpha|}{2}-\frac{|\beta|}{2}}(X)(\langle\xi\rangle^{2}+|x|^{2})^{\frac{|\beta|}{2}}.$
(3.11)

###### Proof.

The Hörmander metric is given by

$g_{X}(dx,d\xi):=m^{-1}(x,\xi)(\langle\xi\rangle^{2}+|x|^{2})dx^{2}+m^{-1}(x,\xi)d\xi^{2}.$
(3.12)

The class $S(m,g)$ consists of all smooth functions $\sigma\in
C^{\infty}(\mathbb{R}^{2n})$ such that for any $k\in\mathbb{N}$ there exists
$C_{k}>0$, such that for all $X,T_{1},...,T_{k}\in\mathbb{R}^{2n}$ we have
that

$|\sigma^{(k)}(X;T_{1}\otimes\cdots\otimes T_{k})|\leq
C_{k}m(X)\prod_{i=1}^{k}g_{X}^{1/2}(T_{i}).$ (3.13)

In consequence, $\sigma\in S(m,g),$ if and only if, for every
$\alpha\in\mathbb{N}_{0}^{n}$ and for every $\beta\in\mathbb{N}_{0}^{n},$ we
have that

$\displaystyle|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}\sigma(x,\xi)|\leq
C_{\alpha,\beta}m(x,\xi)\left(m^{-1}(x,\xi)(\langle\xi\rangle^{2}+|x|^{2})\right)^{\frac{|\beta|}{2}}(m^{-1}(x,\xi))^{\frac{|\alpha|}{2}},$

since any partial derivative with respect to $x,$ in any canonical direction
$x_{i},$ contributes on the right hand side of (3.13) with the factor

$\left(m^{-1}(x,\xi)(\langle\xi\rangle^{2}+|x|^{2})\right)^{\frac{1}{2}}.$

Also, any partial derivative with respect to $\xi,$ in any canonical direction
$\xi_{j},$ contributes on the right hand side of (3.13) with the factor

$\left(m^{-1}(x,\xi))\right)^{\frac{1}{2}}.$

Observing that

$\displaystyle m(x,\xi)$
$\displaystyle\left(m^{-1}(x,\xi)(\langle\xi\rangle^{2}+|x|^{2})\right)^{\frac{|\beta|}{2}}m^{-1}(x,\xi)^{-\frac{|\alpha|}{2}}$
$\displaystyle=m(x,\xi)^{1-\frac{|\beta|}{2}-\frac{|\alpha|}{2}}(\langle\xi\rangle^{2}+|x|^{2})^{\frac{|\beta|}{2}},$

we conclude the proof of Lemma 3.13. ∎

We will now restrict our attention to non-negative symbols $a_{2}$ of
differential operators of order $2$.

######  Definition 3.6.

Let $C_{b}^{\infty}(\mathbb{R}^{n})$ be the class of $C^{\infty}$ functions
over $\mathbb{R}^{n}$ with bounded derivatives of any order. We denote by
$\mbox{Diff }_{+}^{2}(\mathbb{R}^{n})$ the class of differential operators of
order $2$ on $\mathbb{R}^{n}$ with $C_{b}^{\infty}(\mathbb{R}^{n})$
coefficients and non-negative symbol.

######  Lemma 3.7.

Let $a_{2}(x,D)\in{\mbox{Diff }}_{+}^{2}(\mathbb{R}^{n})$. We consider
$a(x,\xi)=a_{2}(x,\xi)+|x|^{2}$ and the corresponding Hörmander metric $g$ and
the weight $m$ as in (3.1) and (3.2), respectively. Then $a,\,m\in S(m,g)$.

###### Proof.

We first consider the membership of $a$ in $S(m,g)$. We want to prove that for
all $k\in\mathbb{N}$ there exists $C_{k}>0$ such that for all
$X,T_{1},\cdots,T_{k}\in\mathbb{R}^{n}\times\mathbb{R}^{n}$:

$|(a^{(k)}(X);T_{1}\otimes T_{2}\otimes\cdots\otimes T_{k})|\leq
C_{k}m(X)\prod\limits_{j=1}^{k}g_{X}^{\frac{1}{2}}(T_{j}).$

As it is customary these estimates are reduced to the ones for canonical
directions which are equivalent to the following symbol inequalities (see
Lemma 3.5): for all $\alpha,\beta$, there exists $C_{\alpha\beta}>0$ such that

$|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|\leq
C_{\alpha\beta}m^{1-\frac{|\alpha|}{2}-\frac{|\beta|}{2}}(X)(\langle\xi\rangle^{2}+|x|^{2})^{\frac{|\beta|}{2}}.$

We are also going to prove that $m\in S(m,g).$ We write

$T=(t,\tau)=(t_{1},\dots,t_{n},\tau_{1},\dots,\tau_{n})\in\mathbb{R}^{n}\times\mathbb{R}^{n}.$

We observe that from the definition of the metric $g$ we have

$\frac{(\langle\xi\rangle^{2}+|x|^{2})}{m(X)}|t|^{2}\leq g_{X}(T);\hskip
11.38092pt\frac{|\tau|^{2}}{m(X)}\leq g_{X}(T).$

Then

$\frac{|t_{i}|}{g_{X}^{\frac{1}{2}}(T)}\leq\frac{m^{\frac{1}{2}}(X)}{(\langle\xi\rangle^{2}+|x|^{2})^{\frac{1}{2}}};\hskip
11.38092pt\frac{|\tau_{j}|}{g_{X}^{\frac{1}{2}}(T)}\leq m^{\frac{1}{2}}(X).$

For $k=1$ we will apply Proposition 3.2 to the partial derivatives of the
symbol $a_{2}\in S^{2}(\mathbb{R}^{n}\times\mathbb{R}^{n})$. Since
$\partial_{x_{i}}a(X)=\partial_{x_{i}}a_{2}(X)+2x_{i}$, we have

$\displaystyle|\partial_{x_{i}}a(X)|\frac{1}{m(X)}\frac{m^{\frac{1}{2}}(X)}{(\langle\xi\rangle^{2}+|x|^{2})^{\frac{1}{2}}}$
$\displaystyle\leq$ $\displaystyle
C\frac{a_{2}^{\frac{1}{2}}(X)\langle\xi\rangle+|x_{i}|}{m^{\frac{1}{2}}(X)(\langle\xi\rangle^{2}+|x|^{2})^{\frac{1}{2}}}$
$\displaystyle\leq$ $\displaystyle
C\frac{a_{2}^{\frac{1}{2}}(X)\langle\xi\rangle}{m^{\frac{1}{2}}(X)\langle\xi\rangle}+C\frac{|x_{i}|}{(1+|x|^{2})^{\frac{1}{2}}}\leq
C.$

On the other hand, since $\partial_{\xi_{j}}a(X)=\partial_{\xi_{j}}a_{2}(X)$
we have

$|\partial_{\xi_{j}}a(X)|\frac{1}{m(X)}m^{\frac{1}{2}}(X)\leq
C\frac{a_{2}^{\frac{1}{2}}(X)}{m^{\frac{1}{2}}(X)}\leq C.$

If $k=|\alpha|+|\beta|\geq 2$, we will use the membership of $a_{2}$ to the
class $S^{2}(\mathbb{R}^{n}\times\mathbb{R}^{n})$.

We first observe that if $|\alpha|=0$, then $|\beta|\geq 2$ and

$|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|=|\partial_{x}^{\beta}(a_{2}(x,\xi)+|x|^{2})|\leq
C_{1}\langle\xi\rangle^{2}+C_{2}\leq C\langle\xi\rangle^{2}.$

If $0<|\alpha|\leq 2$, we have

$|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|=|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a_{2}(x,\xi)|\leq
C\langle\xi\rangle^{2-|\alpha|}.$

Thus, for $0\leq|\alpha|\leq 2$, we obtain

$\displaystyle|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|$
$\displaystyle\leq$ $\displaystyle
C_{\alpha\beta}\langle\xi\rangle^{2-|\alpha|}$ $\displaystyle\leq$
$\displaystyle
C_{\alpha\beta}(\langle\xi\rangle^{2}+|x|^{2})^{1-\frac{|\alpha|}{2}}$
$\displaystyle=$ $\displaystyle
C_{\alpha\beta}\,\,m(X)(\langle\xi\rangle^{2}+|x|^{2})^{\frac{|\beta|}{2}}m^{-\frac{|\beta|}{2}-\frac{|\alpha|}{2}}(\langle\xi\rangle^{2}+|x|^{2})^{1-\frac{|\beta|}{2}-\frac{|\alpha|}{2}}m^{-1+\frac{|\beta|}{2}+\frac{|\alpha|}{2}}$
$\displaystyle\leq$ $\displaystyle
C_{\alpha\beta}\,\,m(X)(\langle\xi\rangle^{2}+|x|^{2})^{\frac{|\beta|}{2}}m^{-\frac{|\beta|}{2}-\frac{|\alpha|}{2}},$

where in the last inequality we have used the fact that
$\left(\frac{m(X)}{\langle\xi\rangle^{2}+|x|^{2}}\right)^{\frac{|\alpha|+|\beta|-2}{2}}\leq
C$, since $m(X)\leq C(\langle\xi\rangle^{2}+|x|^{2}).$ This gives the right
estimate for $0\leq|\alpha|\leq 2$.

For the case $|\alpha|\geq 3$, we note that since $a_{2}(x,D)$ is a
differential operator of order $2$ we have

$\partial_{\xi}^{\alpha}a(x,\xi)=\partial_{\xi}^{\alpha}a_{2}(x,\xi)=0.$

Hence $\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)=0$, and the proof
is concluded.

In order to see that $\langle X\rangle\in S(m,g)$, we observe that $\langle
X\rangle$ is a regular weight with respect to the metric $G_{s}$. Then

$\langle X\rangle\in S(\langle X\rangle,G_{s})\subset S(\langle
X\rangle,g)\subset S(m,g).$ (3.14)

We have also used the fact that $\langle X\rangle$ is a $g-$weight.

Therefore $m=a+\langle X\rangle\in S(m,g)$. ∎

In the special case of
$a_{2}(x,\xi)=\xi_{1}^{2}+\tilde{x_{1}}^{2}\xi_{2}^{2}$, consequently we have
the following property.

######  Corollary 3.8.

The metric $g$ given in (3.1) with
$a_{2}(x,\xi)=\xi_{1}^{2}+\tilde{x_{1}}^{2}\xi_{2}^{2}$, is a Hörmander metric
on $\mathbb{R}^{2}\times\mathbb{R}^{2}$,
$m(X)=\xi_{1}^{2}+\tilde{x_{1}}^{2}\xi_{2}^{2}+|x|^{2}+\langle X\rangle$ is a
$g$-weight and
$\xi_{1}^{2}+\tilde{x_{1}}^{2}\xi_{2}^{2}+|x|^{2},\,\,m=\xi_{1}^{2}+\tilde{x_{1}}^{2}\xi_{2}^{2}+|x|^{2}+\langle
X\rangle\in S(m,g).$

##  4\. Some applications to well-posedness for degenerate Schrödinger and
degenerate parabolic equations

We are now going to establish some implications in the $L^{2}$ and Sobolev
spaces $H(M,g)$ context. Indeed, the construction of the metric $g$ and the
class $S(m,g)$ adapted to our degenerate harmonic oscillators has other
consequences, regarding the well-posedness for degenerate Schrödinger
equations and degenerate parabolic equations in the $L^{2}$-setting. We
introduce the following appropriate class of potentials on $\mathbb{R}^{n}$.

######  Definition 4.1.

We will denote by $\mathcal{P}_{2}(\mathbb{R}^{n})$, the class of Borel
functions $V:\mathbb{R}^{n}\rightarrow\mathbb{R}$ satisying the following
conditions:

(V1) $\displaystyle\mbox{There exist }C,C_{1}>0\mbox{ such that }|V(x)|\leq
C|x|^{2},\mbox{ for a.e. }|x|\geq C_{1}.$ (V2) $\displaystyle\mbox{There
exists }C_{2}>0\mbox{ such that }V(x)\geq-C_{2},\mbox{ for a.e.
}x\in\mathbb{R}^{n}.$

It is worth to point out that no regularity assumption has been imposed on $V$
as is customary in most of the main theorems on the classical spectral theory
of Schrödinger operators where the conditions are more likely to be on the
integrability ([5], [48]). We will consider Hamiltonians of the form
$\mathcal{H}_{V}=a_{2}(x,D)+V$.

######  Theorem 4.2.

Let $a_{2}(x,D)\in\mbox{Diff }_{+}^{2}(\mathbb{R}^{n})$ be formally self-
adjoint, $V\in\mathcal{P}_{2}(\mathbb{R}^{n})$, $g$ the Hörmander metric and
$m$ the $g$-weight associated to $a_{2}(x,D)$ as in (3.1). Then,
$-i\mathcal{H}_{V}$ is the infinitesimal generator of a $C_{0}-$group of
unitary operators. Consequently, the following Cauchy problem for the
corresponding degenerate Schrödinger equation is well-posed on $L^{2}$:

$\left\\{\begin{array}[]{rl}i\partial_{t}u&=\mathcal{H}_{V}u,\\\
u(0)&=f.\end{array}\right.$ (4.1)

Furthermore, if $a_{2}(x,D)=-\sum\limits_{j=1}^{k}X_{j}^{2}$ is a sum of
squares of real vector fields, then $\mathcal{H}=-(a_{2}(x,D)+|x|^{2})$ is the
infinitesimal generator of a $C_{0}-$semigroup (degenerate harmonic semigroup)
of contractions and the corresponding degenerate parabolic equation is well-
posed on $L^{2}$:

$\left\\{\begin{array}[]{rl}\partial_{t}u&=\mathcal{H}u,\\\
u(0)&=f.\end{array}\right.$ (4.2)

###### Proof.

We first state the boundedness of the potential operator. Indeed, we first
note that since $|x|^{2}\in S(m,g)$, we have that $|x|^{2}:H(m,g)\rightarrow
L^{2}(\mathbb{R}^{n})$ is bounded. From (V1) and (V2), we have $|V(x)|\leq
C(1+|x|^{2})$, for a.e. $x\in\mathbb{R}^{n}$. Since $m\geq 1$, it is now clear
that

$\lvert|V(x)u\rvert|_{L^{2}}\leq C\lvert|u\rvert|_{H(m,g)},$ (4.3)

for all $u\in H(m,g)$.

On the other hand, the operator $a_{2}(x,D)+V(x)$ is symmetric, Lemma 3.7,
Theorem 2.6 and the inequality (4.3) imply that the operator
$a_{2}(x,D)+V(x):H(m,g)\rightarrow L^{2}(\mathbb{R}^{n})$ is bounded.
Moreover, since $0\leq a_{2}(x,\xi)\in
S^{2}(\mathbb{R}^{n}\times\mathbb{R}^{n})$, applying Fefferman-Phong theorem
there exists $C>0$ such that

$\mbox{Re}\langle
a_{2}(x,D)v,v\rangle\geq-\frac{C}{2}\lvert|v\rvert|_{L^{2}(\mathbb{R}^{n})}^{2}.$

Hence

$\mbox{Re}\langle(a_{2}(x,D)+C)v,v\rangle\geq\frac{C}{2}\lvert|v\rvert|_{L^{2}(\mathbb{R}^{n})}^{2}.$

Thus, by using (V2) there exists $\widetilde{C}>0$ such that

$\mbox{Re}\langle(a_{2}(x,D)+V(x)+\widetilde{C})v,v\rangle\geq\frac{C}{2}\lvert|v\rvert|_{L^{2}(\mathbb{R}^{n})}^{2},$

and by the Cauchy-Schwarz inequality we have

$\lvert|(a_{2}(x,D)+V(x)+\widetilde{C})v\rvert|_{L^{2}(\mathbb{R}^{n})}\geq
C\lvert|v\rvert|_{L^{2}(\mathbb{R}^{n})}.$ (4.4)

Therefore, Friedrichs extension theorem ensures that
$\mathcal{H}_{V}+\widetilde{C}:H(m,g)\rightarrow L^{2}(\mathbb{R}^{n})$ is
self-adjoint. Thus, $\mathcal{H}_{V}$ is also self-adjoint. Now, an
application of Stone's theorem guarantees that $-i\mathcal{H}_{V}$ is the
infinitesimal generator of a $C_{0}-$group of unitary operators. Therefore,
the Cauchy problem for the corresponding degenerate Schrödinger equation (4.1)
is well-posed in $L^{2}$.

On the other hand since $\mathcal{H}=-(a_{2}(x,D)+|x|^{2})$ is self-adjoint,
$-\mathcal{H}+C$ is semibounded for some $C>0$ as above with $V=|x|^{2}$, and
$-a_{2}(x,D)$ being a sum of squares of real vector fields, $\mathcal{H}$ is
dissipative. Then, the Lummer-Phillips theorem ensures that $\mathcal{H}$ is
the infinitesimal generator of a $C_{0}-$semigroup of contractions and hence
the corresponding degenerate parabolic equation (4.2) is well-posed. ∎

######  Remark 4.3.

We recall that the simultaneous study of the Hamiltonian as an infinitesimal
generator for a Schrödinger semigroup as for Heat semigroup has been an active
field of research in the last decades as it was well pointed out by B. Simon
in [76]. For instance, when looking at the $L^{p}$ properties of the
eigenfunctions of $\mathcal{H}=-\Delta+V(x)$, if $\mathcal{H}\psi=E\psi$ and
$\psi\in L^{2}$, one observes that since $e^{-t\mathcal{H}}\psi=e^{-tE}\psi$,
then $\psi\in e^{-t\mathcal{H}}(L^{2})$. Thus, a mapping property such as
$e^{-t\mathcal{H}}:L^{2}\rightarrow L^{\infty}$ can be used to deduce that
$\psi$ belongs also to $L^{\infty}$. The works of B. Simon and R. Carmona on
Schrödinger semigroups have deeply influenced the research on these
semigroups, their applications are diverse and the literature is currently
huge. See for instance [1], [17], [77], [75], [53], [57], for more accounts on
this matter. Thus, the Schrödinger semigroups are relevant for the study of
$L^{p}$ properties of eigenfunctions of Schrödinger operators and especially
for the so-called ground states.

As a corollary of our previous $L^{2}$ estimates, we can extend the well-
posedness for the positive powers of $\mathcal{H}$, in the case where

$a_{2}(x,D)=-\sum\limits_{j=1}^{k}X_{j}^{2}$

is a sum of squares of real vector fields, where the family of vector fields
satisfies the Hörmander condition of order $2$. A reason for considering a
sub-Laplacian associated to a system of vector fields satisfying the Hörmander
condition of order 2 is that in this case the geometry of the balls associated
to the Carnót-Carathéodory distance of the Hörmander system is very well
understood from the fundamental work of Nagel, Stein and Wainger [69], see
Remark 6.1.

On the other hand, thanks to the Fefferman-Phong inequality, we can define the
positive powers of $\mathcal{H}^{\beta}$, and we have the membership
$\mathcal{H}^{\beta}\in OpS(m^{\beta},g)$. Moreover, the maximal
hypoellipticity of $a_{2}(x,D)$ guarantees that the Sobolev space
$H(m^{\beta},g)$ is characterised by the condition
$(a_{2}(x,D)+|x|^{2}+C)^{\beta}u\in L^{2}$. Hence,
$(a_{2}(x,D)+|x|^{2}+C)^{\beta}$ is semi-bounded from below and consequently
we obtain the following corollary.

######  Corollary 4.4.

Let $\beta>0.$ Let

$a_{2}(x,D)=-\sum\limits_{j=1}^{k}X_{j}^{2}$

be a sum of squares of real vector fields, where $X_{1},\dots,X_{k}$ satisfy
the Hörmander condition of order $2$, $g$ the Hörmander metric and $m$ the
$g$-weight associated to $a_{2}(x,D)$ as in (3.1). Then, there exists $C>0$
such that $\mathcal{H}_{C}=a_{2}(x,D)+|x|^{2}+C$ is semibounded in
$L^{2}(\mathbb{R}^{n})$, and $-i\mathcal{H}_{C}$ is the infinitesimal
generator of a $C_{0}-$semigroup. Consequently, the following Cauchy problem
for the corresponding degenerate Schrödinger equation is well-posed on
$L^{2}$:

$\left\\{\begin{array}[]{rl}i\partial_{t}u+\mathcal{H}_{0}^{\beta}u&=0,\\\
u(0)&=f.\end{array}\right.$ (4.5)

Furthermore, the operator

$\mathcal{H}=-(a_{2}(x,D)+|x|^{2}),$

is the infinitesimal generator of a $C_{0}-$semigroup (degenerate harmonic
semigroup) of contractions and the corresponding degenerate parabolic equation
is well-posed on $L^{2}$:

$\left\\{\begin{array}[]{rl}\partial_{t}u-\mathcal{H}^{\beta}u&=0,\\\
u(0)&=f.\end{array}\right.$ (4.6)

In order to go further with our analysis we need the property of geodesic
temperateness for our metric $g$. This property allows us to obtain the
invertibilitiy of the Hamiltonian as a pseudodifferential operator. This
important notion was introduced by Bony and Chemin in [8], originally calling
it strong temperateness. We start by recalling its definition, see [62], [9].

Let $d^{\sigma}(X,Y)$ be the geodesic distance for the Riemannian metric
$g^{\sigma}$ between $X$ and $Y$, where $g^{\sigma}$ is given by (2.1).

######  Definition 4.5.

A Hörmander metric $g$ is called geodesically temperate if it is temperate and
if there exist $C,N>0$ such that

$\frac{g_{Y}}{g_{X}}\leq C(1+d^{\sigma}(X,Y))^{N}.$ (4.7)

It is known (c.f. [9]) that the condition (4.7) is equivalent to : there exist
$C,N>0$ such that

$C^{-1}(1+d^{\sigma}(X,Y))^{\frac{1}{N}}\leq(1+g^{\sigma}(X-Y))\leq
C(1+d^{\sigma}(X,Y))^{N}.$ (4.8)

Since our Hörmander metric is symmetrically split, it follows that it is
geodesically temperate by a result of Bony (Cf. Theorem 5 (i), [9]). In what
follows we require the following definition. We record that a second order
differential operator $P(x,D)$ is subelliptic of order $0<\tau\leq 2,$ if it
satisfies the a-priori estimate

$\lVert v\rVert_{H^{\tau}}\leq C_{1}(\lVert P(x,D)v\rVert_{L^{2}}+\lVert
v\rVert_{L^{2}}),\,\,\,\forall v\in C_{0}^{\infty}(\mathbb{R}^{n}),$ (4.9)

where $H^{\tau}$ denotes the standard Sobolev space of order $\tau$ on
$\mathbb{R}^{n}.$

######  Corollary 4.6.

Let $a_{2}(x,D)\in\mbox{Diff }_{+}^{2}(\mathbb{R}^{n})$ be formally self-
adjoint and subelliptic with order of subellipticity $0<\tau\leq 2$, $g$ the
Hörmander metric and $m$ the $g$-weight associated to $a_{2}(x,D)$ as in
(3.1). Then, there exist $C>0$ such that the operator

$\mathcal{H}_{C}=a_{2}(x,D)+|x|^{2}+C:H(m,g)\rightarrow
L^{2}(\mathbb{R}^{n}),$

is an isomorphism with inverse a pseudodifferential operator

$\mathcal{H}_{C}^{-1}:L^{2}(\mathbb{R}^{n})\rightarrow H(m,g),$

with symbol in $S(m^{-1},g)$.

###### Proof.

The a priori estimate (4.4) and the self-adjointness of $\mathcal{H}_{C}$
gives the surjectivity. On the other hand, the subellipticity of $a_{2}(x,D)$
gives

$\lVert v\rVert_{H^{\tau}}\leq
C_{1}(\lVert(a_{2}(x,D)+|x|^{2})v\rVert_{L^{2}}+\lVert
v\rVert_{L^{2}}),\,\,\,\forall v\in C_{0}^{\infty}(\mathbb{R}^{n}).$ (4.10)

By the a priori estimate (4.4) again and (4.10), there exists a constant
$C_{2}>0$ such that

$\lVert u\rVert_{H^{\tau}}\leq
C_{2}\lVert\mathcal{H}_{C}u\rVert_{L^{2}},\,\,\,\forall u\in
C_{0}^{\infty}(\mathbb{R}^{n}).$ (4.11)

Now, an analogous argument to the one for the proof of Lemma 3.3 in [25],
splitting the phase-space within an elliptic and a subelliptic zones with
respect to $g$ give us

$\|u\|_{H(m,g)}\leq C_{2}\|\mathcal{H}_{C}u\|_{L^{2}},\,\,\,\forall u\in
C_{0}^{\infty}(\mathbb{R}^{n}).$

Hence $\mathcal{H}_{C}:H(m,g)\rightarrow L^{2}(\mathbb{R}^{n})$ is injective.
Therefore, an application of Corollary 7.7 in [8] concludes the proof. ∎

##  5\. $L^{p}$-bounds, $H^{1}$-$L^{1}$ and $L^{\infty}$-$BMO$ estimates for
degenerate operators

In this section we consider our analysis of pseudo-differential operators
associated to the degenerate class of operators $S(m,g)$ where $g$ is the
metric in (3.1).

###  5.1. $L^{\infty}$-$BMO$ and $H^{1}$-$L^{1}$ boundedness

The next lemma will be essential for us in order to estimate an optimal
fractional power of the weight $m(x,\xi)$, which determines a suitable class
for the $L^{p}$-boundedness. The $L^{\infty}$-$BMO$ boundedness will be
reached by analysing a partition of unity of the symbol of interest. The
estimates for each piece is obtained from the next lemma which is an extension
of a classical Hardy-Littlewood-Sobolev inequality.

######  Lemma 5.1.

Let $\varepsilon_{0}=\frac{3}{4}$ and $\varepsilon_{0}\leq\varepsilon<1,$ and
let $q\in S(m^{-\frac{n}{2}\varepsilon},g)$ be supported in $R\leq
a(x,\xi)+\langle X\rangle\leq 3R$ for $R>1$. Then, for all $l>n/4$ we have

$\lvert|q(x,D)f\rvert|_{\infty}\leq
C\lvert|q\rvert|_{l;S(m^{-\frac{n\varepsilon}{2}},g)}\lvert|f\rvert|_{L^{\infty}},$

where the constant $C$ is independent of $q$ and $f$.

###### Proof.

Let $q\in S(m^{-\frac{n}{2}\varepsilon},g)$ be supported in

$\\{(x,\xi)\in\mathbb{R}^{n}\times\mathbb{R}^{n}\,:R\leq a(x,\xi)+\langle
X\rangle\leq 3R\\},$

for $R>1$ fixed. We observe that

$\displaystyle q(x,D)f(x)=$ $\displaystyle(2\pi)^{-n}\int\int
e^{i(x-y)\cdot\xi}q\left(x,\xi\right)f(y)dyd\xi$ $\displaystyle=$
$\displaystyle\int\hat{q}_{x}(y-x)f(y)dy$ $\displaystyle=$
$\displaystyle\hat{q}_{x}*f(x),$

where $\hat{q}_{x}$ is the Fourier transform $q_{x}(\xi)=q(x,\xi)$ with
respect to the variable $\xi$. We obtain

$|q(x,D)f(x)|\leq||\hat{q}_{x}\rvert|_{L^{1}}||f\rvert|_{L^{\infty}},\,\,x\in\mathbb{R}^{n}.$

It is then enough to prove that for all $x\in\mathbb{R}^{n}$ and $l>n/4$

$||\hat{q}_{x}\rvert|_{L^{1}}\leq
C\lvert|q\rvert|_{l;S(m^{-\frac{n}{2}\varepsilon},g)}.$

Let us consider $b>0$ to be fixed below. By applying the Cauchy-Schwarz
inequality, we obtain

$\displaystyle\int\limits_{|y|<b}|\hat{q}_{x}(y)|dy$ $\displaystyle\leq$
$\displaystyle
Cb^{\frac{n}{2}}\left(\,\int\limits_{|y|<b}|\hat{q}_{x}(y)|^{2}dy\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
Cb^{\frac{n}{2}}\left(\,\int\limits_{\mathbb{R}^{n}}|q(x,\xi)|^{2}d\xi\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C||q\rvert|_{l;S(m^{-\frac{n}{2}\varepsilon},g)}b^{\frac{n}{2}}\left(\,\int\limits_{a(x,\xi)+\langle
x,\xi\rangle\leq 3R}m^{-n\varepsilon}(x,\xi)d\xi\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C||q\rvert|_{l;S(m^{-\frac{n}{2}\varepsilon},g)}b^{\frac{n}{2}}\left(\,\int\limits_{\langle\xi\rangle\leq
3R}\langle\xi\rangle^{-n\varepsilon}d\xi\right)^{\frac{1}{2}}\,.$

Now, by choosing $b=R^{\varepsilon-1}$, we obtain

$\int\limits_{|y|<b}|\hat{q}_{x}(y)|dy\leq
C||q\rvert|_{l;S(m^{-\frac{n}{2}\varepsilon},g)}.$

On the other hand, for the analysis of the integral $\int\limits_{|y|\geq
b}|\hat{q}_{x}(y)|dy,$ we first note that for an integer $k>\frac{n}{4}$, we
have $k>\frac{n}{4}\geq n(1-\varepsilon)$, since $\varepsilon\geq\frac{3}{4}$.
Hence

$\displaystyle\int\limits_{|y|\geq b}|\hat{q}_{x}(y)|dy\leq$ $\displaystyle
Cb^{\frac{n}{2}-k}\left(\,\int\limits_{y\geq
b}|y|^{2k}|\hat{q}_{x}(y)|^{2}dy\right)^{\frac{1}{2}}$ $\displaystyle\leq$
$\displaystyle
Cb^{\frac{n}{2}-k}\left(\,\int\limits_{\mathbb{R}^{n}}|\nabla_{\xi}^{k}q(x,\xi)|^{2}d\xi\right)^{\frac{1}{2}},$

and the Plancherel theorem implies the inequality

$\int\limits_{|y|\geq b}|\hat{q}_{x}(y)|dy\leq
C\lvert|q\rvert|_{k;S(m^{-\frac{n\varepsilon}{2}},g)}b^{\frac{n}{2}-k}\left(\,\int\limits_{R\leq
a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq
3R}m^{-n\varepsilon-k}(x,\xi)d\xi\right)^{\frac{1}{2}}.$ (5.1)

In order to estimate the last integral, we first note that there exists $C>1$
such that for all $(x,\xi)$:

$a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq C(\langle\xi\rangle^{2}+|x|^{2}).$

We split the argument into two cases : $|x|^{2}\leq\frac{R}{2C}$ and
$|x|^{2}>\frac{R}{2C}$.

If $|x|^{2}\leq\frac{R}{2C}$, we get

$R\leq a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq
C(\langle\xi\rangle^{2}+|x|^{2})\leq C\langle\xi\rangle^{2}+\frac{R}{2}.$

Hence $\frac{R}{2}\leq C\langle\xi\rangle^{2}.$ Thus

$b^{\frac{n}{2}-k}\left(\,\int\limits_{R\leq a_{2}(x,\xi)+|x|^{2}+\langle
X\rangle\leq 3R}m^{-n\varepsilon-k}(x,\xi)d\xi\right)^{\frac{1}{2}}$

$\leq Cb^{\frac{n}{2}-k}\left(\,\int\limits_{R\leq
2C\langle\xi\rangle^{2}}\langle\xi\rangle^{-n\varepsilon-k}d\xi\right)^{\frac{1}{2}}\leq
CR^{(\varepsilon-1)(\frac{n}{2}-k)+\frac{1}{4}(n(1-\varepsilon)-k)}\leq C.$

To see the last inequality, we note that

$(\varepsilon-1)(\frac{n}{2}-k)+\frac{1}{4}(n(1-\varepsilon)-k)=\frac{n}{4}(\varepsilon-1)+k(\frac{3}{4}-\varepsilon).$

Since $\varepsilon\geq\frac{3}{4}$, we obtain the desired bound.

On the other hand, if $|x|^{2}>\frac{R}{2C}$, we have $-|x|^{2}<-\frac{R}{2C}$
and

$\langle\xi\rangle\leq a_{2}(x,\xi)+\langle X\rangle\leq 3R-|x|^{2}\leq
3R-\frac{R}{2C}=\frac{(6C-1)}{2C}R.$

Since $6C-1>0$ and by using the fact that $|x|^{2}>\frac{R}{2C}$, we have
$\langle x\rangle^{-k}\leq C^{\prime}R^{-\frac{k}{2}}$ and we obtain

$\displaystyle b^{\frac{n}{2}-k}\left(\,\int\limits_{R\leq
a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq
3R}m^{-n\varepsilon-k}(x,\xi)d\xi\right)^{\frac{1}{2}}=$ $\displaystyle
b^{\frac{n}{2}-k}\left(\,\int\limits_{\langle\xi\rangle\leq
C^{\prime}R}m^{-n\varepsilon}(x,\xi)m^{-k}(x,\xi)d\xi\right)^{\frac{1}{2}}$
$\displaystyle\lesssim$ $\displaystyle
R^{(\frac{n}{2}-k)(\varepsilon-1)}\left(\,\int\limits_{\langle\xi\rangle\leq
C^{\prime}R}\langle\xi\rangle^{-n\varepsilon}\langle
x\rangle^{-2k}d\xi\right)^{\frac{1}{2}}$ $\displaystyle\lesssim$
$\displaystyle R^{(\frac{n}{2}-k)(\varepsilon-1)}\langle
x\rangle^{-k}\left(\,\int\limits_{\langle\xi\rangle\leq
C^{\prime}R}\langle\xi\rangle^{-n\varepsilon}d\xi\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
CR^{(\frac{n}{2}-k)(\varepsilon-1)}R^{-\frac{k}{2}}R^{\frac{n}{2}(1-\varepsilon)}.$

By observing that

$R^{(\frac{n}{2}-k)(\varepsilon-1)+\frac{n}{2}(1-\varepsilon)-\frac{k}{2}}=R^{k(\frac{1}{2}-\varepsilon)}\leq
C,$

since $\varepsilon\geq\frac{3}{4}\geq\frac{1}{2}$, the desired estimate
follows. ∎

We can now state the $L^{\infty}$-$BMO$ bounds as a consequence of our
$L^{\infty}$-$L^{\infty}$ estimates.

######  Theorem 5.2.

Let $\frac{3}{4}\leq\beta<1,$ and let
$\sigma(x,D)\in\textnormal{Op}(S(m^{-\frac{n}{2}\beta},g)).$ Then, for
$\ell>\frac{n}{4},$ we have

$\|\sigma(x,D)f\|_{BMO(\mathbb{R}^{n})}\leq
C\|\sigma(x,\xi)\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}\|f\|_{L^{\infty}(\mathbb{R}^{n})},$

where the constant $C>0,$ is independent of $f\in L^{\infty}(\mathbb{R}^{n}).$
So, $\sigma(x,D):L^{\infty}(\mathbb{R}^{n})\rightarrow BMO(\mathbb{R}^{n})$
extends to a bounded operator.

###### Proof.

Let $f\in L^{\infty}(\mathbb{R}^{n}),$ and let us denote $A=\sigma(x,D).$ Let
us fix a ball $B(x_{0},r)$ where $x_{0}\in\mathbb{R}^{n},$ and $r>0.$ We will
prove that there exists a constant $C>0,$ independent of $f$ and $r,$ such
that

$\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|Af(x)-(Af)_{B(x_{0},r)}|dx\leqslant
C\|\sigma\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}\|f\|_{L^{\infty}(\mathbb{R}^{n})}$
(5.2)

where

$(Af)_{B(x_{0},r)}:=\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}Af(x)dx.$

We will split to $\sigma$ as the sum of two symbols

$\sigma(x,\xi)=\sigma^{0}(x,\xi)+\sigma^{1}(x,\xi),\,\,\,\sigma^{j}(x,\xi)\in
S(m^{-\frac{n}{2}\beta},g),\,j=0,1,$

in a such way that both, $\sigma^{0}(x,\xi)$ and $\sigma^{1}(x,\xi),$ satisfy
the estimate

$\|\sigma^{j}\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}\leqslant
C_{j,\ell}\|\sigma\|_{\ell,S(m^{-\frac{n}{2}\beta},g)},\,\,j=0,1,\,\ell\geqslant
1.$ (5.3)

Now, if $A^{j}:=\textnormal{Op}(\sigma^{j}),$ for $j=0,1,$ then
$A=A^{0}+A^{1}$ on $C^{\infty}(\mathbb{R}^{n}),$ and we only need to prove
that

$\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|A^{j}f(x)-(A^{j}f)_{B(x_{0},r)}|dx\leqslant
C\|\sigma^{j}\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}\|f\|_{L^{\infty}(\mathbb{R}^{n})}.$
(5.4)

Then, if we prove (5.3), we can deduce (5.2). Now, we proceed to prove the
existence of $\sigma^{0}$ and $\sigma^{1}$ satisfying the requested
properties. Let us define

$\sigma^{0}(x,\xi)=\sigma(x,\xi)\tilde{\gamma}(x,\xi),\textnormal{
}\sigma^{1}(x,\xi)=\sigma(x,\xi)-\sigma^{0}(x,\xi),\,(x,\xi)\in\mathbb{R}^{n},$
(5.5)

where

$\tilde{\gamma}(x,\xi):=\gamma(r\times
m(x,\xi))=\gamma(r\times(a_{2}(x,\xi)+|x|^{2}+\langle x,\xi\rangle)),$

and $\gamma\in C^{\infty}_{0}(\mathbb{R},\mathbb{R}^{+}_{0}),$ is a smooth
function supported in $\\{t:|t|\leqslant 1\\},$ satisfying $\gamma(t)=1,$ for
$|t|\leqslant\frac{1}{2}.$

In order to prove (5.3) let us proceed as follows. Since
$\tilde{\gamma}(x,\xi)$ is a cut-off function supported in the set
$\\{(x,\xi):a_{2}(x,\xi)+|x|^{2}+\langle x,\xi\rangle\leq r^{-1}\\},$ in this
region, the symbol $\sigma^{0}(x,\xi)=\sigma(x,\xi)\tilde{\gamma}(x,\xi)$ also
satisfies the $S(m,g)$-estimates in (2.2) but probably depending on $r>0$. To
remove this dependence in $r>0,$ let us use the natural Fréchet topology on
$S(m,g)$ induced by its family of seminorms. This is the smalles topology on
$S(m,g)$ which makes these seminorms continuous. Indeed, we have that

$\lim_{r\rightarrow
0^{+}}\|\sigma(x,\xi)\gamma(r\times(a_{2}(x,\xi)+|x|^{2}+\langle
x,\xi\rangle))\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}=\|\sigma(x,\xi)\|_{\ell,S(m^{-\frac{n}{2}\beta},g)},$

and

$\lim_{r\rightarrow\infty}\|\sigma(x,\xi)\gamma(r\times(a_{2}(x,\xi)+|x|^{2}+\langle
x,\xi\rangle))\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}=0.$

In consequence, there exists $0<\varkappa<1,$ such that for all
$r\in(0,\varkappa)\cup[\varkappa^{-1},\infty),$ we have that

$\|\sigma(x,\xi)\gamma(r\times(a_{2}(x,\xi)+|x|^{2}+\langle
x,\xi\rangle))\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}\leq
2\|\sigma(x,\xi)\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}.$

On the other hand, the function

$r\in[\varkappa,\varkappa^{-1}]\mapsto\sigma(x,\xi)\gamma(r\times(a_{2}(x,\xi)+|x|^{2}+\langle
x,\xi\rangle))\in S(m,g),$ (5.6)

is continuous, and since $S(m,g)$ is a complete topological space, there
exists $r_{0}\in[\varkappa,\varkappa^{-1}]$ such that

$\displaystyle\max_{r\in[\varkappa,\varkappa^{-1}]}\|\sigma(x,\xi)\gamma(r\times(a_{2}(x,\xi)+|x|^{2}+\langle
x,\xi\rangle))\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}$
$\displaystyle=\|\sigma(x,\xi)\gamma(r_{0}\times(a_{2}(x,\xi)+|x|^{2}+\langle
x,\xi\rangle))\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}\lesssim_{r_{0}}\|\sigma(x,\xi)\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}.$

Since the previous estimate depends on $r_{0},$ and then on $\varkappa,$ but
it is independent of $r>0,$ the estimates above prove the estimate in (5.3).

To estimate the integral

$I_{0}:=\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|A^{0}f(x)-(A^{0}f)_{B(x_{0},r)}|dx,$

we will use the mean value theorem. Indeed, observe that

$\displaystyle|A^{0}f(x)-A^{0}f(y)|$ $\displaystyle\leqslant
C_{0}\sum_{k=1}^{n}\|\partial_{x_{k}}A^{0}f\|_{L^{\infty}(\mathbb{R}^{n})}|x-y|$
$\displaystyle\leqslant
Cr\sum_{k=1}^{n}\|\partial_{x_{k}}A^{0}f\|_{L^{\infty}(\mathbb{R}^{n})}.$
(5.7)

In order to estimate the $L^{\infty}$-norm of $\partial_{x_{k}}A^{0}f,$ first
let us observe that the symbol of
$\partial_{x_{k}}A^{0}=\textnormal{Op}(\sigma^{\prime}_{k})$ is given by

$\sigma^{\prime}_{k}(x,\xi):=2\pi
i\xi_{k}\sigma^{0}(x,\xi)+(\partial_{x_{k}}\sigma^{0}(x,\xi)).$ (5.8)

Indeed, the Leibniz law gives

$\displaystyle\partial_{x_{k}}A^{0}f(x)$
$\displaystyle=\int\limits_{\mathbb{R}^{n}}(\partial_{x_{k}}(e^{i2\pi
x\xi}\sigma^{0}(x,\xi))\widehat{f}(\xi))\,d\xi$
$\displaystyle=\int\limits_{\mathbb{R}^{2}}([\partial_{x_{k}}(e^{i2\pi
x\xi})\sigma^{0}(x,\xi)+e^{i2\pi
x\xi}\partial_{x_{k}}\sigma^{0}(x,\xi))]\widehat{f}(\xi))\,d\xi\,,$

and we obtain

$\displaystyle\partial_{x_{k}}A^{0}f(x)$
$\displaystyle=\int\limits_{\mathbb{R}^{2}}([e^{i2\pi x\xi}2\pi
i\xi_{k}\sigma^{0}(x,\xi)+e^{i2\pi
x\xi}\partial_{x_{k}}\sigma^{0}(x,\xi))]\widehat{f}(\xi)),$

which proves (5.8). By using a suitable partition of unity we will decompose
$\sigma^{\prime}_{k}(x,\xi)$ in the following way,

$\sigma^{\prime}_{k}(x,\xi)=\sum_{j=1}^{\infty}\rho_{j,k}(x,\xi).$

To construct the family of operators $\rho_{j,k}(x,\xi)$ we will proceed as
follows. We choose a smooth real function $\eta$ satisfying $\eta(t)\equiv 0$
for $|t|\leqslant 1$ and $\eta(t)\equiv 1$ for $|t|\geqslant 2.$ Set

$\rho(t)=\eta(t)-\eta(\frac{t}{2}).$

The support of $\rho$ satisfies $\textnormal{supp}(\rho)\subset[1,4].$ One can
check that

$1=\eta(t)+\sum_{j=1}^{\infty}\rho(2^{j}t),\,\,\,\,\textnormal{ for all
}t\in\mathbb{R}.$

Indeed,

$\eta(t)+\sum_{j=1}^{\ell}\rho(2^{j}t)=\eta(t)+\sum_{j=1}^{\ell}\left(\eta(2^{j}t)-\eta(2^{j-1}t)\right)=\eta(2^{\ell}t)\rightarrow
1,\,\,\ell\rightarrow\infty.$

For $t=rm(x,\xi)$ we have

$1=\eta(rm(x,\xi))+\sum_{j=1}^{\infty}\rho(2^{j}rm(x,\xi)).$

Observe that

$\sigma^{\prime}_{k}(x,\xi)=\eta(rm(x,\xi))\sigma^{\prime}_{k}(x,\xi)+\sum_{j=1}^{\infty}\rho(2^{j}rm(x,\xi))\sigma^{\prime}_{k}(x,\xi).$

Because $\eta(rm(x,\xi))\sigma^{\prime}_{k}(x,\xi)=0,$ in view that the
support of $\tilde{\gamma}$ and $\eta$ are disjoint sets, we have

$\sigma^{\prime}_{k}(x,\xi)=\sum_{j=1}^{\infty}\rho_{j,k}(x,\xi),\,\,\,\rho_{j,k}(x,\xi):=\rho(2^{j}rm(x,\xi))\sigma^{\prime}_{k}(x,\xi).$

From the mean value theorem we have,

$\displaystyle|A^{0}f(x)-A^{0}f(y)|$ $\displaystyle\leqslant
C_{0}\sum_{k=1}^{n}|y-x|\cdot\|\partial_{x_{k}}A^{0}f\|_{L^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\lesssim
r\sum_{k=1}^{n}\|\partial_{x_{k}}A^{0}f\|_{L^{\infty}(\mathbb{R}^{n})}.$

So, we have,

$\displaystyle I_{0}:$
$\displaystyle=\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|A^{0}f(x)-(A^{0}f)_{B(x_{0},r)}|dx$
$\displaystyle=\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}\left|\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}(A^{0}f(x)-A^{0}f(y))dy\right|dx$
$\displaystyle\leqslant\frac{1}{|B(x_{0},r)|^{2}}\int\limits_{B(x_{0},r)}\int\limits_{B(x_{0},r)}|A^{0}f(x)-A^{0}f(y)|dydx\lesssim
r\sup_{1\leqslant k\leqslant
n}\|\partial_{x_{k}}A^{0}f\|_{L^{\infty}(\mathbb{R}^{n})}$
$\displaystyle=r\sup_{1\leqslant k\leqslant
n}\|\textnormal{Op}(\sigma^{\prime}_{k})f\|_{L^{\infty}(\mathbb{R}^{n})}\leqslant
r\sup_{1\leqslant k\leqslant
n}\sum_{j=1}^{\infty}\|\textnormal{Op}(\rho_{j,k})f\|_{L^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\lesssim
r\sum_{j=1}^{\infty}r^{-1}2^{-j}\|\sigma\|_{\ell,S(m^{-\beta},g)}\|f\|_{L^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\lesssim
rr^{-1}\|\sigma\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}\|f\|_{L^{\infty}(\mathbb{R}^{n})}$
$\displaystyle=\|\sigma\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}\|f\|_{L^{\infty}(\mathbb{R}^{n})},$

where we have used that for $\ell>\frac{n}{4}$

$\displaystyle\|\textnormal{Op}(\rho_{j,k})\|_{\mathscr{B}(L^{\infty}(\mathbb{R}^{n}))}\lesssim\|\rho_{j,k}\|_{\ell,S(m^{-\frac{n}{2}\beta},g)}\lesssim
r^{-1}2^{-j}\|\sigma\|_{\ell,S(m^{-\frac{n}{2}\beta},g)},$

in view of Lemma 5.1 and Proposition 2.5 of [20] applied to $\rho_{j,k}$ with
$s=2^{j}r.$ In order to finish the proof, we need to prove (5.4) for $j=1.$ In
order to obtain a similar $L^{\infty}(\mathbb{R}^{n})$-${BMO}(\mathbb{R}^{n})$
estimate for $A^{1},$ we will proceed as follows. Let $\phi$ be a smooth
function compactly supported in $B(x_{0},2r)$ satisfying that

$\phi(x)=1,\textnormal{ for }\,\,x\in B(x_{0},r),\textnormal{ and
}0\leqslant\phi\leqslant 10.$

Note that

$|B(x_{0},r)|\leqslant\int\limits_{B(x_{0},r)}\phi(x)^{2}dx\leqslant\int\limits_{B(x_{0},2r)}\phi(x)^{2}dx=\|\phi\|_{L^{2}(\mathbb{R}^{n})}^{2}\leqslant
100|B(x_{0},2r)|,$

and consequently

$|B(x_{0},r)|^{\frac{1}{2}}\leqslant\|\phi\|_{L^{2}(\mathbb{R}^{n})}\leqslant
10|B(x_{0},2r)|^{\frac{1}{2}}\leqslant 10C|B(x_{0},r)|^{\frac{1}{2}},$ (5.9)

where in the last inequality we have used that the Lebesgue measure on
$\mathbb{R}^{n}$ satisfies the doubling property. Taking into account that

$\displaystyle\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|A^{1}f(x)-(A^{1}f)_{B(x_{0},r)}|dx\leqslant\frac{2}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|A^{1}f(x)|dx,$

we will estimate the right-hand side. Indeed, let us observe that

$\displaystyle\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|A^{1}f(x)|dx\leqslant\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|\phi(x)A^{1}f(x)|dx$
$\displaystyle\leqslant\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|A^{1}[\phi
f](x)|dx+\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|[M_{\phi},A^{1}]f(x)|dx$
$\displaystyle:$ $\displaystyle=I+II,$

where $M_{\phi}$ is the multiplication operator by $\phi.$ To estimate $I,$
observe that, in view of the Cauchy-Schwartz inequality, we have

$\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|A^{1}[\phi
f](x)|dx\leqslant\frac{1}{|B(x_{0},r)|^{\frac{1}{2}}}\left(\int\limits_{B(x_{0},r)}|A^{1}[\phi
f](x)|^{2}dx\right)^{\frac{1}{2}}.$

Let
$L:=m(x,D)^{\frac{n}{2}\beta\varepsilon}\in\textnormal{Op}(S(m^{\frac{n}{2}\beta\varepsilon},g)),$
where $0<\varepsilon<1.$ Observe that

$\textnormal{Op}(S(m^{-\frac{n}{2}\beta\varepsilon},g))\subset\textnormal{Op}(S(1,g))\subset\mathscr{B}(L^{2}(\mathbb{R}^{n}))\,.$
(5.10)

Because $A^{1}\in S(m^{-\frac{n}{2}\beta},g),$ the Weyl-Hörmander calculus
gives

$A^{1}L\in\textnormal{Op}(S(m^{-\frac{n}{2}\beta(1-\varepsilon)},g)).$

It follows from (5.10) that $A^{1}L$ is bounded on $L^{2}(\mathbb{R}^{n}).$
Consequently,

$\displaystyle\frac{1}{|B(x_{0},r)|^{\frac{1}{2}}}\left(\int\limits_{B(x_{0},r)}|A^{1}L[L^{-1}(\phi
f)](x)|^{2}dx\right)^{\frac{1}{2}}\leqslant\frac{\|A^{1}L[L^{-1}(\phi
f)]\|_{L^{2}(\mathbb{R}^{n})}}{|B(x_{0},r)|^{\frac{1}{2}}}$
$\displaystyle\leqslant\frac{C\|\sigma_{A^{1}L}\|_{k,S(m^{-\frac{n}{2}\beta(1-\varepsilon)},g)}\|L^{-1}(\phi
f)\|_{L^{2}(\mathbb{R}^{n})}}{|B(x_{0},r)|^{\frac{1}{2}}}.$

By observing that

$\|L^{-1}(\phi f)\|_{L^{2}(\mathbb{R}^{n})}=\|\phi
f\|_{H(m^{-\frac{n}{2}\beta\varepsilon},g)},$

where $H(m^{-\frac{n}{2}\beta\varepsilon},g)$ is the Sobolev space of order
$-\frac{n}{2}\beta\varepsilon,$ the embedding
$L^{2}(\mathbb{R}^{n})\hookrightarrow H(m^{-\frac{n}{2}\beta\varepsilon},g),$
implies that

$\|L^{-1}(\phi f)\|_{L^{2}(\mathbb{R}^{n})}=\|\phi
f\|_{H(m^{-\frac{n}{2}\beta\varepsilon},g)}\lesssim\|\phi
f\|_{L^{2}(\mathbb{R}^{n})}.$

Moreover, from (5.9), we deduce the inequality

$\|\phi
f\|_{L^{2}(\mathbb{R}^{n})}\leqslant\|f\|_{L^{\infty}(\mathbb{R}^{n})}\|\phi\|_{L^{2}(\mathbb{R}^{n})}\lesssim
10\|f\|_{L^{\infty}(\mathbb{R}^{n})}|B(x_{0},r)|^{\frac{1}{2}}.$

So, we conclude

$\displaystyle I:=\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|A^{1}[\phi
f](x)|dx$
$\displaystyle\leqslant{C\|\sigma_{A^{1}L}\|_{\ell^{\prime},S(1,g)}}\|f\|_{L^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\leqslant{C\|\sigma_{A^{1}}\|_{\ell^{\prime\prime},S(m^{-\frac{n\beta\varepsilon}{2}},g)}}\|f\|_{L^{\infty}(\mathbb{R}^{n})}$

which is the desired estimate for $I.$ Indeed, since $A^{1}L\in S(1,g),$ we
have that $A^{1}=A^{1}LL^{-1}\in S(m^{-\frac{n\beta\varepsilon}{2}},g).$ Now,
we will estimate $II.$ For this, observe that the symbol of
$[M_{\phi},A^{1}]=\textnormal{Op}(\theta),$ is given by

$\theta(x,\xi)=\int\limits_{\mathbb{R}^{n}}(\phi(x)-\phi(x-y))k(y,x-y)e^{-2\pi
iy\xi}dy,$ (5.11)

where $x\mapsto k(\cdot,\cdot)\in
C^{\infty}(\mathbb{R}^{n})\otimes\mathscr{D}^{\prime}(\mathbb{R}^{n}),$ is the
integral kernel of $A^{1}.$ Indeed, the equality (5.11) was shown in [20, page
364]. Using the Taylor expansion we obtain

$\phi(x-y)=\phi(x)+\sum_{|\alpha|=1}C_{\alpha}(\partial_{x}^{\alpha}\phi)(x)y^{\alpha},$

So, we can write

$\theta(x,\xi)=\sum_{|\alpha|=1}C_{\alpha}{\partial_{x}^{\alpha}\phi(x)}\partial_{\xi}^{{\alpha}}\sigma(x,\xi).$

The hypothesis $\sigma\in S(m^{-\frac{n}{2}\beta},g),$ implies that
$\partial_{\xi}^{{\alpha}}\sigma(x,\xi)\in S(m^{-\frac{n}{2}\beta},g),$ and
the fact that $\phi$ is of compact support, implies the same conclusion for
$\theta:$ $\theta(x,\xi)\in S(m^{-\frac{n}{2}\beta},g).$ So, one has

$\displaystyle\|\textnormal{Op}(\theta)\|_{\mathscr{B}(L^{\infty}(\mathbb{R}^{n}))}\lesssim\left(\sup_{(x,\xi)\in\mathbb{R}^{2n},|\alpha|\leqslant\ell^{\prime}}|[\partial_{\xi}^{\alpha}\theta(x,\xi)]m(x,\xi)^{\frac{n}{2}\beta})|\right)$
$\displaystyle\lesssim\sup_{|\alpha|\leqslant\ell}\|{\partial_{x}^{\alpha}\phi}\|_{L^{\infty}(\mathbb{R}^{n})}\left(\sup_{(x,\xi)\in\mathbb{R}^{2n},|\alpha|\leqslant\ell^{\prime}}|[\partial_{\xi}^{\alpha}\sigma(x,\xi)]m(x,\xi)^{\frac{n}{2}\beta})|\right)$
$\displaystyle\lesssim_{\ell^{\prime}}\|\sigma\|_{\ell^{\prime}+1,S(m^{-\frac{n}{2}\beta},g)}.$

So, we deduce

$\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|[M_{\phi},A^{1}]f(x)|\leqslant\|[M_{\phi},A^{1}]f\|_{L^{\infty}(\mathbb{R}^{n})},$

and consequently,

$\displaystyle\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|[M_{\phi},A^{1}]f(x)|$
$\displaystyle\leqslant\|\textnormal{Op}(\theta)f\|_{L^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\|\sigma\|_{\ell^{\prime}+1,S(m^{-\frac{n}{2}\beta},g)}\|f\|_{L^{\infty}(\mathbb{R}^{n})}.$

Thus, we obtain

$II:=\frac{1}{|B(x_{0},r)|}\int\limits_{B(x_{0},r)}|[M_{\phi},A^{1}]f(x)|dx\leqslant
C\|\sigma_{A}\|_{\ell^{\prime}+1,S(m^{-\frac{n}{2}\beta},g)}\|f\|_{L^{\infty}(\mathbb{R}^{n})}.$

Thus, the proof is complete. ∎

###  5.2. $L^{p}$-boundedness

The previous theorem and classical real interpolation have the following
consequences on the $L^{p}$-boundedness for the classes
$S(m^{-\frac{n}{2}\beta},g)$. The proof of this theorem follows standard
arguments and we omit it.

######  Theorem 5.3.

Let $\varepsilon_{0}=\frac{3}{4}$ and $\varepsilon_{0}\leq\beta<1$ and let
$\sigma(x,D)\in\textnormal{Op}S(m^{-\frac{n}{2}\beta},g).$ Then,
$\sigma(x,D):L^{p}(\mathbb{R}^{n})\rightarrow L^{p}(\mathbb{R}^{n})$ extends
to a bounded operator, for all $1<p<\infty.$ Moreover, for every
$\ell>\frac{n}{4},$ we have

$\|\sigma(x,D)f\|_{L^{p}(\mathbb{R}^{n})}\leq
C\|\sigma(x,\xi)\|_{\ell,S(m^{-n\beta/2},g)}\|f\|_{{L^{p}(\mathbb{R}^{n})}},$

where the constant $C>0,$ is independent of $f\in L^{p}(\mathbb{R}^{n}),$

Regarding the smaller exponents $\varepsilon_{0}$ below, by applying the
Fefferman-Stein interpolation we obtain the following bounds:

######  Theorem 5.4.

Let $\varepsilon_{0}=\frac{3}{4}$ and $0\leq\beta<\varepsilon_{0}$ and let
$\sigma(x,D)\in\textnormal{Op}(S(m^{-\frac{n}{2}\beta},g)).$ Then,
$\sigma(x,D):L^{p}(\mathbb{R}^{n})\rightarrow L^{p}(\mathbb{R}^{n})$ extends
to a bounded operator, provided

$\left|\frac{1}{p}-\frac{1}{2}\right|\leq\frac{n}{2}\beta.$ (5.12)

Moreover, for such $p$ and all $\ell>\frac{n}{4}$ we have

$\|\sigma(x,D)f\|_{L^{p}(\mathbb{R}^{n})}\leq
C\|\sigma(x,\xi)\|_{\ell,S(m^{-n\beta/2},g)}\|f\|_{{L^{p}(\mathbb{R}^{n})}},$

where the constant $C>0,$ is independent of $f\in L^{p}(\mathbb{R}^{n}).$

###### Proof.

Now, having proved Theorem 5.2, the proof is verbatim the proof of Theorem
2.10 of [20]. We will present the argument here, for completeness of our
paper. We will use the complex Fefferman-Stein interpolation theorem. Let us
consider the complex family of operators indexed by $z\in\mathbb{C},$
$\mathfrak{Re}(z)\in[0,1],$ given by

$T_{z}:=\textnormal{Op}(\sigma_{z}),\,\,\,\,\sigma_{z}(x,\xi):=e^{z^{2}}\sigma(x,\xi)m(x,\xi)^{\frac{n\beta}{2}+{\frac{n}{2}(\frac{3}{4})(z-1)}}.$

The family of operators $\\{T_{z}\\},$ defines an analytic family of operators
from $\mathfrak{Re}(z)\in(0,1),$ (resp. continuous for
$\mathfrak{Re}(z)\in[0,1]$), into the algebra of bounded operators on
$L^{2}(\mathbb{R}^{n}).$ Let us observe that
$\sigma_{0}(x,\xi)=\sigma(x,\xi)m(x,\xi)^{{\frac{n\beta}{2}-{\frac{n}{2}(\frac{3}{4})}}},$
and $\sigma_{1}(x,\xi)=e\sigma(x,\xi)m(x,\xi)^{{\frac{n\beta}{2}}}.$ Since
$T_{0}$ is bounded from $L^{\infty}(\mathbb{R}^{n})$ into
${BMO}(\mathbb{R}^{n})$ and $T_{1}$ is bounded on $L^{2}(\mathbb{R}^{n}),$ the
Fefferman-Stein interpolation theorem implies that $T_{t}$ extends to a
bounded operator on $L^{p}(\mathbb{R}^{n}),$ for $p=\frac{2}{t}$ and all
$0<t\leqslant 1.$ Because $0\leqslant\beta<\frac{3}{4},$ there exist
$t_{0}\in(0,1)$ such that $\beta=\frac{3}{4}(1-t_{0}).$ So,
$T_{t_{0}}=e^{t_{0}^{2}}A$ extends to a bounded operator on
$L^{\frac{2}{t_{0}}}.$ The Fefferman-Stein interpolation theorem, the
$L^{2}(\mathbb{R}^{n})$-boundedness and the $L^{\frac{2}{t_{0}}}$-boundedness
of $A$ give the $L^{p}(\mathbb{R}^{n})$-boundedness of $A$ for all $2\leqslant
p\leqslant\frac{2}{t_{0}},$ and interpolating the
$L^{\frac{2}{t_{0}}}(\mathbb{R}^{n})$-boundedness with the
$L^{\infty}(\mathbb{R}^{n})$-${BMO}(\mathbb{R}^{n})$ boundedness of $A$ we
obtain the boundedneess of $A$ on $L^{p}(\mathbb{R}^{n})$ for all
$\frac{2}{t_{0}}\leqslant p<\infty.$ So, $A$ extends to a bounded operator on
$L^{p}(\mathbb{R}^{n})$ for all $2\leqslant p<\infty.$ The
$L^{p}(\mathbb{R}^{n})$-boundedness of $A$ for $1<p\leqslant 2$ now follows by
the duality argument. ∎

The exponent $\varepsilon_{0}$ in Lemma 5.1 can be improved if we dispose of
more information on $a_{2}$. For instance, if $L$ is a second order
differential operator, formally self-adjoint,

$Lf=-\sum a_{ij}(x)\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}f+\mbox{
lower order terms},\quad\quad f\in C_{0}^{\infty}(\mathbb{R}^{n}),$ (5.13)

with the conditions as in (1.2) and for any $x\in\mathbb{R}^{n}$, the matrix
$a_{i,j}(x)$ has rank $r(x)\geq r_{0}$. An important special case of the
situation above arises from the sum of squares

$L=-\sum\limits_{j=1}^{M}X_{j}^{*}X_{j},$

where $X_{j}$ are real vector fields on $\mathbb{R}^{n}$, provided that the
linear span of the family of vector fields $\\{X_{j}(x)\\}_{1\leq j\leq M}$ is
at least $r_{0}$-dimensional, $r_{0}\geq 1$, for each $x$.

We give an example on $\mathbb{R}^{2}$ in the case of the operator

$a(x,D):=-(\partial_{x_{1}}^{2}+\tilde{x}_{1}^{2}\partial_{x_{2}}^{2})+|x|^{2}.$
(5.14)

The exponent $\varepsilon_{0}=\frac{3}{4}$ can be improved by replacing it by
$\frac{3}{2}$ in Lemma 5.1, and as a consequence improving it as well in
Theorems 5.2, 5.3 and 5.6.

We now state the improvement in Lemma 5.1 for the classes corresponding to the
operator (5.14). More precisely, a detailed look at the proof of Lemma 5.1
allows one to improve the parameter $\varepsilon_{0}$ from $3/4$ until $5/12$
as follows.

######  Lemma 5.5.

Let us consider the metric $g$ given in (3.1) with
$a_{2}(x,\xi)=\xi_{1}^{2}+\tilde{x_{1}}^{2}\xi_{2}^{2},$
$(x,\xi)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$ being the symbol of the
degenerate elliptic operator

$a_{2}(x,D):=-(\partial_{x_{1}}^{2}+\tilde{x}_{1}^{2}\partial_{x_{2}}^{2}).$

Let us consider the $g$-weight
$m(X)=\xi_{1}^{2}+\tilde{x_{1}}^{2}\xi_{2}^{2}+|x|^{2}+\langle X\rangle\in
S(m,g).$ Let $\varepsilon_{0}=\frac{5}{12}$ and
$\varepsilon_{0}\leq\varepsilon<1.$ Consider a symbol $q\in
S(m^{-\varepsilon},g)$ supported in $R\leq a(x,\xi)+\langle X\rangle\leq 3R$
for $R>1$. Then, for all $l>1$ we have

$\lvert|q(x,D)f\rvert|_{\infty}\leq
C\lvert|q\rvert|_{l;S(m^{-\varepsilon},g)}\lvert|f\rvert|_{L^{\infty}},$

where the constant $C$ is independent of $q$ and of $f$.

###### Proof.

Wee proceed like in Lemma 5.1. Let $q\in S(m^{-\varepsilon},g)$ be supported
in

$\\{(x,\xi)\in\mathbb{R}^{2}\times\mathbb{R}^{2}\,:R\leq a(x,\xi)+\langle
X\rangle\leq 3R\\},$

for $R>1$ fixed. It is then enough to prove that for all $x\in\mathbb{R}^{2}$
and $l>1,$

$||\hat{q}_{x}\rvert|_{L^{1}}\leq C\lvert|q\rvert|_{l;S(m^{-\varepsilon},g)}.$

Let us consider $b=R^{\varepsilon-1}.$ The Cauchy-Schwarz inequality gives the
estimate

$\int\limits_{|y|<b}|\hat{q}_{x}(y)|dy\leq
C||q\rvert|_{l;S(m^{-\varepsilon},g)}.$

On the other hand, for the integral $\int\limits_{|y|\geq
b}|\hat{q}_{x}(y)|dy,$ we first note that for an integer $k>1$, we have that

$\displaystyle\int\limits_{|y|\geq b}|\hat{q}_{x}(y)|dy\leq$ $\displaystyle
Cb^{1-k}\left(\,\int\limits_{y\geq
b}|y|^{2k}|\hat{q}_{x}(y)|^{2}dy\right)^{\frac{1}{2}}$ $\displaystyle\leq$
$\displaystyle
Cb^{1-k}\left(\,\int\limits_{\mathbb{R}^{2}}|\nabla_{\xi}^{k}q(x,\xi)|^{2}d\xi\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C\lvert|q\rvert|_{k;S(m^{-\varepsilon},g)}b^{1-k}\left(\,\int\limits_{R\leq
a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq
3R}m^{-2\varepsilon-k}(x,\xi)d\xi\right)^{\frac{1}{2}}.$

In order to estimate the last integral we split into two cases :
$|x|^{2}\leq\frac{R}{2C}$ and $|x|^{2}>\frac{R}{2C}$.

If $|x|^{2}\leq\frac{R}{2C}$, we get

$R\leq a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\lesssim
C(\xi_{1}^{2}+|\xi_{2}|+|x|^{2}).$

Since in the subelliptic zone $R\lesssim\xi_{1}^{2}+|\xi_{2}|,$ we have that

$\xi_{1}^{2}+|\xi_{2}|\lesssim a_{2}(x,\xi)+|x|^{2}+\langle
X\rangle=m(x,\xi),$

we can compare

$\int\limits_{R\leq a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq
3R}m^{-2\varepsilon-k}(x,\xi)d\xi\lesssim\int\limits_{\begin{subarray}{c}R\lesssim\xi_{1}^{2}+\xi_{2}\\\
\xi_{1},\xi_{2}>0\end{subarray}}(\xi_{1}^{2}+|\xi_{2}|)^{-2\varepsilon-k}d\xi.$

By making the changes of variables

$(\xi_{1},\xi_{2})=(\sqrt{R}u_{1},Ru_{2}^{2}),\,\,u_{1},u_{2}>0,$

we can estimate

$\int\limits_{\begin{subarray}{c}R\lesssim\xi_{1}^{2}+\xi_{2}\\\
\xi_{1},\xi_{2}>0\end{subarray}}(\xi_{1}^{2}+|\xi_{2}|)^{-2\varepsilon-k}d\xi\lesssim\int\limits_{\begin{subarray}{c}1\lesssim
u_{1}^{2}+u_{2}^{2}\\\
u_{1},u_{2}>0\end{subarray}}(u_{1}^{2}+u_{2}^{2})^{-2\varepsilon-k}R^{-2\varepsilon-k}R^{\frac{3}{2}}u_{2}du_{1}du_{2}=R^{-2\varepsilon-k+\frac{3}{2}}I,$

where the integral

$\displaystyle I=\int\limits_{\begin{subarray}{c}1\lesssim
u_{1}^{2}+u_{2}^{2}\\\
u_{1},u_{2}>0\end{subarray}}(u_{1}^{2}+u_{2}^{2})^{-2\varepsilon-k}u_{2}du_{1}du_{2}<\infty$
(5.15)

converges since when $k>3-2\varepsilon.$ So, putting all the previous
estimates together we deduce that

$\int\limits_{|y|\geq b}|\hat{q}_{x}(y)|dy\lesssim
R^{(\varepsilon-1)(1-k)-\varepsilon-\frac{k}{2}+\frac{3}{4}}\leq C.$

To see the last estimate note that $R>1$ and that

$(\varepsilon-1)(1-k)-\varepsilon-\frac{k}{2}+\frac{3}{4}=k(\frac{1}{2}-\varepsilon)-\frac{1}{4}\leq
0,$ (5.16)

if for example $k$ is the sharp parameter in the convergence of $I$ in (5.15),
namely, $k=3,$ from which the inequality

$3(\frac{1}{2}-\varepsilon)-\frac{1}{4}\leq 0,$

from then we require that $\varepsilon\geq 5/12.$

On the other hand, if $|x|^{2}>\frac{R}{2C}$, we have $-|x|^{2}<-\frac{R}{2C}$
and

$\xi_{1}^{2}+|\xi_{2}|\leq a_{2}(x,\xi)+\langle\xi\rangle\leq 3R-|x|^{2}\leq
3R-\frac{R}{2C}=\frac{(6C-1)}{2C}R.$

Since $6C-1>0$ and by using the fact that $|x|^{2}>\frac{R}{2C}$, we have
$\langle x\rangle^{-k}\leq C^{\prime}R^{-\frac{k}{2}}$ and we obtain

$b^{1-k}\left(\,\int\limits_{R\leq a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq
3R}m^{-2\varepsilon-k}(x,\xi)d\xi\right)^{\frac{1}{2}}=b^{1-k}\left(\,\int\limits_{\xi_{1}^{2}+|\xi_{2}|\leq
C^{\prime}R}m^{-2\varepsilon}(x,\xi)m^{-k}(x,\xi)d\xi\right)^{\frac{1}{2}}$

$\leq b^{\frac{n}{2}-k}\left(\int\limits_{\xi_{1}^{2}+|\xi_{2}|\leq
C^{\prime}R}(\xi_{1}^{2}+|\xi_{2}|)^{-2\varepsilon}\langle
x\rangle^{-2k}d\xi\right)^{\frac{1}{2}}$ $\leq
R^{(\varepsilon-1)(1-k)-\frac{k}{2}}\left(\int\limits_{\xi_{1}^{2}+|\xi_{2}|\leq
C^{\prime}R}(\xi_{1}^{2}+|\xi_{2}|)^{-2\varepsilon}d\xi\right)^{\frac{1}{2}}$
$\lesssim
R^{(\varepsilon-1)(1-k)-\frac{k}{2}}\left(\int\limits_{\begin{subarray}{c}u_{1}^{2}+u_{2}^{2}\lesssim
1\\\
u_{1},u_{2}>0\end{subarray}}(u_{1}^{2}+u_{2}^{2})^{-2\varepsilon}R^{-2\varepsilon}R^{\frac{3}{2}}u_{2}du_{1}du_{2}\right)^{\frac{1}{2}}$
$=R^{(\varepsilon-1)(1-k)-\frac{k}{2}}R^{-\varepsilon+\frac{3}{4}}\sqrt{J},$

where

$J=\int\limits_{\begin{subarray}{c}u_{1}^{2}+u_{2}^{2}\lesssim 1\\\
u_{1},u_{2}>0\end{subarray}}(u_{1}^{2}+u_{2}^{2})^{-2\varepsilon}u_{2}du_{1}du_{2}\lesssim\int\limits_{\begin{subarray}{c}u_{1}^{2}+u_{2}^{2}\lesssim
1\\\
u_{1},u_{2}>0\end{subarray}}(u_{1}^{2}+u_{2}^{2})^{-2\varepsilon}du_{1}du_{2}$

is convergent for any $\varepsilon$ such that $2\varepsilon<2,$ that is when
$\varepsilon<1.$ So, in the case where $|x|^{2}>\frac{R}{2C},$ this analysis
shows that

$\int\limits_{|y|\geq b}|\hat{q}_{x}(y)|dy\lesssim
R^{(\varepsilon-1)(1-k)-\frac{k}{2}}R^{-\varepsilon+\frac{3}{4}}\leq C,$

when e.g. $k=3,$ in view of the inequality

$(\varepsilon-1)(1-k)-\frac{k}{2}-\varepsilon+\frac{3}{4}=3(\frac{1}{2}-\varepsilon)-\frac{1}{4}\leq
0$

for any $\varepsilon\geq 5/12.$ The proof of the lemma is complete. ∎

Regarding the improved exponent below $\varepsilon_{0}$, by applying the
Fefferman-Stein interpolation we obtain the following bounds. Since its proof
follows the same arguments as the ones in the proof of Theorem 5.6 we omit it.

######  Theorem 5.6.

Let $\varepsilon_{0}=\frac{5}{12}$ and $0\leq\beta<\varepsilon_{0}$ and let
$\sigma(x,D)\in\textnormal{Op}(S(m^{-\beta},g)).$ Then,
$\sigma(x,D):L^{p}(\mathbb{R}^{2})\rightarrow L^{p}(\mathbb{R}^{2})$ extends
to a bounded operator, provided

$\left|\frac{1}{p}-\frac{1}{2}\right|\leq\beta.$ (5.17)

Moreover, for such $p$ we have

$\|\sigma(x,D)f\|_{L^{p}(\mathbb{R}^{2})}\leq
C\|\sigma(x,\xi)\|_{3,S(m^{-\beta},g)}\|f\|_{{L^{p}(\mathbb{R}^{2})}},$

where the constant $C>0,$ is independent of $f\in L^{p}(\mathbb{R}^{2}).$

##  6\. Improved $L^{p}$ estimates and Schatten properties for Schrödinger
operators for Hörmander sums of squares

In this section we are going to improve the estimates obtained in Lemma 5.1
for the case of sums of squares $\sum_{j=1}^{k}X_{j}^{2}$ of vector fields
$X_{j}$ satisfying the Hörmander condition of order $2$. This means that the
vector fields $X_{j}$ and their commutators $[X_{i},X_{j}]$ span
$\mathbb{R}^{n}.$ As a consequence all the corresponding $L^{p}$ estimates for
the classes $S(m^{-\frac{n\theta}{2}},g)$ will also be improved. Also in this
case, a sharp index $\mu_{0}$ that allows the membership of the operators
$S(m^{-\mu},g)$ to the Schatten von Neumann classes
$S_{r}(L^{2}(\mathbb{R}^{n})),$ when $\mu>\mu_{0}$ is given.

######  Remark 6.1.

For the next properties we follow Nagel, Stein and Wainger [69], see also
Jerison and Sánchez-Calle [58]. Let $X_{1},X_{2},\cdots,X_{k},$ be a system of
vector fields satisfying the Hörmander condition of order 2. Then, we can
endow $\mathbb{R}^{n}$ with a nilpotent structure as follows. If
$Y_{1},\cdots,Y_{n},$ is an enumeration of the vectors
$X_{1},X_{2},\cdots,X_{k},$ and of their commutators $[X_{i},X_{j}],$ define
$d_{j}=\textnormal{deg}(Y_{j})=1,$ if $Y_{j}$ is one of the vector-fields
$X_{i},$ and define $d_{j}(Y_{j})=2,$ if $Y_{j}$ is a commutator. Let
$Q=\sum_{j}d_{j}$ be the homogeneous dimension associated the Carnot-
Carathéodory distance $|\cdot|_{cc}$ associated with the Hörmander system
$\\{X_{1},X_{2},\cdots,X_{k}\\}.$ For every $j,$ let $X_{j}(x,\xi)$ be the
principal symbol of the vector-field $X_{j},$ $1\leq j\leq k.$ Then:

  * •

The function $(x,\xi)\mapsto a_{2}(x,\xi)=\sum_{j=1}^{k}X_{j}(x,\xi)^{2}$ is
the principal symbol of $a_{2}(x,D).$

  * •

For any $x\in\mathbb{R}^{n},$ there exists a positive-definite matrix
$(a_{i,j}(x))_{i,j=1}^{n}$ in such a way that

$a_{2}(x,\xi)=\sum_{j=1}^{k}X_{j}(x,\xi)^{2}=\sum_{1\leq j\leq k\,,\\\
1\leq\ell\leq n}a_{j,\ell}(x)\xi_{j}\xi_{\ell},\,$

and with rank $r_{0}(x)\geq r_{0}.$

  * •

There exists a non-singular matrix $\theta(x),$ such that $\det(\theta(x))=1$
and with

$a_{2}(x,\xi)=\sum_{j=1}^{r(x)}\lambda_{j}(x)(\theta(x)\xi)_{j},$ (6.1)

where $0<\lambda_{j}(x),$ $1\leq j\leq r(x),$ are the positive eigenvalues of
the matrix $(a_{j,\ell}(x)).$

  * •

The function $\xi\mapsto\overline{\xi}=\theta(x)\xi,$ and the norm

$\|\overline{\xi}\|:=\left(\sum_{j=1}^{r_{0}}\overline{\xi}_{j}^{4}+\sum_{j=r_{0}+1}\overline{\xi}_{j}^{2}\right)^{\frac{1}{4}},$
(6.2)

is homogenous of order one at $\overline{\xi}\neq 0,$ that is,
$\|\delta_{a}\overline{\xi}\|=a\|\overline{\xi}\|,$ where the family of
dilations $(\delta_{a})_{a>0}$,
$\delta_{a}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is defined by

$(C):\quad\delta_{a}(x)=(ax_{1},\cdots,ax_{r_{0}},a^{2}x_{r_{0}+1},\cdots,a^{2}x_{n}).$

One also has the change of variable property $d(\delta_{a}\xi)=a^{Q}d\xi.$

  * •

On has the estimate

$(1+\|\overline{\xi}\|)\leq Cm(x,\theta^{-1}(x)\xi),\,\xi\in\mathbb{R}^{n},$
(6.3)

for some $C>0.$

With the notation in Remark 6.1 we present the following lemma.

######  Lemma 6.2.

Let $\mathbb{X}:=\\{X_{1},X_{2},\dots,X_{k}\\}$ be a family of real vector
fields on $\mathbb{R}^{n}$ satisfying the Hörmander condition of order $2.$
Let $a_{2}(x,D)$ be the Hörmander sub-Laplacian

$a_{2}(x,D)=-\sum\limits_{j=1}^{k}X_{j}^{2},$ (6.4)

and let us consider the metric $g$ and the $g$-weight $m$ defined as in (3.1)
and (3.2), respectively. Let $\varepsilon_{0}=\frac{Q}{2n}$ and let
$\varepsilon_{0}\leq\varepsilon\leq 1.$ Consider a symbol $q\in
S(m^{-\frac{n}{2}\varepsilon},g)$ supported in $R\leq a(x,\xi)+\langle
X\rangle\leq 3R$ for $R>1$. Then, for all $k\geq 1$ we have that

$\lvert|q(x,D)f\rvert|_{\infty}\leq
C\lvert|q\rvert|_{k;S(m^{-n\varepsilon},g)}\lvert|f\rvert|_{L^{\infty}},$

where the constant $C$ is independent of $q$ and $f$.

###### Proof.

We only have to prove the lemma in the borderline case
$\varepsilon=\varepsilon_{0}.$ The result for the other values of
$\varepsilon\in[\varepsilon_{0},1)$ follows from the inclusion properties of
the $S(m,g)$-classes. Let us improve as necessary the proof given for Lemma
5.1. Let $q\in S(m^{-\frac{n}{2}\varepsilon},g)$ be supported in

$\\{(x,\xi)\in\mathbb{R}^{n}\times\mathbb{R}^{n}\,:R\leq a(x,\xi)+\langle
X\rangle\leq 3R\\},$

for $R>1$ fixed. In view of the convolution identity

$\displaystyle q(x,D)f(x)=$ $\displaystyle\hat{q}_{x}*f(x),$

where $\hat{q}_{x}$ is the Fourier transform $q_{x}(\xi)=q(x,\xi)$ with
respect to the variable $\xi,$ we obtain

$|q(x,D)f(x)|\leq||\hat{q}_{x}\rvert|_{L^{1}}||f\rvert|_{L^{\infty}},\,\,x\in\mathbb{R}^{n}.$

It is then enough to prove that for all $x\in\mathbb{R}^{n}$ and $l>n/4$

$||\hat{q}_{x}\rvert|_{L^{1}}\leq
C\lvert|q\rvert|_{l;S(m^{-\frac{n}{2}\varepsilon},g)}.$

Let us consider $b>0$ to be fixed below. We can follow the same argument as in
the proof of Lemma 5.1 to deduce by the use of the Cauchy-Schwarz inequality
in order to obtain the estimate

$\displaystyle\int\limits_{|y|<b}|\hat{q}_{x}(y)|dy\leq
C||q\rvert|_{l;S(m^{-\frac{n}{2}\varepsilon},g)}b^{\frac{n}{2}}\left(\,\int\limits_{\langle\xi\rangle\leq
3R}\langle\xi\rangle^{-n\varepsilon}d\xi\right)^{\frac{1}{2}}.$

Now, by choosing $b=R^{\varepsilon-1}$, we obtain

$\int\limits_{|y|<b}|\hat{q}_{x}(y)|dy\leq
C||q\rvert|_{l;S(m^{-\frac{n}{2}\varepsilon},g)}.$

On the other hand, for the integral $\int\limits_{|y|\geq
b}|\hat{q}_{x}(y)|dy,$ we first note that for an integer $k>\frac{n}{4}$, we
have $k>1$ we have

$\displaystyle\int\limits_{|y|\geq b}|\hat{q}_{x}(y)|dy\leq$ $\displaystyle
Cb^{\frac{n}{2}-k}\left(\,\int\limits_{y\geq
b}|y|^{2k}|\hat{q}_{x}(y)|^{2}dy\right)^{\frac{1}{2}}$ $\displaystyle\leq$
$\displaystyle
Cb^{\frac{n}{2}-k}\left(\,\int\limits_{\mathbb{R}^{n}}|\nabla_{\xi}^{k}q(x,\xi)|^{2}d\xi\right)^{\frac{1}{2}},$

the Plancherel theorem implies the inequality

$\int\limits_{|y|\geq b}|\hat{q}_{x}(y)|dy\leq
C\lvert|q\rvert|_{k;S(m^{-n\varepsilon},g)}b^{\frac{n}{2}-k}\left(\,\int\limits_{R\leq
a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq
3R}m^{-n\varepsilon-k}(x,\xi)d\xi\right)^{\frac{1}{2}}.$ (6.5)

Let $X_{j}(x,\xi)$ be the principal symbol of the vector-field $X_{j},$ $1\leq
j\leq k.$ Then

$a_{2}(x,\xi)=\sum_{j=1}^{k}X_{j}(x,\xi)^{2}$ (6.6)

is the principal symbol of $a_{2}(x,D).$ For any $x\in\mathbb{R}^{n},$ in view
of Remark 6.1 we can find a positive-definite matrix
$(a_{i,j}(x))_{i,j=1}^{n}$ in such a way that

$a_{2}(x,\xi)=\sum_{j=1}^{k}X_{j}(x,\xi)^{2}=\sum_{1\leq j\leq k\,,\\\
1\leq\ell\leq n}a_{j,\ell}(x)\xi_{j}\xi_{\ell},\,$

and with range $r_{0}(x)\geq r_{0}.$ Then, there exists a non-singular matrix
$\theta(x),$ such that $\det(\theta(x))=1$ and with

$a_{2}(x,\xi)=\sum_{j=1}^{r(x)}\lambda_{j}(x)(\theta(x)\xi)_{j},$ (6.7)

where $0<\lambda_{j}(x),$ $1\leq j\leq r(x),$ are the positive eigenvalues of
the matrix $(a_{j,\ell}(x)).$ By defining

$\overline{\xi}:=\theta(x)\xi,$ (6.8)

$\|\overline{\xi}\|:=\left(\sum_{j=1}^{r_{0}}\overline{\xi}_{j}^{4}+\sum_{j=r_{0}+1}^{n}\overline{\xi}_{j}^{2}\right)^{\frac{1}{4}}\,,$
(6.9)

we use the following family of dilations

$\delta_{r}(x):=(r\overline{\xi}_{1},\cdots,r\overline{\xi}_{r_{0}},r^{2}\overline{\xi}_{r_{0}+1},\cdots,r^{2}\overline{\xi}_{n}),\,r>0.$
(6.10)

Observe that $d(\delta_{r}\xi)=r^{Q}d\xi.$ The properties of $\theta(x)$ give
the lower bound

$(1+\|\overline{\xi}\|)^{2}\lesssim a_{2}(x,\theta(x)^{-1}\overline{\xi}).$
(6.11)

In order to estimate the integral in (6.5) we split into two cases :
$|x|^{2}\leq\frac{R}{2C}$ and $|x|^{2}>\frac{R}{2C}$.

If $|x|^{2}\leq\frac{R}{2}$, let us estimate (6.5) as follows:

$\displaystyle\int\limits_{R\leq a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq
3R}m^{-n\varepsilon-k}(x,\xi)d\xi$ $\displaystyle=\int\limits_{R\leq
a_{2}(x,\xi)\leq 3R-|x|^{2}-\langle X\rangle}m^{-n\varepsilon-k}(x,\xi)d\xi$
$\displaystyle\leq\int\limits_{R\leq
a_{2}(x,\xi)}m^{-n\varepsilon-k}(x,\xi)d\xi$ $\displaystyle=\int\limits_{R\leq
a_{2}(x,\theta(x)^{-1}\overline{\xi})}m^{-n\varepsilon-k}(x,\theta(x)^{-1}\overline{\xi})|\det(\theta(x)^{-1})|d\overline{\xi}$
$\displaystyle\lesssim\int\limits_{\mathcal{A}}\left(\sum_{j=1}^{r_{0}}\overline{\xi}_{j}^{2}+\sum_{j=r_{0}+1}^{r(x)}\lambda_{j}(x)|\overline{\xi}_{j}|^{2}+|\overline{\xi}|\right)^{-n\varepsilon-k}d\overline{\xi},$

where

$\mathcal{A}:=\left\\{\overline{\xi}:\sum_{j=1}^{r_{0}}|\overline{\xi}_{j}|^{2}+\sum_{j=r_{0}+1}^{r(x)}\lambda_{j}(x)|\overline{\xi}_{j}|^{2}+|\overline{\xi}|\geq
R\right\\}.$

The change of variables $\overline{\xi}:=\delta_{\sqrt{R}}\xi^{\prime}$ gives
the inequality

$\displaystyle\int\limits_{R\leq a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq
3R}m^{-n\varepsilon-k}(x,\xi)d\xi$
$\displaystyle\lesssim\sqrt{R}^{Q-2(n\varepsilon+k)}\int\limits_{\mathcal{A}^{\prime}}\left(\sum_{j=1}^{r_{0}}\overline{\xi^{\prime}}_{j}^{2}+\sum_{j=r_{0}+1}^{r(x)}R\lambda_{j}(x)|\overline{\xi^{\prime}}_{j}|^{2}+|\overline{\xi^{\prime}}|\right)^{-n\varepsilon-k}d\overline{\xi^{\prime}},$

where

$\mathcal{A}^{\prime}:=\left\\{\overline{\xi}:\sum_{j=1}^{r_{0}}|\overline{\xi}_{j}^{\prime}|^{2}+\sum_{j=r_{0}+1}^{r(x)}R\lambda_{j}(x)|\overline{\xi^{\prime}}_{j}|^{2}+|\overline{\xi^{\prime}}|\geq
1\right\\}.$

Let us consider the set $B=\\{j:\,r_{0}\leq j\leq r(x)\textnormal{ and
}R\lambda_{j}(x)\geq 1\\}.$ Let us apply the change of variables
$\overline{\xi}^{\prime}\mapsto\overline{\xi},$ defined by

$\overline{\xi}_{j}:=\overline{\xi}_{j}^{\prime},\,j\notin B,$

and

$\overline{\xi}_{j}:=\delta_{\sqrt{R\lambda_{j}(x)}}\overline{\xi}_{j}^{\prime},\,j\in
B.$

Since $d\xi^{\prime}_{j}=(R\lambda_{j})^{-1}d\overline{\xi}_{j},$ for $j\in
B,$ the Jacobian of this change is

$d\overline{\xi}^{\prime}_{1}d\overline{\xi}_{2}^{\prime}\cdots
d\overline{\xi}_{r_{0}}^{\prime}\Pi_{j\in
B}d\overline{\xi}_{j}^{\prime}\Pi_{j\in(\\{r_{0}+1,\cdots,r(x)\\}\setminus
B\cup\\{r(x)+1,r(x)+2,\cdots,n\\}}d\overline{\xi}_{j}^{\prime}$ $=\Pi_{j\in
B}({R\lambda_{j}(x)})^{-|B|}d\overline{\xi}.$

The eigenvalue conditions $R\lambda_{j}(x)\geq 1,$ $j\in B,$ imply that

$|\overline{\xi}|\geq|\overline{\xi}^{\prime}|\geq
C\sum_{i=r(x)+1}^{n}|\overline{\xi}^{\prime}_{i}|,\textnormal{ for some }C>0,$

and taking into account that $({R\lambda_{j}(x)})^{-|B|}\leq 1,$ for any $j\in
B,$ we have the estimate

$\displaystyle\int\limits_{\mathcal{A}^{\prime}}\left(\sum_{j=1}^{r_{0}}\overline{\xi^{\prime}}_{j}^{2}+\sum_{j=r_{0}+1}^{r(x)}R\lambda_{j}(x)|\overline{\xi^{\prime}}_{j}|^{2}+|\overline{\xi^{\prime}}|\right)^{-n\varepsilon-k}d\overline{\xi^{\prime}}$
$\displaystyle\leq\int\limits_{\mathcal{A}^{\prime}}\left(\sum_{j=1}^{r_{0}}\overline{\xi^{\prime}}_{j}^{2}+\sum_{j=r_{0}+1}^{r(x)}R\lambda_{j}(x)|\overline{\xi^{\prime}}_{j}|^{2}+\sum_{i=r(x)+1}^{n}|\overline{\xi}^{\prime}_{i}|\right)^{-n\varepsilon-k}d\overline{\xi^{\prime}}$
$\displaystyle\lesssim\int\limits_{\mathcal{A}^{\prime\prime}}\left(\sum_{j=1}^{r_{0}}|\overline{\xi}_{j}|^{2}+\sum_{j=r_{0}+1}^{r(x)}|{\overline{\xi}}_{j}|^{2}+\sum_{i=r(x)+1}^{n}|\overline{\xi}_{i}|\right)^{-n\varepsilon-k}d\overline{\xi}$

where

$\mathcal{A}^{\prime\prime}:=\left\\{\overline{\xi}:\sum_{j=1}^{r_{0}}|\overline{\xi}_{j}|^{2}+\sum_{j=r_{0}+1}^{r(x)}|{\overline{\xi}}_{j}|^{2}+|\overline{\xi}|\geq
1\right\\}.$

Then, we can compare

$\displaystyle\int\limits_{\mathcal{A}^{\prime\prime}}\left(\sum_{j=1}^{r_{0}}|\overline{\xi}_{j}|^{2}+\sum_{j=r_{0}+1}^{r(x)}|{\overline{\xi}}_{j}|^{2}+\sum_{i=r(x)+1}^{n}|\overline{\xi}_{i}|\right)^{-n\varepsilon-k}d\overline{\xi}$
$\displaystyle\lesssim\int\limits_{\\{\xi^{\prime}:\sum_{j=1}^{r(x)}{\xi^{\prime}}_{j}^{2}+|\xi^{\prime}|\geq
1\\}}\left(\sum_{j=1}^{r_{0}}\overline{\xi^{\prime}}_{j}^{2}+|\overline{\xi}|\right)^{-n\varepsilon-k}d\overline{\xi^{\prime}}\leq
C(n,k,Q)<\infty$

when

$n\varepsilon+k>\frac{Q}{2}.$

So, putting all the estimates above together we have

$\int\limits_{|y|\geq b}|\hat{q}_{x}(y)|dy\leq
C\lvert|q\rvert|_{k;S(m^{-n\varepsilon},g)}R^{(\varepsilon-1)(\frac{n}{2}-k)}\left(\,\int\limits_{R\leq
a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq
3R}m^{-n\varepsilon-k}(x,\xi)d\xi\right)^{\frac{1}{2}}$ $\lesssim
C\|q\|_{k;S(m^{-n\varepsilon},g)}R^{(\varepsilon-1)(\frac{n}{2}-k)}\left(\sqrt{R}^{Q-2\left(n\varepsilon+k\right)}\right)^{\frac{1}{2}}$
$\lesssim
C\|q\|_{k;S(m^{-n\varepsilon},g)}R^{(\varepsilon-1)(\frac{n}{2}-k)+\frac{1}{4}({Q-2\left(n\varepsilon+k\right))}}.$

So, to estimate the norm $\int\limits_{|y|\geq b}|\hat{q}_{x}(y)|dy$ with a
bound independent of $R>1,$ we require the following two conditions

$n\varepsilon+k>\frac{Q}{2}$ (6.12)

and

$(\varepsilon-1)(\frac{n}{2}-k)+\frac{1}{4}({Q-2\left(n\varepsilon+k\right))}\leq
0.$ (6.13)

If $k\geq 1$ for $\varepsilon=\varepsilon_{0}=\frac{Q}{2n}$ we have that

$n\varepsilon+k\geq\frac{Q}{2}+1>\frac{Q}{2}$ (6.14)

which shows that for $k\geq 1$ the inequality in (6.12) is automatically
satisfied. Observe that

$\displaystyle(\varepsilon-1)\left(\frac{n}{2}-k\right)+\frac{1}{4}({Q-2\left(n\varepsilon+k\right))}$
$\displaystyle=\left(\frac{Q}{2n}-1\right)\left(\frac{n}{2}-k\right)+\frac{Q}{4}-\frac{Q}{4}-\frac{k}{2}$
$\displaystyle=\frac{Q}{4}-\frac{Qk}{2n}-\frac{n}{2}+\frac{k}{2}\leq 0,$

for any $k\geq 1.$ Indeed, since $Q=2n-r_{0},$ one has

$Q/4-n/2=-r_{0}/4.$

Note also that the inequality $Q\geq n$ implies that $Q/2n\geq 1/2,$ from
which for any $k\geq 1$ we deduce the required condition

$\displaystyle\frac{Q}{4}-\frac{Qk}{2n}-\frac{n}{2}+\frac{k}{2}=\left(\frac{Q}{4}-\frac{n}{2}\right)-k\left(\frac{Q}{2n}-\frac{1}{2}\right)\leq-
r_{0}/4\leq 0.$

On the other hand, if $|x|^{2}>\frac{R}{2C}$, we have $-|x|^{2}<-\frac{R}{2C}$
and

$\langle\xi\rangle\leq a_{2}(x,\xi)+\langle X\rangle\leq 3R-|x|^{2}\leq
3R-\frac{R}{2C}=\frac{(6C-1)}{2C}R.$

Since $6C-1>0$ and by using the fact that $|x|^{2}>\frac{R}{2C}$, we have
$\langle x\rangle^{-k}\leq C^{\prime}R^{-\frac{k}{2}}$ and we obtain

$\displaystyle b^{\frac{n}{2}-k}\left(\,\int\limits_{R\leq
a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq
3R}m^{-n\varepsilon-k}(x,\xi)d\xi\right)^{\frac{1}{2}}=$ $\displaystyle
b^{\frac{n}{2}-k}\left(\,\int\limits_{\langle\xi\rangle\leq
C^{\prime}R}m^{-n\varepsilon}(x,\xi)m^{-k}(x,\xi)d\xi\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
CR^{(\frac{n}{2}-k)(\varepsilon-1)}\left(\,\int\limits_{\langle\xi\rangle\leq
C^{\prime}R}\langle\xi\rangle^{-n\varepsilon}\langle
x\rangle^{-2k}d\xi\right)^{\frac{1}{2}}$ $\displaystyle\leq$ $\displaystyle
CR^{(\frac{n}{2}-k)(\varepsilon-1)}\langle
x\rangle^{-k}\left(\,\int\limits_{\langle\xi\rangle\leq
C^{\prime}R}\langle\xi\rangle^{-n\varepsilon}d\xi\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
CR^{(\frac{n}{2}-k)(\varepsilon-1)}R^{-\frac{k}{2}}R^{\frac{n}{2}(1-\varepsilon)}.$

By observing that

$R^{(\frac{n}{2}-k)(\varepsilon-1)+\frac{n}{2}(1-\varepsilon)-\frac{k}{2}}=R^{k(\frac{1}{2}-\varepsilon)}\leq
C,$

since $\varepsilon={Q}/{2n}\geq 1/2.$ The desired estimate follows. ∎

Note that the index $\varepsilon_{0}=Q/2n$ is the best possible with respect
to the approach that we have developed in the proof of Lemma 6.2. Since the
$L^{p}$-boundedness of the classes $S(m,g)$ for all $1<p<\infty$ depends on
the index $\varepsilon_{0}$ computed in Lemma 6.2, the methods of this paper
imply the following $L^{p}$-boundedness result.

######  Theorem 6.3.

Let $\mathbb{X}:=\\{X_{1},X_{2},\dots,X_{k}\\}$ be a family of real vector
fields on $\mathbb{R}^{n}$ satisfying the Hörmander condition of order $2$.
Let $a_{2}(x,D)$ be the Hörmander sub-Laplacian

$a_{2}(x,D)=-\sum\limits_{j=1}^{k}X_{j}^{2},$ (6.15)

and let us consider the metric $g$ and the $g$-weight $m$ defined as in (3.1)
and (3.2), respectively. Let $\varepsilon_{0}=\frac{Q}{2n}$ and
$\varepsilon_{0}\leq\beta\leq 1.$ Let
$\sigma(x,D)\in\textnormal{Op}S(m^{-\frac{n}{2}\beta},g).$ Then,
$\sigma(x,D):L^{p}(\mathbb{R}^{n})\rightarrow L^{p}(\mathbb{R}^{n})$ extends
to a bounded operator, for all $1<p<\infty.$ Moreover, for every $k\geq 1,$ we
have

$\|\sigma(x,D)f\|_{L^{p}(\mathbb{R}^{n})}\leq
C\|\sigma(x,\xi)\|_{k,S(m^{-n\beta/2},g)}\|f\|_{{L^{p}(\mathbb{R}^{n})}},$

where the constant $C>0,$ is independent of $f\in L^{p}(\mathbb{R}^{n}),$

Regarding the smaller exponents $\varepsilon_{0}$ below, by applying the
Fefferman-Stein interpolation we obtain the following bounds:

######  Theorem 6.4.

Let $\mathbb{X}:=\\{X_{1},X_{2},\dots,X_{k}\\}$ be a family of real vector
fields on $\mathbb{R}^{n}$ satisfying the Hörmander condition of order $2$.
Let $a_{2}(x,D)$ be the Hörmander sub-Laplacian

$a_{2}(x,D)=-\sum\limits_{j=1}^{k}X_{j}^{2},$ (6.16)

and let us consider the metric $g$ and the $g$-weight $m$ defined as in (3.1)
and (3.2), respectively. Let $\varepsilon_{0}=\frac{Q}{2n}$ and
$0\leq\beta<\varepsilon_{0}.$ Let
$\sigma(x,D)\in\textnormal{Op}(S(m^{-\frac{n}{2}\beta},g)).$ Then,
$\sigma(x,D):L^{p}(\mathbb{R}^{n})\rightarrow L^{p}(\mathbb{R}^{n})$ extends
to a bounded operator, provided

$\left|\frac{1}{p}-\frac{1}{2}\right|\leq\frac{n}{2}\beta.$ (6.17)

Moreover, for such $p$ and all $\ell>\frac{n}{4}$ we have

$\|\sigma(x,D)f\|_{L^{p}(\mathbb{R}^{n})}\leq
C\|\sigma(x,\xi)\|_{\ell,S(m^{-n\beta/2},g)}\|f\|_{{L^{p}(\mathbb{R}^{n})}},$

where the constant $C>0,$ is independent of $f\in L^{p}(\mathbb{R}^{n}).$

######  Remark 6.5.

Observe from the argument in the proof of Lemma 6.2 we have the following
estimate

$\int\limits_{R\leq a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq
3R}m^{-s}(x,\xi)d\xi$
$\lesssim\sqrt{R}^{Q-2s}\int\limits_{\\{\xi^{\prime}:\sum_{j=1}^{r(x)}{\xi^{\prime}}_{j}^{2}+|\xi^{\prime}|\geq
1\\}}\left(\sum_{j=1}^{r_{0}}\overline{\xi^{\prime}}_{j}^{2}+||\overline{\xi^{\prime}}||\right)^{-s}d\overline{\xi^{\prime}}<\infty,$

when $s>\frac{Q}{2}$ and $R\geq 1.$ We will use this inequality to give sharp
spectral properties for the $S(m,g)$ classes in the next section.

##  7\. Schatten-von Neumann classes of the Hamiltonians

In this section we present some aspects of the spectral analysis of the
Hamiltonians of the form (1.7). We first discuss some basic properties of the
Schatten-von Neumann classes of operators. A preliminary introduction to the
trace classes of operators can be found in [61], while for the theory of
Schatten-von Neumann classes we refer the reader to [45], [73] or to [78].

Let $H$ be a separable Hilbert space over $\mathbb{C}$ endowed with an inner
product $(\cdot,\cdot)$, and let $T:H\rightarrow H$ be some compact linear
operator with adjoint operator $T^{*}$. Clearly $T^{*}T$ is positive,
symmetric and compact, and we can write $|T|=(T^{*}T)^{\frac{1}{2}}$ for the
absolute value of $T$.

By the spectral theorem there exists an orthonormal basis for $H$ consisting
of eigenfunctions of $|T|$. Let $s_{n}(T)$ denote the corresponding non-zero
eigenvalues- also called the singular values of the operator $T$.

######  Definition 7.1.

A compact operator $T$ on a Hilbert space $H$ into $H$ belongs to the
Schatten-von Neumann class $\mathit{S}_{p}(H)$, $1\leq p<\infty$, if

$\left(\sum_{k=1}^{\infty}(s_{k}(T))^{p}\right)^{\frac{1}{p}}<\infty\,.$ (7.1)

The sum (7.1) is the norm of $T$ in $\mathit{S}_{p}$ denoted by
$\parallel\cdot\parallel_{\mathit{S}_{p}}$.

######  Remark 7.2.

If we endow $\mathit{S}_{p}$, $1\leq p<\infty$, with the norm
$\parallel\cdot\parallel_{\mathit{S}_{p}}$, then $\mathit{S}_{p}$ becomes a
Banach space.

As a particular case of the spaces $\mathit{S}_{p}$, $1\leq p<\infty$,
$\mathit{S}_{2}$ is the space of Hilbert-Schmidt operators. Moreover, the
Schatten-von Neumann classes are nested and we have

$\mathit{S}_{p}\subset\mathit{S}_{q}\,,\quad\text{if}\quad 1\leq p\leq
q<\infty\,.$

The analogy between the Schatten-von Neumann classes of operators and
functions in the Lebesgue spaces $L^{r}(\mathbb{R}^{n})$ in the context of
Weyl-Hörmander calculus, is given in the following proposition by Toft; see
Remark 6.4 [82].

######  Proposition 7.3.

We have the following:

  1. (A)

If $p\in[1,\infty]$, then $S(m,g)\subset s_{t,p}(\mathbb{R}^{2n})$ if and only
if $m\in L^{p}(\mathbb{R}^{n})$.

  2. (B)

If $p\in[1,\infty]$, $a\in L^{p}(\mathbb{R}^{2n})\cap S(m,g)$ and
$h_{g}^{N/2}m\in L^{p}(\mathbb{R}^{2n})$ for some $N\geq 0$, then $a\in
s_{t,p}(\mathbb{R}^{2n})$,

where we have denoted by $s_{\tau,p}(\mathbb{R}^{2n})$ the set of all symbols
in $\mathcal{S}^{{}^{\prime}}(\mathbb{R}^{n})$ such that $a^{\tau}(x,D)$,
$\tau\in\mathbb{R}$, belongs to $S_{p}(L^{2}(\mathbb{R}^{n}))$.

The theorem below is an application of Proposition 7.3 to the Hörmander
classes associated with the Hamiltonians of the form (1.7) that we consider
here, for the special case of sum of squares.

######  Theorem 7.4.

Let $\mathbb{X}:=\\{X_{1},X_{2},\dots,X_{k}\\}$ be a family of real vector
fields on $\mathbb{R}^{n}$ satisfying the Hörmander condition of order $2$.
Let $a_{2}(x,D)$ be the Hörmander sub-Laplacian

$a_{2}(x,D)=-\sum\limits_{j=1}^{k}X_{j}^{2},$ (7.2)

and let us consider the metric $g$ and the $g$-weight $m$ defined as in (3.1)
and (3.2), respectively. Then we have

$m^{-\mu}\in L^{r}(\mathbb{R}^{2n})\,,$

for some $0<r<\infty$, provided that

$\mu>\frac{Q}{r}=\frac{2n-r_{0}}{r}.$ (7.3)

Consequently, for such choices of $\mu$ and for $a\in S(m^{-\mu},g)$ we have

$a^{\tau}(x,D)\in\mathit{S}_{r}(L^{2}(\mathbb{R}^{n}))\,,\quad\text{for
all}\quad\tau\in\mathbb{R}\,.$

###### Proof.

In order to estimate the norm $\|m^{-\mu}\|_{L^{r}(\mathbb{R}^{2n})},$ let us
take a strategy from harmonic analysis and let us make a suitable triadic
decomposition of the phase space. To do this we decompose $\mathbb{R}^{2n}$ in
triadic sectors

$A_{R}:=\\{R\leq a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq 3R\\}$ (7.4)

where $R=3^{k}$ and $k\in\mathbb{Z}.$ Also, note that as $m(x,\xi)\geq 1,$
only the annulus $A_{3^{k}}$ with $k\in\mathbb{N}$ will contribute to the
estimate of the norm $\|m^{-\mu}\|_{L^{r}(\mathbb{R}^{2n})}.$ Indeed, by using
Remark 6.5 with $s=\mu r/2>Q/2$ (from which we have that $\mu r>Q\geq n$) we
can estimate the norm $\|m^{-\mu}\|_{L^{r}(\mathbb{R}^{2n})}$ as follows

$\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}m(x,\xi)^{-\mu
r}\,dxd\xi$ $\displaystyle\lesssim\sum_{k=0}^{\infty}\int\limits_{\\{3^{k}\leq
a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq 3^{k+1}\\}}m(x,\xi)^{-\mu
r/2}m(x,\xi)^{-\mu r/2}dx\,d\xi$
$\displaystyle\lesssim\sum_{k=0}^{\infty}\int\limits_{\\{x\in\mathbb{R}^{n}:\langle
x\rangle\leq 3^{k+1}\\}}\int\limits_{\\{\xi\in\mathbb{R}^{n}:3^{k}\leq
a_{2}(x,\xi)+|x|^{2}+\langle X\rangle\leq 3^{k+1}\\}}m(x,\xi)^{-\mu
r/2}m(x,\xi)^{-\mu r/2}\,d\xi\,dx$
$\displaystyle\lesssim\sum_{k=0}^{\infty}\int\limits_{\\{\langle x\rangle\leq
3^{k+1}\\}}\left(|x|^{2}\right)^{-\mu
r/2}dx\sup_{y\in\mathbb{R}^{n}}\int\limits_{\\{3^{k}\leq
a_{2}(y,\xi)+|y|^{2}+\langle(y,\xi)\rangle\leq 3^{k+1}\\}}m(y,\xi)^{-\mu
r/2}d\xi$ $\displaystyle\lesssim\sum_{k=0}^{\infty}3^{-k\left(\mu
r-n\right)}\int\limits_{\\{\langle x\rangle\leq 1\\}}\langle x\rangle^{-\mu
r}dx\sqrt{3}^{k(Q-\mu
r)}\int\limits_{\\{\xi^{\prime}:\sum_{j=1}^{r(y)}{\xi^{\prime}}_{j}^{2}+|\xi^{\prime}|\geq
1\\}}\left(\sum_{j=1}^{r_{0}}\overline{\xi^{\prime}}_{j}^{2}+||\overline{\xi^{\prime}}||\right)^{-\mu
r}d\overline{\xi^{\prime}}$
$\displaystyle\lesssim\sum_{k=0}^{\infty}3^{-k\left(\mu
r-n\right)}\sqrt{3}^{k(Q-\mu r)}=\sum_{k=0}^{\infty}3^{-k\left(n-\mu
r\right)}3^{k({Q}/{2}-\mu r/2)}.$

Since $\sum_{k=0}^{\infty}3^{-k\left(n-\mu r/2\right)}3^{k({Q}/{2}-\mu
r/2)}\leq\sum_{k=0}^{\infty}3^{k/2({Q}-\mu r)}<\infty$ we have proved that the
condition $\mu>Q/r$ implies that $m^{-\mu}\in L^{r}(\mathbb{R}^{2n})$ and then
an application of condition (A) in Proposition 7.3 finishes the proof of the
theorem. ∎

######  Remark 7.5.

We note that the condition in (7.3) is sharp. Indeed, by testing in the
elliptic case $r_{0}=n$ taking the Laplacian and $r=2$, the condition (7.3)
coincides with the well-known sharp condition $\mu>\frac{n}{2}$ for the
Hilbert-Schmidt class.

We now derive some consequences on the rate of growth of eigenvalues for the
operator $m(x,D)=a_{2}(x,D)+|x|^{2}+J_{\langle X\rangle}$, where $a_{2}(x,D)$
is as in Theorem 7.4 and $J_{\langle
X\rangle}=(-\Delta+|x|^{2}+1)^{\frac{1}{2}}$. To do so, we will apply Theorem
7.4 to the negative powers of $m(x,D)$, then we get the rate of decay of
eigenvalues and consequently the rate of growth.

######  Theorem 7.6.

Let $a_{2}(x,D)$ be a Hörmander sub-Laplacian as in Theorem 7.4. Then
$m(x,D)=a_{2}(x,D)+|x|^{2}+J_{\langle X\rangle}$ satisfies

$m(x,D)^{-\mu}\in\mathit{S}_{r}(L^{2}(\mathbb{R}^{n}))\,,$

provided $\mu>\frac{2n-r_{0}}{r}.$

Moreover, the sequence of eigenvalues of
$\lambda_{j}(a_{2}(x,D)+|x|^{2}+J_{\langle X\rangle})$ of
$a_{2}(x,D)+|x|^{2}+J_{\langle X\rangle}$ has a growth of order at least

$j^{\frac{1}{r}},\mbox{ as }j\rightarrow\infty,$ (7.5)

provided $r>2n-r_{0}$. Consequently, the sequence of eigenvalues of
$\lambda_{j}(a_{2}(x,D)+|x|^{2})$ has a growth at least as (7.5) for
$r>2n-r_{0}$.

###### Proof.

If $\mu>\frac{2n-r_{0}}{r},$ the membership of $m(x,D)^{-\mu}$ into the class
$\mathit{S}_{r}(L^{2}(\mathbb{R}^{n}))$ is an immediate consequence of Theorem
7.4. On the other hand, we note that from the case $\mu=1$, we can deduce the
rate of decay of the eigenvalues of $m(x,D)^{-1}$, obtaining for $r>2n-r_{0}$

$\lambda_{j}(m(x,D)^{-1})=\operatorname{o}(j^{-\frac{1}{r}}),\,\mbox{ as
}j\rightarrow\infty\,.$ (7.6)

From this we will obtain the following estimate for the rate of growth of the
eigenvalues of $m(x,D)=a_{2}(x,D)+|x|^{2}+J_{\langle X\rangle}:\\\ $

For every $L\in\mathbb{N}$ there exists $L_{0}\in\mathbb{N}$ such that

$Lj^{\frac{1}{r}}\leq\lambda_{j}(m(x,D)),\,\,\mbox{ for }j\geq L_{0}.$ (7.7)

Thus, the eigenvalues $\lambda_{j}(a_{2}(x,D)+|x|^{2}+J_{\langle X\rangle})$
have a growth of order at least

$j^{\frac{1}{r}},\mbox{ as }j\rightarrow\infty,$ (7.8)

provided $r>2n-r_{0}$.

For the last part, we observe that since $2n-r_{0}=n+n-r_{0}\geq n$ and the
eigenvalues of $J_{\langle X\rangle}=(-\Delta+|x|^{2}+1)^{\frac{1}{2}}$ are of
order $j^{\frac{1}{r}}$ for $r>n$. It follows that the eigenvalues of
$\lambda_{j}(a_{2}(x,D)+|x|^{2})$ have a growth at least as (7.5) for
$r>2n-r_{0}$. ∎

In order to obtain (7.6) one can apply the Weyl inequality below which relates
the singular values $s_{n}(T)$ and the eigenvalues $\lambda_{n}(T)$ for a
compact operator $T$ on a complex separable Hilbert space:

$\sum\limits_{n=1}^{\infty}|\lambda_{n}(T)|^{p}\leq\sum\limits_{n=1}^{\infty}s_{n}(T)^{p},\quad
p>0.$

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