# Cwikel Estimates and Negative Eigenvalues of Schrödinger Operators on
Noncommutative Tori

Edward McDonald  School of Mathematics and Statistics, University of New South
Wales, Sydney, Australia<EMAIL_ADDRESS>and  Raphaël Ponge
School of Mathematics, Sichuan University, Chengdu, China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we establish Cwikel-type estimates for noncommutative tori for
any dimension $n\geq 2$. We use them to derive Cwikel-Lieb-Rozenblum
inequalities and Lieb-Thirring inequalities for negative eigenvalues of
fractional Schrödinger operators on noncommutative tori. The latter leads to a
Sobolev inequality for noncommutative tori. On the way we establish a
“borderline version” of the abstract Birman-Schwinger principle for the number
of negative eigenvalues of relatively compact form perturbations of a non-
negative semi-bounded operator with isolated 0-eigenvalue.

###### Key words and phrases:

noncommutative geometry; Cwikel estimates; Schrödinger operators

###### 2020 Mathematics Subject Classification:

58B34; 47B10; 81Q10

##  1\. Introduction

The celebrated estimates of Cwikel [22] are a landmark application of trace-
ideal techniques in mathematical physics. They assert that if $f\in
L_{p}(\mathbb{R}^{n})$ and $g$ is in the weak $L_{p}$-space
$L_{p,\infty}(\mathbb{R}^{n})$ with $p>2$, then the operator $f(X)g(-i\nabla)$
on $L_{2}(\mathbb{R}^{n})$ is in the weak Schatten class
$\mathscr{L}_{p,\infty}$, and we have

$\big{\|}f(X)g(-i\nabla)\big{\|}_{\mathscr{L}_{p,\infty}}\leq
c_{np}\big{\|}f\big{\|}_{L_{p}}\big{\|}g\big{\|}_{L_{p,\infty}},$ (1.1)

where the constant $c_{np}$ depends only on $n$ and $p$. Here $f(X)$ is the
multiplication by $f$ in position space and $g(-i\nabla)$ is the
multiplication by $g$ in momentum space. We refer to Section 4 for background
on Schatten classes and weak Schatten classes. Cwikel’s estimates were
extended to $p\in(0,2)$ by Birman-Solomyak [7] and to $p=2$ by Birman-
Karadzhov-Solomyak [5] (who also dealt with more general Lorentz ideals
$\mathscr{L}_{p,q}$). Solomyak [93] substantially improved the understanding
of the $p=2$ case in even dimension.

Cwikel’s estimates in the form stated above were conjectured by Simon [89]. He
pointed out that combining them with the Birman-Schwinger principle [4, 84]
would give an upper-bound for the number of negative eigenvalues (i.e., the
number of bound states) of Schrödinger operators $\Delta+V$ with $V\in
L_{\frac{n}{2}}(\mathbb{R}^{n})$, $n\geq 3$. Namely, there exists a constant
$C_{n}>0$ depending only on $n$ such that

$N^{-}(\Delta+V)\leq C_{n}\int|V_{-}(x)|^{\frac{n}{2}}dx,$ (1.2)

where $N^{-}(\Delta+V)$ is the number of negative eigenvalues of $\Delta+V$
counted with multiplicity and $V_{-}=\frac{1}{2}(|V|-V)$ is the negative part
of $V$.

The main motivation for the inequality (1.2) is establishing a Weyl’s law for
Schrödinger operators $\hbar^{2}\Delta+V$ with a non-smooth potential $V$
under the semi-classical limit $h\rightarrow 0$ (see [6, 11, 89]). As it turns
out, the inequality (1.2) was already established by Rozenblum [78, 79].
Another independent proof was provided by Lieb [54, 56] (see [17, 28, 29, 38,
61] for further alternative proofs). The equality (1.2) is now known as the
Cwikel-Lieb-Rozenblum (CLR) inequality.

A related theme, initiated by Lieb-Thirring [59, 60] concerns bounds for
$\gamma$-moments of negative eigenvalues $\lambda^{-}_{j}(\Delta+V)$, $j\geq
0$. Namely, if $\gamma>0$, then there is a constant $L_{\gamma,n}>0$ depending
only $n$ and $\gamma$ such that, for any real-valued potential $V\in
L_{n+\gamma}(\mathbb{R}^{n})$, we have

$\sum_{j}|\lambda^{-}_{j}(\Delta+V)|^{\gamma}\leq
L_{\gamma,n}\int|V_{-}(x)|^{\frac{n}{2}+\gamma}\,dx.$ (1.3)

The above inequality (1.3) is known as the Lieb–Thirring (LT) inequality. It
was first established for $\gamma=1$ and $n=3$ by Lieb-Thirring [59] as a key
step toward proving the stability of matter in quantum mechanics. They
subsequently extended the inequality in [60] to all $n\geq 2$ and $\gamma>0$
and for $n=1$ and $\gamma>1/2$. The case $n=1$ and $\gamma=1/2$ was settled by
Weidl [102].

A fundamental fact about the LT inequality for $\gamma=1$ is its equivalence
with Sobolev inequality (see [59]). More generally, the LT inequalities have
widespread applications in mathematical physics (see, e.g., [55, 58]). They
also play an important role in the study of Navier-Stokes equations (see,
e.g., [57, 100]).

There are versions of the CLR inequality and the LT inequalities for
fractional Schrödinger operators $\Delta^{n/2p}+V$ (see, e.g., [23, 50, 49,
78, 79, 80, 93]). Such operators naturally appear in the framework of
fractional quantum mechanics [48]; for $p=n$ we get the ultra-relativistic
Schrödinger operator $|\nabla|+V$ (see [23]). We refer to the recent survey
[30] and the book [31] (and the numerous references therein) for a more
thorough account on the CLR and LT inequalities and their various applications
and generalizations.

Recently, Cwikel estimates have been experiencing a revival of interest. A
general abstract operator theoretic framework for Cwikel-type estimates has
been emerging (see [29, 38, 53]). In a recent article [38] Hundertmark _et
al_. used a refinement of Cwikel’s estimates to get some of the best CLR
bounds to date. Furthermore, applications of Cwikel-type estimates to Connes’s
noncommutative geometry program were found [32, 51, 52, 53, 64, 66, 96, 98].
In particular, Cwikel-type estimates for noncommutative Euclidean spaces were
established by Levitina-Sukochev-Zanin [53].

The aim of this paper is to obtain general Cwikel-type estimates and establish
CLR inequalities on noncommutative tori (a.k.a. quantum tori). Noncommutative
tori are arguably the most well known examples of noncommutative spaces. In
particular, they naturally appear in the noncommutative geometry approach to
the quantum Hall effect [2] and to topological insulators [14, 73]. In
addition, noncommutative tori have been considered in the context of string
theory (see, e.g., [20, 86]). Noncommutative 2-tori naturally arise from
actions of $\mathbb{Z}$ on the circle $\mathbb{S}^{1}$ by irrational
rotations. More generally, a noncommutative $n$-tori $\mathbb{T}^{n}_{\theta}$
is generated by unitaries $U_{1},\ldots,U_{n}$ subject to the relations,

$U_{l}U_{j}=e^{2i\pi\theta_{jl}}U_{j}U_{l},\qquad j,l=1,\ldots,n,$

where $\theta=(\theta_{jl})$ is a given real anti-symmetric matrix. We refer
to Section 2 for more background on noncommutative tori.

We establish Cwikel-type estimates on $\mathbb{T}^{n}_{\theta}$ for operators
$\lambda(x)g(-i\nabla)$, where $x$ is in $L_{p}(\mathbb{T}^{n}_{\theta})$ with
$p\geq 2$. Here $\nabla=(\partial_{1},\ldots,\partial_{n})$, where
$\partial_{1},\ldots,\partial_{n}$ are the canonical derivations of
$\mathbb{T}^{n}_{\theta}$, and $\lambda$ is the extension to $L_{p}$-spaces of
the left-regular representation of $L_{\infty}(\mathbb{T}^{n}_{\theta})$ in
$L_{2}(\mathbb{T}^{n}_{\theta})$ (see Section 3). More precisely, we show that
if $x\in L_{q}(\mathbb{T}^{n}_{\theta})$ and
$g\in\ell_{p,\infty}(\mathbb{Z}^{n})$, we have

$\|\lambda(x)g(-i\nabla)\|_{\mathscr{L}_{p,\infty}}\leq
c(p,q)\|x\|_{L_{q}}\|g\|_{\ell_{p,\infty}},$ (1.4)

where $c(p,q)$ is some _explicit_ constant independent of $g$ and $x$. We
establish those Cwikel-type estimates for $q=p>2$ (Theorem 4.4, part (2)),
$q=2<p$ (Theorem 4.8, part (2)), and for $p=2<q$ (Theorem 4.9). In each case,
we obtain explicit bounds for the best constants in those inequalities.

We also have $\mathscr{L}_{p}$-estimates for $p<2$. Namely, if $x\in
L_{q}(\mathbb{T}^{n}_{\theta})$ and $g\in\ell_{p}(\mathbb{Z}^{n})$, then

$\|\lambda(x)g(-i\nabla)\|_{\mathscr{L}_{p}}\leq\|x\|_{L_{2}}\|g\|_{\ell_{p}}.$

The inequalities hold for $q=p\geq 2$ (Theorem 4.4, part (1)) and for $q=2>p$
(Theorem 4.8).

Note that $\mathscr{L}_{p,\infty}$-estimates with $p>2$ and
$\mathscr{L}_{p}$-estimate with $p\geq 2$ were established in [66] with
unspecificed constantsas a special case of general results of Levitina-
Sukochev-Zanin [53]. We further elaborate on the arguments of [53] to get
explicit constants for these estimates.

The $\mathscr{L}_{2,\infty}$-estimate is deduced from the estimates in the
$p>2$ case and $p<2$ case by an interpolation argument. The proof of the
estimates in the $p<2$ case essentially follows the approach of [53] to the
Cwikel-type estimates on NC Euclidean spaces. As it turns out, on
noncommutative tori some simplifications occur, which allows us to get much
sharper results. In fact, as far as Schatten and weak Schatten classes are
concerned, the estimates of this papers are as sharp as in the (commutative)
Euclidean space setting.

The general Cwikel estimates (1.4) specialize to estimates for operators of
the form $\lambda(x)\Delta^{-n/2p}$, where $\Delta=\nabla^{*}\nabla$ is the
(positive) Laplacian on $\mathbb{T}^{n}_{\theta}$ (Theorem 6.1). Set

$\nu_{0}(n):=\sup_{\lambda\geq
1}\lambda^{-\frac{n}{2}}\\#\left\\{k\in\mathbb{Z}^{n}\setminus 0;\
|k|\leq\sqrt{\lambda}\right\\}.$

If $x\in L_{q}(\mathbb{T}^{n}_{\theta})$, and either $p\neq 2$ and
$q=\max(p,2)$, or if $p=2<q$, then

$\big{\|}\lambda(x)\Delta^{-\frac{n}{2p}}\big{\|}_{\mathscr{L}_{p,\infty}}\leq
c(p,q)\nu_{0}(n)^{\frac{1}{p}}\|x\|_{L_{q}}.$ (1.5)

For $p\neq 2$ the constant $c(p,q)$ is the best constant from the
corresponding Cwikel-type inequality (1.4). For $p=2$ the inequality follows
from the case $p>2$ by using Hölder’s inequality, and so we obtain a better
constant.

This allows us to get estimates for operators of the form
$\Delta^{-n/4p}\lambda(x)\Delta^{-n/4p}$ (Theorem 6.5). More precisely, if
$x\in L_{q}(\mathbb{T}^{m}_{\theta})$, and either $p\neq 1$ and $q=\max(p,1)$,
or if $p=1<q$, then

$\big{\|}\Delta^{-\frac{n}{4p}}\lambda(x)\Delta^{-\frac{n}{4p}}\big{\|}_{\mathscr{L}_{p,\infty}}\leq
2^{\frac{1}{p}}c(2p,2q)\nu_{0}(n)^{\frac{1}{p}}\|x\|_{L_{q}}.$ (1.6)

This inequality holds without the extra $2^{1/p}$-factor if $x\geq 0$ (see
Remark 6.6).

In the Euclidean space setting the CLR inequality (1.2) can be deduced from
the Cwikel estimates (1.1) by using the Birman-Schwinger principle [4, 84]. In
its abstract form due to Birman-Solomyak [13] (see also Proposition 7.5 and
Corollary 7.7) the Birman-Schwinger principle implies that if $H$ is a (semi-
bounded) non-negative selfadjoint operator on Hilbert space and $V$ is a non-
positive relatively form-compact perturbation such that
$(H+1)^{-1/2}V(H+1)^{-1/2}\in\mathscr{L}_{p,\infty}$, $p>0$, then

$N(H+V;\lambda)\leq\big{\|}(H+\lambda)^{-\frac{1}{2}}V(H+\lambda)^{-\frac{1}{2}}\big{\|}_{\mathscr{L}_{p,\infty}}^{p}\qquad\forall\lambda<0.$
(1.7)

When $H$ is the Laplacian on $\mathbb{R}^{n}$, $n\geq 3$, the inequality (1.7)
continues to hold for $\lambda=0$, thereby providing an estimate for
$N^{-}(H+V)$. In various examples, including the Laplacians on NC tori, the
origin is in the discrete spectrum, and so the resolvent $(H-\lambda)^{-1}$
has a pole singularity at $\lambda=0$. This prevents us from letting
$\lambda\rightarrow 0^{-}$ in (1.7). To remedy we derive a “borderline”
Birman-Schwinger principle (Theorem 7.9). In particular, we show that if $0$
is in the discrete spectrum of $H$ and $V$ is a non-positive relatively form-
compact perturbation such that $H^{-1/2}VH^{-1/2}\in\mathscr{L}_{p,\infty}$,
$p>0$, then, we have

$0\leq
N^{-}(H+V)-N^{-}(\Pi_{0}V\Pi_{0})\leq\big{\|}H^{-\frac{1}{2}}VH^{-\frac{1}{2}}\big{\|}_{\mathscr{L}_{p,\infty}}^{p}.$
(1.8)

Here $\Pi_{0}$ is the orthogonal projection onto the nullspace of $H$. This
result seems to be new, at least at this level of generality. Its scope of
validity goes beyond the scope of this paper. For instance, it also
encompasses (fractional) Schrödinger operators on closed manifolds or compact
manifolds with boundary under Neumann boundary conditions, as well as
Schrödinger operators on hyperbolic manifolds with infinite volume.

Combining the specific Cwikel estimates (1.6) and the borderline Birman-
Schwinger principle (1.8) allows us to get CLR-type inequalities for
fractional Schrödinger operators $\Delta^{n/2p}+\lambda(V)$ on NC tori
(Theorem 8.1). Namely, if $V=V^{*}\in L_{q}(\mathbb{T}^{n}_{\theta})$ and,
either $p\neq 1$ and $q=\max(p,1)$, or $p=1<q$, then

$N^{-}\big{(}\Delta^{\frac{n}{2p}}+\lambda(V)\big{)}-1\leq
c(2p,2q)^{2p}\nu_{0}(n)\tau\big{[}|V_{-}|^{q}\big{]}^{\frac{p}{q}},$ (1.9)

where $V_{-}=\frac{1}{2}(|V|-V)$ is the negative part of $V$ and $\tau$ is the
standard normalized trace of $L_{\infty}(\mathbb{T}^{n}_{\theta})$. This
inequality is consistent with Lieb’s version of the CLR inequality for closed
manifolds [54, 56]. For $p=n/2$ we get a CLR inequality for Schrödinger
operators $\Delta+\lambda(V)$ with $V=V^{*}\in
L_{n/2}(\mathbb{T}^{n}_{\theta})$ if $n\geq 3$ or with $V=V^{*}\in
L_{q}(\mathbb{T}^{n}_{\theta})$, $q>1$, if $n=2$. The latter condition is
consistent with the CLR inequality for Schrödinger operators on bounded
regions of $\mathbb{R}^{2}$ by Birman-Solomyak [6] (see also [11]). In
particular, unlike in the Euclidean space setting we do get an inequality in
dimension 2.

Under the semiclassical limit $h\rightarrow 0^{+}$ the CLR inequality (1.9)
implies that

$N^{-}\big{(}h^{\frac{n}{p}}\Delta^{\frac{n}{2p}}+\lambda(V)\big{)}\leq
c(2p,2q)^{2p}\nu_{0}(n)h^{-n}\tau\big{[}|V_{-}|^{q}\big{]}^{\frac{p}{q}}+\operatorname{O}(1).$
(1.10)

This leads us to conjecture that if $V=V^{*}\in
L_{q}(\mathbb{T}^{n}_{\theta})$ and, either $p\neq 1$ and $q=\max(p,1)$, or
$p=1<q$, then we have the semi-classical Weyl’s law,

$N^{-}\big{(}h^{\frac{n}{p}}\Delta^{\frac{n}{2p}}+\lambda(V)\big{)}=\operatorname{Vol}(\mathbb{B}^{n})h^{-n}\tau\big{[}|V_{-}|^{p}\big{]}+\operatorname{o}\big{(}h^{-n}\big{)}.$
(1.11)

In the same way as in the Euclidean space setting, the semiclassical CLR
inequality (1.10) allows us to reduce the proof of (1.11) for
$L_{q}$-potentials to that for smooth potentials. We believe a semiclassical
Weyl law for smooth potentials can be established by using semiclassical
pseudodifferential calculus on NC tori. However, such a pseudodifferential
calculus has yet to be set up. As this project falls out of the scope of this
paper we leave (1.11) as a conjecture.

As is well known (see, e.g., [91]), in dimension $\geq 3$ the simplest way to
derive the Lieb-Thirring inequalities (1.3) is to deduce them from the CLR
inequality (1.2) (although this approach does not lead to good bounds for the
best LT constants $L_{\gamma,n}$). Lieb-Thirring inequalities on ordinary tori
(and spheres) have been established by Ilyin [39] in dimension 2 and by
Laptev-Ilyin [40] in dimension $\leq 19$ (see also [41, 42, 43]). We establish
Lieb-Thirring inequalities on NC tori by following the CLR-route

As in [39], we restrict ourselves to the zero mean-value subspace $\\{u;\
\tau(u)=0\\}$, i.e., the orthogonal complement of the nullspace
$\ker\Delta=\mathbb{C}\cdot 1$. Denoting by $\dot{\Delta}$ and
$\dot{\lambda}(V)$ the relevant restrictions of $\Delta$ and $\lambda(V)$, we
have CLR inequalities for the operators
$\dot{\Delta}^{\frac{n}{2p}}+\dot{\lambda}(V)$ for $V=V^{*}\in
L_{q}(\mathbb{T}^{n}_{\theta})$ with $p$ and $q$ as above (Theorem 8.12).
Namely,

$N^{-}\big{(}\dot{\Delta}^{\frac{n}{2p}}+\dot{\lambda}(V)\big{)}\leq
c(2p,2q)^{2p}\nu_{0}(n)\tau\big{[}|V_{-}|^{q}\big{]}^{\frac{p}{q}}.$

This leads to LT inequalities for the operators
$\dot{\Delta}^{\frac{n}{2p}}+\dot{\lambda}(V)$ for $p>1$ (Theorem 8.14). More
precisely, if $\gamma>0$ and $V=V^{*}\in
L_{p+\gamma}(\mathbb{T}^{n}_{\theta})$, then

$\sum_{j}\big{|}\lambda^{-}_{j}(\Delta^{\frac{n}{2p}}+\dot{\lambda}(V))\big{|}^{\gamma}\leq
L_{p,\gamma,n}\tau\big{[}|V_{-}|^{p+\gamma}\big{]},$

where the best constant $L_{p,\gamma,n}$ is such that

$L_{p,\gamma,n}\leq\gamma\frac{\Gamma(p+1)\Gamma(\gamma)}{\Gamma(p+\gamma+1)}c(2p,2p)^{2p}\nu_{0}(n).$

As with the original Lieb-Thiring inequalities (1.3) on $\mathbb{R}^{n}$, for
$p=n/2$ and $\gamma=1$ with $n\geq 3$, the LT inequality (1) is equivalent to
a Sobolev inequality on NC tori (Theorem 8.19). More precisely, there is a
constant $K_{n}>0$, such that, for any orthonormal family
$\\{u_{0},\ldots,u_{N}\\}$ of zero-mean value elements of the Sobolev space
$W_{2}^{1}(\mathbb{T}^{n}_{\theta})$, we have

$\sum_{\ell=0}^{N}\tau\big{[}|\nabla u_{\ell}|^{2}\big{]}\geq
K_{n}\tau\left[\bigg{(}\sum_{\ell=1}^{N}|u_{\ell}|^{2}\bigg{)}^{1+n/2}\right].$
(1.12)

Moreover, the best constant $K_{n}$ is related to the best LT constant
$L_{n}=L_{\frac{n}{2},1,n}$ by

$K_{n}=\frac{n}{n+2}\bigg{(}\frac{n+2}{2}L_{n}\bigg{)}^{-\frac{n}{2}}.$

We refer to [65] for further applications of the Cwikel estimates (1.5)–(1.6)
to curved noncommutative tori, i.e., noncommutative tori equipped with a
Riemannian metric. In this setting the role of the flat Laplacian is played by
the corresponding Laplace-Beltrami operator. In particular, we get “curved”
analogues of the CLR inequalities (1.9) and obtain $L_{2}$-versions of the
Connes’ integration formulas of [67, 70].

The paper is organized as follows. In Section 2, we review the main background
on noncommutative tori. Section 3 contains some technical preliminaries,
including sufficient conditions for the operators appearing in Cwikel-type
estimates to be bounded. In Section 4, we present our main Cwikel-type
estimates for noncommutative tori and show how to deduce the $p=2$ case from
the $p<2$ case. The $p<2$ case is proved in Section 5. In Section 6 we
specialize the main Cwikel estimates into the specific Cwikel estimates
(1.5)–(1.6). In Section 7, we establish the borderline Birman-Schwinger
principle (1.8). In Section 8 we establish our CLR inequalities and LT for
fractional Schrödinger operators on NC tori and use them to derive the Sobolev
inequality (1.12). In Appendix A, for reader’s convenience we reproduce
Birman-Solomyak’s proof of the abstract Birman-Schwinger principle.

### Notation

Throughout the paper we make the convention that $c_{abd}$ and $C_{abd}$ are
positive constants which depend only on the parameters $a$, $b$, $d$, etc.,
and may change from line to line. They do not depend on the other variables
that are floating around.

##  2\. Noncommutative Tori

In this section, we review the main definitions and properties of
noncommutative $n$-tori, $n\geq 2$. We refer to [19, 35, 77], and the
references therein, for a more comprehensive account.

Throughout this paper, we let $\theta=(\theta_{jk})$ be a real anti-symmetric
$n\times n$-matrix, and denote by $\theta_{1},\ldots,\theta_{n}$ its column
vectors. We also let $L_{2}(\mathbb{T}^{n})$ be the Hilbert space of
$L_{2}$-functions on the ordinary torus
$\mathbb{T}^{n}=\mathbb{R}^{n}/(2\pi\mathbb{Z})^{n}$ equipped with the inner
product,

$\left\langle\xi|\eta\right\rangle=(2\pi)^{-n}\int_{\mathbb{T}^{n}}\xi(x)\overline{\eta(x)}dx,\qquad\xi,\eta\in
L_{2}(\mathbb{T}^{n}).$ (2.1)

For $j=1,\ldots,n$, let $U_{j}:L_{2}(\mathbb{T}^{n})\rightarrow
L_{2}(\mathbb{T}^{n})$ be the unitary operator defined by

$\left(U_{j}\xi\right)(x)=e^{ix_{j}}\xi\left(x+\pi\theta_{j}\right),\qquad\xi\in
L_{2}(\mathbb{T}^{n}).$

We then have the relations,

$U_{k}U_{j}=e^{2i\pi\theta_{jk}}U_{j}U_{k},\qquad j,k=1,\ldots,n.$ (2.2)

The _noncommutative torus_ is the noncommutative space whose $C^{*}$-algebra
$C(\mathbb{T}^{n}_{\theta})$ and von Neuman algebra
$L_{\infty}(\mathbb{T}^{n}_{\theta})$ are generated by the unitary operators
$U_{1},\ldots,U_{n}$. For $\theta=0$ we obtain the $C^{*}$-algebra
$C(\mathbb{T}^{n})$ of continuous functions on the ordinary $n$-torus
$\mathbb{T}^{n}$ and the von Neuman algebra $L_{\infty}(\mathbb{T}^{n})$ of
essentially bounded measurable functions on $\mathbb{T}^{n}$. Note that (2.2)
implies that $C(\mathbb{T}^{n}_{\theta})$ (resp.,
$L_{\infty}(\mathbb{T}^{n}_{\theta})$) is the norm closure (resp., weak
closure) in $\mathscr{L}(L_{2}(\mathbb{T}^{n}))$ of the linear span of the
unitary operators,

$U^{k}:=U_{1}^{k_{1}}\cdots U_{n}^{k_{n}},\qquad
k=(k_{1},\ldots,k_{n})\in\mathbb{Z}^{n}.$

###  2.1. GNS representation

Let $\tau:\mathscr{L}(L_{2}(\mathbb{T}^{n}))\rightarrow\mathbb{C}$ be the
state defined by the constant function $1$, i.e.,

$\tau(T)=\left\langle
T1|1\right\rangle=(2\pi)^{-n}\int_{\mathbb{T}^{n}}(T1)(x)dx,\qquad
T\in\mathscr{L}\left(L_{2}(\mathbb{T}^{n})\right).$

This induces a continuous tracial state on the von Neuman algebra
$L_{\infty}(\mathbb{T}^{n}_{\theta})$ such that $\tau(1)=1$ and
$\tau(U^{k})=0$ for $k\neq 0$. The GNS construction then allows us to
associate with $\tau$ a $*$-representation of
$L_{\infty}(\mathbb{T}^{n}_{\theta})$ as follows.

Let $\left\langle\cdot|\cdot\right\rangle$ be the sesquilinear form on
$C(\mathbb{T}^{n}_{\theta})$ defined by

$\left\langle u|v\right\rangle=\tau\left(uv^{*}\right),\qquad u,v\in
C(\mathbb{T}^{n}_{\theta}).$ (2.3)

Note that the family $\\{U^{k};k\in\mathbb{Z}^{n}\\}$ is orthonormal with
respect to this sesquilinear form. We let $L_{2}(\mathbb{T}^{n}_{\theta})$ be
the Hilbert space arising from the completion of $C(\mathbb{T}^{n}_{\theta})$
with respect to the pre-inner product (2.3). The action of
$C(\mathbb{T}^{n}_{\theta})$ on itself by left-multiplication uniquely extends
to a $*$-representation of $L_{\infty}(\mathbb{T}^{n}_{\theta})$ in
$L_{2}(\mathbb{T}^{n}_{\theta})$. When $\theta=0$ we recover the Hilbert space
$L_{2}(\mathbb{T}^{n})$ with the inner product (2.1) and the representation of
$L_{\infty}(\mathbb{T}^{n})$ by bounded multipliers. In addition, as
$(U^{k})_{k\in\mathbb{Z}^{n}}$ is an orthonormal basis of
$L_{2}(\mathbb{T}^{n}_{\theta})$, every $u\in L_{2}(\mathbb{T}^{n}_{\theta})$
can be uniquely written as

$u=\sum_{k\in\mathbb{Z}^{n}}u_{k}U^{k},\qquad u_{k}:=\left\langle
u|U^{k}\right\rangle,$ (2.4)

where the series converges in $L_{2}(\mathbb{T}^{n}_{\theta})$. When
$\theta=0$ we recover the Fourier series decomposition in
$L_{2}(\mathbb{T}^{n})$.

###  2.2. The smooth algebra $C^{\infty}(\mathbb{T}^{n}_{\theta})$

The natural action of $\mathbb{R}^{n}$ on $\mathbb{T}^{n}$ by translation
gives rise to an action on $\mathscr{L}(L_{2}(\mathbb{T}^{n}))$. This induces
a $*$-action $(s,u)\rightarrow\alpha_{s}(u)$ on $C(\mathbb{T}^{n}_{\theta})$
given by

$\alpha_{s}(U^{k})=e^{is\cdot k}U^{k},\qquad\text{for all $k\in\mathbb{Z}^{n}$
and $s\in\mathbb{R}^{n}$}.$

This action is strongly continuous, and so we obtain a $C^{*}$-dynamical
system $(C(\mathbb{T}^{n}_{\theta}),\mathbb{R}^{n},\alpha)$. We are especially
interested in the subalgebra $C^{\infty}(\mathbb{T}^{n}_{\theta})$ of smooth
elements of this $C^{*}$-dynamical system, i.e., $u\in
C(\mathbb{T}^{n}_{\theta})$ such that $\alpha_{s}(u)\in
C^{\infty}(\mathbb{R}^{n};C(\mathbb{T}^{n}_{\theta}))$.

The unitaries $U^{k}$, $k\in\mathbb{Z}^{n}$, are contained in
$C^{\infty}(\mathbb{T}^{n}_{\theta})$, and so
$C^{\infty}(\mathbb{T}^{n}_{\theta})$ is a dense subalgebra of
$C(\mathbb{T}^{n}_{\theta})$. Denote by $\mathscr{S}(\mathbb{Z}^{n})$ the
space of rapid-decay sequences with complex entries. In terms of the Fourier
series decomposition (2.4) we have

$C^{\infty}(\mathbb{T}^{n}_{\theta})=\bigg{\\{}u=\sum_{k\in\mathbb{Z}^{n}}u_{k}U^{k};(u_{k})_{k\in\mathbb{Z}^{n}}\in\mathscr{S}(\mathbb{Z}^{n})\bigg{\\}}.$

When $\theta=0$ we recover the algebra $C^{\infty}(\mathbb{T}^{n})$ of smooth
functions on the ordinary torus $\mathbb{T}^{n}$ and the Fourier-series
description of this algebra.

For $j=1,\ldots,n$, let
$\partial_{j}:C^{\infty}(\mathbb{T}^{n}_{\theta})\rightarrow
C^{\infty}(\mathbb{T}^{n}_{\theta})$ be the derivation defined by

$\partial_{j}(u)=\partial_{s_{j}}\alpha_{s}(u)|_{s=0},\qquad u\in
C^{\infty}(\mathbb{T}^{n}_{\theta}),$

When $\theta=0$ it agrees with the derivation $\partial_{x_{j}}$ on
$C^{\infty}(\mathbb{T}^{n})$. In general, we have

$\partial_{j}(U_{l})=\left\\{\begin{array}[]{ll}iU_{j}&\text{if $l=j$},\\\
0&\text{if $l\neq j$}.\end{array}\right.$

###  2.3. $L_{p}$-Spaces

The $L_{p}$-spaces of $\mathbb{T}^{n}_{\theta}$ are special instances of
noncommutative $L_{p}$ spaces associated with a semi-finite faithful normal
trace on a von Neumann algebra [46, 85] (see also [26, 69]). We refer to [26,
46] for the main background on noncommutative $L_{p}$-spaces needed in this
paper.

By definition $L_{\infty}(\mathbb{T}^{n}_{\theta})$ is a von Neuman algebra of
bounded operators on $L_{2}(\mathbb{T}^{n})$. Thus, a closed densely defined
operator on $L_{2}(\mathbb{T}^{n})$ is
$L_{\infty}(\mathbb{T}^{n}_{\theta})$-affiliated when it commutes with the
commutant of $L_{\infty}(\mathbb{T}^{n}_{\theta})$ in
$\mathscr{L}(L_{2}(\mathbb{T}^{n}))$. Furthermore, as $\tau$ is a finite
faithful positive trace on $L_{\infty}(\mathbb{T}_{\theta}^{n})$ every such
operator is $\tau$-measurable in the sense of [26, 69]. Therefore, these
operators form a $*$-algebra, where the sum and product of such operators are
meant as the closures of their usual sum and product in the sense of unbounded
operators (see [69]).

The space $L_{1}(\mathbb{T}^{n}_{\theta})$ consists of all
$L_{\infty}(\mathbb{T}^{n}_{\theta})$-affiliated operators $x$ on
$L_{2}(\mathbb{T}^{n})$ such that

$\tau\left(|x|\right):=\int_{0}^{\infty}\lambda
d(\tau_{*}E_{\lambda})<\infty,$

where $E_{\lambda}=\mathbbm{1}_{[0,\lambda]}(|x|)$ is the spectral measure of
$|x|$. We obtain a Banach space upon equipping
$L_{1}(\mathbb{T}^{n}_{\theta})$ with the norm,

$\|x\|_{L_{1}}:=\tau\big{(}|x|\big{)},\qquad x\in
L_{1}(\mathbb{T}^{n}_{\theta}).$

We have a continuous inclusion with dense range of
$L_{\infty}(\mathbb{T}^{n}_{\theta})$ into $L_{1}(\mathbb{T}^{n}_{\theta})$.
The trace $\tau$ uniquely extends to a continuous linear functional on
$L_{1}(\mathbb{T}^{n}_{\theta})$ such that

$|\tau(x)|\leq\tau(|x|)=\|x\|_{1}\qquad\forall x\in
L_{1}(\mathbb{T}^{n}_{\theta}).$

For $p>1$, the space $L_{p}(\mathbb{T}^{n}_{\theta})$ consists of all
$L_{\infty}(\mathbb{T}^{n}_{\theta})$-affiliated operators $x$ on
$L_{2}(\mathbb{T}^{n})$ such that $|x|^{p}\in L_{1}(\mathbb{T}^{n}_{\theta})$.
This is a Banach space with respect to the norm,

$\|x\|_{L_{p}}:=\tau(|x|^{p})^{1/p}=\bigg{(}\int_{0}^{\infty}\lambda^{p}d(\tau_{*}E_{\lambda})\bigg{)}^{\frac{1}{p}},\qquad
x\in L_{p}(\mathbb{T}^{n}_{\theta}),$

where as above $E_{\lambda}$, $\lambda\geq 0$, is the spectral measure of
$|x|$. We also have a continuous inclusion with dense range of
$L_{\infty}(\mathbb{T}^{n}_{\theta})$ into $L_{p}(\mathbb{T}^{n}_{\theta})$.
In particular, for $p=2$ the above definition is consistent with the previous
definition of $L_{2}(\mathbb{T}^{n}_{\theta})$, since in both cases we get the
completion of $L_{\infty}(\mathbb{T}^{n}_{\theta})$ with respect to the same
norm.

We have the following version of Hölder’s inequality.

######  Proposition 2.1 ([26, 46]).

Suppose that $p^{-1}+q^{-1}=r^{-1}\leq 1$. If $x\in
L_{p}(\mathbb{T}^{n}_{\theta})$ and $y\in L_{q}(\mathbb{T}^{n}_{\theta})$,
then $xy\in L_{r}(\mathbb{T}^{n}_{\theta})$ with norm inequality,

$\|xy\|_{L_{r}}\leq\|x\|_{L_{p}}\|y\|_{L_{q}}.$ (2.5)

This implies that, for every $p\geq 1$, the multiplication of
$L_{\infty}(\mathbb{T}_{\theta}^{n})$ uniquely extends to continuous bilinear
maps,

$L_{\infty}(\mathbb{T}^{n}_{\theta})\times
L_{p}(\mathbb{T}^{n}_{\theta})\longrightarrow
L_{p}(\mathbb{T}^{n}_{\theta}),\qquad L_{p}(\mathbb{T}^{n}_{\theta})\times
L_{\infty}(\mathbb{T}^{n}_{\theta})\longrightarrow
L_{p}(\mathbb{T}^{n}_{\theta}).$

In particular, for $p=2$ we recover the GNS representation of
$L_{\infty}(\mathbb{T}_{\theta}^{2})$ associated with $\tau$.

If $x$ is an $L_{\infty}(\mathbb{T}^{n}_{\theta})$-affiliated operator on
$\mathscr{L}(L_{2}(\mathbb{T}^{n}_{\theta}))$, its polar decomposition [74,
Theorem VIII.32] takes the form $x=u|x|$, where $u$ is a partial isometry in
$L_{\infty}(\mathbb{T}^{n}_{\theta})$. Moreover, the equality
$x^{*}=u|x|u^{*}$ (see, e.g., [12, Theorem 1.8.3]) and Hölder’s inequality
show that

$x\in L_{p}(\mathbb{T}^{n}_{\theta})\Longrightarrow x^{*}\in
L_{p}(\mathbb{T}^{n}_{\theta})\ \text{and}\ \|x^{*}\|_{L_{p}}=\|x\|_{L_{p}}.$

We may also define $L_{p}$-spaces on $\mathbb{T}^{n}_{\theta}$ for $0<p<1$ as
above. In this case we obtain a quasi-Banach spaces (see [16, 26]).

###  2.4. Sobolev spaces

Given any $s\geq 0$, the Sobolev space $W_{2}^{s}(\mathbb{T}^{n}_{\theta})$ is
defined by

$W_{2}^{s}(\mathbb{T}^{n}_{\theta}):=\Big{\\{}u=\sum_{k\in\mathbb{Z}^{n}}u_{k}U^{k}\in
L_{2}(\mathbb{T}^{n}_{\theta});\sum_{k\in\mathbb{Z}^{n}}(1+|k|^{2})^{s}|u_{k}|^{2}<\infty\Big{\\}}.$
(2.6)

This is a Hilbert space with respect to the inner product and norm,

$\left\langle
u|v\right\rangle_{s}=\sum_{k\in\mathbb{Z}^{n}}(1+|k|^{2})^{s}u_{k}\overline{v}_{k},\qquad\|u\|_{W_{2}^{s}}=\bigg{(}\sum_{k\in\mathbb{Z}^{n}}(1+|k|^{2})^{s}|u_{k}|^{2}\bigg{)}^{\frac{1}{2}}.$

Note that $W_{2}^{0}(\mathbb{T}^{n}_{\theta})=L_{2}(\mathbb{T}^{n}_{\theta})$.

Equivalently, let $\Delta=-(\partial_{1}^{2}+\cdots+\partial_{n}^{2})$ be the
Laplacian on $\mathbb{T}^{n}_{\theta}$. This is a non-negative selfadjoint
operator on $L_{2}^{(}\mathbb{T}^{n}_{\theta})$ with domain
$W_{2}^{2}(\mathbb{T}^{n}_{\theta})$. We have

$\Delta\big{(}U^{k})=|k|^{2}U^{k},\qquad k\in\mathbb{Z}^{n}.$

In particular, $\Delta$ is isospectral to the Laplacian on the ordinary torus
$\mathbb{T}^{n}$. Set $\Lambda=(1+\Delta)^{\frac{1}{2}}$. Given any $s\geq 0$,
we have

$W_{2}^{s}(\mathbb{T}^{n}_{\theta}):=\Big{\\{}u\in
L_{2}(\mathbb{T}^{n}_{\theta});\ \Lambda^{s}u\in
L_{2}(\mathbb{T}^{n}_{\theta})\Big{\\}},\qquad\|u\|_{W_{2}^{s}}=\|\Lambda^{s}u\|_{W_{2}^{0}}.$

Note also (see [94, 101]) that, given an integer $p\geq 0$, we have

$W_{2}^{p}(\mathbb{T}^{n}_{\theta})=\Big{\\{}u\in
L_{2}(\mathbb{T}^{n}_{\theta});\ \delta^{\alpha}u\in
L_{2}(\mathbb{T}^{n}_{\theta})\ \
\forall\alpha\in\mathbb{N}_{0}^{n},\,|\alpha|\leq p\big{\\}}.$

We mention the following versions of Sobolev’s embedding theorems.

######  Proposition 2.2 (see [101, Theorem 6.6]).

Let $p\in[2,\infty)$. For every $s\geq n(1/2-p^{-1})$, we have a continuous
embedding $W_{2}^{s}(\mathbb{T}^{n}_{\theta})\subset
L_{p}(\mathbb{T}^{n}_{\theta})$. This embedding is compact when the inequality
is strict.

######  Proposition 2.3.

For any $s>n/2$, we have a compact embedding
$W^{s}_{2}(\mathbb{T}^{n}_{\theta})\subset C(\mathbb{T}^{n}_{\theta})$.

######  Remark 2.4.

The continuity of the inclusion of the
$W^{s}_{2}(\mathbb{T}^{n}_{\theta})\subset C(\mathbb{T}^{n}_{\theta})$ for
$s>n/2$ is established in [36, p. 67] (see also Remark 3.10 below). We obtain
compactness by factorizing it through any inclusion
$W^{s}_{2}(\mathbb{T}^{n}_{\theta})\subset
W^{s^{\prime}}_{2}(\mathbb{T}^{n}_{\theta})$ with $s>s^{\prime}>n/2$.

##  3\. $L_{p}$-Action and Boundedness of $\lambda(x)g(-i\nabla)$ and
$(1+\Delta)^{-\frac{p}{4n}}\lambda(x)(1+\Delta)^{-\frac{p}{4n}}$

In what follows, we denote by $\lambda$ the left-regular representation of
$L_{\infty}(\mathbb{T}^{n}_{\theta})$ on $L_{2}(\mathbb{T}^{n}_{\theta})$. In
this section, we shall explain how to extend it to
$L_{p}(\mathbb{T}^{n}_{\theta})$, $p\geq 1$. We shall then give sufficient
conditions for the boundedness of operators of the forms
$\lambda(x)g(-i\nabla)$ and
$(1+\Delta)^{-\frac{p}{4n}}\lambda(x)(1+\Delta)^{-\frac{p}{4n}}$.

###  3.1. Left-multipliers $\lambda(x)$

First, we observe that Hölder’s inequality (Proposition 2.1) yields the
following extension result for the left-regular representation
$\lambda:L_{\infty}(\mathbb{T}^{n}_{\theta})\rightarrow\mathscr{L}(L_{2}(\mathbb{T}^{n}))$.

######  Proposition 3.1 ([46]).

Suppose that $p^{-1}+q^{-1}=r^{-1}\geq 1$. Then the left-regular
representation uniquely extends to a continuous linear map,

$\lambda:L_{p}(\mathbb{T}^{n}_{\theta})\longrightarrow\mathscr{L}\big{(}L_{q}(\mathbb{T}^{n}_{\theta}),L_{r}(\mathbb{T}^{n}_{\theta})\big{)}.$

In particular, if $p\geq 2$ and $p^{-1}+q^{-1}=2$, then we get a continuous
linear map,

$\lambda:L_{p}(\mathbb{T}^{n}_{\theta})\longrightarrow\mathscr{L}\big{(}L_{q}(\mathbb{T}^{n}_{\theta}),L_{2}(\mathbb{T}^{n}_{\theta})\big{)}.$

######  Remark 3.2.

The above linear maps are isometries (see [46]).

If $x\in L_{p}(\mathbb{T}^{n}_{\theta})$ with $p\geq 2$, the above corollary
asserts that $\lambda(x)$ a continuous linear operator from
$L_{q}(\mathbb{T}_{\theta}^{n})$ to $L_{2}(\mathbb{T}^{n}_{\theta})$ with
$p^{-1}+q^{-1}=1/2$. In particular, we may regard $\lambda(x)$ as an unbounded
operator on $L_{2}(\mathbb{T}^{n}_{\theta})$ with domain
$L_{q}(\mathbb{T}_{\theta}^{n})$.

Given any $q\geq 1$, we denote by $L_{q}(\mathbb{T}^{n}_{\theta})^{*}$ the
anti-linear dual of $L_{q}(\mathbb{T}^{n}_{\theta})$, i.e., the space of
continuous anti-linear forms on $L_{q}(\mathbb{T}^{n}_{\theta})$. We observe
that the left-regular regular representation of
$L_{\infty}(\mathbb{T}^{n}_{\theta})$ can be regarded as a continuous linear
map
$\lambda:L_{\infty}(\mathbb{T}^{n}_{\theta})\rightarrow\mathscr{L}(L_{2}(\mathbb{T}^{n}_{\theta}),L_{2}(\mathbb{T}^{n}_{\theta})^{*})$
such that

$\left\langle\lambda(x)u,v\right\rangle=\left\langle\lambda(x)u|v\right\rangle=\tau\big{[}v^{*}xu\big{]},\qquad
x\in L_{\infty}(\mathbb{T}^{n}_{\theta}),\quad u,v\in
L_{2}(\mathbb{T}^{n}_{\theta}),$

where
$\left\langle\cdot,\cdot\right\rangle:L_{2}(\mathbb{T}^{n}_{\theta})^{*}\times
L_{2}(\mathbb{T}^{n}_{\theta})\rightarrow\mathbb{C}$ is the duality pairing.

######  Proposition 3.3.

Suppose that $p^{-1}+2q^{-1}=1$. Then the left-regular representation uniquely
extends to a continuous linear map
$\lambda:L_{p}(\mathbb{T}^{n}_{\theta})\rightarrow\mathscr{L}(L_{q}(\mathbb{T}^{n}_{\theta}),L_{q}(\mathbb{T}^{n}_{\theta})^{*})$
such that

$\left\langle\lambda(x)u,v\right\rangle=\tau\big{[}v^{*}xu\big{]}\qquad\forall
x\in L_{p}(\mathbb{T}^{n}_{\theta})\quad\forall u,v\in
L_{q}(\mathbb{T}^{n}_{\theta}).$ (3.1)

###### Proof.

Let $x\in L_{p}(\mathbb{T}^{n}_{\theta})$ and $u,v\in
L_{q}(\mathbb{T}^{n}_{\theta})$. By Hölder’s inequality $xu\in
L_{r}(\mathbb{T}^{n}_{\theta})$ with $r^{-1}=p^{-1}+q^{-1}=1-q^{-1}$, and so
$v^{*}(xu)\in L_{1}(\mathbb{T}^{n}_{\theta})$. Moreover, we have

$\left|\tau(v^{*}xu)\right|\leq\|v^{*}(xu)\|_{L_{1}}\leq\|v^{*}\|_{L_{q}}\|xu\|_{L_{r}}\leq\|x\|_{L_{p}}\|u\|_{L_{q}}\|v\|_{L_{q}}.$

This gives the result. ∎

In particular, if $x\in L_{p}(\mathbb{T}^{n}_{\theta})$ with $1\leq p<2$, then
the above proposition shows that $\lambda(x)$ makes sense as a bounded
operator from $L_{q}(\mathbb{T}^{n}_{\theta})$ to
$L_{q}(\mathbb{T}^{n}_{\theta})^{*}$, with $p^{-1}+2q^{-1}=1$, i.e.,
$q^{-1}=\frac{1}{2}(1-p^{-1})$.

###  3.2. The operators $(1+\Delta)^{-s/2}\lambda(x)(1+\Delta)^{-s/2}$

Given any $s>0$, we denote by $W_{2}^{-s}(\mathbb{T}^{2}_{\theta})$ the anti-
linear dual of the Sobolev space $W_{2}^{s}(\mathbb{T}^{n}_{\theta})$. Set
$\Lambda=(1+\Delta)^{1/2}$. As mentioned above
$\Lambda^{s}:W^{s}_{2}(\mathbb{T}^{n}_{\theta})\rightarrow
L_{2}(\mathbb{T}^{n}_{\theta})$ is an isometric isomorphism. By duality we get
a continuous isomorphism
$\Lambda^{s}:L_{2}(\mathbb{T}^{n}_{\theta})\rightarrow
W_{2}^{-s}(\mathbb{T}^{n}_{\theta})$ such that

$\left\langle\Lambda^{s}u,v\right\rangle=\left\langle
u|\Lambda^{s}v\right\rangle,\qquad u\in L_{2}(\mathbb{T}^{n}_{\theta}),\quad
v\in W_{2}^{s}(\mathbb{T}^{n}_{\theta}),$

where
$\left\langle\cdot,\cdot\right\rangle:W_{2}^{-s}(\mathbb{T}^{n}_{\theta})\times
W_{2}^{s}(\mathbb{T}^{n}_{\theta})\rightarrow\mathbb{C}$ is the duality
pairing. Its inverse
$\Lambda^{-s}:W_{2}^{-s}(\mathbb{T}^{n}_{\theta})\rightarrow
L_{2}(\mathbb{T}^{n}_{\theta})$ is given by

$\left\langle\Lambda^{-s}u|v\right\rangle=\left\langle
u,\Lambda^{-s}v\right\rangle,\qquad u\in
W_{2}^{s}(\mathbb{T}^{n}_{\theta}),\quad v\in L_{2}(\mathbb{T}^{n}_{\theta}).$
(3.2)

######  Lemma 3.4.

Let $x\in L_{p}(\mathbb{T}^{n}_{\theta})$, $p\geq 1$, and assume that, either
$s>n/2p$, or $s=n/2p$ and $p>1$. Then $\lambda(x)$ uniquely extends to a
bounded operator $\lambda(x):W_{2}^{s}(\mathbb{T}^{n}_{\theta})\rightarrow
W_{2}^{-s}(\mathbb{T}^{n}_{\theta})$.

###### Proof.

Suppose that $p^{-1}+2q^{-1}=1$, i.e., $q^{-1}=\frac{1}{2}(1-p^{-1})$. We know
by Proposition 3.3 that $\lambda(x)$ is a bounded operator from
$L_{q}(\mathbb{T}^{n}_{\theta})$ to $L_{q}(\mathbb{T}^{n}_{\theta})^{*}$.

Suppose that $p>1$ and $s\geq n/2p$. Then $q\in[2,\infty)$, and so by
Proposition 2.2 we have a continuous embedding of
$W_{2}^{s}(\mathbb{T}^{n}_{\theta})$ into $L_{q}(\mathbb{T}^{n}_{\theta})$
since $s\geq n/{2p}=\frac{n}{2}(1-q^{-1})$. By duality we get a continuous
embedding of $L_{q}(\mathbb{T}^{n}_{\theta})^{*}$ into
$W_{2}^{-s}(\mathbb{T}^{n}_{\theta})$. It then follows that $\lambda(x)$
induces a bonded operator
$\lambda(x):W_{2}^{s}(\mathbb{T}^{n}_{\theta})\rightarrow
W_{2}^{-s}(\mathbb{T}^{n}_{\theta})$.

Assume now that $p=1$ and $s>n/2$. In this case $q=\infty$, and so
$\lambda(x)$ is a bounded operator from $L_{\infty}(\mathbb{T}^{n}_{\theta})$.
As $s>n/2$, by Proposition 2.3 we have a continuous embedding of
$W_{2}^{s}(\mathbb{T}^{n}_{\theta})$ into $C(\mathbb{T}^{n}_{\theta})$, and
hence we get a continuous embedding into
$L_{\infty}(\mathbb{T}^{n}_{\theta})$. By duality we get a continuous
embedding of $L_{\infty}(\mathbb{T}^{n}_{\theta})^{*}$ into
$W_{2}^{-s}(\mathbb{T}^{n}_{\theta})$. Thus, as above $p>1$, $\lambda(x)$
induces a bounded operator
$\lambda(x):W_{2}^{s}(\mathbb{T}^{n}_{\theta})\rightarrow
W_{2}^{-s}(\mathbb{T}^{n}_{\theta})$. The proof is complete. ∎

Combining the above lemma with the boundedness of the operators
$\Lambda^{-s}:L_{2}(\mathbb{T}^{n}_{\theta})\rightarrow
W_{2}^{s}(\mathbb{T}^{n}_{\theta})$ and
$\Lambda^{-s}:W_{2}^{-s}(\mathbb{T}^{n}_{\theta})\rightarrow
L_{2}(\mathbb{T}^{n}_{\theta})$ we arrive at the following result.

######  Proposition 3.5.

Let $x\in L_{p}(\mathbb{T}^{n}_{\theta})$, $p\geq 1$, and assume that either
$s>n/2p$, or $s=n/2p$ and $p>1$. Then the composition
$\Lambda^{-s}\lambda(x)\Lambda^{-s}$ makes sense as a bounded operator on
$L_{2}(\mathbb{T}^{n}_{\theta})$.

######  Remark 3.6.

The above result holds _verbatim_ if we replace $\Lambda$ by $\sqrt{\Delta}$.

###  3.3. The operators $\lambda(x)g(-i\nabla)$

Recall that, given any $p\in(0,\infty)$, the quasi-Banach space
$\ell_{p}(\mathbb{Z}^{n})$ consists of $p$-summable sequences
$a=(a_{k})_{k\in\mathbb{Z}^{n}}\subset\mathbb{C}$ with quasi-norm,

$\|a\|_{\ell_{p}}:=\bigg{(}\sum_{k\in\mathbb{Z}^{n}}|a_{k}|^{p}\bigg{)}^{1/p},\qquad
a=(a_{k})\in\ell_{p}(\mathbb{Z}^{n}).$

For $p\geq 1$ this is a norm, and so in this case $\ell_{p}(\mathbb{Z}^{n})$
is a Banach space. In addition, we denote by $\ell_{\infty}(\mathbb{Z}^{n})$
the Banach space of bounded sequences with norm,

$\|a\|_{\ell_{\infty}}:=\sup_{k\in\mathbb{Z}^{n}}|a_{k}|,\qquad
a=(a_{k})\in\ell_{\infty}(\mathbb{Z}^{n}).$

For $p\in(0,\infty)$, the weak $\ell_{p}$-space
$\ell_{p,\infty}(\mathbb{Z}^{n})$ is defined as follows. Given any sequence
$a=(a_{k})_{k\in\mathbb{Z}^{n}}\in\ell_{\infty}(\mathbb{Z}^{n})$, let
$\mu(a)=(\mu_{j}(a))_{j\geq 0}$ be its symmetric decreasing re-arrangement,
i.e.,

$\mu_{j}(a):=\sup_{k^{0},\ldots,k^{j}\in\mathbb{Z}^{n}}\min\\{|a_{k^{0}}|,|a_{k^{1}}|,\ldots,|a_{k^{j}}|\\}.$

In other words, $\mu(a)=(\mu_{j}(a))_{j\geq 0}$ is the non-increasing
rearrangement of the sequence $(|a_{k}|)_{k\in\mathbb{Z}^{n}}$. The space
$\ell_{p,\infty}(\mathbb{Z}^{n})$ then consists of sequences
$a=(a_{k})_{k\in\mathbb{Z}^{n}}\in\ell_{\infty}(\mathbb{Z}^{n})$ such that

$\mu_{j}(a)=\operatorname{O}\big{(}j^{-\frac{1}{p}}\big{)}\qquad\text{as
$j\rightarrow\infty$}.$

We equip it with the quasi-norm,

$\|a\|_{\ell_{p,\infty}}:=\sup_{j\geq 0}\,(j+1)^{1/p}\mu_{j}(a),\qquad
a\in\ell_{p,\infty}(\mathbb{Z}^{n}).$

With this quasi-norm $\ell_{p,\infty}(\mathbb{Z}^{n})$ is a quasi-Banach
space. In fact, for $p>1$ the above quasi-norm is equivalent to the norm,

$\|a\|_{\ell_{p,\infty}}^{\prime}:=\sup_{N\geq
1}N^{-1+\frac{1}{p}}\sum_{j<N}\mu_{j}(a),\qquad
a\in\ell_{p,\infty}(\mathbb{Z}^{n}).$

Therefore, for $p>1$ we actually obtain a Banach space.

The canonical derivations $\partial_{1},\ldots,\partial_{n}$ pairwise commute
with each other. Their joint spectrum is $i\mathbb{Z}^{n}$. Given any
$g\in\ell_{\infty}(\mathbb{Z}^{n})$, the operator $g(-i\nabla)$ is given by

$g(-i\nabla)U^{k}=g(k)U^{k},\quad k\in\mathbb{Z}^{n}.$

This is a bounded operator on $L_{2}(\mathbb{T}^{n}_{\theta})$ with norm equal
to $\|g\|_{\ell_{\infty}}$. In fact, if we also denote by $g$ the operator of
multiplication by $g$ on $\ell_{2}(\mathbb{Z}^{n})$, then
$\mu(g(-i\nabla))=\mu(g)$. In particular, this implies that

  * •

If $g\in\ell_{p}$, then $g(-i\nabla)\in\mathscr{L}_{p}$ and
$\|g(-i\nabla)\|_{\mathscr{L}_{p}}=\|g\|_{\ell_{p}}$.

  * •

If $g\in\ell_{p,\infty}$, then $g(-i\nabla)\in\mathscr{L}_{p,\infty}$ and
$\|g(-i\nabla)\|_{\mathscr{L}_{p,\infty}}=\|g\|_{\ell_{p,\infty}}$.

Let us now look at the mapping properties of the operators $g(-i\nabla)$. To
this end, given any $p\in[1,2]$, it is convenient to introduce the space
$\hat{\ell}_{p}(\mathbb{T}^{n}_{\theta})$ that consists of all
$x=\sum\widehat{x}_{k}U^{k}$ in $L_{2}(\mathbb{T}^{n}_{\theta})$ such that
$\sum|\widehat{x}_{k}|^{p}<\infty$. We equip it with the norm,

$\|x\|_{\hat{\ell}_{p}}:=\bigg{(}\sum_{k\in\mathbb{Z}^{n}}|\widehat{x}_{k}|^{p}\bigg{)}^{\frac{1}{p}},\qquad
x\in\hat{\ell}_{p}(\mathbb{T}^{n}_{\theta})$

In other words, $\hat{\ell}_{p}(\mathbb{T}^{n}_{\theta})$ is the inverse image
of $\ell_{p}(\mathbb{Z}^{n})$ under the Fourier transform
$x\rightarrow(\hat{x}_{k})$. Thus, under the Fourier transform the spaces
$\hat{\ell}_{p}(\mathbb{T}^{n}_{\theta})$ and $\ell_{p}(\mathbb{Z}^{n})$ are
isometrically isomorphic. In particular,
$\hat{\ell}_{p}(\mathbb{T}^{n}_{\theta})$ is a Banach space.

For $p=2$ the spaces $\hat{\ell}_{p}(\mathbb{T}^{n}_{\theta})$ and
$L_{2}(\mathbb{T}^{n})$ and their norms agree as well. Moreover, we have a
continuous inclusion $\hat{\ell}_{1}(\mathbb{T}^{n}_{\theta})\subset
C(\mathbb{T}_{\theta}^{n})$, since, for every $x=\sum\hat{x}_{k}U^{k}$ in
$\hat{\ell}_{1}(\mathbb{T}^{n}_{\theta})$, the Fourier series
$\sum\hat{x}_{k}U^{k}$ converges normally in $C(\mathbb{T}_{\theta}^{n})$, and
we have

$\|x\|\leq\sum_{k\in\mathbb{Z}^{n}}|\hat{x}_{k}|\|U^{k}\|=\sum_{k\in\mathbb{Z}^{n}}|\hat{x}_{k}|=\|x\|_{\hat{\ell}_{1}}.$

######  Lemma 3.7.

Suppose that $p^{-1}+q^{-1}=r^{-1}$ and $g\in\ell_{p}(\mathbb{Z}^{n})$ with
$p\geq 2$ and $1\leq q,r\leq 2.$ Then $g(-i\nabla)$ induces a continuous
linear operator
$g(-i\nabla):\hat{\ell}_{q}(\mathbb{T}^{n}_{\theta})\rightarrow\hat{\ell}_{r}(\mathbb{T}^{n}_{\theta})$
with norm inequality,

$\big{\|}g(-i\nabla)x\big{\|}_{\hat{\ell}_{r}}\leq\|g\|_{\ell_{p}}\|x\|_{\hat{\ell}_{q}}\qquad\forall
x\in\hat{\ell}_{q}(\mathbb{T}^{n}_{\theta}).$

###### Proof.

Let $x\in\hat{\ell}_{q}(\mathbb{T}^{n}_{\theta})$. We have $g(-i\nabla)x=\sum
g(k)\hat{x}(k)U^{k}$. By assumption
$(g(k))_{k\in\mathbb{Z}^{n}}\in\ell_{p}(\mathbb{Z}^{n})$ and
$\hat{x}:=(\hat{x}(k))_{k\in\mathbb{Z}^{n}}\in\ell_{q}(\mathbb{Z}^{n})$, and
so by Hölder’s inequality
$(g(k)\hat{x}(k))_{k\in\mathbb{Z}^{n}}\in\ell_{r}(\mathbb{Z}^{n})$, since
$p^{-1}+q^{-1}=r^{-1}$. This means that
$g(-i\nabla)x\in\hat{\ell}_{r}(\mathbb{T}^{n}_{\theta})$. Moreover, we have
the inequalities,

$\big{\|}g(-i\nabla)x\big{\|}_{\hat{\ell}_{r}}=\|(g(k)\hat{x}(k))\|_{\ell_{r}}\leq\|g\|_{\ell_{p}}\|\hat{x}\|_{\ell_{q}}=\|g\|_{\ell_{p}}\|x\|_{\hat{\ell}_{q}}.$

This proves the result. ∎

The following two lemmas give the relationships of the $\hat{\ell}_{p}$-spaces
with the $L_{p}$-spaces and Sobolev spaces.

######  Lemma 3.8 (Hausdorff-Young Inequality).

Suppose that $q\geq 2$ and $q^{-1}+r^{-1}=1$. Then we have a continuous
inclusion $\hat{\ell}_{r}(\mathbb{T}^{n}_{\theta})\subseteq
L_{q}(\mathbb{T}^{n}_{\theta})$ with norm inequality,

$\|x\|_{L_{q}}\leq\|x\|_{\hat{\ell}_{r}}\qquad\forall
x\in\hat{\ell}_{r}(\mathbb{T}^{n}_{\theta}).$ (3.3)

###### Proof.

The proof uses complex interpolation theory essentially in the same way as in
the proof of the classical Hausdorff-Young inequality (see, e.g., [3]). For
background on interpolation theory we refer to the short survey of Connes [19,
Appendix IV.B] and the references therein. The main reference for complex
interpolation theory there is the article of Calderón [15] (see also [3, 45]).

As mentioned above $L_{2}(\mathbb{T}^{n}_{\theta})$ and
$\hat{\ell}_{2}(\mathbb{T}_{\theta}^{n})$ agree as Banach spaces and the
inclusion of $\ell_{1}(\mathbb{T}^{n}_{\theta})$ into
$C(\mathbb{T}^{n}_{\theta})$ gives rise to a contraction into
$L_{\infty}(\mathbb{T}^{n}_{\theta})$. In particular, we have the
inequalities,

$\displaystyle\|x\|_{L_{2}}=\|x\|_{\hat{\ell}_{2}}\qquad\forall
x\in\hat{\ell}_{2}(\mathbb{T}^{n}_{\theta}),$ (3.4)
$\displaystyle\|x\|_{L_{\infty}}\leq\|x\|_{\hat{\ell}_{1}}\qquad\forall
x\in\hat{\ell}_{1}(\mathbb{T}^{n}_{\theta}).$ (3.5)

Suppose that $q\in(2,\infty)$. Then $L_{q}(\mathbb{T}_{\theta}^{n})$ is a
complex interpolation space for the pair of Banach spaces
$(L_{2}(\mathbb{T}^{n}_{\theta}),L_{\infty}(\mathbb{T}^{n}_{\theta}))$.
Namely, in the notation of [19, Appendix IV.B] we have
$L_{q}(\mathbb{T}_{\theta}^{n})=[L_{2}(\mathbb{T}^{n}_{\theta}),L_{\infty}(\mathbb{T}^{n}_{\theta})]_{\theta}$,
where $\theta$ is such that $q=(1-\theta)\frac{1}{2}$, i.e.,
$\theta=1-2q^{-1}$ (see [24, Section 4] and [46]). Note also that, as
$r\in(1,2)$, the space $\ell_{r}(\mathbb{T}^{n}_{\theta})$ is an exact
interpolation space for the pair
$(\ell_{2}(\mathbb{T}^{n}_{\theta}),\ell_{1}(\mathbb{T}^{n}_{\theta}))$.
Namely,
$\ell_{r}(\mathbb{T}^{n}_{\theta})=[\ell_{2}(\mathbb{T}^{n}_{\theta}),\ell_{1}(\mathbb{T}^{n}_{\theta})]_{\theta}$,
since we have

$(1-\theta)\frac{1}{2}+\theta=\frac{1}{2}(1+\theta)=\frac{1}{2}(2-2q^{-1})=1-q^{-1}=r^{-1}.$

As the Fourier transform induces isometric isomorphisms between the spaces
$\ell_{p}(\mathbb{Z}^{n})$ and $\hat{\ell}_{p}(\mathbb{T}^{n}_{\theta})$, we
see that
$\hat{\ell}_{r}(\mathbb{T}^{n}_{\theta})=[\hat{\ell}_{2}(\mathbb{T}^{n}_{\theta}),\hat{\ell}_{1}(\mathbb{T}^{n}_{\theta})]_{\theta}$.
Combining all this with the inequalities (3.4)–(3.5) and using complex
interpolation theory (see [19, Theorem IV.B.1]) then shows we have a
continuous inclusion $\hat{\ell}_{r}(\mathbb{T}^{n}_{\theta})\subseteq
L_{q}(\mathbb{T}^{n}_{\theta})$ with the norm inequality (3.3). The proof is
complete. ∎

######  Lemma 3.9.

Suppose that $1\leq p<2$ and $s>\frac{n}{2}(2p^{-1}-1)$. Then we have a
continuous inclusion
$W_{2}^{s}(\mathbb{T}^{n}_{\theta})\subset\hat{\ell}_{p}(\mathbb{T}^{n}_{\theta})$.

###### Proof.

Let $u\in\sum u_{k}U^{k}$ be in $W_{2}^{s}(\mathbb{T}^{n}_{\theta})$. By
Hölder’s inequality we have

$\displaystyle\sum_{k\in\mathbb{Z}^{n}}|u_{k}|^{p}$
$\displaystyle=\sum_{k\in\mathbb{Z}^{n}}\left(1+|k|^{2}\right)^{-\frac{ps}{2}}\big{[}\left(1+|k|^{2}\right)^{s}|u|_{k}^{2}\big{]}^{\frac{p}{2}}$
$\displaystyle\leq\bigg{[}\sum_{k\in\mathbb{Z}^{n}}\left(1+|k|^{2}\right)^{-\frac{rps}{2}}\bigg{]}^{\frac{1}{r}}\bigg{[}\sum_{k\in\mathbb{Z}^{n}}\left(1+|k|^{2}\right)^{s}|u|_{k}^{2}\bigg{]}^{\frac{p}{2}}$
(3.6)
$\displaystyle\leq\bigg{[}\sum_{k\in\mathbb{Z}^{n}}\left(1+|k|^{2}\right)^{-\frac{1}{2}psr}\bigg{]}^{\frac{1}{r}}\|u\|_{W_{2}^{s}}^{p}.$

Here $r$ is such that $r^{-1}+(2p^{-1})^{-1}=1$, i.e.,
$r^{-1}=1-\frac{1}{2}p$. By assumption
$s>\frac{n}{2}(2p^{-1}-1)=n(1-\frac{1}{2}p)p^{-1}=n(rp)^{-1}$, and hence
$psr>n$ and $\sum(1+|k|^{2})^{-\frac{1}{2}psr}<\infty$. Combining this with
(3.6) shows that $\sum_{k\in\mathbb{Z}^{n}}|u_{k}|^{p}<\infty$, i.e.,
$u\in\hat{\ell}_{p}(\mathbb{Z}^{n})$, and there is a constant $C_{nps}>0$
independent of $u$ such that

$\|u\|_{\hat{\ell}_{p}}=\bigg{[}\sum_{k\in\mathbb{Z}^{n}}|u_{k}|^{p}\bigg{]}^{\frac{1}{p}}\leq
C_{nps}\|u\|_{W_{2}^{s}}.$

This proves the result. ∎

######  Remark 3.10.

For $p=1$ and $s>n/2$, we get a continuous inclusion of
$W_{2}^{s}(\mathbb{T}^{n}_{\theta})$ into
$\hat{\ell}_{1}(\mathbb{T}^{n}_{\theta})$. Combining it with the continuity of
the inclusion of $\hat{\ell}_{1}(\mathbb{T}^{n}_{\theta})$ into
$C(\mathbb{T}^{n}_{\theta})$ mentioned above, we get a continuous inclusion
$W_{2}^{s}(\mathbb{T}^{n}_{\theta})$ into $C(\mathbb{T}^{n}_{\theta})$.

We are now in a position to prove the following result.

######  Proposition 3.11.

Suppose that $p^{-1}+q^{-1}=\frac{1}{2}$.

  1. (1)

If $g\in\ell_{p}(\mathbb{Z}^{n})$, then $g(-i\nabla)$ maps continuously
$L_{2}(\mathbb{T}^{n}_{\theta})$ to $L_{q}(\mathbb{T}^{n})$.

  2. (2)

If $g\in\ell_{p,\infty}(\mathbb{Z}^{n})$, then $g(-i\nabla)$ maps continuously
$W^{s}_{2}(\mathbb{T}^{n}_{\theta})$ to $L_{q}(\mathbb{T}^{n})$ for every
$s>0$.

###### Proof.

Let $g\in\ell_{p}(\mathbb{Z}^{n})$. Lemma 3.7 ensures us that $g(-i\nabla)$
maps continuously $L_{2}(\mathbb{T}^{n}_{\theta})$ to
$\hat{\ell}_{r}(\mathbb{T}^{n}_{\theta})$, where $r^{-1}=\frac{1}{2}+p^{-1}$.
Note that $1-r^{-1}=\frac{1}{2}-p^{-1}=q^{-1}$, and so by Lemma 3.8 we have a
continuous inclusion of $\hat{\ell}_{r}(\mathbb{T}^{n}_{\theta})$ into
$L_{q}(\mathbb{T}^{n}_{\theta})$. It then follows that $g(-i\nabla)$ maps
continuously $L_{2}(\mathbb{T}^{n}_{\theta})$ to
$L_{q}(\mathbb{T}_{\theta}^{n})$. This proves the first part.

To prove the 2nd part let $g\in\ell_{p,\infty}(\mathbb{Z}^{n})$ and $s>0$. In
addition, let $t\in[1,2)$ be such that $s>\frac{n}{2}(2t^{-1}-1)$. Then by
Lemma 3.9 we have continuous embedding of $W_{2}^{s}(\mathbb{T}^{n}_{\theta})$
into $\hat{\ell}_{t}(\mathbb{T}^{n}_{\theta})$. Set
$p_{1}=[p^{-1}-(t^{-1}-\frac{1}{2})]^{-1}$, i.e.,
$p_{1}^{-1}=p^{-1}-(t^{-1}-\frac{1}{2})$. Note that $p_{1}>p$ since $t<2$. We
also observe that
$p_{1}^{-1}+t^{-1}=p^{-1}+\frac{1}{2}=1-(\frac{1}{2}-p^{-1})=1-q^{-1}$. The
fact $p_{1}>p$ implies that $g\in\ell_{p_{1}}(\mathbb{Z}^{n})$, and so Lemma
3.7 ensures us that $g(-i\nabla)$ maps continuously
$\hat{\ell}_{t}(\mathbb{T}^{n}_{\theta})$ into
$\hat{\ell}_{r}(\mathbb{T}^{n}_{\theta})$ with
$r^{-1}=p_{1}^{-1}+t^{-1}=1-q^{-1}$. Furthermore, by Lemma 3.8 we have a
continuous inclusion of $\hat{\ell}_{r}(\mathbb{T}^{n}_{\theta})$ into
$L_{q_{1}}(\mathbb{T}^{n}_{\theta})$. It then follows that $g(-i\nabla)$ maps
continuously $W_{2}^{s}(\mathbb{T}^{n}_{\theta})$ to
$L_{q}(\mathbb{T}^{n}_{\theta})$. This proves the 2nd part and completes the
proof. ∎

By combining Proposition 3.1 and Proposition 3.11 we arrive at the following
result.

######  Proposition 3.12.

Let $p\in[2,\infty)$.

  1. (1)

If $x\in L_{p}(\mathbb{T}^{n}_{\theta})$ and $g\in\ell_{p}(\mathbb{Z}^{n})$,
then the operator $\lambda(x)g(-i\nabla)$ is bounded on
$L_{2}(\mathbb{T}^{n}_{\theta})$.

  2. (2)

If $x\in L_{p}(\mathbb{T}^{n}_{\theta})$ and
$g\in\ell_{p,\infty}(\mathbb{Z}^{n})$, then the domain of
$\lambda(x)g(-i\nabla)$ contains
$\cup_{s>0}W_{2}^{s}(\mathbb{T}^{n}_{\theta})$. In particular,
$\lambda(x)g(-i\nabla)$ is densely defined.

######  Remark 3.13.

The inclusion $\cup_{s>0}W_{2}^{s}(\mathbb{T}^{n}_{\theta})\subset
L_{2}(\mathbb{T}^{n}_{\theta})$ is strict. For instance, if $u=\sum
u_{k}U^{k}$ with $u_{k}=(1+|k|)^{-\frac{n}{2}}[\log(1+|k|)]^{-1}$, then $u\in
L_{2}(\mathbb{T}^{n}_{\theta})$, but $u\not\in
W_{2}^{s}(\mathbb{T}^{n}_{\theta})$ for any $s>0$.

##  4\. Cwikel Estimates on NC Tori

In this section, we establish Cwikel type estimates on NC tori for Schatten
classes and their weak versions.

###  4.1. Schatten classes and weak Schatten classes

We briefly review the main definitions and properties of Schatten classes and
weak Schatten classes on Hilbert space. We refer to [34, 90] for further
details.

In what follows we let $\mathscr{H}$ be a (separable) Hilbert space with inner
product $\left\langle\cdot|\cdot\right\rangle$. We also denote by
$\mathscr{K}$ the (closed) ideal of compact operators on $\mathscr{H}$. Given
any operator $T\in\mathscr{K}$ we let $\mu=(\mu_{j}(T))_{j\geq 0}$ be its
sequence of singular values, i.e., $\mu_{j}(T)$ is the $(j+1)$-the eigenvalue
counted with multiplicity of the absolute value $|T|=\sqrt{T^{*}T}$. By the
min-max principle [34, 76] we have

$\displaystyle\mu_{j}(T)$ $\displaystyle=\min\left\\{\|T_{|E^{\perp}}\|;\ \dim
E=j\right\\},$ (4.1) $\displaystyle=\min\left\\{\|T-R\|;\
\operatorname{rk}(R)\leq j\right\\},\qquad j\geq 0.$ (4.2)

This implies the following properties of singular values (see, e.g., [34,
90]),

$\displaystyle\mu_{j}(T)=\mu_{j}(T^{*})=\mu_{j}(|T|),$ (4.3)
$\displaystyle\mu_{j+k}(S+T)\leq\mu_{j}(S)+\mu_{k}(T),$ (4.4)
$\displaystyle\mu_{j}(ATB)\leq\|A\|\mu_{j}(T)\|B\|,\qquad
A,B\in\mathscr{L}(\mathscr{H}).$ (4.5)

In addition, we have the monotonicity principle,

$0\leq T\leq S\ \Longrightarrow\ \mu_{j}(T)\leq\mu_{j}(S)\quad\forall j\geq
0.$ (4.6)

In what follows, we denote by $\mathscr{L}_{1}$ the trace-class with norm,

$\|T\|_{\mathscr{L}_{1}}:=\operatorname{Tr}|T|=\sum_{j\geq 0}\mu_{j}(T),\qquad
T\in\mathscr{L}_{1}.$

Recall that for $p\in(0,\infty)$ the Schatten class $\mathscr{L}_{p}$ consist
of operators $T\in\mathscr{K}$ such that $|T|^{p}$ is trace-class. It is
equipped with the quasi-norm,

$\|T\|_{\mathscr{L}_{p}}:=\big{(}\operatorname{Tr}|T|^{p}\big{)}^{\frac{1}{p}}=\bigg{(}\sum_{j\geq
0}\mu_{j}(T)^{p}\bigg{)}^{\frac{1}{p}},\qquad T\in\mathscr{L}_{p}.$

We obtain a quasi-Banach ideal. For $p\geq 1$ the $\mathscr{L}_{p}$-quasi-norm
is actually a norm, and so in this case $\mathscr{L}_{p}$ is a Banach ideal.

For $p\in(0,\infty)$, the weak Schatten class $\mathscr{L}_{p,\infty}$ is
defined by

$\mathscr{L}_{p,\infty}:=\left\\{T\in\mathscr{K};\
\mu_{j}(T)=\operatorname{O}\big{(}j^{-\frac{1}{p}}\big{)}\right\\}.$

This is a two-sided ideal. We equip it with the quasi-norm,

$\|T\|_{\mathscr{L}_{p,\infty}}:=\sup_{j\geq
0}\;(j+1)^{\frac{1}{p}}\mu_{j}(T),\qquad T\in\mathscr{L}_{p,\infty}.$ (4.7)

Note that

$T\in\mathscr{L}_{p,\infty}\Longleftrightarrow|T|^{p}\in\mathscr{L}_{1,\infty}\quad\text{and}\quad\|T\|_{\mathscr{L}_{p,\infty}}=\left(\||T|^{p}\|_{\mathscr{L}_{1,\infty}}\right)^{\frac{1}{p}}.$
(4.8)

In addition, for $p>1$, the quasi-norm $\|\cdot\|_{p,\infty}$ is equivalent to
the norm,

$\|T\|_{\mathscr{L}_{p,\infty}}^{\prime}:=\sup_{N\geq
1}N^{-1+\frac{1}{p}}\sum_{j<N}\mu_{j}(T),\qquad T\in\mathscr{L}_{p,\infty}.$

More precisely (see, e.g., [90, §1.7]), we have

$\|T\|_{\mathscr{L}_{p,\infty}}\leq\|T\|_{\mathscr{L}_{p,\infty}}^{\prime}\leq\frac{p}{p-1}\|T\|_{\mathscr{L}_{p,\infty}}.$

Therefore, in this case $\mathscr{L}_{p,\infty}$ is a Banach ideal with
respect to the equivalent norm $\|\cdot\|_{p,\infty}^{\prime}$.

We also have the following version of Hölder’s inequality for weak Schatten
classes.

######  Proposition 4.1 ([34, 90, 99]).

Suppose that $p^{-1}+q^{-1}=r^{-1}$. If $S\in\mathscr{L}_{p,\infty}$ and
$T\in\mathscr{L}_{q,\infty}$, then $ST\in\mathscr{L}_{r,\infty}$ with norm
inequality,

$\|ST\|_{\mathscr{L}_{r,\infty}}\leq\gamma(p,q)\|S\|_{\mathscr{L}_{p,\infty}}\|T\|_{\mathscr{L}_{q,\infty}},\qquad\gamma(p,q):=p^{-\frac{1}{q}}q^{-\frac{1}{p}}(p+q)^{\frac{1}{p}+\frac{1}{q}}.$

Moreover, the constant $\gamma(p,q)$ above is optimal.

######  Remark 4.2.

The fact that the constant $\gamma(p,q)$ is optimal is established in [99].

######  Remark 4.3.

In terms of $p$ and $r$, we have $\gamma(p,q)=p^{1/r}r^{-1/p}(p-r)^{1/p-1/r}$.

Recall that, given non-increasing sequences of non-negative numbers
$a=(a_{j})_{j\geq 0}$ and $b=(b_{j})_{j\geq 0}$, the Hardy-Littlewood-Pólya
submajorization $a\prec\prec b$ means that

$\sum_{j=0}^{N}a_{j}\leq\sum_{j=0}^{N}b_{j}\quad\forall N\geq
0\quad\text{and}\quad\sum_{j=0}^{\infty}a_{j}=\sum_{j=0}^{\infty}b_{j}<\infty.$

The Hardy-Littlewood-Pólya majorization $a\prec b$ means that

$a\prec\prec
b\qquad\text{and}\qquad\sum_{j=0}^{\infty}a_{j}=\sum_{j=0}^{\infty}b_{j}<\infty.$

Given compact operators $S$ and $T$ on Hilbert space we shall write
$S\prec\prec T$ (resp., $S\prec T$) if $\mu(S)\prec\prec\mu(T)$ (resp.,
$\mu(S)\prec\mu(T)$).

The Schatten $\mathscr{L}_{p}$-norms are monotone under submajorization

$T\prec\prec
S\Longrightarrow\|T\|_{\mathscr{L}_{p}}\leq\|S\|_{\mathscr{L}_{p}}.$ (4.9)

Note that if $T\prec\prec S,$ then
$\|T\|_{\mathscr{L}_{p,\infty}}^{\prime}\prec\prec\|S\|_{\mathscr{L}_{p,\infty}}^{\prime}.$
Therefore, for $p>1$, we have

$\displaystyle T\prec\prec
S\Longrightarrow\|T\|_{\mathscr{L}_{p,\infty}}\leq\frac{p}{p-1}\|S\|_{\mathscr{L}_{p,\infty}}.$
(4.10)

###  4.2. Cwikel type estimates

We shall distinguish between the following cases:

  * •

Case I: $\lambda(x)g(-i\nabla)\in\mathscr{L}_{p}$ with $p\geq 2$ and
$\lambda(x)g(-i\nabla)\in\mathscr{L}_{p,\infty}$ with $p>2$.

  * •

Case II: $\lambda(x)g(-i\nabla)\in\mathscr{L}_{p}$ and
$\lambda(x)g(-i\nabla)\in\mathscr{L}_{p,\infty}$ with $0<p<2$.

  * •

Case III: $\lambda(x)g(-i\nabla)\in\mathscr{L}_{2,\infty}$.

The first case is dealt with by the following result.

######  Theorem 4.4 (see also [53, 66]).

The following holds.

  1. (1)

If $x\in L_{p}(\mathbb{T}^{n}_{\theta})$ and $g\in\ell_{p}(\mathbb{Z}^{n})$
with $p\geq 2$, then $\lambda(x)g(-i\nabla)\in\mathscr{L}_{p}$, and we have

$\|\lambda(x)g(-i\nabla)\|_{\mathscr{L}_{p}}\leq\|x\|_{L_{p}}\|g\|_{\ell_{p}}.$
(4.11)

  2. (2)

Assume $p>2$. There is $c_{+}(p)>0$ such that, if $x\in
L_{p}(\mathbb{T}^{n}_{\theta})$ and $g\in\ell_{p,\infty}(\mathbb{Z}^{n})$ with
$p>2$, then $\lambda(x)g(-i\nabla)\in\mathscr{L}_{p,\infty}$, and we have

$\|\lambda(x)g(-i\nabla)\|_{\mathscr{L}_{p,\infty}}\leq
c_{+}(p)\|x\|_{L_{p}}\|g\|_{\ell_{p,\infty}}.$ (4.12)

Moreover, the best constant $c_{+}(p)$ satisfies

$c_{+}(p)\leq\left(\frac{130p}{p-2}\right)^{\frac{1}{2}}.$

###### Proof.

The proof of the Cwikel estimate (4.11) for $p=2$ is elementary and is
presented in the proof of [66, Theorem 3.1]. In fact, in this case we actually
have an equality,

$\|\lambda(x)g(-i\nabla)\|_{\mathcal{L}_{2}}=\|x\|_{L_{2}}\|g\|_{\ell_{2}},\qquad
x\in L_{2}(\mathbb{T}^{n}_{\theta}),\ g\in\ell_{2}(\mathbb{Z}^{n}).$

In particular, Hypothesis 3.1 of [53] is satisfied with a constant equal to 1.

It follows from [53, Lemma 3.3] that if $x\in L_{2}(\mathbb{T}^{n}_{\theta})$
and $g\in\ell_{\infty}(\mathbb{Z}^{n})$, then

$\mu(\lambda(x)g(-i\nabla))^{2}\prec\prec 130\mu(x\otimes g)^{2}.$ (4.13)

Thus, if $x\in L_{\infty}(\mathbb{T}^{n}_{\theta})$ and
$g\in\ell_{p}(\mathbb{Z}^{n})$ are positive, then by using (4.9) we get

$\big{\|}|\lambda(x)g(-i\nabla)|^{2}\big{\|}_{L_{\frac{p}{2}}}\leq
130\big{\|}|x|^{2}\otimes|g|^{2}\big{\|}_{L_{\frac{p}{2}}\big{(}L_{\infty}(\mathbb{T}^{n}_{\theta})\overline{\otimes}\ell_{\infty}(\mathbb{Z}^{n})\big{)}}\leq
130\|x\|_{L_{p}}^{2}\|g\|_{\ell_{p}}^{2}.$

As
$\||\lambda(x)g(-i\nabla)|^{2}\|_{L_{\frac{p}{2}}}=\|\lambda(x)g(-i\nabla)\|_{L_{p}}^{2}$
we get

$\big{\|}\lambda(x)g(-i\nabla)\big{\|}_{L_{p}}\leq\sqrt{130}\|x\|_{L_{p}}\|g\|_{\ell_{p}}.$

Let $N\geq 1,$ and let $\theta^{\oplus N}$ be the $(Nn)\times(Nn)$-matrix
given by the direct sum of $N$ copies of $\theta$. Note that

$\|x\|_{L_{p}(\mathbb{T}^{n}_{\theta})}=\|x^{\otimes
N}\|_{L_{p}\big{(}\mathbb{T}^{Nn}_{\theta^{\oplus
N}}\big{)}}^{\frac{1}{N}},\qquad\|g\|_{\ell_{p}(\mathbb{Z}^{n})}=\|g^{\oplus
N}\|_{\ell_{p}(\mathbb{Z}^{Nn})}^{\frac{1}{N}}.$ (4.14)

Moreover, as $L_{2}(\mathbb{T}^{Nn}_{\theta^{\oplus
N}})=L_{2}(\mathbb{T}^{n}_{\theta})^{\otimes N}$, where $\otimes$ is the
Hilbert space tensor product, we also have

$\|T\|_{\mathscr{L}_{p}(\mathbb{T}^{n}_{\theta})}=\|T\|_{\mathscr{L}_{p}\big{(}L_{2}\big{(}\mathbb{T}^{Nn}_{\theta^{\oplus
N}}\big{)}\big{)}}^{\frac{1}{N}}\qquad\forall
T\in\mathscr{L}_{p}(L_{2}(\mathbb{T}^{n}_{\theta})).$ (4.15)

The inequality (4.2) holds on $\mathbb{T}^{n}_{\theta^{\oplus N}}$. Thus, if
$x\in L_{p}(\mathbb{T}^{n}_{\theta})$ and $g\in\ell_{p}(\mathbb{Z}^{n})$, then

$\displaystyle\big{\|}\ (\lambda(x)g(-i\nabla))^{\otimes
N}\big{\|}_{\mathscr{L}_{p}\big{(}L_{2}\big{(}\mathbb{T}^{nN}_{\theta}\big{)}\big{)}}$
$\displaystyle=\big{\|}\lambda(x^{\otimes N})g^{\otimes
N}(-i\nabla)\big{\|}_{\mathcal{L}_{p}(L_{2}(\mathbb{T}^{nN}_{\theta}))}$
$\displaystyle\leq\sqrt{130}\big{\|}x^{\otimes
N}\big{\|}_{L_{p}\big{(}\mathbb{T}^{Nn}_{\theta^{\oplus
N}}\big{)}}\big{\|}g^{\oplus N}\big{\|}_{\ell_{p}(\mathbb{Z}^{Nn})}.$

Combining this with (4.14)–(4.15) gives

$\big{\|}\lambda(x)g(-i\nabla)\big{\|}_{\mathscr{L}_{p}}\leq
130^{\frac{1}{2N}}\|x\|_{L_{p}}\|g\|_{\ell_{p}}.$

Letting $N\to\infty$ then yields

$\big{\|}\lambda(x)g(-i\nabla)\big{\|}_{\mathscr{L}_{p}}\leq\|x\|_{L_{p}}\|g\|_{\ell_{p}}.$

This is the inequality (4.11).

To estimate the weak Schatten norms we can do a similar argument using (4.10)
in place of (4.9). Let $p>2$. By using (4.10) and (4.13) and arguing as above
shows that if $x\in L_{\infty}(\mathbb{T}^{n}_{\theta})$ and
$g\in\ell_{p,\infty}(\mathbb{Z}^{n})$ are positive, then

$\big{\|}\lambda(x)g(-i\nabla)\big{\|}_{\mathscr{L}_{p,\infty}}\leq\left(\frac{130p}{p-2}\right)^{\frac{1}{2}}\|x\|_{L_{p}}\|g\|_{\ell_{p,\infty}}.$
(4.16)

As above the inequality continues to hold if $x$ and $g$ are not positive.
Thanks to the density of $L_{\infty}(\mathbb{T}^{n}_{\theta})$ in
$L_{p}(\mathbb{T}^{n}_{\theta})$, it follows that, if $x\in
L_{p}(\mathbb{T}^{n}_{\theta})$ and $g\in\ell_{p,\infty}(\mathbb{Z}^{n})$,
then $\lambda(x)g(-i\nabla)\in\mathscr{L}_{p,\infty}$ and the inequality
(4.16) holds. This gives the 2nd part of Theorem 4.4. The proof is complete. ∎

######  Remark 4.5.

The estimates (4.11)–(4.12) are established in [66, Theorem 3.1] up to
unspecified constants depending on $p$. This is a consequence of the results
of [53]. More generally, the results of [53] allows us to get Cwikel-type
estimates for any interpolation space between $\mathscr{L}_{2}$ and
$\mathscr{K}$ (see [66]).

######  Remark 4.6.

The analogue of (4.11) on $\mathbb{R}^{n}$ is known as the Kato-Seiler-Simon
inequality [87].

Combining Theorem 4.4 with Proposition 2.2 we immediately obtain the following
statement.

######  Corollary 4.7.

Let $s\in(2,\infty)$ and set $s=n(1/2-p^{-1})$. The following holds.

  1. (1)

If $x\in W_{2}^{s}(\mathbb{T}^{n}_{\theta})$ and
$g\in\ell_{p}(\mathbb{Z}^{n})$, then
$\lambda(x)g(-i\nabla)\in\mathscr{L}_{p}$, and we have

$\|\lambda(x)g(-i\nabla)\|_{\mathscr{L}_{p}}\leq
c_{p}\|x\|_{W_{2}^{s}}\|g\|_{\ell_{p}}.$

  2. (2)

If $x\in W_{2}^{s}(\mathbb{T}^{n}_{\theta})$ and
$g\in\ell_{p,\infty}(\mathbb{Z}^{n})$, then
$\lambda(x)g(-i\nabla)\in\mathscr{L}_{p,\infty}$, and we have

$\|\lambda(x)g(-i\nabla)\|_{\mathscr{L}_{p,\infty}}\leq
c_{p}\|x\|_{W_{2}^{s}}\|g\|_{\ell_{p,\infty}}.$

Case II is dealt with by the following result, the proof of which is postponed
to next section.

######  Theorem 4.8.

Suppose that $0<p<2$. The following holds.

  1. (1)

Let $x\in L_{2}(\mathbb{T}^{n}_{\theta})$ and $g\in\ell_{p}(\mathbb{Z}^{n})$.
Then $\lambda(x)g(-i\nabla)\in\mathscr{L}_{p}$, and we have

$\|\lambda(x)g(-i\nabla)\|_{\mathscr{L}_{p}}\leq\|x\|_{L_{2}}\|g\|_{\ell_{p}}.$

  2. (2)

There is $c_{-}(p)>0$ such that, if $x\in L_{2}(\mathbb{T}^{n}_{\theta})$ and
$g\in\ell_{p,\infty}(\mathbb{Z}^{n})$, then
$\lambda(x)g(-i\nabla)\in\mathscr{L}_{p,\infty}$, and we have

$\|\lambda(x)g(-i\nabla)\|_{\mathscr{L}_{p,\infty}}\leq
c_{-}(p)\|x\|_{L_{2}}\|g\|_{\ell_{p,\infty}}.$ (4.17)

Moreover, the best constant $c_{-}(p)$ satisfies

$c_{-}(p)\leq 2^{\frac{1}{p}}(2-p)^{-\frac{1}{p}}.$

Finally, for Case III we shall prove the following result.

######  Theorem 4.9.

For any $x\in L_{p}(\mathbb{T}_{\theta})$ with $p>2$ and
$g\in\ell_{2,\infty}(\mathbb{Z}^{n})$, the operator $\lambda(x)g(-i\nabla)$ is
in $\mathscr{L}_{2,\infty}$, and we have

$\|\lambda(x)g(-i\nabla)\|_{\mathscr{L}_{2,\infty}}\leq
c_{p}\|x\|_{L_{p}}\|g\|_{\ell_{2,\infty}}.$ (4.18)

Moreover, the best constant in the inequality is less than or equal to

$\inf_{1<q<2}\left\\{\gamma_{1}(p,q)\gamma_{2}(p,q)c_{+}(p)^{\frac{p(2-q)}{2(p-q)}}c_{-}(q)^{\frac{q(p-2)}{2(p-q)}}\right\\},$
(4.19)

where $c_{+}(p)$ and $c_{+}(q)$ are the best constants in the inequalities
(4.12) and (4.17), respectively, and the constants $\gamma_{1}(p,q)$ and
$\gamma_{2}(p,q)$ arise from real interpolation (see Eqs. (4.22)–(4.23)
below).

###### Proof.

The proof uses real interpolation theory. Once again, for background on
interpolation theory we refer to the survey of Connes [19, Appendix IV.B] and
the references therein. The main reference for real interpolation theory there
is the article of Lions-Peetre [62] (see also [3, 45]).

Let $x\in L_{p}(\mathbb{T}_{\theta}^{n})$, $p>2$. By Theorem 4.4, for every
$g\in\ell_{p,\infty}(\mathbb{Z}^{n})$, the operator $\lambda(x)g(-i\nabla)$ is
in $\mathscr{L}_{p,\infty}$. Thus, we have a linear operator
$\Phi_{x}:\ell_{p,\infty}(\mathbb{Z}^{n})\rightarrow\mathscr{L}_{p,\infty}$
given by

$\Phi_{x}(g)=\lambda(x)g(-i\nabla),\qquad g\in\ell_{p,\infty}.$

Furthermore, the Cwikel-type estimate (4.12) gives

$\|\Phi_{x}(g)\|_{\mathscr{L}_{p,\infty}}\leq
c_{+}(p)\|x\|_{L_{p}}\|g\|_{\ell_{p,\infty}}\qquad\forall
g\in\ell_{p,\infty}.$ (4.20)

In addition, as $L_{p}(\mathbb{T}^{n}_{\theta})\subset
L_{2}(\mathbb{T}^{n}_{\theta})$, it follows from Theorem 4.8 that, given any
$q\in(1,2)$, if $g\in\ell_{q,\infty}(\mathbb{Z}^{n})$, then
$\Phi_{x}(g)\in\mathscr{L}_{q,\infty}$, and we have

$\big{\|}\Phi_{x}(g)\big{\|}_{\mathscr{L}_{q,\infty}}\leq
c_{-}(q)\|x\|_{L_{2}}\|g\|_{\ell_{q,\infty}}\leq
c_{-}(q)\|x\|_{L_{p}}\|g\|_{\ell_{q,\infty}}.$ (4.21)

As $2\in(q,p)$, the spaces $\ell_{2,\infty}(\mathbb{Z}^{n})$ and
$\mathscr{L}_{2,\infty}$ are real interpolation spaces for the pairs of Banach
spaces $(\ell_{q,\infty}(\mathbb{Z}^{n}),\ell_{p,\infty}(\mathbb{Z}^{n}))$ and
$(\mathscr{L}_{q,\infty},\mathscr{L}_{p,\infty})$ with the exact same
exponents. Namely, in the notation of [19, Appendix IV.B], and with the same
norm equivalences (see e.g. [90, Theorem 2.10]), we have
$\ell_{2,\infty}(\mathbb{Z}^{n})=[\ell_{q,\infty}(\mathbb{Z}^{n}),\ell_{p,\infty}(\mathbb{Z}^{n})]_{\theta,\infty}$
and
$\mathscr{L}_{2,\infty}=[\mathscr{L}_{q,\infty},\mathscr{L}_{p,\infty}]_{\theta,\infty}$,
with $\theta\in(0,1)$ such that $1/2=(1-\theta)q^{-1}+\theta p^{-1}$ (see [19,
§IV.2.$\alpha$ & Appendix IV.B] and [24, Section 4]). That is, there are
constants $\gamma_{1}(p,q)>0$ and $\gamma_{2}(p,q)>0$ such that

$\displaystyle\gamma_{2}(p,q)^{-1}\|g\|_{\ell_{2,\infty}}\leq\|g\|_{[\ell_{q,\infty},\ell_{p,\infty}]_{\theta,\infty}}\leq\gamma_{1}(p,q)\|g\|_{\ell_{2,\infty}}\qquad\forall
g\in\ell_{2,\infty}(\mathbb{Z}^{n}),$ (4.22)
$\displaystyle\gamma_{2}(p,q)^{-1}\|T\|_{\mathscr{L}_{2,\infty}}\leq\|T\|_{[\mathscr{L}_{q,\infty},\mathscr{L}_{p,\infty}]_{\theta,\infty}}\leq\gamma_{1}(p,q)\|T\|_{\mathscr{L}_{2,\infty}}\qquad\forall
T\in\mathscr{L}_{2,\infty}.$ (4.23)

We assume that $\gamma_{1}(p,q)$ and $\gamma_{2}(p,q)$ are the best constants
in the above inequalities.

By combining the estimates (4.20)–(4.21) with real interpolation theory (see
[19, Theorem IV.B.2]) and by using the inequalities (4.22)–(4.23) shows that
$\Phi_{x}$ induces a continuous linear map from
$\ell_{2,\infty}(\mathbb{Z}^{n})$ to $\mathscr{L}_{2,\infty}$, and, for all
$g\in\ell_{2,\infty}(\mathbb{Z}^{n})$, we have

$\displaystyle\big{\|}\lambda(x)g(-i\nabla)\big{\|}_{\mathscr{L}_{2,\infty}}=\big{\|}\Phi_{x}(g)\big{\|}_{\mathscr{L}_{2,\infty}}$
$\displaystyle\leq\gamma_{1}(p,q)\gamma_{2}(p,q)(c_{-}(q)\|x\|_{L_{p}})^{1-\theta}(c_{+}(p)\|x\|_{L_{p}})^{\theta}\|g\|_{\ell_{2,\infty}}$
$\displaystyle\leq\gamma_{1}(p,q)\gamma_{2}(p,q)c_{+}(p)^{\theta}c_{-}(q)^{1-\theta}\|x\|_{L_{p}}\|g\|_{\ell_{2,\infty}}.$

This gives the inequality (4.18).

Here $\theta$ is such that $1/2=(1-\theta)q^{-1}+\theta p^{-1}$. That is,

$\theta=\left(\frac{1}{2}-\frac{1}{q}\right)\left(\frac{1}{p}-\frac{1}{q}\right)^{-1}=\frac{p(2-q)}{2(p-q)},\qquad
1-\theta=\frac{q(p-2)}{2(p-q)}.$

Thus, if we denote by $c(p)$ the best constant in (4.18), then

$c(p)\leq\gamma_{1}(p,q)\gamma_{2}(p,q)c_{+}(p)^{\frac{p(2-q)}{2(p-q)}}c_{-}(q)^{\frac{q(p-2)}{2(p-q)}}\qquad\forall
q\in(1,2).$

Taking the infimum over $q\in(1,2)$ shows that (4.19) is an upper bound for
$c(p)$. The proof is complete. ∎

######  Remark 4.10.

Theorem 4.9 does not hold for $x\in L_{2}(\mathbb{T}^{n}_{\theta})$ (see
Remark 6.7).

######  Corollary 4.11.

For any $x\in W_{2}^{s}(\mathbb{T}_{\theta})$ with $s>0$ and
$g\in\ell_{2,\infty}(\mathbb{Z}^{n})$, the operator $\lambda(x)g(-i\nabla)$ is
in $\mathscr{L}_{2,\infty}$, and we have

$\|\lambda(x)g(-i\nabla)\|_{\mathscr{L}_{2,\infty}}\leq
c_{s}\|x\|_{W_{2}^{s}}\|g\|_{\ell_{2,\infty}}.$

###### Proof.

Let $x\in W_{2}^{s}(\mathbb{T}_{\theta})$ with $s>0$ and
$g\in\ell_{2,\infty}(\mathbb{Z}^{n})$. Let $p>2$ be such that $s\geq
n(1/2-p^{-1})$. By Proposition 2.2 this ensures us that
$W_{2}^{s}(\mathbb{T}^{n}_{\theta})$ embeds continuously into
$L_{p}(\mathbb{T}^{n}_{\theta})$. Combining this with Theorem 4.9 then gives
the result. ∎

##  5\. Proof of Theorem 4.8

In this section, we prove Theorem 4.8. The proof attempts to follow the
approach to the proof of Cwikel-type estimates for noncommutative Euclidean
spaces in [53]. However, some significant simplification occurs thanks to
Proposition 5.3 below.

######  Lemma 5.1 ([53]).

Suppose that $0<p<2$, and let $S,T\in\mathscr{L}_{2}$ be such that
$|S|^{2}\prec|T|^{2}$.

  1. (i)

If $S\in\mathscr{L}_{p}$, then $T\in\mathscr{L}_{p}$, and
$\|T\|_{\mathscr{L}_{p}}\leq\|S\|_{\mathscr{L}_{p}}$.

  2. (ii)

If $S\in\mathscr{L}_{p,\infty}$, then $T\in\mathscr{L}_{p,\infty}$, and
$\|T\|_{\mathscr{L}_{p,\infty}}\leq
2^{\frac{1}{p}}(2-p)^{-\frac{1}{p}}\|S\|_{\mathscr{L}_{p,\infty}}$.

###### Proof.

The first part is the contents of the first part of [53, Proposition 2.7]. The
2nd part of [53, Proposition 2.7] gives the 2nd part of the lemma with an
upper bound $\|T\|_{\mathscr{L}_{p,\infty}}\leq
c_{p}\|S\|_{\mathscr{L}_{p,\infty}}$ for an unspecified constant $c_{p}$. We
can get the required constant by a slight modification of the proof in [53].

Suppose that $S\in\mathscr{L}_{p,\infty}$. For $t\geq 0$, set
$f(t)=\mu_{[t]}(S)$ and $g(t)=\mu_{[t]}(T)$, where $[\cdot]$ is the floor
function. Note that $\sup_{t\geq 0}t^{1/p}f(t)=\sup_{j\geq
0}(j+1)^{\frac{1}{p}}\mu_{j}(S)=\|S\|_{\mathscr{L}_{p,\infty}}$. Likewise,
$\|T\|_{\mathscr{L}_{p,\infty}}=\sup_{t\geq 0}t^{1/p}g(t)$. Moreover, as
pointed out in the proof of [53, Proposition 2.7], the fact that $T^{2}\prec
S^{2}$ implies that

$\int_{t}^{2}g(s)^{2}ds\leq\int_{t}^{2}f(s)^{2}ds,\qquad t\geq 0.$ (5.1)

We also observe that in Eq. (14) of [53] the constant is equal to
$p(2-p)^{-1}$. Thus, we have

$\int_{t}^{\infty}f(s)^{2}ds\leq\bigg{(}\frac{p}{2-p}\bigg{)}\|S\|_{\mathscr{L}_{p,\infty}}^{2}t^{1-\frac{2}{p}},\qquad
t>0.$ (5.2)

Given any $a>1$ the fact that $g(t)$ is non-increasing ensures that

$t(a-1)g(at)^{2}\leq\int_{t}^{at}g(s)^{2}ds\leq\int_{t}^{\infty}g(s)^{2}ds,\qquad
t>0.$

Combining this with (5.1)–(5.2) gives

$tg(at)^{2}\leq(a-1)^{-1}\bigg{(}\frac{p}{2-p}\bigg{)}\|S\|_{\mathscr{L}_{p,\infty}}^{2}t^{1-\frac{2}{p}},\qquad
t>0.$

Thus,

$t^{\frac{1}{p}}g(at)\leq(a-1)^{-\frac{1}{2}}\bigg{(}\frac{p}{2-p}\bigg{)}^{\frac{1}{2}}\|S\|_{\mathscr{L}_{p,\infty}},\qquad
t\geq 0$

It follows that

$\sup_{t\geq 0}t^{\frac{1}{p}}g(t)=a^{\frac{1}{p}}\sup_{t\geq
0}t^{\frac{1}{p}}g(t)\leq
a^{\frac{1}{p}}(a-1)^{-\frac{1}{2}}\bigg{(}\frac{p}{2-p}\bigg{)}^{\frac{1}{2}}\|S\|_{\mathscr{L}_{p,\infty}}.$

This shows that $T\in\mathscr{L}_{p,\infty}$, and we have

$\|T\|_{\mathscr{L}_{p,\infty}}\leq
a^{\frac{1}{p}}(a-1)^{-\frac{1}{2}}\bigg{(}\frac{p}{2-p}\bigg{)}^{\frac{1}{2}}\|S\|_{\mathscr{L}_{p,\infty}}\qquad\forall
a>1.$

The function $a\rightarrow a^{\frac{1}{p}}(a-1)^{-\frac{1}{2}}$ reaches its
minimum on $(1,\infty)$ at $a=2(2-p)^{-1}$. The minimum value is
$2^{1/p}p^{-1/2}(2-p)^{1/2-1/p}.$ This yields the inequality
$\|T\|_{\mathscr{L}_{p,\infty}}\leq
2^{1/p}(2-p)^{-1/p}\|S\|_{\mathscr{L}_{p,\infty}}$. The proof is complete. ∎

Recall that a sequence of bounded operators $(T_{j})_{j\geq 0}$ on Hilbert
space is called _right-disjoint_ (resp., _left-disjoint_) when
$T_{j}T_{l}^{*}=0$ (resp., $T_{j}^{*}T_{l}=0$) when $j\neq l$. When all the
operators $T_{j}$ are selfadjoint notions of left-disjointness and right-
disjointness are both equivalent to the condition $T_{j}T_{l}=0$ when $j\neq
0$. In that case we simply that say that we have a _disjoint_ sequence.

######  Lemma 5.2 ([53, Lemma 2.9]).

Let $(T_{j})_{j\geq 0}$ be a sequence in $\mathscr{L}_{2}$ such that
$\sum\|T_{j}\|_{\mathscr{L}_{2}}^{2}<\infty$. Assume further that the sequence
$(T_{j})_{j\geq 0}$ is left-disjoint or right-disjoint. Then, we have

$\mu^{2}\bigg{(}\bigoplus_{j\geq
0}T_{j}\bigg{)}\prec\mu^{2}\bigg{(}\sum_{j\geq 0}T_{j}\bigg{)}.$

The operator-theoretic underpinning of Theorem 4.8 is the following
majorization estimate.

######  Proposition 5.3.

Let $x\in L_{2}(\mathbb{T}^{n}_{\theta})$ and $g\in\ell_{2}(\mathbb{Z}^{n})$.
Then, we have

$\|x\|_{L_{2}}^{2}\mu^{2}\big{(}g(-i\nabla)\big{)}\prec\mu^{2}\big{(}\lambda(x)g(-i\nabla)\big{)}.$
(5.3)

###### Proof.

The fact that $g\in\ell_{2}(\mathbb{Z}^{n})$ ensures us that
$g(-i\nabla)\in\mathscr{L}_{2}$. Furthermore, as $x\in
L_{2}(\mathbb{T}^{n}_{\theta})$ we know by Theorem 4.4 that
$\lambda(x)g(-i\nabla)$ is in $\mathscr{L}_{2}$ as well.

Bearing this in mind, given any $k\in\mathbb{Z}^{n}$, we denote by $\Pi_{k}$
the orthogonal projection onto $\mathbb{C}U^{k}$. Thus, for all
$u=\sum\hat{u}_{k}U^{k}$ in $L_{2}(\mathbb{T}^{n}_{\theta})$, we have

$\Pi_{k}u=\left\langle U^{k}|u\right\rangle U^{k}=\hat{u}_{k}U^{k}.$

Note that $(\Pi_{k})_{k\in Z^{n}}$ is a disjoint family of rank 1 projections.

For $k\in\mathbb{Z}^{n}$, we also set

$T_{k}:=\lambda(x)g(-i\nabla)\Pi_{k}.$

Each operator $T_{k}$ has rank $\leq 1$, and so this is a Hilbert-Schmidt
operator. Moreover, if $k\neq l$, then
$T_{k}T_{l}^{*}=(\lambda(x)g(-i\nabla))\Pi_{k}\Pi_{l}(\lambda(x)g(-i\nabla))^{*}=0$.
Thus, the sequence $(T_{k})_{k\in\mathbb{Z}^{n}}$ is right-disjoint. We also
note that, as $U^{k}$ is in the domain of $\lambda(x)$, for all
$u=\sum\hat{u}_{k}U^{k}$ in $L_{2}(\mathbb{T}^{n}_{\theta})$, we have

$T_{k}u=\lambda(x)g(-i\nabla)\Pi_{k}u=\hat{u}_{k}\lambda(x)g(-\nabla)U^{k}=\hat{u}_{k}g(k)\lambda(x)U^{k}.$

######  Claim.

For all $k\in\mathbb{Z}^{n}$, we have

$|T_{k}|=\|x\|_{L_{2}}|g(k)|\Pi_{k}.$

###### Proof of the claim.

Let $k\in\mathbb{Z}^{n}$, and set $A=\lambda(x)g(-i\nabla)$. Note that $A$ is
a bounded operator by Proposition 3.12. We have

$T_{k}^{*}T_{k}=(A\Pi_{k})^{*}(A\Pi_{k})=\Pi_{k}A^{*}A\Pi_{k}.$

We observe that

$\Pi_{k}A^{*}AU^{k}=\left\langle U^{k}|A^{*}AU^{k}\right\rangle
U^{k}=\left\langle AU^{k}|AU^{k}\right\rangle U^{k}.$

As $AU^{k}=\lambda(x)g(-\nabla)U^{k}=g(k)\lambda(x)U^{k}$, we get

$\left\langle
AU^{k}|AU^{k}\right\rangle=|g(k)|^{2}\left\langle\lambda(x)U^{k}|\lambda(x)U^{k}\right\rangle=|g(k)|^{2}\tau\big{[}xU^{k}(xU^{k})^{*}\big{]}=|g(k)|^{2}\tau[xx^{*}]=|g(k)|^{2}\|x\|_{L_{2}}^{2}.$

Thus,

$\Pi_{k}A^{*}AU^{k}=\left\langle AU^{k}|AU^{k}\right\rangle
U^{k}=|g(k)|^{2}\|x\|_{L_{2}}^{2}U^{k}.$

It then follows that, for all $u=\sum\hat{u}_{k}U^{k}$, we have

$T_{k}^{*}T_{k}=\Pi_{k}A^{*}A\Pi_{k}u=\hat{u}_{k}\Pi_{k}A^{*}AU^{k}=\hat{u}_{k}|g(k)|^{2}\|x\|_{L_{2}}^{2}U^{k}=|g(k)|^{2}\|x\|_{L_{2}}^{2}\Pi_{k}u.$

That is, $T_{k}^{*}T_{k}=|g(k)|^{2}\|x\|_{L_{2}}^{2}\Pi_{k}$. Thus,

$|T_{k}|=\sqrt{T_{k}^{*}T_{k}}=|g(k)|\|x\|_{L_{2}}\sqrt{\Pi_{k}}=|g(k)|\|x\|_{L_{2}}\Pi_{k}.$

This proves the claim. ∎

Combining the claim above with the fact that $g\in\ell_{2}(\mathbb{Z}^{n})$
gives

$\sum_{k\in\mathbb{Z}^{n}}\|T_{k}\|^{2}_{\mathscr{L}_{2}}=\sum_{k\in\mathbb{Z}^{n}}\||T_{k}|\|^{2}_{\mathscr{L}_{2}}=\sum_{k\in\mathbb{Z}^{n}}|g(k)|^{2}\|x\|_{L_{2}}^{2}\|\Pi_{k}\|_{\mathscr{L}_{2}}=\|x\|_{L_{2}}^{2}\|g\|^{2}_{\ell_{2}}<\infty.$

All this allows us to apply Lemma 5.2 to get

$\mu^{2}\bigg{(}\bigoplus_{k\in\mathbb{Z}^{n}}T_{k}\bigg{)}\prec\mu^{2}\bigg{(}\sum_{k\in\mathbb{Z}^{n}}T_{k}\bigg{)}=\mu^{2}\big{(}\lambda(x)g(-i\nabla)\big{)}.$
(5.4)

Set $S=\bigoplus_{k\in\mathbb{Z}^{n}}T_{k}$. The claim above implies that
$|S|=\bigoplus_{k\in\mathbb{Z}^{n}}|T_{k}|=\bigoplus_{k\in\mathbb{Z}^{n}}\|x\|_{L_{2}}|g(k)|\Pi_{k}.$
Therefore, we have

$\mu(S)=\mu\bigg{(}\bigoplus_{k\in\mathbb{Z}^{n}}\|x\|_{L_{2}}|g(k)|\Pi_{k}\bigg{)}=\|x\|_{L_{2}}\mu\big{(}|g(-i\nabla)|\big{)}=\|x\|_{L_{2}}\mu\big{(}g(-i\nabla)\big{)}.$

Combining this with (5.4) then gives the majorization
$\|x\|_{L_{2}}^{2}\mu^{2}(g(-i\nabla))\prec\mu^{2}(\lambda(x)g(-i\nabla))$.
The proof is complete. ∎

We are now in a position to prove Theorem 4.8.

###### Proof of Theorem 4.8.

Let $x\in L_{2}(\mathbb{T}^{n}_{\theta})$ and $g\in\ell_{p,\infty}$, $p<2$.
Recall that $g(-i\nabla)\in\mathscr{L}_{p,\infty}$ and
$\|g(-i\nabla)\|_{\mathscr{L}_{p,\infty}}=\|g\|_{\ell_{p,\infty}}$. Note also
that $g\in\ell_{2}(\mathbb{Z}^{n})$, since $p<2$. Thus, combining the
majorization (5.3) with Lemma 5.1 shows that
$\lambda(x)g(-i\nabla)\in\mathscr{L}_{p,\infty}$, and we have

$\big{\|}\lambda(x)g(-i\nabla)\big{\|}_{\mathscr{L}_{p,\infty}}\leq
2^{\frac{1}{p}}(2-p)^{-\frac{1}{p}}\|x\|_{L_{2}}\|g(-i\nabla)\|_{\mathscr{L}_{p,\infty}}\leq
2^{\frac{1}{p}}(2-p)^{-\frac{1}{p}}\|x\|_{L_{2}}\|g\|_{\ell_{p,\infty}}.$

If $g\in\ell_{p}(\mathbb{Z}^{n})$, then $g(-i\nabla)\in\mathscr{L}_{p}$ and
$\|g(-i\nabla)\|_{\mathscr{L}_{p}}=\|g\|_{\ell_{p}}$. Therefore, in the same
way as above, we deduce that $\lambda(x)g(-i\nabla)\in\mathscr{L}_{p}$, and we
have

$\big{\|}\lambda(x)g(-i\nabla)\big{\|}_{\mathscr{L}_{p}}\leq\|x\|_{L_{2}}\|g(-i\nabla)\|_{\mathscr{L}_{p}}\leq\|x\|_{L_{2}}\|g\|_{\ell_{p}}.$

This completes the proof of Theorem 4.8. ∎

##  6\. Specific Cwikel Estimates

It is worth specializing the Cwikel estimates of Section 4 to operators of the
forms $\lambda(x)\Delta^{-n/2p}$ and $\Delta^{-n/4p}\lambda(x)\Delta^{-n/4p}$,
since these estimates are used in the derivation of the CLR inequalities on NC
tori in Section 8. In the terminology of [98] these Cwikel-type estimates are
called _specific_ Cwikel estimates.

In what follows we set

$\nu_{0}(n):=\sup_{\lambda\geq
1}\lambda^{-\frac{n}{2}}\\#\left\\{k\in\mathbb{Z}^{n}\setminus 0;\
|k|\leq\sqrt{\lambda}\right\\}.$

######  Theorem 6.1.

The following holds.

  1. (1)

If $p>2$ and $x\in L_{p}(\mathbb{T}^{n}_{\theta})$, then
$\lambda(x)\Delta^{-n/2p}\in\mathscr{L}_{p,\infty}$, and we have

$\big{\|}\lambda(x)\Delta^{-\frac{n}{2p}}\big{\|}_{\mathscr{L}_{p,\infty}}\leq
c_{+}(p)\nu_{o}(n)^{\frac{1}{p}}\|x\|_{L_{p}}.$ (6.1)

  2. (2)

If $0<p<2$ and $x\in L_{2}(\mathbb{T}^{n}_{\theta})$, then
$\lambda(x)\Delta^{-n/2p}\in\mathscr{L}_{p,\infty}$, and we have

$\big{\|}\lambda(x)\Delta^{-\frac{n}{2p}}\big{\|}_{\mathscr{L}_{p,\infty}}\leq
c_{-}(p)\nu_{o}(n)^{\frac{1}{p}}\|x\|_{L_{2}}.$

  3. (3)

If $x\in L_{p}(\mathbb{T}^{n}_{\theta})$, $p>2$, then
$\lambda(x)\Delta^{-n/4}\in\mathscr{L}_{2,\infty}$, and we have

$\big{\|}\lambda(x)\Delta^{-\frac{n}{4}}\big{\|}_{\mathscr{L}_{2,\infty}}\leq
c_{2}(p)\nu_{o}(n)^{\frac{1}{2}}\|x\|_{L_{p}}\|x\|_{L_{p}},$

where we have set

$c_{2}(p)=2^{-\frac{1}{p}}p^{-\frac{1}{2}}(p-2)^{\frac{1}{p}-\frac{1}{2}}c_{+}(p).$
(6.2)

Here $c_{+}(p)$ and $c_{-}(p)$ are the best constants in the
$\mathscr{L}_{p,\infty}$-Cwikel estimates (4.12) and (4.17), respectively.

###### Proof.

Suppose that $p>2$, and let $x\in L_{p}(\mathbb{T}^{n}_{\theta})$. Observe
that $\Delta^{-n/2p}=\lambda(x)g_{p}(-i\nabla)$, where $g_{p}(k)=|k|^{-n/p}$
for $k\neq 0$ and $g_{p}(0)=0$. In particular,
$g_{p}\in\ell_{p,\infty}(\mathbb{Z}^{n})$. Thus, by specializing Theorem 4.4
to $g=g_{p}$ we see that $\lambda(x)\Delta^{-n/2p}\in\mathscr{L}_{p,\infty}$,
and we have

$\big{\|}\lambda(x)\Delta^{-\frac{n}{2p}}\big{\|}_{\mathscr{L}_{p,\infty}}\leq
c_{+}(p)\|g_{p}\|_{\ell_{p,\infty}}\|x\|_{L_{p}}.$ (6.3)

Let us arrange the family $\\{|k|^{2};\ k\in\mathbb{Z}^{n}\setminus 0\\}$ as a
non-decreasing sequence
$1=\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{j}\leq\cdots$. We have
$\mu_{j}(g_{p})=\lambda_{j+1}^{-n/2p}$ for all $j\geq 0$, and hence

$\|g_{p}\|_{\ell_{p,\infty}}=\sup_{j\geq
0}(j+1)^{\frac{1}{p}}\mu_{j}(g_{p})=\sup_{j\geq
0}(j+1)^{\frac{1}{p}}\lambda_{j+1}^{-\frac{n}{2p}}=\big{(}\sup_{j\geq
1}j\lambda_{j}^{-n/2}\big{)}^{\frac{1}{p}}.$ (6.4)

Set $N_{0}(\lambda)=\\#\\{j\geq 1;\lambda_{j}\leq\lambda\\}$, $\lambda\geq 1$.
In view of the definition of $\nu_{0}(n)$ we have

$N_{0}(\lambda)=\\#\left\\{k\in\mathbb{Z}^{n}\setminus 0;\
|k|^{2}\leq\lambda\right\\}=\\#\left\\{k\in\mathbb{Z}^{n}\setminus 0;\
|k|\leq\sqrt{\lambda}\right\\}\leq\nu_{0}(n)\lambda^{\frac{n}{2}}.$ (6.5)

It follows that $j\leq N_{0}(\lambda_{j})\leq\nu_{0}(n)\lambda_{j}^{-n/2}$ for
all $j\geq 1$. Together with (6.4) this implies that
$\|g_{p}\|_{\ell_{p,\infty}}\leq\nu_{0}(n)^{1/p}$. Combining this with (6.3)
then gives (6.1). This gives the 1st part.

If $p<2$, then we can similarly prove the 2nd part by using Theorem 4.8
instead of Theorem 4.4.

It remains to prove the 3rd part. Let $x\in L_{p}(\mathbb{T}^{n}_{\theta})$,
$p>2$, and let $q$ be such that $2^{-1}=p^{-1}+q^{-1}$. In particular,
$\lambda(x)\Delta^{-n/4}=\lambda(x)\Delta^{-n/2p}\cdot\Delta^{-n/2q}$. As
$p>2$ the operator $\lambda(x)\Delta^{-n/2p}$ is in $\mathscr{L}_{p,\infty}$
by the 1st part. Note also that
$\Delta^{-n/2q}=g_{q}(-\nabla)\in\mathscr{L}_{q,\infty}$. Therefore, by the
version of Hölder’s inequality provided by Proposition 4.1 the operator
$\lambda(x)\Delta^{-n/4}$ is in $\mathscr{L}_{2,\infty}$, and we have

$\big{\|}\lambda(x)\Delta^{-\frac{n}{4}}\big{\|}_{\mathscr{L}_{2,\infty}}\leq\gamma(p,q)\big{\|}\lambda(x)\Delta^{-\frac{n}{2p}}\big{\|}_{\mathscr{L}_{p,\infty}}\big{\|}\Delta^{-\frac{n}{2q}}\big{\|}_{\mathscr{L}_{q,\infty}},$

where
$\gamma(p,q)=p^{-\frac{1}{q}}q^{-\frac{1}{p}}(p+q)^{\frac{1}{p}+\frac{1}{q}}$.
Here
$\|\Delta^{-\frac{n}{2q}}\|_{\mathscr{L}_{q,\infty}}=\|g_{q}(-\nabla)\|_{\mathscr{L}_{q,\infty}}=\|g_{q}\|_{\ell_{q,\infty}}\leq\nu_{0}(n)^{1/q}$.
Thus, by using (6.1) we get

$\displaystyle\big{\|}\lambda(x)\Delta^{-\frac{n}{4}}\big{\|}_{\mathscr{L}_{2,\infty}}$
$\displaystyle\leq\gamma(p,q)\big{(}c_{+}(p)\nu_{0}(n)^{\frac{1}{p}}(n)\|x\|_{L_{p}}\big{)}\nu_{0}(n)^{1/q}$
$\displaystyle\leq c_{2}(p)\nu_{0}(n)^{\frac{1}{2}}\|x\|_{L_{p}},$

where we have set $c_{2}(p)=\gamma(p,q)c_{+}(p)$. In fact, as
$p^{-1}+q^{-1}=2^{-1}$, by Remark 4.3 we have
$\gamma(p,q)=2^{-1/p}p^{-1/2}(p-2)^{1/p-1/2}$, and so $c_{2}(p)$ is given by
(6.2). This proves the 3rd part. The proof is complete. ∎

######  Remark 6.2.

As (6.5) shows, $N_{0}(\lambda)$ is the number of non-zero integer points in
the ball centered at the origin of radius $\sqrt{\lambda}$. This ball is
contained in the cube $[-\sqrt{\lambda},\sqrt{\lambda}]^{n}$. There are at
most $2\sqrt{\lambda}+1$ integers in the interval
$[-\sqrt{\lambda},\sqrt{\lambda}]$. Thus,

$N_{0}(\lambda)\leq\big{(}2\sqrt{\lambda}+1\big{)}^{n}-1=\lambda^{\frac{n}{2}}\left(\big{(}2+1/\sqrt{\lambda}\big{)}^{n}-\lambda^{-\frac{n}{2}}\right)\leq\lambda^{\frac{n}{2}}(3^{n}-1).$

It then follows that

$\nu_{0}(n)=\sup_{\lambda\geq 1}\lambda^{-\frac{n}{2}}N_{0}(\lambda)\leq
3^{n}-1.$

For $n=2$ it can be shown that $N_{0}(\lambda)\leq 4\lambda$ for $\lambda\geq
1$ (see [41, §2]). Thus, in this case we get $\nu_{0}(2)\leq 4$.

Let us now turn to the Cwikel operators
$\Delta^{-n/4p}\lambda(x)\Delta^{-n/4p}$. Suppose that $p^{-1}+2q^{-1}=1$. As
mentioned in Section 2, if $x\in L_{p}(\mathbb{T}^{n}_{\theta})$, then
$\lambda(x)$ makes sense as a bounded operator from
$L_{q}(\mathbb{T}^{n}_{\theta})$ to its anti-linear dual
$L_{q}(\mathbb{T}^{n}_{\theta})^{*}$. Moreover, by Lemma 3.4, if $s>n/2p$, or
if $s=n/2p$ and $p>1$, then $\lambda(x)$ induces a bounded operator
$\lambda(x):W_{2}^{s}(\mathbb{T}^{n}_{\theta})\rightarrow
W_{2}^{-s}(\mathbb{T}^{n}_{\theta})$. This allows us to makes sense of the
composition $\Delta^{-s/2}\lambda(x)\Delta^{-s/2}$ as a bounded operator on
$L_{2}(\mathbb{T}^{n}_{\theta})$ (_cf_. Proposition 3.5).

Let $y\in L_{2p}(\mathbb{T}^{n}_{\theta})$. As
$(2p)^{-1}+q^{-1}=\frac{1}{2}(p^{-1}+2q^{-1})=\frac{1}{2}$, we know from
Proposition 3.1 that $\lambda(y)$ makes sense as a bounded operator from
$L_{q}(\mathbb{T}^{n}_{\theta})$ to $L_{2}(\mathbb{T}^{n}_{\theta})$. By
duality we get a bounded operator
$\lambda(y)^{*}:L_{2}(\mathbb{T}^{n}_{\theta})\rightarrow
L_{q}(\mathbb{T}^{n}_{\theta})^{*}$ such that

$\left\langle\lambda(y)^{*}u,v\right\rangle=\left\langle
u|\lambda(y)v\right\rangle,\qquad u\in L_{2}(\mathbb{T}^{n}_{\theta}),\ v\in
L_{q}(\mathbb{T}^{n}_{\theta}).$ (6.6)

In addition, if $s>n/2p$, or if $s=n/2p$ and $p>1$, then we have a continuous
embedding of $W_{2}^{s}(\mathbb{T}^{n}_{\theta})$ into
$L_{q}(\mathbb{T}^{n}_{\theta})$, and so the operator
$\lambda(x)\Delta^{-s/2}$ is bounded on $L_{2}(\mathbb{T}^{n}_{\theta})$.

######  Lemma 6.3.

Let $x\in L_{p}(\mathbb{T}^{n}_{\theta})$, $p\geq 1$, be of the form $x=yz$
with $y,z\in L_{2p}(\mathbb{T}^{n}_{\theta})$. In addition, assume that either
$s>n/2p$, or $s=n/2p$ and $p>1$. Then

$\lambda(x)=\lambda(y^{*})^{*}\lambda(z),\qquad\Delta^{-\frac{s}{2}}\lambda(x)\Delta^{-\frac{s}{2}}=\big{[}\lambda(y^{*})\Delta^{-\frac{s}{2}}\big{]}^{*}\lambda(z)\Delta^{-\frac{s}{2}}.$

###### Proof.

Let $u,v\in L_{q}(\mathbb{T}^{n}_{\theta})$, $p+2q^{-1}=1$. In view of (3.1)
we have

$\left\langle\lambda(x)u,v\right\rangle=\tau\big{[}v^{*}xu\big{]}=\tau\big{[}v^{*}yzu\big{]}.$

As $zu=\lambda(z)u\in L_{2}(\mathbb{T}^{n}_{\theta})$ and
$v^{*}y=(\lambda(y^{*})v)^{*}\in L_{2}(\mathbb{T}^{n}_{\theta})$, we get

$\left\langle\lambda(x)u,v\right\rangle=\tau\big{[}\left(\lambda(y^{*})v\right)^{*}\lambda(z)u\big{]}=\big{\langle}\lambda(z)u|\lambda(y^{*})v\big{\rangle}.$
(6.7)

Combining this with (6.6) gives

$\left\langle\lambda(x)u,v\right\rangle=\big{\langle}\lambda(y^{*})^{*}\lambda(z)u,v\big{\rangle}\qquad\forall
u,v\in L_{q}(\mathbb{T}^{n}_{\theta}).$

That is, $\lambda(x)=\lambda(y^{*})^{*}\lambda(z)$.

Suppose that, either $s>n/2p$, or $s=n/2p$ and $p>1$. Using (3.2) and (6.7)
shows that, for all $u,v\in L_{2}(\mathbb{T}^{n}_{\theta})$, we have

$\displaystyle\big{\langle}\Delta^{-\frac{s}{2}}\lambda(x)\Delta^{-\frac{s}{2}}u|v\big{\rangle}$
$\displaystyle=\big{\langle}\lambda(x)\Delta^{-\frac{s}{2}}u,\Delta^{-\frac{s}{2}}v\big{\rangle}$
$\displaystyle=\big{\langle}\lambda(z)\Delta^{-\frac{s}{2}}u|\lambda(y^{*})\Delta^{-\frac{s}{2}}v\big{\rangle}$
$\displaystyle=\big{\langle}\big{[}\lambda(y^{*})\Delta^{-\frac{s}{2}}\big{]}^{*}\lambda(z)\Delta^{-\frac{s}{2}}u|v\big{\rangle}.$

It then follows that
$\Delta^{-s/2}\lambda(x)\Delta^{-s/2}=[\lambda(y^{*})\Delta^{-s/2}]^{*}\lambda(z)\Delta^{-s/2}$.
The proof is complete. ∎

######  Lemma 6.4.

Any $x\in L_{p}(\mathbb{T}^{n}_{\theta})$, $p\geq 1$, can be written in the
form $x=yz$ with $y,z\in L_{2p}(\mathbb{T}^{n}_{\theta})$ such that
$\|y\|_{L_{2p}}=\|z\|_{L_{2p}}=(\|x\|_{L_{p}})^{1/2}$.

###### Proof.

Let $x=u|x|$ be the polar decomposition of $x$. As mentioned in Section 2 the
phase $u$ is a partial isometry in $L_{\infty}(\mathbb{T}^{n}_{\theta})$, and
hence $\|u\|_{L_{\infty}}\leq 1$. Thus, if we set $y=u|x|^{1/2}$ and
$z=|x|^{1/2}$, then $x=yz$ and $z\in L_{2p}(\mathbb{T}^{n}_{\theta})$ with
$\|z\|_{L_{2p}}=(\|x\|_{L_{p}})^{1/2}$. In addition, Hölder’s inequality (see
Proposition 2.1) ensures that $y=u|x|^{1/2}\in
L_{2p}(\mathbb{T}^{n}_{\theta})$, and we have

$\|y\|_{L_{2p}}\leq\|u\|_{L_{\infty}}\||x|^{1/2}\|_{L_{2p}}\leq(\|x\|_{L_{p}})^{1/2}.$

As Hölder’s inequality also gives

$\|x\|_{L_{p}}=\|yz\|_{L_{p}}\leq\|y\|_{L_{2p}}\|z\|_{L_{2p}}\leq\|y\|_{L_{2p}}\big{(}\|x\|_{L_{p}}\big{)}^{1/2},$

we deduce that $\|y\|_{L_{2p}}=(\|x\|_{L_{p}})^{1/2}$. The proof is complete.
∎

We are now in a position to establish Cwikel estimates for the operators
$\Delta^{-n/4p}\lambda(x)\Delta^{-n/4p}$.

######  Theorem 6.5.

The following hold.

  1. (1)

If $x\in L_{p}(\mathbb{T}^{n}_{\theta})$, $p>1$, then
$\Delta^{-{n}/{4p}}\lambda(x)\Delta^{-{n}/{4p}}\in\mathscr{L}_{p,\infty}$, and
we have

$\big{\|}\Delta^{-\frac{n}{4p}}\lambda(x)\Delta^{-\frac{n}{4p}}\big{\|}_{\mathscr{L}_{p,\infty}}\leq
2^{\frac{1}{p}}c_{+}(2p)^{2}\nu_{0}(n)^{\frac{1}{p}}\|x\|_{L_{p}}.$ (6.8)

  2. (2)

If $x\in L_{p}(\mathbb{T}^{n}_{\theta})$, $p>1$, then
$\Delta^{-n/4}\lambda(x)\Delta^{-n/4}\in\mathscr{L}_{1,\infty}$, and we have

$\big{\|}\Delta^{-\frac{n}{4}}\lambda(x)\Delta^{-\frac{n}{4}}\big{\|}_{\mathscr{L}_{1,\infty}}\leq
2c_{2}(2p)^{2}\nu_{0}(n)\|x\|_{L_{p}}.$ (6.9)

  3. (3)

If $x\in L_{1}(\mathbb{T}^{n}_{\theta})$ and $p<1$, then
$\Delta^{-{n}/{4p}}\lambda(x)\Delta^{-{n}/{4p}}\in\mathscr{L}_{p,\infty}$, and
we have

$\big{\|}\Delta^{-\frac{n}{4p}}\lambda(x)\Delta^{-\frac{n}{4p}}\big{\|}_{\mathscr{L}_{p,\infty}}\leq
2^{\frac{1}{p}}c_{-}(2p)^{2}\nu_{0}(n)^{\frac{1}{p}}\|x\|_{L_{1}}.$ (6.10)

Here $c_{+}(2p)$ and $c_{-}(2p)$ are the best constants in the
$\mathscr{L}_{2p,\infty}$-Cwikel estimates (4.12) and (4.17), respectively,
and $c_{2}(2p)$ is given by (6.2).

###### Proof.

Let $p>0$, and suppose that $q=\max(p,1)$ if $p\neq 1$ or $q>1$ if $p=1$. Note
that $q\geq 1$. Set $c(p,q)$ to be equal to $c_{+}(2p)$ (resp., $c_{-}(2p)$,
$c_{2}(2q)$) if $p>1$ (resp., $p<1$, $p=1$). We need to show that if $x\in
L_{q}(\mathbb{T}^{n}_{\theta})$, then
$\Delta^{-{n}/{4p}}\lambda(x)\Delta^{-{n}/{4p}}\in\mathscr{L}_{p,\infty}$, and
we have

$\big{\|}\Delta^{-\frac{n}{4p}}\lambda(x)\Delta^{-\frac{n}{4p}}\big{\|}_{\mathscr{L}_{p,\infty}}\leq
2^{\frac{1}{p}}c(p,q)^{2}\nu_{0}(n)^{\frac{1}{p}}\|x\|_{L_{q}}.$ (6.11)

Let $x\in L_{q}(\mathbb{T}^{n}_{\theta})$. By Lemma 6.4 we may write $x=yz$,
with $y,z\in L_{2q}(\mathbb{T}^{n}_{\theta})$ such that
$\|y\|_{L_{2q}}=\|z\|_{L_{2q}}=(\|x\|_{L_{q}})^{1/2}$. By Lemma 6.3 we have

$\Delta^{-\frac{n}{4p}}\lambda(x)\Delta^{-\frac{n}{4p}}=\big{[}\lambda(y^{*})\Delta^{-\frac{n}{4p}}\big{]}^{*}\lambda(z)\Delta^{-\frac{n}{4p}}.$

It follows from Theorem 6.1 that $[\lambda(y^{*})\Delta^{-\frac{n}{4p}}]^{*}$
and $\lambda(z)\Delta^{-\frac{n}{4p}}$ are both in the weak Schatten class
$\mathscr{L}_{2p,\infty}$, and we have

$\displaystyle\big{\|}\lambda(z)\Delta^{-\frac{n}{24}}\big{\|}_{\mathscr{L}_{2p,\infty}}\leq
c(p,q)\nu_{0}(n)^{\frac{1}{2p}}\|z\|_{L_{2q}}=c(p,q)\nu_{0}(n)^{\frac{1}{2p}}\|x\|^{\frac{1}{2}}_{L_{2q}},$
(6.12)
$\displaystyle\big{\|}\big{[}\lambda(y^{*})\Delta^{-\frac{n}{4p}}\big{]}^{*}\big{\|}_{\mathscr{L}_{2p,\infty}}=\big{\|}\lambda(y^{*})\Delta^{-\frac{n}{2p}}\big{\|}_{\mathscr{L}_{2p,\infty}}\leq
c(p,q)\nu_{0}(n)^{\frac{1}{2p}}\|y^{*}\|_{L_{2p}}=c(p,q)\nu_{0}(n)^{\frac{1}{2p}}\|x\|^{\frac{1}{2}}_{L_{2q}}.$
(6.13)

The Hölder’s inequality for weak Schatten classes provided by Proposition 4.1
then implies that
$\Delta^{-\frac{n}{4p}}\lambda(x)\Delta^{-\frac{n}{4p}}\in\mathscr{L}_{p,\infty}$,
and we have

$\displaystyle\big{\|}\Delta^{-\frac{n}{4p}}\lambda(x)\Delta^{-\frac{n}{4p}}\|_{\mathscr{L}_{p,\infty}}$
$\displaystyle\leq
2^{\frac{1}{p}}\big{\|}\big{[}\lambda(y^{*})\Delta^{-\frac{n}{4p}}\big{]}^{*}\big{\|}_{\mathscr{L}_{2p,\infty}}\big{\|}\lambda(z)\Delta^{-\frac{n}{4p}}\big{\|}_{\mathscr{L}_{2p,\infty}}$
$\displaystyle\leq
2^{\frac{1}{p}}c(p,q)^{2}\nu_{0}(n)^{\frac{1}{p}}\|x\|_{L_{q}}.$

This proves (6.11). The proof is complete. ∎

######  Remark 6.6.

If $x\geq 0$, then the inequalities (6.8)–(6.10) hold without the extra
$2^{\frac{1}{p}}$-factor. Indeed, if $p$ and $q$ are as in the proof of
Theorem 6.5 and $0\leq x\in L_{q}(\mathbb{T}^{n}_{\theta})$, then we may take
$y=z=\sqrt{x}$. In this case
$\Delta^{-{n}/{4p}}\lambda(x)\Delta^{-{n}/{4p}}=\big{[}\lambda(\sqrt{x})\Delta^{-{n}/{4p}}\big{]}^{*}\lambda(\sqrt{x})\Delta^{-{n}/{4p}}=|\lambda(\sqrt{x})\Delta^{-{n}/{4p}}|^{2}$.
Thus,

$\big{\|}\Delta^{-\frac{n}{4p}}\lambda(x)\Delta^{-\frac{n}{4p}}\big{\|}_{\mathscr{L}_{p,\infty}}=\left\|\big{|}\lambda(\sqrt{x})\Delta^{-{n}/{4p}}\big{|}^{2}\right\|_{\mathscr{L}_{p,\infty}}=\big{\|}\lambda(\sqrt{x})\Delta^{-\frac{n}{4p}}\big{\|}^{2}_{\mathscr{L}_{2p,\infty}}$

Combining this with the inequality (6.12) for $y=\sqrt{x}$ then gives

$\big{\|}\Delta^{-\frac{n}{4p}}\lambda(x)\Delta^{-\frac{n}{4p}}\big{\|}_{\mathscr{L}_{p,\infty}}\leq
c(p,q)^{2}\nu_{0}(n)^{\frac{1}{p}}\|x\|_{L_{q}},$

which proves our claim.

######  Remark 6.7.

The estimate (6.9) does not hold for $x\in L_{1}(\mathbb{T}^{n}_{\theta})$
(see [63, Lemma 5.7]).

######  Remark 6.8.

For the ordinary torus $\mathbb{T}^{n}$, i.e., $\theta=0$, by a recent result
of Sukochev-Zanin [98] the estimate (6.9) still holds if $x$ is in the Orlicz
space $L\\!\log\\!L(\mathbb{T}^{n})$ (see also [93]). It would be interesting
to have an analogue of this result for $\theta\neq 0$.

##  7\. Borderline Birman-Schwinger Principle

In this section, we establish a “borderline” version of the abstract Birman-
Schwinger principle for the number of negative values of relatively form-
compact perturbations of non-negative semi-bounded operators on Hilbert space.

To a large extent we follow the original approach of Birman-Solomyak [13],
which we recast in the framework of [92]. However, our ultimate result
(Theorem 7.9) seems to be new, at least at the level of generality it is
stated. In particular, it can be applied to Schrödinger operators
$\Delta_{g}+V$, and more generally fractional Schrödinger operators
$\Delta_{g}^{\alpha}+V$, in the following setups: closed Riemannian manifolds,
compact manifolds with boundary with suitable boundary condition, or even
hyperbolic manifolds with infinite volume.

Throughout this section we let $\mathscr{H}$ be a (separable) Hilbert space.

###  7.1. Glazman’s Lemma

The proof of the abstract Birman-Schwinger principle by Birman-Solomyak [13]
relies on Glazman’s lemma. We shall now briefly recall this result and set
some notation along the way.

Let $A$ be a selfadjoint operator on $\mathscr{H}$ which is bounded or semi-
bounded. We denote by $Q_{A}$ its quadratic form. If $A$ is bounded, then
$Q_{A}$ is the quadratic form on $\mathscr{H}$ defined by

$Q_{A}(\xi,\eta):=\left\langle
A\xi|\eta\right\rangle\qquad\forall\xi,\eta\in\mathscr{H}.$

If $A$ is semi-bounded, then $Q_{A}$ has domain
$\operatorname{dom}(A-\lambda)^{1/2}$ with
$\mu\in\mathbb{R}\setminus\operatorname{Sp}(A)$. We also denote by
$\operatorname{Sp}_{\text{ess}}(A)$ the essential spectrum of $A$, i.e., the
complement of the discrete spectrum (which consists of isolated eigenvalues
with finite multiplicity).

Given any $\lambda\in\mathbb{R}$ we set

$N^{+}(A;\lambda)=\dim\big{(}\operatorname{ran}\mathbbm{1}_{(\lambda,\infty)}(A)\big{)},\qquad
N^{-}(A;\lambda)=\dim\big{(}\operatorname{ran}\mathbbm{1}_{(-\infty,\lambda)}(A)\big{)}.$
(7.1)

Thus, if $[\lambda,\infty)$ (resp., $(-\infty,\lambda]$) does not meet the
essential spectrum of $A$, then $N^{+}(A;\lambda)$ (resp., $N^{-}(A;\lambda)$)
is the number of eigenvalues of $A$ counted with multiplicity that are
$>\lambda$ (resp., $<\lambda$).

In what follows, we denote by $\mathscr{F}^{\pm}(A;\lambda)$ (resp.,
$\mathscr{F}_{0}^{\pm}(A;\lambda)$) the collection of all subspaces
$F\subset\operatorname{dom}(Q_{A})$ such that $\pm
Q_{A}(\xi,\xi)>\pm\lambda\|\xi\|^{2}$ (resp., $\pm
Q_{A}(\xi,\xi)\geq\pm\lambda\|\xi\|^{2}$) on $F\setminus 0$. We have the
following variational principles.

######  Lemma 7.1 (Glazman’s Lemma [12, 33]).

For all $\lambda\in\mathbb{R}$, we have

$\displaystyle N^{\pm}(A;\lambda)=$ $\displaystyle\max\left\\{\dim F;\
F\in\mathscr{F}^{\pm}(A;\lambda)\right\\},$ (7.2) $\displaystyle=$
$\displaystyle\min\left\\{\dim F^{\perp};\
F\in\mathscr{F}^{\mp}_{0}(A;\lambda)\right\\}.$ (7.3)

######  Remark 7.2.

The variational principle (7.3) is due to Glazman [33]. A proof of (7.2) is
given in [12] (see Theorem 10.2.3; see also Theorem 9.2.6 for the compact
case).

In what follows given selfadjoint operators $A$ and $B$ we shall write $A\leq
B$ if $\operatorname{dom}(Q_{A})=\operatorname{dom}(Q_{B})$ and
$Q_{A}(\xi,\xi)\leq Q_{B}(\xi,\xi)$ for all $\xi$ in
$\operatorname{dom}(Q_{A})=\operatorname{dom}(Q_{B})$. Recall that Glazman’s
Lemma implies the following monotonicity principle.

######  Corollary 7.3.

Let $A$ and $B$ be selfadjoint operators on $\mathscr{H}$ such that $A\leq B$.
Then, for all $\lambda\in\mathbb{R}$, we have

$N^{+}(A;\lambda)\leq N^{+}(B;\lambda),\qquad N^{-}(A;\lambda)\geq
N^{-}(B;\lambda).$

###  7.2. The Abstract Birman-Schwinger principle

From now on we let $H$ be a (densely defined) selfadjoint operator on
$\mathscr{H}$ with non-negative spectrum containing $0$. Its quadratic form
$Q_{H}$ has domain
$\operatorname{dom}(Q_{H})=\operatorname{dom}(H+1)^{\frac{1}{2}}$. We denote
by $\mathscr{H}_{+}$ the Hilbert space obtained by endowing
$\operatorname{dom}(Q_{H})$ with the Hilbert space norm,

$\|\xi\|_{+}=\big{(}Q_{H}(\xi,\xi)+\|\xi\|^{2}\big{)}^{\frac{1}{2}}=\big{\|}(1+H)^{1/2}\xi\big{\|},\qquad\xi\in\operatorname{dom}(Q_{H}).$

We also let $\mathscr{H}_{-}$ be the Hilbert space of continuous _anti-linear_
functionals on $\mathscr{H}_{+}$. Note that we have a continuous embedding
$\iota:\mathscr{H}\hookrightarrow\mathscr{H}_{-}$ with dense range given by

$\left\langle\iota(\xi),\eta\right\rangle=\left\langle\xi|\eta\right\rangle,\qquad\xi\in\mathscr{H},\
\eta\in\mathscr{H}_{+}.$

The operator $(H+1)^{1/2}:\mathscr{H}_{+}\rightarrow\mathscr{H}$ is a unitary
isomorphism. By selfadjointness it extends to a unitary isomorphism
$(H+1)^{1/2}:\mathscr{H}\rightarrow\mathscr{H}_{-}$ such that

$\big{\langle}(H+1)^{1/2}\xi,\eta\big{\rangle}=\big{\langle}{\xi}\big{|}{(H+1)^{1/2}\eta}\big{\rangle},\qquad\xi\in\mathscr{H},\
\eta\in\mathscr{H}_{+},$ (7.4)

where
$\left\langle\cdot,\cdot\right\rangle:\mathscr{H}_{-}\times\mathscr{H}_{+}\rightarrow\mathbb{C}$
is the natural duality pairing. In particular, we have bounded inverses
$(H+1)^{-1/2}:\mathscr{H}\rightarrow\mathscr{H}_{+}$ and
$(H+1)^{-1/2}:\mathscr{H}_{-}\rightarrow\mathscr{H}$ such that

$\big{\langle}(H+1)^{-1/2}\xi|\eta\big{\rangle}=\big{\langle}\xi,(H+1)^{-1/2}\eta\big{\rangle},\qquad\xi\in\mathscr{H}_{-},\
\eta\in\mathscr{H}.$ (7.5)

More generally, for any $\lambda<0$, we have bounded operators
$(H-\lambda)^{1/2}:\mathscr{H}_{+}\rightarrow\mathscr{H}$ and
$(H-\lambda)^{1/2}:\mathscr{H}\rightarrow\mathscr{H}_{-}$ with bounded
inverses $(H-\lambda)^{-1/2}:\mathscr{H}\rightarrow\mathscr{H}_{+}$ and
$(H-\lambda)^{-1/2}:\mathscr{H}_{-}\rightarrow\mathscr{H}$.

Similarly, the operator $H$ extends to a bounded operator
$H:\mathscr{H}_{+}\rightarrow\mathscr{H}_{-}$ such that

$\left\langle
H\xi,\eta\right\rangle=Q_{H}(\xi,\eta)\qquad\xi,\eta\in\mathscr{H}_{+}.$ (7.6)

More generally, for any $\lambda<0$, the operator
$H-\lambda=(H-\lambda)^{1/2}(H-\lambda)^{1/2}$ extends to a bounded operator
from $\mathscr{H}_{+}$ to $\mathscr{H}_{-}$ with bounded inverse
$(H-\lambda)^{-1}:\mathscr{H}_{-}\longrightarrow\mathscr{H}_{+}$.

In what follows, we let $V:\mathscr{H}_{+}\rightarrow\mathscr{H}_{-}$ be a
bounded operator. We denote by $Q_{V}$ the corresponding quadratic form with
domain $\mathscr{H}_{+}$ and given by

$Q_{V}(\xi,\eta):=\left\langle
V\xi,\eta\right\rangle,\qquad\xi,\eta\in\mathscr{H}_{+}.$

We assume that $Q_{V}$ is _symmetric_ and _$H$ -form compact_. The latter
condition means that the operator
$V:\mathscr{H}_{+}\rightarrow\mathscr{H}_{-}$ is compact, or equivalently,
$(H+1)^{-1/2}V(H+1)^{-1/2}$ is a compact operator on $\mathscr{H}$.

Our main focus is the operator $H_{V}:=H+V$. It makes sense as a bounded
operator $H_{V}:\mathscr{H}_{+}\rightarrow\mathscr{H}_{-}$. Furthermore, as
the symmetric quadratic form $Q_{V}$ is $H$-form compact, it is $H$-form
bounded with zero $H$-bound (see [92, §7.8]). Therefore, by the KLMN theorem
(see, e.g., [75, 83]) the restriction of $H_{V}$ to
$\operatorname{dom}(H_{V}):=H_{V}^{-1}(\mathscr{H})$ is a bounded from below
selfadjoint operator on $\mathscr{H}$ whose quadratic form is precisely
$Q_{H}+Q_{V}$.

######  Lemma 7.4 (see [92, Theorem 7.8.4]).

The following holds.

  1. (i)

For all $\lambda\not\in\operatorname{Sp}(H)\cup\operatorname{Sp}(H_{V})$, the
operator $(H_{V}-\lambda)^{-1}-(H-\lambda)^{-1}$ is compact.

  2. (ii)

The operators $H$ and $H_{V}$ have the same essential spectrum.

  3. (iii)

If $H$ has compact resolvent, then so does $H_{V}$.

As $H_{V}$ is bounded from below, we define the counting function
$N^{-}(H_{V};\lambda)$ as in (7.1). If
$\lambda<\inf\operatorname{Sp}_{\textup{ess}}(H)$, then $N^{-}(H_{V};\lambda)$
this is the number of eigenvalues of $H_{V}$ which are $<\lambda$. We also set
$N^{-}(H_{V}):=N(H_{V};0)$; this is the number of “bound states” of $H_{V}$.

Given operators $V_{j}:\mathscr{H}_{+}\rightarrow\mathscr{H}_{-}$, $j=1,2$, we
shall write $V_{1}\leq V_{2}$ when $Q_{V_{1}}\leq Q_{V_{2}}$, i.e.,
$Q_{V_{1}}$ and $Q_{V_{2}}$ are both symmetric and $Q_{V_{1}}(\xi,\xi)\leq
Q_{V_{2}}(\xi,\xi)$ for all $\xi\in\mathscr{H}_{+}$. In this case
$Q_{H_{V_{1}}}(\xi,\xi)\leq Q_{H_{V_{2}}}(\xi,\xi)$ for all
$\xi\in\mathscr{H}_{+}$, and so by using Corollary 7.3 we get

$V_{1}\leq V_{2}\ \Longrightarrow\ N(H_{V_{2}};\lambda)\leq
N(H_{V_{1}};\lambda)\qquad\forall\lambda\in\mathbb{R}.$ (7.7)

The Birman-Schwinger principle was established by Birman [4] and Schwinger
[84] for Schrödinger operators $\Delta+V$ on $\mathbb{R}^{n}$, $n\geq 3$. It
is the main impetus for using Cwikel estimates to establish the Cwikel-Lieb-
Rozenblum inequality (see [22, 89]). In our setting it relates the counting
function $N(H_{V};\lambda)$ to the eigenvalues of the Birman-Schwinger
operators,

$K_{V}(\lambda)=-(H-\lambda)^{-\frac{1}{2}}V(H-\lambda)^{-\frac{1}{2}},\qquad\lambda<0.$

The compactness of $V$ ensures us that $K_{V}(\lambda)$ is a compact operator
on $\mathscr{H}$.

Note also that $K_{V}(\lambda)$ is related to the quadratic form $Q_{V}$ by

$\displaystyle\left\langle K_{V}(\lambda)\xi|\eta\right\rangle$
$\displaystyle=-\big{\langle}V(H-\lambda)^{-1/2}\xi,(H-\lambda)^{-1/2}\eta\big{\rangle}$
$\displaystyle=-Q_{V}\big{(}(H-\lambda)^{-1/2}\xi,(H-\lambda)^{-1/2}\eta\big{)},\qquad\xi,\eta\in\mathscr{H}.$
(7.8)

Thus, the fact that $Q_{V}$ is symmetric ensures us that $K_{V}(\lambda)$ is a
selfadjoint compact operator. We then define the counting function
$N^{+}(K_{V}(\lambda);\mu)$, $\mu>0$, as above. If in addition $V\leq 0$, then
$K_{V}(\lambda)\geq 0$, and so in this case the eigenvalues of
$K_{V}(\lambda)$ agree with its singular values.

######  Proposition 7.5 (Abstract Birman-Schwinger Principle [13, Lemma 1.4]).

For all $\lambda<0$, we have

$N(H_{V};\lambda)=N^{+}\left(K_{V}(\lambda);1\right).$ (7.9)

######  Remark 7.6.

For reader’s convenience a proof of Proposition 7.5 is given in Appendix A.

The following is a well known consequence of the Birman-Schwinger principle
(see [22, 89]). For reader’s convenience, a proof has been included in
Appendix A.

######  Corollary 7.7.

Assume $V\leq 0$ and $(H+1)^{-1/2}V(H+1)^{-1/2}\in\mathscr{L}_{p,\infty}$ for
some $q\in(0,\infty)$. Then, for all $\lambda<0$, the operator
$K_{V}(\lambda)$ is in the weak Schatten class $\mathscr{L}_{p,\infty}$, and
we have

$N(H_{V};\lambda)\leq\left\|K_{V}(\lambda)\right\|_{\mathscr{L}_{p,\infty}}^{p}.$
(7.10)

###  7.3. Borderline Birman-Schwinger principle

The proofs of the standard CLR inequality for Schrödinger operators on
$\mathbb{R}^{n}$, $n\geq 3$, uses the fact that for the Laplacian on
$\mathbb{R}^{n}$ with $n\geq 3$ the essential spectrum contains $0$ and the
resolvent $(\Delta-\lambda)^{-1}$ has a weak limit as $\lambda\rightarrow
0^{-}$. This allows us to take $\lambda=0$ in (7.10) and get an upper bound
for $N^{-}(H_{V})$ (see [90, Theorem 7.9.11]).

However, on closed manifolds and NC tori, $0$ is in the discrete spectrum of
the Laplacian, and so the resolvent has a pole singularity at $\lambda=0$.
This prevents us from letting $\lambda\rightarrow 0^{-}$ in (7.10). This
stresses out the need for a “borderline” version of the Birman-Schwinger
(7.10) when $0$ is the discrete spectrum.

Assume that $0$ lies in the discrete spectrum of $H$, i.e., $0$ is an isolated
eigenvalue of $H$. Thus, the essential spectrum of $H$ is contained in some
interval $[a,\infty)$ with $a>0$. As $H$ and $H_{V}$ have same essential
spectrum by Lemma 7.4, it follows that $H_{V}$ has at most finitely many non-
positive eigenvalues.

We denote by $H^{-1}$ the partial inverse of $H$. That is, $H^{-1}$ vanishes
on $\ker H$ and inverts $H$ on $(\ker H)^{\perp}=\operatorname{ran}H$. This is
a selfadjoint bounded operator with non-negative spectrum. We then define
$H^{-1/2}$ to be $(H^{-1})^{1/2}$. Equivalently, $H^{-1}=f(H)$ and
$H^{-1/2}=g(H)$, where $f(t)=\mathbbm{1}_{[\epsilon,\infty)}(t)t^{-1}$ and
$g(t)=\mathbbm{1}_{[\epsilon,\infty)}(t)t^{-1/2}$, with $\epsilon>0$ small
enough so that $\operatorname{Sp}(H)\cap(0,\epsilon]=\emptyset$. We then set

We shall need the following abstract version of Lemma 6.3.

######  Lemma 7.8.

Suppose that $V\leq 0$. Then there is a compact operator
$W:\mathscr{H}_{+}\rightarrow\mathscr{H}$ such that $V=-W^{*}W$, where
$W^{*}:\mathscr{H}\rightarrow\mathscr{H}_{-}$ is the adjoint map.

###### Proof.

Set $K=K_{V}(-1)$. As $V\leq 0$, we know that $K$ is a positive compact
operator. Set $W=K^{1/2}(H+1)^{1/2}$. Then $W$ is a compact operator from
$\mathscr{H}_{+}$ to $\mathscr{H}$. By duality we get a linear operator
$W^{*}:\mathscr{H}\rightarrow\mathscr{H}_{-}$ such that

$\left\langle
W^{*}\xi,\eta\right\rangle=\left\langle\xi|W\eta\right\rangle,\qquad\xi\in\mathscr{H},\
\eta\in\mathscr{H}_{+}.$ (7.11)

In particular, for $\xi,\eta\in\mathscr{H}$, we have

$\displaystyle\left\langle W^{*}W\xi,\eta\right\rangle=\left\langle
W\xi|W\eta\right\rangle$
$\displaystyle=\big{\langle}K^{\frac{1}{2}}(H+1)^{\frac{1}{2}}\xi|K^{\frac{1}{2}}(H+1)^{\frac{1}{2}}\eta\big{\rangle}$
$\displaystyle=\big{\langle}K(H+1)^{\frac{1}{2}}\xi|(H+1)^{\frac{1}{2}}\eta\big{\rangle}$
$\displaystyle=\big{\langle}(H+1)^{\frac{1}{2}}K(H+1)^{\frac{1}{2}}\xi,\eta\big{\rangle}$
$\displaystyle=-\left\langle V\xi,\eta\right\rangle.$

This shows that $V=-W^{*}W$. The proof is complete. ∎

In what follows we denote by $\Pi_{0}$ the orthogonal projection on $\ker H$.
This is a selfadjoint finite rank operator whose range is contained in
$\mathscr{H}_{+}$, and hence we get a bounded operator
$\Pi_{0}:\mathscr{H}\rightarrow\mathscr{H}_{+}$. By duality we obtain a
bounded operator $\Pi_{0}:\mathscr{H}_{-}\rightarrow\mathscr{H}$ such that

$\left\langle\Pi_{0}\xi|\eta\right\rangle=\left\langle\xi,\Pi_{0}\eta\right\rangle,\qquad\xi\in\mathscr{H}_{-},\
\eta\in\mathscr{H}.$

The composition $\Pi_{0}V\Pi_{0}$ then is a selfadjoint finite-rank operator
on $\mathscr{H}$. As above, we denote by $N^{-}(\Pi_{0}V\Pi_{0})$ its number
of negative eigenvalues counted with multiplicity.

######  Theorem 7.9 (Borderline Birman-Schwinger Principle).

Suppose that $0$ is in the discrete spectrum of $H$. Set
$K_{V}=-H^{-1/2}VH^{-1/2}$. Then, the following hold.

  1. (1)

We have

$N^{+}(K_{V};1)\leq N^{-}(H_{V})\leq N^{+}(K_{V};1)+\dim\ker H.$ (7.12)

  2. (2)

Assume further that $V\leq 0$ and $K_{V}\in\mathscr{L}_{p,\infty}$ for some
$p\in(0,\infty)$. Then

$0\leq
N^{-}(H_{V})-N^{-}(\Pi_{0}V\Pi_{0})\leq\big{\|}K_{V}\big{\|}_{\mathscr{L}_{p,\infty}}^{p}.$
(7.13)

In particular, if $V(\ker H)\subset\operatorname{ran}H$, then

$N^{-}(H_{V})\leq\big{\|}K_{V}\big{\|}_{\mathscr{L}_{p,\infty}}^{p}.$ (7.14)

###### Proof.

Set $\mathscr{H}_{1}=\operatorname{ran}H$ and $V_{1}=(1-\Pi_{0})V(1-\Pi_{0})$.
We also denote by $H_{V|\mathscr{H}_{1}}$ and $K_{V|\mathscr{H}_{1}}$ the
operators on $\mathscr{H}_{1}$ induced by
$(1-\Pi_{0})H_{V}(1-\Pi_{0})=H_{V_{1}}$ and $K_{V}$, respectively. By applying
the Birman-Schwinger principle (7.9) to $H_{V|\mathscr{H}_{1}}$ on
$\mathscr{H}_{1}$ we get

$N^{-}\big{(}H_{V|\mathscr{H}_{1}}\big{)}=N^{+}\big{(}K_{V|\mathscr{H}_{1}};1\big{)}=N^{+}\big{(}K_{V};1\big{)}.$
(7.15)

Moreover, as
$\mathscr{F}^{-}(H_{V|\mathscr{H}_{1}};0)\subset\mathscr{F}^{-}(H_{V};0)$ by
using the variational principles (7.2)–(7.3) we get

$N^{-}\big{(}H_{V|\mathscr{H}_{1}}\big{)}\leq N^{-}\big{(}H_{V}\big{)}\leq
N^{-}\big{(}H_{V|\mathscr{H}_{1}}\big{)}+\dim\ker H.$ (7.16)

Combining this with (7.15) gives the inequalities (7.12).

From now on we assume that $V\leq 0$ and $K_{V}\in\mathscr{L}_{p,\infty}$,
$p>0$. Set $A=H^{1/2}(H+1)^{-1/2}$. Then $(H+1)^{-1/2}=H^{-1/2}A+\Pi_{0}$, and
so in the same way as in (A.2) we have

$(H+1)^{-1/2}V(H+1)^{-1/2}=A^{*}K_{V}A+R,$

where $R$ is a finite rank operator. This implies that
$(H+1)^{-1/2}V(H+1)^{-1/2}$ is in the weak Schatten class
$\mathscr{L}_{p,\infty}$. In particular, this is a compact operator. Likewise,
the Birman-Schwinger operators $K_{V}(\lambda)$, $\lambda<0$, are in
$\mathscr{L}_{p,\infty}$ as well.

Bearing this in mind, set $N=N^{-}(H_{V})$ and $N_{0}=N^{-}(\Pi_{0}V\Pi_{0})$.
As $\Pi_{0}V\Pi_{0}=\Pi_{0}H_{V}\Pi_{0}$, we see that
$\mathscr{F}^{-}(\Pi_{0}V\Pi_{0};0)\subset\mathscr{F}^{-}(H_{V};0)$, and so it
follows from (7.2) that $N_{0}\leq N$.

It remains to show the 2nd inequality in (7.13). We may assume $N-N_{0}\geq
1$, since otherwise the inequality is trivially satisfied. Note that
$N=N(H_{V};\lambda)$ as soon as $\lambda<0$ is small enough. The Birman-
Schwinger principle (7.9) then asserts that $N=N^{+}(K_{V}(\lambda);1)$. As
$K_{V}(\lambda)\geq 0$, in the same way as in the proof of Corollary 7.7 this
ensures that $K_{V}(\lambda)$ has exactly $N$ singular values $>1$, and hence
$\mu_{N-1}(K_{V}(\lambda))\geq 1$.

By Lemma 7.8 we can find a compact operator
$W:\mathscr{H}_{+}\rightarrow\mathscr{H}$ such that $V=-W^{*}W$. Set
$T(\lambda)=W(H-\lambda)^{-\frac{1}{2}}$. This is a compact operator on
$\mathscr{H}$. Moreover, by using (7.5) and (7.11) we see that, for all
$\xi,\eta\in\mathscr{H}$, we have

$\big{\langle}(H-\lambda)^{-\frac{1}{2}}W^{*}\xi|\eta\big{\rangle}=\big{\langle}W^{*}\xi,(H-\lambda)^{-\frac{1}{2}}\eta\big{\rangle}=\big{\langle}\xi|W(H-\lambda)^{-\frac{1}{2}}\eta\big{\rangle}=\left\langle\xi|T(\lambda)\eta\right\rangle.$

This shows that $T(\lambda)^{*}=(H-\lambda)^{-1/2}W^{*}$. Thus,

$K_{V}(\lambda)=(H-\lambda)^{-\frac{1}{2}}W^{*}W(H-\lambda)^{-\frac{1}{2}}=T(\lambda)^{*}T(\lambda)=|T(\lambda)|^{2}.$

Set $\tilde{K}_{V}(\lambda)=T(\lambda)T(\lambda)^{*}=W(H-\lambda)^{-1}W^{*}$.
By using (4.3) we get

$\mu_{j}\left(K_{V}(\lambda)\right)=\mu_{j}\left(T(\lambda)\right)^{2}=\mu_{j}\left(T(\lambda)^{*}\right)^{2}=\mu_{j}\big{(}\tilde{K}_{V}(\lambda)\big{)},\quad
j\geq 0.$ (7.17)

Observe that
$\tilde{K}_{V}(\lambda)=T(\lambda)(1-\Pi_{0})T(\lambda)^{*}+T(\lambda)\Pi_{0}T(\lambda)^{*}$.
By using (4.4) we get

$\mu_{N-1}\big{(}\tilde{K}_{V}(\lambda)\big{)}\leq\mu_{N-N_{0}-1}\big{(}T(\lambda)(1-\Pi_{0})T(\lambda)^{*}\big{)}+\mu_{N_{0}}\big{(}T(\lambda)\Pi_{0}T(\lambda)^{*}\big{)}.$

As
$\operatorname{rk}T(\lambda)\Pi_{0}T(\lambda)^{*}\leq\operatorname{rk}\Pi_{0}=N_{0}$,
it follows from (4.2) that $\mu_{N_{0}}(T(\lambda)\Pi_{0}T(\lambda)^{*})=0$.
Thus,

$\mu_{N-1}\big{(}\tilde{K}_{V}(\lambda)\big{)}\leq\mu_{N-N_{0}-1}\big{(}T(\lambda)(1-\Pi_{0})T(\lambda)^{*}\big{)}.$
(7.18)

We also have

$\displaystyle T(\lambda)(1-\Pi_{0})T(\lambda)^{*}$
$\displaystyle=W(H-\lambda)^{-1/2}(1-\Pi_{0})(H-\lambda)^{-1/2}W^{*},$
$\displaystyle=W(H+1)^{-1/2}\cdot(1-\Pi_{0})(H+1)(H-\lambda)^{-1}\cdot(H+1)^{-1/2}W^{*},$
$\displaystyle=T(-1)(1-\Pi_{0})(H+1)(H-\lambda)^{-1}T(-1)^{*}.$

As $(1-\Pi_{0})(H+1)(H-\lambda)^{-1}\leq(H+1)H^{-1}$, it follows that

$T(\lambda)(1-\Pi_{0})T(\lambda)^{*}\leq
T(-1)(H+1)H^{-1}T(-1)^{*}=WH^{-1}W^{*}.$

Therefore, by using (7.18) and the monotonicity principle (4.6) we get

$\mu_{N-1}\big{(}\tilde{K}_{V}(\lambda)\big{)}\leq\mu_{N-N_{0}-1}\big{(}T(\lambda)(1-\Pi_{0})T(\lambda)^{*}\big{)}\leq\mu_{N-N_{0}-1}\big{(}WH^{-1}W^{*}\big{)}.$
(7.19)

As in (7.17) we have
$\mu_{j}(WH^{-1}W^{*})=\mu_{j}(H^{-1/2}VH^{-1/2})=\mu_{j}(K_{V})$ for all
$j\geq 0$. Combining this with (7.19) and the fact that
$\mu_{N-1}(\tilde{K}_{V}(\lambda))\geq 1$ gives

$1\leq\mu_{N-1}\big{(}\tilde{K}_{V}(\lambda)\big{)}\leq\mu_{N-N_{0}-1}\big{(}K_{V}\big{)}.$
(7.20)

In the same way as in (A.3) this implies that

$N-N_{0}\leq(N-N_{0})\mu_{N-N_{0}-1}\big{(}K_{V}\big{)}^{p}\leq\big{\|}K_{V}\big{\|}_{\mathscr{L}_{p,\infty}}^{p}.$

This gives the 2nd inequality in (7.13). The proof is complete. ∎

######  Remark 7.10.

By using (7.12) and arguing along similar lines as that of the proof of
Corollary 7.7 in Appendix A it is easier to get the inequality,

$N^{-}(H_{V})-\dim\ker
H\leq\big{\|}K_{V}\big{\|}_{\mathscr{L}_{p,\infty}}^{p}.$ (7.21)

We also recover this inequality from (7.13). This inequality and the
inequalities (7.12) are known already. They appear in various forms in the
work of Birman and Solomyak, in particular for dealing with Schrödinger
operators on bounded domains under Neuman’s boundary condition (see, e.g., [6,
11, 93]). In most of the instances where it has been used by Birman and
Solomyak, the inequality (7.21) is relevant, since in those situations $\ker
H=1$ and $\Pi_{0}V\Pi_{0}$ has a negative eigenvalue is $V\leq 0$ and $V\neq
0$. However, in general we might have $N^{-}(H_{V})-\dim\ker H<0$ (e.g., if
$V(\ker H)\subset\operatorname{ran}H$), and so in such a case (7.21) is not
sharp (see Example 7.11 below).

######  Example 7.11.

Let $(M^{n},g)$ be a closed Riemannian manifold, and take $H$ to be the
(positive) Laplacian $\Delta_{g}$ on $\mathscr{H}=L_{2}(M;g)$ with domain the
Sobolev space $W_{2}^{2}(M)$. In this case, $\ker\Delta_{g}$ is spanned by the
characteristic functions of the connected components of $M$, and so
$\dim\ker\Delta_{g}=\dim H^{0}(M)$, where $H^{0}(M)$ is the zero-th degree de
Rham cohomology space of $M$. Let $V\in L_{\infty}(M)$. Then,
$\Delta_{g}^{-1/2}V\Delta_{g}^{-1/2}\in\mathscr{L}_{n/2,\infty}$, and so from
(7.21) we get

$N^{-}(\Delta_{g}+V)-\dim
H^{0}(M)\leq\big{\|}\Delta_{g}^{-\frac{1}{2}}V\Delta_{g}^{-\frac{1}{2}}\big{\|}_{\mathscr{L}_{\frac{n}{2},\infty}}^{\frac{n}{2}}.$
(7.22)

Suppose now that $M$ has at least two connected components, and let $M_{1}$ be
such a component. If $V=-\alpha\mathbbm{1}_{M_{1}}$ with $\alpha>0$, then
$\Pi_{0}V\Pi_{0}=-\alpha\Pi_{1}$, where $\Pi_{1}$ is the orthogonal projection
onto $\mathbbm{1}_{M_{1}}$, and hence $N^{-}(\Pi_{0}V\Pi_{0})=1$. Furthermore,
if $\alpha$ is small enough, then

$N^{-}(\Delta_{g}-\alpha\mathbbm{1}_{M_{1}})=N^{-}(\Delta_{g|M_{1}}-\alpha)+N^{-}(\Delta_{g|M\setminus
M_{1}})=N^{-}(\Delta_{g|M_{1}};\alpha)=1<\dim H^{0}(M).$

Thus, in this case the l.h.s. of (7.22) is $<0$.

##  8\. Cwikel-Lieb-Rozenblum Inequalities and Lieb-Thirring Inequalities

In this section, we combine the result of the previous two sections to
establish Cwikel-Lieb-Rozenblum (CLR) inequalities and Lieb-Thirring (LT)
inequalities for (fractional) Schrödinger operators on NC tori. As we shall
see, this will lead us to a Sobolev inequality for NC tori.

###  8.1. CLR inequalities

In the notation of the previous section we let
$\mathscr{H}=L_{2}(\mathbb{T}^{n}_{\theta})$ and let $H$ be the fractional
Laplacian $\Delta^{n/2p}$, $p>0$, with domain
$W_{2}^{n/2p}(\mathbb{T}^{n}_{\theta})$. Note that $\Delta^{n/2p}$ has compact
resolvent and its nullspace is $\mathbb{C}\cdot 1$. In particular, $0$ is an
isolated eigenvalue. Up to equivalent norms, the Hilbert spaces
$\mathscr{H}_{+}$ and $\mathscr{H}_{-}$ are the Sobolev spaces
$W_{2}^{n/4p}(\mathbb{T}^{n}_{\theta})$ and
$W_{2}^{-n/4p}(\mathbb{T}^{n}_{\theta})$, respectively

Suppose that, either $s>n/2q$ and $q\geq 1$, or $s=n/2q$ and $q\geq 1$. In
this case, if $V\in L_{q}(\mathbb{T}^{n}_{\theta})$, then we know by Lemma 3.4
that $\lambda(V)$ induces a bounded operator
$\lambda(V):W^{s}_{2}(\mathbb{T}^{n}_{\theta})\rightarrow
W^{-s}_{2}(\mathbb{T}^{n}_{\theta})$. Thus, it defines a quadratic form on
$W_{2}^{s}(\mathbb{T}^{n}_{\theta})$ by

$Q_{\lambda(V)}(u,v)=\left\langle\lambda(V)u,v\right\rangle=\tau\big{[}v^{*}Vu\big{]},\qquad
u,v\in W_{2}^{s}(\mathbb{T}^{n}_{\theta}).$

In particular, if $V^{*}=V$, then

$Q_{\lambda(V)}(u,v)=\overline{\tau\big{[}u^{*}Vv\big{]}}=\overline{Q_{\lambda(V)}(v,u)}.$

That is, $Q_{\lambda(V)}$ is symmetric. If in addition $V\geq 0$, then

$Q_{\lambda(V)}(u,u)=\tau\big{[}u^{*}Vu]=\tau\big{[}(V^{1/2}u)^{*}V^{1/2}u\big{]}\geq
0.$

Thus, $Q_{\lambda(V)}\geq 0$. In particular, in the notation of Section 7 we
have

$V_{1}\leq V_{2}\Longrightarrow\lambda(V_{1})\leq\lambda(V_{2}).$ (8.1)

Bearing this in mind, Theorem 6.5 implies that the operator
$\lambda(V):W_{2}^{n/2p}(\mathbb{T}^{n}_{\theta})\rightarrow
W_{2}^{-n/2p}(\mathbb{T}^{n}_{\theta})$ is compact under any of the following
conditions:

  1. (i)

$p>1$ and $V\in L_{p}(\mathbb{T}^{n}_{\theta})$.

  2. (ii)

$p=1$ and $V\in L_{q}(\mathbb{T}^{n}_{\theta})$, $q>1$.

  3. (iii)

$p<1$ and $V\in L_{1}(\mathbb{T}^{n}_{\theta})$.

It follows that, if $V$ is selfadjoint and any of the above conditions
(i)–(iii) holds, then the pair $(\Delta^{n/2p},\lambda(V))$ fits into the
framework of the previous section. Thus, we may define the fractional
Schrödinger operator $\Delta^{\frac{n}{2p}}+\lambda(V)$ as a form sum. We
obtain a bounded from below selfadjoint operator on
$L_{2}(\mathbb{T}^{n}_{\theta})$ with compact resolvent. In particular, it has
no essential spectrum. As above we denote by $N^{-}(\Delta^{n/2p}+\lambda(V))$
the number of negative eigenvalues counted with multiplicity.

In what follows, given any selfadjoint element $V\in
L_{p}(\mathbb{T}^{n}_{\theta})$, we let $V_{\pm}=\frac{1}{2}(|V|\pm V)$ be the
positive and negative parts of $V$. Note that $V_{+}$ and $V_{-}$ are positive
elements of $L_{p}(\mathbb{T}^{n}_{\theta})$.

We are now in a position to establish CLR inequalities on NC tori in the
following form.

######  Theorem 8.1 (CLR Inequalities on NC Tori).

Let $n\geq 2$. The following holds.

  1. (i)

If $p>1$ and $V=V^{*}\in L_{p}(\mathbb{T}^{n}_{\theta})$, then

$N^{-}\big{(}\Delta^{\frac{n}{2p}}+\lambda(V)\big{)}-1\leq
c_{+}(2p)^{2p}\nu_{0}(n)s\tau\big{[}|V_{-}|^{p}\big{]}.$ (8.2)

  2. (ii)

If $V=V^{*}\in L_{p}(\mathbb{T}^{n}_{\theta})$, $p>1$, then

$N^{-}\big{(}\Delta^{\frac{n}{2}}+\lambda(V)\big{)}-1\leq
c_{2}(2p)^{2p}\nu_{0}(n)\tau\big{[}|V_{-}|^{p}\big{]}^{\frac{1}{p}}.$ (8.3)

  3. (iii)

If $p<1$ and $V=V^{*}\in L_{1}(\mathbb{T}^{n}_{\theta})$, then

$N^{-}\big{(}\Delta^{\frac{n}{2p}}+\lambda(V)\big{)}-1\leq
c_{-}(2p)^{2p}\nu_{0}(n)\tau\big{[}|V_{-}|\big{]}^{p}.$ (8.4)

Here $c_{+}(2p)$ and $c_{-}(2p)$ are the best constants in the
$\mathscr{L}_{p,\infty}$-Cwikel estimates (4.12) and (4.17), respectively, and
$c_{2}(2p)$ is given by (6.2).

###### Proof.

Suppose that $q=\max(p,1)$ if $p\neq 1$, or $q>1$ if $p=1$. Set $c(p,q)$ to be
equal to $c_{+}(2p)$ (resp., $c_{-}(2p)$, $c_{2}(2q)$) if $p>1$ (resp., $p<1$,
$p=1$). The proof amounts to show that if $V=V^{*}\in
L_{q}(\mathbb{T}^{n}_{\theta})$, then

$N^{-}\big{(}\Delta^{\frac{n}{2p}}+\lambda(V)\big{)}-1\leq
c(p,q)^{2p}\nu_{0}(n)\tau\big{[}|V_{-}|^{q}\big{]}^{\frac{p}{q}}.$ (8.5)

Note also that if $V_{-}=0$, then $V=V_{+}\geq 0$ and $Q_{\lambda(V)}\geq 0$,
and hence $\Delta^{n/2p}+\lambda(V)\geq 0$. Thus, in this case
$N^{-}(\Delta^{n/2p}+\lambda(V))=0$, and so the inequality (8.5) is trivially
satisfied. Therefore, we may assume $V_{-}\neq 0$.

As $V=V_{+}-V_{-}\geq-V_{-}$, by (8.1) we have
$\lambda(V)\geq-\lambda(V_{-})$, and so the monotonicity principle (7.7)
implies that

$N^{-}\big{(}\Delta^{\frac{n}{2p}}+\lambda(V)\big{)}\leq
N^{-}\big{(}\Delta^{\frac{n}{2p}}-\lambda(V_{-})\big{)}.$ (8.6)

Let $\Pi_{0}$ be the orthogonal projection onto
$\ker\Delta^{n/2p}=\mathbb{C}\cdot 1$. It is a rank 1 projection given by

$\Pi_{0}u=\tau(u)1,\qquad u\in L_{2}(\mathbb{T}^{n}_{\theta}).$

The formula continues to make sense for any $u\in
L_{1}(\mathbb{T}^{n}_{\theta})$. We have

$-\Pi_{0}\lambda(V_{-})\Pi_{0}1=-\tau(1)\Pi_{0}V_{-}=-\tau(V_{-})<0.$

Thus, $-\tau(V_{-})$ is a negative eigenvalue of
$-\Pi_{0}\lambda(V_{-})\Pi_{0}$. As $-\Pi_{0}\lambda(V_{-})\Pi_{0}$ has rank 1
this is the unique negative eigenvalue, and hence
$N^{-}(-\Pi_{0}\lambda(V_{-})\Pi_{0})=1$. Combining this with (8.6) gives

$N^{-}\big{(}\Delta^{\frac{n}{2p}}+\lambda(V)\big{)}-1\leq
N^{-}\big{(}\Delta^{\frac{n}{2p}}-\lambda(V_{-})\big{)}-N^{-}(-\Pi_{0}\lambda(V_{-})\Pi_{0}).$
(8.7)

Moreover, as $0\leq V_{-}\in L_{q}(\mathbb{T}^{n}_{\theta})$, Theorem 6.5 and
Remark 6.6 ensure us that
$\Delta^{-n/4p}\lambda(V_{-})\Delta^{-n/4p}\in\mathscr{L}_{p,\infty}$ with
norm estimate,

$\big{\|}\Delta^{-\frac{n}{4p}}\lambda(V)\Delta^{-\frac{n}{4p}}\big{\|}_{\mathscr{L}_{p,\infty}}\leq
c(p,q)^{2}\nu_{0}(n)^{\frac{1}{p}}\|V_{-}\|_{L_{q}}=c(p,q)^{2}\nu_{0}(n)^{\frac{1}{p}}\tau\big{[}|V_{-}|^{q}\big{]}^{\frac{1}{q}}.$
(8.8)

This allows us to apply the borderline Birman-Schwinger principle (Theorem
7.9) to get

$N^{-}\big{(}\Delta^{\frac{n}{2p}}-\lambda(V_{-})\big{)}-N^{-}(-\Pi_{0}\lambda(V_{-})\Pi_{0})\leq\big{\|}\Delta^{-\frac{n}{4p}}\lambda(V_{-})\Delta^{-\frac{n}{4p}}\big{\|}_{\mathscr{L}_{p,\infty}}^{p}.$

Combining this with (8.7) and (8.8) gives the inequality (8.5). The proof is
complete. ∎

######  Remark 8.2.

For the ordinary torus $\mathbb{T}^{n}$, i.e., $\theta=0$, the critical CLR
inequality (8.3) actually hold for potentials $V$ in the Orlicz class
$L\\!\log\\!L(\mathbb{T}^{n})$ (see [72, 81, 82]).

######  Remark 8.3.

Suppose that either $q=\max(p,1)$ and $p\neq 1$, or $q>1=p$. The CLR
inequalities (8.2)–(8.4) are consistent with Lieb’s version of the CLR
inequality for Schrödinger operators on closed manifolds (see [54, 56]). Note
that we cannot expect estimates of the form,

$N^{-}\big{(}\Delta^{\frac{n}{2p}}+\lambda(V)\big{)}\leq
c_{npq}\tau\big{[}|V_{-}|^{q}\big{]}^{\frac{p}{q}}.$ (8.9)

To see this take $V=-\epsilon$ with $\epsilon>0$. In that case the right-hand
side of (8.9) is $c_{npq}\tau[\epsilon^{q}]^{p/q}=c_{npq}\epsilon^{p}$. In
addition $(\Delta^{{n}/{2p}}-\epsilon)1=-\epsilon 1$, and hence $-\epsilon$ is
always a negative eigenvalue of $\Delta^{{n}/{2p}}-\epsilon$. Thus,
$N^{-}(\Delta^{{n}/{2p}}-\epsilon)\geq 1$. Therefore, if we had the inequality
(8.9), then we would have

$1\leq N^{-}\big{(}\Delta^{\frac{n}{2p}}-\epsilon\big{)}\leq
c_{npq}\epsilon^{p}\qquad\forall\epsilon>0,$

which does not hold as soon as $\epsilon$ is small enough.

######  Remark 8.4.

The most interesting case of the above CLR inequalities is for the Schrödinger
operator $\Delta+\lambda(V)$, i.e., $p=n/2$. In this case Theorem 8.1
specializes to the following:

  1. (i)

If $n\geq 3$ and $V=V^{*}\in L_{n/2}(\mathbb{T}^{n}_{\theta})$, then

$N^{-}\big{(}\Delta+\lambda(V)\big{)}-1\leq
c_{+}(n)^{n}\nu_{0}(n)\tau\big{[}|V_{-}|^{\frac{n}{2}}\big{]}.$ (8.10)

  2. (ii)

If $n=2$ and $V=V^{*}\in L_{p}(\mathbb{T}^{2}_{\theta})$, $p>1$, then

$N^{-}\big{(}\Delta+\lambda(V)\big{)}-1\leq
c_{2}(2p)^{2p}\nu_{0}(2)\tau\big{[}|V_{-}|^{p}\big{]}^{\frac{1}{p}}.$ (8.11)

######  Remark 8.5.

The 2-dimensional case (8.11) shows a stark contrast with the Euclidean space
case, since on $\mathbb{R}^{2}$ the Birman-Schwinger principle and the CLR
inequality for Schrödinger operators $\Delta+V$ do not hold. In particular,
recent results of Hoang  _et al._ [37, Theorem 3.1] show that if $V\in
L_{1}(\mathbb{R}^{2})\setminus 0$ is such that $\int V(x)dx\leq 0$, then
$\Delta+V$ always has at least one negative eigenvalue (see also Simon [88]).

######  Remark 8.6.

The cases $p\neq n/2$ are also of interest, since fractional Schrödinger
operators naturally appear in the framework of fractional quantum mechanics
[48]. For $p=n$ we get the hyper-relativistic Schrödinger operator
$|\nabla|+V$ (see, e.g., [23]).

###  8.2. Semiclassical Weyl’s laws

Although the inequality (8.9) does not hold, it holds _semiclassically_.
Namely, we have the following result.

######  Corollary 8.7 (Semiclassical CLR Inequality).

Assume that, either $q=\max(p,1)$ with $p\neq 1$, or $q>p=1$. Let $V=V^{*}\in
L_{q}(\mathbb{T}^{n}_{\theta})$. Then, as $h\rightarrow 0^{+}$ we have

$N^{-}\big{(}h^{\frac{n}{p}}\Delta^{\frac{n}{2p}}+\lambda(V)\big{)}\leq
c(p,q)^{2p}\nu_{0}(n)h^{-n}\tau\big{[}|V_{-}|^{q}\big{]}^{\frac{p}{q}}+\operatorname{O}(1).$
(8.12)

Here, $c(p,q)$ is the same as in the proof of Theorem 8.1.

###### Proof.

As $h^{n/p}\Delta^{n/2p}+\lambda(V)=h^{n/p}(\Delta+\lambda(h^{-n/p}V))$, the
operators $h^{n/p}\Delta^{n/2p}+\lambda(V)$ and $\Delta+\lambda(h^{-n/p}V)$
have the same number of negative eigenvalues counted with multiplicity.
Therefore, the CLR inequality (8.5) for $h^{-n/p}V$ gives

$\displaystyle
N^{-}\big{(}h^{\frac{n}{p}}\Delta^{\frac{n}{2p}}+\lambda(V)\big{)}=N^{-}\big{(}\Delta^{\frac{n}{2p}}+\lambda\big{(}h^{-\frac{n}{p}}V\big{)}\big{)}$
$\displaystyle\leq
c(p,q)^{2p}\nu_{0}(n)\tau\big{[}\big{|}h^{-\frac{n}{p}}V_{-}\big{|}^{q}\big{]}^{\frac{p}{q}}+1$
$\displaystyle\leq
c(p,q)^{2p}\nu_{0}(n)h^{-n}\tau\big{[}|V_{-}|^{q}\big{]}^{\frac{p}{q}}+\operatorname{O}(1).$

The proof is complete. ∎

The semiclassical CLR inequality on $\mathbb{R}^{n}$, $n\geq 3$, is the main
tool to extend the semi-classical Weyl’s law for Schrödinger operators
$h^{2}\Delta+V$ with smooth potentials $V$ to Schrödinger operators with
potentials in $L_{n/2}(\mathbb{R}^{n})$ (see [89, Proposition 5.2]). This
result can be traced back to the work of Birman and his collaborators in the
late 60s and early 70s (see [6, 11]). Likewise, we can extend the
semiclassical Weyl’s law for fractional Schrödinger operators
$h^{\frac{n}{p}}\Delta^{\frac{n}{2p}}+V$ for smooth potentials to potentials
in a suitable $L_{q}$-class (see [93] and the references therein).

Similarly, the semiclassical CLR inequality (8.12) for NC tori would enable us
to extend the semiclassical Weyl law on NC tori for smooth potentials to non-
smooth potentials. However, to date the semiclassical Weyl law for smooth
potentials has still not been established. We believe it can be established in
a similar fashion as in the Euclidean case by setting up a semi-classical
pseudodifferential calculus on NC tori. However, this falls out of the scope
of this paper. Therefore, we only state the semiclassical Weyl’s law on NC
tori as a conjecture.

######  Conjecture 8.8 (Semiclassical Weyl Law on NC Tori).

Let $n\geq 2$. Suppose that, either $p\neq 1$ and $q=\max(p,1)$, or $p=1<q$.
If $V=V^{*}\in L_{q}(\mathbb{T}^{n}_{\theta})$, then as $h\rightarrow 0^{+}$
we have

$N^{-}\big{(}h^{\frac{n}{p}}\Delta^{\frac{n}{2p}}+\lambda(V)\big{)}=c(n)h^{-n}\tau\big{[}|V_{-}|^{p}\big{]}+\operatorname{o}\big{(}h^{-n}\big{)},\quad\text{where}\
c(n)=|\mathbb{B}^{n}|.$ (8.13)

In particular, if $n\geq 3$ and $V=V^{*}\in L_{n/2}(\mathbb{T}^{n}_{\theta})$,
or if $n=2$ and $V=V^{*}\in L_{q}(\mathbb{T}^{n}_{\theta})$ with $q>1$, then

$N^{-}\big{(}h^{2}\Delta+\lambda(V)\big{)}=c_{n}h^{-n}\tau\big{[}|V_{-}|^{\frac{n}{2}}\big{]}+\operatorname{o}\big{(}h^{-n}\big{)}.$

######  Remark 8.9.

In the critical case $p=1$ we may hope that the semiclassical Weyl’s law
(8.13) further hold for potentials in the NC Orlicz class
$L\\!\log\\!L(\mathbb{T}^{n}_{\theta})$. Establishing this would require
extending the Cwikel estimate (6.9) and the CLR inequality (8.3) to this class
of potentials. However, at this point it is still unclear how to do this.

######  Remark 8.10.

An alternative route to establish the semiclassical Weyl’s law (8.13) for
$L_{q}$-potentials is outlined in the sequel article [65]. It is based on the
(borderline) Birman-Schwinger principle (7.12) and a (conjectural) version for
NC tori of the celebrated Weyl’s law for negative order pseudodifferential
operators of Birman-Solomyak [8, 9, 10] (see also [71]). Note that the
conjecture in [65] extends Conjecture 8.8 to _curved_ NC tori.

######  Remark 8.11.

After earlier versions of this article and the sequel [65] were posted on