# Weyl’s Laws and Connes’ Integration Formulas for Matrix-Valued
$L\\!\log\\!L$-Orlicz Potentials

Raphaël Ponge  School of Mathematics, Sichuan University, Chengdu, China
<EMAIL_ADDRESS>
###### Abstract.

Thanks to the Birman-Schwinger principle, Weyl’s laws for Birman-Schwinger
operators yields semiclassical Weyl’s laws for the corresponding Schrödinger
operators. In a recent preprint Rozenblum established quite general Weyl’s
laws for Birman-Schwinger operators associated with pseudodifferential
operators of critical order and potentials that are product of
$L\\!\log\\!L$-Orlicz functions and Alfhors-regular measures supported on a
submanifold. In this paper, we show that, for matrix-valued
$L\\!\log\\!L$-Orlicz potentials supported on the whole manifold, Rozenblum’s
results are direct consequences of the Cwikel-type estimates on tori recently
established by Sukochev-Zanin. As applications we obtain CLR-type inequalities
and semiclassical Weyl’s laws for critical Schrödinger operators associated
with matrix-valued $L\\!\log\\!L$-Orlicz potentials. Finally, we explain how
the Weyl’s laws of this paper imply a strong version of Connes’ integration
formula for matrix-valued $L\\!\log\\!L$-Orlicz potentials.

##  1\. Introduction

The study of Schrödinger operators $h^{2}\Delta+V$, and more generally
fractional Schrödinger operators $h^{n/p}\Delta^{n/2p}+V$ on domains of
$\mathbb{R}^{n}$, as well as on manifolds, is an important focus of interest
in mathematical physics. Of special interest are semiclassical Weyl’s laws
that describe the semiclassical behaviour of the number of bound states, i.e.,
the number of negative eigenvalues $N^{-}(h^{2}\Delta+V)$. Thanks to the
Birman-Schwinger principle this translates into determining the asymptotic
distribution of the negative eigenvalues of Birman-Schwinger operators
$\Delta^{-n/2p}V\Delta^{-n/2p}$. In the 60s and 70s Weyl’s laws for positive
and negative eigenvalues of Birman-Schwinger operators and semiclassical
Weyl’s laws for the corresponding Schrödinger operators were obtained on
$\mathbb{R}^{n}$ and bounded domains of $\mathbb{R}^{n}$ for $p<1$ with $V\in
L_{1}$, and for $p>1$ with $V\in L_{p}$ (see [7, 56, 57]). In particular, we
have semiclassical Weyl’s laws for Schrödinger operators $\Delta+V$ with $V\in
L_{n/2}$ in any dimension $n\geq 3$ (see also [21, 44, 65]).

The critical case $p=1$, including Schrödinger operators in 2D, is the most
challenging case. Significant progress on this case was made in _even_
dimension by Solomyak [69, 70] in the mid-90s. He obtained eigenvalues
estimates and Weyl’s laws for operators $\Delta^{-n/4}V\Delta^{-n/4}$ for
$L\\!\log\\!L$-Orlicz potentials on bounded domains of $\mathbb{R}^{n}$. This
yields semiclassical Weyl’s laws for critical Schrödinger operators
$h^{2}\Delta^{n/2}+V$ in this setting. On $\mathbb{R}^{n}$, once again with
$n$ even, Solomyak further obtained eigenvalues inequalities for operators
$(\Delta^{-n/4}+\mu^{2})V(\Delta^{-n/4}+\mu^{2})$, $\mu>0$, where $V$ is a
locally $L\\!\log\\!L$-Orlicz. Thanks to the Birman-Schwinger principle this
provides bounds on the number of bound states $<-\mu^{2}$ of the critical
Schrödinger operator $\Delta^{n/2}+V$. There are also results for $\mu=0$ by
suitably restricting the domain of $\Delta$ (see [70]).

Solomyak’s results triggered an intensive activity on eigenvalues estimates
for Schrödinger operators in 2D (see, e.g., [5, 17, 31, 38, 39, 40, 41, 51,
71]). We also refer to [6] for further results on critical Schrödinger
operators in even dimension $\geq 2$. However, the odd dimensional case
remained merely unattended until recent articles of Rozenblum-Shargorodsky
[59] and Rozenblum [58]. In particular, by using the eigenvalue inequalities
of [59], Rozenblum [58] established Weyl’s laws in _any_ dimension $n\geq 2$
for operators $P^{*}uP$, where $P$ is a $\Psi$DO of order $-n/2$ on a closed
manifold $M$ and $u$ is a potential of the form $u=f\mu$, where $\mu$ is an
Alfhors-regular measure supported on a regular submanifold $\Sigma\subset M$
and $f$ is a real-valued $L\\!\log\\!L$-Orlicz function on $\Sigma$ with
respect to $\mu$ (see also [61] for the non-critical case). The main results
of [58] further include an extension of Connes’ trace theorem for these
operators. This shows that even singular measures can be recovered from
Connes’ integral. This also exhibits an interesting link between semiclassical
analysis and Connes’ noncommutative geometry.

Meanwhile, Sukochev-Zanin [72] obtained Cwikel-type estimates on any torus
$\mathbb{T}^{n}$, $n\geq 2$, for operators of the form
$(1+\Delta)^{-n/4}f(1+\Delta)^{-n/4}$, where $f$ is an $L\\!\log\\!L$-Orlicz
function on $\mathbb{T}^{n}$. One goal of this paper is to show that, in the
case of smooth measures, the results of Rozenblum [58] are direct consequences
of the Cwikel-type estimates of [72]. Although, we don’t obtain the results at
the level of generality as in [58], for smooth measures this still allows us
to deal with the odd dimensional case, which had been out of focus until
recently. Furthermore, our approach simplifies and supersedes the main results
of [73].

Throughout this paper we work with a closed Riemannian manifold $(M^{n},g)$
and a Hermitian vector bundle $E$ over $M$. Thus, the results of this paper
are established for matrix-valued potentials, i.e., $L\\!\log\\!L$-Orlicz
sections of $\operatorname{End}(E)$. The main ingredient is a Cwikel-type
estimate for operators of the form $QuP$, where $P$ and $Q$ are operators in
$\Psi^{-n/2}(M,E)$ and $u$ is a matrix-valued $L\\!\log\\!L$-Orlicz potential.
In this case, the operator $QuP$ is in the weak trace class
$\mathscr{L}_{1,\infty}$, and we have

(1.1) $\big{\|}QuP\big{\|}_{1,\infty}\leq C_{PQ}\|u\|_{L\\!\log\\!L},$

where the constant $C_{PQ}$ depends only on $P$ and $Q$ (see Proposition
3.13). This extends the “specific” Cwikel-type estimates on tori of [72]. In
fact, thanks to the (pseudo-)local nature of these estimates it is a routine
argument to get the former from the latter (compare [73]).

Once the Cwikel-type estimates (1.1) are established, there is no difficulty
to get Weyl laws for operators $QuP$ as above by combining these estimates
with the perturbation theory of Birman-Solomyak [7] and the Weyl’s laws for
negative order $\Psi$DOs of Birman-Solomyak [10, 11, 12] (see Theorem 4.1 for
the precise statement). In particular, if $P$ and $Q$ are operators in
$\Psi^{-n/2}(M,E)$ and $u$ is a matrix-valued $L\\!\log\\!L$-Orlicz potential,
then

(1.2)
$\lim_{j\rightarrow\infty}j\mu_{j}\left(QuP\right)=\frac{1}{n}(2\pi)^{-n}\\!\int_{S^{*}M}\operatorname{tr}_{E}\big{[}\left|\sigma(Q)(x,\xi)u(x)\sigma(P)(x,\xi)\right|\big{]}dxd\xi,$

where $\mu_{0}(QuP)\geq\mu_{1}(QuP)\geq\cdots$ are the singular values of
$QuP$, and $\sigma(P)$ (resp., $\sigma(Q)$) is the principal symbol of $P$
(resp., $Q$). If $Q=P^{*}$ and $u(x)^{*}=u(x)$, then the operator $P^{*}uP$ is
selfadjoint and satisfies the Weyl’s law,

(1.3)
$\lim_{j\rightarrow\infty}j\lambda^{\pm}_{j}\left(P^{*}uP\right)=\frac{1}{n}(2\pi)^{-n}\\!\int_{S^{*}M}\operatorname{tr}_{E}\left[\big{(}\sigma(P)(x,\xi)^{*}u(x)\sigma(P)(x,\xi)\big{)}_{\pm}\right]dxd\xi,$

where $\lambda_{0}^{\pm}(P^{*}uP)\geq\lambda_{1}^{\pm}(P^{*}uP)\geq\cdots\geq
0$ are the positive/negative eigenvalues of $P^{*}uP$ and the subscript $\pm$
refers to the positive/negative parts in $\operatorname{End}(E_{x})$. If
$\Delta_{E}=\nabla^{*}\nabla$ is the Laplacian of some Hermitian connection on
$E$ and we take $P=Q=\Delta_{E}^{-n/4}$, then we get

(1.4)
$\displaystyle\lim_{j\rightarrow\infty}j\mu_{j}\left(\Delta_{E}^{-\frac{n}{4}}u\Delta_{E}^{-\frac{n}{4}}\right)=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}|u(x)|\big{]}\sqrt{g(x)}dx,$
(1.5)
$\displaystyle\lim_{j\rightarrow\infty}j\lambda^{\pm}_{j}\left(\Delta_{E}^{-\frac{n}{4}}u\Delta_{E}^{-\frac{n}{4}}\right)=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}u(x)_{\pm}\big{]}\sqrt{g(x)}dx\qquad\text{if
$u(x)^{*}=u(x)$}.$

In the form stated above the Weyl’s law (1.2) is new. As mentioned above, in
the scalar case Rozenblum [58] estasblished the Weyl’s law (1.3) for a larger
class of potentials. Still in the scalar case, Sukochev-Zanin [73] obtained
the estimates (1.1) and the Weyl’s law (1.3) in the special case
$P=Q=(1+\Delta_{g})^{-n/4}$. The approach in [73] uses the Cwikel-type
estimates of [72], but it also relies on some deep results on commutators in
weak Schatten classes from [25, 34]. Once again, our prime goal here is
explaining that, for our class of $L\\!\log\\!L$-Orlicz potentials, the
Cwikel-type estimates (1.1) and all the Weyl’s laws (1.2)–(1.5) are direct
consequences of the Cwikel-type estimates of [72]. We also stress that the
Weyl’s laws (1.4)–(1.5) hold _verbatim_ if we replace $\Delta_{E}$ by any
Laplace-type operator. In particular, they hold for squares of Dirac-type
operators (see Proposition 6.6).

We also give applications of the above to the semiclassical analysis of
critical Schrödinger operators $\Delta_{E}^{n/2}+V$, where $V$ is a
(Hermitian) matrix-valued $L\\!\log\\!L$-Orlicz potential. The Cwikel estimate
(1.1) for $Q=P=\Delta_{E}^{-n/2}$ ensures that $V$ is $\Delta_{E}^{n/2}$-form
compact, and so the operator $\Delta_{E}^{n/2}+V$ makes sense as a form sum.
It is selfadjoint and bounded from below and has pure discrete spectrum. As
above we denote by $N^{-}(\Delta_{E}^{n/2}+V)$ its number of negative
eigenvalues.

By a standard application of the abstract Birman-Schwinger principle mentioned
above, the Weyl’s law (1.5) implies the following semiclassical Weyl’s law
(_cf_. Corollary 4.5),

(1.6) $\lim_{h\rightarrow
0^{+}}h^{n}N^{-}\big{(}h^{n}\Delta_{E}^{\frac{n}{2}}+V\big{)}=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}V(x)_{-}\big{]}\sqrt{g(x)}dx.$

Semiclassical Weyl’s laws are not considered in [58]. However, the same
argument as above shows that the Weyl’s laws of [58] similarly yields a
semiclassical Weyl’s law for the type of potentials considered in [58].

Another application concerns a version of the Cwikel-Lieb-Rozenblum (CLR)
inequality for $L\\!\log\\!L$-Orlicz potentials (see Section 4.2 for
background on the CLR inequality). By combining the Cwikel-type estimates
(1.1) for $P=Q=\Delta_{E}^{-n/4}$ with the borderline Birman-Schwinger
principle of [48] we immediately get the following CLR-type inequality,

(1.7)
$N^{-}\big{(}\Delta_{E}^{\frac{n}{2}}+V\big{)}-N^{+}(\Pi_{0}V_{-}\Pi_{0})\leq
C_{\nabla}\|V_{-}\|_{L\\!\log\\!L},$

where the constant $C_{\nabla}$ does not depend on $V$ (see Corollary 4.7).
Here $\Pi_{0}$ is the orthogonal projection onto $\ker\Delta_{E}$ and
$N^{+}(\Pi_{0}V_{-}\Pi_{0})$ is the number of positive eigenvalues of
$\Pi_{0}V_{-}\Pi_{0}$. In particular, we obtain a version of the CLR
inequality for Schrödinger operators $\Delta_{E}+V$ on 2-dimensional closed
manifolds. Note that the original CLR inequality does not hold on
$\mathbb{R}^{2}$.

We also clarify the links between the Weyl’s laws (1.2)–(1.5) and Connes’
integration. In the framework of Connes’ noncommutative geometry [20] the role
of the integral is played by positive traces on the weak trace class
$\mathscr{L}_{1,\infty}$ (see also Section 5 and the references therein). We
define the NC integral of an operator $A\in\mathscr{L}_{1,\infty}$ by

(1.8) $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int A:=\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{j<N}\lambda_{j}(A)\quad\text{provided the limit exists},$

where $(\lambda_{j}(A))_{j\geq 0}$ is any eigenvalue sequence for $A$ such
that $|\lambda_{0}(A)|\geq|\lambda_{1}(A)|\geq\cdots$. An operator for which
the above limit exists is called _measurable_. The measurable operators form a
closed subspace of $\mathscr{L}_{1,\infty}$ which contains the commutator
subspace $\operatorname{Com}(\mathscr{L}_{1,\infty})$ on which the NC integral
is a positive linear trace.

Measurability of weak trace-class operators is equivalently described in terms
of Dixmier traces (see [24, 20, 47, 54]). These traces extends to the Dixmier-
Macaev class $\mathfrak{M}_{1,\infty}$, which is strictly larger than
$\mathscr{L}_{1,\infty}$. There are many traces on $\mathscr{L}_{1,\infty}$
that are not Dixmier or do not extend to $\mathfrak{M}_{1,\infty}$. We say
that an operator $A\in\mathscr{L}_{1,\infty}$ is _strongly measurable_ if, for
every (normalized) positive trace $\varphi$ on $\mathscr{L}_{1,\infty}$, the
value $\varphi(A)$ is given by the limit in (1.8). We stress that this notion
of measurability is stronger than the notion of measurability considered in
[58], which is defined in terms of traces on $\mathfrak{M}_{1,\infty}$.
Moreover, as pointed out in [54], Weyl’s laws for operators in
$\mathscr{L}_{1,\infty}$ imply strong measurability in the sense meant in this
paper.

Let $(M^{n},g)$ be a closed Riemannian manifold and $E$ a Hermitian vector
bundle over $M$. By Connes’s trace theorem [19, 36] any operator
$P\in\Psi^{-n}(M,E)$ is strongly measurable, and we have

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
P=\frac{1}{n}\int_{S^{*}M}\operatorname{tr}_{E}\big{[}\sigma(P)(x,\xi)\big{]}dxd\xi.$

In particular, for $P=\Delta_{E}^{-n/4}f\Delta_{E}^{-n/4}$ with $f\in
C^{\infty}(M)$ we get Connes’ integration formula,

(1.9) $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\Delta_{E}^{-\frac{n}{4}}u\Delta_{E}^{-\frac{n}{4}}=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}u(x)\big{]}\sqrt{g(x)}dx.$

This shows that Connes’ integral recaptures the Riemannian volume measure.

The Weyl’s laws (1.2)–(1.3) and the observations of [54] immediately imply the
following extension of Connes’ trace theorem (see Theorem 6.1). If $P$ and $Q$
are operators in $\Psi^{-n/2}(M,E)$ and $u$ is a matrix-valued potential
$L\\!\log\\!L$-Orlicz potential, then the operators $QuP$ and $|QuP|$ are both
strongly measurable, and we have

(1.10) $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
QuP=\frac{1}{n}(2\pi)^{-n}\\!\int_{S^{*}M}\operatorname{tr}_{E}\big{[}\sigma(Q)(x,\xi)u(x)\sigma(P)(x,\xi)\big{]}dxd\xi,$
(1.11) $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\big{|}QuP\big{|}=\frac{1}{n}(2\pi)^{-n}\\!\int_{S^{*}M}\operatorname{tr}_{E}\big{[}\left|\sigma(Q)(x,\xi)u(x)\sigma(P)(x,\xi)\right|\big{]}dxd\xi.$

In particular, Connes’ integration formula (1.9) holds _verbatim_ for matrix-
valued $L\\!\log\\!L$-potentials. This also leads to semiclassical
interpretation of Connes’ integration formula (see Remark 6.5).

Note that (1.11) yields an integration formula for the absolute value
$|\Delta_{E}^{-\frac{n}{4}}u\Delta_{E}^{-\frac{n}{4}}|$ as well. Namely, if
$u$ is any matrix-valued $L\\!\log\\!L$-Orlicz potential, then

(1.12) $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\bigg{|}\Delta_{E}^{-\frac{n}{4}}u\Delta_{E}^{-\frac{n}{4}}\bigg{|}=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}|u(x)|\big{]}\sqrt{g(x)}dx.$

We also have versions of the trace formulas (1.9) and (1.12) for (squares of)
Dirac-type operators and matrix-valued $L\\!\log\\!L$-Orlicz potentials (see
Proposition 6.6).

The trace formula (1.11) is new. In the scalar case Rozenblum [58] established
the trace formula (1.10) for a larger class of potentials. As mentioned above,
a weaker notion of measurability is used in [58]. However, the same arguments
as above show that the operators under consideration in [58] are strongly
measurable in the sense used in this paper. Still in the scalar case, a weaker
version of the trace formula (1.10) in the special case
$P=Q=(1+\Delta)^{-n/4}$ and $u\in L_{\infty}(M)$, is given in [73].

The remainder of this paper is organized as follows. In Section 2, we review
some facts regarding weak Schatten classes and Birman-Solomyak’s results on
Weyl’s laws for compact operators. In Section 3, we establish the Cwikel-type
estimates (1.1). In Section 4, we derive the Weyl’s laws (1.2)–(1.5) for
matrix-valued $L\\!\log\\!L$-Orlicz potentials. We also establish the
semiclassical Weyl’s law (1.6) and the CLR-type inequality (1.7). In Section
5, we review the main facts regarding Connes’ integration and its relationship
with Weyl’s laws. Finally, in Section 6, we deal with the integration formulas
(1.10)–(1.11) and their consequences.

### Aknowledgements

I wish to thank Grigori Rozenblum, Edward McDonald, and Fedor Sukochev for
various stimulating discussions related to the subject matter of this article.
I also thank the University of Ottawa for its hospitality during the whole
preparation of this paper.

##  2\. Weak Schatten Classes and Weyl’s Laws for Compact Operators

In this section, we recall the main facts regarding weak Schatten classes and
Birman-Solomyak’s results on Weyl’s laws for compact operators.

Throughout this section we let $\mathscr{H}$ be a (separable) Hilbert space
with inner product $\left\langle\cdot|\cdot\right\rangle$. We denote by
$\mathscr{L}(\mathscr{H})$ the $C^{*}$-algebra of bounded operators on
$\mathscr{H}$ with norm $\|\cdot\|$.

###  2.1. Weak Schatten Classes

We briefly review the main definitions and properties regarding weak Schatten
classes on $\mathscr{H}$. We refer to [66, 30] for further details.

Let $\mathscr{K}$ be (closed) the ideal of compact operators on $\mathscr{H}$.
Given any operator $T\in\mathscr{K}$ we denote by $(\mu_{j}(T))_{j\geq 0}$ its
sequence of _singular values_ , i.e., $\mu_{j}(T)$ is the $(j+1)$-th
eigenvalue counted with multiplicity of the absolute value
$|T|=\sqrt{T^{*}T}$. The _min-max principle_ states that

(2.1) $\displaystyle\mu_{j}(T)$
$\displaystyle=\min\left\\{\|T_{|E^{\perp}}\|;\ \dim E=j\right\\}.$

The min-max principle implies the following properties (see, e.g., [30, 66]),

(2.2) $\displaystyle\mu_{j}(T)=\mu_{j}(T^{*})=\mu_{j}(|T|),$ (2.3)
$\displaystyle\mu_{j+k}(S+T)\leq\mu_{j}(S)+\mu_{k}(T),$ (2.4)
$\displaystyle\mu_{j}(ATB)\leq\|A\|\mu_{j}(T)\|B\|,\qquad
A,B\in\mathscr{L}(\mathscr{H}).$

The inequality (2.3) is known as Ky Fan’s inequality [26].

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 and a quasi-Banach ideal with respect to the quasi-
norm,

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

For $p>1$, the quasi-norm is equivalent to the norm,

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

Thus, in this case $\mathscr{L}_{p,\infty}$ is a Banach ideal with respect to
that equivalent norm.

In addition, we denote by $(\mathscr{L}_{p,\infty})_{0}$ the closure in
$\mathscr{L}_{p,\infty}$ of the finite-rank operators. We have

$\big{(}\mathscr{L}_{p,\infty}\big{)}_{0}=\left\\{T\in\mathscr{K};\
\mu_{j}(T)=\operatorname{o}\big{(}j^{-\frac{1}{p}}\big{)}\right\\}.$

We note the continuous inclusions,

$\mathscr{L}_{p}\subsetneq\big{(}\mathscr{L}_{p,\infty}\big{)}_{0}\subsetneq\mathscr{L}_{p,\infty}\subsetneq\mathscr{L}_{q},\qquad
0<p<q.$

Let $\mathscr{H}^{\prime}$ be another Hilbert space, and let
$\iota:\mathscr{H}^{\prime}\rightarrow\mathscr{H}$ be a Hilbert space
embedding, i.e., a continuous linear map which is one-to-one and has closed
range. For instance, we may take $\iota$ to be an isometry. Set
$\mathscr{H}_{1}=\operatorname{ran}\iota$; by assumption this is a closed
subspace of $\mathscr{H}$. We get a continuous linear isomorphism
$\iota:\mathscr{H}^{\prime}\rightarrow\mathscr{H}_{1}$ with inverse
$\iota^{-1}:\mathscr{H}_{1}\rightarrow\mathscr{H}^{\prime}$. Thus, if
$\pi:\mathscr{H}\rightarrow\mathscr{H}_{1}$ is the orthogonal projection onto
$\mathscr{H}_{1}$, then $\iota^{-1}\circ\pi$ is a left-inverse of $\iota$. The
pushforward
$\iota_{*}:\mathscr{L}(\mathscr{H}^{\prime})\rightarrow\mathscr{L}(\mathscr{H})$
is then defined by

(2.6) $\iota_{*}A=\iota\circ A\circ(\iota^{-1}\circ\pi),\qquad
A\in\mathscr{L}(\mathscr{H}^{\prime}).$

This is an embedding of (unital) Banach algebras. If $\iota$ is an isometry,
then
$\iota_{*}:\mathscr{L}(\mathscr{H}^{\prime})\rightarrow\mathscr{L}(\mathscr{H})$
is an isometric $*$-homomorphism.

In what follows, for $p>0$, we denote by $\mathscr{L}_{p,\infty}(\mathscr{H})$
and $\mathscr{L}_{p,\infty}(\mathscr{H}^{\prime})$ the corresponding weak
Schatten classes on $\mathscr{H}$ and $\mathscr{H}^{\prime}$, respectively.

######  Proposition 2.1 (see, e.g., [54, Appendix]).

The following holds.

  1. (1)

If $A$ is a compact operator on $\mathscr{H}^{\prime}$, then $A$ and
$\iota_{*}A$ have the same non-zero eigenvalues with the same algebraic
multiplicities.

  2. (2)

There is $c>0$ such that, for every compact operator $A$ on
$\mathscr{H}^{\prime}$, we have

$c^{-1}\mu_{j}(A)\leq\mu_{j}\big{(}\iota_{*}A)\leq c\mu_{j}(A)\qquad\forall
j\geq 0.$

We may take $c=1$ when $\iota$ is an isometric embedding.

  3. (3)

The pushforward map (2.6) induces a quasi-Banach embedding
$\iota_{*}:\mathscr{L}_{p,\infty}(\mathscr{H}^{\prime})\rightarrow\mathscr{L}_{p,\infty}(\mathscr{H})$
for each $p>0$. This is an isometric embedding when $\iota$ is an isometry.

###  2.2. Weyl operators

In what follows, given any selfadjoint operator $A\in\mathscr{K}$, we denote
by $(\pm\lambda^{\pm}(A))_{j\geq 0}$ its sequences of positive and negative
eigenvalues, so that

$\lambda^{\pm}_{0}(A)\geq\lambda^{\pm}_{1}(A)\geq\cdots\geq 0,$

where each eigenvalue is repeated according to multiplicity. More precisely,
$\lambda_{j}^{\pm}(A)=\mu_{j}(A^{\pm})$, where $A^{\pm}=\frac{1}{2}(|A|\pm A)$
are the positive and negative parts of $A$.

We refer to [14, §9.2] for the main properties of the positive/negative
eigenvalue sequences of selfadjoint compact operators. In particular, we have
the following min-max principle (_cf_. [14, Theorem 9.2.4]),

$\lambda^{\pm}_{j}(A)=\min\bigg{\\{}\max_{0\neq\xi\in
E^{\perp}}\pm\frac{\left\langle
A\xi|\xi\right\rangle}{\left\langle\xi|\xi\right\rangle};\ \dim
E=j\bigg{\\}}.$

This implies the following version of Ky Fan’s inequality (_cf_. [14, Theorem
9.2.8]),

(2.7)
$\lambda^{\pm}_{j+k}(A+B)\leq\lambda^{\pm}_{j}(A)+\lambda^{\pm}_{k}(B),\qquad
j,k\geq 0.$

######  Definition 2.2.

We say that $A\in\mathscr{L}_{p,\infty}$, $p>0$ is a _Weyl operator_ if one of
the following conditions applies:

  1. (i)

$A\geq 0$ and $\lim j^{1/p}\lambda_{j}(A)$ exists.

  2. (ii)

$A^{*}=A$ and $\lim j^{1/p}\lambda_{j}^{+}(A)$ and $\lim
j^{1/p}\lambda_{j}^{-}(A)$ both exist.

  3. (iii)

The real part $\Re A=\frac{1}{2}(A+A^{*})$ and the imaginary part $\Im
A=\frac{1}{2i}(A-A^{*})$ of are both Weyl operators in the sense of (ii).

We denote by $\mathscr{W}_{p,\infty}$ the class of Weyl operators in
$\mathscr{L}_{p,\infty}$. If $A\in\mathscr{W}_{p,\infty}$, $A\geq 0$, we set

$\Lambda(A)=\lim_{j\rightarrow\infty}j^{\frac{1}{p}}\lambda_{j}(A).$

If $A=A^{*}\in\mathscr{W}_{p,\infty}$, we set

$\Lambda^{\pm}(A)=\lim_{j\rightarrow\infty}j^{\frac{1}{p}}\lambda^{\pm}_{j}(A).$

Finally, for a general $A\in\mathscr{W}_{p,\infty}$, we define

$\Lambda^{\pm}(A)=\Lambda^{\pm}\big{(}\Re A\big{)}+i\Lambda^{\pm}\big{(}\Im
A\big{)}.$

######  Remark 2.3.

Given any selfadjoint compact operator $A$ on $\mathscr{H}$, its counting
functions are given by

$N^{\pm}(A;\lambda):=\\#\big{\\{}j;\
\lambda_{j}^{\pm}(A)>\lambda\big{\\}},\qquad\lambda>0.$

If $A\in\mathscr{W}_{p,\infty}$, $p>0$, then (see, e.g., [64, Proposition
13.1]), we have

(2.8) $\lim_{\lambda\rightarrow
0^{+}}\lambda^{p}N^{\pm}(A;\lambda)=\lim_{j\rightarrow\infty}j\lambda_{j}^{\pm}(A)^{p}=\Lambda^{\pm}(A)^{p}.$

It is also convenient to introduce the following class of operators.

######  Definition 2.4.

$\mathscr{W}_{|p,\infty|}$, $p>0$, consists of operators
$A\in\mathscr{L}_{p,\infty}$ such that $|A|\in\mathscr{W}_{p,\infty}$, i.e.,
$\lim j^{1/p}\mu_{j}(A)$ exists.

In particular, if $A\in\mathscr{W}_{|p,\infty|}$, then

$\Lambda\big{(}|A|\big{)}=\lim_{j\rightarrow\infty}j^{\frac{1}{p}}\mu_{j}(A).$

The perturbation theory of Birman-Solomyak [7, Theorem 4.1 & Remark 4.2]
implies the following results (see also [54]).

######  Proposition 2.5.

The following holds.

  1. (1)

$\mathscr{W}_{p,\infty}$ is a closed subset of $\mathscr{L}_{p,\infty}$ on
which $\Lambda^{\pm}:\mathscr{W}_{p,\infty}\rightarrow\mathbb{C}$ are
continuous maps.

  2. (2)

Let $A\in\mathscr{W}_{p,\infty}$ and $B\in(\mathscr{L}_{p,\infty})_{0}$. Then
$A+B\in\mathscr{W}_{p,\infty}$, and we have

$\Lambda^{\pm}(A+B)=\Lambda^{\pm}(A).$

######  Proposition 2.6.

The following holds.

  1. (1)

$\mathscr{W}_{|p,\infty|}$ is a closed subset of $\mathscr{L}_{p,\infty}$ on
which the functional $A\rightarrow\Lambda(|A|)$ is continuous.

  2. (2)

If $A\in\mathscr{W}_{|p,\infty|}$ and $B\in(\mathscr{L}_{p,\infty})_{0}$. Then
$A+B\in\mathscr{W}_{|p,\infty|}$, and we have

$\Lambda^{\pm}\big{(}|A+B|\big{)}=\Lambda^{\pm}\big{(}|A|\big{)}.$

######  Remark 2.7.

The 2nd part of Proposition 2.6 is due originally to Ky Fan [26, Theorem 3]
(see also [30, Theorem II.2.3]).

######  Remark 2.8.

We stress out that Proposition 2.5 and Proposition 2.6 are elementary
consequences of the Ky Fan’s inequalities (2.3) and (2.7) (see [7]; compare
[73]).

###  2.3. Example: Weyl’s laws for negative order $\Psi$DOs

Suppose that $(M^{n},g)$ is a closed Riemannian manifold and $E$ a Hermitian
vector bundle over $M$. Given any $m\in\mathbb{R}$ we denote by
$\Psi^{m}(M,E)$ the space of $m$-th order classical pseudodifferential
operators ($\Psi$DOs) $P:C^{\infty}(M,E)\rightarrow C^{\infty}(M,E)$. For
$P\in\Psi^{m}(M,E)$ we denote by $\sigma(P)(x,\xi)$ its principal symbol; this
is a smooth section of $\operatorname{End}(E)$ over $T^{*}M\setminus 0$.
Recall that any $P\in\Psi^{m}(M,E)$ with $m\leq 0$ extends to a bounded
operator $P:L_{2}(M,E)\rightarrow L_{2}(M,E)$; if $m<0$, then this operator is
actually in the weak Schatten class $\mathscr{L}_{p,\infty}$ with
$p=n|m|^{-1}$.

######  Proposition 2.9 (Birman-Solomyak [10, 11, 12]).

Let $P\in\Psi^{-m}(M,E)$, $m<0$, and set $p=nm^{-1}$.

  1. (1)

$P$ and $|P|$ are Weyl operators in $\mathscr{L}_{p,\infty}$.

  2. (2)

We have

(2.9)
$\lim_{j\rightarrow\infty}j^{\frac{1}{p}}\mu_{j}(P)=\bigg{[}\frac{1}{n}(2\pi)^{-n}\int_{S^{*}M}\operatorname{tr}_{E}\big{[}|\sigma(P)(x,\xi)|^{p}\big{]}dxd\xi\bigg{]}^{\frac{1}{p}}.$

  3. (3)

If $P$ is selfadjoint, then

(2.10)
$\lim_{j\rightarrow\infty}j^{\frac{1}{p}}\lambda^{\pm}_{j}(P)=\bigg{[}\frac{1}{n}(2\pi)^{-n}\int_{S^{*}M}\operatorname{tr}_{E}\big{[}\sigma(P)(x,\xi)_{\pm}^{p}\big{]}dxd\xi\bigg{]}^{\frac{1}{p}}.$

######  Remark 2.10.

In [10, 11] Birman-Solomyak established the above Weyl’s laws for compactly
supported pseudodifferential operators on $\mathbb{R}^{n}$ under very low
regularity assumptions on the symbols. Furthermore, the symbols are allowed to
be anisotropic. This was extended to classical $\Psi$DOs on closed manifolds
in [12]. Unfortunately, the key technical details are somehwat compressed and
contained in the Russian paper [11], the translation of which is unavailable.
We refer to [54] for a soft proof of Proposition 2.9.

######  Remark 2.11.

We refer to [1, 3, 15, 23, 32, 37, 54, 58, 59], and the references therein,
for various generalizations and applications of Birman-Solomyak’s asymptotics.

##  3\. Cwikel-Type Estimates for Matrix-Valued $L\\!\log\\!L$-Orlicz
Potentials

In this section, we establish general $\mathscr{L}_{1,\infty}$-Cwikel type
estimates for operators of the form $QuP$, where $u$ is a matrix-valued
$L\\!\log\\!L$-Orlicz potential and $Q$ and $P$ are $\Psi$DOs of order $-n/2$
on an $n$-dimensional closed manifold. In the scalar case Rozenblum [58]
established such estimates for potentials of the form $V=f\mu$, where $\mu$ is
an Alfhors-regular measure supported on a regular submanifold $\Sigma\subset
M$ and $f$ is a real-valued $L\\!\log\\!L$-Orlicz function on $\Sigma$ with
respect to $\mu$. Our aim is to explain how in the case of matrix-valued
$L\\!\log\\!L$-Orlicz potentials defined on all $M$ the estimates are easy
consequences of the specific Cwikel-type estimates on tori in the recent
preprint of Sukochev-Zanin [72].

Throughout this section we make the convention that $C_{abd}$ are positive
constants which depend only on the parameters $a$, $b$, $d$, etc., and may
change from line to line.

###  3.1. Sobolev spaces

Given $s\geq 0$, the (Bessel potential) Sobolev space
$W^{s}_{2}(\mathbb{R}^{n})$ consists of all $u\in L_{2}(\mathbb{R}^{n})$ such
that $(1+\Delta)^{s/2}u\in L_{2}(\mathbb{R}^{n})$, where
$\Delta=-(\partial_{x_{1}}^{2}+\cdots+\partial_{x_{n}}^{2})$ is the (positive)
Laplacian. This is a Hilbert space with respect to the norm,

$\|u\|_{W^{s}_{2}}=\big{\|}(1+\Delta)^{\frac{s}{2}}u\big{\|}_{L_{2}}=(2\pi)^{-\frac{n}{2}}\bigg{(}\int\big{(}1+|\xi|\big{)}^{s}|\hat{u}(\xi)|^{2}d\xi\bigg{)}^{\frac{1}{2}},\qquad
u\in W^{s}_{2}(\mathbb{R}^{n}).$

Here $\hat{u}(\xi)=\int e^{-ix\cdot\xi}u(x)dx$ is the Fourier transform of
$u$. Recall that by Sobolev’s embedding theorem, for $s<n/2$ we have a
continuous embedding,

(3.1) $W^{s}_{2}(\mathbb{R}^{n})\subset L_{p}(\mathbb{R}^{n}),\qquad
p=\frac{n}{2}-s.$

For $s>n/2$ we have continuous embeddings into Hölder spaces
$C^{k,\alpha}(\mathbb{R}^{n})$. Namely,

(3.2) $W^{s}_{2}(\mathbb{R}^{n})\subset C^{k,\alpha}(\mathbb{R}^{n}),$

where $k\in\mathbb{N}_{0}$ and $\alpha\in(0,1]$ are such that
$s-n/2=k+\alpha$.

If $s>0$, then $W^{-s}_{2}(\mathbb{R}^{n})$ consists of all tempered
distributions $u\in\mathscr{S}^{\prime}(\mathbb{R}^{n})$ such that
$(1+\Delta)^{-s/2}u\in L_{2}(\mathbb{R}^{n})$, i.e.,
$(1+|\xi|^{2})^{-s/2}\hat{u}\in L_{2}(\mathbb{R}^{n})$. This is also a Hilbert
space, which is naturally identified with the anti-linear dual of
$W^{s}_{2}(\mathbb{R}^{n})$, i.e., the Hilbert space of continuous anti-linear
forms on $W^{s}_{2}(\mathbb{R}^{n})$. More precisely, the inner product
$L_{2}(\mathbb{R}^{n})\times L_{2}(\mathbb{R}^{n})\rightarrow\mathbb{C}$
uniquely extends to a nondegenerate continuous sesquilinear pairing
$W^{-s}_{2}(\mathbb{R}^{n})\times
W^{s}_{2}(\mathbb{R}^{n})\rightarrow\mathbb{C}$.

Let $(M^{n},E)$ be a closed manifold and $E^{r}$ a Hermitian vector bundle
over $M$. For $s\geq 0$ the Sobolev space $W^{s}_{2}(\mathbb{R}^{n})$ consists
of all sections $u\in L_{2}(M,E)$ such that, for every local chart
$\kappa:U\rightarrow V\subset\mathbb{R}^{n}$ over which there is a
trivialization $\tau:E_{|U}\rightarrow U\times\mathbb{C}^{r}$ and for every
test function $\varphi\in C^{\infty}_{c}(U)$, the pushforward
$\kappa_{*}\tau_{*}(\varphi u)$ is in
$W^{s}_{2}(\mathbb{R}^{n})\otimes\mathbb{C}^{r}$.

The Sobolev space $W^{s}_{2}(M,E)$ is a Hilbert space. For instance, if
$(\varphi_{i})_{1\leq i\leq N}$ is a finite smooth partition of unity
subordinate to an open cover $(U_{i})_{1\leq i\leq N}$ such that each open
$U_{i}$ is the domain of chart $\kappa_{i}:U_{i}\rightarrow\mathbb{R}^{n}$
over which there is a trivialization $\tau_{i}:E_{|U_{i}}\rightarrow
U_{i}\times\mathbb{C}^{r}$, then a Hilbert norm is given by

(3.3) $\|u\|_{W_{2}^{s}}:=\bigg{(}\sum_{1\leq i\leq
N}\big{\|}(\kappa_{i}\circ\tau_{i})_{*}(\varphi_{i}u)\big{\|}_{W_{2}^{s}}^{2}\bigg{)}^{\frac{1}{2}},\qquad
u\in W_{2}^{s}(M,E).$

The corresponding locally convex topology does not depend on the choice of the
partition of unity.

We define $W^{-s}_{2}(M,E)$ to be the anti-linear dual of $W^{s}_{2}(M,E)$.
Recall that if $P\in\Psi^{m}(M,E)$, $m\in\mathbb{R}$, then $P$ extends to a
bounded map $P:W^{2,s+m}(M,E)\rightarrow W^{s}_{2}(M,E)$ for every
$s\in\mathbb{R}$. In particular, if $\Delta_{E}=\nabla^{*}\nabla$ is the
Laplacian of some Hermitian connection on $E$, then we have

$W^{s}_{2}(M,E)=\big{\\{}u\in\mathscr{D}^{\prime}(M,E);\
(1+\Delta_{E})^{s/2}u\in L_{2}(M,E)\big{\\}},\qquad s\in\mathbb{R}.$

An equivalent Hilbert norm on $W^{s}_{2}(M,E)$ then is
$u\rightarrow\|(1+\Delta_{E})^{s/2}u\|_{L_{2}}$.

###  3.2. Orlicz spaces and critical Sobolev embedding

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space. For our purpose we may
take $\Omega$ to be $\mathbb{R}^{n}$ or a bounded domain in $\mathbb{R}^{n}$
equipped with the Lebesgue measure, or a closed Riemannian manifold equipped
with its Riemannian measure.

A _Young function_ is a convex function $F:[0,\infty)\rightarrow[0,\infty)$
such that

$\lim_{x\rightarrow
0}x^{-1}F(x)=0\quad\text{and}\quad\lim_{x\rightarrow\infty}x^{-1}F(x)=\infty.$

The Orlicz space $L_{F}(\Omega)$ associated to such a function consists of all
(classes of) measurable functions $f:\Omega\rightarrow\mathbb{C}$ for which
$F(\lambda^{-1}|f|)\in L_{1}(\Omega)$ for some $\lambda>0$. We refer to [67,
Chapter 2] for a concise and up to date exposition of Orlicz spaces. In
particular, $L_{F}(\Omega)$ is a Banach space with respect to the norm,

(3.4) $\|f\|_{L_{F}}=\inf\left\\{\lambda>0;\
\|F(\lambda^{-1}|f|)\|_{L_{1}}\leq 1\right\\},\qquad f\in L_{F}(\Omega).$

For $F(t)=t^{p}$, $p\geq 1$, we recover the $L_{p}$-space $L_{p}(\Omega)$. For
$F(t)=(1+t)\log(1+t)-t$ we get Zygmund’s space $L\\!\log\\!L(\Omega)$. We
observe that if $\mu(\Omega)<\infty$, then we have continuous inclusions,

$L_{p}(\Omega)\subset L\\!\log\\!L(\Omega)\subset L_{1}(\Omega),\qquad p>1.$

Moreover, in this case $L_{\infty}(\Omega)$ is dense in
$L\\!\log\\!L(\Omega)$. If we further assume that $\Omega$ is a bounded domain
in $\mathbb{R}^{n}$ or a compact Riemannian manifold, then
$C^{\infty}_{c}(\Omega)$ is a dense subspace of $L\\!\log\\!L(\Omega)$.

For $q=1,2$ we denote by $\exp(L_{q})(\Omega)$ the Orlicz space associated to
$F(t)=\exp(t^{q})-1-t^{q}$. If $\Omega=\mathbb{R}^{n}$ the relevance of
$\exp(L_{2})(\mathbb{R}^{n})$ stems from filling the gap in Sobolev’s
embedding theorems (3.1)-(3.2). More precisely, we have the following version
of Möser-Trudinger’s inequality.

######  Proposition 3.1 (Ozawa [52, Theorem 1]).

The Sobolev space $W^{n/2}_{2}(\mathbb{R}^{n})$ embeds continuously into the
Orlicz space $\exp(L_{2})(\mathbb{R}^{n})$.

Supposed now that $(M^{n},g)$ is a closed Riemannian manifold and
$(E^{r},\left\langle\cdot,\cdot\right\rangle_{E})$ is a Hermitian vector
bundle over $M$. We define $L_{F}(M,E)$ as the set of measurable sections
$u:M\rightarrow E$ such that the function $\|u(x)\|_{E}$ is in
$L\\!\log\\!L(M)$, where $\|\cdot\|_{E}$ is the norm associated with the
Hermitian metric of $E$. We then equip $L\\!\log\\!L(M,E)$ with the norm,

$\|u\|_{L_{F}(M,E)}=\big{\|}\|u\|_{E}\big{\|}_{L_{F}(M)},\qquad u\in
L_{F}(M,E)$

Equivalently, $u\in L_{F}(M,E)$ if and only if, for every local chart
$\kappa:U\rightarrow V\subset\mathbb{R}^{n}$ over which there is a
trivialization $\tau:E_{|U}\rightarrow U\times\mathbb{C}^{r}$ and for every
test function $\varphi\in C^{\infty}_{c}(U)$, the pushforward
$(\kappa\circ\tau)_{*}(\varphi u)$ is in
$L_{F}(\mathbb{R}^{n})\otimes\mathbb{C}^{r}$. If $(\varphi_{i})_{1\leq i\leq
N}$ is a finite smooth partition of unity subordonate to an open cover
$(U_{i})_{1\leq i\leq N}$ such that each open $U_{i}$ is the domain of chart
$\kappa_{i}:U_{i}\rightarrow\mathbb{R}^{n}$ over which there is a
trivialization $\tau_{i}:E_{|U_{i}}\rightarrow U_{i}\times\mathbb{C}^{r}$,
then an equivalent Banach norm is given by

(3.5) $u\longrightarrow\sum_{1\leq i\leq
N}\big{\|}\|(\kappa_{i}\circ\tau_{i})_{*}(\varphi_{i}u)\|_{\mathbb{C}^{r}}\big{\|}_{L_{F}}.$

Note that $C^{\infty}(M,E)$ is a dense subspace of $L_{F}(M,E)$.

In view of the definitions (3.3) and (3.5) of the norms on $W_{2}^{n/2}(M,E)$
and $\exp(L_{2})(M,E)$ we have the following consequence of Proposition 3.1.

######  Corollary 3.2.

The Sobolev space $W^{n/2}_{2}(M,E)$ embeds continuously into
$\exp(L_{2})(M,E)$.

######  Remark 3.3.

In the scalar case, i.e., $E$ is the trivial line bundle, the above result was
obtained by Fontana [28] in any dimension $n\geq 2$ and by Branson-Chang-Yang
[16] in dimension $n=4$. These papers predate [52].

As $e^{t}-1-t$ and $(1+t)\log(1+t)-t$ are convex conjugates of each other (see
[67, Example 2.18]) we have the following version of Hölder’s inequality.

######  Proposition 3.4 (see [67, Theorem 2.21]).

If $u\in L\\!\log\\!L(M)$ and $v\in\exp(L_{1})(M)$, then $uv\in L_{1}(M)$ with
norm inequality,

$\|uv\|_{L_{1}}\leq C_{M}\|u\|_{L\\!\log\\!L}\|v\|_{\exp(L_{1})}.$

If $v\in\exp(L_{2})(M)$, then $v^{2}\in\exp(L_{1})(M)$ and
$\|v^{2}\|_{\exp{L_{1}}}\leq C\|v\|_{\exp{L_{2}}}^{2}$. A polarization
argument shows that if $v,w\in\exp(L_{2})(M)$, then $vw\in\exp(L_{1})(M)$, and
we have

$\|vw\|_{\exp(L_{1})}\leq C\|v\|_{\exp(L_{2})}\|w\|_{\exp(L_{2})}.$

Combining this with Proposition 3.4 shows that, if $u\in L\\!\log\\!L(M)$ and
$v,w\in\exp(L_{2})(M)$, then $uvw\in L_{1}(M)$ with norm inequality,

$\|uvw\|_{L_{1}}\leq
C_{M}\|u\|_{L\\!\log\\!L}\|v\|_{\exp(L_{2})}\|w\|_{\exp(L_{2})}.$

More generally, if $u\in L\\!\log\\!L(M,\operatorname{End}(E))$ and
$v,w\in\exp(L_{2})(M,E)$, then

$\big{|}\left\langle
w(x)|u(x)v(x)\right\rangle_{E}\big{|}\leq\|u(x)\|_{\operatorname{End}(E)}\|v(x)\|_{E}\|w(x)\|_{E}\in
L_{1}(M).$

Thus, $\left\langle w(x)|u(x)v(x)\right\rangle_{E}\in L_{1}(M)$, and we have

$\big{\|}\left\langle w(x)|u(x)v(x)\right\rangle_{E}\big{\|}_{L_{1}}\leq
C_{M}\|u\|_{L\\!\log\\!L}\|v\|_{\exp(L_{2})}\|w\|_{\exp(L_{2})}.$

Therefore, if we denote by $\exp(L_{2})(M,E)^{*}$ the anti-linear dual of
$\exp(L_{2})(M,E)$, then $u$ acts as a bounded operator
$u:\exp(L_{2})(M,E)\rightarrow\exp(L_{2})(M,E)^{*}$ such that

$\left\langle uv,w\right\rangle=\int_{M}\left\langle
w(x)|u(x)v(x)\right\rangle_{E_{x}}\sqrt{g(x)}dx,\qquad
v,w\in\exp(L_{2})(M,E).$

This operator depends continuously on $u$ as $u$ ranges over
$L\\!\log\\!L(M,E)$. By duality the continuous embedding of $W_{2}^{n/2}(M,E)$
into $\exp(L_{2})(M,E)$ provided by Corollary 3.2 gives rise to a continuous
embedding of $\exp(L_{2})(M,E)^{*}$ into $W_{2}^{-n/2}(M,E)$. Therefore, we
arrive at the following result.

######  Proposition 3.5.

If $u\in L\\!\log\\!L(M,\operatorname{End}(E))$, then $u$ gives rise to a
bounded operator,

$u:W_{2}^{\frac{n}{2}}(M,E)\longrightarrow W_{2}^{-\frac{n}{2}}(M,E).$

This operator depends continuously on $u$ with respect to the
$L\\!\log\\!L$-norm.

Combining this with the boundedness properties of $\Psi$DOs, we obtain the
following corollary.

######  Corollary 3.6.

Let $P$ and $Q$ be operators in $\Psi^{-n/2}(M,E)$. If $u\in
L\\!\log\\!L(M,\operatorname{End}(E))$, then the composition $QuP$ is bounded
on $L_{2}(M,E)$ and depends continuously on $u$ with respect to the
$L\\!\log\\!L$-norm.

###  3.3. Cwikel-type estimates

The original estimates of Cwikel [21] imply that if $f\in
L_{p}(\mathbb{R}^{n})$, $n\geq 3$, with $p>1$, then the operator
$(1+\Delta)^{-n/2p}f(1+\Delta)^{-n/2p}$ is in the weak trace class
$\mathscr{L}_{p,\infty}$, and we have

$\big{\|}(1+\Delta)^{-\frac{n}{4}}f(1+\Delta)^{-\frac{n}{4}}\big{\|}_{p,\infty}\leq
C_{n}\|f\|_{L_{p}}.$

We shall now explain how to extend these estimates to operators of the form
$QuP$ where $P,Q\in\Psi^{-n/2}(M,E)$ and $u\in L\\!\log\\!L(M,E)$. Recall that
such operators are bounded by Corollary 3.6.

The starting point is the following specific Cwikel-type estimate on the torus
$\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$.

######  Proposition 3.7 (Sukochev-Zanin [73]; see also [70]).

If $f\in L\\!\log\\!L(\mathbb{T}^{n})$, then
$(1+\Delta)^{-n/4}f(1+\Delta)^{-n/4}$ is in the weak trace class
$\mathscr{L}_{1,\infty}$, and we have

$\big{\|}(1+\Delta)^{-\frac{n}{4}}f(1+\Delta)^{-\frac{n}{4}}\big{\|}_{\mathscr{L}_{1,\infty}}\leq
C_{n}\|f\|_{L\\!\log\\!L}.$

There is no difficulty to extend the above result to pseudodifferential
operators.

######  Corollary 3.8.

Let $P$ and $Q$ be operators in $\Psi^{-n/2}(\mathbb{T}^{n})$. If $f\in
L\\!\log\\!L(\mathbb{T}^{n})$, then the operator $QfP$ is in the weak trace
class $\mathscr{L}_{1,\infty}$, and we have

(3.6) $\big{\|}QfP\big{\|}_{\mathscr{L}_{1,\infty}}\leq
C_{n}\|P\|_{\mathscr{L}(L_{2},W_{2}^{{n}/{2}})}\|Q\|_{\mathscr{L}(L_{2},W_{2}^{{n}/{2}})}\|f\|_{L\\!\log\\!L}.$

###### Proof.

The operator $A=(1+\Delta)^{n/4}P$ and $B=Q(1+\Delta)^{n/4}$ are zeroth order
$\Psi$DOs, and hence are bounded on $L_{2}(M,E)$. As
$QfP=B(1+\Delta)^{-n/4}f(1+\Delta)^{-n/4}A$, by using Proposition 3.7 we
deduce that $QfP\in\mathscr{L}_{1,\infty}$, and we have

$\displaystyle\big{\|}QfP\big{\|}_{\mathscr{L}_{1,\infty}}$
$\displaystyle\leq\|B\|\big{\|}(1+\Delta)^{-\frac{n}{4}}f(1+\Delta)^{-\frac{n}{4}}\big{\|}_{\mathscr{L}_{1,\infty}}\|A\|$
$\displaystyle\leq
C_{n}\|A\|_{\mathscr{L}(L_{2})}\|B\|_{\mathscr{L}(L_{2})}\|f\|_{L\\!\log\\!L}.$

Note that

$\displaystyle\|A\|_{\mathscr{L}(L_{2})}=\big{\|}(1+\Delta)^{n/4}P\big{\|}_{\mathscr{L}(L_{2})}=\|P\|_{\mathscr{L}(L_{2},W_{2}^{{n}/{2}})},$
$\displaystyle\|B\|_{\mathscr{L}(L_{2})}=\|Q(1+\Delta)^{n/4}\|_{\mathscr{L}(L_{2})}=\|Q\|_{\mathscr{L}(W_{2}^{-{n}/{2}},L_{2})}=\|Q\|_{\mathscr{L}(L_{2},W_{2}^{{n}/{2}})}.$

This gives the estimate (3.6). The proof is complete. ∎

As $\Psi$DOs are (pseudo-)local objects it’s a routine argument to extend
results on a given closed manifold to all such manifolds.

From now on, we let $M^{n}$ be a closed Riemannian manifold and $E^{r}$ a
Hermitian vector bundle over $M$. In the following, given any compact
$K\subset M$, we denote by $\Psi_{K}^{m}(M,E)$, $m\in\mathbb{R}$, the class of
operators $P\in\Psi^{m}(M,E)$ whose Schwartz kernels are supported in $K\times
K$. We also denote by $L_{2,K}(M,E)$ (resp.,
$L\\!\log\\!L_{K}(M,\operatorname{End}(E))$) the sections $u$ in $L_{2}(M,E)$
(resp., $L\\!\log\\!L(M,\operatorname{End}(E))$) such that
$\operatorname{supp}u\subset K$.

Corollary 3.8 admits the following local version.

######  Lemma 3.9.

Suppose that $K$ is a compact subset of $M$ which is contained in the domain
of a local chart of $M$ over which $E$ is trivial. Let $P$ and $Q$ be
operators in $\Psi_{K}^{-n/2}(M,E)$. If $u\in
L\\!\log\\!L_{K}(M,\operatorname{End}(E))$, then $QuP$ is in the weak trace
class $\mathscr{L}_{1,\infty}(L^{2}(M,E))$, and we have

(3.7) $\big{\|}QuP\big{\|}_{1,\infty}\leq C_{KPQ}\|u\|_{L\\!\log\\!L}.$

###### Proof.

As $\mathbb{R}^{n}$ can be smoothly embedded as an open set of
$\mathbb{T}^{n}$, the assumptions ensure us there exist an open set $U\subset
M$ containing $K$ over which $E$ is trivializable and a smooth diffeomorphism
$\phi:U\rightarrow V$ where $V$ is an open set of $\mathbb{T}^{n}$. Set
$K^{\prime}=\phi(K)$, and let $\tau:E_{|U}\rightarrow U\times\mathbb{C}^{r}$
be a smooth trivialization of $E$ over $U$. We then have the following
continuous linear embeddings of Hilbert spaces,

$L_{2}(M,E)\xleftarrow[(\iota_{U})_{*}]{~{}}L_{2}(U,E_{|U})\xrightarrow[~{}\tau_{*}~{}]{\sim}L_{2,K}(U)\otimes\mathbb{C}^{r}\xrightarrow[~{}\phi_{*}~{}]{\sim}L_{2,K^{\prime}}(V)\otimes\mathbb{C}^{r}\xrightarrow[(\iota_{V})_{*}]{~{}}L_{2}(\mathbb{T}^{n})\otimes\mathbb{C}^{r}.$

Here $\tau_{*}:L_{2,K}(U,E)\otimes\mathbb{C}^{r}\rightarrow
L_{2,K}(U)\otimes\mathbb{C}^{r}$ is the pushforward-isomorphism given by the
trivialization $\tau$. The linear embeddings
$(\iota_{U})_{*}:L_{2,K}(U,E)\rightarrow L_{2}(M,E)$ and
$(\iota_{V})_{*}:L_{2,K^{\prime}}(V)\otimes\mathbb{C}^{r}\rightarrow
L_{2}(\mathbb{T}^{n})\otimes\mathbb{C}^{r}$ arise from the inclusions
$U\subset M$ and $V\subset\mathbb{T}^{n}$, respectively. They are just given
by the extension to zero outside $K$ and $K^{\prime}$, and hence are isometric
embeddings. Therefore, by Proposition 2.1 in Appendix there is $c>0$ such
that, for any compact operator $A$ on $L_{2,K}(U,E)$, we have

(3.8) $\mu_{j}\big{(}(\iota_{U})_{*}A)=\mu_{j}(A)\leq
c\mu_{j}\big{(}(\iota_{V})_{*}\phi_{*}\tau_{*}A\big{)}\qquad\forall j\geq 0.$

Set $\tilde{P}=(\iota_{V})_{*}\phi_{*}\tau_{*}(P_{|U})$ and
$\tilde{Q}=(\iota_{V})_{*}\phi_{*}\tau_{*}(Q_{|U})$. They both are operators
in $\Psi^{-n/2}(\mathbb{T}^{n})$. If $\psi\in C^{\infty}_{c}(V)$ is such that
$\psi=1$ near $K^{\prime}$, then
$\tilde{P}=\psi(\phi_{*}\tau_{*}(P_{|U}))\psi$ and
$\tilde{Q}=\psi(\phi_{*}\tau_{*}(Q_{|U}))\psi$. If $u\in
L\\!\log\\!L_{K}(M,\operatorname{End}(E))$, then
$\tilde{u}:=(\iota_{V})_{*}\phi_{*}\tau_{*}(u_{|U})\in
L\\!\log\\!L_{K^{\prime}}(\mathbb{T}^{n})\otimes M_{r}(\mathbb{C})$, and so
Corollary 3.8 ensures that
$\tilde{Q}\tilde{u}\tilde{P}\in\mathscr{L}_{1,\infty}(L^{2}(\mathbb{T}^{n})\otimes\mathbb{C}^{r})$,
and we have

$\big{\|}\tilde{Q}\tilde{u}\tilde{P}\big{\|}_{1,\infty}\leq
C_{\tilde{P}\tilde{Q}}\|\tilde{u}\|_{L\\!\log\\!L}\leq
C_{KPQ}\|u\|_{L\\!\log\\!L}.$

Moreover, as $PuQ=(\iota_{U})_{*}(Q_{|U}u_{|U}P_{|U})$ and
$\tilde{Q}\tilde{u}\tilde{P}=(\iota_{V})_{*}\phi_{*}\tau_{*}(Q_{|U}u_{|U}P_{|U})$,
by (3.8) we have

$\mu_{j}(PuQ)\leq c\mu_{j}(\tilde{Q}\tilde{u}\tilde{P})\qquad\forall j\geq 0.$

It then follows that $QuP$ is in the weak Schatten class
$\mathscr{L}_{1,\infty}(L(M,E))$, and the estimate (3.7) holds. The proof is
complete. ∎

######  Remark 3.10.

If $n$ is even, then Lemma 3.9 can also be deduced from the Cwikel-type
estimates of Solomyak [70].

######  Remark 3.11.

Lemma 3.9 continues to holds if $M$ is not compact.

######  Lemma 3.12.

Assume $M$ is a closed manifold. Let $P$ and $Q$ be operators in
$\Psi^{-n/2}(M,E)$ such that one of them is smoothing. If $u\in
L\\!\log\\!L(M,\operatorname{End}(E))$, then $QuP$ is in every weak Schatten
class $\mathscr{L}_{p,\infty}$, $p>0$, and we have

(3.9) $\|QuP\|_{p,\infty}\leq C_{PQp}\|u\|_{L\\!\log\\!L}.$

###### Proof.

Let $p>0$. Assume that $Q$ is smoothing. Let $\Delta_{E}=\nabla^{*}\nabla$ be
the Laplacian of some Hermitian connection on $E$. If $u\in
L\\!\log\\!L(M,\operatorname{End}(E))$, then we have

$QuP=\big{(}Q(1+\Delta_{E})^{\frac{n}{4}}\big{)}\cdot(1+\Delta_{E})^{-\frac{n}{4}}uP.$

Here $Q(1+\Delta_{E})^{\frac{n}{4}}$ is a smoothing operator, and so it is
contained in every weak Schatten class $\mathscr{L}_{p,\infty}$. Moreover, as
$(1+\Delta_{E})^{-\frac{n}{4}}$ is a $\Psi$DO of order $-n/2$, Corollary 3.6
ensures that $(1+\Delta_{E})^{-\frac{n}{4}}uP$ is bounded on $L_{2}(M,E)$ and
depends continuously on $u$. It follows that $QuP$ is in
$\mathscr{L}_{p,\infty}$ and depends continuously on $u$, i.e., we have the
estimate (3.9). Similar arguments give the result when $P$ is smoothing. The
proof is complete. ∎

We are now in a position to get the Cwikel-type estimates we are seeking for.

######  Proposition 3.13.

Suppose that $M^{n}$ is a closed manifold and $E$ is a Hermitian vector bundle
over $M$. Let $P$ and $Q$ be operators in $\Psi^{-n/2}(M,E)$. If $u\in
L\\!\log\\!L(M,\operatorname{End}(E))$, then the operator $QuP$ is in the weak
trace class $\mathscr{L}_{1,\infty}(L^{2}(M,E))$, and we have

(3.10) $\|QuP\|_{1,\infty}\leq C_{PQ}\|u\|_{L\\!\log\\!L}.$

###### Proof.

This is a routine partition of unity argument (see, e.g., [12]). Let
$(\varphi_{i})_{1\leq i\leq N}$ be a smooth partition of unity subordinate to
an open cover $(U_{i})_{1\leq i\leq N}$, where each open $U_{i}$ is the domain
of a chart over which $E$ is trivializable. For $i=1,\ldots,N$, let
$\psi_{i}\in C^{\infty}_{c}(U_{i})$ be such that $\psi_{i}=1$ near
$\operatorname{supp}\varphi_{i}$. Given any $u\in
L\\!\log\\!L(M,\operatorname{End}(E))$ we have

$\displaystyle QuP=\sum_{1\leq i\leq N}Q\varphi_{i}uP=\sum_{1\leq i\leq
N}\bigg{\\{}Q_{i}(\varphi_{i}u)P_{i}+S_{i}uP+Q_{i}uR_{i}\bigg{\\}},$

where we have set

$P_{i}=\psi_{i}P\psi_{i},\quad Q_{i}=\psi_{i}Q\psi_{i},\quad
R_{i}=\varphi_{i}P(1-\psi_{i}),\quad S_{i}=(1-\psi_{i})Q\varphi_{i}.$

Here $P_{i}$ and $Q_{i}$ are operators in $\Psi_{K_{i}}(M,E)$ with
$K_{i}=\operatorname{supp}\psi_{i}\subset U_{i}$. Thus, by Lemma 3.9 each
operator $Q(\varphi_{i}u)P$ is in $\mathscr{L}_{1,\infty}$, and we have

$\|Q(\varphi_{i}u)P\|_{1,\infty}\leq
C_{P_{i}Q_{i}K_{i}}\|\varphi_{i}u\|_{L\\!\log\\!L}\leq
C_{P_{i}Q_{i}K_{i}}\|u\|_{L\\!\log\\!L}.$

Note also that $R_{i}$ and $S_{i}$ are both smoothing operators. Therefore, by
Lemma 3.12 the operators $S_{i}uP$ and $Q_{i}uR_{i}$ are both in
$\mathscr{L}_{1,\infty}$, and we have

$\|S_{i}uP\|_{1,\infty}\leq
C_{S_{i}P}\|u\|_{L\\!\log\\!L},\qquad\|Q_{i}uR_{i}\|_{1,\infty}\leq
C_{Q_{i}R_{i}}\|u\|_{L\\!\log\\!L}.$

Combining all this we deduce that $QuP$ is in the weak trace class
$\mathscr{L}_{1,\infty}(L^{2}(M,E))$, and we have

$\displaystyle\|QuP\|_{1,\infty}$ $\displaystyle\leq\sum_{1\leq i\leq
N}\bigg{\\{}\|Q_{i}(\varphi_{i}u)P_{i}\|_{1,\infty}+\|S_{i}uP\|_{1,\infty}+\|Q_{i}uR_{i}\|_{1,\infty}\bigg{\\}}$
$\displaystyle\leq C\|u\|_{L\\!\log\\!L}.$

The proof is complete. ∎

######  Remark 3.14.

In the scalar case, building on the results of [59], Rozenblum [58, Corollary
3.4] established eigenvalues inequalities for operators of the form $P^{*}uP$,
where $P\in\Psi^{-n/2}(M)$ and $u$ is a (real-valued) potential of the form
$u=f\mu$ such that $\mu$ is an Alfhors-regular measure supported on a regular
submanifold $\Sigma\subset M$ and $f$ is in $L\\!\log\\!L(\Sigma,\mu)$ (see
also [61] for the non-critical case). These eigenvalues inequalities can be
used to get Cwikel-type inequalities for this more general kind of potentials.

Proposition 3.13 implies the following specific Cwikel-type estimates.

######  Corollary 3.15.

Under the assumptions of Proposition 3.13, let $\Delta_{E}=\nabla^{*}\nabla$
be the Laplacian of some Hermitian connection $\nabla$ on $E$. If $u\in
L\\!\log\\!L(M,\operatorname{End}(E))$, then the operator
$\Delta_{E}^{-n/4}u\Delta_{E}^{-n/4}$ is in the weak trace class
$\mathscr{L}_{1,\infty}(L^{2}(M,E))$, and we have

(3.11)
$\big{\|}\Delta_{E}^{-\frac{n}{4}}u\Delta_{E}^{-\frac{n}{4}}\big{\|}_{1,\infty}\leq
C_{\nabla}\|u\|_{L\\!\log\\!L}.$

######  Remark 3.16.

In the scalar case, Sukochev-Zanin [73, Theorem 1.1] obtained Cwikel-type
estimates for operators of the form
$(1+\Delta_{g})^{-\frac{n}{4}}f(1+\Delta_{g})^{-\frac{n}{4}}$ with $f\in
L\\!\log\\!L(M)$. We recover those estimates by specializing Proposition 3.13
to $P=Q=(1+\Delta_{g})^{-\frac{n}{4}}$.

##  4\. Weyl’s Laws and Semiclassical Analysis

In this section, we explain how the Cwikel-type estimates of the previous
section lead to Weyl’s laws for operators of the form $QuP$, where $u$ is a
matrix-valued $L\\!\log\\!L$-Orlicz potential and $P$ and $Q$ are $\Psi$DOs of
order $-n/2$ on an $n$-dimensional closed manifold (compare [73]). Thanks to
the Birman-Schwinger principle this will further allow us to get semiclassical
Weyl’s laws and CLR-type inequalities for the number of negative eigenvalues
(i.e., the number of bound states) of critical Schrödinger operators
$\Delta_{E}^{n/2}+V$, where $V$ is a matrix-valued $L\\!\log\\!L$-Orlicz
potential.

Throughout this section, we let $(M^{n},g)$ be a closed Riemannian manifold
and $E$ a Hermitian vector bundle over $M$.

###  4.1. Weyl’s laws

The following is the main result of this section.

######  Theorem 4.1.

Let $P$ and $Q$ be operators in $\Psi^{-n/2}(M,E)$. If $u\in
L\\!\log\\!L(M,\operatorname{End}(E))$, then the following holds.

  1. (1)

The operators $QuP$ and $|QuP|$ are both Weyl operators in
$\mathscr{L}_{1,\infty}$.

  2. (2)

We have

(4.1)
$\lim_{j\rightarrow\infty}j\mu_{j}\left(QuP\right)=\frac{1}{n}(2\pi)^{-n}\\!\int_{S^{*}M}\operatorname{tr}_{E}\left\\{\big{|}\sigma(Q)(x,\xi)u(x)\sigma(P)(x,\xi)\big{|}\right\\}dxd\xi.$

  3. (3)

Suppose that $Q=P^{*}$ and $u(x)^{*}=u(x)$. Then $P^{*}uP$ is selfadjoint, and
we have

(4.2)
$\lim_{j\rightarrow\infty}j\lambda^{\pm}_{j}\left(P^{*}uP\right)=\frac{1}{n}(2\pi)^{-n}\\!\int_{S^{*}M}\operatorname{tr}_{E}\left\\{\big{[}\sigma(P)(x,\xi)^{*}u(x)\sigma(P)(x,\xi)\big{]}_{\pm}\right\\}dxd\xi.$

###### Proof.

It follows from Proposition 2.5 and Proposition 2.6 that
$\mathscr{W}_{1,\infty}\cap\mathscr{W}_{|1,\infty|}$ is a closed subset of
$\mathscr{L}_{1,\infty}$ on which $A\rightarrow\Lambda_{\pm}(A)$ and
$A\rightarrow\Lambda(|A|)$ are continuous functionals. It also follows from
Proposition 2.9 that $\mathscr{W}_{1,\infty}\cap\mathscr{W}_{|1,\infty|}$
contains the subspace $\mathscr{V}:=\left\\{QvP;v\in
C^{\infty}(M,\operatorname{End}(E))\right\\}$, and hence contains its closure
$\overline{\mathscr{V}}$. Proposition 3.13 implies that $u\rightarrow QuP$ is
a continuous linear map from $L\\!\log\\!L(M,\operatorname{End}(E))$ to
$\mathscr{L}_{1,\infty}$. As $C^{\infty}(M,\operatorname{End}(E))$ is dense in
$L\\!\log\\!L(M,E)$, it follows that if $u\in
L\\!\log\\!L(M,\operatorname{End}(E))$, then $QuP$ is contained in the closure
$\overline{\mathscr{V}}$, and hence is contained in
$\mathscr{W}_{1,\infty}\cap\mathscr{W}_{|1,\infty|}$. That is, $QuP$ and
$|QuP|$ are both Weyl operators in $\mathscr{L}_{1,\infty}$.

Furthermore, it follows from Proposition 2.9 that, for all $u\in
C^{\infty}(M,\operatorname{End}(E))$, we have

$\Lambda\big{(}|QuP|)=\frac{1}{n}(2\pi)^{-n}\\!\int_{S^{*}M}\operatorname{tr}_{E}\left\\{\big{|}\sigma(Q)(x,\xi)u(x)\sigma(P)(x,\xi)\big{|}\right\\}dxd\xi.$

Each side depends continuously on $u$ with respect to the $L\\!\log\\!L$-norm.
As $C^{\infty}(M,\operatorname{End}(E))$ is dense in
$L\\!\log\\!L(M,\operatorname{End}(E))$ the equality continues to hold for any
$u\in L\\!\log\\!L(M,\operatorname{End}(E))$. This gives (4.1). Likewise, if
$Q=P^{*}$, then, again by Proposition 2.9, the equality (4.2) holds for all
$u\in C^{\infty}(M,\operatorname{End}(E))$ and each side depends continuously
on $u$ with respect to the $L\\!\log\\!L$-norm. Therefore, the equality
continues to holds for any $u\in L\\!\log\\!L(M,\operatorname{End}(E))$. The
proof is complete. ∎

######  Remark 4.2.

The Weyl’s law (4.1) is new. In the scalar case, Rozenblum [58] established
the Weyl’s law (4.2) for a larger class of potentials of the form $u=f\mu$,
where $\mu$ is an Alfhors-regular measure supported on a regular submanifold
$\Sigma\subset M$ and $f$ is a real-valued function in
$L\\!\log\\!L(\Sigma,\mu)$ (see also [61] for the non-critical case).

Let $\Delta_{E}=\nabla^{*}\nabla$ be the Laplacian of some Hermitian
connection $\nabla$ on $E$. In particular, $\Delta_{E}$ is (formally)
selfadjoint and has principal symbol $\sigma(\Delta_{E})=|\xi|^{2}$ (where we
denote by $|\cdot|$ the Riemannian metric on $T^{*}M$). Specializing Theorem
4.1 to $P=\Delta_{E}^{-n/4}$ immediately leads to the following statement.

######  Corollary 4.3.

Let $u\in L\\!\log\\!L(M,\operatorname{End}(E))$. The following holds.

  1. (1)

The operators $\Delta_{E}^{-n/4}u\Delta_{E}^{-n/4}$ and
$|\Delta_{E}^{-n/4}u\Delta_{E}^{-n/4}|$ are Weyl operators in
$\mathscr{L}_{1,\infty}$.

  2. (2)

We have

(4.3)
$\lim_{j\rightarrow\infty}j\mu_{j}\left(\Delta_{E}^{-\frac{n}{4}}u\Delta_{E}^{-\frac{n}{4}}\right)=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}|u(x)|\big{]}\sqrt{g(x)}dx.$

  3. (3)

If $u(x)^{*}=u(x)$, then

(4.4)
$\lim_{j\rightarrow\infty}j\lambda^{\pm}_{j}\left(\Delta_{E}^{-\frac{n}{4}}u\Delta_{E}^{-\frac{n}{4}}\right)=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}u(x)_{\pm}\big{]}\sqrt{g(x)}dx.$

######  Remark 4.4.

In the scalar case, Sukochev-Zanin [73, Theorem 1.2] obtained a Weyl’s law
similar to (4.4) for operators of the form
$(1+\Delta_{g})^{-n/4}f(1+\Delta_{g})^{-n/4}$ with $f\in L\\!\log\\!L(M)$. The
approach of [73] involves deep results on commutators in operator ideals from
[25, 34]. This is unnecessary, since, as the proof of Theorem 4.1 displays,
the Weyl’s laws (4.1)–(4.4) are direct consequences of the Birman-Solomyak’s
Weyl’s laws (2.9)–(2.10) and the Cwikel-type estimates (3.10).

###  4.2. Semiclassical Weyl’s law

As above we let $\Delta_{E}=\nabla^{*}\nabla$ be the Laplacian of some
Hermitian connection on $E$. The operator $\Delta_{E}^{n/2}$ with domain
$W_{2}^{n}(M,E)$ is selfadjoint with non-negative and discrete spectrum. Let
$V(x)=V(x)^{*}\in L\\!\log\\!L(M,E)$. This defines a continuous operator
$V:W^{n/2}(M,E)\rightarrow W^{-n/2}(M,E)$ (_cf_. Proposition 3.5). Corollary
3.15 implies that $\Delta_{E}^{-n/4}V\Delta_{E}^{-n/4}$ is a compact operator
(and even is weak trace-class). This means that $V$ is relatively
$\Delta_{E}^{n/2}$-form compact, and so we may define the (fractional)
Schrödinger operator $H_{V}:=\Delta_{E}^{n/2}+V$ as a form sum (see, e.g.,
[68]). Namely, $H_{V}$ is the selfadjoint operator defined by the closed
quadratic form,

$Q_{H_{V}}(\xi,\xi)=\big{\langle}\Delta_{E}^{\frac{n}{2}}\xi,\xi\big{\rangle}+\left\langle
V\xi,\xi\right\rangle,\qquad\xi\in W^{\frac{n}{2}}(M,E),$

where $\left\langle\cdot,\cdot\right\rangle:W^{n/2}(M,E)\rightarrow
W^{-n/2}(M,E)$ is the duality pairing.

The operator $H_{V}$ is bounded from below and has discrete spectrum (see,
e.g., [68]). Therefore, we can arrange its spectrum as a non-decreasing
sequence,

$-\infty<\lambda_{0}\big{(}H_{V}\big{)}\leq\lambda_{1}\big{(}H_{V}\big{)}\leq\lambda_{2}\big{(}H_{V}\big{)}\leq\cdots,$

where each eigenvalue is repeated according to multiplicity. We are interested
in the number of negative eigenvalues, i.e.,

$N^{-}(H_{V}):=\\#\left\\{j;\lambda_{j}\big{(}H_{V}\big{)}<0\right\\}.$

In physicists’ language $N^{-}(H_{V})$ is the number of bound states of
$H_{V}$. In addition, if we denote by $\mathscr{F}^{-}(H_{V})$ the collection
of subspaces $F\subset W^{n/2}(M,E)$ such that $Q_{H_{V}}(\xi,\xi)<0$ on
$F\setminus 0$, then by Glazman’s lemma [14] we have

(4.5) $N^{-}(H_{V})=\max\left\\{\dim F;\ F\in\mathscr{F}^{-}(H_{V})\right\\}.$

This implies the monotonicity principle,

(4.6) $V_{1}\leq V_{2}\Longrightarrow N^{-}(H_{V_{2}})\leq N^{-}(H_{V_{1}}).$

Recall if $A$ is a selfadjoint compact operator, then
$N^{\pm}(A;\lambda)=\\#\\{j;\ \lambda_{j}^{\pm}(A)>\lambda\\}$, $\lambda>0$
(_cf._ Remark 2.3). By the abstract Birman-Schwinger principle [13] (see also
[48]), we have

(4.7)
$N^{-}\big{(}\Delta_{E}^{-\frac{n}{4}}V\Delta_{E}^{-\frac{n}{4}};1\big{)}\leq
N^{-}\big{(}\Delta_{E}^{\frac{n}{2}}+V\big{)}\leq
N^{-}\big{(}\Delta_{E}^{-\frac{n}{4}}V\Delta_{E}^{-\frac{n}{4}};1\big{)}+\dim\ker\Delta_{E}.$

If in addition $V(x)\leq 0$, then by the borderline Birman-Schwinger principle
of [48] we have

(4.8) $0\leq
N^{-}\big{(}\Delta_{E}^{\frac{n}{2}}+V\big{)}-N^{+}(\Pi_{0}V\Pi_{0})\leq\big{\|}\Delta_{E}^{-\frac{n}{4}}V_{-}\Delta_{E}^{-\frac{n}{4}}\big{\|}_{1,\infty},$

where $\Pi_{0}$ is the orthogonal projection onto $\ker\Delta_{E}$.

It follows from (4.7) (see, e.g., [8, 9, 54]) that, under the semiclassical
limit $h\rightarrow 0^{+}$, we get

(4.9)
$N^{-}\big{(}h^{n}\Delta_{E}^{\frac{n}{2}}+V\big{)}=N^{-}\big{(}\Delta_{E}^{-\frac{n}{4}}V\Delta_{E}^{-\frac{n}{4}};h^{n}\big{)}+\operatorname{O}(1).$

Thanks to (2.8) and the Weyl’s law (4.4) we have

$\lim_{\lambda\rightarrow 0^{+}}\lambda
N^{-}\big{(}\Delta_{E}^{-\frac{n}{4}}V\Delta_{E}^{-\frac{n}{4}};\lambda\big{)}=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}V(x)_{-}\big{]}\sqrt{g(x)}dx.$

Combining this with (4.9) we then arrive at the following semiclassical Weyl’s
law.

######  Corollary 4.5.

If $V(x)=V(x)^{*}\in L\\!\log\\!L(M,E)$, then

(4.10) $\lim_{h\rightarrow
0^{+}}h^{n}N^{-}\big{(}h^{n}\Delta_{E}^{\frac{n}{2}}+V\big{)}=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}V(x)_{-}\big{]}\sqrt{g(x)}dx.$

######  Remark 4.6.

The semiclassical Weyl’s law (4.10) does not seem to have appeared elsewhere,
and this sense this is a new result. However, as mentioned in Remark 4.2, in
the scalar case Rozenblum [58] established a version of the Weyl’s law (4.2)
for potentials of the form $V=f\mu$, where $\mu$ is an Alfhors-regular measure
supported on a regular submanifold $\Sigma\subset M$ and $f$ is a real-valued
function in $L\\!\log\\!L(\Sigma,\mu)$. Using this result and arguing as above
it is immediate to obtain we version of the semiclassical Weyl’s law (4.10)
for that class of potentials.

###  4.3. CLR-Type inequality

The celebrated CLR inequality was established by Rozenblum [56, 57], Cwikel
[21] and Lieb [44, 45] for Schrödinger operators $\Delta+V$ on
$\mathbb{R}^{n}$, $n\geq 3$, for potentials $V\in L_{n/2}(\mathbb{R}^{n})$
(see [18, 27, 29, 35, 46] for further alternative proofs). It gives a
universal bound on the number of negative eigenvalues in terms of the
$L_{n/2}$-norm of $V_{-}$. This stem from a conjecture of Simon [65] (who was
unaware of Rozenblum’s results). There are also versions of the CLR inequality
for fractional Schrödinger operators $\Delta^{n/2p}+V$ with $V\in L_{p}$ for
$p>1$ (see, e.g., [22, 43, 56, 57, 60, 61]). In the critical case $p=1$
Solomyak [70] obtained a CLR-type inequality for Schrödinger operators
$\Delta^{n/2p}+V$ associated with $L\\!\log\\!L$-Orlicz potentials $V$ on
bounded domains of $\mathbb{R}^{n}$ in even dimension only. We stress that the
inequality does not hold for any larger class of Orlicz potentials, including
the class of $L_{1}$-potentials (see [70]).

Bearing this in mind, as a further application of the Cwikel estimates (3.11),
we have the following CLR-type inequality for matrix-valued
$L\\!\log\\!L$-Orlicz potentials. Note that this result holds in any
dimension.

######  Corollary 4.7 (CLR-Type Inequality).

Under the above assumptions, there is a constant $C_{\nabla}>0$ such that, for
every potential $V(x)=V(x)^{*}\in L\\!\log\\!L(M,\operatorname{End}(E))$, we
have

(4.11)
$N^{-}\big{(}\Delta_{E}^{\frac{n}{2}}+V\big{)}-N^{+}(\Pi_{0}V_{-}\Pi_{0})\leq
C_{\nabla}\|V_{-}\|_{L\\!\log\\!L},$

where $\Pi_{0}$ is the orthogonal projection onto $\ker\Delta_{E}$.

###### Proof.

As $V\geq-V_{-}$ from (4.6) we get $N^{-}(\Delta_{E}^{\frac{n}{2}}+V)\leq
N^{-}(\Delta_{E}^{\frac{n}{2}}-V_{-})$, and so by using (4.8) we obtain

$N^{-}\big{(}\Delta_{E}^{\frac{n}{2}}+V\big{)}-N^{+}(\Pi_{0}V_{-}\Pi_{0})\leq\big{\|}\Delta_{E}^{-\frac{n}{4}}V_{-}\Delta_{E}^{-\frac{n}{4}}\big{\|}_{1,\infty}.$

Combining this with the Cwikel-type estimate (3.11) then gives the result. ∎

######  Remark 4.8.

As
$N^{+}(\Pi_{0}V_{-}\Pi_{0})\leq\operatorname{ran}\Pi_{0}=\dim\ker\Delta_{E}$,
the inequality (4.11) implies that

$N^{-}\big{(}\Delta_{E}^{\frac{n}{2}}+V\big{)}-\dim\ker\Delta_{E}\leq
C_{\nabla}\|V_{-}\|_{L\\!\log\\!L}.$

This inequality can also be obtained by using the usual abstract Birman-
Schwinger principle of [13]. Note however that the above inequality is not as
sharp as (4.11), since there are many examples for which
$N^{+}(\Pi_{0}V_{-}\Pi_{0})\leq\dim\ker\Delta_{E}-1$ (see, e.g., [48]).

######  Remark 4.9.

The semiclassical Weyl’s law (4.10) and the CLR-type estimate (4.11) hold
_verbatim_ if we replace $\Delta_{E}$ by any Laplace-type operator. In
particular, these results hold for squares of Dirac-type operators.

##  5\. Noncommutative Geometry and Weyl’s Laws

In this section, we review the notion of integral in the framework of Connes’
noncommutative geometry and its relationship to Birman-Solomyak’s Weyl’s laws
for compact operators.

###  5.1. Quantized calculus

The main goal of the quantized calculus of Connes [20] is to translate into a
the Hilbert space formalism of quantum mechanics the main tools of the
classical infinitesimal calculus.

Classical | Quantum
---|---
Complex variable | Operator on $\mathscr{H}$
Real variable | Selfadjoint operator on $\mathscr{H}$
Infinitesimal variable | Compact operator on $\mathscr{H}$
Infinitesimal of order $\alpha>0$ | Compact operator $T$ such that
| $\mu_{j}(T)=\operatorname{O}(j^{-\alpha})$

The first two lines arise from quantum mechanics. Intuitively speaking, an
infinitesimal is meant to be smaller than any real number. For a bounded
operator the condition $\|T\|<\epsilon$ for all $\epsilon>0$ gives $T=0$. This
condition can be relaxed into the following: For every $\epsilon>0$ there is a
finite-dimensional subspace $E$ of $\mathscr{H}$ such that
$\|T_{|E^{\perp}}\|<\epsilon$. This is equivalent to $T$ being a compact
operator.

The order of compactness of a compact operator is given by the order of decay
of its singular values. Namely, an _infinitesimal operator_ of order
$\alpha>0$ is any compact operator such that
$\mu_{j}(T)=\operatorname{O}(j^{-\alpha})$. Thus, if we set $p=\alpha^{-1}$,
then $T$ is infinitesimal operator of order $\alpha>0$ iff
$T\in\mathscr{L}_{p,\infty}$.

The next line of the dictionary is the NC analogue of the integral. As an
Ansatz the NC integral should be a linear functional satisfying the following
conditions:

  1. (1)

It is defined on a suitable class of infinitesimal operators of order 1.

  2. (2)

It vanishes on infinitesimal operators of order $>1$.

  3. (3)

It takes non-negative values on positive operators.

  4. (4)

It is invariant under Hilbert space isomorphisms.

As mentioned above, the infinitesimal operators of order 1 are the operators
in the weak trace class $\mathscr{L}_{1,\infty}$. The condition (3) means that
the functional should be positive. The condition (4) forces the functional to
be a trace, in the sense it is annihilated by the commutator subspace,

$\operatorname{Com}\big{(}\mathscr{L}_{1,\infty}\big{)}:=\operatorname{Span}\left\\{[A,T];\
A\in\mathscr{L}(\mathscr{H}),\ T\in\mathscr{L}_{1,\infty}\right\\}.$

Therefore, the NC integral must be a positive trace which defined on a
suitable subspace of $\mathscr{L}_{1,\infty}$ containing and is annihilated by
$\operatorname{Com}\big{(}\mathscr{L}_{1,\infty}\big{)}$ and infinitesimal
operators of order $>1$.

###  5.2. Construction of the NC integral

If $A$ is a compact operator on $\mathscr{H}$, then its spectrum can be
arranged as a sequence $(\lambda_{j}(A))_{j\geq 0}$ converging to $0$ such
that

$|\lambda_{0}(A)|\geq|\lambda_{1}(A)|\geq\cdots\geq|\lambda_{j}(A)|\geq\cdots\geq
0,$

where eigenvalue is repeating according to its algebraic multiplicity, i.e.,
the dimension of the corresponding root space $E_{\lambda}(A):=\cup_{\ell\geq
0}\ker(A-\lambda)^{\ell}$. If $\lambda\neq 0$, the algebraic multiplicity is
always finite (see, e.g., [30]). It agrees with the dimension of the
eigenspace $\ker(A-\lambda)$ when $A$ is norm.

A sequence as above is called an _eigenvalue sequence_. If the spectrum of $A$
lies on a ray, then $A$ admits a unique eigenvalue sequence. In particular, if
$A\geq 0$, then the eigenvalue sequence agrees with its singular value
sequence $(\mu_{j}(T))_{j\geq 0}$. In general, an eigenvalue sequence need not
be unique.

In what follows we shall denote by $(\lambda_{j}(A))_{j\geq 0}$ any eigenvalue
sequence for $A$. It is understood that all the results involving eigenvalue
sequences do not depend on the choice of the sequence.

We record the Weyl’s inequalities (see, e.g., [30, 66]),

(5.1)
$\big{|}\sum_{j<N}\lambda_{j}(A)\big{|}\leq\sum_{j<N}\left|\lambda_{j}(A)\right|\leq\sum_{j<N}\mu_{j}(A)\qquad\forall
N\geq 1.$

######  Lemma 5.1 ([47, Lemma 5.7.5]).

If $A$ and $B$ are operators in $\mathscr{L}_{1,\infty}$, then

$\sum_{j<N}\lambda_{j}(A+B)=\sum_{j<N}\lambda_{j}(A)+\sum_{j<N}\lambda_{j}(B)+\operatorname{O}(1).$

The above result implies the following property of sums of commutators in
$\mathscr{L}_{1,\infty}$.

######  Corollary 5.2 ([25, 47]; see also [54, Corollary 2.6]).

If $A\in\operatorname{Com}(\mathscr{L}_{1,\infty})$, then

$\sum_{j<N}\lambda_{j}(A)=\operatorname{O}(1).$

######  Remark 5.3.

Corollary 5.2 has a converse. It can be shown (see [25, 47]) that, if $A$ and
$B$ are operators in $\mathscr{L}_{1,\infty}$, then

(5.2)
$A-B\in\operatorname{Com}\big{(}\mathscr{L}_{1,\infty}\big{)}\Longleftrightarrow\sum_{j<N}\lambda_{j}(A)=\sum_{j<N}\lambda_{j}(B)+\operatorname{O}(1).$

In particular, this implies that $\operatorname{Com}(\mathscr{L}_{1,\infty})$
contains the trace class $\mathscr{L}_{1}$, and so it contains all
infinitesimal operators of order $>1$.

Let $\ell_{\infty}$ be the $C^{*}$-algebra of bounded sequences
$(a_{N})_{N\geq 1}\subset\mathbb{C}$ and $\mathfrak{c}_{0}$ the closed ideal
of sequences converging to $0$. If $A\in\mathscr{L}_{1,\infty}$, then the
Weyl’s inequalities (5.1) imply that the Cesàro mean sequence
$\\{\frac{1}{\log N}\sum_{j<N}\lambda_{j}(A)\\}_{N\geq 1}$ is bounded.
Moreover, it follows from Lemma 5.1 that its class in
$\ell_{\infty}/\mathfrak{c}_{0}$ does not depend on the choice of the
eigenvalue sequence.

######  Definition 5.4.

The _NC integral_ of an operator $A\in\mathscr{L}_{1,\infty}$ is defined by

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int A:=\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{j<N}\lambda_{j}(A),$

provided the limit exists. An operator for which the above limit exists is
called _measurable_.

######  Remark 5.5.

The existence and value of the above limit does not depend on the choice of
the eigenvalue sequence.

In what follows we denote by $\mathscr{M}$ the class of measurable operators.
The following shows that $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int$ satisfies the Ansatz for the integral in the
quantized calculus.

######  Proposition 5.6 ([54]).

$\mathscr{M}$ is a subspace of $\mathscr{L}_{1,\infty}$ containing
$\operatorname{Com}(\mathscr{L}_{1,\infty})$ on which
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int:\mathscr{M}\rightarrow\mathbb{C}$ is a positive
linear trace which is annihilated by operators in
$(\mathscr{L}_{1,\infty})_{0}$, including infinitesimals of order $>1$.

We stress out that the above construction of the NC integral is of purely
spectral nature. In particular, we immediately have the following spectral
invariance result.

######  Proposition 5.7 ([54]).

Let $A$ and $B$ be weak trace class operators possibly acting on different
Hilbert spaces. Then one of these operators is measurable if and only if the
other operator is measurable. Moreover, in this case
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int A=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int B$.

We have an alternative description of the NC integral in terms of Dixmier
traces [20, 24]. Let $\omega$ be a state on the quotient $C^{*}$-algebra
$\ell_{\infty}/\mathfrak{c}_{0}$, i.e., a positive linear form such that
$\omega(1)=1$. This lifts to a positive linear functional $\lim_{\omega}$ on
$\ell_{\infty}$. Such a functional is called an _extended limit_. In
particular, if $a=(a_{N})_{N\geq 1}$ is any bounded sequence, then

$\lim_{j\rightarrow\infty}a_{j}=L\ \Longleftrightarrow\
\big{(}{\lim}_{\omega}\,a_{j}=L\quad\forall\omega\big{)}.$

######  Proposition 5.8 ([24, 20, 47, 54]).

The following formula

$\operatorname{Tr}_{\omega}(A)={\lim}_{\omega}\,\frac{1}{\log
N}\sum_{j<N}\lambda_{j}(A),\qquad A\in\mathscr{L}_{1,\infty},$

uniquely defines a positive linear trace
$\operatorname{Tr}_{\omega}:\mathscr{L}_{1,\infty}\rightarrow\mathbb{C}$.

The trace
$\operatorname{Tr}_{\omega}:\mathscr{L}_{1,\infty}\rightarrow\mathbb{C}$ is
called the _Dixmier trace_ associated with the extended limit
${\lim}_{\omega}\,$.

######  Proposition 5.9 ([47, 54]).

If $A\in\mathscr{L}_{1,\infty}$, then

$\lim_{N\rightarrow\infty}\frac{1}{\log N}\sum_{j<N}\lambda_{j}(A)=L\
\Longleftrightarrow\
\big{(}\operatorname{Tr}_{\omega}(A)=L\quad\forall\omega\big{)}.$

In particular, $A$ is measurable if and only if the value of
$\operatorname{Tr}_{\omega}(A)$ is independent of $\omega$. Moreover, in this
case this value is equal to the NC integral
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int A$.

The above proposition shows that we recover the original construction of the
NC integral by Connes [20].

###  5.3. Measurability and Weyl’s laws

In what follows we denote by $T_{0}$ any positive operator in
$\mathscr{L}_{1,\infty}$ such that $\mu_{j}(T_{0})=(j+1)^{-1}$ for all $j\geq
0$. Note that any two such operators are unitary equivalent, and hence agree
up to an element of $\operatorname{Com}(\mathscr{L}_{1,\infty})$.

A trace $\varphi:\mathscr{L}_{1,\infty}\rightarrow\mathbb{C}$ is called
_normalized_ if $\varphi(T_{0})=1$. All the Dixmier traces are normalized
traces. However, there are plenty of positive normalized traces on
$\mathscr{L}_{1,\infty}$ that are not Dixmier traces (see, e.g., [63, Theorem
4.7]). Therefore, it stands for reason to consider a stronger notion of
measurability.

######  Definition 5.10.

An operator $A\in\mathscr{L}_{1,\infty}$ is called _strongly measurable_ when
there is $L\in\mathbb{C}$ such that $\varphi(A)=L$ for every positive
normalized trace $\varphi$ on $\mathscr{L}_{1,\infty}$.

######  Remark 5.11.

In [58] measurable operators are defined as operators in the Dixmier-Macaev
class $\mathfrak{M}_{1,\infty}$ which take the same value on all positive
normalized traces on $\mathfrak{M}_{1,\infty}$. The weak trace class
$\mathscr{L}_{1,\infty}$ is strictly contained in $\mathfrak{M}_{1,\infty}$.
Every Dixmier trace extends to a positive normalized trace on
$\mathfrak{M}_{1,\infty}$. However, there are many traces on
$\mathscr{L}_{1,\infty}$ that do not extend to $\mathfrak{M}_{1,\infty}$.
Therefore, measurability in the sense of [58] is stronger than measurability
in the sense of Definition 5.4, but it is also weaker than strong
measurability in the sense above.

In the scalar case Rozenblum established the trace formula (1.10) for a larger
class of potentials. As mentioned above, a weaker notion of measurability is
used in [58]. However, the same arguments as above show that the operators
under consideration in [58] are strongly measurable in the sense used in this
paper.

We denote by $\mathscr{M}_{\textup{s}}$ the class of strongly measurable
operators. It follows from Proposition 5.9 that if
$A\in\mathscr{M}_{\textup{s}}$, then $A$ is measurable, and
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int A=\varphi(A)$ for every positive normalized trace
$\varphi$ on $\mathscr{L}_{1,\infty}$. Note also that the inclusion of
$\mathscr{M}_{\textup{s}}$ into $\mathscr{M}$ is strict (see [63, Theorem
7.4]).

######  Proposition 5.12 (see, e.g., [54]).

The following holds.

  1. (1)

$\mathscr{M}_{\textup{s}}$ is a closed subspace of $\mathscr{L}_{1,\infty}$
containing $\operatorname{Com}(\mathscr{L}_{1,\infty})$ and
$(\mathscr{L}_{1,\infty})_{0}$. In particular, it contains all infinitesimal
operators of order $>1$.

  2. (2)

If $A\in\mathscr{L}_{1,\infty}$ is such that
$\sum_{j<N}\lambda_{j}(A)=L+\operatorname{O}(1)$, then
$A\in\mathscr{M}_{\textup{s}}$ and $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int A=L$.

  3. (3)

If $A$ and $B$ are operators in $\mathscr{L}_{1,\infty}$ with same non-zero
eigenvalues and same multiplicities. Then one operator is strongly measurable
if and only if the other is.

We have the following relationship between strong measurability and Weyl’s
laws.

######  Proposition 5.13 (see [54]).

The following holds.

  1. (1)

If $A\in\mathscr{W}_{1,\infty}$, then $A$ is strongly measurable, and we have

(5.3) $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int A=\Lambda^{+}(A)-\Lambda^{-}(A).$

In particular, if $A$ is selfadjoint, then

(5.4) $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
A=\lim_{j\rightarrow\infty}j\lambda_{j}^{+}(A)-\lim_{j\rightarrow\infty}j\lambda_{j}^{-}(A).$

  2. (2)

If $A\in\mathscr{W}_{|1,\infty|}$, then $|A|$ is strongly measurable, and we
have

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int|A|=\lim_{j\rightarrow\infty}j\mu_{j}(A)$

###  5.4. Connes’ trace theorem

Suppose that $(M^{n},g)$ is a closed Riemannian manifold and $E$ is a
Hermitian vector bundle over $M$. Let
$\operatorname{Res}:\Psi^{\mathbb{Z}}(M,E)\rightarrow\mathbb{C}$ be the
noncommutative residue trace of Guillemin [33] and Wodzicki [75], where
$\Psi^{\mathbb{Z}}(M,E)=\bigcup_{m\in\mathbb{Z}}\Psi^{m}(M,E)$ is the algebra
of integer order $\Psi$DOs acting on sections of $E$. This functional appears
as the residual trace on integer-order $\Psi$DOs induced by the analytic
extension of the ordinary trace to all non-integer order $\Psi$DOs (see [33,
75]). If $P\in\Psi^{m}(M,E)$, $m\in\mathbb{Z}$, then

$\operatorname{Res}(P)=\int_{M}\operatorname{tr}_{E}[c_{P}(x)],$

where $c_{P}(x)$ is an $\operatorname{End}(E)$-valued 1-density which is given
in local coordinates by

$c_{P}(x)=(2\pi)^{-n}\int_{|\xi|=1}a_{-n}(x,\xi)d^{n-1}\xi,$

where $a_{-n}(x,\xi)$ is the symbol of degree $-n$ of $P$. In particular, if
$P$ has order $-n$, then

$\operatorname{Res}(P)=(2\pi)^{-n}\int_{S^{*}M}\operatorname{tr}_{E}\big{[}\sigma(P)(x,\xi)\big{]}dxd\xi.$

A well known result of Wodzicki [74] further asserts that this is the unique
trace on $\Psi^{\mathbb{Z}}(M,E)$ up to constant multiple if $M$ is connected
and has dimension $n\geq 2$ (see also [42, 53]).

######  Proposition 5.14 (Connes’s Trace Theorem [19, 36]).

Every operator $P\in\Psi^{-n}(M,E)$ is strongly measurable, and we have

(5.5) $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int P=\frac{1}{n}\operatorname{Res}(P).$

Let $\Delta_{E}=\nabla^{*}\nabla$ be the Laplacian of some Hermitian
connection $\nabla$ on $E$. As an application of Connes’ trace theorem we
obtain the following integration formula, which shows that the noncommutative
integral recaptures the Riemannian volume density.

######  Corollary 5.15 (Connes’s Integration Formula [19, 36]).

For all $u\in C^{\infty}(M,\operatorname{End}(E))$, the operator
$\Delta_{E}^{-{n}/{4}}u\Delta_{E}^{-{n}/{4}}$ is strongly measurable, and we
have

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\Delta_{E}^{-\frac{n}{4}}u\Delta_{E}^{-\frac{n}{4}}=c_{n}\int_{M}\operatorname{tr}_{E}\big{[}u(x)\big{]}\sqrt{g(x)}d^{n}x,\qquad
c_{n}:=\frac{1}{n}(2\pi)^{-n}|\mathbb{S}^{n-1}|.$

We observe that Birman-Solomyak’s Weyl’s laws (2.9)–(2.10) and Proposition
5.13 enable us to recover Connes’ trace theorem. In particular, if
$P=P^{*}\in\Psi^{-n}(M,E)$, then by using (2.10) and (5.4) we get

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int P$
$\displaystyle=\lim_{j\rightarrow\infty}j\lambda_{j}^{+}(P)-\lim_{j\rightarrow\infty}j\lambda_{j}^{-}(P)$
$\displaystyle=\frac{1}{n}(2\pi)^{-n}\\!\int_{S^{*}M}\left\\{\operatorname{tr}_{E}\big{[}\sigma(P)(x,\xi)_{+}\big{]}-\operatorname{tr}_{E}\big{[}\sigma(P)(x,\xi)_{-}\big{]}\right\\}dxd\xi,$
$\displaystyle=\frac{1}{n}(2\pi)^{-n}\\!\int_{S^{*}M}\operatorname{tr}_{E}\big{[}\sigma(P)(x,\xi)\big{]}dxd\xi,$

which gives back Connes’ trace theorem (5.5).

In fact, Birman-Solomyak’s Weyl’s law (2.9) further provide us with a trace
formula for absolute values of $\Psi$DOs even though they need not be
$\Psi$DOs. Namely, by combining (2.9) and Proposition 5.13 we get the
following result.

######  Proposition 5.16.

If $P\in\Psi^{-n}(M,E)$, then $|P|$ is strongly measurable, and we have

$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int|P|=\lim_{j\rightarrow\infty}j\mu_{j}(P)=\frac{1}{n}(2\pi)^{-n}\int_{S^{*}M}\operatorname{tr}_{E}\big{[}|\sigma(P)(x,\xi)|\big{]}dxd\xi.$

It is interesting to note that Birman-Solomyak’s Weyl’s laws predate Connes’
trace theorem by a decade.

##  6\. Integration Formulas for $L\\!\log\\!L$-Orlicz Potentials

In this section, we explain how the Weyl’s laws (4.1)–(4.2) allow us to get
stronger forms of Connes’ trace theorem and Connes’s integration formula.

We keep on using the notation of the previous sections. In particular, we let
$(M^{n},g)$ be a closed Riemannian manifold and $E$ a Hermitian vector bundle
over $M$.

By combining Theorem 4.1 and Proposition 5.13 we immediately the following
trace theorem.

######  Theorem 6.1.

Let $P$ and $Q$ be operators in $\Psi^{-n/2}(M,E)$. If $u\in
L\\!\log\\!L(M,\operatorname{End}(E))$, then $QuP$ and $|QuP|$ are both
strongly measurable operators, and we have

(6.1) $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int
QuP=\frac{1}{n}(2\pi)^{-n}\\!\int_{S^{*}M}\operatorname{tr}_{E}\big{[}\sigma(Q)(x,\xi)u(x)\sigma(P)(x,\xi)\big{]}dxd\xi,$
(6.2) $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\big{|}QuP\big{|}=\frac{1}{n}(2\pi)^{-n}\\!\int_{S^{*}M}\operatorname{tr}_{E}\big{[}\left|\sigma(Q)(x,\xi)u(x)\sigma(P)(x,\xi)\right|\big{]}dxd\xi.$

######  Remark 6.2.

In the scalar case, Rozenblum [58] established obtained a trace formula
similar to (6.1) for a larger class of potentials of the form $u=f\mu$, where
$\mu$ is an Alfhors-regular measure supported on a regular submanifold
$\Sigma\subset M$ and $f$ is a real-valued function in
$L\\!\log\\!L(\Sigma,\mu)$. As mentioned in Remark 5.11, in [58] a weaker
notion of measurability is used, but the same arguments as above show that the
operators under consideration in [58] are strongly measurable in the sense
used in this paper.

Specializing the above results to $Q=P=\Delta_{E}^{-n/4}$ allows us to get the
following integration formula.

######  Corollary 6.3.

If $u\in L\\!\log\\!L(M,\operatorname{End}(E))$, then
$\Delta_{E}^{-n/4}u\Delta_{E}^{-n/4}$ and
$|\Delta_{E}^{-n/4}u\Delta_{E}^{-n/4}|$ are both strongly measurable
operators, and we have

(6.3) $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\Delta_{E}^{-\frac{n}{4}}u\Delta_{E}^{-\frac{n}{4}}=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}u(x)\big{]}\sqrt{g(x)}dx,$
(6.4) $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\left|\Delta_{E}^{-\frac{n}{4}}u\Delta_{E}^{-\frac{n}{4}}\right|=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}|u(x)|\big{]}\sqrt{g(x)}dx.$

######  Remark 6.4.

In the scalar case, Sukochev-Zanin [73, Lemma 5.8] got a weaker form of the
integration formula (6.3) for operators of the form
$(1+\Delta_{g})^{-n/4}f(1+\Delta_{g})^{-n/4}$ with $f\in L_{\infty}(M)$. We
recover this formula from (6.3), since the operators
$(1+\Delta_{g})^{-n/4}f(1+\Delta_{g})^{-n/4}$ and
$\Delta_{g}^{-n/4}f\Delta_{g}^{-n/4}$ agree modulo a trace-class operator.
Note that the approach in [73] relies on a deep result of [25] asserting that
$\operatorname{Com}(\mathscr{L}_{1,\infty})$ is spanned by commutators $[A,B]$
with $A,B\in\mathscr{L}_{2,\infty}$.

######  Remark 6.5.

Combining (1.6) and (6.3) shows that, if $V\in
L\\!\log\\!L(M,\operatorname{End}(E))$, $V(x)\geq 0$, then

$\lim_{h\rightarrow
0^{+}}h^{n}N^{-}\big{(}h^{n}\Delta_{E}^{\frac{n}{2}}-V\big{)}=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\Delta_{E}^{-\frac{n}{4}}V\Delta_{E}^{-\frac{n}{4}}=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}Vq(x)\big{]}\sqrt{g(x)}dx.$

This gives a semiclassical interpretation of Connes’ integration formula. The
first equality is also a direct consequence of the fact that
$\Delta_{E}^{-{n}/{4}}u\Delta_{E}^{-{n}/{4}}$ is a Weyl operator in
$\mathscr{L}_{1,\infty}$ (see [54]; see also [50]).

For applications in noncommutative geometry, as well as in mathematical
physics, it is often preferable to use Dirac-type operators instead of
connection Laplacians. As a matter of fact, Connes’ original formulation of
the integration formula was stated in terms of the Dirac operator (see [19,
Theorem 3]). Thus, it is worth reformulating Corollary 4.3 and Corollary 6.3
in terms of Dirac operators.

Assume that $E$ is Clifford module in the sense of [4]. Let
$\not{D}_{E}:C^{\infty}(M,E)\rightarrow C^{\infty}(M,E)$ be the Dirac operator
associated with a Hermitian Clifford connection $\nabla$ on $E$. Thus,
$\not{D}_{E}$ is a selfadjoint and its square $\not{D}_{E}^{2}$ has same
principal symbol as the corresponding connection Laplacian $\Delta_{E}$.

For instance, if $M$ is spin and $E$ is the spinor bundle, then we may take
$\not{D}_{E}$ to be the usual Dirac operator $\not{D}_{M}$. If $E$ is the
bundle of differential forms on $M$, then we may take $\not{D}_{E}$ to be the
de Rham-Dirac operator $d+d^{*}$, so that $\not{D}_{E}$ is the Hodge-de Rham
Laplacian. If $M$ is Kähler manifold and $E$ is the bundle of anti-holomorphic
forms, then we may take $\not{D}_{E}$ to be the Dolbeault-Dirac operator
$\overline{\partial}+\overline{\partial}^{*}$, in which case $\not{D}_{E}^{2}$
is the Dolbeault Laplacian. There are numerous examples (see, e.g., [4]).

Specializing Theorem 4.1 and Theorem 6.1 to $P=Q=|\not{D}_{E}|^{-n/2}$ leads
to the following statement.

######  Proposition 6.6.

Under the above assumptions, let $u\in L\\!\log\\!L(M,\operatorname{End}(E))$.

  1. (1)

The operators $|\not{D}_{E}|^{-n/2}u|\not{D}_{E}|^{-n/2}$ and
$\big{|}|\not{D}_{E}|^{-n/2}u|\not{D}_{E}|^{-n/2}\big{|}$ are both Weyl
operators in $\mathscr{L}_{1,\infty}$, and hence are strongly measurable.

  2. (2)

We have

$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int|\not{D}_{E}|^{-\frac{n}{2}}u|\not{D}_{E}|^{-\frac{n}{2}}=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}u(x)\big{]}\sqrt{g(x)}dx,$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\left||\not{D}_{E}|^{-\frac{n}{2}}u|\not{D}_{E}|^{-\frac{n}{2}}\right|=\lim_{j\rightarrow\infty}j\mu_{j}\left(|\not{D}_{E}|^{-\frac{n}{2}}u|\not{D}_{E}|^{-\frac{n}{2}}\right)=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}|u(x)|\big{]}\sqrt{g(x)}dx.$

  3. (3)

If $u(x)^{*}=u(x)$, then

$\lim_{j\rightarrow\infty}j\lambda^{\pm}_{j}\left(|\not{D}_{E}|^{-\frac{n}{2}}u|\not{D}_{E}|^{-\frac{n}{2}}\right)=\frac{1}{n}(2\pi)^{-n}\\!\int_{M}\operatorname{tr}_{E}\big{[}u(x)_{\pm}\big{]}\sqrt{g(x)}dx.$

######  Remark 6.7.

It would be interesting to obtain analogues of all the results of this paper
for (curved) noncommutative tori. Cwikel-type estimates, CLR-type
inequalities, and integration formulas have been established in this setting
(see [48, 49]). In the critical case the results are established for
potentials in $L_{p}$, $p>1$. It is also expected to have a version for the
Birman-Solomyak’s Weyl’s laws for negative order $\Psi$DOs on NC tori (see
[50, 55]). Therefore, the main task is to establish Cwikel-type estimates for
$L\\!\log\\!L$-Orlicz potentials on NC tori.

## References

  * [1] Agranovich, M.S.; Amosov, B.A.: _Estimates for $s$-numbers, and spectral asymptotics for integral operators of potential type on nonsmooth surfaces_. Funct. Anal. Appl. 30 (1996), no. 2, 75–89
  * [2] Andreev, A.S.: _An estimate of the remainder in the spectral asymptotics of pseudodijferential operators of negative order_. Funct. Anal. Appl. 20 (1986), 124–125.
  * [3] Andreev, A.S.: _Asymptotics of the spectrum of a compact pseudodifferential operators in a Euclidean domain_. Math. USSR Sb. 65 (1990), 205–228.
  * [4] Berline, N.; Getzler, E.; Vergne, M.: _Heat kernels and Dirac operators_. Springer-Verlag, Berlin, 1992.
  * [5] Birman, M.Sh.; Laptev, A.: _The negative discrete spectrum of a two-dimensional Schrödinger operator_. Comm. Pure Appl. Math. 49 (1996), 967–997.
  * [6] Birman, M. Sh.; Laptev, A.; Solomyak, M. _The negative discrete spectrum of the operator $(-\Delta)^{l}-\alpha V$ in $L_{2}(\mathbb{R}^{d})$ for $d$ even and $2l\geq d$_. Ark. Mat. 35 (1997), no. 1, 87–126.
  * [7] Birman, M.Sh.; Solomyak, M.Z.: _The leading term of the spectral asymptotics for “non-smooth” elliptic problems_. Funct. Anal. Appl. 4 (1970), 265–275.
  * [8] Birman, M.Sh.; Solomyak, M.Z.: _Spectral asymptotics of nonsmooth elliptic operators. I_. Trudy Moskov. Mat. Obshch. 27 (1972), 3–52 (Russian). Trans. Moscow Math. Soc. 27 (1972), 1–52 (1975) (English).
  * [9] Birman, M.Sh., Solomyak, M.Z.: _Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory_. Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1974 (Russian). American Mathematical Society Translations, Series 2, 114. American Mathematical Society, Providence, R.I., 1980.
  * [10] Birman, M.S.; Solomyak, M.Z.: _Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols_. Vestnik Leningrad. Univ. 13, no. 3 (1977), 13–21 (Russian). English translation in: Vestn. Leningr. Univ., Math. 10 (1982), 237–247.
  * [11] Birman, M.S.; Solomyak, M.Z.: _Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols. II_. Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 13, no. 3 (1979), 5–10 (Russian).
  * [12] Birman, M.S., Solomyak, M.Z.: _Asymptotics of the spectrum of variational problems on solutions of elliptic equations_. Sib. Math J. 20 (1979), 1–15.
  * [13] Birman M.Sh., Solomyak M.Z.: _Schrödinger operator. Estimates for number of bound states as function-theoretical problem._ Spectral theory of operators (Novgorod, 1989), 1–54. Amer. Math. Soc. Transl. Ser. 2, 150, Amer. Math. Soc., Providence, RI, 1992.
  * [14] Birman, M.Sh., Solomyak, M.Z.: _Spectral theory of self-adjoint operators in Hilbert space_ , D. Reidel, Boston, MA, 1987.
  * [15] Birman, M.S.; Yafaev, D.R.: _Asymptotic behavior of the spectrum of the scattering matrix_. J. Sov. Math. 25 (1984), 793–814.
  * [16] Branson, T.; Chang, S.-Y.A.; Yang, P.: _Estimates and extremals for zeta function determinants on four-manifolds_. Commun. Math. Phys. 149 (1992), 241–262.
  * [17] Chadan, K.; Khuri, N.N. ; Martin, A.; Wu, T.T.: _Bound states in one and two spatial dimensions_. J. Math. Phys. 44 (2003), 406–422.
  * [18] Conlon, J.G.: _A new proof of the Cwikel-Lieb-Rosenbljum bound_. Rocky Mountain J. Math. 15 (1985), 117–122.
  * [19] Connes, A.: _The action functional in non-commutative geometry_. Comm. Math. Phys. 117 (1988), 673–683.
  * [20] Connes, A.: _Noncommutative geometry_. Academic Press, Inc., San Diego, CA, 1994.
  * [21] Cwikel, M.: _Weak type estimates for singular values and the number of bound states of Schrödinger operators_. Ann. of Math. (2), 106(1):93–100, 1977.
  * [22] Daubechies, I.: _An uncertainty principle for fermions with generalized kinetic energy_. Comm. Math. Phys. 90 (1983), 511–520.
  * [23] Dauge, M.; Robert, D.: _Weyl’s formula for a class of pseudodifferential operators with negative order on $L_{2}(\mathbb{R}^{n})$_. Pseudodifferential Operators (Oberwolfach, 1986), Lecture Notes in Math., vol. 1256, Springer-Verlag, 1987, pp. 91–122.
  * [24] Dixmier, J.: _Existence de traces non normales_. C. R. Acad. Sci. Paris Sér. A–B 262 (1966), A1107–A1108.
  * [25] Dykema, K; Figiel, T.; Weiss, G.; Wodzicki, M.: _Commutator structure of operator ideals_. Adv. Math. 185 (2004), 1–79.
  * [26] Fan, K.: _Maximum properties and inequalities for the eigenvalues of completely continuous operators_. Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 760–766.
  * [27] Fefferman, C.L.: _The uncertainty principle_. Bull. Amer. Math. Soc. (N.S.) 9 (1983), 129–206.
  * [28] Fontana, L.: _Sharp borderline Sobolev inequalities on compact Riemannian manifolds_. Comment. Math. Helv. 68 (1993) 415–454.
  * [29] Frank, R.: _Cwikel’s theorem and the CLR inequality_. J. Spectr. Theory 4 (2014), 1–21.
  * [30] Gohberg, I.C.; Krein, M.G.: _Introduction to the theory of linear nonselfadjoint operators_. Translations of Mathematical Monographs, Vol. 18. American Mathematical Society, Providence, R.I., 1969.
  * [31] Grigor’yan, A.; Nadirashvili, N.: _Negative eigenvalues of two-dimensional Schrödinger operators_. Arch. Ration. Mech. Anal. 217 (2015), 975–1028.
  * [32] Grubb, G.: _Spectral asymptotics for nonsmooth singular Green operators_. Comm. Partial Differential Equations 39 (2014), no. 3, 530–573.
  * [33] Guillemin, V.: _A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues_. Adv. Math. 55 (1985), no. 2, 131–160.
  * [34] Huang, J., Sukochev, F., Zanin D.: _Operator $\theta$-Hölder functions with respect to $\|\cdot\|_{p}$, $0<p\leq\infty$_. Preprint.
  * [35] Hundertmark, D.; Kunstmann, P.; Ried, T.; Vulgater, S.: _Cwikel’s bound reloaded_. Preprint, arXiv:1809.05069, 30 pp..
  * [36] Kalton, N.; Lord, S.; Potapov, D.; Sukochev, F.: _Traces of compact operators and the noncommutative residue_. Adv. Math. 235 (2013), 1–55.
  * [37] Ivrii, V.: _Microlocal Analysis, Sharp Spectral Asymptotics and Applications II. Functional Methods and Eigenvalue Asymptotics_. Springer, Cham, 2019.
  * [38] Karuhanga, M.: _On estimates for the number of negative eigenvalues of two-dimensional Schr’́odinger operators with potentials supported by Lipschitz curves_. J. Math. Appl. 456 (2017), 1365–1379.
  * [39] Karuhanga, M; Shargorodsky, E.: _On negative eigenvalues of two-dimensional Schrödinger operators with singular potentials_. J. Math. Phys. 61 (2020), 051509, 26pp..
  * [40] Khuri, N.N.; Martin, A.; Wu, T.T.: _Bound states in $n$ dimensions (especially $n=1$ and $n=2$)_. Few Body Syst. 31 (2002), 83–89.
  * [41] Laptev, A.; Solomyak, M.: _On spectral estimates for two-dimensional Schrödinger operators_. J. Spectr. Theory 3(4) (2013), 505–515.
  * [42] Lesch, M.: _On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols_. Ann. Global Anal. Geom. 17 (1999), no. 2, 151–187.
  * [43] Levin, D.; Solomyak, M.: _The Rozenblum-Lieb-Cwikel inequality for Markov generators_. J. Anal. Math. 71 (1997), 173–193.
  * [44] Lieb, E. H.: _Bounds on the eigenvalues of the Laplace and Schrödinger operators._ Bull. Amer. Math. Soc. 82 (1976), 751–752.
  * [45] Lieb, E. H.: _The number of bound states of one-body Schrödinger operators and the Weyl problem_. Proceedings of Symposia in Pure Mathematics XXXVI. American Mathematical Society, Providence, RI, 1980. 241–252.
  * [46] Li, P.; Yau, S.T.: _On the Schrödinger equation and the eigenvalue problem_. Comm. Math. Phys. 88 (1983), no. 3, 309–318.
  * [47] Lord, S.; Sukochev, F.; Zanin, D.: _Singular traces: theory and applications_. De Gruyter Studies in Mathematics, Vol. 46, Walter de Gruyter, 2012, Theory and applications.
  * [48] McDonald, E; Ponge, R.: _Cwikel estimates and negative eigenvalues of Schrödinger operators on noncommutative tori_. Preprint arXiv:2102.12021, 28 pp..
  * [49] McDonald, E.; Ponge: _Dixmier trace formulas and negative eigenvalues of Schrödinger operators on curved noncommutative tor_. Preprint arXiv:2103.16869, 32 pp..
  * [50] McDonald, E.; Sukochev, F.; Zanin, D.: _Semiclassical Weyl law and exact spectral asymptotics in noncommutative geometry_. Preprint arXiv:2106.02235, 31 pp..
  * [51] Molchanov, S.; Vainberg, B.: _Bargmann type estimates of the counting function for general Schrödinger operators_. J. Math. Sci. 184 (2012), 457–508.
  * [52] Ozawa, T.: _On critical cases of Sobolev’s inequalities_. J. Funct. Anal. 127(2) (1995), 259–269.
  * [53] Ponge, R.: _Traces on pseudodifferential operators and sums of commutators_. J. Anal. Math. 110 (2010), 1–30.
  * [54] Ponge, R.: _Connes’ integration and Weyl’s laws_. Preprint, arXiv:2107.01242, 29pp..
  * [55] Ponge, R.: _Zeta functions and Weyl’s laws on noncommutative tori_. In preparation.
  * [56] Rozenbum, G. V.: _Distribution of the discrete spectrum of singular differential operators._ Soviet Math. Dokl. 13 (1972), 245–249.
  * [57] Rozenblum, G. V.: _Distribution of the discrete spectrum of singular differential operators._ Soviet Math. (Iz. VUZ) 20 (1976), 63–71.
  * [58] Rozenblum, G.: _Eigenvalues of singular measures and Connes noncommutative integration_. Preprint arXiv:2103.02067, 33 pp.. To appear in J. Spectr. Theory.
  * [59] Rozenblum, G; Shargorodsky, E.: _Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case_. Partial differential equations, spectral theory, and mathematical Physics. The Ari Laptev anniversary volume, EMS Series of Congress Reports, Vol. 18, EMS Publishing, 2021, pp. 331–353.
  * [60] Rozenblum, G.; Solomyak, M. _The Cwikel-Lieb-Rozenblyum estimator for generators of positive semigroups and semigroups dominated by positive semigroups_. St. Petersburg Math. J. 9 (1998), no. 6 1195–1211.
  * [61] Rozenblum, G.; Tashchiyan., G.: _Eigenvalues of the Birman-Schwinger operator for singular measures: the noncritical case_. Preprint arXiv:2107.04682, 40pp..
  * [62] Shargorodsky, E.: _On negative eigenvalues of two-dimensional Schrödinger operators_. Proc. London Math. Soc. 108 (2014), 441–483.
  * [63] Semenov, E.; Sukochev, F.; Usachev, A.; Zanin, D.: _Banach limits and traces on $\mathscr{L}_{1,\infty}$_. Adv. Math. 285 (2015), 568–628.
  * [64] Shubin, M.A.: _Pseudodifferential operators and spectral theory_ , 2nd edition. Springer, Berlin, 2001.
  * [65] Simon, B.: _Analysis with weak trace ideals and the number of bound states of Schrödinger operators_. Trans. Amer. Math. Soc. 224 (1976), 367–380.
  * [66] Simon, B.: _Trace ideals and their applications_. Second edition. Mathematical Surveys and Monographs, 120. American Mathematical Society, Providence, RI, 2005.
  * [67] Simon, B.: _Convexity: an analytic viewpoint_. Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2011.
  * [68] Simon, B.: _Operator Theory: A Comprehensive Course in Analysis, Part 4_. American Mathematical Society, Providence, RI, 2015.
  * [69] Solomyak, M.: _Piecewise-polynomial approximation of functions from $H^{l}((0,1)^{d})$, $2l=d$, and applications to the spectral theory of the Schrödinger operator_. Isr. J. Math. 86 (1994), 253–275.
  * [70] Solomyak, M.: _Spectral problems related to the critical exponent in the Sobolev embedding theorem_. Proc. London Math. Soc. 71 (1995), no. 1, 53–75.
  * [71] Stoiciu, M.: _An estimate for the number of bound states of the Schrödinger operator in two dimensions_. Proc. Am. Math. Soc. 132 (2004), 1143–1151.
  * [72] Sukochev, F.; Zanin, D.: _Cwikel-Solomyak estimates on tori and Euclidean spaces_. Preprint arXiv:2008.04494, 31pp..
  * [73] Sukochev, F.; Zanin, D.: _Connes integration formula without singular traces_. Preprint arXiv:2103.08817, 19 pp..
  * [74] Wodzicki, M.: _Spectral asymmetry and noncommutative residue_ (in Russian). Habilitation Thesis, Steklov Institute, Soviet Academy of Sciences, Moscow, 1984.
  * [75] Wodzicki, M.: _Noncommutative residue. I. Fundamentals_. $K$-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math., Vol. 1289, Springer, Berlin, 1987, pp. 320–399.