# On Quantum Sobolev Inequalities

Laurent Lafleche Institut Camille Jordan, UMR 5208 CNRS
& Université Claude Bernard Lyon 1, France<EMAIL_ADDRESS>
###### Abstract.

We investigate the quantum analogue of the classical Sobolev inequalities in
the phase space. These inequalities can be seen as a many body uncertainty
principle, and also lead to new bounds on the Schatten norms of the Weyl
quantization in terms of its symbol. As an intermediate tool, we define a
semiclassical analogue of the convolution together with the corresponding
Young’s and Hardy–Littlewood–Sobolev’s inequalities, and introduce quantum
Besov spaces. Explicit estimates are obtained on the optimal constants.

###### Key words and phrases:

Operator inequalities, trace inequalities, Sobolev inequalities, semiclassical
limit, many body uncertainty principle

###### 2020 Mathematics Subject Classification:

46E35 $\cdot$ 81Q20 $\cdot$ 81S30 $\cdot$ 47A30 (81S07, 46N50).

######  Contents

  1. 1 Introduction
  2. 2 Notations, basic properties and results.
    1. 2.1 Classical Sobolev spaces
    2. 2.2 Quantum Sobolev spaces
    3. 2.3 Quantum Sobolev Inequalities
    4. 2.4 Quantum Besov spaces
    5. 2.5 Boundedness of Riesz transforms in the quantum phase space
  3. 3 Semiclassical convolution
    1. 3.1 Phase space translations
    2. 3.2 Definition of the semiclassical convolution and inequalities
    3. 3.3 Convolution by a Gaussian: Husimi transform and coherent states
  4. 4 Proofs
    1. 4.1 Semiclassical convolution
    2. 4.2 Proofs of Sobolev inequalities
    3. 4.3 Quantum Besov spaces
  5. A Vector valued operators

##  1\. Introduction

Semiclassical analysis allows to understand the similarities between quantum
and classical mechanics, and more generally to obtain the leading behavior of
highly oscillating waves. From a physical viewpoint, this corresponds to take
units such that the constant $\hbar$ becomes negligible, or at least very
small. However, error terms corresponding to high power of $\hbar$ often
involve the knowledge of some uniform in $\hbar$ regularity.

A typical object used to make the link between classical and quantum mechanics
is the Weyl quantization, that associates to a function $f=f(x,\xi)$ of the
phase space an operator

(1)
$\boldsymbol{\rho}_{f}:=\iint_{\mathbb{R}^{2d}}\widehat{f}(y,\xi)\,e^{2i\pi\left(y\cdot
x+\xi\cdot\boldsymbol{p}\right)}\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}\xi$

acting on $L^{2}(\mathbb{R}^{d})$, by analogy with the Fourier inversion
formula. Here, $(y,\xi)\in\mathbb{R}^{2d}$ are the phase space variables,
$\boldsymbol{p}=-i\hbar\nabla$ is the quantum analogue of the momentum, and
$\widehat{f}$ denotes the Fourier transform of $f$, defined by

$\operatorname{\mathcal{F}}\\!\left(f\right)\\!(y,\xi)=\widehat{f}(y,\xi)=\iint_{\mathbb{R}^{2d}}f(y,\xi)\,e^{-2i\pi\left(y\cdot
y^{\prime}+\xi\cdot\xi^{\prime}\right)}\mathop{}\\!\mathrm{d}y^{\prime}\mathop{}\\!\mathrm{d}\xi^{\prime}.$

Equivalently, $\boldsymbol{\rho}_{f}$ is the operator with integral kernel

$\boldsymbol{\rho}_{f}(x,y)=\int_{\mathbb{R}^{d}}e^{-2i\pi\left(y-x\right)\cdot\xi}\,f(\tfrac{x+y}{2},h\xi)\mathop{}\\!\mathrm{d}\xi.$

Another widely used object is the Wigner transform, that is the inverse
operation: it associates to an operator $\boldsymbol{\rho}$ a function on the
phase space

(2)
$f_{\boldsymbol{\rho}}(x,\xi)=\int_{\mathbb{R}^{d}}e^{-i\,y\cdot\xi/\hbar}\,\boldsymbol{\rho}(x+\tfrac{y}{2},x-\tfrac{y}{2})\mathop{}\\!\mathrm{d}y.$

In general, these pointwise identities should be understood in the sense of
distributions. One of the difficulties when trying to prove the propagation of
regularity uniformly with respect to $\hbar$ of the Wigner transform of a
solution for example of the Schrödinger or Hartree equation is the fact that
the Wigner transform is not positive and its Lebesgue norms
$L^{p}(\mathbb{R}^{2d})$ are not conserved except for $p=2$ (see e.g. [34]).
Therefore, it is more convenient to work at the level of operators and to
study conveniently scaled Schatten norms of these operators as a replacement
of the Lebesgue norms, that we will denote by111Notice that the spaces of
operators with finite $\mathcal{L}^{p}$ norm are just the standard Schatten
spaces. If one wants to define a space taking into account the $\hbar$, then
one should rather consider a space $\ell^{\infty}\mathcal{L}^{p}$ of sequences
$(\boldsymbol{\rho}_{\hbar})_{\hbar\in(0,1)}$ of operators indexed by $\hbar$
such that
$\sup_{\hbar\in(0,1)}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}<\infty$.

(3)
$\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}=h^{\frac{d}{p}}\left\|{\boldsymbol{\rho}}\right\|_{p}=h^{\frac{d}{p}}\operatorname{Tr}\\!\left(\left|{\boldsymbol{\rho}}\right|^{p}\right)^{\frac{1}{p}},$

where
$\left|{\boldsymbol{\rho}}\right|=\sqrt{\boldsymbol{\rho}^{*}\boldsymbol{\rho}}$.
The normalization implies in particular that
$\|\boldsymbol{\rho}_{f}\|_{\mathcal{L}^{2}}=\left\|{f}\right\|_{L^{2}(\mathbb{R}^{2d})}$
and
$h^{d}\operatorname{Tr}(\boldsymbol{\rho}_{f})=\int_{\mathbb{R}^{2d}}f(x,\xi)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}\xi$.
For similar reasons, it is convenient to consider the quantum analogue of
gradients in the phase space. The correspondence principle leads to define the
quantum gradients by the following formulas

(4)
$\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}:=\left[{\nabla,\boldsymbol{\rho}}\right]\quad\text{
and
}\quad\boldsymbol{\nabla}_{\\!\xi}\boldsymbol{\rho}:=\left[{\frac{x}{i\hbar},\boldsymbol{\rho}}\right].$

They can also be seen just as the Weyl quantization of the classical phase
space gradients since

(5)
$\boldsymbol{\rho}_{\nabla_{x}f}=\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}_{f}\quad\text{
and
}\quad\boldsymbol{\rho}_{\nabla_{\xi}f}=\boldsymbol{\nabla}_{\\!\xi}\boldsymbol{\rho}_{f}.$

In this paper, we prove the quantum analogue of the classical Sobolev
inequalities. In dimension $d=3$, it implies for example inequalities such as

$\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{3}}^{2}\leq
2\,C^{2}\left\|{\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{2}}\left\|{\boldsymbol{\nabla}_{\\!\xi}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{2}}$

where
$C=\mathcal{C}^{\mathrm{S}}_{1,2}+\frac{1}{2\sqrt{2\pi}}=\frac{1}{2\sqrt{2\pi}}\left((\frac{20}{9})^{1/6}+1\right)=\left(1+(\frac{9}{20})^{1/6}\right)\mathcal{C}^{\mathrm{S}}_{1,2}$
with $\mathcal{C}^{\mathrm{S}}_{1,2}$ the optimal constant in the classical
Sobolev inequality in dimension $6$.

These inequalities can be seen as many-body uncertainty principles. If
$\boldsymbol{\rho}\geq 0$ is an operator such that
$h^{d}\operatorname{Tr}\\!\left(\boldsymbol{\rho}\right)=1$, notice indeed
that by the Cauchy–Schwarz inequality
$\left|{\boldsymbol{\rho}(x,y)}\right|^{2}\leq\boldsymbol{\rho}(x,x)\,\boldsymbol{\rho}(y,y)$,
so that
$D_{x}:=\hbar\,h^{d/2}\left\|{\boldsymbol{\nabla}_{\\!\xi}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{2}}$
verifies

$D_{x}^{2}=h^{2d}\iint_{\mathbb{R}^{2d}}\left|{x-y}\right|^{2}\left|{\boldsymbol{\rho}(x,y)}\right|^{2}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\leq
2\,\sigma_{x}^{2}$

where the standard variation $\sigma_{x}$ is defined by
$\sigma_{x}^{2}=h^{d}\operatorname{Tr}(\boldsymbol{\rho}\left|{x}\right|^{2})-(h^{d}\operatorname{Tr}\\!\left(\boldsymbol{\rho}\,x\right))^{2}$.

The boundedness of the quantum gradients (4) in the above defined quantum
Lebesgue norms (3) independently of $\hbar$ is at the core of several works
dealing with semiclassical estimates. It is used in [8, 7, 39, 26, 18] in the
context of the derivation of the Hartree equation in the mean-field regime, in
[41, 31, 20] in the context of the derivation of the Vlasov equation from the
Hartree–Fock equation in the semiclassical regime, in [25] in the context of
the convergence of numerical schemes for quantum dynamics. Propagation of
these quantities in the case of the Hartree and Hartree–Fock equation with
Coulomb potential was proved in [18, 19] where it was used to derive the
Hartree–Fock and Vlasov equations with singular potentials in a combined mean-
field and semiclassical limit. Other mixed norms of the quantum gradients were
considered in [39, 41].

These semiclassical Schatten bounds on the quantum gradients are natural in
the case when the system is in a state corresponding to a bounded distribution
in the phase space. This type of bounds were proved to hold for certain pure
states (that is projection operators) in [23], and in [21] for thermal states.
Other examples of more general states can be found in [8, 31, 18, 6]. The
regularity of projection operators in terms of the spaces defined in this
paper is studied in [30]. A similar definition of quantum derivatives were
used in [5].

##  2\. Notations, basic properties and results.

###  2.1. Classical Sobolev spaces

Denote by $z=(x,\xi)\in\mathbb{R}^{2d}$ the phase space variable. Then, in the
classical setting, the homogeneous Sobolev spaces of order $1$ can be defined
as set of phase space functions $f$ vanishing at infinity such that the
following norm is finite

$\left\|{f}\right\|_{\dot{W}^{1,p}(\mathbb{R}^{2d})}=\left\|{\nabla_{z}f}\right\|_{L^{p}(\mathbb{R}^{2d})}.$

One can also define the corresponding non-homogeneous space endowed with the
norm
$\left\|{f}\right\|_{W^{1,p}(\mathbb{R}^{2d})}=\left\|{f}\right\|_{L^{p}(\mathbb{R}^{2d})}+\left\|{\nabla_{z}f}\right\|_{L^{p}(\mathbb{R}^{2d})}$.
By convention, we set $\dot{W}^{0,p}:=L^{p}$. We will here mostly work on
homogeneous spaces and not repeat these details.

####  2.1.1. Fractional Sobolev spaces

There is not a unique way to define the generalization to fractional order. We
refer to [44, 48, 1, 35] for details on the extensions of Sobolev spaces to
fractional order, with spaces such as the Besov spaces $B^{s}_{p,q}$ and the
Triebel–Lizorkin spaces $F^{s}_{p,q}$. In the case $p=\infty$ and $s\in(0,1)$,
the most commonly used spaces are the Hölder spaces
$\dot{W}^{s,\infty}=\dot{B}^{s}_{\infty,\infty}$ defined by the norm

$\left\|{f}\right\|_{\dot{W}^{s,\infty}}:=\sup_{(z,w)\in\mathbb{R}^{2d}\times\mathbb{R}^{2d}}\frac{\left|{f(z)-f(w)}\right|}{\left|{z-w}\right|^{s}}.$

More generally, the most commonly called fractional Sobolev spaces are the
Bessel–Sobolev spaces
$\dot{H}^{s,p}(\mathbb{R}^{2d})=\dot{F}^{s}_{p,2}(\mathbb{R}^{2d})$ defined
through the norm

$\left\|{f}\right\|_{\dot{H}^{s,p}(\mathbb{R}^{2d})}:=\left\|{(-\Delta_{z})^{s/2}f}\right\|_{L^{p}(\mathbb{R}^{2d})}$

where $(-\Delta_{z})^{s/2}$ denotes the fractional Laplacian and we will write
$\dot{H}^{s}=\dot{H}^{s,2}$, and the Sobolev–Slobodeckij spaces
$\dot{W}^{s,p}=\dot{B}^{s}_{p,p}$ with norm

(6)
$\left\|{f}\right\|_{\dot{W}^{s,p}}^{p}:=\gamma_{s,p}\,\iint_{\mathbb{R}^{2d}\times\mathbb{R}^{2d}}\frac{\left|{f(z)-f(w)}\right|^{p}}{\left|{z-w}\right|^{2d+sp}}\mathop{}\\!\mathrm{d}w\mathop{}\\!\mathrm{d}z$

for $s\in(0,1)$, where $\gamma_{s,p}=\gamma_{s,p}$ is chosen so that
$\left\|{f}\right\|_{\dot{W}^{s,2}}=\left\|{f}\right\|_{\dot{H}^{s}}$,
$\left\|{f}\right\|_{\dot{W}^{s,p}}\to\left\|{f}\right\|_{\dot{W}^{n,p}}$ when
$s\to n$ with $n=0$ or $n=1$, and
$\left\|{f}\right\|_{\dot{W}^{s,p}}\to\left\|{f}\right\|_{\dot{W}^{s,\infty}}$
when $p\to\infty$.

######  Remark 2.1.

One can take for instance

$\gamma_{s,p}=\frac{p\left|{\omega_{-2s}}\right|}{4\,\omega_{2d+sp}}\left(\frac{\pi\,\omega_{p+1}}{s^{\frac{p-2}{2}}}\right)^{s},$

where $\omega_{d}=|\SS^{d-1}|=\frac{2\,\pi^{d/2}}{\Gamma(d/2)}$. Indeed it
follows from the integral formula for the fractional Laplacian222interpreting
the integral as a principal value when $s\geq 1$, for $s\in(0,2)$ and
$z\in\mathbb{R}^{2d}$

(7)
$(-\Delta_{z})^{\frac{s}{2}}f(z)=c_{2d,s}\int_{\mathbb{R}^{2d}}\frac{f(z)-f(w)}{\left|{z-w}\right|^{2d+s}}\mathop{}\\!\mathrm{d}w$

where
$c_{2d,s}:=\left(2\pi\right)^{s}\frac{\left|{\omega_{-s}}\right|}{\omega_{2d+s}}$,
that
$\gamma_{s,2}=\frac{c_{2d,2s}}{2}=\left(2\pi\right)^{2s}\frac{\left|{\omega_{-2s}}\right|}{2\,\omega_{2d+2s}}$.
Moreover, it is not difficult to check using the Stirling formula that
$\gamma_{s,p}^{1/p}\to 1$ when $p\to\infty$. On the other hand, it was proved
in [12] (see also [13, Equation (44)]) that
$\gamma_{s,p}\sim\frac{p\,\omega_{p+1}}{2\,\omega_{2d+p}}\left(1-s\right)$
when $s\to 1$ and it was proved in [36] that
$\gamma_{s,p}\sim\frac{p\,s}{2\,\omega_{2d}}$ when $s\to 0$.

######  Remark 2.2.

In the case $s=1$ and $p\in(1,\infty)$ as well as in the case $p=2$, the
spaces $\dot{W}^{s,p}$ and $\dot{H}^{s,p}$ coincide (see e.g. [14]). Their
norms are equal in the case when $p=2$.

####  2.1.2. Classical Sobolev inequalities

One of the important feature of Sobolev spaces is the Sobolev inequality,
originally proved in [43] and then generalized to many more spaces (see e.g.
[33, 1]). It tells that regularity implies additional integrability. More
precisely, if $s>0$ and $1\leq p\leq q<\infty$ verify

(8) $\frac{1}{p}-\frac{1}{q}=\frac{s}{2d}$

then there exists a constant $\mathcal{C}^{\mathrm{S}}_{s,p}$ such that for
any $f\in\dot{W}^{s,p}(\mathbb{R}^{2d})$

(9)
$\left\|{f}\right\|_{L^{q}(\mathbb{R}^{2d})}\leq\mathcal{C}^{\mathrm{S}}_{s,p}\left\|{f}\right\|_{\dot{W}^{s,p}(\mathbb{R}^{2d})}.$

When $p>1$, an analogue version holds for the Bessel–Sobolev spaces, i.e.
there exists a constant $\mathcal{C}^{\mathrm{B}}_{s,p}$ such that for any
$f\in\dot{H}^{s,p}(\mathbb{R}^{2d})$

(10)
$\left\|{f}\right\|_{L^{q}(\mathbb{R}^{2d})}\leq\mathcal{C}^{\mathrm{B}}_{s,p}\left\|{f}\right\|_{\dot{H}^{s,p}(\mathbb{R}^{2d})}.$

As found by [3, 45], when $p=1$, the optimal constant in Inequality (9) is
given by

$\mathcal{C}^{\mathrm{S}}_{1,p}=\frac{1}{2d}\left(\frac{q}{p^{\prime}}\right)^{1/p^{\prime}}\left(\frac{\omega_{4d/p}\,\omega_{4d/q^{\prime}}}{\omega_{2d+2}\,\omega_{4d}}\right)^{1/d}.$

When $p=2$, the optimal constant was found in [32] to be

$\mathcal{C}^{\mathrm{B}}_{s,2}=\mathcal{C}^{\mathrm{S}}_{s,2}=\frac{1}{\pi^{s}\,\omega_{2d+1}^{s/(2d)}}\left(\frac{\omega_{2d+2s}}{\omega_{2d-2s}}\right)^{1/2}.$

To conclude this section, notice that replacing $f$ by $(-\Delta)^{t/2}f$ with
$\alpha\in(-d,d)$ shows that more generally, if
$\frac{1}{p}-\frac{1}{q}=\frac{s-t}{2d}$, then

(11)
$\left\|{f}\right\|_{\dot{H}^{t,q}(\mathbb{R}^{2d})}\leq\mathcal{C}^{\mathrm{B}}_{s-t,p}\left\|{f}\right\|_{\dot{H}^{s,p}(\mathbb{R}^{2d})}.$

###  2.2. Quantum Sobolev spaces

The above considerations lead us to define the quantum analogue of Sobolev
spaces in the phase space. We define the homogeneous quantum Sobolev
norms333This is in general a semi-norm, but gives a norm when restricted to
the set of compact operators $\boldsymbol{\rho}$ such that
$\boldsymbol{\nabla}^{n}\boldsymbol{\rho}$ is compact. of integer order by

$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{n,p}}:=\left\|{\boldsymbol{\nabla}^{n}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}.$

This is an equivalent norm to the norm
$\left\|{\boldsymbol{\nabla}_{\\!x}^{n}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}+\big{\lVert}\boldsymbol{\nabla}_{\\!\xi}^{n}\boldsymbol{\rho}\big{\rVert}_{\mathcal{L}^{p}}$
used in [18] (see Appendix A about vector valued operators).

When $s\in(0,1)$, one can define the quantum analogue of the Gagliardo semi-
norms $\dot{W}^{s,p}$ by setting

(12)
$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}^{p}:=\gamma_{s,p}\,h^{d}\int_{\mathbb{R}^{2d}}\frac{\operatorname{Tr}\\!\left(\left|{\mathsf{T}_{z}\boldsymbol{\rho}-\boldsymbol{\rho}}\right|^{p}\right)}{\left|{z}\right|^{2d+sp}}\mathop{}\\!\mathrm{d}z.$

Here, $\mathsf{T}_{z}$ is the quantum phase space translation operator defined
by

(13) $\mathsf{T}_{z_{0}}\boldsymbol{\rho}=e^{i\left(\xi_{0}\cdot
x-x_{0}\cdot\boldsymbol{p}\right)/\hbar}\boldsymbol{\rho}\,e^{i\left(x_{0}\cdot\boldsymbol{p}-\xi_{0}\cdot
x\right)/\hbar}.$

One can then define higher order Sobolev spaces by defining for
$s\in\mathbb{R}_{+}\setminus\mathbb{N}$,
$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}=\left\|{\boldsymbol{\nabla}^{n}\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s-n,p}}$
with $n=\left\lfloor{s}\right\rfloor$. In the particular case $p=\infty$, the
limiting norm corresponding to (12) gives the quantum analogue of the Hölder
norms, that is

(14)
$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,\infty}}:=\sup_{z\in\mathbb{R}^{2d}}\frac{\left\|{\mathsf{T}_{z}\boldsymbol{\rho}-\boldsymbol{\rho}}\right\|_{\mathcal{L}^{\infty}}}{\left|{z}\right|^{s}}.$

To define a quantum analogue of the Bessel-type Sobolev spaces, we start to
define the fractional Laplacian for operators. A convenient way is to go
through the Wigner transform and Weyl quantization, as in Formula (5). Hence
we define

(15)
$(-\boldsymbol{\Delta})^{s}\boldsymbol{\rho}:=\boldsymbol{\rho}_{(-\Delta_{z})^{s}f_{\boldsymbol{\rho}}}.$

One can then verify easily that
$\boldsymbol{\Delta}\boldsymbol{\rho}=\boldsymbol{\nabla}_{\\!x}\cdot\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}+\boldsymbol{\nabla}_{\\!\xi}\cdot\boldsymbol{\nabla}_{\\!\xi}\boldsymbol{\rho}$
and get an analogue of the classical integral formulas such as Equation (7).
If $s\in(0,1)$, then444More precisely, Equation (16) follows formally by
linearity of the Weyl quantization from the definition (15), Equation (7) and
Formula (30) used with $f=f_{\boldsymbol{\rho}}$. Similarly as the Definition
(32) given later, the integral should in general be interpreted in the weak-
sense as defined in Remark 4.1.

(16)
$(-\boldsymbol{\Delta})^{\frac{s}{2}}\boldsymbol{\rho}=c_{2d,s}\int_{\mathbb{R}^{d}}\frac{\boldsymbol{\rho}-\mathsf{T}_{z}\boldsymbol{\rho}}{\left|{z}\right|^{2d+s}}\mathop{}\\!\mathrm{d}z.$

Now, as a quantum analogue of the Bessel-type homogeneous Sobolev norms, one
can define for any $s>-2d$,

$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{H}}^{s,p}}:=\left\|{(-\boldsymbol{\Delta})^{s/2}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}.$

We will use the shortcut notation
$\dot{\mathcal{H}}^{s}=\dot{\mathcal{H}}^{s,2}$. Since
$\lVert\boldsymbol{\rho}_{f}\rVert_{\mathcal{L}^{2}}=\left\|{f}\right\|_{L^{2}(\mathbb{R}^{2d})}$,
one obtains from Remark 2.2 that for any compact operator $\boldsymbol{\rho}$,

$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,2}}=\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{H}}^{s}}.$

To finish this section, let us comment about the case of a negative order of
regularity. A general procedure to define negative Sobolev spaces is to
proceed by duality and define for $s>0$ and an operator $\boldsymbol{\mu}$,

$\left\|{\boldsymbol{\mu}}\right\|_{\dot{\mathcal{W}}^{-s,p}}:=\sup_{\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p^{\prime}}}\leq
1}h^{d}\operatorname{Tr}\\!\left(\boldsymbol{\mu}\,\boldsymbol{\rho}\right)$

where the supremum is taken over all compact operators $\boldsymbol{\rho}$
such that
$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p^{\prime}}}\leq 1$.
One can define similarly the non-homogeneous norm $\mathcal{W}^{-s,p}$ by
taking the supremum over all the compact operators verifying
$\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{W}^{s,p^{\prime}}}\leq 1$
instead. Particular cases of these norms where used in [25, 27] where they are
compared with optimal transport pseudo-norms. For example, it is proved in
[27, Proposition B.2] that

$\mathop{}\\!\mathrm{d}(\boldsymbol{\rho}_{1},\boldsymbol{\rho}_{2}):=\left\|{\boldsymbol{\rho}_{1}-\boldsymbol{\rho}_{2}}\right\|_{\mathcal{W}^{-1,\infty}}$

is a distance on the set of Hilbert–Schmidt operators which verifies
$\mathop{}\\!\mathrm{d}(\boldsymbol{\rho}_{1},\boldsymbol{\rho}_{2})\leq
2^{d}\sup_{\left\|{\varphi}\right\|_{W^{1,\infty}\cap H^{1}}\leq
1}\int_{\mathbb{R}^{2d}}\left(f_{\boldsymbol{\rho}_{1}}(z)-f_{\boldsymbol{\rho}_{2}}(z)\right)\varphi(z)\mathop{}\\!\mathrm{d}z$
and
$\mathop{}\\!\mathrm{d}(\boldsymbol{\rho}_{1},\boldsymbol{\rho}_{2})\leq\left\|{\boldsymbol{\rho}_{1}-\boldsymbol{\rho}_{2}}\right\|_{\mathcal{L}^{\infty}}$.

###  2.3. Quantum Sobolev Inequalities

We are now able to obtain the analogue of the classical Sobolev inequalities
in terms of these Schatten-based Sobolev norms, everything being uniform in
$\hbar$.

######  Theorem 1 (Gagliardo–Sobolev inequalities).

Let $s\in[0,1]$ and $1\leq p\leq q<\infty$ verifying (8). Then there exits a
constant $\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}$ independent of $\hbar$
such that for any compact operator $\boldsymbol{\rho}$,

(17)
$\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}\leq\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}.$

The optimal independent-of-$\hbar$ constant
$\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}$ verifies

(18)
$\displaystyle\mathcal{C}^{\mathrm{S}}_{s,p}\leq\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}$
$\displaystyle\leq\mathcal{C}^{\mathrm{S}}_{s,p}+\frac{\omega_{2d}}{\omega_{2d+1}}$
$\displaystyle\text{ when }s=1$ (19)
$\displaystyle\mathcal{C}^{\mathrm{S}}_{s,p}\leq\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}$
$\displaystyle\leq\mathcal{C}^{\mathrm{S}}_{s,p}+\tfrac{\theta_{s}}{\left(8\pi\right)^{s/2}}$
$\displaystyle\text{ when }p=2$ (20)
$\displaystyle\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}$
$\displaystyle\leq\mathcal{C}^{\mathrm{S}}_{s,p}+\frac{\left(1-2^{\left(s-1\right)r}\right)^{\frac{1}{r}}}{\gamma_{s,p}^{1/p}\left(p^{\prime}\right)^{d+\frac{s}{2}}}\left(\frac{\omega_{2d}}{\omega_{(2d+s)p^{\prime}}}\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle\text{ when }s\in(0,1)$

where $r=\max(p,p^{\prime})$, $\mathcal{C}^{\mathrm{S}}_{s,p}$ is the constant
of the corresponding classical Sobolev inequality (9) and
$\theta_{s}=\sup_{r\geq 0}\frac{1-e^{-r}}{r^{s}}$ verifies $\theta_{s}\leq 1$.
The second inequality in Equation (19) is also verified when $q=2$.

In the case when $s=1$, applying the above theorem (together with Proposition
A.2) to
$\boldsymbol{\rho}_{\lambda}(x,y)=\lambda^{d}\boldsymbol{\rho}(\lambda\,x,\lambda\,y)$
and optimizing with respect to $\lambda$ implies the following result.

######  Corollary 2.1 (Sobolev inequalities).

If $1\leq p\leq q<\infty$ verify $\frac{1}{2d}=\frac{1}{p}-\frac{1}{q}$ then
for any compact operator $\boldsymbol{\rho}$,

(21) $\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}\leq
2^{1/r}\,\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{1,p}\left\|{\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}^{1/2}\left\|{\boldsymbol{\nabla}_{\\!\xi}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}^{1/2}\leq\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{1,p}\left(\left\|{\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}^{r}+\left\|{\boldsymbol{\nabla}_{\\!\xi}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}^{r}\right)^{\frac{1}{r}}$

where $r=\min(p,2)$.

######  Remark 2.3 (Operator norm of the Weyl quantization).

In the case when $q=\infty$, the classical Sobolev embedding breaks, but a
proof similar to the proof of Theorem 1 yields the following interesting bound
for the operator norm of the Weyl quantization (1) in terms of its symbol

$\lVert\boldsymbol{\rho}_{f}\rVert_{\mathcal{L}^{\infty}}\leq\left\|{f}\right\|_{L^{\infty}(\mathbb{R}^{2d})}+C\left\|{f}\right\|_{\dot{B}^{d}_{2,\infty}(\mathbb{R}^{2d})},$

which is a slight improvement, in terms of Besov spaces, with respect to the
classical bound that follows from Formula (1)

$\lVert\boldsymbol{\rho}_{f}\rVert_{\mathcal{L}^{\infty}}\leq\left\|{\operatorname{\mathcal{F}}\\!\left(f\right)}\right\|_{L^{1}(\mathbb{R}^{2d})}\leq
C\left\|{f}\right\|_{\dot{B}^{d}_{2,1}(\mathbb{R}^{2d})}$

since $\dot{B}^{d}_{2,1}(\mathbb{R}^{2d})\subset
L^{\infty}(\mathbb{R}^{2d})\cap\dot{B}^{d}_{2,\infty}(\mathbb{R}^{2d})$. This
complements works such as [29] about the boundedness of operators with
discontinuous symbol. From the above theorem with $p=2$ and $q<\infty$, one
gets the following bound for the Schatten norm of the Weyl quantization

$\lVert\boldsymbol{\rho}_{f}\rVert_{\mathcal{L}^{q}}\leq\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,2}\left\|{f}\right\|_{\dot{H}^{s}(\mathbb{R}^{2d})}\qquad\text{
with }\frac{s}{2d}=\frac{1}{2}-\frac{1}{q}.$

One can also compare the above results with what is obtained using the
Calderòn–Vaillancourt theorem together with the fact that
$\lVert\boldsymbol{\rho}_{f}\rVert_{\mathcal{L}^{2}}=\left\|{f}\right\|_{L^{2}(\mathbb{R}^{2d})}$.
If $n=\left\lfloor{d/2}\right\rfloor+1$, by an improved version of the
Calderòn–Vaillancourt theorem for the Weyl quantization proved by Boulkhemair
[11, Theorem 1.2], the following inequality holds
$\lVert\boldsymbol{\rho}_{f}\rVert_{\mathcal{L}^{\infty}}\leq
C\left\|{f}\right\|_{W^{2n,\infty}(\mathbb{R}^{2d})}$.

As in the classical case, there is an analogue of the quantum Sobolev
inequality in the case of the Bessel–Sobolev spaces whenever $p>1$. One can
also estimate the constant in this case, however we do not give such estimates
in the general case, as the proof given here uses interpolation theory and
equivalent norms, which makes much more involved the computations of the
constants. It is an interesting question to know if it is possible to get a
more elementary proof.

######  Theorem 2 (Bessel–Sobolev inequalities).

Let $s\in[0,1]$ and $1<p\leq q<\infty$ verifying (8). Then there exist a
constant $\widetilde{\mathcal{C}}^{\mathrm{\,B}}_{s,p}$ such that for any
compact operator $\boldsymbol{\rho}$,

(22)
$\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}\leq\widetilde{\mathcal{C}}^{\mathrm{\,B}}_{s,p}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{H}}^{s,p}}.$

When $p=2$ then
$\widetilde{\mathcal{C}}^{\mathrm{\,B}}_{s,p}=\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}$.
When $q=2$ then
$\widetilde{\mathcal{C}}^{\mathrm{\,B}}_{s,p}\leq\mathcal{C}^{\mathrm{S}}_{s,p}+\frac{\theta_{s}}{(8\pi)^{s/2}}$.
More generally, when $p\leq 2\leq q$, a bound on the constant is given in
Proposition 4.3.

######  Remark 2.4.

For the Bessel–Sobolev spaces or the Sobolev spaces of integer regularity, the
analogue of Inequality (11) holds. More generally, replacing
$\boldsymbol{\rho}$ by $\boldsymbol{\nabla}^{n}\boldsymbol{\rho}$ or
$(-\boldsymbol{\Delta})^{\alpha}\boldsymbol{\rho}$, it is not difficult to
obtain inequalities for higher order Sobolev spaces, and from one Sobolev
space to another.

Let us also indicate that in the above Sobolev inequalities, the relation (8)
always implies that $p<\frac{2d}{s}$. In the case when $p>\frac{2d}{s}$, then
a similar proof as the proof of Theorem (1) gives the analogue of the
Morrey–Sobolev inequalities, that is the embedding of Sobolev spaces in Hölder
spaces. In the quantum case, this gives the following inequalities in terms of
the norms (14).

######  Theorem 3 (Morrey–Sobolev inequalities).

Let $0<\theta\leq s\leq 1$ and $1\leq p<\infty$ verify

$\frac{1}{p}=\frac{s-\theta}{2d}.$

Then there exists a constant $\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}$
independent of $\hbar$ such that for any compact operator $\boldsymbol{\rho}$,

(23)
$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{\theta,\infty}}\leq\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}.$

###  2.4. Quantum Besov spaces

More generally, one can also consider the quantum analogue of Besov spaces by
defining for example for $s\in(0,2)$,

$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,r}}:=\Big{\lVert}\frac{\mathsf{T}_{2z}\boldsymbol{\rho}-2\,\mathsf{T}_{z}\boldsymbol{\rho}+\boldsymbol{\rho}}{\left|{z}\right|^{s+2d/r}}\Big{\rVert}_{L^{r}_{z}(\mathcal{L}^{p})}.$

They are in particular natural when considering the maximal regularity of
projection operators [30]. Since $\mathsf{T}_{z}$ preserves the Schatten norm,
one can replace
$\mathsf{T}_{2z}\boldsymbol{\rho}+\boldsymbol{\rho}-2\mathsf{T}_{z}\boldsymbol{\rho}$
by
$\mathsf{T}_{z}\boldsymbol{\rho}+\mathsf{T}_{-z}\boldsymbol{\rho}-2\boldsymbol{\rho}$.
As in the classical case, one can find a Littlewood–Paley decomposition of
these spaces (see Section 4.3.1), using the semiclassical convolution product
presented in Section 3. This gives an easy generalization of these spaces to
other orders, up to the equivalence of norms, and proves the ordering of Besov
spaces with respect to the third index (Corollary 4.1), the comparison of
Besov spaces of order $0$ with Schatten norms (Corollary 4.2) and Sobolev’s
inequalities for Besov spaces (Corollary 4.3). When $s\in(0,1)$, the norms
$\dot{\mathcal{W}}^{s,p}$ and $\dot{\mathcal{B}}^{s,p}$ are equivalent
independently of $\hbar$. From the above given definition, it follows directly
from the triangle inequality that
$\gamma_{s,p}^{1/p}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,p}}\leq
2\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}$. It is not
difficult to get an inequality in the other direction using the well-known
formula
$\mathsf{T}_{z}-1=\frac{1}{2}\left(\mathsf{T}_{2z}-1\right)-\frac{1}{2}\left(\mathsf{T}_{2z}+1-2\mathsf{T}_{z}\right)$.
This yields

(24)
$\left(2-2^{s}\right)\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}\leq\gamma_{s,p}^{1/p}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,p}}\leq
2\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}.$

When $s=1$, we can also apply the fundamental theorem of calculus to the
Wigner transform $f_{\boldsymbol{\rho}}$ of $\boldsymbol{\rho}$ defined by
Equation (2), and then take the Weyl quantization to get

(25)
$\mathsf{T}_{z}\boldsymbol{\rho}-\boldsymbol{\rho}=z\cdot\int_{0}^{1}\mathsf{T}_{\theta
z}\boldsymbol{\nabla}\boldsymbol{\rho}\mathop{}\\!\mathrm{d}\theta$

from which we deduce that

(26)
$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{1}_{p,\infty}}\leq
2\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{1,p}}.$

In the particular case when $p=2$, due to the fact that the Wigner transform
is an isomorphism from $\mathcal{L}^{2}$ to $L^{2}(\mathbb{R}^{2d})$, these
spaces are just the classical Besov spaces for the Wigner transform, i.e.

$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{2,r}}=\left\|{\frac{\left\|{f_{\boldsymbol{\rho}}(\cdot-2z)-2\,f_{\boldsymbol{\rho}}(\cdot-z)+f_{\boldsymbol{\rho}}}\right\|_{L^{2}}}{\left|{z}\right|^{s+2d/r}}}\right\|_{L^{r}_{z}}=:\left\|{f_{\boldsymbol{\rho}}}\right\|_{\dot{B}^{s}_{2,r}}.$

To get better constants in Sobolev inequalities when $s\to 1$, it is useful to
obtain an inequality sharper than Inequality (24), in which the constant
$\gamma_{s,p}$ converges to $0$.

######  Proposition 2.1.

Let $(s,p,q)\in(0,1)\times[2,\infty]^{2}$ and
$\boldsymbol{\rho}\in\dot{\mathcal{W}}^{s,p}$. Then
$\boldsymbol{\rho}\in\dot{\mathcal{B}}^{s}_{p,p}$ and

(27)
$\frac{(2^{p^{\prime}}-2^{sp^{\prime}})^{1/p^{\prime}}}{\gamma_{s,p}^{1/p}}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}\leq\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,p}}\leq\left(\frac{2^{p}-2^{sp}}{\gamma_{s,p}}\right)^{\frac{1}{p}}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}.$

When $p\in[1,2]$, the inequalities are reversed. In particular, when $p=2$,
then

(28)
$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{H}}^{s}}^{2}=\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,2}}^{2}=\frac{\gamma_{s,2}}{4-4^{s}}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{2,2}}^{2}.$

######  Remark 2.5.

These constants are sharper when $s\to 1$. The constant on the right-hand-side
of Inequality (27) converges to
$\frac{2^{p+1}\ln(2)\,\omega_{2d+p}}{\omega_{p+1}}$ when $s\to 1$. In
particular, when $p=2$, this yields

$\displaystyle\big{\lVert}(-\boldsymbol{\Delta})^{\frac{1}{2}}\boldsymbol{\rho}\big{\rVert}_{\mathcal{L}^{2}}^{2}=\left\|{\boldsymbol{\nabla}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{2}}^{2}=\frac{d}{\ln(4)\,\omega_{2d}}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{1}_{2,2}}^{2}.$

As in the classical case, Besov spaces are intermediate spaces between Sobolev
spaces. In particular, the inequality in the next proposition is reminiscent
of the possible definition of classical Besov spaces as real interpolation of
Sobolev spaces.

######  Proposition 2.2 (Comparison of Besov and Sobolev spaces).

For any $(s,p,r)\in(0,1)\times[1,\infty]^{2}$, there exists a constant $C>0$
such that for any $\boldsymbol{\rho}\in\mathcal{L}^{p}$.

$\tfrac{1}{2}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,r}}\leq
C_{s}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}^{1-s}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{1,p}}^{s}.$

The constant verifies $C_{s}\leq
2^{1-s}\left(\tfrac{\omega_{2d}}{s\left(1-s\right)r}\right)^{1/r}$.

By the same proof, taking $r=p$ and using the definition (12) of the
$\dot{\mathcal{W}}^{s,p}$ norm, it yields

$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}\leq
C_{s,p}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}^{1-s}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{1,p}}^{s}$

where using the fact that
$\left|{\omega_{-2s}}\right|=\tfrac{2s\left(1-s\right)}{\pi^{s}\Gamma(2-s)}$,
one obtains that
$C_{s,p}^{p}=\tfrac{2^{\left(1-s\right)p}\omega_{2d}}{s\left(1-s\right)p}\,\gamma_{s,p}\leq\frac{\omega_{2d}\,\omega_{p+1}^{s}}{s^{sp/2}\,\omega_{2d+sp}}$
is uniformly bounded with respect to $s\in[0,1]$.

###  2.5. Boundedness of Riesz transforms in the quantum phase space

As indicated in Remark 2.2, the Bessel potential spaces and the Sobolev spaces
coincide in the case of an integer order of regularity. This can equivalently
be stated in terms of the boundedness of the Riesz transform in Lebesgue
spaces, and more generally the boundedness of integral operators or Fourier
multipliers. In the non-commutative case, similar inequalities exist, see for
example [17, 28, 22] and the references therein. In particular, it follows
from [22, Theorem A] that the Schur multiplier
$(-\boldsymbol{\Delta}_{\xi})^{-1/2}\,\boldsymbol{\nabla}_{\xi}$ which acts on
an operator $\boldsymbol{\rho}$ through the formula

$\left((-\boldsymbol{\Delta}_{\xi})^{-1/2}\,\boldsymbol{\nabla}_{\xi}\boldsymbol{\rho}\right)(x,y)=-i\,\frac{x-y}{\left|{x-y}\right|}\,\boldsymbol{\rho}(x,y)$

is bounded on $\mathcal{L}^{p}$ for $1<p<\infty$. From this result, we deduce
in particular that the quantities
$\left\|{\boldsymbol{\nabla}_{\\!\xi}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}$
and
$\left\|{(-\boldsymbol{\Delta}_{\xi})^{1/2}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}$
are equivalent semi-norms. By conjugation by the Fourier transform, the same
result holds when exchanging $\xi$ by $x$ as in [20, Section 3.2], that is
$\left\|{\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}$
and
$\left\|{(-\boldsymbol{\Delta}_{x})^{1/2}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}$
are also equivalent. In particular, there exists constants $c$ and $C$
independent of $\hbar$ such that

$c\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{1,p}}\leq\left\|{(-\boldsymbol{\Delta}_{x})^{1/2}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}+\left\|{(-\boldsymbol{\Delta}_{\xi})^{1/2}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}\leq
C\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{1,p}}.$

When $p\neq 2$, it is however not clear that these expressions are equivalent
to
$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{H}}^{1,p}}=\left\|{(-\boldsymbol{\Delta})^{1/2}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}$
where $(-\boldsymbol{\Delta})^{s}$ was defined in Equation (15). Hence, it is
not clear that the quantum Bessel and Sobolev norms of order $1$ are
equivalent. We conjecture however that this should be true, and more precisely
that
$(-\boldsymbol{\Delta})^{-1/2}\,\boldsymbol{\nabla}:\mathcal{L}^{p}\to\mathbb{R}^{2d}$
is a bounded operator for any $p\in(1,\infty)$.

##  3\. Semiclassical convolution

One of the important tools in our analysis will be the use of a quantum
analogue of the convolution, actually already introduced in [49], that we
present in this section.

###  3.1. Phase space translations

Let us start by recalling some well-known properties of translation operators
in quantum mechanics. For any $z_{0}=(x_{0},\xi_{0})\in\mathbb{R}^{2d}$, we
define the associated phase space translation operator by
$\tau_{z_{0}}=e^{i\left(\xi_{0}\cdot
x-x_{0}\cdot\boldsymbol{p}\right)/\hbar}$, i.e.
$\tau_{z_{0}}\varphi=e^{i\,\xi_{0}\cdot\left(x-\frac{x_{0}}{2}\right)/\hbar}\,\varphi(x-x_{0})$.
The translation operators are unitary with $\tau_{z}^{-1}=\tau_{-z}$. They
verify a semigroup formula up to a phase factor

(29)
$\tau_{z+z^{\prime}}=e^{-i\pi\left(x\cdot\xi^{\prime}-x^{\prime}\cdot\xi\right)/h}\,\tau_{z}\tau_{z^{\prime}}.$

On the operator level, the phase space translation of a density operator
$\boldsymbol{\rho}$ is defined by Equation (13), which can be written
$\mathsf{T}_{z}\boldsymbol{\rho}=\tau_{z}\,\boldsymbol{\rho}\,\tau_{-z}$. The
semigroup relation (29) now becomes a true semigroup formula

$\mathsf{T}_{z+z^{\prime}}\boldsymbol{\rho}=\mathsf{T}_{z}\mathsf{T}_{z^{\prime}}\boldsymbol{\rho}.$

Notice that these operators translate the position and momentum operators
through the formulas $\mathsf{T}_{z_{0}}\,x=x-x_{0}$ and
$\mathsf{T}_{z_{0}}\,\boldsymbol{p}=\boldsymbol{p}-\xi_{0}$, or equivalently,
if $\boldsymbol{z}=(x,\boldsymbol{\rho})$,
$\mathsf{T}_{z_{0}}\,\boldsymbol{z}=\boldsymbol{z}-z_{0}$. More generally,
they translate any Weyl quantization, i.e.

(30) $\mathsf{T}_{z_{0}}\,\boldsymbol{\rho}_{f}=\boldsymbol{\rho}_{f(\cdot-
z_{0})}.$

###  3.2. Definition of the semiclassical convolution and inequalities

As was already introduced by Werner in [49], we define the semiclassical
convolution of a phase space function $f$ and a bounded operator
$\boldsymbol{\rho}$ as the following operator-valued integral555See Remark 4.1
for the more rigorous meaning of this integral.

(31) $f\star\boldsymbol{\rho}=\boldsymbol{\rho}\star
f:=\int_{\mathbb{R}^{2d}}f(z)\,\mathsf{T}_{z}\boldsymbol{\rho}\mathop{}\\!\mathrm{d}z.$

One of the important property of this definition for the semiclassical
convolution that was proved in [49] is the fact that the analogue of Young’s
convolution inequality holds. It also allows to better understand when the
semiclassical convolution is well defined.

######  Theorem 4 (Semiclassical Young’s convolution inequality [49]).

Let $(p,q,r)\in[1,\infty]^{3}$ be such that
$1+\frac{1}{p}=\frac{1}{q}+\frac{1}{r}$. Then there exists $C>0$ independent
of $\hbar$ such that for any $f\in L^{q}$

(32)
$\left\|{f\star\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}\leq\left\|{f}\right\|_{L^{q}}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{r}}.$

When $r=p$, one can also take $f$ to be a bounded measure, and replace the
$L^{q}$ norm by the total variation norm.

In our case, it will be useful to also have an analogue of the
Hardy–Littlewood–Sobolev’s inequality in the phase space, as given by the
following theorem.

######  Theorem 5 (Semiclassical Hardy–Littlewood–Sobolev’s inequality).

Let $(p,q,r)\in(1,\infty)^{3}$ be such that
$1+\frac{1}{p}=\frac{1}{q}+\frac{1}{r}$ and $r\leq p$. Then there exists $C>0$
independent of $\hbar$ such that for any $f\in L^{q,\infty}$

(33) $\left\|{f\star\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}\leq
C\left\|{f}\right\|_{L^{q,\infty}}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{r}}.$

######  Remark 3.1.

When moreover $(p,q,r)\in(1,\infty)^{3}$, then one more precisely has for any
$(a,b,c)\in[1,\infty]^{3}$ verifying $\frac{1}{c}=\frac{1}{a}+\frac{1}{b}$

$\left\|{f\star\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p,c}}\leq
C\left\|{f}\right\|_{L^{q,a}}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{r,b}}$

where the spaces $L^{p,q}=L^{p,q}(\mathbb{R}^{2d})$ are the Lorentz spaces,
which are the spaces obtained by real interpolation of Lebesgue spaces, and
the $\mathcal{L}^{p,q}$ spaces are their analogue for operators introduced in
[9]. In particular, taking $a=\frac{p+1}{p}\,q$, $b=\frac{p+1}{p}\,r$ and
using the inclusions between $\mathcal{L}^{p,a}$ spaces, one gets
$\left\|{f\star\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p,1}}\leq
C\left\|{f}\right\|_{L^{q}}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{r}}$,
which is slightly stronger than Equation (32).

######  Remark 3.2.

As in the classical case, by using the dual definition of the
$\mathcal{L}^{p}$ norm, Inequality (33) can be written
$h^{d}\operatorname{Tr}\\!\left(\boldsymbol{\mu}\left(f\star\boldsymbol{\rho}\right)\right)\leq
C\left\|{f}\right\|_{L^{r,\infty}}\left\|{\boldsymbol{\mu}}\right\|_{\mathcal{L}^{p}}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}$
for any $(p,q,r)\in(1,\infty)^{3}$ such that
$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=2$. Taking $f(z)=\left|{z}\right|^{-a}$
leads by definition of the semiclassical convolution to

$h^{d}\operatorname{Tr}\int_{\mathbb{R}^{2d}}\frac{\boldsymbol{\mu}\,\mathsf{T}_{z}\boldsymbol{\rho}}{\left|{z}\right|^{a}}\mathop{}\\!\mathrm{d}z\leq
C\left\|{\boldsymbol{\mu}}\right\|_{\mathcal{L}^{p}}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}},$

whenever $\frac{1}{p}+\frac{1}{q}+\frac{a}{2d}=2$. Because of the use of
interpolation techniques, it is however not clear what the value of the
constant $C$ is.

From Formula (30) and the linearity of the Weyl quantization, we deduce that
$f\star\boldsymbol{\rho}_{g}=\int_{\mathbb{R}^{2d}}f(z)\,\boldsymbol{\rho}_{g(\cdot-z)}\mathop{}\\!\mathrm{d}z=\boldsymbol{\rho}_{f*g}$.
In particular, the Wigner transform of the semiclassical convolution is given
by the classical convolution through the formula

$f_{g\star\boldsymbol{\rho}}=g*f_{\boldsymbol{\rho}}$

and the property of commutation of the classical convolution product yields

(34)
$f_{1}\star\boldsymbol{\rho}_{f_{2}}=\boldsymbol{\rho}_{f_{1}*f_{2}}=\boldsymbol{\rho}_{f_{1}}\star
f_{2}.$

The semiclassical convolution has properties close to the classical
convolution. It preserves the positivity: if $\boldsymbol{\rho}\geq 0$ and
$f\geq 0$, then $f\star\boldsymbol{\rho}\geq 0$. Taking $f=1$ yields
$1\star\boldsymbol{\rho}=\boldsymbol{\rho}_{1*f_{\boldsymbol{\rho}}}=h^{d}\operatorname{Tr}\\!\left(\boldsymbol{\rho}\right)\mathbf{1}$
where $\mathbf{1}$ is the identity on $L^{2}$. Taking $f=\delta_{0}$ yields
also as expected

(35)
$\delta_{0}\star\boldsymbol{\rho}=\mathsf{T}_{0}\boldsymbol{\rho}=\boldsymbol{\rho}.$

From the first identity in Equation (34) and the compatibility of the quantum
gradients with the Weyl quantization (5), we also deduce that whenever $\nabla
f\in L^{1}+L^{\infty}$ and $\nabla\boldsymbol{\rho}$ is trace class

$\boldsymbol{\nabla}\\!\left(f\star\boldsymbol{\rho}\right)=\nabla
f\star\boldsymbol{\rho}=f\star\boldsymbol{\nabla}\boldsymbol{\rho}.$

###  3.3. Convolution by a Gaussian: Husimi transform and coherent states

We define the semiclassical Gaussian by

$\displaystyle g_{h}(z)=\left(2/h\right)^{d}e^{-\left|{z}\right|^{2}/\hbar}.$

When $\hbar\to 0$, the convolution by this function is an approximation of the
identity. For a function $f=f(z)$, we will denote by

$\tilde{f}(z):=g_{h}*f(z).$

This converges to $f$ when $\hbar\to 0$. In the particular case when $f$ is a
Wigner transform of an operator $\boldsymbol{\rho}$, we obtain the so called
Husimi transform (see e.g. [34, Eq. (25)])

$\displaystyle\tilde{f}_{\boldsymbol{\rho}}=g_{h}*f_{\boldsymbol{\rho}}$

which is a nonnegative function whenever $\boldsymbol{\rho}$ is a positive
operator.

To get the quantum analogue of the convolution by a Gaussian, we use the above
defined semiclassical convolution. We define

$\,\tilde{\\!\boldsymbol{\rho}}:=g_{h}\star\boldsymbol{\rho}.$

Similarly as above in the particular case when $\boldsymbol{\rho}$ is a Weyl
quantization, we get

$\,\tilde{\\!\boldsymbol{\rho}}_{f}:=g_{h}\star\boldsymbol{\rho}_{f}=\boldsymbol{\rho}_{\widetilde{f}}=f\star\boldsymbol{\rho}_{g_{h}}.$

We also deduce that the Husimi transform of $\boldsymbol{\rho}$ is nothing but
the Wigner transform of $\,\tilde{\\!\boldsymbol{\rho}}$, i.e.
$\tilde{f}_{\boldsymbol{\rho}}=f_{\,\tilde{\\!\boldsymbol{\rho}}}$. It is
well-known (see e.g. [34]) and not difficult to prove that the Weyl
quantization of $g_{h}$ is a projection of the form
$\boldsymbol{\rho}_{g_{h}}=h^{-d}\ket{\psi_{h}}\bra{\psi_{h}}$ on the Gaussian
coherent state

$\psi_{h}(x)=\left(\tfrac{2}{h}\right)^{d/4}e^{-\left|{x}\right|^{2}/(2\hbar)}.$

Hence,

$\,\tilde{\\!\boldsymbol{\rho}}_{f}=h^{-d}\int_{\mathbb{R}^{2d}}f(z)\ket{\tau_{z}\psi_{h}}\bra{\tau_{z}\psi_{h}}\mathop{}\\!\mathrm{d}z$

is nothing but what is sometimes called a superposition of coherent sates, a
Töplitz operator or the (Anti-)Wick quantization of $f$. Let us mention that
links between Töplitz operators and convolutions were studied in [24]. As it
is a special case of a semiclassical convolution and since
$\boldsymbol{\rho}_{g_{h}}\geq 0$, we recover the fact that it maps positive
functions to positive operators. Moreover, as a corollary of the semiclassical
Young’s inequality and the fact that
$\left\|{\boldsymbol{\rho}_{g_{h}}}\right\|_{\mathcal{L}^{1}}=1$, we get the
well-known bound

(36)
$\left\|{\,\tilde{\\!\boldsymbol{\rho}}_{f}}\right\|_{\mathcal{L}^{p}}\leq\left\|{f}\right\|_{L^{p}(\mathbb{R}^{2d})}.$

######  Remark 3.3.

One can more generally prove (see e.g. [34, Eq. (61), (59)]) that for any
convex function $\Phi$ such that $\Phi(0)=0$,
$h^{d}\operatorname{Tr}\\!\left(\Phi\\!\left(\,\tilde{\\!\boldsymbol{\rho}}_{f}\right)\right)\leq\int_{\mathbb{R}^{2d}}\Phi(f)$.
An analogous identity holds for the Husimi transform (see e.g. [34, Eq.
(64)]), that is $\int_{\mathbb{R}^{2d}}\Phi(\tilde{f}_{\boldsymbol{\rho}})\leq
h^{d}\operatorname{Tr}\\!\left(\Phi\\!\left(\boldsymbol{\rho}\right)\right)$.
In particular,

(37)
$\left\|{\tilde{f}_{\boldsymbol{\rho}}}\right\|_{L^{p}(\mathbb{R}^{2d})}\leq\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}.$

##  4\. Proofs

###  4.1. Semiclassical convolution

To prove the semiclassical analogue of the Hardy–Littlewood–Sobolev
inequality, we will use the following bilinear interpolation result (see e.g.
[46, Lemma 28.2] and [15, 10.1]).

######  Lemma 4.1.

Let $B$ be a bilinear operator mapping continuously $E_{0}\times F_{0}\to
G_{0}$, $E_{0}\times F_{1}\to G_{1}$, and $E_{1}\times F_{0}\to G_{1}$. Then
for every $\left(\theta,\theta_{2}\right)\in(0,1)^{2}$ such that
$\theta+\theta_{2}<1$ and any $(a,b,c)\in[1,\infty]^{3}$ verifying
$\frac{1}{c}=\frac{1}{a}+\frac{1}{b}$, it holds

$\displaystyle\left\|{B(u,v)}\right\|_{G_{\theta+\theta_{2},c}}$
$\displaystyle\leq
C\left\|{u}\right\|_{E_{\theta,a}}\left\|{v}\right\|_{F_{\theta_{2},b}}$
$\displaystyle\left\|{B(u,v)}\right\|_{G_{\theta}}$ $\displaystyle\leq
M_{0}^{1-\theta}M_{1}^{\theta}\,\left\|{u}\right\|_{E_{\theta}}\left\|{v}\right\|_{F_{\theta}}$

where $X_{\theta,a}=(X_{0},X_{1})_{\theta,a}$,
$X_{\theta}=[X_{0},X_{1}]_{\theta}$ are respectively the real and complex
interpolation spaces between $X_{0}$ and $X_{1}$, and $M_{k}$ denotes the norm
of $B$ as a bilinear operator from $E_{k}\times F_{k}$ to $G_{k}$.

The proof is then similar to the proof of the semiclassical Young’s inequality
in [49] and consists mainly in obtaining the estimate at the three endpoints.

###### Proof of theorems 4 and 5.

Notice first that if $f\in L^{1}$ and $\boldsymbol{\rho}\in\mathcal{L}^{p}$
for any $p\in[1,\infty]$, then

(38)
$\left\|{f\star\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}\leq\int_{\mathbb{R}^{2d}}\left|{f(z)}\right|\left\|{\mathsf{T}_{z}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}\mathop{}\\!\mathrm{d}z=\left\|{f}\right\|_{L^{1}}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}.$

On another side, if $f$ is bounded and $\boldsymbol{\rho}\in\mathcal{L}^{1}$,
then for any $(\varphi,\phi)\in(L^{2}(\mathbb{R}^{d}))^{2}$, by definition it
holds

(39)
$\Braket{\varphi}{\left(f\star\boldsymbol{\rho}\right)\phi}=\int_{\mathbb{R}^{2d}}f(z)\Braket{\varphi}{\left(\mathsf{T}_{z}\boldsymbol{\rho}\right)\phi}\mathop{}\\!\mathrm{d}z.$

Since $\boldsymbol{\rho}$ is a compact operator, we may write it using its
singular values decomposition (see e.g. [40, 42]) under the form

$\boldsymbol{\rho}=\sum_{j\in J}\mu_{j}\ket{\psi_{j}}\bra{\tilde{\psi}_{j}}$

for some orthonormal families $(\psi_{j})_{j\in J}$ and
$(\tilde{\psi}_{j})_{j\in J}$ of functions in $L^{2}(\mathbb{R}^{d})$, and
where $(\mu_{j})_{j\in J}$ are positive numbers called the singular values of
$\boldsymbol{\rho}$. Since $\boldsymbol{\rho}$ is trace class, the singular
values form an integrable sequence. Thus, applying the Fubini theorem and then
the Cauchy–Schwarz inequality to Equation (39) leads to

$\displaystyle\Braket{\varphi}{\left(f\star\boldsymbol{\rho}\right)\phi}$
$\displaystyle\leq\sum_{j\in
J}\mu_{j}\left\|{f}\right\|_{L^{\infty}}\int_{\mathbb{R}^{2d}}\left|{\Braket{\varphi}{\tau_{z}\psi_{j}}\Braket{\tau_{z}\tilde{\psi}_{j}}{\phi}}\right|\mathop{}\\!\mathrm{d}z$
$\displaystyle\leq\operatorname{Tr}\\!\left(\left|{\boldsymbol{\rho}}\right|\right)\left\|{f}\right\|_{L^{\infty}}\sup_{j}\left\|{\Braket{\varphi}{\tau_{z}\psi_{j}}}\right\|_{L^{2}_{z}}\left\|{\Braket{\tau_{z}\tilde{\psi}_{j}}{\phi}}\right\|_{L^{2}_{z}}.$

Observe that by the definition of the shift operators $\tau_{z}$, the inner
products appearing on the right-hand side of the above inequality can be
written as a Fourier transform under the form

$\Braket{\varphi}{\tau_{z}\psi_{j}}=e^{-i\xi\cdot
x/(2\hbar)}\operatorname{\mathcal{F}}\\!\left(\varphi(\cdot)\,\psi_{j}(\cdot-x)\right)(\tfrac{\xi}{h}).$

Therefore, it follows from the Plancherel theorem and the normalization of
$\psi_{j}$ that

$\left\|{\Braket{\varphi}{\tau_{z}\psi_{j}}}\right\|_{L^{2}_{z}}=h^{d/2}\left\|{\varphi(\xi)\,\psi_{j}(\xi-x)}\right\|_{L^{2}_{x,\xi}}=h^{d/2}\left\|{\varphi}\right\|_{L^{2}}.$

The same inequality holds when replacing $\psi_{j}$ by $\tilde{\psi}_{j}$, and
$\varphi$ by $\phi$. This leads to
$\Braket{\varphi}{\left(f\star\boldsymbol{\rho}\right)\phi}\leq
h^{d}\operatorname{Tr}\\!\left(\left|{\boldsymbol{\rho}}\right|\right)\left\|{f}\right\|_{L^{\infty}}\left\|{\varphi}\right\|_{L^{2}}\left\|{\phi}\right\|_{L^{2}}$.
Hence, $f\star\boldsymbol{\rho}$ is a bounded operator with norm verifying

$\displaystyle\left\|{f\star\boldsymbol{\rho}}\right\|_{\mathcal{L}^{\infty}}\leq\left\|{f}\right\|_{L^{\infty}}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{1}}.$

Theorem 4 now follows by combining the above equation with Inequality (38) and
bilinear complex interpolation as in Lemma 4.1. Theorem 5 follows from the
bilinear real interpolation result of the same lemma with $1-\theta=1/q$ and
$1-\theta_{2}=1/r$. ∎

######  Remark 4.1.

Let us give a comment about the signification of the definition of the
semiclassical convolution. In general, if $f\in
L^{1}(\mathbb{R}^{2d})+L^{\infty}(\mathbb{R}^{2d})$ and $\boldsymbol{\rho}$ is
a bounded operator, then the integral appearing of Formula (31) as to be
considered as a weak (Pettis) integral. Interpreting a bounded operator
$\boldsymbol{\rho}$ acting on $L^{2}$ as a linear form on
$L^{2}(\mathbb{R}^{d})\otimes L^{2}(\mathbb{R}^{d})$ through the formula
$\left\langle\boldsymbol{\rho},\varphi\otimes\phi\right\rangle:=\Braket{\overline{\varphi}}{\boldsymbol{\rho}\phi}$,
then a correct interpretation of Definition (31) is

$\left\langle
f\star\boldsymbol{\rho},\varphi\otimes\phi\right\rangle=\int_{\mathbb{R}^{2d}}\left\langle
f(z)\,\mathsf{T}_{z}\boldsymbol{\rho},\varphi\otimes\phi\right\rangle\mathop{}\\!\mathrm{d}z.$

This is indeed a well defined Lebesgue integral as follows from the above
proof whenever $f\in L^{\infty}$. In the case when $f\in L^{1}$, then by
Inequality (38), the integral in Definition (31) can even better be defined as
a strong (Bochner) integral, and so as a weak integral.

###  4.2. Proofs of Sobolev inequalities

We can now prove our main theorems. We start with the proof of the inequality
for Bessel type spaces without giving an explicit constant. It is the simplest
proof as it is a consequence of the Hardy–Littlewood–Sobolev inequality (33).
The method of proof is actually reminiscent of the original strategy of
Sobolev [43]. It however does not include the case $p=1$.

######  Proposition 4.1.

Let $s>0$ and $1<p\leq q<\infty$ verifying (8). Then there exist a constant
$C$ independent of $\hbar$ such that for any compact operator
$\boldsymbol{\rho}$,

(40) $\displaystyle\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}$
$\displaystyle\leq
C\left\|{(-\boldsymbol{\Delta})^{s/2}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}$
$\displaystyle\text{ if }s\in(0,2d)$ (41)
$\displaystyle\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}$
$\displaystyle\leq
C\left\|{(-\boldsymbol{\Delta})^{\frac{s-1}{2}}\boldsymbol{\nabla}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}$
$\displaystyle\text{ if }s\in(0,2d-1).$

######  Remark 4.2.

As a particular case of (41), when $s=1$ and $1<p\leq q<\infty$ verify
$\frac{1}{2d}=\frac{1}{p}-\frac{1}{q}$, then we already obtain the Sobolev
inequality (21).

###### Proof.

For $s\in\mathbb{R}_{+}\setminus\set{2d}$, define the function acting on
$z\in\mathbb{R}^{2d}$ by

$K_{s}(z)=\frac{c_{2d,-s}}{\left|{z}\right|^{2d-s}}$

with $c_{2d,-s}=\tfrac{\omega_{s}}{\left(2\pi\right)^{s}\omega_{2d-s}}$. In
the case when $s=2d$, define instead $K_{2d}(z)=-\omega_{2d}\ln(\left|{\pi
x}\right|)$. When $s\in(0,2d)$, it is well-known that this function is a
solution in the sense of distributions of

$(-\Delta_{z})^{s/2}K_{s}=\delta_{0}$

as can be checked for example taking the Fourier transform. Since $K_{s}\in
L^{\frac{2d}{2d-s},\infty}$ when $s\in(0,2d)$ and
$1+\frac{1}{q}=\frac{2d-s}{2d}+\frac{1}{p}$ with $p\leq q$, it follows from
Formula (35) and the quantum Hardy–Littlewood–Sobolev inequality (33) that

$\displaystyle\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}$
$\displaystyle=\left\|{(-\Delta_{z})^{s/2}K_{s}\star\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}=\left\|{K_{s}\star(-\boldsymbol{\Delta})^{s/2}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}\leq
C\left\|{(-\boldsymbol{\Delta})^{s/2}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}$

which proves Inequality (40). In the same spirit, when $s\in(0,2d-1)$, the
vector valued function $\nabla K_{s+1}\in L^{\frac{2d}{2d-s},\infty}$ verifies
$\left(-\Delta_{z}\right)^{\frac{s-1}{2}}\nabla_{z}\cdot\left(\nabla_{z}K_{s+1}\right)=-\delta_{0}$,
and one obtains Inequality (41). ∎

We now prove the Sobolev inequalities for the fractional Sobolev spaces
$\dot{\mathcal{W}}^{s,p}$ as defined in Equation (12). This proves Theorem 1,
except for the special bounds on the constant
$\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}$ in the case when $p=2$ or
$q=2$, which will be proved in lemmas 4.2 and 4.3, and the lower bound that
will be proved in Lemma 4.4.

######  Proposition 4.2.

Let $s\in[0,1]$ and $1\leq p\leq q<\infty$ verifying (8). Then

$\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}\leq\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}$

where $\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}$ verifies Inequality (20)
when $s\in(0,1)$ and Inequality (18) when $s=1$.

###### Proof.

It follows from the triangle inequality for Schatten norms that

(42)
$\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}\leq\left\|{\,\tilde{\tilde{\\!\boldsymbol{\rho}}}}\right\|_{\mathcal{L}^{q}}+\left\|{\,\tilde{\tilde{\\!\boldsymbol{\rho}}}-\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}$

where
$\,\tilde{\tilde{\\!\boldsymbol{\rho}}}=g_{h}\star(g_{h}\star\boldsymbol{\rho})=(g_{h}*g_{h})\star\boldsymbol{\rho}=g_{2h}\star\boldsymbol{\rho}$.
To bound the first term, notice that
$\,\tilde{\tilde{\\!\boldsymbol{\rho}}}=\,\tilde{\\!\boldsymbol{\rho}}_{\tilde{f}_{\boldsymbol{\rho}}}$
where $\tilde{f}_{\boldsymbol{\rho}}$ is a smooth function converging to $0$
at infinity. The smoothness comes from the convolution by the Gaussian. The
decay at infinity follows from the fact that compact operators are limit in
$\mathcal{L}^{\infty}$ of finite rank operators, and so their Husimi transform
is the uniform limit of integrable functions. Therefore, by the classical
Sobolev inequality and by the property (36) of the Wick quantization

$\left\|{\,\tilde{\tilde{\\!\boldsymbol{\rho}}}}\right\|_{\mathcal{L}^{q}}\leq\left\|{\tilde{f}_{\boldsymbol{\rho}}}\right\|_{L^{q}}\leq\mathcal{C}^{\mathrm{S}}_{s,p}\left\|{\tilde{f}_{\boldsymbol{\rho}}}\right\|_{\dot{W}^{s,p}}.$

Then, if $s=1$, by the property (37) of the Husimi transform,

(43)
$\left\|{\tilde{f}_{\boldsymbol{\rho}}}\right\|_{\dot{W}^{1,p}}=\left\|{\tilde{f}_{\boldsymbol{\nabla}\boldsymbol{\rho}}}\right\|_{L^{p}}\leq\left\|{\boldsymbol{\nabla}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}.$

Similarly, if $s\in(0,1)$, by the definition of the fractional Sobolev spaces
and property (37),

(44)
$\left\|{\tilde{f}_{\boldsymbol{\rho}}}\right\|_{\dot{W}^{s,p}}^{p}=\gamma_{s,p}\left\|{\frac{\left\|{\tilde{f}_{\mathsf{T}_{z}\boldsymbol{\rho}-\boldsymbol{\rho}}}\right\|_{L^{p}}}{\left|{z}\right|^{s+2d/p}}}\right\|_{L^{p}}^{p}\leq\gamma_{s,p}\left\|{\frac{\left\|{\mathsf{T}_{z}\boldsymbol{\rho}-\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}}{\left|{z}\right|^{s+2d/p}}}\right\|_{L^{p}}^{p}=\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}^{p}.$

We now obtain a bound on the second term in the right-hand-side of Inequality
(42). Since $\int_{\mathbb{R}^{2d}}g_{2h}=1$ and $g_{2h}$ is even, it holds

$\,\tilde{\tilde{\\!\boldsymbol{\rho}}}-\boldsymbol{\rho}=\frac{1}{2}\int_{\mathbb{R}^{2d}}g_{2h}(z)\left(\mathsf{T}_{z}\boldsymbol{\rho}+\mathsf{T}_{-z}\boldsymbol{\rho}-2\,\boldsymbol{\rho}\right)\mathop{}\\!\mathrm{d}z.$

Therefore, taking the $\mathcal{L}^{q}$ norm leads to

$\left\|{\,\tilde{\tilde{\\!\boldsymbol{\rho}}}-\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}\leq\frac{1}{2}\int_{\mathbb{R}^{2d}}g_{2h}(z)\left|{z}\right|^{s+2d/r}\frac{\left\|{\mathsf{T}_{z}\boldsymbol{\rho}+\mathsf{T}_{-z}\boldsymbol{\rho}-2\,\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}}{\left|{z}\right|^{s+2d/r}}\mathop{}\\!\mathrm{d}z.$

By Hölder’s inequality, this leads to

(45)
$\left\|{\,\tilde{\tilde{\\!\boldsymbol{\rho}}}-\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}\leq\tfrac{1}{2}\,C_{d,s,r}\,h^{s/2}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{q,r}}$

where

$C_{d,s,r}=\frac{1}{h^{s/2}}\left\|{g_{2h}(z)\left|{z}\right|^{s+\frac{2d}{r}}}\right\|_{L^{r^{\prime}}_{z}}=\frac{1}{\left(r^{\prime}\right)^{d+\frac{s}{2}}}\left(\frac{\omega_{2d}}{\omega_{(2d+s)r^{\prime}}}\right)^{\frac{1}{r^{\prime}}}.$

Observe now that for any $p\leq q$, by the scaled definition of the Schatten
norms (3), it holds
$h^{s/2}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}=h^{d\left(\frac{1}{q}-\frac{1}{p}\right)+\frac{s}{2}}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}$.
Therefore, since $\frac{1}{p}-\frac{1}{q}=\frac{s}{2d}$, it leads to

$\left\|{\,\tilde{\tilde{\\!\boldsymbol{\rho}}}-\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}\leq\tfrac{1}{2}\,C_{d,s,r}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,r}}.$

When $s=1$, then we use Inequality (26) to deduce that

(46)
$\left\|{\,\tilde{\tilde{\\!\boldsymbol{\rho}}}-\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}\leq
C_{d,s,\infty}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}$

while if $s\in(0,1)$, then we use Proposition 2.1 to get

$\left\|{\,\tilde{\tilde{\\!\boldsymbol{\rho}}}-\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}^{p}\leq\tfrac{\left(1-2^{\left(s-1\right)r}\right)^{p/r}}{\gamma_{s,p}}\,C_{d,s,p}^{p}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}^{p}$

where $r=\max(p,p^{\prime})$. In any of these cases, combining the estimates
on $\,\tilde{\tilde{\\!\boldsymbol{\rho}}}$ and
$\,\tilde{\tilde{\\!\boldsymbol{\rho}}}-\boldsymbol{\rho}$ finishes the proof.
∎

We now get explicit upper bounds for the Bessel–Sobolev inequality in the case
when $p\leq 2\leq q$, i.e. when $q\in[2,\frac{2\,d}{d-1}]$, using the fact
that the Wigner transform is an isometry from $\mathcal{L}^{2}$ to
$L^{2}(\mathbb{R}^{2d})$.

######  Proposition 4.3.

With the notations of theorems 1 and 2, assume $1<p\leq 2\leq q<\infty$ verify
Equation (8), and let $\tilde{s}:=2d\,(\frac{1}{2}-\frac{1}{q})$. Then

$\widetilde{\mathcal{C}}^{\mathrm{\,B}}_{s,p}\leq\widetilde{\mathcal{C}}^{\mathrm{\,B}}_{\tilde{s},2}\,\widetilde{\mathcal{C}}^{\mathrm{\,B}}_{s-\tilde{s},p}\leq\left(\mathcal{C}^{\mathrm{B}}_{\tilde{s},2}+\tfrac{\theta_{\tilde{s}}}{\left(8\pi\right)^{\tilde{s}/2}}\right)\Big{(}\mathcal{C}^{\mathrm{B}}_{s-\tilde{s},p}+\tfrac{\theta_{s-\tilde{s}}}{\left(8\pi\right)^{(s-\tilde{s})/2}}\Big{)}$

and similarly, if $s=1$, then the same inequality holds with
$\widetilde{\mathcal{C}}^{\mathrm{\,B}}$ replaced by
$\widetilde{\mathcal{C}}^{\mathrm{\,S}}$ when $1<p\leq 2\leq q<\infty$.

######  Remark 4.3.

Recall that $\theta_{s}=\sup_{r\geq 0}\frac{1-e^{-r}}{r^{s}}$. In the case
when $s<1$, then $\theta_{s}<1$ and it is reached for some $r>0$ verifying
$e^{r}=1+r/s$.

We start with the special cases $p=2$ and $q=2$. The proposition then follows
by taking the case $\tilde{p}=2$ as an intermediate case, and writing

$\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}\leq\widetilde{\mathcal{C}}^{\mathrm{\,B}}_{\tilde{s},2}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{H}}^{s}}=\widetilde{\mathcal{C}}^{\mathrm{\,B}}_{\tilde{s},2}\left\|{(-\boldsymbol{\Delta})^{\tilde{s}/2}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{2}}\leq\widetilde{\mathcal{C}}^{\mathrm{\,B}}_{\tilde{s},2}\,\widetilde{\mathcal{C}}^{\mathrm{\,B}}_{s-\tilde{s},p}\left\|{(-\boldsymbol{\Delta})^{s/2}\boldsymbol{\rho}}\right\|_{\dot{\mathcal{L}}^{p}}.$

######  Lemma 4.2 (First basis case).

Let $s\in[0,1]$ and $q:=\frac{2\,d}{d-s}$. Then for any
$f\in\dot{H}^{s}(\mathbb{R}^{2d})$,

(47)
$\left\|{\boldsymbol{\rho}_{f}}\right\|_{\mathcal{L}^{q}}\leq\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,2}\left\|{f}\right\|_{\dot{H}^{s}(\mathbb{R}^{2d})}=\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,2}\left\|{\boldsymbol{\rho}_{f}}\right\|_{\dot{\mathcal{H}}^{s}}$

where
$\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,2}=\widetilde{\mathcal{C}}^{\mathrm{\,B}}_{s,2}\leq\mathcal{C}^{\mathrm{B}}_{s,2}+\theta_{s}\left(8\pi\right)^{-\frac{s}{2}}$
with $\mathcal{C}^{\mathrm{S}}_{s,2}=\mathcal{C}^{\mathrm{B}}_{s,2}$ the
optimal constant appearing in the associated classical Sobolev inequality
(10).

###### Proof.

It follows from the definition of $\mathcal{L}^{2}$ given in Equation (3), the
Plancherel theorem and the fact that the Fourier transform of $g_{h}$ is
$\widehat{g_{h}}(w)=e^{-\pi h\left|{w}\right|^{2}/2}$ that

$\left\|{\boldsymbol{\rho}_{f}-\,\tilde{\\!\boldsymbol{\rho}}_{f}}\right\|_{\mathcal{L}^{2}}=\left\|{f-g_{h}*f}\right\|_{L^{2}(\mathbb{R}^{2d})}=\left\|{u(\tfrac{\pi
h\left|{w}\right|^{2}}{2})\left(\tfrac{\pi
h\left|{w}\right|^{2}}{2}\right)^{\frac{s}{2}}\widehat{f}}\right\|_{L^{2}(\mathbb{R}^{2d})},$

where $u:\mathbb{R}_{+}\to\mathbb{R}_{+}$ be defined by
$u(r)=\frac{1-e^{-r}}{r^{s/2}}$. Since $u$ is a bounded function on
$\mathbb{R}_{+}$, it yields

(48)
$\left\|{\boldsymbol{\rho}_{f}-\,\tilde{\\!\boldsymbol{\rho}}_{f}}\right\|_{\mathcal{L}^{2}}\leq\theta_{s}\left(\tfrac{h}{8\pi}\right)^{s/2}\left\|{f}\right\|_{\dot{H}^{s}(\mathbb{R}^{2d})}.$

From the relation $\frac{1}{q}=\frac{1}{2}-\frac{s}{2d}$, the scaling of the
$\mathcal{L}^{p}$ norms and the inequalities between Schatten norms, the above
inequality leads to

$\displaystyle\left\|{\boldsymbol{\rho}_{f}-\,\tilde{\\!\boldsymbol{\rho}}_{f}}\right\|_{\mathcal{L}^{q}}=h^{\frac{d}{q}}\left\|{\boldsymbol{\rho}_{f}-\,\tilde{\\!\boldsymbol{\rho}}_{f}}\right\|_{q}\leq
h^{-\frac{s}{2}}\left\|{\boldsymbol{\rho}_{f}-\,\tilde{\\!\boldsymbol{\rho}}_{f}}\right\|_{\mathcal{L}^{2}}\leq\tfrac{\theta_{s}}{\left(8\pi\right)^{s/2}}\left\|{f}\right\|_{\dot{H}^{s}(\mathbb{R}^{2d})}.$

By the Triangle inequality for the Schatten norms and the bound (36) for the
Wick quantization, it leads to the following inequality

$\left\|{\boldsymbol{\rho}_{f}}\right\|_{\mathcal{L}^{q}}\leq\left\|{\,\tilde{\\!\boldsymbol{\rho}}_{f}}\right\|_{\mathcal{L}^{q}}+\tfrac{\theta_{s}}{\left(8\pi\right)^{s/2}}\left\|{f}\right\|_{\dot{H}^{s}(\mathbb{R}^{2d})}\leq\left\|{f}\right\|_{L^{q}(\mathbb{R}^{2d})}+\tfrac{\theta_{s}}{\left(8\pi\right)^{s/2}}\left\|{f}\right\|_{\dot{H}^{s}(\mathbb{R}^{2d})}$

and we deduce Inequality (47) by the classical Sobolev inequality. ∎

######  Lemma 4.3 (Second basis case).

Let $s\in[0,1]$ and $p>1$ be such that $\frac{1}{p}=\frac{1}{2}+\frac{1}{2d}$.
Then
$\widetilde{\mathcal{C}}^{\mathrm{\,B}}_{s,p}\leq\mathcal{C}^{\mathrm{B}}_{s,p}+\tfrac{\theta_{s}}{\left(8\pi\right)^{s/2}}$,
and if $s=1$ and $p\geq 1$, then
$\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}\leq\mathcal{C}^{\mathrm{S}}_{s,p}+\tfrac{\theta_{s}}{\left(8\pi\right)^{s/2}}$.

###### Proof.

Once again we use Inequality (48), which can be written in terms of the Wigner
transform $f_{\boldsymbol{\rho}}$ of $\boldsymbol{\rho}$ in the form

$\left\|{f_{\boldsymbol{\rho}}-\tilde{f}_{\boldsymbol{\rho}}}\right\|_{L^{2}(\mathbb{R}^{2d})}\leq\theta_{s}\left(\tfrac{h}{8\pi}\right)^{s/2}\left\|{(-\Delta)^{\frac{s}{2}}f_{\boldsymbol{\rho}}}\right\|_{L^{2}(\mathbb{R}^{2d})}=\theta_{s}\left(\tfrac{h}{8\pi}\right)^{s/2}\left\|{(-\boldsymbol{\Delta})^{\frac{s}{2}}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{2}},$

and so by the fact that
$\left\|{\boldsymbol{\mu}}\right\|_{2}\leq\left\|{\boldsymbol{\mu}}\right\|_{p}$
since $p\leq 2$, the definition of $p$, and the definition of
$\mathcal{L}^{p}$ norms, we deduce

$\left\|{f_{\boldsymbol{\rho}}-\tilde{f}_{\boldsymbol{\rho}}}\right\|_{L^{2}(\mathbb{R}^{2d})}\leq\tfrac{\theta_{s}}{\left(8\pi\right)^{s/2}}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{H}}^{s,p}}.$

Then by the classical Bessel–Sobolev inequalities (10) and Inequality (37), we
get

$\left\|{\tilde{f}_{\boldsymbol{\rho}}}\right\|_{L^{2}(\mathbb{R}^{2d})}\leq\mathcal{C}^{\mathrm{B}}_{s,p}\left\|{(-\Delta)^{\frac{s}{2}}\tilde{f}_{\boldsymbol{\rho}}}\right\|_{L^{p}(\mathbb{R}^{2d})}=\mathcal{C}^{\mathrm{B}}_{s,p}\left\|{\tilde{f}_{(-\boldsymbol{\Delta})^{\frac{s}{2}}\boldsymbol{\rho}}}\right\|_{L^{p}(\mathbb{R}^{2d})}\leq\mathcal{C}^{\mathrm{B}}_{s,p}\left\|{(-\boldsymbol{\Delta})^{\frac{s}{2}}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}.$

As in the proof of Lemma 4.2, the triangle inequality and the two previous
inequalities lead to a bound on
$\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{2}}=\left\|{f_{\boldsymbol{\rho}}}\right\|_{L^{2}(\mathbb{R}^{2d})}$.
When $s=1$, the bound on $\widetilde{\mathcal{C}}^{\mathrm{\,S}}$ follows
similarly replacing $(-\boldsymbol{\Delta})^{\frac{s}{2}}$ by the quantum
gradient $\boldsymbol{\nabla}^{s}$. ∎

####  4.2.1. Bound by below for the quantum Sobolev constants

In this section, we finish the proof of Theorem 1 by proving a lower bound for
the constant $\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}$.

######  Lemma 4.4.

Under the hypotheses of Theorem 1, assuming $p=2$ or $s=1$, then the optimal
uniform-in-$\hbar$ constant appearing in inequality (17) verifies

$\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}\geq\mathcal{C}^{\mathrm{S}}_{s,p}.$

###### Proof.

When $p=2$ or $s=1$, it is well-known [45, 32] that the optimizer of the
Sobolev inequality (9) is given by the function

$g_{s,p}(z)=\frac{1}{\left(1+\left|{z}\right|^{p^{\prime}}\right)^{2d/q}}$

up to rotation, translation and dilatation. This means that
$g\in\dot{W}^{s,p}(\mathbb{R}^{2d})$ and
$\left\|{g_{s,p}}\right\|_{L^{q}}=\mathcal{C}^{\mathrm{S}}_{s,p}\left\|{g_{s,p}}\right\|_{\dot{W}^{s,p}}$.
The observation that we will use is that these functions are actually in a
slightly more regular space than $\dot{W}^{s,p}$. Letting $a=p^{\prime}$ and
$b=2d/q$, then by an explicit computation, it is not difficult to see that for
any $n\in\mathbb{N}$ and any $x\neq 0$,

$\left|{\nabla^{n}g}\right|\leq
C\,\frac{\left|{z}\right|^{a-n}+\left|{z}\right|^{n\left(a-1\right)}}{\left(1+\left|{z}\right|^{a}\right)^{b+n}}$

for some constant $C>0$ depending only one $d$, $a$, $b$ and $n$. Therefore,
$g\in\dot{W}^{n,r}(\mathbb{R}^{2d})$ as soon as $a>n-2d/r$, and $ab>2d/r-n$.
We can then estimate by above the $\dot{W}^{\alpha,q}(\mathbb{R}^{2d})$ norm
of $g_{s,p}$ by the classical Sobolev embedding
$\dot{W}^{n,r}(\mathbb{R}^{2d})\subset\dot{W}^{\alpha,q}(\mathbb{R}^{2d})$
with $n=\lceil\alpha\rceil$. This leads to

$g_{s,p}\in\dot{W}^{\alpha,q}(\mathbb{R}^{2d})\text{ for any
}0\leq\alpha<p^{\prime}+\frac{2d}{q}.$

Now consider $\boldsymbol{\rho}:=\,\tilde{\\!\boldsymbol{\rho}}_{g_{s,p}}$.
Then by the properties of the Husimi transform,

(49)
$\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}\geq\left\|{\tilde{f}_{\boldsymbol{\rho}}}\right\|_{L^{q}}=\left\|{\tilde{\tilde{g}}_{s,p}}\right\|_{L^{q}}\geq\left\|{g_{s,p}}\right\|_{L^{q}}-C\,h^{\alpha/2}\left\|{g_{s,p}}\right\|_{\dot{W}^{\alpha,q}}$

where the last inequality follows by the same proof as the proof of Inequality
(45) and together with the triangle inequality. The first term on the right-
hand-side of Equation (49) is bounded below using the fact that similarly as
in (43) and (44)

(50)
$\left\|{g_{s,p}}\right\|_{L^{q}}=\mathcal{C}^{\mathrm{S}}_{s,p}\left\|{g_{s,p}}\right\|_{\dot{W}^{s,p}}\geq\mathcal{C}^{\mathrm{S}}_{s,p}\big{\lVert}\,\tilde{\\!\boldsymbol{\rho}}_{g_{s,p}}\big{\rVert}_{\dot{\mathcal{W}}^{s,p}}=\mathcal{C}^{\mathrm{S}}_{s,p}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{s,p}}.$

To control the second term on the right-hand-side of Equation (49) by
something of the form
$-Ch^{\alpha/2}\left\|{\boldsymbol{\rho}}\right\|_{\dot{W}^{s,p}}$, one
similarly writes

(51)
$\left\|{\boldsymbol{\rho}}\right\|_{\dot{W}^{s,p}}=\big{\lVert}\,\tilde{\\!\boldsymbol{\rho}}_{g_{s,p}}\big{\rVert}_{\dot{W}^{s,p}}\geq\left\|{\tilde{\tilde{g}}_{s,p}}\right\|_{\dot{W}^{s,p}}\geq
C$

where $C$ does not depend on $\hbar$. Combining the above equations (49), (50)
and (51) yields

$\left(\mathcal{C}^{\mathrm{S}}_{s,p}-C\,h^{\alpha/2}\right)\left\|{\boldsymbol{\rho}}\right\|_{\dot{W}^{s,p}}\leq\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}$

for some constant $C$ depending only on $(d,s,p,\alpha)$. Hence, the constant
$\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}$ verifies
$\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}\geq\mathcal{C}^{\mathrm{S}}_{s,p}-C\,h^{\alpha/2}$
for any $h>0$. Since $\widetilde{\mathcal{C}}^{\mathrm{\,S}}_{s,p}$ is by
definition independent of $h$, we deduce the result by letting $h\to 0$. ∎

###  4.3. Quantum Besov spaces

Define
$\delta^{1}_{z}\boldsymbol{\rho}:=\mathsf{T}_{z}\boldsymbol{\rho}-\boldsymbol{\rho}$
and
$\delta^{2}_{z}\boldsymbol{\rho}:=\delta^{1}_{z}\delta^{1}_{z}\boldsymbol{\rho}=\mathsf{T}_{2z}\boldsymbol{\rho}-2\mathsf{T}_{z}\boldsymbol{\rho}+\boldsymbol{\rho}$.
When $s\in(0,1)$, the following norm

$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,r,(1)}}:=\Big{\lVert}\frac{\mathsf{T}_{z}\boldsymbol{\rho}-\boldsymbol{\rho}}{\left|{z}\right|^{s+2d/r}}\Big{\rVert}_{L^{r}_{z}(\mathcal{L}^{p})}$

is equivalent to the $\dot{\mathcal{B}}^{s}_{p,r}$ norm, independently of
$\hbar$. Proposition 2.1 corresponds to the special case $p=q$ in the
following proposition comparing these equivalent norms.

######  Proposition 4.4.

Let $(s,p,q)\in(0,1)\times[1,\infty]^{2}$ and
$\boldsymbol{\rho}\in\dot{\mathcal{B}}^{s}_{p,q}$. Then
$\boldsymbol{\rho}\in\dot{\mathcal{B}}^{s}_{p,q,(1)}$ and

(52)
$(2^{r^{\prime}}-2^{sr^{\prime}})^{\frac{1}{r^{\prime}}}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q,(1)}}\leq\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q}}\leq\left(2^{r}-2^{sr}\right)^{\frac{1}{r}}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q,(1)}}$

where $r=\max(p,p^{\prime},q,q^{\prime})$.

######  Remark 4.4.

We could also have defined $r$ by
$\frac{1}{2}-\frac{1}{r}=\max\\!\left(\left|{\frac{1}{2}-\frac{1}{p}}\right|,\left|{\frac{1}{2}-\frac{1}{q}}\right|\right)$.
Similarly, $r^{\prime}$ verifies $r^{\prime}=\min(p,p^{\prime},q,q^{\prime})$.
In all the cases, $r^{\prime}\leq 2\leq r$, with equality only if $r=2$.

###### Proof of Proposition 4.4 and Proposition 2.1.

Notice that

$\displaystyle\delta_{z}^{2}\boldsymbol{\rho}$
$\displaystyle=\delta_{z}^{1}\mathsf{T}_{z}\boldsymbol{\rho}-\delta_{z}^{1}\boldsymbol{\rho}$
$\displaystyle\delta_{2z}^{1}\boldsymbol{\rho}$
$\displaystyle=\delta_{z}^{1}\mathsf{T}_{z}\boldsymbol{\rho}+\delta_{z}^{1}\boldsymbol{\rho}.$

Let $\boldsymbol{\mu}_{1}=\left|{\delta_{z}^{2}\boldsymbol{\rho}}\right|$,
$\boldsymbol{\mu}_{2}=\left|{\delta_{2z}^{1}\boldsymbol{\rho}}\right|$,
$\boldsymbol{\nu}_{1}=\left|{\delta_{z}^{1}\mathsf{T}_{z}\boldsymbol{\rho}}\right|$
and $\boldsymbol{\nu}_{2}=\left|{\delta_{z}^{1}\boldsymbol{\rho}}\right|$.
Then, defining
$\mu(\mathop{}\\!\mathrm{d}z):=\frac{\mathop{}\\!\mathrm{d}z}{\left|{z}\right|^{d+sq}}$,
and the notation
$\left\|{\boldsymbol{\mu}}\right\|_{L^{q}_{\mu}\mathcal{L}^{p}_{\vphantom{\mu}}}^{q}:=\int_{\mathbb{R}^{d}}\left\|{\boldsymbol{\mu}(z)}\right\|_{\mathcal{L}^{p}}^{q}\mu(\mathop{}\\!\mathrm{d}z)$,
it holds

$\displaystyle\left\|{\boldsymbol{\mu}_{1}}\right\|_{L^{q}_{\mu}\mathcal{L}^{p}_{\vphantom{\mu}}}$
$\displaystyle=\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q}}$
$\displaystyle\left\|{\boldsymbol{\mu}_{2}}\right\|_{L^{q}_{\mu}\mathcal{L}^{p}_{\vphantom{\mu}}}$
$\displaystyle=2^{s}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q,(1)}}$
$\displaystyle\left\|{\boldsymbol{\nu}_{1}}\right\|_{L^{q}_{\mu}\mathcal{L}^{p}_{\vphantom{\mu}}}$
$\displaystyle=\left\|{\boldsymbol{\nu}_{2}}\right\|_{L^{q}_{\mu}\mathcal{L}^{p}_{\vphantom{\mu}}}=\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q,(1)}}.$

Since $r\geq\max(p,p^{\prime})$, with the notation
$\boldsymbol{\mu}=(\boldsymbol{\mu}_{1},\boldsymbol{\mu}_{2})$ and
$\boldsymbol{\nu}=(\boldsymbol{\nu}_{1},\boldsymbol{\nu}_{2})$ we deduce from
the Clarkson–McCarthy inequalities (See Equation (57) in Appendix A) that

$\left\|{\boldsymbol{\mu}}\right\|_{\ell^{r}(\mathcal{L}^{p})}\leq
2^{1/r}\left\|{\boldsymbol{\nu}}\right\|_{\ell^{r^{\prime}}(\mathcal{L}^{p})}.$

By Jensen’s inequality, since $q\leq r$ and $r^{\prime}\leq q$, this leads to

$\left\|{\boldsymbol{\mu}}\right\|_{\ell^{{}^{r}}\\!L^{q}_{\mu}\mathcal{L}^{{}^{p}}}\leq\left\|{\boldsymbol{\mu}}\right\|_{L^{q}_{\mu}\ell^{{}^{r}}\\!\mathcal{L}^{{}^{p}}}\leq
2^{1/r}\left\|{\boldsymbol{\nu}}\right\|_{L^{q}_{\mu}\ell^{{}^{r^{\prime}}}\\!\\!\mathcal{L}^{{}^{p}}}=2^{1/r}\left\|{\boldsymbol{\nu}}\right\|_{\ell^{{}^{r^{\prime}}}\\!\\!L^{q}_{\mu}\mathcal{L}^{{}^{p}}}.$

In terms of $\boldsymbol{\rho}$, this yields

$\left(\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q}}^{r}+2^{rs}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q,(1)}}^{r}\right)^{1/r}\leq
2^{1/r}\left(2\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q,(1)}}^{r^{\prime}}\right)^{1/r^{\prime}}=2\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q,(1)}}$

which proves the second inequality in Equation (52). The first inequality
follows similarly using Jensen’s inequality and the reversed version of
Clarkson–McCarthy inequalities (Formula (58) in Appendix A). ∎

###### Proof of Proposition 2.2.

From the Formula (25) and the fact that the unitary operators
$\mathsf{T}_{\theta z}$ preserve Schatten norms, one obtains

$\left\|{\delta^{2}_{z}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}\leq
2\left|{z}\right|\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{1,p}}$

while using the triangle inequality yields
$\left\|{\delta^{2}_{z}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}\leq
4\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}$. Therefore, cutting the
integral appearing in the definition of the Besov norm into two parts
$\left|{z}\right|<R$ and $\left|{z}\right|\geq R$ leads to

$\int_{\mathbb{R}^{2d}}\frac{\left\|{\delta^{2}_{z}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}^{r}}{\left|{z}\right|^{2d+sr}}\mathop{}\\!\mathrm{d}z\leq
2^{r}\omega_{2d}\left(\frac{R^{\left(1-s\right)r}}{\left(1-s\right)r}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{1,p}}^{r}+\frac{2^{r}}{s\,r\,R^{sr}}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}^{r}\right)$

optimizing with respect to $R$ leads to

$\displaystyle\int_{\mathbb{R}^{2d}}\frac{\left\|{\delta^{2}_{z}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}^{r}}{\left|{z}\right|^{2d+sr}}\mathop{}\\!\mathrm{d}z\leq\frac{2^{\left(2-s\right)r}\,\omega_{2d}}{r\,s\left(1-s\right)}\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{W}}^{1,p}}^{sr}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}^{\left(1-s\right)r}$

which proves the result. ∎

####  4.3.1. Semiclassical Littlewood–Paley decomposition

We refer for example to [48, 4] for details on the Littlewood–Paley
decomposition in the classical case. As in the classical case, we let
$\varphi$ be a smooth function of the phase space compactly supported in an
annulus $\mathcal{C}=\set{z\in\mathbb{R}^{2d},r\leq\left|{z}\right|\leq R}$
such that it is a dyadic partition of the unity, i.e for any $z\neq 0$,

$\sum_{j\in\mathbb{Z}}\varphi_{j}(z)=1$

where $\varphi_{j}(z)=\varphi(2^{-j}\,z)$, so
$\widehat{\varphi_{j}}=2^{jd}\,\widehat{\varphi}(2^{j}\,z)$. To define the
quantum analogue of the dyadic block $\triangle_{j}$ for an operator
$\boldsymbol{\rho}$, we can use the semiclassical convolution (31). It leads
us to define

(53)
$\triangle_{j}\boldsymbol{\rho}=\widehat{\varphi_{j}}\star\boldsymbol{\rho}=\boldsymbol{\rho}_{\widehat{\varphi}_{j}*f_{\boldsymbol{\rho}}}=\boldsymbol{\rho}_{\triangle_{j}f_{\boldsymbol{\rho}}},$

that is nothing but the Weyl quantization of the dyadic block of the Wigner
transform. We can then get a Littlewood–Paley characterization of Besov
spaces. We define the homogeneous semi-norm

$\left\|{\boldsymbol{\rho}}\right\|_{\tilde{\mathcal{B}}^{s}_{p,q}}:=\left\|{2^{js}\left\|{\triangle_{j}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}}\right\|_{\ell^{q}}.$

As in the classical case, it gives an equivalent norm to the
$\dot{\mathcal{B}}^{s}_{p,q}$ norm defined in Section 2.4. The proof follows
the proof given in [4, Theorem (2.36) and (2.37)], and we just give the idea
on how to adapt the proof by writing the proof of one of the inequalities in
the case $s\in(0,1)$.

######  Proposition 4.5.

Let $s\in(0,1)$ and $(p,q)\in[1,\infty]^{2}$. Then there exists a constant
$C>0$ independent of $\hbar$ such that for any $\boldsymbol{\rho}\in$

(54)
$C^{-1}\left\|{\boldsymbol{\rho}}\right\|_{\tilde{\mathcal{B}}^{s}_{p,q}}\leq\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q}}\leq
C\left\|{\boldsymbol{\rho}}\right\|_{\tilde{\mathcal{B}}^{s}_{p,q}}.$

###### Proof of Proposition 4.5.

Assume $q<\infty$. The case $q=\infty$ can be treated similarly. Since
$s\in(0,1)$, it is sufficient to look at
$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q,(1)}}$ instead
of $\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q}}$ by
Proposition 4.4. Notice that

$\mathsf{T}_{z}\\!\left(f\star\boldsymbol{\rho}\right)=\int_{\mathbb{R}^{2d}}f(z^{\prime})\,\mathsf{T}_{z^{\prime}+z}\boldsymbol{\rho}\mathop{}\\!\mathrm{d}z^{\prime}=\int_{\mathbb{R}^{2d}}f(z^{\prime}-z)\,\mathsf{T}_{z^{\prime}}\boldsymbol{\rho}\mathop{}\\!\mathrm{d}z^{\prime}=\left(\mathsf{T}_{z}f\right)\star\boldsymbol{\rho}.$

Therefore, using the fact that for any $j\in\mathbb{Z}$,
$\triangle_{j}=\sum_{j^{\prime},\left|{j-j^{\prime}}\right|\leq
1}\triangle_{j}\triangle_{j^{\prime}}$, one obtains

$\displaystyle\mathsf{T}_{z}\triangle_{j}\boldsymbol{\rho}-\triangle_{j}\boldsymbol{\rho}$
$\displaystyle=\sum_{j^{\prime},\left|{j-j^{\prime}}\right|\leq
1}\int_{\mathbb{R}^{2d}}\left(\widehat{\varphi_{j}}(z^{\prime}-z)-\widehat{\varphi_{j}}(z^{\prime})\right)\mathsf{T}_{z^{\prime}}\triangle_{j^{\prime}}\boldsymbol{\rho}\mathop{}\\!\mathrm{d}z^{\prime}$
$\displaystyle=\sum_{j^{\prime},\left|{j-j^{\prime}}\right|\leq
1}\int_{\mathbb{R}^{2d}}\int_{0}^{1}2^{j}\,z\cdot\left(\nabla\widehat{\varphi}\right)_{j}(z^{\prime}-\theta
z)\mathop{}\\!\mathrm{d}\theta\,\mathsf{T}_{z^{\prime}}\triangle_{j^{\prime}}\boldsymbol{\rho}\mathop{}\\!\mathrm{d}z^{\prime}$

and so since
$\left\|{\left(\nabla\widehat{\varphi}\right)_{j}(z^{\prime}-\theta
z)}\right\|_{L^{1}_{z^{\prime}}}=\left\|{\nabla\widehat{\varphi}}\right\|_{L^{1}}=:D_{\varphi}$,
it leads to

$\displaystyle\left\|{\mathsf{T}_{z}\triangle_{j}\boldsymbol{\rho}-\triangle_{j}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}$
$\displaystyle\leq
D_{\varphi}\,2^{j}\left|{z}\right|\sum_{j^{\prime},\left|{j-j^{\prime}}\right|\leq
1}\left\|{\triangle_{j^{\prime}}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}=:D_{\varphi}\,\left\|{\boldsymbol{\rho}}\right\|_{\tilde{B}^{s}_{p,q}}2^{j\left(1-s\right)}\left|{z}\right|d_{j,s}$

where $\left\|{d_{j,s}}\right\|_{\ell^{q}}\leq 3^{1/q^{\prime}}$. On the other
hand, by the triangle inequality

$\left\|{\mathsf{T}_{z}\triangle_{j}\boldsymbol{\rho}-\triangle_{j}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}\leq
2\left\|{\triangle_{j}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}\leq
2^{-js}\left\|{\boldsymbol{\rho}}\right\|_{\tilde{B}^{s}_{p,q}}\tilde{d}_{j,s}$

where $\left\|{\tilde{d}_{j,s}}\right\|_{\ell^{q}}\leq 2$. Hence, summing over
$j$ and using the two above inequalities we deduce

$\left\|{\mathsf{T}_{z}\boldsymbol{\rho}-\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}\leq\left(D_{\varphi}\left|{z}\right|\sum_{j\leq
j_{0}}2^{j\left(1-s\right)}d_{j,s}+\sum_{j>j_{0}}2^{-js}\tilde{d}_{j,s}\right)\left\|{\boldsymbol{\rho}}\right\|_{\tilde{B}^{s}_{p,q}}.$

The triangle inequality then yields

(55)
$\left\|{\frac{\left\|{\mathsf{T}_{z}\boldsymbol{\rho}-\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}}{\left|{z}\right|^{s+2d/q}}}\right\|_{L^{q}}\leq\left(D_{\varphi}\,I_{1}+I_{2}\right)\left\|{\boldsymbol{\rho}}\right\|_{\tilde{B}^{s}_{p,q}},$

where

$\displaystyle I_{1}^{q}$
$\displaystyle=\int_{\mathbb{R}^{2d}}\frac{1}{\left|{z}\right|^{2d-(1-s)q}}\Big{(}\sum_{j\leq
j_{0}}2^{j\left(1-s\right)}\,d_{j,s}\Big{)}^{q}\mathop{}\\!\mathrm{d}z$
$\displaystyle I_{2}^{q}$
$\displaystyle=\int_{\mathbb{R}^{2d}}\frac{1}{\left|{z}\right|^{2d+sq}}\Big{(}\sum_{j>j_{0}}2^{-js}\,\tilde{d}_{j,s}\Big{)}^{q}\mathop{}\\!\mathrm{d}z.$

By Hölder’s inequality and the explicit formula for the geometric sum

$\displaystyle\Big{(}\sum_{j\leq
j_{0}}2^{j\left(1-s\right)}d_{j,s}\Big{)}^{q}\leq
C_{1-s}^{q-1}\,2^{j_{0}\left(1-s\right)\left(q-1\right)}\sum_{j\leq
j_{0}}2^{j\left(1-s\right)}d_{j,s}^{q}$

where $C_{1-s}=\frac{2^{1-s}}{2^{1-s}-1}$. Hence, $I_{1}$ can be estimated by

$I_{1}^{q}\leq C_{1-s}^{q-1}\int_{\mathbb{R}^{2d}}\sum_{j\leq
j_{0}}\frac{2^{j_{0}\left(1-s\right)\left(q-1\right)}}{\left|{z}\right|^{2d-(1-s)q}}\,2^{j\left(1-s\right)}d_{j,s}^{q}\mathop{}\\!\mathrm{d}z.$

Taking $j_{0}$ such that $\frac{\lambda}{\left|{z}\right|}\leq
2^{j_{0}}\leq\frac{2\lambda}{\left|{z}\right|}$, one finds by Fubini’s theorem

$I_{1}^{q}\leq\frac{\left(3C_{1-s}\right)^{q-1}\left(2\lambda\right)^{\left(1-s\right)q}\,\omega_{2d}}{1-s}.$

Similarly,

$I_{2}^{q}\leq\left|{C_{s}}\right|^{q-1}\sum_{j\in\mathbb{Z}}\int_{\left|{z}\right|>\lambda\,2^{-j}}\frac{\lambda^{-s\left(q-1\right)}}{\left|{z}\right|^{2d+s}}\,2^{-js}\tilde{d}_{j,s}^{q}\mathop{}\\!\mathrm{d}z\leq\frac{2^{q}\left|{C_{s}}\right|^{q-1}\omega_{2d}}{s\,\lambda^{sq}}.$

Combining these two inequalities in Inequality (55) and optimizing with
respect to $\lambda$ yields

$\left\|{\frac{\left\|{\mathsf{T}_{z}\boldsymbol{\rho}-\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}}{\left|{z}\right|^{s+2d/q}}}\right\|_{L^{q}}\leq\frac{2^{\left(1-s\right)\left(s+1\right)}\omega_{2d}^{1/q}D_{\varphi}^{s}\left(3C_{1-s}\right)^{s/q^{\prime}}\left(\left|{C_{s}}\right|\right)^{(1-s)/q^{\prime}}}{s^{s+(1-s)/q}\left(1-s\right)^{1-s+s/q}}\left\|{\boldsymbol{\rho}}\right\|_{\tilde{B}^{s}_{p,q}}$

finishing the proof of the second inequality in Equation (54). ∎

From the above proposition, we deduce the classical embedding between Besov
spaces with different third index.

######  Corollary 4.1 (Comparison of Quantum Besov spaces).

For any $1\leq r<q\leq\infty$,
$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,q}}\leq
C\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s}_{p,r}}$ for some
constant $C>0$ independent of $\hbar$.

From the semiclassical Young inequality, we get a Bernstein-type lemma.

######  Lemma 4.5 (Quantum Bernstein Lemma).

For any $1\leq p\leq q\leq\infty$, there exists a constant $C$ independent of
$\hbar$ such that for any operator $\boldsymbol{\rho}\in\mathcal{L}^{p}$,

(56)
$\left\|{\boldsymbol{\nabla}^{n}\triangle_{j}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{q}}\leq
C\,2^{j\left(n+2d\left(\frac{1}{p}-\frac{1}{p}\right)\right)}\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}.$

The above inequalities are a very useful tool. In particular, they allow to
prove that the Schatten norms can always be compared to Besov norms in the
following way.

######  Corollary 4.2.

For any $1\leq p\leq q\leq\infty$ and $r\in[1,\infty]$, there exists $C>0$
such that for any compact operator $\boldsymbol{\rho}$

$C^{-1}\left\|{\boldsymbol{\rho}}\right\|_{\tilde{\mathcal{B}}^{0}_{p,\infty}}\leq\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}\leq
C\left\|{\boldsymbol{\rho}}\right\|_{\tilde{\mathcal{B}}^{0}_{p,1}}.$

###### Proof.

The first inequality follows from Bernstein lemma with $p-q=n=0$. To get the
second inequality, write
$\boldsymbol{\rho}=\sum_{j\in\mathbb{N}}\triangle_{j}\boldsymbol{\rho}$, so
that

$\left\|{\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}\leq\sum_{j\in\mathbb{N}}\left\|{\triangle_{j}\boldsymbol{\rho}}\right\|_{\mathcal{L}^{p}}=\left\|{\boldsymbol{\rho}}\right\|_{\tilde{\mathcal{B}}^{0}_{p,1}}$

follows by the triangle inequality for Schatten norms. ∎

From Bernstein’s Lemma 4.5, we also obtain the analogue of the Sobolev
embedding for Besov spaces.

######  Corollary 4.3 (Sobolev inequalities for quantum Besov spaces).

For any $1\leq p\leq q\leq\infty$, $r\in[1,\infty]$ and
$(s_{0},s_{1})\in(0,1)^{2}$ such that
$s_{1}-s_{0}=2d\left(\frac{1}{p}-\frac{1}{q}\right)$, there exists $C>0$ such
that for any compact operator $\boldsymbol{\rho}$

$\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s_{0}}_{q,r}}\leq
C\left\|{\boldsymbol{\rho}}\right\|_{\dot{\mathcal{B}}^{s_{1}}_{p,r}}.$

##  Appendix A Vector valued operators

Let $A=(A_{1},\dots,A_{n})$ and $B=(B_{1},\dots,B_{n})$ be vector valued
operators, i.e. for any $k\in\set{1,\dots,n}$, $A_{k}$ is an operator acting
on $L^{2}(\mathbb{R}^{d})$. Then we denote by $A\cdot B$ the operator $A\cdot
B:=\sum_{k=1}^{n}A_{k}B_{k}$ and by $\left|{A}\right|$ the square root of the
positive operator

$\left|{A}\right|^{2}=A^{*}\cdot A=\sum_{k=1}^{n}\left|{A_{k}}\right|^{2}.$

More generally, we define
$\left|{A}\right|_{\ell^{r}}^{r}=\sum_{k=1}^{n}\left|{A_{k}}\right|^{r}$. We
warn the reader about the fact that similarly as the case $r=2$, these
quantities do not verify in general the triangle inequality. We will use the
same notation $\ell^{r}$ to denote
$\left\|{a_{k}}\right\|_{\ell^{r}}^{r}=\sum|a_{k}|^{r}$ for sequences
$(a_{k})_{k\geq 1}$, in which case these are indeed norms. The Schatten norm
of a vector valued operator is given by
$\left\|{A}\right\|_{\mathfrak{S}^{p}}=\left\|{A}\right\|_{p}=\left(\operatorname{Tr}\\!\left(\left|{A}\right|^{p}\right)\right)^{\frac{1}{p}}$.
In the case of vector valued operators, we define the Wigner transform
component by component, i.e. $f_{A}=(f_{A_{1}},\dots,f_{A_{d}})$. One easily
verifies that

$\left\|{f_{A}}\right\|_{L^{2}}=\left\|{A}\right\|_{\mathcal{L}^{2}}.$

######  Proposition A.1.

Let $x\in\mathbb{R}^{n}$. Then for any $p\in[0,2]$ $\left|{x\cdot
A}\right|^{p}\leq\left|{x}\right|^{p}\left|{A}\right|^{p}$ and for any
$p\in[0,\infty]$, $\left\|{x\cdot
A}\right\|_{p}\leq\left|{x}\right|\left\|{A}\right\|_{p}$.

###### Proof.

From the identity

$\left|{x}\right|^{2}\left|{A}\right|^{2}-\left|{x\cdot
A}\right|^{2}=\left|{x}\right|^{-2}\left|{\left|{x}\right|^{2}A-\left(x\cdot
A\right)x}\right|^{2}$

for any $x\neq 0$, we deduce that
$\left|{x}\right|^{2}\left|{A}\right|^{2}\leq\left|{x\cdot A}\right|^{2}$, and
so we deduce the first claimed inequality from the fact that the function
$t\to t^{a}$ is operator monotone for any $a\in[0,1]$ by the Löwner–Heinz
theorem. Taking the trace yields the second inequality for $p\leq 2$. Taking
the operator norm gives the case $p=\infty$. The result for $p\in(2,\infty)$
follows by noticing that $f:t\mapsto\left|{t}\right|^{k}t$ is increasing on
$\mathbb{R}$ for any $k\geq 0$, so
$A\mapsto\operatorname{Tr}\\!\left(f(A)\right)$ is increasing (see e.g. [16]).
∎

######  Proposition A.2.

Let $p\in[2,\infty]$, then

$\left(\sum_{k=1}^{n}\left\|{A_{k}}\right\|_{p}^{p}\right)^{1/p}\leq\left\|{A}\right\|_{p}\leq\left(\sum_{k=1}^{n}\left\|{A_{k}}\right\|_{p}^{2}\right)^{1/2}$

while if $p\in[1,2]$, then

$n^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{k=1}^{n}\left\|{A_{k}}\right\|_{p}^{p}\right)^{1/p}\leq\left\|{A}\right\|_{p}\leq\left(\sum_{k=1}^{n}\left\|{A_{k}}\right\|_{p}^{p}\right)^{1/p}.$

The above proposition follows by taking $r=2$ in the following inequalities.

######  Proposition A.3.

Let $(r,p)\in[1,\infty]^{2}$. Then

$\displaystyle\left\|{A}\right\|_{\ell^{r}(\mathfrak{S}^{r})}$
$\displaystyle\leq\left\|{A}\right\|_{\mathfrak{S}^{p}(\ell^{r})}\leq\left\|{A}\right\|_{\ell^{r}(\mathfrak{S}^{p})}$
$\displaystyle\text{ if }p\geq r$ $\displaystyle
n^{\frac{1}{r}-\frac{1}{p}}\left\|{A}\right\|_{\ell^{p}(\mathfrak{S}^{p})}$
$\displaystyle\leq\left\|{A}\right\|_{\mathfrak{S}^{p}(\ell^{r})}\leq\left\|{A}\right\|_{\ell^{p}(\mathfrak{S}^{p})}$
$\displaystyle\text{ if }p\leq r.$

###### Proof.

If $p\geq r$, then by the triangle inequality for Schatten norms,

$\left\|{\left|{A}\right|_{\ell^{r}}}\right\|_{p}^{r}=\left\|{\sum_{k=1}^{n}\left|{A_{k}}\right|^{r}}\right\|_{p/r}\leq\sum_{k=1}^{n}\left\|{\left|{A_{k}}\right|^{r}}\right\|_{p/r}=\sum_{k=1}^{n}\left\|{A_{k}}\right\|_{p}^{r}.$

Conversely, by [42, Theorem 1.22], it holds

$\sum_{k=1}^{n}\left\|{A_{k}}\right\|_{p}^{p}=\sum_{k=1}^{n}\left\|{\left|{A_{k}}\right|^{r}}\right\|_{p/r}^{p/r}\leq\left\|{\sum_{k=1}^{n}\left|{A_{k}}\right|^{r}}\right\|_{p/r}^{p/r}=\left\|{\left|{A}\right|_{\ell^{r}}}\right\|_{p}^{p}.$

Now let $p\in[1,r]$. Then, $t\mapsto t^{p/r}$ is operator concave on positive
operators, which yields

$\sum_{k=1}^{n}\left|{A_{k}}\right|^{p}\leq
n^{1-\frac{p}{r}}\left(\sum_{k=1}^{n}\left|{A_{k}}\right|^{r}\right)^{\frac{p}{r}}$

and so taking the trace,

$\sum_{k=1}^{n}\left\|{A_{k}}\right\|_{p}^{p}\leq
n^{1-\frac{p}{r}}\left\|{\left|{A}\right|_{\ell^{r}}}\right\|_{p}^{p}.$

Finally, assume $n=2$. Then since
$\left|{A_{1}}\right|^{r}+\left|{A_{2}}\right|^{r}\geq\left|{A_{1}}\right|^{r}$,
and $t\mapsto t^{p/r}$ is operator monotone, we deduce that
$\left(\left|{A_{1}}\right|^{r}+\left|{A_{2}}\right|^{r}\right)^{\frac{p}{r}}-\left|{A_{1}}\right|^{p}\geq
0$. Since by the Birman–Koplienko–Solomyak–Ando inequality [9, 2]

$\operatorname{Tr}\\!\left(\left|{\left(\left|{A_{1}}\right|^{r}+\left|{A_{2}}\right|^{r}\right)^{\frac{p}{r}}-\left(\left|{A_{1}}\right|^{r}\right)^{\frac{p}{r}}}\right|\right)\leq\operatorname{Tr}\\!\left(\left(\left|{A_{2}}\right|^{r}\right)^{\frac{p}{r}}\right),$

we deduce that
$\operatorname{Tr}\\!\left(\left(\left|{A_{1}}\right|^{r}+\left|{A_{2}}\right|^{r}\right)^{\frac{p}{r}}\right)\leq\operatorname{Tr}\\!\left(\left|{A_{1}}\right|^{p}+\left|{A_{2}}\right|^{p}\right)$.
By induction, this generalizes to the case $n\geq 2$ leading to
$\left\|{\left|{A}\right|_{\ell^{r}}}\right\|_{p}^{p}\leq\sum_{k=1}^{n}\left\|{A_{k}}\right\|_{p}^{p}$.
∎

We recall the Clarkson–McCarthy inequalities [37]

$\displaystyle\left\|{A-B}\right\|_{p}^{p^{\prime}}+\left\|{A+B}\right\|_{p}^{p^{\prime}}$
$\displaystyle\leq
2\left(\left\|{A}\right\|_{p}^{p}+\left\|{B}\right\|_{p}^{p}\right)^{\frac{p^{\prime}}{p}}$
$\displaystyle\text{ if }p\in(1,2]$
$\displaystyle\left\|{A-B}\right\|_{p}^{p}+\left\|{A+B}\right\|_{p}^{p}$
$\displaystyle\leq
2\left(\left\|{A}\right\|_{p}^{p^{\prime}}+\left\|{B}\right\|_{p}^{p^{\prime}}\right)^{\frac{p}{p^{\prime}}}$
$\displaystyle\text{ if }p\in[2,\infty)$

More generally, these can be written as follows.

######  Proposition A.4.

Let $(p,r)\in[1,\infty]^{2}$ be such that $r\geq\max(p,p^{\prime})$. Then
defining $M=(A-B,A+B)$ and $N=(A,B)$, it holds

(57) $\displaystyle\left\|{M}\right\|_{\ell^{r}(\mathcal{L}^{p})}$
$\displaystyle\leq
2^{1/r}\left\|{N}\right\|_{\ell^{r^{\prime}}(\mathcal{L}^{p})}$ (58)
$\displaystyle 2^{1/r^{\prime}}\left\|{N}\right\|_{\ell^{r}(\mathcal{L}^{p})}$
$\displaystyle\leq\left\|{M}\right\|_{\ell^{r^{\prime}}(\mathcal{L}^{p})}.$

###### Proof.

Following [10], the result can be proved by interpolation. Notice first that
$\left|{M}\right|_{\ell^{2}}^{2}=2\left|{N}\right|_{\ell^{2}}^{2}$, to deduce
that

$\left\|{M}\right\|_{\ell^{2}(\mathcal{L}^{2})}=\sqrt{2}\left\|{N}\right\|_{\ell^{2}(\mathcal{L}^{2})}.$

On the other hand, for any $p_{1}\in[1,\infty]$, by the triangle inequality
for Schatten norms,

$\left\|{M}\right\|_{\ell^{\infty}(\mathcal{L}^{p_{1}})}\leq\left\|{N}\right\|_{\ell^{1}(\mathcal{L}^{p_{1}})}.$

Therefore, since $r\geq 2$ and $p\in[r^{\prime},r]$, the Inequality (57)
follows using the fact that the spaces $\ell^{r}(\mathcal{L}^{p})$ are an
exact family of complex interpolation (see [38, Satz 8] and [47, 1.18.1]).
Inequality (58) then follows from Equation (57) just by replacing $(A,B)$ by
$(\frac{A+B}{2},\frac{A-B}{2})$. ∎

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