We obtain the combined mean-field and semiclassical limit from the $N$-body Schrödinger equation for fermions interacting via singular potentials. To obtain the result, we first prove the uniformity in Planck's constant $h$ propagation of regularity for solutions to the Hartree–Fock equation with singular pair interaction potentials of the form $\pm|x-y|^{-a}$, including the Coulomb and gravitational interactions.

In the context of mixed states, we use these regularity properties to obtain quantitative estimates on the distance between solutions to the Schrödinger equation and solutions to the Hartree–Fock and Vlasov equations in Schatten norms. For $a\in(0,1/2)$, we obtain local-in-time results when $N^{-1/2} \ll h \leq N^{-1/3}$. In particular, it leads to the derivation of the Vlasov equation with singular potentials. For $a\in[1/2,1]$, our results hold only on a small time scale,	or with an $N$-dependent cutoff.

§ BACKGROUND

§.§ The Equations

We consider a system of $N$ identical quantum particles of mass $m$ interacting through a pair potential $K(x-y)$. The state of the system is described by an $N$-body wave function $\psi_N = \psi_N(t, x_1, x_2, \dots, x_N)$, belonging to the space of square-integrable complex-valued functions $\h=L^2(\R^{3N},\CC)$, with evolution given by the $N$-body Schrödinger equation

\begin{equation}\label{eq:Schrodinger}
i\hbar\,\dpt \psi_N = \sum_{k=1}^N-\frac{\hbar^2}{2\,m}\,\Delta_{x_k} \psi_N + \sum_{1\leq k<l\leq N} K(x_k-x_l)\,\psi_N,
\end{equation}

where $h$ is the Planck constant and $\hbar = \frac{h}{2\,\pi}$ is the reduced Planck constant. In applications, one is typically interested in systems where the number of particles $N$ is large, thus making the microscopic description given by the solution to Equation (<ref>) not suitable for studies. In fact, the high dimensionality of the configuration space presents a formidable barrier for understanding the dynamics of the many-body wave function at a microscopic scale. Instead, one can consider the problem at a mesoscopic scale and look at the classical phase space distributions of particles $f=f(t,x,\xi)$, where $(x,\xi)\in\Rd\times\Rd$ are the spatial and momentum variables. When the collisions are not dominant, the equation governing the dynamics of a large number of interacting particles is given by the Vlasov equation

\begin{equation}\label{eq:Vlasov}
\partial_t f + \xi\cdot\Dx f + E_f\cdot\Dv f = 0,
\end{equation}

where $E_f = -\nabla V_f$ is the force field corresponding to the mean-field potential

\begin{equation*}
V_f(x) = (K * \rho_f)(x) = \intd K(x-y)\,\rho_f(y)\dd y
\end{equation*}

and $\rho_f$ is the spatial distribution of particles defined by

\begin{equation}\label{eq:spatial_density}
\rho_f(x) = \intd f(x,\xi)\dd \xi.
\end{equation}

To explore the connection between the microscopic and the mesoscopic scales of the system, we consider the intermediate case of a large number of particles at a microscopic scale where the effective quantum dynamics of the system can be described by its mean-field behavior. Roughly speaking, this means the many-body effects exerted by the system on each particle is well-approximated by an effective interaction potential obtained by averaging the pair potential $K$ with the underlying spatial density of the system. To draw a parallel with the classical theory, one can consider the mean-field equation called the Hartree equation which is the quantum analogue of the Vlasov equation.
More precisely, let us take a positive self-adjoint trace class operator $\op$ acting on $L^2(\Rd,\CC)$, which can be seen as a positive linear convex combination  of projections onto one-particle wave functions. We use the same notation to denote both the operator $\op$ and its integral kernel $\op(x,y)$. Here, $\op$ plays the role of the quantum one-particle phase space distribution of particles. Moreover, the effective one-particle Hamiltonian is given by $H = -\frac{\hbar^2}{2}\, \Delta + V_{\op}$, called the Hartree Hamiltonian, where $V_{\op}$ is the mean-field potential $V_{\op} = K*\rho(x)$ and $\rho(x)$ is the quantum spatial distribution of particles defined by

\begin{equation}\label{eq:spatial_density_quantum}
\rho(x) = \Diag{\op}(x) := h^3\, \op(x,x).
\end{equation}

With these notations, the Hartree equation reads

\begin{equation*}
i\hbar\,\partial_t \op = \com{H,\op},
\end{equation*}

where $\com{A, B} := AB-BA$ is the commutator of the operators $A$ and $B$. If the particles obey the Fermi–Dirac statistics, a more accurate description of their evolution is given by the Hartree–Fock equation

\begin{align}\label{eq:Hartree--Fock}
i\hbar\,\partial_t \op &= \com{H_{\op},\op}, & H_{\op} &= -\frac{\hbar^2}{2}\, \Delta + V_{\op} - h^3\, \sfX_{\op}\,,
\end{align}

where the exchange term $\sfX_{\op}$ is the operator with integral kernel

\begin{equation}\label{eq:exchange_operator}
\sfX_{\op}(x,y) = K(x-y)\, \op(x,y).
\end{equation}

§.§ The Change of Scaling

Our goal in this paper is to study simultaneously the mean-field limit, corresponding to the approximations made when the number of particles $N$ is large, and the semiclassical limit, corresponding to a change of scaling where the Planck constant $h$ becomes negligible. More precisely, we consider a solution $\psi_N$ to the Schrödinger equation (<ref>) and define the function $\widetilde{\psi}_N(t,x_1, \dots, x_N) = \psi_N(T\,t,L\,x_1, \dots, L\, x_N)$ for some characteristic length $L$ and timescale $T$. Multiplying equation (<ref>) by $\frac{T}{m\,L^2}$ yields

\begin{align*}
i\,\widetilde{\hbar}\,\dpt \widetilde{\psi}_N = \sum_{k=1}^N -\frac{\widetilde{\hbar}^2}{2}\,\Delta_{x_k} \widetilde{\psi}_N + \frac{1}{N}\,\sum_{1\leq k<l\leq N} \widetilde{K}(x_k-x_l) \, \widetilde{\psi}_N,
\end{align*}

where $\widetilde{\hbar} = \frac{\hbar\, T}{m\,L^2}$ and $\widetilde{K}(x) = \frac{N\,T^2}{m\,L^2}\, K(L\,x)$. In particular, in the case of a homogeneous interaction $K(x) = \kappa \n{x}^{-a}$, this gives $\widetilde{K}(x) = \widetilde{\kappa}\, \n{x}^{-a}$ where $\widetilde{\kappa} = \frac{\kappa\,N\,T^2}{m\,L^{2+a}}$. In the remainder of the paper, we drop the tilde and study the equation

\begin{align}\label{eq:Schrodinger_rescaled}
i\hbar\,\dpt \psi_N &= H_N\,\psi_N, & H_N &= \sum_{k=1}^N -\frac{\hbar^2}{2}\Delta_{x_k} + \frac{1}{N}\,\sum_{1\leq k<l\leq N} K(x_k-x_l)
\end{align}

with scalings such that $\hbar\to 0$, $N\to \infty$ and with

\begin{equation}\label{eq:force}
K(x) = \frac{\kappa}{\n{x}^a},
\end{equation}

where $\kappa\in\R$ is of order $1$ and $a\in(0,1]$. In fact, we study the time evolution of $N$-body mixed states, which are self-adjoint, positive trace class operators of rank larger than one. By the spectral theorem, they can be expressed in the following way:

\begin{align}
\op_N = \sum_{j=1}^\infty \lambda_j \ket{\psi_j}\!\bra{\psi_j} \ \text{ with }\ \lambda_j \ge 0
\end{align}

where $\set{\psi_j}_{j\in \N} \subset \h^{\otimes N}$ is an orthonormal set of anti-symmetric wave functions. The operator $\op_N$ is called a pure state provided it is a rank one projection, that is, $\op_N=\ket{\psi_N}\!\bra{\psi_N}$. The time evolution equation for density operators is given by the Liouville–von Neumann equation

\begin{equation}\label{eq:Liouville}
i\hbar\,\dpt \op_{N} = \com{H_N,\op_{N}}
\end{equation}

where the Hamiltonian $H_N$ is given in Equation (<ref>), which is the quantum analogue of the classical Liouville equation, equivalent to the $N$-body Newton laws. The two parameters $\hbar$ and $N$ can a priori be independent, however some restrictions apply in the case of fermions, as shown in Figure <ref> and explained in Section <ref>.

(-4.5,5.8)–(-4.5,5.6)–(-4,5.6)–(-4,5.8) node[midway, right=5]Lower densities of
bosons or fermions;

[pattern=north west lines,pattern color = bleuFonce]
(0.5,6.0)–(0.5,5.7)–(1,5.7)–(1,6.0) node[midway, right=5]Higher densities of bosons;
[line width = 0.8, blue]
(1,5.2)–(0.5,5.5) node[midway, black, right=11]Maximal density of fermions;

[align=left] at (-2.7,4.2) Schrödinger;
[] at (2.8,4.2) Hartree(–Fock);
[] at (-2.5,-0.2) Newton;
[] at (2.5,-0.2) Vlasov;

[pattern=north west lines,pattern color = bleuFonce]
– plot [domain=-2:2, samples=120] (, 4*((2-)/4)^(1/3))
– cycle;

– plot [domain=-2:2, samples=100] (, 4*((2-)/4)^(1/3))
– cycle;

– plot [domain=-2:2, samples=100] (, 4*((2-)/4)^(1/2))
– cycle;

[->,line width = 1](-1.5,4.2)–(1.5,4.2) node[midway, above]$N^{-1}\to 0$ $\(h=1\)$;
[<-,line width = 1](-2.5,0.2)–(-2.5,3.8) node[midway,left]$h\to 0$;
[<-,line width = 1](2.5,0.2)–(2.5,3.8) node[midway,right]$h\to 0$

[->,line width = 1](-1.5,-0.2)–(1.5,-0.2) node[midway, below]$N^{-1}\to 0$ $\(h=0\)$;

[domain=0:4, line width=0.8,blue] plot (-(/4)^3*4+2, );

[domain=0:4, line width=0.8,red] plot (-(/4)^2*4+2, );

The different scalings for the combined mean-field and semiclassical limits. The red line corresponds to the line $h = N^{-1/2}$ and the blue line to $h = N^{-1/3}$.

Notice that the density of particles, i.e. the number of particles per unit of volume, has to be of order $N\,L^{-3}$, which according to our scaling leads to a quantity proportional to $N\,h^{3/2}\(m/T\)^{3/2}$. This explains why the part of Figure (<ref>) that is closer to the Hartree–Fock equation corresponds to relatively higher densities, while the part that is below corresponds to relatively lower densities.

§.§ State of the Art

Both the problems of the mean-field limits and the semiclassical limits are well-known questions that are largely addressed in the literature. However, the derivation of the Vlasov–Poisson equation, i.e. the case of the Coulomb and gravitational potentials, remains an open problem, both in the case of quantum mechanics and in the case of classical Newton's laws.

§.§.§ The Classical Mean-Field Limit
In the context of classical mechanics, the problem of justifying the Vlasov equation (<ref>) starting from the dynamics of $N$-particles obeying Newton's laws  was first proven for twice differentiable potentials in the pioneering works by Neunzert and Wick [55], Braun and Hepp [18], and then by Dobrushin [25] using the Wasserstein–Monge–Kantorovich distance (see also [71]). The class of potentials was then extended to less regular potentials but still locally Hölder continuous by Hauray and Jabin [40, 41], which was later improved by Jabin and Wang using entropy methods in [43], where the potential is only required to be bounded.

From another point of view, it was also proved in [41] that it is possible to obtain the mean-field limit for potentials with a vanishing cutoff, converging to potentials almost as singular as the Coulomb potential when $N\to\infty$ (see also Boers and Pickl [17]). This is in particular interesting from a numerical point of view. These results were then improved by Lazarovici [48], allowing the cutoff potential to converge to the Coulomb potential, and by Lazarovici and Pickl [49], with a $N$-dependent cutoff of the order of the inter-particle distance.

§.§.§ Combined Mean-Field and Semiclassical Limits
The first rigorous derivation of the Vlasov equation (<ref>) from the $N$-body Schrödinger equation (<ref>) was proved by Narnhofer and Sewel [54] in the case of smooth potentials, with $\hbar = N^{-1/3}$. Subsequently, the restriction on the potential was substantially relaxed by Spohn [70] to twice differentiable potentials. For the same kind of potentials, a more explicit rate of convergence without assuming $\hbar = N^{-1/3}$ was later obtained by Graffi, Martinez, and Pulvirenti [37] in the case of weak convergence, and more recently by Golse and Paul [34] in the quantum Wasserstein metrics, and by Chen, Lee and Liew for fermions [21] in the scaling $\hbar = N^{-1/3}$.

§.§.§ Quantum Mean-Field Limit
It is also possible to first look at the mean-field limit with $\hbar=1$, i.e. without taking the semiclassical limit, leading to the Hartree and the Hartree–Fock equations. In this case, the situation is better understood, even for the Coulomb and gravitational potentials. For bosons, weak convergence was proved in [10, 28, 8], and explicit rates in stronger norms were obtained in [63, 38, 59, 23, 44, 53, 22, 56]. For fermions, weak convergence was proved in [9] for bounded potentials, and estimates in trace norm and singular potentials such as the Coulomb potential were obtained in [32, 5, 58, 57].

Some of these results have been extended by taking into account the semiclassical parameter $\hbar$. For fermions, taking $\hbar=N^{-1/3}$, convergence of Husimi transform has been proven in [27] for analytic interactions and short times. Schatten norms estimates have been obtained in  [13, 11, 58] for at least twice differentiable potentials. A conditional result was obtained in the case of pure states and singular potentials in [60, 64].

For bosons, results were obtained for at least twice differentiable potentials in [33, 36, 35]

§.§.§ Semiclassical Limit
Another possible direction is to look only at the semiclassical limit $\hbar\to 0$, either for the number of particles $N$ fixed or in the mean-field regime. This last case corresponds to going from the Hartree or the Hartree–Fock equation to the Vlasov equation. In the case of the Hartree equation, this was proved in [51] in weak topology, but including singular potentials such as the Coulomb interaction (See also [52, 30]). Explicit rates in stronger norms were then obtained in [4, 1, 12, 34] for at least twice differentiable potentials, and then in [45, 65, 66, 46, 47] for singular interactions.

§.§ Constraints on the Scalings

The Fourier transform is defined by

\begin{align}
\widehat g(\xi) = \intd e^{-2\pi i\,x\cdot \xi}\, g(x)\dd x
\end{align}

for $g \in L^2(\R^3)$. We also adopt the following conventions for the time-dependent operator solution $\op = \op(t)$ to the Hartree–Fock equation (<ref>)

\begin{equation}\label{eq:scaling_op}
\begin{split}
\Nrm{\op}{\infty} &= \cC_\infty, \vspace{3pt}
\\
\Tr{\op} &= h^{-3}.
\end{split}
\end{equation}

for some constant $\cC_\infty$. In particular, $\intd \rho(x)\dd x = h^3 \Tr{\op} = 1$. For such an operator, we define its Wigner transform by

\begin{equation*}
f_{\op}(x,\xi) := \intd e^{-i\,y\cdot\xi/\hbar} \,\op(x+\tfrac{y}{2},x-\tfrac{y}{2})\dd y,
\end{equation*}

so that it is a function of the phase space with mass $\iint f_{\op}\dd x\dd\xi = h^3\Tr{\op} = 1$.
It is well known that, under some regularity assumptions, the Wigner transform of solutions $\op$ to the Hartree–Fock equation (<ref>) converge to solutions of the Vlasov equation (<ref>) in the semiclassical limit $h\to 0$ (see e.g. [51]). We refer to [51] for a listing of the properties of the Wigner transform. One of them is the fact that

\begin{equation}\label{eq:L2_norm}
\Nrm{f_{\op}}{L^2(\RRd)} = h^{\frac{3}{2}} \Nrm{\op}{2},
\end{equation}

where we denote by

\begin{align}
\Nrm{\op}{p} = \(\Tr{\n{\op}^p}\)^\frac{1}{p}
\end{align}

the Schatten norm of order $p$. Here, the absolute value of an operator $A$ is defined by $\n{A} = \sqrt{A^*A}$. Since we want to address the case when $f_{\op}$ converges in $L^2(\RRd)$ to a solution $f$ of the Vlasov equation, this implies that $h^{\frac{3}{2}} \Nrm{\op}{2} \underset{h\to 0}{\rightarrow} \Nrm{f}{L^2(\RRd)}$, so that $\Nrm{\op}{2}$ is of size $h^{-3/2}$.

For an $N$-particle density operator $\op_{N}$, we look at its one-particle reduced density operator $\op_{N:1}$ defined as the partial trace with of $\op_{N}$ respect to the variables $2$ to $N$, that is,

\begin{align*}
\op_{N:1} = \tr_{2, \ldots, N}(\op_{N}).
\end{align*}

Since we also want the corresponding Wigner transform $f_{N:1}$ of the operator $\op_{N:1}$ of the $N$-particle density operator to converge to $f$, we have as well

\begin{equation}\label{eq:CV_of_L2_norm}
\Nrm{f_{N:1}}{L^2(\RRd)} = h^{\frac{3}{2}} \Nrm{\op_{N:1}}{2} {\longrightarrow} \Nrm{f}{L^2(\RRd)}\quad \mbox{as } N\to\infty\ \mbox{and } h\to 0.
\end{equation}

However, in the case of fermions, we also know that (as proved for example in <cit.>, or <cit.>)

\begin{align}\label{eq:fermionic_inequality}
0 \leq \op_{N:1} \leq \frac{\Tr{\op_{N:1}}}{N} = \frac{h^{-3}}{N}.
\end{align}

Therefore, by bounding the square of the Hilbert–Schmidt norm by the product of the trace norm and the operator norm, we deduce that

\begin{align}\label{eq:fermionic_inequality2}
\Nrm{\op_{N:1}}{2}^2 \leq \Nrm{\op_{N:1}}{1} \Nrm{\op_{N:1}}{\infty} \leq \frac{h^{-6}}{N}.
\end{align}

Combining Inequality (<ref>) with Formula (<ref>) with $\op = \op_{N:1}$, we obtain

\begin{equation}
h \leq \cC_2^{-2/3} \, N^{-1/3},
\end{equation}

where $\cC_2 = \Nrm{f_{N:1}}{L^2(\RRd)}$ converges to $\Nrm{f}{L^2(\RRd)}$, and so remains of order $1$. Hence, we are mainly interested in the case when $N\,h^3$ is bounded above by a constant independent of $N$ and $h$. This corresponds to the part below the blue line in Figure <ref>.

Observe our analysis still makes sense if $N\,h^3 \to \infty$. However, in this situation, even though the solution to the $N$-body Schrödinger equation and the solution to the Hartree–Fock equation are close, they will not converge in the semiclassical limit to a nontrivial solution of the Vlasov equation, but to zero.

§ FUNCTIONAL SPACES

§.§ Semiclassical Spaces

Since we want to look at the convergence in the semiclassical limit $\hbar\to 0$ towards probability distributions of the phase space, we define the semiclassical versions of the Lebesgue norms of the phase space as the following rescaled Schatten norms

\begin{align}\label{eq:def_norm}
\Nrm{\op}{\L^p} = h^{\frac{3}{p}} \Nrm{\op}{p} = h^{\frac{3}{p}} \Tr{\n{\op}^p}^\frac{1}{p}.
\end{align}

More generally, given some weight $m$, we define the weighted spaces by the norm $\Nrm{\op}{\L^p(m)} = \Nrm{\op\,m}{\L^p}$. To see this definition is compatible with the semiclassical limit, notice that for any operator $\op\geq 0$ verifying the scaling assumptions (<ref>), one obtains

\begin{align}
\Nrm{\op}{\L^1} &= 1, & \Nrm{\op}{\L^2} &= \Nrm{f_{\op}}{L^2(\RRd)},
\Nrm{\op}{\L^\infty} &= \cC_\infty.
\end{align}

One good property of the norm defined by (<ref>) is that it is compatible with taking powers of the operator, in the sense that for any $c>0$, $\Nrm{\op^c}{\L^p} = \Nrm{\op}{\L^{pc}}^c$. In particular, in the rest of the paper we will often work with the operator $\sqrt{\op}$, which verifies, as one would expect, $\Nrm{\sqrt{\op}}{\L^2}= 1$ and $\Nrm{\sqrt{\op}}{\L^\infty} = \sqrt{\cC_\infty}$.

The fact that these norms are good analogues of the classical Lebesgue norms can be better understood in light of particular examples. One class of examples  is when the density operator has the form $f(x)\,g(\opp)$, where $\opp = -i\hbar\nabla$ is the momentum operator. Then the Kato-Seiler-Simon inequality <cit.> reads

\begin{equation}\label{eq:Kato_Seiler_Simon}
\Nrm{f(x)\,g(\opp)}{\L^p} \leq \Nrm{f}{L^p}\Nrm{g}{L^p} \ \text{ if } p\in[2,\infty),
\end{equation}

with equality when $p=2$, and where $L^p = L^p(\Rd)$. It is the analogue of the function identity $\Nrm{f(x)\,g(\xi)}{L^p_{x,\xi}} = \Nrm{f}{L^p}\Nrm{g}{L^p}$. Another class of examples is the class of Töplitz operators, namely when $\op$ is an averaging of coherent states (see <cit.> and <cit.>). If $\op$ is such a state, then its Wigner transform verifies

\begin{equation*}
\Nrm{f_{\op}}{L^p(\RRd)} \leq \Nrm{\op}{\L^p} \ \text{ if } p\in[2,\infty]
\end{equation*}

with equality when $p=2$. More generally, given such states, for any convex function $\Phi$ such that $\Phi(0) = 0$, we have the inequality

\begin{equation*}
\iintd \Phi(f_{\op}) \dd x\dd\xi \leq h^3 \Tr{\Phi\!\(\op\)}.
\end{equation*}

We also want to consider the semiclassical version of Sobolev spaces of the phase space. Thus, as in [47], we introduce the following operators

\begin{equation}
\Dhx \op := \com{\nabla,\op} \quad \text{ and } \quad
\Dhv \op := \com{\frac{x}{i\hbar},\op},
\end{equation}

which can be seen as an application of the correspondence principle of quantum mechanics. More precisely, one can observe that these operators correspond to the gradients of the Wigner transform, since

\begin{equation}
f_{\Dhx \op} = \Dx f_{\op}\quad \text{ and } \quad f_{\Dhv \op} = \Dv f_{\op}.
\end{equation}

In the rest of the paper, we will refer to $\Dhx\op$ and $\Dhv\op$ as the quantum gradients or first-order quantum gradients.

We can then define semiclassical analogues of the weighted kinetic homogeneous Sobolev norms by

\begin{align}
\Nrm{\op}{\dot{\cW}^{1,p}(m)}^p &:= \sum_{\jj=1}^3 \Nrm{\Dhvj{\op}}{\L^p(m)}^p + \Nrm{\Dhxj{\op}}{\L^p(m)}^p,
\\
\Nrm{\op}{\dot{\cW}^{1,\infty}(m)} &:= \sup_{\jj\in \{1,2,3\}} \( \Nrm{\Dhvj{\op}}{\L^\infty(m)}, \Nrm{\Dhxj{\op}}{\L^\infty(m)}\),
\end{align}

and consider the particular case of the weight defined for $n\in \N$ by

\begin{align}\label{eq:weight}
m := 1 + \n{\opp}^n.
\end{align}

where $\opp = -i\hbar\nabla$ so $\n{\opp}^2 = -\hbar^2\Delta$. We also define the inhomogeneous version by

\begin{align}
\Nrm{\op}{\cW^{1,p}(m)}^p &:= \Nrm{\op}{\L^p(m)}^p + \Nrm{\op}{\dot{\cW}^{1,p}(m)}^p,
\end{align}

with the usual modification when $p=\infty$. In particular, for $p=2$, $\Nrm{\op}{\cW^{1,2}} = \Nrm{f_{\op}}{H^1(\RRd)}$.

§.§ Fermionic Fock Space

Let $\h^{\wedge N} := \h\wedge \cdots \wedge \h$ denotes the $n$-fold anti-symmetric tensor product of $\h$. We define the fermionic (anti-symmetric) Fock space over $\h$ to be the closure of

\begin{align}
\cF(\h)= \cF := \CC\oplus \bigoplus_{n=1}^\infty \h^{\wedge n}
\end{align}

with respect to the norm induced by the endowed inner product

\begin{align}
\Inprod{\psi}{\varphi}_{\cF} = \conj{\psi^{(0)}}\, \varphi^{(0)} + \sum_{n\geq 1} \int_{\R^{3n}} \conj{\psi^{(n)}(\underline{x}_n)}\,\varphi^{(n)}(\underline{x}_n) \dd x_1\cdots \dd x_n,
\end{align}

for any pair of vectors $\psi = (\psi^{(0)}, \psi^{(1)}, \ldots)$ and $\varphi = (\varphi^{(0)}, \varphi^{(1)}, \ldots)$ in $\cF$ where $\underline{x}_k = (x_1,\ldots, x_k)\in \RR^{3k}$. For simplicity of notation, we will also denote the closure by $\cF$. The state with no particles describes the vacuum, defined by the vector $\Omega_{\cF} = (1, 0, \ldots)\in\cF$.
We define the number of particles operator by

\begin{equation}\label{eq:def_number_operator}
\cN \psi = \(n\,\psi^{(n)}\)_{n\in\N}
\end{equation}

whose meaning can be interpreted as counting the number of particles in each sector of $\cF$. A class of operators on $\cF$ that is important to our studies is the class of mixed states on $\cF$, which are high rank density matrices on $\cF$. More specifically, we are interested in operators of the form

\begin{equation}\label{eq:diagonalisation_op_N}
\opN := \sum_{\jj\in\N} \lambda_\jj \ket{\psi_\jj}\bra{\psi_\jj},
\end{equation}

for some orthonormal set $\psi_\jj$ of vectors of $\cF$ with the normalization

\begin{align}
\Tr{\opN} = \sum_\jj \lambda_\jj = h^{-3} \ \ \text{ and } \ \ h^3 \Tr{\cN\,\opN} = N.
\end{align}

Here, $N$ is the mean number of particles. Moreover, for each $(n,m)\in\N^2$, the integral kernel of $\opN^{(n,m)}$ has the form

\begin{equation}
\opN^{(n,m)}(\underline{x}_{n},\underline{y}_{m}) = \sum_{\jj\in\N} \lambda_\jj \,\psi_\jj^{(n)}(\underline{x}_{n})\,\conj{\psi_\jj^{(m)}(\underline{y}_{m})}.
\end{equation}

As in the case of the one-particle operator, we define the semiclassical Schatten norms by

\begin{align}
\Nrm{\opN}{\L^p(\cF)} := h^\frac{3}{p} \Tr{\n{\opN}^p}^\frac{1}{p},
\end{align}

so that $\Nrm{\opN}{\L^1(\cF)} = 1$ and $\Nrm{\cN\opN}{\L^1(\cF)} = N$. We also define the one-particle reduced density matrix, i.e. the analogue of the classical first marginal, by

\begin{equation*}
\opNr := \sum_{n\in\N} \frac{n}{N}\,\tr_{2..n}\!\(\opN^{(n,n)}\),
\end{equation*}

where $\tr_{2..n}$ indicates the partial trace with respect to all variables except the first.

§ MAIN RESULTS

§.§ Propagation of Regularity

Our first result gives the local-in-time and uniform in $\hbar$  propagation of the regularity of the solution to the Hartree–Fock equation (<ref>). Remark that there are no constraints on the scaling here since we are only considering the mean-field equation. Moreover, this result also works in the case of the Coulomb potential.

We define

\begin{align}
\fb := \frac{3}{a+1}
\end{align}

which corresponds to the integrability of the force field since $\nabla K \in L^{\fb,\infty}$.

Let $a\in (0,1]$, $m=1+\n{\opp}^n$ with $n\in\N$ verifying $n\geq 3$ and $\op$ be a solution to the Hartree–Fock equation (<ref>) with initial condition $\op^\init\in \L^\infty(m)$ satisfying (<ref>) and such that

\begin{equation}\label{eq:regularity0_0}
\op^\init \in \cW^{1,2}(m)\cap \cW^{1,4}(m).
\end{equation}

Then there exists $T>0$ such that

\begin{align}
\op \in L^\infty([0,T], \cW^{1,2}(m)\cap \cW^{1,4}(m)),
\end{align}

uniformly in $\hbar$.

Define $\varphi(y) = e^{-\pi\n{x}^2/2}$ and $\varphi_{x,\xi}(y) := \frac{1}{h^{9/4}} \varphi\!\(\frac{y-x}{\sqrt{h}}\) e^{i\,y\cdot\xi/\hbar}$. Then one can define an approximation of the Dirac delta on the phase space by $\op_{x,\xi} := \ket{\varphi_{x,\xi}}\bra{\varphi_{x,\xi}}$. Now for any $g : \RRd \to \R$ such that $g \in W^{1,\infty}(1+\n{\xi}^n) \cap W^{1,2}(1+\n{\xi}^n) \cap L^2$, one can define the averaging of coherent states (also called Töplitz operator) as the operator

\begin{align*}
\OP_\varphi(g) := \iintd g(x,\xi)\,\op_{x,\xi} \dd x\dd \xi.
\end{align*}

This defines a positive compact operator such that (see e.g. <cit.>)

\begin{align*}
\Nrm{\OP_\varphi(g)}{\L^\infty} &= \Nrm{g}{L^\infty(\RRd)},
\Nrm{\OP_\varphi(g)}{\L^2} &= \Nrm{g}{L^2(\RRd)}.
\end{align*}

In particular, in Theorem <ref>, we can take $\op^\init = \OP_\varphi(g)$ with $\Nrm{g}{L^2(\RRd)} = 1$ and $\Nrm{g}{L^\infty(\RRd)} = \cC_\infty^{1/2}$, and then $\op^\init$ verifies (<ref>).

However, we can take more general operators than averaging of coherent states. Taking a function $g$ on the phase space, one can indeed associate to it an operator by doing the inverse of the Wigner transform, called the Weyl transform, and defined as the operator with integral kernel

\begin{equation}
\op_g(x,y) = \intd e^{-2i\pi \(y-x\)\cdot\xi}\, g(\tfrac{x+y}{2},h\xi) \dd \xi.
\end{equation}

This operator will verify the hypotheses of the initial condition of Theorem <ref> if $g$ is sufficiently smooth and decaying at infinity, as proved for example in <cit.>.

§.§ Mean-Field and Semiclassical Limits

To state our mean-field results, we assume

\begin{equation}\label{eq:scaling}
N^{-\frac{1}{2}} \ll h < \cC_\infty N^{-\frac{1}{3}},
\end{equation}

where $a\ll b$ means that $\frac{a}{b}\to 0$ as $N\rightarrow \infty$. Observe that the choice $h=O(N^{-\frac{1}{3}})$ is admissible with a different constant in front. We define the following trace class norm over the Fock space weighted by the number operator

\begin{equation}
\Nrm{\opN}{\L^1_k(\cF)} := \Nrm{\(\cN+N\)^k\opN}{\L^1(\cF)}.
\end{equation}

In what follows, for technical reasons related to the well-posedness given in Appendix <ref>, we will assume that the initial quantum spatial distribution of particles (<ref>) verifies

\begin{equation*}
\intd \rho^\init(x) \weight{x}^3\d x \leq C,
\end{equation*}

where $C$ may depend on $h$.

Let $a\in(0,\frac{1}{2})$ and assume Condition (<ref>) is verified. Let $n\in\N$ verifying $n > 3$. Let $\op$ be a solution to the Hartree–Fock equation (<ref>) with initial condition $\op^\init\in \L^\infty(m)$ verifying (<ref>) and such that

\begin{align}\label{eq:regularity_op}
\op^\init &\in \cW^{2,2}(m)\cap \cW^{2,4}(m)
\\\label{eq:regularity_sqrt}
\sqrt{\op^\init} &\in \cW^{1,2}(m)\cap \cW^{1,q}(m)
\end{align}

with $q\in [\tfrac{6}{1-2a},\infty]$. Then there exists $T>0$, $\opNop^\init \in\L^1(\cF)$, $\lambda>0$ and $C > 0$ such that for any $\opN$ solution of (<ref>) with initial condition $\opN^\init\in\L^1(\cF)$ commuting with $\cN$, for any $t\in[0,T]$

\begin{align*}
\Nrm{\opNr - \op}{\L^p} \leq \frac{C\,e^{\lambda\,t}}{\min(N^{1/2},N\,h^{3/p'})} \(1+ \Nrm{\opN^\init - \opNop^\init}{\L^1_k(\cF)}\)
\end{align*}

for any $k \geq \frac{1}{2\,p} + \frac{3}{2}\lceil\frac{\ln N}{p\ln\(N h^2\)}\rceil$.

The $N$-body operator $\opNop^\init$ is explicitly created from $\op^\init$ thanks to the Bogoliubov transformation (see Equation (<ref>) in Section <ref>). One should note that $\opNop^\init$ is constructed so that its one-particle reduced density matrix coincides with the solution $\op$ to the Hartree–Fock equation.

In the particular case $h = N^{-\frac{1}{3}}$, $\Nrm{\opN^\init-\opNop^\init}{\L^1(\cF)} \leq C N^{-4}$ and $\Nrm{\cN^4\(\opN-\opNop\)}{\L^1(\cF)} \leq C$, then for any $t\in[0,T]$, one obtains

\begin{align*}
\Nrm{f_{N:1} - f_{\op}}{L^2} = \Nrm{\opNr - \op}{\L^2} \leq \frac{C_T}{N^{1/2}}.
\end{align*}

where we denote by $f_{N:1}$ the Wigner transform of $\opNr$.

One can combine the above theorem with the result proved in [47] by two of the authors to obtain an estimate directly between the solution of the $N$-body Schrödinger equation (<ref>) and the Vlasov equation (<ref>). To simplify, we restrict our attention to the case when $p\leq 2$.

Assume the conditions of Theorem <ref> and that $f$ is a positive solution of the Vlasov equation (<ref>) with initial condition satisfying

\begin{equation*}
\(1+\n{x}^8 +\n{\xi}^8\) \nabla_x^{\ell_0}\nabla_{\xi}^\ell f^\init \in L^\infty(\RRd)\cap L^2(\RRd) \ \text{ where } \ \ell_0+\ell \le 9.
\end{equation*}

Moreover, assume $\opN^\init \in \cL^1(\cF)$ is such that $[\mathcal{N},\opN^\init]=0$. Then, for any $p\in[1,2]$, there exists $T>0, C_T>0$ and an operator $\opNf^\init\in\L^1(\cF)$ such that for any solution $\opN$  to (<ref>) with initial condition $\opN^\init$, the estimate

\begin{align*}
\Nrm{\opNr - \op_f}{\L^p} \leq C_T\(\frac{1}{N\,h^{3/p'}}+h\) \(1 + \Nrm{\opN^\init - \opNf^\init}{\L^1_k(\cF)}\),
\end{align*}

holds for any $t\in[0,T]$ and any $k \geq \frac{1}{2\,p} + \frac{3}{2}\lceil\frac{\ln N}{p\ln\(N h^2\)}\rceil$.

In particular, when $\Nrm{\opN^\init - \opNf}{\L^1_k(\cF)} \leq C$ and $p=2$, then, by Identity (<ref>), we obtain again a $L^2$ convergence result with the quantitative bound

\begin{align*}
\Nrm{f_{N:1} - f}{L^2(\RRd)} \leq C_T\(\frac{1}{N\,h^{3/2}}+h\),
\end{align*}

where $f_{N:1}$ is the Wigner transform of $\opNr$.

It is interesting to notice that the semiclassical error $h$ is larger than the mean-field error when $N \gg h^{-5/2}$, and smaller when $N \ll h^{-5/2}$. When the two are of the same order, one obtains an error of order $h = N^{-2/5}$, which is the best of the rates in term of the number of particles, while the rate is of order $h=N^{-1/3}$ in the critical scaling.

In the particular case of the Coulomb potential, we can still obtain an estimate for small times or with a $N$ dependent cutoff (see Theorem <ref> and Remark <ref>). Our results are summarized in the following table.

Potential     $a\in(0,1/2)$     $a\in [1/2,1]$

Regularity     $t<T$     $t<T$

Mean-field     $t<T$     $t\ll h^{a-1/2}$ or cutoff

Mean-field + semiclassical     $t<T$     $t\ll h^{a-1/2}$ or cutoff

§ THE STRATEGY AND THE GENERAL RESULT

§.§ Second Quantization

The method of second quantization provides a mathematical framework for studying the notion of quantum fluctuations. The goal of this section is to recast the original Cauchy problem (<ref>) with a mixed state initial data on $\h^{\wedge N}$ to a problem on the Fock space $\cF$. We briefly present the method of second quantization and state the corresponding Hamiltonian evolution problem on $\cF$. We refer the interested reader to [15, 62, 31, 24, 6] for a more complete presentation.

For every $f \in \h$, we define the associated creation operator $a^\ast(f)$ and its adjoint the annihilation operator $a(f)$ on $\cF$ by their actions on the $n$-sector of $\cF$ as follows

\begin{align*}
(a^*(f)\,\psi)^{(n)}(\underline{x}_n) &:= \frac{1}{\sqrt{n}}\sum^n_{j=1}(-1)^{j-1}f(x_j)\, \psi^{(n-1)}(\underline{x}_{n\backslash\, j})
\\
(a(f)\,\psi)^{(n)}(\underline{x}_n) &:= \sqrt{n+1}\intd \conj{f(x)} \, \psi^{(n+1)}(x, \underline{x}_n) \dd x,
\end{align*}

where $ \underline{x}_{n\backslash\, j}:= (x_1, \ldots, \cancel{x}_j, \ldots, x_n)$. Moreover, the action of the annihilation operator on the vacuum of $\cF$ is $a(f)\,\Omega_{\cF} = 0$. Then, we extend the operators linearly to the whole $\cF$. It can easily be checked that the collection of creation and annihilation operators on $\cF$ satisfies the canonical anti-commutation relations (CAR) making it a CAR algebra:

\begin{align}\label{CAR}
\com{a(f), a^*(g)}_+ = \inprod{f}{g}_\h, \ \ \com{a(f), a(g)}_+ = \com{a^*(f), a^*(g)}_+ = 0
\end{align}

for all $f, g \in \h$ where $\com{A, B}_+ = AB + BA$ is the anti-commutator of the operators $A, B$. Moreover, from relation (<ref>), we have the identity

\begin{align}
\Nrm{a(f)\psi}{\cF}^2+\Nrm{a^*(f)\psi}{\cF}^2 = \Nrm{f}{\h}^2\Nrm{\psi}{\cF}^2 \ \ \implies \ \ \Nrm{a^\sharp(f)}{\infty}= \Nrm{f}{\h}
\end{align}

for all $f \in \h$ where $a^\sharp$ is either $a^*$ or $a$. Thus, both the creation and annihilation operators are bounded operators on $\cF$.

At times, it is more convenient to deal with localized object as opposed to $a^*(f)$ and $a(f)$. Thus, it is useful to introduce, at least formally, the fermionic creation and annihilation operator-valued distributions at the position $x$, denoted respectively by $a^*_x$ and $a_x$, as follows

\begin{align}
(a_x^* \psi)^{(n)}(\underline{x}_n) &= \frac{1}{\sqrt{n}}\,\sum_{j=1}^n \(-1\)^{j-1}\delta(x-x_j)\,\psi^{(n-1)}(\underline{x}_{n\backslash j}),\\
(a_x \psi)^{(n)}(\underline{x}_n) &= \sqrt{n+1}\, \psi^{(n+1)}(x,\underline{x}_n).
\end{align}

It is also straightforward to check that $a^*_x$ and $a_x$ satisfy the anticommutation relations:

\begin{align}
\com{a_x,a_y^*}_+ = \delta(x-y),	&\qquad \com{a_x,a_y}_+ = \com{a_x^*,a_y^*}_+ = 0,
\end{align}

and that the creation and annihilation operators can be rewritten as follows

\begin{align}\label{eq:op-val-distr}
a^*(f) = \intd f(x)\, a_x^* \dd x, &\qquad a(f) = \intd \conj{f(x)}\,a_x\dd x.
\end{align}

To every observable $O$ on $\h$ corresponds an induced linear operator $\dG(O):\cF\rightarrow \cF$ called the second quantization of $O$ on $\cF$, defined as

\begin{align}
\DG{O} = 0\oplus\bigoplus_{n=1}^\infty \dG_n(O) &\qquad
\end{align}

where $\dG_n(O)$ is the $N$-particle operator

\begin{align}
\dG_n(O) = \sum_{j=1}^n 1_{\h^{j-1}}\otimes O \otimes 1_{\h^{n-j}}.
\end{align}

An important example of second quantized operator is the number operator which is simply the second quantization of the identity operator. Another relevant class of operators are the trace class operators. It is straightforward to check that the second quantization of trace class operators on $\h$ are also trace class operators on $\cF$, that is, the second quantization of mixed states on $\h$ are mixed states on $\cF$.

If the observable $O$ has a distributional kernel $O(x, y)$, then we could rewrite $\dG(O)$ in terms of the operator-valued distributions $a^*_x$ and $a_x$ as follows

\begin{align}
\dG(O) = \intdd O(x, y)\,a_x^*\, a_y \dd x\dd y.
\end{align}

In particular, the number operator can be rewritten as

\begin{align}
\cN = \intd a_x^*\, a_x \dd x.
\end{align}

§.§ State Purification and Time Evolution

We define the Fock space Hamiltonian by

\begin{align}\label{eq:Fock_Hamiltonian}
\sfH_N = \intd a_x^* \(-\hbar^2\lapl_x\) a_x \dd x+ \frac{1}{2N}\intdd K(x-y)\,a^*_x \,a^*_y \,a_y\, a_x\dd x\dd y.
\end{align}

By direct computation, we see that $\sfH_N$ commutes with the number operator, which implies that the expectation of the number of particles is conserved under the Hamiltonian dynamics.
Moreover, its action on the $n$-sector is given by

\begin{align}
(\sfH_N\psi)^{(n)}= \sfH_N^{(n)} =\sum^n_{k=1}-\frac{\hbar^2}{2}\lapl_{x_k}\psi^{(n)}+\frac{1}{N}\sum^n_{k<l} K(x_l-x_k)\,\psi^{(n)},
\end{align}

which, on the $N$-sector of $\cF$, coincides with the mean-field Hamiltonian defined in Equation (<ref>). We consider the Cauchy problem

\begin{align}\label{eq:density_matrix_Cauchy_prblm_on_Fock}
i\hbar\,\dpt \opN = \com{\sfH_N, \opN}, \ \text{ with }\ \opN(t=0) = \op_N^\init = \sum_\jj \lambda_\jj \ket{\psi_\jj}\!\bra{\psi_j}
\end{align}

where the data are defined as in (<ref>). Following the idea of [11], we reformulate problem (<ref>) as an evolution problem of a pure state[Here, we make the identification of $\ket{\Psi}\!\bra{\Psi}$ with $\Psi \in \cG$. In other words, pure state density matrices are simply vectors.] in the fermionic Fock space

\begin{align}
\cG := \cF(\h\oplus\h)
\end{align}

which hereinafter will be referred to as the double Fock space. This procedure is commonly known as purification of mixed states. For completeness, we devote the remainder of this section to review the state purification process.

For any operator $\opN$ as defined in (<ref>) and any orthonormal basis $\phi_{\jj}$ of $\cF$, we construct the following Hilbert–Schmidt operator on $\cF$

\begin{align}
\opupN := \sum_{\jj\in\N} \eps_\jj \ket{\psi_{\jj}}\bra{\phi_{\jj}},
\end{align}

where $\n{\eps_\jj}^2 = \lambda_\jj$ (cf. Equation (<ref>)). Then, we see that $\opN = \n{\opupN}^2$, which is called the Schmidt decomposition of $\opN$. In particular, its scaled Hilbert–Schmidt norm, defined by $\Nrm{\opup}{\L^2(\cF)}^2 = h^3 \Tr{\n{\opup}^2}$, is

\begin{equation}
\Nrm{\opupN}{\L^2(\cF)}^2 = \Nrm{\opN}{\L^1(\cF)} = 1.
\end{equation}

It is important to notice the decomposition is not unique. In fact, we will need to make a definite choice later.

Recall that the space of Hilbert–Schmidt operators $\L^2(\cF)$ is isomorphic to the tensor product $\cF(\h)\otimes\cF(\h)$, as Hilbert spaces, via the linear mapping $\sfJ_h=\sfJ$ that maps $\ket{\phi}\!\bra{\psi}\mapsto h^{-\frac{3}{2}}\phi\otimes\overline\psi$. One can then associate to $\opupN$ an element of $\cF(\h)\otimes\cF(\h)$ as follows

\begin{align}\label{eq:def_tensor_prod_vector}
\sfJ\opupN = h^{-\frac{3}{2}}\sum_{\jj\in\N} \eps_\jj \, \phi_{\jj}\otimes \conj{\psi_{\jj}}.
\end{align}

Furthermore, we can associate to every element (<ref>) a vector in the double Fock space $\cG$ via the isomorphism $\sfU : \cF\otimes\cF \to \cG$ defined by the following: for $f\in \h^{\wedge n}$ and $g \in \h^{\wedge m}$

\begin{equation}
\sfU\(f\otimes g\) = \sqrt{\frac{\(n+m\)!}{n!\,m!}} \(J_l^{\otimes n}f\)\otimes_a \(J_r^{\otimes m}g\),
\end{equation}

where $J_l, J_r:\h\rightarrow \h\oplus\h$ are respectively the canonical embedding of $\h$ into the left and the right coordinate of $\h\oplus\h$, and $\otimes_a$ is the antisymmetric tensor product. Then extend the mapping linearly to to the whole $\cF\otimes\cF$. The unitary map $\sfU$ is known as the exponential law for Fock spaces and it verifies the following properties (see <cit.> or <cit.>)

\begin{align}
\Omega_\cG &= \sfU(\Omega_{\cF} \otimes \Omega_{\cF})
\\
a_l^\sharp(f) := a^\sharp(f\oplus 0) &= \sfU \(a^\sharp(f)\otimes 1\) \sfU^*
\\
a_r^\sharp(f) := a^\sharp(0\oplus f) &= \sfU \((-1)^\cN \otimes a^\sharp(f)\) \sfU^*,
\end{align}

with $a^\sharp = a$ or $a^*$. The presence of the operator $(-1)^\cN$ ensures that the operators satisfy the CAR. It can also be readily checked that $a^\sharp_l(f)$ anti-commutes with $a^\sharp_r(g)$ for all $f, g \in \h$.

Just like in the case of $\cF$, it is useful to define the left and right creation and annilihation operator-valued distributions at the position $x$ by

\begin{align}
(a_{x, l}^*\Psi)^{(n, m)}(\underline x_n, \underline y_m) &:= \frac{1}{\sqrt{n}}\,\sum_{j=1}^n \(-1\)^{j-1}\delta(x-x_j)\,\Psi^{(n-1, m)}(\underline{x}_{n\backslash j}, \underline y_m),
\\
(a_{x, l}\Psi)^{(n, m)}(\underline x_n, \underline y_m) &:= \sqrt{n+1}\,\Psi^{(n+1, m)}(x, \underline x_n, \underline y_m),
\\
(a_{x, r}^*\Psi)^{(n, m)}(\underline x_n, \underline y_m) &:= \frac{1}{\sqrt{m}}\,\sum_{j=1}^m \(-1\)^{n+j-1}\delta(x-y_j)\,\Psi^{(n, m-1)}(\underline x_n, \underline{y}_{m\backslash j}),
\\
(a_{x, r}\Psi)^{(n, m)}(\underline x_n, \underline y_m) &:= \(-1\)^n\sqrt{m+1}\,\Psi^{(n, m+1)}(\underline x_n, x, \underline y_m).
\end{align}

This allows us to write

\begin{align}
a_\sigma(f) = \intd \overline{f(x)} \, a_{x, \sigma} \dd x \ \ \text{ and } \ \ a_\sigma^*(f) = \intd f(x) a^*_{x, \sigma} \dd x
\end{align}

where $\sigma \in \{l, r\}$. It is again straightforward to check the CAR relations: $\com{a_{x, \sigma}, a^*_{y, \sigma}}_+ = \delta(x-y)$ and $\com{a^\sharp_{x, \sigma}, a^\sharp_{y, \sigma'}}_+ = 0$ where $\sigma, \sigma' \in \{l, r\}$.

For every observable $O$ on $\h$, we can define the left or right induced linear operator $\DGl{O}, \DGr{O} : \cG\rightarrow \cG$

\begin{align}
\DGl{O} &:= \DG{O\oplus 0} = \sfU\(\dG(O)\otimes 1\)\sfU^* =\intdd O(x, y)\,a^*_{x, l}\, a_{y, l} \dd x\dd y, \\
\DGr{O} &:= \DG{0\oplus O} = \sfU\(1\otimes\dG(O)\)\sfU^* =\intdd O(x, y)\,a^*_{x, r}\, a_{y, r} \dd x\dd y.
\end{align}

The number operator on $\cG$ is defined by

\begin{align}
\cN = \cN_l + \cN_r = \sfU \(\cN\otimes 1 + 1\otimes\cN\) \sfU^*.
\end{align}

We will denote by

\begin{equation}\label{eq:def_I_G}
\sfI_\cG := \sfU\sfJ
\end{equation}

the transformation from $\L^2(\cF)$ to $\cG$ mapping density operators to vectors of the double Fock space. Then for an operator $\opupN \in \L^2(\cF)$, the action of the operator $\cN$ in $\cG$ becomes $\cN \,\sfI_\cG(\opupN) = \sfI_\cG(\cN\opupN + \opupN\cN)$.

With the above purification process, we can recast our Cauchy problem for mixed states to a Cauchy problem for states exhibiting in the double Fock space $\cG$ the structure of pure states. Recall the solution to the Cauchy problem (<ref>) in the Schrödinger picture is given by

\begin{align}
\opN = e^{-i(t/\hbar)\, \sfH_N} \opN^\init \, e^{i(t/\hbar)\, \sfH_N}.
\end{align}

We define the time evolution of $\opupN$ with initial state $\opupN^\init$ by

\begin{equation}
\opupN = e^{-i(t/\hbar)\, \sfH_N} \opupN^\init \, e^{i(t/\hbar)\, \sfH_N}.
\end{equation}

Then $\opN = \n{\opupN}^2$ solves Equation (<ref>) with initial data $\opN^\init = \n{\opupN^\init}^2$. In the double Fock space $\cG$, this corresponds to say that the evolution is given by $\Phi = \Phi(t)$ with

\begin{equation}
\Phi := \sfI_\cG(\opupN) = e^{-i(t/\hbar)\, \sfL_N} \sfI_\cG(\opupN^\init) = e^{-i(t/\hbar)\, \sfL_N}\Phi^\init
\end{equation}

where the Liouvillian $\sfL_N$ is defined by $\sfL_N = \sfU\(\sfH_N\otimes 1 - 1 \otimes \sfH_N\)\sfU^*$. In particular, for any observable $\mathsf{O}$ of $\cF$, we have the relation

\begin{align}\label{eq:mixed-state-pure-state-trace}
\tr_{\cF}\(\mathsf{O}\opN\) &= \Inprod{\Phi}{\(\mathsf{O}\otimes 1\)\Phi}_{\cG} = \tr_{\cG}\((\mathsf{O}\otimes 1)\ket{\Phi}\!\bra{\Phi}\).
\end{align}

which allows us to compute the mean value of the observable $\mathsf{O}$ with respect to the mixed state $\op_N$ in terms of the purified state $\Phi$. In particular, we could express the one-particle reduced density matrix of $\op_N$ in terms of $\Phi$, that is, the integral kernel of $\op_{N:1}$ is given by

\begin{align}
\op_{N:1}(x, y)=\frac{1}{N\,h^3}\Inprod{\Phi}{a_{x, l}^*\, a_{y, l}\,\Phi}.
\end{align}

Notice that we are using the normalization $\Tr{\opNr} = h^{-3}$.

§.§ Bogoliubov Transformation and Quasi-Free States

In general, we do not expect the evolution of the Cauchy problem (<ref>) to be well-approximated by its mean-field dynamics. In fact, for initial data that are far from equilibrium, the effects of quantum fluctuation about its mean-field dynamics are not expected to be small. Thus, it is natural to restrict our studies to a subclass of initial data that are in close proximity to equilibrium states. As explained in [11], equilibrium states at finite positive temperature are believed to be well-approximated by mixed quasi-free states. Furthermore, mixed quasi-free states has the important property that it can be represented by a Bogoliubov transformation acting on the vacuum of the double Fock space $\cG$ which is a key object in our study of the mean-field limit.

In this section, we give a brief overview of rudimentary facts about Bogoliubov transformation in the framework of the double Fock space $\cG$ and construct a class quasi-free states exhibiting the structure of pure states in $\cG$, with average number of particles $N$ and pairing density equal to zero. We follow closely to the presentation given in [69].

§.§.§ Bogoliubov Transformation
For the pairs $f=f_1\oplus f_2, g= g_1\oplus g_2 \in \h\oplus \h$, we define the corresponding field operators by

\begin{align}
A(f, g) &:= a(f)+a^*(\overline g)= a_l(f_1)+a_r(f_2)+a^*_l(\conj{g_1})+a^*_r(\conj{g_2})
\\
A^*(f, g) &:= (A(f, g))^* = a_l(\conj{g_1})+a_r(\conj{g_2})+a^*_l(f_1)+a^*_r(f_2).
\end{align}

Notice the field operator $A(f, g)$ and its adjoint satisfy the relation

\begin{align}\label{eq:field_op_conj_relation}
A^*(f, g) = A(C(f, g))
\end{align}

for all $f, g \in \h\oplus \h$ where $C:(\h\oplus\h)\,\oplus\,(\h\oplus\h)\to (\h\oplus\h)\,\oplus\,(\h\oplus\h)$ is the anti-linear map (antiunitary involution) defined by $C(f_1\oplus f_2,g_1\oplus g_2)=(\overline{g_1}\oplus \overline{g_2},\overline{f_1}\oplus \overline{f_2})$.
We can also readily check the collection of field operators satisfy the anticommutation relations:

\begin{align}\label{eq:field_op_anticommutation_relation}
\com{A(f, g), A^*(h, k)}_+ = \Inprod{(f, g)}{(h, k)}_{(\h\oplus \h)\oplus (\h\oplus \h)}, \ \com{A^\sharp(f, g), A^\sharp(h, k)}_+ = 0
\end{align}

where $A^\sharp = A$ or $A^*$ and $f, g, h, k \in \h\oplus \h$.

A linear isomorphism
$\nu:(\h\oplus\h)\,\oplus\,(\h\oplus\h)\,\rightarrow\,(\h\oplus\h)\,\oplus\,(\h\oplus\h)$ is called a Bogoliubov (canonical) transformation of $(\h\oplus\h)\,\oplus\,(\h\oplus\h)$ provided it preserves the anticommutation relations (<ref>), that is, we have that

\begin{align}\label{eq:field_op_anticommutation_relation_preserved}
[A(\nu(f, g)), A^*(\nu(h, k))]_+ = \Inprod{(f, g)}{(h, k)}_{(\h\oplus \h)\oplus (\h\oplus \h)}
\end{align}

for all $f, g, h, k \in \h\oplus \h$ and, likewise, for the other relations. Hence, it follows from (<ref>) and (<ref>) that $\nu$ is a Bogoliubov transformation provided it satisfies the conditions

\begin{equation}\label{eq:Bogolibov-properties}
\nu\, C = C\,\nu\,, \qquad \text{ and } \qquad \nu^*\nu= \nu \nu^* = I\,,
\end{equation}

where $I$ is the corresponding identity map.

It is more convenient to express conditions (<ref>) as follows: $\nu$ is a Bogoliubov transformation on $(\h\oplus\h)\,\oplus\,(\h\oplus\h)$ if there exist operators $U$, $V: \h\oplus\h\rightarrow \h\oplus\h$ satisfying the properties

\begin{equation}\label{eq:Bogoliubov-cond}
U^*U+V^*V=I\qquad \text{ and }\qquad U^*\overline{V}+V^*\overline{U}=0
\end{equation}

such that $\nu$ has the form

\begin{equation}\label{eq:Bogoliubov-expres}
\nu = \(\begin{array}{ll} U & \overline{V} \\ V & \overline{U} \end{array}\).
\end{equation}

Moreover, we say that the Bogoliubov transformation $\nu$ is (unitarily) implementable on $\cG$ if there exists a unitary map $\sfR_{\nu}:\cG\mapsto\cG$ such that

\begin{equation}\label{eq:implementation_of_nu}
\sfR^*_\nu\,A(f,g)\, \sfR_\nu = A(\nu(f,g))
\end{equation}

for all $f, g \in \h\oplus \h$. A necessary and sufficient condition for the transformation $\nu$ to be implementable is given by Shale and Stinespring in [67]: $\nu$ is implementable if and only if $V$ is a Hilbert–Schmidt operator. In particular, if $\Tr{V^*V}$ is finite, then $\nu$ is an implementable Bogoliubov transformation. It is common to refer to $\sfR_\nu$ as the Bogoliubov transformation on $\cG$.

§.§.§ Quasi-Free States
A state $\opN$ on $\cF$ is said to be quasi-free if

\begin{align}
&\tr_{\cF}\(a^{\sharp_1}(f_1)\cdots a^{\sharp_{2n+1}}(f_{2n+1}) \opN\) =0, \\
&\tr_{\cF}\(a^{\sharp_1}(f_1)\cdots a^{\sharp_{2n+1}}(f_{2n}) \opN\)\\
\nonumber &= \sum_{\pi} (-1)^\pi \prod_{j=1}^n\tr_{\cF}\(a^{\sharp_{\pi(j)}}(f_{\pi(j)})a^{\sharp_{\pi(j+n)}}(f_{\pi(j+n)}) \opN\),
\end{align}

where $f_i \in \h$ and the sum is over all permutations $\pi$ satisfying

\begin{align}
\pi(1)<\pi(2)<\ldots <\pi(n) \ \ \text{ and } \ \ \pi(j)<\pi(j+n), \ j=1,\ldots, n.
\end{align}

In short, a state is said to be quasi-free if the higher-order reduced density matrices of $\opN$ are completely determined by the generalized one-particle reduced density matrix. We could also express Conditions (<ref>) in terms of the purified state $\Phi$. This means that any quasi-free mixed state can be viewed as the partial trace of a quasi-free pure state. Moreover, using the fact that pure quasi-free states are completely characterized by its generalized one-particle reduced density matrix, it can be shown that a pure quasi-free state $\Phi$ on $\cG$ can be written as $\Phi = \sfR_{\nu}\Omega$ for some Bogoliubov transformation $\sfR_{\nu}$.

Let us now construct the Bogoliubov transformation and its corresponding class of quasi-free states that we will study in Part <ref> of the paper. Let $\omega$ be a one-particle density operator on $\h$ satisfying the properties: $0\leq \omega \leq 1$ and $\Tr{\omega}=N$. Define the map
$\nu : (\h\oplus\h)\oplus(\h\oplus\h)\mapsto(\h\oplus\h)\oplus(\h\oplus\h)$
given by Equation (<ref>) with $U$ and $V$ having the explicit forms

\begin{equation}
U = \(\begin{array}{ll}
u & 0 \\ 0 & \overline{u}
\end{array}\)
\quad\quad\mbox{and}\quad\quad
V =\(\begin{array}{ll}
0 & \overline{v} \\ -v & 0
\end{array}\)
\end{equation}

\begin{equation}
u := \sqrt{1-\omega} \quad\quad\mbox{and}\quad\quad v := \sqrt{\omega}\,.
\end{equation}

Notice $U$ and $V$ verify Equation (<ref>) which means $\nu$ is a Bogoliubov transformation. Furthermore, $V$ is a Hilbert–Schmidt operator. Indeed, since $\Tr{V^*V}=2\Tr{\omega} = 2\,N$ is clearly finite, then, by the Shale–Stinespring condition [67], $\nu$ is implementable.
Hence, there exists a unitary map $\sfR_{\nu}:\cG\mapsto\cG$ implementing $\nu$. Consequentially, Equation (<ref>) yields the relations

\begin{align}
\sfR^*_{\nu}a_{x, l}\sfR_{\nu} =&\ a_l(u_x)-a^*_r(\conj{v_x}),\\
\sfR^*_{\nu}a_{x, r}\sfR_{\nu} =&\ a_r(\conj{u_x})+a^*_l(v_x).
\end{align}

where we used the notation $u_x(y)= u(y, x), v_x(y)= v(y, x)$.

Let us now use the Bogoliubov transformation to represent quasi-free mixed states. The construction we present here is an example of the well-known Araki–Wyss representation, see [3, 2, 24]. More precisely, we are interested in constructing a quasi-free mixed state with one-particle reduced density $\op$ on the double Fock space $\cG$. To this end, we define $\sfR_{\op}$ as the Bogoliubov transform with $\omega = N\,h^3 \op$, and we let the unitary map $\sfR_{\op}$ act on the vacuum $\Omega_\cG$, i.e.

\begin{equation}\label{eq:quasi-free-mixed}
\Phi_{\op} := \sfR_{\op}\,\Omega_{\cG}\in\cG\,.
\end{equation}

We can now compute the integral kernel of one-particle reduced density matrix associated with
the state $\Phi_{\op}$:

\begin{equation*}
\begin{split}
\op_{N:1}(x,y) &= \frac{1}{N\,h^3} \Inprod{\Phi_{\op}}{a_{l,y}^* \, a_{l,x} \Phi_{\op}} = \frac{1}{N\,h^3} \Inprod{\Omega_\cG}{\sfR^*_{\op}a_{l,y}^*\sfR_{\op}\sfR^*_{\op}a_{l,x}\sfR_{\op}\Omega_{\cG}}
\\
&= \frac{1}{N\,h^3} \Inprod{\Omega_\cG}{a_{l}(\overline{v_y})\,a^*_r(\overline{v_x})\,\Omega_{\cG}} =\frac{1}{N\,h^3} \, (v^*v)(x,y)=\op(x,y).
\end{split}
\end{equation*}

Therefore, the one-particle reduced density matrix associated with $\Phi_{\op}$ corresponds to the operator $\op$. Furthermore, the off-diagonal term associated with the state $\Phi_{\op}$, referred to as pairing density, is zero. Indeed,

\begin{equation*}
\begin{split}
\alpha_{\Phi_{\op}}(x,y)&:= \Inprod{R_{\op}\Omega_\cG}{a_{l,y}\,a_{l,x}\, R_{\op}\Omega_{\cG}} = \Inprod{\Omega_\cG}{a_{l}(u_y)\,a_l(\conj{v_x})\,\Omega_{\cG}} = 0
\end{split}
\end{equation*}

where we used that $[a_l(u_y),a_r^*(\overline{v_x})]_+=0$. Undoing the purification process, we can now define the reference state (mean-field approximation) $\opNop$, associated to the solution $\op$ of the Hartree–Fock equation (<ref>), as stated in Theorem <ref> by

\begin{equation}\label{eq:reference_state}
\opNop = \n{\sfI_\cG^{-1}(\Phi_{\op})}^2.
\end{equation}

§.§ The General Result

In this section, we state our more general result from which our main results follow. Our result is obtained by controlling the growth of the weighted norm

\begin{align*}
\Nrm{\Psi}{\cG_k} := \Nrm{\(\cN+1\)^k\Psi}{\cG}.
\end{align*}

Let $a\in[0,1]$ and assume Condition (<ref>) is verified. Let $(k,n)\in\N^2$ and $\alpha\in \lt[0,1\rt]$ verifying $n > 3$ and $\alpha > a - \frac{1}{2}$. Let $\op$ be a solution of the Hartree–Fock equation (<ref>) with initial condition $\op^\init\in \L^\infty(m)$ satisfying (<ref>) and such that

\begin{align}\label{eq:regularity_op_2}
\op^\init &\in \cW^{2,2}(m)\cap \cW^{2,4}(m)
\\\label{eq:regularity_sqrt_2}
\sqrt{\op^\init} &\in \cW^{1,2}(m)\cap \cW^{1,q}(m)
\end{align}

with $q\in [2,\infty]$ verifying

\begin{equation}\label{eq:scaling_q_2}
\frac{3}{q} \in \lt[2\(\alpha - a - \frac{1}{4}\), \alpha - a + \frac{1}{2}\rt].
\end{equation}

Let $\Psi^\init\in \cG$. Then, there exists $T>0$ and $C>0$ such that for any $t\in[0,T]$ and any $p\in[1,\infty)$

\begin{equation*}
\Nrm{\op_{N:1}-\op}{\L^p} \leq \frac{C\,e^{\lambda\,h^{-\alpha}\,t}}{\min(N^\frac{1}{2}, N\,h^\frac{3}{p'})} \(\Nrm{\Psi^\init}{\cG_{\frac{3}{2}k+\frac{1}{2\,p}}}^2 + \frac{h^{2 k \(\alpha-1\)}}{N^{k-\frac{1}{p}}} t^2 \Nrm{\Psi^\init}{\cG_{\frac{3}{2}k}}^2\).
\end{equation*}

where $\lambda = C_{a,\alpha}\n{\kappa} C_{\op}$ for some constant $C_{\op}$ depending only on $T$ and the initial condition of the Hartree–Fock equation.

In the above theorem, we assumed we know the perturbation of the vacuum, $\Psi^\init$. As done in (<ref>) for the reference state $\opNop$, we can associate to $\Psi^\init$ an operator $\opN = \n{\sfI_\cG^{-1}(\sfR_{\op}\Psi)}^2$ which solves the Schrödinger equation (<ref>).

In particular, notice that

\begin{equation*}
\frac{h^{2 k\(\alpha-1\)}}{N^{k-\frac{1}{p}}} \leq 1 \iff k \geq \frac{\ln N}{p\,\ln\(Nh^{2\(1-\alpha\)}\)}.
\end{equation*}

More specifically, if $N = h^{-c}$, then this is equivalent to $k\geq \frac{c}{p\(c+2 (\alpha-1)\)}$. For instance, take $c=3$. Then for any $a<1/2$, we can take $\alpha = 0$ and $k = 3$, leading to

\begin{equation*}
\Nrm{\op_{N:1}-\op}{\L^p} \leq \frac{C\,e^{\lambda\,t}}{N^{\min(\frac{1}{2},\frac{1}{p})}} \Nrm{\Psi}{\cG_{5}}^2.
\end{equation*}

In the case of the Coulomb potential $a=1$, we can take $k = 2$ and any $\alpha > 1/2$, leading to

\begin{equation*}
\Nrm{\op_{N:1}-\op}{\L^p} \leq \frac{C}{N^{\min(\frac{1}{2},\frac{1}{p})}} \Nrm{\Psi}{\cG_4}^2 e^{\lambda\,t/h^\alpha},
\end{equation*}

which is small only for small times $t\ll N^{-1/6} = h^{1/2}$. This is an improvement in comparison to non semiclassical estimates which are valid only for $t\ll h$.

When $a\geq 1/2$, one can also consider the potential with a $h$-dependent cut-off. For example, a way to get a potential bounded at distance $\n{x} \leq R$ is to take

\begin{equation}\label{eq:cutoff}
K_R(x) = \frac{\omega_a\,\kappa}{2} \int_0^{R^{-2}} s^{\frac{a}{2}-1} e^{-\pi\n{x}^2s}\dd s \,\underset{R\to 0}{\longrightarrow}\, \kappa \, \frac{1}{\n{x}^a},
\end{equation}

which is a radial decreasing potential verifying $K_R(x) \leq \n{\kappa}\max\!\(\frac{1}{\n{x}^a}, \frac{\omega_a}{a R^a}\)$. For the Coulomb interaction potential for example, assuming $R\leq 1$ and $N=h^{-c}$ and taking $c=3$ and $p\leq 2$, this leads to

\begin{align*}
\Nrm{\op_{N:1}-\op}{\L^p} \leq \frac{C\,e^{\lambda\,t/\sqrt{R}}}{N^{1/2}} \Nrm{\Psi}{\cG_{ 5}}^2.
\end{align*}

Thus, one obtains a quantitative convergence result as soon as $R > \frac{4 \lambda^2 t^2}{\(\ln N\)^2}$.

Let $\opNop$ be defined by (<ref>). Then the standard deviation of the number of particles $\sigma_\cN^2 := h^3\Tr{\cN_l^2\,\opNop} - \(h^3\Tr{\cN_l\,\opNop}\)^2$ is given by

\begin{equation*}
\sigma_\cN^2 = \Tr{\omega-\omega^2} = N\(1-\cC_2^2 Nh^3\).
\end{equation*}

In particular, $\sigma_\cN \leq \sqrt{N}$.

Notice also that $\sigma_\cN = 0 \iff \omega = \omega^2 \iff \cC_2^2Nh^3 = 1$. This implies that in order for the reference state $\opNop$ to have a fixed number of particles, it has to be a pure state and the scaling has to be the critical scaling $Nh^3 = \cC_2^{-2}$. In this case, the regularity conditions (<ref>) are not expected to hold. However, it is a good question to know whether it is possible to find a state $\opN = \n{\sfI_\cG^{-1}(\sfR_{\op}\Psi)}^2$ with a fixed number of particles but still closed to $\opNop$, in the sense that the associated $\Psi$ verifies $\Nrm{\Psi}{\cG_5} \ll N^\frac{1}{2}$.

Propagation of Regularity

This part is devoted to the proof of Theorem <ref> about the propagation of the semiclassical regularity of the solutions of the Hartree–Fock equation (<ref>), and also of higher regularity properties needed to obtain Theorem <ref>.

§ THE CLASSICAL CASE: REGULARITY FOR THE VLASOV EQUATION

As a warm-up and an explanation of our strategy, we start by the analogue of our method in the classical case of the kinetic Vlasov equation. We define

\begin{align*}
N_{p,x} := \iintd \n{\Dx f}^p m
&\text{ and } N_{p,\xi} := \iintd \n{\Dv f}^p m.
\end{align*}

Denoting $\sfT:= \xi\cdot\Dx + E\cdot\Dv$, then we have

\begin{align}\label{eq:time-der-grad}
\partial_t(\Dx f) &= -\sfT \Dx f - \nabla E \cdot \Dv f
\\
\partial_t(\Dv f) &= -\sfT\Dv f - \Dx f.
\end{align}

Let $n>3$ and $f$ be a solution of Equation (<ref>)  with initial condition verifying

\begin{equation*}
\nabla_{x,\xi}f^\init \in L^p(1 + \n{\xi}^n)
\end{equation*}

for any $p\in [1,\infty)$. Then there exists a time $T>0$ such that

\begin{align*}
\nabla_{x,\xi}f \in L^\infty((0,T),L^p(1 + \n{\xi}^n)).
\end{align*}

Let $m := 1+\n{\xi}^{np}$. To simplify the computations, we remark that $\sfT^* = -\sfT$ and $\sfT(uv) = u\,\sfT(v) + \sfT(u)\,v$, so that by writing $u^p := \n{u}^{p-1}u$, it holds

\begin{align*}
\iintd \sfT(u)\cdot u^{p-1}m &= -\iintd u\cdot\sfT(u^{p-1})\,m + \n{u}^p\sfT(m).
\end{align*}

\begin{align*}
u\cdot(\sfT(u^{p-1})) &= u^{p-1}\cdot\sfT(u) + (p-2)\(\sfT(u)\cdot u\)\n{u}^{p-2}
\\
&= \(p-1\)u^{p-1}\cdot\sfT(u).
\end{align*}

We deduce

\begin{align*}
-p\iintd \sfT(u)\cdot u^{p-1}m &= \iintd \n{u}^p\sfT(m).
\end{align*}

Therefore, differentiating the weighted $L^p$ norms, we obtain

\begin{align*}
\ddt{N_{p,x}} &= \iintd \n{\Dx f}^p \sfT(m) - p\(\Dx f\)^{p-1}\cdot\nabla E\cdot\Dv f \,m\dd x\dd\xi
\\
\ddt{N_{p,\xi}} &= \iintd \n{\Dv f}^p \sfT(m) - p\(\Dv f\)^{p-1}\cdot\Dx f \,m\dd x\dd\xi.
\end{align*}

Then by Young's inequality for the product

\begin{align*}
\sfT(m) = n\,p\, E\cdot \xi^{np-1} \leq np\Nrm{E}{L^\infty} m.
\end{align*}

We cut $\nabla K = F_1+F_2\in L^{\fb_1} + L^{\fb_2}$. The difficult term is

\begin{align*}
\iintd (\Dx f)^{p-1}\cdot\nabla E\cdot\Dv f\,m\dd x\dd \xi &\leq \Nrm{\nabla K * \nabla\rho}{L^{\infty}} \iintd \n{\Dx f}^{p-1}\n{\Dv f}m\dd x\dd \xi
\\
&\leq \(\Nrm{F_1}{L^{\fb_1}} \Nrm{\nabla\rho}{L^{{\fb_1'}}}+\Nrm{F_2}{L^{\fb_2}} \Nrm{\nabla\rho}{L^{{\fb_2'}}}\) N_{p,x}^{1-1/p} N_{p,\xi}^{1/p}
\\
&\leq B_K \(\Nrm{\nabla\rho}{L^{{\fb_1'}}} + \Nrm{\nabla\rho}{L^{{\fb_2'}}}\) N_{p,x}^{1-1/p} N_{p,\xi}^{1/p},
\end{align*}

with $B_K = \Nrm{\nabla K}{L^{\fb_1}+L^{\fb_2}}$ and where we used three times Hölder's inequality. Then, when $n\geq 3/\fb$, we get

\begin{align*}
\Nrm{\nabla\rho}{L^{\fb'}} = \Nrm{\intd\Dx f\dd \xi}{L^{\fb'}} &\leq \Nrm{\Dx f \weight{\xi}^{n}}{L^{\fb'}_{x,\xi}} \leq C\,N_{\fb',x}^{1/\fb'}.
\end{align*}

Therefore, taking respectively $p=\fb_1'$ and $p = \fb_2'$ and using the notation $u_k = u_k(t) := N_{\fb_k',x}^\frac{1}{\fb_k'}$ and $v_k = v_k(t) := N_{\fb_k',\xi}^\frac{1}{\fb_k'}$, for $k=1$ and $k=2$, it holds

\begin{align*}
\dt u_k &\leq n \Nrm{E}{L^\infty} u_k + B_K\(u_1+u_2\) v_k
\\
\dt v_k &\leq n \Nrm{E}{L^\infty} v_k + u_k v_k,
\end{align*}

so that defining $U := u_1 + u_2 + v_1 + v_2$ we get

\begin{align}\label{eq:U-gronwall}
\dt U \leq n \Nrm{E}{L^\infty} U + \(B_K+\frac{1}{2}\) U^2,
\end{align}

where we used the fact $2uv\leq u^2+v^2$, and we obtain by Grönwall's inequality that $U$ remains finite as long as $t<T$ where $T$ depends on the growth of $\Nrm{E(t,\cdot)}{L^\infty}$, which we can control by

\begin{align*}
\Nrm{E}{L^\infty} \leq C_K \(\Nrm{\nabla\rho}{L^{c_1'}} +\Nrm{\nabla\rho}{L^{c_2'}}\) \leq C_K \(N_{c_1',x}^{1/c_1'} + N_{c_2',x}^{1/c_2'}\),
\end{align*}

with $C_K = \Nrm{K}{L^{c_1}+L^{c_2}}$ .
In particular, for the Coulomb interaction, one can choose $c_1=\fb_1 < 3/2$ and $c_2=\fb_2 > 3$.

§ THE QUANTUM CASE: PROPAGATION OF REGULARITY FOR THE HARTREE–FOCK EQUATION

In this section, we prove the semiclassical analogue of the propagation of regularity for the Vlasov equation shown in Section <ref>. The main difficulty is to close the Grönwall's inequality, which we managed to do by propagating at the same time the $\L^\infty(m)$, the $\cW^{1,2}(m)$, and the $\cW^{1,q}(m)$ norms with $q\geq 2$ and

\begin{equation*}
m = 1+\n{\opp}^n,
\end{equation*}

where $n>0$. This first step allows us to prove that the $\cW^{1,q}(m)$ norm remains bounded on some time interval for $q\in[2,q_a)$ with $q_a := \infty$ if $\fb := \frac{3}{a+1} \geq 2$ and

\begin{equation*}
\frac{1}{q_a} := \frac{1}{\fb} - \frac{1}{2}
\end{equation*}

when $\fb<2$. It is the content of the following proposition, where we only consider $q\in[2,4]$ for simplicity.

Fix $a\in(0,1]$. If ${\op}^{\init}\in \cW^{1,2}(m)\cap \cW^{1,4}(m) \cap \L^\infty(m)$, then there exists $T>0$ such that

\begin{align*}
{\op} &\in L^\infty((0,T),\cW^{1,2}(m)\cap \cW^{1,4}(m) \cap \L^\infty(m)),
\\
{\rho} &\in L^\infty((0,T),H^{1} \cap W^{1,4} \cap L^1 \cap L^\infty).
\end{align*}

Now that we know that the first-order semiclassical Sobolev norms remain bounded for some finite time $T>0$ for $q\in[2,q_a)$, we can use this first result and a similar strategy to prove the propagation of higher Sobolev norms on the same time scale. This is done in the following proposition.

Under the hypotheses of Proposition <ref>, if $\op^\init\in \cW^{2,2}(m)\cap \cW^{2,4}(m)$, then

\begin{align*}
\op &\in L^\infty((0,T),\cW^{2,2}(m)\cap \cW^{2,4}(m)\cap\cW^{1,\infty}(m)),
\\
\rho &\in L^\infty((0,T),H^{2} \cap W^{2,4}).
\end{align*}

The propagation of the second-order Sobolev norms will allow us to remove the constraint $q\in[2,q_a)$ and to get the boundedness of first-order Sobolev norms also for $q\geq q_a$. This is relevant when $a\geq \tfrac{1}{2}$.

In order to perform the mean-field limit, we actually need to prove the propagation of these norms for $\sqrt{\op}$ instead of $\op$ (Cf. Equations (<ref>)-(<ref>)), which works in a similar way.

Under the hypotheses of Proposition <ref>, if $\sqrt{\op^\init}\in \cW^{1,q}(m)$ for some $q\in [2,\infty]$, then

\begin{align*}
\sqrt{\op} &\in L^\infty((0,T),\cW^{1,q}(m)).
\end{align*}

This proposition then implies also the regularity of $\op$ as indicated in next lemma.

Let $\op \geq 0$ be a compact operator. Then for any $q\in[1,\infty]$,

\begin{equation}\label{eq:regu_rho_vs_sqrt}
\Nrm{\op}{\dot{\cW}^{1,q}(m)} \leq 2 \Nrm{\sqrt{\op}}{\L^\infty(m)} \Nrm{\sqrt{\op}}{\dot{\cW}^{1,q}(m)}.
\end{equation}

By the product rule for commutators and Hölder's inequality for Schatten norms, for any $\eta \in \set{x,\xi}$,

\begin{align*}
\Nrm{\Dh_{\!\eta} \op}{\L^q} &= \Nrm{\(\Dh_{\!\eta}(\sqrt{\op})\, \sqrt{\op} + \sqrt{\op}\, \Dh_{\!\eta}(\sqrt{\op})\)m}{\L^q}
\\
&\leq \Nrm{\Dh_{\!\eta}(\sqrt{\op})}{\L^q} \Nrm{\sqrt{\op}\, m}{\L^\infty} + \Nrm{\sqrt{\op}}{\L^\infty} \Nrm{\Dh_{\!\eta}\sqrt{\op}\,m}{\L^q},
\end{align*}

which implies Inequality (<ref>).

§.§ The Strategy

Both the Hartree and the Hartree–Fock equations can be written in the form

\begin{align*}
i\hbar\,\dpt\op = \com{H,\op}
\end{align*}

with $H = \frac{\n{\opp}^2}{2} + V_{\op} - h^3\sfX_{\op}$ (with $\sfX_{\op} = 0$ in the case of the Hartree equation). If we look at the time derivatives of quantum gradients, since $\Dhx H = \nabla V_{\op} = -E_{\op}$ and $ \frac{1}{i\hbar}\com{\Dhv H,\op} = \frac{1}{i\hbar} \com{\opp,\op} = - \Dhx\op$ and since $\com{x,\sfX_{\op}} = \sfX_{\com{x,\op}}$ and $\com{\nabla,\sfX_{\op}} = \sfX_{\com{\nabla,\op}}$ (see Lemma <ref>), we obtain

\begin{align}\label{eq:time-der-grad-op}
\partial_t \Dhx \op &= \frac{1}{i\hbar}\com{H,\Dhx\op} - \frac{1}{i\hbar} \com{h^3\sfX_{\Dhx \op},\op} - \frac{1}{i\hbar} \com{E_{\op},\op}
\\\nonumber
\partial_t \Dhv \op &= \frac{1}{i\hbar}\com{H,\Dhv\op} - \frac{1}{i\hbar} \com{h^3\sfX_{\Dhv \op},\op} - \Dhx\op.
\end{align}

These equations are of the form

\begin{align}\label{eq:abstract_equation}
i\hbar\, \dpt \opmu = \com{\sfA,\opmu} + \com{\sfB,\op},
\end{align}

with $\sfA$ and $\sfB$ self-adjoint. Our goal is to propagate the weighted Schatten norms for solutions of these equations. Computing the time derivative of such quantities, we get the following result.

Let $\op$, $\sfA$ and $\sfB$ be self-adjoint operators and $\opmu = \opmu(t)$ be a family of self-adjoint operators verifying Equation (<ref>). Then, formally, for any even integer $q\geq 2$ we have

\begin{align*}
\dt \Nrm{\opmu\, m}{q} \leq \frac{1}{\hbar}\Nrm{\com{\sfA,m}\opmu}{q} + \frac{1}{\hbar} \Nrm{\com{\sfB,\op}m}{q}.
\end{align*}

Applying this lemma for $\opmu = \op$ solution of the Hartree–Fock equation or for $\opmu = \Dhx{\op}$ or $\opmu = \Dhv{\op}$, and with $m = 1$ or $m = \opp_\jj^n$ for some $\jj\in\set{1,2,3}$, we obtain

\begin{align}\label{eq:rho}
\dt \Nrm{\op\, m}{q} &\leq \frac{1}{\hbar}\Nrm{\com{V_{\op},m}\op}{q} + \frac{1}{\hbar} \Nrm{\com{h^3\sfX_{\op},m}\op}{q},
\\
\dt \Nrm{\Dhx \op\, m}{q} &\leq \frac{1}{\hbar}\Nrm{\com{V_{\op},m}\Dhx \op}{q} + \frac{1}{\hbar} \Nrm{\com{E_{\op},\op}m}{q}\nonumber
\\\label{eq:Dhx}
&\qquad + \frac{1}{\hbar}\Nrm{\com{h^3\sfX_{\op},m}\Dhx \op}{q} + \frac{1}{\hbar} \Nrm{\com{h^3\sfX_{\Dhx \op},\op}m}{q},
\\
\dt \Nrm{\Dhv \op\, m}{q} &\leq \frac{1}{\hbar}\Nrm{\com{V_{\op},m}\Dhv \op}{q} + \Nrm{\Dhx\op\, m}{q}\nonumber
\\\label{eq:Dhv}
&\qquad + \frac{1}{\hbar}\Nrm{\com{h^3\sfX_{\op},m}\Dhv \op}{q} + \frac{1}{\hbar} \Nrm{\com{h^3\sfX_{\Dhv \op},\op}m}{q},
\end{align}

where we used the fact that $\com{H,m} = \com{V_{\op}-h^3\sfX_{\op},m}$ since $\com{\n{\opp}^2,m} = 0$. In the next sections,  we will bound all the weighted Schatten norms of the commutators appearing on the right-hand side of inequalities (<ref>), (<ref>) and (<ref>) in order to get a Grönwall-type inequality.

First notice that

\begin{align*}
\dpt \opmu^2 = \frac{1}{i\hbar}\com{\sfA,\opmu^2} + \frac{1}{i\hbar}\(\com{\sfB,\op}\opmu + \opmu\com{\sfB,\op}\).
\end{align*}

Therefore, using that $2p := q$ is even and the cyclicity of the trace, we get

\begin{align}\nonumber
\dt \Nrm{\opmu\, m}{2p}^{2p} &= \dt \Tr{\(m\opmu^2m\)^p} = p\Tr{m\(\dt(\opmu^2)\) m\(m\opmu^2m\)^{p-1}}
\\\label{eq:IA_IB}
&= \frac{p}{i\hbar}\Tr{m \(\com{\sfA,\opmu^2} +\com{\sfB,\op}\opmu + \opmu\com{\sfB,\op}\) m\(m\opmu^2m\)^{p-1}} =: I_\sfA + I_\sfB.
\end{align}

For $I_\sfA$, we use again the cyclicity of the trace to write

\begin{align*}
I_\sfA &= \frac{p}{i\hbar} \Tr{\sfA\opmu^2m^2 (\opmu^2 m^2)^{p-2} \opmu^2m^2 - \opmu^2 \sfA m^2 (\opmu^2 m^2)^{p-2} \opmu^2m^2 }
\\
&= \frac{p}{i\hbar} \Tr{ m^2 \sfA\opmu^2m^2 (\opmu^2 m^2)^{p-2} \opmu^2 - \sfA m^2 (\opmu^2 m^2)^{p-2} \opmu^2m^2\opmu^2 }
\\
&= \frac{p}{i\hbar} \Tr{ \com{m^2,\sfA}\(\opmu^2 m^2\)^{p-1} \opmu^2 }.
\end{align*}

This can also be written as

\begin{align*}
I_\sfA &= \frac{p}{i\hbar} \Tr{\opmu \(\com{m,\sfA}m+m\com{m,\sfA}\)\opmu \n{m\opmu}^{2(p-1)} }
\\
&= \frac{2p}{i\hbar} \Im{\Tr{\opmu\, m\com{m,\sfA}\opmu \n{m\opmu}^{2(p-1)}}}.
\end{align*}

Therefore, by Hölder's inequality for the trace, we obtain

\begin{align}\nonumber
\n{I_\sfA} &\leq \frac{2p}{\hbar} \Nrm{\opmu\, m}{2p} \Nrm{\com{m,\sfA}\opmu}{2p} \Nrm{m\,\opmu}{2p}^{2(p-1)}
\\\label{eq:IA}
&\leq \frac{q}{\hbar} \Nrm{\opmu\, m}{q}^{q-1} \Nrm{\com{\sfA,m}\opmu}{q},
\end{align}

where we used the fact that since $\opmu$ and $m$ are self-adjoint, and since the Schatten norm is invariant by taking the adjoint, we have $\Nrm{m\,\opmu}{2p} = \Nrm{\opmu\,m}{2p}$. For the $\sfB$ term, we get more easily

\begin{align*}
I_\sfB &= \frac{2p}{i\hbar}\Im{\Tr{m\com{\sfB,\op}\opmu\, m \n{m\opmu}^{2(p-1)}}}\,.
\end{align*}

By using again Hölder's inequality and the commutation in the Schatten norm, we obtain

\begin{align}\label{eq:IB}
\n{I_\sfB} &\leq \frac{q}{\hbar}\Nrm{m\com{\sfB,\op}}{q}\Nrm{\opmu\, m}{q}^{q-1}.
\end{align}

We conclude the proof by combining inequalities (<ref>) and (<ref>) on Formula (<ref>) and using the fact that $\dt \Nrm{\opmu\, m}{q} = \tfrac{1}{q} \Nrm{\opmu\, m}{q}^{1-q} \dt \Nrm{\opmu\, m}{q}^{q}$.

§.§ Preliminary Inequalities

In order to simplify the computations, we will sometimes use weights of the form

\begin{equation*}
m = 1 + \n{\opp}^n
\end{equation*}

\begin{equation*}
\tilde{m} = 1 + \sum_{\jj = 1}^3 \opp_\jj^n.
\end{equation*}

Thanks to the following lemma, these weights define equivalent weighted Schatten norms.

Let $n\in\R$. Then there exists $C>0$ such that for any $p\in [1,\infty]$ and any operator $\op$

\begin{align}\label{eq:equiv_weight_1}
C^{-1}\Nrm{\op\, \tilde{m}}{p} &\leq \Nrm{\op\, m}{p} \leq C \Nrm{\op\, \tilde{m}}{p}
\\\label{eq:equiv_weight_2}
C^{-1}\Nrm{\op\, \opp_{\jj}^n}{p} &\leq \Nrm{\op\, m}{p} \leq C \(\Nrm{\op}{p}+\sum_{\jj = 1}^3 \Nrm{\op\, \opp_{\jj}^n}{p}\).
\end{align}

We remark that $\tilde{m}$ and $m$ commute, $m$ is invertible, and $m^{-1} \tilde{m} = g(\opp)$ with $\Nrm{g}{L^\infty} < C$ uniformly in $\hbar$. Therefore, we obtain

\begin{align*}
\Nrm{\op\, \tilde{m}}{p} = \Nrm{\op\, m \,g(\opp)}{p} \leq C \Nrm{\op\, m}{p},
\end{align*}

which proves the first inequality of Equation (<ref>). The second one follows by reversing the role of $\tilde{m}$ and $m$, and the first inequality of Equation (<ref>) by replacing $\tilde{m}$ by $\opp_{\jj}^n$. The second inequality of Equation (<ref>) follows from the second inequality of Equation (<ref>) and the triangle inequality for Schatten norms.

We will need the following unmixing inequality similar to <cit.>.

Let $p\geq 1$ and $(n,m) \in \N^2$. Then for any self-adjoint operators $A$ and $B$, the following inequality holds

\begin{equation}\label{eq:mixing}
\Nrm{B^nAB^m}{p} \leq 2 \Nrm{AB^{n+m}}{p}.
\end{equation}

Assume that $A\geq 0$. Then by Hölder's inequality, we have

\begin{equation}\label{eq:mixing_0}
\Nrm{B^nAB^m}{p} \leq \Nrm{B^n A^\frac{n}{n+m}}{\frac{n+m}{n}\, p} \Nrm{A^\frac{m}{n+m} B^m}{\frac{n+m}{m}\, p}.
\end{equation}

Now observe that since by definition of the absolute value $\n{BA}=\n{\n{B}A}$, we get $\Nrm{B^n A^\frac{n}{n+m}}{\frac{n+m}{n}\, p} = \Nrm{\n{B}^n A^\frac{n}{n+m}}{\frac{n+m}{n}\, p}$, and since the Schatten norm is invariant by taking the adjoint

\begin{align*}
\Nrm{A^\frac{m}{n+m} B^m}{\frac{n+m}{m}\, p} = \Nrm{B^m A^\frac{m}{n+m}}{\frac{n+m}{m}\, p} = \Nrm{\n{B}^m A^\frac{m}{n+m}}{\frac{n+m}{m}\, p}\,.
\end{align*}

Now, by the Araki-Lieb-Thirring inequality

\begin{align*}
\Nrm{\n{B}^n A^\frac{n}{n+m}}{\frac{n+m}{n}\, p} &\leq \Nrm{\n{B}^{n+m} A}{p}^\frac{n}{n+m} = \Nrm{ AB^{n+m}}{p}^\frac{n}{n+m},
\\
\Nrm{\n{B}^m A^\frac{m}{n+m}}{\frac{n+m}{m}\, p} &\leq \Nrm{\n{B}^{n+m} A}{p}^\frac{m}{n+m} = \Nrm{ AB^{n+m}}{p}^\frac{m}{n+m}.
\end{align*}

Combining these inequalities with Inequality (<ref>) leads to

\begin{equation}\label{eq:mixing_positive}
\Nrm{B^nAB^m}{p} \leq \Nrm{AB^{n+m}}{p}.
\end{equation}

In the more general case of a self-adjoint operator $A$ possibly not nonnegative, we write $A = A_+ - A_-$ with $A_+ = \frac{\n{A}+A}{2}$ and $\frac{\n{A}-A}{2}$. Then by Inequality (<ref>) and the triangle inequality for Schatten norms, we get

\begin{multline*}
\Nrm{B^nAB^m}{p} \leq \Nrm{A_+B^{n+m}}{p} + \Nrm{A_-B^{n+m}}{p}
\\
\leq \frac{1}{2} \(\Nrm{\n{A}B^{n+m}+ AB^{n+m}}{p} + \Nrm{\n{A}B^{n+m}-AB^{n+m}}{p}\)
\\
\leq \Nrm{\n{A}B^{n+m}}{p} + \Nrm{AB^{n+m}}{p}
\end{multline*}

and we conclude by using again the fact that $\n{\n{A}B} = \n{AB}$.

Let us define the adjoint representation of $A$ as

\begin{align*}
\Ad{A}{B} := \com{A,B}.
\end{align*}

Then, using the above Lemma, we can prove the following inequality.

Let $n\in\N$. Then for any self-adjoint operators $A$ and $B$, we have

\begin{equation*}
\Nrm{\ad{B}^n(A)}{p} \leq 2^{n+1} \Nrm{AB^n}{p}.
\end{equation*}

This follows from the expansion

\begin{align*}
\ad{B}^n(A) = \sum_{k=0}^n \binom{n}{k} \(-1\)^{k} B^{n-k} A B^k,
\end{align*}

together with the triangle inequality for the Schatten norms and the unmixing inequality (<ref>).

The next proposition allows us to control $\Nrm{\nabla\rho}{L^p}$ by $\Nrm{\Dhx{\op}\,m}{\L^p}$ for some weight $m$. Notice that it can be viewed as a trace-type inequality.

Let $p\in[1,\infty]$ and $n>\frac{3}{p'}$. Then there exists a constant $C>0$ such that for any compact self-adjoint operator $\opmu$

\begin{align*}
\Nrm{\Diag{\opmu}}{L^p} \leq C \Nrm{\opmu\,m}{\L^p},
\end{align*}

with $m = 1+\n{\opp}^n$.

In particular, since for $k\in\N$, $\nabla^k\rho = \Diag{\Dhx^k{\op}}$, the above estimate implies

\begin{align*}
\Nrm{\nabla^k\rho}{L^p} \leq C \Nrm{\Dhx^k{\op}\,m}{\L^p}.
\end{align*}

Let $\diagopmu(x) := \Diag{\opmu}\!(x) = h^3\opmu(x,x)$. Then, using the dual formulation of the $L^p$ norm and separating $\varphi$ into the sum of its positive and negative parts $\varphi := \varphi_++\varphi_-$, we have

\begin{align*}
\Nrm{\diagopmu}{L^p} \leq \sup_{\Nrm{\varphi}{L^{p'}}\leq 1}\( \n{\intd \diagopmu\,\varphi_-} + \n{\intd \diagopmu\,\varphi_+}\),
\end{align*}

from which we deduce that we can actually restrict ourselves to nonnegative functions $\varphi$ and identifying the function $\varphi$ with the multiplication operator by the function $\varphi$, we get

\begin{align}\label{eq:dual_Lp}
\Nrm{\diagopmu}{L^p} \leq 2\,\sup_{\substack{\varphi\geq 0\\\Nrm{\varphi}{L^{p'}}\leq 1}} \n{\intd \diagopmu\,\varphi} = 2 \sup_{\substack{\varphi\geq 0\\\Nrm{\varphi}{L^{p'}}\leq 1}} \n{h^3 \Tr{\opmu\,\varphi}}.
\end{align}

Taking $m(y) := \sqrt{1+\n{y}^n}$ and $w(y) = m(y)^{-1}$, we see that $\opm := m(\opp)$ is a positive invertible operator and its inverse $\opw := w(\opp)$ is a compact operator. By Hölder's inequality for the trace, we have

\begin{align}\label{eq:weighted_holder}
h^3\Tr{\opmu\,\varphi} &= h^3\Tr{\opm\,\opmu\, \opm\,\opw\,\varphi\,\opw} \leq \Nrm{\opm\,\opmu\,\opm}{\L^p} \Nrm{\opw\,\varphi\,\opw}{\L^{p'}}.
\end{align}

However, since $\varphi$ is a nonnegative function, we get that it is also a positive operator. Hence

\begin{align*}
\Nrm{\opw\,\varphi\,\opw}{\L^{p'}} &= \Nrm{\n{\sqrt{\varphi}\,\opw}^2}{\L^{p'}} = \Nrm{\sqrt{\varphi}\,\opw}{\L^{2p'}}^2 \leq \Nrm{\varphi}{L^{p'}} \Nrm{w}{L^{2p'}}^2
\end{align*}

where to get the last inequality we used the Kato-Seiler-Simon Inequality (<ref>) since $2p'\geq 2$. Combining the above inequality with inequalities (<ref>) and (<ref>) yields

\begin{align*}
\Nrm{\diagopmu}{L^p} \leq C_{p,n}\,\Nrm{\opm\,\opmu\,\opm}{\L^p} \leq C_{p,n}\,\Nrm{\opmu\,\opm^2}{\L^p}
\end{align*}

where the second inequality is a consequence of Lemma <ref>, and $C_{p,n} = 2\Nrm{w}{L^{2p'}}^2$ is finite because $n>\frac{3}{p'}$ by assumption.

§.§ Commutators Involving the Direct Term

In the semiclassical case, instead of $\nabla E_{\op}\cdot\Dv f$ (see Equation (<ref>)), the time derivative of the gradient let appear the term $\frac{1}{i\hbar} \com{E_{\op},\op}$ (see Equation (<ref>)). Hence we will need to get semiclassical estimates on this quantity, which is the purpose of the following proposition.

Let $a\in(0,1]$, $\fb = \frac{3}{a+1}$ and $(q,r)\in[2,\infty]^2$ be such that $\frac{1}{r}+\frac{1}{q} = \frac{1}{2}$. Then for any compact operator $\op_2$ it holds

\begin{align}\label{eq:estim_commutator_Lq_Besov}
\frac{1}{\hbar}\Nrm{\com{E_{\op},\op_2}}{\L^q} \leq C\,\Nrm{\rho}{B^{1-3\(\frac{1}{r'}-\frac{1}{\fb}\)}_{r,1}} \Nrm{\Dhv\op_2}{\L^q}.
\end{align}

When $q=2$ and $r=\infty$, we also have

\begin{align}\label{eq:estim_commutator_L2}
\frac{1}{\hbar}\Nrm{\com{E_{\op},\op_2}}{\L^2} \leq C\,\Nrm{\nabla\rho}{L^{\fb',1}} \Nrm{\Dhv\op_2}{\L^2}
\end{align}

for $a\in\left[\frac{1}{2},1\right]$, and

\begin{align}\label{eq:estim_commutator_L2_slow-tail}
\frac{1}{\hbar}\Nrm{\com{E_{\op},\op_2}}{\L^2} \leq C\,\Nrm{\rho}{L^{\frac{3}{1-a},1}} \Nrm{\Dhv\op_2}{\L^2}
\end{align}

for $a\in\(0,\frac{1}{2}\)$.

From the fact that $(L^r,W^{1,r})_{s,1} = B^s_{r,1}$ for any $r\in[1,\infty)$ and $s\in(0,1)$, we deduce the following inequality in terms of more classical Sobolev spaces.

Let $(q,r)\in(2,\infty]\times (\fb',\infty)$ be such that $\frac{1}{r}+\frac{1}{q} = \frac{1}{2}$. Then it holds

\begin{align*}
\frac{1}{\hbar}\Nrm{\com{E_{\op},\op_2}}{\L^q} \leq C \Nrm{\rho}{L^r}^{1-s} \Nrm{\rho}{W^{1,r}}^s \Nrm{\Dhv\op_2}{\L^q}
\end{align*}

with $s = 1-3\(\frac{1}{r'}-\frac{1}{\fb}\)$.

From the fact that $\frac{1}{r}+\frac{1}{q} = \frac{1}{2}$ and the fact that $r>\fb'$, when $a \geq 1/2$, the above results only work when $q<q_a$ with $\frac{1}{q_a} = \frac{1}{\fb} - \frac{1}{2}$.

First observe that the integral kernel of the operator $\com{E_{\op},\op_2}$ can be written as

\begin{align*}
\com{E_{\op},\op_2}\!(x,y) &= \(E_{\op}(x)-E_{\op}(y)\)\op_2(x,y)
\\
&= \frac{\(E_{\op}(x)-E_{\op}(y)\)\otimes\(x-y\)}{\n{x-y}^2}\cdot \(x-y\)\op_2(x,y).
\end{align*}

Thus, we can explicitly compute its Hilbert–Schmidt norm by computing the $L^2$ norm of the kernel, and since the kernel of the operator $\Dhv{\op_2}$ is $\frac{x-y}{i\hbar}\op_2(x,y)$, we get the following estimate

\begin{multline}\label{eq:estim_commutator_L2_0}
\frac{1}{\hbar}\Nrm{\com{E_{\op},\op_2}}{2} = \(\iintd\n{\frac{\(E_{\op}(x)-E_{\op}(y)\)\otimes\(x-y\)}{\n{x-y}^2}\cdot \Dhv{\op_2}(x,y)}^2 \d x\dd y\)^{\frac{1}{2}}
\\
\leq \Nrm{\nabla E_{\op}}{L^\infty} \Nrm{\Dhv{\op_2}}{2}.
\end{multline}

In particular, for $a\in\left[\frac{1}{2},1\right]$, since $\nabla E_{\op} = \nabla K*\nabla \rho$ with $\nabla K\in L^{\fb,\infty}$, we deduce Inequality (<ref>) using the fact that the dual of $L^{\fb',1}$ is $L^{\fb,\infty}$ (see e.g. [42])

If $a\in\(0,\frac{1}{2}\)$, we use that $\nabla E_{\op} = \nabla^2 K* \rho$ with $\nabla^2 K\in L^{\frac{3}{a+2},\infty}$. Thus Inequality (<ref>) follows from   Hölder's inequality for  Lorentz norms.

A second possibility is to use the fundamental theorem of calculus for $E_{\op}$ and then the Fourier inversion theorem to rewrite the integral kernel of the commutator as

\begin{align*}
\frac{1}{i\hbar}[E_{\op},\op_2](x,y) &= \int_0^1\nabla E_{\op}((1-\theta)x+\theta y)\dd\theta\cdot (\Dhv\op_2)(x,y)
\\
&= \int_{[0,1]\times\R^3} \widehat{\nabla E_{\op}}(z)\, e^{2i\pi z\cdot(1-\theta)x}\cdot (\Dhv\op_2)(x,y)\, e^{2i\pi z\cdot\theta y}\dd\theta\dd z,
\end{align*}

which implies that by writing $e_\omega$ the operator of multiplication by $e^{2i\pi\omega\cdot x}$, it holds

\begin{equation*}
\frac{1}{i\hbar}[E_{\op},\op_2] = \int_{[0,1]\times\R^3} \widehat{\nabla E_{\op}}(z)\, e_{(1-\theta)z} \(\Dhv\op_2\) e_{\theta z} \dd\theta\dd z.
\end{equation*}

Since the operators $e_\omega$ are bounded operators of norm $\Nrm{e_\omega}{\infty} = 1$, we deduce the following estimate on the operator norm of the commutator

\begin{align}\label{eq:estim_commutator_Linfty}
\frac{1}{\hbar}\Nrm{\com{E_{\op},\op_2}}{\infty} \leq \int_{[0,1]\times\R^3} \n{\widehat{\nabla E_{\op}}(z)} \Nrm{\Dhv\op_2}{\infty} \dd\theta\dd z = \Nrm{\widehat{\nabla E_{\op}}}{L^1} \Nrm{\Dhv\op_2}{\infty}.
\end{align}

In order to get a result for a general $q\in[2,\infty]$, we proceed by complex interpolation. Defining the vector-valued Hilbert–Schmidt operator $\opmu := \Dhv{\op_2}$, we remark that $\op_2(x,y) = i\hbar\, \frac{x-y}{\n{x-y}^2}\cdot\opmu(x,y)$ and the commutator can be rewritten as the bilinear operator

\begin{align*}
\Lambda(E,\opmu) := \com{E,\frac{x-y}{\n{x-y}^2}\cdot\opmu} = \frac{1}{i\hbar} \com{E,\op_2}.
\end{align*}

Thus, using the fact that $B^0_{\infty,1}\subset L^\infty$ and $B^\frac{3}{2}_{2,1}\subset \F{L^1}$, inequalities (<ref>) and (<ref>) imply

\begin{align*}
\Nrm{\Lambda(E,\opmu)}{2} &\leq C \Nrm{E}{B^1_{\infty,1}} \Nrm{\opmu}{2},
\\
\Nrm{\Lambda(E,\opmu)}{\infty} &\leq C \Nrm{E}{B^{1+\frac{3}{2}}_{2,1}} \Nrm{\opmu}{\infty}.
\end{align*}

By the same proof, one obtains the inequality for any vector-valued Hilbert–Schmidt operator $\opmu$. Finally, we use the fact that the complex interpolation space between the involved Besov spaces is given by $[B^1_{\infty,1},B^{1+\frac{3}{2}}_{2,1}]_{\frac{2}{r}} = B^{1+\frac{3}{r}}_{r,1}$ (see for example <cit.>), while the complex interpolation of Schatten spaces $\mathfrak{S}^q$ gives $[\mathfrak{S}^2,\mathfrak{S}^\infty]_{1-\frac{2}{q}} = \mathfrak{S}^q$ (see for example <cit.>), so that by bilinear interpolation (see <cit.>), we obtain

\begin{align*}
\Nrm{\Lambda(E,\opmu)}{q} &\leq C \Nrm{E}{B^{1+\frac{3}{r}}_{r,1}} \Nrm{\opmu}{q} && \text{ with } \frac{1}{r} = \frac{1}{2} - \frac{1}{q}.
\end{align*}

If we take $E_{\op} = \nabla K * \rho$ with $\rho\in L^1\cap L^p$ for some $p\in(1,\infty)$ and $\nabla K\in L^{\fb,\infty}$, we know that $E_{\op}\in L^{\tilde{r}}$ for some $\tilde{r}\in(1,\infty)$. Moreover, $E_{\op}$ is proportional to $(-\Delta)^\frac{a-3}{2}\nabla\rho$, therefore we can apply <cit.> and deduce that

\begin{align*}
\Nrm{E_{\op}}{B^{1+\frac{3}{r}}_{r,1}} \leq C \Nrm{\rho}{B^{\frac{3}{r}+a-1}_{r,1}} = \Nrm{\rho}{B^{1-3\(\frac{1}{r'}-\frac{1}{\fb}\)}_{r,1}}.
\end{align*}

Taking $\opmu = \Dhv{\op_2}$ yields the results.

To get estimates with weights, remark that we can write $\com{E_{\op},\op}\opp_\jj^{2n}$ in the following form

\begin{align*}
\frac{1}{i\hbar}\com{E_{\op},\op}\opp_\jj^{2n} &= \frac{1}{i\hbar}\com{E_{\op},\op\, \opp_\jj^{2n}} - \frac{1}{i\hbar}\com{E_{\op},\opp_\jj^{2n}}\op.
\end{align*}

To control the $\L^q$ norm of the first term of the right-hand side we use Proposition <ref>, which gives

\begin{align*}
\frac{1}{\hbar}\Nrm{\com{E_{\op},\op\,\opp_\jj^{2n}}}{\L^q} &\leq C \Nrm{\rho}{B^{1-3\(\frac{1}{r'}-\frac{1}{\fb}\)}_{r,1}} \Nrm{\Dhv(\op\,\opp_\jj^{2n})}{\L^q},
\end{align*}

and we can also replace $\Nrm{\rho}{B^{1-3\(\frac{1}{r'}-\frac{1}{\fb}\)}_{r,1}}$ by $\Nrm{\nabla\rho}{L^{\fb',1}}$ when $q=2$. To bound the second term, we will use the following proposition.

Let $a \in (0,1]$ and $\fb = \frac{3}{a+1}$. For $a\in\left[\tfrac{1}{2},1\right]$ take $(q,r,r_1)\in \left[\frac{3}{2},\infty\right]\times\(\tfrac{3}{3-a},\fb'\)\times[1,\infty]$, for $a\in\left(0,\tfrac{1}{2}\right)$ take $(q,r,r_1)\in \left[\frac{3}{2},\infty\right]\times\(\tfrac{3}{3-a},\tfrac{3}{1-a}\)\times[1,\infty]$, such that

\begin{equation}\label{eq:exponents_weighted_commutator}
\frac{1}{r} + \frac{1}{r_1} = \frac{1}{q} + \frac{1}{\fb'}.
\end{equation}

Let $n\in\N$ and $m := 1+\n{\opp}^{2n}$. Then there exists a constant $C>0$ such that

\begin{align*}
\frac{1}{\hbar}\Nrm{\com{E_{\op},\opp_\jj^{2n}}\opmu}{\L^q} \leq C \Nrm{\Dhxj\op \,m}{\L^r} \Nrm{\opmu \,m}{\L^{r_1}}.
\end{align*}

If $r = \fb'$, we need to replace $\L^{{r}}$ by $\L^{{r}-\eps}\cap \L^{{r}+\eps}$ for some $\eps\in(0,\fb'-1)$.

With a similar proof, one obtains a similar result replacing $E_{\op}$ by $V_{\op} = K*\rho$. This yields the following inequalities.

With the same hypotheses as in Proposition <ref>, it holds

\begin{align*}
\frac{1}{\hbar}\Nrm{\com{V_{\op},\opp_\jj^{2n}}\opmu}{\L^q} \leq C \Nrm{\op \,m}{\L^r} \Nrm{\opmu \,m}{\L^{r_1}},
\end{align*}

with the same additional constraints in the endpoint cases.

Let $E := E_{\op}$. We remark that $\com{E,\opp_\jj} = E\,\opp_\jj-\opp_\jj\,E = i\hbar\,\partial_{\jj} E$ is the operator of multiplication by $x\mapsto i\hbar\,\partial_{\jj} E(x)$, and since $\opp_\jj^2 = \opp_\jj\opp_\jj$, we get

\begin{align*}
\frac{1}{i\hbar}\com{E,\opp_\jj^2} = \frac{1}{i\hbar}\(\com{E,\opp_\jj}\opp_\jj + \opp_\jj\com{E,\opp_\jj}\) = \(\partial_{\jj} E\) \opp_\jj + \opp_\jj\( \partial_{\jj} E\),
\end{align*}

and more generally, for any $n\in\N$

\begin{align*}
\frac{1}{i\hbar}\com{E,\opp_\jj^{2n}} &= \sum_{k=0}^{n-1} \opp_\jj^{2k} \(\partial_{\jj} E\, \opp_\jj + \opp_\jj\,\partial_{\jj} E\) \opp_\jj^{2\(n-k-1\)} = \sum_{k=0}^{2n-1} \opp_\jj^{k}\, \partial_{\jj} E\, \opp_\jj^{2n-1-k}.
\end{align*}

From this formula and the triangle inequality for Schatten norms, we deduce

\begin{align*}
\frac{1}{\hbar}\Nrm{\com{E,\opp_\jj^{2n}}\opmu\,}{\L^q} &\leq \sum_{k=0}^{2n-1}\Nrm{\opp_\jj^{k}\, \partial_{\jj} E\, \opp_\jj^{2n-1-k}\opmu\,}{\L^q}.
\end{align*}

We cannot directly  apply  Hölder's inequality here since $\opp_\jj^{k} E$ is an unbounded operator, therefore we have to make some commutations between $\opp_\jj^{k}$ and $\partial_{\jj} E$. By Leibniz formula

\begin{align*}
\opp_\jj^{k}\, \partial_{\jj} E = \sum_{\ell=0}^k \binom{k}{\ell} g_\ell\, \opp_\jj^{k-\ell},
\end{align*}

where $g_\ell$ is the function defined by $g_\ell(x) = (\opp_\jj^\ell(\partial_{\jj} E))(x)$. Multiplying and dividing by $m_{\ell+1} := 1+\n{\opp}^{\ell+1}$, we deduce that

\begin{align*}
\frac{1}{\hbar}\Nrm{\com{E,\opp_\jj^{2n}}\opmu\,}{\L^q} &\leq \sum_{k=0}^{2n-1}\sum_{\ell=0}^k \binom{k}{\ell} \Nrm{g_\ell\, m_{\ell+1}^{-1}m_{\ell+1}\,\opp_\jj^{2n-1-\ell}\opmu\,}{\L^q}
\\
&\leq \sum_{\ell=0}^{2n-1} C_\ell \Nrm{g_\ell\, m_{\ell+1}^{-1}m_{\ell+1}\,\opp_\jj^{2n-1-\ell}\opmu\,}{\L^q},
\end{align*}

where $C_\ell = \sum_{k=\ell}^{2n-1} \binom{k}{\ell}$.

1. Case $\ell$ small

In this case we define $\frac{1}{\sigma} = \frac{1}{q} - \frac{1}{r_1}$. By the relation (<ref>) and the fact that $r\in(\frac{3}{3-a},\fb']$, we deduce that

\begin{align*}
\frac{1}{\sigma} = \frac{1}{r} - \frac{1}{\fb'} \in \lt[0,\frac{1}{3}\rt),
\end{align*}

or equivalently $\sigma\in ({3},\infty]$. Thus, we can use Hölder's inequality for Schatten norms and   the Kato-Seiler-Simon Inequality (<ref>) to deduce that

\begin{align*}
\Nrm{g_\ell\, m_{\ell+1}^{-1} m_{\ell+1}\,\opp_\jj^{2n-1-\ell}\opmu\,}{\L^q} &\leq \Nrm{g_\ell\, m_{\ell+1}^{-1}}{\L^\sigma} \Nrm{m_{\ell+1}\,\opp_\jj^{2n-1-\ell}\opmu\,}{\L^{r_1}}
\\
&\leq C_m^{1/\sigma}\,\Nrm{g_\ell}{L^\sigma} \Nrm{\opmu\,m_{2n}}{\L^{r_1}},
\end{align*}

where $C_m = \intd \frac{\dd x}{(1+\n{x}^{\ell+1})^\sigma}$ is finite because $\sigma>3$ implies that $\sigma>\frac{3}{\ell+1}$. Then observe that

\begin{align*}
g_\ell = -\nabla K * (\opp_\jj^\ell\partial_{\jj}\rho) = -\nabla K * \Diag{\ad{\opp_\jj}^\ell(\Dhxj{\op})}.
\end{align*}

We first consider $a\in\left[\frac{1}{2},1\right]$. If $r<\fb'$, then $\sigma<\infty$. Therefore, since $\nabla K\in L^{\fb,\infty}$, $(\sigma,\fb)\in(1,\infty)^2$ and from Formula (<ref>), $\frac{1}{r} = \frac{1}{\fb'} + \frac{1}{\sigma}$, by the  Hardy–Littlewood–Sobolev inequality,

\begin{align*}
\Nrm{g_\ell}{L^\sigma} \leq C_K \Nrm{\Diag{\ad{\opp_\jj}^\ell(\Dhxj{\op})}}{L^r},
\end{align*}

where $C_K = \Nrm{\nabla K}{L^{\fb,\infty}}$. By Proposition <ref> and Lemma <ref>, we get that for $\ell < 2n - 3/r'$ we have

\begin{align*}
\Nrm{g_\ell}{L^\sigma} \leq C_K \Nrm{\ad{\opp_\jj}^\ell\!\(\Dhxj{\op}\) m_{2n-\ell }}{\L^r} \leq 2^{\ell+1} C_K \Nrm{\Dhxj{\op}\,m_{2n}}{\L^r}.
\end{align*}

Since $\frac{1}{r} > 1-\frac{2n}{3}$, this includes at least the case $\ell=0$.

If $r=\fb'$, then $\sigma = \infty$ and we use Young's inequality instead of the  Hardy–Littlewood–Sobolev inequality, and the fact that $\nabla K \in L^{\fb+\epsilon} + L^{\fb-\epsilon}$ for any $\epsilon\in(0,\fb-1]$ to get

\begin{align*}
\Nrm{g_\ell}{L^\sigma} \leq C_K \Nrm{\Diag{\ad{\opp_\jj}^\ell(\Dhxj{\op})}}{L^{r+\eps}\cap L^{r-\eps}},
\end{align*}

for any $\eps\in(0,\fb'-1]$, and the proof follows similarly.

Now we consider $a\in\(0,\frac{1}{2}\)$ and rewrite $g_\ell$ as

\begin{align*}
g_\ell = -\nabla^2 K * (\opp_\jj^\ell\rho) = -\nabla^2 K * \Diag{\ad{\opp_\jj}^\ell({\op})}.
\end{align*}

If $r<\fb'$, then $\sigma<\infty$. Therefore, since $\nabla^2 K\in L^{\frac{3}{a+2},\infty}$, $(\sigma,\fb)\in(1,\infty)^2$ and from Formula (<ref>), $\frac{1}{r} = \frac{1}{\fb'} + \frac{1}{\sigma}$, by the  Hardy–Littlewood–Sobolev inequality,

\begin{align*}
\Nrm{g_\ell}{L^\sigma} \leq C_K \Nrm{\Diag{\ad{\opp_\jj}^\ell({\op})}}{L^r},
\end{align*}

where $C_K = \Nrm{\nabla^2 K}{L^{\frac{3}{a+2},\infty}}$. By Proposition <ref> and Lemma <ref>, we get that for $\ell < 2n - 3/r'$ we have

\begin{align*}
\Nrm{g_\ell}{L^\sigma} \leq C_K \Nrm{\ad{\opp_\jj}^\ell\!\({\op}\) m_{2n-\ell }}{\L^r} \leq 2^{\ell+1} C_K \Nrm{{\op}\,m_{2n}}{\L^r}.
\end{align*}

Since $\frac{1}{r} > 1-\frac{2n}{3}$, this includes at least the case $\ell=0$.

If $r=\fb'$, then $\sigma = \infty$ and we use Young's inequality instead of the Hardy–Littlewood–Sobolev inequality, and the fact that $\nabla^2 K \in L^{\frac{3}{a+2}+\epsilon} + L^{\frac{3}{a+2}-\epsilon}$ for any $\epsilon\in(0,\frac{3}{a+2}-1]$ to get

\begin{align*}
\Nrm{g_\ell}{L^\sigma} \leq C_K \Nrm{\Diag{\ad{\opp_\jj}^\ell\!(\op)}}{L^{r+\eps}\cap L^{r-\eps}},
\end{align*}

for any $\eps\in(0,\frac{3}{a+2}-1]$, and the proof follows similarly.

2. Case $\ell>0$  Let $\frac{1}{\tilde{q}} = \frac{1}{q} + \frac{1}{3}$. Then since $\tilde{q} \leq q$,

\begin{align*}
\Nrm{\cdot}{\L^q} = h^{3/q} \Nrm{\cdot}{q} \leq h^{3/q} \Nrm{\cdot}{\tilde{q}} = h^{-1} \Nrm{\cdot}{\L^{\tilde{q}}}.
\end{align*}

Similarly as in the step 1, we define $\frac{1}{\tilde{\sigma}} = \frac{1}{\tilde{q}} - \frac{1}{\tilde{r}_1}$ and by Formula (<ref>) we get

\begin{align*}
\frac{1}{\tilde{\sigma}} = \frac{1}{3} + \frac{1}{\tilde{r}} - \frac{1}{\fb'} \in \(\frac{a}{3},\frac{1}{2}\)
\end{align*}

or equivalently $\tilde{\sigma}\in \(2,\frac{3}{a}\)$, and we deduce

\begin{align*}
\Nrm{g_\ell\, m_{\ell+1}^{-1} \, m_{\ell+1}\,\opp_\jj^{2n-1-\ell}\opmu\,}{\L^q} &\leq h^{-1}\Nrm{g_\ell\, m_{\ell+1}^{-1} m_{\ell+1}\,\opp_\jj^{2n-1-\ell}\opmu\,}{\L^{\tilde{q}}}
\\
&\leq C_m^{1/\tilde{\sigma}} \,h^{-1}\Nrm{g_\ell}{L^{\tilde{\sigma}}} \Nrm{\opmu\,m_{2n}}{\L^{\tilde{r}_1}},
\end{align*}

where $C_m = \intd \frac{\dd x}{(1+\n{x}^{\ell+1})^\sigma}$. Since $\tilde{\sigma}> \frac{3}{2}$ and $\ell \geq 1$, we have $\tilde{\sigma} > \frac{3}{\ell+1}$ and so $C_m < \infty$. Note that since $\ell\geq 1$

\begin{align*}
g_\ell = -\partial_{\jj}\nabla K * \opp_\jj^\ell\rho = i\hbar\(\partial_{\jj}\nabla K\) *  \Diag{\ad{\opp_\jj}^{\ell-1}(\Dhxj{\op})}.
\end{align*}

To control $g_\ell$, when $a=1$, we use the fact that $\partial_{\jj}\nabla K$ is continuous from $L^{\tilde{\sigma}}$ to $L^{\tilde{\sigma}}$ by the  Calderón-Zygmund Theorem (see e.g. <cit.>). When $a\in\lt[\frac{1}{2},1\rt)$, $\n{\partial_{\jj}\nabla K} \leq \tfrac{1}{\n{x}^{a+2}}$ and we use the  Hardy–Littlewood–Sobolev inequality. In both cases, we obtain

\begin{align*}
\Nrm{g_\ell}{L^{\tilde{\sigma}}} \leq C_K \,h \Nrm{\Diag{\ad{\opp_\jj}^{\ell-1}(\Dhxj{\op})}}{L^{\tilde{r}}}
\end{align*}

since $1+\frac{1}{\tilde{\sigma}} = \frac{1}{3} + \frac{1}{\tilde{r}} + \frac{1}{\fb} = \frac{1}{\tilde{r}} + \frac{a+2}{3}$.

By Proposition <ref> and Lemma <ref>, we get that for $\ell < 2n+1-\frac{3}{\tilde{r}'}$ we have

\begin{align*}
h^{-1}\,\Nrm{g_\ell}{L^{\tilde{r}}} \leq C_K \Nrm{\ad{\opp_\jj}^{\ell-1}\!\(\Dhxj{\op}\) m_{2n-\ell+1}}{\L^{\tilde{r}}} \leq 2^\ell C_K \Nrm{\Dhxj{\op}\,m_{2n}}{\L^{\tilde{r}}}.
\end{align*}

Since $\tilde{r} < 3$, we have $\frac{3}{\tilde{r}'} < 2$, so $2n+1-\frac{3}{\tilde{r}'} > 2n-1$, therefore this works for any $1\leq \ell \leq 2n-1$. By choosing $\tilde{r}=r$ we conclude the proof.

When $a\in\(0,\frac{1}{2}\)$, we obtain

\begin{align*}
\Nrm{g_\ell}{L^{\tilde{\sigma}}} \leq C_K \,h \Nrm{\Diag{\ad{\opp_\jj}^{\ell-1}({\op})}}{L^{\tilde{r}}}
\end{align*}

where $C_K=\Nrm{\nabla^2 K}{L^{\frac{3}{a+2},\infty}}$, and $1+\frac{1}{\tilde{\sigma}} = \frac{1}{3} + \frac{1}{\tilde{r}} + \frac{1}{\fb} = \frac{1}{\tilde{r}} + \frac{a+2}{3}$.

By Proposition <ref> and Lemma <ref>, we get that for $\ell < 2n+1-\frac{3}{\tilde{r}'}$ we have

\begin{align*}
h^{-1}\,\Nrm{g_\ell}{L^{\tilde{r}}} \leq C_K \Nrm{\ad{\opp_\jj}^{\ell-1}\!\({\op}\) m_{2n-\ell+1}}{\L^{\tilde{r}}} \leq 2^\ell C_K \Nrm{{\op}\,m_{2n}}{\L^{\tilde{r}}}.
\end{align*}

Since $\tilde{r} < 3$, we have $\frac{3}{\tilde{r}'} < 2$, so $2n+1-\frac{3}{\tilde{r}'} > 2n-1$, therefore this works for any $1\leq \ell \leq 2n-1$. By choosing $\tilde{r}=r$ we conclude the proof.

§.§ Preliminary Properties of the Exchange Operator

§.§.§ Preliminary Identities
Let $\sfX = \sfX_{\op}$ be the operator of integral kernel $\sfX(x,y) = K(x-y)\,\op(x,y)$ with $K(x) = \n{x}^{-a}$ and recall the notation of the adjoint representation of $A$, $\Ad{A}{B} = \com{A,B}$.

Let $a\in(0,1]$. Then the following identities holds true

\begin{align*}
\com{x,\sfX_{\op}} &= \sfX_{\com{x,\op}}, & \com{\nabla,\sfX_{\op}} &= \sfX_{\com{\nabla,\op}},
\end{align*}

and more generally, with the adjoint notation, $\ad{x}^n(\sfX_{\op}) = \sfX_{\ad{x}^n(\op)}$ and $\ad{\nabla}^n(\sfX_{\op}) = \sfX_{\ad{\nabla}^n(\op)}$. In particular, since $\Dhx{\op} = \Ad{\nabla}\op$ and $\Dhv{\op} = \frac{1}{i\hbar}\Ad{x}\op$, it can be written $\Dhv^n(\sfX_{\op}) = \sfX_{\Dhv^n\op}$, and $\Dhx^n(\sfX_{\op}) = \sfX_{\Dhx^n\op}$.

The first identity follows immediately by looking at the integral kernel of the operator

\begin{align*}
\com{x,\sfX_{\op}}(x,y) = \frac{\(x-y\)\r(x,y)}{\n{x-y}^a} = \sfX_{\com{x,\op}}(x,y).
\end{align*}

To get the second we take $\varphi\in C^\infty_c$, integrate by parts and use the fact that $\nabla_x K(x-y) = -\nabla_y K(x-y)$ to get

\begin{multline*}
\com{\nabla,\sfX_{\op}}\!\varphi(x) = \nabla \intd \frac{\op(x,y)}{\n{x-y}^a}\,\varphi(y)\dd y - \intd \frac{\op(x,y)}{\n{x-y}^a}\,\nabla\varphi(y)\dd y
\\
= \intd \(\nabla_x +\nabla_y\)\(\frac{\op(x,y)}{\n{x-y}^a}\)\,\varphi(y)\dd y = \intd \frac{\(\nabla_x +\nabla_y\)\!\(\op(x,y)\)}{\n{x-y}^a}\,\varphi(y)\dd y,
\end{multline*}

and we conclude by remarking that $\(\nabla_x +\nabla_y\)\!\(\op(x,y)\)$ is nothing but the integral kernel of the operator $\com{\nabla,\op}$.

Let $\eta=1$, $\eta=\opp_\jj$ or $\eta=x_\jj$, then we have

\begin{align}\label{eq:commut_X_0}
\eta^n\,\sfX_{\op} &= \sum_{k=0}^n \binom{n}{k} \sfX_{\ad{\eta}^k(\op)}\,\eta^{n-k}
\\\label{eq:commut_X_1}
\com{\eta^n,\sfX_{\op}} &= \sum_{k=1}^n \binom{n}{k} \sfX_{\ad{\eta}^k(\op)}\,\eta^{n-k}.
\end{align}

Since $A^k B = (A^{k-1}B)A + A^{k-1}\Ad{A}{B}$, we easily deduce the following commutator expansion

\begin{align*}
A^nB = \sum_{k=0}^n \binom{n}{k} \ad{A}^k(B) \, A^{n-k}.
\end{align*}

Therefore, we deduce the result by taking $B = \sfX_{\op}$ and by Lemma <ref>.

§.§.§ Preliminary Inequalities
We know from <cit.> that if $a\in[0,3/2)$ and $q=2$, then

\begin{equation}\label{eq:X_HS_bound}
\Nrm{\sfX_{\op}}{q} \leq C\,h^{-a} \Nrm{\op \n{\opp}^a}{2}.
\end{equation}

By the fact that the Schatten norms of smaller order controls the Schatten norms of higher order, we deduce that this inequality actually holds for any $q\in[2,\infty]$. The next proposition will allow us to put the weight $\n{\opp}^a$ on an other operator $\opmu$ instead of $\op$.
Let $\opmu$ and $\tilde{\opmu}$ be compact operators. Then for any $q\in[2,\infty]$ and any $\theta\in\{0,1\}$,

\begin{equation}\label{eq:X_commut_p}
\Nrm{\sfX_{\tilde{\opmu}}\,\opmu}{q} \leq C_{a}\,h^{-a}\Nrm{\tilde{\opmu}\n{\opp}^{a\(1-\theta\)}}{2} \Nrm{\opmu^*\n{\opp}^{\theta a}}{\infty},
\end{equation}

where $\opmu^*$ is the adjoint operator of $\opmu$.

Take $\opmu_2$ a compact but possibly not self-adjoint operator. Then

\begin{align*}
\Nrm{\sfX_{\opmu_2}}{2}^2 &= \Tr{\sfX_{\opmu_2}^*\,\sfX_{\opmu_2}} = \iintd \frac{\opmu_2^*(x,y)\,\opmu_2(y,x)}{\n{x-y}^{2a}}\dd x\dd y
\\
&= \iintd \frac{\n{\opmu_2(x,y)}^2}{\n{x-y}^{2a}}\dd x\dd y = \iintd \frac{\n{\opmu_2(x,y+x)}^2}{\n{y}^{2a}}\dd x\dd y,
\end{align*}

so that by the Hardy-Rellich inequality,

\begin{align*}
\Nrm{\sfX_{\opmu_2}}{2}^2 &\leq \cC_{a}\,\iintd \n{\Delta^{a/2}_y\opmu_2(x,y+x)}^2\dd x\dd y = \cC_{a}\,\iintd \n{\Delta^{a/2}_y\opmu_2(x,y)}^2\dd x\dd y
\\
&\leq C_{a}\,h^{- 2a}\iintd \n{\(\opmu_2 \n{\opp}^a\)(x,y)}^2\dd x\dd y,
\end{align*}

where $\cC_{a}$ is the constant appearing in the Hardy–Rellich inequality and $C_{a} = \(2\pi\)^a\cC_{a}$. From this we deduce the generalization of (<ref>) for possibly not self-adjoint operators

\begin{equation}\label{eq:X_HS_bound_2}
\Nrm{\sfX_{\opmu_2}}{2} \leq C_{a}\,h^{- a}\Nrm{\opmu_2 \n{\opp}^a}{2}.
\end{equation}

By Hölder's inequality, taking $\opmu_2=\tilde{\opmu}$, it implies (<ref>) when $\theta=0$.
Now, remarking that we have the following integration by parts like formula

\begin{align*}
\Tr{\sfX_{\tilde{\opmu}}\,\opmu} = \iintd \frac{\tilde{\opmu}(x,y)\,\opmu(y,x)}{\n{x-y}^a}\dd x\dd y = \Tr{\tilde{\opmu}\, \sfX_{\opmu}},
\end{align*}

and using the cyclicity of the trace, Hölder's inequality and Inequality (<ref>) with $\opmu_2 = \sfX_{\tilde{\opmu}}\,\opmu\opmu^*$, we get

\begin{multline*}
\Nrm{\sfX_{\tilde{\opmu}}\,\opmu}{2}^2 = \Tr{\sfX_{\tilde{\opmu}^*}\,\sfX_{\tilde{\opmu}}\,\opmu\opmu^*} = \Tr{\tilde{\opmu}^*\,\sfX_{(\sfX_{\tilde{\opmu}}\,\opmu\opmu^*)}}
\\
\leq \Nrm{\tilde{\opmu}}{2} \Nrm{\sfX_{(\sfX_{\tilde{\opmu}}\,\opmu\opmu^*)}}{2} \leq C_{a}\,h^{-a}\Nrm{\tilde{\opmu}}{2} \Nrm{\sfX_{\tilde{\opmu}}\,\opmu\opmu^*\n{\opp}^a}{2}.
\end{multline*}

By Hölder's inequality, this leads to

\begin{align*}
\Nrm{\sfX_{\tilde{\opmu}}\,\opmu}{2}^2 \leq C_{a}\,h^{-a}\Nrm{\tilde{\opmu}}{2} \Nrm{\sfX_{\tilde{\opmu}}\,\opmu}{2}\Nrm{\opmu^*\n{\opp}^a}{\infty}.
\end{align*}

We deduce the result by dividing both sides by $\Nrm{\sfX_{\tilde{\opmu}}\,\opmu}{2}$ and then using the fact that for $q\geq2$, $\Nrm{\sfX_{\tilde{\opmu}}\,\opmu}{q}\leq \Nrm{\sfX_{\tilde{\opmu}}\,\opmu}{2}$.

The following lemma will allow us to replace the Hilbert–Schmidt norm on the right-hand side of Inequality (<ref>) by another Schatten norm with higher index at the expense of using a less sharp power on $\n{\opp}$.

Let $\opmu$ be a compact operator. Then for any $\alpha > a$ and any $q\in[2,\infty]$, it holds

\begin{equation}\label{eq:X_p_bound}
\Nrm{\sfX_{\opmu}}{q} \leq C\, h^{-\alpha} \Nrm{\opmu\(1+\n{\opp}^{\alpha}\)}{q},
\end{equation}

for a constant $C$ depending only on $a$ and $\alpha$.

Take $(\varphi,\phi)\in (L^2)^2$. Then one has

\begin{align*}
\Inprod{\varphi}{\sfX_{\opmu}\, \phi}_{L^2} = \iintd \frac{\opmu(x,y)\,\conj{\varphi(x)}\,\phi(y)}{\n{x-y}^a} \dd x \dd y = \(2\pi\)^{3-a} C_{a} \Tr{\opmu\, \varphi \(-\Delta\)^\frac{a-3}{2}\phi},
\end{align*}

where $\varphi$ and $\phi$ are seen as multiplication operators and $C_{a} = \frac{\omega_a}{\omega_{3-a}}$. By the definition of $\opp$, this can be written

\begin{align*}
\Inprod{\varphi}{\sfX_{\opmu}\, \phi}_{L^2} = C_{a} h^{3-a} \Tr{\opmu\, \varphi \n{\opp}^{a-3}\phi} = C_{a} h^{3-a} \Tr{m\,\opmu\,m\,m^{-1} \varphi\, g(\opp)\,\phi\,m^{-1}},
\end{align*}

with $g(x) = \n{x}^{a-3}$ and $m=1+\n{\opp}^{\alpha}$. Now taking $1 \leq \frac{3}{\alpha} < p_0' < \frac{3}{a} < p_1' \leq \infty$ such that $\frac{1}{p_0'}+\frac{1}{p_1'} = \frac{a}{3}$, we have $g \in L^{p_0} + L^{p_1}$, hence we can write $g = g_0+g_1$ with $(g_0,g_1)\in L^{p_0}\times L^{p_1}$. Let $\tilde{g} = g_0$ or $\tilde{g} = g_1$, or more generally, take $\tilde{g} \in L^p$ for some $p\geq 1$ verifying $p'>\frac{3}{\alpha}$. Then, by Hölder's inequality for Schatten norms, Lemma <ref> and the Kato-Seiler-Simon Inequality (<ref>), we have

\begin{align*}
h^{3} &\n{\Tr{m^\frac{1}{2}\,\opmu\,m^\frac{1}{2}\,m^{-\frac{1}{2}} \varphi\, \tilde{g}(\opp)\,\phi\,m^{-\frac{1}{2}}}}
\\
&\leq \Nrm{m^\frac{1}{2}\,\opmu\,m^\frac{1}{2}}{\infty} \Nrm{m^{-\frac{1}{2}}\varphi^\frac{1}{p'}}{\L^{2p'}} \Nrm{\varphi^\frac{1}{p} \tilde{g}(\opp)^\frac{1}{2}}{\L^{2p}} \Nrm{\tilde{g}(\opp)^\frac{1}{2}\phi^\frac{1}{p}}{\L^{2p}} \Nrm{\phi^\frac{1}{p'} m^{-\frac{1}{2}}}{\L^{2p'}}
\\
&\leq C_{p}^\frac{1}{p'} \Nrm{\opmu\,m}{\infty} \Nrm{\varphi}{L^2} \Nrm{\tilde{g}}{L^p} \Nrm{\phi}{L^2},
\end{align*}

where we used the notation $z^b = \n{z}^{b-1} z$ and with $C_{p} = \intd \frac{\d y}{\(1+\n{y}^\alpha\)^{p'}}$. This constant is finite since $\alpha p' > 3$. This proves Inequality (<ref>) when $q=\infty$. When $q=2$, the inequality follows from Formula (<ref>). The other cases follow by complex interpolation.

§.§ Commutators Involving the Exchange Term

Let $a\in[0,1]$. Then there exists $C>0$ such that for any compact self-adjoint operators $\op$ and $\opmu$, any $q\in[1,\infty]$ and any integer $n\geq 2a-1$

\begin{align*}
\frac{1}{\hbar} \Nrm{\com{h^3\sfX_{\op},\opp_\jj^n}\opmu\,}{\L^q} &\leq 3^n\,h^{\frac{3}{2}-a}\,C \Nrm{\Dhxj{\op}\, m}{\L^2} \Nrm{\opmu\, m}{\L^q},
\end{align*}

where $m = 1+\n{\opp}^n$.

By Formula (<ref>), and the triangle inequality, we have

\begin{align*}
\Nrm{\com{\sfX_{\op},\opp_\jj^n}\opmu\,}{q} &\leq \sum_{k=1}^n \binom{n}{k} \Nrm{\sfX_{\ad{\opp_\jj}^k\!(\op)}\,\opp_\jj^{n-k}\,\opmu}{q}.
\end{align*}

Now, Lemma <ref> gives the bound

\begin{equation*}
\Nrm{\sfX_{\ad{\opp_\jj}^k\!(\op)}\,\opp_\jj^{n-k}\,\opmu}{q} \leq C_{a}\,h^{-a}\Nrm{\ad{\opp_\jj}^k\!(\op)\n{\opp}^{a\(1-\theta\)}}{2} \Nrm{\opmu\, \opp_\jj^{n-k}\n{\opp}^{\theta a}}{\infty}.
\end{equation*}

Using the fact that $\ad{\opp_\jj}\!(\op) = -i \hbar \Dhxj{\op}$ and expanding the $k-1$ commutators in $\ad{\opp_\jj}^{k-1}$ by Lemma <ref>, we get

\begin{align*}
\Nrm{\ad{\opp_\jj}^k\!(\op)\n{\opp}^{a\(1-\theta\)}}{2} \leq 2^k\hbar \Nrm{\Dhxj{\op} \n{\opp}^{a\(1-\theta\)+k-1}}{2}.
\end{align*}

Now when $k\geq a$, we take $\theta = 1$ so that $n-k+\theta a \leq n$ and $a\(1-\theta\)+k-1 = k-1 \leq n$. When $k < a$, we take $\theta = 0$ so that $n-k+\theta a = n-k \leq n$ and $a\(1-\theta\)+k-1 \leq 2a-1 \leq n$. In all the cases, this leads to

\begin{align*}
\frac{1}{\hbar} \Nrm{\com{\sfX_{\op},\opp_\jj^n}\opmu\,}{q} &\leq C\,h^{-a}\sum_{k=1}^n \binom{n}{k} 2^k \Nrm{\Dhxj{\op} \,m}{2} \Nrm{\opmu\, m}{\infty}.
\end{align*}

We conclude using the fact that $\Nrm{\opmu\, m}{\infty} \leq \Nrm{\opmu\, m}{q}$ and the definition (<ref>) of the $\L^2$ norm.

Let $a\in[0,1]$, $\fb = \frac{3}{a+1}$ and $n\in\N$ verifying $n\geq 2a$. Then for any $\alpha\in(a,n-a]$ and any $q\in[2,\infty]$

\begin{align}\label{eq:exchange_com_1}
\frac{1}{\hbar} \Nrm{\com{h^3\sfX_{\opmu},\op}\opp_\jj^n}{\L^q} &\leq 3^n\,C\, h^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)}\Nrm{\op \,m}{\L^\infty} \Nrm{\opmu\, m}{\L^{2}}
\\\label{eq:exchange_com_2}
\frac{1}{\hbar} \Nrm{\com{h^3\sfX_{\opmu},\op}\opp_\jj^n}{\L^q} &\leq 3^n\,C \Nrm{\op\,m}{\L^\infty} \(h^{\frac{3}{\beta'}}\Nrm{\opmu\,m}{\L^q} + h^{\frac{3}{2}-a} \Nrm{\Dhxj{\opmu}\,m}{\L^2}\),
\end{align}

where $m = 1+\n{\opp}^n$ and $\beta = \frac{3}{\alpha+1}$.

Note that the power of $h$ in the first formula is nonnegative only for $q\leq q_a$ with $\frac{1}{q_a} = \frac{1}{\fb} - \frac{1}{2}$. In the second formula, this is true for every $q$ but involves a semiclassical derivative of $\opmu$.

Since the exchange term is vanishing when $\hbar\to 0$, we can estimate the two parts of the commutator separately by writing

\begin{align*}
\Nrm{\com{\sfX_{\opmu},\op}\opp_\jj^n}{q} = \Nrm{\opp_\jj^n\com{\sfX_{\opmu},\op}}{q} \leq \Nrm{\opp_\jj^n\,\op\,\sfX_{\opmu}}{q} + \Nrm{\opp_\jj^n\,\sfX_{\opmu}\,\op}{q}.
\end{align*}

The first term in the right-hand side can be bounded using Hölder's inequality for Schatten norms and Lemma <ref> with $\theta = 0$, leading to

\begin{equation}\label{eq:exchange_com_3}
\Nrm{\opp_\jj^n\,\op\,\sfX_{\opmu}}{q} \leq C\,h^{-a}\Nrm{\opp_\jj^n\,\op}{\infty} \Nrm{\opmu \n{\opp}^a}{2} \leq C\, h^{-a} \Nrm{\op\,m}{\infty} \Nrm{\opmu\,m}{2}.
\end{equation}

We can also use Lemma <ref> with $\alpha \in (a,n]$, to get

\begin{equation}\label{eq:exchange_com_3_bis}
\Nrm{\opp_\jj^n\,\op\,\sfX_{\opmu}}{q} \leq C\,h^{-\alpha} \Nrm{\opp_\jj^n\,\op}{\infty} \Nrm{\opmu\(1+\n{\opp}^\alpha\)}{q} \leq C\, h^{-\alpha} \Nrm{\op\,m}{\infty} \Nrm{\opmu\,m}{q}.
\end{equation}

To treat the second term, we want to put the first weight $m$ either on $\opmu$ either on $\op$. To obtain this effect, we use Formula (<ref>) to get

\begin{equation}\label{eq:expansion_pn_X}
\Nrm{\opp_\jj^n\,\sfX_{\opmu}\,\op}{q} \leq \sum_{k=0}^n \binom{n}{k} \Nrm{\sfX_{\ad{\opp_\jj}^k\!(\opmu)}\,\opp_\jj^{n-k}\,\op\,}{q}.
\end{equation}

Now we use Lemma <ref> and then expand the commutators by Lemma <ref> to get for any $\theta\in\{0,1\}$

\begin{align*}
\Nrm{\sfX_{\ad{\opp_\jj}^k\!(\opmu)}\,\opp_\jj^{n-k}\,\op\,}{q} &\leq C\,h^{-a}\Nrm{\ad{\opp_\jj}^k\!(\opmu)\n{\opp}^{a\(1-\theta\)}}{2} \Nrm{\op\,\opp_\jj^{n-k}\n{\opp}^{\theta a}}{\infty}
\\
&\leq 2^k\, C\, h^{-a} \Nrm{\opmu \(1+\n{\opp}^{k+a\(1-\theta\)}\)}{2} \Nrm{\op \(1+\n{\opp}^{n-k+\theta a}\)}{\infty},
\end{align*}

and similarly as in the proof of Proposition <ref>, if $k\geq a$, we take $\theta = 1$ and if $k\leq a$, we take $\theta = 0$ and use the fact that $2a\leq n$. In any cases, the power on $\n{\opp}$ is smaller than $n$. Therefore, recalling Inequality (<ref>), we obtain

\begin{equation}\label{eq:exchange_com_4}
\Nrm{\opp_\jj^n\,\sfX_{\opmu}\,\op}{q} \leq 3^n\,C\,h^{-a} \Nrm{\op\,m}{\infty} \Nrm{\opmu\,m}{2}.
\end{equation}

Combining inequalities (<ref>) and (<ref>) and using the definition (<ref>) of $\L^q$ norms yield Formula (<ref>).

To get Formula (<ref>), we start back from Inequality (<ref>). If $k> a$, so that in particular $k\geq 1$, we use again Lemma <ref> and Lemma <ref> but we use first the fact that $\ad{\opp_j}\!(\opmu) = -i\hbar\,\Dhxj{\opmu}$ to get an additional $\hbar$. This yields

\begin{align*}
\Nrm{\sfX_{\ad{\opp_\jj}^k\!(\opmu)}\,\opp_\jj^{n-k}\,\op\,}{q} \leq 2^k\, C\, h^{1-a} \Nrm{\Dhxj{\opmu} \(1+\n{\opp}^{k-1}\)}{2} \Nrm{\op \(1+\n{\opp}^{n-k+ a}\)}{\infty},
\end{align*}

If $k\leq a$, then we use Lemma <ref> with $\alpha\in(a,n-a]$ to get

\begin{align*}
\Nrm{\sfX_{\ad{\opp_\jj}^k\!(\opmu)}\,\opp_\jj^{n-k}\,\op\,}{q} \leq 2^k\, C\, h^{-\alpha} \Nrm{\opmu \(1+\n{\opp}^{k+\alpha}\)}{q} \Nrm{\op \(1+\n{\opp}^{n-k}\)}{\infty}.
\end{align*}

Therefore, Inequality (<ref>) implies

\begin{equation}\label{eq:exchange_com_5}
\Nrm{\opp_\jj^n\,\sfX_{\opmu}\,\op}{q} \leq 3^n\,C \Nrm{\op\,m}{\infty} \(h^{-\alpha}\Nrm{\opmu\,m}{q} + h^{1-a} \Nrm{\Dhxj{\opmu}\,m}{2}\),
\end{equation}

and together with Inequality (<ref>) and the definition of the $\L^q$ norm, this implies Formula (<ref>).

§.§ Proof of the Propagation of Regularity

The strategy to prove Proposition <ref> is to look at Equations (<ref>), (<ref>) and (<ref>) and find a Grönwall-type inequality on $\Nrm{\op}{\cW^{1,q}(m)}$. In particular we will see that to close the Grönwall argument for $q=2$, we need to estimate $\Nrm{\op}{\cW^{1,q}(m)}$ for $q\in(2,4)$. We will therefore proceed by interpolation and define

\begin{equation*}
M_2(t):= \Nrm{\op}{\cW^{1,2}(m)},
\quad
\quad
M_\infty(t) :=\Nrm{\op\,m}{\L^\infty}.
\end{equation*}

For $a\in \lt[\frac{1}{2},1\rt]$ we will find a Grönwall-type inequality on $M_2(t)+M_4(t)+M_\infty(t)$, whereas for $a\in \lt[0,\frac{1}{2}\rt)$ it suffices to apply Grönwall's Lemma to $M_2(t)+M_4(t)$.

We now look at Equation (<ref>). By Proposition <ref> we have that

\begin{equation*}
\frac{1}{\hbar}\Nrm{\com{V_{\op},m}\op}{\L^q}\leq C\Nrm{\op\,m}{\L^r}\Nrm{\op\,m}{\L^{r_1}}.
\end{equation*}

The contribution given by the exchange term in the right-hand side of (<ref>) can be bounded by Proposition <ref> with $\opmu=\op$. Therefore, we obtain the following bounds on the right-hand side of Equation (<ref>)

\begin{equation*}
\dt\Nrm{\op\,m}{\L^q} \leq C \Nrm{\op\,m}{\L^r}\Nrm{\op\,m}{\L^{r_1}} + h^{\frac{3}{2}-a}\Nrm{\Dhx\op\,m}{\L^2}\Nrm{\op\,m}{\L^q}.
\end{equation*}

In particular, for $q=\infty$ we get

\begin{equation*}
\dt\Nrm{\op\,m}{\L^{\infty}}\leq C \Nrm{\op\,m}{\L^{\frac{3}{2-a}}}\Nrm{\op\,m}{\L^\infty}+h^{\frac{3}{2}-a}\Nrm{\Dhx\op\,m}{\L^2}\Nrm{\op\,m}{\L^\infty}.
\end{equation*}

Note that we cannot close the Grönwall's inequality and we will need bounds on $\Dhx\op$ and $\Dhv\op$. To this end, we look at Equation (<ref>) and Equation (<ref>). We start bounding the right-hand side of Equation (<ref>). By Proposition <ref> we obtain

\begin{equation}\label{eq:com_V_Dhx}
\dfrac{1}{\hbar}\Nrm{\com{V_{\op},m} \Dhx\op}{\L^q} \leq \Nrm{\op\,m}{\L^r}\Nrm{\Dhx\op\,m}{\L^{r_1}}.
\end{equation}

By writing $\com{E_{\op},\op} m = \com{E_{\op},\op\,m} + \op\,\com{E_{\op},m}$, applying Proposition <ref> with $\op_2=\op\,m$ and Proposition <ref> with $\opmu=\op$, we get

\begin{equation}\label{eq:E_com_est}
\dfrac{1}{\hbar}\Nrm{\com{E_{\op},\op} m}{\L^q} \leq C \Nrm{\rho}{L^r}^{1-s}\Nrm{\rho}{W^{1,r}}^s \Nrm{\Dhv\op\,m}{\L^q} + C\Nrm{\Dhx\op\,m}{\L^r} \Nrm{\op\,m}{\L^{r_1}}
\end{equation}

for $q > 2$ and $s=1-3\(\frac{1}{r'}-\frac{1}{\fb'}\)$, where we used the interpolation of Besov spaces stated in Corollary <ref>.

For $q = 2$, we have

\begin{equation}\label{eq:E_com_est>}
\dfrac{1}{\hbar}\Nrm{\com{E_{\op},\op}m}{\L^{2}} \leq C \Nrm{\nabla\rho}{L^{\fb',1}} \Nrm{\Dhv\op\,m}{\L^{2}} +C \Nrm{\Dhx\op\,m}{\L^r} \Nrm{\op\,m}{\L^{r_1}}
\end{equation}

for $a\in\lt[\frac{1}{2},1\rt]$, and

\begin{equation}\label{eq:E_com_est<}
\dfrac{1}{\hbar}\Nrm{\com{E_{\op},\op}m}{\L^{2}} \leq C \Nrm{\rho}{L^{\frac{3}{1-a},1}} \Nrm{\Dhv\op\,m}{\L^{2}} +C \Nrm{\Dhx\op\,m}{\L^r} \Nrm{\op\,m}{\L^{r_1}}
\end{equation}

for $a\in\(0,\frac{1}{2}\)$.

The contributions of the exchange term can be bounded using Proposition <ref> and Proposition <ref>. Combining them with (<ref>) leads to, for $q > 2$,

\begin{multline}\label{eq:Dhx_rho_est}
\dt\Nrm{\Dhx\op\,m}{\L^q} \leq C \Nrm{\op\,m}{\L^{r}}\Nrm{\Dhv\op\,m}{\L^{r_1}}+C\Nrm{\rho}{B_{r,1}^{1-3\(\frac{1}{r'}-\frac{1}{\fb}\)}}\Nrm{\Dhv\op\,m}{\L^q}
\\
+C \Nrm{\Dhx\op\,m}{\L^r}\Nrm{\op\,m}{\L^{r_1}} + Ch^{\frac{3}{2}-a}\Nrm{\Dhx\op\,m}{\L^2}\Nrm{\Dhx\op\,m}{\L^q}
\\
+ Ch^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} \Nrm{\op\,m}{\L^\infty}\Nrm{\Dhx\op\,m}{\L^{2}}.
\end{multline}

To bound the right-hand side of Equation (<ref>), we use Proposition <ref> for the contribution due to the direct term and Proposition <ref> and Proposition <ref> to estimate the contributions of the exchange term. Hence,

\begin{multline*}
\dt\Nrm{\Dhv\op\,m}{\L^q} \leq C \Nrm{\op\,m}{\L^{r}}\Nrm{\Dhv\op\,m}{\L^{r_1}} + C\Nrm{\Dhx\op\,m}{\L^q}
\\
+C h^{\frac{3}{2}-a}\Nrm{\Dhx\op\,m}{\L^2}\Nrm{\Dhv\op\,m}{\L^q}+C h^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)}\Nrm{\op\,m}{\L^\infty}\Nrm{\Dhv\op\,m}{\L^{2}}.
\end{multline*}

To get an estimate in $\L^2$ we need a bound on the $\L^q$ norm for $q\in(2,4)$. Therefore we look for a bound when $q=4$ and proceed by interpolation.

To establish a Grönwall type inequality for $a\geq\frac{1}{2}$, we consider the sum $M_2(t)+M_4(t)+M_\infty(t)$ and observe that it satisfies

\begin{multline}\label{eq:M2-M4-gronwall}
\dt\lt[M_2(t)+M_4(t)+M_\infty(t)\rt] \leq C M_2(t) + C M_4(t) + CM_\infty(t)
\\
+ C\(1 + h^{\frac{3}{2}-a} + h^{\frac{3}{\fb'}} + h^{3\(\frac{3}{4}-\frac{1}{\fb}\)}\) [M_2(t)+M_4(t)+M_{\infty}(t)]^2,
\end{multline}

where we used interpolation inequality with $\theta\in(0,1)$ and Young's inequality for products to bound

\begin{equation*}
\begin{split}
\Nrm{\op\,m}{\L^r}&\leq C\Nrm{\op\,m}{\L^2}^\theta\Nrm{\op\,m}{\L^4}^{1-\theta}\leq C \Nrm{\op\,m}{\L^2}+\Nrm{\op\,m}{\L^4}\,,
\\
\Nrm{\Dhx\op\,m}{\L^r}&\leq C\Nrm{\Dhx\op\,m}{\L^2}^\theta\Nrm{\Dhx\op\,m}{\L^4}^{1-\theta}\leq C \Nrm{\Dhx\op\,m}{\L^2}+ C\Nrm{\Dhx\op\,m}{\L^4}\,,
\\
\Nrm{\Dhv\op\,m}{\L^r}&\leq C\Nrm{\Dhv\op\,m}{\L^2}^\theta\Nrm{\Dhv\op\,m}{\L^4}^{1-\theta}\leq C \Nrm{\Dhv\op\,m}{\L^2}+C\Nrm{\Dhv\op\,m}{\L^4},
\end{split}
\end{equation*}

whenever $r\in(2,4)$. We observe that Equation (<ref>) is a Grönwall-type inequality of the same form as Equation (<ref>). Thus there exists a time $T>0$, depending only in the initial data, such that $M_2(t)+M_4(t)$ is bounded for all $t\in[0,T]$.

For $a<\frac{1}{2}$, we consider the quantity
$M_2(t)+M_4(t)$ and use that

\begin{equation*}
\Nrm{\op\,m}{\L^\infty} \leq C h^{-\frac{3}{q}}\Nrm{\op\,m}{\L^q}.
\end{equation*}

\begin{multline*}
\dt\(M_2(t)+M_4(t)\)
\\
\leq C\(M_2(t)+M_4(t)\) + C \(1+h^{\frac{3}{2}-a}+h^{3\(\frac{1}{\fb'}-\frac{1}{2}\)}\) \(M_2(t)+M_4(t)\)^2.
\end{multline*}

Therefore there exists $T>0$, depending only in the initial data, such that $M_2(t)+M_4(t)$ is bounded for all $t\in[0,T]$, thus $\op\in L^{\infty}((0,T),\cW^{1,2}(m)\cap\cW^{1,4}(m)\cap\L^\infty(m))$. Moreover, ${\rho} \in L^\infty((0,T),H^{1} \cap W^{1,4} \cap L^1 \cap L^\infty)$ thanks to Proposition <ref> and the bounds on $M_2(t),\,M_4(t)$ and $M_\infty(t)$.

Similarly to what we have done for the first-order quantum gradients, we can compute the time derivative of the second order quantum gradients of $\op$

\begin{equation}\label{eq:grad2-moments}
\begin{split}
i\hbar\,\partial_t\,\Dhx^2\op &= \com{H,\Dhx^2\op} - 2\com{E_{\op},\Dhx\op} - \com{\Dhx E_{\op},\op} - 2\com{h^3\sfX_{\Dhx\op},\Dhx\op} - \com{h^3\sfX_{\Dhx^2\op},\op}
\\
i\hbar\,\partial_t\,\Dhv^2\op &= \com{H,\Dhv^2\op} - i\hbar\Dhv\Dhx\op - 2\com{h^3\sfX_{\Dhv\op},\Dhv\op} - \com{h^3\sfX_{\Dhv^2\op},\op}
\\
i\hbar\,\partial_t\,\Dhv\Dhx\op &= \com{H,\Dhv\Dhx\op} - i\hbar\Dhx^2\op - \com{E_{\op},\Dhv\op} - \com{h^3\sfX_{\Dhv\Dhx\op},\op} - \com{h^3\sfX_{\Dhx\op},\Dhv\op},
\end{split}
\end{equation}

that are of the form

\begin{equation}\label{eq:abstract_equation_2}
i\hbar\, \dpt \opmu = \com{\sfA,\opmu} + \com{\sfB,\Dhx\op} + \com{\sfC,\op},
\end{equation}

with $\sfA,\, \sfB$ and $\sfC$ being self-adjoint operators. The proof of Lemma <ref> proves also the following statement.

Let $\op,\, \sfA,\,\sfB,\,\sfC$ be self-adjoint operators and $\opmu = \opmu(t)$ be a family of self-adjoint operators verifying (<ref>). Then, formally, for any even integer $q\geq 2$ we have

\begin{align*}
\dt \Nrm{\opmu\, m}{q} \leq \frac{1}{\hbar}\Nrm{\com{\sfA,m}\opmu}{q} + \frac{1}{\hbar} \Nrm{\com{\sfB,\Dhx\op}m}{q}+\frac{1}{\hbar} \Nrm{\com{\sfC,\op}m}{q}.
\end{align*}

We consider the Identities (<ref>) and bound them by Lemma <ref>. This yields

\begin{multline}\label{eq:grad_xx}
\hbar\,\dt\Nrm{\Dhx^2\op\,m}{q} \leq C \Nrm{\com{V_{\op},m} \Dhx^2\op}{q}
+C \Nrm{\com{E_{\op},\Dhx\op} m}{q} +C \Nrm{\com{\Dhx E_{\op},\op} m}{q}
\\
+C \Nrm{\com{h^3\sfX_{\op},m} \Dhx^2\op}{q} + C\Nrm{\com{h^3\sfX_{\Dhx\op},\Dhx\op} m}{q} + C\Nrm{\com{h^3\sfX_{\Dhx^2\op},\op} m}{q}
\end{multline}

\begin{equation}\label{eq:grad_vv}
\begin{aligned}
\hbar&\,\dt\Nrm{\Dhv^2\op\,m}{q} \leq C \Nrm{\com{V_{\op},m} \Dhv^2\op}{q} + C\Nrm{\Dhv\Dhx\op\,m}{q}
\\
&+ C\Nrm{\com{h^3\sfX_{\op},m}\Dhv^2\op}{q} + C\Nrm{\com{h^3\sfX_{\Dhv\op},\Dhv\op} m}{q} + C\Nrm{\com{h^3\sfX_{\Dhv^2\op},\op} m}{q}
\end{aligned}
\end{equation}

\begin{equation}\label{eq:grad_vx}
\begin{split}
\hbar\,\dt\Nrm{\Dhv\Dhx\op\,m}{q} &\leq C \Nrm{\com{V_{\op},m} \Dhv\Dhx\op}{q} + C \Nrm{\com{E_{\op},\Dhv\op} m}{q}
\\
&\quad+ C \Nrm{\Dhx^2 \op\,m}{q} + C \Nrm{\com{h^3\sfX_{\op},m} \Dhv\Dhx\op}{q}
\\
&\quad+ C \Nrm{\com{h^3\sfX_{\Dhx\op},\Dhv\op}m}{q} + C \Nrm{\com{h^3\sfX_{\Dhv\Dhx\op},\op} m}{q}.
\end{split}
\end{equation}

We now estimate the right-hand side of Equation (<ref>). The first three contributions are related to the direct term in the Hartree equation, whereas in the others the exchange operator appears. By Proposition <ref> we have

\begin{equation}\label{eq:com_V_est}
\frac{1}{\hbar} \Nrm{\com{V_{\op},m} \Dhx^2\op}{\L^q} \leq C \Nrm{\op\,m}{{\L}^{r}}\Nrm{\Dhx^2\op\,m}{{\L}^{r_1}}
\end{equation}

where $\dfrac{1}{r} + \dfrac{1}{r_1} = \dfrac{1}{q} + \dfrac{1}{\fb'}$. As for the second term on the right-hand side of Equation (<ref>), we rewrite it as follows:

\begin{equation*}
\frac{1}{\hbar} \Nrm{\com{E_{\op},\Dhx\op} m}{\L^q} = \frac{1}{\hbar} \Nrm{\com{E_{\op},\Dhx\op\, m}}{\L^q} + \frac{1}{\hbar} \Nrm{\com{E_{\op},m} \Dhx\op}{\L^q}.
\end{equation*}

By Proposition <ref> and Corollary <ref>, we get

\begin{equation*}
\frac{1}{\hbar} \Nrm{\com{E_{\op},\Dhx\op\,m}}{\L^q} \leq C \Nrm{\rho}{L^r}^{1-s} \Nrm{\rho}{W^{1,r}}^s \Nrm{\Dhv\Dhx\op\,m}{\L^q},
\end{equation*}

for $\frac{1}{r}+\frac{1}{q}=\frac{1}{2}$ and $s=1-3\(\frac{1}{r'}-\frac{1}{\fb}\)$. By Proposition <ref>, for $\frac{1}{r}+\frac{1}{r_1} = \frac{1}{q}+\frac{1}{\fb'}$ we have

\begin{equation*}
\frac{1}{\hbar} \Nrm{\com{E_{\op},m} \Dhx\op}{\L^q}\leq C \Nrm{\Dhx\op\,m}{\L^{r}} \Nrm{\Dhx\op\,m}{\L^{r_1}}.
\end{equation*}

Whence, the second term in the right-hand side of Equation (<ref>) is bounded by

\begin{equation*}
\frac{1}{\hbar} \Nrm{\com{E_{\op},\Dhx\op} m}{\L^{q}} \leq C \Nrm{\rho}{L^r}^{1-s}\Nrm{\rho}{W^{1,r}}^s \Nrm{\Dhv\Dhx\op\,m}{\L^{q}} + C\Nrm{\Dhx\op\,m}{\L^{r}}\Nrm{\Dhx\op\,m}{\L^{r_1}}.
\end{equation*}

The third term on the right-hand side of Equation (<ref>) can be dealt in an analogously manner as to the second term by using that $\Dhx E_{\op}=E_{\Dhx\op}$ and Proposition <ref>. This gives

\begin{equation*}
\frac{1}{\hbar} \Nrm{\com{\Dhx E_{\op},\op} m}{\L^{q}} \leq C \Nrm{\nabla\rho}{L^r}^{1-s} \Nrm{\nabla\rho}{W^{1,r}}^s \Nrm{\Dhv\op\,m}{\L^{q}} + C\Nrm{\Dhx^2\op\,m}{\L^r} \Nrm{\op\,m}{\L^{r_1}}\,.
\end{equation*}

We now turn to terms in which the contribution of the exchange term appears. By Proposition <ref> we obtain

\begin{equation}\label{eq:exchange_moments}
\frac{1}{\hbar} \Nrm{\com{h^3\sfX_{\op},m} \Dhx^2\op}{\L^q} \leq C \hbar^{\frac{3}{2}-a} \Nrm{\Dhx\op\,m}{\L^2} \Nrm{\Dhx^2\op\,m}{\L^q}\,.
\end{equation}

By Proposition <ref> we get the bound

\begin{equation}\label{eq:exchange_com}
\frac{1}{\hbar} \Nrm{\com{h^3\sfX_{\Dhx\op},\Dhx\op} m}{\L^q} \leq C \hbar^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} \Nrm{\Dhx\op\,m}{\L^2} \Nrm{\Dhx\op\,m}{\L^\infty}\,.
\end{equation}

Finally, by noticing that $\Dhx\sfX_{\Dhx\op}=\sfX_{\Dhx^2\op}$, we apply Proposition <ref> to the last term on the right-hand side in Equation (<ref>):

\begin{equation}\label{eq:exchange_com_grad}
\frac{1}{\hbar} \Nrm{\com{h^3\sfX_{\Dhx^2\op},\op} m}{\L^q} \leq C \hbar^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} \Nrm{\op\,m}{\L^\infty} \Nrm{\Dhx^2\op\,m}{\L^2}\,.
\end{equation}

Therefore, using Proposition <ref> and estimates (<ref>)–(<ref>)
we obtain

\begin{equation}\label{eq:grad_xx_est1}
\begin{split}
\dt\Nrm{\Dhx^2\op\,m}{\L^{q}} &\leq C \Nrm{\Dhx^2\op\,m}{\L^{r_1}} +C \Nrm{\Dhv\Dhx\op\,m}{\L^{q}}
\\
&\quad +\Nrm{\nabla\rho}{W^{1,r}}^s+ \Nrm{\Dhx^2\op\,m}{\L^r} + \hbar^{\frac{3}{2}-a}\Nrm{\Dhx^2\op\,m}{\L^{q}}
\\
&\quad + \hbar^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} \Nrm{\Dhx\op\,m}{\L^\infty} + \hbar^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} \Nrm{\Dhx^2\op\,m}{\L^{2}}
\end{split}
\end{equation}

for $s=1-3\(\frac{1}{r'}-\frac{1}{\fb}\)$ and with the constraints
$\frac{1}{r}+\frac{1}{r_1} = \frac{1}{q}+\frac{1}{\fb'}$ and $\frac{1}{r}+\frac{1}{q} = \frac{1}{2}$.

We now look at the right-hand side of Equation (<ref>). By using Proposition <ref>, Proposition <ref>, Proposition <ref>, Proposition <ref> and by Proposition <ref> with $\Dhv\sfX_{\Dhv\op}=\sfX_{\Dhv^2\op}$ we obtain

\begin{equation}\label{eq:grad_vv_est1}
\begin{split}
\dt\Nrm{\Dhv^2\op\,m}{\L^{q}} &\leq C \Nrm{\Dhv^2\op\,m}{\L^{r_1}} + C \hbar^{\frac{3}{2}-a} \Nrm{\Dhv^2\op\,m}{\L^{q}}
\\
&\quad+ C\hbar^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} \Nrm{\Dhv\op\,m}{\L^\infty} + C\hbar^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} \Nrm{\Dhv^2\op\,m}{\L^{2}}
\end{split}
\end{equation}

for $s=1-3\(\frac{1}{r'}-\frac{1}{\fb}\)$ and with the constraints
$\frac{1}{r}+\frac{1}{r_1}=\frac{1}{q}+\frac{1}{\fb'}$ and $\frac{1}{r}+\frac{1}{q}=\frac{1}{2}$.

As for the mixed term (<ref>), its right-hand side can be bounded as follows. By Proposition <ref> we get

\begin{equation}\label{eq:com_V_est-vx}
\frac{1}{\hbar} \Nrm{\com{V_{\op},m} \Dhv\Dhx\op}{\L^q}\leq C \Nrm{\op\,m}{\L^r} \Nrm{\Dhv\Dhx\op\,m}{\L^{r_1}}
\end{equation}

where $\frac{1}{r}+\frac{1}{r_1} = \frac{1}{q}+\frac{1}{\fb'}$. As for the second term on the right-hand side of Equation (<ref>), we rewrite it as follows:

\begin{equation*}
\frac{1}{\hbar} \Nrm{\com{E_{\op},\Dhv\op} m}{\L^q} = \frac{1}{\hbar} \Nrm{\com{E_{\op},\Dhv\op\,m}}{\L^q} + \frac{1}{\hbar} \Nrm{\com{E_{\op},m} \Dhv\op}{\L^q}.
\end{equation*}

Proposition <ref> and Corollary <ref> yield

\begin{equation*}
\frac{1}{\hbar} \Nrm{\com{E_{\op},\Dhv\op\,m}}{\L^q} \leq C \Nrm{\rho}{L^r}^{1-s} \Nrm{\rho}{W^{1,r}}^s \Nrm{\Dhv^2\op\,m}{\L^q}.
\end{equation*}

By Proposition <ref>, we get

\begin{equation*}
\frac{1}{\hbar} \Nrm{\com{E_{\op},m} \Dhv\op}{\L^q} \leq C \Nrm{\Dhx\op\,m}{\L^r} \Nrm{\Dhv\op\,m}{\L^{r_1}}.
\end{equation*}

Whence, the second term in the right-hand side of Equation (<ref>) is bounded by

\begin{equation*}
\frac{1}{\hbar} \Nrm{\com{E_{\op},\Dhv\op} m}{\L^q} \leq C \Nrm{\rho}{L^r}^{1-s} \Nrm{\rho}{W^{1,r}}^s \Nrm{\Dhv^2\op\,m}{\L^q} + C\Nrm{\Dhx\op\,m}{\L^r}\Nrm{\Dhv\op\,m}{\L^{r_1}}.
\end{equation*}

We turn now on the terms in which the contribution of the exchange term appears. By Proposition <ref> we obtain

\begin{equation}\label{eq:exchange_moments-vx}
\frac{1}{\hbar} \Nrm{\com{\sfX_{\op},m} \Dhv\Dhx\op}{\L^q} \leq C \hbar^{\frac{3}{2}-a} \Nrm{\Dhx\op\,m}{\L^2} \Nrm{\Dhv\Dhx\op\,m}{\L^q}.
\end{equation}

By Proposition <ref> we get the bounds

\begin{align}\label{eq:exchange_com-vx}
\frac{1}{\hbar} \Nrm{\com{\sfX_{\Dhx\op},\Dhv\op} m}{\L^q} &\leq C \hbar^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} \Nrm{\Dhv\op\,m}{\L^\infty} \Nrm{\Dhx\op\,m}{\L^2}.
\\\label{eq:exchange_com_grad-vx}
\frac{1}{\hbar} \Nrm{\com{\sfX_{\Dhv\Dhx\op},\Dhx\op} m}{\L^q} &\leq C \hbar^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} \Nrm{\Dhx\op\,m}{\L^\infty} \Nrm{\Dhv\Dhx\op\,m}{\L^2}.
\end{align}

Therefore, using Proposition <ref> and estimates (<ref>)–(<ref>), we obtain

\begin{align*}
\dt\Nrm{\Dhv\Dhx\op\,m}{\L^q} &\leq C\Nrm{\Dhv\Dhx\op\,m}{\L^{r_1}} + C\Nrm{\Dhv^2\op\,m}{\L^q} + C\,\hbar^{\frac{3}{2}-a}\Nrm{\Dhv\Dhx\op\,m}{\L^q}
\\
&\quad + C\,\hbar^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} \Nrm{\Dhv\op\,m}{\L^\infty} + C\,\hbar^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} \Nrm{\Dhv\Dhx\op\,m}{\L^2}
\end{align*}

for $s=1-3\(\frac{1}{r'}-\frac{1}{\fb}\)$ and with the constraints
$\frac{1}{r}+\frac{1}{r_1} =\frac{1}{q}+\frac{1}{\fb'}$ and $\frac{1}{r}+\frac{1}{q}=\frac{1}{2}$. Now we define

\begin{equation*}
N_{x,q}(t) := \Nrm{\Dhx^2\op\,m}{\L^{q}},\quad N_{v,q}(t):=\Nrm{\Dhv^2\op\,m}{\L^{q}},\quad N_{xv,q}(t) := \Nrm{\Dhv\Dhx\op\,m}{\L^{q}}
\end{equation*}

and denote by $N_{q}(t)$ the quantity

\begin{equation*}
\end{equation*}

Then we proceed as for the first-order gradients. Using Proposition <ref>, we obtain

\begin{equation}\label{eq:Nx-q}
\begin{split}
\dt N_{x,q}(t)&\leq C\,N_{r_1}(t) + C\,N_{q}(t) + C\,N_{r}(t) + C\,h^{\frac{3}{2}-a}N_{q}(t)
\\
&\quad+ C\,h^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} \(N_{2}(t)+\Nrm{\Dhx\op\,m}{\L^\infty}\).
\end{split}
\end{equation}

\begin{equation}\label{eq:Nv-q}
\begin{split}
\dt N_{v,q}(t)&\leq C\,N_{r_1}(t)+h^{\frac{3}{2}-a}N_{q}(t)
\\
&\quad+ C\,h^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} \(N_{2}(t)+\Nrm{\Dhv\op\,m}{\L^\infty}\).
\end{split}
\end{equation}

\begin{equation}\label{eq:Nxv-q}
\begin{split}
\dt N_{vx,q}(t)&\leq C\,N_{r_1}(t)+C\,N_{q}(t)+C\,h^{\frac{3}{2}-a}N_{q}(t)
\\
&\quad+ C\,h^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)}\(N_{2}(t)+\Nrm{\Dhv\op\,m}{\L^\infty}\).
\end{split}
\end{equation}

For $a\in\left[\frac{1}{2},1\right]$, we consider the quantity
$F_{q,\infty}(t) := N_{q}(t) + \Nrm{\op\,m}{\dot{\cW}^{1,\infty}}$ and look for a Grönwall type inequality. From Equation (<ref>) and Equation (<ref>) with $q=\infty$, we obtain

\begin{align*}
\dt F_{q,\infty}(t) &\leq C\,F_{r_1,\infty}(t) + C\,F_{q,\infty}(t) + C\,F_{r,\infty}(t)
\\
&\quad + C\,h^{\frac{3}{2}-a} F_{q,\infty}(t) + C\,h^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} F_{2,\infty}(t),
\end{align*}

where we used Equation (<ref>) and Proposition <ref> to show that $\Nrm{\com{E_{\op},\op\,m}}{\L^\infty} \leq C\,\Nrm{\rho}{L^2}^{1/2}\Nrm{\nabla^2\rho}{L^2}^{1/2}\leq C\,\Nrm{\Dhx^2\op\,m}{\L^2}^{1/2}$. Since $(r,r_1)\in[2,4]^2$ standard interpolation allows to conclude by Grönwall's Lemma.

For $a\in\(0,\frac{1}{2}\)$, using that $\op\in\cW^{1,4}(m)$ by Proposition <ref> and that

\begin{equation*}
\Nrm{\Dhx\op\,m}{\L^\infty} \leq C\,h^{-\frac{3}{4}}\Nrm{\Dhx\op\,m}{\L^4},\quad\quad \Nrm{\Dhv\op\,m}{\L^\infty}\leq C h^{-\frac{3}{4}}\Nrm{\Dhv\op\,m}{\L^4},
\end{equation*}

we get

\begin{equation}\label{eq:N-q}
\begin{split}
\dt N_{x,q}(t) &\leq C\,N_{r_1}(t)+C\,N_{q}(t)+C\,N_{r}(t)+C\,h^{\frac{3}{2}-a}N_{q}(t)
\\
&\quad + C\,h^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}\)} N_{2}(t) + C\,h^{3\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}-\frac{1}{4}\)} M_4(t),
\end{split}
\end{equation}

where $M_4(t)$ is bounded by Proposition <ref>. We proceed analogously for $N_{v,q}$ and $N_{vx,q}$.
Hence, as for the first-order gradients, we consider
$N_2(t)+N_4(t)$ and close the Grönwall type inequality by observing that $r,\,r_1\in[2,4]$ and that we can actually bound $N_{r_1}(t)$ and $N_r(t)$ by interpolation with $N_2(t)$ and $N_4(t)$.
Notice that we have the constraint

\begin{equation*}
\(\frac{1}{q}+\frac{1}{2}-\frac{1}{\fb}-\frac{1}{4}\)>0,
\end{equation*}

which leads to $a<\frac{1}{2}$ when $q=4$. We conclude that $\op\in\cW^{2,2}(m)\cap\cW^{2,4}(m)$ for $a\in\(0,\frac{1}{2}\)$.

We observe that analogously to (<ref>), (<ref>) and (<ref>), the following bounds hold

\begin{equation*}
\dt\Nrm{\sqrt{\op}\,m}{q} \leq \dfrac{1}{\hbar} \Nrm{\com{V_{\op},m} \sqrt{\op}}{q} + \dfrac{1}{\hbar} \Nrm{\com{h^3X_{\op},m} \sqrt{\op}}{q},
\end{equation*}

\begin{equation}\label{eq:Dhx-sqrt}
\begin{aligned}
\dt\Nrm{\Dhx\sqrt{\op}\,m}{q} &\leq \dfrac{1}{\hbar} \Nrm{\com{V_{\op},m} \Dhx\sqrt{\op}}{q} + \dfrac{1}{\hbar} \Nrm{\com{E_{\op},\sqrt{\op}} m}{q}
\\
&\quad +\dfrac{1}{\hbar} \Nrm{\com{h^3X_{\op},m} \Dhx\sqrt{\op}}{q} + \dfrac{1}{\hbar} \Nrm{\com{h^3 X_{\Dhx\op},\sqrt{\op}} m}{q},
\end{aligned}
\end{equation}

\begin{equation}\label{eq:Dhv-sqrt}
\begin{aligned}
\dt\Nrm{\Dhv\sqrt{\op}\,m}{q} &\leq \frac{1}{\hbar}\Nrm{\com{V_{\op},m}\Dhv \sqrt{\op}}{q} + \Nrm{\Dhx\sqrt{\op}\, m}{q}
\\
&\quad + \frac{1}{\hbar}\Nrm{\com{h^3\sfX_{\op},m}\Dhv \sqrt{\op}}{q} + \frac{1}{\hbar} \Nrm{\com{h^3\sfX_{\Dhv \op},\sqrt{\op}}m}{q}.
\end{aligned}
\end{equation}

As in Proposition <ref>, we look for a Grönwall type inequality. To this end, we define

\begin{equation*}
\widetilde{M}_{q}(t) = \Nrm{\sqrt{\op}}{\L^{2}(m)} + \Nrm{\sqrt{\op}}{\L^{q}(m)},
\end{equation*}

for $q\in[2,\infty]$ and notice that, because of Propositions <ref> and <ref>, we have

\begin{equation*}
\dt\widetilde{M}_{q}(t)\leq C\,M_{r}\,\widetilde{M}_{\tilde{r}} + C\,M_2\,\widetilde{M}_{q}\,,
\end{equation*}

that implies the boundedness of $\widetilde{M}_{q}(t)$ for $q\in[2,\infty]$ thanks to Proposition <ref>.

We now define the quantity

\begin{equation*}
\widetilde{N}_{q}(t) = \Nrm{\sqrt{\op}}{\dot{\cW}^{1,2}(m)} + \Nrm{\sqrt{\op}}{\dot{\cW}^{1,q}(m)},
\end{equation*}

for $q\in[2,\infty]$ and using Equations (<ref>) and (<ref>) we compute

\begin{equation}\label{eq:gronwall-sqrt-1}
\dt\(\widetilde{N}_2(t)+\widetilde{N}_{q}(t)\).
\end{equation}