# Mean-field limits of Riesz-type singular flows with possible multiplicative
transport noise

Quoc Hung Nguyen<EMAIL_ADDRESS>,  Matthew Rosenzweig
<EMAIL_ADDRESS>and  Sylvia Serfaty<EMAIL_ADDRESS>
###### Abstract.

We provide a proof of mean-field convergence of first-order dissipative or
conservative dynamics of particles with Riesz-type singular interaction (the
model interaction is an inverse power $s$ of the distance for any $0<s<d$)
when assuming a certain regularity of the solutions to the limiting evolution
equations. It relies on a modulated-energy approach, as introduced in previous
works where it was restricted to the Coulomb and super-Coulombic cases. The
method also allows us to incorporate multiplicative noise of transport type
into the dynamics for the first time in this generality. It relies in
extending functional inequalities of [Ser20b, Ros20c, Ser20a] to more general
interactions, via a new, robust proof that exploits a certain commutator
structure.

Q.H.N. is supported by the Shanghai Tech University startup fund, the National
Natural Science Foundation of China (12050410257), and NSFC Fund’s staff
support.

M.R. is supported by the Simons Foundation through the Simons Collaboration on
Wave Turbulence.

S.S. is supported by NSF grant DMS-1700278 and by the Simons Foundation
through the Simons Investigator program.

##  1\. Introduction

###  1.1. Main goal and method

In this article, we consider the first-order mean-field dynamics of
interacting particle systems of the form

(1.1) $\begin{cases}\dot{x}_{i}=\displaystyle\frac{1}{N}\sum_{{1\leq j\leq
N}:j\neq i}{\mathbb{M}}\nabla{\mathsf{g}}(x_{i}-x_{j})\\\
x_{i}|_{t=0}=x_{i}^{0},\end{cases}\qquad i\in\\{1,\ldots,N\\}.$

Here, $N\in{\mathbb{N}}$ is the number of particles,
$x_{i}^{0}\in{\mathbb{R}}^{d}$ are the pairwise distinct initial
configurations and ${\mathbb{M}}$ is a matrix satisfying

(1.2)
$\forall\xi\in{\mathbb{R}}^{d},\quad\langle{\mathbb{M}}\xi,\xi\rangle\leq 0,$

where $\langle\cdot,\cdot\rangle$ is the standard Euclidean scalar product.
Taking ${\mathbb{M}}=-\mathbb{I}$ yields _gradient-flow/dissipative_ dynamics,
while taking ${\mathbb{M}}$ to be an antisymmetric matrix yields Hamiltonian
dynamics. General ${\mathbb{M}}$ allow for possible mixed flows. We assume
that the potential ${\mathsf{g}}$ is repulsive, so that particles never
collide and there is a unique global solution to the system of ODEs (1.1). The
model case we have in mind is ${\mathsf{g}}$ a Riesz potential depending on a
parameter $0\leq s<d$ according to the convention

(1.3) ${\mathsf{g}}(x)=\begin{cases}-\log|x|,&{s=0}\\\
\quad|x|^{-s},&{s>0}.\end{cases}$

We will in fact consider a slightly more general class of potentials with
singularity of the type (1.3), as explained in the next subsection. Based on
the value of the parameter $s$ and its relation to the dimension $d$, we
classify the potential ${\mathsf{g}}$ as _sub-Coulombic_ if $0\leq s<d-2$,
_Coulombic_ if $s=d-2$, and _super-Coulombic_ if $d-2<s<d$.

Motivations for studying systems of the form (1.1) are numerous: interacting
particles in physics, particle approximation to PDEs, finding equilibrium
states of interaction energies, biological and sociological models, large
neural networks (see for instance the introduction in [Ser20b]). Riesz
interactions are particularly interesting for physics and approximation
theory, see for instance [DRAW02, BHS19].

By mean-field limit, we mean proving the convergence as $N\to\infty$ of the
empirical measure

(1.4) $\mu_{N}^{t}\coloneqq\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}^{t}}$

associated to a solution
$\underline{x}_{N}^{t}\coloneqq(x_{1}^{t},\dots,x_{N}^{t})$ of the system
(1.1). If the points $x_{i}^{0}$, which themselves depend on $N$, are such
that $\mu_{N}^{0}$ converges to some regular measure $\mu^{0}$, then a formal
derivation leads to the expectation that for $t>0$, $\mu_{N}^{t}$ converges to
the solution of the Cauchy problem with initial data $\mu^{0}$ for the
limiting evolution equation

(1.5)
$\begin{cases}\partial_{t}\mu=-\operatorname{div}(({\mathbb{M}}\nabla{\mathsf{g}}*\mu)\mu)&\\\
\mu(0)=\mu^{0},&\end{cases}\qquad(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$

with $*$ denoting the usual convolution. Proving the convergence of the
empirical measure is closely related to proving propagation of molecular chaos
(see [Gol16, HM14, Jab14] and references therein) which means showing that if
$f_{N}^{0}(x_{1},\dots,x_{N})$ is the initial probability density of seeing
particles at positions $(x_{1},\dots,x_{N})$ and if $f_{N}^{0}$ converges to
some factorized state $(\mu^{0})^{\otimes N}$, then the $k$-point marginals
$f_{N,k}^{t}$ converge for all time to $(\mu^{t})^{\otimes k}$. Our result
implies a convergence of this type as well (see Remark 1.6 below).

We note that we restrict to the case $s<d$ because for $s\geq d$, called the
hypersingular case, the potential ${\mathsf{g}}$ is not integrable near the
origin and the convolution ${\mathsf{g}}*\mu$ no longer makes sense for a
nonnegative measure $\mu$. In fact, the expected evolution equation is no
longer of the form (1.5), see for instance [HSST20]. This is the reason for
restricting ourselves to the potential case $s<d$.

Mean-field limits for systems of particles with regular (typically, Lipschitz)
interactions have been understood for a long time [Dob79, Szn91], with a
trajectorial approach consisting in comparing the positions of the particles
to the characteristics of the limiting PDE. In contrast, systems with singular
interaction are far more challenging and only in recent years have
breakthroughs been made to cover the full potential case $s<d$. Various
approaches have been put forward to study them. They usually involve finding a
good metric to measure the distance of the empirical measure to its evolving
expected limit and showing a Gronwall relation on the time evolution of that
metric. The use of the $\infty$-Wasserstein metric has been successful in the
sub-Coulombic case $s<d-2$, [Hau09, CCH14]. The modulated-energy approach of
[Due16, Ser20b], itself inspired from [Ser17], has, in contrast, been
successful in the Coulomb and super-Coulombic case $d-2\leq s<d$ (in low
dimension and the dissipative case in [Due16], and generally in [Ser20b]).
There, the metric used, called the modulated energy, is a Coulomb/Riesz-based
metric, based on the interaction itself, and which can be understood as a
renormalized negative-order homogeneous Sobolev norm. To be more precise, the
modulated energy or (squared) “distance” between
$\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}$ and $\mu$ is defined to be

(1.6)
${F}_{N}(\underline{x}_{N},\mu)\coloneqq\int_{({\mathbb{R}}^{d})^{2}\backslash\triangle}{\mathsf{g}}(x-y)d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu\right)(x)d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu\right)(y),$

where we excise the diagonal $\triangle$ in order to remove the infinite self-
interaction of each particle. Again, a Gronwall-type inequality can be proved
for this modulated energy.

In parallel, work of Jabin and Wang [JW16, JW18] put forward a relative-
entropy based method which can handle moderately singular interactions, and is
particularly tailored for systems with added noise/diffusion. The two
approaches (modulated energy and relative entropy) were recently combined in
[BJW19b, BJW19a, BJW20] into a _modulated free energy_ method capable of
treating dissipative singular flows with additive noise. In the course of that
combination, these works also relaxed the assumptions on the interaction
(their setting is the torus, which changes the long-range effects though), in
particular allowing even for attractive potentials, such as those of Patlak-
Keller-Segel-type. Note that the one-dimensional case can be handled via a
Wasserstein-gradient-flow approach [CFP12, BO19] using the fact that in (and
only in) dimension 1, the Riesz interaction is convex.

Our present concern is showing that the modulated-energy method functions
beyond the Coulombic or super-Coulombic Riesz case. Proving an exact Gronwall
relation on the modulated energy requires showing that the derivative of the
modulated energy along particle displacement is bounded by the modulated
energy itself. This is in essence a functional inequality, which can be proven
independently of any dynamics. Plugging into (1.6) the solution
$\underline{x}_{N}^{t}$ of (1.1) and the solution $\mu^{t}$ of (1.5), direct
computation (see [Ser20b, Lemma 2.1]) yields the inequality

(1.7)
$\begin{split}\frac{d}{dt}{F}_{N}(\underline{x}_{N}^{t},\mu^{t})&\leq\int_{({\mathbb{R}}^{d})^{2}\setminus\triangle}\left\lparen
u^{t}(x)-u^{t}(y)\right\rparen\cdot\nabla{\mathsf{g}}(x-y)d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}^{t}}-\mu^{t}\right)^{\otimes
2}(x,y),\end{split}$

where $u^{t}\coloneqq{\mathbb{M}}\nabla({\mathsf{g}}*\mu^{t})$. Thus, we need
to estimate the right-hand side, which is done by proving the following
functional inequality: for any configuration
$\underline{x}_{N}\coloneqq(x_{1},\dots,x_{N})$, any suitably decaying,
bounded probability density $\mu$, and any suitably regular vector field
$v:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$,

(1.8)
$\int_{({\mathbb{R}}^{d})^{2}\backslash\triangle}(v(x)-v(y))\cdot\nabla{\mathsf{g}}(x-y)d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu\right)^{\otimes
2}(x,y)\leq C{F}_{N}(\underline{x}_{N},\mu)+CN^{-\alpha}$

for some $C>0$ and $\alpha>0$ depending only on norms of $\mu$ and $v$. The
precise statement of the functional inequality (1.8) is given in 4.1 below.

In the prior works [Due16, Ser20b] that used the modulated-energy method, the
exact Coulomb nature of ${\mathsf{g}}$ and the fact that the potential
$\mathsf{h}\coloneqq{\mathsf{g}}*\mu$ solves a local equation in terms of
$\mu$ was used crucially as it allowed to rewrite the left-hand side of (1.8)
in terms of the stress-energy tensor associated to $\mathsf{h}$. Thanks to a
dimension extension procedure popularized by Caffarelli and Silvestre [CS07],
the super-Coulombic case $d-2<s<d$ could also be handled this way by going
through a local operator and a stress-energy tensor. This approach was rigid
in the exact form of the interaction. Later in [BJW19a], a Fourier-based
approach to proving a relation similar to (1.8) was employed and allows, in
the case of the torus, to give more flexibility on the interaction, which
still needs to have Riesz-type behavior in physical and Fourier space. In
fact, the assumptions placed there are similar to the general assumptions
imposed in this article, a point on which we elaborate more in Section 1.3
below. In [Ser20b], in addition to the stress-tensor structure, a truncation
or smearing procedure with a truncation radius depending on the points and a
monotonicity with respect to the smearing radii were crucially used. This
allows for a “renormalization” of the stress-energy tensor and the energy in
order to account for the removal of the diagonal in (1.6) and (1.8).

Our main achievement here is to show that the stress-energy tensor structure
is not needed to prove (1.8), nor is a Fourier-based approach. Rather, some
underlying commutator structure to the left-hand side of (1.8) can be
exploited for all Riesz interactions (with $s<d$). We write here of commutator
structure because one may view the left-hand side of (1.8) as a
renormalization of the (dualized) commutator

(1.9) $\int_{{\mathbb{R}}^{d}}\left\lparen
v\cdot\nabla({\mathsf{g}}*f)-{\mathsf{g}}*\left(\nabla\cdot(vf)\right)\right\rparen
dx,$

or equivalently

(1.10) $\left\langle
f,\commutator{v}{\frac{\nabla}{(-\Delta)^{\frac{d-s}{2}}}}f\right\rangle_{L^{2}},$

where

$f\coloneqq\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu.$

In fact, in the exact Riesz cases, we could also prove (1.8) via off-the-shelf
commutator estimates from the harmonic analysis literature on paraproducts,
specifically refinements by Li [Li19] of the classical fractional Leibnitz
rules due to Kato-Ponce [KP88] and Kenig-Ponce-Vega [KPV93]. Instead, we
provide a fairly direct and essentially physical-space proof based on
integration by parts. The Coulomb case $s=d-2$ is the most delicate. In that
case–and only to obtain the sharp estimate–we appeal to a deep result of
Christ and Journé [CJ87] (see also [SSS19, Lai20] for extensions) for
commutators of Calderón-Zygmund operators. Following the terminology of Christ
and Journé, such expressions are known as _Calderón $d$-commutators_,
generalizations of Calderón’s classic one-dimensional commutator [Cal80]. With
a little work, the structure in the left-hand side of (1.8) allows for a
reduction to estimating such a Calderón $d$-commutator. We mention that such
commutator expressions often appear in studying fluid problems. See for
example [Mar08, CCC+12, HSSS18].

The truncation or smearing procedure of [Ser20b] is still needed and is done
in a slightly different way depending on whether one is in the sub-
Coulombic/Coulombic or the super-Coulombic case. In the sub-
Coulombic/Coulombic case, the nature of the interaction makes ${\mathsf{g}}$
superharmonic, which makes the smearing monotone in the right way. The way we
handle super-Coulombic interactions is then by noticing that a subharmonic
Riesz potential becomes superharmonic when viewed in a larger dimension. We
thus appeal again to a dimension-extension procedure (different from that of
[Ser20b], however) to perform the smearing of the Dirac masses without other
modification of the method. This thus provides a unified treatment of all
Riesz interactions. Since the stress-energy tensor structure is no longer
used, we do not need the interaction to be the kernel of a local operator and
to be exactly an inverse power. Consequently, we can relax the assumptions in
a way similar to [BJW19a].

As advertised in the title of this article, our approach is robust to the
inclusion of multiplicative noise of transport type. Treating the case with
such noise was done in [Ros20c] for the two-dimensional Coulomb case, with the
proof generalizing to higher dimensions. Due to the nonzero quadratic
variation of the noise, it involves proving another functional inequality (see
6.1 below), which is exactly a second-order version of (1.8). In [Ros20c], the
Coulomb nature of the interaction was strongly used. But with the commutator-
based approach and the new insights of this article, it is then not too
difficult to prove the desired functional inequality in all the Riesz cases we
are considering. With this second-order functional inequality and assuming the
existence of suitably regular solutions to the limiting stochastic PDEs
(SPDEs), one can follow the road map of [Ros20c] to bound the expected
magnitude of the modulated energy. Such a bound, of course, implies
convergence in law of the empirical measure to the solution of the limiting
SPDE.

###  1.2. Assumptions and statement of main results

We now state the precise assumptions for the class of interaction potentials
we consider. In the statements below and throughout this article, the notation
$\mathbf{1}_{(\cdot)}$ denotes the indicator function for the condition
$(\cdot)$.

For $d\geq 2$ and $0\leq s\leq d-2$, we assume the following:

(1.11) ${\mathsf{g}}(x)={\mathsf{g}}(-x)\quad{\forall
x\in{\mathbb{R}}^{d}\setminus\\{0\\}}$ (1.12) $\lim_{x\rightarrow
0}{\mathsf{g}}(x)=+\infty$ (1.13) $\text{$\exists r_{0}>0$ such
that}\quad\Delta{\mathsf{g}}\leq 0\quad{\text{in $B(0,r_{0})$}}$ (1.14)
$\forall k\geq 0,\qquad|\nabla^{\otimes k}{\mathsf{g}}(x)|\leq
C\left\lparen\frac{1}{|x|^{s+k}}+|\log|x||\mathbf{1}_{s=k=0}\right\rparen\quad\forall{x\in{\mathbb{R}}^{d}\setminus\\{0\\}}$
(1.15) $|x||\nabla{\mathsf{g}}(x)|+|x|^{2}|\nabla^{\otimes
2}{\mathsf{g}}(x)|\leq C{\mathsf{g}}(x)\quad\forall x\in
B(0,r_{0}){\setminus\\{0\\}}$ (1.16)
$\frac{C_{1}}{|\xi|^{d-s}}\leq\hat{\mathsf{g}}(\xi)\leq\frac{C_{2}}{|\xi|^{d-s}}\quad{\forall\xi\in{\mathbb{R}}^{d}\setminus\\{0\\}}\\\
$

and

(1.17) $\text{$\exists c_{s}<1$ such
that}\quad{\mathsf{g}}(y)<c_{s}{\mathsf{g}}(x)\quad\forall x,y\in
B(0,r_{0})\setminus\\{0\\}\ \text{with}\ |y|\geq 2|x|.$

In the cases $s=d-2k\geq 0$, for some positive integer $k$, we also assume
that the $({\mathbb{R}}^{d})^{\otimes(2k+2)}$-valued kernel

(1.18)
$\mathsf{k}(x-y)\coloneqq(x-y)\otimes\nabla^{\otimes(2k+1)}{\mathsf{g}}(x-y)$

is associated to a Calderón-Zygmund operator in
${\mathbb{R}}^{d}$.222Sufficient and necessary conditions for this Calderón-
Zygmund property are explained in [Gra14a, Section 5.4]. The reader may check
that this condition is satisfied if ${\mathsf{g}}$ is the genuine Coulomb
potential and more generally if ${\mathsf{g}}$ satisfies $|\nabla^{\otimes
k}\hat{\mathsf{g}}(\xi)|\leq\frac{C}{|\xi|^{d+k-s}}~{}~{}\forall\xi\in{\mathbb{R}}^{d}\setminus\\{0\\}$.
For $s=0$, we replace the assumption (1.17) with

(1.19) $\text{$\exists{c_{0}>0}$ such
that}\quad{\mathsf{g}}(x)-{\mathsf{g}}(y)\geq{c_{0}}\quad\forall x,y\in
B(0,r_{0})\setminus\\{0\\}\ \text{with}\ |y|\geq 2|x|.$

If $\max\\{d-2,0\\}<s<d$ or $d=1$ and $s=0$, then we expect the local
superharmonicity assumption (1.13) to fail. The idea, though, is that
${\mathsf{g}}$ becomes superharmonic when viewing it as an interaction
potential in an extended space of dimension $d+m$. Indeed,
$\Delta|X|^{-s}=-s(d+m-s-2)|X|^{-s-2}$ in $\mathbb{R}^{d+m}$, which becomes
$\leq 0$ if $m$ is large enough. We will denote points in ${\mathbb{R}}^{d+m}$
by $X=(x,z)$ with $x\in{\mathbb{R}}^{d}$ and $z\in{\mathbb{R}}^{m}$. Our
assumption for the case $\max\\{d-2,0\\}<s<d$ or $d=1$ and $s=0$ is thus that
there exists an integer $m\geq 0$,
${\mathsf{G}}:{\mathbb{R}}^{d+m}\rightarrow{\mathbb{R}}$, and constants
$C_{1},C_{2},C>0$ depending only on $d,s,m$, such that

(1.20)
${\mathsf{G}}(x,0)={\mathsf{g}}(x)\quad\forall(x,0)\in{\mathbb{R}}^{d+m}$
(1.21) ${\mathsf{G}}(X)={\mathsf{G}}(-X)\quad{\forall
X\in{\mathbb{R}}^{d+m}\setminus\\{0\\}}$ (1.22) $\lim_{X\rightarrow
0}{\mathsf{G}}(X)=+\infty$ (1.23) $\text{$\exists r_{0}>0$ such
that}\quad\Delta{\mathsf{G}}\leq 0\quad\text{in
$B(0,r_{0})\subset{\mathbb{R}}^{d+m}$}$ (1.24) $\forall k\geq
0,\quad|\nabla^{\otimes k}{\mathsf{G}}(X)|\leq
C\left\lparen\frac{1}{|X|^{s+k}}+|\log|X||\mathbf{1}_{s=k=0\wedge
d=1}\right\rparen\quad\forall{X\in{\mathbb{R}}^{d+m}\setminus\\{0\\}}$ (1.25)
$|X||\nabla{\mathsf{G}}(X)|+|X|^{2}|\nabla^{\otimes 2}{\mathsf{G}}(X)|\leq
C{\mathsf{G}}(X)\quad\forall X\in B(0,r_{0})\setminus\\{0\\}$ (1.26)
$\frac{C_{1}}{|{\Xi}|^{d+m-s}}\leq\hat{\mathsf{G}}({\Xi})\leq\frac{C_{2}}{|{\Xi}|^{d+m-s}}\quad\forall\Xi\in{\mathbb{R}}^{d+m}\setminus\\{0\\}.$

We also add

(1.27) $\begin{cases}\text{$\exists c_{s}<1$ such
that}\enspace{\mathsf{G}}(Y)<c_{s}{\mathsf{G}}(X)\quad\forall X,Y\in
B(0,r_{0})\setminus\\{0\\}\ \text{with}\ |Y|\geq 2|X|,&{s>0}\\\
\text{$\exists{c_{0}}>0$ such that}\enspace{\mathsf{G}}(X)-{\mathsf{G}}(Y)\geq
c_{0}\quad\forall X,Y\in B(0,r_{0})\setminus\\{0\\}\ \text{with}\ |Y|\geq
2|X|,&{s=0}.\end{cases}$

If $s=d-1$ and $m=1$, then we also assume that the
$({\mathbb{R}}^{d})^{\otimes 4}$-valued kernel

(1.28) ${\mathsf{K}}(X-Y)\coloneqq(X-Y)\otimes\nabla^{\otimes
3}{\mathsf{G}}(X-Y),\qquad\forall X\neq Y$

is associated to a Calderón-Zygmund operator in ${\mathbb{R}}^{d+m}$. We can
simultaneously treat the cases $0\leq s\leq d-2$ and $\max\\{d-2,0\\}<s<d$ or
$d=1$ and $s=0$ by setting $m=0$ if $0\leq s\leq d-2$. If ${\mathsf{g}}$
satisfies the assumptions (1.11) – (1.19) in the case $0\leq s\leq d-2$ or
(1.20) – (1.28) in the case $d-2<s<d$, then we call ${\mathsf{g}}$ an
_admissible potential_.

The main result of this article is the following functional inequality for the
modulated energy (cf. [Ser20b, Theorem 1], [BJW19a, Theorem 3.2]).

######  Theorem 1.1.

Let ${\mathsf{g}}$ be an admissible potential. Assume the equation (1.5)
admits a solution $\mu\in L^{\infty}([0,T],\mathcal{P}({\mathbb{R}}^{d})\cap
L^{\infty}({\mathbb{R}}^{d}))$, for some $T>0$, such that

(1.29)
$\||\nabla|^{s-d}\nabla\mu\|_{L^{\infty}([0,T],L^{\infty})}\mathbf{1}_{s\geq
d-1}+\|\nabla^{\otimes
2}{\mathsf{g}}\ast\mu\|_{L^{\infty}([0,T],L^{\infty})}\mathbf{1}_{s=d-2}\\\
+\Big{(}\|\nabla^{\otimes
2}{\mathsf{g}}\ast\mu\|_{L^{\infty}([0,T],C^{0,\alpha})}+\|\nabla{\mathsf{g}}\ast\mu\|_{L^{\infty}([0,T],W^{\frac{d(d+m-s)+2m}{2(d+m)},\frac{2(d+m)}{d+m-2-s}})}\Big{)}\mathbf{1}_{s>d-2}<\infty.$

If $s=0$, then also assume that
$\int_{{\mathbb{R}}^{d}}\log(1+|x|)d\mu^{t}(x)<\infty$ for all $t\in[0,T]$.
Here, $|\nabla|=(-\Delta)^{1/2}$, $m$ is the dimension of the extension
coordinate space ${\mathbb{R}}^{m}$ for ${\mathsf{g}}$, and $0<\alpha\leq 1$.

Let $\underline{x}_{N}$ solve (1.1). Then there exist positive constants
$C_{3},C_{4}$, depending only on $|{\mathbb{M}}|$, $\|\mu^{0}\|_{L^{\infty}}$
and the norms of $\mu$ controlled by (1.29) and on the potential only through
the constants in assumptions (1.13)–(1.28), and an exponent $\beta>0$,
depending only on $d,s$, such that

(1.30)
$F_{N}(\underline{x}_{N}^{t},\mu^{t})\leq\left(F_{N}(\underline{x}_{N}^{0},\mu^{0})+C_{3}N^{-\beta}\right)e^{C_{4}t}\qquad\forall
t\in[0,T].$

In particular, using the notation (1.4), if
$\mu_{N}^{0}\rightharpoonup\mu^{0}$ in the weak-* topology for measures and

(1.31) $\lim_{N\to\infty}F_{N}(\underline{x}_{N}^{0},\mu^{0})=0,$

then

(1.32) $\mu_{N}^{t}\rightharpoonup\mu^{t}\qquad\forall t\in[0,T].$

######  Remark 1.2.

The reader may infer from Section 5 the precise dependence of the constants
$C_{3},C_{4}$ above on the norms of the solution $\mu$ as well as the size of
the exponent $\beta$. We have omitted this precise dependence from the
statement of 1.1 so as to present the result in the most accessible terms.

We have not attempted to optimize the regularity/integrability assumptions on
the solution $\mu$. But we expect that one can proceed in the direction of the
works [Ros20a, Ros20b] to only require that $\mu$ belongs to a function space
which is critical or borderline with respect to the scaling of the equation.
For more discussion on the existence of (necessarily unique) solutions
satisfying the condition (1.29), we refer the reader to [Ser20b, Subsection
1.3].

######  Remark 1.3.

In the case $s=0$, the condition
$\int_{{\mathbb{R}}^{d}}\log(1+|x|)d\mu(x)<\infty$ is to ensure that the
convolution ${\mathsf{g}}\ast\mu$ is a well-defined function. This assumption
is purely qualitative, in the sense that none of our estimates will depend on
it. One may check through a Gronwall argument that if the initial datum
$\mu^{0}$ satisfies this condition, then $\mu^{t}$ also satisfies this
condition for all $t\in[0,T]$.

######  Remark 1.4.

Sufficient conditions for the initial modulated energy
${F}_{N}(\underline{x}_{N}^{0},\mu^{0})$ to tend vanish as
$N\rightarrow\infty$ are explained in [Due16, Remark 1.2(c)]. In particular,
convergence of the energy for (1.1) to the energy of (1.5) together with
weak-* convergence of the initial empirical measure to the initial datum
$\mu^{0}$ suffice.

######  Remark 1.5.

The weak-* convergence of $\mu_{N}^{t}$ to $\mu^{t}$ is quantitative, as shown
in 2.4. In fact, one can also express the convergence in terms of a suitably
negative-order Sobolev norm using 2.4.

######  Remark 1.6.

As is well-known, 1.1 implies propagation of chaos for the marginals of the
system (1.1) with initial data $\underline{x}_{N}^{0}$ randomly chosen
according to the law $(\mu^{0})^{\otimes N}$. See [Ser20b, Remark 3.7] for
details.

###  1.3. Allowable potentials and further extensions

Assumptions (1.20) - (1.28) allow for a large class of interaction potentials
beyond the standard Riesz or log cases. Indeed, the simplest, nontrivial
extension is the Lenard-Jones-type potential

(1.33)
${\mathsf{g}}(x)=|x|^{-s}+\varphi_{1}(x)|x|^{-s_{1}}+\cdots+\varphi_{n}(x)|x|^{-s_{n}},$

where $0<s<d$, $0\leq s_{n}<\cdots<s_{1}<s$, and
$\varphi_{1},\ldots,\varphi_{n}$ are smooth, compactly supported functions–not
necessarily nonnegative. It is straightforward to check that assumptions
(1.20) - (1.28) are satisfied.

In the gradient-flow case, we can even consider genuine Lenard-Jones
potentials, where $\varphi_{1},\ldots,\varphi_{n}$ are identically equal to
real-valued constants $a_{1},\ldots,a_{n}$. Indeed, let $\chi$ be a smooth,
radial bump function adapted to the region $|\xi|\geq 2$. Introducing a small
parameter $\kappa>0$, we split

(1.34) ${\mathsf{g}}={\mathsf{g}}_{good}+{\mathsf{g}}_{bad},$

where

(1.35) $\begin{split}\hat{{\mathsf{g}}}_{bad}(\xi)&\coloneqq
c|\xi|^{s-d}+\chi(\kappa\xi)\left\lparen\sum_{j=1}^{n}\tilde{c}_{j}|\xi|^{s_{j}-d}\right\rparen,\\\
\hat{{\mathsf{g}}}_{good}(\xi)&\coloneqq\left\lparen
1-\chi(\kappa\xi)\right\rparen\left\lparen\sum_{j=1}^{n}\tilde{c}_{j}|\xi|^{s_{j}-d}\right\rparen.\end{split}$

Here, the constants $c,\tilde{c}_{1},\ldots,\tilde{c}_{n}$ are normalization
constants from taking the Fourier transform. If $d-2<s<d$, then we set
${\mathsf{G}}_{bad}(X)\coloneqq{\mathsf{g}}_{bad}(|X|)$ and
${\mathsf{G}}_{good}(X)\coloneqq{\mathsf{g}}_{good}(|X|)$ for
$X\in{\mathbb{R}}^{d+m}$. Since the second term in the definition of
$\hat{{\mathsf{g}}}_{bad}$ is smooth, its inverse Fourier transform and its
derivatives are rapidly decaying. It is straightforward to check that

(1.36)
${\mathsf{g}}_{bad}(x)\geq\frac{1}{|x|^{s}}-C\sum_{j=1}^{n}\frac{1}{|x|^{s_{j}}}-C,\qquad\forall
x\neq 0,$

where $C>0$ is some constant. In particular, there exists $0<r_{0}\ll 1$, such
that for all $|x|\leq r_{0}$,

(1.37) ${\mathsf{g}}_{bad}(x)\geq\frac{1}{2|x|^{s}}.$

Furthermore,

(1.38) $-\Delta{\mathsf{G}}_{bad}(X)\geq
s(d+m-s-2)|X|^{-2-s}-C\sum_{j=1}^{n}|X|^{-s_{j}-2}-C>0,$

for $|X|\leq r_{0}$ (possibly choosing $r_{0}$ smaller) and $d+m-s-2>0$.
Lastly, it follows from the radial symmetry of ${\mathsf{G}}_{bad}$ that

(1.39) $C_{1}|\Xi|^{-d-m+s}\leq|\hat{{\mathsf{G}}}_{bad}(\Xi)|\leq
C_{2}|\Xi|^{-d-m+s},\qquad\forall\Xi\neq 0.$

Since $\hat{{\mathsf{g}}}_{good}$ has compact support and is locally
integrable, it belongs to $C^{\infty}$. Moreover, direct computation shows
that $\nabla{\mathsf{g}}_{good}\in\dot{H}^{\frac{d-s}{2}}$ if $2(s_{n}+1)>s$.
Thus, we can rewrite the system (1.1) as

(1.40) $\begin{cases}\dot{x}_{i}=\displaystyle-\frac{1}{N}\sum_{{1\leq j\leq
N}\atop{j\neq
i}}\left\lparen\nabla{\mathsf{g}}_{bad}(x_{i}-x_{j})+\mathsf{F}(x_{i}-x_{j})\right\rparen\\\
x_{i}(0)=x_{i}^{0},\end{cases}$

where $\mathsf{F}\coloneqq\nabla{\mathsf{g}}_{good}$ is now a smooth added
force. The potential ${\mathsf{g}}_{bad}$ satisfies assumptions (1.20) –
(1.28), and the contribution of $\mathsf{F}$ can be handled using [Ser20b,
Lemma 2.3]. One may modify the preceding argument to allow for Lenard-Jones
potentials with log interactions as well.

Given that Bresch et al. [BJW19b, BJW19a, BJW20] also consider a general class
of singular interaction potentials, including those of Riesz type, let us
compare and contrast our 1.1 to their main result. Bresch et al. consider only
the gradient-flow case of the system (1.1) in the periodic setting with random
initial data. So rather than working with particle trajectories, they work
with the associated Liouville equation. They also allow for additive noise in
the dynamics, but we shall ignore this aspect for the purposes of this
discussion. Through a Gronwall-type argument, they prove a functional
inequality for the modulated free energy. As previously remarked, this
quantity combines the relative entropy of earlier work [JW16, JW18] with the
modulated energy of [Due16, Ser20b]. By standard arguments, their functional
inequality then implies propagation of chaos. Instead of the smearing and
extension procedure used in our article, Bresch et al. introduce a clever
construction of a regularized potential [BJW19a, Lemma 4.1], which tames the
singularity of the Dirac masses. To replace the stress-energy tensor argument
from [Ser20b] based on the Caffarelli-Silvestre extension, they substitute the
regularized potential ${\mathsf{g}}_{reg}$ into the right-hand side of
inequality (1.7), use some elementary Fourier analysis [BJW19a, Lemma 5.2] to
bound the resulting expression, and finally estimate the error from this
substitution. Although it is never mentioned in their work, their Fourier-
analytic argument is, in fact, an elementary commutator estimate, a
perspective that we develop in detail in this paper.

As in our work, the potentials they consider are even, singular at the origin,
and have nonnegative Fourier transforms. Given that the torus is compact, they
only need to assume that the potential is in $L^{p}({\mathbb{T}}^{d})$, for
some $p>1$. Similar to assumptions (1.14), (1.24) and (1.15), (1.25), Bresch
et al. need pointwise control on the potential and its derivatives to
sufficiently high order. They also need a doubling condition analogous to
(1.17), (1.19), (1.27). A key difference, though, is that no pointwise control
on $\hat{{\mathsf{g}}}(\xi)$ is needed, aside from nonnegativity. Bresch et
al. only need some control on the the first derivative of
$\hat{{\mathsf{g}}}$. Also, superharmonicity plays no role in their work, in
sharp contrast to our work

It is worth mentioning that the relative entropy portion of the functional
used by Bresch et al. is unnecessary and in fact one can use their regularized
potential construction with their elementary commutator estimate in order to
treat both the conservative and gradient flow cases of the particle system
(1.1). Moreover, one can work directly with the particle dynamics, as opposed
to with the Liouville equation. In some sense, one of the new contributions of
our paper is the recognition that this is possible, albeit using different
methods. In forthcoming work by the second and third authors [RS21], we will
also show that this is possible when one includes so-called additive noise in
the dynamics, distinct from the multiplicative noise considered in this
article (see Section 6.1 for a comparison); and, in fact, one can obtain
bounds on the modulated energy which are _global_ in time.

Compared to the work of Bresch et al., the advantages of our approach are that
it is not limited to the periodic setting; it is pathwise in the sense that no
randomization of the initial data is needed, avoiding the use of modulated
free energy in favor of the modulated energy; and, perhaps most importantly,
the rate of convergence is explicit. In [BJW19a], for instance, an explicit
rate is never presented. If one does attempt to optimize the rate in their
work, then one sees that the use of the regularized potential construction
[BJW19a, Lemma 4.1] is a bottleneck, and the rate one does obtain is
suboptimal compared to our work. Additionally, the regularity requirements for
the solution $\mu$ to (1.5) of [BJW19a] are more severe than our work. The
cost of these advantages is the need for stronger pointwise control on the
potential and its Fourier transform, in particular we have to worry about
decay at infinity in physical space and blow-up near the origin in Fourier
space.

Finally, as commented in the previous subsection, our method of proof also
works when multiplicative noise is added to the right-hand side of the
$N$-body problem (1.1). The expected limiting equation is now a stochastic
PDE, so that the modulated energy is now a stochastic process. By combining
Itô’s lemma for the modulated energy with pathwise analysis in the form of
first- and second-order commutator estimates, we can perform a similar
Gronwall argument for the expectation of the magnitude of the modulated
energy. In fact, one can bound moments of arbitrarily large degree. We defer a
precise statement of the model under consideration and the new functional
inequality until Section 6.

###  1.4. Organization of article

Before introducing the basic notation of the article, let us briefly comment
on the organization of the remaining body of the article. In Section 2, we
introduce the smearing procedure and establish properties of the modulated
energy following the outline of [Ser20b, Section 3], but for the more general
class of potentials satisfying assumptions (1.20) - (1.27). In Section 3, we
present our new commutator estimate 3.1, and in Section 4, we use the smearing
procedure from Section 2 to renormalize our commuator estimate, obtaining 4.1.
Then in Section 5, we use the analysis of Section 4, specifically 4.1,
together with a delicate optimization of the floating small parameters in
order to close the Gronwall argument for the modulated energy. This then
completes the proof of 1.1. Lastly, in Section 6, we close the paper by
showing how to extend our modulated-energy approach to variants of the system
(1.1) with multiplicative noise added to the dynamics using second-order
commutator estimates.

###  1.5. Notation

We close the introduction by specifying the basic notation used in the body of
the article without further comment.

Given nonnegative quantities $A$ and $B$, we write $A\lesssim B$ if there
exists a constant $C>0$, independent of $A$ and $B$, such that $A\leq CB$. If
$A\lesssim B$ and $B\lesssim A$, we write $A\sim B$. To emphasize the
dependence of the constant $C$ on some parameter $p$, we sometimes write
$A\lesssim_{p}B$ or $A\sim_{p}B$.

We denote the natural numbers excluding zero by ${\mathbb{N}}$ and including
zero by ${\mathbb{N}}_{0}$. Similarly, we denote the positive real numbers by
${\mathbb{R}}_{+}$. Given $N\in{\mathbb{N}}$ and points
$x_{1,N},\ldots,x_{N,N}$ in some set $X$, we will write $\underline{x}_{N}$ to
denote the $N$-tuple $(x_{1,N},\ldots,x_{N,N})$. Given $x\in{\mathbb{R}}^{d}$
and $r>0$, we denote the ball and sphere centered at $x$ of radius $r$ by
$B(x,r)$ and ${\partial}B(x,r)$, respectively. Given a function $f$, we denote
the support of $f$ by $\operatorname{supp}f$.

We make frequent use of tensor notation. $A\otimes B$ denotes the usual tensor
product between $A=(A_{i_{1}\cdots i_{m}}^{j_{1}\cdots j_{n}})$ and
$B=(B_{i_{1}^{\prime}\cdots i_{m^{\prime}}^{\prime}}^{j_{1}^{\prime}\cdots
j_{n^{\prime}}^{\prime}})$ with components

(1.41) $(A\otimes B)_{i_{1}\cdots i_{m}i_{1}^{\prime}\cdots
i_{m^{\prime}}^{\prime}}^{j_{1}\cdots j_{n}j_{1}^{\prime}\cdots
j_{n^{\prime}}^{\prime}}=A_{i_{1}\cdots i_{m}}^{j_{1}\cdots
j_{n}}B_{i_{1}^{\prime}\cdots i_{m^{\prime}}^{\prime}}^{j_{1}^{\prime}\cdots
j_{n^{\prime}}^{\prime}}.$

We identify $2$-tensors as matrices and use the notation $:$ to denote the
Frobenius inner product. Given a function $f$, $\nabla^{\otimes k}f$ denotes
the $({\mathbb{R}}^{d})^{\otimes k}$-valued field whose components are given
by the partial derivatives ${\partial}_{i_{1}\cdots i_{k}}f$, for any $1\leq
i_{1},\ldots,i_{k}\leq d$.

We denote the space of probability measures on ${\mathbb{R}}^{d}$ by
$\mathcal{P}({\mathbb{R}}^{d})$. When $\mu$ is in fact absolutely continuous
with respect to Lebesgue measure on ${\mathbb{R}}^{d}$, we shall abuse
notation by writing $\mu$ for both the measure and its density function. We
denote the Banach space of complex-valued continuous functions on
${\mathbb{R}}^{d}$ by $C({\mathbb{R}}^{d})$ equipped with the uniform norm
$\|\cdot\|_{\infty}$. More generally, we denote the Banach space of $k$-times
continuously differentiable functions with bounded derivatives up to order $k$
by $C^{k}({\mathbb{R}}^{d})$ equipped with the natural norm, and we define
$C^{\infty}\coloneqq\bigcap_{k=1}^{\infty}C^{k}$. Functions $f$ which belong
to $C^{k-1}({\mathbb{R}}^{d})$ with $\nabla^{\otimes k}f$ $\alpha$-Hölder
continuous are denoted by $C^{k,\alpha}({\mathbb{R}}^{d})$, which we equip
with the usual inhomogeneous norm. We use the subscript $c$ to denote the
subspace with compact support. Also, we denote the Schwartz space by
${\mathcal{S}}({\mathbb{R}}^{d})$ and the Bessel potential space by
$W^{s,p}({\mathbb{R}}^{d})$.

##  2\. The modulated energy

In this section, we perform a functional-analytic study of the modulated
energy ${F}_{N}$ for an arbitrary pairwise distinct configuration
$\underline{x}_{N}\in({\mathbb{R}}^{d})^{N}$ and probability density $\mu\in
L^{\infty}$ which has a logarithmic-type decay at infinity in the case $s=0$.
This last assumption is purely qualitative (our estimates will not depend on
it): it serves to ensure that the convolution ${\mathsf{g}}\ast\mu$ is well-
defined.

We start by considering the case where (1.13) holds, which is satisfied by all
the sub-Coulombic Riesz cases $0<s<d-2$ as well as the log case $s=0$ if
$d\geq 2$.

###  2.1. Smearing

Since (1.13) holds, ${\mathsf{g}}$ is superharmonic in $B(0,r_{0})$. For any
sufficiently integrable function $f$, we have

(2.1) $\frac{d}{dr}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\partial
B(x,r)}fd{\mathcal{H}}^{d-1}=\frac{1}{d|B(0,1)|r^{d-1}}\int_{B(x,r)}\Delta
fdy,$

where ${\mathcal{H}}^{d-1}$ denotes the $(d-1)$-dimensional Hausdorff measure
on ${\mathbb{R}}^{d}$. This immediately implies that ${\mathsf{g}}$ satisfies
a mean-value inequality:

(2.2) $\forall B(x,r)\subset
B(0,r_{0}),\qquad{\mathsf{g}}(x)\geq\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\partial B(x,r)}{\mathsf{g}}dy.$

We note that this inequality holds despite the singularity of ${\mathsf{g}}$
at the origin. Indeed, $\Delta{\mathsf{g}}$ is locally integrable if $0\leq
s<d-2$ by assumption (1.14). If $s=d-2$, then $\Delta{\mathsf{g}}$ is locally
integrable away from the origin, in which case (2.1) holds. Using an
approximation argument, (2.2) also follows.

Next, for any $0<\eta<\min\\{\frac{1}{2},\frac{r_{0}}{2}\\}$ and
$x\in{\mathbb{R}}^{d}$, we let $\delta_{x}^{(\eta)}$ be the uniform
probability measure on $\partial B(x,\eta)$ and set

(2.3) ${\mathsf{g}}_{\eta}\coloneqq{\mathsf{g}}\ast\delta_{0}^{(\eta)}.$

It follows directly from (2.2) that

(2.4) ${\mathsf{g}}_{\eta}(x)\leq{\mathsf{g}}(x)\quad\forall x\in
B(0,r_{0}-\eta)\setminus\\{0\\}$

and from (2.1) and (1.14) that

(2.5)
$|{\mathsf{g}}(x)-{\mathsf{g}}_{\eta}(x)|\leq\frac{C\eta^{2}}{|x|^{s+2}}\quad\forall|x|\geq
2\eta,$

where the constant $C>0$ depends on $r_{0}$.

Observe that in view of (2.4) and (1.14), the self-interaction of the smeared
point mass $\delta_{x_{0}}^{(\eta)}$ satisfies the relation

(2.6)
$\int_{({\mathbb{R}}^{d})^{2}}{\mathsf{g}}(x-y)d\delta_{x_{0}}^{(\eta)}(x)d\delta_{x_{0}}^{(\eta)}(y)=\int_{{\mathbb{R}}^{d}}{\mathsf{g}}_{\eta}(x)d\delta_{0}^{(\eta)}(x)\leq\int_{{\mathbb{R}}^{d}}{\mathsf{g}}(x)d\delta_{0}^{(\eta)}(x)={\mathsf{g}}_{\eta}(0)\leq
C\eta^{-s}.$

If ${\mathsf{g}}(x)=|x|^{-s}$ then a direct computation with a rescaling is
that

(2.7)
$\int_{({\mathbb{R}}^{d})^{2}}{\mathsf{g}}(x-y)d\delta_{x_{0}}^{(\eta)}(x)d\delta_{x_{0}}^{(\eta)}(y)=\eta^{-s}\int_{({\mathbb{R}}^{d})^{2}}{\mathsf{g}}(x-y)d\delta_{0}^{(1)}(x)d\delta_{0}^{(1)}(y)=\eta^{-s}{\mathsf{g}}_{1}(1),$

where we abuse notation by letting $1$ denote any choice of unit vector.
Similarly, if ${\mathsf{g}}(x)=-\log|x|$, then

(2.8)
$\int_{({\mathbb{R}}^{d})^{2}}{\mathsf{g}}(x-y)d\delta_{x_{0}}^{(\eta)}(x)d\delta_{x_{0}}^{(\eta)}(y)=-\log\eta+\int_{({\mathbb{R}}^{d})^{2}}{\mathsf{g}}(x-y)d\delta_{0}^{(1)}(x)d\delta_{0}^{(1)}(y)=-\log\eta+{\mathsf{g}}_{1}(1).$

###  2.2. Monotonicity, coerciveness and local interaction control

By definition (1.6) of $F_{N}(\underline{x}_{N},\mu)$ and the relation (2.6),
we may write

(2.9) $\begin{split}{F}_{N}(\underline{x}_{N},\mu)&=\lim_{\alpha_{i}\to
0}\left\lparen\int_{({\mathbb{R}}^{d})^{2}}{\mathsf{g}}(x-y)d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}^{(\alpha_{i})}-\mu\right)^{\otimes
2}(x,y)-\frac{1}{N^{2}}\sum_{i=1}^{N}\int_{{\mathbb{R}}^{d}}{\mathsf{g}}_{\alpha_{i}}d\delta_{0}^{\alpha_{i}}\right\rparen.\end{split}$

This is a way to express ${F}_{N}$ as a renormalized version of the undefined
expression

(2.10)
$\int{\mathsf{g}}(x-y)d\left\lparen\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu\right\rparen^{\otimes
2},$

where the renormalization occurs via the smearing of the Dirac masses.

One may then examine the change in the expression in the limit when increasing
$\alpha_{i}$ to fixed smearing radii $\eta_{i}$. As first noted in [PS17,
LSZ17], the expression is almost decreasing in the radii, with errors that can
be explicitly controlled. In addition, thanks to the Fourier transform bounds
(1.26), once the diagonal has been reinserted in this way we have

(2.11)
$\left\|\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}^{(\eta_{i})}-\mu\right\|_{\dot{H}^{\frac{s-d}{2}}({\mathbb{R}}^{d})}^{2}\sim\int_{({\mathbb{R}}^{d})^{2}}{\mathsf{g}}(x-y)d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}^{(\eta_{i})}-\mu\right)^{\otimes
2}(x,y),$

thus allowing us to control a true Sobolev distance between
$\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}^{(\eta_{i})}$ and $\mu$.

The next proposition expresses this crucial monotonicity property (cf.
[Ser20b, Proposition 3.3]) and shows that the modulated energy is bounded from
below and coercive, and controls the small scale interactions. It covers
potentials with singularity up to and including Coulomb.

######  Proposition 2.1.

Let $d\geq 2$ and $0\leq s\leq d-2$. Suppose that
$\underline{x}_{N}\in({\mathbb{R}}^{d})^{N}$ is pairwise distinct and
$\mu\in\mathcal{P}({\mathbb{R}}^{d})\cap L^{\infty}({\mathbb{R}}^{d})$. In the
case $s=0$, also suppose that
$\int_{{\mathbb{R}}^{d}}\log(1+|x|)d\mu(x)<\infty$. There exists a constant
$C$ depending only on the potential ${\mathsf{g}}$ through assumptions (1.13)
- (1.19), such that for every set of truncation parameters
$0<\eta_{1},\ldots,\eta_{N}<r_{0}/2$, we have

(2.12) $\frac{1}{N^{2}}\sum_{\begin{subarray}{c}1\leq i\neq j\leq N\\\
|x_{i}-x_{j}|\leq\frac{1}{2}r_{0}\end{subarray}}\left\lparen{\mathsf{g}}(x_{j}-x_{i})-{\mathsf{g}}_{\eta_{i}}(x_{j}-x_{i})\right\rparen_{+}+C^{-1}\left\|\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}^{(\eta_{i})}-\mu\right\|_{\dot{H}^{\frac{s-d}{2}}}^{2}\\\
\leq{F}_{N}(\underline{x}_{N},\mu)+\frac{1}{N^{2}}\sum_{i=1}^{N}{\mathsf{g}}_{\eta_{i}}(0)+\frac{C}{N}\sum_{i=1}^{N}\eta_{i}^{2}\\\
+\frac{C}{N}\sum_{i=1}^{N}\|\mu\|_{L^{\infty}}\left\lparen\eta_{i}^{d-s}+(\eta_{i}^{d}|\log\eta_{i}|)\mathbf{1}_{s=0}+(\eta_{i}^{2}|\log\eta_{i}|)\mathbf{1}_{s=d-2}\right\rparen,$

where $r_{0}$ is the constant in (1.13).

###### Proof.

Adding and subtracting $\delta_{x_{i}}^{(\eta_{i})}$ and regrouping terms, we
find that

$\displaystyle{F}_{N}(\underline{x}_{N},\mu)$
$\displaystyle=\int_{({\mathbb{R}}^{d})^{2}}{\mathsf{g}}(x-y)d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}^{(\eta_{i})}-\mu\right)^{\otimes
2}(x,y)-\frac{1}{N^{2}}\sum_{i=1}^{N}\int_{{\mathbb{R}}^{d}}{\mathsf{g}}_{\eta_{i}}d\delta_{0}^{(\eta_{i})}$
$\displaystyle-\frac{2}{N}\sum_{i=1}^{N}\int_{{\mathbb{R}}^{d}}\left({\mathsf{g}}(y-x_{i})-{\mathsf{g}}_{\eta_{i}}(y-x_{i})\right)d\mu(y)$
(2.13) $\displaystyle+\frac{1}{N^{2}}\sum_{1\leq i\neq j\leq
N}\int_{{\mathbb{R}}^{d}}\left({\mathsf{g}}(y-x_{i})-{\mathsf{g}}_{\eta_{i}}(y-x_{i})\right)d(\delta_{x_{j}}+\delta_{x_{j}}^{(\eta)})(y).$

From inequalities (2.4) and (2.5), we see that

(2.14) $\frac{1}{N^{2}}\sum_{1\leq i\neq j\leq
N}\int_{{\mathbb{R}}^{d}}\left({\mathsf{g}}(y-x_{i})-{\mathsf{g}}_{\eta_{i}}(y-x_{i})\right)d(\delta_{x_{j}}+\delta_{x_{j}}^{(\eta_{j})})(y)\\\
>\frac{1}{N^{2}}\sum_{\begin{subarray}{c}1\leq i\neq j\leq N\\\
|x_{i}-x_{j}|<r_{0}-\eta\end{subarray}}\left({\mathsf{g}}(x_{j}-x_{i})-{\mathsf{g}}_{\eta_{i}}(x_{j}-x_{i})\right)_{+}-\frac{C}{N}\sum_{i=1}^{N}\eta_{i}^{2}.$

Next, using (2.4) and (2.5) again, we find that

$\displaystyle\int_{B(0,r_{0}-\eta)}|{\mathsf{g}}(x)-{\mathsf{g}}_{\eta}(x)|dx$
$\displaystyle\leq
2\int_{B(0,2\eta)}{\mathsf{g}}(x)dx+C\eta^{2}\int_{B(0,r_{0})\backslash
B(0,2\eta)}\frac{dx}{|x|^{s+2}}$ (2.15) $\displaystyle\leq
C\left\lparen\eta^{d-s}+(\eta^{d}|\log\eta|)\mathbf{1}_{s=0}+(\eta^{2}|\log\eta|)\mathbf{1}_{s=d-2}\right\rparen.$

Since $\mu$ is a probability density, we deduce from Hölder’s inequality that

(2.16)
$-\frac{2}{N}\sum_{i=1}^{N}\int_{{\mathbb{R}}^{d}}\left\lparen{\mathsf{g}}(y-x_{i})-{\mathsf{g}}_{\eta_{i}}(y-x_{i})\right\rparen
d\mu(y)\geq-{\frac{C}{N}\sum_{i=1}^{N}\eta_{i}^{2}}\\\
-\frac{C}{N}\|\mu\|_{L^{\infty}}\sum_{i=1}^{N}\left\lparen\eta_{i}^{d-s}+(\eta_{i}^{d}|\log\eta_{i}|)\mathbf{1}_{s=0}+(\eta_{i}^{2}|\log\eta_{i}|)\mathbf{1}_{s=d-2}\right\rparen.$

Inserting (2.14) and (2.16) into (2.13) and using (2.11), the result follows.
∎

Since ${\mathsf{g}}_{\eta}(0)\leq C(\eta^{-s}+|\log\eta|\mathbf{1}_{s=0})$,
the right-hand side of (2.12) is easily optimized by choosing
$\eta_{i}=N^{-\frac{1}{2+s}}\leq N^{-1/d}$. Despite not having a sign,
${F}_{N}$ is bounded below:

(2.17)
${F}_{N}(\underline{x}_{N},\mu)\geq-C(1+\|\mu\|_{L^{\infty}})\left\lparen
N^{-\frac{2}{2+s}}+(N^{-1}\log N)\mathbf{1}_{s=0}+(N^{-\frac{2}{d}}\log
N)\mathbf{1}_{s=d-2}\right\rparen,$

which, by triangle inequality, implies that

(2.18)
$|{F}_{N}(\underline{x}_{N},\mu)|\leq{F}_{N}(\underline{x}_{N},\mu)+2C(1+\|\mu\|_{L^{\infty}})\Big{(}N^{-\frac{2}{2+s}}+(N^{-1}\log
N)\mathbf{1}_{s=0}\\\ +(N^{-\frac{2}{d}}\log N)\mathbf{1}_{s=d-2}\Big{)}.$

###  2.3. Renormalization in extended space

Similar to before, we define $\delta_{X}^{(\eta)}$ to be the uniform
probability measure on $\partial B(X,\eta)\subset{\mathbb{R}}^{d+m}$ and
define the mollified potential

(2.19) ${\mathsf{G}}_{\eta}\coloneqq{\mathsf{G}}\ast\delta_{0}^{(\eta)}.$

With the observation that ${\mathsf{G}}$ is superharmonic in
$B(0,r_{0})\subset{\mathbb{R}}^{d+m}$, we have the analogue of (2.4) and
(2.5):

(2.20) ${\mathsf{G}}_{\eta}(X)\leq{\mathsf{G}}(X)\qquad\forall X\in
B(0,r_{0}-\eta)\setminus\\{0\\}$

and, from (2.1) and (1.14),

(2.21)
$|{\mathsf{G}}(X)-{\mathsf{G}}_{\eta}(X)|\leq\frac{C\eta^{2}}{|X|^{s+2}}\qquad\forall|X|\geq
2\eta,$

where $C>0$ depends on $r_{0}$. From (2.20), we obtain the analogue of (2.6):

(2.22)
$\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{G}}(x-y)d\delta_{0}^{(\eta)}(x)d\delta_{0}^{(\eta)}(y)\leq{\mathsf{G}}_{\eta}(0)\leq
C\left\lparen\eta^{-s}+|\log\eta|\mathbf{1}_{d=1\wedge s=0}\right\rparen.$

We may now repeat all of the analysis of Section 2.2 in the extended space
${\mathbb{R}}^{d+m}$. The modulated energy is still defined by (1.6). The only
change is in the mollification procedure which happens in extended space.
Analogous to the lower bound of 2.1, we have a lower bound for the modulated
energy in the super-Coulombic case via this mollification procedure.

######  Proposition 2.2.

Let $d\geq 1$ and $\max\\{d-2,0\\}<s<d$ or $d=1$ and $s=0$. Suppose that
$\underline{x}_{N}\in({\mathbb{R}}^{d})^{N}$ is a pairwise distinct
configuration and $\mu\in\mathcal{P}({\mathbb{R}}^{d})\cap
L^{\infty}({\mathbb{R}}^{d})$. If $s=0$, also assume that
$\int_{{\mathbb{R}}^{d}}\log(1+|x|)d\mu(x)<\infty$. There exists a constant
$C$ depending on $s,d$ and on ${\mathsf{g}}$ only through assumptions (1.23) –
(1.27), such that for every $\eta_{i}<\min\\{\frac{1}{2},\frac{1}{2}r_{0}\\}$,
we have

(2.23) $\frac{1}{N^{2}}\sum_{\begin{subarray}{c}1\leq i\neq j\leq N\\\
|x_{i}-x_{j}|\leq\frac{1}{2}r_{0}\end{subarray}}\left\lparen{\mathsf{g}}(x_{j}-x_{i})-{\mathsf{G}}_{\eta_{i}}(x_{j}-x_{i},0)\right\rparen_{+}+C^{-1}\left\|\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{j}}^{(\eta_{i})}-\tilde{\mu}\right\|_{\dot{H}^{\frac{s-d-m}{2}}({\mathbb{R}}^{d+m})}^{2}\\\
\leq{F}_{N}(\underline{x}_{N},\mu)+\frac{1}{N^{2}}\sum_{i=1}^{N}{\mathsf{G}}_{\eta_{i}}(0)+\frac{C}{N}\sum_{i=1}^{N}\left\lparen\|\mu\|_{L^{\infty}}\left\lparen\eta_{i}^{d-s}+(\eta_{i}^{d}|\log\eta_{i}|)\mathbf{1}_{s=0}\right\rparen+\eta_{i}^{2}\right\rparen,$

where $r_{0}$ is the constant in (1.23), $X_{i}\coloneqq(x_{i},0)$, and

(2.24)
$\tilde{\mu}(X)\coloneqq\mu(x)\delta_{{\mathbb{R}}^{d}\times\\{0\\}}(X).$

###### Proof.

Following the proof of 2.1, we write

(2.25)
$\begin{split}{F}_{N}(\underline{x}_{N},\mu)&=\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{G}}(X-Y)d\left\lparen\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}}^{(\eta_{i})}-\tilde{\mu}\right\rparen^{\otimes
2}(x,y)\\\
&\phantom{=}-\frac{1}{N^{2}}\sum_{i=1}^{N}\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{G}}(X-Y)d(\delta_{0}^{(\eta_{i})})^{\otimes
2}(X,Y)\\\
&\phantom{=}-\frac{2}{N}\sum_{i=1}^{N}\int_{{\mathbb{R}}^{d+m}}\left\lparen{\mathsf{G}}(Y-X_{i})-{\mathsf{G}}_{\eta_{i}}(Y-X_{i})\right\rparen
d\tilde{\mu}(Y)\\\ &\phantom{=}+\frac{1}{N^{2}}\sum_{1\leq i\neq j\leq
N}\int_{{\mathbb{R}}^{d}}\left\lparen{\mathsf{G}}(Y-X_{i})-{\mathsf{G}}_{\eta_{i}}(Y-X_{i})\right\rparen
d({\delta}_{X_{j}}+{\delta}_{X_{j}}^{(\eta_{j})})(Y).\end{split}$

The third line reduces to an integral over ${\mathbb{R}}^{d}$, identical to
those in (2.16). Analogously to (2.2), in view of (2.20), (1.24) and (2.21) we
have

$\displaystyle\int_{B(0,r_{0}-\eta)\subset{\mathbb{R}}^{d}}|{\mathsf{g}}(x)-{\mathsf{G}}_{\eta}(x,0)|dx$
$\displaystyle\leq
2\int_{B(0,2\eta)}{\mathsf{g}}(x)+C\eta^{2}\int_{B(0,r_{0})\backslash
B(0,2\eta)}\frac{dx}{|x|^{2+s}}$ (2.26) $\displaystyle\leq
C\left(\eta^{d-s}+(\eta^{d}|\log\eta|)\mathbf{1}_{s=0}\right).$

The third line in (2.25) is thus bounded below by

(2.27)
$-\frac{C}{N}\sum_{i=1}^{N}\left\lparen\eta_{i}^{2}+\|\mu\|_{L^{\infty}}\left(\eta_{i}^{d-s}+(\eta_{i}^{d}|\log\eta_{i}|)\mathbf{1}_{s=0}\right)\right\rparen$

as in (2.16). For the fourth line, we use (2.21) and that
$\delta_{X_{j}}^{(\eta_{j})}$ is a positive measure on ${\mathbb{R}}^{d+m}$ to
obtain

$\displaystyle\frac{1}{N^{2}}\sum_{1\leq i\neq j\leq
N}\int_{{\mathbb{R}}^{d+m}}\left\lparen{\mathsf{G}}(Y-X_{i})-{\mathsf{G}}_{\eta_{i}}(Y-X_{i})\right\rparen
d({\delta}_{X_{j}}+{\delta}_{X_{j}}^{(\eta_{j})})(Y)$
$\displaystyle\geq\frac{1}{N^{2}}\sum_{\begin{subarray}{c}1\leq i\neq j\leq
N\\\ |x_{i}-x_{j}|\leq
r_{0}-\eta\end{subarray}}\int_{{\mathbb{R}}^{d+m}}\left\lparen{\mathsf{G}}(Y-X_{i})-{\mathsf{G}}_{\eta_{i}}(Y-X_{i})\right\rparen
d\delta_{X_{j}}(Y)-\frac{C}{N}\sum_{i=1}^{N}\eta_{i}^{2}$ (2.28)
$\displaystyle\geq\frac{1}{N^{2}}\sum_{\begin{subarray}{c}1\leq i\neq j\leq
N\\\ |x_{i}-x_{j}|\leq
r_{0}-\eta\end{subarray}}\left\lparen{\mathsf{g}}(x_{j}-x_{i})-{\mathsf{G}}_{\eta_{j}}(X_{j}-X_{i})\right\rparen_{+}-\frac{C}{N}\sum_{i=1}^{N}\eta_{i}^{2}.$

Finally, noting that we have by virtue of (1.26) that

(2.29)
$\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{G}}(X-Y)d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}}^{(\eta_{i})}-\tilde{\mu}\right)^{\otimes
2}(X,Y)\geq
C_{1}\left\|\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}}^{(\eta_{i})}-\tilde{\mu}\right\|_{\dot{H}^{\frac{s-d-m}{2}}({\mathbb{R}}^{d+m})}^{2},$

we arrive at the inequality in the statement of the proposition. ∎

###  2.4. Control of microscale interactions

As recognized for the first time by the third author in [Ser20b], an
interesting choice of parameters $\eta_{i}$ is to take each $\eta_{i}$ equal
to a fixed fraction of the distance from $x_{i}$ to its nearest neighbor(s).
This ensures that the balls $B(x_{i},\eta_{i})$ are pairwise disjoint, while
also keeping the $\eta_{i}$ large enough that the right-hand side error in
(2.12) and (2.23) is kept small. However, in this paper, we are able to
dispense with the requirement that the balls be disjoint; so, we can choose
the $\eta_{i}$ to all be equal to some parameter $\eta$ or $\varepsilon$, to
be optimized at the end.

Inequalities (2.12) and (2.23) allow us to control the small-scale
interactions. Indeed, applying them to $\eta_{i}=3\varepsilon$ we arrive at
the following corollary.

######  Corollary 2.3.

Let $0\leq s<d$. For any pairwise distinct
$\underline{x}_{N}\in({\mathbb{R}}^{d})^{N}$ and
$\mu\in\mathcal{P}({\mathbb{R}}^{d})\cap L^{\infty}({\mathbb{R}}^{d})$ (with
$\int_{{\mathbb{R}}^{d}}\log(1+|x|)d\mu(x)<\infty$ if $s=0$), we have for all
$0<\varepsilon<\min\\{\frac{1}{8},\frac{r_{0}}{8}\\}$,

(2.30) $\frac{1}{N^{2}}\sum_{\begin{subarray}{c}1\leq i\neq j\leq N\\\
|x_{i}-x_{j}|\leq\varepsilon\end{subarray}}\left\lparen{\mathsf{g}}(x_{i}-x_{j})\mathbf{1}_{s>0}+\mathbf{1}_{s=0}\right\rparen\leq
C\Bigg{(}{F}_{N}(\underline{x}_{N},\mu)+C\Big{(}\frac{\varepsilon^{-s}+|\log\varepsilon|\mathbf{1}_{s=0}}{N}+\varepsilon^{2}\\\
+\|\mu\|_{L^{\infty}}(\varepsilon^{d-s}+(\varepsilon^{d}|\log\varepsilon|)\mathbf{1}_{s=0}+(\varepsilon^{2}|\log\varepsilon|)\mathbf{1}_{s=d-2})\Big{)}\Bigg{)},$

where $C$ depends on $s,d$ and on ${\mathsf{g}}$ only through the assumptions
(1.13) – (1.27).

###### Proof.

We use assumption (1.27), which ensures that for
$|x_{i}-x_{j}|\leq\varepsilon$, we have

(2.31)
${{\mathsf{g}}(x_{i}-x_{j})-{\mathsf{g}}_{3\varepsilon}(x_{i}-x_{j})}\geq(1-c_{s}){\mathsf{g}}(x_{i}-x_{j})\mathbf{1}_{s>0}+c_{0}\mathbf{1}_{s=0}.$

∎

###  2.5. Coerciveness

From 2.1 or 2.2, we easily deduce that the modulated energy ${F}_{N}$ controls
$\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu$ in the weak-* topology (cf.
[Ser20b, Proposition 3.6]).

######  Proposition 2.4.

Let $\phi\in{\mathcal{S}}({\mathbb{R}}^{d})$ be a test function. For any
$\eta>0$, any $\underline{x}_{N}$ and $\mu$ as in 2.1 or 2.2, we have

(2.32)
$\left|\int_{{\mathbb{R}}^{d}}\phi\,d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu\right)\right|\leq\|\phi\|_{C^{0,\alpha}}\eta^{\alpha}+C\|\phi\|_{\dot{H}^{\frac{d-s}{2}}}\Bigg{(}{F}_{N}(\underline{x}_{N},\mu)+{C}\frac{\eta^{-s}+|\log\eta|\mathbf{1}_{s=0}}{N}\\\
+C\|\mu\|_{L^{\infty}}(\eta^{d-s}+(\eta^{d}|\log\eta|)\mathbf{1}_{s=0}+(\eta^{2}|\log\eta|)\mathbf{1}_{s=d-2})+\eta^{2}\Bigg{)}^{1/2},$

where $C$ depends on $s,d,m$ and the potential ${\mathsf{g}}$ through
assumptions (1.13) - (1.27).

###### Proof.

Fix $\phi\in{\mathcal{S}}({\mathbb{R}}^{d})$, and let
$\tilde{\phi}\in{\mathcal{S}}({\mathbb{R}}^{d+m})$ be an extension of $\phi$
such that

(2.33)
$\begin{split}\|\tilde{\phi}\|_{\dot{H}^{\frac{d+m-s}{2}}({\mathbb{R}}^{d+m})}&\lesssim\|\phi\|_{\dot{H}^{\frac{d-s}{2}}({\mathbb{R}}^{d})},\\\
\|\tilde{\phi}\|_{C^{0,\alpha}({\mathbb{R}}^{d+m})}&\lesssim\|\phi\|_{C^{0,\alpha}({\mathbb{R}}^{d})}.\end{split}$

For instance, take
$\tilde{\phi}(x,z)\coloneqq(e^{|z|\Delta_{{\mathbb{R}}^{d}}}\phi)(x)$. We may
now write

(2.34) $\left|\int_{{\mathbb{R}}^{d}}\phi
d\left\lparen\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu\right\rparen\right|\leq\left|\int_{{\mathbb{R}}^{d+m}}\tilde{\phi}d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}}^{(\eta)}-\tilde{\mu}\right)\right|+\left|\int_{{\mathbb{R}}^{d+m}}\tilde{\phi}\frac{1}{N}\sum_{i=1}^{N}d(\delta_{X_{i}}-\delta_{X_{i}}^{(\eta)})\right|\\\
\leq\|\tilde{\phi}\|_{\dot{H}^{\frac{m+d-s}{2}}({\mathbb{R}}^{d+m})}\left\|\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}}^{(\eta)}-\tilde{\mu}\right\|_{\dot{H}^{\frac{s-d-m}{2}}({\mathbb{R}}^{d+m})}+\|\phi\|_{C^{0,\alpha}({\mathbb{R}}^{d})}\eta^{\alpha}.$

We conclude after applying the estimate (2.12) or (2.23). ∎

By duality, 2.4 implies that the modulated energy controls inhomogeneous
Sobolev norms: for $\gamma>\frac{d+2}{2}$,

(2.35)
$\left\|\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu\right\|_{H^{-\gamma}}^{2}\leq
C{F}_{N}(\underline{x}_{N},\mu)\\\ +C(1+\|\mu\|_{L^{\infty}})\left\lparen
N^{\frac{-2}{2+s}}+(N^{-1}\log N)\mathbf{1}_{s=0}+(N^{-\frac{2}{d}}\log
N)\mathbf{1}_{s=d-2}\right\rparen.$

Indeed, Sobolev embedding implies that $H^{\gamma}({\mathbb{R}}^{d})\subset
C^{1}({\mathbb{R}}^{d})$, from which the preceding bound follows. Further
refinements of such norm control, such as in terms of Besov spaces, are
possible. For more details, we refer to [Ros20a, Proposition 3.10] and
[Ros20b, Proposition 3.8].

##  3\. Commutator estimates

In this section, we present commutator estimates without renormalization,
which is the most technical component of the paper. This is a purely space-
based functional inequality (cf. [BJW19a, Lemma 5.1]) which dispenses with the
stress-energy tensor structure of [Due16, Ser20b].

######  Proposition 3.1.

Let $0\leq s<d$. Let ${\mathsf{g}}\in
C^{\infty}({\mathbb{R}}^{d}\setminus\\{0\\})$ such that
${\mathsf{g}}(x)={\mathsf{g}}(-x)$ in $B(0,r_{0})$, and

(3.1) $\forall k\geq 1,\qquad|\nabla^{\otimes
k}{\mathsf{g}}(x)|\lesssim|x|^{-k-s}\quad\forall x\neq 0.$

If $s\leq d-2$, then also assume that

(3.2) $|\hat{{\mathsf{g}}}(\xi)|\lesssim|\xi|^{s-d}\qquad\forall\xi\neq 0.$

If $s=d-2k$, for some $k\in{\mathbb{N}}$, also assume that ${\mathsf{g}}$ is
such that the the kernel

(3.3)
$\mathsf{k}(x-y)\coloneqq(x-y)\otimes\nabla^{\otimes(1+2k)}{\mathsf{g}}(x-y)\qquad\forall
x\neq y$

is associated to a Calderón-Zygmund operator. Let $v$ be a Lipschitz
continuous vector field from ${\mathbb{R}}^{d}$ to ${\mathbb{R}}^{d}$. There
exists a constant $C>0$, depending only $s,d$ and the potential ${\mathsf{g}}$
through (3.1) – (3.3), such that for any
$f,g\in{\mathcal{S}}({\mathbb{R}}^{d})$,333If $s=0$, then implicit is the
requirement that the Fourier transforms of $f,g$ vanish sufficiently rapidly
at the origin so that the $\dot{H}^{-d/2}$ norm converges. This may be ensured
by assuming a log-weighted $L^{1}$ condition on $f,g$. we have

(3.4)
$\begin{split}&\left|\int_{({\mathbb{R}}^{d})^{2}}(v(x)-v(y))\cdot\nabla{\mathsf{g}}(x-y)f(x)g(y)dxdy\right|\\\
&\leq C\left\lparen\|\nabla
v\|_{L^{\infty}}+\||\nabla|^{\frac{d-s}{2}}v\|_{L^{\frac{2d}{d-2-s}}}\mathbf{1}_{s<d-2}\right\rparen\|f\|_{\dot{H}^{\frac{s-d}{2}}}\|g\|_{\dot{H}^{\frac{s-d}{2}}}.\end{split}$

Consequently, the integral in the left-hand side of (3.4) extends to a bounded
bilinear form $B_{v}(\cdot,\cdot)$ on
$\dot{H}^{\frac{s-d}{2}}({\mathbb{R}}^{d})$ satisfying the bound (3.4).

###### Proof.

We divide the proof into the super-Coulombic, Coulombic, and sub-Coulombic
cases, beginning with the super-Coulombic. We will not explicitly track the
dependence of the implicit constants; but the reader should keep in mind that
they depend on the parameters $s,d$ as well as the implicit constants in
(3.1), (3.2) as well as the size and smoothness condition constants for the
kernel (3.3) and the $L^{2}$ operator norm of the associated Calderón-Zygmund
operator.

The super-Coulombic case. Define

(3.5)
$\mathsf{k}_{v}(x,y)\coloneqq(v(x)-v(y))\cdot\nabla{\mathsf{g}}(x-y)\qquad\forall
x\neq y.$

By approximation, we may assume that $v\in C^{\infty}$. Furthermore, by
density in $\dot{H}^{\frac{s-d}{2}}$, we may assume without loss of generality
that $f,g\in{\mathcal{S}}({\mathbb{R}}^{d})$ with Fourier support away from
the origin. Additionally, by dominated convergence, we may assume that
${\mathsf{g}}$ has compact support. Write

(3.6) $f=\operatorname{div}\nabla\Delta^{-1}f\eqqcolon\operatorname{div}f_{1}$

and similarly for $g$. Note that $f_{1}=(f_{1}^{i}),g_{1}=(g_{1}^{i})$ are
${\mathbb{R}}^{d}$-valued vector fields. Then

(3.7)
$\int_{({\mathbb{R}}^{d})^{2}}\mathsf{k}_{v}(x,y)f(x)g(y)dxdy=\int_{({\mathbb{R}}^{d})^{2}}\mathsf{k}_{v}(x,y)\operatorname{div}f_{1}(x)\operatorname{div}g_{1}(y)dxdy.$

We want to integrate by parts to move the derivatives onto the kernel
$\mathsf{k}_{v}$. But since $\mathsf{k}_{v}$ is singular along the diagonal
$x=y$, we need to exercise some care.

To this end, we write the preceding right-hand side as

(3.8) $\lim_{\varepsilon\rightarrow
0^{+}}\int_{|x-y|\geq\varepsilon}\mathsf{k}_{v}(x,y)\operatorname{div}f_{1}(x)\operatorname{div}g_{1}(y)dxdy.$

Integrating by parts in $y$, we find that the preceding integral equals

(3.9)
$\begin{split}\int_{|x-y|=\varepsilon}\mathsf{k}_{v}(x,y)f(x)g_{1}(y)\cdot\frac{(x-y)}{|x-y|}d{\mathcal{H}}^{2d-1}(x,y)\\\
-\int_{|x-y|\geq\varepsilon}\nabla_{y}\mathsf{k}_{v}(x,y)f(x)\cdot
g_{1}(y)dxdy,\end{split}$

where ${\mathcal{H}}^{2d-1}$ denotes the $(2d-1)$-dimensional Hausdorff
measure on ${\mathbb{R}}^{2d}$. If $s<d-1$, then we can use the crude kernel
estimate based on (1.24)

(3.10) $|\mathsf{k}_{v}(x,y)|\lesssim\min\left(\frac{\|\nabla
v\|_{L^{\infty}}}{|x-y|^{s}},\frac{\|v\|_{L^{\infty}}}{|x-y|^{s+1}}\right)$

to directly estimate the boundary term by

(3.11)
$\left|\int_{|x-y|=\varepsilon}\mathsf{k}_{v}(x,y)f(x)g_{1}(y)\cdot\frac{(x-y)}{|x-y|}d{\mathcal{H}}^{2d-1}(x,y)\right|\lesssim\|\nabla
v\|_{L^{\infty}}\|f\|_{L^{1}}\|g_{1}\|_{L^{\infty}}\varepsilon^{d-1-s},$

which tends to zero as $\varepsilon\rightarrow 0$. Otherwise, we subtract and
add $g_{1}(x)$ to write

(3.12)
$\begin{split}&\int_{|x-y|=\varepsilon}\mathsf{k}_{v}(x,y)f(x)g_{1}(y)\cdot\frac{(x-y)}{|x-y|}d{\mathcal{H}}^{2d-1}(x,y)\\\
&=\int_{|x-y|=\varepsilon}\mathsf{k}_{v}(x,y)f(x)(g_{1}(y)-g_{1}(x))\cdot\frac{(x-y)}{|x-y|}d{\mathcal{H}}^{2d-1}(x,y)\\\
&\phantom{=}+\int_{|x-y|=\varepsilon}\mathsf{k}_{v}(x,y)f(x)g_{1}(x)\cdot\frac{(x-y)}{|x-y|}d{\mathcal{H}}^{2d-1}(x,y).\end{split}$

In the sequel, it will be convenient to sometimes write in coordinates and
employ the convention of Einstein summation. Writing $z=x-y$, so that by
Taylor’s theorem,

(3.13)
$\mathsf{k}_{v}(x,y)=-z^{i_{1}}{\partial}_{i_{1}}v(x)\cdot\nabla{\mathsf{g}}(z)+O(\|\nabla^{\otimes
2}v\|_{L^{\infty}}|z|^{1-s}),$

we see that

(3.14)
$\int_{|x-y|=\varepsilon}\mathsf{k}_{v}(x,y)f(x)g_{1}(x)\cdot\frac{(x-y)}{|x-y|}d{\mathcal{H}}^{2d-1}(x,y)=O(\|\nabla^{\otimes
2}v\|_{L^{\infty}}\|fg_{1}\|_{L^{1}}\varepsilon^{d-s})\\\
+\int_{{\mathbb{R}}^{d}}\int_{{\partial}B(0,\varepsilon)}(z^{i_{1}}{\partial}_{i_{1}}v(x)\cdot\nabla{\mathsf{g}}(z))f(x)\left\lparen
g_{1}(x)\cdot\frac{z}{|z|}\right\rparen d{\mathcal{H}}^{d-1}(z)dx.$

Note that the first term in the right-hand side vanishes as
$\varepsilon\rightarrow 0$ since $s<d$ and the second term vanishes by
assumption (1.21) and the invariance of the measure under $z\mapsto-z$.

For the non-boundary term, we integrate by parts now in $x$ to obtain

(3.15)
$-\int_{|x-y|\geq\varepsilon}\nabla_{y}\mathsf{k}_{v}(x,y)\operatorname{div}f_{1}(x)\cdot
g_{1}(y)dxdy=\int_{|x-y|\geq\varepsilon}\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y):(f_{1}(x)\otimes
g_{1}(y))dxdy\\\
+\int_{|x-y|=\varepsilon}\left\lparen\frac{(x-y)}{|x-y|}\otimes\nabla_{y}\mathsf{k}_{v}(x,y)\right\rparen:\left\lparen
f_{1}(x)\otimes g_{1}(y)\right\rparen d{\mathcal{H}}^{2d-1}(x,y).$

We subtract and add $f_{1}(y)$ to write

(3.16)
$\begin{split}&\int_{|x-y|=\varepsilon}\left\lparen\frac{(x-y)}{|x-y|}\otimes\nabla_{y}\mathsf{k}_{v}(x,y)\right\rparen:\left\lparen
f_{1}(x)\otimes g_{1}(y)\right\rparen d{\mathcal{H}}^{2d-1}(x,y)\\\
&=\int_{|x-y|=\varepsilon}\left\lparen\frac{(x-y)}{|x-y|}\otimes\nabla_{y}\mathsf{k}_{v}(x,y)\right\rparen:\left\lparen(f_{1}(x)-f_{1}(y))\otimes
g_{1}(y)\right\rparen d{\mathcal{H}}^{2d-1}(x,y)\\\
&\phantom{=}+\int_{|x-y|=\varepsilon}\left\lparen\frac{(x-y)}{|x-y|}\otimes\nabla_{y}\mathsf{k}_{v}(x,y)\right\rparen:\left\lparen
f_{1}(y)\otimes g_{1}(y)\right\rparen d{\mathcal{H}}^{2d-1}(x,y).\end{split}$

By Taylor’s theorem,

(3.17)
$\begin{split}&\int_{|x-y|=\varepsilon}\left\lparen\frac{(x-y)}{|x-y|}\otimes\nabla_{y}\mathsf{k}_{v}(x,y)\right\rparen:\left\lparen(f_{1}(x)-f_{1}(y))\otimes
g_{1}(y)\right\rparen d{\mathcal{H}}^{2d-1}(x,y)\\\
&=\int_{|x-y|=\varepsilon}\left\lparen\frac{(x-y)}{|x-y|}\otimes\nabla_{y}\mathsf{k}_{v}(x,y)\right\rparen:\left\lparen(x-y)^{i_{1}}{\partial}_{i_{1}}f_{1}(y)\otimes
g_{1}(y)\right\rparen d{\mathcal{H}}^{2d-1}(x,y)\\\
&\phantom{=}+O(\|v\|_{W^{1,\infty}}\|\nabla^{\otimes
2}f_{1}\|_{L^{\infty}}\|g_{1}\|_{L^{1}}\varepsilon^{d-s}).\end{split}$

Proceeding similarly as above, we have that

(3.18)
$\begin{split}&\left|\int_{|x-y|=\varepsilon}\left\lparen\frac{(x-y)}{|x-y|}\otimes\nabla_{y}\mathsf{k}_{v}(x,y)\right\rparen:\left\lparen(x-y)^{i_{1}}{\partial}_{i_{1}}f_{1}(y)\otimes
g_{1}(y)\right\rparen d{\mathcal{H}}^{2d-1}(x,y)\right|\\\
&\lesssim\|v\|_{W^{2,\infty}}\|\nabla
f_{1}g_{1}\|_{L^{1}}\varepsilon^{d-s}.\end{split}$

Using that

(3.19)
$\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y)=\nabla_{x}\nabla_{y}\mathsf{k}_{v}(y,x)$

by assumption (1.21), we can swap $x$ and $y$ to obtain

(3.20)
$\begin{split}&\int_{|x-y|\geq\varepsilon}\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y):\left\lparen
f_{1}(x)\otimes g_{1}(y)\right\rparen dxdy\\\
&=-\frac{1}{2}\int_{|x-y|\geq\varepsilon}\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y):\left\lparen(f_{1}(x)-f_{1}(y))\otimes(g_{1}(x)-g_{1}(y))\right\rparen
dxdy\\\
&\phantom{=}+\frac{1}{2}\int_{|x-y|\geq\varepsilon}\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y):\left\lparen
f_{1}(x)\otimes g_{1}(x)\right\rparen dxdy\\\
&\phantom{=}+\frac{1}{2}\int_{|x-y|\geq\varepsilon}\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y):\left\lparen
f_{1}(y)\otimes g_{1}(y)\right\rparen dxdy.\end{split}$

Note that the last two terms are equal by symmetry and by Stokes theorem,

(3.21)
$\begin{split}&\int_{|x-y|\geq\varepsilon}\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y):\lparen
f_{1}(y)\otimes g_{1}(y)\rparen dxdy\\\
&=-\int_{|x-y|=\varepsilon}\left\lparen\frac{(x-y)}{|x-y|}\otimes\nabla_{y}\mathsf{k}_{v}(x,y)\right\rparen:\left\lparen
f_{1}(y)\otimes g_{1}(y)\right\rparen d{\mathcal{H}}^{2d-1}(x,y).\end{split}$

After a little bookkeeping, we realize we have shown that

(3.22)
$\int_{|x-y|\geq\varepsilon}\mathsf{k}_{v}(x,y)f(x)g(y)dxdy=O(\|\nabla^{\otimes
2}v\|_{L^{\infty}}\|fg_{1}\|_{L^{1}}\varepsilon^{d-s})\\\
+O(\|v\|_{W^{2,\infty}}\|\nabla
f_{1}g_{1}\|_{L^{1}}\varepsilon^{d-s})+O(\|v\|_{W^{1,\infty}}\|\nabla^{\otimes
2}f_{1}\|_{L^{\infty}}\|g_{1}\|_{L^{1}}\varepsilon^{d-s})\\\
-\frac{1}{2}\int_{|x-y|\geq\varepsilon}\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y):\left\lparen(f_{1}(x)-f_{1}(y))\otimes(g_{1}(x)-g_{1}(y))\right\rparen
dxdy.$

The first three terms of the right-hand side vanish as $\varepsilon\rightarrow
0$ since $s<d$. For the last term, direct computation reveals

(3.23)
$\begin{split}\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y)&=-(v(x)-v(y))\cdot\nabla^{\otimes
3}{\mathsf{g}}(x-y)-\left\lparen\nabla v(y)+\nabla
v(x)\right\rparen:\nabla^{\otimes 2}{\mathsf{g}}(x-y).\end{split}$

Therefore, using the triangle inequality and mean-value theorem, we find from
assumption (3.1) that

(3.24) $|\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y)|\lesssim\frac{\|\nabla
v\|_{L^{\infty}}}{|x-y|^{s+2}}.$

This bound together with Cauchy-Schwarz implies that

(3.25)
$\begin{split}&\int_{|x-y|\geq\varepsilon}|\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y)||f_{1}(x)-f_{1}(y)||g_{1}(x)-g_{1}(y)|dxdy\\\
&\lesssim\|\nabla
v\|_{L^{\infty}}\left\lparen\int_{({\mathbb{R}}^{d})^{2}}\frac{|f_{1}(x)-f_{1}(y)|^{2}}{|x-y|^{s+2}}dxdy\right\rparen^{1/2}\left\lparen\int_{({\mathbb{R}}^{d})^{2}}\frac{|g_{1}(x)-g_{1}(y)|^{2}}{|x-y|^{s+2}}dxdy\right\rparen^{1/2}.\end{split}$

Provided that $d-2<s<d$, the preceding right-hand side is equivalent to (for
instance, see [DNPV12, Proposition 3.4])

(3.26) $\|\nabla
v\|_{L^{\infty}}\|f_{1}\|_{\dot{H}^{1+\frac{s-d}{2}}}\|g_{1}\|_{\dot{H}^{1+\frac{s-d}{2}}}\lesssim\|\nabla
v\|_{L^{\infty}}\|f\|_{\dot{H}^{\frac{s-d}{2}}}\|g\|_{\dot{H}^{\frac{s-d}{2}}},$

where the ultimate inequality follows from the definition of $f_{1},g_{1}$. In
conclusion,

(3.27) $\begin{split}\lim_{\varepsilon\rightarrow
0}\left|\int_{|x-y|\geq\varepsilon}\mathsf{k}_{v}(x,y)f(x)g(y)dxdy\right|\lesssim\|\nabla
v\|_{L^{\infty}}\|f\|_{\dot{H}^{\frac{s-d}{2}}}\|g\|_{\dot{H}^{\frac{s-d}{2}}}.\end{split}$

The Coulombic case. We can modify the preceding argument to also work in the
Coulombic case $s=d-2$. To see this, we observe from above that

(3.28)
$\int_{({\mathbb{R}}^{d})^{2}}\mathsf{k}_{v}(x,y)f(x)g(y)dxdy=\lim_{\varepsilon\rightarrow
0}\Bigg{(}\int_{|x-y|=\varepsilon}\frac{(x-y)_{i_{1}}}{|x-y|}{\partial}_{y_{i_{2}}}\mathsf{k}_{v}(x,y)f_{1}^{i_{1}}(y)g_{1}^{i_{2}}(y)d{\mathcal{H}}^{2d-1}(x,y)\\\
+\int_{|x-y|=\varepsilon}\frac{(x-y)_{i_{1}}}{|x-y|}{\partial}_{y_{i_{2}}}\mathsf{k}_{v}(x,y)(f_{1}(x)-f_{1}(y))^{i_{1}}g_{1}^{i_{2}}(y)d{\mathcal{H}}^{2d-1}(x,y)\\\
+\int_{|x-y|\geq\varepsilon}{\partial}_{x_{i_{1}}}{\partial}_{y_{i_{2}}}\mathsf{k}_{v}(x,y)f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy\Bigg{)}.$

The second boundary term vanishes as $\varepsilon\rightarrow 0$ by (3.17) and
(3.18). We rewrite the first boundary term as

(3.29)
$\begin{split}&\int_{{\mathbb{R}}^{d}}f_{1}^{i_{1}}(y)g_{1}^{i_{2}}(y)\left\lparen\int_{{\partial}B(0,\varepsilon)}\left\lparen\frac{z_{i_{1}}}{|z|}{\partial}_{y_{i_{2}}}\mathsf{k}_{v}(y+z,y)\right\rparen
d{\mathcal{H}}^{d-1}(z)\right\rparen dy.\end{split}$

It is straightforward to check that as $\varepsilon\rightarrow 0$, this
expression tends to

(3.30)
$\begin{split}&-\int_{{\mathbb{R}}^{d}}f_{1}^{i_{1}}(y)g_{1}^{i_{2}}(y)\left\lparen\varepsilon^{d}\int_{{\partial}B(0,1)}\frac{z_{i_{1}}}{|z|}{\partial}_{i_{3}}v^{i_{4}}(y)z^{i_{3}}{\partial}_{i_{2}i_{4}}{\mathsf{g}}(\varepsilon
z))d{\mathcal{H}}^{d-1}(z)\right\rparen dy\\\
&\phantom{=}-\int_{{\mathbb{R}}^{d}}f_{1}^{i_{1}}(y)g_{1}^{i_{2}}(y)\left\lparen\varepsilon^{d-1}\int_{{\partial}B(0,1)}\frac{z_{i_{1}}}{|z|}({\partial}_{i_{2}}v(y)\cdot\nabla{\mathsf{g}}(\varepsilon
z))d{\mathcal{H}}^{d-1}(z)\right\rparen dy.\end{split}$

Using Cauchy-Schwarz and assumption (3.1), we can bound this term directly (up
to a constant) by

(3.31) $\|\nabla
v\|_{L^{\infty}}\|f_{1}\|_{L^{2}}\|g_{1}\|_{L^{2}}\lesssim\|\nabla
v\|_{L^{\infty}}\|f\|_{\dot{H}^{-1}}\|g\|_{\dot{H}^{-1}}.$

For the non-boundary term, we write out
$\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y)$ to obtain

(3.32)
$\begin{split}&-\int_{|x-y|\geq\varepsilon}\left\lparen{\partial}_{i_{1}}\nabla{\mathsf{g}}(x-y)\cdot{\partial}_{i_{2}}v(y)+{\partial}_{i_{2}}\nabla{\mathsf{g}}(x-y)\cdot{\partial}_{i_{1}}v(x)\right\rparen
f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y))dxdy\\\
&-\int_{|x-y|\geq\varepsilon}(v(x)-v(y))\cdot\nabla{\partial}_{i_{1}i_{2}}{\mathsf{g}}(x-y)f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y))dxdy.\end{split}$

Observe from assumption (3.2) that the kernel $\nabla^{\otimes
2}{\mathsf{g}}(x-y)$ defines a $({\mathbb{R}}^{d})^{\otimes 2}$-valued Fourier
multiplier with $L^{\infty}$ symbol, and therefore by Cauchy-Schwarz,

(3.33)
$\sup_{\varepsilon>0}\left|\int_{|x-y|\geq\varepsilon}\left\lparen{\partial}_{i_{1}}\nabla{\mathsf{g}}(x-y)\cdot{\partial}_{i_{2}}v(y)+{\partial}_{i_{2}}\nabla{\mathsf{g}}(x-y)\cdot{\partial}_{i_{1}}v(x)\right\rparen
f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y))dxdy\right|\\\ \lesssim\|\nabla
v\|_{L^{\infty}}\|f_{1}\|_{L^{2}}\|g_{1}\|_{L^{2}}\lesssim\|\nabla
v\|_{L^{\infty}}\|f\|_{\dot{H}^{-1}}\|g\|_{\dot{H}^{-1}}.$

For the remaining term, we use Taylor’s theorem to write

$\displaystyle(v(x)-v(y))\cdot\nabla{\partial}_{i_{1}i_{2}}{\mathsf{g}}(x-y)$
$\displaystyle=\left\lparen\int_{0}^{1}\nabla
v(y+t(x-y))\cdot(x-y)dt\right\rparen\cdot\nabla{\partial}_{i_{1}i_{2}}{\mathsf{g}}(x-y)$
(3.34) $\displaystyle=\left\lparen\int_{0}^{1}\nabla
v(y+t(x-y))dt\right\rparen\mathsf{k}_{i_{1}i_{2}}(x-y),$

where the $({\mathbb{R}}^{d})^{\otimes 2}$-valued kernel
$\mathsf{k}_{i_{1}i_{2}}$ is defined by

(3.35)
$\mathsf{k}_{i_{1}i_{2}}(x-y)\coloneqq(x-y)\otimes\nabla{\partial}_{i_{1}i_{2}}{\mathsf{g}}(x-y).$

By assumption (3.3), $\mathsf{k}_{i_{1}i_{2}}$ is a standard Calderón-Zygmund
kernel which is associated to a Calderón-Zygmund operator (see [Gra14a,
Theorem 5.4.1]). Consequently, the second term in (3.32) is a Calderón
$d$-commutator, which by the Christ-Journé theorem [CJ87], is $\lesssim$

(3.36) $\|\nabla
v\|_{L^{\infty}}\|f_{1}\|_{L^{2}}\|g_{1}\|_{L^{2}}\lesssim\|\nabla
v\|_{L^{\infty}}\|f\|_{\dot{H}^{-1}}\|g\|_{\dot{H}^{-1}}.$

The sub-Coulombic case. Let us now show that the above integration-by-parts
argument works for the case $0\leq s<d-2$. First, suppose that $d-4\leq
s<d-2$. An examination of the above argument shows that

(3.37)
$\int_{({\mathbb{R}}^{d})^{2}}\mathsf{k}_{v}(x,y)f(x)g(y)dxdy=\lim_{\varepsilon\rightarrow
0}\int_{|x-y|\geq\varepsilon}\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y):f_{1}(x)\otimes
g_{1}(y)dxdy.$

Substituting in identity (3.23) yields

(3.38) $\lim_{\varepsilon\rightarrow
0}\Bigg{(}-\int_{|x-y|\geq\varepsilon}(v(x)-v(y))\cdot\nabla{\partial}_{i_{1}i_{2}}{\mathsf{g}}(x-y)f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy\\\
-\int_{|x-y|\geq\varepsilon}\left\lparen{\partial}_{i_{2}}v(y)\cdot\nabla{\partial}_{i_{1}}{\mathsf{g}}(x-y)+{\partial}_{i_{1}}v(x)\cdot\nabla{\partial}_{i_{2}}{\mathsf{g}}(x-y)\right\rparen
f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy\Bigg{)}.$

For the first term, we can repeat the proof above for the super-Coulombic
($d-4<s<d-2$) and Coulombic ($s=d-4$) cases with ${\mathsf{g}}$ replaced by
${\partial}_{i_{1}i_{2}}{\mathsf{g}}$, and writing $f_{1}$ as the divergence
of a $2$-tensor,

(3.39)
$f_{1}^{i_{1}}=\operatorname{div}\nabla\Delta^{-1}f_{1}^{i_{1}}\eqqcolon\operatorname{div}f_{2}^{i_{1}},$

and similarly for $g_{1}$. For the second term in (3.38), we can pass to the
limit $\varepsilon\rightarrow 0$ (note that $\nabla^{\otimes 2}{\mathsf{g}}$
is locally integrable), so that by Fubini-Tonelli, we now consider

(3.40) $\int_{{\mathbb{R}}^{d}}(\nabla{\partial}_{i_{1}}{\mathsf{g}}\ast
f_{1}^{i_{1}})(y)\cdot{\partial}_{i_{2}}v(y)g_{1}^{i_{2}}(y)dy+\int_{{\mathbb{R}}^{d}}{\partial}_{i_{1}}v(x)f_{1}^{i_{1}}(x)\cdot(\nabla{\partial}_{i_{2}}{\mathsf{g}}\ast
g_{1}^{i_{2}})(x)dx.$

Since $f_{1},g_{1}$ satisfy the same bounds, we can use the $x\leftrightarrow
y$ swapping symmetry of the kernel to only consider the first term in the
preceding expression. Writing
$\mathrm{I}=|\nabla|^{\frac{d-s}{2}-1}|\nabla|^{\frac{s-d}{2}+1}$, the Cauchy-
Schwarz inequality implies

(3.41) $\left|\int_{{\mathbb{R}}^{d}}(\nabla{\partial}_{i_{1}}{\mathsf{g}}\ast
f_{1}^{i_{1}})(x)\cdot{\partial}_{i_{2}}v(x)g_{1}^{i_{2}}(x)dx\right|\lesssim\||\nabla|^{\frac{d-s}{2}-1}(\nabla{\partial}_{i_{1}}{\mathsf{g}}\ast
f_{1}^{i_{1}})\|_{L^{2}}\||\nabla|^{\frac{s-d}{2}+1}({\partial}_{i_{2}}vg_{1}^{i_{2}})\|_{L^{2}}.$

By Plancherel and assumption (3.2), it then follows that

(3.42) $\||\nabla|^{\frac{d-s}{2}-1}(\nabla{\partial}_{i_{1}}{\mathsf{g}}\ast
f_{1}^{i_{1}})\|_{L^{2}}\lesssim\|f\|_{\dot{H}^{\frac{s-d}{2}}},$

and by duality,

$\displaystyle\||\nabla|^{\frac{s-d}{2}+1}({\partial}_{i_{2}}vg_{1}^{i_{2}})\|_{L^{2}}$
$\displaystyle\leq\sup_{h\in{\mathcal{S}}({\mathbb{R}}^{d}):\|h\|_{L^{2}}\leq
2}\left|\int_{{\mathbb{R}}^{d}}|\nabla|^{\frac{s-d}{2}+1}h(x){\partial}_{i_{2}}v(x)g_{1}^{i_{2}}(x)dx\right|$
$\displaystyle\leq\sup_{h\in{\mathcal{S}}({\mathbb{R}}^{d}):\|h\|_{L^{2}}\leq
2}\||\nabla|^{\frac{d-s}{2}-1}(|\nabla|^{\frac{s-d}{2}+1}h{\partial}_{i_{2}}v)\|_{L^{2}}\||\nabla|^{\frac{s-d}{2}+1}g_{1}^{i_{2}}\|_{L^{2}}$
(3.43)
$\displaystyle\lesssim\|g\|_{\dot{H}^{\frac{s-d}{2}}}\sup_{h\in{\mathcal{S}}({\mathbb{R}}^{d}):\|h\|_{L^{2}}\leq
2}\||\nabla|^{\frac{d-s}{2}-1}(|\nabla|^{\frac{s-d}{2}+1}h{\partial}_{i_{2}}v)\|_{L^{2}}.$

By fractional Leibnitz rule (for instance, see [Gra14b, Theorem 7.6.1])

(3.44)
$\displaystyle\||\nabla|^{\frac{d-s}{2}-1}(|\nabla|^{\frac{s-d}{2}+1}h{\partial}_{i_{2}}v)\|_{L^{2}}$
$\displaystyle\lesssim\|h\|_{L^{p_{1}}}\|\nabla
v\|_{L^{q_{1}}}+\||\nabla|^{\frac{s-d}{2}+1}h\|_{L^{p_{2}}}\||\nabla|^{\frac{d-s}{2}-1}\nabla
v\|_{L^{q_{2}}}$

for any $1<p_{1},q_{1},p_{2},q_{2}\leq\infty$ with
$p_{i}^{-1}+q_{i}^{-1}=2^{-1}$. We choose $p_{1}=2$ and
$p_{2}=\frac{2d}{s+2}$. So by Hardy-Littlewood-Sobolev and the $L^{p}$
boundedness of Riesz transforms, the right-hand side is $\lesssim$

(3.45) $\|h\|_{L^{2}}\left\lparen\|\nabla
v\|_{L^{\infty}}+\||\nabla|^{\frac{d-s}{2}}v\|_{L^{\frac{2d}{d-2-s}}}\right\rparen.$

After a little bookkeeping, we arrive at the estimate

(3.46) $\left|\int_{{\mathbb{R}}^{d}}(\nabla{\partial}_{i_{1}}{\mathsf{g}}\ast
f_{1}^{i_{1}})(x)\cdot{\partial}_{i_{2}}v(x)g_{1}^{i_{2}}(x)dx\right|\lesssim\|f\|_{\dot{H}^{\frac{s-d}{2}}}\|g\|_{\dot{H}^{\frac{s-d}{2}}}\left\lparen\|\nabla
v\|_{L^{\infty}}+\||\nabla|^{\frac{d-s}{2}}v\|_{L^{\frac{2d}{d-2-s}}}\right\rparen.$

For the general case $0\leq s<d-2$, we let $k\in{\mathbb{N}}$ be such that
$d-2(k+1)\leq s<d-2k$, and proceed by induction on $k$, using the argument
just given above for the induction step. The existence of the form extension
$B_{v}$ follows by the usual density argument. ∎

##  4\. Renormalization of the commutator estimate

In order to apply 3.1 to $f,g$ which fail to be in $\dot{H}^{\frac{s-d}{2}}$,
such as the Dirac mass, we now have to deal with the nondiagonal
renormalization which is carried out through the smearing procedure. The main
result of this section is the following proposition, which we view as a
_renormalized commutator estimate_ and which is the workhorse of this article.

4.1 stated below should be compared to [Ser20b, Proposition 1.1], which, as
previously discussed, treated the exact Coulomb/super-Coulombic Riesz case, as
well as the $\log$ case in dimension 1. That result only required that the
vector field $v$ be in $W^{1,\infty}$. In contrast, 4.1 requires in the super-
Coulombic case the stronger assumption that $v\in C^{1,\alpha}$, for any
positive $\alpha\in(0,1]$, and belong to the Bessel potential space
$W^{\frac{d(d+m-s)+2m}{2(d+m)},\frac{2(d+m)}{d+m-2-s}}$. The reason for this
difference is precisely the combination of the commutator estimate with the
extension/smearing procedure used in the present article. Namely, we want to
apply 3.1 by using the smearing procedure to renormalize. But this forces us
to work in the extended space ${\mathbb{R}}^{d+m}$ when we apply 3.1 and then
to find a way to control the right-hand side of the proposition in terms of
the modulated energy and norms of $v$ and $\mu$, which live in
${\mathbb{R}}^{d}$. Consequently, we choose an extension
$\tilde{v}:{\mathbb{R}}^{d+m}\rightarrow{\mathbb{R}}^{d+m}$ of $v$ and apply
3.1 with $\tilde{v}$. It is easy to construct an extension satisfying
$\|\tilde{v}\|_{W^{1,\infty}}\lesssim\|v\|_{W^{1,\infty}}$ simply by
multiplying $v$ by a bump function in the new variable. But such an extension
does not satisfy the auxiliary Sobolev condition, which is present (unless
$s=d-1$) since we always choose $m$ so that $d+m-s-2\geq 0$. Finding an
extension which satisfies both conditions essentially amounts to showing that
the trace operator

(4.1) $\Tr:W^{1,\infty}({\mathbb{R}}^{d+m})\cap
W^{\frac{d+m-s}{2},\frac{2(d+m)}{d+m-2-s}}({\mathbb{R}}^{d+m})\rightarrow
W^{1,\infty}({\mathbb{R}}^{d})\cap
W^{\frac{d(d+m-s)+2m}{2(d+m)},\frac{2(d+m)}{d+m-2-s}}({\mathbb{R}}^{d})$

is surjective with a bounded right inverse. If one could show such a result,
which we do not know how to do, then one could drop the $C^{1,\alpha}$
assumption.

######  Proposition 4.1.

Let $d\geq 1$, $0\leq s<d$. Let $\underline{x}_{N}\in({\mathbb{R}}^{d})^{N}$
be pairwise distinct, and $\mu\in\mathcal{P}({\mathbb{R}}^{d})\cap
L^{\infty}({\mathbb{R}}^{d})$. If $s=0$, assume that
$\int_{{\mathbb{R}}^{d}}\log(1+|x|)d\mu(x)<\infty$ and if $s\geq d-1$, assume
that $\||\nabla|^{s-d}\nabla\mu\|_{L^{\infty}}<\infty$. Let $v$ be a
continuous vector field on ${\mathbb{R}}^{d}$ such that

(4.2) $\begin{cases}\|\nabla
v\|_{L^{\infty}}+\||\nabla|^{\frac{d-s}{2}}v\|_{L^{\frac{2d}{d-2-s}}}\mathbf{1}_{s<d-2}<\infty,&{0\leq
s\leq d-2}\\\
\|v\|_{C^{1,\alpha}}+\|v\|_{W^{\frac{d(d+m-s)+2m}{2(d+m)},\frac{2(d+m)}{d+m-2-s}}}<\infty,&{d-2<s<d}\end{cases}$

for some $0<\alpha\leq 1$ and where $m$ is the dimension of the extension
space. There exists a constant $C$ depending only $d,s,\alpha,m$ and on the
potential ${\mathsf{g}}$ through assumptions (1.13) - (1.28), such that

(4.3)
$\left|\int_{({\mathbb{R}}^{d})^{2}\setminus\triangle}(v(x)-v(y))\cdot\nabla{\mathsf{g}}(x-y)d\left\lparen\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu\right\rparen^{\otimes
2}(x,y)\right|\\\ \leq C\Bigg{(}\|\nabla
v\|_{L^{\infty}}+\||\nabla|^{\frac{d-s}{2}}v\|_{L^{\frac{2d}{d-2-s}}}\mathbf{1}_{s<d-2}+\left\lparen\|v\|_{C^{1,\alpha}}+\|v\|_{W^{\frac{d(d+m-s)+2m}{2(d+m)},\frac{2(d+m)}{d+m-2-s}}}\right\rparen\mathbf{1}_{s>d-2}\Bigg{)}\\\
\times\Bigg{(}{F}_{N}(\underline{x}_{N},\mu)+C(1+\|\mu\|_{L^{\infty}})\Big{(}N^{-\frac{\min\\{d-s,2\\}}{\min\\{d,s+2\\}(s+1)}}+(N^{-1}\log
N)\mathbf{1}_{s=0}+(N^{-\frac{2}{d(d-1)}}\log N)\mathbf{1}_{s=d-2}\Big{)}\\\
+CN^{-\frac{\min\\{d+1,s+3\\}}{\min\\{d,s+2\\}(s+1)}}\Big{(}\||\nabla|^{s+1-d}\mu\|_{L^{\infty}}\mathbf{1}_{s<d-1}+\||\nabla|^{s-d}\nabla\mu\|_{L^{\infty}}\mathbf{1}_{s\geq
d-1}\Big{)}\Bigg{)}.$

###### Proof.

We need to introduce an extension of $\mathsf{k}_{v}$ to ${\mathbb{R}}^{d+m}$.
By the surjectivity of the trace theorem for Besov and fractional Sobolev
spaces (see [Tri10, Section 2.7.2]),444This is where the $C^{1,\alpha}$
assumption is used, even though our estimates in the proof will only depend on
the Lipschitz seminorm of the extended vector field $\tilde{v}$. It is
possible to relax $C^{1,\alpha}$ to the Besov space $B_{\infty,1}^{1}$, but
this is still stronger than $W^{1,\infty}$. there exists a vector field
$\tilde{v}$ on ${\mathbb{R}}^{d+m}$ whose $(m+1)$-th through $(m+d)$-th
components vanish identically, $\tilde{v}(x,0)=(v(x),0)$, and

(4.4)
$\begin{split}\|\tilde{v}\|_{C^{1,\alpha}({\mathbb{R}}^{d+m})}&\lesssim\|v\|_{C^{1,\alpha}({\mathbb{R}}^{d})},\\\
\|\tilde{v}\|_{W^{\frac{d+m-s}{2},\frac{2(d+m)}{d+m-2-s}}({\mathbb{R}}^{d+m})}&\lesssim\|v\|_{W^{\frac{d(d+m-s)+2m}{2(d+m)},\frac{2(d+m)}{d+m-2-s}}({\mathbb{R}}^{d})}.\end{split}$

With this notation, we define

(4.5)
${\mathsf{K}}_{v}(X,Y)\coloneqq\nabla{\mathsf{G}}(X-Y)\cdot(\tilde{v}(X)-\tilde{v}(Y))\qquad\forall
X\neq Y\in{\mathbb{R}}^{d+m}.$

We note that from assumption (1.24) that

(4.6)
$|{\mathsf{K}}_{v}(X,Y)|\leq{C\|\nabla\tilde{v}\|_{L^{\infty}}}\left\lparen|X-Y|^{-s}\mathbf{1}_{s>0}+\mathbf{1}_{s=0}\right\rparen$

and

(4.7)
$|\nabla_{X}{\mathsf{K}}_{v}(X,Y)|\leq{C\|\nabla\tilde{v}\|_{L^{\infty}}}{|X-Y|^{-s-1}}.$

For simplicity of notation, we again let $X_{i}$ denote $(x_{i},0)$.

Adding and subtracting the smeared point mass $\delta_{X_{i}}^{(\eta)}$, for
some $\eta$ to be chosen momentarily, we obtain the decomposition

(4.8)
$\int_{({\mathbb{R}}^{d})^{2}\setminus\triangle}\mathsf{k}_{v}(x,y)d\left\lparen\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu\right\rparen^{\otimes
2}(x,y)=\mathrm{Term}_{1}+\mathrm{Term}_{2}+\mathrm{Term}_{3},$

where

(4.9) $\displaystyle\mathrm{Term}_{1}$
$\displaystyle\coloneqq\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{K}}_{v}(X,Y)d\left\lparen\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}}^{(\eta)}-\tilde{\mu}\right\rparen^{\otimes
2}(X,Y),$ (4.10) $\displaystyle\mathrm{Term}_{2}$
$\displaystyle\coloneqq-\frac{2}{N}\sum_{j=1}^{N}\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{K}}_{v}(X,Y)d\tilde{\mu}(X)d(\delta_{X_{j}}-\delta_{X_{j}}^{(\eta)})(Y),$
(4.11) $\displaystyle\mathrm{Term}_{3}$
$\displaystyle\coloneqq\frac{1}{N^{2}}\sum_{1\leq i,j\leq
N}\int_{({\mathbb{R}}^{d+m})^{2}\setminus\triangle}{\mathsf{K}}_{v}(X,Y)d(\delta_{X_{i}}+\delta_{X_{i}}^{(\eta)})(X)d(\delta_{X_{j}}-\delta_{X_{j}}^{(\eta)})(Y).$

As before, $\tilde{\mu}\coloneqq\mu\delta_{{\mathbb{R}}^{d}\times\\{0\\}}$. We
proceed to estimate each of the $\mathrm{Term}_{j}$ individually.

Estimate for $\mathrm{Term}_{1}$. This term is exactly of the form that can be
handled by 3.1 with

(4.12)
$f=g=\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}}^{(\eta)}-\tilde{\mu}\in\dot{H}^{\frac{s-d-m}{2}}({\mathbb{R}}^{d+m}).$

We deduce from that proposition that

(4.13) $|\mathrm{Term}_{1}|\leq
C\left\lparen\|\nabla\tilde{v}\|_{L^{\infty}}+\||\nabla|^{\frac{d+m-s}{2}}\tilde{v}\|_{L^{\frac{2(d+m)}{d+m-2-s}}}\mathbf{1}_{s<d+m-2}\right\rparen\left\|\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}}^{(\eta)}-\tilde{\mu}\right\|_{\dot{H}^{\frac{s-d-m}{2}}}^{2}.$

Applying either 2.1 or 2.2, we can then bound

(4.14) $|\mathrm{Term}_{1}|\leq
C\left\lparen\|\nabla\tilde{v}\|_{L^{\infty}}+\||\nabla|^{\frac{d+m-s}{2}}\tilde{v}\|_{L^{\frac{2(d+m)}{d+m-2-s}}}\mathbf{1}_{s<d+m-2}\right\rparen\\\
\times\left({F}_{N}(\underline{x}_{N},\mu)+\frac{{\mathsf{G}}_{\eta}(0)}{N}+C\left\lparen\eta^{d-s}+(\eta^{d}|\log\eta|)\mathbf{1}_{s=0}+(\eta^{2}|\log\eta|)\mathbf{1}_{s=d-2}\right\rparen\|\mu\|_{L^{\infty}}+C\eta^{2}\right).$

Estimate for $\mathrm{Term}_{2}$. Suppose first that $0\leq s<d-1$. We break
$\mathrm{Term}_{2}$ into a sum over the index $i$ of terms

(4.15)
$\int_{|X-X_{i}|<2\eta}\int_{{\mathbb{R}}^{d+m}}{\mathsf{K}}_{v}(X,Y)d\left(\delta_{X_{i}}-\delta_{X_{i}}^{(\eta)}\right)(Y)d\tilde{\mu}(X)\\\
+\int_{|X-X_{i}|\geq
2\eta}\int_{{\mathbb{R}}^{d+m}}{\mathsf{K}}_{v}(X,Y)d\left(\delta_{X_{i}}-\delta_{X_{i}}^{(\eta)}\right)(Y)d\tilde{\mu}(X).$

For the first type of integral, we use that $\delta_{X_{i}}^{(\eta)}$ is a
probability measure to rewrite

(4.16)
$\int_{{\mathbb{R}}^{d+m}}{\mathsf{K}}_{v}(X,Y)d\left(\delta_{X_{i}}-\delta_{X_{i}}^{(\eta)}\right)(Y)=\int_{{\mathbb{R}}^{d+m}}\left({\mathsf{K}}_{v}(X,X_{i})-{\mathsf{K}}_{v}(X,Y)\right)d\delta_{X_{i}}^{(\eta)}(Y),$

and unpacking the definition (4.5) of ${\mathsf{K}}_{v}$, we have

(4.17) ${\mathsf{K}}_{v}(X,X_{i})-{\mathsf{K}}_{v}(X,Y)\\\
=(\tilde{v}(Y)-\tilde{v}(X_{i}))\cdot\nabla{\mathsf{G}}(X-X_{i})+(\tilde{v}(X)-\tilde{v}(Y))\cdot(\nabla{\mathsf{G}}(X-X_{i})-\nabla{\mathsf{G}}(X-Y)).$

Using (1.24), the mean-value theorem, and triangle inequality

(4.18)
$\int_{{\mathbb{R}}^{d+m}}|{\mathsf{K}}_{v}(X,X_{i})-{\mathsf{K}}_{v}(X,Y)|d\delta_{X_{i}}^{(\eta)}(Y)\\\
\leq
C\eta\|\nabla\tilde{v}\|_{L^{\infty}}\left(|X-X_{i}|^{-s-1}+\int_{{\mathbb{R}}^{d+m}}\left\lparen|X-Y|^{-s-1}\right\rparen
d\delta_{X_{i}}^{(\eta)}(Y)\right)$

for all $|X-X_{i}|\leq 2\eta$. Integrating the left-hand side with respect
$d\tilde{\mu}(X)$ and using Fubini-Tonelli, we find that the first term in
(4.15) is controlled by

(4.19)
$C\eta\|\nabla\tilde{v}\|_{L^{\infty}}\||\nabla|^{s+1-d}\mu\|_{L^{\infty}},$

where we also use that $|\cdot|^{-s-1}$ is (up to a constant) the convolution
kernel of the Fourier multiplier $|\nabla|^{s+1-d}$ in ${\mathbb{R}}^{d}$.

For the second type of contributions we use the mean-value inequality and
(4.7) to bound

$\displaystyle\left|\int_{|X-X_{i}|\geq
2\eta}\int_{{\mathbb{R}}^{d+m}}{\mathsf{K}}_{v}(X,Y)d\left(\delta_{X_{i}}-\delta_{X_{i}}^{(\eta)}\right)(Y)d\tilde{\mu}(X)\right|$
$\displaystyle\leq C\eta\|\nabla\tilde{v}\|_{L^{\infty}}\int_{|X-X_{i}|\geq
2\eta}|X-X_{i}|^{-s-1}d\tilde{\mu}(X)$ (4.20) $\displaystyle\leq
C\eta\|\nabla\tilde{v}\|_{L^{\infty}}{\||\nabla|^{s+1-d}\mu\|_{L^{\infty}}}.$

Now if $d-1\leq s<d$, we integrate by parts in $X$ to write

(4.21)
$\begin{split}&\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{K}}_{v}(X,Y)d\left\lparen\delta_{X_{i}}-\delta_{X_{i}}^{(\eta)}\right\rparen(Y)d\tilde{\mu}(X)\\\
&=-\int_{({\mathbb{R}}^{d+m})^{2}}\tilde{\mathsf{K}}_{v}(X,Y)\cdot
d\left\lparen\delta_{X_{i}}-\delta_{X_{i}}^{(\eta)}\right\rparen(Y)d\widetilde{\nabla\mu}(X)\\\
&\phantom{=}-\int_{{\mathbb{R}}^{d}}\operatorname{div}\tilde{v}(x,0)\left\lparen{\mathsf{G}}(x-x_{i},0)-{\mathsf{G}}_{\eta}(x-x_{i},0)\right\rparen
d\mu(x),\end{split}$

where
$\widetilde{\nabla\mu}(X)\coloneqq\nabla\mu(x)\delta_{{\mathbb{R}}^{d}\times\\{0\\}}(X)$
and we have defined the vector-valued kernel

(4.22)
$\tilde{\mathsf{K}}_{v}(X,Y)\coloneqq\left\lparen\tilde{v}(X)-\tilde{v}(Y)\right\rparen{\mathsf{G}}(X-Y).$

Proceeding as in the case $0\leq s<d-1$ with $\mu$ replaced by $\nabla\mu$,
the first term in the preceding right-hand side is bounded by

(4.23)
$C\eta\|\nabla\tilde{v}\|_{L^{\infty}}\||\nabla|^{s-d}\nabla\mu\|_{L^{\infty}}.$

Using that

(4.24) $\int_{|x-x_{i}|\leq
2\eta}\left|{\mathsf{G}}(x-x_{i},0)-{\mathsf{G}}_{\eta}(x-x_{i},0)\right|dx\leq
C\eta^{d-s}$

by assumption (1.24) and also using the bound (2.21), we see from Hölder’s
inequality that the second term is bounded by

(4.25) $C\|\nabla\tilde{v}\|_{L^{\infty}}\|\mu\|_{L^{\infty}}\eta^{d-s}.$

After a little bookkeeping, we conclude that

(4.26) $|\mathrm{Term}_{2}|\leq
C\|\nabla\tilde{v}\|_{L^{\infty}}\Bigg{(}\eta\||\nabla|^{s+1-d}\mu\|_{L^{\infty}}\mathbf{1}_{s<d-1}\\\
+\left\lparen\eta\||\nabla|^{s-d}\nabla\mu\|_{L^{\infty}}+\eta^{d-s}\|\mu\|_{L^{\infty}}\right\rparen\mathbf{1}_{s\geq
d-1}\Bigg{)}.$

Estimate for $\mathrm{Term}_{3}$. We first remove the self-interaction by
observing that

(4.27)
$\begin{split}&\sum_{i=1}^{N}\int_{({\mathbb{R}}^{d+m})^{2}\setminus\triangle}{\mathsf{K}}_{v}(X,Y)d(\delta_{X_{i}}+\delta_{X_{i}}^{(\eta)})(X)d(\delta_{X_{i}}-\delta_{X_{i}}^{(\eta)})(Y)\\\
&=-\sum_{i=1}^{N}\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{K}}_{v}(X,Y)d(\delta_{X_{i}}^{(\eta)})^{\otimes
2}(X,Y),\end{split}$

so that

(4.28)
$\begin{split}\mathrm{Term}_{3}&=-\frac{1}{N^{2}}\sum_{i=1}^{N}\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{K}}_{v}(X,Y)d(\delta_{X_{i}}^{(\eta)})^{\otimes
2}(X,Y)\\\ &\phantom{=}+\frac{1}{N^{2}}\sum_{1\leq i\neq j\leq
N}\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{K}}_{v}(X,Y)d(\delta_{X_{i}}+\delta_{X_{i}}^{(\eta)})(X)d(\delta_{X_{j}}-\delta_{X_{j}}^{(\eta)})(Y).\end{split}$

For the first sum, we use assumption (1.25) with the estimates (4.6) and (2.6)
to bound it as

$\displaystyle\frac{1}{N^{2}}\left|\sum_{i=1}^{N}\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{K}}_{v}(X,Y)d(\delta_{X_{i}}^{(\eta)})^{\otimes
2}(X,Y)\right|$
$\displaystyle\leq\frac{C}{N}\|\nabla\tilde{v}\|_{L^{\infty}}\left\lparen{\mathsf{G}}_{\eta}(0)\mathbf{1}_{s>0}+\mathbf{1}_{s=0}\right\rparen$
(4.29)
$\displaystyle\leq\frac{C}{N}\|\nabla\tilde{v}\|_{L^{\infty}}\left\lparen\eta^{-s}\mathbf{1}_{s>0}+\mathbf{1}_{s=0}\right\rparen.$

Next, we split the second sum over $1\leq i\neq j\leq N$ into a sum over
“close pairs” for which $|x_{i}-x_{j}|<\varepsilon$ and a sum over “far pairs”
for which $|x_{i}-x_{j}|\geq\varepsilon$, where $\varepsilon>2\eta$ is a
parameter, to be chosen later. For the sum over close pairs, we again use
assumption (1.25) with the bound (4.6) to obtain

$\displaystyle\frac{1}{N^{2}}\sum_{{1\leq i\neq j\leq
N}\atop{|x_{i}-x_{j}|<\varepsilon}}\left|\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{K}}_{v}(X,Y)d(\delta_{X_{i}}+\delta_{X_{i}}^{(\eta)})(X)d(\delta_{X_{j}}-\delta_{X_{j}}^{(\eta)})(Y)\right|$
(4.30)
$\displaystyle\leq\frac{C\|\nabla\tilde{v}\|_{L^{\infty}}}{N^{2}}\sum_{{1\leq
i\neq j\leq
N}\atop{|x_{i}-x_{j}|<\varepsilon}}\Bigg{(}\left({\mathsf{g}}(x_{i}-x_{j})+\int{\mathsf{G}}_{\eta}(X_{i}-Y)d\delta_{X_{j}}^{(\eta)}(Y)\right)\mathbf{1}_{s>0}+\mathbf{1}_{s=0}\Bigg{)}.$

Using (2.20), we then obtain (provided $\varepsilon<\frac{r_{0}}{2}$)

$\displaystyle\frac{1}{N^{2}}\sum_{{1\leq i\neq j\leq
N}\atop{|x_{i}-x_{j}|<\varepsilon}}\left|\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{K}}_{v}(X,Y)d(\delta_{X_{i}}+\delta_{X_{i}}^{(\eta)})(X)d(\delta_{X_{j}}-\delta_{X_{j}}^{(\eta)})(Y)\right|$
$\displaystyle\leq C\frac{\|\nabla\tilde{v}\|_{L^{\infty}}}{N^{2}}\sum_{{1\leq
i\neq j\leq
N}\atop{|x_{i}-x_{j}|<\varepsilon}}\left\lparen{\mathsf{g}}(x_{i}-x_{j})\mathbf{1}_{s>0}+\mathbf{1}_{s=0}\right\rparen$
$\displaystyle\leq
C\|\nabla\tilde{v}\|_{L^{\infty}}\Bigg{(}{F}_{N}(\underline{x}_{N},\mu)+C\Big{(}\frac{\varepsilon^{-s}+|\log\varepsilon|\mathbf{1}_{s=0}}{N}+\varepsilon^{2}$
(4.31)
$\displaystyle\phantom{=}+\|\mu\|_{L^{\infty}}(\varepsilon^{d-s}+(\varepsilon^{d}|\log\varepsilon|)\mathbf{1}_{s=0}+(\varepsilon^{2}|\log\varepsilon|)\mathbf{1}_{s=d-2})\Big{)}\Bigg{)},$

where the ultimate inequality follows from application of 2.3. For the sum
over far pairs, we first note that by symmetry under swapping
$i\leftrightarrow j$,

(4.32) $\begin{split}&\sum_{{1\leq i,j\leq
N}\atop{|x_{i}-x_{j}|\geq\varepsilon}}\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{K}}_{v}(X,Y)d(\delta_{X_{i}}+\delta_{X_{i}}^{(\eta)})(X)d(\delta_{X_{j}}-\delta_{X_{j}}^{(\eta)})(Y)\\\
&=\sum_{{1\leq i,j\leq
N}\atop{|x_{i}-x_{j}|\geq\varepsilon}}\int_{({\mathbb{R}}^{d+m})^{2}}\left\lparen{\mathsf{K}}_{v}(X_{i},X_{j})-{\mathsf{K}}_{v}(X,Y)\right\rparen
d\delta_{X_{i}}^{(\eta)}(X)d\delta_{X_{j}}^{(\eta)}(Y).\end{split}$

Since $\varepsilon>2\eta$, using the mean-value inequality and (4.7), we find

(4.33) $\frac{1}{N^{2}}\sum_{{1\leq i\neq j\leq
N}\atop{|x_{i}-x_{j}|\geq\varepsilon}}\left|\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{K}}_{v}(X,Y)d(\delta_{X_{i}}+\delta_{X_{i}}^{(\eta)})(X)d(\delta_{X_{j}}-\delta_{X_{j}}^{(\eta)})(Y)\right|\leq\frac{C\eta\|\nabla\tilde{v}\|_{L^{\infty}}}{\varepsilon^{s+1}}.$

Putting together the estimates (4.29), (4.31), and (4.33), we obtain

(4.34) $|\mathrm{Term}_{3}|\leq
C\|\nabla\tilde{v}\|_{L^{\infty}}\Bigg{(}{F}_{N}(\underline{x}_{N},\mu)+C\Big{(}\frac{\eta^{-s}+|\log\eta|\mathbf{1}_{s=0}}{N}+\frac{\varepsilon^{-s}+|\log\varepsilon|\mathbf{1}_{s=0}}{N}+\varepsilon^{2}\\\
+\|\mu\|_{L^{\infty}}(\varepsilon^{d-s}+(\varepsilon^{d}|\log\varepsilon|)\mathbf{1}_{s=0}+(\varepsilon^{2}|\log\varepsilon|)\mathbf{1}_{s=d-2})+\eta\varepsilon^{-s-1}\Big{)}\Bigg{)}.$

Conclusion. Combining the estimates (4.14), (4.26), (4.34) obtained above, we
have shown that

(4.35)
$\left|\int_{({\mathbb{R}}^{d})^{2}\setminus\triangle}\mathsf{k}_{v}(x,y)d\left\lparen\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu\right\rparen^{\otimes
2}(x,y)\right|\\\ \leq
C\left\lparen\|\nabla\tilde{v}\|_{L^{\infty}}+\||\nabla|^{\frac{d+m-s}{2}}\tilde{v}\|_{L^{\frac{2(d+m)}{d+m-2-s}}}\mathbf{1}_{s<d+m-2}\right\rparen\Bigg{(}{F}_{N}(\underline{x}_{N},\mu)+C\Big{(}\frac{\eta^{-s}+|\log\eta|\mathbf{1}_{s=0}}{N}+\eta^{2}\\\
+\|\mu\|_{L^{\infty}}(\eta^{d-s}+(\eta^{d}|\log\eta|)\mathbf{1}_{s=0}+(\eta^{2}|\log\eta|)\mathbf{1}_{s=d-2})\Big{)}\Bigg{)}\\\
+C\|\nabla\tilde{v}\|_{L^{\infty}}\Bigg{(}(\eta\||\nabla|^{s+1-d}\mu\|_{L^{\infty}})\mathbf{1}_{s<d-1}+(\eta\||\nabla|^{s-d}\nabla\mu\|_{L^{\infty}}+\eta^{d-s}\|\mu\|_{L^{\infty}})\mathbf{1}_{s\geq
d-1}\\\
+\frac{\eta^{-s}+|\log\eta|\mathbf{1}_{s=0}}{N}+\frac{\varepsilon^{-s}+|\log\varepsilon|\mathbf{1}_{s=0}}{N}+\varepsilon^{2}\\\
+\|\mu\|_{L^{\infty}}(\varepsilon^{d-s}+(\varepsilon^{d}|\log\varepsilon|)\mathbf{1}_{s=0}+(\varepsilon^{2}|\log\varepsilon|)\mathbf{1}_{s=d-2})+\eta\varepsilon^{-s-1}\Bigg{)}.$

We may now optimize over $\eta$ and $\varepsilon$ by taking

(4.36) $\begin{cases}\eta=\varepsilon^{s+2}\ \text{and}\
\varepsilon=N^{-1},&{s=0}\\\ \eta=\varepsilon^{s+3}\ \text{and}\
\varepsilon=N^{-\frac{1}{(s+2)(s+1)}},&{0<s\leq d-2}\\\
\eta=\varepsilon^{d+1}\ \text{and}\
\varepsilon=N^{-\frac{1}{d(s+1)}},&{d-2<s<d}.\end{cases}$

Using (4.4) to control the norms of $\tilde{v}$ in terms of norms of $v$, we
then find

(4.37)
$\left|\int_{({\mathbb{R}}^{d})^{2}\setminus\triangle}\mathsf{k}_{v}(x,y)d\left\lparen\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu\right\rparen^{\otimes
2}(x,y)\right|\leq C\Bigg{(}\|\nabla
v\|_{L^{\infty}}+\||\nabla|^{\frac{d-s}{2}}v\|_{L^{\frac{2d}{d-2-s}}}\mathbf{1}_{s<d-2}+\\\
+\left\lparen\|v\|_{C^{1,\alpha}}+\|v\|_{W^{\frac{d(d+m-s)+2m}{2(d+m)},\frac{2(d+m)}{d+m-2-s}}}\right\rparen\mathbf{1}_{s>d-2}\Bigg{)}\Bigg{(}{F}_{N}(\underline{x}_{N},\mu)\\\
+C(1+\|\mu\|_{L^{\infty}})\Big{(}N^{-\frac{\min\\{d-s,2\\}}{\min\\{d,s+2\\}(s+1)}}+(N^{-1}\log
N)\mathbf{1}_{s=0}+(N^{-\frac{2}{d(d-1)}}\log N)\mathbf{1}_{s=d-2}\Big{)}\\\
+CN^{-\frac{\min\\{d+1,s+3\\}}{\min\\{d,s+2\\}(s+1)}}\Big{(}\||\nabla|^{s+1-d}\mu\|_{L^{\infty}}\mathbf{1}_{s<d-1}+\||\nabla|^{s-d}\nabla\mu\|_{L^{\infty}}\mathbf{1}_{s\geq
d-1}\Big{)}\Bigg{)}.$

Thus, the proof of the proposition is complete. ∎

##  5\. Gronwall argument

We now have all the ingredients for our Gronwall argument on the modulated
energy. Set $u\coloneqq{\mathbb{M}}\nabla{\mathsf{g}}\ast\mu$. Applying the
bound of 4.1 pointwise in time with $v=u^{t}$ to the right-hand side of
inequality (1.7) and then integrating with respect to time, we find that for
$0\leq t\leq T$,

(5.1)
$|{F}_{N}(\underline{x}_{N}^{t},\mu^{t})|\leq|{F}_{N}(\underline{x}_{N}^{0},\mu^{0})|+C\int_{0}^{t}\Bigg{(}\|\nabla
u^{\tau}\|_{L^{\infty}}+\||\nabla|^{\frac{d-s}{2}}u^{\tau}\|_{L^{\frac{2d}{d-2-s}}}\mathbf{1}_{s<d-2}\\\
+\left\lparen\|u^{\tau}\|_{C^{1,\alpha}}+\|u^{\tau}\|_{W^{\frac{d(d+m-s)+2m}{2(d+m)},\frac{2(d+m)}{d+m-2-s}}}\right\rparen\mathbf{1}_{s>d-2}\Bigg{)}\Bigg{(}{F}_{N}(\underline{x}_{N}^{\tau},\mu^{\tau})\\\
+C(1+\|\mu^{\tau}\|_{L^{\infty}})\Big{(}N^{-\frac{\min\\{d-s,2\\}}{\min\\{d,s+2\\}(s+1)}}+(N^{-1}\log
N)\mathbf{1}_{s=0}+(N^{-\frac{2}{d(d-1)}}\log N)\mathbf{1}_{s=d-2}\Big{)}\\\
+CN^{-\frac{\min\\{d+1,s+3\\}}{\min\\{d,s+2\\}(s+1)}}\Big{(}\||\nabla|^{s+1-d}\mu^{\tau}\|_{L^{\infty}}\mathbf{1}_{s<d-1}+\||\nabla|^{s-d}\nabla\mu^{\tau}\|_{L^{\infty}}\mathbf{1}_{s\geq
d-1}\Big{)}\Bigg{)}d\tau.$

If $0\leq s<d-2$, then by Sobolev embedding, assumption (1.16), and Hölder’s
inequality,

(5.2) $\||\nabla|^{\frac{d-s}{2}}u^{\tau}\|_{L^{\frac{2d}{d-2-s}}}+\|\nabla
u^{\tau}\|_{L^{\infty}}\lesssim 1+\|\mu^{\tau}\|_{L^{\infty}}.$

If $0\leq s<d-1$, then it also follows from Hölder’s inequality that

(5.3) $\||\nabla|^{s+1-d}\mu^{\tau}\|_{L^{\infty}}\lesssim
1+\|\mu^{\tau}\|_{L^{\infty}}.$

Applying the Gronwall-Bellman lemma, we conclude that for all $0\leq t\leq T$,

(5.4) $|{F}_{N}(\underline{x}_{N}^{t},\mu^{t})|\leq
A_{N}(t)\exp\Bigg(C\int_{0}^{t}\Big{(}\|\nabla
u^{\tau}\|_{L^{\infty}}+\||\nabla|^{\frac{d-s}{2}}u^{\tau}\|_{L^{\frac{2d}{d-2-s}}}\mathbf{1}_{s<d-2}{\\\
}+\Big{(}\|u^{\tau}\|_{C^{1,\alpha}}+\|u^{\tau}\|_{W^{\frac{d(d+m-s)+2m}{2(d+m)},\frac{2(d+m)}{d+m-2-s}}}\Big{)}\mathbf{1}_{s>d-2}\Big{)}d\tau\Bigg{missing}),$

where the time-dependent prefactor $A_{N}$ is defined by

(5.5) $A_{N}(t)\coloneqq
C\int_{0}^{t}\Bigg{(}(1+\|\mu^{\tau}\|_{L^{\infty}})\Big{(}N^{-\frac{\min\\{d-s,2\\}}{\min\\{d,s+2\\}(s+1)}}+(N^{-1}\log
N)\mathbf{1}_{s=0}+(N^{-\frac{2}{d(d-1)}}\log N)\mathbf{1}_{s=d-2}\Big{)}\\\
+CN^{-\frac{\min\\{d+1,s+3\\}}{\min\\{d,s+2\\}(s+1)}}\Big{(}\||\nabla|^{s+1-d}\mu^{\tau}\|_{L^{\infty}}\mathbf{1}_{s<d-1}+\||\nabla|^{s-d}\nabla\mu^{\tau}\|_{L^{\infty}}\mathbf{1}_{s\geq
d-1}\Big{)}\Bigg{)}d\tau+|{F}_{N}(\underline{x}_{N}^{0},\mu^{0})|.$

Thus, the proof of our bound for the evolution of the modulated energy and
therefore of 1.1 is complete.

##  6\. Multiplicative Noise

###  6.1. Overview

In this last section, we show how to extend our method of proof to treat the
mean-field limit of first-order systems, such as (1.1), with multiplicative
noise added to the dynamics:

(6.1) $\begin{cases}\dot{x}_{i}=\displaystyle\frac{1}{N}\sum_{{1\leq j\leq
N}:j\neq
i}{\mathbb{M}}\nabla{\mathsf{g}}(x_{i}-x_{j})+\sum_{k=1}^{\infty}\sigma_{k}(x_{i})\circ\dot{W}_{k}\\\
x_{i}|_{t=0}=x_{i}^{0},\end{cases}\qquad i\in\\{1,\ldots,N\\}.$

Here, the $\sigma_{k}$ are $C^{\infty}$ vector fields on ${\mathbb{R}}^{d}$,
the $W_{k}$ are independent real Brownian motions, and $\circ$ denotes that
product should be interpreted in the Stratonovich sense.555When the $\circ$ is
not present, the product should be interpreted in the Itô sense throughout
this paper. The correspond mean-field equation is no longer deterministic, but
instead gains a stochastic term:

(6.2)
$\begin{cases}\partial_{t}\mu=\displaystyle-\operatorname{div}(({\mathbb{M}}\nabla{\mathsf{g}}*\mu)\mu)-\sum_{k=1}^{\infty}\operatorname{div}(\sigma_{k}\mu)\circ\dot{W}_{k}&\\\
\mu(0)=\mu^{0}&\end{cases}\qquad(t,x)\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}.$

The model example of a system of the form (6.1) is the stochastic point vortex
model of Flandoli, Gubinelli, and Priola [FGP11], for which ${\mathsf{g}}$ is
the 2D Coulomb potential. The limiting SPDE is now the 2D incompressible Euler
vorticity equation with an additional stochastic transport term, the well-
posedness of which has been studied by Brzeźniak, Flandoli, and Maurelli
[BFM16]. We emphasize that the noise in (6.1) is very different than the so-
called additive noise model,

(6.3) $\dot{x}_{i}=\frac{1}{N}\sum_{{1\leq j\leq N}:j\neq
i}{\mathbb{M}}\nabla{\mathsf{g}}(x_{i}-x_{j})+\sqrt{2\sigma}\dot{\underline{W}}_{i},$

where now $\sigma>0$ is a constant and the $\underline{W}_{i}$ are independent
Brownian motions in ${\mathbb{R}}^{d}$. Indeed, in equation (6.3), the noise
is independent for each particle, whereas in equation (6.1), the spatially
dependent noise acting on each particle is the same. This difference clearly
manifests itself at the limiting level. As has been rigorously shown by
numerous authors [Osa87b, Osa87a, FHM14, LY16, JW18, LLY19, BJW19b, BJW19a,
BJW20], the empirical measure of the system (6.3) converges in law to the
solution of the deterministic diffusive/viscous equation

(6.4)
${\partial}_{t}\mu=-\operatorname{div}(({\mathbb{M}}\nabla{\mathsf{g}}\ast\mu)\mu)+\sigma\Delta\mu.$

As mentioned in the introduction, the second author [Ros20c] was the first to
study the convergence in law of the empirical measure for (6.1) to the
solution (if one exists) of the SPDE (6.2).666An earlier work of Coghi and
Maurelli [CM20] considered this problem but with an $N$-dependent
asymptotically vanishing truncation of the potential in (6.1) to distances
much larger than the typical interparticle distance $N^{-1/d}$. By developing
a stochastic extension of the modulated-energy approach of [Ser20b] and
introducing the commutator perspective adopted in this paper, he proved a
quantitative estimate for the convergence in the 2D Coulomb case corresponding
to the aforementioned stochastic point vortex model. The proof can be extended
to dimensions $d\geq 3$, but, importantly, it very strongly depends on the
Coulomb nature of the interaction. At the time of the present article, this is
the only such result of which we are aware.

The idea of [Ros20c] is to consider the modulated energy
${F}_{N}(\underline{x}_{N}^{t},\mu^{t})$ as before, noting that it is now a
real-valued stochastic process which is almost surely finite. A formal
application of Itô’s lemma (in Stratonovich form) to
${F}_{N}(\underline{x}_{N}^{t},\mu^{t})$ yields the stochastic differential
inequality

(6.5)
$\frac{d}{dt}{F}_{N}(\underline{x}_{N}^{t},\mu^{t})\leq\int_{({\mathbb{R}}^{d})^{2}\setminus\triangle}\left\lparen
u^{t}(x)-u^{t}(y)\right\rparen\cdot\nabla{\mathsf{g}}(x-y)d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}^{t}}-\mu^{t}\right)^{\otimes
2}(x,y)\\\
+\sum_{k=1}^{\infty}\int_{({\mathbb{R}}^{d})^{2}\setminus\triangle}\nabla{\mathsf{g}}(x-y)\cdot\left\lparen\sigma_{k}(x)-\sigma_{k}(y)\right\rparen
d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}^{t}}-\mu^{t}\right)^{\otimes
2}(x,y)\circ\dot{W}_{k},$

where again $u^{t}\coloneqq{\mathbb{M}}\nabla{\mathsf{g}}\ast\mu^{t}$.
Converting from Stratonovich to Itô, the second line becomes

(6.6)
$\sum_{k=1}^{\infty}\int_{({\mathbb{R}}^{d})^{2}\setminus\triangle}\nabla{\mathsf{g}}(x-y)\cdot\left\lparen\sigma_{k}(x)-\sigma_{k}(y)\right\rparen
d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}^{t}}-\mu^{t}\right)^{\otimes
2}(x,y)\dot{W}_{k}\\\
+\frac{1}{2}\sum_{k=1}^{\infty}\int_{({\mathbb{R}}^{d})^{2}\setminus\triangle}\nabla{\mathsf{g}}(x-y)\cdot\lparen\sigma_{k}\cdot\nabla\sigma_{k}(x)-\sigma_{k}\cdot\nabla\sigma_{k}(y)\rparen
d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}^{t}}-\mu^{t}\right)^{\otimes
2}(x,y)\\\
+\frac{1}{2}\sum_{k=1}^{\infty}\int_{({\mathbb{R}}^{d})^{2}\setminus\triangle}\nabla^{\otimes
2}{\mathsf{g}}(x-y):\left\lparen\sigma_{k}(x)-\sigma_{k}(y)\right\rparen^{\otimes
2}d\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}^{t}}-\mu^{t}\right)^{\otimes
2}(x,y).$

The first term in (6.6), which is now an Itô integral, should have zero
expectation, therefore we can ignore it. The second term has the same
commutator structure as in the functional inequality (1.8) with
$v=\sum_{k=1}^{\infty}\sigma_{k}\cdot\nabla\sigma_{k}$. Consequently, we can
treat this expression as a renormalized commutator using 4.1. The third term
is more complicated. It appears similar to the first two terms, except now
there are two derivatives on ${\mathsf{g}}$ and two symmetrized expressions
$\sigma_{k}(x)-\sigma_{k}(y)$ which, in principle, should cancel out the
singularities induced by the derivatives. However, 4.1 does not cover such
expressions. Accordingly, the goal of this section is to prove its second-
order analogue, which is 6.2 stated below. Again, the idea will be to prove a
second-order version of the commutator estimate of 3.1 using integration by
parts and then to obtain a renormalization of this estimate using the smearing
procedure.

The preceding computation to arrive at (6.5) and (6.6) is completely formal.
We cannot directly invoke Itô’s lemma because the potential ${\mathsf{g}}$ is
singular. Also, we have not specified whether there is a solution to the
system (6.1) or in what sense the equation (6.2) holds. The $N$-body problem
is relatively straightforward to make sense of. Using a truncation of the
potential near the origin and a stopping time argument in the spirit of
[FGP11, Section 3], one can show there is a unique strong solution. In
particular, with probability one, the particles never collide. The SPDE (6.2)
is more difficult to make sense of. In fact, we are unaware of any well-
posedness results for this equation aside from the 2D Euler case in [BFM16], a
gap in the literature which should be filled. Accordingly, we will not attempt
to rigorously justify equation (6.5) in this article. Instead, we will only
show in this section how to use our methods to estimate the last term in (6.6)
_pathwise_ in the noise.

The assumptions (1.11) – (1.19) in the case $0\leq s\leq d-2$ and (1.20) –
(1.28) in the case $d-2<s<d$ carry over. But we also need to supplement (1.18)
and (1.28), respectively, with the assumptions that the
$({\mathbb{R}}^{d})^{\otimes(4+2k)}$-valued kernel

(6.7) $\mathsf{k}(x-y)\coloneqq(x-y)^{\otimes
2}\otimes\nabla^{\otimes(2+2k)}{\mathsf{g}}(x-y)$

and the $({\mathbb{R}}^{d})^{\otimes 6}$-valued kernel

(6.8) ${\mathsf{K}}(X-Y)\coloneqq(X-Y)^{\otimes 2}\otimes\nabla^{\otimes
4}{\mathsf{G}}(X-Y)$

are associated to Calderón-Zygmund operators on ${\mathbb{R}}^{d}$ and
${\mathbb{R}}^{d+m}$.

###  6.2. Second-order commutator estimate

Following the strategy of Section 3, we prove a second-order version of the
commutator estimate 3.1. With more work, one can extend the method of proof to
arbitrary $k$-th order, but having no need for such generality, we will not do
so in this work.

######  Proposition 6.1.

Let $0\leq s<d$. Let ${\mathsf{g}}\in
C^{\infty}({\mathbb{R}}^{d}\setminus\\{0\\})$ such that
${\mathsf{g}}(x)={\mathsf{g}}(-x)$ in $B(0,r_{0})$, and

(6.9) $\forall k\geq 1,\qquad|\nabla^{\otimes
k}{\mathsf{g}}(x)|\lesssim|x|^{-k-s}\qquad\forall x\neq 0.$

If $s\leq d-2$, then also assume that

(6.10) $|\hat{{\mathsf{g}}}(\xi)|\lesssim|\xi|^{s-d}\qquad\forall\xi\neq 0.$

If $s=d-2k$, for some $k\in{\mathbb{N}}$, also assume that the
$({\mathbb{R}}^{d})^{\otimes(4+2k)}$-valued kernel

(6.11) $\mathsf{k}(x,y)\coloneqq(x-y)^{\otimes
2}\otimes\nabla^{\otimes(2+2k)}{\mathsf{g}}(x-y)$

is associated to a Calderón-Zygmund operator. Let $v$ be a Lipschitz
continuous vector field on ${\mathbb{R}}^{d}$. There exists a constant $C>0$,
depending only on $s,d$, and the potential ${\mathsf{g}}$ through (3.1) –
(3.3), such that for any $f,g\in{\mathcal{S}}({\mathbb{R}}^{d})$,777If $s=0$,
then it is again implicit that the Fourier transforms of $f,g$ vanish
sufficiently rapidly at the origin so that the $\dot{H}^{-d/2}$ norm
converges. we have

(6.12) $\left|\int_{({\mathbb{R}}^{d})^{2}}(v(x)-v(y))^{\otimes
2}:\nabla^{\otimes 2}{\mathsf{g}}(x-y)f(x)g(y)dxdy\right|\\\ \leq
C\left\lparen\|\nabla
v\|_{L^{\infty}}+\||\nabla|^{\frac{d-s}{2}}v\|_{L^{\frac{2d}{d-2-s}}}\mathbf{1}_{s<d-2}\right\rparen^{2}\|f\|_{\dot{H}^{\frac{s-d}{2}}}\|g\|_{\dot{H}^{\frac{s-d}{2}}}.$

Consequently, the integral in the left-hand side of (6.12) extends to a
bounded bilinear form $B_{v}(\cdot,\cdot)$ on
$\dot{H}^{\frac{s-d}{2}}({\mathbb{R}}^{d})$ satisfying the bound (6.12).

###### Proof.

We follow the outline of the proof of 3.1. Since the arguments are very
similar, we only sketch the proof, focusing on what is different. Again, we
will not explicitly track the dependence of the implicit constants.

The super-Coulombic case. Recycling the kernel notation $\mathsf{k}_{v}$,
define

(6.13) $\mathsf{k}_{v}(x,y)\coloneqq(v(x)-v(y))^{\otimes 2}:\nabla^{\otimes
2}{\mathsf{g}}(x-y)\qquad\forall x\neq y.$

Note that by assumption (1.24) and the mean-value theorem, we have the kernel
estimate

(6.14) $|\mathsf{k}_{v}(x,y)|\leq C\min\left\lparen\frac{\|\nabla
v\|_{L^{\infty}}^{2}}{|x-y|^{s}},\frac{\|v\|_{L^{\infty}}^{2}}{|x-y|^{s+2}}\right\rparen.$

By approximation, we assume that $v\in C^{\infty}$;
$f,g\in{\mathcal{S}}({\mathbb{R}}^{d})$ with Fourier support away from the
origin; and ${\mathsf{g}}$ has compact support. Writing

(6.15)
$f=\operatorname{div}\nabla\Delta^{-1}f\eqqcolon\operatorname{div}f_{1}$

and similarly for $g$, we find

(6.16)
$\int_{({\mathbb{R}}^{d})^{2}}\mathsf{k}_{v}(x,y)f(x)g(y)dxdy=\lim_{\varepsilon\rightarrow
0^{+}}\int_{|x-y|\geq\varepsilon}\mathsf{k}_{v}(x,y)\operatorname{div}f_{1}(x)\operatorname{div}g_{1}(y)dxdy.$

Integrating by parts and using assumption (6.9) to estimate the boundary terms
as in the proof of 3.1, we find that

(6.17)
$\int_{|x-y|\geq\varepsilon}\mathsf{k}_{v}(x,y)f(x)g(y)dxdy=O(\|\nabla^{\otimes
2}v\|_{L^{\infty}}^{2}\|fg_{1}\|_{L^{1}}\varepsilon^{d-s})\\\
+O(\|v\|_{W^{2,\infty}}^{2}\|\nabla
f_{1}g_{1}\|_{L^{1}}\varepsilon^{d-s})+O(\|v\|_{W^{1,\infty}}^{2}\|\nabla^{\otimes
2}f_{1}\|_{L^{\infty}}\|g_{1}\|_{L^{1}}\varepsilon^{d-s})\\\
-\frac{1}{2}\int_{|x-y|\geq\varepsilon}\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y):\left\lparen(f_{1}(x)-f_{1}(y))\otimes(g_{1}(x)-g_{1}(y))\right\rparen
dxdy.$

The first three terms of the right-hand side vanish as $\varepsilon\rightarrow
0$ since $s<d$. For the last term, direct computation reveals

(6.18) $\begin{split}\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y)&=-\nabla^{\otimes
4}{\mathsf{g}}(x-y):(v(x)-v(y))^{\otimes 2}\\\ &\phantom{=}-2\nabla^{\otimes
3}{\mathsf{g}}(x-y):\left\lparen(\nabla v(x)+\nabla
v(y))\otimes(v(x)-v(y))\right\rparen\\\ &\phantom{=}-2\nabla^{\otimes
2}{\mathsf{g}}(x-y):(\nabla v(x)\otimes\nabla v(y)).\end{split}$

So using the triangle inequality and mean-value theorem, we find that

(6.19)
$\left|\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y)\right|\lesssim\frac{\|\nabla
v\|_{L^{\infty}}^{2}}{|x-y|^{s+2}},$

which implies by Cauchy-Schwarz that

$\displaystyle\int_{|x-y|\geq\varepsilon}|\nabla_{x}\nabla_{y}\mathsf{k}_{v}(x,y)||f_{1}(x)-f_{1}(y)||g_{1}(x)-g_{1}(y)|dxdy$
$\displaystyle\lesssim\|\nabla
v\|_{L^{\infty}}^{2}\left\lparen\int_{({\mathbb{R}}^{d})^{2}}\frac{|f_{1}(x)-f_{1}(y)|^{2}}{|x-y|^{s+2}}dxdy\right\rparen^{1/2}\left\lparen\int_{({\mathbb{R}}^{d})^{2}}\frac{|g_{1}(x)-g_{1}(y)|^{2}}{|x-y|^{s+2}}dxdy\right\rparen^{1/2}$
(6.20) $\displaystyle\lesssim\|\nabla
v\|_{L^{\infty}}^{2}\|f\|_{\dot{H}^{\frac{s-d}{2}}}\|g\|_{\dot{H}^{\frac{s-d}{2}}}.$

In conclusion,

(6.21) $\begin{split}\lim_{\varepsilon\rightarrow
0}\left|\int_{|x-y|\geq\varepsilon}\mathsf{k}_{v}(x,y)f(x)g(y)dxdy\right|\lesssim\|\nabla
v\|_{L^{\infty}}^{2}\|f\|_{\dot{H}^{\frac{s-d}{2}}}\|g\|_{\dot{H}^{\frac{s-d}{2}}}.\end{split}$

The Coulombic case. To modify the proof of 3.1 in the Coulombic case $s=d-2$,
we first note that if $|x-y|=\varepsilon$, then writing $x=y+z$,

$\displaystyle\frac{(x-y)^{i_{1}}}{|x-y|}{\partial}_{y_{i_{2}}}\mathsf{k}_{v}(x,y)$
$\displaystyle=\frac{z^{i_{1}}}{|z|}{\partial}_{y_{i_{2}}}\mathsf{k}_{v}(y+z,y)$
$\displaystyle=-\frac{z^{i_{1}}}{|z|}\left\lparen(z^{i_{3}}{\partial}_{i_{3}}v)^{\otimes
2}:\nabla^{\otimes
2}{\partial}_{i_{2}}{\mathsf{g}}(z)+2({\partial}_{i_{2}}v(y)\otimes(z^{i_{3}}{\partial}_{i_{3}}v(y))):\nabla^{\otimes
2}{\mathsf{g}}(z)\right\rparen$ (6.22)
$\displaystyle\phantom{=}+O(\|v\|_{W^{2,\infty}}^{2}\varepsilon^{d}),$

as $\varepsilon\rightarrow 0$, where to obtain the error bound we use
assumption (6.9). So by Cauchy-Schwarz and assumption (6.9),

$\displaystyle\left|\int_{{\mathbb{R}}^{d}}(f_{1}(y)\otimes
g_{1}(y)):\int_{{\partial}B(0,\varepsilon)}\frac{z}{|z|}\otimes\nabla_{y}\mathsf{k}_{v}(y+z,y)d{\mathcal{H}}^{d-1}(z)dy\right|$
$\displaystyle\lesssim\|\nabla
v\|_{L^{\infty}}^{2}\|f_{1}\|_{L^{2}}\|g_{1}\|_{L^{2}}$ (6.23)
$\displaystyle\lesssim\|\nabla
v\|_{L^{\infty}}^{2}\|f\|_{\dot{H}^{-1}}\|g_{1}\|_{\dot{H}^{-1}}.$

This takes care of the first boundary term in the identity (3.28). The second
boundary term vanishes as $\varepsilon\rightarrow 0$ by modifying (3.17),
(3.18) to account for the new definition of $\mathsf{k}_{v}$.

For the non-boundary term in (3.28), we see, using the identity (6.18), that
it is equal to

(6.24) $\begin{split}&-\int_{|x-y|\geq\varepsilon}\nabla^{\otimes
2}{\partial}_{i_{1}i_{2}}{\mathsf{g}}(x-y):(v(x)-v(y))^{\otimes
2}f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy\\\
&-2\int_{|x-y|\geq\varepsilon}\nabla^{\otimes
2}{\partial}_{i_{2}}{\mathsf{g}}(x-y):{\partial}_{i_{1}}v(x)\otimes(v(x)-v(y))f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy\\\
&-2\int_{|x-y|\geq\varepsilon}\nabla^{\otimes
2}{\partial}_{i_{1}}{\mathsf{g}}(x-y):(v(x)-v(y))\otimes{\partial}_{i_{2}}v(y)f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy\\\
&-2\int_{|x-y|\geq\varepsilon}\nabla^{\otimes
2}{\mathsf{g}}(x-y):({\partial}_{i_{1}}v(x)\otimes{\partial}_{i_{2}}v(y))f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy.\end{split}$

By Taylor’s theorem,

$\displaystyle{\partial}_{i_{1}}v(x)\otimes(v(x)-v(y)):\nabla^{\otimes
2}{\partial}_{i_{2}}{\mathsf{g}}(x-y)$
$\displaystyle={\partial}_{i_{1}}v(x)\otimes\left\lparen\int_{0}^{1}\nabla
v(y+t(x-y))\cdot(x-y)dt\right\rparen:\nabla^{\otimes
2}{\partial}_{i_{2}}{\mathsf{g}}(x-y)$ (6.25)
$\displaystyle\eqqcolon{\partial}_{i_{1}}v(x)\otimes\left\lparen\int_{0}^{1}{\partial}_{i_{3}}v(y+t(x-y))dt\right\rparen:\mathsf{k}_{i_{2}}^{i_{3}}(x-y),$

where for fixed $i_{2},i_{3}$, the $({\mathbb{R}}^{d})^{\otimes 2}$-valued
kernel $\mathsf{k}_{i_{2}}^{i_{3}}(x-y)$ is defined by

(6.26) $\mathsf{k}_{i_{2}}^{i_{3}}(x-y)\coloneqq(x-y)^{i_{3}}\nabla^{\otimes
2}{\partial}_{i_{2}}{\mathsf{g}}(x-y),\qquad\forall x\neq y.$

By assumption (6.11), $\mathsf{k}_{i_{2}}^{i_{3}}$ is associated to a
Calderón-Zygmund operator. Hence, the expression

(6.27)
$\left\lparen\int_{0}^{1}{\partial}_{i_{3}}v(y+t(x-y))dt\right\rparen\cdot\mathsf{k}_{i_{2}}^{i_{3}}(x-y)$

is the kernel of an $({\mathbb{R}}^{d})^{\otimes 2}$-valued Calderón
$d$-commutator, which by the Christ-Journé theorem [CJ87] is bounded on
$L^{2}({\mathbb{R}}^{d})$. So by Cauchy-Schwarz,

$\displaystyle\left|\int_{|x-y|\geq\varepsilon}\nabla^{\otimes
2}{\partial}_{i_{2}}{\mathsf{g}}(x-y):{\partial}_{i_{1}}v(x)\otimes(v(x)-v(y))f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy\right|$
$\displaystyle\lesssim\|\nabla v\|_{L^{\infty}}\|{\partial}_{i_{1}}v\otimes
f_{1}^{i_{1}}\|_{L^{2}}\|g_{1}\|_{L^{2}}$ $\displaystyle\lesssim\|\nabla
v\|_{L^{\infty}}^{2}\|f_{1}\|_{L^{2}}\|g_{1}\|_{L^{2}}$ (6.28)
$\displaystyle\lesssim\|\nabla
v\|_{L^{\infty}}^{2}\|f\|_{\dot{H}^{-1}}\|g\|_{\dot{H}^{-1}}.$

By symmetry under swapping the $i_{1}\leftrightarrow i_{2}$, this takes care
of the second and third lines of (6.24). For the fourth line, similarly using
Taylor’s theorem yields

$\displaystyle\left\lparen v(x)-v(y)\right\rparen^{\otimes 2}:\nabla^{\otimes
2}{\partial}_{i_{1}i_{2}}{\mathsf{g}}(x-y)$
$\displaystyle=\left\lparen\int_{0}^{1}\nabla
v(y+t(x-y))\cdot(x-y)dt\right\rparen^{\otimes 2}:\nabla^{\otimes
2}{\partial}_{i_{1}i_{2}}{\mathsf{g}}(x-y)$ (6.29)
$\displaystyle\eqqcolon\left\lparen\int_{0}^{1}{\partial}_{i_{3}}v(y+t(x-y))dt\right\rparen\otimes\left\lparen\int_{0}^{1}{\partial}_{i_{4}}v(y+t(x-y))dt\right\rparen:\mathsf{k}_{i_{1}i_{2}}^{i_{3}i_{4}}(x-y),$

where the $({\mathbb{R}}^{d})^{\otimes 2}$-valued kernel
$\mathsf{k}_{i_{1}i_{2}}^{i_{3}i_{4}}$ is defined by

(6.30) $\mathsf{k}_{i_{1}i_{2}}^{i_{3}i_{4}}(x-y)\coloneqq(x-y)^{\otimes
2,i_{3}i_{4}}\nabla^{\otimes
2}{\partial}_{i_{1}i_{2}}{\mathsf{g}}(x-y)\qquad\forall x\neq y.$

By assumption (6.11), $\mathsf{k}_{i_{1}i_{2}}^{i_{3}i_{4}}$ is associated to
a Calderón-Zygmund operator. Hence, the expression (6.29) is the kernel of a
Calderón $d$-commutator, and we can apply the Christ-Journé theorem [CJ87]
like before to conclude that the modulus of the fourth line of (6.24) is
bounded by

(6.31) $\|\nabla
v\|_{L^{\infty}}^{2}\|f\|_{\dot{H}^{-1}}\|g\|_{\dot{H}^{-1}}.$

The sub-Coulombic case. Finally, in the sub-Coulombic case $0\leq s<d-2$, the
induction argument proceeds exactly as in the proof of 3.1. Instead of
identity (3.38), we now have

(6.32)
$\begin{split}&\int_{({\mathbb{R}}^{d})^{2}}\mathsf{k}_{v}(x,y)f(x)g(y)dxdy\\\
&=\lim_{\varepsilon\rightarrow
0}\Bigg{(}-\int_{|x-y|\geq\varepsilon}\nabla^{\otimes
2}{\partial}_{i_{1}i_{2}}{\mathsf{g}}(x-y):\left\lparen
v(x)-v(y)\right\rparen^{\otimes 2}f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy\\\
&\phantom{=}-2\int_{|x-y|\geq\varepsilon}\nabla^{\otimes
2}{\partial}_{i_{2}}{\mathsf{g}}(x-y):{\partial}_{i_{1}}v(x)\otimes\left\lparen
v(x)-v(y)\right\rparen f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy\\\
&\phantom{=}-2\int_{|x-y|\geq\varepsilon}\nabla^{\otimes
2}{\partial}_{i_{1}}{\mathsf{g}}(x-y):\left\lparen
v(x)-v(y)\right\rparen\otimes{\partial}_{i_{2}}v(y)f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy\\\
&\phantom{=}-\int_{|x-y|\geq\varepsilon}\nabla^{\otimes
2}{\mathsf{g}}(x-y):\left\lparen{\partial}_{i_{1}}v(x)\otimes{\partial}_{i_{2}}v(y)+{\partial}_{i_{2}}v(y)\otimes{\partial}_{i_{1}}v(x)\right\rparen
f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy\Bigg{)}.\end{split}$

Fix $i_{1},i_{2}$. For the first term, we can repeat repeat the proof above
for the super-Coulombic ($d-4<s<d-2$) and Coulombic ($s=d-4$) cases with
${\mathsf{g}}$ replaced by ${\partial}_{i_{1}i_{2}}{\mathsf{g}}$.

The second and third terms are symmetric, so it suffices to consider the
second term. Writing

$\displaystyle\nabla^{\otimes
2}{\partial}_{i_{2}}{\mathsf{g}}(x-y):{\partial}_{i_{1}}v(x)\otimes\left\lparen
v(x)-v(y)\right\rparen={\partial}_{j_{2}}{\partial}_{j_{1}i_{2}}{\mathsf{g}}(x-y){\partial}_{i_{1}}v^{j_{1}}(x)\left\lparen
v(x)-v(y)\right\rparen^{j_{2}}$ (6.33)
$\displaystyle=\nabla{\partial}_{j_{1}i_{2}}{\mathsf{g}}(x-y)\cdot\left\lparen
v(x)-v(y)\right\rparen{\partial}_{i_{1}}v^{j_{1}}(x)$

and therefore

(6.34) $\begin{split}&\lim_{\varepsilon\rightarrow
0}\int_{|x-y|\geq\varepsilon}\nabla^{\otimes
2}{\partial}_{i_{2}}{\mathsf{g}}(x-y):{\partial}_{i_{1}}v(x)\otimes\left\lparen
v(x)-v(y)\right\rparen f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy\\\
&=\lim_{\varepsilon\rightarrow
0}\int_{|x-y|\geq\varepsilon}\nabla{\partial}_{j_{1}i_{2}}{\mathsf{g}}(x-y)\cdot\left\lparen
v(x)-v(y)\right\rparen{\partial}_{i_{1}}v^{j_{1}}(x)f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy,\end{split}$

we see that we can apply 3.1 to this expression with ${\mathsf{g}}$ replaced
by ${\partial}_{j_{1}i_{2}}{\mathsf{g}}$, $f$ replaced by
${\partial}_{i_{1}}v^{j_{1}}f_{1}^{i_{1}}$ and $g$ replaced by
$g_{1}^{i_{2}}$. Thus, its modulus is bounded by

(6.35) $\|\nabla
v\|_{L^{\infty}}\|{\partial}_{i_{1}}v^{j_{1}}f_{1}^{i_{1}}\|_{L^{2}}\|g_{1}^{i_{2}}\|_{L^{2}}\lesssim\|\nabla
v\|_{L^{\infty}}^{2}\|f\|_{\dot{H}^{-1}}\|g\|_{\dot{H}^{-1}}.$

Finally, for the fourth term, $\nabla^{\otimes 2}{\mathsf{g}}$ is locally
integrable, so we can use dominated convergence to pass to the limit
$\varepsilon\rightarrow 0$. By symmetry under swapping $x\leftrightarrow y$,
it suffices to consider the expression

(6.36) $\int_{({\mathbb{R}}^{d})^{2}}\nabla^{\otimes
2}{\mathsf{g}}(x-y):\left\lparen{\partial}_{i_{1}}v(x)\otimes{\partial}_{i_{2}}v(y)\right\rparen
f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy.$

Writing $\mathrm{I}=|\nabla|^{\frac{d-s}{2}-1}|\nabla|^{\frac{s-d}{2}+1}$ and
using Cauchy-Schwarz, the modulus of the preceding expression is bounded by

(6.37) $\||\nabla|^{\frac{d-s}{2}-1}(\nabla^{\otimes
2}{\mathsf{g}}\ast({\partial}_{i_{2}}vg_{1}^{i_{2}}))\|_{L^{2}}\||\nabla|^{\frac{s-d}{2}+1}({\partial}_{i_{1}}vf_{1}^{i_{1}})\|_{L^{2}}$

By Plancherel’s theorem and assumption (6.10),

(6.38) $\||\nabla|^{\frac{d-s}{2}-1}(\nabla^{\otimes
2}{\mathsf{g}}\ast({\partial}_{i_{2}}vg_{1}^{i_{2}}))\|_{L^{2}}\lesssim\||\nabla|^{\frac{s-d}{2}+1}({\partial}_{i_{2}}vg_{1}^{i_{2}})\|_{L^{2}}.$

Using (3.43) and (3.45), we conclude that

(6.39) $\begin{split}&\left|\int_{({\mathbb{R}}^{d})^{2}}\nabla^{\otimes
2}{\mathsf{g}}(x-y):\left\lparen{\partial}_{i_{1}}v(x)\otimes{\partial}_{i_{2}}v(y)\right\rparen
f_{1}^{i_{1}}(x)g_{1}^{i_{2}}(y)dxdy\right|\\\ &\lesssim\left\lparen\|\nabla
v\|_{L^{\infty}}+\||\nabla|^{\frac{d-s}{2}}v\|_{L^{\frac{2d}{d-2-s}}}\right\rparen^{2}\|f\|_{\dot{H}^{\frac{s-d}{2}}}\|g\|_{\dot{H}^{\frac{s-d}{2}}}.\end{split}$

Just as before, the general case $d-2(k+1)\leq s<d-2k$ follows by induction on
$k$, and the existence of the form extension $B_{v}$ by density of
${\mathcal{S}}$ in $\dot{H}^{\frac{s-d}{2}}$. ∎

###  6.3. Renormalization of second-order commutator estimate

We now use the smearing procedure to obtain a renormalization of 6.1, which is
the second-order analogue of 4.1.

######  Proposition 6.2.

Let $d\geq 1$, $0\leq s<d$. Let $\underline{x}_{N}\in({\mathbb{R}}^{d})^{N}$
be pairwise distinct, and $\mu\in\mathcal{P}({\mathbb{R}}^{d})\cap
L^{\infty}({\mathbb{R}}^{d})$. If $s=0$, assume that
$\int_{{\mathbb{R}}^{d}}\log(1+|x|)d\mu(x)<\infty$ and if $s\geq d-1$, assume
that $\||\nabla|^{s-d}\nabla\mu\|_{L^{\infty}}<\infty$. Let be $v$ a
continuous vector field on ${\mathbb{R}}^{d}$ such that

(6.40) $\begin{cases}\|\nabla
v\|_{L^{\infty}}+\||\nabla|^{\frac{d-s}{2}}v\|_{L^{\frac{2d}{d-2-s}}}\mathbf{1}_{s<d-2}<\infty,&{0\leq
s\leq d-2}\\\
\|v\|_{C^{1,\alpha}}+\|v\|_{W^{\frac{d(d+m-s)+2m}{2(d+m)},\frac{2(d+m)}{d+m-2-s}}}<\infty,&{d-2<s<d},\end{cases}$

for some $0<\alpha\leq 1$ and where $m$ is the dimension of the extension
space. There exists a constant $C$ depending only $d,s,\alpha,m$ and on the
potential ${\mathsf{g}}$ through assumptions (1.13) - (1.28), such that

(6.41)
$\left|\int_{({\mathbb{R}}^{d})^{2}\setminus\triangle}(v(x)-v(y))^{\otimes
2}\cdot\nabla^{\otimes
2}{\mathsf{g}}(x-y)d\left\lparen\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu\right\rparen^{\otimes
2}(x,y)\right|\\\ \leq C\Bigg{(}\|\nabla
v\|_{L^{\infty}}+\||\nabla|^{\frac{d-s}{2}}v\|_{L^{\frac{2d}{d-2-s}}}\mathbf{1}_{s<d-2}+\left\lparen\|v\|_{C^{1,\alpha}}+\|v\|_{W^{\frac{d(d+m-s)+2m}{2(d+m)},\frac{2(d+m)}{d+m-2-s}}}\right\rparen\mathbf{1}_{s>d-2}\Bigg{)}^{2}\\\
\times\Bigg{(}{F}_{N}(\underline{x}_{N},\mu)+C(1+\|\mu\|_{L^{\infty}})\Big{(}N^{-\frac{\min\\{d-s,2\\}}{\min\\{d,s+2\\}(s+1)}}+(N^{-1}\log
N)\mathbf{1}_{s=0}\\\ +(N^{-\frac{2}{d(d-1)}}\log
N)\mathbf{1}_{s=d-2}\Big{)}\Bigg{)}\\\
+\Bigg{(}N^{-\frac{\min\\{d+1,s+3\\}}{\min\\{d,s+2\\}(s+1)}}\|v\|_{C^{1,\alpha}}^{2}\left\lparen\||\nabla|^{s+1-d}\mu\|_{L^{\infty}}\mathbf{1}_{s<d-1}+\||\nabla|^{s-d}\nabla\mu\|_{L^{\infty}}\mathbf{1}_{s\geq
d-1}\right\rparen\\\
+\|v\|_{C^{1,\alpha}}\|\operatorname{div}v\|_{L^{\infty}}\Bigg{(}\|\mu\|_{L^{\infty}}N^{-\frac{(d-s)\min\\{d+1,s+3\\}}{\min\\{d,s+2\\}(s+1)}}+N^{-\frac{\min\\{d+1,s+3\\}}{\min\\{d,s+2\\}(s+1)}}\Big{(}\|\mu\|_{L^{\infty}}\log
N+1\Big{)}\mathbf{1}_{s=d-1}\Bigg{)}\Bigg{)}.$

###### Proof.

We follow the outline of the proof of 4.1, focusing on what is different in
the second-order setting. Let $\tilde{v}$ be the same extension of the vector
field $v$ as before. Recycling the kernel notation ${\mathsf{K}}_{v}$, we
define

(6.42) ${\mathsf{K}}_{v}(X,Y)\coloneqq\nabla^{\otimes
2}{\mathsf{G}}(X-Y):(\tilde{v}(X)-\tilde{v}(Y))^{\otimes 2}\qquad\forall X\neq
Y\in{\mathbb{R}}^{d+m}.$

We note that from assumption (1.24) that

(6.43) $|{\mathsf{K}}_{v}(X,Y)|\leq
C\|\nabla\tilde{v}\|_{L^{\infty}}^{2}\left\lparen|X-Y|^{-s}\mathbf{1}_{s>0}+\mathbf{1}_{s=0}\right\rparen$

and

(6.44) $|\nabla_{X}{\mathsf{K}}_{v}(X,Y)|\leq
C\|\nabla\tilde{v}\|_{L^{\infty}}^{2}|X-Y|^{-s-1}.$

We again let $X_{i}$ denote $(x_{i},0)$.

Adding and subtracting the smeared point mass $\delta_{X_{i}}^{(\eta)}$, we
obtain the decomposition

(6.45)
$\int_{({\mathbb{R}}^{d})^{2}\setminus\triangle}\mathsf{k}_{v}(x,y)d\left\lparen\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}-\mu\right\rparen^{\otimes
2}(x,y)=\mathrm{Term}_{1}+\mathrm{Term}_{2}+\mathrm{Term}_{3},$

where $\mathrm{Term}_{1},\ldots,\mathrm{Term}_{3}$ are defined as before. We
proceed to estimate each of the $\mathrm{Term}_{j}$ individually.

Estimate for $\mathrm{Term}_{1}$. We apply 6.1 with

(6.46)
$f=g=\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}}^{(\eta)}-\tilde{\mu}\in\dot{H}^{\frac{s-d-m}{2}}({\mathbb{R}}^{d+m})$

followed by 2.1 or 2.2 to obtain

(6.47) $|\mathrm{Term}_{1}|\leq
C\left\lparen\|\nabla\tilde{v}\|_{L^{\infty}}+\||\nabla|^{\frac{d+m-s}{2}}\tilde{v}\|_{L^{\frac{2(d+m)}{d+m-2-s}}}\mathbf{1}_{s<d+m-2}\right\rparen^{2}\\\
\times\left({F}_{N}(\underline{x}_{N},\mu)+\frac{{\mathsf{G}}_{\eta}(0)}{N}+C\|\mu\|_{L^{\infty}}\left\lparen\eta^{d-s}+(\eta^{d}|\log\eta|)\mathbf{1}_{s=0}+(\eta^{2}|\log\eta|)\mathbf{1}_{s=d-2}\right\rparen+C\eta^{2}\right).$

Estimate for $\mathrm{Term}_{2}$. Suppose first that $0\leq s<d-1$. We break
$\mathrm{Term}_{2}$ into a sum over the index $i$ of terms

(6.48)
$\int_{{\mathbb{R}}^{d+m}}\int_{{\mathbb{R}}^{d+m}}\left\lparen{\mathsf{K}}_{v}(X,X_{i})-{\mathsf{K}}_{v}(X,Y)\right\rparen
d\delta_{X_{i}}^{(\eta)}(Y)d\tilde{\mu}(X).$

Unpacking the definition (6.42) of ${\mathsf{K}}_{v}$, we have

(6.49)
${\mathsf{K}}_{v}(X,X_{i})-{\mathsf{K}}_{v}(X,Y)=\left\lparen\nabla^{\otimes
2}{\mathsf{G}}(X-X_{i})-\nabla^{\otimes
2}{\mathsf{G}}(X-Y)\right\rparen:\left\lparen(\tilde{v}(X)-\tilde{v}(Y))\otimes(\tilde{v}(X)-\tilde{v}(X_{i}))\right\rparen\\\
+\nabla^{\otimes
2}{\mathsf{G}}(X-Y):\left\lparen(\tilde{v}(X)-\tilde{v}(Y))\otimes(\tilde{v}(Y)-\tilde{v}(X_{i}))\right\rparen\\\
+\nabla^{\otimes
2}{\mathsf{G}}(X-X_{i}):\left\lparen(\tilde{v}(Y)-\tilde{v}(X_{i}))\otimes(\tilde{v}(X)-\tilde{v}(X_{i}))\right\rparen.$

By considering the cases $|X-X_{i}|\leq 2\eta$ and $|X-X_{i}|>2\eta$, it
follows from using (1.24), the mean-value theorem, and triangle inequality
that

(6.50) $|{\mathsf{K}}_{v}(X,X_{i})-{\mathsf{K}}_{v}(X,Y)|\leq
C\eta\|\nabla\tilde{v}\|_{L^{\infty}}^{2}\left\lparen|X-X_{i}|^{-s-1}+|Y-X_{i}|^{-s-1}\right\rparen$

for all $Y\in\operatorname{supp}(\delta_{X_{i}}^{(\eta)})$. So, we find that

(6.51) $|\mathrm{Term}_{2}|\mathbf{1}_{s<d-1}\leq
C\eta\|\nabla\tilde{v}\|_{L^{\infty}}^{2}\||\nabla|^{s+1-d}\mu\|_{L^{\infty}}.$

Now if $d-1\leq s<d$, we integrate by parts in $X$ to write

(6.52)
$\int_{({\mathbb{R}}^{d+m})^{2}}{\mathsf{K}}_{v}(X,Y)d\left\lparen\delta_{X_{i}}-\delta_{X_{i}}^{(\eta)}\right\rparen(Y)d\tilde{\mu}(X)\\\
=-\int_{({\mathbb{R}}^{d+m})^{2}}\tilde{\mathsf{K}}_{v}(X,Y)\cdot
d\left\lparen\delta_{X_{i}}-\delta_{X_{i}}^{(\eta)}\right\rparen(Y)d\widetilde{\nabla\mu}(X)\\\
-2\int_{({\mathbb{R}}^{d+m})^{2}}\operatorname{div}\tilde{v}(X)\nabla{\mathsf{G}}(X-Y)\cdot\left\lparen\tilde{v}(X)-\tilde{v}(Y)\right\rparen
d\left\lparen\delta_{X_{i}}-\delta_{X_{i}}^{(\eta)}\right\rparen(Y)d\tilde{\mu}(X),$

where we have defined the vector-valued kernel

(6.53)
$\tilde{\mathsf{K}}_{v}(X,Y)\coloneqq\nabla{\mathsf{G}}(X-Y)\cdot\left\lparen\tilde{v}(X)-\tilde{v}(Y)\right\rparen^{\otimes
2}.$

Proceeding as in the case $0\leq s<d-1$, the first term in the right-hand side
of (6.52) is bounded by

(6.54)
$C\eta\|\nabla\tilde{v}\|_{L^{\infty}}^{2}\||\nabla|^{s-d}\nabla\mu\|_{L^{\infty}}.$

For the second term, we observe that

$\displaystyle\int_{{\mathbb{R}}^{d+m}}\left\lparen\tilde{v}(X)-\tilde{v}(Y)\right\rparen\cdot\nabla
G(X-Y)d(\delta_{X_{i}}-\delta_{X_{i}}^{(\eta)})(Y)$
$\displaystyle=\int_{{\mathbb{R}}^{d+m}}\left\lparen\left\lparen\tilde{v}(X)-\tilde{v}(X_{i})\right\rparen\cdot\nabla{\mathsf{G}}(X-X_{i})-\left\lparen\tilde{v}(X)-\tilde{v}(Y)\right\rparen\cdot\nabla
G(X-Y)\right\rparen d\delta_{X_{i}}^{(\eta)}(Y)$
$\displaystyle=\left\lparen\tilde{v}(X)-\tilde{v}(X_{i})\right\rparen\cdot\left\lparen\nabla{\mathsf{G}}(X-X_{i})-\nabla{\mathsf{G}}_{\eta}(X-X_{i})\right\rparen$
(6.55)
$\displaystyle\phantom{=}-\int_{{\mathbb{R}}^{d+m}}\left\lparen\tilde{v}(X_{i})-\tilde{v}(Y)\right\rparen\cdot\nabla{\mathsf{G}}(X-Y)d\delta_{X_{i}}^{(\eta)}(Y).$

If $|X-X_{i}|\leq 2\eta$, then it follows from the second line using the mean-
value theorem and assumption (1.24) that

(6.56)
$\left|\int_{{\mathbb{R}}^{d+m}}\left\lparen\tilde{v}(X)-\tilde{v}(Y)\right\rparen\cdot\nabla
G(X-Y)d(\delta_{X_{i}}-\delta_{X_{i}}^{(\eta)})(Y)\right|\\\ \leq
C\|\nabla\tilde{v}\|_{L^{\infty}}\left\lparen|X-X_{i}|^{-s}+\int_{{\mathbb{R}}^{d+m}}|X-Y|^{-s}d\delta_{X_{i}}^{(\eta)}(Y)\right\rparen.$

If $|X-X_{i}|>2\eta$, then it follows from the third line using the mean-value
theorem, assumption (1.24), and repeating the proof of (2.21) with
${\mathsf{G}}$ replaced by $\nabla{\mathsf{G}}$ that

(6.57)