# Local well-posedness for the Boltzmann equation with very soft potential and
polynomially decaying initial data

Christopher Henderson  Department of Mathematics, University of Arizona,
Tucson, AZ 85721<EMAIL_ADDRESS>and  Weinan Wang  Department of
Mathematics, University of Arizona, Tucson, AZ 85721
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we address the local well-posedness of the spatially
inhomogeneous non-cutoff Boltzmann equation when the initial data decays
polynomially in the velocity variable. We consider the case of very soft
potentials $\gamma+2s<0$. Our main result completes the picture for local
well-posedness in this decay class by removing the restriction
$\gamma+2s>-3/2$ of previous works. Our approach is entirely based on the
Carleman decomposition of the collision operator into a lower order term and
an integro-differential operator similar to the fractional Laplacian.
Interestingly, this yields a very short proof of local well-posedness when
$\gamma\in(-3,0]$ and $s\in(0,1/2)$ in a weighted $C^{1}$ space that we
include as well.

##  1\. Introduction

The Boltzmann equation is a kinetic equation arising in statistical physics.
Its solution $f(t,x,v)\geq 0$ models the density of particles of a diffuse gas
at time $t\in[0,T]$, at location $x\in\mathbb{T}^{3}$, and with velocity
$v\in\mathbb{R}^{3}$. Roughly, each particle travels with a fixed velocity
until a collision at which time it takes on a new velocity chosen in a way
compatible with physical laws. In this article, we focus on the non-cutoff
version of (1.1) that includes the physically realistic singularity at grazing
collisions. The equation reads

(1.1) $\begin{cases}\partial_{t}f+v\cdot\nabla_{x}f=Q(f,f)\qquad&\text{ in
}[0,T]\times\mathbb{T}^{3}\times\mathbb{R}^{3},\\\ f(0,\cdot,\cdot)=f_{\rm
in}\geq 0\qquad&\text{ in }\mathbb{T}^{3}\times\mathbb{R}^{3}.\end{cases}$

The collision operator $Q$ is defined by

$Q(f,f)=\int_{\mathbb{R}^{3}}\int_{{\mathbb{S}}^{2}}B(v-v_{*},\sigma)\left(f(v_{*}^{\prime})f(v^{\prime})-f(v_{*})f(v)\right)\,\mathrm{d}\sigma\,\mathrm{d}v_{*},$

where $v$ and $v_{*}$ are pre-collisional velocities and $v^{\prime}$ and
$v_{*}^{\prime}$ are post-collisional velocities related by

$v^{\prime}=\frac{v+v_{*}}{2}+\sigma\frac{|v-v_{*}|}{2}\qquad\text{ and
}\qquad v_{*}^{\prime}=\frac{v+v_{*}}{2}-\sigma\frac{|v-v_{*}|}{2}$

and the collision kernel $B$ is given by

$B(v-v_{*},\sigma)=|v-v_{*}|^{\gamma}\theta^{-2-2s}\tilde{b}(\cos\theta),\qquad\text{
where
}\cos\theta=\sigma\cdot\frac{v-v_{*}}{|v-v_{*}|},~{}\gamma\in(-3,1],~{}s\in(0,1),$

and $\tilde{b}$ is a positive bounded function. In this work, we are mostly
interested in the regime of very soft potentials, that is, when $\gamma+2s<0$.

There are several active research directions regarding the well-posedness of
the Boltzmann equation: global well-posedness in the spatially homogeneous
setting (that is, $x$-independent), global well-posedness and regularity of
weak solutions, global well-posedness and convergence of close-to-equilibrium
solutions, and local well-posedness with large initial data. Here, we are
interested in the local well-posedness of (1.1) with large initial data, and,
as such, leave it to other references to detail the extensive history of
research into the first three categories (see, e.g., [3, 6, 9, 10, 11, 14, 15,
16, 17, 18, 19, 20, 21, 23, 27, 34, 37, 38]).

Alexandre, Morimoto, Ukai, Xu, and Yang, often referred to by the acronym
AMUXY, made the first serious progress on the local well-posedness theory for
the (non-cutoff) Boltzmann equation. In particular, in a sequence of seminal
works, by deriving new estimates on the collision operator $Q$, they were able
to establish local well-posedness under the condition that
$e^{\alpha|v|^{2}}f_{\rm in}$ is bounded in certain Sobolev-based spaces [2,
4, 5, 7, 8]. We note that the Gaussian decay plays a large role in their
analysis to compensate for moment loss.

The first results weakening the Gaussian-decay condition on the initial data
are due to Morimoto and Yang [35]. They established local well-posedness in an
$H^{6}$-based space under the assumptions that $\gamma\in(-3/2,0]$ and
$s\in(0,1/2)$. This was later extended by Henderson, Snelson, and Tarfulea
[26], who showed local well-posedness in an $H^{5}$-based space under the
assumption $s\in(0,1)$ and $\max{\\{-3,-\frac{3}{2}-2s\\}}<\gamma<0$. Our
goal, in the present work, is to remove the restriction $\gamma+2s>-3/2$. In
general, the larger $\gamma+2s$ is, the more the decay of $f$ at $|v|=+\infty$
is the issue, and the smaller (more negative) $\gamma+2s$ is, the more
regularity is the issue.

Our interest in establishing local well-posedness with initial data that is
merely polynomially decaying is due to its relationship to the recent
conditional regularity program initiated by Silvestre [36] and continued in
collaboration with Imbert and Mouhot [28, 29, 33, 30, 31, 32]. The goal of the
program is to understand the regularity theory for the Boltzmann equation
conditional to the mass, energy, and entropy densities

$M(t,x)=\int f(t,x,v)dv,\quad E(t,x)=\int
f(t,x,v)|v|^{2}dv\quad\text{and}\quad H(t,x)=\int f(t,x,v)\log f(t,x,v)dv,$

satisfying, uniformly in $(t,x)$,

(1.2) $M,E,H\leq C\quad\text{and}\quad\frac{1}{C}\leq M\qquad\text{ for all
}(t,x),$

where $C$ is a positive constant. When $f$ is $x$-independent, it is well-
known that these conditions are always satisfied. To date, Imbert, Mouhot, and
Silvestre have developed a Harnack inequality and Schauder estimates, obtained
a sharp lower bound on the tail behavior of $f$, and proved a propagation of
polynomial upper bounds of $f$ result, all of which depended only on the
bounds in (1.2).

An upshot of the program of Imbert, Mouhot, and Silvestre is that, roughly,
when a suitable local well-posedness result exists, solutions may be continued
as long as (1.2) holds (see [30, Section 1.1.2] and [26, Corollary 1.2]). In
particular, the local well-posedness result must allow for polynomially
decaying initial data as that type of decay can be propagated forward in time
depending only on the constant $C$ in (1.2). As no such propagation-of-decay
result exists for Gaussian decay, the classical results of AMUXY cannot be
used. It is for this reason that it is important to develop the local well-
posedness theory when $f_{\rm in}$ decays only polynomially.

Our main theorem removes the restriction of previous results [26, 35] that
$\gamma+2s>-3/2$, thereby completing the picture for local well-posedness with
polynomially decaying initial data when $\gamma\in(-3,0)$ and $s\in(0,1)$. In
order to state our result, we define the following two spaces: given
$k,n,m\geq 0$ and $T>0$, let

(1.3) $X^{k,n,m}=H^{k,n}(\mathbb{T}^{3}\times\mathbb{R}^{3})\cap
L^{\infty,m}(\mathbb{T}^{3}\times\mathbb{R}^{3})\qquad\text{and}\qquad
Y_{T}^{k,n,m}=L^{\infty}([0,T];X^{k,n,m}).$

For any $p\geq 1$, we use $L^{p,n}$ to refer to the space of functions $g$
such that $\langle v\rangle^{n}g\in L^{p}$, where $\langle
v\rangle^{2}=1+|v|^{2}$. The weighted Sobolev space $H^{k,n}$ is defined
analogously.

######  Theorem 1.1.

Assume that $\gamma+2s<0$, $k\geq 5$, $n>3/2$, and $m>M=M(k,n,\gamma,s)$
sufficiently large. Suppose $0\leq f_{\rm in}\in X^{k,n,m}$. Then there exists
a time $T>0$, depending only on $\|f_{\rm in}\|_{X^{k,n,m}}$ as well as $n$,
$k$, $m$, $\gamma$, $s$, and $\tilde{b}$, and a unique solution $f\in
Y_{T}^{k,n,m}\cap C([0,T];H^{k,n}_{x,v})$ of (1.1) such that $f\geq 0$ and

$\|f\|_{Y_{T}^{k,n,m}}\lesssim\|f_{\rm in}\|_{X^{k,n,m}}.$

As discussed above, an important motivation of Theorem 1.1 is to extend the
continuation criterion for the Boltzmann equation to the very soft potentials
range. While such a result does not directly follow from Theorem 1.1 and the
Imbert-Mouhot-Silvestre regularity program (as these results only deal with
the regime $\gamma+2s\in[0,2]$), it is likely a straightforward exercise after
adding an additional assumption on the $L^{\infty}_{t,x}L^{p}_{v}$ norm of $f$
to (1.2). Indeed, this has already been accomplished for the closely-related
Landau equation in [24] in the analogous parameter regime. An upshot of such a
continuation criterion, were it established, is the ability to construct
classical solutions from rough initial initial data as accomplished for the
Landau equation [25]. These will be the subject of a future work.

As is typical for nonlinear equations, the main step in the proof of local
well-posedness of (1.1) is to establish a priori estimates on solutions. In
particular, this requires obtaining bounds on the collision operator $Q$ as a
bilinear form from and to various Banach spaces.

In order to explain the strategy and difficulties in obtaining such estimates,
we discuss the restriction $\gamma+2s>-3/2$ in [26]. This is inherited in the
application of the estimates on the collision operator developed in [4]. For
certain key estimates, AMUXY use Fourier analysis, which is most suited
$L^{2}$-based spaces. However, a major lesson from [36] is that one can,
roughly, think of $Q$ as having a coefficient of the form

$\int f(t,x,w)|v-w|^{\gamma+2s}dw,$

and one sees, after applying the Cauchy-Schwarz inequality, that such
coefficients are bounded using the (weighted) $L^{2}$-norm of $f$ only when
$\gamma+2s>-3/2$.

In view of the above, it is required to develop new estimates on the collision
operator in spaces that are not $L^{2}$-based. Our approach is to take
advantage of the Carleman decomposition (see equation (2.1)), which views $Q$
as the sum of an integro-differential operator similar to the fractional
Laplacian and a lower order term. As this is a real space-based approach, it
is possible, through intricate analysis, to obtain estimates on $Q(g,f)$ in
various spaces depending on both $L^{2}$\- and $L^{\infty}$-based norms. This
allows us to circumvent the issues encountered in previous works.

Curiously, this approach makes an extremely simple proof of local well-
posedness in a weighted $C^{1}$ space obvious when $s\in(0,1/2)$. The reason
for this is as follows. First, as observed in [26, Proposition 3.2], the
Carleman decomposition makes it easy to obtain $L^{\infty,m}$ bounds on $f$
from $\|f_{\rm in}\|_{L^{\infty,m}}$ via a simple comparison principle
argument. The important observation is that, roughly, at a maximum of $\langle
v\rangle^{m}f$, the only high order term has a good sign. A straightforward
attempt to repeat this for the $L^{\infty,m}$ norm of $\partial f$ is
complicated by the fact that $\partial f$ solves an equation involving a term
$Q(\partial f,f)$. From the Carleman decomposition of $Q$, we, roughly, see

$Q(\partial f,f)\sim\left(\int\partial
f(w)|v-w|^{\gamma+2s}dw\right)\Delta^{s}f\lesssim\|\partial
f\|_{L^{\infty,m}}\|\langle v\rangle^{m}f\|_{C^{2s+\epsilon}}.$

Fortunately, when $s\in(0,1/2)$, this is lower order and the previous argument
can be repeated.

We now state the result. We first define the $C^{k}$ analogue of the spaces
$X$ and $Y_{T}$ (1.3):

(1.4) $\begin{split}&\tilde{X}^{k,m_{0},m_{1}}=\\{f:\langle
v\rangle^{m_{0}}\nabla^{\ell}f\in
C(\mathbb{T}^{3}\times\mathbb{R}^{3})^{6^{\ell}}\text{ for }0\leq\ell\leq
k-1,\langle v\rangle^{m_{1}}\nabla^{k}f\in
C(\mathbb{T}^{3}\times\mathbb{R}^{3})^{6^{k}}\\}\\\
&\text{and}\quad\tilde{Y}_{T}^{k,m_{0},m_{1}}=L^{\infty}([0,T];\tilde{X}^{k,m_{0},m_{1}}).\end{split}$

######  Theorem 1.2.

Let $k\geq 1$, $\gamma\in(-3,0]$, $s\in(0,1/2)$, $m_{1}>3+\gamma+2s$, and
$m_{0}$ be sufficiently large depending only on $k$, $\gamma$, $s$, and
$m_{1}$. Let the initial data $0\leq f_{\rm
in}\in\tilde{X}^{k,m_{0},m_{1}}$.Then there exists a time $T>0$, depending
only on $\|f_{\rm in}\|_{\tilde{X}^{k,m_{0},m_{1}}}$, $\gamma$, $s$, $m_{0}$,
$m_{1}$, and $\tilde{b}$, and a unique solution $f\geq 0$ of (1.1) such that

$\begin{split}&f\in\tilde{Y}_{T}^{k,m_{0},m_{1}}\cap
C([0,T];C^{k}(\mathbb{T}^{3}\times\mathbb{R}^{3}))\cap
C^{1}([0,T];C^{k-1}(\mathbb{T}^{3}\times\mathbb{R}^{3}))\\\ &\text{and
}\|f\|_{\tilde{Y}_{T}^{k,m_{0},m_{1}}}\lesssim\|f_{\rm
in}\|_{\tilde{X}^{k,m_{0},m_{1}}}.\end{split}$

We note that simply by differentiating the equation, we can obtain further
time regularity when $k>1$. In addition, a careful accounting based on the
estimates of the collision operator and the definition of
$\tilde{Y}^{k,m_{0},m_{1}}_{T}$ yields the decay in velocity of the
$C^{\ell}_{x,v}$ norms. This is not the main interest of the statement above
so we omit it.

The significance of Theorem 1.2, besides having such a short proof, is that it
improves on [35, 26] in two major ways. First, it increases the range of
possible $\gamma$: [35] requires $\gamma\in(-3/2,0]$ and [26] requires
$\gamma\in(-3/2-2s,0)$. Second, it weakens conditions on the initial
regularity: [35] works in an $H^{6}$-based space and [26] works in an
$H^{5}$-based space, both of which embed in $C^{1}$. Note that it also reduces
the regularity required of the initial data in comparison to Theorem 1.1. On
the other hand, like [35] but unlike [26] and Theorem 1.1, it only applies to
$s\in(0,1/2)$.

###  1.1. Notation

We use the notation $A\lesssim B$ if there is a constant $C$ such that $A\leq
CB$. In general, the constant $C$ may depend on $\gamma$, $s$, $n$, $m$, $k$,
$m_{0}$, $m_{1}$, and $\tilde{b}$. Additionally, if an assumption for an
estimate involves a requirement such as $\alpha>\beta$, then the constant $C$
may depend on $\alpha-\beta$. We use $A\approx B$ if $A\lesssim B$ and
$B\lesssim A$. Occasionally, it will be necessary to include a constant, in
which case we use $C$ to represent such a constant and this constant $C$ may
change line-by-line.

Any integral whose domain of integration in $v$ is not specified is understood
to be an integral over $\mathbb{R}^{3}$ and any integral whose domain of
integration in $x$ is not specified is understood to be an integral over
$\mathbb{T}^{3}$. For example, for any measurable $\varphi$ and any measurable
sets $\Omega_{x}\subset\mathbb{T}^{3}$ and $\Omega_{v}\subset\mathbb{R}^{3}$,
we have

$\int\int_{\Omega_{v}}\varphi(x,v)dvdx=\int_{\mathbb{T}^{3}}\int_{\Omega_{v}}\varphi(x,v)dvdx\quad\text{and}\quad\int_{\Omega_{x}}\int\varphi(x,v)dvdx=\int_{\Omega_{x}}\int_{\mathbb{R}^{3}}\varphi(x,v)dvdx.$

Similarly, we often suppress the domain in Lebesgue, Sobolev, and Hölder
spaces when it is clear, writing, e.g., $f\in L^{\infty,m}$ instead of $f\in
L^{\infty,m}(\mathbb{R}^{3})$ if has already been established that
$f:\mathbb{R}^{3}\to\mathbb{R}$.

We use $B_{R}$ to mean a ball of radius $R$ around the origin. Whenever the
ball is not centered at the origin, we denote the center $v_{0}$ as
$B_{R}(v_{0})$.

Finally, when stating estimates on the collision operator $Q(g,f)$, we often
omit the assumptions on the involved functions $g$ and $f$. In these cases,
the estimate holds whenever the right hand side is finite.

###  1.2. Outline

The rest of the paper is organized as follows. In Section 2, we consider
bounds on the collision operator. In particular, we recall useful known
results, prove some easy extensions of them, and state our main new estimates.
Then, in Section 3, we prove the existence and uniqueness of solutions using
the bounds from Section 2. Afterwards, in Section 4, we prove the estimates on
the collision operator $Q$. Finally, in Section 5, we give a simple proof of
local well-posedness for the case of $s\in(0,1/2)$ and $\gamma\in(-3,0]$.

##  2\. Estimates on the collision operator

In this section, we state the key estimates on the collision operator $Q$ that
we use in our proof of well-posedness. We begin with a brief overview of the
Carleman decomposition allowing us to use ideas from the study of integro-
differential operators. Then we state known estimates and their easy
extensions. Finally, we state new estimates whose proof, contained in Section
4, makes up the bulk of this manuscript.

###  2.1. Carleman decomposition

A key tool in our analysis is the Carleman decomposition [12, 13] that views
the Boltzmann collision operator $Q$ as the sum of a non-local diffusion
operator locally similar to $-(-\Delta)^{s}$ and a lower order reaction term.
This decomposition is well-known, see [1] for an early discussion of it and
[36, Sections 4 and 5] for the presentation used here. Indeed,

(2.1) $\begin{split}&Q(g,f)=Q_{\rm s}(g,f)+Q_{\rm ns}(g,f)\\\ &Q_{\rm
s}(g,f)=\int(f(v^{\prime})-f(v))K_{g}(v,v^{\prime})dv^{\prime}\qquad\\\
&Q_{\rm ns}(g,f)=c_{b}(S_{\gamma}*g)f,\end{split}$

where $S_{\gamma}(v)=|v|^{\gamma}$, $c_{b}>0$ is a fixed constant, and $K_{g}$
satisfies, for any $g\geq 0$ and any $v,v^{\prime}\in\mathbb{R}^{3}$,

(2.2)
$\begin{split}&K_{g}(v,v^{\prime})\approx\frac{1}{|v-v^{\prime}|^{3+2s}}\int_{w\in
v+(v^{\prime}-v)^{\perp}}g(w)|v-w|^{\gamma+2s+1}dw\\\ &\text{and}\qquad
K_{g}(v,v+v^{\prime})=K_{g}(v,v-v^{\prime}).\end{split}$

We refer to $Q_{\rm s}$ as the “singular” part and $Q_{\rm ns}$ as the “non-
singular” part.

Actually, to be fully rigorous, $Q_{\rm s}$ should be defined using a
principal value. We abuse notation and suppress this as all our estimates
occur over symmetric domains near the base point $v$ and are, thus, compatible
with the limit involved in the principal value.

###  2.2. Previously established estimates and easy extensions

In this section, we state various estimates on the collision operator that are
well-known or are simple extensions of previous results.

######  Lemma 2.1 (Estimates of the kernel $K_{g}$).

For all $r>0$ and $v\in\mathbb{R}^{3}$,

  1. (i)

$\displaystyle\int\limits_{B_{2r}(v)\backslash
B_{r}(v)}K_{g}(v^{\prime},v)\,dv^{\prime},\int\limits_{B_{2r}(v)\backslash
B_{r}(v)}K_{g}(v,v^{\prime})\,dv^{\prime}\lesssim
r^{-2s}\int|g(z)||z-v|^{\gamma+2s}\,dz$.

  2. (ii)

$\displaystyle\left|\,\int[K_{g}(v,v^{\prime})-K_{g}(v^{\prime},v)]\,dv^{\prime}\right|\lesssim\int|g(z)||z-v|^{\gamma}\,dz.$

  3. (iii)

$\displaystyle\int\limits_{B_{r}(v)}(v^{\prime}-v)K_{g}(v,v^{\prime})\,dv^{\prime}=0.$

  4. (iv)

$\displaystyle\Big{|}\int_{B_{r}(v)}(v^{\prime}-v)K_{g}(v,v^{\prime})\,dv\Big{|}\lesssim\int|g(z)||z-v^{\prime}|^{1+\gamma}\,dz.$

Lemma 2.1 follows from [32, Lemmas 3.4, 3.5, 3.6, and 3.7]. The following
lemma can be regarded as a slight generalization of [29, Proposition 2.1].

######  Lemma 2.2.

For $0<s<1$, $\alpha>2s$, $r>0$, and $g:\mathbb{R}^{3}\to\mathbb{R}_{+}$ there
holds

$\int_{B_{r}(v^{\prime})}K_{g}(v^{\prime},v)|v-v^{\prime}|^{\alpha}dv,\int_{B_{r}(v^{\prime})}K_{g}(v,v^{\prime})|v-v^{\prime}|^{\alpha}dv\lesssim
r^{\alpha-2s}\int|g(w)||v-w|^{\gamma+2s}dw.$

###### Proof.

The proofs of both inequalities are similar, so we show only the latter.
Assume without loss of generality that $g\geq 0$. We proceed with a simple
annular decomposition paired with the existing estimate Lemma 2.1.(i). Indeed,
letting $A_{k}=B_{2^{-k}r}(v^{\prime})\setminus B_{2^{-k-1}r}(v^{\prime})$, we
have

$\begin{split}\int_{B_{r}(v^{\prime})}&K_{g}(v,v^{\prime})|v-v^{\prime}|^{\alpha}dv=\sum_{k=0}^{\infty}\int_{A_{k}}K_{g}(v,v^{\prime})|v-v^{\prime}|^{\alpha}dv\leq\sum_{k=0}^{\infty}2^{-\alpha
k}r^{\alpha}\int_{A_{k}}K_{g}(v,v^{\prime})dv\\\
&\leq\sum_{k=0}^{\infty}2^{-\alpha
k}r^{\alpha}\int_{B_{2^{-k-1}r}^{c}(v^{\prime})}K_{g}(v,v^{\prime})dv\lesssim\sum_{k=0}^{\infty}2^{-k(\alpha-2s)}r^{\alpha-2s}\int
g(w)|v^{\prime}-w|^{\gamma+2s}dw.\end{split}$

The claim then follows due to the fact that the sum over $k$ is finite. ∎

The next lemma concerns bounds on $K_{g}$ via $\|g\|_{L^{\infty,m}}$.

######  Lemma 2.3.

Fix any $m>3+\gamma+2s$, $g\in L^{\infty,m}$, and
$v,v^{\prime}\in\mathbb{R}^{3}$. Then

$|K_{g}(v,v^{\prime})|\lesssim\frac{1}{|v-v^{\prime}|^{3+2s}}\|g\|_{L^{\infty,m}}\langle
v\rangle^{\gamma+2s+1}.$

We omit the proof as this is obvious from (2.2) and a straightforward
parametrization of the 2-dimensional hyperplane that is the domain of
integration.

On the other hand, under a smallness condition on $v^{\prime}$, we can
establish a refined estimate involving the decay of $g$. To our knowledge this
was first observed in [26, equation (4.39)] but not stated as a stand-alone
lemma or given a proof in complete generality. As such, we include it here.

######  Lemma 2.4.

Fix any $m>3+\gamma+2s$, $\theta\in(0,1)$, $g\in L^{\infty,m}$, and
$v,v^{\prime}\in\mathbb{R}^{3}$ with $(1-\theta)|v|\geq|v^{\prime}|$. Then

$|K_{g}(v,v^{\prime})|\lesssim\frac{1}{|v-v^{\prime}|^{3+2s}}\|g\|_{L^{\infty,m}}\langle
v\rangle^{\gamma+2s+3-m}.$

As the proof of Lemma 2.4 is longer than the others of this subsection, we
include it in Section 4; however, it is simply a more careful writing of the
ideas in the proof of [26, equation (4.39)].

The next lemma provides estimates for the non-singular part $Q_{\rm ns}$.
Recall (2.2). Then, we have the following estimates.

######  Lemma 2.5.

Suppose that $f,g:\mathbb{T}^{3}\times\mathbb{R}^{3}\to\mathbb{R}$. Then, for
any $\epsilon>0$ and $n\geq 0$,

$\begin{split}\|Q_{\rm
ns}(g,f)\|_{L^{2,n}}&\lesssim\begin{cases}\|g\|_{L^{\infty,3+\gamma+\epsilon}}\|f\|_{L^{2,n}}\\\
\|g\|_{L^{2,n}}\|f\|_{L^{\infty,n+\epsilon+3/2+\gamma+(3/2-n)_{+}}}.\end{cases}\end{split}$

######  Remark.

The first inequality in Lemma 2.5 is obvious by writing $Q_{\rm
ns}(g,f)=(S*g)f$ and bounding $S*g$ in $L^{\infty}$ using the weighted
$L^{\infty}$ norm of $g$. The second inequality can be easily proved by using
our weighted Young’s inequality Lemma 4.2. As this proof is straightforward
from the statement of Lemma 4.2, we omit the details.

We also require the following from [26, Lemma 2.6]:

######  Lemma 2.6 (Interpolation lemma).

If $n,m\geq 0$, $k^{\prime}\in(0,k)$, and
$l<(m-\frac{3}{2})(1-\frac{k^{\prime}}{k})+n\frac{k^{\prime}}{k}$, then

$\|f\|_{H^{k^{\prime},l}}\lesssim\|f\|_{L^{\infty,m}}^{1-\frac{k^{\prime}}{k}}\|f\|_{H^{k,n}}^{\frac{k^{\prime}}{k}}.$

###  2.3. New estimates

We now state new estimates on the collision operator that are crucial to
allowing us to extend well-posedness to the full range of soft potentials. The
prior similar work [26] relied heavily on [26, Theorem 2.4, Proposition 2.5,
and Proposition 3.1]. The first two come directly from [4, Proposition 2.9 and
2.8], respectively. Each result, unfortunately, requires
$\gamma+2s>-\frac{3}{2}$. Thus, these are not applicable in our setting, and
the main issue of the present work is to obtain suitable replacements, which
we state here.

The first is a commutator estimate (cf. [4, Proposition 2.8], [26, Proposition
2.5]).

######  Proposition 2.7 (Commutator estimate).

For any $\epsilon>0$, $\gamma\in(-3,0]$, $\mu\in((1-2s)_{+},2-2s)$,
$m>\max\\{3+\gamma+2s,\ell+\gamma+\frac{3}{2}\\}$, $\ell>\frac{3}{2}$, and
$f,g:\mathbb{R}^{3}\to\mathbb{R}$, we have

$\displaystyle\|\langle v\rangle^{\ell}Q_{\rm s}(g,f)-Q_{\rm s}(g,\langle
v\rangle^{\ell}f)\|_{L^{2}}\lesssim(\|f\|_{L^{2,\ell+3/2+\epsilon}}+\|f\|_{H^{2s-1+\mu,\mu+\ell+\gamma+2s}})\|g\|_{L^{\infty,m}}.$

The next estimates concern $Q_{\rm s}(g,f)$ and involve only the $v$-variable.

######  Proposition 2.8.

For any $f,g:\mathbb{R}^{3}\to\mathbb{R}$, $\gamma\in(-3,0]$, $\gamma+2s\leq
0$, and $\epsilon>0$:

  1. (i)

If $\theta\in(0,2s)$, then

$\int Q_{\rm
s}(g,f)hdv\lesssim\|g\|_{L^{\infty,3+\gamma+2s+\epsilon}}\|f\|_{H^{2s-\theta}}\|h\|_{H^{\theta}}.$

  2. (ii)

If $\theta>0$, then

$\|Q_{\rm
s}(g,f)\|_{L^{2}}\lesssim\|g\|_{L^{\infty,3+\gamma+2s+\epsilon}}\|f\|_{H^{2s+\theta}}.$

  3. (iii)

If $n>3/2+\gamma+2s$, $m>3/2+\gamma+(3/2-n)_{+}$, and $\alpha>2s$, then

$\|Q_{\rm
s}(g,f)\|_{L^{2,n}}\lesssim\|g\|_{L^{2,n}}\left(\|f\|_{L^{\infty,m}}+\|\langle
v\rangle^{n+5/2+(3/2-n)_{+}+\alpha+\gamma+\epsilon}f\|_{C^{\alpha}_{v}}\right).$

  4. (iv)

If $n\geq 0$ and $m>n+6+\gamma+2s$, we have

$\int\langle v\rangle^{2n}fQ_{\rm
s}(g,f)\,dv\lesssim\|g\|_{L^{\infty,m}}\|f\|_{L^{2,n}}^{2}.$

The first two parts above, (i) and (ii), rely heavily on the work in [32];
however, that reference is focused on local estimates and, as such, is not
concerned with understanding the dependence on weights. Combined they are a
replacement for [26, Theorem 2.4] (see also [4, Proposition 2.9]). The second
two parts above, (iii) and (iv), are new. They are replacements for [26,
Proposition 3.1.(i) and (iii)], respectively.

We make two brief remarks. First, the result (i) is a slight generalization of
the results in [32] as it allows to choose $\theta$ in (i). Second, the result
(ii) almost certainly holds without $\theta=0$; however, as this is not needed
for our purposes and the current statement is easy to derive from [32], we are
content to use (ii) as is.

The final estimate makes use of symmetry properties of $Q_{\rm s}$ in order to
avoid having more than one full derivative “land” on $f$. This is crucial in
case two of the proof of the main a priori estimate Proposition 3.1. It is a
replacement for [26, Proposition 3.1.(iv)].

######  Proposition 2.9.

Suppose that $f,g:\mathbb{T}^{3}\times\mathbb{R}^{3}\to\mathbb{R}$. If
$\gamma\in(-3,0]$, $\epsilon>0$, $\mu\in((1-2s)_{+},2-2s)$,
$\kappa\in(s,\min\\{2s,1\\})$, $n>\frac{3}{2}$, and
$m>\max\\{3+\gamma+2s,n+\gamma+\frac{3}{2}\\}$. Then

$\begin{split}\Big{|}\int&\langle v\rangle^{2n}Q_{\rm s}(g,f)\partial
fdvdx\Big{|}\lesssim\|g\|_{L^{\infty,m}}\left(\|f\|_{L^{2,n+3/2+\epsilon}}+\|f\|_{H^{2s-1+\mu,\mu+n+\gamma+2s}}\right)\|f\|_{H^{1,n}}\\\
&+\|\partial
g\|_{H^{3/2+(2s-1/2)_{+}+\epsilon,3+\gamma+2s+\epsilon}}\|f\|_{H^{1,n}}^{2}+\|g\|_{C^{\kappa,3+\epsilon}}\|f\|_{H^{s,n+3/2+\epsilon+(\gamma+2s+1)_{+}}}\|f\|_{H^{1,n}},\end{split}$

where $\partial=\partial_{x_{i}}$ or $\partial_{v_{i}}$ for some
$i\in\\{1,2,3\\}$.

Recall that we prove the above estimates in Section 4.

##  3\. Existence and uniqueness of solutions: Theorem 1.1

In this section, we prove Theorem 1.1. The majority of the work is in the
proof of existence and our approach for this follows [26] closely. Indeed, the
construction procedure is similar, relying on exhibiting a solution to a
suitably regularized and linearized problem. We then use compactness to
deregularize and a fixed point argument to pass from the linearized problem to
the nonlinear one. The main novelty to the current work as compared to [26] is
in the establishment of a priori estimates in $Y_{T}^{k,n,m}$ of the
regularized and linearized problem. When possible, we omit details that are
unchanged from [26].

###  3.1. Proof of existence in Theorem 1.1

First, we define a smooth cut-off function
$\psi:\mathbb{R}^{3}\rightarrow\mathbb{R}$ with $0\leq\psi\leq 1$,
$\int\psi(v)\,dv=1$,

$\displaystyle\psi=1,\quad\text{on}~{}B_{1/2}\qquad\text{and}\qquad\psi=0\quad\text{on}~{}B_{1}^{c}.$

Next, for any $\phi:\mathbb{T}^{3}\times\mathbb{R}^{3}\rightarrow\mathbb{R}$
and $\epsilon>0$, we define

$\displaystyle\phi^{\epsilon}(x,v)=\frac{1}{\epsilon^{6}}\int\psi\Big{(}\frac{x-y}{\epsilon}\Big{)}\psi\Big{(}\frac{v-w}{\epsilon}\Big{)}\phi(y,w)\,dydw.$

Then we define the regularized collision operator, for any $\delta>0$ and
$(x,v)\in\mathbb{T}^{3}\times\mathbb{R}^{3}$,

$\displaystyle Q_{\epsilon,\delta}(g(x,\cdot),f(x,\cdot))(v)=\psi(\delta
v)Q(g^{\epsilon}(x,\cdot),\psi(\delta\cdot)f(x,\cdot)).$

Finally, for any $\sigma\in[0,1]$, we define the differential operator

(3.1)
$\displaystyle\mathcal{L}_{\sigma,\epsilon,\delta}(f)=\partial_{t}f+\sigma\psi(\delta
v)v\cdot\nabla_{x}f-(\epsilon+(1-\sigma))\Delta_{x,v}f-\sigma
Q_{\epsilon,\delta}(g,f).$

The intuition for the above regularizations and cut-offs is given in [26,
Section 3].

We now establish a priori estimates that hold for both the full equation and
the regularized one above. This is done in the following proposition.

######  Proposition 3.1.

Suppose that $T>0,k\geq
5,n>3/2+(\gamma+2s)_{+},\sigma\in[0,1],\epsilon,\delta\geq 0$, and $m\geq 0$.
Suppose that $R,f\in Y_{T}^{k,n,m}$

(3.2)
$\begin{cases}\mathcal{L}_{\sigma,\epsilon,\delta}f=R,\qquad&\text{in}~{}(0,T)\times\mathbb{T}^{3}\times\mathbb{R}^{3}\\\
f(0,\cdot,\cdot)=f_{\text{in}},\qquad&\text{in}~{}\mathbb{T}^{3}\times\mathbb{R}^{3}.\end{cases}$

For any $\mu>0$, if $\delta=0$ and $m\geq 3/2+\mu$ or if $\delta>0$, then

(3.3)
$\begin{split}\|f\|_{L^{\infty,m}}&\leq\exp\Big{\\{}C\int_{0}^{T}\|g(t)\|_{L^{\infty,\max\\{m,3/2+\mu\\}}}dt\Big{\\}}\Big{(}\|f_{\rm
in}\|_{L^{\infty,m}}+\int_{0}^{T}\|R(t)\|_{L^{\infty,m}}dt\Big{)}.\end{split}$

If $\delta=0$ and $m$ is sufficiently large depending on $k,n,\gamma$, and
$s$, then

(3.4)
$\begin{split}\|f\|_{L^{\infty,m}([0,T];H_{x,v}^{k,n})}&\leq\exp\Big{\\{}C\int_{0}^{T}\|g(t)\|_{X_{x,v}^{k,n,m}}\,dt\Big{\\}}\Big{(}(1+T)\|f_{\rm
in}\|_{X_{x,v}^{k,n,m}}+\int_{0}^{T}\|R(t)\|_{X_{x,v}^{k,n,m}}dt\Big{)}.\end{split}$

Now, we prove Proposition 3.1. The proof follows that of [26, Proposition 3.1]
with small changes due to the new estimates on the collision operator
necessary in our setting.

###### Proof.

The argument of (3.3) goes exactly as that in [26, Proposition 3.1] and hence
we omit the proof here. Now we focus on proving (3.4). First, we let
$\alpha,\beta\in\mathbb{N}_{0}^{3}$ be any multi-indices such that
$|\alpha|+|\beta|=k$. Then, differentiating eq. 3.1, multiplying the resulting
equation by $\langle v\rangle^{2n}\partial_{x}^{\alpha}\partial_{v}^{\beta}f$,
integrating in $x$ and $v$, we get

$\begin{split}\frac{1}{2}\frac{d}{dt}\int|\langle
v\rangle^{n}\partial_{x}^{\alpha}\partial_{v}^{\beta}f|^{2}\,dxdv&=-\sigma\int\left(\sum_{i=1}^{3}\beta_{i}\partial_{x_{i}}\partial_{x}^{\alpha}\partial_{v}^{\beta-
e_{i}}f\right)\langle
v\rangle^{2n}\partial_{x}^{\alpha}\partial_{v}^{\beta}f\,dxdv\\\
&\quad+\sigma\sum_{\begin{subarray}{c}\alpha^{\prime}+\alpha^{\prime\prime}=\alpha\\\
\beta^{\prime}+\beta^{\prime\prime}=\beta\end{subarray}}C_{\alpha^{\prime},\beta^{\prime},\alpha^{\prime\prime},\beta^{\prime\prime}}\int
Q(\partial_{x}^{\alpha^{\prime}}\partial_{v}^{\beta^{\prime}}g,\partial_{x}^{\alpha^{\prime}}\partial_{v}^{\beta^{\prime}}f)\langle
v\rangle^{2n}\partial_{x}^{\alpha}\partial_{v}^{\beta}f\,dxdv\\\
&\quad-(\epsilon+1-\sigma)\int|\nabla_{x,v}\partial_{x}^{\alpha}\partial_{v}^{\beta}f|^{2}+\int\partial_{x}^{\alpha}\partial_{v}^{\beta}R\langle
v\rangle^{2n}\partial_{x}^{\alpha}\partial_{v}^{\beta}f\,dxdv\\\
&=I_{1}+I_{2}+I_{3}+I_{4},\end{split}$

for some constants
$C_{\alpha^{\prime},\beta^{\prime},\alpha^{\prime\prime},\beta^{\prime\prime}}>0$
depending only on the subscripted quantities.

We see that $I_{1}$ is bounded by $\|f\|_{H^{k,n}}^{2}$, $I_{4}$ is bounded by
$\|R\|_{H^{k,n}}^{2}+\|f\|_{H^{k,n}}^{2}$, and $I_{3}$ has a good sign. Thus,
our focus is primarily on $I_{2}$, the term involving the collision operator
$Q$. We argue case by case depending on the size of
$|\alpha^{\prime\prime}|+|\beta^{\prime\prime}|$ in order to establish that

(3.5)
$\begin{split}I_{2}\lesssim\|g\|_{X^{k,n,m}}\|f\|_{X^{k,n,m}}^{2}.\end{split}$

The proof of (3.5) is postponed momentarily while we show how to conclude.
Indeed, assuming (3.5) is proved, we arrive at

(3.6) $\begin{split}\frac{1}{2}\frac{d}{dt}\int|\langle
v\rangle^{n}\partial_{x}^{\alpha}\partial_{v}^{\beta}f|^{2}\,dxdv&\lesssim(\|g\|_{X^{k,n,m}}+1)\|f\|_{X^{k,n,m}}^{2}+\|R\|_{H^{k,n}}^{2}.\end{split}$

Recalling the definition of $X^{k,n,m}$ in (1.3) and using (3.3), we find

(3.7)
$\begin{split}\frac{1}{2}\frac{d}{dt}\|f\|_{H^{k,n}}^{2}&\lesssim(\|g(t)\|_{X^{k,n,m}}+1)\|f\|_{H^{k,n}}^{2}\\\
&\quad+(\|g(t)\|_{X^{k,n,m}}+1)\exp\left\\{C\int_{0}^{T}\|g(t)\|_{X^{k,n,m}}dt\right\\}\left(\|f_{\rm
in}\|_{L^{\infty,m}}^{2}+\|R\|_{X^{k,n,m}}^{2}\right).\end{split}$

Therefore, we conclude the proof of equation (3.4) by applying Grönwall
inequality.

We now establish (3.5). For notational ease, we set

(3.8) $\displaystyle
F=\partial^{\alpha^{\prime\prime}}_{x}\partial^{\beta^{\prime\prime}}_{v}f{\rm,\qquad{}}G=\partial^{\alpha^{\prime}}_{x}\partial^{\beta^{\prime}}_{v}g.$

Thus, we are estimating terms of the form

(3.9) $\begin{split}\int\langle
v\rangle^{2n}&Q(G,F)\partial^{\alpha^{\prime}}_{x}\partial_{v}^{\beta^{\prime}}Fdvdx\\\
&=\int\langle v\rangle^{2n}Q_{\rm
s}(G,F)\partial^{\alpha^{\prime}}_{x}\partial_{v}^{\beta^{\prime}}Fdvdx+\int\langle
v\rangle^{2n}Q_{\rm
ns}(G,F)\partial^{\alpha^{\prime}}_{x}\partial_{v}^{\beta^{\prime}}Fdvdx.\end{split}$

Case one: $|\alpha^{\prime\prime}|+|\beta^{\prime\prime}|=k$, i.e.,
$\alpha^{\prime\prime}=\alpha$, $\beta^{\prime\prime}=\beta$, and in the form
of $\int\langle v\rangle^{2n}Q(g,F)F$.
We estimate the $Q_{\rm s}$ term first. We proceed by using Proposition
2.8.(iv), up to increasing $m$ if necessary,

$\begin{split}\int\langle v\rangle^{2n}Q_{\rm
s}(g,F)F\,dvdx&\lesssim\int\|F\|_{L_{v}^{2,n}}^{2}\|g\|_{L_{v}^{\infty,m}}\,dx\lesssim\|F\|_{L^{2,n}}^{2}\|g\|_{L^{\infty,m}}\\\
&\lesssim\|f\|_{H^{k,n}}^{2}\|g\|_{L^{\infty,m}}\lesssim\|f\|_{X^{k,n,m}}^{2}\|g\|_{X^{k,n,m}},\end{split}$

as desired.

Furthermore, for $\int\langle v\rangle^{2n}Q_{\rm ns}(g,F)F\,dvdx$, we recall
(2.1) and apply Lemma 2.5 to find

$\begin{split}\int\langle v\rangle^{2n}Q_{\rm
ns}(g,F)F\,dvdx&\approx\int\langle
v\rangle^{2n}(S_{\gamma}*g)F^{2}dvdx\lesssim\int\|g\|_{L_{v}^{\infty,n}}\|F\|_{L_{v}^{2,n}}^{2}\,dx\\\
&\lesssim\|g\|_{L^{\infty,m}}\|F\|_{L^{2,n}}\lesssim\|g\|_{L^{\infty,m}}\|f\|_{H^{k,n}}^{2}\leq\|g\|_{X^{k,n,m}}\|f\|_{X^{k,n,m}}^{2}.\end{split}$

This concludes the proof of (3.5) in case one.

Case two: $|\alpha^{\prime\prime}|+|\beta^{\prime\prime}|=k-1$, and in the
form, $\int\langle v\rangle^{2n}Q(\partial g,F)\partial F$. Here we denote
derivative operator
$\partial=\partial_{x}^{\alpha^{\prime}}\partial_{v}^{\beta^{\prime}}$ as
$|\alpha^{\prime}|+|\beta^{\prime}|=1$.

We first estimate the $Q_{\rm s}$ portion. Fix
$\epsilon\in(0,\min\\{s,1-s\\})$. Let $\mu=(1-2s)_{+}+\epsilon$,
$\kappa=s+\epsilon$, and
$\tilde{m}=\epsilon+\max\\{3+\gamma+2s,n+\gamma+3/2\\}$. We then directly
apply Proposition 2.9 to find

$\begin{split}\left|\int\langle v\rangle^{2n}Q_{\rm s}(\partial g,F)\partial
Fdvdx\right|&\lesssim\|\partial
g\|_{L^{\infty,\tilde{m}}}\|F\|_{H^{2s-1+\mu,\mu+n+3/2}}\|F\|_{H^{1,n}}\\\
&\quad+\|\partial^{2}g\|_{H^{3-s,3+\epsilon}}\|F\|_{H^{1,n}}^{2}+\|\partial
g\|_{C^{\kappa,3+\epsilon}}\|F\|_{H^{s,n+5/2+\epsilon}}\|F\|_{H^{1,n}}\\\
&\lesssim\|\partial
g\|_{L^{\infty,\tilde{m}}}\|f\|_{H^{k-2(1-s)+\mu,\mu+n+3/2}}\|f\|_{H^{k,n}}\\\
&\quad+\|\partial^{2}g\|_{H^{3-s,3+\epsilon}}\|f\|_{H^{k,n}}^{2}+\|\partial
g\|_{C^{\kappa,3+\epsilon}}\|f\|_{H^{k-(1-s),n+5/2+\epsilon}}\|f\|_{H^{k,n}}.\end{split}$

Applying the Sobolev embedding theorem on terms involving $g$ and then Lemma
2.6 (up to increasing $m$ if necessary) yields

$\begin{split}\left|\int\langle v\rangle^{2n}Q_{\rm s}(\partial g,F)\partial
Fdvdx\right|&\lesssim\|g\|_{H^{4+\epsilon,\tilde{m}}}\|f\|_{H^{k-(2-2s)+\mu,\mu+n+3/2}}\|f\|_{H^{k,n}}+\|g\|_{H^{5-s,3+\epsilon}}\|f\|_{H^{k,n}}^{2}\\\
&\quad+\|g\|_{H^{4+\kappa,3+\epsilon}}\|f\|_{H^{k-(1-s),n+5/2+\epsilon}}\|f\|_{H^{k,n}}\lesssim\|g\|_{X^{k,n,m}}\|f\|_{X^{k,n,m}}^{2}.\end{split}$

The estimate of the non-singular part $Q_{{\rm ns}}$ is the same as in the
previous case and is thus omitted.

Case three: $|\alpha^{\prime\prime}|+|\beta^{\prime\prime}|=k-2$ and
$|\alpha^{\prime}|+|\beta^{\prime}|=2$. First, we estimate the $Q_{\rm s}$
term. We see

(3.10) $\begin{split}\int\langle v\rangle^{2n}&Q_{\rm
s}(G,F)\partial^{\alpha}_{x}\partial^{\beta}_{v}f\,dvdx\leq\|Q_{\rm
s}(G,F)\|_{L^{2,n}}\|f\|_{H^{k,n}}\\\ &\leq\left(\|Q_{\rm s}(G,\langle
v\rangle^{n}F)-\langle v\rangle^{n}Q_{\rm s}(G,F)\|_{L^{2}}+\|Q_{\rm
s}(G,\langle v\rangle^{n}F)\|_{L^{2}}\right)\|f\|_{H^{k,n}}\\\
&=:(B_{1}+B_{2})\|f\|_{H^{k,n}}.\end{split}$

We estimate $B_{2}$ first. Fix any
$\theta\in(0,\min\\{\frac{2-2s}{3},\frac{3}{4}\\})$ and let $p=3/(4\theta)$
and $q=\frac{3}{3-4\theta}$. Then we apply Proposition 2.8.(ii), Hölder’s
inequality, and the Sobolev embedding theorem to find

$\begin{split}B_{2}&\lesssim\Big{(}\int\|G\|_{L^{\infty,3}_{v}}^{2}\|F\|_{H^{2s+\theta,n}_{v}}^{2}\,dx\Big{)}^{1/2}\lesssim\|G\|_{L_{x}^{2p}L^{\infty,3}_{v}}\|F\|_{L_{x}^{2q}H^{2s+\theta,n}_{v}}\\\
&\lesssim\|G\|_{H_{x}^{3/2-2\theta}H^{3/2+\theta,3}_{v}}\|F\|_{H_{x}^{2\theta}H^{2s+\theta,n}_{v}}\lesssim\|G\|_{H^{3-\theta,3}}\|F\|_{H^{2s+3\theta,n}}\lesssim\|g\|_{X^{k,n,m}}\|f\|_{X^{k,n,m}}.\end{split}$

The last inequality follows by our choice of $\theta$.

For $B_{1}$, for any $\mu\in((1-2s)_{+},2-2s)$ and
$\tilde{m}=1+\max\\{3,n+\gamma+3/2\\}$, we appeal to our commutator estimate
Proposition 2.7, the Cauchy-Schwarz inequality, and the Sobolev embedding
theorem to obtain:

$\begin{split}&B_{1}\lesssim\left(\int(\|F\|_{L_{v}^{2,n+2}}+\|F\|_{H_{v}^{2s-1+\mu,\mu+n}})^{2}\|G\|_{L_{v}^{\infty,\tilde{m}}}^{2}\,dx\right)^{1/2}\\\
&\lesssim\|G\|_{L_{x}^{4}L_{v}^{\infty,\tilde{m}}}\left(\|F\|_{L_{x}^{4}L_{v}^{2,n+2}}+\|F\|_{L_{x}^{4}H_{v}^{2s-1+\mu,\mu+n}}\right)\lesssim\|G\|_{H^{5/2,\tilde{m}}}\left(\|F\|_{H^{3/4,n+2}}+\|F\|_{H_{v}^{2s-1/4+\mu,\mu+n}}\right)\\\
&\lesssim\|g\|_{H^{9/2,\tilde{m}}}(\|f\|_{H^{k-5/4,n+2}}+\|f\|_{H^{k+2s-9/4+\mu,\mu+n}}).\end{split}$

Notice that $2s-9/4+\mu<0$ as $\mu<2-2s$. With this, observe that all three
norms above involve regularity of order strictly less than $k$. Hence,
assuming $m$ is sufficiently large, the interpolation lemma Lemma 2.6 yields

$\begin{split}B_{1}\lesssim\|g\|_{X^{k,n,m}}\|f\|_{X^{k,n,m}}.\end{split}$

This concludes the estimates for the singular part.

For the non-singular part, we apply Lemma 2.5 to find

$\begin{split}&\int\langle v\rangle^{2n}Q_{\rm
ns}(G,F)\partial^{\alpha}_{x}\partial^{\beta}_{v}f\,dvdx\lesssim\|f\|_{H^{k,n}}\|Q_{\rm
ns}(G,F)\|_{L^{2,n}}\lesssim\|f\|_{H^{k,n}}\|G\|_{L^{\infty,3+\gamma+\epsilon}}\|F\|_{L^{2,n}}.\end{split}$

Using the Sobolev embedding theorem and Lemma 2.6, we obtain the desired
estimate

$\int\langle v\rangle^{2n}Q_{\rm
ns}(G,F)\partial^{\alpha}_{x}\partial^{\beta}_{v}f\,dvdx\lesssim\|g\|_{X^{k,n,m}}\|f\|_{X^{k,n,m}}^{2}.$

This concludes the proof of (3.5) in this case.

Case four: $|\alpha^{\prime\prime}|+|\beta^{\prime\prime}|=k-3$ and
$|\alpha^{\prime}|+|\beta^{\prime}|=3$. The proof of (3.5) in this case is
exactly as in case three, except with the choices

$\theta\in\left(0,\min\left\\{\frac{1}{2},\frac{5-4s}{6}\right\\}\right),\quad
p=\frac{3}{1+4\theta},\quad\text{and}\quad q=\frac{3}{2-4\theta}$

in the estimate of $Q_{\rm s}$. As such, we omit the proof.

Case five: $|\alpha^{\prime\prime}|+|\beta^{\prime\prime}|=k-4$ and
$|\alpha^{\prime}|+|\beta^{\prime}|=4$. We begin with the singular term:

$\begin{split}\int\langle v\rangle^{2n}Q_{\rm
s}(G,F)\partial^{\alpha}_{x}\partial^{\beta}_{v}f\,dvdx&\lesssim\|\partial^{\alpha}_{x}\partial^{\beta}_{v}f\|_{L^{2,n}}\|Q_{\rm
s}(G,F)\|_{L^{2,n}}\leq\|f\|_{H^{k,n}}\|Q_{\rm
s}(G,F)\|_{L^{2,n}}.\end{split}$

It is clear that we need only bound the last term above. Recalling Proposition
2.8.(iii), we have, for any $\mu\in((2s-1)_{+},1)$,

$\begin{split}\|Q_{s}(G,F)\|_{L_{v}^{2,n}}&\lesssim\left(\|F\|_{L_{v}^{\infty,3}}+\|\langle
v\rangle^{n+5/2+\mu}F\|_{C_{v}^{1+\mu}}\right)\|G\|_{L_{v}^{2,n}}.\end{split}$

Applying the Sobolev embedding theorem with $\tilde{m}$ sufficiently large
(depending only on $n$), we obtain, for $\epsilon=(1-\mu)/4$,

$\|Q_{s}(G,F)\|_{L_{v}^{2,n}}\lesssim\|F\|_{H_{v}^{5/2+\mu+\epsilon,\tilde{m}}}\|G\|_{L_{v}^{2,n}}.$

Using Hölder’s inequality and the Sobolev embedding theorem yields

$\begin{split}\|Q_{\rm
s}(G,F)\|_{L^{2,n}}^{2}&\lesssim\int\|F\|_{H_{v}^{5/2+\mu+\epsilon,\tilde{m}}}^{2}\|G\|_{L_{v}^{2,n}}^{2}\,dx\leq\|F\|^{2}_{L_{x}^{3}H_{v}^{5/2+\mu+\epsilon,\tilde{m}}}\|G\|^{2}_{L^{6}_{x}L_{v}^{2,n}}\\\
&\lesssim\|F\|_{H_{x}^{1/2}H_{v}^{5/2+\mu+\epsilon,\tilde{m}}}^{2}\|G\|_{H^{1,n}}^{2}\leq\|g\|_{H^{k,n}}^{2}\|f\|_{H^{k-1+\mu+\epsilon,\tilde{m}}}^{2}.\end{split}$

To control the last term, we use the interpolation lemma, Lemma 2.6 and that,
by construction, $\mu+\epsilon<1$, to find

$\|f\|_{H^{k-1+\mu,\tilde{m}}}\lesssim\|f\|_{X^{k,n,m}}$

as long as $m$ is sufficiently large depending only on $n$ and $s$. This
concludes the proof of the bound of the singular term.

We now consider the non-singular part. As above, it is enough to bound
$\|Q_{\rm ns}(G,F)\|_{L^{2,n}}$. To this end, applying Lemma 2.5 yields

$\begin{split}\|Q_{\rm
ns}(G,F)\|_{L^{2,n}}^{2}&\lesssim\left(\int\|F\|_{L_{v}^{\infty,n+3}}^{2}\|G\|_{L_{v}^{2,n}}^{2}\,dx\right)^{1/2}\lesssim\|F\|_{L^{\infty,n+3}}^{2}\|G\|_{L^{2,n}}^{2}\leq\|F\|_{L^{\infty,n+3}}^{2}\|g\|_{X^{k,n,m}}^{2}.\end{split}$

Thus, we need only bound the norm of $F$ on the right hand side. By the
Sobolev embedding theorem and the interpolation lemma Lemma 2.6, we find

$\|F\|_{L^{\infty,n+3}}\lesssim\|F\|_{H^{7/2+1/4,n+3}}\lesssim\|f\|_{H^{k-1/4,n+3}}\lesssim\|f\|_{X^{k,n,m}},$

as long as $m$ is sufficiently large. This concludes the proof of (3.5) in
case five.

Case six: $|\alpha^{\prime\prime}|+|\beta^{\prime\prime}|\leq k-5$ and
$|\alpha^{\prime}|+|\beta^{\prime}|\geq 5$. We begin by bounding the term with
$Q_{\rm s}$. As above, it is enough to bound $Q_{\rm s}(G,F)$ in $L^{2,n}$.
First, by Proposition 2.8.(iii) with $\mu\in((2s-1)_{+},1)$, we find

$\|Q_{\rm
s}(G,F)\|_{L^{2,n}}^{2}\lesssim\int\|G\|_{L_{v}^{2,n}}^{2}\left(\|F\|_{L_{v}^{\infty,m}}^{2}+\|\langle
v\rangle^{n+5/2+\mu}F\|_{C^{1+\mu}}^{2}\right)\,dx.$

Applying the Sobolev embedding theorem and letting $\epsilon=(1-\mu)/2$, we
obtain

$\begin{split}\|Q_{\rm
s}(G,F)\|_{L^{2,n}}^{2}&\lesssim\int\|G\|_{L_{v}^{2,n}}^{2}\|F\|_{H^{5/2+\mu,\tilde{m}}_{v}}^{2}\,dx\leq\|G\|_{L^{2,n}}^{2}\|F\|_{L^{\infty}_{x}H_{v}^{5/2+\mu,\tilde{m}}}^{2}\\\
&\lesssim\|G\|_{L^{2,n}}^{2}\|F\|_{H^{4+\mu+\epsilon,\tilde{m}}}^{2}\lesssim\|g\|_{H^{k,n}}^{2}\|f\|_{H^{k-1+\mu+\epsilon,\tilde{m}}}^{2},\end{split}$

where $\tilde{m}$ is a constant depending only on $n$. The proof concludes as
in the previous case by using the fact that $k-1+\mu+\epsilon<k$ and Lemma
2.6.

The estimate of the non-singular part $Q_{{\rm ns}}$ is the same as in the
previous case and is thus omitted. This concludes the proof of (3.5) in case
six and, thus, all cases. ∎

Having established the bounds above, we now construct a solution.

######  Proposition 3.2 (Construction of solution in the linear equation).

Fix $T>0$, a function $g\in Y^{k,n,m}_{T}$, and the initial data $0\leq f_{\rm
in}\in X^{k,n,m}$. Then there exists $f\in Y^{k,n,m}_{T}$ such that

(3.11) $\displaystyle f_{t}+v\cdot\nabla_{x}f=Q(g,f)$

and $f(0,\cdot,\cdot)=f_{\rm in}$. Moreover, $f\geq 0$.

###### Proof.

The proof of [26, Proposition 3.3] can be adapted verbatim as it requires only
the established bounds in [26, Proposition 3.2] (the analogue of our
Proposition 3.1). The proof is composed of three steps: (1) due to the
Laplacian in $\mathcal{L}_{\sigma,\epsilon,\delta}$, apply the Schauder
estimates to establish boundedness of a linear operator involving of
$\mathcal{L}_{\sigma,\epsilon,\delta}$; (2) apply the method of continuity to
construct the solution of $\mathcal{L}_{\sigma,\epsilon,\delta}f=0$ using the
bounds from the previous step, and (3) use the a priori estimates from
Proposition 3.1 to deregularize. Due to its similarity to [26, Proposition
3.3], we omit the details. ∎

###### Proof of existence in Theorem 1.1.

The idea used to prove [26, Theorem 2.1] is to construct a sequence $f_{i}$
solving

$(\partial_{t}+v\cdot\nabla_{x})f_{i}=Q(f_{i-1},f_{i}),$

establishing the boundedness of this sequence inductively, and taking the
limit $i\to\infty$. Notice that we have the same bounds in Proposition 3.1 as
in [26, Proposition 3.2], which is the crux of argument. Thus, the proof in
our setting will be unchanged and we omit the details. ∎

###  3.2. Proof of uniqueness in Theorem 1.1

We now finish the proof of Theorem 1.1 by establishing uniqueness.

###### Proof of uniqueness in Theorem 1.1.

Consider any two solutions $f$ and $g$ of (1.1) with
$f(0,\cdot,\cdot)=g(0,\cdot,\cdot)=f_{\rm in}$ and set $h=f-g$. We have

(3.12) $\displaystyle h_{t}+v\cdot\nabla_{x}h=Q(f,h)+Q(h,g).$

Then, we multiply (3.12) by $\langle v\rangle^{2n}h$ and integrate with
respect to $v$ and $x$, yielding

(3.13) $\begin{split}\frac{1}{2}\frac{d}{dt}\|h\|_{L^{2,n}}^{2}&=\int\langle
v\rangle^{2n}Q(f,h)h\,dvdx+\int\langle
v\rangle^{2n}Q(h,g)h\,dvdx=I_{1}+I_{2},\end{split}$

where

(3.14) $\begin{split}I_{1}&=\int\langle v\rangle^{2n}Q_{\rm
s}(f,h)h\,dvdx+\int\langle v\rangle^{2n}Q_{\rm
ns}(f,h)h\,dvdx=I_{11}+I_{12}\end{split}$

and

(3.15) $\begin{split}I_{2}&=\int\langle v\rangle^{2n}Q_{\rm
s}(h,g)h\,dvdx+\int\langle v\rangle^{2n}Q_{\rm
ns}(h,g)h\,dvdx=I_{21}+I_{22}.\end{split}$

For $I_{11}$, Proposition 2.8.(iv) yields, for $\tilde{m}$ sufficiently large,

(3.16)
$I_{11}\lesssim\int\|h\|_{L_{v}^{2,n}}^{2}\|f\|_{L_{v}^{\infty,\tilde{m}}}\,dx\lesssim\|h\|_{L^{2}_{x}L_{v}^{2,n}}^{2}\|f\|_{L^{\infty}_{x}L_{v}^{\infty,\tilde{m}}}\lesssim\|h\|_{L^{2,n}}^{2}\|f\|_{X^{k,n,m}}.$

For $I_{12}$, we apply Lemma 2.5 to obtain

(3.17)
$I_{12}\lesssim\|h\|_{L^{2,n}}^{2}\|f\|_{L^{\infty,3+\gamma+\epsilon}}\lesssim\|h\|_{L^{2,n}}^{2}\|f\|_{X^{k,n,m}}.$

For $I_{21}$, fix $\alpha\in(2s,2)$, $\epsilon=(2-\alpha)/2$, and $\tilde{m}$
be sufficiently large and apply Proposition 2.8.(iii) to find

(3.18)
$\begin{split}I_{21}&\lesssim\int\|h\|_{L_{v}^{2,n}}^{2}\left(\|g\|_{L_{v}^{\infty,m}}+\|\langle
v\rangle^{\tilde{m}}g\|_{C^{\alpha}_{v}}\right)dx\lesssim\|h\|_{L^{2,n}}^{2}\|g\|_{L^{\infty}_{x}H_{v}^{3/2+\alpha,\tilde{m}}}\\\
&\lesssim\|h\|_{L^{2,n}}^{2}\|g\|_{H^{3+\alpha+\epsilon,\tilde{m}}}\lesssim\|h\|_{L^{2,n}}^{2}\|g\|_{X^{k,n,m}}.\end{split}$

For $I_{22}$, apply Lemma 2.5 to find

(3.19)
$\begin{split}I_{22}\lesssim\|h\|_{L^{2,n}}^{2}\|g\|_{L^{\infty,n+3}}\lesssim\|h\|_{L^{2,n}}^{2}\|g\|_{X^{k,n,m}}.\end{split}$

Combining the estimates of $I_{11},I_{12},I_{21}$, and $I_{22}$, that is,
(3.16)-(3.19), and recalling that $\|f\|_{X^{k,n,m}},\|g\|_{X^{k,n,m}}\lesssim
1$, we find

$\frac{d}{dt}\|h(t)\|_{L^{2,n}}^{2}\lesssim\|h(t)\|_{L^{2,n}}^{2}.$

The Grönwall inequality and the fact that $h(0,\cdot,\cdot)=0$ implies that
$h=0$. We deduce that $f=g$, concluding the proof. ∎

##  4\. Proof of the estimates on the collision operator $Q$

###  4.1. Proof of the refined estimate on $K_{g}$ Lemma 2.4

###### Proof.

We first show that $|v+w|\approx|v|+|w|$ for any $w\in(v-v^{\prime})^{\perp}$.
The “$\lesssim$” inequality is clear, so we show the other inequality:

$\displaystyle|v+w|^{2}$ $\displaystyle=|v|^{2}+2v\cdot
w+|w|^{2}=|v|^{2}+2v^{\prime}\cdot
w+|w|^{2}\geq|v|^{2}-\frac{1}{1-\theta}|v^{\prime}|^{2}-(1-\theta)|w|^{2}+|w|^{2}$
$\displaystyle\geq|v|^{2}-(1-\theta)|v|^{2}-(1-\theta)|w|^{2}+|w|^{2}=\theta|v|^{2}+\theta|w|^{2}.$

In the second equality, we used that $(v-v^{\prime})\cdot w=0$, in the first
inequality, we used Young’s inequality, and in the second inequality, we used
the hypothesis that $(1-\theta)|v|\geq|v^{\prime}|$.

Recalling (2.2) and changing variables, we have

$|v-v^{\prime}|^{3+2s}K_{g}(v,v^{\prime})\approx\int_{v+(v^{\prime}-v)^{\perp}}g(w)|v-w|^{\gamma+2s+1}\,dw=\int_{(v^{\prime}-v)^{\perp}}g(v+w)|w|^{\gamma+2s+1}\,dw.$

Clearly, it is enough to simply bound the integral on the right hand side.
Using that $|v+w|\approx|v|+|w|$, as established above, we see that

(4.1)
$\begin{split}&\Big{|}\int_{(v^{\prime}-v)^{\perp}}g(v+w)|w|^{\gamma+2s+1}\,dw\Big{|}\lesssim\|g\|_{L^{\infty,m}}\int_{(v^{\prime}-v)^{\perp}}\frac{|v|^{\gamma+2s+1}}{\langle
v+w\rangle^{m}}\,dw\\\
&\quad\lesssim\|g\|_{L^{\infty,m}}\int_{(v^{\prime}-v)^{\perp}}\frac{|w|^{\gamma+2s+1}}{\langle
v\rangle^{m}+\langle w\rangle^{m}}\,dw\\\
&\quad=\|g\|_{L^{\infty,m}}\int_{(v^{\prime}-v)^{\perp}\cap B_{\langle
v\rangle}}\frac{|w|^{\gamma+2s+1}}{\langle v\rangle^{m}+\langle
w\rangle^{m}}\,dw+\|g\|_{L^{\infty,m}}\int_{(v^{\prime}-v)^{\perp}\cap
B^{c}_{\langle v\rangle}}\frac{|w|^{\gamma+2s+1}}{\langle v\rangle^{m}+\langle
w\rangle^{m}}\,dw\\\
&\quad=\|g\|_{L^{\infty,m}}\left(I_{1}+I_{2}\right).\end{split}$

For $I_{1}$, that is, $w\in B_{\langle v\rangle}$, we use the fact that

(4.2) $\frac{|w|^{\gamma+2s+1}}{\langle v\rangle^{m}+\langle
w\rangle^{m}}\lesssim|w|^{\gamma+2s+1}\langle v\rangle^{-m}.$

Thus, we see (recall we are integrating over a subset of a two-dimensional
hyperplane)

(4.3) $\begin{split}I_{1}\lesssim\langle
v\rangle^{-m}\int_{(v^{\prime}-v)^{\perp}\cap B_{\langle
v\rangle}}|w|^{\gamma+2s+1}\,dw\lesssim\langle
v\rangle^{\gamma+2s+3-m}.\end{split}$

For $I_{2}$, that is, $w\in B^{c}_{\langle v\rangle}$, we have

(4.4) $\displaystyle\frac{|w|^{\gamma+2s+1}}{\langle v\rangle^{m}+\langle
w\rangle^{m}}\lesssim\frac{|w|^{\gamma+2s+1}}{\langle
w\rangle^{m}}\lesssim\langle w\rangle^{\gamma+2s+1-m}.$

Therefore, we get (again, recall, we are integrating over a subset of a two-
dimensional hyperplane)

(4.5) $\begin{split}I_{2}\lesssim\int_{(v^{\prime}-v)^{\perp}\cap
B^{c}_{\langle v\rangle}}\langle w\rangle^{\gamma+2s+1-m}\,dw\lesssim\langle
v\rangle^{\gamma+2s+3-m}.\end{split}$

Combining (4.1), (4.3), and (4.5), we obtain the desired inequality,
concluding the proof. ∎

###  4.2. Commutator estimate: proof of Proposition 2.7

Before beginning, we require a helper lemmas concerning the weighted Sobolev
norms. While this result is somewhat elementary, we do not know of a
reference.

######  Lemma 4.1.

For $\tilde{s}\in(0,1)$, $R>0$, $\ell\geq 0$, and
$\mathcal{D}=\\{(v,v^{\prime})\in\mathbb{R}^{6}:|v-v^{\prime}|\leq\langle
v\rangle/R\\}$, we have, for any $f\in H^{\tilde{s},\ell}(\mathbb{R}^{3})$,

(4.6) $\int_{\mathcal{D}}\langle
v\rangle^{\ell}\frac{|f(v)-f(v^{\prime})|^{2}}{|v-v^{\prime}|^{3+2\tilde{s}}}\,dv^{\prime}dv\lesssim\|f\|_{H^{\tilde{s},\ell}}^{2}.$

Before beginning we remark briefly about the content of Lemma 4.1. Recall that
$\|f\|_{\dot{H}^{\tilde{s}}}=\int\frac{|f(v)-f(v^{\prime})|^{2}}{|v-v^{\prime}|^{3+2\tilde{s}}}\,dv^{\prime}dv$
and, hence,

$\|f\|_{H^{\tilde{s},\ell}}^{2}=\|\langle
v\rangle^{\ell}f\|_{H^{\tilde{s}}}^{2}=\|\langle
v\rangle^{\ell}f\|_{L^{2}}^{2}+\int\frac{|\langle v\rangle^{\ell}f(v)-\langle
v^{\prime}\rangle^{\ell}f(v^{\prime})|^{2}}{|v-v^{\prime}|^{3+2\tilde{s}}}dvdv^{\prime}.$

The difference between the quantity above and the left hand side of (4.6) is
now clear.

###### Proof.

To begin, we use the triangle inequality and that $(a+b)^{2}\leq
2a^{2}+2b^{2}$ to find

$\displaystyle\int_{\mathcal{D}}$ $\displaystyle\langle
v\rangle^{\ell}\frac{|f(v)-f(v^{\prime})|^{2}}{|v-v^{\prime}|^{3+2\tilde{s}}}\,dv^{\prime}dv\lesssim\int\limits_{\mathcal{D}}\frac{|\langle
v\rangle^{\ell}f(v)-\langle
v^{\prime}\rangle^{\ell}f(v^{\prime})|^{2}+|\langle v\rangle^{\ell}-\langle
v^{\prime}\rangle^{\ell}|^{2}|f(v^{\prime})|^{2}}{|v-v^{\prime}|^{3+2\tilde{s}}}\,dv^{\prime}dv$
$\displaystyle=\int\limits_{\mathcal{D}}\frac{|\langle
v\rangle^{\ell}f(v)-\langle
v^{\prime}\rangle^{\ell}f(v^{\prime})|^{2}}{|v-v^{\prime}|^{3+2\tilde{s}}}\,dv^{\prime}dv+\int\limits_{\mathcal{D}}\frac{|\langle
v\rangle^{\ell}-\langle
v^{\prime}\rangle^{\ell}|^{2}|f(v^{\prime})|^{2}}{|v-v^{\prime}|^{3+2\tilde{s}}}\,dv^{\prime}dv.$

The first term above is clearly bounded above by
$\|f\|_{\dot{H}^{\tilde{s},\ell}}$ simply by enlarging the domain of
integration. Hence, we consider only the second term.

For $(v,v^{\prime})\in\mathcal{D}$, we find, via Taylor’s theorem, that
$|\langle v\rangle^{\ell}-\langle v^{\prime}\rangle^{\ell}|^{2}\lesssim\langle
v\rangle^{2\ell-2}|v-v^{\prime}|^{2}$. Thus,

$\begin{split}\int\limits_{\mathcal{D}}&\frac{|\langle v\rangle^{\ell}-\langle
v^{\prime}\rangle^{\ell}|^{2}|f(v^{\prime})|^{2}}{|v-v^{\prime}|^{3+2s}}\,dv^{\prime}dv\lesssim\int\limits_{\mathcal{D}}\frac{\langle
v\rangle^{2(\ell-1)}|f(v^{\prime})|^{2}}{|v-v^{\prime}|^{1+2s}}\,dv^{\prime}dv.\end{split}$

Next, clearly there exists $\tilde{R}>0$ depending only on $R$ such that
$\mathcal{D}\subset\\{(v,v^{\prime})\in\mathbb{R}^{6}:|v-v^{\prime}|\leq\langle
v^{\prime}\rangle/\tilde{R}\\}$. Additionally, $(v,v^{\prime})\in\mathcal{D}$
implies that $\langle v\rangle\approx\langle v^{\prime}\rangle$. These two
facts yield

$\begin{split}\int\limits_{\mathcal{D}}&\frac{|\langle v\rangle^{\ell}-\langle
v^{\prime}\rangle^{\ell}|^{2}|f(v^{\prime})|^{2}}{|v-v^{\prime}|^{3+2s}}\,dv^{\prime}dv\lesssim\int\langle
v^{\prime}\rangle^{2(\ell-1)}|f(v^{\prime})|^{2}\int_{B_{\langle
v^{\prime}\rangle/\tilde{R}}(v^{\prime})}\frac{1}{|v-v^{\prime}|^{1+2s}}\,dvdv^{\prime}\\\
&\lesssim\int\langle v^{\prime}\rangle^{2(\ell-1)}|f(v^{\prime})|^{2}\langle
v^{\prime}\rangle^{2-2s}dv^{\prime}\lesssim\int\langle
v^{\prime}\rangle^{2\ell}|f(v^{\prime})|^{2}dv^{\prime}=\|f\|_{L^{2,\ell}},\end{split}$

which concludes the proof. ∎

###### Proof of Proposition 2.7.

We prove this using the characterization of the $L^{2}$-norm via duality; that
is, fix any $h\in L^{2}(\mathbb{R}^{3})$ and we estimate

$\int h\left(\langle v\rangle^{\ell}Q_{\rm s}(g,f)-Q_{\rm s}(g,\langle
v\rangle^{\ell}f)\right)\,dv.$

For any $v$, let $R_{v}=\langle v\rangle/10$ and denote the diagonal strip

(4.7) $\mathcal{D}=\\{(v,v^{\prime}):|v-v^{\prime}|<R_{v}\\}.$

Recalling (2.1), we rewrite the quantity of interest as

(4.8) $\begin{split}\int h&\left(\langle v\rangle^{\ell}Q_{\rm s}(g,f)-Q_{\rm
s}(g,\langle v\rangle^{\ell}f)\right)\,dv=\int
K_{g}(v,v^{\prime})h(v)\left(\langle v\rangle^{l}f(v^{\prime})-\langle
v^{\prime}\rangle^{l}f(v^{\prime})\right)\,dv^{\prime}dv\\\
&=\int\limits_{\mathcal{D}}K_{g}(v,v^{\prime})h(v)\left(\langle
v\rangle^{\ell}f(v^{\prime})-\langle
v^{\prime}\rangle^{\ell}f(v^{\prime})\right)\,dv^{\prime}dv\\\
&\qquad+\int\limits_{\mathcal{D}^{c}}K_{g}(v,v^{\prime})h(v)\left(\langle
v\rangle^{\ell}f(v^{\prime})-\langle
v^{\prime}\rangle^{\ell}f(v^{\prime})\right)\,dv^{\prime}dv=I_{1}+I_{2}.\end{split}$

We estimate each of $I_{1}$ and $I_{2}$ in turn.

Step one: bounding $I_{2}$. We further decompose $I_{2}$ as

(4.9) $I_{2}=\int\limits_{\mathcal{D}^{c}}K_{g}(v,v^{\prime})h(v)\langle
v\rangle^{\ell}f(v^{\prime})\,dv^{\prime}dv-\int\limits_{\mathcal{D}^{c}}K_{g}(v,v^{\prime})h(v)\langle
v^{\prime}\rangle^{\ell}f(v^{\prime})\,dv^{\prime}dv=I_{21}-I_{22}.$

We first consider $I_{22}$. Applying Lemma 2.3, we find

(4.10)
$\begin{split}|I_{22}|&\lesssim\|g\|_{L^{\infty,m}}\int\limits_{\mathcal{D}^{c}}\frac{|h(v)|}{|v-v^{\prime}|^{3+2s}}\langle
v\rangle^{\gamma+2s+1}\langle
v^{\prime}\rangle^{\ell}|f(v^{\prime})|\,dv^{\prime}dv.\end{split}$

Rewriting the limits of integration, using that $|v-v^{\prime}|\gtrsim\langle
v\rangle$, and applying Cauchy-Schwarz in $v^{\prime}$ yields

$\begin{split}\int\limits_{\mathcal{D}^{c}}&\frac{|h(v)|}{|v-v^{\prime}|^{3+2s}}\langle
v\rangle^{\gamma+2s+1}\langle
v^{\prime}\rangle^{\ell}|f(v^{\prime})|\,dv^{\prime}dv=\int\langle
v\rangle^{\gamma+2s+1}|h(v)|\left(\int_{B_{R_{v}}^{c}(v)}\frac{\langle
v^{\prime}\rangle^{\ell}|f(v^{\prime})|}{|v-v^{\prime}|^{3+2s}}dv^{\prime}\right)dv\\\
&\lesssim\int\langle v\rangle^{\gamma-2}h(v)\left(\int\langle
v^{\prime}\rangle^{-3-2\epsilon}dv^{\prime}\right)^{1/2}\left(\int\langle
v^{\prime}\rangle^{\ell+3+2\epsilon}f^{2}(v^{\prime})dv^{\prime}\right)^{1/2}dv.\end{split}$

Noticing that the integral involving $f$ is a weighted $L^{2}$ norm of $f$,
the middle integral is finite, and combining this with (4.10), we obtain

(4.11)
$\begin{split}|I_{22}|&\lesssim\|g\|_{L^{\infty,m}}\|f\|_{L^{2,\ell+1}}\int\langle
v\rangle^{\gamma-\frac{3}{2}}h(v)dv=\|g\|_{L^{\infty,m}}\|f\|_{L^{2,\ell+3/2+\epsilon}}\|h\|_{L^{2}}.\end{split}$

We now consider $I_{21}$. Here, we split the integral as follows:

$\displaystyle I_{21}$
$\displaystyle=\int\limits_{\mathcal{D}^{c}\cap\\{|v^{\prime}|\geq|v|/10\\}}K_{g}(v,v^{\prime})h(v)\langle
v\rangle^{\ell}f(v^{\prime})\,dv^{\prime}dv+\int\limits_{\mathcal{D}^{c}\cap\\{|v^{\prime}|\leq|v|/10\\}}K_{g}(v,v^{\prime})h(v)\langle
v\rangle^{\ell}f(v^{\prime})\,dv^{\prime}dv$ $\displaystyle=I_{211}+I_{212}.$

The estimate of $I_{211}$ reduces to the estimate $I_{22}$:

$\displaystyle|I_{211}|$
$\displaystyle\leq\int\limits_{\mathcal{D}^{c}\cap\\{|v^{\prime}|\geq|v|/10\\}}|K_{g}(v,v^{\prime})h(v)\langle
v\rangle^{\ell}f(v^{\prime})|\,dv^{\prime}dv$
$\displaystyle\lesssim\int\limits_{\mathcal{D}^{c}\cap\\{|v^{\prime}|\geq|v|/10\\}}|K_{g}(v,v^{\prime})h(v)\langle
v^{\prime}\rangle^{\ell}f(v^{\prime})|\,dv^{\prime}dv\leq\int\limits_{\mathcal{D}^{c}}|K_{g}(v,v^{\prime})h(v)\langle
v^{\prime}\rangle^{\ell}f(v^{\prime})|\,dv^{\prime}dv.$

The last term above is exactly the term we estimate in (4.10); hence,

(4.12)
$|I_{211}|\lesssim\|g\|_{L^{\infty,m}}\|f\|_{L^{2,\ell+3/2+\epsilon}}\|h\|_{L^{2}}.$

Turning to $I_{212}$, we get

(4.13)
$\begin{split}|I_{212}|&\lesssim\|g\|_{L^{\infty,m}}\int\limits_{\mathcal{D}^{c}\cap\\{|v^{\prime}|\leq|v|/10\\}}\frac{h(v)\langle
v\rangle^{\ell+\gamma+2s+3-m}f(v^{\prime})}{|v-v^{\prime}|^{3+2s}}\,dv^{\prime}dv.\end{split}$

After applying Cauchy-Schwarz to the integral in $v^{\prime}$, a direct
computation using that $\ell>3/2$ yields

(4.14)
$\begin{split}&\int\limits_{\mathcal{D}^{c}\cap\\{|v^{\prime}|\leq|v|/10\\}}\frac{h(v)\langle
v\rangle^{\ell+\gamma+2s+3-m}f(v^{\prime})}{|v-v^{\prime}|^{3+2s}}\,dv^{\prime}dv\\\
&\qquad\leq\int\langle
v\rangle^{\ell+\gamma+2s+3-m}h(v)\left(\int_{B_{R_{v}}(v)^{c}\cap
B_{|v|/10}}\frac{\langle
v^{\prime}\rangle^{-2\ell}dv^{\prime}}{|v-v^{\prime}|^{6+4s}}\right)^{1/2}\left(\int\langle
v^{\prime}\rangle^{2\ell}f^{2}(v^{\prime})\,dv^{\prime}\right)^{1/2}dv\\\
&\qquad\lesssim\|f\|_{L^{2,\ell}}\int\langle
v\rangle^{\ell+\gamma-m}h(v)dv.\end{split}$

Using that $m>\ell+\gamma+\frac{3}{2}$, we conclude from (4.13) and (4.14)
that

(4.15)
$|I_{212}|\lesssim\|g\|_{L^{\infty,m}}\|f\|_{L^{2,\ell}}\|h\|_{L^{1,\ell+\gamma-m}}\lesssim\|g\|_{L^{\infty,m}}\|f\|_{L^{2,\ell}}\|h\|_{L^{2}}.$

Combining (4.11), (4.12), and (4.15) we deduce that

(4.16) $|I_{2}|\lesssim\|f\|_{L^{2,\ell+1}}\|g\|_{L^{\infty,m}}\|h\|_{L^{2}}.$

This concludes step one.

Step two: bounding $I_{1}$. For notational convenience, let
$W_{\ell}(v)=\langle v\rangle^{\ell}$. For any function $\psi$ and any
velocities $v$ and $v^{\prime}$, let $\delta\psi=\psi(v)-\psi(v^{\prime})$ (we
suppress the dependence on $v$ and $v^{\prime}$ as no confusion will arise).
Then, we rewrite $I_{1}$ as

$\displaystyle I_{1}=\int\limits_{\mathcal{D}}K_{g}(v,v^{\prime})h(v)\delta
f\,\delta
W_{\ell}\,dv^{\prime}dv+\int\limits_{\mathcal{D}}K_{g}(v,v^{\prime})h(v)f(v)\delta
W_{\ell}\,dv^{\prime}dv=I_{11}+I_{12}.$

For $I_{11}$, we see, by the definition of the kernel $K_{g}$ and get that

(4.17)
$I_{11}^{2}\lesssim\|g\|_{L^{\infty,m}}^{2}\left(\int\limits_{\mathcal{D}}\frac{\langle
v\rangle^{\gamma+2s+1}}{|v-v^{\prime}|^{3+2s}}|h(v)||\delta f||\delta
W_{\ell}|\,dv^{\prime}dv\right)^{2}.$

Next, applying the Cauchy-Schwarz inequality yields

(4.18) $\begin{split}&\left(\int\limits_{\mathcal{D}}\frac{\langle
v\rangle^{\gamma+2s+1}}{|v-v^{\prime}|^{3+2s}}|h(v)||\delta f||\delta
W_{\ell}|\,dv^{\prime}dv\right)^{2}\\\
&\quad\leq\left(\int\limits_{\mathcal{D}}\frac{h(v)^{2}}{\langle
v\rangle^{2\mu}}|v-v^{\prime}|^{-3+2\mu}\,dv^{\prime}dv\right)\left(\int\limits_{\mathcal{D}}\langle
v\rangle^{2(\mu+\gamma+2s+1)}\frac{(\delta f)^{2}(\delta
W_{\ell})^{2}}{|v-v^{\prime}|^{3+4s+2\mu}}\,dv^{\prime}dv\right).\end{split}$

We first consider the integral involving $h$. Recalling the definition of
$\mathcal{D}$ (4.7), we find

(4.19)
$\begin{split}\int\limits_{\mathcal{D}}h(v)^{2}|v-v^{\prime}|^{-3+2\mu}\langle
v\rangle^{-2\mu}\,dv^{\prime}dv&\leq\int h(v)^{2}\langle
v\rangle^{-2\mu}\int_{B_{R_{v}}(v)}|v-v^{\prime}|^{-3+2\mu}dv^{\prime}dv\\\
&\lesssim\int h(v)^{2}dv=\|h\|_{L^{2}}^{2}.\end{split}$

Next, we consider the second integral in (4.18). Recall that
$|v-v^{\prime}|<R_{v}$ by the definition of $\mathcal{D}$, (4.7). Hence, by
Taylor’s theorem, we have

$|\delta W_{\ell}|^{2}\lesssim|v^{\prime}-v|^{2}\langle v\rangle^{2\ell-2}.$

Using this and Lemma 4.1, we find

(4.20) $\begin{split}\int\limits_{\mathcal{D}}\frac{\langle
v\rangle^{2(\mu+\gamma+2s+1)}(\delta f)^{2}(\delta
W_{\ell})^{2}}{|v-v^{\prime}|^{3+4s+2\mu}}\,dv^{\prime}dv&\lesssim\int\limits_{\mathcal{D}}\frac{\langle
v\rangle^{2(\mu+\gamma+2s+\ell)}(\delta
f)^{2}}{|v-v^{\prime}|^{3+2(2s-1+\mu)}}\,dv^{\prime}dv\lesssim\|f\|_{H^{2s-1+\mu,\mu+\ell}}^{2}.\end{split}$

We conclude by combining (4.17)-(4.20) to obtain

(4.21)
$|I_{11}|\lesssim\|g\|_{L^{\infty,m}}\|h\|_{L^{2}}\|f\|_{H^{2s-1+\mu,\mu+\ell+\gamma+2s}}$

We consider now $I_{12}$. Using a second order Taylor expansion of
$W_{\ell}(v)=\langle v\rangle^{\ell}$, we see that

$\displaystyle I_{12}$
$\displaystyle=\int\limits_{\mathcal{D}}K_{g}(v,v^{\prime})h(v)f(v)\delta
W_{\ell}\,dv^{\prime}dv$ $\displaystyle=\int
h(v)f(v)\int\limits_{B_{R_{v}(v)}}K_{g}(v,v^{\prime})\left((D_{v}W_{\ell})|_{v}(v-v^{\prime})+\frac{1}{2}(v-v^{\prime})\cdot(D_{v}^{2}W_{\ell})|_{\xi_{v,v^{\prime}}}(v-v^{\prime})\right)\,dv^{\prime}dv$
$\displaystyle=I_{121}+I_{122},$

where $\xi_{v,v^{\prime}}=tv^{\prime}+(1-t)v$ for some $t\in[0,1]$. For
$I_{121}$, we use Lemma 2.1.(iii) to obtain

(4.22) $I_{121}=0.$

For $I_{122}$, we use that
$|(D^{2}_{v}W_{\ell})|_{\xi_{v,v^{\prime}}}|\lesssim\langle
v\rangle^{\ell-2}$, due to the fact that $v^{\prime}\in B_{R_{v}}(v)$, in
order to find

$|I_{122}|\lesssim\int_{\mathbb{R}^{3}}|h(v)f(v)|\int_{B_{R_{v}(v)}}|K_{g}(v,v^{\prime})|\langle
v\rangle^{\ell-2}|v-v^{\prime}|^{2}\,dv^{\prime}dv.$

Thus, we have by appealing to Lemma 2.2

(4.23)
$\begin{split}|I_{122}|&\lesssim\|g\|_{L^{\infty,m}}\int_{\mathbb{R}^{3}}|h(v)f(v)|\langle
v\rangle^{\ell-2+\gamma+2s}\,dv\lesssim\|g\|_{L^{\infty,m}}\|f\|_{L^{2,\ell-2+\gamma+2s}}\|h\|_{L^{2}}.\end{split}$

Combining (4.22) and (4.23) and the fact that
$\ell+3/2+\epsilon>\ell-2+\gamma+2s$, we find

(4.24)
$|I_{12}|\lesssim\|g\|_{L^{\infty,m}}\|f\|_{L^{2,\ell+1}}\|h\|_{L^{2}}.$

Thus, by (4.21) and (4.24),

$|I_{1}|\lesssim\left(\|f\|_{L^{2,\ell+3/2+\epsilon}}+\|f\|_{H^{2s-1+\mu,\mu+\ell+\gamma+2s}}\right)\|g\|_{L^{\infty,m}}\|h\|_{L^{2}}.$

This concludes step two, and, thus, the proof. ∎

###  4.3. Collection of $Q_{\rm s}$ estimates: proof of Proposition
2.8.(i)-(iv)

####  4.3.1. Proof of Proposition 2.8.(i)

###### Proof.

Let

(4.25)
$\hat{K}_{g}(v,v^{\prime})=\frac{1}{\|g\|_{L^{\infty,3+\gamma+2s+\epsilon}}}K_{g}(v,v^{\prime}),$

and we have that $\hat{K}_{g}$ satisfies the conditions (4.2), (4.3), and
(4.4) in [32, Section 4] uniformly in $v$. This allows us to apply their
general estimates, which we do now. For clarity, we adopt their notation as
closely as possible.

Let $\hat{L}_{g}$ be the operator defined by replacing the kernel $K_{g}$ with
$\hat{K}_{g}$ in $Q_{\rm s}$, and let $\hat{L}_{g}^{t}$ be its transpose.
Letting $\Delta_{i}$ be the Littlewood-Paley projectors as in [32, Proof of
Theorem 4.1] and using [32, Theorems 4.3 and 4.6], yields, for any $\theta$,

$\|\hat{L}_{g}\Delta_{i}f\|_{L^{2}}\lesssim
2^{i\theta}\|\Delta_{i}f\|_{H^{2s-\theta}}\quad\text{and}\quad\|\hat{L}_{g}^{t}\Delta_{i}h\|_{L^{2}}\lesssim
2^{i(2s-\theta)}\|\Delta_{i}h\|_{H^{\theta}}.$

Also, recall that $\|\Delta_{i}\phi\|_{H^{\theta}}\approx
2^{i\theta}\|\Delta_{i}\phi\|_{L^{2}}$ for any $\theta$, $i$ and $\phi$.

Using all estimates above for any fixed $\theta\in(0,2s)$ yields

$\begin{split}&\frac{1}{\|g\|_{L^{\infty,3+\gamma+2s+\epsilon}}}\int Q_{\rm
s}(g,f)hdv=\frac{1}{\|g\|_{L^{\infty,3+\gamma+2s+\epsilon}}}\sum_{i,j}\int
Q_{\rm s}(g,\Delta_{i}f)\Delta_{j}hdv\\\ &\qquad=\sum_{i\leq
j}\int(\hat{L}_{g}\Delta_{i}f)\Delta_{j}hdv+\sum_{\theta
i>(2s-\theta)j}\int\Delta_{i}f(\hat{L}_{g}^{t}\Delta_{j}h)dv\\\
&\qquad\lesssim\sum_{\theta i\leq(2s-\theta)j}2^{\theta
i-(2s-\theta)j}\|\Delta_{i}f\|_{H^{2s-\theta}}\|\Delta_{j}g\|_{H^{\theta}}+\sum_{\theta
i>(2s-\theta)j}2^{-\theta
i+(2s-\theta)j}\|\Delta_{i}f\|_{H^{2s-\theta}}\|\Delta_{j}g\|_{H^{\theta}}\\\
&\qquad=\sum_{i,j}2^{-|\theta
i-(2s-\theta)j|}\|\Delta_{i}f\|_{H^{2s-\theta}}\|\Delta_{j}g\|_{H^{\theta}}\\\
&\qquad\leq\Big{(}\sum_{i,j}2^{-|\theta
i-(2s-\theta)j|}\|\Delta_{i}f\|^{2}_{H^{2s-\theta}}\Big{)}^{1/2}\Big{(}\sum_{i,j}2^{-|\theta
i-(2s-\theta)j|}\|\Delta_{j}g\|^{2}_{H^{\theta}}\Big{)}^{1/2}\lesssim\|f\|_{H^{2s-\theta}}\|g\|_{H^{\theta}}.\end{split}$

In the last inequality, we sum first over $j$, using that $\theta,2s-\theta>0$
by assumption, and then recalling that
$\sum_{i}\|\Delta_{i}f\|^{2}_{H^{2s-\theta}}\approx\|f\|_{H^{2s-\theta}}^{2}$
(and similarly for $g$). ∎

####  4.3.2. Proof of Proposition 2.8.(ii)

###### Proof.

We adopt the notation and setting of the proof of Proposition 2.8.(i). Then

$\begin{split}\frac{1}{\|g\|^{1/2}_{L^{\infty,3+\gamma+2s+\epsilon}}}\|Q_{\rm
s}(g,f)\|_{L^{2}}&=\|\hat{L}_{g}f\|_{L^{2}}\leq\sum_{i=0}^{\infty}\|\hat{L}_{g}\Delta_{i}f\|_{L^{2}}\lesssim\sum_{i=0}^{\infty}2^{-i\theta}\|\Delta_{i}f\|_{H^{2s+\theta}}\\\
&\lesssim\Big{(}\sum_{i=0}^{\infty}\|\Delta_{i}f\|^{2}_{H^{2s+\theta}}\Big{)}^{1/2}\approx\|f\|_{H^{2s+\theta}}.\end{split}$

∎

####  4.3.3. Proof of Proposition 2.8.(iii)

In order to establish part (iii) of Proposition 2.8, we require an analogue of
Young’s convolution inequality in the setting of the weighted Lebesgue spaces
in order to handle terms of the form $\int g(w)|v-w|^{\gamma+2s}dw$. These
have been well-studied and are understood in some generality (see, e.g.,
[22]). However, for the convenience of the reader and because we can get a
slightly sharper estimate (due to the specific form considered here), we
include the proof.

######  Lemma 4.2 (Weighted Young’s inequality).

Suppose that $n>3/2+\eta$, $-3<\eta<0$, and $\ell>3/2+\eta+(3/2-n)_{+}$. If
$g\in L^{2,n}$, then

(4.26) $\int\langle v\rangle^{-2\ell}\left(\int
g(\tilde{v})|v-\tilde{v}|^{\eta}\,d\tilde{v}\right)^{2}\,dv\lesssim\|g\|_{L^{2,n}}^{2}.$

###### Proof.

For succinctness, we let $A(v)=|v|^{\eta}$ and, without loss of generality we
assume that $g\geq 0$. First, we decompose the integral on the left hand side
yielding

$\begin{split}\int\langle v\rangle^{-2\ell}(g*A)^{2}\,dv&\leq\int\langle
v\rangle^{-2\ell}\Big{(}\int_{B_{R_{v}}(v)}g(v^{\prime})A(v-v^{\prime})\,dv^{\prime}\Big{)}^{2}\,dv\\\
&\quad+\int\langle
v\rangle^{-2\ell}\Big{(}\int_{B_{R_{v}}^{c}(v)}g(v^{\prime})A(v-v^{\prime})\,dv^{\prime}\Big{)}^{2}\,dv=I_{1}+I_{2}.\end{split}$

For $I_{1}$, we use Cauchy-Schwarz inequality to obtain

$\begin{split}I_{1}&\leq\int\langle
v\rangle^{-2\ell}\Big{(}\int_{B_{|v|/10}(v)}g(v^{\prime})^{2}A(v-v^{\prime})\,dv^{\prime}\Big{)}\Big{(}\int_{B_{|v|/10}(v)}A(v-v^{\prime})\,dv^{\prime}\Big{)}\,dv\\\
&\lesssim\int\langle
v\rangle^{-2\ell+3+\eta}\Big{(}\int_{B_{|v|/10}(v)}g(v^{\prime})^{2}A(v-v^{\prime})\,dv^{\prime}\Big{)}\,dv.\end{split}$

For $v^{\prime}\in B_{|v|/10}(v)$, we have $\langle
v^{\prime}\rangle\approx\langle v\rangle$ and $v\in
B_{|v^{\prime}|/2}(v^{\prime})$. Therefore,

$\begin{split}&I_{1}\lesssim\int|g(v^{\prime})|^{2}\int_{B_{|v^{\prime}|/2}(v^{\prime})}\langle
v\rangle^{-2\ell+3+\eta}A(v-v^{\prime})\,dv\,dv^{\prime}\\\
&\lesssim\int|g(v^{\prime})|^{2}\langle
v^{\prime}\rangle^{-2\ell+3+\eta}\int_{B_{|v^{\prime}|/2}(v^{\prime})}A(v-v^{\prime})\,dv\,dv^{\prime}\lesssim\int
g(v^{\prime})^{2}\langle
v^{\prime}\rangle^{-2\ell+2(3+\eta)}\,dv^{\prime}\lesssim\|g\|_{L^{2,n}}^{2},\end{split}$

where we used that $-\ell+(3+\eta)\leq n$. For $I_{2}$, we apply the Cauchy-
Schwarz inequality to find

$\begin{split}I_{2}&\leq\int\langle
v\rangle^{-2\ell}\Big{(}\int_{B^{c}_{|v|/10}(v)}\langle
v^{\prime}\rangle^{2n}|g(v^{\prime})|^{2}\,dv^{\prime}\Big{)}\Big{(}\int_{B^{c}_{|v|/10}(v)}\langle
v^{\prime}\rangle^{-2n}|v-v^{\prime}|^{2\eta}\,dv^{\prime}\Big{)}\,dv\\\
&\lesssim\|g\|_{L^{2,n}}^{2}\int\langle
v\rangle^{-2\ell}\Big{(}\int_{B^{c}_{|v|/10}(v)}\langle
v^{\prime}\rangle^{-2n}|v-v^{\prime}|^{2\eta}\,dv^{\prime}\Big{)}\,dv\lesssim\|g\|_{L^{2,n}}^{2}\int\langle
v\rangle^{-2\ell+(3-2n)_{+}+2\eta}\,dv.\end{split}$

We conclude by using the conditions on $n$ and $\ell$. These were also used in
the last inequality. Combining the estimates of $I_{1}$ and $I_{2}$ finishes
the proof. ∎

We are now able to prove Proposition 2.8.(iii).

###### Proof of Proposition 2.8.(iii).

The proof is somewhat close to that of [26, Proposition 3.1.(i)], so we omit
details where steps are similar. We may, without loss of generality, assume
that $\alpha\in(0,1)\cup(1,2)$. If not, we may simply take
$\alpha^{\prime}<\alpha$ such that $\alpha^{\prime}\in(0,1)\cup(1,2)$ and use
that $C^{\alpha}\xhookrightarrow{}C^{\alpha^{\prime}}$. Finally, the proof is
simpler when $\alpha<1$; hence, we consider only the case $\alpha\in(1,2)$.

We begin with an annular decomposition: let
$A_{k}(v)=B_{2^{k}|v|}(v)\backslash B_{2^{k-1}|v|}(v)$ and write:

(4.27) $Q_{\rm
s}(g,f)=\sum_{k\in\mathbb{Z}}\int_{A_{k}(v)}K_{g}(v,v^{\prime})(f(v^{\prime})-f(v))\,dv^{\prime}.$

Let $\bar{\mu}=n+5/2+(3/2-n)_{+}+\alpha+\gamma+\epsilon$.

Step One: estimating the sum for any $k\leq 1$. By using a Taylor expansion,
we see

$f(v^{\prime})-f(v)=((Df)(\xi_{v^{\prime},v})-(Df)(v))\cdot(v^{\prime}-v)+(Df)(v)\cdot(v^{\prime}-v),$

where $\xi_{v,v^{\prime}}=tv^{\prime}+(1-t)v$ for some $t\in[0,1]$. Thus, by
Lemma 2.1.(i) and (iii),

$\begin{split}&\Big{|}\int\limits_{A_{k}(v)}K_{g}(v,v^{\prime})(f(v^{\prime})-f(v))\,dv^{\prime}\Big{|}\lesssim\langle
v\rangle^{-\bar{\mu}}(2^{k}|v|)^{\alpha-2s}\|\langle\cdot\rangle^{\bar{\mu}}Df\|_{C^{\alpha}}\int|g(v^{\prime})||v-v^{\prime}|^{\gamma+2s}\,dv^{\prime}.\end{split}$

Recalling that $\alpha-2s>0$, by assumption, we have that
$|v|^{\alpha-2s}\leq\langle v\rangle^{\alpha-2s}$. Hence,

$\displaystyle\int\Big{(}\int_{A_{k}(v)}\langle
v\rangle^{n}K_{g}(v,v^{\prime})(f(v^{\prime})-f(v))\,dv^{\prime}\Big{)}^{2}\,dv$
$\displaystyle\quad\lesssim
2^{2k(1+\alpha-2s)}\|\langle\cdot\rangle^{\bar{\mu}}Df\|_{C^{\alpha}}^{2}\int\langle
v\rangle^{-2(\bar{\mu}-n-1-\alpha+2s)}\left(\int|g(v^{\prime})||v-v^{\prime}|^{\gamma+2s}\,dv^{\prime}\right)^{2}\,dv.$

We are now in a position to apply the weighted Young’s convolution inequality
Lemma 4.2. Indeed, by construction, $\ell:=\bar{\mu}-n-1-\alpha+2s$ and $n$
satisfy the conditions of Lemma 4.2 so that

$\int\Big{(}\int_{A_{k}(v)}\langle
v\rangle^{n}K_{g}(v,v^{\prime})(f(v^{\prime})-f(v))\,dv^{\prime}\Big{)}^{2}\,dv\lesssim
2^{2k(1+\alpha-2s)}\|\langle\cdot\rangle^{\bar{\mu}}Df\|_{C^{\alpha}}^{2}\|g\|_{L^{2,n}}^{2}.$

Step Two: estimating the sum for $k\geq 0$ when $|v^{\prime}|\geq\langle
v\rangle/2$. By Lemma 2.1.(i),

$\begin{split}\Big{|}\int\limits_{A_{k}\backslash B_{\langle
v\rangle/2}}&K_{g}(v,v^{\prime})(f(v^{\prime})-f(v))\,dv^{\prime}\Big{|}\lesssim\langle
v\rangle^{-m}\|f\|_{L^{\infty,m}}\int\limits_{A_{k}\backslash B_{\langle
v\rangle/2}}|K_{g}(v,v^{\prime})|\,dv^{\prime}\\\ &\lesssim\langle
v\rangle^{m}\|f\|_{L^{\infty,m}}(2^{k}\langle
v\rangle)^{-2s}\left(\int|g(v^{\prime})||v-v^{\prime}|^{\gamma+2s}\,dv^{\prime}\right).\end{split}$

Then, similar to Step One, we apply Lemma 4.2 to obtain

$\begin{split}&\int\langle v\rangle^{2n}\left|\sum_{k\geq
0}\int_{A_{k}\backslash B_{\langle
v\rangle/2}}K_{g}(v,v^{\prime})(f(v^{\prime})-f(v))\,dv^{\prime}\right|^{2}\,dv\\\
&\lesssim\|f\|_{L^{\infty,m}}^{2}\Big{(}\sum_{k\geq
0}2^{-2ks}\Big{)}^{2}\int\langle
v\rangle^{2(n-m-2s)}\left(\int|g(v^{\prime})||v-v^{\prime}|^{\gamma+2s}\,dv^{\prime}\right)^{2}\,dv\lesssim\|f\|_{L^{\infty,m}}^{2}\|g\|_{L^{2,n}}^{2},\end{split}$

where we used $n>3/2+\gamma+2s$ and $m>3/2+\gamma+(3/2-n)_{+}$.

Step Three: estimating the sum for $k\geq 0$ when $|v|\leq 10$ and
$|v^{\prime}|\leq\langle v\rangle/2$. This is similar to Step One. The benefit
is we are integrating over a compact set in $v$. As such, we omit the proof
and simply state that

$\displaystyle\int_{B_{10}}\Bigg{(}$ $\displaystyle\sum_{k\geq
0}\int_{A_{k}\cap B_{\langle v\rangle/2}}\langle
v\rangle^{n}K_{g}(v,v^{\prime})(f(v^{\prime})-f(v))\,dv^{\prime}\Bigg{)}^{2}\,dv$
$\displaystyle=\int_{B_{10}}\Bigg{(}\int_{B_{\langle v\rangle/2}\setminus
B_{|v|/2}}\langle
v\rangle^{n}K_{g}(v,v^{\prime})(f(v^{\prime})-f(v))\,dv^{\prime}\Bigg{)}^{2}\,dv\lesssim\|Df\|_{C^{\alpha}}^{2}\|g\|_{L^{2,n}}^{2}.$

Hence, we proved Step Three.

Step Four: estimating the sum for $k\geq 0$ when $|v|\geq 10$ and
$|v^{\prime}|\leq\langle v\rangle/2$. For any $|v|\geq 10$,

$\begin{split}\Big{|}\sum_{k\geq 0}&\int_{A_{k}\cap B_{\langle
v\rangle/2}}K_{g}(v,v^{\prime})(f(v^{\prime})-f(v))\,dv^{\prime}\Big{|}\\\
&\lesssim\int_{B_{\langle
v\rangle/2}}K_{|g|}(v,v^{\prime})|f(v^{\prime})|\,dv^{\prime}+\int_{B_{\langle
v\rangle/2}}K_{|g|}(v,v^{\prime})|f(v)|\,dv^{\prime}=I_{1}+I_{2}.\end{split}$

For $I_{2}$, we notice that $B_{\langle v\rangle/2}\subseteq(B_{2\langle
v\rangle}(v)\backslash B_{\langle v\rangle/4}(v))$ due to the fact that
$|v|\geq 10$. Then by Lemma 2.1.(i), we have

$\begin{split}\int_{B_{\langle
v\rangle/2}}K_{|g|}(v,v^{\prime})\,dv^{\prime}&\lesssim\int_{B_{2\langle
v\rangle}(v)\backslash B_{\langle
v\rangle/4}(v)}K_{|g|}(v,v^{\prime})\,dv^{\prime}\lesssim\langle
v\rangle^{-2s}\int|g(v^{\prime})||v-v^{\prime}|^{\gamma+2s}\,dv^{\prime}.\end{split}$

Applying the weighted Young’s inequality Lemma 4.2 yields

$\begin{split}\int\langle v\rangle^{2n}I_{2}^{2}\,dv&\lesssim\int\langle
v\rangle^{2(n-2s)}|f(v)|^{2}\left(\int|g(v^{\prime})||v-v^{\prime}|^{\gamma+2s}\,dv^{\prime}\right)^{2}\,dv\\\
&\lesssim\|f\|_{L^{\infty,m}}^{2}\int\langle
v\rangle^{-2m+2n-4s}\left(\int|g(v^{\prime})||v-v^{\prime}|^{\gamma+2s}\,dv^{\prime}\right)^{2}\,dv\lesssim\|f\|_{L^{\infty,m}}^{2}\|g\|_{L^{2,n}}^{2},\end{split}$

as desired. For $I_{1}$, the proof is omitted as it is exactly as in [26,
Proposition 3.1.(i)]. This finishes the proof. ∎

####  4.3.4. Proof of Proposition 2.8.(iv)

###### Proof.

Without loss of generality, we assume that $f,g\geq 0$. Let $F=\langle
v\rangle^{n}f(v)$. We see

$\begin{split}\int\langle v\rangle^{2n}Q_{s}(g,f)f\,dv=\int
FQ_{s}(g,F)\,dv+\int F[\langle
v\rangle^{n}Q_{s}(g,f)-Q_{s}(g,F)]\,dv=I_{1}+I_{2}.\end{split}$

We further decompose $I_{1}$ into three parts:

$\begin{split}I_{1}&=-\int[F(v)-F(v^{\prime})]^{2}K_{g}(v,v^{\prime})\,dv^{\prime}dv+\int\int[K_{g}(v,v^{\prime})-K_{g}(v^{\prime},v)]F(v)F(v^{\prime})\,dv^{\prime}\,dv\\\
&\qquad-\int[K_{g}(v,v^{\prime})-K_{g}(v^{\prime},v)]F(v^{\prime})^{2}\,dv^{\prime}\,dv=I_{11}+I_{12}+I_{13}.\end{split}$

The first term, $I_{11}$, has a good sign (and is used for cancellation
below). The integrand in $I_{12}$ is antisymmetric with respect to the “pre-
post change of variables” $(v,v^{\prime})\mapsto(v^{\prime},v)$, so
$I_{12}=0$. To estimate $I_{13}$, we use Lemma 2.1.(ii). Hence, we find

$|I_{12}|\lesssim\int F(v^{\prime})\int
g(z)|z-v^{\prime}|^{\gamma+2s}dzdv^{\prime}\lesssim\|g\|_{L^{\infty,m}}\|F\|_{L^{2}}=\|g\|_{L^{\infty,m}}\|f\|_{L^{2,n}}.$

Here we used that $m>3+\gamma+2s$ and $\gamma+2s\leq 0$. This concludes the
bound on $I_{1}$.

For $I_{2}$, we apply Young’s inequality to find

$\begin{split}&I_{2}=\int F(v)f(v^{\prime})K_{g}(v,v^{\prime})(\langle
v\rangle^{n}-\langle v^{\prime}\rangle^{n})\,dv^{\prime}dv\\\
&=\int(F(v)-F(v^{\prime}))f(v^{\prime})K_{g}(v,v^{\prime})(\langle
v\rangle^{n}-\langle v^{\prime}\rangle^{n})\,dv^{\prime}dv+\int
F(v^{\prime})f(v^{\prime})K_{g}(v,v^{\prime})(\langle v\rangle^{n}-\langle
v^{\prime}\rangle^{n})\,dv^{\prime}dv\\\
&\leq-\frac{1}{2}I_{11}+\frac{1}{2}\int
f^{2}(v^{\prime})K_{g}(v,v^{\prime})(\langle v\rangle^{n}-\langle
v^{\prime}\rangle^{n})^{2}\,dv^{\prime}dv+\int
F(v^{\prime})f(v^{\prime})K_{g}(v,v^{\prime})(\langle v\rangle^{n}-\langle
v^{\prime}\rangle^{n})\,dv^{\prime}dv.\end{split}$

Define the last two integrals to be $I_{21}$ and $I_{22}$. The argument for
$I_{21}$ is similar to and easier than $I_{22}$; hence, we omit it.

We now bound $I_{22}$. To do so, we split the integral into domains of
integration $\mathcal{D}$, $\mathcal{D}^{c}\cap\\{|v|\leq 10|v^{\prime}|\\}$,
and $\mathcal{D}^{c}\cap\\{|v|\geq 10|v^{\prime}|\\}$, where
$\mathcal{D}=\\{(v,v^{\prime}):10|v-v^{\prime}|\leq\min\\{\langle
v\rangle,\langle v^{\prime}\rangle\\}\\}$. We denote the resulting integrals
$I_{221},$ $I_{222}$, and $I_{223}$, respectively.

Considering $I_{221}$ first, we use a Taylor expansion, Lemma 2.1.(iv), Lemma
2.2, and the fact that $\langle v\rangle\approx\langle v^{\prime}\rangle$ to
find, for $\xi$ between $v$ and $v^{\prime}$

$\begin{split}|I_{221}|&\leq\int\frac{F(v^{\prime})^{2}}{\langle
v^{\prime}\rangle}\int\limits_{B_{\langle
v^{\prime}\rangle/2}(v^{\prime})}K_{g}(v,v^{\prime})\left[(v-v^{\prime})\cdot
v^{\prime}n\langle
v^{\prime}\rangle^{n-2}+\frac{n\langle\xi\rangle^{n-2}}{2}(v-v^{\prime})\cdot\left(\operatorname{Id}+\frac{\xi\otimes\xi}{|\xi|^{2}}\right)(v-v^{\prime})\right]dvdv^{\prime}\\\
&\lesssim\int\frac{F(v^{\prime})^{2}}{\langle
v^{\prime}\rangle}\Big{|}\int\limits_{B_{\langle
v^{\prime}\rangle/2}(v^{\prime})}K_{g}(v,v^{\prime})(v-v^{\prime})dv\Big{|}dv^{\prime}+\int\frac{F(v^{\prime})^{2}}{\langle
v^{\prime}\rangle^{2}}\int\limits_{B_{\langle
v^{\prime}\rangle/2}(v^{\prime})}K_{g}(v,v^{\prime})|v-v^{\prime}|^{2}dvdv^{\prime}\\\
&\lesssim\int\frac{F(v^{\prime})^{2}}{\langle v^{\prime}\rangle}\int
g(w)|v^{\prime}-w|^{1+\gamma}dwdv^{\prime}+\int\frac{F(v^{\prime})^{2}}{\langle
v^{\prime}\rangle^{2s}}\int
g(w)|v^{\prime}-w|^{\gamma+2s}dwdv^{\prime}\lesssim\|g\|_{L^{\infty,m}}\|f\|_{L^{2,n}}^{2}.\end{split}$

Above, we used that $m>3+\gamma+2s$.

Next we consider $I_{222}$. In this case $\langle v\rangle\lesssim\langle
v^{\prime}\rangle$; hence, using that Lemma 2.1.(i) and that $m>3+\gamma+2s$
yields

$|I_{222}|\lesssim\int F(v^{\prime})^{2}\int\limits_{B_{\langle
v^{\prime}\rangle/2}^{c}(v^{\prime})}K_{g}(v,v^{\prime})dvdv^{\prime}\lesssim\int\frac{F(v^{\prime})^{2}}{\langle
v^{\prime}\rangle^{2s}}\int
g(w)|v^{\prime}-w|^{\gamma+2s}dwdv^{\prime}\lesssim\|g\|_{L^{\infty,m}}\|f\|_{L^{2,n}}^{2}.$

Finally, we handle $I_{223}$. Indeed, we use that $\langle
v^{\prime}\rangle\lesssim\langle v\rangle$, the definition of $\mathcal{D}$,
and Lemma 2.4 to get

$\begin{split}\Big{|}\int\limits_{\mathcal{D}^{c}\cap\\{|v|\geq
10|v^{\prime}|\\}}&F(v^{\prime})f(v^{\prime})K_{g}(v,v^{\prime})(\langle
v\rangle^{n}-\langle v^{\prime}\rangle^{n})\,dv^{\prime}dv\Big{|}\\\
&\lesssim\int\limits_{\mathcal{D}^{c}\cap\\{|v|\geq
10|v^{\prime}|\\}}F(v^{\prime})f(v^{\prime})K_{g}(v,v^{\prime})\langle
v\rangle^{n}\,dv^{\prime}dv\\\
&\lesssim\int\limits_{\mathcal{D}^{c}\cap\\{|v|\geq
10|v^{\prime}|\\}}F(v^{\prime})f(v^{\prime})\frac{|v-v^{\prime}|^{3+2s}}{\langle
v^{\prime}\rangle^{3+2s}}K_{g}(v,v^{\prime})\langle
v\rangle^{n}\,dv^{\prime}dv\\\ &\lesssim\|g\|_{L^{\infty,m}}\int
F(v^{\prime})f(v^{\prime})\int\limits_{\\{10|v^{\prime}|\leq|v|\\}}\langle
v\rangle^{-m+3+\gamma+2s+n}dvdv^{\prime}\lesssim\|g\|_{L^{\infty,m}}\|f\|_{L^{2,n}}^{2}.\end{split}$

In the last inequality, we used that $m>n+6+\gamma+2s$. This concludes the
proof. ∎

###  4.4. Proof of Proposition 2.9

In order to prove Proposition 2.9, we first state a useful estimate that
follows from work in [32].

######  Lemma 4.3.

For any measurable $g$, if $\gamma+2s\leq 0$ and $\epsilon>0$, then

(4.28) $\begin{split}\Big{|}\int
K_{g}(f^{\prime}-f)^{2}\,dv^{\prime}dv\Big{|}\lesssim\|g\|_{L^{\infty,3+\gamma+2s+\epsilon}}\|f\|_{H^{s}}^{2}.\end{split}$

###### Proof.

Recall that $\hat{K}_{g}$, defined in (4.25), satisfies the conditions (4.2),
(4.3), and (4.4) in [32, Section 4] uniformly in $v$. Thus, applying [32,
Lemma 4.2], we find

$\begin{split}\Big{|}\int
K_{g}(f^{\prime}-f)^{2}\,dv^{\prime}dv\Big{|}=\|g\|_{L^{\infty,3+\gamma+2s+\epsilon}}\Big{|}\int\hat{K}_{g}(v,v^{\prime})(f^{\prime}-f)^{2}\,dv^{\prime}dv\Big{|}\lesssim\|g\|_{L^{\infty,3+\gamma+2s+\epsilon}}\|f\|_{H^{s}}^{2},\end{split}$

which concludes the proof. ∎

Now we prove Proposition 2.9.

###### Proof of Proposition 2.9.

We consider only the case $\partial=\partial_{v_{i}}$ for $i\in\\{1,2,3\\}$.
The case when $\partial=\partial_{x_{i}}$ is similar and simpler as it
commutes with $\langle v\rangle^{2n}$. First, let $F=\langle v\rangle^{n}f$.
Then

$\begin{split}\int\langle v\rangle^{2n}Q_{\rm s}(g,f)\partial
f\,dvdx&=\int[\langle v\rangle^{n}Q_{\rm s}(g,f)-Q_{\rm s}(g,\langle
v\rangle^{n}f)]\langle v\rangle^{n}\partial f\,dvdx-\int Q_{\rm
s}(g,F)fnv_{i}\langle v\rangle^{n-2}\,dvdx\\\ &\quad+\int Q_{\rm
s}(g,F)\partial F\,dvdx=I_{1}+I_{2}+I_{3}.\end{split}$

For $I_{1}$, we apply the commutator estimate Proposition 2.7 to get

(4.29)
$\begin{split}|I_{1}|&\lesssim\int(\|f\|_{L_{v}^{2,n+3/2+\epsilon}}+\|f\|_{H_{v}^{2s-1+\mu,\mu+n+\gamma+2s}})\|g\|_{L_{v}^{\infty,m}}\|\partial
f\|_{L_{v}^{2,n}}\,dx\\\
&\lesssim\int(\|f\|_{L_{v}^{2,n+3/2+\epsilon}}+\|f\|_{H_{v}^{2s-1+\mu,\mu+n+\gamma+2s}})\|g\|_{L_{v}^{\infty,m}}\|f\|_{H_{v}^{1,n}}\,dx\\\
&\lesssim\|g\|_{L^{\infty,m}}\left(\|f\|_{L^{2,n+3/2+\epsilon}}+\|f\|_{H^{2s-1+\mu,\mu+n+\gamma+2s}}\right)\|f\|_{H^{1,n}}.\end{split}$

To estimate $I_{2}$, we apply Proposition 2.8.(i) with $\theta=1$ if $s>1/2$
or $\theta=2s-1+\mu$ if $s\leq 1/2$ to find

(4.30)
$\begin{split}|I_{2}|&\lesssim\int\|g\|_{L_{v}^{\infty,m}}\|F\|_{H_{v}^{\theta}}\|fnv_{i}\langle
v\rangle^{n-2}\|_{H_{v}^{2s-\theta}}\,dx\\\
&\lesssim\int\|g\|_{L_{v}^{\infty,m}}\|f\|_{H_{v}^{\theta,n}}\|f\|_{H_{v}^{2s-\theta,n-1}}\,dx\lesssim\|g\|_{L^{\infty,m}}\|f\|_{H^{\theta,n}}\|f\|_{H^{2s-\theta,n-1}}.\end{split}$

Using the choice of $\theta$, the right hand side above is less than or equal
to (up to a constant) the right hand side of (4.29).

We decompose $I_{3}$ into two parts:

$\begin{split}I_{3}&=\int Q_{\rm s}(g,F)\partial F\,dvdx\\\ &=\int
K_{g}(F^{\prime}-F)(\partial F-(\partial
F)^{\prime})\,dv^{\prime}dvdx+\int(K_{g}-K^{\prime}_{g})(F^{\prime}-F)\partial
F\,dv^{\prime}dvdx=I_{31}+I_{32}.\end{split}$

For $I_{31}$, we manipulate by integration-by-parts and apply Lemma 4.3 to
find

$\begin{split}|I_{31}|&=\Big{|}\int
K_{g}(\partial+\partial^{\prime})(F^{\prime}-F)^{2}\,dv^{\prime}dvdx\Big{|}=\Big{|}\int(\partial+\partial^{\prime})K_{g}(F^{\prime}-F)^{2}\,dv^{\prime}dvdx\Big{|}\\\
&=\Big{|}\int K_{\partial
g}(F^{\prime}-F)^{2}\,dv^{\prime}dvdx\Big{|}\lesssim\int\|\partial
g\|_{L_{v}^{\infty,3+\gamma+2s+\epsilon}}\|F\|_{H_{v}^{s}}^{2}dx=\int\|\partial
g\|_{L_{v}^{\infty,3+\gamma+2s+\epsilon}}\|f\|_{H_{v}^{s,n}}^{2}dx.\end{split}$

Fix the conjugate exponents $p=3/2(1-s)$ and $q=3/(2s+1)$. Applying Hölder’s
inequality and the Sobolev embedding theorem yields

$\begin{split}|I_{31}|&\lesssim\|\partial
g\|_{L^{p}_{x}L_{v}^{\infty,3+\gamma+2s+\epsilon}}\|f\|_{L^{2q}_{x}H_{v}^{s,n}}^{2}\\\
&\lesssim\|\partial
g\|_{H^{(2s-1/2)_{+}}_{x}H_{v}^{3/2+\epsilon,3+\gamma+2s+\epsilon}}\|f\|_{H^{1-s}_{x}H_{v}^{s,n}}^{2}\lesssim\|\partial
g\|_{H^{3/2+\epsilon+(2s-1/2)_{+},3+\gamma+2s+\epsilon}}\|f\|_{H^{1,n}}^{2}.\end{split}$

The term $I_{32}$ is considered in [26, Proposition 3.1.(iv), estimate of
$I_{2}$]. A close inspection of the proof shows that it applies in our
setting. Hence, for simplicity, we cite directly that, for any
$\mu\in(s,\min\\{2s,1\\})$,

$\displaystyle|I_{32}|\lesssim\|g\|_{C^{\mu,3+\epsilon}}\|f\|_{H^{s,n+3/2+\epsilon+(\gamma+2s+1)_{+}}}\|f\|_{H^{1,n}}.$

Combining the above estimates of $I_{31}$ and $I_{32}$ together yields

(4.31) $\begin{split}|I_{3}|\lesssim\|\partial
g\|_{H^{3-s,3}}\|f\|_{H^{1,n}}^{2}+\|g\|_{C^{\mu,3+\epsilon}}\|f\|_{H^{s,n+3/2+\epsilon+(\gamma+2s+1)_{+}}}\|f\|_{H^{1,n}}.\end{split}$

The proof is finished after combining (4.29), (4.30), and (4.31). ∎

##  5\. A simple proof of local well-posedness when $0<s<1/2$: Theorem 1.2

Here we provide a short proof of local well-posedness when $s\in(0,1/2)$,
taken as a standing assumption throughout the section even when not explicitly
stated. As many of the technical details are exactly the same as in the proof
of Theorem 1.1, we only outline the main points. As the proof is the same for
$k>1$, we show only the $k=1$ case. Thus, we simplify the notation using
$\tilde{X}^{m_{0},m_{1}}$ in place of $\tilde{X}^{1,m_{0},m_{0}}$ (the
definition of $\tilde{X}^{k,m_{0},m_{1}}$ is given in (1.4)).

The first step is to obtain a weighted $C^{1}$ estimate of $Q_{s}$.

######  Lemma 5.1.

Let $m_{1}>3+\gamma+2s$ and $m_{0}$ sufficiently large depending only on
$m_{1}$, $s$, and $\gamma$. The following inequality holds

$\|Q_{\rm
s}(g,f)\|_{L^{\infty,m_{1}}}\lesssim\|g\|_{L^{\infty,m_{1}}}\left(\|f\|_{L^{\infty,m_{0}}}+\|\nabla_{v}f\|_{L^{\infty,m_{1}}}\right).$

###### Proof.

Let $\mu=1$ if $\gamma\leq-1$ and $\mu=\frac{-\gamma-2s}{1-2s}$ otherwise. Fix
$r=\langle v\rangle^{\mu}/2$. We first decompose the integral into two parts:

$|Q_{{\rm s}}(g,f)\langle v\rangle^{m_{1}}|=\int\left|\langle
v\rangle^{m_{1}}(f(v^{\prime})-f(v))K_{g}(v,v^{\prime})\right|dv^{\prime}\leq
I_{1}+I_{2},$

where $I_{1}$ and $I_{2}$ are the integrals over $B_{r}(v)$ and
$B_{r}(v)^{c}$, respectively. Applying Lemma 2.2 and using that if $\xi\in
B_{r}(v)$ then $\langle\xi\rangle\approx\langle v\rangle$, we bound $I_{1}$ as

(5.1)
$\begin{split}I_{1}&\lesssim\|\nabla_{v}f\|_{L^{\infty,m_{1}}}\int_{B_{r}(v)}|v-v^{\prime}|K_{|g|}(v,v^{\prime})dv^{\prime}\lesssim\|\nabla_{v}f\|_{L^{\infty,m_{1}}}r^{1-2s}\int|g(w)||v-w|^{\gamma+2s}dw.\end{split}$

We are finished after bounding the integral by $\langle
v\rangle^{\gamma+2s}\|g\|_{L^{\infty,m_{0}}}$ and using the definition of $r$.

The first step to handle $I_{2}$ is to split it into the parts containing
$f(v)$ and $f(v^{\prime})$ via the triangle inequality. Call these integrals
$I_{21}$ and $I_{22}$, respectively. Using Item i again, we see that

$\begin{split}I_{21}=\int_{B_{r}(v)^{c}}\langle
v\rangle^{m_{1}}|f(v)|K_{|g|}(v,v^{\prime})dv^{\prime}\lesssim\|f\|_{L^{\infty,m_{0}}}\langle
v\rangle^{-m_{0}}r^{-2s}\int g(z)|v-z|^{\gamma+2s}dv^{\prime}.\end{split}$

Bounding the last integral using $\|g\|_{L^{\infty,m_{1}}}$ and using the
definition of $r$ finishes the estimate of $I_{21}$.

The last integral, that of $I_{22}$ requires further decomposition into
$I_{221}$ and $I_{222}$ over the domains $B_{r}(v)^{c}\cap B_{\langle
v\rangle/2}^{c}$ and $B_{r}(v)^{c}\cap B_{\langle v\rangle/2}$. The former is
easy to handle using

$|f(v^{\prime})|\leq\|f\|_{L^{\infty,m_{0}}}\langle
v^{\prime}\rangle^{-m_{0}}\lesssim\|f\|_{L^{\infty,m_{0}}}\langle
v\rangle^{-m_{0}}$

where we used that $\langle v^{\prime}\rangle\gtrsim\langle v\rangle$. The
rest of the bound follows exactly as for $I_{21}$.

As for $I_{222}$, notice that for such $v^{\prime}$,
$|v-v^{\prime}|\approx\langle v\rangle$. We use this, along with Lemma 2.4, to
find

(5.2)
$\begin{split}I_{222}&\lesssim\|g\|_{L^{\infty,m_{1}}}\int_{B_{r}(v)^{c}\cap
B_{\langle v\rangle/2}}\frac{\langle v\rangle^{m_{1}}f(v^{\prime})}{\langle
v\rangle^{3+2s}}\langle v\rangle^{3+\gamma+2s-m_{1}}dv^{\prime}\\\
&\lesssim\|g\|_{L^{\infty,m_{1}}}\int_{B_{r}(v)^{c}\cap B_{\langle
v\rangle/2}}\langle
v\rangle^{\gamma}f(v^{\prime})dv^{\prime}\lesssim\|g\|_{L^{\infty,m_{1}}}\|f\|_{L^{\infty,m_{0}}}.\end{split}$

Combining this with the above estimates finishes the proof. ∎

Next we give the key estimate for constructing a solution. To that end, we
present a proposition that plays the role of Proposition 3.1 above. Recall the
space $\tilde{Y}^{m_{0},m_{1}}$ from (1.4).

######  Proposition 5.2 (Propagation of the weighted $C^{1}$ bounds).

Fix any $m_{1}>3+\gamma+2s$ and $m_{0}$ sufficiently large depending only on
$m_{1}$, $\gamma$ and $s$. Suppose that $f_{\rm
in}\in\tilde{X}^{m_{0},m_{1}}$, and $g,R\in\tilde{Y}^{m_{0},m_{1}}_{T}$. If
$f$ solves (3.2) then, there is a constant $C>0$ depending only on $m$, $s$,
and $\gamma$ such that

$\|f\|_{\tilde{Y}^{m_{0},m_{1}}}\lesssim\exp\Big{\\{}C\int_{0}^{T}\|g(t)\|_{\tilde{X}^{m_{0},m_{1}}}dt\Big{\\}}\left(\|f_{\rm
in}\|_{\tilde{X}^{m_{0},m_{1}}}+T\|R\|_{\tilde{Y}_{T}^{m_{0},m_{1}}}\right).$

###### Proof.

First notice that the proof of the bound

(5.3) $\|f\|_{L^{\infty}([0,T];L^{\infty,m_{0}})}\lesssim
e^{C\int_{0}^{T}\|g(t)\|_{L^{\infty,m_{0}}}dt}\Big{(}\|f_{\rm
in}\|_{\tilde{X}^{m_{0},m_{1}}}+\int_{0}^{T}\|R(t)\|_{\tilde{X}^{m_{0},m_{1}}}dt\Big{)}.$

is exactly the same as the (brief) proof in [26, Proposition 3.1] and, hence,
is omitted here. We note that it is a simpler version of the proof of the
bounds on the derivatives that follows.

We now focus instead on bounding $\nabla_{x,v}f$. Fix $\phi(t)$ to be an
increasing function to be determined such that $\phi(0)=\|\nabla_{x,v}f_{\rm
in}\|_{L^{\infty,m_{1}}}$, and let $F(t,x,v)=\phi(t)\langle
v\rangle^{-m_{1}}$. Clearly we have that

(5.4) $F(0,x,v)>\max\left\\{|\partial_{x_{i}}f_{\rm
in}(x,v)|,|\partial_{v_{i}}f(x,v)|:i\in\\{1,2,3\\}\right\\}\quad\text{ for all
}(x,v).$

Let $t_{0}$ be the first time that the above inequality is violated. If
$t_{0}$ does not exist, we are finished. Hence, we argue by contradiction
assuming that there exists $t_{0}\in[0,T]$. Without loss of
generality111Indeed, the only technical issue here is if the inequality is
violated at $|v|=\infty$. One may sidestep this by simply including a cutoff
as a multiplicative factor of the initial data and of $R$. It then follows
from standard facts about the heat equation that $f$ and its derivatives decay
as a Gaussian at high velocities. The cutoff can be removed by a limiting
procedure., we may assume that there exists
$(x_{0},v_{0})\in\mathbb{T}^{3}\times\mathbb{R}^{3}$ such that equality above
holds in (5.4) at the point $(t_{0},x_{0},v_{0})$. Assume momentarily that

(5.5) $F(t_{0},x_{0},v_{0})=\partial_{x_{1}}f(t_{0},x_{0},v_{0}).$

The cases where $i=2,3$ are clearly analogous, as are the case when a negative
sign appears in the equality (i.e., $F=-\partial_{x_{1}}f$). The case when the
derivative is in the $v$ variable is slightly more complicated as new terms
arise, but these new terms can be handled in a straightforward way.

Since $F-\partial_{x_{1}}f\geq 0$ on
$[0,t_{0}]\times\mathbb{T}^{3}\times\mathbb{R}^{3}$, we find

(5.6)
$0\geq\partial_{t}(F-\partial_{x_{1}}f)+v\cdot\nabla_{x}(F-\partial_{x_{1}}f)-(\epsilon+(1-\sigma))\Delta_{x,v}(F-\partial_{x_{1}}f)-\sigma
Q_{\epsilon,\delta}(g,F-\partial_{x_{1}}f).$

We use this to derive a contradiction.

On the one hand, an explicit computation for $F$, along with [26, Proposition
3.1.(v)] yields

(5.7)
$\partial_{t}F+v\cdot\nabla_{x}F-(\epsilon+(1-\sigma))\Delta_{x,v}F-\sigma
Q_{\epsilon,\delta}(g,F)\geq\phi^{\prime}\langle
v_{0}\rangle^{-m_{1}}-C\phi(1+\|g\|_{L^{\infty,m_{1}}})\langle
v_{0}\rangle^{-m_{1}},$

where we used that $m_{1}>3+\gamma+2s$, a condition of the quoted result.

On the other hand, using Lemma 5.1, we find

(5.8)
$\begin{split}\partial_{t}\partial_{x_{1}}f&+v\cdot\nabla_{x}\partial_{x_{1}}f-(\epsilon+(1-\sigma))\Delta_{x,v}\partial_{x_{1}}f-\sigma
Q_{\epsilon,\delta}(g,\partial_{x_{1}}f)=\sigma
Q_{\epsilon,\delta}(\partial_{x_{1}}g,f)+\partial_{x_{1}}R\\\
&\lesssim\left(\|g(t_{0})\|_{\tilde{X}^{m_{0},m_{1}}}\left(\|f\|_{L^{\infty,m_{0}}}+\|\nabla
f\|_{L^{\infty,m_{1}}}\right)+\|R(t_{0})\|_{\tilde{X}^{m_{0},m_{1}}}\right)\langle
v_{0}\rangle^{-m_{1}}\\\
&\leq\left(\|g(t_{0})\|_{\tilde{X}^{m_{0},m_{1}}}\left(\|f\|_{L^{\infty,m_{0}}}+\phi\right)+\|R(t_{0})\|_{\tilde{X}^{m_{0},m_{1}}}\right)\langle
v_{0}\rangle^{-m_{1}}.\end{split}$

Using (5.3), it is clear from (5.7) and (5.8) that we can choose $\phi$ to
obtain a contradiction in (5.6). This yields a contradiction. Hence (5.4)
always holds, finishing the proof. ∎

As usual, once a priori estimates are established, the construction of a
solution follows easily. In fact, in this case, the solution can be
constructed exactly as in [26]. Indeed, one can use the method of continuity
as well as a smoothing argument in order to establish the existence of
solutions to the linear problem. After this, an iteration yields a solution to
the nonlinear problem. As it is exactly the same as in [26], we omit the
details.

One subtle issue that may cause worry is whether the process above provides a
$W^{1,\infty}$ solution instead of $C^{1}$. However, at the level of the
method of continuity, the solutions constructed is smooth. Hence all
quantities $\partial_{t}f$, $\nabla_{x}f$, and $\nabla_{v}f$ are continuous
and such continuity is passed through all (locally uniform) limits.

For uniqueness, one can actually simply use an $L^{2}$-based argument. Indeed,
a quick check of the arguments in Section 3.2 reveals that they can be adapted
in a straightforward way to use only the $Y^{m_{0},m_{1}}$ norms of two
potential solutions $f$ and $g$. Actually, the proof is easier in this case as
there is no need to use the Sobolev embedding theorem.

The above concludes the proof of Theorem 1.2.

## Acknowledgments

CH was partially supportd by NSF grant DMS-2003110. WW was partially supported
by NSF grant DMS-1928930 while participating in the Mathematical Fluid
Dynamics program hosted by the Mathematical Sciences Research Institute in
Berkeley, California, during the Spring 2021 semester.

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