Global regularity estimates for the Boltzmann equation without cut-off
Cyril Imbert Luis Silvestre



  We derive C^∞ a priori estimates for solutions of the inhomogeneous Boltzmann equation without cut-off, conditional to pointwise bounds on their mass, energy and entropy densities. We also establish decay estimates for large velocities, for all derivatives of the solution. 


Quorum-sensing active particles with discontinuous motility
    Thomas Speck
    December 30, 2023
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§ INTRODUCTION


The Boltzmann equation is a fundamental nonlinear evolution model from statistical mechanics. It describes the evolution of a
system made of a very large number of particles at an intermediate scale between the microscopic one (which consists of the trajectory of every single particle and their interactions) and the macroscopic one (the hydrodynamic models like Euler or Navier-Stockes equations).

We consider the space in-homogeneous Boltzmann equation without cut-off,

    ∂_t f + v ·∇_x f = (f,f)     for   (t,x,v) ∈ (0,T) ×^d ×^d.

Boltzmann's collision operator (f,f) is typically written in the following way

    (f,f) = ∫_^d∫_S^d-1 (f(v'_*)f(v') - f(v_*)f(v)) B(|v-v_*|,cosθ)  v_* σ

where v_*' and v' are computed in terms of v_∗ and σ by the formula

    v' = v+v_*/2 + |v-v_*|/2σ   and    
    v'_* = v+v_*/2 - |v-v_*|/2σ .

The angle θ measures the deviation between v and v'. In this case, it is the angle so that

    cosθ := v-v_*/|v-v_*|·σ    (  and   sin (θ/2) := v'-v/|v'-v|·σ) .

We consider the standard non-cutoff collision kernels B. They have the form

    B(r,cosθ) = r^γ b(cosθ)    with   b (cosθ) 
    ≈|sin (θ/2)|^-(d-1)-2s

with γ > -d and s ∈ (0,1). 

In a microscopic model where the particles repel each other by an inverse-power law potential with exponent q>2, the collision kernel has the form (<ref>) with γ = (q-2d+1)/(q-1) and s = 1/(q-1) (See for example <cit.>, chapter 1, Section 1.4). In three dimensions, for inverse-power law potentials, the value of γ+2s would be in the range [-1,1]. Our results in this paper apply to the range γ+2s ∈ [0,2]. In Subsection <ref>, we briefly discuss the problem with the very soft potential case: γ+2s<0.

We define the hydrodynamic quantities

    M (t,x)    := ∫_^df(t,x,v)  v       (mass density), 
    
    E (t,x)     := ∫_^df(t,x,v) |v|^2  v       (energy density), 
    
    H (t,x)     := ∫_^d f ln f (t,x,v)  v       (entropy density).

These hydrodynamic quantities, together with moment density, are the 
quantities associated with the solution of the Boltzmann equation
that are macroscopically observable.

In this article, we are concerned with regularity estimates for the solution of (<ref>). This is intimately related with the well posed-ness problem for smooth classical solutions. The question of existence of global smooth solutions for the Boltzmann equation (<ref>) is a well known and remarkable open problem. There is a warm discussion about it in the first chapter of Cédric Villani's book <cit.>. The Boltzmann equation is a more detailed model for the evolution of a fluid than the hydrodynamic models like Euler or Navier-Stokes equations. Indeed, in certain asymptotic regime (see <cit.>), the hydrodynamic quantities associated to the Boltzmann equation converge to the solution of the compressible Euler equation, that is known to develop singularities in finite time <cit.>. A next order expansion shows that the hydrodynamic quantities approximately solve a compressible Navier-Stokes equation, for which the classical well-posedness problem is not well understood. It makes sense to expect the Boltzmann equation to retain the difficulties of the hydrodynamic models, and add some more. Should we expect singularity formation in finite time then? The answer to this question is not straight forward. There are different types of singularities that emerge from the flow of the compressible Euler equation. Some of them may be compatible with the Boltzmann equation, and others are not. In a shock singularity for the Euler equation, a discontinuity emerges from the flow similarly as in Burgers equation. All the quantities involved stay bounded up to the time of the discontinuity. One would not see this as a singularity for the Boltzmann equation, since the kinetic model allows for different velocities to co-exist at one point in space. The Navier-Stokes equation will not allow for shock singularities either, since the viscosity would smooth out any discontinuity for as long as solutions stay bounded and away from vacuum. A fundamentally different, and much more delicate, kind of singularity is that of an implosion. In that case, the mass and energy concentrate and become unbounded at one point. It was only very recently (in fact, after this paper was initially posted) that smooth implosion profiles for the compressible Euler equation were found and proved to be stable in <cit.> and <cit.>. These implosion singularities exist for the compressible Navier-Stokes equation as well. As of now, we cannot think of any reason to rule out the existence of implossion singularities for the Boltzmann equation. If they did, which seems like a likely scenario, the question that remains is whether this is the only type of singularity that may emerge from the flow of the Boltzmann equation. Our main result in this paper aims at answering that question.

As we explained in the previous paragraph, the unconditional regularity of solutions to the in-homogeneous Boltzmann equation seems to be completely out of reach. The problem that we study is conditional to pointwise bounds on the hydrodynamic quantities. More precisely, we make the following assumption.
[Hydrodynamic quantities under control] 
The following inequalities hold uniformly in t and x,


  * 0 < m_0 ≤ M(t,x) ≤ M_0.

  * E(t,x) ≤ E_0.

  * H(t,x) ≤ H_0.



We do not prove Assumption <ref>. We take it for granted (hence the name assumption).  Conditional to it, we obtain C^∞ estimates that we state in our main theorem. Assumption <ref> is a way to disallow the implossion singularities that we described above. Our result in this paper essentially says that no other types of singularity may exist for the Boltzmann equation other than (potentially) the ones that are hydrodynamically visible.

  
Let f be a solution to the Boltzmann equation in (0,T) ×^d ×^d (as in Definition <ref>) with a collision kernel of the form (<ref>) and γ+2s ∈ [0,2]. If Assumption <ref> holds, then for any multi-index k ∈ℕ^1+2d, τ>0 and q>0,

    (1+|v|)^q D^k f _L^∞([τ,T) ×^d ×^d)≤ C_k,q,τ.

Here D^k is any arbitrary derivative of f of any order, in t, x and/or v.

When γ>0, the constants C_k,q,τ depend only on k, q and τ, and the constants m_0, M_0, E_0 and H_0 from Assumption <ref>, and the parameters s, γ and dimension d.

When γ≤ 0, the constants C_k,q,τ depend in addition on the pointwise decay  of the initial data. That is, on the constants N_r with r ≥ 0, given by

    N_r := sup_x,v (1+|v|)^r f_0(x,v)     for each  r≥ 0.



Note that the upper bounds on energy density and mass density in (<ref>) correspond to upper bounds for mass, moment and temperature density. Moreover, the upper bound in entropy is slightly stronger than a lower bound for temperature (and equivalent in the hydrodynamic limit).

We work with a strong notion of solution that we describe in Definition <ref>. See section <ref> for a discussion about weaker notions of solutions. Moreover, we work with functions f that are periodic in x. It is a convenient, but non-essential, technical assumption. It is only applied for the upper bounds in <cit.>, that require the use of a maximum principle argument (as in “let (t_0,x_0,v_0) be the point where this maximum is achieved”). Theorem <ref> would hold in any other regime where these upper bounds hold. The estimates in Theorem <ref> do not depend on the length of the period. The problem of regularity estimates as in Theorem <ref> in bounded domain with physical boundary conditions would require some further analysis.


The difference between γ>0 and γ≤ 0 in Theorem <ref> has its origin in the decay estimates from <cit.>. The decay of the solution f is self generated when γ>0. However, when γ≤ 0, the function f will decay rapidly only if it initially does.



In the case γ≤ 0, each constant C_k, q, τ depends on one constant N_r in (<ref>) for a specific value of r depending on k and q. However, its explicit dependence is hard to track. Obviously, the larger q, the larger the value of r will be required to be. It turns out that for higher order derivatives D^k, we also need to use larger values of r.




 §.§ Consequences of our main theorem




  §.§.§ Convergence to equilibrium


In a celebrated result <cit.>, Desvillettes and Villani proved that solutions to the non-cutoff in-homogeneous Boltzmann equation, periodic in x (or with other physical boundary conditions), converge to equilibrium faster than any algebraic rate, conditional to the following two main assumptions


  * The solution f stays in C^∞ for all time with uniform bounds as t →∞.

  * The solution f is bounded below by some fixed Maxwellian.


A priori, these two assumptions appeared to be very strong. After Theorem <ref>, they can be reduced to only Assumption <ref>. Indeed, the lower bound by a fixed Maxwellian is obtained in our earlier work with Clément Mouhot <cit.>. 

Since the estimates in Theorem <ref> do not depend on T, we can take T →∞ and deduce a uniform regularity estimate in (τ,∞] ×^d ×^d. As a consequence, we state the following improvement for the Theorem in <cit.>.



Let f be a solution of (<ref>) in (0,∞) ×^d ×^d (as in Definition <ref>, in particular periodic in x). Assume that Assumption <ref> holds globally. Then f converges to a Maxwellian as t →∞ as described in Theorem 2 in <cit.>.




  §.§.§ Continuation criteria



Theorem <ref> also suggests the following continuation criteria. Let f be a solution to the Boltzmann equation (<ref>) in (0,T) ×^d ×^d as in Definition <ref>. Suppose that it cannot be extended further in time, that is, it cannot be extended as a solution in (0,T+) ×^d ×^d for any >0. Then, one of the following must happen


  * lim_t → Tmax_x ∈^d M(t,x) = +∞.

  * lim_t → Tmax_x ∈^d E(t,x) = +∞.

  * lim_t → Tmax_x ∈^d H(t,x) = +∞.

  * lim_t → Tmin_x ∈^d M(t,x) = 0.


This continuation criteria can be immediately justified by combining Theorem <ref> with an appropriate short time existence result. When s ∈ (0,1/2) and γ∈ (-3/2,0], we can use the short time existence from <cit.>. For any s ∈ (0,1) and γ≤ 0, there is a recent appropriate short time existence result in <cit.>.

Note that the short time existence result in <cit.> requires the initial data to have Gaussian decay, which is not propagated to positive times by our estimates in Theorem <ref>.

This continuation criteria says that the only way a singularity can arise in finite time for the Boltzmann equation without cutoff is by one of the hydrodynamic quantities M, E or H to blow up, or by creation of vacuum. There is a recent result in <cit.> saying that this continuation criteria can be reduced to the first two items. That is, either the mass or the energy density should blow up if the solution develops a singularity.  It rules out the case in which there is creation of vacuum or zero temperature while the mass and energy density stay bounded. It is conceivable that this blow up criteria may be relaxed in some other way in the future. As we explained above, a completely unconditional continuation criteria seems to be out of reach with current techniques.

It is natural to expect a similar continuation criteria to hold in the cut-off case as well. However, the reason for it would be fundamentally different. The cut-off Boltzmann equation does not have a regularization effect. One would expect a propagation of regularity provided that Assumption <ref> holds. From the mathematical point of view, it is a very different problem from the one we address here. We will not analyze the cut-off case any further.



 §.§ Future directions and open problems




  §.§.§ Regularity estimates for weak solutions



In this paper we obtain a priori estimates for classical solutions. Working with a weaker notion of solution would entail several technical difficulties. We thought it was not the right time to take on that burden yet. In fact, we consider a very strong notion of solution (see Definition <ref>).  It would be interesting to extend Theorem <ref> as a regularity estimate for renormalized solutions with a defect measure as defined in <cit.>. Below, we analyze the difficulties of this problem.

The biggest challenge of such an extension would be to recover the pointwise estimates from <cit.> and <cit.>. The proofs in these papers use a maximum principle type argument that seems to be difficult to adapt to the setting of <cit.>.

Once a weak solution is proved to be bounded, we can apply the result in <cit.> (Theorem <ref> below) and deduce the Hölder continuity of the solution.

There is a (presumably minor) difficulty in the application of the Schauder estimates from <cit.> in order to derive Corollary <ref> in this paper. This is because the result of <cit.> is not stated for weak solutions. The later applications of the Schauder estimates in our proof of Theorem <ref> in Section <ref> are not problematic. In each step of the iteration we apply the Schauder estimates to increments that are qualitatively as regular as the function itself.



  §.§.§ The grazing collision limit


When s → 1, the Boltzmann equation converges formally to the Landau equation. For that, we need the collision kernel B to satisfy

    B ≈ (1-s) |v-v_∗|^γsin(θ/2)^-d+1-2s.

The normalizing factor (1-s) is transferred into the ellipticity conditions on the Boltzmann kernel K_f (defined in (<ref>)). It is well known in the literature of nonlocal equations that this is the necessary factor to have uniform bounds as s → 1 (see for example <cit.> or <cit.>).

It is to be expected that the estimates of Theorem <ref> would remain uniform as s → 1 if B satisfies (<ref>). However, it is still an open problem. Below, we explain the difficulties with our current approach.

Note that any technique that establishes the estimates from Theorem <ref> uniformly as s → 1, would also imply the corresponding regularity estimates for the Landau equation as a consequence. A method that provides estimates uniform as s → 1 must use techniques that apply both to integro-differential equations and second order parabolic equations.

The most challenging difficulty in proving uniform estimates as s → 1 would be to establish the pointwise bounds from <cit.> and <cit.>. The proofs in these papers use purely nonlocal techniques. The constants obtained in the estimates there certainly blow up as s → 1. The corresponding pointwise upper bound for the Landau equation is established in <cit.> using different methods.

The Hölder estimates from <cit.> are robust as s → 1. We would also expect the Schauder estimates from <cit.> to be robust as s → 1, however it does not follow directly from the current proof in <cit.> because it is non constructive. Some constants are proved to exist under a compactness argument, and by that we loose track of their dependence on s. It is conceivable that a refinement of the proof in <cit.> may lead to robust estimates since the proof in that paper works for second order equations as well.



  §.§.§ Other open problems



The following is a (non exhaustive) list of other open questions related to the main result of this article.


  * Interior estimates: if Assumption <ref> holds for (t,x) ∈ (-1,0] × B_1, can we establish the regularity estimates as in Theorem <ref> for (t,x) ∈ (-1/2,0]× B_1/2?

  * Bounded domains: when the equation is supported in a smooth bounded domain x ∈Ω⊂^d, with physical boundary conditions, do the estimates from Theorem <ref> hold in the full domain Ω?

  * Weaker conditions: can we reduce Assumption <ref> to a weaker condition?

  * Very soft potentials: can we establish regularity estimates when γ+2s < 0? This is a very difficult problem that is open even in the space homogeneous setting. The most challenging step seems to be obtaining the L^∞ estimate as in <cit.>.




 §.§ Previous regularity results for the Boltzmann and Landau equations
.

The well-posedness and regularity of the space homogeneous Boltzmann equation is well understood in the case of hard and moderately soft potentials (i.e. γ+2s ≥ 0). See <cit.>. Note that in the space homogeneous case, Assumption <ref> is trivially satisfied by the conservation of mass and energy and the monotonicity of entropy.

Results on the regularity for the space in-homogeneous Boltzmann equation are scarce. Other than the papers that are part of our program, the most relevant previous result is the C^∞ regularity of solutions conditional to a uniform bound in H^5_x,v, plus infinite bounded moments, plus a lower bound on the mass density. These results were established in <cit.>. We improve these results by significantly lowering the condition for regularity to the bounds of Assumption <ref>, which are physically meaningful. We refer to   <cit.> for further discussion on other results in the literature.

Our program of establishing conditional regularity provided that the hydrodynamic quantities are controlled as in Assumption
<ref> has also been studied for the Landau equation. It is currently fairly well understood in the cases of hard and moderately soft potentials. The local Hölder estimates were obtained in <cit.>. The upper bounds and Gaussian decay bounds (when appropriate) for moderately soft potentials were obtained in <cit.> using the estimates from <cit.> combined with a change of variables that inspires our construction in Section <ref>. The higher regularity of solutions was studied in <cit.> applying a kinetic version of Schauder estimates. These regularity estimates were extended to the case of hard potentials in <cit.>. In <cit.>, they refine the continuation criteria for the in-homogeneous Landau equation as mentioned in Subsection <ref>.



 §.§ Strategy of proof

The result in this paper is obtained as the final step in a program of conditional regularity that started in 2014. Here, we use the previous results by the authors, and also by the authors in collaboration with Clément Mouhot, that were part of this program. Theorem <ref> is proved by combining the following ingredients.



  * An L^∞ estimate for positive time depending only of the hydrodynamic quantities. This holds provided γ+2s ≥ 0. It is proved in <cit.>.

  * A weak Harnack inequality for kinetic integro-differential equations. It gives us local C^α estimates for some α>0 small. It was obtained in <cit.>.

  * Schauder estimates for kinetic integro-differential equations. It gives us local C^2s+α estimates. It was obtained in <cit.>.

  * Pointwise decay estimates. They say f ≲ (1+|v|)^-q for all q>0.
They are self generated if γ > 0 and they propagate from the initial data if γ≤ 0.
It was proved in <cit.>.

  * A change of variables that turns our local Hölder and Schauder estimates into global ones. We develop it in Section <ref>.

  * Some new inequalities for kinetic Hölder spaces (defined in Section <ref>) and how they interact with the Boltzmann collision operator (see Section <ref>) and increments (see Section <ref>).

  * A bootstrapping mechanism by iterating the global version of the Schauder estimates.




In order to obtain regularity estimates like the ones in this article, it is key to think of the Boltzmann equation as a kinetic equation with integral diffusion in the v variable plus a lower order term, in the way that was described in <cit.>. Using Carleman coordinates and the cancellation lemma (as in <cit.>), the Boltzmann equation takes the following form

    ∂_t f + v ·∇_x f = ∫_^d (f' - f) K_f(t,x,v,v')  v' + c (f ∗_v |·|^γ) f.


The kernel K_f depends on the solution f itself. We give more details in Section <ref> and recall the formula for K_f in (<ref>). When Assumption <ref> holds, the kernel K_f satisfies certain ellipticity conditions that allow us to derive regularity estimates.

In <cit.>, we obtained a weak Harnack inequality for kinetic integro-differential equations. It implies a regularity estimate for the local Hölder regularity, for a small exponent, of bounded solutions to (<ref>) that satisfy Assumption <ref>. In <cit.>, we obtained a Schauder estimate for kinetic integro-differential equations. It implies a local estimate of Hölder regularity with exponent 2s+α for some α>0. It is enough regularity to make sense of the equation classically. These are two results for generic kinetic integro-differential equations. They apply to the solution of the Boltzmann equation thanks to the expression (<ref>). They also apply to derivatives of f with respect to t, x and v provided that we can appropriately bound each of the extra error terms that come up in the equation when differentiating the collision and transport terms. In order to turn this procedure into a bootstrap iteration leading to C^∞ estimates we need to turn the local regularity estimates from <cit.> and <cit.> into global ones.

The weak Harnack inequality in <cit.> and the Schauder estimate in <cit.> depend on ellipticity conditions on the kernel K_f in (<ref>). In these papers, we showed how these ellipticity conditions are implied locally by Assumption <ref>. However, they degenerate for large velocities. In order to obtain global estimates from the application of the weak Harnack inequality and Schauder estimates, we construct a special change of variables. It transforms the function f into a solution to a kinetic integro-differential equation whose kernel is uniformly elliptic with parameters that do not degenerate for large velocities. This change of variables is a key ingredient in the proof of this paper. It is described in Section <ref>. It allows us to turn any local (in velocity) estimate for the Boltzmann equation into a global one.

The bootstrap iteration consists in applying the global version of Schauder estimates (via the change of variables) to the equation satisfied by derivatives of the solution f. The extra error terms are handled by careful (and new) estimates in Hölder spaces for the Boltzmann collision operator, increments and derivatives (described in Sections <ref> and <ref>). In each step of the iteration, we gain a regularity estimate for a higher derivative in terms of the estimates already obtained for the lower order ones. There is a loss of decay exponent in each differentiation step. Thus, we need to start with a function with rapid decay at infinity in order for the iteration to continue indefinitely. This initial decay is provided by the result in <cit.>.



 §.§ Notation


We use the notation a ≲ b to denote the fact that there exists a constant C so that a ≤ Cb. The constant C can depend on a variable collection of parameters depending on context. This notation is used mostly inside proofs of lemmas, propositions and theorems. In each statement, we explain what the constants depend on. The implicit constants in each symbol ≲ inside a proof depend on the parameters specified in the corresponding statement. Even though this notation might be arguably confusing at times, it allows us to clean up the computations substantially.

The symbol a ≈ b means that a ≲ b and a ≳ b.

We use the standard notation B_r to denote a ball of radius r in ^d centered at the origin. We also write B_r(w) to denote a ball centered at some point w ∈^d. The kinetic cylinders Q_r ⊂×^d ×^d are explained in Section <ref>.



§ PRELIMINARY ESTIMATES FOR THE BOLTZMANN EQUATION



In this section, we collect some preliminary results for the Boltzmann equation that play a role in the proofs in this paper.

As we mentioned before, we work with a very strong notion of classical solutions. In this way, all the results in the literature are applicable and we avoid technical difficulties. We give the definition below.

 
A function f : (0,T) ×^d ×^d → is a solution to the Boltzmann equation (<ref>) when 


  * It is non-negative everywhere.

  * It is C^∞ in all variables t,x,v.

  * It solves (<ref>) for every value of (t,x,v) in the classical sense.

  * For each value of (t,x), the function f decays rapidly as |v|→∞. More precisely, for any q>0, we have

    lim_|v|→∞f(t,x,v)/(1+|v|)^q = 0,

locall uniformly in (t,x).
For derivatives of f, we only assume that there is sufficient decay so that

    ∫_^d |D^2_vf(t,x,v)| (1+|v|)^γ+2s v < +∞,

for every value of t and x.

For simplicity, we will also consider f to be periodic in x.


The results in this paper certainly apply to a weaker notions of solution as well. We discuss it in Section <ref>. However, by considering a strong notion of solution as in Definition <ref>, we avoid superflous technical difficulties that would make this paper harder to read.

In the last section of <cit.>, we discuss how to replace the rapid decay assumption in the last item of Definition <ref> with a weaker algebraic decay condition. 

The last condition on the integrability of D^2_v f is convenient for Lemmas <ref> and <ref>. These are the only parts of this paper where it plays a role.



 §.§ Decay estimates


We start by reviewing the decay estimates in the velocity variables obtained
in <cit.> for solutions of the Boltzmann equation. When γ>0, these decay estimates are forced by the equation regardless of the initial data. When γ≤ 0, we need to impose the appropriate decay on the initial data, and it is propagated in time by the equation.



  Let the parameters γ,s from (<ref>) satisfy
  γ+2s ∈ [0,2] and let f be a solution of the Boltzmann
  equation (<ref>) in
  (0,T) ×^d ×^d, periodic in x, such that
  f(0,x,v) =  (x,v) in ^d ×^d and such that
  the Assumption <ref> holds. If γ≤ 0, we also
  asssume that for all q >0, there exists a constant N_0,q so that (x,v) ≤ N_0,q (1+|v|)^-q for
  (x,v) ∈^d ×^d. Then the solution f satisfies 
  
    f (t,x,v) ≤ N_q (1+|v|)^-q    in  (0,T) ×^d ×^d,

  for some constant N_q only depending on dimension d, parameters
  γ,s from the collision kernel, see (<ref>), and the
  hydrodynamical bounds m_0,M_0,E_0,H_0 from Assumption <ref>, and, in the case γ≤ 0, also on the constants N_0,q.


We include the following two lemmas about the decay or growth of convolutions of f with different powers of |v|. They will be applied repeatedly in several parts of the paper. The first one gives us an upper bound depending on the mass and energy of f only. The second one is in terms of its pointwise upper bounds.


 
Let f : ^d → [0,∞). Assume that 0 ≤κ≤ 2. Then

    ∫_^d f(v+w) |w|^κ w ≤ C ( (1+|v|)^κ M_0 + E_0 ),

where C is a universal constant and M_0 and E_0 are numbers so that 

    ∫_^d f(v)  v ≤ M_0     and    ∫_^d f(v) |v|^2  v ≤ E_0 .




We compute directly

    ∫_^d f(v+w) |w|^κ w   = ∫_^d f(w) |w-v|^κ w, 
       ≲∫_^d f(w) ( |w|^κ + |v|^κ)  w, 
       ≲∫_^d f(w) ( 1 + |w|^2 + |v|^κ)  w, 
       ≲( ∫_^df(w) |w|^2  w + (1+|v|)^κ∫_^df(w)  w).



 
Let f : ^d → [0,∞) and κ > -d. Assume that f(v) ≤ N (1+|v|)^-q for some q>d+κ_+. Then

    ∫_^d f(v+w) |w|^κ w ≤ C N (1+|v|)^κ,

for some constant C depending on d, κ and q only (neither on N nor v).



We do a different computation depending on whether κ≥ 0 or κ < 0.

For κ≥ 0, it is very similar to Lemma <ref>. We compute

    ∫_^d f(v+w) |w|^κ   = ∫_^d f(w) |w-v|^κ w, 
       ≲∫_^d f(w) ( |w|^κ + |v|^κ)  w, 
       ≲ N ( ∫_^d (1+|w|)^-q |w|^κ w + |v|^κ∫_^d (1+|w|)^-q w).

Since -q+κ < -d and κ > -d, the integrals are computable for each value of q and κ≥ 0.

For κ <0, we estimate the integrals differently. In this case we will use that q>d.

    ∫_^d f(v+w) |w|^κ w    ≤ N ∫_^d (1+|v+w|)^-q |w|^κ w, 
        = N ∫_|w|<|v|/2 (1+|v+w|)^-q |w|^κ w + N ∫_|w|>|v|/2 (1+|v+w|)^-q |w|^κ w , 
       ≲ N ∫_|w|<|v|/2 (1+|v|)^-q |w|^κ w +N  ∫_|w|>|v|/2 (1+|v+w|)^-q |v|^κ w , 
       ≲ N ∫_|w|<|v|/2 (1+|v|)^-q |w|^κ w + N ∫_^d (1+|v+w|)^-q |v|^κ w , 
       ≲ N (1+|v|)^-q |v|^κ+d + |v|^κ
       ≲ N (1+|v|)^κ.







 §.§ A coercivity condition for integro-differential operators


In <cit.>, there is a practical condition to verify if the quadratic form associated with an integro-differential operator is coercive with respect to the H^s semi-norm. The result says that following.

 
Let K: B_2 × B_2 → [0,∞) be a kernel satisfying the following assumption. There exists λ>0 and μ∈ (0,1) such that for any v ∈ B_2 and any ball B ⊂ B_2 that contains v,

    | { v' ∈ B: K(v,v') ≥λ |v'-v|^-d-2s}| ≥μ |B|.

Then, for any function u : B_2 →,

    ∬_B_2 × B_2 (u(v') - u(v))^2 K(v,v')   v'  v ≥ c λ∬_B_1 × B_1(u(v') - u(v))^2/|v'-v|^d+2s v'  v.

The constant c depends on dimension and μ only.


We recall from (<ref>) that the Boltzmann equation can be written as an integro-differential equation with some kernel K_f depending on the solution f itself. The explicit formula for K_f is worked out in <cit.> and we recall it now,

    K_f (v,v') = 2^d-1/|v'-v|∫_w ⊥ v'-v f(v+w) B(r,cosθ) r^-d+2 w    
    with  
    r^2 =|v'-v|^2 + |w|^2, 
    cosθ = w-(v-v')/|w-(v-v')|·w+(v'-v)/|w+(v'-v)| .
 
 This kernel K_f satisfies the assumption of Theorem <ref> in the stronger form of a cone of nondegeneracy as described in <cit.>. We describe it in the following proposition.

 
Let f:^d → be a nonnegative function and K_f be the corresponding Boltzmann kernel as in (<ref>). For any v ∈^d, there exists a symmetric subset of the unit sphere A(v) ⊂ S^d-1 such that


  * |A(v)| ≥μ (1+|v|)^-1.

  * For every σ∈ A(v), |σ· v| ≤ C 
 (i.e. A(v) is concentrated around the equator perpendicular to v with width ≤ C/(1+|v|)).

  * For any σ∈ A(v) and r>0, 

    K_f(v, v+rσ) ≥λ (1+|v|)^1+γ+2s r^-d-2s.


Here, the constants μ, λ and C depend only on dimension and on the hydrodynamic bounds of Assumption <ref>.


The cone of nondegeneracy described in Proposition <ref> immediately implies the assumption of Theorem <ref>. Thus, the Boltzmann kernel K_f satisfies a coercivity inequality restricted to velocities in B_2. Naturally, we can apply a translated and dilated version of Theorem <ref> to derive a coercivity condition for the Boltzmann collision kernel in any bounded set of velocities. It naturally implies a local coercivity inequality. However, as we see in Proposition <ref>, the thickness of the cone of nondegeneracy  degenerates as |v| →∞. This is natural in view of the fact that the optimal global coercivity inequalities for the Boltzmann collision operator depend on certain modified distance and weight that degenerate as |v| →∞ (see <cit.>).

We are able to recover the optimal coercivity estimates for large velocities using Theorem <ref> and Proposition <ref> together with the change of variables described in Section <ref>. See Appendix <ref> for a derivation of the global coercivity estimate in <cit.> along these lines.





§ KINETIC HÖLDER SPACES



Here, following <cit.>, we describe the appropriate formulation of Hölder spaces for kinetic equations. These are, in the context of Hölder spaces, what the spaces described in <cit.> are in the context of Sobolev spaces. They are adapted to the group of translations and dilations that leave the equation in an invariant ellipticity class. In order to motivate and explain all the necessary machinery related to these spaces, it is best to first consider the simpler fractional Kolmogorov equation

    ∂_t f + v ·∇_x f + (-Δ)^s f = 0.

The equation (<ref>) is the simplest kinetic equation with integro-differential diffusion of order 2s and it serves as a model equation to start our analysis. The Boltzmann collision operator is the sum of a nonlinear integro-differential operator of order 2s (which is not the fractional Laplacian) plus a lower order term. This decomposition is precisely given by the two terms in (<ref>).



 §.§ Scaling and translation invariances


Assume that a function f solves (<ref>) in some domain. For any r>0, if we scale the function by

    f_r(t,x,v) = f(r^2s t, r^1+2s x, rv),

then the scaled function f_r satisfies the same equation in the appropriately scaled domain.

The space ×^d ×^d is endowed with the following Lie group structure: for all ξ = (h,y,w) and z= (t,x,v), the operator ξ∘ z is given by the formula

    ξ∘ z = (h+t,x+y+ tw, v+w).

If f is a solution of (<ref>) and z_0 = (t_0,x_0,v_0) ∈^1+2d is arbitrary, then the function

    f̃(z) = f(z_0 ∘ z)

solves the same equation (in a translated domain).

The scaling invariance and left-translation invariance described here are the motivation for the definitions of kinetic cylinders, distance, degree and Hölder spaces given below.

In Sections <ref> and <ref>, we will describe the results from <cit.> and <cit.> which are kinetic integro-differential versions of the classical regularity results of De Giorgi and Schauder for elliptic equations with variable coefficients. These equations are not invariant by scaling or translations individually, but rather as a class. Scaling or left translations of functions solving an equation as in Theorems <ref> or <ref>, will solve an equation with the same structure and the same ellipticity parameters.



 §.§ Cylinders


When working with parabolic equations, one often considers parabolic cylinders
of the form (t_0-r^2,t_0] × B_r(x_0). Because of the invariant
transformations we mentioned above, it is natural and convenient to consider cylinders respecting them. For all z_0 ∈^1+2d, we define

    Q_r (z_0)  = { (t,x,v) : t_0 -r^2s < t ≤ t_0,  |x-x_0 -(t-t_0) v_0| < r^1+2s, |v-v_0| < r }.

Cylinders centered at the origin (0,0,0) and of radius r>0 are
simply denoted by Q_r.

Note that under this definition (t,x,v) belongs to Q_1 if and only if (r^2s t, r^1+2s x, rv) belongs to Q_r. Thus, our cylinders honor the scaling given in (<ref>). Moreover, for any z_0 ∈^1+2d, we have Q_r(z_0) = z_0 ∘ Q_r, where ∘ denotes the Lie group operator given in (<ref>).



 §.§ Kinetic distance


We recall the notion of kinetic distance defined in <cit.>. It is constructed so that it agrees with the scaling given in (<ref>) and the left action of the group (<ref>).

 
The kinetic distance between two points z_1=(t_1,x_1,v_1) and z_2=(t_2,x_2,v_2) in ^1+2d is given by the following formula

    d_ℓ(z_1,z_2) := min_w ∈^d{max( |t_1-t_2|^1/2s,
         |x_1-x_2-(t_1-t_2)w|^1/1+2s , |v_1-w| , |v_2-w| ) }.



We show in <cit.> that d_ℓ is indeed a distance when s ≥ 1/2. For s<1/2, the triangle inequality fails for d_ℓ, however d_ℓ(z_1,z_2)^2s is in fact a distance. We still work with d_ℓ for any value of s ∈ (0,1) in order to keep our formulas consistent.

This distance is scale invariant in the following sense: for any z_1, z_2 ∈^1+2d and r>0, if we scale S_r z_1 := (r^2s t_1, r^1+2s x_1, rv_1) and S_r z_2 := (r^2s t_2, r^1+2s x_2, rv_2), we have

    d_ℓ(S_r z_1,S_r z_2) = r d_ℓ(z_1,z_2).


This distance is also left invariant by (<ref>). Indeed, for any three points ξ, z_1 and z_2 in ^1+2d we have

    d_ℓ(ξ∘ z_1, ξ∘ z_2) = d_ℓ(z_1, z_2).


It is also convenient to define the length of a vector z ∈^1+2d by z := d_ℓ(z,0). Technically, z is not a norm. It is not homogeneous of degree one, but rather it is homogeneous with respect to the scaling in (<ref>). It satisfies the triangle inequality with respect to the group action (<ref>):

    z_1 ∘ z_2≤z_1 + z_2.

There are several convenient equivalent expressions for z that we write below.

    z   = min_w ∈^d{max( |t|^1/2s,
         |x-tw|^1/1+2s , |v-w| , |w| ) }, 
       ≈max( |t|^1/2s,
         |x|^1/1+2s , |v| ) , 
       ≈ |t|^1/2s+ |x|^1/1+2s + |v|


The symbol ≈ denotes in this context that the quantities on both sides are comparable up to a factor depending on s and dimension d only. The last two expressions would not satisfy (<ref>), but they are easier to compute in some cases in which the constant factors do not matter.

Note that due to the left invariance of this distance, d_ℓ(z_1,z_2) = z_2^-1∘ z_1. 

The distance d_ℓ is left invariant but not right invariant. This lack of right invariance is occasionally problematic and results in some loss in some exponents in some inequalities. It is essentially the reason why the exponents α and α' are different in the main result of <cit.>. For example, for all z_1,z_2 ∈^1+2d and w ∈^d, a direct computation shows the following inequalities that will be used repeatedly.

    d_ℓ (z_1 ∘ (0,0,w), z_2 ∘ (0,0,w))    ≤ d_ℓ(z_1,z_2) + |t_1-t_2|^1/(1+2s) |w|^1/(1+2s), 
       ≤ d_ℓ(z_1,z_2) + d_ℓ(z_1,z_2)^2s / (1+2s) |w|^1/(1+2s).

Moreover, when d_ℓ(z_1,z_2) ≤ 1, then the last inequality is also less than or equal to d_ℓ(z_1,z_2)^2s / (1+2s) (1+|w|)^1/(1+2s). We see that the right translations z_1∘ (0,0,w) and z_2 ∘ (0,0,w) may be an order of magniture further apart than the original points z_1 and z_2.



 §.§ Kinetic degree of polynomials


We recall the definition of kinetic degree from <cit.>, for polynomials p in [t,x,v]. Given a monomial m of the form

    m(t,x,v) = c   t^α_0 x_1^α_1… x_d^α_d v_1^α_d+1… v_d^α_2d with  c ≠ 0,

we define its kinetic degree as 

    m = 2s α_0 + (1+2s) ∑_j=1^d α_j + ∑_j=d+1^2dα_j.

That is, the degree of m is computed by counting 2s times
the exponent for the variable t, 1+2s for the exponents in the variables x_i and 1 for variables v_i. This definition is justified by the fact that we want a notion of kinetic degree consistent with the scaling (<ref>). With this definition in mind,

    m(r^2s t, r^1+2s x, rv) = r^ m  m(t,x,v).


Given any non-zero polynomial p in [t,x,v] we define the kinetic degree of p (and we write it p) as the maximum of the kinetic degree of each of its (non-zero) monomials.

The kinetic degree of the zero polynomial is not properly defined above. It is appropriate to make it equal to -∞ (or perhaps -1). The fact that the kinetic degree of the zero polynomial is a negative value is relevant for the definition of the C^0_ℓ norm given below in Definition <ref>.



 §.§ Kinetic Hölder spaces


We recall here the kinetic Hölder spaces introduced in
<cit.>. 


  Given an
  open set D ⊂^1+2d and a parameter  α∈ [0,∞), a continuous function f: D → is  α-Hölder continuous at a point  z_0 ∈^1+2d if there exists a polynomial  p ∈[t,x,v] such that p < α and for  any z ∈ D  
    |f(z) - p(z)| ≤ C d_ℓ(z,z_0)^α.
 When this property  holds at every point z_0 in the domain D, with a uniform  constant C, we say f ∈ C_ℓ^α(D). The semi-norm   [f]_C_ℓ^α(D) is the smallest value of the constant C so  that the inequality above holds for all z_0, z ∈ D. 

Note that with this definition [f]_C_ℓ^0(D) = f_C^0(D) = f_L^∞(D).
We define the norm  f_C_ℓ^α(D) to be [f]_C_ℓ^α(D)+f_C^0(D).



We recall the interpolation inequalities proven in
<cit.>.

  Given
  0 ≤α_1 < α_2 < α_3  so that
  α_2 = θα_1 + (1-θ)α_3, we have for all
  function f ∈ C_ℓ^α_3 (Q_r (z_0)),
  
    [f]_C_ℓ^α_2(Q_r (z_0))≤ C ( [f]_C_ℓ^α_1(Q_r(z_0))^θ [f]_C_ℓ^α_3(Q_r(z_0))^1-θ
        + r^α_1-α_2 [f]_C_ℓ^α_1(Q_r(z_0))),

for some constant C depending on α_1, α_3 and dimension only.



In this article, we will iteratively gain a priori estimates for solutions to the Boltzmann equation on Hölder spaces with increasingly large exponents α. We deal with global estimates that work for all v ∈^d. We need to keep track of the asymptotic behavior of these norms for large velocities. The most convenient way to do it is by considering functions in Hölder spaces with fast decay, that we define below.

  Given α∈ [0,∞), a
  function f: [τ,T] ×^d ×^d lies in ^α
  if, for all q >0 and all r ∈ (0,1], there exists C_q>0
  such that for all z ∈ [τ,T] ×^d ×^d, with
  Q_r (z) ⊂ [τ,T] ×^d ×^d, 
  
    f_C_ℓ^α (Q_r(z))≤C_q/(1+|v|)^q .

This is a locally convex vector space with the following family of semi-norms

    [f]_C_ℓ,q^α([τ,T] ×^d ×^d) := sup{ (1+|v|)^q [f]_C_ℓ^α (Q_r(z)) : r ∈ (0,1]  and  Q_r(z) ⊂ [τ,T] ×^d ×^d }.

We also write f_C^α_ℓ,q := f_C^0_ℓ,q + [f]_C^α_ℓ,q. Thus, a function f belongs to ^α when f_C^α_ℓ,q < ∞ for all q>0. 



Note that [f]_C^α_ℓ,q_1≥  [f]_C^α_ℓ,q_2 if q_1 ≥ q_2. Also, the norm f_C_ℓ^α (Q_r(z)) is monotone increasing with respect to r. It is pointless to consider small values of r in Definition <ref>. In practice, one would only take the largest r that the interval [τ,T] allows, which will often be r=1.



We know that the property of a function being Hölder continuous is local, but its Hölder norm is not (at least for non-integer exponents). The following lemma is useful to obtain a Hölder estimate in a large domain by covering with smaller patches where the Hölder norm is bounded.

 
Let α >0, r_0>0, f : Q_1 → be a bounded continuous function. Assume that for every z_0 ∈ Q_1, there is a polynomial p_z_0 of kinetic degree strictly less than α such that

    |f(z_0 ∘ξ) - p_z_0(ξ)| ≤ C_0 ξ^α,

whenever ξ≤ r_0 and z_0∘ξ∈ Q_1. Then f ∈ C^α_ℓ(Q_1) and

    [f]_C^α_ℓ(Q_1)≤ C_0 + C r_0^-α_Q_1 f,

for a constant C depending on α and dimension only. Here _Q_1 f = sup_Q_1 f - inf_Q_1 f.



The inequality we assume for |f(z_0∘ξ) - p_z_0(ξ)| when ξ<r_0 is identical to the one in Definition <ref>. We need to extend this inequality to every value of ξ so that z_0 ∘ξ∈ Q_1, regardless of whether ξ < r_0 or not.

Without loss of generality, let us assume inf_Q_1 f=0 (otherwise, repeat the proof below for f - inf_Q_1 f). Thus, in this case _Q_1 f = f_C^0(Q_1). In the following we simply write f_C^0 for f_C^0(Q_1).

For any point z_0 ∈ Q_1, let us analyze the polynomial p_z_0. We know that whenever ξ≤ r_0 and z_0 ∘ξ∈ Q_1, |f(z_0 ∘ξ) - p_z_0(ξ)| ≤ C_0 ξ^α. In particular, |p_z_0| ≤ C_0 r_0^α + f_C^0 at those points.

We use Lemma 2.8 in <cit.> (See also the proof of Lemma 2.7 and Proposition 2.10 in <cit.>), and get that for any point z_0 ∈ Q_1, the polynomial p_z_0 has the form

    p_z_0(z) = ∑_j ∈ℕ^1+2d a_j m_j(z),

where a_j ≠ 0 only for multi-indexes so that _k m_j < α, and moreover 

    |a_j| ≤ C ( C_0 r_0^α - _k m_j  + f_C^0 r_0^-_k m_j).


Thus, when z_0 ∘ξ∈ Q_1 but ξ > r_0, we estimate

    |f(z_0 ∘ξ) - p_z_0(ξ)|    ≤f_C^0 + |p_z_0(ξ)| , 
       ≤f_C^0 + ∑_j ∈ℕ^1+2d |a_j| ξ^_k m_j, 
       ≤f_C^0 + C ( C_0 r_0^α + f_C^0)  ∑_j ∈ℕ^1+2d
    _k m_j < α( ξ/r_0)^_k m_j, 
       ≲( C_0 r_0^α + f_C^0) ( ξ/r_0)^α, 
       = ( C_0 + r_0^-αf_C^0) ξ^α.

And we conclude the proof.



Comparing with classical Hölder spaces C^α, we observe that the estimate in Lemma <ref> is not optimal for large values of α. Consider for example the Lipschitz norm, that corresponds to α=1 and is purely local. So, the optimal inequality for the classical Lipschitz space would not have the second term in Lemma <ref>. Hölder norms are non-local, so some dependence on r_0 ought to be retained at least when α is not an integer (or α∉ℕ + 2s ℕ for kinetic Hölder spaces).


 
Let f, g ∈ C^α_ℓ(Q_1). Then fg ∈ C^α_ℓ(Q_1) and

    f g_C^α_ℓ(Q_1)≤ C f_C^α_ℓ(Q_1)g_C^α_ℓ(Q_1) ,

for a constant C depending on dimension and α only.



It is clear that the C^0 norm satisfies the inequality with constant C=1. We are left with verifying the inequality for the semi-norm [·]_C^α_ℓ. To that end, let z ∈ Q_1 and consider the polynomials p and q of kinetic degree less than α so that

    |f(z ∘ξ) - p(ξ)| ≤ [f]_C^α_ℓξ^α    and     |g(z∘ξ) - q(ξ)| ≤ [g]_C^α_ℓξ^α.

Thanks to Lemma <ref>, it is sufficient to consider the case ξ < 1. We have

    |f(z ∘ξ) g (z ∘ξ) - p(ξ) q(ξ)|    ≤ |f(z ∘ξ)||g(z ∘ξ) - q(ξ)| + |f(z∘ξ) - p(ξ)| |q(ξ)|, 
       ≤( [g]_C_ℓ^αf_C^0 + [f]_C^α_ℓq_C^0) ξ^α, 
       ≲g_C^α_ℓf_C^α_ℓξ^α.

The last inequality holds for ξ < 1 due to the identification of the coefficients of q with derivatives of g and   <cit.> (see also  <cit.>).

The polynomial p(ξ)q(ξ) may have a kinetic degree higher than α. In that case, let r(ξ) be the sum of the terms in p(ξ)q(ξ) of kinetic degree larger than or equal to α. We  also see from the application of <cit.>, reasoning  term by term, that |r(ξ)| ≤f_C^α_ℓg_C^α_ℓξ^α whenever ξ < 1. Thus, the lemma follows.





§ KINETIC EQUATIONS WITH INTEGRAL DIFFUSION





 §.§ The kernel associated with the Boltzmann equation



Like in (<ref>), we use the decomposition of the Boltzmann collision operator described in <cit.> and suggested earlier in <cit.>. We split Boltzmann's collision operator (f,f) appearing in (<ref>) as the sum of two terms = _1 + _2. The first term _1(f,f) is an integro-differential operator and _2(f,f) a lower order term,

    _1(f,f)    := ℒ_K_f f, 
    _2(f,f)    := c_b (f ∗ |·|^γ) f

where c_b is a positive constant only depending on the function b
appearing in (<ref>) and where the integro-differential diffusion operator ℒ_K_f is defined
as

    ℒ_K_f g (t,x,v) = ∫_^d (g(t,x,v') -g(t,x,v)) K_f (t,x,v,v') '.

The kernel K_f characterizing the operator ℒ_K_f is
given by the following formula

    K_f (v,v') = 2^d-1/|v'-v|∫_w ⊥ v'-v f(v+w) B(r,cosθ) r^-d+2 w    
    with  
    r^2 =|v'-v|^2 + |w|^2, 
    cosθ = w-(v-v')/|w-(v-v')|·w+(v'-v)/|w+(v'-v)| .
 
The following expressions are easier to handle in computations.

    K_f(v,v')    =  |v'-v|^-d-2s∫_w ⊥ v'-v f(t,x,v+w) A(|v'-v|,|w|)  |w|^γ+2s+1 w  
       ≈ |v-v'|^-d-2s∫_w ⊥ (v'-v) f(t,x,v+w) |w|^γ+2s+1 w

where A ≃ 1 is a bounded function only depending on the collision kernel B.

In Formula (<ref>) we omited the (t,x) dependence in K_f=K_f(t,x,v,v') and f=f(t,x,v). This is because for every fixed value of (t,x), we think of f(t,x,·) as a function of v and compute the kernel K_f accordingly by Formula (<ref>). Thus, if f=f(v) is a function of v only, the kernel K_f=K_f(v,v') depends on v and v'. When f depends on other parameters, so does K_f. In particular, K_f=K_f(t,x,v,v') when f=f(t,x,v). In the same spirit, we occasionally refer to Assumption <ref> for a function f=f(v) depending only on v as a way to state that its mass, energy and entropy are bounded by constants m_0>0, M_0, E_0 and H_0. This abuse of notation is convenient when stating lemmas that relate bounds for f with bounds for K_f.

There are two general regularity results for general kinetic integro-differential equations that we apply in this paper. The first one, given in <cit.>, is a Hölder estimate with a small exponent, in the style of the well known theorem of De Giorgi and Nash. It is, in some sense, the integro-differential version of the result in <cit.>. The second one, given in <cit.>, is a higher order Hölder estimate in the style of the classical result by Schauder for linear elliptic equations. We will iterate these Schauder-type estimates, and combine them with the large-velocity decay estimates from <cit.>, in order to obtain C^∞ estimates. Each of these two regularity results depends on different conditions on the diffusion kernel K_f. In the next two sections, we discuss the assumptions for each of these results.

 
The theorems for kinetic integro-differential equations in the style of De Giorgi/Nash (explained below in Section <ref>) and the Schauder-type results (described in Section <ref>) depend on assumptions on the kernel that look rather different from each other. These assumptions are best understood by comparing them with the corresponding conditions for the Landau equation, in terms of classical second order diffusion, that formally correspond to the limit as s → 1.

The in-homogeneous Landau equation is also a kinetic equation of the form

    ∂_t f + v·∇_x f = 𝒬_L(f,f),

where the collision operator 𝒬_L(f,f) involves second order derivatives of the function f. It can be written in divergence form

    𝒬_L(f,f) = ∂_v_i( a^f_ij∂_v_j f + b^f_i f ).

The Hölder estimates (as in <cit.>), obtained following De Giorgi method, depend on this expression and on the uniform ellipticity conditions on the diffusion matrix a^f_ij (that depend on the solution f itself through its hydrodynamic quantities). The term ∂_v_i (b^f_i f) is of lower order.

The application of a Schauder type estimate (for example as in <cit.>) would depend on the expression of 𝒬_L(f,f) in non-divergence form. In the case of the Landau equation, it takes the form

    𝒬_L(f,f) = a^f_ij∂_v_i v_j f + c^f f.

Here, the lower order term is c^f f. The Schauder estimate depends on the diffusion coefficients (in this case a_ij^f) being uniformly elliptic and Hölder continuous.

The difference between the divergence and nondivergence structures in (<ref>) and (<ref>) translates in different structure assumptions for the diffusion kernel in their integro-differential counterpart. The two terms in the decomposition of the Boltzmann collision kernel (<ref>) correspond more naturally to (<ref>) than to (<ref>).

One can apply divergence-form techniques to integro-differential operators when these have a variational structure. It corresponds to cancellation conditions between  K(t,x,v,v') and K(t,x,v',v). Ideally, the case of symmetry of the form K(t,x,v,v') = K(t,x,v',v) would correspond to an integro-differential operator in divergence form without lower order terms. However, we see in (<ref>) that there is a first order lower order term in the Landau equation. In Section <ref> we will state the precise cancellation conditions for K(t,x,v,v') - K(t,x,v',v) so that the asymmetry in the kernel is of lower order than the diffusion. The Hölder estimate in the style of the theorem of De Giorgi, Nash and Moser in Section <ref> depends on this cancellation condition.

One can apply nondivergence techniques to integro-differential operators when their structure allows us to make pointwise estimates. It corresponds to the cancellation condition K(t,x,v,v+w) = K(t,x,v,v-w). The Boltzmann kernel K_f satisfies this symmetry by construction. Thus, the application of nondivergence techniques to the Boltzmann equation is more direct. This will be reflected in the assumptions of the Schauder-type estimate described in Section <ref>.





 §.§ The local Hölder estimate




A local Hölder estimate for a general class of kinetic equations
with integral diffusion was obtained in <cit.> following classical ideas from De Giorgi. It applies to equations of the form

    ∂_t g  + v ·∇_x g= ℒ_K g +h

where h is a given source term and ℒ_K is an integral operator of the form

    ℒ_K g (t,x,v) = ∫_^d (g(t,x,v') -g(t,x,v)) K
    (t,x,v,v') '

associated with a (non-negative) kernel K(t,x,v,v') defined in
(-1,0] × B_1 × B_2 ×^d. 

The Hölder estimates for kinetic integro-differential equations developed in <cit.> are a result comparable to the theorem of De Giorgi, Nash and Moser for elliptic or parabolic equations in divergence form. These regularity estimates are independent of any well-posedness questions. It does not matter where the kernel K comes from, weather it depends on f or not, or how smooth it is with respect to any of its parameters. It is a result that only requires some uniform ellipticity conditions on the kernel K that we describe below. 

The following list of assumptions must be met uniformly in t and x. In order to keep the formulas short, we omit their dependence on t and x.



  
Non-degeneracy conditions


    For all v ∈ B_2 and r>0, inf_|e| =1∫_B_r(v) ((v'-v) · e)^2_+ K(v,v')  v' ≥λ r^2-2s.
          For any f supported in B_2, ∬_B_2 ×^d f(v)(f(v)-f(v')) K(v,v')  v'  v ≥λf_Ḣ^s(^d)^2 - Λf^2_L^2(^d).

The first non-degeneracy condition (<ref>) is necessary only
for s < 1/2. It is not clear if the second condition
(<ref>) may actually follow from (<ref>). In
practice, (<ref>) is usually much easier to check than
(<ref>).



  
Boundedness conditions


    For all v ∈ B_2 and r>0,    ∫_^d ∖ B_r(v) K(v,v')  v' ≤Λ r^-2s.
          For all v' ∈ B_2 and r>0,    ∫_^d ∖ B_r(v') K(v,v')  v ≤Λ r^-2s.




  
Cancellation conditions


    For all v ∈ B_7/4,    |∫_B_1/4(v)( K(v,v') - K(v',v) )  v' |≤Λ.
          For all r ∈ [0,1/4] and  v ∈ B_7/4,    |∫_B_r (v)( K(v,v') - K(v',v) )(v'-v)  v' |≤Λ (1+r^1-2s).

The second cancellation condition (<ref>) is necessary only for s ≥1/2. 

The cancellation conditions (<ref>) and (<ref>) correspond to the representation of the integral diffusion as a divergence form operator with a lower order asymmetry (see Remark <ref> above).

The nondegeneracy and boundedness conditions (<ref>), (<ref>), (<ref>) and (<ref>) correspond to the uniform ellipticity of the integral diffusion kernel. In practice, the most difficult to verify is the coercivity assumption (<ref>).




  Let
  K:(-1,0] × B_1 × B_2 ×^d →
  [0,+∞)
  be a kernel satisfying the ellipticity conditions (<ref>)
  (only if s < 1/2), (<ref>), (<ref>),
  (<ref>), (<ref>), (<ref>)
  (only if s ≥1/2).  Let f:(-1,0] × B_1 ×^d be
  a solution of (<ref>) in Q_1 for some bounded function
  h. Assume also that f is bounded in
  (-1,0] × B_1 ×^d. Then f is Hölder continuous in
  Q_1/2 and the following estimate holds
  
    [f]_C^α_ℓ (Q_1/2)≤ C ( f_L^∞
        ((-1,0]× B_1 ×^d) + h _L^∞ (Q_1))

  where α∈ (0,1) and C>0 only depend on dimension d, and the
  constants λ and Λ appearing in the assumptions.



In <cit.>, we verified that the Boltzmann kernel K_f (given in (<ref>)) satisfies (locally in v) the assumptions (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) with parameters depending only on the hydrodynamic constants m_0, M_0, E_0 and H_0 of Assumption <ref>. 

 
  In <cit.>, the Hölder estimate is obtained in a
  classical Hölder space for some α>0 sufficiently small. Such a Hölder estimate implies an estimate in the kinetic Hölder space used in the present work at the expense of reducing the exponent α by a factor min(2s,1). This is because for any two points z,z_0 ∈× B_1 ×^d such that d_ℓ(z,z_0)<1, we have |z-z_0| ≤ C d_ℓ(z,z_0)^min (1,2s). Indeed, when |v_0| ≤ 1, we have
 
    |z -z_0|     = |t-t_0| + |x-x_0| + |v-v_0| 
       ≤  (1+ |v_0|)|t-t_0| + |x-x_0 -(t-t_0)v_0 | + |v-v_0|  
       ≤ C( d_ℓ (z,z_0)^2s +d_ℓ (z,z_0)^1+2s+d_ℓ (z,z_0))

  for some constant C only depending on s.  In particular, if for all z_0,z ∈ Q_1/2, we have
  |f(z)-f(z_0)| ≤ C_α |z-z_0|^α, then
  |f(z)-f(z_0)| ≤C̃ C_α d_ℓ (z,z_0)^α̃ with
  α̃ = min (1,2s) α. The constant C̃ only
  depends on s and α.




 §.§ A Schauder estimate for kinetic integro-differential equations




The classical Schauder estimates for elliptic or parabolic equations of second order apply whenever we have an equation with uniformly elliptic and Hölder continuous coefficients. In <cit.>, we obtained a Schauder-type estimate for kinetic equations with integro-differential diffusion like (<ref>) in non-divergence form. The result depends on the kernel satisfying different
ellipticity conditions than the ones ensuring the local Hölder
estimate (Theorem <ref>). In some sense, the conditions described below reflect that the integro-differential equation is in non-divergence form and the kernel has a Hölder continuous dependence with respect to (t,x,v). They should be understood from the perspective described in Remark <ref> above.

There are two types of conditions that are necessary for a Schauder-type estimate: uniform ellipticity and Hölder continuity of coefficients. We should think of the kernel K(t,x,v,v') as a map from the first three variables (t,x,v) to a kernel depending on a single parameter w ∈^d given by K_(t,x,v) (w) := K(t,x,v,v+w). The uniform ellipticity assumption will say that for every value of (t,x,v), the kernel K_(t,x,v) belongs to a certain ellipticity class. The Hölder continuity will say that for two different values z_1=(t_1,x_1,v_1) and z_2 = (t_2,x_2,v_2), the kernels K_z_1 and K_z_2 are in some sense at distance ≲ d_ℓ(z_1,z_2)^α.

Let us recall the ellipticity class of order 2s defined in <cit.>.

 
Let s ∈ (0,1).   A non-negative kernel K: ^d → belongs to the ellipticity class 𝒦 if


  i. K(w) = K(-w), 

  ii. For all r>0, ∫_B_r |w|^2 K(w) ≤Λ r^2-2s, 

  iii. For all R>0 and φ∈ C^2 (B_R), 
  
    ∬_B_R × B_R (φ (v') -φ (v))^2 K (v'-v) ' ≥λ∬_B_R/2× B_R/2 (φ (v') -φ (v))^2 |v'-v|^-d-2s'  .


  iv. For any r>0 and e ∈ S^d-1, 
    ∫_B_r (w · e)_+^2 K(w) ≥λ r^2-2s.




Some remarks are in order. 


  * Definition <ref> is borrowed from <cit.>. However, the definition in that paper is more general since K(w)  w is supposed to be a nonnegative Radon measure that is not necessarily absolutely continuous. For the purpose of this paper, because we deal with classical solutions, our kernel K will always be given by a non-negative density function and we do not need to deal with singular measures.

  * The last two items iii. and iv. might be redundant. Indeed, we do not know any example of a kernel satisfying i. and ii. and either iii. or iv., without satisfying all of them. This is related to the problem of coercivity for integro-differential operators. See the discussion in <cit.>, <cit.>, <cit.> and <cit.>. Item iv. is in practice much easier to verify than iii.

  * Earlier works on integro-differential equations concentrated on a more restricted class of kernels that were pointwise comparable to the fractional Laplacian: K(w) ≈ |w|^-d-2s. This traditional assumption does not suffice to study the Boltzmann equation. The diffusion kernel that appears in the Boltzmann equation belongs to the more general class of Defintion <ref>, with parameters depending on the constants in Assumption <ref>.


The condition on the Hölder dependence of K_z with respect to the point z is given in the assumption (<ref>) below.



  Let s ∈ (0,1) and α∈ (0, min (1,2s)) and
  α' = 2s/1+2sα. Let
  K: (-(2r)^2s,0] × B_(2r)^1+2s×^d ×^d → [0,+∞) such
  that for all z =(t,x,v) ∈ (-(2r)^2s,0] × B_(2r)^1+2s×^d, the
  kernel K_z (w) = K(t,x,v, v+w) belongs to the ellipticity class
  𝒦 from Definition <ref>. Assume moreover that for
  all z_1,z_2 ∈ Q_2r and all ρ >0,
  
    ∫_B_ρ|K_z_1 (w) - K_z_2 (w) |   |w|^2   w
      ≤ A_0 ρ^2-2s d_ℓ (z_1,z_2)^α'

  with z_i =(t_i,x_i,v_i) for i=1,2. 

If f ∈ C_ℓ^α ((-(2r)^2s,0] × B_(2r)^1+2s×^d) solves (<ref>) in Q_2r, then 

    [f]_C_ℓ^2s+α' (Q_r)≤ C (    max( r^-2s-α'+α, A_0^2s+α'-α/α') [f]_C_ℓ^α ((-(2r)^2s,0] × B_(2r)^1+2s×^d)
        + [h]_C_ℓ^α' (Q_2r) + max( r^-α', A_0 )  h_C^0(Q_2r)).

where the constant C depends on d,s, and the constants
λ, Λ from the definition of 𝒦.



The main result in <cit.> is for r=1 and the constant C depends on A_0 in an unspecified way. In order to justify (<ref>) we work out explicitly its dependence on r and A_0. It is a consequence of Theorem 1.6 in <cit.> combined with a scaling argument. Indeed, let S_r(t,x,v) = (r^2s t, r^1+2s x, rv). This is the natural scaling that maps Q_1 into Q_r. The function f ∘ S_r satisfies the scaled equation

    (∂_t + v ·∇_x ) [f ∘ S_r] = ℒ_K̃ + r^2s h ∘ S_r,

where

    K̃_z(w) = r^d+2s K_S_r z (rw).

We point out that K̃ satisfies assumption (<ref>) with r^α' A_0 instead of A_0. Indeed,

    ∫_B_ρ (K̃_z_1(w) - K̃_z_2(w)) |w|^2  w    = ∫_B_ρ r^d+2s( K_S_r z_1 (rw) - K_S_r z_2 (rw) ) |w|^2  w, 
       = r^2s-2∫_B_rρ( K_S_r z_1 (w̅) - K_S_r z_2 (w̅) ) |w̅|^2 w̅, 
       ≤ A_0 r^2s-2 (r ρ)^2-2s d_ℓ(S_r z_1, S_r z_2)^α' = (A_0 r^α') ρ^2-2s d_ℓ(z_1,z_2)^α'.


Provided that r^α' A_0 ≤ 1, we apply Theorem 1.6 in <cit.> (same as Theorem <ref> but for r=1 and constants depending implicitly on A_0). We get

    [f ∘ S_r]_C_ℓ^2s+α' (Q_1)≤ C([f ∘ S_r]_C_ℓ^α ((-2^2s,0] × B_2^1+2s×^d) + r^2s h ∘ S_r _C_ℓ^α' (Q_2)).

We can take a universal constant C (depending on d, λ, Λ and s only) provided that A_0 r^α'≤ 1. In that case, we scale back to express (<ref>) in terms of the original functions f and h to obtain

    [f]_C_ℓ^2s+α' (Q_r)≤ C (r^α-2s-α' [f]_C_ℓ^α ((-(2r)^2s,0] × B_(2r)^1+2s×^d) + [h]_C_ℓ^α' (Q_2r) + r^-α'h_C^0(Q_2r)),

provided that A_0 r^α'≤ 1.

If A_0 r^α' > 1, we should further look at a smaller scale r̃ < r so that A_0 r̃^α' = 1. In that case, the inequality (<ref>) holds with r̃ instead of r, and for any cylinder Q_r̃(z_0) ⊂ Q_r. Taking into account Lemma <ref>, we get

    [f]_C_ℓ^2s+α' (Q_r)≤ C (r̃^α-2s-α' [f]_C_ℓ^α ((-(2r)^2s,0] × B_(2r)^1+2s×^d) + [h]_C_ℓ^α' (Q_2r) + r̃^-α'h_C^0(Q_2r)),


Taking into account that 1/r̃ = A_0^1/α', we get

    [f]_C_ℓ^2s+α' (Q_r)≤ C ( A_0^2s+α'-α/α' [f]_C_ℓ^α ((-(2r)^2s,0] × B_(2r)^1+2s×^d) + [h]_C_ℓ^α' (Q_2r) + A_0 h_C^0(Q_2r)).


Combining (<ref>) with (<ref>) we obtain (<ref>).



The Boltzmann kernel K_f,(t,x,v) as in (<ref>) belongs (locally) to the class 𝒦 provided that Assumption <ref> holds. This follows from computations that are in the literature. Indeed, at least when v stays in a bounded domain, we have


  i. The symmetry of the Boltzmann kernel is immediate by construction. We see in the formula (<ref>) that K_f(t,x,v,v+w) = K_f(t,x,v,v-w).

  ii. The condition ii. in Definition <ref> tells us that the kernels K ∈𝒦 are bounded in an averaged sense. It is a weaker condition than the more classical pointwise bound K(w) ≤Λ |w|^-d-2s. By a simple computation, we can verify that it is equivalent to any of the following two alternative formulations (see  <cit.>)

    ∫_^d ∖ B_r K(w)  w    ≤Λ r^-2s, 
    ∫_B_2r∖ B_r K(w)  w    ≤Λ r^-2s.

In each case, the inequality is supposed to hold for all r>0 and the value of Λ may need to be adjusted by a dimensional constant when passing from one formulation to another.

It is the same inequality as in the assumption (<ref>) for Theorem <ref>.

This boundedness assumption for a kernel K, together with the symmetry condition i. in Definition <ref> allows us to estimate the value of the integro-differential operator ℒ_K f pointwise. See Lemma <ref> below.

  iii. Using that γ + 2s ∈ [0,2], the integral upper bound on item ii. in Definition <ref> holds for the Boltzmann kernel K_f (at least locally) according to <cit.>. Indeed, that lemma says that for any f: ^d →,

    ∫_^d ∖ B_r K_f(v,v+w)  w    ≲( ∫_^d f(v+w) |w|^γ+2s w ) r^-2s, 
    applying Lemma <ref>,   ≲ ((1+|v|)^γ+2s M_0 + E_0) r^-2s.


  iv. The coercivity condition for K_f,(t,x,v) is easier to verify than the usual coercivity estimates for the Boltzmann equation. This is because K_f,(t,x,v) depends on the single variable w ∈^d. We should think of the kernel K_f,(t,x,v) as what we get from the original kernel K_f by freezing coefficients. The coercivity estimate iii. in Definition <ref> is a direct consequence of the existence of a cone of nondegeneracy described in <cit.> combined with the coercivity conditions from <cit.> or from <cit.>.

  v. The last nondegeneracy assumption, i.e. item iv. in Definition <ref>, is a straight-forward consequence of the existence of a cone of nondegeneracy described in <cit.>.


An important difficulty is apparent at this point: the constants Λ in ii. and λ in iii. and iv. deteriorate as |v| →∞. The kernel K_f,(t,x,v) belongs to an ellipticity class only locally in v. In order to control the asymptotic behavior of all our regularity estimates, it will be important to establish precise asymptotics on the ellipticity of the kernel as v →∞. The same difficulty arises in regards to the assumptions for Theorem <ref>. A change of variables will be described in Section <ref> that addresses this difficulty.

Now we state and prove the lemma mentioned above about pointwise bounds for ℒ_K f. For more applicability, we state it for kernels K satisfying only i. and ii. in Definition <ref>, and that might even change sign. It is related to the   <cit.>.

 
Let K: ^d → be a symmetric kernel (i.e. K(w) = K(-w)) so that 

    ∫_^d ∖ B_r |K(w)|  w ≤Λ r^-2s.

Consider the integro-differential operator ℒ_K,

    ℒ_K f(v) = ∫_^d (f(v+w) - f(v)) K(w)  w.

If f is bounded in ^d and C^2s+α at v for some α∈ (0,1), then

    | ℒ_K f(v) | ≤ C Λ |f|_C^0(^d)^α/2s+α [f]_C^2s+α(v)^2s/2s+α.

The constant C depends on dimension, s and α.


We use the standard notation [·]_C^α(v) to denote the smallest value of N ≥ 0 so that there exists a polynomial q of degree strictly less than α so that |f(v+w) - q(w)| ≤ N|w|^α for all w ∈^d.
Note that 2s+α may be larger than 2 in Lemma <ref>.


The fact that f is C^2s+α at the point v means that


  * |f(v+w) - f(v)| ≤ [f]_C^2s+α(v) |w|^2s+α if 2s+α∈ (0,1].

  * |f(v+w) - f(v) - w ·∇ f(v) | ≤ [f]_C^2s+α(v) |w|^2s+α if 2s+α∈ (1,2].

  * |f(v+w) - f(v) - w ·∇ f(v) - 1/2 w_i w_j ∂_ij f(v)| ≤ [f]_C^2s+α(v) |w|^2s+α if 2s+α∈ (2,3].

For some r>0 to be determined later, we use the inequalities above to estimate the part of the integral where w ∈ B_r. Note that the term w ·∇ f(v) is odd in w. Since the kernel K is symmetric, it vanishes in the principal value. We have

    |∫_B_r    (f(v+w) - f(v)) K(w)  w |
       ≤∫_B_r( [f]_C^2s+α(v) |w|^2s+α + {1/2 w_i w_j |∂_ij f(v)| }_if 2s+α>2) |K(w)|  w 
       ≲Λ( [f]_C^2s+α(v) r^α + { |D^2 f(v)|  r^2-2s}_if 2s+α>2).
    If 2s+α > 2, we use the (classical) interpolation inequality in the full space |D^2 f(v)| ≤ [f]_C^2s+α(v)^2/2s+α  |f|_C^0(^d)^2s+α-2/2s+α,   ≲Λ( [f]_C^2s+α(v) r^α + { [f]_C^2s+α(v)^2/2s+α  |f|_C^0(^d)^2s+α-2/2s+α r^2-2s}_if 2s+α>2).

For the part of the integral w ∉ B_r, we bound |f(v+w) - f(v)| by 2|f|_C^0(^d). We get

    |∫_^d ∖ B_r (f(v+w) - f(v)) K(w)  w |   ≤ 2 |f|_C^0(^d)∫_^d ∖ B_r |K(w)|  w, 
       ≲Λ |f|_C^0(^d) r^-2s.

Adding up the two (or three if 2s+α>2) terms,

    |ℒ_K f(v)| ≤Λ( [f]_C^2s+α(v) r^α + { [f]_C^2s+α(v)^2/2s+α  |f|_C^0(^d)^2s+α-2/2s+α r^2-2s}_if 2s+α>2 + |f|_C^0(^d) r^-2s).

We finish the proof by choosing (the optimal) r>0 as

    r = ( |f|_C^0(^d)/[f]_C^2s+α(v))^1/2s+α.
 





§ THE CHANGE OF VARIABLES





 §.§ The change of variables


The ellipticity of Boltzmann's collision operator degenerates for large velocities.  This shows up, for example, in the weights of the well known coercivity estimates from <cit.>. Correspondingly, the constants λ and Λ in the assumptions of Theorem <ref> are bounded for K_f only locally in v. Likewise, if we want to apply Theorem <ref> to the Boltzmann kernel K_f given in (<ref>), the constants λ and Λ in Definition <ref> would only exist for a bounded set of velocities.

This is a major obstruction in order to obtain global regularity estimates using Theorems <ref> and <ref>. Moreover, global regularity estimates are crucial in order to carry out an iterative gain of regularity. The constants in Theorems <ref> and <ref> do not have an explicit dependence on the parameters λ and Λ in the assumptions. There is no hope to obtain a global regularity estimate unless we are able to apply these theorems with fixed values of the ellipticity parameters λ and Λ for all velocities in ^d.

In this section, we describe a change of variables that resolves this difficulty. For any point z_0 = (t_0,x_0,v_0) ∈^1+2d, we construct a function 𝒯_0 that maps the kinetic cylinder Q_1 into a product of ellipsoids centered at z_0. Moreover, the function f ∘𝒯_0 satisfies a kinetic integro-differential equation whose kernel is elliptic with constants λ and Λ (either in the sense of Theorem <ref> or Theorem <ref>) depending only on the constants m_0, M_0, E_0, H_0, s and dimension, but not on v_0.

This change of variables allows us to turn our local regularity results (as in Theorems <ref> and <ref>) into global estimates with precise asymptotics as |v| →∞. It is a key tool for the proofs of the main results in this paper. It was motivated by a similar change of variables for the Landau equation from <cit.>.

In order to illustrate the significance of this change of variables, we show in Appendix <ref> how it can be used to derive the global coercivity estimate with respect to the anisotropic distance obtained by Gressman and Strain in <cit.> and <cit.>.

Given t_0 ∈, x_0 ∈^d and v_0 ∈^d, we consider the transformed
function

    f̅(t,x,v) := f(t̅, x̅, v̅)

with (t̅, x̅, v̅) =𝒯_0(t,x,v). The transformation 𝒯_0 depends on the reference point z_0 = (t_0,x_0,v_0) ∈^1+2d. If |v_0| < 2, we will simply take 𝒯_0 z := z_0 ∘ z. When |v_0| ≥ 2, which is the important case, we define

    (t̅, x̅, v̅) =𝒯_0(t,x,v)    := (t_0+t/|v_0|^γ+2s, x_0 +
    T_0x + t v_0/|v_0|^γ+2s , v_0 + T_0v) , 
       = z_0 ∘ (|v_0|^-γ-2s t, |v_0|^-γ-2s T_0 x, T_0 v)

and  T_0 : ^d →^d is the following transformation:

    T_0(a v_0 + w) := a/|v_0| v_0 + w     for all  w ⊥ v_0, a ∈.

Note that T_0 maps B_1 into an ellipsoid E_1 with radius
1/|v_0| in the direction of v_0 and 1 in the directions
perpendicular to v_0. For consistency, let us also define T_0 as the identity operator whenever |v_0| < 2. The following sets are
naturally associated with the change of variables. For
z_0 ∈^1+2d and r >0, we consider

    ℰ_r (z_0 ) = 𝒯_0 (Q_r),    E_r (v_0) = v_0 +
      T_0 (B_r).

The set of velocities E_r (v_0) is an ellipsoid in ^d. 
The linear operator 𝒯_0 maps Q_1 into
ℰ_1(z_0):= ℰ_1^t,x (z_0)× E_1(v_0) where
ℰ_1^t,x(z_0)={ (t_0+|v_0|^-γ-2s t, x_0 +
|v_0|^-γ-2s (T_0x + t v_0)):t ∈ [-1,0], x ∈ B_1} is a
slanted cylinder. 

By a direct computation, we verify that if f satisfies the Boltzmann
equation in ℰ_1(z_0) then f̅ solves the equation

    ∂_t f̅ + v ·∇_x f̅ = ℒ_K̅_ff̅ + h̅,    (t,x,v) ∈ Q_1
 
where

    K̅_f(t,x,v,v') = |v_0|^-1-γ-2s K_f(t̅, x̅, v̅, v_0+T_0v')

and 

    h̅ (t,x,v)= c_b |v_0|^-γ-2s f(t̅,x̅,v̅) (f ∗  |·|^γ ) (t̅,x̅,v̅).
 


The point of this change of variables is to straighten up the ellipticity of the Boltzmann kernel K_f. The following theorem says that we are able to apply the Hölder estimates of Theorem <ref> with uniform ellipticity constants to f̅.

 
  Let z_0 = (t_0,x_0,v_0) and  ℰ_1(z_0) = ℰ_1^t,x (z_0) × E_1 (v_0) be defined as above. Assume that Assumption <ref> holds for all (t,x) ∈ℰ_1^t,x (z_0), and
  
    if γ<0,     sup_v ∈^d∫_B_1 f(v+u) |u|^γ u ≤ C_γ .

Then the kernel K̅_f satisfies (<ref>) (only if s<1/2), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) (only if s ≥ 1/2), with constants depending on d,s,γ and m_0, M_0, E_0 and H_0 (and C_γ if γ<0) only, uniformly with respect to v_0. 




  Condition (<ref>) is weaker than imposing  an L^∞
  bound for f. Such a bound is proved in <cit.> for
  solutions of the Boltzmann equation for t>0 and it only depends on
  the hydrodynamic quantities appearing in Assumption (<ref>)
  when γ +2s ∈ [0,2].


Note that our computation works for γ+2s ∈ [0,2]. For values
of s and γ away of that range, we would need further
assumptions on either integrability of f (for γ +2s < 0) or
higher moments (if γ+2s>2).


We also have a corresponding result for the ellipticity assumptions of the Schauder-type estimates in Theorem <ref>.

 
  Let z_0 = (t_0,x_0,v_0) with  v_0 ∈^d and ℰ_1(z_0) = ℰ_1^t,x (z_0) × E_1 (v_0)  be defined as above. Assume that Assumption <ref> holds for all (t,x) ∈ℰ_1^t,x (z_0). Then, for every z=(t,x,v) ∈ Q_1, the kernel K̅_f,z(w) = K̅_f(t,x,v,v+w) belongs to the class 𝒦 of Definition <ref>. The constants λ and Λ in Definition <ref> depend on d,s,γ and m_0, M_0, E_0 and H_0 (and C_γ if γ<0) but not on v_0. 


The proofs in this section largely consist in direct computations to verify the claims. However, rather involved manipulations of multiple integrals are needed, especially for the proof of the second cancellation condition in Lemma <ref>.

 
When |v_0|  ≤ 2, there is no need for a change of variables. All our ellipticity conditions hold for any arbitrary (but fixed) bounded set of velocities. The results of Theorems <ref> and <ref> are already established in <cit.> and <cit.> for |v_0| < 2. Here, we need to prove them for |v_0|≥ 2. The purpose of the change of variables 𝒯_0 is to analyze the asymptotic behavior of the estimates for large values of |v_0|. Thus, the case |v_0| ≥ 2 is the important one. Yet, we define 𝒯_0 for any value of v_0 for consistency. The change of variables 𝒯_0 does not modify the equation at all when |v_0|<2.
 




 §.§ Non-degeneracy conditions




The nondegeneracy condition (<ref>) and the coercivity condition (<ref>) are a consequence of the existence of a cone of nondegeneracy described in Proposition <ref>.

Proposition <ref> describes a set of directions A(v) depending on each point v ∈^d, along which the kernel K_f has a lower bound. Using the notation introduced in <cit.>, we call the cone of nondegeneracy (v). Here

    (v) := { w : w/|w|∈ A(v) }.


Proposition <ref> says that for each value of v ∈^d, the set of directions A(v) ⊂ S^d-1 is contained in a strip of width ≈ 1/(1+|v|) around the equator perpendicular to v, with measure ≳ 1/(1+|v|), so that K(v,v+w)  ≳ (1+|v|)^1+γ+2s |w|^-d-2s whenever w belongs to (v).

Naturally, there is a cone of nondegeneracy for K̅_f corresponding to the cone of nondegenerate directions for K_f. Indeed, we write

    (v)    = { w ∈^d : T_0 w ∈(v_0 + T_0 v)}, 
    A̅(v)    = {σ∈ S^d-1 : T_0 σ / |T_0 σ| ∈ A(v_0 + T_0 v)}.

By construction, we have that w ∈(v) if and only if w/|w| ∈A̅(v). Moreover, (v_0 + T_0 v) = T_0((v)).

The following lemma tells us that K̅_f has its nondegenerate directions A̅(v), and both its lower bound K̅_f and the volume of A̅(v) are independent of the center point v_0 of the change of variables.

 
Let f be a function such that Assumption <ref> holds. Let v_0 ∈^d and v ∈ B_2, with A̅(v) and (v) defined as above. Then


  * K̅_f(v,v+w) ≥λ |w|^-d-2s whenever w ∈(v).

  * |A̅(v)| ≥μ̅ for some μ̅>0 depending on the parameters of Assumption <ref> and dimension, but not on v_0.




Proposition <ref> immediately implies the result of this lemma when |v_0| ≤ 2. In order to prove it for |v_0| ≥ 2, we need to analyze the interaction of the change of variables with the bounds in Proposition <ref>.

We first check the first item in the lemma. Pick w such that w ∈ (v), i.e. T_0w ∈(v_0 + T_0 v). Then 

    K̅_f(v,v+w) = 1/|v_0|^1+γ+2s K_f(v_0+T_0 v, v_0+T_0 v + T_0 w) ≥λ |T_0 w|^-d-2s   ≥λ |w|^-d-2s.

For the last inequality, we used the fact that |T_0(w)| ≤ |w|.

We are left with checking the second item. Note that A(v_0 + T_0 v) and A̅(v) are subsets of S^d-1 related by the nonlinear map σ↦ T_0 σ / |T_0 σ|. In order to relate their volumes, we would have to make a computation involving the Jacobian of the map, which in this case is the determinant of the derivatives that act on the tangent space of S^d-1. This kind of computations are confusing to the best of us. So, instead, we opt to estimate the volume of A̅(v) through (v). Indeed, the following elementary formula allows us to relate the volume of a set of directions with the volume of the corresponding cone. For any R>0, 

    | A(v_0 + T_0 v) |_ℋ^d-1 = d/R^d | (v_0 + T_0 v) ∩ B_R|     and      |A̅(v) |_ℋ^d-1 = d/R^d | (v) ∩ B_R|.


Combining this formula with the estimate in Proposition <ref>, we have, for any R>0,

    | (v_0 + T_0 v) ∩ B_R| ≥R^d/dμ (1+|v_0 + T_0v|)^-1.


Let us recall the definition of T_0. Given any w ∈^d, we write it as w = a v_0 + w^⊥ with v_0 · w^⊥ = 0. Then T_0 w = a v_0 / |v_0| + w^⊥. We want to estimate an upper bound for a under the condition that T_0 w ∈(v_0 + T_0 v) ∩ B_R.

According to the width condition in Proposition <ref>, if T_0 w ∈(v_0 + T_0 v), we must have

    |T_0 w/|T_0 w|· (v_0 + T_0 v) |≤ C_0.

Replacing the formula T_0 w = a v_0 / |v_0| + w^⊥, and recalling w^⊥· v_0 = 0, we get

    |a|/|T_0 w| |v_0| ≤ C_0 + |T_0 v| ≤ C_0 + 2.

If in addition we know that |T_0 w| ≤ R, we conclude that

    |a| ≤(C_0 + 2) R /|v_0|.

Now, for every w such that T_0 w ∈(v_0 + T_0 v) ∩ B_R, we have

    |w|    = √(a^2 |v_0|^2 + |w^⊥|^2) , 
       ≤ a |v_0| + |w^⊥| , 
       ≤ R(C_0+3).

Let us pick R = (C_0+3)^-1, which is a constant depending on the parameters of Assumption <ref> and dimension only. We deduce that

    T_0^-1 ( (v_0 + T_0 v) ∩ B_R) ⊂ B_1.

Therefore, 

    |(v) ∩ B_1|    ≥ |T_0^-1 ( (v_0 + T_0 v) ∩ B_R) | , 
       = (  T_0^-1 ) |(v_0 + T_0 v) ∩ B_R | , 
       ≥ |v_0| (1+|v_0 + T_0 v|)^-1μ R^d / d     using (<ref>) and Proposition <ref>, 
       ≥μ̅

for some constant μ̅>0 depending only on the constants μ and C_0 of Proposition <ref> and dimension (and not on v_0).



When f satisfies Assumption <ref>, the kernel K̅_f satisfies
  (<ref>) and also the inequality

    ∬_B_1 × B_1 (g(v') - g(v))^2 K̅_f(v,v')  v'  v ≥λ∬_B_1/2× B_1/2(g(v') - g(v))^2/|v'-v|^d+2s v'  v

for a parameter λ>0 depending on the constants in Assumption <ref> (uniform with respect to (t_0,x_0,v_0)).



The cone of nondegeneracy described in Lemma <ref> trivially implies (<ref>) for K̅_f. It also fulfills the assumption of Theorem <ref>, from which the inequality (<ref>) follows.


Recall that for z=(t,x,v),  K̅_f,z(w) denotes the kernel K̅_f(t,x,v,v+w).

When f satisfies Assumption <ref>,  the kernel K̅_f,z satisfies for any z ∈ Q_1  the coercivity conditions iii. and iv. in Definition <ref> for a parameter λ>0 uniform with respect to (t_0,x_0,v_0).



Note that the statement of Corollary <ref> is not the same as (<ref>). One is for the kernels K̅_f,z : ^d →^d, depending on a single parameter w ∈^d. The other is for the kernels K̅_f as a function of both v and v'. The dependence of t,x is irrelevant for either statement. The coercivity condition given in Corollary <ref> is in some sense simpler than (<ref>) given in Corollary <ref> since it applies to the kernel with frozen coefficients (referring to standard terminology from elliptic PDEs). Theorem <ref> is sufficiently strong that it implies both Corollaries <ref> and <ref>. One could alternatively justify Corollary <ref> using coercivity results that are suitable for translation invariant integro-differential operators like the ones described in <cit.>.


Let K̃(w) := K̅_f,z(w). The cone of nondegeneracy described in Lemma <ref> applies to K̅_f, and therefore also to K̃. In this case, we have

    K̃(w) ≥λ̃|w|^-d-2s     whenever  w ∈(z).

Here, (z) is the cone of nondegeneracy for K̅_f at the point z=(t,x,v). The frozen kernel K̃ has this cone of nondegeneracy at every point v ∈^d.

The coercivity condition iv. from Definition <ref> easily follows from the existence of the cone of non-degeneracy for K̃. 

The properties of condition of the cone of nondegeneracy described in Lemma <ref> imply the assumptions of Theorem <ref>. Therefore, we have

    ∬_B_2 × B_2 (φ(v') - φ(v))^2 K̃(v,v')  v'  v ≳λ̃∬_B_1× B_1 (φ(v') - φ(v))^2 |v'-v|^-d-2s v'  v.

Since the inequality (<ref>) is scale invariant, a standard scaling argument allows us to conclude (<ref>) for any R>0.






 §.§ First boundedness condition




  Let us assume that γ+2s ∈ [0,2] and an upper bound in mass and energy
  
    ∫_^d f(v)  v ≤ M_0,     ∫_^d f(v) |v|^2  v ≤ E_0.

  The kernel K̅_f from (<ref>) satisfies (<ref>) with parameters depending on M_0, E_0,  γ, s and d only.



We start with the following computation. 

 
  Let v_0 ∈^d ∖ B_2 and v ∈ B_2. For any r>0, we have 
  
    ∫_^d ∖ B_r(v)K̅_f(v,v')  v' ≲Λ̅r^-2s
       with    Λ̅=  |v_0|^-γ-2s∫_w∈^d f(v̅+w) ( |v_0|^2 - ( v_0 ·w/|w|)^2 + 1)^s |w|^γ+2s w .




In view of the defintion of K̅_f, we can write

    ∫_^d ∖ B_r(v)K̅_f(v,v')  v'    = ∫_^d ∖ E_r |v_0|^-1-γ-2s K_f(v̅,v̅ + u)  u/ T_0, 
       = |v_0|^-γ-2s∫_^d ∖ E_r K_f(v̅,v̅ + u)  u,

where v̅ = v_0 + T_0v and E_r=  T_0(B_r). The
set E_r is an ellipsoid centered at the origin with radius
r/|v_0| in the direction of v_0 and r in the directions
perpendicular to v_0. 


Using (<ref>), we rewrite the expression above as

    ∫_^d ∖ B_r(v)K̅_f(v,v')  v' 
       ≈ |v_0|^-γ-2s∫_u ∈^d ∖ E_r |u|^-d-2s( ∫_w ⊥ u f(v̅+w) |w|^1+γ+2s w )  u, 
       = |v_0|^-γ-2s∫_w∈^d( ∫_u ⊥ w, 
     u ∈^d ∖ E_r |u|^-d+1-2s u )  f(v̅+w) |w|^γ+2s w.

We used the fact that 

    ∫_u {∫_w⊥ u (…)  w } u = ∫_w {∫_u⊥ w (…) |u|/|w| u } w.


In order to estimate the integral in the inner factor, we analyze the intersection of the ellipsoid E_r with the hyperplane {u:u ⊥ w}. This is of course a (d-1)-dimensional ellipsoid whose dimensions are computable. Its smallest radius ρ equals

    ρ := r/√(|v_0|^2 ( 1- ( v_0 · w/|v_0| |w|)^2 ) + ( v_0 · w/|v_0| |w|)^2 ).

Indeed, it is an elementary planar computation: for u ∈∂ E_r with |u|=ρ and u ⊥ w, we write u = (u ·v̂_0) v̂_0 + u_1 with u_1 ⊥ v_0 and v̂_0 = v_0 / |v_0|. In particular, ρ^2 = (u ·v̂_0)^2 + |u_1|^2. From the definition of E_r and our choice of u, we know that the vector v = (u·v̂_0) |v_0| v̂_0 + u_1 is of norm r. We now write

    r^2    = (u ·v̂_0)^2 |v_0|^2 + |u_1|^2 
       = (u ·v̂_0)^2 (|v_0|^2-1) + ρ^2.

Use next the fact that u ⊥ w and get ρ^-2 (u ·v̂_0)^2 + (ŵ·v̂_0)^2 = 1 with ŵ = w / |w|. In particular,

    (u ·v̂_0)^2 = ρ^2 (1 - (ŵ·v̂_0)^2 ).

Combining the formulas for r^2 and (u ·v̂_0)^2 yields (<ref>).

Therefore

    ∫_u ⊥ w, 
     u ∈^d ∖ E_r |u|^-d+1-2s u
         ≤∫_u ⊥ w, 
     u ∈^d ∖ B_ρ |u|^-d+1-2s u ≲ρ^-2s
        = r^-2s( |v_0|^2 ( 1- ( v_0 · w/|v_0| |w|)^2 ) + ( v_0 · w/|v_0| |w|)^2 )^s 
       ≤ r^-2s( |v_0|^2 - ( v_0 · w/|w|)^2  + 1 )^s .

Substituting in our previous formula, we get the desired result.

We next aim at estimating the constant Λ̅ appearing in Lemma <ref>. 

 
  Let f : ^d → [0,∞), v_0 ∈^d ∖ B_1, |v - v_0|≤ 2 and s ≥ 0, then for all w ∈^d,
 
    |v_0|^-γ-2s∫_w ∈^d f(v+w) ( |v_0|^2 - ( v_0 ·w/|w|)^2 + 1 )^s |w|^γ+2s   ≤ C ∫_^d f(ṽ)( 1+|ṽ|^2s + |ṽ|^γ+2s) ṽ
       ≤ C (M_0+E_0)

for a constant C depending on s and dimension only (not on v_0).



It is enough to prove

    |v_0|^-γ-2s( |v_0|^2 - ( v_0 ·w/|w|)^2 + 1 )^s |w|^γ+2s   ≲  1+|v_0+w|^2s + |v_0+w|^γ+2s, 
       ≲ 1+|v+w|^2s + |v+w|^γ+2s.


  The second of these inequalities follows simply because |v-v_0| ≤ 2. We need to prove the first one.
Note that 

    ( |v_0|^2 - ( v_0 ·w/|w|)^2 + 1 )^s ≲( |v_0|^2 - ( v_0 ·w/|w|)^2 )^s + 1.

Moreover, 

    |v_0|^-γ-2s |w|^γ+2s≲ |v_0|^-γ-2s (|v_0+w|^γ+2s+|v_0|^γ+2s) ≤ 1+|v_0+w|^γ+2s,

using that |v_0|>1. We are left with studying an upper bound for 

    |v_0|^-γ-2s( |v_0|^2 - ( v_0 ·w/|w|)^2 )^s |w|^γ+2s.

It is convenient to write w = α v_0/|v_0| + b, where α∈ and b ∈^d is perpendicular to v_0. With this notation

    |v_0|^2 - ( v_0 ·w/|w|)^2 = |v_0|^2 |b|^2/|w|^2.

Thus, the expression in (<ref>) equals |v_0|^-γ |b|^2s |w|^γ. We need to study an upper bound for it. The key observation here is that |b| satisfies the two inequalities |b| ≤ |w| and |b| ≤ |v_0+w|, both of which are immediate from its definition.

If |w| ≤ 2|v_0| and γ≥ 0, we have |v_0|^-γ |b|^2s |w|^γ≲ |b|^2s≲ |v_0+w|^2s and (<ref>) follows. The same conclusion holds if |v_0| ≤ 2|w| and γ≤ 0.

If γ≥ 0 and 2|v_0|≤ |w|, we use that |b|≤ |v_0+w| and |w|^γ≲ (|v_0+w|^γ+|v_0|^γ ). Thus,

    |v_0|^-γ |b|^2s |w|^γ   ≲ |v_0|^-γ |v_0+w|^2s (|v_0+w|^γ+|v_0|^γ ) ,
       ≲ |v_0+w|^2s + |v_0+w|^γ+2s .


We are left to analyze the case γ < 0 and |v_0| > 2|w|. Using that |b|≤ |w| and γ + 2s ≥ 0, we have

    |v_0|^-γ |w|^γ |b|^2s≤ |v_0|^-γ |w|^γ+2s≲ |v_0|^2s≲ |v_0+w|^2s.


We conclude that the inequality (<ref>) holds in all cases. The proof is now complete. 


From Lemmas <ref> and <ref>, we can derive Lemma <ref>.

We get from Lemma  <ref> that
  
    |v_0|^-γ-2s∫_w∈^d f(v+w) ( |v_0|^2 - ( v_0 ·w/|w|)^2 + 1 )^s |w|^γ+2s w ≤ C ∫_^df(ṽ) (1+ |ṽ|^2s+|ṽ |^γ+2s)ṽ.

Combining this estimate with  Lemma <ref> yields
  
    ∫_^d ∖ B_r(v)K̅_f(v,v')  v'    ≲Λ̅r^-2s
       ≲( ∫_^df(ṽ) (1+ |ṽ|^2s+|ṽ|^γ+2s)ṽ) r^-2s
    then use the fact that γ +2s ∈ [0,2],   ≲( ∫_^df(ṽ) (1+ |ṽ|^2)ṽ) r^-2s
       ≲ (M_0+E_0) r^-2s.




Lemma <ref> is phrased in terms of the condition (<ref>) for Theorem <ref>. As far as Theorem <ref> is concerned, we remark that it is the same condition as the second item in Definition <ref>. We rephrase it in the following corollary. Recall that K̅_f,z(w) denotes K̅_f(t,x,v,v+w) for z = (t,x,v).
 
Let f ∈ [τ,T]×^d ×^d → [0,∞) and K_f be given by (<ref>). Then, for any z ∈ Q_1,

    ∫_B_rK̅_f,z(w) |w|^2  w ≤ C ( ∫_^d (1+|ṽ|^2) f(t̅, x̅, ṽ) ṽ) r^2-2s

with z=(t,x,v) and (t̅,x̅,v̅) = 𝒯_0 (z). 
 The constant C in the right hand side depends on γ, s and dimension only.



In terms of the notation K̅_f,z, Lemmas <ref> and <ref> say exactly that

    ∫_^d ∖ B_rK̅_f,z(w)  w    ≤ C ( ∫_^d (1+|v|^2s+|v|^γ+2s) f(t̅, x̅, v)  v ) r^-2s, 
       ≤ C ( ∫_^d (1+|v|^2) f(t̅, x̅,v)  v ) r^-2s

with (t̅,x̅,v̅) = 𝒯_0 (t,x,v). 
The bound for the integral of the kernel on the complement of any ball ^d ∖ B_r, is equivalent to the bound of the integral of the same kernel times |w|^2 on any ball. Indeed, the inequality above implies that

    ∫_B_2r∖ B_rK̅_f,z(w) |w|^2  w ≲( ∫_^d (1+|v|^2) f(t̅, x̅, v)  v ) r^2-2s.

Applying this inequality to dyadic rings for r̃ = r/2, r/4, r/8, … and summing the resulting estimates, we conclude the proof.




 §.§ Second boundedness condition




Let us assume that γ+2s ∈ [0,2] and let the upper bound in the mass and energy hold

    ∫_^d f(v)  v ≤ M_0,     ∫_^d f(v) |v|^2  v ≤ E_0.

If γ+s<0, we also assume (<ref>).

Then, the kernel K̅_f from (<ref>) satisfies (<ref>) with parameters depending on M_0, E_0, γ, s, d and C_γ (in case γ+s<0).




For |v_0| < 4 (or any other fixed constant) the result follows from <cit.>. Here, we concentrate on the case |v_0| ≥ 4.

Given v' ∈ B_2, recall that v̅' = v_0 + T_0 v' and write

    ∫_^d ∖ B_r (v')K̅_f(v,v')    ≤1/|v_0|^γ + 2s∫_^d ∖ E_r(v̅') K_f(v̅, v̅') v̅
       ≃1/|v_0|^γ + 2s∫_^d ∖ E_r(v̅') |v̅'-v̅|^-d -2s{∫_w ⊥ (v̅'- v̅) f(v̅+w) |w|^γ + 2s +1}v̅, 
    we use polar coordinates for v̅: v̅ - v̅' = ρσ; we denote r_σ to be max{ρ  : ρσ∈ E_r},    = 1/|v_0|^γ + 2s∫_S^d-1∫_r_σ^∞ρ^-d -2s{∫_w ⊥σ f(v̅'+ ρσ +w) |w|^γ + 2s +1}ρ^d-1ρσ, 
    writing u = ρσ + w, we have w ρ =  u and ρ = u ·σ,   ≲1/|v_0|^γ + 2s∫_S^d-1∫_{u:u ·σ > r_σ} f(v̅'+ u) |u|^γ + 2s +1 (u ·σ)^-1-2s u σ, 
        = 1/|v_0|^γ + 2s∫_S^d-1∫_{u:u ·σ > r_σ} f(v̅'+ u) |u|^γ(u/|u|·σ)^-1-2s u σ, 
        = 1/|v_0|^γ + 2s∫_^d f(v̅'+ u) |u|^γ( ∫_{σ : u ·σ > r_σ}(u/|u|·σ)^-1-2sσ)  u.


We computed above r_σ = ρ, see (<ref>) with u = ρσ = r_σσ. In particular, this implies

    r_σ   = r/√((σ· v_0)^2 (1-|v_0|^-2) +1).
    For |v_0|“large” (as a matter of fact |v_0|≥ 2), we have
      r_σ   ≈r/√(1 + (σ· v_0)^2).

Thus, we are left with the inequality,

    ∫_^d ∖ B_r (v')K̅_f(v,v')    ≲1/|v_0|^γ + 2s∫_^d f(v̅'+ u) |u|^γ( ∫_{σ : u ·σ >r/√(1 + (σ· v_0)^2)}(u/|u|·σ)^-1-2sσ)  u.

We need to estimate the inner integral now. For that, it is essential to understand the smallest value of σ· u/|u| in the set {σ : u ·σ >r/√(1 + (σ· v_0)^2)}. Let us write e = u/|u| and v_0 = a e + b. We see that for every σ in the domain of integration we have

    (e ·σ) √( 1 + (e ·σ)^2 a^2 + |b|^2) > r / |u|.

Therefore, either (e·σ) √(1+|b|^2)≳ r / |u| or (e·σ) |e·σ| |a| ≳ r / |u|. In other words,

    e ·σ≳min( r/|u| (1+|b|^2)^1/2 , (  r  / |a| |u| )^1/2) =: ρ_0.

Therefore

    ∫_{u ·σ >r/√(1 + (σ· v_0)^2)}(u/|u|·σ)^-1-2sσ   ≲∫_{ e ·σ≳ρ_0}(e ·σ)^-1-2sσ, 
       ≲ρ_0^-2s,
       = min( r/|u| (1+|b|^2)^1/2 , (  r  / |a| |u| )^1/2)^-2s, 
       ≤ r^-2s |u|^2s (1+|b|^2)^s + r^-s |u|^s |a|^s,
       = r^-2s |u|^2s(1+|v_0|^2 - (v_0 · u)^2/|u|^2)^s + r^-s |u|^s |u/|u|· v_0 |^s.

Thus, we are left with

    ∫_^d ∖ B_r (v')K̅_f(v,v') ≲ I_1 + I_2,

where

    I_1    = r^-2s |v_0|^-γ - 2s∫_^d f(v̅'+ u) |u|^γ+2s(1 + |v_0|^2 - (v_0 · u)^2/|u|^2)^s  u, 
    
    I_2    = r^-s |v_0|^-γ - s∫_^d f(v̅'+ u) |u|^γ+s u.


The term I_1 is bounded thanks to Lemma <ref>.

The term I_2 is lower order in the sense that it has a smaller power of r. Note that only the values of r ∈ (0,2) are relevant for (<ref>), since for larger values of r the domain of integration is empty. We still need to make sure that the factor multiplying r^-s in the definition of I_2 is bounded independently of v_0.

When γ+s ≥ 0, we apply Lemma <ref>,

    ∫_^d f(v̅'+ u) |u|^γ+s u ≲ (1+|v̅'|)^γ+s≈ (1+|v_0|)^γ+s

yielding

    I_2 ≲ r^-s.


The case γ+s < 0 is more involved. We split the integral in I_2 into three subdomains

    D_1 = ^d ∖ B_|v_0|/4,      D_2 = B_|v_0|^-1,      D_3 = ^d ∖ (D_1 ∪ D_2).

We estimate each subintegral separately. In D_1, we have |u|≳ |v_0|. Thus,

    ∫_D_1 f(v̅'+ u) |u|^γ+s u ≤  M_0 |v_0|^γ+s.


In D_2, we have |u|^γ+s≤ |u|^γ |v_0|^-s. Therefore

    ∫_D_2 f(v̅'+ u) |u|^γ+s u    ≲  |v_0|^-s∫_D_2 f(v̅'+ u) |u|^γ u, 
       ≤ |v_0|^-s (M_0 + C_γ), 
       ≤ |v_0|^γ+s (M_0 + C_γ)          using that γ+2s ≥ 0.


We estimate the integral on D_3 using the upper bound on the energy: E_0. We use the fact that in D_3, we have on the one hand |v̅' + u | ≳ |v_0| and on the other hand |u| ≥ |v_0|^-1.  In particular, 

    ∫_D_3 f(v̅'+ u) |u|^γ+s u    ≤ E_0 |v_0|^-2 |v_0|^-γ -s ,
       ≲ E_0 |v_0|^γ+s    using that γ+2s ≥ 0 and 2s ≤ 2.


Adding the three terms together, we conclude that I_2 ≤ C r^-s, for a constant C independent of v_0. The proof is now complete.




 §.§ First cancellation condition


The cancellation condition (<ref>) involves an integral in the principal value sense. We have to be careful when we compute a change of variables of such an integral. There is a delicate cancellation that takes place and we have to make sure that the change of variables does not cause an imbalance around the origin that would ruin this cancellation. The following lemma is precisely what we need in order to carry out the rest of our computations.


 Assuming that D^2 f ∈ L^1(^d, (1+|v|)^γ+2s), we have 

    lim_R→ 0^+∫_B_R ∖ E_R (K_f(v̅,v̅ + w) -K_f(v̅ + w,v̅))  = 0.




The result of Lemma <ref> is certainly to be expected by common sense. However, its rigorous verification requires some work. We prove it under the condition that D^2 f ∈ L^1(^d, (1+|v|)^γ+2s). It holds under much more general conditions on the function f. However, due to the kind of solutions that we work with, the scope of Lemma <ref> is enough for the purpose of this article. A proof of a version of Lemma <ref> under less restrictive assumptions on the function f would require considerably more work. Note that Lemma <ref> is merely a qualitative result. There is no estimate resulting from this lemma in terms of the weighted L^1 norm of D_v^2 f. We invite the reader to skip its proof, that we include below for completeness.


For all v̅ = v_0 + T_0 v with v ∈ B_1, we expand the integral in terms of the formula (<ref>) and symmetrize it.

    (I) :=   ∫_B_R ∖ E_R (K_f(v̅,v̅ + w) -K_f(v̅ + w,v̅))  , 
     
     =    - 1/2∫_B_R ∖ E_R∫_u : u ⊥ w (δ^2 f) (v̅+u, w)  A (|w|,|u|)  |w|^-d-2s |u|^γ+2s+1 u  w

with (δ^2 f) (v,w) =f(v+w) + f(v-w) -2f(v) and A is a bounded. We express this second order differential quotient using  D^2_v f using the elementary formula

    (δ^2 f) (v,w) = ∫_-1^1 (1-|τ|)  D^2f(v+τ w) w · w  τ.


Thus, we bound the integral in terms of D^2_v f_L^1(^d, |v|^γ+2s v).

    (I) ≲∫_B_R ∖ E_R∫_u ⊥ w∫_-1^1  |D^2_vf (v̅+u + τ w)| |w|^-d-2s+2  |u|^γ+2s+1τ u  w.


For each value of w ∈^d, we make the change of variables (τ,u) ↦ z = u +τ w; in particular z = |w|  u τ and τ = (z · w)/|w|^2 and |u| ≤ |z|. Therefore, we estimate the integral above by

    (I)    ≲∫_B_R ∖ E_R∫_-|w|^2 < z · w < |w|^2 |D^2_vf (v̅+z)| |w|^-d-2s+1 |z|^γ+2s+1 z  w, 
    switching the order of integration,   = ∫_z ∈^d |z|^γ+2s+1 |D^2_v f(v̅+z)| {∫_w ∈ B_R, 
     -|w|^2 < z · w < |w|^2 |w|^1-d-2s w } z, 
    we claim that,   ≲ R^2-2s∫_z ∈^d |D^2_vf (v̅+z)| |z|^γ+2s.


In order to justify the last inequality, observe that the set {w :-|w|^2 < z · w < |w|^2} is the complement of two balls of radius |z|/2 centered at z/2 and -z/2 respectively. The intersection of that set with the ball B_R is of volume ≲ R^d+1 |z|^-1. Thus, the following inequality follows by an elementary computation.

    ∫_w ∈ B_R, 
     -|w|^2 < z · w < |w|^2 |w|^1-d-2s w ≲ R^2-2s |z|^-1.


Since s∈ (0,1), R^2-2s converges to zero as R → 0, which concludes the proof.





Let f: ^d → be a function so that Assumption <ref> holds. If γ<0, we assume (<ref>) in addition. Then, the kernel K̅_f given in (<ref>) satisfies (<ref>) with parameters depending on M_0, E_0, γ, s, d, and also C_γ if γ<0. More precisely, for v ∈ B_2,

    | ∫_^d (K̅_f(v,v') - K̅_f(v',v) ) ' |    ≲  |v_0|^-γ -2s∫_^d f(v̅') |v̅'-v̅|^γ v', 
       ≲
    (M_0 + E_0) |v_0|^-2s   if γ∈ [0,2], 
    
    (M_0 + C_γ) |v_0|^-γ-2s   if γ < 0.



The inequality in Lemma <ref> implies (<ref>) because the tail of the integral (i.e. |v'-v|>1/4) is bounded by Lemmas <ref> and <ref>.



When |v_0|<2, the result is proved in <cit.>, and it corresponds to the classical cancellation lemma. Here, we focus on the case |v_0| ≥ 2.

As before, for v ∈ B_2, we write v̅ = v_0 + T_0 v ∈ B_2 (v_0). Using Lemma <ref>, we compute

    ∫_^d (K̅_f(v,v')    - K̅_f(v',v) ) '  
       = |v_0|^-γ -2s-1∫_^d (K_f(v_0+T_0v,v_0+T_0 v') -K_f(v_0+T_0v',v_0+T_0 v)) ' 
        =  |v_0|^-γ -2s-1lim_R → 0^+∫_|z|≥ R (K_f(v̅,v̅ + T_0 w) -K_f(v̅ + T_0 w,v̅)) 
        =  |v_0|^-γ -2slim_R → 0^+∫_^d ∖ E_R (K_f(v̅,v̅ + w̅) -K_f(v̅ + w̅,v̅)) w̅
        =  |v_0|^-γ -2slim_R → 0^+∫_|w̅|≥ R (K_f(v̅,v̅ + w̅) -K_f(v̅ + w̅,v̅)) w̅
        =  |v_0|^-γ -2s∫_^d (K_f(v̅,v̅ + w̅) -K_f(v̅ + w̅,v̅)) w̅.

We now  use <cit.> and get

    | ∫_^d (K̅_f(v,v') - K̅_f(v',v) ) ' | ≤ C |v_0|^-γ -2s∫_^d f(v̅') |v̅'-v̅|^γ.

If γ <0, we use (<ref>) and get 

    | ∫_^d (K̅_f(v,v') - K̅_f(v',v) ) ' | ≲ |v_0|^-γ -2s (M_0 + C_γ).

while for γ >0, we estimate it using Lemma <ref>,

    | ∫_^d (K̅_f(v,v') - K̅_f(v',v) ) ' | ≲ |v_0|^-γ -2s (E_0 + M_0 (1+|v̅|)^γ ) ≲ (M_0 + E_0) |v_0|^-2s.
 
The proof is now complete. 





 §.§ Second cancellation condition


Like in the first cancellation condition, the second cancellation (<ref>) also involves the principal value of an integral. In this case, the following technical lemma is the one that ensures that we can perform the change of variables.


 Assuming that ∇_v f ∈ L^1(^d, (1+|v|)^γ+2s ), we have 

    lim_R→ 0∫_B_R ∖ E_R w K_f(v̅+w,v̅)  w = 0.



Like in Lemma <ref>, the identity (<ref>) is clearly to be expected by common sense but it takes some work to prove it rigorously. As before, the condition D f ∈ L^1(^d, (1+|v|)^γ+2s) is a qualitative requirement that does not affect any of our estimates. 


As in the proof of Lemma <ref>, we expand the integral using (<ref>) and symmetrize it in w.

    ∫_B_R ∖ E_R w    K_f(v̅ + w, v̅)   = 
    ∫_B_R ∖ E_R w {∫_u ⊥ w f(v̅+w  + u) 
    A (|u|,|w|) |u|^γ+2s+1/|w|^d+2s} w, 
       = 1/2∫_B_R ∖ E_R w {∫_u ⊥ w ( f(v̅+w  + u) - f(v̅-w  + u) )
    A (|u|,|w|) |u|^γ+2s+1/|w|^d+2s} w.

We write the increment f(v̅+w  + u) - f(v̅-w  + u) in terms of the integral of the derivative along the segment, and proceed like in the proof of Lemma <ref>.

    |∫_B_R ∖ E_R w    K_f(v̅ + w, v̅) |
       ≲∫_B_R ∖ E_R∫_u ⊥ w
    |u|^γ+2s+1 |w|^2-d-2s∫_-1^1 |∇ f(v̅ + τ w + u)| τ w, 
    for fixed w, we write z = τ w + u and observe that z = |w|  u τ and |u| ≤ |z|,   ≤∫_B_R ∖ E_R∫_-|w|^2 < z · w < |w|^2 |z|^γ+2s+1 |w|^1-d-2s |∇ f(v̅ + z)|  z  w, 
    switching the order of integration,   ≤∫_z ∈^d |z|^γ+2s+1 |∇ f(v̅+z)| {∫_w ∈ B_R, 
     -|w|^2 < z · w < |w|^2 |w|^1-d-2s w } z, 
    using again (<ref>),   ≲ R^2-2s∫_z ∈^d |z|^γ+2s |∇ f(v̅+z)|  z.

This converges to zero as R → 0, so the proof is concluded.


The following auxiliary lemma contains an estimate that will be useful in the next lemma.
 
Let s ∈ [1/2,1). For any v_0 ∈^d ∖ B_2, v ∈ B_1(v_0), and r ∈ (0,1), we have

    ∫_^d f(v+z) |z|^γ+1min(1,r^2-2s |z|^2s-2)  z ≤ C |v_0|^γ+2s-1 r^1-2s,

for some constant C depending on M_0, E_0, s, γ, dimension d, and C_γ if γ<0, but not on v_0.



Note that since s ≥ 1/2 and r<1, we have r^1-2s≥ 1.

If γ+2s ≥ 1 the inequality follows easily applying Lemma <ref>. Indeed,

    ∫_^d f(v+z) |z|^γ+1min(1,r^2-2s |z|^2s-2)  z
         ≤ r^2-2s∫_^d f(v+z) |z|^γ+2s-1 z, 
       ≲ r^2-2s (E_0 + (1+|v|^γ+2s-1) M_0),
       ≲ r^2-2s |v_0|^γ+2s-1
       ≲ r^1-2s |v_0|^γ+2s-1.

We used that r ∈ (0,1) and |v|≈ |v_0|.

So, let us concentrate on the more difficult case γ+2s < 1.

We split the domain of integration between z ∈ B_r and z ∉ B_r. Let us call each term I_1 and I_2,

    I_1    := ∫_|z| ≤ r f (v  +z)  |z|^γ+1, 
    
    I_2    := r^2-2s∫_|z| ≥ r f (v  +z)  |z|^γ+2s-1 z.

We now estimate I_1 and I_2 separately.


When γ+1 ≥ 0, we easily get I_1 ≲ E_0 |v_0|^-2≲ |v_0|^γ+2s-1 r^1-2s, since r ≤ 1 and |v_0| ≥ 2. If γ+1 < 0, we use Hölder inequality and get

    I_1 = ∫_|z|≤ r f(v+z) |z|^γ+1 z     ≤( ∫_|z|≤ r f(v+z) |z|^γ z )^γ/1+γ( ∫_|z|≤ r f(v+z)   z )^-1/γ
       ≲ C_γ^γ/1+γ E_0^-1/γ |v_0|^2/γ.

Since 2/γ≤ -1 and γ+2s ≥ 0, then γ+2s-1 ≥ 2/γ and I_1 ≲ |v_0|^γ+2s-1≤ |v_0|^γ+2s-1 r^1-2s. This concludes the upper bound for I_1. 


As far as I_2 is concerned, we further split the integral between two subdomains. If v+z ∈ B_|v_0|/2 and |z| ≥ r, we use that |z| ≈ |v_0|. If v+z ∉ B_|v_0|/2 and |z| ≥ r, we have,

    ∫_v+z ∉ B_|v_0|/2 f(v+z) |z|^γ+2s z    ≲∫_v+z ∉ B_|v_0|/2 f(v+z) (|v+z|^γ+2s +|v|^γ +2s )  z
       ≲∫_v+z ∉ B_|v_0|/2 f(v+z) (|v+z|^2 |v_0|^γ +2s-2 +|v_0|^γ +2s)  z
       ≲∫_v+z ∉ B_|v_0|/2 f(v+z) (|v+z|^2 +1) |v_0|^γ +2s-2 z 
       ≲ (E_0+M_0)|v_0|^γ+2s-2.

With such an estimate at hand, we can now write,

    I_2    ≲ r^2-2s |v_0|^γ+2s-1∫_v+z ∈ B_|v_0|/2 f(v+z)  z + r^1-2s∫_v+z ∉ B_|v_0|/2 f(v+z) |z|^γ+2s z, 
       ≲ r^2-2s |v_0|^γ+2s-1 M_0 + r^1-2s |v_0|^γ+2s-2 (E_0+M_0) ≲ r^1-2s |v_0|^γ+2s-1.

Adding up the upper bounds for I_1 and I_2, we finish the proof of the lemma.





The main result of this subsection is the following.


Let s ≥ 1/2 and f(v) be a function so that Assumption <ref> holds. If γ<0, we assume (<ref>) as well. Then, the kernel K̅_f from (<ref>) satisfies
  (<ref>) with a parameter Λ depending on M_0, E_0,
  γ, s, d, and  also C_γ if γ<0.


For |v_0|<2, the result was established in <cit.>. Here, we focus on the case |v_0| ≥ 2.

Given r ∈ [0,1/4], note that for s=1/2, r^1-2s = 1, whereas for s>1/2, r^1-2s > 1.

For any  v ∈^d and r>0, we have

    ∫_B_r(v) (v'-v) K̅_f(v,v') ' = 1/|v_0|^γ +2s+1∫_B_r w K_f(v̅, v̅ + T_0w)  w =0.

Here, we write v̅=v_0 + T_0v and we use the symmetry of the Boltzmann kernel: K_f(v̅, v̅+w̅) = K_f(v̅, v̅ - w̅).

Therefore, the proof is reduced to estimating the term in (<ref>) involving K_f(v',v) only. That is, we need to estimate the quantity I(v') for v' ∈ B_7/4 given by

    I(v'):= |∫_B_r( v') ( v'-v) K̅_f( v, v') |.


Let us change variables. As usual, we write v̅' = v_0 + T_0 v' and w̅ = T_0(v'-v). We get

    I( v') = |v_0|^-γ-2s| ∫_E_r (T_0^-1w̅)   K_f(v̅'-w̅,v̅') w̅|.

We used Formula (<ref>), T_0^-1=|v_0|, and Lemma <ref> in order to justify the change of variables under the principal value.

Recall that the domain of integration E_r is an ellipsoid. In order to capture the cancellations correctly, it is better to work with a round ball B_r. We use that I(v') ≤ I_1 + I_2, where

    I_1 := |v_0|^-γ-2s|∫_B_r (T_0^-1w̅)   K_f(v̅'-w̅,v̅') w̅|,      I_2 := |v_0|^-γ-2s|∫_B_r ∖ E_r (…) w̅|.

The rest of the proof is divided into two steps corresponding to establishing the bound for each one of the terms I_1 and I_2. 

Step 1: I_2 ≲ r^1-2s.

In order to estimate I_2, we do not need to take any cancellation into consideration. We simply take absolute values everywhere and estimate each factor separately. Recalling Formula (<ref>), we have

    |v_0|^γ+2s I_2    =  |∫_B_r ∖ E_r (T_0^-1w̅)   K_f(v̅'-w̅,v̅') w̅|, 
       ≲∫_B_r ∖ E_r |T_0^-1w̅| |w̅|^-d-2s∫_u ⊥w̅ f(v̅' - w̅ + u) |u|^γ+2s+1 u w̅.
    Like in the proof of Lemma <ref> about the second boundedness condition, we use polar coordinates -w̅ = ρσ, and we write r_σ≈ r(1+(σ· v_0)^2)^-1/2 for the maximum value of ρ so that ρσ∈ E_R.   = ∫_σ∈ S^d-1∫_r_σ < ρ < rρ^-2s |T_0^-1σ| ∫_u ⊥σ f(v̅' + ρσ + u) |u|^γ+2s+1 u ρσ. 
    We now write z = ρσ + u so that z = ρ u, and observe ρ = z ·σ and |u| ≤ |z|.   ≤∫_σ∈ S^d-1∫_r_σ < z ·σ <r (z ·σ)^-2s |T_0^-1σ| f(v̅' + z) |z|^γ+2s+1 z σ, 
       = ∫_^d f(v̅' + z) |z|^γ+1( ∫_{σ : r_σ < σ· z < r} (σ· z/|z|)^-2s |T_0^-1σ| σ)  z, 
       = ∫_^d f(v̅' + z) |z|^γ+1( ∫_{σ : r_σ < σ· z < r} (σ· z/|z|)^-2sr/r_σσ)  z, 
       ≈∫_^d f(v̅' + z) |z|^γ+1( ∫_{σ : r_σ < σ· z < r} (σ· z/|z|)^-2s√(1+ (σ· v_0)^2)σ)  z.


We analyze the spherical integral similarly as in the proof of Lemma <ref>. We write e = z/|z|, v_0 = ae+b with b ⊥ e and a = v_0 · e. We observe √(1+ (σ· v_0)^2)≲( 1 + a^2 (σ· e)^2 + |b|^2 )^1/2. We divide the domain of integration depending on whether a^2 (σ· e)^2 > 1+|b|^2 or not. The purpose of these two subdomains is to know which term in r_σ to focus on. Let us call S_1 and S_2  these two integrals respectively. We have

    I_2    ≲ |v_0|^-γ-2s∫_^d f(v̅'+z) |z|^γ+1 (S_1+S_2)  z,
    where
      S_1    ≲∫_σ· e ≤ r/|z| (σ· e)^-2s+1 |a| σ
       ≲ |a| min(1,(r/|z|)^2-2s)  
    
      S_2    ≲∫_r/√(2) |z| (1+|b|^2)^1/2≤ e ·σ (e ·σ)^-2s (1+|b|^2)^1/2σ, 
       ≲ r^1-2s |z|^2s-1 (1+|b|^2)^s.


For the first term in I_2, we have |a| ≤ |v_0| and consequently,

    |v_0|^-γ-2s∫_^d f(v̅'+z) |z|^γ+1 S_1  z    ≲ |v_0|^-γ-2s+1∫_^d f(v̅'+z) |z|^γ+1min(1,r^2-2s |z|^2s-2)  z 
       ≲ r^1-2s    thanks to Lemma <ref>.


Note that |b|^2 = |v_0|^2 - (v_0 · e)^2. So, for the second term in I_2, we have

    |v_0|^-γ-2s∫_^d f(v̅'+z) |z|^γ+1 S_2  z    ≲ r^1-2s |v_0|^-γ-2s∫_^d f(v̅'+z) |z|^γ+2s (1+|v_0|^2-(v_0 · z)^2/|z|^2)^s  z, 
       ≲ r^1-2s    thanks to Lemma <ref>.


Adding all the terms, we get the announced estimate on I_2. 

Step 2: I_1 ≲ r^1-2s.

The cancellation inside the integral, and in particular in the principal value, plays a central role in the inequality for I_1. We proceed similarly as in step 1 but without taking absolute values and keeping equalities. We use polar coordinates w̅ = r σ and write z = - r σ + u with z =  r  u.

    |v_0|^γ+2s I_1
    = |∫_z ∈^d f(v̅'+z) T_0^-1{∫_σ∈ S^d-1 : 0 < z ·σ < rσ (σ· z)^-2s ( |z|^2 - (z ·σ)^2)^γ+2s+1/2 A(… ) σ} z | .

Here, A is the function from (<ref>). In this case, its value depends on σ· z/|z| only: A(…) = Ã (σ· z /|z|) ≈ 1.

If we write σ = ρ z / |z| + σ^⊥ with σ^⊥ perpendicular to z, we see that the only factor in the integrand that depends on σ^⊥ is σ. Since σ^⊥ takes values on a (d-2)-dimensional circle, its values cancel out in the integral. Thus, we reduce the integral to

    |v_0|^γ+2s  I_1 
       = | ∫_z ∈^d f(v̅'+z) |z|^-2s T_0^-1(z/|z|) ∫_σ∈ S^d-1 : 0 < z ·σ < r(σ·z/|z|)^1-2s ( |z|^2 - (z ·σ)^2)^γ+2s+1/2Ã(σ· z/|z| ) σ z |. 
     At this point, the cancellations have been taken into account already. All quantities that remain are positive so we continue with inequalities. We use that T_0^-1=|v_0| and that Ã is bounded.   ≤∫_z ∈^d f(v̅'+z) |z|^-2s| T_0^-1z/|z||∫_σ∈ S^d-1 : 0 < z ·σ < r(σ·z/|z|)^1-2s ( |z|^2 - (z ·σ)^2)^γ+2s+1/2Ã(σ· z/|z| ) σ z, 
       ≲ |v_0| ∫_z ∈^d f(v̅'+z) |z|^γ+1( ∫_σ∈ S^d-1 : 0 < z ·σ < r(σ·z/|z|)^1-2sσ)  z , 
       ≲ |v_0| ∫_z ∈^d f(v̅'+z) |z|^γ+1min(1,(r/|z|)^2-2s)  z .

We conclude that I_1 ≲ r^1-2s by applying Lemma <ref>.


The change of variables theorems derive from the series of lemmas established in this section.  

  Combine Corollary <ref> and Lemmas <ref>, <ref>, <ref>,  <ref>. The inequality (<ref>) is a combination of (<ref>) with (<ref>).



Combine Corollary <ref> and Corollary <ref>.



 

 §.§ Hölder spaces through the change of variables



We examine how the change of variables 𝒯_0 defined in (<ref>)
affects the kinetic Hölder spaces introduced in Section <ref>.


  Given z_0 ∈^1+2d and  F : ℰ_R (z_0) →, we define F̅ :Q_R → by
  F̅(z) = F (𝒯_0 (z)). Then, for any β>0,
  
    F̅_C_ℓ^β (Q_R)≲F_C_ℓ^β (ℰ_R (z_0))≲ |v_0|^c̅βF̅_C_ℓ^β (Q_R),

  with c̅ = max( γ+2s/2s,1 ). 




We point out that we only need to prove this Lemma for |v_0|>2 (See Remark <ref>).

We can write 𝒯_0 as z_0 ∘𝒯 with 

    𝒯 (t,x,v) = (|v_0|^-γ -2st ,|v_0|^-γ - 2s (T_0 x),T_0v).


We first prove that for v_0 ∈^d ∖ B_2 and for all z,z_1 ∈^1+2d, 

    d_ℓ (𝒯^-1z,𝒯^-1 z_1)    ≲ |v_0|^c̅ d_ℓ(z,z_1) 
    
      d_ℓ (𝒯 z,𝒯 z_1)    ≲ d_ℓ(z,z_1)

with c̅ = max (γ+2s/2s,γ+2s/1+2s+1). As far as (<ref>) is concerned, using the definition of d_ℓ, we write

    d_ℓ (𝒯^-1z,𝒯^-1 z_1)    = min_w ∈^dmax{ (|v_0|^γ+2s |t-t_1|)^1/(2s), 
       = min_w ∈^dmax{( |v_0|^γ+2s |T_0^-1 (x-x_1-(t-t_1)w)| )^1/(1+2s), 
       = min_w ∈^dmax{ |T_0^-1(v-w)|, |T_0^-1(v_1-w)| }, 
    note that T_0^-1 = |v_0|,   ≤max( |v_0|^γ+2s/2s, |v_0|^γ+2s+1/1+2s,|v_0| ) min_w ∈^dmax{ |t-t_1|^1/2s, 
       ≤max( |v_0|^γ+2s/2s, |v_0|^γ+2s+1/1+2s,|v_0| ) min_w ∈^dmax{ |x-x_1-(t-t_1)w|^1/1+2s,
       ≤max( |v_0|^γ+2s/2s, |v_0|^γ+2s+1/1+2s,|v_0| ) min_w ∈^dmax{ |v-w|, |v_1-w| }, 
       ≤ |v_0|^c̅ d_ℓ(z,z_1).

Note that |v_0|^γ+2s+1/1+2s≤max( |v_0|^γ+2s/2s,|v_0| ). This justifies (<ref>). The verification of (<ref>) is very similar using that T_0=1.

Since d_ℓ is left invariant, (<ref>) implies that for v_0 ∈^d ∖ B_2, z,z_1 ∈^1+2d, and z̅ = 𝒯_0 z, z̅_1 = 𝒯_0 z_1,

    d_ℓ(z̅, z̅_1) ≤  d_ℓ (z,z_1) ≲ |v_0|^c̅ d_ℓ(z̅, z̅_1).


We deduce (<ref>) from (<ref>). Given any z̅_1,z̅∈ Q_R, and z̅ = 𝒯_0 z, z̅_1 = 𝒯_0 z_1, let p be the polynomial expansion of F at the point z̅_1 so that |F(z̅) - p(z̅)| ≤ [F]_C^β_ℓ d_ℓ(z̅, z̅_1)^β and _k p < β. We observe that p ∘𝒯_0 is a polynomial of the same degree. Thus,

    |F̅(z) - p ∘𝒯_0 (z)| = |F(z̅) - p(z̅)| ≤ [F]_C^β_ℓ d_ℓ(z̅, z̅_1)^β≤ [F]_C^β_ℓ d_ℓ(z, z_1)^β.

We deduced the first inequality in (<ref>) from the first inequality in (<ref>). We deduce the second one similarly.




 §.§ Hölder continuity of the kernel


In the next lemma we explore how a Hölder estimate for f of the form f ∈^α results in a Hölder estimate for the kernel K̅_f as in the assumption (<ref>) in Theorem <ref>. 


 

Let f : [τ,T] ×^d ×^d be such that  f ∈^α.  Then  (<ref>) holds true for K̅_f with  α' = 2s/1+2sα and 

    A̅_0 ≤ C ( f_C^0_ℓ,q +  (1+|v_0|)^α/1+2s (1-2s-γ)_+ [f]_C^α_ℓ,q).
 
Here, q can be any number larger than d+2+α/(1+2s). The constant C depends only on dimension, γ, s, min(1,T-τ), and the choice of q.




  Without loss of generality, let us assume T-τ≥ 1. The effect of this assumption is that we take cylinders of the form Q_1(z_0) ⊂ [τ,T] ×^d ×^d. Otherwise we would have to work with cylinders Q_r(z_0) for a smaller r>0 and the choice of r, depending on T-τ, would affect the constants in the lemma.

  As usual, we also focus on |v_0|>2.

Recall that the definition of  says that for all q>0, there exists a constant C_q = [f]_C^α_ℓ,q so that

    f_C^α_ℓ(Q_1(z_0))≤ C_q (1+|v_0|)^-q,

whenever Q_1(z_0) ⊂ [τ,T] ×^d ×^d. In particular, for z=(t,x,v),

    |f(z)| ≤ C_1 (1+|v|)^-q    and     |f(z_1) - f(z_2)| ≤ C_q d_ℓ(z_1,z_2)^α (1+|v_1|)^-q whenever  d_ℓ(z_1,z_2) < 1.


As usual, we write z̅ = 𝒯_0 z. According to the change
of variables formula (<ref>), we thus have for all
z_1,z_2 ∈ Q_1 (in particular d_ℓ(z_1,z_2)<1),

    K̅_f,z_1(w) - K̅_f,z_2(w) = |v_0|^-1-γ-2s( K_f,z̅_1 (T_0 w) - K_f,z̅_2 (T_0 w) ).


From the formula (<ref>), we observe that, for any z̅_1 = 𝒯_0 z_1 and z̅_2 = 𝒯_0 z_2, 

    |K_f,z̅_1 (w)    - K_f,z̅_2(w)| 
        = |w|^-d-2s|∫_u ⊥ w |u|^γ+2s+1 A(|w|,|u|) ( f(z̅_1 ∘ (0,0,u)) - f(z̅_2 ∘ (0,0,u)) )  u |, 
       ≤ |w|^-d-2s∫_u ⊥ w |u|^γ+2s+1 A(|w|,|u|) | f(z̅_1 ∘ (0,0,u)) - f(ξ∘z̅_1∘ (0,0,u)) | u, 
       = K_Δ f, z̅_1(w),

where Δ f(z) = |f(z) - f(ξ∘ z)| and  ξ = z̅_1 ∘z̅_2^-1. Combining with (<ref>), we get

    ∫_B_ρ |K̅_f,z_1(w) - K̅_f,z_2(w)| |w|^2  w    ≤∫_B_ρ |K̅_Δ f,z_1(w)| |w|^2  w, 
    we now apply Corollary <ref> to obtain   ≲( ∫_^d (1+|v|^2) Δ f(t̅_1, x̅_1, v)  v ) ρ^2-2s, 
       = ( ∫_^d (1+|v̅_1+w|^2) | f(z̅_1 ∘ (0,0,w)) - f(z̅_2 ∘ (0,0,w))  |   w ) ρ^2-2s.


In order to estimate the difference | f(z̅_1 ∘ (0,0,w)) - f(z̅_2 ∘ (0,0,w))  |, we use the C^α_ℓ,q semi-norms of f. We use (<ref>) and we get

    | f(z̅_1 ∘ (0,0,w)) - f(z̅_2 ∘ (0,0,w))  |  ≤( d_ℓ(z̅_1,z̅_2) + |t̅_1-t̅_2|^1/(1+2s) |w|^1/(1+2s))^α (1+|v̅_1+w|)^-q [f]_C^α_ℓ,q, 
    { (1+|v̅_1+w|)^-q + (1+|v̅_2+w|)^-q}   [f]_C^0_ℓ,q.

The first line applies whenever d_ℓ(z̅_1 ∘ (0,0,w),z̅_2 ∘ (0,0,w)) < 1. Note that from (<ref>), this distance is less or equal than the factor inside the parenthesis on the first line.

The second line applied whenever d_ℓ(z̅_1 ∘ (0,0,w),z̅_2 ∘ (0,0,w)) ≥ 1. In that case, it is better to estimate the left hand side in (<ref>) by the C^0 norm of each term. 

Note that since v̅_1, v̅_2 ∈ B_1(v_0), then (1+|v̅_1+w|) ≈ (1+|v̅_2+w|) ≈ (1+|v_0+w|). Recall that we write f_C^α_ℓ,q = [f]_C^α_ℓ,q +f_C^0_ℓ,q. Thus, in any case we have,

    | f(z̅_1 ∘ (0,0,w)) - f(z̅_2 ∘ (0,0,w))  |  ≲( d_ℓ(z̅_1,z̅_2) + |t̅_1-t̅_2|^1/(1+2s) |w|^1/(1+2s))^α (1+|v̅_1+w|)^-qf_C^α_ℓ,q.



By the definition of the transformation 𝒯_0, we always have d_ℓ(z̅_1, z̅_2) ≤ d_ℓ(z_1,z_2). Moreover, |t̅_1-t̅_2| = |v_0|^-γ-2s |t_1-t_2|.

We estimate

    ∫    (1+|v̅_1+w|^2) | f(z̅_1 ∘ (0,0,w)) - f(z̅_2 ∘ (0,0,w))  |   w  
       ≲∫( d_ℓ(z̅_1,z̅_2) + |t̅_1-t̅_2|^1/(1+2s) |w|^1/(1+2s))^α (1+|v_0+w|)^2-qf_C^α_ℓ,q w, 
       ≲f_C^α_ℓ,q( d_ℓ(z_1,z_2)^α + |v_0|^-γ+2s/1+2sα |t_1-t_2|^1/1+2sα∫_^d |w|^α/1+2s(1+|v_0+w|)^2-q w ), 
       ≲f_C^α_ℓ,q( d_ℓ(z_1,z_2)^α + d_ℓ(z_1,z_2)^α' |v_0|^(-γ-2s+1) α/1+2s), 
       ≲f_C^α_ℓ,q d_ℓ(z_1,z_2)^α' |v_0|^α/1+2s (1-γ-2s)_+ .

In the last inequality we used d_ℓ(z_1,z_2) < 1 and |v_0| > 2. We also used Lemma <ref> to estimate the last integral, which holds provided that q > d+2+α/(1+2s).





§ BOUNDS FOR THE BILINEAR OPERATOR (·, ·)




The right hand side of the Boltzmann equation (f,f) is a quadratic function of f. Its structure as a bilinear operator (f,g) is relevant when differentiating the equation. In this section we collect several lemmas to evaluate the Hölder regularity of (f,g) in terms of Hölder norms of f and g.

Recall that we write = _1 + _2. We obtain bounds for each of these two terms separately.



 §.§ Bounds for _2


Recall that _2 (f,g) = c_b (f ∗ |· |^γ) g. We start with estimating how the convolution with |·|^γ affects the local Hölder norm of a function. 

  
Let f ∈^α([τ,T] ×^d ×^d) for 0 < α≤min(1,2s). Let us consider the convolution of f with |·|^γ in velocity. That is

    g(z) = ∫_^d f(z ∘ (0,0,w)) |w|^γ w.

Then for all z_0 ∈ [τ,T] ×^d ×^d and r ∈ (0,1) such that Q_r(z_0) ⊂ [τ,T] ×^d ×^d, and any q > d+γ_+ + α/(1+2s),
  
    g _C_ℓ^α'(Q_r(z_0))≤ C  (1+|v_0|)^α/1+2s+γf_C_ℓ,q^α

 with C=C(d,γ,s,α) and α' = 2s/1+2sα.



Let z_0=(t_0,x_0,v_0), z_1=(t_1,x_1,v_1) ∈ [τ,T] ×^d ×^d with d_ℓ(z_0,z_1) < 1. We will show the two inequalities below from which the conclusion follows.

    |g(z_0)|    ≤ C  (1+|v_0|)^γ [f]_C_ℓ,q^0,
    
     |g(z_1) - g(z_0)|    ≤ C  (1+|v_0|)^α/1+2s+γf_C_ℓ,q^α d_ℓ(z_0,z_1)^α'.

Note that the assumption α≤min(1,2s) implies that any polynomial of degree less than α must be  constant. Thus, the Hölder semi-norm of order α involves merely increments.

Let us start with the first inequality. Using Lemma <ref> (since q > d+γ^+), we compute

    | g(z_0)|    ≤∫_^d |f|(t_0,x_0,v_0+w) |w|^γ w, 
       ≤ [f]_C_ℓ,q^0∫_^d (1+|v_0+w|)^-q |w|^γ w, 
       ≤ C [f]_C_ℓ,q^0 (1+|v_0|)^γ .


The second inequality requires a slightly longer computation,

    |g(z_1) - g(z_0)|    ≤∫_^d |f(z_1 ∘ (0,0,w)) - f(z_0 ∘ (0,0,w))| |w|^γ w, 
       ≤∫_|w| < d_ℓ(z_0,z_1)^-2s (…)   + ∫_|w|>d_ℓ(z_0,z_1)^-2s (…)  , 
       =: I + II.


For the first integral I, we use the inequality (<ref>) together with the semi-norm [f]_C^α_ℓ,q. In this domain, we have 

    d_ℓ(z_0 ∘ (0,0,w) , z_1 ∘ (0,0,w))    ≤ d_ℓ(z_0,z_1) + d_ℓ(z_0,z_1)^2s/(1+2s)|w|^1/(1+2s), 
       ≤ d_ℓ(z_0,z_1) + 1 ≤ 2.

Therefore, using again Lemma <ref>,

    I   ≲ [f]_C^α_ℓ,q∫_^d(d_ℓ(z_0,z_1) +  d_ℓ(z_0,z_1)^2s/(1+2s)|w|^1/(1+2s))^α |w|^γ (1+|v_0+w|)^-q w, 
       ≲ [f]_C^α_ℓ,q( d_ℓ(z_0,z_1)^α (1+|v_0|)^γ + d_ℓ(z_0,z_1)^α'∫_^d |w|^α/(1+2s)+γ (1+|v_0+w|)^-q w ), 
    (recall α' = 2s α / (1+2s))   ≲ [f]_C^α_ℓ,q( d_ℓ(z_0,z_1)^α (1+|v_0|)^γ + d_ℓ(z_0,z_1)^α' (1+|v_0|)^α/(1+2s)+γ)   since  q > d+γ_++α/(1+2s).

Naturally, the second term is larger than the first one.

For the second integral II we bound |f(z_0 ∘ (0,0,w)) - f(z_1 ∘ (0,0,w))| using [f]_C^0_ℓ,q. That is

    II   ≤ [f]_C^0_ℓ,q∫_|w|>d_ℓ(z_0,z_1)^-2s  ((1+|v_0 + w|)^-q + (1+|v_1 + w|)^-q ) |w|^γ w.

We analyze two cases depending on whether d_ℓ(z_0,z_1)^-2s > (1+3|v_0|) or not. In the first case, we have |v_1+w| ≳ |w| and |v_1 + w|≳ |w|, therefore

    II   ≲ [f]_C^0_ℓ,q∫_|w|>d_ℓ(z_0,z_1)^-2s |w|^-q+γ w, 
       ≲ [f]_C^0_ℓ,q d_ℓ(z_0,z_1)^2s (q-d-γ), 
       ≲  [f]_C^0_ℓ,q (1+|v_0|)^γ d_ℓ(z_0,z_1)^α'.

The last inequality holds because q > d+γ_++α/(1+2s) and d_ℓ(z_0,z_1)^-2s > (1+3|v_0|). Indeed, in this case we have

    d_ℓ(z_0,z_1)^2s (q-d-γ)/ (1+|v_0|)^γ d_ℓ(z_0,z_1)^α'   = d(z_0,z_1)^2s (q-d-γ - α/(1+2s) ) (1+|v_0|)^-γ, 
       ≲ (1+|v_0|)^-(q-d-γ - α/(1+2s)) - γ, 
       = (1+|v_0|)^-(q-d - α/(1+2s))  ,
       ≲ 1.


In the second case d_ℓ(z_0,z_1)^-2s≤ (1+3|v_0|), which means that d(z_0,z_1) ≥ (1+3|v_0|)^-1/(2s). Therefore

    II   ≤ [f]_C^0_ℓ,q∫_^d ((1+|v_0 + w|)^-q + (1+|v_1 + w|)^-q ) |w|^γ w,   
       ≲ [f]_C^0_ℓ,q  (1+|v_0|)^γ, 
       ≲ [f]_C^0_ℓ,q  (1+|v_0|)^γ+α/1+2s d_ℓ(z_0,z_1)^α',      using  d(z_0,z_1) ≳ (1+|v_0|)^-1/(2s).

Adding the upper bounds for I and II we conclude the proof of Lemma <ref>.


  
Let f, g ∈^α([τ,T] ×^d ×^d) for 0 < α≤min(1,2s). Then Q_2(f,g) ∈^α([τ,T] ×^d ×^d) and the following estimates hold for any q > d+γ_++α/(1+2s),
  
    _2(f,g) _C_ℓ,q^α'≤ C   f_C_ℓ,q^αg_C_ℓ,q+α/(1+2s) + γ^α'

 with C=C(d,γ,s,α) and α' = 2s/1+2sα.



Recall that
_2(f,g)(v) = c_b (∫_^d f(v+w) |w|^γ w ) g(v).
Given Q_r(z_0) ⊂ [τ,T] ×^d ×^d, we combine Lemma <ref> and Lemma <ref> to get

    (1+|v_0|)^q _2 (f,g) _C_ℓ^α'(Q_r(z_0))≲ (1+|v_0|)^q+ γ + α/1+2sf_C_ℓ,q^α g_C_ℓ^α'(Q_r(z_0)).

Taking the supremum over Q_r(z_0) yields the announced estimate. 






 §.§ Bounds for _1

This paragraph is dedicated to the derivation of appropriate bounds for _1 (f,g) when f ∈^α and g ∈^2s+α. For this purpose, we need to localize around a given point z_0. In order to measure the effect of this localization procedure, we need some preparatory lemmas. 

The proof of the following lemma uses ideas introduced in <cit.>. It is used to bound a part of the integral involved in the computation of an integro-differential operator.

 
  Let f be such that for q> d+ (γ +2s) and for all v ∈^d, 
  
    |f(v)| ≤ N_q (1+|v|)^-q.

Let K_f be the Boltzmann kernel given in formula (<ref>) applied to this function f. Then for all v ∈ B_r (v_0) with r ∈ (0,1)  and g ∈ L^1(B_|v_0|/8),
  
    ∫_
    |v'-v| > 1+ |v_0|/8, 
    
    |v'| < |v_0|/8 
     |g(v')| K_f (v,v') ' ≤ C N_q g_L^1(B_|v_0|/8) (1+|v_0|)^-q+γ

for some C depends on q,d,γ,s and the constants in B (the Boltzmann kernel). 


Later on, Lemma <ref> will be applied to large values of |v_0|. In that case, the inequality |v'| < |v_0|/8, together with v ∈ B_1(v_0), implies |v'-v| > 1+|v_0|/8.



  The proof is very similar to the ones of
  <cit.>. 
Using (<ref>), we first write for v ∈ B_r(v_0), 

    ∫_
    |w| > 1+|v_0|/8, 
    
    |v+w| < |v_0|/8 
       |g(v+w)| K_f (v,v+w) , 
        = ∫_
    |w| > 1+|v_0|/8, 
    
    |v+w| < |v_0|/8 
     |g(v+w)| |w|^-d-2s{∫_u ⊥ w f(v+u) A(|w|,|u|)|u|^γ+2s+1},  
       ≲  N_q g_L^1(B_|v_0|/8) (1+|v_0|/8)^-d-2smax_
    |w| > 1+|v_0|/8, 
    
    |v+w| < |v_0|/8 
    {∫_u ⊥ w (1+|v+u|)^-q |u|^γ+2s+1 u  }.


Since |v+w| ≤ |v_0|/8, we also have

    |w| ≥ |v_0| - |v-v_0| - |v+w| ≥7/8 |v_0| - 1.

Thus

    |w|+1/7≥ |v+w|.

Therefore, we get for u ⊥ w,

    1+ |v+u|    ≥ 1 + |u-w| - |v+w|, 
       ≥ 1 + |u-w| - |w|+1/7 , 
       = 6/7  + (|u|^2+|w|^2)^1/2 - |w|/7, 
       ≥6/7 + |u|/√(2) + |w|/√(2)  -|w|/7,
       ≳ (1+|u|+|w|).

We use this inequality to continue our estimate from the beginning of this proof.



    ∫_
    |w| > 1+|v_0|/8, 
    
    |v+w| < |v_0|/8 
       |g(v+w)|  K_f (v,v+w) , 
       ≲ N_q g_L^1(B_|v_0|/8) (1+|v_0|)^-d-2smax_|w| > 1+|v_0|/8{( ∫_u ⊥ w (1+ |u| +|w|)^-q |u|^γ+2s+1 u ) } , 
       ≲ N_q g_L^1(B_|v_0|/8) (1+|v_0|)^-d-2smax_|w| ≥ 1+|v_0|/8 (1+|w|)^-q+γ+2s+d, 
       ≲ N_q g_L^1(B_|v_0|/8) (1+|v_0|)^-q+γ.


The implicit constant in ≲ depends only on d,q,s,γ. The proof is now complete. 


For the next lemmas in this section, we define a cutoff function in the following way. Let φ̅ be a fixed smooth nonnegative bump function supported in B_1/8 so that φ̅=1 in B_1/9. For any given value of v_0 ≠ 0, we define the cutoff function φ as

    φ(v) := φ̅( |v_0|^-1 v).

This function φ is supported in B_|v_0|/8 and is identically 1 in B_|v_0|/9. Note that Lemma <ref> can be reformulated using φ in the following way (at least for |v_0| large, see Remark <ref>), 

    ∀ v ∈ B_1 (v_0),     _1(f,φ g)(v) = ∫_^dg(v') φ(v') K_f(v,v')  v' ≲ N_q φ g_L^1(^d) (1+|v_0|)^-q+γ.

By the definition of g_C^α_ℓ,q, we see that for any α≥ 0 and q>0,

    (1-φ) g_C^α_ℓ(Q_1(z_0))≲ (1+|v_0|)^-qg_C^α_ℓ,q.


In the following lemma, we establish the upper bound and decay as |v|→∞ for _1(f,g) in terms of corresponding norms of f and g.

 
Let f ∈^0([0,T] ×^d ×^d) and g ∈^2s+α([0,T] ×^d ×^d) for some α>0. Then _1(f,g) ∈^0([0,T] ×^d ×^d). Moreover, for any q>d+γ+2s,

    _1(f,g)_C^0_ℓ,q-γ-2s≤ C f_C^0_ℓ,qg_C^2s+α_ℓ,q.

Here, the constant C depends only on α, dimension, s, γ, and the constant in B (the Boltzmann kernel).



Let us start by recalling the formula for _1(f,g).

    _1(f,g)    = ∫_^d (g'-g) K_f(t,x,v,v')  v', 
       = ∫__d (g(z∘ (0,0,w)) - g(z)) K_f,z(w)  w.

Here, K_f is the Boltmann kernel depending on the function f as in (<ref>). As usual, we write K_f,z(w) = K_f(t,x,v,v+w) for z=(t,x,v).

We need to establish an upper bound for _1(f,g)(z_0) for any given z_0 = (t_0,x_0,v_0) ∈ [0,T] ×^d ×^d.

If |v_0| ≤ 2, we use (<ref>) together with Lemma <ref> and conclude the inequality immediately.

If |v_0| > 2, we decompose _1(f,g) = _1(f,φ g)+ _1(f,(1-φ)g). Here, φ is the cutoff function defined in (<ref>).

For _1(f,φ g), we use Lemma <ref> to get (<ref>) and obtain

    |_1(f,φ g)(z_0)| ≲f_C^0_ℓ,qg_C^0_ℓ,q (1+|v_0|)^-q+γ.


For _1(f,(1-φ) g), we apply (<ref>) and Lemma <ref>. We get

    | _1(f,(1-φ) g)(z_0)|    ≲( ∫_^d f(t_0,x_0,v_0+w) |w|^γ+2s w ) (1-φ) g_C_ℓ^2s+α , 
       ≲g_C^2s+α_ℓ,qf_C^0_ℓ,q (1+|v_0|)^-q+γ+2s    using  Lemma <ref>.

Adding the estimates for |_1(f,φ g)(z_0)| and |_1(f,(1-φ) g)(z_0)|, we conclude the proof.


In order to estimate the Hölder semi-norm of _1 (f,g), we need the following lemma that is the C^α version of Lemma <ref>.

 
Let f,g ∈^α for some α∈ (0,min(1,2s)]. Let z_0 = (t_0,x_0,v_0) such that |v_0| > 2 and Q_1(z_0) ⊂ [0,∞) ×^d ×^d. Let φ be the smooth bump function supported in B_|v_0|/8 with φ=1 in B_|v_0|/9. Let h: Q_1(z_0) → be given by

    h(z) := ∫_^dφ(v+w) g(z ∘ (0,0,w)) K_f,z (w)  w.

Then h ∈ C^α'_ℓ(Q_1(z_0)) and for any q>d+γ+2s,

    [h]_C^α'_ℓ(Q_1(z_0)) ≤ C g_C^α_ℓ,q+α/(1+2s)f_C^α_ℓ,q+α/(1+2s) (1+|v_0|)^-q+γ+α/(1+2s),

for a constant C depending on q, d, γ and s.



Let z_1, z_2 ∈ Q_1(z_0). We need to estimate an upper bound for |h(z_1)-h(z_2)|. We write ξ = z_2 ∘ z_1^-1. As usual,  τ_ξ  denotes the right translation operator τ_ξ f(z) := f(ξ∘ z).

Note that d_ℓ(z_1,z_2) ≈z_2^-1∘ z_1 ≠ξ. In fact, ξ can be large.

We have

    h(z_2) - h(z_1)    = ∫_^d( φ(v_2+w) g(z_2 ∘ (0,0,w)) K_f,z_2(w) - φ(v_1+w) g(z_1 ∘ (0,0,w)) K_f,z_1(w) )  w, 
       = ∫_^d( φ(v_2+w) g(z_2 ∘ (0,0,w)) - φ(v_1+w) g(z_1 ∘ (0,0,w)) ) K_f,z_1(w)  w 
       = + ∫_^dφ(v_2+w) g(z_2 ∘ (0,0,w)) ( K_f,z_2(w) - K_f,z_1(w) )  w, 
       = ∫_^d( τ_ξφ g - φ g) (z_1 ∘ (0,0,w))  ) K_f,z_1(w)  w 
       = + ∫_^dφ(v_2+w) g(z_2 ∘ (0,0,w)) ( K_(f - τ_ξ^-1 f),z_2(w))  w. 
    This implies in particular that,
    | h(z_2) - h(z_1) |    ≤∫_^d| (τ_ξφ g - φ g) (z_1 ∘ (0,0,w))  | K_f,z_1(w)  w 
       = + ∫_^d |φ(v_2+w) g(z_2 ∘ (0,0,w))| ( K_ |f - τ_ξ^-1 f| ,z_2(w))  w, 
    applying Lemma <ref> and observing |v_1| ≈ |v_2| ≈ |v_0|, we get   ≲( f_C^0_ℓ,qN_1  + φ g _C^0_ℓ,q N_2 ) (1+|v_0|)^-q+γ,

where

    N_1    := τ_ξφ g - φ g _L^1(B_|v_0|/8), 
    
    N_2    := sup_w ∈^d( |f - τ_ξ^-1 f| (z_2 ∘ (0,0,w)) |v_2 + w|^q ).


Let us first analyze N_2. 

    N_2     = sup_w ∈^d  | f(z_2 ∘ (0,0,w)) - f(z_1 ∘ (0,0,w))| (1+|v_2+w|)^q,  
     using that 1+| v_1| ≈ 1+|v_2| ≈ 1+|v_0|, we compute   ≲sup_w ∈^dmin( 1, d_ℓ( z_1 ∘ (0,0,w), z_2 ∘ (0,0,w))^α) (1+|v_0+w|)^-α/(1+2s)f_C^α_ℓ,q+α/(1+2s),
    using (<ref>),   ≲sup_w ∈^dmin( 1, d_ℓ( z_1, z_2)^α' (1+|w|)^α/(1+2s)) (1+|v_0+w|)^-α/(1+2s)f_C^α_ℓ,q+α/(1+2s),
       ≲  d_ℓ( z_1, z_2)^α' (1+|v_0|)^α/(1+2s)f_C^α_ℓ,q+α/(1+2s).


The factor N_1 is estimated similarly in terms of φ g_C^α_ℓ,q+α/(1+2s)≲g_C^α_ℓ,q+α/(1+2s).

Thus, we conclude

    |h(z_1) - h(z_2)| ≲g_C^α_ℓ,q+α/(1+2s)f_C^α_ℓ,q+α/(1+2s) (1+|v_0|)^-q+γ+α/(1+2s) d_ℓ( z_1,  z_2)^α'

from which we get the desired estimate. 


 
Let f,g ∈^α for some α∈ (0,min(1,2s)]. Let z_0 such that ℰ_1(z_0) ⊂ [0,∞) ×^d ×^d. Let φ be the smooth bump function supported in B_|v_0|/8 with φ=1 in B_|v_0|/9. Let h̅: Q_1 → be given by

    h̅(z) := ∫_^dφ(v̅+w) g((𝒯_0 z) ∘ (0,0,w)) K_f,𝒯_0 z (w)  w

with 𝒯_0 z  =(t̅,x̅,v̅). 
Then h ∈ C^α'_ℓ(Q_1) and for any q>d+(γ+2s),

    [h̅]_C^α'_ℓ(Q_1)≤ C g_C^α_ℓ,q+α/(1+2s)f_C^α_ℓ,q+α/(1+2s) (1+|v_0|)^-q+γ+α/(1+2s),

for a constant C depending on q, d, γ and s.

 Apply Lemma <ref> and observe h̅ = h ∘𝒯_0. Then use Lemma <ref> to conclude.


We are now in position to derive the desired estimate for _1 (f,g). 

 
Let f ∈^α and g ∈^2s+α for some 0<α≤min(1,2s). Then _1(f,g) ∈^α' for α' = 2s α / (1+2s). Moreover, for any q>d+γ+2s,

    _1(f,g)_C^α'_ℓ,q-γ-2s-α/(1+2s)≤ C  f_C^α_ℓ,qg_C^2s+α_ℓ,q .

Here, the constant C depends only on α, dimension, s, γ, and the constants in B (the Boltzmann kernel).



  The norm _1(f,g)_C^0_ℓ,q-γ-2s is already controlled by Lemma <ref>. We are left with estimating the semi-norm [_1(f,g)]_C^α'_ℓ,q-γ-2s.

  Let z_0=(t_0,x_0,v_0) be so that Q_r(z_0) ⊂ [0,T] ×^d ×^d for some r∈ (0,1) like in Definition <ref> and
let φ be the cutoff function as in (<ref>). From Lemma <ref>, we know that 

    [_1(f,φ g)]_C^α'_ℓ(Q_r(z_0))≲g_C^α_ℓ,qf_C^α_ℓ,q (1+|v_0|)^-q+γ+2α/(1+2s).


In order to estimate _1(f,(1-φ) g)_C^α'_ℓ(Q_r(z_0)), let us consider two points z_1, z_2 ∈ Q_r(z_0). We have

    _1(f,(1-φ) g)(z_2) - _1(f,(1-φ) g)(z_1)    = ℒ_K_f,z_2 [(1-φ)g](z_2) -  ℒ_K_f,z_1 [(1-φ)g](z_1), 
       = ( ℒ_K_f,z_2 [(1-φ)g](z_2) - ℒ_K_f,z_2 [(1-φ)g](z_1) )
       = + ( ℒ_K_f,z_2 [(1-φ)g](z_1) -  ℒ_K_f,z_1 [(1-φ)]g(z_1) ).

In the first term we are fixing the kernel K_f,z_2 (freezing coefficients) and evaluating in the function (1-φ)g at two points z_2 and z_1. In the second term, we are evaluating the operator at the same point z_1, for the same function (1-φ)g, and we will obtain cancellation from the two kernels K_f,z_2 - K_f,z_1.

For estimating the first term, we use  <cit.>. It gives us that

    | ℒ_K_f,z_2 [(1-φ)g](z_2) - ℒ_K_f,z_2 [(1-φ)g](z_1)|    ≲Λ_K_f,z_2 [(1-φ) g]_C^2s+α'_ℓ d_ℓ(z_1,z_2)^α', 
       ≲ (1+|v_0|)^-q+γ+2sf_C^0_ℓ,qg_C^2s+α'_ℓ,q d_ℓ(z_1,z_2)^α',

using (<ref>) and Lemma <ref> to get the second inequality.


For the second term, we use Lemma <ref>, and compute

    | ℒ_K_f,z_2   [(1-φ)g](z_1) - ℒ_K_f,z_1 [(1-φ)g](z_1) | ≲Λ_(K_f,z_2-K_f,z_1)(1-φ) g_C^2s+α'_ℓ, 
    using (<ref>) and estimating Λ_(K_f,z_2-K_f,z_1) from (<ref>),   ≲ (1+|v_0|)^-qg_C^2s+α'_ℓ,q( ∫_^d | f(z_2 ∘ (0,0,w))-f(z_1 ∘ (0,0,w)) | |w|^γ+2s w )


We proceed like in the proof of Lemma <ref> to estimate the integral. Using (<ref>), we have that

    | f(z_2 ∘ (0,0,w))-f(z_1 ∘ (0,0,w)) | ≲min( 1, (d_ℓ(z_1,z_2) + d_ℓ(z_1,z_2)^2s/(1+2s) |w|^1/(1+2s))^α ) (1+|v_1+w|)^-qf_C^α_ℓ,q

from which we get,

    ∫_^d | f(z_2 ∘ (0,0,w))-f(z_1 ∘ (0,0,w)) | |w|^γ+2s w ≲ d_ℓ(z_1,z_2)^α'f_C^α_ℓ,q (1+|v_0|)^α/(1+2s),

provided that q > d+γ+2s+α/(1+2s). Incorporating this inequality in the computation above, we get

    | ℒ_K_f,z_2   [(1-φ)g](z_1) - ℒ_K_f,z_1 [(1-φ)g](z_1) |≲
    (1+|v_0|)^-q+α/(1+2s)g_C^2s+α'_ℓ,qf_C^α_ℓ,q d_ℓ(z_1,z_2)^α'.

Collecting the two upper bounds above,

    |_1(f,(1-φ) g)(z_2) - _1(f,(1-φ) g)(z_1)| ≲ (1+|v_0|)^-q+γ+2s+α/(1+2s) d_ℓ(z_1,z_2)^α'f_C^α_ℓ,qg_C^2s+α'_ℓ,q.

Combining with (<ref>) and using that α/(1+2s) < 2s,

    |_1(f,g)(z_2) - _1(f,g)(z_1)| ≲ (1+|v_0|)^-q+γ+2s+α/(1+2s) d_ℓ(z_1,z_2)^α'f_C^α_ℓ,qg_C^2s+α'_ℓ,q.

Thus, [ _1(f,g) ]_C^α'_ℓ(Q_1(z_0))≤ (1+|v_0|)^-q+γ+2s+α/(1+2s)f_C^α_ℓ,qg_C^2s+α'_ℓ,q and we concluded the proof.



The lemmas in Section <ref> allow us to estimate the Höder norm of Q(f,g) in terms of the norms of f and g. When it comes to global Hölder norms, we obtain certain decay exponent “q” in each of the lemmas. The precise value (and loss) of decay exponent is not computed sharply because it was not necessary for the purpose of the result in this paper. It might make sense to investigate sharper version of the lemmas in this section if one tries to obtain C^∞ estimates as in Theorem <ref> for γ≤ 0 but for solutions that do not decay rapidly as |v| →∞.





§ GLOBAL HÖLDER ESTIMATES FOR THE BOLTZMANN EQUATION


In this section, we derive global Hölder estimates and global Schauder estimates from the local ones derived in the previous sections. 



 §.§ Global Hölder estimates


The Hölder estimate in Theorem <ref> applies directly to the Boltzmann equation. However, in doing so, it leads to a Hölder estimate only locally, that is to say for a compact set of velocities. In order to obtain a global estimate (that holds for v ∈^d) we combine Theorem <ref> with the change of variables described in Section <ref>.


  Let f
  be a solution of the Boltzmann
  equation (<ref>) in (0,T) ×^d ×^d so that Assumption <ref> holds in its full domain. Assume that for some q>d, there exists  N_q>0 such that for all (t,x,v) ∈ (0,T) ×^d ×^d, 
  
    f(t,x,v) ≤ N_q (1+|v|)^-q

(this is the same as to say N_q = f_C^0_ℓ,q).

  Let us set z_0=(t_0,x_0,v_0) ∈ (τ,T) ×^d ×^d with
  |v_0| ≥ 2, to be the center of the change of variables 𝒯_0 from Section <ref>.  Then for all r ∈ (0,1) such that
  ℰ_r (z_0) ⊂ (τ,T) ×^d ×^d and
  all z̅_1, z̅_2 ∈ℰ_r/2(z_0),
  
    | f(z̅_1) - f(z̅_2)| ≤ C(N_q)  (1+|v_0|)^-q+γ_+ d_ℓ(z_1,z_2)^α

  where C>0 and α∈ (0,1) only depend on q, N_q, the parameters in Assumption <ref>, dimension d, γ, s from
  (<ref>) and τ. Here z̅_i = 𝒯_0 z_i.



Our solutions f will be in ^0 with semi-norms controlled by Theorem <ref>.



  When t_1 = t_2 and x_1=x_2, d_ℓ(z_1,z_2) is the same as |v_1-v_2|. It is exactly comparable to the non-isotropic
  distance (v̅_1, v̅_2) of Gressman and Strain as defined in <cit.> (since r<1); see Lemma <ref>
  in the appendix.

Proposition <ref> gives us a global Hölder estimate in all variables t, x and v. 


We will use in Corollary <ref> below the following straightforward consequence of the sharp global Hölder estimate from Proposition <ref>.


  Let f be a solution of the Boltzmann
  equation (<ref>) in (0,T) ×^d ×^d so that Assumption <ref> holds for all
  (t,x) ∈ (0,T) ×^d. Moreover, assume f ∈^0.

  Then, there is an α>0 so that for all τ∈ (0,T), f ∈^α with

    f_C_ℓ,q^α( (τ,T) ×^d ×^d)≤ C_q f_C^0_ℓ,q+q̃(γ,s) ( (0,T) ×^d ×^d)     for all q>d.

The constant C_q>0 depends on q, the parameters from Assumption <ref>, dimension d, γ and s from
  (<ref>) and time τ. The value of α>0  depends on the constants in Assumption <ref>, d, γ and  s. The value of q̃(γ,s) depends on s and γ only.


We now turn to the proof of the previous proposition. 


Without loss of generality, we assume τ≥ 1 and take r=1. Otherwise we would have to adjust the choice of r to be ≤τ^1/(2s) so that Q_r(z_0) ⊂ (0,T) ×^d ×^d. The constants in the result would be affected by this value accordingly.

  Let φ be the cutoff function as in (<ref>). In particular φ =0 in E_1(v_0) for |v_0|≥ 2 because E_1 (v_0) ⊂ B_1 (v_0). Let g(t,x,v) = (1-φ(v))  f(t,x,v). Thus, g_L^∞≤ N_q (1+|v_0|)^-q.

By a direct computation, we observe that g solves the equation

    ∂_t g + v ·∇_x g = _K_f(g) 
    + h_1 + h_2      in   (0,T) ×^d × E_1(v_0) ,

where

    h_1 = ∫_^dφ(v') f(v') K_f(v,v')  v'  
        and      
    h_2 = _2(f,f) =c_b (f ∗ |·|^γ)f.


Recall that φ is supported in B_|v_0|/8. Thus, we apply Lemma <ref> and obtain

    |h_1|    ≲φ f_L^1_v N_q  (1+|v_0|)^-q+γ
       ≲ M_0   N_q  (1+|v_0|)^-q+γ

in ℰ_1(z_0) ⊂ Q_1 (z_0). 

In order to estimate h_2, we note that if γ≥ 0, then Lemma <ref> implies that |·|^γ∗_v f ≲ (1+|v_0|)^γ (M_0+E_0). On the other hand, if γ < 0, then Lemma <ref> implies that |·|^γ∗_v f ≲ N_q (1+|v_0|)^γ, provided that q>d. Thus

    |h_2| ≲
    (1+|v_0|)^-q+γ N_q,     if γ≥ 0,
    
    (1+|v_0|)^-q+γ N_q^2     if γ < 0,
 
in ℰ_1(z_0) ⊂ Q_1 (z_0).  

Applying the change of variables  from (<ref>) and Theorem <ref>, we have that the function
g̅ = g ∘ solves

    ∂_t g̅ + v ∇_x g̅ =  _K̅_fg̅ + h̅     in   Q_1

with 

    h̅ := |v_0|^-γ-2s(  h_1((t,x,v)) + h_2( (t,x,v) )

and K̅_f satisfies  ellipticity conditions (<ref>) (only if s < 1/2), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) (only if s ≥1/2). Note that Assumption <ref> in Theorem <ref> holds with C_γ≲ N_q because q > d, recall Lemma <ref>.

Applying Theorem <ref> to g̅, we get for all
z_1, z_2 ∈ Q_1/2,

    |g̅(z_1) - g̅(z_2)|    ≤ C  ( g̅_L^∞([-1,0]× B_1 ×^d) + h̅_L^∞(Q_1)) d_ℓ(z_1, z_2)^α, 
       ≤ C ( N_q (1+|v_0|)^-q + (N_q + N_q^2) (1+|v_0|)^-q+γ) d_ℓ(z_1, z_2)^α
       ≤ C (N_q) (1+|v_0|)^-q+γ_+ d_ℓ(z_1, z_2)^α.


This estimate yields the result since g̅ = g ∘ and z_i = z̅_i for i=1,2.






 §.§ Global Schauder estimates for the Boltzmann equation



Our next task is to use the change of variables in order to derive global Schauder estimates for the Boltzmann equation. In this case, we work with a more general equation, the linear Boltzmann equation,

    (∂_t + v ·∇_x) g = _1(f,g) +h  in  (0,T) ×^d ×^d.

Theorem <ref> gives us local Schauder estimates for the solution g, with a precise exponent, in terms of Hölder norms of h, g and the kernel K_f. We will combine it with the change of variables described in Section <ref> in order to obtain global Schauder estimates.

We should not be deceived by the description of (<ref>) as a linear equation. Proposition <ref> below applies whenever a function g satisfies any equation of that form, for any functions f and h. Whether the functions f, g and h are related to each other or not is irrelevant for the estimates. In particular, if f=g and h = _2(f,f), the estimate in Proposition <ref> applies to the original (nonlinear) Boltzmann equation. Equation (<ref>) will also be satisfied when g is a directional derivative of f or some incremental quotient, for an appropriate h in each case. In that sense, an estimate for (<ref>) as in Proposition <ref> is more general than a Schauder estimate for merely the original Boltzmann equation (<ref>).




  Let f : (0,T) ×^d ×^d → [0,∞) be such that Assumption <ref> holds. Assume also that f ∈^α for some  α∈ (0,min (1,2s)).  Let g ∈^α be a
  solution of (<ref>) with h ∈^α'
  with α' = 2s/1+2sα and 2s+α' ∉{1,2}. Then for all τ > 0, we have the following a priori estimate for g in ^2s+α'([τ,T] ×^d ×^d), for each q > d+2+2s,
  
    g_C_ℓ,q^2s+α' ([τ,T] ×^d ×^d) ≤ C ( g_C_ℓ,q+κ^α ([0,T] ×^d ×^d))
        + h_C_ℓ,q+κ^α' ([0,T] ×^d ×^d) )

  where the constant κ depends on s and γ only, and C depends on r,q, dimension d, parameters
  s, γ in (<ref>), m_0,M_0,E_0,H_0 from Assumption <ref>, τ and f_C_ℓ,q+κ^α(Q_r(z_0)).




Like in the proof of Proposition <ref>, we concentrate on |v_0|>2 and assume without loss of generality that τ≥ 1. Let us pick any z_0 so that ℰ_1(z_0) ⊂ [0,T] ×^d ×^d.

Let φ be the cutoff function as in (<ref>).

We multiply g by (1-φ) in order to concentrate on velocities |v| ≥ |v_0|/9. Then, we change variables by looking at g̅ = [(1-φ) g] ∘𝒯_0. Recall that 𝒯_0 maps Q_1 into the slanted ellipsoidal cylinder ℰ_1(z_0). The function g̅ satisfies the following equation in Q_1,

    ∂_t g̅ + v ·∇_x g̅ = ( ∫_^d(g̅'- g̅) K̅_f(t,x,v,v')  v' ) + h̅ + h̅_2.

Here, K̅_f is the kernel after the change of variables, as in (<ref>). 

The function h̅ corresponds to the source term h after the change of variables. The new source term h̅_2 is the result of  our application of the cutoff factor (1-φ). The functions h̅ and h̅_2 are given by

    h̅   := |v_0|^-γ-2s h ∘𝒯_0, 
    h̅_2    := |v_0|^-γ-2s∫_^dφ(v') g(t̅, x̅, v') K_f(t̅, x̅, v̅, v')  v'.

As usual, we write z̅ = (t̅, x̅, v̅) = 𝒯_0 z.

According to the Schauder estimates of Theorem <ref>, we get

    [g̅]_C^2s+α'_ℓ(Q_1)   ≲(1+A̅_0^2s+α'-α/α') [g̅]_C^α_ℓ([-2^2s,0] × B_2 ×^d)  + [h̅ + h̅_2]_C^α'_ℓ(Q_2) + (1+A̅_0) h̅ + h̅_2_C^0(Q_2) , 
       =: T_1 + T_2 + T_3.


Since we know from Lemma <ref> that [g]_C^2s+α'_ℓ(ℰ_1(z_0))≲|v_0|^c̅ (2s+α')  [g̅]_C^2s+α'_ℓ(Q_1), the proof of this proposition will proceed by estimating the right hand side in the inequality above. Let us estimate the three terms T_1, T_2 and T_3, one by one.

For the first term, let us observe that by the construction of φ and the definition of the norm C^α_ℓ,q,

    [(1-φ) g]_C^α_ℓ([0,T]×^d ×^d)≲ (1+|v_0|)^-qg_C^α_ℓ,q([0,T]×^d ×^d).

Combining with the change of variables and using Lemma <ref>, 

    [g̅]_C^α_ℓ([-2^2s,0] × B_2 ×^d)≤ (1+|v_0|)^-q_1g_C^α_ℓ,q_1([0,T]×^d ×^d).

The estimate in (<ref>) holds for any value of q_1>0.

Using Lemma <ref>, we have that, for any q_1 > d+2+α/(1+2s)

    A̅_0 ≲ (1+|v_0|)^α/1+2s(1-2s-γ)_+f_C^α_ℓ,q_1.

Combining it with (<ref>), we estimate the first term T_1 as

    T_1    ≲ |v_0|^-q_1g_C^α_ℓ,q_1 + |v_0|^-q_1 + α/1+2s(1-2s-γ)_+ 2s+α'-α/α'f_C^α_ℓ,q_1^2s+α'-α/α'g_C^α_ℓ,q_1, 
       = |v_0|^-q_1g_C^α_ℓ,q_1 + |v_0|^-q_1 + (1-2s-γ)_+ ( 1 - α/2s (1+2s)) f_C^α_ℓ,q_1^2s+α'-α/α'g_C^α_ℓ,q_1,
       ≤ |v_0|^-q_1g_C^α_ℓ,q_1 + |v_0|^-q_1 + 1 f_C^α_ℓ,q_1^2s+α'-α/α'g_C^α_ℓ,q_1.

This is true for any q_1 > d+2+α/(1+2s).

For the other terms, we must estimate the C^α'_ℓ norms of h̅ and h̅_2. In the case of h̅, we simply observe that

    h̅_C^0(Q_2)   = |v_0|^-γ-2sh_C^0(ℰ_2(z_0)) , 
       ≤ |v_0|^-γ-2s-q_1h_C^0_ℓ,q_1. 
    
    [ h̅ ]_C^α'_ℓ(Q_2)   ≤ |v_0|^-γ-2s [h]_C^α'_ℓ(ℰ_2(z_0)) , 
       ≤ |v_0|^-γ-2s-q_1 [h]_C^α'_ℓ,q_1.


In the case of h̅_2, we apply Lemma <ref> (with q=q_1) and Corollary <ref> (with q=q_1-α / (1+2s)) and obtain  for any q_1 > d+(γ+2s),

    h̅_2_C^0_ℓ(Q_2)   ≲ |v_0|^-q_1-2sg_C^0_ℓ,q_1f_C^0_ℓ,q_1 , 
    
      [h̅_2]_C^α'_ℓ(Q_2)   ≲ |v_0|^-q_1-2s+2α/(1+2s)f_C^α_ℓ,q_1g_C^α_ℓ,q_1  .

We use these estimates to obtain upper bounds for T_2 and T_3.

    T_2    ≲ |v_0|^-γ-2s-q_1 [h]_C^α'_ℓ,q_1 + |v_0|^-q_1-2s+2α/(1+2s)f_C^α_ℓ,q_1g_C^α_ℓ,q_1 ,
    
    T_3    ≲(1+ |v_0|^α/1+2s(1-2s-γ)_+f_C^α_ℓ,q_1) (  |v_0|^-γ-2s-q_1h_C^0_ℓ,q_1  + |v_0|^-q_1-2sg_C^0_ℓ,q_1f_C^0_ℓ,q_1).


Finally, using Lemma <ref>,

    [g]_C^2s+α'_ℓ (ℰ_1(z_0))≲ |v_0|^c̅(2s+α') [g̅]_C^2s+α'_ℓ(Q_1)≲ |v_0|^c̅(2s+α')(T_1+T_2+T_3).


Note that ℰ_1(z_0) ⊃ Q_|v_0|^-c̃ (z_0) with c̃ = max (1, (γ+2s)/(2s)) and (<ref>) holds at any point z_0. Using Lemma <ref>, we extend the inequality to the larger domain Q_1(z_0),

    [g]_C^2s+α'_ℓ ( Q_1(z_0))   ≲ |v_0|^c̅ (2s+α') (T_1+T_2+T_3) + |v_0|^c̃ (2s+α')g_C^0_ℓ(Q_1(z_0)).  
    Collecting all inequalities, not tracking the dependence on f_C^α_ℓ,q_1, and keeping only the largest exponents of |v_0|, we are left with    ≤ C  (  |v_0|^-q_1+κg_C^α_ℓ,q_1 +  |v_0|^-q_1+κh_C^α'_ℓ,q_1).

Here, the constant C depends on f_C^α_ℓ,q_1 and κ depends on s and γ only.

For any given value of q, we pick q_1 = q + κ and conclude the proof of the lemma.



In Proposition <ref>, we obtain a priori estimates for the norms g_C^2s+α'_ℓ,q in terms of g_C^α_ℓ,q_1, f_C^α_ℓ,q_1 and h_C^α'_ℓ,q_1 for q_1 = q+κ. Note that we gain some regularity in the estimate but we lose some decay from q to q_1. We have made no effort to make the choice of q_1 as q+κ optimal. Since we work with functions that have a rapid decay as |v| →∞, the precise exponent in the loss of decay in the estimate has no consequence for our proof.



Following the proof of Proposition <ref> one can compute how the constant C depends on f_C^α_ℓ,q+κ. We get C ≈f_C^α_ℓ,q+κ^α+2s-α'/α' = f_C^α_ℓ,q+κ^1/(2s) + (1+2s)/α.


 
Let f  be a  solution of the Boltzmann  equation (<ref>) in (0,T) ×^d ×^d so that Assumption <ref> holds. If γ≤ 0, assume further that f(0,x,v) = f_0(x,v) with

    0 ≤ f_0(x,v) ≤ N_q (1+|v|)^-q,

for all non-negative integer q.

Then, for some α>0 and every q ∈ℕ, the norm f_C^2s+α_ℓ,q((τ,T)×^d ×^d) is bounded depending only d, γ, s, τ, the parameters in Assumption <ref>, and the values of N_q (if γ≤ 0).



Applying Theorem <ref>, we get an estimate for the norms f_C^0_ℓ,q(τ/3,T) ×^d ×^d) for any value of q ∈ℕ.

Applying Corollary <ref>, we get an estimate for the norms f_C^α_ℓ,q((τ/2,T) ×^d ×^d) for any value of q ∈ℕ, and some small α>0.

Applying Proposition <ref> to g=f and h=f, we conclude the proof of the corollary.




§ INCREMENTS




In order to bootstrap the regularity estimate from Corollary <ref>, we will apply the global Schauder estimates from Proposition <ref> to derivatives and increments of the solution f iteratively.

Before doing that, we develop some technical lemmas about increments and Hölder norms in this section.

Let us write

    Δ_y f (z) = f(z ∘ (0,y,0)) - f(z)     and   Δ_w f (z) = f(z∘ (0,0,w))-f(z)

for some small increments y ∈^d and w ∈^d.

Roughly speaking, the global Schauder estimate from Proposition <ref> allows us to gain only 2s derivatives at each iteration, which can be less than 1 if s<1/2. In order to gain one full derivative in each variable, we will apply this estimate to increments of f as above.

The following fact about (usual) Hölder spaces is commonly used to study the regularity of solutions to nonlinear equations (see <cit.>). If f: → is a C^α function, and the C^α semi-norm of the increments f(x+h)-f(x) is bounded above by ≲ |h|^β, then f is Hölder continuous with the larger exponent min(α+β,1). In our current context of kinetic equations, the underlying geometry and Galilean invariance make the procedure more complicated. Here, we present two separate lemmas that involve increments in space and velocity respectively. They allow us to transfer a regularity estimate for an increment, into a higher order of differentiation. 

In spite of the apparent simplicity of the statement, the proof is rather involved. The first step in the proof is inspired by <cit.>.



  Let α_1,α_2 >0 and β≥ 0.  Given a cylinder  Q= Q_R(z_0) with R ∈ (0,1) and a bounded continuous function  f defined in Q, we consider for any  y ∈ B_R^1+2s/2  the following  function,
  
    Δ_y f(z) = f(z ∘ (0,y,0)) -f(z).

It is defined in = Q_R/2(z_0).

We assume there exists an N>0 such that for all  y ∈ B_R^1+2s/2

    Δ_y f_C^0()≤ N,       [Δ_y f]_C_ℓ^α_1 + α_2 ()≤ N (0,y,0)^β.


We assume that α_2 ∈ (0,min (1,2s)], α_1+α_2 ≤ 1+  2s, α_1+β≤ 1+  2s.  Then for all  y ∈ B_R^1+2s/2, 
  
    Δ_y f _C_ℓ^α_2 ()≲ N (0,y,0)^α_1+β    if α_1+α_2+β≤ 1+ 2s, 
    Δ_y f _C_ℓ^η ()≲ N (0,y,0)^1+ 2s    if α_1+α_2+β>1+ 2s,

 for some η=η(α_1,α_2,β,s)>0. 



  One might expect that η = α_1 + α_2 + β - (1+2s).
  Our proof gives us a smaller number η>0, with an explicit
  formula. We do not know if the value we obtain is sharp.



  Let α_1,α_2 >0 and β≥ 0.  Given a cylinder   Q= Q_R(z_0) with R ∈ (0,1) and a bounded continuous function  f defined in Q, we consider for any w ∈ B_R/2 the  following function,
  
    Δ_w f(z) = f(z ∘ (0,0,w)) -f(z).

  It is defined in = Q_R/2(z_0).

We assume there exists an N>0 such that for all  w ∈ B_R/2,
  
    Δ_w f_C^0()≤ N,       [Δ_w f]_C_ℓ^α_1 + α_2 ()≤ N (0,0,w)^β.


We assume that α_2 ∈ (0,min (1,2s)], α_1+α_2 ≤ 1, α_1+β≤ 1.  Then  for all  w ∈ B_R/2, 
  
    Δ_w f _C_ℓ^α_2 ()≲ N (0,0,w)^α_1+β    if α_1+α_2+β≤ 1, 
    Δ_w f _C_ℓ^η ()≲ N (0,0,w)    if α_1+α_2+β>1

 for some η=η(α_1,α_2,β,s)>0. 



  Let ι= 1 if a=(0,y,0) and ι=0 if a=(0,0,w). We then consider  Δ_a f (z) = f(z ∘ a) -f(z)  so that Δ_a f equals either Δ_y f or Δ_w f if a =(0,y,0) or a =(0,0,w). 
  
  Let p_z denote the polynomial expansion of f at z of kinetic  degree strictly smaller than α_1 + α_2. The assumptions   (<ref>) and (<ref>) implies in particular the  following: for all z ∈ and ξ such that  z ∘ξ∈,

    |Δ_a f (z ∘ξ) - δ_a p_z (ξ)| ≤ N a^βξ^α_1+α_2

where Δ_a f (z) = f(z ∘ a) -f(z) and where  δ_a p_z is the polynomial expansion of Δ_a f at the point z.

We abuse notation by writing Δ_y f = Δ_(0,y,0) f,  δ_y p_z = δ_(0,y,0) p_z,  Δ_w f = Δ_(0,0,w)f and  δ_w p_z = δ_(0,0,w)p_z. Since α_2 ∈ (0,min (1,2s)), we aim at proving that for  z ∈ and a ∈ Q_R/2 and ξ such that  z ∘ξ∈,

    |Δ_a f (z ∘ξ) - Δ_a f (z)| ≲ N a^α_1+βξ^α_2 ,         if α_1+α_2 + β≤ 1 + ι 2s  
    
        |Δ_a f (z ∘ξ) - Δ_a f (z)| ≲ N a^1 + ι 2sξ^η,         if α_1+α_2 + β > 1 + ι 2s

where ι = 1 for a =(0,y,0) and ι = 0 if a= (0,0,w).

The remainder of the proof proceeds in several steps. The first one is reminiscent of the proof of <cit.>.  

Step 1. We claim that for all  z ∈ and all k ∈ℕ such that z ∘ (2^k a) ∈, we have
  
    |Δ_a f (z) - 2^-kΔ_2^k a f(z) | ≲   N a^β+α_1+α_2 2^k ( β+α_1+α_2/1+ ι 2s - 1 )_+.

  
  In order to get such an estimate, we remark that
  
    Δ_2a f(z) =Δ_a f(z) + Δ_a f (z∘ a).

  Using (<ref>), we thus get
  
    |Δ_2a f(z) - 2 Δ_a f(z)|      = |Δ_a f (z∘ a) - Δ_a f(z)| 
       ≲ N a^α_1+α_2+β + |δ_a p_z (a) - Δ_a f (z)| .

  Since the polynomial p_z is of degree strictly less than α_1+α_2, we have for ξ = (ξ_t,ξ_x,ξ_v) ∈^1+2d, 
  
    δ_a p_z (ξ) = Δ_a f(z) + if α_1 + α_2 > 2s(∂_t + v·∇_x)Δ_a f (z)ξ_t
        + if α_1 + α_2 > 1D_v Δ_a f (z)·ξ_v
        + 1/2if α_1 + α_2 > 2D_v^2 Δ_a f (z) ξ_v ·ξ_v.

  In particular, since α_1 + α_2 ≤ 1 when a = (0,0,w), we remark that, when evaluating the previous expression with a = (0,y,0) at ξ = (0,y,0) or with a =(0,0,w) at ξ =(0,0,w), 
  
    δ_y p_z ((0,y,0)) = Δ_y f(z) .

  In the case a = (0,0,w), we used the assumption α_1+α_2 ≤ 1. 
  
We thus conclude that
  
    |Δ_2a f(z) - 2 Δ_a f(z)| ≲ N a^α_1+α_2+β

  or equivalently
  
    |Δ_a f(z) - 2^-1Δ_ 2a f(z)| ≲ 2^-1 N a^α_1+α_2+β.

  By induction, we get
  
    |Δ_a f(z) - 2^-kΔ_2^k a f(z)|    ≲ N ∑_j=1^k2^-j2^j-1a^α_1+α_2+β
       ≲ N a^α_1+α_2+β∑_j=1^k 2^-j + (j-1)α_1+α_2 + β/1+ι 2s
       ≲ N a^α_1+α_2+β 2^k(α_1+α_2 + β/1+ ι 2s -1)_+.

  This achieves the proof of the claim.
  

  Step 2. We claim now that for z ∈ and a ∈ Q_R/2, 
  
    |Δ_a f(z)| ≲ N a^m

  with m=min (α_1+α_2+β,1+ ι 2s).   It is enough to pick an integer k=k(a) such that  2^k a≃ 1 (or equivalently 2^-k≃a^1+ι 2s), and apply Claim (<ref>) from Step 1.

  Indeed, using the assumption α_1+α_2 ≤ 1 in the case a=(0,0,w), we can write in both cases 
  
    |Δ_a f(z) |    ≲ 2^-k | Δ_2^k a f (z)| + N a^α_1+α_2+β 2^k(α_1+α_2 + β/1+ι 2s -1)_+
       ≲Δ_2^k a f_C^0a^1+ι 2s + N a^min (α_1+α_2+β,1+ ι 2s)
       ≲  N a^min (α_1+α_2+β,1+ι 2s).


 Because α_1 + β≤ 1+ι 2s and a ∈ Q_1, we remark that this implies
  |Δ_a f(z)| ≲ N a^α_1 + β.
We are thus left with estimating the semi-norm.
  

  Step 3.
  We next claim that for z ∈ and  a ∈ Q_R/2,
  
    {
        | (∂_t + v ·∇_x) Δ_a f (z)|    ≲ N  a ^m -  (2s) θ   if α_1 + α_2 > 2s 
    
        | D_v Δ_a f (z)|    ≲ N  a ^m-θ   if α_1 + α_2 > 1 
    
          | D_v^2 Δ_a f (z)|    ≲ N  a ^m-2θ   if α_1 + α_2 > 2
        .

  where
  
    m = min (α_1+α_2 + β, 1+ ι 2s)      and     θ = min(1,1+ι 2s-β/α_1+α_2).


  It is a consequence of the assumption (<ref>), the
  estimate (<ref>) from Step 2 and the interpolation
  inequality given by Proposition <ref>.  For instance, in
  the case a =(0,y,0) and if α_1 + α_2 > 2s, we have
  
    [(∂_t + v ·∇_x) Δ_y f ]_C_ℓ^0 ()   ≲ [Δ_y f]_C_ℓ^2s ()
       ≲ [Δ_y f]_C_ℓ^0 ()^1- 2s/α_1+α_2
                                                                      [Δ_y f]_C_ℓ^α_1+α_2 ()^2s/α_1+α_2 + [Δ_y f]_C_ℓ^0 (). 
       ≲ N  (0,y,0)^(1-2s/α_1+α_2) m + 2s/α_1+α_2β.

  We now remark that (1-2s/α_1+α_2) m + 2s/α_1+α_2β = m -2s θ. 
   The other cases are treated similarly.  
  Step 4.
Let z ∈ and a ∈ Q_R/2 and ξ such that z ∘ξ∈. Assume ξ≤a.  We derive from (<ref>) and the previous step 

    |δ_a p_z (ξ) - Δ_a f (z)|    ≲ N ( a^m-2s θξ^2s + a^m-θξ + a^m-2θξ^2 ),  
        = N a^m ( ( ξ/a^θ)^2s + ( ξ/a^θ) + ( ξ/a^θ)^2 ).



Since   θ≤ 1 and we are now focusing on the case ξ≤a,

    |δ_a p_z (ξ) - Δ_a f (z)| ≲ N a^m  ( ξ/a^θ)^min(1,2s) = N a^m-θmin(1,2s)ξ^min(1,2s).

Here m and θ are given in (<ref>).



Step 5. Assume a≳ξ.

We first use (<ref>) with a≳ξ to get

    |Δ_a f (z∘ξ) - δ_a p_z(ξ)| ≲  N a^α_1+βξ^α_2.


We claim that in the case α_1 + α_2 + β≤ 1+2s for a=(0,y,0) or α_1 + α_2 + β≤ 1 for a=(0,0,w), (<ref>) holds true. Indeed, since  in this case m = α_1+α_2+β and θ=1 and we also have α_2 < min (1,2s), in view of (<ref>) we get 

    |δ_a p_z (ξ) - Δ_a f (z)|  ≲  N a^α_1+α_2+β-min(1,2s)ξ^min(1,2s)≲ N a^β+α_1ξ^α_2.

Adding the previous two inequalities, we get (<ref>) for such a's and ξ's. 

For those values of α_1, α_2 and β so that
θ≠ 1, we obtain a somewhat weaker estimate. In this case, (<ref>) tells us that

    |δ_a p_z (ξ) - Δ_a f (z)|  ≲ N a^1+ι 2s - min(1,2s) θξ^min(1,2s)≲ N a^1+ι 2sξ^min(1,2s)(1-θ) .

The last inequality holds because ξ≲a.

In this case, using again that ξ≲a, (<ref>) implies that

    |Δ_a f (z∘ξ) - δ_a p_z(ξ)| ≲  N a^α_1+βξ^α_2≲ N a^1+ ι 2sξ^α_2 -(1+ι 2s-(β +α_1))≲  N a^1+ ι 2sξ^(1-θ)(α_1+α_2).


Combining the two inequalities above, we get
  
    |Δ_a f (z∘ξ) - Δ_a f(z) | ≲ N a^1+ι 2sξ^η,

  where
    
    η = min(1,2s,α_1+α_2)(1-θ).


  Step 6.  We finally claim that (<ref>),  (<ref>) hold true in all cases. In order to prove  it, we only have to deal with the case a≲ξ in  which we pick k ∈ℕ such that  2^k a≃ξ.  In this case, we can use  (<ref>), (<ref>) with 2^k a and ξ (from Step 5) and  α_1 +β≤ 1+ ι 2s and get
   
    | 2^-kΔ_2^k a f(z ∘ξ) - 2^-kΔ_2^k a f(z)|
         ≲ N  a^α_1+βξ^α_2        if α_1+α_2 + β≤ 1 + ι 2s,  
    
         | 2^-kΔ_2^k a f(z ∘ξ) - 2^-kΔ_2^k a f(z)|
         ≲ N a^1 + ι 2sξ^η,         if α_1+α_2 + β > 1 + ι 2s .

   We now use twice what we obtained in Step 1, to z and z∘ξ, and get for α_1 + α_2 + β≤ 1+ ι 2s
   (using α_1+β≤ 1+ι 2s once again),
  
    |Δ_a f (z ∘ξ) - 2^-kΔ_2^k a f(z ∘ξ ) |    ≲ N a^α_1+βξ^α_2 , 
         |Δ_a f (z) - 2^-kΔ_2^k a f(z) |     ≲ N a^α_1+βξ^α_2  ,

  and for α_1 + α_2 + β > 1 + ι 2s,
  
    |Δ_a f (z ∘ξ) - 2^-kΔ_2^k a f(z ∘ξ ) |    ≲ N a^1 + ι 2sξ^η ,
         |Δ_a f (z) - 2^-kΔ_2^k a f(z) |     ≲ N a^1+ι 2sξ^η .

  
  Summing (<ref>), (<ref>) and (<ref>) yields
  (<ref>) for all y and ξ.  In the same way,
  Summing (<ref>), (<ref>) and (<ref>) yields
  (<ref>) for all y and ξ. This achieves the proof
  of the lemma.




  Given y ∈ B_R^1+2s/2 with R ≤ 1 and  α∈ (0,min (1,2s)] and some cylinder Q=Q_R(z_0), let  f ∈ C_ℓ^2s+α (Q). Then Δ_y f lies in C_ℓ^α () with  = Q_R/2(z_0) and
 
    Δ_y f _C_ℓ^α ()≤ C f_C_ℓ^2s+α (Q)(0,y,0)^2s

 for some constant C only depending on dimension and s.



  This lemma and the following one can be seen as discrete
  counterparts of <cit.>.



  We remark that the assumption of the lemma implies that the
  assumptions of Lemma <ref> holds true with β=0 and
  α_1 =2s and α_2 = α with
  N = 2f_C_ℓ^α (Q). Applying Lemma <ref> yields
  the desired result. 


Lemma <ref> can also be proved directly along the lines of the proof of Lemma <ref> below. The proof would be easier because (0,y,0) belongs to the center of the Lie group and thus z ∘ (0,y,0) ∘ξ = z ∘ξ∘ (0,y,0).




  Given w ∈ B_R/2 with R ≤ 1, and 2s + α < 1 and α≤min (1,2s) and some cylinder Q=Q_R(z_0),  let  f ∈ C_ℓ^2s+α (Q) such that ∇_x f ∈ C^0(Q). Then Δ_w f lies in C_ℓ^α () with = Q_R/2 (z_0) and 
 
    [Δ_w f ]_C_ℓ^α ()≤ C ( [f]_C_ℓ^2s+α (Q) + |w|^1-α∇_x f_C^0(Q) )(0,0,w)^2s

   for some constant C only depending on dimension and s.



  It is convenient to write a=(0,0,w). We need to estimate the quantity

    W := |Δ_a f(z ∘ξ) - Δ_a f(z)| = |f(z ∘ξ∘ a) - f(z ∘ξ) - f(z ∘ a) + f(z)|.

The easiest case is when a≤ξ. In this case, we apply Definition <ref> at the point z and z ∘ξ with increment a. We get

    |f(z ∘ξ∘ a) - p_z ∘ξ(a)| ≤ [f]_C_ℓ^2s+αa^2s+α,      |f(z ∘ a) - p_z(a)| ≤ [f]_C_ℓ^2s+αa^2s+α.

The polynomials p_z and p_z ∘ξ are of kinetic degree less than 2s+α < 1. Thus, they do not have any component in the “v” variable: p_z(0,0,w)=f(z) and p_z ∘ξ(0,0,w) = f(z ∘ξ). Thus,

    W    ≤ 2 [f]_C_ℓ^2s+αa^2s+α≲  [f]_C_ℓ^2s+αa^2sξ^α.

The last inequality holds when a≤ξ. Note that for this case, we did not need a correction in terms of ∇_x f_C^0(Q). For a > ξ, we will need an alternative chain of inequalities.

When a > ξ, we apply Definition <ref> at the point z and z ∘ a with increment ξ. We get

    |f(z ∘ a ∘ξ) - p_z ∘ a(ξ)| ≤ [f]_C_ℓ^2s+αξ^2s+α,      |f(z ∘ξ) - p_z(ξ)| ≤ [f]_C_ℓ^2s+αξ^2s+α.

The polynomials p_z and p_z ∘ a have kinetic degree less than 2s+α < 1. Thus, they have at most two nonzero terms, the constant one, and the one in the “t” variable. They are (see <cit.>),

    p_z(ξ) = f(z) + (∂_t + v ·∇_x) f(z) ξ_t,      p_z∘ a(ξ) = f(z ∘ a) + (∂_t + (v+w) ·∇_x) f(z ∘ a) ξ_t.



 Note that z ∘ a ∘ξ differs
  from z∘ξ∘ a. We estimate this discrepancy. If z = (t,x,v) and ξ = (ξ_t,ξ_x,ξ_v), we have
  
    f (z ∘ a ∘ξ)- f(z∘ξ∘ a)  =      f(t+ξ_t, x + ξ_x + ξ_t (v+w), v + ξ_v + w) - f(t+ξ_t, x+ξ_x + ξ_t v, v + ξ_v + w) 
    
        =    ∫_0^1 ∇_x f (t+ξ_t, x+ξ_x + ξ_t v + θξ_t w,v +ξ_v + w) ·ξ_t w θ.

  This implies that
  
    | f (z ∘ a ∘ξ)    - f(z∘ξ∘ a) - (w ·∇_x f (z ∘ a)) ξ_t | 
       ≤ |ξ_t||w| ∫_0^1 |∇_x f (t+ξ_t, x+ξ_x + ξ_t v + θξ_t w,v +ξ_v + w) - ∇_x f (t,x,v+w)| θ
       ≤ 2 ∇_x f_C^0ξ^2s |w|.
  

We combine this with (<ref>) and (<ref>) to obtain the following upper bound for W,

    W    ≤ 2 [f]_C_ℓ^2s+αξ^2s+α +  ∇_x f_C^0  |w|  ξ^2s
       ≤ + |  (∂_t + v ·∇_x) f(z) -  (∂_t + (v+w) ·∇_x) f(z ∘ a)| |ξ_t|, 
    Using <cit.>,   ≤ 2 [f]_C_ℓ^2s+αξ^2s+α +  ∇_x f_C_0|w|  ξ^2s + [f]_C_ℓ^2s+αa^αξ^2s, 
       ≲( [f]_C_ℓ^2s+α + |w|^1-α∇_x f_C_0) a^2sξ^α,

For the last inequality, we used |w| = a≥ξ and α≤ 2s.






§ THE PROOF OF THEOREM <REF>



This section is devoted to proving Theorem <ref>. By an iterative process, we will establish the following family of inequalities. For all differential operator D = ∂_t^k_t D_x^k_x D_v^k_v with k = (k_t,k_x,k_v) ∈^1+2d, there exists α>0 so that for all τ>0 and q>0 there is a constant C_k,q (depending on k_t,k_x,k_v, q, τ, and the parameters in Theorem <ref>) such that

    Df_C^2s+α_ℓ,q([τ,∞) ×^d ×^d)≤ C_k,q.

The value of α that we obtain in the iteration will also depend on k and it will tend to be smaller as the order of differentiation increases. A posteriori, we obtain a C^∞ estimate for f, so the particular values of α after each iteration do not matter.

In order to be in position to apply the Schauder estimate and gain 2s derivatives, we will always pick α∈ (0,min (1,2s)) such that 2s + α∉{1,2}. To do so, it is convenient to work with exponents α such that α < 1-2s if s < 1/2 and α < 2-2s if s ≥ 1/2.

  We use the (classical) definition D^(k_t,k_x,k_v) = ∂_t^k_t∂_x_1^k_x^1…∂_x_d^k_x^d∂_v_1^k_v^1…∂_v_d^k_v^d if  k_x = (k_x^1,…, k_x^d) and k_v = (k_v^1,…, k_v^d). We recall that the order of a multi-index k ∈ℕ^1+2d is k_t + k_x^1 + … k_x^d + k_v^1 + … + k_v^d and is denoted by |k|.  In this section, when we refer to the order of D, we mean literally the classical order of differentiation (not the kinetic order as defined in <cit.>).

Note that the value of C_k,q depend on several parameters. We stress their dependence with respect to k and q because it affects the order in which these numbers are computed. As we said, we establish Inequalities (<ref>) for every value of k and q iteratively. We first prove if for k=(0,0,0) and any value of q. Then, we will compute C_k,q in terms of the values of C_i,q+κ+3 for multi-indices i ∈^1+2d so that either |i| < |k|, or |i|=|k| and i_x > k_x. In other words, the upper bounds for the differential operator D^k f will depend on the bounds for lower order operators, and on the bounds for operators with the same total order but higher order in x. We observe that the computation of any of these values C_k,q would involve finitely many iterations, starting from the family of inequalities (<ref>) for k=0. Note the addition “+γ” in the decay exponent q+κ+3, which is not problematic since we start with the inequality C_0,q for every value of q. This loss κ+3 only depends on the parameter s and γ from the collision kernel B, see  (<ref>).

There are several sequential orders which we could employ in order to compute all the constants C_k,q. In this proof, we make the following (somewhat arbitrary) choice. We first establish (<ref>) for k=(0,k_x,0), with k_t=|k_v|=0. In the second step, we extend the inequalities (<ref>) to indices of the form k=(k_t,k_x,0), with k_v = 0. In the third and last step, we establish (<ref>) for all values of k ∈^1+2d. By proving Estimates (<ref>) in this order, we ensure that we always have enough previous information to establish the value of C_k,q in each step.

We start with a function f ∈^0 (according to Theorem <ref>). The iteration procedure described below allows us to obtain upper bounds of the form (<ref>) for increasingly higher values of |k|. If we only had an upper bound for f_C_ℓ,q^0 for some finite exponent q, the iteration would provide regularity estimates only up to certain order of differentiation.

The zeroth step of the iteration is to apply Corollary <ref>, which provides Inequality (<ref>) for k_t=0, k_x=0, k_v = 0. This is the case where Df=f.  The remainder of the proof proceeds in three steps, as described above. 



 Step 1. We prove (<ref>) holds true for all
 differential operators of the form D =D_x^k_x. We proceed by
 induction on n=|k_x|. It is convenient to make the inductive statement in terms of increments. More precisely, we are going to prove
 by induction on n ≥ 1 that there exists an α_n such that for any τ>0, there exists a C_n,q>0 so that
 
    ∀ k_x ∈ℕ^d,  q>0,  y ∈ B_1,      |k_x |≤ n-1 ⇒Δ_y D_x^k_x f_C_ℓ,q^2s+α_n ([τ,∞) ×^d ×^d)≤ C_n,q |y|

where we recall that Δ_y f (z) = f (z ∘ (0,y,0)) - f(z).

Note that passing to the limit as y → 0, the inequality above implies that for all |k_x| ≤ n,

    D_x^k_x f_C_ℓ,q^2s+α_n ([τ,∞) ×^d ×^d)≤ C_n,q.

  
Corollary <ref> provides the case n=0 in (<ref>). Note that (<ref>) holds trivially for n=0 since there is no k_x so that |k_x| ≤ -1. In order to proceed by induction, we assume we know (<ref>) and (<ref>) hold up to certain value of n ∈ℕ and we prove it for n+1.

Let |k_x| = n and g = Δ_y D^k_x_x f. By the inductive hypothesis (<ref>) combined with Lemma <ref>, we have that for any value of τ>0 and q>0,

    g_C^α_n_ℓ,q([τ,∞) ×^d ×^d)≲(0,y,0)^2s = |y|^2s/(1+2s).

We want to enhance the exponent 2s/(1+2s) on the right hand side all the way to one. For that, we apply the following lemma successively.


Let g = Δ_y D^k_x_x f (as above), β∈ (0,1+2s) and assume that    (<ref>) holds true. If there exists α̅∈ (0,α_n] such that 2s + α̅' ∉{1,2} and, 

    g_C_ℓ,q+κ+3^α̅( [τ,∞) ×^d ×^d )≲(0,y,0)^β,

then

    g_C_ℓ,q^2s + α̅'( [2τ,∞) ×^d ×^d )≲(0,y,0)^β,

with α̅' = 2s/1+2sα̅. Here, κ is the constant in Proposition <ref>.



The key to this lemma is to differentiate (<ref>) and compute an equation for g. Then, we apply the global Schauder estimate of Proposition <ref> together with the estimates we have for each incremental quotient.

Indeed, by a direct computation, we verify that g verifies the equation

    (∂_t + v ·∇_x) g - _1(f,g) = h,

where

    h = ∑_|i|<n 
     i ≤ k_x{_1(Δ_y D̂_i f, τ_y D_i f) + _1(D̂_i f, Δ_y D_i f) } + ∑_i ≤ k_x{_2(Δ_y D̂_i f, τ_y D_i f) + _2(D̂_i f, Δ_y D_i f) }.

Here, i ∈ℕ^d is a multi-index. When we write i ≤ k_x, we mean that each component of i is less or equal than each component of k_x. We write τ_y f(z) = f(z ∘ (0,y,0)) = Δ_y f + f. We also write D̂_i to denote the differential operator so that D^k_x_x = D̂_i ∘ D_i.

Since the index i in the first sum runs over |i| < n, the inductive hypothesis (<ref>) tells us that both τ_y D_i f and Δ_y D_i f are bounded in C^2s+α_n_ℓ,q+κ+3  by ≲ |y|. Likewise, for every value of i so that i ≤ k_x, we have D_i f, D̂_i f, Δ_y D_i f, Δ_y D̂_i f, all bounded in C^α_n_ℓ,q+κ+3 by ≲ |y| except for the two extreme cases: Δ_y D_i f for i=k_x and Δ_y D̂_i f for i=(0,0,0). Both functions coincide with Δ_y D^k_x_x f. For this reason, the hypothesis of the lemma bounds these two functions in C^α̅_ℓ,q+κ+3 by ≲ (0,y,0)^β.

Taking the previous paragraph into account, we bound each term in h using Lemmas <ref> and <ref>. We obtain a bound for h_C^α̅'_ℓ,q+κ+3-(γ+2s)-α/(1+2s) and consequently a bound for h_C^α̅'_ℓ,q+κ since γ +2s + α / (1+2s) ≤ 3.

Applying Proposition <ref>, we obtain the desired  bound for g_C^2s+α̅'_ℓ,q.


Note that Lemma <ref> provides a gain in regularity at the expense of a loss in decay, from q+κ +3 to q. We did not try to make κ explicit in Proposition <ref> and +3 is a rather rough overestimation of the additional loss in the decay exponent when applying Lemmas <ref> and <ref>.  


Applying Lemma <ref> once, we transform (<ref>) into the following inequality, for every value of q>0,

    g_C^2s+α'_n_ℓ,q([2τ,∞) ×^d ×^d)≲(0,y,0)^2s = |y|^2s/(1+2s).

Note that the time shift τ was updated to 2τ. This is because the application of Proposition <ref> in the proof of Lemma <ref> requires a gap in time. We obtain estimate for every value of τ>0 (with constants depending on τ). So, the difference between τ and 2τ is not relevant for the final estimates. In view of this observation, we will omit the domain dependence in the estimates below as a way to unclutter the expressions and focus on the Hölder and decay exponents.

The estimate (<ref>) can be combined with Lemma <ref> for (α_1,α_2,β)=(2s,α'_n,2s). We get, 

    g_C^α'_n_ℓ,q≲(0,y,0)^4s.


This is an improvement on the exponent in the right hand side of (<ref>) from 2s to 4s (at the expense of reducing α_n to α'_n). We continue applying Lemma <ref> together with Lemma <ref> successively improving the exponent on the right hand side to 6s, 8s, 10s, …for as long as this exponent is strictly less than 1+2s. After j steps, we are left with the inequality

    g_C^α̃_j_ℓ,q≲(0,y,0)^2s (j+1)    where α̃_j := ( 2s/1+2s)^j α_n.

This iteration continues identically until α̃_j + (j+1)(2s)> 1+2s. At that point, Lemma <ref> takes a different form and the next step gives us,

    g_C^α̃_j+1_ℓ,q≲(0,y,0)^1+2s.

If the value of α̃_j + (j+1) (2s) is only barely above 1+2s, the value of α̃_j+1 that we would get applying Lemma <ref> might be tiny. In order to avoid that inconvenience, if α̃_j + j(2s) ∈ (1,1+s], then we can perform an intermediate step gaining s derivatives instead of 2s derivatives, i.e. taking α_1 =s when applying Lemma <ref>. That way, we ensure that the value of α̃_j+1  is bounded below only in terms of the parameters of Theorem <ref>.

One more application of Lemma <ref>  gives us

    g_C^2s+α̃_j+1_ℓ,q≲(0,y,0)^1+2s = |y|.


Recalling that g = Δ_y D^k_x_x f, we finished the proof of (<ref>) with α_n+1 := α̃_j+1≤α_n < 1-2s. This finishes Step 1 in the proof of Theorem <ref>. That is, we obtained (<ref>) when D involves derivatives with respect to x only.

Step 2. 
We next prove that for all k=(k_t,k_x,0), and all q>0 and τ>0, we can control
  ∂_t^k_t D_x^k_x f _C_ℓ,q^2s+α_k for some small
  α_k > 0. That means that for any τ>0, there is a C_k,q so that
  
    ∂_t^k_t D_x^k_x f _C_ℓ,q^2s+α_k ( [τ,∞) ×^d ×^d) ≤ C_k,q.

We are going to derive (<ref>) for all k_x by  induction on n=k_t. Remark that for n=0, we proved these estimates for all k_x  in Step 1. We argue by induction as follows: we  prove that, given any n ∈ℕ, n ≥ 1, and m ∈ℕ, if (<ref>) holds whenever k_t ≤ n-1 and |k_x|≤ m+1, and also for k_t = n and |k_x| < m then it also holds true for k_t = n and |k_x| = m.

Let n ≥ 1 and k_x ∈^d be any multi-index with |k_x| = m. Using the inductive hypothesis (<ref>) with k_t = n-1, we apply
<cit.> and, for any value of q>0, get a bound on

    (∂_t + v ·∇_x)∂_t^n-1 D^k_x_x f
      _C_ℓ,q^α≲ C_k_0, q

with k_0= (n-1,k_x,0). 

Using the induction assumption for k_t = n-1, |k̃_x| = m+1, we also control the
norm of (v·∇_x) ∂_t^n-1 D^k_x_x f,

    (v·∇_x) ∂_t^n-1 D^k_x_x f _C_ℓ,q^2s+α≤∂_t^n-1∇_x D^k_x_x f _C_ℓ,q+1^2s+α≲max_k̃ C_k̃, q+1

with k̃ = (n-1,k̃_x,0) and |k̃_x| = m+1.

Therefore, we combine the last two estimates to obtain the inequality, for some α>0 and some constant C depending on n and m,

    ∂_t^n D^k_x_x f _C_ℓ,q^α≤ C.


Our next objective is to turn the estimate (<ref>) into

    ∂_t^n D^k_x_x f _C_ℓ,q^2s+α'≤ C.


Let g := ∂_t^n D^k_x_x f. We compute an equation for g and get

    (∂_t + v·∇_x) g - _1(f,g) = h,

where

    h = ∑_i ≤ (n,k_x,0) 
     i ≠ (n,k_x,0)_1(D̂_i f, D_i f)  + ∑_i ≤ (n,k_x,0)_2(D̂_i f, D_i f).

Here, i ∈ℕ^1+2d is a multi-index, and like in Step 1, ∂_t^n D^k_x_x = D̂_i ∘ D_i.

An  inspection of the functions involved in h shows that, by applying the inductive hypothesis together with Lemmas <ref> and <ref>, we bound h_C^α'_ℓ,q for all q>0. Finally, (<ref>) follows after applying Proposition <ref>.



Step 3. In the third and last step, we establish the inequality (<ref>) for every differential operator
D=∂_t^k_t D_x^k_xD_v^k_v with k ∈^1+2d, and for all q>0 and τ>0. We will prove that
 
    ∃α_n,m >0   /  ∀ k ∈^1+2d, q>0,w ∈ B_1, 
    { |k_v| ≤ n, k_t + |k_x| ≤ m }⇒  D f_C_ℓ,q^2s+α_n,m≤ C_n,m.

We proceed the proof of Step 3 by a bidimensional induction similar as in Step 2. 


If s ≥ 1/2, we can proceed like in Step 2. Indeed, (<ref>)  implies that (See Proposition <ref>)

    ∂_v_i Df_C^α_n-1,m_ℓ,q≤ C_n-1,m.

Thus, we compute an equation for g = ∇_v Df and argue like in the previous step. 


When s < 1/2, like in Step 1, it is convenient to set up the induction keeping track of the  Hölder regularity of differential operators, and also of increments. Thus, we prove that for all n ≥ 1, m ∈,
 
    ∃α_n,m∈ (0, 1-2s),   /  ∀ k ∈^1+2d, q>0,w ∈ B_1,   
    {  |k_v| ≤ n -1,   k_t + |k_x| ≤ m}⇒  Δ_w D f_C_ℓ,q^2s+α_n,m≤ C_n,m |w|.

The constants α_n,m depend on n, m and the parameters in Assumption <ref>. The constants C_n,m depend in addition on τ and q. By taking w → 0, (<ref>) implies (<ref>).


The case n=0 of (<ref>) was established in Step 2. The inequality (<ref>) holds trivially for n=0.


Now, let n≥ 1 and k be any multi-index with |k_v|=n-1 and k_t + |k_x| = m. From the inductive hypothesis, Df satisfies (<ref>). Remark that we can assume without loss of generality that α_n,m < 1-2s in (<ref>) and (<ref>).  Thus, for any q>0,

    Df_C^2s+α_n-1,m_ℓ,q≤ C_n-1,m.


Let w ∈ B_1. Since 2s+α_n-1,m < 1, we apply Lemma <ref> together with (<ref>) and obtain, for  α=α_n-1,m>0,

    Δ_w Df_C^α_ℓ,q   ≲( Df_C^2s+α_n-1,m_ℓ,q + ∇_x Df_C^α_n-1,m+1_ℓ,q) |w|^2s
       ≲( C_n-1,m + C_n-1,m+1) |w|^2s
       ≲ |w|^2s.



In order to obtain (<ref>) for n and m, we need to enhance the exponent on the right hand side of the inequality above, from 2s all the way to one. We do it through an iterative process similar to Step 1.

  
    Let g = Δ_w Df, β∈ (0,1) and assume that    (<ref>) holds true for smaller values of n+m, or for the same value of n+m with n smaller. If there exists    α̅∈ (0,α_n] such that 2s +α̅' ∉{1,2} and,
    
    g_C_ℓ,q+γ+3^α̅( [τ,∞) ×^d ×^d )≲ |w|^β,

    then
    
    g_C_ℓ,q^2s + α̅'( [2τ,∞) ×^d ×^d )≲ |w|^β,

    with α̅' = 2s/1+2sα̅.
  


  The proof is very similar to the proof of Lemma <ref>.  The only difference is that the equation for g will now have terms involving ∇_x Df.

 The function g̅ = Df satisfies
  
    (∂_t + v ·∇_x) g̅ -_1(f, g̅) = H̅
 
  with
  
    H̅  =  ∑_i ≤ k 
     i ≠ k_1(D̂_i f, D_i f)  + ∑_i ≤ k_2(D̂_i f, D_i f) +  {(∂_t + v ·∇_x),D} f
  
  where {(∂_t + v ·∇_x),D} f=(∂_t + v ·∇_x)Df-D(∂_t + v ·∇_x)f (Poisson bracket) and D_i and D̂_i  are such that D_i ∘D̂_i = D.

Since ∂_t and D commute, {(∂_t + v ·∇_x),D} = { v ·∇_x,D}. Given k = (k_t,k_x,k_v), by a direct computation one verifies that

    { v ·∇_x,D} = ∑_k̃ D^k̃

where the multi-index k̃ runs over all multi-indexes with the same order as k so that D^k̃ = ∂_x_iD̃ and D^k = ∂_v_iD̃ for some differential operator D̃ and i=1,…,d. According to our induction hypothesis, (<ref>) holds for all these indexes k̃, therefore

    Δ_w {(∂_t + v ·∇_x),D} f_C^2s+α_ℓ,q≤  C_n-1,m+1 |w|.


The function g = Δ_w D f satisfies the  equation 

    (∂_t + v ·∇_x) g - _1(f,g) = H    in  (0,T) ×^d ×^d

  where 
  
    H    = Δ_w H̅ - ∑_j=1^d w_j τ_w (∂_x_j Df), 
       = ∑_|i|<n 
     i ≤ k_x{_1(Δ_y D̂_i f, τ_y D_i f) + _1(D̂_i f, Δ_y D_i f) } + ∑_i ≤ k_x{_2(Δ_y D̂_i f, τ_y D_i f) +_2(D̂_i f, Δ_y D_i f) }
       = +  Δ_w {(∂_t + v ·∇_x),D}f - ∑_j=1^d w_j τ_w (∂_x_j Df).

  The last term is the commutator between Δ_w and the transport part (∂_t + v ·∇_x), and it is bounded in C^2s+α_ℓ,q, for all q>0, by the inductive hypothesis. The first two terms are bounded identically as in the proof of Lemma <ref>. And the third term was bounded in (<ref>). The proof finish by applying Proposition <ref> to g, in the same way as in the proof of Lemma <ref>.



Once Lemma <ref> is established, the rest of the proof of Step 3 proceeds similarly  as in Step 1 using Lemma <ref> instead of Lemma <ref>.

This finishes the proof of Theorem <ref>.




§ GRESSMAN-STRAIN COERCIVITY ESTIMATE



In this appendix, we show how the change of variables described in Section <ref>, together with a local coercivity estimate like the one in Theorem <ref>, can be used to recover the global coercivity estimate with respect to the lifted anisotropic distance of Gressman and Strain <cit.> (see also the prequel paper <cit.>).

The transformation T_0 defined in (<ref>) depends on a given point v_0 ∈^d. For any such v_0, let us consider the pushed forward distance: for
v_1,v_2 ∈ E_1 (v_0)= v_0 + T_0 (B_1), 

    (v_1,v_2) = | T_0^-1 (v_1-v_2)|

This distance  depends on the choice of v_0. However, as we will see, for any pair v_1, v_2 ∈^d, all the possible values of (v_1,v_2) are comparable for all possible choices of v_0 so that v_1,v_2 ∈ E_1(v_0).

We also recall the anisotropic distance defined in <cit.>:
for all v_1,v_2 ∈^d,

    (v_1,v_2) =√(1/4(|v_1|^2 - |v_2|^2)^2 + |v_1-v_2|^2).



Given  v_0 ∈^d with |v_0| ≥ 2, we have for all v_1,v_2 ∈ v_0 + T_0 (B_1),

    (v_1,v_2) ≃ (v_1,v_2).

The hidden constants in ≃ do not depend on any parameter, not even  dimension. 



Since T_0 is linear, we have to estimate |T_0^-1(v_1-v_2)|. Let v_1,2 =v_1-v_2. 
We have 

    v_1,2 = λv_0/|v_0| + w    with  w ·
    v_0 =0.

The real number λ satisfies
λ |v_0| = v_1,2· v_0 and
|v_1,2|^2 = λ^2 + |w|^2. Hence we have

    (v_1,v_2) =    | T_0^-1 (v_1,2)|  = √(λ^2 |v_0|^2 + |w|^2)
    
    =    √(λ^2 (|v_0|^2 -1) + |v_1-v_2|^2)
    ≃   √(λ^2 |v_0|^2 + |v_1-v_2|^2)
    
    =    √(((v_1-v_2) · v_0)^2 + |v_1-v_2|^2).

We finally use that T_0 (B_1) is a convex subset of B_1 in order to get

    |v_1+v_2/2 - v_0 | ≤ 1

which allows us to conclude.  


In <cit.>, Gressman and Strain obtained
sharp coercivity estimates for the linear Boltzmann collision operator
under some conditions on f on mass, concentration and moments. In
the next proposition, we prove an inequality of the same nature.

 Let f be non-negative and such that
Assumption <ref> holds. If γ < 0, we also assume (<ref>). Let  g:^d → be an arbitrary function. Then

    -∫_^d (f,g) g 
    ≥ c ∬_(v,v')<ρ  (g(v)-g(v'))^2 (1+|v+ v'|)^γ + 2s +1/(v,v')^d+2s'
    - C ∫_^d g(v)^2 (1+| v|)^max(γ,0)  v

where the constants c, ρ and C only depend on dimension d and m_0,M_0,E_0,H_0 from Assumption <ref> and C_γ in (<ref>) (only if γ<0). We recall that  denotes the Boltzmann collision operator defined in
(<ref>) and  denotes the non-isotropic distance defined in (<ref>).


We recall that the collision operator can be split in a principal part
and a lower order term, see (<ref>).  We prepare the proof of
the proposition by first estimating from below the principal
contribution of the bilinear form
⟨ (f,g), g ⟩_L^2.

  Let f be non-negative and such that
  Assumption <ref> holds, and if γ<0 also (<ref>) holds true. Let g:^d → be an
  arbitrary function. Then

    ∬_(v,v') < R (g(v)-g(v'))^2 K_f(v,v')   v'  v 
    ≥ c ∬_(v,v')<ρ  (g(v)-g(v'))^2 (1+|v+ v'|)^γ + 2s +1/(v,v')^d+2s'.

Here, the constants c>0 and ρ∈ (0,1), R ∈ (2,+∞) only depend on dimension d and
m_0,M_0,E_0,H_0 from Assumption (<ref>) and C_γ in
(<ref>) (only if γ<0).  We recall that K_f is the
kernel defined in (<ref>) and  denotes the non-isotropic
distance defined in (<ref>).



  We are going to use the change of variables from  Section <ref>. We recall that a kernel K̅_f is defined  in (<ref>) and that this kernel satisfies appropriate   ellipticity conditions. 

From Corollary <ref>, we know that the kernel K̅_f satisfies (<ref>), with a constant λ independent of v_0.

Let R_0 ≥ 2 and v_0 such that |v_0|=R_0.   We change variables in (<ref>). Recall that  v̅ = v_0 + T_0  v and v̅' = v_0 + T_0  v'. We also write  g̅(v) = g(v̅). Note that v = |v_0| v̅. Thus,   (<ref>) for K̅_f translates into the following inequality for K_f,

    ∬_ E_1 (v_0) ×  E_1(v_0) (g(v̅')- g(v̅))^2 |v_0|^1-γ-2s K_f(v̅, v̅') v̅' v̅
    ≳  |v_0|^2 ∬_ E_1/2 (v_0)×  E_1/2(v_0) (g(v̅')-  g(v̅))^2  (v̅,v̅')^-d-2sv̅' v̅

where we recall that E_r (v_0) = v_0+T_0 (B_r) for r>0. We used the definition of d_a and Lemma <ref>. Rearranging the powers of |v_0|, we get for any v_0 ∈^d ∖ B_2,

    ∬_E_1 (v_0) × E_1 (v_0)  (g(v̅')- g(v̅))^2 K_f(v̅, v̅') v̅' v̅
    ≳ (1+|v_0|)^1+γ+2s∬_ E_1/2 (v_0) ×  E_1/2 (v_0) (g(v̅')-  g(v̅))^2  (v̅,v̅')^-d-2sv̅' v̅.

We remark that for v̅,v̅' ∈ E_1/2 (v_0), we have 1+|v_0| ≃ 1+|v̅ + v̅'|. Hence, we get 

    ∬_E_1 (v_0) × E_1 (v_0)  (g(v̅')- g(v̅))^2 K_f(v̅, v̅') v̅' v̅
    ≳∬_E_1/2(v_0) × E_1/2 (v_0) (g(v̅')-  g(v̅))^2 (1+|v̅ + v̅'|)^1+γ+2s/ (v̅,v̅')^d+2sv̅' v̅.

We now multiply the previous inequality by |v_0|, integrate with
respect to v_0 ∈^d ∖ B_2. We get

    ∬ (g(v̅')- g(v̅))^2 K_f(v̅, v̅')  W_1 (v̅,v̅') v̅' v̅≳∬ (g(v̅')-  g(v̅))^2 (1+|v̅ + v̅'|)^1+γ+2s/ (v̅,v̅')^d+2s 
    W_1/2 (v̅,v̅') v̅' v̅

with

    W_1 (v,v') := ∫_^d ∖ B_2  |v_0| _v,v' ∈ E_1(v_0)_0    and    
       W_1/2 (v,v') := ∫_^d ∖ B_2 |v_0| _v,v' ∈ E_1/2(v_0)_0

where _A denotes the indicator function of a set A:
_A (v)=1 if v ∈ A and _A (v)=0 if v ∉ A.

We now observe that for some constants R>0 (large) and ρ>0 (small),

    W_1(v,v')    ≲_{ (v,v') < R }
     W_1/2 (v,v')    ≳_{ (v,v') < ρ}_{v ∉ B_2 or  v' ∉ B_2} .


As far as (<ref>) is concerned, if there exists v_0 ∈^d
such that v,v' ∈ E_1(v_0), then (v,v') <2, see
(<ref>). Thus, from Lemma <ref>, (v,v') < R for some universal constant R. Moreover, since we have (v,v_0) < 1, we also have (v,v_0) < R. In particular |v| ≈ |v_0|. The set of points v_0 ∈^d so that (v,v_0) < R has volume ≈ (1+|v|)^-1. Thus, W_1 ≲ |v| (1+|v|)^-1≤ 1, and (<ref>) follows. As far as (<ref>) is concerned, if (v,v') < ρ for ρ small, then the set of v_0 so that v,v' ∈ E_1(v_0) will be indeed of volume ≈ (1+|v|)^-1. If v ∉ B_2 or v' ∉ B_2, we ensure that at least half of this set lies outside B_2. Note that since |v_0| ≈ |v|>2 (or |v'|>2), we have |v_0| / (1+|v|) ≈ 1 and (<ref>) follows.

With (<ref>) and (<ref>) at hand, we can deduce from (<ref>) that

    ∬_{ (v̅,v̅')<R} (g(v̅')- g(v̅))^2 K_f(v̅, v̅') v̅' v̅
    ≳∬_{(v̅, v̅') < ρ} (g(v̅')-  g(v̅))^2 _{|v̅|>2  or  |v̅'| >2}(1+|v̅ + v̅'|)^1+γ+2s/ (v̅,v̅')^d+2sv̅' v̅.


In order to deal with small velocities, the change of variables is not needed: we apply (<ref>) (scaled to B_4) directly to K_f and get, 
  
    ∬_B_4× B_4 (g(v̅')- g(v̅))^2 K_f(v̅, v̅') v̅' v̅   ≳∬_B_2 × B_2 (g(v̅')-  g(v̅))^2 |v̅-v̅'|^-d-2sv̅' v̅
       ≳∬_{ (v̅,v̅') < ρ} (g(v̅')-  g(v̅))^2 _{v̅,v̅' ∈ B_2}(1+|v̅+ v̅'|)^γ + 2s +1/(v̅,v̅')^d+2sv̅' v̅.


We conclude the proof by combining the estimate for large velocities with the one for small velocities. 



We can now prove Proposition <ref>. 


  From Corollary <ref>, we know that K̅_f satisfies (<ref>) with a λ>0 independent of v_0.

We use again the decomposition (<ref>) from <cit.>  After straight-forward arithmetic manipulations, we get

    -∫_^d (f,g) g    = 1/2∬_^d ×^d (g( v')- g( v))^2 K_f( v,  v')   v'   v 
            - 1/2∫_^d g(v)^2 ( ∫_^d (K_f(v,v') - K_f(v',v))  v' )   v, 
            - ∫_^d (f ∗ |·|^γ) g(v)^2  v, 
        = I_1 - I_2 - I_3.


We use Lemma <ref> to estimate the first term. We use <cit.> to estimate the second term. In fact, the classical cancellation lemma from <cit.> (see also  <cit.>) tells us that the second and third terms are
identical. Thus, using (<ref>) if γ <0, 

    I_1    ≳∬_(v,v')<ρ (g( v') - g( v))^2 (1+| v+ v'|)^1+γ+2s/ ( v, v')^d+2s  v'   v, 
    
    I_2    = I_3 = ∫_^d g(v)^2 ( f(v+w) |w|^γ w )  v ≤
    C (1+|v|)^γ∫_^dg(v)^2    if γ≥ 0, with C=C(M_0,E_0),
    
    C_γ∫_^dg(v)^2    if γ < 0.

The proof is now complete. 



  It is possible to justify that the universal constants R>ρ can be  chosen arbitrarily using a covering argument as in  <cit.>.  The norm 𝒩^s_γ in   <cit.> is defined with ρ = 1.



  The coercivity estimate from
  <cit.> and the coercivity
  estimates from <cit.> and in Proposition <ref> involve different operators. Our proposition, as well as the estimate in <cit.>, is for the
  linear operator

    L(g) = -(f,g),

for any given profile f for which the mass, energy and entropy are bounded above, and the mass is bounded below. The estimate in <cit.> is for the linearized Boltzmann operator

    L(g) = - M^-1/2(M, M^1/2 g) - M^-1/2(M^1/2 g, M),

where M is a Maxwellian profile.

The linear operators (<ref>) and (<ref>) are different. The operator (<ref>) is useful to study (so far conditional) regularity estimates for generic solutions of the Boltzmann equation. The operator (<ref>) is useful to study the stability of the equation for small perturbations around a Maxwellian.

Coercivity estimates from <cit.> and from  Proposition <ref> are proved under slightly different sets of  assumptions. It is assumed in <cit.> that f satisfies for  all v ∈^d and a ∈ [γ,γ +2s],
  
    ∫_^d f(w) |w-v|^a (1+|w|)^i  w ≲ (1+|v|)^a

with i=1 if s < 1/2 and i=2 for s ≥1/2.  For γ<0, (<ref>) implies (<ref>) by choosing a=γ. Notice that (<ref>) imply a control of moments  of order 2+γ +2s if s ≥ 1/2 which can be larger than 2.

Note also that Assumption L in <cit.> is slightly more general than the upper bound on the entropy in Assumption <ref>.



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