Luis Silvestre

THE WEAK HARNACK INEQUALITY FOR THE BOLTZMANN EQUATION WITHOUT CUT-OFF

We obtain the weak Harnack inequality and Hölder estimates for a large class of kinetic integrodifferential equations. We prove that the Boltzmann equation without cut-off can be written in this form and satisfies our assumptions provided that the mass density is bounded away from vacuum and mass, energy and entropy densities are bounded above. As a consequence, we derive a local Hölder estimate and a quantitative lower bound for solutions of the (inhomogeneous) Boltzmann equation without cut-off. Contents 1. Introduction 2. Preliminaries 3. The Boltzmann kernel 4. Study of a bilinear form 5. Reduction to global kernels and weak solutions 6. The first lemma of De Giorgi 7. Barrier functions for s ă 1{2 8. The intermediate-value lemma for s ě 1 2 9. The propagation lemma 10. The ink-spots theorem for slanted cylinders 11. Proofs of the main results Appendix A. New proofs of known estimates and technical lemmas References

Introduction

The main result in this article is a version of the weak Harnack inequality, in the style of De Giorgi, Nash and Moser, for kinetic integro-differential equations. As a consequence, we derive local Hölder estimates and a quantitative lower bound for the inhomogeneous Boltzmann equation without cut-off.

Our estimates are local in the sense that they only require the equation to hold in a bounded domain.

The Boltzmann equation has the form f t `v ¨∇x f " Qpf, f q for t P p´1, 0s, x P B 1 , v P B 1 .

Here, the function f " f pt, x, vq must be defined for t P p´1, 0s, x P B 1 and v P R d in order to make sense of the nonlocal right hand side Qpf, f q.

We recall that Boltzmann's collision operator Qpf, f q is defined as follows Qpf, f q " ˆRd ˆSd´1

pf pv 1 ˚qf pv 1 q ´f pv ˚qf pvqqBp|v ´v˚| , cos θq dv ˚dσ where v 1 ˚and v 1 are given by v 1 " v `v2 `|v ´v˚| 2 σ and v 1 ˚" v `v2 ´|v ´v˚| 2 σ and cos θ (and sinpθ{2q) is defined as cos θ :" v ´v| v ´v˚| ¨σ ˆand sinpθ{2q :" v 1 ´v |v 1 ´v| ¨σ˙.

We assume that the cross-section B satisfies (1.1) Bpr, cos θq " r γ bpcos θq with bpcos θq « | sinpθ{2q| ´pd´1q´2s with γ P p´d, 1s and s P p0, 1q.

The equation describes the density of particles at a specific time t, point in space x and with velocity v. This model stands at a mesoscopic level, in between the microscopic description of interactions between individual particles, and the macroscopic models of fluid dynamics.

We define the hydrodynamic quantities (mass density) M pt, xq :" ˆf pt, x, vq dv, (energy density) Ept, xq :" ˆf pt, x, vq|v| 2 dv, (entropy density) Hpt, xq :" ˆf ln f pt, x, vq dv.

These are the only quantities associated with a solution f which are meaningful at a macroscopic scale. Under some asymptotic regime, the hydrodynamic quantities in the Boltzmann equation formally converge to solutions of the compressible Euler equation, which is known to develop singularities in finite time (see for example [START_REF] Bardos | Fluid dynamic limits of kinetic equations. I. Formal derivations[END_REF]). Because of this fact, one could speculate that the Boltzmann equation may develop singularities as well. From this point of view, the best regularity result that one would expect is that if the hydrodynamic quantities are under appropriate control, then the solution f will be smooth. In other words, that every singularity of f would be observable at the macroscopic scale. It is proved in [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF] that when M pt, xq, Ept, xq and Hpt, xq are uniformly bounded above, and in addition M pt, xq is bounded below by a positive constant, then the solution f satisfies the L 8 a priori estimate depending on those bounds only. The result in this paper goes a step further by proving a Hölder modulus of continuity, in all variables, under the same assumptions.

Theorem 1.1 (Hölder continuity). Assume s P p0, 1q, γ P p´d, 1s, γ `2s ď 2 and let f be a non-negative solution of the Boltzmann equation for all t P p´1, 0s, x P B 1 and v P B 1 . Assume that f is essentially bounded in p´1, 0s ˆB1 ˆRd and there are positive constants M 0 , M 1 , E 0 such that for all pt, xq we have M 1 ď M pt, xq ď M 0 and Ept, xq ď E 0 for all pt, xq P p´1, 0s ˆB1 , then f is Hölder continuous in p´1{2, 0s ˆB1{2 ˆB1{2 with }f } C α pp´1{2,0sˆB 1{2 ˆB1{2 q ď C where C ą 0 and α P p0, 1q are constants depending on dimension, the L 8 bound of f , M 0 , M 1 and E 0 . Remark 1.2. Theorem 1.1 also holds true in any cylinder Q Ă R ˆRd ˆRd . In this case, constants C and γ also depends on the center of the cylinder and its radius.

Note that the value of the entropy Hpt, xq is bounded above by some constant H 0 depending only on M 0 , E 0 and }f } L 8 so we do not need to include the hypothesis Hpt, xq ď H 0 in Theorem 1.1 . Recall also that }f } L 8 is bounded above for t ą 0 in terms of M 0 , M 1 , E 0 and H 0 , according to the result in [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF], provided that γ `2s ą 0. So, at least in this range of values of γ, the Hölder modulus of continuity depends on the values of M 0 , M 1 , E 0 and H 0 only.

The best regularity results previously available for the inhomogeneous Boltzmann equation without cut-off give us C 8 regularity depending on the assumption that the solution has infinite moments and belongs to the space H 5 with respect to all variables (v, x and t) [START_REF] Alexandre | Regularity of solutions for the Boltzmann equation without angular cutoff[END_REF], [START_REF] Alexandre | The Boltzmann equation without angular cutoff in the whole space: I, global existence for soft potential[END_REF], [START_REF] Chen | Smoothing estimates for Boltzmann equation with full-range interactions: Spatially inhomogeneous case[END_REF]. Of course this is a much more stringent assumption than what we need for our Theorem 1.1 to hold. We make further comments about these and other related results in Section 1. [START_REF] Alexandre | The Boltzmann equation without angular cutoff in the whole space: I, global existence for soft potential[END_REF].

We also obtain a quantitative lower bound for the solution f . Theorem 1.3 (Lower bound). Let f be a non-negative supersolution of the Boltzmann equation in r0, T s BR ˆBR . Under the same assumptions on γ, s and f as in Theorem 1.1, we have the lower bound inf rT {2,T sˆB R{2 ˆBR{2 f ě cpRq.

The constant cpRq depends on T , R, γ, s, d, M 0 , M 1 , E 0 , and }f } L 8 .

It has been a longstanding issue to find appropriate lower bounds for the solutions of the Boltzmann equation. The best result available is perhaps from the work of Mouhot [START_REF] Mouhot | Quantitative lower bounds for the full Boltzmann equation. I. Periodic boundary conditions[END_REF]. He obtains an explicit exponentially decaying lower bound for the Boltzmann equation without cut-off. He makes strong a priori regularity assumptions on the solution f , in addition to the assumptions that we make in this paper. We do not provide an explicit formula for cpRq. Its precise decay as R Ñ 8 will be the subject of future work.

Remark 1.4. If γ `2s ą 2, similar results can be obtained by further assuming that the pγ `2sq-momentum of the function f is finite at every point pt, xq.

1.1. A linear kinetic integro-differential equation. The main result of this paper concerns a general kinetic integro-differential equation. The results for the Boltzmann equation described above follow as corollaries. We study an equation of the form

(1.2) f t `v ¨∇x f " L v f `h
for t P p´1, 0s, x P B 1 and v P B 1 , where L v f is a linear integro-differential operator in the velocity variable of the following form L v f pt, x, vq " P V ˆRd pf pt, x, v 1 q ´f pt, x, vqqKpt, x, v, v 1 q dv 1 for a locally bounded function h and a measurable kernel K : r´1, 0s ˆB1 ˆB R ˆRd Ñ r0, `8q satisfying appropriate assumptions that we describe below.

For every value of t and x, the kernel Kpt, x, v, wq is a non-negative function of v and w. We assume that the following conditions hold for every value of t and x (we omit t and x dependence to clean up the notation).

Let us fix a R ě 1. We will make assumptions on the kernel Kpv, v 1 q for v P B R. We need to pick R slightly larger than one for technical reasons that will be apparent in Section 5.

Our first assumption is a coercivity condition on L v . We assume that there exists λ ą 0 and Λ ą 0 such that, for any function f : R d Ñ R supported in B R,

(1.3) λ ¨Rd ˆRd |f pvq ´f pv 1 q| 2 |v ´v1 | d`2s dv dv 1 ď ´ˆR d L v f pvq f pvq dv `Λ}f } 2 L 2 pR d q .
This coercivity condition is well known to hold for the Boltzmann equation when the function f has bounded mass, energy and entropy density, and its mass is also away from vacuum (see [START_REF] Lions | Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire[END_REF][START_REF] Villani | Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off[END_REF][START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF] and the discussion below). The proofs in the literature are based on Fourier analysis. We provide a proof in the appendix which follows by a direct geometric computation in physical variables. In the case s ă 1{2, we also make the following nondegeneracy assumption.

(1.4) inf |e|"1 ˆBrpvq ppv 1 ´vq ¨eq 2 `K pv, v 1 q dv 1 ě λr 2´2s for every value of v P B R.

Here, when we write pw ¨eq 2 `, we mean ppw ¨eq `q2 " maxpw ¨e, 0q 2 . The coercivity condition would be obviously true if K is symmetric (i.e. Kpv, v 1 q " Kpv 1 , vq) and Kpv, v 1 q ě λ|v ´v1 | ´d´2s . These assumptions are not satisfied by the Boltzmann kernel a priori.

For some kernels (not necessarily coming from the Boltzmann equation) it might be difficult to check whether the coercivity condition (1.3) holds. The nondegeneracy assumption (1.4) is usually very easy to check in explicit examples of kernels K. We do not know of any example of a kernel which satisfies (1.4) but not (1.3). It is natural to conjecture this implication (modulo adjusting λ by a fixed factor).

The second assumption is a weak upper bound on the kernel K.

(1.5)

$ & % piq ´Rd zBrpvq Kpv, v 1 q dv 1 ď Λr ´2s for any r ą 0 and v P B R piiq ´B R zBrpv 1 q Kpv, v 1 q dv ď Λr ´2s for any r ą 0 and v 1 P B R.

Note that if Kpv, v 1 q À |v ´v1 | ´d´2s , then the assumption (1.5) holds. Our assumption only concerns average values of K on the complementary set of balls. Therefore, a kernel containing a singular part is allowed. We will see that the Boltzmann kernel satisfies (1.5) even though Kpv, v 1 q À |v ´v1 | ´d´2s may not hold a priori. Note that both inequalities in (1.5) would be the same if K were symmetric. But we do not assume symmetry of the kernel. That is Kpv, v 1 q ‰ Kpv 1 , vq in general. The symmetry assumption is very common for integro-differential equations because it represents the fact that the equation is in divergence form. It is equivalent to the operator L v being self adjoint. We explain this concept in Subsection 1.3.3.

The following assumptions provide a mild control on the anti-symmetric part of the kernel. We assume that (1.6) @v P B 7 R{8 , ˇˇˇˇP V ˆB R{8 pvq `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 ˇˇˇˇď Λ.

Moreover, if s ě 1{2, we need to assume the following extra cancellation.

(1.7) @r P p0, R{8s, @v P B 7 R{8 , ˇˇˇˇP V ˆBrpvq pv ´v1 q `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 ˇˇˇˇď Λp1 `r1´2s q.

When K is symmetric, the left hand sides in (1.6) and (1.7) are identically zero and therefore the assumptions trivially hold. When s ą 1{2, if the assumption (1.7) holds for r " R and in addition (1.5) holds, then we observe that (1.7) automatically holds for all r P p0, Rs. The requirement that the inequality (1.7) holds for all r P p0, Rs, as opposed to only r " R, only makes a difference for the case s " 1{2. We discuss the scaling properties of our assumptions in subsections 2.2 and 2.3.

When we apply our results to the Boltzmann equation, the kernel K depends on the solution f and is determined by the formula ˆRd ˆBB1 f pv 1 ˚qpgpv 1 q ´gpvqqBp|v ´v˚| , θq dσ dv ˚" ˆRd pgpv 1 q ´gpvqqK f pv, v 1 q dv 1 .

In this way, Qpf, gq " ˆpgpv 1 q ´gpvqqK f pv, v 1 q dv 1 `plower order termsq.

The constant λ in the assumption (1.3) depend only on the mass, energy and entropy densities of f . The constant Λ in (1.5), (1.6) and (1.7), depends only on the mass and energy density of f when γ P r0, 1s. It depends on further integrability properties of f when γ ă 0 (they are bounded in terms of }f } L 8 for example). All these assumptions will be verified in Section 3.

1.2. Main results. The notion of weak solution will be made precise by the end of Section 5.

Theorem 1.5 (Hölder continuity). Assume the kernel is non-negative and there exist λ ą 0 and Λ ą 0 such that (1.3), (1.5) and (1.6) hold true with R " 2. If s ě 1{2, we also assume (1.7); if s ă 1{2, we also assume (1.4). Let f be a solution of (1.2) for all t P p´1, 0s, x P B 1 and v P B 1 . Assume that f is essentially bounded in p´1, 0s ˆB1 ˆRd . Then f is Hölder continuous in p´1{2, 0s ˆB1{2 ˆB1{2 with }f } C γ pp´1{2,0sˆB 1{2 ˆB1{2 q ď C `}f } L 8 pp´1,0sˆB1ˆR d q `}h} L 8 pp´1,0sˆB1ˆB1q where C ą 0 and γ P p0, 1q are constants only depending on dimension, λ and Λ.

This theorem is in fact derived from the following estimate.

Theorem 1.6 (Weak Harnack inequality). There are constants r 0 , R 1 ą 1, ε and C so that the following proposition holds. Assume the kernel is non-negative and there exist λ ą 0 and Λ ą 0 such that (1.3), (1.5) and (1.6) holds true with R " 2R 1 . If s ě 1{2, we also assume (1.7); if s ă 1{2, we also assume (1.4).

Assume that f is a non-negative supersolution of (1.2) in p´1, 0s ˆBR 1`2s 1 ˆBR1 . Then ˆˆQ ´f ε pt, x, vq dv dx dt

˙1{ε ď C ˆinf Q `f `}h} L 8 pp´1,0sˆB1ˆB1q
ẇhere Q `" p´r 2s 0 , 0s ˆBr 1`2s 0 ˆBr0 and Q ´" p´1, ´1 `r2s 0 s ˆBr 1`2s 0 ˆBr0 r 0 R 1 t P p´1, 0s

Figure 1. The geometric setting of the weak Harnack inequality (see Figure 1) and the constants C ą 0, ε ą 0, only depend on dimension, s, λ and Λ. The constants r 0 and R 1 depend on dimension and s only (not on λ and Λ).

1.3. Comments on the results and related works.

1.3.1. Difficulties related to this problem. This subsection is our attempt to explain and compare the main challenges that we faced proving the main results in this paper, and the new ideas that were introduced. We start by reviewing some recent developments about parabolic kinetic equations in divergence form, with rough coefficients. In some sense, our main theorems are an integro-differential counterpart of these previous results. The equations have the form

f t `v ¨∇x f " B Bv i ˆaij B Bv j f ˙.
The diffusion coefficient a ij " a ij pt, x, vq is assumed to be uniformly elliptic. No regularity assumption should be made in a ij , otherwise the equation may fit into the more classical hypoelliptic theory, and would not imply such interesting results for the Landau equation. Pascucci and Polidoro [START_REF] Pascucci | The Moser's iterative method for a class of ultraparabolic equations[END_REF] obtained the local L 8 estimate for this equation using Moser's method. Continuing in that direction, Wang and Zhang obtained Hölder estimates in [START_REF] Zhang Liqun | The C α regularity of a class of ultraparabolic equations[END_REF], [START_REF] Wang | The C α regularity of a class of non-homogeneous ultraparabolic equations[END_REF] and [START_REF] Wang | The C α regularity of weak solutions of ultraparabolic equations[END_REF]. Their proof is quite involved. A highly nontrivial step is to obtain an appropriate formulation of a Poincaré inequality adapted to the Lie group action related to the equation. A simplified proof, following the method of De Giorgi, was recently obtained by Golse, Imbert, Mouhot and Vasseur [START_REF] Golse | Harnack inequality for kinetic fokker-planck equations with rough coefficients and application to the landau equation[END_REF]. In this paper a version De Giorgi's isoperimetric inequality is obtained by a compactness argument. We use that idea for the case s P r1{2, 1q. This general method is not applicable to the case s ă 1{2, since it uses crucially that the characteristic function of a nontrivial set can never be in H s . We do not use velocity averaging lemmas like in [START_REF] Golse | Harnack inequality for kinetic fokker-planck equations with rough coefficients and application to the landau equation[END_REF] anywhere in this paper. Instead, we take advantage of more elementary properties of the fractional Kolmogorov equation.

The first step in the proof of De Giorgi, Nash and Moser, which consists of a local L 8 estimate, needs to be formulated appropriately to hold for integro-differential equations with degenerate kernels. Our proof in Section 6 follows a properly adapted version of De Giorgi's iteration. We do not use either averaging lemmas, or hypoelliptic estimates for the x variable (like in [START_REF] Golse | Harnack inequality for kinetic fokker-planck equations with rough coefficients and application to the landau equation[END_REF] or [START_REF] Pascucci | The Moser's iterative method for a class of ultraparabolic equations[END_REF]). Instead, we iterate an improvement of integrability obtained directly from the fundamental solution to the fractional Kolmogorov equation.

In the second part of the proof of the theorem of De Giorgi, Nash and Moser we take different strategies depending on whether s P p0, 1{2q or s P r1{2, 1q. In the first case, we construct a barrier function to propagate lower bounds as in the method by Krylov and Safonov for nondivergence equations. When s P r1{2, 1q, the proof is based on a measure estimate of intermediate sets (as in De Giorgi's original work) obtained by compactness (as in [START_REF] Golse | Harnack inequality for kinetic fokker-planck equations with rough coefficients and application to the landau equation[END_REF]), but using a more direct approach based on the fractional Kolmogorov equation instead of hypoelliptic estimates and averaging lemmas. We could not find a single method that works for the full range s P p0, 1q for general integro-differential equations. However, in the case of the Boltzmann equation, the method used for the range s P p0, 1{2q actually works for the full range, as we explain below.

The kernel K f , from the Boltzmann equation, satisfies the extra symmetry condition Kpv, v `wq " Kpv, v ´wq which we do not use in this paper. We chose not to take advantage of this condition in order to have the most natural result for general integro-differential equations. Using this assumption would allow us to simplify some of the proofs. Most importantly, the barriers of Section 7 would hold for the full range s P p0, 1q and therefore the results from Section 8 would be unnecessary. Moreover, the proof of Lemma 6.4 could be done more easily using a similar function g as in the proof of Lemma 6.2. The commutator estimates of Lemmas 4.10 and 4.11 would not be necessary anywhere.

One of the main ideas in the work of Caffarelli, Chan and Vasseur [START_REF] Caffarelli | Regularity theory for parabolic nonlinear integral operators[END_REF] about parabolic integro-differential equations (not kinetic) is how they formulate De Giorgi's isoperimetric lemma in the integro-differential setting. Their original method is purely nonlocal. It does not work for second order equations. It uses crucially that Epg `, g ´q Á }g `}L 1 }g ´}L 1 , where E is a bilinear form like the one we define in Section 4. In our context, this is not true for two reasons. First, because we have the additional variable x that plays no explicit role in the integral diffusion and is not seen by the bilinear form E. Secondly, because the assumptions that we make in the kernels are too mild for this condition to hold even in the space homogeneous case. In [START_REF] Caffarelli | Regularity theory for parabolic nonlinear integral operators[END_REF], they assume that the kernel K is symmetric and Kpv, v 1 q Á λ|v ´v1 | ´d´2s for every value of v and v 1 .

Our equation (1.2) involves three different variables: t, x and v. It reduces to a more standard parabolic integro-differential equation when f is constant in x. The diffusion takes place with respect to the variable v only. The equation includes the kinetic transport term v ¨∇x f , which somehow transfers the regularization effect from the v variable to the x variable. The variable x has to be dealt with differently to the t and v variables. For example it has a different scaling and it is affected by translations of the function with respect to the v variable. One major difficulty that it brings is in the proof of the ink-spots theorem. The original ink-spots covering by Krylov and Safonov was for non-kinetic parabolic equations without the extra variable x. Including this extra variable changes the geometry. The natural parabolic cylinders, which are invariant by the Lie group acting on the equation, are oblique in the variable x. With this geometry, there is no chance to apply a Calderón-Zygmund decomposition like in [START_REF] Imbert | An introduction to fully nonlinear parabolic equations[END_REF] because we cannot tile the space with slanted cylinders with varying slopes. We need a custom made version of the ink-spots covering theorem, which is developed in Section 10. See that section for further explanation on the difficulties and ideas involved in this covering result.

When we apply our main results to the Boltzmann equation in Theorems 1.3 and 1.1, we only want to assume a priori some minimal physically relevant information on f . We assume a control, for all t and x, of the mass, energy and entropy densities. Under these assumptions, there is very little one can say about the Boltzmann collision kernel K f . We are forced to work with very general, non-symmetric, and possibly singular kernels. This paper would be much simpler if we made a convenience assumption like Kpv, v 1 q " Kpv 1 , vq « |v ´v1 | ´d´2s , but it would not suffice to apply the result to the Boltzmann equation. It is not a priori obvious what assumptions the Boltzmann kernel will satisfy. In Section 3, we prove that K f satisfies (1.3), (1.5), (1.6) and (1.7). Our assumptions (1.6) and (1.7) allow us to consider non-symmetric kernels whose anti-symmetric part is as singular as the symmetric one in absolute value, but contains some cancellation. Up to the authors' knowledge, this is the first time such a condition appears in the literature of integro-differential equations.

The estimate for the bilinear form given in Theorem 4.1 is interesting in itself and new. It tells us that the bilinear form xL v f, gy is bounded in H s ˆHs assuming the very mild, and easy to check, conditions on the kernel K given in (1.5), (1.6) and (1.7). Such an estimate is reminiscent of some others proved specifically for the Boltzmann equation, see for instance [START_REF] Alexandre | Regularizing effect and local existence for the non-cutoff Boltzmann equation[END_REF], [START_REF] Alexandre | A review of Boltzmann equation with singular kernels[END_REF], [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous boltzmann equation without angular cutoff[END_REF], [START_REF] Chen | Smoothing estimates for Boltzmann equation with full-range interactions: Spatially inhomogeneous case[END_REF]. Here, the estimate is proved for a very general bilinear form associated with a non-symmetric integro-differential operator. Note that in previous works in integro-differential equations, the upper bound of Theorem 4.1 was included as an assumption together with (1.5) and symmetry (see [START_REF] Kassmann | Regularity results for nonlocal parabolic equations[END_REF]). 1.3.2. Boltzmann without cut-off. The main results of this paper apply to the Boltzmann equation without cut-off in the inhomogeneous setting.

In the case of moderately soft potentials, which corresponds to γ `2s ą ´2, an a priori estimate in L 8 is given in [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF]. In that case, we obtain a Hölder modulus of continuity depending on the bounds on M pt, xq, Ept, xq and Hpt, xq only. For very soft potentials, Theorem 1.1 gives us a Hölder modulus of continuity provided that we know a priori that f is bounded. Note that our estimates do not depend on any further regularity assumption on the initial data.

Since Carlo Cercignani in 1969, it is believed that the Boltzmann collision operator without cut-off has a regularizing effect. Some similarities with the fractional Laplacian operator in the velocity variable have been observed in the form of coercivity estimates. This is the first time that ideas originating in the work of De Giorgi and Nash for parabolic equations are applied in the context of the Boltzmann equation.

The first results for the Boltzmann equations without cut-off that indicate a regularization effect appear in the study of the entropy dissipation. A lower bound for the entropy dissipation with respect to a fractional Sobolev norm is first obtained [START_REF] Lions | Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire[END_REF] and improved in [START_REF] Villani | Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off[END_REF]. The optimal space H s is finally obtained in [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF]. We can also deduce a coercivity estimate from the proof in this paper. The coercivity estimate, which we mention in Proposition 3.3, essentially says that the Boltzmann collision operator satisfies the assumption (1.3). It plays an essential role in most of the works concerning the regularization effect of the Boltzmann equation without cut-off. The proof of the coercivity estimate in [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF] is done using Fourier analysis after reducing the problem to the case of Maxwellian molecules (γ " 0). There is a simplified proof, also using Fourier analysis and in particular the Littlewood-Paley decomposition, in [START_REF] Alexandre | Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff case and Maxwellian molecules[END_REF] and [START_REF] Alexandre | Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. II. Non cutoff case and non Maxwellian molecules[END_REF]. These proofs are considerably easier in the Maxwellian case (γ " 0), because they use Bobylev's formula. We give a new alternative proof in the Appendix A based on the geometric understanding of the Boltzmann kernel. All computations are done in physical variables. Our proof works in the same way for any value of γ. It transparently gives us an estimate with respect to the same anisotropic weighted Sobolev spaces as in [START_REF] Gressman | Global classical solutions of the Boltzmann equation without angular cut-off[END_REF].

The coercivity estimate implies some gain of regularity for the Boltzmann equation without cut-off. In the space homogeneous case, iterating this gain of regularity, it is known that solutions belong to the Schwartz class for all positive times. This result holds under rather general cross section assumptions, including essentially hard and moderately soft potentials in the non-cut-off case. See [START_REF] Desvillettes | Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff[END_REF], [START_REF] Alexandre | Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff case and Maxwellian molecules[END_REF], [START_REF] Alexandre | Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. II. Non cutoff case and non Maxwellian molecules[END_REF], [START_REF] Huo | Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff[END_REF], [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous boltzmann equation without angular cutoff[END_REF] and [START_REF] Chen | Smoothing estimates for Boltzmann equation with full-range interactions: Spatially homogeneous case[END_REF].

For the spacially inhomogeneous case without cut-off, one can also obtain some regularization effect combining the coercivity with hypoelliptic estimates. Iterating such estimates leads to the C 8 regularity of solutions. However, it is necessary to impose significant conditional regularity in order to start the iteration. The best regularity results available require the assumptions that xvy k f pt, x, vq belongs to H 5 pr0, T s, R 3 , R 3 q for all values of k P N, and in addition the mass density is assumed to be bounded below. Under these assumptions, they prove that f belongs to the Schwartz class for positive time in [START_REF] Alexandre | Regularity of solutions for the Boltzmann equation without angular cutoff[END_REF], [START_REF] Alexandre | Regularizing effect and local existence for the non-cutoff Boltzmann equation[END_REF], [START_REF] Chen | Smoothing estimates for Boltzmann equation with full-range interactions: Spatially inhomogeneous case[END_REF].

It may be interesting to compare the current state of the regularity results for the Boltzmann equation with the classical development of nonlinear elliptic equations. Hilbert's 19 th problem consisted in the regularity of minimizers of smooth convex functionals in H 1 (see [START_REF] Wikipedia | Hilbert's nineteenth problem[END_REF]). These minimizers solve a nonlinear elliptic equation in divergence form. From the beginning of the century (starting by the work of Bernstein [START_REF] Bernstein | Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre[END_REF]), people proved that solutions were analytic provided that some conditional regularity assumption was satisfied. The assumptions were progressively improved through the years. By iterating the Schauder estimates, it was possible to prove that solutions were analytic starting from a C 1,α estimate. However, variational techniques only provided a weak solution in H 1 . It was a long standing problem to bridge that gap, and it was finally achieved independently by De Giorgi [START_REF] De | Sulla differenziabilità e lanaliticità delle estremali degli integrali multipli regolari[END_REF] and Nash [START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF]. Our result in this paper plays the role, in the context of the inhomogeneous non-cut-off Botlzmann equation, of the results of De Giorgi and Nash for elliptic and parabolic equations. Unfortunately, there is still a gap between what we prove (C α regularity) and what is necessary to iteratively obtain C 8 regularity of the solution by current methods (H 5 regularity plus infinite moments). So, more work is necessary.

In [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF], results from general integro-differential equations are applied to the Boltzmann equation. There is an L 8 estimate, a Hölder estimate and a lower bound. However, the last two apply only to the space homogeneous case. The results in this paper are proved with different techniques compared to [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF]. In this work, we develop a result in the flavor of De Giorgi, Nash and Moser theorem for equations in divergence form. The results in [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF] use the methods from [START_REF] Russell | Regularity for parabolic integro-differential equations with very irregular kernels[END_REF] which are in the flavor of Krylov-Safonov theory for equations in nondivergence form. The coercivity estimate plays no role in [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF], and it certainly does here. Our result in Theorem 1.5 complements the L 8 estimate from [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF].

In [START_REF] Alexandre | The Boltzmann equation without angular cutoff in the whole space: I, global existence for soft potential[END_REF], the authors prove that if the inital data is sufficiently nice, the Boltzmann equation admits a unique smooth solution locally in time. For small perturbations around a Maxwellian, the equation is known to have global smooth solutions [START_REF] Philip T Gressman | Global classical solutions of the Boltzmann equation with long-range interactions[END_REF], [START_REF] Gressman | Global classical solutions of the Boltzmann equation without angular cut-off[END_REF], [START_REF] Alexandre | Global well-posedness theory for the spatially inhomogeneous Boltzmann equation without angular cutoff[END_REF], [START_REF] Alexandre | Global existence and full regularity of the Boltzmann equation without angular cutoff[END_REF]. As far as existence of weak solutions is concerned, Alexandre and Villani prove in [START_REF] Alexandre | On the Boltzmann equation for long-range interactions[END_REF] the existence of a certain type of renormalized solution. Neither the uniqueness nor the regularity of these solutions is well understood. They prove that the family of solutions is compact using the entropy dissipation estimate.

The study of the regularity of solutions is relevant for most aspects of the qualitative analysis of the Boltzmann equation without cutoff. For example, Desvillettes and Villani prove in [START_REF] Desvillettes | On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation[END_REF] that the solutions converge to equilibrium, at a specific rate, provided that the solution remains smooth.

We consider this paper to be an important step towards a longer term goal to prove the following conjecture. We believe that if f is a solution to the Boltzmann equation with γ `2s P p0, 2s and such that 0 ă M 1 ď M pt, xq ď M 0 , Ept, xq ď E 0 and Hpt, xq ď H 0 , then f should be C 8 for positive time.

It is not at all clear whether the assumption γ `2s ą 0 is necessary to obtain regularity. However, the L 8 estimate for very soft potentials is out of reach by current methods without further assumptions. This is also the case for the space homogeneous Boltzmann equation.

It would be possible to study the precise behavior of the constants λ and Λ for which (1.3), (1.5), (1.6) and (1.7) hold and obtain a global weighted C α estimate using a scaling argument as in Remark A.7. However, this estimate also depends on the L 8 norm of f . It is to be expected that the solution f should decay exponentially for large velocities, in addition to the L 8 bound given in [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF]. See [START_REF] Gamba | On the pointwise propagation of exponential tails for the Boltzmann equation without cutoff[END_REF] for a result in that direction in the space homogeneous case. A better decay in f for large velocities would imply a better C α estimate for large velocities. Because of that, we postpone the analysis of large velocities to future work when the decay of f is better understood. The local result provided in Theorem 1.5 provides the right tool to study the C α estimate for large velocities in terms of the decay of f . 1.3.3. Regularity theory for integro-differential equations. The study of Hölder estimates and the Harnack inequality for integro-differential equations of the form

f t pt, vq " ˆRd pf pt, v 1 q ´f pt, vqqKpt, v, v 1 q dv 1
is a very active area of current research. It developed originally motivated by problems in probability, with applications to mathematical finance [START_REF] Tankov | Financial modelling with jump processes[END_REF] and physics [START_REF] Metzler | The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics[END_REF]. The main technical novelty of this work is our study of a kinetic equation with this kind of diffusion. Our equation has the extra variable x, and the transport term v ¨∇x f , without any explicit diffusion in x. Previous Hölder estimates for integro-differential equations may be applied to the Boltzmann equation, at most, in the space homogeneous case only. Yet, even in the space homogeneous case, the results in this paper present novelties. The assumptions we make on the kernel (1.3), (1.5), (1.6) and (1.7) are more general than in previous works about integro-differential equations. Because of that, our main results in Theorems 1.5 and 1.6 are new even in the space homogeneous case. In this subsection, we review and compare the literature about integro-diferential diffusions. We stress that all previous results apply to the space homogeneous case only.

The interest in Hölder estimates and Harnack inequalities started from the study of regularization properties of classical parabolic equations of second order. For equations in divergence form (like f t " B i a ij pt, vqB j f ), the estimates were originally obtained independently by De Giorgi [START_REF] De | Sulla differenziabilità e lanaliticità delle estremali degli integrali multipli regolari[END_REF] and Nash [START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF], and later reproved by Moser [START_REF] Moser | A harnack inequality for parabolic differential equations[END_REF]. For equations in nondivergence form (like f t " a ij pt, vqB ij f ) the result was obtain much later by Krylov and Safonov [START_REF] Krylov | A property of the solutions of parabolic equations with measurable coefficients[END_REF]. The techniques used for equations in divergence or nondivergence form are very different. In the former case, the equation's structure is amenable to variational methods, and energy estimates in Sobolev spaces. In the latter case, tools like the Alexandroff estimate and explicit barrier functions are used for the proofs. Both types of results, with their corresponding approaches, have their counterparts for integro-differential equations. In this paper, we use the variational structure of the equation and work with localized energy estimates. These are ideas for equations in divergence form. However, we use some ideas that originated in the study of equations in nondivergence form, like the ink-spots theorem and barrier functions. Below, we review other results for integro-differential equations following each approach.

A second order operator in divergence form f Þ Ñ B i pa ij pt, vqB j f q is characterized by the fact that it is self-adjoint in L 2 . For integro-differential operators, this is reflected in a symmetry condition for the kernel: Kpv, v 1 q " Kpv 1 , vq. A second order operator in nondivergence form f Þ Ñ a ij pt, vqB ij f has the convenient property that it returns a bounded function when evaluated in a smooth function f . For integro-differential operators, this is reflected in a different symmetry condition Kpv, v `wq " Kpv, v ´wq. The Boltzmann collision kernel has the symmetry condition that corresponds to equations in nondivergence form. This structure is exploited in [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF] to obtain Hölder estimates in the space homogeneous case, and L 8 estimates for the full equation. In this paper we apply techniques for equations in divergence form. We include assumptions (1.6) and (1.7) which measure how much the kernel K is allowed to depart from being symmetric (as in Kpv, v 1 q " Kpv 1 , vq).

The Harnack inequality and Hölder estimates for integro-differential equations in divergence form has a long history with several major contributions. Some results in this direction are [START_REF] Komatsu | Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms[END_REF], [START_REF] Barlow | Non-local Dirichlet forms and symmetric jump processes[END_REF], [START_REF] Moritz Kassmann | A priori estimates for integro-differential operators with measurable kernels[END_REF], [START_REF] Chen | Global heat kernel estimates for symmetric jump processes[END_REF], [START_REF] Caffarelli | Regularity theory for parabolic nonlinear integral operators[END_REF], [START_REF] Felsinger | Local regularity for parabolic nonlocal operators[END_REF], [START_REF] Kassmann | Regularity results for nonlocal parabolic equations[END_REF] and [START_REF] Dyda | Regularity estimates for elliptic nonlocal operators[END_REF]. There is a small survey on the subject in [START_REF] Kassmann | Regularity results for nonlocal parabolic equations[END_REF]. In these papers the kernel K satisfies the symmetry condition Kpv, v 1 q " Kpv 1 , vq plus some ellipticity assumptions. It is perhaps clear that there is some room in the methods for a lower order asymetric part in K. Our assumptions (1.6) and (1.7) allow us to consider a non-symmetric kernel K whose asymetric part is as singular as the symmetric part. We require a control of the asymetric part in terms of cancellation conditions, which is new.

A natural ellipticity condition on the kernel is to assume that it is comparable with the fractional Laplacian. The classical assumption would be Kpv, v 1 q « |v ´v1 | ´d´2s for every value of v and v 1 . This assumption is made in [START_REF] Komatsu | Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms[END_REF], [START_REF] Barlow | Non-local Dirichlet forms and symmetric jump processes[END_REF], [START_REF] Moritz Kassmann | A priori estimates for integro-differential operators with measurable kernels[END_REF], [START_REF] Chen | Global heat kernel estimates for symmetric jump processes[END_REF] and [START_REF] Caffarelli | Regularity theory for parabolic nonlinear integral operators[END_REF]. The results were extended to a much more general class of kernels in [START_REF] Felsinger | Local regularity for parabolic nonlocal operators[END_REF], [START_REF] Kassmann | Regularity results for nonlocal parabolic equations[END_REF] and [START_REF] Dyda | Regularity estimates for elliptic nonlocal operators[END_REF]. The assumptions there are essentially equivalent to our assumptions (1.3) (the lower bound on the bilinear form in the symmetric case) and (1.5) (the upper bound on the kernel), plus the result of our Lemma 4.2 (the upper bound for the bilinear form). It is a new contribution of this paper that Lemma 4.2 follows from (1.5). We also prove in Theorem 4.1 that the integro-differential operator L v is bounded in H s to H ´s for a non-symmetric kernel satisfying (1.5), (1.6) and (1.7). The proof is significantly more complicated in the non-symmetric case.

The study of integro-differential equations in nondivergence form followed a parallel path using different tools. These are the Hölder estimates and the Harnack inequality for kernels satisfying the other symmetry condition: Kpv, v `wq " Kpv, v ´wq. There are also many important results in this direction including [START_REF] Bass | Transition probabilities for symmetric jump processes[END_REF], [START_REF] Song | Harnack inequality for some classes of Markov processes[END_REF], [START_REF] Bass | Hölder continuity of harmonic functions with respect to operators of variable order[END_REF], [START_REF] Bass | Harnack inequalities for non-local operators of variable order[END_REF], [START_REF] Silvestre | Hölder estimates for solutions of integro-differential equations like the fractional Laplace[END_REF], [START_REF] Caffarelli | Regularity theory for fully nonlinear integro-differential equations[END_REF], [START_REF] Silvestre | On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion[END_REF], [START_REF] Barles | Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations[END_REF], [START_REF] Barles | Lipschitz regularity of solutions for mixed integrodifferential equations[END_REF], [START_REF] Bjorland | Non-local gradient dependent operators[END_REF], [START_REF] Caffarelli | Hölder regularity for generalized master equations with rough kernels[END_REF], [START_REF] Héctor | Regularity for solutions of nonlocal, nonsymmetric equations[END_REF], [START_REF] Héctor | Regularity for solutions of non local parabolic equations[END_REF], [START_REF] Héctor | Hölder estimates for non-local parabolic equations with critical drift[END_REF], [START_REF] Moritz Kassmann | Integro-differential equations with nonlinear directional dependence[END_REF] and [START_REF] Russell | Regularity for parabolic integro-differential equations with very irregular kernels[END_REF]. The majority of these results make the pointwise assumption on the kernel Kpv, v 1 q « |v´v 1 | ´d´2s , and therefore are not directly applicable to the Boltzmann equation. It is only in [START_REF] Bjorland | Non-local gradient dependent operators[END_REF], [START_REF] Moritz Kassmann | Integro-differential equations with nonlinear directional dependence[END_REF] and [START_REF] Russell | Regularity for parabolic integro-differential equations with very irregular kernels[END_REF] that more singular kernels are considered. The assumptions in [START_REF] Russell | Regularity for parabolic integro-differential equations with very irregular kernels[END_REF] are sufficiently general to be applicable to the space homogeneous Boltzmann equation. Our result is for equations in divergence form, and thus none of these papers either implies or follows from ours. Interestingly, we use some of the ideas for nondivergence equations. Most importantly, the ink-spots theorem that we develop in Section 10 is a generalization of a similar covering argument in [START_REF] Russell | Regularity for parabolic integro-differential equations with very irregular kernels[END_REF].

We stress that our main regularity result in Theorem 1.5 requires the equation to hold in a bounded domain only. The parameters λ and Λ in the assumptions (1.3), (1.5), (1.6) and (1.7) will deteriorate as |v| Ñ 8 in the case of the Boltzmann equation.

1.4. Organization of the article. We set our notation and further analyze our assumptions in Section 2. The relationship between our main results and the Boltzmann equation is discussed in Section 3, where we prove in particular that the Boltzmann kernel satisfies the assumptions listed above. The analysis of the operator L v and its associated bilinear form E is done in Section 4. This section should be interesting in itself. This is where the generality of our assumptions on the kernels is reflected. All the results in Section 4 would be straight forward if we assumed that the kernels satisfy Kpv, v 1 q " Kpv 1 , vq and Kpv, v 1 q « |v ´v1 | ´d´2s . The core of the proof of the Weak Harnack inequality and Hölder estimates for integrodifferential equations is done in sections 6, 7, 8, 9, 10 and 11. Section 5 contains fairly unsurprising statements that are technically necessary for the completeness of the rest of our proofs. Experts will probably skim through this section quickly. The appendix A contains a new proof of the coercivity bound for the Boltzmann equation (Subsection A.1) and some technical lemmas (Subsection A.2).

Preliminaries

2.1. Notation. For a real number a, a `" maxpa, 0q.

A constant is called universal if it only depends on dimension and the constants s, λ and Λ in the assumptions (1.3), (1.5), (1.6) and (1.7).

When we write a À b, we mean that there exists a universal constant C, so that a ď Cb. We write a « b when both a À b and b À a hold.

When we write 9 H s pΩq for some Ω Ă R d , we mean the space whose norm is given by }f } 2

9

H s pΩq :"

¨ΩˆΩ |f pv 1 q ´f pvq| 2 |v ´v1 | d`2s dv 1 dv.
The space H s pΩq is the one corresponding to the norm

}f } 2 H s pΩq :" }f } 2 9 H s pΩq `}f } 2 L 2 pΩq .
The space H s 0 pΩq is obtained by completing the space of C 8 functions in R d supported in Ω with respect to the norm } ¨}H s pΩq . When Ω " R d , H s 0 pΩq " H s pΩq. We also define H ´spΩq as the dual of H s 0 pΩq.

It is well known that }f } 2 9 HpR d q " ´Rd |ξ| 2s | f pξq| 2 dξ. Moreover, f P H ´spR d q if and only if f " g 1 p´∆q s{2 g 2 with g 1 , g 2 P L 2 pR d q. Similarly, f is in the dual of 9

H s pR d q if f " p´∆q s{2 g for some function g P L 2 pR d q.

Note also that if f : R d Ñ R is supported in B 1 , then }f } H s pR d q , }f } H s pB2q and }f } 9 H s pB2q are all equivalent.

2.2. First consequences of assumptions. After an obvious readjustment of constants (depending on d and s), the assumption (1.5) is equivalent to the following $ & % piq ´B2rpvqzBrpvq Kpv, v 1 q dv 1 ď Λr ´2s for any r ą 0 and v P B R piiq ´B R XB2rpv 1 qzBrpv 1 q Kpv, v 1 q dv ď Λr ´2s for any r ą 0 and

v 1 P B R.
It is also equivalent to

$ & % piq ´Brpvq |v ´v1 | 2
Kpv, v 1 q dv 1 ď Λr 2´2s for any r ą 0 and v P B R piiq ´B R XBrpv 1 q |v ´v1 | 2 Kpv, v 1 q dv ď Λr 2´2s for any r ą 0 and v 1 P B R.

We use the three forms of the assumption (1.5) indistinctively in different parts of the paper.

As we mentioned before, when s ą 1{2, if the assumption (1.7) holds for some value of r " r 0 and also (1.5) holds, then (1.7) also holds for any other value r P p0, R{8s. The reason is the following computation. We write it for the case r ă r 0 . The case r ą r 0 follows similarly.

ˇˇˇP V ˆBr

pv ´v1 q `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 ˇˇˇď ˇˇˇˇP V ˆBr 0 pv ´v1 q `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 ˇˇˇP V ˆBr 0 zBr pv ´v1 q `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 ˇˇˇˇ, ď Λp1 `r1´2s 0 q `ˆR d zBr |v ´v1 |pKpv, v 1 q `Kpv 1 , vqq dv 1 , ď Λp1 `r1´2s 0 `r1´2s q.

The last inequality is a consequence of (1.5). Note that, for the case s " 1{2, this last integral may be divergent and thus the assumption (1.7) is made so that the inequality holds for all values of r in the range p0, R{8s.

2.3. Invariant transformations. If f satisfies the equation (1.2) for some kernel K satisfying (1.3), (1.5), (1.6) and (1.7), then the scaled function f r pt, x, vq " f pr 2s t, r 2s`1 x, rvq satisfies a modified equation

B t f r `v∇ x f r `L v f r " h r ,
where h r pt, x, vq " r 2s hpr 2s t, r 2s`1 x, rvq, K r pt, x, v, v 1 q " r d`2s Kpr 2s t, r 2s`1 x, rvq.

For any r P r0, 1s, the kernel K r satisfies the assumptions (1.3), (1.5), (1.6) and (1.7) with a larger radius R{r instead of R. Moreover, }h r } L 8 pQ1q ď r 2s }h} L 8 pQ1q ď }h} L 8 pQ1q .

The equation is also invariant under the family of transformations T z0 . Here z 0 " pt 0 , x 0 , v 0 q P RˆR d ˆRd .

T z0 pt, x, vq " pt 0 `t, x 0 `x `tv 0 , v 0 `vq " z 0 ˝z, T ´1 z0 pt, x, vq " pt ´t0 , x ´x0 ´pt ´t0 qv 0 , v ´v0 q " z ´1 0 ˝z (see Figure 2). Indeed, the product ˝induces a Lie group structure on R ˆRd ˆRd . We remark that pT, 0, 0q ˝pt, x, vq " pt `T, x, vq, that is to say, translation in time coincides with a left Lie product. Because of the scaling and the group action that keep our class of equations invariant, we are forced to work with slanted cylinders: for a given center z 0 " pt 0 , x 0 , v 0 q and some radius r ą 0 by the following formula (2.1) Q r pz 0 q " tpt, x, vq : ´r2s ď t ´t0 ď 0, |v ´v0 | ă r, |x ´x0 ´pt ´t0 qv 0 | ă r 1`2s u.

Remark that for z 0 " 0, Q r " Q r p0q " p´r 2s , 0s ˆBr 2s`1 ˆBr .

2.4.

The fractional Kolmogorov equation. In this subsection we review the fractional Kolmogorov's equation:

(2.2) f t `v ¨∇x f `p´∆ v q s f " h.
The previous Lie group structure also preserves this equation. There is a fundamental solution Jpt, x, vq which has the following form

Jpt, x, vq " c d 1 t d`d{s J ´x t 1`1{2s , v t 1{2s ¯.
The function J can be computed explicitly in Fourier variables by the formula Ĵ pϕ, ξq " exp

ˆ´ˆ1 0 |ξ ´τ ϕ| 2s dτ ˙.
In the physical variables x and v, the formula for J is not explicit. However, some simple properties can be deduced from classical considerations. We collect them in the following proposition. Proposition 2.1 (Fundamental solution of the fractional Kolmogorov equation). The functions J and J have the following properties.

(1) The function J is C 8 and decays polynomially at infinity. Moreover, J and all its derivatives are integrable in R 2d . (2) For every t ą 0, ´R2d Jpt, x, vq dv dx " 1.

(3) Both functions are nonnegative: J ě 0 and J ě 0. (4) For any p ě 1, we have }Jpt, ¨, ¨q} L p pR 2d q " t ´dp1`1{sqp1´1{pq }J } L p pR 2d q , }p´∆q s{2 Jpt, ¨, ¨q} L p pR 2d q " t ´dp1`1{sqp1´1{pq´1{2 }p´∆q s{2 J } L p pR 2d q .

In particular, for p ‹ " p2dp1`sq`2sq{p2dp1`sq`sq P p1, 2q, we have }Jpt, ¨, ¨q} L p‹ pR 2d q ď Ct 1{2´1{p‹ and }p´∆q s{2 v Jpt, ¨, ¨q} L p‹ pR 2d q ď Ct ´1{p‹ . The initial value problem (2.2) is solved by the formula (2.3) f pt, x, vq " ˆRd ˆRd f 0 py, wqJpt, x ´y ´tw, v ´wq dw dy `ˆt 0 ˆRd ˆRd hpτ, y, wqJpt ´τ, x ´y ´pt ´τ qw, v ´wq dw dy dτ

We define the modified convolution ˚t by the formula h ˚t jpx, vq " ¨hpy, wqjpx ´y ´tw, v ´wq dw dy.

If we make the change of variables jpx, vq " jpx `tv, vq, then h ˚t jpx, vq " h ˚jpx ´tv, vq. Thus, the modified convolution is the same as the usual convolution conjugated by that change of variables (of Jacobian one). We observe that this convolution satisfies the usual Young's inequality:

(2.4)

› › › › ¨hpy, wqjpx ´y ´tw, v ´wq dw dy › › › › L r x,v ď }h} L p x,v }j} L q x,v
independenly of t.

Here 1 `1{r " 1{p `1{q.

The following proposition is simply a consequence of Young's inequality.

Proposition 2.2 (Gain of integrability). Let f be the solution of (2.2) in r0, T s ˆR2d , with f p0, x, vq "

f 0 px, vq P L 2 pR 2d q. Assume h P L 2 pr0, T s ˆRd , H ´spR d qq. Then }f } L q pr0,T sˆR 2d q ď CpT q `}f 0 } L 2 pR 2d q `}h} L 2 pr0,T sˆR d ,H ´spR d qq for
any q such that 1{q ą 1{p ‹ ´1{2 and p ‹ is the one from Proposition 2.1.

Proof. Since h P L 2 pr0, T s ˆRd , H ´spR d qq, then there exists h 1 and h 2 in L 2 pr0, T s ˆR2d q. so that h "

h 1 `p´∆q s{2 v h 2 and }h 1 } L 2 `}h 2 } L 2 « }h} L 2 pr0,T sˆR d ,H ´spR d qq .
We use the formula (2.3) to solve (2.2). Let us write f pt, x, vq " f 1 pt, x, vq `f2 pt, x, vq `f3 pt, x, vq, where

f 1 pt, ¨, ¨q :" f 0 ˚t Jpt, ¨, ¨q, f 2 pt, ¨, ¨q :" ˆt 0 h 1 pτ q ˚pt´τq Jpt ´τ, ¨, ¨q dτ, f 3 pt, ¨, ¨q :" ˆt 0 h 2 pτ q ˚pt´τq p´∆q s{2 v Jpt ´τ, ¨, ¨q dτ.
Let p P r1, p ‹ q be the number such that 1{q " 1{p ´1{2. Applying Young's inequality for each value of t, we have

}f 1 pt, ¨, ¨q} L q ď }f 0 } L 2 }J } L p t 1{2´α , }f 2 pt, ¨, ¨q} L q ď ˆt 0 }h 1 pτ q} L 2 }J } L p pt ´τ q 1{2´α dτ, }f 3 pt, ¨, ¨q} L q ď ˆt 0 }h 2 pτ q} L 2 }p´∆q s{2 J } L p pt ´τ q ´α dτ.
Here α " dp1 `1{sqp1 ´1{pq `1{2 ă 1{p ‹ ă 1{p since p ă p ‹ . Moreover, qp1{2 ´αq ą ´1 so f 1 P L q pr0, T s ˆR2d q with }f 1 } L q pr0,T sˆR 2d q ď CT p1{2´αq`1{q }f 0 } L 2 . We estimate the other two terms applying Young's inequality once again

}f 2 } L q pr0,T sˆR 2d q ď C}h 1 } L 2 pr0,T s,R 2d q T 1{2`1{p´α , }f 3 } L q pr0,T sˆR 2d q ď C}h 2 } L 2 pr0,T s,R 2d q T 1{p´α .
This finishes the proof.

Remark 2.3. The power p in Lemma 2.2 can also be taken equal to p ‹ by using the weak-type Young's inequality in place of the usual Young's inequality for convolutions and a finer analysis of the L p‹,8 norm of J. Since we do not need a sharp result in this paper, we prefer to keep this lemma as elementary as possible.

The Boltzmann kernel

In this subsection, we explain why the Boltzmann collision operator associated with inverse power-law potentials (see (1.1)) satisfy the assumptions we made on the kernel as soon the quantities M pt, xq, Ept, xq and Hpt, xq defined in the introduction are under control.

3.1.

The collision operator as an integro-differential operator plus a lower order term. It is classical to observe that B can be replaced with any B satisfying for all k, σ P S d´1 , Bpr, k ¨σq `Bpr, ´k ¨σq " Bpr, k ¨σq `Bpr, ´k ¨σq.

For this reason, we can (and do) follow [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF] and assume

(3.1) If k ¨σ ă 0, then Bpr, k ¨σq « r γ | cospθ{2q| γ`2s`1
where cospθ{2q :"

v´v| v´v˚| ¨v´v 1 |v´v˚´| . We split Q in Q 1 and Q 2 as follows: Qpf, gq " Q 1 pf, gq `Q2 pf, gq with # Q 1 pf, gq " ˜f 1 ˚pg 1 ´gqB dv ˚dσ, Q 2 pf, gq " `˜pf 1 ˚´f ˚qB dv ˚dσ ˘g.
Such a decomposition appears for instance in [START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF][START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF].

The term Q 1 can be rewritten using Carleman coordinates [START_REF] Carleman | Sur la théorie de l'équation intégrodifférentielle de Boltzmann[END_REF].

Lemma 3.1 (The integro-differential operator [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF]). The term g Þ Ñ Q 1 pf, gq corresponds to some linear operator L v g with K " K f given by

(3.2) K f pv, v 1 q " 2 d´1 |v 1 ´v| ˆwKv 1 ´v f pv `wqBpr, cos θqr ´d`2 dw where r 2 " |v 1 ´v| 2 `|w| 2 and cos θ " v ´v1 ´w |v ´v1 ´w| ¨v1 ´v ´w |v 1 ´v ´w| .
The proof of the previous lemma is simply a change of variables to Carleman coordinates, see [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF]. It is recalled in Appendix for the reader's convenience, see Lemma A.9. The term Q 2 pf, gq is of lower order because of the cancellation lemma [START_REF] Villani | Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off[END_REF], [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF]. Lemma 3.2 (Cancellation [START_REF] Villani | Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off[END_REF], [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF]). The following formula holds true for any

v P R d , ¨pf 1 ˚´f ˚qB dv ˚dσ " C b | ¨|γ ‹ f pvq with C b " ˆSd´1 " 2 pd`γq{2 p1 `σ ¨eq pd`γq{2
´1* bpσ ¨eq dσ

for any e P S d´1 .

Coercivity bound.

We prove this lower bound in the Appendix A.1. This is a well known result in the Boltzmann literature.

Proposition 3.3 (Lower bound [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF][START_REF] Gressman | Global classical solutions of the Boltzmann equation without angular cut-off[END_REF]). Let g : R d Ñ R be a function supported in B R. Then

c}g} 2 9 H s pR d q ď ´ˆR d Qpf, gqpvqgpvq dv `C}g} 2 L 2 pR d q .
The constants c and C depend on the mass, energy and entropy of f , the dimension d and the radius R. In other words, K f satisfies (1.3) as soon as mass, energy and entropy of f are bounded. In the case of the mass, we also need it to be bounded below.

The assumption (1.4), which we need in the case s P p0, 1{2q is clearly satisfied by the Boltzmann kernel. This follows as consequence of Lemma 4.8 in [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF].

Upper bounds.

In this paragraph, we justify that the Boltzmann kernel satisfies (1.5). We recall that (1.5)-(i) was already proved in [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF]. Recall the equivalent formulations of (1.5) explained in Section 2. Lemma 3.4 (Upper bound (1.5)-(i) [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF]Corollary 4.4]). Assume γ `2s ď 2. Then for all r ą 0 and v

P B R, ˆB2rpvqzBrpvq K f pv, v 1 q dv 1 À r ´2s ˆˆR d f pzq|z ´v| γ`2s dz ˙.
In particular, K f satisfies (1.5)-(i) with Λ that depends only on }f ‹ | ¨|γ`2s } L 8 pB2q . More precisely, if γ `2s P r0, 2s, then Λ in (1.5)-(i) depends only on mass and energy; if γ `2s ď 0, then it depends on mass, dimension, γ, s and }f } L 8 .

We can now derive (1.5)-(ii).

Lemma 3.5 (Upper bound (1.5)-(ii)). Assume γ `2s ď 2. Then for all v 1 P B R and r ą 0,

ˆRd zBrpv 1 q K f pv, v 1 q dv À r ´2s ˆˆR d f pzq|z ´v1 | γ`2s dz ˙.
In particular, K f satisfies (1.5)-(ii) with Λ that only depends on }f ‹ | ¨|γ`2s } L 8 pB R q . More precisely, if γ `2s P r0, 2s, it depends only on mass and energy; if γ `2s ď 0, Λ then it depends on mass, dimension, γ, s and }f } L 8 .

Proof. According to the formula for Kpv, v 1 q in terms of f (Corollary 4.2 in [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF]),

K f pv, v 1 q « |v ´v1 | ´d´2s ˜ˆwKpv´v 1 q f pv `wq|w| γ`2s`1 dw ¸.
Without loss of generality, let us take v 1 " 0 in order to simplify the notation. Therefore

ˆRd zBr K f pv, 0q dv À ˆ8 r ρ ´d´2s
ˆBBρ ˆwKv f pv `wq|w| γ`2s`1 dw dSpvq dρ.

Applying (A.6) from Lemma A.10,

ˆRd zBr K f pv, 0q dv À ˆ8 r ρ ´2s´1 ˆRd zBρ f pzq p|z| 2 ´ρ2 q d´2`γ`2s 2 |z| d´2 dz dρ, " ˆRd zBr f pzq |z| d´2 ˜ˆ|z| r ρ ´2s´1 p|z| 2 ´ρ2 q d´2`γ`2s 2 dρ ¸dz, ď ˆRd zBr f pzq |z| d´2 `r´2s |z| d´2`γ`2s ˘dz, " r ´2s ˆRd zBr f pzq|z| γ`2s dz.
3.4. The cancellation assumptions. In this paragraph, we justify that the kernel associated with the Boltzmann equation satisfies the cancellation assumptions (1.6) and (1.7). The first cancellation condition, assumption (1.6), is essentially the cancellation lemma, which is well known in the kinetic community.

Lemma 3.6 (Classical cancellation lemma). The kernel K f satisfies for all v P R d , ˇˇˇP V ˆRd pK f pv, v 1 q ´Kf pv 1 , vqq dv 1 ˇˇˇď C ˆˆR d f pzq|z ´v| γ dz ˙.
In particular, K f satisfies (1.6) with Λ that only depends on }f ‹ | ¨|γ } L 8 pB R q . More precisely, if γ P r0, 2s, Λ in (1.6) depends only on upper bounds on mass and energy; if γ ď 0, it depends on mass, dimension, γ and }f } L 8 .

Proof. Let P pvq denote P V ´pK f pv 1 , vq ´Kf pv, v 1 qq dv 1 . In view of the definition of K f , we have

P pvq " 2 d´1 ˆRd dv 1 ˜ˆwKv 1 ´v f pv `wq Bpr, cos θq |v 1 ´v|r d´2 dw ´ˆwKv 1 ´v f pv 1 `wq Bpr, cos θq |v 1 ´v|r d´2 dw " 2 d´1
ˆRd ˆwKv 1 ´v pf pv `wq ´f pv 1 `wqq Bpr, cos θq |v 1 ´v|r d´2 dw dv 1 since r " r and cos θ " cos θ. Using now (A.3) from Lemma A.9, we get

P pvq " ˆRd ˆSd´1
pf pv 1 ˚q ´f pv ˚qqBpr, cos θq dσ dv ˚.

The cancellation Lemma 3.2 tells us that

P pvq " c ˆRd |v ´w| γ f pwq dw.
The proof is now complete.

Lemma 3.7 (More subtle cancellation lemma). The two following properties hold true for any R ą 0,

P V ˆBR pvq pv 1 ´vqK f pv, v 1 q dv 1 " 0, (3.3) ˇˇˇˇP V ˆBR pvq pv 1 ´vqK f pv, v 1 q dv ˇˇˇˇď C ˆˆR d f pzq|z ´v1 | 1`γ dz ˙. (3.4)
In particular, the kernel satisfies (1.7) and Λ only depends on }f ‹|¨| 1`γ } L 8 pB R q . More precisely, if γ P r´1, 1s, Λ in (1.7) depends only on upper bounds on mass and energy; if γ ď ´1, it depends on mass, dimension, γ and }f } L 8 .

Proof. The first identity (3.3) is obvious from the symmetry property: K f pv, v `wq " K f pv, v ´wq. The difficulty is thus to justify the second identity (3.4).

Without loss of generality, let us assume v 1 " 0. In view of Lemma 3.1 (coming from [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF]), the kernel K f can be written for v 1 " 0 as follows,

K f pv, 0q " 2 d´1
|v| ˆtw:wKvu f pv `wqBpr, cos θq 1 r d´2 dw where r 2 " |v| 2 `|w| 2 " |z| 2 and z " v `w and

cos θ " { v `w ¨{ w ´v " |w| 2 ´|v| 2 |v `w| 2 " |z| 2 ´2|v| 2 |z| 2 .
The way bpcos θq is modified for cos θ ă 0, implies that

1 |v|r d´2 Bpr, cos θq « |v| ´d´2s |w| γ`2s`1 .
Since r and cos θ only depend on |z| and |v|, this implies that there exists Ap|z|, |v|q such that 1 |v|r d´2 Bpr, cos θq " Ap|z|, |v|q|v| ´d´2s |w| γ`2s`1 and a constant C A ą 1 such that for all z, v,

C ´1

A ď Ap|z|, |v|q ď C A . In the following computation, the definition of r changes. We write r " |v|. We integrate in v first on spheres BB r and then with respect to the radius r.

ˇˇˇP V ˆBR vK f pv, 0q dv ˇˇˇ" ˇˇˇˇˆR 0 r ´d´2s
ˆBBr ˆwKv vAp|v `w|, |v|qf pv `wq|w| γ`2s`1 dw dv dr

ˇˇˇˇ

We use the change of variables (A.7) of Lemma A.10. Note that |w| 2 `r2 " |z| 2 .

ˇˇˇP V ˆBR vK f pv, 0q dv ˇˇˇ" ω d´2 ˇˇˇˇˆR 0 r 1´2s ˆRd zBr Ap|z|, rqzf pzq p|z| 2 ´r2 q d´2`γ`2s 2 |z| d dz dr ˇˇˇ" ω d´2 ˇˇˇˇˆR d zf pzq|z| ´d ˜ˆminp|z|,Rq 0 r 1´2s Ap|z|, rqp|z| 2 ´r2 q d´2`γ`2s 2 dr ¸dz ˇˇˇď ω d´2 C A ˆRd f pzq|z| ´1`γ`2s ˜ˆminp|z|,Rq 0 r 1´2s dr ¸dz ď C ˆRd f pzq|z| 1`γ dz.
The proof is now complete.

Remark 3.8. There is a subtle cancellation that allows this proof to work. The whole point of this lemma is that the principal value of the integral is bounded around the origin. The reader will notice that here we end up with an integral of the form ´|z| 0 r 1´2s dr. In the proof of Lemma 3.5, we end up with an intergrand r ´1´2s which is not integrable around the origin. The difference originates in Lemma A.10 given in Appendix. The third identity in that lemma incorporates an extra cancellation due to the fact that the average values of v P BB r so that v `w " z, for some w K v, is r 2 z{|z| 2 . Remark 3.9. Note that the cancellation condition given in Lemma 3.7 is slightly stronger than (1.7) since the right hand side is bounded independently of R even when s ą 1{2. Moreover, a rate of convergence to zero as R Ñ 0 can be deduced from the proof.

3.5. Proofs of Theorems 1.1 and 1.3. In this subsection we explain how Theorems 1.1 and 1.3 follow from Theorems 1.5 and 1.6. Theorem 1.1 is indeed a straight forward application of Theorem 1.5.

Proof of Theorem 1.1. The Boltzmann equation can be written in the form

f t `v ¨∇x f " ˆˆR d pf pv 1 q ´f pvqqK f pv, v 1 q dv 1 ˙`c ˆˆR d f pv ´wq|w| γ dw ˙f.
Thus, if we define

h :" c ˆˆR d f pv ´wq|w| γ dw ˙f,
then h P L 8 with its norm bounded in terms of }f } L 8 and M 0 . Moreover, from Proposition 3.3 and Lemmas 3.4, 3.5, 3.6 and 3.7, the kernel K f satisfies the assumptions (1.3), (1.5), (1.6) and (1.7). Thus, the proof is finished as a corollary of Theorem 1.5. Theorem 1.3 follows mostly from Theorem 1.6. We use some other results which are presented later in this article which allow us to extend the lower bound to an arbitrary radius R ą 0.

Proof of Theorem 1.3. Without loss of generality, we assume T " 4. The general case follows by scaling.

Like in the proof above of Theorem 1.1, we have that f is a supersolution of (1.2) for some h ě 0. In particular,

f t `v ¨∇x f ě L v f.
According to Lemma A.2, there is an R 0 ą 0, m ą 0 and ą 0 so that for all pt, xq, |tv P B R0 : f pt, x, vq ě u| ě m. Let r 0 be the one from Theorem 1.6. We have that ˆr0,r 2s 0 sˆB

r 1`2s 0 ˆBR 0 f ε dv dx dt ě ε mr 2s`p1`2sqd 0 .
It is possible to cover the set r0, r 2s 0 s ˆBr 1`2s 0 ˆBR0 with N slanted cylinders Q r0 pzq with N ď pR 0 {r 0 q 2d {c for some universal constant c ą 0. This implies that there must be some point z " pr 2s 0 , x, vq P tr 2s 0 u Br 2s 0 pr0`R0q ˆBR0 so that ˆQr 0 pzq f ε dv dx dt ě c ε mr 2s`p3`2sqd 0 {R 2d 0 .

Applying Theorem 1.6 (properly translated), we get inf

Qr 0 pzq f ě c,
for some constant c ą 0 and z P t1u ˆBr 1`2s 0 `R0 ˆBR0 . This bound below in Q r0 pzq is propagated to r2, 4s ˆBR ˆBR , for any arbitrary R ą 0 using the barrier function from Lemma 7.1 if s ă 1{2 or the combination of Lemmas 8.3 and 6.6 if s ě 1{2.

Note that the geometric setting of Lemmas 7.1, 8.3 and 6.6 are independent of the constants λ and Λ. This is important since these ellipticity constants depend on R.

Study of a bilinear form

This section is devoted to the study of a general bilinear form E associated with a kernel K through the following formula, Epϕ, gq " ´ˆpL v ϕqpvqgpvq dv " lim εÑ0 ˜¨|v´v 1 |ąε pϕpvq ´ϕpv 1 qqgpvqKpv, v 1 q dv 1 dv ¸.

In the remainder of this section, we abuse notation by ignoring the limit as ε Ñ 0. This means that some integrals corresponding to the odd part of K may need to be understood in the principal value sense. Indeed, we recall that the operator L v ϕ is given by the formula (4.1) L v ϕpvq :" lim εÑ0 ˆRd zBεpvq pϕpv 1 q ´ϕpvqqKpv, v 1 q dv 1 .

The limit does not necessarily converge for every value of v, even if ϕ is smooth. The correct understanding of L v ϕ as a distribution is obtained through the analysis of the bilinear form E done in this section.

When we study the equation (1.2), the bilinear form E will be computed for functions ϕ and g depending on values of t and x. The kernel, and consequently also the bilinear form, depend on t and x. In this section we study properties of bilinear forms like this that will be applied for every fixed value of t and x.

In this section, we also assume that the kernel K is defined for all values of v P R d and our assumptions hold uniformly. This is convenient for the exposition and some of the proofs. In Section 5, we will show that any kernel satisfying (1.3), (1.5), (1.6) and (1.7) can be extended to all values of v P R d to satisfy a global version of these assumptions. So, a posteriori, our approach is not limiting.

Since it is not necessary for K to be non-negative for the results in this section to hold, and they may be used elsewhere, we restate here our main assumption allowing sign changing kernels. We make the following assumptions for some parameter s P p0, 1q and a constant Λ.

(4.2) @v P R d , @r ą 0, $ & % piq ´Rd zBrpvq |Kpv, v 1 q| dv 1 ď Λr ´2s piiq ´Rd zBrpvq |Kpv 1 , vq| dv 1 ď Λr ´2s .
We also state a global version of the cancellation assumptions (1.6) and (1.7).

(4.3) @v P R d , ˇˇˇP V ˆRd pKpv, v 1 q ´Kpv 1 , vqq dv 1 ˇˇˇď Λ.
In the case s ě 1{2, we also assume that for all R ą 0, (4.4)

@v P R d , ˇˇˇˇP V ˆBR pvq pKpv, v 1 q ´Kpv 1 , vqqpv ´v1 q dv 1 ˇˇˇˇď Λp1 `R1´2s q.
The main result of this section will be that the bilinear form E is bounded in H s ˆHs provided that (4.2), (4.3) and (4.4) hold. We also show some other estimates that we will use.

4.1.

Estimates in H s . The main result of this section is the fact that the bilinear form E is bounded in H s ˆHs as soon as our assumptions (4.2), (4.3) and (4.4) hold. We state it in the following theorem. Theorem 4.1 (Estimate in H s ). Let K satisfy (4.2) and (4.3). If s ě 1{2, we also assume that it satisfies (4.4). There then exists a constant C depending only on s, d and Λ, so that

Epf, gq ď C}f } H s }g} H s .
It is convenient for some of our proofs to spit E into the symmetric and anti-symmetric part of K. Let Epϕ, gq " E sym pϕ, gq `Eskew pϕ, gq with E sym pϕ, gq " 1 2 ¨pϕpvq ´ϕpv 1 qqpgpvq ´gpv 1 qqKpv, v 1 q dv 1 dv,

E skew pϕ, gq " 1 2 P V ¨pϕpvq ´ϕpv 1 qqpgpvq `gpv 1 qqKpv, v 1 q dv 1 dv.
Note that E " E sym and E skew " 0 when the symmetry condition Kpv, v 1 q " Kpv 1 , vq holds. Likewise, when K is anti-symmetric (i.e. Kpv, v 1 q " ´Kpv 1 , vq) then E sym " 0 and E skew " E. Consequently, writing K as the sum of its symmetric plus anti-symmetric part corresponds to writing E as the sum of E sym and E skew . We will prove Theorem 4.1 estimating E sym and E skew separately. Note that, because of the density of smooth functions in H s , it suffices to prove Theorem 4.1 when g and ϕ are smooth. Lemma 4.2 (Estimate of the symmetric part). Let K be a kernel satisfying (4.2). Then, there exists a constant depending only on Λ, s and dimension, so that for any function g P H s pR d q,

E sym pg, gq ď C}g} 2 9 H s .
Proof. Without loss of generality, we can assume K ě 0 here. Otherwise, the value of E sym pg, gq would only increase if we replace Kpv, v 1 q by |Kpv, v 1 q|. We write (4.5) E sym pg, gq "

8 ÿ k"´8 P p2 k q
where, for any r ą 0,

P prq :" ¨tpv,v 1 qPR d ˆRd :rď|v´v 1 |ă2ru
|gpv 1 q ´gpvq| 2 Kpv, v 1 q dv 1 dv.

The key of this proof is to estimate P prq with a similar expression involving the kernel |v ´v1 | ´d´2s instead. For any values of v and v 1 , let m " pv `v1 q{2, we introduce an auxiliary point w P B r{4 pmq. From the triangle inequality |gpv 1 q ´gpvq| 2 ď 2|gpv 1 q ´gpwq| 2 `2|gpwq ´gpvq| 2 . Then

P prq À 1 r d ¨tpv,v 1 qPR d ˆRd :rď|v´v 1 |ă2ru ˆBr{4 pmq `|gpv 1 q ´gpwq| 2 `|gpwq ´gpvq| 2 ˘Kpv, v 1 q dw dv 1 dv,
we change the order of integration for each term in the integrand,

ď 1 r d ¨r{4ď|v 1 ´w|ă5r{4 |gpv 1 q ´gpwq| 2 ˜ˆΩ v 1 ,w Kpv, v 1 q dv ¸dw dv 1 `1 r d ¨r{4ď|v´w|ă5r{4 |gpwq ´gpvq| 2 ˜ˆΩv,w Kpv, v 1 q dv 1 ¸dw dv.
Here the set Ω v,w contains all values of v 1 that correspond to any given pair pv, wq. We will only use that Ω v,w Ă B 2r pvqzB r pvq. Both terms are bounded by the same expression using each line in (4.2). Thus,

P prq À Λ r d`2s ¨r{4ď|v´w|ă5r{4 |gpv 1 q ´gpwq| 2 dw dv À Λ ¨r{4ď|v´w|ă5r{4 |gpv 1 q ´gpwq| 2 |v ´w| d`2s dw dv.
Applying this estimate for each term in (4.5), we get the desired estimate.

}L v f } L 2 ď C}f } 1´2s 1`2s L 2 }f } 4s 1`2s 9 H s`1{2 .
Proof. For some R ą 0, to be determined below, let us write L v f " 0 ` 1 ` 2 , where 0 pvq " ˆBR pvq pf pv 1 q ´f pvqqKpv, v 1 q dv 1 ,

1 pvq " ˆRd zB R pvq f pv 1 qKpv, v 1 q dv 1 , 2 pvq " ´˜ˆR d zB R pvq Kpv, v 1 q dv 1 ¸f pvq.
We prove the estimate for each one of the three terms.

Let us start with 2 , which is the easiest. In this case, obviously,

} 2 } L 2 ď ˜sup v ˆRd zB R pvq Kpv, v 1 q dv 1 ¸}f } L 2 ď ΛR ´2s }f } L 2 .
The estimate for 1 involves the Cauchy-Schwarz inequality and an application of Fubini's theorem. In this case we use the second line of (4.2).

} 1 } 2 L 2 " ˆRd ˜ˆR d zB R pvq f pv 1 qKpv, v 1 q dv 1 ¸2 dv, ď ˆRd ˜ˆR d zB R pvq Kpv, v 1 q dv 1 ¸˜ˆR d zB R pvq f pv 1 q 2 Kpv, v 1 q dv 1 ¸dv, ď ΛR ´2s ˆRd f pv 1 q 2 ˜ˆtv:|v 1 ´v|ąRu Kpv, v 1 q dv ¸dv 1 ď Λ 2 R ´4s }f } 2 L 2 .
We estimate 0 using the Cauchy-Schwarz inequality together with (4.2) and comments from Subsection 2.2.

} 0 } 2 L 2 " ˆRd ˜ˆB R pvq pf pv 1 q ´f pvqqKpv, v 1 q dv 1 ¸2 dv, ď ˆRd ˜ˆB R pvq pf pv 1 q ´f pvqq 2 |v ´v1 | ´1K pv, v 1 q dv 1 ¸˜ˆB R pvq |v ´v1 |Kpv, v 1 q dv 1 ¸dv, ď ΛR 1´2s ¨|v´v 1 |ăr pf pv 1 q ´f pvqq 2 |v ´v1 | ´1K pv, v 1 q dv 1 dv.
The kernel |v ´v1 | ´1K pv, v 1 q satisfies (1.5) with s `1{2 instead of s. Then, we apply Lemma 4.2 to get

} 0 } 2 L 2 À R 1´2s }f } 2 9 H s`1{2 . The proof is finished choosing R " p}f } L 2 {}f } H s`1{2 q 2{p1`2sq .
The estimate for }L v f } L 2 when s ě 1{2 is harder to obtain. We will use the following auxiliary kernel.

Apv, wq " ˆtv 1 PB R pvq:pv 1 ´vq¨pw´vqě|w´v| 2 u |v 1 ´v| d´2 Kpv, v 1 q |w ´v| d´2 |w ´v1 | d´2 dv 1 .
Lemma 4.4 (Estimates on the auxiliary kernel). Let K be a kernel satisfying (4.2) and s ě 1{2. We have Proof. The first of the two inequalities in (4.6) is a relatively straight forward computation using (4.2). Let us choose v " 0 without loss of generality. We have

ˆBR |Ap0, wq| dw ď ˆBR ˆtv 1 PB R :v 1 ¨wě|w| 2 u |v 1 | d´2 |Kp0, v 1 q| |w| d´2 |w ´v1 | d´2 dv 1 dw, " ˆBR |v 1 | d´2 |Kp0, v 1 q| ˜ˆB |v 1 |{2 pv 1 {2q 1 |w| d´2 |w ´v1 | d´2 dw ¸dv 1 , " C ˆBR |Kp0, v 1 q||v 1 | 2 dv 1 À CR 2´2s .
Let us move to the second inequality in (4.6). Assume without loss of generality that w " 0. We have

ˆBR |Apv, 0q| dv ď ˆBR ˆtv 1 PB R pvq:v 1 ¨vď0u |v 1 ´v| d´2 |Kpv, v 1 q| |v| d´2 |v 1 | d´2 dv 1 dv ď I 1 `I2 .
From the triangle inequality |v 1 ´v| d´2 À |v| d´2 `|v 1 | d´2 , we can estimate the above integral by I 1 `I2 , where I 1 and I 2 are defined below. We analyze both terms using (4.2) and Fubini's theorem.

I 1 : " ˆBR ˆtv 1 :|v 1 ´v|ăR and v 1 ¨vď0u |v| d´2 |Kpv, v 1 q| |v| d´2 |v 1 | d´2 dv 1 dv, " ˆB2R ˆtv:|v´v 1 |ăR and v 1 ¨vď0uXB R |Kpv, v 1 q| |v 1 | d´2 dv dv 1 , À R ´2s ˆB2R |v 1 | 2´d dv 1 À R 2´2s .
We now consider

I 2 :" ˆBR ˆtv 1 :|v 1 ´v|ăR and v 1 ¨vď0u |v 1 | d´2 |Kpv, v 1 q| |v| d´2 |v 1 | d´2 dv 1 dv.
The computation that proves that I 2 À R 2´2s is almost identical integrating in v 1 first and in v second. This concludes the estimate for every term involved in (4.6).

We will use the following lemma from multivariate calculus when proving estimates on the operator L v f associated with the kernel K. ˇˇˇf pv 1 q ´f pvq ´∇f pvq `∇f pv 1 q 2 ¨pv 1 ´vq ˇˇˇď 1

dω d |v ´v1 | d´2 ˆBR pmq |D 2 f pwq| |w ´v| d´2 |w ´v1 | d´2 dw. (4.7)
Here R " |v ´v1 |{2 and m " pv `v1 q{2. Thus, B R pmq is the ball with diameter from v to v 1 . Proof. For any w P B R pmq, we write |f pwq ´f pvq ´∇f pvq ¨pw ´vq| ď |w ´v|

ˆ|w´v| 0 |D 2 f |pv `z { w ´vq dz.
where { w ´v " pw ´vq{|w ´v|. In particular, computing with spherical coordinates the integral in the first line below with origin at w " v,

ˇˇˇˇ˜ B R pmq f pwq dw ¸´f pvq ´∇f pvq ¨pv 1 ´vq 2 ˇˇˇˇ" ˇˇˇˇ B R pmq f pwq ´f pvq ´∇f pvq ¨pw ´vq dw ˇˇˇˇ, ď B R pmq |w ´v| # ˆ|w´v| 0 |D 2 f |pv `ζ { w ´vq dζ + dw À ˆBR pmq |D 2 f |pwq |w ´v| d´2 dw.
This implies that

ˇˇˇˇ˜ B R pmq f pwq dw ¸´f pvq ´∇f pvq ¨pv 1 ´vq 2 ˇˇˇˇÀ R d´2 ˆBR pmq |D 2 f |pwq |w ´v| d´2 |w ´v1 | d´2 dw.
Exchanging the role of v and v 1 and subtracting the resulting inequalities yields (4.7).

Lemma 4.6 (Estimate of L v f for s ě 1{2). Assume s P r1{2, 1q. Let K be an anti-symmetric kernel (i.e. Kpv, v 1 q " ´Kpv 1 , vq) satisfying (4.2) and (4.4). The following estimate holds

}L v f } L 2 ď C}f } 1´s L 2 }D 2 f } s L 2 `Λ}∇f } L 2 .
Proof. We write

L v f " 0 ` 1 ` 2 like in Lemma 4.3. The estimates } 1 } L 2 ď ΛR ´2s }f } L 2 and } 2 } L 2 ď ΛR ´2s }f } L 2 follow an identical proof. The estimate for } 0 } L 2 is different. Recall that 0 pvq " ˆBR pvq pf pv 1 q ´f pvqqKpv, v 1 q dv 1
We write 0 " 0 0 ` 1 0 ` 2 0 with 0 0 " 1 2 ˆBR pvq `∇f pv 1 q ´∇f pvq ˘pv 1 ´vqKpv, v 1 q dv 1 , 1 0 " ∇f pvq ˆBR pvq pv 1 ´vqKpv, v 1 q dv 1 , 2 0 " ˆBR pvq ˆf pv 1 q ´f pvq ´∇f pvq `∇f pv 1 q 2 ¨pv 1 ´vq ˙Kpv, v 1 q dv 1 .

The same argument that gives us the upper bound for 0 in Lemma 4.3 gives us in this case

} 0 0 } L 2 ď CR 1´s }∇f } 9 H s À R 1´s }f } p1´sq{2 L 2 }D 2 f } p1`sq{2 L 2
. Indeed, 0 0 equals the same as 0 in the proof of Lemma 4.3 with ∇f instead of f and pv ´v1 qKpv, v 1 q instead of Kpv, v 1 q. The fact that these are vector valued functions does not affect the proof. Note that pv ´v1 qKpv, v 1 q satisfies (4.2) with s ´1{2 instead of s. The second inequality is an elementary interpolation.

The cancellation assumption (4.4) says that

} 1 0 } L 2 ď ΛpR 1´2s `1q}∇f } L 2 ď ΛR 1´2s }f } 1 2 L 2 }D 2 f } 1 2 L 2 `Λ}∇f } L 2 .
In order to estimate 2 0 , we use Lemma 4.5. We write

} 2 0 } 2 L 2 "
ˆRd ˜ˆB R pvq ˆf pv 1 q ´f pvq ´∇f pvq `∇f pv 1 q 2 ¨pv 1 ´vq ˙Kpv, v 1 q dv 1 ¸2 dv,

À ˆRd ˜ˆB R pvq ˆBrpmq |D 2 f pwqq| |v 1 ´v| d´2 Kpv, v 1 q |w ´v| d´2 |w ´v1 | d´2 dw dv 1 ¸2 dv, using Fubini's theorem " ˆRd ˜ˆB R pvq |D 2 f pwq| ˜ˆtv 1 :pv 1 ´vq¨pw´vqě|w´v| 2 u |v 1 ´v| d´2 Kpv, v 1 q |w ´v| d´2 |w ´v1 | d´2 dv 1 ¸dw ¸2 dv.
In view of the definition of Apv, wq, we can use (4.6) and get

} 2 0 } 2 L 2 ď ˆRd ˆˆB R |D 2 f pwq|Apv, wq dw ˙2 dv, ď ˆRd ˆˆB R Apv, wq dw ˙ˆˆB R |D 2 f pwq| 2 Apv, wq dw ˙dv, ď CR 2´2s ˆRd |D 2 f pwq| 2 ˜ˆ|v´w|ăR Apv, wq dv ¸dw ď CR 2p2´2sq }D 2 f } 2 L 2 . Choosing R " }f } 1 2 L 2 {}D 2 f } 1 2
L 2 completes the proof. We can now prove the main result of this section.

Proof of Theorem 4.1. We prove the upper bound applying Lemmas 4.3 and 4.6 to both operators L v and its adjoint L t v , and doing some sort of interpolation. Note that L t v has the same form as L v plus a correction which is bounded from L 2 to L 2 (thanks to the cancellation assumption (4.3)), so Lemmas 4.3 and 4.6 apply to L t v as well. Indeed,

L t v f pvq " ˆRd pf pv 1 q ´f pvqqKpv 1 , vq dv 1 `ˆˆR d Kpv 1 , vq ´Kpv, v 1 q dv 1 ˙f pvq.
The following interpolation is probably classical. We prove it using Littlewood-Paley theory. Since we have already obtained the estimate for E sym in Lemma 4.2, we are only left to prove the estimate for E skew . In the case s P p0, 1{2q, the proof below gives the estimate for E right away. For s P r1{2, 1q the proof below applies to E skew only.

Let ∆ i be the Littlewood-Paley projectors. We use the convention that all low modes are enclosed in ∆ 0 . That is f " ř 8 i"0 ∆ i f , with the index i being non-negative. We use the fact that for any s ě 0,

}∆ i f } H s « 2 is }∆ i f } L 2
Moreover, from Lemma 4.3, if s P p0, 1{2q,

}L v ∆ i f } L 2 À }∆ i f } 1´2s 1`2s L 2 }∆ i f } 4s 1`2s 9 H s`1{2 À 2 si }∆ i f } H s From Lemma 4.6, if s P r1{2, 1q, }L v ∆ i f } L 2 À }∆ i f } 1´s L 2 }∆ i f } s H 2 `}∆ i f } H 1 À 2 si }∆ i f } H s The same estimates hold for L t v in the place of L v . Therefore, Epf, gq " ÿ ij Ep∆ i f, ∆ j gq, " ÿ iďj xL v ∆ i f, ∆ j gy `ÿ iąj x∆ i f, L t ∆ j gy, À ÿ i,j 2 ´s|i´j| }∆ i f } H s }∆ j g} H s , " 8 ÿ k"0 2 ´sk 8 ÿ i"0 }∆ i f } H s }∆ i`k g} H s `}∆ i`k f } H s }∆ i g} H s , ď 8 ÿ k"0 2 ´sk`1 ˜8 ÿ i"0 }∆ i f } 2 H s ¸1{2 ˜8 ÿ i"0 }∆ j g} 2 H s ¸1{2 , À }f } H s }g} H s .
The proof is now complete.

4.2.

A generalized cancellation lemma. As a preparation for the next subsection, we prove the following generalized cancellation lemma.

Lemma 4.7 (Generalized cancellation). Let K be a kernel satisfying (4.2); if s ě 1{2, we also assume that K satisfies (4.4). Let ϕ be a bounded C 2 function. Then

P V ˆRd pϕpv 1 q ´ϕpvqqrKpv, v 1 q ´Kpv 1 , vqs dv 1 ď C}ϕ} C2 ,
for some constant C depending on Λ and dimension.

Proof. The proof is a direct computation. We estimate the tail of the integral using (4.2) together with the boundedness of ϕ. Then, we estimate the integral in B 1 using (4.4) and the smoothness of ϕ. We write the proof for the case 2s ě 1 first, and later indicate its simplification when 2s ă 1.

P V ˆpϕpv 1 q ´ϕpvqqrKpv, v 1 q ´Kpv 1 , vqs dv 1 ďP V ˆB1 pϕpv 1 q ´ϕpvqqrKpv, v 1 q ´Kpv 1 , vqs dv 1 `CΛ}ϕ} L 8 , ďP V ˆB1 pv 1 ´vq∇ϕpvqrKpv, v 1 q ´Kpv 1 , vqs `}D 2 ϕ} 8 |v ´v1 | 2 |Kpv, v 1 q ´Kpv 1 , vq| dv 1 `CΛ}ϕ} L 8 , ďCΛ}ϕ} C 2 .
For the last inequality we used that thanks to (4.2), ˆB1

|v 1 ´v| 2 rKpv, v 1 q ´Kpv 1 , vqs dv 1 À Λ, and thanks to (4.4),

P V ˆB1 pv 1 ´vqrKpv, v 1 q ´Kpv 1 , vqs dv 1 ď Λ.
When s ă 1{2, we do not need to use (4.4). We simply use (4.2) to get ˆB1

pϕpv 1 q ´ϕpvqqrKpv, v 1 q ´Kpv 1 , vqs dv 1 ď ˆB1 |v ´v1 |rϕs C 1 rKpv, v 1 q ´Kpv 1 , vqs dv 1 , ď CΛrϕs C 1 .
Remark 4.8. Lemma 4.7 tells us in particular that when K is anti-symmetric, the operator L v f is well defined pointwise. The same cannot be said for a symmetric kernel of s ě 1{2. When K is a symmetric kernel assuming only (4.2), the value of L v f pvq is not necessarily defined pointwise, even if f is smooth. It is only through E sym that we can define L v as an operator from H s to H ´s.

4.3.

Estimate focusing on the smoothness of only one function. In this section we obtain an estimate for Epϕ, gq taking maximum advantage of the smoothness of ϕ, and not so much on the smoothness of g.

Lemma 4.9 (Second upper bound for E). Let K satisfy (4.2) and (4.3). If s ě 1{2, we also assume (4.4).

For any two functions g P H s pR d q X L 1 pR d q and ϕ P C 2 with g ě 0 and any ε ą 0, we have

(4.8) Epϕ, gq ď ε}g} 2 9 H s `Cε ´1}ϕ} 2 C 1 |tv P R d : gpvq ą 0u| `C}ϕ} C 2 }g} L 1 .
Proof. Recall that E " E sym `Eskew . We estimate each term separately.

In order to estimate E sym , we apply the following elementary identity |gpvq ´gpv 1 q| ď pχ gą0 pvq `χgą0 pv 1 qq|gpvq ´gpv 1 q|;

we then get E sym pϕ, gq ď ¨|ϕpvq ´ϕpv 1 q|χ gą0 pvq|gpvq ´gpv 1 q|Kpv, v 1 q dv 1 dv, ď ε ¨pgpvq ´gpv 1 qq 2 Kpv, v 1 q dv 1 dv `p4εq ´1 ¨pϕpvq ´ϕpv 1 qq 2 χ gą0 pvqKpv, v 1 q dv 1 dv, " εE sym pg, gq `p4εq ´1 ˆχgą0 pvq ˆˆpϕpvq ´ϕpv 1 qq 2 Kpv, v 1 q dv 1 ˙dv, using Lemma 4.2 and the assumption (4.2),

ď εC}g} 2 9 H s `Cε ´1}ϕ} 2 C 1 ˆχgą0 dv.
As far as E skew is concerned, we first rewrite it as follows

E skew pϕ, gq " 1 4 ¨pϕpvq ´ϕpv 1 qqpgpvq `gpv 1 qqpKpv, v 1 q ´Kpv 1 , vqq dv 1 dv " 1 2 ¨pϕpvq ´ϕpv 1 qqgpvq `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 dv " 1 2
ˆgpvq " P V ˆRd pϕpvq ´ϕpv 1 qq `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 * dv, using Lemma 4.7, ď C}ϕ} C 2 ˆgpvq dv.
Combining the upper bounds for E sym and E skew , we conclude the proof. We have the following commutator estimate

L v rϕf s ´ϕL v f " h 1 `h2 , with }h 1 } L 8 pR d q À }ϕ} L 8 }f } L 8 pR d q dpD, R d zΩq ´2s , }h 1 } L 2 pR d zΩq À }ϕ} L 8 }f } L 2 pDq dpD, R d zΩq ´2s , }h 2 } L 2 pR d q À }ϕ} C 1 }f } L 2 pΩq .
Moreover, h 2 " 0 outside Ω. Whenever Ω " R d , we can consider dpD, R d zΩq " `8 and h 1 " 0.

Proof. From the formula (4.1), we get

Crϕ, f spvq :" L v rϕf spvq ´ϕpvqL v f pvq " ˆRd f pv 1 qpϕpv 1 q ´ϕpvqqKpv, v 1 q dv 1 .
Let r " dpD, R d zΩq{2, and let E " D `Br . Thus, we have D Ť E Ť Ω, with dpD, R d zEq " r and dpD, R d zΩq " r.

We define

h 1 pvq " ˆRd zBrpvq f pv 1 qpϕpv 1 q ´ϕpvqqKpv, v 1 q dv 1 , h 2 pvq " ˆBrpvq f pv 1 qpϕpv 1 q ´ϕpvqqKpv, v 1 q dv 1 .
From (4.2), for any value of v P R d , we have

|h 1 pvq| ď 2}f } L 8 }ϕ} L 8 Λr ´2s
which is the first inequality.

When v R D, we have ϕpvq " 0. Therefore, the integrand in Crϕ, f spvq is nonzero only for v 1 P D. We thus have for v R Ω Ą D,

h 1 pvq " ˆD f pv 1 qϕpv 1 qKpv, v 1 q dv 1 Therefore ˆRd zΩ h 1 pvq 2 dv " ˆRd zΩ ˆˆD f pv 1 qϕpv 1 qKpv, v 1 q dv 1 ˙2 dv, ď }ϕ} 2 L 8 ˆRd zΩ ˆˆD f pv 1 q 2 |Kpv, v 1 q| dv 1 ˙ˆˆD |Kpv, v 1 q| dv 1 ˙dv, using (4.2), ď }ϕ} 2 L 8 Λr ´2s ˆRd zΩ ˆD f pv 1 q 2 |Kpv, v 1 q| dv 1 dv, ď }ϕ} 2 L 8 Λr ´2s ˆD f pv 1 q 2 ˜ˆ|v´v 1 |ąr |Kpv, v 1 q| dv ¸dv " Λ 2 r ´4s }ϕ} 2 L 8 }f } 2 L 2 pDq .
This gives us the second inequality.

In order to estimate }h 2 } L 2 , we use Cauchy Schwarz.

}h 2 } 2 L 2 " ˆE ˜ˆBrpvq f pv 1 qpϕpv 1 q ´ϕpvqqKpv, v 1 q dv 1 ¸2 dv, ď ˆE ˜ˆBrpvq f pv 1 q 2 |ϕpv 1 q ´ϕpvq| |Kpv, v 1 q| dv 1 ¸˜ˆB r pvq |ϕpv 1 q ´ϕpvq| |Kpv, v 1 q| dv 1 ¸dv,
Since ϕ is bounded and C 1 , then (4.2) implies (note that s ă 1{2), ˆRd

|ϕpv 1 q ´ϕpvq| |Kpv, v 1 q| dv 1 À }ϕ} C 1 for every value of v P R d , ˆRd |ϕpv 1 q ´ϕpvq| |Kpv, v 1 q| dv À }ϕ} C 1
for every value of

v 1 P R d .
Therefore,

}h 2 } 2 L 2 À }ϕ} C 1 ˆE ˜ˆBrpvq f pv 1 q 2 |ϕpv 1 q ´ϕpvq| |Kpv, v 1 q| dv 1 ¸dv, À }ϕ} C 1 ˆΩ f pv 1 q 2 ˜ˆEXBrpv 1 q |ϕpv 1 q ´ϕpvq| |Kpv, v 1 q| dv ¸dv 1 , À }ϕ} 2 C 1 }f } 2 L 2 pΩq .
Lemma 4.11 (Commutator estimate for s P r1{2, 1q). Let us assume s P r1{2, 1q and that K satisfies (4.2) and (4.4). Let D be a closed set, and Ω open so that D Ť Ω Ă R d . Let ϕ be a smooth function supported in D and f P H s pΩq X L 8 pR d q. We have the following commutator estimate

L v rϕf s ´ϕL v f " h 1 `h2 `p´∆q s{2 h 3 , with }h 1 } L 8 pR d q À }ϕ} L 8 }f } L 8 pR d q pdpD, R d zΩq `dpv, Dqq ´2s , }h 1 } L 2 pR d zΩq À }ϕ} L 8 }f } L 2 pDq dpD, R d zΩq ´2s , }h 2 } L 2 pR d q À }ϕ} C 2 }f } H s pΩq , }h 3 } L 2 pR d q À }ϕ} C 2 }f } L 2 pΩq .
Moreover, h 2 " 0 outside Ω. Whenever Ω " R d , we can consider dpD, R d zΩq " `8 and h 1 " 0.

Proof. We define h 1 and h2 by the expressions of h 1 and h 2 in the proof of Lemma 4.10. The estimates for h 1 follow identically. We will split h2 " h 2 `p´∆q s{2 h 3 , and need to prove the estimate for each term. Note that, by construction, h2 pvq " 0 for any v R E.

Let us write K as the sum of its symmetric plus antisymmetric parts: K " K s `Ka . We start by estimating the antisymmetric contribution.

Because of Lemma 4.7, we have that

}L a v ϕ} L 8 À }ϕ} C 2 . Then } ha 2 } L 2 pΩq :" ˇˇˇˇˆB r pvq f pv 1 qpϕpv 1 q ´ϕpvqqK a pv, v 1 q dv 1 ˇˇˇˇL 2 pΩq , ď ˇˇˇˇˆB r pvq pf pv 1 q ´f pvqqpϕpv 1 q ´ϕpvqqK a pv, v 1 q dv 1 ˇˇˇˇL 2 pΩq `C}ϕ} C 2 }f } L 2 pΩq .
With respect to the first term, we apply Cauchy-Schwarz and Lemma 4.2 to obtain ˇˇˇˆB r pvq pf pv 1 q ´f pvqqpϕpv 1 q ´ϕpvqqK a pv, v 1 q dv 1 ˇˇˇ2

L 2 pΩq ď ˆE ˜ˆBrpvq pf pv 1 q ´f pvqq 2 |K a pv, v 1 q| dv 1 ¸˜ˆB r pvq pϕpv 1 q ´ϕpvqq 2 |K a pv, v 1 q| dv 1 ¸dv, À }ϕ} C 2 ¨ΩˆΩ pf pv 1 q ´f pvqq 2 |K a pv, v 1 q| dv 1 dv À }ϕ} C 2 }f } 2 9
H s pΩq .
Therefore, we conclude the estimate for the antisymmetric contribution } ha 2 } L 2 pR d q ď C}ϕ} C 2 }f } H s pΩq . Now we need to analyse the contribution of K s to h2 , which we call hs

2 . We estimate it by duality. Let g P H s pR d q, recall that h2 s is supported in E and consider ˆE hs 2 pvqgpvq dv " ˆE ˆBrpvq gpvqf pv 1 qpϕpv 1 q ´ϕpvqqK s pv, v 1 q dv 1 dv, " 1 2 ˆE f pvq ˜ˆBrpvq pgpvq ´gpv 1 qqpϕpv 1 q ´ϕpvqqK s pv, v 1 q dv 1 dv ģpvq ˜ˆBrpvq pf pv 1 q ´f pvqqpϕpv 1 q ´ϕpvqqK s pv, v 1 q dv 1 dv ¸dv.

Applying the Cauchy-Schwarz inequality and Lemma 4.2 as above, we get ˆΩ hs

2 pvqgpvq dv À }ϕ} C 2 `}f } 9 H s }g} L 2 `}f } L 2 }g} 9 H s ˘.
Therefore, hs 2 can be written as a sum ĥs

2 `p´∆qh 3 with } ĥs 2 } L 2 pR d q À }ϕ} C 2 }f } H s pΩq and }h 3 } L 2 pR d q À }ϕ} C 2 }f } L 2 pΩq .
We finish the proof by letting h 2 " ha 2 `ĥ s 2 .

Reduction to global kernels and weak solutions

The assumptions (1.3), (1.5), (1.6) and (1.7) are given in terms of values of v P B R only. It is natural that if we consider the equation (1.2) to hold for v P B 1 and we intend to prove local regularity estimates, it should be useless to make assumptions for Kpv, v 1 q when v R B 2 . It is confortable for the proofs of a few lemmas (in particular the results in Section 4 above and Lemma 6.1 below) to have a kernel that is globally defined and satisfies all these assumptions for all values of v and v 1 . In this section we explain how to extend a kernel to the full space in order to have that. ' Kpv, v 1 q " Kpv, v 1 q whenever v and v 1 belong to B 2 R{3 . Moreover Kpv, v 1 q ě 0 for all v, v 1 P R d and for all v P B R{2 ,

(5.1) ˆRd |Kpv, v 1 q ´Kpv, v 1 q| dv 1 ď CΛ.

' For any function f P H s pR d q,

(5.2) λ}f } 2 9 H s ď ´ˆR d Lv f pvq f pvq dv `Λ}f } 2 L 2 pR d q .
Here Lv is the integro-differential operator corresponding to the kernel K. ' The assumptions (4.2), (4.3) and (4.4) hold for K with a constant CΛ instead of Λ, where C depends on s, R, and dimension only.

Proof. Let η : R d Ñ r0, 1s be a smooth radial function so that η " 1 in B 3 R{4 and η " 0 outside B 7 R{8 . We define Kpv, v 1 q " ηpvqηpv 1 qKpv, v 1 q `Λp1 ´ηpvqηpv 1 qq|v ´v1 | ´d´2s .

Note that even though Kpv, v 1 q is not defined when v R B R, since we have the factor ηpvq " 0 there, there is no ambiguity in the definition of Kpv, v 1 q.

The first item in the Proposition is obvious by construction. We start by checking (4.2). For any v P R d and r ą 0, we have ˆRd zBrpvq Kpv, v 1 q dv 1 " ˆRd zBrpvq ηpvqηpv 1 qKpv, v 1 q `Λp1 ´ηpvqηpv 1 qq|v ´v1 | ´d´2s dv 1 , ď ηpvq ˆRd zBrpvq Kpv, v 1 q dv 1 `p1 ´ηpvqqΛ ˆRd zBrpvq |v ´v1 | ´d´2s dv 1 À Λ.

For any v 1 P R d and r ą 0, we do almost the same computation ˆRd zBrpv 1 q Kpv, v 1 q dv " ˆRd zBrpvq ηpvqηpv 1 qKpv, v 1 q `Λp1 ´ηpvqηpv 1 qq|v ´v1 | ´d´2s dv 1 , ď ηpv 1 q ˆB R zBrpv 1 q Kpv, v 1 q dv `p1 ´ηpv 1 qqΛ ˆRd zBrpv 1 q |v ´v1 | ´d´2s dv À Λ.

This justifies (4.2). We now verify (4.4). This only applies when 2s ě 1. Given any r ą 0, we compute ˇˇˇP V ˆBrpvq pv ´v1 q `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 ˇˇ" ηpvq ˇˇˇˇP V ˆBrpvqXB R pv ´v1 qηpv 1 q `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 ˇˇˇˇ, ď ηpvq ˜ηpvqΛp1 `minpr, p R ´|v|qq 1´2s q `ˇˇˇˇˆB r pvqXB R pv ´v1 qpηpv 1 q ´ηpvqq `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 ˇˇˇˇ¸, Note that ηpvq " 0 if |v| ą 7 R{8, therefore ηpvqp R ´|v|q 1´2s ď C for some constant depending on R.

ď Cηpvq ˜Λp1 `r1´2s q `ˆBrpvqXB R |v ´v1 | 2 |Kpv, v 1 q ´Kpv 1 , vq| dv 1 ¸, ď CΛp1 `r1´2s q.
This proves (4.4). We now move on to (4.3). When s P r0, 1{2q the proof is similar to the computation above for (4.4). Indeed,

ˇˇˇP V ˆRd ´Kpv, v 1 q ´Kpv 1 , vq ¯dv 1 ˇˇˇ" ηpvq ˇˇˇˇP V ˆB R ηpv 1 q `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 ˇˇˇˇ, ď ηpvq ˜Λ `ˇˇˇˇP V ˆB R pηpv 1 q ´ηpvqq `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 ˇˇˇˇ¸, ď ηpvq ˜Λ `C ˆB R |v 1 ´v||Kpv, v 1 q ´Kpv 1 , vq|q dv 1 ¸ď CΛηpvq.
The last inequality follows from (1.5) because s P r0, 1{2q. In the case s P r1{2, 1q, we modify the estimate of the last line. We have

ˇˇˇP V ˆRd ´Kpv, v 1 q ´Kpv 1 , vq ¯dv 1 ˇˇˇď ηpvq ˜Λ `ˇˇˇˇP V ˆB R pηpv 1 q ´ηpvqq `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 ˇˇˇˇ¸, ď ηpvq ˆΛ `ˇˇˇˇ∇ ηpvq ¨P V ˆB R pv 1 ´vq `Kpv, v 1 q ´Kpv 1 , vq ˘dv 1 ˇˇˇB R C|v 1 ´v| 2 |Kpv, v 1 q ´Kpv 1 , vq| dv 1 ˙ď CΛηpvq.
For the last inequality, we apply (1.5) and (1.7).

We now justify (5.2). We see that

´ˆL v f pvqf pvq dv " Ẽsym pf, f q `Ẽ skew pf, f q,
" ¨|f pvq ´f pv 1 q| 2 Kpv, v 1 q dv dv 1 `ˆf pvq 2 ˆP V ˆp Kpv, v 1 q ´Kpv 1 , vqq dv 1 ˙dv,

ě ¨R2d |f pvq ´f pv 1 q| 2 Kpv, v 1 q dv dv 1 ´Λ}f } 2 L 2 .
Let 3 R{4 ă r 1 ă r 2 ă R so that ηpvq ă 2{3 if |v| ą r 1 and ηpvq ą 1{3 if |v| ă r 2 . The first term in the definition of Kpv, v 1 q of bounded below by Kpv, v 1 q{9 when both v and v 1 belong to B r2 . When v and v 1 do not belong to B r1 , we can estimate Kpv, v 1 q from below by Λ|v ´v1 | ´d´2s {3. If v and v 1 belong to B r2 zB r1 , the value of Kpv, v 1 q is bounded below by the sum of the two previous terms. We have, ¨R2d

|f pvq ´f pv 1 q| 2 Kpv, v 1 q dv dv 1 ě 1 9 ¨Br 2 ˆBr 2 |f pvq ´f pv 1 q| 2 Kpv, v 1 q dv dv 1 `Λ 3 ¨R2d zpBr 1 ˆBr 1 q |f pvq ´f pv 1 q| 2 |v ´v1 | ´d´2s dv dv 1 .

We need to estimate the first term using (1.3). Let ϕ be a smooth radial function so that ϕ " 1 in B r1 and ϕ " 0 outside B r2 . Using Lemma 4.7 after some arithmetic manipulations, we see that

Epϕf, ϕf q " ¨B R ˆB R ϕpvqϕpv 1 qpf pvq ´f pv 1 qq 2 Kpv, v 1 q dv 1 dv `2 ˆB R f pvq 2 ϕpvq ˜P V ˆB R pϕpvq ´ϕpv 1 qqpKpv, v 1 q ´Kpv 1 , vqq dv 1 ¸dv, `2 ˆB R ϕpvq 2 f pvq 2 ˜ˆR d zB R Kpv, v 1 q dv 1 ¸dv, ď ¨Br 2 ˆBr 2 |f pvq ´f pv 1 q| 2 Kpv, v 1 q dv dv 1 `C}f } 2 L 2 .
Combining the last three displayed inequalities with (1.3), we obtain

Epf, f q " ´ˆL v f pvqf pvq dv, ě ¨R2d |f pvq ´f pv 1 q| 2 Kpv, v 1 q dv dv 1 ´Λ}f } 2 L 2 , ě 1 9 ¨Br 2 ˆBr 2 |f pvq ´f pv 1 q| 2 Kpv, v 1 q dv dv 1 `Λ 3 ¨R2d zpBr 1 ˆBr 1 q |f pvq ´f pv 1 q| 2 |v ´v1 | ´d´2s dv dv 1 ´Λ}f } 2 L 2 , ě 1 9 Epϕf, ϕf q ´C}f } 2 L 2 `Λ 3 ¨R2d zpBr 1 ˆBr 1 q |f pvq ´f pv 1 q| 2 |v ´v1 | ´d´2s dv dv 1 , ě minpλ{9, Λ{3q ˆ¨R d ˆRd |f pvq ´f pv 1 q| 2 |v ´v1 | ´d´2s dv dv 1 ˙´C}f } 2 L 2 .
The extended kernel K can be used to reduce many results to the case of globally defined kernels. The following results, which we will need later, are examples.

Corollary 5.2 (The operator L v maps H s into H ´s). Assume K : B R ˆRd Ñ R is a non-negative kernel that satisfies (1.5) and (1.6); if s ě 1{2, we also assume that K satisfies (1.7). For any f P H s pR d q and g P H s pR d q supported in B R{2 ,

(5.3) Epf, gq " ´ˆB R{2 L v f pvqgpvq dv ď Λ}f } H s pR d q }g} H s pR d q
for some positive constant Λ depending on dimension.

Corollary 5.3 (Second upper bound for E). Let K satisfy (1.5), (1.6). If s ě 1{2, we also assume (1.7).

For any two functions g P H s pB R{2 q X L 1 pR d q and ϕ P C 2 , both compactly supported in B R{2 , with g ě 0 and any ε ą 0, we have The justifications of the two lemmas above are almost identical. We explain the latter one.

(
Proof of Corollary 5.5. Let K be the extended kernel according to Proposition 5.1. Applying Lemma 4.11, we obtain that Lrϕf s ´ϕ Lf " h1 `h2 `p´∆q s{2 h 3 .

For this corollary, we want to replace ϕ Lf by ϕLf . Since ϕ is supported in D, these two expressions only differ when v P D. In this case, we have |ϕpvq Lf pvq ´ϕpvqLf pvq| " ˇˇˇϕ pvq ˆRd rf pvq ´f pv 1 qs ´Kpv, v 1 q ´Kpv, v 1 q ¯dv ˇˇˇď Cϕpvq}f } L 8 δ ´2s . This difference is absorbed by the term h 1 by setting h 1 " h1 `ϕpvq Lf pvq ´ϕpvqLf pvq.

5.2.

Definition of weak solutions. We now discuss the concept of weak solutions. In order to justify the definition we are going to give below, we start with the following preparatory lemma.

Lemma 5.6 (The bilinear form E in the local case). Let supp ϕ Ť B R{2 and ϕ P H s pR d q. Assume K satisfies (1.5), (1.6) and (1.7). Then for all f P L 8 pR d zB R{2 q `Hs pR d q Epf, ϕq ď C}f } L 8 pR d zB R{2 q`H s pR d q }ϕ} H s pR d q ,

where the constant C depends on Λ, d, s and the support of ϕ. Here,

}f } L 8 pR d zB R{2 q`H s pR d q " inf ! }f 1 } L 8 pR d zB R{2 q `}f 2 } H s pR d q : f " f 1 `f2 and f 1 " 0 in B R{2
) .

More precisely, the inequality holds for smooth functions, and therefore it allows the bilinear form to be extended to the appropriate spaces of functions.

Note that the restriction f P L 8 pR d zB R{2 q `Hs pR d q imposes some fractional Sobolev regularity in B R{2 but not so much outside. In particular, any function f P H s pB R{2`ε q X L 8 pR d zB R{2`ε q is in this space.

Proof. As mentioned above, we assume for the proof that both f and ϕ are smooth. Afterwards, the inequality is obtained by density when f P L 8 pR d zB R{2 q `Hs pR d q and ϕ P H s pR d q is compactly supported in B R{2

Let f " f 1 `f2 as in the definition of the norm in L 8 pR d zB R{2 q `Hs pR d q. Applying Corollary 5.2, |Epf 2 , ϕq| À }f 2 } H s }ϕ} H s . We are left to compute Epf 1 , ϕq. We have Epf 1 , ϕq " lim εÑ0 ¨|v 1 ´v|ąε pf 1 pv 1 q ´f1 pvqqϕpvqKpv, v 1 q dv 1 dv,

" lim εÑ0 ˆsupp ϕ ˜ˆR d zBεpvq f 1 pv 1 qKpv, v 1 q dv 1 ¸ϕpvq dv, " ˆsupp ϕ ˜ˆR d zB δ pvq f 1 pv 1 qKpv, v 1 q dv 1 ¸ϕpvq dv.
Here δ is the distance between the support of ϕ and R d zB R{2 .

ď Λδ ´2s }f 1 } L 8 ˆˆsupp ϕ ϕpvq dv ˙ď CΛδ ´2s }f 1 } L 8 }ϕ} H s .

Another way to describe Lemma 5.6 is that L v is a bounded operator from L 8 pR d zB R{2 q `Hs pR d q to H ´spB Rq. We will use this to define the concept of weak solution.

Definition 5.7 (Weak solutions). Assume K satisfies (1.5), (1.6) and (1.7). Given the cylinder Q " p0, T q ˆBp R{2q 1`2s ˆB R{2 , We say that a function f : r0, T s ˆBp R{2q

1`2s ˆRd Ñ R is a subsolution of (1.2) in the cylinder Q if f P C 0 pp0, T q, L 2 pB p R{2q 1`2s ˆB R{2 qq X L 2 pp0, T q ˆBp R{2q 1`2s , L 8 pR d zB R{2 q `Hs pR d qq, f t `v ¨∇x f P L 2 pp0, T q ˆBp R{2q 1`2s , H ´spB R{2 qq,
and for all non-negative test function ϕ P L 2 pp0, T q ˆBp R{2q 1`2s , H s pR d qq so that for every t and x, ϕpt, x¨q is compactly supported in B R{2 , (5.5) ˚pf t `v ¨∇x f qϕ `¨Epf, ϕq ´˚hϕ ď 0.

A function f is a supersolution of (1.2) in Q if ´f is a subsolution of (1.2) in Q. A function f is a solution of (1.2) in Q if it
is both a sub-and a supersolution. Remark 5.8. Assuming that f P C 0 pp0, T q, L 2 pB p R{2q 1`2s ˆB R{2 qq and f P L 2 pp0, T q ˆBp R{2q 1`2s , H s pB R{2 qq is rather natural in view of the energy estimates one can easily get from the coercivity assumption.

Note that the bilinear form ˜Epf, ϕq in (5.5) is well defined because of Lemma 5.6.

The first lemma of De Giorgi

This section is devoted to the first intermediate result in the proof of the weak Harnack inequality. It is referred to as the first lemma of De Giorgi. It consists in controlling a local pointwise bound in the interior of a cylinder by an integral quantity in the cylinder. Its proof (see Subsection 6.2) relies on a global energy estimate (See Subsection 6.1).

For degenerate integral equations, the situation is different than for equations of second order. It is not true that the maximum of a nonnegative subsolution can be bounded by above by a multiple of its L 2 norm. One needs to impose an extra global restriction (in this case we assume 0 ď f ď 1 globally). This is because of nonlocal effects, since the positive values of the function outside of the domain of the equation may pull the maximum upwards. The strong Harnack inequality fails in general. This fact is well documented and there are counterexamples (see [START_REF] Bogdan | Harnack's inequality for stable Lévy processes[END_REF]).

6.1. Energy estimates. The proof of the first lemma of De Giorgi relies on an iteration of energy estimates applied to certain truncated functions. For kinetic equations, the energy estimate naturally gives us some regularization with respect to the v variable. We use the fractional Kolmogorov equation to translate this regularization in v to a higher degree of integrability of the function. Lemma 6.1 (Global energy inequality and gain of integrability). Assume K, and its corresponding operator Lv , satisfy (5.2), (4.2), (4.3) and (4.4). Let G ě 0 be a weak sub-solution of (6.1)

# pB t `v ¨∇x qG ´L v G ď H 1 `p´∆q s{2 v H 2 in r0, T s ˆR2d , Gp0, x, vq " G 0 px, vq in R 2d
with a source terms H 1 , H 2 P L 2 pr0, T s ˆR2d q. Then, (

sup τ Pr0,T s }Gpτ q} 2 L 2 pR 2d q `}G} 2 L 2 pr0,T sˆR d , 6.2)
H s pR d qq ď C ´}G 0 } 2 L 2 pR 2d q `}H 1 } 2 L 2 pr0,T sˆR 2d q `}H 2 } 2 L 2 pr0,T sˆR 2d q ¯. 9
Moreover, there exists p ą 2 (only depending on dimension and s) such that

(6.3) }G} 2 L p pr0,T sˆR 2d q ď C ´}G 0 } 2 L 2 pR 2d q `}H 1 } 2 L 2 pr0,T sˆR 2d q `}H 2 } 2 L 2 pr0,T sˆR 2d q ¯,
for some constant C depending on λ, Λ, d, p, s, and T .

Proof. Multiplying the equation by G and integrating on the time interval r0, τ s for τ P r0, T s, we get

1 2 }Gpτ q} 2 L 2 pR 2d q `ˆτ 0 ˆRd EpG, Gq dx dt ď 1 2 }G 0 } 2 L 2 pR 2d q `ˆr0,T sˆR 2d pH 1 `p´∆q s{2 v H 2 qG.
Using (5.2) from Proposition 5.1, we have (6.4) 1 2 }Gpτ q} 2

L 2 pR 2d q `ˆτ 0 ˆRd λ}G} 2 9 H s ´Λ}G} 2 L 2 dx dt ď 1 2 }G 0 } 2 L 2 pR 2d q `ˆT 0 }H 1 ptq} L 2 }Gptq} L 2 `}H 2 ptq} L 2 }Gptq} 9 H s dt.
Therefore,

1 2 }Gpτ q} 2 L 2 pR 2d q `ˆτ 0 ˆRd ´Λ 2 }G} 2 L 2 dx dt ď 1 2 }G 0 } 2 L 2 pR 2d q `C ˆT 0 }H 1 ptq} 2 L 2 `}H 2 ptq} 2 L 2 dt.
Integrating against expp´Λτ {2q with respect to τ yields

}G} 2 L 2 pr0,T sˆR 2d q ď C ´}G 0 } 2 L 2 pR 2d q `}H 1 } 2 L 2 pr0,T sˆR 2d q `}H 2 } 2 L 2 pr0,T sˆR 2d q ¯.
Using this information back into (6.4), we finally get

(6.5) sup τ Pr0,T s }Gpτ q} 2 L 2 pR 2d q `}G} 2 L 2 t,x 9 H s v prc,bsˆR 2d q ď C ´}G 0 } 2 L 2 pR 2d q `}H 1 } 2 L 2 pr0,T sˆR 2d q `}H 2 } 2 L 2 pr0,T sˆR 2d q ¯.
The function G is also a subsolution of the fractional Kolmogorov equation with an appropriate right hand side

G t `v ¨∇x G `p´∆q s G ď p´∆q s G `L v G `H1 `p´∆q s{2 H 2 .
Thus, G is smaller or equal to the exact solution of this equation. Theorem 4.1 ensures that Lv G P L 2 pr0, T sR d , H ´spR d qq. We then can apply Proposition 2.2 to G with h "

H 1 `p´∆q s{2 H 2 `L v G `p´∆q s G so that }h} L 2 pr0,T sˆR d ,H ´spR d qq ď }H 1 } L 2 `}H 2 } L 2 `C}G} L 2 t,x 9
H s v pr0,T sˆR 2d q . and get (6.3).

Let us analyze a localized version of the energy dissipation. Lemma 6.2 (Local energy dissipation). Let f be a subsolution of (1.2) in r0, T s ˆBR 1`2s ˆBR with h " 0. Assume 0 ď f ď 1 almost everywhere in r0, T s ˆBR 1`2s ˆRd . Assume K satisfies (1.3), (1.5), (1.6) and (1.7) with R " 2R. Then, for any δ P p0, 1q, we have

(6.6) sup tPr0,T s ¨BpR´δq 1`2s ˆBR´δ f pt, x, vq 2 dv dx `ˆT 0 ˆBpR´δq 1`2s }f } 2 H s pB R´δ q dx dt ď ¨BR 1`2s ˆBR f p0, x, vq 2 dv dx `Cδ ´2|tf ą 0u X r0, T s ˆBR 1`2s ˆBR |.
Remark 6.3. The factor in δ ´2 can be improved in terms of s (probably to δ ´2s ). The optimal power is irrelevant for the rest of our proof.

Proof. Let ϕ : R 2d Ñ r0, 1s be C 8 , supported in B R 1`2s ˆBR , so that ϕ " 1 in B pR´δq We used the fact that ∇ x pg 2 q " 2gp´∇ x ϕ `∇x f q. Remarking that Epg, gq ď 0 and using (1.3) and (5.4) from Corollary 5.3 yields for any t 0 P r0, T s,

1 2 ¨g2 pt 0 , x, vq dv dx `λ ˆt0 0 ˆ}g} 2 H s dx dt ď ¨g2 p0, x, vq dv dx `ε ˆt0 0 ˆ}g} 2 H s dx dt `Cε ´1rϕs 2 C 1 |tg ą 0u X t0 ď t ď t 0 u| `C}ϕ} C 2 }g} L 1 `C}g} 2 L 2 `ˆt0 0 ¨gpB t ϕ `v ¨∇x ϕq dv dx dt.
Recall that }ϕ} C 1 À δ ´1 and }ϕ} C 2 À δ ´2. Also gpt, x, vq P r0, 1s for all pt, x, vq, therefore }g} L 1 and }g} 2

L 2
are both bounded by |tg ą 0u| ď |tf ą 0u X r0, T s ˆBR

H s dx dt ``ε ´1δ ´2 `δ´2 `2˘| tf ą 0u X r0, T s ˆBR 1`2s ˆBR |.
Note that g " f in B pR´δq 1`2s ˆBR´δ , g ď f everywhere, and g " 0 outside of B R 1`2s ˆBR . We thus conclude the proof picking ε ą 0 small. Lemma 6.4 (Local gain of integrability). Let f be a subsolution of (1.2) in r0, T s ˆBR 1`2s ˆBR with h " 0. Assume 0 ď f ď 1 almost everywhere in r0, T s ˆBR 1`2s ˆRd . Assume K satisfies (1.3), (1.5), (1.6) and (1.7) with R " 2R. Then for any δ P p0, 1q and δ ă R, (6.7)

˜ˆT 0 ¨BpR´δq 1`2s ˆBR´δ f p dt dv dx ¸2{p ď δ ´2 ˆBR 1`2s ˆBR f p0, x, vq 2 dv dx `Cδ ´4 |tf ą 0u X pr0, T s ˆBR 1`2s ˆBR q|
where p ą 2 is some universal constant (explicit).

Remark 6.5. The exponents in the factors δ ´2 and δ ´4 are most certainly not optimal. This is not important for the rest of our proof.

Proof. Let us start by the following simple observation. Wherever f pt, x, vq " 0, we have f t `v ¨∇x f " 0 (a.e.) and L v f ě 0. In particular, the following equation also holds and contains slightly more information than (1.2).

(6.8)

f t `v ¨∇x f ´Lv f ď ´pL v f qχ tf "0u " ´ˆˆR d f pv 1 qKpv, v 1 q dv 1 ˙χtf"0u .
Let us call

N :" δ ´2 ˆBR 1`2s ˆBR f p0, x, vq 2 dv dx `Cδ ´4 |tf ą 0u X pr0, T s ˆBR 1`2s ˆBR q| .
From Lemma 6.2, we know that

ˆT 0 ˆBpR´δ{2q 1`2s }f } 2 H s pB R´δ{2 q dx dt ď δ 2 N.
Let ϕ : R 2d Ñ r0, 1s be C 8 , supported in B pR´δ{2q 1`2s ˆBR´δ{2 , so that ϕ " 1 in B pR´δq 1`2s ˆBR´δ . It is not hard to check that we can construct such ϕ with }Dϕ} L 8 À δ ´1 and }D 2 ϕ} L 8 À δ ´2.

Let us analyse what equation the funtion g " ϕf satisfies. Combining Corollaries 5.4 and 5.5 with (6.8), we have rB t `v ¨∇x ´L v sg ď f pv ¨∇x ϕq ´ϕpL v f qχ tf "0u ´h1 ´h2 ´p´∆q s{2 h 3 in r0, T s ˆBpR´δ{2q 1`2s ˆBR´δ{2 .

We want to verify that the right hand side belongs to L 2 pr0, T s ˆRd , H ´spR d qq with norm bounded above by N . Following the proofs of Lemmas 4.10 and 4.11 and Corollaries 5.4 and 5.5, we have

h 1 " ˆRd zB δ{2 pvq ϕpvqf pvqrKpv, v 1 q ´Kpv, v 1 qs `f pv 1 qpϕpv 1 qKpv, v 1 q ´ϕpvq Kpv, v 1 qq dv 1 .
Therefore, at the points in tf " 0u we have

´ϕpL v f qχ tf "0u ´h1 ď ˆRd zB δ{2 pvq f pv 1 q ´pϕpvq ´1qKpv, v 1 q ´ϕpv 1 q Kpv, v 1 q ¯dv 1 ď 0.
This allows us to simplify the equation to (6.9) rB t `v ¨∇x ´Lx v sg ď f pv ¨∇x ϕq ´h1 χ tf ą0u ´h2 ´p´∆q s{2 h 3 in r0, T s ˆBpR´δ{2q 1`2s ˆBR´δ{2 .

Corollaries 5.4 and 5.5 tell us that

}h 2 } L 2 , }h 3 } L 2 pr0,T sˆR d ,L 2 pR d qq À δ ´2}f } L 2 pr0,T sˆB R 1`2s ,H s pB R qq ď N.
Since pv ¨∇x ϕq is bounded and supported in B pR´δ{2q 1`2s ˆBR´δ{2 , and 0 ď f ď 1, we clearly have }f pv ¨∇x ϕq} L 2 ď N . Likewise }h 1 χ tf ą0u } L 2 ď N .

We conclude the proof applying Lemma 6.1 to (6.9).

6.2. De Giorgi's iteration. This subsection is devoted to the proof of the following lemma.

Lemma 6.6 (First lemma of De Giorgi). Let Q " r´τ, 0s ˆBR 1`2s 1 ˆBR1 and Q " r´τ , 0s ˆBR 1`2s 2 ˆBR2 with 0 ă τ ă τ and R 1 ď R 2 . There exists ε 0 ą 0 (depending on τ , τ , R 1 , R 2 , dimension, s, λ and Λ) such that for all supersolution f of f t `v ¨∇x f ´Lv f ě 0 in Q such that f ě 0 almost everywhere in r´τ , 0s ˆR2d and (6.10)

ˆQ p2 ´f q 2
`dt dv dx ď ε 0 , we have f ě 1 a.e. in Q. Proof of Lemma 6.6. Let us consider the sequences k " 1 `2´k , r k " R 1 `pR 2 ´R1 q2 ´k, t k " τ ´2´k pτ ´τ q.

We define

A k :" ˆ0 t k ¨Br 1`2s k ˆBr k p k ´f q 2 `dv dx dt.
The assumption (6.10) tells us that A 0 ď ε 0 . The strategy of De Giorgi's iteration is to prove that A k Ñ 0 as k Ñ 8 provided that ε 0 is sufficiently small. The conclusion clearly follows from that.

In order to prove that A k converges towards 0, we are going to prove that (6.11)

A k`1 ď C2 Ck A 1`ε k for some ε ą 0. We first pick t k`1 2 P rt k , t k`1 s such that ¨Br 1`2s k ˆBr k p k ´f pt k`1 2 , x, vqq 2 `dv dx ď 1 t k`1 ´tk ˆtk`1 t k ¨Br 1`2s k ˆBr k p k ´f q 2 `dv dx dt ď C2 k A k .
Note that p k`1 ´f q `is a subsolution with values in r0, 2s (in particular half of it takes values in r0, 1s). We then apply Lemma 6.4, and obtain the following inequality (note that k`1 ď k ) (6.12) ¨ˆ0

t k`1 2 ˆBr 1`2s k`1 ˆBr k`1 p k`1 ´f q p `dv dx dt '2{p ď C4 k A k `C16 k |tf ă k`1 u X prt k`1{2 , 0s ˆBr 1`2s k ˆBr k q|.
We now estimate |tf ă k`1 u X prt k`1{2 , 0s ˆBr k ˆBr k q| in terms of A k . We use Chebyshev inequality and get (6.13)

|tf ă k`1 u X prt k`1{2 , 0s ˆBr 1`2s k ˆBr k q| " |tp k ´f q `ą 2 ´k´1 u X prt k`1{2 , 0s ˆBr 1`2s k ˆBr k q|, ď 16 k`1 A k .
Combining (6.12) and (6.13), we get

¨ˆ0 t k`1 ¨Br 1`2s k`1 ˆBr k`1 p k`1 ´f q p `dt dv dx '2 p ď C2 8k A k
(we used that t k`1 2 ď t k`1 ď 0). We can now combine this estimate with (6.13) and get

A k`1 ď ¨ˆ0 t k`1 ¨Br 1`2s k`1 ˆBr k`1 p k`1 ´f q p `dt dv dx '2 p |tf ă k`1 u X prt k`1 , 0s ˆBr 1`2s k`1 ˆBr k`1 q| 1´2 p ď C2 8k A 1`2 ´p p k .
This yields (6.11) with ε " 2´p p ą 0. The proof is now complete.

Barrier functions for s ă 1{2

A remarkable difference between the range s ă 1{2 and s ě 1{2 is that, in the former, the integral expression in the definition of L v f pvq is computable pointwise for all smooth functions f provided that K satisfies the first line in (1.5). The reason for this is simply that from the Lipschitz continuity of f we get (7.1)

ˆB2rpvqzBrpvq |f pvq ´f pv 1 q|Kpv, v 1 q dv 1 ď r}f } Lip ˆB2rpvq zBrpvq Kpv, v 1 q dv 1 ď Λ}f } Lip r 1´2s .
This is summable for r " 2 ´k as k ranges accross the natural numbers when s ă 1{2.

If we assumed further than K is symmetric in the non-divergence sense Kpv, v `hq " Kpv, v ´hq, then the same analysis as above would hold for s P p0, 1q and f P C 1,1 (instead of f P Lip) and the results in this section could be extended to the full range s P p0, 1q. Note that the Boltzmann kernel satisfies this symmetry, but we do not make that assumption in Theorems 1.6 and 1.5.

We build barrier functions using crucially the assumption (1.4).

Lemma 7.1 (Existence of barriers). For any r ą 0, R ą 0, τ ą 0 and T ą 0, there exist constants θ ą 0 and R 1 ą 0, and a function ϕ : r0, 8q ˆRd ˆRd Ñ r0, 1s such that ' we have ϕ P C 1,1 pr0, 8q ˆR2d q; moreover, ϕ is smooth in the open set tϕ ą 0u; ' for any kernel Kpt, x, vq that satisfies (1.5) and (1.4) with R " R 1 , and all pt, x, vq P Ω Ă r0, 8qˆR 2d , we have ϕ t `v ¨∇x ϕ ´Lv ϕ ď 0 in Ω;

' at the initial time, the support of ϕp0, ¨, ¨q is contained in B r 1`2s ˆBr ; ' we have the following lower bound: ϕpt, x, vq ě θ if t P rτ, T s, x P B R 1`2s and v P B R ; ' the function ϕpt, x, vq vanishes if t P r0, T s and px, vq R B R 1`2s 1 ˆBR1 .

The function ϕ depends on r, R, τ , T , dimension d, λ, Λ and s (which should be in p0, 1q). The radius R 1 depends on r, R, τ , T , dimension d, and s (but not λ and Λ). Lemma 7.1 will be proved by the end of this section. We remark that we only use the first line in (1.5). It is convenient to define the extremal (Pucci type) operators which correspond to the supremum and infimum of all values of L v f pvq for any kernel K satisfying (1.5) and (1.4).

Let us say that a nonnegative kernel K : R d Ñ r0, `8s belongs to the class K 0 if

K P K 0 ô # ´Rd zBr Kpwq dw ď Λr ´2s ,
inf |e|"1 ´Br pw ¨eq 2 `K pwq dw ě λr 2´2s . Correspondingly, we define the extremal operators M `and M

´.

M `f pvq " sup "ˆR d pf pv 1 q ´f pvqqKpv 1 ´vq dv 1 : K P K 0 * , M ´f pvq " inf "ˆR d pf pv 1 q ´f pvqqKpv 1 ´vq dv 1 : K P K 0 * .
Note that the infimum and supremum are taken only with respect to a family of translation invariant linear operators, whose kernels depend only on v 1 ´v. However, the kernel which achieves the extremal value will be different at every value of v. Therefore, effectively, the operators M `f and M ´f correspond to the supremum and infimum value of L v f for all kernels Kpv, v 1 q satifying the first line in (1.5) and (1.4).

We start by pointing out a simple continuity property of M `and M

´.

Lemma 7.2. Let f and g be two bounded functions that are Lipschitz in B r pvq, then

|M ´f pvq ´M´g pvq| ď C r `}f ´g} L 8 pR d q `}f ´g} LippBrpvqq ˘.
The same holds for M `.

Remark 7.3. Note that the norm }f ´g} L 8 pR d q can be weighted. Indeed, the same estimate holds with }p1 `|v|q ´σ pf pvq ´gpvqq} L 8 pR d q instead provided that σ ă 2s.

Proof. It is enough to notice that each linear operator in the infimum of the definition of M ´satisfies the continuity estimate.

Corollary 7.4. If f n is a sequence of functions so that f n Ñ f uniformly in R d and f n Ñ f in LippΩq, then M `fn and M ´fn converge to M `f and M ´f uniformly in compact sets of Ω.

The following is perhaps not strictly a corollary of Lemma 7.2, since it requires a slightly sharper analysis (but standard and elementary). Since the operators M `and M ´are a supremum and infimum of linear ones, then they are also suband super-additive respectively. That means that for any f and g, M ´pf `gqpvq ě M ´f pvq `M´g pvq, M `pf `gqpvq ď M `f pvq `M`g pvq.

Lemma 7.6 (The function ϕ 1 ). Let ϕ 1 : R d Ñ r0, 1s be a nonnegative, radially symmetric function, so that ' tϕ 1 ą 0u " B 1 , ϕ 1 P C 2 pB 1 q, ϕ 1 " 1 in B 1{2 , and v ¨∇ϕ 1 pvq ď 0; ' ϕ 1 P C 2 pB 1 q and ϕ P C 1,1 pR d q; more precisely, there is a discontinuity of D 2 ϕ 1 on BB 1 so that lim rÑ1 ´D2 ϕ 1 preq " e b e for any |e| " 1. Then, there exist two constants δ ą 0 and θ ą 0 so that M ´ϕ1 pvq ě θ for any v P B 1 so that ϕ 1 pvq ă δ.

Remark 7.7. We can choose any function ϕ 1 pxq " Ψp|x|q with Ψ non-increasing in R, positive and C 2 in r0, 1s, supported in r0, 1s, Ψ " 1 in r0, 1{2s, and Ψ 1 p1q " 0 and Ψ 2 p1q " 1.

Proof. Since M ´ϕ1 is continuous in B 1 , it is enough to prove that M ´ϕ1 is strictly positive on BB 1 . From radial symmetry, we are left to show that M ´ϕ1 peq ą 0 for e " p1, 0, . . . , 0q.

Let ε ą 0. From the super-additivity of M ´, we have

M ´ϕ1 peq ě M ´pϕ 1 χ Bεpeq qpeq `M´p ϕ 1 χ R d zBεpeq qpeq.
For any K P K 0 , since K ě 0 and ϕ 1 ě 0, we have ˆRd

ppϕ 1 χ R d zBεpeq qpv 1 q ´pϕ 1 χ R d zBεpeq qpeqqqKpv 1 ´eq dv 1 " ˆRd zBεpeq ϕ 1 pv 1 qKpv 1 ´eq dv 1 ě 0.
Therefore M ´pϕ 1 χ R d zBεpeq qpeq ě 0. We now show that M ´pϕ 1 χ Bεpeq qpeq is bounded below for ε ą 0 small. Essentially this follows because ϕ 1 pv 1 q is approximately ppv 1 ´eq ¨p´eqq 2 `in B ε peq. Indeed, let the scaled function ϕ ε be

ϕ ε pwq " # ε ´2ϕ 1 pe `εwq if |w| ă 1, 0 if |w| ě 1. Thus M ´pϕ 1 χ Bεpeq qpeq " ε 2´2s M ´ϕε p0q.
From the definition of ϕ 1 ,we know that

ϕ ε pwq Ñ qpwq :" # p´w ¨eq 2 `for |w| ă 1, 0 for |w| ě 1,
uniformly in R d and also in LippB 1{2 q. Therefore, using Corollary 7.4,

M ´ϕε p0q Ñ M ´qp0q ě λ.
The last inequality comes from the non-degeneracy condition (1.4). Therefore, choosing ε sufficiently small,

M ´ϕ1 peq ě M ´pϕ 1 χ Bεpeq qpeq ě λ 2 ε 2´2s ą 0.
This concludes the proof.

Lemma 7.8 (The function ϕ 2 ). Let t 0 ą p0, 1q be arbitrary and ϕ 1 be a function as in Lemma 7.6. Let A " `5 `1 2s ˘. Let us define the function ϕ 2 : R 2d Ñ r0, 1s to be ϕ 2 px, vq :" ϕ 1 pxqϕ 1 pv ´Axq.

There exists a constant δ ą 0 so that if at some point px, vq, ϕ 2 px, vq ă δ, then

ˆ´1 ´1 2s ˙x ¨∇x ϕ 2 ´1 2s v ¨∇v ϕ 2 `t0 `v ¨∇x ϕ 2 ´Mv ϕ 2 ˘ď 0. (7.2)
Proof. Since min ϕ 2 " 0, then M v ϕ 2 ě 0 wherever ϕ 2 " 0. Thus, the inequality is trivial wherever ϕ 2 " 0. We are left to verify it at points where ϕ 2 ą 0. Note that this is a bounded set since there |x| ď 1 and |v| ď A|x| `1 ď A `1.

We expand the left hand of (7.2), in terms of ϕ 1 , x and v, as the sum of two terms T 1 `T2 , where

T 1 " ϕ 1 pxq " ∇ϕ 1 pv ´Axq ¨ˆA ˆ1 `1 2s ˙x ´ˆt 0 A `1 2s ˙v˙´t 0 M ´ϕ1 pv ´Axq * , T 2 " ϕ 1 pv ´Axq∇ϕ 1 pxq ¨"´ˆ1 `1 2s ˙x `t0 v * .
We first claim that (7.3) there exist δ 1 ą 0 such that T 1 ď 0 if ϕ 1 pv ´Axq ă δ 1 .

Using Lemma 7.6, we pick δ 1 sufficiently small so that M ´ϕ1 pv ´Axq ě θ whenever ϕ 1 pv ´Axq ă δ 1 . Thanks to the continuity of ∇ϕ 1 , we pick δ 1 smaller if necessary so that whenever ϕ 1 pv ´Axq ă δ 1 ,

∇ϕ 1 pv ´Axq ¨ˆA ˆ1 `1 2s ˙x ´ˆt 0 A `1 2s ˙v˙´t 0 M ´ϕ1 pv ´Axq ă ´t0 θ 2 .
Therefore, we have

T 1 ď ´t0 θϕ 1 pxq 2 whenever ϕ 1 pv ´Axq ă δ 1 .
In particular, (7.3) holds true. We next claim that

(7.4) there exist δ 2 ą 0 such that T 2 if ϕ 1 pxq ă δ 2 .
Because of the second derivative of ϕ 1 of BB 1 , we have the following expansion

∇ϕ 1 pxq " ´p1 ´|x|q x |x| `Opp1 ´|x|q 2 q.
Whenever ϕ 1 pv ´Axq ą 0, also v P B 1 pAxq, and therefore

∇ϕ 1 pxq ¨"´ˆ1 `1 2s ˙x `v* ď p1 ´|x|qp´4|x| `1q `Cp1 ´|x|q 2 ă ´p1 ´|x|q `Cp1 ´|x|q 2 .
Thus, T 2 ď ´ϕ1 pv ´Axqp1 ´|x|q{2 ď 0 whenever ϕ 1 pxq ă δ 2 and δ 2 is sufficiently small. In particular, (7.4) holds true.

In view of (7.3) and (7.4), T 1 `T2 ď 0 if ϕ 1 pv ´Axq ă δ 1 and ϕ 1 pxq ă δ 2 . Let us analyse the case ϕ 1 pxq ě δ 2 ; in this case consider ϕ 1 pv ´Axq ă δ 11 ă δ 1 so that

T 1 `T2 ď ´t0 δ 2 θ{2 `Cδ 11 .
Picking δ 11 sufficiently small (depending on the previous choice of δ 2 ), we assure T 1 `T2 ă 0 in this case.

We are left with the case ϕ 1 pv ´Axq ě δ 1 . In this case we have for ϕ 1 pxq ă δ 21 , T 1 `T2 ď Cϕ 1 pxq ´ϕ1 pv ´Axqp1 ´|x|q{2, ď Cδ 21 ´δ1 p1 ´|x|q{2 ă 0, provided |x| is sufficiently close to 1, which follows if ϕ 1 pxq ă δ 21 ă δ 2 with δ 21 sufficiently small. Finally, we finish the proof picking δ " δ 11 δ 21 to ensure that at least one of the three cases above holds.

Lemma 7.9 (The function ϕ 3 ). Let ϕ 2 be the function from Lemma 7.8 and t 0 ą 0. The function ϕ 3 pt, x, vq given by

ϕ 3 pt, x, vq " t p 0 pt `t0 q p ϕ 2 ˜ˆt 0 t `t0 ˙1`1 2s x, ˆt0 t `t0 ˙1 2s v ¸,
is a subsolution of the equation B t ϕ 3 `v ¨∇x ϕ 3 ´M´ϕ 3 ď 0, provided that p is sufficiently large (depending on ϕ 1 , λ, Λ, s and d, but not t 0 ).

Proof. We write the equation in terms of ϕ 2 . We have

B t ϕ 3 `v ¨∇x ϕ 3 ´M´ϕ 3 " t p 0 pt `t0 q p`1 " ´pϕ 2 pX, V q `ˆ´1 ´1 2s ˙X ¨∇x ϕ 2 pX, V q ´1 2s V ¨∇v ϕ 2 pX, V q `t0 V ¨∇x ϕ 2 pX, V q ´t0 M v ϕ 2 pX, V q * ,
where X " pt 0 {pt `t0 qq 1`1 2s x and V " pt 0 {pt `t0 qq 1 2s v. Let δ ą 0 be as in Lemma 7.8, so that the right hand side is non-positive when ϕ 2 ă δ. We choose p large so that the term pϕ 2 ě pδ is larger than all the others terms when ϕ 2 ě δ. Thus, the right hand side is never positive.

Proof of Lemma 7.1. Note that ϕ 3 p0, x, vq " ϕ 2 px, vq, where ϕ 2 and ϕ 3 are the functions in Lemmas 7.8 and 7.9 respectively. Note that these fuctions depends on the choice of t 0 which will be made below. Also, the value of p depends on t 0 . The function ϕ 2 is supported in B 1 ˆBA`1 . We must rescale ϕ 3 in order to obtain a function so that ϕp0, x, vq is supported in B r ˆBr . We pick ρ ą 0 small and let ϕpt, x, vq " ϕ 3 pρ ´2s t, ρ ´2s´1 x, ρ ´1vq, so that ρpA `1q ď r. This ensures the first three items in Lemma 7.1. Indeed, the function ϕ satisfies ϕ t `v ¨∇x ϕ ´M´ϕ ď 0.

In particular, also

ϕ t `v ¨∇x ϕ ´Lv ϕ ď 0, since L v ϕ ě M ´ϕ in Ω.
In order to obtain the lower bound in rτ, T s ˆBR , we are going to choose the parameter t 0 accordingly. Note that the value of t 0 does not affect ϕp0, x, vq.

From the construction of ϕ 1 and ϕ 2 , we have ϕ 2 px, vq " 1 whenever |x| ă 1 4A and |v| ă 1{4. Picking t 0 sufficiently small, for pt, x, vq P rτ, T s ˆBR 1`2s ˆBR , we have

ρ ´2s´1 ˆt0 t `t0 ˙1`1 2s |x| ď ρ ´2s´1 ˆt0 τ ˙1`1 2s R 1`2s ă 1 4A , ρ ´1 ˆt0 t `t0 ˙1 2s |v| ď ρ ´1 ˆt0 τ ˙1 2s R ă 1 4 .
Therefore, when pt, x, vq P rτ, T s ˆBR ˆBR , we have ϕpt, x, vq " t p 0 pρ ´2s t `t0 q p ě t p 0 pρ ´2s T `t0 q p ": θ ą 0.

This justifies the fourth item in Lemma 7.1. Finally, for the last item, we just pick R 1 sufficiently large. The function ϕ 2 is supported in B 1 ˆB1`A . Depending on our choices of t 0 and ρ above, the function ϕpt, ¨, ¨q is supported inside B R1 ˆBR1 for all t P r0, T s. This achieves the construction of the barrier.

Note that the only parameters in this construction that depend on λ and Λ are p and θ.

The intermediate-value lemma for s ě 1 2

This section is devoted to the statement and proof of a version of De Giorgi's isoperimetric lemma in the case s ě 1 2 . It is inspired by the compactness method in [START_REF] Golse | Harnack inequality for kinetic fokker-planck equations with rough coefficients and application to the landau equation[END_REF]. However, unlike [START_REF] Golse | Harnack inequality for kinetic fokker-planck equations with rough coefficients and application to the landau equation[END_REF], we do not use averaging lemmas. Instead, the analysis of the fractional Kolmogorov equation plays a critical role.

The first lemma of this section concerns a supersolution of the equation (1.2). In this case we add a nonnegative measure to the right hand side in order to have an exact solution. The purpose of this lemma is to provide a basic control of the total measure that we add. Lemma 8.1 (A priori estimate on a nonnegative measure). Let Q " r0, T s ˆBR 1`2s ˆBR , f : r0, T s ˆRd Rd Ñ r0, 1s be supported in Q. Assume also that

f t `v ¨∇x f `p´∆q s f ě h in r0, T s ˆRd ˆRd ,
for some h P L 2 pr0, T s ˆRd , H ´spR d qq. Then

f t `v ¨∇x f `p´∆q s f " h `µ in r0, T s ˆRd ˆRd ,
where µ is a nonnegative measure supported in r0, T s ˆBp2Rq 1`2s ˆB2R such that

µpQq ď Cp1 `}h} L 2 t,x H ´s v q,
and h " h in r0, T s ˆBp2Rq 1`2s ˆB2R and

} h} L 2 t,x H ´s v ď Cp1 `}h} L 2 t,x H ´s v q. Proof. Note that for px, vq R B R 1`2s ˆBR , f t `v ¨∇x f " 0. Moreover, |p´∆q s f pt, x, vq| " c ˇˇˇˆB R f pt, x, wq|w ´v| ´d´2s dw ˇˇˇď p|v| ´Rq ´d´2s |B R |χ |x|ďR . Therefore, f t `v ¨∇x f `p´∆q s f " p´∆q s f is an L 2 function outside of B R 1`2s ˆB3R{2 .
Let ϕ : R 2d Ñ r0, 1s be a smooth bump function so that ϕ " 1 in B R 1`2s ˆB3R{2 and ϕ " 0 outside of B p2Rq 1`2s ˆB2R .

We first need to justify that there is h P L 2 pr0, T s ˆRd , H ´spR d qq so that h " hϕ `p1 ´ϕqp´∆q s f.

We clearly have that p1´ϕqp´∆q s f is in L 2 t,x H ´s v . We are left to justify that hϕ P L 2 pr0, T sˆR d , H ´spR d qq. This follows by duality once we observe that for every g P L 2

t,x H s v , we also have ϕg P H s pR d q. With this definition of h, we still have

f t `v ¨∇x f `p´∆q s f ě h in r0, T s ˆRd ˆRd ,
with equality for px, vq R B p2Rq 1`2s ˆB2R . Let µ be the nonnegative measure, supported in r0, T s ˆBp2Rq 1`2s ˆB2R , defined by µ :" f t `v ¨∇x f `p´∆q s f ´h.

In order to estimate the total measure of µ, we test it against a test function which is identically one on its support. Let φ " 1 in B p2Rq 1`2s ˆB2R and be supported in B p3Rq 1`2s ˆB3R . We have µpr0, T s ˆRd ˆRd q " ˆr0,T sˆR 2d φ dµ,

" ˆr0,T sˆR 2d φ ´ft `v ¨∇x f `p´∆q s f
f t `v ¨∇x f `p´∆q s f ě h in r0, T s ˆRd ˆRd ,
where h P L 2 pr0, T s ˆRd , H ´spR d qq. Let r 1 ą 0, r 2 ą 0, 0 ă r 3 ă r 4 and 0 ă t 1 ă t 2 ă T such that r 3 {2 ą pr 1`2s 1 `r1`2s 2 q{pt 2 ´t1 q. We define

Q 1 " r0, t 1 s ˆBr 1`2s 1 ˆBr1 , Q 2 " rt 2 , T s ˆBr 1`2s 2 ˆBr2 , Q 3 " r0, T s ˆBr 1`2s 3 ˆBr3 , Q 4 " r0, T s ˆBr 1`2s 4 ˆBr4 .
Let us assume that f is supported in Q 4 and f P L 2 pr0, T s ˆRd , H s pR d qq X Cpr0, T s, L 2 pR d ˆRd qq. For every pair of positive numbers δ 1 , δ 2 , there exist θ ą 0 and µ ą 0 so that whenever

|tf " 1u X Q 1 | ě δ 1 and |tf ď θu X Q 2 | ě δ 2 , then |tθ ă f ă 1u X Q 3 | ě µ.
Here, the constants θ and µ depend on Proof. Assume the contrary. Then, there is a sequence of functions f i , uniformly bounded in L 2 pr0, T s Rd , H s pR d qq, h i uniformly bounded in L 2 pr0, T s ˆRd , H ´spR d qq, and sequences of positive numbers θ i Ñ 0 and µ i Ñ 0 so that all hypotheses in the lemma hold, however (8.1)

δ 1 , δ 2 , }h} L 2 t,x H ´s v , }f } L 2 t,x H s v , t 1 , t 2 , T , r 1 , r 2 , r 3 , r 4 , s and d. r 1 r 3 r 2 r 4 t 1 t 2 T t " 0 t px, vq Q 2 Q 1 Q 3 Q 4
|tf i " 1u X Q 1 | ě δ 1 , |tf i ď θ i u X Q 2 | ě δ 2 , |tθ i ă f i ă 1u X Q 3 | ă µ i .
We will find a contradiction by compactness. That is, we will find a subsequence that converges and find a limit function f 8 which only takes the values 1 and 0 in Q 3 . We will derive a contradiction there provided s P r1{2, 1q.

According to Lemma 8.1, there are measures µ i , supported in r0, T s ˆBp2r4q 1`2s ˆB2r4 , and modified right hand sides hi so that

rB t `v ¨∇x `p´∆q s sf i " hi `µi . Moreover, µ i pr0, T s ˆRd ˆRd q ď C, } hi } L 2 t,x H ´s v ď C.

Let us write hi

" h i 1 `p´∆q s{2 h i 2 for h i 1 and h i 1 in L 2 pr0, T s ˆR2d q, with }h i 1 } L 2 ď C and }h i 2 } L 2 ď C.
Up to extracting a subsequence, we can assume that f i p0, ¨, ¨q converges weakly in L 2 pR 2d q, f i , h i 1 and h i 2 converge weakly in L 2 pr0, T s ˆR2d q to f 8 , h 8 1 and h 8 2 , and µ i converges weakly-˚in the space of Radon measures Mpr0, T s ˆR2d q to some measure µ 8 .

Using the formula (2.3), we can write

f i " T 0 f i p0, ¨, ¨q `T1 µ i `T2 h i 1 `T3 h i 2 .
Here, the operators T 0 : L 2 pB r4 q Ñ L 1 pQ 4 q, T 1 : Mpr0, T s ˆR2d q Ñ L 1 pQ 4 q, T 2 , T 3 : L 2 pr0, T s ˆR2d q Ñ L 2 pQ 4 q and are given by

T 0 f 0 :" f 0 ˚t Jpt, ¨, ¨q. T 1 µ :" ˆt 0 µptq ˚t Jpt ´τ, ¨, ¨q dτ, T 2 h 1 :" ˆt 0 h 1 ptq ˚t Jpt ´τ, ¨, ¨q dτ,
T 3 h 2 :" ˆt 0 h 2 ptq ˚t p´∆q s{2 Jpt ´τ, ¨, ¨q dτ.

Note that T 1 , T 2 and T 3 are exactly convolutions in all variables pt, x, vq with respect to the natural Lie group structure. Also T 0 is the same as T 1 applied to a singular measure concentrated on t " 0 with marginal density f 0 . The operators T 1 , T 2 and T 3 are compact simply because they are convolutions with the L 1 functions J and p´∆q s{2 J. Therefore f i " T 0 f i p0, ¨, ¨q`T 1 µ i `T2 h i 1 `T3 h i 2 converges strongly in L 1 pQ 4 q to some function f 8 . Since we have 0 ď f i ď 1, then in fact f i converges strongly to f 8 in L p pQ 4 q for any p P r1, `8q.

The function f 8 solves, in the sense of distributions,

rB t `v ¨∇x `p´∆q s sf 8 ě h 8 1 `p´∆q s{2 h 8 2 .
Moreover, since f i Ñ f 8 in L 1 , from (8.1) we deduce that (8.2)

|tf 8 " 1u X Q 1 | ě δ 1 , |tf 8 " 0u X Q 2 | ě δ 2 , |t0 ă f 8 ă 1u X Q 3 | " 0.
Then, f 8 only takes the values 0 and 1, almost everywhere in Q 3 . Moreover, we have }f 8 } L 2 pr0,T sˆR d ,H s pR d qq ď C. Thus f 8 pt, x, ¨q P H s pB r3 q almost everywhere in r0, T s ˆBr 1`2s

3

. Since s ě 1{2, this implies that f pt, x, ¨q is either constant 1 or constant 0 in B r3 for pt, xq P r0, T s ˆBr 1`2s 3 . From this point on, we write f 8 pt, xq :" f 8 pt, x, vq provided that pt, x, vq is restricted to Q 3 . Note that p´∆q s f 8 is not constant in Q 3 due to the nonlocality of p´∆q s .

Let ϕ : R d Ñ r0, `8q be a smooth bump function supported in B r3{2 , such that ˆRd ϕpvq dv " 1, ˆRd ϕpvq v dv " 0.

For any v 0 P B r3{2 and pt, xq P r0, T s ˆBr3 , we have where H v0 is the function in L 2 pr0, T s ˆBr 1`2s 3 q given by H v0 pt, xq " ˆRd p´f 8 pt, x, vq `h8 2 pt, x, vqqp´∆q s ϕpv ´v0 q `h8 1 ϕpv ´v0 q dv. From (8.2), we know that there exist some τ 1 P r0, t 1 s and τ 2 P rt 2 , T s so that

f
|tx : f 8 pτ 1 , xq " 1u X B r1 | ě δ 1 t 1 , |tx : f 8 pτ 2 , xq " 0u X B r2 | ě δ 2 T ´t2 . Let S 1 " tx : f 8 pτ 1 , xq " 1u X B r 1`2s 1 and S 2 " tx : f 8 pτ 2 , xq " 0u X B r 1`2s 2 . Since }χ S1 ˚χ´S2 } L 1 " |S 1 ||S 2 | ě δ 1 δ 2 t 1 pT ´t2 q ,
then, there exists one vector w 0 P B r ": c 0 .

Let v 0 " w 0 {pτ 2 ´τ1 q. We have |v 0 | ď |w 0 |{pt 2 ´t1 q ď r 3 {2.

Since the right hand side H v0 P L 2 pr0, T s ˆBr 1`2s 3 q, in particular, for almost all x P S 1 X pS 2 ´w0 q, the function t Þ Ñ H v0 pt, x `pt ´τ1 qv 0 q is in L 2 pτ 1 , τ 2 q.

Because of the transport equation that f 8 satisfies in Q 3 , we have d dt f 8 pt, x `pt ´τ1 qv 0 q ě H v0 pt, x `pt ´τ1 qv 0 q.

In particular, for almost every x P B r1 , there is a constant Cpxq ą 0 so that f 8 p t2 , x `pt 2 ´τ1 qv 0 q ´f8 p t1 , x `pt 1 ´τ1 qv 0 q ě ´Cpxqp t2 ´t 1 q 1{2 , for any t 1 ă t1 ă t2 ă t 2 .

However, since f 8 pt, x`pt´τ 1 qv 0 q only takes the values 0 and 1, and f 8 pτ 1 , xq " 1 for every x P S 1 XpS 2 ´w0 q, then f 8 pt, x `pt ´τ1 qv 0 q " 1 for every x P S 1 X pS 2 ´w0 q and t P rτ 1 , T s.

We arrive to a contradiction since f 8 pτ 2 , x `pτ 2 ´τ1 qv 0 q " f 8 pτ 2 , x `w0 q " 0 for every x P S 1 X pS 2 ´w0 q. This achieves the proof. Lemma 8.3 (Intermediate sets for local super-solutions). Let s P r1{2, 1q. Let r 1 , r 2 , r 3 , r 4 , t 1 , t 2 , T , Q 1 , Q 2 , Q 3 and Q 4 be like in Lemma 8.2. Let f : r0, T s ˆBr4 ˆRd Ñ r0, `8q. Assume f is a supersolution of

f t `v ¨∇x f ´Lv f ě 0 in Q 4 .
For every pair of positive numbers δ 1 , δ 2 , there exists θ ą 0 and µ ą 0 so that whenever

|tf ě 1u X Q 1 | ě δ 1 and |tf ď θu X Q 2 | ě δ 2 , then |tθ ă f ă 1u X Q 3 | ě µ.
Here, the constants θ and µ depend on δ 1 , δ 2 , t 1 , t 2 , T , r 1 , r 2 , r 3 , r 4 , λ, Λ, s and d.

Proof. By replacing f with minpf, 1q (see Lemma A.11 in Appendix), we can assume that 0 ď f ď 1 everywhere.

Let ρ ą 0 so that 2ρ ą r 4 ´r3 . Applying Lemma 6.2 to 1 ´f , we obtain that f P L 2 pr0, T s ˆBpr4´ρq From a direct computation, we get rB t `v ¨∇x ´Lv spϕf q ě pv ¨∇x ϕqf ´pL v pϕf q ´ϕL v f q in r0, T s ˆRd ˆRd .

The term pv ¨∇x ϕqf is bounded by one, and supported in B pr4´2ρq 1`2s ˆBr4´2ρ . The second term is a commutator, which is also bounded in L 2 pr0, T s ˆRd , H ´spR d qq because of Lemmas 4.10 and 4.11. Let h 0 :" pv ¨∇x ϕqf ´pL v rϕf s ´ϕL v f q P L 2 pr0, T s ˆRd , H ´spR d qq.

Now, we rewrite the equation for ϕf as a fractional Kolmogorov equation rB t `v ¨∇x `p´∆q s v spϕf q ě h 0 `p´∆q s v pϕf q `Lv pϕf q.

Because of (8.3), there is a function h 1 P L 2 pr0, T s ˆRd ˆRd q such that p´∆q s v pϕf q " p´∆q s{2 v h 1 . Also because of (8.3) and applying Corollary 5.2, L v pϕf q belongs to H ´spR d q. Summarizing, pϕf q is a supersolution to a fractional Kolmogorov equation with a right hand side in L 2 pr0, T s ˆRd , H ´spR d qq, rB t `v ¨∇x `p´∆q s v spϕf q ě h 0 `p´∆q s{2 v h 1 `Lv pϕf q in r0, T s ˆRd ˆRd .

We finish the proof applying Lemma 8.2 with r 4 ´2ρ instead of r 4 .

Lemma 8.4 (Propagation in measure). Under the same assumptions as in Lemma 8.3, For every pair of positive numbers δ 1 , δ 2 , there exists θ ą 0 so that whenever

|tf ě 1u X Q 1 | ě δ 1 then |tf ď θu X Q 2 | ă δ 2 ,
Here, the constant θ depends on δ 1 , δ 2 , t 1 , t 2 , T , r 1 , r 2 , r 3 , r 4 , λ, Λ, s and d.

Proof. Let θ and µ ą 0 be the values from Lemma 8.3. In this lemma we choose θ :" θk where k is the smallest integer larger than |Q 3 |{µ. Assume the conclusion of the lemma was not true. Then for all values of j " 0, 1, . . . , k ´1, the function θ´j f would satisfy the hypothesis of Lemma 8.3. Therefore, for every j " 0, 1, . . . , k ´1,

|t θj`1 ă f ă θj u X Q 3 | ě µ.
This is clearly impossible since all these are disjoint sets contained in Q 3 , so their measures cannot add up to more than |Q 3 |.

The propagation lemma

We call propagation lemma a result that says that as soon as a (super)solution is above a large constant in most of a cylinder, then it is bounded from below by 1, say, for later times.

The difference between this propagation lemma and the first De Giorgi lemma proved in Section 6 lies in the sets of points where the estimates hold. Essentially, the propagation lemma is the result of De Giorgi's first lemma, combined with a propagation of the lower bound to later times and larger sets. This propagation is obtained using the barrier function of section 7 when s P p0, 1{2q or the intermediate-value lemma from Section 8 when s P r1{2, 1q. Lemma 9.1 (Propagation lemma). There exist R 1 ą 0 (large, only depending on dimension and s), δ ą 0 (small, universal) and M ě 1 (large, universal) such that for T " 2 2s , if f is a supersolution

f t `v ¨∇x f ´Lv f ě 0 in r´1, T s ˆBR 1`2s 1 ˆBR1 ,
which is non-negative in r´1, T s ˆR2d and such that

(9.1) |tf ą M u X Q 1 | ě p1 ´δq|Q 1 |
then f ě 1 in Q where Q " r0, T s ˆB2 2s`1 ˆB2 (see Figure 5). Proof. We will prove the equivalent result that if |tf ă 1u X Q 1 | ă δ then f ě 1{M in Q. The proof will be different depending on whether s P p0, 1{2q or s P r1{2, 1q. Let us start with the case s P p0, 1{2q. We combine De Giorgi's first lemma with a barrier function.

We first apply Lemma 6.6 to 2f , shifted in time, with Q " r´1, ´1{2s ˆB1 ˆB1 and Q " r´3{4, ´1{2s B1{2 ˆB1{2 . For δ suffiently small, we obtain that f ě 1{2 in Q. In particular f p´1{2, x, vq ě 1{2 for all px, vq P B 1{2 ˆB1{2 .

Let ϕ be the barrier of Lemma 7.1 with T " 3{2, τ " 1{2 and r " 1{2. Lemme 7.1 also gives us the value of R 1 . We apply the comparison principle to get that f ě 1 2 ϕpt `1{2, ¨, ¨q in r´1{2, T s ˆBR1 ˆBR1 and conclude the proof. In this case M " 2{θ, where θ ą 0 is the constant from Lemma 7.1.

For the case s P r1{2, 1q, we combine the intermediate set lemma with De Giorgi's first lemma. We apply Lemma 8.4 to f pt ´1, x, vq, with r 1 " 1, r 2 " 3, r 3 " 4, r 4 " R 1 " 5, t 1 " 1{2, t 2 " 3{4, T " T `1, arbitrary δ 1 " δ ą 0 and δ 2 ą 0 sufficiently small. We obtain that there is a θ 1 ą 0 so that ˇˇtf ą θ 1 u X pr´1{4, T s ˆB3 1`2s ˆB3 q ˇˇă δ 2 .

Then we apply Lemma 6.6 to 2f {θ 1 (again shifted in time) with Q " r´1{4, T s ˆB3 1`2s ˆB3 and Q " r0, T s ˆB2 1`2s ˆB2 . This concludes the proof.

The propagation lemma implies the following corollaries.

Corollary 9.2 (Stacked propagation). Let R 1 and δ be the constants from Lemma 9.1. Let k ě 1, T k "

ř k i"1 2 2si and R k " 2 k R 1 . If f is a supersolution of (1.2) with h " 0 in r´1, T k s ˆBR 1`2s k ˆBR k and |tf ą M k u X Q 1 | ą p1 ´δq|Q 1 |, then f ě 1 in Qrks :" rT k´1 , T k s ˆB2 p1`2sqk ˆB2 k .
Proof. This is simply an iteration of Lemma 9.1. Indeed, getting f ě 1 in Q implies that |t f ą M u X Q| ą p1 ´δq| Q| where f " M f . Choosing Q as the new cylinder Q in the basic propagation lemma yields that f is bounded from below by M ´1 in a new cylinder. Iterating this estimate, we get the corollary.

Corollary 9.3 (Propagation of minima). Let R 1 and δ as in Lemma 9.1. Let f be a supersolution of (1.2)

with h " 0 in Q " r´1, 0s ˆBR 1`2s 1 ˆBR1 . Let Q r pt 0 , x 0 , v 0 q Ă Q 1 such that |tf ą Au X Q r pt 0 , x 0 , v 0 q| ą p1 ´δq|Q r |.
Then, there exists some p ą 0 and c ą 0 so that

f pt, x, vq Á A ˆ1 `t ´t0 r 2s ˙´p ,
The set S Q r pt 0 , x 0 , v 0 q

Q 1 Qr1s Qr2s Qr3s Qr4s Qr5s Q 1 Figure 6
. Geometric setting of Corollaries 9.2 and 9.3.

whenever pt, x, vq belongs to the set S " Spt 0 , x 0 , v 0 q :" " pt, x, vq : t ą t 0 , |x ´x0 ´pt ´t0 qv 0 | ă `p1 ´2´2s qpt ´t0 q `r2s ˘1`1 2s , and |v ´v0 | ă `p1 ´2´2s qpt ´t0 q `r2s ˘1 2s * .

Proof. Let t k " t 0 `řk i"1 p2 i rq 2s , r k " p2 k rq and Qrks :" Q r k pt k , x 0 `tk v 0 , v 0 q.

The change of variables pt, x, vq Þ Ñ pr 2s pt ´t0 q, r 1`2s px ´x0 ´pt ´t0 qv 0 q, rpv ´v0 qq, which preserves the equation, transforms the cylinder Q 1 into Q r pt 0 , x 0 , v 0 q and the cylinders Qrks of Corollary 9.2 into Qrks. We can easily check that S Ă Ť Qrks. Corollary 9.2 tells us (after the change of variables above) that f ě A{M k in Qrks. Observe that pt ´t0 `r2s q « p2 k rq 2s in Qrks, therefore

f pt, x, vq ě AM ´k Á A ˆt ´t0 `r2s r 2s ˙´p ,
where p " logpM q logp2 2s q .

Remark 9.4. It is possible that in the proof of Corollary 9.3 some cylinder Qrks extends pass the time t " 0 and thus it is not strictly contained in Q. This is not a problem since we are dealing with a parabolic equation and future values of f do not affect earlier values. Indeed, we can readily verify that Lemma 9.1 also holds for any value of T P p0, 2 2s q. The only thing that matters is that R 1 is sufficiently large.

The ink-spots theorem for slanted cylinders

This section is dedicated to the statement and proof of a theorem involving a covering argument in the flavor of Krylov-Safonov growing ink spots theorem, or the Calderón-Zygmund decomposition. Such a theorem is used in the proof of the weak Harnack inequality. The statement of the theorem involves stacked (and slanted) cylinders: (10.1) Qm pz 0 , rq " tpt, x, vq : 0 ă t ´t0 ď mr 2s , |v ´v0 | ă r, |x ´x0 ´pt ´t0 qv 0 | ă pm `2qr 1`2s u (see Figure 7). The cylinder Qm is a delayed version of Q. It starts immediately at the end of Q. Its duration in time is m times as long as Q. Its radius in x is pm `2q times the radius in Q. It shares the same values of velocities v as Q.

Theorem 10.1 (The ink-spots theorem). Let E Ă F be two bounded measurable sets. We make the following assumption for some constant µ P p0, 1q.

' E Ă Q 1 and |E| ă p1 ´µq|Q 1 |. ' If any cylinder Q Ă Q 1 such that Qm Ă Q 1 satisfies |Q X E| ě p1 ´µq|Q|, then Qm Ă F . Then |E| ď m`1
m p1 ´cµq|F | for some constant c P p0, 1q depending on s and dimension only.

Figure 7. Stacked cylinders

There is no chance to adapt the Calderón-Zygmund decomposition to this context. It would require that we split a larger piece into smaller pieces of the same type. Even if we replace balls with cubes, the different slopes, depending on the center velocities, make this tiling condition impossible.

What we do is a variation of the growing ink-spots theorem. The original construction by Krylov and Safonov can be found (in English) in the Appendix 1 of [START_REF] Krylov | Nonlinear elliptic and parabolic equations of the second order[END_REF]. Here, we have one extra dimension, x, which plays a different role and presents additional difficulies. The most significant difficulty is to go from a lower bound on the measure of the union of disjoint cylinders Q (Lemma 10.7) to a lower bound on the measure of the union of their delayed versions Qm (Theorem 10.1). The problem is that if the center velocities of two cubes flow towards each other, they may create extra overlaps in their delayed versions. This is addressed essentially in Lemma 10.9, using that we expand the radius in x only by a fixed factor.

The values of x that belong to a slanted cylinder Q r pt 0 , x 0 , v 0 q change for different values of t. They are contained in a ball with radius r 1`2s which flows in the direction of v 0 and is shifted a total distance |v 0 |r 2s from the initial to the end time. For small values of r, the lenght of this shift is an order of magnitude larger than the radius of the ball r 1`2s . Dealing with this shift is non-trivial, and that is the main difference between the covering argument described in this section and the usual ink-spots theorem.

The following corollary will be used when we need to confine both E and F to stay within a fixed cylinder.

Corollary 10.2 (Ink-spots theorem with leakage). Let E Ă F be two bounded measurable sets. We make the following assumption for some constant µ P p0, 1q.

' E Ă Q 1 . ' If any cylinder Q Ă Q 1 satisfies |Q X E| ě p1 ´µq|Q|, then Qm Ă F and also Q " Q r pt,
x, vq for some r ă r 0 . Then |E| ď m`1 m p1 ´cµqp|F X Q 1 | `Cmr 2s 0 q for some constants c and C depending on s and dimension only. 10.1. Stacked cylinders and scaling. For any factor k, we define the scaled cylinder kQ r by kQ r " Q kr ˆk2s ´1 2 r 2s , 0, 0 ˙.

Here, we scaled the radius r by a factor k and kept the same center of the cylinder. Note that the point pt 0 , x 0 , v 0 q in Q r pt 0 , x 0 , v 0 q refers to the top of the cylinder, not its center. In order to keep the center fixed, we updated the top. Consistently with the Lie group action, we define kQ r pt 0 , x 0 , v 0 q :" T pt0,x0,v0q kQ r , " tpt, x, vq :

´k2s `1 2 r 2s ă t ´t0 , ď k 2s ´1 2 r 2s , |v ´v0 | ă kr, |x ´x0 ´pt ´t0 qv 0 | ă pkrq 1`2s u.
Note that |kQ r pt 0 , x 0 , v 0 q| " k 2psd`s`dq |Q r |.

The first version of the growing ink-spots lemma uses essentially a variation of the Vitali covering lemma together with a generalized Lebesgue differentiation theorem. 10.2. A generalized Lebesgue differentiation theorem. In [START_REF] Imbert | An introduction to fully nonlinear parabolic equations[END_REF], a generalized Lebesgue differentiation theorem was derived for parabolic cylinders. Here, even though we have one additional variable (x), the proof is the essentially the same. It relies on an adaptation of Vitali's covering lemma (Lemma 10.5 below) and a maximal inequality (Lemma 10.6 below). Theorem 10.3 is obtained from Lemma 10.6 exactly as in [START_REF] Imbert | An introduction to fully nonlinear parabolic equations[END_REF]. For the reader's convenience we will provide below a proof of the maximal inequality.

In our setting, the cylinders Q r pt 0 , x 0 , v 0 q are not the balls of any metric. The important properties of cylinders are explicitly given by the following lemma.

Lemma 10.4. Let Q r0 pt 0 , x 0 , v 0 q and Q r1 pt 1 , x 1 , v 1 q be two cylinders with nonempty intersection. Assume that 2r 0 ě r 1 . Then Q r1 pt 1 , x 1 , v 1 q Ă kQ r0 pt 0 , x 0 , v 0 q, for some universal constant k (it depends on s only).

Proof. Since all our definitions are invariant by the action of the Lie group, we can assume without loss of generality that pt 0 , x 0 , v 0 q " 0 (the general case is reduced to this applying T ´1 pt0,x0,v0q ). We need to choose the constant k so that k ě 5,

k 2s ě 1 `2 ¨22s , k 1`2s ě 1 `2 ¨21`2s .
The first inequality implies the other two when s ě 1{2. The second inequality implies the other two when s ď 1{2. In particular the third inequality is always redundant. In any case, we pick the smallest k satisfying these inequalities, which depends only on s.

Let pt 2 , x 2 , v 2 q P Q r0 X Q r1 pt 1 , x 1 , v 1 q. Let pt, x, vq P Q r1 pt 1 , x 1 , v 1 q. Then all the following hold

t ď t 1 ă t 2 `r2s 1 ď p2r 0 q 2s ď k 2s ´1 2 r 2s 0 , t ě t 1 ´r2s 1 ě t 2 ´r2s 1 ě ´r2s 0 ´p2r 0 q 2s ě ´k2s `1 2 r 2s 0 , |v| ď |v ´v1 | `|v 1 ´v2 | `|v 2 | ď 2r 1 `r0 ď 5r 0 ď kr 0 , |x| ď |x ´x1 ´pt ´t1 qv 1 | `|x 2 ´x1 ´pt 2 ´t1 qv 1 | `|x 2 | ď 2r 2s`1 1 `r2s`1 0 ď k 2s`1 r 2s`1 0 .
Thus, we get that pt, x, vq P kQ r0 and we conclude the proof. Lemma 10.5 (Vitali). Let tQ j u jPJ be an arbitrary collection of slanted cylinders with bounded radius. Then, there exists a disjoint countable subcollection tQ ji u so that

ď jPJ Q j " 8 ď i"1 kQ ji .
The proof of Lemma 10.5 is the same as the classical proof of the Vitali coverling lemma using Lemma 10.4 instead of the fact that in any metric space B r1 px 1 q Ă 5B r0 if B r1 px 1 q X B r0 ‰ H and r 1 ď 2r 0 .

We next define the maximal function M f as follows: for px, v, tq P Ω,

M f pt, x, vq " sup QQpx,v,tq QXΩ |f |
where the supremum is taken over cylinders of the form py, w, sq `RQ 1 .

Lemma 10.6 (The maximal inequality). For all λ ą 0,

|tM f ą λu X Ω| ď C λ }f } L 1 pΩq . Proof. For px, v, tq P tM f ą λu X Ω, there exists a cylinder Q Q px, v, tq such that ˆQXΩ |f | ě λ 2 |Q X Ω|.
Then tM f ą λuXΩ is covered with cylinders tQ j u such that the previous inequality holds. From Lemma 10.5, there exists a disjoint countable subcollection tQ ji u so that

tM f ą λu X Ωu Ă 8 ď i"1 kQ ji
for some integer k only depending on s.

We now write

ˆΩ |f | ě ˆΩXYiQj i |f | " ÿ i ˆΩX Qj i |f | ě ÿ i λ 2 |Q ji X Ω| " λ 2 | Y i Q ji X Ω| " λ 2k 2pd`ds`sq | Y i kQ ji X Ω| ě λ 2k 2pd`ds`sq |tM f ą λu X Ω|.
We obtain the desired inequality with C " 2k 2pd`ds`sq . 10.3. Preliminary version without time delay. Lemma 10.7. Let E Ă F Ă Q 1 be two measurable sets. Assume that there is a constant µ ą 0 such that

' |E| ă p1 ´µq|Q 1 |. ' if any cylinder Q Ă Q 1 satisfies |Q X E| ě p1 ´µq|Q|, then Q Ă F .
Then |E| ď p1 ´cµq|F | for some constant c depending on s and dimension only.

Proof. Using the generalized Lebesgue differentiation theorem (see [START_REF] Imbert | An introduction to fully nonlinear parabolic equations[END_REF] for an adaptation of the classical Lebesgue differentiation theorem 10.3, for almost all points x P E, there is some cylinder Q x containing x such that |Q x X E| ě p1 ´µq|Q x |. For all Lebesgue points x P E, let us choose a maximal slanted cylinder Q x Ă Q 1 that contains x and such that |Q x X E| ě p1 ´µq|Q x |. Here Q x " Q r p t, x, vq for some r, t, x and v. From one of the assumptions, we know that Q x ‰ Q 1 for any x. The other assumption tells us that Q x Ă F . We claim that |Q x X E| " p1 ´µq|Q x |. Otherwise, for δ ą 0 small enough, there would be a Q such that

Q x Ă Q Ă p1 `δqQ x , Q Ă Q 1 and | Q X E| ą p1 ´µq| Q|, contradicting the maximality of the choice of Q x .
The family of cylinders Q x covers the set E. By Lemma 10.5, we can select a finite subcollection of non overlapping cylinders Q

j :" Q xj such that E Ă Ť n j"1 kQ j . Since Q j Ă F and |Q j X E| " p1 ´µq|Q j |, we have that |Q j X F zE| " µ|Q j |. Therefore |F zE| ě n ÿ j"1 |Q j X F zE| ě n ÿ j"1 µ|Q j | " k ´2pd`ds`sq µ n ÿ j"1 |kQ j | ě k ´2pd`ds`sq µ|E|.

We thus get

|F | ě p1 `cµq|E| with c " k ´2pds`d`sq . Since cµ P p0, 1q, this implies |E| ď p1 ´cµq|F | with c " c{2. 10.4. Stacked cylinders and leakage. The following lemma can be deduced from Lemma 4.29 in [START_REF] Imbert | An introduction to fully nonlinear parabolic equations[END_REF] (There is a typo in the statement in that note, we embarrassingly apologize).

Lemma 10.8. Consider a (possibly infinite) sequence of intervals pa j ´hk , a j s. Then

ˇˇˇˇď k pa k , a k `mh k q ˇˇˇˇě m m `1 ˇˇˇˇď k pa k ´hk , a k s ˇˇˇˇ.
Proof. We first assume that k " 1, . . . , N for some finite number N . Let

N ď k"1 pa k , a k `mh k q " ď I ,
for a disjoint family of intervals I . Here, each I is a union of intervals of the form pa i , a i `mh i s. Let a 0 ´h0 be the minimum of a i ´hi and a 1 `mh 1 be the maximum of a i `mh i respectively, for all i so that pa i , a i `mh i s Ă I . Naturally, we have

|I | ě pa 1 `mh 1 q ´a0 ě m m `1 ppa 1 `mh 1 q ´pa 0 ´h0 qq ě m m `1 ˇˇˇˇˇď ti:pai,ai`mhisĂI u pa i ´hi , a i s ˇˇˇˇˇ, Therefore ˇˇˇˇN ď k"1 pa k , a k `mh k s ˇˇˇˇ" | ď I | " ÿ |I |, ě m m `1 ÿ ˇˇˇˇˇď ti:pai,ai`mhisĂI u pa i ´hi , a i s ˇˇˇˇˇ, ě m m `1 ˇˇˇˇď i"1,...,N pa i ´hi , a i s ˇˇˇˇ.
It is now enough to let N Ñ 8 to conclude.

Lemma 10.9. Let tQ j u be a collection of slanted cylinders, and Qm j be the corresponding delayed versions as in (10.1)

+ˇˇˇˇˇ.

From now on, let v be any fixed v P R d . Any cylinder Q j corresponds to Q rj pt j , x j , v j q for some choice of r j ą 0, t j P R and x j , v j P R d . If |v ´vj | ă r j , we have pt, xq : pt, x, vq P Qm Let z " x ´tv. The change of variables pt, xq Þ Ñ pt, zq has Jacobian one. We will estimate the measure of the points pt, zq so that pt, z `tvq belongs to the set above. Thus Let Qj be the cylinders in R ˆRd used in the right hand side of the inequality above, Qj " tpt, zq : 0 ă t ´tj ď mr 2s j , |z ´xj `tj v| ă 2r

+ˇˇˇˇˇ.

For the last inequality we used that if ´r2s j ă t ´tj ď 0, then pt j ´tq|v ´vj | ă r 1`2s j . Combining (10.3) with (10.4), we obtain (10.2) and finish the proof.

We can now turn to the proof of the main theorem.

Proof of Theorem 10.1. Let Q be the collection of all cylinders Q Ă Q 1 such that |Q X E| ě p1 ´µq|Q|. Let G :" Ť QPQ Q. By construction, the sets E and G satisfy the hypothesis of the Lemma 10.7. Therefore p1 ´cµq|G| ě |E|.

From our hypothesis Ť QPQ Qm Ă F . We conclude the proof applying Lemma 10.9,

|F | ě ˇˇˇˇď QPQ Qm ˇˇˇˇě m m `1 ˇˇˇˇď QPQ Q ˇˇˇˇ" m m `1 |G|.
Proof of Corollary 10.2. Note that the condition |E| ď p1´δq|Q 1 | is implied by the second assumption when r 0 ă 1. Moreover, the result is trivial for r 0 ě 1 choosing C sufficiently large.

Let Q be the collection of all cylinders Q Ă Q 1 such that |Q X E| ě p1 ´µq|Q|. Let G :" Ť QPQ Qm .

From Theorem 10.1, we have that |E| ď m m`1 p1 ´cµq|G|. Moreover, our hypothesis tell us that G Ă F . In order to conclude the corollary, we will estimate the measure GzQ 1 using the fact that each of the cubes Q " Q r pt, x, vq Ă Q 1 has radius bounded by r 0 . Recall that Qm " tp t, x, vq : 0 ă t ´t ď mr 2s , |v ´v| ă r, |x ´x ´pt ´tqv| ă pm `2qr 1`2s u.

Since Q Ă Q 1 , then t ă 0. So t ď mr 2s 0 . Moreover, |v| ă 1, since the velocities in Qm are the same as in Q. Also, |x| ă 1, so |x| ď 1 `mr 2s 0 . Therefore, Qm Ă p´1, mr 2s 0 s ˆB1`mr 2s 0 ˆB1 . The same thing applies to G. G Ă p´1, mr 2s 0 s ˆB1`mr 2s 0 ˆB1 .

Therefore |F X Q 1 | ě |G X Q 1 | ě |G| ´|GzQ 1 | ě |G| ´Cmr 2s
0 and we conclude the proof.

Proofs of the main results

In this section we complete the proofs of our main results. At this point, the main tools have already been established in previous sections. The weak Harnack inequality is proved combining the propagation lemma (Lemma 9.1) with our special version of the ink-spots theorem (Theorem 10.1). The structure of this proof is inspired by the work of Krylov and Safonov [START_REF] Krylov | A property of the solutions of parabolic equations with measurable coefficients[END_REF] for equations in nondivergence form.

The weak Harnack inequality.

Proof of Theorem 1.6. We choose R 1 to be the radius given in Lemma 9.1. We choose r 0 sufficiently small so that the set Spt 0 , x 0 , v 0 q from Corollary 9.3 contains Q `for any pt 0 , x 0 , v 0 q P Q ´and r P p0, r 0 q.

Replacing f and h with cf and ch where the constant c is choosen as follows c " p2 inf Q `f `2}h} L 8 pQ1q q ´1, we reduce to the case where inf Q `f ď 1{2 and }h} L 8 pQ1q q ď 1{2.

We can further reduce to the case inf Q `f ď 1 and h " 0. Indeed, if the function f is a supersolution of

f t `v ¨∇x f ´Lv f ě ´1{2, then the function f pt, x, vq " f pt, x, vq `pt `1q{2 is a nonnegative function in r´1, 0s ˆR2d which is a supersolution to (1.2) with h " 0. Moreover, inf Q `f ď inf Q `f `1{2 ď 1 and f ε ď f ε .
The proof relies on the application of the propagation lemmas 9.1 and Corollary 9.3. The constants M, δ in the remainder of the proof are chosen so that these propagation lemmas can be applied.

We are going to prove that in this case ˆQ´f ε pt, x, vq dv dx dt ď Cw.h.i. .

In order to do so, it is enough to prove that (11.1) @k ě 1, |tf ą M k u X Q ´| ď C w.h.i. p1 ´δ1 q k for some universal constants M ě 1, C w.h.i. ě 1 and δ 1 P p0, 1q. Estimate (11.1) is proved by induction. For k " 1, we simply choose C w.h.i. and δ 1 so that |Q ´| ď C w.h.i. p1 ´δq and δ 1 ď δ.

Note that by choosing a larger constant C w.h.i. we can make sure the inequality holds for arbitrarily many values of k.

Assume that the inequality holds true up to rank k ě 1 and let us prove it for k `1. We want to apply Corollary 10.2 of the growing ink spots theorem 10.1 with µ " δ, some integer m ě 1 (to be fixed later, only depending on δ), and M " M m where δ and M given by the propagation lemma 9.1, and

E " tf ą M k`1 u X Q ´and F " tf ą M k u X Q 1 .
The sets E and F are bounded and measurable and E Ă F Ă Q 1 . We consider a cylinder Q " Q r pz 0 q Ă Q ´(in particular r P p0, r 0 q) such that |Q X E| ą p1 ´δq|Q|, that is to say (11.2) |tf ą M k`1 u X Q ´| ą p1 ´δq|Q|.

We now prove that r is small. Since we have inf Q `f ď 1 and Spt 0 , x 0 , v 0 q contains Q `, Corollary 9.3 yields M k`1 ˆ1 `1 ´2r 2s 0 r 2s ˙´p À 1.

Therefore r À M ´k{p2spq . In particular Qm Ă Q 1 (at least for k large). Now, we want to prove that, Qm Ă F , that is to say, (11.3) Qm Ă tf ą M k u.

This follows simply from Corollary 9.2, with k " m to the function f " M ´kf ˝Tz0 . Applying Corollary 10.2 to E and F with µ " δ and r 0 " M ´k{p2spq , we get

|tf ą M k`1 u X Q ´| ď m `1 m p1 ´cδq ! |tf ą M k u X Q ´| `Cm M ´k{p )
where C " Cps, dq. We now use the induction hypothesis and get

|tf ą M k`1 u X Q ´| ď m `1 m p1 ´cδq ! C w.h.i. p1 ´δ1 q k `Cm M ´k{p
) .

Choosing δ 1 smaller than M ´1{p , we have

|tf ą M k`1 u X Q ´| ď
|tf ą M k`1 u X Q ´| ď C w.h.i. p1 ´δ1 q k`1 .
This achieves the proof of Estimate (11.1) and of the theorem.

11.2. The Hölder estimate. In order to prove Theorem 1.5, we first prove two preparatory results, Lemma 11.1 has the flavor of a weak Harnack inequality, but for supersolutions that can take (controled) negative values. The lemma then implies Corollary 11.2 which is concerned with the improvement of oscillation of solutions with small forcing terms. Let f : p´1, 0s ˆB1 ˆRd Ñ R be a function satisfying the following assumptions. ' f t `v ¨∇x f ě L v f `h in Q 1 , with h ě ´ε0 ; ' For t P p´1, 0s, x P B 1 , v P B 2 , f pt, x, vq P r0, 1s; ' For t P p´1, 0s, x P B 1 and v P R d zB 2 , f pt, x, vq ě ´´|v| 2 ¯α0

`1;

' |tf ě 1{2u X Q´| ě 1 2 | Q´| . If α 0 ą 0, ε 0 ą 0 and δ ą 0 are sufficiently small, then

f ě δ in Q ρ .
Proof. We can assume that h ď 0 without loss of generality. Let us first scale the function by defining f pt, x, vq " f pR ´2s 1 t, R ´1´2s 1

x, R ´1 1 vq. This function satisfies the equation Then osc Qρ f ď 1 ´δ.

B t f `v ¨∇x f ´L v f ě ´ε0 R ´2s 1 ě ´ε0 , in Q R1 .
Here, ε 0 ą 0, δ ą 0, α 0 ą 0 and ρ ą 0 are the same constants as in Lemma 11.1.

Proof. Let a " essinf p´1,0sˆB1ˆB2 f and b " esssup p´1,0sˆB1ˆB2 f . The values of f pt, x, vq are either above or below the middle value pa `bq{2 in at last half of the points in Q´. Thus, one of the following inequalities holds.

ˇˇˇ" f ě a `b 2 * X Q´ˇˇˇě 1 2 | Q´| or ˇˇˇ" f ď a `b 2 * X Q´ˇˇˇě 1 2 | Q´| .
Assume the former. The opposite case would follow from the same proof upside down. Consider the function f pt, x, vq " 1 ´b `f pt, x, vq.

This choice is made so that esssup p´1,0sˆB1ˆB2 f " 1. Since osc p´1,0sˆB1ˆB2 f ď 1, then f P r0, 1s for pt, x, vq P p´1, 0s ˆB1 ˆB2 .

Since osc p´1,0sˆB1ˆB R f ď `R 2 ˘α0 for R ě 2 and esssup p´1,0sˆB1ˆB2 f " 1, then f pt, x, vq ě 1 ´´|v| 2 ¯α0 for t P p´1, 0s, x P B 1 and v P R d zB 2 . Thus, f satisfies the hypothesis of Lemma 11.1, f P rδ, 1s in B ρ , and the corollary follows.

Proof of Theorem 1.5. Without loss of generality we assume }f } L 8 pp´1,0sˆB1ˆR d q ď 1 and }h} L 8 pQ1q ď ε 0 , where ε 0 is the constant from Lemma 11.1. Otherwise, we replace f by f pt, x, vq " 1 }f } L 8 pp´1,0sˆB1ˆR d q `}h} L 8 pQ1q {ε 0 f pt, x, vq.

We want to prove that there exists some universal constant C so that for all r ą 0, osc Qr f ď Cr α .

We choose α ă minpα 0 , lnp1 ´δq{ lnpρ{2qq, where ρ, δ and α 0 are the constants from Lemma 11.1. Let Aprq :" osc Qr f " esssup Qr f ´essinf Qr f . It is a monotone increasing function. We cannot assume a priori that A is a continuous function, but it is always left continuous. Since |f | ď 1, we also have Aprq ď 2 for all r ą 0. Hence, we can choose C large enough so that Aprq ď Cr α for all r ě ρ.

Assume the theorem is not true, then let r 0 :" suptr : Aprq ą Cr α u P p0, ρq.

Since Aprq is left continuous, Apr 0 q ě Cr α 0 . Let f 0 be the rescaled function

f 0 pt, x, vq " 1 C ˆρ 2r 0 ˙α f ˜ˆr 0 ρ ˙2s t, ˆr0 ρ ˙2s`1 x, r 0 ρ v ¸.
Since Aprq ď Cr α for r ą r 0 , osc

Q R f 0 ď pR{2q α for R ą ρ.
tf ě u

B r

The set A on BB 1

A is contained inside a band of width at most C{|v| Here K f pv, v 1 q is bounded below The second item in Lemma A.3 says that there is a universal lower bound on the density of Ξpvq inside the cone of v 1 given by (A.2). This is all we will use in order to prove the coercivity estimate below.

The third item in the properties of A says that for any v 1 P Ξpvq, (A.2) |v ¨pv 1 ´vq| ď C|v 1 ´v|.

This third point plays no role in the local version of the coercivity estimate. It is useful to understand the global coercivity estimate as explained in Remark A.7.

A.1.2. Proof of the lower bound. It turns out that the conditions on the kernel K f given by Lemma A.3 plus the cancellation lemma is all we need to obtain the bound from below of Proposition A.1. For some arbitrary R ą 0, let us call µ :" cp1 `Rq ´1 to the lower bound on the one-dimensional measure of the intersection of Apvq with big circles as in the last item of Lemma A.3. We start with a few preparatory lemmas. Proof. The projection of the line L ´v on the sphere S d´1 is half of a big circle. According to Lemma A.3, the intersection of this projection with the set of directions A " Apvq has (one-dimensional) measure at least µpvq{2 (recall that Apvq is symmetric). At least half of these directions form an angle with L of at least µpvq{8.

For each of these points z P Apvq Ă S d´1 , there corresponds an actual intersection point in v `αz P Ξpvq X L, with α P rρ, 8µpvq ´1ρs. Thus, the one dimensional measure of the points v `αz P Ξpvq X L X B Cρ is bounded below by cρ, where C " 8µpvq ´1 and c " µpvq{4.

Lemma A.5. Let Ξpvq be the cones corresponding to the directions Apvq as in Lemma A.3. Let v 1 and v 2 be two points in R d . Assume |v 1 | ě |v 2 |. Let µpv 1 q ě µ 0 and µpv 2 q ě µ 0 for some µ 0 ą 0 . We have |Ξpv 1 q X Ξpv 2 q X B r pv 2 q| ě c|v 1 ´v2 | d , where r " C|v 1 ´v2 |, and c and C depend on µ 0 only.

Proof. The lines L contained in Ξpv 1 q are indexed by their directions e P Apv 1 q. At least half of these lines form an angle θpµq ą 0 with the vector v 2 ´v1 . In particular, all such lines are at distance at least c|v 1 ´v2 | from v 2 , where c depends on µ and d only. According to Lemma A. [START_REF] Alexandre | A review of Boltzmann equation with singular kernels[END_REF], for all such L, |Ξpv 2 q X L X B r pv 2 q| ě c|v 1 ´v2 |.

Integrating over all directions e P Apv 1 q we conclude the lemma. The following lemma is the main estimate in the context of integro-differential equations, which implies Proposition A.1 when combined with the cancellation Lemma.

Lemma A.6. Let K : R d ˆRd Ñ R be a non-negative kernel like in Lemma A.3. Let µ " inftµpvq : v P B R u.

Then, there is a constant c ą 0, depending only on µ , λ , d and s, so that for any function g supported in B R , ¨Rd ˆRd pgpvq ´gpv 1 qq 2 Kpv, v 1 q dv dv 1 ě c}g} 2 9 H s .

Proof. Symmetrizing the integral, we can replace Kpv, v 1 q by 1 2 pKpv, v 1 q `Kpv 1 , vqq. Thus, we assume that K is symmetric.

From Lemma A.5, we have that for all v 1 , v 2 P B 2R , there is a constant C 0 (sufficiently large depending on R and the various constants involving f ) so that |Ξpv 1 q X Ξpv 2 q X B C0|v1´v2| pv 2 q| ě c|v 1 ´v2 | d .

Note that B C0|v1´v2| pv 2 q Ă B pC0`1q|v1´v2| pv 1 q. Moreover, we can choose c 0 small enough so that |Ξpv 1 q X Ξpv 2 q X B C0|v1´v2| pv 2 qzB c0|v1´v2| pv 1 qzB c0|v1´v2| pv 2 q| ě c|v 1 ´v2 | d .

For the same choice of constants c 0 and C 0 , let N r pvq :" ˆBC 0 r pvqzBc 0 r pvq |gpvq ´gpwq| 2 Kpv, wq dw.

Therefore, for any v 1 , v 2 P B 2R , using that |gpv 1 q ´gpzq| 2 `|gpv 2 q ´gpzq| 2 ě |gpv 1 q ´gpv 2 q| 2 {2, if we let r " |v 1 ´v2 |, N r pv 1 q `Nr pv 2 q ě c ˜ˆΞpv1qXB C 0 r pv1qzBc 0 r pv1q |gpv 1 q ´gpzq| 2 |v 1 ´z| ´d´2s dz `ˆΞpv2qXB C 0 r pv2qzBc 0 r pv2q |gpv 1 q ´gpzq| 2 |v 2 ´z| ´d´2s dz ¸, ě c ˜ˆΞpv1qXΞpv2qXB C 0 r pv2qzBc 0 r pv1qzBc 0 r pv2q |gpv 1 q ´gpv 2 q| 2 r ´d´2s dz ¸, This finishes the proof.

ě
Once we have Lemma A.6, we can derive Proposition A.1 as a corollary.

Proof of Proposition A.1. It follows from a simple computation using Lemma A.6 and Lemma 3.6.

´ˆL v gpvqgpvq dv " ¨Rd ˆRd pgpvq ´gpv 1 qqgpvqK f pv, v 1 q dv 1 dv, " 1 2 ¨Rd ˆRd pgpvq ´gpv 1 qq 2 K f pv, v 1 q dv 1 dv `1 2 ˆRd gpvq 2 ˆˆR d pKpv 1 , vq ´Kpv, v 1 qq dv 1 ˙dv, ě λ}g} 2 H s ´Λ}g} 2 L 2 . The first term was bounded using Lemma A.6 and the second term using Lemma 3.6.

Remark A.7. We sketch the precise asymptotics of the coercity estimate for large velocities. This computation plays no role in this paper, but it is interesting to see how the metric introduced in [START_REF] Gressman | Global classical solutions of the Boltzmann equation without angular cut-off[END_REF] arises naturally from the geometry described above. We only analyze the symmetric part of the bilinear form as in Lemma A.6. Analyzing the full bilinear form requires another similar computation for the cancellation estimate.

For large values of v, the cone Apvq is approximately of width 1{|v| and perpendicular to v. The lower bound in Lemma A.6 depends only on a lower bound for µpvq in B R and the lower bound for Kpv, v 1 q for v 1 P Ξpvq. It is easy to see how the estimate behaves for large velocities from a scaling argument. Indeed, let v 0 P R d . For every v P B 1 pv 0 q, the cone Ξpvq has measure µpvq Á p1 `|v 0 |q ´1 and it is approximately perpendicular to v 0 in the sense described above. Let T be the linear change of variables T v " p1 `|v 0 |q ´1P v `P K v, where P v " xv, v 0 y |v 0 | 2 v 0 , P K v " v ´P v.

Lemma A.12 (Maximum principle). If f is a weak subsolution of (1.2), with h " 0, in Q " pa, bsˆΩ x ˆΩv , then esssup Q f ď esssup f pt, x, vq : t P pra, bs ˆΩx ˆRd qzpa, bs ˆΩx ˆΩv ( .

Proof. Let m " esssuptf pt, x, vq : t P pra, bs ˆΩx ˆRd qzpa, bs ˆΩx ˆΩv u. The proof follows using pf ´mq às a test function in (5.5).

Figure 2 .

 2 Figure 2. The transformation leaving the equation invariant. On the left, a straight cylinder centered at the origin. On the right a slanted cylinder centered at z 0 .

4. 1 . 1 .

 11 Estimate of the symmetric part.

( 4 . 6 )

 46 ˆBR pvq |Apv, wq| dw À R 2´2s and ˆBR pwq |Apv, wq| dv À R 2´2s .

Lemma 4 . 5 (

 45 The multi-path lemma). Let f : R d Ñ R be any twice differentiable function. The following inequality holds for any pair of points v, v 1 P R d .

4. 4 .

 4 Commutator estimates. Lemma 4.10 (Commutator estimate for s P p0, 1{2q). Let us assume s P p0, 1{2q and that K satisfies (4.2). Let D be a closed set and Ω open so that D Ť Ω Ă R d . Let ϕ be a smooth function supported in D and f P H s pΩq X L 8 pR d q.

5. 1 .

 1 Reduction to global kernels. Proposition 5.1 (A kernel defined globally). Assume that K : B R ˆRd Ñ R satisfies (1.3), (1.5), (1.6) and (1.7). There exists a kernel K : R d ˆRd Ñ R satisfying the following global version of assumptions (1.3), (1.5), (1.6) and (1.7).

Figure 3 .

 3 Figure 3. The cylinders Q and Q

Corollary 7 . 5 .

 75 Let f be a bounded continuous function in R d and Lipchitz in some open set Ω. The functions M ´f and M `f are continuous in Ω.

Figure 4 .

 4 Figure 4. The geometric setting of Lemma 8.2.

Q Q 1 Figure 5 .

 15 Figure 5. Geometric setting of the propagation Lemma 9.1.

Theorem 10 . 3 .

 103 Let f P L 1 pΩ, dx b dv b dtq where Ω is an open set of R 2d`1 . Then for a.e. pt, x, vq P Ω, lim rÑ0 ` Qrpt,x,vq |f ´f pt, x, vq| dx dv dt " 0.

  xq : 0 ă t ´tj ď mr 2s j , |x ´xj ´pt ´tj qv j | ă pm `2qr 1`2s j u.The set in the left hand side would be empty when |v ´vj | ě r j .When |v ´vj | ă r j , we have |pt ´tj qpv j ´vq| ă mr 1`2s j . Therefore, we can switch v j with v in the last term, changing the right hand side, and we obtain a smaller set. pt, xq : pt, x, vq P Qm j ( Ą tpt, xq : 0 ă t ´tj ď mr 2s j , |x ´xj ´pt ´tj qv| ă 2r 1`2s j u.

  |ărj tpt, zq : 0 ă t ´tj ď mr 2s j , |z ´xj `tj v| ă 2r 1`2s j u ˇˇˇ.

Lemma 11 . 1 . 1 `

 1111 Let r 0 and R 1 as in Theorem 1.6 and ρ " r 0 {R 1 . Let Q´b e Q´: " Q ρ p´R ´2s ρ2s , 0, 0q.

Figure 9 .

 9 Figure 9. The geometric setting of Lemma A.3. The cone Ξpvq is generated by the set tf ě lu.

Lemma A. 4 .

 4 (See Figure10) Let Ξpvq be the cones corresponding to the sets of directions Apvq as in Lemma A.3. Let L be a line in R d at distance ρ ą 0 from a point v P R d . Then, for some constants c and C depending only on µpvq, |Ξpvq X L X B Cρ | ě cρ.

Figure 10 . 1 v 2 Figure 11 .

 101211 Figure 10. Intersection of a line L with a cone Ξpvq.

  4.1.2. Estimates of the anti-symmetric part. Finding an appropriate upper bound for E when K is not symmetric is more complicated than for E sym . The cancellation assumptions (4.3) and (4.4) are necessary. We will prove the estimates differently for the case s P p0, 1{2q and s P r1{2, 1q. Note that the hypothesis (4.4) is only used in the later case. Lemma 4.3 (Estimate of L v f for s ă 1{2). Assume s P p0, 1{2q. Let K be a kernel satisfying (4.2). The following estimate holds

  1`2s ˆBR |. Therefore, taking supremum in t 0 ,

	sup tPr0,T s ď ¨g2 p0, x, vq dv dx 1 ¨g2 pt, x, vq dv dx 2 `ε	`ˆT 0 ˆT 0 ˆ}g} 2 ˆ}g} 2 H s dx dt

  ´h ¯dv dx dt, Lemma 8.2 (Intermediate sets for the Kolmogorov equation). Let s P r1{2, 1q. Let f : r0, T s ˆRd ˆRd Ñ r0, 1s. Assume f is a supersolution of the fractional Kolmogorov equation

	ˆR2d
	"	pf pT, x, vq ´f p0, x, vqq φpx, vq dv dx
	`ˆr0,T sˆR 2d	! r´v ¨∇x φ `p´∆q s φs f ´φ	h)	dv dx dt ď C.

  8 pt, xq " ˆRd f 8 pt, xqϕpv ´v0 q dv. P B r3{2 , f 8 pt, xq satisfies the transport equation B t f 8 `v0 ¨∇x f 8 ě H v0 pt, xq.

	Therefore, using the equation ˆRd
	B t f 8 pt, xq ě	"	´v ¨∇x f 8 pt, xq ´p´∆q s f 8 pt, x, vq `h8 1 pt, x, vq `p´∆q s{2 h 8 2 pt, x, vq	‰	ϕpv ´v0 q dv,
	" ´v0 ¨∇x f 8 pt, xq	`ˆR d	tp´f 8 pt, x, vq `h8 2 pt, x, vqqp´∆q s ϕpv ´v0 q `h8 1 ϕpv ´v0 qu dv.
	Thus, for any v 0

  1`2s , H s pB r4´ρ qq, with ˆT 0 ˆBr 4 ´ρ }f } 2 H s pBr 4 ´ρq dx dt ď C, for some constant C depending only on r 4 , ρ, d, λ, Λ and s.Let ϕ : R d ˆRd Ñ r0, 1s be a smooth bump function supported in B pr4´2ρq 1`2s ˆBr4´2ρ and such that ϕ " 1 in B r1`2s

	3	ˆBr3 . We now have
	(8.3)	ˆT 0 ˆRd	}ϕf } 2 H s pR d q dx dt ď C.

.

  Then ˇˇˇˇď

	j	Qm j	ˇˇˇˇě m m `1 ˇˇˇˇď

j Q j ˇˇˇˇ.

Proof. Because of Fubini's theorem, we know that for any set A Ă R ˆRd ˆRd , |A| " ˆ|tpt, xq : pt, x, vq P Au| dv.

Therefore, in order to prove the lemma, it is enough to show that for every v P R d , (10.2) ˇˇˇˇ# pt, xq : pt, x, vq P ď j Qm j +ˇˇˇˇˇě m m `1 ˇˇˇˇ# pt, xq : pt, x, vq P ď j Q j

  C w.h.i.We next pick m large enough (depending on δ) and then C w.h.i. large enough (depending on δ and m) so Now imposing δ 1 ă pc{2qδ, we get the desired inequality:

		m m `1	p1 ´cδq 1 `C´1 w.h.i. Cm (	p1 ´δ1 q k .
	that
	m m `1	p1 ´cδq 1 `C´1 w.h.i. Cm (	ď 1 ´pc{2qδ.

  The rescaled kernel in Lv satisfies the same assumptions (1.3), (1.5),(1.6), (1.4) if s ă 1{2, and (1.7) if s ě 1{2. Note that Q R1 contains p´1, 0s ˆBR 1`2s 1 ˆBR1 . Let f`" maxp f , 0q. This function satisfies the following equation in Q R1 , B t f``v ¨∇x f`ě ˆRd p f`p wq ´f `pvqqK pv, wq dw `h ´ˆ|w|ě2R1 f´p wqKpv, wq dw, Corollary 11.2. Let f be a solution of (1.2) in Q 1 with |h| ď ε 0 . Assume that

	Applying Theorem 1.6 (the weak Harnack inequality), we get
	f`ě	˜ˆQ ´f ε	¸1{ε	´2ε 0	in Q ě
		1 2	| Q´|	1{ε ´2ε 0
	ě δ				for ε 0 and δ sufficiently small.
	Rescaling back to f , we finish the proof.
					osc p´1,0sˆB1ˆB R	f ď	ˆR 2	˙α0	for all R ě 2.

ě ˆRd p f`p wq ´f `pvqqK pv, wq dw ´2ε 0 provided α 0 is small.

  c|gpv 1 q ´gpv 2 q| 2 |v 1 ´v2 | ´2s . ¨B2ˆB2 |gpv 1 q ´gpv 2 q| 2 |v 1 ´v2 | ´d´2s dv 1 dv 2 , ď C ¨B2ˆB2 pN r pv 1 q `Nr pv 2 qq|v 1 ´v2 | ´d dv 1 dv 2 here r " |v 1 ´v2 |, " 2C ¨B2ˆB2 N r pv 1 q|v 1 ´v2 | ´d dv 1 dv 2 ,ď C ˆB2 ˆ8 r"0 ˆSd´1 N r pv 1 qr ´1 dσ dr dv 1 , using polar coordinates for v 2 ,

	Therefore
	}g} 2 H s ď C
	" C	ˆB2 ˆRd	|gpv 1 q ´gpzq| 2 Kpv 1 , zq	˜ˆc ´1 0 |v1´z| C ´1 0 |v1´z|

r ´1 dr ¸dz dv 1 , " C ˆB2 ˆRd |gpv 1 q ´gpzq| 2 Kpv 1 , zq dz dv 1 .

. LS is supported in part by NSF grants DMS-1254332 and DMS-1362525.

In particular, since α ď α 0 , we can apply Corollary 11.2 and obtain that osc Qρ f 0 ď 1 ´δ.

Therefore, in terms of the original function f , Apr 0 q ď C ˆ2r 0 ρ ˙α p1 ´δq.

This contradicts that Apr 0 q ě Cr α 0 since α ă lnp1 ´δq{ lnpρ{2q, and we finish the proof.

Appendix A. New proofs of known estimates and technical lemmas A.1. The coercivity estimate for the Boltzmann kernel. In this appendix we give a geometric proof of the following coercivity estimate for the Boltzmann equation. It says that the Boltzmann kernel satisfies the assumption (1.3).

Proposition A.1. Assume that the function f satisfies the inequalities

Assume also that f ˚|¨| γ is bounded by some constant K 0 . This bound is controlled by M 0 and E 0 if γ P r0, 2s, and it is an extra assumption when γ ă 0.

The constants λ and Λ depend only on M 0 , M 1 , E 0 , H 0 , K 0 the dimension d and the radius R.

In particular, for an appropriately larger constant Λ, ˆRd Qpf, gqpvqgpvq dv ď ´λ}g} 2 9

Note that the extra assumptions about the boundedness of f ˚| ¨|γ comes from the usual condition for the classical cancellation Lemma to give us a bounded function. It is the same assumption as in Lemma 3.6.

The constant c above may go to zero as R Ñ 8 depending on the value of γ. The precise optimal rate for this can be easily deduced from the proof. We explain this in remark A. [START_REF] Alexandre | Regularity of solutions for the Boltzmann equation without angular cutoff[END_REF] The proofs of Proposition A.1 that can be found in the Boltzmann literature are done using Fourier analysis. Here, we present a relatively elementary proof based on a direct computation and a geometric argument in physical variables.

We define K f , Q 1 pf, gq " L v g and Q 2 pf, gq " cpf ˚| ¨|γ q g as described in Section 3.

In [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF], there is an estimate for K f in terms of a simplified integral expression. It says that (A.1) K f pv, v 1 q « ˜ˆtw¨pv 1 ´vq"0u f pv `wq |w| γ`2s`1 dw ¸|v 1 ´v| ´d´2s .

A.1.1. Lower bounds for K f in a cone of directions. We obtain a lower bound for K f pv, v 1 q in a symmetric cone of directions with vertex v. This was done in [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF]. It follows essentially from the following lemma.

Lemma A.2. Let f : R N Ñ R be as in Proposition A.1. There exists an r ą 0, ą 0 and m ą 0 depending on M 1 , E 0 and H 0 such that |tv :

Combining Lemma A.2 with the expression (A.1), we deduce the following statement. It is essentially the same as Lemma 4.8 in [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF], but with a more detailed description of the cone of directions where the lower bound holds.

Lemma A.3. Let f : R N Ñ R be non-negative and

For any v P R d , there exists a set of directions A " Apvq P BB 1 , so that K f pv, v 1 q ě λp1 `|v|q 1`2s`γ |v v1 | ´d´2s for all v 1 so that pv 1 ´vq{|v 1 ´v| P A. Moreover, this set of directions A Ă S d´1 satisfies the following properties.

' A is symmetric: A " ´A. ' Any big circle in S d´1 intersects A on a set of (one dimensional) measure at least cp1 `|v|q ´1. In particular, the pd ´1q dimensional measure of A is at least µpvq :" cp1 `|v|q ´1. ' A is contained on a strip of width ď Cp1 `|v|q ´1 around the equator perpendicular to v.

By a big circle, we mean a closed geodesic in S d´1 . They are the intersection of S d´2 with any 2dimensional subspace.

The proof of Lemma A.3 is similar to the one of Lemma 4.8 in [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF]. Here we have a more precise description than in that paper because we add a lower bound of the measure of the intersection of A with any big circle instead of only its total measure. The proof is relatively easy to explain with a picture on the blackboard, but perhaps somewhat cumbersome to write down.

Proof. Let F " tv : f pvq ą u X B r be the set described in Lemma A.2, which has measure at least m.

From the formula for K f given in (A.1), one immediately sees that σ P Apvq when the hyperplane perpendicular to σ intersects F in a set of measure at least cm{r, with λ " c m{r.

The three properties described in the lemma are simple geometric consequences of this construction using only that the measure of F is bounded below and F Ă B r for some given constant r. Indeed, as σ takes all values on a big circle in BB 1 , its perpendicular hyperplanes swipe the space R d (see Figure 8). σ Figure 8. As σ moves along a big circle in BB 1 , its perpendicular planes swipe the space Because of Fubini's theorem, the points σ on that big circle for which its perpendicular hyperplane intersects F in a set of measure at least cm{p1 `|v|q has to be at least of measure cm{p1 `|v|q.

Note that depending on the direction of the big circle, the lower bound on its intersection with Apvq could be improved. For example, if the big circle is perpendicular to v, the measure of its intersection with Apvq is bounded below independently of v. This fact will not be relevant to any of the computations below.

Figure 9 is taken from [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF] and shows all the elements in Lemma A.3. We may call Ξpvq the symmetric cone of values of v 1 so that pv 1 ´vq{|v 1 ´v| P A. In particular, the lower bound K f pv, v 1 q ě λp1 `|v|q 1`2s`γ |v ´v1 | ´d´2s holds when v 1 P Ξpvq.

Let gpvq " gpT vq. So that ¨Rd ˆRd pgpvq ´gpv 1 qq 2 Kpv, v 1 q dv dv 1 " ¨Rd ˆRd pgpvq ´gpv 1 qq 2 Kpv, v 1 q dv dv 1 , where Kpv, v 1 q " | detpDT q| 2 KpT v, T v 1 q " p1 `|v 0 |q ´2K pT v, T v 1 q.

The point of this change of variables is to make the non-degeneracy cone K bounded below in measure for all v P B 1 pT ´1pv 0 qq, uniformly in v 0 , i.e. μpvq Á 1 for all v P B 1 pT ´1v 0 q. Moreover, for v 1 in this nondegeneracy cone Ξpvq, we have

Therefore, from the computation in the proof of Lemma A.6, for some universal constant r ą 0, and D r " T pB r pT ´1v 0 qq, we get ¨D1ˆD1 pgpvq ´gpv 1 qq 2 Kpv, v 1 q dv dv 1 " ¨B1pT ´1v0qˆB1pT ´1v0q pgpvq ´gpv 1 qq 2 Kpv, v 1 q dv dv 1 ,

Note that |T ´1v ´T ´1v 1 | is equivalent to the metric dpv, v 1 q introduced in [START_REF] Gressman | Global classical solutions of the Boltzmann equation without angular cut-off[END_REF]. The set D r is exactly the ball of radius r centered at v 0 . Therefore, covering R d with these balls D r pv 0 q and adding up, we get ¨dpv,v 1 qă1 pgpvq ´gpv 1 qq 2 Kpv, v 1 q dv dv 1 ě cp1 `|v 0 |q 1`γ`2s ¨dpv,v 1 qăr |gpvq ´gpv 1 q| 2 dpv, v 1 q d`2s dv dv 1 .

The right hand side is the same as the norm }g} 2 N s,γ introduced in [37] minus a lower order correction corresponding to the tails of the integral.

Remark A.8. It is interesting to notice that the estimate of Lemma A.6 depends only on the structure of the kernel described in Lemma A.3. It would be interesting to see whether the result of Lemma A.6 holds for general kernels K (not necessarily arising from the Boltzmann equation) under less restrictive conditions on the cones Apvq. There is an interesting conjecture mentioned in [START_REF] Dyda | Regularity estimates for elliptic nonlocal operators[END_REF] which is related to our condition (1.4).

A.2. Technical lemmas.

A.2.1. Change of variables. We recall here a change of variables from [START_REF] Silvestre | A new regularization mechanism for the boltzmann equation without cut-off[END_REF].

Other changes of variables were used in proofs. Here the constant ω d´2 stands for the surface area of S d´2 . Note that the integrals on the left hand side are on spheres and hyperplanes, thus dw and dσ stand for differential of surface.

A.2.2. Positive part of subsolutions.

Lemma A.11 (Positive part of subsolutions). Let f be a subsolution of (1.2) in a cylinder Q. Then f `" maxpf, 0q is still a subsolution of (1.2) (where h is replaced with h1 f ě0 ) in Q.

Proof. Since we assume that f t `v ¨∇x f P L 2 , then

The equality holds in the sense of distributions.

In order to conclude that f `is a subsolution of (1.2), we need to prove the following inequality in the sense of distributions.

Let γprq " r `and let tγ δ u δ be a smooth approximation of γ such that |γ 1 δ | ď 1 and γ δ is convex. Let ρ ε be an even mollifier and f ε " ρ ε ˚f . Here the mollification is done with respect to the variable v only.

Since f P L 2 pr0, T s, R d , H s pR d qq, it is not hard to see that lim This proves that L v rγ δ pf ε qs converges to L v rf `s in the sense of distributions. Thus, in order to obtain (A.9) and finish the proof, we need to prove that for all δ ą 0 and ε ą 0, (A.10) L v rγ δ pf ε qs ě γ 1 δ pf ε qL v rf ε s. In fact, we can check by a direct computation that the inequality holds pointwise. Indeed, L v rγ δ pf ε qspvq " ˆpγ δ pf ε pv 1 qq ´γδ pf ε pvqqqKpv, v 1 q dv 1 , ě ˆγ1 δ pf ε pvqqpf ε pv 1 q ´fε pvqqKpv, v 1 q dv 1 using the convexity of γ δ , " γ 1 δ pf ε pvqqL v f ε pvq. Taking the limit as δ Ñ 0 and ε Ñ 0 in (A.10) we obtain (A.9). Combining it with (A.8) we finish the proof.