QUANTITATIVE DE GIORGI METHODS IN KINETIC THEORY FOR NON-LOCAL OPERATORS

. We derive quantitatively the weak and strong Harnack inequality for kinetic Fokker- Planck type equations with a non-local diﬀusion operator for the full range of the non-locality exponent s ∈ (0 , 1). This implies H¨older continuity. We give novel proofs on the boundedness of the bilinear form associated to the non-local operator and on the construction of a geometric covering accounting for the non-locality to obtain the Harnack inequalities. Our results apply to the inhomogeneous Boltzmann equation in the non-cutoﬀ case.

1. Introduction 1.1. Problem Formulation. We consider non-local kinetic equations of the form for someR > 0, where we assume h = h(t, x, v) is a real scalar field in L ∞ , and for a non-negative measurable kernel K : in the principal value sense. The question we raise is whether solutions to (1.1)-(1.2) satisfy Hölder continuity and Harnack inequalities.
We make the following assumptions on the kernel for v ∈ BR. We require the following upper bound for r > 0 for some constants 0 < λ < Λ with 0 < s < 1. We also need a coercivity condition Moreover, instead of the usual symmetry assumption K(v, w) = K(w, v), which corresponds to the divergence form, we assume the following cancellation March 29, 2022. and if s ≥ 1 2 we assume that for all R > 0 We remark that the upper bound (1.3) is equivalent to We also note that (1.6) holds for s ∈ (0, 1 2 ) as a consequence of (1.5) and (1.7). These assumptions lean on the work of Imbert and Silvestre in [14]. We want to assume conditions that apply to the Boltzmann collision kernel in the non-cutoff case. We know from section 3 of [14] that the Boltzmann kernel satisfies assumptions (1.3)-(1.6) provided that physically relevant macroscopic quantities stated in Assumption 1.1 of [15] are bounded.
Equation (1.1) is invariant under Galilean transformation, i.e. under the family of transformations z → z 0 • z = (t 0 + t, x 0 + x + tv 0 , v 0 + v) where z 0 = (t 0 , x 0 , v 0 ) ∈ R 1+2d . It is also invariant under scaling f r (t, x, v) = f (r 2s t, r 1+2s x, rv) for r ∈ [0, 1], in the sense that f r solves a modified equation where the scaled kernel satisfies assumptions (1.3)-(1.6) in a larger radiusR r and the scaled source term is bounded by h. This motivates why we consider (1.1) in the cylinder for r > 0 and z 0 = (t 0 , x 0 , v 0 ) ∈ R × R d × BR. For later reference, we also introduce the cylinder shifted to the past Q − r (z 0 ) := Q r (z 0 − (2r 2s , 2r 2s v 0 , 0)), so that in particular for z 0 = 0 Q − r := Q r (−2r 2s , 0, 0) = (−3r 2s , −2r 2s ] × B r 1+2s × B r . Similarly the cylinder shifted to the future is denoted as Q + r (z 0 ) := Q r (z 0 + (2r 2s , 2r 2s v 0 , 0)). Our equation involves a transport term, which transfers some regularity of the velocity variable to the space variable. It also involves a non-local diffusion in the v variable. It is a non-linear equation since the kernel K depends on the solution f in general. Our motivation to study the regularity of this type of equation is linked to the question of well-posedness for smooth classical solutions of the inhomogeneous Boltzmann equation without cut-off. There are linear kinetic equations whose solutions in the hydrodynamic limit are described by a fractional diffusion [2,19,20]. Indeed diffusion limits of the linear Boltzmann equation with a heavy-tailed distribution of infinite variance as an equilibrium distribution give rise to a fractional diffusion equation [20]. Such heavy-tailed distribution functions arise in astrophysical plasmas [21] or also in granular gases through dissipative collision mechanisms [5]. However, the only source of fractional diffusion at the kinetic level stems from long range interactions of the Boltzmann collision kernel.
In the limit case s → 1 equation (1.1) models the local kinetic Fokker-Planck equation, whose study is motivated by applications to the Landau equation [8]. For the local case, there is a nonconstructive method discussed in [8]. A constructive proof first appeared in a series of works [28][29][30] for ultraparabolic equations that has further been developed in [10] to local kinetic Fokker-Planck type equations. The construction is based on a Poincaré-type inequality and Kruzhkov's method [18]. A novel constructive approach for local kinetic Fokker-Planck type equations has been devised by Guerand and Mouhot in [11]. Their method relies on trajectories. For general s ∈ (0, 1) Cyril Imbert and Luis Silvestre (together with Clément Mouhot in [12,13]) made important contributions in a series of papers [12][13][14]16,25] that culminated in the final work [15]. In [25] Silvestre proves for a certain range of s that any solution f is a priori essentially bounded provided that the hydrodynamic quantities of mass, energy and entropy satisfy some uniform bounds. In section 3 of [14] they show that these hydrodynamic bounds imply assumptions (1.3)-(1.6) on the kernel. With an additional non-degeneracy assumption, they obtain the weak Harnack inequality with a quantitative argument in case that s ∈ (0, 1 2 ) [14]. Note that their method, which uses barrier functions, can be extended to s ∈ (0, 1) under the additional symmetry assumption K(v, v + w) = K(v, v − w), cf. section 7 in [14]. In case that s ∈ ( 1 2 , 1) their methods are non-constructive. In [16] Imbert and Silvestre derive Schauder estimates for kinetic equations, which can then be bootstrapped for the noncutoff Boltzmann equation to obtain smooth solutions [15]. To achieve global smooth solutions all these local estimates are turned into global ones by a change of variables, see section 5 in [15]. There has also been a work by Stokols [26] where he combines the method of [8] with fractional estimates from [3] to obtain a non-constructive proof of Hölder continuity in the non-local case. The assumptions he poses on the kernel are stronger than ours. Our assumptions coincide with those in [14] apart from the non-degeneracy assumption (Equation (1.4) in [14]) that is required for the construction of the barrier functions in Imbert and Silvestre's work.
The following work is a constructive proof of Hölder continuity and Harnack inequalities, leaning on the work of Jessica Guerand and Clément Mouhot [11]. We generalise De Giorgi's method [7] in the kinetic context developed in [11] to the non-local case. These methods were originally established for non-linear elliptic equations by De Giorgi [7]. Moser then showed how to deduce a Harnack inequality [22,23]. The weak and strong Harnack inequality are local regularity results. In particular, the weak Harnack inequality implies Hölder continuity [6]. The assumptions we pose on the kernel are weak enough so that they are satisfied by the Boltzmann collision kernel. By using an argument based on trajectories, we simplify the barrier method used in [14] to obtain Hölder continuity for the non-cutoff Boltzmann equation.

1.2.
Contribution. Our contribution consists of a quantitative proof of regularity for fractional Fokker-Planck type equations. Our results are applicable to the non-cutoff Boltzmann equation. We prove Harnack inequalities and Hölder continuity: Then there is C and ζ > 0 depending on s, d, λ, Λ such that the weak Harnack inequality is satisfied: with C depending on d, s, λ, Λ.
where C depends on d, s, λ and Λ.
1.3. Structure of the article. In section 2 we fix the notation and state some preliminary considerations on the definition of weak solutions for (1.1). In particular we state a result on the boundedness in H s × H s of the bilinear form (2.1) corresponding to the non-local diffusion. The theorem was first proved in section 4 of [14]. We give a different proof for the anti-symmetric part of the operator in Theorem 2.1 below. We proceed with the proof in three steps.
For De Giorgi's first lemma we prove an energy estimate 3. For De Giorgi's second lemma we start section 5 with a weak Poincaré inequality in L 1 , which we need in the proof of the intermediate value theorem 5.2. The proof of the former is based on trajectories, as was first employed by [11]. The latter theorem 5.2 then follows just as in [11]. The intermediate value theorem implies together with a direct consequence of De Giorgi's first lemma 4.2 a measure-to-pointwise estimate 5.3, which is De Giorgi's second lemma.
As a last step, we deduce Hölder continuity and the Harnack inequalities in section 6. Hölder continuity follows by standard methods. For the Harnack inequalities, we use a covering argument, which we adapt from [11]. The geometric construction for the covering had to account for the fractional diffusion.

Notation.
A constant is called universal, if it only depends on the dimension, the fractional exponent s and λ, Λ in (1.3)-(1.4). We use the notation a b if there exists a universal constant C such that a ≤ Cb. Moreover, we say that a d b if a ≤ Cb where C = C(d). For a real number a we denote a + = max(a, 0).
For a given domain Ω ⊂ R d we denote withḢ s (Ω) the space that is equipped with the norm The space H s (Ω) is correspondingly equipped with the norm The space H s 0 (Ω) is defined as the closure of the space of smooth functions in R d with compact support contained in Ω, where the closure is taken with respect to the H s (Ω)-norm. We denote the dual of H s 0 (Ω) with H −s (Ω).

Bilinear Form.
We introduce the bilinear form associated to the kernel K In the remainder, we will abuse notation by ignoring the limit as ε → 0 and understanding some integrals in the principal value sense. The following theorem states the same result as Corollary 5.2 in [14]. (1.6).
there holds where C depends on s, d and Λ.
Proof. By density of C ∞ in Sobolev spaces it suffices to consider smooth f, g. We divide the proof into the symmetric and anti-symmetric part of the kernel: we write where the integrals are understood in a principal value sense.
For I 1 we first note that it suffices to only consider the quadratic form due to the polarisation identity I 1 (f, g) = 1 2 I 1 (f + g, f + g) − I 1 (f − g, f − g) . Thus we write as in the proof of Lemma 4.2 of [14] for where for r > 0 where we used Fubini's theorem and writeΣ r := {(v, w) ∈ BR × R d : r 4 ≤ |v − w| < 5r 4 } and Ω v,w for the set containing the u corresponding to any pair (v, w). We note that We can apply this final estimate to each term in the sum (2.2), use polarisation again and Cauchy-Schwarz inequality to obtain For I 2 we distinguish the far and near part for 0 < R <R to be determined below We rewrite I 21 with Fubini's theorem Then for I 21 we use the cancellation assumption (1.5) and (1.6). We get (2.4) For I 212 we use Taylor's theorem to write for some u between v and w. We can bound the first term as above in (2.4). For the second term we use (1.6)ˆB We used Corollary 2.5 in [17] and chose With this choice of R we get for the last termˆB . For the far part I 22 we have for u between v, w We used Taylor's theorem, the same proof as for the symmetric part to deduce the third inequality, repeatedly the Cauchy Schwarz inequality, the upper bound (1.3), Corollary 2.5 in [17] and the choice of R as above. Finally for I 23 we can use the exact same estimates as for I 22 when we first apply Fubini's theorem and expand g(v) instead of g(w): Now we use Littlewood-Paley theory inspired from the proof of Theorem 4.1 in [14]. We denote with ∆ i the Littlewood-Paley projectors. We decompose f = ∞ i=0 ∆ i f , where we use the convention that all low modes are contained in ∆ 0 so that the index i ≥ 0. Note that for s ≥ 0 Moreover, we bound as in [14] (for a justification see [1]) Then (2.6) Note that splitting the sum into j ≤ i and i < j works since the adjoints of the corresponding integral operators satisfy the same bounds as in (2.5). Thus we conclude with (2.3) and (2.6).
To motivate the following definition of weak solutions, we recall Lemma 5.6 of [14], which states that the operator L is bounded from the space (1.6). Then Proof. The proof comes from Lemma 5.6 in [14]. It suffices to consider smooth f, ϕ just as above. We write f = f 1 +f 2 with f 1 and f 2 as in the definition of the norm for the space Using the upper bound on the kernel, we find It is a solution if it is a sub-and super-solution.
We conclude this section with the following useful bound, where the integrals are understood in the principal value sense.
Proposition 2.4. Let v ∈ BR and assume K is a non-negative kernel satisfying (1.3). Then for r > 0 there holds We repeatedly use the non-negativity of the integrands. We can use this estimate for each summand in (2.7) to conclude.

3.1.
Kolmogorov's fundamental solutions. In this subsection, we consider the fractional Kolmogorov equation given by Proof. Equation (3.1) admits a fundamental solution, see for example Theorem 1.1 in [24] or Section 2.4 in [14], given by Since f and J are non-negative, we deduce that We remark that for any r ≥ 1 there holds for t > 0 In particular for r = p * we deduce We define as in Section 2.4 of [14] the modified convolution We remark that the modified convolution satisfies the usual Young inequality independent of t: Following the proof of Proposition 2.2 in [14], we split Let q ∈ [1, p * ) be such that 1 p = 1 q − 1 2 . By Young's inequality we get for α = d(1 For f 1 and f 2 we apply Young's inequality again and get This implies (3.2). To prove (3.3) we follow the idea of Lemma 10 in [11]. We split for (t, where ε > 0 and η is a smooth function on R + such that 0 ≤ η ≤ 1, equal to 1 in [0, 1] and 0 on [2, +∞) (we assume without loss of generality τ ≥ 1). Then we estimate for l ∈ N Now assuming ε < 1 these estimates yield The splitting on J yields a splitting on the solution f = f ε + f ⊥ ε . Young's convolution inequality and the convolution inequality on The decomposition above holds for all ε > 0; thus we can conclude the proof with the same justification as in the proof of Lemma 10 in [11]: Let σ ∈ 0, s 1+2s . Using the notation ζ := (1 + |ζ| 2 ) 1 2 we can decompose Therefore we find for the solution f to the fractional Kolmogorov equation (2s+1)(δ(2s+1)+2−2s) for some small δ > 0. This concludes the proof.
3.2. Energy estimates. The following two lemmas are the analogue of Proposition 9 and Proposition 11 in [11] respectively. The proofs are technically more involved due to the non-locality.

Lemma 3.2 (Local energy estimate). Let f be a non-negative sub-solution to
Proof. Let ϕ be a smooth function such that 0 ≤ ϕ ≤ 1 equal to 1 in Q r (z 0 ) and 0 outside Q R (z 0 ). We integrate (1.1) against f ϕ 2 ≥ 0 up to time τ . . .
First, we note that by choice of ϕ we have that I 2 = 0. Let us now deal with I 3 . We will split it into three parts . . . . . .

=:I33
Then for I 31 we get with Proposition 2.4 Now for the left hand side we find on the one hand For the right hand side we estimate using Cauchy Schwarz and Young's inequality The first term we absorb on the left hand side. Now we consider the error terms. We find We used 0 ≤ f ≤ 1 almost everywhere. For I 3,2 we get Similarly, we estimate I 1 . We obtain On the other hand, we integrate the transport term by partŝ Putting everything together we have We used the bounds |∂ t ϕ| 1 R 2s −r 2s , |∇ x ϕ| 1 (R 2s −r 2s )r . Finally we conclude by taking the supremum in τ ∈ (−r 2 + t 0 , t 0 ).

Lemma 3.3 (Local gain of integrability). Let f be a non-negative sub-solution of
where C ′ s, r, R, v 0 := 1 r s + 1 C(s, r, R, v 0 ) + 1 R 2s −r 2s + |v0|+R (R 2s −r 2s )r with C from Lemma 3.2 and Proof. Let ϕ be a smooth function with values in [0, 1] such that ϕ = 1 on Q r (z 0 ) and ϕ = 0 outside Q r+ R−r 2 (z 0 ). Consider g := f ϕ. Then there is some non-negative measurem such that where h 1 := Lf ϕ + hϕ + f T ϕ , h 2 := (−∆) s 2 v g and m :=mϕ ≥ 0. Note that we have by the energy estimate where in the last step we use Proposition 2.4 and Cauchy-Schwarz inequality. Therefore we get using the energy estimate 3.2 Using Lemma 3.1 we deduce .
To conclude the proof, we consider a second smooth cut-off function 0 ≤φ ≤ 1 such thatφ = 1 in Q r+ R−r 2 (z 0 ) andφ = 0 outside Q R (z 0 ). Integrating (1.1) againstφ yields just as in the energy estimate m

First Lemma of DeGiorgi
Then we have As a consequence, we get the following result.

Weak
Poincaré. This is where we introduce trajectories in order to obtain a hypoelliptic Poincaré-type inequality with an error term. The idea comes from Guerand and Mouhot [11].
For I 2 we find where we used 0 ≤ ϕ ε ≤ 1, the change of variables x → X = x + εw and w → V = x+εw−y t−s and the Cauchy-Schwarz inequality.
We perform the change of variables y → V = x+εw−y t−s , . . .

Then we bound
Now for I 311 we have We used Proposition 2.4, the symmetry of the fractional Laplacian, Fubini's theorem, the Cauchy-Schwarz inequality, some rescaling in the time variable and the bounds on ∇ y ϕ ε and ∇ w ϕ ε . With the same change of variables and rescaling in time, we also deduce Finally, for I 1 we estimate where we use 0 ≤ ϕ ε ≤ 1, the change of variables w → x ′ = x + εw and (3.6).
Putting everything together yields the claim, when we notice that ε σ > ε d(1+s) .
On the one hand, by assumption (5.2) we have On the other hand, we find . . .
For I 2 we get where we used the Cauchy-Schwarz inequality, Lemma 3.2 and 0 < f + < 1. Similarly we find for I 3 Therefore we can estimate the right hand side of (5.4) as Together with (5.5) this gives for some universal constant C. Now we choose ε < 1 and θ so that Cε σ ≤ δ1δ2 4 and δ 2 θ + Then we conclude and denote Just as in [11] we construct z l = (t l , x l , v l ) ∈ Q k+1 and r l > 0, l ≥ 1, m ≥ 3, n ≥ 1 so that C m·r l , l ≥ 1, are disjoint cylinders, A k+1 is covered by the family C 5m·r l [z l ] l≥1 .
Then by choice of z l and r l we have that C 5m·r l [z l ] ⊂ Q k . Note that for example, if s ≥ 1 2 we can choose n = 1.
This finishes the construction of the covering with the desired properties. We can then apply Lemma 5.3 to each C r l [z l ] to obtain C r l [z l ] + ⊂ A k and the (C r l [z l ] + ) l≥1 are disjoint since C r l [z l ] + ⊂ C m·r l [z l ].