Quantitative De Giorgi methods in kinetic theory

We consider hypoelliptic equations of kinetic Fokker-Planck type, also known as Kolmogorov or ultraparabolic equations, with rough coefficients in the drift-diffusion operator. We give novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities. This implies H{\"o}lder continuity with quantitative estimates. The paper is self-contained.

where A = A(t, x, v), B = B(t, x, v) and S = S(t, x, v) satisfy (for some constants 0 < λ < Λ): (1.2) A is a measurable symmetric real matrix field with eigenvalues in [λ, Λ], B is a measurable vector field such that |B| Λ, S is a real scalar field in L ∞ .
This equation naturally appears in kinetic theory where it is refereed to as the kinetic Fokker-Planck equation; it is included in the class considered by Kolmogorov [Kol34] (with constant A and linear B) that inspired the theory of hypoellipticity of Hörmander [Hör67] (see [AP20]).The coefficients are called "rough" because A, B and S in the drift-diffusion operator on the v variable are merely measurable with no further regularity.
Our class (1.1)-(1.2) is invariant under translations in t, x and under Galilean translations, i.e., under z → z 0 • z where z 0 = (t 0 , x 0 , v 0 ), z = (t, x, v) and with the non-commutative group operation Finally for any r > 0 it is invariant under the scaling z = (t, x, v) → rz := (r 2 t, r 3 x, rv).Using the invariances of the equation, we define for z 0 ∈ R 1+2d and r > 0: and we simply write Q r (0) = Q r when z 0 = 0. We denote |E| the Lebesgue measure of a Lebesgue set E. We write a b (resp.a b) when a Cb (resp.a Cb) for some constant C > 0 whose only relevant dependency, if any, is specified in the index, as in parameter .We write a ∼ b if a b and a b.We write ffl for integrals normalized by the volume of the integration domain, and Definition 1 (Weak solution, sub-solution, super-solution) Let U = (a, b) × Ω x × Ω v with −∞ < a < b +∞ and Ω x and and (1.1) is satisfied in the sense of distributions in U .A function f is a weak subsolution of (1.1) if and for all β : R → R in C 2 with β 0 and β 0 both bounded, and any non-negative ϕ ∈ C ∞ c (U ), It is a weak super-solution of (1.1) if −f is a weak sub-solution.
J.É.P. -M., 2022, tome 9 Remark 2. -This definition is equivalent to those in [PP04] and [GIMV19] in the case of solutions, but is weaker than them in the case of sub-and super-solutions.Indeed [PP04,GIMV19] make respectively the extra regularity assumption ).These assumptions were introduced to justify the energy estimates.It is however enough to assume the renormalization formulation above, and it allows to include important sub-solutions such as for instance f = f (t) = 1 t 0 (when S = 0) which were excluded by the definition in [PP04,GIMV19].Our definition is equivalent to that of De Giorgi in the elliptic case (and reminiscent of the definition of solutions in [GV15]).
1.2.Main contributions.-Given the invariances, we only state results in unit centered cylinders.Theorem 3 (Intermediate-Value Lemma).-Given δ 1 , δ 2 ∈ (0, 1), there are explicit constants both in (0, 1), such that any sub-solution f : e., we control the measure of where f is below 0 and above (1 − θ), satisfies 1).Remark 4. -This lemma is the kinetic quantitative counterpart of the quantitative elliptic [DG56,DG57,Vas16] and parabolic [Gue20] intermediate value lemma.As in the parabolic case, past and a future cylinders Q − r0 and Q + r0 are required to be disjoint but contrary to the parabolic case, a gap in time between the two cylinders is also required.This gap is also mentioned in [GIMV19,AP20].Let us explain why it cannot be removed.Consider for instance S = 0 and velocities bounded by |v| V m in the cylinder.Then 1 x+ct<a is a sub-solution for any a ∈ R and |c| > V m .If Q − r0 and Q r0 were too close, a line of discontinuity of the form x + ct = a could cross both and the previous sub-solution would contradict the conclusion of Theorem 3.
Theorem 5 (Harnack inequalities).-There is ζ > 0 depending only λ, Λ such that any non-negative weak super-solution f to (1.1)-(1.2) in Q 1 satisfies the weak Harnack inequality where r 0 = 1/20 and Q − r0/2 := Q r0/2 ((− 19 8 r 2 0 , 0, 0)) (see Figure 1), and any nonnegative weak solution f to (1.1)-(1.2) satisfies the following Harnack inequality (with Remarks 6 (1) The "weak" Harnack inequality, in spite of its name, is not weaker than Harnack inequality since it holds for super-solutions.Combined with the L ζ → L ∞ gain of integrability in Proposition 12, it implies the Harnack inequality for solutions.Supersolutions of the form 1 x+ct a for a ∈ R and |c| > V m (included in our definition) show that the gap in time is required in (1.5).
(2) The Harnack inequality for equation (1.1) was first proved in [GIMV19] by a non-constructive argument.The present paper provides a new constructive De Giorgi approach.Another constructive proof by the Moser-Kružkov approach is proposed in [GI21].The weak Harnack inequality was obtained for the long-range Boltzmann equation in [IS20], and was proved for the kinetic Fokker-Planck equations considered in this paper in [GI21] by the Moser-Kružkov approach.
(3) As compared to that in [GI21], our approach is based on trajectorial arguments and does not require working on the logarithm of the solution or the so-called inkspot lemma.Our Poincaré inequality (Proposition 13) and measure-to-pointwise estimate (Lemma 16) take into account a gap in time which removes the requirement for the sub-solution to be considered in a large domain.Our Poincaré inequality also holds without an information in measure around the center of the cylinder as in [GI21].
Theorem 7 (Hölder continuity).-There is α ∈ (0, 1), computable from the proof and only depending on λ, Λ and S L ∞ , such that any weak solution f of (1.1) Once these steps are proved, it is immediate to prove the Harnack inequality for solutions by combining the weak Harnack inequality for super-solutions and step (1) for sub-solutions.The Hölder continuity follows classically (see Subsection 4.2) from either the measure-to-pointwise estimate applied to both sub-solutions f and −f , or from the Harnack inequality.
Step (1) (Section 2) is semi-novel: it elaborates upon ideas in [PP04] to prove the first Lemma of De Giorgi as well as a gain of Sobolev regularity with the help of Kolmogorov fundamental solutions.Step (2) (Proposition 13) is the most novel step and introduces an argument based on trajectories and the previous Sobolev regularity to "noise" the x-dependency of the trajectories.Step (3) (proof in Subsection 3.2) is novel and based on simple energy estimates.Step (4) (Lemma 16 in Subsection 3.3) is standard and sketched for the sake of obtaining quantitative constants.Step (5) (in Section 4) is semi-novel but immediate when constants are quantified properly in the previous steps.Step (6) (in Section 4) is novel in the context of hypoelliptic equations but inspired from elliptic equations [LZ17]; it uses an induction, Vitali's covering lemma and Step (5) at every scale.
Acknowledgements.-The authors are grateful to C. Imbert for the inspirational interactions, as well as for specific help with the literature and the comparison with the Moser-Kružkov approach in [GI21].The second author would also like to thank L. Silvestre who pointed out several years ago how Kolmogorov fundamental solutions were used in [PP04] to replace averaging lemma, which was the starting point of our Section 2 (and is also used in [IS20]).

Integral estimates revisited
In this section, we briefly revisit estimates from [PP04, GIMV19] on the gain of integrability for sub-solutions (the kinetic counterpart to the first lemma of De Giorgi) and the low-order Sobolev regularity estimate for sub-solutions, first J.É.P. -M., 2022, tome 9 mentioned in [GIMV19].We provide new proofs based on fundamental solutions which, albeit variants of existing ones, seem simpler and optimal.

The energy estimate
Proposition 9 (Energy estimate).-Let f be a non-negative weak sub-solution to In order to use f ϕ 2 as a test function, we argue by density.Introduce where ψ n (t, x, v) = n 4d+2 ψ(n 2 t, n 3 x, nv) and ψ(t, x, v) := π −d−1/2 e −t 2 −|x| 2 −|v| 2 .Then The result follows from Cauchy-Schwarz' inequality and J.É.P. -M., 2022, tome 9 2.2.Integral estimates on Kolmogorov fundamental solutions.-We denote Lemma 10 (Estimates on the fundamental solution with constant coefficients) ) (a non-negative measure with finite mass) and where F 1 , F 2 and m have compact support in time included in some (−τ, 0].Then there for any p Proof of Lemma 10. -We use the fundamental solution computed by Kolmogorov in [Kol34] (see for instance [BDM + 20, App.A] for details): Since f and G are non-negative, we deduce that and since ) and therefore by integration by parts and Young's convolution inequality (which works in unimodular spaces like (R 2d+1 , •) with the Lebesgue measure), we deduce, by tracking down the dependency in p of the (The threshold 2 + 1/d is likely to be optimal.)This proves (2.2).To prove (2.3) split where ε > 0 and χ is a smooth function on R + valued in [0, 1] equal to 1 in [0, 1] and 0 on [2, +∞).We have the following simple estimates for every ∈ N which straightforwardly implies (assuming τ 1 and ε < 1 wlog) The splitting

and the convolution inequality
Since this decomposition holds for all ε > 0, it implies by standard interpolation the estimate (2.3) for any σ ∈ [0, 1/3) (again the exponent is likely to be optimal but in any case our constant degenerates as σ → 1/3).In order to be self-contained let us a give a short proof.Given σ ∈ [0, 1/3), we Fourier-transform and decompose dyadically, defining (2.4) function valued in [0, 1] and equal to 1 in B(0, 1) and 0 outside B(0, 2), and For a given F = F (y) one has by splitting the integrand into |x − y| 2 −k and |x − y| > 2 −k and integrating by parts the operator ∆ /2 ξ with even and strictly greater than d.We then use the decomposition (2.4) in the "a k " form on f ε and in the "B k " form on f ⊥ ε , and with a ε = ε k depending on k defined below: with the choice σ = 1/3 − δ ∈ [0, 1/3) and ε k := 2 −2k(1/3−δ/2) and > 1 + 4/9δ.This concludes the proof.
2.4.Iterated gain of integrability for sub-solutions.-We give a short proof of this result first obtained in [PP04, Th. 1.2] and then proved differently [GIMV19, Th. 12].This is the counterpart of the "first lemma of De Giorgi" for elliptic equations, in the context of kinetic hypoelliptic equations.We allow for an initial integrability L ζ with exponent ζ ∈ (0, 2) (such extension is well-known for elliptic equations).
Proposition 12 (Upper bound for sub-solutions).-Let f be a non-negative weak subsolution to J.É.P. -M., 2022, tome 9 Proof of Proposition 12. -Fix p 0 := 2+1/2d and define q := p 0 /2 and q n := q n .Consider β n,k on R + with β n,k 0 and β n,k 0 both bounded and so that β n,k (z) → z qn as k → ∞ and β n,k (z) z qn and β n,k (z) z qn−1 uniformly in k ∈ N * .Definition 1 implies that β n,k (f ) is a weak sub-solution with source term S n,k := β n,k (f )S.Define r 0 = R and for n 1, which means by taking k → ∞ and coming back to f f L 2q n+1 (Qr n (z0)) assuming by induction f L 2qn (Qr n−1 (z0)) < +∞.The convergence of the infinite product then implies

This proves the claim when ζ
2. To prove it when ζ ∈ (0, 2), we deduce from the previous estimate and thus by Young inequality, the quantity A(r) := f L ∞ (Qr(z0)) + S L ∞ (Qr(z0)) satisfies, for some C > 0, Introducing an (increasing this time) sequence of radii r n := r n−1 + δn −2 we obtain by induction which yields the result by taking n → ∞ in the right hand side.Proposition 13 (Hypoelliptic Poincaré inequality with error).-Given any ε ∈ (0, 1) and σ ∈ (0, 1/3), any non-negative sub-solution f to (1.1)-(1.2) on Q 5 satisfies where Remark 14. -The motivation for the following argument was [Vas16, Lem. 10, p. 11], where a simple quantitative proof of the intermediate value lemma of De Giorgi (also sometimes called De Giorgi's isoperimetric inequality) is sketched in the elliptic case, based on introducing the trajectory between two points of the domain and using the vector field ∇ v to connect them.We have to deal here with the hypoelliptic structure.
Proof.-Consider, for ε ∈ (0, 1), a smooth function ϕ ε = ϕ ε (y, w) which satisfies 0 ϕ ε 1 and has compact support in B 2 1 and such that We then split the integral to be estimated as follows where we have used The first sub-trajectory is estimated by the integral regularity L 1 t,v W σ,1 x proved in (2.6).The other trajectories are estimated directly by the vector fields in the equation.The position x + εw ∈ Q 2 since x, w ∈ B 1 and ε ∈ (0, 1).The velocity (x + εw − y)/(t − s) ∈ Q 3 since x, w, y ∈ B 1 and t − s 1 due to the definitions of Q + 1 and Q − 1 , and this velocity yields a transport line from (t, x + εw) to (s, y).Note that we are implicitly using the Hörmander commutator condition: ∇ v , T , [∇ v , T ] span all the vector fields on R 2d+1 .
Decompose along the previous trajectories and integrate against ϕ ε (y, w) on (s, y, w) ∈ Q − 1 , which gives the four terms Regarding the term I 2 , we use Taylor's formula and 0 ϕ ε 1 to deduce where we have used successively the following changes of variables with bounded Jacobians: J.É.P. -M., 2022, tome 9 The term I 4 is treated like I 2 : x regularity of non-negative sub-solutions proved in (2.6): (3.4) Regarding the term I 3 , we note first that arguing as in proof of Proposition 11).The Taylor formula between (t, x + εw) and (s, y) along T thus holds in weak form against ϕ ε thanks to the latter bounds and the non-singular change of variable (3.7) discussed below: We then use the fact that f is a sub-solution to (1.1) in the distributional sense: Arguing as for I 2 and I 4 , we have where we performed consecutively the changes of variable To estimate the remaining term I 31 , we use the change of variable with Jacobian (ε/(t − s)) d and which maps respective boundaries (to compute the Jacobian easily use the formula We deduce W ε and we integrate by parts in W , using that ϕ ε = 0 on the boundary of E(τ, ε, t, s, x): Using the bounds on the derivatives of ϕ ε then yields The result follows from combining (3.2), (3.3), (3.4), (3.6) and (3.8).
Remark 15. -Note that the regularity W σ,1 x is only used over a small trajectory that "noises" the position variable x in Q 1 with the velocity w in Q − 1 , hence allowing to integrate by parts the diffusion operator using only the variables in Q − 1 .Note also that it is possible to get some W σ ,1 t,x,v regularity in all variable with σ ∈ (0, σ) small by the same method as in Lemma 10, however such regularity is too weak to yield any intermediate value estimate alone.Note also that the gap in time between Q − and Q 1 is used to make sure the intermediate velocity (x + εw − y)/(t − s) remains bounded and the various domains of integration remain bounded along the velocity variable.In fact, the result is false without such gap, see Remark 4.

3.2.
Proof of the Intermediate-Value Lemma.-In this subsection, we prove that Proposition 13 implies Theorem 3. Take f a sub-solution to (1.1)-(1.2) on Q 1 and satisfying (1.3) for some given δ 1 , δ 2 > 0: . Then its positive part g + is a sub-solution to (1.1)-(1.2) in Q 5r0 with zero source term and with g + ∈ [0, 1] since f 1 in Q 1/2 .We set and we apply (3.1) to g + at scale r 0 , for some ε > 0 to be chosen later: (3.10) where we have used the bound g + ∈ [0, 1] to control the L 2 norm.Then (3.9) implies (3.11) (3.12) We then estimate from above the right hand side of the Poincaré inequality (3.10): J.É.P. -M., 2022, tome 9 The first term I 1 = 0 since ∇ v f + = 0 almost everywhere on {f + = 0} (see [EG15, §4.2.2]).Combining the Cauchy-Schwarz inequality, Proposition 9 and the fact that f 1, we get and (using that ∇ v f is zero almost everywhere on {f = cst}, see again [EG15, §4.2.2]) where we have used the energy estimate in Proposition 9 on [f − (1 − θ)] + .The last two estimates on I 2 and I 3 yield the following control on the right hand side of (3.10): Combining (3.12) and (3.13) gives, for some universal constant C 1: We choose ε such that Cε σ δ 1 δ 2 /8 and θ such that , which finally implies the result with

Harnack inequalities and Hölder continuity
This "point-to-measure" estimate controls the decay of the upper level set in the manner of a weak Harnack inequality, although with a "logarithmic" rather than power-law integrability.We shall now improve the integrability to a power-law by going back to (4.1) and performing an inductive argument inspired from the elliptic theory [LZ17].Note that the logarithmic integrability in (4.2) is reminiscent of Moser's approach.
We improve inductively the control of upper level sets in the following decreasing sequence of cylinders We now claim that for δ 0 > 0 small enough (to be chosen later), for any non-negative supersolution h with inf Q r 0 /2 h < 1 we have where M ∼ δ −2(1+δ −10d−16 ) with δ := δ 0 /210 4d+2 as in (4.1).Admitting first (4.3)we deduce by layer-cake representation that there is an explicit J.É.P. -M., 2022, tome 9 This implies the weak Harnack inequality (1.5) on any f non-negative super-solution to (1.1)-(1.2) by applying the previous estimate to h := f + (1 + t) S L ∞ (Q1) .To deduce the Harnack inequality (1.6) we consider f a non-negative solution to (1.1)-(1.2) and combine the previous control with Proposition 12 to get sup Let us now prove the claim (4.3) to conclude the proof.The initialization k = 1 is proved in (4.2).Then define A k+1 := {h > M k+1 } ∩ Q k+1 and denote the following translated centered cylinders Let us construct z = (t , x , v ∈ Q k+1 and r > 0, 1, so that: ( 1, are disjoint, (5) A k+1 is covered by the family C 15r [z ], 1.

Figure 1 .
Figure 1.The different cylinders in the Intermediate-Value Lemma and Harnack inequalities.

2. 3 .
Integral estimates for sub-solutions.-We combine the previous lemma with a localization argument and the energy estimate to get the Proposition 11 (Integral regularization estimates for non-negative sub-solutions) Let f be a non-negative weak sub-solution to (1.1)-(1.2) in an open set U J.É.P. -M., 2022, tome 9 3. Intermediate-Value Lemma and oscillations 3.1.Weak Poincaré inequality.-The adjective 'weak' refers to the small additional L 2 error term below.

1
and the Cauchy-Schwarz inequality.Let us estimate the first term of the previous inequality.Given t, x, v fixed, we decompose the trajectory (t, x, v) → (s, y, w) into four sub-trajectories in Q 5 : a trajectory of length O(ε) along ∇ x , two trajectories of length O(1) along ∇ v , and finally one trajectory of length O(1) along T w) := I 31 + I 32 + I 33 .J.É.P. -M., 2022, tome 9