LOG-TRANSFORM AND THE WEAK HARNACK INEQUALITY FOR KINETIC FOKKER-PLANCK EQUATIONS

Abstract

This article deals with kinetic Fokker–Planck equations with essentially bounded coefficients. A weak Harnack inequality for nonnegative super-solutions is derived by considering their log-transform and adapting an argument due to S. N. Kružkov (1963). Such a result rests on a new weak Poincaré inequality sharing similarities with the one introduced by W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.

1 Introduction

This paper is concerned with local properties of solutions of linear kinetic equations of Fokker–Planck type in some cylindrical domain $Q^{0}$

(1) $$ \begin{align} (\partial_{t} + v \cdot \nabla_{x}) f = \nabla_{v} \cdot ( A \nabla_{v} f) + B \cdot \nabla_{v} f + S \end{align} $$

assuming that the diffusion matrix A is uniformly elliptic and B and S are essentially bounded: There exist $\lambda ,\Lambda>0$ such that, for almost every $(t,x,v) \in Q^{0}$ ,

(2) $$ \begin{align} \begin{cases} \text{eigenvalues of } A(t,x,v) = A^{T}(t,x,v) \text{ lie in } [\lambda,\Lambda], \\ \text{the vector field }B\text{ satisfies: } |B(t,x,v)| \le \Lambda. \end{cases} \end{align} $$

In particular, coefficients do not enjoy further regularity such as continuity, vanishing mean oscillation etc. For this reason, coefficients are said to be rough.

1.1 Main result

We classically reduce the local study to the case where $Q^{0}$ is at unit scale. For some reasons we expose below, $Q^{0}$ takes the form $(-1,0]\times B_{R_{0}} \times B_{R_{0}}$ for some large constant $R_{0}$ only depending on dimension and the ellipticity constants $\lambda ,\Lambda $ in (2).

Before stating our main result, we give the definition of (weak) super-solutions in a cylindrical open set $\Omega $ , that is to say an open set of the form $I \times B^{x} \times B^{v}$ . A function $f \colon \Omega \to \mathbb {R}$ is a weak super-solution of (1) in $\Omega $ if $f \in L^{\infty } (I, L^{2} (B^{x} \times B^{v})) \cap L^{2} (I \times B^{x}, H^{1}(B^{v}))$ and $(\partial _{t} + v \cdot \nabla _{x}) f \in L^{2}(I \times B^{x}, H^{-1}(B^{v}))$ and for all nonnegative $\varphi \in \mathcal {D} (\Omega )$ ,

$$ \begin{align*} - \int_{Q^{0}} f (\partial_{t} + v \cdot \nabla_{x}) \varphi \, \mathrm{d}z \ge - \int_{Q^{0}} A \nabla_{v} f \cdot \nabla_{v} \varphi \, \mathrm{d}z + \int_{Q^{0}} (B \cdot \nabla_{v} f+S) \varphi \, \mathrm{d}z. \end{align*} $$

Theorem 1.1 weak Harnack inequality

Let $Q^{0} = (-1,0] \times B_{R_{0}} \times B_{R_{0}}$ and $A,B$ satisfy (2) and S be essentially bounded in $Q^{0}$ . Let f be a nonnegative super-solution of (1) in some cylindrical open set $\Omega \supset Q^{0}$ . Then

$$ \begin{align*} \left( \int_{Q_{-}} f^{p} (z) \, \mathrm{d}z\right)^{\frac1p} \le C \left( \inf_{Q_{+}}f + \|S\|_{L^{\infty} (Q_{+})} \right)\end{align*} $$

where $Q_{+} = (-\omega ^{2},0] \times B_{\omega ^{3}} \times B_{\omega }$ and $Q_{-} = (-1, -1+\omega ^{2}] \times B_{\omega ^{3}} \times B_{\omega }$ ; the positive constants C, p, $\omega $ and $R_{0}$ only depend on the dimension and the ellipticity constants $\lambda , \Lambda $ .

Remark 1. Combined with the fact that nonnegative subsolutions are locally bounded [33], the weak Harnack inequality implies the Harnack inequality proved in [13]; see Theorem 1.3

Remark 2. The proof of Theorem 1.1 is constructive. As a consequence, it provides a constructive proof of the Harnack inequality from [13].

Remark 3. The (weak) Harnack inequality implies Hölder regularity of weak solutions.

Remark 4. Such a weak Harnack inequality can be generalized to the ultraparabolic equations with rough coefficients considered for instance in [33, 40, 41, 42].

Remark 5. This estimate can be scaled and stated in arbitrary cylinders thanks to Galilean and scaling invariances of the class of equations of the form (1). These invariances are recalled at the end of the introduction.

Remark 6. As in [13, 18], the radius $\omega $ is small enough so that when ‘stacking cylinders’ over a small initial one contained in $Q_{-}$ , the cylinder $Q_{+}$ is captured; see Lemma 4.3. As far as $R_{0}$ is concerned, it is large enough so that it is possible to apply the expansion of positivity lemma (see Lemma 4.1) to every stacked cylinder.

Soon after this work was completed, another constructive proof of the Harnack inequality was proposed by the second author and C. Mouhot in [14]. While in the present article, we start from ideas developed by W. Wang and L. Zhang (see below for further details) and go further than their Hölder bound by deriving a weak Harnack inequality for super-solutions, the first author and C. Mouhot show that De Giorgi’s arguments implemented in [13] can be made constructive. It is worth mentioning that they also derive the weak Harnack inequality.

1.2 Historical background and motivations

The weak Harnack inequality from our main theorem and the techniques we develop to establish it are deeply rooted in the large literature about elliptic and parabolic regularity, both in divergence and nondivergence form.

De Giorgi’s theorem and Harnack inequality

E. De Giorgi proved that solutions of elliptic equations in divergence form with rough coefficients are locally Hölder continuous [7, 8]. This regularity result for linear equations allowed him to solve Hilbert’s 19^th problem by proving the regularity of weak solutions of a nonlinear elliptic equation. The case of parabolic equations was addressed by J. Nash in [32]. Then J. Moser [30, 31] showed that a Harnack inequality can be derived for nonnegative solutions of elliptic and parabolic equations with rough coefficients by considering the logarithm of positive solutions. The proof of E. De Giorgi applies not only to solutions of elliptic equations but also to functions in what is now known as the elliptic De Giorgi class. Parabolic De Giorgi classes were then introduced in particular in [24].

The log-transform

While the proof of the continuity of solutions for parabolic equations by J. Nash [32] includes the study of the ‘entropy’ of the solution, related to its logarithm, the proof of the Harnack inequality for parabolic equations by Moser [31] relies in an essential way on the observation that the logarithm of the solution of a parabolic equation in divergence form satisfies an equation with a dominating quadratic term in the left-hand side. This observation is then combined with a lemma that is the parabolic counterpart of a result by F. John and L. Nirenberg about functions with bounded mean oscillation. Independently, S. N. Kružkov reached Hölder continuity thanks to a Poincaré inequality due to Sobolev; see [21, 22, Eq. (1.18)]. But his method is not adapted to prove the Harnack inequality. This point will be further discussed in Subsection 1.4 below.

Weak Harnack inequality

Moser [31] and then Trudinger [37, Theorem 1.2] proved a weak Harnack inequality for parabolic equations. Lieberman [27] made the following comment: ‘It should be noted […] that Trudinger was the first to recognize the significance of the weak Harnack inequality even though it was an easy consequence of previously known results. […]’ He also mentioned that DiBenedetto and Trudinger [9] showed that nonnegative functions in the elliptic De Giorgi class corresponding to super-solutions of elliptic equations satisfy a weak Harnack inequality and G. L. Wang [38, 39] proved a weak Harnack inequality for functions in the corresponding parabolic De Giorgi class.

Parabolic equations in nondivergence form

N. V. Krylov and M. V. Safonov [23] derived a Harnack inequality for equations in nondivergence form. In order to do so, they introduced a covering argument now known as the ink-spots theorem; see for instance [17]. Such a covering argument will be later used in the various studies of elliptic equations in divergence form; see e.g., [38, 39] or [9].

Expansion of positivity

Ferretti and Safonov [11] established the interior Harnack inequality for both elliptic equations in divergence and nondivergence form by establishing what they called growth lemmas, allowing to control the behavior of solutions in terms of the measure of their super-level sets.

Gianazza and Vespri introduced in [12] suitable homogeneous parabolic De Giorgi classes of order p and proved a Harnack inequality. They shed light on the fact that their main technical point is an expansion of positivity in the following sense: If a solution lies above $\ell $ in a ball $B_{\rho }(x)$ at time t, then it lies above $\mu \ell $ in a ball $B_{2\rho } (x)$ at time $t+C \rho ^{p}$ for some universal constants $\mu $ and C, that is to say constants only depending on dimension and ellipticity constants. They mentioned that G. L. Wang also used some expansion of positivity in [38].

More recently, R. Schwab and L. Silvestre [36] used such ideas in order to derive a weak Harnack inequality for parabolic integro-differential equations with very irregular kernels.

Hypoellipticity

In the case where A is the identity matrix, equation (1) was first studied by Kolmogorov [19]. He exhibited a regularizing effect, despite the fact that diffusion only occurs in the velocity variable. This was the starting point of the hypoellipticity theory developed by Hörmander [15] for equations with smooth variable coefficients (unlike A and B in (1)).

Regularity theory for ultraparabolic equations

The elliptic regularity for degenerate Kolmogorov equations in divergence form with discontinuous coefficients, including (1) with $B=0$ , started at the end of the 1990s with contributions including [5, 29, 34, 35]. As far as the rough coefficients case is concerned, A. Pascucci and S. Polidoro [33] proved that weak (sub)solutions of (1) are locally bounded (from above). This result was later extended in [6, 2]. Then W. Wang and L. Zhang [40, 41, 42] proved that solutions of (1) are Hölder continuous. Even if the authors do not state their result as an a priori estimate, it is possible to derive from their proof the following result for a class of ultraparabolic equations that contains equations of the form (1).

Theorem 1.2 Hölder regularity – [40, 41, 42]

There exist $\alpha \in (0,1)$ only depending on dimension, $\lambda $ and $\Lambda $ such that all weak solution f of (1) in some cylindrical open set $\Omega \supset Q_{1}=(-1,0] \times B_{1} \times B_{1}$ satisfies

$$ \begin{align*} [f]_{C^{\alpha} (Q_{1/2})} \le C (\|f\|_{L^{2} (Q_{1})} + \|S\|_{L^{\infty}(Q_{1})}) \end{align*} $$

with $Q_{1/2} = (-1/4,0] \times B_{1/8} \times B_{1/2}$ ; the constant C only depends on the dimension and the ellipticity constants $\lambda , \Lambda $ .

More recently, M. Litsgård and K. Nyström [28] established existence and uniqueness results for the Cauchy Dirichlet problem for Kolmogorov–Fokker–Planck type equations with rough coefficients.

Linear kinetic equations with rough coefficients

Since the resolution of the 19^th Hilbert problem by E. de Giorgi [8], it is known that being able to deal with coefficients that are merely bounded is of interest for studying nonlinear problems. There are several models from the kinetic theory of gases related to equations of the form (1) with A, B and S depending on the solution itself. The most famous and important example is probably the Landau equation [25].

An alternative proof of the Hölder continuity Theorem 1.2 was proposed by F. Golse, C. Mouhot, A. F. Vasseur and the second author [13] and a Harnack inequality was obtained.

Theorem 1.3 Harnack inequality – [13]

Let f be a nonnegative weak solution of (1) in some cylindrical open set $\Omega $ containing $Q^{0} := (-1,0] \times B_{R_{0}} \times B_{R_{0}}$ . Then

$$ \begin{align*} \sup_{Q_{-}} f \le C \left( \inf_{Q_{+}} f + \| S \|_{L^{\infty}(Q^{0})} \right), \end{align*} $$

where $Q_{+} = (-\omega ^{2},0] \times B_{\omega ^{3}} \times B_{\omega }$ and $Q_{-} = (-1,-1+\omega ^{2}] \times B_{\omega ^{3}} \times B_{\omega }$ ; the positive constants C, $R_{0}$ and $\omega $ only depend on the dimension and the ellipticity constants $\lambda , \Lambda $ .

On the one hand, this new proof is closer to the original argument used by De Giorgi. On the other hand, the second step of the proof from [13] relies on a compactness argument while the proof by W. Wang and L. Zhang is constructive. It is worth mentioning that the second author and C. Mouhot [14] managed to make constructive the De Giorgi argument. We will give further details below.

Such a Harnack inequality implies in particular the strong maximum principle [1] relying on a geometric construction known as Harnack chains. The Hölder regularity result of [40] was extended by Y. Zhu [43] to general transport operators $\partial _{t} + b(v)\cdot \nabla _{v}$ for some nonlinear function b.

To finish, we mention that C. Mouhot and the second author [16] initiated the study of a toy nonlinear model, and F. Anceschi and Y. Zhu continued it in [3]. Both studies rely in an essential way on Hölder continuity of weak solutions to the linear equation (1).

Hypoelliptic estimates

We previously mentioned that in the case where A and B are $C^{\infty }$ , the study of (1) falls into the realm of the hypoellipticity theory developed by Hörmander in [15]. Such a point of view over (1) is adopted in [4]. Let A be the identity matrix for simplicity. We can write (1) by writing $X_{1}^{2}f + X_{0} f + c=0$ , where $X_{1}$ and $X_{0}$ are the following vector fields: $X_{1} = \nabla _{v}$ , $X_{0} = (\partial _{t} + v \cdot \nabla _{x})$ and $c = B \cdot \nabla _{v} f + S$ . Hörmander mentions that ‘the main point of [his] paper is the proof’ of [15, Estimate (3.4)]. In our setting, it reads

$$ \begin{align*} \| f \|_{H^{\epsilon}_{t,x,v}} \le C (\|f\|_{L^{2}_{t,x}H^{1}_{v}} + \|(\partial_{t} + v \cdot \nabla_{x}) f\|_{L^{2}_{t,x} H^{-1}_{v}})\end{align*} $$

for some $\epsilon>0$ .

In order to derive local properties of solutions such as their Hölder continuity by elliptic regularity methods, it is necessary to be able to work with subsolutions of (1). In this case, the source term S is supplemented with $- \mu $ , where $\mu $ is an arbitrary Radon measure. Such a source term cannot be treated directly by the previous estimate. Comparison principles are used in [13] to gain locally some integrability for nonnegative subsolutions.

Kinetic equations with integral diffusions

We would like to conclude this review of literature by mentioning the weak Harnack inequality derived in [18] for kinetic equations. The proof also relies on De Giorgi type arguments that are combined with a covering argument, referred to as an ink-spots theorem and inspired by the elliptic regularity for equations in nondivergence form (see above). The interested reader is referred to the introduction of [18] for further details.

1.3 Weak expansion of positivity

The proof of the main result of this article relies on proving that super-solutions of (1) expand positivity along times (Lemma 4.1) and to combine it with the covering argument from [18] mentioned in the previous paragraph. The derivation of the weak Harnack inequality in the present article from the expansion of positivity follows very closely the reasoning in [18].

In contrast with parabolic equations, it is not possible to apply the Poincaré inequality in v for $(t,x)$ fixed when studying solutions of linear Fokker–Planck equations such as (1). Instead, if a subsolution vanishes enough, then a quantity replacing the average in the usual Poincaré inequality is decreased in the future. See $\theta _{0} M$ in the weak Poincaré inequality in the next paragraph (Theorem 1.4).

We establish the expansion of positivity of super-solutions in the spirit of [11]. Given a small cylinder $Q_{\mathrm {pos}}$ lying in the past of $Q_{1}$ (see Figure 1), Lemma 4.1 states that, if a super-solution f lies above $1$ in ‘a good proportion’ of $Q_{\mathrm {pos}}$ , then it lies above a constant $\ell _{0}>0$ in the whole cylinder $Q_{1}$ . Roughly speaking, such a lemma transforms an information in measure about positivity in the past into a pointwise positivity in the future in a (much) larger cylinder.

Figure 1 Expansion of positivity.

We emphasize the fact that in the classical parabolic case, S. N. Kružkov does not need to prove such an expansion of positivity to get Hölder continuity of solutions. But he cannot reach the weak Harnack inequality. To get an Hölder estimate, he uses an appropriate Poincaré inequality that can be applied at any time $t>0$ . Such an approach is not applicable in the x-dependent case.

Such a weak propagation of positivity was already proved in [13] thanks to a lemma of intermediate values in the spirit of De Giorgi’s original proof. While the proof of this key lemma uses a compactness argument in [13], C. Mouhot and the first author [14] found a constructive proof of it.

1.4 A weak Poincaré inequality

The proof of the expansion of positivity relies on a weak Poincaré inequality. It is worth mentioning that Armstrong and Mourrat proved in [4] a Poincaré inequality. Unfortunately, such a Poincaré inequality cannot be applied to subsolutions (see the discussion on page 5). The geometric setting of the next theorem is shown in Figure 2

Theorem 1.4 Weak Poincaré inequality

Let $\eta \in (0,1)$ . There exist $R> 1$ and $\theta _{0} \in (0,1)$ depending on dimension and $\eta $ such that, if $Q_{\mathrm {ext}} = (-1-\eta ^{2},0] \times B_{8R} \times B_{2R}$ and $Q_{\mathrm {zero}} = (-1-\eta ^{2},-1] \times B_{\eta ^{3}} \times B_{\eta } $ (see Figure 2), then for any nonnegative function $f \in L^{2} (Q_{\mathrm {ext}})$ such that $\nabla _{v} f \in L^{2} (Q_{\mathrm {ext}})$ , $(\partial _{t} +v\cdot \nabla _{x}) f \in L^{2}((-1-\eta ^{2},0] \times B_{8R},H^{-1}(B_{2R}))$ , $f \le M $ in $Q_{\mathrm {ext}}$ and $|\{ f = 0 \} \cap Q_{\mathrm {zero}}| \ge \frac 14 |Q_{\mathrm {zero}}|, $ satisfying

$$ \begin{align*} (\partial_{t} + v \cdot \nabla_{x}) f \le H &\quad \mbox{ in } \mathcal{D}^{\prime} (Q_{\mathrm{ext}}) \quad \mbox{ with } H \in L^{2}_{t,x} H^{-1}_{v}(Q_{\mathrm{ext}}), \end{align*} $$

we have

$$ \begin{align*} \| (f -\theta_{0} M)_{+} \|_{L^{2} (Q_{1})} \le C (\|\nabla_{v} f \|_{L^{2} (Q_{\mathrm{ext}})} + \|H\|_{L^{2}_{t,x} H^{-1}_{v}(Q_{\mathrm{ext}})}) \end{align*} $$

for some constant $C>0$ only depending on dimension.

Remark 7. In the previous statement, $L^{2}_{t,x} H^{-1}_{v}(Q_{\mathrm {ext}})$ is a shorthand notation for $L^{2}((-1-\eta ^{2},0] \times B_{8R}, H^{-1}(B_{2R}))$ .

Figure 2 Geometric setting of the weak Poincaré inequality.

A somewhat similar inequality was introduced by W. Wang and L. Zhang for subsolutions of ultraparabolic equations; see for instance [40, Lemmas 3.3 & 3.4] and the corresponding lemmas in [41, 42]. The statements and proofs share similarities, and we explain next in what they differ. The main difference between statements comes from the fact that the information in measure in Theorem 1.4, the information in measure $|\{f=0\} \cap Q_{\mathrm {zero}}| \ge \frac 14 |Q_{\mathrm {zero}}|$ comes ‘from the past’ since $Q_{\mathrm {zero}}$ is contained in $\{t \le -1\}$ while the functional inequality is stated in $Q_{1} \subset \{t>-1\}$ . This is in contrast with [40, 41, 42] and [21, 22]; indeed, in these works the information in measure is contained in $\{ -1 < t \le 0\}$ and the pointwise bound is derived in the same time interval. For this reason, it is not possible to derive a (weak) Harnack inequality directly from these proofs because no time lap is considered or no positivity is propagated in time. Moreover, we adopt the functional framework from [4] and forget about the equation under study. The main difference in proofs lies on the fact that we avoid using repeatedly the exact form of the fundamental solution of the Kolmogorov equation, and we seek for arguments closer to the classical theory of parabolic equations presented, for instance, in [24] or [27]. In contrast with [40, 41, 42], the information obtained through the log-transform is summarized in only one weak Poincaré inequality (while it is split it several lemmas in [40, 41, 42]) and the geometric settings of the main lemmas are as simple as possible. For instance, it is the same for the weak Harnack inequality and for the lemma of expansion of positivity (Lemma 4.1). We also mostly use cylinders respecting the invariances of the equation (see the defintion of $Q_{r}(z)$ in the paragraph devoted to notation), except the ‘large’ cylinders where the equation is satisfied such as $Q_{\mathrm {ext}}$ in Theorem 1.4.

The Lie group structure

Equation (1) is not translation invariant in the velocity variable because of the free transport term. But this (class of) equation(s) comes from mathematical physics, and it enjoys the Galilean invariance: In a frame moving with constant speed $v_{0}$ , the equation is the same. For $z_{1} =(t_{1},x_{1},v_{1})$ and $z_{2} = (t_{2},x_{2},v_{2})$ , we define the following noncommutative group product

$$ \begin{align*} z_{1} \circ z_{2} = (t_{1}+t_{2},x_{1}+x_{2} + t_{2} v_{1}, v_{1} + v_{2}). \end{align*} $$

In particular, for $z = (t,x,v)$ , the inverse element is $z^{-1} = (-t,-x+tv,-v)$ .

Scaling and cylinders

Given a parameter $r>0$ , the class of equations (1) is invariant under the scaling

$$ \begin{align*} f_{r} (t,x,v) = f(r^{2} t, r^{3}x, r v ). \end{align*} $$

It is convenient to write $S_{r}(z) = (r^{2}t,r^{3}x,rv)$ if $z = (t,x,v)$ . It is thus natural to consider the following cylinders ‘centered’ at $(0,0,0)$ of radius $r>0$ : $Q_{r} = (-r^{2},0] \times B_{r^{3}} \times B_{r} = S_{r} (Q_{1})$ . Moreover, in view of the Galilean invariance, it is then natural to consider cylinders centered at $z_{0} \in \mathbb {R}^{1+2d}$ of radius $r>0$ of the form: $Q_{r} (z_{0}) := z_{0} \circ Q_{r} $ which is

$$ \begin{align*} Q_{r}(z_{0}) &:= \left\{ z \in \mathbb{R}^{1+2d} \ : \ z_{0}^{-1} \circ z \in Q_{r}(0) \right\} \\ &:= \left\{ -r^{2} <t-t_{0} \le 0, \ |x - x_{0}- (t-t_{0})v_{0}| < r^{3}, \ |v-v_{0}| < r \right\}. \end{align*} $$

Organization of the article

In Section 2, the definition of weak subsolutions, super-solutions and solutions for (1) is recalled and two properties of the log-transform are given. Section 3 is devoted to the proof of the weak Poincaré inequality. In Section 4, we explain how to derive the lemma of expansion of positivity from the weak Poincaré inequality and how to prove the weak Harnack inequality from expansion of positivity by using a covering lemma called the ink-spots theorem. This last result is recalled in Appendix A. In another appendix, see B, we recall how Hölder regularity can be derived directly from the expansion of positivity of super-solutions. The proof of a technical lemma about stacked cylinders is given in Appendix C.

Notation

The open ball of the Euclidian space centered at c of radius R is denoted by $B_{R} (c)$ . The measure of a Lebesgue set A of the Euclidian space is denoted by $|A|$ . The z variable refers to $(t,x,v) \in \mathbb {R} \times \mathbb {R}^{d} \times \mathbb {R}^{d} =\mathbb {R}^{1+2d}$ . For $z_{1},z_{2} \in \mathbb {R}^{1+2d}$ , $z_{1} \circ z_{2}$ denotes their Lie group product and $z_{1}^{-1}$ denotes the inverse of $z_{1}$ with respect to $\circ $ . For $r>0$ , $S_{r}$ denotes the scaling operator. A constant is said to be universal if it only depends on dimension and the ellipticity constants $\lambda ,\Lambda $ appearing in (2). The notation $a \lesssim b$ means that $a \le Cb$ for some universal constant $C>0$ .

For an open set $\Omega $ , $\mathcal {D} (\Omega )$ denotes the set of $C^{\infty }$ functions compactly supported in $\Omega $ while $\mathcal {D}^{\prime } (\Omega )$ denotes the set of distributions in $\Omega $ .

2 Weak solutions and log-transform

2.1 Weak solutions

We start with the definition of weak (super- and sub-) solutions of (1).

Definition 2.1 Weak solutions

Let $\Omega = I \times B^{x} \times B^{v}$ be open. A function $f\colon \Omega \to \mathbb {R}$ is a weak super-solution (resp. weak subsolution) of (1) in $\Omega $ if $f \in L^{\infty } (I, L^{2} (B^{x} \times B^{v})) \cap L^{2} (I \times B^{x}, H^{1}(B^{v}))$ and $(\partial _{t} + v \cdot \nabla _{x}) f \in L^{2}(I \times B^{x}, H^{-1}(B^{v}))$ and

$$ \begin{align*} - \int f (\partial_{t} + v \cdot \nabla_{x}) \varphi \, \mathrm{d}z + \int A \nabla_{v} f \cdot \nabla_{v} \varphi \, \mathrm{d}z - \int (B \cdot \nabla_{v} f+S) \varphi \, \mathrm{d}z \ge 0 \quad \text{(resp. }\le 0\text{)} \end{align*} $$

for all nonnegative $\varphi \in \mathcal {D}(\Omega )$ . It is a weak solution of (1) in Q if it is both a weak super-solution and a weak subsolution.

As explained in the introduction, the local boundedness of subsolutions has been known since [33]. We give below the version contained in [13].

Proposition 2.1 Local upper bound for subsolutions – [13]

Consider two cylinders $Q_{\mathrm {int}} = (t_{1},T] \times B_{r_{x}} \times B_{r_{v}}$ and $Q_{\mathrm {ext}} = (t_{0},T] \times B_{R_{x}} \times B_{R_{v}}$ with $t_{1}> t_{0}$ , $r_{x} <R_{x}$ and $r_{v} < R_{v}$ . Assume that f is a subsolution of (1) in some cylindrical open set $\Omega \supset Q_{\mathrm {ext}}$ . Then

$$ \begin{align*} \sup_{Q_{\mathrm{int}}} f \le C ( \| f_{+} \|_{L^{2}(Q_{\mathrm{ext}})} + \|S\|_{L^{\infty} (Q_{\mathrm{ext}})}), \end{align*} $$

where C only depends on $d,\lambda ,\Lambda $ and $(t_{1} -t_{0},R_{x}-r_{x},R_{v}-r_{v})$ .

2.2 Log-transform of subsolutions

For technicals reasons, the positive part of the opposite of the logarithm is replaced with a more regular function G that keeps the important features of $\max (0, {-}\ln )$ . The function $\max (0, {-}\ln )$ was first considered in [21, 22].

Lemma 2.1 A convex function

There exists $G: (0, {+}\infty ) \to [0, {+}\infty )$ nonincreasing and $C^{2}$ such that

• • $G^{\prime \prime } \ge (G^{\prime })^{2} \text { and } G^{\prime } \le 0 \text { in } (0,+\infty )$ ,

• • G is supported in $(0,1]$ ,

• • $G (t) \sim - \ln t$ as $t \to 0^{+}$ ,

• • $-G^{\prime }(t) \le \frac 1t$ for $t \in (0,\frac 14]$ .

Lemma 2.2 Log-transform of solutions

Let $\epsilon \in (0,\frac 14]$ and f be a nonnegative weak super-solution of (1) in a cylinder $Q_{\mathrm {ext}} = (t_{0},T] \times B_{R_{x}} \times B_{R_{v}}$ . Then $g = G(\epsilon +f)$ satisfies

$$ \begin{align*} (\partial_{t} + v \cdot \nabla_{x} ) g + \lambda |\nabla_{v} g|^{2} \le \nabla_{v} \cdot (A \nabla_{v} g) + B \cdot \nabla_{v} g + \epsilon^{-1} |S| \quad \text{ in } Q_{\mathrm{ext}}, \end{align*} $$

i.e., it is a subsolution of the corresponding equation in $Q_{\mathrm {ext}}$ .

Proof. We first note that $g \in L^{\infty } (Q_{\mathrm {ext}})$ since $0 \le g \le G(\epsilon )$ . Moreover, $\nabla _{v} g = G^{\prime }(\epsilon + f) \nabla _{v} f$ with $|G^{\prime }(\epsilon +f)| \le |G^{\prime }(\epsilon )|$ . In particular, $g \in L^{2} ((t_{0},T] \times B_{R_{x}}, H^{1} (B_{R_{v}}))$ . In order to obtain the subequation, it is sufficient to consider the test-function $G^{\prime }(\epsilon +f) \Psi $ in the definition of super-solution for f.

The following observation is key in Moser’s reasoning since the square of the $L^{2}$ -norm of $\nabla _{v} g$ is controlled by the mass of g.

Lemma 2.3 The mass of g

Let f be a nonnegative weak subsolution of (1) in a cylinder $Q_{\mathrm {ext}} = (t_{0},T] \times B_{r_{x}} \times B_{r_{v}}$ and $Q_{\mathrm {int}} = (t_{1},T] \times B_{R_{x}} \times B_{R_{v}}$ with $t_{1}> t_{0}$ , $r_{x} < R_{x}$ and $r_{v} < R_{v}$ . Then

$$ \begin{align*} \frac{\lambda}2 \int_{Q_{\mathrm{int}}} |\nabla_{v} g |^{2} \le C \left( \int_{Q_{\mathrm{ext}}} g(\tau) + 1 + \epsilon^{-1} \|S\|_{L^{\infty}(Q_{\mathrm{ext}})} \right), \end{align*} $$

where C depends on dimension, $\lambda $ , $\Lambda $ , $Q_{\mathrm {int}}$ and $Q_{\mathrm {ext}}$ .

Proof. Consider a smooth cut-off function $\Psi $ valued in $[0,1]$ , supported in $Q_{\mathrm {ext}}$ and equal to $1$ in $Q_{\mathrm {int}}$ , and use $\Psi ^{2}$ as a test-function for the subequation satisfied by g, and get

$$ \begin{align*} \lambda \int |\nabla_{v} g|^{2} \Psi^{2} \le &- 2 \int A \nabla_{v} g \cdot \Psi \nabla_{v} \Psi + \int g (\partial_{t} + v \cdot \nabla_{x}) \Psi^{2} + \int (B \cdot \nabla_{v} g + \epsilon^{-1} |S| ) \Psi^{2} \\ \le & \frac\lambda4 \int |\nabla_{v} g |^{2} \Psi^{2} + C (1+\int_{Q_{\mathrm{ext}}} g) + \frac\lambda4 \int |\nabla_{v} g |^{2} \Psi^{2} + C \epsilon^{-1} \|S\|_{L^{\infty}(Q_{\mathrm{ext}})}. \end{align*} $$

This yields the desired estimate.

2.3 Kružkov’s proof in the parabolic case

In this subsection, we discuss how Kružkov proved in [21, 22] (see also [20]) Hölder continuity of solutions of parabolic equations corresponding to solutions u of (1) such that the solution and the coefficients $A,B$ and S are independent of the x variable. It relies on the local boundedness of nonnegative subsolutions [20, Theorem 2.1] and a Poincaré inequality due to Sobolev and Ilin.

A Poincaré inequality

We start with presenting the functional inequality [22, (1.8)]: for any $u \in H^{1}(B_{r})$ and $N \subset B_{r}$ such that $|N| \ge c_{0} r^{d}$ ,

$$ \begin{align*} r^{-2d/p} \|u\|^{2}_{L^{p}(B_{r})} \le c \left[ r^{-d+2} \| \nabla u\|_{L^{2}(B_{r})}^{2} + r^{-d} \|u\|^{2}_{L^{2}(N)} \right] \end{align*} $$

for $p \in [2, 2d/(d-2)]$ for $d \ge 3$ and $p>0$ arbitrary for $d=2$ and $c=c(d,p)$ . In particular, if $N = \{ u =0\}$ and $p=2$ , this implies

(3) $$ \begin{align} \|u\|_{L^{2}(B_{r})} \le C \| \nabla u \|_{L^{2} (B_{r})} \end{align} $$

for some $C = C(d,r)$ .

Decrease of the oscillation

Kružkov proves a local Hölder estimate [20, Theorem 3.1] by proving that the oscillation of a bounded solution decreases by a universal factor strictly smaller than $1$ . We change here some constants in his proof, but we follow exactly the same reasoning. See [20, p. 187--190].

Consider a nonnegative super-solution $u=u(t,v)$ of (1), and assume that

(4) $$ \begin{align} |\{ u \ge 1\} \cap (-1,0]\times B_{1} | \ge \frac12 |B_{1}|. \end{align} $$

We ¹ then prove that $u \ge \ell _{0}$ in $(-\theta ^{2},0] \times B_{\theta }$ . Such a result implies the decrease of oscillation and in turn yields the Hölder estimate.

A pointwise in time estimate

In order to get a lower bound on u in $(-\theta ^{2},0] \times B_{\theta }$ , Kružkov first proves that there exist $\omega , \beta , h>0$ such that, for a.e. $t \in (-\omega ,0]$ , we have

(5) $$ \begin{align} |\{ x \in B_{\beta} : u \ge h \}| \ge (1/4)|B_{\beta}|. \end{align} $$

In words, the lower bound on the volume of $\{ u \ge 1\}$ is made pointwise in time. It is necessary to apply the Poincaré inequality (3) after freezing the time variable.

To get such a result, Kružkov first considers $\mu (t) = |\{ u(t) \ge 1 \} \cap B_{1}|$ and remarks that (4) implies that there exists $\omega $ small and $\tau \in (-1,-\omega ]$ so that u is defined a.e. in $B_{1}$ and $\mu (\tau ) \ge \frac {1/2-\omega }{1-\omega } |B_{1}|$ . In particular, considering $w = G(u+h)$

$$ \begin{align*} \int_{B_{\beta}} w(\tau,v) \, \mathrm{d}v & \le G(h) |\{ u(\tau) \le 1 \} \cap B_{1}| \\ & \le G(h) \left( 1 - \frac{1/2-\omega}{1-\omega} \right) |B_{1}| \\ & \le \frac23 G(h) |B_{\beta}| \end{align*} $$

the last inequality holding true if $\beta $ is chosen so that $2(1-\omega ) \beta ^{d} = 3/2$ .

He then uses the evolution of the mass of w with time. Let us assume $B=0$ and $S=0$ for simplicity. Using Lemma 2.2, we can get that, for a.e. $t \in (\tau , 0]$ ,

$$ \begin{align*} \int_{B_{\beta}} w(t,v) \, \mathrm{d}v \le \int_{B_{\beta}} w(\tau,v) \, \mathrm{d}v + C .\end{align*} $$

Since for $u(t,v) \le h$ , we also have $w(t,v) \ge G(2h)$ , we can write, for a.e. $t \in (-\omega ,0]$ ,

$$ \begin{align*} G(2h) |\{ u( t) \le h \} \cap B_{\beta} | & \le \int_{B_{\beta}} w(t,v) \, \mathrm{d}v \\ & \le \int_{B_{\beta}} w(\tau,v) \, \mathrm{d}v + C \\ & \le \frac23 G(h) |B_{\beta}| +C. \end{align*} $$

This implies (5) by choosing h small enough.

The decrease of oscillation

With (5) at hand, Kružkov then considers $g = G((u+\epsilon )/h)$ and combines i) local boundedness of nonnegative subsolutions with ii) the Poincaré inequality (3) and iii) the rough estimate of the mass of g given by Lemma 2.3 to get for $\theta $ small enough,

$$ \begin{align*} \sup_{(-\theta^{2},0]\times B_{\theta}} g^{2} & \le C \int_{(-\omega,0] \times B_{\beta}} g^{2}\, \mathrm{d}t \, \mathrm{d}v \quad \text{(local boundedness of subsolutions)} \\ & \le C \int_{(-\omega,0] \times B_{\beta}} |\nabla_{v} g|^{2} \, \mathrm{d}t \, \mathrm{d}v \quad \text{(Poincar}\acute{\rm e}\ \text{inequality)} \\ & \le C \left(\int_{(-\omega,0] \times B_{\beta}} g \, \mathrm{d}t \, \mathrm{d}v +1 \right) \quad \text{(Lemma 2.3)} \\ & \le C G(\epsilon/h). \end{align*} $$

This implies that $G((u+\epsilon )/h) \le C \sqrt {G(\epsilon /h)}$ in $Q_{\theta }$ , and since $G(t) \sim -\ln t$ as $t \to 0^{+}$ , we get the desired lower bound on u in $Q_{\theta }$ by choosing $\epsilon $ small enough.

3 A weak Poincaré inequality

In order to prove Theorem 1.4, we first derive a local Poincaré inequality (Lemma 3.1) with an error function h due to the localization. This function h satisfies

(6) $$ \begin{align} \begin{cases} \mathcal{L}_{K} h = f \mathcal{L}_{K} \Psi, & \text{ in } (a,b) \times \mathbb{R}^{2d},\\ h = 0, & \text{ in } \{a \} \times \mathbb{R}^{2d}, \end{cases} \end{align} $$

where $\mathcal {L}_{K} = (\partial _{t} + v\cdot \nabla _{x}) - \Delta _{v}$ is the Kolmogorov operator and $\Psi $ is a cut-off function equal to $1$ in $Q_{1}$ . We will estimate h in Lemma 3.2 below.

Lemma 3.1 A local estimate

Let $Q_{\mathrm {ext}} = (a,0] \times B_{R_{x}} \times B_{R_{v}}$ be a cylinder such that $\overline {Q_{1}} \subset Q_{\mathrm {ext}}$ , and let $\Psi \colon \mathbb {R}^{2d+1} \to [0,1]$ be $C^{\infty }$ , supported in $Q_{\mathrm {ext}}$ and $\Psi =1$ in $Q_{1}$ . Then for any function $f \in L^{2} (Q_{\mathrm {ext}})$ such that $\nabla _{v} f \in L^{2} (Q_{\mathrm {ext}})$ and $(\partial _{t} + v \cdot \nabla _{x}) f \le H$ in $\mathcal {D}^{\prime } (Q_{\mathrm {ext}})$ with $H \in L^{2} ((a,0] \times B_{R_{x}} ,H^{-1}(B_{R_{v}}))$ , we have

$$ \begin{align*} \| (f -h)_{+} \|_{L^{2} (Q_{1})} \le C (\|\nabla_{v} f \|_{L^{2} (Q_{\mathrm{ext}})} + \|H \|_{L^{2} ((a,0] \times B_{R_{x}} ,H^{-1} (B_{R_{v}}))}), \end{align*} $$

where h satisfies the Cauchy problem (6) and $C = c(a) (1+\|\nabla _{v} \Psi \|_{\infty })$ for some constant $c(a)$ only depending on $|a|$ .

Proof. Since $H \in L^{2}((a, 0]\times B_{R_{x}},H^{-1} (B_{R_{v}}))$ , there exists $H_{0},H_{1} \in L^{2} (Q_{\mathrm {ext}})$ such that $H = \nabla _{v} \cdot H_{1} + H_{0}$ and such that $ \|H_{0} \|_{L^{2}(Q_{\mathrm {ext}})}+\|H_{1} \|_{L^{2}(Q_{\mathrm {ext}})}\leq 2\|H \|_{L^{2} ((a,0] \times B_{R_{x}} ,H^{-1} (B_{R_{v}}))}$ (see for instance [10]). The function $g = f \Psi $ satisfies

$$ \begin{align*} \mathcal{L}_{K} g \le \nabla_{v} \cdot \tilde H_{1} + \tilde H_{0} + f [(\partial_{t} + v \cdot \nabla_{x}) \Psi - \Delta_{v} \Psi] \text{ in } \mathcal{D}^{\prime}((a,0) \times \mathbb{R}^{2d}) \end{align*} $$

with $\tilde H_{1} = (H_{1}- \nabla _{v} f) \Psi $ and $\tilde H_{0} = H_{0} \Psi - H_{1} \nabla _{v} \Psi - \nabla _{v} \Psi \cdot \nabla _{v} f$ . We thus get

$$ \begin{align*} \mathcal{L}_{K} (g-h) \le \tilde H \text{ in } \mathcal{D}^{\prime}((a,0) \times \mathbb{R}^{2d}) \end{align*} $$

with $\tilde H = \nabla _{v} \cdot \tilde H_{1} + \tilde H_{0}$ . We then multiply by $(g-h)_{+}$ to get the natural energy estimate for all $T, T^{\prime }\in (a,0)$ and $\epsilon>0$ ,

$$ \begin{align*} \int (g-h)_{+}^{2} (T,x,v) \, \mathrm{d}x \, \mathrm{d}v & + \int_{a}^{T^{\prime}} \int |\nabla_{v} (g-h)_{+} |^{2} \, \mathrm{d}t \, \mathrm{d}x \, \mathrm{d}v \\ \le & 2 \int_{a}^{0} \int |-\tilde H_{1} \cdot \nabla_{v} (g-h)_{+} + \tilde H_{0} (g-h)_{+}| \, \mathrm{d}t \, \mathrm{d}x \, \mathrm{d}v \\ \le & \frac12 \| \nabla_{v} (g-h)_{+}\|^{2}_{L^{2} ((a,0] \times \mathbb{R}^{2d})}+ 2\|\tilde H_{1}\|^{2}_{L^{2} ((a,0] \times \mathbb{R}^{2d})}\\ &+ 2\epsilon \| (g-h)_{+} \|^{2}_{L^{2} ((a,0] \times \mathbb{R}^{2d})} + \frac1{2\epsilon} \| \tilde H_{0} \|^{2}_{L^{2} ((a,0] \times \mathbb{R}^{2d})}. \end{align*} $$

Remark that we can deal with the two terms of the left-hand side separately so that we can consider the two parameters T and $T^{\prime }$ . Writing $\|\cdot \|_{L^{2}}$ for $\|\cdot \|_{L^{2} ((a,0] \times \mathbb {R}^{2d})}$ , we get after integrating in T from a to $0$ , and choosing $T^{\prime }=0$

$$ \begin{align*} \| (g-h)_{+}\|_{L^{2}}^{2} \le -2\epsilon a \| (g-h)_{+}\|^{2}_{L^{2} } - a \|\tilde H_{1}\|^{2}_{L^{2}} - \frac{a}{2\epsilon} \|\tilde H_{0}\|^{2}_{L^{2} } .\end{align*} $$

Then remark that the function $(g-h)_{+}$ equals $(f -h)_{+}$ in $Q_{1}$ and that

$$ \begin{align*} \| \tilde H_{1} \|_{L^{2}((a,0] \times \mathbb{R}^{2d})} & \le \|H_{1}\|_{L^{2}(Q_{\mathrm{ext}})} + \|\nabla_{v} f \|_{L^{2}(Q_{\mathrm{ext}})}, \\ \| \tilde H_{0} \|_{L^{2}((a,0] \times \mathbb{R}^{2d})} & \le \|H_{0}\|_{L^{2}(Q_{\mathrm{ext}})} + \|\nabla_{v} \Psi\|_{\infty} ( \|H_{1}\|_{L^{2}(Q_{\mathrm{ext}})} + \|\nabla_{v} f \|_{L^{2}(Q_{\mathrm{ext}})} ). \end{align*} $$

We get the desired inequality by combining the three previous inequalities and choosing $\epsilon = -(4a)^{-1}$ .

In view of Lemma 3.1, in order to prove Theorem 1.4, it is sufficient to prove that, if the function f satisfies

$$ \begin{align*}|\{ f = 0 \} \cap Q_{\mathrm{zero}}| \ge \frac14 |Q_{\mathrm{zero}}|, \end{align*} $$

then the function h given by the Cauchy problem (6) is bounded from above by $\theta _{0} M$ for some universal parameter $\theta _{0} \in (0,1)$ .

Lemma 3.2 Control of the localization term

Let $\eta \in (0,1]$ . There exist a (large) constant $R>1$ and a (small) constant $\theta _{0} \in (0,1)$ both depending on the dimension and $\eta $ , and a $C^{\infty }$ cut-off function $\Psi \colon \mathbb {R}^{2d+1} \to [0,1]$ , supported in $Q_{\mathrm {ext}} = (-1-\eta ^{2},0] \times B_{8R} \times B_{2R}$ and equal to $1$ in $Q_{1}$ , such that, for all nonnegative bounded function $f: Q_{\mathrm {ext}} \to \mathbb {R}$ satisfying

(7) $$ \begin{align} |\{ f = 0 \} \cap Q_{\mathrm{zero}}| \ge \frac14 |Q_{\mathrm{zero}}|, \end{align} $$

the solution h of the following initial value problem

$$ \begin{align*} \begin{cases} \mathcal{L}_{K} h = f \mathcal{L}_{K} \Psi & \text{ in } (-1-\eta^{2},0) \times \mathbb{R}^{2d} \\ h = 0 & \text{ in } \{ -1-\eta^{2} \} \times \mathbb{R}^{2d} \end{cases} \end{align*} $$

satisfies $h \le \theta _{0} \|f\|_{L^{\infty }(Q_{\mathrm {ext}})}$ in $Q_{1}$ .

Remark 8. This lemma is related to [40, Lemma 3.4] and [41, Lemma 3.3].

Remark 9. The conclusion of the lemma and its proof are essentially unchanged under the weaker assumption $ |\{ f = 0 \} \cap Q_{\mathrm {zero}}| \ge \alpha _0 |Q_{\mathrm {zero}}| $ for some $\alpha _{0} \in (0,1)$ .

Remark 10. Theorem 1.4 will be used in the proof of Lemma 4.1 about the expansion of positivity of super-solutions. The parameter $\eta $ will be then chosen after choosing $\theta $ .

The proof of Lemma 3.2 requires the following test-function whose construction is elementary.

Lemma 3.3 Cut-off function

Given $\eta \in (0,1]$ and $T \in (0,\eta ^{2})$ , there exists a smooth function $\Psi _{1} : [-1-\eta ^{2},0] \times \mathbb {R}^{d}\times \mathbb {R}^{d} \to [0,1]$ , supported in $[-1-\eta ^{2},0] \times B_{8} \times B_{2}$ , equal to $1$ in $(-1,0] \times B_{1} \times B_{1}$ , such that $(\partial _{t} + v \cdot \nabla _{x}) \Psi _{1} \ge 0$ everywhere and $(\partial _{t} + v \cdot \nabla _{x}) \Psi _{1} \ge 1$ in $(-1-\eta ^{2},-1-T] \times B_{1} \times B_{1}$ .

Proof. Consider $\Psi _{1} (t,x,v) = \varphi _{1} (t) \varphi _{2} (x-tv) \varphi _{3} (v)$ with

• • a smooth function $\varphi _{1}: [-1-\eta ^{2},0] \to [0,1]$ equal to $1$ in $[-1,0]$ with $\varphi _{1} (-1-\eta ^{2}) = 0$ , $\varphi _{1}^{\prime } \ge 0$ in $[-1-\eta ^{2},0]$ and $\varphi _{1}^{\prime }=1$ in $[-1-\eta ^{2},-1-T]$ ;

• • a smooth function $\varphi _{2} : \mathbb {R}^{d} \to [0,1]$ supported in $B_{4}$ and equal to $1$ in $B_{3}$ ;

• • a smooth function $\varphi _{3} : \mathbb {R}^{d} \to [0,1]$ supported in $B_{2}$ and equal to $1$ in $B_{1}$ .

It is then easy to check that the conclusion of the lemma holds true.

We can now turn to the proof of Lemma 3.2.

Proof of Lemma 3.2

If $f=0$ in $Q_{\mathrm {ext}}$ , then $h=0$ . We thus can assume from now on that f is not identically $0$ . We next reduce to the case $\|f\|_{L^{\infty }(Q_{\mathrm {ext}})}=1$ by considering $f/\|f\|_{L^{\infty }(Q_{\mathrm {ext}})}$ .

We introduce a time lap T between the top of the cylinder $Q_{\mathrm {zero}}$ and the bottom of the cylinder $Q_{1}$ ; see Figure 3.

Figure 3 Reducing to the case with a time lap.

Fix $T=\eta ^{2}/8$ . Since $|Q_{\mathrm {zero}} \cap \{ t \ge -1-T\}| = \frac 18|Q_{\mathrm {zero}}|$ , then

(8) $$ \begin{align} |\{ f = 0 \} \cap Q_{\mathrm{zero}} \cap \{t \le -1-T\}| \ge \frac18 |Q_{\mathrm{zero}}|. \end{align} $$

Let $R>1$ to be chosen later. We consider the cut-off function

$$ \begin{align*} \Psi (t,x,v) = \Psi_{1} (t,x/R,v/R). \end{align*} $$

Remark that $\Psi $ is supported in $Q_{\mathrm {ext}}$ and equal to $1$ in $(-1,0] \times B_{R} \times B_{R}$ . Moreover,

$$ \begin{align*} \mathcal{L}_{K} \Psi (t,x,v) = (\partial_{t} + v \cdot \nabla_{x}) \Psi_{1} (t,x/R,v/R) - R^{-2} \Delta_{v} \Psi_{1} (t,x/R,v/R), \end{align*} $$

where in the last equation $v \cdot \nabla _{x}$ means the scalar product of the value of third variable in $\Psi $ (here it is $\frac {v}{R}$ ) with the gradient in the second variable. We then have

$$ \begin{align*} \mathcal{L}_{K} (h- \Psi) = -(1-f) (\partial_{t} + v \cdot \nabla_{x}) \Psi_{1} (t,x/R,v/R) + \frac{1-f}{R^{2}} \Delta_{v} \Psi_{1} (t,x/R,v/R), \end{align*} $$

and we can write

$$ \begin{align*}h - \Psi = -P_{R} + E_{R} \end{align*} $$

( $P_{R}$ for positive and $E_{R}$ for error) with $P_{R}$ and $E_{R}$ solutions of the following Cauchy problems in $(-1-\eta ^{2},0) \times \mathbb {R}^{2d}$ :

$$ \begin{align*} \mathcal{L}_{K} P_{R} &= (1-f) (\partial_{t} + v \cdot \nabla_{x}) \Psi_{1}(t,x/R,v/R), \\ \mathcal{L}_{K} E_{R} &= \frac{1-f}{R^{2}} \Delta_{v} \Psi_{1} (t,x/R,v/R), \end{align*} $$

and $P_{R} = E_{R} =0$ at time $t=-1-\eta ^{2}$ .

We claim that there exist constants $C>0$ and $\delta _{0}>0$ depending on the dimension and $\eta $ (in particular independent of R) such that

(9) $$ \begin{align} E_{R} \le C R^{-2} \quad \text{ and } \quad P_{R} \ge \delta_{0} \quad \text{ in } Q_{1}. \end{align} $$

As far as the estimate of $E_{R}$ is concerned, it is enough to remark that $\mathcal {L}_{K} E_{R} \le C_{0} R^{-2}$ for some constant $C_{0}= \| \Delta _{v} \Psi _{1} \|_{L^{\infty }}$ only depending on $d,\lambda ,\Lambda , \eta $ (in particular not depending on R). The maximum principle then yields the result for some universal constant C. As far as $P_{R}$ is concerned, we remark that

$$ \begin{align*} \mathcal{L}_{K} P_{R} \ge \mathbb{I}_{\mathcal{Z}} \quad \text{ in } (-1-\eta^{2},0] \times \mathbb{R}^{2d}, \end{align*} $$

where $\mathcal {Z} = \{f = 0\} \cap Q_{\mathrm {zero}} \cap \{ t \le -1-T\}$ . We use here the fact that $(\partial _{t} + v \cdot \nabla _{x}) \Psi _{1} \ge 1$ in $(-1-\eta ^{2},-1-T) \times \mathbb {R}^{2d}$ . Let P be such that $\mathcal {L}_{K} P = \mathbb {I}_{\mathcal {Z}}$ in $(-1-\eta ^{2},0] \times \mathbb {R}^{2d}$ and $P=0$ at the initial time $-1-\eta ^{2}$ . The maximum principle implies that $P_{R} \ge P$ in the time interval $(-1-\eta ^{2},0]$ and in particular in $Q_{1}$ .

We now claim that $P \ge \delta _{0}$ in $Q_{1}$ for some constant $\delta _{0}>0$ depending on the dimension and $\eta $ . Indeed, one can use the fundamental solution $\Gamma $ of the Kolmogorov equation and write

$$ \begin{align*} P (t,x,v) = \int \Gamma (z,\zeta) \mathbb{I}_{\mathcal{Z}} (\zeta) \, \mathrm{d} \zeta \ge \frac18 \mathsf{m} |Q_{\mathrm{zero}}| = \delta_{0},\end{align*} $$

with $\mathsf {m}= \min \limits _{Q_{1} \times Q_{\mathrm {zero}}\cap \{ t \le -1-T\}} \Gamma .$ The claim (9) is now proved.

Inequalities from (9) imply that

$$ \begin{align*} h \le 1 - \delta_{0} + C R^{-2} \text{ in } Q_{1}.\end{align*} $$

This yields the desired result with $\theta _{0} = 1-\delta _{0}/2$ for R large enough. Note in particular that R only depends on C and $\delta _{0}$ and consequently depends on the dimension and $\eta $ .

4 The weak Harnack inequality

Before proving the weak Harnack inequality stated in Theorem 1.1, we investigate how equation (1) expands positivity of super-solutions.

Lemma 4.1 Expansion of positivity

Let $\theta \in (0,1]$ and $Q_{\mathrm {pos}} = (-1-\theta ^{2},-1] \times B_{\theta ^{3}} \times B_{\theta }$ , and let R be the constant given by Lemma 3.2 which depends on $\theta $ , d, $\lambda $ , $\Lambda $ . There exist $\eta _{0},\ell _{0} \in (0,1)$ , only depending on $\theta $ , d, $\lambda $ , $\Lambda $ , such that for $\mathcal {Q}_{\mathrm {whi}} := (-1-\theta ^{2},0] \times B_{9 R} \times B_{3 R}$ and any nonnegative super-solution f of (1) in some cylindrical open set $\Omega \supset \mathcal {Q}_{\mathrm {whi}}$ and $\|S\|_{L^{\infty } (\mathcal {Q}_{\mathrm {whi}})} \le \eta _{0}$ and such that $ |\{ f \ge 1 \} \cap Q_{\mathrm {pos}} | \ge \frac 12 |Q_{\mathrm {pos}}|$ we have $f \ge \ell _{0}$ in $Q_{1}$ .

Remark 11. The parameter $\theta $ will be chosen in such a way that the stacked cylinder $\overline {Q_{\mathrm {pos}}}^{m}$ is contained in $Q_{1}$ (see the definition of stacked cylinders in Appendix A). The cylinder ${\overline {Q_{\mathrm {pos}}}}^{m}$ can be thought of as the union of m copies of $Q_{\mathrm {pos}}$ stacked above (in time) of $Q_{\mathrm {pos}}$ . Such cylinders are used in the covering argument used in the proof of the weak Harnack inequality, and the parameter $m \in \mathbb {N}$ only depends on dimension.

Proof of Lemma 4.1

We consider $g = G(f+\epsilon )$ . We remark that $g \le G(\epsilon )$ since f is nonnegative and G is nonincreasing. We also remark that $|G^{\prime }(f+\epsilon )|\le |G^{\prime }(\epsilon )| \le \epsilon ^{-1}$ since $f \ge 0$ ; see Lemma 2.1.

We know from Lemma 2.2 that g is a nonnegative subsolution of (1) with S replaced with $S G^{\prime }(f+\epsilon )$ . In particular, $(\partial _{t} + v \cdot \nabla _{x} )g \le H$ with $H = \nabla _{v} \cdot (A \nabla _{v}g) + B\cdot \nabla _{v} g+ \epsilon ^{-1}|S|$ . Recall that, for a set $Q \subset \mathbb {R}^{2d+1}$ , $S_{(r)}(Q)=\{(r^{2}t,r^{3}x,rv) \quad \mbox {for } z=(t,x,v)\in Q \}. $ We introduce $\eta \in (0,\left (\frac {4}{5}\right )^{\frac {1}{4d+2}}\theta )$ and $\iota>0$ two parameters depending on $\theta $ to be chosen later in the proof.

We are going to apply successively: the $L^{2}-L^{\infty }$ estimate from $Q_{1}$ to a slightly larger cylinder $Q_{1+\iota }$ for an accurate choice of $\iota $ ; the (scaled) weak Poincaré inequality in the big cylinder $\tilde {Q}_{\mathrm {ext}} = S_{(1+\iota )}(Q_{\mathrm {ext}})$ with $Q_{\mathrm {ext}} = (-1-\eta ^{2}, 0] \times B_{8R} \times B_{2R}$ . Then we estimate the $L^{2}$ -norm of $\nabla _{v} g$ by the square root of its mass in a cylinder larger than $\tilde {Q}_{\mathrm {ext}}$ , namely $S_{(1+\iota )^{2}}(Q_{\mathrm {ext}})\subset \mathcal {Q}_{\mathrm {whi}}$ . This is illustrated in Figure 5.

Figure 4 Geometric setting of the expansion of positivity lemma. It is the same as the one of the weak Poincaré inequality, except that $Q_{\mathrm {ext}}$ and $Q_{\mathrm {zero}}$ are replaced with $\mathcal {Q}_{\mathrm {whi}}$ and $Q_{\mathrm {pos}}$ . Here, $Q_{\mathrm {pos}}$ denotes a set ‘in the past’ where $\{f \ge 1\}$ occupies half of it.

Figure 5 Intermediate cylinders in the proof of the expansion of positivity. The great parts are obtained after a scaling with a parameter $1+\iota $ close to $1$ . This is necessary in order to keep f vanishing in a ‘good ratio’ of $S_{(1+\iota )}(Q_{\mathrm {zero}})$ .

The remainder of the proof is split into several steps. We explain in Step 1 how to choose $\iota $ to ensure that cylinders are properly ordered and $\eta $ so that we retain enough information from the assumption $ |\{ f \ge 1 \} \cap Q_{\mathrm {pos}} | \ge \frac 12 |Q_{\mathrm {pos}}|$ . We then apply the aforementioned successive estimates in Step 2 before deriving the lower bound on f in Step 3.

Step 1. Choose $\iota $ small enough so that $Q_{\mathrm {ext}} \subset \tilde {Q}_{\mathrm {ext}} \subset S_{(1+\iota )^{2}}(Q_{\mathrm {ext}})\subset \mathcal {Q}_{\mathrm {whi}}$ . We only need to check the last inclusion. We choose $\iota>0$ small enough so that $(1+\iota )^{4} (1+\eta ^{2})\leq 1+\theta ^{2} $ , $2(1+\iota )^{2}\le 3$ and $8(1+\iota )^{6}\le 9$ . Since $\eta \in (0,\left (\frac {4}{5}\right )^{\frac {1}{4d+2}}\theta )$ , to satisfy the first inequality it is enough to satisfy $(1+\iota )^{4} \left (1+\left (\frac {4}{5}\right )^{\frac {1}{2d+1}}\theta ^{2}\right )\leq 1+\theta ^{2} $ . So we pick $\iota = \min \left (\left (\frac {5^{\frac {1}{2d+1}}(1+\theta ^{2})}{5^{\frac {1}{2d+1}}+4^{\frac {1}{2d+1}}\theta ^{2}}\right )^{1/4}-1,\left (\frac {9}{8}\right )^{1/6}-1,\left (\frac {3}{2}\right )^{1/2}-1\right ).$

Recall that $Q_{\mathrm {zero}} = (-1-\eta ^{2},-1] \times B_{\eta ^{3}} \times B_{\eta }$ . In particular, $S_{(1+\iota )}(Q_{\mathrm {zero}}) = (-(1+\iota )^{2}(1+\eta ^{2}),-(1+\iota )^{2}] \times B_{(1+\iota )^{3}\eta ^{3}} \times B_{(1+\iota ) \eta }$ . We next pick $\eta \in (0,\left (\frac {4}{5}\right )^{\frac {1}{4d+2}}\theta )$ big enough so that

$$ \begin{align*} |Q_{\mathrm{pos}} \setminus S_{(1+\iota)} (Q_{\mathrm{zero}})| \le \frac14 |S_{(1+\iota)} (Q_{\mathrm{zero}})|. \end{align*} $$

It is then enough to satisfy $\theta \le (5/4)^{\frac {1}{4d+2}} (1+\iota ) \eta $ , so we pick $\eta = (4/5)^{\frac {1}{4d+2}} (1+\iota )^{-1}\theta $ . In particular, the previous volume condition implies that

$$ \begin{align*} | \{ g =0 \} \cap S_{(1+\iota)}(Q_{\mathrm{zero}})|\ge |\{f \ge 1 \} \cap S_{(1+\iota)}(Q_{\mathrm{zero}})| \ge \frac14 |S_{(1+\iota)}(Q_{\mathrm{zero}})|. \end{align*} $$

Step 2. With such an information at hand, we know that there exists $\theta _{0} \in (0,1)$ , only depending on $\eta $ and thus only depending on $\theta $ , such that

$$ \begin{align*} \sup_{Q_{1}} (g- \theta_{0} G(\epsilon))_{+} & \lesssim \|(g-\theta_{0} G(\epsilon))_{+}\|_{L^{2} (Q_{1+\iota})} +\eta_{0} & \text{from Proposition 2.1} \\ & \lesssim_{\iota} \|\nabla_{v} g \|_{L^{2}(\tilde{Q}_{\mathrm{ext}})} + (\eta_{0}/\epsilon) + \eta_{0} & \text{from Theorem 1.4} \\ & \lesssim \left( \int_{\mathcal{Q}_{\mathrm{whi}}} g + 1+ \eta_{0} / \epsilon\right)^{\frac12}+ (\eta_{0} /\epsilon ) + \eta_{0} & \text{from Lemma 2.3} \\ & \lesssim \left( G(\epsilon) + 2\right)^{\frac12}+2 & \text{for }\eta_{0} \le \epsilon \le 1 \\ & \lesssim \sqrt{G(\epsilon)} & \text{for }\epsilon\text{ such that }G(\epsilon)\geq 2. \end{align*} $$

Remark that it has been necessary to scale g before applying Theorem 1.4. This generates a constant depending on $\iota $ . This is emphasized by writing $\lesssim _{\iota }$ . But $\iota $ only depends on dimension, $\lambda $ , $\Lambda $ and $\theta $ .

Step 3. The previous computation yields

$$ \begin{align*} g \le C \sqrt{G(\epsilon)} + \theta_{0} G (\epsilon) \quad \text{ in } Q_{1}\end{align*} $$

for some $\theta _{0} \in (0,1)$ depending on universal constants and $\theta $ . Since $G(\epsilon ) \to +\infty $ (we can pick $\epsilon $ and $\eta _{0}$ small enough only depending on the universal constants and $\theta $ ), we thus have

$$ \begin{align*}G(f+\epsilon)= g \le \frac{1+2\theta_{0}}3 G (\epsilon) \quad \text{ in } Q_{1}.\end{align*} $$

Now recall that $G(t) \sim - \ln t$ as $t \to 0+$ and $-G^{\prime }(t)\leq \frac {1}{t}$ for $t\in \left (0,\frac {1}{4}\right ]$ . The previous inequality thus implies that, as $\epsilon \to 0+$ ,

$$ \begin{align*} \ln (f+\epsilon) \ge \frac{2+\theta_{0}}3 \ln \epsilon .\end{align*} $$

This yields the result with $\ell _{0} = \epsilon ^{\frac {2+\theta _{0}}3} - \epsilon>0$ .

Before iterating the lemma of expansion of positivity, we state and prove a straightforward consequence of it that will be used when applying the ink-spots theorem.

Lemma 4.2. Let $m \ge 3$ be an integer and R be given by Lemma 4.1 for $\theta =m^{-1/2}$ . There exists a constant $M>1$ only depending on m, d, $\lambda $ , $\Lambda $ such that for all nonnegative super-solution f (1) with $S=0$ in some cylindrical open set $\Omega \supset (-1,m] \times B_{9Rm^{3/2}} \times B_{3Rm^{1/2}}$ , such that

$$ \begin{align*} |\{ f \ge M \} \cap Q_{1} | \ge \frac12 |Q_{1}|, \end{align*} $$

then $f \ge 1$ in $\bar Q_{1}^{m} = (0,m] \times B_{m+2} \times B_{1}$ .

Proof. Let $\theta =m^{-\frac {1}{2}}$ so that $\bar Q_{1}^{m} \subset (0,\theta ^{-2}] \times B_{\theta ^{-3}} \times B_{\theta ^{-1}}$ . We apply Lemma 4.1 to $\frac {f}{M}$ with $Q_{1}$ and $\bar Q_{1}^{m}$ taking the role of $Q_{\mathrm {pos}}$ and $Q_{1}$ thanks to a rescaling argument. This yields that $f \ge \ell _{0}M$ in $(0,\theta ^{-2}] \times B_{\theta ^{-3}} \times B_{\theta ^{-1}}$ . We then pick $M = 1/\ell _{0}$ , and we conclude the proof.

When deriving the weak Harnack inequality, we will need to estimate how the lower bound deteriorates with time. Indeed, such information is needed in the ink-spots theorem: Since cylinders can ‘leak’ out of the set F, a corresponding error has to be estimated; see the term $Cm r_{0}^{2}$ in Theorem A.1. The geometric setting is the one from Theorem 1.1. In particular, recall that $Q_{+} = (-\omega ^{2},0] \times B_{\omega ^{3}} \times B_{\omega }$ and $Q_{-} = (-1,-1+\omega ^{2}] \times B_{\omega ^{3}} \times B_{\omega }$ , where $\omega $ is small and universal. It has to be small enough so that, when spreading positivity from a cylinder $Q_{r}(z_{0})$ from the past, i.e., included in $Q_{-}$ , the union of the stacked cylinders where positivity is expanded captures $Q_{+}$ . Then the radius $R_{0}$ in the statement of weak Harnack inequality is chosen so that the expansion of positivity lemma can be applied as long as new cylinders are stacked over previous ones. These two facts are stated precisely in the following lemma.

In order to avoid the situation where the last stacked cylinder (see $Q[N+1]$ in the next lemma) leaks out of the domain where the equation is satisfied, we choose it in a way that we can use the information obtained in the previous cylinder $Q[N]$ : the ‘predecessor’ of $Q[N+1]$ is contained in $Q[N]$ (see Figure 6).

Lemma 4.3 Stacking cylinders

Let $\omega < 10^{-2}$ . Given any nonempty cylinder $Q_{r} (z_{0}) \subset Q_{-}$ , let $T_{k} = \sum _{j=1}^{k} (2^{j}r)^{2}$ and $N \ge 1$ such that $ T_{N} \le -t_{0} < T_{N+1}$ . Let

$$ \begin{align*} Q[k] &= Q_{2^{k}r}(z_{k}) \text{ for } k =1,\dots, N,\\ Q [N+1] &= Q_{R_{N+1}}(z_{N+1}), \end{align*} $$

where $z_{k} = z_{0} \circ (T_{k},0,0)$ and, letting $R =|t_{0}+T_{N}|^{\frac 12}$ and $\rho = (4\omega )^{\frac 13}$ , $R_{N+1} = \max (R,\rho )$ and

$$ \begin{align*} z_{N+1} = \begin{cases} z_{N} \circ (R^{2}, 0,0) & \text{ if } R \ge \rho, \\ (0,0,0) & \text{ if } R < \rho. \end{cases} \end{align*} $$

These cylinders satisfy

$$ \begin{align*} Q_{+} \subset Q[N+1], \qquad \bigcup_{k=1}^{N+1} Q[k] \subset (-1,0] \times B_{2} \times B_{2}, \qquad Q[N] \supset \tilde Q[N], \end{align*} $$

where $\tilde Q[N]$ is the ‘predecessor’ of $Q[N+1]$ : $\tilde Q[N] = Q_{R_{N+1}/2} (z_{N+1} \circ (-R_{N+1}^2, 0,0)).$

Figure 6 Stacking cylinders above an initial one contained in $Q_{-}$ . We see that the stacked cylinder obtained after $N+1$ iterations by doubling the radius leaks out of the domain. This is the reason why $Q[N+1]$ is chosen in a way that it is contained in the domain and its ‘predecessor’ is contained in $Q[N]$ . Notice that the cylinders $Q[k]$ are in fact slanted since they are not centered at the origin. We also mention that $Q[N+1]$ is choosen centered if the time $t_{0}+T_{N}$ is too close to the final time $0$ .

The proof of this lemma is posponed until Appendix C.

With such a technical lemma at hand, expansion of positivity for large times follows easily.

Lemma 4.4 Expansion of positivity for large times

Let $R_{1/2}$ given by Lemma 4.1 with $\theta =1/2$ . There exist a universal constant $p_{0}>0$ such that, if f is a nonnegative weak super-solution of (1) with $S=0$ in some cylindrical open set $\Omega \supset Q=(-1,0] \times B_{18 R_{1/2}} \times B_{6 R_{1/2}}$ such that

$$ \begin{align*} |\{ f \ge A \} \cap Q_{r} (z_{0})| \ge \frac12 |Q_{r}(z_{0})| \end{align*} $$

for some $A>0$ and for some cylinder $Q_{r}(z_{0}) \subset Q_{-}$ , then $f \ge A (r^{2}/4)^{p_{0}}$ in $Q_{+}$ .

Proof. We first apply Lemma 4.1 with $\theta =1/2$ to $f/A$ (after rescaling $Q_{r}(z_{0})$ into $Q_{\mathrm {pos}}$ ) and get $f/A \ge \ell _{0}$ in $Q[1]$ . We then apply it to $f / (A \ell _{0})$ and get $f \ge A \ell _{0}^{2}$ in $Q[2]$ . By induction, we get $f \ge A \ell _{0}^{k}$ in $Q[k]$ for $k=1,\dots , N$ .

We then apply Lemma 4.1 one more time and get $f \ge A \ell _{0}^{N+1}$ in $Q[N+1]$ and in particular $f \ge A \ell _{0}^{N+1}$ in $Q_{+}$ . Since $T_{N} \le 1$ , we have $4^{N} r^{2} \le 1$ . Choosing $p_{0}>0$ such that $\ell _{0} = (1/4)^{p_{0}}$ , we get $f \ge A ((1/4)^{N+1})^{p_{0}} \ge A (r^{2}/4)^{p_{0}}$ .

We finally turn to the proof of the main result of this paper, Theorem 1.1.

Proof of Theorem 1.1

We start the proof with general comments about the geometric setting. The proof is going to use a covering argument through the application of the ink-spots theorem. To apply this result, we will consider an arbitrary cylinder Q contained in $Q_{-}$ . The parameter $\omega $ used in the definition of $Q_{-}$ and $Q_{+}$ is chosen small enough ( $\omega \leq 10^{-2}$ ) so that the cylinder $Q_{+}$ is ‘captured’ when stacking cylinders (Lemma 4.3) and propagating positivity (Lemma 4.4). We also pick the parameter $R_{0}$ in the definition of the cylinder $Q^{0}$ large enough so that the stacked cylinders do not leak out of $Q^{0}$ ; we impose $R_{0} \ge 18 R_{1/2}$ , where $R_{1/2}$ is given by Lemma 4.1 for $\theta =1/2$ . We also impose $R_{0} \ge 9 R_{m^{-1/2}} m^{3/2} \omega ^{3}$ , where $R_{m^{-1/2}}$ is given by Lemma 4.1 with $\theta = m^{-1/2}$ in order to be in position to apply Lemma 4.2 to cylinders contained in $Q^{-}$ , hence of radius smaller than $\omega $ .

We first classically reduce to the case

$$ \begin{align*} \inf_{Q_{+}} f \le 1 \quad \text{ and } \quad S=0. \end{align*} $$

Considering $\tilde {f}(t,x,v)=f(t,x,v)+\|S \|_{L^{\infty }} t$ , $\tilde {f}$ is a super-solution of the same equation with no source term ( $S=0$ ), and the weak Harnack inequality for $\tilde {f}$ implies the one for f. So from now we assume $S=0$ . Considering next $\tilde {f} = f / (\inf _{Q_{+}} f +1)$ reduces to the case $\inf _{Q_{+}} f \le 1$ .

We then aim at proving that $\int _{Q_{-}} f^{p} (z) \, \mathrm {d}z$ is bounded from above by a universal constant for some universal exponent p. This amounts to prove that, for all $k \in \mathbb {N}$ ,

$$ \begin{align*} |\{ f> M^{k}\} \cap Q_{-} | \le C_{\text{w.h.i.}} (1-\tilde{\mu})^{k} \end{align*} $$

for some universal parameters $\tilde {\mu } \in (0,1)$ , $M>1$ and $C_{\text {w.h.i.}}>0$ to be determined later. We can see that this property would be enough by transposing it to the continuous case (k real and above $1$ ) and by application of the layer cake formula to $\int _{Q_{-}} f^{p} (z) \, \mathrm {d}z$ .

We are going to apply Theorem A.1 with $\mu =1/2$ . We pick $m \in \mathbb {N}$ such that $\frac {m+1}m (1-c/2) \le 1 -c/4$ . Then the constant $M>1$ is given by Lemma 4.2.

We prove the result by induction. For $k=1$ , we simply choose $\tilde {\mu } \le 1/2$ and $C_{\text {w.h.i.}}$ such that $|Q_{-}| \le \frac {1}{2}C_{\text {w.h.i.}}$ . Now assume that the claim holds true for $k \ge 1$ , and let us prove it for $k+1$ . We thus consider

$$ \begin{align*} E = \{ f> M^{k+1} \} \cap Q_{-} \quad \text{ and } \quad F = \{ f > M^{k} \} \cap Q_{1}. \end{align*} $$

These two sets are bounded and measurable and such that $E \subset F \cap Q_{-}$ . We consider a cylinder $Q =Q_{r} (t,x,v) \subset Q_{-}$ such that $|Q \cap E |> \frac 12 |Q|$ , that is to say

$$ \begin{align*} |\{f> M^{k+1} \} \cap Q | > \frac12 |Q|.\end{align*} $$

We first prove that r is small, i.e., we determine a universal $r_{0}$ which depends on k such that $r < r_{0}$ . Lemma 4.4 (after translation in time) implies that $ f \ge M^{k+1} (r^2/4)^{p_0} $ in $Q_{+}$ . In particular, $1 \ge \inf _{Q_{+}} f \ge M^{k+1} (r^{2}/4)^{p_{0}}$ , so $r^{2p_{0}}\leq 4^{p_{0}} M^{-(k+1)}$ . We thus choose $r_{0} = 2 M^{-\frac {k+1}{2p_{0}}}$ .

We next prove that $\bar Q^{m} \subset F$ , i.e.,

$$ \begin{align*} \bar Q^{m} \subset \{ f> M^{k}\} .\end{align*} $$

In order to do so, we apply Lemma 4.2 to $f/M^{k}$ after rescaling Q in $Q_{1}$ where we assume $\omega \leq (2m+3)^{-1/2}$ to be able to rescale.

By Theorem A.1, we conclude thanks to the induction assumption that

$$ \begin{align*} |\{ f> M^{k+1} \} \cap Q_-| &\le (1-c/4) \bigg(C_{\text{w.h.i.}} (1 -\mu)^k + C m r_0^2 \bigg) \\ & \le (1-c/4) \bigg(C_{\text{w.h.i.}} (1 -\mu)^k + C m M^{-\frac{k+1}{p_0}} \bigg). \end{align*} $$

Then pick $\tilde {\mu }$ small enough so that $M^{-1/p_0} \le (1- \mu )$ and $\tilde {\mu } \le \frac {c}{4}$ , and get

$$ \begin{align*} & \le C_{\text{w.h.i.}} (1-c/4) \bigg(1 + C_{\text{w.h.i.}}^{-1} C m M^{-\frac{1}{p_0}} \bigg) (1 -\tilde{\mu})^k. \end{align*} $$

Now pick $C_{\text {w.h.i.}}$ large enough (depending on c, C, m and $M^{-1/p_0}$ ), and get

$$ \begin{align*} & \le C_{\text{w.h.i.}} (1 - c/4) (1 -\tilde{\mu})^k\\ & \le C_{\text{w.h.i.}} (1 -\tilde{\mu})^{k+1}. \end{align*} $$

The proof is now complete.

The full Harnack inequality is a direct consequence of the local boundedness of subsolutions and the weak Harnack inequality.

Proof of Theorem 1.3

Combine Proposition 2.1 and Theorem 1.1, and rescale to reach the result. See, for example, [26] for more details.