\documentclass[JEP,XML,SOM,Unicode,NoFloatCountersInSection,published]{cedram} \datereceived{2021-03-23} \dateaccepted{2022-06-28} \dateepreuves{2022-07-06} \usepackage{mathrsfs} \let\mathcal\mathscr \multlinegap0pt \def\mto{\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}} \newcommand{\Psfrac}[2]{(\sfrac{#1}{#2})} \newcommand{\psfrac}[2]{\sfrac{(#1)}{#2}} \newcommand{\spfrac}[2]{\sfrac{#1}{(#2)}} \newcommand{\pspfrac}[2]{\sfrac{(#1)}{(#2)}} \newcommand{\bnorm}[1]{\bigl\| #1 \bigr\|} \newcommand{\nnorm}[1]{\| #1\|} \DeclareMathOperator{\osc}{osc} \newcommand{\Changel}{\mathcode`l="7160} \newcommand{\Changelback}{\mathcode`l="716C} \newcommand{\NoChangel}[1]{% \expandafter\let\csname old\string#1\endcsname=#1 \let#1=\relax \newcommand{#1}{\mathcode`l="716C\csname old\string#1\endcsname\mathcode`l="7160 }% } \NoChangel{\log} \NoChangel{\ln} \NoChangel{\lim} \NoChangel{\limsup} \NoChangel{\liminf} \NoChangel{\varlimsup} \NoChangel{\varliminf} \NoChangel{\varinjlim} \NoChangel{\varprojlim} \makeatletter \def\@settitle{% \vspace*{-8mm} \raggedleft\includegraphics[scale=.5]{titre-jep} \vtop to 50 mm{% \parindent=0pt {\abstractfont\article@logo\par} \medskip \hrule \vfil \begin{center} \def\baselinestretch{1.2}\large\vfil {\didottitraille\MakeUppercase\@title\par} \vfil\vfil \begin{minipage}{.8\textwidth}\centering \ifx\@empty\smfbyname\else {\smf@byfont\smfbyname\ifsmf@byauthor\enspace\else\ \fi}% \fi {\smf@authorfont \edef\smfandname{{\noexpand\smf@andfont \smfandname}} \andify\authors\authors\par} \end{minipage} \vfil \vrule height .4pt width .3\textwidth \vfil \end{center}}% \par\enlargethispage{.5\baselineskip}% } \def\@setthanks{\def\thanks##1{\par##1\@addpunct{{\upshape.}}}\vspace*{-5pt}\thankses} \makeatother \usepackage{mathtools} \usepackage{esint} \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{remarks}[theorem]{Remarks} \newtheorem{example}[theorem]{Example} \newcommand{\norm}[1]{\left\| #1 \right\|} \newcommand\N{{\mathbb N}} \newcommand\R{{\mathbb R}} \newcommand\T{{\mathbb T}} \newcommand\CC{{\mathbb C}}\let\C\CC \newcommand\Q{{\mathbb Q}} \newcommand\Z{{\mathbb Z}} \newcommand{\cC}{\mathcal C} \newcommand{\cQ}{\mathcal Q} \newcommand{\cU}{\mathcal{U}} \newcommand{\cK}{\mathcal K} \newcommand{\cT}{\mathcal T} \newcommand{\dd}{{\, \mathrm d}} \let\var\varepsilon \datepublished{2022-07-13} \begin{document} \frontmatter \title[Quantitative De~Giorgi methods in kinetic theory]{Quantitative De~Giorgi methods\\ in kinetic theory} \author[\initial{J.} \lastname{Guerand}]{\firstname{Jessica} \lastname{Guerand}} \address{Université de Montpellier, IMAG\\ 499-554 rue du Truel, 34090 Montpellier, France} \email{jessica.guerand@umontpellier.fr} \urladdr{https://sites.google.com/view/guerand} \author[\initial{C.} \lastname{Mouhot}]{\firstname{Clément} \lastname{Mouhot}} \address{University of Cambridge, Department of Pure Mathematics and Mathematical Statistics\\ Wilberforce Road, Cambridge CB3 0WA, United Kingdom} \email{c.mouhot@dpmms.cam.ac.uk} \urladdr{https://cmouhot.wordpress.com/} \thanks{The authors acknowledge funding by the ERC grant MAFRAN 2017-2022} \begin{abstract} 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 Hölder continuity with quantitative estimates. The paper is self-contained. \end{abstract} \subjclass{35K70, 35Q84, 35R09, 35B45, 35B65} \keywords{Hypoelliptic equations, kinetic theory, Fokker-Planck equation, ultraparabolic equations, Kolmogorov equation, Hölder continuity, De~Giorgi method, Moser iteration, averaging lemma, weak Harnack inequality, trajectories} \altkeywords{Équations hypoelliptiques, théorie cinétique, équation de Fokker-Planck, équations ultraparaboliques, équation de Kolmogorov, continuité höldérienne, méthode de De~Giorgi, itération de Moser, lemme de moyenne, inégalité de Harnack faible, trajectoires} \alttitle{Méthodes à la De~Giorgi quantitatives en théorie cinétique} \begin{altabstract} Nous considérons des équations hypoelliptiques de type Fokker-Planck cinétique, également appelées équations de Kolmogorov ou ultraparaboliques, avec des coefficients sans régularité dans l'opérateur de dérive-diffusion. Nous donnons de nouvelles preuves quantitatives du lemme des valeurs inter\-médiaires de De~Giorgi ainsi que des inégalités de Harnack faibles et fortes. Cela implique la continuité höldérienne avec bornes explicites. L'article ne fait pas appel à des résultats précédents.\end{altabstract} \maketitle \vspace*{-\baselineskip} \tableofcontents \mainmatter \section{Introduction} \label{sec:intro} \subsection{The problem studied} \label{sec:theequation} This paper is concerned with local regularity properties, namely boundedness, Harnack inequalities and Hölder continuity, of solutions $f=f(t,x,v)$ to the following class of hypoelliptic partial differential equations in divergence form \begin{equation} \label{e:main} \partial_t f + v \cdot \nabla_x f = \nabla_v \cdot ( A \nabla_v f) + B \cdot \nabla_v f + S, \quad t \in \R, \ x \in \R^d, \ v \in \R^d, \end{equation} where $A=A(t,x,v)$, $B=B(t,x,v)$ and $S=S(t,x,v)$ satisfy (for some constants $0 < \lambda < \Lambda$): \begin{equation} \label{e:hyp-coef} \begin{dcases} \text{$A$ is a measurable symmetric real matrix field with eigenvalues in } [\lambda,\Lambda],\\ \text{$B$ is a measurable vector field such that } |B| \le \Lambda,\\ \text{$S$ is a real scalar field in $L^\infty$.} \end{dcases} \end{equation} This equation naturally appears in kinetic theory where it is refereed to as the \emph{kinetic Fokker-Planck equation}; it is included in the class considered by Kolmogorov~\cite{kolmogorov} (with constant $A$ and linear $B$) that inspired the theory of hypoellipticity of Hörmander~\cite{MR0222474} (see \cite{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~\eqref{e:main}--\eqref{e:hyp-coef} is invariant under translations in $t$, $x$ and under \emph{Galilean translations}, \ie under $z \mto z_0 \circ z$ where $z_0 =(t_0,x_0,v_0)$, $z = (t,x,v)$ and with the non-commutative group operation \begin{equation*} z_0 \circ z = (t_0+t,x_0+x + t v_0, v_0 + v). \end{equation*} Finally for any $r>0$ it is invariant under the scaling $z=(t,x,v) \mto rz := (r^2t,r^3x,r v)$. Using the invariances of the equation, we define for $z_0 \in \R^{1+2d}$ and $r>0$: \begin{align*} Q_r(z_0) := z_0 \circ \left[ r Q_1 \right] := \left\{ -r^2 0$ whose only relevant dependency, if any, is specified in the index, as in $\lesssim_{\text{parameter}}$. We write $a\sim b$ if $a \lesssim b$ and $a \gtrsim b$. We write $\fint$ for integrals normalized by the volume of the integration domain, and $\cT:=\partial_t+v\cdot \nabla_x$. \skpt \begin{definition}[Weak solution, sub-solution, super-solution] \label{d:weak} Let $\cU = (a,b) \times \Omega_x \times \Omega_v$ with $-\inftyV_m$. If $Q_{r_0}^{-}$ and $Q_{r_0}$ 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~\ref{t:IVL}. \end{remark} \begin{theorem}[Harnack inequalities] \label{t:harnack} There is $\zeta >0$ depending only $\lambda,\Lambda$ such that any non-negative weak super-solution $f$ to~\eqref{e:main}--\eqref{e:hyp-coef} in $Q_1$ satisfies the weak Harnack inequality\vspace*{-3pt} \begin{equation} \label{eq:w-Harnack-stat} \biggl( \int_{\tilde Q_{\sfrac{r_0}{2}}^-} f^\zeta (z) \dd t \dd x \dd v \biggr)^{\sfrac{1}{\zeta}} \lesssim_{\lambda,\Lambda} \inf_{ Q_{\sfrac{r_0}{2}}}f + \|S\|_{L^\infty (Q_1)}, \end{equation} where $r_0=\sfrac{1}{20}$ and $\tilde Q_{\sfrac{r_0}{2}} ^{-}:=Q_{\sfrac{r_0}{2}}((-\frac{19}{8}\,r_0^2,0,0))$ (see Figure~\ref{fig:LVI}), and any non-negative weak solution $f$ to~\eqref{e:main}--\eqref{e:hyp-coef} satisfies the following Harnack inequality (with $\tilde Q_{\sfrac{r_0}{4}}^{-}:=Q_{\sfrac{r_0}{4}}((-\frac{19}{8}\,r_0^2,0,0))$)\vspace*{-3pt} \begin{equation} \label{eq:s-Harnack-stat} \sup_{\tilde Q_{\sfrac{r_0}{4}}^{-}} f \lesssim_{\lambda,\Lambda} \inf_{Q_{\sfrac{r_0}{4}}} f + \| S \|_{L^\infty(Q_1)}. \end{equation} \end{theorem} \skpt \begin{remarks} \begin{enumerate} \item The ``weak'' Harnack inequality, in spite of its name, is not weaker than Harnack inequality since it holds for super-solutions. Combined with the $L^{\zeta} \to L^\infty$ gain of integrability in~Proposition~\ref{prop:1st lemma}, it implies the Harnack inequality for solutions. Super-solutions of the form ${\bf 1}_{x+ct\ge a}$ for $a\in \mathbb{R}$ and $|c|>V_m$ (included in our definition) show that the gap in time is required in~\eqref{eq:w-Harnack-stat}. \item The Harnack inequality for equation~\eqref{e:main} was first proved in~\cite{gimv} by a non-constructive argument. The present paper provides a new constructive De~Giorgi approach. Another constructive proof by the Moser-Kru\v{z}kov approach is proposed in~\cite{Guerand-Imbert}. The weak Harnack inequality was obtained for the long-range Boltzmann equation in~\cite{MR4049224}, and was proved for the kinetic Fokker-Planck equations considered in this paper in~\cite{Guerand-Imbert} by the Moser-Kru\v{z}kov approach. \item As compared to that in~\cite{Guerand-Imbert}, 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~\ref{p:HPI lemma}) and measure-to-pointwise estimate (Lemma \ref{l:increase}) 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~\cite{Guerand-Imbert}. \end{enumerate} \end{remarks} \begin{theorem}[Hölder continuity] \label{t:holder} There is $\alpha \in (0,1)$, computable from the proof and only depending on $\lambda, \Lambda$ and $\|S\|_{L^{\infty}}$, such that any weak solution $f$ of \eqref{e:main}--\eqref{e:hyp-coef} in~$Q_2$ satisfies\vspace*{-3pt} \begin{align*} [f]_{C^\alpha (Q_{1})} & := \sup_{\substack{z_1,z_2 \in Q_1\\ z_1 \not = z_2}} \frac{|f(z_1) - f(z_2)|}{|z_1-z_2|^{\alpha}} \\ & \lesssim_{\lambda,\Lambda} \bigl(1+\|S\|_{L^\infty(Q_2)} \bigr) \bigl( \|f\|_{L^2 (Q_2)} + \|S\|_{L^\infty(Q_2)} \bigr). \end{align*} \end{theorem} \begin{remark} The Hölder continuity was first proved in \cite{wz09,wz11} (with constructive method) and this proof is revisited and simplified in~\cite{Guerand-Imbert}, including ideas and methods from~\cite{moser,kru63,kru64}. An alternative non-constructive proof was proposed in~\cite{gimv} following the De~Giorgi method \cite{DG56}: the non-constructive part was the intermediate-value lemma and we provide here a new constructive argument. \end{remark} \subsection{Structure of the method} The core of our proof is, given $f$ sub-solution to~\eqref{e:main}--\eqref{e:hyp-coef} with $S=0$:\vspace*{-5pt}\enlargethispage{.5\baselineskip}% \begin{align*} f \in L^\zeta, \ \zeta >0 & \quad \xrightarrow{(1)} \quad f \in L^\infty \cap L^1_{t,v}W^{\sfrac13-0,1}_x \\[-5pt] & \quad \xrightarrow{(2)} \quad \text{Weak Poincaré inequality in $L^1$} \\[-5pt] & \quad \xrightarrow{(3)} \quad \text{Intermediate-Value Lemma (Theorem~\ref{t:IVL}})\\[-5pt] & \quad \xrightarrow{(4)} \quad \text{Measure-to-pointwise estimate} \\[-5pt] & \quad \xrightarrow{(5)} \quad \text{Weak log-Harnack estimate} \\[-5pt] & \quad \xrightarrow{(6)} \quad \text{Weak Harnack estimate.} \end{align*} 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~\ref{ss:Holder}) from either the measure-to-pointwise estimate applied to both sub-solutions $f$ and $-f$, or from the Harnack inequality. Step~(1) (Section~\ref{sec:int}) is semi-novel: it elaborates upon ideas in~\cite{pp} 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~\ref{p:HPI lemma}) 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~\ref{ss:proof-ivl}) is novel and based on simple energy estimates. Step~(4) (Lemma~\ref{l:increase} in Subsection~\ref{ss:m2p}) is standard and sketched for the sake of obtaining quantitative constants. Step~(5) (in Section~\ref{sec:Harnack}) is semi-novel but immediate when constants are quantified properly in the previous steps. Step~(6) (in Section~\ref{sec:Harnack}) is novel in the context of hypoelliptic equations but inspired from elliptic equations~\cite{MR3565366}; it uses an induction, Vitali's covering lemma and Step~(5) at every scale. \subsubsection*{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-Kru\v{z}kov approach in \cite{Guerand-Imbert}. The second author would also like to thank L.\,Silvestre who pointed out several years ago how Kolmogorov fundamental solutions were used in~\cite{pp} to replace averaging lemma, which was the starting point of our Section~\ref{sec:int} (and is also used in~\cite{MR4049224}). \section{Integral estimates revisited} \label{sec:int} In this section, we briefly revisit estimates from~\cite{pp,gimv} on the gain of inte\-grability for sub-solutions (the kinetic counterpart to the first lemma of De~Giorgi) and the low-order Sobolev regularity estimate for sub-solutions, first mentioned in~\cite{gimv}. We provide new proofs based on fundamental solutions which, albeit variants of existing ones, seem simpler and optimal. \Subsection{The energy estimate} \begin{proposition}[Energy estimate] \label{prop:EE} Let $f$ be a non-negative weak sub-solution to~\eqref{e:main}--\eqref{e:hyp-coef} in an open set $\cU \in \R^{1+2d}$. Given any $Q_r(z_0) \subset Q_R(z_0) \subset \cU$ with $00,\\ 0 & \text{if } t\leq 0. \end{dcases} \end{align*} Since $f$ and $G$ are non-negative, we deduce that \begin{multline*} 0 \le f(t,x,v) \le\int_{(t',x',v')\in \R^{2d+1}} G(t-t', x-x'-(t-t')v',v-v')\\ \cdot\bigl[(\nabla_{v'} \cdot F_1) (t',x',v')+ F_2 (t',x',v') \bigr] \end{multline*} and since \begin{multline*} \forall \, t \ge 0, \ x,v \in \R^d, \\[-5pt] |\nabla_v G(t,x,v)| +t |\nabla_x G(t,x,v)| \lesssim t^{-2d-\sfrac12} \exp \Bigl[ - \frac{3 \left| x - \Psfrac{t}{2} v \right|^2}{2t^3} - \frac{|v|^2}{8t} \Bigr] \end{multline*} we have $\nabla_v G, t \nabla_x G \in L^{\frac{2d+1}{2d+1/2}-0} (\left(0,\tau\right) \times \R^{2d})$ and therefore by integration by parts \begin{multline*} f(t,x,v) \le \int_{(t',x',v')\in \R^{2d+1}} \nabla_{v'} G(t-t',x-x'-(t-t')v',v-v') F_1(t',x',v')\\ + \int_{(t',x',v') \in \R^{2d+1}} G(t-t',x-x'-(t-t')v',v-v') F_2(t',x',v') \end{multline*} and Young's convolution inequality (which works in unimodular spaces like $(\R^{2d+1},\circ)$ with the Lebesgue measure), we deduce, by tracking down the dependency in $p$ of the constant \begin{multline*} \forall \, p \in \left[2,2+\sfrac1d\right), \\ \| f \|_{L^{p}(\R_- \times \R^{2d})}\lesssim_\tau \left( 2+ \Psfrac1d - p \right)^{-1} \bigl[ \| F_1 \|_{L^2(\R_- \times \R^{2d})} + \left\| F_2 \right\|_{L^2(\R_- \times \R^{2d})} \bigr]. \end{multline*} (The threshold $2+\sfrac1d$ is likely to be optimal.) This proves~\eqref{eq:Kolm1}. To prove~\eqref{eq:Kolm2} split \begin{align*} G = G_\var + G_\var^\bot \quad \text{ with } \quad G_\var(t,x,v) := \chi \left( \sfrac{t}{\var} \right) G(t,x,v), \end{align*} where $\var >0$ and $\chi$ is a smooth function on $\R_+$ valued in $[0,1]$ equal to $1$ in $[0,1]$ and~$0$ on $[2,+\infty)$. \Changel We have the following simple estimates for every $l \in \mathbb{N}$ \begin{align*} \left|\nabla_x^l G_\var^\bot(t,x,v)\right| &\lesssim_l \var^{-\Psfrac32 l}t^{-2d} \exp \Bigl[ - \frac{3 \left| x - \psfrac{t}{2} v \right|^2}{2t^3} - \frac{|v|^2}{8t} \Bigr] \\ \left|\nabla_v \nabla_x^l G_\var^\bot(t,x,v)\right| + t\bigl|\nabla_x \nabla_x^l &G_\var^\bot(t,x,v) \bigr| \\[-3pt] &\lesssim_l \var^{-\Psfrac32 l -\sfrac12} t^{-2d} \exp \Bigl[ - \frac{3 \left| x - \Psfrac{t}{2} v \right|^2}{2t^3} - \frac{|v|^2}{8t} \Bigr] \end{align*} which straightforwardly implies (assuming $\tau \ge 1$ and $\var <1$ wlog) \begin{align*} \left\| G_\var^\bot \right\|_{L^1_{t,v}((0,\tau) \times \R^d; W^{l,1}_x(\R^d)))} + \left\| \nabla_v G_\var^\bot \right\|_{L^1_{t,v}((0,\tau) \times \R^d ; W^{l,1}_x(\R^d))}\hspace*{1mm}& \\ + \left\|t\nabla_x G_\var^\bot \right\|_{L^1_{t,v}((0,\tau) \times \R^d ; W^{l,1}_x(\R^d))} &\lesssim_l \tau \var^{-\Psfrac32 l -\sfrac12} \\ \| G_\var \|_{L^1((0,\tau) \times \R^{2d})} + \left\| \nabla_v G_\var \right\|_{L^1((0,\tau) \times \R^{2d})} + \left\|t\nabla_x G_\var \right\|_{L^1((0,\tau) \times \R^{2d})} &\lesssim \tau \var^{\sfrac12}. \end{align*} The splitting $G=G_\var + G_\var^\bot$ yields $f = f_\var + f_\var^\bot$, and the convolution inequality $M^1 * L^1 \to L^1$ implies \begin{align*} & \| f_\var \|_{L^1(\R_- \times \R^{2d})} \lesssim \tau \var^{\sfrac12} \bigl( \| F_1 \|_{L^1(\R_- \times \R^{2d})} + \left\| F_2 \right\|_{L^1(\R_- \times \R^{2d})} + \| m \|_{M^1(\R_- \times \R^{2d})} \bigr) \\ & \bigl\| f_\var^\bot \bigr\|_{L^1_{t,v}W^{l,1}_x(\R_- \times \R^{2d})} \\ & \hphantom{\| f_\var \|_{L^1(\R_- \times \R^{2d})}} \lesssim \tau \var^{-\Psfrac32 l -\sfrac12} \bigl[ \| F_1 \|_{L^1(\R_- \times \R^{2d})} \!+\! \left\| F_2 \right\|_{L^1(\R_- \times \R^{2d})} \!+\! \| m \|_{M^1(\R_- \times \R^{2d})} \bigr]. \end{align*} Since this decomposition holds for all $\var >0$, it implies by standard interpolation the estimate~\eqref{eq:Kolm2} for any $\sigma \in [0,\sfrac{1}{3})$ (again the exponent is likely to be optimal but in any case our constant degenerates as $\sigma \to \sfrac{1}{3}$). In order to be self-contained let us a give a short proof. Given $\sigma \in [0,\sfrac13)$, we Fourier-transform and decompose dyadically, defining $\langle \xi \rangle:= (1+|\xi|^2)^{\sfrac12}$ \begin{equation} \label{eq:decomp} \begin{aligned} (1-\Delta_x)&^{\sfrac{\sigma}{2}} f(t,x,v) \\ &= \int_{\xi,y \in \R^d} e^{i \xi \cdot (x-y)} \langle \xi \rangle^\sigma f(t,y,v) = \sum_{k \ge -1} \int_{\xi,y \in \R^d} e^{i \xi \cdot (x-y)} a_k(\xi) f(t,y,v) \\ &= \sum_{k \ge -1} \int_{\xi,y \in \R^d} e^{i \xi \cdot (x-y)} B_k(\xi) (1-\Delta_y)^{\sfrac{l}{2}} f(t,y,v), \end{aligned} \end{equation} where $a_k(\xi) := \langle \xi \rangle^\sigma \varphi_k$ and $B_k(\xi) := \langle \xi \rangle^{\sigma-l} \varphi_k$, and where we have defined in the standard way $\varphi_k(\xi) := [ \chi(2^{-k}\xi) - \chi(2^{-k+1} \xi) ]$ for $k \ge 0$ with $\chi$ a smooth function valued in $[0,1]$ and equal to $1$ in $B(0,1)$ and $0$ outside $B(0,2)$, and $\varphi_{-1}(\xi) =\nobreak \sum_{k \le -1} [ \chi(2^{-k}\xi) -\nobreak \chi(2^{-k+1} \xi) ]$. For a given $F=F(y)$ one has \begin{align*} \int_{x \in \R^d} \left| \int_{\xi,y \in \R^d} e^{i \xi \cdot (x-y)} a_k(\xi) F(y) \right| &\lesssim 2^{k\sigma} \| F \|_{L^1}, \\ \int_{x \in \R^d} \left| \int_{\xi,y \in \R^d} e^{i \xi \cdot (x-y)} B_k(\xi) (1-\Delta_y) F(y) \right| &\lesssim 2^{k(\sigma-l)} \| F \|_{W^{l,1}} \end{align*} by splitting the integrand into $|x-y| \le 2^{-k}$ and $|x-y| > 2^{-k}$ and integrating by parts the operator $\Delta_\xi^{\sfrac{\ell}{2}}$ with $\ell$ even and strictly greater than $d$. We then use the decomposition~\eqref{eq:decomp} in the ``$a_k$'' form on $f_\var$ and in the ``$B_k$'' form on $f_\var^\bot$, and with a $\var=\var_k$ depending on $k$ defined below: \begin{align*} \bigl\| (1-\Delta_x)^{\sfrac{\sigma}{2}} f \bigr\|_{L^1(\R_- \times \R^{2d})} & \lesssim \tau \sum_{k \ge -1} \bigl( \var_k ^{\sfrac12} 2^{k\sigma} + \var_k^{-\Psfrac{3}{2}l -\sfrac12} 2^{k(\sigma-l)} \bigr) \\ &\hspace*{3mm} \times \bigl[ \| F_1 \|_{L^1(\R_- \times \R^{2d})} + \left\| F_2 \right\|_{L^1(\R_- \times \R^{2d})} + \| m \|_{M^1(\R_- \times \R^{2d})} \bigr]\\ & \lesssim \frac{\tau}{\delta} \bigl[ \| F_1 \|_{L^1(\R_- \times \R^{2d})} + \left\| F_2 \right\|_{L^1(\R_- \times \R^{2d})} + \| m \|_{M^1(\R_- \times \R^{2d})} \bigr] \end{align*} with the choice $\sigma = \sfrac13 - \delta \in [0,\sfrac13)$ and $\var_k := 2^{-2k(\sfrac13-\sfrac{\delta}{2})}$ and $l> 1+\sfrac{4}{9\delta}$. This concludes the proof. \Changelback \end{proof} \subsection{Integral estimates for sub-solutions} We combine the previous lemma with a localization argument and the energy estimate to get the \begin{proposition}[Integral regularization estimates for non-negative sub-solutions] \label{prop:Gain sol} Let $f$ be a non-negative weak sub-solution to~\eqref{e:main}--\eqref{e:hyp-coef} in an open set $\cU \in \R^{1+2d}$. Given any $Q_r(z_0) \subset Q_R(z_0) \subset \cU$ with $00$, $f$ satisfies \begin{equation*} \| f \|_{L^\infty(Q_r(z_0))} \lesssim_{\lambda,\Lambda} \Bigl( \frac{1+|v_0|}{r^2(R-r)^3}\Bigr)^{\psfrac{1+4d}{\zeta}} \bigl[ \|f\|_{L^\zeta(Q_R(z_0))} + \|S\|_{L^\infty(Q_R(z_0))} \bigr]. \end{equation*} \end{proposition} \begin{proof}[Proof of Proposition~\ref{prop:1st lemma}] Fix $p_0:=2+\sfrac{1}{2d}$ and define $q:=\sfrac{p_0}{2}$ and $q_n := q^n$. Consider $\beta_{n,k}$ on $\R_+$ with $\beta_{n,k}' \ge 0$ and $\beta''_{n,k} \ge 0$ both bounded and so that $\beta_{n,k}(z) \to\nobreak z^{q_n}$ as $k \to \infty$ and $\beta_{n,k}(z) \lesssim z^{q_n}$ and $\beta_{n,k}'(z) \lesssim z^{q_n-1}$ uniformly in $k \in \N^*$. Definition~\ref{d:weak} implies that $\beta_{n,k}(f)$ is a weak sub-solution with source term $S_{n,k} := \beta_{n,k}'(f) S$. Define $r_0=R$ and $r_n := r_{n-1} - \delta n^{-2}$ with $\delta = \frac12( \sum_{k \ge 1} k^{-2} )^{-1}(R-r)$. Since $p_0 \in [2,2+\sfrac1d)$, the estimate~\eqref{eq:gain-int} implies for all $n \ge 1$ \begin{align*} \|\beta_{n,k}(f)\|_{L^{p_0}(Q_{r_n}(z_0))} & \lesssim \cC'(r_n,r_{n-1},v_0) \bigl[ \|\beta_{n,k}(f)\|_{L^{2}(Q_{r_{n-1}}(z_0))} + \|S_{n,k}\|_{L^2(Q_{r_{n-1}}(z_0))} \bigr] \\ & \lesssim \frac{(1+|v_0|)n^6}{r^2(R-r)^3} \bigl[ \|\beta_{n,k}(f)\|_{L^{2}(Q_{r_{n-1}}(z_0))} + \|S_{n,k}\|_{L^2(Q_{r_{n-1}}(z_0))} \bigr] \end{align*} for $n \ge 1$, which means by taking $k \to \infty$ and coming back to $f$ \begin{multline*} \|f\|_{L^{2q_{n+1}}(Q_{r_n}(z_0))} \\ \lesssim \Bigl( \frac{(1+|v_0|)n^6}{r^2(R-r)^3} \Bigr)^{\sfrac{1}{q^n}} 2^{-1+\sfrac{1}{q_n}} \Bigl[ \|f\|_{L^{2q_n}(Q_{r_{n-1}}(z_0))} + \| f\|_{L^{2q_n}(Q_{r_{n-1}}(z_0))}^{1-\sfrac{1}{q_n}} \|S\|_{L^\infty(Q_R(z_0))}^{\sfrac{1}{q_n}} \Bigr] \\ \lesssim \Bigl( \frac{(1+|v_0|)n^6}{r^2(R-r)^3} \Bigr)^{\sfrac{1}{q^n}} \Bigl[ \Bigl( 1 + \frac{1}{q_n} \Bigr) \|f\|_{L^{2q_n}(Q_{r_{n-1}}(z_0))} + \frac{1}{q_n} \|S\|_{L^\infty(Q_R(z_0))} \Bigr], \end{multline*} assuming by induction $\|f\|_{L^{2q_n}(Q_{r_{n-1}}(z_0))} < +\infty$. The convergence of the infinite product then implies \begin{equation*} \|f\|_{L^\infty(Q_r(z_0))} \lesssim \Bigl( \frac{1+|v_0|}{r^2(R-r)^3} \Bigr)^{1+4d} \bigl[ \|f\|_{L^2(Q_R(z_0))} + \|S\|_{L^\infty(Q_R(z_0))} \bigr]. \end{equation*} This proves the claim when $\zeta \ge 2$. To prove it when $\zeta \in (0,2)$, we deduce from the previous estimate \begin{multline*} \|f\|_{L^\infty(Q_r(z_0))} + \|S\|_{L^\infty(Q_r(z_0))} \\ \lesssim \Bigl( \frac{1+|v_0|}{r^2(R-r)^3} \Bigr)^{1+4d} \bigl[ \|f\|_{L^\infty(Q_R(z_0))}^{1-\zeta} \|f\|_{L^\zeta(Q_R(z_0))}^\zeta + \|S\|_{L^\infty(Q_R(z_0))}\bigr] \end{multline*} and thus by Young inequality, the quantity $A(r) := \|f\|_{L^\infty(Q_r(z_0))} + \|S\|_{L^\infty(Q_r(z_0))}$ satisfies, for some $C>0$, \begin{align*} A(r) \le \frac12 A(R) + C \Bigl( \frac{1+|v_0|}{r^2(R-r)^3} \Bigr)^{\psfrac{1+4d}{\zeta}} \bigl[ \|f\|_{L^\zeta(Q_R(z_0))} + \|S\|_{L^\infty(Q_R(z_0))}\bigr]. \end{align*} Introducing an (increasing this time) sequence of radii $r_n := r_{n-1} + \delta n^{-2}$ we obtain by induction \begin{align*} A(r_n) &\le \Psfrac12 A(r_{n+1})\\ & + C n^{\psfrac{2+8d}{\zeta}} \Bigl( \frac{1+|v_0|}{r^2(R-r)^3} \Bigr)^{\psfrac{1+4d}{\zeta}} \bigl[ \|f\|_{L^\zeta(Q_R(z_0))} + \|S\|_{L^\infty(Q_R(z_0))} \bigr], \\ A(r_0) &\le \left( \sfrac12 \right)^n A(r_{n+1}) \\ & \hspace{2mm}+ C \biggl( \sum_{k=1}^n \frac{k^{\psfrac{2+8d}{\zeta}}}{2^k} \biggr) \Bigl( \frac{1+|v_0|}{r^2(R-r)^3} \Bigr)^{\psfrac{1+4d}{\zeta}} \bigl[ \|f\|_{L^\zeta(Q_R(z_0))} + \|S\|_{L^\infty(Q_R(z_0))}\bigr], \end{align*} which yields the result by taking $n \to \infty$ in the right hand side. \end{proof} \section{Intermediate-Value Lemma and oscillations} \subsection{Weak Poincaré inequality} The adjective `weak' refers to the small additional $L^2$ error term below. \begin{proposition}[Hypoelliptic Poincaré inequality with error] \label{p:HPI lemma} Given any $\var\in (0,1)$ and $\sigma \in (0,\sfrac13)$, any non-negative sub-solution $f$ to \eqref{e:main}--\eqref{e:hyp-coef} on $Q_5$ satisfies \begin{multline} \label{HPI inequality} \bnorm{\bigl(f-\langle f \rangle_{Q_1^{-}} \bigr)_+}_{L^{1}(Q_1)}\\ \lesssim_{\lambda,\Lambda} \frac{1}{\varepsilon^{d+2}} \norm{\nabla_v f}_{L^1 (Q_5)} + \varepsilon^\sigma \left( \sfrac13 - \sigma \right)^{-1} \norm{f}_{L^2(Q_5)} + \nnorm{S}_{L^2(Q_5)}, \end{multline} where $Q_1^- := Q_1(-2,0,0) = (-3,-2] \times B_1 \times B_1$ and $\langle f \rangle_{Q_1^{-}} := \fint_{Q_1^-} f := \frac{1}{|Q_1^-|} \int_{Q_1^-} f$. \end{proposition} \begin{remark} The motivation for the following argument was \cite[Lem.\,10, p.\,11]{vasseur}, 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 $\nabla_v$ to connect them. We have to deal here with the hypoelliptic structure. \end{remark} \begin{proof} Consider, for $\var \in (0,1)$, a smooth function $\varphi_\var=\varphi_\var(y,w)$ which satisfies $0\leq \varphi_\var \leq 1$ and has compact support in $B_1^{2}$ and such that $\varphi_\var=1$ in $B_{(1-\var)}\times B_{(1-\var)}$ and with $|\nabla_y \varphi_\var|\lesssim \var^{-1}$ and $|\nabla_w \varphi_\var|\lesssim \var^{-1}$. We then split the integral to be estimated as follows \begin{multline*} \bnorm{\bigl(f-\langle f \rangle_{Q_1^{-}}\bigr)_+}_{L^{1}(Q_1)} \lesssim \bnorm{\bigl(f-\langle f \varphi_\var \rangle_{Q_1^{-}} \bigr)_+}_{L^{1}(Q_1)}\\ \hspace*{-1.13cm}\lesssim \int_{(t,x,v) \in Q_1} \biggl\{ \fint_{(s,y,w) \in Q_1^-} \left[ f(t,x,v)-f(s,y,w) \right] \varphi_\var(y, w) \biggr\}_{\!+} \\ \shoveright{+ \norm{f}_{L^1(Q_1)} \fint_{Q_{1}^{-}}\left( 1 - \varphi_\var(y,w) \right)}\\ \lesssim \int_{(t,x,v) \in Q_1} \biggl\{ \fint_{(s,y,w) \in Q_1^-} \left[ f(t,x,v)-f(s,y,w) \right] \varphi_\var(y, w) \biggr\}_{\!+} + \var^{2d} \nnorm{f}_{L^2(Q_1)}, \end{multline*} where we have used $\langle f \varphi_\var \rangle_{Q_1^{-}}\leq \langle f \rangle_{Q_1^{-}}$ and the Cauchy-Schwarz inequality. Let us estimate the first term of the previous inequality. Given $t,x,v$ fixed, we~decompose the trajectory $(t,x,v)\to (s,y,w)$ into four sub-trajectories in $Q_5$: a trajectory of length $O(\var)$ along $\nabla_x$, two trajectories of length $O(1)$ along $\nabla_v$, and finally one trajectory of length $O(1)$ along $\cT := \partial_t + v \cdot \nabla_x$: \begin{multline*} (t,x,v)\underset{\nabla_x}{\to} (t,x+\varepsilon w,v) \underset{\nabla_v}{\to} \Bigl(t,x+\varepsilon w, \frac{x+\varepsilon w -y}{t-s} \Bigr) \\ \underset{\cT}{\to} \Bigl(s, y,\frac{x+\varepsilon w -y}{t-s} \Bigr) \underset{\nabla_v}{\to} (s,y,w). \end{multline*} The first sub-trajectory is estimated by the {\it integral} regularity $L^1_{t,v} W^{\sigma,1}_{x}$ proved in~\eqref{eq:gain-reg}. The other trajectories are estimated directly by the vector fields in the equation. The position $x+\varepsilon w \in Q_2$ since $x,w \in B_1$ and $\var \in (0,1)$. The velocity $\pspfrac{x+\varepsilon w -y}{t-s} \in Q_3$ since $x,w,y \in B_1$ and $t-s\geq 1$ due to the definitions of~$Q_{1}^+$ and $Q_{1}^{-}$, and this velocity yields a transport line from $(t,x+\varepsilon w)$ to $(s,y)$. Note that we are implicitly using the Hörmander commutator condition: $\nabla_v,\cT, [\nabla_v,\cT]$ span all the vector fields on $\R^{2d+1}$. Decompose along the previous trajectories \begin{align*} f(t&,x,v)-f(s,y,w) \\ & = \bigl[ f(t,x,v)-f(t,x+\varepsilon w,v) \bigr] + \Bigl[ f(t,x+\varepsilon w,v)- f \Bigl(t,x+\varepsilon w,\frac{x+\varepsilon w -y}{t-s} \Bigr) \Bigr] \\ & \qquad + \Bigl[ f \Bigl(t,x+\varepsilon w,\frac{x+\varepsilon w -y}{t-s} \Bigr)- f \Bigl(s,y,\frac{x+\varepsilon w -y}{t-s} \Bigr) \Bigr] \\ & \qquad + \Bigl[ f \Bigl(s,y,\frac{x+\varepsilon w -y}{t-s} \Bigr)-f(s,y,w) \Bigr] \end{align*} and integrate against $\varphi_\var(y,w)$ on $(s,y,w)\in Q_{1}^-$, which gives the four terms \begin{align*} I_1(t,x,v) & := \int_{(s,y,w)\in Q_1^{-}} \bigl[ f(t,x,v)-f(t,x+\varepsilon w,v)\bigr] \varphi_\var(y,w),\\ I_2(t,x,v) & := \int_{(s,y,w)\in Q_1^{-}} \Bigl[ f(t,x+\varepsilon w,v)-f\Bigl(t,x+\varepsilon w,\frac{x+\varepsilon w -y}{t-s}\Bigr)\Bigr] \varphi_\var(y,w), \\ I_3(t,x,v) & := \int_{(s,y,w)\in Q_1^{-}} \Bigl[ f\Bigl(t,x+\varepsilon w,\frac{x+\varepsilon w -y}{t-s}\Bigr)-f\Bigl(s,y,\frac{x+\varepsilon w -y}{t-s}\Bigr)\Bigr] \varphi_\var(y,w), \\ I_4(t,x,v) & := \int_{(s,y,w)\in Q_1^{-}} \Bigl[ f\Bigl(s,y,\frac{x+\varepsilon w -y}{t-s}\Bigr)-f(s,y,w)\Bigr] \varphi_\var(y,w). \end{align*} Regarding the term $I_2$, we use Taylor's formula and $0\leq \varphi_\var\leq 1$ to deduce \begin{align*} I_2(t,x,v) &\leq \int_{(s,y,w) \in Q_1^{-}} \int_{\tau\in [0,1]} \Bigl(v-\frac{x+\varepsilon w-y}{t-s}\Bigr) \cdot \\ & \hspace{3.3cm} \times \nabla_v f \Bigl(t,x+\varepsilon w, \tau v + (1-\tau)\frac{x+\varepsilon w-y}{t-s} \Bigr) \varphi_\var(y,w) \\ &\lesssim \int_{(s,y,w) \in Q_1^{-}} \int_{\tau\in [0,1]} |\nabla_v f| \Bigl(t,x+\varepsilon w, \tau v + (1-\tau)\frac{x+\varepsilon w-y}{t-s} \Bigr). \end{align*} Integrate then on $(t,x,v)\in Q_{1}^+$ to get \begin{multline} \label{estim I2} \fint_{(t,x,v) \in Q_1} I_2 \\ \hspace*{-3mm}\lesssim \int_{(t,X,v)\in (-1,0)\times B_2 \times B_1} \int_{(s,Y,w)\in (-3,-2)\times B_4 \times B_1} \int_{\tau\in (0,1)} |\nabla_v f| \left(t,X, v+ (1-\tau) Y \right) \\ \lesssim \int_{(t,X,V)\in (-1,0)\times B_2 \times B_5} \int_{(s,Y,w)\in (-3,-2)\times B_4 \times B_1} \int_{\tau\in (0,1)} |\nabla_v f| \left(t,X, V\right) \lesssim \int_{Q_5} |\nabla_v f|, \end{multline} where we have used successively the following changes of variables with bounded Jacobians: \begin{equation*} x\to X=x+\varepsilon w \in B_2, \quad y\to Y= \frac{X -y}{t-s} -v\in B_4, \quad v\to V=v+(1-\tau)Y\in B_5. \end{equation*} The term $I_4$ is treated like $I_2$: \begin{align} \label{estim I4} \fint_{(t,x,v) \in Q_1} I_4 \lesssim \int_{Q_5} |\nabla_v f|. \end{align} Regarding the term $I_1 $, we perform the change of variable $w \in B_1 \to x'=x+\varepsilon w \in B_\var(x)$ with Jacobian $\var^{-d}$ and use the $L^1_{t,v} W^{\sigma,1}_{x}$ regularity of non-negative sub-solutions proved in~\eqref{eq:gain-reg}:\vspace*{-3pt} \begin{equation} \label{estim I1} \begin{aligned} \fint_{(t,x,v)\in Q_1} I_1 & \lesssim \int_{(t,x,v)\in Q_1, \, (s,y,w)\in Q_1^{-}} |f(t,x,v)-f(t,x+\varepsilon w,v) | \\ &\lesssim \int_{(t,x,v)\in Q_1, \, w \in B_1} \frac{|f(t,x,v)-f(t,x+\varepsilon w,v)|}{|\varepsilon w |^{d+\sigma}}\,|\varepsilon w|^{d+\sigma} \\ &\lesssim \var^\sigma \int_{(t,x,v)\in Q_1, \, x' \in B_2 } \frac{|f(t,x,v)-f(t,x',v)|}{|x-x'|^{d+\sigma}} \\ & \lesssim \var^\sigma \norm{f}_{L^1_{t,v}W^{\sigma,1}_x(Q_2)} \lesssim \var^\sigma \left( \sfrac13 -\sigma \right)^{-1} \bigl[ \norm{f}_{L^2(Q_3)} + \norm{S}_{L^2(Q_3)} \bigr]. \end{aligned} \end{equation} Regarding the term $I_3$, we note first that $\cT f \in L^2_{t,x} H^{-1}_v + M^1_{t,x,v}$ with finite norm in $Q_R(z_0)$ (arguing as in proof of Proposition~\ref{prop:Gain sol}). The Taylor formula between $(t,x+\varepsilon w)$ and $(s,y)$ along $\cT$ thus holds in weak form against $\varphi_\var$ thanks to the latter bounds and the non-singular change of variable~\eqref{eq:change-variables} discussed below:\vspace*{-5pt} \begin{multline}\label{eq:Taylor-weak} I_3(t,x,v)\\ = \int_{(s,y,w)\in Q_1^{-}} \Bigl[ f\Bigl(t,x+\varepsilon w,\frac{x+\varepsilon w -y}{t-s}\Bigr)-f\Bigl(s,y,\frac{x+\varepsilon w -y}{t-s}\Bigr)\Bigr] \varphi_\var(y,w)\\ \shoveleft{\hspace*{-1mm}\lesssim \int_{(s,y,w)\in Q_1^{-}} \int_{\tau \in [0,1]} (t-s)}\\ \times \cT f\Bigl(\tau t+(1-\tau)s,\tau (x+\varepsilon w)+(1-\tau)y,\frac{x+\varepsilon w -y}{t-s}\Bigr)\varphi_\var(y,w). \end{multline} We then use the fact that $f$ is a sub-solution to~\eqref{e:main} in the distributional sense:\vspace*{-5pt} \begin{align*} I_3(t&,x,v)\\ &\lesssim \int_{(s,y,w)\in Q_1^{-}} \int_{\tau \in [0,1]} (t-s) \\ &\hspace*{1.1cm}\times \nabla_v \cdot (A \nabla_v f)\Bigl(\tau t+(1-\tau)s,\tau (x+\varepsilon w)+(1-\tau)y,\frac{x+\varepsilon w -y}{t-s}\Bigr)\varphi_\var(y,w)\\ &\hspace*{.2cm}+\int_{(s,y,w)\in Q_1^{-}} \int_{\tau \in [0,1]} (t-s) \\ &\hspace*{1.1cm}\times B\cdot \nabla_v f\Bigl(\tau t+(1-\tau)s,\tau (x+\varepsilon w)+(1-\tau)y,\frac{x+\varepsilon w -y}{t-s}\Bigr)\varphi_\var(y,w)\\ &\hspace*{.2cm}+ \int_{(s,y,w)\in Q_1^{-}} \int_{\tau \in [0,1]} (t-s) \\ &\hspace*{1.1cm}\times S\Bigl(\tau t+(1-\tau)s,\tau (x+\varepsilon w)+(1-\tau)y,\frac{x+\varepsilon w -y}{t-s}\Bigr)\varphi_\var(y,w)\\ &:= I_{31}+I_{32}+I_{33}. \end{align*} Arguing as for $I_2$ and $I_4$, we have \begin{align} \label{estim I323} \fint_{(t,x,v) \in Q_1} I_{32} \lesssim \int_{Q_5} \left| \nabla_v f\right| \quad \text{ and } \quad \fint_{(t,x,v) \in Q_1} I_{33} \lesssim \int_{Q_5} |S|, \end{align} where we performed consecutively the changes of variable \begin{gather*} y\to V=\frac{x+\varepsilon w-y}{t-s},\quad x\to X= x+\varepsilon w -(1-\tau)(t-s)V,\\ s\to s'=t-s\quad \text{and}\quad t'\to t-(1-\tau)s'. \end{gather*} To estimate the remaining term $I_{31}$, we use the change of variable \begin{equation} \label{eq:change-variables} (y,w) \mto (Y,W) \quad \text{with } Y:=\tau (x+\varepsilon w)+(1-\tau)y \text{ and } W:=\frac{x+\varepsilon w -y}{t-s} \end{equation} such that $(y,w)\mto (Y,W)$ is a bijection from the set $(B_1)^2$ to the (diamond-shaped) set \begin{equation*} E:=E(\tau,\varepsilon,t,s,x) \subset B\left(\tau x, (1-\tau) + \tau \var\right) \times B\Bigl(\frac{x}{t-s},\frac{1+\var}{t-s}\Bigr) \subset B_2 \times B_3 \end{equation*} with Jacobian $(\spfrac{\var}{t-s})^d$ and which maps respective boundaries (to compute the Jacobian easily use the formula $\det \bigl( \begin{smallmatrix} A & B \\ C & D \end{smallmatrix}\bigr) = \det(A - B D^{-1} C) \det D$). We deduce \begin{multline*} I_{31} = \frac{1}{\varepsilon^d} \int_{\tau\in [0,1]}\int_{s\in (-3,-2), \, (Y,W)\in E} (t-s)^{d+1}\nabla_v\cdot (A \nabla_v f)\left(\tau t+(1-\tau)s,Y,W\right)\\ \times \varphi_{\varepsilon} \Bigl(Y-\tau (t-s)W,\frac{Y-x+(1-\tau)(t-s)W}{\varepsilon} \Bigr) \end{multline*} and we integrate by parts in $W$, using that $\varphi_\var=0$ on the boundary of $E(\tau,\var,t,s,x)$: \begin{multline*} I_{31}= \frac{1}{\varepsilon^d}\int_{\tau \in [0,1]}\int_{s\in (-3,-2), (Y,W)\in E} (t-s)^{d+1}(A \nabla_v f)\left(\tau t+(1-\tau)s,Y,W\right) \\ \times \Bigl[\tau (t-s)\nabla_y\varphi_\var \Bigl(Y - \tau (t-s)W,\frac{Y-x+(1-\tau)(t-s)W}{\varepsilon} \Bigr)\\ -\frac{(1-\tau)(t-s)}{\var}\nabla_w\varphi_\var \Bigl(Y-\tau (t-s)W,\frac{Y-x+(1-\tau)(t-s)W}{\varepsilon} \Bigr)\Bigr]. \end{multline*} Using the bounds on the derivatives of $\varphi_\var$ then yields \begin{multline} I_{31}(t,x,v) \lesssim \frac{1}{ \varepsilon^{d+2}} \int_{\tau \in [0,1]}\int_{s\in (-3,-2), (Y,W)\in E} |\nabla_v f|\left(\tau t+(1-\tau)s,Y,W\right) \\ \implies \label{estim I31} \fint_{Q_1} I_{31} \lesssim \frac{1}{ \varepsilon^{d+2}} \int_{Q_3} |\nabla_v f|. \end{multline} The result follows from combining \eqref{estim I2}, \eqref{estim I4}, \eqref{estim I1}, \eqref{estim I323} and~\eqref{estim I31}. \end{proof} \begin{remark} Note that the regularity $W^{\sigma,1}_{x}$ 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 \emph{only} the variables in $Q_1^-$. Note also that it is possible to get some $W^{\sigma',1}_{t,x,v}$ regularity in all variable with $\sigma' \in (0,\sigma)$ small by the same method as in Lemma~\ref{lem:Kolmogorov}, however such regularity is too weak to yield any intermediate value estimate alone. Note also that the gap in time between $Q_1^-$ and~$Q_1$ is used to make sure the intermediate velocity $\pspfrac{x+\var w-y}{t-s}$ 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~\ref{rem:gap}. \end{remark} \subsection{Proof of the Intermediate-Value Lemma} \label{ss:proof-ivl} In this subsection, we prove that Proposition~\ref{p:HPI lemma} implies Theorem~\ref{t:IVL}. Take $f$ a sub-solution to~\eqref{e:main}--\eqref{e:hyp-coef} on $Q_1$ and satisfying~\eqref{e: hyp IVL} for some given $\delta_1,\delta_2>0$:\vspace*{-3pt} \begin{align} \label{hyp bis IVL} |\{f\leq 0\}\cap Q_{r_0}^{-}| \geq \delta_1 |Q_{r_0}^{-}| \quad \text{ and }\ \quad |\{f\geq 1-\theta\} \cap Q_{r_0}| \geq \delta_2 |Q_{r_0}|. \end{align} Define $g:=f-(t+25r_0^2) \|S\|_{L^\infty(Q_1)}$. Then its positive part $g_+$ is a sub-solution to~\eqref{e:main}--\eqref{e:hyp-coef} in $Q_{5r_0}$ with zero source term and with $g_+ \in [0,1]$ since $f\le 1$ in $Q_{\sfrac{1}{2}}$. We set\vspace*{-3pt} \[ r_0 = \begin{dcases} \Bigl(\dfrac{\delta_1}{400 (1+\|S\|_{L^\infty(Q_1)})}\Bigr)^{\sfrac12} \leq \dfrac{1}{20}&\text{if $S$ non-zero},\\ \dfrac{1}{20}&\text{if $S=0$}, \end{dcases} \] and we apply~\eqref{HPI inequality} to $g_+$ at scale $r_0$, for some $\var>0$ to be chosen later:\vspace*{-3pt} \begin{equation}\label{Poinc in proof} \begin{aligned} \fint_{Q_{r_0}} \bigl(g_+-\langle g_+\rangle_{Q_{r_0}^{-}}\bigr)_{+} & \lesssim \frac{r_0}{\varepsilon^{d+2}} \fint_{Q_{5r_0}} |\nabla_v g_+| + \var^\sigma \biggl( \fint_{Q_{5r_0}} g_+^2 \biggr)^{\sfrac12}\\ & \lesssim \frac{1}{r_0^{4d+1} \varepsilon^{d+2}} \int_{Q_{5r_0}} |\nabla_v g_+| + \var^\sigma, \end{aligned} \end{equation} where we have used the bound $g_+ \in [0,1]$ to control the $L^2$ norm. Then~\eqref{hyp bis IVL} implies\vspace*{-3pt} \begin{equation} \begin{aligned} \label{estim averag} \langle g_+\rangle_{Q_{r_0}^{-}} &= \fint_{(s,y,w) \in Q_{r_0}^-} \big[f(s,y,w) - (s+25r_0^2) \| S \|_{L^\infty(Q_1)}\big]_+\\ &\leq \frac{\left|{\{f>0\}\cap Q_{r_0}^-}\right|}{|Q^{-}_{r_0}|} \leq 1-\delta_1 \end{aligned} \end{equation} and\vspace*{-3pt} \begin{equation}\label{lower bd Poinc} \begin{aligned} \fint_{Q_{r_0}} \bigl(g_+&-\langle g_+ \rangle_{Q_{r_0}^{-}}\bigr)_{+} \\ &\ge \frac{1}{|Q_{r_0}|} \int_{(t,x,v) \in Q_{r_0}} \big[ f(t,x,v) - (t+25r_0^2) \|S\|_{L^\infty(Q_1)} -(1-\delta_1)\big]_{+}\\ &\ge \frac{1}{|Q_{r_0}|} \int_{(t,x,v) \in Q_{r_0}} \big[ f(t,x,v) - 25r_0^2 \|S\|_{L^\infty(Q_1)} -(1-\delta_1)\big]_{+}\\ &\geq \frac{1}{|Q_{r_0}|} \int_{\{f\geq 1-\theta\}\cap Q_{r_0}} \Bigl(\frac{\delta_1}{2}-\theta\Bigr)_{+} \ge \delta_2 \Bigl(\frac{\delta_1}{2}-\theta\Bigr). \end{aligned} \end{equation} We then estimate from above the right hand side of the Poincaré inequality \eqref{Poinc in proof}:\vspace*{-3pt} \begin{align*} \int_{Q_{5r_0}} |\nabla_v g_+| &\le \int_{Q_{5r_0}} |\nabla_v f_+| \\ &\le \int_{\{ f=0 \} \cap Q_{5r_0}} \hspace*{-2mm}\cdots\ +\int_{\{ 0\!0$} such that any sub-solution $f$ to~\eqref{e:main}--\eqref{e:hyp-coef} in $Q_1$ with $S$ such that $\| S\|_{L^{\infty}(Q_1)}\le \mu$ and so that $f\leq 1$ in $Q_{\sfrac{1}{2}}$ and \begin{align} \label{eq:measure-hyp-pointwise} \left| \{ f \leq 0 \} \cap Q_{r_0}^- \right| \ge \delta \left| Q_{r_0}^- \right| \end{align} satisfies $f \le 1-\mu$ in $Q_{\sfrac{r_0}{2}}$, with $Q_{r_0}^{-} := Q_{r_0}(-2r_0^2,0,0) =(-3r_0^2,-2r_0^2] \times B_{r_0^3} \times B_{r_0}$ (see Figure~\ref{fig:LVI}). \end{lemma} \begin{proof} In view of Proposition~\ref{prop:1st lemma} and the scaling invariance, there is $\delta'>0$ depending only on $\lambda$ and $\Lambda$ such that for any $r>0$, any sub-solution $f$ on $Q_{2r}$ so that $\int_{Q_{r}} f_{+}^2 \leq \delta' |Q_{r}|$ satisfies $f\leq \sfrac{1}{2}$ in $Q_{\sfrac{r}{2}}$ (imposing $C\mu\le \sfrac{1}{4}$ with $C$ the universal constant in the estimate of Proposition~\ref{prop:1st lemma} used here). Define then $\nu, \theta>0$ as in~\eqref{eq:choice-theta}--\eqref{eq:choice-nu} with $\delta_1=\delta$ and $\delta_2=\delta'$ and a source term bounded in $L^\infty$ by $1$. Define $f_k := \theta^{-k}[f-(1-\theta^k)]$ for $k \ge 0$. The functions $f_{k}$ are sub-solutions to \eqref{e:main}--\eqref{e:hyp-coef} for all $k \ge 0$ with a source term of $L^{\infty}$ norm less than $1$ as long as $k\leq 1+\sfrac{1}{\nu}$ (assuming $\| S\|_{L^{\infty}(Q_1)}\le \mu$ so that $\| S\|_{L^{\infty}(Q_1)} \le \theta^{1+\sfrac{1}{\nu}}$). The sets $\{ 0 \delta'|Q_{r_0}|$ for any~$k$ such that $0\leq k\leq k_0$. Then for $k$ such that $0\leq k\leq k_0-1$ \begin{align*} & \left|\left\{ f_{k}\geq 1-\theta \right\} \cap Q_{r_0}\right| = \left|\{f_{k+1}\geq 0\} \cap Q_{r_0} \right| \geq \int_{Q_{r_0} } (f_{k+1})_{+}^2 > \delta' |Q_{r_0}|, \\ & \left|\left\{f_{k}\leq 0\right\} \cap Q_{r_0}^{-}\right| \geq \left|\left\{f\leq 0 \right\}\cap Q_{r_0}^{-}\right| \geq \delta |Q_{r_0}^-|. \end{align*} Theorem \ref{t:IVL} for sub-solutions with source term of norm $L^{\infty}$ less than $1$ then implies, choosing $r_0 = (\sfrac{\delta}{800})^{\sfrac12}$, \[ \left| \left\{ 0 M \} \cap Q_{r}(z) \right|}{\left| Q_{r}(z) \right|} > \delta \implies \inf_{Q_{\sfrac{r}{2}}^+(z)} h \ge 1, \end{equation} where $Q_{\sfrac{r}{2}}^{+}(z) = Q_{\sfrac{r}{2}}(z +(2r^2,2r^2v,0))$, for $z=(t,x,v)$, is obtained by inverting the operation $Q_{\sfrac{r}{2}}(z) \to Q_{r}^-(z)$ in Lemma~\ref{l:increase} (noting that $Q_r^-(z)=Q_r(z-\nobreak(2r^2,2r^2v,0))$). It implies (inverting the relation $\delta \to M$ and using the layer-cake representation) that if $\inf_{Q_{\sfrac{r_0}{2}}} h <1$, \begin{multline} \label{eq:initial} \forall \, M \ge 1, \quad \frac{\left| \{ h \ge M \} \cap Q_{r_0}^- \right|}{\left|Q_{r_0}^- \right|} \lesssim \delta(M) =\Bigl( \frac{1}{ \ln (1+M)} \Bigr)^{\spfrac{1}{10d+17}} \\ \implies \int_{Q_{r_0}^- } \left[ \ln \left( 1+ h \right) \right]^{\spfrac{1}{10d+18}} \lesssim 1. \end{multline} This ``point-to-measure'' estimate controls the decay of the upper level set in the manner of a \emph{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~\eqref{eq:scaling-invariant} and performing an inductive argument inspired from the elliptic theory~\cite{MR3565366}. Note that the logarithmic integrability in~\eqref{eq:initial} is reminiscent of Moser's approach. We improve inductively the control of upper level sets in the following decreasing sequence of cylinders \begin{equation*} \cQ^k:= Q_{\Psfrac{r_0}{2}+\alpha_k} \left(-\frac{5}{2}\, r_0^2 + \frac12 \left( \frac{r_0}{2}+\alpha_k \right)^2,0,0 \right)\subset Q_{r_0}^{-} \quad \text{ with } \quad \alpha_k:=\frac{r_0}{2\times 7^{k-1}}. \end{equation*} These cylinders satisfy $\tilde Q^-_{\sfrac{r_0}{2}} \subset \cQ^k \subset \bar \cQ^k \subset \mathring{\cQ}^{k-1} \subset Q^- _{r_0}$ for all $k \ge 1$. We now claim that for $\delta_0>0$ small enough (to be chosen later), for any non-negative super-solution~$h$ with $\inf_{Q_{\sfrac{r_0}{2}}} h <1$ we have \begin{align} \label{eq:claim} \forall \, k \ge 1, \quad \frac{\left| \{ h \ge M^k \} \cap \cQ^k \right|}{\left| \cQ^k \right|} \le \frac{\delta_0}{210^{(4d+2)k}}, \end{align} where $M \sim \delta^{-2 (1+\delta^{-10d-16})}$ with $\delta:=\sfrac{\delta_0}{210^{4d+2}}$ as in~\eqref{eq:scaling-invariant}. Admitting first~\eqref{eq:claim} we deduce by layer-cake representation that there is an explicit $\zeta \gtrsim \delta_0^{10d+17}>0$ such that $\int_{\tilde Q_{\sfrac{r_0}{2}}^{-}} h^\zeta \dd z \lesssim 1$, which implies by linearity \[ \biggl( \int_{\tilde Q_{\sfrac{r_0}{2}}^{-}} h(z)^\zeta \dd z \biggr)^{\sfrac{1}{\zeta}} \lesssim \inf_{Q_{\sfrac{r_0}{2}}} h. \] This implies the weak Harnack inequality~\eqref{eq:w-Harnack-stat} on any $f$ non-negative super-solution to~\eqref{e:main}--\eqref{e:hyp-coef} by applying the previous estimate to $h := f + (1+t) \| S\|_{L^\infty(Q_1)}$. To~deduce the Harnack inequality~\eqref{eq:s-Harnack-stat} we consider $f$ a non-negative solution to~\eqref{e:main}--\eqref{e:hyp-coef} and combine the previous control with Proposition~\ref{prop:1st lemma} to get \begin{align*} \sup_{\tilde Q_{\sfrac{r_0}{4}}^{-}} f &\lesssim \biggl( \int_{\tilde Q_{\sfrac{r_0}{2}}^{-}} f(z)^\zeta \dd z \biggr)^{\sfrac{1}{\zeta}} + \| S\|_{L^\infty(Q_1)}\\ &\lesssim \inf_{Q_{\sfrac{r_0}{2}}} f + \| S\|_{L^\infty(Q_1)} \lesssim \inf_{Q_{\sfrac{r_0}{4}}} f + \| S\|_{L^\infty(Q_1)}. \end{align*} Let us now prove the claim~\eqref{eq:claim} to conclude the proof. The initialization $k=1$ is proved in~\eqref{eq:initial}. Then define $A_{k+1} := \{ h > M^{k+1} \} \cap \cQ^{k+1}$ and denote the following translated centered cylinders \[ \mathfrak C_{r}[z] :=z\circ Q_{2r}((2r^2,0,0))= z \circ \left(-2r^2,2r^2\right]\times B_{(2r)^3} \times B_{2r}. \] Let us construct $z_\ell=(t_\ell,x_\ell,v_\ell)\in \cQ^{k+1}$ and $r_\ell >0$, $\ell \ge 1$, so that: \begin{enumerate} \item\label{enum:1} $\forall \, \ell \ge 1$, $r_\ell \in (0,\sfrac{\alpha_{k+1}}{15})$, \item\label{enum:2} $\forall \, \ell \ge 1$, $|A_{k+1} \cap \mathfrak C_{15r_\ell}[z_\ell]| \le \delta_0 |\mathfrak C_{15 r_\ell}[z_\ell]|$, \item\label{enum:3} $\forall \, \ell \ge 1$, $|A_{k+1} \cap \mathfrak C_{r_\ell}[z_\ell]| > \delta_0 |\mathfrak C_{r_\ell}[z_\ell]|$, \item\label{enum:4} the cylinders $\mathfrak C_{3r_\ell}[z_\ell]$, $\ell \ge 1$, are disjoint, \item\label{enum:5} $A_{k+1}$ is covered by the family $\mathfrak C_{15r_\ell}[z_\ell]$, $\ell \ge 1$. \end{enumerate} Note that inverting the operation $Q_{\sfrac{r}{2}}(z) \to Q_{r}^-(z)$ in Lemma~\ref{l:increase} yields, when starting from $\mathfrak C_{r_\ell}[z_\ell]$, the cylinder $\mathfrak C_{r_\ell}[z_\ell]^+ :=z_\ell\circ Q_{r_\ell}((10r_\ell^2,0,0))= z_\ell \circ \left(9r_\ell^2,10r_\ell^2\right]\times B_{r_\ell^3} \times B_{r_\ell}$. Note also that $\mathfrak C_{r_\ell}[z_\ell]^+ \subset \mathfrak C_{3 r_\ell}[z_\ell]$ and that property \eqref{enum:1} combined with $z_\ell \in \cQ^{k+1}$ imply $\mathfrak C_{15r_\ell}[z_\ell] \subset \cQ^k$. Let us prove that the family $\mathcal F$ of cylinders $\mathfrak C_{r}[z]$ with $z \in \cQ^{k+1}$, $r \in (0,\sfrac{\alpha_{k+1}}{15})$ and so that $|A_{k+1} \cap \mathfrak C_{15r}[z]| \le \delta_0 |\mathfrak C_{15 r}[z]|$ and $|A_{k+1} \cap \mathfrak C_{r}[z]| > \delta_0 |\mathfrak C_{r}[z]|$ cover $A_{k+1}$. We have, using~\eqref{eq:claim} at the previous step $k$, \begin{multline} \label{eq:constraint-varphi} \forall \, r \in \left(\frac{\alpha_{k+1}}{15},\alpha_{k+1}\right), \quad |A_{k+1} \cap \mathfrak C_{r}[z]| \le |A_{k} \cap \mathfrak C_{r}[z]| \le |A_{k} \cap \cQ^k| \\ \le \frac{\delta_0}{210^{(4d+2)k}} |\cQ^k| \le \delta_0 |\mathfrak C_{r}[z]|. \end{multline} If $z \in A_{k+1}$ is not covered by $\mathcal F$ it means that the continuous positive function $\varphi(r) = \sfrac{|A_{k+1} \cap \mathfrak C_{r}[z]|}{|\mathfrak C_{r}[z]|}$ on $(0,+\infty)$ satisfies $\varphi(r) \le \delta_0$ or $\varphi(15r) > \delta_0$ for all $r \in (0,\sfrac{\alpha_{k+1}}{15})$. The constraint~\eqref{eq:constraint-varphi} and the continuity impose $\varphi(r) \le \delta_0$ for all $r \in\nobreak (0,\sfrac{\alpha_{k+1}}{15})$. Taking $r \to 0$, a straightforward variation of the Lebesgue differentiation theorem then implies $z\notin A_{k+1}$ which contradicts the assumption. Hence $A_{k+1}$ is covered by the family $\mathcal F$. It implies in particular that $A_{k+1}$ is covered by the family $\mathcal F'$ of cylinders $\mathfrak C_{3r}[z]$ with $z \in \cQ^{k+1}$, $r \in (0,\sfrac{\alpha_{k+1}}{15})$ and such that $|A_{k+1} \cap \mathfrak C_{15r}[z]| \le \delta_0 |\mathfrak C_{15 r}[z]|$ and $|A_{k+1} \cap \mathfrak C_{r}[z]| > \delta_0 |\mathfrak C_{r}[z]|$. The Vitali covering lemma then gives the existence of a countable sub-family, denoted $(\mathfrak C_{r_\ell}[z_\ell])_{\ell \ge 1}$, such that the $(\mathfrak C_{15 r_\ell}[z_\ell])_{\ell \ge 1}$ cover $A_{k+1}$ and the $(\mathfrak C_{3 r_\ell}[z_\ell])_{\ell \ge 1}$ are disjoint. The Vitali lemma applies thanks to the following property: \begin{equation*} \Big[ \mathfrak C_{r_1}[z_1] \cap \mathfrak C_{r_2}[z_2] \neq \varnothing \ \mbox{ and } \ r_1\le 2r_2 \Big] \implies \mathfrak C_{r_1}[z_1] \subset \mathfrak C_{5r_2}[z_2]. \end{equation*} Take $z_0=(t_0,x_0,v_0)$ in the intersection and $z=(t,x,v)\in \mathfrak C_{r_1}[z_1]$. Inequalities $|t-t_2|\leq 18 r_2^2$ and $|v-v_2|\leq 10r_2$ come naturally and $|x-[x_2+2 r_2^2v_2 +(t-t_2)v_2]| \le 200r_2^3$ follows from \begin{align*} \big| x&- \big[ x_2+ 2r_2^2v_2 + (t-t_2)v_2 \big] \big| \\ &\le \big| x- \big[ x_1+ 2r_1^2v_1 + (t-t_1)v_1 \big] \big|\\ &\hspace*{4cm} + \big| \big[ x_2+ 2r_2^2v_2 + (t-t_2)v_2 \big] - \big[ x_1+ 2r_1^2v_1 + (t-t_1)v_1 \big] \big| \\ & \le r_1^3 + \big| \big[ x_2+ 2r_2^2v_2 + (t_0-t_2)v_2 \big] - \big[ x_1+ 2r_1^2v_1 + (t_0-t_1)v_1 \big] \big|\\ &\hspace*{4cm} + \big| (t-t_0)(v_2-v_1) \big| \\ & \le 128 r_2^3 + \big| x_0- \big[ x_1+ 2r_1^2v_1 + (t_0-t_1)v_1 \big] \big| + \big| x_0- \big[ x_2+ 2r_2^2v_2 + (t_0-t_2)v_2 \big] \big|\\ &\le 200 r_2^3. \end{align*} This finishes constructing the covering with the properties (1)-(2)-(3)-(4)-(5) above. Then Lemma~\ref{l:increase} applied to each $\mathfrak C_{r_\ell}[z_\ell]$ implies $\mathfrak C_{r_\ell}[z_\ell]^+ \subset A_k$, and the $\mathfrak C_{r_\ell}[z_\ell]^+ \subset \mathfrak C_{3 r_\ell}[z_\ell]$ are disjoint. We deduce \begin{align*} |A_{k+1}| & \le \sum_{\ell \ge 1} |A_{k+1} \cap \mathfrak C_{15 r_\ell}[z_\ell]| \le \delta_0 \sum_{\ell \ge 1} |\mathfrak C_{15 r_\ell}[z_\ell]| \\ & \le 15^{4d+2} \delta_0 \sum_{\ell \ge 1} |\mathfrak C_{r_\ell}[z_\ell]| \le 30^{4d+2} \delta_0 \sum_{\ell \ge 1} |\mathfrak C_{r_\ell}[z_\ell]^+| \\ & \le 30^{4d+2} \delta_0 |A_k| \le \frac{30^{4d+2}\delta_0^2}{210^{(4d+2)k}} \le \frac{\delta_0}{210^{(4d+2)(k+1)}} \left| \cQ^{k+1} \right| \end{align*} for $\delta_0$ small enough which proves the induction claim~\eqref{eq:claim} and concludes the proof. \subsection{The Hölder continuity} \label{ss:Holder} De~Giorgi's argument to Hölder continuity uses the measure-to-pointwise Lemma~\ref{l:increase}. We briefly sketch it in order to track the constant. Hölder regularity could also be deduced from the Harnack inequality in Theorem~\ref{t:harnack}. Given $f$ solution to \eqref{e:main}--\eqref{e:hyp-coef} on $Q_2$ and $r_0 =\sfrac{1}{40}$ \begin{equation} \label{eq:reduc-osc} \osc_{Q_{r_0}}f \leq \left( 1- \frac{\mu}{2} \right) \max\bigl(\osc_{Q_{1}}f,e^{2(1+2^{10d+16})} \|S\|_{L^{\infty}(Q_2)}\bigr) \end{equation} follows from~Lemma \ref{l:increase} rescaled to $Q_2$ with $\delta=\sfrac12$ and applied to whichever of $F$ or $-F$ satisfies~\eqref{eq:measure-hyp-pointwise}, where \[\textstyle F : =2 \bigl[\max\bigl(\osc_{Q_{1}} f, e^{2(1+2^{10d+16})}\|S\|_{L^{\infty}(Q_2)} \bigr) \bigr]^{-1} \bigl[f-\frac12 (\sup_{Q_{1}} f+\inf_{Q_{1}} f)\bigr]. \] By iteration we deduce \begin{multline} \label{eq:iter-osc} \forall z_0 \in Q_{1}, \forall \, r\in(0,r_0),\\ \osc_{Q_{r}(z_0)}f \leq r^{\alpha} e^{2(1+2^{10d+16})} \left(1+\|S\|_{L^{\infty}(Q_2)}\right) \max\bigl(e^{2(1+2^{10d+16})} \|S\|_{L^{\infty}(Q_2)},\osc_{Q_1}f \bigr). \end{multline} Indeed, the following sequence of solution of \eqref{e:main}--\eqref{e:hyp-coef} in $Q_1$ \begin{align*} f_{n}(\tau,y,w)= 2 \frac{\left( 1 - \sfrac{\mu}{2} \right)^{1-n} f\left(t_0+ r_0^{2n} \tau, x_0 -r_0^{2n} \tau v_0 + r_0^{3n} y,v_0+ r_0 ^n w \right)}{\max \bigl(\osc_{Q_{1}}f, e^{2(1+2^{10d+16})} \|S\|_{L^{\infty}(Q_2)} \bigr)} \end{align*} satisfies $\osc_{Q_{1}}f_{n} \leq 2 e^{2(1+2^{10d+16})}(1+\|S\|_{L^{\infty}(Q_2)})$ by induction on $n \!\ge\! 1$ (the case $n\!=\!1$ is true by definition of $f_n$ and it propagates thanks to~\eqref{eq:reduc-osc}). If one defines $\alpha\in (0,1)$ such that $1-\sfrac{\mu}{2}=r_0^{\alpha}$ (assuming that $\mu$ is small enough), the previous induction implies~\eqref{eq:iter-osc} by standard arguments. To deduce the Hölder estimate between $z,z'\in\nobreak Q_1$, use intermediate points in $[z,z']$ at distance less than $r_0$ and the estimate~\eqref{eq:iter-osc}. \vspace*{\baselineskip}\enlargethispage{-\baselineskip} \backmatter \bibliographystyle{jepalpha+eid} \bibliography{guerand-mouhot} \end{document}