# A note on the weak regularity theory for degenerate Kolmogorov equations

Francesca Anceschi  Dipartimento di Matematica e Applicazioni ”Renato
Caccioppoli” - Università degli Studi di Napoli ”Federico II”: Via Cintia,
Monte S. Angelo I-80126 Napoli, Italy<EMAIL_ADDRESS>and
Annalaura Rebucci  Dipartimento di Scienze Matematiche, Fisiche e Informatiche
- Università degli Studi di Parma: Parco Area delle Scienze, 7/A 43124 Parma,
Italy<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

The aim of this work is to prove a Harnack inequality and the Hölder
continuity for weak solutions to the Kolmogorov equation $\mathscr{L}u=f$ with
measurable coefficients, integrable lower order terms and nonzero source term.
We introduce a function space $\mathcal{W}$, suitable for the study of weak
solutions to $\mathscr{L}u=f$, that allows us to prove a weak Poincaré
inequality. More precisely, our goal is to prove a weak Harnack inequality for
non-negative super-solutions by considering their Log-transform and following
S. N. Krŭzkov (1963). Then this functional inequality is combined with a
classical covering argument (Ink-Spots Theorem) that we extend for the first
time to the case of ultraparabolic equations.

Key words: Kolmogorov equation, weak regularity theory, Moser iterative
method, weak Poincaré inequality, Harnack inequality, Hölder regularity,
ultraparabolic, ink-spots theorem

AMS subject classifications: 35K70, 35Q84, 35H20, 35B65, 35B09, 35B45

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Local boundedness for weak solutions to $\mathscr{L}u=f$
  4. 4 Weak Poincaré inequality
  5. 5 Main results
  6. A The Ink-Spots Theorem
  7. B Stacked cylinders

##  1\. Introduction

The aim of this work is to study the De Giorgi-Nash-Moser regularity theory
for weak solutions to the second order partial differential equation of
Kolmogorov type of the form

$\begin{split}\mathscr{L}u(x,t):&=\sum_{i,j=1}^{m_{0}}\partial_{x_{i}}\left(a_{ij}(x,t)\partial_{x_{j}}u(x,t)\right)+\sum_{i,j=1}^{N}b_{ij}x_{j}\partial_{x_{i}}u(x,t)-\partial_{t}u(x,t)\\\
&+\sum_{i=1}^{m_{0}}b_{i}(x,t)\partial_{i}u(x,t)+c(x,t)u(x,t)=f,\end{split}$
(1.1)

where $z=(x,t)=(x_{1},\ldots,x_{N},t)\in\mathbb{R}^{N+1}$ and $1\leq m_{0}\leq
N$. Moreover, the matrices $A_{0}=(a_{ij}(x,t))_{i,j=1,\ldots,m_{0}}$ and
$B=(b_{ij})_{i,j=1,\ldots,N}$ satisfy the following structural assumptions.

  * (H1)

The matrix $A_{0}$ is symmetric with real measurable entries. Moreover,
$a_{ij}(x,t)=a_{ji}(x,t)$, for every $i,j=1,\ldots,m_{0}$, and there exist two
positive constants $\lambda$ and $\Lambda$ such that

$\displaystyle\lambda|\xi|^{2}\leq\sum_{i,j=1}^{m_{0}}a_{ij}(x,t)\xi_{i}\xi_{j}\leq\Lambda|\xi|^{2}$
(1.2)

for every $(x,t)\in\mathbb{R}^{N+1}$ and $\xi\in\mathbb{R}^{m_{0}}$. The
matrix B has constant entries.

Note that we allow the operator $\mathscr{L}$ to be strongly degenerate
whenever $m_{0}<N$. However, the first order part of $\mathscr{L}$ may induce
a strong regularizing property. Indeed, it is known that, under suitable
assumptions on the matrix $B$, the operator $\mathscr{L}$ is hypoelliptic,
namely that every distributional solution $u$ to $\mathscr{L}u=f$ defined in
some open set $\Omega\subset\mathbb{R}^{N+1}$ belongs to $C^{\infty}(\Omega)$
and it is a classical solution to $\mathscr{L}u=f$, whenever $f\in
C^{\infty}(\Omega)$. In the sequel, we will therefore rely on the following
assumption.

  * (H2)

The principal part operator $\mathscr{K}$ of $\mathscr{L}$ is hypoelliptic and
homogeneous of degree $2$ with respect to the family of dilations
$\left({\delta}_{r}\right)_{r>0}$ introduced in (2.9), where $\mathscr{K}$ is
defined as

$\displaystyle\mathscr{K}u(x,t):=\sum_{i=1}^{m_{0}}\partial^{2}_{x_{i}}u(x,t)+\sum_{i,j=1}^{N}b_{ij}x_{j}\partial_{x_{i}}u(x,t)-\partial_{t}u(x,t),\qquad(x,t)\in\mathbb{R}^{N+1}.$
(1.3)

It is clear that if $\mathscr{L}$ is uniformly parabolic (i.e. $m_{0}=N$ and
$B\equiv\mathbb{O}$), then assumption (H2) is satisfied. In fact, in this case
the principal part operator $\mathscr{K}$ is simply the heat operator, which
is known to be hypoelliptic. For further information on the hypoellipticity of
$\mathscr{K}$ and on the equivalent structural condition for the matrix B we
refer to Section 2.

In the sequel we will also make use of the following notation in order to
introduce a compact formulation for the operator $\mathscr{L}$. More
precisely, here and in the sequel

$\displaystyle
D=(\partial_{x_{1}},\ldots,\partial_{x_{N}}),\quad\langle\cdot,\cdot\rangle,\quad{\text{\rm
div}}$

respectively denote the gradient, the inner product and the divergence in
$\mathbb{R}^{N}$. In addition,

$D_{m_{0}}=(\partial_{x_{1}},\ldots,\partial_{x_{m_{0}}}),\quad{\text{\rm
div}}_{m_{0}}$

denote the partial gradient and the partial divergence in the first $m_{0}$
components, respectively. Moreover, we introduce the matrix

$\displaystyle A(x,t)=\left(a_{ij}(x,t)\right)_{1\leq i,j\leq N},$

where $a_{ij}$, for every $i,j=1,\ldots,m_{0}$, are the coefficients appearing
in (1.1), while $a_{ij}\equiv 0$ whenever $i>m_{0}$ or $j>m_{0}$, and we let

$\displaystyle
b(x,t):=\left(b_{1}(x,t),\ldots,b_{m_{0}}(x,t),0,\ldots,0\right),\qquad
Y:=\sum_{i,j=1}^{N}b_{ij}x_{j}\partial_{x_{i}}u(x,t)-\partial_{t}u(x,t).$
(1.4)

Now, we are in a position to rewrite the operator $\mathscr{L}$ in the
following compact form

$\displaystyle\mathscr{L}u={\text{\rm div}}(ADu)+Yu+\langle b,Du\rangle+cu$

and to state our last assumption on the integrability of $b$, $c$ and of the
source term $f$.

  * (H3)

$c,f\in L^{q}({\Omega})$ and $b\in\left(L^{q}({\Omega})\right)^{m_{0}}$ for
some $q>\frac{3}{4}\left(Q+2\right)$. Moreover, we assume

${\text{\rm div}}b\geq 0\qquad\text{in}\,\,{\Omega}.$

The physical interpretation of the sign of the divergence of $b$ can be
understood by considering the Vlasov-Poisson-Fokker-Planck equation [36], for
which the lower order term $b$ represents the electrostatic or gravitational
forces. The equations with the term $b$ satisfying the structural assumption
${\text{\rm div}}b\geq 0$ arise also in some other applications, like the ones
contained in [34, 31]. Moreover, the sign assumption on the divergence of $b$
is also quite relevant in the case of parabolic equations, since it has
several applications to, for instance, incompressible flows and
magnetostrophic turbulence models for the Earth’s fluid core, e.g. [27]. In
particular, it is nowadays known that the sign (or either the divergence free,
i.e. ${\text{\rm div}}b=0$) assumption can be considered to relax the
regularity assumptions on $b$ under which one can prove the Harnack inequality
and other results, see [34, 16]. Nevertheless, in our case as in the parabolic
setting presented in [34], there is still the need to require that the
divergence of $b$ exists in the sense of distributions and that $b$ is at
least locally integrable up to a certain power.

###  1.1. Main results

Our aim is to prove the local boundedness and the local Hölder continuity,
alongside with a Harnack inequality, for weak solutions to $\mathscr{L}u=f$
under the assumptions (H1)-(H3). In particular, in order to expose our main
results we first need to introduce some preliminary notation. From now on, we
consider a set ${\Omega}={\Omega}_{m_{0}}\times{\Omega}_{N-m_{0}+1}$ of
$\mathbb{R}^{N+1}$, where ${\Omega}_{m_{0}}$ is a bounded Lipschitz domain of
$\mathbb{R}^{m_{0}}$ and ${\Omega}_{N-m_{0}+1}$ is a bounded Lipschitz domain
of $\mathbb{R}^{N-m_{0}+1}$. This is not restrictive since the cylinders
${\mathcal{Q}}$ introduced in (1.11) that we consider in our local analysis
satisfy the Lipschitz boundary assumption. Then we split the coordinate
$x\in\mathbb{R}^{N}$ as

$x=\big{(}x^{(0)},x^{(1)},\ldots,x^{(\kappa)}\big{)},\qquad
x^{(0)}\\!\in\mathbb{R}^{m_{0}},\quad x^{(j)}\\!\in\mathbb{R}^{m_{j}},\quad
j\in\\{1,\ldots,\kappa\\},$ (1.5)

where we have that in accordance with the scaling of the differential equation
(see (2.9) below) every $m_{j}$ is a positive integer such that

$\sum\limits_{j=1}^{\kappa}m_{j}=N\qquad\text{and}\qquad N\geq m_{0}\geq
m_{1}\geq\ldots\geq m_{\kappa}\geq 1.$

We denote by ${\mathcal{D}}({\Omega})$ the set of $C^{\infty}$ functions
compactly supported in ${\Omega}$ and by ${\mathcal{D}}^{\prime}({\Omega})$
the set of distributions in ${\Omega}$. From now on, $H^{1}_{x^{(0)}}$ denotes
the Sobolev space of functions $u\in L^{2}({\Omega}_{m_{0}})$ with
distribution gradient $D_{m_{0}}u$ lying in
$(L^{2}({\Omega}_{m_{0}}))^{m_{0}}$, i.e.

$H^{1}_{x^{(0)}}:=\left\\{u\in
L^{2}({\Omega}_{m_{0}}):\,D_{m_{0}}u\in(L^{2}({\Omega}_{m_{0}}))^{m_{0}}\right\\},$

and we set

$\|u\|_{H^{1}_{x^{(0)}}}:=\|u\|_{L^{2}({\Omega}_{m_{0}})}+\|D_{m_{0}}u\|_{L^{2}({\Omega}_{m_{0}})}.$

We let $H^{1}_{c,x^{(0)}}$ denote the closure of
$C^{\infty}_{c}({\Omega}_{m_{0}})$ in the norm of $H^{1}_{x^{(0)}}$ and we
recall that $C^{\infty}_{c}(\overline{\Omega}_{m_{0}})$ is dense in
$H^{1}_{x^{(0)}}$ since ${\Omega}_{m_{0}}$ is a bounded Lipschitz domain by
assumption. Moreover, $H^{1}_{c,x^{(0)}}$ is a reflexive Hilbert space and
thus we may consider its dual space

$\left(H^{1}_{c,x^{(0)}}\right)^{*}=H^{-1}_{x^{(0)}}\quad\text{and}\quad\left(H^{-1}_{x^{(0)}}\right)^{*}=H^{1}_{c,x^{(0)}},$

where the notation we consider is the classical one. Hence, from now on we
denote by $H^{-1}_{x^{(0)}}$ the dual of $H^{1}_{c,x^{(0)}}$ acting on
functions in $H^{1}_{c,x^{(0)}}$ through the duality pairing
$\langle\cdot,\cdot\rangle:=\langle\cdot,\cdot\rangle_{H^{1}_{x^{(0)}},H^{1}_{c,x^{(0)}}}$.
In a standard manner, see for instance [4, 26], we let $\mathcal{W}$ denote
the closure of $C^{\infty}(\overline{\Omega})$ in the norm

$\|u\|^{2}_{\mathcal{W}}=\|u\|^{2}_{L^{2}({\Omega}_{N-m_{0}+1};H^{1}_{x^{(0)}})}+\|Yu\|^{2}_{L^{2}({\Omega}_{N-m_{0}+1};H^{-1}_{x^{(0)}})},$
(1.6)

where the previous norm can explicitly computed as follows:

$\displaystyle\|u\|^{2}_{\mathcal{W}}=\left(\int_{{\Omega}_{N-m_{0}+1}}\|u(\cdot,y,t)\|_{H^{1}_{x^{(0)}}}^{2}dy\,dt\right)^{\frac{1}{2}}+\left(\int_{{\Omega}_{N-m_{0}+1}}\|Yu(\cdot,y,t)\|^{2}_{H^{-1}_{x^{(0)}}}dy\,dt\right)^{\frac{1}{2}},$

where $y=(x^{(1)},\ldots,x^{(\kappa)})$. In particular, $\mathcal{W}$ is a
Banach space and we remark that the dual of
$L^{2}({\Omega}_{N-m_{0}+1};H^{1}_{c,x^{(0)}})$ satisfies

$\displaystyle\left(L^{2}({\Omega}_{N-m_{0}+1};H^{1}_{c,x^{(0)}})\right)^{*}=L^{2}({\Omega}_{N-m_{0}+1};H^{-1}_{c,x^{(0)}})\quad\text{and}\quad$
$\displaystyle\qquad\qquad\qquad\qquad\left(L^{2}({\Omega}_{N-m_{0}+1};H^{-1}_{c,x^{(0)}})\right)^{*}=L^{2}({\Omega}_{N-m_{0}+1};H^{1}_{c,x^{(0)}}).$

From now on, we consider the shorthand notation $L^{2}H^{-1}$ to denote
$L^{2}({\Omega}_{N-m_{0}+1};H^{-1}_{c,x^{(0)}})$.

The space of functions $\mathcal{W}$ is the most natural framework for the
study of the weak regularity theory for the operator $\mathscr{L}$. In
particular, it is an extension of the functional setting proposed by Armstrong
and Mourrat in [4] for the study of the kinetic Kolmogorov-Fokker-Planck
equation, where the authors show that it is sufficient to control $u\in L^{2}$
and $Yu\in L^{2}H^{-1}$ in order to derive new Poincaré inequalities, such as
Proposition 4.1, on which we base our analysis later on. Moreover, we refer to
[26, Section 2] for some properties of the space $\mathcal{W}$. Lastly, we
remark that the major issue when dealing with the space $\mathcal{W}$ is that
it requires to handle the duality pairing between $L^{2}H^{1}$ and
$L^{2}H^{-1}$. To this end, we take advantage of the following remark, see
[20, Chapter 4].

######  Remark 1.1.

For every open subset $A\subset\mathbb{R}^{n}$ and for every function $g\in
H^{-1}(A)$ there exist two functions $H_{0}$, $H_{1}\in L^{2}(A)$ such that

$g={\text{\rm
div}}_{m_{0}}H_{1}+H_{0}\qquad\text{and}\qquad\|H_{0}\|_{L^{2}(A)}+\|H_{1}\|_{L^{2}(A)}\leq
2\|g\|_{H^{-1}(A)}.$

Now, we introduce the definition of weak solution we consider in our work.

######  Definition 1.2.

A function $u\in\mathcal{W}$ is a weak solution to (1.1) with source term
$f\in L^{2}({\Omega})$ if for every non-negative test function
${\varphi}\in{\mathcal{D}}({\Omega})$, we have

$\displaystyle\int_{{\Omega}}-\langle
ADu,D{\varphi}\rangle+{\varphi}Yu+\langle
b,Du\rangle{\varphi}+cu{\varphi}=\int_{{\Omega}}f{\varphi}.$ (1.7)

In the sequel, we will also consider weak sub-solutions to (1.1), namely
functions $u\in\mathcal{W}$ that satisfy the following inequality

$\displaystyle\int_{{\Omega}}-\langle
ADu,D{\varphi}\rangle+{\varphi}Yu+\langle
b,Du\rangle{\varphi}+cu{\varphi}\geq\int_{{\Omega}}f{\varphi},$ (1.8)

for every non-negative test function ${\varphi}\in{\mathcal{D}}({\Omega})$. A
function $u$ is a super-solution to (1.1) if $-u$ is a sub-solution.

This framework is quite classical for the study of the weak regularity theory
for the kinetic Kolmogorov-Fokker-Planck equation [4, 11, 13, 14], that can be
recovered from (1.1) by choosing $N=2d$, ${\kappa}=1$, $m_{0}=m_{1}=d$ and
$c\equiv 0$. Still, even if it is the most natural framework for (1.1), to the
best of our knowledge it has never been considered in literature yet. Indeed,
the weak regularity theory for the operator $\mathscr{L}$ has been widely
developed during the last decade starting from the paper [33] by Pascucci and
Polidoro, where the authors worked under the stronger assumption $Yu\in
L^{2}_{{\text{\rm loc}}}({\Omega})$. In the same framework (i.e. $Yu\in
L^{2}({\Omega})$), Wang and Zang [38, 39] lately proved the results for
equation (1.1) with $a=b=f=0$ on the local Hölder continuity for solutions.

The aim of this paper is to prove the local Hölder continuity and a Harnack
inequality for solutions to (1.1) in the sense of Definition 1.2. Our method
is based on the combination of three fundamental ingredients - boundedness of
weak solutions, weak Poincaré inequality and Log-transformation - in the same
spirit of the recent paper [13] for the Fokker-Planck equation. In particular,
we give an answer to Remark $4$, p. 2 of [13] and we are also able to simplify
the proof proposed in [40] to obtain the local Hölder continuity for weak
solutions to (1.1). We classically reduce the local study to the case where
${\mathcal{Q}}^{0}$ is at unit scale and for some reasons we expose below in
Section 5, ${\mathcal{Q}}^{0}$ takes the form $B_{R_{0}}\times
B_{R_{0}}\times\ldots\times B_{R_{0}}\times(-1,0]$ for some large constant
$R_{0}$ only depending on the dimension $Q$ and the ellipticity constants
$\lambda,\Lambda$ in (H1).

As we will see in Section 2, the suitable geometry when dealing with operator
$\mathscr{L}$ is given by an homogeneous Lie group structure defined on
$\mathbb{R}^{N+1}$. Our results naturally reflect this non-Euclidean setting.
Let $"\circ"$ denote the composition law introduced in (2.1) and
$\left(\delta_{r}\right)_{r>0}$ the family of dilations defined in (2.9). We
consider the unit past cylinder

$\displaystyle{\mathcal{Q}}_{1}:=B_{1}\times B_{1}\times\ldots\times
B_{1}\times(-1,0),$ (1.9)

defined through the open balls

$\displaystyle B_{1}=\\{x^{(j)}\\!\in\mathbb{R}^{m_{j}}:|x|\leq 1\\},$ (1.10)

where $j=0,\ldots,\kappa$ and $|\cdot|$ denotes the euclidean norm in
$\mathbb{R}^{m_{j}}$. Now, for every $z_{0}\in\mathbb{R}^{N+1}$ and $r>0$, we
set

$\displaystyle{\mathcal{Q}}_{r}(z_{0}):=z_{0}\circ\left(\delta_{r}\left({\mathcal{Q}}_{1}\right)\right)=\\{z\in\mathbb{R}^{N+1}\,:\,z=z_{0}\circ\delta_{r}(\zeta),\zeta\in{\mathcal{Q}}_{1}\\}.$
(1.11)

We are now in a position to state our main results.

######  Theorem 1.3 (Weak Harnack inequality).

Let ${\mathcal{Q}}^{0}=B_{R_{0}}\times B_{R_{0}}\times\ldots\times
B_{R_{0}}\times(-1,0]$ and let $u$ be a non-negative weak super-solution to
$\mathscr{L}u=f$ in ${\Omega}\supset{\mathcal{Q}}^{0}$ under the assumptions
(H1)-(H3). Then we have

$\displaystyle\left(\int_{{\mathcal{Q}}_{-}}u^{p}\right)^{\frac{1}{p}}\leq
C\left(\inf_{{\mathcal{Q}}_{+}}u+\|f\|_{L^{q}({\mathcal{Q}}^{0})}\right),$
(1.12)

where ${\mathcal{Q}}_{+}=B_{\omega}\times B_{\omega^{3}}\times\ldots\times
B_{\omega^{2\kappa+1}}\times(-\omega^{2},0]$ and
${\mathcal{Q}}_{-}=B_{\omega}\times B_{\omega^{3}}\times\ldots\times
B_{\omega^{2\kappa+1}}\times(-1,-1+\omega^{2}]$. Moreover, the constants $C$,
$p$, $\omega$ and $R_{0}$ only depend on the homogeneous dimension $Q$ defined
in (2.12), $q$ and on the ellipticity constants $\lambda$ and $\Lambda$ in
(1.2).

We remark that as in [33], the radius ${\omega}$ is small enough so that when
“stacking cylinders” over a small initial one contained in
${\mathcal{Q}}^{-}$, the cylinder ${\mathcal{Q}}^{+}$ is captured, see Lemma
B.1. As far as we are concerned with $R_{0}$, it is large enough so that it is
possible to apply the expansion of positivity lemma (see Lemma 5.1) to every
stacked cylinder. By combining the local boundedness of positive weak sub-
solutions proved in Theorem 3.1 and Theorem 1.3 we obtain the following
Harnack inequality, which is an extension of the analogous result contained in
[1, 11].

######  Theorem 1.4 (Harnack inequality).

Let ${\mathcal{Q}}^{0}=B_{R_{0}}\times B_{R_{0}}\times\ldots\times
B_{R_{0}}\times(-1,0]$ and let $u$ be a non-negative weak solution to
$\mathscr{L}u=f$ in ${\Omega}\supset{\mathcal{Q}}^{0}$ under the assumptions
(H1)-(H3). Then we have

$\displaystyle\sup\limits_{{\mathcal{Q}}_{-}}u\,\leq\,C\left(\inf_{{\mathcal{Q}}_{+}}u+\|f\|_{L^{q}({\mathcal{Q}}^{0})}\right),$
(1.13)

where ${\mathcal{Q}}_{+}=B_{\omega}\times B_{\omega^{3}}\times\ldots\times
B_{\omega^{2\kappa+1}}\times(-\omega^{2},0]$ and
${\mathcal{Q}}_{-}=B_{\omega}\times B_{\omega^{3}}\times\ldots\times
B_{\omega^{2\kappa+1}}\times(-1,-1+\omega^{2}]$. Moreover, the constants $C$,
$\omega$ and $R_{0}$ only depend on the homogeneous dimension $Q$ defined in
(2.12), $q$ and on the ellipticity constants $\lambda$ and $\Lambda$ in (1.2).

Since our proof of Theorem 1.3 is constructive, as it is based on the
combination of an expansion of positivity argument with the weak Poincaré
inequality of Proposition 4.1, also the proof of Theorem 1.4 is constructive.
Moreover, the weak Harnack inequality also implies the Hölder regularity of
weak solutions. In order to state this result, we first need to give the
following definition.

######  Definition 1.5.

Let ${\alpha}$ be a positive constant, ${\alpha}\leq 1$, and let ${\Omega}$ be
an open subset of $\mathbb{R}^{N+1}$. We say that a function
$f:{\Omega}\longrightarrow\mathbb{R}$ is Hölder continuous with exponent
${\alpha}$ in ${\Omega}$ with respect to the group
$\mathbb{K}=(\mathbb{R}^{N+1},\circ)$, defined in (2.1), (in short: Hölder
continuous with exponent ${\alpha}$, $f\in C^{\alpha}_{K}({\Omega})$) if there
exists a positive constant $C>0$ such that

$|f(z)-f({\zeta})|\leq C\;d(z,\zeta)^{{\alpha}}\qquad{\rm
for\,every\,}z,{\zeta}\in{\Omega},$

where $d$ is the distance defined in (2.11).

To every bounded function $f\in C^{\alpha}_{K}({\Omega})$ we associate the
semi-norm

$[f]_{C^{{\alpha}}({\Omega})}=\hskip
2.84526pt\sup\limits_{z,{\zeta}\in{\Omega}\atop
z\neq{\zeta}}\frac{|f(z)-f({\zeta})|}{d(z,\zeta)^{{\alpha}}}.$

Moreover, we say a function $f$ is locally Hölder continuous, and we write
$f\in C^{{\alpha}}_{K,{\text{\rm loc}}}({\Omega})$, if $f\in
C^{{\alpha}}_{K}({\Omega}^{\prime})$ for every compact subset
${\Omega}^{\prime}$ of ${\Omega}$.

We are now in a position to state the following result.

######  Theorem 1.6 (Hölder regularity).

There exists ${\alpha}\in(0,1)$ only depending on dimension $Q$, ${\lambda}$,
$\Lambda$ such that all weak solutions $u$ to (1.1) under assumption (H1)\-
(H3) in ${\Omega}\supset{\mathcal{Q}}_{1}$ satisfy

$[u]_{C^{{\alpha}}(Q_{\frac{1}{2}})}\,\leq
C\left(\|u\|_{L^{2}({\mathcal{Q}}_{1})}+\|f\|_{L^{q}({\mathcal{Q}}_{1})}\right),$

where the constant $C$ only depends on the homogeneous dimension $Q$ defined
in (2.12), $q$ and the ellipticity constants ${\lambda}$, $\Lambda$.

######  Remark 1.7.

The estimates presented in Theorem 1.4 and Theorem 1.6 can be scaled and
stated in arbitrary cylinders thanks to the invariance of the operator
$\mathscr{L}$ with respect to the group operations (dilations (2.9) and
translations (2.1)) introduced in Section 2. Moreover, all of our main results
still hold true if we replace assumption (H3) with the following one:

$c,f\in L^{q}({\Omega})$ for some $q>\frac{Q+2}{2}$and
$b\in\left(L^{q}({\Omega})\right)^{m_{0}}$ for some $q>Q+2$ in ${\Omega}$.

This is exactly the integrability requirement assumed by Wang and Zhang in
[40], thus our work is an improvement of this article for the case homogeneous
ultraparabolic operators. Moreover, as we work in the appriopriate functional
setting $\mathcal{W}$, our proofs of the main results are simpler than the
ones proposed in [40].

###  1.2. Comparison with existing results and organization of the paper

The purpose of this paper is to provide a complete characterization of the De
Giorgi-Nash-Moser weak regularity theory for the Kolmogorov equation in
divergence form with measurable coefficients in the more natural space
$\mathcal{W}$ for weak solutions to $\mathscr{L}u=f$. First of all, we improve
the existing result proved by Wang and Zhang in [38, 39, 40] by providing a
simpler and more elegant proof of the weak Harnack inequality and by
explicitly showing the derivation of an invariant Harnack inequality, the
first one available for weak solutions to $\mathscr{L}u=f$. In particular, our
results hold true also under the assumptions of [40] as it is pointed out in
Remark 1.7 and Remark 3.2, and extend the ones obtained in the particular case
of the kinetic Fokker-Planck equation in [13, 11, 26].

Our proof is based on the combination of a weak Poincaré inequality for the
space $\mathcal{W}$ with the Ink-spots theorem in the same spirit of [19, 13].
We remark that, up to this day, the Ink-spot theorem has only been available
just for the Fokker-Planck equation case in $\mathbb{R}^{2n+1}$, and thus also
the technical result that we prove in Appendix A is a complete novelty in this
more general framework. Moreover, we manage to lower the integrability
assumption on the lower order coefficients and on the source term $f$ and to
extend the Moser’s iterative method proposed in [33] to this more general
case.

The structure of the paper is the following. In Section 2 we recall some known
facts about operators $\mathscr{L}$ and we state some preliminary results. The
proofs of some intermediate theorems (a Sobolev type and a Caccioppoli type
inequality), together with the Moser’s iterative method, are presented in
Section 3. Section 4 is devoted to the proof of a weak Poincarè inequality. In
Section 5 we derive the weak Harnack inequality by combining the expansion of
positivity and a covering argument known as Ink-spots theorem, whose proof is
contained in Appendix A. Finally, in Appendix B we state a technical lemma
regarding stacked cylinders.

### Acknowledgements

The authors are grateful to Prof. Sergio Polidoro for suggesting the question
and the fruitful discussions. The first author is funded by the research grant
PRIN2017 2017AYM8XW “Nonlinear Differential Problems via Variational,
Topological and Set-valued Methods”.

##  2\. Preliminaries

In this Section we recall notation and results we need in order to deal with
the non-Euclidean geometry underlying the operators $\mathscr{L}$ and
$\mathscr{K}$. We refer to the articles [33, 24] and to the survey [2] for a
comprehensive treatment of this subject.

As first observed by Lanconelli and Polidoro in [24], the operator
$\mathscr{K}$ is invariant with respect to left translation in the group
$\mathbb{K}=(\mathbb{R}^{N+1},\circ)$, where the group law is defined by

$(x,t)\circ(\xi,\tau)=(\xi+E(\tau)x,t+\tau),\hskip
14.22636pt(x,t),(\xi,\tau)\in\mathbb{R}^{N+1},$ (2.1)

and

$E(s)=\exp(-sB),\qquad s\in\mathbb{R}.$ (2.2)

Then $\mathbb{K}$ is a non-commutative group with zero element $(0,0)$ and
inverse

$(x,t)^{-1}=(-E(-t)x,-t).$

For a given $\zeta\in\mathbb{R}^{N+1}$ we denote by $\ell_{{\zeta}}$ the left
traslation on $\mathbb{K}=(\mathbb{R}^{N+1},\circ)$ defined as follows

$\ell_{{\zeta}}:\mathbb{R}^{N+1}\rightarrow\mathbb{R}^{N+1},\quad\ell_{{\zeta}}(z)={\zeta}\circ
z.$

Then the operator $\mathscr{K}$ is left invariant with respect to the Lie
product $\circ$, that is

$\mathscr{K}\circ\ell_{{\zeta}}=\ell_{{\zeta}}\circ\mathscr{K}\qquad{\rm\textit{or,
equivalently,}}\qquad\mathscr{K}\left(u({\zeta}\circ
z)\right)=\left(\mathscr{K}u\right)\left({\zeta}\circ z\right),$

for every $u$ sufficiently smooth.

We recall that, by [24] (Propositions 2.1 and 2.2), assumption (H2) is
equivalent to assume that, for some basis on $\mathbb{R}^{N}$, the matrix $B$
takes the following form

$B=\begin{pmatrix}\mathbb{O}&\mathbb{O}&\ldots&\mathbb{O}&\mathbb{O}\\\
B_{1}&\mathbb{O}&\ldots&\mathbb{O}&\mathbb{O}\\\
\mathbb{O}&B_{2}&\ldots&\mathbb{O}&\mathbb{O}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\mathbb{O}&\mathbb{O}&\ldots&B_{\kappa}&\mathbb{O}\end{pmatrix}$ (2.3)

where every $B_{k}$ is a $m_{k}\times m_{k-1}$ matrix of rank $m_{j}$,
$j=1,2,\ldots,{\kappa}$ with

$m_{0}\geq m_{1}\geq\ldots\geq m_{\kappa}\geq 1\hskip
14.22636pt\text{and}\hskip 14.22636pt\sum\limits_{j=0}^{\kappa}m_{j}=N.$

In the sequel we will assume that $B$ has the canonical form (2.3). We remark
that assumption (H2) is implied by the condition introduced by Hörmander in
[15]:

${\rm rank\
Lie}\left(\partial_{x_{1}},\dots,\partial_{x_{m_{0}}},Y\right)(x,t)=N+1,\qquad\forall\,(x,t)\in\mathbb{R}^{N+1},$
(2.4)

where ${\rm Lie}\left(\partial_{x_{1}},\dots,\partial_{x_{m_{0}}},Y\right)$
denotes the Lie algebra generated by the first order differential operators
$\left(\partial_{x_{1}},\dots,\partial_{x_{m_{0}}},Y\right)$ computed at
$(x,t)$. Yet another condition equivalent to (H2), (see [24], Proposition
A.1), is that

$\displaystyle C(t)>0,\quad\text{for every $t>0$},$ (2.5)

where

$C(t)=\int_{0}^{t}\hskip 2.84526ptE(s)\,A_{0}\,E^{T}(s)\,ds,$

and $E(\cdot)$ is the matrix defined in (2.2).

We now recall that, under the the hypothesis of hypoellipticity, Hörmander in
[15] constructed the fundamental solution of $\mathscr{K}$ as

$\Gamma(z,\zeta)=\Gamma(\zeta^{-1}\circ z,0),\hskip 11.38109pt\forall
z,\zeta\in\mathbb{R}^{N+1},\hskip 2.84526ptz\neq\zeta,$ (2.6)

where

$\Gamma((x,t),(0,0))=\begin{cases}\frac{(4\pi)^{-\frac{N}{2}}}{\sqrt{\text{det}C(t)}}\exp\left(-\frac{1}{4}\langle
C^{-1}(t)x,x\rangle-t\,{\text{\rm tr}}(B)\right),\hskip
8.53581pt&\text{if}\hskip 2.84526ptt>0,\\\ 0,&\text{if}\hskip
2.84526ptt<0.\end{cases}$ (2.7)

Note that condition (2.5) implies that $\Gamma$ in (2.7) is well-defined.

Let us now consider the second part of assumption (H2). We say that
$\mathscr{K}$ is invariant with respect to $(\delta_{r})_{r>0}$ if

$\mathscr{K}\left(u\circ\delta_{r}\right)=r^{2}\delta_{r}\left(\mathscr{K}u\right),\quad\text{for
every}\quad r>0,$ (2.8)

for every function $u$ sufficiently smooth. It is known (see Proposition 2.2
of [24]) that this dilation invariance property can be read in the expression
of the matrix $B$ in (2.3). More precisely, $\mathscr{K}$ satisfies (2.8) if
and only if the matrix $B$ takes the form (2.3). In this case, we have

${\delta}_{r}=\text{diag}(r\mathbb{I}_{m_{0}},r^{3}\mathbb{I}_{m_{1}},\ldots,r^{2{\kappa}+1}\mathbb{I}_{m_{\kappa}},r^{2}),\qquad\qquad
r>0.$ (2.9)

We next introduce a homogeneous norm of degree $1$ with respect to the
dilations $({\delta}_{r})_{r>0}$ and a corresponding quasi-distance which is
invariant with respect to the group operation (2.1).

######  Definition 2.1.

Let ${\alpha}_{1},\ldots,{\alpha}_{N}$ be positive integers such that

${\text{\rm
diag}}\left(r^{{\alpha}_{1}},\ldots,r^{{\alpha}_{N}},r^{2}\right)={\delta}_{r}.$
(2.10)

If $\|z\|=0$ we set $z=0$ while, if $z\in\mathbb{R}^{N+1}\setminus\\{0\\}$ we
define $\|z\|=r$ where $r$ is the unique positive solution to the equation

$\frac{x_{1}^{2}}{r^{2{\alpha}_{1}}}+\frac{x_{2}^{2}}{r^{2{\alpha}_{2}}}+\ldots+\frac{x_{N}^{2}}{r^{2{\alpha}_{N}}}+\frac{t^{2}}{r^{4}}=1.$

Accordingly, we define the quasi-distance $d$ by

$d(z,w)=\|z^{-1}\circ w\|,\hskip 14.22636ptz,w\in\mathbb{R}^{N+1}.$ (2.11)

######  Remark 2.2.

As $\det E(t)=e^{t\hskip 2.84526pt\text{\rm trace}\,B}=1$, the Lebesgue
measure is invariant with respect to the translation group associated to
$\mathscr{K}$. Moreover, since $\det{\delta}_{r}=r^{Q+2}$, we also have

${\text{\rm meas}}\left({\mathcal{Q}}_{r}(z_{0})\right)=r^{Q+2}{\text{\rm
meas}}\left({\mathcal{Q}}_{1}(z_{0})\right),\qquad\forall\
r>0,z_{0}\in\mathbb{R}^{N+1},$

where

$Q=m_{0}+3m_{1}+\ldots+(2{\kappa}+1)m_{\kappa}.$ (2.12)

The natural number $Q+2$ is called the homogeneous dimension of
$\mathbb{R}^{N+1}$ with respect to $({\delta}_{r})_{r>0}$ and the spatial
homogeneous dimension $Q$ is the hypoelliptic counterpart of the space
dimension $N$ usually considered in the parabolic setting, see [32]. This
denomination is proper since the Jacobian determinant of ${\delta}_{r}$ equals
to $r^{Q+2}$.

######  Remark 2.3.

The semi-norm $\|\cdot\|$ is homogeneous of degree $1$ with respect to
$({\delta}_{r})_{r>0}$, i.e.

$\|{\delta}_{r}(x,t)\|=r\|(x,t)\|\qquad\forall r>0\hskip
5.69054pt\text{and}\hskip 5.69054pt(x,t)\in\mathbb{R}^{N+1}.$

Since in $\mathbb{R}^{N+1}$ all the norms which are $1$-homogeneous with
respect to $({\delta}_{r})_{r>0}$ are equivalent, the norm introduced in
Definition 2.1 is equivalent to the following one

$\|(x,t)\|_{1}=|t|^{\frac{1}{2}}+|x|,\quad|x|=\sum_{j=1}^{N}|x_{j}|^{\frac{1}{\alpha_{j}}}$

where the exponents $\alpha_{j}$, for $j=1,\ldots,N$ were introduced in
(2.10). We prefer the norm of Definition 2.1 to $\parallel\cdot\parallel_{1}$
because its level sets (spheres) are smooth surfaces.

######  Remark 2.4.

We recall that for every cylinder ${\mathcal{Q}}_{r}(z_{0})$ defined in (1.11)
there exists a positive constant $\overline{c}$ [39, equation (21)] such that

$\displaystyle B_{r}(x^{(0)}_{0})$ $\displaystyle\times
B_{r_{1}^{3}}(x^{(1)}_{0})\times\ldots\times
B_{r_{1}^{2{\kappa}+1}}(x^{({\kappa})}_{0})\times(-r_{1}^{2},0]$
$\displaystyle\subset{\mathcal{Q}}_{r}(z_{0})\subset
B_{r_{2}}(x^{(0)}_{0})\times B_{r_{2}^{3}}(x^{(1)}_{0})\times\ldots\times
B_{r_{2}^{2{\kappa}+1}}(x^{({\kappa})}_{0})\times(-r_{2}^{2},0],$

where $r_{1}=r/\overline{c}$ and $r_{2}=\overline{c}r$. From now on, by abuse
of notation we will sometimes consider the newly introduced ball
representation instead of the definition of cylinder in (1.11).

Since $\mathscr{K}$ is dilation invariant with respect to
$({\delta}_{r})_{r>0}$, also its fundamental solution ${\Gamma}$ is a
homogeneous function of degree $-Q$, namely

$\Gamma\left({\delta}_{r}(z),0\right)=r^{-Q}\hskip
2.84526pt\Gamma\left(z,0\right),\hskip 14.22636pt\forall
z\in\mathbb{R}^{N+1}\setminus\\{0\\},\hskip 2.84526ptr>0.$

This property implies a $L^{p}$ estimate for Newtonian potential (c. f. for
instance [9]).

######  Theorem 2.5.

Let ${\alpha}\in]0,Q+2[$ and let $G\in C(\mathbb{R}^{N+1}\setminus\\{0\\})$ be
a ${\delta}_{\lambda}-$homogeneous function of degree ${\alpha}-Q-2$. If $f\in
L^{p}(\mathbb{R}^{N+1})$ for some $p\in]1,+\infty[$, then the function

$G_{f}(z):=\int_{\mathbb{R}^{N+1}}G({\zeta}^{-1}\circ z)f({\zeta})d{\zeta},$

is defined almost everywhere and there exists a constant $c=c(Q,p)$ such that

$\|G_{f}\|_{L^{q}(\mathbb{R}^{N+1}}\leq
c\max_{\|z\|=1}|G(z)|\|f\|_{L^{p}(\mathbb{R}^{N+1})},$

where $q$ is defined by

$\frac{1}{q}=\frac{1}{p}-\frac{{\alpha}}{Q+2}.$

Now, we are able to define the ${\Gamma}-$potential of the function $f\in
L^{1}(\mathbb{R}^{N+1})$ as follows

${\Gamma}(f)(z)=\int_{\mathbb{R}^{N+1}}{\Gamma}(z,{\zeta})f({\zeta})d{\zeta},\qquad
z\in\mathbb{R}^{N+1}.$ (2.13)

We also remark that the potential
${\Gamma}(D_{m_{0}}f):\mathbb{R}^{N+1}\longrightarrow\mathbb{R}^{m_{0}}$ is
well-defined for any $f\in L^{p}(\mathbb{R}^{N+1})$, at least in the
distributional sense, that is

${\Gamma}(D_{m_{0}}f)(z):=-\int_{\mathbb{R}^{N+1}}D^{({\xi})}_{m_{0}}{\Gamma}(z,{\xi})\,f({\xi})\,d{\xi},$

where $D^{({\xi})}_{m_{0}}{\Gamma}(x,t,{\xi},{\tau})$ is the gradient with
respect to ${\xi}_{1},\ldots,{\xi}_{m_{0}}$. Based on Theorem 2.5, we derive
the following explicit potential estimates by substituting ${\alpha}=1$ and
${\alpha}=2$ when considering the $\Gamma$-potential for $f$ and $D_{0}f$,
respectively. For the proof of this corollary we refer to [33, Corollary 2.2].

######  Corollary 2.6.

Let $f\in L^{p}({\mathcal{Q}}_{r})$. There exists a positive constant
$c=c(T,B)$ such that

$\displaystyle\|{\Gamma}(f)\|_{L^{p**}({\mathcal{Q}}_{r})}$ $\displaystyle\leq
c\|f\|_{L^{p}({\mathcal{Q}}_{r})},$ (2.14)
$\displaystyle\|{\Gamma}(D_{m_{0}}f)\|_{L^{p*}({\mathcal{Q}}_{r})}$
$\displaystyle\leq c\|f\|_{L^{p}({\mathcal{Q}}_{r})},$ (2.15)

where $\frac{1}{p*}=\frac{1}{p}-\frac{1}{Q+2}$ and
$\frac{1}{p**}=\frac{1}{p}-\frac{2}{Q+2}$.

Lastly, we show that it is possible to use the fundamental solution ${\Gamma}$
as a test function in the definition of sub and super-solution. The following
result extends [33, Lemma 2.5], [6, Lemma 3] and [3, Lemma 2.6] to the
functional setting $\mathcal{W}$.

######  Lemma 2.7.

Let (H1)-(H2) hold. Let $c\in L^{q}({\Omega})$ and
$b\in\left(L^{q}({\Omega})\right)^{m_{0}}$ for some $q>\frac{Q+2}{2}$ and let
$f\in L^{2}({\Omega})$. Moreover, let us assume that ${\text{\rm div}}b$ in
${\Omega}$. Let $v$ be a non-negative weak sub-solution to $\mathscr{L}v=f$ in
${\Omega}$. For every ${\varphi}\in{\mathcal{D}}({\Omega})$, ${\varphi}\geq
0$, and for almost every $z\in\mathbb{R}^{N+1}$, we have

$\displaystyle\int_{{\Omega}}-\langle
ADv,D({\Gamma}(z,\cdot){\varphi})\rangle$
$\displaystyle+{\Gamma}(z,\cdot){\varphi}Yv+\langle
b,Dv\rangle{\Gamma}(z,\cdot){\varphi}+cv{\Gamma}(z,\cdot){\varphi}-{\Gamma}(z,\cdot){\varphi}f\geq
0.$

An analogous result holds for weak super-solutions to $\mathscr{L}u=f$.

###### Proof.

For every $\varepsilon>0$, we set

$\displaystyle\psi_{\varepsilon}(z,{\zeta})=1-\chi_{\varepsilon,2\varepsilon}\left(\|{\zeta}^{-1}\circ
z\|\right)$ (2.16)

where $\chi_{{\rho},r}\in C^{\infty}([0,+\infty))$ is the cut-off function
defined by

$\chi_{{\rho},r}(s)=\left\\{\begin{array}[]{ll}0,&\textrm{if $s\geq r$},\\\
1,&\textrm{if $0\leq
s\leq{\rho}$},\end{array}\right.\quad|\chi^{\prime}_{{\rho},r}|\leq\frac{2}{r-{\rho}},$
(2.17)

with $\frac{1}{2}\leq{\rho}<r\leq 1$. As $v$ is a weak-subsolution, for every
$\varepsilon>0$ and $z\in\mathbb{R}^{N+1}$, we have

$\displaystyle 0\leq-
I_{1,\varepsilon}(z)+I_{2,\varepsilon}(z)-I_{3,\varepsilon}(z)+I_{4,\varepsilon}(z)+I_{5,\varepsilon}(z)+I_{6,\varepsilon}(z)$

where

$\begin{split}I_{1,\varepsilon}(z)&=\int_{\Omega}[\langle
ADv,D{\Gamma}(z,\cdot)\rangle{\varphi}\psi_{\varepsilon}(z,\cdot)]({\zeta})d{\zeta}\qquad\qquad\qquad
I_{3,\varepsilon}(z)=\int_{\Omega}[\langle
ADv,D\psi_{\varepsilon}(z,\cdot)\rangle{\varphi}{\Gamma}(z,\cdot)]({\zeta})d{\zeta}\\\
I_{2,\varepsilon}(z)&=\int_{\Omega}[{\Gamma}(z,\cdot)\psi_{\varepsilon}(z,\cdot)(-\langle
ADv,D{\varphi})\rangle+{\varphi}Yv)]({\zeta})d{\zeta}\quad
I_{4,\varepsilon}(z)=\int_{\Omega}\langle
b,Dv\rangle{\Gamma}(z,\cdot){\varphi}\psi_{\varepsilon}(z,\cdot)]({\zeta})d{\zeta}\\\
I_{5,\varepsilon}(z)&=\int_{\Omega}[cv{\Gamma}(z,\cdot){\varphi}\psi_{\varepsilon}(z,\cdot)]({\zeta})d{\zeta}\qquad\qquad\quad\qquad\qquad
I_{6,\varepsilon}(z)=-\int_{\Omega}[{\Gamma}(z,\cdot){\varphi}\psi_{\varepsilon}(z,\cdot)f]({\zeta})d{\zeta}\end{split}$

Keeping in mind Corollary 2.6, it is clear that the integrals that define
$I_{i,\varepsilon}(z)$, $i=1,2,3$ are potentials and therefore convergent for
almost every $z\in\mathbb{R}^{N+1}$. Thus, by a similar argument to the one
used in [33] in the proof of Lemma 2.5, we infer that for almost every
$z\in\mathbb{R}^{N+1}$

$\begin{split}\lim_{\varepsilon\to
0^{+}}I_{1,\varepsilon}(z)&=\int_{\Omega}[\langle
ADv,D{\Gamma}(z,\cdot)\rangle{\varphi}]({\zeta})d{\zeta}\qquad\qquad\lim_{\varepsilon\to
0^{+}}I_{3,\varepsilon}(z)=0\\\ \lim_{\varepsilon\to
0^{+}}I_{2,\varepsilon}(z)&=\int_{\Omega}[{\Gamma}(z,\cdot)(-\langle
ADv,D{\varphi})\rangle+{\varphi}Yv)]({\zeta})d{\zeta},\end{split}$

where the passage to the limit for the term $I_{2,{\varepsilon}}$ by a density
argument. We now take care of the term $I_{4,\varepsilon}$. Integrating by
parts and taking advantage of the assumption ${\text{\rm div}}b\geq 0$, we
obtain

$\displaystyle I_{4,\varepsilon}(z)$ $\displaystyle=-\int_{\Omega}[{\text{\rm
div}}b\,{\Gamma}(z,\cdot){\varphi}\psi_{\varepsilon}(z,\cdot)v](\zeta)d\zeta-\int_{\Omega}[\langle
b,D\left({\Gamma}(z,\cdot){\varphi}\psi_{\varepsilon}(z,\cdot)\right)\rangle\,v](\zeta)d\zeta$
$\displaystyle\leq-\int_{\Omega}[\langle
b,D\left({\Gamma}(z,\cdot){\varphi}\psi_{\varepsilon}(z,\cdot)\right)\rangle\,v](\zeta)d\zeta$

We are left with the estimate of a potential and therefore we exploit once
again Corollary 2.6. Since we have $b\in\left(L^{q}({\Omega})\right)^{m_{0}}$
and $v\in L^{2}({\Omega})$, we get

$|{\Gamma}(z,\cdot)||{\varphi}||b||Dv|\in L^{2\alpha}({\Omega}),$

where

$\alpha=\alpha(q)=\frac{q(Q+2)}{q(Q-2)+2(Q+2)}>1\qquad\text{if and only
if}\qquad q>\frac{Q+2}{2}.$

Hence, $|\langle
b,D\left({\Gamma}(z,\cdot){\varphi}\psi_{\varepsilon}(z,\cdot)\right)\rangle
v|\leq|\langle b,D\left({\Gamma}(z,\cdot){\varphi}\right)\rangle v|\in
L^{1}({\Omega})$. Thus, the Lebesgue convergence theorem yields

$\displaystyle\lim_{\varepsilon\to 0^{+}}I_{4,\varepsilon}(z)$
$\displaystyle=\lim_{\varepsilon\to 0^{+}}-\int_{\Omega}[\langle
b,D\left({\Gamma}(z,\cdot){\varphi}\psi_{\varepsilon}(z,\cdot)\right)\rangle\,v](\zeta)d\zeta$
$\displaystyle=\int_{\Omega}[\langle b,D{\Gamma}(z,\cdot){\varphi}\rangle
v]({\zeta})d{\zeta}.$

Similarly, we can estimate the term $I_{5,\varepsilon}$ noting that
$|c||v||{\Gamma}(z,\cdot)||{\varphi}|\in L^{2\alpha}({\Omega})$ with $\alpha$
as above. As a consequence, we have

$|cv{\Gamma}(z,\cdot){\varphi}\psi_{\varepsilon}(z,\cdot)|\leq|cv{\Gamma}(z,\cdot){\varphi}|\in
L^{1}({\Omega}),\qquad\text{and thus}\,\,\lim_{\varepsilon\to
0^{+}}I_{5,\varepsilon}(z)=\int_{\Omega}[cv{\Gamma}(z,\cdot){\varphi}]({\zeta})d{\zeta}.$

Now, we are left with the estimate of term $I_{6,\varepsilon}$, which is again
a ${\Gamma}$-potential such that

$|{\Gamma}(z,\cdot)||{\varphi}||f|\in L^{2\kappa}({\Omega}),$

where $\kappa=\frac{Q+2}{Q-2}$. Thus, we infer
$|{\Gamma}(z,\cdot){\varphi}\psi_{\varepsilon}(z,\cdot)f|\leq|{\Gamma}(z,\cdot){\varphi}f|\in
L^{1}({\Omega})$. Therefore we conclude the proof by applying the dominated
convergence theorem to $I_{6,\varepsilon}(z)$. ∎

We conclude this Section by recalling the following lemma for whose proof we
refer to [6, Lemma 6].

######  Lemma 2.8.

There exists a positive constant $\overline{c}\in(0,1)$ such that

$z\circ{\mathcal{Q}}_{\overline{c}r(r-{\rho})}\subseteq{\mathcal{Q}}_{r},$
(2.18)

for every $0<{\rho}<r\leq 1$ and $z\in{\mathcal{Q}}_{\rho}$.

##  3\. Local boundedness for weak solutions to $\mathscr{L}u=f$

Other than the new framework for the study of the weak regularity theory of
(1.1), the local Hölder continuity for solutions and the Harnack inequality,
the Moser’s iterative scheme for weak solutions to (1.1) with unbounded source
term $f$, to whose proof this Section is devoted, is in itself one of the main
novelties of our work.

Indeed, this procedure was firstly introduced by Moser in [28, 30] and it is
based on the iterative combination of a Caccioppoli and a Sobolev inequality.
When considering the classical uniformly elliptic and parabolic settings, the
Caccioppoli inequality provides us with a priori estimates for the $L^{2}$
norm of the complete gradient of the solution in terms of the $L^{2}$ norm of
the solution and we are able to consider the classical Sobolev embedding to
obtain a gain of integrability for the solution.

This is not the case when dealing with operator (1.1). In fact, the degeneracy
of the diffusion part allows us to estimate only the partial gradient
$D_{m_{0}}u$ of the solution to $\mathscr{L}u=f$ (see Theorem 3.4). In
addition, according to our definition of weak solution, $u$ does not lie in a
classical Sobolev space. In order to overcome these issues, we adopt a
technique based on the representation of a solution $u$ to $\mathscr{L}u=f$ in
terms of the fundamental solution ${\Gamma}$ (see (2.7)) of the principal part
operator $\mathscr{K}$. Indeed, following the idea presented for the first
time in [33] and later on applied in [3, 6], we have that if $u$ is a solution
to $\mathscr{L}u=f$ then

$\mathscr{K}u=\left(\mathscr{K}-\mathscr{L}\right)u-f={\text{\rm
div}}_{m_{0}}\left(\left(\mathbb{I}_{m_{0}}-A_{0}\right)D_{m_{0}}u\right)-\langle
Bx,Du\rangle-\partial_{t}u.$ (3.1)

Thus, by combining this representation formula with the potential estimates
presented in Corollary 2.6, we are able to prove a Sobolev inequality (see
Theorem 3.3) and a Caccioppoli inequality (see Theorem 3.4) for weak solutions
to $\mathscr{L}u=f$. We remark that in literature there is no similar result
available for weak solutions in the sense of Definition 1.2 to the Kolmogorov
equation $\mathscr{L}u=f$, with $f$ non zero source term. Indeed, in [11, 13,
14] the authors consider the case of the kinetic Fokker-Planck operator,
whereas in [33, 6, 3] the authors consider the Kolmogorov equation
$\mathscr{L}u=0$ under the “strong” notion of weak solution, i.e. $Yu\in
L^{2}$.

######  Theorem 3.1.

Let $u$ be a non-negative weak sub-solution to $\mathscr{L}u=f$ in ${\Omega}$
under the assumptions (H1)-(H3). Let $z_{0}\in{\Omega}$ and $r,{\rho}$, with
$0<{\rho}<r$, be such that
$\overline{{\mathcal{Q}}_{r}(z_{0})}\subseteq{\Omega}$. Then for every
$p\in\mathbb{R}$ such that $p>1/2$ there exist a positive constant $C$ such
that

$\displaystyle\sup_{{\mathcal{Q}}_{\rho}(z_{0})}u^{p}\leq\frac{C}{(r-{\rho})^{4\mu}}\parallel
u^{p}\parallel_{L^{{\beta}}({\mathcal{Q}}_{r})}$

where $C=C\left(p,{\lambda},\Lambda,Q,\parallel
b\parallel_{L^{q}({\mathcal{Q}}_{r}(z_{0}))},\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{r}(z_{0}))},\parallel
f\parallel_{L^{q}({\mathcal{Q}}_{r}(z_{0}))}\right)$ and

$\displaystyle\mu:=\frac{{\alpha}}{{\alpha}-{\beta}},\qquad\text{where
}\quad\alpha:=\frac{q(Q+2)}{q(Q-2)+2(Q+2)}\quad\text{and}\quad\beta:=\frac{q}{q-1}.$
(3.2)

######  Remark 3.2.

Theorem 3.1 holds true under the assumptions of Remark 1.7. In particular, in
this case the constant $\alpha$ is replaced by $\alpha=1+\frac{2}{Q}$, that
was firstly obtained in [33]. This is due the fact that the Sobolev
inequality, Theorem 3.3, holds true with a greater exponent if we assume more
integrability for the coefficient $b$. Thus, this allows us to obtain the
local boundedness for weak solutions to $\mathscr{L}u=f$ with lower
integrability for $c$ and $f$, i.e. $c,f\in L^{q}({\Omega})$, with
$q>\frac{Q+2}{2}$.

###  3.1. Sobolev Inequality

This subsection is devoted to the proof of a Sobolev type inequality for weak
solutions to $\mathscr{L}u=f$. Our approach is inspired by the paper [33] and
allows us to construct an “ad hoc” Sobolev embedding for weak solutions to
$\mathscr{L}u=f$ by overcoming the difficulties due to the degeneracy of the
second order part of the operator $\mathscr{L}$ at the cost of lowering the
Sobolev exponent, that in our case depends on $q$ and it is defined in (3.2).
We remark that the following statement holds true under lower integrability
assumption than the one required in (H3).

######  Theorem 3.3 (Sobolev Type Inequality for sub-solutions).

Let (H1)-(H2) hold. Let $c\in L^{q}({\Omega})$ and
$b\in\left(L^{q}({\Omega})\right)^{m_{0}}$ for some $q>\frac{Q+2}{2}$ and let
$f\in L^{2}({\Omega})$. Let $v$ be a non-negative weak sub-solution of
$\mathscr{L}v=f$ in ${\mathcal{Q}}_{1}$. Then there exists a constant
$C=C(Q,{\lambda},\Lambda)>0$ such that $v\in
L^{2{\alpha}}({\mathcal{Q}}_{1})$, and the following inequality holds

$\displaystyle\parallel
v\parallel_{L^{2{\alpha}}({\mathcal{Q}}_{\rho}(z_{0}))}\leq$ $\displaystyle
C\cdot\left(\parallel
b\parallel_{L^{q}({\mathcal{Q}}_{r}(z_{0}))}+\frac{r-{\rho}+1}{r-{\rho}}\right)\parallel
D_{m_{0}}v\parallel_{L^{2}({\mathcal{Q}}_{r}(z_{0}))}+$
$\displaystyle\qquad\qquad+C\cdot\left(\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{r}(z_{0}))}+\frac{{\rho}+1}{{\rho}(r-{\rho})}\right)\parallel
v\parallel_{L^{2}({\mathcal{Q}}_{r}(z_{0}))}+C\cdot\parallel
f\parallel_{L^{2}({\mathcal{Q}}_{r}(z_{0}))}$

for every ${\rho},r$ with $\frac{1}{2}\leq{\rho}<r\leq 1$ and for every
$z_{0}\in{\Omega}$, where ${\alpha}={\alpha}(q)$ is defined as (3.2). The same
statement holds for non-negative super-solutions.

###### Proof.

Let $v$ be a non-negative weak sub-solution to $\mathscr{L}v=f$. We represent
$v$ in terms of the fundamental solution ${\Gamma}$. To this end, we consider
the cut-off function ${\chi}_{{\rho},r}$ defined in (2.17) for
$\frac{1}{2}\leq{\rho}<r\leq 1$. Then, if we consider the test function

$\psi(x,t)={\chi}_{{\rho},r}(\parallel(x,t)\parallel),$ (3.3)

the following estimates hold true

$|Y\psi|\leq\frac{c_{0}}{{\rho}(r-{\rho})},\hskip
22.76219pt|\partial_{x_{j}}\psi|\leq\frac{c_{1}}{r-{\rho}}\hskip
5.69054pt\text{for }j=1,\ldots,m_{0}$ (3.4)

where $c_{0}$, $c_{1}$ are dimensional constants. For every
$z\in{\mathcal{Q}}_{\rho}$, we have

$\displaystyle v(z)=v\psi(z)$
$\displaystyle=\int_{{\mathcal{Q}}_{r}}\left[\langle
A_{0}D(v\psi),D{\Gamma}(z,\cdot)\rangle-{\Gamma}(z,\cdot)Y(v\psi)\right]({\zeta})d({\zeta})$
(3.5) $\displaystyle=I_{0}(z)+I_{1}(z)+I_{2}(z)+I_{3}(z)$

where

$\displaystyle I_{0}(z)$ $\displaystyle=\int_{{\mathcal{Q}}_{r}}\left[\langle
b,Dv\rangle{\Gamma}(z,\cdot)\psi\right]({\zeta})d{\zeta}\hskip
2.84526pt\,+\,\int_{{\mathcal{Q}}_{r}}\left[cv{\Gamma}(z,\cdot)\psi\right]({\zeta})d{\zeta}-\int_{{\mathcal{Q}}_{r}}[{\Gamma}(z,\cdot)\psi
f]({\zeta})d{\zeta}$ $\displaystyle I_{1}(z)$
$\displaystyle=\int_{{\mathcal{Q}}_{r}}\left[\langle
A_{0}D\psi,D{\Gamma}(z,\cdot)\rangle v\right]({\zeta})d{\zeta}\hskip
2.84526pt-\hskip
2.84526pt\int_{{\mathcal{Q}}_{r}}\left[{\Gamma}(z,\cdot)vY\psi\right]({\zeta})d{\zeta}=I_{1}^{{}^{\prime}}+I_{1}^{{}^{\prime\prime}},$
$\displaystyle I_{2}(z)$
$\displaystyle=\int_{{\mathcal{Q}}_{r}}\left[\langle(A_{0}-A)Dv,D{\Gamma}(z,\cdot)\rangle\psi\right]({\zeta})d{\zeta}\hskip
2.84526pt-\hskip
2.84526pt\int_{{\mathcal{Q}}_{r}}\left[{\Gamma}(z,\cdot)\langle
ADv,D\psi\rangle\right]({\zeta})d{\zeta}$ $\displaystyle I_{3}(z)$
$\displaystyle=\int_{{\mathcal{Q}}_{r}}\left[\langle
ADv,D({\Gamma}(z,\cdot)\psi)\rangle\right]({\zeta})d{\zeta}\hskip
2.84526pt-\hskip
2.84526pt\int_{{\mathcal{Q}}_{r}}\left[\left({\Gamma}(z,\cdot)\psi\right)Yv\right]({\zeta})d{\zeta}\hskip
2.84526pt$ $\displaystyle-\int_{{\mathcal{Q}}_{r}}\left[\langle
b,Dv\rangle{\Gamma}(z,\cdot)\psi\right]({\zeta})d{\zeta}-\,\int_{{\mathcal{Q}}_{r}}\left[cv{\Gamma}(z,\cdot)\psi\right]({\zeta})d{\zeta}+\int_{{\mathcal{Q}}_{r}}[{\Gamma}(z,\cdot)\psi
f]({\zeta})d{\zeta}$

Since $v$ is a non-negative weak sub-solution to $\mathscr{L}v=f$, it follows
from Lemma 2.7 that $I_{3}\leq 0$, then

$0\leq v(z)\leq I_{0}(z)+I_{1}(z)+I_{2}(z)\hskip 11.38109pt\text{for a.e.
}z\in{\mathcal{Q}}_{\rho}.$

To prove our claim is sufficient to estimate $v$ by a sum of
${\Gamma}-$potentials. We start by estimating $I_{0}$. In order to do so, we
recall that

$\langle b,Dv\rangle,\,cv\,\in L^{2\frac{q}{q+2}}\hskip 11.38109pt\text{for
}b,c\in L^{q},\hskip 5.69054ptq>\frac{Q+2}{2}\hskip 5.69054pt\text{and }v\in
L^{2}.$

Thus by Corollary 2.6 we get

${\Gamma}*\langle b,Dv\rangle,{\Gamma}*(cv)\in L^{2{\alpha}},$

where ${\alpha}={\alpha}(q)$ is defined in (3.2). In addition, for $f\in
L^{2}$, we have

${\Gamma}*f\in L^{2\kappa},\quad\kappa=\frac{Q+2}{Q-2}.$

Observing that $\kappa>\alpha$, we obtain that ${\Gamma}*f\in L^{2\alpha}$ and
therefore

$\displaystyle\parallel
I_{0}({\zeta})\parallel_{L^{2{\alpha}}({\mathcal{Q}}_{\rho})}$
$\displaystyle\leq{\Gamma}*\left(\langle
b,D_{m_{0}}v\rangle\psi\right)+{\Gamma}*\left(cv\psi\right)\parallel_{L^{2\alpha}({\mathcal{Q}}_{\rho})}+\parallel{\Gamma}*f\parallel_{L^{2\alpha}({\mathcal{Q}}_{\rho})}$
$\displaystyle\leq C\cdot\left(\parallel
b\parallel_{L^{q}({\mathcal{Q}}_{\rho})}\parallel
D_{m_{0}}v\parallel_{L^{2}({\mathcal{Q}}_{\rho})}+\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{\rho})}\parallel
v\parallel_{L^{2}({\mathcal{Q}}_{\rho})}+\parallel
f\parallel_{L^{2}({\mathcal{Q}}_{\rho})}\right).$

We now deal with the $I_{1}$. $I_{1}^{\prime}$ can be estimated by (2.15) of
Corollary 2.6 as follows

$\parallel I_{1}^{\prime}\parallel_{L^{2{\alpha}}({\mathcal{Q}}_{\rho})}\leq
C\parallel I_{1}^{\prime}\parallel_{L^{2^{*}}({\mathcal{Q}}_{\rho})}\leq
C\parallel
vD_{m_{0}}\psi\parallel_{L^{2}(\mathbb{R}^{N+1})}\leq\frac{C}{r-{\rho}}\parallel
v\parallel_{L^{2}({\mathcal{Q}}_{\rho})},$

where the last inequality follows from (3.4). To estimate
$I_{1}^{\prime\prime}$ we use (2.14)

$\displaystyle\parallel
I_{1}^{\prime\prime}\parallel_{L^{2{\alpha}}({\mathcal{Q}}_{\rho})}$
$\displaystyle\leq C\parallel
I_{1}^{\prime\prime}\parallel_{L^{2^{*}}({\mathcal{Q}}_{\rho})}\leq\text{meas}({\mathcal{Q}}_{\rho})^{2/Q}\parallel
I_{1}^{\prime\prime}\parallel_{L^{2^{**}}({\mathcal{Q}}_{\rho})}$
$\displaystyle\leq C\parallel
vY\psi\parallel_{L^{2}(\mathbb{R}^{N+1})}\leq\frac{C}{{\rho}(r-{\rho})}\parallel
v\parallel_{L^{2}({\mathcal{Q}}_{\rho})}.$

We can use the same technique to prove that

$\parallel I_{2}\parallel_{L^{2{\alpha}}({\mathcal{Q}}_{\rho})}\leq
C\left(1+\frac{1}{r-{\rho}}\right)\parallel
Dv\parallel_{L^{2}({\mathcal{Q}}_{\rho})},$

for some constant $C=C(Q,{\lambda},\Lambda)$. A similar argument proves the
thesis when $v$ is a super-solution to $\mathscr{L}v=f$. In this case we
introduce the following auxiliary operator

$\mathscr{K}={\text{\rm
div}}(A_{0}\,D)+\widetilde{Y},\qquad\qquad\widetilde{Y}\equiv-\langle
x,BD\rangle-\partial_{t}\,.$ (3.6)

Then we proceed analogously as in [33], Section 3, proof of Theorem 3.3. ∎

###  3.2. Caccioppoli inequality

The aim of this subsection is to prove a Caccioppoli type inequality for
powers $v=u^{p}$ of weak sub-solutions to $\mathscr{L}u=f$.

######  Theorem 3.4 (Caccioppoli type inequality for sub-solutions).

Let (H1)-(H3) hold. Let $r,{\rho}$ be such that $\frac{1}{2}\leq{\rho}<r\leq
1$. Then for every weak sub-solutions to $\mathscr{L}u=f$ we have that for
every $p>1/2$ it holds

$\displaystyle\parallel
D_{m_{0}}v\parallel^{2}_{L^{2}({\mathcal{Q}}_{\rho})}\leq\frac{4p}{{\lambda}(2p-1)}$
$\displaystyle\Big{(}\frac{2p}{2p-1}\frac{c_{1}^{2}\Lambda}{(r-{\rho})^{2}}\,+\,\frac{c_{0}}{{\rho}(r-{\rho})}\,+\frac{\parallel
b\parallel^{2}_{L^{q}({\mathcal{Q}}_{r})}}{2{\varepsilon}_{b}}$ (3.7)
$\displaystyle\qquad\qquad\qquad+p\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{r})}+p\parallel
f\parallel_{L^{q}({\mathcal{Q}}_{r})}\Big{)}\parallel
u^{p}\parallel_{L^{2{\beta}}({\mathcal{Q}}_{r})}^{2}$

where ${\beta}={\beta}(q)=\frac{q}{q-1}$,
${\varepsilon}_{b}=\frac{|2p-1|{\lambda}}{2|p|}$ and $c_{0},c_{1}$ are defined
in (3.4).

######  Remark 3.5.

If $f\in L^{2}({\Omega})$ then the estimate (3.7) holds true for every
$\frac{1}{2}<p\leq 1$.

For the sake of completeness and in order to simplify the exposition of the
proof of Theorem 3.4 we consider the following lemma, which is the analogous
to [3, Proposition 3.2] when considering Definition 1.2 of weak solution.

######  Lemma 3.6.

Let (H1)-(H3) hold. Let $p\in\mathbb{R}$, $p\neq 0$, $p\neq 1/2$ and let
$r,{\rho}$ be such that $\frac{1}{2}\leq{\rho}<r\leq 1$. Then for every weak
sub-solution to $\mathscr{L}u=0$ the following estimate holds true

$\displaystyle{\lambda}\Big{|}\frac{2p-1}{p}\Big{|}\,\parallel
D_{m_{0}}u^{p}\parallel^{2}_{L^{2}({\mathcal{Q}}_{\rho})}$
$\displaystyle\leq\Big{(}\Big{|}\frac{2p}{2p-1}\Big{|}\frac{c_{1}^{2}\lambda}{(r-{\rho})^{2}}\,+\,\frac{c_{0}}{{\rho}(r-{\rho})}\,+$
(3.8) $\displaystyle\qquad\qquad\qquad\quad\,+\frac{c_{1}}{r-{\rho}}\parallel
b\parallel^{2}_{L^{q}({\mathcal{Q}}_{r})}+|p|\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{r})}\Big{)}\parallel
u^{p}\parallel_{L^{2{\beta}}({\mathcal{Q}}_{r})}^{2},$

where ${\beta}={\beta}(q)=\frac{q}{q-1}$ and $c_{0},c_{1}$ are defined in
(3.4).

###### Proof.

Let us consider $p<1$, $p\neq 0$, $p\neq 1/2$. First of all, we consider a
non-negative weak sub-solution $u$ to $\mathscr{L}u=0$. For every $\psi\in
C^{\infty}_{0}({\mathcal{Q}}_{r})$ we consider the function
${\varphi}=u^{2p-1}\psi^{2}$. Note that ${\varphi},D_{m_{0}}{\varphi}\in
L^{2}({\mathcal{Q}}_{r})$, then we can use ${\varphi}$ as a test function in
the weak formulation (1.7):

$\displaystyle 0$ $\displaystyle\leq\int_{{\mathcal{Q}}_{r}}-\langle
ADu,D(u^{2p-1}\psi^{2})\rangle+u^{2p-1}\psi^{2}Yu+\langle b,Du\rangle
u^{2p-1}\psi^{2}+cu^{2p}\psi^{2}$

Let $v=u^{p}$. Since $u$ is a weak sub-solution to $\mathscr{L}u=0$ and $u\geq
u_{0}$, thanks to Proposition 3.3 we have that $v,D_{m_{0}}v\in
L^{2}({\mathcal{Q}}_{r})$ and $Yv\in L^{2}H^{-1}({\mathcal{Q}}_{r})$ and thus:

$\displaystyle 0\leq$
$\displaystyle-\frac{2p-1}{p}\int_{{\mathcal{Q}}_{r}}\langle
ADv,Dv\rangle\psi^{2}\,-2\int_{{\mathcal{Q}}_{r}}\langle ADv,D\psi\rangle
v\psi\,+\int_{{\mathcal{Q}}_{r}}v\,Yv\,\psi^{2}\,+$ (3.9)
$\displaystyle+\,\int_{{\mathcal{Q}}_{r}}v\,\langle
b,D_{m_{0}}v\rangle\,\psi^{2}\,+p\,\int_{{\mathcal{Q}}_{r}}cv^{2}\psi^{2}.$

First of all, we estimate of the terms involving the matrix $A$. In
particular, by applying Young’s inequality we have that for every
${\varepsilon}>0$:

$\displaystyle 2\int_{{\mathcal{Q}}_{r}}|\langle ADv,D\psi\rangle
v\psi|\leq{\varepsilon}\int_{{\mathcal{Q}}_{r}}\langle
ADv,Dv\rangle\psi^{2}\,+\,\frac{1}{{\varepsilon}}\int_{{\mathcal{Q}}_{r}}\langle
AD\psi,D\psi\rangle v^{2}.$ (3.10)

Thus, taking the absolute value on the right-hand side of (3.9) and
considering ${\varepsilon}=|(2p-1)/2p|$ in (3.10), for every $p<1$, $p\neq
0,1/2$ we obtain:

$\displaystyle\Big{|}\frac{2p-1}{p}\Big{|}{\lambda}\int_{{\mathcal{Q}}_{{\rho}}}|D_{m_{0}}v|^{2}\,$
$\displaystyle\leq\Big{|}\frac{2p\lambda}{2p-1}\Big{|}\frac{c_{1}^{2}}{(r-{\rho})^{2}}\int_{{\mathcal{Q}}_{r}}v^{2}\,+\,\frac{c_{0}}{{\rho}(r-{\rho})}\int_{{\mathcal{Q}}_{r}}v^{2}\,+$
(3.11) $\displaystyle+\,\boxed{\int_{{\mathcal{Q}}_{r}}v\,\langle
b,D_{m_{0}}v\rangle\,\psi^{2}}_{B}\,+\,\boxed{|p|\int_{{\mathcal{Q}}_{r}}|c|v^{2}\psi^{2}}_{C},$

where we have considered the definition (3.3) of the cut-off function $\psi$
in order to estimate the first integral on the right-hand side. By considering
the bounds on (3.4), assumption (H3) and the Hölder’s inequality with exponent
${\beta}=\frac{q}{q-1}$ we are in position to estimate the term B

$\displaystyle\boxed{\frac{1}{2}\int_{{\mathcal{Q}}_{r}}\langle
b,D_{m_{0}}v^{2}\rangle\,\psi^{2}}_{B}$
$\displaystyle=-\frac{1}{2}\int_{{\mathcal{Q}}_{r}}{\text{\rm div}}\cdot
b\,v^{2}\,\psi^{2}-\int_{{\mathcal{Q}}_{r}}\langle b,\psi D_{m_{0}}\psi\rangle
v^{2}$ $\displaystyle\leq\,\frac{C}{r-{\rho}}\parallel
b\parallel_{L^{q}({\mathcal{Q}}_{r})}\,\parallel
v\parallel^{2}_{L^{2{\beta}}({\mathcal{Q}}_{r})}$

and the term C:

$\boxed{|p|\int_{{\mathcal{Q}}_{r}}|c|v^{2}\psi^{2}}_{C}\leq|p|\,\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{r})}\,\parallel
v\parallel_{L^{2{\beta}}({\mathcal{Q}}_{r})}^{2}.$

Thus, by by choosing ${\varepsilon}_{a}=\lambda/|p|$ and
${\varepsilon}_{b}=(|2p-1|{\lambda})/(2|p|)$ we recover (3.8).

Now, let us consider the case $p\geq 1$. For any $n\in\mathbb{N}$, we define
the function $g_{n,p}$ on $]0,+\infty[$ as follows

$g_{n,p}(s)=\begin{cases}s^{p},&\text{if}\hskip 5.69054pt0<s\leq n,\\\
n^{p}+pn^{p-1}(s-n),\hskip 11.38109pt&\text{if}\hskip
5.69054pts>n,\end{cases}$

then we let $v_{n,p}=g_{n,p}(u)$. Note that $g_{n,p}\in C^{1}(\mathbb{R}^{+})$
and $g_{n,p}^{{}^{\prime}}\in L^{\infty}(\mathbb{R}^{+})$. Thus, since $u$ is
a weak sub-solution to $\mathscr{L}u=0$, we have

$v_{n,p}\in L^{2},\hskip 5.69054ptD_{m_{0}}v_{n,p}\in L^{2},\hskip
5.69054ptYv_{n,p}\in L^{2}H^{-1}.$

We also note that the function

$g^{{}^{\prime\prime}}_{n,p}(s)=\begin{cases}p(p-1)s^{p-2},\hskip
11.38109pt&\text{if}\hskip 5.69054pt0<s<n\\\ 0,&\text{if}\hskip 5.69054pts\geq
n,\end{cases}$

is the weak derivative of $g_{n,p}^{{}^{\prime}}$, then
$D_{m_{0}}g_{n,p}^{{}^{\prime}}(u)=g^{{}^{\prime\prime}}_{n,p}(u)D_{m_{0}}(u)$
(for the detailed proof of this assertion, we refer to [10], Theorem 7.8).
Hence, by considering

${\varphi}=g_{n,p}(u)\hskip 2.84526ptg_{n,p}^{{}^{\prime}}(u)\hskip
2.84526pt\psi^{2},\hskip 11.38109pt\psi\in C^{\infty}_{0}({\mathcal{Q}}_{r})$
(3.12)

as a test function in the weak formulation (1.7) and recalling that we find
$v_{n,p}=g_{n,p}(u)$ we have

$\displaystyle 0$
$\displaystyle\leq\int\limits_{{\mathcal{Q}}_{r}}-\psi^{2}\langle
ADv_{n,p},Dv_{n,p}\rangle-\int\limits_{{\mathcal{Q}}_{r}}\frac{v^{{}^{\prime\prime}}_{n,p}(u)\,v_{n,p}(u)}{\left(v^{{}^{\prime}}_{n,p}(u)\right)^{2}}\psi^{2}\langle
ADv_{n,p},Dv_{n,p}\rangle+$ (3.13)
$\displaystyle\qquad-\int\limits_{{\mathcal{Q}}_{r}}2\psi\langle
ADv_{n,p},D\psi\rangle
v_{n,p}+\int\limits_{{\mathcal{Q}}_{r}}\psi^{2}\,v_{n,p}Y(v_{n,p})+$
$\displaystyle\qquad+\boxed{\int\limits_{{\mathcal{Q}}_{r}}\langle
b,D_{m_{0}}v_{n,p}\rangle
v_{n,p}\,\psi^{2}}_{B}+\boxed{\int\limits_{{\mathcal{Q}}_{r}}cuv_{n,p}(u)\,v^{{}^{\prime}}_{n,p}(u)\psi^{2}}_{C}.$

Let us firstly estimate the term $B$ as in the previous case:

$\displaystyle\boxed{\frac{1}{2}\int\limits_{{\mathcal{Q}}_{r}}\langle
b,D_{m_{0}}v^{2}_{n,p}\rangle\,\psi^{2}}_{B}\leq\,\int_{{\mathcal{Q}}_{r}}|\langle
b,\psi D_{m_{0}}\psi\rangle|v_{n,p}^{2}\,\leq\,\frac{C}{r-{\rho}}\parallel
b\parallel_{L^{q}({\mathcal{Q}}_{r})}\,\parallel
v_{n,p}\parallel^{2}_{L^{2{\beta}}({\mathcal{Q}}_{r})}$

Then the term $C$ is treated as follows:

$\begin{split}\boxed{\int\limits_{{\mathcal{Q}}_{r}}cuv_{n,p}(u)\,v^{{}^{\prime}}_{n,p}(u)\psi^{2}}_{C}\leq
p\int\limits_{{\mathcal{Q}}_{r}}|c|\,|u^{2p}|\,\psi^{2}\leq p\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{r})}\parallel
u^{p}\parallel^{2}_{L^{2{\beta}_{2}}({\mathcal{Q}}_{r})}.\end{split}$

Thus we rewrite equation (3.13) as follows

$\displaystyle\int\limits_{{\mathcal{Q}}_{r}}\psi^{2}\langle
ADv_{n,p},Dv_{n,p}\rangle\left(1+\frac{v^{{}^{\prime\prime}}_{n,p}(u)\,v_{n,p}(u)}{\left(pu^{p-1}\right)^{2}}-\varepsilon\right)$
(3.14)
$\displaystyle\qquad\leq\left(\frac{c_{1}^{2}\Lambda}{{\varepsilon}(r-{\rho})^{2}}+\frac{c_{0}}{{\rho}(r-{\rho})}+\parallel
b\parallel_{L^{q}({\mathcal{Q}}_{r})}\,+p\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{r})}\right)\parallel
u^{p}\parallel^{2}_{L^{2{\beta}_{2}}({\mathcal{Q}}_{r})}.$ (3.15)

We now observe that

$\left|\psi^{2}\langle
ADv_{n,p},Dv_{n,p}\rangle\frac{v^{{}^{\prime\prime}}_{n,p}(u)\,v_{n,p}(u)}{p^{2}u^{2p-2}}\right|\leq\frac{p-1}{p}\langle
ADv_{n,p},Dv_{n,p}\rangle\in L^{1}({\mathcal{Q}}_{r})$ (3.16)

and thus, keeping in mind that $|Dv_{n,p}|\,\uparrow\,|Du^{p}|$ as
$n\rightarrow\infty$, we can apply the dominated convergence theorem to
(3.14). The proof is complete if we choose ${\varepsilon}$ as in the previous
case. ∎

Proof of Theorem 3.4. We are now in a position to consider weak sub-solutions
to $\mathscr{L}u=f$, with $f$ a non zero source term under assumption (H3).
First of all, let us consider the case where $1/2<p\leq 1$. Following the
lines of the first part of this proof, for every $\psi\in
C^{\infty}_{0}({\mathcal{Q}}_{r})$ we consider the function
${\varphi}=u^{2p-1}\psi^{2}$ as a test function in the weak formulation of
(1.7) with non zero source term. Indeed, by denoting $v=u^{p}$ we have that
also in this case $v,D_{m_{0}}v\in L^{2}({\mathcal{Q}}_{r})$ and $Yv\in
L^{2}H^{-1}$. Thus, we get exactly estimate (3.11) with the following
additional term regarding the source term $f$:

$\boxed{|p|\int_{{\mathcal{Q}}_{r}}|f|u^{2p-1}\psi^{2}}_{F}$

In order to conclude the proof, we therefore need to properly estimate
$u^{2p-1}$. In particular, if

  * •

$0\leq u<1$ we have that $u^{2p-1}<1$, thus

$\boxed{|p|\int_{{\mathcal{Q}}_{r}}|f|u^{2p-1}\psi^{2}}_{F}\leq|p|\int_{{\mathcal{Q}}_{r}}|f|\psi^{2}$
(3.17)

  * •

$u\geq 1$ we have that $u^{2p-1}<u^{p}=v$. By combining Hölder’s inequality
and Young’s inequality (3.10) for every ${\varepsilon}_{f}>0$ we get

$\boxed{|p|\int_{{\mathcal{Q}}_{r}}|f|u^{2p-1}\psi^{2}}_{F}\leq|p|\|f\|_{L^{2}({\mathcal{Q}}_{r})}\|v\|_{L^{2}({\mathcal{Q}}_{r})}$
(3.18)

Thus, by combining (3.17) with (3.18) and considering ${\varepsilon}_{f}=1$ we
get (3.7).

We now address the case $p>1$. Reasoning as above, we set $v_{n,p}=g_{n,p}(u)$
and we choose the test function as in (3.12). Then the weak formulation of
(1.7) for $v_{n,p}$ reads exactly as (3.13) with the additional term

$\boxed{-p\int\limits_{{\mathcal{Q}}_{r}}fv_{n,p}(u)\,v^{{}^{\prime}}_{n,p}(u)\psi^{2}}_{F}.$

We deal with the boxed term $F$ by distinguishing two different cases. In
particular, if

  * •

$0\leq u<1$ we have that $u^{2p-1}<1$, thus we exactly recover estimate
(3.17).

  * •

$u\geq 1$ we have that $u^{2p-1}<u^{2p}$. We now observe that $u^{p}$ is a
non-negative weak super-solution to $\mathscr{L}u=f$ and therefore $u^{p}\in
L^{2\beta}$ in virtue of Theorem 3.3. By applying Hölder’s inequality, we then
infer

$\boxed{-p\int\limits_{{\mathcal{Q}}_{r}}fv_{n,p}(u)\,v^{{}^{\prime}}_{n,p}(u)\psi^{2}}_{F}\leq
p\int_{{\mathcal{Q}}_{r}}|f|u^{2p-1}\psi^{2}\leq
p\|f\|_{L^{q}({\mathcal{Q}}_{r})}\|u^{p}\|^{2}_{L^{2{\beta}}({\mathcal{Q}}_{r})}.$
(3.19)

By combining (3.17) and (3.19) we conclude the proof (3.7). $\hfill\square$

###  3.3. Proof of Theorem 3.1

Keeping in mind Lemma 2.8, it suffices to give a proof in the case $z_{0}=0$
and to consider $\frac{1}{2}\leq{\rho}<r\leq 1$, since the general statement
is obtained by applying to (3.23) the group dilations (2.9) and translations
(2.1). Combining Theorems 3.3 and 3.6 for a non-negative sub-solution $u$, we
obtain the following estimate. If $s>1/2$, ${\delta}>0$ verify the condition

$|s-1/2|\geq{\delta},$

then, for every ${\rho},r$ such that $\frac{1}{2}\leq{\rho}<r\leq 1$ we have

$\displaystyle\parallel
u^{s}\parallel_{L^{2{\alpha}}({\mathcal{Q}}_{\rho})}\leq\widetilde{C}\left(s,{\lambda},\Lambda,Q,\parallel
b\parallel_{L^{q}({\mathcal{Q}}_{r})},\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{r})},\parallel
f\parallel_{L^{q}({\mathcal{Q}}_{r})}\right)\parallel
u^{s}\parallel_{L^{2{\beta}}({\mathcal{Q}}_{r})}$ (3.20)

for a positive constant $\widetilde{C}$ that we be estimated as follows

$\displaystyle\widetilde{C}(s,{\lambda},\Lambda,Q,\parallel
b\parallel_{L^{q}({\mathcal{Q}}_{r})}$ $\displaystyle,\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{r})},\parallel
f\parallel_{L^{q}({\mathcal{Q}}_{r})})$ (3.21)
$\displaystyle\leq\frac{K(s,{\lambda},\Lambda,Q,\parallel
b\parallel_{L^{q}({\mathcal{Q}}_{r})},\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{r})},\parallel
f\parallel_{L^{q}({\mathcal{Q}}_{r})})}{\left(r-{\rho}\right)^{2}},$

since for every ${\rho},r$ of our choice the following estimates hold

$\begin{split}\frac{1}{\left({\rho}(r-{\rho})\right)^{\frac{1}{2}}(r-{\rho})}\leq\frac{1}{\left(r-{\rho}\right)^{2}},\qquad\frac{1}{{\rho}(r-{\rho})}\leq\frac{1}{\left(r-{\rho}\right)^{2}},\qquad\frac{1}{\left({\rho}(r-{\rho})\right)^{2}}\leq\frac{1}{(r-{\rho})^{2}}.\end{split}$

Then we set $v=u^{\frac{p}{2}}$. Fixed a suitable ${\delta}>0$, that we will
specify later on, and a suitable $p\geq 1$ such that

$p\,\left(\frac{{\alpha}}{{\beta}}\right)^{n}-1\geq 2{\delta},\qquad\forall
n\in\mathbb{N}\cup\\{0\\},$ (3.22)

we iterate inequality (3.20) by choosing

${\rho}_{n}={\rho}+\frac{1}{2^{n}}\left(r-{\rho}\right),\qquad
p_{n}=\left(\frac{{\alpha}}{{\beta}}\right)^{n}\,\frac{p}{2},\hskip
17.07164ptn\in\mathbb{N}\cup\\{0\\}.$

Thus, by combining (3.20) and (3.21), for every $n\in\mathbb{N}\cup\\{0\\}$
the following holds

$\displaystyle\parallel v^{(\frac{{\alpha}}{{\beta}})^{n}}$
$\displaystyle\parallel_{L^{2{\alpha}}({\mathcal{Q}}_{{\rho}_{n+1})}}\leq\frac{K(p,{\lambda},\Lambda,Q,\parallel
b\parallel_{L^{q}({\mathcal{Q}}_{r})},\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{r})},\parallel
f\parallel_{L^{q}({\mathcal{Q}}_{r})})}{\left({\rho}_{n}-{\rho}_{n+1}\right)^{2}}\parallel
v^{(\frac{{\alpha}}{{\beta}})^{n}}\parallel_{L^{2{\beta}}({\mathcal{Q}}_{{\rho}_{n}})}.$

From now on we denote $K=K(p,{\lambda},\Lambda,Q,\parallel
b\parallel_{L^{q}({\mathcal{Q}}_{r})},\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{r})},\parallel
f\parallel_{L^{q}({\mathcal{Q}}_{r})})$. Since

$\parallel
v^{\left(\frac{{\alpha}}{{\beta}}\right)^{n}}\parallel_{L^{2{\alpha}}}=\left(\parallel
v\parallel_{L^{2{\alpha}\left(\frac{{\alpha}}{{\beta}}\right)^{n}}}\right)^{\left(\frac{{\alpha}}{{\beta}}\right)^{n}}\qquad{\rm
and}\qquad\parallel
v^{\left(\frac{{\alpha}}{{\beta}}\right)^{n}}\parallel_{L^{2{\beta}}}=\left(\parallel
v\parallel_{L^{2{\beta}\left(\frac{{\alpha}}{{\beta}}\right)^{n}}}\right)^{\left(\frac{{\alpha}}{{\beta}}\right)^{n}}$

we are able to rewrite the previous estimate in the following form for every
$n\in\mathbb{N}\cup\\{0\\}$

$\displaystyle\parallel
v\parallel_{L^{2{\alpha}\left(\frac{{\alpha}}{{\beta}}\right)^{n}}\left({\mathcal{Q}}_{{\rho}_{n+1}}\right)}\leq\left(\frac{K}{\left({\rho}_{n}-{\rho}_{n+1}\right)^{2}}\right)^{\left(\frac{{\beta}}{{\alpha}}\right)^{n}}\parallel
v\parallel_{L^{2{\beta}\left(\frac{{\alpha}}{{\beta}}\right)^{n}}({\mathcal{Q}}_{{\rho}_{n}})}.$

Iterating this inequality and letting $n$ go to infinity, we get

$\sup_{{\mathcal{Q}}_{{\rho}}}v\,\leq\,\frac{\widetilde{K}}{(r-{\rho})^{2{\mu}}}\parallel
v\parallel_{L^{2{\beta}}({\mathcal{Q}}_{r})},\qquad\text{where
}\,{\mu}=\frac{{\alpha}}{{\alpha}-{\beta}}\quad\text{and}\quad\widetilde{K}=\prod_{j=0}^{+\infty}\left[K2^{2(j+2)}\right]^{\left(\frac{{\beta}}{{\alpha}}\right)^{j}}.$

We remark that $\widetilde{K}$ is a finite constant depending on ${\delta}$,
since the product over $j$ corresponds to a converging series. Thus, for every
$p\geq 1$ which verifies condition (3.22) we have proved that

$\sup_{{\mathcal{Q}}_{{\rho}}}u^{p}\,\leq\,\frac{\widetilde{K}^{2}}{(r-{\rho})^{4\mu}}\parallel
u^{p}\parallel_{L^{{\beta}}({\mathcal{Q}}_{r})},$ (3.23)

and thus the statement is proved. We now make a suitable choice of
${\delta}>0$, only dependent on the homogeneous dimension $Q$, and on $q$ in
order to show that (3.22) holds for every $p\geq 1$. We notice that, if $p$ is
a number of the form

$p_{m}=\frac{1}{2}\left(\frac{{\alpha}}{{\beta}}\right)^{m}\left(\frac{{\alpha}}{{\beta}}+1\right),\qquad
m\in{\mathbb{Z}},$

then (3.22) is satisfied with the following choice of ${\delta}$ for every
$m\in{\mathbb{Z}}$

${\delta}=\frac{{\alpha}-{\beta}}{8{\beta}}.$

Therefore (3.23) holds for such a choice of $p$, with $\widetilde{K}$ only
dependent on $Q,q,{\lambda}$, $\Lambda$ and $\parallel
a\parallel_{L^{q}({\mathcal{Q}}_{r})}$, $\parallel
b\parallel_{L^{q}({\mathcal{Q}}_{r})}$, $\parallel
c\parallel_{L^{q}({\mathcal{Q}}_{r})}$. On the other hand, if $p$ is an
arbitrary positive number, we consider $m\in{\mathbb{Z}}$ such that

$p_{m}\leq p<p_{m+1}.$

Hence, the proof is complete. $\square$

##  4\. Weak Poincaré inequality

This section is devoted to the proof of a weak Poincaré inequality (see
Theorem 4.1) that holds true for every function $u\in\mathcal{W}$. As one
immediately understands, this Poincaré inequality is independent of the
equation $\mathscr{L}u=f$ and only relies on the structure of the space
$\mathcal{W}$. Its importance lies in the fact that it is a necessary tool in
the proof of a Harnack inequality (see Theorem 1.4) and of the local Hölder
continuity (see Theorem 1.6) of a solution $u$ to (1.1). In order to state our
result, we first need to introduce the following sets

$\displaystyle{\mathcal{Q}}_{zero}$
$\displaystyle=\\{(x,t):|x_{j}|\leq\eta^{\alpha_{j}},j=1,\ldots,N,-1-\eta^{2}<t\leq-1\\},$
(4.1) $\displaystyle{\mathcal{Q}}_{ext}$ $\displaystyle=\\{(x,t):|x_{j}|\leq
2^{\alpha_{j}}R,j=1,\ldots,N,-1-\eta^{2}<t\leq 0\\},$

where $R>1$, $\eta\in(0,1]$ and the exponents $\alpha_{j}$, for
$j=1,\ldots,N$, are defined in (2.10). Indeed, ${\mathcal{Q}}_{zero}$ and
${\mathcal{Q}}_{ext}$ are completely equivalent to (1.11) thanks to Remark
2.4, but they are more convenient for the construction of the cut-off function
$\psi_{1}$ introduced in Lemma 4.3. Now, we state our weak Poincaré
inequality.

######  Theorem 4.1 (Weak Poincaré inequality).

Let $\eta\in(0,1)$ and let ${\mathcal{Q}}_{zero}$ and ${\mathcal{Q}}_{ext}$ be
defined as in (4.1). Then there exist $R>1$ and $\vartheta_{0}\in(0,1)$ such
that for any non-negative function $u\in\mathcal{W}$ such that $u\leq M$ in
${\mathcal{Q}}_{1}$ for a positive constant $M$, we have

$\|(u-\vartheta_{0}M)_{+}\|_{L^{2}({\mathcal{Q}}_{1})}\leq
C\left(\|D_{m_{0}}u\|_{L^{2}({\mathcal{Q}}_{ext})}+\|Yu\|_{L^{2}H^{-1}({\mathcal{Q}}_{ext})}\right),$
(4.2)

where $C>0$ is a constant only depending on $Q$.

The notation we consider here needs to be intended in the sense of (1.6). In
particular, we have that $L^{2}H^{-1}({\mathcal{Q}}_{ext})$ is short for

$L^{2}(B_{2^{3}R}\times\ldots\times
B_{2^{2\kappa+1}R}\times(-1-\eta^{2},0],H^{-1}_{x^{(0)}}(B_{2R})),$ (4.3)

where we have split $x=\big{(}x^{(0)},x^{(1)},\ldots,x^{(\kappa)}\big{)}$
according to (1.5).

We observe that a somewhat similar Poincarè inequality was already introduced
by W. Wang and L. Zhang for subsolutions of ultraparabolic equations, see for
instance [38, Lemmas 3.3 & 3.4] and the corresponding lemmas in [39], for the
“strong” notion of weak solution introduced in [33], i.e. $Yu\in L^{2}$.
However, the statement of our Theorem 4.1 differs substantially from the ones
in the articles mentioned above. This is mainly due to the fact that in our
treatment we extend the functional framework introduced in [4] to study the
Kolmogorov-Fokker-Planck operator. As already mentioned, this framework seems
to be the appropriate one when dealing with operator $\mathscr{L}$ and allows
us to forget the equation under study. Moreover, the proof we propose here is
different from the ones in [38, 39], as we avoid using repeatedly the exact
form of the fundamental solution of $\mathscr{L}$ and exploit arguments closer
to the classical theory of parabolic equations (developed, for instance, in
[25]). Additionally, the information obtained through the log-transform is
here summarized in only one weak Poincaré inequality (this is in contrast with
[38, 39], where it is split in several lemmas). Finally, we remark that in
this paper the geometric settings of the main lemmas are simpler than the ones
in [38], [39].

In order to prove Theorem 4.1, the idea is to firstly derive a local Poincaré
inequality in terms of an error function $h$, that is defined as the solution
to the following Cauchy problem

$\left\\{\begin{array}[]{ll}\mathscr{K}h=u\mathscr{K}\psi,&\textit{in
$\mathbb{R}^{N}\times(-{\rho}^{2},0)$}\\\ h=0,&\textit{in
$\mathbb{R}^{N}\times\\{-{\rho}^{2}\\}$}\end{array}\right.$ (4.4)

where $\mathscr{K}$ is the operator defined in (1.3) and $\psi$ is a given
cut-off function, and then to explicitly control the error $h$ through the
$L^{\infty}$ norm of the function $u$ (see Lemma 4.4).

######  Lemma 4.2.

Let ${\mathcal{Q}}_{ext}$ be as defined in (4.1) and let
$\psi:\mathbb{R}^{N+1}\rightarrow[0,1]$ be a $C^{\infty}$ function, with
support in ${\mathcal{Q}}_{ext}$ and such that $\psi=1$ in
${\mathcal{Q}}_{1}$. Then for any $u\in\mathcal{W}$, the following holds

$\|(u-h)_{+}\|_{L^{2}({\mathcal{Q}}_{1})}\leq
C\left(\|D_{m_{0}}u\|_{L^{2}({\mathcal{Q}}_{ext})}+\|Yu\|_{L^{2}H^{-1}({\mathcal{Q}}_{ext})}\right)$
(4.5)

where $h$ is defined in (4.4), $C$ is a constant only depending on
$|{\rho}^{2}|$ and $\|D_{m_{0}}\psi\|_{L^{\infty}({\mathcal{Q}}_{ext})}$, and
the notation we consider needs to be intended in the sense of (4.3).

This local weak Poincaré inequality is an extension to operator $\mathscr{L}$
and to the space $\mathcal{W}$ of the one proved in [13] and a simplification
of the one proved in [38, 39]. Moreover, the result holds true for any
cylinder of the form ${\mathcal{Q}}_{ext}=B_{R_{0}}\times\ldots\times
B_{R_{\kappa}}\times(-{\rho}^{2},0]$, provided that
$\overline{{\mathcal{Q}}_{1}}\subset{\mathcal{Q}}_{ext}$. The proof of Lemma
4.2 is mainly based on the properties of the principal part operator
$\mathscr{K}$ and of $\mathcal{W}$.

Proof of Theorem 4.2. In virtue of Remark 1.1, the function $g:=u\psi$
satisfies the following equation in the sense of distributions

$\mathscr{K}g=u\mathscr{K}\psi+{\text{\rm
div}}_{m_{0}}\widetilde{H}_{1}+\widetilde{H}_{0},\quad\textrm{in
$\mathcal{D}^{\prime}(\mathbb{R}^{N}\times(-{\rho}^{2},0))$},$

where $\widetilde{H}_{1}=(H_{1}+D_{m_{0}}u)\psi$ and
$\widetilde{H}_{0}=H_{0}\psi-H_{1}D_{m_{0}}\psi+\langle
D_{m_{0}}\psi,D_{m_{0}}u\rangle$. Thus, owing to (4.4), we obtain

$\mathscr{K}(g-h)={\text{\rm
div}}_{m_{0}}\widetilde{H}_{1}+\widetilde{H}_{0}=:\widetilde{H},\quad\textrm{in
$\mathcal{D}^{\prime}(\mathbb{R}^{N}\times(-{\rho}^{2},0))$}.$ (4.6)

Now, choosing $2(g-h)_{+}\psi^{2}$ as a test function in (4.6) and integrating
on the domain $D=\mathbb{R}^{N}\times[-{\rho}^{2},0]$, we get

$\displaystyle
2\int_{D}\left|D_{m_{0}}\right|^{2}\psi^{2}+\boxed{4\int_{D}\psi(g-h)_{+}\langle
D_{m_{0}}((g-h)_{+}),D_{m_{0}}(\psi)\rangle}_{A}-\boxed{2\int_{D}(g-h)_{+}\psi^{2}Y((g-h)_{+})}_{B}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\boxed{2\int_{D}(g-h)_{+}\psi^{2}D_{m_{0}}\cdot\widetilde{H}_{1}}_{C}+\boxed{2\int_{D}(g-h)_{+}\psi^{2}\widetilde{H}_{0}}_{D}=0.$
(4.7)

We estimate the boxed term $A$ by applying Young’s inequality (3.10) and
choosing $\varepsilon=1$. As far as we are concerned with the boxed term $B$,
we rewrite it as

$\displaystyle-\boxed{2\int_{D}(g-h)_{+}\psi^{2}Y((g-h)_{+})}_{B}$
$\displaystyle=-\int_{D}\psi^{2}Y_{0}((g-h)_{+}^{2})+\int_{D}\psi^{2}\partial_{t}((g-h)_{+}^{2})$
$\displaystyle=-\int_{D}[\psi^{2}Y_{0}((g-h)_{+}^{2})+\int_{D}\partial_{t}((g-h)_{+}^{2}\psi^{2})-(g-h)_{+}^{2}\partial_{t}(\psi^{2})]$
$\displaystyle=\int_{D}(g-h)_{+}^{2}Y(\psi^{2})+\int_{D}\partial_{t}((g-h)_{+}^{2}\psi^{2}),$

where in the first line we defined $Y_{0}=\langle Bx,D\rangle$, in the second
line we used the equality
$\partial_{t}((g-h)_{+}^{2})\psi^{2}=\partial_{t}((g-h)_{+}^{2}\psi^{2})-\partial_{t}(\psi^{2})(g-h)_{+}^{2}$
and in the third we integrated by parts the term involving $Y_{0}$. Finally,
we take care of the boxed term $C$ and $D$ using Young’s inequality as follows

$\displaystyle\boxed{2\int_{D}(g-h)_{+}\psi^{2}D_{m_{0}}\cdot\widetilde{H}_{1}}_{C}$
$\displaystyle=-2\int_{D}\langle
D_{m_{0}}((g-h)_{+}\psi^{2}),\widetilde{H}_{1}\rangle$
$\displaystyle\leq\frac{1}{2}\|\psi
D_{m_{0}}(g-h)_{+}\|_{L^{2}(D)}+\frac{1}{2}\|(g-h)_{+}D_{m_{0}}\psi\|_{L^{2}(D)}+10\|\widetilde{H}_{1}\|_{L^{2}(D)},$
$\displaystyle\boxed{2\int_{D}(g-h)_{+}\psi^{2}\widetilde{H}_{0}}_{D}$
$\displaystyle\leq
2\varepsilon\|(g-h)_{+}\|_{L^{2}(D)}+\frac{1}{2\varepsilon}\|\widetilde{H}_{0}\|_{L^{2}(D)}.$

Combining the previous estimates, for every $T\in(-{\rho}^{2},0)$ and
$\varepsilon>0$ to be chosen later, we rewrite (4.7) as

$\displaystyle\int_{-{\rho}^{2}}^{T}\int_{D}\partial_{t}((g-h)_{+}^{2}\psi^{2})dxdt+2\int_{D}\left|D_{m_{0}}(g-h)_{+}\right|^{2}\psi^{2}+\int_{D}(g-h)_{+}^{2}Y(\psi^{2})$
$\displaystyle\qquad\qquad\leq\frac{1}{2}\|D_{m_{0}}(g-h)_{+}\psi\|_{L^{2}(D)}+10\|\widetilde{H}_{1}\|_{L^{2}(D)}+2\varepsilon\|(g-h)_{+}\|_{L^{2}(D)}+\frac{1}{2\varepsilon}\|\widetilde{H}_{0}\|_{L^{2}(D)}.$

We now apply the fundamental theorem of calculus to the term involving the
time derivative and we infer

$\displaystyle\int_{D}((g-h)_{+}^{2}\psi^{2})(x,T)dx+2\int_{D}\left|D_{m_{0}}(g-h)_{+}\right|^{2}\psi^{2}+\int_{D}(g-h)_{+}^{2}Y(\psi^{2})$
(4.8)
$\displaystyle\quad\leq\frac{1}{2}\|D_{m_{0}}(g-h)_{+}\psi\|_{L^{2}(D)}+10\|\widetilde{H}_{1}\|_{L^{2}(D)}+2\varepsilon\|(g-h)_{+}\|_{L^{2}(D)}+\frac{1}{2\varepsilon}\|\widetilde{H}_{0}\|_{L^{2}(D)}.$

We then integrate in $T$ from $-{\rho}^{2}$ to 0 and we obtain

$\displaystyle\|(g-h)_{+}\psi\|_{L^{2}(D)}+{\rho}^{2}\int_{D}(g-h)_{+}^{2}Y(\psi^{2})$
(4.9)
$\displaystyle\quad\leq-\frac{3}{2}{\rho}^{2}\|D_{m_{0}}(g-h)_{+}\psi\|_{L^{2}(D)}+10{\rho}^{2}\|\widetilde{H}_{1}\|_{L^{2}(D)}+2{\rho}^{2}\varepsilon\|(g-h)_{+}\|_{L^{2}(D)}+\frac{{\rho}^{2}}{2\varepsilon}\|\widetilde{H}_{0}\|_{L^{2}(D)}$
$\displaystyle\quad\leq
10{\rho}^{2}\|\widetilde{H}_{1}\|_{L^{2}(D)}+2{\rho}^{2}\varepsilon\|(g-h)_{+}\|_{L^{2}(D)}+\frac{{\rho}^{2}}{2\varepsilon}\|\widetilde{H}_{0}\|_{L^{2}(D)}.$

We later observe that $(g-h)_{+}\psi$ equals $(u-h)_{+}$ and $Y(\psi^{2})$
equals 0 in ${\mathcal{Q}}_{1}$. In addition, the following estimates hold

$\displaystyle\|\widetilde{H}_{0}\|_{L^{2}(D)}$
$\displaystyle\leq\|H_{0}\|_{L^{2}({\mathcal{Q}}_{ext})}+\|D_{m_{0}}\psi\|_{L^{\infty}({\mathcal{Q}}_{ext})}\left(\|D_{m_{0}}u\|_{L^{2}({\mathcal{Q}}_{ext})}+\|H_{1}\|_{L^{2}({\mathcal{Q}}_{ext})}\right),$
(4.10) $\displaystyle\|\widetilde{H}_{1}\|_{L^{2}(D)}$
$\displaystyle\leq\|H_{1}\|_{L^{2}({\mathcal{Q}}_{ext})}+\|D_{m_{0}}u\|_{L^{2}({\mathcal{Q}}_{ext})}.$

By combining (4.8), (4.9) and (4.10) and choosing
$\varepsilon=\frac{1}{4{\rho}^{2}}$ the claim is proved. $\hfill\square$

Given the local Poincaré inequality proved in Lemma 4.2, we just need to
estimate the function $h$ defined in (4.4) in order to complete the proof of
Theorem 4.1. In particular, our aim is to show that the error function $h$ is
bounded from above by $\vartheta_{0}M$, where $\vartheta_{0}\in(0,1)$ is a
constant only depending on $Q$, ${\lambda}$ and $\Lambda$. In order to prove
such result, we first need to explicitly construct an appropriate cut-off
function, that differs from the one considered in [13, Lemma 3.3] due to the
more involved structure of our drift term $Y$. We note that our construction
of the suitable cut-off function is constructive, in contrast with the one
proposed in [13].

######  Lemma 4.3.

Given $\eta\in(0,1]$ and $T\in(0,\eta^{2})$, there exists a smooth function
$\psi_{1}:\mathbb{R}^{N}\times[-1-\eta^{2},0]$, supported in
$\\{(x,t):|x_{j}|\leq 2^{\alpha_{j}},j=1,\ldots,N,t\in[-1-\eta^{2},0]\\}$,
equal to 1 in ${\mathcal{Q}}_{1}$, and such that the following conditions hold

$\begin{split}Y\psi_{1}&\leq 0\quad\textrm{everywhere}\\\
Y\psi_{1}&\leq-1\quad\textrm{if $t\in(-1-\eta^{2},-1-T]$}.\end{split}$ (4.11)

###### Proof.

Let us consider the cut-off function $\chi\in C^{\infty}([0,+\infty))$ defined
by

$\chi(s)=\left\\{\begin{array}[]{ll}0,&\textrm{if $s>\frac{2}{\sqrt{2}}$},\\\
1,&\textrm{if $0\leq s\leq C+1$},\end{array}\right.\quad\chi^{\prime}\leq 0,$
(4.12)

where $C>1$ is a constant we shall specify later on. In addition, we consider
a smooth function $\chi_{t}:[-1-\eta^{2},0]\rightarrow[0,1]$ equal to 1 in
$[-1,0]$, with $\chi_{t}(-1-\eta^{2})=0$, $\chi_{t}^{\prime}\geq 0$ in
$[-1-\eta^{2},0]$ and $\chi_{t}^{\prime}=1$ in $[-1-\eta^{2},-1-T]$. Now,
setting

$\displaystyle\chi_{0}(x,t)=\chi\left(\sum_{j=m_{0}}^{N}\frac{2x_{j}^{2}}{2^{2\alpha_{j}}\sqrt{2}}-C\,t\right),$

we define the cut-off function $\psi_{1}$ as follows

$\displaystyle\psi_{1}(x,t)=\chi(\|(x_{1},\ldots,x_{m_{0}})\|)\chi_{0}(x,t)\chi_{t}(t).$

We only have to check that conditions (4.11) hold, as the other desired
properties immediately follow from the definition of $\psi_{1}$. To this end,
we compute the following derivative

$\displaystyle
Y\chi_{0}=\chi^{\prime}((...))\left[\sum_{i=1}^{N}\sum_{j>m_{0}}2x_{i}b_{ij}x_{j}2^{-2\alpha_{j}-1/2}+C\right],$

where $(...)$ denotes
$\left(\sum_{j=m_{0}}^{N}\frac{2x_{j}^{2}}{2^{2\alpha_{j}}\sqrt{2}}-C\,t\right)$.
It can be shown (see [39]) that it is well defined a certain constant $C>1$
such that

$\displaystyle
C\geq\sum_{i=1}^{N}\sum_{j>m_{0}}2x_{i}b_{ij}x_{j}2^{-2\alpha_{j}-1/2}.$

Thus, with such a choice of $C$ and keeping in mind that
$\chi_{t}^{\prime}\geq 0$ in $[-1-\eta^{2},0]$ and $\chi_{t}^{\prime}=1$ in
$[-1-\eta^{2},-1-T]$, we have

$\displaystyle
Y\psi_{1}=\chi\,\chi_{t}\,Y\chi_{0}-\chi\,\chi_{0}\,\chi^{\prime}_{t}\leq
0\qquad\textrm{everywhere},$ $\displaystyle
Y\psi_{1}=\chi\,\chi_{t}\,Y\chi_{0}-\chi\,\chi_{0}\,\chi^{\prime}_{t}\leq-1\quad\textrm{if
$t\in(-1-\eta^{2},-1-T]$}.$

∎

Thus, we are now in a position to state and prove the following result
regarding the control of the localization term $h$ defined in (4.4).

######  Lemma 4.4.

Let $\eta\in(0,1]$ and let ${\mathcal{Q}}_{ext}$ be as defined in (4.1). then
there exist $R=R(Q,\eta)>1$, $\vartheta_{0}=\vartheta_{0}(Q,\eta)\in(0,1)$ and
a $C^{\infty}$ cut-off function $\psi:\mathbb{R}^{N+1}\rightarrow[0,1]$, with
support in ${\mathcal{Q}}_{ext}$ and equal to 1 in ${\mathcal{Q}}_{1}$, such
that for all $u\in\mathcal{W}$ non-negative bounded functions defined on
${\mathcal{Q}}_{ext}$, the function $h$ solution to the Cauchy problem (4.4)
with ${\rho}^{2}=1+\eta^{2}$ satisfies

$h\leq\vartheta_{0}\|u\|_{L^{\infty}({\mathcal{Q}}_{ext})}.$ (4.13)

###### Proof.

We assume that $u$ is not identically vanishing in ${\mathcal{Q}}_{ext}$.
Indeed, if $u=0$ in ${\mathcal{Q}}_{ext}$, then $h=0$ and inequality (4.13) is
trivially satisfied. Moreover, we can reduce to the case of a function $u$
with $L^{\infty}$\- norm equal to 1 by taking
$u/\|u\|_{L^{\infty}({\mathcal{Q}}_{ext})}$.

We now fix $T=\eta^{2}/2$; then,

$|{\mathcal{Q}}_{zero}\cap\\{t\leq-1-T\\}|=\frac{1}{2}|{\mathcal{Q}}_{zero}|.$

We now consider the cut-off function

$\displaystyle\psi(x,t)=\psi_{1}(x/R,t),$

whew $R>1$ is a constant we will specify later and $\psi_{1}$ is given by
Lemma 4.3. We observe that, by definition of $\psi_{1}$, $\psi$ is supported
in ${\mathcal{Q}}_{ext}$ and equal to 1 in $\\{(x,t):|x_{j}|\leq
R,j=1,\ldots,N,t\in(-1,0]\\}$. In addition, it satifies

$\displaystyle\mathscr{K}\psi(x,t)=R^{-2}\Delta_{m_{0}}\psi_{1}(x/R,t)+Y\psi_{1}(x/R,t).$

Thus, in virtue of (4.4), we have

$\displaystyle\mathscr{K}(h-\psi)=\frac{u-1}{R^{2}}\Delta_{m_{0}}\psi_{1}(x/R,t)+(u-1)Y\psi_{1}(x/R,t)$

and we can write the difference $h-\psi$ as

$\displaystyle h-\psi=E_{R}+N_{R},$ (4.14)

where $E_{R}$ and $N_{R}$ are solutions in
$\mathbb{R}^{N}\times(-1-\eta^{2},0)$ to the following Cauchy problems

$\displaystyle\mathscr{K}E_{R}$
$\displaystyle=\frac{u-1}{R^{2}}\Delta_{m_{0}}\psi_{1}(x/R,t),$
$\displaystyle\mathscr{K}N_{R}$ $\displaystyle=(u-1)Y\psi_{1}(x/R,t),$

with $E_{R}=P_{R}=0$ at the initial time. We first focus on the term involving
$E_{R}$, and we remark that

$\displaystyle\mathscr{K}E_{R}\leq\frac{C^{\prime}}{R^{2}},$ (4.15)

where $C^{\prime}=\|\Delta_{m_{0}}\psi_{1}\|_{L^{\infty}}$ is a constant only
depending on $Q$, $\lambda$, $\Lambda$ and $\eta$. As far as the term
involving $N_{R}$ is concerned, we observe that, owing to $Y\psi_{1}\leq-1$
for $t\in(-1-\eta^{2},-1-T)$, we have

$\displaystyle\mathscr{K}N_{R}\leq-\mathbb{I}_{\mathcal{Z}},\quad\textrm{in
$\mathbb{R}^{N}\times(-1-\eta^{2},0)$},$

where $\mathcal{Z}:={\mathcal{Q}}_{zero}\cap\\{t\leq-1-T\\}$. Let $N$ such
that $\mathscr{K}N=-\mathbb{I}_{\mathcal{Z}}$ in
$\mathbb{R}^{N}\times(-1-\eta^{2},0)$ and $N=0$ at the initial time
$t=-1-\eta^{2}$. Then, the maximum principle [5] for the principle part
operator $\mathscr{K}$ yields

$\displaystyle P\leq N_{R},\quad\textrm{in ${\mathcal{Q}}_{1}$}.$

We now represent $N$ in using the fundamental solution $\Gamma$ of
$\mathscr{K}$ and we infer

$\displaystyle
N(z)=\int\Gamma(z,\zeta)\left(-\mathbb{I}_{\mathcal{Z}}(\zeta)\right)d\zeta\leq-\frac{1}{2}m|{\mathcal{Q}}_{zero}|=:-\delta_{0},$

where
$m=\min_{{\mathcal{Q}}_{1}\times{\mathcal{Q}}_{zero}\cap\\{t\leq-1-T\\}}\Gamma(z,\zeta)$.
As a consequence,

$\displaystyle N_{R}\leq-\delta_{0},$ (4.16)

for a constant $\delta_{0}$ only depending on $Q$ and $\eta$. Using estimates
(4.15) and (4.16) in (4.14), we finally obtain

$\displaystyle h\leq 1-\delta_{0}+\frac{C^{\prime}}{R^{2}}.$

We now observe that for $R$ large enough we have $C/R^{2}\leq\delta_{0}/2$.
Thus, setting $\vartheta_{0}=1-\delta_{0}/2<1$, we get the desired inequality
(4.13). ∎

##  5\. Main results

This section is devoted to the proof of our main results. The approach we
present here is an extension of the method inspired by [21, 22] and then
followed by Guerand and Imbert in [13] for the kinetic Kolmogorov-Fokker-
Planck equation. In particular, we remark that an analogous approach based on
a weak Poincaré type inequality was firstly introduced by Wang and Zhang in
[38, 39] for the Kolmogorov equation $\mathscr{L}u=0$ under the assumption
$Yu\in L^{2}$, and thus with a stronger notion of weak solution, and with a
different $\log-$transform. The main advantage of our approach is that it only
relies on the structure of the function space $\mathcal{W}$ to which every
weak solution belongs to and on the non-Euclidean geometrical structure
presented in Section 2 behind the operator $\mathscr{L}$. To our knowledge, it
is the first time that the study of the weak regularity theory is carried on
replacing the classical $L^{Q+2}$ integrability assumptions for the lower
order coefficients $b$, $c$ and the non zero right-hand side $f$ with (H3).

###  5.1. Weak Harnack inequality

First of all we address the proof of the weak Harnack inequality (Theorem 1.3)
that relies on combining the fact that super-solutions to (1.1) expand
positivity along times (Lemma 5.1) with the covering argument presented in
Appendix B.

The derivation of the weak Harnack inequality in the present paper from the
expansion of positivity follows very closely the reasoning in [19] and extends
the results presented in [13] for the Kolmogorov-Fokker- Planck case. For
reader’s convenience, we here state (and adapt to our more involved case) the
results contained in [13, Section 4], sketching their proofs only when they
differ from the ones contained in the aforementioned paper.

We observe that, in contrast with parabolic equations, it is not possible to
apply a classical Poincaré inequality in the spirit of [29]. Indeed, in our
case there is a positive quantity replacing the avarage in the usual Poincarè
inequality (see the statement of Theorem 4.1). Following [13] we circumvent
this difficulty by estabilishing a weak expansion of positivity of super-
solutions to (1.1). More precisely, given a small cylider
${\mathcal{Q}}_{pos}$ lying in the past of ${\mathcal{Q}}_{1}$ (see Definition
(5.1)), we show that the positivity of a non-negative super solution $u$ lying
above 1 in a ”big” part of ${\mathcal{Q}}_{pos}$ is spread to the whole
${\mathcal{Q}}_{1}$ (see Lemma 5.1). In other words, a positivity in measure
in a smaller cylinder ${\mathcal{Q}}_{pos}$ is transformed into a pointwise
positivity in a bigger cylinder ${\mathcal{Q}}_{1}$. We emphasize that such a
weak expansion of positivity was already proved in [11] thanks to an
intermediate value lemma, following De Giorgi’s original proof, but the
argument was not constructive and specific for the Fokker-Planck equation
case.

Lastly, we mention that Moser [29] and Trudinger [35] proved a weak Harnack
inequality in the spirit of Theorem 1.3 in the setting of parabolic equations.
We also recall that Di Benedetto and Trudinger [7] proved that non-negative
functions in the elliptic De Giorgi class, which corresponds to super-
solutions to elliptic equations, satisfy a weak Harnack inequality. Moreover,
let us emphasize that in [37] it is proved a weak Harnack for the
corresponding parabolic case, i.e. for functions in the parabolic De Giorgi’s
class. We conclude by observing that quantitative interior Hölder regularity
estimates for functions in the parabolic De Giorgi class (and for parabolic
equations with rough coefficients) can be found in [12].

We first study how equation (1.1) spreads positivity of super-solutions. More
precisely, we state the upcoming Lemma 5.1, given in terms of the cylinders

$\displaystyle{\mathcal{Q}}_{pos}$ $\displaystyle=B_{\theta}\times
B_{\theta^{3}}\times\ldots\times
B_{\theta^{2\kappa+1}}\times(-1-\theta^{2},-1],$ (5.1)
$\displaystyle\widetilde{{\mathcal{Q}}}_{ext}$ $\displaystyle=B_{3R}\times
B_{3^{3}R}\times\ldots\times B_{3^{2\kappa+1}R}\times(-1-\theta^{2},0],$

where $R=R(\theta,Q,\lambda,\Lambda)$ is the constant given by Lemma 4.4 and
$\theta\in(0,1]$ is a parameter we will choose later on. In particular,
$\theta$ will be chosen such that the stacked cylinder
$\overline{\mathcal{Q}}_{pos}^{m}$ (see definition A.1) is contained in
${\mathcal{Q}}_{1}$. To stress the dependence of $R$ on $\theta$, we will
sometimes write $R_{\theta}$ instead of $R$.

######  Lemma 5.1.

Let $\theta\in(0,1]$ and ${\mathcal{Q}}_{pos}$,
$\widetilde{{\mathcal{Q}}}_{ext}$ be the cylinders defined in (5.1). Then
there exist a small constant
$\eta_{0}=\eta_{0}(\theta,Q,\lambda,\Lambda)\in(0,1)$ such that for any non-
negative super-solution $u$ of (1.1) in some cylindrical open set
$\Omega\supset\widetilde{{\mathcal{Q}}}_{ext}$ such that $\left|\\{u\geq
1\\}\cap{\mathcal{Q}}_{pos}\right|\geq\frac{1}{2}\left|{\mathcal{Q}}_{pos}\right|$,
we have $u\geq\eta_{0}$ in ${\mathcal{Q}}_{1}$.

###### Proof.

For the sake of completeness we hereby state a sketch of the proof, that is an
extension of [13, Lemma 4.1]. We consider $g=G(u+{\varepsilon})$, where $G$ is
the convex function defined in [13, Lemma 2.1]. In particular, $G$ is such
that

  * •

$G^{\prime\prime}\geq(G^{{}^{\prime}})^{2}$ and $G^{{}^{\prime}}\leq 0$ in
$]0,+\infty[$,

  * •

$G$ is supported in $]0,1]$,

  * •

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

  * •

$-G^{{}^{\prime}}(t)\leq\frac{1}{t}$ for $t\in]0,\frac{1}{4}]$.

Thus, we have that
$|G^{\prime}(u+{\varepsilon})|\leq|G^{\prime}({\varepsilon})|\leq{\varepsilon}^{-1}$
as $u$ is non-negative. Moreover, adapting [13, Lemma 2.2] to our case, we
find that $g$ is a non-negative sub-solution to (1.1) with $f$ replaced by
$fG^{\prime}(u+{\varepsilon})$. This implies in particular that the drift term
$Yg$ is bounded, i.e. $Yg\leq{\text{\rm div}}(ADg)+\langle
b,Dg\rangle+cg+\varepsilon^{-1}\|f\|_{{L^{q}}(\widetilde{{\mathcal{Q}}}_{ext})}$.
The rest of the proof follows very closely the one of [13, Lemma 4.1], with
the only difference that we consider our Theorem 3.1 instead of the classical
$L^{2}-L^{\infty}$ estimate and the weak Poincarè inequality 4.1 instead of
[13, Theorem 1.4]. ∎

As a straightforward consequence of Lemma 5.1 we have the following result,
whose proof is obtained reasoning exactly as in [13, Lemma 4.2].

######  Lemma 5.2.

Let $m\geq 3$ and let $R$ be the constant given in Lemma 5.1 for $\theta\leq
m^{-1/2}$. Then there exists a constant $M=M(m,Q,\lambda,\Lambda)>1$ such that
for any non-negative super-solution $u$ to (1.1) with $f$ equal to 0
satisfying $\left|\\{u\geq
M\\}\cap{\mathcal{Q}}_{1}\right|\geq\frac{1}{2}|{\mathcal{Q}}_{1}|$, we have
$u\geq 1$ in $\overline{{\mathcal{Q}}}_{1}^{m}$ (see (A.1)).

Before proving the weak Harnack inequality, we need to show that we can spread
positivity along ”suitable” cylinders. More precisely, recalling the
definition of the open ball in (1.10), we set

${\mathcal{Q}}_{+}=B_{\omega}\times B_{\omega^{3}}\times\ldots\times
B_{\omega^{2\kappa+1}}\times(-\omega^{2},0],\quad{\mathcal{Q}}_{-}=B_{\omega}\times
B_{\omega^{3}}\times\ldots\times
B_{\omega^{2\kappa+1}}\times(-1,-1+\omega^{2}],$ (5.2)

where $\omega$ is a small positive constant. In particular, we will choose
$\omega$ small enough so that, when expanding positivity from a given cylinder
${\mathcal{Q}}_{r}(z_{0})$ in the past, the union of the stacked cylinders
where the positivity is spread includes ${\mathcal{Q}}_{+}$. Moreover, we will
choose the radius $R_{0}$ in the statement of Theorem 1.3 so that Lemma 5.1
can be applied to every stacked cylinder. The two previous statements are
specified in Lemma B.1. The staking cylinders Lemma B.1, combined with Lemma
5.1, implies the following result regarding the expansion of positivity for
large times.

In the sequel, we will largely use the cylinders ${\mathcal{Q}}_{r}[k]$, for
$k=1,\ldots,N$ and ${\mathcal{Q}}_{R_{N+1}}[N+1]$, whose definition and
properties are presented in Appendix B.

######  Lemma 5.3.

Let $R_{1/2}$ be the constant given by Lemma 5.1 for $\theta=1/2$ and let $u$
be any non-negative super-solution to (1.1) with $f=0$ in
${\Omega}\supset{\mathcal{Q}}$ such that $\left|\\{u\geq
M\\}\cap{\mathcal{Q}}_{r}(z_{0})\right|\geq\frac{1}{2}\left|{\mathcal{Q}}_{r}(z_{0})\right|$
for some $M>0$ and for some cylinder
${\mathcal{Q}}_{r}(z_{0})\subset{\mathcal{Q}}_{-}$. Then there exists a
positive constant $p_{0}$, only depending on $Q$, $\lambda$, $\Lambda$, such
that

$\displaystyle u\geq M\left(\frac{r^{2}}{4}\right)^{p_{0}},\quad\textit{in
${\mathcal{Q}}_{+}$}.$ (5.3)

###### Proof.

We apply Lemma 5.1 for $\theta=\frac{1}{2}$ to the function $u/M$, with
${\mathcal{Q}}_{r}(z_{0})$ and ${\mathcal{Q}}_{r}[1]$ taking the role of
${\mathcal{Q}}_{pos}$ and ${\mathcal{Q}}_{1}$ (this is achieved through a
rescaling argument) and obtain $u/M\geq\eta_{0}$ in ${\mathcal{Q}}_{r}[1]$. We
then apply it to $u/(M\eta_{0})$ and get $u\geq M\eta_{0}^{2}$ in
${\mathcal{Q}}_{r}[2]$. Reasoning by induction on $k=1,\ldots,N$ we infer
$u\geq M\eta_{0}^{k}$ in ${\mathcal{Q}}[k]$.

By exploiting Lemma 5.1 again, we get $u\geq M\eta_{0}^{N+1}$ in
${\mathcal{Q}}_{R_{N+1}}[N+1]$, which implies that the same inequality holds
true in ${\mathcal{Q}}_{+}$. As $T_{N}\leq-t_{0}<1$, we have in particular
$4^{N}r^{2}\leq 1$. Picking $p_{0}>0$ so that
$\eta_{0}=\left(\left(\frac{1}{4}\right)^{\frac{N}{N+1}}\right)^{p_{0}}$, we
finally obtain

$u\geq
M\left(\left(\left(1/4\right)^{\frac{N}{N+1}}\right)^{N+1}\right)^{p_{0}}=M\left(\left(1/4\right)^{N}\right)^{p_{0}}\geq
M\left(r^{2}/4\right)^{p_{0}},$

which concludes the proof. ∎

From now on we will assume $\omega<1/\sqrt{2\kappa+1}$, where ${\kappa}$ is
defined in (1.5). We are in a position to prove the main result of this
Section, Theorem 1.3.

Proof of Theorem 1.3. We start the proof by fixing the parameters $\omega$ and
$R_{0}$ in order to select the appropriate geometric setting. More precisely,
we choose $\omega$ so that we capture ${\mathcal{Q}}_{+}$ when applying Lemma
B.1, namely we fix $\omega<\frac{1}{\sqrt{2\kappa+1}}$. In addition, we choose
the radius $R_{0}$ so that the stacked cylinders do not leak out of
${\mathcal{Q}}^{0}$, i.e. $R_{0}\geq 6\left(2\kappa+1\right)R_{1/2}$, where
$R_{1/2}$ is the constant given by Lemma 5.1 when $\theta=1/2$. As we want to
apply Lemma 5.2 to cylinders contained in ${\mathcal{Q}}_{-}$, we also assume
$R_{0}\geq 3(2\kappa+1)R_{m^{-1/2}}m^{(2\kappa+1)/2}\omega^{2\kappa+1}$, where
$R_{m^{-1/2}}$ is the constant given by Lemma 5.1 for $\theta=m^{-1/2}$.

Our aim is to reduce ourselves to the case where

$\displaystyle\inf_{{\mathcal{Q}}_{+}}u\leq 1,\quad\textit{and}\quad f=0.$
(5.4)

On one hand, if $\inf_{{\mathcal{Q}}_{+}}u>1$ we can simply consider
$\bar{u}=u/\left(\inf_{{\mathcal{Q}}_{+}}u+1\right)$ and reduce to the case
where $\inf_{{\mathcal{Q}}_{+}}u\leq 1$. On the other hand, if $f\neq 0$ and
$c=0$ we have that $\widetilde{u}:=u+\vartheta
t\|f\|_{L^{q}({\mathcal{Q}}^{0})}$ is a super-solution to equation (1.1) with
source term equal to 0, provided that we choose $\vartheta$ such that

$\vartheta=R_{0}^{-\frac{Q+2}{q}}.$ (5.5)

Indeed, exploiting the fact that $u$ is non-negative super-solution to (1.1)
in ${\mathcal{Q}}^{0}$, we infer

$\displaystyle\int_{{\mathcal{Q}}^{0}}-\langle
AD\widetilde{u},D{\varphi}\rangle+{\varphi}Y\widetilde{u}+\langle
b,D\widetilde{u}\rangle{\varphi}$
$\displaystyle\quad=\int_{{\mathcal{Q}}^{0}}-\langle
ADu,D{\varphi}\rangle+{\varphi}Yu-{\varphi}\vartheta\|f\|_{L^{q}({\mathcal{Q}}^{0})}+\langle
b,Du\rangle{\varphi}$ (5.6)
$\displaystyle\quad\leq\int_{{\mathcal{Q}}^{0}}f{\varphi}-\int_{{\mathcal{Q}}^{0}}{\varphi}\vartheta\|f\|_{L^{q}({\mathcal{Q}}^{0})}.$

We now observe that the last line in (5.1) can be estimated as follows

$\displaystyle\int_{{\mathcal{Q}}^{0}}f{\varphi}-\int_{{\mathcal{Q}}^{0}}{\varphi}\vartheta\|f\|_{L^{q}({\mathcal{Q}}^{0})}$
$\displaystyle\leq\left(R_{0}^{Q+2}\right)^{1-1/q}\|f\|_{L^{q}({\mathcal{Q}}^{0})}\|{\varphi}\|_{L^{\infty}({\mathcal{Q}}^{0})}-\vartheta\|f\|_{L^{q}({\mathcal{Q}}^{0})}R_{0}^{Q+2}\|{\varphi}\|_{L^{\infty}({\mathcal{Q}}^{0})}$
$\displaystyle=\|f\|_{L^{q}({\mathcal{Q}}^{0})}R_{0}^{Q+2}\|{\varphi}\|_{L^{\infty}({\mathcal{Q}}^{0})}\left(\left(R_{0}^{Q+2}\right)^{1-1/q}-\vartheta
R_{0}^{Q+2}\right),$

which is equal to 0 when $\vartheta=R_{0}^{-\frac{Q+2}{q}}$. Thus,
$\widetilde{u}$ is a super-solution of equation (1.1) with $f=0$ and the weak
Harnack inequality for $\widetilde{u}$ implies the one for $u$. Lastly, if
$c\neq 0$ the reasoning follows by replacing the value of $\vartheta$ in (5.5)
by

$\vartheta=\frac{1}{(R_{0}^{Q+2})^{1/q}-\|c\|_{L^{q}({\mathcal{Q}}^{0})}}.$

We now want to prove that for all $k\in\mathbb{N}$, the following inequality
holds

$\displaystyle\left|\\{u>M^{k}\\}\cap{\mathcal{Q}}_{1}\right|\leq\widetilde{C}(1-\widetilde{\mu})^{k},$
(5.7)

for some constants $\widetilde{\mu}\in(0,1)$ $M>1$ and $\widetilde{C}>0$ that
only depend on $Q$, $\lambda$ and $\Lambda$. The proof of this fact is carried
out by induction. For $k=1$ it is sufficient to choose
$\widetilde{\mu}\leq\frac{1}{2}$ and $\widetilde{C}$ such that
$|{\mathcal{Q}}_{-}|\leq\frac{1}{2}\widetilde{C}$. We now assume that (5.7)
holds true for $k\geq 1$ and we prove it for $k+1$. To this end, we consider
the sets

$\displaystyle E:=\\{u>M^{k+1}\\}\cap{\mathcal{Q}}_{-},\qquad
F:=\\{u>M^{k}\\}\cap{\mathcal{Q}}_{1}.$ (5.8)

We observe that $E$ and $F$ satisfy the assumptions of Corollary A.3 with
${\mathcal{Q}}_{1}$ replaced by ${\mathcal{Q}}_{-}$ and $\mu=1/2$. Indeed, by
definition $E$ and $F$ are bounded measurable sets such that
$E\subset{\mathcal{Q}}_{-}\cap F$. We now consider a cylinder
${\mathcal{Q}}={\mathcal{Q}}_{r}(z)\subset{\mathcal{Q}}_{-}$ such that
$|{\mathcal{Q}}\cap E|>\frac{1}{2}|{\mathcal{Q}}|$, i.e.

$\displaystyle|\\{u>M^{k+1}\\}\cap{\mathcal{Q}}|>\frac{1}{2}|{\mathcal{Q}}|.$

We show that $r$ needs to be small, that is to say $r$ is less than some
parameter $r_{0}=r_{0}(Q,\lambda,\Lambda,k)$. Indeed, applying Lemma 5.3 to
$u$, we obtain $u\geq M^{k+1}(r^{2}/4)^{p_{0}}$ in ${\mathcal{Q}}_{+}$. Thus,
owing to $\inf_{{\mathcal{Q}}_{+}}u\leq 1$, we infer $1\geq
M^{k+1}(r^{2}/4)^{p_{0}}$ and therefore it is sufficient to choose $r_{0}\leq
2M^{-k-1/2p_{0}}$. In order to apply Corollary A.3, we are left with proving
that $\overline{{\mathcal{Q}}}^{m}\subset F$, which holds true if
$\overline{{\mathcal{Q}}}^{m}\subset\\{u>M^{k}\\}$. To this end, we apply
Lemma 5.2 to $u/M^{k}$ after rescaling the cylinder ${\mathcal{Q}}$ in
${\mathcal{Q}}_{1}$.

In virtue of Corollary A.3, there exist $c_{\rm is}\in(0,1)$ and $C_{\rm
is}>0$ such that

$\displaystyle|E|=\left|\\{u>M^{k+1}\\}\cap{\mathcal{Q}}_{-}\right|$
$\displaystyle\leq\frac{m+1}{m}\left(1-\frac{c_{\rm
is}}{2}\right)\left(\left|\\{u>M^{k}\\}\cap{\mathcal{Q}}_{1}\right|+C_{\rm
is}mr_{0}^{2}\right)$ $\displaystyle\leq\left(1-\frac{c_{\rm
is}}{4}\right)\left(\left|\\{u>M^{k}\\}\cap{\mathcal{Q}}_{1}\right|+C_{\rm
is}mr_{0}^{2}\right),$

provided that we chose $m\in\mathbb{N}$ so that
$\frac{m+1}{m}\left(1-\frac{c_{\rm is}}{2}\right)\leq 1-\frac{c_{\rm is}}{4}$.
Thanks to the induction assumption and our choice of $r_{0}$ we get

$\displaystyle|E|$ $\displaystyle\leq\left(1-\frac{c_{\rm
is}}{4}\right)\left(\widetilde{C}(1-\widetilde{\mu})^{k}+C_{\rm
is}mr_{0}^{2}\right)$ $\displaystyle\leq\left(1-\frac{c_{\rm
is}}{4}\right)\left(\widetilde{C}(1-\widetilde{\mu})^{k}+C_{\rm
is}mM^{-\frac{k+1}{p_{0}}}\right).$

Picking then $\widetilde{\mu}$ small enough so that
$M^{-1/p_{0}}\leq(1-\widetilde{\mu})$ and $\widetilde{\mu}\leq\frac{c_{\rm
is}}{4}$, we obtain

$\displaystyle|E|$ $\displaystyle\leq\widetilde{C}\left(1-\frac{c_{\rm
is}}{4}\right)(1-\widetilde{\mu})^{k}\left(1+\widetilde{C}^{-1}mM^{-\frac{1}{p_{0}}}\right)$
$\displaystyle\leq\widetilde{C}(1-\widetilde{\mu})^{k+1}\left(1+\widetilde{C}^{-1}mM^{-\frac{1}{p_{0}}}\right).$

Picking $\widetilde{C}$ large enough so that
$\left(1+\widetilde{C}^{-1}mM^{-\frac{1}{p_{0}}}\right)\leq 2$ we conclude the
proof of (5.7). By extending estimate (5.7) to the continuous case (i.e.
$k\in\mathbb{R}$ and $k\geq 1$) and applying the layer cake formula to
$\int_{{\mathcal{Q}}_{-}}f^{p}$ for some exponent $p$, we obtain that
$\int_{{\mathcal{Q}}_{-}}f^{p}$ is bounded from above by a constant that only
depends on $Q$, $\lambda$ and $\Lambda$. $\hfill\square$

###  5.2. Harnack inequality and local Hölder continuity

Proof of Theorem 1.4. The full Harnack inequality is a direct consequence of
the combination of the local boundedness of weak sub-solutions proved in
Theorem 3.1 and the weak Harnack inequality of Theorem 1.3. $\hfill\square$

Proof of Theorem 1.6. The Hölder continuity of weak solutions is classically
obtained by proving that the oscillation of the solution decays by a universal
factor. This can be achieved in two different ways. Either by applying Lemma
5.1 with $\theta=1$ in the same spirit of [13, Appendix B], or by directly
applying the weak Harnack inequality, Theorem 1.3, following a standard
argument, for further reference see [10]. $\hfill\square$

##  Appendix A The Ink-Spots Theorem

For the sake of completeness, we provide here the proof of the Ink Spots
Theorem for the case of ultraparabolic equations. This theorem involves a
covering argument in the spirit of Krylov and Safonov [23] growing ink spots
theorem, or the Calderón-Zygmund decomposition, and it is a fundamental
ingredient for the proof of the weak Harnack inequality (see Theorem 1.3). In
order to give its statement in our setting, we introduce the delayed cylinder

$\displaystyle\overline{\mathcal{Q}}_{r}^{m}(z_{0})$
$\displaystyle=\left((0,\ldots,0,mr^{2})\circ{\mathcal{Q}}_{r}(z_{0})\right)\cap\left(\mathbb{R}^{N+1}\times(t_{0},+\infty)\right)$
(A.1)

where
$z_{0}=(x_{0},t_{0})=(x^{(0)}_{0},\ldots,x^{({\kappa})}_{0},t_{0})\in\mathbb{R}^{N+1}$.
We remark that $\overline{\mathcal{Q}}_{r}^{m}(z_{0})$ starts immediately at
the end of ${\mathcal{Q}}_{r}(z_{0})$, with which shares the same values for
$x^{(0)}$, and its structure follows the non Euclidean geometry presented in
Section 2 associated to the principal part operator $\mathscr{K}$. The aim of
this section is to prove the following statement.

######  Theorem A.1.

Let $E\subset F$ be two bounded measurable sets. We assume there exists a
constant $\mu\in]0,1[$ such that

  * •

$E\subset{\mathcal{Q}}_{1}$ and $|E|<(1-\mu)|{\mathcal{Q}}_{1}|$;

  * •

moreover, there exist an integer $m$ such that for any cylinder
${\mathcal{Q}}\subset{\mathcal{Q}}_{1}$ such that
$\overline{\mathcal{Q}}^{m}\subset{\mathcal{Q}}_{1}$ and $|{\mathcal{Q}}\cap
E|\geq(1-\mu)|{\mathcal{Q}}|$, we have that $\overline{\mathcal{Q}}^{m}\subset
F$.

Then for some universal constant $c_{\rm is}\in(0,1)$ only depending on $N$,
there holds

$|E|\leq\frac{m+1}{m}(1-c_{\rm is}\mu)|F|.$

######  Remark A.2.

Theorem A.1 still holds true if we replace ${\mathcal{Q}}_{1}$ with
${\mathcal{Q}}_{-}$ defined in (5.2).

As it has already been pointed out by Imbert and Silvestre in [18], there is
no chance to adapt the Calderón-Zygmund decomposition to this context, because
it would require to split a larger piece into smaller ones of the same type
and this is impossible due to the non Euclidean nature of our geometry. What
we do is a generalization of the procedure proposed in [18], that is in fact
an adaptation of the growing ink-spots theorem, whose original construction in
the parabolic case dates back to Krylov and Safonov [23, Appendix A].

Moreover, when we need to confine both $E$ and $F$ to stay within a fixed
cylinder, the following corollary directly follows.

######  Corollary A.3.

Let $E\subset F$ be two bounded measurable sets. We assume

  * •

$E\subset{\mathcal{Q}}_{1}$;

  * •

there exist two constants $\mu,r_{0}\in]0,1[$ and an integer $m$ such that for
any cylinder ${\mathcal{Q}}\subset{\mathcal{Q}}_{1}$ of the form
$Q_{r}(z_{0})$ such that $|{\mathcal{Q}}\cap E|\geq(1-\mu)|{\mathcal{Q}}|$, we
have $\overline{\mathcal{Q}}^{m}\subset F$ and also $r<r_{0}$.

Then for some universal constants $c_{\rm is}$ and $C_{\rm is}$ only depending
on $N$

$|E|\leq\frac{m+1}{m}(1-c_{\rm is}\mu)\left(|F\cap{\mathcal{Q}}_{1}|+C_{\rm
is}mr^{2}_{0}\right).$

###  A.1. Stacked cylinders

First of all we recall some important properties of the following family of
stacked cylinders

$\displaystyle
k{\mathcal{Q}}_{r}=\left(0,\ldots,0,\frac{k^{2}-1}{2}r^{2}\right)\circ{\mathcal{Q}}_{kr}\quad\text{and}\quad
k{\mathcal{Q}}_{r}(x_{0},t_{0})=\left(0,\ldots,0,\frac{k^{2}-1}{2}r^{2}\right)\circ{\mathcal{Q}}_{kr}(x_{0},t_{0}),$

where $(x_{0},t_{0})\in\mathbb{R}^{N+1}$, that are defined starting from the
unit cylinder (1.9) for a certain $k>0$. By definition, it is clear that
$|k{\mathcal{Q}}_{r}(x_{0},t_{0})|=k^{Q+2}|{\mathcal{Q}}_{r}(x_{0},t_{0})|$,
and that the cylinders ${\mathcal{Q}}_{r}(x_{0},t_{0})$ are not the balls of
any metric. Thus, the important properties of the cylinders are explicitly
given by the following lemmas.

######  Lemma A.4.

Let ${\mathcal{Q}}_{r_{0}}(x_{0},t_{0})$ and
${\mathcal{Q}}_{r_{1}}(x_{1},t_{1})$ be two cylinders with non empty
intersection, with $(x_{0},t_{0}),(x_{1},t_{1})\in\mathbb{R}^{N+1}$ and
$2r_{0}\geq r_{1}>0$. Then

${\mathcal{Q}}_{r_{1}}(x_{1},t_{1})\subset
k{\mathcal{Q}}_{r_{0}}(x_{0},t_{0})$

for some universal constant $k$.

###### Proof.

Without loss of generality, we may assume $(x_{0},t_{0})=(0,0)$. Then we need
to choose the constant $k$ in order to satisfy our statement. In particular,
if we consider the ball associated to the first $m_{0}$ variables we get that
$B_{r_{1}}(x_{1}^{(0)})\subset B_{kr_{0}}$ if

$kr_{0}\geq r_{0}+2r_{1}\quad\implies\quad k\geq 5.$

By repeating the same argument for all the ${\kappa}$ blocks of variables, we
get that $k$ must satisfy the following conditions:

$k^{2j+1}\geq 1+2\cdot 2^{2j+1}\qquad\text{for }j=0,\ldots,{\kappa}.$

As far as we are concerned with the condition regarding the time interval, we
need $k$ to be such that

$-\frac{k^{2}+1}{2}r^{2}_{0}\leq-r_{0}-2r_{1}^{2}\quad\implies\quad k^{2}\geq
9.$

All of these inequalities are satisfied when the first one, i.e. the one
corresponding to $j=0$, is satisfied. We choose $k$ to be the smallest
parameter satisfying these inequalities. ∎

######  Lemma A.5.

Let $\\{{\mathcal{Q}}_{j}\\}_{j\in J}$ be an arbitrary collection of slanted
cylinders with bounded radius. Then there exists a disjoint countable
subcollection $\\{{\mathcal{Q}}_{j_{i}}\\}_{i\in I}$ such that

$\bigcup\limits_{j\in
J}{\mathcal{Q}}_{j}=\bigcup\limits_{i=1}^{\infty}k{\mathcal{Q}}_{j_{i}}.$

The proof of Lemma A.5 is the same as the classical proof of the Vitali
covering lemma, where we employ Lemma A.4 instead of the fact that in any
metric space $B_{r_{1}}(x_{1})\subset 5B_{r_{0}}$, if $B_{r_{1}}(x_{1})\cap
B_{r_{0}}\neq\emptyset$ and $r_{1}\leq 2r_{0}$.

###  A.2. A generalized Lebesgue differentiation theorem

For the readers convenience, we also recall the definition of maximal
function:

$Mf(x,t)=\sup\limits_{{\mathcal{Q}}:(x,t)\in{\mathcal{Q}}}\frac{1}{|{\mathcal{Q}}|}\int\limits_{{\mathcal{Q}}\cap\Omega}|f(y,s)|\,dy\,ds,$

where the supremum is taken over cylinders of the form
$(y,s)+R{\mathcal{Q}}_{1}$.

######  Lemma A.6.

For every ${\lambda}>0$ and $f\in L^{1}({\Omega})$ , we have

$|\\{Mf<{\lambda}\\}\cap\Omega|\leq\frac{C}{{\lambda}}\|f\|_{L^{1}({\Omega})}.$

###### Proof.

Let us consider $(x,t)\in\\{Mf<{\lambda}\\}\cap\Omega$. Then there exists a
cylinder ${\mathcal{Q}}$ such that $(x,t)\in{\mathcal{Q}}$ and

$\int\limits_{{\mathcal{Q}}\cap\Omega}|f(y,s)|\,dy\,ds\geq\frac{{\lambda}}{2}|{\mathcal{Q}}\cap{\Omega}|.$

Then $\\{Mf<{\lambda}\\}\cap\Omega$ is covered with cylinders
$\\{{\mathcal{Q}}_{j}\\}$ such that the previous inequality holds. From Lemma
A.5, there exists a disjoint countable subcollection $\\{Q_{j_{i}}\\}$ so that

$\\{Mf<{\lambda}\\}\cap\Omega=\bigcup\limits_{j=1}^{\infty}Q_{j}\subset\bigcup\limits_{i=1}^{\infty}kQ_{j_{i}},$

for some integer $k$. Thus, we get

$\displaystyle\|f\|_{L^{1}({\Omega})}\geq\int\limits_{{\Omega}\cap\cup_{i}Q_{j_{i}}}|f|\geq\frac{{\lambda}}{2}\sum\limits_{i=1}^{\infty}|{\mathcal{Q}}_{j_{i}}\cap\Omega|=\frac{{\lambda}}{2k^{Q+2}}\Big{|}\bigcup\limits_{i=1}^{\infty}k{\mathcal{Q}}_{j_{i}}\cap\Omega\Big{|}\geq\frac{{\lambda}}{2k^{Q+2}}\Big{|}\\{Mf<{\lambda}\\}\cap\Omega\Big{|}.$

Thus, the claim is proved when $C=2k^{Q+2}$. ∎

The following generalized version of the Lebesgue differentiation theorem
holds.

######  Theorem A.7 (Genaralized Lebesgue Differentiation Theorem).

Let $f\in L^{1}({\Omega},dx\otimes dt)$, where ${\Omega}$ is an open subset of
$\mathbb{R}^{N+1}$. Then for a.e. $(x,t)\in{\Omega}$

$\lim\limits_{r\to
0^{+}}\frac{1}{|{\mathcal{Q}}_{r}(x,t)|}\int\limits_{{\mathcal{Q}}_{r}(x,t)}|f(y,s)-f(x,t)|\,dy\,ds=0.$

Theorem A.7 is obtained from the following Lemma A.6 exactly as in [17,
Theorem 2.5.1] by considering Lemma A.6.

###  A.3. Ink-spots theorem without time delay

######  Lemma A.8.

Let $E\subset F\subset{\mathcal{Q}}_{1}$ be two bounded measurable sets. We
make the following assumptions for some constant $\mu\in(0,1)$:

  * •

$E<(1-\mu)|{\mathcal{Q}}_{1}|$;

  * •

if for any cylinder ${\mathcal{Q}}\subset{\mathcal{Q}}_{1}$ such that
$|{\mathcal{Q}}\cap E|\geq(1-\mu)|{\mathcal{Q}}|$, then ${\mathcal{Q}}\subset
F$.

Then $|E|\leq(1-c\mu)|F|$ for some universal constant $c$ only depending on
$N$.

###### Proof.

Thanks to Theorem A.7, for almost all points $z\in E$ there is some cylinder
${\mathcal{Q}}^{z}$ containing $z$ such that $|{\mathcal{Q}}^{z}\cap
E|\geq(1-\mu)|{\mathcal{Q}}^{z}|$. Thus, for all Lebesgue points $z\in E$ we
choose a maximal cylinder ${\mathcal{Q}}^{z}\subset{\mathcal{Q}}_{1}$ that
contains $z$ and such that $|{\mathcal{Q}}^{z}\cap
E|\geq(1-\mu)|{\mathcal{Q}}^{z}|$. Here
${\mathcal{Q}}^{z}={\mathcal{Q}}_{\overline{r}}(\overline{x},\overline{t})$
for some $\overline{r}>0$ and
$(\overline{x},\overline{t})\in{\mathcal{Q}}_{1}$. In particular, we have that
${\mathcal{Q}}^{z}$ differs from ${\mathcal{Q}}_{1}$ and
${\mathcal{Q}}^{z}\subset F$ by our assumption.

First of all we prove that $|{\mathcal{Q}}^{z}\cap
E|=(1-\mu)|{\mathcal{Q}}^{z}|$. By contradiction, let us suppose that is not
true. Then there exists ${\delta}>0$ small enough and $\overline{\mathcal{Q}}$
such that
${\mathcal{Q}}^{z}\subset\overline{\mathcal{Q}}\subset(1+{\delta}){\mathcal{Q}}^{z}$,
$\overline{\mathcal{Q}}\subset{\mathcal{Q}}_{1}$ and
$|\overline{\mathcal{Q}}\cap E|>(1-\mu)|{\mathcal{Q}}^{z}|$, and this
contradicts the maximality of the choice of ${\mathcal{Q}}^{z}$.

Then we recall that the family of cylinders $\\{{\mathcal{Q}}^{z}\\}_{z\in E}$
covers the set $E$. Thanks to Lemma A.5 and considering that $E$ is a bounded
set, we can extract a finite subfamily of non overlapping cylinders
${\mathcal{Q}}_{j}:={\mathcal{Q}}^{z_{j}}$ such that
$E\subset\cup_{j=1}^{n}k{\mathcal{Q}}_{j}$. Since ${\mathcal{Q}}_{j}\subset F$
and $|{\mathcal{Q}}_{j}\cap E|=(1-\mu)|{\mathcal{Q}}_{j}|$, we have that
$|{\mathcal{Q}}_{j}\cap F\setminus E|=\mu|{\mathcal{Q}}^{z}|$. Therefore,

$\displaystyle|F\setminus E|\geq\sum\limits_{j=1}^{n}|{\mathcal{Q}}_{j}\cap
F\setminus
E|\geq\sum\limits_{j=1}^{n}\mu|{\mathcal{Q}}_{j}|=k^{-(Q+2)}\mu\sum\limits_{j=1}^{n}|k{\mathcal{Q}}_{j}|\geq
k^{-(Q+2)}\mu|E|.$

Thus, we get that $|F|\geq(1+\overline{c}\mu)|E|$, with
$\overline{c}=k^{-(Q+2)}$. Since $\overline{c}\mu\in(0,1)$, we complete the
proof by choosing $c=\overline{c}/2$. ∎

###  A.4. Proof of Theorem A.1 and Corollary A.3

In order to proceed with the proof of the Ink Spots Theorem, we first need to
recall two preliminary results.

######  Lemma A.9.

Consider a (possibly infinite) sequence of intervals $(a_{j}-h_{k},a_{j}]$.
Then

$\Big{|}\bigcup\limits_{k}(a_{k},a_{k}+mh_{k}]\Big{|}\geq\frac{m}{m+1}\Big{|}\bigcup\limits_{k}(a_{k}-h_{k},a_{k}]\Big{|}.$

The proof of Lemma A.9 can be found in [19, Lemma 10.8]. Here, we only report
the proof of the following lemma, that is an extension of Lemma 10.9 [19].

######  Lemma A.10.

Let $\\{{\mathcal{Q}}_{j}\\}$ be a collection of slanted cylinders and let
$\overline{\mathcal{Q}}_{j}^{m}$ be the corresponding versions as in (A.1).
Then

$\Big{|}\bigcup\limits_{j}\overline{\mathcal{Q}}_{j}^{m}\Big{|}\geq\frac{m}{m+1}\Big{|}\bigcup\limits_{j}{\mathcal{Q}}_{j}\Big{|}.$

###### Proof.

Because of Fubini’s Theorem we know that for any set
${\Omega}\subset\mathbb{R}^{N+1}$

$|{\Omega}|=\int|\\{(x^{(1)},\ldots,x^{({\kappa})},t):(x^{(0)},x^{(1)},\ldots,x^{({\kappa})},t)\in{\Omega}\\}|\,dx^{(0)}.$

Therefore, in order to prove our statement it is sufficient to show that for
every $x^{(0)}\in\mathbb{R}^{m_{0}}$

$\displaystyle\Big{|}\Big{\\{}(x^{(1)},\ldots,x^{({\kappa})},t)$
$\displaystyle:(x^{(0)},x^{(1)},\ldots,x^{({\kappa})},t)\in\bigcup\limits_{j}\overline{\mathcal{Q}}_{j}^{m}\Big{\\}}\Big{|}$
$\displaystyle\geq\frac{m}{m+1}\Big{|}\Big{\\{}(x^{(1)},\ldots,x^{({\kappa})},t):(x^{(0)},x^{(1)},\ldots,x^{({\kappa})},t)\in\bigcup\limits_{j}{\mathcal{Q}}_{j}\Big{\\}}\Big{|}$

From now on, let us consider a fixed $\overline{x}\in\mathbb{R}^{m_{0}}$. Any
cylinder ${\mathcal{Q}}_{j}$ is a cylinder with center

$(x_{j}^{(0)},x_{j}^{(1)},\ldots,x_{j}^{({\kappa})},t_{j})\in\mathbb{R}^{N+1}$

and radius $r_{j}>0$. $\overline{\mathcal{Q}}_{j}^{m}$ is its delayed version
(A.1), that thanks to Remark 2.4 can equivalently be represented as follows

$\overline{\mathcal{Q}}^{m}_{j}=(t_{0},t_{0}+mr_{j}^{2}]\times
B_{r}(x^{(0)}_{j})\times B_{(m+2)r_{j}^{3}}(x^{(1)}_{j})\times\ldots\times
B_{(m^{{\kappa}}+2\sum_{i=0}^{{\kappa}}m^{i})r_{j}^{2{\kappa}+1}}(x^{({\kappa})}_{j}).$

On one hand, when $|\overline{x}-x^{(0)}_{j}|\geq r_{j}$ the set

$\Big{\\{}(x^{(1)},\ldots,x^{({\kappa})},t):(x^{(0)},x^{(1)},\ldots,x^{({\kappa})},t)\in\overline{\mathcal{Q}}_{j}^{m}\Big{\\}}\quad\text{is
empty}.$

On the other hand, when $|\overline{x}-x^{(0)}_{j}|<r_{j}$ we have that

$\displaystyle\Big{\\{}(x^{(1)},\ldots,x^{({\kappa})},t)$
$\displaystyle:(\overline{x},x^{(1)},\ldots,x^{({\kappa})},t)\in\overline{\mathcal{Q}}_{j}^{m}\Big{\\}}$
$\displaystyle\supset\,\widetilde{\mathcal{Q}}_{j}:=\,(t_{j},t_{j}+mr^{2}_{j}]\times
B_{2r_{j}^{3}}(x^{(1)}_{j})\times\ldots\times
B_{2\sum_{i=0}^{{\kappa}-1}m^{i}r_{j}^{2{\kappa}+1}}(x^{({\kappa})}_{j}).$

Based on these last observations, we have that

$\displaystyle\Big{|}\Big{\\{}(x^{(1)},\ldots,x^{({\kappa})},t)$
$\displaystyle:(x^{(0)},x^{(1)},\ldots,x^{({\kappa})},t)\in\bigcup\limits_{j}\overline{\mathcal{Q}}_{j}^{m}\Big{\\}}\Big{|}\geq\Big{|}\bigcup\limits_{j:|\overline{x}-x^{(0)}_{j}|<r_{j}}\widetilde{\mathcal{Q}}_{j}\Big{|}.$

Now, thanks to Fubini’s Theorem and Lemma A.9 we have

$\displaystyle\Big{|}\Big{\\{}(x^{(1)},\ldots,x^{({\kappa})},t)$
$\displaystyle:(x^{(0)},x^{(1)},\ldots,x^{({\kappa})},t)\in\bigcup\limits_{j}\overline{\mathcal{Q}}_{j}^{m}\Big{\\}}\Big{|}$
$\displaystyle\geq\frac{m}{m+1}\Big{|}\bigcup\limits_{j:|\overline{x}-x^{(0)}_{j}|<r_{j}}(t_{j}-r^{2}_{j},0]\times
B_{2r_{j}^{3}}(x^{(1)}_{j})\times\ldots\times
B_{2\sum_{i=0}^{{\kappa}-1}m^{i}r_{j}^{2{\kappa}+1}}(x^{({\kappa})}_{j})\Big{|}$
$\displaystyle\geq\frac{m}{m+1}\Big{|}\bigcup\limits_{j:|\overline{x}-x^{(0)}_{j}|<r_{j}}(t_{j}-r^{2}_{j},0]\times
B_{r_{j}^{3}}(x^{(1)}_{j})\times\ldots\times
B_{r_{j}^{2{\kappa}+1}}(x^{({\kappa})}_{j})\Big{|}$
$\displaystyle=\frac{m}{m+1}\Big{|}\Big{\\{}(x^{(1)},\ldots,x^{({\kappa})},t):(\overline{x},x^{(1)},\ldots,x^{({\kappa})},t)\in\bigcup\limits_{j}{\mathcal{Q}}_{j}\Big{\\}}\Big{|}.$

Combining all of the above results, the proof is complete. ∎

Proof of Theorem A.1. Let $Q$ be the collection of all cylinders
${\mathcal{Q}}\subset{\mathcal{Q}}_{1}$ such that $|{\mathcal{Q}}\cap
E|\geq(1-\mu)|{\mathcal{Q}}|$. Let $G:=\cup_{{\mathcal{Q}}\in
Q}{\mathcal{Q}}$. By construction, the sets $E$ and $G$ satisfy the
assumptions of Lemma A.8. Therefore $(1-c_{\rm is}\mu)|G|\geq|E|$. Combining
the assumptions of the theorem with Lemma A.10 we conclude the proof.
$\hfill\square$

Proof of Corollary A.3. The condition $|E|\leq(1-{\delta})|{\mathcal{Q}}_{1}|$
is implied by the second assumption when $r_{0}<1$. Moreover, the result is
trivial when $r_{0}\geq 1$ choosing $C$ sufficiently large. Let $Q$ be the
collection of all cylinders ${\mathcal{Q}}\subset{\mathcal{Q}}_{1}$ such that
$|{\mathcal{Q}}\cap E|\geq(1-\mu)|{\mathcal{Q}}|$. Let
$G:=\cup_{{\mathcal{Q}}\in Q}\overline{\mathcal{Q}}^{m}$. From Theorem A.1 we
have that $|E|\leq\frac{m}{m+1}(1-c\mu)|G|$. Moreover, our assumptions tell us
$G\subset F$. In order to conclude the proof is sufficient to estimate the
measure $G\setminus{\mathcal{Q}}_{1}$ by considering that each cylinder
${\mathcal{Q}}={\mathcal{Q}}_{r}(x,t)\subset{\mathcal{Q}}_{1}$ has radius
bounded by $r_{0}$ (see [19, Corollary 10.2]). $\hfill\square$

##  Appendix B Stacked cylinders

For the sake of completeness, we here state the stacking cylinders lemma for
our operator $\mathscr{L}$. Such a result is used when applying the Ink-Spots
Theorem in the proof of the weak Harnack inequality, Theorem 1.3.

######  Lemma B.1.

Let $\omega<\frac{1}{\sqrt{2\kappa+1}}$ and
${\rho}=\left((3\kappa+1)\omega\right)^{\frac{1}{2\kappa+1}}$. We consider any
non-empty cylinder ${\mathcal{Q}}_{r}(z_{0})\subset{\mathcal{Q}}_{-}$ and we
set $T_{k}=\sum_{j=1}^{k}(2^{k}r)^{2}$. Let $N\geq 1$ such that $T_{N}\leq-
t_{0}<T_{N+1}$ and let

$\displaystyle{\mathcal{Q}}_{r}[k]$
$\displaystyle:={\mathcal{Q}}_{2^{k}r}(z_{k}),\quad k=1,\ldots,N$
$\displaystyle{\mathcal{Q}}_{R_{N+1}}[N+1]$
$\displaystyle:={\mathcal{Q}}_{R_{N+1}}(z_{N+1}),$

where $z_{k}=z_{0}\circ(0,\ldots,0,T_{k})$ and
$R=|t_{0}+T_{N}|^{\frac{1}{2}}$, $R_{N+1}=\max(R,{\rho})$, and

$z_{N+1}=\left\\{\begin{array}[]{ll}z_{N}\circ(0,\ldots,0,R^{2}),&\textit{if
$R\geq{\rho}$}\\\ (0,0),&\textit{if $R<{\rho}$}\end{array}\right.$

These cylinders satisfy

$\displaystyle{\mathcal{Q}}_{+}\subset{\mathcal{Q}}_{R_{N+1}}[N+1],\qquad\cup_{k=1}^{N+1}{\mathcal{Q}}_{r}[k]\subset(-1,0]\times
B_{2},\qquad\widetilde{{\mathcal{Q}}}[N]\subset{\mathcal{Q}}_{r}[N],$

where
$\widetilde{{\mathcal{Q}}}[N]={\mathcal{Q}}_{R_{N+1}/2}\left(z_{N+1}\circ(0,\ldots,0,-R_{N+1}^{2})\right)$.

###### Proof.

As our derivation of the previous lemma follows very closely the one contained
in [13, Appendix C], we here do not write explicitly the proof. Indeed, the
proof of the result is merely geometric and the main difference with [13] lies
in the fact that in our case we exploit the more general composition law and
dilations defined in (2.1) and in (2.9), respectively. This explains why here
the constants $\omega$ and ${\rho}$ differ from the ones in [13]. ∎

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