# Schauder type estimates for degenerate Kolmogorov equations with Dini
continuous coefficients

Sergio Polidoro Dipartimento di Scienze Fisiche, Informatiche e Matematiche,
Università di Modena e Reggio Emilia, via Campi 213/b, 41115 Modena (Italy).
E-mail<EMAIL_ADDRESS>Annalaura Rebucci Dipartimento di Scienze
Fisiche, Matematiche e Informatiche, Università di Parma, Parco Area delle
Scienze 7/A, 43124 Parma (Italy). Email<EMAIL_ADDRESS>Bianca
Stroffolini  Dipartimento di Ingegneria Elettrica e delle Tecnologie
dell’Informazione, Università di Napoli Federico II, Via Claudio 25, 80125
Napoli, Italy. E-mail<EMAIL_ADDRESS>
###### Abstract

We study the regularity properties of the second order linear operator in
${\mathbb{R}}^{N+1}$:

$\mathscr{L}u:=\sum_{j,k=1}^{m}a_{jk}\partial_{x_{j}x_{k}}^{2}u+\sum_{j,k=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u,$

where
$A=\left(a_{jk}\right)_{j,k=1,\dots,m},B=\left(b_{jk}\right)_{j,k=1,\dots,N}$
are real valued matrices with constant coefficients, with $A$ symmetric and
strictly positive. We prove that, if the operator $\mathscr{L}$ satisfies
Hörmander’s hypoellipticity condition, and $f$ is a Dini continuous function,
then the second order derivatives of the solution $u$ to the equation
$\mathscr{L}u=f$ are Dini continuous functions as well. We also consider the
case of Dini continuous coefficients $a_{jk}$’s. A key step in our proof is a
Taylor formula for classical solutions to $\mathscr{L}u=f$ that we establish
under minimal regularity assumptions on $u$.

2000 Mathematics Subject Classification. 35K70, 35K65, 35B65.

Keywords and phrases: Degenerate Kolmogorov equations, regularity theory,
classical solutions, Dini continuity, Taylor formula.

##  1 Introduction

We consider second order linear differential operators of the form

$\mathscr{L}:=\sum_{i,j=1}^{m}a_{ij}\partial^{2}_{x_{i}x_{j}}+\sum_{i,j=1}^{N}b_{ij}x_{j}\partial_{x_{i}}-\partial_{t},$
(1.1)

where $(x,t)\in{\mathbb{R}}^{N+1}$, and $1\leq m\leq N$. The matrices
$A:=(a_{ij})_{i,j=1,\ldots,m}$ and $B:=(b_{ij})_{i,j=1,\ldots,N}$ have real
constant entries. The first order part of the operator $\mathscr{L}$ will be
denoted by $Y$

$Y:=\sum_{i,j=1}^{N}b_{ij}x_{j}\partial_{x_{i}}-\partial_{t}=\langle
Bx,D\rangle-\partial_{t},$ (1.2)

and $Yu$ will be understood as the _Lie derivative_

$Yu(x,t):=\lim_{s\to 0}\frac{u(\text{exp}(sB)x,t-s)-u(x,t)}{s}.$ (1.3)

Note that $Yu$ is the derivative of $u$ along the characteristic trajectory of
$Y$, if we identify the directional derivative $Y$ with the vector valued
function $Y(x,t)=(Bx,-1)$. The standing assumption of this article is:

[H.1] The matrix $A$ is symmetric and strictly positive, the matrix $B$ has
the form

$B=\begin{pmatrix}B_{0,0}&B_{0,1}&\ldots&B_{0,{\kappa}-1}&B_{0,{\kappa}}\\\
B_{1}&B_{1,1}&\ldots&B_{1,{\kappa}-1}&B_{1,{\kappa}}\\\
{\mathbb{O}}&B_{2}&\ldots&B_{2,{\kappa}-1}&B_{2,{\kappa}}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
{\mathbb{O}}&{\mathbb{O}}&\ldots&B_{\kappa}&B_{{\kappa},{\kappa}}\end{pmatrix}=\begin{pmatrix}\ast&\ast&\ldots&\ast&\ast\\\
B_{1}&\ast&\ldots&\ast&\ast\\\ {\mathbb{O}}&B_{2}&\ldots&\ast&\ast\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
{\mathbb{O}}&{\mathbb{O}}&\ldots&B_{{\kappa}}&\ast\end{pmatrix}$ (1.4)

where every block $B_{j}$ is an $m_{j}\times m_{j-1}$ matrix of rank $m_{j}$
with $j=1,2,\ldots,{\kappa}$. Moreover, the $m_{j}$s are positive integers
such that

$\displaystyle m_{0}\geq m_{1}\geq\ldots\geq m_{\kappa}\geq
1,\quad\textrm{and}\quad m_{0}+m_{1}+\ldots+m_{\kappa}=N.$ (1.5)

We agree to let $m_{0}:=m$ to have a consistent notation, moreover
${\mathbb{O}}$ denotes a block matrix whose entries are zeros, whereas the
coefficients of the blocks “$\ast$” are arbitrary. Note that we allow the
operator $\mathscr{L}$ to be strongly degenerate, when $m<N$. However, the
assumption [H.1] implies that the first order part $Y$ of $\mathscr{L}$
induces a strong regularity property. Indeed, it is known that $\mathscr{L}$
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 is a classical solution to
$\mathscr{L}u=f$, whenever $f\in C^{\infty}(\Omega)$. In Section 2 we will
recall several known results about the operators $\mathscr{L}$ that will be
used in the sequel.

In this article we study the local regularity of the classical solution $u$ to
$\mathscr{L}u=f$ when $f$ is Dini continuous. For this reason we require as
few conditions as possible for the definition of $\mathscr{L}u$.

######  Definition 1.1.

Let $\Omega$ be an open subset of ${\mathbb{R}}^{N+1}$. We say that a function
$u$ belongs to $C^{2}_{\mathscr{L}}(\Omega)$ if $u$, its derivatives
$\partial_{x_{i}}u,\partial_{x_{i}x_{j}}u$ ($i,j=1,\dots,m$) and the Lie
derivative $Yu$ defined in (1.3) are continuous functions in $\Omega$. We also
require, for $i=1,\dots,m$, that

$\lim_{s\to 0}\frac{\partial_{x_{i}}u(\text{\rm
exp}(sB)x,t-s)-\partial_{x_{i}}u(x,t)}{|s|^{1/2}}=0,$ (1.6)

uniformly for every $(x,t)\in K$, where $K$ is a compact set $K\subset\Omega$.

Let $f$ be a continuous function defined in $\Omega$. We say that a function
$u$ is a classical solution to $\mathscr{L}u=f$ in $\Omega$ if $u$ belongs to
$C^{2}_{\mathscr{L}}(\Omega)$, and the equation $\mathscr{L}u=f$ is satisfied
at every point of $\Omega$.

Note that, as $\mathscr{L}$ is a linear second order differential operator, it
is natural to consider $Y$ as a second order derivative, and (1.6) can be
interpreted as a condition on the second order mixed derivative of the form
$Y^{1/2}\partial_{x_{i}}u$. This condition will be used to prove that $u$ is
approximated by its intrinsic Taylor polynomial of degree 2, which is one of
the main achievements of this article. We recall that the _nth_ -order
intrinsic Taylor polynomial of a function $u$ around the point $z$ can be
defined as the unique polynomial function $P_{z}^{n}u$ of order $n$ such that

$\displaystyle
u(\zeta)-P_{z}^{n}u(\zeta)=o(\|z^{-1}\circ{\zeta}\|_{K}^{n})\quad\text{as}\quad\zeta\to
z,$

where $\|\cdot\|_{K}$ denotes the semi-norm as defined in (1.14) below.

######  Theorem 1.2.

Let $\mathscr{L}$ be an operator in the form (1.1) satisfying hypothesis
[H.1]. Let $\Omega$ be an open subset of ${\mathbb{R}}^{N+1}$ and let $u$ be a
function in $C_{\mathscr{L}}^{2}(\Omega)$. For every $z:=(x,t)\in\Omega$ we
define the second order Taylor polynomial of $u$ around $z$ as

$\begin{split}T^{2}_{z}u({\zeta})&:=u(z)+\sum_{i=1}^{m}\partial_{x_{i}}u(z)({\xi}_{i}-x_{i})\\\
&+\frac{1}{2}\sum_{i,j=1}^{m}\partial^{2}_{x_{i}x_{j}}u(z)({\xi}_{i}-x_{i})({\xi}_{j}-x_{j})-Yu(z)({\tau}-t),\end{split}$
(1.7)

for any ${\zeta}=({\xi},{\tau})\in\Omega$. Moreover, we have

$\displaystyle
u({\zeta})-T^{2}_{z}u({\zeta})=o(\|z^{-1}\circ{\zeta}\|_{K}^{2})\quad\text{as}\quad\zeta\to
z.$ (1.8)

Our main result is the local regularity of the classical solution $u$ to
$\mathscr{L}u=f$ when $f$ is Dini continuous. In order to define a modulus of
continuity which is suitable for the operator $\mathscr{L}$ we recall the Lie
group structure $\mathbb{K}=({\mathbb{R}}^{N+1},\circ)$ introduced by
Lanconelli and Polidoro in [11], and some related notation. In Section 2 we
will explain its connection with $\mathscr{L}$. We let

$E(t):=\text{exp}(-tB),$ (1.9)

and we define

$\mathbb{K}=({\mathbb{R}}^{N+1},\circ),\quad(x,t)\circ({\xi},{\tau})=({\xi}+E({\tau})x,t+{\tau}),\quad(x,t),({\xi},{\tau})\in{\mathbb{R}}^{N+1}.$
(1.10)

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

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

In [11] it is proved that the operator $\mathscr{L}$ is invariant with respect
to a family of dilations $(\delta_{r})_{r>0}$ if, and only if, the matrix $B$
in (1.4) agrees with $B_{0}$ defined as:

$B_{0}=\begin{pmatrix}{\mathbb{O}}&{\mathbb{O}}&\ldots&{\mathbb{O}}&{\mathbb{O}}\\\
B_{1}&{\mathbb{O}}&\ldots&{\mathbb{O}}&{\mathbb{O}}\\\
{\mathbb{O}}&B_{2}&\ldots&{\mathbb{O}}&{\mathbb{O}}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
{\mathbb{O}}&{\mathbb{O}}&\ldots&B_{{\kappa}}&{\mathbb{O}}\end{pmatrix}$
(1.11)

In other words, every block denoted by $*$ in (1.4) has zero entries. In this
case the dilation is defined for every positive $r$ as

$\delta_{r}:=\textrm{diag}(rI_{m},r^{3}I_{m_{1}},\ldots,r^{2\kappa+1}I_{m_{\kappa}},r^{2}),$
(1.12)

where $I_{k}$, $k\in{\mathbb{N}}$, is the $k$-dimensional unit matrix.

In the sequel we let $\mathscr{L}_{0}$ be the operator obtained from
$\mathscr{L}$ by replacing its matrix $B$ with $B_{0}$ defined in (1.11), and
we base our blow-up argument on the family of dilations $(\delta_{r})_{r>0}$.
Hence we take advantage of the invariant structure of $\mathscr{L}_{0}$ in the
study of the regularity of $\mathscr{L}$. This fact is quite natural as
$\mathscr{L}_{0}$ is the _blow-up limit of_ $\mathscr{L}$, as it is shown in
Section 2.2 of [1].

We now introduce a homogeneous semi-norm of degree 1 with respect to the
family of dilations $(\delta_{r})_{r>0}$ in (1.12) and a quasi-distance which
is invariant with respect to the group operation in (1.10). We first rewrite
the matrix $\delta_{r}$ with the equivalent notation

$\delta_{r}:=\textrm{diag}(r^{\alpha_{1}},\ldots,r^{\alpha_{N}},r^{2}),$
(1.13)

where
$\alpha_{1},\dots,\alpha_{m_{0}}=1,\alpha_{m_{0}+1},\dots,\alpha_{m_{0}+m_{1}}=3,\alpha_{N-m_{\kappa}},\dots,\alpha_{N}=2\kappa+1$.

######  Definition 1.3.

For every $(x,t)\in{\mathbb{R}}^{N+1}$ we set

$\|(x,t)\|_{K}:=\max\left\\{|x_{1}|^{\frac{1}{\alpha_{1}}},\dots,|x_{N}|^{\frac{1}{\alpha_{N}}},|t|^{\frac{1}{2}}\right\\}.$
(1.14)

Note that the semi-norm is homogeneous of degree $1$ with respect to the
family of dilations $(\delta_{r})_{r>0}$, namely
$\|\delta_{r}(x,t)\|_{K}=r\|(x,t)\|_{K}$ for every $r>0$ and
$(x,t)\in{\mathbb{R}}^{N+1}$. Moreover, the following pseudo-triangular
inequality holds: for every bounded set $H\subset{\mathbb{R}}^{N+1}$ there
exists a positive constant ${\bf c}_{H}$ such that

$\|(x,t)^{-1}\|_{K}\leq{\bf
c}_{H}\|(x,t)\|_{K},\qquad\|(x,t)\circ({\xi},{\tau})\|_{K}\leq{\bf
c}_{H}(\|(x,t)\|_{K}+\|({\xi},{\tau})\|_{K}),$ (1.15)

for every $(x,t),({\xi},{\tau})\in H$. We then define the quasi-distance
$d_{K}$ by setting

$d_{K}((x,t),({\xi},{\tau})):=\|({\xi},{\tau})^{-1}\circ(x,t)\|_{K},\qquad(x,t),({\xi},{\tau})\in{\mathbb{R}}^{N+1},$
(1.16)

and the ball

$\mathcal{Q}_{r}(x_{0},t_{0}):=\\{(x,t)\in{\mathbb{R}}^{N+1}\mid
d_{K}((x,t),(x_{0},t_{0}))<r\\}.$ (1.17)

Note that from (1.15) it directly follows

$d_{K}((x,t),({\xi},{\tau}))\leq{\bf
c}_{H}(d_{K}((x,t),(y,s))+d_{K}((y,s),({\xi},{\tau}))),$

for every $(x,t),({\xi},{\tau}),(y,s)\in{\mathbb{R}}^{N+1}$. We eventually
define the modulus of continuity of a function $f$ defined on any set
$H\subset{\mathbb{R}}^{N+1}$ as follows

$\omega_{f}(r):=\sup_{\begin{subarray}{c}(x,t),({\xi},{\tau})\in H\\\
d_{K}((x,t),({\xi},{\tau}))<r\end{subarray}}|f(x,t)-f({\xi},{\tau})|.$ (1.18)

######  Definition 1.4.

A function $f$ is said to be Dini-continuous in $H$ if

$\int_{0}^{1}\frac{\omega_{f}(r)}{r}dr<+\infty.$

We are now in position to state our main result.

######  Theorem 1.5.

Let $\mathscr{L}$ be an operator in the form (1.1) satisfying hypothesis
[H.1]. Let $u\in C^{2}_{\mathscr{L}}(\mathcal{Q}_{1}(0,0))$ be a classical
solution to $\mathscr{L}u=f$. Suppose that $f$ is Dini continuous. Then there
exists a positive constant $c$, only depending on the operator $\mathscr{L}$,
such that:

i)

$|\partial^{2}u(0,0)|\leq
c\left(\sup_{\mathcal{Q}_{1}(0,0)}|u|+|f(0,0)|+\int_{0}^{1}\frac{\omega_{f}(r)}{r}\right);$

ii)

for any points $(x,t)$ and $({\xi},{\tau})\in\mathcal{Q}_{\frac{1}{4}}(0,0)$
we have

$|\partial^{2}u(x,t)-\partial^{2}u({\xi},{\tau})|\leq
c\left(d\sup_{\mathcal{Q}_{1}(0,0)}|u|+d\sup_{\mathcal{Q}_{1}(0,0)}|f|+\int_{0}^{d}\frac{\omega_{f}(r)}{r}+d\int_{d}^{1}\frac{\omega_{f}(r)}{r^{2}}\right).$

where $d:=d_{K}((x,t),({\xi},{\tau}))$ and $\partial^{2}$ stands either for
$\partial^{2}_{x_{i}x_{j}}$, with $i,j=1,\ldots,m$, or for $Y$.

We emphasize that Theorem 1.5 fails even in the simplest Euclidean setting if
we don’t assume any regularity condition on the function $f$. Consider for
instance the function

$u(x,y)=xy(\log(x^{2}+y^{2}))^{\alpha},\qquad\text{with}\quad 0<\alpha<1.$

A direct computation shows that

$\Delta
u(x,y)=8\alpha\frac{xy}{x^{2}+y^{2}}(\log(x^{2}+y^{2}))^{\alpha-1}+4\alpha(\alpha-1)\frac{xy}{x^{2}+y^{2}}(\log(x^{2}+y^{2}))^{\alpha-2},$

so that $f(x,y):=\Delta u(x,y)$ extends to a continuous function on
${\mathbb{R}}^{2}$, which is not Dini continuous at the point $(0,0)$. On the
other hand, the derivative $\partial_{x}\partial_{y}u(x,y)$ is unbounded near
the origin. We also point out that, when $\alpha=1$, the function $u$ is a
counterexample for the $L^{\infty}$ bounds of the second order derivatives of
weak solutions to $\Delta u=f$. 111We acknowledge that this counterexample was
pointed out to one of the authors by Andreas Minne during the Workshop “New
trends in PDEs”, held in Catania on 29-30 May 2018.

We finally consider the non-constant coefficients operator
${\widetilde{\mathscr{L}}}$ defined as follows

${\widetilde{\mathscr{L}}}:=\sum_{i,j=1}^{m}a_{ij}(x,t)\partial^{2}_{x_{i}x_{j}}+\sum_{i,j=1}^{N}b_{ij}x_{j}\partial_{x_{i}}-\partial_{t},$
(1.19)

We assume that the coefficients $a_{ij}$ are Dini continuous functions and, in
order to simplify the notation, we write

$\omega_{a}(r):=\max_{i,j=1,\dots,m}\sup_{\begin{subarray}{c}(x,t),({\xi},{\tau})\in
H\\\
d_{K}((x,t),({\xi},{\tau}))<r\end{subarray}}|a_{ij}(x,t)-a_{ij}({\xi},{\tau})|.$
(1.20)

We assume that the following condition on the matrix
$A(x,t):=(a_{ij}(x,t))_{i,j=1,\ldots,m}$ is satisfied.

[H.2] For every $(x,t)\in{\mathbb{R}}^{N+1}$, the matrix $A(x,t)$ is symmetric
and satisfies

$\lambda|\xi|^{2}\leq\langle
A(x,t)\xi,\xi\rangle\leq\Lambda|\xi|^{2},\qquad\text{for
every}\quad\xi\in{\mathbb{R}}^{m},$ (1.21)

for some positive constants $\lambda,\Lambda$.

######  Theorem 1.6.

Let ${\widetilde{\mathscr{L}}}$ be an operator in the form (1.19) satisfying
the hypotheses [H.1] and [H.2]. Let $u\in
C^{2}_{\mathscr{L}}(\mathcal{Q}_{1}(0,0))$ be a classical solution to
${\widetilde{\mathscr{L}}}u=f$. Suppose that $f$ and the coefficients
$a_{ij}$, $i,j=1,\dots,m$, are Dini continuous. Then for any points $(x,t)$
and $({\xi},{\tau})\in\mathcal{Q}_{\frac{1}{2}}(0,0)$ the following holds:

$\begin{split}|\partial^{2}u(x,t)-\partial^{2}u({\xi},{\tau})|\leq&c\Big{(}d\sup_{\mathcal{Q}_{1}(0,0)}|u|+d\sup_{\mathcal{Q}_{1}(0,0)}|f|+\int_{0}^{d}\frac{\omega_{f}(r)}{r}+d\int_{d}^{1}\frac{\omega_{f}(r)}{r^{2}}\Big{)}\\\
+&c\bigg{(}\sum_{i,j=1}^{m}\sup_{\mathcal{Q}_{1}(0,0)}|\
\partial^{2}_{x_{i}x_{j}}u|\bigg{)}\Big{(}\int_{0}^{d}\frac{\omega_{a}(r)}{r}+d\int_{d}^{1}\frac{\omega_{a}(r)}{r^{2}}\Big{)}.\end{split}$

where $d=d_{K}((x,t),({\xi},{\tau}))$ and $\partial^{2}$ stands either for
$\partial^{2}_{x_{i}x_{j}}$,$i,j=1,\ldots,m$, or for $Y$.

We now compare our main findings with the current literature on this subject.
We first consider functions $f$ defined on $H\subset{\mathbb{R}}^{N+1}$ that
are Hölder continuous with respect to the distance (1.16), that is

$|f(x,t)-f({\xi},{\tau})|\leq
M\,d_{K}((x,t),({\xi},{\tau}))^{\alpha},\qquad\text{for
every}\quad(x,t),({\xi},{\tau})\in H,$ (1.22)

for some constants $M>0$ and $\alpha\in]0,1]$. In this case we say that $f\in
C^{0,\alpha}_{L}(H)$ and we let

$\|f\|_{C^{0,\alpha}_{L}(H)}\>=\sup_{H}|f|+\inf\big{\\{}M\geq 0\mid\eqref{eq-
Holder}\ \text{holds}\big{\\}}.$

When $\alpha<1$ we write $C^{\alpha}_{L}(H)$ instead of $C^{0,\alpha}_{L}(H)$.
As a direct consequence of Theorem 1.6 we have

######  Corollary 1.7.

Let $u\in C^{2}_{\mathscr{L}}(\mathcal{Q}_{1}(0,0))$ be a classical solution
to ${\widetilde{\mathscr{L}}}u=f$. Suppose that $f$ and the coefficients
$a_{ij}$, $i,j=1,\dots,m$, belong to $C^{0,\alpha}_{L}(\mathcal{Q}_{1}(0,0))$.
Then for any points $(x,t)$ and
$({\xi},{\tau})\in\mathcal{Q}_{\frac{1}{2}}(0,0)$ the following holds:

$\begin{split}|\partial^{2}u(x,t)-\partial^{2}u({\xi},{\tau})|\leq&c\,d^{\alpha}\bigg{(}\sup_{\mathcal{Q}_{1}(0,0)}|u|+\frac{\|f\|_{C^{\alpha}_{L}(\mathcal{Q}_{1}(0,0))}}{\alpha(1-\alpha)}\\\
&+\sum_{i,j=1}^{m}\sup_{\mathcal{Q}_{1}(0,0)}|\
\partial^{2}_{x_{i}x_{j}}u|\frac{\|a\|_{C^{\alpha}_{L}(\mathcal{Q}_{1}(0,0))}}{\alpha(1-\alpha)}\bigg{)},\qquad\text{if}\quad\alpha<1,\\\
|\partial^{2}u(x,t)-\partial^{2}u({\xi},{\tau})|\leq&c\,d\bigg{(}\sup_{\mathcal{Q}_{1}(0,0)}|u|+\|f\|_{C^{0,1}_{L}(\mathcal{Q}_{1}(0,0))}|\log\;d|\\\
&+\bigg{(}\sum_{i,j=1}^{m}\sup_{\mathcal{Q}_{1}(0,0)}|\
\partial^{2}_{x_{i}x_{j}}u|\bigg{)}\|a\|_{C^{0,1}_{L}(\mathcal{Q}_{1}(0,0))}|\log\;d|\bigg{)},\quad\text{if}\quad\alpha=1.\end{split}$

Note that, for $\alpha<1$, Corollary 1.7 restores the Schauder estimates
previously proved by Manfredini in [14], and by Di Francesco and Polidoro in
[6]. Note that, in this case, an interpolation inequality allows us to state a
bound for the $C^{\alpha}_{L}$ norm of the derivatives $\partial^{2}u$ in
terms of
$\|a\|_{C^{\alpha}_{L}(\mathcal{Q}_{1}(0,0))},\|f\|_{C^{\alpha}_{L}(\mathcal{Q}_{1}(0,0))}$,
and $\sup_{\mathcal{Q}_{1}(0,0)}|u|$ only. We also recall that Schauder
estimates in the framework of semigroups have been proved by Lunardi [13],
Lorenzi [12], Priola [19]. Theorems 1.5 and 1.6 improve the previous ones, not
only because we weaken the regularity assumption on $f$ and on the
coefficients $a_{ij}$’s, but also because the Schauder estimate for $\alpha=1$
is not given in the aforementioned articles. We also quote analogous results
obtained in the framework of stochastic theory (see Menozzi [15] and its
bibliography).

The proof of our main results is based on the method introduced by Wang [21]
in the study of the Poisson equation, which greatly simplifies the other
approaches previously used in literature. Wang considers in [21] a solution
$u$ to the equation $\Delta u=f$ in some open set $\Omega$. Without loss of
generality, he assumes that the unit ball $B_{1}(0)$ is contained in $\Omega$
and considers a sequence of Dirichlet problems as follows. Let $B_{r_{k}}(0)$
be the Euclidean ball centered at the origin and of radius
$r_{k}=\frac{1}{2^{k}}$, and let $u_{k}$ be the solution to the Dirichlet
problem

$\Delta u_{k}=f(0),\quad\text{in}\quad B_{r_{k}}(0),\qquad
u_{k}=u\quad\text{in}\quad\partial B_{r_{k}}(0).$

Quantitative information on the derivatives of every solution $u_{k}$ is
obtained by using only the elementary properties of the Laplace equation,
namely the weak maximum principle, and the standard apriori estimates of the
derivatives, that are obtained in [21] via mean value formulas. The bounds for
the derivatives of $u$ are obtained as the limit of the analogous bounds for
$u_{k}$. The Taylor expansion in this step is crucial to conclude the proof.

In this work we apply the method described above to degenerate Kolmogorov
operators $\mathscr{L}$, by adapting Wang’s approach to the non-Euclidean
structure defined in (1.10). In particular, the ball $B_{r_{k}}(0)$ is
replaced by the box $\mathcal{Q}_{r_{k}}(0,0)$ defined through the dilation
${\delta}_{r_{k}}$ introduced in (1.12). Concerning the Taylor expansion, we
recall the results due to Bonfiglioli [4] and the ones proved by Pagliarani,
Pascucci and Pignotti [17]. We emphasize that the authors of the above
articles assume that the second order derivatives of the function $u$ are
Hölder continuous, while we only require that $u$ belongs to the space
$C^{2}_{\mathscr{L}}(\Omega)$ introduced in Definition 1.1. As the regularity
of the second order derivatives of $u$ is the very subject of this note, we
don’t assume extra conditions on them and we prove in Proposition 1.2 the
Taylor approximation under the minimal requirement that $u\in
C^{2}_{\mathscr{L}}(\Omega)$.

We conclude this introduction with a short discussion about the applicative
and theoretical interest in the operator $\mathscr{L}$. A simple meaningful
example is the operator introduced by Kolmogorov in [10], defined for
$(x,t)=(v,y,t)\in{\mathbb{R}}^{m}\times{\mathbb{R}}^{m}\times{\mathbb{R}}$ as
follows

$\mathscr{K}:=\sum_{j=1}^{m}\partial^{2}_{x_{j}}-\sum_{j=1}^{m}x_{j}\partial_{x_{m+j}}-\partial_{t}=\Delta_{v}-\langle
v,D_{y}\rangle-\partial_{t}.$ (1.23)

The operator $\mathscr{K}$ can be written in the form (1.1) with
$\kappa=1,m_{1}=m$, and

$B=\begin{pmatrix}{\mathbb{O}}&{\mathbb{O}}\\\
-I_{m}&{\mathbb{O}}\end{pmatrix}$ (1.24)

It arises in several areas of application of PDEs. In particular, in kinetic
theory the density $u$ of particles, with velocity $v$ and position $y$ at
time $t$, satisfies equation $\mathscr{K}u=0$. In this setting, the Lie group
has a quite natural intepretation. Indeed the composition law (1.10) agrees
with the _Galilean_ change of variables

$(v,y,t)\circ(v_{0},y_{0},t_{0})=(v_{0}+v,y_{0}+y+tv_{0},t_{0}+t),\qquad(v,y,t),(v_{0},y_{0},t_{0})\in{\mathbb{R}}^{2m+1}.$

It is easy to see that $\mathscr{K}$ is invariant with respect to the above
change of variables. Specifically, if
$w(v,y,t)=u(v_{0}+v,y_{0}+y+tv_{0},t_{0}+t)$ and
$g(v,y,t)=f(v_{0}+v,y_{0}+y+tv_{0},t_{0}+t)$, then

$\mathscr{K}u=f\quad\iff\quad\mathscr{K}w=g\quad\text{for
every}\quad(v_{0},y_{0},t_{0})\in{\mathbb{R}}^{2m+1}.$

As the matrix $B$ in (1.24) is in the form (1.11), $\mathscr{K}$ is invariant
with respect to the dilatation $\delta_{r}(v,y,t):=(rv,r^{3}y,r^{2}t)$. Note
that the dilatation acts as the usual parabolic scaling with respect to the
variable $v$ and $t$. The term $r^{3}$ in front of $y$ is due to the fact that
the velocity $v$ is the derivative of the position $y$ with respect to time
$t$. For a more exhaustive description of the operator $\mathscr{L}$, and of
its applications, we refer to the survey article [1] by Anceschi and Polidoro
and to its bibliography.

After the work of Kolmogorov [10] where (1.23) was introduced, and Hörmander’s
celebrated article [9] on the hypoellipticity of second order degenerate
linear operators, the regularity theory for operators that are invariant with
respect to a Lie group structure has been widely developed by many authors. We
quote here the seminal works by Folland [7], Folland and Stein [8], Rotschild
and Stein [20], Nagel, Stein and Wainger [16]. We also refer to the monograph
by Bonfiglioli, Lanconelli and Uguzzoni [3] that contains an updated
description of this theory. Wei, Jiang, and Wu adapt in [22] the method
introduced by Wang [21] and prove Schauder estimates for hypoelliptic
degenerate operators on the Heisenberg group. The Taylor formula used in [22]
is proved by Arena, Caruso and Causa in [2]. In a different framework, Wang’s
method has been used by Bucur and Karakhanyan [5] in the study of fractional
operators.

This paper is structured as follows. In Section 2, we recall the basic facts
about the analysis on Lie groups we need in our treatment. It also contains
some recalls about the fundamental solution of the operator $\mathscr{L}$. In
Section 3 we prove some preliminary results. In particular, we obtain some _a
priori_ estimates of the derivatives of the solutions $u$ to $\mathscr{L}u=0$
in terms of the $L^{\infty}$ norm of $u$. In Section 4 we prove our main
result on the Taylor approximation of any function $u\in
C^{2}_{\mathscr{L}}(\Omega)$. Section 5 contains the proof of Theorem 1.5,
while Section 6 contains the proof of Theorem 1.6.

##  2 Lie Group Invariance and Fundamental Solution

Here we discuss the invariance properties of Kolmogorov operators with respect
to the Lie Group structure $\mathbb{K}=({\mathbb{R}}^{N+1},\circ)$ introduced
in (1.10). Moreover, we recall some known results concerning the fundamental
solution of $\mathscr{L}$.

We first introduce some further notation. As the constant matrix $A$ is
symmetric and positive, there exists a symmetric and positive matrix
$A^{1/2}=\left(\overline{a}_{ij}\right)_{i,j=1,\dots,m}$ such that
$A=A^{1/2}A^{1/2}$. In order to check the hypothesis [H.1], we write
$\mathscr{L}$ in terms of vector fields as follows

$\mathscr{L}=\sum_{i=1}^{m}X_{i}^{2}+Y,$

where

$X_{i}:=\sum_{j=1}^{m}\overline{a}_{ij}\partial_{x_{j}},\quad
i=1,\ldots,m,\qquad Y:=\langle Bx,D\rangle-\partial_{t},$ (2.1)

We recall that assumption [H.1] is implied by Hörmander’s condition (see [9]):

${\rm rank\
Lie}\left(X_{1},\dots,X_{m},Y\right)(x,t)=N+1,\qquad\forall\,(x,t)\in{\mathbb{R}}^{N+1}.$
(2.2)

Yet another condition, equivalent to [H.1], (see [11]), is that

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

where

$C(t)=\int_{0}^{t}E(s)\,\begin{pmatrix}A&{\mathbb{O}}\\\
{\mathbb{O}}&{\mathbb{O}}\end{pmatrix}\,E^{T}(s)\,ds.$

We now recall that, under the the hypothesis of hypoellipticity, Hörmander
constructed the fundamental solution of $\mathscr{L}$ as

$\Gamma(x,t,{\xi},{\tau})=\Gamma(x-E(t-{\tau}){\xi},t-{\tau}),$

where $\Gamma(x,t)=\Gamma(x,t,0,0)$ and $\Gamma(x,t)=0$ for every $t\leq 0$,
while

$\Gamma(x,t)=\frac{(4\pi)^{-\frac{N}{2}}}{\sqrt{\det
C(t)}}\text{exp}\Big{(}-\frac{1}{4}\langle
C^{-1}(t)x,x\rangle-t\,\text{tr}(B)\Big{)},\quad t>0.$

As a fundamental solution to $\mathscr{L}$, the following representation
formula holds true: for every $u\in C_{0}^{\infty}({\mathbb{R}}^{N+1})$ we
have

$\displaystyle
u(z)=-\int_{{\mathbb{R}}^{N+1}}[\Gamma(z,\cdot)\mathscr{L}(u)]({\zeta})d{\zeta}.$
(2.3)

Here and in the sequel $z=(x,t)$ and ${\zeta}=({\xi},{\tau})$ denote points of
${\mathbb{R}}^{N+1}$.

We now conclude the analysis of the Lie Group $\mathbb{K}$, providing tools
that will be very useful to prove our main results. We adopt the notation of
[11] and we quote the results therein. For a given
${\zeta}\in{\mathbb{R}}^{N+1}$, we denote by $\ell_{{\zeta}}$ the left
translation 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 vector fields $X_{1},\dots,X_{m}$ and $Y$ are left-invariant, with
respect to the group law (1.10), in the sense that

$X_{j}\left(u(\zeta\circ\,\cdot\,)\right)=\left(X_{j}u\right)(\zeta\circ\,\cdot\,),\quad
j=1,\dots,m,\qquad
Y\left(u(\zeta\circ\,\cdot\,)\right)=\left(Yu\right)(\zeta\circ\,\cdot\,),$
(2.4)

for every $\zeta\in{\mathbb{R}}^{N+1}$ and every $u$ sufficiently smooth.
Hence, in particular,

$\displaystyle\mathscr{L}\circ\ell_{{\zeta}}=\ell_{{\zeta}}\circ\mathscr{L}\quad\textrm{or,
equivalently,}\quad\mathscr{L}\left(u(\zeta\circ\,\cdot\,)\right)=\left(\mathscr{L}u\right)(\zeta\circ\,\cdot\,).$

Regarding the invariance with respect to the dilation introduced in (1.12), we
recall that the operator $\mathscr{L}_{0}$, obtained from $\mathscr{L}$ by
replacing its matrix $B$ with $B_{0}$ in (1.11), satisfies

$\mathscr{L}_{0}\left(u\circ\delta_{r}\right)=r^{2}\delta_{r}\left(\mathscr{L}_{0}u\right),\quad\text{for
every}\quad r>0,$ (2.5)

for every function $u$ sufficiently smooth (see Proposition 2.2 in [11]). In
this case, we say that
$\mathbb{K}=\left({\mathbb{R}}^{N+1},\circ,\left(\delta_{r}\right)_{r>0}\right)$
is a _homogeneous Lie group_ , and we have

$\delta_{r}\left(z\circ{\zeta}\right)=\left(\delta_{r}z\right)\circ\left(\delta_{r}{\zeta}\right),\quad\text{for
every}\quad z,\zeta\in{\mathbb{R}}^{N+1}\ \text{and}\ r>0.$

As we rely on a blow-up argument, we also apply the dilation (1.12) to the
general operator $\mathscr{L}$ satisfying [H.1]. Specifically, we define
$\mathscr{L}_{r}$ as the scaled operator of $\mathscr{L}$ in terms of
$(\delta_{r})_{r>0}$ as follows

$\mathscr{L}_{r}:=r^{2}(\delta_{r}\circ\mathscr{L}\circ\delta_{\frac{1}{r}}),$
(2.6)

and we write its explicit expression in terms of the matrix $B$ and
$(\delta_{r})$ as

$\mathscr{L}_{r}=\sum_{i,j=1}^{m}a_{ij}\partial^{2}_{x_{i}x_{j}}+Y_{r},\quad
r\in(0,1]$ (2.7)

where

$Y_{r}:=\langle B_{r}x,D\rangle-\partial_{t}$ (2.8)

and $B_{r}:=r^{2}\delta_{r}B\delta_{\frac{1}{r}}$, i.e.,

$B_{r}=\begin{pmatrix}r^{2}B_{0,0}&r^{4}B_{0,1}&\ldots&r^{2{\kappa}}B_{0,{\kappa}-1}&r^{2{\kappa}+2}B_{0,{\kappa}}\\\
B_{1}&r^{2}B_{1,1}&\ldots&r^{2{\kappa}-2}B_{1,{\kappa}-1}&r^{2{\kappa}}B_{1,{\kappa}}\\\
{\mathbb{O}}&B_{2}&\ldots&r^{2{\kappa}-4}B_{2,{\kappa}-1}&r^{2{\kappa}-2}B_{2,{\kappa}}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
{\mathbb{O}}&{\mathbb{O}}&\ldots&B_{\kappa}&r^{2}B_{{\kappa},{\kappa}}\end{pmatrix}.$
(2.9)

Clearly, $\mathscr{L}_{r}=\mathscr{L}$ for every $r>0$ if and only if
$B=B_{0}$, and the principal part operator $\mathscr{L}_{0}$ is obtained as
the limit of (2.6) as $r\to 0$.

Setting $E_{r}(t)=\exp(-tB_{r})$, we define the translation group related to
$\mathscr{L}_{r}$ as

$(x,t)\circ_{r}(\xi,\tau)=(\xi+E_{r}(\tau)x,t+\tau),\quad(x,t),(\xi,\tau)\in\mathbb{R}^{N+1}.$
(2.10)

######  Remark 2.1.

As it will be useful in the blow-up limit procedure, we point out that the
composition law defined in (2.10) depends continuously on $r\in(0,1]$.
Moreover, taking $r=0$ in (2.9) we find the matrix $B_{0}$ and
“$\;\circ_{r}\\!\\!$” in (2.10) simply becomes the composition law related to
the dilation-invariant operator $\mathscr{L}_{0}$. Thus, “$\;\circ_{r}\\!\\!$”
is a continuous function on the compact set $[0,1]$.

The _homogeneous dimension of ${\mathbb{R}}^{N+1}$ with respect to
$(\delta_{r})_{r>0}$_ is the integer $Q+2$, where $Q$ is the so called
_spatial homogeneous dimension of ${\mathbb{R}}^{N+1}$_

$Q:=m+3m_{1}+\ldots+(2{\kappa}+1)m_{\kappa}.$ (2.11)

We observe that the following equation holds true

$\det\delta_{r}=r^{Q+2}\qquad\text{for every}\ r>0.$

We now recall the notion of _homogeneous function_ in a homogeneous group. We
say that a function $u$ defined on ${\mathbb{R}}^{N+1}$ is homogeneous of
degree $\alpha\in{\mathbb{R}}$ if

$u(\delta_{r}(z))=r^{\alpha}u(z)\qquad\text{for every
}\,z\in{\mathbb{R}}^{N+1}.$

A differential operator $X$ will be called homogeneous of degree
${\beta}\in{\mathbb{R}}$ with respect to $(\delta_{r})_{r\geq 0}$ if

$Xu(\delta_{r}(z))=r^{{\beta}}\left(Xu\right)(\delta_{r}(z))\qquad\text{for
every }\,z\in{\mathbb{R}}^{N+1},$

and for every sufficiently smooth function $u$. Note that, if $u$ is
homogeneous of degree $\alpha$ and $X$ is homogeneous of degree $\beta$, then
$Xu$ is homogeneous of degree $\alpha-\beta$.

As far as we are concerned with the vector fields of the Kolmogorov operator
$\mathscr{L}_{0}$ under the invariance assumption (2.5), we have that
$X_{1},\ldots,X_{m}$ are homogeneous of degree $1$ and $Y$ is homogeneous of
degree $2$ with respect to $(\delta_{r})_{r\geq 0}$. In particular,
$\mathscr{L}_{0}=\sum_{j=1}^{m}X_{j}^{2}+Y$ is homogeneous of degree $2$, and
its fundamental solution $\Gamma_{0}$ is a homogeneous function of degree
$-Q$. As a direct consequence, the estimate
$\Gamma_{0}(z,\zeta)\leq\frac{c}{\|\zeta^{-1}\circ z\|_{K}^{Q}}$ holds for
every $z,\zeta\in{\mathbb{R}}^{N+1}$, with $z\neq\zeta$. Analogous bounds hold
for the first order and second order derivatives of $\Gamma_{0}$, as they are
homogeneous of degree $-Q-1$ and $-Q-2$, respectively.

In the sequel, as we also consider the non dilation-invariant operator
$\mathscr{L}$, we rely on the following estimates (see Proposition 2.7 in
[6]). Let $z_{0}\in{\mathbb{R}}^{N+1}$ and $R_{0}>0$ be a given point and a
given constant. Assume that all the eigenvalues of the matrix $A$ belong to
some interval $[\lambda,\Lambda]\subset{\mathbb{R}}^{+}$. Then there exists a
positive constant $c$, only depending on $\lambda,\Lambda$ and on the matrix
$B$, such that the following bounds hold

$\begin{split}\Gamma(z,\zeta)&\leq\frac{c}{\|\zeta^{-1}\circ z\|_{K}^{Q}},\\\
|\partial_{x_{j}}\Gamma(z,\zeta)&|\leq\frac{c}{\|\zeta^{-1}\circ
z\|_{K}^{Q+1}},\qquad|\partial_{{\xi}_{j}}\Gamma(z,\zeta)|\leq\frac{c}{\|\zeta^{-1}\circ
z\|_{K}^{Q+1}},\\\
|\partial_{x_{i}x_{j}}\Gamma(z,\zeta)&|\leq\frac{c}{\|\zeta^{-1}\circ
z\|_{K}^{Q+2}},\qquad|\partial_{{\xi}_{i}{\xi}_{j}}\Gamma(z,\zeta)|\leq\frac{c}{\|\zeta^{-1}\circ
z\|_{K}^{Q+2}},\\\ |Y\Gamma(z,\zeta)&|\leq\frac{c}{\|\zeta^{-1}\circ
z\|_{K}^{Q+2}},\qquad|Y^{*}\Gamma(z,\zeta)|\leq\frac{c}{\|\zeta^{-1}\circ
z\|_{K}^{Q+2}},\end{split}$ (2.12)

for every $i,j=1,\dots,m$, $z,\zeta\in\mathcal{Q}_{R_{0}}(z_{0})$ with
$z\neq\zeta$. Here $Y^{*}$ denotes the transposed operator of $Y$, defined as
follows

$\displaystyle\int_{{\mathbb{R}}^{N+1}}\phi(x,t)Y^{*}\psi(x,t)\,dxdt=\int_{{\mathbb{R}}^{N+1}}\psi(x,t)Y\phi(x,t)\,dxdt,$

for every $\psi$, $\phi\in C^{\infty}_{0}(\mathbb{R}^{N+1})$.

A similar result holds for the derivatives $\partial_{x_{j}}\Gamma(z,\zeta)$
and $\partial_{\xi_{j}}\Gamma(z,\zeta)$ for $j=m+1,\dots,N$. These functions
need to be considered as derivatives of order $\alpha_{j}$, where the integer
$\alpha_{j}$ has been introduced in (1.13). We have

$\begin{split}|\partial_{x_{j}}\Gamma(z,\zeta)|\leq\frac{c}{\|\zeta^{-1}\circ
z\|_{K}^{Q+\alpha_{j}}},&\qquad|\partial_{\xi_{k}}\Gamma(z,\zeta)|\leq\frac{c}{\|\zeta^{-1}\circ
z\|_{K}^{Q+\alpha_{k}}},\\\
|\partial_{x_{j}}\partial_{\xi_{k}}\Gamma(z,\zeta)|\leq&\frac{c}{\|\zeta^{-1}\circ
z\|_{K}^{Q+\alpha_{j}+\alpha_{k}}},\end{split}$ (2.13)

for every $j,k=1,\dots,N$, $z,\zeta\in\mathcal{Q}_{R_{0}}(z_{0})$ with
$z\neq\zeta$. Note that, as $\alpha_{1}=\dots=\alpha_{m}=1$, the bounds in the
first line of (2.13) agree with the second line of (2.12). The proof of (2.13)
directly follows from the bound (2.59) and (2.60) in [6].

We conclude this Section with the following corollary of the estimates (2.12)
and (2.13), which will be useful in the sequel.

######  Lemma 2.2.

Assume that all the eigenvalues of the matrix $A$ belong to some interval
$[\lambda,\Lambda]\subset{\mathbb{R}}^{+}$. Then there exist two positive
constants $C$, only depending on $\lambda,\Lambda$ and on the matrix $B$, such
that the following holds true. For every $R\in]0,1]$ we have that

$\sup\left\\{\Gamma(z,\zeta):z\in\mathcal{Q}_{\frac{R}{2}}(0),\zeta\in\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)\right\\}\leq\frac{C}{R^{Q}}.$
(2.14)

Moreover

$\sup\left\\{\left|\partial_{x_{j}}\partial_{\xi_{k}}\Gamma(z,\zeta)\right|:z\in\mathcal{Q}_{\frac{R}{2}}(0),\zeta\in\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)\right\\}\leq\frac{C}{R^{Q+\alpha_{j}+\alpha_{k}}}.$
(2.15)

and

$\sup\left\\{\left|Y\partial_{\xi_{k}}\Gamma(z,\zeta)\right|:z\in\mathcal{Q}_{\frac{R}{2}}(0),\zeta\in\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)\right\\}\leq\frac{C}{R^{Q+2+\alpha_{k}}}.$
(2.16)

for every $j,k=1,\dots,N$.

###### Proof.

We first choose $R_{0}>0$ such that $\|\zeta^{-1}\circ z\|_{K}\leq R_{0}$
whenever $z\in\mathcal{Q}_{\frac{1}{2}}(0)$, and $\zeta\in\mathcal{Q}_{1}(0)$.
The existence of such a positive number follows from the pseudo-triangular
inequality (1.15). With this choice of $R_{0}$, we apply (2.12), and we find

$\begin{split}\sup&\left\\{\Gamma(z,\zeta):z\in\mathcal{Q}_{\frac{R}{2}}(0),\zeta\in\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)\right\\}\leq\\\
&c\left(\inf\left\\{\|\zeta^{-1}\circ
z\|_{K}:z\in\mathcal{Q}_{\frac{R}{2}}(0),\zeta\in\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)\right\\}\right)^{-Q}.\end{split}$
(2.17)

We therefore need to estimate the infimum of $\|\zeta^{-1}\circ z\|_{K}$ for
$z\in\mathcal{Q}_{\frac{R}{2}}(0)$ and
$\zeta\in\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)$. We first
consider the points $\bar{z}:=\delta_{\frac{1}{R}}(z)$ and
$\bar{\zeta}:=\delta_{\frac{1}{R}}(\zeta)$ which belong to
$\mathcal{Q}_{\frac{1}{2}}(0)$ and
$\mathcal{Q}_{1}(0)\setminus\mathcal{Q}_{\frac{3}{4}}(0)$, respectively. We
now define the function
$g(\bar{z},\bar{\zeta}):=\|\bar{\zeta}^{-1}\circ_{R}\bar{z}\|_{K}$, which is
continuous on the compact set
$E:=\overline{\mathcal{Q}_{\frac{1}{2}}(0)}\times\overline{\mathcal{Q}_{1}(0)\setminus\mathcal{Q}_{\frac{3}{4}}(0)}\times[0,1]$,
as observed in Remark 2.1. Thus, by Weierstrass’s Theorem, $g$ attains a
minimum $m$ on $E$, i.e.,

$\displaystyle\|\bar{\zeta}^{-1}\circ_{R}\bar{z}\|_{K}\geq
m,\quad\forall\bar{z}\in\overline{\mathcal{Q}_{\frac{1}{2}}(0)},\quad\forall\bar{\zeta}\in\overline{\mathcal{Q}_{1}(0)\setminus\mathcal{Q}_{\frac{3}{4}}(0)},\quad\forall
R\in[0,1].$

Going back to the box of radius $R$, i.e. applying dilation $\delta_{R}$ to
the points $\bar{z}$ and $\bar{\zeta}$ yields

$\displaystyle\|\zeta^{-1}\circ z\|_{K}\geq mR,\quad
z\in\overline{\mathcal{Q}_{\frac{R}{2}}(0)},\quad\zeta\in\overline{\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)},$
(2.18)

and therefore (2.17) becomes

$\begin{split}\sup\left\\{\Gamma(z,\zeta):z\in\mathcal{Q}_{\frac{R}{2}}(0),\zeta\in\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)\right\\}\leq\frac{c}{m^{Q}R^{Q}}=\colon\frac{C}{R^{Q}}\end{split}$
(2.19)

where the constant $C$ does not depend on $R$.

To obtain (2.15) and (2.16), we use the bounds for the derivatives of $\Gamma$
in (2.12) and apply the same arguments as above. ∎

##  3 Preliminary results

In this Section we list some preliminary facts, which are useful in proving
our main results. First, we prove a priori estimates for the derivatives of
$u$ solution to the Kolmogorov equation with right-hand side equal to $0$. To
this end, we represent solutions to $\mathscr{L}u=0$ as convolutions with the
fundamental solution $\Gamma$ of $\mathscr{L}$ and its derivatives
$\partial_{x_{1}}\Gamma,...,\partial_{x_{N}}\Gamma$.

We then prove a mean-value formula for $u$, which is based on the Euclidean
mean-value theorem and on the homogeneity of the fundamental solution.

In order to state the first result of this Section, we recall the notation
introduced in (1.13), that is
$\delta_{r}=\textrm{diag}(r^{\alpha_{1}},\ldots,r^{\alpha_{N}},r^{2})$. In the
sequel we assume that all the eigenvalues of the constant matrix $A$ belong to
some interval $[\lambda,\Lambda]\subset{\mathbb{R}}^{+}$. We are now in
position to state our result.

######  Proposition 3.1.

Let $u$ be a solution to $\mathscr{L}u=0$ in $\mathcal{Q}_{R}(z_{0})$, with
$R\in]0,1]$. Then

$\displaystyle|\partial_{x_{j}}u|(z)\leq\frac{C}{R^{\alpha_{j}}}\|u\|_{L^{\infty}(\mathcal{Q}_{R}(z_{0}))},\quad\text{for
every}\quad z\in\mathcal{Q}_{{\frac{R}{2}}}(z_{0}),\quad j=1,\ldots,N,$

for some positive constant $C$ only depending on $\lambda,\Lambda$ and on the
matrix $B$.

###### Proof.

Without loss of generality, we can assume $z_{0}=0$. Let $\eta_{R}\in
C^{\infty}_{0}(\mathbb{R}^{N+1})$ be a cut-off function such that

$\displaystyle\eta_{R}(x,t)=\chi(\|(x,t)\|_{K}),$ (3.1)

where $\chi\in C^{\infty}([0,+\infty),[0,1])$ is such that $\chi(s)=1$ if
$s\leq\frac{3R}{4}$, $\chi(s)=0$ if $s\geq R$ and
$|\chi^{\prime}|\leq\frac{c}{R}$, $|\chi^{\prime\prime}|\leq\frac{c}{R^{2}}$.
Then, for every $z\in\mathcal{Q}_{R}(0)$ and for $i=1,\ldots,N$, there exists
a constant $c$, only depending on $B$, such that

$\displaystyle|\partial_{x_{i}}\eta_{R}(z)|\leq\frac{c}{R^{\alpha_{i}}},\qquad|\partial_{t}\,\eta_{R}(z)|\leq\frac{c}{R^{2}}.$
(3.2)

Consequently, for every $z\in\mathcal{Q}_{R}(0)$ and $i,j=1,\ldots,m$, we have
$|\partial_{x_{i}x_{j}}^{2}\eta_{R}(z)|\leq\frac{c}{R^{2}}$ and therefore we
obtain a bound for the second order part of $|\mathscr{L}\eta_{R}(z)|$.

Since $\eta_{R}\equiv 1$ in $\in\mathcal{Q}_{\frac{3R}{4}}(0)$, for every
$z\in\mathcal{Q}_{\frac{R}{2}}(0)$ we represent a solution $u$ to
$\mathscr{L}u=0$ as follows

$\displaystyle
u(z)=(\eta_{R}u)(z)=-\int_{\mathcal{Q}_{R}(0)}[\Gamma(z,\cdot)\mathscr{L}(\eta_{R}u)]({\zeta})d{\zeta}.$
(3.3)

Since $\mathscr{L}={\text{\rm div}}(AD_{m})+Y$ and $\mathscr{L}u=0$ by
assumption, (3.3) can be rewritten as

$\begin{split}u(z)=(\eta_{R}u)(z)&=-\int_{\mathcal{Q}_{R}(0)}[\Gamma(z,\cdot){\text{\rm
div}}(AD_{m}(\eta_{R}))u]({\zeta})d{\zeta}\\\
&\quad-\int_{\mathcal{Q}_{R}(0)}[\Gamma(z,\cdot)Y(\eta_{R})u]({\zeta})d{\zeta}\\\
&\quad-2\int_{\mathcal{Q}_{R}(0)}[\Gamma(z,\cdot)\langle
D_{m}u,AD_{m}\eta_{R}\rangle]({\zeta})d{\zeta}.\end{split}$ (3.4)

Integrating by parts the last integral in (3.4), we obtain, for every
$z\in\mathcal{Q}_{\frac{R}{2}}(0)$

$\begin{split}u(z)=(\eta_{R}u)(z)&=\int_{\mathcal{Q}_{R}(0)}[\Gamma(z,\cdot){\text{\rm
div}}(AD_{m}(\eta_{R}))u]({\zeta})d{\zeta}\\\
&\quad-\int_{\mathcal{Q}_{R}(0)}[\Gamma(z,\cdot)Y(\eta_{R})u]({\zeta})d{\zeta}\\\
&\quad+2\int_{\mathcal{Q}_{R}(0)}[\langle
D_{m}^{\zeta}\Gamma(z,\cdot),AD_{m}\eta_{R}\rangle
u]({\zeta})d{\zeta},\end{split}$ (3.5)

where $D_{m}$ is the gradient with respect to $x_{1},\ldots,x_{m}$ and the
superscript in $D_{m}^{\zeta}$ indicates that we are differentiating w.r.t the
variable $\zeta$.

Since $z\in\mathcal{Q}_{\frac{R}{2}}(0)$ and
$\partial_{x_{i}}\eta_{R},Y(\eta_{R})=0$ ($i=1,\ldots,m$) in
$\mathcal{Q}_{\frac{3R}{4}}(0)$, after differentiating under the integral sign
(3.5), we find

$\begin{split}\partial_{x_{j}}u(z)=\partial_{x_{j}}(\eta_{R}u)(z)&=\int_{\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)}[\partial_{x_{j}}\Gamma(z,\cdot){\text{\rm
div}}(AD_{m}(\eta_{R}))u]({\zeta})d{\zeta}\\\
&\quad-\int_{\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)}[\partial_{x_{j}}\Gamma(z,\cdot)Y(\eta_{R})u]({\zeta})d{\zeta}\\\
&\quad+2\int_{\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)}[\langle\partial_{x_{j}}D_{m}^{\zeta}\Gamma(z,\cdot),AD_{m}\eta_{R}\rangle
u]({\zeta})d{\zeta},\\\ \end{split}$

for every $j=1,...,N$. Thus, we obtain

$\begin{split}|\partial_{x_{j}}u(z)|=|\partial_{x_{j}}(\eta_{R}u)(z)|&\leq\int_{\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)}\big{|}[\partial_{x_{j}}\Gamma(z,\cdot){\text{\rm
div}}(AD_{m}(\eta_{R}))u]({\zeta})\big{|}d{\zeta}\\\
&\quad+\int_{\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)}\big{|}[\partial_{x_{j}}\Gamma(z,\cdot)Y(\eta_{R})u]({\zeta})\big{|}d{\zeta}\\\
&\quad+2\int_{\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)}\big{|}[\langle\partial_{x_{j}}D_{m}^{\zeta}\Gamma(z,\cdot),AD_{m}\eta_{R}\rangle
u]({\zeta})\big{|}d{\zeta}\\\
&=:\widetilde{I}_{1}(z)+\widetilde{I}_{2}(z)+\widetilde{I}_{3}(z),\end{split}$

We estimate $\widetilde{I}_{1}(z)$ and $\widetilde{I}_{2}(z)$, for
$z\in\mathcal{Q}_{\frac{R}{2}}(0)$. We have

$\displaystyle\widetilde{I}_{1}(z)$
$\displaystyle\leq\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))}\sup_{\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)}|{\text{\rm
div}}(AD_{m}(\eta_{R}))|\textrm{meas}(\mathcal{Q}_{R}(0))\\!\\!\\!\\!\sup_{\begin{subarray}{c}z\in\mathcal{Q}_{\frac{R}{2}}(0),\\\
\zeta\in\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)\end{subarray}}\\!\\!\\!\\!\big{|}\partial_{x_{j}}\Gamma(z,\zeta)\big{|},$
$\displaystyle\widetilde{I}_{2}(z)$
$\displaystyle\leq\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))}\sup_{\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)}|Y(\eta_{R})|\textrm{meas}(\mathcal{Q}_{R}(0))\\!\\!\\!\sup_{\begin{subarray}{c}z\in\mathcal{Q}_{\frac{R}{2}}(0),\\\
\zeta\in\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)\end{subarray}}\\!\\!\\!\big{|}\partial_{x_{j}}\Gamma(z,\zeta)\big{|}.$

We now apply Lemma 2.2 and obtain

$\sup_{\begin{subarray}{c}z\in\mathcal{Q}_{\frac{R}{2}}(0),\\\
\zeta\in\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)\end{subarray}}\big{|}\partial_{x_{j}}\Gamma(z,\zeta)\big{|}\leq\frac{\widetilde{C}}{R^{Q+\alpha_{j}}}.$
(3.6)

Moreover, by our choice of the cut-off function $\eta_{R}$, we have

$|{\text{\rm
div}}(AD_{m}(\eta_{R}))|\leq\frac{\Lambda\,c}{R^{2}}\quad\text{in}\quad\mathcal{Q}_{R}(0),$
(3.7)

where $\Lambda$ is the largest eigenvalue of $A$. Finally, combining
inequalities (3.6) and (3.7) with
$\textrm{meas}(\mathcal{Q}_{R}(0))=R^{Q+2}\textrm{meas}(\mathcal{Q}_{1}(0))$,
we obtain

$\displaystyle\widetilde{I}_{1}(z)\leq\frac{C}{R^{\alpha_{j}}}\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))},\quad
z\in\mathcal{Q}_{\frac{R}{2}}(0),$ (3.8)

We now estimate $|Y(\eta_{R})|\leq|\langle
Bx,D\eta_{R}\rangle|+|\partial_{t}\eta_{R}|$ in
$\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)$. The bound for the
derivative with respect to time of $\eta_{R}$ is obtained using (3.2).
Moreover

$\displaystyle|\langle
Bx,D\eta_{R}(\zeta)\rangle|\leq\sum_{i,k=1}^{N}|b_{ik}||x_{k}||\partial_{x_{i}}\eta_{R}(\zeta)|\leq
c\sum_{i,k=1}^{N}|b_{ik}|R^{\alpha_{k}-\alpha_{i}},$ (3.9)

where $\zeta\in\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)$.
Notice that in sum (3.9) the exponent $\alpha_{k}-\alpha_{i}$ is always
greater or equal to $-2$, because of the form of the matrix $B$. Since by
assumption $R\leq 1$, we estimate (3.9) as follows

$\displaystyle|\langle
Bx,D\eta_{R}\rangle|\leq\frac{C^{\prime}}{R^{2}},\quad\textrm{in
$\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)$},$ (3.10)

where $C^{\prime}$ is a constant that only depends on the matrix $B$ and on
the constant $c$ in (3.2).

Finally, using again
$\textrm{meas}(\mathcal{Q}_{R}(0))=R^{Q+2}\textrm{meas}(\mathcal{Q}_{1}(0))$,
together with (3.6) and (3.10), we obtain

$\displaystyle\widetilde{I}_{2}(z)\leq\frac{C}{R^{\alpha_{j}}}\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))},\quad
z\in\mathcal{Q}_{\frac{R}{2}}(0),$ (3.11)

where $C$ depends only on the constants $c$ and $\widetilde{C}$ in (3.2) and
(3.6) and on the matrix $B$.

By the same argument we prove that, for a point
$z\in\mathcal{Q}_{\frac{R}{2}}(0)$, we have

$\displaystyle\widetilde{I}_{3}(z)\leq\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))}\frac{c}{R}\
\textrm{meas}(\mathcal{Q}_{R}(0))\\!\\!\\!\\!\\!\\!\\!\\!\sup_{\begin{subarray}{c}z\in\mathcal{Q}_{\frac{R}{2}}(0),\\\
\zeta\in\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)\end{subarray}}\\!\\!\\!\\!\\!\\!\big{|}\partial_{x_{j}}D_{m}^{\zeta}\Gamma(z,\zeta)\big{|}\leq\frac{C}{R^{\alpha_{j}}}\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))}.$

where $C$ denotes once again a constant depending only on $c$, $\widetilde{C}$
and $B$. Combining the inequality above with (3.8) and (3.11), we finally
obtain

$\|\partial_{x_{j}}u\|_{L^{\infty}(\mathcal{Q}_{{\frac{R}{2}}}(0))}\leq\frac{C}{R^{\alpha_{j}}}\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))},\quad
j=1,...,N.$

∎

We state a result analogous to Proposition 3.1, written in terms of the vector
fields $X_{1},\ldots,X_{m},Y$ introduced in (2.1).

######  Proposition 3.2.

Let $u$ be a solution to $\mathscr{L}u=0$ in $\mathcal{Q}_{R}(0)$, for
$R\in]0,1[$, then for any $X_{i},X_{j}\in\\{X_{1},...,X_{m}\\}$, there exists
a constant $C$, only depending on $\lambda,\Lambda$ and on the matrix $B$,
such that

$\displaystyle|X_{i}u|(z)\leq\frac{C}{R}\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))},\quad
z\in\mathcal{Q}_{{\frac{R}{2}}}(0),$
$\displaystyle|X_{i}X_{j}u|(z)\leq\frac{C}{R^{2}}\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))},\quad
z\in\mathcal{Q}_{{\frac{R}{2}}}(0).$

Similarly, we have that

$\displaystyle|Yu|(z)\leq\frac{C}{R^{2}}\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))},\quad
z\in\mathcal{Q}_{{\frac{R}{2}}}(0).$

###### Proof.

The estimate of $X_{1},\dots,X_{m}$ has been proved in Proposition 3.1. The
proof of the remaining estimates is obtained by reasoning as in Proposition
3.1, and using estimates (2.15) and (2.16), respectively. We omit the details
here. ∎

In the sequel, we will need to estimate the second order derivatives of a
solution to $\mathscr{L}u=g$, where $g$ is a polynomial of degree at most two.
To this end, we let

$g_{1}(z)=\langle v,x\rangle,\qquad g_{2}(z)=\langle Mx,x\rangle,$ (3.12)

be two polynomial functions, where $v$ and $M$ denote a constant vector of
${\mathbb{R}}^{N}$ and a $N\times N$ constant matrix, respectively.

######  Lemma 3.3.

Let $\eta_{R}$ be the cut-off function introduced in (3.1) and let $g_{1}$ and
$g_{2}$ be the functions defined in (3.12). Then there exists a positive
constant $C$, only depending on $\lambda,\Lambda$ and on the matrix $B$, such
that

$\displaystyle\bigg{|}\partial^{2}_{x_{i}x_{j}}\int_{\mathcal{Q}_{R}(0)}\Gamma(z,{\zeta})\eta_{R}({\zeta})d{\zeta}\bigg{|}\leq
C,$ (3.13)
$\bigg{|}\partial^{2}_{x_{i}x_{j}}\int_{\mathcal{Q}_{R}(0)}\Gamma(z,{\zeta})\eta_{R}({\zeta})g_{1}({\zeta})d{\zeta}\bigg{|}\leq
CR,$ (3.14)
$\bigg{|}\partial^{2}_{x_{i}x_{j}}\int_{\mathcal{Q}_{R}(0)}\Gamma(z,{\zeta})\eta_{R}({\zeta})g_{2}({\zeta})d{\zeta}\bigg{|}\leq
CR^{2},$ (3.15)

for every $z\in\mathcal{Q}_{\frac{R}{2}}(0)$, $R\in]0,1]$ and for any
$i,j=1,\ldots,m$.

###### Proof.

Reasoning as in the proof of Proposition 2.11 in [6], we write the right-hand
side of (3.13) as

$\begin{split}\partial^{2}_{x_{i}x_{j}}\int_{\mathcal{Q}_{R}(0)}\Gamma({\zeta}^{-1}\circ
z)\eta_{R}({\zeta})d{\zeta}&=\lim_{\varepsilon\to
0}\int_{\mathcal{Q}_{R}(0)\cap\\{\|{\zeta}^{-1}\circ
z\|_{K}\geq\varepsilon\\}}\partial^{2}_{x_{i}x_{j}}\Gamma({\zeta}^{-1}\circ
z)\eta_{R}({\zeta})d{\zeta}\\\
&\quad+\eta_{R}(z)\int_{\|{\zeta}\|_{K}=1}\partial_{x_{i}}\Gamma_{0}({\zeta})\nu_{j}d\sigma({\zeta})\\\
&=:\lim_{\varepsilon\to 0}I_{1}^{0}(\varepsilon,z)+I_{2}^{0}(z).\end{split}$
(3.16)

We rewrite $I_{1}^{0}(\varepsilon,z)$ as

$\begin{split}I^{0}_{1}(\varepsilon,z)&=\int_{\mathcal{Q}_{R}(0)\cap\\{\|{\zeta}^{-1}\circ
z\|_{K}\geq\varepsilon\\}}\partial^{2}_{x_{i}x_{j}}\Gamma({\zeta}^{-1}\circ
z)\left(\eta_{R}({\zeta})-\eta_{R}(z)\right)d{\zeta}\\\
&\quad+\eta_{R}(z)\int_{\mathcal{Q}_{R}(0)\cap\\{\|{\zeta}^{-1}\circ
z\|_{K}\geq\varepsilon\\}}\partial^{2}_{x_{i}x_{j}}\Gamma({\zeta}^{-1}\circ
z)d{\zeta}.\\\ \end{split}$ (3.17)

By the definition of $\eta_{R}$, we have

$\displaystyle 0\leq\eta_{R}\leq
1,\quad\eta_{R}({\zeta})-\eta_{R}(z)=0,\quad\textrm{for any
${\zeta}\in\mathcal{Q}_{\frac{3R}{4}}(0)$,
$z\in\mathcal{Q}_{\frac{R}{2}}(0)$}.$ (3.18)

Thus, taking advantage of Lemma 2.2, we infer

$\begin{split}&\bigg{|}\int_{\mathcal{Q}_{R}(0)\cap\\{\|{\zeta}^{-1}\circ
z\|_{K}\geq\varepsilon\\}}\partial^{2}_{x_{i}x_{j}}\Gamma({\zeta}^{-1}\circ
z)\left(\eta_{R}({\zeta})-\eta_{R}(z)\right)d{\zeta}\bigg{|}\\\
&\quad=\bigg{|}\int_{\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}(0)}}\partial^{2}_{x_{i}x_{j}}\Gamma({\zeta}^{-1}\circ
z)\left(\eta_{R}({\zeta})-\eta_{R}(z)\right)d{\zeta}\bigg{|}\leq\frac{C}{R^{Q+2}}R^{Q+2}=C\end{split}$
(3.19)

Thus we find

$\begin{split}&I^{0}_{2}(z)+\lim_{\varepsilon\to
0}\eta_{R}(z)\int_{\mathcal{Q}_{R}(0)\cap\\{\|{\zeta}^{-1}\circ
z\|_{K}\geq\varepsilon\\}}\partial^{2}_{x_{i}x_{j}}\Gamma({\zeta}^{-1}\circ
z)d{\zeta}=C.\end{split}$ (3.20)

Combining estimates (3.19) and (3.20) we conclude the proof of (3.13).

We now prove (3.14). Reasoning as in (3.16) and exploiting the definition of
$g_{1}$, we can rewrite the right-hand side of (3.14) as

$\begin{split}\partial^{2}_{x_{i}x_{j}}\int_{\mathcal{Q}_{R}(0)}\\!\\!\\!\\!\\!\\!\Gamma({\zeta}^{-1}\circ
z)\eta_{R}({\zeta})\langle v,\xi\rangle d{\zeta}&=\lim_{\varepsilon\to
0}\int_{\mathcal{Q}_{R}(0)\cap\\{\|{\zeta}^{-1}\circ
z\|_{K}\geq\varepsilon\\}}\\!\\!\\!\\!\\!\\!\partial^{2}_{x_{i}x_{j}}\Gamma({\zeta}^{-1}\circ
z)\eta_{R}({\zeta})\langle v,\xi\rangle d{\zeta}\\\ &\quad+\langle
v,x\rangle\eta_{R}(z)\int_{\|{\zeta}\|_{K}=1}\partial_{x_{i}}\Gamma_{0}({\zeta})\nu_{j}d\sigma({\zeta})\\\
&=:\lim_{\varepsilon\to 0}I_{1}^{1}(\varepsilon,z)+I_{2}^{1}(z).\end{split}$
(3.21)

We prove that the first integral in (3.21) uniformly converges as
$\varepsilon\to 0^{+}$. We first rewrite $I_{1}^{1}(\varepsilon,z)$ as

$\begin{split}I_{1}^{1}(\varepsilon,z)&=\int_{\mathcal{Q}_{R}(0)\cap\\{\|{\zeta}^{-1}\circ
z\|_{K}\geq\varepsilon\\}}\partial^{2}_{x_{i}x_{j}}\Gamma({\zeta}^{-1}\circ
z)\left(\eta_{R}({\zeta})-\eta_{R}(z)\right)\langle v,\xi\rangle d{\zeta}\\\
&\quad+\int_{\mathcal{Q}_{R}(0)\cap\\{\|{\zeta}^{-1}\circ
z\|_{K}\geq\varepsilon\\}}\partial^{2}_{x_{i}x_{j}}\Gamma({\zeta}^{-1}\circ
z)\left(\eta_{R}(z)\right)\langle v,\xi-x\rangle d{\zeta}\\\ &\quad+\langle
v,x\rangle\eta_{R}(z)\int_{\mathcal{Q}_{R}(0)\cap\\{\|{\zeta}^{-1}\circ
z\|_{K}\geq\varepsilon\\}}\partial^{2}_{x_{i}x_{j}}\Gamma({\zeta}^{-1}\circ
z)d{\zeta}\\\
&=:I^{\prime}_{1}(\varepsilon,z)+I^{\prime}_{2}(\varepsilon,z)+I^{\prime}_{3}(\varepsilon,z).\end{split}$
(3.22)

To estimate $I^{\prime}_{1}(\varepsilon,z)$ we use the same argument as in
(3.19), with the only difference that now in the integral we have the
additional term $\langle v,\xi\rangle$. We find a bound for this term
observing that

$|\langle v,\xi\rangle|\leq\|v\|\cdot\|{\zeta}\|_{K}\leq\|v\|\cdot R,$ (3.23)

where $\|v\|$ denotes the norm of $v$ in ${\mathbb{R}}^{N}$. Therefore, we
obtain

$\displaystyle|I^{\prime}_{1}(\varepsilon,z)|\leq
C\int_{\mathcal{Q}_{R}(0)\cap\\{\|{\zeta}^{-1}\circ
z\|_{K}\geq\varepsilon\\}}\frac{d{\zeta}}{\|{\zeta}^{-1}\circ
z\|_{K}^{Q+1}}\leq c\,R,$ (3.24)

where $C$ is a constant that depends only on $\lambda,\Lambda,B$ and $v$.

We now show that the same bound holds for $I^{\prime}_{2}(\varepsilon,z)$. We
first observe that

$\displaystyle|\langle v,x-\xi\rangle|\leq\|v\|\cdot\|{\zeta}^{-1}\circ
z\|_{K},$ (3.25)

As a consequence, using again (3.18) and (2.12), we infer

$\displaystyle|I^{\prime}_{2}(\varepsilon,z)|\leq\int_{\mathcal{Q}_{R}(0)\cap\\{\|{\zeta}^{-1}\circ
z\|_{K}\geq\varepsilon\\}}\frac{d{\zeta}}{\|{\zeta}^{-1}\circ
z\|_{K}^{Q+1}}\leq c(R-\varepsilon)\leq c\,R.$ (3.26)

Using (3.24) and (3.26) we obtain

$\displaystyle\lim_{\varepsilon\to
0^{+}}I^{\prime}_{1}(\varepsilon,z)=O(R),\qquad\lim_{\varepsilon\to
0^{+}}I^{\prime}_{2}(\varepsilon,z)=O(R),\quad\textrm{as $R\to 0$}.$ (3.27)

Finally, as for $I^{\prime}_{3}(\varepsilon,z)$, we compute

$\begin{split}&\lim_{\varepsilon\to
0}\int_{\mathcal{Q}_{R}(0)\cap\\{\varepsilon\leq\|{\zeta}^{-1}\circ
z\|_{K}\leq\textbf{c}R\\}}\partial^{2}_{x_{i}x_{j}}\Gamma({\zeta}^{-1}\circ
z)d{\zeta}\\\ &=\lim_{\varepsilon\to
0}\int_{\mathcal{Q}_{R}(0)\cap\\{\varepsilon\leq\|{\zeta}^{-1}\circ
z\|_{K}\leq\textbf{c}R\\}}\partial^{2}_{w_{i}w_{j}}\Gamma(w)e^{-\tau\,{\rm
tr}B}dw\\\ &=-\lim_{\varepsilon\to
0}\int_{\|w\|_{K}=\varepsilon}\partial_{w_{i}}\Gamma(w)e^{-\tau\,{\rm
tr}B}\nu_{j}d\sigma_{j}(w)+\lim_{\varepsilon\to
0}\int_{\|w\|_{K}=\textbf{c}R}\partial_{w_{i}}\Gamma(w)e^{-\tau\,{\rm
tr}B}\nu_{j}d\sigma_{j}(w)\\\
&=-\int_{\|w\|_{K}=1}\partial_{w_{i}}\Gamma_{0}(w)\nu_{j}d\sigma_{j}(w)+\int_{\|w\|_{K}=\textbf{c}R}\partial_{w_{i}}\Gamma(w)e^{-\tau\,{\rm
tr}B}\nu_{j}d\sigma_{j}(w).\end{split}$

We then obtain,

$\begin{split}I_{2}^{1}(z)+\lim_{\varepsilon\to
0}I^{\prime}_{3}(\varepsilon,z)&=\langle
v,x\rangle\eta_{R}(z)\int_{\|{\zeta}\|_{K}=1}\partial_{x_{i}}\Gamma_{0}({\zeta})\nu_{j}d\sigma({\zeta})\\\
&\quad-\langle
v,x\rangle\eta_{R}(z)\int_{\|w\|_{K}=1}\partial_{w_{i}}\Gamma_{0}(w)\nu_{j}d\sigma_{j}(w)\\\
&\quad+\langle
v,x\rangle\eta_{R}(z)\int_{\|w\|_{K}=\textbf{c}R}\partial_{w_{i}}\Gamma(w)e^{-\tau\,{\rm
tr}B}\nu_{j}d\sigma_{j}(w)\\\ &=\langle
v,x\rangle\eta_{R}(z)\int_{\|w\|_{K}=\textbf{c}R}\partial_{w_{i}}\Gamma(w)e^{-\tau\,{\rm
tr}B}\nu_{j}d\sigma_{j}(w).\end{split}$ (3.28)

Keeping in mind that

$\displaystyle\lim_{R\to
0}\int_{\|w\|_{K}=\textbf{c}R}\partial_{w_{i}}\Gamma(w)e^{-\tau\,{\rm
tr}B}\nu_{j}d\sigma_{j}(w)=\int_{\|w\|_{K}=1}\partial_{w_{i}}\Gamma_{0}(w)\nu_{j}d\sigma_{j}(w)=c^{\prime},$

we finally find

$I_{2}^{1}(z)+\lim_{\varepsilon\to
0}I^{\prime}_{3}(\varepsilon,z)=O(R),\quad\textrm{as $R\to 0$}.$ (3.29)

Identity (3.14) follows from (3.24), (3.26) and (3.29).

By the same argument, we obtain

$\partial^{2}_{x_{i}x_{j}}\int_{\mathcal{Q}_{R}(0)}\Gamma({\zeta}^{-1}\circ
z)\eta_{R}({\zeta})\langle M\xi,\xi\rangle d{\zeta}=O(R^{2}).$ (3.30)

We omit the details here as the procedure is analogous. ∎

From Proposition 3.1 and Lemma 3.3, we derive the following result.

######  Lemma 3.4.

Let $w$ be a solution to $\mathscr{L}w=\langle v,\xi\rangle+\langle
M\xi,\xi\rangle$ in $\mathcal{Q}_{R}(0)$, where $v$ and $M$ are as in (3.12).
Then

$\displaystyle|\partial^{2}_{x_{i}x_{j}}w(z)|\leq\frac{C}{R^{2}}\|w\|_{L^{\infty}(\mathcal{Q}_{R}(0))}+CR,$
(3.31)

for every $i,j=1,\ldots,m,0<R\leq 1$ and for any
$z\in\mathcal{Q}_{\frac{R}{2}(0)}$.

###### Proof.

Reasoning as in Proposition 3.1, we obtain

$\begin{split}|\partial^{2}_{x_{i}x_{j}}w(z)|&\leq\int_{\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)}\big{|}[\partial^{2}_{x_{i}x_{j}}\Gamma(z,\cdot){\text{\rm
div}}(AD_{m}(\eta_{R}))w]({\zeta})\big{|}d{\zeta}\\\
&\quad+\int_{\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{(3/4)\varrho^{k}}(0)}\big{|}[\partial^{2}_{x_{i}x_{j}}\Gamma(z,\cdot)Y(\eta_{R})w]({\zeta})\big{|}d{\zeta}\\\
&\quad+2\int_{\mathcal{Q}_{R}(0)\setminus\mathcal{Q}_{\frac{3R}{4}}(0)}\big{|}[\partial^{2}_{x_{i}x_{j}}\Gamma(z,\cdot)\langle
D_{m}w,AD_{m}\eta_{R}\rangle]({\zeta})\big{|}d{\zeta}\\\
&\quad+\bigg{|}\Big{[}\partial^{2}_{x_{i}x_{j}}\int_{\mathcal{Q}_{R}(0)}[\Gamma(z,\cdot)\eta_{R}]({\zeta})\langle
v,\xi\rangle d{\zeta}\Big{]}\bigg{|}\\\
&\quad+\bigg{|}\Big{[}\partial^{2}_{x_{i}x_{j}}\int_{\mathcal{Q}_{R}(0)}[\Gamma(z,\cdot)\eta_{R}]({\zeta})\langle
M\xi,\xi\rangle d{\zeta}\Big{]}\bigg{|}\\\
&=:\overline{I}_{1}(z)+\overline{I}_{2}(z)+\overline{I}_{3}(z)+\overline{I}_{4}(z)+\overline{I}_{5}(z).\end{split}$
(3.32)

The terms $\overline{I}_{1}(z),\overline{I}_{2}(z),\overline{I}_{3}(z)$ were
already estimated in Proposition 3.1 as

$\displaystyle\overline{I}_{1}(z),\overline{I}_{2}(z),\overline{I}_{3}(z)\leq\frac{C}{R^{2}}\|w\|_{L^{\infty}(\mathcal{Q}_{R}(0))},\quad
z\in\mathcal{Q}_{\frac{R}{2}(0)}.$

Additionally, $\overline{I}_{4}(z)$ and $\overline{I}_{5}(z)$ are $O(R)$ in
virtue of Lemma 3.3 and thus (3.31) is proved. ∎

We now prove a mean value theorem for solutions $u$ to $\mathscr{L}u=0$ in
cylinders $\mathcal{Q}_{R}({\zeta})$.

######  Proposition 3.5 (Mean value theorem).

Let ${\zeta}$ be any point of ${\mathbb{R}}^{N+1}$, and let $u$ be a solution
to $\mathscr{L}u=0$ in $\mathcal{Q}_{R}({\zeta})$, with $R\in]0,1]$. Then the
following estimate holds

$\displaystyle|u(z)-u({\zeta})|\leq\frac{C}{R}d_{K}(z,{\zeta})\|u\|_{L^{\infty}(\mathcal{Q}_{R}({\zeta}))},$
(3.33)

for every $z\in\mathcal{Q}_{\frac{R}{2}}({\zeta})$. Here $C$ is a constant
that only depends on $\lambda,\Lambda$ and on the matrix $B$.

###### Proof.

Thanks to the left-invariance of the operator $\mathscr{L}$, it is not
restrictive to assume ${\zeta}=0$, then we need to prove

$\displaystyle|u(z)-u(0)|\leq\frac{C}{R}\|z\|_{K}\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))}.$

Consider $z=(x,t)\in\mathcal{Q}_{\frac{R}{2}}(0)$, and apply the standard
mean-value theorem

$\begin{split}|u(z)-u(0)|&=|u(x_{1},\ldots,x_{N},t)-u(0,\ldots,0,0)|\\\
&\leq\sum_{i=1}^{N}|x_{i}|\,|\partial_{x_{i}}u(\vartheta_{1}x_{1},\ldots,\vartheta_{N}x_{N},t)|+|t|\,|Yu(0,\ldots,0,\vartheta
t),\end{split}$ (3.34)

where $\vartheta_{1},\ldots,\vartheta_{N},\vartheta\in]0,1[$. For every
$i=1,\ldots,N$, we have $|x_{i}|\leq\|z\|_{K}^{\alpha_{i}}\leq
R^{\alpha_{i}}$, and
$(\vartheta_{1}x_{1},\ldots,\vartheta_{N}x_{N},t)\in\mathcal{Q}_{\frac{R}{2}}(0)$.
Then, by Proposition 3.1, we find

$\displaystyle|\partial_{x_{i}}u(\vartheta_{1}x_{1},\ldots,\vartheta_{N}x_{N},t)|\leq\frac{c}{R^{\alpha_{i}}}\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))}.$

so that

$\displaystyle|x_{i}|\,|\partial_{x_{i}}u(\vartheta_{1}x_{1},\ldots,\vartheta_{N}x_{N},t)|\leq\frac{c}{R}\|z\|_{K}\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))}.$

Analogously, we have that $|\vartheta t|\leq|t|\leq\|z\|_{K}^{2}\leq R^{2}$,
and from Proposition 3.2 it follows that

$\displaystyle|Yu(0,\dots,0,\vartheta
t)|\leq\frac{c}{R^{2}}\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))},$

thus

$\displaystyle|t|\,|Yu(0,\dots,0,\vartheta
t)|\leq\frac{c}{R}\|z\|_{K}\|u\|_{L^{\infty}(\mathcal{Q}_{R}(0))}.$

The proof of the proposition can be obtained by combining the above estimates.
∎

##  4 Taylor formula

In this Section we prove Theorem 1.2. The proof is based on the method
introduced by Pagliarani, Pascucci and Pignotti in [17] for the dilation-
invariant operator $\mathscr{L}_{0}$ and then generalized by Pagliarani and
Pignotti in [18] to the non dilation-invariant one. In both articles the
function $u$ is assumed to belong to some Hölder space
$C^{2+\alpha}_{\mathscr{L}}(\Omega)$ and the following bound

$\displaystyle
u({\zeta})-T^{2}_{z}u({\zeta})=O(\|z^{-1}\circ{\zeta}\|_{K}^{2+\alpha})\quad\text{as}\quad\zeta\to
z$ (4.1)

is proved instead of (1.8). Note that the case $\alpha=0$ is covered in (4.1),
but (1.8) contains a stronger statement. We follow the same procedure
introduced in [17] and [18] and we point out the modifications needed to deal
with our slightly different situation.

We next introduce some further notation. We define the spaces
$V_{0},\dots,V_{\kappa}$ as the vector subspaces of $\mathbb{R}^{N}$ which are
invariant with respect to dilation $(\delta_{r})_{r>0}$ introduced in (1.12).
Specifically, for $n=0,\ldots,\kappa$, we set

$V_{n}:=\\{0\\}^{\bar{m}_{n-1}}\times\mathbb{R}^{m_{n}}\times\\{0\\}^{N-\bar{m}_{n}},$

where $\bar{m}_{n}:=m_{0}+\ldots m_{n}$, with $m_{-1}\equiv 0$. Moreover, we
let $x^{[n]}$ be the projection of $x\in\mathbb{R}^{N}$ on $V_{n}$. Note that

$\displaystyle\mathbb{R}^{N}=\bigoplus_{n=0}^{\kappa}V_{n},\qquad
x=x^{[0]}+\dots+x^{[\kappa]},$ (4.2)

for every $x\in{\mathbb{R}}^{N}$. Moreover, in accordance with the dilation
$(\delta_{r})_{r>0}$, we have

$\displaystyle\delta_{r}(x^{[n]})=r^{2n+1}x^{[n]},\quad\forall x^{[n]}\in
V_{n},$ (4.3)

for every $n=0,\ldots,\kappa$. In virtue of assumption [H.1], the linear
application $B^{n}:V_{0}\rightarrow V_{n}$ is surjective; however, it is in
general not injective. Thus, we define the subspaces $V_{0,n}\subset V_{0}$ as
follows

$\displaystyle V_{0,n}:=\ker(B^{n})^{\perp}.$

The linear map $B^{n}:V_{0,n}\rightarrow V_{n}$ is now bijective.

The method of the proof relies on the construction of a finite sequence of
points which connect $z=(x,t)$ and ${\zeta}=(\xi,\tau)$ and are located along
suitable trajectories. More precisely, we start from $z$ and choose
$z_{1}=(x_{1},t_{1})$ as the point along the integral curve of the drift $Y$
satisfying the condition $t_{1}=\tau$. We then move along the integral paths
of $X_{1},\ldots,X_{m}$ to a point $z_{2}=(x_{2},t_{2})$ such that
$x_{2}^{[0]}=\xi^{[0]}$ and $t_{2}=\tau$. This allows us to exploit the
regularity of $u$ along the vector fields $X_{1},\ldots,X_{m},Y$ and estimate
the remainder in (1.8) in terms of the homogeneous norm of the new points.

Since we have no apriori regularity of $u$ with respect to other vector
fields, we increment the higher level coordinates $x^{[1]},\dots,x^{[\kappa]}$
by moving along trajectories defined as concatenations of integral curves of
$X_{1},\ldots,X_{m},Y$. Specifically, for any $z\in\mathbb{R}^{N+1}$ and
$s\in\mathbb{R}$ we define iteratively the family of trajectories
$(\gamma_{v,s}^{(n)}(z))_{n=0,\ldots,\kappa}$ as follows

$\begin{split}\gamma_{v,s}^{(0)}(z)&=e^{sX_{v}}(z)=(x+sv,t)\\\
\gamma_{v,s}^{(n+1)}(z)&=e^{-s^{2}Y}(\gamma_{v,-s}^{(n)}(e^{s^{2}Y}(\gamma_{v,s}^{(n)}(z)))),\end{split}$
(4.4)

where $v$ is a suitable vector in $V_{0}$, and
$X_{v}=v_{1}\partial_{x_{1}}+\dots+v_{m}\partial_{x_{m}}$.

At this point we need to distinguish the dilation-invariant operators from the
non dilation-inviariant ones. In the first case, the trajectories
$(\gamma_{v,s}^{(n)}(z))_{n=0,\ldots,\kappa}$ have the remarkable property of
modifying the components $x^{[n]}+\dots+x^{[\kappa]}$ leaving unchanged the
components $x^{[0]}+\dots+x^{[n-1]}$; thus, we reach the point $\zeta$ after
$\kappa$ steps. The proof of Theorem 1.2, for dilation-invariant operators,
follows by exploiting the regularity of $u$ with respect to
$X_{1},\dots,X_{m},Y$, as we connect $z$ to $\zeta$ along integral curves of
the vector fields $X_{1},\dots,X_{m},Y$. The next example illustrates the
geometric construction in the simplest case, corresponding to $\kappa=1$.

######  Example 4.1.

We consider the degenerate Kolmogorov operator

$\mathcal{K}_{0}:=\partial^{2}_{xx}+x\partial_{y}-\partial_{t}$

and show how to use the trajectories defined in (4.4) to connect an arbitrary
point $z\in\mathbb{R}^{3}$ with the origin. In this case, we have

$B=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}$

and thus

$\displaystyle e^{sX}(x,y,t)=(x+s,y,t),\quad e^{sY}(x,y,t)=(x,y+sx,t-s).$

Moreover,

$\displaystyle\mathbb{R}^{2}=V_{0}\oplus
V_{1}=\textrm{\rm{span}}\\{e_{1}\\}\oplus\textrm{\rm{span}}\\{e_{2}\\},\quad
V_{0,0}=V_{0,1}=\textrm{\rm{span}}\\{e_{1}\\}.$

Let $z=(x,y,t)$ be a point in $\mathbb{R}^{3}$, and consider for simplicity
$\zeta=(0,0,0)$. We first adjust the temporal component by moving along the
drift $Y$, and we reach the point

$\displaystyle z_{1}=e^{tY}(z)=(x,y+tx,0).$

We then move along the integral curve of the vector field $X$ to make $x$
equal to $0$:

$\displaystyle
z_{2}=e^{s_{0}X}(z_{1})=(x+s_{0},y+tx,0)=(0,y+tx,0),\quad\text{by
choosing}\quad s_{0}=-x.$

We reached the point $z_{2}\in V_{1}$ and we plan to steer it to $(0,0,0)$. We
move along a curve defined as concatenation of integral paths of $X$ and $Y$
as follows:

$\begin{split}z_{3}&=e^{s_{1}X}(z_{2})=(s_{1},y+tx,0),\\\
z_{4}&=e^{s_{1}^{2}Y}(z_{3})=(s_{1},y+tx+s_{1}^{3},-s_{1}^{2}),\\\
z_{5}&=e^{-s_{1}X}(z_{4})=(0,y+tx+s_{1}^{3},-s_{1}^{2}),\\\
z_{6}&=e^{-s_{1}^{2}Y}(z_{5})=(0,y+tx+s_{1}^{3},0),\end{split}$ (4.5)

and we reach the point $\zeta=(0,0,0)$ if we choose
$s_{1}=(-tx-y)^{\frac{1}{3}}$.

When considering a non dilation-invariant operator $\mathscr{L}$, the method
illustrated above fails. Indeed, in this case the trajectory
$(\gamma_{v,s}^{(n)}(z))$ may affect the components $x^{[0]}+\dots+x^{[n-1]}$,
as the following example shows.

######  Example 4.2.

We consider the degenerate Kolmogorov operator

$\displaystyle\mathcal{K}:=\partial_{xx}^{2}+x\partial_{y}+x\partial_{x}-\partial_{t}.$
(4.6)

In this case, $B$ takes the form

$B=\begin{pmatrix}1&0\\\ 1&0\end{pmatrix}.$

and therefore the operator $\mathcal{K}$ is non dilation-invariant. Let us
emphasize the differences with the dilation-invariant case studied in Example
4.1. We denote again the points in $\mathbb{R}^{3}$ by $z=(x,y,t)$ and
consider $\zeta=(0,0,0)$. The first two steps of the procedure used in Example
4.1 allow us to move from $z$ to some point $z_{1}=(x_{1},y_{1},0)$, then to
some other point $z_{2}=(0,y_{2},0)$. The difference with the homogeneous case
arises in the third step, i.e. when we are dealing with the $y$-variable.

Let us suppose we want to move from any point $z=(0,y,0)\in V_{1}$ to the
origin $(0,0,0)$. If we reproduce the same construction as in (4.5), we find:

$\begin{split}z_{1}&=e^{sX}(z)=(s,y,0),\\\
z_{2}&=e^{s^{2}Y}(z_{1})=(se^{s^{2}},-s+se^{s^{2}}+y,-s^{2}),\\\
z_{3}&=e^{-sX}(z_{2})=(se^{s^{2}}-s,-s+se^{s^{2}}+y,-s^{2}),\\\
z_{4}&=e^{-s^{2}Y}(z_{3})=(s(1-e^{-s^{2}}),s(1-e^{-s^{2}})+y,0).\end{split}$
(4.7)

If we choose $s$ such that $s(1-e^{-s^{2}})=-y$, we obtain $z_{4}=(-y,0,0)$,
so that its second component is zero but, in constrast with the previous
Example 4.1, we have that $z_{4}$ doesn’t agree with our target point
$\zeta=(0,0,0)$.

In order to reach the point $\zeta=(0,0,0)$ also in the case of non dilation-
invariant operators, we rely on the method introduced by Pagliarani and
Pignotti in [18]. In the case of the operator $\mathcal{K}$ in (4.6) it is
sufficient to use once more the integral curve of the vector field
$X=\partial_{x}$. In the case of more general operators a further topological
argument is needed to conclude the construction. We refer to [18] for a
detailed description of this construction.

We are now ready to prove our result.

###### Proof of Theorem 1.2.

Let $z=(x,t),{\zeta}=(\xi,\tau)$ be two given points of $\Omega$. As explained
above, the proof relies on a finite sequence of integral paths of the vector
fields $X_{1},\ldots,X_{m}$ and $Y$ connecting $z$ to ${\zeta}$. We use the
construction made by Pagliarani, Pascucci and Pignotti [17] for a dilation-
invariant operator $\mathscr{L}$. In this case the trajectories
$(\gamma_{v,s}^{(n)}(z))_{n=0,\ldots,\kappa}$ defined in (4.4) are explicitely
given and we prove that (1.8) holds. We then discuss the modifications needed
to deal with any non dilation-invariant operator $\mathscr{L}$, as introduced
by Pagliarani and Pignotti in [18].

As a preliminary result, we prove our claim (1.8) under the assumption that
the points $z=(x,t)$ and $\zeta=(\xi,\tau)$ have the same temporal component
$t=\tau$, by a finite iteration on $n=0,\dots,\kappa$. We remove this
assumption in the last part of the proof.

Base case $n=0$. In this case, we are only changing the variables $x_{i}$, for
$i=1,\ldots,m$, moving along the direction $e^{s_{0}X_{v_{0}}}$ where
$v_{0}=(v_{0,1},\ldots,v_{0,m},0\ldots,0)$ is a suitable unit vector in
$V_{0}$. Thus, equation (1.7), with $z=(x,t)$ and $\zeta=(x+s_{0}v_{0},t)$,
rewrites as

$\displaystyle
T^{2}_{z}u({\zeta})=u(x,t)+\sum_{i=1}^{m}\partial_{{x}_{i}}u(x,t)s_{0}v_{0,i}+\frac{s_{0}^{2}}{2}\sum_{i,j=1}^{m}\partial^{2}_{x_{i},x_{j}}u(x,t)v_{0,i}v_{0,j}.$
(4.8)

We observe that $\|z^{-1}\circ\zeta\|^{2}_{K}=|s_{0}|^{2}$ and therefore we
want to show that

$\displaystyle
u({\zeta})-T^{2}_{z}u({\zeta})=o(|s_{0}|^{2})\quad\text{as}\quad s_{0}\to 0.$
(4.9)

By the multidimensional euclidean mean-value theorem, there exist
$(\bar{v}_{i,j})_{1\leq i,j\leq m}$, with
$\bar{v}_{i,j}\in{\rm{span}}\\{e_{1},\ldots,e_{m}\\}$ and
$|\bar{v}_{i,j}|\leq|v_{0}|$ such that

$\begin{split}u({\zeta})-T^{2}_{z}u(\zeta)&=\frac{s_{0}^{2}}{2}\sum_{i,j=1}^{m}(\partial^{2}_{x_{i},x_{j}}u(x+s_{0}\bar{v}_{i,j},t)v_{0,i}v_{0,j}-\partial^{2}_{x_{i},x_{j}}u(x,t))v_{0,i}v_{0,j}\\\
&=o(|s_{0}|^{2})\quad\textit{as $s_{0}\to 0$},\end{split}$ (4.10)

where we have used the continuity of the second order derivatives of $u$.
Thus, we have proved (4.9) and we are done.

Let us remark that we don’t need the dilation-invariance property for $Y$, as
we don’t make use of the vector field $Y$ in this part of the construction.

Inductive step. We now suppose the thesis true for a given non negative
$n<\kappa$ and we prove it for $n+1$. For every $z,\zeta\in{\mathbb{R}}^{N+1}$
we set

$\widetilde{T}^{2}_{z}u({\zeta}):=T^{2}_{z}u({\zeta})-u(z).$ (4.11)

We define the points

$\begin{split}z&=(x,t),z_{1}=\gamma_{v,s}^{(n)}(z),z_{2}=e^{s^{2}Y}(z_{1})\\\
z_{3}&=\gamma_{v,-s}^{(n)}(z_{2}),z_{4}=e^{-s^{2}Y}(z_{3})=\gamma_{v,s}^{(n+1)}(z)\end{split}$

where $v$ is the unique unitary vector in $V_{0,n+1}\subset V_{0}$, defined as
$v=\frac{w}{|w|}$, where $w$ is the vector in $V_{0,n+1}$ such that
$B^{n+1}w={\zeta}^{[n+1]}-z^{[n+1]}$ and $s=|w|^{\frac{1}{2(n+1)+1}}$. We aim
to prove that

$\displaystyle u(z_{4})-T^{2}_{z}u(z_{4})=o(\|z^{-1}\circ
z_{4}\|_{K}^{2})=o(|s|^{2})\quad\text{as}\quad s\to 0.$ (4.12)

We now rewrite (4.12) by using the notation (4.11) as follows

$\begin{split}u(z_{4})-T^{2}_{z}u(z_{4})&=\boxed{u(z_{4})-u(z_{3})}_{(1)}\\\
&+\boxed{u(z_{3})-u(z_{2})-\widetilde{T}^{2}_{z_{2}}u(z_{3})}_{(2)}\\\
&+\boxed{u(z_{2})-u(z_{1})}_{(3)}\\\
&+\boxed{\widetilde{T}^{2}_{z_{1}}u(z)+u(z_{1})-u(z)}_{(4)}\\\
&+\boxed{\widetilde{T}^{2}_{z_{2}}u(z_{3})-\widetilde{T}^{2}_{z_{1}}u(z)}_{(5)}\\\
&-\boxed{\widetilde{T}^{2}_{z}u(z_{4})}_{(6)}.\end{split}$ (4.13)

By the inductive hypothesis, the second and the forth difference are
$o(|s|^{2})$ as $s\to 0$. Moreover, recalling (1.7), we have that
$\widetilde{T}^{2}_{z}u(z_{4})\equiv 0$, being $x_{4}^{[0]}=x^{[0]}$ and
$t_{4}=t$.

We next apply definition (1.7) to the fifth difference, and we find

$\displaystyle\begin{split}\widetilde{T}^{2}_{z_{2}}u(z_{3})-\widetilde{T}^{2}_{z_{1}}u(z)=&-s\sum_{i=1}^{m}(\partial_{x_{i}}u(z_{2})-\partial_{x_{i}}u(z_{1}))v_{i}\\\
&-\frac{s^{2}}{2}\sum_{i,j=1}^{m}(\partial^{2}_{x_{i}x_{j}}u(z_{2})-\partial^{2}_{x_{i}x_{j}}u(z_{1}))v_{i}v_{j}.\end{split}$
(4.14)

As a consequence of condition (1.6), we obtain the following equation

$\partial_{x_{i}}u(z_{2})-\partial_{x_{i}}u(z_{1})=\partial_{x_{i}}u(e^{s^{2}Y}(z_{1}))-\partial_{x_{i}}u(z_{1})=o(|s|).$

Using the previous equation and the continuity of second order derivatives of
$u$, we find that (4.14) is equal to $o(|s|^{2})$.

We now observe that

$\displaystyle u(z_{4})-u(z_{3})=u(e^{-s^{2}Y}(z_{3}))-u(z_{3}).$

By applying the mean value theorem along the direction of the drift, we find
that there exists $\bar{s}$ such that

$\displaystyle u(e^{-s^{2}Y}(z_{3}))-u(z_{3})=-s^{2}Yu(e^{\bar{s}Y}(z_{3})),$

where $|\bar{s}|\leq|s|$. Similarly we obtain that

$\displaystyle u(z_{2})-u(z_{1})=s^{2}Yu(e^{\widetilde{s}Y}(z_{1})),$

where again $\widetilde{s}$ verifies $|\widetilde{s}|\leq|s|$.

By letting $s\to 0$, we find that $\bar{s},\widetilde{s}\to 0$, and therefore,
using the continuity of $Yu$, we have showed that the sum of the first and the
third difference in (4.13) is again equal to $o(|s|^{2})$ as $s\to 0$. This
proves (4.12) and therefore concludes the proof of the inductive step.

As already pointed out, the construction of the trajectories in the case of
non dilation-invariant operators requires the adjustments introduced in [18],
to deal with the fact that the term $\widetilde{T}^{2}_{z}u(z_{4})$ in (4.13)
fails to vanish. Indeed, with the notation (4.2), $x$ writes as
$x=x^{[0]}+x^{[1]}+\dots+x^{[\kappa]}$, and we have
$\widetilde{T}^{2}_{z}u(z_{4})\neq 0$ whenever $x_{4}^{[0]}\neq x^{[0]}$. To
overcome this problem, we define a new point $z_{5}=\left(x_{5},t_{5}\right)$
as follows:

$\displaystyle x_{5}^{[0]}=x^{[0]},\quad
x_{5}^{[1]}=x_{4}^{[1]},\dots,x_{5}^{[\kappa]}=x_{4}^{[\kappa]},\quad
t_{5}=t_{4}.$

Note that in [18] it is proved that

$\displaystyle\left|x^{[0]}-x_{4}^{[0]}\right|\leq
C\|z^{-1}\circ{\zeta}\|_{K},$

for some positive constant $C$ only depending on the matrix $B$. Then

$u(z_{5})-u(z_{4})=o(\|z^{-1}\circ{\zeta}\|_{K}^{2})\quad\text{as}\quad\zeta\to
z.$ (4.15)

With this modification, expression (4.13) is replaced by

$u(z_{5})-T^{2}_{z}u(z_{5})=u(z_{5})-u(z_{4})-\widetilde{T}^{2}_{z}u(z_{5})+o(\|z^{-1}\circ{\zeta}\|_{K}^{2})\quad\text{as}\quad\zeta\to
z.$

Moreover, $x_{5}^{[0]}=x^{[0]}$ and $t_{5}=t$ yield
$\widetilde{T}^{2}_{z}u(z_{5})=0$. From (4.15) it then follows that

$u(z_{5})-T^{2}_{z}u(z_{5})=o(\|z^{-1}\circ{\zeta}\|_{K}^{2})\quad\text{as}\quad\zeta\to
z.$

We are now in position to prove (1.8). We first consider the point
$\bar{z}=e^{(t-\tau)Y}(z)=(e^{(t-\tau)B}x,\tau)$ and write

$\displaystyle
u({\zeta})-T^{2}_{z}u({\zeta})=u({\zeta})-T^{2}_{\bar{z}}u({\zeta})+T^{2}_{\bar{z}}u({\zeta})-T^{2}_{z}u({\zeta}).$
(4.16)

Thanks to the previous steps, the first difference is
$o(\|\bar{z}^{-1}\circ{\zeta}\|^{2}_{K})=o(\|z^{-1}\circ{\zeta}\|_{K}^{2})$ as
$\|z^{-1}\circ{\zeta}\|_{K}^{2}\to 0$, since $\zeta$ and $\bar{z}$ have the
same temporal component $\tau$. At the same time, the second difference in
(4.16) can be rewritten as

$\begin{split}T^{2}_{\bar{z}}u({\zeta})-T^{2}_{z}u({\zeta})&=u(\bar{z})-u(z)+\sum_{i=1}^{m}\left(\partial_{x_{i}}u(\bar{z})-\partial_{x_{i}}u(z)\right)(\xi_{i}-x_{i})\\\
&\quad+\frac{1}{2}\sum_{i,j=1}^{m}\left(\partial^{2}_{x_{i}x_{j}}u(\bar{z})-\partial^{2}_{x_{i}x_{j}}u(z)\right)(\xi_{i}-x_{i})(\xi_{j}-x_{j})+Yu(z)(\tau-t).\end{split}$
(4.17)

Using the mean value theorem along the drift, we can rewrite difference
$u(\bar{z})-u(z)$ in (4.17) as

$\displaystyle u(e^{(t-\tau)Y}(z))-u(z)=(t-\tau)Yu(e^{\delta Y}(z)),$ (4.18)

where $\delta$ is such that $|\delta|\leq|t-\tau|$. Hence, we obtain

$\displaystyle u(\bar{z})-u(z)-Y(z)(t-\tau)=(t-\tau)(Yu(e^{\delta
Y}(z))-Y(z)),$ (4.19)

which is $o(|t-\tau|)=o(\|z^{-1}\circ{\zeta}\|_{K}^{2})$ as
$\|z^{-1}\circ{\zeta}\|_{K}^{2}\to 0$, thanks to the continuity of $Yu$.

Finally, using again condition (1.6) and the continuity of the second
derivatives of $u$, we obtain that the second and third difference in (4.17)
are also $o(|t-\tau|)=o(\|z^{-1}\circ{\zeta}\|_{K}^{2})$ as
$\|z^{-1}\circ{\zeta}\|_{K}^{2}\to 0$.

Combining all the previous estimates, we obtain

$T^{2}_{\bar{z}}u({\zeta})-T^{2}_{z}u({\zeta})=o(\|z^{-1}\circ{\zeta}\|_{K}^{2}),\quad\textrm{as
$\|z^{-1}\circ{\zeta}\|_{K}^{2}\to 0$}.$ (4.20)

and therefore (4.16) is equal to $o(\|z^{-1}\circ{\zeta}\|_{K}^{2})$ as
$\|z^{-1}\circ{\zeta}\|_{K}^{2}\to 0$. This concludes the proof. ∎

##  5 Proof of Theorem 1.5

We first prove a preliminary lemma, which is a straightforward consequence of
the maximum principle.

######  Lemma 5.1.

Given $\varphi\in C(\partial\mathcal{Q}_{R}(z_{0}))$ and $g\in
C_{b}(\mathcal{Q}_{R}(z_{0}))$, we let $v$ be the solution to the following
Dirichlet problem

$\left\\{\begin{array}[]{ll}\mathscr{L}v=g,&\textit{in
$\mathcal{Q}_{R}(z_{0})$},\\\ v=\varphi,&\textit{in
$\partial\mathcal{Q}_{R}(z_{0})$}.\end{array}\right.$

Then, the following holds

$\displaystyle\|v\|_{{L}^{\infty}(\mathcal{Q}_{R}(z_{0}))}\leq\|\varphi\|_{{L}^{\infty}(\mathcal{Q}_{R}(z_{0}))}+|t-t_{1}|\|g\|_{{L}^{\infty}(\mathcal{Q}_{R}(z_{0}))},$
(5.1)

where $t_{1}=t_{0}-R^{2}$ is the time coordinate of the basis of the cylinder
$\mathcal{Q}_{R}(z_{0})$.

###### Proof.

We introduce the function
$w(x,t):=(t-t_{1})\|g\|_{{L}^{\infty}(\mathcal{Q}_{R}(z_{0}))}+\|\varphi\|_{{L}^{\infty}(\mathcal{Q}_{R}(z_{0}))}$
and we let $u:=v-w$. Clearly, $u$ satisfies
$\mathscr{L}u=g+\|g\|_{{L}^{\infty}(\mathcal{Q}_{R}(z_{0}))}\geq 0$ in
$\mathcal{Q}_{R}(z_{0})$. Moreover, as $v\equiv\varphi$ on the boundary of
$\mathcal{Q}_{R}(z_{0})$, we have
$u=\varphi-(t-t_{1})\|g\|_{{L}^{\infty}(\mathcal{Q}_{R}(z_{0}))}-\|\varphi\|_{{L}^{\infty}(\mathcal{Q}_{R}(z_{0}))}\leq
0$ in $\partial\mathcal{Q}_{R}(z_{0})$. By the strong maximum principle, it
follows that $u(x,t)\leq 0$ in $\mathcal{Q}_{R}(z_{0})$. Replacing $v$ by
$-v$, estimate (5.1) follows at once. ∎

###### Proof of Theorem 1.5.

We first prove assertion _(ii)_. We denote
$\mathcal{Q}_{k}=\mathcal{Q}_{\varrho^{k}}(0)$, $\varrho=\frac{1}{2}$ and we
consider the following sequence of Dirichlet problems:

$\left\\{\begin{array}[]{ll}\mathscr{L}u_{k}=f(0),\quad\textit{in
$\mathcal{Q}_{k}$}\\\ u_{k}=u,\quad\textit{in
$\partial\mathcal{Q}_{k}$}\end{array}\right.$ (5.2)

For any point $z=(x,t)$ satisfying $\|z\|_{K}\leq\frac{1}{2}$, we want to
estimate the quantity

$\displaystyle I(z):=|\partial^{2}u(z)-\partial^{2}u(0)|,$

where $\partial^{2}u(z)$ stands for either $\partial^{2}_{x_{i}x_{j}}u(z)$,
with $i,j=1,\ldots,m$, or $Yu(z)$. To this end, we write $I$ as the sum of
three terms:

$\displaystyle I(z)$
$\displaystyle\leq|\partial^{2}u_{k}(z)-\partial^{2}u_{k}(0)|+|\partial^{2}u_{k}(0)-\partial^{2}u(0)|+$
$\displaystyle\qquad\qquad\quad+|\partial^{2}u(z)-\partial^{2}u_{k}(z)|=:I_{1}(z)+I_{2}(z)+I_{3}(z).$

We first estimate $I_{2}$. Following [21], we prove that
$\left(\partial^{2}u_{k}(0)\right)_{k\in{\mathbb{N}}}$ is a Cauchy sequence
and that its limit agrees with $\partial^{2}u(0)$. The same assertion holds
for $I_{3}$ of course.

First, we let $v_{k}:=u-u_{k}$ and we observe that $v_{k}$ satisfies the
Dirichlet boundary value problem

$\left\\{\begin{array}[]{ll}\mathscr{L}v_{k}=f-f(0),\quad\textit{in
$\mathcal{Q}_{k}$}\\\ v_{k}=0,\quad\textit{in
$\partial\mathcal{Q}_{k}$}\end{array}\right.$ (5.3)

From Lemma 5.1 it follows that

$\displaystyle\|v_{k}\|_{\infty}\leq 4\varrho^{2k}\|f-f(0)\|_{\infty}\leq
4\varrho^{2k}\omega_{f}(\varrho^{k}).$ (5.4)

Moreover, since $\mathscr{L}(u_{k}-u_{k+1})=0$ in $\mathcal{Q}_{k+1}$, we
apply Proposition 3.2 and Lemma 5.1, and we find

$\displaystyle\|\partial_{x_{i}}(u_{k}-u_{k+1})\|_{L^{\infty}(\mathcal{Q}_{k+2})}$
$\displaystyle\leq C\varrho^{-k-2}\sup_{\mathcal{Q}_{k+1}}|u_{k}-u_{k+1}|$
$\displaystyle\leq
C\varrho^{-k}\Big{(}\sup_{\mathcal{Q}_{k+1}}|v_{k}|+\sup_{\mathcal{Q}_{k+1}}|v_{k+1}|\Big{)}$
$\displaystyle\leq
C\varrho^{-k}\varrho^{2k}\omega_{f}(\varrho^{k})=C\varrho^{k}\omega_{f}(\varrho^{k}),$
(5.5)

for any $i=1,\dots,m$. In the same way, we obtain

$\displaystyle\|\partial_{x_{i}x_{j}}^{2}(u_{k}-u_{k+1})\|_{L^{\infty}(\mathcal{Q}_{k+2})}$
$\displaystyle\leq C\varrho^{-2k-4}\sup_{\mathcal{Q}_{k+1}}|u_{k}-u_{k+1}|$
$\displaystyle\leq
C\varrho^{-2k}\varrho^{2k}\omega_{f}(\varrho^{k})=C\omega_{f}(\varrho^{k})$
(5.6)

for $i,j=1,\dots,m$, and

$\displaystyle\|Y(u_{k}-u_{k+1})\|_{L^{\infty}(\mathcal{Q}_{k+2})}$
$\displaystyle\leq C\varrho^{-2k-4}\sup_{\mathcal{Q}_{k+1}}|u_{k}-u_{k+1}|$
$\displaystyle\leq
C\varrho^{-2k}\varrho^{2k}\omega_{f}(\varrho^{k})=C\omega_{f}(\varrho^{k}).$
(5.7)

Let $k\geq 1$ such that $\varrho^{k+4}\leq\|z\|_{K}\leq\varrho^{k+3}$ , then
we have:

$\sum_{l=k}^{\infty}|\partial^{2}u_{l}(0)-\partial^{2}u_{l+1}(0)|\leq
C\sum_{l=k}^{\infty}\omega_{f}(\varrho^{l})\leq
C\int_{0}^{\|z\|_{K}}\frac{\omega_{f}(r)}{r}dr.$ (5.8)

We next identify the sum of the series
$\sum_{l=k}^{\infty}\left(\partial^{2}u_{l}(0)-\partial^{2}u_{l+1}(0)\right)$
as

$\sum_{l=k}^{\infty}\left(\partial^{2}u_{l}(0)-\partial^{2}u_{l+1}(0)\right)=\partial^{2}u_{k}(0)-\partial^{2}u(0).$
(5.9)

To this aim, we first consider the derivative $\partial^{2}_{x_{i}x_{j}}u_{k}$
and we prove that

$\displaystyle\lim_{k\to+\infty}\partial^{2}_{x_{i}x_{j}}u_{k}(0)=\partial^{2}_{x_{i}x_{j}}T_{0}^{2}u(0),$
(5.10)

where $T_{0}^{2}u({\zeta})$ is the second-order Taylor polynomial of $u$
around the origin, computed at some point
${\zeta}=({\xi},{\tau})\in\mathcal{Q}_{k}$:

$\displaystyle
T_{0}^{2}u({\zeta})=u(0)+\sum_{i=1}^{m}\partial_{x_{i}}u(0){\xi}_{i}+\frac{1}{2}\sum_{i,j=1}^{m}\partial^{2}_{x_{i}x_{j}}u(0){\xi}_{i}{\xi}_{j}-Yu(0){\tau}.$

Thus, by applying Theorem 1.2 to $u\in
C^{2}_{\mathscr{L}}(\mathcal{Q}_{1}(0))$, we obtain from (5.10) that

$\displaystyle\lim_{k\to+\infty}\partial^{2}_{x_{i}x_{j}}u_{k}(0)=\partial^{2}_{x_{i}x_{j}}u(0).$
(5.11)

We compute $\mathscr{L}T_{0}^{2}u$ in $\zeta=(\xi,\tau)$ as

$\begin{split}\mathscr{L}T_{0}^{2}u(\zeta)&=\\!\\!\sum_{i,j=1}^{m}\partial^{2}_{\xi_{i}{\zeta}_{j}}u(0)-\partial_{t}u(0)+\sum_{j=1}^{N}\\!\\!\bigg{(}\\!\sum_{i=1}^{m}b_{ij}\partial_{\xi_{i}}u(0)\bigg{)}\xi_{j}+\sum_{l,j=1}^{N}\\!\\!\bigg{(}\\!\sum_{i=1}^{m}b_{il}\partial^{2}_{\xi_{j}\xi_{i}}u(0)\bigg{)}\xi_{l}\xi_{j}\\\
&=\\!\\!\sum_{i,j=1}^{m}\partial^{2}_{\xi_{i}{\zeta}_{j}}u(0)-\partial_{t}u(0)+\langle
v,\xi\rangle+\langle M\xi,\xi\rangle,\end{split}$

where
$v=\left(v_{j}\right)_{j=1,\ldots,N}=\left(\sum_{i=1}^{m}b_{ij}\partial_{\xi_{i}}u(0)\right)_{j=1,\ldots,N}$
is a constant vector of ${\mathbb{R}}^{N}$ and
$M=\left(m_{lj}\right)_{l,j=1,\ldots,N}=\left(\sum_{i=1}^{m}b_{il}\partial^{2}_{\xi_{j}\xi_{i}}u(0)\right)_{l,j=1,\ldots,N}$
is a $N\times N$ constant matrix.

In addition, as $\mathscr{L}u=f$ in $\mathcal{Q}_{k}$, we have that

$\displaystyle\sum_{i,j=1}^{m}\partial^{2}_{\xi_{i}{\zeta}_{j}}u(0)-\partial_{t}u(0)=\mathscr{L}_{0}u(0)=\mathscr{L}u(0)=f(0)$
(5.12)

and thus

$\displaystyle\mathscr{L}T_{0}^{2}u(\zeta)=f(0)+\langle v,\xi\rangle+\langle
M\xi,\xi\rangle.$ (5.13)

Thus, the definition of $u_{k}$ in (5.2) gives us

$\displaystyle\mathscr{L}\left(T_{0}^{2}u-u_{k}\right)(\zeta)=\langle
v,\xi\rangle+\langle M\xi,\xi\rangle,\quad\zeta\in\mathcal{Q}_{k}.$ (5.14)

We now apply Lemma 3.4 to $T_{0}^{2}u-u_{k}$ for $R=\varrho^{k}$ and infer

$|\partial^{2}_{x_{i}x_{j}}(u_{k}-T_{0}^{2}u)(0)|\leq
C\varrho^{-2k}\sup_{\mathcal{Q}_{k}}|u_{k}-T_{0}^{2}u|+O(\varrho^{k}).$ (5.15)

Moreover, since $T_{0}^{2}u$ is the second-order Taylor polynomial of $u$, we
have $u({\zeta})=T_{0}^{2}u({\zeta})+o(\|{\zeta}\|_{K}^{2})$. It follows that

$\displaystyle\sup_{{\zeta}\in\mathcal{Q}_{k}}|u-T_{0}^{2}u|=o(\varrho^{2k})$
(5.16)

Thus, from estimates (5.16) and (5.4), we obtain

$\displaystyle\sup_{\mathcal{Q}_{k}}|u_{k}-T_{0}^{2}u|\leq\sup_{\mathcal{Q}_{k}}|v_{k}|+\sup_{\mathcal{Q}_{k}}|u-T_{0}^{2}u|\leq
4\omega_{f}(\varrho^{k})\varrho^{2k}+o(\varrho^{2k})\leq o(\varrho^{2k}).$
(5.17)

Estimates (5.15) and (5.17) finally yield

$|\partial^{2}_{x_{i}x_{j}}(u_{k}-T_{0}^{2}u)(0)|\leq
C\varrho^{-2k}\sup_{\mathcal{Q}_{k}}|u_{k}-T_{0}^{2}u|+O(\varrho^{k})\leq
C\varrho^{-2k}o(\varrho^{2k})+O(\varrho^{k})\leq o(1),$

where, as usual, the indexes $i$ and $j$ range from $1$ to $m$. Thus, for any
$i,j=1,\ldots,m$ we have showed that (5.10) holds true. Repeating the same
argument for the vector field $Y$, and using again Theorem 1.2, we obtain:

$\displaystyle\lim_{k\to+\infty}Yu_{k}(0)=YT_{0}^{2}u(0)=Yu(0).$

In conclusion, using (5.8), we obtain:

$I_{2}\leq\sum_{l=k}^{\infty}|\partial^{2}u_{l}(0)-\partial^{2}u_{l+1}(0)|\leq
C\int_{0}^{\|z\|_{K}}\frac{\omega_{f}(r)}{r}dr,$ (5.18)

for $k\geq 1$ such that $\varrho^{k+4}\leq\|z\|_{K}\leq\varrho^{k+3}$.
Similarly, we can estimate $I_{3}$ through the solution of $\mathscr{L}v=f(z)$
in $\mathcal{Q}_{j}(z)$ and $v=u$ on $\partial\mathcal{Q}_{j}(z)$ and obtain

$I_{3}\leq\sum_{l=k}^{\infty}|\partial^{2}u_{l}(z)-\partial^{2}u_{l+1}(z)|\leq
C\int_{0}^{\|z\|_{K}}\frac{\omega_{f}(r)}{r}dr.$ (5.19)

Finally, let us estimate $I_{1}$. Since $h_{k}=u_{k}-u_{k+1}\in
C^{\infty}(\mathcal{Q}_{k+2})$, we can apply Proposition 3.5 to the functions
$\partial^{2}_{x_{i}x_{j}}h_{k}$ and $Yh_{k}$:

$\begin{split}|\partial^{2}_{x_{i}x_{j}}h_{k}(z)-\partial^{2}_{x_{i}x_{j}}h_{k}(0)|\leq\frac{C}{\varrho^{k}}\|z\|_{K}\|\partial^{2}_{x_{i}x_{j}}\|_{L^{\infty}(\mathcal{Q}_{k+1})}\end{split}$

and

$\begin{split}|Yh_{k}(z)-Yh_{k}(0)|\leq\frac{C}{\varrho^{k}}\|z\|_{K}\|Yh_{k}\|_{L^{\infty}(\mathcal{Q}_{k+1})},\end{split}$

for $i,j=1,\ldots,m$. We can now apply once again (5.6) to obtain

$\begin{split}|\partial^{2}_{x_{i}x_{j}}h_{k}(z)-\partial^{2}_{x_{i}x_{j}}h_{k}(0)|\leq\frac{C}{\varrho^{k}}\|z\|_{K}\|\partial^{2}_{x_{i}x_{j}}h_{k}\|_{L^{\infty}(\mathcal{Q}_{k+1})}\leq
C\|z\|_{K}\varrho^{-k}\omega_{f}(\varrho^{k}).\end{split}$

In addition, thanks to (5.7), we infer

$\begin{split}|Yh_{k}(z)-Yh_{k}(0)|\leq\frac{C}{\varrho^{k}}\|z\|_{K}\|Yh_{k}\|_{L^{\infty}(\mathcal{Q}_{k+1})}\leq
C\|z\|_{K}\varrho^{-k}\omega_{f}(\varrho^{k}).\end{split}$

Hence, since
$u_{k}(z)-u_{k}(0)=u_{0}(z)-u_{0}(0)+\sum_{j=0}^{k-1}\left(h_{j}(0)-h_{j}(z)\right)$,
we have

$\displaystyle I_{1}$
$\displaystyle\leq|\partial^{2}u_{0}(z)-\partial^{2}u_{0}(0)|+\sum_{j=0}^{k-1}|\partial^{2}h_{j}(z)-\partial^{2}h_{j}(0)|$
$\displaystyle\leq
C\|z\|_{K}\big{(}\|u_{0}\|_{L^{\infty}(\mathcal{Q}_{0})}+C\sum_{j=0}^{k-1}\varrho^{-j}\omega_{f}(\varrho^{j})\big{)}$
$\displaystyle\leq
C\|z\|_{K}\big{(}\|u\|_{L^{\infty}(\mathcal{Q}_{1}(0))}+\|f\|_{L^{\infty}(\mathcal{Q}_{1}(0))}+C\int_{\|z\|_{K}}^{1}\frac{\omega_{f}(r)}{r^{2}}\big{)}.$

Combining the above estimate with (5.18) and (5.19), we complete the proof of
_(ii)_.

We now prove assertion _(i)_. We consider $u_{1}$ solution to the following
Dirichlet problem

$\displaystyle\left\\{\begin{array}[]{ll}\mathscr{L}u_{1}=f(0),&\textit{in
$\mathcal{Q}_{1/2}(0)$}\\\ u_{1}=u,&\textit{in
$\partial\mathcal{Q}_{1/2}(0)$}\end{array}\right.$

Then, we have

$\displaystyle|\partial^{2}u(0)|\leq|\partial^{2}u(0)-\partial^{2}u_{1}(0)|+|\partial^{2}u_{1}(0)|$
(5.21)

Thanks to (5.18), we can estimate the first term in (5.21) as

$\displaystyle|\partial^{2}u(0)-\partial^{2}u_{1}(0)|\leq
C\int_{0}^{1}\frac{\omega_{f}(r)}{r}dr.$ (5.22)

To estimate the second term in (5.21), we consider the function
$v(z):=u_{1}(z)\eta_{1/2}(z)$, where $\eta_{1/2}$ is the cut-off function
introduced in (3.1) with $R=\frac{1}{2}$. Reasoning as in the proof of
Proposition 3.1, we obtain

$\begin{split}u(z)=v(z)&=\int_{\mathcal{Q}_{\frac{1}{2}}(0)}[\Gamma(z,\cdot){\text{\rm
div}}(AD_{m}(\eta_{1/2}))u_{1}]({\zeta})d{\zeta}\\\
&\quad-\int_{\mathcal{Q}_{\frac{1}{2}}(0)}[\Gamma(z,\cdot)Y(\eta_{1/2})u_{1}]({\zeta})d{\zeta}\\\
&\quad-\int_{\mathcal{Q}_{\frac{1}{2}}(0)}[\Gamma(z,\cdot)\eta_{1/2}\mathscr{L}(u_{1})]({\zeta})d{\zeta}\\\
&+2\int_{\mathcal{Q}_{\frac{1}{2}}(0)}[\langle
D_{m}^{\zeta}\Gamma(z,\cdot),AD_{m}\eta_{1/2}\rangle
u_{1}]({\zeta})d{\zeta},\end{split}$

where $z\in\mathcal{Q}_{\frac{1}{4}}(0)$. Thanks to Lemma 5.1, we estimate

$\displaystyle\sup_{\mathcal{Q}_{\frac{1}{2}}(0)}{|u_{1}|}\leq\sup_{\mathcal{Q}_{\frac{1}{2}}(0)}{|u|}+4|f(0)|.$

As the derivatives of $\eta_{1/2}$ vanish in $\mathcal{Q}_{3/8}(0)$, for any
$i,j=1,\ldots,m$, we obtain

$\begin{split}|\partial^{2}_{x_{i}x_{j}}u_{1}(z)|&\leq\int_{\mathcal{Q}_{\frac{1}{2}}(0)\setminus\mathcal{Q}_{\frac{3}{8}}(0)}\big{|}[\partial^{2}_{x_{i}x_{j}}\Gamma(z,\cdot){\text{\rm
div}}(AD_{m}(\eta_{1/2}))u_{1}]({\zeta})\big{|}d{\zeta}\\\
&\quad+\int_{\mathcal{Q}_{\frac{1}{2}}(0)\setminus\mathcal{Q}_{\frac{3}{8}}(0)}\big{|}[\partial^{2}_{x_{i}x_{j}}\Gamma(z,\cdot)Y(\eta_{1/2})u_{1}]({\zeta})\big{|}d{\zeta}\\\
&\quad+2\int_{\mathcal{Q}_{\frac{1}{2}}(0)\setminus\mathcal{Q}_{\frac{3}{8}}(0)}\big{|}[\langle\partial^{2}_{x_{i}x_{j}}D_{m}^{\zeta}\Gamma(z,\cdot),AD_{m}\eta_{1/2}\rangle
u_{1}]({\zeta})\big{|}d{\zeta}\\\
&\quad+\bigg{|}f(0)\Big{[}\partial^{2}_{x_{i}x_{j}}\int_{\mathcal{Q}_{\frac{1}{2}}(0)}[\Gamma(z,\cdot)\eta_{1/2}]({\zeta})d{\zeta}\Big{]}\bigg{|}\\\
&\quad=:\overline{I}_{1}(z)+\overline{I}_{2}(z)+\overline{I}_{3}(z)+\overline{I}_{4}(z).\end{split}$
(5.23)

Moreover, as the derivatives of $\eta_{1/2}$ are bounded, we estimate the
first and second integral in (5.23) as

$\displaystyle\overline{I}_{1}(z)\leq
C\big{[}\sup_{\mathcal{Q}_{\frac{1}{2}}(0)}{|u|}+4|f(0)|\big{]},$
$\displaystyle\overline{I}_{2}(z)\leq
C\big{[}\sup_{\mathcal{Q}_{\frac{1}{2}}(0)}{|u|}+4|f(0)|\big{]},$
$\displaystyle\overline{I}_{3}(z)\leq
C\big{[}\sup_{\mathcal{Q}_{\frac{1}{2}}(0)}{|u|}+4|f(0)|\big{]}.$

Finally, by taking advantage of (3.13), we obtain that $\overline{I}_{4}(z)$
is bounded by a constant $C$ that only depends on $B$, $\lambda$ and
$\Lambda$.

By using the same argument we can estimate $|Yu_{1}(0)|$ and thus

$\displaystyle|\partial^{2}u_{1}(0)|\leq
C\big{[}\sup_{\mathcal{Q}_{\frac{1}{2}}(0)}{|u|}+4|f(0)|\big{]}.$ (5.24)

Combining estimates (5.21) and (5.24), we conclude the proof of Theorem 1.5. ∎

##  6  Dini continuous coefficients

This Section is devoted to the proof of Theorem 1.6. We therefore consider a
solution $u$ to the equation

${\widetilde{\mathscr{L}}}u=f,$

where the operator ${\widetilde{\mathscr{L}}}$ does satisfy the hypotheses
[H.1] and [H.2] and $f$ is assumed to be Dini continuous, and we proceed as in
the proof of Theorem 1.5. Specifically, we denote
$\mathcal{Q}_{k}=\mathcal{Q}_{\varrho^{k}}(0)$, $\varrho=\frac{1}{2}$ and we
consider the following sequence of Dirichlet problems:

$\left\\{\begin{array}[]{ll}\sum\limits_{i,j=1}^{m}a_{ij}(0,0)\partial^{2}_{x_{i}x_{j}}u_{k}+Yu_{k}=f(0),\quad\text{in
$\mathcal{Q}_{k}$}\\\ u_{k}=u,\quad\text{on
$\partial\mathcal{Q}_{k}$}.\end{array}\right.$ (6.1)

Note that the bounds given in Propositions 3.1, 3.2 and 3.5 only depend on the
constants $\lambda,\Lambda$ in [H.2] and on the matrix $B$. Keeping in mind
this fact, the proof of Theorem 1.6 is given by the same argument used in the
proof of Theorem 1.5.

###### Proof of Theorem 1.6.

Consider, for every $k\in{\mathbb{N}}$, the auxiliary function
$v_{k}:=u-u_{k}$, and note that it is a solution to the boundary value problem

$\left\\{\begin{array}[]{lll}\sum\limits_{i,j=1}^{m}a_{ij}(0,0)\partial^{2}_{x_{i}x_{j}}v_{k}+Yv_{k}\\\
\quad=f-f(0)+\sum\limits_{i,j=1}^{m}(a_{ij}(0)-a_{ij}(x,t))\partial^{2}_{x_{i}x_{j}}u,\quad\text{in
$\mathcal{Q}_{k}$}\\\ v_{k}=0,\quad\textit{in
$\partial\mathcal{Q}_{k}$}\end{array}\right.$ (6.2)

In order to simplify the notation, we let

$\eta:=\max_{i,j=1,\dots,m}\|\partial^{2}_{x_{i}x_{j}}u\|_{L^{\infty}(\mathcal{Q}_{1})}.$
(6.3)

From Lemma 5.1 it follows that

$\displaystyle\|v_{k}\|_{L^{\infty}(\mathcal{Q}_{k})}\leq
C\varrho^{2k}[\omega_{f}(\varrho^{k})+\omega_{a}(\varrho^{k})\eta].$

Hence

$\displaystyle\|u_{k}-u_{k+1}\|_{L^{\infty}(\mathcal{Q}_{k+1})}\leq
C\varrho^{2k}[\omega_{f}(\varrho^{k})+\omega_{a}(\varrho^{k})\eta].$

As already observed, we can apply Corollary 3.2 and obtain estimates for the
second order derivatives of $v_{k}$. In fact, for any $i,j=1,\ldots,m$, we
have

$\displaystyle\|\partial^{2}_{x_{i}x_{j}}(u_{k}-u_{k+1})\|_{L^{\infty}(\mathcal{Q}_{k+2})}$
$\displaystyle\leq C(\varrho^{k})^{-2}\sup_{\mathcal{Q}_{k+1}}|u_{k}-u_{k+1}|$
$\displaystyle\leq
C\varrho^{-2k}\varrho^{2k}[\omega_{f}(\varrho^{k})+\omega_{a}(\varrho^{k})\eta]=C[\omega_{f}(\varrho^{k})+\omega_{a}(\varrho^{k})\eta]$
(6.4)

and

$\displaystyle\|Y(u_{k}-u_{k+1})\|_{L^{\infty}(\mathcal{Q}_{k+2})}$
$\displaystyle\leq C(\varrho^{k})^{-2}\sup_{\mathcal{Q}_{k+1}}|u_{k}-u_{k+1}|$
$\displaystyle\leq
C\varrho^{-2k}\varrho^{2k}[\omega_{f}(\varrho^{k})+\omega_{a}(\varrho^{k})\eta]=C[\omega_{f}(\varrho^{k})+\omega_{a}(\varrho^{k})\eta]$
(6.5)

To estimate the second order derivatives of the function $u$, we apply Theorem
1.2 and proceed as in the proof of Theorem 1.5. Since there are no significant
differences, we omit the details here. ∎

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