text:
\section{Introduction and statement of the main result}
The fundamental role of Schauder estimates (both interior and at the boundary) in the theory of elliptic and parabolic partial differential equations is well-known. In this paper we are interested in $\Gamma^{k,\alpha}$ Schauder estimates at the boundary in the Dirichlet problem for a class of second order partial differential equations in stratified, nilpotent Lie groups, nowadays known as Carnot groups. The central position of such Lie groups in the analysis of the hypoelliptic operators introduced by H\"ormander in his famous paper \cite{H} was established in the 1976 work of Rothschild and Stein on the so-called \emph{lifting theorem}, see \cite{RS}. One should also see Stein's visionary address at the 1970 ICM in Nice \cite{Snice}, and also E. Cartan's seminal address at the 1928 ICM in Bologna \cite{Ca}.

To provide the reader with some historical background on our main result we mention that in his 1981 works \cite{Je1, Je2} D. Jerison first analysed the question of Schauder estimates at the boundary for the horizontal Laplacian in the Heisenberg group $\mathbb{H}^{n}$ (see also \cite{Je3} for a further extension to CR manifolds)\footnote{Long known to physicists as the Weyl's group, $\mathbb{H}^{n}$ is an important model of a (non-Abelian) Carnot group of step $r=2$, see Definition \ref{D:CG} below for the general notion.}. Jerison divided his analysis into two parts, according to whether or not the relevant portion of the boundary contains so-called \emph{characteristic points}, a notion that goes back to the pioneering works of Fichera \cite{Fi1, Fi2} (see Definition \ref{D:char} below). At such points the vector fields that form the relevant differential operator become tangent to the boundary  and thus one should expect a sudden loss of differentiability,  somewhat akin to what happens in the classical setting with oblique derivative problems. Remarkably, Jerison proved that there exist no Schauder boundary estimates at characteristic points! He did so by constructing a domain in $\mathbb{H}^{n}$ with real-analytic boundary that support solutions of the horizontal Laplacian $\Delta_H u = 0$ vanishing near a characteristic boundary point, and which near such point possess no better regularity than H\"older. For a detailed analysis of these aspects in connection with the subelliptic Dirichlet problem we refer the reader to the papers \cite{LU, CG, GV, CGNajm, CGN}.

On the positive side, Jerison proved in \cite{Je1} that at a non-characteristic portion of the boundary it is possible to develop a Schauder theory based on the non-isotropic Folland-Stein H\"older classes $\Gamma^{k,\alpha}$. He achieved this by adapting to $\mathbb{H}^{n}$ tools from Fourier and microlocal analysis. However, if we leave the Heisenberg group $\mathbb{H}^{n}$ and we move to a general Carnot group $\mathbb G$, then there exists no known counterpart of the results from \cite{Je1}, with the exception of the following partial ones. In \cite{BGM} we derived $\Gamma^{1, \alpha}$ boundary Schauder estimates near a $C^{1, \alpha}$ portion of the non-characteristic boundary and for variable coefficient operators such as \eqref{dp0} below. Our method employed a delicate adaptation of Caffarelli's compactness arguments.
In \cite{BCC},  via an adaptation of the classical method of the Levi parametrix, the authors obtained $\Gamma^{2, \alpha}$ boundary Schauder estimates  near a $C^\infty$ portion of the non-characteristic boundary, and for constant coefficient operators, under the additional geometric hypothesis that, roughly speaking, the boundary has a H\"ormander structure (see assumption (1.5) in \cite{BCC}). As it was noted in the subsequent related work \cite{CGS}, where the authors obtained an inversion result for a double layer potential at non-characteristic points in $\mathbb H^1$, even in the model Heisenberg group, their geometric hypothesis is not generically verified.

In the present work we develop a complete, coordinate-free theory of the higher-order  boundary Schauder estimates for general Carnot groups.  Before we state our main result we introduce some notations, referring the reader to Sections \ref{S:prel} and \ref{S:back}
 for a detailed account.
For $k\in \mathbb N\cup\{0\}$ and $0<\alpha< 1$ we indicate with $\Gamma^{k,\alpha}$ the Folland-Stein non-isotropic H\"older classes, see Definition \ref{hf} and Definition \ref{higherholder} below. If $\Omega\subset \mathbb G$ is a bounded open set, then given a point $p\in \partial \Omega$, for any $s>0$ we set for simplicity
\begin{equation}\label{VS}
\mathscr V_s(p) = \Omega\cap B(p,s),\ \ \ \ \ \ \mathscr S_s(p) = \partial\Omega\cap B(p,s),
\end{equation}
where $B(p,s)$ is the open ball in the pseudodistance  \eqref{pseudo} below. When $p=e$, the group identity, we simply write $\mathscr V_s$ and $\mathscr S_s$. We suppose that $\mathbb{A}= [a_{ij}]$ be a given $m\times m$ symmetric matrix-valued function with real coefficients and satisfying the following ellipticity condition for some $\lambda > 0$,
\begin{equation}\label{ea0}
 \lambda \mathbb{I}_m \leq \mathbb{A}(p) \leq \lambda^{-1} \mathbb{I}_m,\ \ \ \ \ \ \ \ \ p\in \mathbb G,
 \end{equation}
where $\mathbb{I}_m$ denotes the $m \times m$ identity matrix. The hypothesis \eqref{ea0} will be assumed throughout the paper, without further reference. Furthermore, for a given $k\in \mathbb N$ and a multi-index $I$ of order $|I|=k$,  the $k$-th order horizontal derivative $X^I f$  is defined  as in \eqref{hder} below. Our main result is as follows.

\begin{theorem}\label{main}
Suppose that $u\in \Gamma^2(\mathscr V_s(p_0)) \cap C(\overline{\mathscr V_s}(p_0))$ be a weak solution to
\begin{equation}\label{dp0}
\sum_{i,j=1}^m  a_{ij}(p) X_iX_j u = f\ \ \ \text{in}\ \mathscr V_s(p_0),\ \ \ \ \ u  = \phi\ \ \ \text{on}\ \mathscr{S}_s(p_0),
\end{equation}
for given $p_0 \in \partial \Omega$ and $s>0$. Assume that $\mathscr S_s(p_0)$ be non-characteristic and of class $C^{k,\alpha}$, for some $k \in \mathbb N$ such that $k\geq 2$ and $\alpha \in (0,1)$, and that
\begin{equation}\label{assump1}
a_{ij} \in \Gamma^{k-2, \alpha}(\overline{\mathscr V_s}(p_0)),\ \ f \in \Gamma^{k-2,\alpha}(\overline{\mathscr V_s}(p_0)),\  \ \phi \in \Gamma^{k,\alpha}(\overline{\mathscr V_s}(p_0)).
\end{equation}
Then, $u\in \Gamma^{k,\alpha}(\overline{\mathscr V_{s/2}}(p_0))$, and we have the following a priori estimate for the $k$-th order H\"older seminorm of $u$
\begin{align}\label{ap}
&\underset{|I| =k}{\sup} [X^I u]_{\Gamma^{k,\alpha}(\mathscr V_{s/2}(p_0))} \\& \leq \frac{C}{s^{k+\alpha}} \bigg[||u||_{L^{\infty}(\mathscr V_s(p_0))} + \sum_{|I| \leq k-2} s^{2+|I|} ||X^I f||_{L^{\infty}(\mathscr V_s(p_0))} \notag \\ &+ \sum_{|I|=k-2} s^{k+\alpha} [X^If]_{\Gamma^{0,\alpha}(\mathscr V_s(p_0))}+\sum_{|I| \leq k} s^{|I|} ||X^I \phi||_{L^{\infty}(\mathscr V_s(p_0))} + s^{k+\alpha} \sum_{|I|=k}  [X^I \phi]_{\Gamma^{2,\alpha}(\mathscr V_s(p_0))} \bigg],\notag
\end{align}
where $C = C(\mathbb G, \lambda, \Omega, k, \alpha, [a_{ij}]_{\Gamma^{k-2,\alpha}(\mathscr V_s(p_0))})>0$.
\end{theorem}

\begin{remark}
Concerning Theorem \ref{main}, we emphasise that we have considered boundary value problems of the type \eqref{dp0} solely for the simplicity of exposition. With standard modifications, our method can be adapted  to infer Schauder estimates for more general equations of the type
\[
\sum_{i,j=1}^m a_{ij} X_i X_j u + \sum_{i=1}^m b_i X_i u + c u= f,
\]
where $b_i$'s and $c$ are in $\Gamma^{k-2, \alpha}(\overline{\mathscr V_s}(p_0))$.
\end{remark}

Our approach to Theorem \ref{main} is inspired to the groundbreaking 1989 paper of Caffarelli \cite{Ca}. In this connection, we mention that the homogeneous  structure of a Carnot group, based on the presence of a family of non-isotropic dilations, plays a critical role. We have also been influenced by De Silva and Savin's beautiful results on the higher-order boundary Harnack inequality and Schauder theory for slit domains, in connection with thin obstacle problems, see \cite{DS1, DS2}. Having said this, the reader will see that, in the present sub-Riemannian framework, our coordinate-free approach, which completely avoids flattening of the boundary, has entailed some very delicate adaptation of the Euclidean ideas in the above mentioned works.

We next provide the reader with some perspective on Theorem \ref{main}. If $u$ is a solution to \eqref{dp0} vanishing on the boundary (after subtracting off the boundary datum), then via compactness arguments we deduce that, at a non-characteristic boundary point $p_0$, $u$ can be approximated up to order $"k+\alpha"$ by  $\Gamma^{k, \alpha}$ functions of the type $d P$, where $d$ is the signed Riemannian distance function from the boundary, and $P$ is an appropriate stratified polynomial of homogeneous degree at most $k-1$. This step crucially utilises, for the limiting ``constant coefficient problem", the non-characteristic boundary regularity of Kohn-Nirenberg in \cite{KN}. Finally, in our analysis the $k-$th order  Taylor polynomial of $dP$ at $p_0 \in \partial \Omega$ takes the role of the $k-$th order Taylor polynomial of $u$ at $p_0$. We note here that, for the case $k=1$ treated in \cite{BGM}, our approach was instead to  approximate  the solution $u$ with affine functions that came from the limiting problem, and then at each step we had to  ensure  the $"1+\alpha"$  scale invariant decay of the boundary datum to eventually derive the existence of the first order limiting Taylor polynomial at the boundary point.

Differently from \cite{BGM}, in the present work we   approximate the solutions  to \eqref{dp0} (corresponding to  $\phi=0$)  with $\Gamma^{k, \alpha}$ functions of the type $dP$   in order to ensure  homogeneous (zero) boundary conditions at each step, and then eventually we pass to the  $k$-th Taylor polynomial at the end of the iteration process. Although this gets rid of the difficulty of ensuring $k+\alpha$  decay of the boundary datum at every step, a serious new difficulty arises. In this approach the functions $dP$  are not in the kernel of the ``constant coefficient operator" $\sum_{i,j=1}^m a_{ij}(p_0) X_i X_j$. Therefore, in order to obtain the appropriate invariant decay of the right-hand side one has to incorporate the right correction factors. The latter turn out to be small perturbations of the Taylor polynomial of the limiting problem. This is precisely where we borrow some beautiful ideas from the works \cite{DS1, DS2}. In this endeavor the stratified nature of the Lie algebra plays a very crucial role and allows us to adapt the Euclidean type algorithms in \cite{DS1, DS2} to the sub-Riemannian framework. This represents one key novelty of our work. Once the $k+\alpha$ boundary decay is obtained, we combine it with the  interior Schauder estimates in \cite{CH, Xu} to deduce that the $k$-th order derivatives are H\"older continuous up to the boundary. In the present work, however, combining the interior estimates with the boundary decay is considerably subtler than the case $k=1$ treated in \cite{BGM}.

The paper is organised as follows. In Sections \ref{S:prel} and \ref{S:back} we introduce some basic notations and notions and gather some  known results that are relevant to our work. In Section \ref{S:main} we  prove our main result.

\section{Preliminaries on Carnot groups}\label{S:prel}

In this section we collect the definition and the basic properties of Carnot groups which will be used in the rest of the paper, referring the reader to \cite[Chap. 1 \& 2]{Gems} for a short and self-contained introduction to these geometric ambients.

\begin{definition}\label{D:CG}
Given $r\in \mathbb N$, a \emph{Carnot group} of step $r$ is a simply-connected real Lie group $(\mathbb G, \circ)$ whose Lie algebra $\mathfrak{g}$ is stratified and $r$-nilpotent. This means that there exist vector spaces $\mathfrak{g}_1,...,\mathfrak{g}_r$ such that:
\begin{itemize}
\item[(i)] $\mathfrak{g}=\mathfrak{g}_1\oplus \dots\oplus\mathfrak{g}_r$;
\item[(ii)] $[\mathfrak{g}_1,\mathfrak{g}_j] = \mathfrak{g}_{j+1}$, $j=1,...,r-1,\ \ \ [\mathfrak{g}_1,\mathfrak{g}_r] = \{0\}$.
\end{itemize}
\end{definition}
The identity element in $\mathbb G$ will be routinely denoted by $e$. We assume that $\mathfrak{g}$ is endowed with a
scalar product $\langle\cdot,\cdot\rangle$ with respect to which the vector spaces $\mathfrak{g}_j's$, $j=1,...,r$, are mutually orthogonal.  We let $m_j =$ dim\ $\mathfrak{g}_j$, $j=
1,...,r$, and denote by $N = m_1 + ... + m_r$ the topological
dimension of $\mathbb G$.  From the assumption (ii) on the Lie
algebra it is clear that any basis of the first layer $\mathfrak{g}_1$ bracket generates the whole Lie algebra $\mathfrak{g}$. Because of such special role $\mathfrak{g}_1$ is usually called the horizontal
layer of the stratification. For ease of notation we henceforth write $m = m_1$. In the case in which $r =1$ we are in the Abelian situation in which $\mathfrak{g} = \mathfrak{g}_1$, and thus $\mathbb G$ is isomorphic to $\mathbb{R}^m$. We are primarily interested in the genuinely non-Riemannian setting $r>1$. The exponential map $\exp : \mathfrak{g} \to \mathbb G$ defines an analytic
diffeomorphism of the Lie algebra $\mathfrak{g}$ onto $\mathbb G$, see e.g. \cite[Sec. 2.10 forward]{V}, or also \cite{CGr}. Using such diffeomorphism, whenever convenient we will routinely identify a point $p = \exp \xi \in \mathbb G$ with its logarithmic image $\xi = \exp^{-1} p\in \mathfrak{g}$. More explicitly, this means that if $p= \exp(\xi_1+...+\xi_r)$, where $\xi_1 = x_1 e_1+...+x_m e_m$, $\xi_2 = x_{2,1} e_{2,1}+...+x_{2,m_2}e_{2,m_2}$, ... , $\xi_r = x_{r,1} e_{r,1},...,x_{r,m_r} e_{r,m_r}$, we identify $p$ with
\begin{equation}\label{p}
p \cong (\xi_1,...,\xi_r) = (x_1,...,x_m,x_{2,1},...,x_{r,1},...,x_{r,m_r}).
\end{equation}
At times, it will be expedient to break the variables by grouping those in the horizontal layer into a single one as follows
$x = x(p) \cong \xi_1 = (x_1,\ldots,x_m)$, and indicate with $y= y(p)$ the $(N - m)-$dimensional vector
\[
y \cong (\xi_2,\xi_3,\ldots,\xi_r) = (x_{2,1},\ldots,x_{2,m_2},\ldots,x_{r,1},\ldots,x_{r,m_r}).
\]
In this case, we will write $w = (x,y)$.
Given $\xi, \eta\in \mathfrak{g}$, the Baker-Campbell-Hausdorff formula reads
\begin{equation}\label{BCH}
\exp(\xi) \circ \exp(\eta) = \exp{\bigg(\xi + \eta + \frac{1}{2}
[\xi,\eta] + \frac{1}{12} \big\{[\xi,[\xi,\eta]] -
[\eta,[\xi,\eta]]\big\} + ...\bigg)},
\end{equation}
where the dots indicate commutators of order four and higher, see \cite[Sec. 2.15]{V}. Since by (ii) in Definition \ref{D:CG} all commutators of order $r$ and higher are trivial, in every Carnot group the series in the right-hand side of \eqref{BCH} is finite.
Using \eqref{BCH}, with $p = \exp \xi,  p' = \exp \xi'$, one can recover the group law $p \circ p'$ in $\mathbb G$ from the knowledge of the algebraic commutation relations between the elements of its Lie algebra. We respectively denote
by $L_p(p') = p \circ p'$ and $R_p(p') = p'\circ p$ the left- and right-translation operator by an element $p\in
\mathbb G$. We indicate by $dp$ the bi-invariant
Haar measure on $\mathbb G$ obtained by lifting via the exponential map
 the Lebesgue measure on $\mathfrak{g}$.

The stratification (ii) induces in $\mathfrak{g}$ a natural one-parameter family of non-isotropic dilations by assigning to each
element of the layer $\mathfrak{g}_j$ the formal degree $j$.
Accordingly, if $\xi = \xi_1 + ... + \xi_r \in \mathfrak{g}$, with $\xi_j\in \mathfrak{g}_j$,
one defines dilations on $\mathfrak{g}$ by the rule
$\Delta_\lambda \xi = \lambda \xi_1 + ... + \lambda^r
\xi_r,$
and then use the exponential map to transfer such
anisotropic dilations to the group $\mathbb G$ as
follows
\begin{equation}\label{dil}
\delta_\lambda(p) = \exp \circ \Delta_\lambda \circ
\exp^{-1} p.
\end{equation}
Given $f:\mathbb G\to \mathbb{R}$, the action of $\{\delta_\lambda\}_{\lambda>0}$ on $f$ is defined by
\[
\delta_\lambda f(p) = f(\delta_\lambda(p)),\ \ \ \ \ \ \ \ \ \ \ p\in \mathbb G.
\]
A function $f:\mathbb G \to \mathbb{R}$ is called homogeneous of degree $\kappa\in \mathbb{R}$ if for every $\lambda >0$ one has
$\delta_\lambda f = \lambda^\kappa f$.
A vector field $Y$ on $\mathbb G$ is homogeneous of degree $\kappa$ if for every $f\in C^\infty(\mathbb G)$ one has $Y(\delta_\lambda f) = \lambda^\kappa \delta_\lambda(Yf)$.
The dilations \eqref{dil} are group automorphisms, and thus we have for any $p, p'\in \mathbb G$ and $\lambda>0$
\begin{equation}\label{ga}
(\delta_\lambda(p))^{-1} = \delta_\lambda(p^{-1}),\ \ \ \ \ \ \ \ \ \ \delta_\lambda(p) \circ \delta_\lambda(p') = \delta_\lambda(p\circ p').
\end{equation}

The homogeneous dimension of $\mathbb G$ with respect to \eqref{dil} is the number $Q = \sum_{j=1}^r j m_j.$ Such number plays an important role in the analysis of Carnot groups.
The motivation for this name comes from the equation
$$(d\circ\delta_\lambda)(p) = \lambda^Q dp,$$
where $dp$ denotes the push forward to $\mathbb G$ of Lebesgue measure on $\mathfrak{g}$ (this defines a Haar measure on $\mathbb G$, see \cite{CGr}). In the
non-Abelian case $r>1$, one clearly has $Q>N$. A non-isotropic gauge in $\mathfrak{g}$ which respects the dilations $\Delta_\lambda$, is given by $|\xi|_\mathfrak{g} = \left(\sum_{j=1}^r ||\xi_j||^{2r!/j}\right)^{1/2r!}$, see \cite{F}. It is obvious that $|\Delta_\lambda \xi|_\mathfrak{g} = \lambda |\xi|_\mathfrak{g}$ for $\lambda>0$.
One defines a non-isotropic gauge in the group $\mathbb G$ by letting $|p| = |\xi|_\mathfrak{g}$ for $p = \exp \xi$. Clearly, $|\cdot|\in C^\infty(\mathbb G\setminus\{e\})$, and moreover $|\delta_\lambda p| = \lambda |p|$ for every $p\in \mathbb G$ and $\lambda>0$. The pseudodistance
\begin{equation}\label{pseudo}
d(p,p')  = |(p')^{-1} \circ p|,
\end{equation}
generated by such gauge is equivalent to the intrinsic, or Carnot-Carath\'eodory distance $d_C(g,g')$ on $\mathbb G$, i.e., there exists a universal constant $c_1>0$ such that for every $p, p'\in \mathbb G$
\begin{equation}\label{pseudoe}
c_1 d(p,p')  \le d_C(p,p') \le c_1^{-1} d(p,p'),
\end{equation}
see \cite{F}, \cite{NSW}. We will almost always work with the pseudodistance \eqref{pseudo}, and denote by $B(p,s) = \{p'\in \mathbb G\mid d(p',p)<s\}$ the relative open balls. Hereafter, we indicate with $|E| = \int_E dg$ the Haar measure of a set $E\subset \mathbb G$.
One easily recognises that, with $\omega_C  = |B_C(1)|>0$ and $\omega = |B(1)|>0$, one has for every $p \in \mathbb G$ and $r > 0$,
\begin{equation}\label{volb}
|B_C(p,r)| =\omega_C r^Q,\ \ \ \ \ \ \ \ \ \ \ |B(p,r)| =\omega r^Q.
\end{equation}
We will use the following result from \cite{NSW}. Denote by $d_e(p,p')$ the Riemannian distance in $\mathbb G$.   We will denote by $B_e(p,r)$, the corresponding ball  centered at $p$ of radius $r$  with respect to the Riemannian distance $d_e$.
For every connected  $\Omega \subset\subset \mathbb G$ there exist $C,\varepsilon
>0$ such that for $p, p' \in \Omega$ one has
\begin{equation}\label{xy}
C d_e(p,p') \leq d_C(p,p')\leq C^{-1} d_e(p,p')^\varepsilon.
\end{equation}

Given a orthonormal basis $\{e_1,...,e_m\}$ of the horizontal layer $\mathfrak{g}_1$ one associates corresponding left-invariant $C^\infty$ vector fields on $\mathbb G$ by the formula $X_i(p) = (L_p)_\star(e_i)$, $i=1,...,m$, where $(L_p)_\star$ indicates the differential of $L_p$. We note explicitly that, given a smooth function $u$ in $\mathbb G$, the derivative of $u$ in $p\in \mathbb G$ along the vector field $X_i$ is given by the Lie formula
\begin{equation}\label{lie}
X_i u(p)  = \frac{d}{dt} u(p \exp t e_i)\big|_{t=0}.
\end{equation}
The left-invariant vector fields defined by \eqref{lie} are homogeneous of degree $\kappa = 1$. We also note that, if $X_i^\star$ indicates the formal adjoint of $X_i$ in $L^2(\mathbb G)$, then $X_i^{\star} = -X_i$, see \cite{F}. If $\{e_{j,1},...,e_{j,m_j}\}$ denotes an orthonormal basis of the layer $\mathfrak g_j$, $j=2,...,r$, then as in \eqref{lie} we can define left-invariant vector fields $\{X_{j,k}\}_{k=1}^{m_j}$. With respect to this notation, we have the identification $X_{1,i}=X_i$, $i=1,...,m$.

The \emph{carr\'e du champ} and the \emph{horizontal Laplacian} associated with the orthonormal basis $\{e_1,...,e_m\}$  are respectively given by
\begin{equation}\label{lhg}
|\nabla_{H} f|^2 = \sum_{j=1}^m (X_j f)^2,
\end{equation}
and
\begin{equation}\label{subl}
 \Delta_H = -\sum^m_{i=1}X^{\star}_{i}X_i=\sum^m_{i=1}X^2_i.
\end{equation}
It is clear that for every $f\in C^\infty(\mathbb G)$ one has
$\Delta_H(\delta_\lambda f) = \lambda^2 \delta_\lambda(\Delta_H f)$.
Furthermore, by the assumptions (i) and (ii) in Definition \ref{D:CG} one immediately sees that the system $\{X_1,...,X_m\}$ satisfies the finite rank condition \begin{equation}\label{rank}\text{rank Lie} [X_1,\ldots,X_m] \equiv N, \end{equation}
therefore by  H\"ormander's theorem \cite{H} the operator $\Delta_H$ is hypoelliptic. However, when the step $r$ of $\mathbb G$ is $>1$ this operator fails to be elliptic at every point $p\in \mathbb G$.

In the proof of Theorem \ref{main} the specific structure of the $X_i$ in \eqref{lie} will be important, as well as the so-called stratified Taylor formula. With this in mind, we recall next some facts from \cite[Sec. C, p. 20]{FS}. A multi-index  $J\in (\mathbb N \cup \{0\})^N$ can be represented as
\begin{equation}\label{I}
J = (\beta^J_1,\ldots,\beta^J_r) = (\beta^J_{1,1},\ldots,\beta^J_{1,m},\ldots,\beta^J_{r,1},\ldots,\beta^J_{r,m_r}).
\end{equation}
Whenever the context is clear, for $i=1,...,r$, we will simply write $\beta_i$ instead of $\beta^J_i$. As customary, the length of $J$ is the number $|J| = \sum_{j=1}^r  |\beta_j| = \sum_{j=1}^r \sum_{s=1}^{m_j} \beta_{j,s}$. The \emph{weighted degree} of $J$ is instead defined by the equation
\begin{equation}\label{hl}
d(J) = \sum_{j=1}^r j |\beta_j| = \sum_{j=1}^r j (\sum_{s=1}^{m_j} \beta_{j,s}).
\end{equation}
Given $J$ as in \eqref{I}, and keeping \eqref{p} in mind, we consider the monomial $z^J$ defined by
\[
z^J = \xi_1^{\beta_1} \cdots\ \xi_r^{\beta_r} = \prod_{j=1}^r x_{j,1}^{\beta_{j,1}} \cdots\ x_{j,m_j}^{\beta_{j,m_j}}.
\]
It is clear that the function $f(p) = z^J$ is homogeneous of weighted degree $\kappa = d(J)$.
A \emph{stratified polynomial} in $\mathbb G$  is a function $P:\mathbb G\rightarrow \mathbb{R}$ which, in the logarithmic coordinates $z = (x,y)$, can be expressed as
\[
P(z)=\sum_J a_J z^J,
\]
where $a_J\in \mathbb{R}$. The \emph{weighted degree} of $P$ is the largest $d(J)$ for which the corresponding $a_J\not= 0$. For any  $\kappa\in \mathbb{N}\cup \{0\}$ we denote by $\mathscr{P}_\kappa$ the set of stratified polynomials in $\mathbb G$ of weighted degree less or equal to $\kappa$.
The space $\mathscr{P}_\kappa$ is invariant under left- and right-translation.

We recall that by the Baker-Campbell-Hausdorff formula \eqref{BCH} one obtains from \eqref{lie} the following expression
\begin{equation}\label{fdG}
X_i  = \frac{\partial }{\partial{x_i}} +
\sum_{j=2}^{r}\sum_{s=1}^{m_j} b^s_{j,i}(\xi_1,...,\xi_{j-1})
\frac{\partial }{\partial{x_{j,s}}},
\end{equation}
where each $b^s_{j,i}$ is a homogeneous polynomial of
weighted degree $j-1$. This means that for $j=2,...,r$  we can write
\[
b^s_{j,i}(\xi_1,...,\xi_{j-1})=\sum_{ d(I^{(j-1)})= j-1}b^s_{i,I^{(j-1)}}z^{I^{(j-1)}},
\]
where $I^{(j-1)}$  denotes a multi-index with zeroes in the last $N- \sum_{i=1}^{j-1} m_{i}$ coordinates, and $b^s_{i,I^{(j-1)}}\in \mathbb{R}$. More explicitly, if $I^{(j-1)}=(\beta_1,\ldots,\beta_{(j-1)},0,\ldots,0)$, then $z^{I^{(j-1)}}=\xi_1^{\beta_1}\cdots\xi^{\beta_{j-1}}_{j-1}$, and \eqref{fdG} can be alternatively written as
\begin{equation}\label{xiex2}
X_i= \frac{\partial}{\partial x_i}+\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{ d(I^{(j-1)})= j-1}b^s_{i,I^{(j-1)}}z^{I^{(j-1)}}\right)\frac{\partial}{\partial x_{j,s}}.
\end{equation}
Using \eqref{xiex2} it will be useful in the proof of Theorem \ref{main} to have the following expanded expression of \eqref{subl}. For $j\in\{1,...,r\}$ and $s\in \{1,...,m_j\}$, we will denote by $\overline{(j,s)}$ a multi-index which has 1 in the $(j,s)$ position and zeroes everywhere else.  When $j=1$, for convenience we will denote $\overline{(1,i)}$ with $\overline{i}$. With this notation in place, a tedious but trivial bookkeeping based on \eqref{xiex2} gives for \eqref{subl}
\begin{align}\label{subLap}
\Delta_H
&= \sum^{m}_{i=1}\frac{\partial^2}{\partial x^2_i}+2\sum^{m}_{i=1}\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{d(I^{(j-1)})= j-1}b^s_{i,I^{(j-1)}}z^{I^{(j-1)}}\frac{\partial^2}{\partial x_{j,s}\partial x_i}\right)
\\
 & + \sum^{m}_{i=1}\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{d(I^{(j-1)})= j-1}\alpha_{1,i} b^s_{i,I^{(j-1)}}z^{I^{(j-1)- \overline{i}}}\frac{\partial}{\partial x_{j,s}}\right)
 \notag\\
 &+\sum^{m}_{i=1}\sum^r_{j,j'=2}\sum^{m_j}_{s,s'=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1\\  d(I^{(j'-1)})= j'-1 }}b^s_{i,I^{(j-1)}}b^{s'}_{i,I^{(j'-1)}}z^{I^{(j-1)}+I^{(j'-1)}}\right)\frac{\partial^2}{\partial x_{j,s}\partial x_{j',s'}}
\notag \\
&+\sum^{m}_{i=1}\sum^r_{\substack{j,j'=2\\ j'\leq j-1}}\sum^{m_j}_{s,s'=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1\\  d(I^{(j'-1)})= j'-1 }}\alpha_{j',s'}b^s_{i,I^{(j-1)}}b^{s'}_{i,I^{(j'-1)}}z^{I^{(j-1)}-\overline{(j',s')}+I^{(j'-1)}}\right)\frac{\partial}{\partial x_{j,s}}.
\notag
\end{align}
We mention that the reason for introducing the expanded expression \eqref{subLap} of $\Delta_H$ is because, in \emph{Step 1} of the proof of Theorem \ref{main}, we eventually have to determine an appropriate approximating polynomial  by solving a linear system of the type \eqref{alg} below. In order to do so, it is crucial to have a precise knowledge regarding  the value of the $(1,m)$ coordinate of the various multi-indices  involved in the  expression \eqref{alg}.

\section{Further background material}\label{S:back}

In this section we collect some further known results that will be needed in the proof of Theorem \ref{main}. We begin with the following weak maximum principle, see \cite{BU}.

\begin{lemma}\label{comp}
Let $\Omega\subset \mathbb G$ be a bounded open set, and $a_{ij}, b_i\in C(\Omega)$. If $u\in \Gamma^2(\Omega)\cap C(\overline \Omega)$ satisfies $\sum_{i,j=1}^m a_{ij}(p) X_i X_j u + \sum_{i=1}^m b_i(p) X_i u \ge 0$ in $\Omega$ and $u\le 0$ on $\partial \Omega$, then $u\le 0$ in $\Omega$.
\end{lemma}

Another basic tool in this paper are the intrinsic H\"older classes $\Gamma^{k, \alpha}$ of those functions that are H\"older continuous, together with their derivatives up to order $k$ along the vector fields \eqref{lie}, with respect to the Carath\'eodory distance $d_C(p,p')$, or equivalently with respect to the pseudodistance $d(p,p')$ in \eqref{pseudo}, see \eqref{pseudoe}. For the proofs of the relevant results we refer the reader to \cite[Sec. C on p. 20]{FS} and also \cite[Chap. 20]{BLU}.

\begin{definition}\label{hf}
Let $0<\alpha\leq 1$. Given an open set $\Omega \subset \mathbb G$ we say that $f:\Omega\to \mathbb{R}$ belongs to $\Gamma^{0, \alpha}(\Omega)$ if $f\in L^\infty(\Omega)$ and there exists a constant $M>0$ such that for every $p, p' \in \Omega$ one has
$|f(p) - f(p')| \leq M\ d(p,p')^{\alpha}$.
We define the seminorm
\begin{equation}\label{semi}
[f]_{\Gamma^{0,\alpha}(\Omega)}= \underset{\underset{p \neq p'}{p, p'\in \Omega}}{\sup} \frac{|f(p)-f(p')|}{d(p,p')^{\alpha}}.
\end{equation}
\end{definition}
From \eqref{pseudoe} and \eqref{xy} it is obvious that if $f\in \Gamma^{0, \alpha}(\Omega)$, then $f\in C(\Omega)$.
We now define the higher order H\"older spaces. If $k\in \mathbb N$ and $I = (i_1,...,i_k)$, where $1\le i_j\le m$, and $j=1,...,k$. We set $|I| = k$, and let
\begin{equation}\label{hder}
X^I = X_{i_1}...X_{i_k}.
\end{equation}

\begin{definition}\label{higherholder}
Let $k \in \mathbb N \cup \{0\}$, $0 < \alpha \leq 1$ and $\Omega \subset G$ be an open connected set. The space $\Gamma^{k, \alpha}(\Omega)$ denotes the space of all functions $f$ for which $X^{I} f$ exists for $|I| \leq k$ and such that the following quantity is finite
\[
||f||_{\Gamma^{k, \alpha}(\Omega)} \overset{def}= \sum_{|I| \leq k} ||X^If||_{L^{\infty}(\Omega)} + \sum_{|I|=k} [X^I f]_{\Gamma^{0, \alpha}(\Omega)} < \infty.\]
\end{definition}

\subsection{Taylor polynomials and stratified Taylor inequality}
We first introduce the notion of a Taylor polynomial, see \cite[pages 26-27]{FS}.
\begin{definition}\label{tp}
Let $f$ be a function whose derivatives $X^{I} f$ are continuous for $|I| \leq k$. The $k$-th Taylor polynomial of $f$ at $g_0 \in \mathbb G$ is the unique polymomial $P_{g_0}$ of homogeneous degree at most $k$ such that
\[
X^I f(g_0) =  X^I P_{g_0} (e),
\]
for all $I$ such that $|I| \leq k$.

\end{definition}
\begin{remark}
We mention that in view of the bracket generating condition in \eqref{rank} above,  the notion of a Taylor polynomial in Definition \ref{tp} is the same  as  that in \cite[pages 26-27]{FS}. \end{remark}

We now state the relevant stratified Taylor inequality which can be found in \cite[pages 33-35]{FS}.
\begin{theorem}[Stratified Taylor inequality]\label{stp}
For each positive integer $k$, there exist constants $c_k, b>0$ such that for all $f$ for which $X^{I} f$ is continuous for $|I| \leq k$, one has
\begin{equation}\label{stp1}
|f(g_0 g) - P_{g_0} (g)| \leq c_k |g|^k \operatorname{sup}_{|p| \leq b^k |g|, |I|=k} |X^I f(g_0 p ) - X^I f(g_0)|,
\end{equation}
where $P_{g_0}$ is the $k$-th order Taylor polynomial of $f$ at $g_0$.
\end{theorem}

\subsection{The characteristic set}\label{SS:char}

We recall that an open set $\Omega \subset \mathbb G$ is said to be of class $C^1$ if for every $p_0\in \partial\Omega$ there exist a neighborhood $U_{p_0}$ of $p_0$, and a function $\varphi_{p_0} \in C^1(U_{p_0})$, with $|\nabla \varphi_{p_0}| \geq \alpha > 0$ in $U_{p_0}$, such that
\begin{equation}\label{ome}
\Omega \cap U_{p_0} = \{p\in U_{p_0} \mid \varphi_{p_0}(p)<0\}, \quad\partial\Omega\cap U_{p_0} =\{p\in U_{p_0} \mid \varphi_{p_0}(p)=0\}.
\end{equation}
At every point $p\in \partial \Omega \cap U_{p_0}$ the outer unit normal is given by
\[
\nu(p) = \frac{\nabla \varphi_{p_0}(p)}{|\nabla \varphi_{p_0}(p)|},
\]
where $\nabla$ denotes the Riemannian gradient.

\begin{definition}\label{D:char}
Let  $\Omega \subset \mathbb G$ be an open set of class $C^1$. A point $p_0\in \partial\Omega$ is called \emph{characteristic} if  one has
\begin{equation}\label{cp}
\nu(p_0) \perp \operatorname{linear\ span\ of}\ \{X_1,..,X_m\}.
\end{equation}
The \emph{characteristic set} $\Sigma = \Sigma_{\Omega}$ is the collection of all characteristic points of $\Omega$. A boundary point $p_0 \in \partial \Omega \setminus \Sigma$ will be referred to as a \emph{non-characteristic} boundary point.
\end{definition}

We note explicitly that \eqref{cp} is equivalent to saying that,
given any couple $U_{p_0}, \varphi_{p_0}$ as in \eqref{ome}, and the family of horizontal vector fields $\{X_1,...,X_m\}$ as in \eqref{lie} above, one has
\begin{equation}\label{cp2}
X_1\varphi_{p_0}(p_0) = 0, \ldots, X_m\varphi_{p_0}(p_0) = 0,
\end{equation}
or equivalently
\begin{equation}\label{cp3}
 |\nabla_H \varphi_{p_0}(p_0)| = 0.
\end{equation}
Bounded domains generically have non-empty characteristic set if they have a trivial topology. For instance, in the Heisenberg group $\mathbb{H}^{n}$, with logarithmic coordinates $(x,y,t)\in \mathbb{R}^{2n+1}$, every bounded $C^1$ open set which is homeomorphic to the sphere $\mathbb S^{2n}\subset \mathbb{R}^{2n+1}$ must have at least one characteristic point. A torus obtained by revolving around the $t$-axis a closed simple curve in $\mathbb H^1$ non intersecting the $t$-axis itself, provides an example of a non-characteristic domain. For instance, the domain bounded by $\{(x,y,t)\in \mathbb H^1\mid 16 t^2 + (x^2 +y^2 - 2)^2 = 1\}$ is non-characteristic. For a discussion of these aspects we refer the reader to \cite{CG} and \cite{CGN}. Examples of unbounded non-characteristic domains are the following. In $\mathbb H^1$ consider the unbounded domain $\Omega= \{(x,y,t)\in \mathbb H^1\mid x < yt\}$. Its boundary $S = \partial \Omega$ is an entire graph which is non-characteristic. We can in fact describe $S$ using the global defining function $\varphi(x,y,t) = x - yt$, for which we have $X_1 \phi = \phi_x - \frac{y}2 \phi_t = 1 + \frac{y^2}2\ge 1$, for every $(x,y,t)\in S$. In view of \eqref{cp2} this shows that $\Sigma_\Omega = \varnothing$. Furthermore, $S$ is an $H$-minimal surface that was shown to be unstable in \cite{DGN}. If instead, for a fixed vector $a \in \mathfrak{g}_1 \setminus \{0\}$ and for $\lambda \in \mathbb{R}$, we consider the so-called \emph{vertical half-space}
 \[
H^+_a = \{p\in\mathbb G\mid \langle x(p),a\rangle > \lambda\}.
 \]
Then, using the global defining function $\varphi(p) = \lambda - \langle x(p),a\rangle$, one has $|\nabla_H \varphi(p)| \equiv |a|>0$, and thus in view of \eqref{cp3} one has $\Sigma = \Sigma_{H^a_+} = \varnothing$. In $\mathbb H^n$ such half-spaces were shown to be stable $H$-minimal surfaces in \cite{DGNP}.

In the proof of Theorem \ref{main} we will need the following by now classical smoothness result at non-characteristic points of Kohn and Nirenberg, see \cite{KN} and also \cite{De}. We recall the notation \eqref{VS}.

\begin{theorem}\label{KN}
  Assume that $\Omega$ be a $C^\infty$ domain, and let $u\in \mathscr{L}^{1,2}_{loc}(\Omega)\cap C(\overline{\Omega})$ be a weak solution of
$\Delta_{H} u=0$ in $\mathscr V_{r_0}(p_0)$, $|u| \leq M$, $u=0$ on $\mathscr S_{r_0}(p_0)$, where $\mathscr S_{r_0}(p_0)$ consists of non-characteristic  boundary points. Then given any $k \in \mathbb N$,  there exists  a positive constant $C^\star=C^\star(\Omega, M, p_0, k)>0$ such that
 \begin{equation}
 \label{c2u}
 \|u\|_{C^k(\overline{\Omega}\cap V)}\leq C^\star.
 \end{equation}
\end{theorem}

We will also need the following interior regularity result which can be found in \cite{CH, Xu}.
\begin{theorem}\label{intreg}
Let $u$ be a solution to $\sum_{i,j=1}^m a_{ij}(p) X_i X_j u=f$
in a gauge ball $B(p_0,1)$. Suppose furthermore that for some $k \in \mathbb N$ such that $k \geq 2$ and  $0<\alpha<1$,  one has $a_{ij}, f \in \Gamma^{k-2, \alpha}(B(p_0,1))$. Then $u \in \Gamma^{k,\alpha}(B(p_0,1/2))$, and there exists a constant $C = C(\mathbb G,\lambda,[a_{ij}]_{\Gamma^{k-2, \alpha}(B(p_0,1))})>0$ such that the following estimates hold
\begin{equation}\label{i10}
 \underset{|J|=k}{\sup} ||X^J u||_{L^{\infty}(B(p_0,1/2))}  \leq  C \left\{||u||_{L^{\infty}(B(p_0,1))} + \sum_{|I| \leq k-2} ||X^I f||_{L^{\infty}(B(p_0, 1))} + \sum_{|I|=k-2} [X^I f]_{\Gamma^{0, \alpha}(B(p_0, 1))} \right\},
\end{equation}
and
\begin{equation}\label{i100}
\underset{|J|= k}{\sup} [X^J u]_{\Gamma^{0,\alpha}(B(p_0,1/2))}  \leq C \left\{||u||_{L^{\infty}(B(p_0,1))} + \sum_{|I| \leq k-2} ||X^I f||_{L^{\infty}(B(p_0, 1))} + \sum_{|I|=k-2} [X^I f]_{\Gamma^{0, \alpha}(B(p_0, 1))} \right\},
\end{equation}
where  $[\ \cdot\ ]_{\Gamma^{0, \alpha}(B(p_0,1))}$ represents the  seminorm  defined by \eqref{semi} above.
\end{theorem}
As a corollary, we have the following rescaled estimate.

\begin{corollary}\label{intreg1}
Let $u$ be a solution to
$\sum_{ij} a_{ij}(p) X_i X_j u=f$
in a gauge ball $B(p_0,s)$. Suppose furthermore that for some $k \in \mathbb N$ such that $k \geq 2$ and  $0<\alpha<1$,  one has $a_{ij}, f \in \Gamma^{k-2, \alpha}(B(p_0,s))$. Then $u \in \Gamma^{k,\alpha}(B(p_0,s/2))$, and there exists a constant $C = C(\mathbb G,\lambda,[a_{ij}]_{\Gamma^{k-2, \alpha}(B(p_0,1))})>0$ such that the following estimates hold
\begin{align}\label{rei1}
& \operatorname{sup}_{|J|=k} ||X^J u||_{L^{\infty}(B(p_0,s/2))} \\& \leq  \frac{C}{s^k} \left\{||u||_{L^{\infty}(B(p_0,s))} + \sum_{|I| \leq k-2} s^{2+|I|} ||X^I f||_{L^{\infty}(B(p_0, s))} + \sum_{|I|=k-2} s^{2+|I|+\alpha}[X^I f]_{\Gamma^{0, \alpha}(B(p_0, s))} \right\}\notag,
\end{align}
and
\begin{align}\label{rei2}
&\underset{|J|= k}{\sup} [X^J u]_{\Gamma^{0,\alpha}(B(p_0,s/2))}\\&  \leq \frac{C}{s^{k+\alpha}} \left\{||u||_{L^{\infty}(B(p_0,s))} + \sum_{|I| \leq k-2} s^{2+|I|} ||X^I f||_{L^{\infty}(B(p_0, s))} + \sum_{|I|=k-2} s^{2+|I|+\alpha}[X^I f]_{\Gamma^{0, \alpha}(B(p_0, s))} \right\}.\notag
\end{align}
\end{corollary}
\begin{proof}
The proof of \eqref{rei1} and \eqref{rei2} is analogous to that of \cite[Corollary 3.2]{BGM} and therefore we skip the details.
\end{proof}
\section{Proof of Theorem \ref{main}}\label{S:main}

We begin with the following Lipschitz regularity result near a portion of the non-characteristic boundary. This constitutes the equicontinuous estimate that is required  in our compactness argument in the proof of Lemma \ref{MCL} below. Henceforth, given a symmetric uniformly elliptic matrix $\mathbb{A}=[a_{ij}]$ satisfying \eqref{ea0} for some $\lambda >0$ and all $p\in\Omega$,
 we use the notation
\begin{equation}\label{L}
\mathscr L\overset{def}{=}  \sum_{i,j=1}^m a_{ij}(p) X_iX_j.
\end{equation}

\begin{proposition}\label{Lip}
Let $\Omega\subset \mathbb{G}$ be a $C^{k,\alpha}$ domain, $k\geq 2$,  such that $p_0  \in \partial \Omega$ is a non-characteristic point.   Let $u\in \Gamma^{2}(\Omega)\cap C(\overline{\Omega})$ be a solution of $\mathscr L u = f$ in $\Omega$, with $u=0$ on $\partial \Omega$.
Furthermore, assume that $a_{ij}$ and $f$ are in $ \Gamma^{\alpha}(\overline{\Omega})$ for some $\alpha>0$. Then there exist $r_0$ and $C>0$,  depending on $\Omega, \lambda, \alpha$, such that
 \begin{equation}\label{lipunif}
\underset{\underset{p\neq p'}{p ,p'\in\overline{\Omega \cap B_e(p_0,r_0)}}}{\sup} \frac{|u(p)-u(p')|}{d(p,p')} < C ( \|f\|_{L^{\infty}} + \underset{\Omega}{\sup}\ |u|).\end{equation}
\end{proposition}
\begin{proof}
We let  $M \overset{def}{=}\underset{\Omega}{\sup}\ |u|.$ Since $p_0$ is non-characteristic, by using the $C^{2}$ character of $\Omega$ we can find a sufficiently small $0<r_1<1$, a local defining function $\varphi\in C^2(B_e(p_0,r_1))$ for $\Omega$, and a number $\gamma_0>0$ such that
\begin{equation}\label{unf0}
|\nabla_H \varphi(p)| \ge \gamma_0
\end{equation}
for every $p\in \partial \Omega \cap B_e(p_0,r_1)$. Notice that, in view of \eqref{cp3}, the condition \eqref{unf0} implies that all points $p\in \partial \Omega \cap B_e(p_0, r_1) $ are non-characteristic. We can further assume that for every such $p$ there exists a point $\tilde p\in \mathbb G\setminus \overline \Omega$ such that the Riemannian ball $B_e(\tilde p,r_1)$ be tangent from outside to $\partial \Omega$ in $p$. This means that
\begin{equation}\label{unf1}
\overline{B}_e(\tilde p,r_1) \cap \overline{\Omega}= \{p\}.
\end{equation}
Moreover  by choosing a  smaller $\gamma_0$ if needed, we can ensure that  for all $q \in \Omega\cap \partial B_e(p,r_1)$ one has
\begin{equation}\label{unf2}
d_e(q,\tilde p) \geq (1+\gamma_0) r_1.
\end{equation}
Given any $p\in \partial \Omega \cap B_e(p_0,r_1)$, consider the function $\psi(q) = d_e(q,\tilde p)^2$, where $\tilde p$ is as in \eqref{unf1}. Notice that it must be $\nabla \psi(p) = \alpha(p) \nu(p)$, for some $\alpha(p)\in \mathbb{R}$. Since $\nu(p)$ is also parallel to $\nabla \varphi(p)$, we conclude that $\nabla \psi(p) = \beta(p) \nabla \varphi(p)$ for some $\beta(p)\in \mathbb{R}$ such that $|\beta(p)| \ge \varepsilon_0>0$. By this observation and \eqref{unf0}, by possibly choosing a smaller $r_1$, we infer the existence of $\delta_0>0$ such that for all $p \in \Omega \cap B_e(p_0,r_1)$ one has
\begin{equation}\label{t1}
|\nabla_H \psi(p)| = |\beta(p)| |\nabla_H \phi(p)| \geq \delta_0.
\end{equation}
In order to establish \eqref{lipunif} we only need to show that, given any $p \in \partial \Omega \cap B_e(p_0,r_1)$, the following estimate hold for all $q \in \Omega\cap  B_e(p,r_1)$
\begin{equation}\label{lipu}
|u(q)|\leq K_1 d_e(q,p),
\end{equation}
where $K_1$ depends on $r_1, \gamma_0, \delta_0$  as in \eqref{unf0}, \eqref{t1}. By continuity, it suffices to show \eqref{lipu} for  $p= p_0$.  We denote by $\tilde p_0\in \mathbb G\setminus \overline \Omega$ the point corresponding to $p_0$ as in \eqref{unf1} above, and we now let $\psi(q) =  d_e(q,\tilde p_0)^2$. We note that $\psi\in C^\infty(\mathbb G)$. We next  introduce the function
\begin{equation}\label{funf}
g(q) \overset{def}{=} \left(\overline{M} + ||f||_{\infty} \right) \left(1-\left(\frac{r_1^2}{\psi(q)}\right)^k\right)
\end{equation}
for $q\in \Omega\cap B_e(p_0,r_1),$ where $\overline{M},k$ and $L$  will be chosen later. We clearly have
\begin{equation}\label{fgu1}
g\ge 0 = u\quad \ \ \ \ \ \ \text{on} \  \partial\Omega\cap B_e(p_0,r_1).
\end{equation}
If instead $q\in \Omega\cap \partial B_e(p_0,r_1)$, then it follows from \eqref{unf2}  that
\[
g(q) \geq \overline{M}\left(1-\left(\frac{1}{(1+\gamma_0)^2}\right)^k\right) \geq \frac{\overline{M}}{2},
\]
provided that $k\ge k_0 \overset{def}{=} \frac{\log 2}{\log(1+\gamma_0)}$. If we thus let $\overline{M}= 2 M$ and $k\ge k_0$, then we conclude that
\begin{equation}\label{fgu2}
g\ge u \quad \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\  \Omega\cap \partial B_e(p_0,r_1).
\end{equation}
If we could show that $\mathscr{L} g\leq \mathscr{L} u$ in $\Omega\cap  B_e(p_0,r_1)$, then from (\ref{fgu1}), (\ref{fgu2}) and the comparison principle in Lemma \ref{comp}, we can conclude that
\begin{equation}\label{bony}
u\leq g \quad \ \ \ \ \ \ \ \ \text{in}\ \Omega\cap  B_e(p_0,r_1).
\end{equation}
We now observe that if $h\in C^2(\mathbb{R})$, $v\in C^2(\mathbb G)$, then
\[
\mathscr{L} h(v) = h'(v)\mathscr{L} v +h''(v)\sum^m_{i,j=1}a_{ij}X_ivX_jv.
\]
Using this formula with $v=\psi$ and $h(t)= \overline M\left(1-(r_1^2/t)^k\right)$, we obtain
\begin{eqnarray*}
\mathscr{L} g &=&\left( \overline M + ||f||_{\infty}\right) k\frac{r_1^{2k}}{\psi^{k+2}}\left(\psi \mathscr{L} \psi - (k+1)\sum^m_{i,j=1}a_{ij}X_i\psi X_j\psi\right)\\
&\leq &\left( \overline M + ||f||_{\infty}\right) k\frac{r_1^{2k}}{\psi^{k+2}}\left(\psi \mathscr{L} \psi - \lambda(k+1)|\nabla_{H}\psi|^2\right).
\end{eqnarray*}
Note that in $\Omega\cap  B_e(p_0,r_1)$ we have from \eqref{t1}
\[
|\nabla_H \psi|^2 \ge \delta_0^2.
\]
Therefore, if $k$ is large enough, one can ensure that in $\Omega\cap  B_e(p_0,r_1)$
\[
\mathscr{L} g \leq  - ||f||_{\infty}.
\]
We thus have  $\mathscr{L} g \leq \mathscr{L} u$ in $\Omega\cap  B_e(p_0,r_1)$, and thus \eqref{bony} holds. In a similar way, if we take $-g$ as a lower barrier we can ensure that
\begin{equation}\label{bony2}
-g \leq u \quad\ \ \ \ \ \ \ \ \  \text{in} \ \Omega\cap  B_e(p_0,r_1).
\end{equation}
Combining \eqref{bony} and \eqref{bony2}, we conclude
\[
|u|\leq g \quad \ \ \ \ \ \ \ \ \  \text{in} \ \Omega\cap  B_e(p_0,r_1).
\]
Since $\psi\ge r_1^2$ in $\Omega\cap  B_e(p_0,r_1)$, by \eqref{funf} we see that the function $g$ is Lipschitz continuous in $\Omega\cap  B_e(p_0,r_1)$. From this fact, and $u(p_0)=g(p_0)=0$, we thus conclude that there exists a constant $K_1>0$ such that \eqref{lipu} holds for $p=p_0$, and therefore for all $p\in \partial \Omega\cap  B_e(p_0,r_1)$, with $q \in \Omega\cap  B_e(p,r_1)$.

If we now combine \eqref{lipu} with the left-hand side of \eqref{xy}, we can assert that there exists a constant $K_2>0$ such that
\begin{equation}\label{lipu2}
|u(q)|\leq K_2 d_C(q,p) \quad \text{for all}\ q \in \Omega\cap  B_e(p,r_1),\ p\in \partial \Omega\cap  B_e(p_0,r_1)
\end{equation}
where $K_2>0$ depends on $r_1, \gamma_0, \delta_0$  as in \eqref{unf0}, \eqref{t1}. Now the desired estimate \eqref{lipunif} follows by combining the boundary decay estimate \eqref{lipu2} with the interior estimates in \cite{CH}.

\end{proof}

\subsection{Proof of Theorem \ref{main}}
Recall that we are assuming that $\Omega \in C^{k, \alpha}$, and that, for $s>0$ we have agreed to write $\mathscr V_s = \mathscr V_s(e)$ and $\mathscr S_s = \mathscr S_s(e)$. After subtracting off the boundary datum $\phi$ and a left-translation, it suffices to  look at the following boundary value problem
\begin{equation}\label{bp}
\mathscr{L} u = f\ \text{in}\ \mathscr V_s,\ \ \ \ \ \ \ \ u=0\ \text{on}\ \mathscr S_s,
\end{equation}
where $\mathscr Lu$ is as in \eqref{L},
with $a_{ij}, f \in \Gamma^{k-2, \alpha}(\overline{\Omega})$.    Then, by scaling with respect to the family of dilations $\{\delta_{\lambda} \}_{\lambda>0}$ in \eqref{dil}, and an appropriate  change of coordinates in the horizontal layer $\mathfrak{g}_1$, we may also assume without loss of generality that $s=1$ and:
\begin{equation}\label{smn}
a_{ij}(e)=\delta_{ij},\ \ \ \ \  ||a_{ij} - \delta_{ij}||_{\Gamma^{k-2, \alpha}}, ||f||_{\Gamma^{k-2, \alpha}} \leq \delta.\end{equation}
Furthermore  we can also assume that
 $\mathscr V_1$ can be expressed in the logarithmic coordinates as
\begin{equation}\label{omega}
\Big\{(x',x_m,y)\mid x_m>h(x',y)\Big\}, \quad \text{with $h(0,0)=0$,  $\nabla_{x'}h(0,0)=0$},
\end{equation}
where $y=(\xi_2, ..., \xi_r)$, and $h\in C^{k,\alpha}$.
In what follows we will work with the distance $d_e(p,q)$ in $\mathbb G$. By this we mean the push forward  of the Euclidean distance on Lie algebra via the exponential map.
To simplify the notation we will write $\mathrm{dist}\,(p,\partial\Omega) = \inf \{d_e(p,q)\mid q\in\partial\Omega\}$,
instead of $\operatorname{dist}_e(p,\Omega)$, and accordingly write $d(p) = \mathrm{dist}\,(p,\partial\Omega), p\in B(1)$, instead of $d_e(p)$.
In logarithmic coordinates we have
\begin{equation}\label{distres}
d(p) = |\nabla_{H} d(e)| x_m + P_d + d_0(p),
\end{equation}
where $P_d$ is a polynomial of homogeneous degree at least 2 and $d_0 \in C^{k, \alpha}$ with $|d_0(p)|= O(|p|^{k+\alpha})$. Formula \eqref{distres} will play an important role and requires an explanation. The inner unit normal along the graph is given by
\[
\nu(x',x_m,y) = \left(- \frac{\nabla_{x'} h}{\sqrt{1+|\nabla_{(x',y)} h|^2}}, \frac{1}{\sqrt{1+|\nabla_{(x',y)} h|^2}}, - \frac{\nabla_{y} h}{\sqrt{1+|\nabla_{(x',y)} h|^2}}\right),
\]
where we have let $h = h(x',x_m,y)$. Since by \eqref{omega} we have $\nabla_{x'} h(0,0) =0\in \mathbb{R}^{m-1}$, the previous formula, combined with the fact $\nabla d(0) = \nu(0)$, gives
\[
\nabla d(0) = \left(0, \frac{1}{\sqrt{1+|\nabla_{y} h(0)|^2}}, - \frac{\nabla_{y} h(0)}{\sqrt{1+|\nabla_{y} h(0)|^2}}\right).
\]
In view of \eqref{fdG} this gives
\[
X_1 d(0) = 0,..., X_{m-1} d(0) = 0,\ X_m d(0) = \frac{1}{\sqrt{1+|\nabla_{y} h(0)|^2}},
\]
and therefore
\[
\nabla_H d(0) = \left(0, \frac{1}{\sqrt{1+|\nabla_{y} h(0)|^2}}\right).
\]
Finally, since $d\in C^{k,\alpha}$ (see \cite[Lemma 14.16]{GT}), from Taylor formula we have
\[
d(p) = d(0) + \langle\nabla d(0),p\rangle + O(|p|^2)
= x_m |\nabla_H d(0)| + \langle\nabla_y d(0),y\rangle + \text{higher order terms},
\]
from which \eqref{distres} follows.
Having said this, for every $0<\sigma\leq 1$ consider  the domain  $\Omega_{\sigma}=  \delta_{\sigma^{-1}}(\Omega)$. We set
\[
\mathscr U_\sigma \overset{def}{=}  \Omega_\sigma \cap B(\sigma^{-1}),\ \ \ \ \mathscr T_\sigma\overset{def}{=}  \partial \Omega_\sigma \cap B(\sigma^{-1}).
\]
In logarithmic coordinates $\Omega_{\sigma}$ is given by
\begin{equation}\label{an1}
\Omega_{\sigma}=\{(x',x_m,y_2,\ldots,y_r)\mid (\sigma x',\sigma x_m,\sigma^2y_2,\ldots,\sigma^ry_r)\in\Omega\}.
\end{equation}
Observe that $\partial\Omega_\sigma$ is given by
\begin{equation}\label{xms}
x_m=h_{\sigma}(x',y)=h_{\sigma}(x',y_2,\ldots,y_r)\overset{def}{=}\frac 1\sigma h(\sigma x',\sigma^{2}y_2,\ldots,\sigma^{r} y_r).
\end{equation}
With $\Omega_\sigma$ as in \eqref{an1} we let $d_{\sigma}(p) = \inf \{d_{e}(p,q)\mid q\in\partial\Omega_\sigma\}$.
From \eqref{omega} and \eqref{xms} it follows that given $\delta>0$, if $\sigma$ is small enough, one can ensure that
\begin{equation}\label{sm0}
|| h_{\sigma}||_{C^{k, \alpha}(B(r))} \leq \delta,
\end{equation}
which, combined with \eqref{distres}, implies that
\begin{equation}\label{sm1}
||d_{\sigma}- |\nabla_{H} d_{\sigma}(e)| x_m||_{\Gamma^{k, \alpha}(B(1))} \leq \delta.
\end{equation}
Moreover, keeping in mind that $e \in \partial \Omega_\sigma$, and setting $u_{\sigma}(\cdot) = u(\delta_{\sigma} \cdot)$, the boundary value  problem \eqref{bp} becomes
\begin{equation}\label{bp1}
\sum a_{ij} (\delta_{\sigma} p) X_i X_j u_{\sigma}(p) = \sigma^2 f(\delta_{\sigma} p) \ \ \text{in}\ \ \mathscr V(\sigma^{-1}),\ \ \ \ \ \ \ u=0\ \ \text{on}\ \ \mathscr S(\sigma^{-1}).
\end{equation}
Therefore, with the hypothesis \eqref{smn} in place, after substituting $\Omega$ with $\Omega_{\sigma}$, $u$ with $u_\sigma$ and so on, we can assume that \eqref{sm0} and \eqref{sm1} hold, where $\delta$ will be chosen later.
Our proof will be divided into the following three steps:
\begin{itemize}
\item[1.] existence of the $k$-th order Taylor polynomial at every boundary point of $\mathscr S_{1/2}$ via compactness arguments;
\item[2.] H\"older continuity of the $k$-th order derivatives at the boundary;
\item[3.] H\"older continuity of the $k$-th order derivatives up to the boundary.
\end{itemize}
Inspired by ideas in \cite{DS1, DS2}, we next introduce the relevant notion of approximating polynomial.
\begin{definition}
We say that a  polynomial $P$ of  degree less than or equal to  $k-1$ is a \emph{$k$-th order approximating polynomial}  for $\mathscr{L} u= f$ at $e$ if for all $I$ such that $|I| \leq k-2$ one has
\[
X^I\left(\mathscr{L}(d P)\right)(e)=X^I f(e).
\]
\end{definition}
Let $P$ be a given stratified polynomial of the type $\sum_{J} a_J z^J$. Then  one has
\[
\mathscr{L} (dP)=P \mathscr{L} d+d\mathscr{L} P+2\langle A(p) \nabla_H d,\nabla_H P\rangle.
\]
Since $\mathscr{L} x_m = 0$, in view of \eqref{distres} we have
\begin{eqnarray*}
P\mathscr{L} d+d\mathscr{L} P&=& |\nabla_{H}d(e)|x_m\mathscr{L} P+(P_d+d_0)\mathscr{L} P+P\mathscr{L}(P_d+d_0).
\end{eqnarray*}
By adding and subtracting the identity matrix to $A(p)$, we obtain
\begin{equation}
\label{xmLP}
 |\nabla_{H} d(e)|x_m\mathscr{L} P= |\nabla_{H} d(e)|x_m\Delta_{H}P+ |\nabla_{H} d(e)|x_m(\mathscr{L}-\Delta_{H}) P,
\end{equation}
and also
\begin{align}
\label{aijXdXP}
2\langle A(p) \nabla_H d,\nabla_H P\rangle & = 2 \langle\nabla_{H}\left(|\nabla_{H} d(e)|x_m+P_d+d_0\right),\nabla_{H}P\rangle
\\
& +2\langle(A(p)-I)\nabla_H d,\nabla_H P\rangle
\notag\\
& = 2|\nabla_{H} d(e)|X_mP + 2 \langle\nabla_{H}\left(P_d+d_0\right),\nabla_{H}P\rangle
\notag\\
& +2\langle(A(p)-I)\nabla_H d,\nabla_H P\rangle,
\notag
\end{align}
where in the last equality we have used that $\nabla_{H}x_m = e_m$. We now recall that, for $j\in\{1,...,r\}$ and $s\in \{1,...,m_j\}$, we denote by $\overline{(j,s)}$ a multi-index which has 1 in the $(j,s)$ position and zeroes everywhere else, with the agreement that, when $j=1$, we indicate $\overline{(1, i)}$ with $\overline{i}$.
If we then assume that, for a given multi-index $J\in (\mathbb N \cup \{0\})^N$ of the type  $J= (\beta_{1,1},\ldots,\beta_{1,m},\ldots,\beta_{r,1},\ldots,\beta_{r,m_r})$  (see \eqref{I} above)  with $d(J)\leq k-1$, we take $P(z)=z^J$,  then \eqref{xiex2} gives
\begin{eqnarray}
\label{XmP}
X_mP&=& \beta_{1,m}z^{J-\bar{m}}+\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{0\leq d(I^{(j-1)})= j-1}b^s_{m,I^{(j-1)}}\beta_{j,s}z^{I^{(j-1)}+J-\overline{(j,s)}}\right)
\end{eqnarray}
Using the stratified Taylor's inequality  \eqref{stp1}, as well as \eqref{smn} and  \eqref{sm1}, it follows that we may write
\begin{align}
\label{l.o.t.}
 &|\nabla_{H} d(e)|\left(x_m(\mathscr{L}-\Delta_{H}) P+2\sum^m_{i,j}(a_{i,j}-\delta_{ij})X_idX_jP\right)\\
& +  (P_d+d_0)\mathscr{L} P+P\mathscr{L}(P_d+d_0)+2\langle\nabla_{H}\left(P_d+d_0\right),\nabla_{H}P\rangle\notag \\
= &\sum_{I}c^J_Iz^I+w_J\notag,
\end{align}
where $c_{I}^J= O(\delta)$,   $c_{I}^J \neq 0$ provided $d(J) \leq d(I) \leq k-2$, and $w_J= O(\delta |p|^{k-2+\alpha})$.
Therefore, using \eqref{xmLP}-\eqref{l.o.t.}, and  computing $\Delta_{H} P$ using the expression in \eqref{subLap}, we find
\begin{align}\label{ex1}
& \mathscr{L}(dP)  = |\nabla_{H} d(e)| \beta_{1,m}(\beta_{1,m} +1) z^{J- \bar{m}}  + |\nabla_{H} d(e)|\sum_{i=1}^{m-1}  \beta_{1, i} (\beta_{1, i} -1) z^{J-2\bar{i} +\bar{m}} \\
  &+ |\nabla_{H} d(e)| \sum^{m}_{i=1}\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{d(I^{(j-1)})= j-1}\alpha_{1,i} \beta_{j,s} b^s_{i,I^{(j-1)}}z^{I^{(j-1)- \overline{i} +J - \overline{(j,s)} + \overline{m}}}\right)\notag\\& +2|\nabla_{H} d(e)|\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{ d(I^{(j-1)})= j-1}b^s_{m,I^{(j-1)}}\beta_{j,s}z^{I^{(j-1)}+J-\overline{(j,s)}}\right)\notag\\
& +2|\nabla_{H} d(e)|\sum^{m}_{i=1}\beta_{1,i}\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{d(I^{(j-1)})= j-1}b^s_{i,I^{(j-1)}}\beta_{j,s}z^{I^{(j-1)}+J-\overline{(j,s)}-\bar{i}+\bar{m}}\right)\notag \\
& +|\nabla_{H} d(e)|\sum^{m}_{i=1}\sum^r_{j,j'=2}\sum^{m_j}_{s,s'=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1\\  d(I^{(j'-1)})= j'-1 }}\beta_{j,s}\beta_{j',s'}b^s_{i,I^{(j-1)}}b^{s'}_{i,I^{(j'-1)}}z^{I^{(j-1)}+I^{(j'-1)}+J-\overline{(j,s)}-\overline{(j',s')}+\bar{m}}\right)\notag\\
&  +|\nabla_{H} d(e)|\sum^{m}_{i=1}\sum^r_{\substack{j,j'=2\\ j'\leq j-1}}\sum^{m_j}_{s,s'=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1\\  d(I^{(j'-1)})= j'-1 }}\alpha_{j',s'}\beta_{j,s}b^s_{i,I^{(j-1)}}b^{s'}_{i,I^{(j'-1)}}z^{I^{(j-1)}-\overline{(j',s')}+I^{(j'-1)}+J-\overline{(j,s)}+\bar{m}}\right)\notag\\
& +\sum_{I} c_{I}^J z^{I}  +w_J(p)\notag.
\end{align}
Moreover, in view of \eqref{smn} and \eqref{sm1}, we have that $|c_{I}^J|, ||w_J||_{\Gamma^{k-2+\alpha}} \leq C\delta$. In general, if $P$ is of the form
\[
P(z)=\sum_{0\leq d(J)\leq k-1}b_Jz^J,
\]
then  from \eqref{ex1} it follows that
\begin{equation}\label{apro}
\mathscr{L} (dP) = R(z) + w,
\end{equation}
with $w$ such that
\begin{equation}\label{w}
||w||_{\Gamma^{k-2+\alpha}} \leq C\delta,\ \ \  |w(p)| = O(\delta|p|^{k-2+\alpha}),
\end{equation}
and
\begin{equation}\label{R}
R= \sum_{\{J\mid d(J) \leq k-2\}}  e_J z^J,
\end{equation}
where the coefficients $e_J$ satisfy the following algebraic condition
\begin{align}\label{alg}
 e_J =& |\nabla_{H} d(e)| (\beta_{1,m} + 1) (\beta_{1,m}+2) b_{J+\bar{m}}  +|\nabla_{H} d(e)|\sum_{i \neq m}  (\beta_{1,i} + 1) (\beta_{1,i}+2) b_{J+2\bar{i} - \bar{m}}\\
  &+ |\nabla_{H} d(e)| \sum^{m}_{i=1}\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{d(I^{(j-1)})= j-1}a_1(J, i, j, s) b_{J- I^{(j-1)} +\overline{(j,s)} + \bar{i} - \bar{m}}\right)\notag\\
 &+2|\nabla_{H} d(e)|\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{ \substack{d(I^{(j-1)})= j-1}} a_2 (J,m, i, j, s) b_{J +\overline{(j,s)}-I^{(j-1)}}\right)\notag \\
 & +2|\nabla_{H} d(e)|\sum^{m}_{i=1}\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1}} a_3(i, J,s)b_{J+\overline{(j,s)}+\bar{i}-\bar{m}-I^{(j-1)}}\right)\notag \\
 &+|\nabla_{H} d(e)|\sum^{m}_{i=1}\sum^r_{j,j'=2}\sum^{m_j}_{s,s'=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1\\ d(I^{(j'-1)})= j'-1}} a_4(J,j,s,j',s',i)b_{J+\overline{(j,s)}+\overline{(j',s')}-\bar{m}-I^{(j-1)}-I^{(j'-1)}}\right)\notag\\
 &+|\nabla_{H} d(e)|\sum^{m}_{i=1}\sum^r_{\substack{j,j'=2\\ j'\leq j-1}}\sum^{m_j}_{s,s'=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1\\  d(I^{(j'-1)})= j'-1}}a_5(J,j,s,j',s',i)b_{J+\overline{(j,s)}+\overline{(j',s')}-\bar{m}-I^{(j-1)}-I^{(j'-1)}}\right)\notag\\
& +\sum_I c_{J}^I b_I. \notag
\end{align}
We now introduce the following.
\newline
\textbf{\underline{Algorithm}:}\label{A}
Given a stratified  polynomial $R= \sum_{J: d(J) \leq k-2}  e_J z^J$,  our objective is finding a polynomial of degree at most $k-1$, i.e., a polynomial of the form $P= \sum_{J: d(J) \leq k-1}  b_J z^J$, such that \eqref{alg} holds.
This will be possible if, given constants $e_J$, we can solve the linear system \eqref{alg} above for the unknowns $b_I$. Notice that \eqref{alg} is an equation where $b_{J+\bar{m}}$ is a linear combination  of  $b_I$, such that either
\begin{itemize}
\item $d(I)<d(J)+1$ or
\item$d(I)=d(J)+1$, then $\beta^I_{1,m} < \beta^J_{1,m}+1$.
\end{itemize}
Since this is always the case, it follows that we can solve \eqref{alg} by arbitrarily assigning all the coefficients $b_I$ when $I$  is a multi-index having $ \beta^I_{1,m}= 0$.

We next proceed with the proof of \emph{Step 1}.

\textbf{Step 1:}
In this step, we show the existence of a $k-$th order Taylor polynomial at every boundary point.
The next compactness lemma is crucial for the proof of $\Gamma^{k,\alpha}$ decay estimate at the boundary.
\begin{lemma}[Compactness Lemma]
\label{MCL}
Let $u\in \Gamma^2(\mathscr V_1)\cap C(\overline{\mathscr V_1})$ be a solution of \eqref{dp0}. Assume that for some small $0<r\leq 1$ there exists an approximating polynomial $P= \sum_{d(I) \leq k-1} b_I z^I$, with  $|b_I|\leq  1$, such that
\begin{equation}
\label{udP1}
\|u-dP\|_{L^{\infty}(\Omega\cap B(r))}\leq r^{k+\alpha}.
\end{equation}
Then, if $\delta>0$ is small enough in \eqref{sm0}, there exists a universal $0<\rho<1$ and $\widetilde{P}$ approximating  polynomial of degree at most  $k-1$ such that
\[
\|u-d\widetilde{P}\|_{L^{\infty}(\Omega\cap B(\rho r))}\leq (\rho r)^{k+\alpha},
\]
and
\[
\|P-\widetilde{P}\|_{L^{\infty}(\Omega\cap B( r))}\leq Cr^{k-1+\alpha}.
\]
\end{lemma}
\begin{proof} Let
\begin{equation}
v(p)\overset{def}{=} \frac{(u-(dP))\circ(\delta_{r}(p))}{r^{k+\alpha}},\quad p\in \widetilde{\Omega}\cap B(1),
\end{equation}
where $\widetilde{\Omega}=\Omega_{r}$. We immediately observe that the validity of \eqref{udP1} implies  that
\begin{equation}
\label{vbdd1}
\|v\|_{L^{\infty}(\widetilde{\Omega}\cap B(1))}\leq 1.
\end{equation}
Furthermore, we obtain
\begin{equation}\label{newl}
\begin{cases}
\tilde \mathscr{L} v \overset{def}= \sum a_{ij} (\delta_r \cdot ) X_i X_j v= r^{-k+2-\alpha}\left(f(\delta_r(p))-\mathscr{L}(dP)(\delta_r(p))\right) \quad &\text{in $\widetilde{\Omega}\cap B(1)$},\\
v=0\quad &\text{on $\partial\widetilde{\Omega}\cap B(1)$}.
\end{cases}
\end{equation}
Since $f\in \Gamma^{k-2,\alpha}$, we have that
\begin{equation}
\label{Tayf}
|f(p)-P^f_e(p)|\leq C\delta d(p,e)^{k-2+\alpha},
\end{equation}
where $P^f_e$ is the Taylor  polynomial of $f$ at $e$  of homogeneous degree at most $k-2$. We note that \eqref{Tayf} is a consequence of the stratified Taylor inequality \eqref{stp1} combined with the reduction in \eqref{smn} that guarantees $||f||_{\Gamma^{k-2, \alpha}}\le \delta$. Recall that $P$ is an approximating polynomial with respect to $\mathscr{L}$  and $f$, hence in \eqref{apro} we must have
\begin{equation}\label{PfR}
P^f_e=R.
\end{equation}
To clarify \eqref{PfR}, we note first that from \eqref{apro} it follows that $R$ is the $(k-2)$-th order Taylor polynomial  of $\mathscr{L} (dP)$ at $e$ because the remainder $w$ in \eqref{apro} is $O(\delta|p|^{k-2+\alpha})$, see \eqref{w}. Furthermore,  by the definition of our approximating polynomial, we have that $X^I (\mathscr{L} (dP)(e)= X^{I} f(e)$ for all $|I| \leq k-2$, and thus (by the definition of the Taylor polynomial) both $f$ and $\mathscr{L} (dP)$ have the same Taylor polynomial of order $k-2$ at $e$. Noting that $P^f_e$ is the Taylor polynomial of $f$ at $e$ of order $(k-2)$, this ensures that \eqref{PfR} hold. Continuing further,  if we combine \eqref{apro} (in which $R$ is now replaced by $P^f_e$) with \eqref{Tayf}, we find that that  term in the right-hand side  of \eqref{newl} can estimated as follows
\begin{align*}
& \ \ \  \left| r^{-k+2-\alpha}\left(f(\delta_r(p))-\mathscr{L}(dP)(\delta_r(p))\right) \right|
\\
& = \left| r^{-k+2-\alpha}\left(f(\delta_r(p))- P^f_e(\delta_r(p))- w(\delta_r(p))\right) \right|
\\
& \leq r^{-k+2-\alpha} \left( |f(\delta_r(p))- P^f_e(\delta_r(p))| + |w(\delta_r(p))| \right)\leq C \delta,
\end{align*}
where in the last inequality we have used \eqref{Tayf} and \eqref{w}.
This ensures the validity of the following estimate
\begin{equation}
\label{bdLv}
|\tilde \mathscr{L} v|\leq C\delta.
\end{equation}
If $\delta>0$ is sufficiently small, Proposition \ref{Lip} implies the existence of a constant $C > 0$, depending exclusively on $\lambda$ but not on $\delta$,  such that
\[
\|v\|_{\Gamma^{0,1}(\widetilde{\Omega}\cap B(4/5)))}\leq C.
\]
Similarly to what we did in \cite{BGM}, in the logarithmic coordinates we now extend  $v$ to a function $V: B(4/5)\to \mathbb{R}$  such that $V\equiv v$ in $\widetilde{\Omega}\cap B(4/5)$ (we use the classical procedure in e.g. \cite[p. 14]{LM}):
\begin{equation}\label{ext}
V(x',\tilde{x}_m,y)=
\begin{cases}
v(x',\tilde{x}_m,y)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \tilde x_m \overset{def}= x_m - h(x', y) \ge 0,
\\
\sum^{k+2}_{i=1}c_iv\left(x',-\frac{\tilde{x}_m}{i},y\right)\ \quad \tilde{x}_m<0,
\end{cases}
\end{equation}
where the constants $c_i$'s are determined by the system of equations
\begin{equation}
\label{constants}
\sum^{k+2}_{i=1}c_i(-1/i)^m=1,\quad m=0,1,2,..., k, k+1.
\end{equation}
It follows that
\begin{equation}\label{ext}
\|V\|_{\Gamma^{{0,1}}(B(4/5))}\leq C'\ \|v\|_{\Gamma^{{0,1}}(\widetilde{\Omega}\cap B(4/5)))}\leq C_1,
\end{equation}
for some $C_1 = C_1(\lambda,\alpha)>0$.
By a compactness argument analogous to that in the proof of \cite[Lemma 4.1]{BGM}, using  the equicontinuous Lipschitz estimate in Proposition \ref{Lip}, we can assert that  given $\varepsilon>0$ there exists a sufficiently small $\delta>0$  in \eqref{sm0}, for which there is a $v_0$ such that
\begin{equation}\label{lim}
\begin{cases}
\Delta_{H}v_0=0 \quad &\text{in  $B(4/5)\cap \{x_m>0\}$},\\
v_0=0\quad &\text{on $ B(4/5)\cap \{x_m=0\}$},
\end{cases}
\end{equation}
with
\[
\|v-v_0\|_{L^{\infty}(\widetilde{\Omega}\cap B(1/2))}\leq \varepsilon.
\]
Said differently, when $\Omega \cap B(1)$ (and consequently $\tilde \Omega \cap B(1)$) is flat enough, we can approximate $v$ by a solution to a ``flat" boundary value problem.
Moreover, it follows from \eqref{ext} that  in the region $\{x_m <0\}$,  $v_0$  satisfies
\begin{equation}\label{v0}
v_0(x', x_m, y) = \sum_{i=1}^{k+2} c_i v\left(x', -\frac{x_m}{i}, y\right),
\end{equation}
where $c_i$'s satisfy \eqref{constants}.
From the   boundary regularity in Theorem \ref{KN}, and the fact that \eqref{v0} determines a $C^{k+1}$ extension across $\{x_m=0\}$, we find
\[
\|v_0\|_{C^{k+1}(B(1/2))}\leq C.
\]
Therefore, there exists a   polynomial  $Q$, with homogeneous degree less than or equal to $k-1$, such that
\begin{equation}
\label{Polv0}
\|v_0-x_mQ\|_{L^{\infty}(B(\rho))}\leq C_0 \rho^{k+1},\quad \text{with $0<\rho<\frac{1}{2}$}.
\end{equation}
Given $0<\alpha<1$, we now fix $\rho>0$ such that
\begin{equation}
\label{rhofix}
C_0\rho^{k+1}=\frac{\rho^{k+\alpha}}{4}.
\end{equation}
Then, we take
\begin{equation}
\label{epsilonfix}
\varepsilon\overset{def}{=}\frac{\rho^{k+\alpha}}{16},
\end{equation}
which in turn decides the choice of $\delta$.
We may thus conclude that
\begin{equation}\label{fr1}
\|v-x_mQ\|_{L^{\infty}(\widetilde{\Omega}\cap B(\rho))}\leq \left(\frac{1}{4}+\frac{1}{16}\right)\rho^{k+\alpha}.
\end{equation}
Since
$$
\tilde d (\cdot) \overset{def}= \frac{d(\delta_r \cdot)}{r |\nabla_{H} d(e)|} = x_m + O(\delta),
$$
for a possibly smaller $\delta$, we obtain from \eqref{fr1} that the following holds
\begin{equation}\label{ki}
\|v- \tilde d Q\|_{L^{\infty}(\widetilde{\Omega}\cap B(\rho))}\leq \left(\frac{3}{4} \right)\rho^{k+\alpha}.\end{equation}
Rescaling back to $u$, we find
 \begin{align}
 \label{udPQ}
\Bigg\|u-d\left(P+\frac{r^{k-1+\alpha}}{|\nabla_{H}d(e)|}Q\circ \delta_{r^{-1}}\right)\Bigg\|_{L^{\infty}(\Omega\cap B(\rho r))}\leq\left(\frac{3}{4}\right)(r\rho)^{k+\alpha}.
\end{align}
We emphasise here that
\[
P+\frac{r^{k-1+\alpha}}{|\nabla_{H}d(e)|}Q\circ \delta_{r^{-1}}
\]
 might not be an approximating polynomial for $u$. Hence,  we need to further modify $Q$ to some other polynomial $\widetilde{Q}$, without essentially
modifying the estimate \eqref{udPQ}.
With this in mind, we write
\[
Q=\sum_{0\leq d(I)\leq k-1}b_Iz^I\quad \text{and}\quad \widetilde{Q}=\sum_{0\leq d(I)\leq k-1}\widetilde{b}_Iz^I.
\]
In view of \eqref{alg}, after a computation, we find that for $P+ r^{k-1+\alpha} \tilde Q \circ \delta_{r^{-1}}$ to be approximating,   the coefficients $\widetilde{b}_I$ have to satisfy the following algebraic conditions:

\begin{align}\label{alg1}
 &0=|\nabla_{H} d(e)| (\beta_{1,m} + 1) (\beta_{1,m}+2) \widetilde{b}_{J+\bar{m}}  +|\nabla_{H} d(e)| \sum_{i \neq m}  (\beta_{1,i} + 1) (\beta_{1,i}+2) \widetilde{b}_{J+2\bar{i} - \bar{m}}\\
  &+ |\nabla_{H} d(e)| \sum^{m}_{i=1}\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{d(I^{(j-1)})= j-1}a_1(J, i, j, s) \widetilde{b}_{J- I^{(j-1)} +\overline{(j,s)} + \bar{i} - \bar{m}}\right)\notag\\
 &+2|\nabla_{H} d(e)|\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{ \substack{d(I^{(j-1)})= j-1}} a_2 (J,m, i, j, s) \widetilde{b}_{J +\overline{(j,s)}-I^{(j-1)}}\right)\notag \\
 & +2|\nabla_{H} d(e)|\sum^{m}_{i=1}\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1}} a_3(i, J,s)\widetilde{b}_{J+\overline{(j,s)}+\bar{i}-\bar{m}-I^{(j-1)}}\right)\notag \\
 &+|\nabla_{H} d(e)|\sum^{m}_{i=1}\sum^r_{j,j'=2}\sum^{m_j}_{s,s'=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1\\ d(I^{(j'-1)})= j'-1}} a_4(J,j,s,j',s',i)\widetilde{b}_{J+\overline{(j,s)}+\overline{(j',s')}-\bar{m}-I^{(j-1)}-I^{(j'-1)}}\right)\notag\\
 &+|\nabla_{H} d(e)|\sum^{m}_{i=1}\sum^r_{\substack{j,j'=2\\ j'\leq j-1}}\sum^{m_j}_{s,s'=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1\\  d(I^{(j'-1)})= j'-1}}a_5(J,j,s,j',s',i)\widetilde{b}_{J+\overline{(j,s)}+\overline{(j',s')}-\bar{m}-I^{(j-1)}-I^{(j'-1)}}\right)\notag\\
& +\sum_I \widetilde{c}_{J}^I \widetilde{b}_I,\notag
\end{align}

where \begin{equation}\label{smo0} \widetilde{c}_{J}^I= r^{d(J) +1 -d(I)} c_{J}^I.\end{equation}

Moreover, since $c_{J}^I \neq 0$ provided that $d(I) \leq d(J)$ and $r \leq 1$, it follows  from \eqref{smo0} that
\begin{equation}\label{smo}
|\widetilde{c}_{J}^I| \leq C \delta.
\end{equation}
Since $x_mQ$ is $\Delta_{H}-$harmonic, we find that the  coefficients $b_I$ must satisfy
\begin{align}\label{alg2}
 &0= (\beta_{1,m} + 1) (\beta_{1,m}+2) b_{J+\bar{m}}  +\sum_{i \neq m}  (\beta_{1,i} + 1) (\beta_{1,i}+2) b_{J+2\bar{i} - \bar{m}}\\
  &+  \sum^{m}_{i=1}\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{d(I^{(j-1)})= j-1}a_1(J, i, j, s) b_{J- I^{(j-1)} +\overline{(j,s)} + \bar{i} - \bar{m}}\right)\notag\\
 &+2\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{ \substack{d(I^{(j-1)})= j-1}} a_2 (J,m, i, j, s) b_{J +\overline{(j,s)}-I^{(j-1)}}\right)\notag \\
 & +2\sum^{m}_{i=1}\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1}} a_3(i, J,s) b_{J+\overline{(j,s)}+\bar{i}-\bar{m}-I^{(j-1)}}\right)\notag \\
 &+\sum^{m}_{i=1}\sum^r_{j,j'=2}\sum^{m_j}_{s,s'=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1\\ d(I^{(j'-1)})= j'-1}} a_4(J,j,s,j',s',i)b_{J+\overline{(j,s)}+\overline{(j',s')}-\bar{m}-I^{(j-1)}-I^{(j'-1)}}\right)\notag\\
 &+\sum^{m}_{i=1}\sum^r_{\substack{j,j'=2\\ j'\leq j-1}}\sum^{m_j}_{s,s'=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1\\  d(I^{(j'-1)})= j'-1}}a_5(J,j,s,j',s',i)b_{J+\overline{(j,s)}+\overline{(j',s')}-\bar{m}-I^{(j-1)}-I^{(j'-1)}}\right).\notag
\end{align}
From \eqref{alg1} and \eqref{alg2} we find that $d_J = \widetilde{b}_J - \frac{b_J}{|\nabla_{H} d(e) |}$ should satisfy the following  system
\begin{align}\label{alg4}
 &-\sum_I  \widetilde{c}_{J}^I \frac{b_I}{|\nabla_{H} d(e)|}=|\nabla_{H} d(e)| (\beta_{1,m} + 1) (\beta_{1,m}+2) d_{J+\bar{m}}  +|\nabla_{H} d(e)| \sum_{i \neq m}  (\beta_{1,i} + 1) (\beta_{1,i}+2) d_{J+2\bar{i} - \bar{m}}\\
  &+ |\nabla_{H} d(e)| \sum^{m}_{i=1}\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{d(I^{(j-1)})= j-1}a_1(J, i, j, s) d_{J- I^{(j-1)} +\overline{(j,s)} + \bar{i} - \bar{m}}\right)\notag\\
 &+2|\nabla_{H} d(e)|\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{ \substack{d(I^{(j-1)})= j-1}} a_2 (J,m, i, j, s) d_{J +\overline{(j,s)}-I^{(j-1)}}\right)\notag \\
 & +2|\nabla_{H} d(e)|\sum^{m}_{i=1}\sum^r_{j=2}\sum^{m_j}_{s=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1}} a_3(i, J,s)d_{J+\overline{(j,s)}+\bar{i}-\bar{m}-I^{(j-1)}}\right)\notag \\
 &+|\nabla_{H} d(e)|\sum^{m}_{i=1}\sum^r_{j,j'=2}\sum^{m_j}_{s,s'=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1\\ d(I^{(j'-1)})= j'-1}} a_4(J,j,s,j',s',i)d_{J+\overline{(j,s)}+\overline{(j',s')}-\bar{m}-I^{(j-1)}-I^{(j'-1)}}\right)\notag\\
 &+|\nabla_{H} d(e)|\sum^{m}_{i=1}\sum^r_{\substack{j,j'=2\\ j'\leq j-1}}\sum^{m_j}_{s,s'=1}\left(\sum_{\substack{d(I^{(j-1)})= j-1\\  d(I^{(j'-1)})= j'-1}}a_5(J,j,s,j',s',i)d_{J+\overline{(j,s)}+\overline{(j',s')}-\bar{m}-I^{(j-1)}-I^{(j'-1)}}\right)\notag\\
& +\sum_I \widetilde{c}_{J}^I d_I.\notag
\end{align}
From \eqref{smo} it follows that the left-hand side of the linear system in \eqref{alg4} is bounded by $C\delta$. Therefore, using the above \textbf{Algorithm}, we can determine $d_I$'s such that $|d_I| \leq C\delta$. It thus  follows that for a new constant $C>0$ we have
\[
|| |\nabla_{H} d(e)| \tilde Q - Q||_{L^{\infty}(\tilde \Omega \cap B(\rho))} \leq C\delta.
\]
Using this inequality in the estimate \eqref{ki}, we obtain
\begin{equation}\label{ki1}
\|v- \tilde d |\nabla_{H} d(e)|  \tilde Q\|_{L^{\infty}(\widetilde{\Omega}\cap B(\rho))}\leq \left(\frac{3}{4} \right)\rho^{k+\alpha} + C\delta  \leq \rho^{k+\alpha},\end{equation}
provided $\delta$ is chosen smaller if necessary.
Rescaling back to $u$, we obtain
 \begin{align}
 \label{newu}
\Bigg\|u-d\left(P+r^{k-1+\alpha}\tilde Q\circ \delta_{r^{-1}}\right)\Bigg\|_{L^{\infty}(\Omega\cap B(\rho r))}\leq(r\rho)^{k+\alpha}.
\end{align}
The conclusion of the lemma thus follows with
\[
\tilde P=P+r^{k-1+\alpha}\tilde Q\circ \delta_{r^{-1}}.\]

\end{proof}

Using Lemma \ref{MCL}, we next show the existence of the $k$-th order Taylor polynomial at $e \in \partial \Omega$.

\begin{lemma}\label{btaylor}
Let $u$ be as in Lemma \ref{MCL}. Then there exists a polynomial $P_e$ at $e$ of homogeneous degree less than or equal to $k$  such that for all $p \in \Omega \cap B(1/2)$
\begin{equation}\label{bdry}
|u(p)- P_e(p)| \leq C |p|^{k+\alpha}.
\end{equation}
\end{lemma}
\begin{proof}
Using the above \textbf{Algorithm}, we  first find an approximating polynomial $P_1$. Then by multiplying $u$ and $P_1$ with a small constant, we may assume that the hypothesis of the Lemma \ref{MCL} is satisfied for $r=1/2$. Iterating Lemma \ref{MCL} at  scales $\rho/2, \rho^2/2$, and so on, we obtain a limiting  approximating polynomial  $P_0$ such that
\begin{equation}\label{bdry1}
|u(p)- dP_0(p)| \leq C |p|^{k+\alpha},
\end{equation}
for all $p \in \Omega \cap B(1/2)$. Since $d P_0 \in \Gamma^{k, \alpha}(\overline{\Omega \cap B(1)})$,  by letting $P_e$ to be the $k$-th order Taylor polynomial of $dP_0$ at $e$, we find that \eqref{bdry} follows.

\end{proof}

\begin{remark}\label{dec1}
From the proof of Lemma \ref{btaylor} it follows that since $P_e$ equals the $k$-th order Taylor polynomial of $dP_0$, where $P_0$ is approximating, one has for all $I$ such that $|I| \leq k-2$
\begin{equation}
X^{I} ( \mathscr LP_e) (e) = X^I f(e).
\end{equation}
Then from the stratified Taylor inequality \eqref{stp1} we have
\begin{equation}\label{dec2}
|X^I (\mathscr LP_e) (p) - X^I f(p) | \leq C |p|^{k-2 - |I| +\alpha}
\end{equation}
for all $I$ such that $|I| \leq k-2$.  Furthermore, by left translation it follows that for every $g \in \partial \Omega \cap B(1/2)$, there exists a polynomial $P_g$ of degree at most $k$ such that
\begin{equation}\label{dec0}
\begin{cases}
|u(g \circ p) - P_g(p)| \leq C|p|^{k+\alpha},
\\
|X^I (\sum_{i,j=1}^m a_{ij}(g \circ \cdot) X_iX_j P_g)(p) - X^I f( g \circ p) | \leq  C|p|^{k-2 - |I| + \alpha}, \ |I| \leq k-2,\\
\operatorname{sup}_{|I|=k-2} [X^I (\sum_{i,j=1}^m a_{ij}(g \circ \cdot) X_i X_jP_g)(\cdot) - X^I f( g \circ \cdot)]_{\Gamma^{0, \alpha}} \leq C.
\end{cases}
\end{equation}

\end{remark}

This completes the proof of \emph{Step 1}. We now proceed with that of \emph{Step 2}.

\textbf{Step 2:} In our subsequent discussion given any boundary point $ p\in \mathscr S_{1/2}$ we will denote by $P_{p}$ the corresponding Taylor polynomial of $u$ at $p$ whose existence has been established in Lemma \ref{btaylor}. Our main objective in this step is to show that,   given $p_1, p_2 \in \mathscr S_{1/2}$, the following estimate holds for some $K_0>0$ universal:
\begin{equation}\label{bdcmp}
\operatorname{sup}_{|J|=k} |X^J P_{p_1} (e) - X^J P_{p_2}(e)| \leq  K_0\ d(p_1,p_2)^{\alpha}.
\end{equation}

Let $t= d(p_1,p_2)$. We consider a ``non-tangential" point $p_3 \in \mathscr V_1$ at a (pseudo-)distance from $p_1$ comparable to $t$,  i.e., let $p_3$ be such that
\begin{equation}\label{nont}
d(p_3, p_1) \sim t,\ d(p_3, \partial \Omega)  \sim t,
\end{equation}
where we have let $d(p,\partial \Omega)= \underset{p' \in \partial \Omega}{\inf} \ d(p,p')$.
Since $\mathscr S_1$ is a non-characteristic $C^{1,\alpha}$ portion of $\partial \Omega$, it is possible to find such a point $p_3$. Arguing as in the proof of Theorem 7.6 in \cite{DGP}, at any scale $t$ one can find a non-tangential pseudo-ball from inside centered at $p_3$. The non-tangentiality of $B(p_3,at)$ means that there exists a universal $a>0$ sufficiently small (which can be seen to depend on the Lipschitz character of $\partial \Omega$ near the non-characteristic portion $\mathscr S_1$), such that for some $c_0$ universal one has for all $p \in B(p_3,a t)$
\[
d(p,\partial \Omega) \geq  c_0 t.
\]
For $s=1,2$, we have that in $B(p_3, at)$, $v^s= u( \cdot) - P_{p_s}( p_s^{-1} \circ \cdot)$, solves
\begin{equation}\label{tu}
\sum_{ij} a_{ij} X_i X_j v^s = \tilde f^s \overset{def} = f - \sum_{i,j} a_{ij} X_i X_j P_{p_s} (p_s^{-1} \circ \cdot).\end{equation}
From the  estimates  in \eqref{dec0} we have for $s=1,2$,
\begin{equation}\label{p1}
\begin{cases}
||v^s||_{L^{\infty}(B(p_3, at))} \leq C t^{k+\alpha},\\
|X^I \tilde f^s (p) | \leq  Ct^{k-2 - |I| + \alpha}, \ |I| \leq k-2,\ \text{for $p \in B(p_3, at)$},\\
\sum_{|I|=k-2} [X^I \tilde f^s]_{\Gamma^{0,\alpha}} \leq C.
\end{cases}
\end{equation}
Using now the interior estimate \eqref{rei1}, \eqref{p1} and the fact that $P_{p_s} \in \mathcal{P}_k$,  we find
\begin{align}\label{rei5}
&\operatorname{sup}_{|J|=k}|X^J v^s (p_3)| =\operatorname{sup}_{|J|=k} |X^J u (p_3) - X^J P_{p_s}(e) | \\ &\leq \frac{C}{t^k} \left(||v^s||_{L^{\infty}(B(p_3, at))}+\sum_{|I| \leq k-2} t^{2+|I|} || X^I \tilde f^s||_{L^{\infty}(B(p_3, at)} + t^{k+\alpha} \sum_{|I|=k-2} [X^I \tilde f^s]_{\Gamma^{0,\alpha}} \right)
\notag
\\
& \leq C t^{\alpha},
\notag
\end{align}
where in the last inequality we have used \eqref{p1}. From \eqref{rei5} and the triangle inequality we thus have for any $J$ such that $|J|=k$,
\begin{equation}\label{result}
|X^J P_{p_1} (e) - X^J P_{p_2}(e) | \leq |X^J u (p_3) - X^J P_{p_1}(e) | + |X^J u (p_3) - X^J P_{p_2}(e) |  \leq Ct^{\alpha},
\end{equation}
which finishes the proof of the claim \eqref{bdcmp}.

\textbf{Step 3:} (Proof of Theorem \ref{main})
In this final step it suffices to prove that the $k$-th order horizontal derivative of $u$  is in $\Gamma^{0,\alpha}$ up to the boundary. As a first step we observe that we can find $\varepsilon>0$ sufficiently small such that for any $p\in \mathscr V_\varepsilon$ there exists $\bar p \in \mathscr S_{1/2}$ for which
\begin{equation}\label{eqt}
d(p,\bar p) = d(p,\partial \Omega).
\end{equation}
To finish the proof of the theorem we will show that for all  $p,p' \in \mathscr V_\varepsilon$ and $J$ such that $|J|=k$,  we have   \begin{equation}\label{fop}
|X^J u(p)- X^J u(p')|\leq  C^\star d(p,p')^{\alpha},
\end{equation}
for some  universal constant $C^\star >0$. The sought for conclusion  follows from \eqref{fop} by a standard covering argument.

Given two points  $p, p' \in \mathscr V_\varepsilon$ we denote by $\bar p, \bar p'$  the corresponding points in $\mathscr S_{1/2}$ for which \eqref{eqt} holds.
Henceforth, we use the notation $\delta(p) = d(p,\partial \Omega)$ for  $p \in \Omega$.
Without loss of generality we assume that $\delta(p)= \min \{\delta(p),\delta(p')\}$. By Lemma \ref{btaylor} and Remark \ref{dec1}  in Step 2, given $\bar p$ as in \eqref{eqt} we know that there exists a $k$-th polynomial $P_{\bar{p}}$ such that for every $q \in \mathscr V_1$ one has
\begin{equation}\label{fopb}
\begin{cases}
|u(q) - P_{\bar{p}}(\bar{p}^{(-1)} \circ q)|\leq C_2d(\bar{p},q)^{k+\alpha},\\
|X^I (\sum_{i,j=1}^m a_{ij}(\bar{p} \circ \cdot) X_i X_jP_{\bar{p}})(q) - X^I f( \bar{p} \circ q) | \leq  C|q|^{k-2 - |I| + \alpha}, \ |I| \leq k-2,\\
\operatorname{sup}_{|I|=k-2} [X^I (\sum_{i,j=1}^m a_{ij}(\bar{p} \circ \cdot)X_i X_j P_{\bar{p}})(\cdot) - X^I f( \bar{p} \circ \cdot)]_{\Gamma^{0, \alpha}} \leq C.\end{cases}
\end{equation}
There exists two possibilities:
\begin{itemize}
\item[(i)] $d(p, p') \leq \frac{\delta(p)}{2}$;
\item[(ii)] $d(p, p') > \frac{\delta(p)}{2}$.
\end{itemize}
 \textbf{Case (i)}:  We first observe that we have $B(p, \delta(p)) \subset \Omega$. Therefore, if we let $v(\cdot)=u(\cdot)-P_{\bar p}(\bar{p}^{(-1)} \circ \cdot)$, then similarly to Step 2 we see that $v$ satisfies an equation of the following type  in $B(p, \delta(p)) \subset \Omega$
\[
\sum_{ij} a_{ij} X_i X_j v(\cdot) = \tilde f,
\]
where $\tilde f$ is as in  \eqref{tu} and moreover  estimates as in \eqref{p1} hold, with $t$ replaced by $\delta(p)$.
In particular, it   follows from \eqref{fopb} that the following estimate holds for some $\tilde C_2>0$
\begin{equation}\label{sup}
||v||_{L^{\infty}(B(p,\delta(p))} \leq \tilde{C}_2 \delta(p)^{k+\alpha}.
\end{equation}
Since $p' \in B(p, \delta(p)/2)$, using the interior regularity estimate \eqref{rei2}  in Corollary \ref{intreg1} and also  \eqref{p1}, we conclude that given $J$ such that $|J|=k$,  there exists some $\tilde C$ depending also on $\tilde{C}_2$ such that the following inequality holds
\begin{align}\label{sup2}
& |X^Jv(p) - X^J v(p')|
\\
& \leq \frac{\tilde C d(p, p')^\alpha}{\delta(p)^{k+\alpha}} \left(||v||_{L^{\infty}(B(p,\delta(p))}+\sum_{|I| \leq k-2} \delta(p)^{2+|I|} ||X^I \tilde f||_{L^{\infty}(B(p, \delta(p))}+\delta(p)^{k+\alpha} \sum_{|I|=k-2} [X^I \tilde f]_{\Gamma^{0, \alpha}(B(p, \delta(p))} \right)
\notag\\
&\leq  \frac{\tilde C}{\delta(p)^{k+\alpha}} \delta(p)^{k+\alpha} d(p, p')^{\alpha}\ \text{(using \eqref{sup} and \eqref{p1})}\notag\\ & \leq   \tilde{C_1} d(p, p')^{\alpha}.
\notag
\end{align}
Since $P_{\bar p} \in \mathcal{P}_k$,  we have that $X^J P_{\bar p}$ is constant for any $J$ such that $|J|=k$. From the definition of $v$ it thus follows that
\[
|X^J u(p) - X^Ju(p')| = |X^J v(p) - X^J v(p')|.
\]
From this observation and \eqref{sup2} we conclude that \eqref{fop} holds.

\noindent \textbf{Case (ii)}: The hypothesis $d(p, p') > \frac{\delta(p)}{2}$ and \eqref{eqt} imply
\[
d(p,\bar{p}) = d(p,\partial\Omega) = \delta(p) < 2 d(p, p').
\]
Combining this observation with \eqref{pseudo}, which in particular implies a pseudo-triangle inequality,  gives
\begin{equation}\label{n101}
d(p', \bar{p}) \leq C_0(d(p',p) + d(p,\bar{p})) \leq  C_0 (d(p',p) + 2 d(p',p)) = 3 C_0 d(p, p').
\end{equation}
Since $d(p',\bar{p}) \geq d(p',\partial \Omega) = \delta(p')$, we immediately find from \eqref{n101}
\begin{equation}\label{n103}
\delta(p') \leq 3C_0 d(p,p').
\end{equation}
From  \eqref{n101}, \eqref{n103}, and an application of the pseudo-triangle inequality, we finally have
\begin{equation}\label{dist}
 d(\bar{p}, \bar{p}') \leq C_0 (d(\bar p, p') + d(p', \bar{p}')) = C_0 (d(\bar p,p') + \delta(p')) \leq 6 C_0^2 d(p, p').
\end{equation}
Let now $a$ be the universal constant in the existence of a non-tangential (pseudo)-ball in Step 2. From the estimate \eqref{rei5} in  Step 2 we infer that the following holds for $J$ such that $|J|=k$,
\begin{equation}\label{imo2}
|X^Ju(p) - X^J P_{\bar{p}}(e)|  \leq C \delta(p)^{\alpha} \leq C d(p, p')^{\alpha}.
\end{equation}
Note that, since we are in Case (ii), in the last inequality in \eqref{imo2} we have used $\delta(p) \leq 2 d(p, p')$.
Arguing similarly to \eqref{imo2}, we find
\begin{equation}\label{imo1}
|X^J  u(p') - X^J P_{\bar{p}'}(e)| \leq C \delta(p')^{\alpha} \leq C d(p, p')^{\alpha},
\end{equation}
where in the last inequality in \eqref{imo1} we have this time used \eqref{n103}.
From \eqref{bdcmp} and  \eqref{dist} we now have
\begin{equation}\label{finali}
|X^J P_{\bar p}(e) - X^J P_{\bar{p}'}(e)| \leq K_0 d(\bar p, \bar{p}')^{\alpha} \le C d(p, p')^{\alpha}.
\end{equation}
Applying the triangle inequality with the estimates \eqref{imo2}, \eqref{imo1} and \eqref{finali}, we finally conclude that \eqref{fop} holds.  This completes the proof of the theorem.

redpajama_set_name: RedPajamaArXiv


The practice of using Neutral Mask to train actors began with Jacques Copeau and was developed by Jacques Lecoq. This male neutral is loosely based on the Sartori masks used at the Lecoq School in Paris.
This male neutral is made out of neoprene and has an adjustable strap. The mask is easily sanitized and comes with a care sheet that explains how to best maintain the mask.
If you have questions about how the masks are used in actor training feel free to check out the resources on the About Masks in Actor Training page or just give Jonathan a call.
With the donning of the neutral mask, the actor must leave behind his or her personality, everything that he or she knows, all the physical qualities that make the individual distinctive, and start in a state of simple readiness for what is to come next.
The neutral mask admonishes the actor to reach for a state of not knowing in which discovery of what is new is the only way to move forward through the world.
There is no better tool to provide for basic understanding of the actors process than the neutral mask.


Every time you go to the track, you should be digging for something to take from the day. I watched, as you did, the first day of the Keeneland meet. It was rainy, and looked miserable. On Saturday, the weather calls for a much cooler day. So, wear a jacket if you are heading that way, and know that the turf course should have a great deal of give to it. You can always count on the poly track being fast….
Here are a few nuggets to keep in mind before making your wagers. For this day, the graded stakes have yielded a sweet 7-1 or better payoff average. That is the kind of day that can break the bank, if used properly. Only 2 of 15 have been more than 15-1, and winning chalksters have taken a nose dive. So, steer clear of the low price plays, and find that fertile middle ground runner.
I would look for a turf runner who has run in Europe, or enjoys a soft to yielding course. I feel the weather will keep the damp in the ground, and breeding may play a factor. You will want to keep track of your tier levels to find value. I also think it is a safe bet to change your form for a soft turf course. This will give you different predictions.
Trainer watch…. Keep your eye peeled for Ken McPeek, Bill Mott, Richard Dutrow, and Kiaran McLaughlin. There are a bevy of year-to-year suspects: Rusty Arnold, Mike Maker, and Neil Howard.
Jockey alert… Bet your pockets on Jamie Theriot on turf events going long. He loves the course, and has the number dialed in already. Play Robby Alabarado, Kent D, Julien Leparoux, and Rajiv Maragh if you want to find solid payoffs. You can feel safe with Jose Lezcano, Miguel Mena, and Shaun Bridgmohan when he is town.
Overall, the track plays closers. I would bank on this going in, but with the rain and cooler temps, I would bet a tight track where they lay a little closer than usual. So, there it is.. A little nugget or two, and all you need now is your data in hand. I would be sure and keep my eye on the weather, and watch the icons. They will guide you well during this short meet where history has held good value.


text:
10 Most famous landmarks in Asia that should be on your bucket list
by Esther | Last updated Oct 30, 2020 | Published on Oct 30, 2020 | Asia | 0 comments
Are you looking for places to visit in Asia? Here are the most famous landmarks in Asia that you should be on your bucket list.
Asia is the biggest and the most populous continent in the world. It is not only gifted with amazing people, culture, and history but also some of the most famous landmarks in the world.
Whether it is natural landmarks that showcase the beautiful landscapes or the manmade landmarks that live to celebrate history and culture, there are so many landmarks in Asia that are worth visiting.
From Mt Fuji in Japan that receives dedicated climbers, the Great Wall of China that was constructed to act as protection now the most visited attraction in China to Angkor Wat that celebrates history and religion, here are the most famous landmarks of Asia.
Famous landmarks in Asia
1. Taj Mahal, India
Taj Mahal is not only the icon and symbol of India but also one of the most famous landmarks of Asia that should be on every traveler's bucket list.
Located on the banks of Yamuna River in the city of Agra, the Taj Mahal is an Ivory-white marble mausoleum that was constructed in 1632 to house the tomb of Mumtaz Mahal who was the favorite wife of Mughal Emperor Shah Jahan – a story that brings out the true meaning of love.
The Taj Mahal receives over 7-8 million visitors every year making it the most visited attraction in India.
It might be the impressive dome-shaped structure atop the tomb of the beloved wife that gets all the attention, but this incredible Asian landmark consists of a number of complexes that are worth exploring.
Also, its magnificent beauty and importance led it to not only be inscribed as a UNESCO World Heritage Site in 1983 but also one of the New 7 Wonders of the world.
The Infusion of the Muslim, Indian and Persian architecture makes the Taj Mahal one f the most beautiful buildings in Asia – no wonder it is one of the most photographed monuments of Asia.
And what makes the Taj Mahal even more impressive is the ability to change color depending on the time of the day.
So if you're looking for places to visit in India, Taj Mahal should be the first on your list.
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2. The Great Wall of China, China
Constructed over 2000 years ago with the initial intent of protecting China's territory from its enemies, the great wall of China has become one of the major landmarks in Asia receiving over 10 million visitors every year. It's actually the most visited attraction in China.
The great wall of China covers an area of 21,196 kilometers (13,171 miles) long, making it the longest structure ever built by humans – now that's an impressive structure that it's even hard to see every inch of just one section in a day.
The impressive views from standing at one of the many watchtowers make the various hikes to the wall worth it. The hikes present various difficulties but the end is rewarding as you get to appreciate this incredible Asian landmark which is also one of the New 7 wonders of the world.
Whether you visit to remember the thousands of workers that died during construction and buried within its walls or to appreciate the engineering they put in place, the Great Wall of China is one of the famous Asian landmarks that must not be missed while in China.
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3. Mount Everest, Nepal and China
Located in both Nepal and China, Mount Everest is the world's highest mountain above sea level with an elevation of 8,848 m (29,029 ft), making it one of the best natural Asian landmarks that should be on every traveler's crazy bucket list.
The most fascinating fact about Mount Everest is that it's still growing at a height of about 40 centimeters (16 inches) per century creating almost a new elevation level every new century.
Though Mount Everest has a number of hiking trails (17 to be exact), it still presents a certain level of difficulty that is considered the hardest in the entire world.
But even with the harsh conditions at the Mountain that has led to the death of over 300 people due to altitude sickness, avalanches, or falling into crevasses, a number of mountain hikers still make their way to tackle this hiking giant.
However, among the thousands that try to reach the summit every other year, half of them always give up due to the difficulty it presents.
Regardless, Mount Everest is one of the famous landmarks in Asia that always just stays a bucket list item that is never ticked off for many people not only because of its difficulty level but also the huge price tag attached to it.
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4. Angkor Wat, Cambodia
Sitting on 162.6 hectares (1.626 km2; 402 acres), the Angkor Wat Complex in Cambodia is the largest religious monument in the world making it one of the most famous landmarks of Asia that is worth visiting.
This enormous Buddhist temple that was originally constructed in the 12th century as a dedication to Hindu god, Vishnu was later transformed into a Buddhist temple by the end of the same century making it the largest Buddhist temple not only in Asia but also the entire world.
Currently, it is no longer an active temple but has become one of the famous monuments in Asia receiving over 2. 6million international visitors a year.
Other than the religious importance that Angkor Wat has, the architecture of the entire complex is one to marvel about.
From the thousands of buildings that perch through the sky to roaming around the large compound while admiring the stonework on the walls, it's truly a beautiful sight.
Angkor Wat is a true symbol of Cambodia as it even appears on the country's National flag making it one of the important landmarks in Asia.
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5. Chocolate Hills, The Philippines
Covering an area of 50 square kilometers in the middle of the island of Bohol in the Philippines, Chocolate Hills are a group of over 1200 hills that are extraordinarily shaped to create an amazing landscape.
Though their formation is still not yet very clear, what is clear is where their name came from. In the rainy season, the hills put on a beautiful green display but in the dry season, the grass dries out creating chocolate-colored grasses – hence their name.
The individual hills themselves might not be high with most of them ranging from 30 to 50 meters high, but the amazing scenery they create together is awe-inspiring.
There may not be much to do as it's not possible to climb the hills but rather to take in the views from the observation deck, but the Chocolate Hills are still one of the most impressive natural landmarks in Asia.
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Bagan, an ancient city in Myanmar is one of the largest archeological sites in the world. Relativity similar to the Angkor Wat of Cambodia in a way, Bagan is an entire city with over 3000 temples, Pagodas, and Stupas even though it is estimated that over 13,000 temples sat there in the early years before they were destroyed by the raiders.
However, even with the grand appearance and historic significance attached to Bagan, it was not initially enlisted as a UNESCO site as the organization believed that not all temples were built in the historic era but it was later included on the list in 2019.
But despite all that, Bagan is still one of the most impressive landmarks of South East Asia. What's even more exciting about Bagan is that each construction has a spiritual meaning in the Buddhism religion.
But with many temples and Pagodas in the city, it can be hard to decide on which one to visit and which one not to.
So some of the most notable ones, not miss include; Ananda Temple, Gawdawpalin Temple, Bupaya Pagoda, Dhammayangyi Temple, and Dhammayazika Pagoda all of which were constructed differently yet so similar.
Though there are many ways to enjoy the ancient city and appreciate the Burmese kind of architecture either by renting a scooter or hiring a taxi, the best way to take in the amazing views and the designs displayed by the Myanmar craftsmen in handicrafts is by going for a hot air balloon over the thousand of temples and Pagodas.
Bagan may be confused with Angkor Wat by many people (which is clear why), it's also one of the famous southeast Asia landmarks just like its counterpart.
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7. Petronas Towers, Malaysia
Located in the Malaysian capital of Kuala Lumpur, the Petronas Towers which are also known as the Petronas twin towers are the tallest structure in the country and also one of the most visited.
The towers are actually the icon of Malaysia internationally making it worth being on the list of the famous buildings in Asia.
Rising at a height of 451.9 m (1,483 ft.) with 88 floors, Petronas towers are the tallest twin towers in the world since 1996 and also the second tallest buildings in the world after being surpassed by Taipei 101 in Taiwan in 2004 after having held the title of tallest since 1998.
Besides the impressive height of the Petronas, the architecture itself is quite magnificent with a tube in tube design meant to give them that tube-like look.
One of the other fascinating things about the Petronas other than the incredible views from the observation deck is the sky bridge that connects both towers – it's actually the highest 2 story bridge in the world making this Asian landmark even grander.
Whether it is being mesmerized by the detail that went into designing the exterior or the weaving patterns that resemble Malaysia's local handicrafts and the beautiful Islamic patterns, the Petronas twin towers are indeed one of the top landmarks in Southeast Asia that are worth a visit when in Malaysia.
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8. Mount Fuji, Japan
Standing tall at an elevation of 3,776.24 m (12,389.2 ft.), Mt. Fuji is the highest mountain in Japan and also one of the Three Holy mountains in the country.
This active volcano that last erupted in 1707 and 1708 is not only a true international icon of Japan but also one of the famous landmarks of Asia that brings in over 200,000 hikers and climbers every year that want to reach the summit making it the most visited attraction in Japan.
A hike to the summit of the mountain at sunrise brings out the true meaning of the "Land of the raising sun" – something many hikers want to experience at least once in a life time.
Climbing Mt. Fiji might present a few challenges to beginner hikers but as the old Japanese saying goes, " A wise person will climb Mt. Fuji once in their lifetime…….", so even if you're just starting out as a mountain climber, this is a challenge you should take on.
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9. Borobudur, Indonesia
Constructed in the 9th century, Borobudur which is located in central Java in Indonesia is the largest Buddhist temple occupying an area of over 123 x 123 meters with over 500 Buddha statues dotted around the complex.
Even though the original reason for its construction is still not yet known up to date, Borobudur took on a number of purposes in the current days by acting as a pilgrimage place for all Buddhist to observe Vesak as they celebrate the life and achievements of their Buddha, Siddhārtha Gautama during the first night of the full moon.
It is also the single most visited tourist attraction in Indonesia receiving over 2 million visitors per year which has really impacted the country's economy in an upward direction.
Besides the many Buddha statues and relief scenes around the complex, the architecture of Borobudur is one to marvel at.
It was constructed without any kind of cement and mortar but looks like interlocking blocks that create a stepped pyramid look. And the views of the overlooking green lush fields add more to the beauty of the temple.
So if you're looking for Asian landmarks to add to your Asia bucket list, Borobudur in Indonesia is one not to miss.
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10. Ha Long Bay, Vietnam
Located in one of the cheapest countries in Asia, Ha Long Bay in Northern Vietnam is one of the most beautiful natural landmarks in Asia.
Sprinkled with over 1600 limestone islands and islets surrounded by rainforests, Ha Long is one natural landmark that depicts beauty like nothing else.
From the magical natural landscape, floating villages to little beautiful beaches, Ha Long Bay is one attraction that should be on every traveler's bucket list.
What even makes a trip to Halong Bay even more interesting is cruising on the "wooden junk" as you take in all the beauty you can.
With Ha Long Bay receiving over 6 million visitors per year, is not only the most visited attraction in Vietnam but also one of the famous Asian landmarks.
There you have it friends, I hope that this list of the top landmarks in Asia gave you more ideas to add to your long travel bucket list.
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I am Esther, an avid traveler from Uganda. When I am not traveling, I am either thinking of my previous trip, dreaming of my next trip, or sharing travel tips and experiences on my blog. I hope i can inspire you to travel more through this blog. Want to learn more about me? Just click on the photo above to read the full story.
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