text:
\section{Introduction}
In this article, we study viscosity solutions to a class of mixed operators defined by super-positioning Pucci-Heisenberg maximal operator with the fractional sub-Laplacian. We establish the comparison principle, existence and regularity of solutions to the following problem:
\begin{align}\label{eq 2.1}
	\begin{cases}
	\a\mathcal{M}^+_{\lambda,\Lambda}\big(D_{\mathbbm{H}^N,S}^2u\big)-\b\big(-\Delta_{\mathbbm{H}^N}\big)^su=f &\text{in } {\Omega},\\
		u=g &\text{in }\mathbbm{H}^N\setminus\Omega,
	\end{cases}
\end{align}
where $\Omega$ is a bounded domain (open, connected set) in Heisenberg group $\mathbbm{H}^N\simeq \mathbb{R}^{2N+1}$, $f\in C(\overline{\Omega}$)$,\,g\in C(\mathbbm{H}^N\setminus\Omega)$ are bounded and  $\a\geq0$ \& $\b> 0$ are constants. Here,
\begin{align}\label{mathcal L}
	\mathcal{L}u:=\a\mathcal{M}^+_{\lambda,\Lambda}\big(D_{\mathbbm{H}^N,S}^2u\big)-\b\big(-\Delta_{\mathbbm{H}^N}\big)^su
\end{align}
is a mixed operator on $\mathbbm{H}^N,$ where $\mathcal{M}^+_{\lambda,\Lambda}$ is the Pucci's extremal (maximal) operator which is the fully nonlinear operator and $(-\Delta_{\mathbbm{H}^N})^s$ is the fractional sub-Laplacian.

Recently, mixed operators have been studied in the Euclidean framework by several researchers. These operators occur naturally, for instance, in the study of plasma physics \cite{Blaze}, population dynamics \cite{Dipierro} and many others. One may see \cite{Barles2, Biagi, Biagi2, Biagi3} for the works on mixed operators in the Euclidean setting. In the above works, the local term is  second order linear elliptic operator and nonlocal term is the fractional Laplacian. There are several articles, where authors have established the existence of solution of the Dirichlet problem using the classical Perron's method \cite{Ishii}. For instance, one may see \cite{Lara, Mou1} and the reference therein for  the Euclidean setting.

In recent years, there has been a significant amount of works on the existence and qualitative properties of solutions to PDEs
in non-Euclidean settings, for e.g., Heisenberg group, and more generally on sub-Riemannian, Carnot-Carath\'eodory spaces. T. Bieske \cite{Bieske} studied infinite harmonic functions in the Heisenberg group using the notion of viscosity solutions. C. Y. Wang \cite{Wang} established the uniqueness of viscosity solution of $\Delta_\infty$ (infinity Laplacian) equation on Carnot groups. B. Wang \cite{Wang 2} investigated the removable singularities for viscosity subsolutions to degenerate elliptic Pucci operators in the Heisenberg group.

It is easy to see that when $\a=0,$ the operator in $\eqref{mathcal L}$ is the fractional sub-Laplacian. Very recently, G. Palatucci and M. Piccinini \cite{Palatucci} considered a large class of nonlinear integro-differential operators in the Heisenberg group setting $\mathbbm{H}^N.$ They proved the general Harnack inequalities for the solutions to Dirichlet problem concerning nonlinear integro-differential operators. It is worth noticing that these problems have connections in quantum mechanics \cite{quantum}, ferromagnetic trajectories \cite{ferro}, image segmentation models \cite{image}, non-Markovian coupling for Brownian motions \cite{non-markov} and many others. We recall that in the above contexts, the non-Euclidean geometry occurs naturally.

F. Ferrari and E. Vecchi \cite{Ferrai} established the H\"older regularity of uniformly continuous and bounded viscosity solutions of degenerate fully nonlinear equations in $\mathbbm{H}^1$. We also refer to \cite{Cutri} for the existence results and Liouville, Harnack type qualitative properties of fundamental solutions. Y. Y. Li and B. Wang \cite{Wang 2} established a form of comparison principle for sub-elliptic equations in the Heisenberg setting. One may also see Manfredini \cite{Manfredi} et al. for the H\"older continuity and boundedness estimates for nonlinear fractional equations in $\mathbbm{H}^N,$ where authors considered equations driven by integro-differential operators whose model is the fractional $p$-Laplacian on Heisenberg group given by
\begin{align}
	\mathcal{L}u(\xi):=P.V.\int_{\mathbbm{H}^N}\frac{\big|u(\xi)-u(\eta)\big|^{p-2}(u(\xi)-u(\eta))}{d_0(\eta^{-1}\,o\,\xi)^{Q+sp}}d\eta, \,\,\xi\in\mathbbm{H}^N.
\end{align}
Here the symbol $P.V.$ in the expression stands for \enquote{in the principal value sense} and $d_0$ denotes a homogeneous norm on $\mathbbm{H}^N$(see Definition \ref{homo}). $Q=2N+2$ is the homogeneous dimension of $\mathbbm{H}^N.$

To the best of our knowledge, the mixed operators of type \eqref{mathcal L} have not been yet considered in the Heisenberg group setting. Motivated by the above works on the mixed operators and recent works on non-Euclidean setting, we consider a class of mixed operators \eqref{mathcal L} on $\mathbbm{H}^N.$

The main results of this paper are the following theorems:
\begin{thm}\label{stability}\textbf{\textit{(Stability).}}
	Let $u_n\in$ LSC$(\Omega)$ be bounded in $\mathbbm{H}^N$ and satisfy
	\begin{align}
		\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}u_n\big)-\b\big(-\Delta_{\mathbb{H}^N}\big)^su_n \leq f_n \text{ in }\Omega
	\end{align}
	in the viscosity sense. Let the following be hold: \\
	(i) $u_n$ converges to $u$ in the $\Gamma$-sense in $\Omega,$\\
	(ii) $u_n$ converges to $u$ a.e. in $\mathbbm{H}^N,$\\
	(iii) $f_n\longrightarrow f$ locally uniformly in $\Omega$.

	Then
	\begin{align}
		\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}u\big)-\b\big(-\Delta_{\mathbb{H}^N}\big)^su \leq f \text{ in }\Omega.
	\end{align}
\end{thm}
Next, we state the comparison principle for viscosity solutions of PDE concerning operators given by \eqref{mathcal L}. The proof is immediate by making use of Lemma \ref{Lemma 1.6} and Lemma \ref{comp} (see, next).
\begin{thm} \label{Comparison}\textbf{\textit{(Comparison Principle).}}
Let $\Omega$ be a bounded domain in $\mathbbm{H}^N.$ Let $f\in C(\Omega)$ and  $u,\,v$ be bounded USC and LSC functions in $\Omega,$ respectively, which satisfy
\begin{align}
	\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}u\big)-\b\big(-\Delta_{\mathbb{H}^N}\big)^su \geq f
\end{align}
and
\begin{align}
	\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}v\big)-\b\big(-\Delta_{\mathbb{H}^N}\big)^sv \leq f
\end{align}
in the viscosity sense in $\Omega.$ Also, if $u\leq v$ in $\mathbbm{H}^N\setminus\Omega$. Then $u\leq v$ in $\mathbbm{H}^N.$
\end{thm}

Further, adapting the standard techniques of using comparison principle for sub and super-solutions and then following  Perron's method, we have the following existence result.
\begin{thm}\label{Theorem 1.3}
	Let $\Omega\subset\mathbbm{H}^N$ be satisfy the exterior Heisenberg ball condition. Let $g\in C(\mathbbm{H}^N\setminus\Omega)$ and bounded in $\mathbbm{H}^N.$ Then there exists a viscosity solution $u\in C(\overline{\Omega})$ of
	\begin{align}
		\begin{cases}
			\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}u\big)-\b\big(-\Delta_{\mathbb{H}^N}\big)^su = 0 &\text{ in }\Omega\\
			u=g &\text{ in }\mathbbm{H}^N\setminus\Omega.
		\end{cases}
	\end{align}
\end{thm}
As mentioned earlier, several authors have established the existence of solutions to Dirichlet problems using the Perron's method. For the Heisenberg group setting, we refer to \cite{Ochoa}, where the authors prove the existence of a viscosity solution for a class of linear second order equations in Heisenberg group. Making use of the comparison principle (Theorem \ref{Comparison}), the proof of Theorem \ref{Theorem 1.3} follows using the standard arguments.

We mention that F. Ferrari and E. Vecchi \cite{Ferrai} studied the H\"older behaviour of fully nonlinear local equations. In particular, we have the following result due to \cite{Ferrai}:
\begin{cor}(Theorem 1.3 \cite{Ferrai})
	Let $u\in C(\mathbbm{H}^1)$ be a bounded and uniformly continuous viscosity solution of
	\begin{align}
		\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}u(\xi)\big)-c(\xi)u(\xi)=f(\xi) \text{ in }\mathbbm{H}^1.
	\end{align}
Let $L_1,$ $L_2,$ $\gamma_1,$ and $\gamma_2$ be positive constants s. t. $\gamma_i\in(0,1],\,i=1,2$ and for any $ \xi,\,\eta\in \mathbbm{H}^1,$
\begin{align}
	\big|c(\xi)-c(\eta)\big|\leq L_1\big|\xi\,o\,\eta^{-1}\big|^{\gamma_1}_{\mathbbm{H}^1},\qquad \big|f(\xi)-f(\eta)\big|\leq L_2\big|\xi\,o\,\eta^{-1}\big|^{\gamma_2}_{\mathbbm{H}^1}.
\end{align}
Let $c(\xi)$ be positive for all $\xi\in\mathbbm{H}^1$ and
\begin{align}
	\displaystyle{\inf_{{\xi\in B_R^{\mathbbm{H}^1}}(P)}}c(\xi):= c_0>0.
\end{align}
Then there exists $\gamma:=\gamma(c_0,p,L_1,L_2,\Lambda)\in (0,1], \gamma\leq \min\{\gamma_1,\gamma_2\}$ such that
\begin{align}
	\big|u(\xi)-u(\eta)\big|\leq L\big|\xi\,o\,\eta^{-1}\big|_{\mathbbm{H}^1}^\gamma,\,\text{ for }\xi\in\mathbbm{H}^1,
\end{align}
for some $L=L(c_0,P,L_1,L_2,\Lambda)>0.$
\end{cor}

For more such results on regularity in the Euclidean setting, when there is no local term and integro-differential operators are of fractional Laplacian type, we refer to \cite{Silvestre, Lara}. We also mention a very recent work by Biagi et al. \cite{Biagi2}, where authors established the interior Sobolev regularity as well as boundary regularity of Lipschitz type for mixed local and nonlocal operators in the Euclidean framework. More precisely, authors proved the following interior regularity result:
\begin{cor}(Theorem 1.4 \cite{Biagi2})
	Let $\Omega\subset\mathbb{R}^N$ be a bounded $C^1$ domain. Let $f\in H^k(\Omega)\big(W^{2,k}(\Omega)\big)$ for some integer $k\geq 0.$ Let $u\in H^1(\mathbb{R}^N)$ be a weak solution of
	\begin{align}
		-\Delta u+(-\Delta)^s u=f \text{ in }\Omega.
	\end{align}
Then $u\in H^{k+2}_{\text{loc}}(\Omega)$.
\end{cor}
Motivated by the above results, we establish the following interior H\"older regularity result.

\begin{thm}\label{Holder}\textbf{(Interior regularity)}
	Let $u$ be a bounded function in $\mathbbm{H}^N$ and viscosity solution to
	\begin{align}
		\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}u\big)-\b\big(-\Delta_{\mathbb{H}^N}\big)^su=0 \text{ in }B_{1}^{\mathbbm{H}^N}.
	\end{align}
Then there exist constants $C$ and $\c=\c(\lambda,\Lambda,N)\in (0,1)$ such that
\begin{align}
	\big|u(\xi)-u(0)\big|\leq C|\xi|_{\mathbbm{H}^N}^\c\|u\|_{\infty,\mathbbm{H}^N},\,\forall\xi\in B_{\frac{1}{2}}^{\mathbbm{H}^N},
\end{align}
where $B_{r}^{\mathbbm{H}^N}$ denotes the ball of radius $r$ with center at the origin.
\end{thm}

The organization of this paper is as follows. In Section 2, we list the basic definitions and introduce the framework in which we work. Section 3 is dedicated to the proofs of our main results.
\section{Preliminaries}
We first recall the briefs about the Heisenberg group $\mathbbm{H}^N.$ The points in $\mathbbm{H}^N$ are denoted by
\begin{align}
	\xi:=(z,t)=(x_1,\dots,x_N,y_1,\dots,y_N,t)
\end{align}
and the group $\mathbbm{H}^N$ is defined as the triplet $\big(\mathbb{R}^{N+1},o,\{\Phi_\lambda\}\big)$, where the group law $o$ is defined as follows:
\begin{align}
	\xi\, o\,\xi'&=\big(x+x',y+y',t+t'+2\langle y,x'\rangle- 2\langle x,y'\rangle\big)\\
	&=\bigg(x_1+x_1',\dots,x_N+x_N',y_1+y_1',\dots,y_N+y_N',t+t'+2\sum_{i=1}^{N}\big(y_ix_i'-x_iy_i'\big)\bigg),
\end{align}
where $\langle.,\,.\rangle$ denotes the standard inner product in $\mathbb{R}^N$.
$(\mathbb{R}^{2N+1},o)$ is a Lie group with identity element the origin $\textbf{0}$ and inverse $\xi^{-1}=-\xi$. The dilation group $\{\Phi_\lambda\}_{\lambda>0}$ is given by
\begin{align}
	\Phi(\lambda): \mathbb{R}^{2N+1}\longrightarrow\mathbb{R}^{2N+1}
\end{align}
such that
\begin{align}
	\xi\mapsto\Phi_\lambda(\xi):=\big(\lambda x,\lambda y,\lambda^2 t\big).
\end{align}
$\mathbbm{H}^N$ is also known as Heisenberg-Weyl group in $\mathbb{R}^{2N+1}$. The Jacobian basis of the Heisenberg Lie algebra of $\mathbbm{H}^N$ is given by
\begin{align}
	X_i=\partial_{x_i}+2y_{i}\partial_t, Y_{i}=\partial _{y_i}-2x_i\partial_t, \, 1\leq i\leq N,\, T=\partial_t.
\end{align}
Given a domain $\Omega\subset \mathbbm{H}^N,$ for $u\in C^1(\Omega,\mathbb
{R}),$ the subgradient or the Heisenberg gradient $\nabla_{\mathbbm{H}^N} u$ is defined as follows:
\begin{align}
	\nabla_{\mathbbm{H}^N} u(\xi):=\big(X_1u(\xi),\dots,X_{N}u(\xi),Y_1u(\xi),\dots,Y_Nu(\xi)\big).
\end{align}
Also,
\begin{align}
	D^2_{\mathbbm{H}^N,S}u:=\begin{bmatrix}
		X_1X_1u &\cdots &X_NX_1u &Y_1X_1u &\cdots &Y_NX_1u\\
		\vdots   &\ddots &\vdots &\vdots &\ddots &\vdots \\
		X_1X_Nu &\dots &X_NX_Nu &Y_1X_Nu &\dots &Y_NX_Nu\\
		X_1Y_1u &\dots &X_NY_1u &Y_1Y_1u &\dots &Y_NY_1u\\
		\vdots   &\ddots &\vdots &\vdots &\ddots &\vdots \\
		X_1Y_Nu &\dots &X_NY_Nu &Y_1Y_Nu &\dots &Y_NY_Nu
	\end{bmatrix}_{Sym},
\end{align}
where
\begin{align}
	A_{\text{Sym}}=\frac{1}{2}\big[A+A^T\big],	\text{ for any matrix }A,
\end{align}
i.e., symmetric part of the matrix $A$.
Now, since
\begin{align}
	[X_i,Y_{i}]&=X_iY_{i}-Y_{i}X_i\\
	&=(\partial_{x_i}+2y_{i}\partial_t)(\partial_{y_{i}}-2x_i\partial_t)-(\partial_{y_{i}}-2x_i\partial_t)(\partial_{x_i}+2y_{i}\partial_t)\\
	&=-4\partial_t,
\end{align}
so it follows that
\begin{align}
	\text{rank}\big(\text{Lie}\{X_1,X_2,\dots,X_{2N},T\}(0,0)\big)=2N+1,
\end{align}
which is the Euclidean dimension of $\mathbbm{H}^N$. We denote by $Q$, the \textit{homogeneous dimension} of $\mathbbm{H}^N$, which is $Q=2N+2.$ The norm on $\mathbbm{H}^N$ is defined by
\begin{align}
	|\xi|_{\mathbbm{H}^N}:=\bigg[\bigg(\sum_{i=1}^N \big(x_i^2+y_i^2\big)^2\bigg)+t^2\bigg]^{\frac{1}{4}}.
\end{align}
The corresponding distance on $\mathbbm{H}^N$ is defined as follows:
\begin{align}
	d_{\mathbbm{H}^N}(\xi,\hat{\xi}):=|\hat{\xi}^{-1}o\,\xi|_{\mathbbm{H}^N},
\end{align}
where $\hat{\xi}^{-1}$ is the inverse of $\hat{\xi}$ w. r. to $o$, i.e., $\hat{\xi}^{-1}=-\hat{\xi}$.

The sub-Laplacian or the Heisenberg Laplacian (also known as Laplacian-Kohn operator), $\Delta_{\mathbbm{H}^N}$ is the self-adjoint operator defined as
\begin{align}
	\Delta_{\mathbbm{H}^N}u&:=\sum_{i=1}^NX_i^2+Y_i^2\\
	&=\sum_{i=1}^N\frac{\partial^2}{\partial x_i^2}+\frac{\partial^2}{\partial y_i^2}+4y_i\frac{\partial^2}{\partial x_i\partial t}-4x_i\frac{\partial^2}{\partial y_i\partial t}+4\big(x_i^2+y_i^2\big)\frac{\partial^2}{\partial t^2}.
\end{align}
It is useful to observe that
\begin{align}
	\Delta_{\mathbbm{H}^N}=\text{div}\big(\sigma^T\sigma\nabla u\big),
\end{align}
where
\begin{align}
	\sigma=
	\begin{bmatrix}
		I_N &0 &2y\\
		0 &I_N &-2x
	\end{bmatrix}
\end{align}
and $\sigma^T$ is its transpose. Note that
\begin{align}
	A=\sigma^T\sigma=\begin{bmatrix}
		I_N &0 &2y\\
		0 &I_N &-2x\\
		2y &-2x &4\big(|x|^2+|y|^2\big)
	\end{bmatrix}
\end{align}
is a positive semi-definite matrix with det$(A)=0,\,\, \forall \xi\in \mathbbm{H}^N$.

Let us recall the definition of \textit{viscosity sub/super-solution} of (\ref{eq 2.1}) by evaluating the operators in $C^2$ test function $\phi$ $(\psi)$ touching $u$ locally from above (below) and then the final test function $v$ $(w)$ is defined by taking $v=\phi$ $(v=\psi)$ in a small ball and $v=u$ outside. One may see \cite{Barles, Silvestre}, where the notion of viscosity solution for second order elliptic integro-differential equations and fully nonlinear integro-differential equations is given in the Euclidean setting, respectively. For the analogous definitions in the Heisenberg group setting, we refer to \cite{Cutri}. We give the definition of viscosity solution for the operator under consideration in the Heisenberg group setting which is consistent with that given in the above mentioned articles.
\begin{defn}
	A function $u:\mathbbm{H}^N\longrightarrow\mathbb{R},$ upper semicontinuous (USC) in $\overline{\Omega}\subset \mathbbm{H}^N$ is called a \textit{viscosity subsolution} of (\ref{eq 2.1}) if for any $\xi\in\Omega$ and $C^2$ function $\varphi:\overline{U}\longrightarrow\mathbb{R},$ for some neighborhood $U$ of $\xi$ in $\Omega$ such that $\varphi(\xi)=u(\xi)$ and $\varphi(\eta)>u(\eta)$ for $\eta\in U\setminus \{\xi\},$ we have
	\begin{align}\label{Sub}
		\a\mathcal{M}^+_{\lambda,\Lambda}\big(D_{\mathbbm{H}^N,S}^2v(\xi)\big)-\b(-\Delta_{\mathbbm{H}^N})^sv(\xi) \geq f(\xi),
	\end{align}
	where
	\begin{align}
		v:=
		\begin{cases}
			\varphi &\text{ in }U,\\
			u &\text{ in } \mathbbm{H}^N\setminus U.
		\end{cases}
	\end{align}
	Moreover, we say $u$ satisfies $\a\mathcal{M}^+_{\lambda,\Lambda}(D_{\mathbbm{H}^N,S}^2u)-\b(-\Delta_{\mathbbm{H}^N})^su \geq f$ in $\Omega$ in the viscosity sense.
\end{defn}

\begin{defn}
	A function $u:\mathbbm{H}^N\longrightarrow\mathbb{R},$ lower semicontinuous (LSC) in $\overline{\Omega}\subset\mathbbm{H}^{N}$ is called a \textit{viscosity supersolution} of (\ref{eq 2.1}) if for any $\xi\in\Omega$ and $C^2$ function $\psi:\overline{U}\longrightarrow\mathbb{R},$ for some neighborhood $U$ of $\xi$ in $\Omega$ such that $\psi(\xi)=u(\xi)$ and $\psi(\eta)<u(\eta)$ for $\eta\in U \setminus \{\xi\},$ we have
	\begin{align}
		\a\mathcal{M}^+_{\lambda,\Lambda}\big(D_{\mathbbm{H}^N,S}^2w(\xi)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw(\xi)\leq f(\xi),
	\end{align}
  where
	\begin{align}
		w:=
		\begin{cases}
			\psi &\text{ in }U,\\
			u &\text{ in } \mathbbm{H}^N\setminus U.
		\end{cases}
	\end{align}
 Moreover, we say $u$ satisfies $\a\mathcal{M}^+_{\lambda,\Lambda}(D_{\mathbbm{H}^N,S}^2u)-\b(-\Delta_{\mathbbm{H}^N})^su \leq f$ in $\Omega$ in the viscosity sense.
\end{defn}
\begin{defn}\label{Vis sol}
	A continuous function $u$ is said to be a \textit{viscosity solution} of (\ref{eq 2.1}) if it is a subsolution as well as a supersolution of (\ref{eq 2.1}).
\end{defn}

Now, we recall the \textit{exterior Heisenberg ball condition}. For the analogous condition in the Euclidean setting, one may see, for instance \cite{Mou1}.
\begin{defn}\cite{Cutri}\label{exterior}
	$\Omega$ is said to satisfy the exterior Heisenberg ball condition if there exists $R>0$ such that for any $\xi\in\partial\Omega$ and $0<r\leq R,$ there exists $\eta_\xi^r\in\Omega^c$ satisfying $\overline{B_r^{\mathbbm{H}^N}(\eta_\xi^r)}\cap\overline{\Omega}=\{\xi\}.$
\end{defn}
Next, we recall the definition of sup (inf)-convolution. The construction of convolutions was done by Jensen et al. \cite{JLS} and further developed on Carnot groups by C. Y. Wang \cite{Wang}.
\begin{defn}
	For an USC function $u,$ the sup-convolution approximation $u^\varepsilon$ is given by
	\begin{align}
		u^\varepsilon(\xi)=\displaystyle{\sup_{\eta\in \mathbbm{H}^N}}\bigg(u(\eta)-\frac{|\xi\,o\,\eta^{-1}|^4_{\mathbbm{H}^N}}{\mathlarger\varepsilon}\bigg).
	\end{align}
\end{defn}
\begin{defn}
	For an LSC function $u$, the inf-convolution approximation $u_\varepsilon$ is given by
	\begin{align}
		u_\varepsilon(\xi)=\displaystyle{\inf_{\eta\in \mathbbm{H}^N}}\bigg(u(\eta)+\frac{|\xi\,o\,\eta^{-1}|^4_{\mathbbm{H}^N}}{\mathlarger\varepsilon}\bigg).
	\end{align}
\end{defn}
It is easy to see that $u^\varepsilon\geq u$ and $u_\varepsilon\leq u.$

\begin{defn}
	A sequence of LSC functions, $u_k$ is said to $\Gamma$-converge to $u$ in a set $\Omega$ if the following hold:\\
	(i) For every sequence $\xi_n\longrightarrow \xi$ in $\Omega,$ we have
	\begin{align}
		\displaystyle{\liminf_{n\longrightarrow \infty}u_n(\xi_n)}\geq u(\xi).
	\end{align}
	(ii) For every $\xi\in\Omega,$ there exists a sequence $\{\xi_n\}$ converging to $\xi$ in $\Omega$ (known as $\Gamma$-realising sequence) such that
	\begin{align}
		\displaystyle{\limsup_{n\longrightarrow\infty}u_n(\xi_n)}=u(\xi).
	\end{align}
\end{defn}
This is known as $\Gamma$-limit in literature, see \cite{Silvestre, Progress}. Note that uniform convergence $\implies$ $\Gamma$-convergence. Also, an important property of $\Gamma$-limits is that if $u_n$ converges to $u,$ which has a strict local minimum at $\xi,$ then $u_n$ would have local minimum at $\xi_n$ for a sequence $\xi_n\longrightarrow \xi.$

\begin{defn}\label{homo}\cite{Palatucci}
	A homogeneous norm on $\mathbbm{H}^N$ is a continuous function (w. r. to Euclidean topology)
	\begin{align}
		d_0:\mathbbm{H}^N\longrightarrow [0,\infty) \text{ s.t. }
	\end{align}
(i) $d_0(\Phi_\lambda(\xi))=\lambda d_0(\xi),\,\forall\lambda>0\, \text{ and } \,\xi\in \mathbbm{H}^N$\\
(ii) $d_0(\xi)=0 \text{ iff }\xi=0.$
\end{defn}
Moreover, we say that the homogeneous norm is symmetric if
\begin{align}
	d_0(\xi^{-1})=d_0(\xi), \,\forall\xi\in \mathbbm{H}^N.
\end{align}
Throughout the paper, we consider the standard homogeneous norm on $\mathbbm{H}^N.$ For fixed $\xi_0\in \mathbbm{H}^N\text{ and }\, R>0,$
\begin{align}
	B_R^{\mathbbm{H}^N}(\xi_0):=\bigg\{\xi\in \mathbbm{H}^N\,;\,|\xi_0^{-1}\,o\,\xi|_{\mathbbm{H}^N}<R\bigg\}
\end{align}
denotes the \textit{Kor\'anyi} ball of radius $R$ around $\xi_0$ in $\mathbbm{H}^N.$

One can see that the Jacobian determinant of the dilation $\Phi_\lambda$ is $\lambda^Q$, where $Q=2N+2$, which is the homogeneous dimension of the Heisenberg group. Let us consider $2N\times(2N+1)$ matrix whose rows are the coefficients of the vector fields $X_i$, i.e.,
\begin{align}
	\sigma=\begin{bmatrix}
		I_N &0 &2y\\
		0 &I_N &-2x
	\end{bmatrix}
\end{align}
for $x=(\xi_1,\dots,\xi_N)^T$ and $y=(\xi_{N+1},\dots,\xi_{2N})^T$. Then the Heisenberg gradient of a function $\Phi:\mathbb{R}^{2N+1} \longrightarrow \mathbb{R}$ is given by
\begin{align}
	\nabla_{\mathbbm{H}^N}\Phi=(X_1\Phi,\dots,X_{2N}\Phi)=\sigma(\xi)\nabla\Phi,
\end{align}
where $\nabla$ denotes the usual gradient. Also, the Heisenberg Hessian of $\Phi$ is given by
\begin{align}\label{sigma T}
	D^2_{\mathbbm{H}_N,S}\Phi&=(X_iX_j\Phi)_{Sym}\\
	&=\sigma(\xi)D^2\Phi\sigma^T(\xi),
\end{align}
where \textit{Sym} denotes the symmetrized matrix.
We recall that the fractional sub-Laplacian operator is defined as
\begin{align}
	(-\Delta_{\mathbbm{H}^N})^su(\xi)=-\frac{1}{2}c(N,s)\int_{\mathbbm{H}^N}\frac{u(\xi\,o\,\eta)+u(\xi\,o\,\eta^{-1})-2u(\xi)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta,\,\,u\in H^{s}(\mathbbm{H}^N),\,\,\,\xi\in \mathbbm{H}^N; Q=2N+2,
\end{align}
see 5.1 \cite{Palatucci}. Here, $c(N,s)$ is a positive constant depending on $N$\,\&\,$s.$ For given two parameters $0<\lambda\leq\Lambda,$ Pucci-Heisenberg operators are defined by the composition of Pucci's extremal operators $\mathcal{M}_{\lambda,\Lambda}^{\pm}$ (see 2.2 \cite{Cabre}) with the Heisenberg Hessian
 as follows:
\begin{align}\label{M+}
	\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}u\big):=\Lambda\displaystyle{\sum_{e_i\geq 0}e_i}+\lambda\displaystyle{\sum_{e_i< 0}e_i}
\end{align}
\&
\begin{align}
	\mathcal{M}_{\lambda,\Lambda}^-\big(D^2_{\mathbbm{H}^N,S}u\big):=\Lambda\displaystyle{\sum_{e_i\leq 0}e_i}+\lambda\displaystyle{\sum_{e_i> 0}e_i},
\end{align}
where $\{e_i\}_{i=1}^{2N}$ are the eigenvalues of the symmetrized horizontal Hessian matrix $D^2_{\mathbbm{H}^N,S}u.$ Let $\mathcal{S}^{2N}$ be denote the set of all real symmetric $2N\times 2N$ matrices. Consider a subset $\mathcal{S}_{\lambda,\Lambda}^{2N}$ of $\mathcal{S}^{2N}$ whose eigenvalues are in $[\lambda,\Lambda].$ These operators are also defined as
\begin{align}\label{max}
\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}u\big):=\displaystyle{\max_{M\in \mathcal{S}_{\lambda,\Lambda}^N}}tr\big(MD^2_{\mathbbm{H}^N,S}u\big)
\end{align}
\&
\begin{align}\label{min}
\mathcal{M}_{\lambda,\Lambda}^-\big(D^2_{\mathbbm{H}^N,S}u\big):=\displaystyle{\min_{M\in \mathcal{S}_{\lambda,\Lambda}^N}}tr\big(MD^2_{\mathbbm{H}^N,S}u\big).
\end{align}
Furthermore, consider a set $K$ consisting of $2N\times 2N$ matrices $\gamma$ such that $\gamma\gamma^T\in \mathcal{S}^{2N}_{\lambda,\Lambda}.$ Using this, we can re-write \eqref{max} as
\begin{align}
	\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}u\big)=\displaystyle{\max_{\gamma\in K}}\,\langle \gamma\gamma^T, D^2_{\mathbbm{H}^N,S}u\rangle.
\end{align}
Now, for any fixed $\gamma\in K,$ we have the following linear operator, say, $L_\gamma$ given by
\begin{align}\label{L beta}
	L_\gamma\big(D^2_{\mathbbm{H}^N,S}u\big):=\langle \gamma\gamma^T, D^2_{\mathbbm{H}^N,S}u\rangle.
\end{align}
It is easy to observe that when $\lambda=\Lambda=1,$ the above mentioned operators reduce to the Heisenberg Laplacian $\Delta_{\mathbbm{H}^N,S}.$

\section{Proofs of main results}
\noindent \textbf{Proof of Theorem \ref{stability}.} Consider a $C^2$ function $\psi$ that touches $u$ from below at $\xi$ in a neighbourhood $U$ in $\Omega.$ By definition of $\Gamma$-convergence, there exists a sequence $\xi_n\longrightarrow \xi$ such that
\begin{align}
\big(u_n-\psi\big)(\xi_n)=\displaystyle{\inf_U}\big(u_n-\psi\big)=\delta_n.
\end{align}
Therefore, $\delta_n\longrightarrow 0$ as $n\longrightarrow\infty$ and $\psi+\delta_n$ touches $u_n$ at $\xi_n.$ Now, since
	\begin{align}
		\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}u_n\big)-\b\big(-\Delta_{\mathbb{H}^N}\big)^su_n \leq f_n \text{ in }\Omega,
	\end{align}
so for
\begin{align}
	\xi_n=\begin{cases}
		\psi+\delta_n &\text{ in }U,\\
		u_n&\text{ in }\mathbbm{H}^N\setminus U,
	\end{cases}
\end{align}
we have by the definition of viscosity solution,
\begin{align}
	\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}u_n(\xi_n)\big)-\b\big(-\Delta_{\mathbb{H}^N}\big)^su_n(\xi_n) \leq f_n(\xi_n) \text{ in }\Omega.
\end{align}
Further, take $\tau\in U$ such that
\begin{align}
	d\big(\tau,\partial U\big)=\displaystyle{\inf_{\widetilde{\tau}\in \partial U}{{\big|\widetilde{\tau}}^{-1}\,o\,\tau\big|_{\mathbbm{H}^N}}}>\rho>0.
\end{align}
Then
\begin{align}
	\big|&\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w_n(\tau)\big)-\b\big(-\Delta_{\mathbb{H}^N}\big)^sw_n(\tau)\big)-\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w(\tau)\big)-\b\big(-\Delta_{\mathbb{H}^N}\big)^sw(\tau)\big)\big|\\
	&\qquad\leq \big|\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w_n(\tau)\big)-\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w(\tau)\big)\big)-\b\big((-\Delta_{\mathbb{H}^N})^sw_n(\tau)-(-\Delta_{\mathbb{H}^N})^sw(\tau)\big)\big|\\
	&\qquad\leq \a \,\displaystyle{\max\bigg(\big|\max_{M\in \mathcal{S}_{\lambda,\Lambda}^N}}tr\big(MD^2_{\mathbbm{H}^N,S}(w_n(\tau)-w(\tau))\big)\big|,\big|\max_{M\in \mathcal{S}_{\lambda,\Lambda}^N}tr\big(MD^2_{\mathbbm{H}^N,S}(w(\tau)-w_n(\tau))\big)\big|\bigg)\\
	&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\b\big|(-\Delta_{\mathbb{H}^N})^sw_n(\tau)-(-\Delta_{\mathbb{H}^N})^sw(\tau)\big|\text{ (by \eqref{L beta})}\\
	&\qquad\leq \a \max_{\gamma\in K}\big|L_\gamma\big(D^2(w_n-w)(\tau)\big)\big|+\b\big|\big(-\Delta_{\mathbb{H}^N}\big)^s(w_n-w)(\tau)\big|\\
	&\qquad= \a \max_{\gamma\in K}\big|L_\gamma\big(D^2\delta_n(\tau)\big)\big|+\b\big|\big(-\Delta_{\mathbb{H}^N}\big)^s(w_n-w)(\tau)\big|\\
    &\qquad\leq \frac{\b}{2}c(N,s)\int_{\mathbbm{H}^N}\frac{\big|{(w_n-w)}(\tau\,o\,\eta)+{(w_n-w)}(\tau\,o\,\eta^{-1})-2{(w_n-w)}(\tau)\big|}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
    &\qquad=\frac{\b}{2}c(N,s)\int_{B_\rho^{\mathbbm{H}^N}}\frac{\big|{(w_n-w)}(\tau\,o\,\eta)+{(w_n-w)}(\tau\,o\,\eta^{-1})-2{(w_n-w)}(\tau)\big|}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
    &\qquad\qquad\qquad+\frac{\b}{2}c(N,s)\int_{\mathbbm{H}^N\setminus B_\rho^{\mathbbm{H}^N}}\frac{\big|{(w_n-w)}(\tau\,o\,\eta)+{(w_n-w)}(\tau\,o\,\eta^{-1})-2{(w_n-w)}(\tau)\big|}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
    &\qquad=\frac{\b}{2}c(N,s)\int_{B_\rho^{\mathbbm{H}^N}}\frac{\big|{\delta_n}(\tau\,o\,\eta)+{\delta_n}(\tau\,o\,\eta^{-1})-2{\delta_n}(\tau)\big|}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
    &\qquad\qquad\qquad+\frac{\b}{2}c(N,s)\int_{\mathbbm{H}^N\setminus B_\rho^{\mathbbm{H}^N}}\frac{\big|{(w_n-w)}(\tau\,o\,\eta)+{(w_n-w)}(\tau\,o\,\eta^{-1})-2{(w_n-w)}(\tau)\big|}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
    &\qquad=\frac{\b}{2}c(N,s)\int_{\mathbbm{H}^N\setminus B_\rho^{\mathbbm{H}^N}}\frac{\big|{(w_n-w)}(\tau\,o\,\eta)+{(w_n-w)}(\tau\,o\,\eta^{-1})-2{(w_n-w)}(\tau)\big|}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta.
\end{align}
Now, using the fact that sequence $w_n$ is bounded and that $(w_n-w)(\tau\,o\,\eta)+(w_n-w)(\tau\,o\,\eta^{-1})-(w_n-w)(\tau)\longrightarrow 0$ a.e. along with the fact that $\frac{1}{|\eta|^{Q+2s}}\in L^1(\mathbbm{H}^N\setminus B_\rho),$ we have by the dominated convergence theorem that
\begin{align}\label{eq 3.2}
\big|\big(\a\mathcal{M}^+_{\lambda,\lambda}\big(D^2_{\mathbbm{H}^N,S}(w_n)(\tau)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw_n(\tau)\big)-\big(\a\mathcal{M}^+_{\lambda,\lambda}\big(D^2_{\mathbbm{H}^N,S}(w)(\tau)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw(\tau)\big)\big|\longrightarrow 0 \text{ as }n\longrightarrow \infty \text{ uniformly in }\tau.
\end{align}
Equivalently,
\begin{align}
	\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}(w_n)(\tau)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw_n(\tau)\longrightarrow\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}(w)(\tau)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw(\tau) \text{ locally uniformly in } U.
\end{align}
Also,
\begin{align}\label{eq 3.3}
	&\big|\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}(w_n)(\xi_n)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw_n(\xi_n)\big)-\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}(w)(\xi)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw(\xi)\big)\big|\\
	&\qquad\leq \big|\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w_n(\xi_n)\big)- \b(-\Delta_{\mathbbm{H}^N})^sw_n(\xi_n)\big)-\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w(\xi_n)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw(\xi_n)\big)\big|\\
	&\qquad\qquad+\big|\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}\psi(\xi_n)\big)- \b(-\Delta_{\mathbbm{H}^N})^sw(\xi_n)\big)-\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}\psi(\xi)\big)- \b(-\Delta_{\mathbbm{H}^N})^sw(\xi)\big)\big|\\
	&\qquad\leq \big|\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w_n(\xi_n)\big)- \b(-\Delta_{\mathbbm{H}^N})^sw_n(\xi_n)\big)-\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w(\xi_n)\big)- \b(-\Delta_{\mathbbm{H}^N})^sw(\xi_n)\big)\big|\\
	&\qquad\qquad+\big|\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}\psi(\xi_n)\big)-\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}\psi(\xi)\big)\big|+\big|\b(-\Delta_{\mathbbm{H}^N})^sw(\xi_n)-\b(-\Delta_{\mathbbm{H}^N})^sw(\xi)\big|.
\end{align}
Further, the first term in the R.H.S. of \eqref{eq 3.3} vanishes as $n\longrightarrow\infty$ by \eqref{eq 3.2}. The third term goes to zero as $n\longrightarrow\infty$ by the continuity of $Iw$ in $U.$
Also, the first term vanishes by an application of Theorem VI.2.1 \cite{Bhatia}. Thus, we have that
\begin{align}
\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w_n(\xi_n)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw_n(\xi_n)\longrightarrow\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}(w)(\xi)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw(\xi) \text{ as }n\longrightarrow\infty.
\end{align}
Further, we get
\begin{align}
	\a\mathcal{M}^+_{\lambda,\Lambda}&\big(D^2_{\mathbbm{H}^N,S}w_n(\xi_n)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw_n(\xi)\\
	&\leq \a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w_n(\xi_n)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw_n(\xi)\\
	&\qquad\qquad+\big|\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w_n(\xi_n)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw_n(\xi_n)-\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w(\xi)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw(\xi)\big)\big|\\
	&\leq f_n(\xi_n)+\big|\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w_n(\xi_n)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw_n(\xi_n)-\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w(\xi)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw(\xi)\big)\big|\\
	&\leq f(\xi)+|f_n(\xi_n)-f(\xi)|\\
	&\qquad\qquad+\big|\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w_n(\xi_n)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw_n(\xi_n)-\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}w(\xi)\big)-\b(-\Delta_{\mathbbm{H}^N})^sw(\xi)\big)\big|.
\end{align}
Finally, using \eqref{eq 3.3} together with the fact that $f_n(\xi_n)\longrightarrow f(\xi),$ since $f_n\longrightarrow f$ locally uniformly and $\xi_n\longrightarrow \xi$ proves the claim.\qed

\begin{rem}
	One can see that an analogous result can be obtained for subsolutions using the $\Gamma$-convergence of USC functions.
\end{rem}

More precisely, we have the following immediate corollary:
\begin{cor}
	Let $u_n\in C(\Omega)$ be bounded in $\mathbbm{H}^N$ and satisfy
	\begin{align}
		\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}u_n\big)-\b\big(-\Delta_{\mathbb{H}^N}\big)^su_n=f_n \text{ in }\Omega
	\end{align}
	in the viscosity sense. Let the following be hold:\\
	(i) $u_n$ converges to $u$ in the $\Gamma$-sense in $\Omega,$\\
	(ii) $u_n$ converges to $u$ a.e. in $\mathcal{H}^N,$\\
	(iii) $f_n\longrightarrow f$ locally uniformly in $\Omega$.

	Then
	\begin{align}
		\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbbm{H}^N,S}u\big)-\b\big(-\Delta_{\mathbb{H}^N}\big)^su=f \text{ in }\Omega.
	\end{align}
\end{cor}

\begin{lem}\label{convo}
	Let  $f\in C(\Omega)$ and $u\in \text{USC}(\overline{\Omega})$ be a viscosity subsolution of \eqref{eq 2.1}. Then
	\begin{align}
		\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}u^\varepsilon\big)-\big(-\Delta_{\mathbb{H}^N}\big)^su^\varepsilon\geq f-d_\mathlarger\varepsilon \text{ in } \Omega_1\Subset\Omega,
	\end{align}
	where $d_\mathlarger\varepsilon\longrightarrow 0$ in $\Omega_1$ as $\mathlarger{\mathlarger\varepsilon}\longrightarrow 0$ and depends on the modulus of continuity of $f$.
\end{lem}
\noindent \textbf{Proof of Lemma \ref{convo}.}
Let $x_0\in \Omega_1$ and $\varphi$ be a $C^2$ smooth function touching $u^\varepsilon$ from above in some neighbourhood $B^{\mathbbm{H}^N}_r(\xi_0)\subset \Omega_1$ of $\xi_0$. Let us define
\begin{align}
	\Psi:=\begin{cases}
		\varphi &\text{ in }B^{\mathbbm{H}^N}_r(\xi_0)\\
		u^\varepsilon &\text{ in }\mathbbm{H}^N\setminus B_r^{\mathbbm{H}^N}(\xi_0).
	\end{cases}
\end{align}
Next, consider a function
\begin{align}\label{Phi}
\Phi(\xi)=\Psi\big(\xi\,o\,{\xi_0^*}^{-1}\,o\,\xi_0\big)+\frac{1}{\mathlarger{\mathlarger\varepsilon}}|\xi_0\,o\,{\xi_0^*}^{-1}|^4_{\mathbbm{H}^N},
\end{align}
where $\xi^*=(x^{1*},x^{2*},\dots,x^{2N*},t^*)\in\Omega$ such that
\begin{align}
	u^\varepsilon(\xi_0)=u(\xi_0^*)-\frac{|\xi_0\,o\,{\xi_0^*}^{-1}|^4_{\mathbbm{H}^N}}{\mathlarger{\mathlarger\varepsilon}}\text{ (see p. 9 \cite{Li})}.
\end{align}
Now, by definition, we have
\begin{align}\label{def1}
	u^\varepsilon\big(\xi_0\,o\,{\xi_0^*}^{-1}\,o\,\xi\big)&\geq u(\xi)-|\big(\xi_0\,o\,{\xi_0^*}^{-1}\,o\,\xi\big)\,o\,\xi^{-1}|_{\mathbbm{H}^N}.
\end{align}
Since
\begin{align}
\big(\xi_0\,o\,{\xi_0^*}^{-1}\,o\,\xi\big)\,o\,\xi^{-1}=\xi_0\,o\,{\xi_0^*}^{-1},
\end{align}
so it implies
\begin{align}
   u^\varepsilon\big(\xi_0\,o\,{\xi_0^*}^{-1}\,o\,\xi\big)&\geq u(\xi)-|\xi_0\,o\,{\xi_0^*}^{-1}|^4_{\mathbbm{H}^N}.
\end{align}
In other words,
\begin{align}
	u(\xi)\leq u^\varepsilon\big(\xi_0\,o\,{\xi_0^*}^{-1}\,o\,\xi\big)+\frac{|\xi_0\,o\,{\xi_0^*}^{-1}|^4_{\mathbbm{H}^N}}{\varepsilon}.
\end{align}
Let $\xi$ be close enough to $\xi_0^*,$ then by definition,
\begin{align}
	u(\xi)&\leq \Psi\big(\xi_0\,o\,{\xi_0^*}^{-1}\,o\,\xi\big)+\frac{|\xi_0\,o\,{\xi_0^*}^{-1}|^4_{\mathbbm{H}^N}}{\varepsilon}\\
	&=\Phi(\xi)
\end{align}
and $u(\xi_0^*)=\Phi(\xi_0^*).$ It implies that $\Phi(\xi)$ touches $u$ from above at $\xi_0^*$ in $B_r^{\mathbbm{H}^N}(\xi_0^*)$. Next, we consider a function
\begin{align}
	v:=\begin{cases}
		\Phi &\text{ in }B_r^{\mathbbm{H}^N}(\xi_0^*)\\
		u &\text{ in }\mathbbm{H}^N\setminus B_r^{\mathbbm{H}^N}(\xi_0^*).
	\end{cases}
\end{align}
It yields by the definition of viscosity subsolution,
\begin{align}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}v(\xi_0^*)\big)-\b(-\Delta_{\mathbbm{H}^N})^sv(\xi_0^*)\geq f(\xi_0^*).
\end{align}
In other words,
\begin{align}\label{eq 3.6}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{H^N,s}v(\xi_0^*)\big)+\frac{\b}{2}c(N,s)\int_{\mathbbm{H}^N}\frac{v(\xi\,o\,\eta)+v(\xi\,o\,\eta^{-1})-2v(\xi)}{|\eta^{-1}\,o\,\xi|^{Q+2s}_{\mathbbm{H}^N}}d\eta\geq f(\xi_0^*).
\end{align}
It is easy to see that \eqref{Phi} yields
\begin{align}\label{eq 3.7}
	v(\xi^*_0)=\Phi(\xi_0^*)=\Psi(\xi_0)+\frac{1}{\mathlarger{\mathlarger\varepsilon}}|\xi_0\,o\,{\xi_0^*}^{-1}|^4_{\mathbbm{H}^N},
\end{align}
\begin{align}
	\Phi(\xi_0^*\,o\,\eta)=\Psi(\xi_0\,o\, \eta)+\frac{1}{\mathlarger{\mathlarger\varepsilon}}|\xi_0\,o\,{\xi_0^*}^{-1}|^4_{\mathbbm{H}^N}
\end{align}
\&
\begin{align}
	\Phi(\xi_0^*\,o\,\eta^{-1})=\Psi(\xi_0\,o\,\eta^{-1})+\frac{1}{\mathlarger{\mathlarger\varepsilon}}|\xi_0\,o\,{\xi_0^*}^{-1}|^4_{\mathbbm{H}^N}.
\end{align}
It gives
\begin{align}\label{eq 3.8}
	\Phi(\xi_0^*\,o\,\eta)-\Phi(\xi_0^*)=\Psi(\xi_0\,o\, \eta)-\Psi(\xi_0)
\end{align}
\&
\begin{align}
	\Phi(\xi_0^*\,o\,\eta^{-1})-\Phi(\xi_0^*)=\Psi(\xi_0\,o\,\eta^{-1})-\Psi(\xi_0).
\end{align}
Now, we claim that
\begin{align}\label{nabla H}
	\nabla_{\mathbbm{H}^N}\Phi(\xi_0^*)&=\nabla_{\mathbbm{H}^N}\Psi(\xi_0).
\end{align}
Since
\begin{align}
	X_i\Phi(\xi)&=\partial_{x_i}\Phi(\xi)+2y_{i}\partial_t\Phi(\xi)\\
	&=\partial_{x_i}\Phi\big(x^1,\dots,x^{N},y^1,\dots,y^N,t\big)+2y_{i}\partial_t\Phi\big(x^1,\dots,x^{N},y^1,\dots,y^N,t\big)\\
	&=\partial_{x_i}\Psi\big(\xi\,o\,{\xi_0^*}^{-1}\,o\,\xi_0\big)+2y_{i}{\partial_t}\Psi\big(\xi\,o\,{\xi_0^*}^{-1}\,o\,\xi_0\big) \text{ for }1\leq i\leq N,
\end{align}
so it gives
\begin{align}
	X_i\Phi(\xi_0^{*})=\partial_{x_i}\Psi(\xi_0)+2y_{i}{\partial_t}\Psi(\xi_0)\text{ for }1\leq i\leq N.
\end{align}
Similarly, one may see that
\begin{align}
	Y_i\Phi(\xi_0^*)=\partial_{y_i}\Psi(\xi_0)-2x_i\partial_t\Psi(\xi_0),\,1\leq i\leq N.
\end{align}
Therefore, \eqref{nabla H} holds. Also,
\begin{align}
D^2_{H^N,S}\Phi(\xi_0^*)=D^2_{\mathbbm{H}^N,S}\Psi(\xi_0)
\end{align}
and therefore
\begin{align}\label{eq 3.9}
	\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\Phi(\xi_0^*)\big)=\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\Psi(\xi_0)\big)=\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi(\xi_0)\big).
\end{align}
Using \eqref{eq 3.8} \& \eqref{eq 3.9} in \eqref{eq 3.6} yields
\begin{align}\label{eq 3.10}
	\qquad\qquad f(\xi_0^*)&\leq \a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}v(\xi_0^*)\big)+\frac{\b}{2}c(N,s)\int_{\mathbbm{H}^N}\frac{v(\xi_0^*\,o\,\eta)+v(\xi_0^*\,o\,\eta^{-1})-2v(\xi_0^*)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&=\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi(\xi_0)\big)+\frac{\b}{2}c(N,s)\int_{\{\eta\in \mathbbm{H}^N;\,\xi_0^*\,o\,\eta,\,\xi_0^*\,o\,\eta^{-1}\in B_r^{\mathbbm{H}^N}(\xi_0^*)\}}\frac{\Phi(\xi_0^*\,o\,\eta)+\Phi(\xi_0^*\,o\,\eta^{-1})-2\Phi(\xi_0^*)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad\qquad\qquad\qquad\qquad\qquad +\frac{\b}{2}c(N,s)\int_{\mathbbm{H}^N\setminus\{\eta\in \mathbbm{H}^N;\,\xi_0^*\,o\,\eta,\,\xi_0^*\,o\,\eta^{-1}\in B_r^{\mathbbm{H}^N}(\xi_0^*)\}}\frac{u(\xi_0^*\,o\,\eta)+u(\xi_0^*\,o\,\eta^{-1})-2u(\xi_0^*)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&=\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi(\xi_0)\big)+\frac{\b}{2}c(N,s)\int_{\{\eta\in \mathbbm{H}^N;\,\xi_0^*\,o\,\eta,\,\xi_0^*\,o\,\eta^{-1}\in B_r^{\mathbbm{H}^N}(\xi_0^*)\}}\frac{\Phi(\xi_0^*\,o\,\eta)+\Phi(\xi_0^*\,o\,\eta^{-1})-2\Phi(\xi_0^*)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad\qquad\qquad\qquad\qquad\qquad +\frac{\b}{2}c(N,s)\int_{\mathbbm{H}^N\setminus\{\eta\in \mathbbm{H}^N;\,\xi_0^*\,o\,\eta,\,\xi_0^*\,o\,\eta^{-1}\in B_r^{\mathbbm{H}^N}(\xi_0^*)\}}\frac{u(\xi_0^*\,o\,\eta)+u(\xi_0^*\,o\,\eta^{-1})-2\Phi(\xi_0^*)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta.
\end{align}
By the definition of $u^\varepsilon$,
\begin{align}
	u(\eta)\leq u^\varepsilon(\xi)+\frac{\big|\xi\,o\,{\eta}^{-1}\big|^4_{\mathbbm{H}^N}}{\mathlarger{\mathlarger\varepsilon}} \text{ for all }\xi,\,\eta\in \mathbbm{H}^N.
\end{align}
It further implies
\begin{align}\label{use compo}
	u(\xi_0^*\,o\,\eta)\leq u(\xi_0\,o\,\eta)+\frac{\big|{(\xi_0\,o\,\eta)\,o\,(\xi_0^*\,o\,\eta)}^{-1}\big|^4_{\mathbbm{H}^N}}{\mathlarger{\mathlarger\varepsilon}}
\end{align}
and
\begin{align}\label{use compo 2}
	u(\xi_0^*\,o\,\eta^{-1})\leq u(\xi_0\,o\,\eta^{-1})+\frac{\big|(\xi_0\,o\,\eta^{-1})\,o\,{(\xi_0^*\,o\,\eta^{-1})}^{-1}\big|^4_{\mathbbm{H}^N}}{\mathlarger{\mathlarger\varepsilon}}.
\end{align}
Let
\begin{align}
\xi_0^*&=(x_0^*,y_0^*,t_0^*)=\big(x_0^{1*},\dots,x_0^{N*},y_0^{1*},\dots,y_0^{N*},t_0^*\big),\\
\xi_0&=(x_0,y_0,z_0,t)=\big(x^1_0,\dots,x^N_0,y^1_0,\dots,y^N_0,t_0\big),\\
\eta&=(x,y,z,t)=\big(x_1,\dots,x_N,y_1,\dots,y_N,t\big).
\end{align}
It implies that
\begin{align}
	\xi_0\,o\,\eta&=\big(x_0^1+x^1,\dots,x_0^N+x^N,y_0^1+y^1,\dots,y_0^N+y^N,t_0+t+2\langle y_0,x\rangle-2\langle x_0,y\rangle\big),\\
	\xi_0^*\,o\,\eta&=\big(x_0^{1*}+x^{1},\dots,x_0^{N*}+x^{N},y_0^{1*}+y^{1},\dots,y^{N*}_0+y^{N},t_0^*+t+2\langle y_0^*,x\rangle-2\langle x_0^*,y\rangle\big).
\end{align}
It immediately gives
\begin{align}
	{(\xi_0^*\,o\,\eta)}^{-1}=\big(-x_0^{1*}-x^{1},\dots,-x_0^{N*}-x^{N},-y_0^{1*}-y^{1},\dots,-y_0^{N*}-y^{N},-t_0^*-t-2\langle y_0^*,x\rangle+2\langle x_0^*,y\rangle\big).
\end{align}
It yields
\begin{align}\label{compo}
	\qquad\qquad(\xi_0\,o\,\eta)\,o\,{(\xi_0^*\,o\,\eta)}^{-1}&=\big(x_0^1-x_0^{1*},\dots,x_0^N-x_0^{N*},y_0^1-y_0^{1*},\dots,y_0^N-y_0^{N*},t_0-t_0^*\\
	&\qquad+2\langle y,x_0^*\rangle-2\langle x,y_0^*\rangle-2\langle y,x_0\rangle+2\langle x,y_0\rangle+2\langle y_0+y,-x_0^*-x\rangle-2\langle x_0+x,-y_0^*-y\rangle\big)\\
	&=\big(x_0^1-x_0^{1*},\dots,x_0^N-x_0^{N*},y_0^1-y_0^{1*},\dots,y_0^N-y_0^{N*},t_0-t_0^*\\
	&\qquad+2\langle y,x_0^*\rangle-2\langle x,y_0^*\rangle-2\langle y,x_0\rangle+2\langle x,y_0\rangle+2\langle y_0,-x_0^*-x\rangle\\
	&\qquad+2\langle y,-x_0^*-x\rangle-2\langle x_0,-y_0^*-y\rangle-2\langle x,-y_0^*-y\rangle\big)\\
	&=\big(x_0^1-x_0^{1*},\dots,x_0^N-x_0^{N*}+,y_0^1-y_0^{1*},\dots,y_0^N-y_0^{N*},t_0-t_0^*\\
	&\qquad+2\langle y,x_0^*\rangle-2\langle x,y_0^*\rangle-2\langle y,x_0\rangle+2\langle x,y_0\rangle+2\langle y_0,-x_0^*\rangle
	+2\langle y_0,-x\rangle+2\langle y,-x_0^*\rangle
	+2\langle y,-x\rangle\\
	&\qquad-2\langle x_0,-y_0^*\rangle-2\langle x_0,-y\rangle-2\langle x,-y_0^*\rangle-2\langle x,-y\rangle\big)\\
	&=\big(x_0^1-x_0^{1*},\dots,x_0^N-x_0^{N*}+,y_0^1-y_0^{1*},\dots,y_0^N-y_0^{N*},t_0-t_0^*\\
	&\qquad+2\langle y,x_0^*\rangle-2\langle x,y_0^*\rangle-2\langle y,x_0\rangle+2\langle x,y_0\rangle+2\langle x_0,y_0^*\rangle+2\langle x_0,y\rangle+2\langle x,y_0^*\rangle+2\langle x,y\rangle\\
	&\qquad-2\langle y_0,x_0^*\rangle
	-2\langle y_0,x\rangle-2\langle y,x_0^*\rangle
	-2\langle y,x\rangle\big)\\
	&=\big(x_0-x_0^*,y_0-y_0^*,t_0-t_0^*+2\langle x_0,y_0^*\rangle-\langle y_0,x_0^* \rangle\big)\\
	&=\big(x_0-x_0^*,y_0-y_0^*,t_0-t_0^*+2\langle y_0,-x_0^*\rangle-2\langle x_0,-y_0^* \rangle\big)\\
	&=\xi_0\,o\,{\xi_0^*}^{-1}.
\end{align}
Similarly, one may see that
\begin{align}\label{compo 2}
	(\xi_0\,o\,\eta^{-1})\,o\,{(\xi_0^*\,o\,\eta^{-1})}^{-1}=\xi_0\,o\,{\xi_0^*}^{-1}.
\end{align}
Using \eqref{compo} \& \eqref{compo 2} in \eqref{use compo} \& \eqref{use compo 2}, respectively infers
\begin{align}\label{eq 3.11}
	\qquad\quad u(\xi_0^*\,o\,\eta)+u(\xi_0^*\,o\,\eta^{-1})-2\Phi(\xi_0^*)&\leq u^\varepsilon(\xi_0\,o\,\eta)+u^\varepsilon(\xi_0\,o\,\eta^{-1})+\frac{2\big|\xi_0\,o\,{\xi_0^*}^{-1}\big|^4_{\mathbbm{H}^N}}{\mathlarger{\mathlarger\varepsilon}}-2\Phi(\xi_0^*)\\
	&=u^\varepsilon(\xi_0\,o\,\eta)+u^\varepsilon(\xi_0\,o\,\eta^{-1})+\frac{2\big|\xi_0\,o\,{\xi_0^*}^{-1}\big|^4_{\mathbbm{H}^N}}{\mathlarger{\mathlarger\varepsilon}}-2\bigg(\Psi(\xi_0)+\frac{1}{\mathlarger{\mathlarger\varepsilon}}\big|\xi_0\,o\,{\xi_0^*}^{-1}\big|^4_{\mathbbm{H}^N}\bigg)\text{ (by \eqref{eq 3.7})}\\
	&=u^\varepsilon(\xi_0\,o\,\eta)+u^\varepsilon(\xi_0\,o\,\eta^{-1})-2\Psi(\xi_0).
\end{align}
We also have
\begin{align}\label{eq 3.12}
	\big|\xi\,o\,{\xi_0^*}^{-1}\big|^4_{\mathbbm{H}^N}\leq \mathlarger\varepsilon\,\underset{\Omega}{\text{osc}}~u,
\end{align}
see \cite{Li, Wang} for the details. It immediately grants that $|\xi\,o\,{\xi_0^*}^{-1}|_{\mathbbm{H}^N}$ can be made as small as possible by the suitable choice of $\mathlarger{\mathlarger\varepsilon}$. Using \eqref{eq 3.11} in \eqref{eq 3.10}, we get
\begin{align}\label{eq 3.13}
\qquad\qquad f(\xi_0^*)&\leq \a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi(\xi_0)\big)+\frac{\beta}{2}c(N,s)\int_{\{\eta\in \mathbbm{H}^N;\,\xi_0^*\,o\,\eta,\,\xi_0^*\,o\,\eta^{-1}\in B_r^{\mathbbm{H}^N}(\xi_0^*)\}}\frac{\Phi(\xi_0^*\,o\,\eta)+\Phi(\xi_0^*\,o\,\eta^{-1})-2\Phi(\xi_0^*)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\frac{\beta}{2}c(N,s)\int_{\mathbbm{H}^N\setminus\{\eta\in \mathbbm{H}^N;\,\xi_0^*\,o\,\eta,\,\xi_0^*\,o\,\eta^{-1}\in B_r^{\mathbbm{H}^N}(\xi_0^*)\}}\frac{u(\xi_0^*\,o\,\eta)+u(\xi_0^*\,o\,\eta^{-1})-2\Phi(\xi_0^*)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
&\leq \a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi(\xi_0)\big)+c(N,s)\int_{\{\eta\in \mathbbm{H}^N;\,\xi_0^*\,o\,\eta,\,\xi_0^*\,o\,\eta^{-1}\in B_r^{\mathbbm{H}^N}(\xi_0^*)\}}\frac{\varphi(\xi_0\,o\,\eta)+\varphi(\xi_0\,o\,\eta^{-1})-2\varphi(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\frac{\beta}{2}c(N,s)\int_{\mathbbm{H}^N\setminus\{\eta\in \mathbbm{H}^N;\,\xi_0^*\,o\,\eta,\,\xi_0^*\,o\,\eta^{-1}\in B_r^{\mathbbm{H}^N}(\xi_0^*)\}}\frac{u^\varepsilon(\xi_0\,o\,\eta)+u^\varepsilon(\xi_0\,o\,\eta^{-1})-2\Psi(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
&=\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi(\xi_0)\big)-\b\big(-\Delta_{\mathbbm{H}^N}\big)^s\Psi(\xi_0).
\end{align}
Now, adding and subtracting $f(\xi_0)$ in the L.H.S. of \eqref{eq 3.13} gives
\begin{align}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi(\xi_0)\big)-\b\big(-\Delta_{\mathbbm{H}^N}\big)^s\Psi(\xi_0)&\geq f(\xi_0)-\big(f(\xi_0)-f(\xi_0^*)\big)\\
	&\geq f(\xi_0)-|f(\xi_0)-f(\xi_0^*)|.
\end{align}
Next, using \eqref{eq 3.12} together with the continuity of $f$, we define
\begin{align}
	d_\varepsilon=\displaystyle{\sup_{B^{\mathbbm{H}^N}_{\delta\sqrt{\varepsilon}}(\xi_0)}}|f(\xi_0)-f(\xi_0^*)|,
\end{align}
for some $\delta=\delta(\underset{\Omega}{\text{osc}}\,u)$. Clearly, $d_\varepsilon\longrightarrow 0$ as $\mathlarger\varepsilon\longrightarrow 0.$ Hence the claim.\qed

\begin{rem}
	Using the similar arguments, one may see that an analogous result also holds for supersolutions.
\end{rem}
\begin{lem}\label{Lemma 1.6}
	Let $f$ $\&$ $g$ be two continuous functions. Let $u$ $\&$ $v$ be bounded USC and $LSC$ functions in $\mathbb{H}^N$, respectively. Let
	\begin{align}
		\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbb{H}^N,S}u\big)-\b(-\Delta_{\mathbb{H}^N})^su\geq f
	\end{align}
	$\&$
	\begin{align}
		\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbb{H}^N,S}v\big)-\b(-\Delta_{\mathbb{H}^N})^sv\leq g
	\end{align}
	be hold in the viscosity sense in $\Omega.$ Then
	\begin{align}
		\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbb{H}^N,S}(u-v)\big)-\b(-\Delta_{\mathbb{H}^N})^s(u-v)\geq f-g \text{ in }\Omega \text{ in the viscosity sense.}
	\end{align}
\end{lem}

\noindent \textbf{Proof of Lemma \ref{Lemma 1.6}.}
By Lemma \ref{convo}, we have
\begin{align}\label{1}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}u^\varepsilon\big)-\b(-\Delta_{\mathbbm{H}^N})^su^\varepsilon\geq f-d_\mathlarger\varepsilon
\end{align}
$\&$
\begin{align}\label{2}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}v_\varepsilon\big)-\b(-\Delta_{\mathbbm{H}^N})^sv_\varepsilon\leq g+d_\mathlarger\varepsilon.
\end{align}
Our aim is to show that
\begin{align}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}(u^\varepsilon-v_\varepsilon)\big)-\b(-\Delta_{\mathbbm{H}^N})^s(u^\varepsilon-v_\varepsilon)\geq f-g-2d_\mathlarger\varepsilon,
\end{align}
so that further using the stability result (Theorem \ref{stability}) yields the claim.

Let $P\in C_b^2(\mathbbm{H}^N)$ be a paraboloid in $\overline{B^{\mathbbm{H}^N}_r}\subset \Omega_1$ such that
\begin{align}
	P(\xi_0)=u^\varepsilon(\xi_0)-v_\varepsilon(\xi_0)
\end{align}
and
\begin{align}
	P(\tau)\geq u^\varepsilon(\tau)-v_\varepsilon(\tau) \text{ for all }\tau \in B^{\mathbbm{H}^N}_r(\xi_0).
\end{align}
We assume that $\overline{B^{\mathbbm{H}^N}_{2r}}(\xi_0)\subset\Omega.$ Consider
\begin{align}
\Phi(x)=\begin{cases}
	P &\text{ in }B_r^{\mathbbm{H}^N}(\xi_0)\\
	u^\varepsilon-v_\varepsilon &\text{ in }\mathbbm{H}^N\setminus B_r^{\mathbbm{H}^N}(\xi_0).
\end{cases}
\end{align}
Take $\delta>0$ and let us define
\begin{align}
	w(\xi)=v_\varepsilon(\xi)-u^\varepsilon(\xi)+\Phi(\xi)+\delta\left(\big|{\xi_0}^{-1}o\,\xi\,\big|_{\mathbbm{H}^N}\land r\right)^4-\delta r_1^4,
\end{align}
for $0<r_1<\delta\land\frac{r}{2}.$ We observe that $w\geq 0$ on $\partial B_{r_1}^{\mathbbm{H}^N}(\xi_0)$ and $w(\xi_0)=-\delta r^2<0$. By Theorem 5.1 \cite{Cabre}, we have that for any $\xi\in \overline{B_{r_1}^{\mathbbm{H}^N}}(\xi_0),$ there exists a convex paraboloid $P^\xi$ of opening $K$ (some constant independent of $\xi$) which touches $w$ from above at $\xi \in B_{r_1}^{\mathbbm{H}^N}(\xi_0)$. Further, by Lemma 3.5 \cite{Cabre} and $w(\xi_0)<0,$ we have that
\begin{align}\label{eq det}
	0<\int_{B_r^{\mathbbm{H}^N}(\xi_0)\cap\{w=\Gamma_w\}}\det D^2\Gamma_w,
\end{align}
where $\Gamma_w$ is the convex envelope of $w$ in $A$ given by
\begin{align}
	\Gamma_w(\xi)=\displaystyle{\sup_v}\bigg\{v(\xi)\,;\,v\leq w \text{ in }A, v \text{ convex in }A\bigg\},
\end{align}
for $\xi\in A$. Moreover, $u^\varepsilon$ and $v_\varepsilon$ are punctually second order differentiable in $A$ (see 2.2 (ii) \cite{Li}). It gives that $\mathcal{M}_{\lambda,\Lambda}^+(D^2_{\mathbbm{H}^N,S}u^\varepsilon)-(-\Delta_{\mathbbm{H}^N})^su^\varepsilon$ and $\mathcal{M}_{\lambda,\Lambda}^+(D^2_{\mathbbm{H}^N,S}v_\varepsilon)-(-\Delta_{\mathbbm{H}^N})^sv_\varepsilon$ are defined in the classical sense for $\xi\in A.$

By the convexity of $\Gamma_w$ and that $\Gamma_w\leq w,$ we have that the Hessian matrix $D^2w(\xi)$ is semi-positive definite for $\xi\in A\cap\{w=\Gamma_w\}.$ It yields that
\begin{align}\label{eq 3.18}
\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}w(\xi)\big)\geq 0
\end{align}
\&
\begin{align}\label{eq 3.20}
	\\ \quad\frac{1}{2}c(N,s)\int_{\{\eta\in \mathbbm{H}^N;\,\xi\,o\,\eta\in B_r^{\mathbbm{H}^N}(\xi_0)\}}&\frac{w(\xi\,o\,\eta)+w(\xi\,o\,\eta^{-1})-2w(\xi)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad=\frac{1}{2}c(N,s)\int_{\{\eta\in \mathbbm{H}^N;\,\xi\,o\,\eta\in B_r^{\mathbbm{H}^N}(\xi_0)\}}\frac{w(\xi\,o\,\eta)-w(\xi)-\mathbbm{1}_{\{|\eta|_{\mathbbm{H}^N}\leq 1\}}\eta.(\nabla_{\mathbbm{H}^N}w(\xi),\,\partial_tw(\xi))}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad\geq 0,
\end{align}
for $\xi\in A\cap\{w=\Gamma_w\}$ using Proposition 2.4 \cite{Palatucci} and
\begin{align}
	w(\xi\,o\,\eta)-w(\xi)-\mathbbm{1}_{\{|\eta|_{\mathbbm{H}^N}\leq 1\}}\eta.(\nabla_{\mathbbm{H}^N}w(\xi),\,\partial_tw(\xi))&\geq \Gamma_w(\xi\,o\,\eta)-\Gamma_w(\xi)-\mathbbm{1}_{\{|\eta|_{\mathbbm{H}^N}\leq 1\}}\eta.(\nabla_{\mathbbm{H}^N}\Gamma_w(\xi),\,\partial_t\Gamma_w(\xi))\\
	&\geq 0
\end{align}
along with $\nabla_w(\xi)=\nabla\Gamma_w(\xi)$ for $\xi
\in \{w=\Gamma_w\}.$
It is clear from \eqref{eq det} and $\big|B_r^{{\mathbbm{H}^N}}(\xi_0)\setminus A\big|=0$ that
\begin{align}
	\big|\{w=\Gamma_w\}\cap A\big|>0,
\end{align}
i.e., there is a point $\xi_\delta\in\{w=\Gamma_w\}\cap A$ such that \eqref{1} and \eqref{2} hold classically. Let $I_N$ be denote the identity matrix of order $N$. We have
\begin{align}\label{11}
	\\
	f(\xi_\delta)-d_\varepsilon&\leq 	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}u^\varepsilon(\xi_\delta)\big)-\b(-\Delta_{\mathbbm{H}^N})^su^\varepsilon(\xi_\delta)\\
	&\leq \a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}v_\varepsilon(\xi_\delta)-D^2_{\mathbbm{H}^N,S}w(\xi_\delta)+D^2_{\mathbbm{H}^N,S}\Phi(\xi_\delta)+D^2_{\mathbbm{H}^N,S}\big(\delta\big|{\xi_0}^{-1}o\,\xi\,\big|^4_{\mathbbm{H}^N}(\xi_\delta)\big)\big)-\b(-\Delta_{\mathbbm{H}^N})^su^\varepsilon(\xi_\delta)\\
	&=\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}v_\varepsilon(\xi_\delta)+D^2_{\mathbbm{H}^N,S}\Phi(\xi_\delta)+\delta D^2_{\mathbbm{H}^N,S}\big(\big|{\xi_0}^{-1}o\,\xi\,\big|^4_{\mathbbm{H}^N}(\xi_\delta)\big)\big)-D^2_{\mathbbm{H}^N,S}w(\xi_\delta)\big)-\b(-\Delta_{\mathbbm{H}^N})^su^\varepsilon(\xi_\delta)\\
	&\leq \a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}v_\varepsilon(\xi_\delta)+D^2_{\mathbbm{H}^N,S}\Phi(\xi_\delta)+\delta D^2_{\mathbbm{H}^N,S}\big(\big|{\xi_0}^{-1}o\,\xi\,\big|^4_{\mathbbm{H}^N}(\xi_\delta)\big)\big)+\a\mathcal{M}_{\lambda,\Lambda}^+\big(-D^2_{\mathbbm{H}^N,S}w(\xi_\delta)\big)-\b(-\Delta_{\mathbbm{H}^N})^su^\varepsilon(\xi_\delta)\\
	&=\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}v_\varepsilon(\xi_\delta)+D^2_{\mathbbm{H}^N,S}\Phi(\xi_\delta)+\delta D^2_{\mathbbm{H}^N,S}\big(\big|{\xi_0}^{-1}o\,\xi\,\big|^4_{\mathbbm{H}^N}(\xi_\delta)\big)\big)-\a\mathcal{M}_{\lambda,\Lambda}^-\big(D^2_{\mathbbm{H}^N,S}w(\xi_\delta)\big)-\b(-\Delta_{\mathbbm{H}^N})^su^\varepsilon(\xi_\delta)\\
	&\leq \a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}v_\varepsilon(\xi_\delta)+D^2_{\mathbbm{H}^N,S}\Phi(\xi_\delta)+\delta D^2_{\mathbbm{H}^N,S}\big(\big|{\xi_0}^{-1}o\,\xi\,\big|^4_{\mathbbm{H}^N}(\xi_\delta)\big)\big)-\b(-\Delta_{\mathbbm{H}^N})^su^\varepsilon(\xi_\delta)\text{ (by \eqref{eq 3.18})},
\end{align}
where in the second last step, we used the relation
\begin{align}
	\mathcal{M}_{\lambda,\Lambda}^+(-M)=-\mathcal{M}_{\lambda,\Lambda}^-(M).
\end{align}
Now, since $\big|{\xi_0}^{-1}o\,\xi\,\big|^4_{\mathbbm{H}^N}$ is a radial function so using Lemma 3.2 \cite{Cutri} together with sub-additivity property of Pucci's maximal operator and \eqref{11} yields
\begin{align}
	f(\xi_\delta)-d_\varepsilon&\leq \a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}v_\varepsilon(\xi_\delta)\big)+\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\Phi(\xi_\delta)\big)+\a\delta \mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\big(\big|{\xi_0}^{-1}o\,\xi_\delta\,\big|^4_{\mathbbm{H}^N}\big)\big)-\b(-\Delta_{\mathbbm{H}^N})^su^\varepsilon(\xi_\delta)\\
	&=\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}v_\varepsilon(\xi_\delta)\big)+\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\Phi(\xi_\delta)\big)\\
	&\qquad\qquad+\a\Lambda\delta\left((2N-2)\bigg(4\sum_{i=1}^N(x^i_\delta-x^i_0)^2+(y^i_\delta-y^i_0)^2\bigg)+2\bigg(12\sum_{i=1}^N(x_\delta^i-x^i_0)^2+(y_\delta^i-y^i_0)^2\bigg)\right)-\b(-\Delta_{\mathbbm{H}^N})^su^\varepsilon(\xi_\delta),
\end{align}
where $(x_\delta^1,\dots,x_\delta^N,y_\delta^1,\dots,y_\delta^N)$ and $(x_0^1,\dots,x_0^N,y_0^1,\dots,y_0^N)$ are the first $2N$ coordinates of $\xi_\delta$ and $\xi_0$, respectively. Next, since $w(\xi\,o\,\eta)-w(\xi)>0$ for $\xi\,o\,\eta\in B_r^c(\xi)$ and $r_1$ small enough so we have by \eqref{eq 3.20},
\begin{align}\label{eq 3.21}
	-(-\Delta_{\mathbbm{H}^N})^su^\varepsilon(\xi_\delta)&=-(-\Delta_{\mathbbm{H}^N})^sv_\varepsilon(\xi_\delta)+(-\Delta_{\mathbbm{H}^N})^sw(\xi_\delta)-(-\Delta_{\mathbbm{H}^N})^s\Phi(\xi_\delta)-\delta(-\Delta_{\mathbbm{H}^N})^s(|\xi_0^{-1}\,o\,\xi|^4)(\xi_\delta)\\
	&\leq -(-\Delta_{\mathbbm{H}^N})^sv_\varepsilon(\xi_\delta)-\int_{\{\eta\in \mathbbm{H}^N;\,\xi_\delta\,o\,\eta\in \mathbbm{H}^N\setminus B_r^{\mathbbm{H}^N}(\xi_\delta)\}}\frac{\mathbbm{1}_{\{|\eta|_{\mathbbm{H}^N}\leq 1\}}\eta.(\nabla_{\mathbbm{H}^N}w(\xi_\delta),\partial_tw(\xi_\delta))}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad-(-\Delta_{\mathbbm{H}^N})^s\Phi(\xi_\delta)-\delta(-\Delta_{\mathbbm{H}^N})^s(|\xi_0^{-1}\,o\,\xi|_{\mathbbm{H}^N}^4)(\xi_\delta)\\
	&=-(-\Delta_{\mathbbm{H}^N})^sv_\varepsilon(\xi_\delta)-\int_{\{\eta\in \mathbbm{H}^N;\,r\leq |\eta|_{\mathbbm{H}^N}\leq 1\}}\frac{\eta.(\nabla_{\mathbbm{H}^N}w(\xi_\delta),\partial_tw(\xi_\delta))}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad-(-\Delta_{\mathbbm{H}^N})^s\Phi(\xi_\delta)-\delta(-\Delta_{\mathbbm{H}^N})^s\big(|\xi_0^{-1}\,o\,\xi|_{\mathbbm{H}^N}^4\big)(\xi_\delta).
\end{align}
Further, using \eqref{eq 3.21} together with \eqref{eq 3.20} yields
\begin{align}
	f(\xi_\delta)-d_\varepsilon&\leq \a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}v_\varepsilon(\xi_\delta)\big)-\b(-\Delta_{\mathbbm{H}^N})^sv_\varepsilon(\xi_\delta)+\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\Phi(\xi_\delta)\big)-\b(-\Delta_{\mathbbm{H}^N})^s\Phi(\xi_\delta)\\
	&\qquad +\a\Lambda\delta\left((2N-2)\bigg(4\sum_{i=1}^N(x_\delta^i-x^i_0)^2+(y_\delta^i-y^i_0)^2\bigg)+2\bigg(12\sum_{i=1}^N(x_\delta^i-x^i_0)^2+(y_\delta^i-y^i_0)^2\bigg)\right)\\
	&\qquad -\frac{\b}{2}c(N,s)\int_{\{\eta\in \mathbbm{H}^N;\,r\leq |\eta|_{\mathbbm{H}^N}\leq 1\}}\frac{\eta.(\nabla_{\mathbbm{H}^N}w(\xi_\delta),\partial_tw(\xi_\delta))}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta-\b\delta(-\Delta_{\mathbbm{H}^N})^s(|\xi_0^{-1}\,o\,\xi|_{\mathbbm{H}^N}^4)(\xi_\delta)\\
	&\leq g(\xi_\delta)+d_\varepsilon+\a\Lambda\delta\left((2N-2)\bigg(4\sum_{i=1}^N(x_\delta^i-x^i_0)^2+(y_\delta^i-y^i_0)^2\bigg)+2\bigg(12\sum_{i=1}^N(x^i_\delta-x^i_0)^2+(y^i_\delta-y^i_0)^2\bigg)\right)\\
	&\qquad-\frac{\b}{2}c(N,s)\int_{\{\eta\in \mathbbm{H}^N;\,r\leq |\eta|_{\mathbbm{H}^N}\leq 1\}}\frac{\eta.(\nabla_{\mathbbm{H}^N}w(\xi_\delta),\partial_tw(\xi_\delta))}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta-\delta(-\Delta_{\mathbbm{H}^N})^s(|\xi_0^{-1}\,o\,\xi|_{\mathbbm{H}^N}^4)(\xi_\delta)\\
	&\qquad+\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\Phi(\xi_\delta)\big)-\b(-\Delta_{\mathbbm{H}^N})^s\Phi(\xi_\delta) \text{ (using \eqref{2})}.
\end{align}
It is easy to observe that $\xi_\delta\longrightarrow \xi_0$ and $\nabla\Gamma_w(\xi_\delta)\longrightarrow\nabla\Gamma_w(\xi_0)=0$ as $r_1\longrightarrow 0$. Now, letting $\delta\longrightarrow 0$ gives
\begin{align}
	f(\xi_0)-d_\varepsilon\leq g(\xi_0)+d_\varepsilon+\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\Phi(\xi_0)\big)-\b(-\Delta_{\mathbbm{H}^N})^s\Phi(\xi_0).
\end{align}
In other words,
\begin{align}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\Phi(\xi_0)\big)-\b(-\Delta_{\mathbbm{H}^N})^s\Phi(\xi_0)\geq f(\xi_0)-g(\xi_0)-2d_\varepsilon.
\end{align}
The above equation clearly implies that
\begin{align}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}(u^\varepsilon-v_\varepsilon)\big)-\b(-\Delta_{\mathbbm{H}^N})^s\Phi(u^\varepsilon-v_\varepsilon)\geq f-g-2d_\varepsilon \text{ in }\Omega_1
\end{align}
in the viscosity sense. Finally, letting $\varepsilon\longrightarrow 0$ along with using Theorem \ref{stability} yields the claim.\qed\\

In order to derive comparison principle, we next state and prove the following lemma. We mention that a similar lemma has been proven in the Euclidean setting (see Lemma 5.5 \cite{Mou}).

\begin{lem}\label{Mou}
	There exists a $C^2(\overline{\Omega})\cap C_b(\mathbbm{H}^N)$ function $\varphi_h$ such that
\begin{align}
		\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi_h\big)-\b\big(-\Delta_{\mathbbm{H}^N}\big)^s\varphi_h\leq -1, \text{ in }\Omega.
\end{align}
\end{lem}

\noindent \textbf{Proof of Lemma \ref{Mou}.} Let diam$(\Omega)=R.$ We may assume without of loss of generality that $\Omega\subset B^{R}_{\mathbbm{H}^N}(\xi^R),$ where $\xi^R:=(2R,0,\dots,0).$ Let us define
\begin{align}
	\varphi_h(\xi)=\begin{cases}
		2-e^{-C\xi_1} &\text{ for }\xi_1\geq 0\\
		\frac{1}{2}+\frac{1}{4}\left(\frac{1}{1-C\xi_1}\right)+\frac{1}{4}\big(\text{sin } 3C\xi_1+ \text{cos }\sqrt{6}\,C\xi_1\big) &\text{ for } \xi_1<0,
	\end{cases}
\end{align}
where $\xi_1$ is the first coordinate of $\xi=(\xi_1,\dots,\xi_N,\xi_{N+1},\dots,\xi_{2N},t)\in \Omega$ and $C>0$ is some constant.
It is easy to calculate for $\xi_1>0,$
\begin{align}
	\partial_{x_i}\varphi_h=\begin{cases}
		Ce^{-C\xi_1} &\text{ if }i=1,\\
		0 &\text{ if }2\leq i\leq 2N,
	\end{cases}
\end{align}
and $\partial_t \varphi_h=0.$ Also, for $\xi_1>0$,
\begin{align}
	\partial^2_{x_ix_j}\varphi_h=\begin{cases}
		-C^2e^{-C\xi_1} &\text{ if }i=j=1\\
		0 &\text{ otherwise, }
	\end{cases}
\end{align}
and $\partial^2_{tx_i}=0=\partial^2_{tt}$ for $i=1,2,\dots,2N.$
Using this, we first compute $\nabla_{\mathbbm{H}^N}\varphi_h(\xi)$ and $D^2_{\mathbbm{H}^N,S}\varphi_h(\xi)$ as follows:
\begin{align}
	\nabla_{\mathbbm{H}^N}\varphi_h(\xi)=\sigma(\xi)\nabla\varphi_h(\xi)=
	\begin{bmatrix}
		I_N &0_N &2y\\
		0_N &I_N &-2x
	\end{bmatrix}_{2N\times (2N+1)}
	\begin{bmatrix}
		Ce^{-C\xi_1}\\
		0\\
		\vdots\\
		0
	\end{bmatrix}_{(2N+1)\times 1}=
	\begin{bmatrix}
		Ce^{-C\xi_1}\\
		0\\
		\vdots\\
		0
	\end{bmatrix}_{2N\times 1}.
\end{align}
Also,
\begin{align}
	D^2_{\mathbbm{H}^N}\varphi_h(\xi)=\sigma(\xi)D^2\varphi_h\sigma^T(\xi)&=\begin{bmatrix}
		I_N &0_N &2y\\
		0_N &I_N &-2x
	\end{bmatrix}_{2N\times(2N+1)}
	\begin{bmatrix}
		-C^2e^{-C\xi_1} &0 &\dots &0\\
		0 &0 &\dots &0\\
		\vdots &\vdots &\ddots &\vdots \\
		0 &0 &\dots &0
	\end{bmatrix}_{(2N+1)\times(2N+1)}
	\begin{bmatrix}
		I_N &0_N\\
		0_N &I_N\\
		2y &-2x
	\end{bmatrix}_{(2N+1)\times 2N}\\
	&=\begin{bmatrix}
		-C^2e^{-C\xi_1} &0 &\dots &0\\
		0 &0 &\dots &0\\
		\vdots &\vdots &\ddots &\vdots \\
		0 &0 &\dots &0
	\end{bmatrix}_{2N\times 2N}.
\end{align}
Now, let $\delta=\displaystyle{\min\{1,R\}}.$ Then for any $\xi\in\Omega,$ we get
\begin{align}\label{111}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi_h(\xi)\big)&-\b(-\Delta_{\mathbbm{H}^N})^s\varphi_h(\xi)\\
	&=-\lambda\a C^2e^{-C\xi_1}+\frac{\b}{2}c(N,s)\int_{\mathbbm{H}^N}\frac{\varphi_h(\xi\,o\,\eta)-\varphi_h(\xi)-\mathbbm{1}_{\{|\eta|_{\mathbbm{H}^N}\leq 1\}}\eta.(\nabla_{\mathbbm{H}^N}\varphi_h(\xi),\partial_t\varphi_h(\xi))}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&=-\lambda \a C^2e^{-C\xi_1}+\frac{\b}{2}c(N,s)\int_{B_\delta^{\mathbbm{H}^N}}\frac{\varphi_h(\xi\,o\,\eta)-\varphi_h(\xi)-\mathbbm{1}_{\{|\eta|_{\mathbbm{H}^N}\leq 1\}}\eta.(\nabla_{\mathbbm{H}^N}\varphi_h(\xi),\partial_t\varphi_h(\xi))}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\quad+\frac{\b}{2}c(N,s)\int_{\mathcal{C}B_\delta^{\mathbbm{H}^N}\cap\{\eta;\eta_1\leq 0\}}\frac{\varphi_h(\xi\,o\,\eta)-\varphi_h(\xi)-\mathbbm{1}_{\{|\eta|_{\mathbbm{H}^N}\leq 1\}}\eta.(\nabla_{\mathbbm{H}^N}\varphi_h(\xi),\partial_t\varphi_h(\xi))}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\quad+\frac{\b}{2}c(N,s)\int_{B^{\mathbbm{H}^N}_1\cap \,\mathcal{C}B_\delta^{\mathbbm{H}^N}\cap\,\{\eta;\eta_1>0\}}\frac{\varphi_h(\xi\,o\,\eta)-\varphi_h(\xi)-\mathbbm{1}_{\{|\eta|_{\mathbbm{H}^N}\leq 1\}}\eta.(\nabla_{\mathbbm{H}^N}\varphi_h(\xi),\partial_t\varphi_h(\xi))}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\quad+\frac{\b}{2}c(N,s)\int_{\mathcal{C}B^{\mathbbm{H}^N}_1\cap\,\{\eta;\eta_1>0\}}\frac{\varphi_h(\xi\,o\,\eta)-\varphi_h(\xi)-\mathbbm{1}_{\{|\eta|_{\mathbbm{H}^N}\leq 1\}}\eta.(\nabla_{\mathbbm{H}^N}\varphi_h(\xi),\partial_t\varphi_h(\xi))}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta.
\end{align}
where $\mathcal{C}B_r^{\mathbbm{H}^N}$ denotes the complement of $B_r^{\mathbbm{H}^N}$ in $\mathbbm{H}^N,$ i.e., $\mathbbm{H}^N\setminus B_r^{\mathbbm{H}^N}.$ We can re-write the above equation as
\begin{align}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi_h(\xi)\big)&-\b\big(-\Delta_{\mathbbm{H}^N}\big)^s\varphi_h(\xi)\\
	&=-\lambda \a C^2e^{-C\xi_1}+\frac{\b}{2}c(N,s)\int_{B_\delta^{\mathbbm{H}^N}}\frac{\varphi_h(\xi\,o\,\eta)-\varphi_h(\xi)-\eta.(\nabla_{\mathbbm{H}^N}\varphi_h(\xi),\partial_t\varphi_h(\xi))}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\qquad+\frac{\b}{2}c(N,s)\int_{\mathcal{C}B_\delta^{\mathbbm{H}^N}\cap\{\eta;\eta_1\leq 0\}}\frac{\varphi_h(\xi\,o\,\eta)-\varphi_h(\xi)}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta+\frac{\b}{2}c(N,s)\int_{B_1^{\mathbbm{H}^N}\cap \,\mathcal{C}B_\delta^{\mathbbm{H}^N}\cap\,\{\eta;\eta_1>0\}}\frac{\varphi_h(\xi\,o\,\eta)-\varphi_h(\xi)}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\qquad+\frac{\b}{2}c(N,s)\int_{B^{\mathbbm{H}^N}_1\cap\mathcal{C}B_1^{\mathbbm{H}^N}}-\frac{\eta.(\nabla_{\mathbbm{H}^N}\varphi_h(\xi),\partial_t\varphi_h(\xi))}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta+\frac{\b}{2}c(N,s)\int_{\mathcal{C}B^{\mathbbm{H}^N}_1\cap\,\{\eta;\eta_1>0\}}\frac{\varphi_h(\xi\,o\,\eta)-\varphi_h(\xi)}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta.
\end{align}
It further gives
\begin{align}\label{112}
	\qquad\quad
	\a\mathcal{M}_{\lambda,\Lambda}^+(D^2_{\mathbbm{H}^N,S}\varphi_h(\xi))-\b(-\Delta_{\mathbbm{H}^N})^s\varphi_h(\xi)&=-\lambda \a C^2e^{-C\xi_1}\\
	&\qquad+\frac{\b}{2}c(N,s)\int_{B_\delta^{\mathbbm{H}^N}}\frac{\big(2-e^{-C(\xi_1+\eta_1)}\big)-\big(2-e^{-C\xi_1}\big)-\eta_1Ce^{-C\xi_1}}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\qquad+\frac{\b}{2}c(N,s)\int_{\mathcal{C}B_\delta^{\mathbbm{H}^N}\cap\{\eta;\eta_1\leq 0\}}\frac{\big(2-e^{-C(\xi_1+\eta_1)}\big)-\big(2-e^{-C\xi_1}\big)}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\qquad+\frac{\b}{2}c(N,s)\int_{B_1^{\mathbbm{H}^N}\cap \,\mathcal{C}B_\delta^{\mathbbm{H}^N}\cap\,\{\eta;\eta_1>0\}}\frac{\big(2-e^{-C(\xi_1+\eta_1)}\big)-\big(2-e^{-C\xi_1}\big)}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\qquad+\frac{\b}{2}c(N,s)\int_{B^{\mathbbm{H}^N}_1\cap\mathcal{C}B_\delta^{\mathbbm{H}^N}}-\frac{\eta_1Ce^{-C\xi_1}}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\qquad+\frac{\b}{2}c(N,s)\int_{\mathcal{C}B_1^{\mathbbm{H}^N}\cap\,\{\eta;\eta_1>0\}}\frac{\big(2-e^{-C(\xi_1+\eta_1)}\big)-\big(2-e^{-C\xi_1}\big)}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta.
\end{align}
Similarly,
\begin{align}\label{112}
\qquad\quad
\a\mathcal{M}_{\lambda,\Lambda}^+(D^2_{\mathbbm{H}^N,S}\varphi_h(\xi))&-\b(-\Delta_{\mathbbm{H}^N})^s\varphi_h(\xi)\\
&=-\lambda\alpha C^2e^{-C\xi_1}+\frac{\b}{2}c(N,s)\int_{B_\delta^{\mathbbm{H}^N}}\frac{\big(-e^{-C(\xi_1+\eta_1)}+e^{-C\xi_1}\big)-\eta_1Ce^{-C\xi_1}}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\qquad+\frac{\b}{2}c(N,s)\int_{\mathcal{C}B_\delta^{\mathbbm{H}^N}\cap\{\eta;\eta_1\leq 0\}}\frac{\big(-e^{-C(\xi_1+\eta_1)}+e^{-C\xi_1}\big)}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\qquad+\frac{\b}{2}c(N,s)\int_{B_1^{\mathbbm{H}^N}\cap \,\mathcal{C}B_\delta^{\mathbbm{H}^N}\cap\,\{\eta;\eta_1>0\}}\frac{\big(-e^{-C(\xi_1+\eta_1)}+e^{-C\xi_1}\big)}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\qquad+\frac{\b}{2}c(N,s)\int_{B^{\mathbbm{H}^N}_1\cap\mathcal{C}B_\delta^{\mathbbm{H}^N}}-\frac{\eta_1Ce^{-C\xi_1}}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta+\b\int_{\mathcal{C}B_1^{\mathbbm{H}^N}\cap\,\{\eta;\eta_1>0\}}\frac{\big(-e^{-C(\xi_1+\eta_1)}+e^{-C\xi_1}\big)}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta.
\end{align}
Now, since $e^{-C\xi_1}$ is a convex function so we have
\begin{align}\label{int 1}
	0&\leq e^{-C(\xi_1+\eta_1)}-e^{-C\xi_1}+\eta.(\nabla_{\mathbbm{H}^N}\varphi_h(\xi),\partial_t\varphi_h(\xi))\\
	&= e^{-C(\xi_1+\eta_1)}-e^{-C\xi_1}+C\eta_1e^{-C\xi_1}.
\end{align}
Also, if $\eta_1\leq 0$ then
\begin{align}\label{int 2}
	e^{-C(\xi_1+\eta_1)}-e^{-C\xi_1}\geq 0.
\end{align}
Using \eqref{int 1} and \eqref{int 2} in the first and second integrals in the R.H.S. of \eqref{112}, respectively, it confers that these integrals are non-positive. We further have that
\begin{align}
	|e^{-C(\xi_1+\eta_1)}-e^{-C\xi_1}|= e^{-C\xi_1}|e^{-C\eta_1}-1|\leq Ce^{-C\xi_1}|\eta|_{\mathbbm{H}^N}.
\end{align}
It infers
\begin{align}\label{int 3}
	\left|\b\int_{B_1^{\mathbbm{H}^N}\cap \,\mathcal{C}B_\delta^{\mathbbm{H}^N}\cap\,\{\eta;\eta_1>0\}}\frac{\left(-e^{-C(\xi_1+\eta_1)}+e^{-C\xi_1}\right)}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\right|&\leq \b\int_{B_1^{\mathbbm{H}^N}\cap \,\mathcal{C}B_\delta^{\mathbbm{H}^N}}\frac{Ce^{-C\xi_1}|\eta|_{\mathbbm{H}^N}}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\leq \b\int_{B_1^{\mathbbm{H}^N}\cap \,\mathcal{C}B_\delta^{\mathbbm{H}^N}}\frac{Ce^{-C\xi_1}|\eta|^2_{\mathbbm{H}^N}}{\delta|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\leq \b e^{-C\xi_1}\frac{C}{\delta}\int_{B^{\mathbbm{H}^N}_1\cap \,\mathcal{C}B_\delta^{\mathbbm{H}^N}}\frac{|\eta|^2_{\mathbbm{H}^N}}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\leq \b e^{-C\xi_1}\frac{C}{\delta}\int_{B_1^{\mathbbm{H}^N}}\frac{|\eta|^2_{\mathbbm{H}^N}}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta.
\end{align}
It is easy to see that for $\eta_1>0,$ we have $\big|e^{-C(\xi_1+\eta_1)}-e^{-C\xi_1}\big|\leq e^{-C\xi_1}$ so
\begin{align}\label{int 5}
	\left|\b\int_{\mathcal{C}B_1^{\mathbbm{H}^N}\cap\,\{\eta;\eta_1>0\}}\frac{\big(-e^{-C(\xi_1+\eta_1)}+e^{-C\xi_1}\big)}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\right|\leq \b e^{-C\xi_1}\int_{\mathcal{C}B_1^{\mathbbm{H}^N}}\frac{1}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta.
\end{align}
Also, it is easy to observe that
\begin{align}\label{int 4}
	\left|\b\int_{B^{\mathbbm{H}^N}_1\cap\mathcal{C}B_\delta^{\mathbbm{H}^N}}-\frac{\eta_1Ce^{-C\xi_1}}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\right|&\leq \b\int_{B^{\mathbbm{H}^N}_1\cap\mathcal{C}B_\delta^{\mathbbm{H}^N}}\frac{|\eta|_{\mathbbm{H}^N}Ce^{-C\xi_1}}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta\\
	&\leq \b e^{-C\xi_1}\frac{C}{\delta}\int_{B^{\mathbbm{H}^N}_1}\frac{|\eta|^2_{\mathbbm{H}^N}}{|\eta|_{\mathbbm{H}^N}^{Q+2s}}d\eta.
\end{align}
Using \eqref{int 3}, \eqref{int 5} \& \eqref{int 4} in \eqref{112}
yields that for sufficiently large enough $C>0$ we can make the L.H.S. of \eqref{112} less than $-1$, i.e.,
\begin{align}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi_h\big)-\b\big(-\Delta_{\mathbbm{H}^N}\big)^s\varphi_h&=\leq -1.
\end{align}\qed

Further, using the above lemma, we prove the comparison principle. Lemma \ref{Lemma 1.6} together with a standard approximation argument produces the following lemma.

\begin{lem}\label{comp}
	Let $u$ be bounded in $\mathbbm{H}^N$ and USC in $\overline{\Omega}$ such that
	\begin{align}
		\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2_{\mathbb{H}^N,S}u\big)-\b(-\Delta_{\mathbb{H}^N})^su\geq 0 \text{ in }\Omega \text{ in the viscosity sense.}
	\end{align}
	Then
	\begin{align}
		\displaystyle{\sup_\Omega}\,u\leq \displaystyle{\sup_{\mathbbm{H}^N\setminus\Omega}u}.
	\end{align}
\end{lem}
\noindent \textbf{Proof of Lemma \ref{comp}.} By Lemma \ref{Mou}, we have a function $\varphi_h\in C^2(\overline{\Omega})\cap C_b(\mathbbm{H}^N)$ such that
\begin{align}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi_h\big)-\b\big(-\Delta_{\mathbbm{H}^N}\big)^s\varphi_h\leq -1.
\end{align}
Now, let us define a function
\begin{align}
	\varphi_M(x)=M+{\mathlarger\varepsilon}\varphi_h(x)\text{ for }\mathlarger\varepsilon>0.
\end{align}
Then
\begin{align}\label{contra}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi_M\big)-\b\big(-\Delta_{\mathbbm{H}^N}\big)^s\varphi_M&=\alpha\mathlarger\varepsilon\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi_h\big)-\b\mathlarger\varepsilon\big(-\Delta_{\mathbbm{H}^N}\big)^s\varphi_h\\
	&\leq -\varepsilon \text{ in }\Omega.
\end{align}
Further, let $M_0$ be the smallest value of $M$ such that $\varphi_M\geq u.$ We aim to prove $M_0\leq \displaystyle{\sup_{{\mathbbm{H}^N}\setminus\Omega}}u$ by the method of contradiction. Let us assume that
\begin{align}
	M_0>\displaystyle{\sup_{{\mathbbm{H}^N}\setminus\Omega}}u,
\end{align}
then we have that there exists a point $\xi_0\in \Omega$ such that $u(\xi_0)=\varphi_{M_0}(\xi_0).$ It immediately implies that $\varphi_{M_0}$ touches $u$ at $\xi_0$ from above and by the definition, it gives
\begin{align}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N,S}\varphi_{M_0}\big)(\xi_0)-\b\big(-\Delta_{\mathbbm{H}^N}\big)^s\varphi_{M_0}(\xi_0)\geq 0.
\end{align}
This contradicts \eqref{contra}. Thus, we have
\begin{align}
	M_0\leq \displaystyle{\sup_{{\mathbbm{H}^N}\setminus\Omega}}u,
\end{align}
 which further entails that
 \begin{align}
 	u(\xi)&\leq \varphi_{M_0}(\xi_0)\\
 	 &\leq M_0+\mathlarger\varepsilon\,\displaystyle{\sup_{\mathbbm{H}^N}\varphi_h}\\
 	 &\leq \displaystyle{\sup_{\mathbbm{H}^N}u}+\mathlarger\varepsilon\,\displaystyle{\sup_{\mathbbm{H}^N}}\,\varphi_h,\, \xi\in \mathbbm{H}^N.
 \end{align}
Finally, letting $\mathlarger\varepsilon\longrightarrow 0$ offers the claim.\qed\\

\noindent \textbf{Proof of Theorem \ref{Holder}.} We follow the similar arguments as in the Euclidean setting, see for instance, \cite{Silvestre, Lara, Silvestre 2}. Without loss of generality, we may assume that $\|u\|_{\infty,\mathbbm{H}^N}=1.$ We first show the existence of a universal constant $0<\delta<1$ such that
\begin{align}
	\underset{{B_{2^{-k}}^{\mathbbm{H}^N}}}{\text{osc}}u\leq 2(1-\delta)^k.
\end{align}
We use the principle of mathematical induction to show the claim. Now, since $\|u\|_{\infty,\mathbbm{H}^N}=1$ so the claim holds trivially for $k\leq 0.$ Let the above inequality holds true up to some $k$, we need to show that it also holds for $k+1.$ For this, consider a function
\begin{align}
	v(\xi):=\frac{u(2^{-k}\xi)}{(1-\delta)^k
	}-a_k\,,
\end{align}
where $a_k$ is a constant chosen such that $-\frac{1}{2}\leq v\leq \frac{1}{2}$ in $B_1^{\mathbbm{H}^N}.$ Also, by the induction hypothesis, we have
\begin{align}
	\underset{{B_{2^{-j}}^{\mathbbm{H}^N}}}{\text{osc}}u\leq 2(1-\delta)^j \text{ for all }j\leq k,
\end{align}
i.e.,
\begin{align}
	\underset{{B_{2^{j}}^{\mathbbm{H}^N}}}{\text{osc}}u\leq 2(1-\delta)^{-j} \text{ for all }j\geq 0.
\end{align}
We aim to show that
\begin{align}
	\underset{{B_{\frac{1}{2}}^{\mathbbm{H}^N}}}{\text{osc}}v\leq \underset{{B_{1}^{\mathbbm{H}^N}}}{\text{osc}}v.
\end{align}
One may show this either by showing that supremum of $v$ in $B_{\frac{1}{2}}^{\mathbbm{H}^N}$ is smaller than that in $B_{1}^{\mathbbm{H}^N}$ or the infimum is larger. It is trivial that either $v\geq 0$ or $v\leq 0$ for atleast half of the points (in measure) in $B_1^{\mathbbm{H}^N}$. Without loss of generality, we may assume that
\begin{align}
	|N|\geq \frac{1}{2}\big|B_1^{\mathbbm{H}^N}\big|,
\end{align}
where $N:=\{v\leq 0\}\cap B_1^{\mathbbm{H}^N}.$ We show now that the induction hypothesis implies that
\begin{align}
	v(\xi)\leq \big(2|\xi|_{{\mathbbm{H}^N}}\big)^\a-\frac{1}{2} \text{ for all } \xi\in B_1^{\mathbbm{H}^N}.
\end{align}
Let $\xi\in {B_{2^{j+1}}^{\mathbbm{H}^N}}\setminus{B_{2^{j}}^{\mathbbm{H}^N}}$ for some $j>0.$ That is
\begin{align}
	2^{j+1}>|\xi|_{\mathbbm{H}^N}>2^j.
\end{align}
In other words,
\begin{align}
	2^{-k+j+1}>|2^{-k}\xi|_{\mathbbm{H}^N}>2^{-k+j}.
\end{align}
\begin{align}
	v(x)=(1-\delta)^{-k}u(2^{-k}x)-a_k.
\end{align}
For $\xi\in {B_{1}^{\mathbbm{H}^N}},$
\begin{align}
	-\frac{1}{2}\leq (1-\delta)^{-k}u(2^{-k}\xi)-a_k\leq \frac{1}{2}.
\end{align}
Now, for $\xi\notin {B_{1}^{\mathbbm{H}^N}}$, we have for any index $j\geq 0$,
\begin{align}\label{eq 3.23}
	v(\xi)\leq \big(2|\xi|_{\mathbbm{H}^N}\big)^{\c}-\frac{1}{2},
\end{align}
where $\c$ is a number such that $(1-\delta)=2^{-\c}.$
Also, for any $\xi\in B_1^{\mathbbm{H}^N},$
\begin{align}
	\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N}v\big)-\b(-\Delta_{\mathbb{H}^N})^sv=(1-\delta)^k\left(\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N}u\big)-\b\big(-\Delta_{\mathbb{H}^N}\big)^su\right)=0.
\end{align}
Now, we show that the following three points\\
$\bullet\,\, v(\xi)\leq \big(2|\xi|_{\mathbbm{H}^N}\big)^\c-\frac{1}{2}$ for $\xi\notin B_1^{\mathbbm{H}^N}$\\
$\bullet\,\, |N|=\big|\{v\leq 0\}\cap B_1^{\mathbbm{H}^N}\big|\geq \frac{1}{2}\big|B_1^{\mathbbm{H}^N}\big|$\\
$\bullet\,\, \a\mathcal{M}_{\lambda,\Lambda}^+(D^2_{\mathbbm{H}^N}v)-\b(-\Delta_{\mathbb{H}^N})^sv=0$ in $B_1^{\mathbbm{H}^N}$\\
imply that $v\leq \big(\frac{1}{2}-\delta\big)$ in $B_\frac{1}{2}^{\mathbbm{H}^N}.$ Let us assume the contrary. Let $v(\xi)>\left(\frac{1}{2}-\delta\right)$ for some $\xi\in B_{\frac{1}{2}}^{\mathbbm{H}^N}.$ Consider a smooth radial function whose support is contained in $B_{\frac{3}{4}}^{\mathbbm{H}^N}$ and $\rho\equiv 1$ in $B_{\frac{1}{2}}^{\mathbbm{H}^N}.$ It immediately gives that $v+\delta\rho$ attains a local maximum at some point $\xi_0\in B_{\frac{3}{4}}^{\mathbbm{H}^N}$ such that $\big(v+\delta\rho
\big)(\xi_0)>\frac{1}{2},$\, i.e.,\, $\displaystyle{\max_{B_1^{\mathbbm{H}^N}}}(v+\delta\rho)=(v+\delta\rho)(\xi_0)>\frac{1}{2}.$

Let us evaluate $\mathcal{L}(v+\delta\rho)$, which further entails getting a contradiction.
\begin{align}
	\mathcal{L}\big(v+\delta\rho\big)(\xi_0)&=\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N}(v+\delta\rho)\big)(\xi_0)-\b\big(-\Delta_{\mathbb{H}^N}\big)^s(v+\delta\rho)(\xi_0)\\
	&\geq \a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N}v\big)(\xi_0)+\delta\a\mathcal{M}_{\lambda,\Lambda}^-\big(D^2_{\mathbbm{H}^N}\rho\big)(\xi_0)-\b\big(-\Delta_{\mathbb{H}^N}\big)^sv(\xi_0)-\delta\b\big(-\Delta_{\mathbb{H}^N}\big)^s\rho(\xi_0)\\
	&=\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N}v\big)(\xi_0)-\b\big(-\Delta_{\mathbb{H}^N}\big)^sv(\xi_0)+\delta\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N}\rho\big)(\xi_0)-\delta\b\big(-\Delta_{\mathbb{H}^N}\big)^s\rho(\xi_0)\\
	&=\delta\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N}\rho\big)(\xi_0)-\delta\b\big(-\Delta_{\mathbb{H}^N}\big)^s\rho(\xi_0)\\
	&\geq \delta\,\displaystyle{\min_{\xi\in B^{\mathbbm{H}^N}_\frac{3}{4}}}\big(\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2_{\mathbbm{H}^N}\rho\big)(\xi)-\b(-\Delta_{\mathbb{H}^N})^s\rho(\xi)\big).
\end{align}
On the other hand,
\begin{align}\label{eq 3.24}
\\	\mathcal{L}(v+\delta\rho)(\xi_0)&=\a\mathcal{M}_{\lambda,\Lambda}^+(D^2_{\mathbbm{H}^N}(v+\delta\rho))(\xi_0)-\b(-\Delta_{\mathbb{H}^N})^s(v+\delta\rho)(\xi_0)\\
	&\leq \a\mathcal{M}_{\lambda,\Lambda}^+(D^2_{\mathbbm{H}^N}(v+\delta\rho))(\xi_0)+\frac{\b}{2}c(N,s)\int_{\mathbbm{H}^N}\frac{(v+\delta\rho)(\xi_0\,o\,\eta)+(v+\delta\rho)(\xi_0\,o\,\eta^{-1})-2(v+\delta\rho)(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&=\a\mathcal{M}_{\lambda,\Lambda}^+(D^2_{\mathbbm{H}^N}(v+\delta\rho))(\xi_0)\\
	&\qquad\qquad+
	\frac{\b}{2}c(N,s)\int_{\mathbbm{H}^N}\frac{(v+\delta\rho)(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)-\mathbbm{1}_{\{|\eta|_{\mathbbm{H}^N}\leq 1\}}\eta.\big(\nabla_{\mathbbm{H}^N}(v+\delta\rho)(\xi_0),\,\partial_t(v+\delta\rho)(\xi_0)\big)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&=\a\mathcal{M}_{\lambda,\Lambda}^+(D^2_{\mathbbm{H}^N}(v+\delta\rho))(\xi_0)\\
	&\,\,+
	\frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\in B_1^{\mathbbm{H}^N}\big\}}\frac{(v+\delta\rho)(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)-\mathbbm{1}_{\{|\eta|_{\mathbbm{H}^N}\leq 1\}}\eta.\big(\nabla_{\mathbbm{H}^N}(v+\delta\rho)(\xi_0),\,\partial_t(v+\delta\rho)(\xi_0)\big)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\,\,+
	\frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\notin B_1^{\mathbbm{H}^N}\big\}}\frac{(v+\delta\rho)(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)-\mathbbm{1}_{\{|\eta|_{\mathbbm{H}^N}\leq 1\}}\eta.\big(\nabla_{\mathbbm{H}^N}(v+\delta\rho)(\xi_0),\,\partial_t(v+\delta\rho)(\xi_0)\big)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&=\a\mathcal{M}_{\lambda,\Lambda}^+(D^2_{\mathbbm{H}^N}(v+\delta\rho))(\xi_0)\\
	&\quad\qquad+\frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\in B_1^{\mathbbm{H}^N}\big\}}\frac{(v+\delta\rho)(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad\quad+
	\frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\notin B_1^{\mathbbm{H}^N}\big\}}\frac{(v+\delta\rho)(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\leq \frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\in B_1^{\mathbbm{H}^N}\big\}}\frac{(v+\delta\rho)(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad\quad+\frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\notin B_1^{\mathbbm{H}^N}\big\}}\frac{(v+\delta\rho)(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta.
\end{align}
Note that in the last two steps, we used the fact that $v+\delta\rho$ has a local maximum at $\xi_0$ which also gives the non-positivity of integrand in the last integral. Moreover, we have
\begin{align}
	\frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\in B_1^{\mathbbm{H}^N}\big\}}&\frac{(v+\delta\rho)(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad\qquad\qquad\qquad=\frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\in N\big\}}\frac{(v+\delta\rho)(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad\qquad\qquad\qquad\qquad+\frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\in B_1^{\mathbbm{H}^N}\setminus N\big\}}\frac{(v+\delta\rho)(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad\qquad\qquad\qquad\leq \frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\in N\big\}}\frac{(v+\delta\rho)(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad\qquad\qquad\qquad\leq \frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\in N\big\}}\frac{\delta\rho(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad\qquad\qquad\qquad<\frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\in N\big\}}\frac{\big(\delta\rho(\xi_0\,o\,\eta)-\frac{1}{2}\big)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
	&\qquad\qquad\qquad\qquad\leq \frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\in N\big\}}\frac{\big(\delta M-\frac{1}{2}\big)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta,
\end{align}
for $M=\displaystyle{\max_{B^{\mathbbm{H}^N}_{\frac{3}{4}}}}\,\rho.$ Taking $\delta<\frac{1}{2M}$ along with using $|N|\geq \frac{\big|B_1^{\mathbbm{H}^N}\big|}{2},$ we get
\begin{align}\label{eq 3.25}
	\frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\in N\big\}}\frac{(v+\delta\rho)(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\leq -C \text{ for some }C>0.
\end{align}
Next, using \eqref{eq 3.23}, we get a bound on the integrand of first integral in the last line of \eqref{eq 3.24} . In particular, we have
\begin{align}\label{eq 3.26}
\\
\frac{1}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\notin B_1^{\mathbbm{H}^N}\big\}}\frac{(v+\delta\rho)(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta&=\frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\notin B_1^{\mathbbm{H}^N}\big\}}\frac{v(\xi_0\,o\,\eta)-(v+\delta\rho)(\xi_0)}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
&\leq \frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\notin B_1^{\mathbbm{H}^N}\big\}}\frac{|\xi_0\,o\,\eta|^\c_{\mathbbm{H}^N}-\frac{1}{2}-\frac{1}{2}}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta\\
&=\frac{\b}{2}c(N,s)\int_{\big\{\eta\in\mathbbm{H}^N;\xi_0\,o\,\eta\notin B_1^{\mathbbm{H}^N}\big\}}\frac{|\xi_0\,o\,\eta|^\c_{\mathbbm{H}^N}-1}{|\eta|^{Q+2s}_{\mathbbm{H}^N}}d\eta.
\end{align}
Further, taking small enough $\gamma$, we can make the above integral in the R.H.S. much less than $1.$ Therefore, by using together \eqref{eq 3.25} and \eqref{eq 3.26} for small enough $\delta$ and $\gamma$, we can make the L.H.S. of \eqref{eq 3.24} arbitrarily small. Hence, we can make it smaller than $\delta \displaystyle{\min_{\xi\in B_{\frac{3}{4}}^{\mathbbm{H}^N}}\big(\a\mathcal{M}_{\lambda,\Lambda}^-(D^2_{\mathbbm{H}^N}\rho)(\xi)-\b(-\Delta_{\mathbb{H}^N})^s\rho(\xi)\big)}$, which yield a contradiction. Thus, we have
\begin{align}
	v\leq \frac{1}{2}-\delta \text{ in }B_{\frac{1}{2}}^{\mathbbm{H}^N}.
\end{align}
It gives
\begin{align}
	\underset{{B_{\frac{1}{2}}^{\mathbbm{H}^N}}}{\text{osc}}v\leq (1-\delta)\underset{{B_{1}^{\mathbbm{H}^N}}}{\text{osc}}v.
\end{align}
More precisely, we have
\begin{align}
	\frac{1}{(1-\delta)^k}\displaystyle{\underset{{B_{\frac{1}{2}}^{\mathbbm{H}^N}}}{\text{osc}}u(2^{-k}\xi)}=\frac{1}{(1-\delta)^k}\displaystyle{\underset{{B_{2^{-k-1}}^{\mathbbm{H}^N}}}{\text{osc}}u(\xi)}\leq \frac{(1-\delta)}{(1-\delta)^k}\displaystyle{\underset{{B_{1}^{\mathbbm{H}^N}}}{\text{osc}}u(2^{-k}\xi)}=\frac{(1-\delta)}{(1-\delta)^k}\displaystyle{\underset{{B_{2^{-k}}^{\mathbbm{H}^N}}}{\text{osc}}u(\xi)},
\end{align}
which immediately offers
\begin{align}
	\displaystyle{\underset{{B_{2^{-k-1}}^{\mathbbm{H}^N}}}{\text{osc}}u(\xi)}&\leq {(1-\delta)}\displaystyle{\underset{{B_{2^{-k}}^{\mathbbm{H}^N}}}{\text{osc}}u(\xi)}\\
	&\leq 2(1-\delta)^{k+1}\\
	&=2.2^{-\c(k+1)}.
\end{align}
Let for some $k>0,\, \xi\in B^{\mathbbm{H}^N}_{2^{-k}}\setminus B^{\mathbbm{H}^N}_{2^{-(k+1)}}.$ Then we get
\begin{align}
	|u(\xi)-u(0)|\leq 2.2^{-\c k}=2.2^\c2^{-\c(k+1)}\leq C|\xi|_{\mathbbm{H}^N}^\c,
\end{align}
for $C=2.2^\c.$ Hence the claim. \qed
\section{Funding and/or Conflicts of interests/Competing interests}
The research of Priyank Oza was financially supported by Council of Scientific $\&$ Industrial Research (CSIR) under the grant no. 09/1031(0005)/2019--EMR--I. The second author thanks DST/SERB for the financial support under the grant CRG/2020/000041.\\

There are no conflict of interests of any type. This manuscript does not use any kind of data.

redpajama_set_name: RedPajamaArXiv


Visualizing the visions of Messiaen's colours.
The preliminary stage of the project was devoted to a performance of the Trois Petites Liturgies de la Présence Divine at the closing concert of a Messiaen festival organised by the group. The musical part of this performance requires a female choir, a string orchestra of specific size, various percussion instruments including celesta, and, beside the piano solo part, an ondes martenot solo that is essential to the colouring of the piece. Valérie Hartmann-Clavérie played this on the ondes martenot. In addition to playing the piano solo part, Håkon Austbø, the author of this article, undertook the development of colour projections.
The resulting colour part, to be played from an electronic keyboard, was performed at the concert by the pianist Sanae Yoshida. With the collaboration of the Oslo Sinfonietta, conducted by Christian Eggen, and the Norwegian Soloist Choir with conductor Grete Pedersen, an ensemble of students and professionals was put together. This turned out to be the first performance of this piece in Norway and took place at the Norwegian Academy on 28 February 2013.
The project was ultimately aimed at realizing a colour part for the gigantic piano cycle Vingt Regards sur l'Enfant-Jésus. This cycle was performed on 10 August 2013, at the Jakob church in Oslo, in the framework of the Oslo International Chamber Music Festival. Again, Håkon Austbø played the piano and prepared the colour part, which was again performed by Sanae Yoshida. During the whole project, Ricardo del Pozo provided expertise on the software environment as well as on the artistic content.


Interested in law and famous lawyers? Do you scour TV listings for Perry Mason and Matlock re-runs? If so, there's a place you might want to be on Wednesday, Nov. 15.
Playwright and actor Gary Anderson will present his one-man play entitled "Naked Darrow," a look into the private life of Clarence Darrow, a polarizing attorney who gained prominence for taking on death penalty cases in the courtroom.
Anderson will perform at 6:30 p.m. Wednesday, Nov. 15, at Loucks Auditorium at the Salem Public Library. Tickets are $20 and available in advance by calling 503-990-7060 or online at www.oadp.org.
Darrow became a controversial figure when people questioned his success rate in defending those clients. Anderson said Darrow sought out death penalty cases because of his personal stance against it.
The play looks at Darrow's flaws, including a drinking problem and his jury tampering trials, Anderson said.
The show performed Off-Broadway in New York for four weeks in 2013 and Anderson has taken his show around the country since.
More information about the show can be found at www.oadp.org.


text:
Showcasing Innovative Programs at the Statehouse
Several central Ohio school districts showcased their Innovation Generation programs February 5 as part of an Ohio Straight A Fund grant spotlight at the Ohio Statehouse.
Canal Winchester Local Schools, Marysville Exempted Village Schools, Reynoldsburg City Schools, and Westerville City School District all staffed booths designed to show the work to link career education, college credits and industry credentials to the region's need for skilled professionals in advanced manufacturing and robotics, logistics, health care and IT.
More than 50 students and educators provided Statehouse visitors with a glimpse into how the Straight A Fund grants are leading to new innovative programs in Ohio's schools. Partners, legislators, industry thought leaders and business leaders, participated as well.
"The Straight A Fund has inspired creativity, efficiency and collaboration in Ohio's schools," said Richard A. Ross, superintendent of public instruction at the Ohio Department of Education, before the event. "We are pleased with the overwhelming response and with the innovative ideas that are now being implemented throughout the state."
Created in 2013, the Straight A Fund invested $88.6 million in round one funding for the 2014 fiscal year and an additional $144.6 million in round two funding for 2015. Nearly 300 school districts have now used the money to launch innovative, locally created programs around the state.
It's Fun, It's Free, It's the Future
Students in high schools across central Ohio already are making plans for their future by enrolling in classes for the 2015-16 school year, and a multitiered promotional campaign is helping to spread the word. The campaign highlights career pathway opportunities available to students in some of the region's most important and needed industries.
The promotional campaign builds awareness about options available to students at their local high schools, and supports the districts' enrollment efforts. The approach includes ads at movie theaters in urban and suburban communities; banner ads on Pandora; and ads and PSAs on Radio One, the number one urban contemporary station in Columbus. The campaign messages tout options available to students and underscores that these options are free, fun and are pathways to the future. In addition, the campaign incorporates a bold visual display about Innovation Generation on the doors at key entrances at Polaris Fashion Place.
Teacher Spotlight: Integrating Pathways into the Classroom
Brad Gintert, the industrial technology teacher at Grandview Heights High School, teaches robotics, engineering, technology and woodworking to high school students and eighth-graders. Gintert has worked diligently over the course of the school year to integrate the manufacturing and robotics pathway into his classroom. And while he notes that this can sometimes be challenging, the reward has been great, as his students are enthusiastic about the hands-on coursework and real-world skills they're gaining.
Gintert says of the pathway, "Innovation Generation has offered my students the opportunity to problem-solve using current technology." He adds, "The greatest success of the program is when students can go from conceptualization to design, and then to prototype creation. Before, we could design it, but now we can build it with the industry-grade equipment in our MIT Mobile Fab Lab."
Recently, Gintert's high school students in the engineering and robotics class designed and printed – using a 3D printer – smartphone cases. Gintert notes that, "The students are proud to share their products with their peers, and it serves as a great way to promote all the cool things they are doing in these courses to those students who are not in the program."
When asked about the biggest challenge he has faced incorporating the pathway into his classroom, Gintert says it was the timeline to get the program up and running.
"For me, training and getting up to speed on how to work the equipment was a challenge," he says. "I don't always have as much time as I'd like to work on the equipment and to get a deeper understanding of how the different pieces all work."
Gintert has overcome this challenge by creating a more collaborative environment with the students.
"I let them know early on that I might not have all the answers, but we'll solve the problems together," he says. "This has been very rewarding in my classroom as the students are receptive to that additional level of engagement and teamwork. I see myself as a facilitator of their learning rather than just instructing."
Making Headlines: Innovation Generation in the News
Media has a keen interest in the programs and pathways that are changing the way students are learning and teachers are teaching in today's classrooms. From a recent editorial in The Columbus Dispatch, to a ThisWeek article about the Westerville program, to the WTVN radio interview with Scott Reeves, executive director of Secondary Academic Affairs at Westerville City Schools, the coverage has been informative and showcases how schools are transforming career education in the classroom.
Early Exposure, Early Engagement
Elementary school students at New Albany-Plain Local Schools district are learning that making dolls and airplanes is way cooler than it used to be. It's all part of the district's focus to reinvent education and expose students early on to careers in engineering, advanced manufacturing and robotics. Watch this video to learn what the students think about Innovation Generation.
Resources Just For You
School districts participating in Innovation Generation have a number of resources available to them to help counselors, teachers and others communicate about the initiative. All the materials can be customized to individual districts. These include:
A general FAQ
Overall message points
A communications piece for parents
An Innovation Generation infographic that makes the case for the initiative
A letterhead template in Word
A PowerPoint presentation template
Display materials for schools or for external meetings:
− Pull-up banners
− Ceiling banners
− Window clings
− Floor clings
To request any of these materials, please contact Paul Werth Associates at info@innovationgenerationohio.com.
About Innovation Generation
Innovation Generation, a collaboration of 15 central Ohio school districts, is made possible by a $14.4 million Ohio Straight A Fund grant from the Ohio Department of Education. Innovation Generation gives students options to gain credentials AND earn college credits while still in high school. Each participating district tailors courses to meet the unique needs of its community and students, while highlighting the rewarding careers available in advanced manufacturing/robotics, business logistics, health care, or IT, some of the region's fastest-growing and most important industries.
Visit us at InnovationGenerationOhio.com to learn more.

redpajama_set_name: RedPajamaCommonCrawl


text:
Our Firm's History
From home to town and then home again - Over 30 years in law practice
In 1978, Priscilla Gruber received her law degree with honors from Northern Illinois University College of Law and established a law practice, Riley and Gruber, with Francis X. Riley, her law professor and mentor. The partners first saw clients at Priscilla's newly built home on Ridge Road in Plainfield. After two years, Frank Riley retired from the practice.
Priscilla continued serving clients from her downtown Plainfield office as Priscilla H. Gruber, P. C., which was later renamed the Gruber Law Office, Ltd. after it began expanding. The first addition was Priscilla's daughter, Jeanette, who began her CPA practice from the same location, then moved her business to Peoria Heights, Illinois.
The second addition was the younger Gruber daughter, Patricia, who graduated with honors from the University of Wisconsin – Madison College of Law in 1992 and passed the Illinois Bar Exam that year. After working at the Illinois Supreme Court as a research attorney for two years, Patricia, joined the firm in 1994.
In 2006, the Grubers built a new wing addition to Priscilla's home, now known as 8380 Old Ridge Road, in Plainfield to allow them to practice there together with much shorter daily commutes for both attorneys. The practice came back home where it began over 40 years earlier and continues there today under Patricia's management after Priscilla's 2018 retirement.


text:
Complete List of Amy Coney Barrett's Legal Qualifications
by Jaco	 2020-09-26 2020-09-27
Amy Coney Barrett is a federal judge initially appointed by Donald J. Trump in 2017.
Tell the Senate: Delay the nomination of Amy Coney Barrett
Clerkships and private practice
After law school Barrett spent two years as a judicial law clerk, first for Judge Laurence Silberman of the U.S. Court of Appeals for the D.C. Circuit from 1997 to 1998,[13] then for Justice Antonin Scalia of the U.S. Supreme Court from 1998 to 1999.[13]
From 1999 to 2002, she practiced law at Miller, Cassidy, Larroca & Lewin in Washington, D.C.[11][14]
Teaching and scholarship
Barrett served as a visiting associate professor and John M. Olin Fellow in Law at George Washington University Law School for a year before returning to her alma mater, Notre Dame Law School, in 2002.[15] At Notre Dame she taught federal courts, constitutional law, and statutory interpretation. Barrett was named a Professor of Law in 2010, and from 2014 to 2017 held the Diane and M.O. Miller Research Chair of Law.[16] Her scholarship focuses on constitutional law, originalism, statutory interpretation, and stare decisis.[12] Her academic work has been published in journals such as the Columbia, Cornell, Virginia, Notre Dame, and Texas Law Reviews.[15]
At Notre Dame, Barrett received the "Distinguished Professor of the Year" award three times.[15] She taught Constitutional Law, Civil Procedure, Evidence, Federal Courts, Constitutional Theory Seminar, and Statutory Interpretation Seminar.[15] Barrett has continued to teach seminars as a sitting judge.[17][failed verification]
Barrett's academic publications have included:[clarification needed]
Barrett, Amy Coney (2017). "Originalism and Stare Decisis". Notre Dame Law Review. 92: 1921–1944.
Barrett, Amy Coney; Nagle, John Copeland (2016). "Congressional Originalism". University of Pennsylvania Journal of Constitutional Law. 19: 1–44.
Barrett, Amy Coney (2017). "Countering the Majoritarian Difficulty [review]". Constitutional Commentary. 32: 61–84.
Barrett, Amy Coney (2014). "Suspension and Delegation". Cornell Law Review. 99: 251–326.
Barrett, Amy Coney (2013). "Precedent and Jurisprudential Disagreement" (PDF). Texas Law Review. 91: 1711–1737.
Barrett, Amy Coney (2010). "Substantive Canons and Faithful Agency" (PDF). Boston University Law Review: 109–182.
Barrett, Amy Coney (2008). "Introduction [Symposium: Stare Decisis and Nonjudicial Actors]". Notre Dame Law Review. 83: 1147–1172.
Barrett, Amy Coney (2008). "Procedural Common Law" (PDF). Virginia Law Review. 94: 813–888.
Barrett, Amy Coney (2006). "The Supervisory Power of the Supreme Court". Columbia Law Review. 106: 324–387.
Barrett, Amy Coney (2005). "Statutory Stare Decisis in the Courts of Appeals". The George Washington Law Review. 73: 317–352.
Barrett, Amy Coney (2003). "Stare Decisis and Due Process". University of Colorado Law Review. 74: 1011–1074.
Barrett, Amy Coney; Garvey, John H. (1998). "Catholic Judges in Capital Cases". Marquette Law Review. 81: 303–350.
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