\section{Introduction} We establish interior $C^{1,\delta}$ regularity of viscosity solutions to the following mixed local and nonlocal degenerate equations:
	\begin{align}\label{eq 0.1}
		|Du|^{\gamma(x)}\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2u\big)-\b\big(-\Delta\big)^su\big)+b(x).Du|Du|^{\gamma(x)}=f &\text{ in } \,{B_1},
	\end{align}
	where $\a$ and $\b$ are non-negative constants, $B_1$ is a unit ball in $\mathbb{R}^N$ centered at origin, $\gamma\in C(\overline{B}_1,\mathbb{R})$ with $\displaystyle{\inf_{x\in B_1}}\gamma(x)\geq 0,$ $f\in L^\infty({B}_1,\mathbb{R})$,\,$Du$ denotes the gradient of $u,$ $\mathcal{M}_{\lambda,\Lambda}^+$ is the extremal Pucci's operator (fully nonlinear), $(-\Delta)^s$ denotes the fractional Laplacian with  order $s\in (0,1)$ and $b:B_1\longrightarrow\mathbb{R}^N$ is a bounded Lipschitz continuous function which is specified later. 
	


	
	


	
	

Results concerning regularity of solutions to PDEs have gained much attention since last several decades. Equations of the form \eqref{eq 0.1} have been investigated widely in the last decade. We begin with mentioning the pioneering works of I. Birindelli and F. Demengel in a series of papers for degenerate equations pertaining fully nonlinear uniformly elliptic local operators: comparison principle and Liouville-type results \cite{Birin 1}, regularity and uniqueness of eigenvalues and eigenfunctions \cite{Birin 3, Birin 2}, $C^{1,\alpha}$ regularity in the radial case \cite{Birin 4}. One may also see \cite{Davila, Davila 2, Imbert 1} for ABP estimates and Harnack's inequalities for degenerate, singular fully nonlinear equations. It is worth mentioning that when $\gamma(x)\equiv 0,$ $\beta=0$ and $b\equiv 0$ in \eqref{eq 0.1}, the regularity results are well known, for instance, see \cite{Cabre}.
	
	

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When $\beta=0,\,b\equiv 0$ in $B_1,$ $\gamma(x)\equiv\gamma\geq0$ is a constant and $\mathcal{M}_{\lambda,\Lambda}^+$ is replaced by a general fully nonlinear uniformly elliptic operator, $F$, \eqref{eq 0.1} reads as follows:
	\begin{align}\label{a=1}
		|Du|^\gamma F\big(D^2u\big)=f &\text{ in } \,{B_1}.
	\end{align}
	These equations are investigated by C. Imbert and L. Silvestre \cite{Imbert}. In this paper, they proved interior regularity of solutions  for a class of equations given by \eqref{a=1}. One may also see \cite{Arau, Birin Leo}, which are closely related to \cite{Imbert}. Ara\'ujo et al. \cite{Arau} established the sharp regularity of solutions to \eqref{a=1}, when $\gamma(x)\equiv\gamma>0$ is a constant. We refer to the works of E. V. Teixeira \cite{Teixeira 1, Teixeira 2, Teixeira 3}, for the related regularity results in different context. Birindelli et al. \cite{Birin Leo} established $C^{1,\delta}$ regularity of fully nonlinear degenerate equations with superlinear and subquadratic Hamoltonian terms. More precisely, authors considered equations of the form:
	\begin{align}
		-|Du|^\gamma F(D^2u)+b(x).|Du|^\beta=f(x) \text{ in }\Omega,
	\end{align} 
	where $b,f$ are continuous, $\Omega\subset \mathbb{R}^N$ with $C^2$-boundary, $\gamma>-1$ and $\gamma+1<\beta\leq \gamma+2.$
	
	
	There are several other works, which are devoted to the degenerate elliptic equations with gradient degeneracy. We first specially mention a few of them, which are closely related to this paper. 
	
	
	\noindent$\bullet$ When $\b=0,\,b\equiv0,\,\gamma(x)\equiv\gamma\geq0$ is a constant and $\mathcal{M}_{\lambda,\Lambda}^+$ is replaced by uniformly elliptic operator, $F,$ C. Imbert and L. Silvestre \cite{Imbert} established interior $C^{1,\delta}$ regularity of solutions to \eqref{eq 0.1}.\\
	$\bullet$ For $\a=0,$ $b\equiv 0, \gamma(x)\equiv\gamma\geq0$ is a constant, D. dos Prazeres and E. Topp \cite{Topp} investigated interior $C^{1,\delta}$ regularity of solutions to \eqref{eq 0.1}, where the nonlocal term is nonlinear.\\
	$\bullet$ Bronzi et al. \cite{Bronzi} studied \eqref{eq 0.1}, where $\gamma(x)$ is a continuous function with ${\inf_{B_1}}\,\gamma(x)>-1$, $\b=0$ and $b\equiv 0$. They established interior $C^{1,\delta}$ regularity. Inspired by the results of E. V. Teixeira \cite{Teixeira H, Teixeira 1}, they also established sharp geometric estimates.\\
	$\bullet$ I. Birindelli and F. Demengel \cite{C1beta} discussed \eqref{eq 0.1} in case when $\b=0,\gamma(x)\equiv\gamma\geq0$ is a constant, i.e., fully nonlinear local equation with a first order term. 
	
	Very recently, C. De Filippis \cite{Filippis} introduced degeneracies of double phase type to fully nonlinear equation
	\begin{align}\label{Filippis}
		\left[|Du|^{\gamma_1}+a(x)|Du|^{\gamma_2}\right]F\big(D^2u\big)=f(x) \text{ in } \Omega,	
	\end{align}
	for some non-negative continuous function $a,$ $\gamma_2\geq \gamma_1>0$ and bounded domains $\Omega$ in $\mathbb{R}^N,$ $N\geq 2.$ De Filippis proved $C^{1,\alpha}$ local regularity of viscosity solutions of \eqref{Filippis}. In another paper, De Filippis \cite{Filippis 1} studied the existence and regularity of solutions to free transmission problems, see \cite{Filippis 1}.
	One may also see \cite{Huaroto, Jesus} for the related works. In particular, Huaroto et al. \cite{Huaroto} considered the degenerate free transmission problem:
	\begin{align}
		|Du|^{\theta_1\mathbbm{1}_{\{u>0\}}+\theta_2\mathbbm{1}_{\{u<0\}}}F(D^2u)=f(x) \text{ in }\Omega \text{ with }\theta_1,\,\theta_2> 0.	
	\end{align} 
	D. Jesus \cite{Jesus} studied the following free transmission problem:
	\begin{align}
		|Du|^{\beta(x,u,Du)}F(D^2u)=f(x) \text{ in }B_1 \text{ with }\beta\geq 0.
	\end{align}
	We also mention the works of da Silva et al. \cite{da Silva} and Fang et al. \cite{Fang}, who established the local $C^{1,\alpha}$ regularity of solutions to fully nonlinear local equations with gradient degeneracy of the form 
	\begin{align}
		|Du|^{\gamma_1(x)}+a(x)|Du|^{\gamma_2(x)} \text{ with } \gamma_2(x)\geq \gamma_1(x)\geq 0 \text{ in }\Omega\subset\mathbb{R}^N,\,N\geq 2.
	\end{align}
	
	We recall that the operator $G(Du,D^2u)=|Du|^\gamma F(D^2u)$ can be viewed as a non-variational extension of the $p$-Laplacian, see \cite{Attouchi, Banerjee}, which occur in the theory of stochastic games and has applications to image processing. From the application point of view, degenerate equations with variable exponent are quite important. These equations appear in the study of image enhancing, see, Chen et al. \cite{image en} and the references therein. They basically proposed a new model for image restoration allowing the exponent to be a variable instead of a fixed constant. The variational approach to image restoration problem consists of minimizing the functional
	\begin{align}
		\int_{B_1}\bigg(|Du|^{\gamma(x)}+\frac{\sigma}{2}|u-I|^2\bigg)dx,
	\end{align}
	where $u:B_1\subset\mathbb{R}^2\longrightarrow\mathbb{R}$ is the true image and $I:B_1\subset\mathbb{R}^2\longrightarrow\mathbb{R}$ is the observed image. Here, $I:=u(x)+\xi,$ with $\xi$ being a noise term. For the different choices of $\gamma,$ different algorithms need to be followed, see \cite{image en} for the details. For instance, the method for the case when $\gamma\equiv 1,$ is known as total variation (TV) based regularization, which was proposed by Rudin et al. \cite{ROF}. Each of these algorithms have their own drawbacks. More precisely, when $\gamma(x)\equiv 1,$ TV-based solutions are piecewise constant. Although the case when $\gamma(x)\equiv 2,$ resolves this issue but it does not preserve the edges. On the other hand, taking fixed $1< \gamma < 2$ may destroy the edges, see \cite{Obliterate} for the details. This elaborates the importance of study of variable exponent degenerate equations. These equations also have applications in the study of electrorheological fluids \cite{electro 1, electro 2, electro 3} as well as in the thermistor problem \cite{thermo}. 
	
	It is important to note that the presence of fully nonlinear Pucci's maximal operator restricts the use of \enquote{linearization} techniques and bootstrap arguments to get the regularity results, which have been exploited in several papers. For a brief overview on bootstrap technique, one may see the book by A. Ambrosetti and A. Malchiodi \cite{Boot}, in particular, Theorem 1.16 therein. We refer to \cite{Felmer 18, Felmer 19, Oza Hopf, Pozza, Quaas, Quaas 2,Tyagi 3, Tyagi 1, Tyagi 2} for the existence and qualitative questions pertaining to extremal Pucci's equations. One may also see \cite{Tyagi degenerate} for fully nonlinear degenerate equations pertaining partial trace operators, $\mathcal{P}_k^{\pm}$ with sublinear gradient term.
	
	
	In a pioneering work in the direction of establishing $C^{1,\delta}$ interior regularity of degenerate fully nonlinear equations, i.e., when $\beta=0,b\equiv 0$ and $\gamma(x)\equiv\gamma\geq0$ is a constant in \eqref{eq 0.1}, we refer to the work of C. Imbert and L. Silvestre \cite{Imbert}. In their proof, Lemma \ref{Lem Imbert} (see next) plays a major role. 
	
We mention that even in a very particular case, i.e., $\gamma(x)\equiv\gamma$ a positive constant, $\a=0$ and $b\equiv 0,$ the comparison principle does not hold for \eqref{eq 0.1}. One may see \cite{Topp} for a counterexample. Therefore one can not use the classical Perron's method to establish the existence of viscosity solution. 
	
Recently, the elliptic equations involving mixed local (linear) and non-local operators without gradient degeneracy have gained much attention, see \cite{Biagi, Biagi 1, Biagi 2, Chen, Chen 2} and \cite{Anup mixed} for mixed local (fully nonlinear) and nonlocal equations. We mention the work \cite{Oza Heisenberg}, where we established the existence of viscosity solutions to mixed fully nonlinear local and nonlocal equations in the Heisenberg group using the Perron's method. We further proved the H\"older regularity of the viscosity solutions. These operators occur naturally in the study of plasma physics \cite{Plasma}, population dynamics \cite{Dipierro} and many more. C. Mou \cite{Mou} established the H\"older regularity of solutions to second order non-translation invariant integro-PDE. For the regularity results for mixed local and nonlocal $p$-Laplace equations, we refer to \cite{Biagi p, Prashanta 2, Prashanta 1} and the references therein. In addition to the above works, we refer to a survey paper by X. Ros-Oton \cite{Ros-Oton} on the existence, regularity and qualitative properties of solutions to a class of elliptic nonlocal equations.\\
	
Motivated by the above works, it is natural to ask the following question:\\ 
	
\noindent Can we establish the regularity of viscosity solutions to \eqref{eq 0.1}?\\
	
The aim of this paper is to answer this question affirmatively. To the best of our knowledge, we are not aware of such results for mixed fully nonlinear local and nonlocal operator with gradient degeneracy.\\
The main challenges in this study are the following:
	\begin{enumerate}[label=\upshape{(C\arabic*)}, ref=(C\arabic*)]
		\item\label{C1} Due to the absence of the comparison principle, the classical Perron's method is not applicable here, which is quite standard and has been exploited in different contexts, see, for instance \cite{Barles Busca, Filippis 1, Oza Heisenberg, Tyagi degenerate}.
		\item\label{C2} As mentioned earlier, the second challenge is that the non-local version of Lemma \ref{Lem Imbert} (see next) is not there. This prevents us to follow the classical approach of C. Imbert and L. Silvestre \cite{Imbert} in our case.
	\end{enumerate}
	
We follow the arguments of I. Birindelli and F. Demengel \cite{C1beta, Birin 1, Birin 4}, where the authors used the Ishii-Lions method \cite{Ishii} to establish the H\"older and Lipschitz regularity of the solutions to degenerate elliptic equations of type \eqref{a=1}. Further, using regularity results established in Theorem \ref{Main 1}, we get compactness of the family of solutions of the approximated problem:
\begin{align}
		\mathlarger\varepsilon\big(\Delta u-(-\Delta)^{s}u\big)+|Du|^{\gamma(x)}\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2u\big)-\b\big(-\Delta\big)^su\big)+b(x).Du|Du|^{\gamma(x)}=f \text{ in }B_1,
	\end{align} 
	which further yields the existence of solution to \eqref{eq 0.1} by letting $\mathlarger\varepsilon\longrightarrow 0^+.$ The similar arguments have been used effectively in \cite{Topp}. In this way, we overcome the challenge \ref{C1}. In dealing with \ref{C2} mentioned above, the order of the fractional Laplacian, $s\in(\frac{1}{2},1)$ plays a crucial role. Here, we make use of the fact that $(-\Delta)^s$ approximates $-\Delta$ as $s\to 1^-,$ see \cite{Hitchhiker} for the details. \\
	
The novelty with respect to the existing literature is due to the following:
	
	\noindent(i) We deal with variable-exponent degenerate mixed fully nonlinear local and nonlocal operator with first-order term $b(x).Du.$\\
	\noindent (ii) The results are new even in case when $\gamma(x)\equiv\gamma>0$ is a constant and $b\equiv 0$ in \eqref{eq 0.1}.\\ 
		
	Now, we list the assumptions on $\gamma(x)$ and $b(x),$ which we use in our main results:
	\begin{enumerate}[label=\upshape {(H\arabic*)}, ref=(H\arabic*)]
		\item\label{H1} (Lower bound on $\gamma(x)$). $\gamma\in C(\overline{B}_1,\mathbb{R})$ with $\displaystyle{\inf_{B_1}\gamma}\geq 0.$
		\item\label{H2} (Condition on $b(x)$). $b:B_1\longrightarrow\mathbb{R}^N$ is bounded Lipschitz with
		\begin{align}\label{b lip}
			(b(x)-b(y)).(x-y)\leq 0, \text{ for }x,y\in B_1.		
		\end{align}
	\end{enumerate}
	
	The main results of this paper are the following theorems:
	
	\begin{thm}\label{Main 1}
		Let \ref{H1} and \ref{H2} be hold. Let $f\in L^\infty(B_1,\mathbb{R})$ with $\|f\|_{\infty,B_1}\leq \varepsilon$ for some $\varepsilon>0$ and $s\in \big(\frac{1}{2},1\big).$
		Then for each $p\in\mathbb{R}^N,$ any bounded viscosity solution $u$ of 
		\begin{align}
			\big|Du+p\big|^{\gamma(x)}\big(\a\mathcal{M}_{\lambda,\Lambda}^+(D^2u)-\b(-\Delta)^su\big)+b.Du|Du+p|^{\gamma(x)}= f \text{ in }B_1,
		\end{align}
		with $\|u\|_{\infty,B_1}\leq 1$ is Lipschitz continuous with Lipschitz constant independent of $p.$ Moreover, the constant is uniformly bounded as $s\longrightarrow 1^-.$ 
	\end{thm}
	
	\begin{thm}\label{main 2}
		Let \ref{H1} and \ref{H2} be hold. Let $f\in L^\infty(B_1,\mathbb{R}).$ Then there exists $s_0\in(\frac{1}{2},1)$ close enough to $1$ such that for any $s\in (s_0,1),$ every bounded viscosity solution of \eqref{eq 0.1} is $C^{1,\delta}$ for some $\delta\in (0,1)$ with
		\begin{align}
			[u]_{C^{1,\delta}(B_\frac{1}{2})}\leq	\widetilde{C}\left(\|u\|_{\infty,B_1}+\left(\|f\|_\infty^{\frac{2s-1}{1+\inf_{B_1}\gamma}}\right)\right)
		\end{align} 
		for some positive constant $\widetilde{C}.$
	\end{thm}
	
	We recall the following results (Corollary \ref{C1bt} to \ref{cor to}) due to several authors, which are closely related to Theorem \ref{main 2}.
	
	\begin{cor}(Remark 3.5 \cite{C1beta})\label{C1bt}
		Let $\Omega\subset\mathbb{R}^N$ be a $C^2$ domain. Let $F$ be a uniformly elliptic operator and let $\gamma\geq 0$ be a constant. Let $h\in C(\overline{\Omega},\mathbb{R}^N)$ and $f\in C(\overline{\Omega},\mathbb{R}).$ Let $u$ be a viscosity solution of
		\begin{align}
			|Du|^\gamma(F(D^2u)+h.Du)=f\text{ in }B_1	
		\end{align}
		with $\underset{B_1}{\text{osc}}~u\leq 1$ and $\|f\|_\infty\leq \varepsilon,$ for some $\varepsilon\in [0,1].$ Then there exists $\tau,\delta\in(0,1)$ such that for all $n\in\mathbb{N},$ there exists $p_n\in\mathbb{R}^N$ such that
		\begin{align}
			\underset{B_{\tau^n}}{\text{osc}}~(u(x)-p_n.x)\leq \tau^{n(1+\delta)}.
		\end{align} 
	\end{cor}
	
	\begin{cor}(Theorem 2.1 \cite{Bronzi})
		Let $F$ be a uniformly elliptic operator. Let \ref{H1} be hold and $f$ be bounded. Let $u$ be a viscosity solution of 
		\begin{align}
			|Du|^{\gamma(x)}F(D^2u)=f \text{ in }B_1.
		\end{align}
		Then $u$ is in $C^{1,\delta}$ for some $\delta\in(0,1).$	
	\end{cor}





	
	
	
	
	
	
	
		
		The following result is due to D. dos Prazeres and E. Topp \cite{Topp}. They considered a nonlocal nonlinear operator $\mathcal{I},$ defined using a class of nonlocal operators whose kernels are comparable to that of fractional Laplacian, i.e., $(-\Delta)^s.$ For the details, see \cite{Topp}.
		
		\begin{cor}(Theorem 1.2 \cite{Topp})\label{cor to}
			Let $\gamma\geq 0$ be a constant and $f\in L^\infty(B_1)$. There exists $s_0\in\big(\frac{1}{2},1\big)$ sufficiently close to $1$ such that for any $s\in (s_0,1),$ every bounded viscosity solution $u$ to
			\begin{align}
				|Du|^\gamma \mathcal{I}(u)=f \text{ in }B_1
			\end{align}
			is in $C^{1,\delta}$ for some $\delta\in (0,1).$
		\end{cor}
		
		\begin{rem}
			Let us recall from \cite{Imbert} that even in case when $\lambda=\Lambda=1,$ i.e., $F(D^2u)=\Delta u,$ $\b=0,$ $b\equiv0$ and $\gamma(x)\equiv\gamma>0$, $f$ are constants, the best regularity one can expect for the solution of \eqref{eq 0.1} is $C^{1,\delta}.$
		\end{rem}
		
		\begin{rem}
			In the spirit of \cite{Arau}, in our case we have the following restriction on $\delta:$
			\begin{align}
				\delta< \displaystyle{\min}\left\{\overline{\delta},\frac{1}{1+\|\gamma^+\|_{\infty,B_1}+\|\gamma^-\|_{\infty,B_1}}\right\},
			\end{align}
			where $\overline{\delta}>0$ is such that the solutions of 
			\begin{align}
				\mathcal{M}^+_{{\a \lambda+\b},{\a \Lambda+\b}}\big(D^2u\big)+b(x).Du=0 \text{ in }B_1
			\end{align}
			are $C^{1,\overline{\delta}}$ regular.
		\end{rem}
		The organization of this paper is as follows. In Section 2, we list the basic definitions and several important results from the literature. Section 3 is dedicated to the proofs of our main results.
		\section{Preliminaries}
		Let $\mathcal{S}^N$ be the set of all real symmetric $N\times N$ matrices. Here, we have the usual partial ordering: $A\leq B$ in $\mathcal{S}^N$ means that $\langle A\xi,\xi\rangle \leq \langle B\xi, \xi\rangle$ for any $\xi \in \mathbb{R}^N$. In other words, $B-A$ is positive semidefinite, where $\|B\|$ is the largest eigenvalue of $B$.
		Let $S\in \mathcal{S}^N$ and for given two parameters $\Lambda\geq\lambda>0,$ extremal Pucci's operators are defined as follows:
		\begin{align}\label{M+}
			\mathcal{M}_{\lambda,\Lambda}^+(S):=\Lambda\displaystyle{\sum_{e_i\geq 0}e_i}+\lambda\displaystyle{\sum_{e_i< 0}e_i
			},
		\end{align}
		where $\{e_i\}_{i=1}^N$ are the eigenvalues of $S$.
		We now define the notion of viscosity solution, which we consider for \eqref{eq 0.1}. For $p\in\mathbb{R}^N,$ consider the auxiliary problem:
		\begin{align}\label{aux eq}
			\big|p+Du\big|^{\gamma(x)}\big(\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2u\big)-\b(-\Delta)^su\big)+b.Du\big|p+Du\big|^{\gamma(x)}= f.
		\end{align}
		
		\begin{defn}\label{subsol}
			Let $u:\mathbb{R}^N\longrightarrow\mathbb{R}$ be an upper semicontinuous (USC) function in $\overline{B}_1.$ Then $u$ is called a \textit{viscosity subsolution} of \eqref{aux eq} if for any $x\in B_1$ and $C^2$ function $\varphi:\overline{B}_\rho\longrightarrow\mathbb{R},$ for some $\rho\in(0,1]$ such that $\varphi(x)=u(x),$ $\varphi(y)>u(y)$ for $y\in \overline{B}_\rho\setminus \{x\}$ and $D\varphi(x)\neq-p,$ we have
			\begin{align}
				\big|p+Dv(x)\big|^{\gamma(x)}\big(\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2v(x)\big)-\b(-\Delta)^sv(x)\big)+b(x).Dv(x)\big|p+Dv(x)\big|^{\gamma(x)}\geq f(x),
			\end{align}
			where
			\begin{align}
				v:=\begin{cases}
					\varphi &\text{ in }B_\rho,\\
					u &\text{ in }\mathbb{R}^N\setminus B_\rho.
				\end{cases}
			\end{align}
			Moreover, we say $u$ satisfies $\big|p+Du\big|^{\gamma(x)}\big(\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2u\big)-\b(-\Delta)^su\big)+b.Du\big|p+Du\big|^{\gamma(x)}\geq f \text{ in }B_1$ in the viscosity sense. 
		\end{defn}
		
		\begin{defn}
			Let $u:\mathbb{R}^N\longrightarrow\mathbb{R}$ be a lower semicontinuous (LSC) function in $\overline{B}_1.$ Then $u$ is called a \textit{viscosity supersolution} of \eqref{aux eq} if for any $x\in B_1$ and $C^2$ function $\psi:\overline{B}_\rho\longrightarrow\mathbb{R},$ for some $\rho\in(0,1]$ such that $\psi(x)=u(x),$ $\psi(y)<u(y)$ for $y\in \overline{B}_\rho\setminus \{x\}$ and $D\psi\neq -p,$ we have
			\begin{align}
				\big|p+Dw(x)\big|^{\gamma(x)}\big(\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2w(x)\big)-\b(-\Delta)^sw(x)\big)+b(x).Dw(x)|p+Dw(x)|^{\gamma(x)}\leq f(x),
			\end{align}
			where
			\begin{align}
				w:=\begin{cases}
					\psi &\text{ in }B_\rho,\\
					u &\text{ in }\mathbb{R}^N\setminus B_\rho.
				\end{cases}
			\end{align}
			Moreover, we say $u$ satisfies $\big|p+Du\big|^{\gamma(x)}\big(\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2u\big)-\b(-\Delta)^su\big)+b.Du\big|p+Du\big|^{\gamma(x)}\leq f \text{ in }B_1$ in the viscosity sense.
		\end{defn}	
		
		We next	define the super/sub jets in order to demonstrate an equivalent definition of viscosity solution. Let us start with recalling the definitions of superjets and subjets.
		\begin{defn}\cite{Katzo}
			The \textit{superjet} $J^{2,+}u(x)$ of an USC function $u$ at $x\in \mathbb{R}^N$ consists of all the couples of the form $(p,X)\in \mathbb{R}^N\times \mathcal{S}^N$ for which
			\begin{align}
				u(x+z)\leq u(x)+p.z+\frac{1}{2}\left\langle Xz,z\right\rangle+o(|z|^2). 
			\end{align} 
			Similarly, 
			The \textit{subjet} $J^{2,-}u(x)$ of a LSC function $u$ at $x\in \mathbb{R}^N$ consists of all the couples of the form $(p,X)\in\mathbb{R}^N\times \mathcal{S}^N$ for which
			\begin{align}
				u(x+z)\geq u(x)+p.z+\frac{1}{2}\left\langle Xz,z\right\rangle+o(|z|^2). 
			\end{align}
		\end{defn}
		We next recall the definitions of the closures of the set-valued mappings, which will be required in the ensuing section.
		\begin{defn}\cite{User}
			The \textit{limiting superjet} $\overline{J}^{2,+}u(x)$ of a USC function $u$ is the set of couples of the form $(p,X)\in\mathcal{S}^N\times\mathbb{R}^N$ for which
			\begin{align}
				\exists (x_n,p_n,X_n)\longrightarrow(x,p,X) \text{ such that } (p_n,X_n) \text{ is a superjet of }u \text{ at }x_n \text{ and }u(x_n)\longrightarrow u(x). 
			\end{align}
			Similarly, the \textit{limiting subjet} $\overline{J}^{2,-}u(x)$ of an LSC function $u$ is the set of couples of the form $(p,X)\in\mathcal{S}^N\times\mathbb{R}^N$ for which
			\begin{align}
				\exists (x_n,p_n,X_n)\longrightarrow(x,p,X) \text{ such that } (p_n,X_n) \text{ is a subjet of }u \text{ at }x_n \text{ and }u(x_n)\longrightarrow u(x). 
			\end{align}
		\end{defn}
		
		C. Imbert and L. Silvestre \cite{Imbert} proved the following result using an improvement of flatness technique:
		\begin{thm}(Theorem 1 \cite{Imbert})\label{Thm Imbert}
			Let $\gamma\geq 0$ be a constant, $F$ be a uniformly elliptic operator with $F(0)=0$ and $f$ be bounded in $B_1.$ Then for any $U\Subset B_1,$ viscosity solution $u$ of
			\begin{align}
				|Du|^\gamma F(D^2u)=f \text{ in }B_1
			\end{align}   
			is in $C^{1,\alpha}(U)$ with
			\begin{align}
				[u]_{1+\a,U}\leq C\left(\|u\|_{\infty,B_1}+\|f\|_{\infty,B_1}^{\frac{1}{1+\gamma}}\right).
			\end{align}
		\end{thm}
		The proof of the above theorem depends on the equicontinuity properties for the auxiliary equation:
		\begin{align}
			|p+Du|^\gamma F\big(D^2u\big)=f \text{ in } B_1,
		\end{align}
		in terms of $p\in\mathbb{R}^N.$ In the proof of Theorem \ref{Thm Imbert}, the following lemma plays a crucial role. 
		\begin{lem}(Lemma 6 \cite{Imbert})\label{Lem Imbert}
			Let $u$ be a viscosity solution of 
			\begin{align}
				|p+Du|^\gamma F\big(D^2u\big)=0 \text{ in }B_1.
			\end{align}
			Then $u$ is a viscosity solution of 
			\begin{align}\label{cab}
				F\big(D^2u\big)=0 \text{ in }B_1.
			\end{align}
		\end{lem}
		Thanks to L. Caffarelli and X. Cabr\'e \cite{Cabre}, we have interior $C^{1,\alpha}$ estimate for the solutions of \eqref{cab}. C. Imbert and L. Silvestre \cite{Imbert} also constructed a counterexample to justify that one can not expect better regularity than $C^{1,\alpha}$ (see, Example 1 therein).
		
		We mention here that the presence of nonlocal term in \eqref{eq 0.1} prevents us to get a nonlocal version of Lemma \ref{Lem Imbert}. In case, when $\a=0$ and $\b=1$, in \eqref{eq 0.1}, one may see D. dos Prazeres and E. Topp \cite{Topp}, where they constructed a function, which immediately gives that for fractional Laplacian operator, $(-\Delta)^s,$ in place of $F,$ the analogue of Lemma \ref{Lem Imbert} does not hold, in general. For the sake of clarity, we mention below the same counterexample, which also works in our case, i.e., \eqref{eq 0.1} and is as follows. Consider a function $u:\mathbb{R}\longrightarrow\mathbb{R}$ given by  
		\begin{align}
			u(x):=
			\begin{cases}
				x+1, &\text{ if }x\leq -1\\
				0, &\text{ if }x\in (-1,1)\\
				x-1, &\text{ if }x\geq 1.
			\end{cases}
		\end{align}
		Let $s\in \big(\frac{1}{2},1\big).$ Since $u\equiv 0$ in $(-1,1),$ so we have $u_x=Du=0$ and $u_{xx}=D^2u=0$ in $(-1,1).$
		One may see that
		\begin{align}
			|Du|^{\gamma(x)}\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2u\big)-\b\big(-\Delta\big)^su\big)+b(x).Du|Du|^{\gamma(x)}=0 &\text{ in } (-1,1).
		\end{align}
		Now, let $x\in(-1+\delta,1-\delta)\Subset(-1,1),$ for some $\delta>0.$  It yields
		\begin{align}
			\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2u(x)\big)-\b\big(-\Delta\big)^su(x)\big)+b(x).Du&=-\beta\big(-\Delta\big)^su(x)\\
			&=\beta C(N,s) \,P.V.\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy,
		\end{align}
		where $P.V.$ stands for the principal value and $C(N,s)$ is a constant depending on $N$ and $s,$ for the details see \cite{Hitchhiker}. It further gives
		\begin{align}\label{counter 1}\\
			\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2u(x)\big)-\b\big(-\Delta\big)^su(x)+b(x).Du(x)&=-\beta\,C(N,s) \,\displaystyle{\lim_{\varepsilon\to 0^+}}\left[\int_{-\infty}^{x-\varepsilon}\frac{u(x)-u(y)}{|x-y|^{1+2s}}dy+\int_{x+\varepsilon}^{\infty}\frac{u(x)-u(y)}{|x-y|^{1+2s}}dy\right]\\
			&=-\beta\,C(N,s) \,\displaystyle{\lim_{\varepsilon\to 0^+}}\left[-\int_{-\infty}^{x-\varepsilon}\frac{u(y)}{|x-y|^{1+2s}}dy-\int_{x+\varepsilon}^{\infty}\frac{u(y)}{|x-y|^{1+2s}}dy\right]\\
			&=-\beta\,C(N,s) \,\displaystyle{\lim_{\varepsilon\to 0^+}}\bigg[-\int_{-\infty}^{-1}\frac{u(y)}{|x-y|^{1+2s}}dy-\int_{-1}^{x-\varepsilon}\frac{u(y)}{|x-y|^{1+2s}}dy\\
			&\qquad\qquad\qquad\qquad\qquad-\int_{x+\varepsilon}^{1}\frac{u(y)}{|x-y|^{1+2s}}dy-\int_{1}^{\infty}\frac{u(y)}{|x-y|^{1+2s}}dy\bigg]\\
			&=-\beta\,C(N,s) \,\displaystyle{\lim_{\varepsilon\to 0^+}}\bigg[-\int_{-\infty}^{-1}\frac{y+1}{|x-y|^{1+2s}}dy-\int_{1}^{\infty}\frac{y-1}{|x-y|^{1+2s}}dy\bigg].
		\end{align}
		For the first term in the last step of R. H. S. of \eqref{counter 1}, $x-y\in(0,\infty)$ and for the second term $x-y\in (-\infty,0)$. This
		infers 
		\begin{align}\label{counter 2}\\
			\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2u(x)\big)-\b\big(-\Delta\big)^su(x)+b(x).Du&=-\beta\,C(N,s)\,\displaystyle{\lim_{\varepsilon\rightarrow 0^+}}\bigg[-\int_{-\infty}^{-1}\frac{y+1}{(x-y)^{1+2s}}dy-\int_{1}^{\infty}\frac{y-1}{(y-x)^{1+2s}}dy\bigg]\\
			&=-\b C(N,s)\bigg[-\frac{2sy-x+2s-1}{(4s^2-2s)(x-y)^{2s}}\bigg|_{-\infty}^{-1}+\frac{2sy-x-2s+1}{(4s^2-2s)(y-x)^{2s}}\bigg|_{1}^{\infty}\bigg].
		\end{align} 
		Now, since $s\in\big(\frac{1}{2},1\big),$ it gives that R. H. S. of \eqref{counter 2} is finite for $x\in (-1,1).$ Whereas, it blows up for $x\longrightarrow 1$ as well as for $x\longrightarrow -1.$ This counterexample elaborates the effect of nonlocal term present with the gradient degeneracy. It is well known that $(-\Delta)^s\longrightarrow -\Delta$ as $s\longrightarrow 1^-,$ see \cite{Hitchhiker}. This property plays a crucial role in our analysis.
		
		Next, we state the non-local version of Ishii-Lions lemma \cite{Barles}, which we use in Theorem \ref{Main 1} to establish the Lipschitz regularity of viscosity solution to \eqref{eq 0.1}. It has also been exploited, for instance, in \cite{Barles 2, Barles 1}.
		\begin{lem}(Corollary 4.3.1 \cite{Barles})\label{Ishii nonlocal}
			Let $F: \mathbb{R}^N\times \mathbb{R}^N\times\mathcal{S}^N\times\mathbb{R}\longrightarrow\mathbb{R}$ be a continuous function satisfying the local and nonlocal degenerate ellipticity conditions. Let $u$ be a viscosity solution of
			\begin{align}
				F(x,Du,D^2u, \mathcal{I}[x,u])=0\text{ in }B_1.
			\end{align}
			Consider a function $v:B_1\times B_1\longrightarrow\mathbb{R}$ given by
			\begin{align}
				v(x,y):=u(x)-u(y).
			\end{align} 
			Let $\Gamma\in C^2(B_1\times B_1)$ and $(x_1,y_1)\in B_1\times B_1$ be a point of global maxima of $v(x,y)-\Gamma(x,y)$ in $B_1\times B_1.$ Then there exists $i>0$ small enough and matrices $X,\,Y\in\mathcal{S}^N$ such that
			\begin{align}
				F\left(x_1,D_x\Gamma(.,y_1),X,\mathcal{I}^{1,\delta}[x_1,\Gamma(.,y_1)],\mathcal{I}^{2,\delta}[x_1,D_x\Gamma(.,y_1),u]\right)\geq 0
			\end{align}
			and
			\begin{align}
				F\left(y_1,-D_y\Gamma(x_1,.),Y,\mathcal{I}^{1,\delta}[y_1,-\Gamma(x_1,.)],\mathcal{I}^{2,\delta}[y_1,-D_y\Gamma(x_1,.),u]\right)\leq 0,
			\end{align}
			where
			\begin{align}
				\mathcal{I}^{1,\delta}[x,\phi]=\int_{B_\delta}\big(\phi(x+y)-\phi(x)-D\phi(x).z\big)\mu(dy),
			\end{align}
			\begin{align}
				\mathcal{I}^{2,\delta}[x,p,w]=\int_{\mathbb{R}^N\setminus{B_\delta}}\big(w(x+y)-w(x)-\mathbbm{1}_{\{z\leq 1\}}z.p\big)\mu(dy),
			\end{align}
			for $\delta\ll 1$ and suitable measure $\mu.$ Also, we have the following matrix inequality:
			\begin{align}
				\begin{bmatrix}
					X &0\\
					0 &-Y
				\end{bmatrix}\leq \begin{bmatrix}
					D_x^2\Gamma(x_1,y_1) &-D_{xy}^2\Gamma(x_1,y_1)\\
					-D_{xy}^2\Gamma(x_1,y_1) &D_y^2\Gamma(x_1,y_1)
				\end{bmatrix}+iI_{2N},	
			\end{align}
			where $I_{2N}$ denotes $2N\times 2N$ identity matrix.
		\end{lem}
		
		\section{Proofs of main results}
		\noindent \textbf{Proof of Theorem \ref{Main 1}.}
		We first state and prove the following two lemmas. The proof of Theorem \ref{Main 1} follows by combining these lemmas. 
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
		
		\begin{lem}\label{lemma 4}
			Let \ref{H1} and \ref{H2} be hold and $s\in\big(\frac{1}{2},1\big).$ Let $u$ be a viscosity solution of 
			\begin{align}\label{eq lem 4}
				\big|Du+p\big|^{\gamma(x)}\big(\a\mathcal{M}_{\lambda,\Lambda}^+(D^2u)-\b(-\Delta)^su\big)+b.Du|Du+p|^{\gamma(x)}= f \text{ in }B_1
			\end{align}
			with $\underset{B_1}{\text{osc}}~u\leq 1$ and $\|f\|_{\infty,B_1}\leq \varepsilon$ for some $\varepsilon>0.$ Then there exists some $a>0$ such that if $|p|\geq \frac{1}{a},$ then $u$ is Lipschitz continuous with
			\begin{align}
				[u]_{C^{0,1}(B_{r})}\leq C=C(\lambda,\Lambda,N,\gamma,r,\|f\|_{\infty,B_1},\|u\|_{\infty,B_1})
			\end{align}
			for each $r\in(0,\frac{1}{2}).$
		\end{lem}
		\noindent \textbf{Proof of Lemma \ref{lemma 4}.}
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
		Let $a_0=\frac{1}{|p|}\in [0,a].$
		We use the viscosity method introduced by H. Ishii and P.--L. Lions \cite{Ishii}. Fix $r\in \big(0,1\big).$ For $x_0\in B_{\frac{r}{2}}$, define the auxiliary function
		\begin{align}\label{eq Phi}
			\Phi(x,y)&:=u(x)-u(y)-\Gamma(x,y)
		\end{align}
		with
		\begin{align}
			\Gamma(x,y)=L_1\phi(|x-y|)-L_2|x-x_0|^2-L_2|y-x_0|^2,
		\end{align}
		for $L_1>0,$ $L_2>0$ (to be chosen later) and
		\begin{align}\label{phi defn}
			\phi(t)=
			\begin{cases}
				t-\phi_0t^{1+\theta} &\text{ if }0<t\leq t_0\\
				\phi(t_0) &\text{ if }t> t_0,
			\end{cases}
		\end{align}
		for the choice of $\theta\in(0,1)$ small enough. Here, we choose $\phi_0\in(0,\frac{1}{1+\theta})$ such that for some $t_0\geq 1$, $\phi(t_0)>0$ and $\phi_0(1+\theta)t^\theta<1$ for all $t\in (0,2).$ We look for $L_1,\, L_2$ such that $M:=\displaystyle{\sup_{B_1\times B_1}}~\Phi\leq 0.$ Consider 
		\begin{align}
			L_2=\frac{16}{r^2}
		\end{align}
		for some fixed $r\in (0,1).$ We assume the contrary that $\Phi$ has a positive maximum in $B_1\times B_1,$ i.e., $M>0$ for some $x_0\in B_{\frac{r}{2}}$ and this leads to a contradiction for $L_1$ large enough. Let $\Phi$ be attain its maximum at some point $(x_1,y_1) \in \overline{B}_1\times\overline{B}_1.$ By the continuity of $u$ along with the fact that $\underset{B_1}{\text{osc}}\,u\leq 1,$ we have
		\begin{align}
			L_1\phi(|x_1-y_1|)+L_2|x_1-x_0|^2+L_2|y_1-x_0|^2\leq \underset{B_1}{\text{osc}}\,u\leq 1.
		\end{align}
		Since $\phi$ given by \eqref{phi defn} is positive, it infers
		\begin{align}
			L_2|x_1-x_0|^2+L_2|y_1-x_0|^2\leq 1.
		\end{align}
		This implies
		\begin{align}
			|x_1-x_0|^2+|y_1-x_0|^2\leq \frac{1}{L_2}=\frac{r^2}{16}. 
		\end{align}
		Using $x_0\in B_\frac{r}{2},$ it further concludes that $x_1, y_1$ belong to the interior of $B_1,$ in particular, in $B_{\frac{3}{4}}.$ Moreover, $x_1\neq y_1,$ otherwise $M\leq 0.$
		Also, it is easy to see that
		\begin{align}
			\phi'(t)=\begin{cases}
				1-(1+\theta) t^{\theta}\phi_0 &\text{ if }0< t\leq t_0\\
				0 &\text{ if }t>t_0.
			\end{cases}
		\end{align}
	
	
	
	
		Observe that $x\mapsto u(x)-\Gamma(x,y_1)$ attains a maximum in $B_1$ at $x_1.$ Similarly, $y\mapsto -u(y)-\Gamma(x_1,y)$ attains a maximum in $B_1$ at $y_1.$ In other words, $u(y)-\big(-\Gamma(x_1,y)\big)$ attains a minimum in $B_1$ at $y_1.$ By the definition of viscosity solution, $\Gamma(x_1,.)$ and $-\Gamma(.,y_1)$ work as test functions for viscosity subsolution and supersolution, respectively. 
		Also, we have
	
	
	
	
	
	
		\begin{align}
			D_x\Gamma(x_1,y_1)&=L_1\phi'\big(|x_1-y_1|\big)\frac{(x-y)}{|x-y|}+2L_2(x_1-x_0),
		
		\end{align}
		and
	
	
	
	
		\begin{align}
			D_y\Gamma(x_1,y_1)&=-L_1\phi'(|x_1-y_1|)\frac{(x_1-y_1)}{|x_1-y_1|}+2L_2(y_1-x_0).
		
		\end{align}
	
	
	
	
	
	
	
	
		From the above equations, note that
		\begin{align}\label{est D Gamma}
			\big|D_x\Gamma(x_1-y_1)-\big(-D_y\Gamma(x_1,y_1)\big)\big|\leq 4L_2.	
		\end{align}
		Also,
		\begin{align}\label{eq cL1}
			|D_x\Gamma(x_1,y_1)|\leq L_1|\phi'(x_1-y_1)|+2L_2\leq cL_1,	
		\end{align}
		for some universal constant $c>0.$ Next, our aim is to write two viscosity inequalities, for which we construct limiting subjet $(D_x\Gamma(.,y_1),X)$ of $u$ at $x_1$ and a limiting superjet $(-D_y\Gamma(x_1,.),Y)$ of $u$ at $y_1$ using nonlocal version of Jensen-Ishii's Lemma (Lemma \ref{Ishii nonlocal}). We have the following matrix inequality for small enough $i>0:$
		\begin{align}
			\begin{bmatrix}
				X &0\\
				0 &-Y
			\end{bmatrix}
		
			
			
			
			&\leq \begin{bmatrix}
				L_1D^2\left(\phi(|.|)\right)(x_1-y_1)+2L_2I_N &-L_1D^2\left(\phi(|.|)\right)(x_1-y_1)\\
				-L_1D^2\left(\phi(|.|)\right)(x_1-y_1) &L_1D^2\left(\phi(|.|)\right)(x_1-y_1)+2L_2I_N
			\end{bmatrix}+iI_{2N}\\
			&= \begin{bmatrix}
				L_1D^2\left(\phi(|.|)\right)(x_1-y_1) &-L_1D^2\left(\phi(|.|)\right)(x_1-y_1)\\ 
				-L_1D^2\left(\phi(|.|)\right)(x_1-y_1) &L_1D^2\left(\phi(|.|)\right)(x_1-y_1)
			\end{bmatrix}+\begin{bmatrix}
				2L_2I_N &0\\
				0 &2L_2I_N\\
			\end{bmatrix}+iI_{2N}\\
		
			
			
			
			
		
			
			
			
			&=\begin{bmatrix}
				Z &-Z\\
				-Z &Z
			\end{bmatrix}+2L_2\begin{bmatrix}
				I_N &0\\
				0 &I_N
			\end{bmatrix}+iI_{2N},
		\end{align}
		where \begin{align}
			Z=L_1\bigg(\phi''(|x_1-y_1|)\frac{(x_1-y_1)}{|x_1-y_1|}\otimes\frac{(x_1-y_1)}{|x_1-y_1|}+\frac{\phi'(|x_1-y_1|)}{|x_1-y_1|}\left(I_N-\frac{(x_1-y_1)}{|x_1-y_1|}\otimes\frac{(x_1-y_1)}{|x_1-y_1|}\right)\bigg),
		\end{align}
		and $I_N,$ $I_{2N}$ denote $N\times N$ and $2N\times 2N$ identity matrices, respectively. Next, applying the above inequality to any vector $(\xi,\xi)$ with $|\xi|=1,$ we have that
		\begin{align}
			\left\langle (X-Y)\xi,\xi\right\rangle\leq 4L_2+2i.
		\end{align}
		Also, applying the matrix inequality to the vector $\left(\frac{(x_1-y_1)}{|x_1-y_1|},-\frac{(x_1-y_1)}{|x_1-y_1|}\right),$ we get
		\begin{align}
			\left\langle (X-Y)\frac{(x_1-y_1)}{|x_1-y_1|},\frac{(x_1-y_1)}{|x_1-y_1|} \right\rangle&\leq 4\left\langle Z\frac{(x_1-y_1)}{|x_1-y_1|},\frac{(x_1-y_1)}{|x_1-y_1|}\right\rangle+(4L_2+2i)\left\langle I_N\frac{(x_1-y_1)}{|x_1-y_1|}, \frac{(x_1-y_1)}{|x_1-y_1|}\right\rangle\\
			&\leq \bigg(-4(1+\theta)\theta L_1|x_1-y_1|^{\theta-1}\phi_0+4L_2+2i\bigg)\left|\frac{x_1-y_1}{|x_1-y_1|}\right|^2\\
			&\leq \bigg(-4(1+\theta)\theta L_12^{\theta-1}\phi_0+4L_2+2i\bigg)\left|\frac{x_1-y_1}{|x_1-y_1|}\right|^2. 
		\end{align}
		It gives that the matrix $X-Y$ has atleast one eigenvalue less than $\left(-4(1+\theta)\theta L_12^{\theta-1}\phi_0+4L_2+i\right).$ This quantity will be negative for sufficiently large values of $L_1.$ Let $\{e_i\}_{i=1}^N$ be denote the eigenvalues of the matrix $X-Y.$ We observe that, by definition,
		\begin{align}\label{in eq 1}
			\mathcal{M}_{\lambda,\Lambda}^+(X-Y)&=\Lambda\sum_{e_i\geq 0}e_i+\lambda\sum_{e_i<0}e_i\\
			&\leq \lambda\left(-4(1+\theta)\theta L_12^{\theta-1}\phi_0+4L_2+2i\right)+\Lambda(N-1)(4L_2+2i).
		\end{align}
		We choose $a:=\frac{1}{5cL_1},$ where $c$ is the same constant appearing in \eqref{eq cL1}. This gives
		\begin{align}
			|{p}+D_x\Gamma(x_1,y_1)|&\geq |p|-|D_x\Gamma(x_1,y_1)|\\
			&\geq 5cL_1-cL_1\\
			&=4cL_1\\
			&=\frac{4}{5a}.
		\end{align}
	
	
	
	
		Next, we write the viscosity inequalities for $u$ at $x_1$ and $y_1:$
		\begin{align}\label{in eq1}
			{f}(x_1)&\leq \big|{p}+D_x\Gamma(x_1,y_1)\big|^{\gamma(x)}\big(\a\mathcal{M}_{\lambda,\Lambda}^+(X)-\b(-\Delta)^s{u}(x_1)+b(x_1).D_x\Gamma(x_1,y_1)\big)\\&=
			\big|{p}+D_x\Gamma(x_1,y_1)\big|^{\gamma(x)}\bigg(\a\mathcal{M}_{\lambda,\Lambda}^+(X)+\b C(N,s)\int_{B_\delta}\frac{\big(\Gamma(x_1+z,y_1)-\Gamma(x_1,y_1)-{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\quad\qquad\qquad\qquad\qquad\qquad+\b C(N,s)\int_{\mathbb{R}^N\setminus B_\delta}\frac{\big(u(x_1+z)-u(x_1)-{\mathbbm{1}_{\{z\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz+b(x_1).D_x\Gamma(x_1,y_1)\bigg),
		\end{align}
		and
		\begin{align}\label{in eq2}
			{f}(y_1)&\geq \big|{p}-D_y\Gamma(x_1,y_1)\big|^{\gamma(x)}\big(\a\mathcal{M}_{\lambda,\Lambda}^+(Y)-\b(-\Delta)^s{u}(y_1)\big)\\
			&=\big|{p}-D_y\Gamma(x_1,y_1)\big|^{\gamma(x)}\bigg(\a\mathcal{M}_{\lambda,\Lambda}^+(Y)+\b C(N,s)\int_{B_\delta}\frac{\big(-\Gamma(x_1,y_1+z)+\Gamma(x_1,y_1)+{\mathbbm{1}_{\{|z|\le 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\quad\qquad\qquad\qquad\qquad\qquad+\b C(N,s)\int_{\mathbb{R}^N\setminus B_\delta}\frac{\big(u(y_1+z)-u(y_1)+{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz-b(y_1).D_y\Gamma(x_1,y_1)\bigg)
		\end{align}
		for small enough $\delta>0.$ Above inequalities further infer
		\begin{align}\label{in eq11}
			\quad\quad\quad \big|{p}+D_x\Gamma(x_1,y_1)\big|^{\gamma(x)}&\bigg(\a\mathcal{M}_{\lambda,\Lambda}^+(X)+\b C(N,s)\int_{B_\delta}\frac{\big(\Gamma(x_1+z,y_1)-\Gamma(x_1,y_1)-{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\quad+\b C(N,s)\int_{\mathbb{R}^N\setminus B_\delta}\frac{\big(u(x_1+z)-u(x_1)-{\mathbbm{1}_{\{z\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\bigg)\\
			&\quad+\big|{p}+D_x\Gamma(x_1,y_1)\big|^{\gamma(x)}b(x_1).D_x\Gamma(x_1,y_1)\geq -\|{f}\|_{\infty,B_1},
		\end{align}
		and
		\begin{align}\label{in eq21}
			\quad\quad\quad \big|{p}-D_y\Gamma(x_1,y_1)\big|^{\gamma(x)}&\bigg(\a\mathcal{M}_{\lambda,\Lambda}^+(Y)+\b C(N,s)\int_{B_\delta}\frac{\big(-\Gamma(x_1,y_1+z)+\Gamma(x_1,y_1)+{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\quad+\b C(N,s)\int_{\mathbb{R}^N\setminus B_\delta}\frac{\big(u(y_1+z)-u(y_1)+{\mathbbm{1}_{\{|z|\le 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\bigg)\\
			&\quad-\big|{p}-D_y\Gamma(x_1,y_1)\big|^{\gamma(x)}b(y_1).D_y\Gamma(x_1,y_1)\leq \|{f}\|_{\infty,B_1}. 
		\end{align}
		Subtracting \eqref{in eq21} from \eqref{in eq11} yields
		\begin{align}\label{in eq 3}
		
			-2\|{f}\|_{\infty,B_1}&\leq\big|{p}+D_x\Gamma(x_1,y_1)\big|^{\gamma(x)}\a\mathcal{M}_{\lambda,\Lambda}^+(X)-\big|{p}-D_y\Gamma(x_1,y_1)\big|^{\gamma(x)}\a\mathcal{M}_{\lambda,\Lambda}^+(Y)\\
			&\qquad\qquad+\b C(N,s)\big|{p}+D_x\Gamma(x_1,y_1)\big|^{\gamma(x)}\bigg(\int_{B_\delta}\frac{\big(\Gamma(x_1+z,y_1)-\Gamma(x_1,y_1)-{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_x\Gamma
				(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\int_{\mathbb{R}^N\setminus B_\delta}\frac{\big(u(x_1+z)-u(x_1)-{\mathbbm{1}_{\{z\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\bigg)\\
			&\qquad\qquad-\b C(N,s)\big|{p}-D_y\Gamma(x_1,y_1)\big|^{\gamma(x)}\bigg(\int_{B_\delta}\frac{\big(-\Gamma(x_1,y_1+z)+\Gamma(x_1,y_1)+{\mathbbm{1}_{\{|z|\le 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\int_{\mathbb{R}^N\setminus B_\delta}\frac{\big(u(y_1+z)-u(y_1)+{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\bigg).	
		\end{align} 
	
		
		
		
			
			
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
		
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		\noindent We next use the idea, which is quite standard and has been exploited in \cite{Barles 2, Barles 1, Topp}. For $|z|\leq\frac{1}{4},$ we have $x_1+z,\,y_1+z\in \overline{B}_1.$
		Define
		\begin{align}
			I_1&:=\b C(N,s)\int_{B_{\frac{1}{4}}\setminus B_\delta}\frac{\big(u(x_1+z)-u(x_1)-{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\qquad\qquad\qquad-\b C(N,s)\int_{B_{\frac{1}{4}}\setminus B_\delta}\frac{\big(u(y_1+z)-u(y_1)+{\mathbbm{1}_{\{|z|\le 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			I_2&:=\b C(N,s)\int_{B_\delta}\frac{\big(\Gamma(x_1+z,y_1)-\Gamma(x_1,y_1)-{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\qquad\qquad\qquad-\b C(N,s)\int_{B_\delta}\frac{\big(-\Gamma(x_1,y_1+z)+\Gamma(x_1,y_1)+{\mathbbm{1}_{\{|z|\le 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz.	
		\end{align}
		We define a set 
		\begin{align}
			\mathcal{C}=\bigg\{z\in B_\rho:(1-\eta)|z||x_1-y_1|\leq |(x_1-y_1).z|\bigg\}
		\end{align}
		for some fixed $\rho\in(0,\frac{1}{4}).$ Consider
		\begin{align}
			I_3:=&\b C(N,s)\int_{\mathcal{C}}\big(u(x_1+z)-u(x_1)-{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)dz\\
			&-\b C(N,s)\int_{\mathcal{C}}\big(u(y_1+z)-u(y_1)+{\mathbbm{1}_{\{|z|\le 1\}}}z.D_y\Gamma(x_1,y_1)\big)dz.
		\end{align}
		Since $(x_1,y_1)$ is a point of maxima for $\Phi$ in $B_1\times B_1$ and $M$ is the maximum, so we have
		\begin{align}
			u(x_1+z)-u(y_1+\overline{z})-\Gamma(x_1+z,y_1+\overline{z})\leq u(x_1)-u(y_1)-\Gamma(x_1,y_1),
		\end{align}
		for any $z,\,\overline{z}\in B_{\frac{1}{4}}.$ 
		It further gives
		\begin{align}\label{zz'}
			u(x_1+z)-u(x_1)-D_x\Gamma(x_1,y_1).z&\leq u(y_1+\overline{z})-u(y_1) +D_y\Gamma(x_1,y_1).\overline{z}\\
			&\quad+\Gamma(x_1+z,y_1+\overline{z})-\Gamma(x_1,y_1)-D\Gamma(x_1,y_1).(z,\overline{z}),
		\end{align}
		where $D\Gamma(x_1,y_1)=\big(D_x\Gamma(x_1,y_1),D_y\Gamma(x_1,y_1)\big).$ Taking, in particular $\overline{z}=0$ yields
		\begin{align}
			u(x_1+z)-u(x_1)-D_x\Gamma(x_1,y_1).z\leq \Gamma(x_1+z,y_1)-\Gamma(x_1,y_1)-D_x\Gamma(x_1,y_1).z.
		\end{align}
		Similarly, putting $z=0$ gives
		\begin{align}
			-\big(u(y_1+\overline{z})-u(y_1) +D_y\Gamma(x_1,y_1).\overline{z}\big)&\leq \Gamma(x_1,y_1+\overline{z})-\Gamma(x_1,y_1)-D_y\Gamma(x_1,y_1).\overline{z}.
		\end{align}
		Using above inequalities, we have
		\begin{align}
			I_3&= \beta C(N,s)\int_{\mathcal{C}}\frac{\big(u(x_1+z)-u(x_1)-z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\qquad-\beta C(N,s)\int_{\mathcal{C}}\frac{\big(u(y_1+z)-u(y_1)+z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\leq \beta C(N,s)\int_{\mathcal{C}}\frac{\big(\Gamma(x_1+z,y_1)-\Gamma(x_1,y_1)-z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\qquad+\beta C(N,s)\int_{\mathcal{C}}\frac{\big(\Gamma(x_1,y_1+z)-\Gamma(x_1,y_1)-z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz.
		\end{align}
		Now, since $\Gamma(x,y)=L_1\phi(|x-y|)+L_2|x-x_0|^2+L_2|y-x_0|^2.$ It yields
		\begin{align}
			I_3
			&\leq L_1\beta C(N,s)\int_{\mathcal{C}}\frac{\big(\phi(|x_1-y_1+z|)-\phi(|x_1-y_1|)-z.D\phi(|x_1-y_1|)\big)}{|z|^{N+2s}}dz\\
			&\quad+L_2\beta C(N,s)\int_{\mathcal{C}}\frac{\big(|x_1+z-x_0|^2-|x_1-x_0|^2-z.2(x-x_0)\big)}{|z|^{N+2s}}dz\\
			&\quad+L_1\beta C(N,s)\int_{\mathcal{C}}\frac{\big(\phi(|x_1-y_1-z|)-\phi(|x_1-y_1|)+z.D\phi(|x_1-y_1|)\big)}{|z|^{N+2s}}dz\\
			&\quad+L_2\beta C(N,s)\int_{\mathcal{C}}\frac{\big(|y_1+z-x_0|^2-|y_1-x_0|^2-z.2(y-x_0)\big)}{|z|^{N+2s}}dz\\
			&=L_1\beta C(N,s)\int_{\mathcal{C}}\frac{\big(\phi(|x_1-y_1+z|)-\phi(|x_1-y_1|)-z.D\phi(|x_1-y_1|)\big)}{|z|^{N+2s}}dz+L_2\beta C(N,s)\int_{\mathcal{C}}\frac{|z|^2}{|z|^{N+2s}}dz\\
			&\qquad +L_1\beta C(N,s)\int_{\mathcal{C}}\frac{\big(\phi(|x_1-y_1-z|)-\phi(|x_1-y_1|)+z.D\phi(|x_1-y_1|)\big)}{|z|^{N+2s}}dz+L_2\beta C(N,s)\int_{\mathcal{C}}\frac{|z|^2}{|z|^{N+2s}}dz.
		\end{align}
		Further, using a second-order Taylor expansion in the above integrals, we get
		\begin{align}\label{I2 est}
			\\I_3&\leq \frac{L_1\beta C(N,s)}{2}\left(\int_{\mathcal{C}}\sup_{t\in(0,1)}\frac{D^2\phi(|x_1-y_1+tz|)z.z}{|z|^{N+2s}}dz+\int_{\mathcal{C}}\sup_{t\in(-1,0)}\frac{D^2\phi(|x_1-y_1+tz|)z.z}{|z|^{N+2s}}dz\right)\\
			&\qquad+2L_2\beta C(N,s)\int_{\mathcal{C}}\frac{|z|^2}{|z|^{N+2s}}dz\\	&\leq\frac{L_1\beta C(N,s)}{2}\int_{\mathcal{C}}\sup_{t\in(-1,1)}\frac{D^2\phi(|x_1-y_1+tz|)z.z}{|z|^{N+2s}}dz+2L_2\beta C(N,s)\int_{\mathcal{C}}\frac{|z|^2}{|z|^{N+2s}}dz\\
			&=\frac{L_1\beta C(N,s)}{2}\int_{\mathcal{C}}\sup_{t\in(-1,1)}\frac{\bigg(\phi''(|x_1-y_1+tz|)\frac{(x_1-y_1+tz)}{|x_1-y_1+tz|}\otimes\frac{(x_1-y_1+tz)}{|x_1-y_1+tz|}+\frac{\phi'(|x_1-y_1+tz|)}{|x_1-y_1+tz|}\left(I-\frac{(x_1-y_1+tz)}{|x_1-y_1+tz|}\otimes\frac{(x_1-y_1+tz)}{|x_1-y_1+tz|}\right)\bigg)z.z}{|z|^{N+2s}}dz\\
			&\qquad+2L_2\beta C(N,s)\int_{\mathcal{C}}\frac{|z|^2}{|z|^{N+2s}}dz\\
			&=\frac{L_1\beta C(N,s)}{2}\int_{\mathcal{C}}\sup_{t\in(-1,1)}\frac{\phi''(|x_1-y_1+tz|)\bigg|\left(\frac{x_1-y_1+tz}{|x_1-y_1+tz|}\right).z\bigg|^2+\frac{\phi'(|x_1-y_1+tz|)}{|x_1-y_1+tz|}\left(|z|^2-\bigg|\left(\frac{x_1-y_1+tz}{|x_1-y_1+tz|}\right).z\bigg|^2\right)}{|z|^{N+2s}}dz\\
			&\qquad+2L_2\beta C(N,s)\int_{\mathcal{C}}\frac{|z|^2}{|z|^{N+2s}}dz.
		\end{align}
		Let $\rho=\nu_0|x_1-y_1|<|x_1-y_1|$ for $\nu_0\in(0,1)$ is small enough to be chosen later. Now, we estimate $|x_1-y_1+tz|$ and $(x_1-y_1+tz).z$ on the set $\mathcal{C}$ as follows:
		\begin{align}
			|x_1-y_1+tz|&\leq |x_1-y_1|+|t||z|\leq|x_1-y_1|+\rho=|x_1-y_1|(1+\nu_0).\\
			|x_1-y_1+tz|&\geq |x_1-y_1|-|t||z|\geq |x_1-y_1|-\rho=|x_1-y_1|(1-\nu_0).\\ 
			|(x_1-y_1+tz).z|&=|(x_1-y_1).z+t|z|^2|
			\\
			&\geq|(x_1-y_1).z|-\rho|z|\\
			&\geq(1-\eta)|z||x_1-y_1|-\rho|z|\\
			&=(1-\eta)|z||x_1-y_1|-|x_1-y_1|\nu_0|z|\\
			&=(1-\eta-\nu_0)|x_1-y_1||z|.
		\end{align}
		Using the above inequalities, we get 
		\begin{align}
			\left|\frac{x_1-y_1+tz}{|x_1-y_1+tz|}.z\right|&\geq \frac{1}{|x_1-y_1+tz|}(1-\eta-\nu_0)|x_1-y_1||z|\\
			&\geq \frac{(1-\eta-\nu_0)|x_1-y_1||z|}{|x_1-y_1|(1+\nu_0)}\\
			&=\frac{(1-\eta-\nu_0)|z|}{(1+\nu_0)},
		\end{align}
		for $t\in(-1,1).$ Using \eqref{phi defn} in \eqref{I2 est} infers
		\begin{align}\label{est2 I2}
			I_3&\leq \frac{L_1\beta}{2}\int_{\mathcal{C}}\sup_{t\in(-1,1)}\frac{-(1+\theta)\theta|x_1-y_1+tz|^{\theta-1}\phi_0\left(\frac{(1-\eta-\nu_0)^2|z|^2}{(1+\nu_0)^2}\right)}{|z|^{N+2s}}dz\\
			&\qquad+\frac{L_1\beta}{2}\int_{\mathcal{C}}\sup_{t\in(-1,1)}\frac{\frac{1}{|x_1-y_1+tz|}\bigg({1}-(1+\theta)\phi_0|x_1-y_1+tz|^{\theta}\bigg)\left(|z|^2-\bigg|\left(\frac{x_1-y_1+tz}{|x_1-y_1+tz|}\right).z\bigg|^2\right)}{|z|^{N+2s}}dz\\
			&\qquad+2L_2\beta C(N,s)\int_{\mathcal{C}}\frac{|z|^2}{|z|^{N+2s}}dz\\
		
		
		
		
			&\leq L_1\beta C(N,s)\int_{\mathcal{C}}\sup_{t\in(-1,1)}\frac{-(1+\theta)\theta|x_1-y_1+tz|^{\theta-1}\phi_0\bigg(\frac{(1-\eta-\nu_0)^2|z|^2}{(1+\nu_0)^2}\bigg)}{|z|^{N+2s}}dz\\
			&\qquad+L_1\beta C(N,s)\int_{\mathcal{C}}\sup_{t\in(-1,1)}\frac{\frac{1}{|x_1-y_1+tz|}\bigg(1-(1+\theta)\phi_0|x_1-y_1+tz|^{\theta}\bigg)\left(1-\frac{(1-\eta-\nu_0)^2}{(1+\nu_0)^2}\right)|z|^2}{|z|^{N+2s}}dz\\
			&\qquad+2L_2\beta C(N,s)\int_{\mathcal{C}}\frac{|z|^2}{|z|^{N+2s}}dz.
		\end{align}
		Note that $\frac{(1-\eta-\nu_0)^2}{(1+\nu_0)^2}\leq 1$. Taking $\eta=c_1|x_1-y_1|^{2\theta}$ and $\nu_0=c_1|x_1-y_1|^{\theta}$ for some constant $c_1>0,$ we obtain $\frac{1-\eta-\nu_0}{1+\nu_0}\simeq 1,$ see Remark $7$ \cite{Barles 2}. Hence the term $I_3$ can be made as negative as we please. 
	
		Also, we have
		\begin{align}
			I_2=\beta C(N,s)\int_{B_{\frac{1}{4}}\setminus B_\delta}\frac{\big(\Gamma(x_1+z,y_1+z)-\Gamma(x_1,y_1)-D\Gamma(x_1,y_1)(z,z)\big)}{|z|^{N+2s}}dz.
		\end{align}
		We now use a second order Taylor expansion for $\Gamma$ and get
		\begin{align}
			I_2\leq \frac{\beta C(N,s)}{2} \int_{B_\frac{1}{4}\setminus B_{\delta}}\frac{D^2\Gamma(x_1,y_1)(z,z).(z,z)}{|z|^{N+2s}}dz.
		\end{align}
		It yields taking $L_1$ large enough and using the concavity of $\phi$ that $I_2\leq 0.$
		Similarly, we have $I_1\leq 0.$ Combining the above estimates, we have that the sum $I_1+I_2+I_3$ can be made as negative as we want, let us say $-P,$ where $P=P(L_1)$ is a large enough positive number. Using the above estimates, we deduce
	
	
	
	
	
		\begin{align}
			\frac{{f}(x_1)}{\big|{p}+D_x\Gamma(x_1,y_1)\big|^{\gamma(x_1)}}&\leq \bigg(\a\mathcal{M}_{\lambda,\Lambda}^+(X)+\b C(N,s)\int_{B_\delta}\frac{\big(\Gamma(x_1+z,y_1)-\Gamma(x_1,y_1)-{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\qquad+\b C(N,s)\int_{\mathbb{R}^N\setminus B_\delta}\frac{\big(u(x_1+z)-u(x_1)-{\mathbbm{1}_{\{z\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\bigg)+b(x_1).D_x\Gamma(x_1,y_1)\\	
			&=\bigg(\a\mathcal{M}_{\lambda,\Lambda}^+(X)+\b C(N,s)\int_{B_\delta}\frac{\big(\Gamma(x_1+z,y_1)-\Gamma(x_1,y_1)-{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\qquad+\b C(N,s)\int_{\mathbb{R}^N\setminus B_\delta}\frac{\big(u(x_1+z)-u(x_1)-{\mathbbm{1}_{\{z\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\bigg)\\	
			&\qquad+\big(b(x_1)-b(y_1)\big).D_x\Gamma(x_1,y_1)-b(y
			_1).D_y\Gamma(x_1,y_1)+b(y_1).\big(D_x\Gamma(x_1,y_1)+D_y\Gamma(x_1,y_1)\big)\\
			&\leq \bigg(\a\mathcal{M}_{\lambda,\Lambda}^+(X)+\b C(N,s)\int_{B_\delta}\frac{\big(\Gamma(x_1+z,y_1)-\Gamma(x_1,y_1)-{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\qquad+\b C(N,s)\int_{\mathbb{R}^N\setminus B_\delta}\frac{\big(u(x_1+z)-u(x_1)-{\mathbbm{1}_{\{z\leq 1\}}}z.D_x\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\bigg)\\	
			&\qquad+L_1\big(b(x_1)-b(y_1)\big).\frac{\phi'(x_1-y_1)}{|x_1-y_1|}(x_1-y_1)-b(y_1).D_y\Gamma(x_1,y_1)+4L_2\|b\|_{\infty,B_1},
		\end{align}
		where in the last step, we have used \eqref{est D Gamma}. Further, using \ref{H2}, we have
		\begin{align}
			\frac{{f}(x_1)}{\big|{p}+D_x\Gamma(x_1,y_1)\big|^{\gamma(x_1)}}&\leq\bigg(\a\mathcal{M}_{\lambda,\Lambda}^+(Y)+\b C(N,s)\int_{B_\delta}\frac{\big(-\Gamma(x_1,y_1+z)+\Gamma(x_1,y_1)+{\mathbbm{1}_{\{|z|\le 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\qquad+\b C(N,s)\int_{\mathbb{R}^N\setminus B_\delta}\frac{\big(u(y_1+z)-u(y_1)-{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\bigg)-b(y_1).D_y\Gamma(x_1,y_1)\\
			&\qquad+\a\mathcal{M}_{\lambda,\Lambda}^+(X-Y)-P+4L_2\|b\|_{\infty,B_1}\\
			&\leq\bigg(\a\mathcal{M}_{\lambda,\Lambda}^+(Y)+\b C(N,s)\int_{B_\delta}\frac{\big(-\Gamma(x_1,y_1+z)+\Gamma(x_1,y_1)+{\mathbbm{1}_{\{|z|\le 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\qquad+\b C(N,s)\int_{\mathbb{R}^N\setminus B_\delta}\frac{\big(u(y_1+z)-u(y_1)-{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\bigg)-b(y_1).D_y\Gamma(x_1,y_1)\\
			&\qquad+\a\mathcal{M}_{\lambda,\Lambda}^+(X-Y)-P+4L_2\|b\|_{\infty,B_1}.
		\end{align}
		Next, using the estimate \eqref{in eq 1} in above inequality yields
		\begin{align}
			\frac{{f}(x_1)}{\big|{p}+D_x\Gamma(x_1,y_1)\big|^{\gamma(x_1)}}&\leq \bigg(\a\mathcal{M}_{\lambda,\Lambda}^+(Y)+\b C(N,s)\int_{B_\delta}\frac{\big(\Gamma(x_1,y_1+z)-\Gamma(x_1,y_1)+{\mathbbm{1}_{\{|z|\le 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\\
			&\qquad+\b C(N,s)\int_{\mathbb{R}^N\setminus B_\delta}\frac{\big(u(y_1+z)-u(y_1)-{\mathbbm{1}_{\{|z|\leq 1\}}}z.D_y\Gamma(x_1,y_1)\big)}{|z|^{N+2s}}dz\bigg)-b(y_1).D_y\Gamma(x_1,y_1)\\ &\qquad+\alpha\Lambda(N-1)(4L_2+2i)+\alpha\lambda\left(-4(1+\theta)\theta L_12^{\theta-1}\phi_0+4L_2+2i\right)-P+4L_2\|b\|_{\infty,B_1}\\
			&\leq\frac{{f}(y_1)}{\big|{p}-D_y\Gamma(x_1,y_1)\big|^{\gamma(y_1)}}+\alpha\Lambda(N-1)(4L_2+2i)+\alpha\lambda\left(-4(1+\theta)\theta L_12^{\theta-1}\phi_0+4L_2+2i\right)\\
			&\qquad-P+4L_2\|b\|_{\infty,B_1}.
		\end{align}
		For some positive constant $C,$ we have
		\begin{align}
			-Ca^{({\inf_{ B_1}\gamma)}}\varepsilon&\leq -\frac{\varepsilon}{{\big|{p}+D_x\Gamma(x_1,y_1)\big|^{(\inf_{B_1}\gamma)}}}\\
			&\leq-\frac{\varepsilon}{{\big|{p}+D_x\Gamma(x_1,y_1)\big|^{\gamma(x_1)}}}\\
			&\leq -\frac{\|{f}\|_{\infty,B_1}}{{\big|{p}+D_x\Gamma(x_1,y_1)\big|^{\gamma(x_1)}}}\\
			&\leq \frac{{f}(x_1)}{\big|{p}+D_x\Gamma(x_1,y_1)\big|^{\gamma(x_1)}}\\
			&\leq \frac{{f}(y_1)}{\big|{p}-D_y\Gamma(x_1,y_1)\big|^{\gamma(y_1)}}\\
			&\quad+\alpha\Lambda(N-1)(4L_2+2i)+\alpha\lambda\left(-4(1+\theta)\theta L_12^{\theta-1}+4L_2+2i\right)-P+4L_2\|b\|_{\infty,B_1}\\
			&\leq \frac{\varepsilon}{\big|{p}-D_y\Gamma(x_1,y_1)\big|^{\gamma(y_1)}}\\
			&\quad+\alpha\Lambda(N-1)(4L_2+2i)+\alpha\lambda\left(-4(1+\theta)\theta L_12^{\theta-1}\phi_0+4L_2+2i\right)-P+4L_2\|b\|_{\infty,B_1}\\
			&\leq \frac{\varepsilon}{\big|{p}-D_y\Gamma(x_1,y_1)\big|^{(\inf_{B_1}\gamma)}}\\
			&\quad+\alpha\Lambda(N-1)(4L_2+2i)+\alpha\lambda\left(-4(1+\theta)\theta L_12^{\theta-1}\phi_0\phi_0+4L_2+2i\right)-P+4L_2\|b\|_{\infty,B_1}\\
			&\leq Ca^{(\inf_{B_1}\gamma)}\varepsilon+\alpha\Lambda(N-1)(4L_2+2i)+\alpha\lambda\left(-4(1+\theta)\theta L_12^{\theta-1}\phi_0+4L_2+2i\right)-P+4L_2\|b\|_{\infty,B_1}.
		\end{align}
		This further implies
		\begin{align}
			4\alpha(1+\theta)\theta\lambda L_12^{\theta-1}\phi_0&\leq 2Ca^{(\inf_{B_1}\gamma)}\varepsilon+\alpha\Lambda(N-1)\big(4L_2+2i\big)+\alpha\lambda\big(4L_2+2i\big)-P+4L_2\|b\|_{\infty,B_1}\\
			&=2Ca^{(\inf_{B_1}\gamma)}\varepsilon+\alpha\big(\Lambda(N-1)+\lambda\big)\big(4L_2+2i\big)-P+4L_2\|b\|_{\infty,B_1}\\
			&=2C_1{L_1}^{-(\inf_{B_1}\gamma)}\varepsilon+\alpha\big(\Lambda(N-1)+\lambda\big)\big(4L_2+2i\big)-P+4L_2\|b\|_{\infty,B_1},
		\end{align} 
		for some $C_1>0.$ Now, choosing $L_1$ large enough, depending only on $\lambda,\Lambda,N,$ $L_2$, and using the condition $\inf_{B_1}\gamma\geq 0,$ we get a contradiction. This implies that $u$ is Lipschitz continuous in $B_r,$ for $r\in(0,\frac{1}{2}).$
		\qed\\
		
		Next, we consider the case when $p\leq\frac{1}{a}.$
		\begin{lem}\label{lemma 4'}
			Let \ref{H1} and \ref{H2} be hold and $s\in\big(\frac{1}{2},1\big).$ Let $u$ be a viscosity solution of 
			\begin{align}
				|Du+p|^{\gamma(x)}\big(\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2u\big)-\b(-\Delta)^su\big)+b.Du|Du+p|^{\gamma(x)}=f \text{ in }B_1
			\end{align}
			with $\underset{B_1}{\text{osc}}~u\leq 1$ and $\|f\|_{\infty,B_1}\leq\varepsilon$ for some $\varepsilon>0.$ If $|p|\leq 
			\frac{1}{a},$ 
		
			then $u$ is Lipschitz continuous in $B_r$ with
			\begin{align}
				[u]_{1,B_r}\leq C=C(\lambda,\Lambda,N,\gamma,r,\|f\|_{\infty,B_1},\|u\|_{\infty,B_1}).
			\end{align}
		\end{lem} 
		\noindent \textbf{Proof of Lemma \ref{lemma 4'}.}
		The proof has exactly the similar lines of proof as in the previous lemma. In this case, we have $|p|\leq \frac{1}{a}=5cL_1$ (as fixed in Lemma \ref{lemma 4}). Taking $L_1$ large enough so that 
		\begin{align}
			|p+D_x\Gamma|,\,|p-D_y\Gamma|\geq 1.
		\end{align}
		The rest of the proof follows as before. For the sake of brevity, we omit the details.\qed
		
		\begin{lem}\label{lemma .5}
			For every $M>0,\,\varepsilon,\,\delta\in(0,1)$ and a modulus of continuity $\omega:[0,+\infty)\longrightarrow\mathbb{R}_+,$ there exist $k\in(0,1)$ and $s_0\in(\frac{1+\delta}{2},1)$ such that for each $p\in\mathbb{R}^N,$ $s\in (s_0,1)$ and a viscosity solution $u$ to 
			\begin{align}
				-k\leq |p+Du|^{\gamma(x)}\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2u\big)-\b\big(-\Delta\big)^su\big)+b.Du|p+Du|^{\gamma(x)}\leq k \text{ in }B_1,
			\end{align}
			satisfying\\
			(i) $|u(x)-u(y)|\leq \omega(|x-y|)$ for $x,\,y\in\overline{B}_1,$\\
			(ii) $|u(x)|\leq M(1+|x|^{1+\delta}),\,x\in\mathbb{R}^N,$\\
			and given any $\varepsilon>0$, there exists $\eta(\varepsilon)>0$ such that if
			\begin{align}
				\|k(1+|p|)^{-\gamma(x)}\|_{\infty,B_1}\leq \eta,
			\end{align}
			then
			\begin{align}
				\displaystyle{\sup_{B_\frac{1}{2}}}\,|u-h|\leq \varepsilon,
			\end{align}
			for a viscosity solution $h\in C^{1,\overline{\delta}}(B_{\frac{3}{4}})$ of
			\begin{align}\label{1.10}
				\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2u\big)+\b\Delta u\big)+b(x).Du=\mathcal{M}^+_{{\a \lambda+\b},{\a \Lambda+\b}}\big(D^2u\big)+b(x).Du=0 \text{ in }B_1
			\end{align}
		for some $\overline{\delta}\in (0,1)$ 
		\end{lem}
		
		\noindent \textbf{Proof of Lemma \ref{lemma .5}.}
		Let if possible, we assume that there exists some $M>0,\varepsilon,\delta\in(0,1)$ a modulus of continuity $\omega,$ and sequence $\{p_n\}$ in $\mathbb{R}^N,$ $k_n\longrightarrow 0,$ $s_n\longrightarrow 1$ with a sequence of functions $\{u_n\}$ satisfying (i) and (ii) listed above along with
		\begin{align}
			\|k_n(|1+|p_n|)^{-\gamma_n(x)}\|_{\infty,B_1}\leq \frac{1}{n},
		\end{align}
		where $0\leq \gamma_n\leq \sup \gamma.$ Moreover, $\{u_n\}$ be satisfy the following:
		\begin{align}\label{lim h'}
			-k_n\leq|p_n+Du_n|^{\gamma_n(x)}\big(\a\mathcal{M}^+_{{ \lambda},{ \Lambda}}\big(D^2u_n\big)-\b\big(-\Delta\big)^{s_n}u_n\big)+b(x).Du_n|Du_n+p_n|^{\gamma_n(x)}\leq k_n \text{ in }B_1, 
		\end{align}  
		in the viscosity sense such that
		\begin{align}\label{eq contra}
			\displaystyle{\sup_{B_\frac{1}{2}}}\,|u_n-h|> \varepsilon,
		\end{align}
		for any function $h\in C^{1,\alpha}(B_{\frac{3}{4}}).$
		
		
		Note that using (i), we have by Arzel\'a-Ascoli theorem that $u_n$ converges locally uniformly in $B_1$ up to a subsequence to a function, say $u_\infty.$ Also, (ii) assures the uniform integrability condition and we have by Vitali's theorem that one can pass the limit. We need to show that the limiting function $u_\infty$ is a solution of
		\begin{align}\label{eq lim h}
			\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2u\big)+\b\Delta u\big)+b(x).Du=\mathcal{M}^+_{{\a \lambda+\b},{\a \Lambda+\b}}\big(D^2u\big)+b(x).Du=0 \text{ in }B_1.
		\end{align}
		
		We closely follow the arguments of Bronzi et al. \cite{Bronzi}, who considered the fully nonlinear local operator with gradient degeneracy, $|\nabla u|^{\gamma(x)}F(D^2u).$ These arguments have been also exploited by Fang et al. \cite{Fang}. We give the complete details for the sake of clarity since in our case, we also have a nonlocal term $(-\Delta)^s$ and a first order term $b(x).Du$ present. We re-write \eqref{lim h'} as follows:
		\begin{align}\label{any n}
			-k_n\leq [1+|p_n|]^{\gamma_n(x)}\left|\frac{p_n}{1+|p_n|}+\frac{1}{1+|p_n|}Du_n\right|^{\gamma_n(x)}\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2u_n\big)-\b\big(-\Delta\big)^{s_n}u_n+b(x).Du_n\big)\leq k_n.
		\end{align}
		It is easy too see that
		\begin{align}
			\xi_n:=\frac{p_n}{1+|p_n|}\longrightarrow \xi,\,\,\,\,\, \text{ and }\,\,\,\,\, \mu_n:=\frac{1}{1+|p_n|}\longrightarrow \mu
		\end{align}
		up to a subsequence with $|\xi|\leq 1$ and $\mu\in [0,1].$ Let $A$ be a real symmetric matrix and $v$ be some constant vector. Consider
		\begin{align}
			P(x):=\frac{1}{2}\langle A(x-y),(x-y)\rangle+v.(x-y)+u_\infty(y)
		\end{align}
		be a quadratic polynomial which touches $u_\infty$ from below at some point $y$ in $B_{\frac{3}{4}}$. Without any loss of generality, let $|y|=u_\infty(y)=0.$ Define
		\begin{align}
			\widetilde{P}=\begin{cases}
				P &\text{ in }B_{\delta'}\\
				u_\infty &\text{ in }\mathbb{R}^N\setminus B_{\delta'},		
			\end{cases}
		\end{align}
		for small enough $\delta'>0.$ We only aim to show that $u_\infty$ is a supersolution to \eqref{eq lim h}. The subsolution part can be proved using analogous arguments. Fix $r\in (0,1)$, we define 
		\begin{align}
			P_j(x):=\frac{1}{2}\langle A(x-x_j),(x-x_j)\rangle+v.(x-x_j)+u_j(x_j)
		\end{align}
		with $x_j$ satisfying
		\begin{align}
			P_j(x_j)-u_j(x_j):=\displaystyle{\max_{x\in B_r}}(P_j(x)-u_j(x)).
		\end{align}
		This is possible since $u_n\longrightarrow u_\infty$ locally uniformly in $B_1.$ Consider
		\begin{align}
			\widetilde{P}_j=\begin{cases}
				P_j &\text{ in }B_{\delta'}(x_j)\\
				u_j &\text{ in }\mathbb{R}^N\setminus B_{\delta'}(x_j)		
			\end{cases}
		\end{align}
		We have for the choice $n=j$ in \eqref{any n},
		\begin{align}\label{n=j 12}
			(1+|p_j|)^{\gamma_j(x_j)}\left|\frac{p_j}{1+|p_j|}+\frac{1}{1+|p_j|}v\right|^{\gamma_j(x_j)}\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(A\big)-\b\big(-\Delta\big)^{s_j}\widetilde{P}_j(x_j)+b(x_j).DP_j(x_j)\big)\leq k_j.
		\end{align}
		It gives
		\begin{align}
			\left|\frac{p_j}{1+|p_j|}+\frac{1}{1+|p_j|}v\right|^{\gamma_j(x_j)}\big(\a\mathcal{M}^+_{\lambda,\Lambda}\big(A\big)-\b\big(-\Delta\big)^{s_j}\widetilde{P}_j(x_j)+b.DP_j(x_j)\big)\leq \frac{k_j}{[1+|p_j|]^{\gamma_j(x_j)}}.
		\end{align}
		Let 
		\begin{align}
			\left|\frac{p_j}{1+|p_j|}+\frac{1}{1+|p_j|}v\right|^{\gamma_j(x_j)}\longrightarrow \beta\in [0,\infty],
		\end{align}
		up to a subsequence. Consider the first case, i.e., $\beta>0,$ then taking the limit $j\longrightarrow\infty$ in \eqref{n=j 12}, we get
		\begin{align}
			\big(\mathcal{M}^+_{{\a \lambda+\b},{\a \Lambda+\b}}\big(A\big)+b(0).DP(0)\big)\leq 0.
		\end{align}
		This further yields
		\begin{align}\label{leq 0}
			\big(\mathcal{M}^+_{{\a \lambda+\b},{\a \Lambda+\b}}\big(A\big)+b.Du_\infty\big)\leq 0 \text{ in }B_{\frac{3}{4}}.
		\end{align}
		For the complementary case, $v=0,$ $|p_n|\longrightarrow 0.$ Consider $v_n=u_n+\langle p_n,x\rangle,$ we only need to analyze the case $v=0=p_n.$ Also, we consider only the case when $A$ is not negative semi-definite, since for $A\leq 0,$ we immediately have
		\eqref{leq 0} (since $v=0$). Thus, we only need to consider the case when the invariant space formed by eigenvectors associated with positive eigenvalues, say $E:=\,$span$\,(e_1,e_2,\dots e_n)\in S^{N-1}$ is non-empty. Consider an orthogonal sum, $\mathbb{R}^N=E\oplus G.$ Let $P_Ex$ be denote the orthogonal projection of $x$ on $E$. Define the test function:
		\begin{align}
			\Phi(x):=\frac{1}{2}\langle Ax,x\rangle+m\displaystyle{\sup_{e\in S^{N-1}}}\langle P_Ex,e\rangle,
		\end{align}  
		for $m>0$ small enough. Now, since we have that $u_j\longrightarrow u_\infty$ locally uniformly and $\frac{1}{2}\langle Ax,x\rangle$ touches $u_\infty$ at $0$ from below so we can choose $m$ small enough such that $\Phi$ touches $u_j$ from below at some point $x_j^m\in B_r.$  Moreover, if $x_j^m\in G,$ then
		\begin{align}
			\Phi(x):=\frac{1}{2}\langle Ax,x\rangle+m\langle P_Ex,e\rangle
		\end{align}
		touches $u_j$ at $x^m_j$ for any choice of $e.$ We define
		\begin{align}
			\widetilde{\Phi}:=\begin{cases}
				\Phi &\text{ in }B_{\delta'}(x_j^m)\\
				u_j &\text{ in }\mathbb{R}^N\setminus B_{\delta'}(x_j^m).	
			\end{cases}
		\end{align}
		Note that here $\delta'$ is a small number which we can choose accordingly smaller than the previous choice. It is easy to observe that
		\begin{align}
			D\Phi(x_j^m)=Ax_j^m+kP_Ee\,\text{ and }\,D^2\Phi(x_j^m)=A.
		\end{align}
		For the choice $n=j,$ we get
		\begin{align}
			k_j&\geq |D\Phi(x_j^m)|^{\gamma_j(x_j^m)}\big(\a\mathcal{M}^+_{{\lambda},{ \Lambda}}\big(D^2\Phi(x_j^m)\big)-\b\big(-\Delta\big)^{s_j}\widetilde{\Phi}(x_j^m)\big)+b.D\Phi(x_j^m)|D\Phi(x_j^m)|^{\gamma_j(x_j^m)}\\
			&=|Ax_j^m+me|^{\gamma_j(x_j^m)}\big(\a\mathcal{M}^+_{{ \lambda},{ \Lambda}}\big(D^2\Phi(x_j^m)\big)-\b\big(-\Delta\big)^{s_j}\widetilde{\Phi}(x_j^m)+b.D\Phi(x_j^m)\big).
		
		\end{align}
		Next, following the arguments of \cite{Bronzi} and taking supremum over all $e\in S^{N-1}$ followed by letting $j\longrightarrow\infty$ gives
		\begin{align}
			\mathcal{M}^+_{{ \a\lambda+\beta},{ \a\Lambda+\beta}}\big(D^2\Phi(x_j^m)\big)+b.D\Phi(x_j^m)\leq 0.	
		\end{align}  
		Finally, consider the case when $P_E(x_j^m)\neq 0.$ Consider
		\begin{align}
			\rho_j^m=\frac{P_E(x_j^m)}{|P_E(x_j^m)|},
		\end{align} 
		where $|P_E(x_j^m)|=\displaystyle{\sup_{e\in S^{N-1}}\langle P_E(x_j^m),e\rangle}$ denotes the Euclidean norm of $P_E(x_j^m).$ It is easy to see that $|P_E(x)|$ is a smooth function with
		\begin{align}
			D(|P_E(x)|)_{x={x_j^m}}=\rho_j^m \,\,\,\,\,\,\,\,\, D^2(|P_E(x)|)_{x={x_j^m}}=I-\rho_j^m\otimes\rho_j^m.
		\end{align}
		Hence, we have for the choice $n=j$, following the arguments of \cite{Bronzi} that
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
		\begin{align}\label{kId}
			\big(\a\mathcal{M}^+_{{ \lambda},{ \Lambda}}(A+m(I-\rho_j^m\otimes \rho_j^m))-\b\big(-\Delta\big)^{s_j}\widetilde{\Phi}(x_j^m)\big)+b(x_j^m).D\Phi(x_j^m)\leq k_j|Ax_j^m+m\rho_j^m|^{-\gamma_j(x_j^m)}.
		\end{align}
		Now, using the fact that $m(I-\rho_j^k\otimes \rho_j^k)$ is positive semi-definite, we get
		\begin{align}
			\mathcal{M}_{\lambda,\Lambda}^+(A+m(I-\rho_j^m\otimes \rho_j^m))\geq \mathcal{M}_{\lambda,\Lambda}^+(A).
		\end{align}
		Using this in \eqref{kId} yields
		\begin{align}
			\big(\a\mathcal{M}^+_{{ \lambda},{ \Lambda}}(A)-\b\big(-\Delta\big)^{s_j}\widetilde{\Phi}(x_j^m)\big)+b(x_j^m).D\Phi(x_j^m)\leq k_j|Ax_j^m+m\rho_j^m|^{-\gamma_j(x_j^m)}.
		\end{align}
		Finally, letting $j\longrightarrow\infty,$ concludes
		\begin{align}
			\mathcal{M}^+_{{ \a\lambda+\beta},{ \a\Lambda+\beta}}(A)+b(0).D\Phi(0)\leq 0.	
		\end{align} 
		Hence, $u_\infty$ is a viscosity subsolution of
		\begin{align}
			\mathcal{M}^+_{{ \a\lambda+\beta},{ \a\Lambda+\beta}}(D^2u)+b.Du=0.	
		\end{align} 
		Similarly, one may check the subsolution part. Hence, we have that $u_\infty$ is a viscosity solution of \eqref{eq lim h}. By Remark 3.5 \cite{C1beta}, we have that $u_\infty\in C^{1,\overline{\delta}}(B_{\frac{3}{4}})$ for some $0<\overline{\delta}<1.$ Finally, taking $h=u_\infty,$ we get a contradiction to \eqref{eq contra} for all $n$ large enough. This proves the claim.\qed
		
	
	
	
		
		
	
	
		
		
	
		
		\begin{lem}\label{Lemma .6}
			Let \ref{H1} and \ref{H2} be hold. There exist $\tau,\delta,k\in (0,1),$ $s_0\in\big(\frac{1}{2},1\big)$ such that for any bounded viscosity solution $u$ of
			\begin{align}\label{in B1}
				-k\leq|Du|^{\gamma(x)}\left(\a\mathcal{M}_{\lambda,\Lambda}^+\big(D^2u\big)-\b(-\Delta)^su\right)+b.Du|Du|^{\gamma(x)}\leq k \text{ in }B_1,
			\end{align} 
			for $s\in (s_0,1)$ with $\|u\|_{\infty,\mathbb{R}^N}\leq 1,$ there exists a sequence of affine functions $t_n=a_n+p_n.x$ with $\{a_n\}\subset\mathbb{R}$ and $\{p_n\}\subset\mathbb{R}^N$ such that
			
			\begin{align}\label{.2}
				\begin{cases}
					\displaystyle{\sup_{B_{\tau^n}}}\,|u-t_n|&\leq \tau^{n(1+\delta)}\\
					|a_{n+1}-a_n|&\leq C\tau^{(1+\delta)n}\\
					|p_{n+1}-p_n|&\leq C\tau^{\delta n},
				\end{cases}
			\end{align} 
			for some positive constant $C.$ 
		\end{lem}
		\noindent \textbf{Proof of Lemma \ref{Lemma .6}.}
		Let $h\in C(B_1)$ (see Lemma \ref{lemma .5} for such a function) be viscosity solution
		\begin{align}
			\mathcal{M}^+_{\alpha\lambda+\beta,\alpha\Lambda+\beta}\big(D^2u\big)+b(x).Du=0,	
		\end{align} 
		with $\|h\|_{\infty,B_1}\leq C'$ for some $C'> 0.$ We set
		\begin{align}\label{t_1}
			t_1=h(0)+Dh(0).x.	
		\end{align} 
		Let us consider universal constants $\overline{A},\,\overline{\delta}\in (0,1)$ such that 
		\begin{align}
			|h(0)|,|Dh(0)|,\|h\|_{C^{1,\overline{\delta}}(B_\tau)}\leq \overline{A}\,\,\,\,\,\, \text{ and }\,\,\,\,\,\, \displaystyle{\sup_{B_\tau}}|h(x)-h(0)-Dh(0).x|\leq \overline{A}|x|^{1+\overline{\delta}},  
		\end{align}
		for $B_r\Subset B_{\frac{3}{4}}.$ Moreover, since $\|u-h\|_{\infty,B_{\frac{1}{2}}}\leq \varepsilon$, so we have \begin{align}
			\displaystyle{\sup_{B_r}}\,|u(x)-h(0)-b.Dh(0)|\leq \varepsilon+\overline{A}|x|^{1+\overline{\delta}}.
		\end{align}
		Fix $\delta\in (0,\overline{\delta})\cap\left(0,\frac{1}{1+\|\gamma^+\|_{\infty}+\|\gamma^-\|_{\infty}}\right)$ such that $1-\delta(1+\sup_{B_1}\gamma)>0,$ and $0<\tau<1$ small enough such that
		\begin{align}\label{A bar}
			\tau^\delta\leq \frac{1}{4},\, \tau^{\overline\delta-{\delta}}(1+\overline{A})\leq \varepsilon_1\,\,\, \text{ and }\,\, (1+2\overline{A})\tau^{1-\delta(1+\|\gamma^+\|_\infty+\|\gamma^-\|_\infty)}\leq 1
		\end{align}
		for a small number $\varepsilon_1>0.$ We take $k$ small enough and $s_0$ close enough to $1$ such that for any solution $u$ of $\eqref{.2}$ satisfying the condition (ii) of Lemma \ref{lemma .5}, we have the existence of a function $h,$ which solves \eqref{1.10} such that $\|u-h\|_{\infty,B_\frac{1}{2}}\leq\varepsilon.$ Recall that the condition (i) of Lemma \ref{lemma .5} follows by Theorem \ref{Main 1}.
		Also, set 
		\begin{align}
			\varepsilon:=\frac{\tau^{1+\delta}}{2}.
		\end{align}
		We prove the result by induction. Consider the auxiliary function:
		\begin{align}\label{w u}
			w_{n+1}(x):=\frac{u(\tau^nx)-t_n(\tau^nx)}{\tau^{n(1+\delta)}},
		\end{align}
		with $t_0=0$ and $t_n(x)=a_n+p_n.x$ to be defined later. It can be re-written as
		\begin{align}\label{u w}
			u(\tau^nx)=\tau^{n(1+\delta)}w_{n+1}(x)+t_n(\tau^nx).
		\end{align}
		It is easy to see that
		\begin{align}
			\tau^nDu(\tau^nx)=\tau^{n(1+\delta)}Dw_{n+1}(x)+\tau^np_n,
		\end{align}
		i.e.,
		\begin{align}\label{grad u w}
			Du(\tau^nx)=\tau^{n\delta}Dw_{n+1}(x)+p_n.
		\end{align}
		Also,
		\begin{align}
			D^2u(\tau^nx)=\tau^{n(\delta-1)}D^2w_{n+1}(x).
		\end{align}
		This gives
		\begin{align}\label{pcci u w}
			\mathcal{M}_{\lambda,\Lambda}^+(D^2u(\tau^nx))=\mathcal{M}_{\lambda,\Lambda}^+\big(\tau^{n(\delta-1)}D^2w_{n+1}(x)\big).
		\end{align}
		On the other hand, since
		\begin{align}
			-(-\Delta)^su(\tau^nx)=\frac{C(N,s)}{2}\int_{\mathbb{R}^N}\frac{u(\tau^nx+y')+u(\tau^nx-y')-2u(\tau^nx)}{|y'|^{N+2s}}dy',
		\end{align}
		so substituting $y'=\tau^ny$ yields
		\begin{align}\label{u taun}
			-(-\Delta)^su(\tau^nx)&=\frac{C(N,s)}{2}\int_{\mathbb{R}^N}\frac{u(\tau^nx+\tau^ny)+u(\tau^nx-\tau^ny)-2u(\tau^nx)}{|\tau^ny|^{N+2s}}\tau^{nN}dy\\
			&=\frac{C(N,s)}{2}\tau^{-2ns}\int_{\mathbb{R}^N}\frac{u(\tau^n(x+y))+u(\tau^n(x-y))-2u(\tau^nx)}{|y|^{N+2s}}dy.
		\end{align}
		Using \eqref{u w} in \eqref{u taun} infers
		\begin{align}\label{3.23}
			-(-\Delta)^su(\tau^nx)&=\frac{C(N,s)}{2}\tau^{n(1+\delta)}\tau^{-2ns}\int_{\mathbb{R}^N}\frac{w_{n+1}(x+y)+w_{n+1}(x-y)-2w_{n+1}(x)}{|y|^{N+2s}}dy\\
			&=-\tau^{n(1+\delta-2s)}(-\Delta)^sw_{n+1}(x).
		\end{align}
		Using \eqref{grad u w}, \eqref{pcci u w} and \eqref{3.23} in \eqref{in B1} yields
		\begin{align}
			-k\leq \big|\tau^{n\delta}Dw_{n+1}(x)+p_n\big|^{\gamma_n(x)}\bigg(&\alpha\mathcal{M}_{\lambda,\Lambda}^+\big(\tau^{n(\delta-1)}D^2w_{n+1}(x)\big)-\tau^{n(1+\delta-2s)}\beta(-\Delta)^sw_{n+1}(x)\bigg)\\
			&\qquad+b(\tau^nx).\big(\tau^{n\delta}Dw_{n+1}(x)+p_n\big)\big|\tau^{n\delta}Dw_{n+1}(x)+p_n\big|^{\gamma_n(x)}\leq k \text{ in }B_{\tau^{-n}},
		\end{align}
		where $\gamma_n(x)=\gamma{(\tau^n(x))}.$ In other words,
		{\begin{align}\label{w solves}
				\quad-k\leq
				\tau^{n\delta\gamma_n(x)}\big|Dw_{n+1}(x)+\tau^{-n\delta}p_n\big|^{\gamma_n(x)}\bigg(&{\alpha\mathcal{M}}_{\lambda,\Lambda}^+\big(\tau^{n(\delta-1)}D^2w_{n+1}(x)\big)-\tau^{n(1+\delta-2s)}\beta(-\Delta)^sw_{n+1}(x)\\
				&\qquad\qquad\qquad\quad+b(\tau^nx).\big(\tau^{n\delta}Dw_{n+1}(x)+p_n\big)\bigg)\leq k \text{ in }B_{\tau^{-n}}.
			\end{align}
			It deduces that $w_{n+1}$ given by \eqref{w u} solves \eqref{w solves} in $B_{\tau^{-n}}.$} Now, multiplying $\tau^{n(1-\delta-\delta\gamma_n(x))}$ in both sides of \eqref{w solves}, we get 
		\begin{align}
			-\tau^{n(1-\delta-\delta\gamma_n(x))}k&\leq \big|Dw_{n+1}(x)+\tau^{-n\delta}p_n\big|^{\gamma_n(x)}\bigg({\tau^{n(1-\delta)}\alpha\mathcal{M}}_{\lambda,\Lambda}^+\big(\tau^{n(\delta-1)}D^2w_{n+1}(x)\big)\\
			&\qquad\qquad\qquad\qquad\qquad-\tau^{n(1-\delta)}\tau^{n(1+\delta-2s)}\beta(-\Delta)^sw_{n+1}(x)+\tau^{n(1-\delta)}b(\tau^nx).\big(\tau^{n\delta}Dw_{n+1}(x)+p_n\big)\bigg)\\
			&=\big|Dw_{n+1}(x)+\tau^{-n\delta}p_n\big|^{\gamma_n(x)}\bigg({\tau^{n(1-\delta)}\tau^{n(\delta-1)}\alpha\mathcal{M}}_{\lambda,\Lambda}^+\big(D^2w_{n+1}(x)\big)\\
			&\qquad\qquad\qquad\qquad\qquad-\tau^{n(2-2s)}\beta(-\Delta)^sw_{n+1}(x) +\tau^{n(1-\delta)}b(\tau^nx).\big(\tau^{n\delta}Dw_{n+1}(x)+p_n\big)\bigg)\\
			&=\big|Dw_{n+1}(x)+\tau^{-n\delta}p_n\big|^{\gamma_n(x)}\bigg({\alpha\mathcal{M}}_{\lambda,\Lambda}^+\big(D^2w_{n+1}(x)\big)\\
			&\qquad\qquad\qquad\qquad\qquad-\tau^{n(2-2s)}\beta(-\Delta)^sw_{n+1}(x)+\tau^{n(1-\delta)}b(\tau^nx).\big(\tau^{n\delta}Dw_{n+1}(x)+p_n\big)\bigg)\\
			&\leq \tau^{n(1-\delta-\delta\gamma_n(x))}k, \text{ for } x\in B_{\tau^{-n}}.
		\end{align}
		This further gives
		\begin{align}\label{More w}
			\\\quad-k	&\leq -\tau^{(1-\delta(1+\sup_{ B_{1}}\gamma(x)))}k\\
			&=-\tau^{(1-\delta(1+\sup_{ B_{\tau^{-n}}}\gamma_n(x)))}k\\
			&\leq -\tau^{(1-\delta(1+\gamma_n(x)))}k\\&\leq \big|Dw_{n+1}(x)+\tau^{-n\delta}p_n\big|^{\gamma_n(x)}\bigg(\alpha{\mathcal{M}}_{\lambda,\Lambda}^+\big(D^2w_{n+1}(x)\big)-\tau^{n(2-2s)}\beta(-\Delta)^sw_{n+1}(x)+\tau^{n(1-\delta)}b(\tau^nx).\big(\tau^{n\delta}Dw_{n+1}(x)+p_n\big)\bigg)\\
			&\leq \tau^{(1-\delta(1+\gamma_n(x)))}k\\
			&\leq \tau^{(1-\delta(1+\sup_{ B_{\tau^{-n}}}\gamma_n(x)))}k\\
			&=\tau^{(1-\delta(1+\sup_{ B_1}\gamma(x)))}k\\
			&\leq k\,\,\,\,\,\, \text{ for } x\in B_{\tau^{-n}}.
		\end{align}
		It implies that $w_{n+1}$ satisfies the inequality \eqref{More w} in the viscosity sense. Next, our aim is to define $t_n$ in such a way that $w_{n+1}$ satisfies the conditions of Lemma \ref{lemma .5}. This would give us the existence of a viscosity solution of \eqref{1.10}, say $\widetilde{h}\in C^{1,\overline{\delta}}(B_{\frac{3}{4}})$ such that $\sup_{B_{\frac{1}{2}}}|w_{n+1}-\widetilde{h}|\leq \frac{\tau^{1+\delta}}{2}.$ We follow the arguments of \cite{Topp}. Similar arguments have also been used, for instance, see \cite{Bronzi, Filippis, Huaroto, Jesus}. We define 
		\begin{align}
			\widetilde{t}=\widetilde{h}(0)+D\widetilde{h}(0).x
		\end{align} 
		and 
	
	
	
	
		\begin{align}\label{eq t tau}
			{t}_{n+1}(x)=t_n(x)+\tau^{n(1+\delta)}\widetilde{t}(\tau^{-n}x).
		\end{align} 
		In other words, ${a}_{n+1}=a_n+\tau^{n(1+\delta)}\widetilde{h}(0)$ and ${p}_{n+1}=p_n+\tau^{n\delta}D\widetilde{h}(0).x$. 
		Taking $w_0=u$ and $t_0=0,$ we have the sequences $\{w_n\}$ (defined by \eqref{w u}) and $\{t_n\}$(defined by \eqref{eq t tau}). Let the conditions of Lemma \eqref{lemma .5}, in particular, (ii) holds, i.e.,
		\begin{align}\label{grth}
			|u(x)|\leq (1+|x|^{1+\overline{\delta}}),\,x\in\mathbb{R}^N,
		\end{align}  
		 for $\{w_i\}$ for $i=1,2,\dots n.$ We prove the same for $i=n+1.$ We first re-write $w_{n+1}$ from \eqref{w u} in terms of $\widetilde{t}$ and $w_n.$
		\begin{align}\label{w ow}
			w_{n+1}(x)&=\frac{u(\tau^nx)-t_n(\tau^nx)}{\tau^{n(1+\delta)}}\\
			&=\frac{u(\tau^nx)-\big(t_{n-1}(\tau^nx)+{\tau^{(n-1)(1+\delta)}}\widetilde{t}{(\tau^{-(n-1)}\tau^nx)}\big)}{\tau^{n(1+\delta)}}\\
		
		
			&=\frac{u(\tau^nx)-\big(t_{n-1}(\tau^nx)+{\tau^{{(n-1)}(1+\delta)}}\widetilde{t}{(\tau x)}\big)}{\tau^{n(1+\delta)}}\\
		
		
			&=\frac{\frac{u(\tau^{n-1}\tau x)-t_{n-1}(\tau^{n-1}\tau x)}{\tau^{(n-1)(1+\delta)}}-\widetilde{t}\big(\tau x\big)}{\tau^{(1+\delta)}}\\
			&=\frac{w_n(\tau x)-\widetilde{t}{(\tau x)}}{\tau^{1+\delta}}.
		\end{align}
		Now, if $\tau x\in \overline{B^c}_{\frac{1}{2}},$ then by \eqref{w ow}, we have
		\begin{align}
			|w_{n+1}(x)|&\leq \frac{\big|w_n(\tau x)\big|+\big|\widetilde{t}{(\tau x)}\big|}{\tau^{1+\delta}}\\
			&=\frac{\big(1+|\tau x|^{1+\overline{\delta}}\big)+\overline{A}(1+\tau|x|)}{\tau^{1+\delta}}\\
			&\leq\frac{(2|x|\tau+|\tau x|^{1+\overline{\delta}})+\overline{A}(3\tau|x|)}{\tau^{1+\delta}}\\
			&\leq \frac{(2^{1+\overline{\delta}}|x|^{1+\overline{\delta}}\tau^{1+\overline{\delta}}+\tau^{1+\overline{\delta}}| x|^{1+\overline{\delta}})+\overline{A}(3^{1+\overline{\delta}}\tau^{1+\overline{\delta}}|x|^{1+\overline{\delta}})}{\tau^{1+\delta}}\\
			&\leq \frac{5|x|^{1+\overline{\delta}}\tau^{1+\overline{\delta}}+9\overline{A}|x|^{1+\overline{\delta}}\tau^{1+\overline{\delta}}}{\tau^{1+\delta}}\\
			&\leq \big(5+9\overline{A}\big)\tau^{\overline{\delta}-\delta}|x|^{1+\overline{\delta}}.
		\end{align}
		Further, using \eqref{A bar}, we get
		$|w_{k+1}|\leq |x|^{1+\overline{\delta}}.$
		In the case when $\tau x\in B_{\frac{1}{2}},$ we have
		\begin{align}
			|w_{n+1}(x)|&\leq \frac{\big|w_n(\tau x)\big|+\big|\widetilde{t}{(\tau x)}\big|}{\tau^{1+\delta}}\\
			&\leq \frac{\big|w_n(\tau x)-\widetilde{h}(\tau x)\big|+\big|\widetilde{h}(\tau x)-\widetilde{t}{(\tau x)}\big|}{\tau^{1+\delta}}\\
			&\leq \frac{\frac{\tau^{1+\delta}}{2}+\overline{A}|\tau x|^{1+\overline{\delta}}}{\tau^{1+\delta}}\text{ (using Lemma \ref{lemma .5})}\\
			&\leq \frac{1}{2}+\overline{A}|\tau |^{\overline{\delta}-\delta}|x|^{1+\overline{\delta}}\\
			&\leq 1+|x|^{1+\overline{\delta}} \text{ (using \eqref{A bar})}.
		\end{align}
		Thus, we have that $w_{n+1}$ satisfies \eqref{grth} for each $n\in\mathbb{N}.$ Moreover, notice that
		\begin{align}
			|p_n|&\leq|p_1|+\sum_{i=2}^n|p_i-p_{i-1}|\\
			&\leq |p_1|+\sum_{i=2}^n Cr^{(i-1)\delta}\\
			&\leq C+\frac{C\tau^\gamma}{1-\tau^\gamma} \text{ (using \ref{t_1})}\\
			&\leq \frac{4}{3}C.
		\end{align}
		We choose $C=\overline{A}.$ Thanks to \eqref{A bar} and (pp. 21 \cite{Bronzi}), we have
		\begin{align}
			\big\|k(1+|r^{-n\delta}p_n|)^{-\gamma_n(x)}\big\|_{\infty}\leq k.
		\end{align}
		Taking $k$ small enough, we have by Lemma \ref{lemma .5} that there exists a function $\widetilde{h}$ for each $w_{n+1}$ such that $\widetilde{h}$ solves \eqref{1.10} and $\sup_{B_{\tau}}|w_{n+1}-\widetilde{h}|\leq \frac{\tau^{(1+\delta)}}{2}.$ Also, we see that
		\begin{align}\label{w tilde t}
			\sup_{x\in B_\tau}|w_{n+1}-\overline{t}|&\leq \sup_{x\in B_\tau}|w_{n+1}-\widetilde{h}|+\sup_{x\in B_\tau}|\widetilde{t}-\widetilde{h}|\\
			&\leq \frac{\tau^{(1+\delta)}}{2}+Cr^{1+\overline{\delta}}\\
			&\leq \tau^{(1+\delta)},
		\end{align}
		using $\eqref{A bar}$ for $C=\overline{A}.$
		Further, we notice that 
		\begin{align}
			\sup_{B_{\tau^{n+1}}}|u(x)-t_{n+1}|&=\sup_{B_{\tau^{n+1}}}|\tau^{n(1+\delta)}w_{n+1}(\tau^{-n}x)+t_n(x)-t_{n+1}(x)|\text{ (using \eqref{w u})}\\
			&=\sup_{B_{\tau}}|\tau^{n(1+\delta)}w_{n+1}(x)+t_n(\tau^n x)-t^{n+1}(\tau^nx)|\\
			&=\sup_{B_{\tau}}|\tau^{n(1+\delta)}w_{n+1}(x)-\tau^{n(1+\delta)}\widetilde{t}(x)|\text{ (using \eqref{eq t tau})}\\
			&=\tau^{n(1+\delta)}\sup_{B_{\tau}}|w_{n+1}(x)-\widetilde{t}(x)|\\
			&\leq \tau^{(n+1)(1+\delta)},
		\end{align}
		where in the last line, we used \eqref{w tilde t}. Also, we see that 
		\begin{align}
			|a_{n+1}-a_n|\leq \tau^{n(1+\delta)}|\widetilde{h}(0)|\leq C\tau^{n(1+\delta)} \text{ and }|b_{n+1}-b_n|\leq \tau^{n\delta}|D\widetilde{h}(0)|\leq C\tau^{n\delta}.
		\end{align}
		Hence the claim follows.\qed\\
		
		\noindent \textbf{Proof of Theorem \ref{main 2}.} 
		Consider a function
		\begin{align}\label{u tilde u}
			\widetilde{u}:=Ku=\frac{u}{\|u\|_{\infty,B_1}+\left(\nu^{-1}\|f\|_{\infty,B_1}^{\frac{2s-1}{1+\inf_{B_1}\gamma}}\right)},
		\end{align}
		where $K$ is a positive constant and $\nu\in(0,1)$ to be chosen later. Now, if $K\geq 1,$ we immediately have
		\begin{align}
			\|u\|_{\infty,B_1}\leq 1.
		\end{align}
		and
		\begin{align}
			\nu^{-1}\|f\|^{\frac{2s-1}{1+\inf_{B_1}\gamma}}_{\infty,B_1}\leq 1.
		\end{align}
		This further gives
		\begin{align}
			\|f\|_{\infty,B_1}\leq \nu<1.
		\end{align}
		It implies that one can use Lemma \ref{Lemma .6} for $u.$ Hence, we may assume that $K\leq 1.$ By \eqref{eq 0.1}, we have
		\begin{align}
			\|f\|_{\infty,B_1}\geq \frac{|D\widetilde{u}|^{\gamma(x)}}{\left(\|u\|_{\infty,B_1}+\left(\nu^{-1}\|f\|^{\frac{2s-1}{1+\inf_{B_1}\gamma}}_{\infty,B_1}\right)\right)^{-(1+\gamma(x))}}\left(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2\widetilde{u}\big)-\b\big(-\Delta\big)^s\widetilde{u}+b(x).D\widetilde{u}\right)\geq -\|f\|_{\infty,B_1}.
		\end{align}
		Note that for $\nu\in(0,1)$ small enough, we get $K\ll1.$
	
	
	
	
		It further implies
		\begin{align}
			1\gg K^{1+\inf_{B_1}\gamma(x)}\|f\|_{\infty,B_1} 
			&\geq K^{1+\gamma(x)}\|f\|_{\infty,B_1}\\
			&\geq {|D\widetilde{u}|^{\gamma(x)}}\left(\a\mathcal{M}^+_{\lambda,\Lambda}\big(D^2\widetilde{u}\big)-\b\big(-\Delta\big)^s\widetilde{u}\right)+b(x).D\widetilde{u}|D\widetilde{u}|^{\gamma(x)}\\
			&\geq -K^{1+\gamma(x)}\|f\|_{\infty,B_1}\\
			&\geq -K^{1+\inf_{B_1}\gamma(x)}\|f\|_{\infty,B_1}\\
			&\gg -1.	
		\end{align}
		Hence, we have that $\widetilde{u}$ satisfies the conditions of Lemma \ref{Lemma .6}. It is immediate to see from \eqref{.2} that $\{a_n\}$ and $\{p_n\}$ are Cauchy sequences and hence convergent. Let $a^*$ and $p^*$ be the limits of $\{a_n\}$ and $\{p_n\},$ respectively. Also, $t^*=a^*+b^*.x.$ It is immediate that
		\begin{align}
			|t^*-t_n|&\leq \sum_{i=n}^\infty|t_{i+1}-t_i|\\
			&\leq \sum_{i=n}^\infty|a_{i+1}-a_i|+\sum_{i=n}^\infty|p_{i+1}-p_i||x|\\
			&\leq \sum_{i=n}^\infty C\tau^{i(1+\delta)}+\sum_{i=n}^\infty C\tau^{i\delta}|x|.
		\end{align}
		Now, letting $\tau^{n+1}\leq |x|<\tau^{n}$ for some $n\in\mathbb{N}$ gives
		\begin{align}
			|\widetilde{u}-t^*|&\leq |\widetilde{u}-t_n|+|t^*-t_n|\\
			&\leq C_0\tau^{n(1+\delta)}+\sum_{i=n}^\infty C\tau^{i(1+\delta)}+\sum_{i=n}^\infty C\tau^{i\delta}|x|\text{ (using \eqref{.2})}\\
			&\leq C_0\tau^{n(1+\delta)}+\sum_{i=n}^\infty C\tau^{i(1+\delta)}+\sum_{i=n}^\infty C\tau^{i(1+\delta)}\\
			&\leq C_0\tau^{n(1+\delta)}+ C_1\tau^{n(1+\delta)}+ C_1\tau^{n(1+\delta)}\\
			&= C_2\tau^{n(1+\delta)}\\
			&=\frac{C_2}{\tau^{1+\delta}}\tau^{(n+1)(1+\delta)}\\
			&\leq {C^*}|x|^{1+\delta},
		\end{align}
		where $C_1,$ $C_2\,(=2C_1+C_0)$ and $C^*=\frac{C_2}{\tau^{1+\delta}}$ are positive constants. This implies
		\begin{align}
			[\widetilde{u}]_{C^{1,\delta}(B_{\frac{1}{2}})}\leq C^*.
		\end{align}
		This together with \eqref{u tilde u} concludes
		\begin{align}
			[\widetilde{u}]_{C^{1,\delta}(B_{\frac{1}{2}})}&\leq C^*\left(\|u\|_{\infty,B_1}+\left(\nu^{-1}\|f\|_\infty^{\frac{2s-1}{1+\inf_{B_1}\gamma}}\right)\right)\\
			&\leq \widetilde{C}\left(\|u\|_{\infty,B_1}+\left(\|f\|_\infty^{\frac{2s-1}{1+\inf_{B_1}\gamma}}\right)\right),
		\end{align} 
		where $\widetilde{C}$ is a positive constant depending on $\nu$ appearing in \eqref{u tilde u}.
		\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.
		
		
		