\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 120, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/120\hfil Lorentz estimates]
{Lorentz estimates for asymptotically regular fully nonlinear
elliptic equations}

\author[Y. Wang, J. Zhang, S. Zheng \hfil EJDE-2017/120\hfilneg]
{Yongyong Wang, Junjie Zhang, Shenzhou Zheng}

\address{Yongyong Wang \newline
Department of Mathematics,
Beijing Jiaotong University,
Beijing 100044, China}
\email{yongyongwang@bjtu.edu.cn}

\address{Junjie Zhang \newline
Department of Mathematics,
 Beijing Jiaotong University, Beijing 100044,  China}
\email{junjiezhang@bjtu.edu.cn}

\address{Shenzhou Zheng (Corresponding author) \newline
Department of Mathematics,
Beijing Jiaotong University,
Beijing 100044,  China}
\email{shzhzheng@bjtu.edu.cn}

\dedicatory{Communicated by Zhasheng Feng}

\thanks{Submitted October 25, 2016. Published May 4, 2017.}
\subjclass[2010]{35J60, 35B65, 35D30}
\keywords{Nonlinear elliptic equations; Lorentz estimate; asymptotically regular; 
\hfill\break\indent small BMO coefficients}


\begin{abstract}
 We prove a global Lorentz estimate of the Hessian of strong solutions
 to a class of asymptotically regular fully nonlinear elliptic equations
 over a $C^{1,1}$ smooth bounded domain. Here, the approach of the main
 proof is based on the Possion's transform from an asymptotically regular
 elliptic equation to the regular one.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Let $\Omega$ be a bounded domain with $\partial\Omega\in C^{1,1}$  in
 $\mathbb{R}^n$  for $n\geq2$. The main purpose of this paper is to attain 
a global Calder\'on-Zygmund type estimate  in the scale of Lorentz spaces
for  the Hessian of strong solutions to the Dirichlet problem of asymptotical 
regular fully nonlinear elliptic equations of nondivergence form. 
The studied problem is
\begin{equation}\label{Dirichlet prob}
\begin{gathered}
F(x,D^2u)=f  \quad \text{in } \Omega,\\
u=0 \quad \text{on } \partial \Omega,
\end{gathered}
\end{equation}
where the real valued function 
$F(x,D^2u):\Omega\times \mathcal{S}(n)\to \mathbb{R}$ is an asymptotically 
regular elliptic operator which $\mathcal{S}(n)$ is the space of real 
$n\times n$ symmetric matrices, and $f$ is any given function in Lorentz spaces 
$L^{\gamma,q}(\Omega)$ with $\gamma>n$ and $0<q\leq\infty$.


The Calder\'on-Zygmund estimate is a popular research to various
 elliptic and parabolic problems in recent decades. In the settings of 
discontinuous coefficients, an interior and boundary $W^{2,p}$ estimate 
for linear elliptic equations with VMO (Vanishing Mean Oscillations) 
discontinuous coefficients was first proved by  Chiarenza, Frasca,
 and Longo \cite{ChFL1,ChFL2}. Since then, there have been a lot
 of research activities on the Calder\'on-Zygmund theory of elliptic
 and parabolic equations and systems in divergence or non-divergence form. 
Regarding fully nonlinear elliptic equations \eqref{Dirichlet prob}, 
an interior $W^{2,p}$ estimate was first obtained by Caffarelli in \cite{Caf} 
if $f\in L^{p}$ with $p>n$ under the assumption of a small measure to the 
oscillation of $F(x,M)$ in the variable $x$ uniformly for $M$. 
Later, Caffarelli and Huang \cite{CaH} further showed that under the
 assumptions of $F(x, M)$ with a small multiplier of BMO in $x$ and the 
prerequisite of Evans-Krylov estimates \cite{Eva}, if $f$ belongs to the 
generalized Campamato-John-Nirenberg spaces, then $D^2u$ correspondingly 
belongs to the same spaces as $f$. Recently, Winter \cite{Win} also used 
a similar technique to establish the corresponding boundary estimate so as 
to get a global $W^{2,p}$-solvability of the associated boundary-value problem. 
All these papers showed that a small oscillation assumption in the $L^\infty$ 
or $L^n$ integral average sense is imposed on the operators $F(x,M)$ in $x$. 
Recently, Krylov et al \cite{DoKL,Kry1, Kry2}  developed $W^{2,p}$-solvability 
for fully nonlinear elliptic and parabolic equations with VMO ``coefficients" 
whose local oscillations are measured in a certain average sense allowing 
rather rough discontinuity. After that, Dong-Krylov-Li \cite{DoKL} demonstrated  
an interior solvability in $W^{2,p} $ for $p>n$ and $W^{2,1}_{p}$ for $p>n+1$, 
respectively, to fully nonlinear elliptic and parabolic equations with 
VMO ``coefficients" in bounded domains or cylinders. 
Moreover, Byun et al \cite{ByLP} also attained the global weighted $W^{2,p}$ 
estimates of the Hessian for fully nonlinear elliptic equations with small BMO 
``coefficients" in a bounded $C^{1,1}$ domain via rather different geometrical 
approaches.

On the other hand, Lorentz spaces are a two-parameter scale of spaces which 
refine Lebesgue spaces in some sense. It is an important observation that 
requiring the principle coefficients to have small mean oscillations in the 
integral average sense is sufficient to achieve higher integrability and 
Lorentz regularity. Since the pioneering work of Talenti \cite{Tal} based 
on symmetrization, there were a large of literature on the topic of Lorentz 
regularity to elliptic and parabolic PDEs. Recently, Mengesha-Phuc in \cite{MeP1}  
used a kind of geometrical approach to prove the weighted Lorentz regularity of 
the gradient for quasilinear elliptic $p$-Laplacian equations, and 
Zhang-Zhou \cite{ZhZ} extended their results to the setting of quasilinear 
$p(x)$-Laplacian. Meanwhile, Baroni in \cite{Bar1,Bar2} made use of so-called 
Large-M-inequality principle introduced by Acerbi-Mingione to show the Lorentz 
estimates of gradient for evolutionary $p$-Laplacian systems and obstacle 
parabolic $p$-Laplacian, respectively.

We would like to point out that another key ingredient is that $F(x,D^2u)$ 
is assumed to be an asymptotically regular. It was Chipot and Evans 
\cite{ChE} to first introduce the notion of asymptotically
regular in the elliptic framework, and Raymond \cite{Ray} further considered  
a Lipschitz regularity  to asymptotically regular problems with $p$-growth. 
Since then there are a large of literatures on the topic of asymptotically 
regular problems. In particular, Scheven and Schmidt in  \cite{ScS1,ScS2} 
recently obtained a local higher integrability and a local partial Lipschitz 
continuity with a singular set of small positive measure for the gradient 
$Du$ to the system which exhibits a certain kind of elliptic behavior near 
infinity, respectively. Furthermore, a global Lipschitz regularity result 
was extended by Foss in \cite{Fos}. Very recently, Byun-Oh-Wang \cite{ByOW} 
proved global Calder\'on-Zygmund estimates for nonhomogeneous asymptotically
regular elliptic and parabolic problems in divergence form in the Reifenberg 
flat domain by covering the given asymptotically regular problems to suitable 
regular problems. Furthermore, Byun-Cho-Oh \cite{ByCO} extended the same 
conclusions to the setting of nonlinear obstacle elliptic problems. 
Zhang-Zheng \cite{ZhZhe1,ZhZhe} also further extended the work of 
Byun-Oh-Wang \cite{ByOW} to the case of obstacle parabolic problems 
in the scale of Lorentz spaces.

Inspired by those recent works  mentioned above, in this paper we consider 
a global Lorentz estimate of the Hessian of strong solutions to the Dirichlet 
problem  \eqref{Dirichlet prob} for asymptotical regular fully nonlinear 
elliptic equations over a $C^{1,1}$ bounded domain.
More precisely, our aim is to attain a global Lorentz estimate of the second 
derivative to the Dirichlet problem  \eqref{Dirichlet prob} with asymptotically 
regular nonlinearity. Indeed, it is a natural refined outgrowth of Byun et al's
 recent papers \cite{ByOW}. In particular, the Lebesgue space $L^{\gamma}$ is
 a special case of Lorentz space $L^{\gamma,q}$ when $q=\gamma$. Before stating 
the main result, let us give some basic concepts and facts.

Let us first recall that the Lorentz space $L^{\gamma,q}(\Omega)$ with 
$1\leq \gamma<\infty$ and $0<q<\infty$, which is the set of measurable function 
$g:\Omega\to \mathbb{R}$ such that
\[
\|g\|^{q}_{L^{\gamma,q}(\Omega)}:=q\int^{\infty}_{0}
\Big(\mu^{\gamma}|\{\xi \in \Omega: |g(\xi)|>\mu\}|\Big)^{q/\gamma}
\frac{d\mu}{\mu} <+\infty.
\]
While the Lorentz space $L^{\gamma,\infty}$ for $1\leq \gamma<\infty, q=\infty$
is set to be the usual Marcinkiewicz space $\mathcal{M}^{\gamma}(\Omega)$ with
quasinorm
\[
  \|g\|_{L^{\gamma,\infty}}=\|g\|_{\mathcal{M}^{\gamma}(\Omega)}
:=\underset{\mu>0}{\sup}\Big(\mu^{\gamma}|
\{\xi \in \Omega: |g(\xi)|>\mu\}|\Big)^{\frac{1}{\gamma}}<+\infty.
\]
The local variant of such spaces is defined in the usual way.
Moreover, we note that by Fubini's theorem there holds
\[
  \|g\|^{\gamma}_{L^{\gamma}(\Omega)}
=\gamma\int^{\infty}_{0}\Big(\mu^{\gamma}|\{\xi \in \Omega:
|g(\xi)|>\mu\}|\Big)\frac{d\mu}{\mu}=\|g\|^{\gamma}_{L^{\gamma,\gamma}(\Omega)},
\]
so that $L^{\gamma}(\Omega)=L^{\gamma,\gamma}(\Omega)$; cf.\ \cite{Bar1,Bar2}.
In this context, we denote by $C(n,\lambda,\Lambda,\dots)$ a universal
constant depending only on prescribed quantities and possibly varying from
line to line.

In this article, we are interested in the case that $F(x,M)$ is asymptotically 
elliptic. This is to say that it is getting closer to some real-valued function 
$G(x,M)$ as $\|M\|$ goes to infinity, where $G(x,M)$ satisfies the following 
uniformly elliptic assumption.

\begin{definition}\label{uniformly ellipticity} \rm
(uniformly ellipticity)\ We say $G(x,M):\Omega\times \mathcal{S}(n)\to \mathbb{R} $
 is uniformly elliptic if there exist constants $0 < \lambda \leq \Lambda < \infty$ 
such that for any $x \in\Omega$ and any $M \in \mathcal{S}(n)$, there holds
\begin{equation}\label{eq1.2}
\lambda \|N\|\leq G(x,M+N)-G(x,M)\leq\Lambda \|N\| , \quad   \forall N\geq 0 .
\end{equation}
\end{definition}


\begin{remark} \rm
(i) We write $N\geq 0$ whenever $N$ is a non-negative definite symmetric matrix.
 $ \|N\|$ denotes the $(L^2,L^2)$-norm of $N$, that is,
 $\|N\|=\sup_{|x|=1}  |Nx|.$ Therefore $\|N\|$ is equal to the maximum 
eigenvalue of $N$ whenever $N \geq 0$.

(ii) The uniformly elliptic assumption implies that $G(x,M)$ is monotone 
increasing and Lipschitz in $M \in \mathcal{S}(n)$.
\end{remark}

We next introduce the definition of asymptotically elliptic operators.


\begin{definition} \label{asymptotically ellipticity}\rm
$F(x,M)$ is asymptotically elliptic if there exists a uniformly elliptic 
operator $G(x,M)$ and a bounded function $\omega : \mathbb{R}^+\to\mathbb{R}^+$ 
with $\underset{r\to\infty}\lim\omega(r)=0$ such that
\begin{equation}
0\leq |F(x,M)-G(x,M)|\leq\omega(\|M\|)(1+\|M\|)
\end{equation}
for all $M\in \mathcal{S}(n)$ and any $x\in\Omega.$
\end{definition}

By a direct calculation, we conclude that
\begin{equation}
\lim_{\|M\|\to \infty} \frac{F(x,M)-G(x,M)}{\|M\|}=0,
\end{equation}
uniformly with respect to $x\in\Omega.$

We define function $\beta_G$  to measure the oscillation of $G(x,M)$ in the variable
 $x$.
Let $G:\Omega\times \mathcal{S}(n)\to\mathbb{R}$ and let $x_0\in \Omega$ be fixed. 
For $x\in\Omega$, we define
\begin{equation}
\beta_G(x,x_0):=\sup_{M\in \mathcal{S}(n)\setminus{\{0\}}} 
\frac{|G(x,M)-G(x_0,M)|}{\|M\|}.
\end{equation}


Now, let us summarize our main results as follows.

\begin{theorem}\label{main result}
 Assume $\gamma>n$ and $0<q\leq \infty$ . Let $u\in W^{2,n}(\Omega)$ be a 
strong solution to \eqref{Dirichlet prob} with $f \in L^{\gamma,q}(\Omega)$ and 
$\partial\Omega\in C^{1,1}$.
Then there exists a small positive constant 
$\beta_0=\beta_0(n,\lambda,\Lambda,\gamma,q)$ such that if $F(x,D^2u)$ 
is asymptotically elliptic with $G(x,D^2u)$ satisfying uniformly elliptic 
condition and
\begin{equation}\label{small consition }
\Big(\frac{1}{|B_r(x_0)\cap\Omega|}\int_{B_r(x_0)\cap\Omega}\beta_G(x,x_0)^ndx
\Big)^{1/n}\leq\beta_0,
\end{equation}
for any $x_0\in\Omega$ and $0<r<R_0$. We suppose that $G(x,D^{2}u)$ is 
convex and positive homogeneous of degree one in $D^{2}u$. 
Then we have $D^2u\in L^{\gamma,q} (\Omega)$, satisfying the estimate
\begin{equation}\label{desired estimate}
\|D^2u\|_{L^{\gamma,q}(\Omega)}\leq C(\|f\|_{L^{{\gamma,q}}(\Omega)}+1),
\end{equation}
where $C=C(n,\lambda,\Lambda,\gamma,q,\Omega)$. 
In the case $q=\infty$ the constant $C$ depends only on 
$n,\lambda,\Lambda,\gamma,\Omega$.
\end{theorem}

To realize our aim, some ideas from \cite{ByOW} are employed 
in our main proof. For example, to get the global Lorentz estimate we 
use  an equivalent representation of Lorentz norm, the Hardy-Littlewood maximal 
functions, and the Poisson formula by constructing a regular problem 
from the given irregular problem.

The rest of this article is organized as follows. 
In section 2, we first prove the global Lorentz estimates of the corresponding
 regular problem, and then we give a proof of the main result by taking 
a transformation from given asymptotically regular problem to a suitable 
regular problem.

\section{Proof of Theorem \ref{main result}}


We prove Theorem \ref{main result} by employing an appropriate transformation 
to construct a uniformly elliptic operator from a given asymptotically elliptic 
operator. To this end, we assume that real-valued function $F(x,M)$ is 
asymptotical to a uniformly elliptic $G(x,M)$, which satisfies convex and
 positive homogeneous of degree one in $M$ and
\[
\Big(\hbox{--}\hskip-9pt\int _{B_r(x_0)\cap\Omega}\beta_G(x,x_0)^ndx\Big)^{1/n}\leq\beta_0,
\]
for any $x_0\in\Omega$ and $0<r<R_0$, where $\beta_0>0$ will be determined later.
By Definition \ref{asymptotically ellipticity}, we have
\[
\lim_{\|M\|\to \infty} \frac{F(x,M)-G(x,M)}{\|M\|}=0.
\]
Now we write a real-valued function $H(x,M):\Omega\times \mathcal{S}(n)$ by
\begin{equation}
\|M\|H(x,M):=F(x,M)-G(x,M);
\end{equation}
then there exists an $K=K(\beta_0)>1$ such that
\begin{equation}
\|M\|\geq K\Rightarrow |H(x,M)|\leq \beta_0,\quad \forall x\in\Omega.
\end{equation}

For any fixed point $x\in \Omega$, we next define a new real-valued 
function $\tilde H(x,M)$ by
\begin{equation}
\tilde H(x,M):=
\begin{cases}
H(x,M)&\text{if } \|M\|\geq K, \\
\frac{\|M\|}{K}H\big(x,\frac{K}{\|M\|}M\big)& \text{if }  0<\|M\|<K,\\
0&\text{if } M=0.
\end{cases}
\end{equation}
It follows that $\tilde H(x,M)$ is also convex in $M$,  positive homogeneous 
of degree one in $M$, and
 \begin{equation}\label{new-H-range}
\tilde H(x,M)\leq\beta_0,\quad \forall M\in \mathcal{S}(n),
\end{equation}
uniformly with respect to $x\in\Omega$.

Note that $\tilde H(x,M)=H(x,M)$ if $\|M\|\geq K$. Therefore, for 
$M\neq0$ we have
\begin{equation}\label{a-equ-1}
\begin{aligned}
F(x,M)&= G(x,M)+\|M\|H(x,M)\\
&=  G(x,M)+\|M\|\tilde H(x,M)+\|M\|(H(x,M)-\tilde H(x,M))\\
&=  G(x,M)+\|M\|\tilde H(x,M)\\
&\quad + \|M\|_{\chi\{M\in \mathcal{S}(n):\|M\|<K\}}(H(x,M)-\tilde H(x,M)),
\end{aligned}
\end{equation}
where $\chi\{M\in \mathcal{S}(n):\|M\|<K\}$ denotes the characteristic 
function on the set $\{M\in \mathcal{S}(n):\|M\|<K\}$. In the setting of $M=0$, 
we define $\|M\|H(x,M)|_{M=0}:=F(x,0)-G(x,0)$, then the formula \eqref{a-equ-1} 
still holds for all $M\in \mathcal{S}(n)$.

Let $u\in W^{2,n}$ be a strong solution of the Dirichlet problem 
\eqref{Dirichlet prob}. Define $\tilde G:\Omega\times \mathcal{S}(n)\to \mathbb{R}$ 
by
 \begin{equation}\label{new F defi}
\tilde G(x,M):=G(x,M)+\|M\|\tilde H(x,D^2u).
 \end{equation}
Then, by  \eqref{a-equ-1} and \eqref{new F defi}, it  yields
 \begin{equation}
F(x,D^2u)=\tilde G(x,D^2u)+\|D^2u\|_{\chi\{\|D^2u\|<K\}}(H(x,D^2u)-\tilde H(x,D^2u)),
 \end{equation}
where $\chi\{\|D^2u\|<K\}=\chi\{x\in \Omega:\|D^2u\|<K\}$ denotes the 
characteristic function on the set $\{x\in \Omega:\|D^2u(x)\|<K\}$. 
Thus, from \eqref{Dirichlet prob} it implies that $u$ is a strong solution of
 \begin{equation}\label{new regular equation}
\tilde G(x,D^2u)
= f+\|D^2u\|_{\chi\{\|D^2u\|<K\}}(\tilde H(x,D^2u)- H(x,D^2u))
:= g,  \quad x\in\Omega.
 \end{equation}


To prove Theorem \ref{main result}, we also need to show that the new nonlinearity 
$\tilde{G}$ satisfies uniformly ellipticity and the oscillation condition 
\eqref{small consition } in the $L^n$ integral average sense with small constant 
$3\beta_{0}$. More precisely, we have the following lemma.

\begin{lemma}\label{new-regular}
 Let $u\in W^{2,n}(\Omega)$ be a strong solution of the Dirichlet problem 
\eqref{Dirichlet prob}. Assume that $F(x,M)$ is asymptotically elliptic
with $G(x,M)$ satisfying
\begin{equation}\label{a-equ-38}
\Big(\hbox{--}\hskip-9pt\int _{B_r(x_0)\cap\Omega}\beta_G(x,x_0)^ndx\Big)^{1/n}\leq\beta_0,
\end{equation}
for any $x_0\in\Omega$ and $0<r<R_0$. Then we have the following conclusions:
\begin{itemize}
\item[(i)] If $0<\beta_0\leq \lambda/3$, then $\tilde G(x,M)$ is 
 uniformly elliptic.
\item[(ii)]  For any $x_0\in\Omega$ and $0<r<R_0$, $\tilde G(x,M)$ satisfies
\begin{equation}
\Big(\hbox{--}\hskip-9pt\int _{B_r(x_0)\cap\Omega}\beta_{\tilde G}(x,x_0)^ndx\Big)^{1/n}
 \leq 3 \beta_0.
\end{equation}
\end{itemize}
\end{lemma}

\begin{proof} (i) Let $0<\beta_0\leq\frac{\lambda}{3}$. For any 
$M,N\in \mathcal{S}(n)$ with $N \geq0$, we have
\[
\tilde G(x,M+N)-\tilde G(x,M)=G(x,M+N)-G(x,M)+(\|M+N\|-\|M\|)\tilde H(x,M)
\]
because of  \eqref{new F defi}. From \eqref{new-H-range} it follows that
\begin{equation}\label{a-equ-2}
\tilde H(x,D^2u)\leq\beta_0,
\end{equation}
uniformly with respect to $x\in\Omega$. We have the triangle inequality
\begin{equation}\label{a-equ-3}
|\ \|M+N\|-\|M\|\ |\leq\|N\|\,.
\end{equation}
Hence, by \eqref{eq1.2}, \eqref{a-equ-2} and \eqref{a-equ-3}, we find that
\[
\tilde G(x,M+N)-\tilde G(x,M)\geq\lambda\|N\|-\beta_0\|N\|
=(\lambda-\beta_0)\|N\|\geq\frac{2\lambda}{3}\|N\|,
\]
and
\[
\tilde G(x,M+N)-\tilde G(x,M)\leq\Lambda\|N\|+\beta_0\|N\|
=(\Lambda+\beta_0)\|N\|\leq(\Lambda+\frac{\lambda}{3})\|N\|,
\]
since $0<\beta_0\leq\frac{\lambda}{3}$.  Namely,
\begin{equation}
\tilde\lambda\|N\|\leq\tilde G(x,M+N)-\tilde G(x,M)\leq\tilde\Lambda\|N\|,
\end{equation}
where $\tilde\lambda=\frac{2}{3}\lambda$ and
$\tilde\Lambda=\Lambda+\frac{\lambda}{3}$.
So the assertion (i) is proved.

(ii) Let $x_0\in\Omega$ and $0<r<R_0$. For any $x\in B_r(x_0)\cap\Omega$, 
it follows from \eqref{new-H-range} and \eqref{new F defi} that
\begin{equation}
|\tilde G(x,M+N)-\tilde G(x,M)|\leq|G(x,M)-G(x_0,M)|+2\beta_0\|M\|,
\end{equation}
which implies
\begin{equation}\label{a-equ-314}
\begin{aligned}
\beta_{\tilde G}(x,x_0)&= \sup_{M\in \mathcal{S}(n)\setminus{\{0\}}}
 \frac{|\tilde G(x,M)-\tilde G(x_0,M)|}{\|M\|}\\
&\leq \sup_{M\in \mathcal{S}(n)\setminus{\{0\}}} 
 \frac{|G(x,M)-G(x_0,M)|}{\|M\|}+2\beta_0\\
&= \beta_G(x,x_0)+2\beta_0.
\end{aligned}
\end{equation}
Therefore, by \eqref{a-equ-38}, \eqref{a-equ-314} and the Minkowski 
inequality we obtain
\begin{equation}
\begin{aligned}
&\Big(\frac{1}{|B_r(x_0)\cap\Omega|}
\int_{B_r(x_0)\cap\Omega}\beta_{\tilde G}(x,x_0)^ndx\Big)^{1/n} \\
&\leq \Big(\frac{1}{|B_r(x_0)\cap\Omega|}\int_{B_r(x_0)\cap\Omega}
 \beta_G(x,x_0)^ndx\Big)^{1/n}+2\beta_0\\
&\leq \beta_0+2\beta_0=3 \beta_0,
\end{aligned}
\end{equation}
which implies the assertion (ii).
\end{proof}

We  recall an interior Lorentz estimate of strong 
solutions to fully nonlinear uniformly elliptic equations, whose proof 
can be found in \cite{ZhZhe} by using the approach of large-M-inequality 
principle originated from Acerbi-Mingione's work. More precisely, 
let us consider the  fully nonlinear uniformly elliptic equations
\begin{equation}\label{regular_eq.}
G(x,D^{2}u)=f(x), \quad \text{in $\Omega$}
\end{equation}
with $f\in L^{\gamma,q}(\Omega)$.

\begin{lemma}[{\cite[Corollary 1.4]{ZhZhe}}] \label{interior result-Lorentz}
  Assume $\gamma>n$ and $0<q\leq \infty$. Let $u\in W^{2,n}(\Omega)$ be a 
strong solution to \eqref{regular_eq.} satisfying uniformly ellipticity and
 $f \in L^{\gamma,q}(\Omega)$.  If $G(x,D^{2}u)$ is convex and positive 
homogeneous of degree one in $D^{2}u$, then there exists a small positive 
$\beta_0=\beta_0(n,\lambda,\Lambda,\gamma,q)$ such that if $G(x,D^2u)$
 satisfies
\[
\Big(\frac{1}{|B_r(x_0)\cap\Omega|}\int_{B_r(x_0)\cap\Omega}\beta_G(x,x_0)^ndx
\Big)^{1/n}\leq\beta_0,
\]
for every $x_0\in\Omega$ and $0<r<R_0$; then we have
$D^2u\in L^{\gamma,q}_{loc} (\Omega)$.  Moreover,
there exists a radii $R_1=R_1(n,\lambda,\Lambda,\gamma,q)$ such that for each
ball $B_{2R}(x_0)\subset\subset\Omega$ and $0<R\leq R_1$ with the estimate
\begin{equation}\label{interior result-Lorentz estimate}
\|D^2u\|_{L^{\gamma,q}(B_R)}\leq C(\|D^2u\|_{L^n(B_{2R})}
+\|f\|_{L^{{\gamma,q}}(B_{2R})}),
\end{equation}
where $C=C(n,\lambda,\Lambda,\gamma,q)$. In the case $q=\infty$ the constant
$C$ and $R_1$ above depend only on $n,\lambda,\Lambda,\gamma$.
\end{lemma}


Next, we establish a local boundary estimate in the scale of Lorentz spaces
 by using the idea of odd/even extensions over the flat boundary.
Fixed a point $x_0\in\partial\Omega$, without loss of generality let us write
\[
\text{$\partial\Omega$ is flat near $x_{0}$ lying in the plane $\{x^{1}=0\}$}.
\]
Then we may assume there exists an open ball $B_{2R}(x_{0})$ with center 
$x_{0}$ and radius $2R$ such that
\begin{gather*}
B^{+}_{2R}:=B_{2R}(x_{0})\cap \{x^{1}>0\}\subset \overline{\Omega},\\
B^{-}_{2R}:=B_{2R}(x_{0})\cap \{x^{1}<0\}\subset \mathbb{R}^{n}-\overline{\Omega}.
\end{gather*}
We also set $\Gamma_{2R}=B_{2R}(x_{0})\cap \{x^{1}=0\}$.

\begin{lemma}\label{flat boundary result-Lorentz}
 For $\gamma>n$ and $0<q\leq \infty$, let $u\in W^{2,n}(\Omega)$ be a 
strong solution of local boundary value problem
\begin{equation}\label{local boundary value problem}
\begin{gathered}
G(x,D^{2}u)=f(x), \quad \text{in $B^{+}_{2R}$},\\
u=0, \quad \text{on $\Gamma_{2R}$}
\end{gathered}
\end{equation}
with $f\in L^{\gamma,q}(\Omega)$.
Then there exist small positive constants $\delta$ and $R_{0}$ 
depending only on $n,\lambda,\Lambda,\gamma,q$  such that, $G(x,D^2u)$ 
satisfying uniformly elliptic, $G(x,D^2u)$ is convex and positive homogeneous 
of degree one in $D^2u$ and
\[
\Big(\hbox{--}\hskip-9pt\int _{B_r(x_0)\cap\Omega}\beta_G(x,x_0)^ndx\Big)^{1/n}\leq\beta_0,
\]
we have
\begin{equation}\label{boundary Lorentz estimate inequality}
 \|D^{2}u\|_{L^{\gamma,q}(B^{+}_{R})}
\leq C\left(\|D^{2}u\|_{L^{n}(B^{+}_{2R})}+\|f\|_{L^{\gamma,q}(B^{+}_{2R})}\right),
\end{equation}
for each half ball $B^{+}_{2R}$ with $0<R \leq R_{0}$, where
$C=C(n,\lambda,\Lambda,\gamma,q)$. In the case $q=\infty$, the constant
$C$ above depends only on $n,\lambda,\Lambda,\gamma$.
\end{lemma}

\begin{proof}
  Note that $G(x,D^2u)$ is convex and positive homogeneous of degree one in 
$D^2u$ and
\[
\Big(\hbox{--}\hskip-9pt\int _{B_r(x_0)\cap\Omega}\beta_G(x,x_0)^ndx\Big)^{1/n}\leq\beta_0,
\]
which implies
\[
  G(x,D^{2}u)=G_{D^{2}_{ij}u}(x,D^{2}u)D^{2}_{ij}u:=a_{ij}(x)D^{2}_{ij}u.
\]

Let us now define $\hat{u}$ in $B_{2R}(x_{0})$ with $x_{0}$ on the flat boundary by
\[
 \hat{u}(x^{1},x')=
  \begin{cases}
   u(x^{1},x') & \text{if } x^{1}\geq 0,\\
  u(-x^{1},x') & \text{if } x^{1}< 0,
  \end{cases}
\]
and extend $a_{ij}(x)=a_{ij}(x^{1},x')$ from $\{x^{1}\geq0\}$ to $\{x^{1}<0\}$
by even or odd reflection, depending on the indices $i$ and $j$. Specifically,
when $x^{1}\geq 0$, $\hat{a}_{ij}(x)=a_{ij}(x)$; when $x^{1}<0$,
\[
  \hat{a}_{ij}(x)= \begin{cases}
a_{ij}(-x^{1},x'),  & \text{if $i=j=1$ or $i,j\in \{2,\dots,n\}$},\\
-a_{ij}(-x^{1},x'), & \text{if $i\in \{2,\dots,n\}$ and $j=1$}.
\end{cases}
\]
Also set $\hat{a}_{1j}=\hat{a}_{j1}$. We see that the nonlinearity
$\widehat{G}(x,D^{2}\hat{u})$ satisfy uniformly elliptic, convex and positive
homogeneous of degree one in $D^{2}\hat{u}$ and
\[
\Big(\hbox{--}\hskip-9pt\int _{B_r(x_0)\cap\Omega}\beta_{\widehat{G}}(x,x_0)^ndx\Big)^{1/n}
\leq\beta_0,
\]
Let $\hat{f}$ be the odd extension of $f$ with respect to $x^{1}$,
then it is easy to check that $\hat{f}\in L^{\gamma,q}(B_{2R}(x_{0}))$.
By Lemma \ref{interior result-Lorentz} it implies that the extended $\hat{u}$
is a strong solution of $\widehat{G}(x,D^{2}\hat{u})=\hat{f}$ in $B_{2R}(x_{0})$,
then it gives rise to the local Lorentz estimate
\eqref{interior result-Lorentz estimate}. Therefore, the desired estimate
\eqref{boundary Lorentz estimate inequality} is obtained by restricting
$\hat{u}$ from $B_{2R}(x_{0})$ to $B^{+}_{2R}$.
\end{proof}

Using the standard flattening and covering arguments, we can derive a global 
Lorentz estimate as follows.

\begin{theorem}\label{main result-1}
For $\gamma>n$ and $0< q\le \infty$, let $u\in W^{2,n}(\Omega)$ be a strong 
solution to the following Dirichlet problem
\begin{equation}\label{regular Dirichlet prob}
\begin{gathered}
G(x,D^{2}u)=f(x),  \quad \text{in $\Omega$},\\
u=0,  \quad \text{on $\partial\Omega$},\\
\end{gathered}
\end{equation}
$G(x,D^2u)$ satisfying uniformly elliptic, convex and positive homogeneous 
of degree one in $D^2u$ with the following oscillation on ``coefficients''
\[
\Big(\hbox{--}\hskip-9pt\int _{B_r(x_0)\cap\Omega}\beta_G(x,x_0)^ndx\Big)^{1/n}\leq\beta_0
\]
for some small positive constant $\beta_0=\beta_0(n,\lambda,\Lambda,\gamma,q)$.
If $f\in L^{\gamma,q}(\Omega)$ and $\partial\Omega\in C^{1,1}$, then
$D^{2}u\in L^{\gamma,q}(\Omega)$ and there exists a positive constant
$C=C(n,\lambda,\Lambda,\gamma,q,\Omega)$ with the estimate
\begin{equation}\label{regular main result-Lorentz inequality}
  \|D^{2}u\|_{L^{\gamma,q}(\Omega)}\leq C\|f\|_{L^{\gamma,q}(\Omega)}.
\end{equation}
While $q=\infty$, the constant $C$ depends only on
$n,\lambda,\Lambda,\gamma,\Omega$.
\end{theorem}

\begin{proof}  (1)  For fixed any point $x_{0}\in \partial\Omega$, we now 
flatten the boundary near $x_{0}$ in order to apply the flat boundary estimates 
\eqref{boundary Lorentz estimate inequality}. Thanks to the assumption 
$\partial\Omega \in C^{1,1}$, there exists a neighborhood 
$\mathcal{N}_{0}\ni x_{0}$ and a $C^{1,1}$-diffeomorphism 
$\Phi:\mathcal{N}_{0}\to B_{2R}$ such that
\[
  \Phi(x_{0})=0, \quad\text{and}\quad  \Phi(\mathcal{N}_{0}\cap \Omega)=B^{+}_{2R}(0).
\]
We write $y=\Phi(x), x\in \mathcal{N}_{0}\cap \Omega$, and define $\Psi=\Phi^{-1}$,
then $x=\Psi(y)$. Define $\tilde{u}(y)=u(\Psi(y))=u(x)$ for $y\in B^{+}_{2R}$.
Then it is readily checked that $\tilde{u}\in W^{2,n}(B^{+}_{2R})$ is a strong
solution of the  flat initial-boundary problem
\begin{gather*}
\widetilde{F}(y,D^{2}\tilde{u})=\tilde{f}(y) \quad \text{in $B^{+}_{2R}$},\\
\tilde{u}=0 \quad \text{on $\Gamma_{2R}\cup B^{+}_{2R}$},
\end{gather*}
where
\begin{gather*}
\widetilde{F}(y,D^{2}\tilde{u}):=F\left(\Psi(y),\left(D\Phi^{T}
\circ \Psi\right)D^{2}\tilde{u}\left(D\Phi \circ \Psi\right)
+D\tilde{u}\left(D^{2}\Phi\circ \Psi\right)\right), \\
  \tilde{f}(y):=f(\Psi(y)).
\end{gather*}
It is obvious that $\widetilde{F}$ is convex in $D^{2}\tilde{u}$ and
$\widetilde{F}(y,0)=0$. Moreover, we readily see that
$\beta_{\widetilde{F}}(y,y_{0})\leq C(\Phi)\beta_{F}(\Psi(y),\Psi(y_{0}))$
for any $y,y_{0}\in B^{+}_{2R}$; and $\widetilde{F}$ satisfies the similar
assumptions of Lemma \ref{flat boundary result-Lorentz} with
different positive constants. Therefore, it yields
\[
 \|D^{2}\tilde{u}\|_{L^{\gamma,q}(B^{+}_{R})}
\leq C\left(\|D^{2}\tilde{u}\|_{L^n(B^{+}_{2R})}
+\|\tilde{f}\|_{L^{\gamma,q}(B^{+}_{2R})}\right).
\]
Converting back to the original $x$-variables, we conclude
\begin{equation}\label{boundary estimate Vi}
\|D^{2}u\|_{L^{\gamma,q}(\Psi(B^{+}_{R}))}
\leq C\left(\|D^{2}u\|_{L^n(\Psi(B^{+}_{2R}))}
 +\|f\|_{L^{\gamma,q}(\Psi(B^{+}_{2R}))}\right),
\end{equation}
From this estimate, along with the interior bound
\eqref{interior result-Lorentz estimate} in Lemma \ref{interior result-Lorentz},
the standard covering arguments lead to
\begin{equation}
  \|D^{2}u\|_{L^{\gamma,q}(\Omega)}
\leq C\left(\|D^{2}u\|_{L^{n}(\Omega)}+\|f\|_{L^{\gamma,q}(\Omega)}\right),
\end{equation}
for some positive constant $C$ depending on $n,\lambda,\Lambda,\gamma,q,\Omega$,
which implies
\begin{equation}\label{global result initial-1}
  \|u\|_{W^2L^{\gamma,q}(\Omega)}
\leq C\left( \|Du\|_{L^{\gamma,q}(\Omega)}+\|D^{2}u\|_{L^{n}(\Omega)}
+\|f\|_{L^{\gamma,q}(\Omega)}\right).
\end{equation}


(2) At this point, the desired estimate 
\eqref{regular main result-Lorentz inequality} follows from the uniqueness 
property of the homogeneous equation. Indeed, if 
\eqref{regular main result-Lorentz inequality} is not true, there exists a 
sequence $\{u_{k}\}_{k=1}^{\infty}$ and $\{f_{k}\}_{k=1}^{\infty}$ such that 
$u_{k}$ for each $k$ is a strong solution of the problem
\begin{gather*}%\label{fk and uk problem}
F(x,D^{2}u_{k})=f_{k}(x) \quad \text{in $\Omega$},\\
u_{k}=0 \quad \text{on $\partial\Omega$},
\end{gather*}
with the estimate
\begin{equation}\label{global result initial-proof contradiction}
  \|u_{k}\|_{W^2L^{\gamma,q}(\Omega)}> k\|f_{k}\|_{L^{\gamma,q}(\Omega)}, 
\quad \text{for all }  k\geq 1.
\end{equation}
Without loss of generality, we may suppose that
\begin{equation}\label{global result initial-proof contradiction assumption uk}
  \|u_{k}\|_{W^2L^{\gamma,q}(\Omega)}=1.
\end{equation}
Then it follows from \eqref{global result initial-proof contradiction} that
\begin{equation}\label{global result initial-proof-f-limiting}
  \|f_{k}\|_{L^{\gamma,q}(\Omega)}<\frac{1}{k}\to 0, \quad \text{as }  k\to \infty.
\end{equation}
Since $\{u_{k}\}_{k=1}^{\infty}$ is uniformly bounded in $W^2L^{\gamma,q}(\Omega)$, 
there exists a subsequence, which be still denoted by $\{u_{k}\}_{k=1}^{\infty}$, 
and a function $u_{0}\in W^2L^{\gamma,q}(\Omega)$, such that
\begin{equation}\label{uk limiting}
  u_{k}\rightharpoonup u_{0}  \text{ weakly in }  W^2L^{\gamma,q}(\Omega), \quad 
u_{k}\to u_{0}  \text{ in }  L^{\gamma,q}(\Omega), \quad \text{as }  k\to\infty.
\end{equation}
It is easy to check that $u_{0}$ is a strong solution of
\begin{equation}\label{u0 problem}
\begin{gathered}
F(x,D^{2}u_{0})=0, \quad \text{in $\Omega$},\\
u_{0}=0, \quad \text{on $\partial\Omega$}.
\end{gathered}
\end{equation}
Accordingly, $u_{0}=0$ due to the uniqueness of strong solutions to zero 
initial-boundary problem \eqref{u0 problem}, so it follows 
from \eqref{global result initial-proof-f-limiting} and \eqref{uk limiting} that
\begin{equation}\label{f-limiting-1}
\begin{gathered}
  f_{k}\to 0 \quad \text{in }  L^{\gamma,q}(\Omega), \\
 u_{k}\rightharpoonup 0 \quad \text{weakly in }  W^2L^{\gamma,q}(\Omega), \\
 u_{k}\to 0 \quad \text{in }  L^{\gamma,q}(\Omega),
\end{gathered}
\end{equation}
as $k\to\infty$. Note that  $W^2L^{\gamma,q}(\Omega)\hookrightarrow W^2L^n(\Omega)$ 
because $\gamma>n$, hence
\begin{equation}\label{u-bounded-limiting}
  \|u_{k}\|_{L^n(\Omega)} \to 0, \quad \|Du_{k}\|_{L^{n}(\Omega)} \to 0, \quad 
\text{as }  k\to \infty.
\end{equation}
Moreover, letting the measure $\nu=dx$, we see that
\[
  Du_{k}\to 0 \quad \text{$\nu$-a.e. in }  \Omega \quad \text{as }
k\to \infty \quad \text{(up  to  subsequence)},
\]
which implies
\[
  |\{x\in\Omega:|Du_{k}|>\mu\}|\to 0 \quad \text{for all } \mu>0 \quad
\text{as }  k\to\infty,
\]
so by the Lebesgue Dominated Convergence Theorem we obtain
\begin{equation}\label{Duk limiting}
 Du_{k}\to 0 \quad \text{in }  L^{\gamma,q}(\Omega) \quad \text{as }  k\to\infty.
\end{equation}
Combining \eqref{global result initial-1},
\eqref{global result initial-proof contradiction assumption uk},
 \eqref{f-limiting-1}, \eqref{u-bounded-limiting} and \eqref{Duk limiting}, it yields
\[
  1\leq C\left( \|Du_{k}\|_{L^{\gamma,q}(\Omega)}+\|D^{2}u_{k}\|_{L^n(\Omega)}
+\|f_{k}\|_{L^{\gamma,q}(\Omega)}\right)\to 0 \quad  \text{as }  k\to \infty,
\]
which is a contradiction. This completes the proof.
\end{proof}


In view of the lemma \ref{new-regular}, the asymptotically elliptic equation 
\eqref{Dirichlet prob} turns out to be a uniformly elliptic equation 
\eqref{new regular equation}.
For the asymptotically elliptic equation, lemma \ref{new-regular} and 
the existing theory for fully nonlinear, uniformly elliptic equation, 
see Lemma \ref{interior result-Lorentz}, will be employed to finally derive 
the required estimate.
We are now ready to prove our main result.

\begin{proof}[Proof of theorem \ref{main result}]
From \eqref{new regular equation}, for any given positive constant 
$\beta_0$, as in Lemma \ref{interior result-Lorentz}, we define a new data 
$\beta_1:=\min\{\frac{\lambda}{3},1\}$, and set
\[
\tilde\beta_0:=\frac{1}{3}\min\{\beta_0,\beta_1\}>0.
\]
Then there exists a real-valued function $\tilde{H}(x,D^{2}u)$ such that
\[
  |\tilde{H}(x,D^{2}u)|\leq  \beta_{0}<1 \quad \text{for all }  x\in \Omega.
\]

Now let $u\in W^{2,p}$ be a strong solution of the problem \eqref{Dirichlet prob} 
and assume that $F(x,D^2u)$ is asymptotically elliptic with $G(x,D^2u)$ 
which satisfies
\[
\Big(\frac{1}{|B_r(x_0)\cap\Omega|}\int_{B_r(x_0)\cap\Omega}
 \beta_G(x,x_0)^ndx\Big)^{1/n}\leq\tilde\beta_0,
\]
for any $x_0\in\Omega$ and $0<r<R_0$.
Then by Lemma \ref{interior result-Lorentz}, $\tilde G(x,D^2u)$ is
uniformly elliptic and satisfies
\[
\Big(\frac{1}{|B_r(x_0)\cap\Omega|}
\int_{B_r(x_0)\cap\Omega}\beta_{\tilde G}(x,x_0)^ndx\Big)^{1/n}
\leq 3\tilde\beta_0\leq\beta_0,
\]
for any $x_0\in\Omega$ and $0<r<R_0$.

According to $f\in L^{\gamma,q}(\Omega)$ and equality \eqref{new regular equation}, 
we have
\[
|g(x)|\leq |f(x)|+2|D^2u(x)|_{\chi\{\|D^2u\|<K\}}.
\]
which implies that
\begin{align*}
&|\{x\in\Omega:|g(x)>\mu|\}| \\
&\leq|\{x\in\Omega:|f(x)|>\frac{\mu}{2}\}|+|\{x\in\Omega:
2|D^2u(x)|_{\chi\{\|D^2u\|<K\}}>\frac{\mu}{2}\}|.
\end{align*}
Therefore,
\begin{align*}
  \|g\|^q_{L^{\gamma,q}(\Omega)}
&\leq q\int^{\infty}_{0}\left(\mu^{\gamma}|\{x\in \Omega:|f(x)|
 >\frac{\mu}{2}\}|\right)^{q/\gamma}\frac{d\mu}{\mu}\\
&\quad + q\int^{\infty}_{0}\left(\mu^{\gamma}|\{x\in \Omega:
 2|D^2u(x)|\chi_{\{\|D^2u\|<K\}}>\frac{\mu}{2}\}|\right)^{q/\gamma}
 \frac{d\mu}{\mu}\\
  &=  2^qq\int^{\infty}_{0}\left(\mu^{\gamma}|\{x\in \Omega:
 |f(x)|>\mu\}|\right)^{q/\gamma}\frac{d\mu}{\mu}\\
  &\quad + 2^qq\int^{\infty}_{0}\left(\mu^{\gamma}|\{x\in \Omega:
 2|D^2u(x)|\chi_{\{\|D^2u\|<K\}}>\mu\}|\right)^{q/\gamma}\frac{d\mu}{\mu}.
\end{align*}
Since
\[
  |\{x\in \Omega:2|Du^2(x)|\chi_{\{\|D^2u\|<K\}}>\mu\}|
\leq |\{x\in \Omega:2K>\mu\}|,
\]
we can derive that
\begin{align*}%\label{G-Lorentz-esti}
\|g\|^q_{L^{\gamma,q}(\Omega)}
&\leq 2^q\|f\|^q_{L^{\gamma,q}(\Omega)}+2^qq\int^{\infty}_{0}
 \left(\mu^{\gamma}|\{x\in
  \Omega: 2K>\mu\}|\right)^{q/\gamma}\frac{d\mu}{\mu}\\
&= 2^q\|f\|^q_{L^{\gamma,q}(\Omega)}+2^qq\int^{2K}_{0}
 \left(\mu^{\gamma}|\{x\in
  \Omega: 2K>\mu\}|\right)^{q/\gamma}\frac{d\mu}{\mu}\\
&\quad +  2^qq\int^{\infty}_{2K}\left(\mu^{\gamma}|\{x\in
  \Omega: 2K>\mu\}|\right)^{q/\gamma}\frac{d\mu}{\mu}\\
&\leq 2^q\|f\|^q_{L^{\gamma,q}(\Omega)}+2^qq\int^{2K}_{0}
 \left(\mu^{\gamma}|\Omega|\right)^{q/\gamma}\frac{d\mu}{\mu}+0\\
&=  2^q\|f\|^q_{L^{\gamma,q}(\Omega)}+2^qq|\Omega|^{q/\gamma}
 \int^{2K}_{0}\mu^{q-1}d\mu\\
&=  2^q\|f\|^q_{L^{\gamma,q}(\Omega)}+2^q|\Omega|^{q/\gamma}(2K)^{q}\\
&\leq  C(\|f\|^q_{L^{\gamma,q}(\Omega)}+1).
\end{align*}
Thus, we have
\[
\|g\|_{L^{\gamma,q}(\Omega)}\leq C(\|f\|_{L^{\gamma,q}(\Omega)}+1),
\]
for some positive constant $C=C(\beta_0,K,n,\gamma,q,|\Omega|)$.


Considering
\[
  H(x,M)=\frac{F(x,M)-G(x,M)}{\|M\|}\geq 0
\]
for all $M\in \mathcal{S}(n)$ and $x\in \Omega$, we have $\tilde{H}(x,M)\geq 0$
for all $x\in \Omega$ due to the definition of $\tilde{H}$.
This shows that $\tilde{H}(x,D^{2}u)\geq 0$ for all $x\in \Omega$.
On the other hand, we know $G(x,M)$ and $\|M\|$ are convex in $M$;
therefore $\tilde{F}(x,M)=G(x,M)+\tilde{H}(x,D^{2}u)\|M\|$ is also
convex with respect to $M$.
We then apply Theorem \ref{main result-1} to $g\in L^{\gamma,q}(\Omega)$
and $\tilde G(x,D^2u)$ to discover $u\in W^{2,n}$ with the estimate
\begin{equation}
  \|D^2u\|_{L^{\gamma,q}(\Omega)} \leq C\|g\|_{L^{{\gamma,q}}(\Omega)}
  \leq C(\|f\|_{L^{{\gamma,q}}(\Omega)}+1),
\end{equation}
where $C=C(n,\lambda,\Lambda,R_0,\Omega,\gamma,q)$ is a positive constant.
This completes the proof.
\end{proof}

\subsection*{Acknowledgements} 
 This work is supported by the  NSF of China under grants 
No. 11371050 and No. 11401165.

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\end{document}